Wyniki wyszukiwania - BazEkon

1

The Generalization of Gaussians and Leonardo's Octonions

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Veira R. P. M., dos Santos Mangueira M. C., Vieira Alves F. R., Machado Cruz Catarino P. M.

Annales Mathematicae Silesianae

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2023

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37 (1)

117-137

In order to explore the Leonardo sequence, the process of complexification of this sequence is carried out in this work. With this, the Gaussian and octonion numbers of the Leonardo sequence are presented. Also, the recurrence, generating function, Binet's formula, and matrix form of Leonardo's Gaussian and octonion numbers are defined. The development of the Gaussian numbers is performed from the insertion of the imaginary component i in the one-dimensional recurrence of the sequence. Regarding the octonions, the terms of the Leonardo sequence are presented in eight dimensions. Furthermore, the generalizations and inherent properties of Leonardo's Gaussians and octonions are presented.(original abstract)

2

Sine Subtraction Laws on Semigroups

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Ebanks B.

Annales Mathematicae Silesianae

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2023

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37 (1)

49-66

We consider two variants of the sine subtraction law on a semi-groupS. The main objective is to solvef(xy∗) =f(x)g(y)-g(x)f(y)forunknown functionsf,g:S→C, wherex7→x∗is an anti-homomorphic invo-lution. Until now this equation was not solved even whenSis a non-Abeliangroup andx∗=x-1. We find the solutions assuming thatfis central. A sec-ondary objective is to solvef(xσ(y)) =f(x)g(y)-g(x)f(y), whereσ:S→Sis a homomorphic involution. Until now this variant was solved assuming thatShas an identity element. We also find the continuous solutions of these equa-tions on topological semigroups.(original abstract)

3

A Variant of D'Alembert's Functional Equation on Semigroups with Endomorphisms

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Akkaoui A., El Fatini M., Fadli B.

Annales Mathematicae Silesianae

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2022

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36 (1)

1-14

Let S be a semigroup, and let φ, ψ: S → S be two endomorphisms (which are not necessarily involutive). Our main goal in this paper is to solve the following generalized variant of d'Alembert's functional equation f(xϕ(y))+f(ψ(y)x)=2f(x)f(y), x,y ∈ S, where f : S → ℂ is the unknown function by expressing its solutions in terms of multiplicative functions. Some consequences of this result are presented.(original abstract)

4

On the Two-Dimensional Version of the Sperner Lemma and Brouwer's Theorem

100%

Barcz E.

Annales Mathematicae Silesianae

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2022

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36 (2)

106-114

In this work the Brouwer fixed point theorem for a triangle wasproved by two methods based on the Sperner Lemma. One of the two proofs of Sperner's Lemma given in the paper was carried out using the so-called index(original abstract)

5

Alienation of Drygas' and Cauchy's Functional Equations

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Aissi Y., Zeglami D., Fadli B.

Annales Mathematicae Silesianae

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2021

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35 (2)

131-148

Inspired by the papers [2, 10] we will study, on 2-divisible groupsthat need not be abelian, the alienation problem between Drygas' and theexponential Cauchy functional equations, which is expressed by the equation f(x+y) +g(x+y) +g(x-y) =f(x)f(y) + 2g(x) +g(y) +g(-y). We also consider an analogous problem for Drygas' and the additive Cauchyfunctional equations as well as for Drygas' and the logarithmic Cauchy func-tional equations. Interesting consequences of these results are presented. (original abstract)

6

The Ninth Katowice-Debrecen Winter Seminaron Functional Equations and Inequalities,Będlewo (Poland), February 4-7, 2009

75%

Szostok T.

Annales Mathematicae Silesianae

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2009

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23

103-117

The Ninth Katowice-Debrecen Winter Seminar on Functional Equationsand Inequalities was held in the Mathematical Research and Conference Center Będlewo, Poland, from February 4 to 7, 2009. It was organized by theInstitute of Mathematics of the Silesian University from Katowice.24 participants came from the University of Debrecen (Hungary) and theSilesian University of Katowice (Poland) at 12 from each of both cities. The25thparticipant of the Seminar was Professor Peter Volkmann from the University of Karlsruhe who is at present a visting professor at the Silesian University of Katowice.Professor Roman Ger opened the Seminar and welcomed the participantsto Będlewo.The scientific talks presented at the Seminar focused on the following topics: equations in a single variable and in several variables, iteration theory,equations on algebraic structures, regularity properties of the solutions of cer-tain functional equations, functional inequalities, Hyers-Ulam stability, func-tional equations and inequalities involving mean values, generalized convexity.Interesting discussions were generated by the talks.(original abstract)

7

Stability of the Pexider Functional Equation

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Badora R., Przebieracz B., Volkmann P.

