The Central Approximation Theorems for the Method of Left Gamma Quasi-Interpolants in L p spaces

The optimal degree of approximation of the method of Gammaoperators G _n in L _p spaces is O(n ^-1). In order to obtain much faster convergence, quasi-interpolants G _n ^(k) of G _n in the sense of Sablonnière are considered. We show that for fixed k the operator-norms G _n ^(k) _p are uniformly bounded in n. In addition to this, for the first time in the theory of quasi-interpolants, all central problems for approximation methods (direct theorem, inverse theorem, equivalence theorem) could be solved completely for the L _p metric. Left Gamma quasi-interpolants turn out to be as powerful as linear combinations of Gammaoperators [6].     

Dimension Theory and Nonstable K1 of Quadratic Modules

Employing Bak's dimension theory, we investigate the nonstable quadratic K-group K _1,2n (A, ) = G _2n (A, )/E _2n (A, ), n 3, where G _2n (A, ) denotes the general quadratic group of rank n over a form ring (A, ) and E _2n (A, ) its elementary subgroup. Considering form rings as a category with dimension in the sense of Bak, we obtain a dimension filtration G _2n (A, ) G _2n ⁰(A, ) ; G _2n ¹(A, ) ... E _2n (A, ) of the general quadratic group G _2n (A, ) such that G _2n (A, )/G _2n ⁰(A, ) is Abelian, G _2n ⁰(A, ) G _2n ¹(A, ) ... is a descending central series, and G _2n ^d(A)(A, ) = E _2n (A, ) whenever d(A) = (Bass–Serre dimension of A) is finite. In particular K _1,2n (A, ) is solvable when d(A) < .     

On Some Sets of Group Functions

Let G be a group, let A be an Abelian group, and let n be an integer such that n –1. In the paper, the sets _n (G,A) of functions from G into A of degree not greater than n are studied. In essence, these sets were introduced by Logachev, Sal'nikov, and Yashchenko. We describe all cases in which any function from G into A is of bounded (or not necessarily bounded) finite degree. Moreover, it is shown that if G is finite, then the study of the set _n (G,A) is reduced to that of the set n(G/O ^p (G),A _p ) for primes p dividing G/G. Here O ^p (G) stands for the p-coradical of the group G, A _p for the p-component of A, and G for the commutator subgroup of G.     

Some greedyt-intersecting families of finite sequences

Letn, s ₁,s ₂, ... ands _n be positive integers. Assume is an integer for eachi}. For , , and , denotes _p (a)={j|1jn,a _j p}, , and . is called anI _t ^p -intersecting family if, for any a,b ,a _i b _i =min(a _i ,b _i )p for at leastt i's. is called a greedyI _t ^P -intersecting family if is anI _t ^p -intersecting family andW _p (A)W _p (B+A ^c ) for anyAS _p ( ) and any with |B|=t–1.In this paper, we obtain a sharp upper bound of | | for greedyI _t ^p -intersecting families in for the case 2ps _i (1in) ands ₁>s ₂>...>s _n .This project is partially supported by the National Natural Science Foundation of China (No.19401008) and by Postdoctoral Science Foundation of China.     

On Finite Groups with a Given Number of Centralizers

For a finite group G, let Cent(G) denote the set of centralizers of single elements of G and #Cent(G) = |Cent(G)|. G is called an n-centralizer group if #Cent(G) = n, and a primitive n-centralizer group if #Cent(G) = #Cent(G/Z(G)) = n. In this paper, we compute #Cent(G) for some finite groups G and prove that, for any positive integer n 2, 3, there exists a finite group G with #Cent(G) = n, which is a question raised by Belcastro and Sherman [2]. We investigate the structure of finite groups G with #Cent(G) = 6 and prove that, if G is a primitive 6-centralizer group, then G/Z(G) A₄, the alternating group on four letters. Also, we prove that, if G/Z(G) A₄, then #Cent(G) = 6 or 8, and construct a group G with G/Z(G) A₄ and #Cent(G) = 8.This research was in part supported by a grant from IPM.2000 Mathematics Subject Classification: 20D99, 20E07     

On Hua-Tuan’s conjecture

Let G be a finite group and |G| = pn, p be a prime. For 0 m n, sm(G) denotes the number of subgroups of of order pm of G. Loo-Keng Hua and Hsio-Fu Tuan have ever conjectured: for an arbitrary finite p-group G, if p > 2, then sm(G) ≡ 1, 1 + p, 1 + p + p2 or 1 + p + 2p2 (mod p3). In this paper, we investigate the conjecture, and give some p-groups in which the conjecture holds and some examples in which the conjecture does not hold.     

