Non-trivial Identities Research Articles

Semigroup identities of supertropical matrices

We prove that, for any n, the monoid of all \(n \times n\) supertropical matrices extending tropical matrices satisfies nontrivial semigroup identities; in particular, such supertropical triangular matrices admit exactly the same identities satisfied by tropical triangular matrices. These semigroup identities are carried over to labeled-weighted digraphs with double arcs.

The complement of the intersection graph of ideals of a poset

Let [Formula: see text] be an atomic poset with the least element [Formula: see text]. The complement of the intersection graph of ideals of [Formula: see text], denoted by [Formula: see text], is defined to be a graph whose vertices are all non-trivial ideals of [Formula: see text] and two distinct vertices [Formula: see text] and [Formula: see text] are adjacent if and only if [Formula: see text]. In this paper, we consider the complement of the intersection graph of ideals of a poset. We prove that [Formula: see text] is totally disconnected or [Formula: see text], where [Formula: see text] is the set of all isolated vertices of [Formula: see text]. We show that [Formula: see text]. Also, we characterize all posets whose complement of the intersection graph is forest, unicyclic or complete [Formula: see text]-partite graph. Among other results, we prove that [Formula: see text] is weakly perfect; and it is perfect if and only if [Formula: see text]. Finally, we show that [Formula: see text] is class [Formula: see text], where [Formula: see text].

Toeplitz algebras of semigroups

To each submonoid P P of a group we associate a universal Toeplitz C ∗ \mathrm {C}^* -algebra T u ( P ) \mathcal {T}_u(P) defined via generators and relations; T u ( P ) \mathcal {T}_u(P) is a quotient of Li’s semigroup C ∗ \mathrm {C}^* -algebra C s ∗ ( P ) \mathrm {C}^*_s(P) and they are isomorphic iff P P satisfies independence. We give a partial crossed product realization of T u ( P ) \mathcal {T}_u(P) and show that several results known for C s ∗ ( P ) \mathrm {C}^*_s(P) when P P satisfies independence are also valid for T u ( P ) \mathcal {T}_u(P) when independence fails. At the level of the reduced semigroup C ∗ \mathrm {C}^* -algebra T λ ( P ) \mathcal {T}_\lambda (P) , we show that nontrivial ideals have nontrivial intersection with the reduced crossed product of the diagonal subalgebra by the action of the group of units of P P , generalizing a result of Li for monoids with trivial unit group. We characterize when the action of the group of units is topologically free, in which case a representation of T λ ( P ) \mathcal {T}_\lambda (P) is faithful iff it is jointly proper. This yields a uniqueness theorem that generalizes and unifies several classical results. We provide a concrete presentation for the covariance algebra of the product system over P P with one-dimensional fibers in terms of a new notion of foundation sets of constructible ideals. We show that the covariance algebra is a universal analogue of the boundary quotient and give conditions on P P for the boundary quotient to be purely infinite simple. We discuss applications to a numerical semigroup and to the a x + b ax+b -monoid of an integral domain. This is particularly interesting in the case of nonmaximal orders in number fields, for which we show independence always fails.

Geometric deduction of the solutions to modular equations

In his notebooks, Ramanujan presented without proof many remarkable formulae for the solutions to generalized modular equations. Much later, proofs of the formulae were provided by making use of highly nontrivial identities for theta series and hypergeometric functions. We offer a geometric approach to the proof of those formulae. We emphasize that our proofs are geometric and independent of such identities.

