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We identify a sufficient condition, treewidth-pliability, that gives a polynomial-time algorithm for an arbitrarily good approximation of the optimal value in a large class of Max-2-CSPs parameterised by the class of allowed constraint graphs (with arbitrary constraints on an unbounded alphabet). Our result applies more generally to the maximum homomorphism problem between two rational-valued structures. The condition unifies the two main approaches for designing a polynomial-time approximation scheme. One is Baker's layering technique, which applies to sparse graphs such as planar or excluded-minor graphs. The other is based on Szemerédi's regularity lemma and applies to dense graphs. We extend the applicability of both techniques to new classes of Max-CSPs. However, we prove that the condition cannot be used to find solutions (as opposed to approximating the optimal value) in general. Treewidth-pliability turns out to be a robust notion that can be defined in several equivalent ways, including characterisations via size, treedepth, or the Hadwiger number. We show connections to the notions of fractional-treewidth-fragility from structural graph theory, hyperfiniteness from the area of property testing, and regularity partitions from the theory of dense graph limits. These may be of independent interest. In particular, we show that a monotone class of graphs is hyperfinite if and only if it is fractionally-treewidth-fragile and has bounded degree. [ABSTRACT FROM AUTHOR]

In this article, we aim to provide a satisfactory algebraic description of the set of affiliated operators for von Neumann algebras. Let |$\mathscr{M}$| be a von Neumann algebra acting on a Hilbert space |$\mathcal{H}$|⁠ , and let |$\mathscr{M}_{\text{aff}}$| denote the set of unbounded operators of the form |$T = AB^{\dagger }$| for |$A, B \in \mathscr{M}$| with |$\text{null} (B) \subseteq \text{null} (A)$|⁠ , where |$(\cdot)^{\dagger }$| denotes the Kaufman inverse. We show that |$\mathscr{M}_{\text{aff}}$| is closed under sum, product, Kaufman-inverse, and adjoint, and has the structure of a (right) near-semiring. Moreover, the above quotient representation of an operator in |$\mathscr{M}_{\text{aff}}$| is essentially unique. Thus, we may view |$\mathscr{M}_{\text{aff}}$| as the multiplicative monoid of unbounded operators on |$\mathcal{H}$| generated by |$\mathscr{M}$| and |$\mathscr{M}^{\dagger }$|⁠. We further show that our definition of affiliation, as reflected in |$\mathscr{M}_{\text{aff}}$|⁠ , subsumes the traditional one. Let |$\Phi $| be a unital normal *-homomorphism between represented von Neumann algebras |$(\mathscr{M}; \mathcal{H})$| and |$(\mathscr{N}; \mathcal{K})$|⁠. Using the quotient representation, we obtain a canonical extension of |$\Phi $| to a mapping |$\Phi _{\text{aff}}: \mathscr{M}_{\text{aff}} \to \mathscr{N}_{\text{aff}}$| which is a near-semiring homomorphism that respects Kaufman-inverse and adjoint; in addition, |$\Phi _{\text{aff}}$| respects Murray–von Neumann affiliation of operators and also respects strong sum and strong product. Thus, |$\mathscr{M}_{\text{aff}}$| is intrinsically associated with |$\mathscr{M}$| and transforms functorially as we change representations of |$\mathscr{M}$|⁠. Furthermore, |$\Phi _{\text{aff}}$| preserves operator properties such as being symmetric, or positive, or accretive, or sectorial, or self-adjoint, or normal, and also preserves the Friedrichs and Krein–von Neumann extensions of densely defined closed positive operators. As a proof of concept, we transfer some well-known results about closed unbounded operators to the setting of closed affiliated operators for properly infinite von Neumann algebras, via "abstract nonsense". [ABSTRACT FROM AUTHOR]

In this work, we use the concept of G -weak graded rings and G -weak graded modules, which are based on grading by a set G of left coset representatives for the left action of a subgroup H of a finite X on X , to define the conjugation action of the set G and to generalize and prove some results from the literature. In particular, we prove that a G -weak graded ring R is strongly graded if and only if each G -weak graded R-module V is induced by an R e G -module. Moreover, we prove that the additive induction functor (−) R and the restriction functor (−) e G form an equivalence between the categories G - GrR - Mod and R e G - Mod when R is strongly G -weak graded. Furthermore, some related results and illustrative examples of G -weak graded R-modules and their morphisms are provided. [ABSTRACT FROM AUTHOR]

