two stage optimization problems with multivariate stochastic order constraints we propose a two stage risk averse stochastic optimization problem with a stochastic order constraint on a vector valued function of the second stage decisions this model is motivated by a multiobjective second stage problem we formulate optimality conditions for the problem and analyse the lagrangian relaxation of the order constraint we propose two decomposition methods to solve the problems and prove their convergence the methods are based on lagrangian relaxation of the order constraints and on a construction of successive risk neutral two stage problems additionally we propose a new combinatorial method for verification of the multivariate order relation which is a key part of both methods we analyse a supply chain problem using our model and we apply our methods to solve the optimization problem numerical results confirm the efficiency of the proposed methods

an algebro geometric proof of witten s conjecture we present a new proof of witten s conjecture the proof is based on the analysis of the relationship between intersection indices on moduli spaces of complex curves and hurwitz numbers enumerating ramified coverings of the 2 sphere

d outcome measurement for a nonlocality test for the purpose of a nonlocality test we propose a general correlation observable of two parties by utilizing local d outcome measurements with su d transformations and classical communications generic symmetries of the su d transformations and correlation observables are found for the test of nonlocality it is shown that these symmetries dramatically reduce the number of numerical variables which is important for numerical analysis of nonlocality a linear combination of the correlation observables which is reduced to the clauser horne shimony holt chsh bell s inequality for two outcome measurements leads to the collins gisin linden massar popescu cglmp nonlocality test for d outcome measurement as a system to be tested for its nonlocality we investigate a continuous variable cv entangled state with d measurement outcomes it allows the comparison of nonlocality based on different numbers of measurement outcomes on one physical system in our example of the cv state we find that a pure entangled state of any degree violates bell s inequality for d 2 measurement outcomes when the observables are of su d transformations

looking for symmetric bell inequalities finding all bell inequalities for a given number of parties measurement settings and measurement outcomes is in general a computationally hard task we show that all bell inequalities which are symmetric under the exchange of parties can be found by examining a symmetrized polytope which is simpler than the full bell polytope as an illustration of our method we generate 238 885 new bell inequalities and 1085 new svetlichny inequalities we find in particular facet inequalities for bell experiments involving two parties and two measurement settings that are not of the collins gisin linden massar popescu type

generalized circulant densities and a sufficient condition for separability in a series of papers with kossakowski the first author has examined properties of densities for which the positive partial transpose ppt property can be readily checked these densities were also investigated from a different perspective by baumgartner hiesmayr and narnhofer in this paper we show how the support of such densities can be expressed in terms of lines in a finite geometry and how the same structure lends itself to checking the necessary ppt condition and to a novel sufficient condition for separability

bounding the set of quantum correlations we introduce a hierarchy of conditions necessarily satisfied by any distribution p ensuremath alpha ensuremath beta representing the probabilities for two separate observers to obtain outcomes ensuremath alpha and ensuremath beta when making local measurements on a shared quantum state each condition in this hierarchy is formulated as a semidefinite program among other applications our approach can be used to obtain upper bounds on the quantum violation of an arbitrary bell inequality it yields for instance tight bounds for the violations of the collins et al inequalities

the gromov witten potential of a point hurwitz numbers and hodge integrals hurwitz numbers which count certain covers of the projective line or equivalently factorizations of permutations into transpositions have been extensively studied for over a century the gromov witten potential f of a point the generating series for descendent integrals on the moduli space of curves is a central object of study in gromov witten theory we define a slightly enriched gromov witten potential g including integrals involving one lambda class and show that after a non trivial change of variables g h in positive genus where h is a generating series for hurwitz numbers we prove a conjecture of goulden and jackson on higher genus hurwitz numbers which turns out to be an analogue of a genus expansion ansatz of itzykson and zuber as consequences we have new combinatorial constraints on f and a much more direct proof of the ansatz of itzykson and zuber we can produce recursions and explicit formulas for hurwitz numbers the algorithm presented proves all such recursions as examples we present surprisingly simple new recursions in genus 0 to 3 similar recursions should exist for all genera as we expect this paper also to be of interest to combinatorialists we have tried to make it as self contained as possible including reviewing some results and definitions well known in algebraic and symplectic geometry and mathematical physics 2000 mathematical subject classification primary 14h10 81t40 secondary 05c30 58d29

