uniqueness of positive radial solutions for infinite semipositone p laplacian problems in exterior domains abstract we prove uniqueness and asymptotic behavior of positive radial solutions to the p laplacian problem p u k x f u in x r 0 u 0 on x r 0 u x 0 as x where x r n x r 0 0 n p f 0 r is continuous f u u q at for some q 0 p 1 with possible infinite semipositone structure at 0 and is a large parameter

uniqueness results for the one dimensional p laplacian this paper is concerned with positive solutions of the boundary value problem y p 2y f y 0 y b 0 y b where p 1 b is a positive parameter assume that f is continuous on 0 changes sign from nonpositive to positive and f y yp 1 is nondecreasing in the interval of f 0 the uniqueness results are proved using a time mapping analysis

monotone approximations of unstable solutions assume that for a nonlinear two point problem we have a subsolution lying above a supersolution it is well known that in general one cannot conclude the existence of a solution in between however if one restricts the growth of the nonlinearity then under some restrictions on the super and subsolution one gets both existence of a solution and two sequences of monotone approximations in addition we can assert some qualitative property of the solution

semipositone problems with falling zeros on exterior domains abstract we study boundary value problems of the form u k x f u x u 0 if x r 0 u 0 as x where is a positive parameter u div u is the laplacian of u x r n n 2 x r 0 k belongs to a class of c 1 functions such that lim r k r 0 and f belongs to a class of c 1 functions which are negative at the origin and have falling zeros we discuss the existence and uniqueness of nonnegative radial solutions when is large

exact multiplicity results for a class of two point boundary value problems abstract this paper is concerned with the exact number of positive solutions for boundary value problems y p 2 y f y 0 and y 1 y 1 0 where p 1 and 0 is a positive parameter we consider the case in which the nonlinearity f is positive on 0 and p 1 f u u f u changes sign from negative to positive

uniqueness of positive radial solutions for a class of semipositone problems on the exterior of a ball abstract we study positive radial solutions to u k x f u x e where 0 is a parameter e x r n x r 0 r 0 0 n 2 is the laplacian operator k c r 0 0 satisfies k r 1 r n 0 for r 1 and f c 1 0 r is a concave increasing function satisfying lim s f s s 0 and f 0 0 semipositone we are interested in solutions u such that u 0 as x and satisfy the nonlinear boundary condition u c u u 0 if x r 0 where is the outward normal derivative and c c 0 0 is an increasing function we will establish the uniqueness of positive radial solutions for large values of the parameter

non negative solutions for a class of non positone problems in the recent past many results have been established on non negative solutions to boundary value problems of the form where 0 f 0 0 positone problems in this paper we consider the impact on the non negative solutions when f 0 f u to be convex to guarantee uniqueness of positive solutions and f u to be appropriately concave for multiple positive solutions this is in contrast to the case of positone problems where the roles of convexity and concavity were interchanged to obtain similar results we further establish the existence of non negative solutions with interior zeros which did not exist in positone problems

diffusive logistic equations with indefinite weights population models in disrupted environments the dynamics of a population inhabiting a strongly heterogeneous environment are modelledby diffusive logistic equations of the form u t d u m x cu u in 0 where u represents the population density c d 0 are constants describing the limiting effects of crowding and the diffusion rate of the population respectively and m x describes the local growth rate of the population if the environment is bounded and is surrounded by uninhabitable regions then u 0 on 0 the growth rate m x is positive on favourablehabitats and negative on unfavourable ones the object of the analysis is to determine how the spatial arrangement of favourable and unfavourable habitats affects the population being modelled the models are shown to possess a unique stable positive steady state implying persistence for the population provided l d where is the principle positive eigenvalue for the problem m x in 0 on analysis of how depends on m indicates that environments with favourable and unfavourable habitats closely intermingled are worse for the population than those containing large regions of uniformly favourable habitat in the limit as the diffusion rate d 0 the solutions tend toward the positive part of m x c and if m is discontinuous develop interior transition layers the analysis uses bifurcation and continuation methods the variational characterisation of eigenvalues upper and lower solution techniques and singular perturbation theory