In this paper we study the set of MV-algebras with given prime spectrum and we introduce the class of spectral MV-algebras. An MV-algebra is spectral if it is generated by the union of all its prime ideals (or proper ideals, or principal ideals, or maximal ideals). Among spectral MV-algebras, special attention is devoted to bipartite MV-algebras. An MV-algebra is bipartite if it admits an homomorphism onto the MV-algebra of two elements. We prove that both bipartite MV-algebras and spectral MV-algebras can be finitely axiomatized in first order logic. We also prove that there is only, up to isomorphism, a set of MV-algebras with given prime spectrum. A further part of the paper is devoted to some relations between bipartite MV-algebras and their states. Recall that a state on an MV-algebra is a generalization of a probability measure on a Boolean algebra. Particular states are the states with Bayes' property. We show that an MV-algebra admits a state with the Bayes' property if and only if it is bipartite. [ABSTRACT FROM AUTHOR]

We prove formulae for the F 2 -Rasmussen invariant of satellite knots of patterns with wrapping number 2, using the multicurve technology for Khovanov and Bar-Natan homology developed by Kotelskiy, Watson, and the second author. A new concordance homomorphism, which is independent of the Rasmussen invariant, plays a central role in these formulae. We also explore whether similar formulae hold for the Ozsváth–Szabó invariant 휏. [ABSTRACT FROM AUTHOR]

In this paper, the group homomorphism graph is introduced and investigated. The group homomorphism graph, denoted by H I (G , G ′) , is an undirected graph in which the vertex set contains all homomorphisms excluding the monomorphisms and the zero homomorphism from the group G to the group G ′ , and two distinct vertices are adjacent if and only if the intersection of their kernels is non-trivial. We investigate the interplay between the graph-theoretic properties of this graph with algebraic properties of the groups. In this work, connectedness, diameter, clique number, chromatic number, domination number, independence number, etc. are found. [ABSTRACT FROM AUTHOR]

We characterise the slices of the category of graphs that are algebraically universal in terms of the structure of the slicing graph. In particular, we show that algebraic universality is obtained if, and only if, the slicing graph contains one of four fixed graphs as a subgraph. [ABSTRACT FROM AUTHOR]

We study rings whose modules and module homomorphisms display behavior similar to that of sets and their maps. For example, whenever there is an epimorphism A → B , there is a monomorphism B → A (Artinian principal ideal rings (PIR) satisfy this property and its dual for all modules). As a byproduct of this framework, we prove that a ring every factor ring of which cogenerates its cyclic right modules (one-sided version of Kaplansky's dual rings) is right Artinian and right serial. Consequently, R is an Artinian PIR if and only if every factor ring of R cogenerates its finitely generated right modules. These results can be viewed as partial answers to the CF problem, the FGF problem due to Faith and a question of Faith and Menal on strongly Johns rings. Some known results and the above one yield the following: A ring R is a direct sum of right Artinian right chain rings and Artinian PIR's if and only if every factor ring of R cogenerates its (uniform) cyclic right modules (with nonzero socle); so, such rings coincide with the right CES-rings of Jain and Lopez-Pérmouth, rings whose factors are right CF and rings that satisfy the above mentioned property for their cyclic right modules A and B. Finally, a ring is either simple Artinian or a right Artinian right chain ring if and only if one of any two cyclic right modules embeds in the other. [ABSTRACT FROM AUTHOR]

A regular matroid M on a finite set E is represented by a totally unimodular matrix. The set of vectors from Z E orthogonal to rows of the matrix form a regular chain group N. Assume that ψ is a homomorphism from N into a finite additive Abelian group A and let A ψ [ N ] be the set of vectors g from (A − 0) E , such that ∑ e ∈ E g (e) · f (e) = ψ (f) for each f ∈ N (where · is a scalar multiplication). We show that | A ψ [ N ] | can be evaluated by a polynomial function of | A | . In particular, if ψ (f) = 0 for each f ∈ N , then the corresponding assigning polynomial is the classical characteristic polynomial of M. [ABSTRACT FROM AUTHOR]

The mod-2 arithmetic Milnor invariants, introduced by Morishita, provide a decomposition law for primes in canonical Galois extensions of Q with unitriangular Galois groups and contain the Legendre and Rédei symbols as special cases. Morishita further proposed a notion of mod-q arithmetic Milnor invariants, where q is a prime power, for number fields containing the qth roots of unity and satisfying certain class field theory assumptions. We extend this theory from the number field context to general fields, by introducing a notion of a linking invariant for discrete valuations and orderings. We further express it as a Magnus homomorphism coefficient and relate it to Massey product elements in Galois cohomology. [ABSTRACT FROM AUTHOR]

