steady state shear bands in thermo plasticity i vanishing yield stress abstract this paper concerns the construction and stability properties of steady state solutions of a system of partial differential equations that model simple shearing of a slab of thermo plastic material the class of constitutive laws that give rise to a variational formulation of the steady state problem is identified and a phase plane argument is used to construct time independent solutions that may be interpreted as steady state shear bands our variational framework captures several commonly adopted constitutive laws techniques from bifurcation theory for variational problems arc applied to classify stable and unstable solutions merely in terms of the shape of the solution branch in the distinguished bifurcation diagram that arises when average strain rate is plotted against shearing force there are two novel features to our approach first the two problems in which loading is imposed by either stress boundary conditions or velocity boundary conditions are treated by one analysis and the differing stability properties of solutions are explained naturally second the stability analysis is based upon a symmetric eigenvalue problem arising from the appropriate second variation the link with dynamic behavior is made through a lyapunov functional and the linearized dynamics are not considered directly provided the proper existence theorems for the time dependent problem can be proven or are assumed our lyapunov approach yields the appropriate nonlinear dynamic stability properties of steady state solutions in this paper we shall consider the case in which vanishing strain rate implies zero stress i e there is no residual or yield stress present in our model but our analysis can be extended to encompass constitutive laws modelling nonzero yield stress

rigidity of c 2 infinitely renormalizable unimodal maps given c 2 infinitely renormalizable unimodal maps f and g with a quadratic critical point and the same bounded combinatorial type we prove that they are c 1 conjugate along the closure of the corresponding forward orbits of the critical points for some 0

the application of the differential operator method to the blume emery griffiths model abstract within an effective field approximation general expressions for evaluating the second order phase transition line and the tricritical point of the anisotropic blume emery griffiths model are obtained by the use of the differential operator technique the phase diagrams and the behavior of the tricritical point are investigated numerically for the honeycomb lattice z 3 and square lattice z 4 we find a new disordered phase which may correspond to the staggered quadrupolar phase predicted in the monte carlo simulation when some conditions are satisfied the phase diagrams for z 3 and z 4 systems exhibit a reentrant behavior for positive values of the uniaxial anisotropy parameter the change of the tricritical point with the value of the reduced biquadratic parameter is also studied for the system with z 3

finite element approximation of an evolution problem modeling shear band formation abstract an error analysis is provided for the spatial approximation by the finite element method of a nonlinear time dependent problem which models the antiplane shear deformations of a thermoplastic material this problem consists of a system of two coupled partial differential equations with velocity and temperature unknowns optimal order error estimates in the l 2 norm are given for each of these

global hyperbolicity of renormalization for c r unimodal mappings in this paper we extend m lyubichi s recent results on the global hyperbolicity of renormalization of quadratic like germs to the space of cr unimodal maps with quadratic critical point we show that in this space the boundedtype limit sets of the renormalization operator have an invariant hyperbolic structure provided r i 2 a with a close to one as an intermediate step between lyubichi s results and ours we prove that the renormalization operator is hyperbolic in a banach space of real analytic maps we construct the local stable manifolds and prove that they form a continuous lamination whose leaves are c1 codimension one banach submanifolds of the ambient space and whose holonomy is c1 for some 0 we also prove that the global stable sets are c1 immersed codimension one submanifolds as well provided r i 3 a with a close to one as a corollary we deduce that in generic oneparameter families of cr unimodal maps the set of parameters corresponding to infinitely renormalizable maps of bounded combinatorial type is a cantor set with hausdorff dimension less than one

smooth classification of geometrically finite one dimensional maps the scaling function of a one dimensional markov map is defined and studied we prove that the scaling function of a non critical geometrically finite one dimensional map is holder continuous while the scaling function of a critical geometrically finite one dimensional map is discontinuous we prove that scaling functions determine lipschitz conjugacy classes and moreover that the scaling function and the exponents and asymmetries of a geometrically finite one dimensional map are complete c1 invariants within a mixing topological conjugacy class

