equivariance brst symmetry and superspace the structure of equivariant cohomology in non abelian localization formulas and topological field theories is discussed equivariance is formulated in terms of a nilpotent becchi rouet stora tyutin brst symmetry and another nilpotent operator which restricts the brst cohomology onto the equivariant or basic sector a superfield formulation is presented and connections to reducible batalin fradkin vilkovisky bfv quantization of topological yang mills theory are discussed

equivariant cohomology fock space and loop groups equivariant de rham cohomology is extended to the infinite dimensional setting of a loop subgroup acting on a loop group using hida supersymmetric fock space for the weil algebra and malliavin test forms on the loop group the mathai quillen isomorphism in the brst formalism of kalkman is defined so that the equivalence of various models of the equivariant de rham cohomology can be established

localization and diagonalization a review of functional integral techniques for low dimensional gauge theories and topological field theories localization techniques for functional integrals which have recently been used to perform calculations in and gain insight into the structure of certain topological field theories and low dimensional gauge theories are reviewed these are the functional integral counterparts of the mathai quillen formalism the duistermaat heckman theorem and the weyl integral formula respectively in each case the necessary mathematical background euler classes of vector bundles equivariant cohomology topology of lie groups is introduced and the finite dimensional integration formulae described then some applications to path integrals are discussed and an overview of the relevant literature is given the applications include supersymmetric quantum mechanics cohomological field theories phase space path integrals and two dimensional yang mills theory

semiclassical analysis of low lying eigenvalues ii tunneling we discuss the leading asymptotics of eigenvalue splittings of 2 x2v in the limit as x o and where v is a non negative potential with several zeros for example if eo x e1 x are the two lowest eigenvalues in a situation where v has precisely two zeros a and b related by a symmetry then limx x lin e1 x eo x is given as the distance from a to b in a certain riemann metric

a piecewise linear model for the zones of instability of an area preserving map abstract in this note we study the global behavior of the piecewise linear area preserving transformation x 1 1 y 0 x 0 y 1 x 0 of the plane we show that there are infinitely many invariant polygons surrounding an elliptic fixed point the regions between these invariant polygons serve as models for the zones of instability in the corresponding smooth case for our model we show that some of these annular zones contain only finitely many elliptic islands the map is hyperbolic on the complement of these islands and hence exhibits stochastic behavior in this region unstable periodic points are dense in this region

rendezvous search on the labeled line the rendezvous search problem is the problem of finding optimal search strategies for two people who are placed randomly on a known search region and want to meet each other in minimal expected time we focus on initial location distributions that are centrally symmetric and nonincreasing as one moves away from the center including the discretized and or truncated gaussian densities when the search region is a discrete or a continuous interval and the interval is labeled so that the searchers know their own location at all times we prove that the optimal strategy for both searchers is to go directly to the center and wait there the same result also holds for rendezvous search on the infinite line

searching a variable speed network a point lies on a network according to some unknown probability distribution starting at a specified root of the network a searcher moves to find this point at speeds that depend on his location and direction he seeks the randomized search algorithm that minimizes the expected search time this is equivalent to modeling the problem as a zero sum hide and seek game whose value is called the search value of the network we make a new and direct derivation of an explicit formula for the search value of a tree proving that it is equal to half the sum of the minimum tour time of the tree and a quantity called its incline the incline of a tree is an average over the leaf nodes of the difference between the time taken to travel from the root to a leaf node and the time taken to travel from a leaf node to the root this difference can be interpreted as height of a leaf node assuming uphill is slower than downhill we then apply this formula to obtain numerous results for general networks we also introduce a new general method of comparing the search value of networks that differ in a single arc some simple networks have very complicated optimal strategies that require mixing of a continuum of pure strategies many of our results generalize analogous ones obtained for constant velocity in both directions by s gal but not all of those results can be extended

kramers equation and supersymmetry hamilton s equations with noise and friction possess a hidden supersymmetry valid for time independent as well as periodically time dependent systems it is used to derive topological properties of critical points and periodic trajectories in an elementary way from a more practical point of view the formalism provides new tools to study the reaction paths in systems with separated time scales a reduced current which contains the relevant part of the phase space probability current is introduced together with strategies for its computation

the brezis nirenberg problem for the laplacian with a singular drift in rn and sn abstract we consider the brezis nirenberg problem for the laplacian with a singular drift for a geodesic ball in both r n and s n 3 n 5 the singular drift we consider derives from a potential which is symmetric around the center of the geodesic ball here the potential is given by a parameter say times the logarithm of the distance to the center of the ball in both cases we determine the exact region in the parameter space for which positive smooth solutions of this problem exist and the exact region for which there are no solutions the parameter space is characterized by the geodesic radius of the ball and the coupling constant of the linear term of the brezis nirenberg problem

