Regularity of Rees Algebras

Let B = k[x₁, ..., x_n] be a polynomial ring over a field k,and let A be a quotient ring of B by a homogeneous ideal J.Let m denote the maximal graded ideal of A. Then the Rees algebraR = A[m t] also has a presentation as a quotient ring of thepolynomial ring k[x₁, ..., x_n, y₁, ..., y_n] by a homogeneousideal J^*. For instance, if A = k[x₁, ..., x_n], then Rk[x₁,...,x_n,y₁,...,y_n]/(x_iy_j–x_jy_i|i, j=1,...,n). In this paper we want to compare the homological propertiesof the homogeneous ideals J and J^*.     

Hilbert Coefficients and the Depths of Associated Graded Rings

This work was motivated in part by the following general question:given an ideal I in a Cohen–Macaulay (abbreviated to CM)local ring R such that dim R/I=0, what information about I andits associated graded ring can be obtained from the Hilbertfunction and Hilbert polynomial of I? By the Hilbert (or Hilbert–Samuel)function of I, we mean the function H_I(n)=(R/Iⁿ) for all n1,where denotes length. Samuel [23] showed that for large valuesof n, the function H_I(n) coincides with a polynomial P_I(n) ofdegree d=dim R. This polynomial is referred to as the Hilbert,or Hilbert–Samuel, polynomial of I. The Hilbert polynomialis often written in the form where e₀(I), ..., e_d(I) are integers uniquely determined byI. These integers are known as the Hilbert coefficients of I.     

Shapiro's Cyclic Sum

Shapiro's cyclic sum is defined by , If K is the cone in Rⁿ of points withnon-negative coordinates, it is shown that the minimum of Ein K is a fixed point of T², where T is the non-linear operatordefined by (Tx)_i = x_n–i+1/(x_n–i+2 + x_n–i+3)²for i = 1,2,...,n. It is conjectured that Tx = S^kx, where Sis the shift operator in Rⁿ, and a proof is given under someadditional hypotheses. One of the consequences is a simple proofthat at the minimum point, a_i(x) = a_n–i+1–k(x) fori = 1,2,...,n.     

Sharp Results Concerning the Expression of Functions as Sums of Finite Differences

Let * denote convolution and let _x denote the Dirac measureat a point x. A function in L²(R)) is called a difference oforder 1 if it is of the form g-_x * g for some x R and g L²(R)).Also, a difference of order 2 is a function of the form for some x R and g L²(R)). In fact,the concept of a ‘difference of order s’ may bedefined in a similar manner for each s 0. If f denotes the Fouriertransform of f, it is known that a function f in L²(R)) is afinite sum of differences of order s if and only if , and the vector space of all suchfunctions is denoted by D_s (L²(R)). Every function in D_s (L²(R))is a sum of int(2s) + 1 differences of order s, where int(t)denotes the integer part of t. Thus, every function in D₁ (L²(R))is a sum of three first order differences, but it was provedin 1994 that there is a function in D₁ (L(R)) which is neverthe sum of two first order differences. This complemented, forthe group R, the corresponding result for first order differencesobtained by Meisters and Schmidt in 1972 for the circle group.The results show that there is a function in L₂ R such that,for each s 1/2, this function is a sum of int (2s) + 1 differencesof order s but it is never the sum of int (2s) differences oforder s. The proof depends upon extending to higher dimensionsthe following result in two dimensions obtained by Schmidt in1972 in connection with Heilbronn's problem: if x₁, x__n arepoints in the unit square, Following on from the work of Meisters and Schmidt, this workfurther develops a connection between certain estimates in combinatorialgeometry and some questions of sharpness in harmonic analysis.2000 Mathematics Subject Classification 42A38 (primary), 52A40(secondary).     

