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Let D = (V , A) be a digraph. For an integer k ≥ 1 , a k-arc-connected flip is an arc subset of D such that after reversing the arcs in it the digraph becomes (strongly) k-arc-connected. The first main result of this paper introduces a sufficient condition for the existence of a k-arc-connected flip that is also a submodular flow for a crossing submodular function. More specifically, given some integer τ ≥ 1 , suppose d A + (U) + (τ k - 1) d A - (U) ≥ τ for all U ⊊ V , U ≠ ∅ , where d A + (U) and d A - (U) denote the number of arcs in A leaving and entering U, respectively. Let C be a crossing family over ground set V, and let f : C → Z be a crossing submodular function such that f (U) ≥ k τ (d A + (U) - d A - (U)) for all U ∈ C . Then D has a k-arc-connected flip J such that f (U) ≥ d J + (U) - d J - (U) for all U ∈ C . The result has several applications to Graph Orientations and Combinatorial Optimization. In particular, it strengthens Nash-Williams' so-called weak orientation theorem, and proves a weaker variant of Woodall's conjecture on digraphs whose underlying undirected graph is τ -edge-connected. The second main result of this paper is even more general. It introduces a sufficient condition for the existence of capacitated integral solutions to the intersection of two submodular flow systems. This sufficient condition implies the classic result of Edmonds and Giles on the box-total dual integrality of a submodular flow system. It also has the consequence that in a weakly connected digraph, the intersection of two submodular flow systems is totally dual integral. [ABSTRACT FROM AUTHOR]

This paper is concerned with the periodic solutions for a coupled system of wave equations with x-dependent coefficients. Such a model arises naturally when two waves propagate simultaneously in the nonisotrpic media. In this paper, for the periods having the form T = 2 a − 1 b (a , b are positive integers) and some types of boundary conditions, we obtain the existence of the time periodic solutions and analyze the asymptotic behaviors as the coupled parameter goes to zero, when the nonlinearities are superlinear and monotone, by using the variational method. In particular, the condition ess inf ηϱ(x) > 0 is not required. [ABSTRACT FROM AUTHOR]

The question of when a vertex algebra is a quantization of the arc space of its associated scheme has recently received a lot of attention in both the mathematics and physics literature. This property was first studied by Tomoyuki Arakawa and Anne Moreau (see their paper in the references), and was given the name \lq\lq classical freeness" by Jethro van Ekeren and Reimundo Heluani [Comm. Math. Phys. 386 (2021), no. 1, pp. 495-550] in their work on chiral homology. Later, it was extended to vertex superalgebras by Hao Li [Eur. J. Math. 7 (2021), pp. 1689–1728]. In this note, we prove the classical freeness of the simple affine vertex superalgebra L_n(\mathfrak {o}\mathfrak {s}\mathfrak {p}_{m|2r}) for all positive integers m,n,r satisfying -\frac {m}{2} + r +n+1 > 0. In particular, it holds for the rational vertex superalgebras L_n(\mathfrak {o}\mathfrak {s}\mathfrak {p}_{1|2r}) for all positive integers r,n. [ABSTRACT FROM AUTHOR]

The primary aim of this paper is to investigate the asymptotic distribution of the zeros of certain classes of hypergeometric $ {}_{q+1}F_{q} $ q + 1 F q polynomials. We employ classical analytical techniques, including Watson's lemma and the method of steepest descent, to understand the asymptotic behavior of these polynomials: $$\begin{align*} & _{q+1}F_{q}\left(-n,kn+\alpha,\ldots, kn+\alpha+\frac{q-1}{q};kn+\beta,\ldots,kn+\beta+\frac{q-1}{q};z\right)\\ &\quad (n\rightarrow \infty), \end{align*} $$ q + 1 F q (− n , kn + α , ... , kn + α + q − 1 q ; kn + β , ... , kn + β + q − 1 q ; z) (n → ∞) , where n is a nonnegative integer, q is a positive integer and the constant parameters α and β are constrained by $ \alpha { α < β. By applying the general results established in this paper, we generate numerical evidence and graphical illustrations using Mathematica to show the clustering of zeros on certain curves. [ABSTRACT FROM AUTHOR]

