Search

This paper presents a multi-dimensional variable-kinematics finite element model for nonlinear static analyses of structures with complex geometries. The approach incorporates higher-order beam models and classical solid finite elements in a unified framework, enabling refined modeling of complex geometries. The finite element procedure proposed follows the Carrera Unified Formulation (CUF) and uses a pure displacement-based methodology. The governing equations are derived within the classical continuum mechanics framework, and weak-form equilibrium equations are established using the Principle of Virtual Displacements (PVD). Within the CUF framework, higher-order beam and hexahedral solid models are defined in a unified manner, and the governing equations are written in terms of invariants of mathematical models used and the theory of structures approximation. A coupling technique is used between the beam and solid elements at the nodal level using superposition. The capabilities of fully nonlinear variable-kinematics models are investigated for the static analysis of various rectangular and curved structures. The numerical results are compared with solutions obtained using commercial software. Finally, the proposed methodology is applied to analyze more complex geometries in engineering applications. The results show the capabilities of variable-kinematics models in terms of both accuracy and computational efficiency for the computation of highly nonlinear deformed states and localized phenomena, such as stress concentrations and buckling. [ABSTRACT FROM AUTHOR]

Let ${\mathfrak g}$ be a complex simple Lie algebra and ${\mathfrak n}$ the nilradical of a parabolic subalgebra of ${\mathfrak g}$. We consider some properties of the coadjoint representation of ${\mathfrak n}$ and related algebras of invariants. This includes (i) the problem of existence of generic stabilizers, (ii) a description of the Frobenius semiradical of ${\mathfrak n}$ and the Poisson center of the symmetric algebra , (iii) the structure of as -module, and (iv) the description of square integrable (= quasi-reductive) nilradicals. Our main technical tools are the Kostant cascade in the set of positive roots of ${\mathfrak g}$ and the notion of optimization of ${\mathfrak n}$. [ABSTRACT FROM AUTHOR]

The Ames room illusion is one of the best-known geometrical illusions but its geometrical properties are often misunderstood. This study discusses the differences in the geometrical properties between the original Ames room and what have been often referred to as "Ames rooms" in recent studies. [ABSTRACT FROM AUTHOR]

We define and study refined Gopakumar-Vafa invariants of contractible curves in complex algebraic threefolds, alongside the cohomological Donaldson-Thomas theory of finite-dimensional Jacobi algebras. These Gopakumar-Vafa invariants can be constructed one of two ways: as cohomological BPS invariants of contraction algebras controlling the deformation theory of these curves, as defined by Donovan and Wemyss, or by feeding the moduli spaces that Sheldon Katz used to define genus zero Gopakumar-Vafa invariants into the machinery developed by Joyce et al. The conjecture that the two definitions give isomorphic results is a special case of a kind of categorified version of the strong rationality conjecture due to Pandharipande and Thomas, that we discuss and propose a means of proving. We prove the positivity of the cohomological/refined BPS invariants of all finite-dimensional Jacobi algebras. This result supports this strengthening of the strong rationality conjecture, as well as the conjecture of Brown and Wemyss stating that all finite-dimensional Jacobi algebras for appropriate symmetric quivers are isomorphic to contraction algebras. [ABSTRACT FROM AUTHOR]

The article offers a general account of the genesis of the absolute differential calculus (ADC), paying special attention to its links with the history of differential geometry. In relatively recent times, several historians have described the development of the ADC as a direct outgrowth either of the theory of algebraic and differential invariants or as a product of analytical investigations, thus minimizing the role of Riemann's geometry in the process leading to its discovery. Our principal aim consists in challenging this historiographical tenet and analyzing the intimate connection between the development of Riemannian geometry and the birth of tensor calculus. [ABSTRACT FROM AUTHOR]

In this paper, we define extensions of the classical eigenvalues of the matrix A ∈ Mm(C). These extensions are eigenvalues matrices for the block matrix A ∈ Mm(Mn), where Mm(Mn) is the set of all m×m block complex matrices with each block in Mn(C), they are.....Among other equalities and inequalities, we prove equalities which relate our new definitions with tr1(A) and tr2(A) as follows,... These relations are extensions of the classical relation tr(A) = Σmh=1λh(A), where A ∈Mm(C). Several new relations and properties of our new definitions are also given. [ABSTRACT FROM AUTHOR]