on brillouin zones brillouin zones were introduced by brillouin br in the thirties to describe quantum mechanical properties of crystals that is in a lattice in n they play an important role in solid state physics it was shown by bieberbach bi that brillouin zones tile the underlying space and that each zone has the same area we generalize the notion of brillouin zones to apply to an arbitrary discrete set in a proper metric space and show that analogs of bieberbach s results hold in this context

stability and folds it is known that when one branch of a simple fold in a bifurcation diagram represents linearly stable solutions the other branch represents unstable solutions the theory developed here can predict instability of some branches close to folds without knowledge of stability of the adjacent branch provided that the underlying problem has a variational structure first one particular bifurcation diagram is identified as playing a special role the relevant diagram being specified by the choice of functional plotted as ordinate the results are then stated in terms of the shape of the solution branch in this distinguished bifurcation diagram in many problems arising in elasticity the preferred bifurcation diagram is the loaddisplacement graph the theory is particularly useful in applications where a solution branch has a succession of folds

exact results of a general spin 1 model on 2d decorated lattices by the decoration iteration transformation we find a general relation between interaction parameters of a general spin 1 model on any lattice with unique coordination number and on its decorated lattice we obtain three exact phase transition points of the general spin 1 model on two dimensional decorated lattices in terms of the exact results of the model on corresponding two dimensional regular lattices including the triangle square kagome and honeycomb lattices we find the critical temperatures have no exact relation with the average coordination number and the type of coordination number for both the ising model and the general spin 1 model

universal properties of the transition from quasi periodicity to chaos in dissipative systems abstract an exact renormalization group transformation is developed for dissipative systems which describes how the transition to chaos may occur in a continuous and universal manner if the frequency ratio in the quasi periodic regime is held at a fixed irrational value our approach is a natural extension of k a m theory to strong coupling most of our analysis is for analytic circle maps we have found a strong coupling fixed point where invertibility is lost which describes the universal features of the transition to chaos we find numerically that any two such critical maps with the same winding number are c 1 conjugate it follows that the low frequency peaks in an experimental spectrum are universal and we determine how their envelope scales with frequency when the winding number has a periodic continued fraction our renormalization transform has a fixed point and spectra are self similar in addition for a set of non periodic winding numbers with full measure our renormalization transformation yields an ergodic trajectory in a sub space of all critical maps physically one finds singular and universal spectra that do not scale

renormalization horseshoe for critical circle maps we prove that the invariant horseshoe for the renormalization of critical circle maps is globally uniformly hyperbolic with one dimensional unstable direction this extends our earlier result on hyperbolicity of periodic orbits of the renormalization and completes the proof of lanford s program

spin 1 2 ising heisenberg model with the pair xyz heisenberg interaction and quartic ising interactions as the exactly soluble zero field eight vertex model the spin 1 2 ising heisenberg model with the pair xyz heisenberg interaction and quartic ising interactions is exactly solved by establishing a precise mapping relationship with the corresponding zero field symmetric eight vertex model it is shown that the ising heisenberg model with the ferromagnetic heisenberg interaction exhibits a striking critical behavior which manifests itself through re entrant phase transitions as well as continuously varying critical exponents the changes in critical exponents are in accordance with the weak universality hypothesis in spite of a peculiar singular behavior that emerges at a quantum critical point of the infinite order which occurs at the isotropic limit of the heisenberg interaction on the other hand the ising heisenberg model with the antiferromagnetic heisenberg interaction surprisingly exhibits less significant changes in both critical temperatures and critical exponents upon varying the strength of the exchange anisotropy in the heisenberg interaction

exact solution of the ising model on a pentagonal lattice it is shown that the ising model on a two dimensional lattice with pentagonal tiling and nearest neighbour interactions is equivalent to the ising model on the union jack lattice with nearest and second nearest neighbour non crossing interactions besides the critical temperature exact expressions are obtained for the spontaneous magnetization for both types of lattice sites

