The Łojasiewicz exponent of an analytic function at an isolated zero

Let f be a real analytic function defined in a neighborhood of 0 ? \Bbb Rⁿ 0 \in {\Bbb R}^n such that f^-1(0)={0} f^{-1}(0)=\{0\} . We describe the smallest possible exponents !, #, / for which we have the following estimates: |f(x)| 3 c|x|^a |f(x)|\geq c|x|^{\alpha} , |grad f(x)| 3 c|x|^b |{\rm grad}\,f(x)|\geq c|x|^{\beta} , |grad f(x)| 3 c|f(x)|^q |{\rm grad}\,f(x)|\geq c|f(x)|^{\theta} for x near zero with c > 0 c > 0 . We prove that a = b+1 \alpha=\beta+1, q = b/a\theta=\beta/\alpha . Moreover b = N+a/b \beta=N+a/b where $ 0 h a < b h N^{n-1} $ 0 h a < b h N^{n-1} . If f is a polynomial then |f(x)| 3 c|x|^(degf-1)n+1 |f(x)|\geq c|x|^{(\deg f-1)^n+1} in a small neighborhood of zero.     

An Erdös-Ko-Rado theorem for restricted signed sets

A restricted signed r-set is a pair （A, f）, where A lohtain in [n] = {1, 2,…, n} is an r-set and f is a map from A to [n] with f（i） ≠ i for all i ∈ A. For two restricted signed sets （A, f） and （B, g）, we define an order as （A, f） ≤ （B, g） if A C B and g｜A ： f A family .A of restricted signed sets on [n] is an intersecting antiehain if for any （A, f）, （B, g） ∈ A, they are incomparable and there exists x ∈ A ∩ B such that f（x） = g（x）. In this paper, we first give a LYM-type inequality for any intersecting antichain A of restricted signed sets, from which we then obtain ｜A｜≤ （r-1^n-1）（n-1）^r-1 if A. consists of restricted signed r-sets on [n]. Unless r = n = 3, equality holds if and only if A consists of all restricted signed r-sets （A, f） such that x0∈ A and f（x0） =ε0 for some fixed x0 ∈ [n], ε0 ∈ [n] ／ {x0}.     

Multiple recurrence and nilsequences

Aiming at a simultaneous extension of Khintchine(X,X,m,T)(X,\mathcal{X},\mu,T) and a set A ? XA\in\mathcal{X} of positive measure, the set of integers n such that A T^2nA T^knA)(A)^k+1-\mu(A{\cap} T^{n}A{\cap} T^{2n}A{\cap} \ldots{\cap} T^{kn}A)>\mu(A)^{k+1}-\epsilon is syndetic. The size of this set, surprisingly enough, depends on the length (k+1) of the arithmetic progression under consideration. In an ergodic system, for k=2 and k=3, this set is syndetic, while for kòf(x)f(Tⁿx)f(T²ⁿx)? f(T^knx)  dm(x)\int{f(x)f(T^{n}x)f(T^{2n}x){\ldots} f(T^{kn}x) \,d\mu(x)} , where k and n are positive integers and f is a bounded measurable function. We also derive combinatorial consequences of these results, for example showing that for a set of integers E with upper Banach density d^*(E)>0 and for all {n ? \mathbbZ\colon d^*(E?(E+n)?(E+2n)?(E+3n)) > d^*(E)⁴-e}\big\{n\in\mathbb{Z}{\colon} d^*\big(E\cap(E+n)\cap(E+2n)\cap(E+3n)\big) > d^*(E)^4-\epsilon\big\}     

Quermassintegrals of a random polytope in a convex body

Let K be a convex body in \mathbbRⁿ \mathbb{R}^n with volume |K| = 1 |K| = 1 . We choose N 3 n+1 N \geq n+1 points x₁,?, x_N x_1,\ldots, x_N independently and uniformly from K, and write C(x₁,?, x_N) C(x_1,\ldots, x_N) for their convex hull. Let f : \mathbbR⁺ ? \mathbbR⁺ f : \mathbb{R^+} \rightarrow \mathbb{R^+} be a continuous strictly increasing function and 0 ￡ i ￡ n-1 0 \leq i \leq n-1 . Then, the quantity¶¶E (K, N, f °W_i) = ò_K ?ò_K f[W_i(C(x₁, ?, x_N))]dx_N ?dx₁ E (K, N, f \circ W_{i}) = \int\limits_{K} \ldots \int\limits_{K} f[W_{i}(C(x_1, \ldots, x_N))]dx_{N} \ldots dx_1 ¶¶is minimal if K is a ball (W_i is the i-th quermassintegral of a compact convex set). If f is convex and strictly increasing and 1 ￡ i ￡ n-1 1 \leq i \leq n-1 , then the ball is the only extremal body. These two facts generalize a result of H. Groemer on moments of the volume of C(x₁,?, x_N) C(x_1,\ldots, x_N) .     