In this paper, we prove some singular value inequalities for sums and products of matrices. Some of our inequalities will give several generalizations of recent known inequalities. Among other inequalities, we prove that if A,B,C,D,X,Y are n×n complex matrices such that X and Y are positive semidefinite, then... for j = 1,2,. . ., n, which is a generalization of an inequality in [12]. Here, s j and ||·|| denote the singular value and the spectral norm of matrices, respectively. [ABSTRACT FROM AUTHOR]

The harmonic status index of a simple connected graph G is defined as the sum of the weights 2 sG(u)+sG(v) over all edges uv of G, where sG(u) denotes the status of the vertex u in G which is the sum of distances between u and all other vertices of G. In this paper, we present upper bounds on the harmonic status index of some families of graph products in terms of certain structural invariants such as the order, size, maximum degree, inverse status and harmonic status index of their components. The graph products considered here are sum, disjunction, symmetric difference, Indu-Bala product, corona product, Cartesian product, lexicographic product, and strong product. Some applications of the obtained results are also presented as corollaries. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a unified method for generating ideals of Möbius-invariant Banach-valued Bloch mappings on the complex open unit disc D, through the composition with the members of a Banach operator ideal I. Using the linearization of derivatives of Banach-valued normalized Bloch mappings on D, this composition method yields the so-called ideals of I-factorizable normalized Bloch mappings I o B, where B denotes the class of normalized Bloch mappings on D. We present new examples of them as ideals of separable (Rosenthal, Asplund) normalized Bloch mappings and p-integral (strictly p-integral, p-nuclear) normalized Bloch mappings for any p ∈ [1,∞). Moreover, the Bloch dual ideal I B-dual of an operator ideal I is introduced and shown that it coincides with the composition ideal Idual o B. [ABSTRACT FROM AUTHOR]

In this paper, we study Ricci-Yamabe and gradient Yamabe solitons on the Lie group H2 × R with a left-invariant metric. We prove that the kind of the Ricci-Yamabe soliton is only related with one variable existing in the definition. [ABSTRACT FROM AUTHOR]

In the present paper, the complete system of projective invariants of a point shape and the complete system of invariants under simultaneous projective and permutation transformations of a point shape are obtained. [ABSTRACT FROM AUTHOR]

We show that the total variation mixing time is not quasi-isometry invariant, even for Cayley graphs. Namely, we construct a sequence of pairs of Cayley graphs with maps between them that twist the metric in a bounded way, while the ratio of the two mixing times goes to infinity. The Cayley graphs serving as an example have unbounded degrees. For non-transitive graphs, we construct bounded degree graphs for which the mixing time from the worst starting point for one graph is asymptotically smaller than the mixing time from the best starting point of the random walk on a network obtained by increasing some of the edge weights from 1 to $1+o(1)$. [ABSTRACT FROM AUTHOR]

Consider a minimal-free topological dynamical system $(X, \mathbb Z^d)$. It is shown that the radius of comparison of the crossed product C*-algebra $\mathrm {C}(X) \rtimes \mathbb Z^d$ is at most half the mean topological dimension of $(X, \mathbb Z^d)$. As a consequence, the C*-algebra $\mathrm {C}(X) \rtimes \mathbb Z^d$ is classified by the Elliott invariant if the mean dimension of $(X, \mathbb Z^d)$ is zero. [ABSTRACT FROM AUTHOR]

In his 1957 paper, John Milnor introduced link invariants which measure the homotopy class of the longitudes of a link relative to the lower central series of the link group. Consequently, these invariants determine the lower central series quotients of the link group. This work has driven decades of research with profound influence. One of Milnor's original problems remained unsolved: to extract similar invariants from the transfinite lower central series of the link group. We reformulate and extend Milnor's invariants in the broader setting of 3-manifolds, with his original invariants as special cases. We present a solution to Milnor's problem for general 3-manifold groups, developing a theory of transfinite invariants and realizing nontrivial values. [ABSTRACT FROM AUTHOR]

