singularly perturbed critical choquard equations abstract in this paper we study the semiclassical limit for the singularly perturbed choquard equation e 2 u v x u e 3 r 3 q y g u y x y d y q x g u in r 3 where 0 3 e is a positive parameter v q are two continuous real function on r 3 and g is the primitive of g which is of critical growth due to the hardy littlewood sobolev inequality under suitable assumptions on g we first establish the existence of ground states for the critical choquard equation with constant coefficients next we establish existence and multiplicity of semi classical solutions and characterize the concentration behavior by variational methods

multiplicity and concentration results for fractional choquard equations doubly critical case abstract in this paper we consider the following fractional choquard equation e 2 s s u v x u e i f u f u x r n where n 3 s 0 1 and 0 n e 0 is a parameter i is the riesz potential and f u 1 2 u 2 1 2 u 2 where 2 n n and 2 n n 2 s are lower and upper critical exponents in the sense of the hardy littlewood sobolev inequality firstly by the refined sobolev inequality with morrey norm we show a generalization of lions type theorem secondly combining this theorem with variational methods we show the multiplicity and concentration of positive solutions for above equation moreover the multiplicity and concentration results are obtained in the case where f u has the lower critical exponent 2 which remains unsolved in the existing literature

regularity theory for fully nonlinear integro differential equations we consider nonlinear integro differential equations like the ones that arise from stochastic control problems with purely jump levy processes we obtain a nonlocal version of the abp estimate harnack inequality and interior c 1 regularity for general fully nonlinear integro differential equations our estimates remain uniform as the degree of the equation approaches two so they can be seen as a natural extension of the regularity theory for elliptic partial differential equations

on nonlocal choquard equations with hardy littlewood sobolev critical exponents abstract we consider the following nonlinear choquard equation with dirichlet boundary condition u u 2 x y d y u 2 2 u f u in where is a smooth bounded domain of r n 0 n 3 0 n and 2 is the critical exponent in the sense of the hardy littlewood sobolev inequality under suitable assumptions on different types of nonlinearities f u we are able to prove some existence and multiplicity results for the equation by variational methods

existence and multiplicity of solutions for fractional choquard equations abstract we consider the following fractional order choquard equation 2 u x v x u x u 2 u 2 2 u x r n with the nonlinearity in the critical growth where 0 2 n 3 r and 2 2 n n using the variational method we establish the existence and multiplicity of weak solutions

cryptanalysis of short rsa secret exponents a cryptanalytic attack on the use of short rsa secret exponents is described the attack makes use of an algorithm based on continued fractions that finds the numerator and denominator of a fraction in polynomial time when a close enough estimate of the fraction is known the public exponent e and the modulus pq can be used to create an estimate of a fraction that involves the secret exponent d the algorithm based on continued fractions uses this estimate to discover sufficiently short secret exponents for a typical case where e pq gcd p 1 q 1 is small and p and q have approximately the same number of bits this attack will discover secret exponents with up to approximately one quarter as may bits as the modulus ways to combat this attack ways to improve it and two open problems are described this attack poses no threat to the normal case of rsa where the secret exponent is approximately the same size as the modulus this is because the attack uses information provided by the public exponent and in the normal case the public exponent can be chosen almost independently of the modulus

on the multiplicity and concentration of positive solutions for a p fractional choquard equation in rn abstract in this paper we deal with the following fractional choquard equation e s p p s u v x u p 2 u e n 1 x f u f u in r n u w s p r n u 0 in r n where e 0 is a small parameter s 0 1 p 1 n s p p s is the fractional p laplacian v is a positive continuous potential 0 s p and f is a continuous superlinear function with subcritical growth using minimax arguments and the ljusternik schnirelmann category theory we obtain the existence multiplicity and concentration of positive solutions for e 0 small enough

random subgraphs of finite graphs ii the lace expansion and the triangle condition in a previous paper we defined a version of the percolation triangle condition that is suitable for the analysis of bond percolation on a finite connected transitive graph and showed that this triangle condition implies that the percolation phase transition has many features in common with the phase transition on the complete graph in this paper we use a new and simplified approach to the lace expansion to prove quite generally that for finite graphs that are tori the triangle condition for percolation is implied by a certain triangle condition for simple random walks on the graph the latter is readily verified for several graphs with vertex set 0 1 r 1 n including the hamming cube on an alphabet of r letters the n cube for r 2 the n dimensional torus with nearest neighbor bonds and n sufficiently large and the n dimensional torus with n 6 and sufficiently spread out long range bonds the conclusions of our previous paper thus apply to the percolation phase transition for each of the above examples

