Decoupling and simplifying of discrete dynamical systems in the ...

Contents Preface

1

Organizing Committee

3

Scientific Committee

5

ISDE Advisory Committee

7

Welcome from the ISDE President

9

ISDE Board of Directors

11

Schedule

13

Monday, July 21 (One-Hour Talks) . . . . . . . . . . . . . . . . . . . . . .

15

Monday, July 21 (Contributed Talks) . . . . . . . . . . . . . . . . . . . . .

17

Tuesday, July 22 (One-Hour Talks) . . . . . . . . . . . . . . . . . . . . . .

19

Tuesday, July 22 (Contributed Talks) . . . . . . . . . . . . . . . . . . . . .

21

Thursday, July 24 (One-Hour Talks) . . . . . . . . . . . . . . . . . . . . . .

23

Thursday, July 24 (Contributed Talks) . . . . . . . . . . . . . . . . . . . .

25

Friday, July 25 (One-Hour Talks) . . . . . . . . . . . . . . . . . . . . . . .

27

Friday, July 25 (Contributed Talks) . . . . . . . . . . . . . . . . . . . . . .

29

One-Hour Speakers

31

Abstracts of One-Hour Talks

35

Agarwal, Ravi (USA) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

36

Akın-Bohner, Elvan (USA) . . . . . . . . . . . . . . . . . . . . . . . . . . .

37

Alsed`a, Llu´ıs (Spain) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

38

iii

Doˇsly, ´ Ondˇrej (Czech Republic) . . . . . . . . . . . . . . . . . . . . . . . .

39

Gesztesy, Fritz (USA) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

40

Gy˝ori, Istv´an (Hungary) . . . . . . . . . . . . . . . . . . . . . . . . . . . .

41

Hilger, Stefan (Germany) . . . . . . . . . . . . . . . . . . . . . . . . . . . .

42

Kloeden, Peter (Germany) . . . . . . . . . . . . . . . . . . . . . . . . . . .

43

¨ Koc¸ak, Huseyin (USA) . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

44

Ladas, Gerasimos (USA) . . . . . . . . . . . . . . . . . . . . . . . . . . . .

45

Mawhin, Jean (Belgium) . . . . . . . . . . . . . . . . . . . . . . . . . . . .

46

Peterson, Allan (USA) . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

47

Smith, Hal (USA) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

48

Vanderbauwhede, Andr´e (Belgium) . . . . . . . . . . . . . . . . . . . . .

49

Yorke, James A. (USA) . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

50

˘ Zafer, Agacık (Turkey) . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

51

Zeidan, Vera (USA) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

52

Abstracts of Contributed Talks

53

´ (Spain) . . . . . . . . . . . . . . . . . . . . . . . . . . . Abderram´an, Jesus

54

Adıvar, Murat (Turkey) . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

55

Afshar Kermani, Mozhdeh (Iran) . . . . . . . . . . . . . . . . . . . . . . .

56

Aghazadeh, Nasser (Iran) . . . . . . . . . . . . . . . . . . . . . . . . . . .

57

˙ ı (Turkey) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Albayrak, Inc˙

58

Aldea Mendes, Diana (Portugal) . . . . . . . . . . . . . . . . . . . . . . .

59

Al-Sharawi, Ziyad (Oman) . . . . . . . . . . . . . . . . . . . . . . . . . . .

60

Alzabut, Jehad (Turkey) . . . . . . . . . . . . . . . . . . . . . . . . . . . .

61

Appleby, John (Ireland) . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

62

Aseeri, Samar (Saudi Arabia) . . . . . . . . . . . . . . . . . . . . . . . . .

63

Atasever, Nur˙ıye (Turkey) . . . . . . . . . . . . . . . . . . . . . . . . . . .

64

Atay, Fat˙ıhcan M. (Germany) . . . . . . . . . . . . . . . . . . . . . . . . .

65

Atıcı, Ferhan (USA) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

66

Awerbuch Friedlander, Tamara (USA) . . . . . . . . . . . . . . . . . . . .

67

¨ Batıt, Ozlem (Turkey) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

68

Bernhardt, Chris (USA) . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

69

iv

Bodine, Sigrun (USA) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

70

Bolat, Yas¸ar (Turkey) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

71

C ¸ akmak, Devr˙ım (Turkey) . . . . . . . . . . . . . . . . . . . . . . . . . . .

72

Camouzis, Elias (Greece) . . . . . . . . . . . . . . . . . . . . . . . . . . . .

73

C´anovas, Jose S. (Spain) . . . . . . . . . . . . . . . . . . . . . . . . . . . .

74

C ¸ et˙ın, Erb˙ıl (Turkey) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

75

C ¸ ıbıkd˙ıken, Al˙ı Osman (Turkey) . . . . . . . . . . . . . . . . . . . . . . . .

76

Costa, Sara (Spain) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

77

Cushing, J. M. (USA) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

78

Dannan, Fozi (Syria) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

79

Doˇsl´a, Zuzana (Czech Republic) . . . . . . . . . . . . . . . . . . . . . . . .

80

Duman, Ahmet (Turkey) . . . . . . . . . . . . . . . . . . . . . . . . . . . .

81

Erbe, Lynn (USA) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

82

Erol, Meltem (Turkey) . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

83

Esty, Norah (USA) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

84

Fernandes, Sara (Portugal) . . . . . . . . . . . . . . . . . . . . . . . . . . .

85

Gomes, Orlando (Portugal) . . . . . . . . . . . . . . . . . . . . . . . . . . ¨ ¨ us ¨ ¸ , Ozlem Gum Ak (Turkey) . . . . . . . . . . . . . . . . . . . . . . . . . .

86 87

Guseinov, Gusein (Turkey) . . . . . . . . . . . . . . . . . . . . . . . . . . .

88

¨ Guven˙ ıl˙ır, A. Feza (Turkey) . . . . . . . . . . . . . . . . . . . . . . . . . .

89

Guzowska, Małgorzata (Poland) . . . . . . . . . . . . . . . . . . . . . . .

90

Hashemiparast, Moghtada (Iran) . . . . . . . . . . . . . . . . . . . . . . .

91

Heim, Julius (USA) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

92

Hilscher, Roman (Czech Republic) . . . . . . . . . . . . . . . . . . . . . .

93

´ Jim´enez Lopez, V´ıctor (Spain) . . . . . . . . . . . . . . . . . . . . . . . . .

94

Kalabuˇsi´c, Senada (Bosnia and Herzegovina) . . . . . . . . . . . . . . . .

95

Karpuz, Bas¸ak (Turkey) . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

96

Keller, Christian (USA) . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

97

Kent, Candace (USA) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

98

Kharkov, Vitaliy (Ukraine) . . . . . . . . . . . . . . . . . . . . . . . . . . .

99

Kipnis, Mikhail (Russia) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 Kostrov, Yevgeniy (USA) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 v

Kulik, Tomasia (Australia) . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 Laitochov´a, Jitka (Czech Republic) . . . . . . . . . . . . . . . . . . . . . . 103 Lawrence, Bonita (USA) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 Lu´ıs, Rafael (Portugal) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 Matthews, Thomas (USA) . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 McCarthy, Michael (Ireland) . . . . . . . . . . . . . . . . . . . . . . . . . . 107 Mendes, Vivaldo (Portugal) . . . . . . . . . . . . . . . . . . . . . . . . . . 108 Mert, Raz˙ıye (Turkey) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 Mesgarani, Hamid (Iran) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 Michor, Johanna (USA) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 Migda, Małgorzata (Poland) . . . . . . . . . . . . . . . . . . . . . . . . . . 112 Morales, Leopoldo (Spain) . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 Oban, Volkan (Turkey) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 Oberste-Vorth, Ralph (USA) . . . . . . . . . . . . . . . . . . . . . . . . . . 115 Oliveira, Henrique (Portugal) . . . . . . . . . . . . . . . . . . . . . . . . . 116 ¨ urk, ¨ Ozt Ruk˙ıye (Turkey) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 Papaschinopoulos, Garyfalos (Greece) . . . . . . . . . . . . . . . . . . . . 118 Park, Choonkil (South Korea) . . . . . . . . . . . . . . . . . . . . . . . . . 119 Pinelas, Sandra (Portugal) . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 Pituk, Mih´aly (Hungary) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 Pop, Nicolae (Romania) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 Popescu, Emil (Romania) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 Popescu, Nedelia Antonia (Romania) . . . . . . . . . . . . . . . . . . . . . 124 Posp´ısˇ il, Zdenˇek (Czech Republic) . . . . . . . . . . . . . . . . . . . . . . 125 ¨ Potzsche, Christian (Germany) . . . . . . . . . . . . . . . . . . . . . . . . 126 Predescu, Mihaela (USA) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 Rabbani, Mohsen (Iran) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 Rachidi, Mustapha (France) . . . . . . . . . . . . . . . . . . . . . . . . . . 129 Radin, Michael (USA) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 Rasmussen, Martin (Germany) . . . . . . . . . . . . . . . . . . . . . . . . 131 ˇ ak, Pavel (Czech Republic) . . . . . . . . . . . . . . . . . . . . . . . . 132 Reh´ Reinfelds, Andrejs (Latvia) . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 vi

Rodkina, Alexandra (Jamaica) . . . . . . . . . . . . . . . . . . . . . . . . . 134 Romero i S´anchez, David (Spain) . . . . . . . . . . . . . . . . . . . . . . . 135 Saker, Samir (Saudi Arabia) . . . . . . . . . . . . . . . . . . . . . . . . . . 136 Sanchez-Moreno, Pablo (Spain) . . . . . . . . . . . . . . . . . . . . . . . . 137 Schinas, Christos (Greece) . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 Schmeidel, Ewa (Poland) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 S¸ekerc˙ı, Nurcan (Turkey) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 Shahrezaee, Mohsen (Iran) . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 Siddikov, Bakhodirzhon (USA) . . . . . . . . . . . . . . . . . . . . . . . . 142 Simon, Moritz (Germany) . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 Sırma, Al˙ı (Turkey) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144 Stefanidou, Gesthimani (Greece) . . . . . . . . . . . . . . . . . . . . . . . 145 Stehlik, Petr (Czech Republic) . . . . . . . . . . . . . . . . . . . . . . . . . 146 Teschl, Gerald (Austria) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 T˙ıryak˙ı, Aydın (Turkey) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 Tlemc¸ani, Mouhaydine (Portugal) . . . . . . . . . . . . . . . . . . . . . . 149 Topal, Fatma Serap (Turkey) . . . . . . . . . . . . . . . . . . . . . . . . . . 150 Yantır, Ahmet (Turkey) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 Yıldırım, Ahmet (Turkey) . . . . . . . . . . . . . . . . . . . . . . . . . . . 152 Zaidi, Atiya (Australia) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 Zakeri, Ali (Iran) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 Zem´anek, Petr (Czech Republic) . . . . . . . . . . . . . . . . . . . . . . . 155 Other Participants

157

Abdeljawad, Thabet (Turkey) . . . . . . . . . . . . . . . . . . . . . . . . . 158 Adıyaman, Meltem (Turkey) . . . . . . . . . . . . . . . . . . . . . . . . . . 158 Akman, Murat (Turkey) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158 Altunkaynak, Meltem (Turkey) . . . . . . . . . . . . . . . . . . . . . . . . 159 Aydın, Kemal (Turkey) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 ¨ Bas¸, Mujgan (Turkey) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 Bohner, Martin (USA) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160 ˙ Bozok, Ilknur (Turkey) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160 vii

¨ Budakc¸ı, Gulter (Turkey) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160 Can, Canan (Turkey) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 C ¸ aylak, Duygu (Turkey) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 C ¸ elebi, Okay (Turkey) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 C ¸ el˙ık, Cem (Turkey) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162 ¨ C ¸ el˙ık Kızılkan, Gulnur (Turkey) . . . . . . . . . . . . . . . . . . . . . . . . 162 C ¸ ınar, Ceng˙ız (Turkey) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162 Das¸, Sebahat Ebru (Turkey) . . . . . . . . . . . . . . . . . . . . . . . . . . 163 Den˙ız, Aslı (Turkey) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 Dong, Zhaoyang (Spain) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 Duman, Melda (Turkey) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164 Elaydi, Saber (USA) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164 Getimane, M´ario (Mozambique) . . . . . . . . . . . . . . . . . . . . . . . 164 ˙ ¨ us ¨ ¸ , Ibrah˙ Gum ım Hal˙ıl (Turkey) . . . . . . . . . . . . . . . . . . . . . . . . 165 ˘ Hat˙ıpoglu, Veysel Fuat (Turkey) . . . . . . . . . . . . . . . . . . . . . . . . 165 ˙ ¨ ¸ e (Turkey) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 Intepe, Gokc Jantarakhajorn, Khajee (Thailand) . . . . . . . . . . . . . . . . . . . . . . . 166 Kara, Ruk˙ıye (Turkey) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166 Kayar, Zeynep (Turkey) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166 ˆ (USA) . . . . . . . . . . . . . . . . . . . . . . . . . . 167 Kaymakc¸alan, B˙ıllur Kıyak Uc¸ar, Yel˙ız (Turkey) . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 Kongnuan, Supachara (Thailand) . . . . . . . . . . . . . . . . . . . . . . . 167 Kosareva, Natalia (Russia) . . . . . . . . . . . . . . . . . . . . . . . . . . . 168 Kulik, Yakov (Australia) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168 Kutay, V˙ıldan (Turkey) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168 Leonhardt, Andreas (Germany) . . . . . . . . . . . . . . . . . . . . . . . . 169 Lesaja, Goran (USA) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169 Marsh, Robert L. (USA) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169 Mısır, Ad˙ıl (Turkey) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170 Nurkanovi´c, Mehmed (Bosnia and Herzegovina) . . . . . . . . . . . . . . 170 Nurkanovi´c, Zehra (Bosnia and Herzegovina) . . . . . . . . . . . . . . . . 170 ¨ ¨ Ocalan, Ozkan (Turkey) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 viii

˙ Okumus¸, Israf˙ ıl (Turkey) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 ¨ Ozkan, Umut Mutlu (Turkey) . . . . . . . . . . . . . . . . . . . . . . . . . 171 ¨ Ozpınar, F˙ıgen (Turkey) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172 ¨ urk, ¨ Ozt Serm˙ın (Turkey) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172 ¨ gurlu, ˘ Ozu Ers˙ın (Turkey) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172 Reankittiwat, Paramee (Thailand) . . . . . . . . . . . . . . . . . . . . . . . 173 Ruffing, Andreas (Germany) . . . . . . . . . . . . . . . . . . . . . . . . . . 173 ˙ Savun, Ipek (Turkey) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 ˘ ˘ ¸ en (Turkey) . . . . . . . . . . . . . . . . . . . . . . . 174 Selmanogulları, Tugc Seneetantikul, Soporn (Thailand) . . . . . . . . . . . . . . . . . . . . . . . 174 Seyhan, G˙ızem (Turkey) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174 ˘ S¸ı˙ms¸ek, Dagıstan (Turkey) . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 Sizer, Walter (USA) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 Suhrer, Andreas (Germany) . . . . . . . . . . . . . . . . . . . . . . . . . . 175 Tas¸kara, Necat˙ı (Turkey) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176 Thongjub, Nawalax (Thailand) . . . . . . . . . . . . . . . . . . . . . . . . 176 Tollu, D. Turgut (Turkey) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176 Uc¸ar, Den˙ız (Turkey) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177 ¨ Unal, Mehmet (Turkey) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177 Vesarachasart, Sirichan (Thailand) . . . . . . . . . . . . . . . . . . . . . . 177 Vu, Dominik (Austria) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178 ¨ ¸ ı˙n (Turkey) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178 Yalazlar, Gulc ˙ Yalc¸ınkaya, Ibrah˙ ım (Turkey) . . . . . . . . . . . . . . . . . . . . . . . . . 178 ˘ ıder, Muhammed (Turkey) . . . . . . . . . . . . . . . . . . . . . . . . . 179 Y˙ıg˙ Yıldız, Mustafa Kemal (Turkey) . . . . . . . . . . . . . . . . . . . . . . . . 179 ¨ Yılmaz, Ozlem (Turkey) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 ¨ uk, ¨ Fulya (Turkey) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180 Yor Local Organization Assistants

181

Aydın, M. Aslı (Turkey) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182 Bayat, Kemal (Turkey) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182 ˘ Dagyar, Nazlı Ceren (Turkey) . . . . . . . . . . . . . . . . . . . . . . . . . 182 ix

¨ Yakup (Turkey) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 Emul, ¨ Erkal, Durdane (Turkey) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 ¨ ¸ e (Turkey) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 Karahan, Gokc Karakelle, Musa (Turkey) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184 ¨ ¨ Ozdem˙ ır, Husey˙ ın (Turkey) . . . . . . . . . . . . . . . . . . . . . . . . . . 184 ¨ Ozen, Bahad˙ır (Turkey) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184 Conference Proceedings

185

Social Program

187

Maps

189

Istanbul

197

Useful Information

205

Index and E-mail Addresses

209

x

Preface Dear Colleague: It is our great pride and pleasure to offer our warmest greetings to you, the participants of the “14th International Conference on Difference Equations and Applications (ICDEA2008)” at the Bes¸iktas¸ campus of Bahc¸es¸ehir University in Istanbul, Turkey. This confer¨ ITAK ˙ ence is sponsored by TUB (Scientific and Technical Research Council of Turkey), Bah¸ce¸sehir University, Dentur Avrasya, Duran Sandwiches, Pırıl Pırıl, and the Turkish Ministry of Culture and Tourism. The purpose of the conference is to bring together both experts and novices in the theory and applications of difference equations and discrete dynamical systems. The main theme of the meeting is dynamic equations on time scales. The previous ICDEA conferences were held in Lisbon (2007), Kyoto (2006), Munich (2005), Los Angeles (2004), Brno (2003), Changsha (2002), Augsburg (2001), Temuco (2000), Poznan (1998), Taipei (1997), Veszpr´em (1995), and San Antonio (1994). In addition to attending the conference’s exciting sessions, we encourage each of the participants to take advantage of our historic city of Istanbul, which is the cradle of many civilizations, to share beauty and scientific knowledge. Last but not least we want to extend our best wishes to all of the conference participants and to its Scientific and Organizing Committee members. Sincerely,

¨ Dr. Mehmet Unal Chair of Organizing Committee Bahc¸es¸ehir University TR-34538 Bahc¸es¸ehir/Istanbul, Turkey Conference web site: http://icdea.bahcesehir.edu.tr 1

2

Organizing Committee ¨ Mehmet Unal (Chair) Bahc¸es¸ehir University Istanbul, Turkey

Martin Bohner (Co-Chair) Missouri S&T Rolla, Missouri, USA

Okay C ¸ elebi Yeditepe University Istanbul, Turkey

Gerasimos Ladas University of Rhode Island Kingston, Rhode Island, USA

Aydın T˙ıryak˙ı Gazi University Ankara, Turkey

Agacık ˘ Zafer Middle East Technical University Ankara, Turkey

3

4

Scientific Committee Martin Bohner (Chair) Missouri S&T Rolla, Missouri, USA Zuzana Doˇsl´a (Co-Chair) Masaryk University Brno, Czech Republic

Saber Elaydi Trinity University San Antonio, Texas, USA

Metin Gurses ¨ Bilkent University Ankara, Turkey

Gusein Guseinov Atılım University Ankara, Turkey

B˙ıllur ˆ Kaymak¸calan Georgia Southern University Statesboro, Georgia, USA

Peter Kloeden Johann Wolfgang Goethe University Frankfurt am Main, Germany 5

Werner Kratz University of Ulm Ulm, Germany

Donald Lutz San Diego State University San Diego, California, USA

Jean Mawhin Universit´e Catholique de Louvain Louvain-la-Neuve, Belgium

Donal O’Regan National University of Ireland Galway, Ireland

Allan Peterson University of Nebraska–Lincoln Lincoln, Nebraska, USA

Alexander Sharkovsky National Academy of Sciences Kiev, Ukraine Gerald Teschl University of Vienna Vienna, Austria

6

ISDE Advisory Committee Kazuo Nishimura (Chair) Kyoto University Kyoto, Japan Andreas Ruffing (Co-Chair) Technical University Munich Munich, Germany

Henrique Oliveira Instituto Superior T´ecnico Lisbon Lisbon, Portugal

Robert J. Sacker University of Southern California Los Angeles, California, USA

7

8

Welcome from the ISDE President Dear Colleagues and ISDE Members: It is an honor to welcome you to the annual meeting of the International Society of Difference Equations in Istanbul, Turkey. The Fourteenth International Conference on Difference Equations and Applications ICDEA2008 is held on the campus of Bahc¸es¸ehir University, July 21–25, 2008. I welcome you to this historical meeting, where west meets east; you may be able to cross on foot from Asia to Europe and vice versa. Not only we have an excellent scientific program, but we have a splendid social program; don’t forget your camera. The ISDE Board of Directors meets on Tuesday, July 22th, 2008, 5:45 pm in the auditorium BFSAY. The general assembly of the society meets on Thursday, July 24th, 2008, 5:45 pm in the auditorium BFSAY. The main event is the presentation of the prize for the best paper published in the Journal of the Society (JDEA) in 2007. The prize carries the amount of £500 granted by Taylor & Francis. ¨ Finally, I would like to thank, on behalf of all of you, Dr. Mehmet Unal of the University of Bahc¸es¸ehir for his relentless efforts to make ICDEA2008 a reality. I am grateful to all members of the organizing, scientific, and advisory committees for their hard work and efforts to make this conference the best it can be. Sincerely,

Dr. Saber Elaydi President of ISDE Trinity University San Antonio, Texas, USA

9

10

ISDE Board of Directors Saber Elaydi (President) Trinity University San Antonio, Texas, USA

George Sell (Vice President) University of Minnesota Minneapolis, Minnesota, USA

Martin Bohner Missouri S&T Rolla, Missouri, USA J. M. Cushing University of Arizona Tucson, Arizona, USA

Istv´an Gyori ˝ University of Pannonia Veszpr´em, Hungary

11

Gerasimos Ladas University of Rhode Island Kingston, Rhode Island, USA

Allan Peterson University of Nebraska–Lincoln Lincoln, Nebraska, USA Andreas Ruffing Technical University Munich Munich, Germany

Robert J. Sacker University of Southern California Los Angeles, California, USA

12

Schedule

13

Time

July 20

July 21

July 22

July 23

July 24

July 25

Sunday

Monday

Tuesday

Wednesday

Thursday

Friday

Plenary Talk

Plenary Talk

5

7

8:00–9:00

Registration

Istanbul

9:00–9:45

Opening

Tour

9:45–10:40

Plenary Talk

Plenary Talk

1

3

Refreshment Break

10:45–11:00 11:00–11:55

Plenary Talk

Plenary Talk

2

4 Lunch

12:00–13:00 13:00–13:55

14:00–14:25

Registration

14:30–14:55

15:00–15:25

15:25–16:15

Registration

Istanbul

Refreshment Break Plenary Talk

Plenary Talk

6

8

Tour

Main Talks

Main Talks

1–3

4–6

Talks

Talks

1–4

29–32

Talks

Talks

5–8

33–36

Talks

Talks

9–12

37–40

Refreshment Break

Lunch Main Talks

Plenary Talk

Istanbul

7–8

9

Tour

Talks

Talks

57–60

81–84

Istanbul

Talks

Talks

Tour

61–64

85–88

Talks

Talks

65–68

89–92

Istanbul Tour

Refreshment Break

Talks

13–16

41–44

Talks

Talks

17–20

45–48

Talks

Talks

21–24

49–52

Talks

Talks

25–28

53–56

Yacht

Meeting

Welcome

Sightseeing

Sightseeing

Tour

Farewell

Sightseeing

Party

free time

free time

Dinner

free time

16:45–17:10

Registration

17:45–18:10

Evening

Tour

Talks

16:15–16:40

17:15–17:40

Istanbul

14

Talks

Talks

Istanbul

69–72

93–96

Tour

Talks

Talks

73–76

97–100

Istanbul

Talks

Closing

Tour

77–80 ISDE

Monday, July 21 (One-Hour Talks) Time

BFSAY

8:00–9:00

Registration

9:00–9:45

Opening

Chair

M. Bohner

9:45–10:40

Allan Peterson (USA) page 47

A205

Refreshment Break

10:45–11:00

11:00–11:55

A101

Ravi Agarwal (USA) page 36 Lunch Break

12:00–13:00 Chair

G. Ladas

R. Hilscher

B. Kaymakc¸alan

13:00–13:55

Istv´an Gyori ˝ (Hungary) page 41

Vera Zeidan (USA) page 52

Stefan Hilger (Germany) page 42

15

? Agarwal, Ravi: Discrete Lidstone boundary value problems ? Gy˝ori, Istv´an: Asymptotic representation of solutions of difference equations and limit formulas ? Hilger, Stefan: Difference equations appearing in ladder theory ? Peterson, Allan: An overview of dynamic equations on time scales ? Zeidan, Vera: Variational problems over time scales

