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The purpose of the present paper is to show that in certain classes of real (or complex) functions, Bernoulli polynomials are essentially the only ones satisfying the Raabe functional equation. For the class of real 1-periodic functions which are expandable as Fourier series, we point out new solutions of the Raabe functional equation, not related to Bernoulli polynomials. Furthermore, we will give for the considered classes various proofs, making the mathematical content of the paper quite rich. [ABSTRACT FROM AUTHOR]

Let G be a connected graph of order n, where n is a positive integer. A spanning subgraph F of G is called a path-factor if every component of F is a path of order at least 2. A P ≥ k -factor means a path-factor in which every component admits order at least k ( k ≥ 2 ). The distance matrix D (G) of G is an n × n real symmetric matrix whose (i, j)-entry is the distance between the vertices v i and v j . The distance signless Laplacian matrix Q (G) of G is defined by Q (G) = T r (G) + D (G) , where Tr(G) is the diagonal matrix of the vertex transmissions in G. The largest eigenvalue η 1 (G) of Q (G) is called the distance signless Laplacian spectral radius of G. In this paper, we aim to present a distance signless Laplacian spectral radius condition to guarantee the existence of a P ≥ 2 -factor in a graph and claim that the following statements are true: (i) G admits a P ≥ 2 -factor for n ≥ 4 and n ≠ 7 if η 1 (G) < θ (n) , where θ (n) is the largest root of the equation x 3 - (5 n - 3) x 2 + (8 n 2 - 23 n + 48) x - 4 n 3 + 22 n 2 - 74 n + 80 = 0 ; (ii) G admits a P ≥ 2 -factor for n = 7 if η 1 (G) < 25 + 161 2 . [ABSTRACT FROM AUTHOR]

The paper concerns an analogue of the familiar Tverberg theorem, using certain staircase convex sets in place of convex hulls. For d ≥ 1 and k ≥ 2 , we obtain a number t(d, k) for which the following is true: Given a collection P of at least t(d, k) points in R d , there is a partition of P into k (nonempty, pairwise disjoint) sets S 1 , ... , S k such that the corresponding smallest boxes have a nonempty intersection. Moreover, the constructive proof offers a way to find appropriate sets S 1 , ... , S k . [ABSTRACT FROM AUTHOR]

In this paper, we present the central solutions f , g : G × G → C of the functional Eq. f (x 1 y 1 , x 2 y 2) + g (x 1 y 1 - 1 , x 2) + f (x 1 , x 2 y 2 - 1) = f (x 1 y 1 - 1 , x 2 y 2 - 1) + g (x 1 y 1 , x 2) + f (x 1 , x 2 y 2) for all x 1 , x 2 , y 1 , y 2 ∈ G , where G is a 2-divisible group. [ABSTRACT FROM AUTHOR]

The functional equation f (x + y) = g (x y) is investigated with unknown functions f : A + A → Y , g : A · A → Y in the following cases: A : = α , β ⊆ F + where F is an Archimedean ordered field; A is the set of all positive integers; A is the set of all positive dyadic rational numbers. The set Y is an arbitrarily fixed (infinite) set. The main result of the paper shows that there exists a set A ⊆ R + that is closed under addition and multiplication and there exist functions f, g : A → Y which satisfy the equation f (x + y) = g (x y) for all x , y ∈ A such that the range of the function f is infinite. Finally, some application of the above results is also given. [ABSTRACT FROM AUTHOR]

It was found that a function with exactly one discontinuity may have a continuous iterate of second order, indicating that a discontinuity may be repaired to be a continuous one by its adjacent pair of functions of second order, called second order sui-repair. If a function has more than one discontinuities, examples show that some discontinuities may be repaired to be continuous ones by the other's adjacent pair of functions of second order, called second order C 0 homi-repair. In this paper we investigate second order C 0 homi-repairs of removable and jumping discontinuities for functions having more than one but finitely many discontinuities. We give necessary and sufficient conditions for removable and jumping discontinuities to be C 0 repaired by the second order iteration. [ABSTRACT FROM AUTHOR]

