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Assume that $$\Omega \subset \mathbb {R}^k$$ Ω ⊂ R k is an open set, V is a real separable Banach space and $$f_1,\ldots ,f_N :\Omega \rightarrow \Omega $$ f 1 , … , f N : Ω → Ω , $$g_1,\ldots , g_N:\Omega \rightarrow \mathbb {R}$$ g 1 , … , g N : Ω → R , $$h_0:\Omega \rightarrow V$$ h 0 : Ω → V are given functions. We are interested in the existence and uniqueness of solutions $$\varphi :\Omega \rightarrow V$$ φ : Ω → V of the linear equation $$\varphi =\sum _{k=1}^{N}g_k\cdot (\varphi \circ f_k)+h_0$$ φ = ∑ k = 1 N g k · ( φ ∘ f k ) + h 0 in generalized Orlicz spaces.

We deal with an alienation problem for an Euler–Lagrange type functional equation $$\begin{aligned} f(\alpha x + \beta y) + f(\alpha x - \beta y) = 2\alpha ^2f(x) + 2\beta ^2f(y) \end{aligned}$$ f ( α x + β y ) + f ( α x - β y ) = 2 α 2 f ( x ) + 2 β 2 f ( y ) assumed for fixed nonzero real numbers $$\alpha ,\beta ,\, 1 \ne \alpha ^2 \ne \beta ^2$$ α , β , 1 ≠ α 2 ≠ β 2 , and the classic quadratic functional equation $$\begin{aligned} g(x+y) + g(x-y) = 2g(x) + 2g(y). \end{aligned}$$ g ( x + y ) + g ( x - y ) = 2 g ( x ) + 2 g ( y ) . We were inspired by papers of Kim et al. (Abstract and applied analysis, vol. 2013, Hindawi Publishing Corporation, 2013) and Gordji and Khodaei (Abstract and applied analysis, vol. 2009, Hindawi Publishing Corporation, 2009), where the special case $$g = \gamma f$$ g = γ f was examined.

The classical result of L. Székelyhidi states that (under some assumptions) every solution of a general linear equation must be a polynomial function. It is known that Székelyhidi’s result may be generalized to equations where some occurrences of the unknown functions are multiplied by a linear combination of the variables. In this paper we study the equations where two such combinations appear. The simplest nontrivial example of such a case is given by the equation$$\begin{aligned} F(x + y) - F(x) - F(y) = yf(x) + xf(y) \end{aligned}$$F(x+y)-F(x)-F(y)=yf(x)+xf(y)considered by Fechner and Gselmann (Publ Math Debrecen 80(1–2):143–154, 2012). In the present paper we prove several results concerning the systematic approach to the generalizations of this equation.

Under some simple conditions on the real functions f and g defined on an interval I ⊂ (0, ∞), the two-place functions Af (x, y) = f (x) + y − f (y) and G g ( x , y ) = g ( x ) g ( y ) y {G_g}\left({x,y} \right) = {{g\left(x \right)} \over {g\left(y \right)}}y generalize, respectively, A and G, the classical weighted arithmetic and geometric means. In this note, basing on the invariance identity G ∘ (H, A) = G (equivalent to the Pythagorean harmony proportion), a suitable weighted extension Hf,g of the classical harmonic mean H is introduced. An open problem concerning the symmetry of Hf,g is proposed. As an application a method of effective solving of some functional equations involving means is presented.

Fix $N\in\mathbb{N}$ , and assume that, for every $n\in\{1,\ldots,N\}$ , the functions $f_{n}\colon[0,1]\to[0,1]$ and $g_{n}\colon[0,1]\to\mathbb{R}$ are Lebesgue-measurable, $f_{n}$ is almost everywhere approximately differentiable with $|g_{n}(x)|\lt |f'_{n}(x)|$ for almost all $x\in[0,1]$ , there exists $K\in\mathbb{N}$ such that the set $\{x\in[0,1]:\operatorname{card}{f_{n}^{-1}(x)}\gt K\}$ is of Lebesgue measure zero, $f_{n}$ satisfy Luzin’s condition N, and the set $f_{n}^{-1}(A)$ is of Lebesgue measure zero for every set $A\subset\mathbb{R}$ of Lebesgue measure zero. We show that the formula $Ph=\sum_{n=1}^{N}g_{n}\cdot (h\circ f_{n})$ defines a linear and continuous operator $P\colon L^{1}([0,1])\to L^{1}([0,1])$ , and then we obtain results on the existence and uniqueness of solutions $\varphi\in L^{1}([0,1])$ of the equation $\varphi=P\varphi+g$ with a given $g\in L^{1}([0,1])$ .

