binary convexities and distributive lattices a convex structure is binary if every finite family of pairwise intersecting convex sets has a non empty intersection distributive lattices with the convexity of all order convex sublattices are a prominent type of example because they correspond exactly to the intervals of a binary convex structure which has a certain separation property in one direction this result relies on a study of so called base point orders induced by a convex structure these orderings are used to construct an intrinsic topology for binary convexities certain basic questions are answered with the aid of some results on completely distributive lattices several applications are given dimension problems are studied in a subsequent paper 0 introduction a convex structure consists of a set x together with a collection of subsets of x henceforth called convex sets such that 1 1 the empty set and the set x are convex 1 2 the intersection of convex sets is convex 1 3 the union of an updirected collection of convex sets is convex the collection itself is called a convexity on x axiom 1 2 allows the construction of an associated convex hull operator usually denoted by h in the obvious way the hull of a finite set is called a polytope and the hull of a two point set is also called an interval a half space is a convex set with a convex complement the following separation axioms comparable with the axioms tl 5 t4 in topology are used frequently sx singletons are convex which we will assume throughout s2 two distinct points are in complementary half spaces s3 a convex set is an intersection of half spaces s4 two disjoint convex sets extend to complementary half spaces in this paper we will concentrate largely on a particular though fundamental class of convexities with the following binarity property each finite collection of pairwise intersecting convex sets has a non empty intersection the basic types of examples are described in 1 many of these examples arise in a topological context for other examples a natural topology can be constructed binary convex structures on a topological space have been studied extensively in the past years 15 17 19 22 29 34 38 the topology and the convexity are always assumed to be compatible in the sense that polytopes are closed a triple consisting of a set x a convexity and a compatible topology st is called a topological convex proc london math soc 3 48 1984 1 33 5388 3 48 a

levy walk with multiple internal states levy walk with multiple internal states can effectively model the motion of particles that don t immediately move back to the directions or areas which they come from when the levy walk behaves superdiffusion it is discovered that the non immediately repeating property characterized by the constructed transition matrix has no influence on the particle s mean square displacement msd or pearson coefficient this is a kind of stable property of levy walk however if the levy walk shows the dynamical behaviors of normal diffusion then the effect of non immediately repeating emerges for the levy walk with some particular transition matrices it may display nonsymmetric dynamics in these cases the behaviors of their variances are detailedly discussed especially some comparisons with the ones of the continuous time random walks are made a striking difference is the changes of the exponents of the variances the first passage time distribution and its average of levy walks are simulated the results of which turn out that the first passage time can distinguish levy walks with different transition matrices while the msd can not

risk sensitive and robust escape criteria the problem of controlling a noisy process so as to prevent it from leaving a prescribed set has a number of interesting applications in this paper new approaches to this problem are considered first a risk sensitive criterion for a stochastic diffusion process model is examined and it is shown that the value is a classical solution of a related pde the qualitative properties of this criterion are favorably contrasted with those of existing criteria in the risk averse limit it is proved that in the risk averse limit the value of the risk sensitive criterion converges to a viscosity solution of a first order pde it is then demonstrated that the value function of a deterministic differential game is also a viscosity solution to the pde this game gives a robust control formulation of the escape time problem and is analogous to h infty control in particular the opposing player attempts to push the process out of the prescribed set and suffers an l2 cost for his efforts lower bounds on the escape time as a function of this cost are obtained

basic theory of f bounded quantification abstract system f bounded is a second order typed lambda calculus where the basic features of object oriented languages can be naturally modelled f bounded extends the better known system f in a way that provides an immediate solution for the treatment of the so called binary methods although more powerful than f and also quite natural system f bounded has only been superficially studied from a foundational perspective and many of its essential properties have been conjectured but never proved in the literature the aim of this paper is to give a solid foundation to f bounded by addressing and proving the key properties of the system in particular transitivity elimination completeness of the type checking semi algorithm the subject reduction property for reduction conservativity with respect to system f and antisymmetry of a full subsystem are considered and various possible formulations for system f bounded are compared finally a semantic interpretation of system f bounded is presented based on partial equivalence relations

