Gorenstein injective dimension and Tor-depth of modules

The main aim of this paper is to obtain a dual result to the now well known Auslander-Bridger formula for G-dimension. We will show that if R is a complete Cohen-Macaulay ring with residue field k, and M is a non-injective h-divisible Ext-finite R-module of finite Gorenstein injective dimension such that for each i 3 1i \geq 1 Extⁱ (E,M) = 0 for all indecomposable injective R-modules E 1 E(k)E \neq E(k), then the depth of the ring is equal to the sum of the Gorenstein injective dimension and Tor-depth of M. As a consequence, we get that this formula holds over a d-dimensional Gorenstein local ring for every nonzero cosyzygy of a finitely generated R-module and thus in particular each such n^th cosyzygy has its Tor-depth equal to the depth of the ring whenever n 3 dn \geq d.     

Quadratic matrix polynomials with Hamiltonian spectrum and oscillatory damped systems

We consider quadratic matrix polynomials of the form L(l) = l²A + elB + CL(\lambda) = \lambda^{2}A + \epsilon\lambda B + C, where e\epsilon is a real parameter, A is positive definite and B and C are symmetric. The main results of the paper are the characterization of the class of symmetric matrices B for which the spectrum of the polynomial is symmetric with respect to the imaginary axis and solutions of the corresponding differential equation oscillate in time. We also extend the results in [2] to allow us to study the asymptotic behaviour of the eigenvalues for large e\epsilon.     

Hall polynomials and the fixed space of an automorphism of a finite abelian p-group

Recent work on Hall polynomials is used to study the fixed space of a random automorphism of a finite abelian p-group. An expression is found for the chance that an automorphism of an abelian p-group of type l\lambda fixes only the identity. A formula is obtained for the chance that a given subgroup H of type n\nu is the fixed space of an automorphism of an abelian p-group of type (k^l). There results generalize work of Rudvalis and Shinoda on the fixed space of an element of GL (n, q).     

On the singularities of residual intertwining operators

Let G be a reductive algebraic group defined over \Bbb Q {\Bbb Q} . Let P, P' be parabolic subgroups of G, defined over \Bbb Q {\Bbb Q} , and let _boxclose_boxclose, a_P') t \in W({\frak a}_{P}, {\frak a}_{P'}) . In this paper we study the intertwining operator M_P￠|P(t,l), l ? \frak a^*_{P,\Bbb C} M_{P' \vert P}(t,\lambda),\,\lambda \in {\frak a}^*_{P,{\Bbb C}} , acting in corresponding spaces of automorphic forms. One of the main results states that each matrix coefficient of M_P￠|P(t,l) M_{P' \vert P}(t,\lambda) is a meromorphic function of order ￡ n + 1 \le n + 1 , where n = dim G. Using this result, we further investigate the rank one intertwining operators, in particular, we study the distribution of their poles.     

On standard Auslander-Reiten sequences for finite groups

Let G be a finite group and O{\cal O} a complete discrete valuation ring of characteristic zero with maximal ideal (p)(\pi ) and residue field k = O/(p)k = {\cal O}/(\pi ) of characteristic p > 0. Let S be a simple kG-module and Q_S a projective O G{\cal O} G-lattice such that Q_S / pQ_SQ_S / \pi Q_S is a projective cover of S. We show that if S is liftable and Q_S belongs to a block of O G{\cal O} G of infinite representation type, then the standard Auslander-Reiten sequence terminating in W^-1S\Omega ^{-1}S is a direct summand of the short exact sequence obtained from some Auslander-Reiten sequence of OG{\cal O}G-lattices by reducing each term mod (p)(\pi ).     

Solvability of systems of partial differential equations for functions defined on nonconvex sets

We consider systems of partial differential equations with constant coefficients of the form ( R(D_x, D_y)f = 0, P(D_x)f = g), f,g ? C^￥(W),\big ( R(D_x, D_y)f = 0, P(D_x)f = {g}\big ), f,g \in {C}^{\infty}(\Omega),, where R (and P) are operators in (n + 1) variables (and in n variables, respectively), g satisfies the compatibility condition R(D_x, D_y)g = 0  and  W ì \Bbb Rⁿ⁺¹R(D_x, D_y){g} = 0 \ {\rm and} \ \Omega \subset {\Bbb R}^{n+1} is open. Let R be elliptic. We show that the solvability of such systems for certain nonconvex sets W\Omega implies that any localization at ￥\infty of the principle part P_m of P is hyperbolic. In contrast to this result such systems can always be solved on convex open sets W\Omega by the fundamental principle of Ehrenpreis-Palamodov.     