In this study, we introduce the Sheffer stroke L-algebra and prove some fundamental theorems, propositions and lemmas of Sheffer Stroke L-algebras. The notions of filter and ultrafilter for Sheffer stroke L-algebra are studied. We give subalgebra and normal subset definitions of a Sheffer stroke L-algebras. Moreover, a homomorphism between Sheffer stroke L-algebras is introduced and isomorphism theorems are presented. Finally, we give three new algorithms for Sheffer stroke L-algebras. Thus, it is contributed to researchers on different application areas by presenting an algorithmic approach on this subject, for the first time in the literature. [ABSTRACT FROM AUTHOR]

The minimum spanning tree problem in graphs under budget-type degree constraints (DCMST) is a well-known NP-hard problem. Spanning trees do not always exist, and the optimum can not be approximated within a constant factor. Recently, solutions have been proposed to solve degree-constrained spanning problems in the case of limited momentary capacities of the nodes. For a given node, the constraint represents a limited degree of the node for each visit. Finding the solution with minimum cost is NP-hard and the related algorithms are not trivial. This paper focuses on this new spanning problem with heterogeneous capacity-like degree bounds. The minimum cost solution corresponds to a graph-related structure, i.e., a hierarchy. We study the conditions of its existence, and we propose its exact computation, a heuristic algorithm, and its approximation. [ABSTRACT FROM AUTHOR]

We construct a class of Gorenstein-projective modules over Milnor squares of rings. As an application, we obtain Gorenstein-projective modules over Morita context rings with two bimodule homomorphisms zero in the general setting instead of Artin algebras or Noetherian rings. [ABSTRACT FROM AUTHOR]

This article presents a new idea for an extension of the fuzzy INK algebra called bipolar fuzzy INK subalgebra. The objective of this study is to define the features that distinguish bipolar fuzzy INK-subalgebras of INK-algebras. The algebraic operations on these sub-algebras are also studied. The thorough examination allows us to prove a number of theorems that shed light on the connections between the higher and lower-level sets related to these ideas. In addition, several related topics are thoroughly examined, and the idea of homomorphism for bipolar fuzzy INK sub-algebras is introduced. [ABSTRACT FROM AUTHOR]

In this paper, we consider Morita rings with zero bimodule homomorphisms. We establish necessary and sufficient conditions for all strongly complete injective resolutions over a Morita ring Δ (0 , 0) = A A N B B M A B . We also obtain necessary and sufficient conditions for all strongly Gorenstein-injective modules over Δ (0 , 0) . [ABSTRACT FROM AUTHOR]

Semantic search encompasses advanced technological approaches to information discovery and retrieval, employing semantic techniques to extract information from intricately structured data sources. An effective search engine must have the ability of accessing and retrieving information of interest by employing reasoning with conceptual models. However, the structural and semantic information intrinsic in conceptual models is not readily amenable to reasoning and AI-enhanced semantic processing. Therefore, the chosen model should enable understanding the meaning of concepts and the relationships between them. Il should also carefully consider the context of the search, ultimately enhancing the accuracy of the returned results. Among the models that fulfill these objectives, conceptual graphs stand out as particularly interesting. They are built upon a robust theoretical framework that spans multiple domains, including philosophical, psychological, linguistic, and artificial intelligence disciplines. In this paper, we describe a method for semantic search driven by conceptual graph-based representation, and a powerful matching reasoning supported by a projection operation between the semantic annotations associated with the document and the submitted query. [ABSTRACT FROM AUTHOR]

The middle graph M(G) of a graph G is an intersection graph on the vertex set V (G) of any graph G. Let E(G) be an edge set of G and F = V 0(G) [ E(G), where V 0(G) indicates the family of all one vertex subsets of the set V (G). This concept was introduced by T. Hamada and I. Yoshimura [4]. M. Chandramouleeswaran et al., studied isomorphism and automorphism groups for semiring valued graph (S-valued graph). In this paper, we study the morphisms and its properties of middle graph of S-valued graphs. [ABSTRACT FROM AUTHOR]

A term which is a formal expression defined by variables and operation symbols can be described by tree diagram. The class of terms under which the longest distance from the root to each vertex is equal is called completely expanded. In this paper, we consider the partial many-sorted operation defined on the family of all completely expanded terms of type τ and construct the partial algebra satisfying the clone axioms. The partial operations on arbitrary algebra generated by completely expanded terms and their related structures are obtained. [ABSTRACT FROM AUTHOR]

