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This article is a follow-up contribution that extends the conference paper (Nxumalo, in: Laarman, Sokolova (eds)Model Checking Software - 27th International Symposium, 2021) which presented the spotlight abstraction method to enable an efficient model checking of schedulability properties of real-time operating systems. Our approach is based on iterative spotlight abstraction whereby the real-time task queue is divided into a so-called spotlight and a shade. Initially, an abstract state space model is generated from the spotlight, which contains a small number of tasks that appear at the front of the queue and the shade contains the remaining tasks. The schedulability of the spotlight tasks is model checked, whereas the behavior of the shade is summarized, and the result is saved for later re-use. If this result is still inconclusive, more tasks are iteratively brought from the shade into the spotlight, with which the model checker can proceed. These steps are repeated until a decisive schedulability result is obtained. In this article, we enrich the encoding of real-time systems into timed automata by separating operations over clock variables from operations based on other variables such as Boolean and integers. We enforce to maintain a reasonable size of the spotlight model by applying counter abstraction to the tasks that were previously confirmed schedulable in the previous iterations. We compare our approach against Timestool and RTLib, an Uppaal template, in the analysis of the schedulability of real-time tasks for the First-In-First-Out and Shortest-Deadline-First scheduling policies. Empirical results show that our approach can handle more tasks than Timestool and RTLib. [ABSTRACT FROM AUTHOR]

Scientific papers, as pivotal tools for academic communication, should be articulated with clarity and precision to ensure the effective conveyance of scholarly ideas and to prevent reader confusion. Yet, many such papers conspicuously lack in-depth research, and their core content is often ambiguously presented. This pattern poses a significant impediment to the progressive evolution of science and technology. While numerous researchers have recognized this widespread challenge, a holistic theoretical or methodological solution remains elusive in the academic realm. To bridge this gap, we introduce the INTEGrity vERification (INTEGER) task. This task aids researchers in assessing the integrity of their papers by verifying the clarity of each knowledge unit. To implement this task on text, we propose a multi-task learning model that utilizes the Tucker decomposition and span-level attention mechanism to identify terms and their integrity precisely. More specifically, to provide insights into the INTEGER task and validate the effectiveness of the proposed model, we collect 8076 sentences and construct three new datasets containing various types of terms and descriptions in different domains. Extensive experimental results show that our proposed model has an average performance improvement of 1.1% F1 over the three datasets compared to a series of state-of-the-art baseline methods. [ABSTRACT FROM AUTHOR]

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]

For nonnegative integers h , g with h ≥ 2 , a set A of nonnegative integers is defined as a B h [ g ] sequence if, for every nonnegative integer n , the number of representations of n with the form n = a 1 + a 2 + ⋯ + a h is no larger than g , where a 1 ≤ ⋯ ≤ a h and a i ∈ A for i = 1 , 2 , ⋯ , h. Let Z be the set of integers and N be the set of positive integers. In 2013, by introducing the method of Inserting Zeros Transformation , Cilleruelo and Nathanson obtained the following result: let f : Z → N ⋃ { 0 , ∞ } be any function such that lim inf | n | → ∞ f (n) ≥ g and let B be any B h [ g ] sequence. Then, for any decreasing function ϵ (x) → 0 as x → ∞ , there exists a sequence A of integers such that r A , h (n) = f (n) for all n ∈ Z and A (x) ≫ B (x ϵ (x)). Recently, Nathanson further considered Sidon sets for linear forms. In this paper, we apply the Inserting Zeros Transformation into Sidon sets for linear forms and generalize the above result related to the inverse problem of representation functions. For a video summary of this paper, please visit https://youtu.be/KRtew2rLbXY. [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 presents an extension of Bhargava's theory of factorials associated to any nonempty subset S of Z. Bhargava's factorials k ! S are invariants, constructed using the notion of p -orderings of S where p is a prime. This paper defines b -orderings of any nonempty subset S of Z for all integers b ≥ 2 , as well as "extreme" cases b = 1 and b = 0. It defines generalized factorials k ! S , T and generalized binomial coefficients ( k + ℓ k ) S , T as nonnegative integers, for all nonempty S and allowing only b in T ⊆ N. It computes b -ordering invariants when S is Z and when S is the set of all primes. [ABSTRACT FROM AUTHOR]

