Algebraic Characterization Research Articles

A new characterization of quasi-hereditary Nakayama algebras and applications

We call a finite dimensional algebra A S-connected if the projective dimensions of the simple A-modules form an interval. We prove that a Nakayama algebra A is S-connected if and only if A is quasi-hereditary. We apply this result to improve an inequality for the global dimension of quasi-hereditary Nakayama algebras due to Brown. We furthermore classify the Nakayama algebras where equality is attained in Brown’s inequality and show that they are enumerated by the even indexed Fibonacci numbers if the algebra is cyclic and by the odd indexed Fibonacci numbers if the algebra is linear.

Completeness of Wronskian Bethe Equations for Rational $${\mathfrak {\mathfrak {gl}}_{{{\mathsf {m}}}|{{\mathsf {n}}}}}$$ Spin Chains

We consider rational integrable supersymmetric $${\mathfrak {\mathfrak {gl}}_{{{\mathsf {m}}}|{{\mathsf {n}}}}}$$ spin chains in the defining representation and prove the isomorphism between a commutative algebra of conserved charges (the Bethe algebra) and a polynomial ring (the Wronskian algebra) defined by functional relations between Baxter Q-functions that we call Wronskian Bethe equations. These equations, in contrast to standard nested Bethe equations, admit only physical solutions for any value of inhomogeneities and furthermore we prove that the algebraic number of solutions to these equations is equal to the dimension of the spin chain Hilbert space (modulo relevant symmetries). Both twisted and twist-less periodic boundary conditions are considered, the isomorphism statement uses, as a sufficient condition, that the spin chain inhomogeneities $${{\theta }_{\ell }}$$ , $$\ell =1,\ldots ,L$$ satisfy $${{\theta }_{\ell }}+\hbar \ne {{\theta }_{\ell '}}$$ for $$\ell <\ell '$$ . Counting of solutions is done in two independent ways: by computing a character of the Wronskian algebra and by explicitly solving the Bethe equations in certain scaling regimes supplemented with a proof that the algebraic number of solutions is the same for any value of $$\theta _\ell $$ . In particular, we consider the regime $$\theta _{\ell +1}/\theta _{\ell }\gg 1$$ for the twist-less chain where we succeed to provide explicit solutions and their systematic labelling with standard Young tableaux.

Characterizing and computing weight-equitable partitions of graphs

Weight-equitable partitions of graphs, which are a natural extension of the well-known equitable partitions, have been shown to be a powerful tool to weaken the regularity assumption in several classic eigenvalue bounds. In this work we aim to further our algebraic and computational understanding of weight-equitable partitions. We do so by showing several spectral properties and algebraic characterizations, and by providing a method to find coarse weight-equitable partitions.

On the Dynamics of Higgins–Selkov, Selkov and Brusellator Oscillators

A complete algebraic characterization of the first integrals of the Higgins–Selkov, Selkov and Brusellator oscillators is given here. The existence of symmetries sometimes forces the existence of such first integrals. The nonexistence of centers for such oscillators is also proved. In order to determine the Puiseux integrability of such systems, the multiple Puiseux solutions are also studied.