Let X and Y be Banach spaces. When A and B are linear relations in X and Y, respectively, we denote by M C the linear relation in X × Y of the form A C 0 B , where 0 is the zero operator from X to Y and C is a bounded operator from Y to X. In this paper, by using properties of the SVEP, we study the defect set (Σ (A) ∪ Σ (B)) \ Σ (M C) , where Σ is the spectrum, the approximate point spectrum, the surjective spectrum, the Fredholm spectrum, the Weyl spectrum, the Browder spectrum, the generalized Drazin spectrum and the Drazin spectrum. [ABSTRACT FROM AUTHOR]

The paper is devoted to some extremal problems, related to convex polygons in the Euclidean plane and their perimeters. We present a number of results that have simple formulations, but rather intricate proofs. Related and still unsolved problems are also discussed. [ABSTRACT FROM AUTHOR]

It is shown that removable and jumping discontinuities for functions having more than one but finitely many discontinuities have a second order C 0 homi-repair. In this paper we study second order C 0 homi-repair of oscillating discontinuities and pre-discontinuities for those functions and give necessary and sufficient conditions for C 0 repair by second order iteration. [ABSTRACT FROM AUTHOR]

In this paper we determine the solutions (φ , f 1 , f 2) of the Pexider functional equation φ ( x + y 2) (f 1 (x) - f 2 (y)) = 0 , (x , y) ∈ I 1 × I 2 , where I 1 and I 2 are nonempty open subintervals. Special cases of the above equation regularly arise in problems with two-variable means. We show that, under a rather weak regularity condition, the coordinate-functions of a typical solution of the equation are constant over several subintervals of their domain. The regularity condition in question will be that the set of zeros of φ is closed. We also discuss particular solutions where this condition is not met. [ABSTRACT FROM AUTHOR]

In its additive version, Bohr–Mollerup's remarkable theorem states that the unique (up to an additive constant) convex solution f(x) to the equation Δ f (x) = ln x on the open half-line (0 , ∞) is the log-gamma function f (x) = ln Γ (x) , where Δ denotes the classical difference operator and Γ (x) denotes the Euler gamma function. In a recently published open access book, the authors provided and illustrated a far-reaching generalization of Bohr–Mollerup's theorem by considering the functional equation Δ f (x) = g (x) , where g can be chosen from a wide and rich class of functions that have convexity or concavity properties of any order. They also showed that the solutions f(x) arising from this generalization satisfy counterparts of many properties of the log-gamma function (or equivalently, the gamma function), including analogues of Bohr–Mollerup's theorem itself, Burnside's formula, Euler's infinite product, Euler's reflection formula, Gauss' limit, Gauss' multiplication formula, Gautschi's inequality, Legendre's duplication formula, Raabe's formula, Stirling's formula, Wallis's product formula, Weierstrass' infinite product, and Wendel's inequality for the gamma function. In this paper, we review the main results of this new and intriguing theory and provide an illustrative application. [ABSTRACT FROM AUTHOR]

In this paper, we utilize some known Steffensen-type identities, obtained by using Lidstone's interpolating polynomial, to prove new generalizations of Steffensen's inequality. We obtain these new generalizations by using the weighted Hermite-Hadamard inequality for (2 n + 2) - convex and (2 n + 3) - convex functions. Further, the newly obtained inequalities can be observed as an upper- and lower-bound for utilized Steffensen-type identities. [ABSTRACT FROM AUTHOR]

This present paper is inspired by one of the questions posed by Okeke (Results Math 78(96):1-30, 2023, see Remark 2.10b). In particular, we aim to develop a robust computer code based on the theoretical results obtained in Okeke (2023), which determines the polynomial solutions of the following functional equation, 0.1 ∑ i = 1 n γ i F (a i x + b i y) = ∑ j = 1 m (α j x + β j y) f (c j x + d j y) , for all x , y ∈ R , γ i , α j , β j ∈ R , and a i , b i , c j , d j ∈ Q , and their special forms. The primary motivation for writing such a computer code is that solving even simple equations belonging to class (0.1) needs long and tiresome calculations. Therefore, one of the advantages of such a computer code is that it allows us to solve complicated problems quickly, easily, and efficiently. Additionally, the computer code will significantly improve the level of accuracy in calculations. Along with that, there is also the factor of speed. We point out that the computer code will operate with symbolic calculations provided by the programming language Python, which means that it does not contain any numerical or approximate methods, and it yields the exact solutions of the equations considered. [ABSTRACT FROM AUTHOR]