bergman kernels and local holomorphic morse inequalities let x be a hermitian manifold and let lk be a high power of a hermitian holomorphic line bundle over x local versions of demailly s holomorphic morse inequalities that give bounds on the dimension of the dolbeault cohomology groups associated to lk are presented after integration they give the usual holomorphic morse inequalities the local weak inequalities hold on any hermitian manifold x regardless of compactness and completeness the proofs which are elementary are based on a new approach to pointwise bergman kernel estimates where the kernels are estimated by a model kernel in open image in new window

quantum classical correspondence via a deformed kinetic operator we propose an approach to quantum classical correspondence based on a deformation of the momentum and kinetic operators of quantum mechanics making use of the factorization method we construct classical versions of the momentum and kinetic operators which in addition to the standard quantum expressions contain terms that are functionals of the n particle density we show that this implementation of quantum classical correspondence is related to witten s deformation of the exterior derivative and laplacian introduced in the context of supersymmetric quantum mechanics the corresponding deformed action is also shown to be related to the fisher information finally we briefly consider the possible relevance of our approach to the construction of kinetic energy density functionals

abcd of instantons we solve open image in new window 2 supersymmetric yang mills theories for an arbitrary classical gauge group i e su n so n sp n in particular we derive the prepotential of the low energy effective theory and the corresponding seiberg witten curves we manage to do this without resolving singularities of the compactified instanton moduli spaces

finite volume flows and morse theory in this paper we present a new approach to morse theory based on the de rham federer theory of currents the full classical theory is derived in a transparent way the methods carry over uniformly to the equivariant and the holomorphic settings moreover the methods are substantially stronger than the classical ones and have interesting applications to geometry they lead for example to formulas relating characteristic forms and singularities of bundle maps

supersymmetry and morse theory it is shown that the morse inequalities can be obtained by consideration of a certain supersymmetric quantum mechanics hamiltonian some of the implications of modern ideas in mathematics for supersymmetric theories are discussed

factorization little or great algorithm the progress of the factorization method since the 1935 work of dirac is briefly reviewed though linked with older mathematical theories the factorization seems an autonomous driving force offering substantial support to the present day darboux and backlund approaches

search for an immobile hider in a known subset of a network abstract a unit speed searcher constrained to start in a given closed set s wishes to quickly find a point x known to be located in a given closed subset h of a metric network q this defines a game g g q h s where the payoff to the maximizing hider is the time for the searcher path to reach x lengths on q are defined by a measure which then defines distance as least length of connecting path for trees q we find that the minimax search time value v of g is given by v h d h s 2 where d h s is what we call the h diameter of s and equals the usual diameter d s of s in the case h q for the classical case of gal where the s is a singleton and h q our formula reduces to his result v q if s h q our formula gives dagan and gal s result v q d q 2 in all other cases our result is new optimal searches consist of minimum length paths covering h which start and end at points of s traversed equiprobably in either direction

the infinitesimal lefschetz formulas a heat equation proof abstract in this paper the lefschetz formulas of atiyah and bott ann of math 86 1967 374 407 88 1968 451 491 and atiyah and singer ann of math 87 1968 546 604 93 1971 119 138 are proved by heat equation methods in infinitesimal form i e without using the usual localization on the fixed point set the proof is obtained by a homotopic deformation of the dirac operator and uses probabilistic methods the main applications are potentially in noncompact situations where fixed point formulas do not always hold while their infinitesimal analogues are still valid

a mathematical model for the dynamics and synchronization of cows we formulate a mathematical model for the daily activities of a cow eating lying down and standing in terms of a piecewise linear dynamical system we analyze the properties of this bovine dynamical system representing the single animal and develop an exact integrative form as a discrete time mapping we then couple multiple cow oscillators together to study synchrony and cooperation in cattle herds we comment on the relevant biology and discuss extensions of our model with this abstract approach we not only investigate equations with interesting dynamics but also develop biological predictions in particular our model illustrates that it is possible for cows to synchronize less when the coupling is increased

heat equation derivative formulas for vector bundles we use martingale methods to give bismut type derivative formulas for differentials and co differentials of heat semigroups on forms and more generally for sections of vector bundles the formulas are mainly in terms of weitzenbock curvature terms in most cases derivatives of the curvature are not involved in particular our results improve b k driver s formula in 1997 j math pures appl 9 76 703 737 for logarithmic derivatives of the heat kernel measure on a riemannian manifold our formulas also include the formulas of k d elworthy and x m li 1998 c r acad sci paris ser i math 327 87 92