uniqueness of nonnegative solutions for semipositone problems on exterior domains abstract we consider the problem u k x f u x u 0 if x r 0 u 0 as x where is a positive parameter u div u is the laplacian of u x r n n 2 x r 0 k c 1 r 0 0 is such that lim r k r 0 and f c 1 0 r is a concave function which is sublinear at and f 0 0 we establish the uniqueness of nonnegative radial solutions when is large

uniqueness of positive radial solutions of δu f u 0 in ℝ n ii we prove a uniqueness result for the positive solution of u f u 0 in r n which goes to 0 at the result applies to a wide class of nonlinear functions f including the important model case f u u u p 1 p n 2 n 2 the result is proved by reducing to an initial boundary problem for the ode u n 1 r f u 0 and using a shooting method

compact range performance evaluation using aperture near field angular spectrums for a compact range cr the near field angular spectrum can be obtained by making an inverse fourier transform of its aperture spatial near field the characteristics of the aperture design have been carried by the aperture near field angular spectrum the direct wave and edge diffracted waves from the aperture can be separated and distinguished clearly in the angular domain therefore the cr performance can be evaluated and improved accordingly in this paper based on the aperture field convolution method and plane wave spectrum pws theory near field angular spectrums for different apertures have been computed the general criteria for the cr aperture design have been discussed the selections of the whole aperture shape the serration height the serration base length the serration number and the serration shape including isosceles and right angled triangles have been compared at last the near fields of an optimized serrated edge aperture and an original circular aperture in different transverse planes have been computed and compared the width and height of the apertures are all 30 wavelengths in free space in order to suppress the sidelobes a chebyshev window with 50 db sidelobes has been adopted in the spatial angular transform

exact multiplicity for degenerate two point boundary value problems with p convex nonlinearity the exact number of positive solutions of a degenerate quasilinear two point boundary value problem is investigated for the generalization of earlier results concerning the nondegenerate case with convex nonlinearity suitably defined p convex nonlinearities are considered strictly p convex c2 functions having a non negative root are classified according to the shape of the bifurcation diagram of positive solutions versus the length of the interval we have uniqueness when f 0 0 and 0 1 or 2 solutions when f 0 0 similarly to the non degenerate case provided that the number of solutions is finite however now there may also occur a continuum of solutions connected to a dead core type phenomenon the proof of our results relies on the shooting method for the characterization of the shape of the time map in contrast to the non degenerate case the shooting method does not determine directly the number of solutions owing to the lack of uniqueness of the corresponding ivp exact conditions on the uniqueness of the ivp and in the case of non uniqueness the number and types of its local solutions are given based on this all the positive solutions of the bvp can be compiled

positive radial solutions to classes of nonlinear elliptic systems on the exterior of a ball abstract we study positive radial solutions to n n nonlinear elliptic systems of the form u i k i x f i u i 1 in x e u i 0 on x r 0 u i 0 as x where i 1 2 n u 1 u n 1 u div u 0 n 2 r 0 0 e x r n x r 0 k i c r 0 0 and f i c 0 r we establish existence multiplicity and uniqueness results for various nonlinearities f i

on a diffusive logistic equation we study a reaction diffusion version on all of rnof the logistic equation of population growth in which the birth rate depends on the spatial variable and may assume both positive and negative values our results which are obtained by the construction of sub and supersolutions and the study of asymptotic properties of solutions show the interplay between the birth rate of the species and the extent of diffusion in determining the existence or nonexistence of nontrivial steady state distributions of population

effects of a degeneracy in the competition model part i classical and generalized steady state solutions abstract we study the competition model where the coefficient functions are strictly positive over the underlying spatial region except b x which vanishes in a nontrivial subdomain of and is positive in the rest of we show that there exists a critical number such that if then the model behaves similarly to the well studied classical competition model where all the coefficient functions are positive constants but when new phenomena occur our results demonstrate the fact that heterogeneous environmental effects on population models are not only quantitative but can be qualitative as well in part i here we mainly study two kinds of steady state solutions which determine the dynamics of the model one consists of finite functions while the other consists of generalized functions which satisfy u v 0 on the part of the domain that b x vanishes but are positive and finite on the rest of the domain and are determined by certain boundary blow up systems the research is continued in part ii where these two kinds of steady state solutions will be used to determine the dynamics of the model

