Search

Abstract: We show that paper-folding words contain arbitrarily large abelian powers. [Copyright &y& Elsevier]

Abstract: The paper gives proofs of some results just claimed in [R. Massy, Galois averages, J. Number Theory 113 (2005) 244–275]. For instance, it is proved that for a finite non-trivial separable extension , , of Hilbertian fields finitely generated over their prime field, the quotient group , for the corresponding multiplicative groups of non-zero elements, cannot be a torsion group of finite exponent. [Copyright &y& Elsevier]

Abstract: In 1.9 of the paper [A. Hahn, A Hermitian Morita theorem for algebras with anti-structure, J. Algebra 93 (1985) 215–235], should be replaced by . This leads to minor changes in the rest of the paper where the ring should be replaced by its opposite and vice versa. [Copyright &y& Elsevier]

Modelling the substitution of nucleotides along a phylogenetic tree is usually done by a hidden Markov process. This allows to define a distribution of characters at the leaves of the trees and one might be able to obtain polynomial relationships among the probabilities of different characters. The study of these polynomials and the geometry of the algebraic varieties defined by them can be used to reconstruct phylogenetic trees. However, not all points in these algebraic varieties have biological sense. In this paper, we explore the extent to which adding semi-algebraic conditions arising from the restriction to parameters with statistical meaning can improve existing methods of phylogenetic reconstruction. To this end, our aim is to compute the distance of data points to algebraic varieties and to the stochastic part of these varieties. Computing these distances involves optimization by nonlinear programming algorithms. We use analytical methods to find some of these distances for quartet trees evolving under the Kimura 3-parameter or the Jukes-Cantor models. Numerical algebraic geometry and computational algebra play also a fundamental role in this paper. [ABSTRACT FROM AUTHOR]

Semantic image manipulation is a complex problem defined as the ability to change high-level features while keeping the final result visually similar to the original. Recent deep generative solutions show that manipulating semantic features in latent space produces compelling results. Even if we consider those advances, the question remains: Can we algebraically operate high-level visual semantic features in images with meaningful operations? In this paper, we demonstrate the feasibility of interpreting and manipulating image pseudovectors (n − 1 -dimensional subspaces) as the union of visual features (k -dimensional subspaces, for 0 < k < n) operated using Geometric Algebra (GA). Depending on how the latent space is organized, any GA operation would be applicable, enabling the solution to handle an open set of problems without retraining generative models for specific tasks. As a proof of concept, in this paper, we demonstrate how GA operations can be applied to manipulate subspaces in the latent space of faces to perform operations like putting on or taking off clothing accessories, transferring age characteristics, changing hairstyles, and performing semantic queries in sets of images. [Display omitted] • Interpret and manipulate latent codes of images as n − 1 -D subspaces using any Geometric Algebra (GA) equations. • Use the contraction concept from GA to remove some features and build up a subspace with the remaining desired ones. • Compute semantic ratios between two images with GA and map the remaining visual energy to another image. • Perform semantic image retrieval with the GA's orthogonal rejection concept. [ABSTRACT FROM AUTHOR]

The category XSq of crossed squares is equivalent to the category Cat2 of cat2-groups. Functions for computing with these structures have been developed in the package XMod written using the GAP computational discrete algebra programming language. This paper contains a table listing the numbers ι G 2 of isomorphism classes of cat2-groups on groups G of order at most 30 – a total of 1007 cat2-groups. Secondly, it contains general formulae for ι G 2 in a number of special cases. [ABSTRACT FROM AUTHOR]

Computing with non-linear polynomial constraints over the real numbers has been an important area within Computer Algebra since the field's inception. More recently, satisfiability for this theory has become an important problem in the SMT-solving community. Inspired by the S C 2 project, this paper combines ideas from both communities to produce a fast "partial solver" for the theory — i.e. a solver that can sometimes decide the satisfiability of formulas, but sometimes fails to make a determination. The partial solver is based on existing "black-box/white-box" simplification algorithms, which this paper extends to allow for the production of UNSAT cores (and stronger simplifications). The production of UNSAT cores is required if the partial solver is to be used in SMT-solving, which is one of the primary motivations for this work. The authors have implemented the partial solver in the Image 1 system, and this paper also reports on experiments based on that implementation. [ABSTRACT FROM AUTHOR]

In this paper we investigate the image of the center Z of the distribution algebra D i s t (G L (m | n)) of the general linear supergroup over a ground field of positive characteristic under the Harish-Chandra morphism h : Z → D i s t (T) obtained by the restriction of the natural map D i s t (G L (m | n)) → D i s t (T). We define supersymmetric elements in D i s t (T) and show that each image h (c) for c ∈ Z is supersymmetric. The central part of the paper is devoted to a description of a minimal set of generators of the algebra of supersymmetric elements over Frobenius kernels T r. [ABSTRACT FROM AUTHOR]