search for an immobile entity on a network we consider the problem of searching for a single uniformly distributed immobile entity on an undirected network this problem differs from edge covering problems e g the chinese postman problem cpp since the objective here is not to find the minimum length tour that covers all the edges at least once but instead to minimize the expected time to find the entity we introduce a heuristic algorithm to deal with the search process given that the entity is equally likely to be at any point on the network computational results are presented

border collision bifurcations in two dimensional piecewise smooth maps recent investigations on the bifurcations in switching circuits have shown that many atypical bifurcations can occur in piecewise smooth maps that cannot be classified among the generic cases like saddle node pitchfork or hopf bifurcations occurring in smooth maps in this paper we first present experimental results to establish the need for the development of a theoretical framework and classification of the bifurcations resulting from border collision we then present a systematic analysis of such bifurcations by deriving a normal form the piecewise linear approximation in the neighborhood of the border we show that there can be eleven qualitatively different types of border collision bifurcations depending on the parameters of the normal form and these are classified under six cases we present a partitioning of the parameter space of the normal form showing the regions where different types of bifurcations occur this theoretical framework will help in explaining bifurcations in all systems which can be represented by two dimensional piecewise smooth maps

an evasion game on a finite tree the paper considers the following two person zero sum multistage game an evader starts at a given vertex of a tree and at discrete intervals of time chooses either to move to one of the vertices adjacent to him or stay where he is a gunner with a single bullet may at each of the same discrete intervals of time either fire the bullet at any of the vertices of the tree or hold his fire the gunner always hits the vertex at which he aims and the bullet takes one unit of time to reach its target the payoff to the gunner is 1 if he hits the evader mu where mu 1 if he fires and misses and 0 if he never fires it is shown that whatever vertex the evader starts at the value of the game is 1 mu 2

morse theory for lagrange multipliers and adiabatic limits abstract given two morse functions f on a compact manifold m we study the morse homology for the lagrange multiplier function on m r which sends x to f x x take a product metric on m r and rescale its r component by a factor 2 we show that generically for large the morse smale witten chain complex is isomorphic to the one for f and the metric restricted to 1 0 with grading shifted by one on the other hand in the limit 0 we obtain another chain complex which is geometrically quite different but has the same homology as the singular homology of 1 0 the isomorphism between the chain complexes is provided by the homotopy obtained by varying our proofs use both the implicit function theorem on banach manifolds and geometric singular perturbation theory

a morse complex for infinite dimensional manifolds part i abstract in this paper and in the forthcoming part ii we introduce a morse complex for a class of functions f defined on an infinite dimensional hilbert manifold m possibly having critical points of infinite morse index and co index the idea is to consider an infinite dimensional subbundle or more generally an essential subbundle of the tangent bundle of m suitably related with the gradient flow of f this part i deals with the following questions about the intersection w of the unstable manifold of a critical point x and the stable manifold of another critical point y finite dimensionality of w possibility that different components of w have different dimension orientability of w and coherence in the choice of an orientation compactness of the closure of w classification up to topological conjugacy of the gradient flow on the closure of w in the case dim w 2

on the approximation ratio of the random chinese postman tour for network search abstract we consider a classic search problem first proposed by gal in which a searcher randomizes between unit speed paths on a network aiming to find a hidden point in minimal expected time in the worst case this can be viewed as a zero sum game between the searcher and a time maximizing hider it is a natural model for many search problems such as search and rescue operations the search for an enemy a bomb or weapons in a military context or predator prey search a chinese postman tour cpt is a minimal time tour of the network that searches all the arcs and a random chinese postman tour rcpt is an equiprobable choice of any given cpt and its reverse the full class of networks for which a rcpt is optimal is known but otherwise little is known about the solution of the game except in some special cases that have complicated optimal strategies that would be impractical to implement the question of how well a rcpt or any other search strategy performs for general networks has never been analyzed we show that a rcpt has an approximation ratio of 4 3 that is the maximum expected time it takes to find a point on the network is no greater than 4 3 times that of the optimal search strategy we then examine the performance of a rcpt in a related search game recently proposed by alpern in which the searcher must return to his starting point after finding the hider