We classify homomorphisms from mapping class groups by using arguments involving finite subgroups. First, we give a new proof of a result of Aramayona–Souto that all homomorphisms between certain mapping class groups of closed surfaces are trivial. Second, we show that only finitely many mapping class groups of closed surfaces have nontrivial homomorphisms to Homeo (S n) for any n, where S n is the n-sphere. We also effectivize this result for small values of n; for instance, we prove that every homomorphism from Mod (S g) to Homeo (S 2) or Homeo (S 3) is trivial if g ≥ 3 , extending a result of Franks–Handel. [ABSTRACT FROM AUTHOR]

Let S be a discrete semigroup and let S S denote the collection of all functions f : S → S . If (P , ∘) is a subsemigroup of S S by composition operation, then P induces a natural topological dynamical system. In fact, (β S , { T f } f ∈ P) is a topological dynamical system, where β S is the Stone–Čech compactification of S, x ↦ T f (x) = f β (x) : β S → β S and f β is a unique continuous22 extension of f. In this paper, we concentrate on the dynamical system (β S , { T f } f ∈ P) , when S is an arbitrary discrete semigroup and P is a subsemigroup of S S and obtain some relations between subsets of S and subsystems of β S with respect to P. As a consequence, we prove that if (S , +) is an infinite commutative discrete semigroup and C is a finite partition of S, then for every finite number of arbitrary homomorphisms g 1 , ⋯ , g l : N → S , there exist an infinite subset B of the natural numbers and C ∈ C such that for every finite summations n 1 , ⋯ , n k of B there exists s ∈ S such that { s + g i (n 1) , s + g i (n 2) , ⋯ , s + g i (n k) } ⊆ C , ∀ i ∈ { 1 , ⋯ , l }. [ABSTRACT FROM AUTHOR]

We show that quantum graph parameters for finite, simple, undirected graphs encode winning strategies for all possible synchronous non-local games. Given a synchronous game G = (I , O , λ) with | I | = n and | O | = k , we demonstrate what we call a weak ∗ -equivalence between G and a 3-coloring game on a graph with at most 3 + n + 9 n (k - 2) + 6 | λ - 1 ({ 0 }) | vertices, strengthening and simplifying work implied by Ji [16] for winning quantum strategies for synchronous non-local games. As an application, we obtain a quantum version of Lovász's reduction [21] of the k-coloring problem for a graph G with n vertices and m edges to the 3-coloring problem for a graph with 3 + n + 9 n (k - 2) + 6 m k vertices. Moreover, winning strategies for a synchronous game G can be transformed into winning strategies for an associated graph coloring game, where the strategies exhibit perfect zero knowledge for an honest verifier. We also show that, for "graph of the game" X (G) associated with G from Atserias et al. [1], the independence number game Hom (K | I | , X (G) ¯) is hereditarily ∗ -equivalent to G , so that the possibility of winning strategies is the same in both games for all models, except the game algebra. Thus, the quantum versions of the chromatic number, independence number and clique number encode winning strategies for all synchronous games in all quantum models. [ABSTRACT FROM AUTHOR]

The notion of local rings with quasi-decomposable maximal ideal was formally introduced by Nasseh and Takahashi. In separate works, the authors of the present paper showed that such rings have rigid homological properties; for instance, they are both Ext- and Tor-friendly. One point of this paper is to further explore the homological properties of these rings and also introduce new classes of such rings from a combinatorial point of view. Another point is to investigate how far some of these homological properties can be pushed along certain diagrams of local ring homomorphisms. [ABSTRACT FROM AUTHOR]

This work introduces the notion of a hesitant bipolar-valued intuitionistic fuzzy graph (HBVIFG), which reflects four different characterizations: membership with positive/negative aspects and non-membership with positive/negative aspects, incorporating multi-dimensional alternatives in all of its information. HBVIFG generalizes both HBVFG and BVHFG due to its diversified nature in observing four perspectives along with multiple attributes in a piece of information. Numerous studies, examples, and graphical representations emphasize the concept's distinctiveness and importance. The following graph theory terms are defined: strong directed HBVIFG, full directed HBVIFG, directed spanning HBVIFSG, directed HBVIFSG, and partial directed hesitant bipolar-valued intuitionistic fuzzy subgraph (HBVIFSG). Examples of operations utilizing two HBVIFGs are Cartesian, direct, lexicographical, and strong products. A scenario is used to generate the mapping of relations, which includes homomorphism, isomorphism, weak isomorphism, and co-weak isomorphism. We describe a directed HBVIFG application that employs an algorithm to determine the most dominant person and self-persistent person in a social system and a comparative study is also provided. The proposed method provides a more detailed framework for assessing the most dominant and self-persistent individual in a social network across multi-level attributes along with positive and negative side membership and non-membership grades in each element of a network. [ABSTRACT FROM AUTHOR]