large deviations estimates for systems with small noise effects and applications to stochastic systems theory for typical stochastic systems e g tracking systems estimates of the behavior are hard to get if the noise effects are small then asymptotic methods are appealing to get for example estimates of times required to lose track etc in this paper systems with small noise effects and wide bandwidth noise inputs are analyzed via large deviations methods inputs to many systems in control communication or in physics are not white noise but of certain wide bandwidth types since large deviations results can be sensitive to the actual noise model used working with a model that is close to the physical form is important several such models are dealt with where the bandwidth is large but the intensity small for the models chosen the action functionals turn out to be the same as for the small gaussian white noise models thus it is actually feasible to do computations with them estimates of the probability that the path lies in various sets are obtained the formula for the mean esca

the kakutani property and the fixed point property of topological spaces with abstract convexity abstract the paper deals with the usual fixed point property and the following kakutani property of a space x for every upper semicontinuous function from x to non empty closed convex subsets of x there exists x0 such that x0 x0 we derive this property of x from various separation properties of convex subsets of x and a kind of local convexity of x convexity in our setup is given in an abstract axiomatic way special emphasis is given to the case where x has the form of a product the obtained results cover several known fixed point theorems ky fan glicksberg wallace and special cases of eilenberg montgomery we also discuss an open problem concerning the fixed point property of finite posets and its role in proving more advanced theorems

a further characteristic of abstract convexity structures on topological spaces abstract in this paper we give a characteristic of abstract convexity structures on topological spaces with selection property we show that if a convexity structure c defined on a topological space has the weak selection property then c satisfies h 0 condition moreover in a compact convex subset of a topological space with convexity structure the weak selection property implies the fixed point property

numerical approximations for stochastic differential games the markov chain approximation method is a widely used robust relatively easy to use and efficient family of methods for the bulk of stochastic control problems in continuous time for reflected jump diffusion type models it has been shown to converge under broad conditions and there are good algorithms for solving the numerical problems if the dimension is not too high versions of these methods have been used in applications to various two player differential and stochastic dynamic games for a long time and proofs of convergence are available for some cases mainly using pde type techniques in this paper purely probabilistic proofs of convergence are given for a broad class of such problems where the controls for the two players are separated in the dynamics and cost function and which cover a substantial class not dealt with in previous works discounted and stopping time cost functions are considered finite horizon problems and problems where the process is stopped on first hitting an a priori given boundary can be dealt with by adapting the methods of h j kushner and p dupuis numerical methods for stochastic control problems in continuous time 2nd ed springer verlag berlin new york 2001 as done in this paper for the treated problems the essential conditions are the weak sense existence and uniqueness of solutions an almost everywhere continuity condition and that a weak local consistency condition holds almost everywhere for the numerical approximations just as for the control problem there are extensions to problems with controlled variance and jumps

h sup infinity control for nonlinear systems with output feedback the basic question of nonlinear h sup infinity control theory is to decide for a given two port system when feedback that makes the full system dissipative and internally stable exists this problem can also be viewed as a question about circuits and after translation also has a game theoretic statement several necessary conditions for solutions to exist are presented and sufficient conditions for a certain construction to lead to a solution are given

the interpretation of discontinuous state feedback control laws as nonanticipative control strategies in differential games in differential games one player chooses a feedback strategy to maximize a payoff the other player counters by applying a minimizing open loop control classical notions of feedback strategies based on state feedback control laws for which the corresponding closed loop dynamics uniquely define a state trajectory are too restrictive for many problems owing to the absence of minimizing classical feedback strategies or because consideration of classical feedback strategies fails to define in a useful way the value of the game a number of feedback strategy concepts have been proposed to overcome this difficulty that of elliot and kalton according to which a feedback strategy is a nonanticipative mapping between control functions for the two players has been widely taken up because it provides a value of the game which connects via the hamilton jacobi isaacs equation with other fields of systems science heuristic analysis of specific games problems often points to discontinuous optimal feedback strategies these cannot be regarded as classical feedback control strategies because the associated state trajectories are not in general unique we give general conditions under which they can be interpreted as generalized feedback strategies in the sense of elliot and kalton

a partial differential inequality for dissipative nonlinear systems abstract in this paper we show that storage functions for dissipative systems can be characterised by a partial differential inequality in the weak or viscosity sense the storage functions need not be differentiable this characterisation is stable with respect to perturbations we formulate a version of the positive real lemma for passive systems using the weak sense framework

whitney continua of curves in this paper we prove several theorems relating shape properties of whitney continua of curves in particular we investigate the fundamental dimension and the shape type of whitney continua of curves

