Bodan Slobodan Sekulovic

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Sekulovic, S. Segmentation of the Market of Consumer Goods by the Function of the Strength of Preference for One Brand over the Other, Year: 1983

Economic Analysis and Workers's Management 17(1983)4. Editor: Branko Horvat. Indexed in: Journal of Economic Literature. YU ISSN 0013-3213.

Sekulovic, S. The Complete or Total Coefficient of Aggregate Demand Elasticity, Year: 1989

Economic Analysis and Worker's Management, 23(1989)2. Editor: Branko Horvat. Indexed in: Journal of Economic Literature. YU ISSN 0351-286 X

This paper brings out the concept of the Average Base Index (ABI), an economic indicator that synthesizes in itself the simultaneous price movements of a set of homogeneous and brand name goods/services on a pencil of straight lines, in real Euclidean space $E^n$. Follows the construction of the Complete or Total Coefficient of Aggregate Demand Elasticity on a pencil of straight lines in $E^n$ and its interpretations.

Sekulovic, S. The Generalization of the Complete or Total Coefficient of Aggregate Demand Elasticity, Year: 1990

Economic Analysis and Worker's Management, 24(1990)4. Editor: Branko Horvat. Indexed in: Journal of Economic Literature. YU ISSN 0351-286 X

This research introduces the Average Chain Index (ACI), an economic indicator which incorporates the simultaneous price movements of a set of homogeneous and brand name goods on a polygonal line, in real Euclidean space $E^n$. Also, the Complete or Total Coefficient of Aggregate Demand Elasticity is constructed on the polygonal line in $E^n$. The Coefficient is decomposed and its constituents are revealed. In addition, the convex linear combination coefficients are related to the competitors' market shares and their formal presentation is derived.

Sekulovic, S. The Complete or Total Coefficient of Aggregate Demand Elasticity on a Smooth Arc of the Hyper Curve, Year: 1991

Economic Analysis and Worker's Management, 25(1991)2. Editor: Branko Horvat. Indexed in: Journal of Economic Literature. YU ISSN 0351-286 X

In this work, the construction and interpretation of three economic indicators is laid out. First, the Average Chain Index (ACI) is constructed on a smooth arc of the hyper-curve $\Gamma$. Second, index of real income (which incorporates not only the simultaneous price movements of a set of homogeneous and brand name goods, but also of the nominal income from which their demand is financed) is constructed on a smooth arc of hyper-curve ${\Gamma }^{*}$. Finally, based on the previous findings the primary task is realized: the construction and interpretation of the Complete or Total Coefficient of Aggregate Demand Elasticity on a smooth arc of hyper-curve ${\Gamma }^{*}$.

Fatkic H., Sekulovic S. On ergodic transformations on metric spaces, means by Kolmogorov and Nagumo and means by M. Bajraktarevic, Year: 2007

The Second Bosnian-Herzegovinian Mathematical Conference, International University, June 23, 2007, Sarajevo, B&H in: Resume of the Bosnian-Herzegovinian Mathematical Conference, Sarajevo J. Math, Vol 3., No. 2 (2007), 277-278.

Notions of generalized diameters and their connections with ergodic transformations are extended from geometric mean to the means satisfying well-known postulates of A. N. Kolmogorov and M. Nagumo from 1930 and some forms of means introduced by M. Bajraktarevic in 1963 by $$M_{\phi }\left(t_1,\dots ,t_n;f;q_1,\dots ,q_n\right)={\phi }^{-1}\left(\frac{{\displaystyle \sum _{i=1}^n{q}_{i}f\left(t_i\right)\phi \left(t_i\right)}}{{\displaystyle \sum _{i=1}^n{q}_{i}f\left(t_i\right)}}\right)$$ where $t_1,\dots ,t_n\in [\alpha ,\beta ](\subset ℝ)$; $q_1,\dots ,q_n\in (0,\infty )$; $\phi \in \Phi ,f\in F$ (wherein $\phi$ respectively $F$ is the set of all continuous strictly monotonic function respectively of all non-negative functions that are different from zero at points $\alpha$ and $\beta$, defined on the segment $[\alpha ,\beta ]$) $$\sum _{i=1}^{n}f\left(t_i\right)>0$$

The results presented here extend and/or complement the previous results of T. Erber, B. Schweizer and A. Sklar, 1973 , R. E. Rice, 1978, C. Sempi, 1985, H. Fatkic, 1987, 1992, 2005, 2006 and also E. Hille, 1965.

