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This paper addresses the problem of computing valuations of the Dieudonne determinants of matrices over discrete valuation skew fields (DVSFs). This problem is an extension of computing degrees of determinants of polynomial matrices and has an application to the analysis of degrees of freedom of linear time-varying systems. Under a reasonable computational model, we propose two algorithms for a kind of DVSFs, called split. Our algorithms are extensions of the combinatorial relaxation of Murota (1995) and the matrix expansion by Moriyama-Murota (2013), both of which are based on combinatorial optimization. While our algorithms require an upper bound on the output, we show that the bound is easily obtained if the input is skew polynomial matrices.

Subfield codes of linear codes over finite fields have recently received a lot of attention, as some of these codes are optimal and have applications in secrete sharing, authentication codes and association schemes. In this paper, a class of binary subfield codes is constructed from a special family of MDS codes, and their parameters are explicitly determined. The parameters of their dual codes are also studied. Some of the codes presented in this paper are optimal or almost optimal.

In this paper we introduce a new species of evolution algebras that we call Cayley evolution algebras. We show that if a field $k$ contains sufficiently many elements (for example if $k$ is infinite) then every finite group $G$ is isomorphic to $Aut(X)$ where $X$ is a finite-dimensional absolutely simple Cayley evolution $k$-algebra., Comment: This version of the article has been accepted for publication, after peer review but is not the Version of Record and does not reflect post-acceptance improvements, or any corrections. The Version of Record is published under the Creative Commons Attribution license and can be freely dowloaded it from https://doi.org/10.1007/s13398-023-01414-w

International audience; The zero locus of a bivariate polynomial P(x,y)=0P(x,y)=0P(x,y)=0 defines a compact Riemann surface Σ\SigmaΣ. The fundamental second kind differential is a symmetric 1⊗11\otimes 11⊗1-form on Σ×Σ\Sigma\times\SigmaΣ×Σ that has a double pole at coinciding points and no other pole. As its name indicates, this is one of the most important geometric objects on a Riemann surface. Here we give a rational expression in terms of combinatorics of the Newton's polygon of PPP, involving only integer combinations of products of coefficients of PPP. Since the expression uses only combinatorics, the coefficients are in the same field as the coefficients of PPP.

A p-indigent module is one that is subprojective only to projective modules. An RD-projective module is subprojective to any torsionfree (and flat) module. An RD-projective module $T$ is called rdp-indigent if it is subprojective only to torsionfree modules. In this work, we consider the structure of SRDP rings whose (simple) RD-projective right $R$-modules are rdp-indigent or torsionfree. Moreover, new characterizations of P-coherent rings and torsionfree rings are presented by subprojectivity domains.

This paper introduces Riordan-Krylov matrices. These matrices naturally generalize Riordan matrices by using Krylov matrices and a more general class of algebras in place of formal power series. Fundamental properties of Riordan-Krylov matrices that are analogous to those of Riordan matrices are developed. These results and techniques are used to study incidence algebras of a poset as well as the chain and zeta-multichain polynomials of the poset. Throughout the paper applications to enumeration problems in combinatorics and number theory are provided.

We give explicit isomorphisms between simple modules of degenerate cyclotomic Hecke algebras defined via various cellular bases. A special case gives a generalized Mullineux involution in the degenerate case., Comment: 21 pages

Arveson's extension theorem guarantees that every completely positive map defined on an operator system can be extended to a completely positive map defined on the whole C*-algebra containing it. An analogous statement where complete positivity is replaced by positivity is known to be false. A natural question is whether extendibility could still hold for positive maps satisfying stronger conditions, such as being unital and norm 1. Here we provide three counterexamples showing that positive norm-one unital maps defined on an operator subsystem of a matrix algebra cannot be extended to a positive map on the full matrix algebra. The first counterexample is an unextendible positive unital map with unit norm, the second counterexample is an unextendible positive unital isometry on a real operator space, and the third counterexample is an unextendible positive unital isometry on a complex operator space., Comments are welcome

This paper classifies Lagrangian fibrations over surfaces with compact total spaces up to fibrewise symplectomorphism identical on the base. Bibliography: 15 titles.

