virtual classes and virtual motives of quot schemes on threefolds abstract for a simple rigid vector bundle f on a calabi yau 3 fold y we construct a symmetric obstruction theory on the quot scheme quot y f n and we solve the associated enumerative theory we discuss the case of other 3 folds exploiting the critical structure on the local model quot a 3 o r n we construct a virtual motive in the sense of behrend bryan szendr i for quot y f n for an arbitrary vector bundle f on a smooth 3 fold y we compute the associated motivic partition function we obtain new examples of higher rank motivic donaldson thomas invariants

same decay rate of second order evolution equations with or without delay abstract we consider abstract second order evolution equations with unbounded feedback with delay if the delay term is small enough we rigorously prove the fact that the system with delay has the same decay rate than the one without delay some old and new results easily follow

gopakumar vafa invariants via vanishing cycles in this paper we propose an ansatz for defining gopakumar vafa invariants of calabi yau threefolds using perverse sheaves of vanishing cycles our proposal is a modification of a recent approach of kiem li which is itself based on earlier ideas of hosono saito takahashi we conjecture that these invariants are equivalent to other curve counting theories such as gromov witten theory and pandharipande thomas theory our main theorem is that for local surfaces our invariants agree with pt invariants for irreducible one cycles we also give a counter example to the kiem li conjectures where our invariants match the predicted answer finally we give examples where our invariant matches the expected answer in cases where the cycle is non reduced non planar or non primitive

a classical model for derived critical loci let f u to mathbb a 1 be a regular function on a smooth scheme u over a field mathbb k pantev toen vaquie and vezzosi arxiv 1111 3209 arxiv 1109 5213 define the derived critical locus crit f an example of a new class of spaces in derived algebraic geometry which they call 1 shifted symplectic derived schemes they show that intersections of algebraic lagrangians in a smooth symplectic mathbb k scheme and stable moduli schemes of coherent sheaves on a calabi yau 3 fold over mathbb k are also 1 shifted symplectic derived schemes thus their theory may have applications in algebraic symplectic geometry and in donaldson thomas theory of calabi yau 3 folds this paper defines and studies a new class of spaces we call algebraic d critical loci which should be regarded as classical truncations of 1 shifted symplectic derived schemes they are simpler than their derived analogues we also give a complex analytic version of the theory complex analytic d critical loci and an extension to artin stacks d critical stacks in the sequels arxiv 1305 6302 arxiv 1211 3259 arxiv 1305 6428 arxiv 1312 0090 we will define truncation functors from 1 shifted symplectic derived schemes or stacks to algebraic d critical loci or d critical stacks and we will apply d critical loci to motivic and categorified donaldson thomas theory and to intersections of derived complex lagrangians in complex symplectic manifolds we will show that the important structures one wants to associate to a derived critical locus virtual cycles perverse sheaves and mixed hodge modules of vanishing cycles and motivic milnor fibres can be defined for oriented d critical loci and oriented d critical stacks

sharp decay rates for the weakly coupled hyperbolic system with one internal damping this paper analyzes the longtime behavior of a system of two hyperbolic equations that are coupled by a localized zero order term with a coupling function c cdot with either dirichlet boundary

the multidimensional wave equation with generalized acoustic boundary conditions i strong stability we study the wave equation on a domain of mathbb r d d geq 2 with dynamical boundary control applied on a part of the boundary and a dirichlet boundary condition on the remaining part we prove the asymptotic stability and nonuniform stability of the associated semigroup

stability and instability results of the wave equation with a delay term in the boundary or internal feedbacks in this paper we consider in a bounded and smooth domain the wave equation with a delay term in the boundary condition we also consider the wave equation with a delayed velocity term and mixed dirichlet neumann boundary condition in both cases under suitable assumptions we prove exponential stability of the solution these results are obtained by introducing suitable energies and by using some observability inequalities if one of the above assumptions is not satisfied some instability results are also given by constructing some sequences of delays for which the energy of some solutions does not tend to zero

two examples of ill posedness with respect to time delays revisited we revisit two examples of elastic control systems which are stabilized by boundary feedback and then destabilized when time delays occur in the feedback in the first example a wave equation with a feedback gain greater than or equal to one arbitrary divergence rates are obtained for small time delays in the second example a euler bernoulli beam any time delay results in an arbitrary divergence rate

conditions for robustness and nonrobustness of thestability of feedback systems with respect to small delays inthe feedback loop it has been observed that for many stable feedback control systems the introduction of arbitrarily small time delays into the loop causes instability in this paper we present a systematic frequency domain treatment of this phenomenon for distributed parameter systems we consider the class of all matrix valued transfer functions which are bounded on some right half plane and which have a limit at spinfty along the real axis such transfer functions are called regular under the assumption that a regular transfer function is stabilized by unity output feedback we give sufficient conditions for the robustness and for the nonrobustness of the stability with respect to small time delays in the loop these conditions are given in terms of the high frequency behavior of the open loop system moreover we discuss robustness of stability with respect to small delays for feedback systems with dynamic compensators in particular we show that if a plant with infinitely many poles in the closed right half plane is stabilized by a controller then the stability is not robust with respect to delays we show that the instability created by small delays is itself robust to small delays three examples are given to illustrate these results

