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Abstract: Abstract In a category \({\mathcal {C}}\) with an ( \({\mathcal {E}}\) , \({\mathcal {M}}\) )-factorization structure for morphisms, we prove that any subclass \({\mathcal {N}}\) of \({\mathcal {M}}\) which is closed under pullbacks determines a transitive quasi-uniform structure on \({\mathcal {C}}\) . 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 \({\mathcal {C}}\) . These Galois connections lead to the description of subcategories of \({\mathcal {C}}\) determined by quasi-uniform structures. Several examples considered at the end of the paper illustrate our results. PubDate: 2023-01-11
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Abstract: Abstract Investigating the future value F(K, s, t) of a capital K invested between dates s and t, the “natural” condition \(F(K,s,t)\ge K\) 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 \(s\le t\le u\) where \(f(s,t)=0\) may happen. In this paper we solve this task and discuss connections to the theory of investments. PubDate: 2023-01-03
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Abstract: Abstract We provide a list of equivalent conditions under which an additive operator acting on a space of smooth functions on a compact real interval is a multiple of the derivation. PubDate: 2022-12-30
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Abstract: Abstract 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. PubDate: 2022-12-27
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Abstract: Abstract This article establishes a few sufficient conditions of the forward-order law for the core inverse of elements in rings with involution. It also presents the forward-order law for the weighted core inverse and the triple forward-order law for the core inverse. Additionally, we discuss the hybrid forward-order law involving different generalized inverses like the Moore–Penrose inverse, the group inverse, and the core inverse. PubDate: 2022-12-26
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Abstract: Abstract Probably the best strategy to give a computer-free proof of the 4-color theorem is to show that the chromatic polynomial of any planar graph evaluated at 4 is nonzero. After making assumptions like no danglers and adding edges if necessary, a connected graph is a ”web” of ”interlocking wheels”. If G is a web of k interlocking wheels obtained from a planar graph, it is conjectured that its chromatic polynomial \(P(G,t) = P_1(t)+ \cdots + P_k(t)\) with all \(P_j(4) >0\) . In this paper, we prove the conjecture in the base case of the interlocking wheels \(W_m\wedge _2W_n\) for \(k = 2\) . As a byproduct, in the last section we apply our result to exponentially simplify the computations to find some interesting facts about a real life graph. PubDate: 2022-12-17
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Abstract: Abstract We consider the sequence of integers whose nth term has base-p expansion given by the nth row of Pascal’s triangle modulo p (where p is a prime number). We first present and generalize well-known relations concerning this sequence. Then, with the great help of Sloane’s On-Line Encyclopedia of Integer Sequences, we show that it appears naturally as a subsequence of a 2-regular sequence. Its study provides interesting relations and surprisingly involves odious and evil numbers, Nim-sum and even Gray codes. Furthermore, we examine similar sequences emerging from prime numbers involving alternating sum-of-digits modulo p. This note ends with a discussion about Pascal’s pyramid built with trinomial coefficients. PubDate: 2022-12-15
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Abstract: Abstract In this paper, we study the transform by Stirling numbers with higher level, and give several concrete results. When \(s=2\) , we consider the transform of rational sequences. In particular, poly-Cauchy numbers of the second kind with level 2 are introduced in order to achieve some extended results. We also give several properties of poly-Cauchy numbers of the second kind with level 2, which are related to those of poly-Bernoulli numbers with level 2 and analogous to those of poly-Cauchy numbers of the first kind with level 2. PubDate: 2022-12-11
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Abstract: Abstract A graceful labeling of a graph G is an injective function from the vertex set of G to the set \(\{0,1,\dots , E(G) \}\) such that the induced edge labels are all different, where an induced edge label is defined as the absolute value of the difference between the labels of its end vertices. If the induced edge labeling is simultaneously antimagic, i.e., the sums of labels of all edges incident to a given vertex are pairwise distinct for different vertices, we say that the graceful labeling is graceful antimagic. In this paper we deal with the problem of finding some classes of graceful antimagic graphs. PubDate: 2022-12-08
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Abstract: Abstract 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 \((0,mf-m+1)\) -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. PubDate: 2022-12-05
