distributed coverage games for energy aware mobile sensor networks inspired by current challenges in data intensive and energy limited sensor networks we formulate a coverage optimization problem for mobile sensors as a constrained repeated multiplayer game each sensor tries to optimize its own coverage while minimizing the processing energy cost the sensors are subject to the informational restriction that the environmental distribution function is unknown a priori we present two distributed learning algorithms where each sensor only remembers its own utility values and actions played during the last plays these algorithms are proven to be convergent in probability to the set of constrained nash equilibria and global optima of a certain coverage performance metric respectively numerical examples are provided to verify the performance of our proposed algorithms

on the partitioning capabilities of feedforward neural networks with sigmoid nodes in this letter the capabilities of feedforward neural networks fnns on the realization and approximation of functions of the form g rl a which partition the rl space into polyhedral sets each one being assigned to one out of the c classes of a are investigated more specifically a constructive proof is given for the fact that fnns consisting of nodes having sigmoid output functions are capable of approximating any function g with arbitrary accuracy also the capabilities of fnns consisting of nodes having the hard limiter as output function are reviewed in both cases the two class as well as the multiclass cases are considered

coverage control by multi robot networks with limited range anisotropic sensory this article considers the deployment of a network of robotic agents with limited range communication and anisotropic sensing capabilities we encode the environment coverage provided by the network by means of an expected value objective function this function has a gradient which is not amenable to distributed computation we provide a constant factor approximation of this measure via an alternative aggregate objective function whose gradient is spatially distributed over the limited range delaunay proximity graph we characterise the smoothness properties of the aggregate expected value function and propose a distributed deployment algorithm that enables the network to optimise it simulations illustrate the results

distributed collaborative coverage control schemes for non convex domains the coverage control problem for mobile sensor networks in non convex environments is addressed in this article the sensing model of each node is assumed identical and radial either in the geodesic or the euclidean sense depending on the selected metric the partitioning of the non convex domain can result either in the geodesic or the euclidean voronoi diagram the coverage problem is examined from a two fold aspect based on a proper selection of the sensing model and b the partitioning scheme as directed by the application itself distributed collaborative control schemes are developed further validated via extensive numerical studies

on the existence and profile of nodal solutions for a two dimensional elliptic problem with large exponent in nonlinearity we study the existence of nodal solutions to the boundary value problem u u p 1u in a bounded smooth domain in ir with homogeneous dirichlet boundary condition when p is a large exponent we prove that for p large enough there exist at least two pairs of solutions which change sign exactly once and whose nodal lines intersect the boundary of

global compactness properties of semilinear elliptic equations with critical exponential growth abstract sequences of positive solutions to semilinear elliptic equations of critical exponential growth in the plane either are precompact in the sobolev h 1 topology or concentrate at isolated points of the domain for energies allowing at most single point blow up we establish a universal blow up pattern near the concentration point and uniquely characterize the blow up energy in terms of a geometric limiting problem

the role of the green s function in a non linear elliptic equation involving the critical sobolev exponent this paper is concerned with non linear elliptic problems of the type pe u u n 2 n 2 eu u 0 on u 0 on where is a smooth and bounded domain in rn n 4 and e 0 we show that if the ue are solutions of pe which concentrate around a point as e 0 then this point cannot be on the boundary of and is a critical point of the regular part of the green s function conversely we show that for n 5 and any non degenerate critical point x0 of the regular part of the green s function there exist solutions of pe concentrating around x0 as e 0

nonuniform line coverage from noisy scalar measurements we study the problem of distributed coverage control in a network of mobile agents arranged on a line the goal is to design distributed dynamics for the agents to achieve optimal coverage positions with respect to a scalar density field that measures the relative importance of each point on the line unlike previous work which implicitly assumed the agents know this density field we only assume that each agent can access noisy samples of the field at points close to its current location we provide a simple randomized protocol wherein every agent samples the scalar field at three nearby points at each step and which guarantees convergence to the optimal positions we further analyze the convergence time of this protocol and show that under suitable assumptions the squared distance to the optimal coverage configuration decays as o 1 t with the number of iterations t where the constant scales polynomially with the number of agents n we illustrate these results with simulations

