We study the symmetric subquotient decomposition of the associated graded algebras A⁎ of a non-homogeneous commutative Artinian Gorenstein (AG) algebra A. This decomposition arises from the stratification of A⁎ by a sequence of ideals A⁎=CA(0)⊃CA(1)⊃⋯ whose successive quotients Q(a)=C(a)/C(a+1) are reflexive A⁎ modules. These were introduced by the first author [46], [47], developed in the Memoir [48]

We investigate when a commutative ring spectrum R satisfies a homotopical version of local Gorenstein duality, extending the notion previously studied by Greenlees. In order to do this, we prove an ascent theorem for local Gorenstein duality along morphisms of k-algebras. Our main examples are of the form R=C⁎(X;k), the ring spectrum of cochains on a space X for a field k. In particular, we establish

Let X be a normal variety endowed with an algebraic torus action. An additive group action α on X is called vertical if a general orbit of α is contained in the closure of an orbit of the torus action and the image of the torus normalizes the image of α in Aut(X). Our first result in this paper is a classification of vertical additive group actions on X under the assumption that X is proper over an

We compute the minimal log discrepancies of determinantal varieties of square matrices, and more generally of pairs (Dk,∑αiDki) consisting of a determinantal variety (of square matrices) and an R-linear sum of determinantal subvarieties. Our result implies the semicontinuity conjecture for minimal log discrepancies of such pairs. For these computations, we use the description of minimal log discrepancies

Let R be a Noetherian ring. We prove that R has global dimension at most two if, and only if, every prime ideal of R is of linear type. Similarly, we show that R has global dimension at most three if, and only if, every prime ideal of R is syzygetic. As a consequence, we derive a characterization of these rings using the André-Quillen homology.

The notion of Hochschild homology of a dg algebra admits a natural dualization, the coHochschild homology of a dg coalgebra, introduced in [28] as a tool to study free loop spaces. In this article we prove “agreement” for coHochschild homology, i.e., that the coHochschild homology of a dg coalgebra C is isomorphic to the Hochschild homology of the dg category of appropriately compact C-comodules, from

We establish geodesic normal forms for the general series of complex reflection groups G(de,e,n) by using the presentations of Corran–Picantin and Corran–Lee–Lee of G(e,e,n) and G(de,e,n) for d>1, respectively. This requires the elaboration of a combinatorial technique in order to explicitly determine minimal word representatives of the elements of G(de,e,n). Using these geodesic normal forms, we construct

In this paper we study algebras with trace and their trace polynomial identities over a field of characteristic 0. We consider two commutative matrix algebras: D2, the algebra of 2×2 diagonal matrices and C2, the algebra of 2×2 matrices generated by e11+e22 and e12. We describe all possible traces on these algebras and we study the corresponding trace codimensions. Moreover we characterize the varieties

We develop a new spectral sequence in order to calculate the Hochschild homology of smash biproducts (also called the twisted tensor products) of unital associative algebras A#B provided one of A or B has Hochschild dimension less than 2. We use this spectral sequence to calculate Hochschild homology of the algebra Mq(2) of quantum 2×2-matrices.

In this article, we prove that if R is a finitely generated ring over Z of dimension d,d≥2,1d!∈R, then any unimodular row over R[X] of length d+1 can be mapped to a factorial row by elementary transformations.

In this paper, we investigate the relative dominant dimension with respect to an injective module and characterize the algebras with finite relative dominant dimension. As an application, we introduce a generalization of n-precluster tilting modules and use this to characterize a class of algebras including the class of n-minimal Auslander-Gorenstein algebras. Moreover, we give a description of the

A split extension of monoids with kernel k:N→G, cokernel e:G→H and splitting s:H→G is Schreier if there exists a unique set-theoretic map q:G→N such that for all g∈G, g=kq(g)⋅se(g). Schreier extensions have a complete characterization and have been shown to correspond to monoid actions of H on N. If the uniqueness requirement of q is relaxed, the resulting split extension is called weakly Schreier

The paper takes advantage of the digital images exported from the digital visualization models of geotechnical engineering and the meshless method based on Shepard function and Partition of Unity (MSPU) to develop a new solution for the automatic digital-numerical integrated analysis. The proposed method fully utilizes the information of pixels to substitute geometric models with digital images, which

We study the moduli space of stable sheaves of Euler characteristic one, supported on curves of degree (3,3) contained in a smooth quadric surface. We compute its Hodge numbers by investigating the variation of the moduli spaces of α-semi-stable coherent systems. We find the classification of the semi-stable sheaves by means of extensions or resolutions.

