Effect of the initial estimator on the asymptotic behavior of one-step M-estimator

For a general (real) parameter, let M _nbe the M-estimator and M _n ⁽¹⁾ be its one-step version (based on a suitable initial estimator M _n ⁽⁰⁾). It is known that, under certain regularity conditions, n(M _n ⁽¹⁾-M _n)=O _p(1). The asymptotic distribution of n(M _n ⁽¹⁾-M _n) is studied; it is typically non-normal and it reveals the role of the initial estimator M _n ⁽⁰⁾.Work of this author was partially supported by the Office of Naval Research, Contract No. N00014-83-K-0387     

A note on the growth of velocities in a collisionless plasma

We consider the initial value problem for the classical Vlasov–Poisson system with smooth compactly supported initial data. In the electrostatic case, we show that the size of the velocity support of the distribution function grows at most like t^2/7+bfε for any ε> 0 . Copyright © 2010 John Wiley & Sons, Ltd.     

The Generalized Dirichlet to Neumann Map for the KdV Equation on the Half-Line

For the two versions of the KdV equation on the positive half-line an initial-boundary value problem is well posed if one prescribes an initial condition plus either one boundary condition if q _t and q _xxx have the same sign (KdVI) or two boundary conditions if q _t and q _xxx have opposite sign (KdVII). Constructing the generalized Dirichlet to Neumann map for the above problems means characterizing the unknown boundary values in terms of the given initial and boundary conditions. For example, if {q(x,0),q(0,t)} and {q(x,0),q(0,t),q _x (0,t)} are given for the KdVI and KdVII equations, respectively, then one must construct the unknown boundary values {q _x (0,t),q _xx (0,t)} and {q _xx (0,t)}, respectively. We show that this can be achieved without solving for q(x,t) by analysing a certain “global relation” which couples the given initial and boundary conditions with the unknown boundary values, as well as with the function Φ ^(t)(t,k), where Φ ^(t) satisfies the t-part of the associated Lax pair evaluated at x=0. The analysis of the global relation requires the construction of the so-called Gelfand–Levitan–Marchenko triangular representation for Φ ^(t). In spite of the efforts of several investigators, this problem has remained open. In this paper, we construct the representation for Φ ^(t) for the first time and then, by employing this representation, we solve explicitly the global relation for the unknown boundary values in terms of the given initial and boundary conditions and the function Φ ^(t). This yields the unknown boundary values in terms of a nonlinear Volterra integral equation. We also discuss the implications of this result for the analysis of the long t-asymptotics, as well as for the numerical integration of the KdV equation.     

First Order Differential Operators Associated to the Cauchy—Riemann Operator in the Plane

The article deals with initial value problems of type δw/δt = Fw, w(0, ·) = φ where t is the time and F is a linear first order operator acting in the z = x ? iy-plane. In view of the classical Cauchy-Kovalevkaya Theorem, the initial value problem is solvable provided F has holomorphic coefficients and the initial function is holomorphic. On the other hand, the Lewy example [H. Lewy (1957). An example of a smooth linear partial differential equation without solution. Ann. of Math., 66, 155–158.] shows that there are equations of the above form with infinitely differentiable coefficients not having any solutions. The article in hand constructs, conversely, all linear operators F for which the initial value problem with an arbitrary holomorphic initial function is always solvable. In particular, we shall see that there are equations of that type whose coefficients are only continuous.     

Positive definite temperature functions on the Heisenberg group

We characterize positive definite temperature functions, i.e., positive definite solutions of the heat equation, on the Heisenberg group in terms of the initial values. We also obtain an integral representation for positive definite and U(n)-invariant temperature functions with polynomial growth, where U(n) is the group of all n× n unitary matrices.     

On the regularization mechanism for the periodic Korteweg–de Vries equation

In this paper we develop and use successive averaging methods for explaining the regularization mechanism in the the periodic Korteweg–de Vries (KdV) equation in the homogeneous Sobolev spaces Ḣ^s for s ≥ 0. Specifically, we prove the global existence, uniqueness, and Lipschitz‐continuous dependence on the initial data of the solutions of the periodic KdV. For the case where the initial data is in L₂ we also show the Lipschitz‐continuous dependence of these solutions with respect to the initial data as maps from Ḣ^s to Ḣ^s for s ∈(−1,0]. © 2010 Wiley Periodicals, Inc.     