Formal Theories of Occurrences and Substitutions

Formal Theories of Occurrences and Substitutions

Symbolic Conversion of Holonomic Functions to Hypergeometric Type Power Series

A term $a_n$ is $m$-fold hypergeometric, for a given positive integer $m$, if the ratio $a_{n+m}/a_n$ is a rational function over a field $K$ of characteristic zero. We establish the structure of holonomic recurrence equation, i.e. linear and homogeneous recurrence equations having polynomial coefficients, that have $m$-fold hypergeometric term solutions over $K$, for any positive integer $m$. Consequently, we describe an algorithm, say $mfoldHyper$, that extends van Hoeij's algorithm (1998) which computes a basis of the subspace of hypergeometric $(m=1)$ term solutions of holonomic recurrence equations to the more general case of $m$-fold hypergeometric terms. We generalize the concept of hypergeometric type power series introduced by Koepf (1992), by considering linear combinations of Laurent-Puiseux series whose coefficients are $m$-fold hypergeometric terms. Thus thanks to $mfoldHyper$, we deduce a complete procedure to compute these power series; indeed, it turns out that every linear combination of power series with $m$-fold hypergeometric term coefficients, for finitely many values of $m$, is detected. On the other hand, we investigate an algorithm to represent power series of non-holonomic functions. The algorithm follows the same steps of Koepf's algorithm, but instead of seeking holonomic differential equations, quadratic differential equations are computed and the Cauchy product rule is used to deduce recurrence equations for the power series coefficients. This algorithm defines a normal function that yields together with enough initial values normal forms for many power series of non-holonomic functions. Therefore, non-trivial identities are automatically proved using this approach. This paper is accompanied by implementations in the Computer Algebra Systems (CAS) Maxima 5.44.0 and Maple 2019.

Ladder Operators and Hidden Algebras for Shape Invariant Nonseparable and Nondiagonalizable Modelswith Quadratic Complex Interaction. II. Three-Dimensional Model

A shape invariant nonseparable and nondiagonalizable three-dimensional model with quadratic complex interaction was introduced by Bardavelidze, Cannata, Ioffe, and Nishnianidze. However, the complete hidden symmetry algebra and the description of the associated states that form Jordan blocks remained to be studied. We present a set of six operators $\{A^{\pm},B^{\pm},C^{\pm}\}$ that can be combined to build a ${\mathfrak{gl}}(3)$ hidden algebra. The latter can be embedded in an ${\mathfrak{sp}}(6)$ algebra, as well as in an ${\mathfrak{osp}}(1/6)$ superalgebra. The states associated with the eigenstates and making Jordan blocks are induced in different ways by combinations of operators acting on the ground state. We present the action of these operators and study the construction of an extended biorthogonal basis. These rely on establishing various nontrivial polynomial and commutator identities. We also make a connection between the hidden symmetry and the underlying superintegrability property of the model. Interestingly, the integrals generate a cubic algebra. This work demonstrates how various concepts that have been applied widely to Hermitian Hamiltonians, such as hidden symmetries, superintegrability, and ladder operators, extend to the pseudo-Hermitian case with many differences.

Combinatorics of KP hierarchy structural constants

We investigate the structural constants of the KP hierarchy, which appear as universal coefficients in the paper of Natanzon–Zabrodin arXiv:1509.04472. It turns out that these constants have a combinatorial description in terms of transport coefficients in the theory of flow networks. Considering its properties we want to point out three novel directions of KP combinatorial structure research: connection with topological recursion, eigenvalue model for the structural constants and its deformations, possible deformations of KP hierarchy in terms of the structural constants. Firstly, in this paper we study the internal structure of these coefficients which involves: (1) construction of generating functions that have interesting properties by themselves; (2) restrictions on topological recursion initial data; (3) construction of integral representation or matrix model for these coefficients with non-trivial Ward identities. This shows that these coefficients appear in various problems of mathematical physics, which increases their value and significance. Secondly, we discuss their role in integrability of KP hierarchy considering possible deformation of these coefficients without changing the equations on tau -function. We consider several plausible deformations. While most failed even very basic checks, one deformation (involving Macdonald polynomials) passes all the simple checks and requires more thorough study.

Combinatorics of continuants of continued fractions with 3 limits

We give combinatorial descriptions of the terms occurring in continuants of general continued fractions that diverge to three limits. Equating this combinatorics with the usual combinatorial description due to Euler induces nontrivial identities. Special cases and applications to counting sequences are given.

