the monodromy of weierstrass points we show that the monodromy of the family of curves riemann surfaces acts as the full symmetric group on the weierstrass points of a general curve the proof uses a degeneration to certain reducible curves and the theory of limit series developed in our 1986 1987a b some of the monodromy is actually constructed by fixing a reducible curve and varying its canonical series

option pricing with markov modulated dynamics markov modulated models for equity prices have recently been extensively studied in the literature in this paper we apply some old results on the wiener hopf factorization of markov processes to a range of option pricing problems for such models the first example is the perpetual american put where the exact numerical solution is obtained without discretizing any pde we then show how the methodology of rogers and stapleton finance stoch 2 1997 pp 3 17 can be used to tackle finite horizon problems and illustrate the methodology by pricing european american single barrier and double barrier options under markov modulated dynamics

thue equations and the method of chabauty coleman let ok be any domain with field of fractions k let f x y ok x y be a homogeneous polynomial of degree n coprime to y and assumed to have unit content i e the coefficients of f generate the unit ideal in ok assume that gcd n char k 1 let h ok and assume that the polynomial hzn f x y is irreducible in k x y z we denote by x f h k the nonsingular complete model of the projective plane curve cf h k defined by the equation hzn f x y 0 we shall assume in this article that g x f h 2 when k is a number field mordell s conjecture now faltings theorem implies that x f h k caporaso harris and mazur chm 1 1 have shown that if lang s conjecture for varieties of general type is true then for any number field k the size x k of the set of k rational points of any curve x k of genus g x 2 can be bounded by a constant depending only on g x prior to the paper chm mazur and others had asked whether x k can be bounded by a constant depending only on g x and the mordell weil rank of x k over k that is the rank of the group j k of k rational points of the jacobian j k of x k these farreaching questions are totally open as we shall recall in sect 1 the method of chabauty coleman sometimes yields a bound for x f h k depending only on g x f h when it is known in advance that the mordell weil rank of x f h k is small unfortunately the chabauty coleman method does not yield a bound for x f h k independent of the coefficients of hzn f x y for all curves of the form x f h it does however produce such a nice bound for the number of primitive integral solutions of f x y h as we now explain let k q and ok z a classical thue equation is an equation f x y h where f x 1 does not have repeated roots thue showed in 1909 that such an equation has finitely many solutions x y z2 if n 3

computational complexity of a cost allocation approach to a fixed cost spanning forest problem we present a computational analysis of a game theoretic approach to a cost allocation problem arising from a graph optimization problem referred to as the fixed cost spanning forest fcsf problem the customers in the fcsf problem represented by nodes in a graph g are in need of service that can be produced at some facilities yet to be constructed the cost allocation problem is concerned with the fair distribution of the cost of providing the service among customers we formulate this cost allocation problem as a cooperative game referred to as the fcsf game in general the core of a fcsf game may be empty however for the case when g is a tree it is shown that the core is not empty moreover we prove that in this case core points can be generated in strongly polynomial time we further provide a nonredundant characterization of the core of the fcsf game defined over a tree in the special case when all nodes are communities this is shown to lead in some instances to a strongly polynomial algorithm for computing the nucleolus

riemann roch theory for weighted graphs and tropical curves abstract we define a divisor theory for graphs and tropical curves endowed with a weight function on the vertices we prove that the riemann roch theorem holds in both cases we extend baker s specialization lemma to weighted graphs

riemann roch and abel jacobi theory on a finite graph abstract it is well known that a finite graph can be viewed in many respects as a discrete analogue of a riemann surface in this paper we pursue this analogy further in the context of linear equivalence of divisors in particular we formulate and prove a graph theoretic analogue of the classical riemann roch theorem we also prove several results analogous to classical facts about riemann surfaces concerning the abel jacobi map from a graph to its jacobian as an application of our results we characterize the existence or non existence of a winning strategy for a certain chip firing game played on the vertices of a graph