fractional kirchhoff type equation with hardy littlewood sobolev critical exponent abstract in this paper we consider the following fractional kirchhoff type equation a b r n r n u x u y 2 x y n 2 s d x d y s u i u 2 h u 2 h 2 u in r n where n 3 s 0 1 a 0 b 0 0 n and 2 h n n 2 s is the hardy littlewood sobolev critical exponent and i is the riesz potential we establish the existence nonexistence and multiplicity of nontrivial solutions to above equation

factoring polynomials with rational coefficients in this paper we present a polynomial time algorithm to solve the following problem given a non zero polynomial fe q x in one variable with rational coefficients find the decomposition of f into irreducible factors in q x it is well known that this is equivalent to factoring primitive polynomials fez x into irreducible factors in z x here we call f z x primitive if the greatest common divisor of its coefficients the content of f is 1 our algorithm performs well in practice cf 8 its running time measured in bit operations is o nl2 n9 log fd3

a generalised inductive approach to the lace expansion the lace expansion is a powerful tool for analysing the critical behaviour of self avoiding walks and percolation it gives rise to a recursion relation which we abstract and study using an adaptation of the inductive method introduced by den hollander and the authors we give conditions under which the solution to the recursion relation behaves as a gaussian both in fourier space and in terms of a local central limit theorem these conditions are shown elsewhere to hold for sufficiently spread out models of networks of self avoiding walks in dimensions d 4 and for sufficiently spread out models of critical oriented percolation in dimensions d 1 5 providing a unified approach and an essential ingredient for a detailed analysis of the branching behaviour of these models

existence and nonexistence results for kohn laplacian with hardy littlewood sobolev critical exponents abstract in this article we are study the following dirichlet problem with choquard type non linearity h u a u u q 1 d u q 2 u in u 0 on where is a smooth bounded subset of the heisenberg group h n n n with c 2 boundary and h is the kohn laplacian on the heisenberg group h n here q 2 q q 2 q 2 n 2 and a is a positive real parameter we derive the brezis nirenberg type result for the above problem moreover we also prove the regularity of solutions and nonexistence of solutions depending on the range of a

gap exponents for percolation processes with triangle condition a study is made of the gap exponents for percolation processes with the triangle condition in the subcritical region it is show that the gaps are given by t 2 fort 2 3 scaling theory predicts thatp p c 0 s p p c p ande p 1 c 0 c 0 s p p c p 3 wheres p is the typical cluster size it is found that p c p p p c 0 s p 1 p c p 1 2 and p c p 3 e p 1 c 0 c 0 s p 1 p c p 3 4

the evans krylov theorem for nonlocal fully nonlinear equations we prove a regularity result for solutions of a purely integro differential bellman equation this regularity is enough for the solutions to be understood in the classical sense if we let the order of the equation approach two we recover the theorem of evans and krylov about the regularity of solutions to concave uniformly elliptic partial differential equations

regularity results on a class of doubly nonlocal problems abstract the purpose of this article is twofold first an issue of regularity of weak solution to the problem p see below is addressed secondly we investigate the question of h s versus c 0 weighted minimizers of the functional associated to problem p and then give applications to existence and multiplicity results

the better bound of private key in rsa with unbalanced primes at asiacrypt 99 sun yang and laih proposed three different schemes of rsa cryptosystem to avoid all known attacks including boneh durfee attack one year after durfee and nguyen attacked two out of these three schemes based on coppersmith s lattice technique for finding small roots to trivariate modular polynomial equations the bounds of private key for the first and third schemes were much improved but the bound of the second scheme was not good enough our result improves durfee and nguyen s bound of the private exponent d from n 0 4 8 3 to n 0 4 8 6 for the second scheme this implies the system is insecure if the private exponent d

groundstates of nonlinear choquard equations existence qualitative properties and decay asymptotics we consider a semilinear elliptic problem delta u u i alpha ast abs u p abs u p 2 u quad text in r n where i alpha is a riesz potential and p 1 this family of equations includes the choquard or nonlinear schr odinger newton equation for an optimal range of parameters we prove the existence of a positive groundstate solution of the equation we also establish regularity and positivity of the groundstates and prove that all positive groundstates are radially symmetric and monotone decaying about some point finally we derive the decay asymptotics at infinity of the groundstates