Annales Mathematicae Silesianae

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2010

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24

7-13

A stability result for the Pexider equation will be derived from a stability theorem published in [9] for the Cauchy functional equation. Then we discuss the quality of some constants occuring in this context; as a model case we consider functions defined on the multiplicative semigroup {1,0}. (original abstract)

8

A Functional Equation Characterizing Homographic Functions

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Matkowski J.

Annales Mathematicae Silesianae

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2010

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24

61-74

Some functional equations related to homographic functions and their characterization are presented. (original abstract)

9

A Levi-Civita Equation on Monoids, Two Ways

75%

Ebanks B.

Annales Mathematicae Silesianae

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2022

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36 (2)

151-166

We consider the Levi-Civita equation f(xy) = g_1(x)h_1(y) + g_2(x)h_2(y) for unknown functions f, g_1, g_2, h_1, h_2 : S→ℂ, where S is a monoid. This functional equation contains as special cases many familiar functional equations, including the sine and cosine addition formulas. In a previous paper we solved this equation on groups and on monoids generated by their squares under the assumption that f is central. Here we solve the equation on monoids by two different methods. The first method is elementary and works on a general monoid, assuming only that the function f is central. The second way uses representation theory and assumes that the monoid is commutative. The solutions are found (in both cases) with the help of the recently obtained solution of the sine addition formula on semigroups. We also find the continuous solutions on topological monoids.(original abstract)

10

A Functional Equation with Biadditive Functions

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Łukasik R.

Annales Mathematicae Silesianae

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2022

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36 (2)

193-205

Let S, H, X be groups. For two given biadditive functions A:S^2→X, B:H^2→X and for two unknown mappings T:S→H, g:S→S we will study the functional equation B(T(x),T(y)) = A(x,g(y)), x,y∈S, which is a generalization of the orthogonality equation in Hilbert spaces.(original abstract)

11

Remarks About the Square Equation. Functional Square Root of a Logarithm

75%

Maciuk A., Smoluk A.

Mathematical Economics

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2016

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nr 12 (19)

39-52

The study shows that the functional equation f (f (x)) = ln(1 + x) has a unique result in a semigroup of power series with the intercept equal to 0 and the function composition as an operation. This function is continuous, as per the work of Paulsen [2016]. This solution introduces into statistics the law of the one-and-a-half logarithm. Sometimes the natural growth processes do not yield to the law of the logarithm, and a double logarithm weakens the growth too much. The law of the one-and-a-half logarithm proposed in this study might be the solution.(original abstract)

12

Solutions and Stability of Generalized Kannappan's and Van Vleck's Functional Equations

75%

Elqorachi E., Redouani A.

Annales Mathematicae Silesianae

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2018

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32

169-200

We study the solutions of the integral Kannappan's and Van Vleck's functional equations ∫Sf(xyt)dµ(t)+∫Sf(xσ(y)t)dµ(t)= 2f(x)f(y), x,y ∈ S; ∫Sf(xσ(y)t)dµ(t)-∫Sf(xyt)dµ(t)= 2f(x)f(y), x,y ∈ S; where S is a semigroup, σ is an involutive automorphism of S and µ is a linear combination of Dirac measures ( ᵟ zi)I ∈ I, such that for all i ∈ I, ziis in the center of S. We show that the solutions of these equations are closely related to the solutions of the d'Alembert's classic functional equation with an involutive automorphism. Furthermore, we obtain the superstability theorems for these functional equations in the general case, where σ is an involutive morphism. (original abstract)

13

Stability of Functional Equations and Properties of Groups

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Forti G. L.

Annales Mathematicae Silesianae

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2019

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33

77-96

Investigating Hyers-Ulam stability of the additive Cauchy equation with domain in a group $G$, in order to obtain an additive function approximating the given almost additive one we need some properties of $G$, starting from commutativity to others more sophisticated. The aim of this survey is to present these properties and compare, as far as possible, the classes of groups involved(original abstract)

14

Report of Meeting. The Twentieth Debrecen-Katowice Winter Seminar on Functional Equations and Inequalities Hajdúszoboszló (Hungary), January 29-February 1, 2020

75%

Nagy G.