Multivariate Orthogonal Laurent Polynomials and Integrable Systems

An ordering for Laurent polynomials in the algebraic torus (\mathbb{C}^*)^D , inspired by the Cantero–Moral–Vel´azquez approach to orthogonal Laurent polynomials in the unit circle, leads to the construction of a moment matrix for a given Borel measure in the unit torus \mathbb{T}^D . The Gauss–Borel factorization of this moment matrix allows for the construction of multivariate biorthogonal Laurent polynomials in the unit torus, which can be expressed as last quasi-determinants of bordered truncations of the moment matrix. The associated second-kind functions are expressed in terms of the Fourier series of the given measure. Persymmetries and partial persymmetries of the moment matrix are studied and Cauchy integral representations of the second-kind functions are found, as well as Plemelj-type formulae. Spectral matrices give string equations for the moment matrix, which model the three-term relations as well as the Christoffel–Darboux formulae. Christoffel-type perturbations of the measure given by the multiplication by Laurent polynomials are studied. Sample matrices on poised sets of nodes, which belong to the algebraic hypersurface of the perturbing Laurent polynomial, are used to find a Christoffel formula that expresses the perturbed orthogonal Laurent polynomials in terms of a last quasi-determinant of a bordered sample matrix constructed in terms of the original orthogonal Laurent polynomials. Poised sets exist only for prepared Laurent polynomials, which are analyzed from the perspective of Newton polytopes and tropical geometry. Then, an algebraic geometrical characterization of prepared Laurent polynomial perturbation and poised sets is given; full-column-rankness of the corresponding multivariate Laurent–Vandermonde matrices and a product of different prime prepared Laurent polynomials leads to such sets. Some examples are constructed in terms of perturbations of the Lebesgue–Haar measure. Discrete and continuous deformations of the measure lead to a Toda-type integrable hierarchy, being the corresponding flows described through Lax and Zakharov–Shabat equations; bilinear equations and vertex operators are found. Varying size matrix nonlinear partial difference and differential equations of two-dimensional Toda lattice type are shown to be solved by matrix coefficients of the multivariate orthogonal polynomials. The discrete flows are connected with a Gauss–Borel factorization of the Jacobi-type matrices and its quasi-determinants allow for expressions for the multivariate orthogonal polynomials in terms of shifted quasi-tau matrices, which generalize those that relate the Baker functions with ratios of Miwa shifted \tau -functions in the one-dimensional scenario. It is shown that the discrete and continuous flows are deeply connected and determine nonlinear partial difference-differential equations that involve only one site in the integrable lattice behaving as a Kadomtsev–Petviashvili-type system.

Transformations of Multi-Player Normal form Games by Preplay Offers between Players

The paper deals with multiplayer normal form games which are preceded by a ‘preplay negotiation phase’ consisting of exchange of preplay offers by players for payments of utility to other players conditional on them playing designated in the offers strategies. The game-theoretic effect of such preplay offers is a transformation of the payoff matrix of the game, obtained by transferring the offered payments between the payoffs of the respective players; thus, certain groups of game matrix transformations naturally emerge. The main result is an explicit and rather transparent algebraic characterization of the possible transformations of the payoff matrix of any given N-person normal form game induced by preplay offers for transfer of payments. That result can be used to describe the ‘bargaining space’ of the game and to determine the mutually optimal game transformations that rational players can achieve by exchange of preplay offers.

Surjective isometries between unitary sets of unital JB⁎-algebras

This paper is, in a first stage, devoted to establishing a topological–algebraic characterization of the principal component, U0(M), of the set of unitary elements, U(M), in a unital JB⁎-algebra M. We arrive to the conclusion that, as in the case of unital C⁎-algebras,U0(M)=M1−1∩U(M)={Ueihn⋯Ueih1(1):n∈N,hj∈Msa∀1≤j≤n}={u∈U(M): there exists w∈U0(M) with ‖u−w‖<2} is analytically arcwise connected. Actually, U0(M) is the smallest quadratic subset of U(M) containing the set eiMsa. Our second goal is to provide a complete description of the surjective isometries between the principal components of two unital JB⁎-algebras M and N. Contrary to the case of unital C⁎-algebras, we shall deduce the existence of connected components in U(M) which are not isometric as metric spaces. We shall also establish necessary and sufficient conditions to guarantee that a surjective isometry Δ:U(M)→U(N) admits an extension to a surjective linear isometry between M and N, a conclusion which is not always true. Among the consequences it is proved that M and N are Jordan ⁎-isomorphic if, and only if, their principal components are isometric as metric spaces if, and only if, there exists a surjective isometry Δ:U(M)→U(N) mapping the unit of M to an element in U0(N). These results provide an extension to the setting of unital JB⁎-algebras of the results obtained by O. Hatori for unital C⁎-algebras.

