Intersective polynomials and the polynomial Szemerédi theorem

Let be a family of polynomials such that , i=1,…,r. We say that the family P has the PSZ property if for any set with there exist infinitely many such that E contains a polynomial progression of the form {a,a+p₁(n),…,a+p_r(n)}. We prove that a polynomial family P={p₁,…,p_r} has the PSZ property if and only if the polynomials p₁,…,p_r are jointly intersective, meaning that for any there exists such that the integers p₁(n),…,p_r(n) are all divisible by k. To obtain this result we give a new ergodic proof of the polynomial Szemerédi theorem, based on the fact that the key to the phenomenon of polynomial multiple recurrence lies with the dynamical systems defined by translations on nilmanifolds. We also obtain, as a corollary, the following generalization of the polynomial van der Waerden theorem: If are jointly intersective integral polynomials, then for any finite partition of , there exist i{1,…,k} and a,nE_i such that {a,a+p₁(n),…,a+p_r(n)}E_i.     

Instance-optimality in probability with an -minimization decoder

Let Φ(ω), ωΩ, be a family of n×N random matrices whose entries _i,j are independent realizations of a symmetric, real random variable η with expectation and variance . Such matrices are used in compressed sensing to encode a vector by y=Φx. The information y holds about x is extracted by using a decoder . The most prominent decoder is the ℓ₁-minimization decoder Δ which gives for a given the element which has minimal ℓ₁-norm among all with Φz=y. This paper is interested in properties of the random family Φ(ω) which guarantee that the vector will with high probability approximate x in to an accuracy comparable with the best k-term error of approximation in for the range kan/log₂(N/n). This means that for the above range of k, for each signal , the vector satisfies

with high probability on the draw of Φ. Here, Σ_k consists of all vectors with at most k nonzero coordinates. The first result of this type was proved by Wojtaszczyk [P. Wojtaszczyk, Stability and instance optimality for Gaussian measurements in compressed sensing, Found. Comput. Math., in press] who showed this property when η is a normalized Gaussian random variable. We extend this property to more general random variables, including the particular case where η is the Bernoulli random variable which takes the values with equal probability. The proofs of our results use geometric mapping properties of such random matrices some of which were recently obtained in [A. Litvak, A. Pajor, M. Rudelson, N. Tomczak-Jaegermann, Smallest singular value of random matrices and geometry of random polytopes, Adv. Math. 195 (2005) 491–523].     

Markov-type inequalities on certain irrational arcs and domains

Let denote the set of real algebraic polynomials of d variables and of total degree at most n. For a compact set KR^d set Then the Markov factors on K are defined by (Here, as usual, S^d-1 stands for the Euclidean unit sphere in R^d.) Furthermore, given a smooth curve ΓR^d, we denote by D_TP the tangential derivative of P along Γ (T is the unit tangent to Γ). Correspondingly, consider the tangential Markov factor of Γ given by Let . We prove that for every irrational number α>0 there are constants A,B>1 depending only on α such that for every sufficiently large n.Our second result presents some new bounds for M_n(Ω_α), where (d=2,α>1). We show that for every α>1 there exists a constant c>0 depending only on α such that M_n(Ω_α)n^clogn.     

On a generalization of Jentzsch’s theorem

Let E be a compact subset of with connected, regular complement and let G(z) denote Green’s function of Ω with pole at ∞. For a sequence (p_n)_nΛ of polynomials with degp_n=n, we investigate the value-distribution of p_n in a neighbourhood U of a boundary point z₀ of E if G(z) is an exact harmonic majorant of the subharmonic functions

in . The result holds for partial sums of power series, best polynomial approximations, maximally convergent polynomials and can be extended to rational functions with a bounded number of poles.     

On the nonlinear wave equation with the mixed nonhomogeneous conditions: Linear approximation and asymptotic expansion of solutions

In this paper, we consider the following nonlinear wave equation

(1)

where , , μ, f, g are given functions. To problem (1), we associate a linear recursive scheme for which the existence of a local and unique weak solution is proved by applying the Faedo–Galerkin method and the weak compact method. In the case of , , μ(z)≥μ₀>0, μ₁(z)≥0, for all , and , , , a weak solution u_ε1,ε2(x,t) having an asymptotic expansion of order N+1 in two small parameters ε₁, ε₂ is established for the following equation associated to (1)_2,3:

(2)

     