minimizing the probability of lifetime ruin under ambiguity aversion we determine the optimal robust investment strategy of an individual who targets at a given rate of consumption and seeks to minimize the probability of lifetime ruin when she does not have perfect confidence in the drift of the risky asset using stochastic control we characterize the value function as the unique classical solution of an associated hamilton jacobi bellman hjb equation obtain feedback forms for the optimal investment and drift distortion and discuss their dependence on various model parameters in analyzing the hjb equation we establish the existence and uniqueness of a viscosity solution using perron s method and then upgrade regularity by working with an equivalent convex problem obtained via the cole hopf transformation we show the original value function may lose convexity for a class of parameters and the isaacs condition may fail numerical examples are also included to illustrate our results

matching games the least core and the nucleolus a matching game is a cooperative game defined by a graph g n e the player set is n and the value of a coalition s n is defined as the size of a maximum matching in the subgraph induced by s we show that the nucleolus of such games can be computed efficiently the result is based on an alternative characterization of the least core which may be of independent interest the general case of weighted matching games remains unsolved

optimal strategies for risk sensitive portfolio optimization problems for general factor models we consider constructing optimal strategies for risk sensitive portfolio optimization problems on an infinite time horizon for general factor models where the mean returns and the volatilities of individual securities or asset categories are explicitly affected by economic factors the factors are assumed to be general diffusion processes in studying the ergodic type bellman equations of the risk sensitive portfolio optimization problems we introduce some auxiliary classical stochastic control problems with the same bellman equations as the original ones we show that the optimal diffusion processes of the problem are ergodic and that under some condition related to integrability by the invariant measures of the diffusion processes we can construct optimal strategies for the original problems by using the solution of the bellman equations

fibers of tropicalization we use functoriality of tropicalization and the geometry of projections of subvarieties of tori to show that the fibers of the tropicalization map are dense in the zariski topology for subvarieties of tori over fields of generalized power series points in each tropical fiber are obtained constructively using kedlaya s transfinite version of newton s method

divisors on graphs binomial and monomial ideals and cellular resolutions we study various binomial and monomial ideals arising in the theory of divisors orientations and matroids on graphs we use ideas from potential theory on graphs and from the theory of delaunay decompositions for lattices to describe their minimal polyhedral cellular free resolutions we show that the resolutions of all these ideals are closely related and that their mathbb z graded betti tables coincide as corollaries we give conceptual proofs of conjectures and questions posed by postnikov and shapiro by manjunath and sturmfels and by perkinson perlman and wilmes various other results related to the theory of chip firing games on graphs also follow from our general techniques and results

nonzero sum risk sensitive stochastic games on a countable state space the infinite horizon risk sensitive discounted cost and ergodic cost nonzero sum stochastic games for controlled markov chains with countably many states are analyzed for the discounted cost game we prove the existence of nash equilibrium strategies in the class of markov strategies under fairly general conditions under an additional weak geometric ergodicity condition and a small cost criterion the existence of nash equilibrium strategies in the class of stationary markov strategies is proved for the ergodic cost game the key nontrivial contributions in the ergodic part are to prove the existence of a particular form of a relative value function solution to a player s bellman equation and the continuity of this solution with respect to the opponent s strategies

a stochastic calculus model of continuous trading optimal portfolios the problem of choosing a portfolio of securities so as to maximize the expected utility of wealth at a terminal planning horizon is solved via stochastic calculus and convex analysis this problem is decomposed into two subproblems with security prices modeled as semimartingales and trading strategies modeled as predictable processes the set of terminal wealths is identified as a subspace in a space of integrable random variables the first subproblem is to find the terminal wealth that maximizes expected utility convex analysis is used to derive necessary and sufficient conditions for optimality and an existence result the second subproblem of finding the admissible trading strategy that generates the optimal terminal wealth is a martingale representation problem the primary advantage of this approach is that explicit formulas can readily be derived for the optimal terminal wealth and the corresponding expected utility as is shown for the case of an exponential utility function and a risky security modeled as geometric brownian motion

risk sensitive asset management and cascading defaults we consider an optimal risk sensitive portfolio allocation problem accounting for the possibility of cascading defaults default events have an impact on the distress state of the surviving stocks in the portfolio we study the recursive system of non lipschitz quasilinear parabolic hjb pdes associated with the value function of the control problem in the different default states of the economy we show the existence of a classical solution to this system via super sub solution techniques and give an explicit characterization of the optimal feedback strategy in terms of the value function we prove a verification theorem establishing the uniqueness of the solution a numerical analysis indicates that the investor significantly accounts for contagion effects when making investment decisions and that his strategy depends nonmonotonically on the aggregate risk level