One important algebraic invariant in networks is complexity. This invariant ensures the accuracy and dependability of the network. In this paper, we employ a combinatorial approach to determine the graph's complexity. A fundamental set of building blocks (basic graphs) will serve as the foundation for all the graphs we investigate, after which we will analyze the individual blocks and the ways in which they are connected. We compute the spectrum and complexity of a number of fundamental graphs and then we employ the novel duplication corona and Cartesian product operations to construct advanced networks from these graphs. Specifically, straightforward formulas are derived for the complexity of the networks created by the new duplicating corona of the regular graphs (prism, diagonal prism, cycle, complete graph, shadow of the cycle, and Petersen graph) with some families of graphs. Furthermore, using Cartesian product operation, evident and specific formulas for the complexity of the prism of the grid graph Gl,κ, the prism of the stacked book graph Bl,s, the diagonal plane prism grid graph, and the prism of the cylindrical graph Cl,κ are derived. [ABSTRACT FROM AUTHOR]

In this expository essay, we introduce some elements of the study of groups by analysing the braid pattern on a knitted blanket. We determine that the blanket features pure braids with a minimal number of crossings. Moreover, we determine polynomial invariants associated to the links obtained by closing the braid patterns of the blanket. [ABSTRACT FROM AUTHOR]

We apply the basic Lie symmetry method to investigate the time-dependent negative-order Calogero-Bogoyavlenskii-Schiff (vnCBS) equation. In this case, the symmetry classification problem is answered. We obtain symmetry algebra and create the optimal system of Lie sub- algebras. We obtain the symmetry reductions and invariant solutions of the considered equation using these vector fields. Finally, we determine the conservation laws of the vnCBS equation via the Bluman-Anco ho- motopy formula [ABSTRACT FROM AUTHOR]

Conjecturing has been considered to inspire the need for proof, enhance the understanding of proofs, and construct a valid proof. This study describes students' processes of formulating a Euclidean geometry conjecture in the form of a conditional statement through constructing a geometric figure and using measuring and dragging modalities of a dynamic geometry system (DGS). To accomplish this aim, we adapted the existing conjecturing model by Baccaglini-Frank and Mariotti ([2010]. Generating conjectures in dynamic geometry: The maintaining dragging model. International Journal of Computers for Mathematical Learning, 15(3), 225–253. https://doi.org/10.1007/s10758-010-9169-3) and used it to analyse students' conjecturing processes. Our participants were prospective mathematics teachers (PMTs) during their first year at an Indonesian university, but the findings can be useful for secondary school students in other countries. We selected and categorized episodes from task-based interviews with eight PMTs a week after a teaching intervention. We interpreted their identification of robust invariants during constructing, dragging, and measuring. Our findings indicated that the adapted model was appropriate to describe PMTs' processes of conjecturing, which emerged through an exploration that involved robust invariants. We found that PMTs determined these invariants as premises or conclusion of the conjecture by observing the measure of parts of the constructed figure during dragging. The findings indicated that the measuring and dragging modalities of DGS supported PMTs in conjecturing. [ABSTRACT FROM AUTHOR]

In this work, we show that if F is a positive integer, then Sat (F) = { S ∣ S is a saturated numerical semigroup with Frobenius number F} is a covariety. As a consequence, we present two algorithms: one that computes Sat (F) , and another which computes all the elements of Sat (F) with a fixed genus. If X ⊆ S \ Δ (F) for some S ∈ Sat (F) , then we see that there exists the least element of Sat (F) containing X. This element is denoted by Sat (F) [ X ]. If S ∈ Sat (F) , then we define the Sat (F) -rank of S as the minimum of { c a r d i n a l i t y (X) ∣ S = Sat (F) [ X ] }. In this paper, we present an algorithm to compute all the elements of Sat (F) with a given Sat (F) -rank. [ABSTRACT FROM AUTHOR]

