multilinear schur multipliers and schatten properties of operator taylor remainders abstract we establish sharp conditions on scalar functions and perturbations that guarantee schatten summability of n th order operator taylor remainders in the special case of dimension one our estimates of these remainders deliver well known classical estimates of scalar taylor remainders we prove that if a scalar function f is in the set c n and a perturbation is in the p th schatten class s p p n then the respective n th order operator taylor remainder is an element of s p n and has an estimate like the one in 16 we construct examples of f c n and perturbations in s n such that the n th order taylor remainder of the respective operator function is not in s 1 our construction relies in particular on novel dimension dependent estimates for schatten norms of multilinear schur multipliers from below that are of interest in their own right our results apply to both self adjoint and unitary operators

homogenization of some degenerate pseudoparabolic variational inequalities abstract multiscale analysis of a degenerate pseudoparabolic variational inequality modelling the two phase flow with dynamical capillary pressure in a perforated domain is the main topic of this work regularisation and penalty operator methods are applied to show the existence of a solution of the nonlinear degenerate pseudoparabolic variational inequality defined in a domain with microscopic perforations as well as to derive a priori estimates for solutions of the microscopic problem the main challenge is the derivation of a priori estimates for solutions of the variational inequality uniformly with respect to the regularisation parameter and to the small parameter defining the scale of the microstructure the method of two scale convergence is used to derive the corresponding macroscopic obstacle problem

a pseudo parabolic variational inequality and stefan problem abstract we study existence and uniqueness questions concerning non linear variational inequalities of the type of 1 1a 1 1b in the introduction applications to free boundary problems are also given author

weak type commutator and lipschitz estimates resolution of the nazarov peller conjecture let m be a semi finite von neumann algebra and let f r c be a lipschitz function if a b m are self adjoint operators such that a b l1 m then

resolution of peller s problem concerning koplienko neidhardt trace formulae a formula for the norm of a bilinear schur multiplier acting from the cartesian product s 2 s 2 of two copies of the hilbert schmidt classes into the trace class s 1 is established in terms of linear schur multipliers acting on the space s 1 of all compact operators using this formula we resolve peller s problem on koplienko neidhardt trace formulae namely we prove that there exist a twice continuously differentiable function f with a bounded second derivative a self adjoint unbounded operator a and a self adjoint operator b 2 s 2 such that f a b f a d dt f a tb t 0 2 s 1

estimates and trace formulas for unitary and resolvent comparable perturbations abstract we prove estimates for schatten norms of multiple operator integrals with symbols arising from fourier series and as a consequence derive higher order trace formulas for functions given by absolutely convergent fourier series of unitary operators and for rational functions bounded on the real line of resolvent comparable self adjoint operators these results substantially extend the main results of 12 14 13 where the estimates were proved for multiple operator integrals with symbols arising from polynomials the trace formulas in the unitary case for analytic functions on the unit disc and the trace formulas in the resolvent comparable self adjoint case for rational functions bounded and analytic in the lower half plane our results rely crucially on an algebraic approach to symbols of multiple operator integrals developed in this paper to replace the prior restrictive analytic approach of 12

higher order spectral shift abstract we construct higher order spectral shift functions extending the perturbation theory results of m g krein m g krein on a trace formula in perturbation theory mat sb 33 1953 597 626 in russian and l s koplienko l s koplienko trace formula for perturbations of nonnuclear type sibirsk mat zh 25 1984 62 71 in russian translation in trace formula for nontrace class perturbations siberian math j 25 1984 735 743 on representations for the remainders of the first and second order taylor type approximations of operator functions the higher order spectral shift functions represent the remainders of higher order taylor type approximations they can be expressed recursively via the lower order in particular krein s and koplienko s ones we also obtain higher order spectral averaging formulas generalizing the birman solomyak spectral averaging formula the results are obtained in the semi finite von neumann algebra setting with the perturbation taken in the hilbert schmidt class of the algebra

degenerate quasilinear pseudoparabolic equations with memory terms and variational inequalities abstract the existence of solutions of degenerate quasilinear pseudoparabolic equations where the term t u is replace by t b u with memory terms and quasilinear variational inequalities is shown the existence of solutions of equations is proved under the assumption that the nonlinear function b is monotone and a gradient of a convex continuously differentiable function the uniqueness is proved for lipschitz continuous elliptic parts the existence of solutions of quasilinear variational inequalities is proved under stronger assumptions namely the nonlinear function defining the elliptic part is assumed to be a gradient and the function b to be lipschitz continuous

stability and uniqueness properties of taylor approximations of matrix functions abstract we consider taylor approximations of matrix functions and establish bounds controlling the size of a perturbation in terms of the size of the taylor remainder or equivalently the associated spectral shift function s this can be viewed as a stability property for the inverse problem of recovering the perturbation from the spectral shift function s from our bounds we derive a uniqueness result for spectral sums if the trace of the taylor remainder of order n 2 equals zero for a finite number of monomials then the perturbation is zero this uniqueness result is in contrast with nonuniqueness of the first order taylor remainder as an application we characterize equality of two graphs in terms of self returning walks