packing spanning trees we given an algorithm for packing spanning trees in a graph g v e with capacities on the edges the problem reduces to o v 2 maximum flow computations the algorithm is based on nash williams s proof of a min max relation for this problem

dimension counts for limit linear series on curves not of compact type we first prove a generalized brill noether theorem for linear series with prescribed multivanishing sequences on smooth curves we then apply this theorem to prove that spaces of limit linear series have the expected dimension for curves of pseudocompact type whenever the gluing conditions in the definition of limit linear series impose the maximal codimension finally we investigate these gluing conditions in specific families of curves showing expected dimension in several cases each with different behavior one of these families sheds new light on the work of cools draisma payne and robeva in tropical brill noether theory and suggests directions of further work in that setting

risk sensitive optimal control for markov decision processes with monotone cost the existence of an optimal feedback law is established for the risk sensitive optimal control problem with denumerable state space the main assumptions imposed are irreducibility and anear monotonicity condition on the one step cost function a solution can be found constructively using either value iteration or policy iteration under suitable conditions on initial feedback law

risk sensitive portfolio management with cox ingersoll ross interest rates the hjb equation this paper presents an application of risk sensitive control theory in financial decision making the investor has an infinite horizon objective that can be interpreted as maximizing the portfolio s risk adjusted exponential growth rate there are two assets a stock and a bank account and two underlying brownian motions so this model is incomplete the novel feature here is that the interest rate for the bank account is governed by cox ingersoll ross dynamics this is significant for risk sensitive portfolio management because the factor process unlike in the gaussian and all other cases treated in the literature cannot be negative under appropriate parameterization

optimal stochastic regulators with state dependent weights abstract we introduce two optimal regulators for linear stochastic systems the first is of a linear state feedback form and it generalises the linear quadratic regulator by introducing state dependent weights in the cost functional the second is a certain risk sensitive version of the first and it is of a nonlinear state feedback form both regulators are applied to the optimal investment problem

limit linear series basic theory in this paper we introduce techniques for handling the degeneration of linear series on smooth curves as the curves degenerate to a certain type of reducible curves curves of compact type the technically much simpler special case of 1 dimensional series was developed by beauville 2 knudsen 21 23 harris and mumford 17 in the guise of admissible covers it has proved very useful for studying the moduli space of curves the above papers and harris 16 and the simplest sorts of weierstrass points diaz 4 with our extended tools we are able to prove for example that in this paper we present the basic theory of limit linear series necessary for proving these results the results themselves will be taken up in our forthcoming papers 8 12 simpler applications not requiring the tools developed in this paper but perhaps clarified by them have already been given in our papers 5 7

zhang s conjecture and the effective bogomolov conjecture over function fields we prove the effective bogomolov conjecture and so the bogomolov conjecture over a function field of characteristic 0 by proving zhang s conjecture about certain invariants of metrized graphs in the function field case these conjectures were previously known to be true only for curves of good reduction for curves of genus at most 4 and a few other special cases we also either verify or improve the previous results we relate the invariants involved in zhang s conjecture to the tau constant of metrized graphs then we use and extend our previous results on the tau constant by proving another conjecture of zhang we obtain a new proof of the slope inequality for faltings heights on moduli space of curves

risk sensitive control for a class of nonlinear systems with multiplicative noise abstract in this paper we consider the problem of optimal control for a class of nonlinear stochastic systems with multiplicative noise the nonlinearity consists of quadratic terms in the state and control variables the optimality criteria are of a risk sensitive and generalised risk sensitive type the optimal control is found in an explicit closed form by the completion of squares and the change of measure methods as applications we outline two special cases of our results we show that a subset of the class of models which we consider leads to a generalised quadratic affine term structure model qatsm for interest rates we also demonstrate how our results lead to generalisation of exponential utility as a criterion in optimal investment

