Resolving Singularities in a Ramsey-type Growth Model

The aim of this paper is to discuss a mathematical solution procedure to solve a Ramsay-type growth model that explains the fundamentals of consumption and capital accumula-tion in a dynamic equilibrium setting. The problem is formulated as a system of recursive equations and studied through some numerical experiments for the time path of the different variables of the model under some alternative assumption for the steady-state equilibrium of the labour market conditioning the possible singularity of the model.     

Instabilities in Concave, Dynamic, Economic Optimization

An important and numerous literature argues that nonconcavity (often convexity with respect to the state) of the Hamiltonian leads to multiple steady states, instability, and a threshold. This threshold property provides a powerful paradigm to explain history dependency and hysteresis. This paper shows that economically relevant properties (in particular, multiple steady states and thresholds) are possible in strict concave models too. Two corresponding necessary conditions with intuitive economic interpretation are derived.     

Vector optimization: Singularities, regularizations

We discuss three scalarizations of the multiobjectie optimization from the point of view of the parametric optimization. We analyze three important aspects:

Isolated Singularities,Smooth Orders,and Auslander Regularity

Abstract

In this note we prove that a smooth order satisfying the reverse geometric engineering conditions in stringtheory of Berenstein (Berenstein, D. Reverse geometric engineering of singularities,arXiv:hep-th/0201093, Sec. 2) (in any compactifying dimension) is Auslander regular. Moreover, we classify the étale local structure of smooth orders over an isolated central singularity.     

Singularities in variational calculus

A survey is given of theories of singularities of systems of rays and wave fronts, that is, singularities of systems of extremals of variational problems and solutions of the Hamilton-Jacobi equations near caustics. The problem of passing about an obstacle bounded by a smooth surface of general position is studied in detail. Theorems are proved on the normal forms of Lagrangian manifolds with singularities formed by rays of the system of extremals of a variational problem in the symplectic space of all oriented lines which tear off from the surface of the obstacle as well as theorems on Legendre manifolds with singularities formed by contact elements of a wave front and 1-jets of a solution of the Hamilton-Jacobi equation.Translated from Itogi Nauki i Tekhniki, Seriya Sovremennye Problemy Matematiki, Vol. 22, pp. 3–55, 1983.     

Singularities in derived categories

Let A be a finite dimensional algebra over an algebraically closed field k and let M and N be two complexes in the bounded derived category D^b(A) of finitely generated A-modules. Together with Alexander Zimmermann we have defined a notion of degeneration for derived categories. We say that M degenerates to N if there is a complex Z and an exact triangle N→M⊕Z→Z→N[1]. In this paper we define and study the type of singularity at every degeneration in the bounded derived categrory.     

Motivic Integration, Quotient Singularities and the McKay Correspondence

The present work is devoted to the study of motivic integration on quotient singularities. We give a new proof of a form of the McKay correspondence previously proved by Batyrev. The paper contains also some general results on motivic integration on arbitrary singular spaces.(UMR 8553 du CNRS)|. E-mail:     

Stress Singularities in Viscoelastic Media 2. Plane-Strain Stress Singularities at Corners

The stress singularities that evolve at the corner of a notchedviscoelastic angular plate subject to mode I deformation isdiscussed when prescribed, but arbitrary, displacements aresymmetrically applied to both radial edges of the sector. Thesolution procedure, based on Laplace and Mellin transforms (withtransform parameters p and s, respectively), leads to an eigenvalueproblem in the complex p-plane, which is dependent on Poisson'sratio and characterizes the singular behaviour of the stressfields. Although simple solutions to the transcendental eigenequationare not available, the real-time evolution of the stress concentrationsis obtained by monitoring a particular branch of the eigenvalueequation as it moves in the complex p-plane. A correspondingpath is traced in an appropriately cut strip in the s-plane,in which solutions to the eigenequation are single-valued. Analyticcontinuation in the p-plane thus allows the Laplace and Mellininversions to be performed and the real-time behaviour of theplane-stress components to be expressed as contour integralswithin the strips in the complex s-plane. Cast in this form,the stress components are evaluated numerically when the viscoelasticmaterial is represented as a standard linear solid. Their dependenceon the angular variation within the plate, the applied load,and the effects of the viscoelastic material properties is exhibitedfor a number of situations, and in each case contrasted withshort- and long-time asymptotic curves based on Tauberian theorems.     

Instabilities in elastomers and in soft tissues

Biological soft tissues exhibit elastic properties that canbe dramatically different from rubber-type materials (elastomers).To gain a better understanding of the role of constitutive relationshipsin determining material responses under loads we compare threedifferent types of instabilities (two in compression, one inextension) in hyperelasticity for various forms of strain-energyfunctions typically used for elastomers and for soft tissues.Surprisingly, we find that the strain-hardening property ofsoft tissues does not always stabilize the material. In particularwe show that the stability analyses for a compressed half-spaceand for a compressed spherical thick shell can lead to oppositeconclusions: a soft tissue material is more stable than an elastomerin the former case and less stable in the latter case.     

Long-Wave Interactions in Morphological and Convective Instabilities

A binary liquid that undergoes directional solidification issusceptible to morphological and solutal-convective instabilitiesthat cause the solid/liquid interface to change from a planarto a cellular state. This paper gives derivations for thoselong-wave evolution equations that describe the weak couplingsbetween convection and interface morphology and gives some analyticalresults obtainable from these.     

For Ideal Fluids,Eulerian and Lagrangian Instabilities are Equivalent

No Abstract.  .     

Singularities in Gaussian random fields

This paper studies a class of Gaussian random fields defined on lattices that arise in pattern analysis. Phase transitions are shown to exist at a critical temperature for these Gaussian random fields. These are established by showing discontinuous behavior for certain field random variables as the lattice size increases to infinity. The discontinuities in the statistical behavior of these random variables occur because the growth rates of the eigenvalues of the inverse of the variance-covariance matrix at the critical temperature are different from the growth rates at noncritical temperatures. It is also shown that the limiting specific heat has a phase transition with a power law behavior. The critical temperature occurs at the end point of the available values of temperature. Thus, although the critical behavior is not extreme, caution should be exercised when using such models near critical temperatures.Research supported by AFOSR Grant No. 91-0048 and by USARO Grant No. DAAL03-90-G-0103.     

Singularities in positive characteristic, stratification and simplification of the singular locus

We introduce an upper semi-continuous function that stratifies the highest multiplicity locus of a hypersurface in arbitrary characteristic. The blow-up along the maximum stratum defined by this function leads to a form of simplification of the singularities, known as the reduction to the monomial case.     

Singularities and noncommutative Frobenius manifolds

We prove that the quaternionic miniversal deformations of an A _n singularity have the structure of a noncommutative Frobenius manifold in the sense of the extended cohomological field theory. Original Russian Text ? S.M. Natanzon, 2007, published in Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2007, Vol. 259, pp. 143–155. Dedicated to V.I. Arnold