abstract volterra integrodifferential equations with applications to parabolic models with memory in this paper we are concerned with local existence regularity and continuous dependence upon the initial data of epsilon regular mild solutions for the abstract integrodifferential equation 1 2 we also present a result on unique continuation and a blow up alternative for an epsilon regular mild solution of 1 2 finally we apply our results to three interesting models navier stokes equations with memory diffusion equations with memory and a strongly damped plate equation with memory

facial reduction in partially finite convex programming we consider the problem of minimizing an extended valued convex function on a locally convex space subject to a finite number of linear in equalities when the standard constraint qualification fails a reduction technique is needed to derive necessary optimality conditions facial reduction is usually applied in the range of the constraints in this paper it is applied in the domain space thus maintaining any structure and in particular lattice properties of the underlying domain applications include constrained approximation and best entropy estimation

equivalent classes of critical circles we present numerical evidence that the fractal properties of the critical invariant circles of a typical area preserving twist map as summarized by the spectrum and the generalized dimensions d q depend only on the tails in the continued fraction expansion of the corresponding rotation numbers and d q are numerically the same for all critical invariant circles of the standard and sine maps which have the rotation numbers with the same periodic tail

rational rotation numbers for maps of the circle we consider families of maps of the circle of degree 1 which are homeomorphisms but not diffeomorphisms that is maps like x to x t frac c 2 pi sin 2 pi x bmod 1 withc 1 we prove that the set of parameter values corresponding to irrational rotation numbers has lebesgue measure 0 in other words the intervals on which frequency locking occurs fill up the set of full measure

partially finite convex programming part ii explicit lattice models in part i of this work we derived a duality theorem for partially finite convex programs problems for which the standard slater condition fails almost invariably our result depended on a constraint qualification involving the notion ofquasi relative interior the derivation of the primal solution from a dual solution depended on the differentiability of the dual objective function the differentiability of various convex functions in lattices was considered at the end of part i in part ii we shall apply our results to a number of more concrete problems including variants of semi infinite linear programming l1 approximation constrained approximation and interpolation spectral estimation semi infinite transportation problems and the generalized market area problem of lowe and hurter 1976 as in part i we shall use lattice notation extensively but as we illustrated there in concrete examples lattice theoretic ideas can be avoided if preferred by direct calculation

on the mathematical model for linear elastic systems with analytic damping in the present paper we investigate linear elastic systems with damping ddot y b dot y ay 0 in hilbert spaces where a is a positive definite unbounded linear operator and b is a closed linear operator related in various ways to a alpha frac 1 2 leqq alpha leqq 1 we discuss the spectral property of these systems and obtain some fundamental results for the holomorphic property and the exponential stability of the semigroups associated with these systems

partially finite programming in l_1 and the existence of maximum entropy estimates best entropy estimation is a technique that has been widely applied in many areas of science it consists of estimating an unknown density from some of its moments by maximizing some measure of the entropy of the estimate this problem can be modelled as a partially finite convex program with an integrable function as the variable a complete duality and existence theory is developed for this problem and for an associated extended problem which allows singular measure theoretic solutions this theory explains the appearance of singular components observed in the literature when the burg entropy is used it also provides a unified treatment of existence conditions when the burg boltzmann shannon or some other entropy is used as the objective some examples are discussed

linear wave equations with time dependent propagation speed and strong damping abstract we consider a second order linear equation with a time dependent coefficient c t in front of the elastic operator for these equations it is well known that a higher space regularity of initial data compensates a lower time regularity of c t in this paper we investigate the influence of a strong dissipation namely a friction term which depends on a power of the elastic operator what we discover is a threshold effect when the exponent of the elastic operator in the friction term is greater than 1 2 the damping prevails and the equation behaves as if the coefficient c t were constant when the exponent is less than 1 2 the time regularity of c t comes into play if c t is regular enough once again the damping prevails on the contrary when c t is not regular enough the damping might be ineffective and there are examples in which the dissipative equation behaves as the non dissipative one as expected the stronger is the damping the lower is the time regularity threshold we also provide counterexamples showing the optimality of our results

