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In two previous papers [Katsanikas & Wiggins, 2021a, 2021b], we developed two methods for the construction of periodic orbit dividing surfaces for Hamiltonian systems with three or more degrees of freedom. We applied the first method (see [Katsanikas & Wiggins, 2021a]) in the case of a quadratic Hamiltonian system in normal form with three degrees of freedom, constructing a geometrical object that is the section of a 4D toroidal structure in the 5D energy surface with the space x = 0. We provide a more detailed geometrical description of this object within the family of 4D toratopes. We proved that this object is a dividing surface and it has the no-recrossing property. In this paper, we extend the results for the case of the full 4D toroidal object in the 5D energy surface. Then we compute this toroidal object in the 5D energy surface of a coupled quadratic normal form Hamiltonian system with three degrees of freedom. [ABSTRACT FROM AUTHOR]

The Drazin inverse is connected with the notion of index and core-nilpotent decomposition whenever it is discussed in the context of ring of matrices over complex field. In the absence of Drazin inverse for a given element from an arbitrary associative ring (not necessarily with unity), in this paper, the notion of right (left) core-nilpotent decomposition has been introduced and established its relations with right (left) π -regular property. In fact, the class of such decomposition has been characterized. In case of regular ring, observed that an element is right (left) π -regular if and only if it has a right (left) core-nilpotent decomposition. In the process, several properties of sharp order in an associative ring are studied and with the help of the same, new characterizations of Drazin inverse over an associative ring are obtained and the relation between core-nilpotent decomposition and the Drazin inverse is obtained. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new class of ring called w p - S -Noetherian ring, which is a weak version of S -Noetherian ring property and study the transfer of this notion to various context of commutative ring extensions such as direct product, trivial ring extensions and amalgamation of rings. Furthermore, we define the concept of nonnil w p - S -Noetherian ring property which is a generalization of the w p - S -Noetherian domain property and establish a characterization of this notion using pullbacks. [ABSTRACT FROM AUTHOR]

Let R be a commutative ring with identity and S be a multiplicatively closed subset of R. The purpose of this paper is to introduce the concept of weakly S -primary ideals as a new generalization of weakly primary ideals. An ideal I of R disjoint with S is called a weakly S -primary ideal if there exists s ∈ S such that whenever 0 ≠ a b ∈ I for a , b ∈ R , then s a ∈ I or s b ∈ I. The relationships among S -prime, S -primary, weakly S -primary and S - n -ideals are investigated. For an element r in any general ZPI-ring, the (weakly) S r -primary ideals are characterized where S r = { 1 , r , r 2 , ... }. Several properties, characterizations and examples concerning weakly S -primary ideals are presented. The stability of this new concept with respect to various ring-theoretic constructions such as the trivial ring extension and the amalgamation of rings along an ideal are studied. Furthermore, weakly S -decomposable ideals and S -weakly Laskerian rings which are generalizations of S -decomposable ideals and S -Laskerian rings are introduced. [ABSTRACT FROM AUTHOR]

In this paper, we introduce and study a generalization of coretractable modules. An R -module M is called coretractable relative to an R-module N if Hom R (M / X , N) ≠ 0 , for any proper submodule X of M. It is shown that a right R -module M is coretractable relative to ⊕ i ∈ I N i if and only if M is coretractable relative to N i , for some i ∈ I. Also we show that if R is a right quasi-duo ring, then there exists a right R -module that it is coretractable relative to N , for any nonzero right R -module N if and only if R is a left perfect local ring. [ABSTRACT FROM AUTHOR]

In this paper, we calculate electrical resistance scaling exponents for analogues of "carpet" and "sponge" fractals in higher-dimensional embedding spaces. The calculation idealizes the voids that define such fractals as spherical, and exploits the elementary theory of dielectrics. Possible applications include models of elementary particles with "extra" dimensions, and new large-dimension methods for the theory of fractals. [ABSTRACT FROM AUTHOR]

The idea of nondeterministic typical hesitant fuzzy automata is a generalization of the fuzzy automata presented by Costa and Bedregal. This paper, presents the sufficient and necessary conditions for a typical hesitant fuzzy language to be computed by nondeterministic typical hesitant fuzzy automata. Besides, the paper introduces a new class of typical hesitant fuzzy automata with crisp transitions, and we will show that this new class is equivalent to the original class introduced by Costa and Bedregal. [ABSTRACT FROM AUTHOR]