approximation of escape time for lagrangian systems with fast noise this note is concerned with the large deviations asymptotics estimate of the mean escape time for lagrangian systems subjected to fast gaussian noise the solution to the hamilton jacobi equation of the associated variational problem is derived as a sum of two terms dependent on kinetic and potential energy of the system respectively a closed form solution for classes of linear and nonlinear systems is obtained an application to a controlled pointing problem is discussed as an example

higher order subtyping abstract system f is an extension with subtyping of the higher order polymorphic calculus an orthogonal combination of girard s system f with cardelli and wegner s kernel fun variant of system f we develop the fundamental metatheory of this calculus decidability of conversion on well kinded types elimination of the cut rule of transitivity from the subtype relation and the soundness completeness and termination of algorithms for subtyping and typechecking

shape properties of whitney maps for hyperspaces in this paper some shape properties of whitney maps for hyperspaces are investigated in particular the following are proved 1 let x be a continuum and let k be the hyperspace 2x or c x of x with the hausdorff metric then if x is any whitney map for for any 0 s s t w x t is an approximate strong deformation retract of w s t in particular sh w 1 t sh w 1 s t 2 pointed 1 movability is a whitney property 3 for any given n oo the property of cohomological dimension n is a sequential strong whitney reversible property 4 the property of being chainable or circle like is a sequential strong whitneyreversible property 5 the property of being an far is a whitney property for 1 dimensional continua property 2 is an affirmative answer to a problem of j t rogers 16 112 properties 3 and 4 are affirmative answers to problems of s b nadler 20 14 57 and 21 0 introduction by a continuum we mean a compact connected metric space for a given continuum x by the hyperspaces of x we mean 2x a c x i a is a nonempty closed subset of x c x a e 2x i a is connected which have the hausdorff metric pg see 20 the term mapping is a continuous function let a 2xor c x a mapping w 0 o x is called a whitney map for t 20 or 31 provided that the following conditions are satisfied a if a c b and a b then w a w b b o x 0 for each x e x in 1932 h whitney 31 showed that whitney maps always exist the notion of whitney map is a convenient and important tool for hyperspace theory a topological property p is said to be a whitney property 20 or 21 provided that whenever a continuum x has property p so does w f t for each whitney map w for c x and 0 t o x it is well known that connectibility is a whitney property if x is a received by the editors april 9 1985 and in revised form october 16 1985 1980 mathematics subject classification primary 54b20 secondary 54c55 54c56 54f45 kev words anid phrases hyperspaces whitney maps whitney property sequential strong whitneyreversible property approximate strong deformation retract pointed 1 movable far anr chainable circle like e mapping indecomposable 1986 american mathematical society 0002 9947 86 1 00 25 per page

diffusion across characteristic boundaries with critical points we consider the problems of the effect of small white noise perturbations on a deterministic dynamical system in the plane with i an asymptotically stable equilibrium point or limit cycle and ii an equilibrium point surrounded by closed trajectories the mean exit time and the distribution of exit points for each problem is determined by solving singularly perturbed elliptic boundary value problems in domains with closed characteristic boundaries with critical points uniformly valid asymptotic solutions are constructed for each of the problems for the asymptotically stable equilibrium point the method of matched asymptotic expansions with the integral condition of matkowsky and schuss is employed a method of averaging combined with boundary layer analysis is used for the problem of an equilibrium point surrounded by closed trajectories the influence on the solutions of the critical points on the boundary is exhibited and explained an application to the physical pendulum is given finally our r