An algebra A over a field K is said to be invertible if it has a basis B consisting only of units; if B − 1 is again a basis, A is invertible-2 , or I2. The question of when an invertible algebra is necessarily I2 arises naturally. The study of these algebras was recently initiated by López-Permouth, Moore, Szabo, Pilewski [13] , [14]. In this paper, we prove several positive results on this problem, answering also some questions and generalizing a few results from these papers. We show that every field is an I2 algebra over any subfield, and that any subalgebra of the rational functions field K (x) which strictly contains K [ x ] , with K an algebraically closed field, has a symmetric basis B = B − 1. Using this, we expand the class of examples of algebras known to be invertible or I2 with several classes, such as semiprimary rings over fields K ≠ F 2 satisfying some additional mild condition. We also show that every commutative affine invertible algebra is almost I2 in the sense that it becomes I2 after localization at a single element. [ABSTRACT FROM AUTHOR]

Polynomial reduction is one of the main tools in computational algebra with innumerable applications in many areas, both pure and applied. Since many years both the theory and an efficient design of the related algorithm have been solidly established. This paper presents a definition of the generic concept of polynomial reduction structure, studies its features and highlights the aspects needed in order to grant and to efficiently test the main properties (Noetherianity, confluence, ideal membership). The most significant aspect of this analysis is a negative reappraisal of the role of the notion of term ordering which is usually considered a central and crucial tool in the theory. In fact, as it was already established in the computer science context in relation with termination of algorithms, most of the properties can be obtained simply considering a well founded order, while the classical requirement that it be preserved by multiplication is irrelevant. The last part of the paper shows how the polynomial basis concepts present in literature are interpreted in our language and their properties are consequences of the general results established in the first part of the paper. [ABSTRACT FROM AUTHOR]

Signature-based algorithms have become a standard approach for computing Gröbner bases in commutative polynomial rings. However, so far, it was not clear how to extend this concept to the setting of noncommutative polynomials in the free algebra. In this paper, we present a signature-based algorithm for computing Gröbner bases in precisely this setting. The algorithm is an adaptation of Buchberger's algorithm including signatures. We prove that our algorithm correctly enumerates a signature Gröbner basis as well as a Gröbner basis of the module generated by the leading terms of the generators' syzygies, and that it terminates whenever the ideal admits a finite signature Gröbner basis. Additionally, we adapt well-known signature-based criteria eliminating redundant reductions, such as the syzygy criterion, the F5 criterion and the singular criterion, to the case of noncommutative polynomials. We also generalise reconstruction methods from the commutative setting that allow to recover, from partial information about signatures, the coordinates of elements of a Gröbner basis in terms of the input polynomials, as well as a basis of the syzygy module of the generators. We have written a toy implementation of all the algorithms in the Mathematica package OperatorGB and we compare our signature-based algorithm to the classical Buchberger algorithm for noncommutative polynomials. [ABSTRACT FROM AUTHOR]

We study linear PDE with constant coefficients. The constant rank condition on a system of linear PDEs with constant coefficients is often used in the theory of compensated compactness. While this is a purely linear algebraic condition, the nonlinear algebra concept of primary decomposition is another important tool for studying such system of PDEs. In this paper we investigate the connection between these two concepts. From the nonlinear analysis point of view, we make some progress in the study of weak lower semicontinuity of integral functionals defined on sequences of PDE constrained fields, when the PDEs do not have constant rank. [ABSTRACT FROM AUTHOR]

Each nontrivial shell (i.e., subconstituent) of the Johnson association scheme J (v , k) is known to be a commutative association scheme which is the product of two smaller Johnson association schemes. The concept of t -designs in one shell of J (v , k) was naturally defined and studied by Martin in the context of mixed t -designs. The purpose of this paper is to try to push this study a bit further. First we give a direct proof of the theorem that if (Y , w) is a relative t -design in J (v , k) on p shells then the part in each shell must be a weighted (t − p + 1) -design in the shell. In particular, it is a mixed (t − p + 1) -design in one shell if the weight function is constant on each shell. We also present another approach that makes use of the Terwilliger algebra, based on the work of Tanaka. This result is essentially proved by Martin in 1998, but the proof there contains a small gap which is repaired in this paper. We also study the existence problems of tight 2-, 3- and 4-designs on one shell of Johnson association scheme J (v , k) with small parameters, say v ≤ 1000 , thereby expanding the range of search in the original paper of Martin in 1998. [ABSTRACT FROM AUTHOR]

In this paper we construct the Shimura canonical model for the quaternion algebra B of discriminant 6 over Q in the sense of the original paper [SmrB] (see also [Shg2]). The "unit group" of B becomes to be a quadrangle group Γ = □ (0 ; 2 , 2 , 3 , 3) acting on the complex upper half space H . For our purpose we use the defining equation of the Shimura curve V = H / □ (0 ; 2 , 2 , 3 , 3) given by Y. Ihara which found in the work of A. Kurihara (1979). We are requested to find a modular function ψ defined on H with respect to Γ which realizes V satisfying several arithmetic conditions stated later. Basically our ψ is the modular function for the family of abelian surfaces with quaternion multiplication by B , so called false elliptic curves, with some arithmetic conditions. Our main tool is the hypergeometric modular function given by M. Petkova and H. Shiga (2011). This is the full paper of the survey report [Shg2] (2018). [ABSTRACT FROM AUTHOR]