on the security of pseudorandomized information theoretically secure schemes in this paper we discuss a naive method of randomness reduction for cryptographic schemes which replaces the required perfect randomness with output distribution of a computationally secure pseudorandom generator prg we propose novel ideas and techniques for evaluating the indistinguishability between the random and pseudorandom cases even against an adversary with computationally unbounded attack algorithm hence the prg based randomness reduction can be effective even for information theoretically secure cryptographic schemes especially when the amount of information received by the adversary is small in comparison to a preceding result of dubrov and ishai stoc 2006 our result removes the requirement of generalized notion of nb prgs and is effective for more general kinds of protocols we give some numerical examples to show the effectiveness of our result in practical situations and we also propose a further idea for improving the effect of the prg based randomness reduction

exact multiplicity of positive solutions for a class of semilinear problem ii we consider the positive solutions to the semilinear problem u f u 0 in b n u 0 on b n where bn is the unit ball in rn n 1 and is a positive parameter it is well known that if is a smooth function then any positive solution to the equation is radially symmetric and all solutions can be parameterized by their maximum values we develop a unified approach to obtain the exact multiplicity of the positive solutions for a wide class of nonlinear functions u and the precise shape of the global bifurcation diagrams are rigorously proved our technique combines the bifurcation analysis stability analysis and topological methods we show that the shape of the bifurcation curve depends on the shape of the graph of function u u as well as the growth rale of

exact number of positive solutions for semipositone problems this paper is concerned with the exact number of positive solutions for the boundary value problem y p 2y f y 0 and y 1 y 1 0 where p 1 and 0 is a positive parameter we consider the case in which both f u and g u p 1 f u uf u change sign exactly once from negative to positive on 0

a variational inequality theory in reflexive smooth banach spaces and applications to p laplacian elliptic inequalities abstract we develop a theory of variational inequality for demicontinuous s contractive maps in reflexive smooth banach spaces variational inequality theorems are proved and applied to study existence of nonzero positive weak solutions for p laplacian elliptic inequalities and a population model of one species arising in mathematical biology

evolution of positive solution curves in semipositone problems with concave nonlinearities abstract we study the existence multiplicity and stability of positive solutions to u x f u x for x 1 1 u 1 0 u 1 where 0 and f 0 r is monotonically increasing and concave with f 0 0 we obtain the exact number of positive solutions as a function of f t t we follow several families of nonlinearities f for which f limt f t 0 and study how the positive solution curves to the above problem evolve also we give examples where our results apply this work extends the work of a castro and r shivaji 1988 proc roy soc edinburgh sect a108 291 302 and s h wang 1994 proc roy soc edinburgh124 no 3 507 515 by obtaining sharper results and also gives a complete study of positive solutions for concave semipositone nonlinearities

positive solutions of elliptic equations involving supercritical growth positive radial solutions of elliptic equation involving supercritical growth are analysed as their supremum norm tends to infinity it is shown that they converge uniformly away from the origin as well as in h 1 to the unique singular solution

serrated edge design and field analysis in spatial time and angular domains for improving the field uniformity of beam combiner edge diffraction of a beam combiner will adversely affect the field uniformity in the observation plane in order to suppress the edge diffraction the serrated edge design method is proposed according to the equivalence principle a field uniformity analysis method was proposed to compute the spatial field distribution radiated from the aperture of a beam combiner in order to separate the direct wave from diffracted wave the spatial field is transformed into the time and angular domain spectrum by inverse fourier transform first the height of serrations is designed by analog of the equivalent trapezoidal amplitude taper then the number of serrations was designed by the assumption of isosceles and right triangles finally the shape of serrations was designed by changing the interior angle of serrations a serrated edge design example is also proposed and analyzed by comparing the field uniformity in spatial time and angular domains before and after serrating the edges the amplitude and phase ripples decrease from 1 85 db and 12 16 to 0 53 db and 3 98 respectively the results show that the field uniformity is significantly improved