16

Monday, July 21 (Contributed Talks)

Time

A101

A205

A206

A207

Chair

V. Zeidan

C. Kent

J. Appleby

N. Aghazadeh

Roman Hilscher (Czech Republic) page 93 Lynn Erbe (USA) page 82 Norah Esty (USA) page 84

Tamara Awerbuch Friedlander (USA) page 67 Mihaela Predescu (USA) page 127 Garyfalos Papaschinopoulos (Greece) page 118

Orlando Gomes (Portugal) page 86 Vivaldo Mendes (Portugal) page 108 Senada Kalabuˇsi´c (Bosnia/Herz.) page 95

Nicolae Pop (Romania) page 122 Volkan Oban (Turkey) page 114 Bakhodirzhon Siddikov (USA) page 142

14:00–14:25

14:30–14:55

15:00–15:25

Refreshment Break

15:25–16:15 Chair

16:15–16:40

16:45–17:10

17:15–17:40

17:45–18:10

S. Bodine

E. Camouzis

L. Alsed`a

Ralph Oberste-Vorth (USA) page 115 Ahmet Yantır (Turkey) page 151 Petr Stehlik (Czech Republic) page 146 Murat Adıvar (Turkey) page 55

Christos Schinas (Greece) page 138 V´ıctor Jim´enez Lopez ´ (Spain) page 94 Sandra Pinelas (Portugal) page 120 Nur˙ıye Atasever (Turkey) page 64

Diana Aldea Mendes (Portugal) page 59 Chris Bernhardt (USA) page 69 Jose S. C´anovas (Spain) page 74 Sara Costa (Spain) page 77

17

N. Popescu ˙ ı Inc˙ Albayrak (Turkey) page 58 Mohsen Shahrezaee (Iran) page 141 Moghtada Hashemiparast (Iran) page 91 Nasser Aghazadeh (Iran) page 57

? Adıvar, Murat: Periodicity in nonlinear systems of dynamic equations ? Aghazadeh, Nasser: Using B-spline scaling functions for solving integro-differential equations ˙ ı: A trace formula for an abstract Sturm–Liouville operator ? Albayrak, Inc˙ ? Aldea Mendes, Diana: Periodic and eventually periodic orbits for skew-product maps ? Atasever, Nur˙ıye: On a class of rational difference equations ? Awerbuch Friedlander, Tamara: Constructing difference equations models for public health policy ? Bernhardt, Chris: A Sharkovsky theorem for maps on trees ? C´anovas, Jose S.: A characterization of k-cycles ? Costa, Sara: Study of attractors for two-dimensional skew-products whose basis is a Denjoy counterexample ? Erbe, Lynn: Comparison and oscillation theorems for linear and half-linear dynamic equations on time scales ? Esty, Norah: Convergence of hyperspaces under the Fell topology, especially of time scales ? Gomes, Orlando: Optimal monetary policy with partially rational agents ? Hashemiparast, Moghtada: Solving systems of nonlinear equations by using difference equations ? Hilscher, Roman: Riccati equations for linear Hamiltonian systems ´ ? Jim´enez Lopez, V´ıctor: On the global stability of xn+1 =

p+qxn 1+xn−1

? Kalabuˇsi´c, Senada: Period-two trichotomies of a difference equation of order higher than two ? Mendes, Vivaldo: Learning to play Nash in deterministic uncoupled dynamics ? Oban, Volkan: Numerical solutions of nonlinear differential-difference equations by the variational iteration method ? Oberste-Vorth, Ralph: Solution spaces of dynamic equations over time scales space ? Papaschinopoulos, Garyfalos: Boundedness, attractivity, stability of a rational difference equation with two periodic coefficients ? Pinelas, Sandra: Bounded solutions of a rational difference equation ? Pop, Nicolae: Generalized Jacobians for solving nondifferentiable equations arising from contact problems ? Predescu, Mihaela: A nonlinear system of difference equations ? Schinas, Boundedness, periodicity, attractivity of the difference equation xn+1 = p  Christos: An + xxn−1 n ? Shahrezaee, Mohsen: Heat solutions by using Fibonacci tane function ? Siddikov, Bakhodirzhon: Applications of finite difference methods in the field of magnetic refrigeration ? Stehlik, Petr: Basic properties of partial dynamic operators ? Yantır, Ahmet: Positive solutions of a second-order m-point BVP on time scales

18

Tuesday, July 22 (One-Hour Talks) Time

BFSAY

Chair

Z. Doˇsl´a

9:45–10:40

¨ Huseyin Ko¸cak (USA) page 44

10:45–11:00

11:00–11:55

A101

A205

Refreshment Break Andr´e Vanderbauwhede (Belgium) page 49 Lunch Break

12:00–13:00 Chair

¨ C. Potzsche

A. Peterson

G. Teschl

13:00–13:55

Llu´ıs Alsed`a (Spain) page 38

Elvan Akın-Bohner (USA) page 37

Fritz Gesztesy (USA) page 40

19

? Alsed`a, Llu´ıs: A lower bound for the maximum topological entropy of 4k + 2-cycles ? Akın-Bohner, Elvan: Quasilinear dynamic equations ? Gesztesy, Fritz: Borg–Marchenko-type uniqueness results for CMV operators and elements of Weyl–Titchmarsh theory ¨ ? Koc¸ak, Huseyin: Rigorous computations in chaotic dynamical systems ? Vanderbauwhede, Andr´e: Stability of bifurcating periodic orbits of reversible maps

20

Tuesday, July 22 (Contributed Talks) Time

A101

A205

A206

A207

Chair

F. Atıcı

S. Pinelas

M. Rasmussen

V. Mendes

Thomas Matthews (USA) page 106 Julius Heim (USA) page 92 Bonita Lawrence (USA) page 104

Elias Camouzis (Greece) page 73 Yevgeniy Kostrov (USA) page 101 Mustapha Rachidi (France) page 129

Sara Fernandes (Portugal) page 85 Leopoldo Morales (Spain) page 113 Henrique Oliveira (Portugal) page 116

John Appleby (Ireland) page 62 Alexandra Rodkina (Jamaica) page 134 Michael McCarthy (Ireland) page 107

14:00–14:25

14:30–14:55

15:00–15:25

Refreshment Break

15:25–16:15 Chair

16:15–16:40

16:45–17:10

17:15–17:40

17:45–18:10

L. Erbe

M. Predescu

H. Oliveira

A. Rodkina

Tomasia Kulik (Australia) page 102 Christian Keller (USA) page 97 Atiya Zaidi (Australia) page 153 Ferhan Atıcı (USA) page 66

Michael Radin (USA) page 130 Candace Kent (USA) page 98 Gesthimani Stefanidou (Greece) page 145 Nurcan S¸ekerc˙ı (Turkey) page 140

Christian Potzsche ¨ (Germany) page 126 Andrejs Reinfelds (Latvia) page 133 Martin Rasmussen (Germany) page 131 Mouhaydine Tlem¸cani (Portugal) page 149

Jitka Laitochov´a (Czech Republic) page 103 ´ Jesus Abderram´an (Spain) page 54 Al˙ı Osman C ¸ ıbıkd˙ıken (Turkey) page 76 Ahmet Duman (Turkey) page 81

21

´ General solution of linear homogeneous difference equations with ? Abderram´an, Jesus: variable coefficients ? Appleby, John: Growth, long memory and heavy tails in difference equation models of inefficient financial markets ? Atıcı, Ferhan: Initial value problems in discrete fractional calculus ? Camouzis, Elias: On the global character of solutions of a rational system of difference equations ? C ¸ ıbıkd˙ıken, Al˙ı Osman: Effect of floating point on computation of monodromy matrix ? Duman, Ahmet: Sensitivity of Schur stable linear systems with periodic coefficients ? Fernandes, Sara: Systoles and topological entropy in discrete dynamical systems ? Heim, Julius: The dynamic multiplier-accelerator model in economics ? Keller, Christian: Dynamic equations with piecewise continuous argument ? Kent, Candace: A cardiac loop reentry model with thresholds ? Kostrov, Yevgeniy: Existence of unbounded solutions in rational equations ? Kulik, Tomasia: Solution to integral equations on time scales: Existence, uniqueness and successive approximations ? Laitochov´a, Jitka: Linear kth order functional and difference equations in the space of strictly monotonic functions ? Lawrence, Bonita: The Marshall differential analyzer: Dynamic equations in motion! ? Matthews, Thomas: Ostrowski inequalities on time scales ? McCarthy, Michael: Numerical detection of explosions and asymptotic behaviour of delaydifferential equations ? Morales, Leopoldo: An example of a strongly invariant, pinched core strip ? Oliveira, Henrique: Bifurcations for nonautonomous interval maps ¨ ? Potzsche, Christian: Nonautonomous continuation and bifurcation, revisited! ? Rachidi, Mustapha: On some rational difference equations via linear recurrence equations properties ? Radin, Michael: Multiple periodic solutions of a second-order nonautonomous rational difference equation ? Rasmussen, Martin: Morse spectrum for linear nonautonomous difference equations ? Reinfelds, Andrejs: Decoupling and simplifying of discrete dynamical systems in the neighbourhood of invariant manifold ? Rodkina, Alexandra: On oscillation of solutions of stochastically perturbed difference equations ? S¸ekerc˙ı, Nurcan: On the behaviour of the difference equation x(n + 1) = max{1/x(n), min{1, A/x(n)}} ? Stefanidou, Gesthimani: On a system of max-difference equations ? Tlemc¸ani, Mouhaydine: Analysis of a nonlinear discrete dynamical system, signal coding and reconstruction ? Zaidi, Atiya: A result on successive approximation of solutions to dynamic equations on time scales

22

Thursday, July 24 (One-Hour Talks) Time

BFSAY

Chair

J. Cushing

9:45–10:40

James A. Yorke (USA) page 50

10:45–11:00

11:00–11:55

12:00–13:00

A101

Refreshment Break Hal Smith (USA) page 48 Lunch Break

Chair

F. Gesztesy

B. Lawrence

13:00–13:55

Jean Mawhin (Belgium) page 46

˘ Agacık Zafer (Turkey) page 51

23

? Mawhin, Jean: Boundary value problems for nonlinear difference equations with discrete singular φ-Laplacian ? Smith, Hal: Some persistence results for discrete-time dynamical systems and applications ? Yorke, James A.: Period doubling cascades in one-parameter families of maps in high dimensions ˘ ? Zafer, Agacık: Interval criteria for second-order super-half-linear functional dynamic equations

24

Thursday, July 24 (Contributed Talks)

Time

A101

A205

A206

A207

Chair

N. Esty

S. Hilger

T. Awerbuch Friedlander

A. T˙ıryak˙ı

Devr˙ım C ¸ akmak (Turkey) page 72 Pavel ˇ ak Reh´ (Czech Republic) page 132 Raz˙ıye Mert (Turkey) page 109

Zuzana Doˇsl´a (Czech Republic) page 80 Vitaliy Kharkov (Ukraine) page 99 Gusein Guseinov (Turkey) page 88

Moritz Simon (Germany) page 143 Małgorzata Guzowska (Poland) page 90 Rafael Lu´ıs (Portugal) page 105

Mikhail Kipnis (Russia) page 100 ¨ Ozlem Ak Gum ¨ u¸ ¨s (Turkey) page 87 Choonkil Park (South Korea) page 119

14:00–14:25

14:30–14:55

15:00–15:25

Refreshment Break

15:25–16:15 Chair

16:15–16:40

16:45–17:10

17:15–17:40

17:45–18:10

M. Adıvar

J. Michor

A. Reinfelds

G. Guseinov

Bas¸ak Karpuz (Turkey) page 96 Erb˙ıl C ¸ et˙ın (Turkey) page 75 Zdenˇek Posp´ısˇ il (Czech Republic) page 125

Mih´aly Pituk (Hungary) page 121 Gerald Teschl (Austria) page 147 Ruk˙ıye ¨ urk Ozt ¨ (Turkey) page 117

Ziyad Al-Sharawi (Oman) page 60 J. M. Cushing (USA) page 78 Fozi Dannan (Syria) page 79

Jehad Alzabut (Turkey) page 61 Aydın T˙ıryak˙ı (Turkey) page 148 Sigrun Bodine (USA) page 70

ISDE General Meeting (BFSAY)

25

? Al-Sharawi, Ziyad: The effect of harvesting strategies on the discrete Beverton–Holt model ? Alzabut, Jehad: Asymptotic behavior of linear impulsive delay difference equations ? Bodine, Sigrun: Exponentially asymptotically constant systems of difference equations with applications ? C ¸ akmak, Devr˙ım: On the equivalence of Rolle’s and generalized mean value theorems on time scales ? C ¸ et˙ın, Erb˙ıl: Higher-order boundary value problems on time scales ? Cushing, J. M.: Difference equations arising in dynamic models of biological evolution ? Dannan, Fozi: A new proof for the Levin–May criterion of asymptotic stability ? Doˇsl´a, Zuzana: On nonoscillation of Emden–Fowler difference equations ¨ ¨ us ¨ ¸ , Ozlem ? Gum Ak: Stability boundary for asymptotic stability of scalar equations ? Guseinov, Gusein: Spectral analysis of a non-selfadjoint second-order difference operator ? Guzowska, Małgorzata: Discrete Haavelmo growth cycle model ? Karpuz, Bas¸ak: Iterated oscillation criteria for delay dynamic equations of first order ? Kharkov, Vitaliy: Asymptotic behavior of one class solutions of the second-order Emden– Fowler difference equation ? Kipnis, Mikhail: Stability via Convexity ? Lu´ıs, Rafael: Nonautonomous periodic systems with Allee effect ? Mert, Raz˙ıye: Time scale extensions of a theorem of Wintner on systems with asymptotic equilibrium ¨ urk, ¨ ? Ozt Ruk˙ıye: On the spectrum of normal difference operators of first order ? Park, Choonkil: Classification and stability of functional equations ? Pituk, Mih´aly: Nonoscillatory solutions of a second-order difference equation of Poincar´e type ? Posp´ısˇ il, Zdenˇek: Dynamic replicator equation and its transformation ˇ ak, Pavel: Power type comparison theorems for half-linear dynamic equations ? Reh´ ? Simon, Moritz: Spectral theory of birth-and-death processes ? Teschl, Gerald: Relative oscillation theory for Jacobi operators ? T˙ıryak˙ı, Aydın: Reducibility and stability results for linear systems of difference equations

26

Friday, July 25 (One-Hour Talks) Time

BFSAY

Chair

O. C ¸ elebi

9:45–10:40

Ondˇrej Doˇsly´ (Czech Republic) page 39

10:45–11:00

11:00–11:55

Refreshment Break Peter Kloeden (Germany) page 43

12:00–13:00

Lunch Break

13:00–13:55

Gerasimos Ladas (USA) page 45

27

? Doˇsly, ´ Ondˇrej: Symplectic difference systems ? Kloeden, Peter: Spatial discretisation of dynamical systems ? Ladas, Gerasimos: Open problems and conjectures in difference equations

28

Friday, July 25 (Contributed Talks)

Time

A101

A205

A206

A207

Chair

A. Zafer ¨ Ozlem

M. Pituk

B. Siddikov

J. Laitochov´a

Małgorzata Migda (Poland) page 112 Ewa Schmeidel (Poland) page 139 Yas¸ar Bolat (Turkey) page 71

Nedelia Antonia Popescu (Romania) page 124 David Romero i S´anchez (Spain) page 135 Al˙ı Sırma (Turkey) page 144

Johanna Michor (USA) page 111 Pablo Sanchez-Moreno (Spain) page 137 Samar Aseeri (Saudi Arabia) page 63

14:00–14:25

14:30–14:55

15:00–15:25

Batıt (Turkey) page 68 Petr Zem´anek (Czech Republic) page 155 Fatma Serap Topal (Turkey) page 150

Refreshment Break

15:25–16:15 Chair

16:15–16:40

16:45–17:10

17:15–17:40

F. Topal

¨ Ocalan ¨ O.

A. Sırma

S. Aseeri

Mozhdeh Afshar Kermani (Iran) page 56 Fat˙ıhcan M. Atay (Germany) page 65

A. Feza Guven˙ ¨ ıl˙ır (Turkey) page 89 Mohsen Rabbani (Iran) page 128

Ahmet Yıldırım (Turkey) page 152 Ali Zakeri (Iran) page 154

Meltem Erol (Turkey) page 83 Hamid Mesgarani (Iran) page 110

Closing (BFSAY)

29

? Afshar Kermani, Mozhdeh: A new method for solving fuzzy partial differential equations ? Aseeri, Samar: Asymptotic formulas for Laplace integrals ? Atay, Fat˙ıhcan M.: Stability of coupled difference equations with delays ¨ ? Batıt, Ozlem: Fredholm integral equations on time scales ? Bolat, Yas¸ar: Necessary and sufficient conditions for oscillation of certain higher order partial difference equations ? Erol, Meltem: The structure of the spectrum for normal operators ¨ ? Guven˙ ıl˙ır, A. Feza: Interval oscillation of second-order difference equations with oscillatory potentials ? Mesgarani, Hamid: A new approach for solving Fredholm integro-difference equations ? Michor, Johanna: Algebro-geometric solutions of the Ablowitz–Ladik hierarchy ? Migda, Małgorzata: Oscillatory and asymptotic properties of solutions of nonlinear neutraltype difference equations ? Popescu, Nedelia Antonia: Finite size scaling technique and applications ? Rabbani, Mohsen: Galerkin method for solving nonlinear Fredholm–Hammerstein integral equations with multiwavelet basis ? Romero i S´anchez, David: Invariant objects through wavelets ? Sanchez-Moreno, Pablo: Discrete densities and Fisher information ? Schmeidel, Ewa: Oscillation of nonlinear three-dimensional difference systems ¨ ? Sırma, Al˙ı: Numerical solution of nonlocal boundary value problems for the Schrodinger equation ? Topal, Fatma Serap: Multiple positive solutions for a system of higher-order boundary value problems on time scales ? Yıldırım, Ahmet: Numerical solutions of nonlinear differential-difference equations by the homotopy perturbation method ? Zakeri, Ali: Application of the WKB estimation method for determining heat flux on the boundary ? Zem´anek, Petr: Trigonometric and hyperbolic systems on time scales

30

One-Hour Speakers

Ravi Agarwal Florida Institute of Technology Melbourne, Florida, USA

Elvan Akın-Bohner Missouri S&T Rolla, Missouri, USA

Llu´ıs Alsed`a ` Universitat Autonoma de Barcelona Cerdanyola del Vall`es, Spain

Ondˇrej Doˇsly´ Masaryk University Brno, Czech Republic

Fritz Gesztesy University of Missouri Columbia, Missouri, USA

Istv´an Gyori ˝ University of Pannonia Veszpr´em, Hungary

31

Stefan Hilger Catholic University of Eichst¨att Eichst¨att, Germany

Peter Kloeden Johann Wolfgang Goethe University Frankfurt am Main, Germany

Huseyin ¨ Ko¸cak University of Miami Miami, Florida, USA

Gerasimos Ladas University of Rhode Island Kingston, Rhode Island, USA

Jean Mawhin Universit´e Catholique de Louvain Louvain-la-Neuve, Belgium

Allan Peterson University of Nebraska–Lincoln Lincoln, Nebraska, USA

32

Hal Smith Arizona State University Tempe, Arizona, USA

Andr´e Vanderbauwhede Ghent University Ghent, Belgium

James A. Yorke University of Maryland College Park, Maryland, USA

Agacık ˘ Zafer Middle East Technical University Ankara, Turkey

Vera Zeidan Michigan State University East Lansing, Michigan, USA

33

34

Abstracts of One-Hour Talks

35

Discrete Lidstone boundary value problems R AVI A GARWAL Florida Institute of Technology Department of Mathematics Melbourne, Florida, USA [email protected] http://cos.fit.edu/math/faculty/agarwal We shall provide sufficient conditions for the existence of single and multiple positive solutions of higher order smooth as well as singular difference equations involving Lidstone boundary conditions. As an application, we shall investigate the existence of radial solutions of certain partial difference equations. To show how easily our results can be applied in practice we shall illustrate many examples.

36

Quasilinear dynamic equations E LVAN A KIN -B OHNER Missouri University of Science and Technology Department of Mathematics and Statistics Rolla, Missouri, USA [email protected] http://web.mst.edu/˜akine We consider a quasilinear dynamic equation reducing to a half-linear equation, Emden– Fowler equation or a Sturm–Liouville equation on a time scale, which is a nonempty closed subset of the real numbers. Any nontrivial solution of a quasilinear equation is eventually monotone. In other words, it can be either positive decreasing (negative increasing) or positive increasing (negative decreasing). We shall provide certain integral conditions to classify solutions and investigate their asymptotic behaviors.

AMS Subject Classification: 39A10. Keywords: Time scales, quasilinear, half-linear equation.

37

A lower bound for the maximum topological entropy of 4k + 2-cycles L LU´I S A LSED A` Universitat Aut`onoma de Barcelona Departament de Matem`atiques Cerdanyola del Vall`es, Spain [email protected] http://www.mat.uab.cat/˜alseda For continuous interval maps we formulate a conjecture on the shape of the cycles of maximum topological entropy of period 4k + 2. We also present numerical support for the conjecture. This numerical support is of two different kinds. For periods 6, 10, 14 and 18 we are able to compute the maximum entropy cycles by using nontrivial, ad hoc numerical procedures and the known results of Jungreis (1991). In fact, the conjecture we formulate is based on these results. For periods n = 22, 26 and 30 we compute the maximum entropy cycle of a restricted subfamily of cycles denoted by Cn∗ . The obtained results agree with the conjectured ones. The conjecture that we can restrict our attention to Cn∗ is motivated theoretically. On the other hand, it is ∗ ∗ ∗ , C26 and C30 is much less than worth noticing that the complexity of examining all cycles in C22 the complexity of computing the entropy of each cycle of period 18 in order to determine the ones with maximal entropy, therefore making it a feasible problem.

AMS Subject Classification: 37B40, 37E15, 37M99. Keywords: Combinatorial dynamics, interval map. This is joint work with David Juher and Deborah King.

38

Symplectic difference systems ˇ D O Sˇ L Y´ O ND REJ Masaryk University Department of Mathematics and Statistics Brno, Czech Republic [email protected] http://www.muni.cz/people/Ondrej.Dosly Symplectic diference systems are first order systems with the property that their fundamental matrix is symplectic whenever it is symplectic at one point. From this point of view, they can be regarded as a discrete counterpart of linear Hamiltonian differential systems. Symplectic systems cover a large variety of difference equations and systems, among them the second order Sturm–Liouville difference equation whose oscillation theory is deeply developed. We will present recent results of the oscillation and spectral theory of symplectic systems. In particular, it will be shown that the classical Sturmian separation and comparison theory can be extended to symplectic systems. The presented results have been achieved in the joint research with Martin Bohner (Univ. Rolla, Missouri, USA) an Werner Kratz (Univ. Ulm, Germany).

AMS Subject Classification: 39A10. Keywords: Sturmian theory, focal point, Picone’s identity.

