Algebraic Point Of View Research Articles

O princípio da inclusão-exclusão e o cálculo de permanentes

In this article we review the inclusion-exclusion principle (PIE) from set theoretical and algebraic points of view and discuss its application to the calculation of permanents, a subject that is not normally covered in undergraduate courses. The presentation is intended to be rigorous but elementary, accessible to first-year students in mathematics, science, and engineering programs, requiring only familiarity with set notation, arithmetic and matrix algebra. In dealing with the calculation of permanents, we present the Ryser algorithm, one of the most spectacular developments in the approach to difficult combinatorial problems, whose computational complexity we briefly discuss. The article contains examples, complementary notes, and a program in Python that implements Ryser's algorithm using Gray codes to calculate permanents, accompanied by its discussion.

Beyond directed hypergraphs: heterogeneous hypergraphs and spectral centralities

Abstract The study of hypergraphs has received a lot of attention over the past few years, however up until recently there has been no interest in systems where higher order interactions are not undirected. In this article, we introduce the notion of heterogeneous hypergraphs from an algebraic point of view, which have traditional directed hypergraphs as a particular case. We furthermore analytically study the spectral centralities associated to some types of heterogeneous hypergraphs, extending previously defined eigenvector-like centrality measures to this new realm. We supplement the analytical arguments with some numerical comparisons of pairwise and higher order rankings, and we construct directed higher order networks from real data, which we then use for discussion and analysis.

T-duality of emergent gravities on nilmanifolds

We study the transport of generalized metrics between topological T-dual nil-manifolds through a Lie algebraic point of view. Emergent gravities are generalized metrics with symplectic B-fields. But this additional property might not be preserved by the aforementioned transport. We describe a necessary condition for it to happen and provide working examples on self-T-dual nilmanifolds with zero H-flux in both 4 and 6 dimensions. We also discuss how this procedure fails in the presence of a non-zero H-flux.

Crossed products, conditional expectations and constraint quantization

Recent work has highlighted the importance of crossed products in correctly elucidating the operator algebraic approach to quantum field theories. In the gravitational context, the crossed product simultaneously promotes von Neumann algebras associated with subregions in diffeomorphism covariant quantum field theories from type III to type II, and provides the necessary ingredients to gravitationally dress operators, thereby enforcing the constraints of the theory. In this note we enhance the crossed product construction to the context of general gauge theories with arbitrary combinations of internal and spacetime local symmetries. This is done by leveraging the correspondence between the crossed product and the extended phase space. We then undertake a detailed study of constraint quantization from the perspective of a generic crossed product algebra. We study and compare four distinct approaches to constraint quantization from this point of view: refined algebraic quantization, BRST quantization, path integral quantization, and the commutation theorem for crossed products. Far from simply reproducing existing analyses, the operator algebraic viewpoint sheds new light on old problems by reformulating the dressing of operators in terms of conditional expectations and other closely related projection maps. We conclude by applying our approach to the constraint quantization of three distinct gauge theories including a discussion of gravity on null hypersurfaces.

A distant descendant of the six-vertex model

In this paper we present a new solution of the star-triangle relation having positive Boltzmann weights. The solution defines an exactly solvable two-dimensional Ising-type (edge interaction) model of statistical mechanics where the local “spin variables” can take arbitrary integer values, i.e., the number of possible spin states at each site of the lattice is infinite. There is also an equivalent “dual” formulation of the model, where the spins take continuous real values on the circle. From algebraic point of view this model is closely related to the 6-vertex model. It is connected with the construction of an intertwiner for two infinite-dimensional representations of the quantum affine algebra Uq(slˆ(2)) without the highest and lowest weights. The partition function of the model in the large lattice limit is calculated by the inversion relation method. Amazingly, it coincides with the partition function of the off-critical 8-vertex free-fermion model.

The fundamental theorem of calculus in differential rings

In this paper, we study the consequences of the fundamental theorem of calculus from an algebraic point of view. For functions with singularities, this leads to a generalized notion of evaluation. We investigate properties of such integro-differential rings and discuss many examples. We also construct corresponding integro-differential operators and provide normal forms via rewrite rules. They are then used to derive several identities and properties in a purely algebraic way, generalizing well-known results from analysis. In identities like shuffle relations for nested integrals and the Taylor formula, additional terms are obtained that take singularities into account. Another focus lies on treating basics of linear ODEs in this framework of integro-differential operators. These operators can have matrix coefficients, which allow to treat systems of arbitrary size in a unified way. In the appendix, using tensor reduction systems, we give the technical details of normal forms and prove them for operators including other functionals besides evaluation.