stochastic dominance constrained markov decision processes we are interested in risk constraints for infinite horizon discrete time markov decision processes mdps starting with average reward mdps we show that increasing concave stochastic dominance constraints on the empirical distribution of reward lead to linear constraints on occupation measures an optimal policy for the resulting class of dominance constrained mdps is obtained by solving a linear program we compute the dual of this linear program to obtain average dynamic programming optimality equations that reflect the dominance constraint in particular a new pricing term appears in the optimality equations corresponding to the dominance constraint we show that many types of stochastic orders can be used in place of the increasing concave stochastic order we also carry out a parallel development for discounted reward mdps with stochastic dominance constraints a portfolio optimization example is used to motivate the paper

a special simplex in the state space for entangled qudits the focus is on two parties with hilbert spaces of dimension d i e qudits in the state space of these two possibly entangled qudits an analogue to the well known tetrahedron with the four qubit bell states at the vertices is presented the simplex analogue to this magic tetrahedron includes mixed states each of these states appears to each of the two parties as the maximally mixed state some studies on these states are performed and special elements of this set are identified a large number of them are included in the chosen simplex which fits exactly into conditions needed for teleportation and other applications its rich symmetry related to that of a classical phase space helps to study entanglement to construct witnesses and perform partial transpositions this simplex has been explored in detail for d 3 in this paper the mathematical background and extensions to arbitrary dimensions are analysed

geometry of two qubit states with negative conditional entropy we review the geometric features of negative conditional entropy and the properties of the conditional amplitude operator proposed by cerf and adami for two qubit states in comparison with entanglement and nonlocality of the states we identify the region of negative conditional entropy in the tetrahedron of locally maximally mixed two qubit states within this set of states negative conditional entropy implies nonlocality and entanglement but not vice versa and we show that the cerf adami conditional amplitude operator provides an entanglement witness equivalent to the peres horodecki criterion outside of the tetrahedron this equivalence is generally not true

virasoro constraints for target curves we prove generalized virasoro constraints for the relative gromov witten theories of all nonsingular target curves descendents of the even cohomology classes are studied first by localization degeneration and completed cycle methods descendents of the odd cohomology are then controlled by monodromy and geometric vanishing relations as an outcome of our results the relative theories of target curves are completely and explicitly determined

maximal violation of the collins gisin linden massar popescu inequality for infinite dimensional states we present a much simplified version of the collins gisin linden massar popescu inequality for the 2 ifmmode times else texttimes fi 2 ifmmode times else texttimes fi d bell scenario numerical maximization of the violation of this inequality over all states and measurements suggests that the optimal state is far from maximally entangled while the best measurements are the same as conjectured best measurements for the maximally entangled state for very large values of d the inequality seems to reach its minimal value given by the probability constraints this gives numerical evidence for a tight quantum bell inequality or generalized csirelson inequality for the 2 ifmmode times else texttimes fi 2 ifmmode times else texttimes fi ensuremath infty scenario

towards the geometry of double hurwitz numbers abstract double hurwitz numbers count branched covers of cp 1 with fixed branch points with simple branching required over all but two points 0 and and the branching over 0 and specified by partitions of the degree with m and n parts respectively single hurwitz numbers or more usually hurwitz numbers have a rich structure explored by many authors in fields as diverse as algebraic geometry symplectic geometry combinatorics representation theory and mathematical physics the remarkable elsv formula relates single hurwitz numbers to intersection theory on the moduli space of curves this connection has led to many consequences including okounkov and pandharipande s proof of witten s conjecture in this paper we determine the structure of double hurwitz numbers using techniques from geometry algebra and representation theory our motivation is geometric we give evidence that double hurwitz numbers are top intersections on a moduli space of curves with a line bundle a universal picard variety in particular we prove a piecewise polynomiality result analogous to that implied by the elsv formula in the case m 1 complete branching over one point and n is arbitrary we conjecture an elsv type formula and show it to be true in genus 0 and 1 the corresponding witten type correlation function has a richer structure than that for single hurwitz numbers and we show that it satisfies many geometric properties such as the string and dilaton equations and an itzykson zuber style genus expansion ansatz we give a symmetric function description of the double hurwitz generating series which leads to explicit formulae for double hurwitz numbers with given m and n as a function of genus in the case where m is fixed but not necessarily 1 we prove a topological recursion on the corresponding generating series which leads to closed form expressions for double hurwitz numbers and an analogue of the goulden jackson polynomiality conjecture an early conjectural variant of the elsv formula in a later paper faber s intersection number conjecture and genus 0 double hurwitz numbers 2005 in preparation the formulae in genus 0 will be shown to be equivalent to the formulae for top intersections on the moduli space of smooth curves m g for example three formulae we give there will imply faber s intersection number conjecture in moduli of curves and abelian varieties aspects of mathematics vol e33 vieweg braunschweig 1999 pp 109 129 in arbitrary genus with up to three points

