Analyticity of classical steady needle crystals

This paper is a continuation of our previous work “Rigorous results in selection of steady needle crystals, J. Differential Equations 197 (2004) 349-426”. It concerns analyticity of a classical steadily translating needle crystal. It is proved that any classical solution to the needle crystal problem with sufficiently small but nonzero surface tension, if its slope deviation is close to some Ivantsov zero-surface-tension solution and if its curvature satisfies some algebraic decay conditions at ∞, must belong to the analytic function space A₀ defined in §1 and chosen in the previous study mentioned above. The analyticity result implies that there can be no classical steady needle crystal solution when anisotropy is zero.     

Statistical properties of topological Collet-Eckmann maps

We study geometric and statistical properties of complex rational maps satisfying a non-uniform hyperbolicity condition called “Topological Collet-Eckmann”. This condition is weaker than the “Collet-Eckmann” condition. We show that every such map possesses a unique conformal probability measure of minimal exponent, and that this measure is non-atomic, ergodic, and that its Hausdorff dimension is equal to the Hausdorff dimension of the Julia set. Furthermore, we show that there is a unique invariant probability measure that is absolutely continuous with respect to this conformal measure, and that this invariant measure is exponentially mixing (it has exponential decay of correlations) and satisfies the Central Limit Theorem.We also show that for a complex rational map the existence of such invariant measure characterizes the Topological Collet-Eckmann condition: a rational map satisfies the Topological Collet-Eckmann condition if, and only if, it possesses an exponentially mixing invariant measure that is absolutely continuous with respect to some conformal measure, and whose topological support contains at least 2 points.     

Exponential decay for a von Kármán equations of memory type with acoustic boundary conditions

In this paper, we consider the dynamical von Kármán equations of memory type with acoustic boundary conditions. We show an exponential decay result of solutions under weaker assumption than the ones frequently used in the literature. In particular, the kernel we are considering is not necessarily exponentially decaying to zero as was assumed before. Copyright © 2014 John Wiley & Sons, Ltd.     

Conditions that cause risk pooling to increase inventory

We say product A is a partial substitute for product B if a fraction of the customers who prefer B are willing to accept A when B is out of stock. When demand is uncertain, it is intuitive and true that a larger “willing to substitute” fraction implies larger expected profits. A higher “willing to substitute” fraction allows one to pool the risk of individual products. It may also be intuitive that a larger “willing to substitute” fraction might result in lower optimal total inventory. For the full substitution structure, several researchers have shown that for certain distributions such as the exponential, this latter intuition is not true. We show that this full substitution anomaly can occur with any right skewed demand distribution. We assume i.i.d. demand distributions unless we indicate otherwise. We also show that the anomaly can occur for a number of realistic situations of partial substitution with commonly used demand distributions such as Normal, exponential, Poisson, and uniform. We also demonstrate the anomaly for more than one period, with backlogging, lost sales, more than two products, and with setup costs.     

Stabilization of linear systems by rotation

We introduce the concept of “stabilization by rotation” for deterministic linear systems with negative trace. This concept encompasses the well-known concept of “vibrational stabilization” introduced by Meerkov in the 1970s and is a deterministic version of ‘stabilization by noise’ for stochastic systems as introduced by Arnold and coworkers in the 1980s. It is shown that a linear system with negative trace can be stabilized by adding a skew-symmetric matrix, multiplied by a suitable scalar so-called “gain function” (possibly a constant) which is sufficiently large. To overcome the problem of what is “sufficiently large”, we also present a servo mechanism which tunes the gain function by learning from the trajectory until finally the trajectory tends to zero. This approach allows to show that one of Meerkov's assumptions for vibrational stabilization is superfluous. Moreover, while Meerkov as well as Arnold and coworkers assume that a stabilizing periodic function or the noise has sufficiently large frequency and amplitude, we also provide a servo mechanism to determine this function dynamically in a deterministic setup.     

Polynomial stability without polynomial decay of the relaxation function

We consider a linear viscoelastic problem and prove polynomial asymptotic stability of the steady state. This work improves previous works where it is proved that polynomial decay of solutions to the equilibrium state occurs provided that the relaxation function itself is polynomially decaying to zero. In this paper we will not assume any decay rate of the relaxation function. In case the kernel has some flat zones then we prove polynomial decay of solutions provided that these flat zones are not too big. If the kernel is strictly decreasing then there is no need for this assumption. Copyright © 2008 John Wiley & Sons, Ltd.     

Almost sure rates of mixing for i.i.d. unimodal maps

It has been known since the pioneering work of Jakobson and subsequent work by Benedicks and Carleson and others that a positive measure set of quadratic maps admit an absolutely continuous invariant measure. Young and Keller-Nowicki proved exponential decay of its correlation functions. Benedicks and Young [8], and Baladi and Viana [4] studied stability of the density and exponential rate of decay of the Markov chain associated to i.i.d. small perturbations. The almost sure statistical properties of the sample stationary measures of i.i.d. itineraries are more difficult to estimate than the “averaged statistics”. Adapting to random systems, on the one hand partitions associated to hyperbolic times due to Alves [1], and on the other a probabilistic coupling method introduced by Young [26] to study rates of mixing, we prove stretched exponential upper bounds for the almost sure rates of mixing.     