We study the alienation problem for two general linear equations i.e. we compare the solutions of the system of equations $$\begin{aligned} \left\{ \begin{array}{l}\sum _{i=1}^n\alpha _if(p_ix+q_iy)=0\\ \sum _{j=1}^m\beta _jg(s_jx+t_jy)=0\end{array}\right. \end{aligned}$$ with the solutions of the single equation $$\begin{aligned} \sum _{i=1}^n\alpha _if(p_ix+q_iy)=\sum _{j=1}^m\beta _jg(s_jx+t_jy). \end{aligned}$$ To this end we introduce the notion of l-alienation—alienation in the class of monomial functions of order l. We use our results among others to study the alienation properties of two monomial functional equations.

We continue our study started in Morawiec and Zürcher (Aequ Math 92(4):601-615, 2018) of the functional equation φ(x)=∑n=0Nφ(fn(x))-∑n=0Nφ(fn(0))and its increasing and continuous solutions φ:[0,1]→[0,1] such that φ(0)=0 and φ(1)=1. In this paper we assume that f0,...,fN:[0,1]→[0,1] are strictly increasing contractions such that 0≤f0(0)

This paper is devoted to finding out some realization of the concept of b-metric like space. First, we attain a fixed point for two fuzzy mappings satisfying a suitable requirement of contractiveness. Subsequently, we apply such a result to graphic contractions. Also, we attain a unique solution for a system of integral equations, and lastly we give an application to ensure that there exists a common bounded solution of a suitable functional equation in dynamic programming.

In this paper we will get a class of functional equations involving a countable set of terms, summed by the well known Bochner–Fejér summation procedure, which are closely associated with the set of almost periodic functions. We will show that the zeros of a prefixed almost periodic function determine analytic solutions of such a functional equation associated with it, and we will obtain other solutions which are analytic or meromorphic on a certain domain. J. M. Sepulcre was supported by PGC2018-097960-B-C22 (MCIU/AEI/ERDF, UE).

Vortex-sheet models of jets are widely used to describe the dynamics of modes, such as the Kelvin-Helmholtz instability and guided acoustic waves. However, it is seldom pointed out in the literature the absence of the free-stream acoustic modes in the vortex-sheet spectrum. This indicates that free-stream sound waves are not eigensolutions of the parallel jet. This family of modes is important if, for example, one is interested in problems of sound emission or flow-acoustic interactions. In this work we show how a distantly-confined jet may be used as a surrogate problem for the free jet, in which free-stream acoustic waves appear as a set of discrete modes. Comparing the modes observed in the free jet with those of the distantly-confined jet, we show that, other than the free-stream acoustic modes, the eigenvectors and eigenvalues converge with wall distance. The proposed surrogate problem thus efficiently reproduces the dynamics of the original problem, while allowing to account for the dynamics of free-stream acoustic modes., Comment: 9 pages, 13 figures

We investigate the Hyers–Ulam stability of an equation involving a single variable of the form ∥f(x)−αf(kn(x))−βf(kn+1(x))∥⩽u(x) where f is an unknown operator from a nonempty set X into a Banach space Y, and it preserves the addition operation, besides other certain conditions. The theory is employed and stability theorems are proven for various functional equations involving several variables. By comparing this method with the available techniques, it was noticed that this method does not require any restriction on the parity, on the domain, and on the range of the function. Our findings suggest that it is very much easy and more appropriate to apply the proposed method while investigating the stability of functional equations, in particular for several variables.

In this paper, we focus on solvable prehomogeneous vector spaces associated with homogeneous cones, and consider the associated zeta functions in several variables. We discuss $\mathbb{Q}$-structures of these prehomogeneous vector spaces, and give explicit formulas of functional equations of the zeta functions by using the data of homogeneous cones. The associated $b$-functions are also described explicitly.

