on dimension of porous measures we introduce a notion of mean porosity for measures and obtain dimensional bounds for mean porous and porous measures as a corollary we generalize to all porous borel measures the estimate obtained recently by j p eckmann e jarvenpaa and m jarvenpaa for porous measures satisfying the doubling condition we also discuss various generalizations of this notion and possible applications

new families of binary sequences with optimal three level autocorrelation in this correspondence we give several new families of binary sequences of period n with optimal three level autocorrelation where n spl equiv 2 mod 4 these sequences are either balanced or almost balanced our construction is based on cyclotomy

uniform rectifiability and harmonic measure ii poisson kernels in l p imply uniform rectifiability the first author was supported by nsf grants dms 0801079 and dms 1101244 the second author was supported by micinn grant mtm2010 16518 and by csic pie 200850i015 the third author was partially supported by grants dms 0901524 career dms 1056965 us nsf sloan research foundation and mtm2010 16232 mtm2009 14694 c02 01 spain

spectral convergence and nonlinear dynamics of reaction diffusion equations under perturbations of the domain abstract in this paper we obtain the continuity of attractors for semilinear parabolic problems with neumann boundary conditions relatively to perturbations of the domain we show that if the perturbations on the domain are such that the convergence of eigenvalues and eigenfunctions of the neumann laplacian is granted then we obtain the upper semicontinuity of the attractors if moreover every equilibrium of the unperturbed problem is hyperbolic we also obtain the continuity of attractors we also give necessary and sufficient conditions for the spectral convergence of neumann problems under perturbations of the domain

perturbations of elliptic operators in 1 sided chord arc domains part i small and large perturbation for symmetric operators the first and third authors acknowledge financial support from the spanish ministry of economy and competitiveness through the severo ochoa programme for centres of excellence in r d sev 2015 0554 they also acknowledge that the research leading to these results has received funding from the european research council under the european union s seventh framework programme fp7 2007 2013 erc agreement no 615112 hapdegmt the second author was supported by nsf grant dms 1664047

continuity of the dynamics in a localized large diffusion problem with nonlinear boundary conditions abstract this paper is concerned with singular perturbations in parabolic problems subjected to nonlinear neumann boundary conditions we consider the case for which the diffusion coefficient blows up in a subregion 0 which is interior to the physical domain r n we prove under natural assumptions that the associated attractors behave continuously as the diffusion coefficient blows up locally uniformly in 0 and converges uniformly to a continuous and positive function in 1 0

harmonic measure and quantitative connectivity geometric characterization of the l p solvability of the dirichlet problem it is well known that quantitative scale invariant absolute continuity more precisely the weak a infty property of harmonic measure with respect to surface measure on the boundary of an open set omega subset mathbb r n 1 with ahlfors david regular boundary is equivalent to the solvability of the dirichlet problem in omega with data in l p partial omega for some p infty in this paper we give a geometric characterization of the weak a infty property of harmonic measure and hence of solvability of the l p dirichlet problem for some finite p this characterization is obtained under background hypotheses an interior corkscrew condition along with ahlfors david regularity of the boundary that are natural and in a certain sense optimal we provide counter examples in the absence of either of them or even one of the two upper or lower ahlfors david bounds moreover the examples show that the upper and lower ahlfors david bounds are each quantitatively sharp

high rate self synchronizing codes self synchronization under the presence of additive noise can be achieved by allocating a certain number of bits of each codeword as markers for synchronization difference systems of sets are combinatorial designs which specify the positions of synchronization markers in codewords in such a way that the resulting error tolerant self synchronizing codes may be realized as cosets of linear codes ideally difference systems of sets should sacrifice as few bits as possible for a given code length alphabet size and error tolerance capability however it seems difficult to attain optimality with respect to known bounds when the noise level is relatively low in fact the majority of known optimal difference systems of sets are for exceptionally noisy channels requiring a substantial amount of bits for synchronization to address this problem we present constructions for difference systems of sets that allow for higher information rates while sacrificing optimality to only a small extent our constructions utilize optimal difference systems of sets as ingredients and when applied carefully generate asymptotically optimal ones with higher information rates we also give direct constructions for optimal difference systems of sets with high information rates and error tolerance that generate binary and ternary self synchronizing codes