Abstract In this paper we establish the relation between key polynomials (as defined in [12]) and minimal pairs of definition of a valuation. We also discuss truncations of valuations on a polynomial ring K [ x ]. We prove that a valuation ν is equal to its truncation on some polynomial if and only if ν is valuation-transcendental. Another important result of this paper is that if μ is any extension of ν to K ‾ [ x ] and Λ is a complete sequence of key polynomials for ν , without last element, then for each Q ∈ Λ there exists a suitable root a Q ∈ K ‾ of Q such that { a Q } Q ∈ Λ is a pseudo-convergent sequence defining μ. [ABSTRACT FROM AUTHOR]

Abstract Hyperovals in PG (2 , GF (q)) with even q are maximal arcs and an interesting research topic in finite geometries and combinatorics. Hyperovals in PG (2 , GF (q)) are equivalent to [ q + 2 , 3 , q ] MDS codes over GF (q) , called hyperoval codes, in the sense that one can be constructed from the other. Ovals in PG (2 , GF (q)) for odd q are equivalent to [ q + 1 , 3 , q − 1 ] MDS codes over GF (q) , which are called oval codes. In this paper, we investigate the binary subfield codes of two families of hyperoval codes and the p -ary subfield codes of the conic codes. The weight distributions of these subfield codes and the parameters of their duals are determined. As a byproduct, we generalize one family of the binary subfield codes to the p -ary case and obtain its weight distribution. The codes presented in this paper are optimal or almost optimal in many cases. In addition, the parameters of these binary codes and p -ary codes seem new. [ABSTRACT FROM AUTHOR]

Abstract We establish a new theory of regularity for elliptic complex valued second order equations of the form L = div A (∇ ⋅) , when the coefficients of the matrix A satisfy a natural algebraic condition, a strengthened version of a condition known in the literature as L p -dissipativity. Precisely, the regularity result is a reverse Hölder condition for L p averages of solutions on interior balls, and serves as a replacement for the De Giorgi–Nash–Moser regularity of solutions to real-valued divergence form elliptic operators. In a series of papers, Cialdea and Maz'ya studied necessary and sufficient conditions for L p -dissipativity of second order complex coefficient operators and systems. Recently, Carbonaro and Dragičević introduced a condition they termed p -ellipticity, and showed that it had implications for boundedness of certain bilinear operators that arise from complex valued second order differential operators. Their p -ellipticity condition is exactly our strengthened version of L p -dissipativity. The regularity results of the present paper are applied to solve L p Dirichlet problems for L = div A (∇ ⋅) + B ⋅ ∇ when A and B satisfy a Carleson measure condition, which previously was known only in the real valued case. We show solvability of the L 2 Dirichlet problem, as well as solvability of the L p Dirichlet boundary value problem for p in the range where A is p -elliptic. [ABSTRACT FROM AUTHOR]

In most cases in geometry, applying analytic or algebraic tools on coordinates helps to solve some difficult problems. For instance, proving that a geometrical construction problem is solvable using ruler and compass is often impossible within a synthetic geometry framework. But in an analytic geometry framework, it is a direct application of Galois theory after performing triangularizations. However, these algebraic tools lead to a large amount of computation. Their implementation in modern Computer Algebra Systems (CAS) are still too time consuming to provide an answer in a reasonable time. In addition, they require a lot of memory space which can grow exponentially with the size of the problem. Fortunately, some geometrical properties can be used to setup the algebraic systems so that they can be more efficiently computed. These properties turn polynomials into new ones so as to reduce both the degrees and the number of monomials. The present paper promotes this approach by considering two corpora of geometric construction problems, namely Wernick's and Connely's lists. These lists contain about 280 problems. The purpose is to determine their status i.e. whether they are constructible or not with ruler and compass. Some of these problems had unknown status that will be settled in this paper. More generally, the status of all problems of these corpora are fully automatically given by an approach combining geometry and algebra. [ABSTRACT FROM AUTHOR]

