honey pot constrained searching with local sensory information in this paper we investigate the problem of searching for a hidden target in a bounded region of the plane using an autonomous robot which is only able to use local sensory information the problem is naturally formulated in the continuous domain but the solution proposed is based on an aggregation refinement approach in which the continuous search space is partitioned into a finite collection of regions on which we define a discrete search problem a solution to the original problem is obtained through a refinement procedure that lifts the discrete path into a continuous one we show that the discrete optimization is computationally difficult np hard but there are computationally efficient approximation algorithms for solving it the resulting solution to the continuous problem is in general not optimal but one can construct bounds to gauge the cost penalty incurred due to i the discretization of the problem and ii the attempt to approximately solve the np hard problem in polynomial time numerical simulations show that the algorithms proposed behave significantly better than naive approaches such as a random walk or a greedy algorithm c 2006 elsevier ltd all rights reserved

twisting the fake monster superalgebra abstract we calculate the twisted denominator identities of the fake monster superalgebra corresponding to elements of 2 aut e 8 of odd order they give many new infinite product identities and new examples of generalized kac moody superalgebras we also obtain generalizations of jacobi s identity on theta functions

concentrations of solutions to time discretizied compressible navier stokes equations the compactness properties of solutions to time discretization of compressible navier stokes equations are investigated in three dimensions the existence of generalized solutions is established

on the incompressible fluid limit and the vortex motion law of the nonlinear schrodinger equation the nonlinear schrodinger equation nls has been a fundamental model for understanding vortex motion in superfluids the vortex motion law has been formally derived on various physical grounds and has been around for almost half a century we study the nonlinear schrodinger equation in the incompressible fluid limit on a bounded domain with dirichlet or neumann boundary condition the initial condition contains any finite number of degree 1 vortices we prove that the nls linear momentum weakly converges to a solution of the incompressible euler equation away from the vortices if the initial nls energy is almost minimizing we show that the vortex motion obeys the classical kirchhoff law for fluid point vortices similar results hold for the entire plane and periodic cases and a related complex ginzburg landau equation we treat as well the semi classical wkb limit of nls in the presence of vortices in this limit sound waves propagate through steady vortices

hardy spaces and the two dimensional euler equations with nonnegative vorticity we prove that certain quadratic expressions involving the gradient of a weakly superharmonic function in r belong to a local hardy space as an application we provide a new proof of j m delort s convergence theorem for solutions of the two dimensional euler equations with vorticities of one sign department of mathematics university of california berkeley california 94720 sfb 256 institut fur angewandte mathematik beringstrasse 4 d5300 bonn 1 ger many this content downloaded from 207 46 13 101 on sat 08 oct 2016 05 44 35 utc all use subject to http about jstor org terms

convergence of the viscosity method for isentropic gas dynamics a convergence theorem for the method of artificial viscosity applied to the isentropic equations of gas dynamics is established convergence of a subsequence in the strong topology is proved without uniform estimates on the derivatives using the theory of compensated compactness and an analysis of progressing entropy waves

renormalized solutions of the 2d euler equations in this paper we prove that solutions of the 2d euler equations in vorticity formulation obtained via vanishing viscosity approximation are renormalized

the complexity of the optimal searcher path problem in this note we show that the problem of finding an optimal searcher path that maximizes the probability of detecting a stationary target by the end of a fixed time is np complete we also demonstrate that the problem of finding a path that minimizes mean time to detection is np hard

oscillations and concentrations in weak solutions of the incompressible fluid equations the authors introduce a new concept of measure valued solution for the 3 d incompressible euler equations in order to incorporate the complex phenomena present in limits of approximate solutions of these equations one application of the concepts developed here is the following important result a sequence of leray hopf weak solutions of the navier stokes equations converges in the high reynolds number limit to a measure valued solution of 3 d euler defined for all positive times the authors present several explicit examples of solution sequences for 3 d incompressible euler with uniformly bounded local kinetic energy which exhibit complex phenomena involving both persistence of oscillations and development of concentrations an extensions of the concept of young measure is developed to incorporate these complex phenomena in the measure valued solutions constructed here

on the kodaira dimension of orthogonal modular varieties we prove that up to scaling there are only finitely many integral lattices l of signature 2 n with n ge 21 or n 17 such that the modular variety defined by the orthogonal group of l is not of general type in particular when n ge 108 every modular variety defined by an arithmetic group for a rational quadratic form of signature 2 n is of general type we also obtain similar finiteness in n ge 9 for the stable orthogonal groups as a byproduct we derive finiteness of lattices admitting reflective modular form of bounded vanishing order which proves a conjecture of gritsenko and nikulin