stochastic dynamic optimization with discounted stochastic dominance constraints we introduce a stochastic dynamic optimization problem where risk aversion is expressed by a stochastic ordering constraint the constraint requires that a random reward sequence depending on our decisions dominates a given benchmark random sequence the dominance is defined by discounting both processes with a family of discount sequences and by applying a univariate order we describe the generator of this order we develop necessary and sufficient conditions of optimality for convex stochastic control problems with the new ordering constraint and we derive an equivalent control problem featuring implied utility functions furthermore we prove the existence of an optimal random discount sequence such that the solution of the risk averse problem is also a solution of an expected value problem with this discount finally we derive a version of the maximum principle for the problem with discounted dominance constraints

entanglement reflected in wigner functions we construct an abelian algebra built by maximally entangled pure states the partial transposition maps this algebra for odd dimensions into a full matrix algebra ppt states are in one to one correspondence to states with a positive definite wigner function special extremal ppt states correspond to projections of various dimensions in particular we recover the projections corresponding to a complete set of mutually unbiased bases in prime dimensions

a composite parameterization of unitary groups density matrices and subspaces unitary transformations and density matrices are central objects in quantum physics and various tasks require to introduce them in a parameterized form in this paper we present a parameterization of the unitary group of arbitrary dimension d which is constructed in a composite way we show explicitly how any element of can be composed of matrix exponential functions of generalized anti symmetric matrices and one dimensional projectors the specific form makes it considerably easy to identify and discard redundant parameters in several cases in this way redundancy free density matrices of arbitrary rank k can be formulated our construction can also be used to derive an orthonormal basis of any k dimensional subspaces of with the minimal number of parameters as an example it is shown that this feature leads to a significant reduction of parameters in the case of investigating distillability of quantum states via lower bounds of an entanglement measure the m concurrence

a geometric comparison of entanglement and quantum nonlocality in discrete systems we compare entanglement with quantum nonlocality employing a geometric structure of the state space of bipartite qudits the central object is a regular simplex spanned by generalized bell states the collins gisin linden massar popescu bell inequality is used to reveal states of this set that cannot be described by local realistic theories optimal measurement settings necessary to ascertain nonlocality are determined by means of a recently proposed parameterization of the unitary group combined with robust numerical methods the main results of this paper are descriptive geometric illustrations of the state space that emphasize the difference between entanglement and quantum nonlocality namely it is found that the shape of the boundaries of separability and bell inequality violation are essentially different moreover it is also shown that for mixtures of states sharing the same amount of entanglement bell inequality violations and entanglement measures are non monotonically related

on the tautological ring of mg we prove that any product of tautological classes of g of degreed in the chow ring of g with rational coefficients vanishes ford g 2 and is proportional to the class of the hyperelliptic locus in degreeg 2

composite parameterization and haar measure for all unitary and special unitary groups we adopt the concept of the composite parameterization of the unitary group u d to the special unitary group su d furthermore we also consider the haar measure in terms of the introduced parameters we show that the well defined structure of the parameterization leads to a concise formula for the normalized haar measure on u d and su d with regard to possible applications of our results we consider the computation of high order integrals over unitary groups

nineteen dubious ways to compute the exponential of a matrix twenty five years later in principle the exponential of a matrix could be computed in many ways methods involving approximation theory differential equations the matrix eigenvalues and the matrix characteristic polyn

bloch vectors for qudits we present three different matrix bases that can be used to decompose density matrices of d dimensional quantum systems so called qudits the generalized gell mann matrix basis the polarization operator basis and the weyl operator basis such a decomposition can be identified with a vector the bloch vector i e a generalization of the well known qubit case and is a convenient expression for comparison with measurable quantities and for explicit calculations avoiding the handling of large matrices we present a new method to decompose density matrices via so called standard matrices consider the important case of an isotropic two qudit state and decompose it according to each basis in the case of qutrits we show a representation of an entanglement witness in terms of expectation values of spin 1 measurements which is appropriate for an experimental realization

bell s theorem for general n qubit states we derive a single general bell inequality which is a sufficient and necessary condition for the correlation function for n particles to be describable in a local and realistic picture for the case in which measurements on each particle can be chosen between two arbitrary dichotomic observables we also derive a necessary and sufficient condition for an arbitrary n qubit mixed state to violate this inequality this condition is a generalization and reformulation of the horodecki family condition for two qubits