whitney levels in hyperspaces of certain peano continua let x be a peano continuum let 2x resp c x be the space of all nonempty compacta resp subcontinua of x with the hausdorff matric let xo be a whitney map defined on sc 2x or c x such that x is admissible this requires the existence of a certain type of deformation of sc if sc c x assume x contains no free arc then for any to c 0 o x it is proved that w to o to and wf to o x are hilbert cubes this is an analogue of the curtis schori theorem for k a general result for the existance of admissible whitney maps is proved which implies that these maps exist when x is starshaped in a banach space or when x is a dendrite using these results it is shown for example that being an ar an anr an lc space or an lc space is not strongly whitney reversible introduction a continuum is a nonempty compact connected metric space a peano continuum is a locally connected continuum the hyperspaces of a continuum x are the spaces 2x a c x a is nonempty and compact and c x a c 2x a is connected metrized with the hausdorff metric 21 let f1 x x x c x a whitney map for a hyperspace sc of x is a continuous function w sc 0 xo such that o x 0 for each x c f1 x and if a b c c and a c b a then a b intuitively a whitney map is a size function if w is a whitney map for sc and 0 t o x then w l t is called a whitney level if 0 t x then w l t is called a positive whitney level thus a whitney level consists of compacta or continua in x of the same size basic facts about whitney maps and whitney levels are in 21 we note that whitney levels are coverings of x which as t gets close to zero converge to l 0 and consist of compacta of small diameter note that l 0 f1 x and hence l 0 is homeomorphic to x thus as t gets close to zero the whitney levels approximate x it is of interest to obtain information about the structure of positive whitney levels and to determine those properties which are preserved by the convergence of positive whitney levels to the zero level resultsa about this could be used to determine properties of x this is in analogy with the use of nerves of covers the nerves correspond to polyhedra these polyhedra approximate the original space as the mesh of the covers approaches zero and properties of these polyhedra determine various properties of the original space

dimension of binary convex structures it is shown that for compact spaces with a normal binary convexity the dimension functions dim ind ind and cohomological dimension are all equal also an h dimensional compact space x with a normal binary convexity embeds in a product of an m dimensional connected quotient of x with the space of xcomponents both results are obtained with the aid of an auxiliary theory of convex dimension some applications to completely distributive lattices are given 0 introduction for a connected and completely distributive lattice it is known 2 3 4 that the lebesgue covering dimension dim and the cohomological dimension cdg where g is an abelian coefficient group are equal by 3 corollary 1 the small inductive dimension ind equals cdg provided that in addition the set of points at which the lattice attains its small inductive dimension has non void interior we have taken up the equality problem for the various dimension functions at the following general level let x be a normal binary that is with helly number at most 2 convex structure with compact polytopes according to 32 all intervals of x are completely distributive lattices under suitably chosen base point orders on the other hand the space x with its weak topology embeds as a median stable subset of a tychonov cube our first main result is as follows 0 1 theorem x is a compact space with a normal binary convexity then for each coefficient group g cdgx dim x ind x ind x if instead of being compact x is only required to have compact polytopes and if a is lindelof in its weak topology then it is still true that dimx indx in 34 a result similar to theorem 0 1 has been obtained for a restricted class of lawson semilattices also there is a theorem of pasynkov 23 p 722 of the english translation asserting that dim ind ind for locally compact groups one is tempted to conclude that dimension functions tend to coincide on topological algebras see 32 3 6 for the algebraicity of binary convexities the equality theorem 0 1 actually involves one other dimension function the socalled convex dimension cind this function defined on topological convex structures only was studied in 28 for convex structures with connected convex sets and it will be the key tool in proving theorem 0 1 for convexities with connected convex sets but with an unrestricted helly number an equality theorem of the above type has already been obtained in 22 where cohomological dimension was not considered to eliminate the connectedness restriction we have re examined cind in the case of non connected convex sets and it appears that the main results of 28 survive for proc london math soc 3 48 1984 34 54 dimension of binary convex structures 35 binary convex structures this almost self contained treatment of cind is given in 1 a proof of theorem 0 1 is given in 2 together with some consequences for instance we use results of 29 concerning convexity invariants to re prove the well known equality of breadth and dimension for connected completely distributive lattices see 2 corollary 3 4 3 corollary 1 17 corollary 2 4 our second main result is the following 0 2 theorem let x be an n dimensional compact space with a normal binary convexity then there exists a convexity preserving quotient q x qx onto an n dimensional connected space qx with a normal binary convexity such that q together with the decomposition map d x dx the space of components gives an embedding of topological convex structures

kakutani property of the polytopes implies kakutani property of the whole space abstract this paper deals with the following property of a space x for every upper semicontinuous function from x to nonempty closed convex subsets of x there exists x 0 which is in x 0 we derive this property of x from the same property assumed to hold for the polytopes in x convexity in our setup is given in an abstract axiomatic way the results are also reformulated in the order theoretical language of continuous lattices our results directly generalize a an extension of the kakutani fixed point theorem to compact convex sets in locally convex spaces due to ky fan and i l glicksberg and b the fixed point theorem of wallace concerning set valued functions on trees

subtyping recursion and parametric polymorphism in kernel fun we study subtype checking for recursive types in system kernel fun a typed calculus with subtyping and bounded second order polymorphism along the lines of acm transactions on programming languages and systems 15 4 1993 575 we define a subtype relation over kernel fun recursive types and prove it to be transitive we then show that the natural extension of the algorithm introduced in acm transactions on programming languages and systems 15 4 1993 575 to compare first order recursive types yields a non complete algorithm finally we prove the completeness and correctness of a different algorithm which lends itself to efficient implementations