This research aims to introduce and explore the theory of neutrosophic soft hyperalgebras (N S H A s), focusing on their core principles and potential applications in decision-making under uncertainty. By defining key operations such as intersection and union, we clarify the foundational characteristics of N S H A s and their relationship to soft hyperalgebras. The concepts of ξ β -identity N S H A and ξ -absolute N S H A are also examined to better understand their properties. The practical relevance of N S H A is demonstrated through applications in various fields, highlighting its adaptability in addressing complex decision-making scenarios. This approach offers a novel, more precise method for navigating uncertainty in areas such as project methodology selection, sensitivity analysis, and AI chatbot selection. [ABSTRACT FROM AUTHOR]

We study pairs of non-constant maps between two integral schemes of finite type over two (possibly different) fields of positive characteristic. When the target is quasi-affine, Tamagawa showed that the two maps are equal up to a power of Frobenius if and only if they induce the same homomorphism on their étale fundamental groups. We extend Tamagawa's result by adding a purely topological criterion for maps to agree up to a power of Frobenius. [ABSTRACT FROM AUTHOR]

A more advanced form of complex fuzzy sets is q‐rung orthopair complex fuzzy (q-ROCF) sets, which offer an additional and broad visualization of the uncertainty present in the unit disk. In comparison to complex intuitionistic fuzzy sets and complex bipolar fuzzy sets, the q-ROCF set is more appropriate and versatile. Their ability to describe a broader range of unclear information makes them distinguishable because the real part of a complex‐valued membership degree and the real part of a complex‐valued nonmembership degree have a sum of qth powers that is equal to or less than one (similarly for the imaginary part of a complex‐valued). In terms of the characteristics of the q-ROCF set, we propose the concept of q-ROCF subgroups and investigate some fundamental features under the q-ROCF set. Moreover, we show that every intuitionistic complex fuzzy subgroup is a q-ROCF subgroup. Also, we use this approach to define q-ROCF level subgroups, q-ROCF cosets, and q-ROCF normal subgroups of a certain group as well as to investigate some of their algebraic characteristics. Furthermore, we develop the concept of group homomorphism, images, and preimages under the influence of the q-ROCF subgroup. [ABSTRACT FROM AUTHOR]

This paper presents the novel concept of rough intuitionistic fuzzy ideals within the realm of BCK-algebras and investigates their fundamental properties. Furthermore, we introduce a set-valued homomorphism over a BCK-algebra, laying the foundation for the establishment of T-rough intuitionistic fuzzy ideals. The characterization of these innovative ideals is accomplished by employing the (α , β) -cut of intuitionistic fuzzy sets in the context of BCK-algebras. [ABSTRACT FROM AUTHOR]

We introduce a class of circuits that solve a particular case of the Bernstein–Vazirani recursive problem for second-level recursion. This class of circuits allows for the implementation of the oracle using a number of T -gates that grows linearly with the number of qubits in the problem. We find an application of this scheme to quantum homomorphic encryption (QHE), which is an important cryptographic technology useful for delegated quantum computing, allowing a remote server to perform quantum computations on encrypted quantum data, so that the server cannot know anything about the client's data. Liang's QHE schemes are suitable for circuits with a polynomial number of gates T / T † . Thus, the simplified circuits we have constructed can be evaluated homomorphically in an efficient manner. [ABSTRACT FROM AUTHOR]

Following the idea of a group with action in a higher dimension, we consider crossed modules with a given crossed-module action upon themselves. A category of these objects XMod* is given by replacing the set of morphisms in the category Gr* of groups with action by the set of φ-crossed homomorphisms of a modified form. Moreover, the connections of XMod* with some other known categories, such as Gr* and XMod, are given by describing some related functors. [ABSTRACT FROM AUTHOR]

Non-Tarskian interpretations of many-valued logics have been widely explored in the logic literature. The development of non-tarskian conceptions of logical consequence set the theoretical foundations for rediscovering well-known (Tarskian) many-valued logics. One may find in distinct authors many novel interpretations of many-valued systems. They are produced through a type of procedure which consists in altering the semantic structure of Tarskian many-valued logics in order to output a non-Tarskian interpretation of these logics. Through this type of transformation the paper explores a uniform way of transforming finitely many-valued Tarskian logics into their non-Tarskian interpretation. Some general properties of carrying out this type of procedure are studied, namely the dualities between these logics and the conditions under which negation-explosive and negation-complete Tarskian logics become non-explosive. [ABSTRACT FROM AUTHOR]

