equivalent spatial and material descriptions of finite deformation elastoplasticity in principal axes abstract a spatial description of the theory of rate independent finite deformation elastoplasticity in which the stress tensor is defined through the strain energy function is discussed the main assumption of isotropic elastic response and invariance requirements under superposed rigid body motion restrict the acceptable forms of the strain energy function to those given in terms of principal values of the strain measure of elastic distortion the formulation is developed on a manifold and the corresponding material description is obtained simply by pull back of the derived spatial form by appealing to the notion of covariance the method of principal axes is systematically exploited to derive the explicit expression for the stress tensor computation for an arbitrary form of the strain energy function and the explicit form of the evolution equation for an arbitrary form of the yield function a model problem of volume preserving plastic flow is discussed in the closure

a constitutive frame of elastoplasticity at large strains based on the notion of a plastic metric abstract the article presents a constitutive framework of large strain elastoplasticity in both the lagrangian and the eulerian geometric setting which takes into account anisotropic material response in summary the key ingredients of this framework are i the introduction of a plastic metric which is assumed to describe locally the history dependent inelastic material response in the sense of an internal variable formulation ii the definition of a convex elastic domain in the space of the local stress like variable conjugate to the plastic metric denoted as the plastic force iii an equivalent lagrangian and eulerian representation of all constitutive functions as isotropic tensor functions in terms of an extended set of arguments denoted as anisotropy variables iv the set up of normality rules for the evolution of the plastic metric and the anisotropy variables yielding a canonical symmetric form of the elastoplastic tangent moduli v a geometrically exact decomposition of the set of constitutive equations into possibly decoupled volumetric and isochoric contributions applications of the constitutive framework are demonstrated by means of several conceptual model problems which cover isotropic initial anisotropic and induced anisotropic elastoplastic response

geometry of the modulational instability iii homoclinic orbits for the periodic sine gordon equation in this paper the homoclinic geometric structure of the integrable sine gordon equation under periodic boundary conditions is developed specifically focus is given to orbits homoclinic to n tori simple examples of such homoclinic orbits are constructed and a physical interpretation of these states is given a labeling is provided which identifies and catalogues all such orbits these orbits are related in a one to one manner to linearized instabilities explicit formulas for all homoclinic orbits are given in terms of backlund transformations

stress rate and the lagrangian formulation of the finite strain plasticity for a von mises kinematic hardening model abstract the use of the jaumann stress rate in kinematic hardening finite strain plasticity in the case of simple shear generates oscillations in the stress field alternate theories have been introduced to define other stress rates by making use of the polar decomposition and consequently producing increasing shear stress in this work a lagrangian formulation is introduced which for the case of simple shear produces monotonically increasing stress strain relationships the yield criterion is originally expressed in terms of the cauchy stress and subsequently transformed to the lagrangian reference frame the associated flow rule used here preserves the normality rule in the second piola kirchhoff stress space and is equivalent to that of the cauchy stress space this approach preserves the accuracy of the interpretation of the material behavior in the eulcrian reference frame and it also bypasses the use of the jaumann stress rate in the formulation of kinematic hardening finite strain plasticity

spectral instability of the semiclassical zakharov shabat eigenvalue problem abstract we present some brief numerical results indicating that in the semiclassical limit the zakharov shabat eigenvalue problem exhibits severe spectral instability and that small changes in the potential can result in large changes in the spectrum

on constitutive relations at finite strain hypo elasticity and elasto plasticity with isotropic or kinematic hardening abstract the question of generalization to finite strains of the constitutive relations of infinitesimal strain theory of elasticity and classical elasto plasticity with isotropic and kinematic hardening is critically examined simple generalizations which lead to physically plausible material behaviour are presented the current controversies surrounding i the choice of stress rate in the above generalization and ii the anomaly of oscillatory stresses in kinematic hardening plasticity as discussed by e h lee and others are analyzed it is found that these controversies are easily resolvable

the semiclassical limit of the defocusing nls hierarchy we establish the semiclassical limit of the one dimensional defocusing cubic nonlinear schrodinger nls equation complete integrability is exploited to obtain a global characterization of the weak limits of the entire nls hierarchy of conserved densities as the field evolves from reflectionless initial data under all the associated commuting flows consequently this also establishes the zero dispersion limit of the modified korteweg de vries equation that resides in that hierarchy we have adapted and clarified the strategy introduced by lax and levermore to study the zero dispersion limit of the korteweg de vries equation expanding it to treat entire integrable hierarchies and strengthening the limits obtained a crucial role is played by the convexity of the underlying log determinant with respect to the times associated with the commuting flows c 1999 john wiley sons inc

