Algebraic Identities Research Articles

On an identity of Sylvester

We discuss an algebraic identity, due to Sylvester, as well as related algebraic identities and applications.

Elliptic modular graphs, eigenvalue equations and algebraic identities

We obtain eigenvalue equations satisfied by various elliptic modular graphs with five links where two of the vertices are unintegrated. Solving them leads to several nontrivial algebraic identities between these graphs.

On isomorphisms between ideals of Fourier algebras of finite abelian groups

On isomorphisms between ideals of Fourier algebras of finite abelian groups

Fake Z

Recently introduced connections between quantum codes and Narain CFTs provide a simple ansatz to express a modular-invariant function Zleft(tau, overline{tau}right) in terms of a multivariate polynomial satisfying certain additional properties. These properties include algebraic identities, which ensure modular invariance of Zleft(tau, overline{tau}right) , and positivity and integrality of coefficients, which imply positivity and integrality of the \U0001d532(1)n × \U0001d532(1)n character expansion of Zleft(tau, overline{tau}right) . Such polynomials come naturally from codes, in the sense that each code of a certain type gives rise to the so-called enumerator polynomial, which automatically satisfies all necessary properties, while the resulting Zleft(tau, overline{tau}right) is the partition function of the code CFT — the Narain theory unambiguously constructed from the code. Yet there are also “fake” polynomials satisfying all necessary properties, that are not associated with any code. They lead to Zleft(tau, overline{tau}right) satisfying all modular bootstrap constraints (modular invariance and positivity and integrality of character expansion), but whether they are partition functions of any actual CFT is unclear. We consider the group of the six simplest fake polynomials and denounce the corresponding Z’s as fake: we show that none of them is the torus partition function of any Narain theory. Moreover, four of them are not partition functions of any unitary 2d CFT; our analysis for other two is inconclusive. Our findings point to an obvious limitation of the modular bootstrap approach: not every solution of the full set of torus modular bootstrap constraints is due to an actual CFT. In the paper we consider six simple examples, keeping in mind that thousands more can be constructed.

The noncommutative geometry of matrix polynomial algebras

This work investigates the noncommutative affine geometry of matrix polynomial algebra extensions of a coefficient algebra by (elementary) matrix variables. A precise description of the spectrum (of maximal one-sided or bilateral ideals) of general matrix algebras is required. It results that the Zariski space of the irreducible representations of a matrix algebra is obtained by a natural gluing of the Zariski spaces of the irreducible representations of its diagonal components. An important step for the geometry of matrix polynomial algebras in commuting variables is achieved by a generalization of the Amitsur–Small Nullstellensatz, from which follows a precise description of their primitive quotients. We also characterize which of them are geometric algebras (in the sense of noncommutative deformation theory), reconstructible as algebras of observables from the scheme of irreducible representations. We then prove that each diagonal component of a matrix polynomial algebra in commuting variables is a Jacobson ring, whose non-Noetherian commutative geometry is efficiently described by the geometry of an affine essential subextension. And in the spirit of nonlocal algebraic geometry and addressing an open question by Charlie Beil, we obtain a class of non-Noetherian commutative monoid rings admitting closed points with positive geometric dimension.

Energy-conserving successive multi-stage method for the linear wave equation with forcing terms

We propose a high-order time-discretized method for a non-homogeneous linear wave equation with a forcing term. The method conserves the accumulated discrete energy with the external term. We provide detailed proofs of unique solvability and unconditional energy conservation of the proposed successive multi-stage (SMS) method. We also present reduced order conditions up to the fourth order with aid of some important algebraic identities from the features of the SMS methods. We demonstrate the accuracy and stability of the SMS methods using numerical experiments. In addition, to show the applicability of the proposed method, we extend the method to solve quasi-linear wave equations and provide numerical simulations for sine-Gordon and Boussinesq-type equations.