The Dunkl--Dirac operator is a deformation of the Dirac operator by means of Dunkl derivatives. We investigate the symmetry algebra generated by the elements supercommuting with the Dunkl--Dirac operator and its dual symbol. This symmetry algebra is realised inside the tensor product of a Clifford algebra and a rational Cherednik algebra associated with a reflection group or root system. For reducible root systems of rank three, we determine all the irreducible finite-dimensional representations and conditions for unitarity. Polynomial solutions of the Dunkl--Dirac equation are given as a realisation of one family of such irreducible unitary representations., Comment: v3 40p. Final version accepted in J. Algebra. See v2 for proof of Thm 4.1

Although the notion of geometric-arithmetic index has been introduced in the chemical graph theory these past years, it has already proved to be useful. The objective of the work we present here is twofold: First, obtaining new relations connecting the geometric-arithmetic index with other topological indices; second, to characterize graphs which are extremal with respect to those relations.

In this paper, we give the free field realization of the baby TKK algebra of Tan [14] arisen as one extended affine Lie algebra of type A 1 with nullity 2. Moreover, the Hermitian form on the module is given and the conditions for the unitary are determined.

Proposition 5.16 of the author's paper ‘Mutations of frozen Jacobian algebras’ [11] is false. We give several possible remedies with different applications.

We construct an integral form for the universal enveloping algebra of the Onsager algebra and an explicit integral basis for this integral form. We also formulate straightening identities among some products of basis elements.

The concept of generalized path algebras was introduced in (Coelho and Liu, 2000). It was shown in (Ib\'a\~nez Cobos et al., 2008) how to obtain the Gabriel quiver of a given generalized path algebra. In this article, we generalize the concept of generalized path algebra to allow them to have relations, and we extend the result in (Ib\'a\~nez Cobos et al., 2008) to this new setting. Moreover, we use the extended result mentioned above to address the inverse problem: that is, the problem of determining when a given algebra is isomorphic to a generalized path algebra in a non-trivial way., Comment: Published online, Comm. Algebra, 13 Nov 2021

Every (full) finite Gabor system generated by a unit-norm vector g∈Cd is a finite unit-norm tight frame (FUNTF) and can thus be associated with a (Gabor) positive operator valued measure (POVM). Su...

In this paper, we firstly present a new criterion of linear complementary pairs (abbreviated to LCP) of codes over finite fields. Our result for the linear complementary pairs of codes extends the ...

In this paper, we apply the concept of fuzzy set to hyper UP-subalgebras. We introduce the notions of fuzzy hyper UP-subalgebra and fuzzy hyper UP-filter and establish some of their properties. Furthermore, some properties of hyper homomorphism in relation to fuzzy hyper UP-subalgebras and fuzzy hyper UP-filter are also presented.

We give criteria when a full subcategory [Formula: see text] of the category of groups has [Formula: see text]-universal factorization property ([Formula: see text]-UFP) or [Formula: see text]-strong universal factorization property ([Formula: see text]-SUFP) for a certain category of groups [Formula: see text]. As a byproduct, we give affirmative answers to three unsettled questions in [S.W. Kim, J.B. Lee, Universal factorization property of certain polycyclic groups, J. Pure Appl. Algebra 204 (2006) 555–567].

Reed–Solomon codes are widely used to establish a reliable channel to transmit information in digital communication which has a strong error correction capability and a variety of efficient decoding algorithm. Usually we use the maximum likelihood decoding (MLD) algorithm in the decoding process of Reed–Solomon codes. MLD algorithm relies on determining the error distance of received word. Dür, Guruswami, Wan, Li, Hong, Wu, Yue and Zhu et al. got some results on the error distance. For the Reed–Solomon code [Formula: see text], the received word [Formula: see text] is called an ordinary word of [Formula: see text] if the error distance [Formula: see text] with [Formula: see text] being the Lagrange interpolation polynomial of [Formula: see text]. We introduce a new method of studying the ordinary words. In fact, we make use of the result obtained by Y.C. Xu and S.F. Hong on the decomposition of certain polynomials over the finite field to determine all the ordinary words of the standard Reed–Solomon codes over the finite field of [Formula: see text] elements. This completely answers an open problem raised by Li and Wan in [On the subset sum problem over finite fields, Finite Fields Appl. 14 (2008) 911–929].