39

Borg–Marchenko-type uniqueness results for CMV operators and elements of Weyl–Titchmarsh theory F RITZ G ESZTESY University of Missouri Department of Mathematics Columbia, Missouri, USA [email protected] http://www.math.missouri.edu/˜fritz We review local and global versions of Borg–Marchenko-type uniqueness theorems for halflattice and full-lattice CMV operators (CMV for Cantero, Moral, and Vel´azquez) with matrixvalued Verblunsky coefficients. While our half-lattice results are formulated in terms of matrixvalued Weyl–Titchmarsh functions, our full-lattice results involve the diagonal and main offdiagonal Green’s matrices. We also hint at the basics of Weyl–Titchmarsh theory for CMV operators with matrix-valued Verblunsky coefficients as this is of independent interest and an essential ingredient in proving the corresponding Borg–Marchenko-type uniqueness theorems. This is based on joint work with Steve Clark and Maxim Zinchenko.

AMS Subject Classification: Primary 34E05, 34B20, 34L40; Secondary 34A55. Keywords: CMV operators, (inverse) spectral theory.

40

Asymptotic representation of solutions of difference equations and limit formulas ˝ ´ G Y ORI I STV AN University of Pannonia Department of Mathematics and Computing Veszpr´em, Hungary [email protected] http://www.szt.vein.hu/˜gyori In this lecture we investigate the growth/decay rate of solutions of linear and quasilinear difference equations. The results can be applied to a particular kind of weight sequences which can be either exponential or slowly decaying. Examples are given to illustrate the sharpness of the results.

AMS Subject Classification: 39A12. Keywords: Limit formulas, difference equations.

41

Difference equations appearing in ladder theory S TEFAN H ILGER Catholic University of Eichst¨att Mathematisch-Geographische Fakult¨at Eichst¨att, Germany [email protected] http://www.ku-eichstaett.de/Fakultaeten/MGF/ ,→Didaktiken/dphys/Mitarbeiter.de − A ladder consists of a sequence of vector spaces (Vn ) and linear operators (A+ n ), (An ), depending on n, acting between these vector spaces in ascending and descending direction. The + job in ladder theory is to find SIE-subladders, on which the intrinsic endomorphisms A− n An and − A+ n An act as scalars αn . A fundamental ladder theorem will provide conditions on the (generalized) commutators or anticommutators assuring the existence of SIE-subladders. Elementary difference operators will enter into those conditions. The second part contains examples of ladders from classical quantum mechanics or orthogonal polynomials. In a final part we point out how to generalize the notion of a ladder to higher dimensional settings with corresponding bidirectional operators.

AMS Subject Classification: 81S05, 39A10, 42C05, 46L65, 34L40. Keywords: Ladder theory, canonical commutator relations.

42

Spatial discretisation of dynamical systems P ETER K LOEDEN Johann Wolfgang Goethe University Department of Mathematics Frankfurt am Main, Germany [email protected] http://www.math.uni-frankfurt.de/˜numerik/kloeden We consider the effects of spatial discretisation on the dynamical behavior of discrete time dynamical systems which are generated by difference equations. This is important, for example when we simulate such systems on computers which have only finite number fields. What is th effect of round-off? A simple example is the chaotic behavior of the tent mapping on the unit interval, which collapses when the mapping is restricted to a the subset of N -dyadic numbers. We will show that invariant measures are more robust to approximation. We consider a Lebesgue measure preserving mapping on torus and its approximation by a permutation of a uniform grid on the torus. Then, more generally, we show how Markov chains can be used to obtain approximations to the invariant measures of discrete time dynamical systems.

Keywords: Discretisation, invariant measures, Markov chains.

43

Rigorous computations in chaotic dynamical systems ¨ H USEYIN K OC¸ AK University of Miami Department of Mathematics / Department of Computer Science Miami, Florida, USA [email protected] http://www.math.miami.edu/˜hk Numerical simulations are indespensible in the investigation of specific dynamical systems. Unfortunately, since chaotic dynamical systems amplify small errors at an exponential rate, the results of most simulations are unreliable. In this talk, we will descibe the medhod of shadowing for extracting mathematically rigorous results from numerical computations. In particular, we will present a computer-assisted procedure for proving the existence of transversal homoclinic and heteroclinic orbits. The talk will be illustrated with computer simulations.

44

Open problems and conjectures in difference equations G ERASIMOS L ADAS University of Rhode Island Department of Mathematics Kingston, Rhode Island, USA [email protected] http://www.math.uri.edu/˜gladas We present some open problems and conjectures about some interesting types of difference equations. We are primarily interested in the boundedness nature of solutions, the periodic character of the equation, the global stability behavior of the equilibrium points, and with convergence to periodic solutions including periodic trichotomies.

45

Boundary value problems for nonlinear difference equations with discrete singular φ-Laplacian J EAN M AWHIN Universit´e Catholique de Louvain Department of Mathematics Louvain-la-Neuve, Belgium [email protected] We study the existence and multiplicity of solutions for boundary value problems of the type ∇[φ(∆xk )] + fk (xk , 4xk ) = 0

(2 ≤ k ≤ n − 1),

l(x, 4x) = 0,

where φ : (−a, a) → R denotes an increasing homeomorphism such that φ(0) = 0 and 0 < a < ∞, l(x, 4x) = 0 denotes the Dirichlet, periodic or Neumann boundary conditions and fk (2 ≤ k ≤ n − 1) are continuous functions. Our main tool is Brouwer degree together with fixed point reformulations of the above problems.

AMS Subject Classification: 39A12, 55M25. Keywords: Nonlinear difference equations, φ-Laplacian. This is joint work with Cristian Bereanu.

46

An overview of dynamic equations on time scales A LLAN P ETERSON University of Nebraska–Lincoln Department of Mathematics Lincoln, Nebraska, USA [email protected] http://www.math.unl.edu/˜apeterson1 The talk will be an overview of dynamic equations on time scales. We will discuss the importance of this emerging area of mathematics and discuss some important results in this area. Some introductory results will also be presented.

AMS Subject Classification: 39. Keywords: Time scales, dynamic equations.

47

Some persistence results for discrete-time dynamical systems and applications H AL S MITH Arizona State University Department of Mathematics Tempe, Arizona, USA [email protected] http://math.la.asu.edu/˜halsmith The theory of persistence focuses on identifying sufficient conditions for certain subsets of the state space to be repellers for the considered dynamics. In an ecological setting, these subsets are often extinction states for one or more populations while in an epidemiological setting they may be disease-free states. We survey some recent results in this area and apply them to models in population biology and epidemiology.

Keywords: Persistence theory.

48

Stability of bifurcating periodic orbits of reversible maps A NDR E´ VANDERBAUWHEDE Ghent University Department of Mathematics Ghent, Belgium [email protected] http://cage.ugent.be/˜avdb The Lyapunov–Schmidt method for the bifurcation of periodic orbits of local diffeomorphisms results in a reduced problem with an additional cyclic symmetry. We show how the method can be refined such that it also gives information on the stability of the bifurcating periodic orbits. We apply this approach (via a Poincar´e map) to the problem of subharmonic bifurcations in continuous reversible systems, discussing both the generic case and a particular degenerate case. A numerical study of a model example for this degenerate situation reveals some nongeneric stability behaviour in the presence of certain first integrals. We describe the results of a detailed analysis for this conservative case, including the transition scenario to the nonconservative case. ˜ This is joint work with Francisco Javier Munoz-Almar´ az (Valencia), and Jorge Gal´an and Emilio Freire (Sevilla).

49

Period doubling cascades in one-parameter families of maps in high dimensions J AMES A. Y ORKE University of Maryland Department of Mathematics College Park, Maryland, USA [email protected] http://yorke.umd.edu Evelyn Sander and I show infinite period-doubling cascades exist for high-dimensional systems.

50

Interval criteria for second-order super-half-linear functional dynamic equations ˘ A GACIK Z AFER Middle East Technical University Department of Mathematics Ankara, Turkey [email protected] http://www.metu.edu.tr/˜zafer Interval oscillation criteria are established for second-order forced super half-linear dynamic equations on time scales containing both delay and advance arguments, where the potentials are allowed to change sign. Examples are given to illustrate the relevance of the results. The theory can be applied to second-order dynamic equations regardless of the choice of delta or nabla derivatives.

AMS Subject Classification: 34K11, 34C10, 39A11, 39A13. Keywords: Time scales, oscillation, functional, half-linear. This is joint work with Douglas R. Anderson.

51

Variational problems over time scales V ERA Z EIDAN Michigan State University Department of Mathematics East Lansing, Michigan, USA [email protected] http://www.math.msu.edu/˜zeidan This talk focuses on the study of variational problems over time scale which encompasses both nonlinear optimal control and calculus of variations problems. The main goal is centered on the question of deriving necessary and sufficient optimality criteria of first and second order. The special feature resides in the fact that these conditions are formulated in terms of a certain “Hamiltonian” corresponding to the nonlinear problem. The second order conditions are obtained in terms of the accessory problem. However, Reid roundabout theorems, that are recently obtained with R. Hilscher, allow these conditions to be equivalently phrased in terms of conjoined basis and Riccati equations corresponding to the accessory problem.

52

Abstracts of Contributed Talks

53

General solution of linear homogeneous difference equations with variable coefficients ´ A BDERRAM AN ´ J ES US Universidad Politecnica de Madrid, Campus Montegancedo Department of Applied Mathematics, Faculty of Computer Science Madrid, Spain [email protected] http://www.dma.fi.upm.es/jesus A constructive theory for the general solution of kth-order difference equation x(k) (n + 1) =

k−1 X

pi+1 (n)x(k) (n − i)

i=0

is given as in a forthcoming paper of the author. As complement of the analytical theory [George D. Birkhoff, General theory of linear difference equations, Transactions of the American Mathematical Society, volume 12, number 2, pages 243–284, 1911], this constructive approach permits us an explicit and nonrecurrent representation of the general solution, for any initial conditions, n−1 x−1 , x0 , . . . , xk−2 , and any sequences of complex numbers, {pi+1 (j)}j=k−2 , i = 0, . . . , k − 1. If k = 1, then the solution is straightforward. For k > 1, a simple change of variable produces an equivalent kth-order linear difference equation that permits us to solve x(k) (n), n ≥ k − 1, by induction on n. Since the representation for the general case is too long, the solution for k = 2 is provided here as an illustration: x(2) (n) =

n−1 Y

  (2,1) p1 (i) c0 Φ(2,0) (α2 (1), . . . , α2 (n − 1)) + c−1 α2 (0)Φn−1 (α2 (2), . . . , α2 (n − 1)) , n

i=0 (2,j)

p2 (k) , α2 (0) = pp21 (0) α) where c−1 , c0 are arbitrary numbers, n ≥ 1, j = 0, 1, α2 (k) = p1 (k−1)p (0) . Φn−j (~ 1 (k) are:      b n−j kl−1 −2 kX n−1 1 −2 2 c X X X (2,j)  Φn−j (~ α) = α2 (k1 )  α2 (k2 ) · · ·  α2 (kl ) . l=0

k1 =2l−1+j

k2 =2(l−1)−1+j

When l = 0, the sum is 1 by convention. AMS Subject Classification: 39A05. Keywords: Linear difference equations.

54

kl =1+j

Periodicity in nonlinear systems of dynamic equations M URAT A DIVAR ˙ Izmir University of Economics ˙ Izmir, Turkey [email protected] http://homes.ieu.edu.tr/˜madivar By means of Schaefer’s fixed point theorem, we show the existence of periodic solutions of a nonlinear system of Volterra-type integro-dynamic equations. Furthermore, we provide several applications to scalar equations, where we develop a time scale analogue of Lyapunov’s direct method and prove an analogue of Sobolev’s inequality on time scales to arrive at a priori bound on all periodic solutions.

AMS Subject Classification: 39A10. Keywords: Time Scale, dynamic equation, fixed point theorems. This is joint work with Youssef Raffoul.

55

A new method for solving fuzzy partial differential equations M OZHDEH A FSHAR K ERMANI Islamic Azad University North Tehran Branch Department of Mathematics Tehran, Iran mog [email protected] In this talk a new method for solving “fuzzy partial differential equations” (FPDE) is considered. This numerical method based on the definition of derivative that considered by Y. ChalcoCano, H. Roman-Flores. We present a difference method to solve FPDEs such as the fuzzy hyperbolic equation and fuzzy parabolic equation, then see if stability of this method exist, and conditions for stability are given. Examples are presented showing the Hausdorff distance between the exact solution and approximate solution is small.

Keywords: Fuzzy partial differential equation, difference method.

56

Using B-spline scaling functions for solving integro-differential equations N ASSER A GHAZADEH Azarbaijan University of Tarbiat Moallem Department of Mathematics Tabriz, Iran [email protected] http://www.azaruniv.edu/˜aghazadeh In this talk, quadratic semiorthogonal B-spline scaling functions together with their dual functions are developed to approximate the solutions of linear second-order Fredholm integrodifferential equations. The quadratic B-spline scaling functions, their properties and the operational matrices of derivative for B-spline scaling functions are presented and are utilized to reduce the solution of Fredholm integro-differential to the solution of algebraic equations. The method is computationally attractive, some numerical examples are presented to support our work.

AMS Subject Classification: 45B05, 45A05, 65D07, 42C05. Keywords: Quadratic spline, Fredholm integro-differential equation.

57

A trace formula for an abstract Sturm–Liouville operator I˙ NC I˙ A LBAYRAK Yıldız Technical University Mathematical Engineering Department Istanbul, Turkey [email protected] http://www.mtm.yildiz.edu.tr/cvler/ialbayrak In this talk we investigate and obtain a regularized trace formula for the operator in the Hilbert space L2 ([0, 1], H) generated by the expression −y 00 (x) + Q(x)y(x) with the boundary conditions y(0) = 0,

y 0 (1) + Ay(1) = 0,

where H is a separable Hilbert space, for x ∈ [0, 1], Q(x) is a self-adjoint nuclear operator defined in H, and A is a real number.

This is joint work with Kevser Koklu and Azad Bayramov

58

Periodic and eventually periodic orbits for skew-product maps D IANA A LDEA M ENDES IBS-ISCTE Business School, ISCTE Department of Quantitative Methods Lisbon, Portugal [email protected] http://iscte.pt/˜deam In this talk we consider triangular (or skew-product) maps of the real plane that admit periodic and eventually periodic critical orbits. A corresponding Markov partition will be constructed for these maps. We also show that there exist an invariant probability measures, namely the Parry measure. In order to obtain these, we apply some tensor products between the invariants associated with the one-dimensional components of the triangular map. An immediate consequence is the computation of the topological and metric entropy for these maps.

AMS Subject Classification: Primary 37B10, 37B40, 37E30; Secondary 15A69. Keywords: Skew product, periodic orbits, Markov partition.

59

The effect of harvesting strategies on the discrete Beverton–Holt model Z IYAD A L -S HARAWI Sultan Qaboos University Department of Mathematics and Statistics Al-Khod, Muscat, Oman [email protected] We discuss the effect of constant, periodic and conditional harvesting strategies on the discrete Beverton–Holt model. We find that for large initial populations, constant harvesting gives the maximum sustainable yield. Periodic harvesting has a short term advantage when the initial population is small, and conditional harvest has the advantage of lowering the risk of extinction. Also, we discuss the periodic character in each case, and show that periodic harvesting drives population cycles to be multiples (period wise) of the harvesting period.

AMS Subject Classification: 39A11, 92D25, 92B99. Keywords: Beverton–Holt model, optimal harvesting. This is joint work with Mohamed Rhouma.

60

Asymptotic behavior of linear impulsive delay difference equations J EHAD A LZABUT C ¸ ankaya University Department of Mathematics and Computer Science Ankara, Turkey [email protected] http://math.cankaya.edu.tr/˜jehad In this talk, it is shown that if a linear impulsive delay difference equation satisfies Perron’s condition, then its trivial solution is asymptotically stable.

AMS Subject Classification: 39A13, 34K45. Keywords: Impulse, delay, adjoint, Perron, stability. This is joint work with Thabet Abdeljawad.

61

Growth, long memory and heavy tails in difference equation models of inefficient financial markets J OHN A PPLEBY Dublin City University School of Mathematical Sciences Dublin, Ireland [email protected] http://webpages.dcu.ie/˜applebyj In this talk we explore the asymptotic behaviour of a stochastic difference equation model of a financial market in which traders use the past behaviour of prices to guide their investment decisions. For a class of affine and maximum type functional difference equations we find an exact exponential rate of growth, just as is seen in classical efficient market models. We also show that these models possess the property of “long memory” in that the autocovariance function of the returns decays so slowly that it is nonintegrable. Furthermore, the asset returns are seen to exhibit “heavy tails”, in that the distribution function of the returns decay polynomially. All the results will be shown to be dynamically consistent with corresponding continuoustime functional differential equation models. The work is joint with Huizhong Wu and Catherine Swords and is supported by the SFI RFP Grant 05/MAT/0018 “Stochastic Functional Differential Equations with Long Memory”.

Keywords: Stochastic difference equation, long memory. This is joint work with Catherine Swords and Huizhong Wu.

62

Asymptotic formulas for Laplace integrals S AMAR A SEERI Umm Al-Qura University Department of Mathematics Makkah, Saudi Arabia [email protected] Solutions of boundary value problems of mathematical physics often involve infinite integrals containing a term consisting of a trigonometrical or Bessel function, with the aid of Laplace integral transform, as an integral equation of the first kind, the solution of the integral equation is obtained. In this talk, many applications on this manner are discussed and solved.

AMS Subject Classification: 65R10. Keywords: Laplace transforms, boundary value problems.

63

On a class of rational difference equations N UR I˙ YE ATASEVER Sel¸cuk University Department of Mathematics, Education Faculty Konya, Turkey atasever [email protected] In this talk we study the behaviour of the positive solutions of the nonlinear difference equation xn+1 = ((xn−k )/(1 + xn−1 xn−3 . . . xn−(k−2) )),

n = 0, 1, 2, . . . ,

where k > 2 is an odd integer.

AMS Subject Classification: 39A10. Keywords: Difference equation, positive solutions. ˙ ˘ This is joint work with Cengiz Cinar, Dagıstan S¸ims¸ek, and Ibrahim Yalc¸ınkaya.

64

Stability of coupled difference equations with delays FAT I˙ HCAN M. ATAY Max Planck Institute Mathematics in the Sciences Leipzig, Germany [email protected] http://personal-homepages.mis.mpg.de/fatay Networks of diffusively-coupled scalar maps are considered with weighted connections which may include a time delay. The stability of equilibria is studied with respect to the delays and connection structure. It is shown that the largest eigenvalue of the graph Laplacian determines the effect of the connection topology on stability. The stability region in the parameter plane shrinks with increasing values of the largest eigenvalue, or of the time delay of the same parity. In particular, all bipartite graphs have an identical stability region, regardless of the delay or graph size, which is also the smallest stability region among those of all graphs. Furthermore, for certain parameter ranges, unstable (and possibly chaotic) maps can be stabilized via diffusive coupling with an odd time delay, provided that the network does not have a nontrivial and connected bipartite component. On the other hand, stabilization is not possible for even values of the delay or for bipartite networks. Reference: F. M. Atay and O. Karabacak. Stability of coupled map networks with delays. SIAM Journal on Applied Dynamical Systems, 5:508–527, 2006.

AMS Subject Classification: 39A11, 37E05, 94C15. Keywords: Network, delay, stability, synchronization, chaos, Laplacian.

65

Initial value problems in discrete fractional calculus F ERHAN ATICI Western Kentucky University Department of Mathematics Bowling Green, Kentucky, USA [email protected] http://www.wku.edu/˜ferhan.atici This paper is devoted to the study of discrete fractional calculus; the particular goal is to define and solve well-defined discrete fractional difference equations. For this purpose we first carefully develop the commutativity properties of the fractional sum and the fractional difference operators. Then a ν-th (0 < ν ≤ 1) order fractional difference equation is defined. A nonlinear problem with an initial condition is solved and the corresponding linear problem with constant coefficients is solved as an example. Further, the half-order linear problem with constant coefficients is solved with a method of undetermined coefficients and with a transform method.

AMS Subject Classification: 39A12, 34A25, 26A33. Keywords: Discrete fractional calculus. This is joint work with Paul Eloe.

66

Constructing difference equations models for public health policy TAMARA AWERBUCH F RIEDLANDER Harvard School of Public Health Department of Global Health and Population Boston, Massachusetts, USA [email protected] http://www.hsph.harvard.edu/research/ ,→tamara-awerbuchfriedlander Difference equations modeling exploits the natural connection between events occurring at discrete intervals and the inherent discrete nature of difference equations. We will show examples used for understanding complex interactions among ecological components that lead to the spread of diseases transmitted by vectors such as ticks and mosquitoes. The emergence of Lyme disease and its early stages was represented and analyzed as a linear system; the system representing later stages of the tick population growth rendered a delay equation with two parameters which are real numbers representing biological characteristics of the tick life-cycle. The mathematical analysis enables us to detect parameter regions of local and global stability, boundedness and oscillatory behavior of solutions. Another example is the construction of nonlinear systems describing community intervention in mosquito control through management of their habitats. One system consists of two equations; representing a more complex intervention resulted in a system of three difference equations. The work has been carried out by a collaboration of an interdisciplinary team of mathematicians, biologists, ecologist, sociologists.

Keywords: System, difference equations, infectious diseases.

67

Fredholm integral equations on time scales ¨ ZLEM B ATIT O Ege University Department of Mathematics ˙ Izmir, Turkey [email protected] http://sci.ege.edu.tr/˜math/index.php? ,→option=com content&task=view&id=55 In this talk, we study linear and nonlinear Fredholm integral equations on time scales. First separable kernels and then symmetric kernels are considered for the linear case. For the nonlinear case, we use the monotone iterative technique to obtain approximations to a unique solution and give some applications.

AMS Subject Classification: 45B05. Keywords: Time scales, Fredholm integral equations.

68

A Sharkovsky theorem for maps on trees C HRIS B ERNHARDT Fairfield University Department of Mathematics and Computer Science Fairfield, Connecticut, USA [email protected] http://cs.fairfield.edu/˜bernhardt The proof of Sharkovsky’s theorem is combinatorial in nature. This means that instead of viewing it as a theorem about maps of the interval one can view it as a theorem about maps on trees that permute the vertices in the special case when the tree is topologically an interval. This way of viewing Sharkovsky’s theorem leads to the natural question of whether there is such a theorem for trees in general. In this talk we give a Sharkovsky-type ordering for trees in general. We also show the converse — that given any n there is a tree and a map that has exactly the periods given by the theorem.

69

Exponentially asymptotically constant systems of difference equations with applications S IGRUN B ODINE University of Puget Sound Department of Mathematics and Computer Science Tacoma, Washington, USA [email protected] We consider the asymptotic behavior of solutions of systems of linear difference equations of the form y(n + 1) = [A + R(n)] y(n) , n ≥ n0 , where A is a constant, invertible matrix and R(n) is an exponentially small perturbation, i.e., |R(n)| ≤ Kn for some 0 <  < 1. While classical results yield, for n sufficiently large, the existence of a fundamental matrix of the form Y (n) = [I + o(1)]An

as n → ∞,

we want to find more precise estimates of the error term o(1). In particular, we are interested in its dependence on  and the eigenvalues of A. We also present an application to nonlinear autonomous dynamical systems with hyperbolic equilibria. Our results were motivated by a recent paper by R. Agarwal and M. Pituk who studied scalar linear difference equations with exponentially small perturbations.

AMS Subject Classification: 39A11. Keywords: Exponentially small perturbations, asymptotics. This is joint work with D. A. Lutz.

70

Necessary and sufficient conditions for oscillation of certain higher order partial difference equations YAS¸ AR B OLAT Afyon Kocatepe University Department of Mathematics Afyonkarahisar, Turkey [email protected] http://www2.aku.edu.tr/˜yasarbolat In this talk, some necessary and sufficient conditions for the oscillation of a certain higher order partial difference equation are obtained.

AMS Subject Classification: 39A11, 34K11, 34C10. Keywords: Partial difference equation, oscillation, oscillatory. ¨ This is joint work with Omer Akın.