fisher information and kinetic energy functionals a dequantization approach we strengthen the connection between information theory and quantum mechanical systems using a recently developed dequantization procedure whereby quantum fluctuations latent in the quantum momentum are suppressed the dequantization procedure results in a decomposition of the quantum kinetic energy as the sum of a classical term and a purely quantum term the purely quantum term which results from the quantum fluctuations is essentially identical to the fisher information the classical term is complementary to the fisher information and in this sense it plays a role analogous to that of the shannon entropy we demonstrate the kinetic energy decomposition for both stationary and nonstationary states and employ it to shed light on the nature of kinetic energy functionals

dynamics of a map with a power law tail we analyze a one dimensional piecewise continuous discrete model proposed originally in studies on population ecology the map is composed of a linear part and a power law decreasing piece and has three parameters the system presents both regular and chaotic behavior we study numerically and in part analytically different bifurcation structures particularly interesting is the description of the abrupt order to chaos transition mediated by an attractor made of an infinite number of limit cycles with only a finite number of different periods it is shown that the power law piece in the map is at the origin of this type of bifurcation the system exhibits interior crises and crisis induced intermittency

a path integral for turbulence in incompressible fluids in this paper a classical path integral is formulated for incompressible fluids that evolve according to the navier stokes equation the path integral propagates probability distributions deterministically on the space gvol of solenoidal velocity fields we construct a set of isp 2 charges associated with the geometry of gvol and its poisson structure and a pair of supersymmetry charges connected with the hamiltonian these charges generate exact symmetries of the classical path integral when the viscosity is set equal to zero when the effect of dissipation is included the charges associated with the poisson structure and the hamiltonian are no longer conserved charges that generate kolmogorov scaling and galilean transformations are also constructed the classical path integral is formulated in terms of vorticity as well

border collision bifurcations including period two to period three for piecewise smooth systems abstract we examine bifurcation phenomena for maps that are piecewise smooth and depend continuously on a parameter in the simplest case there is a surface in phase space along which the map has no derivative or has two one sided derivatives is the border of two regions in which the map is smooth as the parameter is varied a fixed point e may collide with the border and we may assume that this collision occurs at 0 a variety of bifurcations occur frequently in such situations but never or almost never occur in smooth systems in particular e may cross the border and so will exist for 0 but it may be a saddle in one case say 0 for 0 there may be a stable period 3 orbit which similarly shrinks to e0 as 0 hence one observes the following stable periodic orbits a stable period 2 orbit collapses to a point and is reborn as a stable period 3 orbit we also see analogously stable period 2 to stable period p orbit bifurcations with p 5 11 52 or period 2 to quasi periodic or even to a chaotic attractor we believe this phenomenon will be seen in many applications

variational approach to dequantization we present a dequantization procedure based on a variational approach whereby quantum fluctuations latent in the quantum momentum are suppressed this is done by adding generic local deformations to the quantum momentum operator which give rise to a deformed kinetic term quantifying the amount of fuzziness caused by such fluctuations considered as a functional of such deformations the deformed kinetic term is shown to possess a unique minimum which is seen to be the classical kinetic energy furthermore we show that extremization of the associated deformed action functional introduces an essential nonlinearity to the resulting field equations which are seen to be the classical hamilton jacobi and continuity equations thus a variational procedure determines the particular deformation that has the effect of suppressing the quantum fluctuations resulting in dequantization of the system

quantization from an exponential distribution of infinitesimal action abstract a statistical model of quantization based on an exponential distribution of infinitesimal action is proposed a trajectory which does not extremize the action along an infinitesimal short segment of path is allowed to occur with a very small probability following an exponential law planck s constant is argued to give the average deviation from the infinitesimal stationary action

asymptotic expansion of the witten deformation of the analytic torsion given a compact riemannian manifold md g a finite dimensional representation 1 m gl v of the fundamental group 1 m on a vector spacevof dimensionland a hermitian structure on the flat vector bundle formula associated to ray singer rs have introduced the analytic torsiont t m g 0 witten s deformationdq t of the exterior derivativedq dq t e htdqeht withh m ra smooth morse function can be used to define a deformationt h t 0 of the analytic torsiontwitht h 0 t the main results of this paper are to provide assuming that grad ghis morse smale an asymptotic expansion for log t h t fort of the form formula and to present two different formulae fora0 as an application we obtain a shorter derivation of results due to ray singer rs cheeger ch muller mu1 2 which in increasing generality concern the equality for odd dimensional manifolds of the analytic torsion with the average of the reidemeister torsion corresponding to the triangulation t h g and the dual triangulation td d h g

schrodinger equation from an exact uncertainty principle an exact uncertainty principle formulated as the assumption that a classical ensemble is subject to random momentum fluctuations of a strength which is determined by and scales inversely with uncertainty in position leads from the classical equations of motion to the schrodinger equation

