a class of spectral moran measures abstract let d k k 1 be a sequence of digit sets in n and let b k k 1 be a sequence of integer numbers bigger than 1 we call the family f k d k x b k 1 x d d d k k 1 a moran iterated function system ifs which is a natural generalization of an ifs we prove under certain conditions in terms of b k d k that the associated moran measure is a spectral measure i e there exists a countable set n such that e 2 i x is an orthonormal basis for l 2

nonconventional limit theorems the polynomial ergodic theorem pet which appeared in bergelson ergod th dynam sys 7 337 349 1987 and attracted substantial attention in ergodic theory studies the limits of expressions having the form 1 n sum n 1 nt q 1 n f 1 cdots t q ell n f ell where t is a weakly mixing measure preserving transformation f i s are bounded measurable functions and q i s are polynomials taking on integer values on the integers motivated partially by this result we obtain a central limit theorem for even more general expressions of the form 1 sqrt n sum n 1 n f x 0 n x 1 q 1 n x 2 q 2 n ldots x ell q ell n overline f where x i s are exponentially fast mixing bounded processes with some stationarity properties f is a lipschitz continuous function overline f int fd mu 0 times mu 1 times cdots times mu ell j is the distribution of x j 0 and q i s are positive functions taking on integer values on integers with some growth conditions which are satisfied for instance when q i s are polynomials of growing degrees when f x 0 x 1 x l x 0 x 1 x 2 x l exponentially fast mixing already suffices this result can be applied in the case when x i n t n f i where t is a mixing subshift of finite type a hyperbolic diffeomorphism or an expanding transformation taken with a gibbs invariant measure as well as in the case when x i n f i n where n is a markov chain satisfying the doeblin condition considered as a stationary process with respect to its invariant measure

on the spectra of a class of self affine measures on r2 abstract for i 1 2 let 0 i 1 for an expanding real matrix m d i a g 1 1 2 1 and a three element digit set d z 2 with cardinality d let m d be the self affine measure defined by m d 1 d d d m d m d let f 3 2 1 3 l 1 l 2 t l 1 l 2 n 0 l 1 l 2 2 and z m d x r 2 d d e 2 i d x 0 if a f 3 2 z m d a mod z 2 and x 0 1 2 d d e 2 i d x d 0 0 t then l 2 m d has an exponential orthonormal basis if and only if i 1 3 z 0 for i 1 2

spectrality of a class of infinite convolutions for a finite set d z and an integer b 2 we say that b d is compatible with c z if e 2 idc b d d c c is a hadamard matrix let e 1 e a e a denote the uniformly discrete probability measure on e we prove that the class of infinite convolution moran measure b dk b 1d1 b 2d2 is a spectral measure provided that there is a common c z compatible to all the b dk and c c 0 1 b 1 we also give some examples to illustrate the hypotheses and results in particular the last condition on c is essential

analyticity of periodic traveling free surface water waves with vorticity we prove that the prole of a periodic traveling wave propagating at the surface of water above a at bed in a ow with a real analytic vorticity must be real analytic provided the wave speed exceeds the horizontal uid velocity throughout the ow the real analyticity of each streamline be

estimation in spin glasses a first step the sherrington kirkpatrick model of spin glasses the hopfield model of neural networks and the ising spin glass are all models of binary data belonging to the one parameter exponential family with quadratic sufficient statistic under bare minimal conditions we establish the n consistency of the maximum pseudolikelihood estimate of the natural parameter in this family even at critical temperatures since very little is known about the low and critical temperature regimes of these extremely difficult models the proof requires several new ideas the author s version of stein s method is a particularly useful tool we aim to introduce these techniques into the realm of mathematical statistics through an example and present some open questions

spectral property of a class of moran measures on r abstract let b 2 be a positive integer let d be a finite subset of z and n k k 1 n be a sequence of strictly increasing numbers a moran measure b d n k is a borel probability measure generated by the moran iterated function system moran ifs f k d x b n k 1 n k x d d d k n n 0 0 in this paper we study one of the basic problems in fourier analysis associated with b d n k more precisely we give some conditions under which the measure b d n k is a spectral measure i e there exists a discrete subset r such that e e 2 i x is an orthonormal basis for l 2 b d n k