This paper introduces three novel approaches to size geothermal energy piles in a MILP, offering fresh perspectives and potential solutions. The research overlooks MILP models that incorporate the sizing of a geothermal borefield. Therefore, this paper presents a new model utilizing a g-function model to regulate the power limits. Geothermal energy is an essential renewable source, particularly for heating and cooling. Complex energy systems, with their diverse sources of heating and cooling and intricate interactions, are crucial for a climate-neutral energy system. This work significantly contributes to the integration of geothermal energy as a vital energy source into the modelling of such complex systems. Borehole heat exchangers help generate heat in low-temperature energy systems. However, optimizing these exchangers using mixed-integer-linear programming (MILP), which only allows for linear equations, is complex. The current research only uses R-C, reservoir, or g-function models for pre-sized borefields. As a result, borehole heat exchangers are often represented by linear factors such as 50 W/m for extraction or injection limits. A breakthrough in the accuracy of borehole heat exchanger sizing has been achieved with the development of a new model, which has been rigorously compared to two simpler models. The geothermal system was configured for three energy systems with varying ground and bore field parameters. The results were then compared with existing geothermal system tools. The new model provides more accurate depth sizing with an error of less than 5 % compared to simpler models with an error higher than 50 %, although it requires more calculation time. The new model can lead to more accurate borefield sizing in MILP applications to optimize energy systems. This new model is especially beneficial for large-scale projects that are highly dependent on borefield size. [ABSTRACT FROM AUTHOR]

Consider a simple graph G without K2 component with vertex set V and edge set E. A local antimagic labeling f of G is a one-to-one mapping of edges to distinct positive integers 1, 2, ..., |E| such that the weights of adjacent vertices are distinct, where weight of a vertex is sum of labels assigned to the edges incident to it. These weights of the vertex induced by local antimagic labeling result in a proper vertex coloring of the graph G. The local antimagic chromatic number of G, denoted as χla(G), as the smallest number of distinct weights obtained across all possible local antimagic labelings of G. In this paper, we explore the local antimagic chromatic numbers of various classes of graphs, including the union of certain graph families, the corona product of graphs, and the necklace graph. In addition, we provide constructions for infinitely many graphs for which χla(G) equals the chromatic number χ(G) of the graph. [ABSTRACT FROM AUTHOR]

We shall give an elementary proof for Lemma 2.4 and correct some errors in Table 1 of the author's paper of the title. Moreover, we shall extend this table up to integers below 232. [ABSTRACT FROM AUTHOR]

Let n be a positive integer. An NFSR of n stages is called irreducible if the family of output sequences of any NFSR of stages less than n is not included in that of the NFSR. In this paper, we prove that the density of the irreducible NFSRs of n stages is larger than 0.39. This implies that it is expected to find an irreducible NFSR of n stages among three randomly chosen NFSRs of n stages. [ABSTRACT FROM PUBLISHER]

This paper presents a modified version of Leibman's group-theoretic generalizations of the difference calculus for polynomial maps from nonempty commutative semigroups to groups, and proves that it has many desirable formal properties when the target group is locally nilpotent and also when the semigroup is the set of nonnegative integers. We will apply it to solve Waring's problem for general discrete Heisenberg groups in a sequel to this paper. [ABSTRACT FROM AUTHOR]

The aim of this paper is to introduce and study the concept of a hyperideal-based zero-divisor graph associated with a general hyperring. This is a generalized version of the zero-divisor graph associated with a commutative ring. For any general hyperring R having a hyperideal I, the I-based zero-divisor graph Γ(I)(R) associated with R is the simple graph whose vertices are the elements of R∖I having their hyperproduct in I, and two distinct vertices are joined by an edge when their hyperproduct has a non-empty intersection with I. In the first part of the paper, we concentrate on some general properties of this graph related to absorbing elements, while the second part is dedicated to the study of the I-based zero-divisor graph associated to the general hyperring Zn of the integers modulo n, when n=2pmq, with p and q two different odd primes, and fixing the hyperideal I. [ABSTRACT FROM AUTHOR]

The first part of this work provides explicit solutions for two integral equations; both are solved by means of Fourier transform. In the second part of this paper, sufficient conditions for the existence and uniqueness of the solutions satisfying sandwich constraints for two types of full moment problems are provided. The only given data are the moments of all positive integer orders of the solution and two other linear, not necessarily positive, constraints on it. Under natural assumptions, all the linear solutions are continuous. With their value in the subspace of polynomials being given by the moment conditions, the uniqueness follows. When the involved linear solutions and constraints are positive, the sufficient conditions mentioned above are also necessary. This is achieved in the third part of the paper. All these conditions are written in terms of quadratic expressions. [ABSTRACT FROM AUTHOR]