71

On the equivalence of Rolle’s and generalized mean value theorems on time scales D EVR I˙ M C ¸ AKMAK Gazi University Department of Mathematics Education Ankara, Turkey [email protected] http://websitem.gazi.edu.tr/dcakmak In this talk, by using elementary time scale calculus, we recall the equivalence between wellknown Rolle’s and Generalized Mean Value Theorems on time scales.

AMS Subject Classification: 26A24, 39A12. Keywords: Rolle’s theorem, mean value theorem, time scales.

72

On the global character of solutions of a rational system of difference equations E LIAS C AMOUZIS American College of Greece Department of Mathematics and Natural Sciences Athens, Greece [email protected] In this talk we study the global character of solutions of a rational system of difference equations. In particular, we examine the boundedness of solutions, the stability of the equilibrium points, and the periodic character of solutions.

AMS Subject Classification: 39A10. Keywords: Rational system, boundedness, stability, periodicity.

73

A characterization of k-cycles ´ J OSE S. C ANOVAS Technical University of Cartagena Department of Applied Mathematics and Statistics Cartagena, Spain [email protected] http://filemon.upct.es/˜jose We study global periodicity for the difference equation of order l given by xn+l = f (xn+l−1 , xn+l−2 , . . . , xn ), where f : (0, ∞)l → (0, ∞) is a continuous map, l ∈ Z+ . Our main results are the following. We prove that if any solution of the equation is periodic, then there is a minimal k ∈ N such that the period of any solution divides k (and therefore f is called a k-cycle). In addition, if l = 2, then for any k > 2 there are, up to conjugacy, only a k-cycle. Finally, if l > 2 and f gives a (l + 1)-cycle, then f is conjugated to: • xn+l =

1 xn ·xn+1 ·...·xn+l−1 ,

if l is even. (l+1)/2 Q

• The previous equation or xn+l =

xn+2j−2

j=1 (l−1)/2 Q

, if l is odd. xn+2j−1

j=1

AMS Subject Classification: 39A05. Keywords: Cycles, conjugacy. This is joint work with Antonio Linero and Gabriel Soler.

74

Higher-order boundary value problems on time scales E RB I˙ L C ¸ ET I˙ N Ege University Department of Mathematics ˙ Izmir, Turkey [email protected] http://sci.ege.edu.tr/˜math/index.php? ,→option=co m content&task=view&id=63 In this talk, we give the existence of positive solutions of the Lidstone boundary value problem (LBVP)  (−1)n y 42n (t) = f (t, y σ (t)), t ∈ [0, 1], y 42i (0) = y 42i (σ(1)) = 0,

0 ≤ i ≤ n − 1,

where n ≥ 1 and f : [0, σ(1)] × R → R is continuous. Firstly, by using the Schauder fixed point theorem in a cone, we obtain the existence of solutions to a Lidstone boundary value problem (LBVP). Secondly, an existence result for this problem is also given by the monotone method. Finally, by using the Krasnosel’skii fixed point theorem, it is proved that the LBVP has a positive solution.

AMS Subject Classification: 39A10. Keywords: Positive solutions, upper and lower solutions. This is joint work with Fatma Serap Topal.

75

Effect of floating point on computation of monodromy matrix A L I˙ O SMAN C ¸ IBIKD I˙ KEN Sel¸cuk University Department of Computer Technology and Programming Konya, Turkey [email protected] asp.selcuk.edu.tr/asp/personel/ ,→web/goster.asp?sicil=5377 Let A(n) be a matrix of dimension N × N with period T and consider the difference equation system x(n + 1) = A(n)x(n), n ∈ Z. (1) With X(T ) being the monodromy matrix of the system (1), it is well known that ∞ X ω1 (A, T ) = (X ∗ (T ))k (X(T ))k < ∞

(2)

k=0

implies Schur stability of the system (1) [Kemal Aydın, A. Ya. Bulgakov, Gennadii Demidenko, Numerical characteristics of asymptotic stability of solutions of linear difference equations with periodic coefficients, Siberian Mathematical Journal, volume 41, number 6, pages 1005–1014, 2000]. By the spectral criterion, each eigenvalue of the monodromy matrix X(T ) belongs to the unit disk [Saber Elaydi, An introduction to difference equations, third edition, undergraduate texts in mathematics, Springer, New York, 2005]. Schur stability of the system (1) depends on the monodromy matrix X(T ) in both cases. Therefore, Schur stability of the system (1) and quality of Schur stability are related to the results of computation errors on computation of the monodromy matrix X(T ). In this study, the effect of floating point on computation of the monodromy matrix X(T ) is investigated. The bound is obtained for ||X(T ) − Y (T )|| in which the matrix Y (T ) is the computed value of the monodromy matrix.

AMS Subject Classification: 39A11, 65G50. Keywords: Schur stability, monodromy matrix, roundoff error. This is joint work with Kemal Aydın.

76

Study of attractors for two-dimensional skew-products whose basis is a Denjoy counterexample S ARA C OSTA Universitat Aut`onoma de Barcelona Departament de Matem`atiques Bellaterra (Cerdanyola del Vall`es), Spain [email protected] Since Keller studied, in 1996, the existence of strange nonchaotic attractors in a particular kind of two-dimensional quasiperiodically forced skew-products defined on M + := S1 × [0, ∞), several extensions of his results have been published. All these extensions have in common that the system are defined on M + , and the component on the basis is always an irrational rotation. We extend the Keller and Haro results to similar systems defined on S1 × R, in this case we can have two attractors given by the graph of a map defined on S1 or only one given by the graph of a two-valued correspondence. If we exchange the irrational rotation by a Denjoy counterexample, the results are quite similar with the difference that the map, or correspondence, whose graph gives the attractor is defined on P ⊂ S1 , where P is the support of the unique invariant measure of the Denjoy counterexample.

This is joint work with Llu´ıs Alsed`a.

77

Difference equations arising in dynamic models of biological evolution J. M. C USHING University of Arizona Department of Mathematics Tucson, Arizona, USA [email protected] http://math.arizona.edu/˜cushing I will describe some nonlinear difference equation models that arise in modeling the evolution of biological populations. The state variables are mean phenotypic traits of species as well as the usual population densities, and consequently the models involve systems of (or higher order) nonlinear difference equations. The models typically have several equilibria and a fundamental question concerns which are stable. I will give some stability results and some open problems and conjectures.

AMS Subject Classification: 37N25, 92D25. Keywords: Difference equations, models of evolution.

78

A new proof for the Levin–May criterion of asymptotic stability F OZI D ANNAN Arab International University Department of Mathematics Damascus, Syria [email protected] Levin and May obtaind an easy necessary and sufficient condition for the asymptotic stability of the difference equation x(n + 1) − x(n) + qx(n − k) = 0. In this talk we give a new proof for this condition.

AMS Subject Classification: 39A11. Keywords: Levin–May, asymptotic stability, difference equation.

79

On nonoscillation of Emden–Fowler difference equations Z UZANA D O Sˇ L A´ Masaryk University Department of Mathematics and Statistics Brno, Czech Republic [email protected] http://www.math.muni.cz/˜dosla Asymptotic properties of nonoscillatory solutions of the Emden–Fowler equation ∆(an |∆xn |α sgn ∆xn ) + bn |xn+1 |β sgn xn+1 = 0,

α 6= β,

(1)

are investigated using the half-linearization technique. Some interesting discrepancies concerning oscillation and nonoscillation of (1) and its continuous counterpart will be given.

This is joint work with Mariella Cecchi and Mauro Marini.

80

Sensitivity of Schur stable linear systems with periodic coefficients A HMET D UMAN Afyon Kocatepe University Department of Mathematics Afyonkarahisar, Turkey [email protected] http://www.akademi.aku.edu.tr/ ,→frmCvler.aspx?SicilNo=KA0992 Let A(n) be an N × N -matrix with period T and consider the difference equation system x(n + 1) = A(n)x(n),

n ∈ Z.

(1)

With X(T ) being the monodromy matrix of (1), it is well known that ∞ X k k (X ∗ (T )) (X(T )) < ∞ w1 (A, T ) = k=0

implies Schur stability of the system (1) [Kemal Aydın, A. Ya. Bulgakov, Gennadii Demidenko, Numerical characteristics of asymptotic stability of solutions of linear difference equations with periodic coefficients, Siberian Mathematical Journal, volume 41, number 6, pages 1005–1014, 2000]. Let B(n) be a matrix of dimension N × N with period T . There are some results given on the Schur stability of the perturbated system y(n + 1) = [A(n) + B(n)]y(n),

n ∈ Z,

(2)

where B(n) is the perturbation matrix [Kemal Aydın, Haydar Bulgak, Gennadii Demidenko, Continuity of numeric characteristics for asymptotic stability of solutions to linear difference equations with periodic coefficients, Selc¸uk Journal of Applied Mathematics, volume 2, number 2, pages 5–10, 2001]. Note: Haydar Bulgak is the same person as A. Ya. Bulgakov. In this talk, we give new results for Schur stability of the system (2) and compare these new results with the existing ones in the literature. The results are supported with numerical applications too. AMS Subject Classification: 39A11. Keywords: Schur stability, monodromy matrix, sensitivity. This is joint work with Kemal Aydın.

81

Comparison and oscillation theorems for linear and half-linear dynamic equations on time scales LYNN E RBE University of Nebraska–Lincoln Department of Mathematics Lincoln, Nebraska, USA [email protected] http://www.math.unl.edu/˜lerbe2 We obtain some new oscillation and comparison results for the second-order linear (or halflinear) dynamic equation of the form (r(x∆ )α )∆ (t)+p(t)xα (σ(t)) = 0. We are primarily interested in the case when the coefficient p(t) changes sign for arbitrarily large values of t. The results improve and extend some earlier criteria, in both the continuous and discrete cases, as well as for more general time scales.

AMS Subject Classification: 34K11, 39A10. Keywords: Comparison theorems, oscillation, half-linear. This is joint work with Jia Baoguo and Allan Peterson.

82

The structure of the spectrum for normal operators M ELTEM E ROL Karadeniz Technical University Department of Mathematics Trabzon, Turkey [email protected] We have investigated the structure of the spectrum for normal operators on a Hilbert space with a new method and asymptotic behavior of its eigenvalues. The obtained results in this work can be applied to a normal extension of minimal operators generated by a linear differential operator expression in a Hilbert space of vector functions in finite intervals.

AMS Subject Classification: 47A10. Keywords: Normal operators, spectrum. This is joint work with Zameddin Ismailov.

83

Convergence of hyperspaces under the Fell topology, especially of time scales N ORAH E STY Stonehill College Department of Mathematics Boston, Massachusetts, USA [email protected] In this talk we will examine various topologies on hyperspaces, and in particular those which are most useful in the context of time scales. After demonstrating that the Fell topology is the most appropriate, we will state (and time permitting, prove) several theorems about convergence in hyperspaces of locally compact metric spaces under the Fell topology. Finally we will state/prove analogous theorems for the particular case of time scales, where the hyperspace in questions is CL(R).

AMS Subject Classification: 54B20. Keywords: Hyperspaces, time scales, Fell topology. This is joint work with Stefan Hilger.

84

Systoles and topological entropy in discrete dynamical systems S ARA F ERNANDES Universidade de Evora / CIMA-UE Research Centre in Mathematics and Application Evora, Portugal [email protected] http://evunix.uevora.pt/˜saf The fruitful relationship between the geometry and the graph theory has been explored by several authors in the sense of bringing important results for the discrete dynamical systems seen as Markov chains in graphs. In this work we will explore the relationship between the topological entropy and systoles in the context of maps on the interval.

AMS Subject Classification: 37A35, 37B10. Keywords: Dynamical systems, topological entropy, systole. This is joint work with Clara Gracio and Carlos Ramos.

85

Optimal monetary policy with partially rational agents O RLANDO G OMES Instituto Polit´ecnico de Lisboa, UNIDE/ISCTE Escola Superior de Comunica¸ca˜ o Social Lisbon, Portugal [email protected] We explore the dynamic behavior of a New Keynesian monetary policy problem with expectations formed, partially, under adaptive learning. We consider two alternative cases: on the first setting, the private economy has the ability to predict rationally real economic conditions (the output gap) but it needs to learn about the future values of the nominal variable (the inflation rate); on the second setup, private agents are fully aware of future inflation rates, however they lack the ability to predict instantly the correct values of the output gap (learning is attached to this variable). In both cases, we find a simple condition indicating the required learning quality that is needed to guarantee local stability. To achieve convergence to the steady state, the economy does not need to attain full learning efficiency; it just has to secure a minimum learning quality in order to attain the desired long run result.

Keywords: Optimal monetary policy, adaptive learning. This is joint work with Vivaldo M. Mendes and Diana A. Mendes.

86

Stability boundary for asymptotic stability of scalar equations ¨ ZLEM A K G UM ¨ US ¨¸ O Sel¸cuk University Department of Mathematics, Faculty of Science and Literature Konya, Turkey [email protected] http://asp.selcuk.edu.tr/asp/personel/web/ ,→goster.asp?sicil=6644 We consider the scalar equation of the form x(n + 2k) + px(n + k) + qx(n) = 0 and obtain stability regions in the plane by using the Schur–Cohn criterion. In the case of p = 1 or p = −1, the obtained stability region is restricted to a narrow area by the found values of q when k is a positive even integer.

Keywords: Stability, discrete-time system, stability criteria. This is joint work with Necati Tas¸kara.

87

Spectral analysis of a non-selfadjoint second-order difference operator G USEIN G USEINOV Atılım University Department of Mathematics Ankara, Turkey [email protected] http://www.atilim.edu.tr/˜guseinov Non-Hermitian (non-selfadjoint) Hamiltonians and complex extension of quantum mechanics have recently received a lot of attention [C. M. Bender, Making sense of non-Hermitian Hamiltonians, Rep. Progr. Phys. 70(2007), 947–1018]. In this study, we develop spectral analysis of the discrete problem −∆2 yn−1 + qn yn = λρn yn , y−1 = y1 ,

n ∈ {. . . , −3, −2, −1} ∪ {2, 3, 4, . . .}, ∆y−1 = e2iδ ∆y1 ,

(1) (2)

in the Hilbert space l2 , where (yn ), n ∈ Z = {0, ±1, ±2, ±3, . . .}, is a desired solution, ∆ denotes the forward difference operator defined by ∆yn = yn+1 −yn (so that ∆2 yn−1 = yn−1 −2yn +yn+1 ), qn ≥ 0, λ is a spectral parameter, and ( e2iδ , n ≤ −1, ρn = (3) e−2iδ , n ≥ 0, for a δ ∈ [0, π/2). The main distinguishing features of problem (1), (2) are that it involves a complex coefficient ρn of the form (3) and that transition conditions (impulse conditions) of the form (2) are presented which also involve a complex coefficient. Such a problem is not self adjoint with respect to the usual inner product of space l2 and it arises as a discrete version of some quantum systems on a complex contour.

AMS Subject Classification: 39A70. Keywords: Difference operator, spectrum, completely continuous operator. ¨ This is joint work with Ebru Ergun.

88

Interval oscillation of second-order difference equations with oscillatory potentials ¨ A. F EZA G UVEN I˙ L I˙ R Ankara University Department of Mathematics Ankara, Turkey [email protected] Interval oscillation criteria are established for second-order difference equations of the form ∆(k(n)∆x(n)) + p(n)x(g(n)) + q(n)|x(g(n))|α−1 x(g(n)) = e(n), where n ≥ n0 , n0 ∈ N = {0, 1, . . .}, α > 1; {k(n)}, {p(n)}, {q(n)}, {g(n)}, and {e(n)} are sequences of positive real numbers, k(n) > 0 is nondecreasing, g(n) is nondecreasing with g(n) → ∞ as n → ∞, ∆ is the forward difference operator defined as usual by ∆x(n) = x(n + 1) − x(n).

AMS Subject Classification: 34K11, 34C15. Keywords: Interval oscillation, second order, delay argument.

89

Discrete Haavelmo growth cycle model M AŁGORZATA G UZOWSKA University of Szczecin Department of Econometrics and Statistics Szczecin, Poland [email protected] A discretization method attributed to Kahan is used to approximate the Haavelmo growth cycle model. The local dynamics of this discrete-time Haavelmo growth cycle model are analyzed.

Keywords: Kahan’s method, discrete time, Haavelmo model.

90

Solving systems of nonlinear equations by using difference equations M OGHTADA H ASHEMIPARAST K. N. T. University of Technology Department of Mathematics and Statistics Tehran, Iran [email protected] http://www.math.kntu.ac.ir/hashemiparast.htm There are many numerical methods in obtaining the solution of integral equations, system of integral equations or integro-differential equations which reduce to a system of nonlinear equations. These problems are often ill posed, and are difficult to be solved. In this talk, by using a moment characteristic function, these systems are transferred to a set of difference equations. The solution is obtained, by referring to the applied characteristic function. Finally, numerical examples are given.

Keywords: Characteristic function, difference equation.

91

The dynamic multiplier-accelerator model in economics J ULIUS H EIM Missouri University of Science and Technology Department of Mathematics and Statistics Rolla, Missouri, USA [email protected] http://math.mst.edu In this work we derive a linear second-order dynamic equation which describes multiplieraccelerator models in economics on time scales. After we provide the general form of the dynamic equation, which considers both taxes and foreign trade, i.e., imports and exports, we give four special cases of this general multiplier-accelerator model: (1) Samuelson’s basic multiplieraccelerator model. (2) We extend this model with the assumption that taxes are raised by the government and that these taxes are immediately reinvested by the government. (3) We give Hicks’ extension of the basic multiplier-accelerator model as an example and (4) extend this model by allowing foreign trade in the next step. For each of these models we present the dynamic equation in both expanded and self-adjoint form and give examples for particular time scales. Finally we present a criterion under which each solution of the dynamic equation oscillates.

AMS Subject Classification: 91B62, 34C10, 39A10, 39A11, 39A12, 39A13. Keywords: Time scales, multiplier-accelerator, dynamic equation, self adjoint, economics. This is joint work with Martin Bohner.

92

Riccati equations for linear Hamiltonian systems R OMAN H ILSCHER Masaryk University Department of Mathematics and Statistics Brno, Czech Republic [email protected] http://www.math.muni.cz/˜hilscher In this talk we will discuss Riccati matrix differential and difference equations for (possibly abnormal) linear Hamiltonian and symplectic systems. The abormality is reflected in the (possible) noninvertibility of the corresponding principal solution. We show that even in this case one can characterize the nonnegativity and positivity of the associated quadratic functional via certain implicit Riccati equations. These results are derived through the general time scales theory and extend the known classical continuous time results e.g., by Reid and Coppel and recent discrete ˚ ziˇckov´a. time results e.g., by Bohner, Doˇsly, ´ Kratz, and Ruˇ

AMS Subject Classification: 34C10, 39A12. Keywords: Time scale, Riccati equation, generalized inverse. This is joint work with Vera Zeidan.

93

On the global stability of xn+1 =

p+qxn 1+xn−1

´ V´I CTOR J IM E´ NEZ L OPEZ Universidad de Murcia Departamento de Matem´aticas Murcia, Spain [email protected] http://www.um.es/docencia/vjimenez For a long time it has been conjectured that the unique positive equilibrium of the equation from the title attracts all its positive solutions. The conjecture is known to be true in the cases q < 1 (Kulenovi´c and Ladas, 2001) and p ≤ q (Kocic and Ladas, 1993). Under the assumptions q ≥ 1 and q < p it has been proved in progressively more general settings by Kocic, Ladas and Rodrigues (1993), Ou Tang and Luo (2000) and Nussbaum (2007). A paper by Li, Zhang and Su (2005) purportedly provides a full proof of the conjecture but in fact has a rather basic mistake. In this work we use a modified version of the so-called dominance condition, a tool recently introduced by H. El-Morshedy and the author (“Global attractors for difference equations dominated by one-dimensional maps”, J. Difference Equ. Appl. 14 (2008), 391–410) to give a unified proof on the conjecture in the cases listed above and improve Nussbaum’s bounds.

AMS Subject Classification: 39A11, 37C70. Keywords: Global attractor, rational difference equation.

94

Period-two trichotomies of a difference equation of order higher than two S ENADA K ALABU Sˇ I C´ University of Sarajevo Department of Mathematics, Faculty of Science Sarajevo, Bosnia and Herzegovina [email protected] http://www.pmf.unsa.ba We investigate the period-two trichotomies of solution of the equation xn+1 = f (xn , xn−1 , xn−2 ),

n = 0, 1 . . . ,

where the function f satisfies certain monotonicity conditions. We give fairly general conditions for period-two trichotomies to occur and illustrate the results with numerous examples.

AMS Subject Classification: 39A10, 39A11. Keywords: Attractivity, period two solution, unbounded. This is joint work with Dˇz. Burgi´c and M. R. S. Kulenovi´c.

95

Iterated oscillation criteria for delay dynamic equations of first order B AS¸ AK K ARPUZ Afyon Kocatepe University Department Mathematics Afyonkarahisar, Turkey [email protected] http://www.akademi.aku.edu.tr/ ,→frmCvler.aspx?SicilNo=KA1798 We obtain new sufficient conditions for the oscillation of all solutions of first-order delay dynamic equations on arbitrary time scales, hence combining and extending results for corresponding differential and difference equations.

AMS Subject Classification: 39A10, 34C10. Keywords: Oscillation, first-order delay dynamic equations. ¨ ¨ This is joint work with Martin Bohner and Ozkan Ocalan.

96

Dynamic equations with piecewise continuous argument C HRISTIAN K ELLER Missouri University of Science and Technology Department of Mathematics and Statistics Rolla, Missouri, USA [email protected] We extend the theory of differential equations with piecewise continuous argument to general time scales. Systems with alternating retarded and advanced argument will be investigated and conditions for globally asymptotic stability of those systems will be stated. Furthermore we study the oscillatory behaviour for several dynamic equations with piecewise continuous argument.

AMS Subject Classification: 34K11, 39A10, 39A11, 39A12, 39A13. Keywords: Dynamic equation, time scale, piecewise continuous, retarded, advanced, delay. This is joint work with Martin Bohner.

97

A cardiac loop reentry model with thresholds C ANDACE K ENT Virginia Commonwealth University Department of Mathematics and Applied Mathematics Richmond, Virginia, USA [email protected] http://www.math.vcu.edu/faculty/kent.html We investigate the two-dimensional, multiple threshold map, or bimodal system, ( G(x , y) , if (x , y) ∈ T F (x , y) = H(x , y) , if (x , y) ∈ / T, where G : R2+ → R2+ and H : R2+ → R2+ are continuous and T is the intersection of five threshold regions. Sufficient conditions are placed on G and H that guarantee that either every orbit under F that begins in T leaves T and never returns or there exist orbits under F that begin in T and pass between T and its complement infinitely often. Our bimodal system is intended to serve as a simple discrete model of the dynamics of a circulating pulse of depolarization in a ring of two cardiac cells within the context of cardiac arrhythmias or irregular heartbeat.

This is joint work with Hassan Sedaghat.

98

Asymptotic behavior of one class solutions of the second-order Emden–Fowler difference equation V ITALIY K HARKOV I. I. Mechnikov Odessa National University Department of Mathematics Odessa, Ukraine kharkov v [email protected] In this talk we investigate and obtain necessary and sufficient conditions for existence of one nontrivial class solutions of the second-order Emden–Fowler difference equation. Moreover, asymptotic representations of solutions from this class are established.

AMS Subject Classification: 34D05. Keywords: Asymptotics, Emden–Fowler equation.

99

Stability via Convexity M IKHAIL K IPNIS Chelyabinsk State Pedagogical University Department of Mathematics Chelyabinsk, Russia [email protected] Stability analysis of the Volterra difference equations xn +

n X

am xn−m = 0,

n = 1, 2 . . .

