On Sums of Two Coprime k-th Powers(II)

For fixed k ≥ 3, let E_k(x) denote the error term of the sum ?_{n ￡ x}r_k(n)\sum_{n\le x}\rho_k(n) , where r_k(n) = ?_{n=|m|^k+|l|^k, g.c.d.(m,l)=1}\rho_k(n) = \sum_{n=|m|^k+|l|^k, g.c.d.(m,l)=1} 1. It is proved that if the Riemann hypothesis is true, then E₃(x) << x^331/1254+eE_3(x)\ll x^{331/1254+\varepsilon} , E₄(x) << x^37/184+eE_4(x)\ll x^{37/184+\varepsilon} . A short interval result is also obtained.     

Approximation by the Bernstein–Durrmeyer Operator on a Simplex

For the Jacobi-type Bernstein–Durrmeyer operator M _n,κ on the simplex T ^d of ℝ ^d , we proved that for f∈L ^p (W _κ ;T ^d ) with 1<p<∞,

Comonotone approximation of twice differentiable periodic functions

In the case where a 2π-periodic function f is twice continuously differentiable on the real axis ℝ and changes its monotonicity at different fixed points y _i ∈ [− π, π), i = 1,…, 2s, s ∈ ℕ (i.e., on ℝ, there exists a set Y := {y _i } _i∈ℤ of points y _i = y _i+2s + 2π such that the function f does not decrease on [y _i , y _i−1] if i is odd and does not increase if i is even), for any natural k and n, n ≥ N(Y, k) = const, we construct a trigonometric polynomial T _n of order ≤n that changes its monotonicity at the same points y _i ∈ Y as f and is such that

On the log-concavity of Hilbert series of Veronese subrings and Ehrhart series

For every positive integer n, consider the linear operator U _n on polynomials of degree at most d with integer coefficients defined as follows: if we write ${\frac{h(t)}{(1 - t)^{d + 1}}=\sum_{m \geq 0} g(m) \, t^{m}}For every positive integer n, consider the linear operator U _n on polynomials of degree at most d with integer coefficients defined as follows: if we write \frach(t)(1 - t)^{d + 1}=?_{m 3 0} g(m)  t^m{\frac{h(t)}{(1 - t)^{d + 1}}=\sum_{m \geq 0} g(m) \, t^{m}} , for some polynomial g(m) with rational coefficients, then \fracU_nh(t)(1- t)^d+1 = ?_{m 3 0}g(nm)  t^m{\frac{{\rm{U}}_{n}h(t)}{(1- t)^{d+1}} = \sum_{m \geq 0}g(nm) \, t^{m}} . We show that there exists a positive integer n _d , depending only on d, such that if h(t) is a polynomial of degree at most d with nonnegative integer coefficients and h(0) = 1, then for n ≥ n _d , U _n h(t) has simple, real, negative roots and positive, strictly log concave and strictly unimodal coefficients. Applications are given to Ehrhart δ-polynomials and unimodular triangulations of dilations of lattice polytopes, as well as Hilbert series of Veronese subrings of Cohen–Macauley graded rings.     

Lines, Circles, Planes and Spheres

Let S be a set of n points in ℝ³, no three collinear and not all coplanar. If at most n−k are coplanar and n is sufficiently large, the total number of planes determined is at least 1+k\binomn-k2-\binomk2(\fracn-k2)1+k\binom{n-k}{2}-\binom{k}{2}(\frac{n-k}{2}). For similar conditions and sufficiently large n, (inspired by the work of P.D.T.A. Elliott in Acta Math. Sci. Hung. 18:181–188, 1967) we also show that the number of spheres determined by n points is at least 1+\binomn-13-t₃^orchard(n-1)1+\binom{n-1}{3}-t_{3}^{\mathrm{orchard}}(n-1), and this bound is best possible under its hypothesis. (By t₃^orchard(n)t_{3}^{\mathrm{orchard}}(n), we are denoting the maximum number of three-point lines attainable by a configuration of n points, no four collinear, in the plane, i.e., the classic Orchard Problem.) New lower bounds are also given for both lines and circles.     