Hypoelliptic heat kernel inequalities on the Heisenberg group

We study the existence of “L^p-type” gradient estimates for the heat kernel of the natural hypoelliptic “Laplacian” on the real three-dimensional Heisenberg Lie group. Using Malliavin calculus methods, we verify that these estimates hold in the case p>1. The gradient estimate for p=2 implies a corresponding Poincaré inequality for the heat kernel. The gradient estimate for p=1 is still open; if proved, this estimate would imply a logarithmic Sobolev inequality for the heat kernel.     

Asymptotic behavior for coupled abstract evolution equations with one infinite memory

In this paper, we consider two coupled abstract linear evolution equations with one infinite memory acting on the first equation. Our work is motivated by the recent results of [42], where the authors considered the case of two wave equations with one convolution kernel converging exponentially to zero at infinity, and proved the lack of exponential decay. On the other hand, the authors of [42] proved that the solutions decay polynomially at infinity with a decay rate depending on the regularity of the initial data. Under a boundedness condition on the past history data, we prove that the stability of our abstract system holds for convolution kernels having much weaker decay rates than the exponential one. The general and precise decay estimate of solution we obtain depends on the growth of the convolution kernel at infinity, the regularity of the initial data, and the connection between the operators describing the considered equations. We also present various applications to some distributed coupled systems such as wave-wave, Petrovsky-Petrovsky, wave-Petrovsky, and elasticity-elasticity.     

Toroidal automorphic forms for some function fields

Zagier introduced toroidal automorphic forms to study the zeros of zeta functions: an automorphic form on GL₂ is toroidal if all its right translates integrate to zero over all non-split tori in GL₂, and an Eisenstein series is toroidal if its weight is a zero of the zeta function of the corresponding field. We compute the space of such forms for the global function fields of class number one and genus g?1, and with a rational place. The space has dimension g and is spanned by the expected Eisenstein series. We deduce an “automorphic” proof for the Riemann hypothesis for the zeta function of those curves.     

Localized Polynomial Frames on the Interval with Jacobi Weights

As is well known the kernel of the orthogonal projector onto the polynomials of degree n in L²(w_α,β, [−1, 1]), w_α,β(t) = (1 − t)^α(1 + t)^β, can be written in terms of Jacobi polynomials. It is shown that if the coefficients in this kernel are smoothed out by sampling a compactly supported C^∞ function then the resulting function has nearly exponential (faster than any polynomial) rate of decay away from the main diagonal. This result is used for the construction of tight polynomial frames for L²(w_α,β) with elements having almost exponential localization.     

Representations of trigonometric Cherednik algebras of rank one in positive characteristic

In this paper, we classify the irreducible representations of the trigonometric Cherednik algebras of rank 1 in characteristic p>0. There are two cases. One is the “quantum” case, where “Planck’s constant” is nonzero and generic irreducible representations have dimension 2p. In this case, smaller representations exist if and only if the “coupling constant” k is in ; namely, if k is an even integer such that 0≤k≤p−1, then there exist irreducible representations of dimensions p−k and p+k, and if k is an odd integer such that 1≤k≤p−2, then there exist irreducible representations of dimensions k and 2p−k. The other case is the “classical” case, where “Planck’s constant” is zero and generic irreducible representations have dimension 2. In that case, one-dimensional representations exist if and only if the “coupling constant” k is zero.     

The Carlitz shtuka

Recently we have used the Carlitz exponential map to define a finitely generated submodule of the Carlitz module having the right properties to be a function field analogue of the group of units in a number field. Similarly, we constructed a finite module analogous to the class group of a number field. In this short note more algebraic constructions of these “unit” and “class” modules are given and they are related to Ext modules in the category of shtukas.     

Bases for kernel-based spaces

Since it is well-known (De Marchi and Schaback (2001) [4]) that standard bases of kernel translates are badly conditioned while the interpolation itself is not unstable in function space, this paper surveys the choices of other bases. All data-dependent bases turn out to be defined via a factorization of the kernel matrix defined by these data, and a discussion of various matrix factorizations (e.g. Cholesky, QR, SVD) provides a variety of different bases with different properties. Special attention is given to duality, stability, orthogonality, adaptivity, and computational efficiency. The “Newton” basis arising from a pivoted Cholesky factorization turns out to be stable and computationally cheap while being orthonormal in the “native” Hilbert space of the kernel. Efficient adaptive algorithms for calculating the Newton basis along the lines of orthogonal matching pursuit conclude the paper.     

