some remarks on quasiconvexity and rank one convexity we explore some necessary conditions for quasiconvexity in an attempt to show that rank one convexity does not imply quasiconvexity when the target space for deformations is two dimensional an interesting construction is presented showing how rank one directions may fit with each other making the task harder than in higher dimensions

sharp inequalities that generalize the divergence theorem an extension of the notion of quasi convexity subject to suitable boundary conditions being imposed sharp inequalities are obtained on integrals over a region of certain special quadratic functions f e where e x derives from a potential u

a numerical method for detecting singular minimizers of multidimensional problems in nonlinear elasticity in this paper we describe and analyse a numerical method that detects singular minimizers and avoids the lavrentiev phenomenon for three dimensional problems in nonlinear elasticity this method extends to three dimensions the corresponding one dimensional method of ball and knowles

fractional equivalent lagrangian densities for a fractional higher order equation in this communication we show that the equivalent lagrangian densities elds of a fractional higher order nonlinear schrodinger equation with stable soliton like solutions can be related in a hitherto unknown way this new relationship is described in terms of a new fractional operator that includes both left and right sided fractional derivatives using this operator it is possible to generate new elds that contain different fractional parts in addition to the already known elds which only differ by a sum of first order partial derivatives of two arbitrary functions

computational methods for risk averse undiscounted transient markov models the total cost problem for discrete time controlled transient markov models is considered the objective functional is a markov dynamic risk measure of the total cost two solution methods value and policy iteration are proposed and their convergence is analyzed in the policy iteration method we propose two algorithms for policy evaluation the nonsmooth newton method and convex programming and we prove their convergence the results are illustrated on a credit limit control problem

young measure solutions of a class of forward backward diffusion equations abstract this paper is devoted to young measure solutions of a class of forward backward diffusion equations inspired by the idea from a recent work of demoulini we first discuss the regular case by introducing the young measure solutions and prove the existence for such solutions and then approximate the extreme case by the approach of regularization and establish the existence of young measure solutions in the class of functions with bounded variation

towards characterization of all 3 3 extremal quasiconvex quadratic forms given a d d quasiconvex quadratic form d 3 we prove that if the determinant of its acoustic tensor is an irreducible extremal polynomial that is not identically zero then the form itself is an extremal quasiconvex quadratic form i e it loses its quasiconvexity whenever a convex quadratic form is subtracted from it in the special case d 3 we slightly weaken the condition namely we prove that if the determinant of the acoustic tensor of the quadratic form is an extremal polynomial that is not a perfect square then the form itself is an extremal quadratic form in the case d 3 we also prove that if the determinant of the acoustic tensor of the form is identically zero then the form is either extremal or polyconvex also if the determinant of the acoustic tensor of the form is a perfect square then the form is either extremal or polycovex or is a sum of a rank 1 form and an extremal whose acoustic tensor determinant is identically zero here we use the notion of extremality introduced by milton in 1990 2017 wiley periodicals inc

on consistency of stochastic dominance and mean semideviation models we analyze relations between two methods frequently used for modeling the choice among uncertain outcomes stochastic dominance and mean risk approaches new necessary conditions for stochastic dominance are developed these conditions compare values of a certain functional which contains two components the expected value of a random outcome and a risk term represented by the central semideviation of the corresponding degree if the weight of the semideviation in the composite objective does not exceed the weight of the expected value maximization of such a functional yields solutions which are efficient in terms of stochastic dominance the results are illustrated graphically

explicit solutions for multivariate discrete time control problems under uncertainty we consider discrete time control problems where only the support sets of the initial condition and of the disturbances are known we study the applicability of the dp method and as a counterpart of the lq problem of the stochastic setting we present a problem that admits an explicit analytic solution in particular we characterize a property of compatibility between the system dynamics and the norms of the spaces that is crucial to obtain the analytic solution also in the general multidimensional case