packing algorithms for arborescences and spanning trees in capacitated graphs in a digraph with real valued edge capacities we pack the greatest number of arborescences in time o n 3m log n 2 m the packing uses at mostm distinct arborescences heren andm denote the number of vertices and edges in the given graph respectively similar results hold for integral packing we pack the greatest number of arborescences in time o min n logn n 2m log n 2 using at mostm n 2 distinct arborescences heren denotes the largest integral capacity of an edge these resuts improve the best previous bounds for capacitated digraphs the algorithm extends to several related problems including packing spanning trees in an undirected capacitated graph and covering such graphs by forests the algorithm provides a new proof of edmonds theorem for arborescence packing for both integral and real capacities based on a laminar family of sets derived from the packing 1998 the mathematical programming society inc published by elsevier science b v

smooth p adic analytic spaces are locally contractible 0 introduction 293 1 piecewise rs linear spaces 298 2 r colored polysimplicial sets 308 3 r colored polysimplicial sets of length l 313 4 the skeleton of a nondegenerate pluri stable formal scheme 327 5 a colored polysimplicial set associated with a nondegenerate poly stable fibration 336 6 p adic analytic and piecewise linear spaces 346 7 strong local contractibility of smooth analytic spaces 355 8 cohomology with coefficients in the sheaf of constant functions 362

brief multiple objective risk sensitive control and its small noise limit this paper is concerned with a minimizing multiple objective risk sensitive control problem asymptotic analysis leads to the introduction of a new class of two player zero sum deterministic differential games the distinguishing feature of this class of games is that the cost functional is multiple objective in nature being composed of the risk neutral integral costs associated with the original risk sensitive problem more precisely the opposing player in such a game seeks to maximize the most vulnerable member of a given set of cost functionals while the original controller seeks to minimize the worst damage that the opponent can do over this set it is then shown that the problem of finding an efficient risk sensitive controller is equivalent asymptotically to solving this differential game surprisingly this differential game is proved to be independent of the weights on the different objectives in the original multiple objective risk sensitive problem as a by product our results generalize the existing results for the single objective risk sensitive control problem to a substantially larger class of nonlinear systems including those with control dependent diffusion terms

bellman equations of risk sensitive control risk sensitive control problems are considered existence of a nonnegative solution to the bellman equation of risk sensitive control is shown the result is applied to prove that no breaking down occurs asymptotic behaviour of the nonnegative solution is studied in relation to ergodic control problems and the relationship between the asymptotics and the large deviation principle is noted

limit linear series and the amini baker construction we draw comparisons between the author s recent construction of limit linear series for curves not of compact type and the amini baker theory of limit linear series on metrized complexes as well as the related theories of divisors on discrete graphs and on metric graphs from these we conclude that the author s theory like the others satisfies the riemann and clifford inequalities motivated by our comparisons we also develop negative results on brill noether generality for certain families of metric graphs companion work of he develops our comparisons further and uses them to prove new results on smoothability of amini baker limit linear series and of divisors on metric graphs

risk sensitive control on an infinite time horizon stochastic control problems on an infinite time horizon with exponential cost criteria are considered the donsker varadhan large deviation rate is used as a criterion to be optimized the optimum rate is characterized as the value of an associated stochastic differential game with an ergodic expected average cost per unit time cost criterion if we take a small noise limit a deterministic differential game with average cost per unit time cost criterion is obtained this differential game is related to robust control of nonlinear systems

network strength games the core and the nucleolus the maximum number of edge disjoint spanning trees in a network has been used as a measure of the strength of a network it gives the number of disjoint ways that the network can be fully connected this suggests a game theoretic analysis that shows the relative importance of the different links to form a strong network we introduce the network strength game as a cooperative game defined on a graph g v e the player set is the edge set e and the value of a coalition s subseteq e is the maximum number of disjoint spanning trees included in s we study the core of this game and we give a polynomial combinatorial algorithm to compute the nucleolus when the core is non empty

on the structure of solutions of ergodic type bellman equation related to risk sensitive control bellman equations of ergodic type related to risk sensitive control are considered we treat the case that the nonlinear term is positive quadratic form on first order partial derivatives of solution which includes linear exponential quadratic gaussian control problem in this paper we prove that the equation in general has multiple solutions we shall specify the set of all the classical solutions and classify the solutions by a global behavior of the diffusion process associated with the given solution the solution associated with ergodic diffusion process plays particular role we shall also prove the uniqueness of such solution furthermore the solution which gives us ergodicity is stable under perturbation of coefficients finally we have a representation result for the solution corresponding to the ergodic diffusion