This paper studies the distinctness of primitive sequences over \bf Z/(M) modulo 2, where M is an odd integer that is composite and square-free, and \bf Z/(M) is the integer residue ring modulo M. A new sufficient condition is given for ensuring that primitive sequences generated by a primitive polynomial f\left(x\right) over \bf Z/(M) are pairwise distinct modulo 2. Such result improves a recent result obtained in our previous paper, and consequently, the set of primitive sequences over \bf Z/(M) that can be proven to be distinct modulo 2 is greatly enlarged. [ABSTRACT FROM PUBLISHER]

This paper studies binary reverse multi-delimiter (RMD) data compression codes. RMD codes have a range of useful properties, such as unique decodability, completeness, universality, synchronizability, recognition using a finite automaton, and the ability for rapid data retrieval within an encoded file. The authors have constructed a simple monotonic mapping from the set of non-negative integers to the codeword set. Based on this mapping, they have developed a fast byte-aligned decoding algorithm. Computer experiments demonstrate that we can decode RMD codes nearly as quickly as the SCDC code and several times faster than the Fibonacci code. Compared to known codes of a similar type, RMD codes exhibit a better compression ratio for natural language texts (more than four times closer to the entropy bound than SCDC). Additionally, the paper describes a technology for preprocessing natural language texts, which, combined with encoding using RMD codes, enhances the efficiency of powerful modern archivers. [ABSTRACT FROM AUTHOR]

Let p ≥ 2 be a prime number and let a, b,m be positive integers such that p - m. In a recent paper [1], we discussed the maximal prime power pe, which divides the numerator of the fraction ... when written in reduced form. This short note may be regarded as an addendum to paper [1] for the case where p = 2, b = 1, m > 1 and 2a∥m - 1, which was left open. [ABSTRACT FROM AUTHOR]

The present paper considers a discrete analog of the parking problem. Let n be an integer. If n > 1, then we randomly locate an interval (t, t + 1) with integer endpoints on a segment [0, n]. Thus, the original segment is divided into two: [0, t] and [t + 1, n], and each of them is further considered separately likewise the original one. The phrase "randomly" in this problem means that t is a uniformly distributed on a set {1,... , n − 1} random variable. The process of location of the intervals finishes when the lengths of all the remaining intervals are less than 2. Define as Xn the total amount of the located intervals. In the present paper, the expectations {Xn} are calculated. The process described above can be interpreted as a parking process of cars with handlebars on the left. Hence, the driver is able to leave his car only if the place on his left is free. This is exactly the case when the driver cannot take the left end place of any free segment. In this case, Xn stands for the amount of the parked cars. [ABSTRACT FROM AUTHOR]

In this paper, we prove that the Diophantine equation 147x + 741y = z² has no non-negative integer solution. [ABSTRACT FROM AUTHOR]

In 2019, Burshtein [1] showed that the equation px+py = z² has no positive integer solution, except for p = 2, 3. For p = 2, all solutions of the equation in the Fibonacci and Lucas numbers was investigated by Hashim [2]. In this paper, we prove that (x, y, z) = (F5,L3,L6) is the unique solution of the equation, when p = 3. [ABSTRACT FROM AUTHOR]

This paper addresses the uniqueness problem concerning the j-th derivative of a meromorphic function f(z) and the k-th derivative of its shift, f(z + c), where j, k are integers with 0 ≤ j < k. In this regard, our work surpasses the achievements of [2], as we have improved upon the existing results and provided a more refined understanding of this specific aspect. We give some illustrative examples to enhance the realism of the obtained outcomes. Denote by E(a, f) the set of all zeros of f-a, where each zero with multiplicity m is counted m times. In the paper proved, in particular, the following statement: Let f(z) be a non-constant meromorphic function of finite order, c be a non-zero finite complex number and j, k be integers such that 0 ≤ j < k. If f(j)(z) and f(k)(z + c) have the same a-points for a finite value a(̸= 0) and satisfy conditions... [ABSTRACT FROM AUTHOR]