In this paper, we generalize the concepts of (perfect) Roman and Italian dominations to (perfect) strong Roman and Roman k -domination for arbitrary positive integer k. These generalizations cover some of previous ones. After some comparison of their domination numbers, as a first study of these concepts, we study the (perfect, strong, perfect strong) Roman k -domination numbers of complete bipartite graphs. [ABSTRACT FROM AUTHOR]

Let V ⊂ q d be a regular variety, k ≥ 3 be an integer and A ⊆ V. Covert et al. proved the following generalization of the Erdős–Falconer distance problem that if | A | ≫ q d − 1 2 + 1 k − 1 , then we have Δ k (A) = { | x 1 + ⋯ + x k | : x i ∈ A , 1 ≤ i ≤ k } ⊇ q ∗. In this paper, we provide improvements and extensions of their result. [ABSTRACT FROM AUTHOR]

Let G and A be finite groups of relative coprime orders and A act on G via automorphisms. In this paper, we prove that when every maximal A-invariant subgroup of G that contains the normalizer of some Sylow subgroup has prime index, then G is supersolvable; if every non-nilpotent maximal A-invariant subgroup of G has prime index or is normal in G, then G is a Sylow tower group. [ABSTRACT FROM AUTHOR]

In this paper, we define the multi-Stirling numbers of the first kind by means of the multiple logarithm and as a generalization of the Stirling numbers of the first kind. Then we introduce two additional special numbers by using the multiple logarithm, namely the modified multi-Bernoulli numbers as a generalization of the higher-order Bernoulli numbers and the multi-Lah numbers of type 2 as a generalization of the Lah numbers. For both of these special numbers, we derive explicit expressions for them which involve the multi-Stirling numbers of the first kind. [ABSTRACT FROM AUTHOR]

Let G be a group with identity e and R be a graded multiplicative hyperring. In this paper, we introduce and study the notion of graded n -absorbing hyperideals of R as a generalization of prime hyperideals. Many properties, examples and characterizations of graded n -absorbing hyperideals of R are introduced. Moreover, we investigate the behavior of this structure under homogeneous components, graded hyperring homomorphisms, Cartesian product and fundamental relations. We consider the fundamental relation γ ∗ on R as the smallest equivalence relation such that the quotient R / γ ∗ is a ring. Also, we show that if R is a graded hyperring, then R / γ ∗ is a graded ring. [ABSTRACT FROM AUTHOR]

The generalization is one of the main concerns of machine learning method. k-SVCR is an important multi-class classification algorithm, but there is no theoretical analysis on the generalization of k-SVCR algorithm up to now. Therefore, in this paper, we consider multi-class support vector classification and regression (MSVCR) algorithm. We first establish the generalization bounds of MSVCR based on uniformly ergodic Markovian chain (u.e.M.c.) samples, and obtain its fast learning rate of MSVCR. As applications, we estimate the generalization of MSVCR for independent and identically distributed (i.i.d.) observations and strongly mixing observations, respectively. We also propose a new MSVCR algorithm based on q-times Markovian resampling (MSVCR-Mar-q). The experimental studies indicate that compared to the classical k-SVCR and other multi-class SVM algorithms, the proposed algorithm not only has smaller misclassification rate, but also has less sampling and training total time. [ABSTRACT FROM AUTHOR]

In this paper, we consider a generalization of the theory of Higgs bundles over a smooth complex projective curve in which the twisting of the Higgs field by the canonical bundle of the curve is replaced by a rank 2 vector bundle. We define a Hitchin map and give a spectral correspondence. We also state a Hitchin–Kobayashi correspondence for a generalization of Hitchin's equations to this situation. In a certain sense, this theory lies halfway between the theories of Higgs bundles on a curve and on a higher-dimensional variety. [ABSTRACT FROM AUTHOR]

The longest increasing subsequence (LIS) problem aims to find the subsequence exhibiting an increasing trend in a numeric sequence with the maximum length. In this paper, we generalize the LIS problem to the longest wave subsequence (LWS) problem, which encompasses two versions: LWSt and LWSr. Given a numeric sequence A of distinct values and a target trend sequence T, the LWSt problem aims to identify the longest subsequence of A that preserves the trend of the prefix of T. And, the LWSr problem aims to find the longest subsequence of A within r segments, alternating increasing and decreasing subsequences. We propose two efficient algorithms for solving the two versions of the LWS problem. For the LWSt problem, the time complexity of our algorithm is O(nlog n), where n represents the length of the given numeric sequence A. Additionally, we propose an O(rnlog n)-time algorithm for solving the LWSr problem. In both algorithms, we utilize the priority queues for the insertion, deletion, and successor operations. [ABSTRACT FROM AUTHOR]

