Homoclinic solutions for a class of the second order Hamiltonian systems

We study the existence of homoclinic orbits for the second order Hamiltonian system , where q∈Rⁿ and V∈C¹(R×Rⁿ,R), V(t,q)=-K(t,q)+W(t,q) is T-periodic in t. A map K satisfies the “pinching” condition b₁|q|²?K(t,q)?b₂|q|², W is superlinear at the infinity and f is sufficiently small in L²(R,Rⁿ). A homoclinic orbit is obtained as a limit of 2kT-periodic solutions of a certain sequence of the second order differential equations.     

Global attractors for the wave equation with nonlinear damping

Based on a new a priori estimate method, so-called asymptotic a priori estimate, the existence of a global attractor is proved for the wave equation utt+kg(ut)−Δu+f(u)=0 on a bounded domain Ω⊂R³ with Dirichlet boundary conditions. The nonlinear damping term g is supposed to satisfy the growth condition C₁(|s|−C₂)?|g(s)|?C₃(1+p|s|), where 1?p<5; the damping parameter is arbitrary; the nonlinear term f is supposed to satisfy the growth condition |f^′(s)|?C₄(1+q|s|), where q?2. It is remarkable that when 2<p<5, we positively answer an open problem in Chueshov and Lasiecka [I. Chueshov, I. Lasiecka, Long-time behavior of second evolution equations with nonlinear damping, Math. Scuola Norm. Sup. (2004)] and improve the corresponding results in Feireisl [E. Feireisl, Global attractors for damped wave equations with supercritical exponent, J. Differential Equations 116 (1995) 431-447].     

Dirichlet operators with convex potentials on the half-line

We consider the Dirichlet operator H _t =?d ²/dx ²+q(x) on L ²([t,??)), where q is a convex potential with q(x)???? as x????. We show that the eigenvalue gap ??(t) of H _t is monotone increasing as t increases from ??? to ??. We also show that ??(t) is strictly increasing if q is not linear at infinity. An asymptotic estimate of ??(t) for quadratic potentials is obtained.     

Some big sets of mutually disjoint subgeometries of PG(d, q t )

In PG(d, q ^t ) we construct a set ? of mutually disjoint subgeometries isomorphic to PG(d, q) almost partitioning the point set of PG(d, q ^t ) such that there is a group of collineations of PG(d, q ^t ) operating simultaneously as a Singer cycle on all elements of ?. In PG(t?1,q ^t ) we construct big subsets ? of ? whose elements are far away from each other in the following sense:

u

? If P ₁, P ₂ ∈ ? _k , then no point of P ₁ lies on ak-dimensional subspace of P ₂.

For example, we get a set ofq - 1 subplanes of orderq of PG(2,q ³) such that no point of one subplane lies on a line of another subplane, and such that no three points of three different subplanes are collinear.     

Characterizing omega-limit sets which are closed orbits

Let X be a vector field in a compact n-manifold M, n?2. Given Σ⊂M we say that q∈M satisfies (P)_Σ if the closure of the positive orbit of X through q does not intersect Σ, but, however, there is an open interval I with q as a boundary point such that every positive orbit through I intersects Σ. Among those q having saddle-type hyperbolic omega-limit set ω(q) the ones with ω(q) being a closed orbit satisfy (P)_Σ for some closed subset Σ. The converse is true for n=2 but not for n?4. Here we prove the converse for n=3. Moreover, we prove for n=3 that if ω(q) is a singular-hyperbolic set [C. Morales, M. Pacifico, E. Pujals, On C¹ robust singular transitive sets for three-dimensional flows, C. R. Acad. Sci. Paris Sér. I 26 (1998) 81-86], [C. Morales, M. Pacifico, E. Pujals, Robust transitive singular sets for 3-flows are partially hyperbolic attractors or repellers, Ann. of Math. (2) 160 (2) (2004) 375-432], then ω(q) is a closed orbit if and only if q satisfies (P)_Σ for some Σ closed. This result improves [S. Bautista, Sobre conjuntos hiperbólicos-singulares (On singular-hyperbolic sets), thesis Uiversidade Federal do Rio de Janeiro, 2005 (in Portuguese)] and [C. Morales, M. Pacifico, Mixing attractors for 3-flows, Nonlinearity 14 (2001) 359-378].     