strongly damped wave problems bootstrapping and regularity of solutions abstract the aim of the article is to present a unified approach to the existence uniqueness and regularity of solutions to problems belonging to a class of second order in time semilinear partial differential equations in banach spaces our results are applied next to a number of examples appearing in literature which fall into the class of strongly damped semilinear wave equations the present work essentially extends the results on the existence and regularity of solutions to such problems previously these problems have been considered mostly within the hilbert space setting and with the main part operators being selfadjoint in this article we present a more general approach involving sectorial operators in reflexive banach spaces

dual methods in entropy maximization application to some problems in crystallography this paper is devoted to some infinite dimensional optimization problems with finitely many constraints these deal with entropy maximization and this paper is particularly concerned with those or

critical circle maps near bifurcation we estimate harmonic scalings in the parameter space of a one parameter family of critical circle maps these estimates lead to the conclusion that the hausdorff dimension of the complement of the frequency locking set is less than 1 but not less than 1 3 moreover the rotation number is a holder continuous function of the parameter

continuation and asymptotics of solutions to semilinear parabolic equations with critical nonlinearities abstract in this paper we discuss continuation properties and asymptotic behavior of regular solutions to abstract semilinear parabolic problems in case when the nonlinear term satisfies critical growth conditions a necessary and sufficient condition for global in time existence of regular solutions is given we also formulate sufficient conditions to construct a piecewise regular solutions continuation beyond maximal time of existence for regular solutions applications to strongly damped wave equations and to higher order semilinear parabolic equations are finally discussed in particular global solvability and the existence of a global attractor for u t t d 1 2 u t d u f u in h 0 1 l 2 is achieved in case when a nonlinear term f satisfies a critical growth condition and a dissipativeness condition similar result is obtained for a 2 m th order semilinear parabolic initial boundary value problem in a hilbert space h 2 b j m

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

quasiperiodicity in dissipative systems a renormalization group analysis abstract dynamical systems with quasiperiodic behavior i e two incommensurate frequencies may be studied via discrete maps which show smooth continuous invariant curves with irrational winding number in this paper these curves are followed using renormalization group techniques which are applied to a one dimensional system circle and also to an area contracting map of an annulus two fixed points are found representing different types of universal behavior a trivial fixed point for smooth motion and a nontrivial fixed point the latter representsthe incipient breakup of a quasiperiodic motion with frequency ratio the golden mean into a more chaotic flow fixed point functions are determined numerically and via an e expansion and eigenvalues are calculated

pointwise pressure observations of a canonical structural acoustics model in this paper we study regularity properties of a physically motivated observation associated with a controlled partial differential equation pde model on a rectangular domain this particular pde describes the active control of acoustic pressure in a chamber by means of piezoelectric activation and involves a wave coupled to a structurally damped beam equation with the coupling being accomplished by boundary trace terms the observation here consists of pointwise in space measurements of the velocity of the wave component with this velocity representing the acoustic pressure of the system as defined then the observation of the system like the control imput is modeled by an unbounded operator the main result of this paper is the discovery of an energy space such that for initial data taken from this space the desired observation of the acoustic pressure is well defined

parabolic problems with nonlinear boundary conditions and critical nonlinearities we prove existence uniqueness and regularity of solutions for heat equations with nonlinear boundary conditions we study these problems with initial data in l q omega w 1 w q omega 1 q infinity or measures and with critically growing non linearities

