cone valued lyapunov functions and stability for impulsive functional differential equations abstract the stability criteria in terms of two measures for impulsive functional differential equations are established via cone valued lyapunov functions and razumikhin technique the stability can be deduced from the q 0 q stability of comparison impulsive differential equations an example is given to illustrate the advantages of the results obtained

on the φ0 stability of comparison differential systems for many years now the variety of problems of the qualitative properties of differential equations in the context of lyaponov second method has been successfully investigated in a unified way using the comparison principle s in this method the qualitative properties of the system of differential equations are inferred from the corresponding properties of the solutions of the system of comparison equations in order to successfully employ this method it is generally required that the comparison systems possess some special properties the conditions under which the scalar comparison equations possess positivity and stability behaviour have been investigated using scalar lyapunov function method by brauer l siljak and grujic 9 and others ladde 3 gave suficient conditions for nonnegativity and stability of solutions of systems of comparison equations in his investigation of stability of comparison differential systems ladde 3 considered the asymptotic stability and exponential asymptotic stability of the comparison differential system using the method of vector lyapunov functions however imbedded in the method of vector lyapunov function is the requirement of quasimonotone nondecreasing property of the comparison system but quasimonotonicity of the comparison system is not a necessary condition for the system to be stable thus the limitation of the application potential of this general and effective method is due to the fact that

extremality conditions for generalized channels a generalized channel is a completely positive map that preserves trace on a given subspace we find conditions under which a generalized channel with respect to a positively generated subspace j is an extreme point in the set of all such generalized channels as a special case this yields extremality conditions for quantum protocols in particular we obtain new extremality conditions for quantum 1 testers with 2 outcomes which correspond to yes no measurements on the set of quantum channels

efficient elementary and restricted non elementary route pricing abstract column generation is involved in the current most efficient approaches to routing problems set partitioning formulations model routing problems by considering all possible routes and selecting a subset that visits all customers these formulations often produce tight lower bounds and require column generation for their pricing step the bounds in the resulting branch and price are tighter when elementary routes are considered but this approach leads to a more difficult pricing problem balancing the pricing with route relaxations has become crucial for the efficiency of the branch and price for routing problems recently the ng routes relaxation was proposed as a compromise between elementary and non elementary routes the ng routes are non elementary routes with the restriction that when following a customer the route is not allowed to visit another customer that was visited before if they belong to a dynamically computed set the larger the size of these sets the closer the ng route is to an elementary route this work presents an efficient pricing algorithm for ng routes and extends this algorithm for elementary routes therefore we address the shortest path problem with resource constraint spprc and the elementary shortest path problem with resource constraint espprc the proposed algorithm combines the decremental state space relaxation technique dssr with completion bounds we apply this algorithm for the generalized vehicle routing problem gvrp and for the capacitated vehicle routing problem cvrp demonstrating that it is able to price elementary routes for instances up to 200 customers a result that doubles the size of the espprc instances solved to date

generalized channels channels for convex subsets of the state space let k be a convex subset of the state space of a finite dimensional c algebra we study the properties of channels on k which are defined as affine maps from k into the state space of another algebra extending to completely positive maps on the subspace generated by k we show that each such map is the restriction of a completely positive map on the whole algebra called a generalized channel we characterize the set of generalized channels and also the equivalence classes of generalized channels having the same value on k moreover if k contains the tracial state the set of generalized channels forms again a convex subset of a multipartite state space this leads to a definition of a generalized supermap which is a generalized channel with respect to this subset we prove a decomposition theorem for generalized supermaps and describe the equivalence classes the set of generalized supermaps having the same value on equivalent generalized channels is also characterized special cases include quantum

stability of functional differential equations with impulses abstract in this paper the stability of functional differential equations fde with impulses is investigated some comparison theorems are given several lyapunov razumikhin functions of partial components of the state variable x which can be much easier constructed are used so that the conditions ensuring that stability are simpler and less restrictive the results improve and generalize the ones in the literature an example is also given to illustrate the importance of our results

stability criteria of impulsive differential systems the notion of eventual stability has been recently discussed we extend this notion to impulsive systems of differential equations our technique depends on liapunov s direct method