Finite and p-adic Polylogarithms

The finite nth polylogarithm li _n (z) /p(z) is defined as _k=1 ^p–1 z ^k /k ⁿ . We state and prove the following theorem. Let Li _k : _{p p} be the p-adic polylogarithms defined by Coleman. Then a certain linear combination F _n of products of polylogarithms and logarithms, with coefficients which are independent of p, has the property that p ^1–n DF _n (z) reduces modulo p>n+1 to li _n–1((z)), where D is the Cathelineau operator z(1–z)d/dz and is the inverse of the p-power map. A slightly modified version of this theorem was conjectured by Kontsevich. This theorem is used by Elbaz-Vincent and Gangl to deduce functional equations of finite polylogarithms from those of complex polylogarithms.     

Generalized Hadamard Matrices

The paper studies a generalized Hadamard matrix H = (g _{i j}) of order n with entries gi j from a group G of order n. We assume that H satisfies: (i) For m k, G = {g _{m i} g k i ^-1 i = 1,...., n} (ii) g _1i = g _i1 = 1 for each i; (iii) g _ij ^-1 = g _ji for all i, j. Conditions (i) and (ii) occur whenever G is a(P, L) -transitivity for a projective plane of order n. Condition (iii) holds in the case that H affords a symmetric incidence matrix for the plane. The paper proves that G must be a 2-group and extends previous work to the case that n is a square.     

Some Criteria for Multivalently Starlikeness

Let S_n(p)(p, n N) be the class of functions f() = ^p + a_p+n^p+n + which are p-valently starlike in the unit disk. Some sufficient conditions for a function f() to be in the class S_n(p) are given.AMS Subject Classification (2000): primary 30C45     

Two Commuting Involutions Fixing F n ∪ F n-1

Consider G=Z ₂² as the group generated by two commuting involutions, and let be a smooth G-action on a smooth and closed manifold M. Suppose that the fixed point set of Φ consists of two connected components, F ⁿ and F ^n-1, with dimensions n and n−1, respectively. In this paper we prove that, if in the fixed data of Φ at least two eigenbundles over F ⁿ have dimension greater than n, and at least one eigenbundle over F ^n-1 has dimension greater than n−1, then the action (M,Φ) bounds equivariantly.It is well known that, if is a smooth involution on a smooth and closed m-dimensional manifold M ^m such that the fixed point set of T has constant dimension n, and if m > 2n, then (M ^m ,T) bounds equivariantly; this fact was proved by R. E. Stong and C. Kosniowski 27 years ago. As a consequence of our result, we will see that the same fact is true when, besides n-dimensional components, the fixed point set contains additional (n−1)-dimensional components.     

Subspaces of Lp Isometric to Subspaces of ℓ p

We present three results on isometric embeddings of a (closed, linear) subspace X of L_p=L_p[0,1] into _p . First we show that if p 2N, then X is isometrically isomorphic to a subspace of _p if and only if some, equivalently every, subspace of L_p which contains the constant functions and which is isometrically isomorphic to X, consists of functions having discrete distribution. In contrast, if p 2N; and X is finite-dimensional, then X is isometrically isomorphic to a subspace of _p , where the positive integer N depends on the dimension of X, on p , and on the chosen scalar field. The third result, stated in local terms, shows in particular that if p is not an even integer, then no finite-dimensional Banach space can be isometrically universal for the 2-dimensional subspaces of L_p .     

Mean-Field Conditions for Percolation on Finite Graphs

Let {G _n } be a sequence of finite transitive graphs with vertex degree d = d(n) and |G _n | = n. Denote by p ^t (v, v) the return probability after t steps of the non-backtracking random walk on G _n . We show that if p ^t (v, v) has quasi-random properties, then critical bond-percolation on G _n behaves as it would on a random graph. More precisely, if $\mathop {\rm {lim\, sup\,}} \limits_{n} n^{1/3} \sum\limits_{t = 1}^{n^{1/3}} {t{\bf p}^t(v,v) < \infty ,}$ then the size of the largest component in p-bond-percolation with ${p =\frac{1+O(n^{-1/3})}{d-1}}Let {G _n } be a sequence of finite transitive graphs with vertex degree d = d(n) and |G _n | = n. Denote by p ^t (v, v) the return probability after t steps of the non-backtracking random walk on G _n . We show that if p ^t (v, v) has quasi-random properties, then critical bond-percolation on G _n behaves as it would on a random graph. More precisely, if