Let G be a group. Let X be a connected algebraic group over an algebraically closed field K. Denote by A=X(K) the set of K-points of X. We study a class of endomorphisms of pro-algebraic groups, namely algebraic group cellular automata over (G,X,K). They are cellular automata τ:AG→AG whose local defining map is induced by a homomorphism of algebraic groups XM→X where M⊂G is a finite memory set of τ

Let k be an algebraically closed field. Let C be an irreducible smooth projective curve over k. Let E be a locally free sheaf on C of rank ≥2. Fix an integer d≥2. Let Q denote the Quot scheme parameterizing torsion quotients of E of degree d. In this article we compute the S-fundamental group scheme of Q.

We study the characters of simple modules in the parabolic BGG category of the affine Lie algebra in Deligne's category. More specifically, we take the limit of the Weyl-Kac formula to compute the character of the irreducible quotient L(X,k) of the parabolic Verma module M(X,k) of level k, where X is an indecomposable object of Deligne's category Re_p(GLt), Re_p(Ot), or Re_p(Spt), under conditions

We demonstrate equivalence between two definitions of lower finite highest weight categories. We also show that, in the presence of a duality, a lower finite highest weight structure on a category is unique. Finally, we give a proof for the known fact that any abelian category is extension full in its ind-completion.

For a positive integer N, let C(N) be the subgroup of J0(N) generated by the equivalence classes of cuspidal divisors of degree 0 and C(N)(Q):=C(N)∩J0(N)(Q) be its Q-rational subgroup. Let also CQ(N) be the subgroup of C(N)(Q) generated by Q-rational cuspidal divisors. We prove that when N=n2M for some integer n dividing 24 and some squarefree integer M, the two groups C(N)(Q) and CQ(N) are equal.

Over a commutative noetherian ring R of finite Krull dimension, we show that every complex of flat cotorsion R-modules decomposes as a direct sum of a minimal complex and a contractible complex. Moreover, we define the notion of a semi-flat-cotorsion complex as a special type of semi-flat complex, and provide functorial ways to construct a quasi-isomorphism from a semi-flat complex to a semi-flat-cotorsion

Let k be an algebraically closed field and A a Z-graded finitely generated simple k-algebra which is a domain of Gelfand–Kirillov dimension 2. We show that the category of Z-graded right A-modules is equivalent to the category of quasicoherent sheaves on a certain quotient stack. The theory of these simple algebras is closely related to that of a class of generalized Weyl algebras (GWAs). We prove

For an n-dimensional Leibniz/Lie algebra h over a field k we introduce a new invariant A(h), called the universal algebra of h, as a quotient of the polynomial algebra k[Xij|i,j=1,⋯,n] through an ideal generated by n3 polynomials. We prove that A(h) admits a unique bialgebra structure which makes it an initial object among all commutative bialgebras coacting on h. The new object A(h) is the key tool

We consider a block B of a finite group with defect group D≅(C2m)n and inertial quotient E containing a Singer cycle (an element of order 2n−1). This implies E=E⋊F, where E≅C2n−1, F≤Cn, and E acts transitively on the elements in D of order 2. We classify the basic Morita equivalence classes of B over a complete discrete valuation ring O: when m=1, B is basic Morita equivalent to the principal block

We construct canonical strongly bicovariant differential graded algebra structures on all four flavours of cross product Hopf algebras, namely double cross products A↪A⋈H↩H, double cross coproducts , biproducts and bicrossproducts on the assumption that the factors have strongly bicovariant calculi Ω(A),Ω(H) (or a braided version Ω(B)). We use super versions of each of the constructions. Moreover,