Positive solutions for mixed problems of singular fractional differential equations

We investigate the existence of positive solutions to the singular fractional boundary value problem: $^c\hspace{-1.0pt}D^{\alpha }u +f(t,u,u^{\prime },^c\hspace{-2.0pt}D^{\mu }u)=0$, u′(0) = 0, u(1) = 0, where 1 < α < 2, 0 < μ < 1, f is a L^q‐Carathéodory function, $q > \frac{1}{\alpha -1}$, and f(t, x, y, z) may be singular at the value 0 of its space variables x, y, z. Here $^c \hspace{-1.0pt}D$ stands for the Caputo fractional derivative. The results are based on combining regularization and sequential techniques with a fixed point theorem on cones.     

Smooth 2-homogeneous polynomials on Hilbert spaces

We determine the smooth points of the unit ball of the space of 2-homogeneous polynomials on a Hilbert space H. Working separately for the real and the complex cases we show that a smooth polynomial attains its norm. We deduce that the polynomial P is smooth if and only if there exists a unit vector x₀ in H such that P(x)=± á x,x₀ ñ ²+P₁(x₁)P(x)=\pm \left \langle x,x_{0}\right \rangle ^{2}+P_{1}(x_{1}) where x= á x,x₀ ñ x₀+x₁ x=\left \langle x,x_{0}\right \rangle x_{0}+x_{1} is the decomposition of x in H=span{ x₀} ?H₁H={\rm {span}}\{ x_{0}\} \oplus H_{1} and P₁ is a 2-homogeneous polynomial on H₁ of norm strictly less than 1.     

Fast Growing Solutions of Linear Differential Equations in the Unit Disc

For $n \in \mathbb{N}$ , the n-order of an analytic function f in the unit disc D is defined by $$\sigma _{{{M,n}}} (f) = {\mathop {\lim \sup }\limits_{r \to 1^{ - } } }\frac{{\log ^{ + }_{{n + 1}} M(r,f)}} {{ - \log (1 - r)}},$$ where log⁺ x  =  max{log x, 0}, log ⁺ ₁ x  =  log ⁺ x, log ⁺ _n+1 x  =  log ⁺ log ⁺ _n x, and M(r, f) is the maximum modulus of f on the circle of radius r centered at the origin. It is shown, for example, that the solutions f of the complex linear differential equation $$f^{{(k)}} + a_{{k - 1}} (z)f^{{(k - 1)}} + \cdots + a_{1} (z)f^{\prime} + a_{0} (z)f = 0,\quad \quad \quad (\dag)$$ where the coefficients are analytic in D, satisfy σ _M,n+1(f)  ≤  α if and only if σ _M,n (a _j )  ≤  α for all j  =  0, ..., k ? 1. Moreover, if q ∈{0, ..., k ? 1} is the largest index for which $\sigma _{M,n} ( a_{q}) = {\mathop {\max }\limits_{0 \leq j \leq k - 1} }{\left\{ {\sigma _{{M,n}} {\left( {a_{j} } \right)}} \right\}}$ , then there are at least k ? q linearly independent solutions f of ( $\dag$ ) such that σ _M,n+1(f) = σ _M,n (a _q ). Some refinements of these results in terms of the n-type of an analytic function in D are also given.     

Fractional Modified Dyadic Integral and Derivative on ℝ<Subscript>+</Subscript>

For functions in the Lebesgue space L(ℝ₊), a modified strong dyadic integral J _α and a modified strong dyadic derivative D ^(α) of fractional order α > 0 are introduced. For a given function f ∈ L(ℝ₊), criteria for the existence of these integrals and derivatives are obtained. A countable set of eigenfunctions for the operators J _α and D ^(α) is indicated. The formulas D ^(α)(J ^α(f)) = f and J _α(D ^(α)(f)) = f are proved for each α > 0 under the condition that . We prove that the linear operator is unbounded, where is the natural domain of J _α. A similar statement for the operator is proved. A modified dyadic derivative d ^(α)(f)(x) and a modified dyadic integral j _α(f)(x) are also defined for a function f ∈ L(ℝ₊) and a given point x ∈ ℝ₊. The formulas d ^(α)(J _α(f))(x) = f(x) and j _α(D ^(α)(f)) = f(x) are shown to be valid at each dyadic Lebesgue point x ∈ ℝ₊ of f.__________Translated from Funktsional’nyi Analiz i Ego Prilozheniya, Vol. 39, No. 2, pp. 64–70, 2005Original Russian Text Copyright © by B. I. GolubovSupported by the Russian Foundation for Basic Research (grant no. 05-01-00206).     