lim sup  n n1/3 ?t = 1n1/3 tpt(v,v) < ￥,\mathop {\rm {lim\, sup\,}} \limits_{n} n^{1/3} \sum\limits_{t = 1}^{n^{1/3}} {t{\bf p}^t(v,v) < \infty ,}      13. Vector subdivision schemes in (Lp(?s))r(1?p?∞) spaces      Song Li 《中国科学A辑(英文版)》2003,46(3):364-375 The purpose of this paper is to investigate the refinement equations of the form where the vector of functions ϕ=(ϕ 1..., ϕ r ) T is in (L p (ℝ s )) r , 1⩽p⩽∞, a(α), α∈ℤ s is a finitely supported sequence of r × r matrices called the refinement mask, and M is an s × s integer matrix such that lim→∞ M-n = 0. In order to solve the refinement equation mentioned above, we start with a vector of compactly supported functions φ 0∈(L p (ℝ s )) r and use the iteration schemes f n :=Q a n φ 0, n=1,2,..., where Q n is the linear operator defined on (L p (ℝ s )) r given by This iteration scheme is called a subdivision scheme or cascade algorithm. In this paper, we characterize the Lp-convergence of subdivision schemes in terms of the p-norm joint spectral radius of a finite collection of some linear operators determined by the sequence a and the set B restricted to a certain invariant subspace, where the set B is a complete set of representatives of the distinct cosets of the quotient group ℤs/Mℤs containing 0.      14. The minimality of the map \frac{x}{\|{x}\|} for weighted energy      Jean-Christophe Bourgoin 《Calculus of Variations and Partial Differential Equations》2006,25(4):469-489 In this paper, we investigate the minimality of the map from the Euclidean unit ball Bn to its boundary 핊n−1 for weighted energy functionals of the type Ep,f = ∫Bn f(r)‖∇ u‖p dx, where f is a non-negative function. We prove that in each of the two following cases: i)  p = 1 and f is non-decreasing, ii)  p is integer, p ≤ n−1 and f = rα with α ≥ 0, the map minimizes Ep,f among the maps in W1,p(Bn, 핊n−1) which coincide with on ∂ Bn. We also study the case where f(r) = rα with −n+2 < α < 0 and prove that does not minimize Ep,f for α close to −n+2 and when n ≥ 6, for α close to 4−n. Mathematics Subject Classification (2000) 58E20; 53C43      15. Approximation by Reciprocals of Polynomials with Positive Coefficients in L p Spaces   总被引：5，自引：0，他引：5   Y. Zhau  S.P. Zhou 《Acta Mathematica Hungarica》2001,92(3):205-217 We prove that, if f(x) L p [0,1], 1 < p < , f(x) 0, x [0,1], f 0, then there is a polynomial p(x) + n such that f - 1/p LP C(p)(f,n -1/2) LP where + n indicates the set of all polynomials of degree n with positive coeficients (see the definition (1) in the text).      16. <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript> approximation capability of RBF neural networks      Dong Nan  Wei Wu  Jin Ling Long  Yu Mei Ma  Lin Jun Sun 《数学学报(英文版)》2008,24(9):1533-1540 L p approximation capability of radial basis function (RBF) neural networks is investigated. If g: R +1 → R 1 and ∈ L loc p (R n ) with 1 ≤ p < ∞, then the RBF neural networks with g as the activation function can approximate any given function in L p (K) with any accuracy for any compact set K in R n , if and only if g(x) is not an even polynomial. Partly supported by the National Natural Science Foundation of China (10471017)      17. On commutative twisted group rings      Todor Zh. Mollov  Nako A. Nachev 《Czechoslovak Mathematical Journal》2005,55(2):371-392       18. On sylowizers in finite <Emphasis Type="Italic">p</Emphasis>-soluble groups      M. J. Iranzo  Julio P. Lafuente  F. Pérez-Monasor 《Ricerche di matematica》2007,56(2):189-194 In general, given a finite group G, a prime p and a p-subgroup R of G, the sylowizers of R in G are not conjugate. In this paper we afford some conditions to achieve the conjugation of the sylowizers of R in a p-soluble group G, among others 1.  p = 2 and the Sylow 2-subgroups of G are dihedral or quaternion. 2.  The Sylow p-subgroups of G have order at most p 3. 