interaction of localized large diffusion and boundary conditions abstract in this paper we derive the limiting pdes for parabolic problems where localized large diffusion touches the boundary of the domain hence the limit problem reflects the interaction of large diffusion and boundary conditions the limit problem is a pde coupled with a system of odes through some nonlocal terms we study the well posedness and dissipativity of the nonstandard limit problem as well as the continuity of the dynamics as diffusion becomes large

dahlberg s theorem in higher co dimension abstract in 1977 the celebrated theorem of b dahlberg established that the harmonic measure is absolutely continuous with respect to the hausdorff measure on a lipschitz graph of dimension n 1 in r n and later this result has been extended to more general non tangentially accessible domains and beyond in the present paper we prove the first analogue of dahlberg s theorem in higher co dimension on a lipschitz graph of dimension d in r n d n 1 with a small lipschitz constant we construct a linear degenerate elliptic operator l such that the corresponding harmonic measure l is absolutely continuous with respect to the hausdorff measure on more generally we provide sufficient conditions on the matrix of coefficients of l which guarantee the mutual absolute continuity of l and the hausdorff measure

σ finiteness of elliptic measures for quasilinear elliptic pde in space abstract in this paper we study the hausdorff dimension of a elliptic measure f in space associated to a positive weak solution to a certain quasilinear elliptic pde in an open subset and vanishing on a portion of the boundary of that open set we show that this measure is concentrated on a set of finite n 1 dimensional hausdorff measure for p n and the same result holds for p n with an assumption on the boundary we also construct an example of a domain in space for which the corresponding measure has hausdorff dimension n 1 for p n for some which depends on various constants including p the first result generalizes the authors previous work in 3 when the pde is the p laplacian and the second result generalizes the well known theorem of wolff in 24 when p 2 and n 2

rectifiability and elliptic measures on 1 sided nta domains with ahlfors david regular boundaries let omega subset mathbb r n 1 n geq 2 be 1 sided nta domain aka uniform domain i e a domain which satisfies interior corkscrew and harnack chain conditions and assume that partial omega is n dimensional ahlfors david regular we characterize the rectifiability of partial omega in terms of the absolute continuity of surface measure with respect to harmonic measure we also show that these are equivalent to the fact that partial omega can be covered mathcal h n a e by a countable union of portions of boundaries of bounded chord arc subdomains of omega and to the fact that partial omega possesses exterior corkscrew points in a qualitative way mathcal h n a e our methods apply to harmonic measure and also to elliptic measures associated with real symmetric second order divergence form elliptic operators with locally lipschitz coefficients whose derivatives satisfy a natural qualitative carleson condition

capacity and principal eigenvalues the method of enlargement of obstacles revisited we describe a coarse graining method which provides lower bounds on the principal dirichlet eigenvalue of the laplacian in regions receiving small obstacles and sharpens the previous method of enlargement of obstacles based on a quantitative wiener criterion one replaces the actual obstacles by obstacles of a much larger size controls on the shift of principal eigenvalues and capacity estimates on the locus where the wiener criterion breaks down are derived the results are written in a self contained fashion

on the hausdorff dimension of harmonic measure in higher dimension assume a i n e a domain in f n where e is a compact set denote o a a x the harmonic measure for a of a evaluated at x r d according to 0ksendal s theorem o o e o a x is singular with respect to d dimensional lebesgue measure for d 2 and general domains this result seemed to be so far the only known localization property recently for d 2 it has been shown by p jones and t wolff j w that s o9 has dimension at most 1 this result completes previous work due to n g makarov and l carleson see i m and c2 let g be the green s function of a r with pole at some point of c r o9 in both the carleson and jones wolff arguments the integral g g nn log nn as boundary

continuity of attractors for parabolic problems with localized large diffusion abstract in this paper we study the continuity of asymptotics of semilinear parabolic problems of the form u t div p x u u f u in a bounded smooth domain r n with dirichlet boundary conditions when the diffusion coefficient p becomes large in a subregion 0 which is interior to the physical domain we prove under suitable assumptions that the family of attractors behave upper and lower semicontinuously as the diffusion blows up in 0