Polynomial contractions and Lyapunov regularity

The purpose of this note is twofold: to introduce the notion of polynomial contraction for a linear nonautonomous dynamics with discrete time, and to show that it persists under sufficiently small linear and nonlinear perturbations. The notion of polynomial contraction mimics the notion of exponential contraction, but with the exponential decay replaced by a polynomial decay. We show that this behavior is exhibited by a large class of dynamics, by giving necessary conditions in terms of “polynomial” Lyapunov exponents. Finally, we establish the persistence of the asymptotic stability of a polynomial contraction under sufficiently small linear and nonlinear perturbations. We also consider the case of nonuniform polynomial contractions, for which the Lyapunov stability is not uniform.     

Slow motion of gradient flows

We present sufficient conditions on an energy landscape in order for the associated gradient flow to exhibit slow motion or “dynamic metastability.” The first condition is a weak form of convexity transverse to the so-called slow manifold, N. The second condition is that the energy restricted to N is Lipschitz with a constant δ?1. One feature of the abstract result that makes it of broader interest is that it does not rely on maximum principles.As an application, we give a new proof of the exponentially slow motion of transition layers in the one-dimensional Allen-Cahn equation. The analysis is more nonlinear than previous work: It relies on the nonlinear convexity condition or “energy-energy-dissipation inequality.” (Although we do use the maximum principle for convenience in the application, we believe it may be removed with additional work.) Our result demonstrates that a broad class of initial data relaxes with an exponential rate into a δ-neighborhood of the slow manifold, where it is then trapped for an exponentially long time.     

Exponential decay for a viscoelastic problem with a singular kernel

In this paper we consider a problem which arises in viscoelasticity. We prove exponential decay of solutions for the problem with a memory term involving a kernel which is singular at zero. This is established by introducing an appropriate Lyapunov type functional and using the energy method. This work extends earlier results.      

The Oshima-Sekiguchi and Liouville theorems on Heintze groups

Let L be an elliptic operator on a Riemannian manifold M. A function F annihilated by L is said to be L-harmonic. F is said to have moderate growth if and only if F grows at most exponentially in the Riemannian distance. If M is a rank-one symmetric space and L is the Laplace-Beltrami operator for M, the Oshima-Sekiguchi theorem [T. Oshima, J. Sekiguchi, Eigenspaces of invariant differential operators on an affine symmetric space, Invent. Math. 57 (1980) 1-81] states that a L-harmonic function F has moderate growth if and only if F is the Poisson integral of a distribution on the Furstenberg boundary. In this work we prove that this result generalizes to a very large class of homogeneous Riemannian manifolds of negative curvature. We also (i) prove a Liouville type theorem that characterizes the “polynomial-like” harmonic functions which vanish on the boundary in terms of their growth properties, (ii) describe all “polynomial-like” harmonic functions, and (iii) give asymptotic expansions for the Poisson kernel. One consequence of this work is that every Schwartz distribution on the boundary is the boundary value for a L-harmonic function F which is uniquely determined modulo “polynomial-like” harmonic functions.     

Some examples for the Poincaré and Painlevé problems

In 1891, Poincaré started a series of three papers in which he tried to answer the following question (cf. [21-23]): “Is it possible to decide if an algebraic differential equation in two variables is algebraically integrable?” (in the sense that it has a rational first integral). More or less at the same time P. Painlevé asked the following question: “Is it possible to recognize the genus of the general solution of an algebraic differential equation in two variables which has a rational first integral?”. In this paper we give examples of one-parameter families which show that both problems have a negative answer. With some of the families we can also answer a question posed by M. Brunella in [5].     

Criterions for detecting the existence of the exponential dichotomies in the asymptotic behavior of the solutions of variational equations

We prove that the admissibility of any pair of vector-valued Schäffer function spaces (satisfying a very general technical condition) implies the existence of a “no past” exponential dichotomy for an exponentially bounded, strongly continuous cocycle (over a semiflow). Roughly speaking the class of Schäffer function spaces consists in all function spaces which are invariant under the right-shift and therefore our approach addresses most of the possible pairs of admissible spaces. Complete characterizations for the exponential dichotomy of cocycles are also obtained. Moreover, we involve a concept of a “no past” exponential dichotomy for cocycles weaker than the classical concept defined by Sacker and Sell (1994) in [23]. Our definition of exponential dichotomy follows partially the definition given by Chow and Leiva (1996) in [4] in the sense that we allow the unstable subspace to have infinite dimension. The main difference is that we do not assume a priori that the cocycle is invertible on the unstable space (actually we do not even assume that the unstable space is invariant under the cocycle). Thus we generalize some known results due to O. Perron (1930) [14], J. Daleckij and M. Krein (1974) [7], J.L. Massera and J.J. Schäffer (1966) [11], N. van Minh, F. Räbiger and R. Schnaubelt (1998) [26].