ALGEBRAIC STRUCTURES ON A SET OF DISCRETE DYNAMICAL SYSTEM AND A SET OF PROFILE

A discrete dynamical system is represented as a directed graph with graph nodes called states that can be seen on the dynamical map. This discrete dynamical system is symbolized by , where is a finite set of states and the function g is a function from to . In the dynamical map, the discrete dynamical system has a height where the number of states in each height is called a profile. The set of discrete dynamical systems has an addition operation defined as a disjoint union on the graph and a multiplication operation defined as a tensor product on the graph. The set of discrete dynamical systems and the set of profiles are very interesting to observe from the algebraic point of view. Considering operation on the set of discrete dynamical systems and the set of profiles, we can see their algebraic structure. By recognizing the algebraic structure, it will be easy to solve the polynomial equation in the discrete dynamical system and in the profile. In this research, we will investigate the algebraic structure of discrete dynamical systems and the set of profiles. This research shows that the set of discrete dynamical system has an algebraic structure, which is a commutative semiring and the set of profiles has an algebraic structure, which is a commutative semiring and -semimodule. Moreover, both sets have the same property, which is isomorphic to the set of non-negative integers.

The completion of d-abelian categories

Let A be a finite-dimensional algebra, and M be a d-cluster tilting subcategory of mod A. From the viewpoint of higher homological algebra, a natural question to ask is when M induces a d-cluster tilting subcategory in Mod A. In this paper, we investigate this question in a more general form. We consider M as an essentially small d-abelian category, known to be equivalent to a d-cluster tilting subcategory of an abelian category A. The completion of M, denoted by Ind(M), is defined as the universal completion of M with respect to filtered colimits. We explore Ind(M) and demonstrate its equivalence to the full subcategory Ld(M) of ModM, comprising left d-exact functors. Notably, Ind(M) as a subcategory of ModMEff(M) falls short of being a d-cluster tilting subcategory since it satisfies all properties of a d-cluster tilting subcategory except d-rigidity. For a d-cluster tilting subcategory M of mod A, M→ consists of all filtered colimits of objects from M, is a generating-cogenerating, functorially finite subcategory of Mod A. The question of whether M→ is a d-rigid subcategory remains unanswered. However, if it is indeed d-rigid, it qualifies as a d-cluster tilting subcategory. In the case d=2, employing cotorsion theory, we establish that M→ is a 2-cluster tilting subcategory if and only if M is of finite type. Thus, the question regarding whether M→ is a d-cluster tilting subcategory of Mod A appears to be equivalent to Iyama's question about the finiteness of M. Furthermore, for general d, we address the problem and present several equivalent conditions for Iyama's question.

Structural and universal completeness in algebra and logic

In this work we study the notions of structural and universal completeness both from the algebraic and logical point of view. In particular, we provide new algebraic characterizations of quasivarieties that are actively and passively universally complete, and passively structurally complete. We apply these general results to varieties of bounded lattices and to quasivarieties related to substructural logics. In particular we show that a substructural logic satisfying weakening is passively structurally complete if and only if every classical contradiction is explosive in it. Moreover, we fully characterize the passively structurally complete varieties of MTL-algebras, i.e., bounded commutative integral residuated lattices generated by chains.

Jet schemes, quantum dilogarithm and Feigin-Stoyanovsky's principal subspaces

We analyze the structure of Feigin-Stoyanovsky's principal subspaces of affine Lie algebra from the jet algebra viewpoint. For type A level one principal subspaces, we show that their shifted multi-graded Hilbert series can be expressed either using the quantum dilogarithm or as certain generating functions “counting” finite-dimensional representations of A-type quivers. This notably results in novel fermionic character formulas for these principal subspaces. Moreover, our result implies that all level one principal subspaces of type A are “classically free” as vertex algebras.We also analyze infinite jet algebras associated to principal subspaces of affine vertex algebras L1(so5), L1(so8) and L1(g2). We derive a new character formula for the principal subspace of L1(so5), proving that it is classically free, and present evidence that the principal subspaces of L1(so8) and of L1(g2) are also classically free.