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

subtyping dependent types the need for subtyping in type systems with dependent types has been realized for some years but it is hard to prove that systems combining the two features have fundamental properties such as subject reduction here we investigate a subtyping extension of the system p which is an abstract version of the type system of the edinburgh logical framework lf by using an equivalent formulation we establish some important properties of the new system p6 including subject reduction our analysis culminates in a complete and terminating algorithm which establishes the decidability of type checking copyright 2001 elsevier science b v

bounded existentials and minimal typing abstract we study an extension of the second order calculus of bounded quantification system f with bounded existential types surprisingly the most natural formulation of this extension lacks the important minimal typing property of f which ensures that the set of types possessed by a typeable term can be characterized by a single least element we consider alternative formulations and give an algorithm computing minimal types for the slightly weaker kernel fun variant of f

stochastic systems with small noise analysis and simulation a phase locked loop example systems with wide bandwidth noise inputs are a common occurrence in stochastic control and communication theory and elsewhere e g tracking or synchronization systems such as phase locked loops pll one is often interested in calculating such quantities as the probability of escape from a desired error set in some time interval or the mean time for such escape diffusion approximations the system obtained in the limit bandwidth as the to infty are often used for this since they are easier to analyze when the noise effects in the physical system are small one is tempted to do an asymptotic analysis noise intensity to 0 on the diffusion approximation and use this for the desired estimates on the original system such a procedure does not work in general the double limit bandwidth to infty intensity to 0 is not always justified under quite broad conditions on the noise processes it is justified for the systems studied here we study a particular form of the pll owing to

survey paper a partial history of the early development of continuous time nonlinear stochastic systems theory this article is a survey of the early development of selected areas in nonlinear continuous time stochastic control key developments in optimal control and the dynamic programming principle existence of optimal controls under complete and partial observations nonlinear filtering stochastic stability the stochastic maximum principle and ergodic control are discussed issues concerning wide bandwidth noise for stability modeling filtering and ergodic control are dealt with the focus is on the earlier work but many important topics are omitted for lack of space

residence time control the problem of residence time controllability in dynamical systems with stochastic perturbations is formulated the solution is given for linear systems with small additive white noise perturbation it is shown that the existence of the desired residence time controller depends on the relationship between the column spaces of the control and noise matrices if the former includes the latter any residence time is possible if this inclusion does not occur the achievable residence time is bounded lower and upper estimates of this bound are given for each of these cases controller design techniques are suggested and illustrative examples are considered the development is based on an asymptotic version of the large deviations theory

divergence of f type checking abstract system f is an extension of second order typed lambda calculus where a subtype hierarchy among types is defined and bounded second order lambda abstraction is allowed this language is a basis for much of the current research on integration of typed functional languages with subtypes and inheritance an algorithm to perform type checking for f expressions has been known since the language fun was defined the algorithm has been proved complete by the author and curien which means that it is a semi decision procedure for the type checking problem in this paper we show that this algorithm is not a decision procedure by exhibiting a term which makes it diverge this result was the basis of pierce s proof of undecidability of typing for f we study the behavior of the algorithm to show that our diverging judgement is in some sense contained in any judgement which makes the algorithm diverge on the basis of this result and of other results in the paper we claim that the chances that the algorithm will loop while type checking a real program are negligible hence the undecidability of f type checking should not be considered as a reason to prevent the adoption of f as a basis for defining programming languages of practical interest finally we show the undecidability of an important subsystem of f