• Billingsley, P., Ergodic Theory and Information, John Wiley & Sons, Inc., New York - London - Sydney, 1965.
• Brown, J. R., Ergodic theory and topological dynamics, Academics Press, New York, San Francisco - London, 1976.
• Choe, G. H., Computational Ergodic Theory, Springer, Berlin, 2005.
• Cornfeld, I. P., Fomin, S. V., Sinai, Ya. G., Ergodic Theory, Springer Verlag, New York - Heidelberg - Berlin, 1982.
• Erber, T., Schweizer, B., Sklar, A., Mixing transformations on metric spaces, Comm. Math. Phys., 29(1973), 311-317.
• Erber, T., Sklar, A., Macroscope irreversibility as a manifestation of micro-instabilities . In: Modern Developments in Thermodynamics (B. Gal - Or. e.) pp. 281 - 301. Israel Universities Press and J. Wiley and Sons, Jerusalem - New York, 1974.
• Fatkic, H., On mean values and mixing transformations, in: Proc. Twenty-fourth Internat. Symposium on Functional Equations, Aequationes Math. 32(1987), 104 - 105.
• Fatkic, H., Note on weakly mixing transformations, Aequationes Math. 42(1992), 38 - 44.
• Fatkic, H., On probabilistic metric spaces and ergodic transformations, Ph.D. dissertation, Univ. Sarajevo, 2000, 170 pp; Rad. Mat. 10 (2001)2, 261-263.
• Fatkic, H., Characterizations of measurability-preserving ergodic transformations , Sarajevo J. Math. 1(13)(2005), 49 - 58.
• Fatkic, H., Measurability-preserving weakly mixing transformations, Sarajevo J. Math. Vol. 2, No. 2(2006), 159 - 172.
• Hille, E., Topics in classical analysis, in: Lecture on Modern Mathematics (T. L. Saaty, ed.) Vol 3., Wiley, New York, 1965, pp. 1 - 57.
• Landkof, N. S., Foundations of modern potential theory. Translated from the Russian by A. P. Doohovskoy. Die Grundlehren der mathematischen Wissenschaften, Band 180. Springer - Verlag, New York - Heidelberg, 1972.
• Lesigne, E., Rittaud, B., de la Rue, T., Weak disjointness of measure-preserving dynamical systems , Ergodic Theory Dyn. Syst., 23(4)(2003), 1173 - 1198.
• Martin, N., F., G., On ergodic properties of restriction of inner functions, Ergodic Theory Dyn. Syst., 9(1)(1989), 137 - 151.
• Martin, N., F., G., Conformal isomorphism of inner functions, Ergodic Theory Dyn. Syst., 17(3)(1997), 663 - 666.
• Rice, R., E., On mixing transformations, Aequations Math. 17(1978), 104 - 108.
• Schweizer, B., Sklar, A., Probabilistic metric spaces, Elsevier North - Holand, New York, 1983.
• Sempi, C., On weakly mixing transformations on metric spaces, Rad. Mat. 1(1985), 3 - 7.
• Walters, P., An Introduction to Ergodic Theory, (GTM, Vol. 79), Springer - Verlag, New York - Heidelberg - Berlin, 1982.

Fatkic, H., Sekulovic, S., Fatkic, Hana. Further result on the ergodic transformations that are both strongly mixing and measurability preserving, Year: 2011

The sixth Bosnian-Herzegovinian Mathematical Conference, International University of Sarajevo, June 17, 2011, Sarajevo, B&H in: Resume of the Bosnian-Herzegovinian Mathematical Conference, Sarajevo J. Math, Vol 7, No. 2 (2011), 310-311.

In the broadest sense ergodic theory is the study of the qualitative properties of actions of groups on spaces (e.g., measure spaces, or topological spaces, or smooth manifolds). In this work we shall study actions of the group $\mathbb{Z}$ of integers on a measure space $S$, i.e., we study a transformation $\phi :S\to S$ and its iterates ${\phi }_n,n\in \mathbb{Z}$. It is customary in ergodic theory to assume that the underlying space is either a finite or $\sigma$ - finite measure space. We shall assume that the measure is finite. It is commonly further assumed that the measure space is separable. However, we shall not make this assumption, principally because it would rule out some of our principal structure theorems.

Suppose $(S,A,\mu )$ is a finite measure space. A transformation $\phi :S\to S$ is called:

1. measurable ($\mu$-measurable) if, for any $A$ in $\mathbf{A}$, the inverse image ${\phi }^{-1}$ is in $\mathbf{A}$;
2. measure-preserving if $\phi$ is measurable and $\mu \left({\phi }^{-1}\right(A\left)\right)=\mu \left(A\right)$ for any $A$ in $\mathbf{A}$;
3. ergodic if the only members $A$ of $\mathbf{A}$ with ${\phi }^{-1}\left(A\right)=A$ satisfy or ;
4. (strong-) mixing (with respect to $\mu$) if $\phi$ is $\mu$-measurable and $$\underset{n\to \infty }{\lim }\mu \left({\phi }^{\mathrm{-n}}\right(A)\cap B)=\frac{\mu \left(A\right)\mu \left(B\right)}{\mu \left(S\right)}$$ for any two $\mu$-measurable subsets $A,B$ of $S$. We say that the transformation $\phi :S\to S$ is measurability - preserving (preserves $\mu$-measurability) if, for any $A$ in $\mathbf{A}$, the image $\phi \left(A\right)$ is in $\mathbf{A}$.