stochastically weighted stochastic dominance concepts with an application in capital budgeting abstract the problem of comparing random vectors arises in many applications we propose three new concepts of stochastically weighted dominance for comparing random vectors x and y the main idea is to use a random vector v to scalarize x and y as v t x and v t y and subsequently use available concepts from stochastic dominance and stochastic optimization for comparison for the case where the distributions of x y and v have finite support we give mixed integer linear inequalities that can be used for random vector comparison as well as for modeling of optimization problems where one of the random vectors depends on decisions to be optimized some advantages of the proposed new concepts are illustrated with the help of a capital budgeting example

robust decision making using a general utility set abstract we address the problem of ambiguity and inconsistency in a decision maker s dm assessed utility function by using a maxmin framework in this framework dm s utility function belongs to a set of functions the member functions of the set are nondecreasing and satisfy additional boundary and auxiliary conditions the maxmin framework provides robustness in decision making alternatively it allows us to perform sensitivity or parametric analysis for the optimal decision when a reference utility function is known for this purpose we use a cost of ambiguity concept and show that this cost is increasing and concave when extent of ambiguity is parametrically increased next we develop a lagrangian based solution approach for the decision problem we show that under suitable conditions a sample average approximation saa of the lagrangian model can be solved using a mixed integer linear program milp we also show that the set of optimal solutions of the saa converges to that of its true counterpart and the optimum objective value of the saa converges to the true objective value at an exponential rate we use this convergence property to develop a heuristic for identifying a solution of the saa milp with increasing sample size we illustrate the properties of the maxmin model using two examples of portfolio investment and streaming bandwidth we provide a discussion on the performance of a commercial solver used to solve saa milps and the quality of solutions generated by the proposed heuristic

a short proof of the λg conjecture without gromov witten theory hurwitz theory and the moduli of curves we give a short and direct proof of the g conjecture the approach is through the ekedahl lando shapiro vainshtein theorem which establishes the polynomiality of hurwitz numbers from which we pick off the lowest degree terms the proof is independent of gromovwitten theory we briefly describe the philosophy behind our general approach to intersection numbers and how it may be extended to other intersection number conjectures

two party bell inequalities derived from combinatorics via triangular elimination we establish a relation between the two party bell inequalities for two valued measurements and a high dimensional convex polytope called the cut polytope in polyhedral combinatorics using this relation we propose a method triangular elimination to derive tight bell inequalities from facets of the cut polytope this method gives two hundred million inequivalent tight bell inequalities from currently known results on the cut polytope in addition this method gives general formulae which represent families of infinitely many bell inequalities these results can be used to examine general properties of bell inequalities

hurwitz numbers and intersections on moduli spaces of curves 1 department of mathematics university of stockholm s 10691 stockholm sweden e mail teke matematik su se 2 higher college of mathematics independent university of moscow and institute for system research ras moscow russia e mail lando mccme ru 3 department of mathematics royal institute of technology s 10044 stockholm sweden e mail mshapiro math kth se 4 department of mathematics and department of computer science university of haifa haifa 31905 israel e mail alek mathcs haifa ac il

bell inequalities for arbitrarily high dimensional systems we develop a novel approach to bell inequalities based on a constraint that the correlations exhibited by local variable theories must satisfy this is used to construct a family of bell inequalities for bipartite quantum systems of arbitrarily high dimensionality which are strongly resistant to noise in particular our work gives an analytic description of previous numerical results and generalizes them to arbitrarily high dimensionality

the extended bloch representation of quantum mechanics explaining superposition interference and entanglement an extended bloch representation of quantum mechanics was recently derived to offer a possible hidden measurements solution to the measurement problem in this article we use this representation to investigate the geometry of superposition and entangled states explaining interference effects and entanglement correlations in terms of the different orientations a state vector can take within the generalized bloch sphere we also introduce a tensorial determination of the generators of su n which we show to be particularly suitable for the description of multipartite systems from the viewpoint of the sub entities we then use it to show that non product states admit a general description where sub entities can remain in well defined states even when entangled this means that the completed version of quantum mechanics provided by the extended bloch representation where density operators are also considered to be representative of genuine states providing a complete description not only offers a pl

ambiguity in risk preferences in robust stochastic optimization abstract we consider robust stochastic optimization problems for risk averse decision makers where there is ambiguity about both the decision maker s risk preferences and the underlying probability distribution we propose and analyze a robust optimization problem that accounts for both types of ambiguity first we derive a duality theory for this problem class and identify random utility functions as the lagrange multipliers second we turn to the computational aspects of this problem we show how to evaluate our robust optimization problem exactly in some special cases and then we consider some tractable relaxations for the general case finally we apply our model to both the newsvendor and portfolio optimization problems and discuss its implications