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

a new unitary operator for a pair of coupled oscillators using the iwop integration within ordered product technique a new unitary operator is found that is useful in diagonizing the hamiltonian of a pair of coupled harmonic oscillators the coordinate representation of the unitary operator is presented and is well applied to obtain the wave function of energy eigenstate of the coupled oscillator

quasi coherent states for unitary groups the author extends a recent construction of isotropic spin coherent states to unitary groups expectation values in such a quasi coherent state can be obtained from a generating functional which is given explicitly for the fundamental and the adjoint representations of the unitary groups su n and u n the close connection between the corresponding generating functional and the external field problem in qcd is pointed out free quantum fields in such a condensed colour singlet coherent state are formally related to two dimensional lattice gauge theories with in general a mixed action in the large n limit the author exhibits a phase transition for a free quantum field in a singlet quasi coherent state a path integral representation of the transition amplitude in terms of quasi coherent states is also given the corresponding effective action describes dynamics on a complex kahler manifold

numerical approximations of algebraic riccati equations for abstract systems modelled by analytic semigroups and applications this paper provides a numerical approximation theory of algebraic riccati operator equations with unbounded coefficient operators a and b such as arise in the study of optimal quadratic cost problems over the time interval 0 oo for the abstract dynamics j ay bu here a is the generator of a strongly continuous analytic semigroup and b is an unbounded operator with any degree of unboundedness less than that of a convergence results are provided for the riccati operators as well as for all the other relevant quantities which enter into the dynamic optimization problem the present numerical theory is the counterpart of a known continuous theory several examples of partial differential equations with boundary point control where all the required assumptions are verified illustrate the theory they include parabolic equations with l2 dirichlet control as well as plate equations with a strong degree of damping and point control

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

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

new complex function space related to both entangled state representation and spin coherent state a new complex function space whose basis is the single variable hermite polynomial h2j 2 is constructed which is related to both entangled state representation and spin coherent state in schwinger bosonic realization new binomial theorem involving two variable hermite polynomial is derived which helps to constitute the new complex function space we also present a new integration transformation of the basis h2j 2 with its reciprocal transformation which is useful to deriving some operator identities

normally ordered bose fermi operator realization of the su n m group the technique of integration within an ordered product which is now generalized to the supersymmetric case together with the boson fermion coherent states is used to derive the normally ordered bose fermi operator realization of the group su n m the mapping of the su n m fundamental representation in classical complex number grassmann number space into the realization in hilbert space is manifestly shown in our formalism the corresponding antinormally ordered realization is also discussed

matrix methods for wave equations in analogy to a characterisation of operator matrices generating c0 semigroups due to r nagel 13 we give conditions on its entries in order that a 2 2 operator matrix generates a cosine operator function we apply this to systems of wave equations to second order initial boundary value problems and to overdamped wave equations

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

finite parts of the spectrum of a riemann surface let s and s be compact riemann surfaces of the same genus g g ge 2 endowed with the poincare metric of constant negative curvature 1 par the authors show that for every epsilon 0 there exists an integer m m g epsilon with the property assume that 1 injectivity radius of s and s ge epsilon and 2 the first m m g epsilon eigenvalues of the laplacian of s and s coincide then s and s are isospectral par the authors conjecture that the integer m will not depend on epsilon that is the theorem holds with an integer which depends only on the genus par for the proof a model of teichmuller space is described and the analyticity of the resolvent of laplacian on this space is proved also the authors note that the injectivity radius is estimated by a finite part of the spectrum the number of eigenvalues of the laplacian in the interval 1 4 1

local and global smoothing effects for some linear hyperbolic equations with a strong dissipation we consider an abstract second order linear equation with a strong dissipation namely a friction term which depends on a power of the elastic operator in the homogeneous case we investigate the phase spaces in which the initial value problem gives rise to a semigroup and the further regularity of solutions in the nonhomogeneous case we study how the regularity of solutions depends on the regularity of forcing terms and we characterize the spaces where a bounded forcing term yields a bounded solution what we discover is a variety of different regimes with completely different behaviors depending on the exponent in the friction term we also provide counterexamples in order to show the optimality of our results mathematics subject classification 2010 msc2010 35l10 35l15 35l20

simplified conditions of initial observability for infinite dimensional second order damped dynamical systems abstract in the paper the initial observability of abstract second order infinite dimensional dynamical systems is considered two cases of observation operator are explored first the observation operator is assumed to be bounded then results are extended to more general class of observations related to the observation operators that are assumed to be bounded with respect to square root of elasticity operator which covers the case of some unboundedness in observation using the frequency domain method it is proved that initial observability of second order system under discussion can be verified by the initial observability conditions for the corresponding simplified first order system general results are then applied for initial observability investigation of infinite dimensional systems described by partial differential equations modelling flexible mechanical structures some special cases are also discussed and practical comments and remarks are given the paper extends earlier results on initial observability for a class of infinite dimensional second order abstract dynamical systems