Bootstrap percolation on the hypercube

In the bootstrap percolation on the n-dimensional hypercube, in the initial position each of the 2ⁿ sites is occupied with probability p and empty with probability 1−p, independently of the state of the other sites. Every occupied site remains occupied for ever, while an empty site becomes occupied if at least two of its neighbours are occupied. If at the end of the process every site is occupied, we say that the (initial) position spans the hypercube. We shall show that there are constants c₁,c₂>0 such that for the probability of spanning tends to 1 as n→∞, while for the probability tends to 0. Furthermore, we shall show that for each n the transition has a sharp threshold function. J. Balogh: work was done while at The University of Memphis, USA Research supported in part by NSF grant DMS0302804 Research supported in part by NSF grant ITR 0225610 and DARPA grant F33615-01-C-1900     

The Hyperbolic Geometric Flow on Riemann Surfaces

In this paper the authors study the hyperbolic geometric flow on Riemann surfaces. This new nonlinear geometric evolution equation was recently introduced by the first two authors, motivated by Einstein equation and Hamilton's Ricci flow. We prove that, for any given initial metric on ?² in certain class of metrics, one can always choose suitable initial velocity symmetric tensor such that the solution exists for all time, and the scalar curvature corresponding to the solution metric g _ij keeps uniformly bounded for all time; moreover, if the initial velocity tensor is suitably “large", then the solution metric g _ij converges to the flat metric at an algebraic rate. If the initial velocity tensor does not satisfy the condition, then the solution blows up at a finite time, and the scalar curvature R(t, x) goes to positive infinity as (t, x) tends to the blowup points, and a flow with surgery has to be considered. The authors attempt to show that, comparing to Ricci flow, the hyperbolic geometric flow has the following advantage: the surgery technique may be replaced by choosing suitable initial velocity tensor. Some geometric properties of hyperbolic geometric flow on general open and closed Riemann surfaces are also discussed.     

Remarks on the uniqueness of second order ODEs

We are concerned with the uniqueness problem for solutions to the second order ODE of the form x″+f(x, t) = 0, subject to appropriate initial conditions, under the sole assumption that f is non-decreasing with respect to x, for each t fixed. We show that there is non-uniqueness in general; on the other hand, several types of reasonable additional assumptions make the problem uniquely solvable.     

A complete upper estimate on the localization for the degenerate parabolic equation with nonlinear source

This paper deals with the Cauchy problem for the degenerate parabolic equation with a strongly nonlinear source where N ≥ 1, p > 2, q ≥ p ? 1, and the blow‐up time T < ∞ . It has been shown that the solution u(x,t) is strictly localized for q ≥ p ? 1, provided that the initial function u₀(x) has a compact support by Liang and Zhao. In addition, if q > 2p ? 1, an upper estimate on the localization in terms of the initial support and the blow‐up time T is partially derived by Liang. In this work, by using the De Giorgi‐type iteration technique, we give a complete estimate on the localization for all q ≥ p ? 1. Copyright © 2014 John Wiley & Sons, Ltd.     

Remarks on the long time behaviour of the solutions of hamilton-jacobi equations

We study the Navier-Stokes equations, written in a domain of R ³ which is invariant under rotation around the vertical axis, or in the whole space R ³; the solutions seeked are also invariant by that rotation, and we look for conditions on the initial data which are close to the natural assumptions in the case of two space dimensions.     

A note on the complexity and tractability of the heat equation

We wish to solve the heat equation u_t=Δu-qu in I^d×(0,T), where I is the unit interval and T is a maximum time value, subject to homogeneous Dirichlet boundary conditions and to initial conditions u(·,0)=f over I^d. We show that this problem is intractable if f belongs to standard Sobolev spaces, even if we have complete information about q. However, if f and q belong to a reproducing kernel Hilbert space with finite-order weights, we can show that the problem is tractable, and can actually be strongly tractable.     

Randomization in the first hitting time problem

In this paper, we consider the following inverse problem for the first hitting time distribution: given a Wiener process with a random initial state, probability distribution, F(t), and a linear boundary, b(t)=μt, find a distribution of the initial state such that the distribution of the first hitting time is F(t). This problem has important applications in credit risk modeling where the process represents the so-called distance to default of an obligor, the first hitting time represents a default event and the boundary separates the healthy states of the obligor from the default state. We show that randomization of the initial state of the process makes the problem analytically tractable.     

The point-vortex method for periodic weak solutions of the 2-D Euler equations

Almost-sure convergence of a subsequence of the vorticity to a weak solution is proven for the point-vortex method for 2-D, inviscid, incompressible fluid flow. Here “almost-sure” is with respect to sequences of random components included in the initial position and strength of each vortex. The initial vorticity is assumed to be periodic and, depending on the initialization scheme, to lie in L log L or L^p with p > 2. The randomization of the initial data is not needed when the initial vorticity is nonnegative; such initial data also need not be periodic, and is only required to be a bounded measure lying in H⁻¹. All these results are also valid for the “vortex-blob” method with the smoothing parameter vanishing at an arbitrary rate. The sense in which solutions of point-vortex dynamics are weak solutions of the Euler equations is also discussed.     