exact algorithms for the vehicle routing problem based on spanning tree and shortest path relaxations we consider the problem of routing vehicles stationed at a central facility depot to supply customers with known demands in such a way as to minimize the total distance travelled the problem is referred to as the vehicle routing problem vrp and is a generalization of the multiple travelling salesman problem that has many practical applications we present tree search algorithms for the exact solution of the vrp incorporating lower bounds computed from i shortest spanning k degree centre tree k dct and ii q routes the final algorithms also include problem reduction and dominance tests computational results are presented for a number of problems derived from the literature the results show that the bounds derived from the q routes are superior to those from k dct and that vrps of up to about 25 customers can be solved exactly

on the convex structure of process positive operator valued measures measurements on quantum channels are described by so called process positive operator valued measures or process povms we study implementing schemes of extremal process povms as it turns out the corresponding measurement must satisfy certain extremality property which is stronger than the usual extremality given by the convex structure this property motivates the introduction and investigation of the a convex structure of povms which generalizes both the usual convex and c convex structures we show that extremal points and faces of the set of process povms are closely related to a extremal points and a faces of povms for a certain subalgebra a we also give a characterization of a extremal and a pure povms

completely positive divisibility does not mean markovianity in the classical domain it is well known that divisibility does not imply that a stochastic process is markovian however for quantum processes divisibility is often considered to be synonymous with markovianity we show that completely positive divisible quantum processes can still involve non markovian temporal correlations that we then fully classify using the recently developed process tensor formalism which generalizes the theory of stochastic processes to the quantum domain

existence and multiplicity of solutions for some dirichlet problems with impulsive effects abstract in this paper we consider the existence and multiplicity of solutions for some dirichlet impulsive problems and some existence and multiplicity results are obtained the solutions are sought by means of lax milgram theorem and some critical theorems

stronger multi commodity flow formulations of the capacitated vehicle routing problem the capacitated vehicle routing problem is a much studied and strongly np hard combinatorial optimization problem for which many integer programming formulations have been proposed we present two new multi commodity flow mcf formulations and show that they dominate all of the existing ones in the sense that their continuous relaxations yield stronger lower bounds moreover we show that the relaxations can be strengthened in pseudo polynomial time in such a way that all of the so called knapsack large multistar klm inequalities are satisfied the only other relaxation known to satisfy the klm inequalities based on set partitioning is strongly np hard to solve computational results demonstrate that the new mcf relaxations are significantly stronger than the previously known ones

an efficient transformation of the generalized vehicle routing problem abstract the generalized vehicle routing problem gvrp is the problem of designing optimal delivery or collection routes subject to capacity restrictions from a given depot to a number of predefined mutually exclusive and exhaustive clusters in this paper we describe an efficient transformation of the gvrp into a capacitated arc routing problem carp for which an exact algorithm and several approximate procedures are reported in literature it constitutes the only known approach for solving the gvrp

extending quantum operations for a given set of input output pairs of quantum states or observables we ask the question whether there exists a physically implementable transformation that maps each of the inputs to the corresponding output the physical maps on quantum states are trace preserving completely positive maps but we also consider variants of these requirements we generalize the definition of complete positivity to linear maps defined on arbitrary subspaces then formulate this notion as a semidefinite program and relate it by duality to approximative extensions of this map this gives a characterization of the maps which can be approximated arbitrarily well as the restriction of a map that is completely positive on the whole algebra also yielding the familiar extension theorems on operator spaces for quantum channel extensions and extensions by probabilistic operations we obtain semidefinite characterizations and we also elucidate the special case of abelian inputs or outputs finally revisiting a theorem by alberti and uhlmann we provide simpler and more widely applicable conditions for certain extension problems on qubits and by using a semidefinite programming formulation we exhibit counterexamples to seemingly reasonable but false generalizations of the alberti uhlmann theorem

stability of abstract linear switched impulsive differential equations linear impulsive switched differential equations in hilbert space are studied we propose a new method of stability investigation which is based in constructing an equivalent impulsive system without switching using operator valued and scalar lyapunov functions sufficient conditions for lyapunov stability of linear switched impulsive equations in a banach space are established the examples of studies of this class of equations with periodic switching are given