Motivated by a paper of Dragomir, we give new refinements for both discrete and integral Jensen inequalities using the Jensen's gap. As applications, we give refinements of various inequalities verifiable by Jensen's inequality. Topics covered: norms, quasi-arithmetic means, Hölder's inequality and f-divergences in information theory. [ABSTRACT FROM AUTHOR]

In the paper, we will present some important results from the theory of vector-valued invariant means. Discussing the results contained in the author's paper Badora (Ann Pol Math 58:147–159, 1993), we will add new facts that were previously not published. [ABSTRACT FROM AUTHOR]

In this paper, we consider the affine variety codes obtained evaluating the polynomials b y = a k x k + ... + a 1 x + a 0 , b , a i ∈ F q r , at the affine F q r -rational points of the Norm-Trace curve. In particular, we investigate the weight distribution and the set of minimal codewords. Our approach, which uses tools of algebraic geometry, is based on the study of the absolute irreducibility of certain algebraic varieties. [ABSTRACT FROM AUTHOR]

In this research paper, we investigate a generalization of Vincze's type functional equations involving several (up to four) unknown functions in connection with the maximum functional equation as max { ψ (x y) , ψ (x y - 1) } = ψ (x) η (y) + ψ (y) , max { ψ (x y) , ψ (x y - 1) } = ψ (x) η (y) + χ (y) , max { ψ (x y) , ψ (x y - 1) } = ϕ (x) η (y) , max { ψ (x y) , ψ (x y - 1) } = ϕ (x) η (y) + χ (y) , where G is an arbitrary group, x , y ∈ G , and ψ , η , χ , ϕ : G → R are unknown functions. [ABSTRACT FROM AUTHOR]

A double Roman dominating function (DRDF) on a graph G = (V , E) is a function f : V → { 0 , 1 , 2 , 3 } having the property that if f (v) = 0 , then vertex v must have at least two neighbors assigned 2 under f or one neighbor w with f (w) = 3 , and if f (v) = 1 , then vertex v must have at least one neighbor w with f (w) ≥ 2 . The weight of a DRDF is the sum of its function values over all vertices, and the double Roman domination number γ dR (G) is the minimum weight of a DRDF on G. Khoeilar et al. (Discrete Appl. Math. 270:159–167, 2019) proved that if G is a connected graph of order n with minimum degree two different from C 5 and C 7 , then γ dR (G) ≤ 11 10 n. Moreover, they presented an infinite family of graphs G attaining the upper bound, and conjectured that G is the only family of extremal graphs reaching the bound. In this paper, we disprove this conjecture by characterizing all extremal graphs for this bound. [ABSTRACT FROM AUTHOR]

Although there are plentiful results on the dynamics of monotone mappings, the problem becomes difficult in the non-monotone case. For PM functions (i.e., for piecewise monotone functions), it is known that the non-monotonicity height is an important index to describe their complexity, and the dynamical properties for such functions are complicated when the non-monotonicity height is infinity. In this paper, we consider the non-monotonity height for a general PM function, and give a classification for the non-monotonicity height of those functions. Our results present a full description of their complexity under iteration. [ABSTRACT FROM AUTHOR]

In this paper we study the achromatic arboricity of the complete graph. This parameter arises from the arboricity of a graph as the achromatic index arises from the chromatic index. The achromatic arboricity of a graph G, denoted by A α (G) , is the maximum number of colors that can be used to color the edges of G such that every color class induces a forest but any two color classes contain a cycle. In particular, if G is a complete graph we prove that 1 4 n 3 2 - c 1 n ≤ A α (G) ≤ 1 2 n 3 2 - c 2 n where c 1 and c 2 are constants such that 0 < c 1 , c 2 < 1 2 . [ABSTRACT FROM AUTHOR]