new effective field theory for the blume capel model abstract a new type of effective field theory that has recently been used with success for many applications concerning the usual two state ising model is herein extended to the blume capel model the method which can explicitly and systematically include correlation effects is illustrated in several lattice structures by employing its simplest approximate version in which spin spin correlations are neglected it is shown that this approximative procedure although analytically simple provides in particular a vanishing critical temperature for one dimensional systems and leads to results quite superior to those currently obtained within the molecular field approximation for higher dimensional systems within this framework we discuss as functions of temperature and for any values of the anisotropy parameter d the order parameter the thermal average of the square of the single site spin variable the internal energy and the specific heat as well as the ferromagnetic phase stability limit whenever comparison is possible a satisfactory qualitative and to a certain extent quantitative agreement is observed with results available in the literature in which more sophisticated treatments are used in particular our result for the tricritical point at which the system exhibits a first order phase transition is in quite good agreement with those obtained by using series expansion methods

exact solution of the ising model on a 4 6 lattice we consider an ising model on a lattice of squares and hexagons the 4 6 lattice which does not appear to have been studied previously the critical temperature is obtained as the root of a sixth order equation an exact expression is obtained for the spontaneous magnetization

feigenbaum coullet tresser universality and milnor s hairiness conjecture we prove the feigenbaum coullet tresser conjecture on the hyperbolicity of the renormalization transformation of bounded type this gives the first computer free proof of the original feigenbaum observation of the universal parameter scaling laws we use the hyperbolicity theorem to prove milnor s conjectures on self similarity and hairiness of the mandelbrot set near the corresponding parameter values we also conclude that the set of real infinitely renormalizable quadratics of type bounded by some n 1 has hausdorff dimension strictly between 0 and 1 in the course of getting these results we supply the space of quadratic like germs with a complex analytic structure and demonstrate that the hybrid classes form a complex codimension one foliation of the connectedness locus

an asymptotic universal focal decomposition for non isochronous potentials galileo in the xvii century observed that the small os cillations of a pendulum seem to have constant period in fact the taylor expansion of the period map of the pendulum is constant up to second order in the initial angular velocity around the stable equi librium it is well known that for small oscillations of the pendulum and small intervals of time the dynamics of the pendulum can be ap proximated by the dynamics of the harmonic oscillator we study the dynamics of a family of mechanical systems that includes the pendulum at small neighbourhoods of the equilibrium but after long intervals of time so that the second order term of the period map can no longer be neglected we analyze such dynamical behaviour through a renor malization scheme acting on the dynamics of this family of mechanical systems the main theorem states that the asymptotic limit of this renormalization scheme is universal it is the same for all the elements in the considered class of mechanical systems as a consequence we obtain an universal asymptotic focal decomposition for this family of mechanical systems this paper is intended to be the first of a series of articles aiming at a semiclassical quantization of systems of the pendu lum type as a natural application of the focal decomposition associated to the two point boundary value problem

a renormalization approach to invariant circles in area preserving maps abstract kadanoff and shenker introduced a renormalisation approach to invariant circles in area preserving maps this paper makes more precise the connection between invariant circles and the renormalisation operator restricting attention to noble rotation numbers the stability of a simple fixed point of the renormalisation is analysed corresponding to a linear twist map it is found to be essentially attracting so that noble circles persist under perturbation giving a new view on kam theory shenker and kadanoff found evidence for another fixed point corresponding to a map with a non smooth noble circle further evidence is given in this paper it has essentially only one unstable direction and its stable manifold is believed to give the boundary of the set of twist maps with a noble circle finally noble circles are shown to be locally most robust in an important sense

