Propagation of polarization in elastodynamics with residual stress and travel times

 We show that knowing the displacement-to-traction map associated to the equations of isotropic elastodynamics with residual stress we can determine the lens maps of compressional and shear waves. We derive several consequences of this for the inverse problem of determining the residual stress and the Lamé parameters from the displacement-to-traction map. Received: 6 December 2001 / Revised version: 29 October 2002 / Published online: 8 April 2003 RID="⋆" ID="⋆" The author thanks the Department of Mathematics at the University of Washington for its hospitality during his visit in fall 2000. RID="⋆⋆" ID="⋆⋆" Partly supported by NSF grant DMS-0070488 and a John Simon Guggenheim fellowship. The author also thanks MSRI for partial support and for providing a very stimulating environment during the inverse problems program in fall 2001.     

Splitting ρκλ into maximally many stationary sets

Letκ >ω be a regular cardinal and λ >κ a cardinal. Solovay’s classical result for κ[So] led Menas [Me] to conjecture that a stationary subset ofP _κλ would split into λ^<κ stationary set of size κ⁺ (see[BT]), the conjecture implies that the size is (κ⁺)^<κ as well. Part of this work was done during the author’s stay at Boston University as one of the Japanese Overseas Research Fellows. He gratefully acknowledge Professor Akihiro Kanamori’s hospitality. He also wishes to thank members of the set theory seminar at Waseda University for their interest at the early stage.     

Mollification formulas and implicit smoothing

This paper develops some mollification formulas involving convolutions between popular radial basis function (RBF) basic functions Φ, and suitable mollifiers. Polyharmonic splines, scaled Bessel kernels (Matern functions) and compactly supported basic functions are considered. A typical result is that in ℛ^d the convolution of |{•}|^β and (•²+c ²)^−(β+2d)/2 is the generalized multiquadric (•²+c ²)^β/2 up to a multiplicative constant. The constant depends on c>0, β, where ℜ(β)>−d, and d. An application which motivated the development of the formulas is a technique called implicit smoothing. This computationally efficient technique smooths a previously obtained RBF fit by replacing the basic function Φ with a smoother version Ψ during evaluation.     

Index information algorithm with local tuning for solving multidimensional global optimization problems with multiextremal constraints

 Multidimensional optimization problems where the objective function and the constraints are multiextremal non-differentiable Lipschitz functions (with unknown Lipschitz constants) and the feasible region is a finite collection of robust nonconvex subregions are considered. Both the objective function and the constraints may be partially defined. To solve such problems an algorithm is proposed, that uses Peano space-filling curves and the index scheme to reduce the original problem to a H?lder one-dimensional one. Local tuning on the behaviour of the objective function and constraints is used during the work of the global optimization procedure in order to accelerate the search. The method neither uses penalty coefficients nor additional variables. Convergence conditions are established. Numerical experiments confirm the good performance of the technique. Received: April 2002 / Accepted: December 2002 Published online: March 21, 2003 RID="⋆" ID="⋆" This research was supported by the following grants: FIRB RBNE01WBBB, FIRB RBAU01JYPN, and RFBR 01–01–00587. Key Words. global optimization – multiextremal constraints – local tuning – index approach     

Homotopes of (−1, 1)-algebras with two generators

We present an example of a (−1, 1)-algebra that has an isotope which is not an (−1, 1)-algebra. We prove that the defining relation is preserved by the homotopes of 2-generated (−1, 1)-algebras and, moreover, that the variety generated by a free (−1, 1)-algebra of rank 2 is stable under the operation of taking a homotope. Translated fromMatematicheskie Zametki, Vol. 59, No. 4, pp. 550–556, April, 1996. I wish to thank Professor S. V. Pchelintsev for his help during the preparation of this paper.     

Lorentz Gas with Thermostatted Walls

In a planar periodic Lorentz gas, a point particle (electron) moves freely and collides with fixed round obstacles (ions). If a constant force (induced by an electric field) acts on the particle, the latter will accelerate, and its speed will approach infinity (Chernov and Dolgopyat in J Am Math Soc 22:821–858, 2009; Phys Rev Lett 99, paper 030601, 2007). To keep the kinetic energy bounded one can apply a Gaussian thermostat, which forces the particle’s speed to be constant. Then an electric current sets in and one can prove Ohm’s law and the Einstein relation (Chernov and Dolgopyat in Russian Math Surv 64:73–124, 2009; Chernov et al. Comm Math Phys 154:569–601, 1993; Phys Rev Lett 70:2209–2212, 1993). However, the Gaussian thermostat has been criticized as unrealistic, because it acts all the time, even during the free flights between collisions. We propose a new model, where during the free flights the electron accelerates, but at the collisions with ions its total energy is reset to a fixed level; thus our thermostat is restricted to the surface of the scatterers (the ‘walls’). We rederive all physically interesting facts proven for the Gaussian thermostat in Chernov, Dolgopyat (Russian Math Surv 64:73–124, 2009) and Chernov et al. (Comm Math Phys 154:569–601, 1993; Phys Rev Lett 70:2209–2212, 1993), including Ohm’s law and the Einstein relation. In addition, we investigate the superconductivity phenomenon in the infinite horizon case.     