Relations between ergodic transformations and Chebyshev functions (as generalizations of Chebyshev polynomials) are investigated. In doing so, notion of $\mathbf{M}$-radius of Chebyshev order $k$ and $\mathbf{M}$-constant of Chebyshev (where $\mathbf{M}≔\left(M_n\right)$ is the power mean) introduced by E. Hille in 1961 (and improved in 1965) are used to prove that even for these quantities the corresponding analogs for ergodic transformations that are both strongly mixing and measurability preserving, proved for diameters in (T. Erber, B. Schweizer and A. Sklar, 1973, R. E. Rice, 1978 and, for geometric diameters, in H. Fatkic; 1987, 2000), hold.

• Billingsley, P., Ergodic Theory and Information, John Wiley & Sons, Inc., New York - London - Sydney, 1965.
• Brown, J. R., Ergodic theory and topological dynamics, Academics Press, New York, San Francisco - London, 1976.
• Choe, G. H., Computational Ergodic Theory, Springer, Berlin, 2005.
• Cornfeld, I. P., Fomin, S. V., Sinai, Ya. G., Ergodic Theory, Springer Verlag, New York - Heidelberg - Berlin, 1982.
• Erber, T., Schweizer, B., Sklar, A., Mixing transformations on metric spaces, Comm. Math. Phys., 29(1973), 311-317.
• Fatkic, H., On mean values and mixing transformations, in: Proc. Twenty-fourth Internat. Symposium on Functional Equations, Aequationes Math. 32(1987), 104 - 105.
• Fatkic, H., Some results on measure-preserving transformations with weak mixing property , Proc. Internat. Conf. on Functional Equations and Inequalities, Koninki, Poland (1991.), Rocznik Naukowo-Dydaktyczny WSP w Krakowie 135, Prace Matematyczne XVII(1992), 9-10.
• Fatkic, H., Note on weakly mixing transformations, Aequationes Math. 42(1992), 38 - 44.
• Fatkic, H., On probabilistic metric spaces and ergodic transformations, Ph.D. dissertation, Univ. Sarajevo, 2000, 170 pp; Rad. Mat. 10 (2001)2, 261-263.
• Fatkic, H., Characterizations of measurability-preserving ergodic transformations , Sarajevo J. Math. 1(13)(2005), 49 - 58.
• Fatkic, H., Measurability-preserving weakly mixing transformations, Sarajevo J. Math. Vol. 2, No. 2(2006), 159 - 172.
• Hewit, E., Stromberg, K., Real and Abstract Analysis, Springer - Verlag, New York - Berlin - Heidelberg - London - Paris - Tokyo - Hong Kong - Barcelona, 1965.
• Hille, E., Topics in classical analysis, in: Lecture on Modern Mathematics (T. L. Saaty, ed.) Vol 3., Wiley, New York, 1965, pp. 1 - 57.
• Landkof, N. S., Foundations of modern potential theory. Translated from the Russian by A. P. Doohovskoy. Die Grundlehren der mathematischen Wissenschaften, Band 180. Springer - Verlag, New York - Heidelberg, 1972.
• Iams, S., Katz, B., Silva, C. E., Street, B., Wickelgren, K., On weakly mixing and double ergodic nonsingular action , Colloq. Math., 103(2)(2005), 247-264.
• James, J., Koberda, T., Lindsey, K., Silva, C. E. Speh, P., On ergodic transformations that are both weakly mixing and uniformly rigid , New York J. Math. 15(2009), 393-403.
• Lesigne, E., Rittaud, B., de la Rue, T., Weak disjointness of measure-preserving dynamical systems , Ergodic Theory Dyn. Syst., 23(4)(2003), 1173 - 1198.
• Rice, R., E., On mixing transformations, Aequations Math. 17(1978), 104 - 108.
• Schweizer, B., Sklar, A., Probabilistic metric spaces, Elsevier North - Holand, New York, 1983.
• Sekulovic, S., Fatkic, H., Mastilovic, A., Structural Identity Formulas and Series with Application to Elasticity with the Statistical Index , in: Zbornik radova Naučnog skupa - Prva matematička konferencija Republike Srpske (s međunarodnim učešćem), 21. i 22. maj 2011., Pale, B&H.; Naučni skupovi, Vol. 6/3 (2012), pp. 227-242.
• Sempi, C., On weakly mixing transformations on metric spaces, Rad. Mat. 1(1985), 3 - 7.
• Walters, P., An Introduction to Ergodic Theory, (GTM, Vol. 79), Springer - Verlag, New York - Heidelberg - Berlin, 1982.