Various forms of boundedness in rectangular fuzzy n-normed linear spaces of type (X, N,∗) are examined in this work, where ∗ represents an arbitrary t-norm. Some of these highly general notions of boundedness have no counterpart in the traditional topological metrizable linear spaces. We list the properties of these bounded sets and conduct a comparative analysis of the various forms of boundedness. Among these are a number of ideas on the symmetrical characteristics of the objects under study that come from the classical context and are relevant to the issues of this magazine. We define their consequences and provide instances to show how different these notions are from one another. [ABSTRACT FROM AUTHOR]

From the generalized Riemann hypothesis for motivic L-functions, we derive an effective version of the Sato-Tate conjecture for an abelian variety A defined over a number field k with connected Sato-Tate group. By effective we mean that we give an upper bound on the error term in the count predicted by the Sato-Tate measure that only depends on certain invariants of A. We discuss three applications of this conditional result. First, for an abelian variety defined over k, we consider a variant of Linnik's problem for abelian varieties that asks for an upper bound on the least norm of a prime whose normalized Frobenius trace lies in a given interval. Second, for an elliptic curve defined over k with complex multiplication, we determine (up to multiplication by a nonzero constant) the asymptotic number of primes whose Frobenius traces attain the integral part of the Hasse--Weil bound. Third, for a pair of abelian varieties A and A0 defined over k with no common factors up to k-isogeny, we find an upper bound on the least norm of a prime at which the respective Frobenius traces of A and A0 have opposite sign. [ABSTRACT FROM AUTHOR]

In this paper we propose a definition of torsion refined Gopakumar–Vafa (GV) invariants for Calabi–Yau threefolds with terminal nodal singularities that do not admit Kähler crepant resolutions. Physically, the refinement takes into account the charge of five-dimensional BPS states under a discrete gauge symmetry in M-theory. We propose a mathematical definition of the invariants in terms of the geometry of all non-Kähler crepant resolutions taken together. The invariants are encoded in the A-model topological string partition functions associated to non-commutative (nc) resolutions of the Calabi–Yau. Our main example will be a singular degeneration of the generic Calabi–Yau double cover of P 3 and leads to an enumerative interpretation of the topological string partition function of a hybrid Landau–Ginzburg model. Our results generalize a recent physical proposal made in the context of torus fibered Calabi–Yau manifolds by one of the authors and clarify the associated enumerative geometry. [ABSTRACT FROM AUTHOR]

We introduce a polynomial invariant Vτ​∈Z[H1​(M)/torsion] associated to a veering triangulation τ of a 3-manifold M. In the special case where the triangulation is layered, i.e. comes from a fibration, Vτ​ recovers the Teichmüller polynomial of the fibered face canonically associated to τ. Via Dehn filling, this gives a combinatorial description of the Teichmüller polynomial for any hyperbolic fibered 3-manifold. For a general veering triangulation τ, we show that the surfaces carried by τ determine a cone in homology that is dual to its cone of positive closed transversals. Moreover, we prove that this is equal to the cone over a (generally non-fibered) face of the Thurston norm ball, and that τ computes the norm on this cone in a precise sense. We also give a combinatorial description of Vτ​ in terms of the flow graph for τ and its Perron polynomial. This perspective allows us to characterize when a veering triangulation comes from a fibration, and more generally to compute the face of the Thurston norm ball determined by τ. [ABSTRACT FROM AUTHOR]

The symmetry of space and time properties allows us to use the principle of least action to determine the invariants in the processes occurring on the interface of metals. The invariants established by the author and the criterion allowing to estimate the state of the interface are proposed. [ABSTRACT FROM AUTHOR]

In the additive topological group (R, +) of real numbers, we construct families of sets for which elements are not measurable in the Lebesgue sense. The constructed families have algebraic structures of being semigroups (i.e., closed under finite unions of sets), and invariant under the action of the group Φ (R) of all translations of R onto itself. Those semigroups are constructed by using Vitali selectors and Bernstein subsets on R. In particular, we prove that the family (S(B) ν S(V)) * N0 : = {((U1 ∪ U2) \N) ∪ M : U1 ∈ S(B), U2 ∈ S(V), N,M ∈ N0} is a semigroup of sets, invariant under the action of Φ (R) and consists of sets which are not measurable in the Lebesgue sense. Here, S(B) is the collection of all finite unions of some type of Bernstein subsets of R, S(V) is the collection of all finite unions of Vitali selectors of R, and N0 is the σ-ideal of all subsets of R having the Lebesgue measure zero. [ABSTRACT FROM AUTHOR]

