Non-real zeroes of real entire derivatives

A real entire function belongs to the Laguerre-Pólya class LP if it is the limit of a sequence of real polynomials with real zeroes. By building upon results that resolved a long-standing conjecture of Wiman, a number of conditions are established under which a real entire function f must belong to the class LP, or to one of the related classes U _2p ^*. These conditions typically involve the non-real zeroes of f and its derivatives, or those of the differential polynomial f f″−a(f′)².     

On the real zeroes of the Hurwitz zeta-function and Bernoulli polynomials

The behaviour of real zeroes of the Hurwitz zeta-function

     

Polynomial approximation with an exponential weight on the real semiaxis

We consider the polynomial approximation on (0,+∞), with the weight $u(x)= x^{\gamma}e^{-x^{-\alpha}-x^{\beta}}$ , α>0, β>1 and γ≧0. We introduce new moduli of smoothness and related K-functionals for functions defined on the real semiaxis, which can grow exponentially both at 0 and at +∞. Then we prove the Jackson theorem, also in its weaker form, and the Stechkin inequality. Moreover, we study the behavior of the derivatives of polynomials of best approximation.     

Fractional set systems with few disjoint pairs

Ahlswede (1980) [1] and Frankl (1977) [5] independently found a result about the structure of set systems with few disjoint pairs. Bollobás and Leader (2003) [3] gave an alternate proof by generalizing to fractional set systems and noting that the optimal fractional set systems are {0,1}-valued. In this paper we show that this technique does not extend to t-intersecting families. We find optimal fractional set systems for some infinite classes of parameters, and we point out that they are strictly better than the corresponding {0,1}-valued fractional set systems. We prove some results about the structure of an optimal fractional set system, which we use to produce an algorithm for finding such systems. The run time of the algorithm is polynomial in the size of the ground set.     

Random inequality constraint systems with few variables

We show that a system of many linear inequality constraints will have a high proportion of redundant constraints with high probability. Implications for expected time of algorithms are indicated.     

Polynomial Hamiltonian systems with movable algebraic singularities

The singularity structure of solutions of a class of Hamiltonian systems of ordinary differential equations in two dependent variables is studied. It is shown that for any solution, all movable singularities obtained by analytic continuation along a rectifiable curve are at most algebraic branch points.     

Polynomial recognition of equal unions in hypergraphs with few vertices of large degree

A family of nonempty sets has the equal union property if there exist two nonempty disjoint subfamilies having equal unions. If every point belongs to the unions, then we say the family has the full equal union property. Recognition of both properties is NP-complete even when restricted to families for which the degree of every point is at most three. In this paper we show that both recognition problems can be solved in polynomial time for families in which there is a bound on the number of points whose degree exceeds two.     

Random polynomials having few or no real zeros

Consider a polynomial of large degree whose coefficients are independent, identically distributed, nondegenerate random variables having zero mean and finite moments of all orders. We show that such a polynomial has exactly real zeros with probability as through integers of the same parity as the fixed integer . In particular, the probability that a random polynomial of large even degree has no real zeros is . The finite, positive constant is characterized via the centered, stationary Gaussian process of correlation function . The value of depends neither on nor upon the specific law of the coefficients. Under an extra smoothness assumption about the law of the coefficients, with probability one may specify also the approximate locations of the zeros on the real line. The constant is replaced by in case the i.i.d. coefficients have a nonzero mean.

     

Polynomial first integrals of Hamiltonian systems with exponential interaction

Moscow Institute of Radio Engineering, Electronics, and Automation. Translated from Funktsional'nyi Analiz i Ego Prilozheniya, Vol. 25, No. 3, pp. 88–89, July–September, 1991.     

Polynomial approximation of nonlinear differential systems with prefixed accuracy

This paper presents a method for constructing polynomial approximations of the solutions of nonlinear initial value systems of differential equations. Given an a priori chosen accuracy, the degree of the vector polynomial can be adapted so that the approximate solution has the required precision. The method is based on the AI-method of Dzyadyk developed for the scalar case, and the computational cost is shown to be competitive with other methods.     

Holomorphic vector fields with totally degenerate zeroes

A zero set of a holomorphic vector field is totally degenerate, if the endomorphism of the conormal sheaf induced by the vector field is identically zero. By studying a class of foliations generalizing foliations of C^*-actions, we show that if a projective manifold admits a holomorphic vector field with a smooth totally degenerate zero component,then the manifold is stably birational to that component of the zero set.When the vector field has an isolated totally degenerate zero, we prove that the manifold is rational. This is a special case of Carrell's conjecture.     

Polynomial entropies and integrable Hamiltonian systems

We introduce two numerical conjugacy invariants of dynamical systems — the polynomial entropy and the weak polynomial entropy — which are well-suited for the study of “completely integrable” Hamiltonian systems. These invariants describe the polynomial growth rate of the number of balls (for the usual “dynamical” distances) of covers of the ambient space. We give explicit examples of computation of these polynomial entropies for generic Hamiltonian systems on surfaces.     

Polynomial upwind schemes for hyperbolic systems

In this paper, we present an upwinding methodology for systems of conservation laws. Our aim is to construct upwind schemes which do not require the extraction of the eigensystem of the Jacobian matrix (but just the knowledge of the eigenvalues) and rely on the introduction of an appropriate polynomial approximation.     

Polynomial stability of positive switched homogeneous systems with time delay

Based on the logarithm contraction average dwell-time method, this paper investigates the polynomial stability of positive switched homogeneous time-delay systems whose vector fields are of different degrees with respect to a dilation map. Using the analytical skills developed in positive systems, an explicit polynomial stability criterion is established for the first time for the involved system under the logarithm contraction average dwell-time switching. Moreover, the main result is applied to the polynomial stability of Persidskii-type switched systems.     

Normal forms of real symmetric systems with multiplicity

A normal form is given for real symmetric systems of linear partial differential equations, at points where the principal symbol has a two-dimensional kernel under assumptions which apply to the generic case.     

Polynomial entropies for Bott integrable Hamiltonian systems

In this paper, we study the entropy of a Hamiltonian flow in restriction to an energy level where it admits a first integral which is nondegenerate in the sense of Bott. It is easy to see that for such a flow, the topological entropy vanishes. We focus on the polynomial and the weak polynomial entropies h_pol and h _pol ^* . We show that, under natural conditions on the critical levels of the Bott first integral and on the Hamiltonian function H, h _pol ^* ∈ {0, 1} and h_pol ∈ {0, 1, 2}. To prove this result, our main tool is a semi-global desingularization of the Hamiltonian system in the neighborhood of a polycycle.