Free Resolution Invariants for Finite Groups

Given a finite group G and a G-free resolution F _* of Z, then d _G (Im(F _m+1F _m ))–(–1) ^m–i d _G (F _i ) is almost always an invariant of G.     

Intrinsic metrics preserving maps on Abelian lattice-ordered groups

Swamy studied the natural metric ¦x–y¦ on Abelian lattice-ordered groupsG, and he proved that the stable isometries which preserve this metric have to be automorphisms ofG. Holland proved that the only intrinsic metrics on lattice-ordered groups, i.e., invariant and symmetric metrics, are the multiples n¦x –y¦ for some integern. We show that iff is an arbitrary surjection from an Abelian lattice-ordered groupG ₁ onto an Archimedean Abelian lattice-ordered groupG ₂ such that f(0)]0 and, for some non-zero intrinsic metricsD andd, D(f(x),f(y)) depends functionally on d(x,y), thenf is a homomorphism of G₁ onto G₂.Presented by R. S. Pierce.     

Fault-tolerant routing in circulant networks and cycle prefix networks

Reliability and efficiency are important criteria in the design of interconnection networks. Connectivity is a widely used measurement for network fault-tolerance capacities, while diameter determines routing efficiency along individual paths. In practice, we are interested in high-connectivity, small-diameter networks. Recently, Hsu introduced the notion ofw-wide diameter, which unifies diameter and connectivity. This paper investigates thew-wide diameterd _w (G) and two related parameters:w-fault diameterD _w (G) andw-Rabin numberr _w (G). In particular, we determined _w (G) andD _w (G) for 2wK(G) andG is a circulant digraphG(d ⁿ ; 1,d,...,d ^n–1) or a cycle prefix digraph.Supported in part by the National Science Council under grant NSC86-2115-M009-002.     

Critical points for spread-out self-avoiding walk,percolation and the contact process above the upper critical dimensions

We consider self-avoiding walk and percolation in ^d, oriented percolation in ^d×₊, and the contact process in ^d, with p D(·) being the coupling function whose range is proportional to L. For percolation, for example, each bond is independently occupied with probability p D(y–x). The above models are known to exhibit a phase transition when the parameter p varies around a model-dependent critical point p_c. We investigate the value of p_c when d>6 for percolation and d>4 for the other models, and L1. We prove in a unified way that p_c=1+C(D)+O(L^–2d), where the universal term 1 is the mean-field critical value, and the model-dependent term C(D)=O(L^–d) is written explicitly in terms of the random walk transition probability D. We also use this result to prove that p_c=1+cL^–d+O(L^–d–1), where c is a model-dependent constant. Our proof is based on the lace expansion for each of these models.     

A technique for constructing divisible difference sets

An (m, n, k, ₁,₂) divisible difference set in a groupG of ordermn relative to a subgroupN of ordern is ak-subsetD ofG such that the list {xy^–1:x, y D} contains exactly ₁ copies of each nonidentity element ofN and exactly ₂ copies of each element ofG N. It is called semi-regular ifk > ₁ and k²=mn₂. We develop a method for constructing a divisible difference set as a product of a difference set and a relative difference set or a difference set and a subset ofG which we call a relative divisible difference set. The method results in several parametrically new families of semi-regular divisible difference sets.     

On reflecting diffusion processes and Skorokhod decompositions

Summary LetG be ad-dimensional bounded Euclidean domain, H¹ (G) the set off in L²(G) such that f (defined in the distribution sense) is in L²(G). Reflecting diffusion processes associated with the Dirichlet spaces (H¹(G), ) on L²(G, dx) are considered in this paper, where A=(a^ij is a symmetric, bounded, uniformly ellipticd×d matrix-valued function such thata ^ij H¹(G) for eachi,j, and H¹(G) is a positive bounded function onG which is bounded away from zero. A Skorokhod decomposition is derived for the continuous reflecting Markov processes associated with (H¹(G), ) having starting points inG under a mild condition which is satisfied when G has finite (d–1)-dimensional lower Minkowski content.     

Removable singularities of weak solutions to linear partial differential equations

Suppose that P(x, D) is a linear differential operator of order m > 0 with smooth coefficients whose derivatives up to order m are continuous functions in the domain G ⁿ (n 1), 1 < p > , s > 0, and q=p/(p – 1). In this paper, we show that if n, m, p, and s satisfy the two-sided bound 0 n – q(m – s)< n, then for a weak solution of the equation P(x, D)u=0 from the Sharpley-DeVore class C _p ^s (G)_loc, any closed set in G is removable if its Hausdorff measure of order n – q(m – s) is finite. This result strengthens the well-known result of Harvey and Polking on removable singularities of weak solutions to the equation P(x, D)_u=0 from the Sobolev classes and extends it to the case of noninteger orders of smoothness.Translated from Matematicheskie Zametki, vol. 77, no. 4, 2005, pp. 584–591.Original Russian Text Copyright © 2005 by A. V. Pokrovskii.This revised version was published online in April 2005 with a corrected issue number.     