(1)

m=1

P∞ is presented, with assumption that the series m=1 am is convergent, and inequalities am ≥ 0 and ∆2 am = am − 2am+1 + am+2 ≥ 0 hold for all m ∈ N. For example, let am = β/ms for real β and s > 1. The criterion for asymptotic stability of equation (1) is given by −

1 1 = − P∞ ζ(s) m=1

1 ms

1 1 = , 1−s ) ζ(s) m+1 1 (1 − 2 (−1) s m=1 m

< β < P∞

where ζ(s) is Riemann’s zeta function. Results obtained for the difference equations of the k-th ¨ order were compared with the Enestrom–Kakeya stability conditions.

AMS Subject Classification: 39A11, 34K20. Keywords: Volterra equations, stability, convexity. This is joint work with Vitaliy Gilyazev.

100

Existence of unbounded solutions in rational equations Y EVGENIY K OSTROV University of Rhode Island Department of Mathematics Kingston, Rhode Island, USA [email protected] We exhibit a range of parameters and a set of initial conditions where the rational difference equation 2k X α+ βi xn−i i=0

xn+1 = A+

k X

B2j xn−2j

j=0

has unbounded solutions.

AMS Subject Classification: 39A10, 39A11. Keywords: Existence of unbounded solutions, rational difference equation. This is joint work with E. Camouzis, E. A. Grove, and G. Ladas.

101

Solution to integral equations on time scales: Existence, uniqueness and successive approximations T OMASIA K ULIK University of New South Wales School of Mathematics and Statistics Sydney, Australia [email protected] http://web.maths.unsw.edu.au/˜tomasia I will present my research on applying Banach’s and Granas’s fixed point theory to establish theorems with sufficient conditions for existence, uniqueness of solutions to integral equations on time scales, as well as methods of successive approximation to find the solution to any desired accuracy. In particular, I will discuss integral equations on time scales over unbounded intervals and applications of the results to examining and finding solutions of dynamic or integro-differential equations on time scales and the additional conditions requited for existence and uniqueness in these problems. I will discuss applications of dynamic equations on time scales, to modeling various dynamical systems with complex dynamics, which varies continuously part of the time and discretely part of the time.

This is joint work with Christopher C Tisdell.

102

Linear kth order functional and difference equations in the space of strictly monotonic functions J ITKA L AITOCHOV A´ Palack´y University Department of Mathematics, Faculty of Education Olomouc, Czech Republic [email protected] Abel functional equations are associated to a linear homogeneous functional equation with constant coefficients. The work uses the space S of continuous strictly monotonic functions Φ : (−∞, ∞) → (a, b) equipped with a multiplication f ◦ g = f X −1 g, the symbol X being a preselected canonical function in S. Because of the space S, classical terms like composite function, iterates of a function, Abel functional equation and linear homogeneous functional equation, must be re-defined. We consider the functional equation ak f ◦ Φk (x) + · · · + a0 f ◦ Φ0 (x) = 0, which is solved using roots of the characteristic equation and a continuous solution of the Abel functional equation α ◦ Φ(x) = X(x + 1) ◦ α(x). The classical theory of linear homogeneous functional and difference equations is obtained as a specialization of the theory in space S. All functional equation results apply to difference equations.

AMS Subject Classification: 39B05, 39B12. Keywords: Difference equation, functional equation.

103

The Marshall differential analyzer: Dynamic equations in motion! B ONITA L AWRENCE Marshall University Department of Mathematics Huntington, West Virginia, USA [email protected] http://www.science.marshall.edu/lawrence The Marshall University differential analyzer team has completed the construction of a four integrator, primarily mechanical differential analyzer. Machines of this type, first built in the late 1920s and the early 1930s, were designed to solve differential equations and plot the solutions. Our machine, known to the team as “Art”, is modeled after the first differential analyzer built in England and named for its builder, Dr. Arthur Porter. It is built almost exclusively from Meccano or (Meccano type) components and was constructed by a team of undergraduate and graduate students from Marshall University. This talk will begin with a discussion of the construction of the machine and its fundamental components. The machine offers a fantastic physical and visual interpretation of a differential equation. How this visualization is achieved through the fundamental mechanics and the programming of the machine will then be addressed. Examples of machine setups will be presented.

AMS Subject Classification: 34. Keywords: Dynamic equations, differential analyzer.

104

Nonautonomous periodic systems with Allee effect R AFAEL L U´I S Center for Mathematical Analysis and Dynamical Systems Department of Mathematics Lisbon, Portugal [email protected] http://members.netmadeira.com/rafaelluis We introduce a new class of maps, called unimodal Allee maps (UAM). These maps arise in the study of population dynamics in which the system has three fixed points, a stable fixed point zero, an unstable positive fixed point (Allee point) and a stable positive fixed point (carrying capacity). We analyse the properties of the Allee points and the carrying capacity and establish their stability, for nonautonomous periodic systems formed by unimodal Allee maps.

Keywords: Allee effect, Allee point, carrying capacity, UAM. This is joint work with Saber Elaydi and Henrique Oliveira.

105

Ostrowski inequalities on time scales T HOMAS M ATTHEWS Missouri University of Science and Technology Department of Mathematics and Statistics Rolla, Missouri, USA [email protected] The presentation contains proofs of Ostrowski inequalities (regular and weighted cases) on time scales and thus unifies and extends corresponding continuous and discrete versions from the literature. An application to the quantum calculus case will also be provided.

This is joint work with Martin Bohner.

106

Numerical detection of explosions and asymptotic behaviour of delay-differential equations M ICHAEL M C C ARTHY Dublin City University School of Mathematical Sciences Dublin, Ireland [email protected] http://student.dcu.ie/˜mccarm29/index.html In this talk we study scalar delay-differential equations whose solutions explode in finite time. Our goal is to devise a discretisation of the equation such that: (i) the discrete equation “explodes”; (ii) the rate at which the explosion occurs is preserved by discretising; (iii) the explosion time can be approximated arbitrarily well by making a sufficiently large computational effort. We show that these goals are all achieved by making an adaptive time-discretisation where the length of the step size tends to zero as the explosion time is approached. The same adaptive method also reproduces the asymptotic behaviour of rapidly growing solutions of a similar class of nonexploding equations: Therefore, the method does not induce spurious explosions not present in continuous time. The work is joint with John Appleby and is supported by the IRCSET Embark Initiative under the project “Explosions in stochastic dynamical systems applied to finance”.

Keywords: Delay differential equation, explosions. This is joint work with John Appleby.

107

Learning to play Nash in deterministic uncoupled dynamics V IVALDO M ENDES ISCTE Department of Economics Lisbon, Portugal [email protected] In a boundedly rational game, where players cannot be as super-rational as in Kalai and Leher (1993), are there simple adaptive heuristics or rules that can be used in order to secure convergence to Nash equilibria? Young (2008) argues that if an adaptive learning rule obeys three conditions – (i) it is uncoupled, (ii) each player’s choice of action depends solely on the frequency distribution of past play, and (iii) each player’s choice of action, conditional on the state, is deterministic – no such rule leads the players’ behavior to converge to Nash equilibra. In this paper we present a counterexample, showing that there are in fact simple adaptive rules that secure convergence in a fully deterministic and uncoupled game. We used the Cournot model with nonlinear costs and incomplete information for this purpose and also illustrate that the convergence to Nash equilibria can be achieved with or without any coordination of the players actions.

AMS Subject Classification: 91A25, 91A26, 91A50. Keywords: Uncoupled dynamics, Nash equilibrium, convergence. This is joint work with Orlando Gomes and Diana Mendes.

108

Time scale extensions of a theorem of Wintner on systems with asymptotic equilibrium R AZ I˙ YE M ERT Middle East Technical University Department of Mathematics Ankara, Turkey [email protected] http://www.metu.edu.tr/˜raziye We consider quasilinear dynamic systems of the form x∆ = A(t)x + f (t, x),

t ∈ [a, ∞)T ,

where T is a time scale, and extend theorems obtained for differential equations by Trench [SIAM J. Math. Anal.] to dynamic equations on time scales; thus provide extensions of a theorem of Wintner on systems with asymptotic equilibrium to time scales. In particular, we give sufficient conditions for the asymptotic equilibrium of the above system in the sense that there is a solution satisfying lim x(t) = c t→∞

for any given constant vector c. Our results are new for difference and q-difference equations even though their analogues for differential equations have been known for some time.

Keywords: Asymptotic equilibrium, dynamic system, time scales. ˘ This is joint work with Agacık Zafer.

109

A new approach for solving Fredholm integro-difference equations H AMID M ESGARANI Shahid Rajaee University Department of Mathematics Tehran, Iran [email protected] The Taylor expansion approach to solve higher-order linear difference equations has been given by Sezer. In this paper, we modify and develop for solving the Fredholm integro-difference equation. Also, examples that illustrate the pertinent features of the method are presented and the results are discussed.

Keywords: Integro-difference, Fredholm, Taylor expansion. This is joint work with M. Shahrezaee.

110

Algebro-geometric solutions of the Ablowitz–Ladik hierarchy J OHANNA M ICHOR New York University Courant Insitute of Mathematical Sciences New York, New York, USA [email protected] http://www.mat.univie.ac.at/˜jmichor Algebro-geometric solutions of soliton equations are a class of solutions which can be constructed explicitly using tools from algebraic geometry. We present a derivation of all algebrogeometric finite-band solutions of the Ablowitz–Ladik equation, which is a complexified version ¨ of the discrete nonlinear Schrodinger equation. In addition, we survey a recursive construction of the associated Ablowitz–Ladik hierarchy, a completely integrable sequence of systems of nonlinear evolution equations on the lattice Z. This is done by means of a zero-curvature and Lax approach.

AMS Subject Classification: 37K10, 37K15, 35Q55. Keywords: Discrete NLS, algebro-geometric solution, Lax pair. This is joint work with Fritz Gesztesy, Helge Holden, and Gerald Teschl.

111

Oscillatory and asymptotic properties of solutions of nonlinear neutral-type difference equations M AŁGORZATA M IGDA Poznan University of Technology Institute of Mathematics Poznan, Poland [email protected] http://www.math.put.poznan.pl/˜mmigda We consider higher-order neutral difference equations of the form ∆m (xn + pn xn−τ ) = f (n, xn , xσ(n) ) + hn , where m ≥ 2, (pn ), (hn ) are sequences of real numbers, τ is a nonnegative integer, (σ(n)) is an integer sequence with σ(n) ≤ n and lim σ(n) = ∞, f : N × R × R → R. n→∞

The study of asymptotic behavior of solutions of nonlinear equations of this type often requires that the sequence (pn ) satisfies pn > 0 or pn < 0. We examine the case when (pn ) is an oscillatory sequence.

AMS Subject Classification: 39A10, 39A11. This is joint work with Janusz Migda.

112

An example of a strongly invariant, pinched core strip L EOPOLDO M ORALES Universitat Aut`onoma de Barcelona Departament de Matem`atiques Barcelona, Spain [email protected] In [Roberta Fabbri, Tobias J¨ager, Russel Johnson, Gerhard Keller, A Sharkovskii-type theorem for minimally forced interval maps, Topological Methods in Nonlinear Analysis, volume 26, number 1, pages 163–188, 2005], the authors define the concept of Pinched core strip. So far it has not been given an example of such an object that is strongly invariant under a quasi-periodic triangular function and it is not a curve. In this talk we will describe how to construct such an example.

˜ This is joint work with Llu´ıs Alsed`a and Francesc Manosas.

113

Numerical solutions of nonlinear differential-difference equations by the variational iteration method V OLKAN O BAN Ege University Department of Mathematics ˙ Izmir, Turkey [email protected] We extend He’s variational iteration method to find approximate solutions for nonlinear differential-difference equations such as Volterra’s equation. A simple but typical example is applied to illustrate the validity and great potential of the generalized variational iteration method in solving nonlinear differential-difference equations. The results reveal that the method is very effective and simple. We find the extended method for nonlinear differential-difference equations is of good accuracy.

Keywords: He’s variational iteration method, differential-difference, Volterra equation. This is joint work with Ahmet Yıldırım.

114

Solution spaces of dynamic equations over time scales space R ALPH O BERSTE -V ORTH Marshall University Department of Mathematics Huntington, West Virginia, USA [email protected] http://www.marshall.edu/math/contact.asp We prove a recent conjecture characterizing the Fell topology on the space of time scales. We pursue basic questions of how a changes in time scale may affect the solutions of a given dynamic equation. Insight into these questions are of interest both for applications as well as in theory.

AMS Subject Classification: 37. Keywords: Time scales, dynamic equation, Fell topology.

115

Bifurcations for nonautonomous interval maps H ENRIQUE O LIVEIRA Instituto Superior T´ecnico Lisbon Department of Mathematics Lisbon, Portugal [email protected] http://www.math.ist.utl.pt/˜holiv In this work we investigate attracting periodic orbits for nonautonomous discrete dynamical systems with two maps using a new approach. We study some types of bifurcation in these systems. We show that the pitchfork bifurcation plays an important role in the creation of attracting orbits in families of alternating systems with negative Schwarzian derivative and it is central in the geometry of the bifurcation diagrams.

AMS Subject Classification: Primary: 37E05; Secondary: 37E99. Keywords: Nonautonomous system, bifurcation. This is joint work with Emma D’Aniello.

116

On the spectrum of normal difference operators of first order ¨ ZT URK ¨ R UK I˙ YE O Karadeniz Technical University Department of Mathematics Trabzon, Turkey [email protected] In this talk the normality and spectrum of some first order difference operators in the Hilbert space of sequences l2 (N) are investigated. For example, a result has been established in the following form. Let S and A be respectively a right shift and a linear self-adjoint operator in the space l2 (N) and (ImS)l2 (N) ⊂ D(A). Then 1. The operator L = 1 − S + A, L : D(A) ⊂ l2 (N) → l2 (N) is normal in l2 (N) if and only if A = f (ImS) (here f is a function from σ(ImS) to R). 2. If L = 1 − S + A, L : D(A) ⊂ l2 (N) → l2 (N) is a normal operator and A − S = h(ImS), h : σ(ImS) → C, h ∈ C(σ(ImS)), then the spectrum of the operator L is the form σ(L) = (1 + h)([−1, 1]).

AMS Subject Classification: 47A10. Keywords: Space of sequences, difference operators. ˙ This is joint work with Zameddin Ismailov.

117

Boundedness, attractivity, stability of a rational difference equation with two periodic coefficients G ARYFALOS PAPASCHINOPOULOS Democritus University of Thrace Department of Environmental Engineering Xanthi, Greece [email protected] We study the boundedness, the attractivity and the stability of the positive solutions of the rational difference equation xn+1 =

pn xn−2 + xn−3 , qn + xn−3

n = 0, 1, . . . ,

where pn , qn , n = 0, 1, . . . are positive sequences of period 2.

AMS Subject Classification: 39A10. Keywords: Difference equation, boundedness, stability. This is joint work with G. Stefanidou and C. J. Schinas.

118

Classification and stability of functional equations C HOONKIL PARK Hanyang University Department of Mathematics Seoul, South Korea [email protected] In this talk, we classify and prove the generalized Hyers–Ulam stability of linear, quadratic, cubic, quartic and quintic functional equations in complex Banach spaces.

AMS Subject Classification: 39B72. Keywords: Fixed point, functional equations, stability. This is joint work with Young Hak Kwon.

119

Bounded solutions of a rational difference equation S ANDRA P INELAS Azores University Mathematical Department Ponta Delgada, Portugal [email protected] http://www.uac.pt/˜spinelas This talk studies the existence of bounded solutions of the rational difference equation xn+1 =

βn xn + xn−1 , xn−2

n = 1, 2, . . .

with initial conditions x−2 , x−1 , x0 ∈ R+ and 0 < βn < 1.

120

Nonoscillatory solutions of a second-order difference equation of Poincar´e type ´ P ITUK M IH ALY University of Pannonia Department of Mathematics and Computing Veszpr´em, Hungary [email protected] http://www.szt.vein.hu/˜pitukm Poincar´e’s classical theorem about the convergence of the ratios of successive values of the solutions of linear homogeneous difference equations applies if the characteristic values of the limiting equation are simple and have different moduli. In this talk we show that for the nonoscillatory solutions the conclusion of Poincar´e’s theorem is also true in the case when the limiting equation has a double positive characteristic value.

This is joint work with Rigoberto Medina.

121

Generalized Jacobians for solving nondifferentiable equations arising from contact problems N ICOLAE P OP North University of Baia Mare Department of Mathematics and Computer Science Baia Mare, Romania [email protected] http://www.ubm.ro The aim of this talk is to give an algorithm for solving nondifferentiable equations using generalized Jacobians with applications in contact problems. In contact problems, the functional which describes the influence of the friction is nondifferentiable. For solving the discretized contact problem, the Newton method for linearization is employed, where generalized Jacobians must be used. The generalized Jacobians and the generalized gradient coincide. This method can be used to apply the conjugate gradient method for solving of the equation that contains a nondifferentiable nonlinear operator which is reduced to the successive solution of auxiliary linear equations. This linear operator (equations) can be regarded as a special kind of preconditioner. See also Axelson, O., Chronopoulos, A. T., On nonlinear generalized conjugate gradient methods, Numer. Math., 69 (1994), No. 1, pp. 1–15 and Clarke, F. H., Optimization and nonsmooth analysis, Wiley and Sons, 1983.

AMS Subject Classification: 35J85, 74G15. Keywords: Generalized Jacobian, contact problems.

122

Integro-difference equation associated to a reaction-diffusion equation E MIL P OPESCU Technical University of Bucharest Civil Engineering Bucharest, Romania [email protected] Using a product formula and the discretization of the time for a reaction-diffusion equation, we present a sequential splitting schema which gives corresponding discrete time integrodifference equation.

This is joint work with Nedelia Antonia Popescu.

123

Finite size scaling technique and applications N EDELIA A NTONIA P OPESCU Romanian Academy of Sciences Astronomical Institute Bucharest, Romania [email protected] The finite size scaling technique is applied on the Ulysses/VHM data in order to study the scaling of the magnetic field magnitude (B) and energy density (B 2 ) fluctuations of the interplanetary magnetic field. The basic considered quantity is the change in the normalized B, B(t)/hBi, at different scales (time lags) τn = 2n (days), n = 0, 1, 2, . . . as follows: dBn = dBn (ti , τn ) = [B(ti + τn ) − B(ti )] /hBi, where ti is the time (day); < B > is the average of B over 1 year at a specific distance; B(ti ) is the daily average of B.

This is joint work with Emil Popescu.

124

Dynamic replicator equation and its transformation Z DEN Eˇ K P OSP´I Sˇ IL Masaryk University Department of Mathematics and Statistics Brno, Czech Republic [email protected] http://www.math.muni.cz/˜pospisil The replicator equation is a vector ordinary differential equation with a cubic nonlinearity. It provides a description of game dynamics as well as evolutionary models for population genetics.
The contribution introduces a dynamic replicator equation for Sn valued function x(t) = xi (t) : ! n X ∆ σ T σ xi (t) = xi (t) aik xi (t) − x(t) Ax (t) , i = 1, 2, . . . , n; k=1

here A = (aij ) is an n × n real matrix and Sn is the n-dimensional probability simplex. Basic qualitative properties of the solution will be shown. The main result is that under some assumptions, there exists a time scale such that the replicator equation is equivalent to the Lotka–Volterra dynamic equation ! n−1 X ˜ ∆ σ ˜ yj (τ ) = yj (τ ) rj + bjk yk (τ ) , j = 1, 2, . . . , n − 1 k=1


for y(τ ) = yj (τ ) from positive (n − 1)-dimensional orthant.

AMS Subject Classification: 34A34, 39A12, 92B05. Keywords: Dynamic nonlinear equation, transformation.

125

Nonautonomous continuation and bifurcation, revisited! ¨ C HRISTIAN P OTZSCHE Munich University of Technology Center for Mathematical Sciences Munich, Germany [email protected] http://www-m12.ma.tum.de/poetzsche We investigate local continuation and bifurcation properties for nonautonomous difference equations. Due to their arbitrary time dependence, equilibria or periodic solutions typically do not exist and are replaced by bounded globally defined solutions. Following this leitmotiv, hyperbolicity properties are formulated via the Sacker–Sell spectrum and exponential dichotomies yield a robust framework for local continuation arguments using the (surjective) implicit function theorem. Dichotomies in variation also provide a Fredholm theory. Thus, we employ a Lyapunov–Schmidt-reduction to deduce nonautonomous versions of the classical fold, transcritical and pitchfork bifurcation patterns. Finally, Sacker–Sell spectral intervals crossing the stability boundary give rise to a new 2-step bifurcation pattern not present in the autonomous situation.

Keywords: Nonautonomous bifurcation, Sacker–Sell spectrum.

126

A nonlinear system of difference equations M IHAELA P REDESCU Bentley College Department of Mathematical Sciences Waltham, Massachusetts, USA [email protected] http://web.bentley.edu/empl/p/mpredescu We investigate the global stability character and the behavior of solutions of the nonlinear system of difference equations    M = aMn + bHn (1 − e−Mn )   n+1 cHn 1 n = 0, 1, . . . . Hn+1 = 1+pA + 1+qA n n     An+1 = rAn + Mn , The initial conditions are nonnegative, the parameters are positive and a, c, r ∈ (0, 1).

AMS Subject Classification: 39A11. This is joint work with T. Awerbuch, E. Camouzis, E. A. Grove, G. Ladas, and R. Levins.

127

Galerkin method for solving nonlinear Fredholm–Hammerstein integral equations with multiwavelet basis M OHSEN R ABBANI Islamic Azad University, Sari Branch Department of Mathematics Sari, Iran [email protected] In this talk, we solve nonlinear Fredholm–Hammerstein integral equations by using multiwavelets constructed from Legendre polynomials. For reducing the operations in comparing with similar works, we used some modifications in approximation coefficients calculating scheme. The numerical examples for the method are of good accuracy.

Keywords: Multiwavelet, Fredholm–Hammerstein, nonlinear.

128

On some rational difference equations via linear recurrence equations properties M USTAPHA R ACHIDI LEGT - F. Arago. Academie de Reims Mathematics Section Reims, France [email protected] The purpose of this talk is to examine the local stability of the following class of rational difference equations Pk−1 ai xn−i−1 xn+1 = Pi=0 , (1) k−1 i=0 bi xn−i−1 where ai ≥ 0, bi ≥ 0 (i = 0, 1, . . . , k) and the initial conditions x−k , x−k+1 , . . . , x0 are arbitrary real numbers. The approach used here is based on the properties of the linear recurrence part associated to equation (1). More precisely, we consider some properties on the convergence of linear recursive sequences, which permits us to obtain some new results on the local stability of equation (1). In addition, for a particular case of equation (1), a straightforward computation leads to the extension of some recent results concerning the global attractivity and boundedness of this equation.

Keywords: Difference equations, stability, recursiveness. This is joint work with Rajae Ben Taher and Mohamed El Fetnassi.

129

Multiple periodic solutions of a second-order nonautonomous rational difference equation M ICHAEL R ADIN Rochester Institute of Technology School of Mathematical Sciences Rochester, New York, USA [email protected] http://www.rit.edu/cos/math/Directory/ ,→Standard/marsma.html We will investigate the existence of multiple periodic solutions of a second order nonautonomous rational difference equation. We will discover the necessary and sufficient conditions for existence of multiple periodic solutions, the pattern of the periodic solutions and convergence to zero and to multiple periodic solutions.

AMS Subject Classification: 39A. Keywords: Convergence, periodic solutions, boundedness. This is joint work with Nicholas Batista.

130

Morse spectrum for linear nonautonomous difference equations M ARTIN R ASMUSSEN University of Augsburg Department of Mathematics Augsburg, Germany [email protected] http://www.math.uni-augsburg.de/˜rasmusse In this talk, the concept of a Morse spectrum is introduced for nonautonomous linear difference equations. In contrast to other spectral notions such as the Sacker-Sell spectrum (which yields a linear decomposition), the Morse spectrum is based on a linear decomposition, the finest Morse decomposition. The existence of such a Morse decomposition is reviewed, and basic properties of the Morse spectrum are discussed. The content of this talk is based on joint work with Fritz Colonius (University of Augsburg) and Peter Kloeden (University of Frankfurt).

This is joint work with Fritz Colonius and Peter Kloeden.