indirect boundary stabilization of weakly coupled hyperbolic systems this work is concerned with the boundary stabilization of an abstract system of two coupled second order evolution equations wherein only one of the equations is stabilized indirect damping see e g j math anal appl 173 1993 pp 339 358 we show that under a condition on the operators of each equation and on the boundary feedback operator the energy of smooth solutions of this system decays polynomially at infty we then apply this abstract result to several systems of partial differential equations wave wave systems kirchhoff petrowsky systems and wave petrowsky systems

not all feedback stabilized hyperbolic systems are robust with respect to small time delays in their feedbacks we present two examples of hyperbolic partial differential equations which are stabilized by boundary feedback controls and then destabilized by small delays in these controls we show that in a general case when the controls are distributed stabilized hyperbolic systems possess nontrivial periodic solutions if small time delays are introduced into their feedbacks we also indicate by means of an example that the general case of this phenomenon is harder to demonstrate for boundary control problems

perverse t structure on artin stacks in this paper we develop the theory of perverse sheaves on artin stacks continuing our earlier study of lisse etale sheaves on stacks in laszlo and olsson the six operations for sheaves on artin stacks i finite coefficients publ math ihes 2008 the six operations for sheaves on artin stacks ii adic coefficients publ math ihes 2008

on the energy decay of the linear thermoelastic plate with memory abstract we analyze the longterm properties of a c 0 semigroup describing the solutions to a linear evolution system that models a thermoelastic thin plate with memory effects in the heat flux law under reasonably general assumptions on the memory kernel all single trajectories are shown to decay to zero in spite of that the semigroup is not exponentially stable

decay of solutions of wave equations in a bounded region with boundary dissipation abstract an energy decay rate is obtained for solutions of wave type equations in a bounded region in rn whose boundary consists partly of a nontrapping reflecting surface and partly of an energy absorbing surface

modeling simulation and optimization of traffic flow networks a new model for highway traffic networks based on a detailed description of the junctions is presented to obtain suitable conditions at the junctions multilane equations are introduced and investigated the new model is compared with currently known models for traffic flow networks for several situations finally the model is used for network simulation and optimization

spectral conditions for admissibility of evolution equations in hilbert space abstract we study properties of solutions of the evolution equation u t b u t f t where b is a closable operator on the space ap r h of almost periodic functions with values in a hilbert space h such that b commutes with translations the operator b generates a family b of closed operators on h such that b e i t x e i t b x whenever e i t x d b for a closed subset r we prove that the following properties i and ii are equivalent i for every function f ap r h such that f there exists a unique mild solution u ap r h of eq such that u ii b i is invertible for all and sup b i 1

sharp sufficient conditions for the observation control and stabilization of waves from the boundary for the observation or control of solutions of second order hyperbolic equation in mathbb r t times omega ralston s construction of localized states comm pure appl math 22 1969 pp

orientability for gauge theories on calabi yau manifolds abstract we study orientability issues of moduli spaces from gauge theories on calabi yau manifolds our results generalize and strengthen those for donaldson thomas theory on calabi yau manifolds of dimensions 3 and 4 we also prove a corresponding result in the relative situation which is relevant to the gluing formula in dt theory

stability of bright solitary wave solutions to perturbed nonlinear schrodinger equations abstract the propagation of pulses in ideal nonlinear optical fibers without loss is governed by the nonlinear schrodinger equation nls when considering realistic fibers one must examine perturbed nls equations with the particular perturbation depending on the physical situation that is being modeled a common example is the complex ginzburg landau equation cgl which is a dissipative perturbation it is known that some of the stable bright solitons of the nls survive a dissipative perturbation such as the cgl given that a wave persists it is then important to determine its stability with respect to the perturbed nls a major difficulty in analyzing the stability of solitary waves upon adding dissipative terms is that eigenvalues may bifurcate out of the essential spectrum since the essential spectrum of the nls is located on the imaginary axis such eigenvalues may lead to an unstable wave in fact eigenvalues can pop out of the essential spectrum even if the unperturbed problem has no eigenvalue embedded in the essential spectrum here we present a technique which can be used to track these bifurcating eigenvalues as a consequence we are able to locate the spectrum for bright solitary wave solutions to various perturbed nonlinear schrodinger equations and determine precise conditions on parameters for which the waves are stable in addition we show that a particular perturbation the parametrically forced nls equation supports stable multi bump solitary waves the technique for tracking eigenvalues which bifurcate from the essential spectrum is very general and should therefore be applicable to a larger class of problems than those presented here

polynomial decay rate for the dissipative wave equation using fourier integral operators with special amplitude functions we analyze the stabilization of the wave equation in a three dimensional bounded domain on which exists a trapped ray bouncing up and down infinitely between two parallel parts of the boundary

stability and stabilizability of infinite dimensional systems in this paper we are concerned with stability problems for infinite dimensional systems first we review the theory for linear systems where the dynamics are governed by strongly continuous semigro