Semiabelian rings, defined by the property that each of their idempotents is either left semicentral or right semicentral, are one among several natural generalizations of abelian rings. In this semi-expository paper, we review a number of interesting properties of semiabelian (and other closely allied) rings that are so far well known only for abelian rings. For instance, semiabelian rings R are always "J-abelian" in the sense that each idempotent of R maps onto a central idempotent in R /rad(R). On the other hand, J-abelian rings turn out to be precisely the "strongly perspective rings" as well as the "strongly IC rings", and the von Neumann regular elements in such rings are automatically strongly regular and are closed under taking n -th powers. In addition, all J-abelian exchange rings have idempotent stable range one, and are in particular clean (although not necessarily strongly clean) rings. [ABSTRACT FROM AUTHOR]

We provide three generalizations of all best known generating sets for S n and A n : (1) m -cycles with consecutive entries; (2) m -cycles with fixed first ℓ entries; (3) two cycles, either (1 , ... , m) , (1 , ... , n) or (1 , ... , m) , (2 , ... , n). Our work on (3) fills a gap in the history of the problem, going back to a 1939 paper by S. Piccard [Sur les bases du groupe symétrique et du groupe alternant, Math. Ann. 116(1) (1939) 752–767]. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new convexity notion for inter-valued functions, known as Geometrically–Arithmetically Cr-h-convex functions (abbreviated as GA-Cr-h-CFs) and explore its properties. The family of GA-Cr-h-CFs simultaneously covers the family of GA-CFs, GA-h-CFs and GA-Cr-CFs. On the other hand, we demonstrate its equivalence with convex functions (CFs), h-CFs, Cr-CFs and Cr-h-CFs via exponential function. We also investigate necessary and sufficient conditions for GA-Cr-h-CFs via two corresponding real-valued GA-h-CFs. Due to significance and applications of mathematical inequalities, we apply our findings and propose several inequalities such as Jensen–Mercer inequality (JMI), Hermite–Hadamard inequality (HHI) and weighted Hermite–Hadamard-type inequalities (wHHIs) for GA-Cr-h-CFs. Through our results, we re-capture many known results and inequalities for subclasses of GA-Cr-h-CFs studied in the recent literature. [ABSTRACT FROM AUTHOR]

Incremental learning is one of the effective methods of learning from the accumulated training samples and the large-scale dataset. The main advantages of incremental learning consist of making full use of historical information, reducing the training scale greatly and saving space and time consumption. Despite extensive research on incremental support vector machine (SVM) learning algorithms, most of them are based on independent and identically distributed samples (i.i.d.). Not only that, there has been no theoretical analysis of incremental SVM learning algorithms. In this paper, we mainly study the generalization bounds of this incremental SVM learning algorithm whose samples are based on uniformly geometric Markov chains, and exponentially strongly mixing sequence. As a special case, we also obtain the generalization bounds of i.i.d. samples. [ABSTRACT FROM AUTHOR]

Cayley graph is a graph that encodes the abstract structure of a group. It gives a way of encoding information about a group in a graph. On the other hand, hypergroup is a generalization of group in which the composition of any two elements is a non-empty set. The purpose of this paper is to find a suitable generalization of Cayley graphs to cover hypergroups. More precisely, we introduce generalized Cayley graphs over hypergroups, study their properties and find a simple tool to construct large connected GCH-graphs from smaller GCH-graphs by using graph product. [ABSTRACT FROM AUTHOR]

In this paper, we construct two kinds of q -differential integrable hierarchies. The first one is the q -universal character of B-type (BUC) and lattice q -BUC hierarchies expressed by q -shift operators. It is also shown that the lattice q -BUC hierarchy reduces to the q -BUC hierarchy with certain conditions. The other one is the q -deformed generalized symplectic Schur functions and the q -deformed generalized Q-functions defined by q -derivative operators. By means of the q -deformed vertex operators, the q -deformed symplectic universal character (SUC) and BUC hierarchies have been presented. Moreover, it is proved that q -deformed generalized symplectic Schur functions and generalized Q-functions are solutions of the q -deformed SUC and BUC hierarchies, respectively. Meanwhile, by using multi-component tau functions, we establish the multi-component q -deformed SUC and BUC hierarchies. In addition, based upon the discrete tau functions, the multi-component modified q -deformed SUC and BUC hierarchies have been discussed. [ABSTRACT FROM AUTHOR]

