Periodic Atomic Quasiinterpolation

We consider the approximation of periodic functions by using the atomic quasiinterpolation of the second and the first order. We obtain expressions for the coefficients of quasiinterpolants and present estimates for errors in the uniform metric.     

Approximation by radial basis functions with finitely many centers

Interpolation by translates of “radial” basis functions Φ is optimal in the sense that it minimizes the pointwise error functional among all comparable quasiinterpolants on a certain “native” space of functions $\mathcal{F}_\Phi $ . Since these spaces are rather small for cases where Φ is smooth, we study the behavior of interpolants on larger spaces of the form $\mathcal{F}_{\Phi _0 } $ for less smooth functions Φ₀. It turns out that interpolation by translates of Φ to mollifications of functionsf from $\mathcal{F}_{\Phi _0 } $ yields approximations tof that attain the same asymptotic error bounds as (optimal) interpolation off by translates of Φ₀ on $\mathcal{F}_{\Phi _0 } $ .     

The truncated Witten genus

In this paper we define and examine the truncated Witten genus. It is defined as the equivariant index of the Dirac operator on the manifold Map(C_p,M) with its natural C_p-action. Here, Map(C_p,M) is the space of maps from the cyclic group of order p into a closed, connected, spin manifold, M. By applying the Atiyah-Singer index theorem we give a topological formula for the truncated Witten genus which is related to the formula for the Witten genus by truncation of the infinite products. We also show that the equivariant index of the Dirac operator on the projective space Map(C_p,ⁿ⁺¹) is closely related to the truncated Witten genus of ⁿ. The spaces Map(C_p,ⁿ⁺¹) define a filtration of the space Map(S¹,ⁿ⁺¹) which has been used to study equivariant objects on the smooth loop space of ⁿ.Acknowledgement This work was mainly done while the author was studying for a Ph.D. at Warwick University under the supervision of J. D. S. Jones. The author would like to thank John Jones for originally suggesting the problem and for his guidance whilst the author was at Warwick, to thank the EPSRC for funding his Ph.D., and to thank the referee for his detailed comments which helped greatly with the writing of this paper.     

On hardware and software implementation of arithmetic in finite fields of characteristic 7 for calculation of pairings

We study scheme (hardware) and program (software) methods of multiplication of polynomials over fields of characteristic 7 in order to apply them to pairing-based cryptographic protocols on hyperelliptic curves of genus three. We consider hardware and software implementations of arithmetic in GF(7), GF(7²), GF(7 ⁿ ), GF(7⁷ⁿ ), and GF(7¹⁴ⁿ ) and estimate the complexity of corresponding schemes and programs.     

Twice Q-polynomial distance-regular graphs are thin

Let Γ be a distance-regular graph of diameter d 3. For each vertex χ of Γ, let T(χ) denote the subconstituent algebra for Γ with respect to χ. An irreducible T(χ)-module W is said to be thin if dim E_i^*(χ) W 1 for 0 i d, where E_i^*(χ) is the projection onto the ith subconstituent for Γ with respect to χ. The graph Γ is said to be thin if, for each vertex χ of Γ, very irreducible T(χ)-module is thin. Our main result is the following Theorem: If Γ has two Q-polynomial structures, then Γ is thin.     

Equal sums of four seventh powers

In this paper, the method used to find the smallest, nontrivial, positive integer solution of is discussed. The solution is

Factors enabling this discovery are advances in computing power, available workstation memory, and the appropriate choice of optimized algorithms.

     

Cyclic homology of algebras with one generator

We compute the cyclic homology of A = k[X]/X ⁿ for an arbitrary commutative ring k, and we apply this result to compute the cyclic homology of k[X]/f, when k is a field and f is an arbitrary polynomial.This work was presented at the IX-ELAM, Santigo de Chile, July 1988, and was partially supported by CONICET.The following pepole participated in this research: Jorge A. Guccione, juan José guccione, Maria Julia Redondo, Andrea Solotar, and Orlando E. Villamayor.     

Coefficients of half-integral weight modular forms modulo ℓ<Superscript><Emphasis Type="Italic">j</Emphasis></Superscript>

Suppose that 5 is prime, that j0 is an integer, and that F(z) is a half-integral weight modular form with integral Fourier coefficients. We give some general conditions under which the coefficients of F are well-distributed modulo ^j. As a consequence, we settle many cases of a classical conjecture of Newman by proving, for each prime power ^j with 5, that the ordinary partition function p(n) takes each value modulo ^j infinitely often.Mathematics Subject Classification (2000): 11F33, 11P83The first author thanks the National Science Foundation for its support through grant DMS 01-34577. The second author thanks the National Science Foundation for its support through a VIGRE postdoctoral fellowship.Acknowledgement The authors thank the referee for suggestions which improved the exposition in this paper.     