Sekulovic, S., Fatkic, H., Mastilovic, A., Structural Identity Formulas and Series with Application to Elasticity with the Statistical Index, Year: 2011

in: Proceedings of the 1st Mathematical Conference of Republic of Srpska - Section of Applied Mathematics, Pale, B&H, pp. 226-242, 2011

Excerpt from the review: The work is very interesting, mathematically correct, large enough, very applicable and therefore it is strongly recommended for publishing. The authors first define formulas that are called the Structural Identity Formulas (SIF), and then have proved SIF Theorem: $(\forall m\in \mathbb{N}\setminus \{1\left\}\right)$ 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) the statistical interpretation ... There is a considerable evidence to support a conjecture that the result obtained in this work (tightly related to the concept of the Complete or Total Coefficient of Aggregate Demand Elasticity, initially developed in ${ℝ}^n$ 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. Serstnev Dokl. Akad. Nauk SSSR 149 (1963), 280-283, ...

Available at: Research Gate

Fatkic, H., Sekulovic, S., Fatkic, Hana, Probabilistic Metric Spaces Determined by Weakly Mixing Transformations, Year: 2012

The Second Mathematical Conference of Republic of Srpska, Trebinje, 8^th and 9^th June, 2012

In this paper we investigate the probabilistic metric spaces determined by the weakly mixing (WM) transformations (a probabilistic metric space is a generalization of metric space (briefly a PM space), in which the "distance" between any two points is a probability distribution function rather than the number). Continuing the work begun by B. Schweizer and N. Sklar (Probabilistic metric spaces determined by measure-preserving transformations , Z. Wahrsch. Verw. Geb. 26(1973), 235-239), we construct a new class of PM spaces. Specifically, we prove that if $(S,d)$ is a separable metric space endowed with probability measure $P$ and if $T$ is a transformation on $S$, that is weakly mixing with respect to $P$, than for any $x>0$ and almost all pairs of points $(p,q)$ in $S^2$ there is a distribution function $F$ such that the average number of times in the first $(n-1)$ iterations of $T$ that the distance between the points $T^n\left(p\right)$ and $T^n\left(q\right)$ is less than $x$ converges to $F\left(x\right)$ as $n$ goes to infinite. The collection of these distribution functions is almost an equilateral probabilistic pseudometric space and the transformation $T$ is (probabilistic) distance-preserving on this space.

The above ideas play important role in the distributional chaos theory. The results presented here also extend and/or complement the previous results of T. Erber, B. Schweizer and A. Sklar (Mixing transformations on metric spaces, Comm. Math. Phys., 29(1973), 311-317), R. E. Rice (On mixing transformations, Aequations Math. 17(1978), 104-108), H. Fatkic ( Note on weakly mixing transformations , Aequations Math. 42(1998)38-44; Characterization of measurability-preserving ergodic transformations, Sarajevo, J. Math. 1(13)(2005), 49-58; Measurability-preserving weakly mixing transformations, Sarajevo J. Math., Vol. 2, No. 2 (2006), 159-172), H. Fatkic and S. Sekulovic ( On ergodic transformations on metric spaces, means by Kolmogorov and Nagumo and means by M. Bajraktarevic ), and R. Pikula (On some notions of chaos in dimension zero, Colloq. Math. 107(2007), 167-177).

Huse Fatkic, Slobodan Sekulovic, Hana Fatkic On harmonic Mean Values and Weakly Mixing Transformations, Year: 2014

Huse Fatkic, Department of Mathematics, Faculty of Electrical Engineering, University of Sarajevo, B&H; Slobodan Sekulovic, Hot Springs National Park, Arkansas, USA; Hana Fatkic, Department of Computer Science and Informatics, Faculty of Electrical Engineering, University of Sarajevo, B&H; hfatkic@etf.unsa.ba, slobodan.sekulovic@gmail.com, fatkic.hanna@gmail.com

Let $\left(S,d\right)$ be an extended metric space, and let $A$ be a subset of $S$. For any positive integer $k(k\ge 2)$, we define the harmonic diameter of order $k$ of $A$, denoted by $D_k\left(A\right)$, by Note that $D_2\left(A\right)$ is the ordinary diameter of the set $A$. The main result of this work is the following theorem: For the above situation if $\left(S,A,P\right)$ is a probability space with the property that every open ball in $\left(S,d\right)$ is $P$-measurable and has positive measure, and if $\phi$ is transformation on $S$, that is weakly mixing with respect to $P$, and $A$ is $P$-measurable with $P\left(A\right)>0$, than $\underset{x\to \infty }{\lim }\sup D_k\left({\phi }^n\left(A\right)\right)=D_k\left(S\right)$.