In this paper points of lower strict monotonicity, upper strict monotonicity, lower local uniform monotonicity and upper local uniform monotonicity of Orlicz function spaces equipped with a Mazur–Orlicz F-norm and generated by a monotone (non-convex in general) Orlicz function are studied. We show that both the necessary and sufficient conditions and the suitable theorems are formulated in full generality. Moreover, lower (upper) strict monotonicity and lower (upper) local uniform monotonicity theorems of the paper https://doi.org/10.1007/s00010-018-0615-y are revealed as corollaries in this paper. [ABSTRACT FROM AUTHOR]

In Szirmai (Ars Math Contemp 16:349–358, 2019) we proved that to each saturated congruent hyperball packing there exists a decomposition of the 3-dimensional hyperbolic space H 3 into truncated tetrahedra. Therefore, in order to get a density upper bound for hyperball packings, it is sufficient to determine the density upper bound of hyperball packings in truncated simplices. In this paper we prove, using the above results and results of the papers Miyamoto (Topology 33(4): 613–629, 1994) and Szirmai (Mat Vesn 70(3): 211–221, 2018), that the density upper bound of the saturated congruent hyperball (hypersphere) packings related to the corresponding truncated tetrahedron cells is realized in regular truncated tetrahedra with density ≈ 0.86338 . Furthermore, we prove that the density of locally optimal congruent hyperball arrangement in a regular truncated tetrahedron is not a monotonically increasing function of the height (radius) of the corresponding optimal hyperball, unlike the ball (sphere) and horoball (horosphere) packings. [ABSTRACT FROM AUTHOR]

In the paper there are two kinds of functional symmetry considered, i.e., (1)-power symmetry and (- 1) -power symmetry, in a rich subfamily B of the family of all complex valued functions on a symmetric set G ⊂ C . The first one defines the values f (- z) as identical with f(z) and the other as the inverse of the f(z) in G . The announced notions allow a unique decomposition of functions f ∈ B into a product of two factors f 1 , f - 1 having the (1)- and the (- 1) -power symmetry property, respectively. In the paper there are also given examples of such partitions f = f 1 · f - 1 , for various f ∈ B , a solution of the problem of invariance of the above functional symmetries with respect to some one- and two-argument function operations and two applications of the mentioned function decomposition. [ABSTRACT FROM AUTHOR]

In this paper we consider generalized monomial functions f , g : F → C (of possibly different degree) that also fulfill f (P (x)) = Q (g (x)) x ∈ F , where P ∈ F [ x ] and Q ∈ C [ x ] are given (classical) polynomials. [ABSTRACT FROM AUTHOR]

In this paper we consider the problem of finding iterative roots of order less than the number of forts of continuous piecewise monotone(PM) functions with nonmonotonicity height greater than one. We present sufficient conditions to extend iterative roots of PM functions from the characteristic interval which determines the behavior of the function under iteration. [ABSTRACT FROM AUTHOR]

In this paper, a computer program developed in the computer algebra system Maple is presented, which investigates alienness and strong alienness of linear functional equations. [ABSTRACT FROM AUTHOR]

In the present paper we prove a generalized version of the famous decomposition theorem of Ng. We also focus on the problem posed by Zsolt Páles concerning the Wright-convex functions. [ABSTRACT FROM AUTHOR]

László Székelyhidi, a mathematician, was born in Budapest, Hungary in 1952. He initially wanted to pursue a career in music but ultimately decided to study mathematics at Loránd Eötvös University in Budapest. After facing challenges during his entrance exam due to political factors, he redirected his application to Lajos Kossuth University in Debrecen. He continued to play music throughout his university years and became involved in research, publishing papers and attending international conferences. He held various teaching positions and received numerous awards for his research. Székelyhidi's sons also became mathematicians, and he continues to pursue his passion for music. [Extracted from the article]

In this paper we will study a connection between some generalization of AC-subspaces, vector valued invariant λ -means and λ -complementability. [ABSTRACT FROM AUTHOR]