Type II Hermite�CPad�� Approximations of Generalized Hypergeometric Series

We shall present short proofs for type II (simultaneous) Hermite–Padé approximations of the generalized hypergeometric and q-hypergeometric series

On numerical experiments with symmetric semigroups generated by three elements and their generalization

We give a simple explanation of numerical experiments of V. Arnold with two sequences of symmetric numerical semigroups, S(4,6+4k,87−4k) and S(9,3+9k,85−9k) generated by three elements. We present a generalization of these sequences by numerical semigroups S(r₁²,r₁r₂+r₁²k,r₃-r₁²k)\mathsf{S}(r_{1}^{2},r_{1}r_{2}+r_{1}^{2}k,r_{3}-r_{1}^{2}k), k∈ℤ, r ₁,r ₂,r ₃∈ℤ⁺, r ₁≥2 and gcd(r ₁,r ₂)=gcd(r ₁,r ₃)=1, and calculate their universal Frobenius number Φ(r ₁,r ₂,r ₃) for the wide range of k providing semigroups be symmetric. We show that this type of semigroups admit also nonsymmetric representatives. We describe the reduction of the minimal generating sets of these semigroups up to {r₁²,r₃-r₁²k}\{r_{1}^{2},r_{3}-r_{1}^{2}k\} for sporadic values of k and find these values by solving the quadratic Diophantine equation.     

Prime radicals of constants of algebraic derivations

Let R be a prime ring with extended centroid F and let δ be an F-algebraic continuous derivation of R with the associated inner derivation ad(b). Factorize the minimal polynomial μ(λ) of b over F into distinct irreducible factors m(l)=?_ip_i(l)^n_i{\mu(\lambda)=\prod_i\pi_i(\lambda)^{n_i}} . Set ℓ to be the maximum of n _i . Let R^(d)=^def.{x ? R | d(x)=0}{R^{(\delta)}{\mathop{=}\limits^{{\rm def.}}}\{x\in R\mid \delta(x)=0\}} be the subring of constants of δ on R. Denote the prime radical of a ring A by P(A){{\mathcal{P}}(A)} . It is shown among other things that

Wavelet expansions of fractal measures

Let μ be a measure on ℝⁿ that satisfies the estimate μ(B _r(x))≤cr ^α for allx ∈ ℝⁿ and allr ≤ 1 (B _r(x) denotes the ball of radius r centered atx. Let ϕ _j,k ^(ɛ) (x)=2 ^nj2ϕ^(ɛ)(2 ^j x-k) be a wavelet basis forj ∈ ℤ, κ ∈ ℤⁿ, and ∈ ∈E, a finite set, and letP _j (T)=Σ_ɛ,k <T,ϕ _j,k ^(ɛ) >ϕ _j,k ^(ɛ) denote the associated projection operators at levelj (T is a suitable measure or distribution). Iff ∈Ls ^p(dμ) for 1 ≤p ≤ ∞, we show thatP _j(f dμ) ∈ L^p(dx) and ||P _j (fdμ)||L _p(dx)≤c2 ^{j((n-α)/p′))}||f||L _p(dμ) for allj ≥ 0. We also obtain estimates for the limsup and liminf of ||P _j (fdμ)||L _p(dx) under more restrictive hypotheses. Communicated by Guido Weiss     

Laws of the iterated logarithm for high-dimensional wiener sausage

Let {β(s), s ≥ 0} be the standard Brownian motion in ℝ ^d with d ≥ 4 and let |W _r (t)| be the volume of the Wiener sausage associated with {β(s), s ≥ 0} observed until time t. From the central limit theorem of Wiener sausage, we know that when d ≥ 4 the limit distribution is normal. In this paper, we study the laws of the iterated logarithm for | W_r (t) | - \mathbbE| W_r (t) |\left| {W_r (t)} \right| - \mathbb{E}\left| {W_r (t)} \right| in this case.     