Sparse inertially arbitrary patterns

An n-by-n sign pattern is a matrix with entries in {+,-,0}. An n-by-n nonzero pattern is a matrix with entries in {*,0} where * represents a nonzero entry. A pattern is inertially arbitrary if for every set of nonnegative integers n₁,n₂,n₃ with n₁+n₂+n₃=n there is a real matrix with pattern having inertia (n₁,n₂,n₃). We explore how the inertia of a matrix relates to the signs of the coefficients of its characteristic polynomial and describe the inertias allowed by certain sets of polynomials. This information is useful for describing the inertia of a pattern and can help show a pattern is inertially arbitrary. Britz et al. [T. Britz, J.J. McDonald, D.D. Olesky, P. van den Driessche, Minimal spectrally arbitrary sign patterns, SIAM J. Matrix Anal. Appl. 26 (2004) 257–271] conjectured that irreducible spectrally arbitrary patterns must have at least 2n nonzero entries; we demonstrate that irreducible inertially arbitrary patterns can have less than 2n nonzero entries.     

Asymptotics of the orthogonal polynomials for the Szegő class with a polynomial weight

Let p be a trigonometric polynomial, non-negative on the unit circle . We say that a measure σ on belongs to the polynomial Szegő class, if , σ_s is singular, and

For the associated orthogonal polynomials {_n}, we obtain pointwise asymptotics inside the unit disc . Then we show that these asymptotics hold in L²-sense on the unit circle. As a corollary, we get an existence of certain modified wave operators.     

Positive periodic solutions of periodic neutral Lotka–Volterra system with state dependent delays

By using a fixed point theorem of strict-set-contraction, some new criteria are established for the existence of positive periodic solutions of the following periodic neutral Lotka–Volterra system with state dependent delays

where (i,j=1,2,…,n) are ω-periodic functions and (i=1,2,…,n) are ω-periodic functions with respect to their first arguments, respectively.     

An upper bound on Jacobi polynomials

Let be an orthonormal Jacobi polynomial of degree k. We will establish the following inequality:

where δ_-1<δ₁ are appropriate approximations to the extreme zeros of . As a corollary we confirm, even in a stronger form, T. Erdélyi, A.P. Magnus and P. Nevai conjecture [T. Erdélyi, A.P. Magnus, P. Nevai, Generalized Jacobi weights, Christoffel functions, and Jacobi polynomials, SIAM J. Math. Anal. 25 (1994) 602–614] by proving that

in the region .     

Widths of weighted Sobolev classes on the ball

We study the Kolmogorov n-widths and the linear n-widths of weighted Sobolev classes on the unit ball B^d in L_q,μ, where L_q,μ, 1≤q≤∞, denotes the weighted L_q space of functions on B^d with respect to weight . Optimal asymptotic orders of and as n→∞ are obtained for all 1≤p,q≤∞ and μ≥0.     

Interlacing and spacing properties of zeros of polynomials, in particular of orthogonal and -minimal polynomials,

Let be a sequence of polynomials with real coefficients such that uniformly for [α-δ,β+δ] with G(eⁱ)≠0 on [α,β], where 0α<βπ and δ>0. First it is shown that the zeros of are dense in [α,β], have spacing of precise order π/n and are interlacing with the zeros of p_n+1(cos) on [α,β] for every nn₀. Let be another sequence of real polynomials with uniformly on [α-δ,β+δ] and on [α,β]. It is demonstrated that for all sufficiently large n the zeros of p_n(cos) and strictly interlace on [α,β] if on [α,β]. If the last expression is zero then a weaker kind of interlacing holds. These interlacing properties of the zeros are new for orthogonal polynomials also. For instance, for large n a simple criteria for interlacing of zeros of Jacobi polynomials on [-1+,1-], >0, is obtained. Finally it is shown that the results hold for wide classes of weighted L_q-minimal polynomials, q[1,∞], linear combinations and products of orthogonal polynomials, etc.     

On the set of limit points of the partial sums of series rearranged by a given divergent permutation

We give a new characterization of divergent permutations. We prove that for any divergent permutation p, any closed interval I of (the 2-point compactification of ) and any real number sI, there exists a series ∑a_n of real terms convergent to s such that I=σa_p(n) (where σa_p(n) denotes the set of limit points of the partial sums of the series ∑a_p(n)). We determine permutations p of for which there exists a conditionally convergent series ∑a_n such that ∑a_p(n)=+∞. If the permutation p of possesses the last property then we prove that for any and there exists a series ∑a_n convergent to α and such that σa_p(n)=[β,+∞]. We show that for any countable family P of divergent permutations there exist conditionally convergent series ∑a_n and ∑b_n such that any series of the form ∑a_p(n) with pP is convergent to the sum of ∑a_n, while for every pP.     