a discussion of stress rates in finite deformation problems abstract it has recently been shown that the finite elastic plastic solution of the simple shear problem exhibits an oscillatory stress response when kinematic hardening is employed while the solution for isotropic hardening gives a monotonically increasing stress this paper analyzes this response on the basis of continuum mechanical descriptions of the problem three objective stress rates are recalled and spatial descriptions of plasticity at finite deformation are reviewed for the usual generalization of the infinitesimal theory as well as a theory based on an invariant measure of true stress in light of the equations for the evolution of the yield surface the hypoelastic solution to the simple shear problem for each of the three stress rates is presented it is shown that the use of the jaumann rate in the generalization of the infinitesimal theory leads to an oscillation in the evolution of the yield surface in simple shear which is explained on the basis of the hypoelastic solution an alternative theory which makes use of the polar decomposition predicts a monotonically increasing shear stress

the semiclassical modified nonlinear schrodinger equation i modulation theory and spectral analysis abstract we study an integrable modification of the focusing nonlinear schrodinger equation from the point of view of semiclassical asymptotics in particular i we establish several important consequences of the mixed type limiting quasilinear system including the existence of maps that embed the limiting forms of both the focusing and defocusing nonlinear schrodinger equations into the framework of a single limiting system for the modified equation ii we obtain bounds for the location of the discrete spectrum for the associated spectral problem that are particularly suited to the semiclassical limit and that generalize known results for the spectrum of the nonselfadjoint zakharov shabat spectral problem and iii we present a multiparameter family of initial data for which we solve the associated spectral problem in terms of special functions for all values of the semiclassical scaling parameter we view our results as part of a broader project to analyze the semiclassical limit of the modified nonlinear schrodinger equation via the noncommutative steepest descent procedure of deift and zhou and we also present a selfcontained development of a riemann hilbert problem of inverse scattering that differs from those given in the literature and that is well adapted to semiclassical asymptotics

some remarks on a wkb method for the nonselfadjoint zakharov shabat eigenvalue problem with analytic potentials and fast phase abstract a formal method for approximating eigenvalues of the nonselfadjoint zakharov shabat eigenvalue problem in the semiclassical scaling is described analyticity of the potential is assumed and appears to be crucial the method involves finding appropriate paths between pairs of complex turning points and reproduces the y shaped spectra observed by bronski physica d 97 1996 376 an application to all optical ultrashort pulse generation is briefly described and the kind of tools that are required to make the results rigorous are indicated

fourier analysis and optimal hardy adams inequalities on hyperbolic spaces of any even dimension abstract recently the second and third authors established the sharp hardy adams inequalities of second order derivatives with the best constants on the hyperbolic space b 4 of dimension four 40 adv math 2017 a key method used in dimension four in 40 is that the following hardy operator when n 4 and k 2 b n k u 2 d x i 1 k 2 i 1 2 b n u 2 1 x 2 2 k d x can be decomposed exactly as the product of fractional laplacians however such a decomposition in dimension four is no longer possible in the case n 4 and 2 k n 2 and the situation here is thus considerably more difficult and complicated therefore it remains open as whether a sharp hardy adams inequality still holds on higher dimensional hyperbolic space b n for n 4 the main purpose of this paper is to use a substantially new method of estimating the hardy operator to establish the sharp hardy adams inequalities on hyperbolic spaces b n for all even dimension n and n 4 as applications of such inequalities we will improve substantially the known adams inequalities on hyperbolic space b n in the literature and also strengthen the classical adams inequality and the hardy inequality on euclidean balls in any even dimension the later inequality can be viewed as the borderline case of the sharp hardy sobolev maz ya inequalities for higher order derivatives in high dimensions obtained recently by the second and third authors 41

sharp moser trudinger inequality on the heisenberg group at the critical case and applications let h cn r be the n dimensional heisenberg group q 2n 2 be the homogeneous dimension of h q qq 1 and z 4 t2 14 be the homogeneous norm of z t h then we prove the following sharp moser trudinger inequality on h theorem 1 6 there exists a positive constant q q 2 n 12 q 12 q2 1 n 1 q 1 such that for any pair satisfying 0 q 0 q 1 q there holds sup u 1 1 h1 exp u q q 1 k 0q 2 kk u kq q 1 c q the constant q 1 q is best possible in the sense that the supremum is infinite if q 1 q here is any positive number and u 1 h hu q h u q 1 q our result extends the sharp moser trudinger inequality by cohn and lu 2001 19 on domains of finite measure on h and sharpens the recent result of cohn et al 2012 18 where such an inequality was studied for the subcritical case q 1 q we carry out a completely different and much simpler argument than that in cohn et al 2012 18 to conclude the critical case our method avoids using the rearrangement argument which is not available in an optimal way on the heisenberg group and can be used in more general settings such as riemanian manifolds appropriate metric spaces etc as applications we establish the existence and multiplicity of nontrivial nonnegative solutions to certain nonuniformly subelliptic equations of q laplacian type on the heisenberg group theorems 1 8 1 9 1 10 and 1 11 divh hu q 2 hu v u q 2u f u eh with nonlinear terms f of maximal exponential growth exp u qq 1 as u in particular when e 0 the existence of a nontrivial solution is also given