Let X be an abelian metric group with an invariant metric, Y be a real normed space and K be a convex cone in Y. We prove that a K-subadditive (K-superadditive) compact- and convex-valued map F : X → C C (Y) , for which the functionals f y ∗ (x) = inf y ∗ (F (x)) are lower (upper, resp.) semicontinuous for any real continuous and non-negative on K functional y ∗ , has to be locally K-bounded on X. Our results refer to the papers Banakh and Jabłońska (Israel J Math 230:361–386, 2019), Jabłońska and Nikodem (Math Inequal Appl 22:1081–1089, 2019) and Nikodem (Aequationes Math 62:175–183, 2001). [ABSTRACT FROM AUTHOR]

In the second part of his fifth problem Hilbert asks for functional equations "In how far are the assertions which we can make in the case of differentiable functions true under proper modifications without this assumption." In the case of the general functional equation f (x) = h (x , y , (g 1 (x , y)) , ... , (g n (x , y))) for the unknown function f under natural condition for the given functions it is proved on compact manifolds that f ∈ C - 1 implies f ∈ C ∞ and practically the general case can also be treated. The natural conditions imply that the dimension of x cannot be larger than the dimension of y. If we remove this condition, then we have to add another condition. In this survey paper a new problem for this second case is formulated and results are summarised for both cases. [ABSTRACT FROM AUTHOR]

In this paper we give necessary conditions for quadratic functions f : R → R that satisfy the additional equation y 2 f (x) = x 2 f (y) under the condition x y = 1 . [ABSTRACT FROM AUTHOR]

In Shulman (J Funct Anal 263:1468–1484, Theorem 2.2) the author estimated the cardinality of the global image of a subadditive map from a group to an arbitrary power set and applied the result to functional equations of Levi-Civita type (the relations of this result to covering a group by subgroups will be shortly discussed in the present paper). Here we will establish the metric stability of this theorem. Namely it will be shown that if a set-valued map F from a group G to a metric space (X , d) satisfies the condition d F (g h) , F (g) ∪ F (h) < δ , for any g , h ∈ G and some δ > 0 , and the cardinality of each F(g) does not exceed n ∈ N , then the set F (G) : = ∪ g ∈ G F (g) can be covered by n (n + 3) / 2 balls of radius (3 · 6 n - 1 - 1) δ . [ABSTRACT FROM AUTHOR]

In this paper we will study Hyers-Ulam stability for a general linear partial differential equation of first order in a Banach space. [ABSTRACT FROM AUTHOR]

The author, together with M. Veerapazham, S. Wang and W. Zhang, considered the existence and uniqueness of continuous solutions to an iterative equation involving the multiplication of iterates in Gopalakrishna et al. (Sci. China Math., to appear). In this paper we continue to investigate this equation for differentiable solutions. Similarly to continuous solutions until Gopalakrishna et al. (Sci. China Math., to appear), no result on differentiable solutions of this equation on non-compact intervals of R has been obtained until now. Although our strategy here is to use the logarithmic conjugation to reduce the equation to the well-known polynomial-like iterative equation, all known results on differentiable solutions of the latter are given on compact intervals. We revisit polynomial-like iterative equations on the whole of R and use the Banach contraction principle to prove the existence and uniqueness of differentiable solutions to our equation on R + . We then technically extend our results on R + to R - and show that none of the pairs of solutions obtained on R + and R - can be combined at the origin to obtain even a continuous solution of the equation on the whole R . [ABSTRACT FROM AUTHOR]

In a category C with an (E , M )-factorization structure for morphisms, we prove that any subclass N of M which is closed under pullbacks determines a transitive quasi-uniform structure on C . In addition to providing a categorical characterization of all transitive quasi-uniform structures compatible with a topology, this result also permits us to establish a number of Galois connections related to quasi-uniform structures on C . These Galois connections lead to the description of subcategories of C determined by quasi-uniform structures. Several examples considered at the end of the paper illustrate our results. [ABSTRACT FROM AUTHOR]

