Search

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]

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]

Integer variables are among the neo-logistic programming issues that most frequently pertain to system reliability improvement. If 'k' out of 'n' elements are present in each technology, then the configuration can be employed with 'k' out of 'n' systems. Only particular objective function structures and restrictions may be used in a heuristic technique to precisely address a dependability optimization problem. As more constraints are placed on it, the more useful it becomes; it is worthless for boosting dependability in a large system. This paper analyzes existing research on integrated reliability models with redundancy and explores system reliability optimization before proposing new recommendations. [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]

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 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]

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

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]

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]

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]

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]

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]

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]

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]

Let Fq be the finite field of order q where q = pk, k is a positive integer and p is an odd prime. Let F*q represent the nonzero elements of Fq. For f(x1,···,xn) ∊ Fq[x1,···,xn], we used N(f(x1,···,xn) = 0) to denote the number of Fq-rational points of the affine hypersurface f(x1,···,xn) = 0. In 2020, Zhao et al. obtained the explicit formulae for N(x41 + x42 = c), N(x41 + x42+ x43 = c) and N(41+ x42+ x43+ x44 = c) over Fq, with c ∊ F*q. In this paper, by using Jacobi sums and an analog of the Hasse-Davenport theorem, we arrived at explicit formulae for N(a1x41 + a2x42 = c) and N(b1x41 + b2x42 + b3x43 = c) with ai, bj ∊ F*q(1 ≤ i ≤ 2, 1 ≤ j ≤ 3) and c ∊ Fq. Furthermore, by using the reduction formula for Jacobi sums, the number of rational points of the quartic diagonal hypersurface a1x41 + a2x42 +···+ anx4n = c of n ≥ 4 variables with ai ∊ F*q (1 ≤ i ≤ n), c ∊ Fq and p = 1(mod 4), can also be deduced. These extended and improved earlier results. [ABSTRACT FROM AUTHOR]

In this paper, we show that the cover ideals of Jahangir's graphs are normal, and also satisfy the (symbolic) (strong) persistence property. In addition, we determine when the unique homogeneous maximal ideal appears in the set of associated primes of powers of the cover ideals of Jahangir's graphs. [ABSTRACT FROM AUTHOR]

In 2012, Ponraj et al. defined k-product cordial labeling as follows: Let f be a map from V (G) to { 0 , 1 , ... , k − 1 } where k is an integer, 1 ≤ k ≤ | V (G) |. For each edge u v assign the label f (u) f (v) (mod  k). f is called a k-product cordial labeling if | v f (i) − v f (j) | ≤ 1 , and | e f (i) − e f (j) | ≤ 1 , i , j ∈ { 0 , 1 , ... , k − 1 } , where v f (x) and e f (x) denote the number of vertices and edges, respectively, labeled with x (x = 0 , 1 , ... , k − 1). A graph that admits k-product cordial labeling is called k-product cordial graph. Later, we proved that several families of graphs are k-product cordial graphs. In this paper, we show that the product of graphs admit k-product cordial labeling. [ABSTRACT FROM AUTHOR]

Let p and q be two different prime integers such that p Ξ q Ξ 3 (mod 8) with (p/q) = 1, and l a positive odd square-free integer relatively prime to p and q. In this paper we investigate the unit groups of number fields L = Q(√2, √p, √q, √-1). [ABSTRACT FROM AUTHOR]

For an integer k ≥ 2 , let (P n (k)) n be the k-generalized Pell sequence which starts with 0 , ... , 0 , 1 (k terms), and each term afterwards is given by P n (k) = 2 P n − 1 (k) + P n − 2 (k) + ⋯ + P n − k (k) . In this paper, we determine all solutions of the Markov equation x 2 + y 2 + z 2 = 3 x y z , with x, y, and z being k-generalized Pell numbers. This paper continues and extends a previous work of Kafle, Srinivasan and Togbé, who found all Markov triples with Pell components. [ABSTRACT FROM AUTHOR]

Let K3 be a non-normal cubic extension over Q. In this paper, we investigate the higher moments of coefficients aK3 (n) of the Dedekind zeta function of the following type Σ(n≤x) al K3 (n), where l ≥ 4 is any fixed positive integer. [ABSTRACT FROM AUTHOR]

According to Dastanpour and Ghorbani, a ring R is said to satisfy divisibility on ascending chains of right ideals (ACCd) if, for every ascending chain of right ideals I1 ⊆ I2 ⊆ I3 ⊆ I4 ⊆ ... of R, there exists an integer k ∈ N such that for each i ≥ k, there exists an element ai ∈ R such that Ii = aiIi+1. In this paper, we examine the transfer of the ACCd-condition on ideals to trivial ring extensions. Moreover, we investigate the connection between the ACCd on ideals and other ascending chain conditions. For example we will prove that if R is a ring with ACCd on ideals, then R has ACC on prime ideals. [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]