Our goal of this paper is to find a construction of all ℓ-quasi-cyclic self-dual codes over a finite field $ {\mathbb F}_q $ F q of length $ m\ell $ mℓ for every positive even integer ℓ. In this paper, we study the case where $ x^m-1 $ x m − 1 has an arbitrary number of irreducible factors in $ {\mathbb F}_q [x] $ F q [ x ] ; in the previous studies, only some special cases where $ x^m-1 $ x m − 1 has exactly two or three irreducible factors in $ {\mathbb F}_q [x] $ F q [ x ] , were studied. Firstly, the binary code case is completed: for any even positive integer ℓ, every binary ℓ-quasi-cyclic self-dual code can be obtained by our construction. Secondly, we work on the q-ary code cases for an odd prime power q. We find an explicit method for construction of all ℓ-quasi-cyclic self-dual codes over $ {\mathbb F}_q $ F q of length $ m\ell $ mℓ for any even positive integer ℓ, where we require that $ q \equiv 1 \pmod {4} $ q ≡ 1 (mod 4) if the index $ \ell \ge 6 $ ℓ ≥ 6. By implementation of our method, we obtain a new optimal binary self-dual code $ [172, 86, 24] $ [ 172 , 86 , 24 ] , which is also a quasi-cyclic code of index 4. [ABSTRACT FROM AUTHOR]

We study the square root computation by Leonardo Fibonacci (or Leonardo of Pisa) in his MSS Liber Abaci from c1202 and c1228 and De Practica Geometrie from c1220. In this MSS, Fibonacci systematically describes finding the integer part of the square root of an integer in numerous examples with three to seven decimal digits. The results of these examples are summarized in a table in the paper. Liber Abaci also describes in detail finding an approximation to the fractional part of the square root. However, in other examples in Liber Abaci and De Practica Geometrie, only the approximate values of the fractional part of the square roots are stated. This paper further explores these approximate values using techniques like reverse engineering. Contrary to many claims that Fibonacci also used other methods or approximations, we show that all examples can be explained using one digit-by-digit method to compute the integer part of the square root and one approximation scheme for the fractional part. Further, it is shown that the approximation scheme is tied to the method to compute the integer part of the square root. [ABSTRACT FROM AUTHOR]

In this paper we resolve a conjecture of Kresch and Tamvakis [Duke Math. J. 110 (2001), pp. 359–376]. Our result is the following. Theorem : For any positive integer D and any integers i,j (0\leq i,j \leq D), \; the absolute value of the following hypergeometric series is at most 1: \begin{equation*} {_4F_3} \left [ \begin {array}{c} -i, \; i+1, \; -j, \; j+1 \\ 1, \; D+2, \; -D \end{array} ; 1 \right ]. \end{equation*} To prove this theorem, we use the Biedenharn-Elliott identity, the theory of Leonard pairs, and the Perron-Frobenius theorem. [ABSTRACT FROM AUTHOR]

In this paper, we consider the problem of computing the entire sequence of the maximum degree of minors of a block-structured symbolic matrix (a generic partitioned polynomial matrix) A = (A α β x α β t d α β ) , where A α β is a 2 × 2 matrix over a field F , x α β is an indeterminate, and d α β is an integer for α = 1 , 2 , ⋯ , μ and β = 1 , 2 , ⋯ , ν , and t is an additional indeterminate. This problem can be viewed as an algebraic generalization of the maximum weight bipartite matching problem. The main result of this paper is a combinatorial -time algorithm for computing the entire sequence of the maximum degree of minors of a (2 × 2) -type generic partitioned polynomial matrix of size 2 μ × 2 ν . We also present a minimax theorem, which can be used as a good characterization (NP ∩ co-NP characterization) for the computation of the maximum degree of minors of order k. Our results generalize the classical primal-dual algorithm (the Hungarian method) and minimax formula (Egerváry's theorem) for the maximum weight bipartite matching problem. [ABSTRACT FROM AUTHOR]

Let O K = Z [ i ] . For each positive integer n , denote ξ K (n) as the number of integral ideals whose norm divides n in O K . In this paper, we studied the distribution of ideals whose norm divides n in O K by using the Selberg-Delange method. This is a natural variant of a result studied by Deshouillers, Dress, and Tenenbaum (often called the DDT Theorem), and we found that the distribution function was subject to beta distribution with density 3 / (2 π u 2 (1 − u) 3) . Let . For each positive integer , denote as the number of integral ideals whose norm divides in . In this paper, we studied the distribution of ideals whose norm divides in by using the Selberg-Delange method. This is a natural variant of a result studied by Deshouillers, Dress, and Tenenbaum (often called the DDT Theorem), and we found that the distribution function was subject to beta distribution with density . [ABSTRACT FROM AUTHOR]