Abstract Spatially distributed networks of large size arise in a variety of science and engineering problems, such as wireless sensor networks and smart power grids. Most of their features can be described by properties of their state-space matrices whose entries have indices in the vertex set of a graph. In this paper, we introduce novel algebras of Beurling type that contain matrices on a connected simple graph having polynomial off-diagonal decay, and we show that they are Banach subalgebras of B (ℓ p) , 1 ≤ p ≤ ∞ , the space of all bounded operators on the space ℓ p of all p -summable sequences. The ℓ p -stability of state-space matrices is an essential hypothesis for the robustness of spatially distributed networks. In this paper, we establish the equivalence among ℓ p -stabilities of matrices in Beurling algebras for different exponents 1 ≤ p ≤ ∞ , with quantitative analysis for the lower stability bounds. Admission of norm-control inversion plays a crucial role in some engineering practice. In this paper, we prove that matrices in Beurling subalgebras of B (ℓ 2) have norm-controlled inversion and we find a norm-controlled polynomial with close to optimal degree. Polynomial estimate to powers of matrices is important for numerical implementation of spatially distributed networks. In this paper, we apply our results on norm-controlled inversion to obtain a polynomial estimate to powers of matrices in Beurling algebras. The polynomial estimate is a noncommutative extension about convolution powers of a complex function and is applicable to estimate the probability of hopping from one agent to another agent in a stationary Markov chain on a spatially distributed network. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we discuss a method of constructing separable representations of the C ⁎ -algebras associated to strongly connected row-finite k -graphs Λ. We begin by giving an alternative characterization of the Λ-semibranching function systems introduced in an earlier paper, with an eye towards constructing such representations that are faithful. Our new characterization allows us to more easily check that examples satisfy certain necessary and sufficient conditions. We present a variety of new examples relying on this characterization. We then use some of these methods and a direct limit procedure to identify a faithful separable representation for any row-finite source-free k -graph. [ABSTRACT FROM AUTHOR]

The first Weyl algebra over k , A 1 = k 〈 x , y 〉 / ( x y − y x − 1 ) admits a natural Z -grading by letting deg ⁡ x = 1 and deg ⁡ y = − 1 . Smith showed that gr ­ A 1 is equivalent to the category of quasicoherent sheaves on a certain quotient stack. Using autoequivalences of gr ­ A 1 , Smith constructed a commutative ring C , graded by finite subsets of the integers. He then showed gr ­ A 1 ≡ gr ­ ( C , Z fin ) . In this paper, we prove analogues of Smith's results by using autoequivalences of a graded module category to construct rings with equivalent graded module categories. For certain generalized Weyl algebras, we use autoequivalences defined in a companion paper so that these constructions yield commutative rings. [ABSTRACT FROM AUTHOR]

In this paper, we study quasilinear elliptic equations with the nonlinearity modelled after the p ( x ) -Laplacian on nonsmooth domains and obtain sharp Calderón–Zygmund type estimates in the variable exponent setting. In a recent work of [12] , the estimates obtained were strictly above the natural exponent and hence there was a gap between the natural energy estimates and estimates above p ( x ) , see (1.3) and (1.4) . Here, we bridge this gap to obtain the end point case of the estimates obtained in [12] , see (1.5) . In order to do this, we have to obtain significantly improved a priori estimates below p ( x ) , which is the main contribution of this paper. We also improve upon the previous results by obtaining the estimates for a larger class of domains than what was considered in the literature. [ABSTRACT FROM AUTHOR]

Text Let k , l , a , b , c be positive integers such that gcd ⁡ ( k a , l b ) = 1 , min ⁡ { a , b , c } > 1 , a ≠ 3 , b ≠ 3 and 2 ∤ c . In this paper, we prove that there are at most four solutions in positive integers ( x , y , z ) to the equation k a x + l b y = c z and at most two solutions when 2 ∤ ( u ( l / k ) ) , where u ( m ) is the least positive integer t with m t ≡ 1 ( mod c ) . Video For a video summary of this paper, please visit https://youtu.be/Dt3Y7TDMxlg . [ABSTRACT FROM AUTHOR]

This paper examines the Euler–Rodrigues formula in dual 3-space D 3 by analyzing its variations such as vectorial form, exponential map, point–line theory and quaternions which have some intrinsic relations. Contrary to the Euclidean case, dual rotation in dual 3-space corresponds to a screw motion in Euclidean 3-space. This paper begins by explaining dual motion in terms of the given dual axis and angle. It will then go on to express dual Euler–Rodrigues formula with algebraic methods. Furthermore, an application of dual Euler–Rodrigues formula to point–line geometry is accomplished and point–line displacement operator is obtained by dual Euler–Rodrigues formula. Finally, dual Euler–Rodrigues formula is presented with the help of dual Euler–Rodrigues parameters that is expressed as a dual quaternion. [ABSTRACT FROM AUTHOR]