In recent years, the study of social-based recommender systems has become an active research topic. We incorporate a combination of static and dynamic interest characteristics to predict users' real-time dynamic interests, which has rarely been considered in previous studies. In this paper, we propose a graph neural network social recommendation model that integrates static and dynamic feature relationships (FSDFR-GNNSR). The model uses a graph embedding algorithm to extract static features of users and movies, and takes the static features as input to gated recurrent unit (GRU), so that the model can take static features into consideration while modeling user dynamic behavior. Finally, we use graph attention networks to represent the dynamic influence of friends, simplify the update strategy of second-order neighbor nodes. We apply graph pooling operations to improve the generalization ability of the algorithm. Empirical analyses on real datasets show that the proposed approach achieves superior performance to existing approaches. [ABSTRACT FROM AUTHOR]

Our goal in this paper is to establish some second main theorems for holomorphic mappings from angular domains into projective varieties intersecting arbitrary families of moving hypersurfaces with truncated counting functions. Our results generalize and also improve all previous results for the case of holomorphic mappings from the complex plane. [ABSTRACT FROM AUTHOR]

In quantum mechanics, quantum discord is a valuable quantum resource. As a generalization of the initial quantum discord, the multipartite quantum discord (MQD) exhibits excellent properties in the complete monogamy relation. In this paper, the dynamics of MQD are investigated for the Greenberger–Horne–Zeilinger (GHZ) state in Markovian environments. More specifically, it indicates that MQD of the GHZ state drops significantly over time in the depolarizing noise channel. Interestingly, quantum decoherence has no influence on MQD of the GHZ state in the Pauli channel σ z . Subsequently, the analytical formulas of MQD are derived with six parameters in the Pauli channels σ x and σ y , respectively. This is conducive to more efficient use of the GHZ state for various quantum tasks. [ABSTRACT FROM AUTHOR]

In this paper, we prove a criterion for the irrationality of certain constants which arise from the Ramanujan summation of a family of infinite divergent sums. As an application, we provide a sufficient criterion for the irrationality of the values of the Riemann zeta function in the interval (0 , 1). We further see that our discussion leads to a natural generalization of a result of Sondow on the irrationality criterion for the Euler–Mascheroni constant. [ABSTRACT FROM AUTHOR]

Let A (G) and D (G) be the adjacency matrix and the diagonal matrix of vertex degrees of a graph G , respectively. For any real α ∈ [ 0 , 1 ] , Nikiforov defined the matrix A α (G) as A α (G) = α D (G) + (1 − α) A (G). Let G ¯ be complement of G. Define S α (G) = A α (G ¯) − A α (G) = (1 − α) J + (α n − 1) I − 2 A α (G) , α ∈ [ 0 , 1 ] , where I and J are the identity matrix and the all-ones matrix, respectively. Since S 0 (G) = A (G ¯) − A (G) = J − I − 2 A (G) is the Seidel matrix of G , this new matrix is the generalization of Seidel matrix. Further, S α (G) = (1 − α) J + (α n − 1) I − 2 α D (G) − 2 (1 − α) A (G) is a subset of universal adjacency matrices defined by Haemers and Omidi. In this paper, S α -eigenvalues and S α -energy of G , respectively, are investigated, which not only extends Seidel matrix but also enriches the theory of universal adjacency matrices. [ABSTRACT FROM AUTHOR]

In this paper, by considering the notion of fuzzy hyperoperation, we define the concept of fuzzy hyper B C K (B C I) -algebras as a generalization of B C K (B C I) -algebras and study related properties. Moreover, we introduce the notions of fuzzy hyper B C I -ideal and weak (strong) fuzzy hyper B C I -ideal, and then examine the relations among them. Finally, we investigate the relation between fuzzy hyper B C I -algebras with fuzzy hypersemigroup, fuzzy hypergroup and hyper B C I -algebras. [ABSTRACT FROM AUTHOR]