bernstein nagumo conditions and solutions to nonlinear differential inequalities abstract for an open bounded subset of r n with smooth boundary and 1 p we establish w 1 p a priori bounds and prove the compactness of solution sets to differential inequalities of the form div a x u f x u u which are bounded in l the main point in this work is that the nonlinear term f may depend on u and may grow as fast as a power of order p in this variable such growth conditions have been used extensively in the study of boundary value problems for nonlinear ordinary differential equations and are known as bernstein nagumo growth conditions in addition we use these results to establish a sub supersolution theorem

optimal attack and reinforcement of a network in a nonnegative edge weighted network the weight of an edge represents the effort required by an attacker to destroy the edge and the attacker derives a benefit for each new component created by destroying edges the attacker may want to minimize over subsets of edges the difference between or the ratio of the effort incurred and the benefit received this idea leads to the definition of the strength of the network a measure of the resistance of the network to such attacks efficient algorithms for the optimal attack problem the problem of computing the strength and the problem of finding a minimum cost reinforcement to achieve a desired strength are given these problems are also solved for a different model in which the attacker wants to separate vertices from a fixed central vertex

optimal control of partially observable stochastic systems with an exponential of integral performance index the stochastic control problem with linear stochastic differential equations driven by brownian motion processes and as cost functional the exponential of a quadratic form is considered the soluti

on the optimal control of stochastic systems with an exponential of integral performance index abstract in this paper a theory of optimal control is developed for stochastic systems whose performance is measured by the exponential of an integral form such a formulation of the cost function is shown to be not only general and useful but also analytically tractable starting with very general classes of stochastic systems optimality conditions are obtained which exploit the multiplicative decomposability of the exponential of integral form specializing to partially observed systems of stochastic differential equations with brownian motion disturbances optimality conditions are obtained which parallel those for systems with integral costs also treated are the special cases of linear systems with exponential of quadratic costs for which explicit optimal controls are obtainable in addition several general results of independent interest are obtained which concern optimality of stochastic systems

liouville properties and critical value of fully nonlinear elliptic operators abstract we prove some liouville properties for sub and supersolutions of fully nonlinear degenerate elliptic equations in the whole space our assumptions allow the coefficients of the first order terms to be large at infinity provided they have an appropriate sign as in ornstein uhlenbeck operators we give two applications the first is a stabilization property for large times of solutions to fully nonlinear parabolic equations the second is the solvability of an ergodic hamilton jacobi bellman equation that identifies a unique critical value of the operator

linear series on metrized complexes of algebraic curves a metrized complex of algebraic curves over an algebraically closed field kappa is roughly speaking a finite metric graph gamma together with a collection of marked complete nonsingular algebraic curves c v over kappa one for each vertex v of gamma the marked points on c v are in bijection with the edges of gamma incident to v we define linear equivalence of divisors and establish a riemann roch theorem for metrized complexes of curves which combines the classical riemann roch theorem over kappa with its graph theoretic and tropical analogues from amini and caporaso adv math 240 1 23 2013 baker and norine adv math 215 2 766 788 2007 gathmann and kerber math z 259 1 217 230 2008 and mikhalkin and zharkov tropical curves their jacobians and theta functions contemporary mathematics 203 231 2007 providing a common generalization of all of these results for a complete nonsingular curve x defined over a non archimedean field mathbb k together with a strongly semistable model mathfrak x for x over the valuation ring r of mathbb k we define a corresponding metrized complex mathfrak c mathfrak x of curves over the residue field kappa of mathbb k and a canonical specialization map tau mathfrak c mathfrak x from divisors on x to divisors on mathfrak c mathfrak x which preserves degrees and linear equivalence we then establish generalizations of the specialization lemma from baker algebra number theory 2 6 613 653 2008 and its weighted graph analogue from amini and caporaso adv math 240 1 23 2013 showing that the rank of a divisor cannot go down under specialization from x to mathfrak c mathfrak x as an application we establish a concrete link between specialization of divisors from curves to metrized complexes and the theory of limit linear series due to eisenbud and harris invent math 85 337 371 1986 using this link we formulate a generalization of the notion of limit linear series to curves which are not necessarily of compact type and prove among other things that any degeneration of a mathfrak g r d in a regular family of semistable curves is a limit mathfrak g r d on the special fiber