an optimal branch and bound procedure for the constrained path moving target search problem a searcher and target move among a finite set of cells c 1 2 n in discrete time at the beginning of each time period one cell is searched if the target is in the selected cell j it is detected with probability qj if the target is not in the cell searched it cannot be detected during the current time period after each search a target in cell j moves to cell k with probability pjk the target transition matrix p pjk is known to the searcher the searcher s path is constrained in that if the searcher is currently in cell j the next search cell must be selected from a set of neighboring cells cj the object of the search is to minimize the probability of not detecting the target in t searches

existence and uniqueness of solutions of an asymptotic equation arising from a variational wave equation with general data we establish here the global existence and uniqueness of admissible both dissipative and conservative weak solutions to a canonical asymptotic equation for weakly nonlinear solutions of a class of nonlinear variational wave equations with any l 2 initial datum we use the method of young measures and mollification techniques

a criterion for the equivalence of the birkhoff rott and euler descriptions of vortex sheet evolution in this article we consider the evolution of vortex sheets in the plane both as a weak solution of the two dimensional incompressible euler equations and as a weak solution of the birkhoff rott equations we begin by discussing the classical birkhoff rott equations with respect to arbitrary parametrizations of the sheet we introduce a notion of weak solution to the birkhoff rott system and we prove consistency of this notion with the classical formulation of the equations our main purpose in this paper is to present a sharp criterion for the equivalence of the weak euler and weak birkhoff rott descriptions of vortex sheet dynamics

vorticity measures and the inviscid limit we consider a sequence of leray hopf weak solutions of the 2d navier stokes equations on a bounded domain in the vanishing viscosity limit we provide sufficient conditions on the associated vorticity measures away from the boundary which ensure that as the viscosity vanishes the sequence converges to a weak solution of the euler equations the main assumptions are local interior uniform bounds on the l 1 norm of vorticity and the local uniform convergence to zero of the total variation of vorticity measure on balls in the limit of vanishing ball radii

un phenomene de concentration evanescente pour des flots non stationnaires incompressibles en dimension deux we consider a sequenceve of non stationary solutions of the incompressible 2d euler equation locally bounded inl2 we prove that if the defect measure is supported in a one dimensional set 3 of some special type which we call finite type the weak limitv ofve is a solution of the euler equations our theorem is of the type concentration cancellation

global regularity and convergence of a birkhoff rott α approximation of the dynamics of vortex sheets of the two dimensional euler equations we present an regularization of the birkhoff rott equation br equation induced by the two dimensional euler equations for the vortex sheet dynamics we show the convergence of the solutions of euler equations to a weak solution of the euler equations for initial vorticity being a finite radon measure of fixed sign which includes the vortex sheets case we also show that provided the initial density of vorticity is an integrable function over the curve with respect to the arc length measure i an initially lipschitz chord arc vortex sheet curve evolving under the br equation remains lipschitz for all times ii an initially holder c1 0 1 chord arc curve remains in c1 for all times and finally iii an initially holder cn n 1 0 1 closed chord arc curve remains so for all times in all these cases the weak euler and the br descriptions of the vortex sheet motion are equivalent 2009 wiley periodicals inc

existence of vortex sheets with reflection symmetry in two space dimensions the main purpose of this work is to establish the existence of a weak solution to the incompressible 2d euler equations with initial vorticity consisting of a radon measure with distinguished sign in h 1 compactly supported in the closed right half plane superimposed on its odd reflection in the left half plane we make use of a new a priori estimate to control the interaction between positive and negative vorticity at the symmetry axis we prove that a weak limit of a sequence of approximations obtained by either regularizing the initial data or by using the vanishing viscosity method is a weak solution of the incompressible 2d euler equations we also establish the equivalence at the level of weak solutions between mirror symmetric flows in the full plane and flows in the half plane finally we extend our existence result to odd l 1 perturbations without distinguished sign of our original initial vorticity

on the euler equations of incompressible fluids euler equations of incompressible fluids use and enrich many branches of mathematics from integrable systems to geometric analysis they present important open physical and mathematical problems examples include the stable statistical behavior of ill posed free surface problems such as the rayleightaylor and kelvin helmholtz instabilities the paper describes some of the open problems related to the incompressible euler equations with emphasis on the blowup problem the inviscid limit and anomalous dissipation some of the recent results on the quasigeostrophic model are also mentioned