The above theorem shows that, under iteration, all weakly mixing transformations tend to spread the sets out, not only in terms of the ordinary diameter, but also in terms of the harmonic diameter of any finite order. Thus, even though some set A may not spread out in "volume" (measure), there is a very definite sense that $A$ does not remain small. This mixing character of the weakly mixing transformations (e. g., of the polynomials $C_n(n=0,1,2,\dots )$, defined on the interval $\left[2,2\right]$ by $C_n\left(x\right)=2\cos \left(n\arccos \right(x/2\left)\right)$, which are related to the standard Chebyshev polynomials), inter alia, can be used for developing an efficient method for generating sequences of pseudo-random numbers (computer runs on selected pairs of points; the results obtained with $C_{10}$ as transformation $\phi$ indicate that the convergence of the corresponding sequences of the second moments about the mean (variances) to the second moment of corresponding distribution function is fairly rapid).

The obtained results extend and/or complement the previous results due to:

• Rice, Roy E., On mixing transformations, Aequationes Math. 17 (1978), 104-108.
• Erber, Thomas; Schweizer, Berthold; Sklar, Abe, Mixing transformations on metric spaces , Comm. Math. Phys. 29 (1973) 311-317. Grigor'evich.
• Fatkic, Huse; Brkic, Mehmed, Strongly mixing transformations and geometric diameters, Sarajevo J. Math 8 (21) (2012), no. 2, 245-257.
• Landkof, Neum S. (Samoilovich), Foundations of modern potential theory. Translated from Russian by A. P. Doohovskoy. Die Grundlehren der Wissenschaften, Band 180. Springer-Verlag, New York - Heidelbrg, 1972.
• Saff, Edward B., Logarithmic potential theory with applications to approximation theory , Surv. Approx. Theory 5 (2010), 165-200.
• Sempi, Carlo, On weakly mixing transformations on metric spaces, Rad.Mat. 1 (1985), 3-7.

There is considerable evidence to support a conjecture that our result (for weakly mixing transformations/weakly mixing dynamical systems with discrete time) can be extended to weakly mixing dynamical systems with continuous time.

Sekulovic S., Fatkic Hana, On Some Elements of the Theory of Composite Index Numbers, Year: 2015

The fifth mathematical conference of the Republic of Srpska, Trebinje, Bosnia and Herzegovina, June 05-06, 2015 Slobodan Sekulovic, Hot Springs National Park, Arkansas, USA Hana Fatkic, Department of Computer Science and Informatics, Faculty of Electrical Engineering University of Sarajevo, B&H

In this paper we investigate the notions and the properties of three Composite index numbers (Composite indices are unweighted price indices (UPI) designed for monitoring the general price-level movements of an industry, or a part of it: (i) Average Base Index (ABI), (ii) Average Chain Index (ACI), and (iii) Average of Relative-Variation Price Relatives (ARVPR)). The exposition goes to the extent from which further evolution of the said indices becomes interwoven with the evolution of the concept name The Complete or Total Coefficient of Aggregate Demand Elasticity (see [5] - [7]). In the that respect, we define said indices on the set of real numbers, and in normed vector space, we analyze their complex structure which springs from the relative - variation price relatives, and establish their dependability (j) from the perspective of their "end-use domain", and (jj) in terms of meeting the mathematical tests: the time-reversal test (ABI & ACI are reversible) and cyclical or circular test.

The results presented in this paper extend and/or complement the results presented by Blazic (see Milos R. Blazic, Opšta Statistika, Osnovi i Analiza, Savremena Administracija, Beograd, 1977 , pp 187-192).

References:

• Blazic, M., Opšta Statistika - Osnovi i Analiza, Savremena Administracija, Beograd, 1977.
• Fatkic, H., Fatkic, B., Karahodzic, V., On generalized harmonic diameters and some classes of measure-preserving transformations The Fifth Mathematical Conference of Republic of Srpska - Section for Analysis and Probability & Statistics, Trebinje, Bosnia and Herzegovina, June 5&6, 2015.
• Fatkic, H., Fatkic, J., Brkic, M., On the various notions of recurrence for some classes of nonsingular transformations , The Fifth Mathematical Conference of Republic of Srpska - Section for Analysis and Probability & Statistics, Trebinje, Bosnia and Herzegovina, June 5&6, 2015.
• Fatkic, Hana, Sklar, A., Fatkic, H., Dispersion Under Iteration of Strongly Mixing Transformations on Metric Spaces , International Journal of Applied Mathematics, Electronics and Computers (IJAMEC), 3(3)(2015), 150-154.
• Sekulovic, S., The Complete or Total Coefficient of Aggregate Demand Elasticity, Economic Analysis and Workers' Management 23(1989)2. (Editor: Branko Horvat)
• Sekulovic, S., The Generalization of the Complete or Total Coefficient of Aggregate Demand Elasticity, Economic Analysis and Workers' Management, 25(1991)3. (Editor: Branko Horvat)
• Sekulovic, S., The Complete or Total Coefficient of Aggregate Demand Elasticity on a Smooth Arc of the Hyper Curve, Economic Analysis and Workers' Management 25(1991)3 (Editor: Branko Horvat)
• Sekulovic, S., Fatkic, H., Mastilovic, A., Structural Identity Formulas and Series With Application to Elasticity With the Statistical Index , In: Proceedings of the 1^st Mathematical Conference of Republic of Srpska - Section of Applied Mathematics, Pale, B&H, pp. 226-242, 2011.