on the hamilton jacobi bellman equation for an optimal consumption problem i existence of solution we consider a consumption problem with an infinite time horizon to optimize the discounted expected power utility the returns and volatilities of the assets are random and affected by some economic factors modeled as diffusion process the problem becomes a standard control problem we derive the hamilton jacobi bellman hjb equation and study its solutions in part i under some weak conditions we prove the existence of a solution for this hjb equation when we assume the existence of an ordered pair of sub supersolution to construct an ordered pair of sub supersolution we also consider the risk sensitive portfolio optimization problem in part ii hata and sheu siam j control optim 50 2012 pp 2401 2430 we consider the uniqueness of the solution and prove the verification theorem

limit linear series for curves not of compact type we introduce a notion of limit linear series for nodal curves which are not of compact type we give a construction of a moduli space of limit linear series which works also in smoothing families and we prove a corresponding specialization result for a more restricted class of curves which simultaneously generalizes two component curves and curves of compact type we give an equivalent definition of limit linear series which is visibly a generalization of the eisenbud harris definition finally for the same class of curves we prove a smoothing theorem which constitutes an improvement over known results even in the compact type case

on the complexity of cooperative solution concepts we study from a complexity theoretic standpoint the various solution concepts arising in cooperative game theory we use as a vehicle for this study a game in which the players are nodes of a graph with weights on the edges and the value of a coalition is determined by the total weight of the edges contained in it the shapley value is always easy to compute the core is easy to characterize when the game is convex and is intractable np complete otherwise similar results are shown for the kernel the nucleolus the e core and the bargaining set as for the von neumann morgenstern solution we point out that its existence may not even be decidable many of these results generalize to the case in which the game is presented by a hypergraph with edges of size k 2

a learning algorithm for risk sensitive cost a linear function approximation based reinforcement learning algorithm is proposed for markov decision processes with infinite horizon risk sensitive cost its convergence is proved using the o d e method for stochastic approximation the scheme is also extended to continuous state space processes

a model for optimally advertising and launching a product we formulate and solve a problem that combines the features of the so called monotone follower of singular stochastic control theory with optimal stopping in particular we consider a stochastic system whose uncontrolled state dynamics are modelled by a general one dimensional ito diffusion the aim of the problem that we solve is to maximise the utility derived from the system s state at the discretionary time when the system s control is terminated this objective is reflected by the performance criterion that we maximise which also penalises control expenditure as well as waiting the model that we study is motivated by the so called goodwill problem a variant of which is concerned with how to optimally raise a new product s image e g through advertising and with determining the best time to launch the product into the market in the presence of the rather general assumptions that we make we fully characterise the optimal strategy which can take one of three qualitatively different forms depending on the problem data

a weak dynamic programming principle for zero sum stochastic differential games with unbounded controls we analyze a zero sum stochastic differential game between two competing players who can choose unbounded controls the payoffs of the game are defined through backward stochastic differential equations we prove that each player s priority value satisfies a weak dynamic programming principle and thus solves the associated fully nonlinear partial differential equation in the viscosity sense

maximum principles for a class of partial information risk sensitive optimal controls in this technical note we study a class of partial information risk sensitive optimal controls the main results are some necessary and sufficient conditions for these partial information control problems for illustration one example is proposed and then solved using our results

a stochastic portfolio optimization model with bounded memory this paper considers a portfolio management problem of merton s type in which the risky asset return is related to the return history the problem is modeled by a stochastic system with delay the investor s goal is to choose the investment control as well as the consumption control to maximize his total expected discounted utility under certain situations we derive the explicit solutions in a finite dimensional space

minimizing escape probabilities a large deviations approach this paper considers the problem of controlling a possibly degenerate diffusion process so as to minimize the probability of escape over a given time interval it is assumed that the control acts on the process through the drift coefficient and that the noise coefficient is small developing a large deviations type of theory for the controlled diffusion produces several results the limit of the normalized log of the minimum exit probability is identified as the value i of an associated deterministic differential game furthermore we identify a deterministic and varepsilon independent mapping g from the sample values varepsilon w s 0 leqq s leqq t into the control space such that if we define the control used at time t by u t g varepsilon w s 0 leqq s leqq t then the resulting control process is progressively measurable and delta optimal in the sense that the limit of the normalized log of the exit probability is within delta of i