comparison of quantum statistical models equivalent conditions for sufficiency a family of probability distributions i e a statistical model is said to be sufficient for another if there exists a transition matrix transforming the probability distributions in the former to the probability distributions in the latter the blackwell sherman stein bss theorem provides necessary and sufficient conditions for one statistical model to be sufficient for another by comparing their information values in statistical decision problems in this paper we extend the bss theorem to quantum statistical decision theory where statistical models are replaced by families of density matrices defined on finite dimensional hilbert spaces and transition matrices are replaced by completely positive trace preserving maps i e coarse grainings the framework we propose is suitable for unifying results that previously were independent like the bss theorem for classical statistical models and its analogue for pairs of bipartite quantum states recently proved by shmaya an important role in this paper is played by statistical morphisms namely affine maps whose definition generalizes that of coarse grainings given by petz and induces a corresponding criterion for statistical sufficiency that is weaker and hence easier to be characterized than petz s

note on the complexity of the shortest path models for column generation in vrptw in this note we prove that the relaxation approach in designing the subproblem of pricing out only the feasible routes for the set partition formulation of the vrptw is justified on complexity grounds that is the first dynamic programming model presented in m desrochers j desrosiers and m solomon 1992 that is able to price out all feasible routes is np hard in the strong sense

new route relaxation and pricing strategies for the vehicle routing problem in this paper we describe an effective exact method for solving both the capacitated vehicle routing problem cvrp and the vehicle routing problem with time windows vrptw that improves the method proposed by baldacci et al baldacci r n christofides a mingozzi 2008 an exact algorithm for the vehicle routing problem based on the set partitioning formulation with additional cuts math programming115 2 351 385 for the cvrp the proposed algorithm is based on the set partitioning sp formulation of the problem we introduce a new route relaxation called ng route used by different dual ascent heuristics to find near optimal dual solutions of the lp relaxation of the sp model we describe a column and cut generation algorithm strengthened by valid inequalities that uses a new strategy for solving the pricing problem the new ng route relaxation and the different dual solutions achieved allow us to generate a reduced sp problem containing all routes of any optimal solution that is finally solved by an integer programming solver the proposed method solves four of the five open solomon s vrptw instances and significantly improves the running times of state of the art algorithms for both vrptw and cvrp

dividing quantum channels we investigate the possibility of dividing quantum channels into concatenations of other channels thereby studying the semigroup structure of the set of completely positive trace preserving maps we show the existence of indivisible channels which can not be written as non trivial products of other channels and study the set of infinitesimal divisible channels which are elements of continuous completely positive evolutions for qubit channels we obtain a complete characterization of the sets of indivisible and infinitesimal divisible channels moreover we identify those channels which are solutions of time dependent master equations for both positive and completely positive evolutions for arbitrary finite dimension we prove a representation theorem for elements of continuous completely positive evolutions based on new results on determinants of quantum channels and markovian approximations

divisibility and information flow notions of quantum markovianity for noninvertible dynamical maps we analyze the relation between completely positive cp divisibility and the lack of information backflow for an arbitrary not necessarily invertible dynamical map it is well known that cp divisibility always implies a lack of information backflow moreover these two notions are equivalent for invertible maps in this letter it is shown that for a map which is not invertible the lack of information backflow always implies the existence of a completely positive propagator which however needs not be trace preserving interestingly for a wide class of image nonincreasing dynamical maps this propagator becomes trace preserving as well and hence the lack of information backflow implies cp divisibility this result sheds new light into the structure of the time local generators giving rise to cp divisible evolutions we show that if the map is not invertible then positivity of dissipation decoherence rates is no longer necessary for cp divisibility

improved lower bounds and exact algorithm for the capacitated arc routing problem in the capacitated arc routing problem carp a subset of the edges of an undirected graph has to be serviced at least cost by a fleet of identical vehicles in such a way that the total demand of the edges serviced by each vehicle does not exceed its capacity this paper describes a new lower bounding method for the carp based on a set partitioning like formulation of the problem with additional cuts this method uses cut and column generation to solve different relaxations of the problem and a new dynamic programming method for generating routes an exact algorithm based on the new lower bounds was also implemented to assess their effectiveness computational results over a large set of classical benchmark instances show that the proposed method improves most of the best known lower bounds for the open instances and can solve several of these for the first time