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

on the dual boundary conditions and the nonnegative square root of the second derivative operator abstract a dual relationship between the boundary conditions of the second derivative operator au u is established some dual properties determined by the dual boundary conditions are considered furthermore the singularities of the nonnegative square root of the above second derivative operator are studied our results show that although a 1 2 u is not u it is a harmonic conjugate of u in some sense

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

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

carleman estimate for an adjoint of a damped beam equation and an application to null controllability abstract in this article we consider a control problem of a linear euler bernoulli damped beam equation with potential in dimension one with periodic boundary conditions we derive a new carleman estimate for an adjoint of the equation under consideration then using a well known duality argument we obtain explicitly the control function which can be used to drive the solution trajectory of the control problem to zero state

a non autonomous strongly damped wave equation existence and continuity of the pullback attractor abstract in this paper we consider the strongly damped wave equation with time dependent terms u t t u t u t e t u t f u in a bounded domain r n under some restrictions on e t t and growth restrictions on the nonlinear term f the function e t depends on a parameter e e t e 0 0 we will prove under suitable assumptions local and global well posedness using the uniform sectorial operators theory the existence and regularity of pullback attractors a e t t r uniform bounds for these pullback attractors characterization of these pullback attractors and their upper and lower semicontinuity at 0

canonical coherent state representation of some squeeze operators using the iwop technique integration within an ordered product the authors find canonical coherent state representations of two kinds of squeeze operators these representations manifestly show that they are quantum maps imaged by certain symplectic transformations in coordinate momentum phase space in terms of this formalism the coherent state propagator of parametric amplifiers is easily obtained

counting and effective rigidity in algebra and geometry the purpose of this article is to produce effective versions of some rigidity results in algebra and geometry on the geometric side we focus on the spectrum of primitive geodesic lengths resp complex lengths for arithmetic hyperbolic 2 manifolds resp 3 manifolds by work of reid this spectrum determines the commensurability class of the 2 manifold resp 3 manifold we establish effective versions of these rigidity results by ensuring that for two incommensurable arithmetic manifolds of bounded volume the length sets resp the complex length sets must disagree for a length that can be explicitly bounded as a function of volume we also prove an effective version of a similar rigidity result established by the second author with reid on a surface analog of the length spectrum for hyperbolic 3 manifolds these effective results have corresponding algebraic analogs involving maximal subfields and quaternion subalgebras of quaternion algebras to prove these effective rigidity results we establish results on the asymptotic behavior of certain algebraic and geometric counting functions which are of independent interest

on the nonnegative square root of the fourth derivative operators abstract the singularities of the nonnegative square root of the fourth derivative operator au u 4 are studied

existence results for some damped second order volterra integro differential equations abstract in this paper we make a subtle use of operator theory techniques and the well known schauder fixed point principle to establish the existence of pseudo almost automorphic solutions to some second order damped integro differential equations with pseudo almost automorphic coefficients in order to illustrate our main results we will study the existence of pseudo almost automorphic solutions to a structurally damped plate like boundary value problem

normally ordered fermi operator realization of the sun group the mapping of the sun group transformation in grassmann number space into unitary normally ordered and antinormally ordered fermi operator realizations in hilbert space are investigated the unitary fermi operators are evaluated in fermion coherent state representation using the integration within ordered product technique for a fermionic system some new fermi operator identities are thus obtained

ergodic boundary point control of stochastic semilinear systems a controlled markov process in a hilbert space and an ergodic cost functional are given for a control problem that is solved where the process is a solution of a parameter dependent semilinear stochastic differential equation and the control can occur only on the boundary or at discrete points in the domain the linear term of the semilinear differential equation is the infinitesimal generator of an analytic semigroup the noise for the stochastic differential equation can be distributed boundary and point some ergodic properties of the controlled markov process are shown to be uniform in the control and the parameter the existence of an optimal control is verified to solve the ergodic control problem the optimal cost is shown to depend continuously on the system parameter

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