Let $f$ be a map with bounded set of singular values for which periodic dynamic rays exist and land. We prove that each non-repelling cycle is associated to a singular orbit which cannot accumulate on any other non-repelling cycle. When $f$ has finitely many singular values this implies a refinement of the Fatou-Shishikura inequality. Our approach is combinatorial in the spirit of the approach used by [Ki00], [BCL+16] for polynomials., Comment: 13 pages, 1 figure

The research of the first author was supported by the Silesian University Mathematics Department (Iterative Functional Equations and Real Analysis program). Furthermore, the research leading to these results has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP/2007-2013) / ERC Grant Agreement No. 291497 while the second author was a postdoctoral researcher at the University of Warwick., We study the problem of the existence of increasing and continuous solutions φ: [0 , 1] → [0 , 1] such that φ(0) = 0 and φ(1) = 1 of the functional equation φ(x)=∑n=0Nφ(fn(x))-∑n=1Nφ(fn(0)),where N∈ N and f0, … , fN: [0 , 1] → [0 , 1] are strictly increasing contractions satisfying the following condition 0 = f0(0) < f0(1) = f1(0) < ⋯ < fN - 1(1) = fN(0) < fN(1) = 1. In particular, we give an answer to the problem posed in Matkowski (Aequationes Math. 29:210–213, 1985) by Janusz Matkowski concerning a very special case of that equation.

International Conference on Approximation and Computation - Theory and Applications (ACTA) -- NOV 30-DEC 02, 2017 -- Belgrade, SERBIA KUCUKOGLU, IREM/0000-0001-9100-2252 WOS: 000463447500004 The first aim of this paper is to give identities and relations for a new family of the combinatorial numbers and the Apostol-Euler type numbers of the second kind, the Stirling numbers, the Apostol-Bernoulli type numbers, the Bell numbers and the numbers of the Lyndon words by using some techniques including generating functions, functional equations and inversion formulas. The second aim is to derive some derivative formulas and combinatorial sums by applying derivative operators including the Caputo fractional derivative operators. Moreover, we give a recurrence relation for the Apostol-Euler type numbers of the second kind. By using this recurrence relation, we construct a computation algorithm for these numbers. In addition, we derive some novel formulas including the Stirling numbers and other special numbers. Finally, we also some remarks, comments and observations related to our results. Serbian Acad Sci & Arts, Math Inst, Univ Belgrade, Fac Mech Engn, Univ Belgrade, Fac Math, Univ Belgrade, Sch Elect Engn, Univ Nis, Univ Kragujevac, Fac Sci, Univ Novi Sad, Fac Sci Scientific Research Project Administration of Akdeniz UniversityAkdeniz University [FBA-2018-3292] The present paper was supported by Scientific Research Project Administration of Akdeniz University (with Project Number: FBA-2018-3292).

This work is related to inequalities in the approximation theory. Mainly, we study numerical solutions of delay Volterra integral equations by using a collocation method based on sigmoidal function approximation. Error estimation and convergence analysis are provided. At the end of the paper we display numerical simulations verifying our results.

For given $p\in\lbrack1,\infty]$ and $g\in L^{p}\mathbb{(R)}$, we establish the existence and uniqueness of solutions $f\in L^{p}(\mathbb{R)}$, to the equation \[ f(x)-af(bx)=g(x), \] where $a\in\mathbb{R}$, $b\in\mathbb{R} \setminus \{0\}$, and $\left\vert a\right\vert \neq\left\vert b\right\vert ^{1/p}$. Solutions include well-known nowhere differentiable functions such as those of Bolzano, Weierstrass, Hardy, and many others. Connections and consequences in the theory of fractal interpolation, approximation theory, and Fourier analysis are established., Comment: 16 pages, 3 figures

We study a functional equation first proposed by T. Popoviciu [15] in 1955. It was solved for the easiest case by Ionescu [9] in 1956 and, for the general case, by Ghiorcoiasiu and Roscau [7] and Radó [17] in 1962. Our solution is based on a generalization of Radó’s theorem to distributions in a higher dimensional setting and, as far as we know, is different than existing solutions. Finally, we propose several related open problems.