rapid boundary stabilization of the wave equation the boundary stabilization of the wave equation in bounded domains is studied it is shown that a particular choice of the feedback leads to fast energy decay explicit decay rate estimates are obt

frequency domain approach for the polynomial stability of a system of partially damped wave equations in this paper we study the stability of a system of wave equations which are weakly coupled and partially damped using a frequency domain approach based on the growth of the resolvent on the imaginary axis we establish the polynomial energy decay rate for smooth initial data we show that the behavior of the system is sensitive to the arithmetic property of the ratio of the wave propagation speeds of the two equations

logarithmic well posed approximation of the backward heat equation in banach space abstract in this paper we apply a logarithmic approximation introduced by boussetila and rebbani in order to study an ill posed cauchy problem in banach space the proposed method induces a well posed problem whose approximation parameter yields continuous dependence on modeling throughout properties of holomorphic semigroups are utilized culminating in the motivating application which is the backward heat equation in l p r 1 p

note on boundary stabilization of wave equations an energy decay rate is obtained for solutions of wave type equations in a bounded region in mathbb r n whose boundary consists partly of a nontrapping reflecting surface and partly of an energy absorbing surface unlike most previous results on this subject the results presented here are potentially valid for regions having connected boundaries

robustness with respect to delays for exponential stability of distributed parameter systems in this paper we address the question of whether the open loop exponential growth rate of a linear system can be improved by a feedback in such a way that this improvement is robust with respect to small delays in the feedback loop when the input operator is admissible and the class of possible feedbacks consists of compact operators we find that if a feedback can improve the exponential growth rate then it can do so robustly furthermore we find that if the control space is finite dimensional and a bounded feedback cannot be found to improve exponential stability then a large class of unbounded feedbacks cannot improve the exponential growth rate robustly even if such feedbacks can improve the exponential growth rate in the absence of delays

new frequency averaged approximations to the equations of radiative heat transfer we develop new direction and frequency averaged approximations to the equations of radiative heat transfer in glass for optically thick diffusive regimes these approximations which are based on the rm sp n approach given in e larsen g thommes a klar m seaid and t gotz j comput phys 83 2002 pp 652 675 represent asymptotic corrections to the familiar rosseland or equilibrium diffusion approximation numerical results for realistic problems in the simulation of radiative heat transfer in glass cooling confirm the accuracy and efficiency of the new approximations

sensitivity analysis of permeability parameters for flows on barcelona networks abstract we consider the problem of optimizing vehicular traffic flows on an urban network of barcelona type i e square network with streets of not equal length in particular we describe the effects of variation of permeability parameters that indicate the amount of flow allowed to enter a junction from incoming roads on each road a model suggested by helbing et al 2007 11 is considered free and congested regimes are distinguished characterized by an arrival flow and a departure flow the latter depending on a permeability parameter moreover we provide a rigorous derivation of the model from fluid dynamic ones using recent results of bretti et al 2006 3 for solving the dynamics at nodes of the network a riemann solver maximizing the through flux is used see coclite et al 2005 4 and helbing et al 2007 11 the network dynamics gives rise to complicate equations where the evolution of fluxes at a single node may involve time delayed terms from all other nodes thus we propose an alternative hybrid approach introducing additional logic variables finally we compute the effects of variations on permeability parameters over the hybrid dynamics and test the obtained results via simulations

optimal polynomial decay of functions and operator semigroups we characterize the polynomial decay of orbits of hilbert space c 0 semigroups in resolvent terms we also show that results of the same type for general banach space semigroups and functions obtained recently in batty and duyckaerts j evol eq 8 765 780 2008 are sharp this settles a conjecture posed in batty and duyckaerts 2008

locally distributed control and damping for the conservative systems in this paper we note the equivalence between exact controllability and exponential stabilizability for an abstract conservative system with bounded control this enables us to establish a frequency domain characterization for the exact controllability uniform exponential decay property of second order elastic systems such as the wave equation and the petrovsky equation with locally distributed control damping a piecewise multiplier method for frequency domain is introduced for several classes of pdes on regions which are not necessarily smooth we obtain a sufficient condition for the subregion on which the application of control damping will yield the exact controllability uniform exponential decay property this result provides useful information for designing the location of controllers dampers for distributed systems with a law of conservation

a note on stabilization of locally damped wave equations with time delay abstract in this paper we consider the wave equation with internal distributed time delay and local damping in a bounded and smooth domain r n when the local damping acts on a neighborhood of a suitable part of the boundary of we show that an exponential stability result holds if the coefficient of the delay term is sufficiently small

an example on the effect of time delays in boundary feedback stabilization of wave equations this note is concerned with the erect of time delays in boundary feedback stabilization schemes for wave equations the question to be addressed is whether such delays can destabilize a system which is uniformly asymptotically stable in the absence of delays

stability of perturbed linear distributed parameter systems we present an estimate for a class of unbounded perturbations of the generator of an exponentially stable semigroup when this estimate is satisfied then the perturbed operator still generates an exponentially stable semigroup for generators of holomorhic semigroups the class of allowed perturbations contains unbounded operators of the form ha with the estimate is given in terms of the norm of the solution of the lyapunov equation which corresponds to the unperturbed system