Characterizations of Buchsbaum complexes

Letk be a field and an abstract simplicial complex with vertex set . In this article we study the structure of the Ext modules Ext _a ⁱ (A/m _(l ,k[]) of the Stanley-Reisner ringk[] whereA=k[x ₁,...,x _n ] andm _l =(x _l ¹ ,...,x _l ⁿ ). Using this structure theorem we give a characterization of Buchsbaumness ofk[] by means of the length of the modules Ext _A ⁱ (A/m _l ,k[]). That isk[] is Buchsbaum if and only if for allik[], the length of the modules Ext _A ⁱ (A/m _l ,k[]) is independent ofl.     

On a Problem of Karpilovsky

Let G be a finite elementary group. Let ⁿ (G) denote the nth power of the augmentation ideal (G) of the integral group ring G. In this paper, we give an explicit basis of the quotient group Q_n(G) = ⁿ(G)/ⁿ⁺¹ (G) and compute the order of Qⁿ (G).2000 Mathematics Subject Classification: 16S34, 20C05     

Parametrization of Triangle Groups as Subgroups of PSL(2, q)

In this paper we are interested in triangle groups (j, k, l) where j = 2 and k = 3. The groups (j, k, l) can be considered as factor groups of the modular group PSL(2, Z) which has the presentation x, y : x² = y³ = 1. Since PSL(2,q) is a factor group of G^k,l,m if -1 is a quadratic residue in the finite field F_q, it is therefore worthwhile to look at (j, k, l) groups as subgroups of PSL(2, q) or PGL(2, q). Specifically, we shall find a condition in form of a polynomial for the existence of groups (2, 3, k) as subgroups of PSL(2, q) or PGL(2, q).Mathematics Subject Classification: Primary 20F05 Secondary 20G40.     

Upper bound theorems for homology manifolds

In this paper we prove the Upper Bound Conjecture (UBC) for some classes of (simplicial) homology manifolds: we show that the UBC holds for all odd-dimensional homology manifolds and for all 2k-dimensional homology manifolds Δ such that β _k (Δ)⩽Σ{β _i (Δ):i ≠k-2,k,k+2 and 1 ⩽i⩽2k-1}, where β _i (Δ) are reduced Betti numbers of Δ. (This condition is satisfied by 2k-dimensional homology manifolds with Euler characteristic χ≤2 whenk is even or χ≥2 whenk is odd, and for those having vanishing middle homology.) We prove an analog of the UBC for all other even-dimensional homology manifolds. Kuhnel conjectured that for every 2k-dimensional combinatorial manifold withn vertices, . We prove this conjecture for all 2k-dimensional homology manifolds withn vertices, wheren≥4k+3 orn≤3k+3. We also obtain upper bounds on the (weighted) sum of the Betti numbers of odd-dimensional homology manifolds.     

Nonlinear programming and an optimization problem on infinitely differentiable functions

A nonnegative, infinitely differentiable function ø defined on the real line is called a Friedrichs mollifier function if it has support in [0, 1] and ₀ ¹ ø(t)dt=1. In this article the following problem is considered. Determine _k =inf ₀ ¹ vø^(k)(t)dt, k=1,..., where ø^(k) denotes thekth derivative of ø and the infimum is taken over the set of all mollifier functions. This problem has applications to monotone polynomial approximation as shown by this author elsewhere. In this article, the structure of the problem of determining _k is analyzed, and it is shown that the problem is reducible to a nonlinear programming problem involving the minimization of a strictly convex function of [(k–1)/2] variables, subject to a simple ordering restriction on the variables. An optimization problem on the functions of bounded variation, which is equivalent to the nonlinear programming problem, is also developed. The results of this article and those from approximation of functions theory are applied elsewhere to derive numerical values of various mathematical quantities involved in this article, e.g., _k =k~2^2k–1 for allk=1, 2, ..., and to establish certain inequalities of independent interest. This article concentrates on problem reduction and equivalence, and not numerical value.This research was supported in part by the National Science Foundation under Grant No. GK-32712.     