The aim of this paper was a generalization of independence property proposed by J. Kampe de Feriet and B. Forte in Information Theory without probability, called general information. Therefore, its application to fuzzy sets has been presented.

Nowadays \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$(p,q)$\end{document}(p,q)-Bernstein polynomials have been studied in many different fields such as operator theory, CAGD, and number theory. In order to obtain the fundamental properties and results of Bernstein polynomials by using \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$(p,q)$\end{document}(p,q)-calculus, we give basic definitions and results related to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$(p,q)$\end{document}(p,q)-calculus. The main purpose of this study is to investigate a generating function for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$(p,q)$\end{document}(p,q)-Bernstein polynomials. By using an approach similar to that of Goldman et al. in (SIAM J. Discrete Math. 28(3):1009-1025, 2014), we derive some new identities, relations, and formulas for the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$(p,q)$\end{document}(p,q)-Bernstein polynomials. Also, we give a plot generating function of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$(p,q)$\end{document}(p,q)-Bernstein polynomials for some selected p and q values.

Given a stochastic differential equation with path-dependent coefficients driven by a multidimensional Wiener process, we show that the support of the law of the solution is given by the image of the Cameron-Martin space under the flow of mild solutions to a system of path-dependent ordinary differential equations. Our result extends the Stroock-Varadhan support theorem for diffusion processes to the case of stochastic differential equations with path-dependent coefficients. The proof is based on functional Ito calculus., Comment: 42 pages

The functional equation of composite type M(M(x; M(x; y)); M(M(x; y); y)) = M(x; y) arose in the course of the studies on the problem of extension and restriction of the number of arguments of a mean M performed by G. Aumann at the third decade of the past century. A solution to (1) in the analytic case was ulteriorly obtained by Aumann himself and remained as a noteworthy characterization of analytic quasiarithmetic means. An ample generalization of equation (1) which involves general weighting operators is considered in this paper. Under mild conditions on the regularity of the involved means, the general solution to this generalized equation is obtained for a particularly tractable class of weighting operators. Fil: Berrone, Lucio Renato. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Cientifico Tecnológico Rosario; Argentina. Universidad Nacional de Rosario. Facultad de Ciencias Exactas, Ingenieria y Agrimensura. Escuela de Ingenieria Electrónica. Laboratorio de Acústica y Electroacústica; Argentina

Agraïments: The second author is partially supported by the Polish NCN grant decision DEC-2012/06/M/ST1/00168 and the MDM-2014-445 Maria de Maeztu. In 2000 McMullen proved that copies of generalized Mandelbrot set are dense in the bifurcation locus for generic families of rational maps. We develop an algo- rithm to an effective computation of the location and size of these generalized Mandelbrot sets in parameter space. We illustrate the effectiveness of the algorithm by applying it to concrete families of rational and entire maps.

In this paper, we introduce the notion of cyclic ( &alpha, &beta, ) - ( &psi, &phi, ) s -rational-type contraction in b-metric spaces, and using this contraction, we prove common fixed point theorems. Our work generalizes many existing results in the literature. In order to highlight the usefulness of our results, applications to functional equations are given.

The asymptotic behavior of solutions of the system of difference equations with continuous time and lag function between two known real functions is studied. The cases when the lag function is between two linear delay functions, between two power delay functions and between two constant delay functions are observed and illustrated by examples. The asymptotic estimates of solutions of the considered system are obtained.

Through several equivalence binary relations, in this paper we identify, on the one hand, groups of exponential polynomials with the same set of zeros, and on the other, groups of functional equations of the form a1f(1z) + a2f(2z) + : : : + anf(nz) = 0; z 2 C that lead to equivalent exponential polynomials with the same set of zeros.