the dimension of chaotic attractors dimension is perhaps the most basic property of an attractor in this paper we discuss a variety of different definitions of dimension compute their values for a typical example and review previous work on the dimension of chaotic attractors the relevant definitions of dimension are of two general types those that depend only on metric properties and those that depend on the frequency with which a typical trajectory visits different regions of the attractor both our example and the previous work that we review support the conclusion that all of the frequency dependent dimensions take on the same value which we call the dimension of the natural measure and all of the metric dimensions take on a common value which we call the fractal dimension furthermore the dimension of the natural measure is typically equal to the lyapunov dimension which is defined in terms of lyapunov numbers and thus is usually far easier to calculate than any other definition because it is computable and more physically relevant we feel that the dimension of the natural measure is more important than the fractal dimension

controllability of second order infinite dimensional systems in the paper the approximate controllability of linear abstract second order infinite dimensional dynamical systems is considered it is proved using the frequency domain method that approximate controllability of second order system can be verified by the approximate controllability conditions for the corresponding simplified first order system general results are then applied for approximate controllability investigation of a vibratory dynamical system modeling flexible mechanical structure some special cases are also considered moreover remarks and comments on the relationships between different concepts of controllability are given the paper extends earlier results on approximate controllability of second order abstract dynamical systems

regularity of solutions on the global attractor for a semilinear damped wave equation we consider attractors a 0 1 corresponding to a singularly perturbed damped wave equation utt 2 a 1 2 ut aut au f u

longtime dynamics of the kirchhoff equations with fractional damping and supercritical nonlinearity abstract the paper investigates the global well posedness and the longtime dynamics of the kirchhoff equations with fractional damping and supercritical nonlinearity u t t m u 2 u u t f u g x with 1 2 1 the main results are focussed on the relationships among the growth exponent p of the nonlinearity f u the global well posedness and the longtime dynamics of the equations we show that i even if p is up to the supercritical range that is 1 p n 4 n 4 the well posedness and the longtime behavior of the solutions of the equation are of the characters of the parabolic equation ii when n 4 n 4 p n 4 n 4 the corresponding subclass g of the limit solutions exists and possesses a weak global attractor

dissipative mechanism of a semilinear higher order parabolic equation in r n it is known that the concept of dissipativeness is fundamental for understanding the asymptotic behavior of solutions to evolutionary problems in this paper we investigate the dissipative mechanism for some semilinear fourth order parabolic equations in the spaces of bessel potentials and discuss some weak conditions that lead to the existence of a compact global attractor while for second order reaction diffusion equations the dissipativeness mechanism has already been satisfactorily understood see arrieta et al 2004 doi 10 1142 s0218202504003234 for higher order problems in unbounded domains it has not yet been fully developed as shown throughout the paper one of the main differences from the case of reaction diffusion equations stems from the lack of a maximum principle thus we have to rely here on suitable energy estimates for the solutions as in the case of second order reaction diffusion equations we show here that both linear and nonlinear terms have to collaborate in order to produce dissipativeness thus the dissipative mechanisms in second order and fourth order equations are similar although the lack of a maximum principle makes the proofs more difficult and the results not as complete finally we make essential use of the sharp results of cholewa and rodriguez bernal 2012 doi 10 1016 j na 2011 08 022 on linear fourth order equations with a very large class of linear potentials

regularity of some structurally damped problems with point control and with boundary control abstract we study the regularity of structurally damped abstract systems with application to point control and boundary control problems for plate equations in general multidimensional domains in rn our treatment includes in particular both the cases where the damping operator is equal to the square root of the original differential operator and the case where the damping operator is only comparable to such a square root

renormalization for golden circles we prove that twist maps of the cylinder that are attracted by any fixed point of mackay s renormalization operator have a transitive invariant golden circle provided the fixed point satisfies a few simple purely topological conditions these conditions can be verified by finite precision arithmetics they are fulfilled for the simple fixed point and seem to be fulfilled for the critical fixed point taking existence and hyperbolicity of the critical fixed point for granted we conclude that the standard map has a critical invariant golden circle the induced map on the circle is topologically conjugate to a rigid rotation we can show that the conjugator is holder continuous moreover it is not differentiable on a dense set of points