A new algebraic structure was introduced, called an eGE-algebra, which is a generalisation of a GE-algebra and investigated its properties. We explore the definition of filters and the quotient algebra associated with such filters.

We introduce and discuss an elementary tool from representation theory of finite groups for constructing linear codes invariant under a given permutation group G. The tool gives theoretical insight as well as a recipe for computations of generator matrices and weight distributions. In some interesting cases a classification of code vectors under the action of G can be obtained. As an explicit example a class of binary codes is studied extensively which is closely related to the class of binary codes associated to triangular graphs. A second explicit application is related to the action of the Mathieu simple group M 24 on the set of octads giving many binary codes of length 759 with interesting properties. We also obtain new alternative proofs for several other theorems and construct several new codes invariant under various subgroups of the Conway simple group Co 1 .

We show that the Gelfand character χ G \chi_{G} of a finite group 𝐺 (i.e. the sum of all irreducible complex characters of 𝐺) may be realized as a “twisted trace” g ↦ Tr ⁡ ( ρ g ∘ T ) g\mapsto\operatorname{Tr}(\rho_{g}\circ T) for a suitable involutive linear automorphism 𝑇 of L 2 ⁢ ( G ) L^{2}(G) , where ( L 2 ⁢ ( G ) , ρ ) (L^{2}(G),\rho) is the right regular representation of 𝐺. Moreover, we prove that, under certain hypotheses, we have T ⁢ ( f ) = f ∘ L T(f)=f\circ L ( f ∈ L 2 ⁢ ( G ) f\in L^{2}(G) ), where 𝐿 is an involutive anti-automorphism of 𝐺. The natural representation 𝜏 of 𝐺 associated to the natural 𝐿-conjugacy action of 𝐺 in the fixed point set Fix G ⁡ ( L ) \operatorname{Fix}_{G}(L) of 𝐿 turns out to be a Gelfand model for 𝐺 in some cases. We show that ( L 2 ⁢ ( Fix G ⁡ ( L ) ) , τ ) (L^{2}(\operatorname{Fix}_{G}(L)),\tau) fails to be a Gelfand model if 𝐺 admits non-trivial central involutions.

Consider the algebraic function $\Phi_{g,n}$ that assigns to a general $g$-dimensional abelian variety an $n$-torsion point. A question first posed by Kronecker and Klein asks: What is the minimal $d$ such that, after a rational change of variables, the function $\Phi_{g,n}$ can be written as an algebraic function of $d$ variables? Using techniques from the deformation theory of $p$-divisible groups and finite flat group schemes, we answer this question by computing the essential dimension and $p$-dimension of congruence covers of the moduli space of principally polarized abelian varieties. We apply this result to compute the essential $p$-dimension of congruence covers of the moduli space of genus $g$ curves, as well as its hyperelliptic locus, and of certain locally symmetric varieties., Comment: 26 pages. Minor revisions. Comments welcome!

We introduce the notion of support equivalence for (co)module algebras (over Hopf algebras), which generalizes in a natural way (weak) equivalence of gradings. We show that for each equivalence class of (co)module algebra structures on a given algebra A, there exists a unique universal Hopf algebra H together with an H-(co)module structure on A such that any other equivalent (co)module algebra structure on A factors through the action of H. We study support equivalence and the universal Hopf algebras mentioned above for group gradings, Hopf-Galois extensions, actions of algebraic groups and cocommutative Hopf algebras. We show how the notion of support equivalence can be used to reduce the classification problem of Hopf algebra (co)actions. In particular, we apply support equivalence to study the asymptotic behaviour of codimensions of H-identities of a certain class of H-module algebras. This result proves the analogue (formulated by Yu.A. Bahturin) of Amitsur's conjecture which was originally concerned with ordinary polynomial identities., preprint submitted for publication, info:eu-repo/semantics/inPress

We describe an algorithm for computing p p -adic L-functions of characters of totally real fields, using the Fourier expansions of diagonal restrictions of Hilbert modular forms.