On matrix near-rings

Let R be a right near-ring with identity and M_n(R) be the near-ring of n 2 n matrices over R in the sense of Meldrum and Van der Walt. In this paper, M_n(R) is said to be s\sigma-generated if every n 2 n matrix A over R can be expressed as a sum of elements of X_n(R), where X_n(R)={f_ij^r | 1\leqq i, j\leqq n, r ? R}X_n(R)=\{f_{ij}^r\,|\,1\leqq i, j\leqq n, r\in R\}, is the generating set of M_n(R). We say that R is s\sigma-generated if M_n(R) is s\sigma-generated for every natural number n. The class of s\sigma-generated near-rings contains distributively generated and abstract affine near-rings. It is shown that this class admits homomorphic images. For abelian near-rings R, we prove that the zerosymmetric part of R is a ring, so the class of zerosymmetric abelian s\sigma-generated near-rings coincides with the class of rings. Further, for every n, there is a bijection between the two-sided subgroups of R and those of M_n(R).     

Blocks with cyclic defect of Hecke orders of Coxeter groups

Let B\cal B be a p-block of cyclic defect of a Hecke order over the complete ring \Bbb Z[q] _{áq-1,p ?}\Bbb {Z}[q] _{\langle q-1,p \rangle}; i.e. modulo áq-1 ?\langle q-1 \rangle it is a p-block B of cyclic defect of the underlying Coxeter group G. Then B\cal B is a tree order over \Bbb Z[q]_{áq-1, p ?}\Bbb {Z}[q]_{\langle q-1, p \rangle } to the Brauer tree of B. Moreover, in case B\cal B is the principal block of the Hecke order of the symmetric group S(p) on p elements, then B\cal B can be described explicitly. In this case a complete set of non-isomorphic indecomposable Cohen-Macaulay B\cal B-modules is given.     

A rigidity result for highest order local cohomology modules

Let M be a finitely generated faithful module over a noetherian ring R of dimension d < ￥ \infty and let \mathfrak a \subseteqq R {\mathfrak a} \subseteqq R be an ideal. We describe the (finite) set Supp_R(H_{\mathfrak a}^d (M)) = Ass_R(H_{\mathfrak a}^d (M)) \textrm{Supp}_R(H_{\mathfrak a}^d (M)) = \textrm{Ass}_R(H_{\mathfrak a}^d (M)) of primes associated to the highest local cohomology module H_{\mathfrak a}^d (M) H_{\mathfrak a}^d (M) in terms of the local formal behaviour of \mathfrak a {\mathfrak a} . If R is integral and of finite type over a field, Supp_R(H_{\mathfrak a}^d (M)) \textrm{Supp}_R(H_{\mathfrak a}^d (M)) is the set of those closed points of X = Spec(R) whose fibre under the normalization morphism n: X￠? X \nu : X' \rightarrow X contains points which are isolated in n^-1(Spec(R/\mathfrak a)) \nu^{-1}(\textrm{Spec}(R/{\mathfrak a})) .     

The dimension monoid of a lattice

We introduce the dimension monoid of a lattice L, denoted by Dim L. The monoid Dim L is commutative and conical, the latter meaning that the sum of any two nonzero elements is nonzero. Furthermore, Dim L is given along with the dimension map, D\Delta from L2L to Dim L, which has the intuitive meaning of a distance function. The maximal semilattice quotient of Dim L is isomorphic to the semilattice Con_c L of compact congruences of L; hence Dim L is a precursor of the congruence lattice of L. Here are some additional features of this construction: ¶¶ (1) Our dimension theory provides a generalization to all lattices of the von Neumann dimension theory of continuous geometries. In particular, if L is an irreducible continuous geometry, then Dim L is either isomorphic to \Bbb Z⁺\Bbb Z^+ or to \Bbb R⁺\Bbb R^+.¶ (2) If L has no infinite bounded chains, then Dim L embeds (as an ordered monoid) into a power of \Bbb Z⁺è{￥}.{\Bbb Z}^{+}\cup \{\infty\}.¶ (3) If L is modular or if L has no infinite bounded chains, then Dim L is a refinement monoid.¶ (4) If L is a simple geometric lattice, then Dim L is isomorphic to \Bbb Z⁺\Bbb Z^+, if L is modular, and to the two-element semilattice, otherwise.¶ (5) If L is an à₀\aleph_0-meet-continuous complemented modular lattice, then both Dim L and the dimension function D\Delta satisfy (countable) completeness properties.¶¶ If R is a von Neumann regular ring and if L is the lattice of principal right ideals of the matrix ring M₂ (R), then Dim L is isomorphic to the monoid V (R) of isomorphism classes of finitely generated projective right R-modules. Hence the dimension theory of lattices provides a wide lattice-theoretical generalization of nonstable K-theory of regular rings.     