principality and type inference for intersection types using expansion variables principality of typings is the property that for each typable term there is a typing from which all other typings are obtained via some set of operations type inference is the problem of finding a typing for a given term if possible we define an intersection type system which has principal typings and types exactly the strongly normalizable terms more interestingly every finite rank restriction of this system using leivant s first notion of rank has principal typings and also has decidable type inference this is in contrast to system f where the finite rank restriction for every finite rank at 3 and above has neither principal typings nor decidable type inference furthermore the notion of principal typings for our system involves only one operation substitution rather than several operations not all substitution based as in earlier presentations of principality for intersection types without rank restrictions in our system the earlier notion of expansion is integrated in the form of expansion variables which are subject to substitution as are ordinary variables a unification based type inference algorithm is presented using a new form of unification unification

polymorphic type inference and containment abstract type expressions may be used to describe the functional behavior of untyped lambda terms we present a general semantics of polymorphic type expressions over models of untyped lambda calculus and give complete rules for inferring types for terms some simplified typing theories are studied in more detail and containments between types are investigated

finite time horizon risk sensitive control and the robust limit under a quadratic growth assumption the finite time horizon risk sensitive limit problem for continuous nonlinear systems is considered previous results are extended to cover more typical examples in particular the cost may grow quadratically and the diffusion coefficient may depend on the state it is shown that the risk sensitive value function is the solution of the corresponding dynamic programming equation it is also shown that this value converges to the value of the robust control problem as the cost becomes infinitely risk averse with corresponding scaling of the diffusion coefficient

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

comparing object encodings recent years have seen the development of several foundational models for statically typed object oriented programming but despite their intuitive similarity differences in the technical machinery used to formulate the various proposals have made them difficult to compare using the typed lambda calculus f as a common basis we now offer a detailed comparison of four models 1 a recursive record encoding similar to the ones used by cardelli car84 reddy red88 kr94 cook coo89 chc90 and others 2 hofmann pierce and turner s existential encoding pt94 hp95 3 bruce s model based on existential and recursive types bru94 and 4 abadi cardelli and viswanathan s type theoretic encoding acv96 of a calculus of primitive objects

exit time problems in optimal control and vanishing viscosity method the authors study the connections between deterministic exit time control problems and possibly discontinuous viscosity solutions of a first order hamilton jacobi hj equation up to the boundary this equation admits a maximum and a minimum solution that are the value functions associated to stopping time problems on the boundary when these solutions are equal they can be obtained through the vanishing viscosity method finally when the hj equation has a continuous solution it is proved to be the value function for the first exit time of the domain it is also the vanishing viscosity limit arising in particular in some large deviations problems

nash points ky fan inequality and equilibria of abstract economies in max plus and b convexity abstract b convexity was introduced in w briec c horvath b convexity optimization 53 2004 103 127 separation and hahn banach like theorems can be found in g adilov a m rubinov b convex sets and functions numer funct anal optim 27 2006 237 257 and w briec c d horvath a rubinov separation in b convexity pacific j optim 1 2005 13 30 we show here that all the basic results related to fixed point theorems are available in b convexity ky fan inequality existence of nash equilibria and existence of equilibria for abstract economies are established in the framework of b convexity monotone analysis or analysis on maslov semimodules v n kolokoltsov v p maslov idempotent analysis and its applications math appl vol 401 kluwer academic 1997 v p litvinov v p maslov g b shpitz idempotent functional analysis an algebraic approach math notes 69 2001 696 729 v p maslov s n samborski eds idempotent analysis advances in soviet mathematics amer math soc providence ri 1992 is the natural framework for these results from this point of view max plus convexity and b convexity are isomorphic maslov semimodules structures over isomorphic semirings therefore all the results of this paper hold in the context of max plus convexity

some properties of abstract convexity structures on topological spaces abstract in this paper we establish some relationships among abstract convexity the selection property and the fixed point property we show that if a convexity structure c defined on a topological space has the selection property then c satisfies the h condition moreover in an l c compact metric space the selection property implies the fixed point property

set relations and set systems induced by some families of integral domains abstract in this paper given an integral domain u we investigate the main properties of a relation m o d which is based on the interrelation between subdomains of u and finitely generated unitary submodules of u we shall characterize it in terms of a second relation between n tuples u 1 u n of elements of u and subdomains d of u defined by the vanishing in u 1 u n of some polynomial p z 1 z n belonging to a specific subset of the polynomial ring in several variables d z 1 z n such an equivalence shall be used in order to introduce three specific collections of subdomains x u b u and p u whose algebraic properties present a close connection with geometrical and combinatorial properties induced by m o d on the other hand the characterization of the subdomains of x u leads to the more general problem of finding a map associating with a subdomain d of u a collection d of subdomains of k u such that the intersection of some or of any member of d gives d in this perspective in the present paper we shall study two further collections of subdomains of u denoted respectively by e u and l u whose main properties are related to those of the families p u and b u finally our investigation of all the aforementioned subdomain families shall be also related to the study of pairs e where e u 0 and is an idempotent ring endomorphism of u whose kernel agrees with the ideal of u generated by e we shall exhibit several results concerning the membership of u and of u e to the above subdomain families

