Zero separation results for solutions of second order linear differential equations

The oscillation of solutions of f^″+Af=0$f^{″}+Af=0$ is discussed by focusing on four separate situations. In the complex case A$A$ is assumed to be either analytic in the unit disc D$\mathbb{D}$ or entire, while in the real case A$A$ is continuous either on (−1,1)$(-1,1)$ or on (0,∞)$(0,\infty )$. In all situations A$A$ is expected to grow beyond bounds that ensure finite oscillation for all (non-trivial) solutions, and the separation between distinct zeros of solutions is considered.     

Cone isomorphisms and almost surjective operators

Let E$E$ be a Banach lattice and F$F$ a Banach space. A bounded linear operator T:E→F$T:E\to F$ is an isomorphism on the positive cone of E$E$ if and only if T^∗$T^{\ast }$ is almost surjective. A dual version of this theorem holds also. A bounded linear operator T:F→E$T:F\to E$ is almost surjective if and only if T^∗$T^{\ast }$ is an isomorphism on the positive cone of F^∗$F^{\ast }$.     

Upper bounds for the number of zeroes for some Abelian integrals

Consider the vector field x^′=−yG(x,y),y^′=xG(x,y)$x^{\prime }=-yG(x,y),y^{\prime }=xG(x,y)$, where the set of critical points {G(x,y)=0}$\{G(x,y)=0\}$ is formed by K$K$ straight lines, not passing through the origin and parallel to one or two orthogonal directions. We perturb it with a general polynomial perturbation of degree n$n$ and study the maximum number of limit cycles that can bifurcate from the period annulus of the origin in terms of K$K$ and n$n$. Our approach is based on the explicit computation of the Abelian integral that controls the bifurcation and on a new result for bounding the number of zeroes of a certain family of real functions. When we apply our results for K≤4$K\le 4$ we recover or improve some results obtained in several previous works.     

Unilateral small deviations of processes related to the fractional Brownian motion

Let x(s)$x\left(s\right)$, s∈R^d$s\in R^d$ be a Gaussian self-similar random process of index H$H$. We consider the problem of log-asymptotics for the probability _pT$p_T$ that x(s)$x\left(s\right)$, x(0)=0$x\left(0\right)=0$ does not exceed a fixed level in a star-shaped expanding domain T⋅Δ$T\cdot \Delta$ as T→∞$T\to \infty$. We solve the problem of the existence of the limit, θ?lim(−log_pT)/(logT)^D$\theta ?lim(-log{p}_T)/{(logT)}^D$, T→∞$T\to \infty$, for the fractional Brownian sheet x(s)$x\left(s\right)$, s∈[0,T]²$s\in {[0,T]}^2$ when D=2$D=2$, and we estimate θ$\theta$ for the integrated fractional Brownian motion when D=1$D=1$.     

Estimation for stochastic differential equations with a small diffusion coefficient

We consider a multidimensional diffusion X$X$ with drift coefficient b(_Xt,α)$b(X_t,\alpha )$ and diffusion coefficient εa(_Xt,β)$\epsilon a(X_t,\beta )$ where α$\alpha$ and β$\beta$ are two unknown parameters, while ε$\epsilon$ is known. For a high frequency sample of observations of the diffusion at the time points k/n$k/n$, k=1,…,n$k=1,\dots ,n$, we propose a class of contrast functions and thus obtain estimators of (α,β)$(\alpha ,\beta )$. The estimators are shown to be consistent and asymptotically normal when n→∞$n\to \infty$ and ε→0$\epsilon \to 0$ in such a way that ε⁻¹n^−ρ${\epsilon }^{-1}{n}^{-\rho }$ remains bounded for some ρ>0$\rho >0$. The main focus is on the construction of explicit contrast functions, but it is noted that the theory covers quadratic martingale estimating functions as a special case. In a simulation study we consider the finite sample behaviour and the applicability to a financial model of an estimator obtained from a simple explicit contrast function.     

Two remarks on frequent hypercyclicity

We show that if T:X→X$T:X\to X$ is a continuous linear operator on an F$F$-space X≠{0}$X\ne \left\{0\right\}$, then the set of frequently hypercyclic vectors of T$T$ is of first category in X$X$, and this answers a question of A. Bonilla and K.-G. Grosse-Erdmann. We also show that if T:X→X$T:X\to X$ is a bounded linear operator on a Banach space X≠{0}$X\ne \left\{0\right\}$ and if T$T$ is frequently hypercyclic (or, more generally, syndetically transitive), then the T^∗$T^{\ast }$-orbit of every non-zero element of X^∗$X^{\ast }$ is bounded away from 0, and in particular T^∗$T^{\ast }$ is not hypercyclic.     