Let P (x) : = a d x d + ⋯ + a 0 ∈ ℚ [ x ] , a d > 0 , be a polynomial of degree d ≥ 2. Let (x n) be a sequence of integers satisfying x n + 1 = P (x n) for all  n = 0 , 1 , 2 , ... and x n → ∞ as  n → ∞. Set α : = lim n → ∞ x n d − n . Then, under the assumption a d 1 / (d − 1) ∈ ℚ , in a recent result by [A. Dubickas, Transcendency of some constants related to integer sequences of polynomial iterations, Ramanujan J. 57 (2022) 569–581], either α is transcendental or α can be an integer or a quadratic Pisot unit with α − 1 being its conjugate over ℚ. In this paper, we study the nature of such α without the assumption that a d 1 / (d − 1) is in ℚ , and we prove that either the number α is transcendental, or α h is a Pisot number with h being the order of the torsion subgroup of the Galois closure of the number field ℚ α , a d − 1 d − 1 . Other results presented in this paper investigate the solutions of the inequality | | q 1 α 1 n + ⋯ + q k α k n + β | | < n in (n , q 1 , ... , q k) ∈ ℕ × (K ×) k , considering whether β is rational or irrational. Here, K represents a number field, and ∈ (0 , 1). The notation | | x | | denotes the distance between x and its nearest integer in ℤ. [ABSTRACT FROM AUTHOR]

In order to meet ever-increasing demands for reliable parallel and distributed systems, it is crucial to evaluate the reliability and fault tolerance of their underlying interconnection networks. Such an interconnection network is usually modeled as a connected graph G, where the vertex set and edge set represent the processors and links between processors in the network, respectively. In this paper, we combine Fàbrega's idea about h-extra edge-connectivity and Sampathkumar's concept about r-component edge-connectivity to introduce a more refined parameter for characterizing fault tolerance of interconnection networks, named as h-extra r-component edge-connectivity. Given a connected graph G, for two integers h ≥ 1 and r ≥ 2 , the h-extra r-component edge-connectivity of G, denoted as c λ r h (G) , is the minimum cardinality among all edge subsets F ⊂ E (G) , if any, such that G - F has at least r components, and each component has at least h vertices. As an enhancement on hypercube, the n-dimensional augmented cube AQ n , introduced by Choudum and Sunitha in 2002, reserves several excellent topological properties. As | V (AQ n) | = 2 n , the h-extra three-component edge-connectivity of AQ n is well-defined for each integer h with 1 ≤ h ≤ ⌊ 2 n / 3 ⌋ . In this paper, a generalization of Xu et al.'s conclusion is obtained that finds an upper bound for the exact value of general h-extra three-component edge-connectivity of AQ n and shows that it is sharp for 1 ≤ h ≤ 2 n 2 - 1 and h = 2 c where 1 ≤ c ≤ n - 2 . Let h = ∑ i = 0 s 2 t i be a positive integer with t 0 > t 1 > ⋯ > t s ≥ 0 . Let δ = 0 if h is even and δ = 1 if h is odd. Specifically, c λ 3 h (AQ n) = (4 n - 4) h - 2 ∑ i = 0 s (2 t i - 1) 2 t i - 2 ∑ i = 0 s 4 i · 2 t i - δ for n ≥ 4 , h ≤ 2 n 2 - 1 , and c λ 3 2 c (AQ n) = (2 n - 2 c - 1) 2 c + 1 for n ≥ 4 and 1 ≤ c ≤ n - 2 . [ABSTRACT FROM AUTHOR]

Nowadays, researchers are very interested in studying various Diophantine equations due to their importance in Cryptography, Chemistry, Knot Theory, Astronomy, Geometry, Trigonometry, Biology, Algebra, Electrical Engineering, Economics, and Astrology. The present paper is about the non-negative integer solution of the exponential Diophantine equation nx + 43y = z², where are non-negative integers, is a positive integer with nx + 43y = z², where n = 2 (mod 129) and n + 1 and is not a perfect square. The authors use the famous Catalan conjecture for this purpose. Results of the present paper indicate that 2, 3, 0, and 3 are the only required values of and respectively, that satisfy the exponential Diophantine equation, where are non-negative integers, is a positive integer with nx + 43y = z², where n = 2 (mod 129) and n + 1 and is not a perfect square. The present technique of this paper proposes a new approach to solving the Diophantine equations, which is the main scientific contribution of this study, and it is very beneficial, especially for researchers, scholars, academicians, and people interested in the same field. [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]