Orbits of permutation groups on the power set

Let G be a permutation group on a finite set W\Omega . If G does not involve A_n for n \geqq 5 n \geqq 5 , then there exist two disjoint subsets of W\Omega such that no Sylow subgroup of G stabilizes both and four disjoint subsets of W\Omega whose stabilizers in G intersect trivially.     

Lagrangian barriers and symplectic embeddings

We prove that every symplectic Kähler manifold (M;W) (M;\Omega) with integral [W] [\Omega] decomposes into a disjoint union (M,W) = (E,w₀) \coprod D (M,\Omega) = (E,\omega_0) \coprod \Delta , where (E,w₀) (E,\omega_0) is a disc bundle endowed with a standard symplectic form w₀ \omega_0 and D \Delta is an isotropic CW-complex. We perform explicit computations of this decomposition on several examples.¶As an application we establish the following symplectic intersection phenomenon: There exist symplectically irremovable intersections between contractible domains and Lagrangian submanifolds. For example, we prove that every symplectic embedding j:B²ⁿ(l) ? \Bbb CPⁿ \varphi:B^{2n}(\lambda) \to {\Bbb C}P^n of a ball of radius l² 3 1/2 \lambda^2 \ge 1/2 must intersect the standard Lagrangian real projective space \Bbb RPⁿ ì \Bbb CPⁿ {\Bbb R}P^n \subset {\Bbb C}P^n .     

The fixed points of coprime action

Let P be an odd p\pi -group that acts as a group of automorphisms on the soluble p￠\pi '-group G. We obtain generators for the fixed points of P on [G, P].     

Idempotents in a complex group algebra, projective modules, and the von Neumann algebra

The complex group algebra \Bbb CG{\Bbb C}G of a countable group G can be imbedded in the von Neumann algebra NG of G. If G is torsion-free, and if P is a finitely generated projective module over \Bbb CG{\Bbb C}G it is proved that the central-valued trace of NG?_{\Bbb CG}PNG\otimes _{{\Bbb C}G}P, i.e. of an idempotent \Bbb CG{\Bbb C}G-matrix A defining P is equal to the canonical trace k(P)\kappa (P) times identity I. It follows that k(P)\kappa (P) characterizes the isomorphism type of NG?_{\Bbb CG}PNG\otimes _{{\Bbb C}G}P.¶If k(P)\kappa (P) is an integer, e.g., if the weak Bass conjecture holds for G then NG?_{\Bbb C G}PNG\otimes _{{\Bbb C} G}P is free. It is also shown that for certain classes of groups geometric arguments can be used to prove the Bass conjecture.     

Efficient Testing of Large Graphs

Let P be a property of graphs. An e\epsilon -test for P is a randomized algorithm which, given the ability to make queries whether a desired pair of vertices of an input graph G with n vertices are adjacent or not, distinguishes, with high probability, between the case of G satisfying P and the case that it has to be modified by adding and removing more than en²\epsilon n^2 edges to make it satisfy P. The property P is called testable, if for every e\epsilon there exists an e\epsilon -test for P whose total number of queries is independent of the size of the input graph. Goldreich, Goldwasser and Ron [8] showed that certain individual graph properties, like k-colorability, admit an e\epsilon -test. In this paper we make a first step towards a complete logical characterization of all testable graph properties, and show that properties describable by a very general type of coloring problem are testable. We use this theorem to prove that first order graph properties not containing a quantifier alternation of type ``"$\forall \exists ' are always testable, while we show that some properties containing this alternation are not.     