This note deals with some flaws in a recent paper by Gambo and Olarewaju (CJMS. 9(1), 2020, 68-85). It pinpoints the logic error in the proof of Theorem 3.2 in that paper and discusses some corrective works. [ABSTRACT FROM AUTHOR]

Riemannian curvature invariants on hypersurfaces of an almost product-like manifold with constant curvature v are computed. Various relationships involving sectional curvatures and Ricci curvatures have been obtained. Using the Chen-Ricci inequality, some characterizations are presented. [ABSTRACT FROM AUTHOR]

We develop a strategy to classify the components of the space of sections of a del Pezzo fibration over P¹ . In particular, we prove the Movable Bend-and-Break lemma for del Pezzo fibrations. Our approach is motivated by Geometric Manin’s Conjecture and proves upper bounds on the associated counting function. We also give applications to enumerativity of Gromov–Witten invariants and to the study of the Abel–Jacobi map. [ABSTRACT FROM AUTHOR]

By J.F. Nash's Theorem, any Riemannian manifold can be embedded into a Euclidean ambient space with dimension sufficiently large. In 1968, S.-S. Chern pointed out that a key technical element in applying Nash's Theorem effectively is finding useful relationships between intrinsic and extrinsic elements that are characterizing immersions. After 1993, when a groundbreaking work written by B.-Y. Chen on this theme was published, many explorations pursued this important avenue of inquiry. Bearing in mind this historical context, in our present paper we point out a new relationship involving intrinsic and extrinsic curvature invariants, under natural geometric conditions. We conclude our work with an obstruction to minimality, in the spirit of S.-S. Chern's reflection from 1968. [ABSTRACT FROM AUTHOR]

In this paper, we consider the notion of neutrosophic nil radicals of neutrosophic ideals in commutative rings and some properties of such nil radicals. Finally, we study the properties of semiprime neutrosophic ideals of rings. [ABSTRACT FROM AUTHOR]

Copyright of Rad HAZU: Matematicke Znanosti is the property of Croatian Academy of Sciences & Arts (HAZU) and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Copyright of Rad HAZU: Matematicke Znanosti is the property of Croatian Academy of Sciences & Arts (HAZU) and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Copyright of Rad HAZU: Matematicke Znanosti is the property of Croatian Academy of Sciences & Arts (HAZU) and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

We prove that, under a simple condition on the cohomology ring, every closed 4-manifold has mod 2 Seiberg-Witten simple type. This result shows that there exists a large class of topological 4-manifolds such that all smooth structures have mod 2 simple type, and yet some have non-vanishing (mod 2) Seiberg-Witten invariants. As corollaries, we obtain adjunction inequalities and show that, under a mild topological condition, every geometrically simply connected closed 4-manifold has the vanishing mod 2 Seiberg-Witten invariant for at least one orientation. [ABSTRACT FROM AUTHOR]

With this work, we present the asymptotical strongly p-deferred invariant and asymptotical deferred invariant statistical equivalence of order a (0 < a ≤ 1) for sequences of sets in the Wijsman sense. Furthermore, we investigate the connections between these concepts and conduct their properties. [ABSTRACT FROM AUTHOR]