Let S be a monoid (= semigroup with identity), and let σ : S → S be a homomorphism such that σ ∘ σ = i d . In an earlier paper we solved the Pexiderized d'Alembert functional equation (PDFE) f (x y) + g (σ (y) x) = h (x) k (y) for unknown f , g , h , k : S → C , assuming that S is either regular or generated by its squares and that one of the unknown functions is central. The present paper has two main results. The first describes the solutions of PDFE on a general monoid in terms of multiplicative functions, solutions of a special case of the sine subtraction law, and solutions of other functional equations with just one unknown function. The second main result uses the first one to give a more detailed solution of PDFE on a larger class of monoids than has been treated previously. We also find the continuous solutions on topological monoids. Examples are given to illustrate the results. [ABSTRACT FROM AUTHOR]

We investigate the functional equation f (p (x)) = q (f (x)) where p and q are given real functions. In the paper "On solvability of f (p (x)) = q (f (x)) for given real functionsp, q, Aequat. Math. 90 (2016), 471 - 494", we solved the problem of the solvability of f (p (x)) = q (f (x)) under the assumption that p, q are strictly increasing continuous real functions. Now, we extend the solutions of this problem for any strictly monotonous continuous real functions p, q. Thereby, we use the methods of the just mentioned paper. Further, we present computations of the so called characteristics of the given functions p, q using the results of this paper and, finally, present a quite short algorithm with input p, q and output 'solvable/not solvable'. [ABSTRACT FROM AUTHOR]

Consider a real vector space denoted as X, and let cc(Y) represent the collection of all convex and compact subsets of a real Hausdorff topological vector space Y. This paper investigates set-valued solutions of the Pexiderized quadratic functional equation f1(x+y)+f2(x-y)=f3(x)+f4(y),\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\begin{aligned} f_1(x+y)+f_2(x-y)=f_3(x)+f_4(y), \end{aligned}$$\end{document}for unknown functions f1,f2,f3,f4:X→cc(Y)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$f_1,f_2,f_3,f_4:X\rightarrow cc(Y)$$\end{document}. This functional equation incorporates many functional equations including the quadratic, Cauchy’s and Drygas’ equations. A characterization for set-valued solutions of this functional equation is presented in this paper. [ABSTRACT FROM AUTHOR]

The Hyers-Ulam stability of the iterative equation f n = F for continuous functions F was studied under the assumptions that F is a homeomorphism on its range, and the equation has stability on its range. It is important to study the stability of the equation for homeomorphisms on intervals. In this paper, theorems on stability are obtained using the properties of monotonic approximate solutions. The method is based on the stability of two derived iterative equations. [ABSTRACT FROM AUTHOR]

Investigating the future value F(K, s, t) of a capital K invested between dates s and t, the "natural" condition F (K , s , t) ≥ K has lost its naturality because of the strange fact of negative interest rates. This leads to the task of describing the possible solutions of the multiplicative Sincov equation f (s , u) = f (s , t) f (t , u) for s ≤ t ≤ u where f (s , t) = 0 may happen. In this paper we solve this task and discuss connections to the theory of investments. [ABSTRACT FROM AUTHOR]

In this paper, we determine the central solutions f : G × G → C of the functional equation f (x 1 σ y 1 , x 2 τ y 2) - f (x 1 σ y 1 , x 2) - f (x 1 , x 2 τ y 2) = f (x 1 y 1 , x 2 y 2) - f (x 1 y 1 , x 2) - f (x 1 , x 2 y 2) for all x 1 , x 2 , y 1 , y 2 ∈ G , where G is a group, σ , τ : G → G are involutions. [ABSTRACT FROM AUTHOR]

Let G be a graph with vertex set V(G) and edge set E(G), and let f be an integer-valued function defined on V(G). It is proved in this paper that every bipartite (0 , m f - m + 1) -graph has a (0, f)-factorization randomly r-orthogonal to n vertex-disjoint mr-subgraphs of G, which is a generalization of the known result with n = 1 given by Zhou and Wu. [ABSTRACT FROM AUTHOR]

The equality problem of two-variable Bajraktarević means can be expressed as the functional equation f g - 1 ( f (x) + f (y) g (x) + g (y) ) = h k - 1 ( h (x) + h (y) k (x) + k (y) ) (x , y ∈ I) , where I is a nonempty open real interval, f , g , h , k : I → R are continuous functions, g, k are positive and f/g, h/k are strictly monotone. This functional equation, for the first time, was solved by Losonczi in 1999 under 6th-order continuous differentiability assumptions. Additional and new characterizations of this equality problem have been found recently by Losonczi, Páles and Zakaria under the same regularity assumptions in 2021. In this paper it is shown that the same conclusion can be obtained under substantially weaker regularity conditions, namely, assuming only first-order differentiability. [ABSTRACT FROM AUTHOR]