In this paper, we provide generating functions and counting formulas for spanning trees and spanning forests in hypergraphs in two different ways: (1) We represent spanning trees and spanning forests in hypergraphs through Berezin-Grassmann integrals on Zeon algebra and hyper-Hafnians (orders and signs are not considered); (2) We establish a Hyper-Pfaffian-Cactus Spanning Forest Theorem through Berezin-Grassmann integrals on Grassmann algebra (orders and signs are considered), which generalizes the Hyper-Pfaffian-Cactus Theorem by Abdesselam (2004) [1] and Pfaffian matrix tree theorem by Masbaum and Vaintrob (2002) [15]. • We provide generating functions and counting formulas for spanning trees and spanning forests in hypergraphs through Berezin-Grassmann integrals on Zeon algebra and hyper- Hafnians when orders and signs are not considered. • We establish a Hyper-Pfaffian-Cactus Spanning Forest Theorem through Berezin-Grassmann integrals on Grassmann algebra (orders and signs are considered), which generalizes the Hyper-Pfaffian-Cactus Theorem by Abdesselam [Advances in Applied Mathematics (2004) Vol. 33: 51-70] and Pfaffian matrix tree theorem by Masbaum and Vaintrob [Internat. Math. Res. Notices (2002) Vol. 27: 1397-1426]. [ABSTRACT FROM AUTHOR]

Text Let m be an integer with m ≥ 2 . For A ⊆ Z m and n ∈ Z m , let R 1 ( A , n ) , R 2 ( A , n ) , R 3 ( A , n ) denote the number of solutions of the equation a + a ′ = n with ordered pairs ( a , a ′ ) ∈ A × A , unordered pairs ( a , a ′ ) ∈ A × A ( a ≠ a ′ ) and unordered pairs ( a , a ′ ) ∈ A × A , respectively. In this paper, for i ∈ { 1 , 2 , 3 } , we determine all sets A , B ⊆ Z m such that R i ( A , n ) = R i ( B , n ) for all n ∈ Z m when the cardinality of the symmetric difference of A and B is small or large. These extend some previous results. We also pose some problems for further research. Video For a video summary of this paper, please visit https://youtu.be/stBa9Uy5U0I . [ABSTRACT FROM AUTHOR]

Text Let K be a complete, algebraically closed, non-Archimedean valued field, and let ϕ ∈ K ( z ) with deg ⁡ ( ϕ ) ≥ 2 . In this paper we consider the family of functions ord Res ϕ n ( x ) , which measure the resultant of ϕ n at points x in P K 1 , the Berkovich projective line, and show that they converge locally uniformly to the diagonal values of the Arakelov–Green's function g μ ϕ ( x , x ) attached to the canonical measure of ϕ . Following this, we are able to prove an equidistribution result for Rumely's crucial measures ν ϕ n , each of which is a probability measure supported at finitely many points whose weights are determined by dynamical properties of ϕ . Video For a video summary of this paper, please visit https://youtu.be/YCCZD1iwe00 . [ABSTRACT FROM AUTHOR]

As a fundamental notion, the free differential algebra on a set is concretely constructed as the polynomial algebra on the differential variables. Such a construction is not known for the more general notion of the free differential algebra on an algebra, from the left adjoint functor of the forgetful functor from differential algebras to algebras, instead of sets. In this paper we show that a generator-relation presentation of a base algebra can be extended to the free differential algebra on this base algebra. More precisely, a Gröbner-Shirshov basis property of the base algebra can be extended to the free differential algebra on this base algebra, allowing a Poincaré-Birkhoff-Witt type basis for these more general free differential algebras. Examples are provided as illustrations. [ABSTRACT FROM AUTHOR]

The goal of this paper is to study primitive groups that are contained in the union of maximal (in the symmetric group) imprimitive groups. The study of types of permutations that appear inside primitive groups goes back to the origins of the theory of permutation groups. However, this is another instance of a situation common in mathematics in which a very natural problem turns out to be extremely difficult. Fortunately, the enormous progresses of the last few decades seem to allow a new momentum on the attack to this problem. In this paper we prove that there are infinite families of primitive groups contained in the union of imprimitive groups and propose a new hierarchy for primitive groups based on that fact. In addition we introduce some algorithms to handle permutations, provide the corresponding GAP implementation, solve some open problems, and propose a large list of open problems. [ABSTRACT FROM AUTHOR]

Let E be an elliptic curve defined over an algebraically closed field k whose characteristic is not 2 or 3. Let τ be a translation automorphism of E that is not of order 2. In a previous paper we studied an algebra A = A ( E , τ ) that depends on this data: A ( E , τ ) = ( S ( E , τ ) ⊗ M 2 ( k ) ) Γ where S ( E , τ ) is the 4-dimensional Sklyanin algebra associated to ( E , τ ) , M 2 ( k ) is the ring of 2 × 2 matrices over k , and Γ is ( Z / 2 ) × ( Z / 2 ) acting in a particular way as automorphisms of S and M 2 ( k ) . The action of Γ on S is compatible with the translation action of the 2-torsion subgroup E [ 2 ] on E . Following the ideas and results in papers of Artin–Tate–Van den Bergh, Smith–Stafford, and Levasseur–Smith, this paper examines the line modules, point modules, and fat point modules, over A , and their incidence relations. The right context for the results is non-commutative algebraic geometry: we view A as a homogeneous coordinate ring of a non-commutative analogue of P 3 that we denote by Proj n c ( A ) . Point modules and fat point modules determine “points” in Proj n c ( A ) . Line modules determine “lines” in Proj n c ( A ) . Line modules for A are in bijection with certain lines in P ( A 1 ⁎ ) ≅ P 3 and therefore correspond to the closed points of a certain subscheme L of the Grassmannian G ( 1 , 3 ) . Shelton–Vancliff call L the line scheme for A . We show that L is the union of 7 reduced and irreducible components, 3 quartic elliptic space curves and 4 plane conics in the ambient Plücker P 5 , and that deg ⁡ ( L ) = 20 . The union of the lines corresponding to the points on each elliptic curve is an elliptic scroll in P ( A 1 ⁎ ) . Thus, the lines on that elliptic scroll are in natural bijection with a corresponding family of line modules for A . [ABSTRACT FROM AUTHOR]