intersection equivalence of brownian paths and certain branching processes we show that sample paths of brownian motion and other stable processes intersect the same sets as certain random cantor sets constructed by a branching process with this approach the classical result that two independent brownian paths in four dimensions do not intersect reduces to the dying out of a critical branching process and estimates due to lawler 1982 for the long range intersection probability of several random walk paths reduce to kolmogrov s 1938 law for the lifetime of a critical branching process extensions to random walks with long jumps and applications to hausdorff dimension are also derived

continuity of the attractors in a singular problem arising in composite materials abstract in this paper we consider the second order parabolic problem pe u t e p e x u x e x c x u e f u e in 0 1 u e n b x u e g u e for x 0 1 where the diffusion coefficient becomes large in a subset which is interior to the interval 0 1 as e 0 we prove the existence of invariant manifolds for pe and using the transversality of invariant manifolds of hyperbolic points obtained in v l carbone j g ruas filho transversality of stable and unstable manifolds for parabolic problems arising in composite materials j math anal appl 303 2005 220 241 we prove that equation pe and its limit problem are topologically equivalent on the attractors

upper semicontinuity for attractors of parabolic problems with localized large diffusion and nonlinear boundary conditions the motivations to study the problem considered in this paper come from the theory of composite materials where the heat diffusion properties can change from one part of the domain to another mathematically this leads to a nonlinear second order parabolic equation for which the diffusion coefficient becomes large in a subdomain 0 the equation is supplemented by a nonlinear boundary condition on and an initial condition the authors determine the form of the limit problem the so called shadow system which involves an evolution equation for the averages of the density over 0 the main results include global in time existence of solutions and upper semicontinuity of the associated global attractors when the system approaches the shadow system

parabolic problems with nonlinear boundary conditions and critical nonlinearities we prove existence uniqueness and regularity of solutions for heat equations with nonlinear boundary conditions we study these problems with initial data in l q omega w 1 w q omega 1 q infinity or measures and with critically growing non linearities

critical large deviations of one dimensional annealed brownian motion in a poissonian potential we derive a large deviation principle for the position at large times t of a one dimensional annealed brownian motion in a poissonian potential in the critical spatial scale t 1 3 here annealed means that averages are taken with respect to both the path and environment measures in contrast to the d dimensional case for d 2 in the critical scale t d d 2 as treated by sznitman the rate function which measures the large deviations exhibits three different regimes these regimes depend on the position of the path at time t our large deviation principle has a natural application to the study of a one dimensional annealed brownian motion with a constant drift in a poissonian potential

brownian asymptotics in a poissonian environment we consider a brownian motion moving in a random potential obtained by translating a given fixed non negative shape function at the points of a poisson cloud we derive the almost sure principal long time behavior of the expectation of the natural feynman kac functional which is insensitive to the detail of the shape function we also study the situation of hard obstacles where brownian motion is killed once it comes within distancea of a point of the cloud the nature of the results then changes between the case whena is small or large in connection with the presence or absence of an infinite component in the complement of the obstacles

biased random walk conditioned on survival among bernoulli obstacles subcritical phase we consider a discrete time biased random walk conditioned to avoid bernoulli obstacles on mathbb z d d ge 2 up to time n this model is known to undergo a phase transition for a large bias the walk is ballistic whereas for a small bias it is sub ballistic we prove that in the sub ballistic phase the random walk is contained in a ball of radius o n 1 d 2 which is the same scale as for the unbiased case as an intermediate step we also prove large deviation principles for the endpoint distribution for the unbiased random walk at scales between n 1 d 2 and o n d d 2 these results improve and complement earlier work by sznitman ann sci ecole norm sup 4 28 3 345 370 371 390 1995

a new construction of zero difference balanced functions and its applications in this paper a new construction of zero difference balanced functions defined on is given where is an odd positive integer based on the generic constructions proposed by ding optimal constant composition codes and perfect difference systems of sets with new parameters can be generated from the zero difference balanced functions constructed in this paper