131

Power type comparison theorems for half-linear dynamic equations ˇ EH AK ´ PAVEL R Academy of Sciences of the Czech Republic Institute of Mathematics Brno, Czech Republic [email protected] http://www.math.muni.cz/˜rehak We establish conditions which guarantee that oscillatory properties of a half-linear dynamic equation are preserved when the power in the nonlinearities is changed. We discuss the discrepancies between the results on different time scales. The results are original also in the differential and difference equations cases.

132

Decoupling and simplifying of discrete dynamical systems in the neighbourhood of invariant manifold A NDREJS R EINFELDS University of Latvia Institute of Mathematics and Computer Science Riga, Latvia [email protected] http://home.lanet.lv/˜reinf In a Banach space X × E, the discrete dynamical system  x(t + 1) = g(x(t)) + G(x(t), p(t)), p(t + 1) = A(x(t))p(t) + Φ(x(t), p(t))

(1)

is considered. Sufficient conditions under which there is an Lipschitzian invariant manifold u : X → E are obtained. Using this result we find sufficient conditions of conjugacy of (1) and  x(t + 1) = g(x(t)) + G(x(t), u(x(t)), p(t + 1) = A(x(t))p(t). Relevant results concerning partial decoupling and simplifying of the semidynamical systems are given also.

AMS Subject Classification: 39A, 37D30, 34C31. Keywords: Conjugacy, dynamical systems, invariant manifold.

133

On oscillation of solutions of stochastically perturbed difference equations A LEXANDRA R ODKINA University of the West Indies Department of Mathematics and Computer Science Kingston, Jamaica [email protected] http://www.mona.uwi.edu/dmcs/staff/ ,→arodkina/alya.htm We discuss the path-wise oscillatory behavior of the scalar nonlinear stochastic difference equation X(n + 1) = X(n) − f (X(n)) + g(n, X(n))ξ(n + 1), n = 0, 1, . . . , with nonrandom initial value X0 ∈ R. Here (ξ(n))n≥0 is a sequence of independent random variables with zero mean and unit variance. The functions f : R → R and g : N × R → R are presumed to be continuous. We consider state-independent perturbation, when g does not depend on the second variable, as well as the state-dependent perturbation.

AMS Subject Classification: 37H10, 39A11, 60H10, 34F05, 65C20. Keywords: Stochastic difference equations, oscillation.

134

Invariant objects through wavelets ´ D AVID R OMERO I S ANCHEZ Universitat Aut`onoma de Barcelona Departament de Matem`atiques Bellaterra (Cerdanyola del Vall`es), Spain [email protected] http://www.gsd.uab.cat/personal/dromero A standard approach used in the literature to compute and work with invariant objects of systems exhibiting periodic or quasi-periodic behaviour is to use finite Fourier approximations, namely N X F(ξ) = a0 + (an cos(nξ) + bn sin(nξ)) . n=1

Finite wavelet expansions could be used instead, F(ξ) =

Nj N X X

cj,n ψj,n (ξ),

j=0 n=0

where ψj,n (ξ) is obtained by translation and dilation of a mother wavelet ψ(x). Since wavelets can capture different frequencies at different regions of the space, this approach is expected to be more efficient than the Fourier one. The aim of this talk is to compare the (computional) efficiency of both approaches. For that, we will briefly introduce the necessary tools for wavelet basis and multiresolution analysis.

This is joint work with Llu´ıs Alsed`a and Josep M. Mondelo.

135

Compatibility of local and global stability conditions for some discrete population models S AMIR S AKER King Saud University Department of Mathematics Riyadh, Saudi Arabia [email protected] In this talk, we consider a model that has been proposed to study the growth of bobwhite quail populations of Northern Wisconsin and prove that the local stability implies the global stability. We will prove the results by using a suitable Lyapunov function and for illustration we apply the results on the Hassell and Smith models. We will show that for different values of the parameters, the population will exhibit some time varying dynamics. For parameters close to stable region, this will be a simple two-cycle and if the system is moved in a direction away from stability, by increasing the parameters then the dynamics become more complex and the system undergoes a series of bifurcations which leading to increasingly longer periodic cycles and finally deterministic chaos. Some illustrative examples and graphs are included to demonstrate the validity and applicability of the results.

AMS Subject Classification: 39A10, 92D25. Keywords: Local, global stability, population dynamics.

136

Discrete densities and Fisher information PABLO S ANCHEZ -M ORENO University of Granada Institute Carlos I for Theor. and Comput. Physics Granada, Spain [email protected] http://www.ugr.es/˜pablos Analytic information theory of discrete distributions was initiated in 1998 by C. Knessel, P. Jacquet and S. Szpankowski who addressed the precise evaluation of the Renyi and Shannon entropies of the Poisson, Pascal (or negative binomial) and binomial distributions. They were able to derive various asymptotic approximations and, at times, lower and upper bounds for these quantities. Here we extend these investigations in a twofold way. First, we consider a much larger class of distributions, involving discrete hypergeometric-type polynomials which are orthogonal with respect to the weight function of Poisson, Pascal, binomial and hypergeometric types; that is the polynomials of Charlier, Meixner, Kravchuck and Hahn. Second we compute, at times explicitly, the Fisher informations of the four families of these Rakhmanov distributions.

AMS Subject Classification: 62B10, 30G25. Keywords: Fisher information, discrete densities. ˜ This is joint work with J. S. Dehesa, R. J. Yanez.

137

Boundedness, periodicity, attractivity  ofthe difference equation xn+1 = An +

xn−1 xn

p

C HRISTOS S CHINAS Democritus University of Thrace Department of Electrical and Computer Engineering Xanthi, Greece [email protected] http://utopia.duth.gr/˜cschinas This talk studies the boundedness, the persistence, the periodicity and the stability of the positive solutions of the nonautonomous difference equation  p xn−1 xn+1 = An + , n = 0, 1, . . . , xn where An is a positive bounded sequence, p ∈ (0, 1) ∪ (1, ∞) and x−1 , x0 ∈ (0, ∞).

AMS Subject Classification: 39A10. Keywords: Boundedness, persistence, periodicity, stability. This is joint work with G. Papaschinopoulos and G. Stefanidou.

138

Oscillation of nonlinear three-dimensional difference systems E WA S CHMEIDEL Poznan University of Technology Institute of Mathematics Poznan, Poland [email protected] http://www.put.poznan.pl/˜schmeide Oscillatory properties of solutions are investigated usually for two-dimensional difference systems only, but we have not seen too many oscillatory results for three-dimensional systems of the general form. This observation motivated us to consider nonlinear three-dimensional difference systems and to investigate its oscillatory or almost oscillatory behavior. Moreover, it is an interesting problem to extend oscillation criteria for third-order nonlinear difference equations to the case of nonlinear three-dimensional difference systems since such systems include, in particular, third-order nonlinear difference equations as a special case. We shall provide sufficient conditions under which the considered system is oscillatory or almost oscillatory.

AMS Subject Classification: 39A10, 39A11. Keywords: Nonlinear difference system, oscillation.

139

On the behaviour of the difference equation x(n + 1) = max{1/x(n), min{1, A/x(n)}} N URCAN S¸ EKERC I˙ Sel¸cuk University Department of Mathematics Konya, Turkey [email protected] We study the behavior of the solution of the difference equation x(n + 1) = max{1/x(n), min{1, A/x(n)}}, where A is a real number and the initial condition x(0) is a nonzero real number. In the cases of A > 0 and A < 0 we determine the behaviour of the equation with A, x0 .

AMS Subject Classification: 39A10, 39A11. Keywords: Difference equation, periodicity, behaviour. This is joint work with Necati Tas¸kara and D. Turgut Tollu.

140

Heat solutions by using Fibonacci tane function M OHSEN S HAHREZAEE Imam Hossein University Department of Mathematics Tehran, Iran [email protected] In this talk we introduce and use symmetrical Fibonacci tane for solving heat equation. We know the symmetrical Fibonacci tane is constructed according to the symmetrical Fibonacci sine and cosine in the model of αx − α−x √ SF S(x) = 5 CF S(x) =

αx + α−x √ 5

and tFS will be defined by αx − α−x . αx + α−x As one of its applications an algorithm is devised to obtain exact traveling heat solutions for the differential-difference equations by means of the property of function tane. In fact, we have devised a straightforward algorithm to compute traveling heat solutions without using explicit integration. tF S(x) =

141

Applications of finite difference methods in the field of magnetic refrigeration B AKHODIRZHON S IDDIKOV Ferris State University Department of Mathematics Big Rapids, Michigan, USA [email protected] http://www.ferris.edu/htmls/colleges/ ,→artsands/faculty desc.cfm?FSID=174 Magnetic refrigeration is rapidly developing and becoming competitive with conventional gas compression technology, primarily because the most inefficient component of the refrigerator – the compressor – is eliminated. In this talk, we will discuss a time-dependent one-dimensional model of the active magnetic regenerator which was developed as a highly nonlinear system of partial differential equations. One of the difficulties in the numerical simulations of the active magnetic regenerator is determination of the heat capacity of the magnetic material (gadolinium), C = C(T, H), which depends on the temperature of the material, T = T (x, t), as well as on the magnetic induction, H = H(t), where x is a spatial coordinate and t is a chronological coordinate. I will present an approximation surface for C = C(T, H), which was obtained by using the least-squares surface fitting technique and experimental measurements at 460 data points. We developed the numerical scheme for the computer simulations of the active magnetic regenerator by using a finite-difference method. We will analyze the performance of the numerical scheme for stability and convergence.

AMS Subject Classification: 47N. Keywords: Finite difference method, magnetic refrigeration.

142

Spectral theory of birth-and-death processes M ORITZ S IMON Munich University of Technology Department of Mathematics Munich, Germany [email protected] http://ibb.gsf.de/person.php?name=Moritz+Simon This talk gives an outline of the author’s PhD thesis about birth–and–death processes with killing [Moritz Simon, Spectral Theory of Birth-and-Death Processes, PhD thesis (TUM), Sierke Ver¨ lag, Gottingen, 2008]. Such stationary Markov processes admit a representation of their transition probabilities via orthogonal polynomials (OP) with respective spectral measure. The recursion of the OP depends purely on the birth, death and killing rates in the population process. Linear rates for instance admit an explicit computation of the OP and their spectral measure, which in turn allow to determine the stochastic dynamics of the process. Problems come in as soon as the rates are sufficiently complicated: explicit methods are no more tractable then! Anyway, the use of regular perturbation theory for corresponding Jacobi operators enables one to determine the spectrum in qualitative and approximate respects, at least under a certain domination of killing.

143

Numerical solution of nonlocal boundary value problems for the Schrodinger ¨ equation A L I˙ S IRMA Bah¸ce¸sehir University Department of Mathematics and Computer Sciences Istanbul, Turkey [email protected] In this talk the numerical solution of the multipoint nonlocal boundary value problem  m P   iut − (ar (x)uxr )xr + σuf (t, x), 0 < t < T, x ∈ Ω,    r=1  p P u(0, x) = αj u(λj , x) + ϕ(x), x ∈ Ω,   j=1    u(t, x) = 0, ∂u(t,x) = 0, x ∈ S, 0 ≤ t ≤ T, − ∂→ n ¨ for the Schrodinger equation is considered. Here, ar (x) (x ∈ Ω), ϕ(x) (x ∈ Ω), f (t, x) (t ∈ [0, T ], x ∈ Ω) are smooth functions and σ > 0 is a constant. Ω is the unit cube in the m-dimensional → Euclidean space Rm (0 < xk < 1, 1 ≤ k ≤ m) with boundary S and Ω = Ω ∪ S, − n denotes the normal vector to boundary S.

AMS Subject Classification: 65N14. ¨ Keywords: Schrodinger equation, stability. This is joint work with Allaberen Ashyralyev.

144

On a system of max-difference equations G ESTHIMANI S TEFANIDOU Democritus University of Thrace Department of Electrical and Computer Engineering Xanthi, Greece [email protected] In this talk we study the periodic nature of the positive solutions of the system of difference equations     A1 B1 C1 A2 B2 C2 yn = max , , , zn = max , , , n ≥ 0, zn−1 zn−3 zn−5 yn−1 yn−3 yn−5 where Ai , Bi , Ci , i ∈ {1, 2}, are positive real constants and the initial values yi , zi , i ∈ {−5, −4, . . . , −1} are positive numbers. In addition, we give conditions so that the solutions of this system are unbounded.

AMS Subject Classification: 39A10. Keywords: Difference equations, periodicity, unboundedness. This is joint work with G. Papaschinopoulos and C. J. Schinas.

145

Basic properties of partial dynamic operators P ETR S TEHLIK University of West Bohemia Department of Mathematics Pilsen, Czech Republic [email protected] http://www.kma.zcu.cz/stehlik Motivated by the importance of maximum principles in the theory of partial differential equations and in numerical analysis, we establish simple maximum principles for basic partial dynamic operators on multidimensional time scales. As in the case of ordinary dynamic operators we reveal a set of results and counterexamples which illustrate the distinct behaviour in the continuous and discrete cases. Finally, we provide some immediate consequences and prove uniqueness results to problems involving partial dynamic operators.

This is joint work with Bevan Thompson.

146

Relative oscillation theory for Jacobi operators G ERALD T ESCHL University of Vienna Faculty of Mathematics Vienna, Austria [email protected] http://www.mat.univie.ac.at/˜gerald Classical oscillation theory establishes the connection between the number of eigenvalues and sign flips of certain solutions of a Jacobi operator respectively matrix. We add a new wrinkle to this theory by showing how the number of sign flips of Wronski (resp. Casorati) determinants of solutions can be connected to differences of numbers of eigenvalues.

AMS Subject Classification: 39A10, 39A12. Keywords: Oscillation theory, Jacobi operators.

147

Reducibility and stability results for linear systems of difference equations AYDIN T I˙ RYAK I˙ Gazi University Department of Mathematics Ankara, Turkey [email protected] http://websitem.gazi.edu.tr/tiryaki In this talk, we first give a theorem on the reducibility of a linear system of difference equations of the form x(n + 1) = A(n)x(n). Next, by means of Floquet theory, we obtain some stability results. Moreover, some examples are given to illustrate the importance of the results.

AMS Subject Classification: 39A05, 39A11. Keywords: Reducibility, periodic matrix, Floquet exponents. This is joint work with Adil Mısır.

148

Analysis of a nonlinear discrete dynamical system, signal coding and reconstruction M OUHAYDINE T LEMC¸ ANI ´ Universidade de Evora ´ Centro de Geof´ısica de Evora (CGE) ´ Evora, Portugal [email protected] In this talk, we present a study of different iterated maps in which we are looking for invariants that link their dynamics. Various approaches of conductivity of dynamical systems are analyzed looking for real physical examples. The notion of conductance of a discrete nonlinear dynamical system is linked to a physical time dependent example. The time series issued from a physical system behaviour are processed from a new point of view in order to extract hidden information.

AMS Subject Classification: 37B10, 37A35. Keywords: Dynamical systems, conductance, time series. This is joint work with Sara Fernandes.

149

Multiple positive solutions for a system of higher-order boundary value problems on time scales FATMA S ERAP T OPAL Ege University Department of Mathematics ˙ Izmir, Turkey [email protected] http://sci.ege.edu.tr/˜math/index.php? ,→option=com content&task=view&id=48 In this talk, by applying fixed point theorems in cones and under suitable conditions, we present the existence of single and multiple solutions for the following system of higher-order boundary value problems:  2n   (−1)n y14 (t) = f1 (t, y1σ (t), y2σ (t)), t ∈ [0, 1],     (−1)m y 42m (t) = f (t, y σ (t), y σ (t)), t ∈ [0, 1], 2

2

1

2i 2i   y14 (0) = y14 (σ(1)) = 0,     y 42j (0) = y 42j (σ(1)) = 0, 2 2

2

0 ≤ i ≤ n − 1, 0 ≤ j ≤ m − 1.

AMS Subject Classification: 39A10, 34B15, 34A40. Keywords: Positive solutions, cone, fixed point theorems. This is joint work with Erbil C ¸ etin.

150

Positive solutions of a second-order m-point BVP on time scales A HMET YANTIR Atılım University Department of Mathematics Ankara, Turkey [email protected] http://www.atilim.edu.tr/˜ayantir In this study, we are concerned with proving the existence of multiple positive solutions of a general second-order nonlinear m-point boundary value problem u∆∇ (t) + a(t)u∆ (t) + b(t)u(t) + λh(t)f (t, u) = 0, u(ρ(0)) = 0,

u(σ(1)) =

m−2 X

t ∈ [0, 1],

αi u(ηi )

i=1

on time scales. The proofs are based on fixed point theorems in a Banach space. We present an example to illustrate how our results work.

AMS Subject Classification: 39A10, 34B18, 34B40, 45G10. Keywords: Multi-point BVPs, positive solutions, time scales. This is joint work with Fatma Serap Topal.

151

Numerical solutions of nonlinear differential-difference equations by the homotopy perturbation method A HMET Y ILDIRIM Ege University Department of Mathematics ˙ Izmir, Turkey [email protected] http://sci.ege.edu.tr/˜math/index.php? ,→option=com content&task=view&id=58 A new scheme, deduced from He’s homotopy perturbation method, is presented for solving differential-difference equations. A simple but typical example is applied to illustrate the validity and great potential of the generalized homotopy perturbation method in solving differentialdifference equations. The results reveal that the method is very effective and simple.

Keywords: He’s homotopy perturbation method, differential-difference, Volterra equation. ¨ ¸ in Yalazlar. This is joint work with Gulc

152

A result on successive approximation of solutions to dynamic equations on time scales ATIYA Z AIDI University of New South Wales School of Mathematics and Statistics Sydney, Australia [email protected] http://www.maths.unsw.edu.au/˜atiya ¨ theorem for first order initial value problems on time scales, We establish a Picard–Lindelof where a time scale is a nonempty closed subset of reals. The theorem involves sufficient conditions under which a problem will have a unique solution. At the heart of the approach is the method of successive approximations. The investigation relies on ideas from classical analysis rather than functional analysis. The results guarantee that the “error” estimate between the actual and the approximate solution goes to zero as the number of iterations are increased indefinitely. An example regarding the application of the above method to a nonlinear dynamic equation on time scales is also presented. Several open questions will be posed that concern successive approximations in the time scale setting. This talk will be suitable particularly for graduate students.

Keywords: Time scales, successive approximations, dynamic equation. This is joint work with Christopher Tisdell.

153

Application of the WKB estimation method for determining heat flux on the boundary A LI Z AKERI K. N. Toosi University Department of Mathematics Tehran, Iran [email protected] This talk considers a linear one-dimensional inverse heat conduction problem with nonconstant thermal diffusivity. It has been associated with the estimation of an unknown boundary heat flux. For this purpose, by using temperature measurements taken below the boundary surface and using a semi-implicit finite difference method, the problem will be converted to a system of ordinary differential equations of second order depending on a small parameters with initial conditions. Then WKB estimation method gives asymptotic solutions for this system. The solutions that are produced in this algorithm make the process ill-posed. Then by choosing suitable values of small parameters, this algorithm is modified. Finally, a numerical experiment will be presented.

AMS Subject Classification: 35R30. Keywords: Inverse problem, implicit finite difference method.

154

Trigonometric and hyperbolic systems on time scales ´ P ETR Z EM ANEK Masaryk University Department of Mathematics and Statistics Brno, Czech Republic [email protected] http://www.math.muni.cz/˜xzemane2 In this talk we discuss trigonometric and hyperbolic systems on time scales. These systems generalize and unify their corresponding continuous-time and discrete-time analogues, namely the systems known in the literature as trigonometric and hyperbolic linear Hamiltonian systems and discrete symplectic systems. We provide time scale matrix definitions of the usual trigonometric and hyperbolic functions and show that many identities known from the basic calculus extend to this general setting, including the time scale differentiation of these functions.

AMS Subject Classification: 39A12. Keywords: Time scale, Hamiltonian system, trigonometric system. This is joint work with Roman Hilscher.

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Other Participants

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T HABET A BDELJAWAD C ¸ ankaya University Department of Mathematics Ankara, Turkey [email protected] http://math.cankaya.edu.tr/˜thabet

M ELTEM A DIYAMAN Dokuz Eylul ¨ University Department of Mathematics ˙ Izmir, Turkey [email protected]

M URAT A KMAN Middle East Technical University Department of Mathematics Ankara, Turkey [email protected]

158

M ELTEM A LTUNKAYNAK Dokuz Eylul ¨ University Department of Mathematics Ankara, Turkey [email protected]

K EMAL AYDIN Sel¸cuk University Department of Mathematics Konya, Turkey [email protected]

¨ M UJGAN B AS¸ Afyon Kocatepe University Department of Mathematics Afyonkarahisar, Turkey [email protected]

159

M ARTIN B OHNER Missouri University of Science and Technology Department of Mathematics and Statistics Rolla, Missouri, USA [email protected] http://web.mst.edu/˜bohner

I˙ LKNUR B OZOK Atılım University Department of Mathematics Ankara, Turkey kuzu [email protected]

¨ G ULTER B UDAKC¸ I Dokuz Eylul ¨ University Department of Mathematics ˙ Izmir, Turkey [email protected]

160

C ANAN C AN Atılım University Department of Mathematics Ankara, Turkey canan can [email protected]

D UYGU C ¸ AYLAK Dokuz Eylul ¨ University Department of Mathematics ˙ Izmir, Turkey duygu [email protected]

O KAY C ¸ ELEBI Yeditepe University Department of Mathematics Istanbul, Turkey [email protected] http://www.math.metu.edu.tr/˜celebi

161

C EM C ¸ EL I˙ K Dokuz Eylul ¨ University Department of Mathematics ˙ Izmir, Turkey [email protected]

¨ G ULNUR C ¸ EL I˙ K K IZILKAN Sel¸cuk University Department of Mathematics Konya, Turkey [email protected] http://asp.selcuk.edu.tr/asp/personel/ ,→web/goster.asp?sicil=6228

C ENG I˙ Z C ¸ INAR Sel¸cuk University Department of Mathematics, Education Faculty Konya, Turkey [email protected]

162

S EBAHAT E BRU D AS¸ Yıldız Technical University Department of Mathematics Istanbul, Turkey [email protected]

A SLI D EN I˙ Z ˙ Izmir Institute of Technology Department of Mathematics ˙ Izmir, Turkey [email protected]

Z HAOYANG D ONG Universitat Aut`onoma de Barcelona Departament de Matem`atiques Barcelona, Spain [email protected]

163

M ELDA D UMAN Dokuz Eylul ¨ University Department of Mathematics ˙ Izmir, Turkey [email protected]

S ABER E LAYDI Trinity University Department of Mathematics San Antonio, Texas, USA [email protected] http://www.trinity.edu/selaydi

´ M ARIO G ETIMANE Instituto Superior de Transportes e Communica¸co˜ es Department of Mathematics Maputo, Mozambique [email protected]

164

¨ US ¨¸ I˙ BRAH I˙ M H AL I˙ L G UM Sel¸cuk University Department of Mathematics Konya, Turkey [email protected]

˘ V EYSEL F UAT H AT I˙ PO GLU Mu˘gla University Department of Mathematics Mu˘gla, Turkey [email protected]

¨ ¸ E I˙ NTEPE G OKC Dokuz Eylul ¨ University Department of Mathematics ˙ Izmir, Turkey [email protected]

165

K HAJEE J ANTARAKHAJORN Thammasat University Department of Mathematics and Statistics Phatumthani, Thailand [email protected] http://math.sci.tu.ac.th/people 001.html

R UK I˙ YE K ARA Mimar Sinan University Department of Mathematics Istanbul, Turkey [email protected]

Z EYNEP K AYAR Middle East Technical University Department of Mathematics Ankara, Turkey [email protected]

166

ˆ K AYMAKC¸ ALAN B I˙ LL UR Georgia Southern University Department of Mathematical Sciences Statesboro, Georgia, USA [email protected] http://math.georgiasouthern.edu/˜billur

Y EL I˙ Z K IYAK U C¸ AR Afyon Kocatepe University Department of Mathematics Afyonkarahisar, Turkey [email protected]