Stationary Critical Points of the Heat Flow in Spaces of Constant Curvature

The paper considers stationary critical points of the heat flowin sphere S^N and in hyperbolic space H^N, and proves severalresults corresponding to those in Euclidean space R^N which havebeen proved by Magnanini and Sakaguchi. To be precise, it isshown that a solution u of the heat equation has a stationarycritical point, if and only if u satisfies some balance lawwith respect to the point for any time. In Cauchy problems forthe heat equation, it is shown that the solution u has a stationarycritical point if and only if the initial data satisfies thebalance law with respect to the point. Furthermore, one point,say x₀, is fixed and initial-boundary value problems are consideredfor the heat equation on bounded domains containing x₀. It isshown that for any initial data satisfying the balance law withrespect to x₀ (or being centrosymmetric with respect to x₀)the corresponding solution always has x₀ as a stationary criticalpoint, if and only if the domain is a geodesic ball centredat x₀ (or is centrosymmetric with respect to x₀, respectively).     

An Euler-Type Version of the Local Steiner Formula for Convex Bodies

The purpose of this note is to establish a new version of thelocal Steiner formula and to give an application to convex bodiesof constant width. This variant of the Steiner formula generalizesresults of Hann [3] and Hug [6], who use much less elementarytechniques than the methods of this paper. In fact, Hann askedfor a simpler proof of these results [4, Problem 2, p. 900].We remark that our formula can be considered as a Euclideananalogue of a spherical result proved in [2, p. 46], and thatour method can also be applied in hyperbolic space. For some remarks on related formulas in certain two-dimensionalMinkowski spaces, see Hann [5, p. 363]. For further information about the notions used below, we referto Schneider's book [9]. Let Kⁿ be the set of all convex bodiesin Euclidean space Rⁿ, that is, the set of all compact, convex,non-empty subsets of Rⁿ. Let S^n–1 be the unit sphere.For KKⁿ, let NorK be the set of all support elements of K, thatis, the pairs (x, u)RⁿxS^n–1 such that x is a boundarypoint of K and u is an outer unit normal vector of K at thepoint x. The support measures (or generalized curvature measures)of K, denoted by ₀(K.), ..., _n–1(K.), are the unique Borelmeasures on RⁿxS^n–1 that are concentrated on NorK andsatisfy [formula] for all integrable functions f:RⁿR; here denotes the Lebesguemeasure on Rⁿ. Equation (1), which is a consequence and a slightgeneralization of Theorem 4.2.1 in Schneider [9], is calledthe local Steiner formula. Our main result is the following.1991 Mathematics Subject Classification 52A20, 52A38, 52A55.     

On Artin's Conjecture, I: Systems of Diagonal Forms

As a special case of a well-known conjecture of Artin, it isexpected that a system of R additive forms of degree k, say [formula] with integer coefficients a_ij, has a non-trivial solution inQ_p for all primes p whenever [formula] Here we adopt the convention that a solution of (1) is non-trivialif not all the x_i are 0. To date, this has been verified onlywhen R=1, by Davenport and Lewis [4], and for odd k when R=2,by Davenport and Lewis [7]. For larger values of R, and in particularwhen k is even, more severe conditions on N are required toassure the existence of p-adic solutions of (1) for all primesp. In another important contribution, Davenport and Lewis [6]showed that the conditions [formula] are sufficient. There have been a number of refinements of theseresults. Schmidt [13] obtained N>>R²k³ log k, and Low,Pitman and Wolff [10] improved the work of Davenport and Lewisby showing the weaker constraints [formula] to be sufficient for p-adic solubility of (1). A noticeable feature of these results is that for even k, onealways encounters a factor k³ log k, in spite of the expectedk² in (2). In this paper we show that one can reach the expectedorder of magnitude k². 1991 Mathematics Subject Classification11D72, 11D79.     