In the present article, we investigate the following deformation problem. Let (R,m) be a local (graded local) Noetherian ring with a (homogeneous) regular element y∈m and assume that R/yR is quasi-Gorenstein. Then is R quasi-Gorenstein? We give positive answers to this problem under various assumptions, while we present a counter-example in general. We emphasize that absence of the Cohen-Macaulay condition

In this paper, we study derived versions of the fusion category associated to Lusztig's quantum group Uq. The categories that arise in this way are not semisimple but recovers the usual fusion ring when passing to Grothendieck rings. On the derived level it turns out that it is possible to define fusion for Uq without using the notion of tilting modules. Hence, we arrive at a definition of the fusion

We show that a profinite group G is virtually pro-p for some prime p if and only if for each nontrivial x∈G there is a prime p (depending on x) such that CG(x) is virtually pro-p. Further, if G is a profinite group in which each element has either finite or prime power (possibly infinite) order, then G is either torsion or virtually pro-p for some prime p. A detailed description of profinite groups

An integral domain (or a commutative cancellative monoid) is atomic if every nonzero nonunit element factors into atoms, and it satisfies the ACCP if every ascending chain of principal ideals eventually stabilizes. The interplay between these two properties has been investigated since the 1970s. An atomic domain (or monoid) satisfies the finite factorization property (FFP) if every element has only

In this paper, we construct a family of algebras each of whose members is a multiplicity free sum of irreducible rational representations of the general linear group GLn(C). We then use the properties of a generalized version of the Hodge dual operator to determine an explicit basis for each of these algebras, and by restriction, we obtain an explicit basis for each of the irreducible rational representations

An ideal I of a commutative ring R with identity is called an SFT (strong finite type) ideal if there exist a finitely generated ideal J of R with J⊆I and a positive integer k such that ak∈J for each a∈I. A ring R is called an SFT ring if every ideal of R is an SFT ideal. Arnold showed that if R is a non-SFT ring, then the Krull dimension of the power series ring R〚X〛 is infinite. Let C be the class

Let B be a block of a finite group G with defect group D. We prove that the exponent of the center of D is determined by the character table of G. In particular, we show that D is cyclic if and only if B contains a “large” family of irreducible p-conjugate characters. More generally, for abelian D we obtain an explicit formula for the exponent of D in terms of character values. In small cases even

We describe the minimal number of critical points and the minimal number s of singular fibres for a non isotrivial fibration of a surface S over a curve B of genus 1, exhibiting several examples and in particular constructing a fibration with s=1 and irreducible singular fibre with 4 nodes. Then we consider the associated factorizations in the mapping class group and in the symplectic group. We describe

Let g ( G ) denote the girth of a graph G. In this paper, we determine the unique graph with the maximum least Q-eigenvalue among all unicyclic graphs of order n = 6k ( k ≥ 8 ) with perfect matchings. For the cases when n = 6k + 2 and n = 6k + 4, we prove that g ( G ) = 3 if G is a graph with the maximum least Q-eigenvalue, and provide a conjecture and a problem on the sharp upper bound of the least

Let ( A , m ) be an alternative algebra with maximal ideal m which is complete and separated for the m -adic topology. Assuming that A / m : = k is a perfect field of positive characteristic and that the associated graded algebra is a k-algebra, we show that the reduction map W ( k ) → k from the Witt ring W ( k ) lifts canonically to a morphism W ( k ) → A thereby giving A the structure of a unital

(2020). Correction to: Characterization of multiplication commutative rings with finitely many minimal prime ideals. Communications in Algebra. Ahead of Print.