Now Available: Sekulovic, S. New ... Measurement and Positioning Against the Competitors: Part I

General Theory of the Complete or Total Coefficient of Aggregate Demand Elasticity and Some Composite Indices

Book Title

New ... Measurement and Positioning Against the Competitors: Part I

Subtitle

General Theory of the Complete or Total Coefficient of Aggregate Demand Elasticity and Some Composite Indices

Author

Slobodan Bodan Sekulovic

ISBN

978-620-6-14757-2

Field

Economics (mathematical)

Book language

English

Published on

03–22–2023

Publishing House

LAP LAMBERT Academic Publishing

Number of pages

140

Book available now at

MoreBooks.shop

Contents

1. The complete or total coefficient of aggregate demand elasticity
   1. Introduction ... 3
   2. The Connection Between the Parametric Form of the Equation of the Line in $n$-Dimensional Real Euclidian Space $E^n$ and the Average Base Indices ... 4
   3. Construction of the Complete or Total Coefficient of Aggregate Demand Elasticity ... 8
   4. Analysis of the Complete or Total Coefficient of Aggregate Demand Elasticity ... 13
   5. Concluding Remarks ... 17
2. The generalization of the complete or total coefficient of aggregate demand elasticity
   1. Introduction ... 21
   2. The Derivation of Basic Equations of Aggregate Demand Elasticity ... 25
   3. The Analysis of the Basic Equations of Aggregate Demand Elasticity ... 30
   4. Expressing Market Shares of the Competitors Through the Convex Linear Combination Coefficients ... 34
      1. Fractional Market Shares (FMS) ... 35
      2. Expressing Convex Linear Combination Coefficients by the Algebraic Formulas ... 38
      3. The Geometrical Interpretation of SIF ... 40
      4. The Connection Between FMS and SIF ... 43
   5. Concluding Remarks ... 45
3. The complete or total coefficient of aggregate demand elasticity on a smooth arc of the hyper curve
   1. Introduction ... 49
   2. The Average Chain Index on a Smooth Arc of the Hyper Curve $\Gamma$ ... 51
      1. Construction of the Index ... 51
      2. The Problem of Weights with the Average Chain Index $\left(I_{\mathrm{tl}}^{\mathrm{(\Gamma )}}\right)$ ... 57
      3. The Infinitesimal Form of the Average Chain Index Which is Defined on a Smooth Arc of the Curve $\Gamma$ ... 62
   3. Index of Real Income from which Aggregate Demand for Competitive Products is Financed ... 64
      1. Construction of the Index on a Smooth Arc of the Curve $\hat{\Gamma }$ ... 64
      2. Infinitesimal or Marginal form of $\left(I_{\mathrm{rd}}^{\mathrm{(<mover>\; <mrow>\; <mi>\Gamma </mi>\; </mrow>\; <mo>^</mo>\; </mover>)}}\right)$ Index which is Defined on a Smooth Arc of the Hyper Curve $\hat{\Gamma }$ ... 67
   4. Construction of The Complete or Total Coefficient of Aggregate Demand Elasticity On a Smooth Arc of the Hyper Curve $\hat{\Gamma }$ ... 69
   5. Concluding Remarks ... 76
4. Appendices ... 77
5. Appendix A On Some Elements of the Theory of Composite Index Numbers ... 79
   1. Background ... 79
   2. Average Base Index (ABI) – Concept and Interpretation ... 81
   3. Average Link Index (ALI) ... 85
   4. Average of Relative-Variation Price Relatives (ARVPR) ... 86
   5. Dependability of Composite Index-Number Formulae ... 87
   6. Concluding Remarks ... 88
   7. Appendix (Chart I & II – Case of U.S. airline industry) ... 89
6. Appendix B Differentiability of Functions of Two Variables ... 92
   1. Concept and Interpretation ... 93
   2. Tangent Plane ... 96
   3. The Consequences of Differentiability ... 99
   4. Sufficient Conditions for Differentiability ... 103
7. Appendix C Equalities Which Bind up the Total Coefficients of Industry’s and Firm’s Demand Elasticity and the Fractional Market Shares ... 108
8. The Greek Alphabet ... 111
9. List of Notations, Acronyms and Some Formulas ... 112
10. Bibliography ... 116
11. Review of the book ... 119

Preface

The research New Methods of Measurements and Positioning Against Competitors is composed of two parts, the first part of which is finished and presented here under the title General Theory of the Complete or Total Coefficient of Aggregate Demand Elasticity and some Composite Indices. Part II comes under the title Theory of Fractional Market Shares and it is still work in progress.