Let n be an integer greater than 1 . In this paper, we obtained the stability of the multivariable Cauchy-Jensen functional equation n f (x 1 + ⋯ + x n , y 1 + ⋯ + y n n) = ∑ 1 ≤ i , j ≤ n f (x i , y j) in Banach spaces, quasi-Banach spaces, and normed 2 -Banach spaces. Let be an integer greater than . In this paper, we obtained the stability of the multivariable Cauchy-Jensen functional equation in Banach spaces, quasi-Banach spaces, and normed -Banach spaces. [ABSTRACT FROM AUTHOR]

This paper aims to explore the arithmetic properties of Fu's k dots bracelet partition where k = pα, p is a prime number with p ≥ 5 and α is an integer with α ≥ 0. For pα dots bracelet partitions with p = 5, 7 and 11, we found several exciting Ramanujan-like congruences modulo p. We also used Newman's theorems to demonstrate certain congruence modulo p. [ABSTRACT FROM AUTHOR]

Many papers in the field of integer linear programming (ILP, for short) are devoted to problems of the type max { c ⊤ x : A x = b , x ∈ Z ≥ 0 n } , where all the entries of A, b, c are integer, parameterized by the number of rows of A and ‖ A ‖ max . This class of problems is known under the name of ILP problems in the standard form, adding the word "bounded" if x ≤ u , for some integer vector u. Recently, many new sparsity, proximity, and complexity results were obtained for bounded and unbounded ILP problems in the standard form. In this paper, we consider ILP problems in the canonical form max { c ⊤ x : b l ≤ A x ≤ b r , x ∈ Z n } , where b l and b r are integer vectors. We assume that the integer matrix A has the rank n, (n + m) rows, n columns, and parameterize the problem by m and Δ (A) , where Δ (A) is the maximum of n × n sub-determinants of A, taken in the absolute value. We show that any ILP problem in the standard form can be polynomially reduced to some ILP problem in the canonical form, preserving m and Δ (A) , but the reverse reduction is not always possible. More precisely, we define the class of generalized ILP problems in the standard form, which includes an additional group constraint, and prove the equivalence to ILP problems in the canonical form. We generalize known sparsity, proximity, and complexity bounds for ILP problems in the canonical form. Additionally, sometimes, we strengthen previously known results for ILP problems in the canonical form, and, sometimes, we give shorter proofs. Finally, we consider the special cases of m ∈ { 0 , 1 } . By this way, we give specialised sparsity, proximity, and complexity bounds for the problems on simplices, Knapsack problems and Subset-Sum problems. [ABSTRACT FROM AUTHOR]

The singleton defect of an [n, k, d] linear code C is defined as s (C) = n - k + 1 - d . Codes with s (C) = s (C ⊥) = 1 are called near maximum distance separable (NMDS) codes. It is known that an [ n , 3 , n - 3 ] NMDS code is equivalent to an (n, 3)-arc in PG(2, q). In this paper, by adding some suitable projective points into some known (q + 5 , 3) -arcs in PG(2, q), we obtain two families of [ q + 7 , 3 , q + 4 ] NMDS codes for even prime power q and a family of [ q + 6 , 3 , q + 3 ] NMDS codes for odd prime power q. In addition, when q = 2 m and m is odd, by adding m suitable projective points into the maximum arcs in PG(2, q), we obtain a family of [ q + m + 2 , 3 , q + m - 1 ] NMDS codes over F q , from which we further induce a family of NMDS codes with parameters [ q t + m + 2 , 3 , q t + m - 1 ] over the extension field F q t for any odd integer t. All the resulting NMDS codes in this paper are shown to be linearly inequivalent to the NMDS codes constructed from elliptic curves, and their weight distributions are completely determined. [ABSTRACT FROM AUTHOR]

Let R be a ring and let n ≥ 1 be a fixed integer. An additive mapping h of a ring R into itself is called an n-Jordan homoderivation if h ⁢ (x n) = (h ⁢ (x) + x) n - x n holds for all x ∈ R . In this paper, we initiate the study of n-Jordan homoderivations on rings. Precisely, we characterize n-Jordan homoderivations in terms of homoderivations and anti-homomorphisms under certain conditions. Finally, we conclude our paper with a direction for further research. [ABSTRACT FROM AUTHOR]

Congruences mod p for a prime p and partial products of the numbers 1,..., p − 1 are obtained. Bibliography: 2 titles. [ABSTRACT FROM AUTHOR]