The aim of this paper is to generalize the results related to fuzzy filters of ordered semigroups. In this regard, the concept of fuzzy (m , n) -filters of ordered semigroups is initiated, nontrivial examples are presented, and its properties are investigated. It is known that a fuzzy set of an ordered semigroup is a fuzzy filter if and only its complement is a fuzzy prime ideal. As a generalization of the latter result, it is proved that a fuzzy set of an ordered semigroup is a fuzzy (m , n) -filter if and only if its complement is a prime generalized fuzzy (m , n) -ideal. Furthermore, it is shown that in an (m , n) -regular semigroup, fuzzy (m , n) -filters and fuzzy bi-filters coincide. [ABSTRACT FROM AUTHOR]

In this paper, we prove some existence results of a critical p -Laplacian equation involving critical and subcritical Hardy potential on compact Riemannian manifolds. [ABSTRACT FROM AUTHOR]

In this paper, we generalize the Mandelbrot set using quaternions and spherical coordinates. In particular, we use pure quaternions to define a spherical product. This product, which is inspired by the product of complex numbers, adds the angles and multiplies the radii of the spherical coordinates. We show that the algebraic structure of pure quaternions with the spherical product is a commutative unital magma. Then, we present several generalizations of the Mandelbrot set. Among them, we present a set that is visually identical to the so-called Mandelbulb. We show that this set is bounded and that it can be generated by an escape time algorithm. We also define another generalization, the bulbic Mandelbrot set. We show that one of its 2D cuts has the same dynamics as the Mandelbrot set and that we can generate this set only with a quaternionic product, without using the spherical product. [ABSTRACT FROM AUTHOR]

In this paper, we review the properties of six families of orthogonal polynomials that form the main bulk of the collection called the Askey–Wilson scheme of polynomials. We give connection coefficients between them as well as the so-called linearization formulae and other useful important finite and infinite expansions and identities. An important part of the paper is the presentation of probabilistic models where most of these families of polynomials appear. These results were scattered within the literature in recent 15 years. We put them together to enable an overall outlook on these families and understand their crucial rôle in the attempts to generalize Gaussian distributions and find their bounded support generalizations. This paper is based on 65 items in the predominantly recent literature. [ABSTRACT FROM AUTHOR]

Since the study by Jacobi and Hecke, Hecke-type identities have received a lot of attention. In this paper, we obtain a general transformation formula for q -series. Based on this formula, we derive a new q -formula in this paper, which clearly includes infinitely many q -identities. Other identities of Andrews and Liu and new identities for Hecke-type series are also discussed. As applications, we utilize some of these Hecke-type identities to establish families of inequalities for several partition functions. [ABSTRACT FROM AUTHOR]

In this paper, we study the real interpolation theory for variable Lorentz–Karamata spaces as well as for the corresponding martingale Hardy spaces. As a consequence, the generalization of Doob's maximal inequality will be proved. Moreover, the atomic decompositions of the variable martingale Hardy–Lorentz–Karamata spaces are also presented. The results obtained here are new even for martingale Hardy–Lorentz–Karamata spaces. [ABSTRACT FROM AUTHOR]

It is well known that Frobenius groups can be defined by their complement subgroups. But until now we cannot use a complement subgroup to define a modular Frobenius group. In the present paper, a generalization of Frobenius complements is used as a characterization of a class of modular Frobenius groups. In fact, we build a connection between modular Frobenius groups and Frobenius–Wielandt groups. [ABSTRACT FROM AUTHOR]

The aim of the paper is to characterize the equation 1 + Δ (R) = U (R) , where J (R) ⊆ Δ (R) = { x ∈ R : x + u ∈ U (R)   for all   u ∈ U (R) } that is the largest Jacobson radical subring of R and U (R) is the set of invertible elements of a ring R. We show that this equation is closely related to U J -rings and rings whose elements can be written as the sum of an idempotent and an element from Δ (R). After presenting several characterizations and properties of this equation, we consider the rings satisfying the equation 1 + Δ (R) = U (R) within many well-studied classes of rings. Finally, we close the paper with group rings. [ABSTRACT FROM AUTHOR]