Uniform estimates for order statistics and Orlicz functions

We establish uniform estimates for order statistics: Given a sequence of independent identically distributed random variables ξ ₁, … , ξ _n and a vector of scalars x = (x ₁, … , x _n ), and 1 ≤ k ≤ n, we provide estimates for \mathbb E   k-min_{1 ￡ i ￡ n} |x_ix_i|{\mathbb E \, \, k-{\rm min}_{1\leq i\leq n} |x_{i}\xi _{i}|} and \mathbb E k-max_{1 ￡ i ￡ n}|x_ix_i|{\mathbb E\,k-{\rm max}_{1\leq i\leq n}|x_{i}\xi_{i}|} in terms of the values k and the Orlicz norm ||y_x||_M{\|y_x\|_M} of the vector y _x  = (1/x ₁, … , 1/x _n ). Here M(t) is the appropriate Orlicz function associated with the distribution function of the random variable |ξ ₁|, G(t) = \mathbb P ({ |x₁| ￡ t}){G(t) =\mathbb P \left(\left\{ |\xi_1| \leq t\right\}\right)}. For example, if ξ ₁ is the standard N(0, 1) Gaussian random variable, then G(t) = ?{\tfrac2p}ò₀^t e^-\fracs22ds {G(t)= \sqrt{\tfrac{2}{\pi}}\int_{0}^t e^{-\frac{s^{2}}{2}}ds }  and M(s)=?{\tfrac2p}ò₀^se^-\frac12t2dt{M(s)=\sqrt{\tfrac{2}{\pi}}\int_{0}^{s}e^{-\frac{1}{2t^{2}}}dt}. We would like to emphasize that our estimates do not depend on the length n of the sequence.     

Asymptotic Upper Bounds for Ramsey Functions

 We show that for any graph G with N vertices and average degree d, if the average degree of any neighborhood induced subgraph is at most a, then the independence number of G is at least Nf _{a +1}(d), where f _{a +1}(d)=∫₀ ¹(((1−t)^{1/( a +1)})/(a+1+(d−a−1)t))dt. Based on this result, we prove that for any fixed k and l, there holds r(K _k + _l ,K _n )≤ (l+o(1))n ^k /(logn) ^{k −1}. In particular, r(K _k , K _n )≤(1+o(1))n ^{k −1}/(log n) ^{k −2}. Received: May 11, 1998 Final version received: March 24, 1999     

Spectral Conditions for Strong Local Nondeterminism and Exact Hausdorff Measure of Ranges of Gaussian Random Fields

Let X={X(t),t∈ℝ ^N } be a Gaussian random field with values in ℝ ^d defined by

A Generalized Radon Transform on the Plane

A new generalized Radon transform R _α, β on the plane for functions even in each variable is defined which has natural connections with the bivariate Hankel transform, the generalized biaxially symmetric potential operator Δ _α, β , and the Jacobi polynomials P_k^(b, a)(t)P_{k}^{(\beta,\,\alpha)}(t). The transform R _α, β and its dual R_a, b^*R_{\alpha,\,\beta}^{\ast} are studied in a systematic way, and in particular, the generalized Fuglede formula and some inversion formulas for R _α, β for functions in L_a, b^p(\mathbbR²₊)L_{\alpha,\,\beta}^{p}(\mathbb{R}^{2}_{+}) are obtained in terms of the bivariate Hankel–Riesz potential. Moreover, the transform R _α, β is used to represent the solutions of the partial differential equations Lu:=?_j=1^ma_jD_a, b^ju=fLu:=\sum_{j=1}^{m}a_{j}\Delta_{\alpha,\,\beta}^{j}u=f with constant coefficients a _j and the Cauchy problem for the generalized wave equation associated with the operator Δ _α, β . Another application is that, by an invariant property of R _α, β , a new product formula for the Jacobi polynomials of the type P_k^(b, a)(s)C_2k^a+b+1(t)=còòP_k^(b, a)P_{k}^{(\beta,\,\alpha)}(s)C_{2k}^{\alpha+\beta+1}(t)=c\int\!\!\int P_{k}^{(\beta,\,\alpha)} is obtained.     