oxidation of self duality to 12 dimensions and beyond using partial curvature flows and the transitive action of subgroups of rm o d mathbb z on the indices 1 ldots d of the components of the yang mills curvature in an orthonormal basis we obtain a nested system of equations in successively higher dimensions d each implying the yang mills equations on d dimensional riemannian manifolds possessing special geometric structures this matryoshka of self duality equations contains the familiar self duality equations on riemannian fourfold as well as their generalisations on complex kahler threefold and on seven and eight dimensional manifolds with g2 and spin 7 holonomy the matryoshka allows enlargement oxidation to a remarkable system in 12 dimensions invariant under sp 3 there are hints that the underlying geometry is related to the sextonions a six dimensional algebra between the quaternions and octonions

an extension of a theorem by cheeger and muller to spaces with isolated conical singularities the aim of this article is to extend a theorem by cheeger and muller to spaces with isolated conical singularities by generalizing the proof of bismut and zhang to the singular setting the main tools in this approach are the witten deformation and local index techniques

an informational characterization of schrodinger s uncertainty relations heisenberg s uncertainty relations employ commutators of observables to set fundamental limits on quantum measurement the information concerning incompatibility non commutativity of observables is well included but that concerning correlation is missing schrodinger s uncertainty relations remedy this defect by supplementing the correlation in terms of anti commutators however both heisenberg s uncertainty relations and schrodinger s uncertainty relations are expressed in terms of variances which are not good measures of uncertainty in general situations e g when mixed states are involved by virtue of the wigner yanase skew information we will establish an uncertainty relation along the spirit of schrodinger from a statistical inference perspective and propose a conjecture the result may be interpreted as a quantification of certain aspect of the celebrated wigner araki yanase theorem for quantum measurement which states that observables not commuting with a conserved quantity cannot be measured exactly

the princess and monster game on an interval a minimizing searcher s and a maximizing hider h move at unit speed on a closed interval until the first capture or payoff time t min t s t h t that they meet this zero sum princess and monster game or less colorfully search game with mobile hider was proposed by rufus isaacs for general networks q while the existence and finiteness of the value v v q has been established for such games only the circle network has been solved value and optimal mixed strategies it seems that the interval network q 1 1 had not been studied because it was assumed to be trivial with value 3 2 and obvious searcher mixed strategy going equiprobably from one end to the other we establish that this game is in fact nontrivial by showing that v 3 2 using a combination of continuous and discrete mixed strategies for both players we show that 15 11 leq v leq 13 9 the full solution of this very simple game is still open and appears difficult though many properties of the optimal strategies are derived here

holomorphic morse inequalities for orbifolds we prove that demailly s holomorphic morse inequalities hold true for complex orbifolds by using a heat kernel method then we introduce the class of moishezon orbifolds and as an application of our inequalties we give a geometric criterion for a compact connected orbifold to be a moishezon orbifolds thus generalizing siu s and demailly s answers to the grauert riemenschneider conjecture to the orbifold case

find and fetch search on a tree we introduce a new type of search game called the find and fetch game f q o the hider simply picks any point h in the network q the searcher starts at time zero at a given point o of q moving at unit speed until he reaches h finds the hider then he returns at a given speed along the shortest path back to o arriving at time r the payoff this models the problem faced in many types of search including search and rescue problems and foraging problems of animals where food must be found and returned to the lair when q is a binary tree we derive optimal probabilities for the searcher to branch at the nodes these probabilities give a positive bias towards searching longer branches first we show that the minimax value of the return time r the game value of f q o is d where is the total length of q and d is the mean distance from the root o to the leaves terminal nodes of q where the mean is taken with respect to what is known as the equal branch density distribution as goes to infinity our problem reduces to the search game model where the payoff is simply the time to reach the hider and our results tend to those obtained by gal gal s 1979 search games with mobile and immobile hider siam j control optim 17 1 99 122 and anderson and gal anderson e j s gal 1990 search in a maze probab engrg inform sci 4 3 311 318 for that model we also apply our return time formula d to determine the ideal location for the root lair or rescue center o assuming it can be moved in the traditional find only model the location of o does not matter

stochastic equivariant cohomologies and cyclic cohomology we give two stochastic diffeologies on the free loop space which allow us to define stochastic equivariant cohomology theories in the chen souriau sense and to establish a link with cyclic cohomology with the second one we can establish a stochastic fixed point theorem