convergence in competition models with small diffusion coefficients abstract it is well known that for reaction diffusion 2 species lotka volterra competition models with spatially independent reaction terms global stability of an equilibrium for the reaction system implies global stability for the reaction diffusion system this is not in general true for spatially inhomogeneous models we show here that for an important range of such models for small enough diffusion coefficients global convergence to an equilibrium holds for the reaction diffusion system if for each point in space the reaction system has a globally attracting hyperbolic equilibrium this work is planned as an initial step towards understanding the connection between the asymptotics of reaction diffusion systems with small diffusion coefficients and that of the corresponding reaction systems

existence and uniqueness of nontrivial radial solutions for k hessian equations abstract we prove existence and uniqueness of nontrivial radial solutions to the k hessian problem s k d 2 u f u in u 0 on where s k d 2 u is the k hessian operator of u is the open unit ball in r n f is a continuous function and may have k sublinear growth at with possible k superlinear growth at 0 and is a large parameter

singular semilinear elliptic problems with asymptotically linear reaction terms abstract we consider the problem u k x f u in b 1 c u 0 on b 1 u x 0 as x where b 1 c x r n x 1 n 2 is a positive parameter k belongs to a class of functions which satisfy certain decay assumptions and f belongs to a class of functions which are asymptotically linear and may be singular at the origin we prove the existence of positive solutions to such problems for certain values of parameter existence results to similar problems in r n are also obtained our existence results are proved using the schauder fixed point theorem and the method of sub and super solutions

realization of prescribed patterns in the competition model we demonstrate that for any prescribed set of finitely many disjoint closed subdomains d1 dm of a given spatial domain in rn if d1 d2 a1 a2 c d e are positive continuous functions on and b x is identically zero on d d1 dm and positive in the rest of then for suitable choices of the parameters and all small e 0 the competition model ut x t d1 x u x t a1 x u e 1b x 1 u2 c x uv vt x t d2 x v x t a2 x v e x v2 e 1d x uv x t 0 under natural boundary conditions on possesses an asymptotically stable positive steady state solution ue ve that has pattern d that is roughly speaking as e 0 ue converges to a positive function over d while it converges to 0 over the rest of on the other hand ve converges to 0 over d but converges to some positive function in the rest of in other words the two competing species ue and ve become spatially segregated as e 0 with ue concentrating on d and ve concentrating on d

on the monotonicity of the time map the equation au y u y 0 on the unit ball with homogeneous linear boundary conditions has been much studied cf 1 2 and the references therein the change of variable x l i transforms this equation to mx a 0 1x1 r 1 1 where r fi for radial solutions of this equation lying in say the kth nodal class r depends only on p u o r t p this function which describes how the size of the balls on which radial solutions exist vary with the quantity u o turns out to be an important function determining much of the behavior of the solutions thus if we consider 1 1 together with the boundary conditions cu x i du x dn 0 1x1 r

a diffusive competition model with a protection zone this paper is concerned with a two species diffusive competition model with a protection zone for the weak competitor our mathematical results imply that when the protection zone is above a certain critical patch size determined by the birth rate of the weak competitor the weak species almost always survives but it cannot survive when the protection zone is below the critical size and its competitor is strong enough while this is the main feature of the model the actual dynamical behavior of the reaction diffusion system is more complicated the key to reveal the main feature of the system lies in a detailed analysis of the attracting regions of its steady state solutions our mathematical analysis shows that compared with the predator prey model discussed in yihong du junping shi a diffusive predator prey model with a protect zone j differential equations 226 2006 63 91 the protection zone has some essentially different effects on the fine dynamics of the competition model

the positive solutions of boundary value problems for a class of one dimensional p laplacians abstract this paper is concerned with the exact number of positive solutions for boundary value problems u p 2 u f u 0 and u 1 u 1 0 where p 1 and 0 is a positive parameter we consider the case in which the nonlinearity f changes sign from positive to negative on 0 for example f u u u where

a uniqueness result for a semipositone p laplacian problem on the exterior of a ball abstract we consider steady state reaction diffusion equations on the exterior of a ball namely boundary value problems of the form p u k x f u in e u 0 on x r 0 u 0 when x where p z div z p 2 z 1 p n is a positive parameter r 0 0 and e x r n x r 0 here the weight function k c 1 r 0 satisfies k r 0 for r r 0 lim r k r 0 and the reaction term f c 0 c 1 0 is strictly increasing and satisfies f 0 0 semipositone lim sup s 0 s f s lim s f s lim s f s s p 1 0 and f s s q is nonincreasing on a for some a 0 and q 0 p 1 for a class of such steady state equations it turns out that every nonnegative radial solution is strictly positive in the exterior of a ball and exists for 1 we establish the uniqueness of this positive radial solution for 1