E-near rings

If R is a left near-ring in which all endomorphisms of R⁺ are given by¶ a_l : x ? ax (with  a, x ? R) a_{\ell} : x \rightarrow ax ({\rm with} \, a, x \in R) then (R, +) is shown to be abelian.¶ If also a_l = b_l a_{\ell} = b_{\ell} implies a = b, then R is even a ring.     

Kazhdan's property (T), L^2-spectrum and isoperimetric inequalities for locally symmetric spaces

Let V = G\G/KV =\Gamma\backslash G/K be a Riemannian locally symmetric space of nonpositive sectional curvature and such that the isometry group G of its universal covering space has Kazhdan's property (T). We establish strong dichotomies between the finite and infinite volume case. In particular, we characterize lattices (or, equivalently, arithmetic groups) among discrete subgroups G ì G\Gamma\subset G in various ways (e.g., in terms of critical exponents, the bottom of the spectrum of the Laplacian and the behaviour of the Brownian motion on V).     

Linear independence of time-frequency shifts under a generalized Schrödinger representation

Let r_\mathbbR \rho_{\mathbb{R}} be the classical Schrödinger representation of the Heisenberg group and let L \Lambda be a finite subset of \mathbbR ×\mathbbR \mathbb{R} \times \mathbb{R} . The question of when the set of functions {t ? e^{2 pi y t} f(t + x) = (r_\mathbbR(x, y, 1) f)(t) : (x, y) ? L} \{t \mapsto e^{2 \pi i y t} f(t + x) = (\rho_{\mathbb{R}}(x, y, 1) f)(t) : (x, y) \in \Lambda\} is linearly independent for all f ? L²(\mathbbR), f 1 0 f \in L^2(\mathbb{R}), f \neq 0 , arises from Gabor analysis. We investigate an analogous problem for locally compact abelian groups G. For a finite subset L \Lambda of G ×[^(G)] G \times \widehat{G} and r_G \rho_G the Schrödinger representation of the Heisenberg group associated with G, we give a necessary and in many situations also sufficient condition for the set {r_G (x, w, 1)f : (x, w) ? L} \{\rho_G (x, w, 1)f : (x, w) \in \Lambda\} to be linearly independent for all f ? L²(G), f 1 0 f \in L^2(G), f \neq 0 .     

On minimal p-degrees in 2-transitive permutation groups

Suppose G is a transitive permutation group on a finite set W\mit\Omega of n points and let p be a prime divisor of |G||G|. The smallest number of points moved by a non-identity p-element is called the minimal p-degree of G and is denoted m_p (G). ¶ In the article the minimal p-degrees of various 2-transitive permutation groups are calculated. Using the classification of finite 2-transitive permutation groups these results yield the main theorem, that m_p(G) 3 [(p-1)/(p+1)] ·|W|m_{p}(G) \geq {{p-1} \over {p+1}} \cdot |\mit\Omega | holds, if Alt(W) \nleqq G {\rm Alt}(\mit\Omega ) \nleqq G .¶Also all groups G (and prime divisors p of |G||G|) for which m_p(G) ￡ [(p-1)/(p)] ·|W|m_{p}(G)\le {{p-1}\over{p}} \cdot |\mit\Omega | are identified.     

On groups acting freely on a tree

Suppose we are given a group G\mit\Gamma and a tree X on which G\mit\Gamma acts. Let d be the distance in the tree. Then we are interested in the asymptotic behavior of the numbers a_d: = # {w ? vertX : w=gv, g ? G , d(v₀,w)=d }a_d:= \# \{w\in {\rm {vert}}X : w=\gamma {v}, \gamma \in {\mit\Gamma} , d({v}_0,w)=d \} if d? ￥d\rightarrow \infty , where v, v_o are some fixed vertices in X.¶ In this paper we consider the case where G\mit\Gamma is a finitely generated group acting freely on a tree X. The growth function ?a_d x^d\textstyle\sum\limits a_d x^d is a rational function [3], which we describe explicitely. From this we get estimates for the radius of convergence of the series. For the cases where G\mit\Gamma is generated by one or two elements, we look a little bit closer at the denominator of this rational function. At the end we give one concrete example.