For the actual circuit with fractional order characteristics, the fractional order circuit model is more concise and precise than the classical circuit model. So studying the synthesis of fractional order immittance is of great significance. This paper investigates the passive synthesis method of the three-element-kind fractional order immittance function. It means designing a network with three kinds of passive fractional-order elements that exhibit a specific impedance function. This paper proposes a variable substitution method to solve this problem. The method consists of three steps: The first step of the process is to establish the correspondence between the fractional order and lossy reactance. This way, the fractional order immittance function is transformed into an integer order immittance function. Then the Brune procedure realizes the immittance function as a network with lossy reactance. Finally, a fractional order network is obtained by replacing the lossy reactance with the corresponding fractional reactance. A simulation example and an experiment are also given for verification. This method applies to all three-element-kind network impedance functions that meet the passive condition. [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 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 λ = λ0 + uλ1 + ⋯ + uk−1λ k−1 be a Type 1 unit in ℜk = 픽pm + u픽pm + ⋯ + uk−1픽 pm(uk = 0), where p is an odd prime, m is a positive integer and λ0,λ1,…,λk−1 ∈ 픽pm,λ0≠0,λ1≠0. In this paper, we give the complete structure of all Type 1 λ-constacyclic codes and their duals of length 8ps over the finite commutative chain ring ℜk in terms of their generator polynomials. Using this structure, we determine the Hamming distance and the Rosenbloom–Tsfasman (RT) distance of all Type 1 λ-constacyclic codes. For pm ≡ 1(mod 4) and a unit λ ∈ ℜ2, we determine the b -symbol distances of all λ-constacyclic codes of length 8ps over ℜ2, where b ≤ 8. As illustrations, we provide several λ-constacyclic codes with new parameters with respect to Hamming, RT and b-symbol metrics. MDS codes are widely recognized for their optimal error-correction capability, and MDS b-symbol codes are generalization of MDS codes. We found some MDS b-symbol constacyclic codes of length 8ps over ℜ2. Additionally, for pm ≡ 1(mod 4), we provide a decoding algorithm for Type 1 constacyclic codes of length 8ps over ℜk with respect to the Hamming, RT and b-symbol metrics. [ABSTRACT FROM AUTHOR]

AbstractLet m, n be integers with m>n . Denote Mm,n by the set of all m×n matrices over the field F of characteristic zero. Let I be an n×m matrix with (i,i) -position 1 for any 1≤i≤n , and 0 in other position. Define a bracket [A,B]=AIB−BIA , where A,B∈Mm,n . Then Mm,n with this bracket is a Lie algebra, called non-square linear Lie algebra, denoted by gl(m×n,F) . In this paper, all derivations and biderivations of gl(m×n,F) are determined. As applications of biderivations, the linear commuting maps and commutative post-Lie algebra structures on gl(m×n,F) are given. [ABSTRACT FROM AUTHOR]

Traditional single image super-resolution (SISR) methods, which focus on integer scale super-resolution, often require separate training for each scale factor, leading to increased computational resource consumption. In this paper, we propose MambaSR, a novel arbitrary-scale super-resolution approach integrating Mamba with Fast Fourier Convolution Blocks. MambaSR leverages the strengths of the Mamba state-space model to extract long-range dependencies. In addition, Fast Fourier Convolution Blocks are proposed to capture the global information in the frequency domain. The experimental results demonstrate that MambaSR achieves superior performance compared to different methods across various benchmark datasets. Specifically, on the Urban100 dataset, MambaSR outperforms MetaSR by 0.93 dB in PSNR and 0.0203 dB in SSIM, and on the Manga109 dataset, it achieves an average PSNR improvement of 1.00 dB and an SSIM improvement of 0.0093 dB. These results highlight the efficacy of MambaSR in enhancing image quality for arbitrary-scale super-resolution. [ABSTRACT FROM AUTHOR]

Famous Zaremba's conjecture (1971) states that for each positive integer q⩾2$q\geqslant 2$, there exists a positive integer 1⩽a

Let V $V$ be a vector space over the finite field Fq ${{\mathbb{F}}}_{q}$. A q $q$‐Steiner system, or an S(t,k,V)q $S{(t,k,V)}_{q}$, is a collection ℬ ${\rm{{\mathcal B}}}$ of k $k$‐dimensional subspaces of V $V$ such that every t $t$‐dimensional subspace of V $V$ is contained in a unique element of ℬ ${\rm{{\mathcal B}}}$. A large set of q $q$‐Steiner systems, or an LS(t,k,V)q $LS{(t,k,V)}_{q}$, is a partition of the k $k$‐dimensional subspaces of V $V$ into S(t,k,V)q $S{(t,k,V)}_{q}$ systems. In the case that V $V$ has infinite dimension, the existence of an LS(t,k,V)q $LS{(t,k,V)}_{q}$ for all finite t,k $t,k$ with 1

It is well known that a square complex matrix is called EP if it commutes with its Moore–Penrose inverse. In this paper, new classes of matrices which extend this concept are characterized. For that, we consider commutative equalities given by matrices of arbitrary index and generalized inverses recently investigated in the literature. More specifically, these classes are characterized by expressions of type A m X = X A m , where X is an outer inverse of a given complex square matrix A and m is an arbitrary positive integer. The relationships between the different classes of matrices are also analyzed. Finally, a picture presents an overview of the overall studied classes. [ABSTRACT FROM AUTHOR]