persistently optimal policies in stochastic dynamic programming with generalized discounting in this paper we study a markov decision process with a nonlinear discount function first we define a utility on the space of trajectories of the process in the finite and infinite time horizon and then take their expected values it turns out that the associated optimization problem leads to a nonstationary dynamic programming and an infinite system of bellman equations which result in obtaining persistently optimal policies our theory is enriched by examples

integral estimates for null lagrangians sobolev spaces of differential forms are studied lp projections onto exact forms are introduced as a tool to obtain integral estimates for null lagrangians new results on compensated compactness are given and mean coercive variational integrals are found existence of minima of certain mean coercive functionals is established

markov decision problems where means bound variances we identify a rich class of finite horizon markov decision problems mdps for which the variance of the optimal total reward can be bounded by a simple linear function of its expected value the class is characterized by three natural properties reward nonnegativity and boundedness existence of a do nothing action and optimal action monotonicity these properties are commonly present and typically easy to check implications of the class properties and of the variance bound are illustrated by examples of mdps from operations research operations management financial engineering and combinatorial optimization

quasiconvexity and relaxation of nonconvex problems in the calculus of variations keywords quasiconvexity relaxation reference caa article 1982 002doi 10 1016 0022 1236 82 90046 5view record in web of science record created on 2008 11 25 modified on 2017 05 12

unified entropy entanglement measures and monogamy of multi party entanglement we show that restricted shareability of multi qubit entanglement can be fully characterized by unified q s entropy we provide a two parameter class of bipartite entanglement measures namely unified q s entanglement with its analytic formula in two qubit systems for q 1 0 s 1 and qs 3 using unified q s entanglement we establish a broad class of the monogamy inequalities of multi qubit entanglement for q 2 0 s 1 and qs 3

w1 p quasiconvexity and variational problems for multiple integrals abstract variational problems for the multiple integral i u g u x dx where r m and u r n are studied a new condition on g called w1 p quasiconvexity is introduced which generalizes in a natural way the quasiconvexity condition of c b morrey it being shown in particular to be necessary for sequential weak lower semicontinuity of i in w 1 p r n and for the existence of minimizers for certain related integrals counterexamples are given concerning the weak continuity properties of jacobians in w 1 p r n p n m an existence theorem for nonlinear elastostatics is proved under optimal growth hypotheses

berge s theorem for noncompact image sets abstract for an upper semi continuous set valued mapping from one topological space to another and for a lower semi continuous function defined on the product of these spaces berge s theorem states lower semi continuity of the minimum of this function taken over the image sets it assumes that the image sets are compact for hausdorff topological spaces this paper extends berge s theorem to set valued mappings with possible noncompact image sets and studies relevant properties of minima

unified entropy trade off relations for a single quantum channel many important properties of quantum channels are quantified by means of entropic functionals characteristics of such a kind are closely related to different representations of a quantum channel in the jamiolkowski choi representation the given quantum channel is described by the so called dynamical matrix entropies of the rescaled dynamical matrix known as map entropies describe a degree of introduced decoherence within the so called natural representation the quantum channel is formally posed by another matrix obtained as the reshuffling of the dynamical matrix the corresponding entropies characterize an amount with which the receiver a priori knows the channel output as was previously shown the map and receiver entropies are mutually complementary characteristics indeed there exists a non trivial lower bound on their sum first we extend the concept of receiver entropy to the family of unified entropies developing the previous results we further derive non trivial lower bounds on the sum of the map and receiver q s entropies the derivation is essentially based on some inequalities with the schatten norms and anti norms

on the relation between extremal elasticity tensors with orthotropic symmetry and extremal polynomials we prove that an elasticity tensor with orthotropic symmetry is extremal if the determinant of its acoustic tensor is an extremal polynomial that is not a perfect square