reduced hausdorff dimension and concentration cancellation for two dimensional incompressible flow where v t v1 v2 is the fluid velocity p is the scalar pressure dv dt av at v v v and vo is an initial incompressible velocity field i e div vo 0 in this paper we study the detailed limiting behavior of approximate solution sequences for 2 d euler with vortex sheet initial data a sequence of smooth velocity fields v8 x t is an approximate solution sequence for 2 d euler provided that v is incompressible i e div v 0 and satisfies the following properties 1 the velocity fields v6 have uniformly bounded local kinetic energy i e

concentrations in the one dimensional vlasov poisson equations i temporal development and non unique weak solutions in the single component case abstract weak solutions of the one component vlasov poisson equation in a single space dimension are proposed and studied here as a simpler analogue problem for the behavior of weak solutions of the two dimensional incompressible euler equations with non negative vorticity the physical structural and functional analytic analogies between these two problems are developed in detail here with this background explicit solutions for electron sheet initial data the analogue of vortex sheet initial data are presented which display the phenomena of singularity formation in finite time as well as the explicit temporal development of charge concentrations other rigorous explicit examples with charge concentration are developed where there are non unique weak solutions with the same initial data in one of these non unique weak solutions an electron sheet completely collapses to a point charge in finite time the detailed limiting behavior of regularizations such as the diffusive fokker planck equation are developed through a very efficient numerical method which yields extremely high resolution for these simpler analogue problems a striking consequences of the numerical results reported here is the fact that there is not a selection principle for a unique weak solution in some situations where there are several weak solutions with charge concentration for the same initial data in particular two explicit weak solutions with the same initial data are constructed here where it is demonstrated that the zero smoothing limit of time reversible particle methods converges to one of these solutions while the zero diffusion limit of the fokker planck equation converges to the other weak solution

functional and measure valued solutions of the euler equations for flows of incompressible fluids we consider the notion of a functional solution of the euler equations for incompressible fluid flows we show that a functional solution can be constructed under very weak a priori estimates on approximate solution sequences of the equation an estimate uniform in lloc1 together with weak consistency with the equation is sufficient to construct a solution we prove that if we have an estimate uniform in lloc2 available for the approximate solution sequence then the structured functional solution just described becomes a measure valued solution in the sense of diperna majda we also show that a functional solution can be obtained from a measure valued solution we give an example showing that a much higher concentration of energy than in the case of measure valued solutions is allowed by the approximation procedure of a functional solution

detection of an anomalous cluster in a network we consider the problem of detecting whether or not in a given sensor network there is a cluster of sensors which exhibit an unusual behavior formally suppose we are given a set of nodes and attach a random variable to each node we observe a realization of this process and want to decide between the following two hypotheses under the null the variables are i i d standard normal under the alternative there is a cluster of variables that are i i d normal with positive mean and unit variance while the rest are i i d standard normal we also address surveillance settings where each sensor in the network collects information over time the resulting model is similar now with a time series attached to each node we again observe the process over time and want to decide between the null where all the variables are i i d standard normal and the alternative where there is an emerging cluster of i i d normal variables with positive mean and unit variance the growth models used to represent the emerging cluster are quite general and in particular include cellular automata used in modelling epidemics in both settings we consider classes of clusters that are quite general for which we obtain a lower bound on their respective minimax detection rate and show that some form of scan statistic by far the most popular method in practice achieves that same rate within a logarithmic factor our results are not limited to the normal location model but generalize to any one parameter exponential family when the anoma lous clusters are large enough

an optimal algorithm for euclidean shortest paths in the plane we propose an optimal time algorithm for a classical problem in plane computational geometry computing a shortest path between two points in the presence of polygonal obstacles our algorithm runs in worst case time o n log n and requires o n log n space where n is the total number of vertices in the obstacle polygons the algorithm is based on an efficient implementation of wavefront propagation among polygonal obstacles and it actually computes a planar map encoding shortest paths from a fixed source point to all other points of the plane the map can be used to answer single source shortest path queries in o log n time the time complexity of our algorithm is a significant improvement over all previously published results on the shortest path problem finally we also discuss extensions to more general shortest path problems involving nonpoint and multiple sources

concentration cancellation for the velocity fields in two dimensional incompressible fluid flows we show that the weak l2 limit of a sequence of solutions of the two dimensional incompressible euler equation is still a solution provided that a strong concentration set for the reduced defect measure has locally finite one dimensional hausdorff measure in space and time