Understanding the concept of group theory and to apply it in other field of sciences has been a problem among undergraduate students. Although there are some attempt by Dubinsky et al, [1], where some finite groups were discussed among students, the paper was limited to theoretical aspect of the topic. This research is therefore designed to explore clearly the procedure for constructing finite groups for better understanding of the subject area in the given domain. The research is channeled towards the use of group theory in Network Dynamics, which serves as concrete application of finite groups to the students. Some minor open problems with regards to algebraic graph theory was discussed which lead to network dynamics and groupoid formalism. [ABSTRACT FROM AUTHOR]

Let A be a unital C*-algebra and B a unital C*-algebra with a faithful trace τ. Let n be a positive integer. We give the definition of weakly approximate diagonalization (with respect to τ) of a unital homomorphism ϕ : A → M n (B) . We give an equivalent characterization of McDuff II1 factors. We show that, if A is a unital nuclear C*-algebra and B is a type II1 factor with faithful trace τ, then every unital *-homomorphism ϕ : A → M n (B) is weakly approximately diagonalizable. If B is a unital simple infinite dimensional separable nuclear C*-algebra, then any finitely many elements in M n (B) can be simultaneously weakly approximately diagonalized while the elements in the diagonals can be required to be the same. [ABSTRACT FROM AUTHOR]

We say that a signed graph is k $k$‐critical if it is not k $k$‐colorable but every one of its proper subgraphs is k $k$‐colorable. Using the definition of colorability due to Naserasr, Wang, and Zhu that extends the notion of circular colorability, we prove that every 3‐critical signed graph on n $n$ vertices has at least 3n−12 $\frac{3n-1}{2}$ edges, and that this bound is asymptotically tight. It follows that every signed planar or projective‐planar graph of girth at least 6 is (circular) 3‐colorable, and for the projective‐planar case, this girth condition is best possible. To prove our main result, we reformulate it in terms of the existence of a homomorphism to the signed graph C3* ${C}_{3}^{* }$, which is the positive triangle augmented with a negative loop on each vertex. [ABSTRACT FROM AUTHOR]

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In this paper, we introduce a new integral transform, namely, Sumudu transform and we apply the transform to investigate the Hyers-Ulam stability, Hyers-Ulam-Rassias stability, Mittag-Leffler-Hyers-Ulam stability and Mittag-Leffler-Hyers-Ulam-Rassias stability of second order linear differential equations. [ABSTRACT FROM AUTHOR]

In this article, we focus on exploring the fuzzy version of the Hyers-Ulam-Rassias stability of nn-ring homomorphisms and nn-ring derivations in the context of fuzzy Banach algebras. Our investigation utilizes the direct method.

Neutrosophic exact sequence and its construction is presented as algebraic structure in this article. A mathematical construction known as a neutrosophic exact sequence expands the implications of an exact sequence from classical algebra to the domain of neutrosophic sets. This work introduces a full description of neutrosophic exact sequence of R-modules as a structural preserving tools and discusses the fundamental features. Additionally we examine the role of beta level sets in neutrosophic sets and investigate the neutrosophic split exact sequence. [ABSTRACT FROM AUTHOR]

This study discusses the use of t-neutrosophic graphs to express and understand complex interactions. It also shows how these graphs might manage intricate relationships with a wide range of variables or scenario elements. Additionally, it offers fundamental t-neutrosophic graph operations and talks about concepts in these graphs such as homomorphism and isomorphism. Additionally, the study addresses how this approach may be applied in practical contexts to address circular economies and sustainability growth. Through the use of t-neutrosophic graphs, various components, and their connections, the research demonstrates that managing the several circular dimensions is feasible. financial system. This example shows how adaptable and practical t-neutrosophic graphs are as a tool for developing policies and making decisions about complicated social issues that involve some numbers principles. [ABSTRACT FROM AUTHOR]

In this paper, we first develop a general theory of Johnson filtrations and Johnson homomorphisms for a group G acting on another group K equipped with a filtration indexed by a "good" ordered commutative monoid. Then, specializing it to the case where the monoid is the additive monoid ℕ 2 of pairs on non-negative integers, we obtain a theory of double Johnson filtrations and homomorphisms. We apply this theory to the mapping class group ℳ of a surface Σ g , 1 with one boundary component, equipped with the normal subgroups X ̄ , Ȳ of π 1 (Σ g , 1) associated to a standard Heegaard splitting of the 3 -sphere. We also consider the case where the group G is the automorphism group of a free group. [ABSTRACT FROM AUTHOR]