AbstractIn this paper, we consider a class of nonlinear p-k-Hessian systems with the gradient termwhere γ > 0, p > 1, and 1 ≤ k ≤ N is an integer. We show some existence, boundedness and blow-up results of the entire positive p-k-convex radial solutions based on the monotone iterative method. Finally, we present an example to illustrate our result. [ABSTRACT FROM AUTHOR]

We proceed with the study of ultimate periodicity properties related to overlaps between the suffixes of a left-infinite word λ and the prefixes of a right-infinite word ρ . For a positive integer n, let g(n) be n minus the maximum length of overlaps between the suffix of λ and the prefix of ρ of length n. In a recent publication we have shown that the function g has finite image if and only if λ and ρ are ultimately periodic words with a same root. In this paper we give an asymptotic characterization of words λ and ρ for which the function g has finite image. We prove that this condition is true if and only if the sequence (g (n) / n) n tends to zero [ABSTRACT FROM AUTHOR]

Let m and N be positive integers and k be a negative real number. Let χ be a multiplier system of weight k on Γ 0 (N). In this paper, we give an exact formula for the Fourier coefficients of the holomorphic part of a harmonic Maass–Jacobi form of weight k, index m, and multiplier system χ on Γ 0 (N). As an application, we investigate the number of ratios of the Fourier coefficients of the holomorphic parts of two harmonic Maass–Jacobi forms. Moreover, we obtain the equivalent conditions for the principal part of a harmonic Maass–Jacobi form to be the principal part of a weak Jacobi form with poles only at infinity. [ABSTRACT FROM AUTHOR]

This paper is concerned with finite sequences of integers that may be written as sums of squares of two nonzero integers. We first find infinitely many integers n such that n , n + h and n + k are all sums of two squares where h and k are two arbitrary integers, and as an immediate corollary obtain, in parametric terms, three consecutive integers that are sums of two squares. Similarly we obtain n in parametric terms such that all the four integers n , n + 1 , n + 2 , n + 4 are sums of two squares. We also find infinitely many integers n such that all the five integers n , n + 1 , n + 2 , n + 4 , n + 5 are sums of two squares, and finally, we find infinitely many arithmetic progressions, with common difference 4, of five integers all of which are sums of two squares. [ABSTRACT FROM AUTHOR]

Let f be a normalized Hecke eigenform of weight k for the full modular group S L 2 (ℤ). In this paper, we obtain the asymptotic of higher moments of general divisor functions associated to the Fourier coefficients of Rankin–Selberg L-functions R (s , f × f) , supported at the integers represented by primitive integral positive-definite binary quadratic forms (reduced forms) of a fixed discriminant D < 0. We improve previous results in the case when the reduced form is given by (x 1 , x 2) = x 1 2 + x 2 2 . [ABSTRACT FROM AUTHOR]

Let n , k and t be non-negative integers. The generalized Johnson graph G (n , k , t) is the graph whose vertices are the k -subsets of the set { 1 , 2 , ... , n } , and two vertices are adjacent if and only if they intersect with t elements. Special cases of generalized Johnson graph include the Kneser graph G (n , k , 0) and the Johnson graph G (n , k , k − 1). These graphs play an important role in coding theory, Ramsey theory, combinatorial geometry and hypergraphs theory. In this paper, we discuss the connectivity properties of the Kneser graph G (n , k , 0) and G (n , k , 1) by their symmetric properties. Specifically, with the help of some properties of vertex/edge-transitive graphs we prove that G (n , k , 0) (5 ≤ 2 k + 1 ≤ n) and G (n , k , 1) (4 ≤ 2 k ≤ n) are super restricted edge-connected. Besides, we obtain the exact value of the restricted edge-connectivity and the cyclic edge-connectivity of the Kneser graph G (n , k , 0) (5 ≤ 2 k + 1 ≤ n) and G (n , k , 1) (4 ≤ 2 k ≤ n) , and further show that the Kneser graph G (n , k , 0) (5 ≤ 2 k + 1 ≤ n) is super vertex-connected and super cyclically edge-connected. [ABSTRACT FROM AUTHOR]

Let k , t be coprime integers, and let 1 ≤ r ≤ t. We let D k × (r , t ; n) denote the total number of parts among all k -indivisible partitions (i.e., those partitions where no part is divisible by k) of n which are congruent to r modulo t. In previous work of the authors [3] , an asymptotic estimate for D k × (r , t ; n) was shown to exhibit unpredictable biases between congruence classes. In the present paper, we confirm our earlier conjecture in [3] that there are no "ties" (i.e., equalities) in this asymptotic for different congruence classes. To obtain this result, we reframe this question in terms of L -functions, and we then employ a nonvanishing result due to Baker, Birch, and Wirsing [1] to conclude that there is always a bias towards one congruence class or another modulo t among all parts in k -indivisible partitions of n as n becomes large. [ABSTRACT FROM AUTHOR]