the topology of reduces phase spaces of the motion of vortices on a sphere abstract the motion of point vortices in a perfect incompressible fluid on a two dimensional sphere can be represented by a hamiltonian flow which is invariant under the action of the special orthogonal group so 3 following the reduction method of marsden and weinstein this motion can be described by hamiltonian flows on reduced phase spaces some topological invariants of these reduced phase spaces are calculated for suitable vorticities the results obtained can be used to give lower bounds on numbers of relative equilibria of the original system

rendezvous search on the interval and the circle two people are placed randomly and independently on a street of unit length they attempt to find each other in the shortest possible expected time we solve this problem assuming each searcher knows where he or she is on the street for monotonic density functions for the initial placement this includes the uniform pdf as a special case this gives an example of a rendezvous search problem where there is no advantage in being allowed to use asymmetric strategies we also solve some corresponding problems for the circle when asymmetric strategies are permitted one of these shows that it can sometimes be optimal for one player to wait for the other to find him

ising model and n 2 supersymmetric theories we establish a direct link between massive ising model and arbitrary massiven 2 supersymmetric qft s in two dimensions this explains why the equations which appear in the computation of spin correlations in the non critical ising model are the same as those describing the geometry of vacua inn 2 theories the tau function appearing in the ising model i e the spin correlation function is reinterpreted in then 2 context as a new index in special cases this new index is related to the ray singer analytic torsion and can be viewed as a generalization of that to the loop space of kahler manifolds

incompleteness of trajectory based interpretations of quantum mechanics trajectory based approaches to quantum mechanics include the de broglie bohm interpretation and nelson s stochastic interpretation it is shown that the usual route to establishing the validity of such interpretations via a decomposition of the schr dinger equation into a continuity equation and a modified hamilton jacobi equation fails for some quantum states a very simple example is provided by a quantum particle in a box described by a wavefunction that is initially uniform over the interior of the box for this example there is no corresponding continuity or modified hamilton jacobi equation and the space time dependence of the wavefunction has a known fractal structure examples with finite average energies are also constructed

n 2 supersymmetric quantum mechanics on riemann surfaces with meromorphic superpotentials we constructn 2 supersymmetric quantum hamiltonians with meromorphic superpotentials on compact riemann surfaces and investigate the topological properties of these hamiltonians l2 cohomology groups for supercharge a deformed bar partial operator are considered and the witten index for the supersymmetric hamiltonian with meromorphic superpotential is calculated in terms of euler characteristic of the riemann surface and the degree of a divisor of poles for the differential of the superpotential

the fixed point theorem in equivariant cohomology in this paper we study the s equivariant de rham cohomology of infinite dimensional s manifolds our main example is the free loop space lx where x is a finite dimensional manifold with the circle acting by rotating loops we construct a new form of equivariant cohomology h which agrees with the usual periodic equivariant cohomology in finite dimensions and we prove a suitable analogue of the classical fixed point theorem which is valid for loop spaces lx this gives a cohomological framework for studying differential forms on loop spaces and we apply these methods to various questions which arise from the work of witten 16 atiyah 2 and bismut 5 in particular we show following atiyah in 2 that the a polynomial of x arises as an equivariant characteristic class in the theory h of the normal bundle to x considered as the space of constant loops in lx

bifurcations in the lozi map we study the presence in the lozi map of a type of abrupt order to order and order to chaos transitions which are mediated by an attractor made of a continuum of neutrally stable limit cycles all with the same period

orientability for gauge theories on calabi yau manifolds abstract we study orientability issues of moduli spaces from gauge theories on calabi yau manifolds our results generalize and strengthen those for donaldson thomas theory on calabi yau manifolds of dimensions 3 and 4 we also prove a corresponding result in the relative situation which is relevant to the gluing formula in dt theory

on the semi classical analysis of the ground state energy of the dirichlet pauli operator we complete and improve the results of a recent paper by ekholm kova ik and portmann in connection with a question of c guillarmou about the semiclassical expansion of the lowest eigenvalue of the pauli operator with dirichlet conditions we exhibit connections with the properties of the torsion function in mechanics the exit time of a brownian motion and the analysis of the low eigenvalues of some witten laplacian

stochastic formulation of the renormalization group supersymmetric structure and topology of the space of couplings the exact or wilson renormalization group equations can be formulated as a functional fokker planck equation in the infinite dimensional configuration space of a field theory suggesting a stochastic process in the space of couplings indeed the ordinary renormalization group differential equations can be supplemented with noise making them stochastic langevin equations furthermore if the renormalization group is a gradient flow the space of couplings can be endowed with a supersymmetric structure a la parisi sourlas the formulation of the renormalization group as supersymmetric quantum mechanics is useful for analysing the topology of the space of couplings by means of morse theory we present simple examples with one or two couplings