geometrically non linear and elastoplastic three dimensional shear flexible beam element of von mises type hardening material a three dimensional elastoplastic beam element being capable of incorporating large displacement and large rotation is developed and examined elastoplastic constitutive equations are applied to the beam element based upon the assumption of small deformational strain leading to a material formulation which is completely objective for the application of stress update procedures the continuum type equations of plastic model of j2 mixed hardening are transformed into the beam equations by satisfying beam hypotheses an effective stress update algorithm is proposed to integrate elastoplastic rate equations by means of the so called multistep method which is a method of successive control of residuals on yield surfaces it avoids severe divergence when the displacement increments become large which is usual for the continuation methods material tangent stiffness matrix is derived by using consistent elastoplastic modulus resulting from the integration algorithm and is combined with geometric tangent stiffness matrix different from other elements the present element is shear flexible and can satisfy the plasticity condition in a pointwise fashion a great number of numerical examples are analysed and compared with the literature the proposed beam element is verified to be not only quite accurate but also very effective for the analyses of pre buckling and large deflection collapse of spatial framed structures

on the semi classical limit for the focusing nonlinear schrodinger equation sensitivity to analytic properties of the initial data we present the results of a large number of careful numerical experiments carried out to investigate the way that the solution of the integrable focusing nonlinear schroodinger equation with fixed initial data when taken to be close to the semiclassical limit depends on analyticity properties of the data in particular we study a family of initial data that have complex singularities at an adjustable distance from the real axis we also make use of a simple relation that provides the exact solution at the centre of the wave field of the elliptic quasilinear system that appears formally as a leading order model for the semi classical dynamics among other things we conclude that the semi classical limit cannot be expected to be continuous with respect to the initial data even for real analytic data if there are certain complex singularities present we argue that in order to have well posedness of the semiclassical limit the correct setting is a physically relevant space of functions with compactly supported fourier transforms

a new approach to sharp moser trudinger and adams type inequalities a rearrangement free argument abstract the main purpose of this paper is two fold on the one hand we will develop a new approach to establish sharp singular moser trudinger and adams type inequalities in unbounded domains of euclidean spaces without using the standard symmetrization on the other hand we will prove the sharp singular adams type inequality on high order sobolev spaces w m n m r n of arbitrary integer order m theorem 1 1 which improves the results of ruf and sani 2013 48 where sharp adams inequalities were established for even m and those of the authors lam and lu 2012 28 29 for odd m but with different and more restricted norms we first establish the sharp local singular adams inequality on domains in r n of finite measure theorem 1 4 we take a perspective that any function in the high order sobolev spaces w m n m r n can be represented as a bessel potential thus we can fully use the tools from harmonic analysis and the kernel properties of the polyharmonic operators i m 2 once we have established this sharp local adams inequality then we can adapt the rearrangement free method we will develop in this paper to derive a global sharp adams inequality from a local one our argument substantially simplifies those in ruf and sani 2013 48 and lam and lu 2012 28 29 and avoids the use of rather deep and complicated comparison principle of solutions to polyharmonic operators used in ruf and sani 2013 48 lam and lu 2012 28 29 moreover our theorem holds on sobolev spaces w n r n of any positive fractional order n theorems 1 6 and 1 7 as consequences we can also establish sharp adams inequalities with less restrictions on the sobolev norms see theorems 1 2 1 3 and 1 5 our approach is surprisingly simple and general and can be easily applied to scenarios such as riemannian and sub riemannian manifolds where symmetrization argument does not work see e g on the heisenberg group lam and lu 2012 30

on semiclassical zero dispersion limit solutions of the focusing nonlinear schrodinger equation we calculate the leading order term of the solution of the focusing nonlinear cubic schrodinger equation nls in the semiclassical limit for a certain one parameter family of initial conditions this family contains both solitons and pure radiation in the pure radiation case our result is valid for all times t 0 we utilize the riemann hilbert problem formulation of the inverse scattering problem to obtain the leading order term of the solution error estimates are provided 2004 wiley periodicals inc