The aim of this paper is to delineate a set of new classes of natural exponential families and their associated exponential dispersion models whose probability distributions are supported on the set of nonnegative integers with positive mass on 0 and 1. The new classes are obtained by considering a specific form of their variance functions. We show that the distributions of all these classes are supported on nonnegative integers, that they are infinitely divisible, and that they are skewed to the right, leptokurtic, over-dispersed, and zero-inflated (relative to the Poisson class). Accordingly, these new classes significantly enrich the set of probability models for modeling zero-inflated and over-dispersed count data. Furthermore, we elaborate on numerical techniques how to compute the distributions of our classes, and apply these to an actual data experiment. [ABSTRACT FROM AUTHOR]

In this paper, we design and simulate a Single-Ended Colpitts Oscillator (CO) in integer and fractional order domains. The oscillators are realized in 32nm node conventional MOSFET and carbon nanotube field effect transistor (CNTFET) technologies. Therefore, four COs have been designed, simulated and rigorously compared. These include integer order conventional MOSFET-based CO, fractional order MOSFET-based CO, CNTFET-based integer order CO and CNTFET-based fractional order CO, all based on 32-nm technology nodes. The fractional order approach has been used as it results in better control over the phase and frequency of the oscillator. Herein, fractional order capacitors of various orders, used in realizing the fractional order COs, are realized and their frequency responses are studied. This is being done to ensure whether the designed pseudo-capacitances have fractional behavior or not in the desired frequency and phase spectrum. It has been observed that the variation of fractional order (α) from 0.4 to 0.81 has resulted in a slight reduction of oscillation frequency from 1.68GHz (α = 0.4) to 1.351GHz (α = 0.81) keeping the pseudo-capacitance same at 0.3nF in MOS-based topology. Since the fractional order realization increases the circuit complexity and power consumption, therefore, CNTFET-based integer order as well as fractional order COs have been designed. The CNTFET-based fractional order CO retains the advantages of the fractional order domain as well as the power efficiency of CNTFETs. Further, it has been observed that integrating fractional-order capacitor (FOC) with the CNTFET CO results in a much larger constant phase zone (CPZ), an important performance measuring parameter. A rigorous comparative analysis of the four COs designed in this work has been performed. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the minimum number of triangles, denoted by t(n,k) $t(n,k)$, in n $n$‐vertex k $k$‐regular graphs, where n $n$ is an odd integer and k $k$ is an even integer. The well‐known Andrásfai–Erdős–Sós Theorem has established that t(n,k)>0 $t(n,k)\gt 0$ if k>2n5 $k\gt \frac{2n}{5}$. In a striking work, Lo has provided the exact value of t(n,k) $t(n,k)$ for sufficiently large n $n$, given that 2n5+12n5

AbstractLet D be an integral domain and D[​[x]​] be the power series ring over D. In this paper, we study the primality of x−α in D[​[x]​], where α∈D. For this purpose, we generalize the definition of a prime element as follows. For a positive integer N, a nonzero nonunit α∈D is called an N-prime element if for any a,b∈D, αN|ab implies α|a or α|b. We prove that if α is an N-prime element for some N, then x−α is a prime element in D[​[x]​]. Surprisingly, it is shown that the converse also holds when D is a PID, a valuation domain, or a Dedekind domain. In other words, when D is a PID, a valuation domain, or a Dedekind domain, a necessary and sufficient condition for x−α to be a prime element in D[​[x]​] is α is an N-prime element in D for some N. This however does not hold for other types of integral domains such as UFDs or Krull domains. We also investigate the N-prime property in an arbitrary integral domain and give other (equivalent) conditions for an element α in the aforementioned types of integral domains to be an N-prime element. [ABSTRACT FROM AUTHOR]

Let L be a nonnegative integer and ΓL the power set of the set {0, 1,…,L}. Then there is an adjacency relation in ΓL such that ΓL together with the relation forms a regular graph. In this paper, we propose a model of continuous-time magnetic quantum walk (MQW) on the graph ΓL, and investigate its properties from a viewpoint of probability and quantum information. We first introduce a magnetic Laplacian ΔL(휃) on the graph ΓL and examine its spectrum. And then, with ΔL(휃) as the Hamiltonian, we construct our model of continuous-time MQW on the graph ΓL. We find that the model has probability distributions that are completely independent of the magnetic potential at all times. And we show that it has perfect state transfer at time t = π 2 when the magnetic potential satisfies some mild conditions. Some other interesting results are also obtained. [ABSTRACT FROM AUTHOR]