Binary Quadratic Forms with Large Discriminants and Sums of Two Squareful Numbers II

Let F = (F₁, ..., F_m) be an m-tuple of primitive positive binaryquadratic forms and let U_F(x) be the number of integers notexceeding x that can be represented simultaneously by all theforms F_j, j = 1, ... , m. Sharp upper and lower bounds for U_F(x)are given uniformly in the discriminants of the quadratic forms. As an application, a problem of Erds is considered. Let V(x)be the number of integers not exceeding x that are representableas a sum of two squareful numbers. Then V(x) = x(log x)^–+o(1)with = 1 – 2^–1/3 = 0.206....     

On the Algorithmic Complexity of Minkowski's Reconstruction Theorem

In 1903 Minkowski showed that, given pairwise different unitvectors µ₁, ..., µ_m in Euclidean n-space Rⁿ whichspan Rⁿ, and positive reals µ₁, ..., µ_m such that^m_i=1µ_iµ_i = 0, there exists a polytope P in Rⁿ, uniqueup to translation, with outer unit facet normals µ₁, ...,µ_m and corresponding facet volumes µ₁, ..., µ_m.This paper deals with the computational complexity of the underlyingreconstruction problem, to determine a presentation of P asthe intersection of its facet halfspaces. After a natural reformulationthat reflects the fact that the binary Turing-machine modelof computation is employed, it is shown that this reconstructionproblem can be solved in polynomial time when the dimensionis fixed but is #P-hard when the dimension is part of the input. The problem of ‘Minkowski reconstruction’ has variousapplications in image processing, and the underlying data structureis relevant for other algorithmic questions in computationalconvexity.     

Euler Characteristics of Real Varieties

In this paper we show how to associate to any real projectivealgebraic variety Z RP^n–1 a real polynomial F₁:Rⁿ,0 R, 0 with an algebraically isolated singularity, having theproperty that (Z) = (1 – deg (grad F₁), where deg (gradF₁ is the local real degree of the gradient grad F₁:Rⁿ, 0 Rⁿ,0. This degree can be computed algebraically by the method ofEisenbud and Levine, and Khimshiashvili [5]. The variety Z neednot be smooth. This leads to an expression for the Euler characteristic ofany compact algebraic subset of Rⁿ, and the link of a quasihomogeneousmapping f: Rⁿ, 0 Rⁿ, 0 again in terms of the local degree ofa gradient with algebraically isolated singularity. Similar expressions for the Euler characteristic of an arbitraryalgebraic subset of Rⁿ and the link of any polynomial map aregiven in terms of the degrees of algebraically finite gradientmaps. These maps do involve ‘sufficiently small’constants, but the degrees involved ar (theoretically, at least)algebraically computable.     

Correction: the Embedding of Radical Rings in Simple Radical Rings

As G. M. Bergman has pointed out, in the proof of the lemmaon p. 187, we cannot conclude that $$\stackrel{\¯}{S}$$is universal in the sense stated. However, the proof can becompleted as follows: Any element of $$\stackrel{\¯}{S}$$can be obtained as the first component of the solution u ofa system (A–I)u+a = 0, (1) where A S_n, a ⁿS and A–I has an inverse over L. SinceS is generated by R and k{s}, A can (by the last part of Lemma3.2 of [1]) be taken to be linear in these arguments, say A= A₀ + sA1, where A₀ R_n, A₀ R_n, A₁ K_n. Multiplying by (I–sA₁)^–1,we reduce this equation to the form (S_vB_v–I)u+a=0, (2) with the same solution u as before, where B_v R_n, s_v k{s}¹and a ⁿS. Now consider the retraction S k{s} (3) obtained by mapping R 0. If we denote its effect by x x^*,then (2) goes over into an equation –I.v + a^* 0, (4) which clearly has a unique solution v in k{s}; therefore theretraction (3) can be extended to a homomorphism $$\stackrel{\¯}{S}$$ k{s}, again denoted by x x^*, provided we can show that u₁^*does not depend on the equation (1) used to define it. Thisamounts to showing that if an equation (1), or equivalently(2), has the solution u₁ = 0, then after retraction we get v₁= 0 in (4), i.e. a₁^* = 0. We shall use induction on n; if u₁= 0 in (2), then by leaving out the first row and column ofthe matrix on the left of (2), we have an equation for u₂,...,u_n and by the induction hypothesis, their values after retractionare uniquely determined. Now from (2) we have where B = (b_ij^v). Applying ^* and observing that b_ij^vR, we seethat a₁ ^* = 0, as we wished to show. The proof still appliesfor n = 1, so we have a well-defined mapping $$\stackrel{\¯}{S}$$ k{s}, which is a homomorphism. Now the proof of the lemma canbe completed as before.     

