Algebraic Point Of View Research Articles

Covariant mass and geometrical setup in Euclidean gauge theories

A nonlocal mass operator can be consistently set in local form through the introduction of a set of additional fields with geometrical appropriated properties. A local and polynomial gauge invariant action is thus identified. Equations compatible with the study of renormalization, from the algebraic point of view, are presented in the Landau Gauge.

On a Weak Conditional

Abstract It is well-known that adding to a lattice the usual relative meet complement is not conservative, in the sense that distributivity is implied. In this paper we consider a weak relative meet complement that does not have the mentioned effect. We mostly study the mentioned operation from an algebraic point of view. However, we also provide a Hilbert-style axiomatization for its corresponding assertional logic.

Signature cocycles on the mapping class group and symplectic groups

Werner Meyer constructed a cocycle in H2(Sp(2g,Z);Z) which computes the signature of a closed oriented surface bundle over a surface. By studying properties of this cocycle, he also showed that the signature of such a surface bundle is a multiple of 4. In this paper, we study signature cocycles both from the geometric and algebraic points of view. We present geometric constructions which are relevant to the signature cocycle and provide an alternative to Meyer's decomposition of a surface bundle. Furthermore, we discuss the precise relation between the Meyer and Wall-Maslov index. The main theorem of the paper, Theorem 6.6, provides the necessary group cohomology results to analyze the signature of a surface bundle modulo any integer N. Using these results, we are able to give a complete answer for N=2,4, and 8, and based on a theorem of Deligne, we show that this is the best we can hope for using this method.

On formations of monoids

A formation of monoids is a class of finite monoids closed under taking quotients and subdirect products. Formations of monoids were first studied in connection with formal language theory, but in this paper, we come back to an algebraic point of view. We give two natural constructions of formations based on constraints on the minimal ideal and on the maximal subgroups of a monoid. Next we describe two sublattices of the lattice of all formations, and give, for each of them, an isomorphism with a known lattice of varieties of monoids. Finally, we study formations and varieties containing only Clifford monoids, completely describe such varieties and discuss the case of formations.

The Terwilliger algebra of the Grassmann scheme Jq(N,D) revisited from the viewpoint of the quantum affine algebra [formula omitted

Let Δ be the set of all quadruples (ν,μ,d,e) of integers that satisfy0≤D−d2≤ν≤μ≤D−d≤D,e+d+Dis even,|e|≤2ν−D+d,d∈{e+D−2ν,min⁡{D−μ,e+D−2ν+2(N−2D)}}. In [6], it is shown for the Terwilliger algebra T of the Grassmann scheme Jq(N,D), N≥2D, that the isomorphism classes of irreducible T-modules W are determined by their endpoint ν, dual endpoint μ, diameter d, and auxiliary parameter e, that come from the Leonard system attached to W, and it is claimed without proof that the quadruples (ν,μ,d,e) belong to Δ, if d≥1.Let Λ be the set of triples (α,β,ρ) of non-negative integers that satisfy0≤α≤D−ρ2,0≤β≤N−D−ρ2,0≤α+β≤D−ρ. We construct a mapping from Λ to Δ which is bijective if N>2D and 2:1 if N=2D.We show that the set Λ naturally parameterizes the isomorphism classes of irreducible T-modules, by embedding the standard module of Jq(N,D) in a bigger space that allows a Uq(slˆ2)-module structure [9].As a byproduct we have the following: for a fixed ρ, 0≤ρ≤D, set N′=N−2ρ, D′=D−ρ, and Λρ={(α,β)|(α,β,ρ)∈Λ}. Then Λρ is precisely the set that parameterizes the isomorphism classes of irreducible T-modules for the Johnson scheme J(N′,D′)[3].

Isomorphism Theorems for Groupoids and Some Applications

Using an algebraic point of view we present an introduction to the groupoid theory; that is, we give fundamental properties of groupoids as uniqueness of inverses and properties of the identities and study subgroupoids, wide subgroupoids, and normal subgroupoids. We also present the isomorphism theorems for groupoids and their applications and obtain the corresponding version of the Zassenhaus Lemma and the Jordan-Hölder theorem for groupoids. Finally, inspired by the Ehresmann-Schein-Nambooripad theorem we improve a result of R. Exel concerning a one-to-one correspondence between partial actions of groups and actions of inverse semigroups.