AbstractLet R be a commutative noetherian ring and a an ideal of R. The goal of this paper is to establish the local-global principle for the artinianness dimension ra(M), where ra(M) is the smallest integer such that the local homology module of M is not artinian. For an artinian R-module M with the set CoassRHra(M)a(M) finite, we show that ra(M)=inf{raRp(HomR(Rp,M))|p∈SpecR}. And the class of all modules N such that CoassRN is finite is studied. [ABSTRACT FROM AUTHOR]

Given two graphs G , H $G,H$ and a positive integer q $q$, an ( H , q ) $(H,q)$‐coloring of G $G$ is an edge‐coloring of G $G$ such that every copy of H $H$ in G $G$ receives at least q $q$ distinct colors. The bipartite Erdős–Gyárfás function r ( K n , n , K s , t , q ) $r({K}_{n,n},{K}_{s,t},q)$ is defined to be the minimum number of colors needed for K n , n ${K}_{n,n}$ to have a ( K s , t , q ) $({K}_{s,t},q)$‐coloring. For balanced complete bipartite graphs K p , p ${K}_{p,p}$, the function r ( K n , n , K p , p , q ) $r({K}_{n,n},{K}_{p,p},q)$ was studied systematically in Axenovich et al. In this paper, we study the asymptotic behavior of this function for complete bipartite graphs K s , t ${K}_{s,t}$ that are not necessarily balanced. Our main results deal with thresholds and lower and upper bounds for the growth rate of this function, in particular for (sub)linear and (sub)quadratic growth. We also obtain new lower bounds for the balanced bipartite case, and improve several results given by Axenovich, Füredi and Mubayi. Our proof techniques are based on an extension to bipartite graphs of the recently developed Color Energy Method by Pohoata and Sheffer and its refinements, and a generalization of an old result due to Corrádi. [ABSTRACT FROM AUTHOR]

In this paper, we study skew constacyclic codes over a class of non-chain rings T = F q [ u 1 , u 2 , ... , u r ] / ⟨ f i (u i) , u i u j - u j u i ⟩ i , j = 1 r , where q = p m , p is some odd prime, m is a positive integer, and f i (u i) are non-constant, monic polynomials that split into distinct linear factors. We discuss the structural properties of skew constacyclic codes over T and their dual. We characterize Euclidean and Hermitian dual-containing skew constacyclic codes. These characterizations serve as a foundational framework for the development of techniques to construct quantum codes. Consequently, we derive plenty of new quantum codes including many maximum distance separable (MDS) quantum codes, and many quantum codes with better parameters than existing ones. Our work further extends to the characterization of skew constacyclic Euclidean and Hermitian linear complementary dual (LCD) codes over T , and we establish that their Gray images also preserve the LCD property. From this analysis, we derive numerous MDS codes and best known linear codes over F q . [ABSTRACT FROM AUTHOR]

This paper analyzes a novel use of the shifted Fibonacci polynomials (SFPs) to treat the time-fractional Burgers equation (TFBE). We first develop the fundamental formulas of these polynomials, which include their power series representation and the inversion formula. We establish other new formulas for the SFPs, including integer and fractional derivatives, in order to design the collocation approach for treating the TFBE. These derivative formulas serve as tools that aid in constructing the operational metrics for the integer and fractional derivatives of the SFPs. We use these matrices to transform the problem and its underlying conditions into a system of nonlinear equations that can be treated numerically. An error analysis is analyzed in detail. We also present three illustrative numerical examples and comparisons to test our proposed algorithm. These results showed that the proposed algorithm is advantageous since highly accurate approximate solutions can be obtained by choosing a few terms of retained modes of SFPs. [ABSTRACT FROM AUTHOR]

Let { X n } n ∈ Z be a stationary process with values in a finite set. In this paper, we present a moving average version of the Shannon–McMillan–Breiman theorem; this generalize the corresponding classical results. A sandwich argument reduced the proof to direct applications of the moving strong law of large numbers. The result generalizes the work by Algoet et. al., while relying on a similar sandwich method. It is worth noting that, in some kind of significance, the indices a n and ϕ (n) are symmetrical, i.e., for any integer n, if the growth rate of (a n) n ∈ Z is slow enough, all conclusions in this article still hold true. [ABSTRACT FROM AUTHOR]

A set S ⊆ V is referred to as a k-fault-tolerant power-dominating set of a given graph G = (V , E) if the difference S ∖ F remains a power-dominating set of G for any F ⊆ S with | F | ≤ k , where k is an integer with 0 ≤ k < | V | . The lowest cardinality of a k-fault-tolerant power-dominating set is the k-fault-tolerant power-domination number of G, denoted by γ P k (G) . Generalized Petersen graphs G P (m , k) and generalized cylinders S G are two well-known graph classes. In this paper, we calculate the k-fault-tolerant power-domination number of the generalized Petersen graphs G P (m , 1) and G P (m , 2) . Also, we obtain γ P k (G) for the subclasses of cylinders S C m and S B m . [ABSTRACT FROM AUTHOR]