hide and seek games on a network using combinatorial search paths this paper introduces a new search paradigm to hide and seek games on networks the hider locates at any point on any arc the searcher adopts a combinatorial path when searching the network a sequence of arcs each adjacent to the last and traced out at unit speed in previous literature the searcher was allowed simple motion any unit speed path including ones that turn around inside an arc the new approach more closely models real problems such as search for improvised explosive devices using vehicles that can only turn around at particular locations on a road the search game is zero sum with the time taken to find the hider as the payoff using a lemma giving an upper bound for the expected search time on a semi eulerian network we solve the search game on a network q3 consisting of two nodes connected by three arcs of arbitrary lengths when two q3 networks with unit length arcs are linked by two small central arcs incident at the start node one of these arcs must be traversed at least th

supersymmetric path integrals the supersymmetric path integral is constructed for quantum mechanical models on flat space as a supersymmetric extension of the wiener integral it is then pushed forward to a compact riemannian manifold by means of a malliavin type construction the relation to index theory is discussed

weakly non local fluid mechanics the schrodinger equation a weakly non local extension of ideal fluid dynamics is derived from the second law of thermodynamics it is proved that in the reversible limit the additional pressure term can be derived from a potential the requirement of the additivity of the specific entropy function determines the quantum potential uniquely the relation to other known derivations of the schrodinger equation stochastic fisher information exact uncertainty is clarified

new topological field theories in two dimensions it is shown that two 1 1 dimensional 2d free abelian and self interacting non abelian gauge theories without any interaction with matter fields belong to a new class of topological field theories tfts these new theories capture together some of the key features of witten and schwarz types of tft because they are endowed with symmetries that are reminiscent of the schwarz type theories but their lagrangian density has the appearance of the witten type theories the topological invariants for these theories are computed on a 2d compact manifold and their recursion relations are obtained these new theories are shown to provide a class of tractable field theoretical models for the hodge theory in two dimensions of flat minkowski spacetime where there are no propagating degrees of freedom associated with the 2d gauge boson

equivariant brst quantization and reducible symmetries working from first principles quantization of a class of hamiltonian systems with reducible symmetry is carried out by constructing first the appropriate reduced phase space and then the brst cohomology the constraints of this system correspond to a first class set for a group g and a second class set for a subgroup h the brst operator constructed is equivariant with respect to h using algebraic techniques analogous to those of equivariant de rham theory the brst operator is shown to correspond to that obtained by bv quantization of a class of systems with reducible symmetry the ghosts for ghosts correspond to the even degree two generators in the cartan model of equivariant cohomology as an example of the methods developed a topological model is described whose brst quantization relates to the equivariant cohomology of a manifold under a circle action

spreading lengths of hermite polynomials the renyi shannon and fisher spreading lengths of the classical or hypergeometric orthogonal polynomials which are quantifiers of their distribution all over the orthogonality interval are defined and investigated these information theoretic measures of the associated rakhmanov probability density which are direct measures of the polynomial spreading in the sense of having the same units as the variable share interesting properties invariance under translations and reflections linear scaling and vanishing in the limit that the variable tends towards a given definite value the expressions of the renyi and fisher lengths for the hermite polynomials are computed in terms of the polynomial degree the combinatorial multivariable bell polynomials which are shown to characterize the finite power of an arbitrary polynomial play a relevant role for the computation of these information theoretic lengths indeed these polynomials allow us to design an error free computing approach for the entropic moments weighted l q norms of hermite polynomials and subsequently for the renyi and tsallis entropies as well as for the renyi spreading lengths sharp bounds for the shannon length of these polynomials are also given by means of an information theoretic based optimization procedure moreover the existence of a linear correlation between the shannon length as well as the second order renyi length and the standard deviation is computationally proved finally the application to the most popular quantum mechanical prototype system the harmonic oscillator is discussed and some relevant asymptotical open issues related to the entropic moments mentioned previously are posed

search games on trees with asymmetric travel times a point h is hidden in a rooted tree q which is endowed with asymmetric distances travel times between nodes we determine the randomized search strategy starting from the root which minimizes the expected time to reach h in the worst case this is equivalent to a zero sum search game gamma left q right with minimizing searcher maximizing hider and payoff equal to the capture time the worst hiding distribution over the leaves from the searcher s viewpoint is one where at every node i the probability of each branch is proportional to the minimum time required to tour it from i the optimal randomized search is a mixture over depth first searches we also consider briefly some other networks and the possibility of a mobile hider our formulation with asymmetric travel times generalizes that of gal siam j control optim 17 1979 pp 99 122 for symmetric travel times and also the search games of kikuta j oper res 38 1995 pp 70 88 and kikuta and ruckle naval res logist 41 1994 pp 821 831 who posited search costs c i at each node i which were added to the travel time to obtain the payoff we also briefly consider what happens if we allow the searcher hider to start hide at any leaf node we determine when properties found by dagan and gal networks 52 2008 pp 156 161 for the symmetric version of such games hold in our asymmetric context