risk averse control of undiscounted transient markov models we use markov risk measures to formulate a risk averse version of the undiscounted total cost problem for a transient controlled markov process using the new concept of a multikernel we derive conditions for a system to be risk transient that is to have finite risk over an infinite time horizon we derive risk averse dynamic programming equations satisfied by the optimal policy and we describe methods for solving these equations we illustrate the results on an optimal stopping problem and an organ transplantation problem

generalized quantum entropy the generalized quantum entropies are introduced in this paper some important properties such as nonnegativity continuity and concavity are proved but different from the von neumann entropy the subadditivity and additivity fail for the quantum unified r s entropy in general

a new route to finding bounds on the generalized spectrum of many physical operators here we obtain bounds on the generalized spectrum of that operator whose inverse when it exists gives green s function we consider the wide range of physical problems that can be cast in a form where a constitutive equation j x l x e x h x with a source term h x holds for all x in some domain and relates fields e and j that satisfy appropriate differential constraints symbolized by e e 0 and j j where e 0 and j are orthogonal spaces that span the space h of square integrable fields in which h lies boundedness and coercivity conditions on the moduli l x ensure that there exists a unique e for any given h i e e g h which then establishes the existence of green s function g we show that the coercivity condition is guaranteed to hold if weaker conditions involving generalized quasiconvex functions are satisfied the advantage is that these weaker conditions are easier to verify and for multiphase materials they can be independent of the geometry of the phases for l x depending linearly on a vector of parameters z z1 z2 zn we obtain constraints on z that ensure that green s function exists and hence which provide bounds on the generalized spectrum here we obtain bounds on the generalized spectrum of that operator whose inverse when it exists gives green s function we consider the wide range of physical problems that can be cast in a form where a constitutive equation j x l x e x h x with a source term h x holds for all x in some domain and relates fields e and j that satisfy appropriate differential constraints symbolized by e e 0 and j j where e 0 and j are orthogonal spaces that span the space h of square integrable fields in which h lies boundedness and coercivity conditions on the moduli l x ensure that there exists a unique e for any given h i e e g h which then establishes the existence of green s function g we show that the coercivity condition is guaranteed to hold if weaker conditions involving generalized quasiconvex functions are satisfied the advantage is that these weaker conditions are easier to verify and for multiphase materials they can be independent of the geometry of the phases for l x depend

conditional risk mappings we introduce an axiomatic definition of a conditional convex risk mapping and we derive its properties in particular we prove a representation theorem for conditional risk mappings in terms of conditional expectations we also develop dynamic programming relations for multistage optimization problems involving conditional risk mappings

quasiconvexity at the boundary positivity of the second variation and elastic stability for one dimensional problems of the calculus of variations there is a well developed theory relating the positivity of the second variation at a critical point i e linearized stability to strong relative minima these results largely due to weierstrass are usually proved by the method of fields of extremals as is described in bolza 1904 and morrey 1966 or by an argument involving a careful use of taylor series as in hestenes 1966 chapter 3 14

bounds on the volume fractions of two materials in a three dimensional body from boundary measurements by the translation method using the translation method of tartar murat lurie and cherkaev bounds are derived on the volume occupied by an inclusion in a three dimensional conducting body the bounds assume that electrical impedance tomography measurements have been made for three sets of pairs of current flux and voltage measurements around the boundary additionally the conductivity of the inclusion and the conductivity of the surrounding medium are assumed to be known if the boundary data dirichlet or neumann is special i e such that the fields inside the body would be uniform were the body homogeneous then the bounds reduce to those of milton and thus when the volume fraction is small to those of capdeboscq and vogelius

