On the Analytic Type of Artin Algebras

In this paper we give a complete analytic classification of Artin Gorenstein almost stretched algebras, i.e., Artin Gorenstein algebras with Hilbert function {1, n, 2,…, 2, 1,…, 1}.     

On modules of finite projective dimension over complete intersections

Recently Avramov and Miller proved that over a local complete intersection ring in characteristic 0$">, a finitely generated module has finite projective dimension if for some 0$"> and for some 0$">, being the frobenius map repeated times. They used the notion of ``complexity' and several related theorems. Here we offer a very simple proof of the above theorem without using ``complexity' at all.

     

The Hilbert Series of the Ring Associated to an Almost Alternating Matrix

We give an explicit formula for the Hilbert Series of an algebra defined by a linearly presented, standard graded, residual intersection of a grade three Gorenstein ideal.     

On the Cayley-Bacharach Property

The Cayley-Bacharach Property (CBP), which has been classically stated as a property of a finite set of points in an affine or projective space, is extended to arbitrary 0-dimensional affine algebras over arbitrary base fields. We present characterizations and explicit algorithms for checking the CBP directly, via the canonical module, and in combination with the property of being a locally Gorenstein ring. Moreover, we characterize strict Gorenstein rings by the CBP and the symmetry of their affine Hilbert function, as well as by the strict CBP and the last difference of their affine Hilbert function.     

On the Hilbert Transform in Bergman Space

In this paper we describe the image of the Hilbert transform operator for Bergman space.     

The hilbert function and the minimal free resolution of some gorenstein ideals of codimension 4

We give a construction for Gorenstein ideals of codimension 4 which we believe have maximal graded Betti numbers for their Hilbert function.     

A three shuffle case of the compositional parking function conjecture

We prove here that the polynomial 〈∇_Cp1,_eahbhc〉$〈\mathrm{\nabla }{\mathbf{C}}_{p}1,e_a{h}_b{h}_c〉$q,t$q,t$-enumerates, by the statistics dinv and area  , the parking functions whose supporting Dyck path touches the main diagonal according to the composition p?a+b+c$p?a+b+c$ and has a reading word which is a shuffle of one decreasing word and two increasing words of respective sizes a, b, c  . Here _Cp1${\mathbf{C}}_{p}1$ is a rescaled Hall–Littlewood polynomial and ∇ is the Macdonald eigen-operator introduced by Bergeron and Garsia. This is our latest progress in a continued effort to settle the decade old shuffle conjecture   of Haglund et al. This result includes as special cases all previous results connected with the shuffle conjecture such as the q,t$q,t$-Catalan, Schröder and two shuffle results of Haglund as well as their compositional refinements recently obtained by the authors. It also confirms the possibility that the approach adopted by the authors has the potential to yield a resolution of the shuffle parking function conjecture as well as its compositional refinement more recently proposed by Haglund, Morse and Zabrocki.     

群的Gorenstein同调维数

设$k$是一个弱维数有限的交换环, $G$是一个群. 本文讨论了群$G$具有有限的Gorenstein同调维数的标准.证明了群$G$的Gorenstein同调维数的有限性与群环$kG$的Gorenstein弱维数的有限性是一致的.进一步,我们给出了Serre定理的一个Gorenstein类比.推广了整环上$G$的Gorenstein同调维数的一些已知结果.     

On the K-stability of complete intersections in polarized manifolds

We consider the problem of existence of constant scalar curvature Kähler metrics on complete intersections of sections of vector bundles. In particular we give general formulas relating the Futaki invariant of such a manifold to the weight of sections defining it and to the Futaki invariant of the ambient manifold. As applications we give a new Mukai–Umemura–Tian like example of Fano 5-fold admitting no Kähler–Einstein metric, and a strong evidence of K-stability of complete intersections in Grassmannians.     

On the higher delta invariants of a Gorenstein local ring

Let be a Gorenstein complete local ring. Auslander's higher delta invariants are denoted by for each module and for each integer . We propose a conjecture asking if for any positive integers and . We prove that this is true provided the associated graded ring of has depth not less than . Furthermore we show that there are only finitely many possibilities for a pair of positive integers for which .

     

关于纤维锥的深度和Hilbert级数

Let (R,m) be a Cohen-Macaulay local ring of dimension d with infinite residue field, I an m-primary ideal and K an ideal containing I. Let J be a minimal reduction of I such that, for some positive integer k, KIn ∩ J = JKIn-1 for n ≤ k ? 1 and λ( JKKIIkk-1 ) = 1. We show that if depth G(I) ≥ d-2, then such fiber cones have almost maximal depth. We also compute, in this case, the Hilbert series of FK(I) assuming that depth G(I) ≥ d - 1.     

On the probability distribution of data at points in real complete intersection varieties

We show several estimates on the probability distribution of some data at points in real complete intersection varieties: norms of real affine solutions, condition number of real solution of real systems of multi-variate polynomial equations and convergence radius of Newton's operator for under-determined system of multi-variate polynomial equations.     

On some properties of the gamma function

In this paper we prove a complete monotonicity theorem and establish some upper and lower bounds for the gamma function in terms of digamma and polygamma functions.     

On the Riemann‐Hilbert boundary value problem for generalized analytic functions in the framework of variable exponent spaces

Let Γ be a simple closed curve that bounds the finite domain D , z =z (ζ )=z (r e ^{i ?} ) be the conformal mapping of the circle {ζ :|ζ |<1} onto the domain D . Furthermore, let the functions A (z ), B (z ) be given on D and U ^{s ,2}(A ;B ;D ) be the set of regular solutions of the equation . We call the Smirnov class E ^{p (t )}(A ;B ;D ) the set of those generalized functions W in D for which where p (t ) is a positive measurable function on Γ. We consider the Riemann‐Hilbert problem: Define a function W (z ) from the class E ^{p (t )}(A ;B ;D ) for which the equality, is fulfilled almost everywhere on Γ. It is assumed that Γ is a piecewise‐smooth curve without external peaks; , p is Log Hölder continuous and the function belongs to the class A (p (t );Γ), which is the generalization of the well‐known Simonenko class A (p ;Γ), where p is constant. The solvability conditions are established, and solutions are constructed.     

On sharply vertex transitive 2-factorizations of the complete graph

We introduce the concept of a 2-starter in a group G of odd order. We prove that any 2-factorization of the complete graph admitting G as a sharply vertex transitive automorphism group is equivalent to a suitable 2-starter in G. Some classes of 2-starters are studied, with special attention given to those leading to solutions of some Oberwolfach or Hamilton-Waterloo problems.