Convergence to equilibria for compressible navier-stokes equations with large data

We show that for any sequence t_n→∞ and any global weak solution (ρ(t), u(t)) of barotropic compressible Navier-Stokes equations in 2 and 3 space dimensions with potential force there exists a subsequence {s_n} such that ϱ(s_n) in L _ω ^r (Ω) of spatially periodic functions, where ϱ_∞ is an equilibrium density, r>1 suitable. If the equilibrium is unique then the convergence holds for all t→∞. No smallness of data is assumed. Entrata in Redazione il 22 febbraio 1999. Ricevuta nuova versione il 24 agosto 1999. This work was started and completed during the stay of the second author at the University of Toulon in March 1997 and April 1998, which is acknowledged for the financial support, and also partially supported by the Grant Agency of the Czech Republic, Grant No. 201/96/0313.     

Scattering at Zero Energy for Attractive Homogeneous Potentials

We compute up to a compact term the zero-energy scattering matrix for a class of potentials asymptotically behaving as −γ|x|^−μ with 0 < μ < 2 and γ > 0. It turns out to be the propagator for the wave equation on the sphere at time . The research of J.D. is supported in part by the grant N N201 270135. Part of the research was done during a visit of both authors to the Erwin Schr?dinger Institute. Submitted: November 28, 2008. Accepted: March 2, 2009.     

A homological characterization of piecewise hereditary algebras

Let Λ be a finite dimensional algebra over a field k. We will show here that Λ is piecewise hereditary if and only if its strong global dimension is finite. Dedicated to Otto Kerner on the occasion of his 65th birthday. The second author is supported by a grant from NSA. These results were obtained during a visit of the first author at Syracuse University. He would like to thank the second author for the hospitality during his stay.     

On visibility and covering by convex sets

A setX⊆ℝ ^d isn-convex if among anyn of its points there exist two such that the segment connecting them is contained inX. Perles and Shelah have shown that any closed (n+1)-convex set in the plane is the union of at mostn ⁶ convex sets. We improve their bound to 18n ³, and show a lower bound of order Ω(n ²). We also show that ifX⊆ℝ² is ann-convex set such that its complement has λ one-point path-connectivity components, λ<∞, thenX is the union ofO(n ⁴+n ²λ) convex sets. Two other results onn-convex sets are stated in the introduction (Corollary 1.2 and Proposition 1.4). Research supported by Charles University grants GAUK 99/158 and 99/159, and by U.S.-Czechoslovak Science and Technology Program Grant No. 94051. Part of the work by J. Matoušek was done during the author’s visits at Tel Aviv University and The Hebrew University of Jerusalem. Part of the work by P. Valtr was done during his visit at the University of Cambridge supported by EC Network DIMANET/PECO Contract No. ERBCIPDCT 94-0623.     

Instability,complexity, and evolution

In this paper, we consider a new class of random dynamical systems that contains, in particular, neural networks and complicated circuits. For these systems, we consider the viability problem: we suppose that the system survives only the system state is in a prescribed domain Π of the phase space. The approach developed here is based on some fundamental ideas proposed by A. Kolmogorov, R. Thom, M. Gromov, L. Valiant, L. Van Valen, and others. Under some conditions it is shown that almost all systems from this class with fixed parameters are unstable in the following sense: the probability P _t to leave Π within the time interval [0, t] tends to 1 as t → ∞. However, it is allowed to change these parameters sometimes (“evolutionary” case), then it may happen that P _t  < 1 − δ  < 1 for all t (“stable evolution”). Furthermore, we study the properties of such a stable evolution assuming that the system parameters are encoded by a dicsrete code. This allows us to apply complexity theory, coding, algorithms, etc. Evolution is a Markov process of modification of this code. Under some conditions we show that the stable evolution of unstable systems possesses the following general fundamental property: the relative Kolmogorov complexity of the code cannot be bounded by a constant as t → ∞. For circuit models, we define complexity characteristics of these circuits. We find that these complexities also have a tendency to increase during stable evolution. We give concrete examples of stable evolution. Bibliography: 80 titles. To the memory of A. N. Livshitz Published in Zapiski Nauchnykh Seminarov POMI, Vol. 360, 2008, pp. 31–69.     