Only some elements of the both theories emerged some five years apart at the very end of the 20th century. My thanks are due to Professors Branko Horvat and Ivo Gjenero of the University of Zagreb and Hasan Hanić of the University of Belgrade who, at that time, read the elements of what, back then, was the manuscript on three different Total Coefficients of Aggregate Demand Elasticity in its inception phase. The result was three follow-up papers which came out in succession in Horvat’s Journal of Economic Analysis and Worker’s Management. As for the elements of Theory of Fractional Market Shares, Professor Robert Drago of the University of Wisconsin in Milwaukee read the entire manuscript. My sincere gratitude goes to him for his careful reading of the manuscript and numerous insightful comments and valuable suggestions. At that point in time it became clear that two roughly parallel developments were in place, and that eventually the two theories would merge. It had to wait for some mathematical tools to be developed (see Sekulovic et al. [20]), and for the research on Composite Index Numbers to be completed (due to the reasons discussed at great length in Appendix A) in order to launch the preparations of the manuscript for the Part I of the book.

As mentioned on the outset, Part II is still work in progress. It brings a new concept called the Fractional Market Shares, and develops tools for positioning against the competitors using dynamic analysis. A particular class of demand functions and the time path, which spring from the non-linear budget constraint, is to be revealed. The research will demonstrate how the two concepts, that of the Fractional Market Shares and the Complete or Total Coefficient of Aggregate Demand Elasticity, permeate each other thereby immensely simplifying all the computations involved. More concretely put, the approach of the Part II is to develop the theory, first in the simple setting of $n$-dimensional Euclidean space, and later on to introduce more general treatment in probabilistic metric spaces. This ultimately increases the realism of all assumptions made, and moves Part I and Part II towards the application.

Invaluable in all stages of the research and in the preparation of this book was the continued generous help extended by Professor Huse Fatkić from the Faculty of Electrical Engineering of the University of Sarajevo. Almost all chapters bear the imprint of his valuable comments, suggestions and corrections. I also wish to express my deep appreciation to Hana Fatkić, Ph.D. candidate at the Department of Computer Science and Informatics of the University of Sarajevo, Software Engineer, CEO and co-owner of IT company DOERS, Prague, Czech Republic, for the patient cooperation and skillful handling of a complex manuscript.

Finally, my thanks go to Garland County Library in Hot Springs, Arkansas, and in particular to Greg Wallace for his helpful assistance in providing different reports and research papers in timely fashion.

S.S.

Hot Springs, Arkansas

www.BodanSekulovic.com

Review of the book:

New methods of measurements and positioning against the competitors

Part I
General Theory of the Complete or Total Coefficient of Aggregate Demand Elasticity
by
Slobodan Sekulovic

In this excellent book the author has undertaken a difficult task of bringing together in a unified manner three new theoretical concepts:

• The Complete or Total Coefficient of Aggregate Demand Elasticity (for short CTCADE);
• Composite Index Numbers;
• Fractional Market Shares (FMS).

As known, in partial elasticity coefficients “… the impact of all other variables (except for the one in consideration) which influence demand over time is held constant. In the real world such conjecture cannot stand up under scrutiny” (author, p. 3). Generalization of the concept, in spite of the appeal of the idea, has not been explored (if it has the reviewer is not familiar of the fact) in the manner demonstrated in this book until now. The author has devised a straightforward formula which, unlike the partial elasticity coefficients, measures intensity of dependent-variable responsiveness (the relative variation in aggregate demand) relative to simultaneous variation of all independent variables (rival prices). “This was the motivation for naming it the CTCADE” (author, p. 3).

Theoretical concept of Composite Index numbers, in particular that of Average Base Index (ABI) and Average Chain Index (ACI), underpins the CTCADE. Namely, simultaneous variation of all rival prices is expressed through one single summary figure which is then used in the construction of CTCADE. The author has laid the groundwork on Composite Index numbers in Appendix A. It represents the original work with rigorous mathematical formulation of the said indices in a generalized setting. He employs a case of U.S. airline industry in order to demonstrate how the theoretical concept is translated into a vast amount of information not found elsewhere. Reading Appendix A is a prerequisite for in depth understanding of two roughly parallel developments, that of CTCADE and that of ABI and ACI. As previously mentioned, evolution of the latter underpins the evolution of CTCADE.

The algebraic formulas of ABI and ACI, defined in Appendix A, are not fit for the limit process. In a very inspirational manner, the author employs (in Chapter I) the formula for a pencil of straight lines in ${ℝ}^{\mathrm{n}}$, and performs certain mathematical operations over it in order to derive two logically equivalent algebraic formulas of ABI: One has the analytical expression as presented in Appendix A, whereas the other one is fit for the limit process in ${ℝ}^{\mathrm{n}}$. Clearly, this step was the precondition for defining CTCADE on a pencil of straight lines. Although it is not explicitly stated, it is in evidence throughout the book that author considers ${ℝ}^{\mathrm{n}}$ a space with its Euclidian metric/norm and inner product.