Bounds on the degree of APN polynomials: the case of x ?1?+?g(x)

In this paper we consider APN functions ${f:\mathcal{F}_{2^m}\to \mathcal{F}_{2^m}}$ of the form f(x) = x ^?1 + g(x) where g is any non ${\mathcal{F}_{2}}$ -affine polynomial. We prove a lower bound on the degree of the polynomial g. This bound in particular implies that such a function f is APN on at most a finite number of fields ${\mathcal{F}_{2^m}}$ . Furthermore we prove that when the degree of g is less than 7 such functions are APN only if m ?? 3 where these functions are equivalent to x ³.     

Constructing quasi-random subsets of ZN by using elliptic curves

Let ε:y2 =x3 + Ax + B be an elliptic curve defined over the finite field Zp(p ＞ 3)and G be a rational point of prime order N on ε.Define a subset of ZN,the residue class ring modulo N,as S ∶={n ∶n ∈ZN,...     

Multiple solutions for a superlinear and periodic elliptic system on \mathbbRN{\mathbb{R}^N}

This paper is concerned with the following periodic Hamiltonian elliptic system

Homogenization in the scattering problem

The scattering problem is studied, which is described by the equation (-Δ _x +q(x,x/ɛ)−E)ψ = f(x), where ψ = ψ (x,ɛ) ∈ ℂ, x ℂ ℝ ^d , ɛ > 0, E > 0, the function q(x,y) is periodic with respect to y, and the function f is compactly supported. The solution satisfying radiation conditions at infinity is considered, and its asymptotic behavior as ɛ → O is described. The asymptotic behavior of the scattering amplitude of a plane wave is also considered. It is shown that in principal order both the solution and the scattering amplitude are described by the homogenized equation with potential

The evaluation of a quartic integral via Schwinger, Schur and Bessel

We provide additional methods for the evaluation of the integral

Ramanujan’s identities and representation of integers by certain binary and quaternary quadratic forms

We revisit old conjectures of Fermat and Euler regarding the representation of integers by binary quadratic form x ²+5y ². Making use of Ramanujan’s ₁ ψ ₁ summation formula, we establish a new Lambert series identity for $\sum_{n,m=-\infty }^{\infty}q^{n^{2}+5m^{2}}We revisit old conjectures of Fermat and Euler regarding the representation of integers by binary quadratic form x ²+5y ². Making use of Ramanujan’s ₁ ψ ₁ summation formula, we establish a new Lambert series identity for ?_n,m=-￥^￥q^n2+5m2\sum_{n,m=-\infty }^{\infty}q^{n^{2}+5m^{2}} . Conjectures of Fermat and Euler are shown to follow easily from this new formula. But we do not stop there. Employing various formulas found in Ramanujan’s notebooks and using a bit of ingenuity, we obtain a collection of new Lambert series for certain infinite products associated with quadratic forms such as x ²+6y ², 2x ²+3y ², x ²+15y ², 3x ²+5y ², x ²+27y ², x ²+5(y ²+z ²+w ²), 5x ²+y ²+z ²+w ². In the process, we find many new multiplicative eta-quotients and determine their coefficients.     

Degree-uniform lower bound on the weights of polynomials with given sign function

A Boolean function f: {0, 1} ⁿ → {0, 1} is called the sign function of an integer polynomial p of degree d in n variables if it is true that f(x) = 1 if and only if p(x) > 0. In this case the polynomial p is called a threshold gate of degree d for the function f. The weight of the threshold gate is the sum of the absolute values of the coefficients of p. For any n and d ≤ D ≤ $\frac{{\varepsilon n^{1/5} }} {{\log n}} $\frac{{\varepsilon n^{1/5} }} {{\log n}} we construct a function f such that there is a threshold gate of degree d for f, but any threshold gate for f of degree at most D has weight 2^{(dn)d /D4d} 2^{(\delta n)^d /D^{4d} } , where ɛ > 0 and δ > 0 are some constants. In particular, if D is constant, then any threshold gate of degree D for our function has weight 2^{W(nd )}2^{\Omega (n^d )} . Previously, functions with these properties have been known only for d = 1 (and arbitrary D) and for D = d. For constant d our functions are computable by polynomial size DNFs. The best previous lower bound on the weights of threshold gates for such functions was 2^Ω(n). Our results can also be translated to the case of functions f: {−1, 1} ⁿ → {−1, 1}.     

On the boundedness of one recurrent sequence in a banach space

We establish necessary and sufficient conditions under which a sequence x ₀ = y ₀ , x _n+1 = Ax _n  + y _n+1 , n ≥ 0, is bounded for each bounded sequence { y_n :n \geqslant 0 } ì { x ? è_{n = 1}^￥ D( Aⁿ ) |sup_{n \geqslant 0} || Aⁿ x || < ￥ }\left\{ {y_n :n \geqslant 0} \right\} \subset \left\{ {\left. {x \in \bigcup\nolimits_{n = 1}^\infty {D\left( {A^n } \right)} } \right|\sup _{n \geqslant 0} \left\| {A^n x} \right\| < \infty } \right\}, where A is a closed operator in a complex Banach space with domain of definition D(A) .     