The Gabor quaternionic offset linear canonical transform (GQOLCT) is defined as a generalization of the quaternionic offset linear canonical transform (QOLCT). In this paper, we investigate the 2D GQOLCT. A new definition of the GQOLCT is provided along with its several important properties, such as boundedness, orthogonality relation, Plancherel and inversion formulas, are derived based on the spectral representation of the GQOLCT. Further, we establish a version of Lieb’s and logarithmic inequalities. Finally we will prove a type of the Heisenberg inequality by using local uncertainty principle.

We study multiplication modules. The rings are not assumed to be commutative. Several criteria with some applications are given for a direct sum of modules to be a multiplication module.

Let $F$ be an infinite field and let $f$ be a nonzero multilinear polynomial with coefficients in $F$. We prove that for every positive integer $d$ there exists a positive integer $s$ such that $f(M_{s}(F))$, the image of $f$ in $M_{s}(F)$, contains all trace zero $d \times d$ matrices. In particular, the image of $f$ in the algebra of all finitary matrices contains all trace zero finitary matrices., 11 pages, 0 figures, submited to Linear Algebra and Its Applications

We discuss properties of the detS2 map, present a few explicit computations, and give a geometrical interpretation for the condition detS2((vi,j)1≤i

This paper is concerned with the lower and upper redundancy of infinite frames in a separable Hilbert space. Using unit norm linear operators, a new representation of upper redundancy is given. Als...

In this paper we study matching in equational theories that specify counterparts of associativity and commutativity for variadic function symbols. We design a procedure to solve a system of matching equations and prove its termination, soundness, completeness, and minimality. The minimal complete set of matchers for such a system can be infinite, but our algorithm computes its finite representation in the form of solved set. From the practical side, we identify two finitary cases and impose restrictions on the procedure to get an incomplete algorithm, which, based on our experiments, describes the input-output behavior and properties of Mathematica's flat and orderless pattern matching.

In this paper, we characterize hypercyclic sequences of weighted translation operators on an Orlicz space in the context of locally compact hypergroups., 18 pages, Typos are corrected and one reference added

Max-plus algebra is a set ℝmax = ∪ ℝ {ϵ}, with ℝ is the set of all real numbers and e = -∞ equipped with ⊕ (maximum) and ⊗ (plus) operation. Another algebra that can be learned is the min-plus alge...

Max-plus algebra is the set of ℝϵ = ℝ ∪ {ϵ} where ℝ is a set of all real numbers and ϵ = -∞ which is equipped with max (⊕) and plus (⊗) operations. The structure of max-plus algebra is semi-field. ...

In his work on modularity theorems, Wiles proved a numerical criterion for a map of rings $R\to T$ to be an isomorphism of complete intersections. He used this to show that certain deformation rings and Hecke algebras associated to a mod $p$ Galois representation at non-minimal level were isomorphic and complete intersections, provided the same was true at minimal level. In this paper we study Hecke algebras acting on cohomology of Shimura curves arising from maximal orders in indefinite quaternion algebras over the rationals localized at a semistable irreducible mod $p$ Galois representation $\overline \rho$. If $\overline \rho$ is scalar at some primes dividing the discriminant of the quaternion algebra, then Hecke algebra is still isomorphic to the deformation ring, but is not a complete intersection, or even Gorenstein, so the Wiles numerical criterion cannot apply. We consider a weight 2 newform $f$ which contributes to the cohomology of the Shimura curve and gives rise to an augmentation $\lambda_f$ of the Hecke algebra. We quantify the failure of the Wiles numerical criterion at $\lambda_f$ by computing the associated {\it Wiles defect} purely in terms of the local behavior at primes dividing the discriminant of the global Galois representation $\rho_f$ which $f$ gives rise to by the Eichler--Shimura construction. One of the main tools used in the proof is Taylor--Wiles--Kisin patching., Comment: revision after comments of referees, to appear in Compositio Mathematica

A cyclic code has been one of the most active research topics in coding theory because they have many applications in data storage systems and communication systems. They have efficient encoding and decoding algorithms. This paper explains the construction of a family of cyclic codes from sequences generated by a trace of a monomial over finite fields of odd characteristics. The parameter and some examples of the codes are presented in this paper.