improved hardy adams inequality on hyperbolic space of dimension four abstract in this paper we establish a singular hardy adams inequality on the hyperbolic space of dimension four as our main result we will prove that for any 0 0 4 there exists a constant c 0 such that for all u c 0 b 4 satisfying b 4 h 9 4 h u u d v 1 then the following inequality holds b 4 exp 32 2 1 4 u 2 1 32 2 1 4 u 2 x d v c where x log 1 x 1 x is the hyperbolic distance from x to the origin our results improve that of lu and yang 2017

pseudospectra of linear operators if a matrix or linear operator a is far from normal its eigenvalues or more generally its spectrum may have little to do with its behavior as measured by quantities such as an or exp ta more may be learned by examining the sets in the complex plane known as the pseudospectra of a defined by level curves of the norm of the resolvent zi a 1 five years ago the author published a paper that presented computed pseudospectra of thirteen highly nonnormal matrices arising in various applications since that time analogous computations have been carried out for differential and integral operators this paper a companion to the earlier one presents ten examples each chosen to illustrate one or more mathematical or physical principles

semiclassical limit of focusing nls for a family of square barrier initial data the small dispersion limit of the focusing nonlinear schrodinger equation nls exhibits a rich structure of sharply separated regions exhibiting disparate rapid oscillations at microscopic scales the non self adjoint scattering problem and ill posed limiting whitham equations associated to focusing nls make rigorous asymptotic results difficult previous studies have focused on special classes of analytic initial data for which the limiting elliptic whitham equations are wellposed in this paper we consider another exactly solvable family of initial data the family of square barriers 0 x q l l for real amplitudes q using riemann hilbert techniques we obtain rigorous pointwise asymptotics for the semiclassical limit of focusing nls globally in space and up to an o 1 maximal time in particular we show that the discontinuities in our initial data regularize by the immediate generation of genus one oscillations emitted into the support of the initial data to the best of our knowledge this is the first case in which the genus structure of the semiclassical asymptotics for focusing nls have been calculated for nonanalytic initial data 2013 wiley periodicals inc

a sharp inequality of j moser for higher order derivatives in fact go nwl 1 where wnis the area of the surface of the unit n ball also he notes that if 3 exceeds go then 1 can no longer hold with some co independent of the u satisfying 2 the fact that this result holds for some 3 0 independent of u has been observed now by several authors with a variety of proofs see 24 20 23 22 10 4 1 2 13 14 for example the earliest of these arguments expands the exponential function in a power series then observing that the norm in the imbedding

the wentzel kramers brillouin approximation of semiclassical eigenvalues of the zakharov shabat problem the wkb approximation of semiclassical eigenvalues of the non self adjoint zakharov shabat problem is a standard element in the theory of the integrable focusing nonlinear schrodinger equation this approximation is based on a reformulation of the eigenvalue problem as the eigenvalue problem for a self adjoint schrodinger operator with a correction term that depends on the spectral parameter the approximation results from neglecting the correction term we perform a numerical experiment which gives new evidence to support the validity of this procedure in the context of the semiclassical limit problem for the focusing nonlinear schrodinger equation in particular our results suggest that the rate of convergence of the approximate eigenvalues to the true ones is of the order of the square of the small parameter this information is relevant to the task of rigorously incorporating this approximation into the asymptotic analysis of the singular limit for the focusing nonlinear schrodinger equation

adams inequality on the hyperbolic space in this article we establish the following adams type inequality in the hyperbolic space hn supu cc hn hn pku udvg 1 hn e u2 1 dvg iff 0 n k where 2k n pk is the critical gjms operator in hn and 0 n k is as defined in 1 3 as an application we prove the asymptotic behaviour of the best constants in sobolev inequalities when 2k n and also prove some existence results for the qk curvature type equation in hn

the sharp sobolev type inequalities in the lorentz sobolev spaces in the hyperbolic spaces abstract let w 1 l p q h n 1 q p denote the lorentz sobolev spaces of order one in the hyperbolic spaces h n our aim in this paper is three fold first of all we establish a sharp poincare inequality in w 1 l p q h n with 1 q p which generalizes the result in 41 to the setting of lorentz sobolev spaces second we prove several sharp poincare sobolev type inequalities in w 1 l p q h n with 1 q p n which generalize the results in 45 to the setting of lorentz sobolev spaces finally we provide the improved moser trudinger type inequalities in w 1 l n q h n in the critical case p n with 1 q n which generalize the results in 43 and improve the results in 57 in the proof of the main results we shall prove a polya szego type principle in w 1 l p q h n with 1 q p which maybe is of independent interest