We study the equation $M_\Psi(z+1)=\frac{-z}{\Psi(-z)}M_\Psi(z), M_\Psi(1)=1$ defined on a subset of the imaginary line and where $\Psi$ is a negative definite functions. Using the Wiener-Hopf method we solve this equation in a two terms product which consists of functions that extend the classical gamma function. These functions are in a bijection with Bernstein functions and for this reason we call them Bernstein-gamma functions. Via a couple of computable parameters we characterize of these functions as meromorphic functions on a complex strip. We also establish explicit and universal Stirling type asymptotic in terms of the constituting Bernstein function. The decay of $|M_{\Psi}(z)|$ along imaginary lines is computed. Important quantities for theoretical and applied studies are rendered accessible. As an application we investigate the exponential functionals of Levy Processes whose Mellin transform satisfies the recurrent equation above. Although these variables have been intensively studied, our new perspective, based on a combination of probabilistic and complex analytical techniques, enables us to derive comprehensive and substantial properties and strengthen several results on the law of these random variables. These include smoothness, regularity and analytical properties, large and small asymptotic behaviour, including asymptotic expansions, bounds, and Mellin-Barnes representations for the density and its successive derivatives. We also study the weak convergence of exponential functionals on a finite time horizon when the latter expands to infinity. As a result of new factorizations of the law of the exponential functional we deliver important intertwining relation between members of the class of positive self-similar semigroups. The derivation of our results relies on a mixture of complex-analytical and probabilistic techniques.

International audience; Whether Fitts’ law, a well-known model of human pointing movement, is a logarithmic law or a power law has remained unclear so far. In two widely cited papers, Meyer & al. have claimed that the power model they derived from their celebrated stochastic optimized- submovement theory encompasses the logarithmic model as a limiting case, when the number of submovements grows large. We review the Meyer & al. submovement theory and show that this claim is questionable mathematically. Our analysis reveals that Meyer & al.’s theory implies in fact a quasi-logarithmic, rather than quasi-power model, the two models not being equivalent. Awareness that the two classes of candidate mathematical descriptions of Fitts’ law are not equivalent should stimulate experimental research in the field.; Modèle de puissance vs. logarithmique de la loi de Fitts : une analyse mathématique. Après bientôt soixante années d’études, il reste toujours à déterminer si la loi de Fitts, un modèle célèbre du mouvement de pointage humain, est une loi logarithmique ou de puissance. Dans deux articles abondamment cités, Meyer & al. ont avancé l’idée que le modèle de puissance qu’ils ont déduit de leur théorie stochastique des sous-mouvements optimisés englobe le modèle logarithmique comme un cas limite atteint lorsque le nombre de sous-mouvements devient grand. Reconsidérant la théorie des sous-mouvements de Meyer et al., nous montrons que cette proposition est mathématiquement inexacte. La théorie de Meyer et al. implique en réalité un modèle quasi-logarithmique plutôt que de puissance, le premier n’étant pas équivalent au second. Une pleine conscience que les deux classes possibles de description mathématique de la loi de Fitts ne sont pas équivalentes nous semble de nature à stimuler la recherche expérimentale dans ce domaine.

We mathematically analyze the diffusion-based algorithm to produce maps with a given Jacobian, introduced independently by M. T. Gastner and M. E. J. Newman (2004) and ourselves (2003), but in particular cases where the initial density has line or angle discontinuities in the plane. In this situation, the conclusion reinforces the conjecture that the algorithm is always well-posed, in accordance with its extensive numerical use in some areas of applied sciences (cartograms, sensor networks, computational grids, or image registration) This work was partially supported by grants from the Spanish Government (MTM2008-06349-C03-01, MTM2011-27739-C04-01, and MTM2011-27739-C04-03) and the Catalan Government (2009SGR345)

We propose two types of extensions to Hamburger’s theorems on the Dirichlet series with a functional equation like the one of the Riemann zeta function, under weaker hypotheses. This builds upon the dictionary between the moderate meromorphic functions with the functional equation and the tempered distributions with an extended S -support condition.

Motivated by some investigations of Babbage and a method of solving certain functional equations arising in competition problems, we investigate a class of functional equations and prove a local existence and uniqueness theorem for them. The main tools of the proof are the Inverse Function Theorem and the Global Existence and Uniqueness Theorem.

The aim of the paper is to construct a solution for the equation f ⋆ μ = g , where f ⋆ μ is the convolution of f and μ given by ∫ R 2 f ( x ¯ − y ¯ ) d μ ( y ¯ ) , g is an arbitrary continuous function and μ is a finitely supported measure on the plane.

