structural identity formulas and series with application

Slobodan Sekulović* Hot Springs USA Huse Fatkić** Aleksandar Mastilović*** University of Sarajevo

STRUCTURAL IDENTITY FORMULAS AND SERIES WITH APPLICATION TO ELASTICITY WITH THE STATISTICAL INDEX This paper is dedicated to V. Perić and B. Schweizer who have passed away recently

Abstract In this paper we have defined the formulas simply termed the Structural Identity Formulas or SIF. SIF Theorem demonstrates that (∀ 𝑚 ∈ 𝑁) (𝑚 > 1) the sum of all SIF is equal to one. Four interpretations of the SIF are elaborated and they include: (i) the SIF and the Figurate numbers (ii) the geometrical interpretation (iii) a technical interpretation and (iv) stastistic interpretation with aplication in economics. The principle motivation for this paper, under the conditions of the Fractional Market Shares Theorem, is to prove that SIF are analogon of the Fractional Market Shares the presence of which was built in the Complete or Total Coefficient of Aggregate Demand Elasticity together with the statistical composite Average Chain Index (Sekulović, [3]). The limit case of SIF is defined by the Series with Probability Normed Coefficients, simply called the Structural Series or SS. Under the conditions of SS Theorem, SS always converges either to one or zero. Follows three corollaries which demonstrate partitioning of ∏∞𝑖 =1 𝑅0+ ( 𝑅0+ = [0, ∞) and ∏∞𝑖=1 𝑅 induced by SS. The results obtained in this paper show that the concept of the complete or Total Coefficient of Aggregate Demand Elasticity, initially developed in Rn with the usual or standard metric, can be applied and extended to more generalized situation (e.g., ultra probabilistic metric spaces and probabilistic normed spaces).

1. Background It is almost impossible to find domain, whether it be in technical or social sciences, where the concept of elasticity is not present. -----------------*slobodan.sekulovic@gmail. **[email protected] ***[email protected] 227

This all-out presence springs from the fact that all measures of elasticity whether they be analytic (elasticity coefficient) or graphic have advantage over other variability measures: they represent statistical invariants.The theory of „The Complete or Total Coefficient of the Aggregate Demand Elasticity“ was introduced and published in three follow-up papers. With the Average Base Index and Average Chain Index (the final form of which was constructed on the smooth arc of the hypercurve (S. Sekulovic, [4]) this theory was developed in with the usual or standard metric/norm. In it, only implicit definition of the Fractional Market Shares with their inception formula in non-analytic form was introduced. Need for its improvement towards easier practical use has become imperative.

2. Structural Identity Formulas – Concept and Interpretation

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5. Final comments There is considerable evidence to support a conjecture that the results obtained in this paper for the concept of the Complete or Total Coefficient of Aggregate Demand Elasticity, initially developed in Rn with the usual or standard metric/norm, can be applied and extended to more generalized situation - to probabilistic normed (PN) spaces in the sense of C. Alsina, B. Schweizer and A. Sklar. PN spaces are real linear spaces in which the norm of each vector is an appropriate probability distribution function rather than a number. Such spaces were first introduced by A. N. Šerstnev [Dokl. Akad. Nauk SSSR 149 (1963), 280 - 283]. But C. Alsina, B. Schweizer nd A. Sklar [Aequationes Math. 46 (1993), no. 1-2, 91 - 98] developed a new definition of PN spaces which includes the definition of Šerstnev as a special case (see [6 8]).

References [1] S.V. Fomin, B. M. Budak, Multiple Integrals, Field Theory and Series, English translation, Mir Publishers, Moscow, 1973. [2] R. Bartkowiak, Electric Circuit Analysis, John Wiley & Son, Inc., New York, 1985. [3] S. Sekulović, The Generalization of the Complete or Total Coefficient of Aggregate Demand Elasticity, Economic Analysis and Worker's Management, 24 (1990) 4. [4] S. Sekulovic, The Complete or Total Coefficient of Aggregate Demand Elasticity on the Smooth Arc of the Hypercurve, Economic Analysis and Worker's Management, 25(1991), 3. [5] J. G. Stigler, Memoirs of an Unregulated Economist, Basic Books, Inc., Publishers, 1988. [6] B. Schweizer, A. Sklar, Probabilistic metric spaces, North-Holland Publishing Co., New York, 1983.; 2nd ed. Dover Publ., Mineola, NY, 2005. [7] H. Fatkić, A. Mastilović, Further results on the completness and compactness in probabilistic normed spaces, the Fifth Bosnian-Herzegovinian Mathematical Conference, July 7 and 8, 2010, Tuzla (Bosnia & Herzegovina), in : Resume of the Bosnian-Herzegovinian Mathematical Conference, Sarajevo J. Math. 6 (2010), no. 2, 298 - 299. [8] B. Lafuerza-Guillen, C. Sempi, G. Zhang, A study of boundedness in probabilistic normed spaces, Nonlinear Anal. 13 (2010), no. 5. 1127-1135. ----------

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