An (n, m)-graph is a graph with n types of arcs and m types of edges. A homomorphism of an (n, m)-graph G to another (n, m)-graph H is a vertex mapping that preserves the adjacencies along with their types and directions. The order of a smallest (with respect to the number of vertices) such H is the (n, m)-chromatic number of G. Moreover, an (n, m)-relative clique R of an (n, m)-graph G is a vertex subset of G for which no two distinct vertices of R get identified under any homomorphism of G. The (n, m)-relative clique number of G, denoted by ω r (n , m) (G) , is the maximum |R| such that R is an (n, m)-relative clique of G. In practice, (n, m)-relative cliques are often used for establishing lower bounds of (n, m)-chromatic number of graph families. Generalizing an open problem posed by Sopena (Discr Math 339(7):1993–2005, 2016) in his latest survey on oriented coloring, Chakroborty et al. (Discr Appl Math 324:29–40, 2023) conjectured that ω r (n , m) (G) ≤ 2 (2 n + m) 2 + 2 for any triangle-free planar (n, m)-graph G and that this bound is tight for all (n , m) ≠ (0 , 1) . In this article, we positively settle this conjecture by improving the previous upper bound of ω r (n , m) (G) ≤ 14 (2 n + m) 2 + 2 to ω r (n , m) (G) ≤ 2 (2 n + m) 2 + 2 , and by finding examples of triangle-free planar graphs that achieve this bound. As a consequence of the tightness proof, we also establish a new lower bound of 2 (2 n + m) 2 + 2 for the (n, m)-chromatic number of the family of triangle-free planar graphs for 2 n + m ≥ 3. [ABSTRACT FROM AUTHOR]

With the goal of transferring dg algebra structures on complexes along contractions, we introduce a new condition on the associated homotopy, namely a generalized version of the Leibniz rule. We prove that, with this condition, the transfer works to yield a dg algebra (with vanishing descended higher A ∞ products) and prove that it works also after an application of the Perturbation Lemma even though the new homotopy may no longer satisfy that condition. We also extend these results to the setting of A ∞ algebras. Then we return to our original motivation from commutative algebra. We apply these methods to find a new method for building a dg algebra structure on a well-known resolution, obtaining one that is both concrete and permutation invariant. The naturality of the construction enables us to find dg algebra homomorphisms between these as well, enabling them to be used as inputs for constructing bar resolutions. [ABSTRACT FROM AUTHOR]

Fully homomorphic encryption (FHE) is capable of handling sensitive encrypted data in untrusted computing environments. The efficient application of FHE schemes in secure outsourced computation can effectively address security and privacy concerns. This paper presents a novel fully homomorphic encryption scheme called GMS, based on the n-secret learning with errors (LWE) assumption. By utilizing block matrix and decomposition technology, GMS achieves shorter encryption and decryption times and smaller ciphertext sizes compared to existing FHE schemes. For secure outsourced matrix multiplication A m × n · B n × l with arbitrary dimensions, GMS only requires O (max { m , n , l }) rotations and one homomorphic multiplication. Compared to the state-of-the-art methods, our approach stands out by achieving a significant reduction in the number of rotations by a factor of O (log max { n , l }) , along with a decrease in the number of homomorphic multiplications by a factor of n and O (min { m , n , l }) . The experimental results demonstrate that GMS shows superior performance for secure outsourced matrix multiplication of any dimension. For example, when encrypting a 64 × 64 -dimensional matrix, the size of the ciphertext is only 1.27 MB. The encryption and decryption process takes approximately 0.2 s. For matrix multiplication A 64 × 64 · B 64 × 64 , the runtime of our method is 39.98 s, achieving a speedup of up to 5X and 2X. [ABSTRACT FROM AUTHOR]

The notion of complex interval-valued q-neutrosophic subbisemiring (CIVqNSBS) is developed and examined. Additionally, we examine the homomorphic features and significant attributes of CIVqNSBS. We suggest the CIVqNSBS level sets for bisemirings. Consider a complex neutrosophic subset of bisemiring Δ, denoted as-→ℵ, if and only if every non-empty level set--→Z(∂,♭) is a subbisemiring, where ∂, ♭ ∈ D[0, 1], then-→Z = (-→Z⊺ℵ,-→Z גℵ,-→ZℲℵ) is a CIVqNSBS of Δ. Let-→ℵ be the strongest complex neutrosophic relation of bisemiring Δ, and let -→Ψ be a CIVqNSBS of bisemiring Δ, if and only if -→Ψ is a CIVqNSBS of Δ × Δ, then -→ℵ is a CIVqNSBS of bisemiring Δ. We show that homomorphic images of all CIVqNSBSs are CIVqNSBSs, and homomorphic pre-images of all CIVqNSBSs are CIVqNSBSs. There are examples given to illustrate our results. [ABSTRACT FROM AUTHOR]