a simple constraint qualification in infinite dimensional programming a new simple constraint qualification for infinite dimensional programs with linear programming type constraints is used to derive the dual program see theorem 3 1 applications include a proof of the explicit solution of the best interpolation problem presented in 8

finite dimensional attractors for the quasi linear strongly damped wave equation abstract we present a new method of investigating the so called quasi linear strongly damped wave equations t 2 u t x u x u f u x x u g in bounded 3d domains this method allows us to establish the existence and uniqueness of energy solutions in the case where the growth exponent of the non linearity is less than 6 and f may have arbitrary polynomial growth rate moreover the existence of a finite dimensional global and exponential attractors for the solution semigroup associated with that equation and their additional regularity are also established in a particular case 0 which corresponds to the so called semi linear strongly damped wave equation our result allows to remove the long standing growth restriction f u c 1 u 5

return time entropies for a class of circle homeomorphisms poincare recurrence for a class of circle maps is used to study the properties of the corresponding invariant measures in the subcritical case when the map is a diffeomorphism the return time measure is smooth and in the critical case when the map is only a homeomorphism the measure is only continuous furthermore in the considered class of critical maps the behaviour of the return time entropy depends only on the tail in the continued fraction expansion of the rotation number

strong rotundity and optimization standard techniques from the study of well posedness show that if a fixed convex objective function is minimized in turn over a sequence of convex feasible regions converging mosco to a limiting feasible region then the optimal solutions converge in norm to the optimal solution of the limiting problem certain conditions on the objective function are needed as is a constraint qualification if as may easily occur in practice the constraint qualification fails stronger set convergence is required together with stronger analytic geometric properties of the objective function strict convexity to ensure uniqueness weakly compact level sets to ensure existence and weak convergence and the kadec property to deduce norm convergence by analogy with the l p norms such properties are termed pstrong rotundity a very simple characterization of strongly rotund integral functionals on l 1 is presented that shows for example that the boltzmann shannon entropy int x log x is strongly rotund

singularity spectrum of a critical kam torus the authors numerically determine the f alpha spectrum of scaling indices for critical kam tori in the area preserving standard map similarities and differences between the corresponding orbits of circle maps are discussed

the dimension spectrum of some dynamical systems we analyze the dimension spectrum previously introduced and measured experimentally by jensen kadanoff and libchaber using large deviation theory we prove for some invariant measures of expanding markov maps that the hausdorff dimensionf of the set on which the measure has a singularity is a well defined concave and regular function in particular we show that this is the case for the accumulation of period doubling and critical mappings of the circle with golden rotation number we also show in these particular cases that the functionf is universal

linear and semilinear higher order parabolic equations in rn abstract in this paper we consider some fourth order linear and semilinear equations in r n and make a detailed study of the solvability of the cauchy problem for the linear equation we consider some weakly integrable potential terms and for any 1 p prove that for a suitable family of bessel potential spaces h p r n the linear equation defines a strongly continuous analytic semigroup using this result we prove that the nonlinear problems we consider can be solved for initial data in l p r n and in h p 2 r n we also find the corresponding critical exponents that is the largest growth allowed for the nonlinear terms for these classes of initial data

dynamics of parabolic problems with memory subcritical and critical nonlinearities in this paper we study the long time behavior of the solutions of non autonomous parabolic equations with memory in cases when the nonlinear term satisfies subcritical and critical growth conditions in order to do this we show that the family of processes associated to original systems with heat source f x t being translation bounded in lloc2 r l2 is dissipative in higher energy space m 0 1 and possesses a compact uniform attractor in m0

singular measures in circle dynamics critical circle homeomorphisms have an invariant measure totally singular with respect to the lebesgue measure we prove that singularities of the invariant measure are of holder type the hausdorff dimension of the invariant measure is less than 1 but greater than 0

