Algebraic Invariants Research Articles

Statistical–mechanical theory of topological indices

Topological indices (TI) are algebraic invariants of molecular graphs representing the topology of a molecule, which are very valuable in quantitative structure–property relations (QSPR). Here we prove that TI are the partition functions of such molecules when the temperature of the thermal bath at which they are submerged is very high. These partition functions are obtained by describing molecular electronic properties through tight-binding Hamiltonians (TBH), where the hopping parameters are topological properties describing atom–atom interactions. We prove that the TBH proposed here are non-Hermitian diagonalizable Hamiltonians which can be replaced by symmetric ones. In this way we propose a statistical–mechanical theory for TI, which is exemplified by deriving the Randić, Zagreb, Balaban, Wiener and ABC indices. The work also illuminates how to improve QSPR models using the current theoretical framework as well as how to derive statistical–mechanical parameters of molecular graphs.

Differential invariants of curves in G2 flag varieties

We compute the algebra of differential invariants of unparametrized curves in the homogeneous G2 flag varieties, namely in G2/P. This gives a solution to the equivalence problem for such curves. We consider the cases of integral and generic curves and relate the equivalence problems for all three choices of the parabolic subgroup P.

Projective dimension and Castelnuovo–Mumford regularity of t-spread ideals

In this paper, we study some algebraic invariants of t-spread ideals, [Formula: see text], such as the projective dimension and the Castelnuovo–Mumford regularity, by means of well-known graded resolutions. We state upper bounds for these invariants and, furthermore, we identify a special class of t-spread ideals for which such bounds are optimal.

Review of gravitational wave solutions in quadratic metric-affine gauge gravity

In this paper, we consider the quadratic Metric-Affine Gauge gravity Lagrangian, which contains all the algebraic invariants up to quadratic order in torsion, nonmetricity and curvature. The goal is to collect the known exact solutions for this theory that describe the propagation of a gravitational wave and some useful properties. We concentrate on solutions whose connections are different from the Levi-Civita one and provide also the extension of already known axial torsion solutions to more general Lagrangians. In the end, we will briefly discuss colliding waves and Riemannian solutions.

Derived equivalence classification of Brauer graph algebras

We classify Brauer graph algebras up to derived equivalence by showing that the set of derived invariants introduced by Antipov is complete. These algebras first appeared in representation theory of finite groups and can be defined for any suitably decorated graph on an oriented surface. Motivated by the connection between Brauer graph algebras and gentle algebras we consider A∞-trivial extensions of partially wrapped Fukaya categories associated to surfaces with boundary. This construction naturally enlarges the class of Brauer graph algebras and provides a way to construct derived equivalences between Brauer graph algebras with the same derived invariants. As part of the proof we provide an interpretation of derived invariants of Brauer graph algebras as orbit invariants of line fields under the action of the mapping class group.

Multimodal image registration for power equipment using clifford algebraic geometric invariance

Imaging inspection and video monitoring are important means for power equipment condition monitoring and diagnosis. However, single model image cannot supply sufficient information for power equipment condition monitoring, such as infrared images and visible images. The image registration can bridge this gap. This paper develops an image registration method fused by infrared and visible images, which constructs the principal direction of corner points using neighborhood contour point geometric invariance under Clifford algebraic space. Simultaneously, we also design a law for fitting optimal spatial information in the main direction of corner points, where the Clifford rotation operator is utilized to realize the spatial transformation of the image to be registered. The experiment reveals that the average registration error is less than 3 pixels by this developed method, reaching sub-pixel registration, subsequently obtaining satisfactory fusion effect of power equipment. This algorithm enriches the multimodal image registration scheme and meets the requirements of monitoring power equipment.

A signature invariant geometric algebra framework for spacetime physics and its applications in relativistic dynamics of a massive particle and gyroscopic precession

A signature invariant geometric algebra framework for spacetime physics is formulated. By following the original idea of David Hestenes in the spacetime algebra of signature (+,-,-,-), the techniques related to relative vector and spacetime split are built up in the spacetime algebra of signature (-,+,+,+). The even subalgebras of the spacetime algebras of signatures (pm ,mp ,mp ,mp ) share the same operation rules, so that they could be treated as one algebraic formalism, in which spacetime physics is described in a signature invariant form. Based on the two spacetime algebras and their “common” even subalgebra, rotor techniques on Lorentz transformation and relativistic dynamics of a massive particle in curved spacetime are constructed. A signature invariant treatment of the general Lorentz boost with velocity in an arbitrary direction and the general spatial rotation in an arbitrary plane is presented. For a massive particle, the spacetime splits of the velocity, acceleration, momentum, and force four-vectors with the normalized four-velocity of the fiducial observer, at rest in the coordinate system of the spacetime metric, are given, where the proper time of the fiducial observer is identified, and the contribution of the bivector connection is considered, and with these results, a three-dimensional analogue of Newton’s second law for this particle in curved spacetime is achieved. Finally, as a comprehensive application of the techniques constructed in this paper, a geometric algebra approach to gyroscopic precession is provided, where for a gyroscope moving in the Lense-Thirring spacetime, the precessional angular velocity of its spin is derived in a signature invariant manner.