Established molecular machine learning models process individual molecules as inputs to predict their biological, chemical, or physical properties. However, such algorithms require large datasets and have not been optimized to predict property differences between molecules, limiting their ability to learn from smaller datasets and to directly compare the anticipated properties of two molecules. Many drug and material development tasks would benefit from an algorithm that can directly compare two molecules to guide molecular optimization and prioritization, especially for tasks with limited available data. Here, we develop DeepDelta, a pairwise deep learning approach that processes two molecules simultaneously and learns to predict property differences between two molecules from small datasets. On 10 ADMET benchmark tasks, our DeepDelta approach significantly outperforms two established molecular machine learning algorithms, the directed message passing neural network (D-MPNN) ChemProp and Random Forest using radial fingerprints, for 70% of benchmarks in terms of Pearson's r, 60% of benchmarks in terms of mean absolute error (MAE), and all external test sets for both Pearson's r and MAE. We further analyze our performance and find that DeepDelta is particularly outperforming established approaches at predicting large differences in molecular properties and can perform scaffold hopping. Furthermore, we derive mathematically fundamental computational tests of our models based on mathematical invariants and show that compliance to these tests correlates with overall model performance — providing an innovative, unsupervised, and easily computable measure of expected model performance and applicability. Taken together, DeepDelta provides an accurate approach to predict molecular property differences by directly training on molecular pairs and their property differences to further support fidelity and transparency in molecular optimization for drug development and the chemical sciences. [ABSTRACT FROM AUTHOR]

Based on classical Lie symmetry method, the one-dimensional non-linear wave equation is investigated. By using four-dimensional subalgebras of the equation, the invariant groups and commutator table are constructed. Furthermore, optimal system of the equation is obtained, and the exact solutions can be gained by solving reduced equations. Finally, a complete derivation of the conservation law is given by using conservation multipliers. [ABSTRACT FROM AUTHOR]

This commentary paper is a pedagogical reflection on the interplay between variation and invariant. It begins with a brief discussion on the concept of Unity of Opposites as an ancient philosophical theme. Ancient thinking systems regarded the variation and invariant pair as a Unity of Opposites. Next, the use of variation as a pedagogical approach in mathematics education is briefly examined under Marton's variational theory of learning, Gu's bianshi jiaoxue, and the related research done by the author in the context of Dynamic Geometry Environment (DGE). These lead to the formation of the concept of variational thinking, the main contribution of this paper, which is presented and explained. A DGE task design sequence example is presented to illustrate how variational thinking can be used to frame a process of geometrical reasoning and argumentation. [ABSTRACT FROM AUTHOR]

We find a combinatorial formula which computes the first cotangent cohomology module of Stanley-Reisner rings associated to matroids. For arbitrary simplicial complexes we provide upper bounds for the dimensions of the multigraded components of T¹. For specific degrees we prove that these bounds are reached if and only if the simplicial complex is a matroid, obtaining thus a new characterization for matroids. Furthermore, the graded first cotangent cohomology turns out to be a complete invariant for nondiscrete matroids. [ABSTRACT FROM AUTHOR]

Building on the algebraic framework developed by Hendricks, Manolescu, and Zemke, we introduce and study a set of Floer-theoretic invariants aimed at detecting corks. Our invariants obstruct the extension of a given involution over any homology ball, rather than a particular contractible manifold. Unlike previous approaches, we do not utilize any closed 4-manifold topology or contact topology. Instead, we adapt the formalism of local equivalence coming from involutive Heegaard Floer homology. As an application, we define a modification ΘιZ of the homology cobordism group which takes into account an involution on each homology sphere, and prove that this admits a Z1-subgroup of strongly nonextendable corks. The group ΘιZ can also be viewed as a refinement of the bordism group of diffeomorphisms. Using our invariants, we furthermore establish several new families of corks and prove that various known examples are strongly nonextendable. Our main computational tool is a monotonicity theorem which constrains the behavior of our invariants under equivariant negative-definite cobordisms, and an explicit method of constructing such cobordisms via equivariant surgery. [ABSTRACT FROM AUTHOR]

In this article, pseudoparallel submanifolds for almost Kenmotsu (κ, μ, ν) --space are investigated. The almost Kenmotsu (κ, μ, ν) --space is considered on the concircular curvature tensor. Submanifolds of these manifolds with properties such as concircular pseudoparallel, concircular 2--pseudoparallel, concircular Ricci generalized pseudoparallel, and concircular 2--Ricci generalized pseudoparallel has been characterized. Necessary and sufficient conditions are given for the invariant submanifolds of almost Kenmotsu (κ, μ, ν) --space to be total geodesic according to the behavior of the κ, μ, ν functions. [ABSTRACT FROM AUTHOR]