a homotopic deformation along p of a leray schauder degree result and existence for u p 2u ƒ t u 0 u 0 u t 0 p 1 soit f 0 1 r r une fonction continue et soit o p r r definie par o p s s p 2 s pour p fixe 1 on considere le probleme de dirichlet suivant o p u f t u 0 u 0 u t 0 ou t est un nombre reel positif

diffusive logistic equation with constant yield harvesting i steady states we consider a reaction diffusion equation which models the constant yield harvesting to a spatially heterogeneous population which satisfies a logistic growth we prove the existence uniqueness and stability of the maximal steady state solutions under certain conditions and we also classify all steady state solutions under more restricted conditions exact global bifurcation diagrams are obtained in the latter case our method is a combination of comparison arguments and bifurcation theory

on a singular nonlinear semilinear elliptic problem where k x c2 b v9 a p 0 1 and l is a real parameter such singular elliptic problems arise in the contexts of chemical heterogeneous catalysts nonnewtonian fluids and also the theory of heat conduction in electrically conducting materials see 3 5 8 9 for a detailed discussion obviously 1 1 cannot have a solution u c2 v9 if k x is not vanishing near v however under various appropriate assumptions on k x we will obtain classical solutions of 1 1 for l belonging to a certain range and we will also obtain some uniqueness criteria here a classical solution is a solution u of 1 1 which belongs to c2 v mc v9 with u 0 in v we also study the boundary behaviour of solutions of 1 1 and we will show that the solution u of 1 1 lies in a certain holder class the special case when k x is negative and l 0 has been studied by several authors the existence and uniqueness of the solution were established by crandall rabinowitz and tartar 6 del pino 7 gomes 10 and lazer and mckenna

unbiased bits from sources of weak randomness and probabilistic communication complexity a new model for weak random physical sources is presented the new model strictly generalizes previous models e g the santha and vazirani model 27 the sources considered output strings according to probability distributions in which no single string is too probable the new model provides a fruitful viewpoint on problems studied previously such as extracting almost perfect bits from sources of weak randomness the question of possibility as well as the question of efficiency of such extraction schemes are addressed probabilistic communication complexity it is shown that most functions have linear communication complexity in a very strong probabilistic sense robustness of bpp with respect to sources of weak randomness generalizing a result of vazirani and vazirani 32 33

persistence and bifurcation of degenerate solutions abstract we consider a nonlinear equation f e u 0 where f is a differentiable mapping from r r x to y and x y are banach spaces when e varies from a fixed e e 0 bifurcation occurs to the solution curve s u s we study the degenerate solutions of the equation and we obtain several bifurcation theorems on the degenerate solutions which can be applied in many nonlinear problems to obtain precise global bifurcation diagrams

on the stability properties of nonnegative solutions of semilinear problems with convex or concave nonlinearity 497 abstract we investigate the stability of nontrivial nonnegative stationary solutions of semilinear initial boundary value problems with convex or concave nonlinearity in the convex case f 0 0 implies instability in the concave case f 0 0 implies stability we also discuss the necessity of the sign condition on f 0

positive solution curves of semipositone problems with concave nonlinearities we consider the positive solutions to the semilinear equation where denotes a smooth bounded region in n n 1 and 0 here f 0 is assumed to be monotonically increasing concave and such that f 0 f lim t f t 0 we establish the stability and uniqueness of large positive solutions in terms of f t t when is a ball we determine the exact number of positive solutions for each 0 we also obtain the geometry of the branches of positive solutions completely and establish how they evolve this work extends and complements that of 3 7 where f 0