Free Lie-admissible algebras and an analogue of the PBW theorem

We first construct linear bases for free Lie-admissible algebras and develop a new theory of Gröbner–Shirshov bases for Lie-admissible algebras. Then we prove an analogue of the Poincaré–Birkhoff–Witt theorem, that is, every Lie algebra L can be embedded into its universal enveloping Lie-admissible algebra U(L), where the basis of U(L) does not depend on the multiplication table of L. Finally, we show that the basic rank of the variety of Lie-admissible algebras is 1. As a corollary, the universal enveloping Lie-admissible algebra of an abelian Lie algebra does not satisfy any nontrivial identity in the variety of Lie-admissible algebras.

Radiative contributions to gravitational scattering

The linear-order effects of radiation-reaction on the classical scattering of two point masses, in General Relativity, are derived by a variation-of-constants method. Explicit expressions for the radiation-reaction contributions to the changes of 4-momentum during scattering are given to linear order in the radiative losses of energy, linear-momentum and angular momentum. The polynomial dependence on the masses of the 4-momentum changes is shown to lead to non-trivial identities relating the various radiative losses. At order $G^3$ our results lead to a streamlined classical derivation of results recently derived within a quantum approach. At order $G^4$ we compute the needed radiative losses to next-to-next-to-leading-order in the post-Newtonian expansion, thereby reaching the absolute fourth and a half post-Newtonian level of accuracy in the 4-momentum changes. We also provide explicit expressions, at the absolute sixth post-Newtonian accuracy, for the radiation-graviton contribution to {\it conservative} $O(G^4)$ scattering. At orders $G^5$ and $G^6$ we derive explicit theoretical expressions for the last two hitherto undetermined parameters describing the fifth-post-Newtonian dynamics. Our results at the fifth-post-Newtonian level confirm results of [Nucl. Phys. B \textbf{965}, 115352 (2021)] but exhibit some disagreements with results of [Phys. Rev. D \textbf{101}, 064033 (2020)].

Green fields

We introduce Green fields, as commutative Green biset functors with no non-trivial ideals. We state some of their properties and give examples of known Green biset functors which are Green fields. Among the properties, we prove some criterions ensuring that a Green field is semisimple. Finally, we describe a type of Green field for which its category of modules is equivalent to a category of vector spaces over a field.

Six-vertex model on a finite lattice: Integral representations for nonlocal correlation functions

We consider the problem of calculation of correlation functions in the six-vertex model with domain wall boundary conditions. To this aim, we formulate the model as a scalar product of off-shell Bethe states, and, by applying the quantum inverse scattering method, we derive three different integral representations for these states. By suitably combining such representations, and using certain antisymmetrization relation in two sets of variables, it is possible to derive integral representations for various correlation functions. In particular, focusing on the emptiness formation probability, besides reproducing the known result, obtained by other means elsewhere, we provide a new one. By construction, the two representations differ in the number of integrations and their equivalence is related to a hierarchy of highly nontrivial identities.

A survey on the Intersection graphs of ideals of rings

Let L(R) denote the set of all non-trivial left ideals of a ring R. The intersection graph of ideals of a ring R is an undirected simple graph denoted by G(R) whose vertices are in a one-to-one correspondence with L(R) and two distinct vertices are joined by an edge if and only if the corresponding left ideals of R have a non-zero intersection. The ideal structure of a ring reflects many ring theoretical properties. Thus much research has been conducted last few years to explore the properties of G(R). This is a survey of the developments in the study on the intersection graphs of ideals of rings since its introduction in 2009.

Line graph associated to the intersection graph of ideals of rings

Let R be a ring with unity and I(R) ∗ are all non-trivial left ideals of R. The intersection graph of ideals of R is denoted by G(R) is an undirected simple graph with vertex set I(R) ∗ and two distinct vertices I and J are adjacent if and only if I∩J ≠ 0. In this article, we investigate some basic properties of the line graph associated to G(R), denoted by L(G(R)). Moreover, we investigate completeness, unicyclicness, bipartiteness, planarity, outerplanarity, ring graph, diameter, girth and clique of L(G(Z n )). We also investigate some basic properties of L(G(R)) for left Artinian ring and finally, we determine the domination number and bondage number of L(G(Z n )).