Mechanism singularities and shakiness from an algebraic viewpoint

Kinematic singularities are configurations where the configuration space (c-space) V is not smooth or configurations where the rank of the constraint Jacobian drops while the c-space is locally smooth. Both can be analyzed by means of a higher-order analysis, for which the kinematic tangent cone provides a conceptual framework. It is desirable to complement such an analysis with more insight into the origin of specific singular phenomena. Such additional insight is obtained by investigating the kinematics from an algebraic viewpoint. Points of interest are then the singularities of a variety, referred to as algebraic singularities. Algebraic singularities are reflected as kinematic singularities or as shakiness. It is shown that these algebraic singularities can be attributed to properties of the ideal generated by the loop constraints — the constraint ideal. Particular situations arise depending on whether this constraint ideal is radical. If it is not radical, a singular branch is locally defined by a primary ideal. If it is radical, it must be distinguished whether real and complex dimensions of intersecting branches are different. Real intersections indicate kinematic singularities; in all other cases, the mechanism may be in a singularity or may be shaky. Various examples are presented.

Equations of tensor eigenschemes

We study schemes of tensor eigenvectors from an algebraic and geometric viewpoint. We characterize determinantal defining equations of such eigenschemes via linear equations in their coefficients, both in the general and in the symmetric case. We give a geometric necessary condition for a 0-dimensional scheme to be an eigenscheme.

Improved cosine similarity and distance measures-based TOPSIS method for cubic Fermatean fuzzy sets

Similarity and distance measures play important roles in fuzzy environments, helping to quantify the degree of similarity or concepts that may not have clear limits. They are used in various fields, including fuzzy logic, fuzzy clustering, and fuzzy decision-making. The cubic Fermatean fuzzy set (CFFS), which is a type of fuzzy set (FS), is highly favoured as an extension for expressing uncertainty through degrees of membership (η) and non-membership (υ). This article introduces novel measures for cosine similarity and distance in CFFSs. These measures are designed to improve the accuracy and efficiency of similarity and distance calculations in CFFSs. Also, a novel method is introduced for developing alternate similarity measures for CFFSs utilizing the proposed similarity measures that adhere to the similarity measures axiom. In addition, the connection between similarity and distance measures is utilized to construct a cosine distance metric for CFFSs. This newly suggested cosine similarity measure can not only provide solutions to decision-making problems from a geometric perspective but also from an algebraic point of view. To conclude, a case study is presented to showcase the practicality and effectiveness of the proposed approach, followed by a comparison of the outcomes of the suggested technique with some existing methodologies. This analysis helps to validate the proposed method and demonstrates its potential for outperforming other available approaches in terms of efficiency and accuracy.

Operadic actions on long knots and 2–string links

In the present work, we realize the space of string 2-links $\mathcal{L}$ as a free algebra over a colored operad denoted $\mathcal{SCL}$ (for "Swiss-Cheese for links"). This result extends works of Burke and Koytcheff about the quotient of $\mathcal{L}$ by its center and is compatible with Budney's freeness theorem for long knots. From an algebraic point of view, our main result refines Blaire, Burke and Koytcheff's theorem on the monoid of isotopy classes of string links. Topologically, it expresses the homotopy type of the isotopy class of a string 2-link in terms of the homotopy types of the classes of its prime factors.

The varieties of bifocal Grassmann tensors

Grassmann tensors arise from classical problems of scene reconstruction in computer vision. In particular, bifocal Grassmann tensors, related to a pair of projections from a projective space onto view spaces of varying dimensions, generalize the classical notion of fundamental matrices. In this paper, we study in full generality the variety of bifocal Grassmann tensors focusing on its birational geometry. To carry out this analysis, every object of multi-view geometry is described both from an algebraic and geometric point of view, e.g., the duality between the view spaces, and the space of rays is explicitly described via polarity. Next, we deal with the moduli of bifocal Grassmann tensors, thus showing that this variety is both birational to a suitable homogeneous space and endowed with a dominant rational map to a Grassmannian.