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Abstract: Abstract Let S be a monoid ( \(=\) semigroup with identity), and let \(\sigma :S \rightarrow S\) be a homomorphism such that \(\sigma \circ \sigma = id\) . In an earlier paper we solved the Pexiderized d’Alembert functional equation (PDFE) \(f(xy) + g(\sigma (y)x) = h(x)k(y)\) for unknown \(f,g,h,k:S \rightarrow {{\mathbb {C}}}\) , 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. PubDate: 2022-12-01
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Abstract: Abstract This article introduces the novel notion of dimension preserving approximation for continuous functions defined on [0, 1] and initiates the study of it. Restrictions and extensions of continuous functions in regards to fractal dimensions are also investigated. PubDate: 2022-12-01
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Abstract: Abstract Let S be a semigroup that need not be abelian, \(\sigma :S\rightarrow S\) be an involutive endomorphism and let \((H,+)\) be a uniquely 2-divisible abelian group. We study the solutions \(f,g:S\rightarrow H\) of each of the functional equations $$\begin{aligned} f(xy)+g(xy)+g(x\sigma (y))=f(x)+f(y)+2g(x)+2g(y), \end{aligned}$$ and $$\begin{aligned} f(xy)+g(xy)+g(x\sigma (y))=f(x)+f(y)+2g(x)+g(y)+g(\sigma (y)). \end{aligned}$$ Moreover, we show that the Jensen type equation \(f(xy)+f(x\sigma (y))=2f(x)\ \) and the generalized quadratic functional equation \(g(xy)+g(x\sigma (y))=2g(x)+2g(y)\) are strongly alien in the sense of Dhombres. PubDate: 2022-12-01
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Abstract: Abstract In this paper we introduce the concept of translation invariant functions: considering an arbitrary set \(\emptyset \ne S \subset {\mathbb {R}}^n\) , the function \(F : S \longrightarrow {\mathbb {R}}\) is translation invariant if \(F(x) = F(y)\) implies \(F(x+t)=F(y+t)\) for any vectors \(x,y,t \in {\mathbb {R}}^n\) such that \(x ,y , x+t ,y+t \in S\) . In our main results we shall consider an open, connected set \(\emptyset \ne D \subset {\mathbb {R}}^n\) . We prove that if \(F : D \longrightarrow {\mathbb {R}}\) is a translation invariant, continuous function, then there exists a vector \(a = (a_1, \dots , a_n) \in {\mathbb {R}}^n\) and a strictly monotone, continuous function f such that $$\begin{aligned} F(x_1, \dots , x_n) = f (a_1 x_1 + \dots + a_n x_n) \end{aligned}$$ holds for all \((x_1, \dots , x_n) \in D \,\) . Using this result we also show that continuous solutions \( F : D \longrightarrow {\mathbb {R}}\) of the system of functional equations $$\begin{aligned} F(x_1 , \dots , x_j + t_j , \dots , x_n) = \Psi _j (F (x_1 , \dots , x_j , \dots , x_n), t_j) \ \ (j=1,\dots ,n) \end{aligned}$$ can be represented as the composition of a strictly monotone, continuous function and a linear functional as well. Applying the latter theorem, we give a characterization of Cobb–Douglas type utility functions. PubDate: 2022-12-01
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Abstract: Abstract In the present paper we consider the problem of characterization of maps which can be expressed as an affine difference i.e. a three-place map of the form $$\begin{aligned} tf(x)+(1-t)f(y)-f(tx+(1-t)y). \end{aligned}$$ We give a general solution of a functional equation associated with this problem. PubDate: 2022-12-01
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Abstract: Abstract We give a characterization of continuous functions in a complex plane domain satisfying a weighted polygonal mean value property. PubDate: 2022-12-01
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Abstract: Abstract In this paper we prove the following result: Let R be a prime ring with \(char(R)\ne 2,3,5\) and let \(T:R\rightarrow R\) be an additive mapping satisfying the relation \( T(x^{4})=xT(x^2)x\) for all \(x\in R\) . In this case T is of the form \(T(x)=\lambda x\) for all \(x\in R\) and some fixed element \(\lambda \in C\) , where C is the extended centroid of R. PubDate: 2022-12-01
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Abstract: Abstract Being a universal space for weight \(\left A\right \ge \aleph _{0}\) metric spaces Lipscomb’s space \(J_{A}\) has a central role in topological dimension theory. There exists a strong connection between topological dimension theory and fractal set theory since on the one hand, some classical fractals play the role of universal spaces and on the other hand the universal space \(J_{A}\) is a generalized Hutchinson–Barnsley fractal (i.e. the attractor of a possibly infinite iterated function system). In this paper we introduce a generalization of \(J_{A}\) , namely the concept of graph Lipscomb’s space \(J_{A}^{{\mathcal {G}}}\) associated with a graph \({\mathcal {G}}\) on the set A, and we prove that its imbedded version in \(l^{2}(A^{\prime })\) , where \(A^{\prime }=A\setminus \{z\}\) , z being a fixed element of the set A having at least two elements, is a generalized Hutchinson–Barnsley fractal. PubDate: 2022-10-29 DOI: 10.1007/s00010-022-00918-x
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Abstract: Abstract In this paper, we establish some inequalities involving both \(h_1\) -convex functions and \(h_2\) -concave functions such that \( (h_1,h_2) \) is an increasing pair of functions. We apply the obtained results to Breckner’s s-convex functions, P-convex functions and Godunova–Levin functions. In order to unify the results, we introduce generalized s-convex functions of the second kind. PubDate: 2022-10-18 DOI: 10.1007/s00010-022-00915-0
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