On algebraic values of Weierstrass $\sigma$-functions

Suppose that \Omega is a lattice in the complex plane and let \sigma be the corresponding Weierstrass \sigma -function. Assume that the point \tau associated with \Omega in the standard fundamental domain has imaginary part at most 1.9. Assuming that \Omega has algebraic invariants g_2,g_3 we show that a bound of the form c d^m (\log H)^n holds for the number of algebraic points of height at most H and degree at most d lying on the graph of \sigma . To prove this we apply results by Masser and Besson. What is perhaps surprising is that we are able to establish such a bound for the whole graph, rather than some restriction. We prove a similar result when, instead of g_2,g_3 , the lattice points are algebraic. For this we naturally exclude those (z,\sigma(z)) for which z\in\Omega .

Computational approximations of compact metric spaces

Given a compact metric space X, we associate to it an inverse sequence of finite T0 topological spaces. The inverse limit of this inverse sequence contains a homeomorphic copy of X that is a strong deformation retract of the inverse limit. We provide a method to approximate the homology groups of X and other algebraic invariants. Finally, we study computational aspects and the implementation of this method.

Homological Algebra and Its Application: A Descriptive Study

Algebra has been used to define and answer issues in almost every field of mathematics, science, and engineering. Homological algebra depends largely on computable algebraic invariants to categorise diverse mathematical structures, such as topological, geometrical, arithmetical, and algebraic (up to certain equivalences). String theory and quantum theory, in particular, have shown it to be of crucial importance in addressing difficult physics questions. Geometric, topological and algebraic algebraic techniques to the study of homology are to be introduced in this research. Homology theory in abelian categories and a category theory are covered. the n-fold extension functors EXTn (-,-) , the torsion functors TORn (-,-), Algebraic geometry, derived functor theory, simplicial and singular homology theory, group co-homology theory, the sheaf theory, the sheaf co-homology, and the l-adic co-homology, as well as a demonstration of its applicability in representation theory.

Path homologies of motifs and temporal network representations

Path homology is a powerful method for attaching algebraic invariants to digraphs. While there have been growing theoretical developments on the algebro-topological framework surrounding path homology, bona fide applications to the study of complex networks have remained stagnant. We address this gap by presenting an algorithm for path homology that combines efficient pruning and indexing techniques and using it to topologically analyze a variety of real-world complex temporal networks. A crucial step in our analysis is the complete characterization of path homologies of certain families of small digraphs that appear as subgraphs in these complex networks. These families include all digraphs, directed acyclic graphs, and undirected graphs up to certain numbers of vertices, as well as some specially constructed cases. Using information from this analysis, we identify small digraphs contributing to path homology in dimension two for three temporal networks in an aggregated representation and relate these digraphs to network behavior. We then investigate alternative temporal network representations and identify complementary subgraphs as well as behavior that is preserved across representations. We conclude that path homology provides insight into temporal network structure, and in turn, emergent structures in temporal networks provide us with new subgraphs having interesting path homology.

Symmetric polynomials in free associative algebras

By a result of Margarete Wolf in 1936, we know that the algebra $K\langle X_d\rangle^{Sym(d)}$ of symmetric polynomials in noncommuting variables is not finitely generated. In 1984, Koryukin proved that if we equip the homogeneous component of degree $n$ with the additional action of $Sym(n)$ by permuting the positions of the variables, then the algebra of invariants $K\langle X_d\rangle^G$ of every reductive group $G$ is finitely generated. First, we make a short comparison between classical invariant theory of finite groups and its noncommutative counterpart. Then, we expose briefly the results of Wolf. Finally, we present the main result of our paper, which is, over a field of characteristic 0 or of characteristic $p>d$, the algebra $K\langle X_d\rangle^{Sym(d)}$ with the action of Koryukin is generated by the elementary symmetric polynomials.