exact multiplicity of positive solutions of a semipositone problem with concave convex nonlinearity abstract we study the exact multiplicity of positive solutions and bifurcation diagrams of the dirichlet boundary value problem u x f u 0 1 x 1 u 1 u 1 0 where 0 is a bifurcation parameter f c 0 c 2 0 satisfies f 0 0 semipositone and f is concave convex on 0 and is asymptotic superlinear assuming additional suitable conditions on f on the u plane we give a classification of totally three qualitatively different bifurcation diagrams a reversed s shaped curve a broken reversed s shaped curve and a monotone decreasing curve our results improve those in j shi r shivaji exact multiplicity of solutions for classes of semipositone problems with concave convex nonlinearity discrete contin dyn syst 7 2002 559 571 we also give an application to determine completely the exact multiplicity of positive solutions and bifurcation diagrams of the problem with cubic nonlinearity u x u a u b u c 0 1 x 1 u 1 u 1 0 0 a b c

new exact multiplicity results with an application to a population model we obtain some new exact multiplicity results for the dirichlet boundary value problem on a unit ball b n in r n we consider several classes of nonlinearities f u including both positive and sign changing cases a crucial part of the proof is to establish positivity of solutions for the corresponding linearized problem as an application we obtain exact multiplicity results for the holling tanner population model

symmetry breaking for solutions of semilinear elliptic equations with general boundary conditions we study the bifurcation of radially symmetric solutions of f u 0 onn balls into asymmetric ones we show that ifu satisfies homogeneous neumann boundary conditions the asymmetric components in the kernel of the linearized operators can have arbitrarily high dimension for general boundary conditions we prove some theorems which give bounds on the dimensions of the set of asymmetric solutions and on the structure of the kernels of the linearized operators

global bifurcation of steady state solutions in this paper we study the bifurcation of steady state solutions of a reaction diffusion equation in one space variable the steady state solutions satisfy the equation a f u 0 on the interval l 0 or 0 for some nonzero function c c a for the neumann problem we use entirely different techniques to prove that s is never critical for any cubic polynomial j this implies at once that the neumann problem can have at most one nonconstant solution having a given number of maxima or minima this solution is necessarily strongly nondegenerate in the sense that zero is not contained in the spectrum of the linearized operator see 2 for the dirichlet problem the situation is far more complicated and the bifurcation diagrams undergo qualitative changes depending on the positions of the roots off for example fig 1 shows the bifurcation diagram for the positive solutions for two different cubic functions of the form f u a u u b u c it is interesting to note however that in every case we study there are at most three solutions to the dirichlet problem for each

combined effects of concave and convex nonlinearities in some elliptic problems abstract this paper deals with a class of semilinear elliptic dirichlet boundary value problems where the combined effects of a sublinear and a superlinear term allow us to establish some existence and multiplicity results

unbounded connected component of the positive solution set of nonlinear operator equation and its applications abstract in this paper by using global bifurcation theories we obtain some results for structure of positive solution set of some nonlinear equations with parameters as a result we obtain some existence results for positive solutions of the nonlinear operator equation the main result can be applied to various of differential boundary value problems to obtain the existence results for positive solutions without the assumption that the nonlinearities are positone

branches of radial solutions for semipositone problems abstract we consider the radially symmetric solutions to the equation u x u x for x u x 0 for x where denotes the unit ball in r n n 1 centered at the origin and 0 here r r is assumed to be semipositone 0 0 there exists at most one positive solution u such that u belongs to the unbounded branch of positive solutions further when t tp k k 0 and 1

exact multiplicity of positive solutions for a class of semilinear problems we establish an exact multiplicity result of positive solutions of u f u 0inbn u 0on bn wherebnis the unit ball fsatisfiesf changes sign only once and asymptotically sublinear or linear the nonlinearities which we are concerned with here includef u u u b c u for 0 2b candf u up uqfor 1 p q precise global bifurcation diagrams are obtained

bifurcation phenomena associated to the p laplace operator on determine la structure de l ensemble des solutions u de u x p 2 u x x f u u p 2 u sur 0 1 telles que u 0 u 1 0 ou p 1 et r on demontre que les solutions avec k zeros sont uniques quand 1 2

positive solutions of quasilinear boundary value problems abstract this paper deals with the existence of positive solutions for the boundary value problem formula formula where f is either superlinear or sublinear at and f t 0 may be negative and p is a positive continuous function the results extend several known results for semilinear equations our approach is based on fixed point theory for completely continuous operators which leave invariant a suitable cone in a banach space of continuous functions

