We consider the problem of testing whether a Boolean function has Fourier degree ≤ k or it is ǫ-far from any Boolean function with Fourier degree ≤ k, we improve the known lower bound of Ω(k) [4, 6], to Ω(k/ √ ǫ). The lower bound uses the recently discovered connections between property testing and communication complexity by Blais et. al. [4]

Definition 2. [1,2] Let x = (x1, ..., xn) and y = (y1, ..., yn) Î R . (i) x is said to be majorized by y (in symbols x ≺ y) if ∑k i=1 x[i] ≤ ∑k i=1 y[i] for k = 1, 2,..., n 1 and ∑n i=1 xi = ∑n i=1 yi , where x [1] ≥ · · · ≥ x[n] and y[1] ≥ · · · ≥ y[n] are rearrangements of x and y in a descending order. (ii) Let Ω ⊂ R, : Ω ® R is said to be a Schur-convex function on Ω if x ≺ y on Ω implies (...

We give a proof of the isoperimetric inequality for quermassintegrals of non-convex starshaped domains, using a result of Gerhardt [6] and Urbas [13] on an expanding geometric curvature flow. The Alexandrov-Fenchel inequalities [1, 2] for the quermassintegrals of convex domains are fundamental in classical geometry. For a bounded domain Ω ⊂ Rn+1, we denote M = ∂Ω the boundary of Ω. We will assu...

Some arguments are given, related to the apparent universality with which Bloch's famous T 3/2-law is not only found in d=3-dimensional ferromagnetic systems, but also in lower dimensions. It is argued that • one should not simply apply the usual isotropic dispersion relation ω(k) = D · k 2 known to almost everyone but only valid for circular precession of the spins (i.e. where only the exchang...

A compactly supported radially symmetric function Φ : R → R is said to have Sobolev regularity k if there exist constants B ≥ A > 0 such that the Fourier transform of Φ satisfies A(1 + ‖ω‖) ≤ b Φ(ω) ≤ B(1 + ‖ω‖), ω ∈ R. Such functions are useful in radial basis function methods because the resulting native space will correspond to the Sobolev space W k 2 (R). For even dimensions d and integers ...

A vertex cover of a hypergraph is set vertices which intersects each hyperedge. possesses property C(k,ρ) iff |⋂E′|<ρ for k element E′ hyperedges. Komjáth proved that every uniform possessing C(2,r) some r∈ω has minimal cover. In this paper we will relax the assumption uniformity to an cardinalities hyperedges “small” infinite cardinals, e.g. it countable, or does not contain uncountably many l...

Let ω(n) denote the number of distinct prime factors of n. Then for any given K ≥ 2, small ǫ > 0 and sufficiently large (only depending on K and ǫ) x, there exist at least x integers n ∈ [x, (1+K)x] such that ω(nj±ak) ≥ (log log log x) 1 3 −ǫ for all 2 ≤ a ≤ K, 1 ≤ j, k ≤ K and 0 ≤ h ≤ K log x.

We show lower bounds in the multi-party quantum communication complexity model. In this model, there are t parties where the ith party has input Xi ⊆ [n]. These parties communicate with each other by transmitting qubits to determine with high probability the value of some function F of their combined input (X1, . . . , Xt). We consider the class of boolean valued functions whose value depends o...

The purpose of this study is to develop a system analyzing cell activity by the dielectrophoresis method. Our previous studies revealed a correlation between the growth activity and dielectric property (Re[K(ω)]) of mouse hybridoma 3-2H3 cells using dielectrophoretic levitation. Furthermore, it was clarified that the differentiation activity of many stem cells could be evaluated by the Re[K(ω)]...

An approach to the Schottky problem of specification of periods of holomorphic differentials on Riemann surfaces (or, equivalently, specification of Jacobians among all principaly polarized Abelian varieties) based on the theory of Kadomtsev Petviashvili equation, is discussed. Introduction. Dispersion relations for linear and nonlinear waves. One of the first exercises in a course of PDE is in...