This paper introduces the concept of hesitant bipolar-valued fuzzy graph (HBVFG), which captures the two opposing perspectives, namely the positive and negative opinions. The novelty, importance and implications of this concept are illustrated by some results, examples, and graphical representations. There are, respectively, some theoretical terms of graphs such as partial directed hesitant bipolar-valued fuzzy subgraph (HBVFSG), directed HBVFSG, directed spanning HBVFSG, strong directed HBVFG and complete directed HBVFG which are introduced. The operations, such as Cartesian, direct, lexicographical, and strong products, are also defined between two HBVFGs with examples. The mapping relations, such as homomorphism, isomorphism, weak isomorphism, and co-weak isomorphism, are derived with an example. The applications of directed HBVFGs with algorithms for finding the optimal path in a network and the dominant node and influence of index with the self-persistence degree of a node in a social network are presented. For each problem, an algorithm is developed and its effectiveness is demonstrated by examples. The proposed concept is assessed in terms of theory and practice. The benefits of the proposed solution are highlighted and a clear comparison is made with the existing methods. [ABSTRACT FROM AUTHOR]

We prove that every properly edge-colored n-vertex graph with average degree at least 32 (log 5 n) 2 contains a rainbow cycle, improving upon the (log n) 2 + o (1) bound due to Tomon. We also prove that every properly edge-colored n-vertex graph with at least 10 5 k 3 n 1 + 1 / k edges contains a rainbow 2k-cycle, which improves the previous bound 2 c k 2 n 1 + 1 / k obtained by Janzer. Our method using homomorphism inequalities and a lopsided regularization lemma also provides a simple way to prove the Erdős–Simonovits supersaturation theorem for even cycles, which may be of independent interest. [ABSTRACT FROM AUTHOR]

In this paper, we consider the Reshetikhin-Turaev invariants of knots in the three-sphere obtained from a twisted Drinfeld double of a Hopf algebra, or equivalently, the relative Drinfeld center of the crossed product \text {Rep}(H)\rtimes \text {Aut}(H). These are quantum invariants of knots endowed with a homomorphism of the knot group to \text {Aut}(H). We show that, at least for knots in the three-sphere, these invariants provide a non-involutory generalization of the Fox-calculus-twisted Kuperberg invariants of sutured manifolds introduced previously by the author, which are only defined for involutory Hopf algebras. In particular, we describe the SL(n,\mathbb {C})-twisted Reidemeister torsion of a knot complement as a Reshetikhin-Turaev invariant. [ABSTRACT FROM AUTHOR]

This paper is inspired by a class of infinite order differential operators arising in quantum mechanics. They turned out to be an important tool in the investigation of evolution of superoscillations with respect to quantum fields equations. Infinite order differential operators act naturally on spaces of holomorphic functions or on hyperfunctions. Recently, infinite order differential operators have been considered and characterized on the spaces of entire monogenic functions, i.e. functions that are in the kernel of the Dirac operators. The focus of this paper is the characterization of infinite order differential operators that act continuously on a different class of hyperholomorphic functions, called slice hyperholomorphic functions with values in a Clifford algebra or also slice monogenic functions. This function theory has a very reach associated spectral theory and both the function theory and the operator theory in this setting are subjected to intensive investigations. Here we introduce the concept of proximate order and establish some fundamental properties of entire slice monogenic functions that are crucial for the characterization of infinite order differential operators acting on entire slice monogenic functions. [ABSTRACT FROM AUTHOR]

An essential and quite different class of functions is the triangular norm and its related operators (uninorms, nullnorms, and associative copulas). They are used widely in many disciplines, including fuzzy set theory, probability and statistics, decision sciences, and others. This paper proposes the notion of hybrid Ξ -norm and defines the concepts of Ξ -hybrid ideals, Ξ -hybrid h -ideals in a hemiring . Some equivalent conditions are obtained for a hybrid structure to be a Ξ -hybrid left h -ideal, and it is proved that every imaginable Ξ -hybrid left h -ideal of hemiring is a hybrid left h -ideal, but not conversely by an example. [ABSTRACT FROM AUTHOR]