Free-knot Splines Approximation of s-monotone Functions

Let I be a finite interval and r,sN. Given a set M, of functions defined on I, denote by ₊ ^s M the subset of all functions yM such that the s-difference ^s y() is nonnegative on I, >0. Further, denote by ₊ ^s W _p ^r , the class of functions x on I with the seminorm x ^(r)L _p 1, such that ^s x0, >0. Let M _n (h _k ):={ _i=1 ⁿ c _i h _k (w _i t– _i )c _i ,w _i , _i R, be a single hidden layer perceptron univariate model with n units in the hidden layer, and activation functions h _k (t)=t ₊ ^k , tR, kN ₀. We give two-sided estimates both of the best unconstrained approximation E( ₊ ^s W _p ^r ,M _n (h _k ))L _q , k=r–1,r, s=0,1,...,r+1, and of the best s-monotonicity preserving approximation E( ₊ ^s W _p ^r , ₊ ^s M _n (h _k ))L _q , k=r–1,r, s=0,1,...,r+1. The most significant results are contained in theorem 2.2.     

On the distribution of square-full and cube-full integers

LetN _r (x) be the number ofr-full integers x and let _r (x) be the error term in the asymptotic formula forN _r (x). Under Riemann's hypothesis, we prove the estimates ₂(x)x^1/7+, ₃(x)x^97/804+(>0), which improve those of Cao and Nowak. We also investigate the distribution ofr-full andl-free numbers in short intervals (r=2,3). Our results sharpen Krätzel's estimates.     

Control of stochastic distributed-parameter systems

A scheme is proposed for the feedback control of distributed-parameter systems with unknown boundary and volume disturbances and observation errors. The scheme consists of employing a nonlinear filter in the control loop such that the controller uses the optimal estimates of the state of the system. A theoretical comparison of feedback proportional control of a styrene polymerization reactor with and without filtering is presented. It is indicated how an approximate filter can be constructed, greatly reducing the amount of computing required.Notation a(t) l-vector noisy dynamic input to system - A(t, a) l-vector function - A frequency factor for first-order rate law (5.68×10⁶ sec^–1) - b distance to the centerline between two coil banks in the reactor (4.7 cm) - B k-vector function defining the control action - c(, ) concentration of styrene monomer, molel ^–1 - C _p heat capacity (0.43 cal · g^–1 · K^–1) - C _ij constants in approximate filter, Eqs. (49)–(52) - E activation energy (20330 cal · mole^–1) - expectation operator - f(t,...) n-vector function - g ₀,g ₁(t,...) n-vector functions - h(t, u) m-vector function relating observations to states - H(t) function defined in Eq. (36) - k dimensionality of control vectorv(x, t) - k _i constants in approximate filter, Eqs. (49)–(52) - K dimensionless proportional gain - l dimensionality of dynamic inputa(t) - m dimensionality of observation vectory(t) - n dimensionality of state vectoru(x, t) - P ^(vv)(x, s, t) n×n matrix governed by Eq. (9) - P ^(va)(x, t) n×l matrix governed by Eq. (10) - P ^(aa)(t) l×l matrix governed by Eq. (11) - q _i (t) diagonal elements ofm×m matrixQ(x, s, t) - Q(x, s, t) m×m weighting matrix - R universal gas constant (1.987 cal · mole^–1 · K^–1) - R(x, s, t) n×n weighting matrix - R _i (t) n×n weighting matrix - s dimensionless spatial variable - S(x, s, t) matrix defined in Eq. (11) - t dimensionless time variable - T(, ) temperature, K - u(x, t) n-dimensional state vector - u _c (t) wall temperature - u _d desired value ofu ₁(1,t) - u _c * reference control value ofu _c - u _c ^max maximum value ofu _c - u _c ^min minimum value of _c - v(x, t) k-dimensional control vector - W(t) l×l weighting matrix - x dimensionless spatial variable - y(t) m-dimensional observation vector - _i constants in approximate filter, Eqs. (49)–(52) - dimensionless parameter defined in Eq. (29) - H heat of reaction (17500 cal · mole^–1) - dimensionless activation energy, defined in Eq. (29) - (x) Dirac delta function - (x, t) m-dimensional observation noise - thermal conductivity (0.43×10^–3 cal · cm^–1 · sec^–1 · K^–1) - density (1 g · cm^–3) - time, sec - dimensionless parameter defined in Eq. (29) - spatial variable, cm - * reference value - ^ estimated value     