3.  p is odd, R is abelian and every element of order p in C G (R) lies in R. This research has been supported by Grants: MTM2004-06067-C02-01 and MTM 2004-08219-C02-01, MEC (Spain) and FEDER (European Union).      19. Commutative Group Algebras of Cardinality {\cal N} 1      Peter Danchev 《Southeast Asian Bulletin of Mathematics》2002,25(4):589-598 We let FG be the group algebra of an abelian group G over a field F with characteristic p. Also, we define Gp and S(FG) as the groups of all p-primary normed elements in G and FG, respectively. We prove that if Gp is Hausdorff and both F and G have cardinalities not exceeding 1, then S(FG)/Gp is a direct sum of cyclics. Thus Gp is a direct factor of S(FG), and in particular G is a direct factor of the group of all normalized units V(FG), provided that the torsion part of G is a p-group. This answers a question posed by us in Hokkaido Math. J. (2000). Moreover we establish that if G is p-splitting, then any F-isomorphism of the group algebras FG and FH implies that H is p-splitting. We also show that if G is of power 1 whose p-component Gp is a direct sum of torsion-complete groups and F has power p, then the F-isomorphism of FG and FH for any group H yields an isomorphism between Gp and Hp. In particular, when G is of power 1 and is p-mixed of torsion-free rank 1 whose Gp is torsion-complete, we have G H. If F is in power p and G, with cardinality 1, is a direct sum of p-local algebraically compact groups such that FG FH as F-algebras for some group H, then G H. These statements extend results due to Beers-Richman-Walker (1983), and also partially solve a well-known question raised by May in 1979.      20. Sub-Laplacians of Holomorphic L-type on Rank One AN-Groups and Related Solvable Groups      J. Ludwig  D. Müller 《Journal of Functional Analysis》2000,170(2):2767 Consider a right-invariant sub-Laplacian L on an exponential solvable Lie group G, endowed with a left-invariant Haar measure. Depending on the structure of G and possibly also that of L, L may admit differentiable Lp-functional calculi, or may be of holomorphic Lp-type for a given p≠2, as recent studies of specific classes of groups G and sub-Laplacians L have revealed. By “holomorphic Lp-type” we mean that every Lp-spectral multiplier for L is necessarily holomorphic in a complex neighborhood of some point in the L2-spectrum of L. This can only arise if the group algebra L1(G) is non-symmetric. In this article we prove that, for large classes of exponential groups, including all rank one AN-groups, a certain Lie algebraic condition, which characterizes the non-symmetry of L1(G) [37], also suffices for L to be of holomorphic L1-type. Moreover, if this condition, which was first introduced by J. Boidol [6] in a different context, holds for generic points in the dual * of the Lie algebra of G, then L is of holomorphic Lp-type for every p≠2. Besides the non-symmetry of L1(G), also the closedness of coadjoint orbits plays a crucial role. We also discuss an example of a higher rank AN-group. This example and our results in the rank one case suggest that sub-Laplacians on exponential Lie groups may be of holomorphic L1-type if and only if there exists a closed coadjoint orbit Ω * such that the points of Ω satisfy Boidol's condition. In the course of the proof of our main results, whose principal strategy is similar as in [8], we develop various tools which may be of independent interest and largely apply to more general Lie groups. Some of them are certainly known as “folklore” results. For instance, we study subelliptic estimates on representation spaces, the relation between spectral multipliers and unitary representations, and develop some “holomorphic” and “continuous” perturbation theory for images of sub-Laplacians under “smoothly varying” families of irreducible unitary representations.