on a weak solution for a transonic flow problem on essaie d etablir l existence de solutions faibles pour un ecoulement stationnaire bidimensionnel irrotationnel compressible non visqueux dans un canal de longueur infinie ou autour d un profil aerodynamique la vitesse de l ecoulement etant donnee a l infini

the optimal search for a moving target when the search path is constrained a search is conducted for a target moving in discrete time among a finite number of cells according to a known markov process the searcher must choose one cell in which to search in each time period the set of cells available for search depends upon the cell chosen in the last time period the problem is to find a searcher path i e a sequence of search cells that maximizes the probability of detecting the target in a fixed number of time periods we formulate the problem as a partially observable markov decision process and present a finite time horizon pomdp solution technique which is simpler than the standard linear programming methods

hamilton paths in grid graphs a grid graph is a node induced finite subgraph of the infinite grid it is rectangular if its set of nodes is the product of two intervals given a rectangular grid graph and two of its nodes we give necessary and sufficient conditions for the graph to have a hamilton path between these two nodes in contrast the hamilton path and circuit problem for general grid graphs is shown to be np complete this provides a new relatively simple proof of the result that the euclidean traveling salesman problem is np complete

on strong convergence to 3d steady vortex sheets in this paper we first establish a strong convergence criterion of approximate solutions for the 3d steady incompressible euler equations for axisymmetric flows under the assumption that the vorticity is of one sign and uniformly bounded in l 1 space we obtain a sufficient and necessary condition for the strong convergence in l 2 r 3 of approximate solutions furthermore for one sign and l 1 bounded vorticity it is shown that if a sequence of approximate solutions concentrates at an isolated point in r z plane then the concentration point can appear neither in the region near the axis including the symmetry axis itself nor in the region far away from the axis finally we present an example of approximates solutions which converge strongly in l 2 r 3 by using hill s spherical vortex

discrete sequential search with positive switch cost suppose one object is hidden in one of n boxes according to a known probability distribution p1 pn the boxes are to be searched sequentially the probability of overlooking the object if we search box k and if the object is in box k is denoted by qk the cost of a search of box k amounts to ck if we decide to search box k directly after box k we have to pay some extra cost called switch cost tk k the problem is under some reasonable assumptions to construct a strategy with minimal expected cost if t 0 the problem has a well known solution unfortunately there are many reasons which establish the conjecture that there does not exist an easy construction rule for optimal strategies in the general case we give necessary and sufficient conditions for the existence of optimal strategies which are ultimately periodic u p since we state bounds on the length not only of the transient phase but also of the period of optimal u p strategies we may construct these optimal strategies by comparing finitely many strategies if optimal u p strategies exist we prove some other results to reduce the number of strategies which we have to compare at last we present a numerical example

a construction of antisymmetric modular forms for weil representations we give coefficient formulas for antisymmetric vector valued cusp forms with rational fourier coefficients for the weil representation associated to a finite quadratic module the forms we construct always span all cusp forms in weight at least three these formulas are useful for computing explicitly with theta lifts

lagrangian solutions to the 2d euler system with l1 vorticity and infinite energy we consider solutions to the two dimensional incompressible euler system with only integrable vorticity thus with possibly locally infinite energy with such regularity we use the recently developed theory of lagrangian flows associated to vector fields with gradient given by a singular integral in order to define lagrangian solutions for which the vorticity is transported by the flow we prove strong stability of these solutions via strong convergence of the flow under the only assumption of l1 weak convergence of the initial vorticity the existence of lagrangian solutions to the euler system follows for arbitrary l1 vorticity relations with previously known notions of solutions are established

a remark on vorticity maximal function in this note the following inequality is proved for any nonnegative measure h 1 r2 x r2 and 0 r 1 there holds 1 b x r c ln1r 12 h 1 where c is a positive constant using 1 an estimate for the vorticity maximal function similar to the estimate in majda a majda remarks on weak solutions for vortex sheets with a distinguished sign indiana univ math j 42 1993 921 939 is established without assuming the initial vorticity having compact support from this a more simple proof of the delort s celebrated theorem j m delort existence de mappes de fourbillon en dimension deux j amer math soc 4 1991 553 586 is presented