An algorithmic approach to Ramanujan’s congruences

In this paper we present an algorithm that takes as input a generating function of the form $\prod_{\delta|M}\prod_{n=1}^{\infty}(1-q^{\delta n})^{r_{\delta}}=\sum_{n=0}^{\infty}a(n)q^{n}In this paper we present an algorithm that takes as input a generating function of the form ?_d|M?_n=1^￥(1-q^dn)^r_d=?_n=0^￥a(n)qⁿ\prod_{\delta|M}\prod_{n=1}^{\infty}(1-q^{\delta n})^{r_{\delta}}=\sum_{n=0}^{\infty}a(n)q^{n} and three positive integers m,t,p, and which returns true if a(mn+t) o 0 mod p,n 3 0a(mn+t)\equiv0\pmod{p},n\geq0, or false otherwise. Our method builds on work by Rademacher (Trans. Am. Math. Soc. 51(3):609–636, 1942), Kolberg (Math. Scand. 5:77–92, 1957), Sturm (Lecture Notes in Mathematics, pp. 275–280, Springer, Berlin/Heidelberg, 1987), Eichhorn and Ono (Proceedings for a Conference in Honor of Heini Halberstam, pp. 309–321, 1996).     

Possible limit laws for entrance times of an ergodic aperiodic dynamical system

LetG denote the set of decreasingG: ℝ→ℝ withGэ1 on ]−∞,0], and ƒ ₀ ^∞ G(t)dt⩽1. LetX be a compact metric space, andT: X→X a continuous map. Let μ denone aT-invariant ergodic probability measure onX, and assume (X, T, μ) to be aperiodic. LetU⊂X be such that μ(U)>0. Let τ _U (x)=inf{k⩾1:T ^k xεU}, and defineG _U (t)=1/u(U)u({xεU:u(U)τU(x)>t),tεℝ We prove that for μ-a.e.x∈X, there exists a sequence (U _n ) _n≥1 of neighbourhoods ofx such that {x}=∩ _n U _n , and for anyG ∈G, there exists a subsequence (n _k ) _k≥1 withG _{U n k} ↑U weakly. We also construct a uniquely ergodic Toeplitz flowO(x ^∞,S, μ), the orbit closure of a Toeplitz sequencex ^∞, such that the above conclusion still holds, with moreover the requirement that eachU _n be a cylinder set. In memory of Anzelm Iwanik     

Asymptotic behaviour for a nonlocal diffusion equation on a lattice

In this paper we study the asymptotic behaviour as t → ∞ of solutions to a nonlocal diffusion problem on a lattice, namely, u^￠_n(t) = ?_{j ? \mathbbZ^d} J_n-ju_j(t)-u_n(t)u^{\prime}_{n}(t) = \sum_{{j\in}{{{\mathbb{Z}}}^{d}}} J_{n-j}u_{j}(t)-u_{n}(t) with t ≥ 0 and n ? \mathbbZ^dn \in {\mathbb{Z}}^{d}. We assume that J is nonnegative and verifies ?_{n ? \mathbbZ^d}J_n = 1\sum_{{n \in {\mathbb{Z}}}^{d}}J_{n}= 1. We find that solutions decay to zero as t → ∞ and prove an optimal decay rate using, as our main tool, the discrete Fourier transform.     

Compact embeddings of besov spaces involving only slowly varying smoothness

We characterize compact embeddings of Besov spaces B _p,r ^0,b (ℝ ⁿ ) involving the zero classical smoothness and a slowly varying smoothness b into Lorentz-Karamata spaces L<sub>p,q;[`(b)]</sub> {L_{p,q;\overline b }}(Ω), where is a bounded domain in ℝ <sup>n</sup> and [`(b)]\overline b is another slowly varying function.     