Bands of groups with universal properties

For a nonempty setX, a bandB, and a mapping :XB, we construct a band of groups, here called a cryptogroup,F(X,,B) which exhibits some remarkable properties. The first of these is a universal property relative to the classCG of all cryptogroups. In fact,CG is a variety with the operations of multiplication and inversion. For a varietyV of bands, we find a varietyV ⁰ of cryptogroups such that wheneverB is a band free inV ⁰ on the setX with embedding :XB, F(X,,B) is free inV ⁰. IfB is a normal band given as a strong semilattice of rectangular bands, we construct an isomorphic copy ofF(X,, B) which is a strong semilattice of completely simple semigroups. The objectsX, , B) admit the structure of a category, which is then related to the category of cryptogroups and their homomorphisms.This research was supported, in part by, NSERC Grant A4044.     

The Conditions of Analytic Extendability of Functions Defined on a Part of the Boundary of Circular Domains

The paper is devoted to the conditions of existence of the holomorphic expansion, in a domain D, of a function f C (M) defined on a set M with a positive measure of the Shilov boundary S of the complete circular domain D C ⁿ . These conditions are as follows: integrals of the product of f by certain polynomials that are orthogonal holomorphic functions tend to 0.     

Absolute values of the coefficients of the polynomials in Weierstrass's approximation theorem

The following problem, bound up with Weierstrass's classical approximation theorem, is solved definitively: to determine the sequence of positive numbersM _k such that, for anyf(z)c[0,1] and > 0 there exists the polynomial that f–P< and _k <M _k ,k=1, ...,n.Translated from Matematicheskii Zametki, Vol. 22, No. 2, pp. 269–276, August, 1977.     

q-Skew Derivations and Polynomial Identities

Let R be a prime ring and its left Martindale quotient ring. Assume that a q-skew -derivation of R satisfies the identity relationfor all x R, where the subring of constants of on R. It is proved that if R⁽⁾ satisfies nontrivial polynomial identities, then so does R. This answers affirmatively a problem raised in Bergen and Grzeszczuk [2] by removing the assumption on the algebraicity of .Mathematics Subject Classification (2000): 16W20, 16W25, 16W55Members of Mathematics Division, National Center for Theoretical Sciences at Taipei.Acknowledgement The authors are thankful to the referee for her/his useful suggestions and comments. This research was supported by the National Science Council of Taiwan.     

Skorokhod decomposition of reflected diffusions on bounded Lipschitz domains with singular non-reflection part

Let be a compact set with interior G. Let L ¹ (G,dx), >0 dx-a.e. on G, and m:=dx. Let A=(a _ij ) be symmetric, and globally uniformly strictly elliptic on G. Let be such that ; f, , is closable in L ² (G,m) with closure ( ^r ,D( ^r )). The latter is fulfilled if satisfies the Hamza type condition, or _i L ¹ _loc (G,dx), 1id. Conservative, non-symmetric diffusion processes X _t related to the extension of a generalized Dirichlet form where satisfies are constructed and analyzed. If G is a bounded Lipschitz domain, H ^1,1 (G), and a _ij D( ^r ), a Skorokhod decomposition for X _t is given. This happens through a local time that is uniquely associated to the smooth measure 1_{{ Tr ()>0}} d, where Tr denotes the trace and the surface measure on G.This research has been financially supported by TMR grant HPMF-CT-2000-00942 of the European Union. Mathematics Subject Classification (2000): 60J60, 60J55, 31C15, 31C25, 35J25     

Randomized online graph coloring

We study the problem of coloring graphs in an online manner. The only known deterministic online graph coloring algorithm with a sublinear performance function was found by [9.], 319–325). Their algorithm colors graphs of chromatic number χ with no more than (2χn)/log^* n colors, where n is the number of vertices. They point out that the performance can be improved slightly for graphs with bounded chromatic number. For three-chromatic graphs the number of colors used, for example, is O(n log log log n/log log n). We show that randomization helps in coloring graphs online. We present a simple randomized online algorithm to color graphs with expected number of colors O(2^χχ²n^{(χ−2)/(χ−1)}(log n)^1/(χ−1)). For three-colorable graphs the expected number of colors our algorithm uses is . All our algorithms run in polynomial time. It is interesting to note that our algorithm compares well with the best known polynomial time offline algorithms. For instance, the best polynomial time algorithm known for three-colorable graphs, due to [4.] pp. 554–562). We also prove a lower bound of Ω((1/(χ − 1))((log n/(12(χ + 1))) − 1)^χ−1) for the randomized model. No lower bound for the randomized model was previously known. For bounded χ, our result improves even the best known lower bound for the deterministic case: Ω((log n/log log n)^χ−1), due to Noga Alon (personal communication, September 1989).     