algorithmic aspects of the core of combinatorial optimization games we discuss an integer programming formulation for a class of cooperative games we focus on algorithmic aspects of the core one of the most important solution concepts in cooperative game theory central to our study is a simple but very useful observation that the core for this class is nonempty if and only if an associated linear program has an integer optimal solution based on this we study the computational complexity and algorithms to answer important questions about the cores of various games on graphs such as maximum flow connectivity maximum matching minimum vertex cover minimum edge cover maximum independent set and minimum coloring

a new risk sensitive maximum principle in this paper a new maximum principle for the risk sensitive control problem is established one important feature of this result is that it applies to systems in which the diffusion term may depend on the control such control dependence gives rise to interesting phenomena not observed in the usual setting where control independence of the diffusion term is assumed in particular there is an additional second order adjoint equation and additional terms in the maximum condition that involve this second order process as well as the risk sensitive parameter moreover contrary to a conventional maximum principle the first order adjoint equation involved in our maximum principle is a nonlinear equation an advantage of considering this new type of adjoint equation is that the risk sensitive maximum principle derived is similar in form to its risk neutral counterpart the approach is based on the logarithmic transformation and the relationship between the adjoint variables and the value function as an example a linear quadratic risk sensitive problem is solved using the maximum principle derived

on the core and nucleolus of minimum cost spanning tree games we develop two efficient procedures for generating cost allocation vectors in the core of a minimum cost spanning tree m c s t game the first procedure requires o n 2 elementary operations to obtain each additional point in the core wheren is the number of users the efficiency of the second procedure which is a natural strengthening of the first procedure stems from the special structure of minimum excess coalitions in the core of an m c s t game this special structure is later used i to ease the computational difficulty in computing the nucleolus of an m c s t game and ii to provide a geometric characterization for the nucleolus of an m c s t game this geometric characterization implies that in an m c s t game the nucleolus is the unique point in the intersection of the core and the kernel we further develop an efficient procedure for generating fair cost allocations which in some instances coincide with the nucleolus finally we show that by employing sterns transfer scheme we can generate a sequence of cost vectors which converges to the nucleolus

totally balanced games and games of flow a class of characteristic function games arising from maximum flow problems is introduced and is shown to coincide with the class of totally balanced games the proof relies on the max flow min cut theorem of ford and fulkerson and on the observation that the class of totally balanced games is the span of the additive games with the minimum operation

discounted approximations for risk sensitive average criteria in markov decision chains with finite state space this work concerns markov decision processes with finite state space and compact action set the performance of a control policy is measured by a risk sensitive average cost criterion and under standard continuity compactness conditions it is shown that the discounted approximations converge to the optimal value function and that the superior and inferior limit average criteria have the same optimal value function these conclusions are obtained for every nonnull risk sensitivity coefficient and regardless of the communication structure induced by the transition law

state space formulae for all stabilizing controllers that satisfy and h norm bound and relations to risk sensitivity abstract given a linear system all stabilizing controllers such that a specified closed loop transfer function has h norm less than a given scalar are parametrized this characterization involves the solution to two algebraic riccati equations each with the same order as the system and further gives gives feasible controllers also with this order the relationship to the risk sensitive lqg stochastic control problem is established giving an equivalence between robust and stochastic control

a tropical proof of the brill noether theorem we produce brill noether general graphs in every genus confirming a conjecture of baker and giving a new proof of the brill noether theorem due to griffiths and harris over any algebraically closed field

limit canonical systems on curves with two components in the 80 s d eisenbud and j harris considered the following problem what are the limits of weierstrass points in families of curves degenerating to stable curves but for the case of stable curves of compact type treated by them this problem remained wide open since then in the present article we propose a concrete approach to this problem and give a quite explicit solution for stable curves with just two irreducible components meeting at points in general position

topology of nonarchimedean analytic spaces and relations to complex algebraic geometry this note surveys basic topological properties of nonarchimedean analytic spaces in the sense of berkovich including the recent tameness results of hrushovski and loeser we also discuss interactions between the topology of nonarchimedean analytic spaces and classical algebraic geometry