On the Darling-Mandelbrot probability density and the zeros of some incomplete gamma functions

Recently, Mandelbrot has encountered and numerically investigated a probability densityp _d (t) on the nonnegative reals, where, 0D<1. this=" density=" has=" fourier=" transform=">f _d (-is), wheref _d (z)=–Dz ^d (–D, z) and (·.·) is an incomplete gamma function. Previously, Darling had met this density, but had not studied its form. We expressf _d (z) as a confluent hypergeometric function, then locate and approximate its zeros, thereby improving some results of Buchholz. Via properties of Laplace transforms, we approximatep _d (t) asymptotically ast0+ and +, then note some implications asD0+ and 1–.Communicated by Mourad Ismail.     

Indivisible plexes in latin squares

In this paper, we present two constructions of divisible difference sets based on skew Hadamard difference sets. A special class of Hadamard difference sets, which can be derived from a skew Hadamard difference set and a Paley type regular partial difference set respectively in two groups of orders v ₁ and v ₂ with |v ₁ − v ₂| = 2, is contained in these constructions. Some result on inequivalence of skew Hadamard difference sets is also given in the paper. As a consequence of Delsarte’s theorem, the dual set of skew Hadamard difference set is also a skew Hadamard difference set in an abelian group. We show that there are seven pairwisely inequivalent skew Hadamard difference sets in the elementary abelian group of order 3⁵ or 3⁷, and also at least four pairwisely inequivalent skew Hadamard difference sets in the elementary abelian group of order 3⁹. Furthermore, the skew Hadamard difference sets deduced by Ree-Tits slice symplectic spreads are the dual sets of each other when q ≤ 3¹¹.      

A lower bound for n-diameters

Let D be a subset of the s-dimensional lattice Z^S, M=M(D) the number of elements in D, _Dthe space of trigonometric polynomials on the torus T^S with spectrum concentrated in D and having unit norm in L₂(T^S). In this paper we give the following bound for the Gel'fand diameter:d ⁿ( _D,C(T^s))M/2–N/2. This bound is subsequently used for actual functional classes.Translated from Matematicheskie Zametki, Vol. 12, No. 4, pp. 413–419, October, 1972.     

Quantum groups in the higgs phase

In the Higgs phase we may be left with a residual finite symmetry groupH of the condensate. The topological interactions between the magnetic and electric excitations in these so-called discreteH gauge theories are completely described by the Hopf algebra or quantum groupD(H). In 2+1 dimensional space time we may add a Chern-Simons term to such a model. This deforms the underlying Hopf algebraD(H) into the quasi-Hopf algebraD (H) by means of a 3-cocycle onH. Consequently, the finite number of physically inequivalent discreteH gauge theories obtained in this way are labelled by the elements of the cohomology groupH ³(H,U(1)). We briefly review the above results in these notes. Special attention is given to the Coulomb screening mechanism operational in the Higgs phase. This mechanism screens the Coulomb interactions, but not the Aharonov-Bohm interactions.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 98. No. 3, pp. 509–523, March, 1994     

Total outer-connected domination numbers of trees

Let G=(V,E) be a graph without an isolated vertex. A set DV(G) is a total dominating set if D is dominating, and the induced subgraph G[D] does not contain an isolated vertex. The total domination number of G is the minimum cardinality of a total dominating set of G. A set DV(G) is a total outer-connected dominating set if D is total dominating, and the induced subgraph G[V(G)−D] is a connected graph. The total outer-connected domination number of G is the minimum cardinality of a total outer-connected dominating set of G. We characterize trees with equal total domination and total outer-connected domination numbers. We give a lower bound for the total outer-connected domination number of trees and we characterize the extremal trees.     

Infinite families of non-embeddable quasi-residual hadamard designs

LetD be a quasi-residual Hadamard design with =(2m + 1)2ⁿ–1, wherem andn are positive integers. IfD contains a pair of blocks intersecting in m2ⁿ⁺¹ points together with a third block intersecting each of the first two blocks in (m + 1)2ⁿ points thenD is non-embeddable. Using this result together with a recursive construction for quasi-residual Hadamard designs the existence of a previously unknown infinite family of non-embeddable quasi-residual Hadamard designs with =5(2ⁿ)–1 is established. An additional infinite family of non-embeddable quasi-residual Hadamard designs is given. This family has = 2ⁿ–1 with each design in the family having a pair of blocks meeting in (3 + 3)/4 points and a third block meeting each of the first two blocks in (5 + 5)/8 points.     

Number of no nisomorphic subgraphs in an n-point graph

LetG(n) be the set of all nonoriented graphs with n enumerated points without loops or multiple lines, and let v_k(G) be the number of mutually nonisomorphic k-point subgraphs of G G(n). It is proved that at least |G(n)| (1–1/n) graphs G G(n) possess the following properties: a) for any k [6log₂n], where c=–c log₂c–(1–c)×log₂(1–c) and c>1/2, we havev _k(G) > C _n ^k (1–1/n²); b) for any k [cn + 5 log₂n] we havev _k(G) = C _n ^k . Hence almost all graphs G G(n) containv(G) 2ⁿ pairwise nonisomorphic subgraphs.Translated from Matematicheskie Zametki, Vol. 9, No. 3, pp. 263–273, March, 1971.     