Combinatorial Bernoulli factories

A Bernoulli factory is an algorithmic procedure for exact sampling of certain random variables having only Bernoulli access to their parameters. Bernoulli access to a parameter p∈[0,1] means the algorithm does not know p, but has sample access to independent draws of a Bernoulli random variable with mean equal to p. In this paper, we study the problem of Bernoulli factories for polytopes: given Bernoulli access to a vector x∈P for a given polytope P⊂[0,1]n, output a randomized vertex such that the expected value of the i-th coordinate is exactly equal to xi. For example, for the special case of the perfect matching polytope, one is given Bernoulli access to the entries of a doubly stochastic matrix [xij] and asked to sample a matching such that the probability of each edge (i,j) be present in the matching is exactly equal to xij. We show that a polytope P admits a Bernoulli factory if and and only if P is the intersection of [0,1]n with an affine subspace. Our construction is based on an algebraic formulation of the problem, involving identifying a family of Bernstein polynomials (one per vertex) that satisfy a certain algebraic identity on P. The main technical tool behind our construction is a connection between these polynomials and the geometry of zonotope tilings. We apply these results to construct an explicit factory for the perfect matching polytope. The resulting factory is deeply connected to the combinatorial enumeration of arborescences and may be of independent interest. For the k-uniform matroid polytope, we recover a sampling procedure known in statistics as Sampford sampling.1 1 A preliminary conference version of this work has appeared in the proceeding of the 53rd Annual ACM SIGACT Symposium on Theory of Computing (STOC’21) [24]. The current version presents all the missing proofs and technical details, as well as new results, alternative proofs, and more explanations.

On Lie ideals and multiplicative generalized (α, β)-reverse derivations of *-prime rings

In this paper, the notion of multiplicative generalized (α, β)-reverse derivations associated with (α, β)-reverse derivations of *-prime rings is characterized. The action of these derivations on *-Lie ideals of *-prime rings is also investigated. Moreover, the commutativity of *-prime rings admitting multiplicative generalized (α, β)-reverse derivations associated with (α, β)-reverse derivations satisfying certain algebraic identities on *-Lie ideals is explored.

A Characterization of Semiprime Rings with Homoderivations

This paper is focused on the commutativity of the laws of semiprime rings, which satisfy some algebraic identities involving homoderivations on ideals. It provides new and notable results that will interest researchers in this field, such as “R contains a nonzero central ideal if R admits a nonzero homoderivation δ on I such that δ(I)⊆Z where R is a semiprime ring with center Z and I a nonzero ideal of R”. Moreover, the research also generalizes some results previously published in the literature, including derivation on prime rings using homoderivation semiprime rings. It also demonstrates the necessity of hypotheses operationalized in theorems by an example. Finally, the paper discusses how the results herein can be further developed in future research.

Burst error-correcting quantum stabilizer codes designed from idempotents

Certain classical codes can be viewed isomorphically as ideals of group algebras, while studying their algebraic structures help extracting the code properties. Research has shown that this was remarkably efficient in the case when the code generators are idempotents. In quantum error correction, the theory of stabilizer formalism requires classical self-orthogonal additive codes over the finite field GF(4), which, via the lens of group algebras, are essentially F_2-submodules over GF(4). Therefore, this paper provides a classification on idempotents in commutative group algebra GF(4)G, followed by a criterion that allows idempotents to generate stabilizer subgroups. Later, the construction of quantum stabilizer codes is done in the case when G is a cyclic group C_n, for n=2^m-1 and n=2^m+1. Quantum bounds on their burst error minimum distance are subsequently determined.

Closed and Open Maps for Partial Frames

This paper concerns the notions of closed and open maps in the setting of partial frames, which, in contrast to full frames, do not necessarily have all joins. Examples of these include bounded distributive lattices, sigma - and kappa -frames and full frames. We define closed and open maps using geometrically intuitively appealing conditions involving preservation of closed, respectively open, congruences under certain maps. We then characterize them in terms of algebraic identities involving adjoints. We note that partial frame maps need have neither right nor left adjoints whereas frame maps of course always have right adjoints. The embedding of a partial frame in either its free frame or its congruence frame has proved illuminating and useful. We consider the conditions under which these embeddings are closed, open or skeletal. We then look at preservation and reflection of closed or open maps under the functors providing the free frame or the congruence frame. Points arise naturally in the construction of the spectrum functor for partial frames to partial spaces. They may be viewed as maps from the given partial frame to the 2-chain or as certain kinds of filters; using the former description we consider closed and open points. Any point of a partial frame extends naturally to a point on its free frame and a point on its congruence frame; we consider the closedness or openness of these.