Composite implicit iteration process for common fixed points of a finite family of strictly pseudocontractive maps

Let E be a real Banach space and let K be a nonempty closed convex subset of E. Let be N strictly pseudocontractive self-maps of K such that , where F(T_i)={xK:T_ix=x} and let be two real sequence satisfying the conditions:

where L1 is common Lipschitz constant of . For x₀K, let be new implicit process defined by

x_n=α_nx_n−1+(1−α_n)T_ny_n,

y_n=β_nx_n−1+(1−β_n)T_nx_n

where T_n=T_n mod N, then

The results of this paper generalize and improve the results of Osilike in 2004. In this paper, the proof methods of the main results are also different from that of Osilike.     

Blow-up analysis for a system of heat equations with nonlinear flux which obey different laws

We consider a system of heat equations u_t=Δu and v_t=Δv in Ω×(0,T) completely coupled by nonlinear boundary conditions

We prove that the solutions always blow up in finite time for non-zero and non-negative initial values. Also, the blow-up only occurs on ∂Ω with

for p,q>0, 0≤α<1 and 0≤β<p.     

Uniqueness of limit cycles for polynomial first-order differential equations

We study the uniqueness of limit cycles (periodic solutions that are isolated in the set of periodic solutions) in the scalar ODE in terms of {i_k}, {j_k}, {n_k}. Our main result characterizes, under some additional hypotheses, the exponents {i_k}, {j_k}, {n_k}, such that for any choice of the equation has at most one limit cycle. The obtained results have direct application to rigid planar vector fields, thus, planar systems of the form x^′=y+xR(x,y), y^′=−x+yR(x,y), where . Concretely, when the set has at least three elements (or exactly one) and another technical condition is satisfied, we characterize the exponents {i_k}, {j_k} such that the origin of the rigid system is a center for any choice of and also when there are no limit cycles surrounding the origin for any choice of .     

Pointwise universal trigonometric series

A series is called a pointwise universal trigonometric series if for any , there exists a strictly increasing sequence of positive integers such that converges to f(z) pointwise on . We find growth conditions on coefficients allowing and forbidding the existence of a pointwise universal trigonometric series. For instance, if as |n|→∞ for some ε>0, then the series S_a cannot be pointwise universal. On the other hand, there exists a pointwise universal trigonometric series S_a with as |n|→∞.     

The statistical distribution of the zeros of random paraorthogonal polynomials on the unit circle

The orthogonal polynomials on the unit circle are defined by the recurrence relation

where for any k0. If we consider n complex numbers and , we can use the previous recurrence relation to define the monic polynomials Φ₀,Φ₁,…,Φ_n. The polynomial Φ_n(z)=Φ_n(z;α₀,…,α_n-2,α_n-1) obtained in this way is called the paraorthogonal polynomial associated to the coefficients α₀,α₁,…,α_n-1.We take α₀,α₁,…,α_n-2 i.i.d. random variables distributed uniformly in a disk of radius r<1 and α_n-1 another random variable independent of the previous ones and distributed uniformly on the unit circle. For any n we will consider the random paraorthogonal polynomial Φ_n(z)=Φ_n(z;α₀,…,α_n-2,α_n-1). The zeros of Φ_n are n random points on the unit circle.We prove that for any the distribution of the zeros of Φ_n in intervals of size near e^iθ is the same as the distribution of n independent random points uniformly distributed on the unit circle (i.e., Poisson). This means that, for large n, there is no local correlation between the zeros of the considered random paraorthogonal polynomials.     

Existence results for -point boundary value problem of second order ordinary differential equations

This study concerns the existence of positive solutions to the boundary value problemwhere ξ_i(0,1) with 0<ξ₁<ξ₂<<ξ_n-2<1, a_i, b_i[0,∞) with and . By applying the Krasnoselskii's fixed-point theorem in Banach spaces, some sufficient conditions guaranteeing the existence of at least one positive solution or at least two positive solutions are established for the above general n-point boundary value problem.     

Monotone Jacobi parameters and non-Szegő weights

We relate asymptotics of Jacobi parameters to asymptotics of the spectral weights near the edges. Typical of our results is that for a_n≡1, b_n=−Cn^−β (), one has on (−2,2), and near x=2, where

     

A note on the distance between two consecutive zeros of m-orthogonal polynomials for a generalized Jacobi weight

Let , with

-1=x_0n<x_1n<<x_nn<x_n+1,n=1