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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]

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]

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]

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]

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]

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 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]

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]

The turn of the 1980s and 1990s brought a number of changes for the whole Soviet world, including the Hungarian community in Transcarpathia, which obviously had an impact on the language of the press, as elements of political or economic jargon appear in the daily press just as much as the language of sport or the argot. The majority of linguists agree that press language is a mirror of language use in which linguistic changes are detectable and traceable almost immediately. In this paper, we have attempted to present an overview of the 1989-1990 issues of the Berehovo District Council newspaper, the Vörös Zászló [Red Flag]--i.e. more than 300 issues and more than 1,200 pages of text for the two years--presenting the major events and problems of the period, and the linguistic information related to them, especially the state language loanwords that appeared in the paper as a result of bilingualism. [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]

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]

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 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 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]

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]

Satellite navigation positioning has become an indispensable component of everyday life, where precise pinpointing and rapid convergence are crucial in delivering timely and accurate location information. However, due to the damping of integer ambiguities and system residual errors, the rapid convergence of Precise Point Positioning (PPP) implementation is a significant challenge. To address this, this paper proposes a novel Carrier Phase Zero-Baseline Self-Differencing Precise Point Positioning (CZS-PPP) technique and its ionosphere-free fusion model. By employing the proposed CZS-PPP approach in separate scenarios involving BDS-3, GPS, and dual-system settings, we systematically validate the efficacy of the method. The experimental results indicate that the convergence time of the method is less than 4 min in a single-system scenario. Furthermore, in a dual-system scenario, the method can achieve rapid convergence in less than 3 min. The CZS-PPP technique presented demonstrates the elimination of integer ambiguities and the effective suppression of system residuals, in comparison to the conventional method. The proposed approach has demonstrated remarkable performance across different systems, offering a promising new pathway for achieving PPP fast convergence in BDS/GNSS. [ABSTRACT FROM AUTHOR]

Connectivity along with its extensions are important metrices to estimate the fault-tolerance of interconnection networks. The classic connectivity κ (G) of a graph G is the minimum cardinality of a vertex set F ⊆ V (G) such that G − F is connected or a single vertex. For any subset S ⊆ V (G) with | S | ≥ 2 , a tree T in G is called an S -tree if S ⊆ V (T). Furthermore, any two S -tree T 1 and T 2 are internally disjoint if E (T 1) ∩ E (T 2) = ∅ and V (T 1) ∩ V (T 2) = S. We denote by κ G (S) the maximum number of pairwise internally disjoint S -trees in G. For an integer k ≥ 2 , the generalized k -connectivity of a graph G is defined as κ k (G) = min { κ G (S) | S ⊆ V (G) and | S | = k }. For the n -dimensional folded divide-and-swap cubes, F D S C n , we show the upper bound and the lower bound of κ 4 (F D S C n) , that is d ≤ κ 4 (F D S C n) ≤ d + 1 , where n = 2 d and d ≥ 1 in this paper. [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]

Let n ≥ 2 be a fixed integer and A be a C ∗ -algebra. A permuting n-linear map G : A n → A is known to be symmetric generalized n-derivation if there exists a symmetric n-derivation D : A n → A such that G ς 1 , ς 2 , ... , ς i ς i ′ , ... , ς n = G ς 1 , ς 2 , ... , ς i , ... , ς n ς i ′ + ς i D (ς 1 , ς 2 , ... , ς i ′ , ... , ς n) holds ∀ ς i , ς i ′ ∈ A . In this paper, we investigate the structure of C ∗ -algebras involving generalized linear n-derivations. Moreover, we describe the forms of traces of linear n-derivations satisfying certain functional identity. [ABSTRACT FROM AUTHOR]

For non-negative integers d 1 and d 2 , if V 1 and V 2 are two partitions of a graph G 's vertex set V G , such that V 1 and V 2 induce two subgraphs of G , called G V 1 with maximum degree at most d 1 and G V 2 with maximum degree at most d 2 , respectively, then the graph G is said to be improper d 1 , d 2 -colorable, as well as d 1 , d 2 -colorable. A class of planar graphs without C 3 , C 4 , and C 6 is denoted by C. In 2019, Dross and Ochem proved that G is 0 , 6 -colorable, for each graph G in C. Given that d 1 + d 2 ≥ 6 , this inspires us to investigate whether G is d 1 , d 2 -colorable, for each graph G in C. In this paper, we provide a partial solution by showing that G is (3, 3)-colorable, for each graph G in C. [ABSTRACT FROM AUTHOR]