spectral property of cantor measures with consecutive digits abstract we consider equally weighted cantor measures q b arising from iterated function systems of the form b 1 x i i 0 1 q 1 where q b we classify the q b so that they have infinitely many mutually orthogonal exponentials in l 2 q b in particular if q divides b the measures have a complete orthogonal exponential system and hence spectral measures improving the construction by dutkay et al 2009 3 we characterize all the maximal orthogonal sets when q divides b via a maximal mapping on the q adic tree in which all elements in are represented uniquely in finite b adic expansions and we can separate the maximal orthogonal sets into two types regular and irregular sets for a regular maximal orthogonal set we show that its completeness in l 2 q b is crucially determined by the certain growth rate of non zero digits in the tail of the b adic expansions of the elements furthermore we exhibit complete orthogonal exponentials with zero beurling dimensions these examples show that the technical condition in theorem 3 5 of dutkay et al 2011 4 cannot be removed for an irregular maximal orthogonal set we show that under some conditions its completeness is equivalent to that of the corresponding regularized mapping

on the cauchy problem for the two component camassa holm system in this paper we establish the local well posedness for the two component camassa holm system in a range of the besov spaces we also derive a wave breaking mechanism for strong solutions in addition we determine the exact blow up rate of such solutions to the system

stability of peakons the peakons are peaked solitary wave solutions of a certain nonlinear dispersive equation that is a model in shallow water theory and the theory of hyperelastic rods we give a very simple proof of the orbital stability of the peakons in the h1 norm 2000 john wiley sons inc

estimating and understanding exponential random graph models we introduce a method for the theoretical analysis of exponential random graph models the method is based on a large deviations approximation to the normalizing constant shown to be consistent using theory developed by chatterjee and varadhan european j combin 32 2011 1000 1017 the theory explains a host of difficulties encountered by applied workers many distinct models have essentially the same mle rendering the problems practically ill posed we give the first rigorous proofs of degeneracy observed in these models here almost all graphs have essentially no edges or are essentially complete we supplement recent work of bhamidi bresler and sly 2008 ieee 49th annual ieee symposium on foundations of computer science focs 2008 803 812 ieee showing that for many models the extra sufficient statistics are useless most realizations look like the results of a simple erd s renyi model we also find classes of models where the limiting graphs differ from erd s renyi graphs a limitation of our approach inherited from the limitation of graph limit theory is that it works only for dense graphs

nonconventional large deviations theorems we obtain large deviations theorems for both discrete time expressions of the form sum n 1 nf big x q 1 n ldots x q ell n big and similar expressions of the form int 0 tf big x q 1 t ldots x q ell t big dt in continuous time here x n n ge 0 or x t t ge 0 is a markov process satisfying doeblin s condition f is a bounded continuous function and q i n in for i le k while for i k they are positive functions taking on integer values on integers with some growth conditions which are satisfied for instance when q i s are polynomials of increasing degrees applications to some types of dynamical systems such as mixing subshifts of finite type and hyperbolic and expanding transformations will be obtained as well

the extremal index hitting time statistics and periodicity abstract the extremal index appears as a parameter in extreme value laws for stochastic processes characterising the clustering of extreme events we apply this idea in a dynamical systems context to analyse the possible extreme value laws for the stochastic process generated by observations taken along dynamical orbits with respect to various measures we derive new easily checkable conditions which identify extreme value laws with particular extremal indices in the dynamical context we prove that the extremal index is associated with periodic behaviour the analogy of these laws in the context of hitting time statistics as studied in the authors previous works on this topic is explained and exploited extensively allowing us to prove for the first time the existence of hitting time statistics for balls around periodic points moreover for very well behaved systems uniformly expanding we completely characterise the extremal behaviour by proving that either we have an extremal index less than 1 at periodic points or equal to 1 at any other point this theory then also applies directly to general stochastic processes adding both useful tools to identify the extremal index and giving deeper insight into the periodic behaviour it suggests

on a shallow water model with the coriolis effect abstract in the present study an asymptotic model for wave propagation in shallow water with the effect of the coriolis force is derived from the governing equation in two dimensional flows the transport equation theory is then applied to investigate the local well posedness and wave breaking phenomena for this model the nonexistence of the camassa holm type peaked solution and classification of various traveling wave solutions to the new system are also established moreover it is shown that all the symmetric waves to this model are traveling waves