Nonlocal Traffic Models with General Kernels: Singular Limit, Entropy Admissibility, and Convergence Rate

Nonlocal conservation laws (the signature feature being that the flux function depends on the solution through the convolution with a given kernel) are extensively used in the modeling of vehicular traffic. In this work we discuss the singular local limit, namely the convergence of the nonlocal solutions to the entropy admissible solution of the conservation law obtained by replacing the convolution kernel with a Dirac delta. Albeit recent counter-examples rule out convergence in the general case, in the specific framework of traffic models (with anisotropic convolution kernels) the singular limit has been established under rigid assumptions, i.e. in the case of the exponential kernel (which entails algebraic identities between the kernel and its derivatives) or under fairly restrictive requirements on the initial datum. In this work we obtain general convergence results under assumptions that are entirely natural in view of applications to traffic models, plus a convexity requirement on the convolution kernels. We then provide a general criterion for entropy admissibility of the limit and a convergence rate. We also exhibit a counter-example showing that the convexity assumption is necessary for our main compactness estimate.

On Multiplicative (Generalized)-Derivation Involving Semiprime Ideals

Let A be any arbitrary associative ring, P a semiprime ideal, and J a nonzero ideal of A . In this study, using multiplicative (generalized)-derivations, we explore the behavior of semiprime ideals that satisfy certain algebraic identities. Moreover, examples are provided to demonstrate that the restrictions imposed on the hypotheses of the various theorems are necessary.

Divisibility of Weights for Ideals in Group Algebras

Divisibility of Weights for Ideals in Group Algebras

Heredity of Radicals and Ideals of Algebras Generated by Subideals and Subinvariant Subalgebras

This paper collects in a unified form well-known versions of the Anderson–Divinsky–Sulinski lemma for algebras that are nearly associative, and gives a number of its analogues for Lie algebras.

Identities on additive mappings in semiprime rings

Consider a ring $R$, which is semiprime and also having $k$-torsion freeness. If $F, d : R\to R$ are two additive maps fulfilling the algebraic identity $$F(x^{n+m})=F(x^m) x^n+ x^m d(x^n)$$ for each $x$ in $R.$ Then $F$ will be a generalized derivation having $d$ as an associated derivation on $R$. On the other hand, in this article, it is also derived that $f$ is a generalized left derivation having a linked left derivation $\delta$ on $R$ if they satisfy the algebraic identity $$f(x^{n+m})=x^n f(x^m)+ x^m \delta(x^n)$$ for each $x$ in $R$ and $k\in \{2, m, n, (n+m-1)!\}$ and at last an application on Banach algebra is presented.

Operator identities on Lie algebras, rewriting systems and Gröbner-Shirshov bases

Motivated by the pivotal role played by linear operators, many years ago Rota proposed to determine algebraic operator identities satisfied by linear operators on associative algebras, later called Rota's Program on Algebraic Operators. Recent progresses on this program have been achieved in the contexts of operated algebra, rewriting systems and Gröbner-Shirshov bases. These developments also suggest that Rota's insight can be applied to determine operator identities on Lie algebras, and thus to put the various linear operators on Lie algebras in a uniform perspective. This paper carries out this approach, utilizing operated polynomial Lie algebras spanned by nonassociative Lyndon-Shirshov bracketed words. The Lie algebra analog of Rota's program is formulated in terms convergent rewriting systems and equivalently in terms of Gröbner-Shirshov bases. The relation of this Lie algebra analog of Rota's program with Rota's program for associative algebras is established. Applications are given to modified Rota-Baxter operators, differential type operators and Rota-Baxter type operators.

Partial-fraction decomposition of a rational function and its application

In this paper, by using the residue method of complex analysis, we obtain an explicit partial fraction decomposition for the general rational function frac{x^{M}}{(x+1)^{lambda}_{n}} (M is any nonnegative integer, λ and n are any positive integers). As applications, we deduce the corresponding algebraic identities and combinatorial identities which are the corresponding extensions of Chu’ results. We also give some explicit formulas of Apostol-type polynomials and harmonic Stirling numbers of the second kind.

B3-subbalancing and b3-Lucas subbalancing numbers

In this paper, we derive some new algebraic identities on b3-subbalancing numbers and obtain the functions that generate b3-subbalancing numbers. We also introduce b3-Lucas subbalancing numbers and derive some algebraic identities between b3-Lucas subbalancing numbers and b3-subbalancing numbers. Further, we give the Binet formulas for b3-subbalancing and b3-Lucas subbalancing numbers.

Hölder continuity of solutions for a class of drift-diffusion equations

<p style='text-indent:20px;'>We provide several regularity results for non-homogeneous drift-diffusion equations with applications to general dissipative SQG. Our results unify in a rather simple way several previously known results. We build the estimates on an algebraic identity (for the refined energy argument) which relates any integral operator with pure powers of the laplacian.</p>