On Positive Multipeak Solutions of a Nonlinear Elliptic Problem

In this paper we continue our investigation in [5, 7, 8] onmultipeak solutions to the problem –²u+u=Q(x)|u|^q–2u, xR^N, uH¹(R^N) (1.1) where = ^N_i=1²/x²_i is the Laplace operator in R^N, 2 < q < for N = 1, 2, 2 < q < 2N/(N–2) for N3, and Q(x)is a bounded positive continuous function on R^N satisfying thefollowing conditions. (Q₁) Q has a strict local minimum at some point x₀R^N, that is,for some > 0 Q(x)>Q(x₀) for all 0 < |x–x₀| < . (Q₂) There are constants C, > 0 such that |Q(x)–Q(y)|C|x–y| for all |x–x₀| , |y–y₀| . Our aim here is to show that corresponding to each strict localminimum point x₀ of Q(x) in R^N, and for each positive integerk, (1.1) has a positive solution with k-peaks concentratingnear x₀, provided is sufficiently small, that is, a solutionwith k-maximum points converging to x₀, while vanishing as 0 everywhere else in R^N.     

Sums of Connectivity Functions on Rn

A function f: Rⁿ R is a connectivity function if the graphof its restriction f|C to any connected C Rⁿ is connected inRⁿ x R. The main goal of this paper is to prove that every functionf: Rⁿ R is a sum of n + 1 connectivity functions (Corollary2.2). We will also show that if n > 1, then every functiong: Rⁿ R which is a sum of n connectivity functions is continuouson some perfect set (see Theorem 2.5) which implies that thenumber n + 1 in our theorem is best possible (Corollary 2.6). Toprove the above results, we establish and then apply the followingtheorems which are of interest on their own. For every dense G-subset G of Rⁿ there are homeomorphisms h₁,..., h_n of Rⁿ such that Rⁿ = G h₁(G) ... h_n(G) (Proposition2.4). For every n > 1 and any connectivity function f: Rⁿ R, ifx Rⁿ and > 0 then there exists an open set U Rⁿ such thatx U Bⁿ(x, ), f|bd(U) is continuous, and |(x) – f(y)|< for every y bd(U) (Proposition 2.7). 1991 MathematicsSubject Classification: 26B40, 54C30, 54F45.     

On the Local and Superlinear Convergence of Quasi-Newton Methods

This paper presents a local convergence analysis for severalwell-known quasi-Newton methods when used, without line searches,in an iteration of the form to solve for x^* such that Fx^* = 0. The basic idea behind theproofs is that under certain reasonable conditions on x_o, Fand x_o, the errors in the sequence of approximations {H_k} toF'(x^*)^–1 can be shown to be of bounded deterioration inthat these errors, while not ensured to decrease, can increaseonly in a controlled way. Despite the fact that H_k is not shownto approach F'(x^*)^–1, the methods considered, includingthose based on the single-rank Broyden and double-rank Davidon-Fletcher-Powellformulae, generate locally Q-superlinearly convergent sequences{x_k}.     

A pointwise estimate on the 1-order approximation of G{varepsilon}x0

On the basis of Avellaneda & Hua-Lin (1991, Commun. PureAppl. Math., 44 897–910), a pointwise error estimate onthe 1-order approximation of Green function G_x0 defined in R²is shown at first. Then based on this estimate and using asymptoticexpansion method, an improved approximation of G_x0 and its pointwiseerror estimate are obtained.     