clean positive operator valued measures in quantum mechanics the statistics of the outcomes of a measuring apparatus is described by a positive operator valued measure povm a quantum channel transforms povms into povms generally irreversibly thus losing some of the information retrieved from the measurement this poses the problem of which povms are undisturbed i e they are not irreversibly connected to another povm we will call such povms clean in a sense the clean povms would be perfect since they would not have any additional extrinsical noise quite unexpectedly it turns out that such a cleanness property is largely unrelated to the convex structure of povms and there are clean povms that are not extremal and vice versa in this article we solve the cleannes classification problem for number n of outcomes n d d dimension of the hilbert space and we provide a set of either necessary or sufficient conditions for n d along with an iff condition for the case of informationally complete povms for n d2

base norms and discrimination of generalized quantum channels we introduce and study norms in the space of hermitian matrices obtained from base norms in positively generated subspaces these norms are closely related to discrimination of so called generalized quantum channels including quantum states channels and networks we further introduce generalized quantum decision problems and show that the maximal average payoffs of decision procedures are again given by these norms we also study optimality of decision procedures in particular we obtain a necessary and sufficient condition under which an optimal 1 tester for discrimination of quantum channels exists such that the input state is maximally entangled

hilbert s projective metric in quantum information theory we introduce and apply hilbert s projective metric in the context of quantum information theory the metric is induced by convex cones such as the sets of positive separable or positive partial transpose operators it provides bounds on measures for statistical distinguishability of quantum states and on the decrease of entanglement under protocols involving local quantum operations and classical communication or under other cone preserving operations the results are formulated in terms of general cones and base norms and lead to contractivity bounds for quantum channels for instance improving ruskai s trace norm contraction inequality a new duality between distinguishability measures and base norms is provided for two given pairs of quantum states we show that the contraction of hilbert s projective metric is necessary and sufficient for the existence of a probabilistic quantum operation that maps one pair onto the other inequalities between hilbert s projective metric and the chernoff bound the

on cone perturbing liapunov function for impulsive differential systems the concept of f 0 stability and f 0 boundedness recently were introduced for systems of ordinary differential equations odes perturbing liapunov function method was discussed for systems of odes and extend to systems of functional differential equations fdes in this paper we extend these notions to impulsive systems of differential equations via cone perturbing liapunov function method

constructive method for detecting the information backflow of non markovian dynamics we investigate the relation between two approaches to the characterization of quantum markovianity divisibility and lack of information backflow we show that a bijective dynamical map is completely positive divisible if and only if a monotonic nonincrease of distinguishability is observed for two equiprobable states of the evolving system and an ancilla moreover our proof is constructive given any such map that is not completely positive divisible we give an explicit construction of two states that when taken with the same a priori probability exhibit information backflow finally while an ancilla is necessary for the equivalence to hold in general we show that it is always possible to witness the non markovianity of bijective maps without using any entanglement between the system and ancilla

branch price and cut for the mixed capacitated general routing problem with time windows abstract the mixed capacitated general routing problem with time windows mcgrptw is defined over a mixed graph for which some nodes arcs and edges have strictly positive demand and must be serviced the problem consists of determining a set of least cost vehicle routes that satisfy this requirement while respecting pre specified time windows and without exceeding the vehicle capacity in this work we transform the mcgrptw into an equivalent node routing problem over a directed graph thus we solve the equivalent problem by using a branch price and cut algorithm which relies on some effective techniques introduced in the field computational experiments over instances derived from the capacitated arc routing problem with time windows and from the mixed capacitated general routing problem are presented the article also describes experiments over benchmark instances

quantum non markovian processes break conditional past future independence for classical markovian stochastic systems past and future events become statistically independent when conditioned to a given state at the present time memory non markovian effects break this condition inducing a nonvanishing conditional past future correlation here this classical memory indicator is extended to a quantum regime which provides an operational definition of quantum non markovianity based on a minimal set of three time ordered quantum system measurements and postselection the detection of memory effects through the measurement scheme is univocally related to departures from born markov and white noise approximations in quantum and classical environments respectively