from stochastic dominance to mean risk models semideviations as risk measures abstract two methods are frequently used for modeling the choice among uncertain outcomes stochastic dominance and mean risk approaches the former is based on an axiomatic model of risk averse preferences but does not provide a convenient computational recipe the latter quantifies the problem in a lucid form of two criteria with possible trade off analysis but cannot model all risk averse preferences in particular if variance is used as a measure of risk the resulting mean variance markowitz model is in general not consistent with stochastic dominance rules this paper shows that the standard semideviation square root of the semivariance as the risk measure makes the mean risk model consistent with the second degree stochastic dominance provided that the trade off coefficient is bounded by a certain constant similar results are obtained for the absolute semideviation and for the absolute and standard deviations in the case of symmetric or bounded distributions in the analysis we use a new tool the outcome risk o r diagram which appears to be particularly useful for comparing uncertain outcomes

on unified entropy characterization of quantum channels we consider properties of quantum channels with the use of unified entropies extremal unravelings of quantum channel with respect to these entropies are examined the concept of map entropy is extended in terms of the unified entropies the map q s entropy is naturally defined as the unified q s entropy of a rescaled dynamical matrix of given quantum channel inequalities of fannes type are obtained for introduced entropies in terms of both the trace and frobenius norms of difference between corresponding dynamical matrices additivity properties of introduced map entropies are discussed the known inequality of lindblad with the entropy exchange is generalized to many of the unified entropies for the tensor product of a pair of quantum channels we derive a two sided estimate on the output entropy of a maximally entangled input state

risk forms representation disintegration and application to partially observable two stage systems we introduce the concept of a risk form which is a real functional of two arguments a measurable function on a polish space and a measure on that space we generalize the duality theory and the kusuoka representation to this setting for a risk form acting on a product of polish spaces we define marginal and conditional forms and we prove a disintegration formula which represents a risk form as a composition of its marginal and conditional forms we apply the proposed approach to two stage stochastic programming problems with partial information and decision dependent observation distribution

addendum to sharp inequalities that generalize the divergence theorem an extension of the notion of quasi convexity the paper sharp inequalities that generalize the divergence theorem an extension of the notion of quasi convexity published in proc r soc a 2013 469 20130075 doi 10 1098 rspa 2013 0075 is c

possible generalization of boltzmann gibbs statistics with the use of a quantity normally scaled in multifractals a generalized form is postulated for entropy namelys q k 1 i 1 w p i q q 1 whereq characterizes the generalization andp i are the probabilities associated withw microscopic configurations w the main properties associated with this entropy are established particularly those corresponding to the microcanonical and canonical ensembles the boltzmann gibbs statistics is recovered as theq 1 limit

maximum and minimum entropy states yielding local continuity bounds given an arbitrary quantum state we obtain an explicit construction of a state e respectively e which has the maximum respectively minimum entropy among all states which lie in a specified neighborhood e ball of computing the entropy of these states leads to a local strengthening of the continuity bound of the von neumann entropy i e the audenaert fannes inequality our bound is local in the sense that it depends on the spectrum of the states e and e depend only on the geometry of the e ball and are in fact optimizers for a larger class of entropies these include the renyi entropy and the minimum and maximum entropies providing explicit formulas for certain smoothed quantities this allows us to obtain local continuity bounds for these quantities as well in obtaining this bound we first derive a more general result which may be of independent interest namely a necessary and sufficient condition under which a state maximizes a concave

relations for certain symmetric norms and anti norms before and after partial trace changes of some unitarily invariant norms and anti norms under the operation of partial trace are examined the norms considered form a two parametric family including both the ky fan and schatten norms as particular cases the obtained results concern operators acting on the tensor product of two finite dimensional hilbert spaces for any such operator we obtain upper bounds on norms of its partial trace in terms of the corresponding dimensionality and norms of this operator similar inequalities but in the opposite direction are obtained for certain anti norms of positive matrices through the stinespring representation the results are put in the context of trace preserving completely positive maps we also derive inequalities between the unified entropies of a composite quantum system and one of its subsystems where traced out dimensionality is involved as well