Multiplier family of harmonic univalent functions

The aim of this paper is to study a multiplier family of harmonic univalent functions using the sequences {c_n} and {d_n} of positive real numbers. By specializing {c_n} and {d_n}, the generalized Bernardi–Libera–Livingston integral operator is modified for such functions and the closure of the multiplier family under the modified integral operator is determined. Also, convolution products, closure properties, distortion theorems, convex combinations and neighborhoods for such functions are given.     

A class of self-complementary graphs and lower bounds of some ramsey numbers

A method is described of constructing a class of self-complementary graphs, that includes a self-complementary graph, containing no K₅, with 41 vertices and a self-complementary graph, containing no K₇, with 113 vertices. The latter construction gives the improved Ramsey number lower bound r(7, 7) ≥ 114.     

Projective embeddings of projective schemes blown up at subschemes

Suppose X is a nonsingular projective scheme, Z a nonsingular closed subscheme of X. Let X be the blowup of X centered at Z, E ₀ the pull-back of a general hyperplane in X, and E the exceptional divisor. In this paper, we study projective embeddings of X given by divisors . When X satisfies a necessary condition, we give explicit values of d and such that for all e>0 and embeds X as a projectively normal and arithmetically Cohen-Macaulay scheme. We also give a uniform bound for the regularities of the ideal sheaves of these embeddings, and study their asymptotic behaviour as t gets large compared to e. When X is a surface and Z is a 0-dimensional subscheme, we further show that these embeddings possess property N _p for all te>0. Mathematics Subject Classification (2000):14E25, 14M05, 13H10.Dedicated to the sixtieth birthday of Prof. A.V. Geramita     

General system of -maximal relaxed monotone variational inclusion problems based on generalized hybrid algorithms

In this paper, a new system of nonlinear (set-valued) variational inclusions involving (A,η)-maximal relaxed monotone and relative (A,η)-maximal monotone mappings in Hilbert spaces is introduced and its approximation solvability is examined. The notion of (A,η)-maximal relaxed monotonicity generalizes the notion of general η-maximal monotonicity, including (H,η)-maximal monotonicity (also referred to as (H,η)-monotonicity in literature). Using the general (A,η)-resolvent operator method, approximation solvability of this system based on a generalized hybrid iterative algorithm is investigated. Furthermore, for the nonlinear variational inclusion system on hand, corresponding nonlinear Yosida regularization inclusion system and nonlinear Yosida approximations are introduced, and as a result, it turns out that the solution set for the nonlinear variational inclusion system coincides with that of the corresponding Yosida regularization inclusion system. Approximation solvability of the Yosida regularization inclusion system is based on an existence theorem and related Yosida approximations. The obtained results are general in nature.     

A construction of Spin(7)-instantons

Joyce constructed examples of compact eight-manifolds with holonomy Spin(7), starting with a Calabi–Yau four-orbifold with isolated singular points of a special kind. That construction can be seen as the gluing of ALE Spin(7)-manifolds to each singular point of the Calabi–Yau four-orbifold divided by an anti-holomorphic involution fixing only the singular points. On the other hand, there are higher-dimensional analogues of anti-self-dual instantons in four dimensions on Spin(7)-manifolds, which are called Spin(7)-instantons. They are minimizers of the Yang–Mills action, and the Spin(7)-instanton equation together with a gauge fixing condition forms an elliptic system. In this article, we construct Spin(7)-instantons on the examples of compact Spin(7)-manifolds above, starting with Hermitian–Einstein connections on the Calabi–Yau four-orbifolds and ALE spaces. Under some assumptions on the Hermitian–Einstein connections, we glue them together to obtain Spin(7)-instantons on the compact Spin(7)-manifolds. We also give a simple example of our construction.     

Products of Quasi-p-Pseudocompact Spaces

Given p () , we determine when a product of quasi-p-pseudocompact spaces preserves this property. In particular, we analyze the product of quasi-p-pseudocompact subspaces of () containing . We give examples of spaces X, Y, X _s , Ys which are quasi-p-pseudocompact for every p *, but X Y is not pseudocompact, and X _s Y _s is pseudocompact and it is not quasi-s-pseudocompact for each s *. Besides, we prove that every pseudocompact space X of with X, is quasi-p-pseudocompact for some p *. Finally, we introduce, for each p *, the class P _p of all spaces X such that X × Y is quasi-p-pseudocompact when so is Y; and we prove: (1) the intersection of classes P _p ( _p *) coincides with the Frol"ik class; (2) every class P _p is closed under arbitrary products; (3) the partial ordered set ( P _p p ,) is isomorphic to the set of equivalence classes of free ultrafilters on with the Rudin–Keisler order. A topological characterization of RK-minimal ultrafilters is also given.