The mathematical relations between the regular Coulomb function Fηℓ(ρ) and the irregular Coulomb functions H±ηℓ(ρ) and Gηℓ(ρ) are obtained in the complex plane of the variables η and ρ for integer or half-integer values of ℓ. These relations, referred to as “connection formulas,” form the basis of the theory of Coulomb wave functions and play an important role in many fields of physics, especially in the quantum theory of charged particle scattering. As a first step, the symmetry properties of the regular function Fηℓ(ρ) are studied, in particular, under the transformation ℓ ↦ −ℓ − 1, by means of the modified Coulomb function Φηℓ(ρ), which is entire in the dimensionless energy η−2 and the angular momentum ℓ. Then, it is shown that, for integer or half-integer ℓ, the irregular functions H±ηℓ(ρ) and Gηℓ(ρ) can be expressed in terms of the derivatives of Φη,ℓ(ρ) and Φη,−ℓ−1(ρ) with respect to ℓ. As a consequence, the connection formulas directly lead to the description of the singular structures of H±ηℓ(ρ)and Gηℓ(ρ) at complex energies in their whole Riemann surface. The analysis of the functions is supplemented by novel graphical representations in the complex plane of η−1., SCOPUS: ar.j, info:eu-repo/semantics/published

In this paper, we explore the distributive equations of implications, both independently and along with other equations. In detail, we consider three classes of equations. (1) By means of the section of I, we give out the sufficient and necessary conditions of solutions for the distributive equation of implication I(x,T(y,z))=T(I(x,y),(x,z)) based on a nilpotent triangular norm T and an unknown function I, which indicates that there are no continuous solutions satisfying the boundary conditions of implications. Under the assumptions that I is continuous except the vertical section I(0,y), y∈[0,1), we get its complete characterizations. (2) We prove that there are no solutions for the functional equations I(x,T(y,z))=T(I(x,y),I(x,z)),I(x,I(y,z))=I(T(x,y),z). (3) We obtain the sufficient and necessary conditions on T and I to be solutions of the functional equations I(x,T(y,z))=T(I(x,y),I(x,z)), I(x,y)=I(N(y),N(x)).

In the context of the relativistic Doppler effect [DE] and the Lorentz-Fitzgerald contraction [LF], we investigate the consequences of two abstract axioms [R] and [M] expressed in terms of an operation generalizing the addition of velocities and a function. The latter can represent either the Doppler effect or the Lorentz- Fitzgerald Contraction. In words, these axioms state the following: [R] iterating the function L has the same effect as adding velocities; [M] adding a velocity via the operation preserves the order of the function L. We prove that these axioms are equivalent to each other, and also to generalized forms of the Doppler effect and the standard expression [AV] for the relativistic addition of velocities, taken jointly. We also show that if [AV] holds, then the axioms [R] and [M] are inconsistent with [LF]. © 2010 The Author(s)., SCOPUS: ar.j, info:eu-repo/semantics/published

Let T,U be two linear operators mapped onto the function f such that U(T(f))=f, but T(U(f))≠f. In this paper, we first obtain the expansion of functions T(U(f)) and U(T(f)) in a general case. Then, we introduce four special examples of the derived expansions. First example is a combination of the Fourier trigonometric expansion with the Taylor expansion and the second example is a mixed combination of orthogonal polynomial expansions with respect to the defined linear operators T and U. In the third example, we apply the basic expansion U(T(f))=f(x) to explicitly compute some inverse integral transforms, particularly the inverse Laplace transform. And in the last example, a mixed combination of Taylor expansions is presented. A separate section is also allocated to discuss the convergence of the basic expansions T(U(f)) and U(T(f)).

We propose a decomposition principle for the Riemann–Stieltjes integration and study some of its implications which lead to three distinct forms for the Riemann integral for monotonically varying differentiable functions.

We apply the theory of Differential and Integral Calculus in Riesz Spaces introduced in [1] and [4] to investigate some properties of the g-calculus and to solve some types of differential, functional and stochastic equations.