stability results for stochastic programs and sensors allowing for discontinuous objective functions the paper examines the stability of the optimal value and the solutions of stochastic programming problems stability is checked with respect to variations in both the problem formulation and the probability distribution that describes the uncertainty of particular interest is the case where the payoff functions may be discontinuous these results are applied to analyze the stability of sensors that model the possibility of making inquiries to improve the probabilistic information available about the uncertain quantities

null lagrangians weak continuity and variational problems of arbitrary order abstract we consider the problem of minimizing integral functionals of the form i u f x k u x dx where r p u r and k u denotes the set of all partial derivatives of u with orders k the method is based on a characterization of null lagrangians l ku depending only on derivatives of order k applications to elasticity and other theories of mechanics are given

dual monogamy inequality for entanglement we establish duality for monogamy of entanglement whereas monogamy of entanglement inequalities provide an upper bound for bipartite sharability of entanglement in a multipartite system as quantified by linear entropy we prove that the same quantity namely linear entropy provides a lower bound for distribution of bipartite entanglement in a multipartite system our theorem for monogamy of entanglement is used to establish relations between bipartite entanglement that separate one qubit from the rest versus separating two qubits from the rest

brief risk sensitive and minimax control of discrete time finite state markov decision processes this paper analyzes a connection between risk sensitive and minimax criteria for discrete time finite state markov decision processes mdps we synthesize optimal policies with respect to both criteria both for the finite horizon and the discounted infinite horizon problem a generalized decision making framework is introduced which includes as special cases a number of approaches that have been considered in the literature the framework allows for discounted risk sensitive and minimax formulations leading to stationary optimal policies on the infinite horizon we illustrate our results with a simple machine replacement problem

on characterizing the set of possible effective tensors of composites the variational method and the translation method a general algebraic framework is developed for characterizing the set of possible effective tensors of composites a transformation to the polarization problem simplifies the derivation of the hashin shtrikman variational principles and simplifies the calculation of the effective tensors of laminate materials a general connection is established between two methods for bounding effective tensors of composites the first method is based on the variational principles of hashin and shtrikman the second method due to tartar murat lurie and cherkaev uses translation operators or equivalently quadratic quasiconvex functions a correspondence is established between these translation operators and bounding operators on the relevant non local projection operator t1 an important class of bounds namely trace bounds on the effective tensors of two component media are given a geometrical interpretation after a suitable fractional linear transformation of the tensor space each bound corresponds to a tangent plane to the set of possible tensors a wide class of translation operators that generate these bounds is found the extremal translation operators in this class incorporate projections onto spaces of antisymmetric tensors these extremal translations generate attainable trace bounds even when the tensors of the two components are not well ordered in particular they generate the bounds of walpole on the effective bulk modulus the variational principles of gibiansky and cherkaev for bounding complex effective tensors are reviewed and used to derive some rigorous bounds that generalize the bounds conjectured by golden and papanicolaou an isomorphism is shown to underlie their variational principles this isomorphism is used to obtain dirichlet type variational principles and bounds for the effective tensors of general non selfadjoint problems it is anticipated that these variational principles which stem from the work of gibiansky and cherkaev will have applications in many fields of science

algorithmic aspects of mean variance optimization in markov decision processes we consider finite horizon markov decision processes under performance measures that involve both the mean and the variance of the cumulative reward we show that either randomized or history based policies can improve performance we prove that the complexity of computing a policy that maximizes the mean reward under a variance constraint is np hard for some cases and strongly np hard for others we finally offer pseudopolynomial exact and approximation algorithms

process based risk measures and risk averse control of discrete time systems for controlled discrete time stochastic processes we introduce a new class of dynamic risk measures which we call process based their main feature is that they measure risk of processes that are functions of the history of a base process we introduce a new concept of conditional stochastic time consistency and we derive the structure of process based risk measures enjoying this property we show that they can be equivalently represented by a collection of static law invariant risk measures on the space of functions of the state of the base process we apply this result to controlled markov processes and we derive dynamic programming equations we also derive dynamic programming equations for multistage stochastic programming with decision dependent distributions