Multiplicative relations between the roots of a polynomial in Q [ x ] have drawn much attention in the field of arithmetic and algebra, while the problem of computing these relations is interesting to researchers in many other fields. In this paper, a sufficient condition is given for a polynomial f ∈ Q [ x ] to have only trivial multiplicative relations between its roots, which is a generalization of those sufficient conditions proposed in Smyth (1986) , Baron et al. (1995) and Dixon (1997). Based on the new condition, a subset E ⊂ Q [ x ] is defined and proved to be generic (i.e. , the set Q [ x ] \ E is very small). We develop an algorithm deciding whether a given polynomial f ∈ Q [ x ] is in E and returning a basis of the lattice consisting of the multiplicative relations between the roots of f whenever f ∈ E. The numerical experiments show that the new algorithm is very efficient for the polynomials in E. A large number of polynomials with much higher degrees, which were intractable before, can be handled successfully with the algorithm. [ABSTRACT FROM AUTHOR]

The isotropic Grassmannian parametrizes isotropic subspaces of a vector space equipped with a quadratic form. In this paper, we show that any maximal isotropic Grassmannian in its Plücker embedding can be defined by pulling back the equations of G r iso (3 , 7) or G r iso (4 , 8). [ABSTRACT FROM AUTHOR]

The paper consists of two parts. The first part introduces the representation ring for the family of compact unitary groups U ( 1 ) , U ( 2 ) , … . This novel object is a commutative graded algebra R with infinite-dimensional homogeneous components. It plays the role of the algebra of symmetric functions, which serves as the representation ring for the family of finite symmetric groups. The purpose of the first part is to elaborate on the basic definitions and prepare the ground for the construction of the second part of the paper. The second part deals with a family of Markov processes on the dual object to the infinite-dimensional unitary group U ( ∞ ) . These processes were defined in a joint work with Alexei Borodin (2012) [5] . The main result of the present paper consists in the derivation of an explicit expression for their infinitesimal generators. It is shown that the generators are implemented by certain second order partial differential operators with countably many variables, initially defined as operators on R . [ABSTRACT FROM AUTHOR]

This paper is chiefly concerned with qualitative properties of some reaction–diffusion fronts. The recently defined notions of transition fronts generalize the standard notions of traveling fronts. In this paper, we show the existence and the uniqueness of the global mean speed of bistable transition fronts in R N . This speed is proved to be independent of the shape of the level sets of the fronts. The planar fronts are also characterized in the more general class of almost-planar fronts with any number of transition layers. These qualitative properties show the robustness of the notions of transition fronts. But we also prove the existence of new types of transition fronts in R N that are not standard traveling fronts, thus showing that the notions of transition fronts are broad enough to include other relevant propagating solutions. [ABSTRACT FROM AUTHOR]

We prove a version of the Deligne conjecture for n -fold monoidal abelian categories A over a field k of characteristic 0, assuming some compatibility and non-degeneracy conditions for A . The output of our construction is a weak Leinster ( n , 1 ) -algebra over k , a relaxed version of the concept of Leinster n -algebra in A lg ( k ) . The difference between the Leinster original definition and our relaxed one is apparent when n > 1 , for n = 1 both concepts coincide. We believe that there exists a functor from weak Leinster ( n , 1 ) -algebras over k to C • ( E n + 1 , k ) -algebras, well-defined when k = Q , and preserving weak equivalences. For the case n = 1 such a functor is constructed in [31] by elementary simplicial methods, providing (together with this paper) a complete solution for 1-monoidal abelian categories. Our approach to Deligne conjecture is divided into two parts. The first part, completed in the present paper, provides a construction of a weak Leinster ( n , 1 ) -algebra over k , out of an n -fold monoidal k -linear abelian category (provided the compatibility and non-degeneracy condition are fulfilled). The second part (still open for n > 1 ) is a passage from weak Leinster ( n , 1 ) -algebras to C • ( E n + 1 , k ) -algebras. As an application, we prove in Theorem 8.1 that the Gerstenhaber–Schack complex of a Hopf algebra over a field k of characteristic 0 admits a structure of a weak Leinster ( 2 , 1 ) -algebra over k extending the Yoneda structure. It relies on our earlier construction [30] of a 2-fold monoidal structure on the abelian category of tetramodules over a bialgebra. [ABSTRACT FROM AUTHOR]