Thermodynamic formalism of transcendental entire maps of finite singular type

We build a version of a thermodynamic formalism for maps of the form f(z) = ∑ _{j = 0} ^{p + q} a _j e ^(j−p)z where p, q > 0 and . We show in particular the existence and uniqueness of (t,α)-conformal measures and that the Hausdorff dimension HD(J _f ^r ) = h is the unique zero of the pressure function t ↦ P(t) for t > 1, where the set J _f ^r is the radial Julia set. Partially supported by NSF Grant DMS 0100078. Partially supported by Warsaw University of Technology Grant No. 504G11200023000, Polish KBN Grant No. 2PO3A03425 and Chilean FONDECYT Grant No. 11060280.     

A second-order nonlinear difference equation: Oscillation and asymptotic behavior

Discrete analogues are investigated for well-known results on oscillation, growth, and asymptotic behavior of solutions of y″ + q(t) y^γ = 0, for q(t) ? 0 and for q(t) ? 0. The analogue of Atkinson's oscillation criterion is shown to be true for Δ²y_{n ? 1} + q_ny_n^γ = 0, but the analogue for Atkinson's nonoscillation criterion is shown to be false.     

An inequality for a sum of quadratic forms with applications to probability theory

Let {x_t} be a sequence of p-component vectors, and let A_t=∑^T_s=1x_tx′_t with A_n nonsingular for some n?p. It is shown that ∑^t_t=q+1x′_tA^-2_tx_t?trA^-1_q for n?q<T. An application of this proposition is the convergence of a certain martingale with probability 1. A matrix version of Kronecker's lemma then leads to strong consistency of least-squares estimates under a certain condition.     

A proof of the q,t-Catalan positivity conjecture

We present here a proof that a certain rational function C_n(q,t) which has come to be known as the “q,t-Catalan” is in fact a polynomial with positive integer coefficients. This has been an open problem since 1994. The precise form of the conjecture is given in Garsia and Haiman (J. Algebraic Combin. 5(3) (1996) 191), where it is further conjectured that C_n(q,t) is the Hilbert series of the diagonal harmonic alternants in the variables (x₁,x₂,…,x_n;y₁,y₂,…,y_n). Since C_n(q,t) evaluates to the Catalan number at t=q=1, it has also been an open problem to find a pair of statistics a(π),b(π) on Dyck paths π in the n×n square yielding C_n(q,t)=∑_πt^a(π)q^b(π). Our proof is based on a recursion for C_n(q,t) suggested by a pair of statistics a(π),b(π) recently proposed by Haglund. Thus, one of the byproducts of our developments is a proof of the validity of Haglund's conjecture. It should also be noted that our arguments rely and expand on the plethystic machinery developed in Bergeron et al. (Methods and Applications of Analysis, Vol. VII(3), 1999, p. 363).     

Spectral and modulational stability of periodic wavetrains for the nonlinear Klein–Gordon equation

This paper is a detailed and self-contained study of the stability properties of periodic traveling wave solutions of the nonlinear Klein–Gordon equation _utt−_uxx+V^′(u)=0$u_{tt}-u_{xx}+V^{\prime }(u)=0$, where u is a scalar-valued function of x and t  , and the potential V(u)$V(u)$ is of class C²$C^2$ and periodic. Stability is considered both from the point of view of spectral analysis of the linearized problem (spectral stability analysis) and from the point of view of wave modulation theory (the strongly nonlinear theory due to Whitham as well as the weakly nonlinear theory of wave packets). The aim is to develop and present new spectral stability results for periodic traveling waves, and to make a solid connection between these results and predictions of the (formal) modulation theory, which has been developed by others but which we review for completeness.     