On logical and extensional characterizations of attributed feature models

Software-intensive systems can have thousands of interdependent configuration options across different subsystems. Feature models (FMs) allow designers to organize the configuration space by describing configuration options using interdependent features: a feature is a name representing some functionality and each software variant is identified by a set of features. Attributed feature models (AFMs) extend FMs to describe the, possibly constrained, choice of a value from domains such as integers or strings: each attribute is associated to one feature, and when the feature is selected then the attribute brings some additional information relative to the selected features. Different representations of FMs and AFMs have been proposed in the literature. In this paper we focus on the logical representation (which works well in practice) and the extensional representation (which has been recently shown well suited for theoretical investigations). We provide an algebraic and a logical characterization of operations and relations on FMs and AFMs, and we formalize the connection between the two characterizations as monomorphisms from lattices of logical FMs and AFMs to lattices of extensional FMs and AFMs, respectively. This formalization sheds new light on the correspondence between the algebraic and logical characterizations of operations and relations for FMs and AFMs. It aims to foster the development of a formal framework for supporting practical exploitation of future theoretical developments on FMs, AFMs and multi software product lines.

Integrable LCK manifolds

We study a natural class of LCK manifolds that we call integrable LCK manifolds: those where the anti-Lee form $\eta$ corresponds to an integrable distribution. As an application we obtain a characterization of unimodular integrable LCK Lie algebras as K\"ahler Lie algebras equipped with suitable derivations.

Bezout-like polynomial equations associated with dual univariate interpolating subdivision schemes

The algebraic characterization of dual univariate interpolating subdivision schemes is investigated. Specifically, we provide a constructive approach for finding dual univariate interpolating subdivision schemes based on the solutions of certain associated polynomial equations. The proposed approach also makes it possible to identify conditions for the existence of the sought schemes.

Median prime ideals of pseudo-complemented distributive lattices

Coherent ideals, strongly coherent ideals, and $\tau $-closed ideals are introduced in pseudo-complemented distributive lattices and their characterization theorems are derived. A set of equivalent conditions is derived for every ideal of a pseudo-complemented distributive lattice to become a coherent ideal. The notion of median prime ideals is introduced and some equivalent conditions are derived for every maximal ideal of a pseudo-complemented distributive lattice to become a median prime ideal which leads to a characterization of Boolean algebras.

Amenability, proximality and higher-order syndeticity

AbstractWe show that the universal minimal proximal flow and the universal minimal strongly proximal flow of a discrete group can be realized as the Stone spaces of translation-invariant Boolean algebras of subsets of the group satisfying a higher-order notion of syndeticity. We establish algebraic, combinatorial and topological dynamical characterizations of these subsets that we use to obtain new necessary and sufficient conditions for strong amenability and amenability. We also characterize dense orbit sets, answering a question of Glasner, Tsankov, Weiss and Zucker.

Sectional curvature and Weitzenbock formulae

We establish a new algebraic characterization of sectional curvature bounds $\sec\geq k$ and $\sec\leq k$ using only curvature terms in the Weitzenb\"ock formulae for symmetric $p$-tensors. By introducing a symmetric analogue of the Kulkarni-Nomizu product, we provide a simple formula for such curvature terms. We also give an application of the Bochner technique to closed $4$-manifolds with indefinite intersection form and $\sec>0$ or $\sec\geq0$, obtaining new insights into the Hopf Conjecture, without any symmetry assumptions.

An Algebraic Characterization of Prefix-Strict Languages

An Algebraic Characterization of Prefix-Strict Languages

Rewriting systems, plain groups, and geodetic graphs

We take a new step towards an algebraic characterisation of groups presented by length-reducing rewriting systems. We prove that a group is presented by finite convergent length-reducing rewriting systems where each rule has left-hand side of length three if and only if the group is plain.Our proof rests on proving a new result about embedded circuits in geodetic graphs, whose proof may also be of independent interest to graph theorists.