This paper studies closed 3-manifolds which are the attractors of a system of finitely many affine contractions that tile R 3 . Such attractors are called self-affine tiles. Effective characterization and recognition theorems for these 3-manifolds as well as theoretical generalizations of these results to higher dimensions are established. The methods developed build a bridge linking geometric topology with iterated function systems and their attractors. A method to model self-affine tiles by simple iterative systems is developed in order to study their topology. The model is functorial in the sense that there is an easily computable map that induces isomorphisms between the natural subdivisions of the attractor of the model and the self-affine tile. It has many beneficial qualities including ease of computation allowing one to determine topological properties of the attractor of the model such as connectedness and whether it is a manifold. The induced map between the attractor of the model and the self-affine tile is a quotient map and can be checked in certain cases to be monotone or cell-like. Deep theorems from geometric topology are applied to characterize and develop algorithms to recognize when a self-affine tile is a topological or generalized manifold in all dimensions. These new tools are used to check that several self-affine tiles in the literature are 3-balls. An example of a wild 3-dimensional self-affine tile is given whose boundary is a topological 2-sphere but which is not itself a 3-ball. The paper describes how any 3-dimensional handlebody can be given the structure of a self-affine 3-manifold. It is conjectured that every self-affine tile which is a manifold is a handlebody. [ABSTRACT FROM AUTHOR]

This paper tackles the history of tactics, a field of investigation at the crossroads of algebra, combinatorics and recreational mathematics. Tactics was only taken up by mathematicians now and then between the 1850s and the 1900s, and its emergence was a process of mathematization of questions linked to the notions of “order” and “position”. To understand the long-term history of this field of investigation—one that became neither a theory nor a discipline—the paper analyzes the different historical configurations in which tactics took on its scientific meaning. It thus investigates how, under the banner of tactics, a continuity could be claimed by mathematicians that were, finally, working in very different scientific and historical context. [ABSTRACT FROM AUTHOR]

In this paper we describe the method which we applied to successfully compute the primary decomposition of a certain ideal coming from applications in combinatorial algebra and algebraic statistics regarding conditional independence statements with hidden variables. While our method is based on the algorithm for primary decomposition by Gianni, Trager and Zacharias, we were not able to decompose the ideal using the standard form of that algorithm, nor by any other method known to us. [ABSTRACT FROM AUTHOR]

In this paper we introduce a class of noncommutative (finitely generated) monomial algebras whose Hilbert series are algebraic functions. We use the concept of graded homology and the theory of unambiguous context-free grammars for this purpose. We also provide examples of finitely presented graded algebras whose corresponding leading monomial algebras belong to the proposed class and hence possess algebraic Hilbert series. [ABSTRACT FROM AUTHOR]

This paper presents an algorithm which, given a point and a set of polynomials, constructs a single cylindrical cell containing the point, such that the given polynomials are sign-invariant in the computed cell. To represent a single cylindrical cell, a novel data structure is introduced. The algorithm works with this representation and proceeds incrementally, i.e., first constructing a cell representing the whole real space, and then iterating over the input polynomials, refining the cell at each step to ensure the sign-invariance of the current input polynomial. A merge procedure realizing this refinement is described, and its correctness is proven. The merge procedure is based on McCallum's projection operator, but uses geometric information relative to the input point to reduce the number of projection polynomials computed. The use of McCallum's operator implies the incompleteness of the algorithm in general. However, the algorithm is complete for well-oriented sets of polynomials. Moreover, the incremental approach described in this paper can be easily adapted to a different projection operator. The cell construction algorithm presented in this paper is an alternative to the “model-based” method described by Jovanović and de Moura in a 2012 paper. It generalizes the algorithm described by the first author in a 2013 paper, which could only produce full-dimensional cells, to allow for the construction of cells of arbitrary dimension. While a thorough comparison, whether analytical or empirical, of the new algorithm and the “model-based” approach will be the subject of future work, this paper provides preliminary justification for asserting the superiority of the new method. [ABSTRACT FROM AUTHOR]

In Di Nasso (2015) and Luperi Baglini (2012) it has been introduced a technique, based on nonstandard analysis, to study some problems in combinatorial number theory. In this paper we review such a technique and we present three of its applications: the first one is a new proof of a known result regarding the algebra of β N , namely that the center of the semigroup ( β N , ⊕ ) is N ; the second one is a generalization of a theorem of Bergelson and Hindman on arithmetic progressions of length three; the third one regards the study of which polynomials in several variables with integers coefficients have a monochromatic solution for every finite coloring of N . We will study this last application in more detail: we will prove some algebraical properties of the set P of such polynomials and we will present a few examples of nonlinear polynomials in P . In the first part of the paper we will recall the main results of the nonstandard technique that we want to use, which is based on a characterization of ultrafilters by means of nonstandard analysis. [ABSTRACT FROM AUTHOR]