stability for implicit explicit schemes for non equilibrium kinetic systems in weighted spaces with symmetrization abstract we consider kinetic systems and prove their stability working in weighted spaces in which the systems are symmetric we prove stability for various explicit and implicit semi discrete and fully discrete schemes the applications include advective and diffusive transport coupled to the accumulation of immobile components governed by non equilibrium relationships we also discuss extensions to nonlinear relationships and multiple species

domains of infinitesimal generators of automorphism flows we present a systematic characterization of the domain of a generator of a one parameter group on certain c subalgebras of l h via finite dimensional estimates our approach yields an example of a densely defined closed symmetric derivation on a c subalgebras of l h whose domain is not closed with respect to the c1 functional calculus this completes and complements the earlier example of mcintosh j funct anal 30 1977 264 our methods are partly based on the theory of adjoint c0 semigroups

spectral shift function of higher order this paper resolves affirmatively koplienko s sib mat zh 25 62 71 1984 conjecture on existence of higher order spectral shift measures moreover the paper establishes absolute continuity of these measures and thus existence of the higher order spectral shift functions a spectral shift function of order n is the function n n h v such that operatorname tr biggl f h v sum k 0 n 1 frac 1 k frac d k dt k bigl f h tv bigr bigg t 0 biggr int mathbb r f n t eta n t dt for every sufficiently smooth function f where h is a self adjoint operator defined in a separable hilbert space and v is a self adjoint operator in the n th schatten von neumann ideal sn existence and summability of 1 and 2 were established by krein mat sb 33 597 626 1953 and koplienko sib mat zh 25 62 71 1984 respectively whereas for n 2 the problem was unresolved we show that n h v exists integrable and vert eta n vert l 1 mathbb r leq c n vert v vert s n n for some constant cn depending only on n our results for n rely on estimates for multiple operator integrals obtained in this paper our method also applies to the general semi finite von neumann algebra setting of the perturbation theory

hessian matrix vs gauss newton hessian matrix in this paper we investigate how the gauss newton hessian matrix affects the basin of convergence in newton type methods although the newton algorithm is theoretically superior to the gauss newton algorithm and the levenberg marquardt lm method as far as their asymptotic convergence rate is concerned the lm method is often preferred in nonlinear least squares problems in practice this paper presents a theoretical analysis of the advantage of the gauss newton hessian matrix it is proved that the gauss newton approximation function is the only nonnegative convex quadratic approximation that retains a critical property of the original objective function taking the minimal value of zero on an n 1 dimensional manifold or affine subspace due to this property the gauss newton approximation does not change the zero on n 1 d structure of the original problem explaining the reason why the gauss newton hessian matrix is preferred for nonlinear least squares problems especially when the initial point is far from the solution

differentiation of operator functions in non commutative lp spaces the principal results in this paper are concerned with the description of differentiable operator functions in the non commutative lp spaces 1 p associated with semifinite von neumann algebras for example it is established that if f r r is a lipschitz function then the operator function f is g teaux differentiable in l2 m for any semifinite von neumann algebra m if and only if it has a continuous derivative furthermore if f r r has a continuous derivative which is of bounded variation then the operator function f is g teaux differentiable in any lp m 1 p

dixmier traces on noncompact isospectral deformations abstract we extend the isospectral deformations of connes landi and dubois violette to the case of riemannian spin manifolds carrying a proper action of the noncompact abelian group r l under deformation by a torus action a standard formula relates dixmier traces of measurable operators to integrals of functions on the manifold we show that this relation persists for actions of r l under mild restrictions on the geometry of the manifold which guarantee the dixmier traceability of those operators

practical matrix completion and corruption recovery using proximal alternating robust subspace minimization low rank matrix completion is a problem of immense practical importance recent works on the subject often use nuclear norm as a convex surrogate of the rank function despite its solid theoretical foundation the convex version of the problem often fails to work satisfactorily in real life applications real data often suffer from very few observations with support not meeting the randomness requirements ubiquitous presence of noise and potentially gross corruptions sometimes with these simultaneously occurring this paper proposes a proximal alternating robust subspace minimization method to tackle the three problems the proximal alternating scheme explicitly exploits the rank constraint on the completed matrix and uses the ell 0 l 0 pseudo norm directly in the corruption recovery step we show that the proposed method for the non convex and non smooth model converges to a stationary point although it is not guaranteed to find the global optimal solution in practice we find that our algorithm can typically arrive at a good local minimizer when it is supplied with a reasonably good starting point based on convex optimization extensive experiments with challenging synthetic and real data demonstrate that our algorithm succeeds in a much larger range of practical problems where convex optimization fails and it also outperforms various state of the art algorithms

multiple operator integrals and higher operator derivatives abstract in this paper we consider the problem of the existence of higher derivatives of the function t a tk where is a function on the real line a is a self adjoint operator and k is a bounded self adjoint operator we improve earlier results by sten kin in order to do this we give a new approach to multiple operator integrals this approach improves the earlier approach given by sten kin we also consider a similar problem for unitary operators