Leaves in Representation Diagrams of Bipartite Distance-Regular Graphs

Let denote a bipartite distance-regular graph with diameter D 3 and valency k 3. Let ₀ > ₁ ··· > _D denote the eigenvalues of and let q ^h _ij (0 h, i, j D) denote the Krein parameters of . Pick an integer h (1 h D – 1). The representation diagram = _h is an undirected graph with vertices 0,1,...,D. For 0 i, j D, vertices i, j are adjacent in whenever i j and q ^h _ij 0. It turns out that in , the vertex 0 is adjacent to h and no other vertices. Similarly, the vertex D is adjacent to D – h and no other vertices. We call 0, D the trivial vertices of . Let l denote a vertex of . It turns out that l is adjacent to at least one vertex of . We say l is a leaf whenever l is adjacent to exactly one vertex of . We show has a nontrivial leaf if and only if is the disjoint union of two paths.     

A System of Schrödinger Equations in the Critical Case

A system of nonlinear Schrödinger equations u _k } / t=ia _k u _k+f _k (u,u ^*), t>0, k=1,... ,m; u _k (0,x)=u _k0 (x), where f _k are homogeneous functions of order 1+4/n, is considered. Sufficient conditions for the globality of the solution are obtained. The existence of the explicit blow-up solution is proved.     

Existence and construction of Δ1-splines of class Ck on a three-directional mesh

Let ^* be the equilateral triangulation of the plane and let ₁ ^* be the equilateral triangle formed by four triangles of ^*. We study the space of piecewise polynomial functions in C ^k (R ²) with support ₁ ^*, having a sufficiently high degree n and which are invariant with respect to the group of symmetries of ₁ ^*. Such splines are called ₁ ^*-splines. We first compute the dimension of this space in function of n and k. Then, for any fixed k0, we prove the existence of ₁ ^*-splines of class C ^k and minimal degree, but these splines are not unique. Finally, we describe an algorithm computing the Bernstein–Bézier coefficients of these splines.     

Gröbner bases and triangulations of the second hypersimplex

The algebraic technique of Gröbner bases is applied to study triangulations of the second hypersimplex (2,n). We present a quadratic Gröbner basis for the associated toric idealK(K _n ). The simplices in the resulting triangulation of (2,n) have unit volume, and they are indexed by subgraphs which are linear thrackles [28] with respect to a circular embedding ofK _n . Forn6 the number of distinct initial ideals ofI(K _n ) exceeds the number of regular triangulations of (2,n); more precisely, the secondary polytope of (2,n) equals the state polytope ofI(K _n ) forn5 but not forn6. We also construct a non-regular triangulation of (2,n) forn9. We determine an explicit universal Gröbner basis ofI(K _n ) forn8. Potential applications in combinatorial optimization and random generation of graphs are indicated.Research partially supported by a doctoral fellowship of the National University of Mexico, the National Science Foundation, the David and Lucile Packard Foundation and the U.S. Army Research Office (through ACSyAM/MSI, Cornell).     