Previous studies have shown that sampling processing and identification methods for neural networks damage identification in multi-channel and multi-source structural dynamic response data exhibit diversity, and the robustness and generalization issues of the model have not been effectively solved. This paper proposes a sample preprocessing technique and mixed training method suitable for multi-channel and multi-source dynamic response data to optimize the current structural damage identification methods based on neural networks. Through multi-dimensional discrete autocorrelation processing and Fourier transform, the preprocessed datasets from multiple sensors with multi-channel can access multi-channel CNN. Furthermore, the measured datasets from the actual structures are mixed with numerical simulation datasets before being used for neural network model training to address the model calibration and the imbalanced sample size under each classification label. The results of extensive model experiments and finite element verification of specimens show that the method performs outstandingly in detecting and identifying damage to simply supported beam structures using multi-channel CNN. The neural network model trained with preprocessed samples exhibits excellent robustness. The model using the mixed training method still performs well in identifying the accuracy of damage location and degree in simply supported beam structures after changing beam section and initial excitation. The method has certain generalization ability in detecting unknown damage in simply supported beam structures. [ABSTRACT FROM AUTHOR]

Soft set theory is a general mathematical method for dealing with uncertain data. It is a classification of elements of the universe with respect to some given set of parameters. Semigraph is a generalization of graph which is different from hypergraph. Soft semigraph was introduced by applying the concept of soft set in semigraph. Connectedness is one of the basic concepts of graph theory. In this paper, we introduce the concepts of s -walk, s -trail, s -path and s -cycle in soft semigraphs and we define an s -connected soft semigraph. We explain the procedures of vertex and f p -edge deletions from a soft semigraph. Also, we introduce p -part cut vertex, s -cut vertex, p -part bridge and s -bridge of a soft semigraph and investigate some of their properties. [ABSTRACT FROM AUTHOR]

The concept of k -independent set, introduced by Fink and Jacobson in 1986, is a natural generalization of classical independence set. A k -independent set is a set of vertices whose induced subgraph has maximum degree at most k. The k -independence number of G , denoted by α k (G) , is defined as the maximum cardinality of a k -independent set of G. As a natural counterpart of the k -independence number, we introduced the concept of k -edge-independence number. An edge set M in E (G) is called k -edge-independent if the maximum degree of the subgraph induced by the edges in M is less or equal to k. The k -edge-independence number, denoted β k (G) , is defined as the maximum cardinality of a k -edge-independent set. In this paper, we study the Nordhaus–Gaddum-type results for the parameter α k (G) and β k (G). We obtain sharp upper and lower bounds of α k (G) + α r (G ¯) , α k (G) ⋅ α r (G ¯) , β k (G) + β r (G ¯) and β k (G) ⋅ β r (G ¯) for a graph G of order n. Some graph classes attaining these bounds are also given. [ABSTRACT FROM AUTHOR]

We establish a family of coercive Korn-type inequalities for generalized incompatible fields in the superlinear growth regime under sharp criteria. This extends and unifies several previously known inequalities that are pivotal to the existence theory for a multitude of models in continuum mechanics in an optimal way. Different from our preceding work [F. Gmeineder, P. Lewintan and P. Neff, Optimal incompatible Korn–Maxwell–Sobolev inequalities in all dimensions, Calc. Var. PDE 62 (2023) 182], where we focused on the case p = 1 and incompatibilities governed by the matrix curl, the case p > 1 considered in this paper gives us access to substantially stronger results from harmonic analysis but conversely deals with more general incompatibilities. Especially, we obtain sharp generalizations of recently proved inequalities by P. Lewintan, S. Müller and P. Neff [Korn inequalities for incompatible tensor fields in three space dimensions with conformally invariant dislocation energy, Calc. Var. PDE 60 (2021) 150] in the realm of incompatible Korn-type inequalities with conformally invariant dislocation energy. However, being applicable to higher-order scenarios as well, our approach equally gives the first and sharp inequalities involving Kröner's incompability tensor inc. [ABSTRACT FROM AUTHOR]

In this paper, we work out modules with the property that every z -closed submodules are essentially embedded in direct summands. This class of modules properly contains the class of CS-modules as well as some of their generalizations. It is shown that the class of modules with former property is closed under direct sums under a certain condition. Taking into account this case, we obtain several results on the inheritance of the latter closure property. Moreover, we consider the behavior of this new property for extensions of modules. [ABSTRACT FROM AUTHOR]

Suppose is an alternative ring containing a non-trivial idempotent and χ m be a mapping from into itself. In this paper, we study the Jordan n-higher derivations on alternative rings and prove that under some mild conditions every multiplicative Jordan n-higher derivations on is additive. It is to be noted that the similarity to the notions is only in its written form, but not in its theoretical structures, because the Pierce decomposition used in the results for alternative rings is the generalization of the Pierce decomposition for associative rings. [ABSTRACT FROM AUTHOR]