Tropical representations and identities of plactic monoids

We exhibit a faithful representation of the plactic monoid of every finite rank as a monoid of upper triangular matrices over the tropical semiring. This answers a question first posed by Izhakian and subsequently studied by several authors. A consequence is a proof of a conjecture of Kubat and Okniński that every plactic monoid of finite rank satisfies a non-trivial semigroup identity. In the converse direction, we show that every identity satisfied by the plactic monoid of rank n n is satisfied by the monoid of n × n n \times n upper triangular tropical matrices. In particular this implies that the variety generated by the 3 × 3 3 \times 3 upper triangular tropical matrices coincides with that generated by the plactic monoid of rank 3 3 , answering another question of Izhakian.

Entanglement bootstrap approach for gapped domain walls

We develop a theory of gapped domain wall between topologically ordered systems in two spatial dimensions. We find a new type of superselection sector -- referred to as the parton sector -- that subdivides the known superselection sectors localized on gapped domain walls. Moreover, we introduce and study the properties of composite superselection sectors that are made out of the parton sectors. We explain a systematic method to define these sectors, their fusion spaces, and their fusion rules, by deriving nontrivial identities relating their quantum dimensions and fusion multiplicities. We propose a set of axioms regarding the ground state entanglement entropy of systems that can host gapped domain walls, generalizing the bulk axioms proposed in [B. Shi, K. Kato, and I. H. Kim, Ann. Phys. 418, 168164 (2020)]. Similar to our analysis in the bulk, we derive our main results by examining the self-consistency relations of an object called information convex set. As an application, we define an analog of topological entanglement entropy for gapped domain walls and derive its exact expression.

Some non-trivial two-term dilogarithm identities

Some non-trivial two-term dilogarithm identities

Line graph associated to the intersection graph of ideals of rings

Let R be a ring with unity and I(R)∗ are all non-trivial left ideals of R. The intersection graph of ideals of R is denoted by G(R) is an undirected simple graph with vertex set I(R)∗ and two distinct vertices I and J are adjacent if and only if I∩J ≠ 0. In this article, we investigate some basic properties of the line graph associated to G(R), denoted by L(G(R)). Moreover, we investigate completeness, unicyclicness, bipartiteness, planarity, outerplanarity, ring graph, diameter, girth and clique of L(G(Zn)). We also investigate some basic properties of L(G(R)) for left Artinian ring and finally, we determine the domination number and bondage number of L(G(Zn)).

Characteristic decomposition of polynomial sets

An ordered pair $(\pset{G},\,~\pset{C})$ is called the characteristic pair of a polynomial ideal if $\pset{G}$ is the reduced lexicographic grobner basis of the ideal and $\pset{C}$ is the minimal triangular set contained in $\pset{G}$. A characteristic pair $(\pset{G},\,~\pset{C})$ is said to be regular or normal if $\pset{C}$ is regular or normal; it is said to be strong if the ideal generated by $\pset{G}$ and the saturated ideal of $\pset{C}$ are the same. A strong regular or normal characteristic decomposition of a polynomial set means a decomposition of the polynomial set into finitely many (strong) regular or normal characteristic pairs with an associated zero or ideal decomposition. This paper presents two algorithms for characteristic decomposition of any given polynomial set, one producing strong normal characteristic pairs with ideal splitting according to pseudo-divisibility relations between polynomials in reduced lexicographic grobner bases and the other producing strong regular characteristic pairs by constructing nontrivial ideals that can be divided out of given ideals by using ideal quotient. Novel theorems are established for regularizing characteristic pairs and strengthening regular and normal characteristic pairs by using ideal saturation. Strong regular or normal characteristic pairs produced by the algorithms possess nice properties and furnish two representations, one in terms of lexicographic grobner bases and the other in terms of triangular sets, for the zeros of the input set of multivariate polynomials. The paper includes a brief review on triangular decompositions of various kinds and discussions about inherent connections among triangular sets, Ritt characteristic sets, and reduced lexicographic grobner bases. Examples and experimental results are also provided to illustrate the method of characteristic decomposition and its applicability and performance.