an escape time criterion for queueing networks asymptotic risk sensitive control via differential games we consider the problem of risk sensitive control of a stochastic network in controlling such a network an escape time criterion can be useful if one wishes to regulate the occurrence of large buffers and buffer overflow in this paper a risk sensitive escape time criterion is formulated which in comparison to the ordinary escape time criteria penalizes exits that occur on short time intervals more heavily the properties of the risk sensitive problem are studied in the large buffer limit and related to the value of a deterministic differential game with constrained dynamics we prove that the game has value and that the value is the viscosity solution of a pde for a simple network the value is computed demonstrating the applicability of the approach

large deviations results for the exit problem with characteristic boundary abstract we consider the exit problem for an asymptotically small random perturbation of a stable dynamical system in a region d we show the standard large deviations results for the exit distribution and mean exit time as obtained by wentzell and freidlin under the assumption of nontangential drift b n

limiting exit location distributions in the stochastic exit problem consider a two dimensional continuous time dynamical system with an attracting fixed point s if the deterministic dynamics are perturbed by white noise random perturbations of strength epsilon the system state will eventually leave the domain of attraction omega of s we analyze the case when as epsilon to0 the exit location on the boundary partial omega is increasingly concentrated near a saddle point h of the deterministic dynamics we show using formal methods that the asymptotic form of the exit location distribution on partial omega is generically non gaussian and asymmetric and classify the possible limiting distributions a key role is played by a parameter mu equal to the ratio lambda s h lambda u h of the stable and unstable eigenvalues of the linearized deterministic flow at h if mu 1 then the exit location distribution is generically asymptotic as epsilon to0 to a weibull distribution with shape parameter 2 mu on the cal o epsilon mu 2

a selection theorem for topological convex structures a continuous selection theorem has been obtained for multivalued functions the values of which are convex sets of certain synthetic convex structures applications are given related with superextensions semi lattices spaces of order arcs trees whitney levels in hyperspaces and geometric topology applications to traditional convexity in vector spaces involve beer s approximation theorem and a fixed point theorem of dugundji granas some other applications a o an invariant arc theorem appear elsewhere

a singular perturbation method for the computation of the mean first passage time in a nonlinear filter we give a new application of recently developed singular perturbation methods in the area of mathematical theory of nonlinear filtering we consider the phenomenon of cycle slipping in a second order phase locked loop pll which serves as demodulator for a random fm message we introduce new scaling parameters into the ito system of stochastic differential equations describing the pll thus identifying the phenomenon of cycle slipping with kolmogorov s exit problem we use singular perturbation methods to obtain an explicit expression for the mean time between cycle slips furthermore we describe the mechanism of cycle slips and identify new parameters which determine the probability of their occurrence

bounded quantification is undecidable f is a typed calculus with subtyping and bounded second order polymorphism first introduced by cardelli and wegner it has been widely studied as a core calculus for type systems with subtyping we use a reduction from the halting problem for two counter turing machines to show that the subtyping and typing relations of f are undecidable

the exit distribution on the stochastic separatrix in kramers exit problem kramers exit problem is concerned with noise activated escape from a potential well in the case when the noise strength varepsilon temperature measured in units of potential barrier height is small this becomes a singular perturbation problem unexpectedly the noise induced escape from the domain of attraction of the stable equilibrium at the bottom of the well does not occur at points on the separatrix where the energy is minimal rather the distribution of exit points on the separatrix in the phase plane is shown to be spread away from the saddle point where the energy is minimal it is shown that actually most of the probability of the exit points on the separatrix is located at a distance o sqrt varepsilon from the saddle point and the probability vanishes altogether at the saddle point in this problem large deviations theory fails to predict the distribution of the exit point for finite noise we construct an asymptotic approximation to the solution of the fokker planck eq