In this paper, we propose a way to partition any constacyclic code over a finite field in its equivalence classes according to the algebraic structure of the code. Such a method gives the generalization of cyclically permutable codes (CPCs), which are called constacyclically permutable codes (CCPCs), and it is useful to derive a CCPC from a given constacyclic code. Moreover, we present an enumerative formula for the code size of such a CCPC, with all of the terms being positive integers, and we provide an algebraic method to produce such a CCPC. [ABSTRACT FROM AUTHOR]

Reliable probabilistic primality tests are fundamental in public-key cryptography. In adversarial scenarios, a composite with a high probability of passing a specific primality test could be chosen. In such cases, we need worst-case error estimates of the test. However, in many scenarios, the numbers are randomly chosen and thus have a significantly smaller error probability. We are hence interested in average-case error estimates. In this paper we establish such bounds for the strong Lucas primality test, as there exist only worst-case, but no average-case error bounds. This allows us to use this test with more confidence. Let us examine an algorithm that draws odd k-bit integers uniformly and independently, runs t independent iterations of the strong Lucas test with randomly chosen parameters, and outputs the first number that passes all t consecutive rounds. We attain numerical upper bounds on the probability that a composite is returned. Moreover, we examine a slight modification of this algorithm that only considers integers that are not divisible by small primes, yielding improved bounds. In addition, we classify the numbers that contribute most to our estimate. [ABSTRACT FROM AUTHOR]

Entanglement-assisted quantum error-correcting (EAQEC) codes are a generalization of quantum error-correcting (QEC) codes, which can be constructed from arbitrary classical linear codes by relaxing the dual-containing condition and by using preshared entangled states between the sender and the receiver. In this paper, we investigate EAQEC codes of length n = 2 (q 2 + 1) a , where q is an odd prime power, a = m 2 + 1 and m is an odd integer. The resulting EAQEC codes are entanglement-assisted quantum maximum-distance-separable (EAQMDS) codes when the minimum distance d ≤ n + 2 2 . [ABSTRACT FROM AUTHOR]

Let K = Q(α) be a pure number field generated by a complex root α of a monic irreducible polynomial F(x) = x2r · 3k·7s− m ∈ Z[x], where r, k, s are three positive natural integers. The purpose of this paper is to study the monogenity of K. Our results are illustrated by some examples. [ABSTRACT FROM AUTHOR]

A non-parallel submodules graph of M, denoted by G ∦ (M), is an undirected simple graph whose vertices are in one-to-one correspondence with all non-zero proper submodules of M and two distinct vertices are adjacent if and only if they are not parallel to each other. In this paper, we investigate the interplay between some of the module-theoretic properties of M and the graph-theoretic properties of G ∦ (M). It is shown that if G ∦ (M) is connected, then diam(G ∦ (M)) ≤ 3 and if G ∦ (M) is not connected, then G ∦ (M) is a null graph. It is proved that G ∦ (M) is null if and only if M contains a unique simple submodule. In particular, M is a strongly semisimple R -module if and only if G ∦ (M) is a complete graph, and from this it follows that if G ∦ (M) is complete, then every R-module with finite Goldie dimension is Artinian and Noetherian. In addition, G ∦ (M) is a finite star graph if and only if M∼= Zpq, for some distinct prime numbers p and q. [ABSTRACT FROM AUTHOR]