Embedding of a pseudo-residual design into a Möbius plane

Let $\text{U}$ be a class of subsets of a finite set X. Elements of $\text{U}$ are called blocks. Let v, t and λ₁, 0 ? i ? t, be nonnegative integers, and K be a subset of nonnegative integers such that every member of K is at most v. A pair (X, U) is called a (λ₀, λ₁,…, λ_t; K, υ)t-design if (1) |X| = υ, (2) every i-subset of X is contained in exactly λ_t blocks, 0 ? i ? t, and (3) for every block A in $\text{U}$, |A| ?K. It is well-known that if K consists of a singleton k, then λ₀,…, λ_{t ? 1} can be determined from υ, t, k and λ_t. Hence, we shall denote a (λ₀,…, λ_t; {k}, υ)t-design by S_λ(t, k, υ), where λ = λ_t. A Möbius plane M is an S₁(3, q + 1, q² + 1), where q is a positive integer. Let A be a fixed block in M. If A is deleted from M together with the points contained in A, then we obtain a residual design M′ with parameters λ₀ = q³ + q ? 1, λ₁ = q² + q, λ₂ = q + 1, λ₃ = 1, K = {q + 1, q, q ? 1}, and υ = q² ? 1. We define a design to be a pseudo-block-residual design of order q (abbreviated by PBRD(q)) if it has these parameters. We consider the reconstruction problem of a Möbius plane from a given PBRD(q). Let B and B′ be two blocks in a residual design M′. If B and B′ are tangent to each other at a point x, and there exists a block C of size q + 1 such that C is tangent to B at x and is secant to B′, then we say B is r-tangent to B′ at x. A PBRD(q) is said to satisfy the r-tangency condition if for every block B of size q, and any two points x and y not in B, there exists at most one block which is r-tangent to B and contains x and y. We show that any PBRD(q)D can be uniquely embedded into a Möbius plane if and only if D satisfies the r-tangency condition.     

Note on a conjecture of sierksma

LetS(q, d) be the maximal numberv such that, for every general position linear maph: Δ^(q?1)(d+1) →R ^d, there exist at leastv different collections {Δ ^t1, ..., Δ ^t _q} of disjoint faces of Δ(q?1)(d+1) with the property thatf(Δ ^t1) ∩ ... ∩f(Δ ^t _q) ≠ Ø. Sierksma's conjecture is thatS(q, d)=((q?1)!) ^d . The following lower bound (Theorem 1) is proved assuming thatq is a prime number: $$S(q,d) \geqslant \frac{1}{{(q - 1)!}}\left( {\frac{q}{2}} \right)^{{{((q - 1)(d + 1))} \mathord{\left/ {\vphantom {{((q - 1)(d + 1))} 2}} \right. \kern-\nulldelimiterspace} 2}} .$$ Using the same technique we obtain (Theorem 2) a lower bound for the number of different splittings of a “generic” necklace.     

Semi-stable Markov processes in rn

A Markov process in Rⁿ{x_t} with transition function P_t is called semi-stable of order α>0 if for every a>0, P_t(x, E) = P_at(a^ax, a^aE). Let ?_t(ω)=∫^t₀|x_s(ω)|^-1/α ds, T(t) be its inverse and {y_t}={x_T(t)}.Theorem 1: {Y_t} is a multiplicative invariant process; i.e., it has transition function q_t satisfying q_t(x,E)=q_t(ax,aE) for all a > 0.Theorem 2: If {x_t} is Feller, right continuous and uniformly stochastic continuous on a neighborhood of the origin, then {y_t} is Feller.     

On a comparison theorem

This paper gives a generalization of the Sturm comparison theorem for differential equations (p): y″ = p(t)y, (q): y″ = q(t)y under the assumption that the function p ? q changes its sign exactly once on [a, b] or ∝_t^bp ? q, ∝_a^tp ? q maintain the sign on [a, b]. The results are used for investigating the distributions of zeros of solutions and the derivative of solutions of (p), (q).     