Dimension reduction is a technique used to transform data from a high-dimensional space into a lower-dimensional space, aiming to retain as much of the original information as possible. This approach is crucial in many disciplines like engineering, biology, astronomy, and economics. In this paper, we consider the following dimensionality reduction instance: Given an n-dimensional probability distribution p and an integer m < n , we aim to find the m-dimensional probability distribution q that is the closest to p, using the Kullback–Leibler divergence as the measure of closeness. We prove that the problem is strongly NP-hard, and we present an approximation algorithm for it. [ABSTRACT FROM AUTHOR]

A vector is dyadic if each of its entries is a dyadic rational number, i.e. of the form a 2 k for some integers a, k with k ≥ 0 . A linear system A x ≤ b with integral data is totally dual dyadic if whenever min { b ⊤ y : A ⊤ y = w , y ≥ 0 } for w integral, has an optimal solution, it has a dyadic optimal solution. In this paper, we study total dual dyadicness, and give a co-NP characterization of it in terms of dyadic generating sets for cones and subspaces, the former being the dyadic analogue of Hilbert bases, and the latter a polynomial-time recognizable relaxation of the former. Along the way, we see some surprising turn of events when compared to total dual integrality, primarily led by the density of the dyadic rationals. Our study ultimately leads to a better understanding of total dual integrality and polyhedral integrality. We see examples from dyadic matroids, T-joins, cycles, and perfect matchings of a graph. [ABSTRACT FROM AUTHOR]

For a 3-manifold M, the genus of M, denoted by g(M), is defined to be the minimal Heegaard genus among all the Heegaard splittings of M. In this paper, we prove that for any two integers g ≥ 2 and n ≥ 2, there is a 3-manifold M with g(M) = g such that the minimal Heegaard splittings of M are unique up to isotopy, where the distance of the Heegaard splitting is n. [ABSTRACT FROM AUTHOR]

For any positive integer m, let ℤm be the set of residue classes modulo m. For A ⊆ ℤm and n ¯ ∈ Z m , let the representation function R A (n ¯) denote the number of solutions of the equation n ¯ = a ¯ + a ′ ¯ with unordered pairs (a ¯ , a ′ ¯) ∈ A × A . For any integer a with (a, m) = 1, let ordm(a) be the least positive integer h such that ah ≡ 1 (mod m). Let m = 2αM, where α is an integer with α ≥ 2 and M is an odd integer with M ≥ 3. In this paper, we prove that if 2 ∣ ordp (2) for some odd prime p with p ≥ M, then there exist two distinct sets A, B ⊆ ℤm with A ∪ B = ℤm, ∣A ∩ B∣ = 2 and B ≠ A + 2 α − 1 M ¯ such that R A (n ¯) = R B (n ¯) for all n ¯ = Z m . We also prove that if 2 ∤ ordp(2) for any odd prime p with p ∣ M and A, B ⊆ ℤm with A ∪ B = ℤm, ∣A ∩ B∣ =2, then R A (n ¯) = R B (n ¯) for all n ¯ ∈ Z m if and only if B = A + 2 α − 1 M ¯ . [ABSTRACT FROM AUTHOR]

In this paper, we study the combinatorial relations between the cokernels cok ⁡ (A n + p ⁢ x i ⁢ I n ) ( 1 ≤ i ≤ m ), where A n is an n × n matrix over the ring of p-adic integers ℤ p , I n is the n × n identity matrix and x 1 , ... , x m are elements of ℤ p whose reductions modulo p are distinct. For a positive integer m ≤ 4 and given x 1 , ... , x m ∈ ℤ p , we determine the set of m-tuples of finitely generated ℤ p -modules (H 1 , ... , H m) for which (cok ⁡ (A n + p ⁢ x 1 ⁢ I n ) , ... , cok ⁡ (A n + p ⁢ x m ⁢ I n )) = (H 1 , ... , H m) for some matrix A n . We also prove that if A n is an n × n Haar random matrix over ℤ p for each positive integer n, then the joint distribution of cok ⁡ (A n + p ⁢ x i ⁢ I n ) ( 1 ≤ i ≤ m ) converges as n → ∞ . [ABSTRACT FROM AUTHOR]