extremal equilibria for monotone semigroups in ordered spaces with application to evolutionary equations abstract we consider monotone semigroups in ordered spaces and give general results concerning the existence of extremal equilibria and global attractors we then show some applications of the abstract scheme to various evolutionary problems from odes and retarded functional differential equations to parabolic and hyperbolic pdes in particular we exhibit the dynamical properties of semigroups defined by semilinear parabolic equations in r n with nonlinearities depending on the gradient of the solution we consider as well systems of reaction diffusion equations in r n and provide some results concerning extremal equilibria of the semigroups corresponding to damped wave problems in bounded domains or in r n we further discuss some nonlocal and quasilinear problems as well as the fourth order cahn hilliard equation

dynamics of the viscous cahn hilliard equation abstract we study generalized viscous cahn hilliard problems with nonlinearities satisfying critical growth conditions in w 0 1 p where is a bounded smooth domain in r n n 3 in the critical growth case we prove that the problems are locally well posed and obtain a bootstrapping procedure showing that the solutions are classical for p 2 and almost critical dissipative nonlinearities we prove global well posedness existence of global attractors in h 0 1 and uniformly with respect to the viscosity parameter l bounds for the attractors finally we obtain a result on continuity of regular attractors which shows that if n 3 4 the attractor of the cahn hilliard problem coincides in a sense to be specified with the attractor for the corresponding semilinear heat equation

duality relationships for entropy like minimization problems this paper considers the minimization of a convex integral functional over the positive cone of an l p space subject to a finite number of linear equality constraints such problems arise in spectral estimation where the bjective function is often entropy like and in constrained approximation the lagrangian dual problem is finite dimensional and unconstrained under a quasi interior constraint qualification the primal and dual values are equal with dual attainment examples show the primal value may not be attained conditions are given that ensure that the primal optimal solution can be calculated directly from a dual optimum these conditions are satisfied in many examples

on the metric properties of the feigenbaum attractor the two standard literature definitions of the function associated with the feigenbaum attractor are not equivalent the method due to vulet al and feigenbaum is used to calculate the haussdorff dimension of the feigenbaum attractor using as input the trajectory scaling functions the two calculations yield the same hausdorff dimensiond 0 5380451435 to within the accuracy of the computation

fractal properties of critical invariant curves we examine the dimension of the invariant measure for some singular circle homeomorphisms for a variety of rotation numbers through both the thermodynamic formalism and numerical computation the maps we consider include those induced by the action of the standard map on an invariant curve at the critical parameter value beyond which the curve is destroyed our results indicate that the dimension is universal for a given type of singularity and rotation number and that among all rotation numbers the golden mean produces the largest dimension

characterization of domains of fractional powers of certain operators arising in elastic systems and applications with x a hilbert space where a the elastic operator and b the damping operator are closed operators in x with dense domains q a and 9 b respectively moreover a is throughout assumed to be self adjoint and strictly positive stimulated by two conjectures raised in crl the most general of which referred to the case lx f described below we have already studied the operators and sb defined respectively by

universal scaling of critical quasiperiodic orbits in a class of twist maps recently we have shown that the fractal properties of the critical invariant circles of the standard map as summarized by the spectrum and the generalized dimensions d q depend only on the tails in the continued fraction expansion of the corresponding rotation numbers in buric n mudrinic m and todorovic k 1997 30 l161 in the present paper this result is extended on the whole class of sufficiently smooth area preserving twist maps of cylinders we present numerical evidence that the and d q are the same for all critical invariant circles of any such map which have the rotation numbers with the same tail

asymptotic regularity for some dissipative equations abstract this paper is devoted to proving some asymptotic regularity for both reaction diffusion equation with a polynomially growing nonlinearity of arbitrary order and strongly damped wave equation with critical nonlinearity which excel the sharp regularity allowed by the corresponding stationary equations equilibrium points based on this regularity the existence of the finite dimensional global and exponential attractors can be obtained easily