on diperna majda concentration sets for two dimensional incompressible flow abstract the existence of weak solutions to the incompressible euler equations for vortex sheet initial data is an open problem two approaches to this problem are the smoothing of vortex sheets by approximated identifies which provides a sequence of smooth initial conditions converging to vortex sheet initial conditions for which classical solutions exist and viscous smoothing of vortex sheets these ideas are among the principal motivations for the recent series of papers in which the nature of the limiting behavior of sequences of solutions of euler s equations are examined the concept of generalized young measure valued solution of the euler equations is introduced in two dimensions approximate solution sequences always have subsequences which converge in an appropriate sense to non oscillatory generalized young measure valued solutions of euler s equations infinitesimal vortices of zero circulation were described in and named phantom vortices the vortices in our example are somewhat different in that they have two length scales the two length scales are important for one cannot pack phantom vortices on successively finer lattices as above keeping the vorticity absolutely summable without at the same time obtaining strong convergence in l sq reprints

discounted mean bound for the optimal searcher path problem with non uniform travel times we consider an extension of the optimal searcher path problem osp where a searcher moving through a discretised environment may now need to spend a non uniform amount of time travelling from one region to another before being able to search it for the presence of a moving target in constraining not only where but when the search of each cell can take place the problem more appropriately models the search of environments which cannot be easily partitioned into equally sized cells an existing osp bounding method in literature the mean bound is generalised to provide bounds for solving the new problem in a branch and bound framework the main contribution of this paper is an enhancement discounted mean dmean which greatly tightens the bound for the new and existing problems alike with almost no additional computation we test the new algorithm against existing osp bounding methods and show it leads to faster solution times for moving target search problems

a singular perturbation approach to modeling and control of markov chains finite state continuous time markov processes with weak interactions are modeled as singularly perturbed systems aggregate states are obtained using a grouping algorithm two time scale expansions simplify cost equations and lead to decentralized optimization algorithms

concentrations in the one dimensional vlasov poisson equations ii screening and the necessity for measure valued solutions in the two component case abstract weak and measure valued solutions for the two component vlasov poisson equations in a single space dimension are proposed and studied here as a simpler analogue problem for the limiting behavior of approximations for the two dimensional euler equations with general vorticity of two signs from numerical experiments and mathematical theory it is known that much more complex behavior can occur in limiting processes for vortex sheets with general vorticity of two signs as compared with non negative vorticity here such behavior is confirmed rigorously for the simpler analogue problem through explicit examples involving singular charge concentration for the two component vlasov poisson equations the concepts of measure valued and weak solution are introduced explicit examples with charge concentration establish that the limit of weak solutions in a dynamic process is necessarily a measure valued solution in some cases rather than the anticipated weak solution i e no concentration cancellation occurs the limiting behavior of computational regularizations involving high resolution particle methods is presented here both for the instances with measure valued solutions and also for new examples with non unique weak solutions the authors demonstrate that different computational regularizations can exhibit completely different limiting behavior in situations with measure valued and or non unique weak solutions

on strong convergence to 3 d axisymmetric vortex sheets abstract we consider the 3 d axisymmetric incompressible euler equations without swirls with vortex sheets initial data it is proved that the approximate solutions generated by smoothing the initial data converge strongly in l 2 0 t l loc 2 r 3 provided that they have strong convergence in the region away from the symmetry axis this implies that if there would appear singularity or energy lost in the process of limit for the approximate solutions it then must happen in the region away from the symmetry axis there is no restriction on the signs of initial vorticity here in order to exclude the possible concentrations on the symmetry axis we use the special structure of the equations for axisymmetric flows and careful choice of test functions

smooth approximations and exact solutions of the 3d steady axisymmetric euler equations in this paper we prove that a class of c1 smooth approximate solutions ue pe to the 3d steady axisymmetric euler equations will converge strongly to 0 in l 2 loc r 3 the main assumptions are that the approximate solutions have uniformly finite energy and approach a constant state at far fields we also show a liouville type theorem that there are no non trivial c1 smooth exact solutions with finite energy and uniform constant state at far fields

automorphic products of singular weight for simple lattices we classify the simple even lattices of square free level and signature 2 n n ge 4 a lattice is called simple if the space of cusp forms of weight 1 n 2 for the dual weil representation of the lattice is trivial for a simple lattice every formal principal part obeying obvious conditions is the principal part of a vector valued modular form using this we determine all holomorphic borcherds products of singular weight arising from vector valued modular forms with non negative principal part for the simple lattices we construct the corresponding vector valued modular forms by eta products and compute expansions of the automorphic products at different cusps