Let R be a noncommutative prime ring of characteristic different from 2 , with right Martindale quotient ring Q r and extended centroid C. Let m , n ≥ 1 be fixed integers and F a nonzero generalized skew derivation of R. In this paper, we investigate the set S = { F (x m) x n + x n F (x m) : x ∈ R } and prove that its left annihilator in R is identically zero. Using the above result, we also study the identity F (x) x n + x n F (x) = 0 for semiprime rings. [ABSTRACT FROM AUTHOR]

This paper is related to the classic but still being examined issue of approximation of functions by polynomials with integer coefficients. Let r, n be positive integers with n ≥ 6r. Let Pn ∩ Mr be the space of polynomials of degree at most n on [0, 1] with integer coefficients such that P(k)(0)/k! and P(k)(1)/k! are integers for k = 0,. . ., r - 1 and let PZn ∩ Mr be the additive group of polynomials with integer coefficients. We explore the problem of estimating the minimal distance of elements of PZn ∩ Mr from Pn ∩ Mr in L2(0, 1). We give rather precise quantitative estimations for successive minima of PZn in certain specific cases. At the end, we study properties of the shortest polynomials in some hyperplane in Pn ∩ Mr. [ABSTRACT FROM AUTHOR]

Let b ≥ 2 be an integer and A = (a n) n = 1 ∞ be a strictly increasing subsequence of positive integers with η : = lim sup n → ∞ a n + 1 a n < + ∞. For each irrational real number ξ , we denote by v ˆ b , A (ξ) the supremum of the real numbers v ˆ for which, for every sufficiently large integer N , the equation ‖ b a n ξ ‖ < (b a N ) − v ˆ has a solution n with 1 ≤ n ≤ N. For every v ˆ ∈ [ 0 , η ] , let V ˆ b , A (v ˆ) ( V ˆ b , A ⁎ (v ˆ)) be the set of all real numbers ξ such that v ˆ b , A (ξ) ≥ v ˆ ( v ˆ b , A (ξ) = v ˆ) respectively. In this paper, we give some results of the Hausdorfff dimensions of V ˆ b , A (v ˆ) and V ˆ b , A ⁎ (v ˆ). When η = 1 , we prove that the Hausdorfff dimensions of V ˆ b , A (v ˆ) and V ˆ b , A ⁎ (v ˆ) are equal to (1 − v ˆ 1 + v ˆ ) 2 for any v ˆ ∈ [ 0 , 1 ]. When η > 1 and lim n → ∞ ⁡ a n + 1 a n exists, we show that the Hausdorfff dimension of V ˆ b , A (v ˆ) is strictly less than (η − v ˆ η + v ˆ ) 2 for some v ˆ , which is different with the case η = 1 , and we give a lower bound of the Hausdorfff dimensions of V ˆ b , A (v ˆ) and V ˆ b , A ⁎ (v ˆ) for any v ˆ ∈ [ 0 , η ]. Furthermore, we show that this lower bound can be reached for some v ˆ. [ABSTRACT FROM AUTHOR]

In a graph G : = (V (G) , E (G)) , a subset M of the vertex set V (G) is a module (or interval, clan) of G if every vertex outside M is adjacent to all or none of M. The empty set, the singleton sets, and the full set of vertices are trivial modules. The graph G is indecomposable (or prime) if all its modules are trivial. If G is indecomposable, we say that an edge e of G is a removable edge if G − e is indecomposable (here G − e : = (V (G) , E (G) − { e })). The graph G is said to be unstable if it has no removable edges. For a positive integer k , the k th power G k of a graph G is the graph obtained from G by adding an edge between all pairs of vertices of G with distance at most k. A graph G of order n (i.e., | V (G) | = n) is said to be p -placeable into G k , if G k contains p edge-disjoint copies of G. In this paper, we answer a question, suggested by Boudabbous in a personal communication, concerning unstable graphs and packing into their fifth power as follows: First, we give a characterization of the unstable graphs which is deduced from the results given by Ehrenfeucht, Harju and Rozenberg (the theory of 2 -structures: a framework for decomposition and transformation of graphs). Second, we prove that every unstable graph G is 2 -placeable into G 5 . [ABSTRACT FROM AUTHOR]

Serre showed that for any integer m , a (n) ≡ 0 (mod m) for almost all n , where a (n) is the n th Fourier coefficient of any modular form with rational coefficients. In this paper, we consider a certain class of cuspforms and study # { a (n) (mod m) } n ≤ x over the set of integers with O (1) many prime factors. Moreover, we show that any residue class a ∈ ℤ / m ℤ can be written as the sum of at most 13 Fourier coefficients, which are polynomially bounded as a function of m. [ABSTRACT FROM AUTHOR]

In this paper, we show that the Diophantine equation 3x +121y = z² has precisely two solutions in non-negative integers; namely, (1, 0, 2) and (5, 2, 122). [ABSTRACT FROM AUTHOR]