Spaces of Harmonic Functions

It is important and interesting to study harmonic functionson a Riemannian manifold. In an earlier work of Li and Tam [21]it was demonstrated that the dimensions of various spaces ofbounded and positive harmonic functions are closely relatedto the number of ends of a manifold. For the linear space consistingof all harmonic functions of polynomial growth of degree atmost d on a complete Riemannian manifold Mⁿ of dimension n,denoted by H_d(Mⁿ), it was proved by Li and Tam [20] that thedimension of the space H₁(M) always satisfies dimH₁(M) dimH₁(Rⁿ)when M has non-negative Ricci curvature. They went on to askas a refinement of a conjecture of Yau [32] whether in generaldim H_d(Mⁿ) dimH_d(Rⁿ)for all d. Colding and Minicozzi made animportant contribution to this question in a sequence of papers[5–11] by showing among other things that dimH_d(M) isfinite when M has non-negative Ricci curvature. On the otherhand, in a very remarkable paper [16], Li produced an elegantand powerful argument to prove the following. Recall that Msatisfies a weak volume growth condition if, for some constantA and , (1.1) for all x M and r R, where V_x(r) is the volume of the geodesicball B_x(r) in M; M has mean value property if there exists aconstant B such that, for any non-negative subharmonic functionf on M, (1.2) for all p M and r > 0.     

Hurwitz Groups with Given Centre

A Hurwitz group is any non-trivial finite group that can be(2,3,7)-generated; that is, generated by elements x and y satisfyingthe relations x² = y³ = (xy)⁷ = 1. In this short paper a completeanswer is given to a 1965 question by John Leech, showing thatthe centre of a Hurwitz group can be any given finite abeliangroup. The proof is based on a recent theorem of Lucchini, Tamburiniand Wilson, which states that the special linear group SL_n(q)is a Hurwitz group for every integer n 287 and every prime-powerq. 2000 Mathematics Subject Classification 20F05 (primary);57M05 (secondary).     

A Relation between Hausdorff Dimension and a Condition on Time Sets for the Image by the Brownian Sheet to Possess Interior Points

We prove that if W_{N, d} is a Brownian sheet mapping to R^d and E is a set in (0, )^N of Hausdorff dimensiongreater than , then for almost every rotation about a point x and translation _x such that _x(E) (0, )^N, the set _x(E) is such that almost surely W(E) containsinterior points. The techniques are adapted from Kahane andRosen and generalize to higher dimensional time and range.     

Reduction of CalderON Commutators

Let A₁,..., A_n be Lipschitz functions on R such that A'₁,...,A'_nVMO. We show that on any bounded interval, the Calderóncommutator associated with the kernel (A₁(x)–A₁(y)) ...(A_n(x) – A_n(y))/(x–y) ⁿ¹ is a compact perturbationof , where H is the Hilberttransform. 1991 Mathematics Subject Classification 47B38, 47B47,47G10, 45E99.     

Bounding the number of stable homotopy types of a parametrized family of semi-algebraic sets defined by quadratic inequalities

We prove a nearly optimal bound on the number of stable homotopytypes occurring in a k-parameter semi-algebraic family of setsin R, each defined in terms of m quadratic inequalities. Ourbound is exponential in k and m, but polynomial in . More precisely,we prove the following. Let R be a real closed field and let = {P₁, ... , P_m} R[Y₁, ... ,Y,X₁, ... ,X_k], with deg_Y(P_i) 2, deg_X(P_i) d, 1 i m. Let S R^+k be a semi-algebraic set,defined by a Boolean formula without negations, with atoms ofthe form P 0, P 0, P . Let : R^+k R^k be the projection onthe last k coordinates. Then the number of stable homotopy typesamongst the fibers S_x = ^–1(x) S is bounded by (2^mkd)^O(mk).