Relationship between the vertex-centered linearity-preserving scheme and the lowest-order virtual element method for diffusion problems on star-shaped polygons

As a generalization of the finite element method, the virtual element method (VEM) has made remarkable achievements in the algorithm and analysis of simulating various problems recently. Vertex-centered linearity-preserving scheme (VLPS, Wu et al., 2016) is a linear finite volume scheme which is currently only applied to the numerical simulation of diffusion problems. In this paper, we study the relationship between VLPS and the lowest-order VEM for diffusion problems on star-shaped polygonal meshes from an algebraic point of view. The global stiffness matrix of VLPS with a special stabilization term coincides with that of the lowest-order VEM while the load terms are generally different. Specifically, we find that the global stiffness matrices of the two methods can be split as the consistency parts and the stability parts, and the consistency parts are the same while the stability parts coincide under some assumptions. As a by-product, a new stability term is obtained for VLPS. Besides, a post-processing procedure is suggested for the lowest-order VEM to preserve the positivity of the cell-centered unknowns and maintain at the same time the local conservation on the primary meshes. The positivity, existence and uniqueness of the cell-centered unknowns are studied. Numerical experiments confirm the theoretical findings and demonstrate the efficiency of the post-processing procedure on various polygonal meshes.

On generalized inverse sum indeg index and energy of graphs

Topological indices are used to predict certain phsio-chemical properties of the chemical compounds. Among all indices, degree based indices are of vital importance. In this paper, we introduce generalized inverse sum indeg index and generalized inverse sum indeg energy of graphs. We study the generalized inverse sum indeg index and energy from an algebraic point of view. Extremal values of this index for some graph classes are determined. Some spectral properties of generalized inverse sum indeg matrix are studied. We also find Nordhaus-Gaddum-type results for generalized inverse sum indeg index spectral radius and energy.

A spectral Newton-Schur algorithm for the solution of symmetric generalized eigenvalue problems

This paper proposes a numerical algorithm based on spectral Schur complements to compute a few eigenvalues and the associated eigenvectors of symmetric matrix pencils. The proposed scheme follows an algebraic domain decomposition viewpoint and transforms the generalized eigenvalue problem into one of computing roots of scalar functions. These scalar functions are defined so that their roots are equal to the eigenvalues of the original pencil, and these roots are computed by Newton's method. We describe the theoretical aspects of the proposed scheme and demonstrate its performance on a few test problems.

Pure semisimple n-cluster tilting subcategories

From the viewpoint of higher homological algebra, we introduce pure semisimple n-abelian categories, which are analogs of pure semisimple abelian categories. Let Λ be an Artin algebra and M be an n-cluster tilting subcategory of Mod-Λ. We show that M is pure semisimple if and only if each module in M is a direct sum of finitely generated modules. Let m be an n-cluster tilting subcategory of mod-Λ. We show that Add(m) is an n-cluster tilting subcategory of Mod-Λ if and only if m has an additive generator if and only if Mod(m) is locally finite. This generalizes Auslander's classical results on pure semisimplicity of Artin algebras.

Domain decomposition methods in a geometrical multiscale domain using finite volume schemes

SummaryThe heat equation is solved by using a finite volume discretization in a domain that consists of a two‐dimensional central node and several one‐dimensional outgoing branches. Several interface connection options to match the submodels set on the node and on the branches, with or without continuity, are looked at. For each of them, a monolithic scheme is defined, and existence and uniqueness of the solution is proved. New schemes are deduced, which are obtained through domain decomposition methods in the form of interface systems, with one or two unknowns per interface. A comparative systematic study is carried out from an algebraic and numerical point of view according to the interface conditions: Dirichlet, Neumann, or Robin. An efficient diagonal preconditioning is proposed.

On the computation of the Möbius transform

The Möbius transform is a crucial transformation into the Boolean world; it allows to change the Boolean representation between the True Table and Algebraic Normal Form. In this work, we introduce a new algebraic point of view of this transformation based on the polynomial form of Boolean functions. It appears that we can perform a new notion: the Möbius computation variable by variable and new computation properties. As a consequence, we propose new algorithms which can produce a huge speed up of the Möbius computation for sub-families of Boolean function. Furthermore we compute directly the Möbius transformation of some particular Boolean functions. Finally, we show that for some of them the Hamming weight is directly related to the algebraic degree of specific factors.

Two first-order logics of permutations

We consider two orthogonal points of view on finite permutations, seen as pairs of linear orders (corresponding to the usual one line representation of permutations as words) or seen as bijections (corresponding to the algebraic point of view). For each of them, we define a corresponding first-order logical theory, that we call TOTO (Theory Of Two Orders) and TOOB (Theory Of One Bijection) respectively. We consider various expressibility questions in these theories.Our main results go in three different directions. First, we prove that, for all k≥1, the set of k-stack sortable permutations in the sense of West is expressible in TOTO, and that a logical sentence describing this set can be obtained automatically. Previously, descriptions of this set were only known for k⩽3. Next, we characterize permutation classes inside which it is possible to express in TOTO that some given points form a cycle. Lastly, we show that sets of permutations that can be described both in TOOB and TOTO are in some sense trivial. This gives a mathematical evidence that permutations-as-bijections and permutations-as-words are somewhat different objects.