In this paper, the concepts of fuzzy translation and fuzzy multiplication on a BRK-algebra are introduced. We investigated fuzzy translation and fuzzy multiplication (BRK-subalgebras & BRK-ideals) in BRK-algebras and discussed related properties. Finally, we presented the nation fuzzy magnified--translation on BRK-algebra.

This paper introduces the notion of neutrosophic B-algebra. Several results on properties of neutrosophic B-algebras and neutrosophic subalgebras are presented and proved.

Support τ-tilting modules are in bijection with two-term tilting complexes for any finite-dimensional symmetric algebra. This fact motivates us to classify support τ-tilting modules over blocks of finite groups. We classify support τ-tilting modules over particular blocks of finite groups using modular representation theoretical approaches including Clifford theory and Green's indecomposability theorem.

General computational methods based on descriptor state-space realizations are proposed to compute coprime factorizations of rational matrices with minimum degree denominators. The new methods rely on recursive pole dislocation techniques, which allow to successively place all poles of the factors into a "good" region of the complex plane. The resulting McMillan degree of the denominator factor is equal to the number of poles lying in the complementary "bad" region and therefore is minimal. The developed pole dislocation techniques are instrumental for devising numerically reliable procedures for the computation of coprime factorizations with proper and stable factors of arbitrary improper rational matrices and coprime factorizations with inner denominators. Implementation aspects of the proposed algorithms are discussed and illustrative examples are given., Comment: 18 pages; article accepted for publication in the Linear Algebra and Its Applications

In this work we approach three-dimensional evolution algebras from certain constructions performed on two-dimensional algebras. More precisely, we provide four different constructions producing three-dimensional evolution algebras from two-dimensional algebras. Also we introduce two parameters, the annihilator stabilizing index and the socle stabilizing index, which are useful tools in the classification theory of these algebras. Finally, we use moduli sets as a convenient way to describe isomorphism classes of algebras. Funding for open access charge: Universidad de Málaga / CBUA. Spanish Ministerio de Ciencia e Innovación through the project PID2019-104236GB-I00. Junta de Andalucía through the projects FQM-336 and UMA18-FEDERJA-119, all of them with FEDER funds. Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) grant number 307910/2020-2 and Capes-PrInt grant number 88881.310538/2018-01 - Brazil.

The motivation of this research is to introduce some new fractional operators called “the improved fractional (IF) operators”. The originality of these fractional operators comes from the fact that they repeat the method on general forms of conformable integration and differentiation rather than on the traditional ones. Hence the convolution kernels correlating with the IF operators are served in conformable abstract forms. This extends the scientific application scope of their fractional calculus. Also, some results are acquired to guarantee that the IF operators have advantages analogous to the familiar fractional integral and differential operators. To unveil the inverse and composition properties of the IF operators, certain function spaces with their characterizations are presented and analyzed. Moreover, it is remarkable that the IF operators generalize some fractional and conformable operators proposed in abundant preceding works. As scientific applications, the resistor–capacitor electrical circuits are analyzed under some IF operators. In the case of constant and periodic sources, this results in novel voltage forms. In addition, the overall influence of the IF operators on voltage behavior is graphically simulated for certain selected fractional and conformable parameter values. From the standpoint of computation, the usage of new IF operators is not limited to electrical circuits; they could also be useful in solving scientific or engineering problems.

A determinant of Sylvester resurfaced first in a paper by Kac in 1947, and again in a paper by Askey in 2005. Since then several authors have discussed related determinants. We evaluate two further extensions of Sylvester’s determinant.