Annales Mathematicae Silesianae

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2020

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34 (2)

286-304

The Twentieth Debrecen-Katowice Winter Seminar on Functional Equations and Inequalities was held in Hotel Aurum, Hajdúszoboszló, Hungary,from January 29 to February 1, 2020. It was organized by the Department of Analysis of the Institute of Mathematics of the University of Debrecen.The 30 participants came from the University of Silesia (Poland), the University of Debrecen (Hungary), the University of Rzeszów (Poland), the Pedagogical University of Cracow (Poland), the University of Zielona Góra (Poland), The John Paul II Catholic University of Lublin (Poland) and the Karlsruhe Institute of Technology (Germany), 10 from the first, 15 from thesecond and 1 from each of the other universities. [...] The scientific talks presented at the Seminar focused on the following topics: equations in a single variable and in several variables, iterative equations, equations on algebraic structures, functional inequalities, Hyers-Ulam stability, functional equations and inequalities involving mean values, generalized convexity and Walsh-Fourier analysis. Interesting discussions were generatedby the talks. (fragment of text)

15

An Extension of a Ger's Result

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Marinescu D. S., Monea M.

Annales Mathematicae Silesianae

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2018

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32

263-274

The aim of this paper is to extend a result presented by Roman Ger during the 15th International Conference on Functional Equations and Inequalities. First, we present some necessary and sufficient conditions for a continuous function to be convex. We will use these to extend Ger's result. Finally, we make some connections with other mathematical notions, as g-convex dominated function or Bregman distance. (original abstract)

16

On a Functional Equation Related to Two-Sided Centralizers

75%

Kosi-Ulbl I.

Annales Mathematicae Silesianae

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2018

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32

227-235

The main aim of this manuscript is to prove the following result. Let n > 2 be a fixed integer and R be a k-torsion free semiprime ring with identity, where k ∈ {2, n - 1, n}. Let us assume that for the additive mapping T : R→ R 3T(xn) = T(x)xn-1+ xT(xn-2)x + xn-1T(x), x ∈R, is also fulfilled. Then T is a two-sided centralizer. (original abstract)

17

On the General and Measurable Solutions of some Functional Equations

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Nath P., Singh D. K.

Annales Mathematicae Silesianae

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2018

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32

285-294

The general solutions of two functional equations, without imposing any regularity condition on any of the functions appearing, have been obtained. From these general solutions, the Lebesgue measurable solutions have been deduced by assuming the function(s) to be measurable in the Lebesgue sense. (original abstract)

18

Note on an Iterative Functional Equation

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Baron K., Morawiec J.

Annales Mathematicae Silesianae

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2024

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38 (1)

12-17

We study the problem of solvability of the equation φ(x) = ∫_Ω (ω)φ(f(x,ω))P(dω) + F(x), where P is a probability measure on a σ-algebra of subsets of Ω, assuming Hölder continuity of F on the range of f. (original abstract)

19

Report of Meeting: The Twenty-second Debrecen-Katowice Winter Seminar on Functional Equations and Inequalities Hajdúszoboszló (Hungary), February 1-4, 2023

75%

Pénzes E.

Annales Mathematicae Silesianae

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2023

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37 (2)

315-334

The Twenty-second Debrecen-Katowice Winter Seminar on Functional E-quations and Inequalities was held in Hotel Aurum, Hajdúszoboszló, Hungary,from February 1 to February 4, 2023. It was organized by the Department ofAnalysis of the Institute of Mathematics of the University of Debrecen.The Winter Seminar was supported by the Institute of Mathematics, Uni-versity of Debrecen and by the project 2019-2.1.11-TÁT-2019-00049.The 27 participants came from the University of Silesia (Poland), the Uni-versity of Debrecen (Hungary), the University of Rzeszów (Poland) and theKazimierz Wielki University (Poland), 8 from the first, 16 from the second, 2from the third and 1 from the fourth university.(original abstract)

20

Families of Commuting Formal Power Series and Formal Functional Equations

63%

Fripertinger H., Reich L.

Annales Mathematicae Silesianae

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2021

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35 (1)

55-76

In this paper we describe families of commuting invertible formal power series in one indeterminate over ℂ, using the method of formal functional equations. We give a characterization of such families where the set of multipliers (first coefficients) σ of its members F (x) = σx + . . . is infinite, in particular of such families which are maximal with respect to inclusion, so called families of type I. The description of these families is based on Aczél-Jabotinsky differential equations, iteration groups, and on some results on normal forms of invertible series with respect to conjugation. (original abstract)