Accelerating Solution of Volume-Surface Integral Equations With Multiple Right-Hand Sides by Improved Skeletonization Techniques

Skeletonization is an effective method to accelerate the solution of linear systems with multiple right-hand sides (RHSs) by exploiting the rank-deficiency property of the RHS matrix. However, when the size of the RHS matrix is very large, the memory requirement would become a bottleneck of the skeletonization technique. To reduce the memory cost, this letter presents the theoretical basis of how to find out the skeleton basis functions (SBFs) to construct the skeleton RHS matrix more efficiently from a purely algebraic point of view and proposes a multilevel algorithm to figure out the SBFs for solving the integral equations with multiple RHSs. The numerical results show that the proposed method accelerates the solution of the volume–surface integral equation with multiple RHSs and reduces the memory cost significantly.

Introduction to Dependence Relations and Their Links to Algebraic Hyperstructures

The aim of this paper is to study, from an algebraic point of view, the properties of interdependencies between sets of elements (i.e., pieces of secrets, atmospheric variables, etc.) that appear in various natural models, by using the algebraic hyperstructure theory. Starting from specific examples, we first define the relation of dependence and study its properties, and then, we construct various hyperoperations based on this relation. We prove that two of the associated hypergroupoids are H v -groups, while the other two are, in some particular cases, only partial hypergroupoids. Besides, the extensivity and idempotence property are studied and related to the cyclicity. The second goal of our paper is to provide a new interpretation of the dependence relation by using elements of the theory of algebraic hyperstructures.

Reducing Subspaces of Analytic Toeplitz Operators on the Bergman Space of the Annulus

Let $$\mathbb {A}_{r}$$ be the annulus $$\{z\mid \,|z|<r<1\}$$ in the complex plane, $$L_{a}^{2}(\mathbb {A}_{r})$$ be the Bergman space on $$\mathbb {A}_{r}$$ , B be a finite Blaschke product $$B(z)=e^{i\theta }\prod \nolimits _{i=1}^{N}\frac{z-\alpha _{i}}{1-\overline{\alpha _{i}}z}$$ with $$|\alpha _{i}|<r$$ for $$1\le i\le N$$ . In this case, local inverses of B on $$\mathbb {A}_{r}$$ consist of a cyclic group with order N. It is shown that there is an one-to-one correspondence between a minimal reducing subspace of the Toeplitz operator $$T_{B}$$ on $$L_{a}^{2}(\mathbb {A}_{r})$$ and a character of the cyclic group, reducing subspaces of Toeplitz operators are studied from an algebraic point of view and Douglas and Kim’s result is generalized.

The Rough Veronese variety

We study signature tensors of paths from an algebraic geometric viewpoint. The signatures of a given class of paths parametrize a variety inside the space of tensors, and these signature varieties provide both new tools to investigate paths and new challenging questions about their behavior. This paper focuses on signatures of rough paths. Their signature variety shows surprising analogies with the Veronese variety, and our aim is to prove that this so-called Rough Veronese is toric. The same holds for the universal variety. Answering a question of Améndola, Friz and Sturmfels, we show that the ideal of the universal variety does not need to be generated by quadrics.

On a Class of Infinite Systems of Linear Equations Originating in Statistical Physics

We show that the problem of prescribing a system of random variables by means of conditional distributions can be considered from the algebraic point of view as a problem of consistency of an appropriate infinite system of linear equations. We demonstrate also that a potential energy (transition energy field) and probability (specification) are connected as the solutions of corresponding adjoint infinite systems of linear equations.

On the Resolution of Third and Fourth Degree Equations

In this paper we deal with the resolution of third and fourth degree equations by means of radicals. We present some historical details about this fundamental problem. We also discuss practical methods for the resolution of third and fourth degree equations from the algebraic point of view. Finally, we introduce some results about the equations of greater degree.

Spectral Theory, Jacobi Matrices, Continued Fractions and Difference Operators

The aim of this paper is to describe some connections between spectral theory in infinite dimensional Lie algebras, deformation theory and linearization of nonlinear dynamical systems. We explain how results from isospectral deformations, cohomology groups and algebraic geometry can be used to obtain insight into integrable systems. Another part will be dedicated to the study of infinite continued fractions and isospectral deformation of periodic Jacobi matrices and general difference operators from an algebraic geometrical point of view. Also, the notion of algebraically completely integrable systems is explained and techniques to solve such systems are presented. Several nonlinear problems in mathematical physics illustrate these results.