Algebraic approach to annihilation and repulsion of bound states in the continuum in finite systems

We present an algebraic approach to the description of bound states in the continuum (BICs) in finite systems with a discrete energy spectrum coupled to several decay channels. General estimations and bounds on the number of linearly independent BICs are derived. We show that the algebraic point of view provides straightforward and illustrative interpretations of typical well-known results, including the Friedrich–Wintgen mechanism and the Pavlov-Verevkin model. Pair-wise annihilation and repulsion of BICs in the energy–parameter space are discussed within generic two- and three-level models. An illustrative algebraic interpretation of such phenomena in Hilbert space is presented.

Differential–algebraic systems with dissipative Hamiltonian structure

Different representations of linear dissipative Hamiltonian and port-Hamiltonian differential–algebraic equations (DAE) systems are presented and compared. Using global geometric and algebraic points of view, translations between different representations are presented. Characterizations are also derived when a general DAE system can be transformed into one of these structured representations. Approaches for computing the structural information and the described transformations are derived that can be directly implemented as numerical methods. The results are demonstrated with a large number of examples.

Invariants of formal pseudodifferential operator algebras and algebraic modular forms

We study from an algebraic point of view the question of extending an action of a group Γ on a commutative domain R to a formal pseudodifferential operator ring B = R((x ; d)) with coefficients in R, as well as to some canonical quadratic extension C = R((x 1/2 ; 1 2 d))2 of B. We give a necessary and sufficient condition of compatibility between the action and the derivation d of R for such an extension to exist, and we determine all possible extensions of the action to B and C. We describe under suitable assumptions the invariant subalgebras B Γ and C Γ as Laurent series rings with coefficients in R Γ. The main results of this general study are applied in a numbertheoretical context to the case where Γ is a subgroup of SL(2, C) acting by homographies on an algebra R of functions in one complex variable. Denoting by Mj the vector space of algebraic modular forms in R of weight j (even or odd), we build for any nonnegative integer k a linear isomorphism between the subspace C Γ k of invariant operators of order ≥ k in C Γ and the product space M k = j≥k Mj, which can be identified with a space of algebraic Jacobi forms of weight k. It results in particular a structure of noncommutative algebra on M0 and an algebra isomorphism Ψ : M0 → C Γ 0 , whose restriction to the particular case of even weights was previously known in the litterature. We study properties of this correspondence combining arithmetical arguments and the use of the algebraic results of the first part of the article.

Arrhenius Crossover Temperature of Glass-Forming Liquids Predicted by an Artificial Neural Network.

The Arrhenius crossover temperature, TA, corresponds to a thermodynamic state wherein the atomistic dynamics of a liquid becomes heterogeneous and cooperative; and the activation barrier of diffusion dynamics becomes temperature-dependent at temperatures below TA. The theoretical estimation of this temperature is difficult for some types of materials, especially silicates and borates. In these materials, self-diffusion as a function of the temperature T is reproduced by the Arrhenius law, where the activation barrier practically independent on the temperature T. The purpose of the present work was to establish the relationship between the Arrhenius crossover temperature TA and the physical properties of liquids directly related to their glass-forming ability. Using a machine learning model, the crossover temperature TA was calculated for silicates, borates, organic compounds and metal melts of various compositions. The empirical values of the glass transition temperature Tg, the melting temperature Tm, the ratio of these temperatures Tg/Tm and the fragility index m were applied as input parameters. It has been established that the temperatures Tg and Tm are significant parameters, whereas their ratio Tg/Tm and the fragility index m do not correlate much with the temperature TA. An important result of the present work is the analytical equation relating the temperatures Tg, Tm and TA, and that, from the algebraic point of view, is the equation for a second-order curved surface. It was shown that this equation allows one to correctly estimate the temperature TA for a large class of materials, regardless of their compositions and glass-forming abilities.

On The Symbolic 6-Plithogenic and 7-Plithogenic Rings

Symbolic n-plithogenic algebraic structures are considered as a direct application of fuzzy generalized systems in pure algebra, where the symbolic n-plithogenic set is used to generalize algebraic structures by adding logical generators. In this paper, we study the concept of symbolic 6-plithogenic rings and 7-plithogenic rings from an algebraic point of view, where the main substructures formed by them will be presented such as AH-ideals, AHS-isomorphisms, and AH-kernels. Also, many theorems that explain their algebraic behaviors and classifications will be proved and illustrated.