This paper introduced a novel semiring structure involving nonnegative integers, where operations depended on the comparison of the magnitudes of decimal digit sums. Consequently, a corresponding matrix semiring can be established on this commutative semiring. We showed that the 3-satisfiability problem can be polynomial-time reduced to solving systems of quadratic polynomial equations over this semiring. We proposed a key exchange protocol based on this matrix semiring, with its security relying on the two-sided digital circulant matrix action problem over this semiring. This scheme provides a novel cryptographic primitive for post-quantum cryptography. [ABSTRACT FROM AUTHOR]

The fixed-length Levenshtein (FLL) distance between two words x , y ∈ Z m n is the smallest integer t such that x can be transformed to y by t insertions and t deletions. The size of a ball in the FLL metric is a fundamental yet challenging problem. Very recently, Bar-Lev, Etzion, and Yaakobi explicitly determined the minimum, maximum and average sizes of the FLL balls with radius one, respectively. In this paper, based on these results, we further prove that the size of the FLL balls with radius one is highly concentrated around its mean by Azuma's inequality. [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]

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]

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]

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]

Yamamoto’s integral was the integral associated with 2-posets, which was first introduced by Yamamoto. In this paper, we obtained the values of infinite series involving harmonic numbers and reciprocal of binomial coefficients by using some techniques of Yamamoto’s integral. We determine the value of infinite series of the form: ∑ m1,...,mn,ℓ1,...,ℓk≥1 H(a1)m1 · · · H(an)mn/mb11 · · · mbnn ℓc11 · · · ℓckk (m1+···+mn+ℓ1+···+ℓkℓk), in terms of a finite sum of multiple zeta values, for positive integers a1, . . ., an, b1, . . ., bn, c1, . . ., ck. [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]

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]

In this paper, we first consider the existence conditions, the construction and the enumeration of Hermitian self-dual θ -cyclic and θ -negacyclic codes over I F p 2 , where p is a prime number and θ is the Frobenius automorphism over I F p 2 . We then give necessary and sufficient conditions for the existence of Hermitian self-dual θ -cyclic and θ -negacyclic codes over I F p e where e is an even integer greater than 2. [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]

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]

Let p ≥ 5 be a prime number such that p ≡ 3 (mod 8) and d an odd positive squarefree integer. Put K = Q (2 , - p , d) . In this paper, we compute the 2-rank of the class group of the fields K = Q (2 , - p , d) . Furthermore, we give the list of these fields whose 2-class groups are trivial, cyclic and of type (2, 2) or (2, 4). [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]

Let a , b , c , and n be positive integers such that a + b = c 2 , 2 ∤ c and n > 1 . In this paper, we prove that if gcd (c , n) = 1 and n ≥ 117.14 c , then the equation (a n 2 + 1) x + (b n 2 − 1) y = (c n) z has only the positive integer solution (x , y , z) = (1 , 1 , 2) under the assumption gcd (a n 2 + 1 , b n 2 − 1) = 1 . Thus, we affirm that the conjecture proposed by Fujita and Le is true in this case. Moreover, combining the above result with some existing results and a computer search, we show that, for any positive integer n , if (a , b , c) = (12 , 13 , 5) , (18 , 7 , 5) , or (44 , 5 , 7) , then this equation has only the solution (x , y , z) = (1 , 1 , 2) . This result extends the theorem of Terai and Hibino gotten in 2015, that of Alan obtained in 2018, and Hasanalizade's theorem attained recently. [ABSTRACT FROM AUTHOR]

A periodic lattice in Euclidean space is the infinite set of all integer linear combinations of basis vectors. Any lattice can be generated by infinitely many different bases. This ambiguity was partially resolved, but standard reductions remain discontinuous under perturbations modelling atomic displacements. This paper completes a continuous classification of 2-dimensional lattices up to Euclidean isometry (or congruence), rigid motion (without reflections), and similarity (with uniform scaling). The new homogeneous invariants allow easily computable metrics on lattices considered up to the equivalences above. The metrics up to rigid motion are especially non-trivial and settle all remaining questions on (dis)continuity of lattice bases. These metrics lead to real-valued chiral distances that continuously measure lattice deviations from higher-symmetry neighbours. The geometric methods extend the past work of Delone, Conway, and Sloane. [ABSTRACT FROM AUTHOR]

Let (Fn)n≥0 and (Ln)n≥0 be the Fibonacci and Lucas sequences, respectively. In this paper we determine all Fibonacci numbers which are mixed concatenations of a Fibonacci and a Lucas numbers. By mixed concatenations of a and b, we mean the both concatenations ab and ba together, where a and b are any two nonnegative integers. So, the mathematical formulation of this problem leads us searching the solutions of two Diophantine equations Fn = 10dFm + Lk and Fn = 10dLm + Fk in nonnegative integers (n, m, k), where d denotes the number of digits of Lk and Fk, respectively. We use lower bounds for linear forms in logarithms and reduction method in Diophantine approximation to get the results. [ABSTRACT FROM AUTHOR]