In this paper we employ a C-algebra approach to the study of Fredholm properties of differential and pseudodifferential operators on the half line. The algebra investigated in this paper has compact commutators, so the Gelfand theory applies to the quotient algebra, and we obtain an explicit description of the corresponding maximal ideal space and necessary and sufficient conditions for Fredholmness of operators in the algebra. [ABSTRACT FROM AUTHOR]

The concept of a relatively weakly injective pair of operator systems is introduced and studied in this paper, motivated by relative weak injectivity in the C*-algebra category. E. Kirchberg [11] proved that the C\*-algebra C\*(F∞) of the free group F∞ on countably many generators characterises relative weak injectivity for pairs of C\*-algebras by means of the maximal tensor product. One of the main results of this paper shows that C\*(F∞) also characterises relative weak injectivity in the operator system category. A key tool is the theory of operator system tensor products [9,10]. [ABSTRACT FROM AUTHOR]

Leibniz algebras are certain generalizations of Lie algebras. It is natural to generalize concepts in Lie algebras to Leibniz algebras and investigate whether the corresponding results still hold. In this paper we introduce the notion of complete Leibniz algebras as a generalization of complete Lie algebras. Then we study properties of complete Leibniz algebras and their holomorphs. [ABSTRACT FROM AUTHOR]

In this paper, we study an impact of Leibniz algebras on the algebraic structure of N -graded vertex algebras. We provide easy ways to characterize indecomposable non-simple N -graded vertex algebras ⊕ n = 0 ∞ V (n) such that dim ⁡ V (0) ≥ 2. Also, we examine the algebraic structure of N -graded vertex algebras V = ⊕ n = 0 ∞ V (n) such that dim ⁡ V (0) ≥ 2 and V (1) is a (semi)simple Leibniz algebra that has s l 2 as its Levi factor. We show that under suitable conditions this type of vertex algebra is indecomposable but not simple. Along the way we classify vertex algebroids associated with (semi)simple Leibniz algebras that have s l 2 as their Levi factor. [ABSTRACT FROM AUTHOR]

Let A be a finite dimensional algebra over a field k. In this paper, we study dominant dimension from the point of view of the idempotent ideals. The canonical A -bimodule V : = Hom A (D A , A) was studied in [8,11,9] , where D = Hom k (− , k). Under certain condition, we give a new understanding that D A ⊗ A V is isomorphic to an idempotent ideal tensor product itself as A -bimodules and show that the double dual functor is defined by D A ⊗ A V. We also give a characterization of dominant dimension for A in terms of vanishing of certain extension groups over A , which just uses A and DA as ingredients and generalizes the corresponding characterizations given in [8,9]. Moreover, we show that A is a Motita algebra if and only if there is a natural isomorphism ν ν ν − 1 ≅ ν on A -mod where ν is the Nakayama functor. [ABSTRACT FROM AUTHOR]

In this paper we study the combinatorial consequences of the relationship between rational Cherednik algebras of type G (l , 1 , n) , cyclic quiver varieties and Hilbert schemes. We classify and explicitly construct C ⁎ -fixed points in cyclic quiver varieties and calculate the corresponding characters of tautological bundles. Furthermore, we give a combinatorial description of the bijections between C ⁎ -fixed points induced by the Etingof-Ginzburg isomorphism and Nakajima reflection functors. We apply our results to obtain a new proof as well as a generalization of the q -hook formula. [ABSTRACT FROM AUTHOR]

A hom-associative algebra is an algebra whose associativity is twisted by an algebra homomorphism. In this paper, we introduce a strongly homotopy version of hom-associative algebras (H A ∞ -algebras in short) on a graded vector space. We describe 2-term H A ∞ -algebras in detail. In particular, we study 'skeletal' and 'strict' 2-term H A ∞ -algebras. We also introduce hom-associative 2-algebras as categorification of hom-associative algebras. The category of 2-term H A ∞ -algebras and the category of hom-associative 2-algebras are shown to be equivalent. Next, we define a suitable Hochschild cohomology theory for H A ∞ -algebras which control the deformation of the structures. Finally, we visit H L ∞ -algebras introduced by Sheng and Chen and show that an appropriate skew-symmetrization of H A ∞ -algebras give rise to H L ∞ -algebras. [ABSTRACT FROM AUTHOR]

In this paper, we begin with the classification of Harish-Chandra imprimitive representations in non-defining characteristic. We recall the connection of this problem to certain generalizations of Iwahori-Hecke algebras and show that Harish-Chandra induction is compatible with the Morita equivalence by Bonnafé and Rouquier, thus reducing the classification problem to quasi-isolated blocks. Afterwards, we consider imprimitivity of unipotent representations of certain classical groups. In the case of general linear and unitary groups, our reduction methods then lead to results for arbitrary Lusztig series. [ABSTRACT FROM AUTHOR]