Recently, in the complex context, several results were obtained concerning functional equations of the form P ( f ) = Q ( g ) where P and Q are polynomials of only two or three terms whose coefficients are small functions: in certain cases the equation does not admit any pair of admissible solutions and in other cases it only admits pairs of solutions that are of a very particular type. Here we consider similar questions when the ground field is a p-adic complete algebraically closed field of characteristic 0 and we derive results that are often analogous. For instance, if f n + a 1 f n − m + b 1 = c ( g − n + a 2 g n − m + b 2 ) , with a i , b j small functions with regard to f, g and a 2 b 2 non-identically 0, then c = b 1 b 2 and f = h g with h m = a 1 a 2 . However, contrary to the complex context, here results apply not only to meromorphic functions defined in the whole field but also to unbounded meromorphic functions defined inside an open disc. The main tool is the p-adic Nevanlinna theory.

We present an algorithm for finding a system of recurrence relations for the number of k-ary words of length n that satisfy a certain set of conditions. A rewriting of these relations automatically gives a system of functional equations satisfied by the multivariate generating function that counts k-ary words by their length and the indices of the corresponding recurrence relations. We propose an approach to describing such equations. In several interesting cases the algorithm recovers and refines results on τ-avoiding k-ary words and k-ary words containing τ exactly once, where τ is either a classical, a generalized, or a distanced pattern of length three.

In this paper we study the distribution of zeros of each entire function of the sequence { G n ( z ) ≡ 1 + 2 z + ⋯ + n z : n ⩾ 2 } , which approaches the Riemann zeta function for Re z − 1 , and is closely related to the solutions of the functional equations f ( z ) + f ( 2 z ) + ⋯ + f ( n z ) = 0 . We determine the density of the zeros of G n ( z ) on the critical strip where they are situated by using almost-periodic functions techniques. Furthermore, by using a theorem of Kronecker, we also establish a formula for the number of zeros of G n ( z ) inside certain rectangles in the critical strip.

J. R. Meginniss modified expected utility to accommodate a concept of the utility of gambling that led to a representation composed of a utility expectation term plus an entropy of degree κ term. He imposed several apparently strong assumptions. One of these is that a number of unknown generating functions are identical. A second is that he assumed he was working with given probabilities. Here we follow his general framework but weaken considerably those assumptions. Our problem is reduced to solving some functional equations induced by gamble decomposition. From the solutions, we obtain the representation of the utility function. Further axiomatic restrictions are imposed that lead ultimately to Meginniss' earlier result.

In this paper, we give a more direct proof of the results by Clair and Mokhtari-Sharghi on the zeta functions of periodic graphs. In particular, using appropriate operator-algebraic techniques, we establish a determinant formula in this context and examine its consequences for the Ihara zeta function. Moreover, we answer in the affirmative one of the questions raised by Grigorchuk and Zuk. Accordingly, we show that the zeta function of a periodic graph with an amenable group action is the limit of the zeta functions of a suitable sequence of finite subgraphs., 21 pages, 4 figures

We specialize our results on entropy-modified representations of event-based gambles to representations of probability-based gambles by assuming an implicit event structure underlying the probabilities, and adding assumptions linking the qualitative properties of the former and the latter. Under segregation and under duplex decomposition, we obtain numerical representations consisting of a linear weighted utility term plus a term corresponding to information-theoretical entropies. These representations accommodate the Allais paradox and most of the data due to Birnbaum and associates. A representation of mixed event-and probability-based gambles accommodates the Ellsberg paradox. We suggest possible extensions to handle the data not accommodated.

Given a convex function f defined for positive real variables, the so-called Csiszar f -divergence is a function I f defined for two n -dimensional probability vectors p = ( p 1 , … , p n ) and q = ( q 1 , … , q n ) as I f ( p , q ) : = ∑ i = 1 n q i f ( p i q i ) . For this generalized measure of entropy to have distance-like properties, especially symmetry, it is necessary for f to satisfy the following functional equation: f ( x ) = x f ( 1 x ) for all x > 0 . In the present paper we determine all the convex solutions of this functional equation by proposing a way of generating all of them. In doing so, existing usual f -divergences are recovered and new ones are proposed.