Building Symmetric Designs With Building Sets

We introduce a uniform technique for constructing a family of symmetric designs with parameters (v(q ^m+1-1)/(q-1), kq ^m ,q ^m), where m is any positive integer, (v, k, ) are parameters of an abelian difference set, and q = k ²/(k - ) is a prime power. We utilize the Davis and Jedwab approach to constructing difference sets to show that our construction works whenever (v, k, ) are parameters of a McFarland difference set or its complement, a Spence difference set or its complement, a Davis–Jedwab difference set or its complement, or a Hadamard difference set of order 9 · 4 ^d , thus obtaining seven infinite families of symmetric designs.     

Approximate factorizations with S/P consistently ordered M-factors

The behaviour of PCG methods for solving a finite difference or finite element positive definite linear systemAx=b with a (pre)conditioning matrixB=U ^TP^–1 U (whereU is upper triangular andP=diag(U)) obtained from a modified incomplete factorization, isunpredictable in the present status of knowledge whenever the upper triangular factor is not strictly diagonally dominant and 2P –D, whereD=diag(A), is not symmetric positive definite. The origin of this rather surprising shortcoming of the theory is that all upper bounds on the associated spectral condition number (B ^–1 A) obtained so far require either the strict diagonal dominance of the upper triangular factor or the strict positive definiteness of 2P –D. It is our purpose here to improve the theory in this respect by showing that, when the triangular factors are S/P consistently orderedM-matrices, nonstrict diagonal dominance is generally a sufficient requirement, without additional condition on 2P –D. As a consequence, the new analysis does not require diagonal perturbations (otherwise needed to keep control of the diagonal dominance ofU or of the positive definiteness of 2P –D). Further, the bounds obtained here on (B ^–1 A) are independent of the lower spectral bound ofD ^–1 A meaning that quasi-singular problems can be solved at the same speed as regular ones, an unexpected result.     

Reflecting Brownian motions and a deletion result for Sobolev spaces of order (1, 2)

Let D be an open set in ^d and E be a relatively closed subset of D having zero Lebesgue measure. A necessary and sufficient integral condition is given for the Sobolev spaces W ^1,2 (D) and W ^1,2(D\E) to be the same. The latter is equivalent to (normally) reflecting Brownian motion (RBM) on being indistinguishable (in distribution) from RBM on . This integral condition is satisfied, for example, when E has zero (d–1)-dimensional Hausdorff measure. Therefore it is possible to delete from D a relatively closed subset E having positive capacity but nevertheless the RBM on is indistinguishable from the RBM on , or equivalently, W ^1,2(D\E)=W^1,2(D). An example of such kind is: D=² and E is the Cantor set. In the proof of above mentioned results, a detailed study of RBMs on general open sets is given. In particular, a semimartingale decomposition and approximation result previously proved in [3] for RBMs on bounded open sets is extended to the case of unbounded open sets.Research supported in part by NSF Grant DMS 86-57483.     

Density of Gabor Frames

A Gabor system is a set of time-frequency shifts S(g, Λ) ={e^{2 π ibx}g(x − a)}_{(a, b) Λ} of a function g L²(R^d). We prove that if a finite union of Gabor systems _{k = 1}^rS(g_k, Λ_k) forms a frame for L²(R^d) then the lower and upper Beurling densities of Λ = _{k = 1}^r Λ_k satisfy D⁻(Λ) ≥ 1 and D ⁺ (Λ) < ∞. This extends recent work of Ramanathan and Steger. Additionally, we prove the conjecture that no collection _{k = 1}^r{g_k(x − a)}_{a Γk} of pure translates can form a frame for L²(R^d).     

Upper Bounds for Neumann–Schatten Norms

Let H and H _aux be Hilbert spaces, H a nonnegative self-adjoint operator in H,,s>0 and J a bounded linear transformation from the Hilbert space D(H ^s/2) (equipped with the graph scalar product of H ^s/2) to H _aux. It is shown that the operator J(H+)^–t belongs to the Neumann–Schatten class of order p=2+2(u–t)/(t–s/2) provided s/2<t<u,t–s/2<u–t and J(H+)^–u is Hilbert–Schmidt operator. An upper bound for the pth order Neumann–Schatten norm of J(H+)^–t is derived. If J is a closed operator from D(H ^1/2) to H _aux and D(J)D(H) then there exists a unique self-adjoint operator H ^J in H such that D(H ^J )D(J) and ( . Conditions which are sufficient in order that the operator (H ^J +)^–1–(H+)^–1 is compact and conditions which are sufficient in order that the wave operators W ^±(H ^J ,H) exist and are complete are derived. Instead of (Jf,Jg)_aux also certain other perturbation terms, not by necessity nonnegative, are considered. The special case when H equals the operator (–) ^r in L ²(R ^d ) for any strictly positive real number and H ^J equals (–) ^r + for some suitably chosen measure is discussed in detail. In particular, new results on existence and completeness of the wave operators W ^±(–+,–) are obtained.