nonuniform sampling and reconstruction in shift invariant spaces this article discusses modern techniques for nonuniform sampling and reconstruction of functions in shift invariant spaces it is a survey as well as a research paper and provides a unified framework for uniform and nonuniform sampling and reconstruction in shift invariant subspaces by bringing together wavelet theory frame theory reproducing kernel hilbert spaces approximation theory amalgam spaces and sampling inspired by applications taken from communication astronomy and medicine the following aspects will be emphasized a the sampling problem is well defined within the setting of shift invariant spaces b the general theory works in arbitrary dimension and for a broad class of generators c the reconstruction of a function from any sufficiently dense nonuniform sampling set is obtained by efficient iterative algorithms these algorithms converge geometrically and are robust in the presence of noise d to model the natural decay conditions of real signals and images the sampling theory is developed in weighted l p spaces

on stratified steady periodic water waves with linear density distribution and stagnation points abstract we use bifurcation theory to construct small periodic gravity stratified water waves with density which depends linearly upon the pseudostream function as a special feature the density may also decrease with depth and the waves we obtain may posses two different critical layers with cat s eye vortices within the vortex the density of the fluid has an extremum at the stagnation point

persistence properties and unique continuation of solutions of the camassa holm equation it is shown that a strong solution of the camassa holm equation initially decaying exponentially together with its spacial derivative must be identically equal to zero if it also decays exponentially at a later time in particular a strong solution of the cauchy problem with compact initial profile can not be compactly supported at any later time unless it is the zero solution

on the spectra of sierpinski type self affine measures abstract for a class of sierpinski type self affine measures on r 2 a characterization for all maximal orthogonal sets of exponentials is given by this characterization a sufficient condition for a maximal orthogonal set to be a basis of l 2 is given

taming correlations through entropy efficient measure decompositions with applications to mean field approximation the analysis of various models in statistical physics relies on the existence of decompositions of measures into mixtures of product like components where the goal is to attain a decomposition into measures whose entropy is close to that of the original measure yet with small correlations between coordinates we prove a related general result for every measure mu on mathbb r n and every varepsilon 0 there exists a decomposition mu int mu theta d m theta such that h mu mathbb e theta sim m h mu theta le mathrm tr mathrm cov mu varepsilon and mathbb e theta sim m mathrm cov mu theta preceq mathrm id varepsilon as an application we derive a general bound for the mean field approximation of ising and potts models which is in a sense dimension free in both continuous and discrete settings in particular for an ising model on pm 1 n or on 1 1 n we show that the deficit between the mean field approximation and the free energy is at most c frac 1 p p left n vert j vert s p right frac p 1 p for all p 0 where vert j vert s p denotes the schatten p norm of the interaction matrix for the case p 2 this recovers the result of jain et al mean field approximation convex hierarchies and the optimality of correlation rounding a unified perspective arxiv 1808 07226 2018 but for an optimal choice of p it often allows to get almost dimension free bounds

analysis of orthogonality and of orbits in affine iterated function systems we introduce a duality for affine iterated function systems aifs which is naturally motivated by group duality in the context of traditional harmonic analysis our affine systems yield fractals defined by iteration of contractive affine mappings we build a duality for such systems by scaling in two directions fractals in the small by contractive iterations and fractals in the large by recursion involving iteration of an expansive matrix by a fractal in the small we mean a compact attractor x supporting hutchinson s canonical measure and we ask when is a spectral measure i e when the hilbert space l 2 mu has an orthonormal basis onb of exponentials e lambda lambda in lambda we further introduce a fourier duality using a matched pair of such affine systems using next certain extreme cycles and positive powers of the expansive matrix we build fractals in the large which are modeled on lacunary fourier series and which serve as spectra for x our two main results offer simple geometric conditions allowing us to decide when the fractal in the large is a spectrum for x our results in turn are illustrated with concrete sierpinski like fractals in dimensions 2 and 3

global weak solutions for a two component camassa holm shallow water system abstract in this paper we prove the existence of global weak solution for an integrable two component camassa holm shallow water system provided the initial data satisfying some certain conditions