Concerning the wave equation on asymptotically Euclidean manifolds

We obtain KSS, Strichartz and certain weighted Strichartz estimates for the wave equation on (ℝ ^d , g), d ≥ 3, when the metric g is non-trapping and approaches the Euclidean metric like 〈x〉^−ρ with ρ > 0. Using the KSS estimate, we prove almost global existence for quadratically semilinear wave equations with small initial data for ρ > 1 and d = 3. Also, we establish the Strauss conjecture when the metric is radial with ρ > 1 for d = 3.     

On the validity of the Ginzburg-Landau equation

Summary The famous Ginzburg-Landau equation describes nonlinear amplitude modulations of a wave perturbation of a basic pattern when a control parameterR lies in the unstable regionO(ε ²) away from the critical valueR _c for which the system loses stability. Hereε>0 is a small parameter. G-L's equation is found for a general class of nonlinear evolution problems including several classical problems from hydrodynamics and other fields of physics and chemistry. Up to now, the rigorous derivation of G-L's equation for general situations is not yet completed. This was only demonstrated for special types of solutions (steady, time periodic) or for special problems (the Swift-Hohenberg equation). Here a mathematically rigorous proof of the validity of G-L's equation is given for a general situation of one space variable and a quadratic nonlinearity. Validity is meant in the following sense. For each given initial condition in a suitable Banach space there exists a unique bounded solution of the initial value problem for G-L's equation on a finite interval of theO(1/ε²)-long time scale intrinsic to the modulation. For such a finite time interval of the intrinsic modulation time scale on which the initial value problem for G-L's equation has a bounded solution, the initial value problem for the original evolution equation with corresponding initial conditions, has a unique solutionO(ε²) — close to the approximation induced by the solution of G-L's equation. This property guarantees that, for rather general initial conditions on the intrinsic modulation time scale, the behavior of solutions of G-L's equation is really inherited from solutions of the original problem, and the other way around: to a solution of G-L's equation corresponds a nearby exact solution with a relatively small error.     

Existence and asymptotic behavior to the incompressible nematic liquid crystal flow in the whole space

In this article, first of all, the global existence and asymptotic stability of solutions to the incompressible nematic liquid crystal flow is investigated when initial data are a small perturbation near the constant steady state (0,δ₀); here, δ₀ is a constant vector with |δ₀|=1. Precisely, we show the existence and asymptotic stability with small initial data for . The initial data class of us is not entirely included in the space BMO^?1×BMO and contains strongly singular functions and measures. As an application, we obtain a class of asymptotic existence of a basin of attraction for each self‐similar solution with homogeneous initial data. We also study global existence of a large class of decaying solutions and construct an explicit asymptotic formula for ∣x∣→∞, relating the self‐similar profile (U(x),D(x)) to its corresponding initial data (u₀,d₀). In two dimensions, we obtain higher‐order asymptotics of (u(x),d(x)). Copyright © 2016 John Wiley & Sons, Ltd.     

Two Remarks on Independent Sets

In the first part we generalize the notion of strongly independent sets, introduced in [10] for polynomial ideals, to submodules of free modules and explain their computational relevance. We discuss also two algorithms to compute strongly independent sets that rest on the primary decomposition of squarefree monomial ideals.Usually the initial ideal in(I) of a polynomial ideal I is worse than I. In [9] the authors observed that nevertheless in(I) is not as bad as one should expect, showing that in(I) is connected in codimension one if I is prime.In the second part of the paper we add more evidence to that observation. We show that in(I) inherits (radically) unmixedness, connectedness in codimension one and connectedness outside a finite set of points from I and prove the same results also for initial submodules of free modules. The proofs use a deformation from I to in(I ).     

Properties of solutions to the initial-boundary value problem for a porous media-type equation

The initial-boundary value problem in a semi-infinite strip (0, ∞)×(0, T) for a degenerate parabolic equation of the form u, _t= φ(u)_xx + b(x)φ(u)_x is considered. The properties of solutions in the case where the initial function is compactly supported and for constant initial and boundary data are investigated.     

On the recurrence of edge-reinforced random walk on ℤ×G

Let G be a finite tree. It is shown that edge-reinforced random walk on ℤ×G with large initial weights is recurrent. This includes recurrence on multi-level ladders of arbitrary width. For edge-reinforced random walk on {0,1, . . . ,n}×G, it is proved that asymptotically, with high probability, the normalized edge local times decay exponentially in the distance from the starting level. The estimates are uniform in n. They are used in the recurrence proof.