China is the only country that extensively cultivates the indica and japonica rice varieties, with the largest japonica rice production area being in northeast China. A study of the relationship between the yield and quality of japonica rice and the effect of nitrogen fertilizer application on this relationship is important. In this paper, we aimed to assess the current yield and quality of japonica rice in northeast China. We selected erect-panicle varieties as the test materials. Field experiments were conducted using different nitrogen fertilizer levels for two consecutive years to analyze the rice varieties' yield, quality, interrelationship, and nitrogen fertilizer response. The average yield following high- and low-nitrogen treatments exceeded 10,000.00 kg/hm2, with a maximum of 12,285.63 kg/hm2. The high-yield–high-nitrogen treatment group had more panicles, a higher seed-setting rate, and a higher 1000-grain weight than the other groups. The high-yield–low-nitrogen group had a higher number of panicles and seed-setting rate than the other groups. The low-yield–high-nitrogen group had a lower number of whole grains, grain length-to-width ratio, and taste value than the other groups. The low-yield–low-nitrogen group had fewer primary branches than the other groups; excluding the primary branch-setting rate and 1000-grain weight, the values of the other panicle traits of the group were significantly higher than those of the other groups. The high-nitrogen–high-flavor group had lower panicle and spikelet numbers and higher spikelet fertility rates than the other groups. The low-nitrogen–high-flavor group had higher spikelet fertility rates and 1000-grain weight than the other groups. Compared to the other groups, the low-nitrogen–high-flavor group had a higher head rice yield, and the high-nitrogen–high-flavor group had a lower chalkiness rate. The main goal of the breeding and cultivation of high-yield and high-quality erect-panicle japonica rice in northern China is to achieve "dual high, dual low, and one high and one low" conditions, signifying a high yield with high or low nitrogen levels, low protein and amylose contents, high head rice rates, and low chalkiness. This study provides a new technique for enhancing the taste of northern erect-panicle japonica rice to promote the sustainable, high-yield, and high-quality development of japonica rice in northern China. [ABSTRACT FROM AUTHOR]

This paper aims to explore an alternative reversible digital watermarking solution to guarantee the integrity of and detect tampering with data of probative importance. Since the payload for verification is embedded in the contents, algorithms for reversible embedding and extraction, imperceptibility, payload capacity, and computational time are issues to evaluate. Thus, we propose a reversible and imperceptible audio information-hiding algorithm based on modified integer discrete cosine transform (intDCT) coefficient expansion. In this work, the original signal is segmented into fixed-length frames, and then intDCT is applied to each frame to transform signals from the time domain into integer DCT coefficients. Expansion is applied to DCT coefficients at a higher frequency to reserve hiding capacity. Objective evaluation of speech quality is conducted using listening quality objective mean opinion (MOS-LQO) and the segmental signal-to-noise ratio (segSNR). The audio quality of different frame lengths and capacities is evaluated. Averages of 4.41 for MOS-LQO and 23.314 [dB] for segSNR for 112 ITU-T test signals were obtained with a capacity of 8000 bps, which assured imperceptibility with the sufficient capacity of the proposed method. This shows comparable audio quality to conventional work based on Linear Predictive Coding (LPC) regarding MOS-LQO. However, all segSNR scores of the proposed method have comparable or better performance in the time domain. Additionally, comparing histograms of the normalized maximum absolute value of stego data shows a lower possibility of overflow than the LPC method. A computational cost, including hiding and transforming, is an average of 4.884 s to process a 10 s audio clip. Blind tampering detection without the original data is achieved by the proposed embedding and extraction method. [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]

In this paper, by utilizing d-dimensional single-particle states, three semiquantum cryptography protocols, i.e., the multi-party semiquantum private comparison (MSQPC) protocol, the multi-party semiquantum multiplication (MSQM) protocol and the multi-party semiquantum summation (MSQS) protocol, can be achieved simultaneously under the assistance of two semi-honest quantum third parties (TPs). Here, the proposed MSQPC scheme is the only protocol which is devoted to judging the size relationship of secret integers from more than two semiquantum participants without a pre-shared key. And the proposed MSQM protocol absorbs the innovative concept of semiquantumness into quantum multiplication for the first time, which can calculate the modulo d multiplication of private inputs from more than two semiquantum users. As for the proposed MSQS protocol, it is the only semiquantum summation protocol which aims to accomplish the modulo d addition of more than three semiquantum users' private integers. Neither quantum entanglement swapping nor unitary operations are necessary in the three proposed protocols. The security analysis verifies in detail that both the external attacks and the internal attacks can be resisted in the three proposed protocols. [ABSTRACT FROM AUTHOR]

For an odd prime p and a positive integer α , let g be of multiplicative order τ modulo q and q = p α . Denote by N (h , g , q) the number of a such that h ∤ (a + (g a ) q) for any 1 ≤ a ≤ τ and a fixed integer h ≥ 2 with (h , q) = 1. The main purpose of this paper is to give a sharp asymptotic formula for N (k , h , g , q) = ∑ a = 1 h ∤ (a + (g a ) q) τ ⁡ | a − (g a ) q | 2 k where k is any nonnegative integer and (a) q denotes the smallest positive residue of a modulo q. In addition, we know that N (h , g , q) = N (0 , h , g , q). [ABSTRACT FROM AUTHOR]