Let $f$ and $g$ be real-valued continuous injections defined on a non-empty real interval $I$, and let $(X, \mathscr{L}, \lambda)$ and $(Y, \mathscr{M}, \mu)$ be probability spaces in each of which there is at least one measurable set whose measure is strictly between $0$ and $1$. We say that $(f,g)$ is a $(\lambda, \mu)$-switch if, for every $\mathscr{L} \otimes \mathscr{M}$-measurable function $h: X \times Y \to \mathbf{R}$ for which $h[X\times Y]$ is contained in a compact subset of $I$, it holds $$ f^{-1}\!\left(\int_X f\!\left(g^{-1}\!\left(\int_Y g \circ h\;d\mu\right)\right)d \lambda\right)\! = g^{-1}\!\left(\int_Y g\!\left(f^{-1}\!\left(\int_X f \circ h\;d\lambda\right)\right)d \mu\right)\!, $$ where $f^{-1}$ is the inverse of the corestriction of $f$ to $f[I]$, and similarly for $g^{-1}$. We prove that this notion is well-defined, by establishing that the above functional equation is well-posed (the equation can be interpreted as a permutation of generalized means and raised as a problem in the theory of decision making under uncertainty), and show that $(f,g)$ is a $(\lambda, \mu)$-switch if and only if $f = ag + b$ for some $a,b \in \mathbf R$, $a \ne 0$., Comment: 9 pages, no figures. Fixed minor details. Final version to appear in Indagationes Mathematicae

Treballs Finals de Grau de Matemàtiques, Facultat de Matemàtiques, Universitat de Barcelona, Any: 2015, Director: Pilar Bayer i Isant, The first objective of this work it to present a detailed study on the Riemann $\zeta$ function as a complex function, which includes its analytic continuation, functional equation and those necessary properties to proof in the following chapter the prime number theorem. The second objective is to write a self contained proof of the primer number theorem, discarding the superfluous results found in the references to achieve a clear and concise proof. In that regard, we also include our own graphics to help the understanding of the behavior of the functions that appear in the proof. Lastly, we consider the Riemann hypothesis, qualified as one of the Millennium Problems by the Clay Institute and a key point in the development of modern number theory. The second part of this work aims to find physical applications of the mathematical methods used in this work of analytic number theory. On this subject we present the Casimir effect, which is a first example on renormalization in quantum theories and in which we show that the analytic continuation of the $\zeta$ function plays an important role. The second case is Lee-Yang theory. Although in this last case there is no direct application of the methods studied in the mathematical part of the work, it shows the importance of studying the distribution of zeros of certain functions. Therefore, Lee-Yang theory could be a very interesting bridge between two apparently very different disciplines such as statistical mechanics and number theory. In fact, in the development of this work, we unexpectedly discovered that Lee and Yang based a part of their proof of the circle theorem presented here on a Polya article ([11]) on the integral representations of the Riemann $\zeta$ function, as Kac mentions in a comment of the same article. Lastly, we have applied the Lee-Yang theory to the one and two dimensional Ising models. We have been able to compute the distribution of the zeros of the partition functions, which has enabled us the study of the phase transitions of these thermodynamic systems.

In this paper, it is showed that, given an integer number n ≥ 2, each zero of an exponential polynomial of the form $${w_1a_1^{z}+w_2a_2^{z}+\cdots+w_na_n^{z}}$$ , with non-null complex numbers w 1,w 2,…,w n and a 1,a 2,…,a n , produces analytic solutions of the functional equation w 1 f(a 1 z) + w 2 f(a 2 z) + ... + w n f(a n z) = 0 on certain domains of $${\mathbb{C}}$$ , which represents an extension of some existing results in the literature on this functional equation for the case of positive coefficients a j and w j .

We consider the dynamical zeta functions of Selberg and Ruelle associated with the geodesic flow on a compact odd-dimensional hyperbolic manifold. These dynamical zeta functions are defined for a complex variable $s$ in some right-half plane of $\mathbb{C}$. In [Spi18], it was proved that they admit a meromorphic continuation to the whole complex plane. In this paper, we establish functional equations for them, relating their values at $s$ with those at $-s$. We prove also a determinant representation of the zeta functions, using the regularized determinants of certain twisted differential operators., Comment: Revisions