In this paper, we introduce and study a one-parameter generalization of bihyperbolic Jacobsthal numbers. We give some identities and the generating function for these numbers. [ABSTRACT FROM AUTHOR]

Graph Convolution Networks (GCNs) have been shown to be very effective in utilizing pair-wise relationships across samples. They have been successfully applied to solve various machine learning problems in practice. In many applications, the construction of GCNs involves more than one layer. However, their generalization and stability analysis are limited. The main aim of this paper is to analyze GCNs with two layers. The formulation is based on transductive semi-supervised learning and the filtering is done in the eigen-domain. We show the uniform stability of the neural network and the convergence of the generalization gap to zero. The analysis of two-layer GCN is more involved than the single-layer case and requires some new estimates of the neural network's quantities. The analysis confirms the usefulness of GCNs. It also sheds light on the design of the neural network, for instance, how the data should be scaled to achieve the uniform stability of the learning process. Some experimental results on benchmark datasets are presented to illustrate the theory. [ABSTRACT FROM AUTHOR]

Twisted links are a generalization of classical links and correspond to stably equivalence classes of links in thickened surfaces. In this paper, we introduce twisted intersection colorings of a diagram and construct two invariants of a twisted link using such colorings. As an application, we show that there exist infinitely many pairs of twisted links such that for each pair the two twisted links are not equivalent but their double coverings are equivalent. We also introduce a method of constructing a pair of twisted links whose double coverings are equivalent. [ABSTRACT FROM AUTHOR]

We introduce new classes of higher-order functional, termed higher-order (F , α , ρ , d) − convex and higher-order (C , α , ρ , d) − convex functionals. These classes are illustrated by nontrivial examples. A pair of higher-order multiobjective symmetric fractional variational programs with cone constraints and fixed boundary conditions is formulated. Appropriate duality results are discussed utilizing the aforementioned assumptions. The results in this paper are generalizations of the results already existing in literature. [ABSTRACT FROM AUTHOR]

Extriangulated categories were introduced by Nakaoka and Palu as a simultaneous generalization of exact categories and triangulated categories. In this paper, we show that the idempotent completion of an extriangulated category admits a natural extriangulated structure. As an application, we prove that the idempotent completion of a recollement of extriangulated categories is still a recollement. [ABSTRACT FROM AUTHOR]

Let R be a commutative ring and S a given multiplicative closed subset of R. In this paper, we introduce the new concept of S -torsion exact sequences (respectively, S -torsion commutative diagrams) as a generalization of exact sequences (respectively, commutative diagrams). As an application, they can be used to characterize two classes of modules that are generalizations of projective modules. [ABSTRACT FROM AUTHOR]

In this paper, we first obtain Hermite–Hadamard–Fejér inequalities for co-ordinated trigonometrically ρ -convex functions. By using these obtained inequalities, we give some inequalities involving Riemann–Liouville fractional integrals. The inequalities presented in this study provide generalizations of some results given in earlier works. [ABSTRACT FROM AUTHOR]

Let S be a commutative semiring with unity different than zero. In this paper, we study the concept of 2-absorbing ideals of S which can be considered as a generalization of prime ideals. Among others, it is shown that the radical of a 2-absorbing ideal is also a 2-absorbing ideal and there are at most 2 prime k-ideals of S that are minimal over a 2-absorbing ideals. [ABSTRACT FROM AUTHOR]

Let , , and be all positive divisors of . For , define . In this paper, by combining ideas from the finite Fourier transform of arithmetic functions and Ramanujan sums, we give a short proof for the following result: the number of solutions of the linear congruence , with , , is where is a Ramanujan sum. Some special cases and other forms of this problem have been already studied by several authors. The problem has recently found very interesting applications in number theory, combinatorics, computer science, and cryptography. The above explicit formula generalizes the main results of several papers, for example, the main result of the paper by Sander and Sander [ J. Number Theory 133 (2013) 705-718], one of the main results of the paper by Sander [ J. Number Theory 129 (2009) 2260-2266], and also gives an equivalent formula for the main result of the paper by Sun and Yang [ Int. J. Number Theory 10 (2014) 1355-1363]. [ABSTRACT FROM AUTHOR]