how to turn loaded dice into fair coins we present a new technique for simulating fair coin flips using a biased stationary source of randomness sequences of random numbers are of pervasive importance in cryptography and vital to many other computing applications many sources of randomness such as radioactive or quantum mechanical sources possess the property of stationarity in other words they produce independent outputs over fixed probability distributions the output of such sources may be viewed as the result of rolling a biased or loaded die while a biased die may be a good source of entropy many applications require input in the form of unbiased bits rather than biased ones for this reason von neumann 1951 presented a now well known and extensively investigated technique for using a biased coin to simulate a fair coin we describe a new generalization of von neumann s algorithm distinguished by its high level of practicality and amenability to analysis in contrast to previous efforts we are able to prove our algorithm optimally efficient in the sense that it simulates the maximum possible number of fair coin flips for a given number of die rolls in fact we are able to prove that in an asymptotic sense our algorithm extracts the full entropy of its input moreover we demonstrate experimentally that our algorithm achieves a high level of computational and output efficiency in a practical setting

towards 3 query locally decodable codes of subexponential length a q query locally decodable code ldc encodes an n bit message x as an n bit codeword c x such that one can probabilistically recover any bit xi of the message by querying only q bits of the codeword c x even after some constant fraction of codeword bits has been corrupted we give new constructions of three query ldcs of vastly shorter length than that of previous constructions specifically given any mersenne prime p e 2t 1 we design three query ldcs of length n e exp o n1 t for every n based on the largest known mersenne prime this translates to a length of less than exp o n10 7 compared to exp o n1 2 in the previous constructions it has often been conjectured that there are infinitely many mersenne primes under this conjecture our constructions yield three query locally decodable codes of length n e exp no 1 log log n for infinitely many n we also obtain analogous improvements for private information retrieval pir schemes we give 3 server pir schemes with communication complexity of o n10 7 to access an n bit database compared to the previous best scheme with complexity o n1 5 25 assuming again that there are infinitely many mersenne primes we get 3 server pir schemes of communication complexity no 1 log logn for infinitely many n previous families of ldcs and pir schemes were based on the properties of low degree multivariate polynomials over finite fields our constructions are completely different and are obtained by constructing a large number of vectors in a small dimensional vector space whose inner products are restricted to lie in an algebraically nice set

a diffusive predator prey model with a protection zone in this paper we study the effects of a protection zone 0 for the prey on a diffusive predator prey model with holling type ii response and no flux boundary condition we show the existence of a critical patch size described by the principal eigenvalue d 0 of the laplacian operator over 0 with homogeneous dirichlet boundary conditions if the protection zone is over the critical patch size i e if d 0 is less than the prey growth rate then the dynamics of the model is fundamentally changed from the usual predator prey dynamics in such a case the prey population persists regardless of the growth rate of its predator and if the predator is strong then the two populations stabilize at a unique coexistence state if the protection zone is below the critical patch size then the dynamics of the model is qualitatively similar to the case without protection zone but the chances of survival of the prey species increase with the size of the protection zone as generally expected our mathematical approach is based on bifurcation theory topological degree theory the comparison principles for elliptic and parabolic equations and various elliptic estimates

global structure of positive solutions for superlinear second order periodic boundary value problems in this paper we consider the nonlinear eigenvalue problems gather u lambda h t f u 0 quad 0 0 for i 1 ldots m 2 with sum i 1 m 2 alpha i eta i 0 for t 0 in 0 1 f in c 0 infty 0 infty and f s 0 for s 0 and f 0 infty where f 0 lim s rightarrow 0 f s s we investigate the global structure of positive solutions by using the nonlinear krein rutman theorem

existence of solutions for semilinear problems with prescribed number of zeros on exterior domains abstract in this paper we prove the existence of an infinite number of radial solutions of u k r f u 0 on the exterior of the ball of radius r centered at the origin in r n such that lim r u r 0 with prescribed number of zeros where f r r is odd and there exists a 0 with f 0 on 0 f 0 on with f superlinear for large u and k r r with 0 n