a implies nta for a class of variable coefficient elliptic operators abstract we consider a certain class of second order variable coefficient divergence form elliptic operators in a uniform domain with ahlfors regular boundary and we show that the a property of the elliptic measure associated to any such operator and its transpose imply that the domain is in fact nta and hence chord arc the converse was already known and follows from work of kenig and pipher 30

dynamics in dumbbell domains i continuity of the set of equilibria we analyze the dynamics of a reaction diffusion equation with homogeneous neumann boundary conditions in a dumbbell domain we provide an appropriate functional setting to treat this problem and as a first step we show in this paper the continuity of the set of equilibria and of its linear unstable manifolds

almost difference sets and their sequences with optimal autocorrelation almost difference sets have interesting applications in cryptography and coding theory we give a well rounded treatment of known families of almost difference sets establish relations between some difference sets and some almost difference sets and determine the numerical multiplier group of some families of almost difference sets we also construct six new classes of almost difference sets and four classes of binary sequences of period n spl equiv 0 mod 4 with optimal autocorrelation we have also obtained two classes of relative difference sets and four classes of divisible difference sets ddss we also point out that a result due to jungnickel 1982 can be used to construct almost difference sets and sequences of period 4l with optimal autocorrelation

on the structure of partial balayage abstract we define partial balayage construct it with classical potential theoretic methods and establish the so called structure formula in full generality finally we give a stochastic construction of partial balayage

some new classes of zero difference balanced functions zero difference balanced zdb functions were introduced recently by ding for the construction of optimal constant composition codes and optimal and perfect difference systems of sets they are closely related to partitioned difference families in this paper we present generic constructions of zdb functions from functions with difference balanced property in particular two classes of zdb functions with new and flexible parameters are reported employing these new zdb functions we obtain at the same time optimal 1 constant composition codes 2 constant weight codes and 3 perfect difference systems of sets all with new and flexible parameters

the harmonic measure of balls in random trees we study properties of the harmonic measure of balls in typical large discrete trees for a ball of radius nn centered at the root we prove that although the size of the boundary is of order nn most of the harmonic measure is supported on a boundary set of size approximately equal to n n where 0 78 0 78 is a universal constant to derive such results we interpret harmonic measure as the exit distribution of the ball by simple random walk on the tree and we first deal with the case of critical galton watson trees conditioned to have height greater than nn an important ingredient of our approach is the analogous continuous model related to aldous continuum random tree where the dimension of harmonic measure of a level set of the tree is equal to whereas the dimension of the level set itself is equal to 11 the constant is expressed in terms of the asymptotic distribution of the conductance of large critical galton watson trees

asymptotic survival probabilities in the random saturation process we consider a model of diffusion in random media with a two way coupling i e a model in which the randomness of the medium influences the diffusing particles and where the diffusing particles change the medium in this particular model particles are injected at the origin with a time dependent rate and diffuse among random traps each trap has a finite random depth so that when it has absorbed a finite random number of particles it is saturated and it no longer acts as a trap this model comes from a problem of nuclear waste management however a very similar model has been studied recently by gravner and quastel with different tools hydrodynamic limits we compute the asymptotic behavior of the probability of survival of a particle born at some given time both in the annealed and quenched cases and show that three different situations occur depending on the injection rate for weak injection the typical survival strategy of the particle is as in sznitman and the asymptotic behavior of this survival probability behaves as if there was no saturation effect for medium injection rate the picture is closer to that of internal dla as given by lawler bramson and griffeath for large injection rates the picture is less understood except in dimension one

optimal partitioned cyclic difference packings for frequency hopping and code synchronization optimal partitioned cyclic difference packings pcdps are shown to give rise to optimal frequency hopping sequences and optimal comma free codes new constructions for pcdps based on almost difference sets and cyclic difference matrices are given these produce new infinite families of optimal pcdps and hence optimal frequency hopping sequences and optimal comma free codes the existence problem for optimal pcdps in bbz3m with m base blocks of size three is also solved for all m 8 16 mod 24