more risk sensitive markov decision processes we investigate the problem of minimizing a certainty equivalent of the total or discounted cost over a finite and an infinite horizon that is generated by a markov decision process mdp in contrast to a risk neutral decision maker this optimization criterion takes the variability of the cost into account it contains as a special case the classical risk sensitive optimization criterion with an exponential utility we show that this optimization problem can be solved by an ordinary mdp with extended state space and give conditions under which an optimal policy exists in the case of an infinite time horizon we show that the minimal discounted cost can be obtained by value iteration and can be characterized as the unique solution of a fixed point equation using a sandwich argument interestingly it turns out that in the case of a power utility the problem simplifies and is of similar complexity than the exponential utility case however has not been treated in the literature so far we also establish the validity and convergence of the policy improvement method a simple numerical example namely the classical repeated casino game is considered to illustrate the influence of the certainty equivalent and its parameters finally the average cost problem is also investigated surprisingly it turns out that under suitable recurrence conditions on the mdp for convex power utility the minimal average cost does not depend on the parameter of the utility function and is equal to the risk neutral average cost this is in contrast to the classical risk sensitive criterion with exponential utility

monogamy of multi qubit entanglement using renyi entropy we introduce a class of bipartite entanglement measures based on renyi entropy namely renyi entanglement with an analytic formula in two qubit systems for 1 we also show that multi qubit entanglement has a monogamy inequality in terms of renyi entanglement for all 2

dual stochastic dominance and related mean risk models we consider the problem of constructing mean risk models which are consistent with the second degree stochastic dominance relation by exploiting duality relations of convex analysis we develop the quantile model of stochastic dominance for general distributions this allows us to show that several models using quantiles and tail characteristics of the distribution are in harmony with the stochastic dominance relation we also provide stochastic linear programming formulations of these models

variance penalized markov decision processes we consider a markov decision process with both the expected limiting average and the discounted total return criteria appropriately modified to include a penalty for the variability in the stream of rewards in both cases we formulate appropriate nonlinear programs in the space of state action frequencies averaged or discounted whose optimal solutions are shown to be related to the optimal policies in the corresponding variance penalized mdp the analysis of one of the discounted cases is facilitated by the introduction of a cartesian product of two independent mdps

risk sensitive control of discrete time markov processes with infinite horizon in this paper we study existence of solutions to the bellman equation corresponding to risk sensitive ergodic control of discrete time markov processes using three different approaches also for particular classes of systems asymptotics for vanishing risk factor is investigated showing that in the limit the optimal value for an average cost per unit time is obtained

risk averse dynamic programming for markov decision processes we introduce the concept of a markov risk measure and we use it to formulate risk averse control problems for two markov decision models a finite horizon model and a discounted infinite horizon model for both models we derive risk averse dynamic programming equations and a value iteration method for the infinite horizon problem we develop a risk averse policy iteration method and we prove its convergence we also propose a version of the newton method to solve a nonsmooth equation arising in the policy iteration method and we prove its global convergence finally we discuss relations to min max markov decision models

generalized w class state and its monogamy relation we generalize the w class of states from n qubits to n qudits and prove that their entanglement is fully characterized by their partial entanglements even for the case of the mixture that consists of a w class state and a product state

risk sensitive markov control processes we introduce a general framework for measuring risk in the context of markov control processes with risk maps on general borel spaces that generalize known concepts of risk measures in mathematical finance operations research and behavioral economics within the framework applying weighted norm spaces to incorporate unbounded costs also we study two types of infinite horizon risk sensitive criteria discounted total risk and average risk and solve the associated optimization problems by dynamic programming for the discounted case we propose a new discount scheme which is different from the conventional form but consistent with the existing literature while for the average risk criterion we state lyapunov like stability conditions that generalize known conditions for markov chains to ensure the existence of solutions to the optimality equation