Estimates of Weyl’s exponential sums for very small values of the variable

We consider Weyls exponential sums for very small values of the variable. In this case we can give an asymptotic transformation formula. Weyls exponential sums will be usually estimated by means of Weyls or Vinogradovs method. Here we use van der Corputs method and obtain sufficiently good results in the present case.Received: 14 January 2002     

Asymptotics of diagonal Hermite–Padé polynomials for the collection of exponential functions

Mathematical Notes - In the paper, the semigroup of weak limits of the powers of an infinite transformation of rank one of Chacon type is completely described.     

An exponential lower bound for Cunningham’s rule

In this paper we give an exponential lower bound for Cunningham’s least recently considered (round-robin) rule as applied to parity games, Markov decision processes and linear programs. This improves a recent subexponential bound of Friedmann for this rule on these problems. The round-robin rule fixes a cyclical order of the variables and chooses the next pivot variable starting from the previously chosen variable and proceeding in the given circular order. It is perhaps the simplest example from the class of history-based pivot rules. Our results are based on a new lower bound construction for parity games. Due to the nature of the construction we are also able to obtain an exponential lower bound for the round-robin rule applied to acyclic unique sink orientations of hypercubes (AUSOs). Furthermore these AUSOs are realizable as polytopes. We believe these are the first such results for history based rules for AUSOs, realizable or not. The paper is self-contained and requires no previous knowledge of parity games.     

An exponential lower bound for Zadeh’s pivot rule

Mathematical Programming - The question whether the Simplex Algorithm admits an efficient pivot rule remains one of the most important open questions in discrete optimization. While many natural,...     

Sharp estimates for the convergence rate of “hyperbolic” partial sums of double fourier series in orthogonal polynomials

Two-variable functions f(x, y) from the class L ₂ = L ₂((a, b) × (c, d); p(x)q(y)) with the weight p(x)q(y) and the norm $$\left\| f \right\| = \sqrt {\int\limits_a^b {\int\limits_c^d {p(x)q(x)f^2 (x,y)dxdy} } }$$ are approximated by an orthonormal system of orthogonal P _n (x)Q _n (y), n, m = 0, 1, ..., with weights p(x) and q(y). Let $$E_N (f) = \mathop {\inf }\limits_{P_N } \left\| {f - P_N } \right\|$$ denote the best approximation of f ?? L ₂ by algebraic polynomials of the form $$\begin{array}{*{20}c} {P_N (x,y) = \sum\limits_{0 < n,m < N} {a_{m,n} x^n y^m ,} } \\ {P_1 (x,y) = const.} \\ \end{array}$$ . Consider a double Fourier series of f ?? L ₂ in the polynomials P _n (x)Q _m (y), n, m = 0, 1, ..., and its ??hyperbolic?? partial sums $$\begin{array}{*{20}c} {S_1 (f;x,y) = c_{0,0} (f)P_o (x)Q_o (y),} \\ {S_N (f;x,y) = \sum\limits_{0 < n,m < N} {c_{n,m} (f)P_n (x)Q_m (y), N = 2,3, \ldots .} } \\ \end{array}$$ A generalized shift operator Fh and a kth-order generalized modulus of continuity ?? _k (A, h) of a function f ?? L ₂ are used to prove the following sharp estimate for the convergence rate of the approximation: $\begin{gathered} E_N (f) \leqslant (1 - (1 - h)^{2\sqrt N } )^{ - k} \Omega _k (f;h),h \in (0,1), \hfill \\ N = 4,5,...;k = 1,2,... \hfill \\ \end{gathered} $ . Moreover, for every fixed N = 4, 9, 16, ..., the constant on the right-hand side of this inequality is cannot be reduced.     

Higher order derivative tests for exponential sums incorporating the Discrete Hardy–Littlewood method

This paper provides estimates for exponential sums, combining classic tools of Van der Corput type with a deep result from the modern “discrete Hardy–Littlewood method”. As an application, an improved bound for the lattice point discrepancy of a large ellipsoid of rotation is deduced.     

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.     

Orthogonal polynomials for Jacobi-exponential weights (1−x 2) ρ e −Q(x) on (−1,1)

This paper deals with orthogonal polynomials for Jacobi-exponential weights (1?x ²) ^ρ e ^?Q(x) on (?1,1) and gives bounds on orthogonal polynomials, zeros, and Christofel functions. In addition, restricted range inequalities are also obtained.     

Lower bound for the area of an analytic curve inside a cube in the space ℂ n

The exact lower bound of the area of an analytic curve which lies inside a cube in the space and passes through its center is obtained.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 45, No. 1, pp. 145–148, January, 1993.     

Density lower bound estimates for local minimizers of the 2d Mumford–Shah energy

We prove, using direct variational arguments, an explicit energy-treshold criterion for regular points of 2-dimensional Mumford-Shah energy minimizers. From this we infer an explicit constant for the density lower bound of De Giorgi, Carriero and Leaci.     

Modifications of Newton’s method for even-grade palindromic polynomials and other twined polynomials

The paper describes some modifications of Newton??s method for refining the zeros of even-grade f(x)-twined (f(x)-egt) polynomials, defined as polynomials whose roots appear in pairs {x _i ,f(x _i )}. Particular attention is given to even-grade palindromic (egp) polynomials. The algorithms are derived from certain symmetric division processes for computing a symmetric quotient and a symmetric remainder of two given f(x)-egt polynomials. Numerical results indicate that the presented algorithms can be more accurate than other methods which do not take into consideration the symmetry of the coefficients.     

Lower bound of measure and embeddings of Sobolev,Besov and Triebel–Lizorkin spaces

In this article, we study the relation between Sobolev-type embeddings for Sobolev spaces or Hajłasz–Besov spaces or Hajłasz–Triebel–Lizorkin spaces defined on a doubling and geodesic metric measure space and lower bound for measure of balls either in the whole space or in a domain inside the space.     

Convergence of the Vallée-Poussin means for Fourier-Jacobi sums

Letf εC[?1, 1], ?1<α,β≤0, let $f \in C[ - 1, 1], - 1< \alpha , \beta \leqslant 0$ , letS _n ^{α, β} (f, x) be a partial Fourier-Jacobi sum of ordern, and let $$\nu _{m, n}^{\alpha , \beta } = \nu _{m, n}^{\alpha , \beta } (f) = \nu _{m, n}^{\alpha , \beta } (f,x) = \frac{1}{{n + 1}}[S_m^{\alpha ,\beta } (f,x) + ... + S_{m + n}^{\alpha ,\beta } (f,x)]$$ be the Vallée-Poussin means for Fourier-Jacobi sums. It was proved that if 0<a≤m/n≤b, then there exists a constantc=c(α, β, a, b) such that ‖ν _{m, n} ^{α, β} ‖ ≤c, where ‖ν _{m, n} ^{α, β} ‖ is the norm of the operator ν _{m, n} ^{α, β} inC[?1,1].     

Local connectivity of Julia sets for a family of biquadratic polynomials

The local connectivity of Julia sets for the family of biquadratic polynomials f_c(z)= (z~2-2c~2)z~2 with a parameter c is discussed.It is proved that for any parameter c,the boundary of the immediately attracting domain of f_c is a Jordan curve.