a mixed strategy minimax theorem without compactness minimax theorems for infinite games generally require that both players choose their pure strategies from compact sets and have semicontinuity requirements in both variables this paper proves the following theorem let x be a compact hausdorfl space and let y a be a measurable space let f x times y be a measurable function which is bounded below and lower semicontinuous on x for all fixed y in y let m be any convex set of probability measures mixed strategies on y a then mathop min limits mu in b x mathop sup limits rho in m iint f x y d mu d rho mathop sup limits rho in m mathop min limits mu in b x iint f x y d mu d rho where b x denotes the regular borel probability measures on x

localization formulas superconnections and the index theorem for families in this paper we give a new proof of the localization formulas of berline and vergne 9 and duistermaat and heckman 18 when interpreted in the framework of atiyah 2 the probabilistic heat equation proof of the index theorem given in our paper 12 appears as the rigorous infinite dimensional version of this new proof of the localization formulas in finite dimensions the results of quillen 25 on superconnections are briefly presented the heat equation proofs 15 of the index theorem for families are described it is shown that in this framework the superconnections formalism is the operator theoretic description of integration along the fiber in the loop space

supersymmetry and the formal loop space abstract for any algebraic super manifold m we define the super ind scheme lm of formal loops and study the transgression map radon transform on differential forms in this context applying this to the super manifold m s x the spectrum of the de rham complex of a manifold x we obtain in particular that the transgression map for x is a quasi isomorphism between the 2 3 truncated de rham complex of x and the additive part of the 1 2 truncated de rham complex of l x the proof uses the super manifold ss x and the action of the lie super algebra sl 1 2 on this manifold this quasi isomorphism result provides a crucial step in the classification of sheaves of chiral differential operators in terms of geometry of the formal loop space

nonlocal invariants in index theory in its original form the atiyah singer index theorem equates two global quantities of a closed manifold one analytic the index of an elliptic operator and one topological a characteristic number because it relates invariants from different branches of mathematics the index theorem has many applications and extensions to differential geometry k theory mathematical physics and other fields this report focuses on advances in geometric aspects of index theory for operators naturally associated to a riemannian metric on a closed manifold the topological side of the index theorem can often be expressed as the integral of local i e pointwise curvature expression we will first discuss these local refinements in 1 which arise naturally in heat equation proofs of the index theorem in 2 3 we discuss further developments in index theory which lead to spectral invariants the eta invariant and the determinant of an elliptic operator that are definitely nonlocal finally in 4 we point out some recent connections among these nonlocal invariants and classical index theory

unique additive information measures boltzmann gibbs shannon fisher and beyond abstract it is proved that the only additive and isotropic information measure that can depend on the probability distribution and also on its first derivative is a linear combination of the boltzmann gibbs shannon and fisher information measures power law equilibrium distributions are found as a result of the interaction of the two terms the case of second order derivative dependence is investigated and a corresponding additive information measure is given

a rigorous path integral for supersymmetic quantum mechanics and the heat kernel in a rigorous construction of the path integral for supersymmetric quantum mechanics on a riemann manifold based on bar and pfaffle s use of piecewise geodesic paths the kernel of the time evolution operator is the heat kernel for the laplacian on forms the path integral is approximated by the integral of a form on the space of piecewise geodesic paths which is the pullback by a natural section of mathai and quillen s thom form of a bundle over this space in the case of closed paths the bundle is the tangent space to the space of geodesic paths and the integral of this form passes in the limit to the supertrace of the heat kernel

geometry of the schrodinger equation and stochastic mass transportation the schrodinger equation is deduced from a geometric principle lagrange hamilton jacobi and hamilton formalisms are defined almost analogously to the deterministic case which can be identified as geometric optic the form of these formalisms is identical to the deterministic formalisms furthermore it will be shown how the deterministic case is superficial to the stochastic one if planck s constant is very small an elementary proof of heisenberg s uncertainty relation finishes the paper

convergent expansions for tunneling a new method to compute effects of tunneling in one dimensional multiple well is developed a tunneling parameter built with physical quantities is introduced to measure the splitting between eigenvalues due to tunneling these splittings are given by convergent series in term of this tunneling parameter for a wide class of double well