Vector subdivision schemes in (Lp(?s))r(1?p?∞) spaces

The purpose of this paper is to investigate the refinement equations of the form

The minimality of the map \frac{x}{\|{x}\|} for weighted energy

In this paper, we investigate the minimality of the map from the Euclidean unit ball Bⁿ to its boundary 핊ⁿ⁻¹ for weighted energy functionals of the type E_p,f = ∫_Bⁿ f(r)‖∇ u‖^p dx, where f is a non-negative function. We prove that in each of the two following cases:

Approximation by Reciprocals of Polynomials with Positive Coefficients in L p Spaces

We prove that, if f(x) L ^p _[0,1], 1 < p < , f(x) 0, x [0,1], f 0, then there is a polynomial p(x) ⁺ _n such that f - 1/p _LP C(p)(f,n ^-1/2) _LP where ⁺ _n indicates the set of all polynomials of degree n with positive coeficients (see the definition (1) in the text).     

<Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript> approximation capability of RBF neural networks

L ^p approximation capability of radial basis function (RBF) neural networks is investigated. If g: R ₊¹ → R ¹ and ∈ L _loc ^p (R ⁿ ) with 1 ≤ p < ∞, then the RBF neural networks with g as the activation function can approximate any given function in L ^p (K) with any accuracy for any compact set K in R ⁿ , if and only if g(x) is not an even polynomial. Partly supported by the National Natural Science Foundation of China (10471017)     

On sylowizers in finite <Emphasis Type="Italic">p</Emphasis>-soluble groups

In general, given a finite group G, a prime p and a p-subgroup R of G, the sylowizers of R in G are not conjugate. In this paper we afford some conditions to achieve the conjugation of the sylowizers of R in a p-soluble group G, among others

Commutative Group Algebras of Cardinality {\cal N} 1

We let FG be the group algebra of an abelian group G over a field F with characteristic p. Also, we define G_p and S(FG) as the groups of all p-primary normed elements in G and FG, respectively. We prove that if G_p is Hausdorff and both F and G have cardinalities not exceeding ₁, then S(FG)/G_p is a direct sum of cyclics. Thus G_p is a direct factor of S(FG), and in particular G is a direct factor of the group of all normalized units V(FG), provided that the torsion part of G is a p-group. This answers a question posed by us in Hokkaido Math. J. (2000). Moreover we establish that if G is p-splitting, then any F-isomorphism of the group algebras FG and FH implies that H is p-splitting. We also show that if G is of power ₁ whose p-component G_p is a direct sum of torsion-complete groups and F has power p, then the F-isomorphism of FG and FH for any group H yields an isomorphism between G_p and H_p. In particular, when G is of power ₁ and is p-mixed of torsion-free rank 1 whose G_p is torsion-complete, we have G H. If F is in power p and G, with cardinality ₁, is a direct sum of p-local algebraically compact groups such that FG FH as F-algebras for some group H, then G H. These statements extend results due to Beers-Richman-Walker (1983), and also partially solve a well-known question raised by May in 1979.     

Sub-Laplacians of Holomorphic L-type on Rank One AN-Groups and Related Solvable Groups

Consider a right-invariant sub-Laplacian L on an exponential solvable Lie group G, endowed with a left-invariant Haar measure. Depending on the structure of G and possibly also that of L, L may admit differentiable L^p-functional calculi, or may be of holomorphic L^p-type for a given p≠2, as recent studies of specific classes of groups G and sub-Laplacians L have revealed. By “holomorphic L^p-type” we mean that every L^p-spectral multiplier for L is necessarily holomorphic in a complex neighborhood of some point in the L²-spectrum of L. This can only arise if the group algebra L¹(G) is non-symmetric. In this article we prove that, for large classes of exponential groups, including all rank one AN-groups, a certain Lie algebraic condition, which characterizes the non-symmetry of L¹(G) [37], also suffices for L to be of holomorphic L¹-type. Moreover, if this condition, which was first introduced by J. Boidol [6] in a different context, holds for generic points in the dual * of the Lie algebra of G, then L is of holomorphic L^p-type for every p≠2. Besides the non-symmetry of L¹(G), also the closedness of coadjoint orbits plays a crucial role. We also discuss an example of a higher rank AN-group. This example and our results in the rank one case suggest that sub-Laplacians on exponential Lie groups may be of holomorphic L¹-type if and only if there exists a closed coadjoint orbit Ω * such that the points of Ω satisfy Boidol's condition. In the course of the proof of our main results, whose principal strategy is similar as in [8], we develop various tools which may be of independent interest and largely apply to more general Lie groups. Some of them are certainly known as “folklore” results. For instance, we study subelliptic estimates on representation spaces, the relation between spectral multipliers and unitary representations, and develop some “holomorphic” and “continuous” perturbation theory for images of sub-Laplacians under “smoothly varying” families of irreducible unitary representations.