a constructive approach to a class of ergodic hjb equations with unbounded and nonsmooth cost we consider a class of ergodic hamilton jacobi bellman hjb equations related to long time asymptotics of nonsmooth multiplicative functional of diffusion processes under suitable ergodicity assumptions on the underlying diffusion we show existence of these asymptotics and that they solve the related hjb equation in the viscosity sense

logarithmic picard groups chip firing and the combinatorial rank illusie has suggested that one should think of the classifying group of m x gp torsors on a logarithmically smooth curve x over a standard logarithmic point as a logarithmic analogue of the picard group of x this logarithmic picard group arises naturally as a quotient of the algebraic picard group by lifts of the chip firing relations of the associated dual graph we connect this perspective to baker and norine s theory of ranks of divisors on a finite graph and to amini and baker s metrized complexes of curves moreover we propose a definition of a combinatorial rank for line bundles on x and prove that an analogue of the riemann roch formula holds for our combinatorial rank our proof proceeds by carefully describing the relationship between the logarithmic picard group on a logarithmic curve and the picard group of the associated metrized complex this approach suggests a natural categorical framework for metrized complexes namely the category of logarithmic curves

riemann roch theory for graph orientations abstract we develop a new framework for investigating linear equivalence of divisors on graphs using a generalization of gioan s cycle cocycle reversal system for partial orientations an oriented version of dhar s burning algorithm is introduced and employed in the study of acyclicity for partial orientations we then show that the baker norine rank of a partially orientable divisor is one less than the minimum number of directed paths which need to be reversed in the generalized cycle cocycle reversal system to produce an acyclic partial orientation these results are applied in providing new proofs of the riemann roch theorem for graphs as well as luo s topological characterization of rank determining sets we prove that the max flow min cut theorem is equivalent to the euler characteristic description of orientable divisors and extend this characterization to the setting of partial orientations furthermore we demonstrate that p i c g 1 g is canonically isomorphic as a p i c 0 g torsor to the equivalence classes of full orientations in the cycle cocycle reversal system acted on by directed path reversals efficient algorithms for computing break divisors and constructing partial orientations are presented

a riemann roch theorem in tropical geometry recently baker and norine have proven a riemann roch theorem for finite graphs we extend their results to metric graphs and thus establish a riemann roch theorem for divisors on abstract tropical curves

large time behavior of solutions to semilinear equations with quadratic growth in the gradient this paper studies the large time behavior of solutions to semilinear cauchy problems with quadratic nonlinearity in gradients the cauchy problem considered has a general state space and may degenerate on the boundary of the state space two types of large time behavior are obtained i pointwise convergence of the solution and its gradient and ii convergence of solutions to associated backward stochastic differential equations when the state space is mathbb r d or the space of positive definite matrices both types of convergence are obtained under growth conditions on coefficients these large time convergence results have direct applications in risk sensitive control and long term portfolio choice problems

zero sum risk sensitive stochastic differential games we study zero sum risk sensitive stochastic differential games on the infinite horizon with discounted and ergodic payoff criteria under certain assumptions we establish the existence of values and saddle point equilibria we obtain our results by studying the corresponding hamilton jacobi isaacs equations finally we show that the value of the ergodic payoff criterion is a constant multiple of the maximal eigenvalue of the generators of the associated nonlinear semigroups

geometric formulas on rumely s weight function and crucial measure in non archimedean dynamics we introduce the f crucial function text crucial f associated to a rational function f in k z of degree 1 over an algebraically closed field k of possibly positive characteristic that is complete with respect to a non trivial and non archimedean absolute value and give a global and explicit expression of rumely s resultant function text ordres f in terms of the hyperbolic metric rho on the berkovich upper half space mathsf h 1 in the berkovich projective line mathsf p 1 mathsf p 1 k we also obtain geometric formulas for rumely s weight function w f and crucial measure nu f on mathsf p 1 associated to f as well as improvements of rumely s principal results as an application to dynamics we obtain a quantitative equidistribution of the sequence nu f n n of f n crucial measures towards the f equilibrium or canonical measure mu f on mathsf p 1