Positive Cubature Formulas and Marcinkiewicz–Zygmund Inequalities on Spherical Caps

Let Π _n ^d denote the space of all spherical polynomials of degree at most n on the unit sphere $\mathbb{S}^{d}Let Π _n ^d denote the space of all spherical polynomials of degree at most n on the unit sphere \mathbbS^d\mathbb{S}^{d} of ℝ ^d+1, and let d(x,y) denote the geodesic distance arccos x⋅y between x,y ? \mathbbS^dx,y\in\mathbb{S}^{d} . Given a spherical cap

B(e,a)={x ? \mathbbSd:d(x,e) ￡ a}    (e ? \mathbbSd, a ? (0,p) is bounded awayfrom p),B(e,\alpha)=\big\{x\in\mathbb{S}^{d}:d(x,e)\leq\alpha\big\}\quad \bigl(e\in\mathbb{S}^{d},\ \alpha\in(0,\pi)\ \mbox{is bounded awayfrom}\ \pi\bigr),      20. Ramsey Numbers of Some Bipartite Graphs Versus Complete Graphs      Tao Jiang  Michael Salerno 《Graphs and Combinatorics》2011,27(1):121-128 The Ramsey number r(H, K n ) is the smallest positive integer N such that every graph of order N contains either a copy of H or an independent set of size n. The Turán number ex(m, H) is the maximum number of edges in a graph of order m not containing a copy of H. We prove the following two results: (1) Let H be a graph obtained from a tree F of order t by adding a new vertex w and joining w to each vertex of F by a path of length k such that any two of these paths share only w. Then r(H,Kn) ￡ ck,t [(n1+1/k)/(ln1/k n)]{r(H,K_n)\leq c_{k,t}\, {n^{1+1/k}\over \ln^{1/k} n}} , where c k,t is a constant depending only on k and t. This generalizes some results in Li and Rousseau (J Graph Theory 23:413–420, 1996), Li and Zang (J Combin Optim 7:353–359, 2003), and Sudakov (Electron J Combin 9, N1, 4 pp, 2002). (2) Let H be a bipartite graph with ex(m, H) = O(m γ ), where 1 < γ < 2. Then r(H,Kn) ￡ cH ([(n)/(lnn)])1/(2-g){r(H,K_n)\leq c_H ({n\over \ln n})^{1/(2-\gamma)}}, where c H is a constant depending only on H. This generalizes a result in Caro et al. (Discrete Math 220:51–56, 2000).

Ramsey Numbers of Some Bipartite Graphs Versus Complete Graphs

The Ramsey number r(H, K _n ) is the smallest positive integer N such that every graph of order N contains either a copy of H or an independent set of size n. The Turán number ex(m, H) is the maximum number of edges in a graph of order m not containing a copy of H. We prove the following two results: (1) Let H be a graph obtained from a tree F of order t by adding a new vertex w and joining w to each vertex of F by a path of length k such that any two of these paths share only w. Then r(H,K_n) ￡ c_k,t [(n^1+1/k)/(ln^1/k n)]{r(H,K_n)\leq c_{k,t}\, {n^{1+1/k}\over \ln^{1/k} n}} , where c _k,t is a constant depending only on k and t. This generalizes some results in Li and Rousseau (J Graph Theory 23:413–420, 1996), Li and Zang (J Combin Optim 7:353–359, 2003), and Sudakov (Electron J Combin 9, N1, 4 pp, 2002). (2) Let H be a bipartite graph with ex(m, H) = O(m ^γ ), where 1 < γ < 2. Then r(H,K_n) ￡ c_H ([(n)/(lnn)])^1/(2-g){r(H,K_n)\leq c_H ({n\over \ln n})^{1/(2-\gamma)}}, where c _H is a constant depending only on H. This generalizes a result in Caro et al. (Discrete Math 220:51–56, 2000).