Equilibria and fixed points of set-valued maps with nonconvex and noncompact domains and ranges

Let C$C$ be a nonempty subset of a topological vector space E$E$. We state and prove new various fixed point theorems of Fan–Browder type for set-valued maps F:C→2^E$F:C\to 2^E$ such that C⊂F(C)$C\subset F\left(C\right)$ (called expansive), without assuming that the sets C$C$ and F(C)$F\left(C\right)$ are convex or compact or equal, and E$E$ is Hausdorff. Let K$K$ be a convex subset of E$E$ and let C$C$ be a nonempty subset of K$K$. Our proofs use a technique based on the investigations of the images of maps and restated for maps f:C×K→R∪{−∞,+∞}$f:C\times K\to \mathbb{R}\cup \{-\infty ,+\infty \}$ of G.X.-Z. Yuan’s results concerning the existence of equilibrium points and minimax inequalities for maps f:K×K→R∪{−∞,+∞}$f:K\times K\to \mathbb{R}\cup \{-\infty ,+\infty \}$. Examples are provided.     

A limit theorem for the time of ruin in a Gaussian ruin problem

For certain Gaussian processes X(t)$X\left(t\right)$ with trend −ct^β$-c{t}^{\beta }$ and variance V²(t)$V^2\left(t\right)$, the ruin time is analyzed where the ruin time is defined as the first time point t$t$ such that X(t)−ct^β≥u$X\left(t\right)-c{t}^{\beta }\ge u$. The ruin time is of interest in finance and actuarial subjects. But the ruin time is also of interest in other applications, e.g. in telecommunications where it indicates the first time of an overflow. We derive the asymptotic distribution of the ruin time as u→∞$u\to \infty$ showing that the limiting distribution depends on the parameters β$\beta$, V(t)$V\left(t\right)$ and the correlation function of X(t)$X\left(t\right)$.     

Iterative approximation of solutions of nonlinear equations of Hammerstein type

Suppose X$X$ is a real q$q$-uniformly smooth Banach space and F,K:X→X$F,K:X\to X$ are Lipschitz ?$?$-strongly accretive maps with D(K)=F(X)=X$D\left(K\right)=F\left(X\right)=X$. Let u^∗$u^{\ast }$ denote the unique solution of the Hammerstein equation u+KFu=0$u+KFu=0$. An iteration process recently introduced by Chidume and Zegeye is shown to converge strongly to u^∗$u^{\ast }$. No invertibility assumption is imposed on K$K$ and the operators K$K$ and F$F$ need not be defined on compact subsets of X$X$. Furthermore, our new technique of proof is of independent interest. Finally, some interesting open questions are included.     

Boundedness of extremal solutions in dimension 4

In this paper we establish the boundedness of the extremal solution u^∗$u^{\ast }$ in dimension N=4$N=4$ of the semilinear elliptic equation −Δu=λf(u)$-\Delta u=\lambda f\left(u\right)$, in a general smooth bounded domain Ω⊂R^N$\Omega \subset {\mathbb{R}}^N$, with Dirichlet data u_|∂Ω=0$u{|}_{\partial \Omega }=0$, where f$f$ is a C¹$C^1$ positive, nondecreasing and convex function in [0,∞)$[0,\infty )$ such that f(s)/s→∞$f\left(s\right)/s\to \infty$ as s→∞$s\to \infty$.     

On the length of an external branch in the Beta-coalescent

In this paper, we consider Beta(2−α,α)$(2-\alpha ,\alpha )$ (with 1<α<2$1<\alpha <2$) and related Λ$\Lambda$-coalescents. If T⁽ⁿ⁾$T^{\left(n\right)}$ denotes the length of a randomly chosen external branch of the n$n$-coalescent, we prove the convergence of n^α−1T⁽ⁿ⁾$n^{\alpha -1}{T}^{\left(n\right)}$ when n$n$ tends to ∞$\infty$, and give the limit. To this aim, we give asymptotics for the number σ⁽ⁿ⁾${\sigma }^{\left(n\right)}$ of collisions which occur in the n$n$-coalescent until the end of the chosen external branch, and for the block counting process associated with the n$n$-coalescent.     

Estimating the ground state energy of the Schrödinger equation for convex potentials

In 2011, the fundamental gap conjecture for Schrödinger operators was proven. This can be used to estimate the ground state energy of the time-independent Schrödinger equation with a convex potential and relative error ε$\epsilon$. Classical deterministic algorithms solving this problem have cost exponential in the number of its degrees of freedom d$d$. We show a quantum algorithm, that is based on a perturbation method, for estimating the ground state energy with relative error ε$\epsilon$. The cost of the algorithm is polynomial in d$d$ and ε⁻¹${\epsilon }^{-1}$, while the number of qubits is polynomial in d$d$ and logε⁻¹$log{\epsilon }^{-1}$. In addition, we present an algorithm for preparing a quantum state that overlaps within 1−δ,δ∈(0,1)$1-\delta ,\delta \in (0,1)$, with the ground state eigenvector of the discretized Hamiltonian. This algorithm also approximates the ground state with relative error ε$\epsilon$. The cost of the algorithm is polynomial in d$d$, ε⁻¹${\epsilon }^{-1}$ and δ⁻¹${\delta }^{-1}$, while the number of qubits is polynomial in d$d$, logε⁻¹$log{\epsilon }^{-1}$ and logδ⁻¹$log{\delta }^{-1}$.     