On the sets of branch points of mappings more general than quasiregular

It is shown that if a point x ₀ ∊ ℝ ⁿ , n ≥ 3, is an essential isolated singularity of an open discrete Q-mapping f : D → [`(\mathbb R<sup>n</sup>)] \overline {\mathbb {R}^n} , B <sub>f</sub> is the set of branch points of f in D; and a point z <sub>0</sub> ∊ [`(\mathbb Rⁿ)] \overline {\mathbb {R}^n} is an asymptotic limit of f at the point x ₀; then, for any neighborhood U containing the point x ₀; the point z ₀ ∊ [`(f( B<sub>f</sub> ?U ))] \overline {f\left( {B_f \cap U} \right)} provided that the function Q has either a finite mean oscillation at the point x <sub>0</sub> or a logarithmic singularity whose order does not exceed n − 1: Moreover, for n ≥ 2; under the indicated conditions imposed on the function Q; every point of the set [`(\mathbb Rⁿ)] \overline {\mathbb {R}^n} \ f(D) is an asymptotic limit of f at the point x ₀. For n ≥ 3, the following relation is true: [`(\mathbbR<sup>n</sup> )] \f( D ) ì [`(f B_f )] \overline {\mathbb{R}^n } \backslash f\left( D \right) \subset \overline {f\,B_f } . In addition, if ￥ ? f( D ) \infty \notin f\left( D \right) , then the set f B _f is infinite and x₀ ? [`(B_f )] x_0 \in \overline {B_f } .     

A convexity theorem for multiplicative functions

We generalize a well known convexity property of the multiplicative potential function. We prove that, given any convex function g : \mathbbR^m ? [0, ￥]{g : \mathbb{R}^m \rightarrow [{0}, {\infty}]}, the function ${({\rm \bf x},{\rm \bf y})\mapsto g({\rm \bf x})^{1+\alpha}{\bf y}^{-{\bf \beta}}, {\bf y}>{\bf 0}}${({\rm \bf x},{\rm \bf y})\mapsto g({\rm \bf x})^{1+\alpha}{\bf y}^{-{\bf \beta}}, {\bf y}>{\bf 0}}, is convex if β ≥ 0 and α ≥ β ₁ + ··· + β _n . We also provide further generalization to functions of the form (x,y₁, . . . , y_n)? g(x)^1+af₁(y₁)^-b₁ ···f_n(y_n)^-b_n{({\rm \bf x},{\rm \bf y}_1, . . . , {y_n})\mapsto g({\rm \bf x})^{1+\alpha}f_1({\rm \bf y}_1)^{-\beta_1} \cdot \cdot \cdot f_n({\rm \bf y}_n)^{-\beta_n} } with the f _k concave, positively homogeneous and nonnegative on their domains.     

Distribution of the maxima of random Takagi functions

This paper concerns the maximum value and the set of maximum points of a random version of Takagi’s continuous, nowhere differentiable function. Let F(x):=∑ _n=1^∞ ε _n ϕ(2 ⁿ⁻¹ x), x ∈ R, where ɛ ₁, ɛ ₂, ... are independent, identically distributed random variables taking values in {−1, 1}, and ϕ is the “tent map” defined by ϕ(x) = 2 dist (x, Z). Let p:= P (ɛ ₁ = 1), M:= max {F(x): x ∈ R}, and := {x ∈ [0, 1): F(x) = M}. An explicit expression for M is given in terms of the sequence {ɛ _n }, and it is shown that the probability distribution μ of M is purely atomic if p < , and is singular continuous if p ≧ . In the latter case, the Hausdorff dimension and the multifractal spectrum of μ are determined. It is shown further that the set is finite almost surely if p < , and is topologically equivalent to a Cantor set almost surely if p ≧ . The distribution of the cardinality of is determined in the first case, and the almost-sure Hausdorff dimension of is shown to be (2p − 1)/2p in the second case. The distribution of the leftmost point of is also given. Finally, some of the results are extended to the more general functions Σa ^{n − 1} ɛ _n ϕ(2 ^{n − 1} x), where 0 < a < 1.      

Asymptotics and weighted estimates of meixner polynomials orthogonal on the gird {0, δ, 2δ, ...}

Suppose that 0<δ≤1,N=1/δ, and α, ga≥0, is an integer. For the classical Meixner polynomials orthonormal on the gird {0, δ, 2δ, ...} with weight ρ(x)=(1-e ^−δ)^αг(Nx+α+ 1)/г(Nx+1), the following asymptotic formula is obtained: . The remainderv _n,N ^α (z) forn≤λN satisfies the estimate