On the existence of Hadamard matrices

Given any natural number q > 3 we show there exists an integer t ? [2log₂(q ? 3)] such that an Hadamard matrix exists for every order 2^sq where s > t. The Hadamard conjecture is that s = 2.This means that for each q there is a finite number of orders 2^υq for which an Hadamard matrix is not known. This is the first time such a statement could be made for arbitrary q.In particular it is already known that an Hadamard matrix exists for each 2^sq where if q = 2^m ? 1 then s ? m, if q = 2^m + 3 (a prime power) then s ? m, if q = 2^m + 1 (a prime power) then s ? m + 1.It is also shown that all orthogonal designs of types (a, b, m ? a ? b) and (a, b), 0 ? a + b ? m, exist in orders m = 2^t and 2^t+2 · 3, t ? 1 a positive integer.     

The behavior of solutions of second order delay differential equations

In this paper, we study the behavior of solutions of second order delay differential equation

y^″(t)=p₁y^′(t)+p₂y^′(t−τ)+q₁y(t)+q₂y(t−τ),     

Embedding of a pseudo-point residual design into a Möbius plane

Let $\text{U}$ be a class of subsets of a finite set X. Elements of $\text{U}$ are called blocks. Let υ, t, λ and k be nonnegative integers such that υ?k?t?0. A pair (X, $\text{U}$) is called a (υ, k, λ) t-design, denoted by S_λ(t, k, υ), if (1) |X| = υ, (2) every t-subset of X is contained in exactly λ blocks and (3) for every block A in $\text{U}$, |A| = k. A Möbius plane M is an S₁(3, q+1, q²+1) where q is a positive integer. Let ∞ be a fixed point in M. If ∞ is deleted from M, together with all the blocks containing ∞, then we obtain a point-residual design M^*. It can be easily checked that M^* is an S_q(2, q+1, q²). Any S_q(2, q+1, q²) is called a pseudo-point-residual design of order q, abbreviated by PPRD(q). Let A and B be two blocks in a PPRD(q)M^*. A and B are said to be tangent to each other at z if and only if A∩B={z}. M^* is said to have the Tangency Property if for any block A in M^*, and points x and y such that x?A and y?A, there exists at most one block containing y and tangent to A at x. This paper proves that any PPRD(q)M^* is uniquely embeddable into a Möbius plane if and only if M^* satisfies the Tangency Property.     

Generalized double affine Hecke algebras of rank 1 and quantized del Pezzo surfaces

Let D be a simply laced Dynkin diagram of rank r whose affinization has the shape of a star (i.e., D₄,E₆,E₇,E₈). To such a diagram one can attach a group G whose generators correspond to the legs of the affinization, have orders equal to the leg lengths plus 1, and the product of the generators is 1. The group G is then a 2-dimensional crystallographic group: G=Z??Z², where ? is 2, 3, 4, and 6, respectively. In this paper, we define a flat deformation H(t,q) of the group algebra C[G] of this group, by replacing the relations saying that the generators have prescribed orders by their deformations, saying that the generators satisfy monic polynomial equations of these orders with arbitrary roots (which are deformation parameters). The algebra H(t,q) for D₄ is the Cherednik algebra of type C^∨C₁, which was studied by Noumi, Sahi, and Stokman, and controls Askey-Wilson polynomials. We prove that H(t,q) is the universal deformation of the twisted group algebra of G, and that this deformation is compatible with certain filtrations on C[G]. We also show that if q is a root of unity, then for generic t the algebra H(t,q) is an Azumaya algebra, and its center is the function algebra on an affine del Pezzo surface. For generic q, the spherical subalgebra eH(t,q)e provides a quantization of such surfaces. We also discuss connections of H(t,q) with preprojective algebras and Painlevé VI.     

When all solutions of x′ = −∑ qi(t) x(t − Ti(t)) oscillate

In this paper the long-term behavior of solutions to the equation in the title are examined, where q_i(t) and T_i(t) are positive. In particular, it is shown that if lim inf_{t → ∝} ∑_{i = 1}ⁿT_i(t) q_i(t) > 1/e, all solutions oscillate about 0 infinitely often.