existence and nonexistence of positive solutions for a class of superlinear semipositone systems abstract we consider an elliptic system of the form u f v in v g u in u 0 v on where 0 is a parameter is a bounded domain in r n with smooth boundary here the nonlinearities f g 0 r are c loc 0 0 1 functions that are superlinear at infinity and satisfy f 0 0 and g 0 0 we prove that the system has a positive solution for small when is convex with c 3 boundary and no positive solution for large when is a general bounded domain with c 2 boundary moreover we show that there exists a closed connected subset of positive solutions bifurcating from infinity at 0

on certain second order ordinary differential equations associated with sobolev poincare type inequalities on considere l equation e i 1 n u x i p 2 u x i x i u q 2 u 0 dans u x 0 sur on donne les proprietes des solutions de e pour le cas n 1 et a b mais pour p q 1 avec p q regularite existence et unicite de solutions positives

existence of positive radial solutions for a semipositone elliptic equation abstract in this paper we study the existence of positive radial solutions for a semipositone elliptic equation u f u in u 0 on where is a ball or an annulus in r n with n 2 0 is a parameter and f is a continuous function satisfying the semipositone condition f 0 0 we give a weak and general sufficient condition on f for the existence of positive radial solutions for 0 large we show that the linearized operator at each solution has the nonnegative first eigenvalue these solutions are obtained as minimizers of a lagrangian functional

existence of positive solutions for semilinear elliptic equations in general domains the main purpose of this paper is to prove some new existence theorems for positive solutions to the dirichlet problem elta u x f u x 0 x n mega 1 1 u x 0 x n artial mega 1 2

instability of nonnegative solutions for a class of semilinear abstract we consider the boundary value problem u x f u x x bu x 0 x where is a bounded region in rn with smooth boundary bu x h x u 1 u n where 0 1 h r with h 1 when 1 0 f is a smooth function such that u 0 for u 0 u for u 0 and u 0 for u for some 0 we provide a simple proof to establish that every non trivial nonnegative solution is unstable

existence uniqueness and stability of oscillations in differential equations with asymmetric nonlinearities we give conditions for the existence uniqueness and asymptotic stability of periodic solutions of a second order differential equation with piecewise linear restoring and 27r periodic forcing where the range of the derivative of the restoring term possibly contains the square of an integer with suitable restrictions on the restoring and forcing in the undamped case we give a necessary and sufficient condition

compact range reflector analysis using the plane wave spectrum approach with an adjustable sampling rate an improved method for determining the test zone field of compact range reflectors is presented the plane wave spectrum pws approach is used to obtain the test zone field from knowledge of the reflector aperture field distribution the method is particularly well suited to the analysis of reflectors with a linearly serrated rim for reduced edge diffraction computation of the pws of the reflector aperture field is facilitated by a closed form expression for the fourier transform of a polygonal window function inverse transformation in the test zone region is accomplished using a fast fourier transform fft algorithm with a properly adjusted sampling rate which is a function of both the reflector size and the distance from the reflector the method is validated by comparison with results obtained using surface current and aperture field integration techniques the performance of several serrated reflectors is evaluated in order to observe the effects of edge diffraction on the test zone fields

positive solutions for infinite semipositone problems with falling zeros abstract we consider the positive solutions to the singular problem p u a u f u c u in u 0 on where 0 1 a 0 and c 0 are constants is a bounded domain with smooth boundary and f 0 r is a continuous function we assume that there exist m 0 a 0 p 1 such that a u m f u a u p for all u 0 a simple example of f satisfying these assumptions is f u u p for any p 1 we use the method of sub supersolutions to prove the existence of a positive solution of p when a 2 1 1 and c is small here 1 is the first eigenvalue of operator with dirichlet boundary conditions we also extend our result to classes of infinite semipositone systems

optimal probabilistic fingerprint codes we construct binary codes for fingerprinting digital documents our codes for n users that are e secure against c pirates have length o c2log n e this improves the codes proposed by boneh and shaw l1998r whose length is approximately the square of this length the improvement carries over to works using the boneh shaw code as a primitive for example to the dynamic traitor tracing scheme of tassa l2005r by proving matching lower bounds we establish that the length of our codes is best within a constant factor for reasonable error probabilities this lower bound generalizes the bound found independently by peikert et al l2003r that applies to a limited class of codes our results also imply that randomized fingerprint codes over a binary alphabet are as powerful as over an arbitrary alphabet and the equal strength of two distinct models for fingerprinting