a new branch and cut algorithm for the capacitated vehicle routing problem we present a new branch and cut algorithm for the capacitated vehicle routing problem cvrp the algorithm uses a variety of cutting planes including capacity framed capacity generalized capacity strengthened comb multistar partial multistar extended hypotour inequalities and classical gomory mixed integer cuts for each of these classes of inequalities we describe our separation algorithms in detail also we describe the other important ingredients of our branch and cut algorithm such as the branching rules the node selection strategy and the cut pool management computational results for a large number of instances show that the new algorithm is competitive in particular we solve three instances b n50 k8 b n66 k9 and b n78 k10 of augerat to optimality for the first time

operational characterization of divisibility of dynamical maps in this work we show the operational characterization to the divisibility of dynamical maps in terms of the distinguishability of quantum channels it is proven that the distinguishability of any pair of quantum channels does not increase under divisible maps in which the full hierarchy of divisibility is isomorphic to the structure of entanglement between system and environment this shows that i channel distinguishability is the operational quantity signifying detecting divisibility indivisibility of dynamical maps and ii the decision problem for the divisibility of maps is as hard as the separability problem in entanglement theory we also provide the information theoretic characterization to the divisibility of maps with conditional min entropy

stability analysis for a class of functional differential equations and applications abstract the problem of lyapunov stability for functional differential equations in hilbert spaces is studied the system to be considered is non autonomous and the delay is time varying known results on this problem are based on the gronwall inequality yielding relative conservative bounds on nonlinear perturbations in this paper using more general lyapunov krasovskii functional neither model variable transformation nor bounding restriction on nonlinear perturbations is required to obtain improved conditions for the global exponential stability of the system the conditions given in terms of the solution of standard riccati differential equations allow to compute simultaneously the two bounds that characterize the stability rate of the solution the proposed method can be easily applied to some control problems of nonlinear non autonomous control time delay systems

robust branch and cut and price for the capacitated vehicle routing problem the best exact algorithms for the capacitated vehicle routing problem cvrp have been based on either branch and cut or lagrangean relaxation column generation this paper presents an algorithm that combines both approaches it works over the intersection of two polytopes one associated with a traditional lagrangean relaxation over q routes the other defined by bound degree and capacity constraints this is equivalent to a linear program with exponentially many variables and constraints that can lead to lower bounds that are superior to those given by previous methods the resulting branch and cut and price algorithm can solve to optimality all instances from the literature with up to 135 vertices this more than doubles the size of the instances that can be consistently solved

a branch and cut algorithm for the symmetric generalized traveling salesman problem we consider a variant of the classical symmetric traveling salesman problem in which the nodes are partitioned into clusters and the salesman has to visit at least one node for each cluster this np hard problem is known in the literature as the symmetric generalized traveling salesman problem gtsp and finds practical applications in routing scheduling and location routing in a companion paper fischetti et al fischetti m j j salazar p toth 1995 the symmetric generalized traveling salesman polytope networks 26 113 123 we modeled gtsp as an integer linear program and studied the facial structure of two polytopes associated with the problem here we propose exact and heuristic separation procedures for some classes of facet defining inequalities which are used within a branch and cut algorithm for the exact solution of gtsp heuristic procedures are also described extensive computational results for instances taken from the literature and involving up to 442 nodes are reported

extremal generalized quantum measurements a measurement on a section k of the set of states of a finite dimensional c algebra is defined as an affine map from k to a probability simplex special cases of such sections are used in description of quantum networks in particular quantum channels measurements on a section correspond to equivalence classes of the so called generalized povms which are called quantum testers in the case of networks we find extremality conditions for measurements on k and characterize generalized povms such that the corresponding measurement is extremal these results are applied to the set of channels we find explicit extremality conditions for two outcome measurements on qubit channels and give an example of an extremal qubit 1 tester such that the corresponding measurement is not extremal

joint measurability of quantum effects and the matrix diamond in this work we investigate the joint measurability of quantum effects and connect it to the study of free spectrahedra free spectrahedra typically arise as matricial relaxations of linear matrix inequalities an example of a free spectrahedron is the matrix diamond which is a matricial relaxation of the l1 ball we find that joint measurability of binary positive operator valued measures is equivalent to the inclusion of the matrix diamond into the free spectrahedron defined by the effects under study this connection allows us to use results about inclusion constants from free spectrahedra to quantify the degree of incompatibility of quantum measurements in particular we completely characterize the case in which the dimension is exponential in the number of measurements conversely we use techniques from quantum information theory to obtain new results on spectrahedral inclusion for the matrix diamond