A note on Schweizer–Smital chaos

In this paper, we consider a continuous map f:X→X$f:X\to X$, where X$X$ is a compact metric space, and prove that for any positive integer N$N$, f$f$ is Schweizer–Smital chaotic if and only if f^N$f^N$ is too.     

Nonautonomous equations with arbitrary growth rates: A Perron-type theorem

For a linear equation x^′=A(t)x$x^{\prime }=A\left(t\right)x$, we show that the asymptotic behavior of its solutions is reproduced by the solutions of the nonlinear equation x^′=A(t)x+f(t,x)$x^{\prime }=A\left(t\right)x+f(t,x)$ for any sufficiently small perturbation f$f$. More precisely, we show that if the Lyapunov exponents of the linear equation are limits, even for general exponential rates e^cρ(t)$e^{c\rho \left(t\right)}$ for an arbitrary function ρ$\rho$, then the same happens with the Lyapunov exponents of the solutions of the nonlinear equations, without introducing new values. Our approach is based on Lyapunov’s theory of regularity.     

The spectral method for high order problems with proper simulations of asymptotic behaviors at infinity

In this paper, we investigate new spectral and multidomain spectral methods for high order problems. We introduce a family of new generalized Laguerre functions, which are mutually orthogonal with the weight function x^α(δ+x)^−γ$x^{\alpha }{(\delta +x)}^{-\gamma }$, δ>0$\delta >0$,α$\alpha$ and γ$\gamma$ being arbitrary real numbers. The corresponding quasi-orthogonal approximation and Laguerre–Gauss–Radau type interpolation are proposed. The spectral and multidomain spectral schemes are provided for several model problems, which not only fit the mixed inhomogeneous boundary conditions on the fixed boundary exactly, but also match the asymptotic behaviors at infinity reasonably. Numerical results demonstrate the efficiency of suggested algorithms, and confirm the analysis well.     

Random walk in a high density dynamic random environment

The goal of this note is to prove a law of large numbers for the empirical speed of a green particle that performs a random walk on top of a field of red particles which themselves perform independent simple random walks on Z^d${\mathbb{Z}}^d$, d≥1$d\ge 1$. The red particles jump at rate 1 and are in a Poisson equilibrium with density μ$\mu$. The green particle also jumps at rate 1, but uses different transition kernels p^′$p^{\prime }$ and p^″$p^{″}$ depending on whether it sees a red particle or not. It is shown that, in the limit as μ→∞$\mu \to \infty$, the speed of the green particle tends to the average jump under p^′$p^{\prime }$. This result is far from surprising, but it is non-trivial to prove. The proof that is given in this note is based on techniques that were developed in Kesten and Sidoravicius (2005) to deal with spread-of-infection models. The main difficulty is that, due to particle conservation, space–time correlations in the field of red particles decay slowly. This places the problem in a class of random walks in dynamic random environments for which scaling laws are hard to obtain.     

Extension of positive definite functions

Let Ω⊂Rⁿ$\Omega \subset {\mathbb{R}}^n$ be an open, connected subset of Rⁿ${\mathbb{R}}^n$, and let F:Ω−Ω→C$F:\Omega -\Omega \to \mathbb{C}$, where Ω−Ω={x−y:x,y∈Ω}$\Omega -\Omega =\{x-y:x,y\in \Omega \}$, be a continuous positive definite function. We give necessary and sufficient conditions for F   to have an extension to a continuous positive definite function defined on the entire Euclidean space Rⁿ${\mathbb{R}}^n$. The conditions are formulated in terms of existence of a unitary representations of Rⁿ${\mathbb{R}}^n$ whose generators extend a certain system of unbounded Hermitian operators defined on a Hilbert space associated to F. Different positive definite extensions correspond to different unitary representations.     

Simultaneous bifurcation of limit cycles from a linear center with extra singular points

The period annuli of the planar vector field x^′=−yF(x,y)$x^{\prime }=-yF(x,y)$, y^′=xF(x,y)$y^{\prime }=xF(x,y)$, where the set {F(x,y)=0}$\{F(x,y)=0\}$ consists of k   different isolated points, is defined by k+1$k+1$ concentric annuli. In this paper we perturb it with polynomials of degree n and we study how many limit cycles bifurcate, up to a first order analysis, from all the period annuli simultaneously in terms of k and n  . Additionally, we prove that the associated Abelian integral is piecewise rational and, when k=1$k=1$, the provided upper bound is reached. Finally, the case k=2$k=2$ is also treated.