For finite permutation groups, simplicity of the augmentation submodule is equivalent to 2-transitivity over the field of complex numbers. We note that this is not the case for transformation monoids. We characterize the finite transformation monoids whose augmentation submodules are simple for a field 𝔽 (assuming the answer is known for groups, which is the case for ℂ, ℝ, and ℚ) and provide many

A geometric algebra provides a single environment in which geometric entities can be represented and manipulated and in which transforms can be applied to these entities. A number of versions of geometric algebra have been proposed and the aim of the paper is to investigate one of these as it has a number of advantageous features. Points, lines and planes are presented naturally by element of grades

The set of permutations on a finite set can be given the lattice structure known as the weak Bruhat order. This lattice structure is generalized to the set of words on a fixed alphabet Σ={x,y,z,…}, where each letter has a fixed number of occurrences. These lattices are known as multinomial lattices and, when card(Σ)=2, as lattices of lattice paths. By interpreting the letters x,y,z,… as axes, these

Let f be the Fq-linear map over Fq2n defined by x↦x+axqs+bxqn+s with gcd⁡(n,s)=1. It is known that the kernel of f has dimension at most 2, as proved by Csajbók et al. in “A new family of MRD-codes” (2018). For n big enough, e.g. n≥5 when s=1, we classify the values of b/a such that the kernel of f has dimension at most 1. To this aim, we translate the problem into the study of some algebraic curves

We study rational curves on general Fano hypersurfaces in projective space, mostly by degenerating the hypersurface along with its ambient projective space to reducible varieties. We prove results on existence of low-degree rational curves with balanced normal bundle, and reprove some results on irreducibility of spaces of rational curves of low degree.

In this paper, a novel space-time generalized finite difference method (GFDM) is proposed for solving transient heat conduction problems by integrating a direct space-time discretization technique into the meshless GFDM. The spatial and temporal dimensions are simultaneously discretized by randomly distributed nodes in the coupled space-time continuum. Transient heat conduction in homogenous and heterogeneous

A full coupled multi-scale modelling using the Boundary Element Method for analysing the 2D problem of stretched plates composed of heterogeneous materials, where dissipative phenomena can be considered, is presented. Both the macro-scale and the micro-scale are modelled by BEM formulations where the consistent tangent operator (CTO) is used to achieve the equilibrium of the iterative procedures. The

For a graph G = (V, E) with no isolated vertices, a set \(D\subseteq V\) is called a semipaired dominating set of G if (i)D is a dominating set of G, and (ii)D can be partitioned into two element subsets such that the vertices in each two element set are at distance at most two. The minimum cardinality of a semipaired dominating set of G is called the semipaired domination number of G, and is denoted

We determine the structure of the fibered biset functor sending a finite group G to the complex vector space of complex valued class functions of G. Previously, it is studied as a biset functor by Bouc and as a C×-fibered biset functor by Boltje and the second author. In this paper, we complete the study by considering proper non-trivial subgroups of C× as the fiber group. As a corollary, we obtain

When a pairing \(e: {\mathbb {G}}_1 \times {\mathbb {G}}_2 \rightarrow {\mathbb {G}}_{T}\), on an elliptic curve E defined over a finite field \({\mathbb {F}}_q\), is exploited for an identity-based protocol, there is often the need to hash binary strings into \({\mathbb {G}}_1\) and \({\mathbb {G}}_2\). Traditionally, if E admits a twist \({\tilde{E}}\) of order d, then \({\mathbb {G}}_1=E({\mathbb

In this paper we present a symplectic analogue of the Fueter theorem. This allows the construction of special (polynomial) solutions for the symplectic Dirac operator \(D_s\), which is defined as the first-order \(\mathfrak {sp}(2n)\)-invariant differential operator acting on functions on \(\mathbb {R}^{2n}\) taking values in the metaplectic spinor representation.

In recent years, the interaction between the local positivity of divisors and Okounkov bodies has attracted considerable attention, and there have been attempts to find a satisfactory theory of positivity of divisors in terms of convex geometry of Okounkov bodies. Many interesting results in this direction have been established by Choi–Hyun–Park–Won [4] and Küronya–Lozovanu [17], [18], [19] separately

This paper proposes the topology optimization for steady-state heat conduction structures by incorporating the meshless-based generalized finite difference method (GFDM) and the solid isotropic microstructures with penalization interpolation model. In the meshless GFDM numerical scheme, the explicit formulae of the partial differential equation are expressed by the Taylor series expansions and the

Let k be an infinite field and I⊂k[x1,…,xn] be a non-zero ideal such that dim V(I)=q≥0. Denote by (f1,…,fs) a set of generators of I. One can see that in the set I∩k[x1,...,xq+1] there exist non-zero polynomials, depending only on these q+1 variables. We aim to bound the minimal degree of the polynomials of this type, and of a Bézout (i.e. membership) relation expressing such a polynomial as a combination

In this paper we present the concept of weak link and weak link topology on lattice ordered groups and MV-algebras. We show that it is a locally solid group topology. In addition, we investigate the metrizability of a link topology.