We give the relationship between indecomposable modules over the finite dimensional k-algebra A and the smash product A♯G respectively, where G is a finite abelian group satisfying G ⊆ Aut(A), and k is an algebraically closed field with the characteristic not dividing the order of G. More precisely, we construct all indecomposable A♯G-modules from indecomposable A-modules and prove that an A♯G-module is indecomposable if and only if it is an indecomposable G-stable module over A. Besides, we give the relationship between simple, projective and injective modules in modA and those in modA♯G. [ABSTRACT FROM AUTHOR]

In this paper, we study the complete convergence and the complete integration convergence for weighted sums of m-extended negatively dependent (m-END) random variables under sub-linear expectations space with the condition of Ê|X|p 6 CV(|X|p) < ∞, p > 1/α and α > 3/2. We obtain the results that can be regarded as the extensions of complete convergence and complete moment convergence under classical probability space. In addition, the Marcinkiewicz-Zygmund type strong law of large numbers for weighted sums of m-END random variables under the sub-linear expectations space is proved. [ABSTRACT FROM AUTHOR]

Let Gbe a finite abelian group with exponent exp(G) and S be a sequence with elements of G. We say S is a zero-sum sequence if the sum of the elements in S is the zero element of G. For a positive integer t, let stexp(G)(G) (respectively, s'texp(G)(G)) denote the smallest integer ℓ such that every sequence (respectively, zero-sum sequence) S over G with |S|≥ℓ contains a zero-sum subsequence of length texp(G). The invariant stexp(G)(G) (respectively, s'texp(G)(G)) is called the Generalized Erdős-Ginzburg-Ziv constant (respectively, Modified Erdős-Ginzburg-Ziv constant) of G. In this paper, we discuss the relationship between Generalized Erdős-Ginzburg-Ziv constant and Modified Erdős-Ginzburg-Ziv constant, and determine s'texp(G)(G) for some finite abelian groups. [ABSTRACT FROM AUTHOR]

S. Ramanujan documented several modular equations of degrees in his notebooks. These identities are used to evaluate Weber's class in variants, continued fractions and many more. In the present work, we establish modular equations of composite degrees using the known identities. [ABSTRACT FROM AUTHOR]

The Sombor index and reduced Sombor index, introduced by mathematical chemist Ivan Gutman [MATCH Commun. Math. Comput. Chem. 86 (2021) 11-16], are the recently proposed degree-based graph invariants that attained a lot of attention from researchers in a very short time. In this paper, the best possible upper bounds on the both aforementioned indices for molecular trees are obtained in terms of order and number of branching vertices or vertices of degree 2. The optimal molecular trees achieving the obtained bounds are also completely characterized. [ABSTRACT FROM AUTHOR]

A graph invariant is a number that can be easily and uniquely calculated through a graph. Recently, part of mathematical graph invariants has been portrayed and utilized for relationship examination. Nevertheless, no reliable appraisal has been embraced to pick, how much these invariants are associated with a network graph in interconnection networks of various fields of computer science, physics, and chemistry. In this paper, the study talks about sudoku networks will be networks of fractal nature having some applications in computer science like sudoku puzzle game, intelligent systems, Local area network (LAN) development and parallel processors interconnections, music composition creation, physics like power generation interconnections, Photovoltaic (PV) cells and chemistry, synthesis of chemical compounds. These networks are generally utilized in disorder, fractals, recursive groupings, and complex frameworks. Our outcomes are the normal speculations of currently accessible outcomes for specific classes of such kinds of networks of two unmistakable sorts with two invariants K-banhatti sombor (KBSO) invariants, Irregularity sombor (ISO) index, Contraharmonic-quadratic invariants (CQIs) and dharwad invariants with their reduced forms. The study solved the Sudoku network used in mentioned systems to improve the performance and find irregularities present in them. The calculated outcomes can be utilized for the modeling, scalability, introduction of new architectures of sudoku puzzle games, intelligent systems, PV cells, interconnection networks, chemical compounds, and extremely huge scope in very large-scale integrated circuits (VLSI) of processors. [ABSTRACT FROM AUTHOR]