S UPACHARA K ONGNUAN Thammasat University Department of Mathematics and Statistics Phatumthani, Thailand [email protected] http://math.sci.tu.ac.th/people 017.html

167

N ATALIA K OSAREVA Moscow Institute of Electronics and Mathematics Cybernetics Department Moscow, Russia [email protected]

YAKOV K ULIK University of New South Wales School of Physics Sydney, Australia [email protected]

V I˙ LDAN K UTAY Ankara University Department of Mathematics Ankara, Turkey vildan [email protected]

168

A NDREAS L EONHARDT Technical University Munich Department of Mathematics Munich, Germany [email protected]

G ORAN L ESAJA Georgia Southern University Department of Mathematical Sciences Statesboro, Georgia, USA [email protected]

R OBERT L. M ARSH East Georgia College Mathematics / Science Division Statesboro, Georgia, USA [email protected] http://personal.georgiasouthern.edu/˜rmarsh

169

A D I˙ L M ISIR Gazi University Department of Mathematics Ankara, Turkey [email protected]

M EHMED N URKANOVI C´ University of Tuzla Department of Mathematics Tuzla, Bosnia and Herzegovina [email protected] http://www.pmf.untz.ba

Z EHRA N URKANOVI C´ University of Tuzla Department of Mathematics Tuzla, Bosnia and Herzegovina [email protected] http://www.pmf.untz.ba

170

¨ ZKAN O ¨ CALAN O Afyon Kocatepe University Department of Mathematics Afyonkarahisar, Turkey [email protected] http://www2.aku.edu.tr/˜ozkan

I˙ SRAF I˙ L O KUMUS¸ Erzincan University Department of Mathematics Erzincan, Turkey [email protected]

¨ ZKAN U MUT M UTLU O Afyon Kocatepe University Department of Mathematics Afyonkarahisar, Turkey umut [email protected]

171

¨ ZPINAR F I˙ GEN O Afyon Kocatepe University Department of Mathematics Afyonkarahisar, Turkey [email protected]

¨ ZT URK ¨ S ERM I˙ N O Afyon Kocatepe University Department of Mathematics Afyonkarahisar, Turkey [email protected]

¨ ZU GURLU ˘ E RS I˙ N O Bah¸ce¸sehir University Department of Mathematics Istanbul, Turkey [email protected]

172

PARAMEE R EANKITTIWAT Thammasat University Department of Mathematics and Statistics Phatumthani, Thailand [email protected] http://math.sci.tu.ac.th/people 010.html

A NDREAS R UFFING Technical University Munich Department of Mathematics Munich, Germany [email protected] http://www-m6.ma.tum.de/˜ruffing

I˙ PEK S AVUN Dokuz Eylul ¨ University Department of Mathematics ˙ Izmir, Turkey ipek [email protected]

173

˘ ¸ EN S ELMANO GULLARI ˘ T U GC Mimar Sinan University Department of Mathematics Istanbul, Turkey [email protected]

S OPORN S ENEETANTIKUL Thammasat University Department of Mathematics and Statistics Phatumthani, Thailand [email protected] http://math.sci.tu.ac.th/people 018.html

G I˙ ZEM S EYHAN Ankara University Department of Mathematics Ankara, Turkey [email protected]

174

˘ D A GISTAN S¸ I˙ MS¸ EK Sel¸cuk University Department of Mathematics, Education Faculty Konya, Turkey [email protected] http://asp.selcuk.edu.tr/asp/personel/ ,→web/goster.asp?sicil=5960

WALTER S IZER Minnesota State University Department of Mathematics Moorhead, Minnesota, USA [email protected] http://www.mnstate.edu/sizer

A NDREAS S UHRER Technical University Munich Department of Mathematics Munich, Germany [email protected]

175

N ECAT I˙ TAS¸ KARA Sel¸cuk University Department of Mathematics, Education Faculty Konya, Turkey [email protected]

N AWALAX T HONGJUB Thammasat University Department of Mathematics and Statistics Phatumthani, Thailand [email protected] http://math.sci.tu.ac.th/people 006.html

D. T URGUT T OLLU Sel¸cuk University Department of Mathematics, Education Faculty Konya, Turkey hasan [email protected]

176

D EN I˙ Z U C¸ AR U¸sak University Department of Mathematics U¸sak, Turkey [email protected]

¨ NAL M EHMET U Bah¸ce¸sehir University Department of Software Engineering Istanbul, Turkey [email protected] http://web.bahcesehir.edu.tr/munal

S IRICHAN V ESARACHASART Thammasat University Department of Mathematics and Statistics Phatumthani, Thailand [email protected] http://math.sci.tu.ac.th/people 020.html

177

D OMINIK V U Vienna University of Technology Institute of Analysis and Scientific Computing Vienna, Austria [email protected]

¨ ¸ I˙ N YALAZLAR G ULC Ege University Department of Mathematics ˙ Izmir, Turkey sugulu [email protected]

I˙ BRAH I˙ M YALC¸ INKAYA Sel¸cuk University Department of Mathematics, Education Faculty Konya, Turkey [email protected] http://asp.selcuk.edu.tr/asp/personel/ ,→web/goster.asp?sicil=5925

178

M UHAMMED Y I˙ G˘ I˙ DER Erzincan University Department of Mathematics Erzincan, Turkey m.yigider [email protected]

M USTAFA K EMAL Y ILDIZ Afyon Kocatepe University Department of Mathematics Afyonkarahisar, Turkey [email protected]

¨ ZLEM Y ILMAZ O Ege University Department of Mathematics ˙ Izmir, Turkey [email protected]

179

¨ UK ¨ F ULYA Y OR Ege University Department of Mathematics ˙ Izmir, Turkey fulya [email protected]

180

Local Organization Assistants

181

M. A SLI AYDIN Bah¸ce¸sehir University Faculty of Arts and Sciences Istanbul, Turkey [email protected]

K EMAL B AYAT Bah¸ce¸sehir University Faculty of Engineering Istanbul, Turkey [email protected]

˘ N AZLI C EREN D A GYAR Bah¸ce¸sehir University Faculty of Arts and Sciences Istanbul, Turkey [email protected]

182

¨ YAKUP E M UL Bah¸ce¸sehir University Faculty of Arts and Sciences Istanbul, Turkey [email protected]

¨ D URDANE E RKAL Bah¸ce¸sehir University Faculty of Arts and Sciences Istanbul, Turkey [email protected]

¨ ¸ E K ARAHAN G OKC Bah¸ce¸sehir University Faculty of Arts and Sciences Istanbul, Turkey [email protected]

183

M USA K ARAKELLE Bah¸ce¸sehir University Faculty of Engineering Istanbul, Turkey [email protected]

¨ ZDEM I˙ R ¨ H USEY I˙ N O Bah¸ce¸sehir University Faculty of Engineering Istanbul, Turkey [email protected]

¨ ZEN B AHAD I˙ R O Bah¸ce¸sehir University Faculty of Engineering Istanbul, Turkey [email protected]

184

Conference Proceedings The conference publishes refereed proceedings of accepted papers. The Proceedings are pub˘ – Bahc¸es¸ehir University Publishing Company (ISBN 978-975-6437-80-3). Contriblished by Ugur utors receive the proceedings free of charge. The deadline to receive submissions prepared using the style file available from the conference website is October 31, 2008. The maximum page limit for contributed talk papers is 8 printed pages. Please send the manuscript to the e-mail of the conference [email protected] or directly to any of the following editors.

Martin Bohner Missouri S&T Rolla, Missouri, USA

Zuzana Doˇsl´a Masaryk University Brno, Czech Republic

Gerasimos Ladas University of Rhode Island Kingston, Rhode Island, USA

¨ Mehmet Unal Bahc¸es¸ehir University Istanbul, Turkey

Agacık ˘ Zafer Middle East Technical University Ankara, Turkey

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186

Social Program Sunday, July 20, 2008, 6 pm: Bahc¸es¸ehir University invites you to join the Welcome Party at the roof of the Bes¸iktas¸ building overlooking the Bosporus. This event is included in the registration fee.

Monday, July 21, 2008, 6:15 pm: Sightseeing, free time. Suggestions (on participants’ expenses): Visit to Dolmabahc¸e Palace, Or¨ Taksim, C takoy, ¸ ic¸ek Pasajı, and dinner in the Galata Tower.

Tuesday, July 22, 2008, 6:15 pm: Sightseeing, free time (on participants’ expenses).

Wednesday, July 23, 2008, 9 am: Istanbul tour (Topkapı Palace – Ayasofya Mosque – Archeology Museum). The Bosporus yacht tour (on private yacht) including dinner starts at 7 pm and will take about 5 hours. The entire day trip including all admission tickets and including the yacht tour is covered by the registration fee.

Thursday, July 24, 2008, 8 pm: Bahc¸es¸ehir University invites you to join the Farewell Dinner at the roof of the Bes¸iktas¸ building overlooking the Bosporus. This event is included in the registration fee.

Friday, July 25, 2008, 6:15 pm: More sightseeing, free time (on participants’ expenses).

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Maps

¨ International Airport Terminal from 7:00 to ICDEA08 Staff meets you at the exit gate of Ataturk 23:30 on July 18–20, 2008 to help your transfer.

189

The conference site is on the Bes¸iktas¸ Campus of Bahc¸es¸ehir University, on the European shores ¨ udar ¨ of the Bosporus, a short walk from the ferry landing of the Bes¸iktas¸ (Europe) – Usk (Asia) connection (Bes¸iktas¸ Vapur Iskelesi).

190

The address of the Tas¸lık Hotel is Suleyman ¨ Seba Caddesi No:75.

191

The address of the Yurdum Guest House (female) is Tavuk¸cu Fethi Sokak No:29.

192

The address of the Yurdum Guest House (male) is Ta¸s Basamak Sokak No:20.

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196

About Istanbul “There, God and human, nature and art are together, they have created such a perfect place that it is valuable to see.” Lamartine’s famous poetic line reveals his love for Istanbul, describing the embracing of two continents, with one arm reaching out to Asia and the other to Europe. Istanbul, once known as the capital of capital cities, has many unique features. It is the only city in the world to straddle two continents, and the only one to have been a capital during two consecutive empires – Christian and Islamic. Once capital of the Ottoman Empire, Istanbul still remains the commercial, historical and cultural pulse of Turkey, and its beauty lies in its ability to embrace its contradictions. Ancient and modern, religious and secular, Asia and Europe, mystical and earthly all co-exist here. Its variety is one of Istanbul’s greatest attractions: The ancient mosques, palaces, museums and bazaars reflect its diverse history. The thriving shopping area of Taksim buzzes with life and entertainment. And the serene beauty of the Bosporus, Princes Islands and parks bring a touch of peace to the otherwise chaotic metropolis. Districts: Adalar, Avcilar, Bagcilar, Bahc¸elievler, ¨ Bes¸iktas¸, Bayrampasa, Beykoz, Beyoglu, ˘ ¨ u, ¨ Eyup, ¨ Fatih, GaziBakirkoy, Eminon ¨ Kˆagithane, ˘ ¨ ¸ ukc ¨ ¸ ekmece, Pendik, Sarıyer, Sisli, osmanpasa, Kadıkoy, Kartal, Kuc ¨ ¨ udar, ¨ ¨ ukc ¨ ¸ ekmece, C Umraniye, Usk Zeytinburnu, Buy ¸ atalca, Silivri, S¸ile, Esenler, ¨ oren, ¨ Gung Maltepe, Sultanbeyli, Tuzla. Golden Horn: This horn-shaped estuary divides European Istanbul. One of the best natural harbours in the world, it was once the centre for the Byzantine and Ottoman navies and commercial shipping interests. Today, attractive parks and promenades line the shores, a picturesque scene especially as the sun goes down over the water. At Fener and Balat, neighbourhoods midway up the Golden Horn, there are entire streets filled with old wooden houses, churches, and synagogues dating from Byzantine and Ottoman times. The Orthodox Patriarchy resides at 197

¨ are some wonderful exFener and a little further up the Golden Horn at Eyup, amples of Ottoman architecture. Muslim pilgrims from all over the world visit ¨ Camii and Tomb of Eyup, ¨ the Prophet Mohammed’s standard bearer, and Eyup it is one of the holiest places in Islam. The area is still a popular burial place, and the hills above the mosque are dotted with modern gravestones interspersed with ornate Ottoman stones. The Pierre Loti Cafe, atop the hill overlooking the shrine and the Golden Horn, is a wonderful place to enjoy the tranquility of the view. ˘ is an interesting example of a district with EuropeanBeyoglu ˘ and Taksim: Beyoglu influenced architecture, from a century before. Europe’s second oldest subway, Tunel was built by the French in 1875, must be also one of the shortest offering a one-stop ride to start of Taksim. Near to Tunel is the Galata district, whose Galata Tower became a famous symbols of Istanbul, and the top of which offers a tremendous 180 degree view of the city. From the Tunel area to Taksim square is one of the city’s focal points for shopping, entertainment and urban promenading: Istiklal Cadesi is a fine example of the contrasts and compositions of Istanbul; fashion shops, bookshops, cinemas, markets, restaurants and even hand-carts selling trinkets and simit (sesame bread snack) ensure that the street is packed throughout the day until late into the night. The old tramcars re-entered into service, which shuttle up and down this fascinating street, and otherwise the street is entirely pedestrianised. There are old embassy buildings, Galatasaray High School, the colourful ambience of Balık Pazarı (Fish Bazaar) and restaurants in C ¸ ic¸ek Pasajı (Flower Passage). Also on this street is the oldest church in the area, St. Mary’s Draperis dating back to 1789, and the Franciscan Church of St. Antoine, demolished and then rebuilt in 1913. The street ends at Taksim Square, a huge open plaza, the hub of modern Istanbul and always crowded, crowned with an imposing monument celebrating ¨ and the War of Independence. The main terminal of the new subway is Ataturk under the square, adjacent is a noisy bus terminal, and at the north end is the ¨ Cultural Centre, one of the venues of the Istanbul Theatre Festival. SevAtaturk eral five-star hotels are dotted around this area, like the Hyatt, Intercontinental and Hilton (the oldest of its kind in the city). North of the square is the Istanbul Military Museum. ˘ have for centuries been the centre of nightlife, and now Taksim and Beyoglu there are many lively bars and clubs off Istiklal Cadesi, including some of the ˘ is also the centre of the more bohemian arts only gay venues in the city. Beyoglu scene. Sultanahmet: Many places of tourist interest are concentrated in Sultanahmet, heart of the Imperial Centre of the Ottoman Empire. The most important places in this area, all of which are described in detail in the Places of Interest section, 198

are Topkapı Palace, Aya Sofia, Sultan Ahmet Camii (the Blue Mosque), the Hippodrome, Kapalı Cars¸ı (Covered Market), Yerebatan Sarnıcı and the Museum of Islamic Art. In addition to this wonderful selection of historical and architectural sites, Sultanahmet also has a large concentration of carpet and souvenir shops, hotels and guesthouses, cafes, bars and restaurants, and travel agents. ¨ was a resort for the OtOrtakoy: ¨ Ortakoy toman rulers because of its attractive location on the Bosporus, and is still a popular spot for residents and visitors. The village is within a ˘ Palace, Kabatas¸ triangle of a mosque, church and synagogue, and is near C ¸ ıragan High School, Feriye, Princess Hotel. ¨ reflects the university students and teachers who would The name Ortakoy gather to drink tea and discuss life, when it was just a small fishing village. These days, however, that scene has developed into a suburb with an increasing amount of expensive restaurants, bars, shops and a huge market. The fishing, however, lives on and the area is popular with local anglers, and there is now a huge waterfront tea-house which is crammed at weekends and holidays. Sarıyer: The first sight of Sarıyer is where the Bosporus connects with the Black Sea, after the bend in the river after Tarabya. Around this area, old summer houses, embassies and fish restaurants line the river, and a narrow road which ¨ ukdere, ¨ separates it from Buy continues along to the beaches of Kilyos. ˘ are the final wharfs along the European side visSariyer and Rumeli Kavagı ited by the Bosporus boat trips. Both these districts, famous for their fish restau˘ get very crowded at weekends and holidays rants along with Anadolu Kavagı, with Istanbul residents escaping the city. After these points, the Bosporus is lined with tree-covered cliffs and little habitation. The Sadberk Hanım Museum, just before Sarıyer, is an interesting place to visit; a collection of archaeological and ethnographic items, housed in two wooden houses. A few kilometres away is the huge Belgrade Forest, once a haunting ground of the Ottomans, and now a popular weekend retreat into the largest forest area in the city. ¨ udar: Usk ¨ Relatively unknown to tourists, the ¨ udar, ¨ suburb of Usk on the Asian side of the Bosporus, is one of the most attractive suburbs. Religiously conservative in its background, it has a tranquil atmosphere and some fine examples of imperial and domestic architecture. The Iskele, or Mihrimah Camii is opposite the main ferry pier, on a high platform with a huge covered porch in front, often occupied by older local men watching life around them. Opposite this is Yeni Valide Camii, built in 199

1710, and the Valide Sultan’s green tomb rather like a giant birdcage. The Cinili Mosque takes its name from the beautiful tiles which decorate the interior, and was built in 1640. ¨ udar ¨ Apart from places of religious interest, Usk is also well known as a shopping area, with old market streets selling traditional local produce, and a good fleamarket with second hand furniture. There are plenty of good restaurants and cafes with great views of the Bosporus and the rest of the city, along the quayside. In the direction of Haydarpas¸a is the lhe Karaca Ahmet Cemetery, the largest Muslim graveyard in Istanbul. The front of the C ¸ amlıca hills lie at the ridge of area and also offer great panoramic views of the islands and river. ¨ Kadıkoy: ¨ Further south along the Bosporus towards the Sea of Marmara, Kadıkoy has developed into a lively area with up-market shopping, eating and entertainment making it popular especially with wealthy locals. Once prominent in the history of Christianity, the 5th century hosted important consul meetings here, but there are few reminders of that age. It is one of the improved districts of Istanbul over the last century, and fashionable area to promenade along the waterfront in the evenings, especially around the marinas and yacht clubs. ˘ Bagdat Caddesi is one of the most trendy and label-conscious fashion shop¨ uz ¨ Caddesi ping streets, and for more down-to-earth goods, the Gen Azım Gund ¨ is the best place for clothes, and the bit pazarı on Ozelellik Sokak is good for browsing through junk. In the district of Moda is the Benadam art gallery, as well as many foreign cuisine restaurants and cafes. ¨ is Haydarpas¸a, and the train station built Haydarpa¸sa: To the north of Kadıkoy in 1908 with Prussain-style architecture which was the first stop along the Baghdad railway. Now it is the main station going to eastbound destinations both within Turkey, and internationally. There are tombs and monuments dedicated to the English and French soldiers who lost their lives during the Crimean War (1854–56), near the military hospital. The north-west wing of the 19th Century Selimiye Barracks once housed the hospital, used by Florence Nightingale to care for soldiers, and remains to honour her memory. ¨ although still within the city, is 25 km away from the Polonezkoy: ¨ Polonezkoy, centre and not easy to reach by public transport. Translated as village of the Poles, the village has a fascinating history: It was established in 1848 by Prince Czartorisky, leader of the Polish nationals who was granted exile in the Ottoman Empire to escape oppression in the Balkans. During his exile, he succeeded in establishing a community of Balkans, which still survives, on the plot of land sold to him by a local monastery. Since the 1970s the village has become a popular place with local Istanbulites, who buy their pig meat there (pig being forbidden under Islamic law and therefore difficult to get elsewhere). All the Poles have since left the village, and the place is inhabited now by wealthy city people, living in the few remaining Central European style wooden houses with pretty balconies. ¨ is its vast green expanse, which What attracts most visitors to Polonezkoy was designated Istanbul’s first national park, and the walks though forests with 200

streams and wooden bridges. Because of its popularity, it gets crowded at weekends and the hotels are usually full. Kilyos: Kilyos is the nearest beach resort to the city, on the Black Sea coast on the European side of the Bosporus. Once a Greek fishing village, it has quickly been developed as a holiday-home development, and gets very crowded in summer. Because of its ease to get there, 25 km and plenty of public transport, it is good for a day trip, and is a popular weekend getaway with plenty of hotels, and a couple of campsites. ¨ udar ¨ S¸ile: A pleasant, small holiday town, S¸ile lies 50 km from Usk on the Black Sea coast and some people even live here and commute into Istanbul. The white sandy beaches are easily accessible from the main highway, lying on the west, as well as a series of small beaches at the east end. The town itself if perched on a clifftop over looking the bay tiny island. There is an interesting Frenchbuilt black-and-white striped lighthouse, and 14th century Genoese castle on the nearby island. Apart from its popular beaches, the town is also famous for its craft; S¸ile bezi, a white muslin fabric a little like cheesecloth, which the local women embroider and sell their products on the street, as well as all over Turkey. The town has plenty of accommodation available, hotels, guest houses and pansiyons, although can get very crowded at weekends and holidays as it is very popular with people from Istanbul for a getaway, especially in the summer. There are small restaurants and bars in the town. Prince’s Islands: Also known as Istanbul Islands, there are eight within one hour from the city, in the Marmara Sea. Boats ply the islands from Sirkeci, Kabatas¸ and Bostancı, with more services during the summer. These islands, on which monasteries were established during the Byzantine period, were a popular summer retreat for palace officials. It is still a popular escape from the city, with wealthier owning summer houses. ¨ ukada ¨ The largest and most popular is Buy (the Great Island). Large wooden mansions still remain from the 19th century when wealthy Greek and Armenian bankers built them as holiday villas. The island has always been a place predominantly inhabited by minorities, hence Islam has never had a strong ¨ ukada ¨ presence here. Buy has long had a history of people coming here in exile or retreat; its most famous guest being Leon Trotsky, who stayed for four years writing ‘The History of the Russian Revolution’. The monastery of St. George also played host to the granddaughter of Empress Irene, and the royal princess Zoe, in 1012. The island consists of two hills, both surmounted by monasteries, with a valley between. Motor vehicles are banned, so getting around the island can be done by graceful horse and carriage, leaving from the main square off Isa C ¸ elebi Sokak. ¨ Tepe, is the quieter of the two Bicycles can also be hired. The southern hill, Yule and also home of St. George’s Monastery. It consists of a series of chapels on three 201

levels, the site of which is a building dating back to the 12th century. In Byzantine times it was used as an asylum, with iron rings on the church floors used to restrain patients. On the northern hill is the monastery Isa Tepe, a 19th century house. The entire island is lively and colourful, with many restaurants, hotels, tea houses and shops. There are huge well-kept houses, trim gardens, and pine groves, as well as plenty of beach and picnic areas. Smaller and less of a tourist infrastructure is Burgazada. The famous Turkish novelist, Sait Faik Abasiyanik lived here, and his house has been turned into a museum dedicated to his work, and retains a remarkable tranquil and hallowed atmosphere. Heybeliada, ‘Island of the Saddlebag’, because of its shape, is loved for its natural beauty and beaches. It also has a highly prestigious and fashionable watersports club in the northwest of the island. One of its best-known landmarks is the Greek Orthodox School of Theology, with an important collection of Byzantine manuscripts. The school sits loftily on the northern hill, but permission is needed to enter, from the Greek Orthodox Patriarchate in Fener. The Deniz Harp Okulu, the Naval High School, is on the east side of the waterfront near the jetty, which was originally the Naval War Academy set up in 1852, then a high school since 1985. Walking and cycling are popular here, plus isolated beaches as well as the public Yoruk Beach, set in a magnificent bay. There are plenty of good local restaurants and tea houses, especially along Ayyıldız Caddesi, and the atmosphere is one of a close community. Environment: Wide beaches of Kilyos at European side of Black Sea at 25th km outside Istanbul, are attracting Istanbul residents during summer months. Belgrade Forest, inside from Black Sea, at European Side is the widest forest around Istanbul. Istanbul residents, at weekends, come here for family picnic with brazier at its shadows. 7 old water tank and some natural resources in the region ˘ compose a different atmosphere. Moglova Aqueduct, which is constructed by Mimar Sinan during 16th century among Ottoman aqueducts, is the greatest one. ¨ 800 m long Sultan Suleyman Aqueduct, which is passing over Golf Club, and also a piece of art of Mimar Sinan is one of the longest aqueducts within Turkey. ¨ which is 25 km away from Istanbul, is founded at Asia coast Polonezkoy, ¨ for walking in village during 19th century by Polish immigrants. Polonezkoy, atmosphere, travels by horse, and tasting traditional Polish meals served by relatives of initial settlers, is the resort point of Istanbul residents. Beaches, restau¨ udar, ¨ rants and hotels of S¸ile at Black Sea coast and 70 km away from Usk are turning this place into one of the most cute holiday places of Istanbul. Region which is popular in connection with tourism, is the place where famous S¸ile cloth is produced. ˘ - Darica Bird Paradise and Botanic Park is a unique resort place Bayramoglu 202

38 km away from Istanbul. This gargantuan park with its trekking roads, restaurants is full of bird species and plants, coming from various parts of the world. Sweet Eskihisar fisherman borough, to whose marina can be anchored by yachtsmen after daily voyages in Marmara Sea is at south east of Istanbul. Turkey’s 19th century famous painter, Osman Hamdi Bey’s house in borough is turned into a museum. Hannibal’s tomb between Eskihisar and Gebze is one of the sites around a Byzantium castle. There are lots of Istanbul residents’ summer houses in popular holiday place 65 km away from Istanbul, Silivri. This is a huge holiday place with magnificent restaurants, sports and health centers. Conference center is also attracting businessmen, who are escaping rapid tempo of urban life for “cultural tourism” and business - holiday mixed activities. Scheduled sea bus service is connecting Istanbul to Silivri. Islands within Marmara Sea, which is adorned with nine islands, was the banishing place of the Byzantium princes. Today they are now wealthy Istanbul residents’ escaping places for cool winds during summer months and 19th cen¨ ukada. ¨ tury smart houses. The biggest one of the islands is Buy You can have a marvelous phaeton travel between pine trees or have a swim within one of the numerous bays around islands! Other popular islands are Kınali, Sedef, Burgaz and Heybeliada. Regular ferry voyages are connecting islands to both Europe and Asia coasts. There is a rapid sea bus service from Kabatas¸ during summers.