distributed partitioning algorithms for multi agent networks with quadratic proximity metrics and sensing constraints abstract we consider a class of voronoi like partitioning problems in which a multi agent network seeks to subdivide a subset of an affine space into a finite number of cells in the presence of sensing constraints the cell of this subdivision that is assigned to a particular agent consists exclusively of points that can be sensed by this agent and are closer to it than to any other agent that can also sense them the proximity between an agent and an arbitrary point is measured in terms of a non homogeneous quadratic generalized distance function which does not in general enjoy the triangle inequality and the symmetry property one of the consequences of this fact is that the structure of the sublevel sets of the utilized proximity metric does not conform with that of the sensing region of an agent due to this mismatch it is possible that a point may be assigned to an agent which is different from its nearest agent simply because the nearest agent cannot sense this point unless special care is taken we propose a distributed partitioning algorithm that enables each agent to compute its own cell independently from the other agents when the only information available to it is the positions and the velocities of the agents that lie inside its sensing region the algorithm is based on an iterative process that adjusts the size of the sensing region of each agent until the associated cell of the latter corresponds to the intersection of its sensing region with the cell that would have been assigned to it in the absence of sensing constraints the correctness of the proposed distributed algorithm which successfully handles the aforementioned issues is studied in detail

new solutions for trudinger moser critical equations in r2 abstract let be a bounded smooth domain in r 2 we consider critical points of the trudinger moser type functional j u 1 2 u 2 2 e u 2 in h 0 1 namely solutions of the boundary value problem u u e u 2 0 with homogeneous dirichlet boundary conditions where 0 is a small parameter given k 1 we find conditions under which there exists a solution u which blows up at exactly k points in as 0 and j u 2 k we find that at least one such solution always exists if k 2 and is not simply connected if has d 1 holes in addition d 1 bubbling solutions with k 1 exist these results are existence counterparts of one by druet in o druet multibump analysis in dimension 2 quantification of blow up levels duke math j 132 2 2006 217 269 which classifies asymptotic bounded energy levels of blow up solutions for a class of nonlinearities of critical exponential growth including this one as a prototype case

weak solutions to a parabolic elliptic system of chemotaxis abstract we study a parabolic elliptic system of partial differential equations which describes the chemotactic feature of slime molds it is known that the blowup solution forms singularities such as delta functions referred to as the collapses here we study the case that the domain is a flat torus and show that the post blowup continuation of the solution is possible only when those collapses are quantized with the mass 8

concentrating solutions for an exponential nonlinearity with robin boundary condition abstract we study the following robin boundary value problem 0 1 u 2 u p 1 e u p 0 u 0 in u n u 0 on where is a bounded domain in r 2 with smooth boundary as 0 and simultaneously 0 p 2 by using a lyapunov schmidt reduction procedure we extend for the whole range of exponents 0 p 2 the results of davila and topp 2012 who constructed bubbling solutions to problem 0 1 for p 1

bubbling solutions for an exponential nonlinearity in r2 abstract we study the following boundary value problem 0 1 u u p 1 e u p 0 u 0 in u 0 on where is a bounded domain in r 2 with smooth boundary 0 is a small parameter and 0 p 2 we construct bubbling solutions to problem 0 1 using a lyapunov schmidt reduction procedure

blow up solutions for a liouville equation with henon term abstract we study the following boundary value problem 0 1 u x 2 u p 1 e u p 0 u 0 in u 0 on where is a bounded domain in r 2 with smooth boundary 0 n 0 is a small parameter and 0 p 2 we construct the bubbling solution to problem 0 1 which blows up at the origin and at some other m points

mixed interior and boundary bubbling solutions for neumann problem in r 2 let be a bounded domain in r2 with smooth boundary we consider the following problem u u up 1eup u 0 in with neumann boundary condition u 0 on where 0 is a small parameter 0 p 2 and denotes the outer normal vector to we construct solutions of this problem with k interior bubbling points and l boundary bubbling points for any k 1 and l 1

phenomena of critical exponent in ℝ2 in this paper we make an attempt to explain the critical phenomena in r 2 we do this by exhibiting a class of functions having e u 2 growth and for which u u in b r u 0 u 0 on b r do not admit a solution when r is sufficiently small where b r denotes the ball of radius r in r 2

the existence of non topological multivortex solutions in the relativistic self dual chern simons theory we construct a general type of multivortex solutions of the self duality equations the bogomol nyi equations of 2 1 dimensional relativistic chern simons model with the non topological boundary condition near infinity for such construction we use a perturbation argument around the explicit solutions of the liouville equation

