Testing low‐degree polynomials over prime fields

We present an efficient randomized algorithm to test if a given function f : ?? → ??_p (where p is a prime) is a low‐degree polynomial. This gives a local test for Generalized Reed‐Muller codes over prime fields. For a given integer t and a given real ε > 0, the algorithm queries f at O( ) points to determine whether f can be described by a polynomial of degree at most t. If f is indeed a polynomial of degree at most t, our algorithm always accepts, and if f has a relative distance at least ε from every degree t polynomial, then our algorithm rejects f with probability at least . Our result is almost optimal since any such algorithm must query f on at least points. © 2009 Wiley Periodicals, Inc. Random Struct. Alg., 2009     

Extremes of Shepp statistics for Gaussian random walk

Let (ξ _i , i ≥ 1) be a sequence of independent standard normal random variables and let be the corresponding random walk. We study the renormalized Shepp statistic and determine asymptotic expressions for when u,N and T→ ∞ in a synchronized way. There are three types of relations between u and N that give different asymptotic behavior. For these three cases we establish the limiting Gumbel distribution of when T,N→ ∞ and present corresponding normalization sequences.      

Determination of trace impurities in high-purity zirconium dioxide by inductively coupled plasma atomic emission spectrometry using microwave-assisted digestion and wavelet transform-based correction procedure

This paper describes a rapid, accurate and precise method for the determination of trace Fe, Hf, Mn, Na, Si and Ti in high-purity zirconium dioxide (ZrO₂) powders by inductively coupled plasma atomic emission spectrometry (ICP-AES). The samples were dissolved by a microwave-assisted digestion system. Four different digestion programs with various reagents were tested. It was found that using a mixture of sulfuric acid (H₂SO₄) and ammonium sulfate ((NH₄)₂SO₄), the total sample dissolution time was 30 min, much shorter than that required for conventional digestion in an opening system. The determination of almost all of the target analytes suffered from spectral interferences, since Zr shows a line-rich atomic emission spectrometry. The wavelet transform (WT), a recently developed mathematical technique was applied to the correction of spectral interference, and more accurate and precise results were obtained, compared with traditional off-peak background correction procedure. Experimental work revealed that a high Zr concentration would result in a significant decrease in peak height of the analyte lines, which was corrected by standard addition method. The performance of the developed method was evaluated by using synthetic samples. The recoveries were in the range of 87-112% and relative standard deviation was within 1.1-3.4%. The detection limits (3σ) for Fe, Hf, Mn, Na, Si and Ti were found to be 1.2, 13.3, 1.0, 4.5, 5.8 and 2.0 μg g⁻¹, respectively. The results showed that with the microwave-assisted digestion and the WT correction, the detection limits have improved by a factor of about 5 for Fe, 4 for Mn and Ti, 3 for Si, and 2 for Hf and Na, respectively, in comparison with conventional open-system digestion and off-peak correction. The proposed technique was applied to the analysis of trace elements above-mentioned in three types of ZrO₂ powders.     

Asymptotic expression of the linear discrete best -approximation

Let h_p, 1<p<∞, be the best ℓ_p-approximation of the element from a proper affine subspace K of , hK, and let denote the strict uniform approximation of h from K. We prove that there are a vector and a real number a, 0a1, such that

for all p>1, where with γ_p=o(a^p/p).     

Classification of blow-up with nonlinear diffusion and localized reaction

We study the behaviour of nonnegative solutions of the reaction-diffusion equation

     

On (k,\varepsilon)-saddle submanifolds of Riemannian manifolds

The paper continues our (in collaboration with A. Borisenko [J. Differential Geom. Appl. 20 p., to appear]) discovery of the new classes of $(k,\varepsilon)The paper continues our (in collaboration with A. Borisenko [J. Differential Geom. Appl. 20 p., to appear]) discovery of the new classes of -saddle, -parabolic, and -convex submanifolds ( ). These are defined in terms of the eigenvalues of the 2nd fundamental forms of each unit normal of the submanifold, extending the notion of k-saddle, k-parabolic, k-convex submanifolds ( ). It follows that the definition of -saddle submanifolds is equivalent to the existence of -asymptotic subspaces in the tangent space. We prove non-embedding theorems of -saddle submanifolds, theorems about 1-connectedness and homology groups of these submanifolds in Riemannian spaces of positive (sectional or qth Ricci) curvature, in particular, spherical and projective spaces. We apply these results to submanifolds with ‘small’ normal curvature, , and for submanifolds with extrinsic curvature (resp., non-positive) and small codimension, and include some illustrative examples. The results of the paper generalize theorems about totally geodesic, minimal and k-saddle submanifolds by Frankel; Borisenko, Rabelo and Tenenblat; Kenmotsu and Xia; Mendon?a and Zhou.      

An elementary derivation of the asymptotics of partition functions

Let S_a,b = {an+b:n ≥ 0 } where n is an integer. Let P_a,b(n) denote the number of partitions of n into elements of S_a,b. In particular, we have the generating function,

On the Hénon equation: asymptotic profile of ground states, I

This paper is concerned with the qualitative property of the ground state solutions for the Hénon equation. By studying a limiting equation on the upper half space , we investigate the asymptotic energy and the asymptotic profile of the ground states for the Hénon equation. The limiting problem is related to a weighted Sobolev type inequality which we establish in this paper.     

Dynamics of a stochastic Lotka-Volterra model perturbed by white noise

This paper continues the study of Mao et al. investigating two aspects of the equation

     

Vanishing solutions of anisotropic parabolic equations with variable nonlinearity

We study the property of finite time vanishing of solutions of the homogeneous Dirichlet problem for the anisotropic parabolic equations