GROUP CLASSIFICATION OF ONE CLASS (2+1)-DIMENSIONAL LINEAR EQUATIONS OF ASIAN OPTIONS PRICING

A group classification of one class of (2+1)-dimensional linear equations of Asian options pricing was carried out. As a result, the kernel of maximal invariance algebras and continuous equivalence transformations of this class of equations were found. Using equivalence transformations, all non-equivalent subclasses of equations that have an invariance algebra wider than the kernel of maximal invariance algebras are selected. For each such subclass of equations, Lie algebras of symmetry operators of dimensions four, five, and eight are found.

Connection between the ring of Ad*-invariant polynomials and the Jordan-Kronecker invariants of nilpotent low-dimensional Lie algebras

Connection between the ring of Ad*-invariant polynomials and the Jordan-Kronecker invariants of nilpotent low-dimensional Lie algebras

Invariants of symplectic and orthogonal groups acting on $GL(n,CC)$-modules

Let $GL(n) = GL(n, CC)$ denote the complex general linear group and let $G \subset GL(n)$ be one of the classical complex subgroups $OO(n)$, $SO(n)$, and $Sp(2k)$ (in the case $n = 2k$). We take a finite dimensional polynomial $GL(n)$-module $W$ and consider the symmetric algebra $S(W)$. Extending previous results for $G=SL(n)$, we develop a method for determining the Hilbert series $H(S(W)^G, t)$ of the algebra of invariants $S(W)^G$. Our method is based on simple algebraic computations and can be easily realized using popular software packages. Then we give many explicit examples for computing $H(S(W)^G, t)$. As an application, we consider the question of regularity of the algebra $S(W)^{OO(n)}$. For $n=2$ and $n=3$ we give a complete list of modules $W$, so that if $S(W)^{OO(n)}$ is regular then $W$ is in this list. As a further application, we extend our method to compute also the Hilbert series of the algebras of invariants $Λ(S^2 V)^G$ and $Λ(Λ^2 V)^G$, where $V = CC^n$ denotes the standard $GL(n)$-module.

Algebraic invariants of gravity

Newton's mechanics is simple. His equivalence principle is simple, as is the inverse square law of gravitational force. A simple theory should have simple solutions to simple models. A system of n particles, given their initial speed and positions along with their masses, is such a simple model. Yet, solving for n>2 is not simple. This paper discusses what could be done to overcome that problem.

Classification of Cohen–Macaulay t-spread lexsegment ideals via simplicial complexes

We study the minimal primary decomposition of completely t-spread lexsegment ideals via simplicial complexes. We determine some algebraic invariants of such a class of t-spread ideals. Hence, we classify all t-spread lexsegment ideals which are Cohen–Macaulay.

Arithmetic invariants of pseudoreflection groups and regular graded algebras

The main result of this paper is a generalization of the theorem of Chevalley-Shephard-Todd to the rings of invariants of pseudoreflection groups over Dedekind domains. In the special case of a principal ideal domain in which the group order is invertible it is proved that this ring of invariants is isomorphic to a polynomial ring. An intermediate result is that every finitely generated regular graded algebra over a Dedekind domain is isomorphic to a tensor product of blowup algebras.

The relations for Poincaré series of algebras of invariants of binary forms

The relations for Poincaré series of algebras of invariants of binary forms

Finite gentle repetitions of gentle algebras and their Avella-Alaminos–Geiss invariants

Among finite dimensional algebras over a field K, the class of gentle algebras is known to be closed by derived equivalences. Although a classification up to derived equivalences is usually a difficult problem, Avella-Alaminos and Geiss have introduced derived invariants for gentle algebras A, which can be calculated combinatorially from their bound quivers, applicable to such classification. Ladkani has given a formula to describe the dimensions of the Hochschild cohomologies of A in terms of some values of its Avella-Alaminos–Geiss invariants. This in turn implies a cohomological meaning of these values. Since most of the other values do not appear in this formula, it will be a natural question to ask if there is a similar cohomological meaning for such values. In this article, we construct a sequence of gentle algebras indexed by positive integers by a procedure which we call finite gentle repetitions, in order to relate these values of Avella-Alaminos–Geiss invariants of A to the dimensions of Hochschild cohomologies of On the way we will see that the Avella-Alaminos–Geiss invariants of are completely determined by those of A. Therefore one may expect that the finite gentle repetitions would preserve derived equivalences in a nice situation. In the last part of this article, we will see how the generalized Auslander–Platzeck–Reiten reflection can be related to the finite gentle repetition.