peller s problem concerning koplienko neidhardt trace formulae the unitary case we prove the existence of a c2 function f t c defined on the unit circle a unitary operator u and a self adjoint operator z in the hilbert schmidt class s2 such that f eizu f u ddt f eitzu t 0 s1 the space of trace class operators this resolves a problem of peller concerning the validity of the koplienko neidhardt trace formula for unitaries

operator lipschitz functions in schatten von neumann classes this paper resolves a number of problems in the perturbation theory of linear operators linked with the 45 year old conjecure of m g kre n in particular we prove that every lipschitz function is operator lipschitz in the schatten von neumann ideals s 1 0 such that left f a f b right alpha leqslant c alpha left f right text lip 1 left a b right alpha where f is a lipschitz function with left f right text lip 1 mathop sup limits lambda ne mu lambda mu in mathbb r left frac f left lambda right f left mu right lambda mu right infty left cdot right alpha is the norm is s and a and b are self adjoint linear operators such that a b in s alpha

higher order s2 differentiability and application to koplienko trace formula abstract let a be a selfadjoint operator in a separable hilbert space k a selfadjoint hilbert schmidt operator and f c n r we establish that t f a t k f a is n times continuously differentiable on r in the hilbert schmidt norm provided either a is bounded or the derivatives f i i 1 n are bounded this substantially extends the results of 3 on higher order differentiability of in the hilbert schmidt norm for f in a certain wiener class as an application of the second order s 2 differentiability we extend the koplienko trace formula from the besov class b 1 2 r 20 to functions f for which the divided difference f 2 admits a certain hilbert space factorization

an extension of the koplienko neidhardt trace formulae abstract koplienko sibirsk mat zh 25 5 1984 62 found a trace formula for perturbations of self adjoint operators by operators of hilbert schmidt class s 2 a similar formula in the case of unitary operators was obtained by neidhardt math nachr 138 1988 7 in this paper we improve their results and obtain sharp conditions under which the koplienko neidhardt trace formulae hold

tracial bounds for multilinear schur multipliers abstract we derive new types of two sided bounds for traces of multilinear schur multipliers that sharpen existing bounds in a number of situations we also characterize the tracial positivity of the n linear multipliers on symmetric tuples assuming their symbols are mildly sparse when n 5

functions of unitary operators derivatives and trace formulas abstract let a be a bounded self adjoint operator on a separable hilbert space h and u 0 an arbitrary unitary operator we derive formulas for higher order derivatives of the function s e i s a u 0 for sufficiently smooth functions we compare our formulas with earlier results in this area and apply them to establish higher order analogs of trace formulas of krein koplienko neidhardt

a search grid for parameter optimization as a byproduct of model sensitivity analysis inverse problem solving i e the retrieval of optimal values of model parameters from experimental data remains a bottleneck for modelers therefore a large variety of heuristic optimization algorithms has been developed to deal with the inverse problem however in some cases the use of a grid search may be more appropriate or simply more practical in this paper an approach is presented to improve the selection of the grid points to be evaluated and which does not depend on the knowledge or availability of the underlying model equations it is suggested that using the information acquired through a sensitivity analysis can lead to better grid search results using the sensitivity analysis information a gauss newton like matrix is constructed and the eigenvalues and eigenvectors of this matrix are employed to transform naive search grids into better thought out ones after a theoretical analysis of the approach some computational experiments are performed using a simple linear model as well as more complex nonlinear models

micro structure models of diffusion in fissured media abstract two diffusion models are developed which recognize the local geometry of the individual cells or storage sites and the exchange of flux on the micro scale of these cells the cauchy problems for these model systems are shown to be resolved by holomorphic semigroups and various classical models are obtained as limits of these disributed micro structure models

moyal planes are spectral triples axioms for nonunital spectral triples extending those introduced in the unital case by connes are proposed as a guide and for the sake of their importance in noncommutative quantum field theory the spaces r2n endowed with moyal products are intensively investigated some physical applications such as the construction of noncommutative wick monomials and the computation of the connes lott functional action are given for these noncommutative hyperplanes

a free boundary problem for a degenerate parabolic system abstract the degenerate parabolic system 1 1 in the introduction serves as a model for heat conduction in a heterogeneous medium consisting of two components the first component is made up of small pieces suspended in the second component and the second component undergoes a change of phase at a prescribed temperature this phenomenon occurs in a mixture of gravel and wet soil for example melting of frozen soil existence and uniqueness results of weak solutions of the degenerate parabolic problem are shown by employing monotone operator theory local regularity such as continuity and boundedness of the solution is studied a discussion is provided about the mutual interplay of the thermodynamic temperature the temperature in the first component and the conductive temperature the temperature in the second component author

initial boundary value problems for a class of pseudoparabolic equations with integral boundary conditions in this paper we deal with a class of pseudoparabolic problems with integral boundary conditions we will first establish an a priori estimate then we prove the existence uniqueness and continuous dependence of the solution upon the data finally some extensions of the problem are given