existence and non existence of radial solutions for elliptic equations with critical exponent in ℝ2 where 2 2 n n 2 is the critical sobolev exponent for the embedding h 52 c l2 0 criticality and subcriticality play important rbles concerning the solvability of equation 1 1 using by now classical variational methods one sees that equation 1 1 is solvable i f f has subcritical growth and satisfies some additional hypotheses while for f u iuip u the well known pohoiaev identity shows that the solvability of equation 1 1 is lost if p reaches 2 and r is starshaped in their important paper 6 brezis nirenberg showed that for n e 4 solvability is regained if one adds a lower order linear term to the critical term more precisely they showed that for f r ar lr12 2r one has a denotes the 1st eigenvalue of a on ha r

separating the vertices of n cubes by hyperplanes and its application to artificial neural networks a new sufficient condition that a region be classifiable by a two layer feedforward network using threshold activation functions is found briefly it is either a convex polytope or that minus the removal of convex polytope from its interior or recursively the author refers to these sets as convex recursive deletion regions the proof of implementability exploits the equivalence of this problem with that of characterizing two set partitions of the vertices of a hypercube which are separable by a hyperplane for which a new result is obtained

multiple blow up solutions for an exponential nonlinearity with potential in r 2 abstract we study the following boundary value problem 0 1 u a x u p 1 e u p 0 u 0 in u 0 on where is a bounded domain in r 2 with smooth boundary 0 is a small parameter the function a x 0 is a smooth potential and the exponent p satisfies 0 p 2 we construct a family of solutions to problem 0 1 which blows up as 0 at some points of which stay outside the zero set of a x we relate the number of possible blow up points with the zero set of a x

concentrating solutions for a planar elliptic problem involving nonlinearities with large exponent abstract we consider the boundary value problem u u p 0 in a bounded smooth domain in r 2 with homogeneous dirichlet boundary condition and p a large exponent we find topological conditions on which ensure the existence of a positive solution u p concentrating at exactly m points as p in particular for a nonsimply connected domain such a solution exists for any given m 1

concentrating solutions in a two dimensional elliptic problem with exponential neumann data abstract we consider the elliptic equation u u 0 in a bounded smooth domain in r 2 subject to the nonlinear neumann boundary condition u e u here 0 is a small parameter we prove that any family of solutions u for which e u is bounded develops up to subsequences a finite number m of peaks i in the sense that e u 2 k 1 m i as 0 reciprocally we establish that at least two such families indeed exist for any given m 1

mapping boolean functions with neural networks having binary weights and zero thresholds in this paper the ability of a binary neural network comprising only neurons with zero thresholds and binary weights to map given samples of a boolean function is studied a mathematical model describing a network with such restrictions is developed it is shown that this model is quite amenable to algebraic manipulation a key feature of the model is that it replaces the two input and output variables with a single normalized variable the model is then used to provide a priori criteria stated in terms of the new variable that a given boolean function must satisfy in order to be mapped by a network having one or two layers these criteria provide necessary and in the case of a one layer network sufficient conditions for samples of a boolean function to be mapped by a binary neural network with zero thresholds it is shown that the necessary conditions imposed by the two layer network are in some sense minimal

bubbling solutions for moser trudinger type equations on compact riemann surfaces abstract we study an elliptic equation related to the moser trudinger inequality on a compact riemann surface s g g u u e u 2 1 s s u e u 2 d v g 0 in s s u d v g 0 where 0 is a small parameter s is the area of s g is the laplace beltrami operator and d v g is the area element given any integer k 1 under general conditions on s we find a bubbling solution u which blows up at exactly k points in s as 0 when s is a flat two torus in rectangular form we find that either seven or nine families of such solutions do exist for k 2 in particular in any square flat two torus actually nine families of bubbling solutions with two bubbling points do exist if s is a riemann surface with non constant robin s function then at least two bubbling solutions with k 1 exists

centroidal voronoi tessellations applications and algorithms a centroidal voronoi tessellation is a voronoi tessellation whose generating points are the centroids centers of mass of the corresponding voronoi regions we give some applications of such tessellations to problems in image compression quadrature finite difference methods distribution of resources cellular biology statistics and the territorial behavior of animals we discuss methods for computing these tessellations provide some analyses concerning both the tessellations and the methods for their determination and finally present the results of some numerical experiments