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Useful Information Airport ¨ International Airport on the EuroIstanbul has two airports, the major Ataturk ¨ ¸ en Airport on the east side pean shore of the Sea of Marmara and Sabiha Gokc ¨ of the Bosporus. Most long-haul international flights to Turkey land at Ataturk ¨ International Airport (IST) 23 km (14 miles) west of the city center at Yesilkoy. ¨ International Airport ICDEA08 Staff will meet you at the exit gate of Ataturk Terminal and help your transfer. The modern International Terminal (Dis Hatlar Terminali) is spacious and efficient, with all the expected services including ATMs (cash machines) from which you can obtain New Turkish Liras, currency exchange offices, restaurants, cafes, shops, Emanet (Baggage Check, Left Luggage). An underground passage (15-minute walk) connects the International Terminal with the older Domestic Terminal (Ic¸ Hatlar Terminali) and also the Istanbul Metro, called the Hafif Metro (”Light rail system”) on airport terminal signs. You can board a Metro train right from the airport and ride to Zeytinburnu, where you can transfer to the Zeytinburnu-Besiktas tram for the ride to Sultanahmet ¨ u¨ ferry and Sea Bus docks, the Galata Bridge, Square, Sirkeci Station, the Eminon ¨ and its ferry docks, and the Kabatas ferry docks and Funik ¨ ¨ Karakoy uler to Taksim Square. A faster way to Taksim Square is by express city bus 96T, stopping at Yenikapi, Aksaray and Taksim. A taxi from the airport to Sultanahmet costs about US$18 to $25; to Taksim Square, about US$21 to $26; add 50% if you travel between 24:00 (midnight) and 06:00 am. The trip takes between 35 and 75 minutes, depending on traffic. Havas airport buses, long the mainstay of airport-city transfers, are being phased out. Traditionally, they departed the Arrivals level of both the International and Domestic terminals. The trip to Taksim takes between 45 and 65 minutes, depending upon traffic. 205

Passport and visa Most of the travelers to Turkey require a visa. For most of them visas can be obtained at the port of entry in Turkey or from the Turkish Consulate General or Turkish Diplomatic Missions of their home countries. Sticker type visas are issued at the port of entry and allow staying in Turkey for up to 90 days. It is advisable to have a minimum of six months validity on your passport from the date of your entry into Turkey.

Banking and currency The currency of Turkey is New Turkish Lira (YTL) as of 1 January 2005. 1YTL equals to 100 New Kurus (YKR) Banknotes come in 1YTL, 5YTL, 10YTL, 20YTL, 50YTL & 100YTL and coins come in 1, 5YKR, 10YKR, 25YKR and 50YKR and 1YTL. Currency exchange facilities are available in all banks, hotels and airports. 24 hour cash machines providing banking services by different banks are located at suitable points throughout the 3 terminals of Antalya Airport. US dollars and Euros are also widely accepted. Credit cards are accepted at most restaurants and shops, the most widely used being MasterCard & Visa. Please kindly note that American Express, Diners Club and JCB Cards are not commonly accepted.

Business hours Banks are generally open from 09:30–16:00 hours Monday–Friday. General office hours are 09:00–17:00 Monday–Friday. Post offices operate within these hours, however stamps are often available from hotels.

Electricity Turkey operates on 220 volts, 50 Hz, with round-prong European-style plugs that fit into recessed wall sockets/points. Check your appliances before leaving home to see what you’ll need to plug in when you travel in Turkey. Many appliances such as laptop computers and digital cameras with their own power adapters can be plugged into either 120-volt or 220-volt sockets/points and will adapt to 206

the voltage automatically. But you will need a plug adaptor that can fit into the recessed wall socket/point. Read the technical stuff on your power adapter to see “INPUT: A.C. 100-240V”. If it reads that way, it can operate on either 120 or 220 voltage. If it says something like “INPUT: 100-125V”, then it can’t run on Turkey’s 220 volts and you’ll need to bring a voltage converter.

Shopping Shops are usually open between 8:30–19:00 and usually closed on Sunday. Turkey, as a result of its geographical location, is a treasure-house of hand-made products. These range from carpets and kilims, to gold and silver jewelry, ceramics, leather and suede clothing, ornaments fashioned from alabaster, onyx, copper, and meerschaum. When purchasing carpets, jewellery or leather products, it is advisable to consult your guide or do your shopping at a reputable store rather than in the street from vendors.

Tax refund All goods and services in Turkey are applicable to 18% Value Added Tax. You can receive a tax refund for the goods you purchased in Turkey. Refunds will be made to travelers who do not reside in Turkey. All goods are included in the refunds with the exclusion of services rendered and the minimum amount of purchase that qualifies for refund is 5YTL. Retailers that qualify for tax refunds must be “AUTHORIZED FOR REFUND”. These retailers must display a permit received from their respective tax office. The retailer will make four copies of the receipt for your refund, three of which will be received by the purchaser. If photocopies of the receipt are received the retailer must sign and stamp the copies to validate them. If you prefer the refund to be made by check, a Tax-free Shopping Check for the amount to be refunded to the customer must be given along with the receipt. For the purchaser to benefit from this exemption he must leave the country within three months with the goods purchased showing them to Turkish customs officials along with the appropriate receipts and or check. There are four ways to receive your refund: 1. If the retailer gives you a check, it can he cashed at a bank in the customs area at the airport. 207

2. If customer sends a copy of the receipt to the retailer showing that the goods have left the country within one month, the retailer will send a bank transfer to the purchaser’s bank or credit card within ten days upon receiving the receipt. 3. If the certified receipt and check are brought back to the retailer on a subsequent visit thin one-month of the date of customs certification, the refund can be made directly to the purchaser. 4. The refund may be made by the organization of those companies that are authorized to make tax refunds.

Geography The summer months in Istanbul are generally hot and quite humid. The winters can be cold and wet, although not as extreme as other areas of the country. The sea temperature is creep up to 30 degrees in June, July and August, with very little rain. Spring and autumn are popular times to visit because of the comfortable climate, good for lots of walking and sightseeing, with highs between 15–25 degrees C, in April, May, September and October. By the winter, the dry cold air mass from the Black Sea and cold damp front from the Balkans brings a chilly season with daytime highs of between 10–15 degrees C, and nights much colder. Although rarely falling to freezing point, there is the occasional light snow in the city.

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Index and E-mail Addresses

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A Abdeljawad, Thabet (Turkey), [email protected] 158 ´ (Spain), [email protected] Abderram´an, Jesus 21, 22, 54 Adıvar, Murat (Turkey), [email protected] 17, 18, 25, 55 Adıyaman, Meltem (Turkey), [email protected] 158 Afshar Kermani, Mozhdeh (Iran), mog [email protected] 29, 30, 56 Agarwal, Ravi (USA), [email protected] 15, 16, 31, 36 Aghazadeh, Nasser (Iran), [email protected] 17, 18, 57 Akın-Bohner, Elvan (USA), [email protected] 19, 20, 31, 37 Akman, Murat (Turkey), [email protected] 158 ˙ ı (Turkey), [email protected] Albayrak, Inc˙ 17, 18, 58 Aldea Mendes, Diana (Portugal), [email protected] 17, 18, 59 Alsed`a, Llu´ıs (Spain), [email protected] 17, 19, 20, 31, 38 Al-Sharawi, Ziyad (Oman), [email protected] 25, 26, 60 Altunkaynak, Meltem (Turkey), [email protected] 159 Alzabut, Jehad (Turkey), [email protected] 25, 26, 61 Appleby, John (Ireland), [email protected] 17, 21, 22, 62 Aseeri, Samar (Saudi Arabia), [email protected] 29, 30, 63 Atasever, Nur˙ıye (Turkey), atasever [email protected] 210

17, 18, 64 Atay, Fat˙ıhcan M. (Germany), [email protected] 29, 30, 65 Atıcı, Ferhan (USA), [email protected] 21, 22, 66 Awerbuch Friedlander, Tamara (USA), [email protected] 17, 18, 25, 67 Aydın, Kemal (Turkey), [email protected] 159 Aydın, M. Aslı (Turkey), [email protected] 182 B ¨ Bas¸, Mujgan (Turkey), [email protected] 159 ¨ Batıt, Ozlem (Turkey), [email protected] 29, 30, 68 Bayat, Kemal (Turkey), [email protected] 182 Bernhardt, Chris (USA), [email protected] 17, 18, 69 Bodine, Sigrun (USA), [email protected] 17, 25, 26, 70 Bohner, Martin (USA), [email protected] 3, 5, 11, 15, 160, 185 Bolat, Yas¸ar (Turkey), [email protected] 29, 30, 71 ˙ Bozok, Ilknur (Turkey), kuzu [email protected] 160 ¨ Budakc¸ı, Gulter (Turkey), [email protected] 160 C C ¸ akmak, Devr˙ım (Turkey), [email protected] 25, 26, 72 Camouzis, Elias (Greece), [email protected] 17, 21, 22, 73 211

Can, Canan (Turkey), canan can [email protected] 161 C´anovas, Jose S. (Spain), [email protected] 17, 18, 74 C ¸ aylak, Duygu (Turkey), duygu [email protected] 161 C ¸ elebi, Okay (Turkey), [email protected] 3, 27, 161 C ¸ el˙ık, Cem (Turkey), [email protected] 162 ¨ C ¸ el˙ık Kızılkan, Gulnur (Turkey), [email protected] 162 C ¸ et˙ın, Erb˙ıl (Turkey), [email protected] 25, 26, 75 C ¸ ıbıkd˙ıken, Al˙ı Osman (Turkey), [email protected] 21, 22, 76 C ¸ ınar, Ceng˙ız (Turkey), [email protected] 162 Costa, Sara (Spain), [email protected] 17, 18, 77 Cushing, J. M. (USA), [email protected] 11, 23, 25, 26, 78 D ˘ Dagyar, Nazlı Ceren (Turkey), [email protected] 182 Dannan, Fozi (Syria), [email protected] 25, 26, 79 Das¸, Sebahat Ebru (Turkey), [email protected] 163 Den˙ız, Aslı (Turkey), [email protected] 163 Dong, Zhaoyang (Spain), [email protected] 163 Doˇsl´a, Zuzana (Czech Republic), [email protected] 5, 19, 25, 26, 80, 185 212

Doˇsly, ´ Ondˇrej (Czech Republic), [email protected] 27, 28, 31, 39 Duman, Ahmet (Turkey), [email protected] 21, 22, 81 Duman, Melda (Turkey), [email protected] 164 E Elaydi, Saber (USA), [email protected] 5, 9, 11, 164 ¨ Yakup (Turkey), [email protected] Emul, 183 Erbe, Lynn (USA), [email protected] 17, 18, 21, 82 ¨ Erkal, Durdane (Turkey), [email protected] 183 Erol, Meltem (Turkey), [email protected] 29, 30, 83 Esty, Norah (USA), [email protected] 17, 18, 25, 84 F Fernandes, Sara (Portugal), [email protected] 21, 22, 85 G Gesztesy, Fritz (USA), [email protected] 19, 20, 23, 31, 40 Getimane, M´ario (Mozambique), [email protected] 164 Gomes, Orlando (Portugal), [email protected] 17, 18, 86 ˙ ¨ us ¨ ¸ , Ibrah˙ Gum ım Hal˙ıl (Turkey), [email protected] 165 ¨ ¨ us ¨ ¸ , Ozlem Gum Ak (Turkey), [email protected] 25, 26, 87 213

¨ Gurses, Metin (Turkey), [email protected] 5 Guseinov, Gusein (Turkey), [email protected] 5, 25, 26, 88 ¨ Guven˙ ıl˙ır, A. Feza (Turkey), [email protected] 29, 30, 89 Guzowska, Małgorzata (Poland), [email protected] 25, 26, 90 Gy˝ori, Istv´an (Hungary), [email protected] 11, 15, 16, 31, 41 H Hashemiparast, Moghtada (Iran), [email protected] 17, 18, 91 ˘ Hat˙ıpoglu, Veysel Fuat (Turkey), [email protected] 165 Heim, Julius (USA), [email protected] 21, 22, 92 Hilger, Stefan (Germany), [email protected] 15, 16, 25, 32, 42 Hilscher, Roman (Czech Republic), [email protected] 15, 17, 18, 93 I ˙ ¨ ¸ e (Turkey), [email protected] Intepe, Gokc 165 J Jantarakhajorn, Khajee (Thailand), [email protected] 166 ´ Jim´enez Lopez, V´ıctor (Spain), [email protected] 17, 18, 94 K Kalabuˇsi´c, Senada (Bosnia/Herz.), [email protected] 17, 18, 95 214

¨ ¸ e (Turkey), [email protected] Karahan, Gokc 183 Karakelle, Musa (Turkey), [email protected] 184 Kara, Ruk˙ıye (Turkey), [email protected] 166 Karpuz, Bas¸ak (Turkey), [email protected] 25, 26, 96 Kayar, Zeynep (Turkey), [email protected] 166 ˆ (USA), [email protected] Kaymakc¸alan, B˙ıllur 5, 15, 167 Keller, Christian (USA), [email protected] 21, 22, 97 Kent, Candace (USA), [email protected] 17, 21, 22, 98 Kharkov, Vitaliy (Ukraine), kharkov v [email protected] 25, 26, 99 Kipnis, Mikhail (Russia), [email protected] 25, 26, 100 Kıyak Uc¸ar, Yel˙ız (Turkey), [email protected] 167 Kloeden, Peter (Germany), [email protected] 5, 27, 28, 32, 43 ¨ Koc¸ak, Huseyin (USA), [email protected] 19, 20, 32, 44 Kongnuan, Supachara (Thailand), [email protected] 167 Kosareva, Natalia (Russia), [email protected] 168 Kostrov, Yevgeniy (USA), [email protected] 21, 22, 101 Kratz, Werner (Germany), [email protected] 6 Kulik, Tomasia (Australia), [email protected] 21, 22, 102 215

Kulik, Yakov (Australia), [email protected] 168 Kutay, V˙ıldan (Turkey), vildan [email protected] 168 L Ladas, Gerasimos (USA), [email protected] 3, 12, 15, 27, 28, 32, 45, 185 Laitochov´a, Jitka (Czech Republic), [email protected] 21, 22, 29, 103 Lawrence, Bonita (USA), [email protected] 21–23, 104 Leonhardt, Andreas (Germany), [email protected] 169 Lesaja, Goran (USA), [email protected] 169 Lu´ıs, Rafael (Portugal), [email protected] 25, 26, 105 Lutz, Donald (USA), [email protected] 6 M Marsh, Robert L. (USA), [email protected] 169 Matthews, Thomas (USA), [email protected] 21, 22, 106 Mawhin, Jean (Belgium), [email protected] 6, 23, 24, 32, 46 McCarthy, Michael (Ireland), [email protected] 21, 22, 107 Mendes, Vivaldo (Portugal), [email protected] 17, 18, 21, 108 Mert, Raz˙ıye (Turkey), [email protected] 25, 26, 109 Mesgarani, Hamid (Iran), [email protected] 29, 30, 110 216

Michor, Johanna (USA), [email protected] 25, 29, 30, 111 Migda, Małgorzata (Poland), [email protected] 29, 30, 112 Mısır, Ad˙ıl (Turkey), [email protected] 170 Morales, Leopoldo (Spain), [email protected] 21, 22, 113 N Nishimura, Kazuo (Japan), [email protected] 7 Nurkanovi´c, Mehmed (Bosnia/Herz.), [email protected] 170 Nurkanovi´c, Zehra (Bosnia/Herz.), [email protected] 170 O Oban, Volkan (Turkey), [email protected] 17, 18, 114 Oberste-Vorth, Ralph (USA), [email protected] 17, 18, 115 ¨ ¨ Ocalan, Ozkan (Turkey), [email protected] 29, 171 ˙ Okumus¸, Israf˙ ıl (Turkey), [email protected] 171 Oliveira, Henrique (Portugal), [email protected] 7, 21, 22, 116 O’Regan, Donal (Ireland), [email protected] 6 ¨ ¨ Ozdem˙ ır, Husey˙ ın (Turkey), [email protected] 184 ¨ Ozen, Bahad˙ır (Turkey), [email protected] 184 ¨ Ozkan, Umut Mutlu (Turkey), umut [email protected] 171 217

¨ Ozpınar, F˙ıgen (Turkey), [email protected] 172 ¨ urk, ¨ Ozt Ruk˙ıye (Turkey), [email protected] 25, 26, 117 ¨ urk, ¨ Ozt Serm˙ın (Turkey), [email protected] 172 ¨ gurlu, ˘ Ozu Ers˙ın (Turkey), [email protected] 172 P Papaschinopoulos, Garyfalos (Greece), [email protected] 17, 18, 118 Park, Choonkil (South Korea), [email protected] 25, 26, 119 Peterson, Allan (USA), [email protected] 6, 12, 15, 16, 19, 32, 47 Pinelas, Sandra (Portugal), [email protected] 17, 18, 21, 120 Pituk, Mih´aly (Hungary), [email protected] 25, 26, 29, 121 Popescu, Emil (Romania), [email protected] 123 Popescu, Nedelia Antonia (Romania), [email protected] 17, 29, 30, 124 Pop, Nicolae (Romania), [email protected] 17, 18, 122 Posp´ısˇ il, Zdenˇek (Czech Republic), [email protected] 25, 26, 125 ¨ Potzsche, Christian (Germany), [email protected] 19, 21, 22, 126 Predescu, Mihaela (USA), [email protected] 17, 18, 21, 127 R Rabbani, Mohsen (Iran), [email protected] 29, 30, 128 218

Rachidi, Mustapha (France), [email protected] 21, 22, 129 Radin, Michael (USA), [email protected] 21, 22, 130 Rasmussen, Martin (Germany), [email protected] 21, 22, 131 Reankittiwat, Paramee (Thailand), [email protected] 173 ˇ Reh´ak, Pavel (Czech Republic), [email protected] 25, 26, 132 Reinfelds, Andrejs (Latvia), [email protected] 21, 22, 25, 133 Rodkina, Alexandra (Jamaica), [email protected] 21, 22, 134 Romero i S´anchez, David (Spain), [email protected] 29, 30, 135 Ruffing, Andreas (Germany), [email protected] 7, 12, 173 S Sacker, Robert J. (USA), [email protected] 7, 12 Saker, Samir (Saudi Arabia), [email protected] 136 Sanchez-Moreno, Pablo (Spain), [email protected] 29, 30, 137 ˙ Savun, Ipek (Turkey), ipek [email protected] 173 Schinas, Christos (Greece), [email protected] 17, 18, 138 Schmeidel, Ewa (Poland), [email protected] 29, 30, 139 S¸ekerc˙ı, Nurcan (Turkey), [email protected] 21, 22, 140 Sell, George (USA), [email protected] 11 219

˘ ˘ ¸ en (Turkey), [email protected] Selmanogulları, Tugc 174 Seneetantikul, Soporn (Thailand), [email protected] 174 Seyhan, G˙ızem (Turkey), [email protected] 174 Shahrezaee, Mohsen (Iran), [email protected] 17, 18, 141 Sharkovsky, Alexander (Ukraine), [email protected] 6 Siddikov, Bakhodirzhon (USA), [email protected] 17, 18, 29, 142 Simon, Moritz (Germany), [email protected] 25, 26, 143 ˘ S¸ı˙ms¸ek, Dagıstan (Turkey), [email protected] 175 Sırma, Al˙ı (Turkey), [email protected] 29, 30, 144 Sizer, Walter (USA), [email protected] 175 Smith, Hal (USA), [email protected] 23, 24, 33, 48 Stefanidou, Gesthimani (Greece), [email protected] 21, 22, 145 Stehlik, Petr (Czech Republic), [email protected] 17, 18, 146 Suhrer, Andreas (Germany), [email protected] 175 T Tas¸kara, Necat˙ı (Turkey), [email protected] 176 Teschl, Gerald (Austria), [email protected] 6, 19, 25, 26, 147 Thongjub, Nawalax (Thailand), [email protected] 176 220

T˙ıryak˙ı, Aydın (Turkey), [email protected] 3, 25, 26, 148 Tlemc¸ani, Mouhaydine (Portugal), [email protected] 21, 22, 149 Tollu, D. Turgut (Turkey), hasan [email protected] 176 Topal, Fatma Serap (Turkey), [email protected] 29, 30, 150 U Uc¸ar, Den˙ız (Turkey), [email protected] 177 ¨ Unal, Mehmet (Turkey), [email protected] 1, 3, 177, 185 V Vanderbauwhede, Andr´e (Belgium), [email protected] 19, 20, 33, 49 Vesarachasart, Sirichan (Thailand), [email protected] 177 Vu, Dominik (Austria), [email protected] 178 Y ¨ ¸ ı˙n (Turkey), sugulu [email protected] Yalazlar, Gulc 178 ˙ Yalc¸ınkaya, Ibrah˙ ım (Turkey), [email protected] 178 Yantır, Ahmet (Turkey), [email protected] 17, 18, 151 ˘ ıder, Muhammed (Turkey), m.yigider [email protected] Y˙ıg˙ 179 Yıldırım, Ahmet (Turkey), [email protected] 29, 30, 152 Yıldız, Mustafa Kemal (Turkey), [email protected] 179 221

¨ Yılmaz, Ozlem (Turkey), [email protected] 179 Yorke, James A. (USA), [email protected] 23, 24, 33, 50 ¨ uk, ¨ Fulya (Turkey), fulya [email protected] Yor 180 Z ˘ Zafer, Agacık (Turkey), [email protected] 3, 23, 24, 29, 33, 51, 185 Zaidi, Atiya (Australia), [email protected] 21, 22, 153 Zakeri, Ali (Iran), [email protected] 29, 30, 154 Zeidan, Vera (USA), [email protected] 15–17, 33, 52 Zem´anek, Petr (Czech Republic), [email protected] 29, 30, 155

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