The (Coarse) Fine-Grained Structure of NP-Hard SAT and CSP Problems

We study the fine-grained complexity of NP-complete satisfiability (SAT) problems and constraint satisfaction problems (CSPs) in the context of the strong exponential-time hypothesis (SETH) , showing non-trivial lower and upper bounds on the running time. Here, by a non-trivial lower bound for a problem SAT (Γ) (respectively CSP (Γ)) with constraint language Γ, we mean a value c 0 &gt; 1 such that the problem cannot be solved in time O ( c n ) for any c &lt; c 0 unless SETH is false, while a non-trivial upper bound is simply an algorithm for the problem running in time O ( c n ) for some c &lt; 2. Such lower bounds have proven extremely elusive, and except for cases where c 0 =2 effectively no such previous bound was known. We achieve this by employing an algebraic framework, studying constraint languages Γ in terms of their algebraic properties. We uncover a powerful algebraic framework where a mild restriction on the allowed constraints offers a concise algebraic characterization. On the relational side we restrict ourselves to Boolean languages closed under variable negation and partial assignment, called sign-symmetric languages. On the algebraic side this results in a description via partial operations arising from system of identities, with a close connection to operations resulting in tractable CSPs, such as near unanimity operations and edge operations . Using this connection we construct improved algorithms for several interesting classes of sign-symmetric languages, and prove explicit lower bounds under SETH. Thus, we find the first example of an NP-complete SAT problem with a non-trivial algorithm which also admits a non-trivial lower bound under SETH. This suggests a dichotomy conjecture with a close connection to the CSP dichotomy theorem: an NP-complete SAT problem admits an improved algorithm if and only if it admits a non-trivial partial invariant of the above form.

An algebraic characterisation of ample type I groupoids

We give algebraic characterisations of the type I and CCR properties for locally compact second countable, ample Hausdorff groupoids in terms of subquotients of its Boolean inverse semigroup of compact open local bisections. It yields in turn algebraic characterisations of both properties for inverse semigroups with meets in terms of subquotients of their Booleanisation.

Enhancing Linear Algebraic Computation of Logic Programs Using Sparse Representation

Algebraic characterization of logic programs has received increasing attention in recent years. Researchers attempt to exploit connections between linear algebraic computation and symbolic computation to perform logical inference in large-scale knowledge bases. In this paper, we analyze the complexity of the linear algebraic methods for logic programs and propose further improvement by using sparse matrices to embed logic programs in vector spaces. We show its great power of computation in reaching the fixed point of the immediate consequence operator. In particular, performance for computing the least models of definite programs is dramatically improved using the sparse matrix representation. We also apply the method to the computation of stable models of normal programs, in which the guesses are associated with initial matrices, and verify its effect when there are small numbers of negation. These results show good enhancement in terms of performance for computing consequences of programs and depict the potential power of tensorized logic programs.

A note on ensemble holography for rational tori

We study simple examples of ensemble-averaged holography in free compact boson CFTs with rational values of the radius squared. These well-known rational CFTs have an extended chiral algebra generated by three currents. We consider the modular average of the vacuum character in these theories, which results in a weighted average over all modular invariants. In the simplest case, when the chiral algebra is primitive (in a sense we explain), the weights in this ensemble average are all equal. In the non-primitive case the ensemble weights are governed by a semigroup structure on the space of modular invariants.These observations can be viewed as evidence for a holographic duality between the ensemble of CFTs and an exotic gravity theory based on a compact U(1) × U(1) Chern-Simons action. In the bulk description, the extended chiral algebra arises from soliton sectors, and including these in the path integral on thermal AdS3 leads to the vacuum character of the chiral algebra. We also comment on wormhole-like contributions to the multi-boundary path integral.

On the structure of asymptotic expanders

In this paper, we study several analytic and graph-theoretic properties of asymptotic expanders. We show that asymptotic expanders can be characterised in terms of their uniform Roe algebra. Moreover, we use asymptotic expanders to provide uncountably many new counterexamples to the coarse Baum–Connes conjecture. Finally, we show that vertex-transitive asymptotic expanders are actually expanders. In particular, this gives a C⁎-algebraic characterisation of expanders for vertex-transitive graphs. We achieve these results by showing that a sequence of asymptotic expanders always admits a “uniform exhaustion by expanders”. This also implies that asymptotic expanders cannot be coarsely embedded into any Lp-space.