regularity of the obstacle problem for a fractional power of the laplace operator given a function and s 0 1 we will study the solutions of the following obstacle problem u in r n s u 0i nr n s u x 0 for those x such that u x x lim x u x 0 we show that when is c 1 s or smoother the solution u is in the space c 1 for every s in the case where the contact set u is convex we prove the optimal regularity result u c 1 s when is only c 1 for a s we prove that our solution u is c 1 for every c 2006 wiley periodicals inc

existence of groundstates for a class of nonlinear choquard equations we prove the existence of a nontrivial solution h n to the nonlinear choquard equation i in n where i is a riesz potential under almost necessary conditions on the nonlinearity in the spirit of berestycki and lions this solution is a groundstate if moreover is even and monotone on 0 then is of constant sign and radially symmetric

positive solutions to a fractional equation with singular nonlinearity abstract in this paper we study the positive solutions to the following singular and non local elliptic problem posed in a bounded and smooth domain r n n 2 s p s u k x u f u in u 0 in u 0 in r n here 0 s 1 0 0 and f r r is a positive c 2 function k r is a holder continuous function in which behave as dist x near the boundary with 0 2 s first for any 0 and for small enough we prove the existence of solutions to p next for a suitable range of values of we show the existence of an unbounded connected branch of solutions to p emanating from the trivial solution at 0 for a certain class of nonlinearities f we derive a global multiplicity result that extends results proved in 2 to establish the results we prove new properties which are of independent interest and deal with the behavior and holder regularity of solutions to p

the diffusion of self avoiding random walk in high dimensions we use the brydges spencer lace expansion to prove that the mean square displacement of at step strictly self avoiding random walk in thed dimensional square lattice is asymptotically of the formdt ast approaches infinity ifd is sufficiently large the diffusion constantd is greater than one

mean field critical behaviour for percolation in high dimensions the triangle condition for percolation states that sum limits x y tau 0 x tau 0 y cdot tau y 0 is finite at the critical point where x y is the probability that the sitesx andy are connected we use an expansion related to the lace expansion for a self avoiding walk to prove that the triangle condition is satisfied in two situations i for nearest neighbour independent bond percolation on thed dimensional hypercubic lattice ifd is sufficiently large and ii in more than six dimensions for a class of spread out models of independent bond percolation which are believed to be in the same universality class as the nearest neighbour model the class of models in ii includes the case where the bond occupation probability is constant for bonds of length less than some large number and is zero otherwise in the course of the proof an infrared bound is obtained the triangle condition is known to imply that various critical exponents take their mean field bethe lattice values gamma beta 1 delta delta t 2 t underset raise0 3em hbox smash scriptscriptstyle geqslant 2 and that the percolation density is continuous at the critical point we also prove thatv2 in i and ii wherev2 is the critical exponent for the correlation length

regularity results for nonlocal equations by approximation we obtain c1 regularity estimates for nonlocal elliptic equations that are not necessarily translation invariant using compactness and perturbative methods and our previous regularity results for the translation invariant case

sharpness of the phase transition in percolation models the equality of two critical points the percolation threshold ph and the point p where the cluster size distribution ceases to decay exponentially is proven for all translation invariant independent percolation models on homogeneous d dimensional lattices d 1 the analysis is based on a pair of new nonlinear partial differential inequalities for an order parameter m h which for h q reduces to the percolation density p at the bond density p l e in the single parameter case these are 1 m hdm dh m2 mdm d and 2 dm d j mdm dh inequality 1 is intriguing in that its derivation provides yet another hint of a 3 structure in percolation models moreover through the elimination of one of its derivatives 1 yields a pair of ordinary differential inequalities which provide information on the critical exponents and one of these resembles an ising model inequality of fr hlich and sokal and yields the mean field bound 5 2 and the other implies the result of chayes and chayes that an inequality identical to 2 is known for ising models where it provides the basis for newman s universal relation 5 1 1 and for certain extrapolation principles which are now made applicable also to independent percolation these results apply to both finite and long range models with or without orientation and extend to periodic and weakly inhomogeneous systems