Let p be a fixed odd prime number. Throughout this paper, we always make use of the following notations: Z denotes the ring of rational integers, Zp denotes the ring of padic rational integer, Qp denotes the ring of p-adic rational numbers, and Cp denotes the completion of algebraic closure of Qp, respectively. Let N be the set of natural numbers and Z N {0}. Let Cpn {ζ | ζpn 1} be the cyclic g...

as χ varies over all multiplicative characters of k. For each χ, S(χ) is real, and (by Weil) has absolute value at most 2. Evans found empirically that, for large q = #k, these q − 1 sums were approximately equidistributed for the “Sato-Tate measure” (1/2π) √ 4− x2dx on the closed interval [−2, 2], and asked if this equidistribution was provably true. Rudnick had done numerics on the sums T (χ)...

Hi-C experiments capturing the 3D genome architecture have led to the discovery of topologically-associated domains (TADs) that form an important part of the 3D genome organization and appear to play a role in gene regulation and other functions. Several histone modifications have been independently suggested as the possible explanations of TAD formation, but their combinatorial effects on doma...

We introduce the notion of strongly t-convex set-valued maps and present some properties of it. In particular, a Bernstein–Doetsch and Sierpiński-type theorems for strongly midconvex set-valued maps, as well as a Kuhn-type result are obtained. A representation of strongly t-convex set-valued maps in inner product spaces and a characterization of inner product spaces involving this representatio...

Throughout his life Erdős showed a particular fascination with inequalities for constrained polynomials. One of his favorite type of polynomial inequalities was Markovand Bernstein-type inequalities. For Erdős, Markovand Bernstein-type inequalities had their own intrinsic interest. He liked to see what happened when the polynomials are restricted in certain ways. Markovand Bernstein-type inequa...

Bernstein and Markov-type inequalities are discussed for the derivatives of trigonometric and algebraic polynomials on general subsets of the real axis and of the unit circle. It has recently been proven by A. Lukashov that the sharp Bernstein factor for trigonometric polynomials is the equilibrium density of the image of the set on the unit circle under the mapping t → e. In this paper Lukasho...

The paper contains two theorems on approximation of functions with bounbed mixed derivative. These theorems give some progress in two old open problems. The first one gives, in particular, an upper estimate in the Bernstein L1-inequality for trigonometric polynomials on two variables with harmonics in hyperbolic crosses. The second one gives the order of the entropy numbers and Kolmogorov’s wid...

In this note we determine the Bernstein-Sato polynomial bQ(s) of a generic central arrangement Q = ∏k i=1 Hi of hyperplanes. We establish a connection between the roots of bQ(s) and the degrees of the generators for the top cohomology of the corresponding Milnor fiber. This connection holds for all homogeneous polynomials. We also introduce certain subschemes of the arrangement determined by th...

Various important weighted polynomial inequalities, such as Bernstein, Marcinkiewicz, Nikolskii, Schur, Remez, etc. inequalities, have been proved recently by Giuseppe Mastroianni and Vilmos Totik under minimal assumptions on the weights. In most of the cases this minimal assumption is the doubling condition. Here, based on a recently proved Bernstein-type inequality by D.S. Lubinsky, we establ...

Tensorial Bernstein bases can be used to compute sharp ranges of the values of polynomials over a box, and to solve systems of polynomial equations in Computer Graphics, Geometric Modelling, Geometric Constraints Solving. Two kinds of solvers are presented. The first is classical, and applies to small systems, up to 6 or 7 unknowns. The second is new and applies to systems of arbitrary size. It...