asymptotic differentiable structure on cantor set we study hyperbolic maps depending on a parameter e each of them has an invariant cantor set as e tends to zero the map approaches the boundary of hyperbolicity we analyze the asymptotics of scaling function of the invariant cantor set as e goes to zero we show that there is a limiting scaling function of the limiting map and this scaling function has dense jump discontinuities because the limiting map is not expanding removing these discontinuities by continuous extension we show that we obtain the scaling function of the limiting map with respect to a ulam von neumann type metric

the theory of caustics and wave front singularities with physical applications this is intended as an introduction to and review of the theory of lagrangian and legendrian submanifolds and their associated maps developed by arnold and his collaborators the theory is illustrated by applications to hamilton jacobi theory and the eikonal equation with an emphasis on null surfaces and wave fronts and their associated caustics and singularities

on the inversion of the vandermonde matrix abstract the inversion of the vandermonde matrix has received much attention for its role in the solution of some problems of numerical analysis and control theory this work deals with the problem of getting an explicit formula for the generic element of the inverse we derive two algorithms in o n2 and o n3 and compare them with the parker traub and the bjorck pereyra algorithms

the periodic points of renormalization it will be shown that the renormalization operator acting on the space of smooth unimodal maps with critical exponent a 1 has periodic points of any combinatorial type a central question in the theory of dynamical systems is whether small scale geometrical properties of dynamical systems are determined by the combinatorial properties of the system indeed such universality of small scale geometry was discovered by coullet tresser and feigenbaum they studied infinitely renormalizable unimodal maps of the period doubling type and observed that the geometry of the invariant cantor set of such maps converges when looking at smaller and smaller scales furthermore they observed that the limiting geometry was universal in the sense that the small scale geometry of these cantor sets depends only on the local behavior of the map around the critical point this local behavior is specified by the critical exponent to explain the universality of geometry they defined the period doubling renormalization operator on a suitable space of unimodal maps this operator acts like a microscope the image under the renormalization operator is a unimodal map describing the geometry and dynamics on a smaller scale the universality of geometry was understood by conjecturing that the renormalization operator has a unique hyperbolic fixed point in particular the infinitely renormalizable unimodal maps form the stable manifold of the fixed point of the renormalization operator the first step in proving these conjectures is showing the existence of a renormalization fixed point

order from disorder effect in the exactly solved mixed spin 1 2 1 ising model on fully frustrated triangles in triangles lattices abstract the mixed spin 1 2 1 ising model on two fully frustrated triangles in triangles lattices is exactly solved with the help of the generalized star triangle transformation which establishes a rigorous mapping correspondence with the equivalent spin 1 2 ising model on a triangular lattice it is shown that the mutual interplay between the spin frustration and single ion anisotropy gives rise to various spontaneously ordered and disordered ground states which differ mainly in an occurrence probability of the non magnetic spin state of the integer valued decorating spins we have convincingly evidenced a possible coexistence of the spontaneous long range order with a partial disorder within the striking ordered disordered ground state which manifests itself through a non trivial criticality at finite temperatures as well a rather rich critical behavior including the order from disorder effect and reentrant phase transitions with either two or three successive critical points is also found

a bifurcation theorem for potential operators abstract a sufficient condition for bifurcation of real solutions of g u 0 where g r d e d e is given in terms of the crossing number of the eigenvalue perturbation of the linearization along the trivial solutions 0 this number describes how many eigenvalues leave the negative real axis wen the parameter varies from negative to positive values whereas in general only an odd crossing number entails bifurcation for potential operators in a hilbert space e a nonzero crossing number is sufficient for bifurcation the proof uses a lyapunov schmidt reduction and conley s index theory

characteristic initial data and wavefront singularities in general relativity the structure of singularities caustics self intersections of wavefronts null hypersurfaces and wavefront families null coordinates in arbitrary space times is discussed in detail and illustrated by explicit examples of stable wavefront singularities in minkowski space it is shown how characteristic initial data determine the caustics and the self intersections of the characteristics of einstein s field equations