On the Function w( x)=|{1= s= k : x= a s (mod n s)}|

For a finite system of arithmetic sequences the covering function is w(x) = |{1 s k : x a_s (mod n_s)}|. Using equalities involving roots of unity we characterize those systems with a fixed covering function w(x). From the characterization we reveal some connections between a period n₀ of w(x) and the moduli n₁, . . . , n_k in such a system A. Here are three central results: (a) For each r=0,1, . . .,n_k/(n₀,n_k)–1 there exists a Jc{1, . . . , k–1} such that . (b) If n₁ ···n_k–l <n_k–l+1 =···=n_k (0 < l < k), then for any positive integer r < n_k/n_k–l with r 0 (mod n_k/(n₀,n_k)), the binomial coefficient can be written as the sum of some (not necessarily distinct) prime divisors of n_k. (c) max_(xw(x) can be written in the form where m₁, . . .,m_k are positive integers.The research is supported by the Teaching and Research Award Fund for Outstanding Young Teachers in Higher Education Institutions of MOE, and the National Natural Science Foundation of P. R. China.     

Hodge decompositions of Loday symbols inK-theory and cyclic homology

In this paper, we study the Hodge decompositions ofK-theory and cyclic homology induced by the operations ^k and ^k , and in particular the decomposition of the Loday symbols x,y, ...z. Except in special cases, these Loday symbols do not have pure Hodge index. InK _n (A) they can project into every componentK _n ⁽ⁱ⁾ for 2in, and the projection of the Loday symbol x,y, ...,z intoK _n ⁽ⁿ⁾ is a multiple of the generalized Dennis-Stein symbol x,y, ...,z. Our calculations disprove conjectures of Beilinson and Soulé inK-theory, and of Gerstenhaber and Schack in Hochschild homology.Partially supported by National Security Agency grant MDA904-90-H-4019.Partially supported by National Science Foundation grant DMS-8803497.     

The binding number of a graph and its circuits

In this paper we prove the following main results: Theorem A. If bind (G)3/2, thenG–u has a Hamiltonian circuit for every vertexu of graphG _i, unlessG belongs either to two classesH ₁ andH ₂ of graphs or to some smaller order graphs with |V(G)|17. Theorem B. If bind (G)3/2 and the maximum degree (G)>(n–1)/2, |V(G)|=n>17, thenG is pancyclic (i.e., it contains a circuit of every lengthm, 3m|V(G)|).     

Explicit Formulae for the Wave Kernels for the Laplacians Δαβ in the Bergman Ball Bn,n≥1

For the perturbed Bergman Laplacians given by in the unit ball B ⁿ of C ⁿ we establish explicit formulae for the corresponding wave equations in B _n. The formulae obtained generalise, for arbitrary , the formulae given in [2] and [5] for the wave equation associated to the shifted Bergman Laplacian =₀₀ in B _n. Moreover, using an analytic continuation argument, we are able to give explicit formulae for the solutions of the wave equation associated to a two parameter family of Laplacians _, on C ⁿ which are natural deformations of the Fubini-Study Laplacian on the Projective space P ⁿ(C) , n 1, viewed as the dual space of the Bergman ball B _n.     

k-Weakly almost convex groups and π 1 ∞ \tilde M^3

We extend Cannon's notion ofk-almost convex groups which requires that for two pointsx, y on then-sphere in the Cayley graph which can be joined by a pathl ₁ of length k, there is a second pathl ₂ in then-ball, joiningx andy, of bounded length N(k). Ourk-weakly almost convexity relaxes this condition by requiring only thatl ₁ l ₂ bounds a disk of area C ₁(k)n ^{1 - (k)} +C ₂(k). IfM ³ is a closed 3-manifold with 3-weakly almost convex fundamental group, then ₁ .     

Strong bounds for multidimensional spacings

Summary Let U ₁, U ₂,..., U _n be independent random vectors uniformly distributed on (0, 1) ^d . We define the k-th maximal spacing _n ^(k) associated to U ₁, ...,U₁ as the k-th maximal possible value of the length of a side of a d-dimensional square block in (0,1) ^d , which does not intersect the sample, and which cannot be enlarged without doing so. Our main result is that _n ^(k) ={n^–1(Log n+0(Log₂n))} ^1/d almost surely as n. Other bounds are proposed for the limiting almost sure behavior of _n ^(k) as n.