Let f : c 0 + ∪ - c 0 + → Y be a standard ε -isometry, where c 0 + is the positive cone of c 0 and Y is a Banach space. In this paper, we study stability properties of f and its symmetrization Θ = f (·) - f (- ·) 2 . We show that every standard isometry f is always stable. If f is a standard ε -isometry for ε > 0 , then f is stable for every separable space Y, i.e. there exists a bounded linear operator T : Y → c 0 with ‖ T ‖ ≤ 2 such that ‖ T f (x) - x ‖ ≤ 6 ε , for all x ∈ c 0 + ∪ - c 0 +. If f (c 0 + ∪ - c 0 +) contains a reproducing wedge of Y, then there exists a linear surjective isometry U : c 0 → Y such that f - U is uniformly bounded by 2 ε on c 0 + ∪ - c 0 + . We also obtain a series of new sharp stability results for Θ . [ABSTRACT FROM AUTHOR]

Spectral analysis and synthesis studies translation invariant function spaces, so-called varieties over topological groups. The basic building blocks are the finite dimensional varieties. In the commutative case finite dimensional varieties are spanned by exponential polynomials. In non-commutative situations no relevant results exist. In this paper we consider finite dimensional left translation invariant linear spaces of continuous complex valued functions over the Heisenberg group. Using basic knowledge about Lie algebra we describe all left varieties of this type. In particular, it turns out that those function spaces are spanned by exponential polynomials as well. [ABSTRACT FROM AUTHOR]

In this paper, we present an application of the viscosity approximation type iterative method introduced by Nandal et al. (Iteration Process for Fixed Point Problems and Zeros of Maximal Monotone Operators, Symmetry, 2019) to visualize and analyse the Julia and Mandelbrot sets for a complex polynomial of the type T (z) = z n + p z + r , where p , r ∈ C , and n ≥ 2 . This iterative method has many applications in solving various fixed point problems. We derive an escape criterion to visualize Julia and Mandelbrot sets via the proposed viscosity approximation type method. Moreover, we present several graphical examples of the fractals generated with the proposed iteration method. [ABSTRACT FROM AUTHOR]

In the present paper we describe the family of all closed convex plane curves of class C 1 which have circles as their isoptics. In the first part of the paper we give a certain characterization of all ellipses based on the notion of isoptic and we give a geometric characterization of curves whose orthoptics are circles. The second part of the paper contains considerations on curves which have circles as their isoptics and we show the form of support functions of all considered curves. [ABSTRACT FROM AUTHOR]

The aim of this paper is to fill in the gaps in the formulation and the proof of a theorem contained in the paper by K.Ozeki (Aequ Math 25:247–252, 1982) published in this journal. We also give a short proof of this theorem and use it to obtain certain information about the factorization of polynomials of the form f (x) - f (y) . [ABSTRACT FROM AUTHOR]

A sequence (v 1 , ... , v k) of vertices in a graph G without isolated vertices is called a total dominating sequence if every vertex v i in the sequence totally dominates at least one vertex that was not totally dominated by { v 1 , ... , v i - 1 } and { v 1 , ... , v k } is a total dominating set of G. The length of a shortest such sequence is the total domination number of G ( γ t (G) ), while the length of a longest such sequence is the Grundy total domination number of G ( γ gr t (G) ). In this paper we study graphs with equal total and Grundy total domination numbers. We characterize bipartite graphs with both total and Grundy total dominations number equal to 4, and show that there is no connected chordal graph G with γ t (G) = γ gr t (G) = 4 . The main result of the paper is a characterization of bipartite graphs with γ t (G) = γ gr t (G) = 6 proved by establishing a surprising correspondence between the existence of such graphs and a classical but still open problem of the existence of certain finite projective planes. [ABSTRACT FROM AUTHOR]