critical couplings of mixed spin 1 2 spin s ising model a free fermion approximation a free fermion approximation procedure is applied to the mixed spin 1 2 spin s ising models the author s results for the critical coupling kc are in very satisfactory agreement with the high temperature series results of yousif and bowers 1984

thermodynamical properties of the random field mixed spin ising model abstract a theoretical study of a mixed spin ising system consisting of spin 1 2 and spin 1 with a random field is studied by the use of the finite cluster approximation within the framework of a single site cluster theory in this approach the phase diagrams and magnetizations are investigated for the simple cubic lattice when the random field is bimodally and trimodally distributed the internal energy specific heat and susceptibility are also calculated for both kinds of the probability distributions

phase diagrams and tricritical effects in the beg model abstract the critical properties of the spin 1 ising model on the bethe lattice with dipolar and quadrupolar exchange interactions are studied exactly this method is based on recursion relations which are exact in the infinite dimensional case and as a result the phase diagrams are constructed the line ending in the tricritical point is defined exactly it is shown that when the quadrupolar exchange interaction is large enough the tricritical point disappears the tricritical exponents are calculated exactly

generalized transformation for decorated spin models abstract the paper discusses the transformation of decorated ising models into an effective undecorated spin model using the most general hamiltonian for interacting ising models including a long range and high order interactions the inverse of a vandermonde matrix with equidistant nodes s s is used to obtain an analytical expression of the transformation this kind of transformation is very useful to obtain the partition function of decorated systems the method presented by fisher is also extended in order to obtain the correlation functions of the decorated ising models transforming into an effective undecorated ising model we apply this transformation to a particular mixed spin 1 2 1 and 1 2 2 square lattice with only nearest site interaction this model could be transformed into an effective uniform spin s square lattice with nearest and next nearest interaction furthermore the effective hamiltonian also includes combinations of three body and four body interactions in particular we considered spin 1 and 2

reentrant phase transitions and multicompensation points in the mixed spin ising ferrimagnet on a decorated bethe lattice abstract a mixed spin ising model on a decorated bethe lattice is rigorously solved by combining the decoration iteration transformation with the method of exact recursion relations exact results for critical lines compensation temperatures total and sublattice magnetizations are obtained from a precise mapping relationship with the corresponding spin 1 2 ising model on a simple undecorated bethe lattice the effect of next nearest neighbour interaction and single ion anisotropy on magnetic properties of the ferrimagnetic model is investigated in particular it is shown that the total magnetization may exhibit multicompensation phenomenon and the critical temperature vs the single ion anisotropy dependence basically changes with the coordination number of the underlying bethe lattice the possibility of observing reentrant phase transitions is related to a high enough coordination number of the underlying bethe lattice

mixed spin 1 2 and spin 1 ising model in transverse random fields abstract we investigate the critical properties of a mixed spin 1 2 and spin 1 ising model in a transverse random field in this study we use two approaches the pair approximation and the mean field renormalization group each combined with the discretized path integral representation it is found that both methods predict quantitatively the same phase diagram critical exponents of the system are also calculated

exact curie temperature for the ising model on archimedean and laves lattices using the feynman vdovichenko combinatorial approach to the two dimensional ising model we determine the exact curie temperature for all two dimensional archimedean lattices by means of duality we extend our results to cover all two dimensional laves lattices for those lattices where the exact critical temperatures are not yet exactly known we compare them with monte carlo simulations

correlated effective field treatment of spin one ising models abstract a new version of correlated effective field theory is applied to a study of the spin one ising models namely the blume capel and blume emery griffiths modesl using the honmura kaneyoshi differential operator technique this method is illustrated in a honeycomb lattice by investigating the phase diagram the magnetization the correlated effective field parameter and the short range order parameter the theory correctly accounts for many physical quantities and leads to results that are quite superior and in some aspects different to those of the previous works