confinement of brownian motion among poissonian obstacles in ℝ d d 3 consider brownian motion among random obstacles obtained by translating a fixed compact nonpolar subset of d d 1 at the points of a poisson cloud of constant intensity v 0 assume that brownian motion is absorbed instantaneously upon entering the obstacle set in szn conf sznitman has shown that in d 2 conditionally on the event that the process does not enter the obstacle set up to time t the probability that brownian motion remains within distance t1 4 from its starting point is going to 1 as t goes to infinity we show that the same result holds true for d 3 with t1 4 replaced by t1 d 2 the proof is based on sznitmans refined method of enlargement of obstacles 10 as well as on a quantitative isoperimetric inequality due to hall 4

optimal frequency hopping sequences a combinatorial approach frequency hopping multiple access fhma spread spectrum communication systems employing multiple frequency shift keying mfsk as data modulation technique are investigated from a combinatorial approach a correspondence between optimal frequency hopping fh sequences and partition type difference packings is first established by virtue of this correspondence fhma systems with a single optimal fh sequence each are constructed from various combinatorial structures such as affine geometries cyclic designs and difference families combinatorial recursive constructions are also presented many new infinite series of optimal fh sequences are thus obtained these new fh sequences are also useful in ultra wideband uwb communication systems

asymptotic properties in parabolic problems dominated by a p laplacian operator with localized large diffusion abstract this paper is concerned with upper semicontinuity of the family of attractors associated with nonlinear reaction diffusion equations with principal part governed by a degenerate p laplacian in which the diffusion d blows up in localized regions inside the domain

spectrum bottom and largest vacuity the paper considers a scalar second order elliptic operator with a non negative random potential term v x x w x x 0 corresponding to a poisson cloud x of soft obstacles the operator acts on functions vanishing outside a large cubic open box rq 0 r r d d d 2 the paper develops a method of estimating from below the spectrum bottom of the operator through the volume of the largest connected set that can be made of smaller blocks containing relatively few obstacles in the case of constant coefficients the principal eigenvalue v r of v in r 0 is shown to satisfy with high probability the estimate where is the infimum of principal value of operator with zero potential term v 0 under the dirichlet condition on the boundary of a regular set of volume not exceeding one

hausdorff dimension and σ finiteness of p harmonic measures in space when p n abstract in this paper we study a measure associated with a positive p harmonic function u defined in an open set o r n and vanishing on a portion of o if p n we show is concentrated on a set of finite h n 1 measure while if p n the same conclusion holds provided is uniformly fat in the sense of n capacity our work nearly answers in the affirmative a conjecture in lewis 2015 and also appears to be the natural extension of jones and wolff 1988 wolff 1993 to higher dimensions

cyclotomic constructions of zero difference balanced functions with applications based on a recent work of ding wang and xiong we present some cyclotomic constructions of zero difference balanced zdb functions which interpret the previous construction proposed by cai zeng helleseth tang and yang the resulted zdb functions can be used to generate optimal constant composition codes and perfect difference systems of sets with new parameters

fluctuations of principal eigenvalues and random scales we investigate the principal dirichlet eigenvalue of the laplacian with soft poissonian obstacles in large boxes of d 2 with the help of our recent version of the method of enlargement of obstacles 18 we derive quantitative confidence intervals for these eigenvalues we also provide less quantitative estimates which however point out the correct size of fluctuations and indicate a stiffness in their behavior in the two dimensional case we derive geometric controls which relate these eigenvalues to certain empty circular droplets our results also have natural applications to the study of the location of minima of certain intermittent random variational problems motivated by 13 17

dirichlet forms quasiregular functions and brownian motion given a quasiregular function on an open setu in n it is shown that there exists a diffusionx t inu such that mapsx t inton dimensional brownian motion the process is constructed from a dirichlet form which can be described explicitly this enables us to apply stochastic methods in the investigation of quasiregular mappings some examples of applications are given including boundary behaviour and value distribution

dissipation in laplacian fields across irregular boundaries the entropy production associated to a laplacian field distributed across irregular boundaries is studied in the context of the active zone approximation an explicit expression is given for the entropy production in terms of geometry whose relation to the variational formulation is discussed it is shown that the entropy production diminishes for successive prefractal generations of the same fractal generator so that the final fractal object is expected to dissipate less than all previous ones the relevance of this result in the abundance of fractal surfaces or interfaces observed in nature is discussed