Iterated forcing and normal ideals onω 1

We prove that suitable iteration does not collapse ℵ₁ [and does not add reals], i.e., that in such iteration, certain sealing of maximal antichains of stationary subsets ofω ₁ is allowed. As an application, e.g., we prove from supercompact hypotheses, mainly, the consistency of: ZFC + GCH + “for some stationary setS ⊆ω ₁, {ie345-1}(ω ₁)/(D ω ₁ +S) is the Levy algebra” (i.e., the complete Boolean Algebra corresponding to the Levy collapse Levy (ℵ₀,<ℵ₂) (and we can add “a variant of PFA”) and the consistency of the same, with “Ulam property” replacing “Levy algebra”). The paper assumes no specialized knowledge (if you agree to believe in the semi-properness iteration theorem and RCS iteration). This research was partially supported by the NSF. This paper was largely written during the author’s visit at Cal Tech around the end of April 1985. The author would like to thank M. Foreman, A. Kekris and H. Woodin for their hospitality.     

The zeros of functions in Nevanlinna’s area class

Let {Z_n=1{( _n ^∓ ) bea sequence of points in the unit open disk, and letNϕ(U) denote the class of functionsf analytic in the unit disk U such that |f|∈L ( _ϕ ¹ )(U). For ϕ ≡ 1, the necessary and sufficient conditions for the existence off εN(U) and vanishing atz _n is Σ( _n=1 ^∓ ) (1–|Z_n|)² ∞. Also we estimate a large family of canonical products. These results are extended to ϕ(z)=(1-|z|)^ϕ. This represents a part of a Ph.D. thesis conducted at the Technion — Israel Institute of Technology, Department of Mathematics, by Dr. C. A. Horowitz. His help during the preparation of this paper is gratefully acknowledged.     

Gorenstein liaison of 0-dimensional schemes

 The theory of Gorenstein liaison has been developed during the last 3 years to generalize liaison theory of codimension 2 schemes to schemes of codimension ≥ 3 in a projective space. One of the main open questions in Gorenstein liaison theory is whether any arithmetically Cohen-Macaulay subscheme of ℙ ⁿ is in the Gorenstein liaison class of a complete intersection. In this paper we prove that any set of general points lying on a rational normal scroll surface is in the Gorenstein liaison class of a complete intersection. Received: 21 November 2001 Published online: 24 April 2003 Mathematics Subject Classification (2000): 14M06, 14C20, 14M05     

<Emphasis Type="Italic">K</Emphasis>-theory from the point of view of <Emphasis Type="Italic">C</Emphasis><Superscript>*</Superscript>-algebras and Fredholm representations

These notes represent the subject of five lectures which were delivered as a minicourse during the VI conference in Krynica, Poland, “Geometry and Topology of Manifolds”, May, 2–8, 2004.     

Martingale transforms and L p -norm estimates of Riesz transforms on complete Riemannian manifolds

Under the condition that the Bakry–Emery Ricci curvature is bounded from below, we prove a probabilistic representation formula of the Riesz transforms associated with a symmetric diffusion operator on a complete Riemannian manifold. Using the Burkholder sharp L ^p -inequality for martingale transforms, we obtain an explicit and dimension-free upper bound of the L ^p -norm of the Riesz transforms on such complete Riemannian manifolds for all 1 < p < ∞. In the Euclidean and the Gaussian cases, our upper bound is asymptotically sharp when p→ 1 and when p→ ∞. Research partially supported by a Delegation in CNRS at the University of Paris-Sud during the 2005–2006 academic year.     

On property (β) in Banach lattices, Calderón-Lozanowskii and Orlicz-Lorentz spaces

The geometry of Calderón-Lozanowskii spaces, which are strongly connected with the interpolation theory, was essentially developing during the last few years (see [4, 9, 10, 12, 13, 17]). On the other hand many authors investigated property (β) in Banach spaces (see [7, 19, 20, 21, 25, 26]). The first aim of this paper is to study property (β) in Banach function lattices. Namely a criterion for property (β) in Banach function lattice is presented. In particular we get that in Banach function lattice property (β) implies uniform monotonicity. Moreover, property (β) in generalized Calderón-Lozanowskii function spaces is studied. Finally, it is shown that in Orlicz-Lorentz function spaces property (β) and uniform convexity coincide.     

Assessment of hospital efficiency under health transformation program in Turkey

The purpose of this research is to explore the operational performance of the Ministry of Health’s general public hospitals following the implementation of the Health Transformation Program in Turkey. This study adopts the Malmquist index to analyze the operational performance of the 352 Ministry of Health’s general hospitals during the period 2005–2008. The result show that the operational performance of these hospitals regardless of the attributes present a common tendency that the performance of 2005–2007 progressed as compared to the previous year, while that of 2008 has regressed as compared to 2007.     

New asymptotic expansion for the Gamma function

Using a series transformation, the Stirling-De Moivre asymptotic series approximation to the Gamma function is converted into a new one with better convergence properties. The new formula is being compared with those of Stirling, Laplace, and Ramanujan for real arguments greater than 0.5 and turns out to be, for equal number of “correction” terms, numerically superior to all of them. As a side benefit, a closed-form approximation has turned up during the analysis which is about as good as 3rd order Stirling’s (maximum relative error smaller than 1e − 10 for real arguments greater or equal to 24).