Let n be a positive integer with n > 1 and let a , b be fixed coprime positive integers with min { a , b } > 2. In this paper, using the Baker method, we proved that, for any n , if a > max { 15064 b , b 3 / 2 } , then the equation (a n) x + (b n) y = ((a + b) n) z has no positive integer solutions (x , y , z) with x > z > y. Further, let A , B be coprime positive integers with min { A , B } > 1 and 2 | B. Combining the above conclusion with some existing results, we deduced that, for any n , if (a , b) = (A 2 , B 2) , A > max { 123 B , B 3 / 2 } and B ≡ 2 (mod 4) , then this equation has only the positive integer solution (x , y , z) = (1 , 1 , 1). Thus, we proved that the conjecture proposed by Yuan and Han is true for this case. [ABSTRACT FROM AUTHOR]

Outer approximation (OA) for solving convex mixed-integer nonlinear programming (MINLP) problems is heavily dependent on the convexity of functions and a natural issue is to relax the convexity assumption. This paper is devoted to OA for dealing with a pseudo-convex MINLP problem. By solving a sequence of nonlinear subproblems, we use Lagrange multiplier rules via Clarke subdifferentials of subproblems to introduce a parameter and then equivalently reformulate such MINLP as the mixed-integer linear program (MILP) master problem. Then, an OA algorithm is constructed to find the optimal solution to the MNILP by solving a sequence of MILP relaxations. The OA algorithm is proved to terminate after a finite number of steps. Numerical examples are illustrated to test the constructed OA algorithm. [ABSTRACT FROM AUTHOR]

Let k = 1 be an integer, and let D be a finite and simple digraph with vertex set V (D). A weak signed Roman k-dominating function (WSRkDF) on a digraph D is a function f: V (D) {-1, 1, 2} satisfying the condition that P xN-[v] f(x) = k for each v V (D), where N-[v] consists of v and all vertices of D from which arcs go into v. The weight of a WSRkDF f is w(f) = P vV (D) f(v). The weak signed Roman k-domination number kw sR(D) is the minimum weight of a WSRkDF on D. In this paper we initiate the study of the weak signed Roman k-domination number of digraphs, and we present different bounds on kw sR(D). In addition, we determine the weak signed Roman k-domination number of some classes of digraphs. Some of our results are extensions of well-known properties of the weak signed Roman domination number wsR(D) = 1w sR(D) and the signed Roman k-domination number k sR(D). [ABSTRACT FROM AUTHOR]

Let n ≥ 2 be an integer. The [ 1 , n ] -factor of a graph G is a spanning subgraph F if 1 ≤ deg F (x) ≤ n for all x ∈ V (G) , and the { K 2 , C i } -factor is a subgraph whose each component is either K 2 or C i . In this paper, we give the lower bounds with regard to tight toughness, isolated toughness and binding number to guarantee the existence of the [ 1 , n ] -factors and { K 2 , C i | i ≥ 4 } -factors for a graph. [ABSTRACT FROM AUTHOR]

The Ruler function or the Gros sequence is a classical infinite integer sequence that underlies some interesting mathematical problems. In this paper, we provide four new problems containing this type of sequence: (i) demographic discrete dynamical automaton, (ii) the middle interval Cantor set, (iii) construction by duplication of polygons and (iv) the horizontal visibility sequence at the accumulation point of the Feigenbaum cascade. In all of them, the infinite sequence is obtained through a recursive procedure of duplication. The properties of the ruler sequence, in particular, those relating to recursiveness and self-containing, are used to achieve a deeper understanding of these four problems. These new representations of the ruler sequence could inspire new studies in the field of discrete mathematics. [ABSTRACT FROM AUTHOR]

For S ⊆ V (G) , κ G (S) denotes the maximum number k of edge disjoint trees T 1 , T 2 , ... , T k in G, such that V (T i) ∩ V (T j) = S for any i , j ∈ { 1 , 2 , ... , k } and i ≠ j . For an integer 2 ≤ r ≤ | V (G) | , the generalized r-connectivity of G is defined as κ r (G) = min { κ G (S) | S ⊆ V (G) and | S | = r } . In fact, κ 2 (G) is the traditional connectivity of G. Hence, the generalized r-connectivity is an extension of traditional connectivity. The exchanged folded hypercube E F H (s , t) , in which s ≥ 1 and t ≥ 1 are positive integers, is a variant of the hypercube. In this paper, we find that κ 3 (E F H (s , t)) = s + 1 with 3 ≤ s ≤ t . [ABSTRACT FROM AUTHOR]