Chapter II brings out ABI’s complementary expression: It is ACI, devised in such a way that a single summary figure indicates the simultaneous price movements of all rival prices on the polygonal line. Then, CTCADE is defined on a polygonal line. The author demonstrates how the limit process in this case yields the developed form of CTCADE in the form of the basic equations of aggregate demand elasticity. He analyses them, proves the basic theorem, and arrives at the basic matrix equation. The results cast new light on the partial elasticity coefficients. They extend the concept and contribute to greater precision of elasticity coefficients. And not only that. The basic matrix equation indicates presence of the competitors’ market shares. To state it more precisely: competitors’ market shares constitute the developed form of CTCADE. This gives rise to the author’s motivation to go beyond the current exposition on CTCADE, and introduce a new concept called Fractional Market Shares (FMS) in section 3 of the same chapter. Approach is rigorous. Definition of FMS is clearly stated, and the FMS theorem identifies the market structure which underpins the FMS. Original technique is demonstrated in connecting FMS and the convex linear combination coefficients. However, the scope of this exposition does not provide a deeper insight into FMS and CTCADE relationship. But this is in line with the author’s emphasis in the Preface that Part II is designated to “… demonstrate how the two concepts … permeate each other thereby immensely simplifying all the computation involved”. Also, the relevance of FMS concept for positioning against the competitors using dynamic analysis is left for Part II.

It is not until the end of Chapter III that reader gets the unparalleled glimpse into the concept of Composite Index numbers and CTCADE. The author sets out to cast new light on CTCADE and ACI by taking into account the simultaneous movements of all rival prices, and a nominal income from which the aggregate demand is financed. The exposition comprises indicators vital for a market balancing of supply and demand. It cannot escape one’s notice a very intensive use of a strong mathematical apparatus of the vector analysis which “sends” a strong and unequivocal statement: many laws of economic analysis can be defined with due precision, and thereby move ever closer to its implementation. At this point in time, the research remains in the realm of application of mathematics to obtaining the necessary generalizations of the relative economic indicators and definitions. In the conclusion the author indicates the forthcoming empirical investigation. Perhaps the most interesting part of the Chapter III exposition is the proof of Theorem 1.2.1., section 1.2., which demonstrates that ACI represents a weighted price index under the conditions which have economic logic.

Appendix B contains a familiar mathematical concept of differentiability. The exposition is well written, thorough, and systematic with strong emphasis on use of the statement and the predicate calculus. Clearly, with aid of it, rigorous mathematical formulations of the relative definitions and theorems have gained high precision and clarity. Carefully chosen examples are solved step-by-step followed by excellent illustrations of the functions in question.

Conclusion: This book will captivate anyone whose mind is generalization prone and interested in new, market-competition-oriented concepts. It is well-written and to be recommended, it represents a significant timely contribution to the field.
As the author suggests in the Preface, Part I is only a prelude to Part II entitled Theory of Fractional Market Shares where all involved concepts will “exhaust”, on a yet-to-be-seen level, a dynamic theory named New Methods of Measurements and Positioning Against the Competitors.

Reviewed by Huse Fatkić, Ph.D.
Professor of Mathematics
University of Sarajevo

American Mathematical Society reviewer (2007 – present) on a broad range of topics in: Advanced Numerical and Statistical Methods, Applied Mathematics (including Mathematical Economic), Probability and Statistics (Dynamic Systems, Fuzzy Analysis in Statistics, Iteration Theory), Probabilistic and Statistical Metric Spaces, Measure and Measurability-Preserving Transformations

Sekulovic, S., Fatkic, H., Elements of the Theory of Fractional Market Shares - Expressing Market Shares of the Competitors Through the Convex Linear Combination Coefficients

The Sixth Mathematical Conference of Republic of Srpska - Section for Applied Mathematics, Pale, Bosnia and Herzegovina, May 21^st, 2016

This paper constitutes our attempt to cast an unconventional viewpoint upon the firm's market share. More concretely put, the competitors' market share constitute the "Complete or Total Coefficient of Aggregate Demand Elasticity" (Sekulovic, [3] and [4]), and except for simple analogy with convex linear combination coefficients (e.g., each one is larger than zero and their sum is equal to one) no deeper bridge-building effort between these two notions has been made. Therefore, we pursue this investigation by introducing the Fractional Market Shares (FMS) concept first, and then proceed to elaboration of the analytical formulas through which the convex linear combination coefficients can be expressed. The revealed properties demonstrate how seemingly unrelated and distant notions such as the firms' market shares and the convex linear combination coefficients can be linked.

The paper represents a segment of the research in progress by the name: The Theory of Fractional Market Shares.

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• Sekulovic, S., Fatkic, H., Mastilovic, A. (2011), Structural Identity Formulas and Series with Application to Elasticity with the Statistical Index , in: Proceedings of the 1^st Mathematical Conference of Republic of Srpska - Section of Applied Mathematics, Pale, B&H, pp. 226-242.

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