The aim of this paper is to extend the notions of geometric lattices, semimodularity and matroids in the framework of finite posets and related systems of sets. We define a geometric poset as one which is atomistic and which satisfies particular conditions connecting elements to atoms. Next, by using a suitable partial closure operator and the corresponding partial closure system, we define a partial

The problem of determining (up to lattice isomorphism) the lattices that are sublattices of free lattices is in general an extremely difficult and an unsolved problem. A notable result towards solving this problem was established by Galvin and Jónsson when they classified (up to lattice isomorphism) all of the distributive sublattices of free lattices in 1959. In this paper, we weaken the requirement

The present paper deals with new modification of Gamma operators preserving polynomials in Bohman-Korovkin sense and study their approximation properties: Voronovskaya type theorems, weighted approximation and rate of convergence are captured. The effectiveness of the newly modified operators according to classical ones are presented in certain senses as well. Numerical examples are also presented

Let R be a commutative ring with identity. Denote by F P R ( R ) the set of all R-modules admitting a finite projective resolution consisting of finitely generated projective modules. Then the small finitistic dimension of R is defined as fPD ( R ) = sup { pd R M | M ∈ F P R ( R ) } . Cahen et al. posed an open problem as follows: Let R be a Prüfer ring. Is fPD ( R ) ⩽ 1 ? In this paper, we show that

Dynamic Gröbner Basis algorithms aim to obtain small reduced Gröbner Bases in terms of number of polynomials and monomials. Dynamic algorithms allowing previous leading monomials to change during the execution, called unrestricted algorithms, are underexplored in the literature and the only previous unrestricted algorithm was not implemented. In this paper, we introduce a definition of neighborhoods

The aim of this work is to extend to finite potent endomorphisms the notion of G-Drazin inverse of a finite square matrix. Accordingly, we determine the structure and the properties of a G-Drazin inverse of a finite potent endomorphism and, as an application, we offer an algorithm to compute the explicit expression of all G-Drazin inverses of a finite square matrix.

In this paper we prove among others that f y z − z f x ≤ 1 2 π y z − z x ∫ γ f ξ R y ξ R x ξ d ξ ≤ 1 2 π y z − z x ∫ γ f ξ d ξ ξ − y ξ − x , where f : D ⊂ C → C is an analytic function on the domain D, x, y, z ∈ B with σ x , σ y ⊂ D , R y ⋅ , R x ⋅ are the resolvent functions for the elements y and x, and γ is a closed rectifiable path in D and such that σ x , σ y ⊂ i n s   γ . Applications for the

Let Π be an equitable partition of a graph G with k cells. If G has no equitable partition with fewer cells than Π, then Π is called a minimal partition of G and G is called a k-equitable graph. It is clear that 1-equitable graphs are regular graphs. In this paper, we investigate 2-equitable graphs with four distinct eigenvalues.

Let X and Y be compact Hausdorff spaces, E and F be real or complex normed spaces and A ( X , E ) be a subspace of C ( X , E ) . For a function f ∈ C ( X , E ) , let coz ( f ) be the cozero set of f. A pair of additive maps S , T : A ( X , E ) ⟶ C ( Y , F ) is said to be jointly separating if coz ( T f ) ∩ coz ( S g ) = ∅ whenever coz ( f ) ∩ coz ( g ) = ∅ . In this paper, first, we give a partial

Let G be a reductive split p-adic group and let U be the unipotent radical of a Borel subgroup. We study the cohomology with trivial Zp-coefficients of the profinite nilpotent group N=U(OF) and its Lie algebra n, by extending a classical result of Kostant to our integral p-adic setup. The techniques used are a combination of results from group theory, algebraic groups and homological algebra.