semiclassical eigenvalue distribution of the zakharov shabat eigenvalue problem abstract in this paper we consider the semi classical limit of the non self adjoint zakharov shabat eigenvalue problem we conduct a series of careful numerical experiments which provide strong evidence that the number of eigenvalues scales like 1 just as in the self adjoint case and that the eigenvalues appear to approach a limiting curve one general choice of potential functions produces a y shaped spectrum we give an asymptotic argument which predicts a critical value for the phase for which the straight line spectra bifurcates to produce the y shaped spectra this asymptotic prediction agrees quite well with numerical experiments the asymptotic argument also predicts a symmetry breaking in the eigenfunctions which we are able to observe numerically we also show that the number of eigenvalues living away from the real axis for a restricted class of potentials is bounded by c 1 where c is an explicit constant a complete theory of the shape of the eigenvalue curve and a general bound on the number of eigenvalues is still lacking

on the relationship between the eulerian and lagrangian descriptions of finite rigid plasticity the motivation for the present paper is to clarify certain unresolved issues pertaining to the relation between the lagrangian or referential and eulerian or spatial strain space formulations of finite plasticity for conceptual simplicity attention is confined to rigid plastic materials it is shown first that for constitutive equations in which the hardening parameter is a scalar the lagrangian and eulerian descriptions are equivalent and that additionally the choice of objective stress rate is immaterial in the light of these developments the role of objective rates is further explored in connection with more general anisotropic hardening laws which contain a shift tensor a form of the constitutive equation for the rate of the shift tensor is motivated in which the choice of objective rate is arbitrary it is then demonstrated that the structure of the constitutive equations of the theory in both the eulerian and lagrangian descriptions is form invariant under arbitrary transformations of objective rate the approach taken here contrasts with that adopted in a number of recent papers in which preference is given to one particular objective rate or another

spectral instability for some schrodinger operators we show that the condition numbers of isolated eigenvalues of typical non self adjoint differential operators such as the harmonic oscillator may be extremely large we describe a stable procedure for computing the condition numbers for schrodinger operators in one dimension and apply it to the complex resonances of a typical operator with a dilation analytic potential

trudinger moser inequalities on harmonic an groups under lorentz norms abstract the main purpose of this paper is to establish sharp trudinger moser type inequalities on harmonic a n groups s for functions whose gradient is in the lorentz space l n q our results show that even if s is not with strictly negative sectional curvature one can also replace the norm s s u n d v s u n d v by s s u n d v in the trudinger moser type inequalities

trudinger moser inequalities on complete noncompact riemannian manifolds abstract let m g be a complete noncompact riemannian n manifold n 2 if there exist positive constants and such that sup u w 1 n m u 1 1 m e u n n 1 k 0 n 2 k u n k n 1 k d v g where u 1 g u l n m u l n m then we say that the trudinger moser inequality holds suppose the trudinger moser inequality holds we prove that there exists some positive constant such that vol g b x 1 for all x m also we give a sufficient condition under which the trudinger moser inequality holds say the ricci curvature of m g has lower bound and its injectivity radius is positive moreover the adams inequality is discussed in this paper for application of the trudinger moser inequality we obtain existence results for some quasilinear equations with nonlinearity of exponential growth

sharp hardy adams inequalities for bi laplacian on hyperbolic space of dimension four abstract we establish sharp hardy adams inequalities on hyperbolic space b 4 of dimension four namely we will show that for any 0 there exists a constant c 0 such that b 4 e 32 2 u 2 1 32 2 u 2 d v 16 b 4 e 32 2 u 2 1 32 2 u 2 1 x 2 4 d x c for any u c 0 b 4 with b 4 h 9 4 h u u d v 1 as applications we obtain a sharpened adams inequality on hyperbolic space b 4 and an inequality which improves the classical adams inequality and the hardy inequality simultaneously the later inequality is in the spirit of the hardy trudinger moser inequality on a disk in dimension two given by wang and ye in 46 and on any convex planar domain by lu and yang in 33 the fourier analysis techniques on hyperbolic and symmetric spaces play an important role in our work

on the lagrangian description of vorticity this paper exploits a lagrangian form of the vorticity field that is based on a formula of beltrami the merits of which do not seem to have been fully appreciated a number of new results are readily derived by using the lagrangian description and the classical vorticity theorems are also included the results being purely kinematical apply to all deformable media including viscous fluids and may also be of value in computations