on spectral n bernoulli measures abstract for 0 1 and n 1 an integer let be the self similar measure defined by i 0 n 1 1 n 1 i we prove that l 2 has an exponential orthonormal basis if and only if 1 q for some q 0 and n divides q the special case is the cantor measure with 1 2 k and n 2 16 which was proved recently to be the only spectral measure among the bernoulli convolutions with 0 1 4

instability criteria for the flow of an inviscid incompressible fluid we present a geometric estimate from below on the growth rate of a small perturbation of a three dimensional steady flow of an ideal fluid and thus we obtain effective criteria for local instability for euler s equations we use these criteria to demonstrate the instability of several simple flows and to show that any flow with a hyperbolic stagnation point is unstable

on a shallow water approximation to the green naghdi equations with the coriolis effect abstract we consider an asymptotic 1d in space rotation camassa holm r ch model which could be used to describe the propagation of long crested shallow water waves in the equatorial ocean regions with allowance for the weak coriolis effect due to the earth s rotation this model equation has similar wave breaking phenomena as the camassa holm equation it is analogous to the rotation green naghdi r gn equations with the weak earth s rotation effect modeling the propagation of wave allowing large amplitude in shallow water we provide here a rigorous justification showing that solutions of the r gn equations tend to associated solution of the r ch model equation in the camassa holm regime with the small amplitude and the larger wavelength furthermore we demonstrate that the r gn model equations are locally well posed in a sobolev space by the refined energy estimates

phase transitions in edge weighted exponential random graphs near degeneracy and universality conventionally used exponential random graphs cannot directly model weighted networks as the underlying probability space consists of simple graphs only since many substantively important networks are weighted this limitation is especially problematic we extend the existing exponential framework by proposing a generic common distribution for the edge weights minimal assumptions are placed on the distribution that is it is non degenerate and supported on the unit interval by doing so we recognize the essential properties associated with near degeneracy and universality in edge weighted exponential random graphs

the cardinality of orthogonal exponentials of planar self affine measures with three element digit sets abstract in this paper we consider the planar self affine measures m d generated by an expanding matrix m m 2 z and an integer digit set d 0 0 1 2 1 2 with 1 2 2 1 0 we show that if det m 3 z then the mutually orthogonal exponential functions in l 2 m d is finite and the exact maximal cardinality is given

poisson and compound poisson approximations in conventional and nonconventional setups it was shown in kifer israel j math 2013 that for any subshift of finite type considered with a gibbs invariant measure the numbers of multiple recurrencies to shrinking cylindrical neighborhoods of almost all points are asymptotically poisson distributed here we not only extend this result to all psi mixing shifts with countable alphabet but actually show that for all points the distributions of these numbers are asymptotically close either to poisson or to compound poisson distributions it turns out that for all nonperiodic points a limiting distribution is always poisson while at the same time for periodic points there may be no limiting distribution at all unless the shift invariant measure is bernoulli in which case the limiting distribution always exists thus we describe essentially completely limiting distributions of multiple recurrence numbers in this setup as a corollary we obtain also that the first occurence time of the multiple recurrence event is asymptotically exponentially distributed most of the results are new also for the widely studied single recurrencies case see for instance haydn and vaienti discret contin dyn syst a 10 589 616 2004 probab theory relat fields 144 517 542 2009 abadi and saussol stoch process appl 121 314 323 2011 abadi and vergne nonlinearity 21 2871 2885 2008 as well

on the spectra of a cantor measure abstract we analyze all orthonormal bases of exponentials on the cantor set defined by jorgensen and pedersen in j anal math 75 1998 185 228 a complete characterization for all maximal sets of orthogonal exponentials is obtained by establishing a one to one correspondence with the spectral labelings of the infinite binary tree with the help of this characterization we obtain a sufficient condition for a spectral labeling to generate a spectrum an orthonormal basis this result not only provides us an easy and efficient way to construct various of new spectra for the cantor measure but also extends many previous results in the literature in fact most known examples of orthonormal bases of exponentials correspond to spectral labelings satisfying this sufficient condition we also obtain two new conditions for a labeling tree to generate a spectrum when other digits digits not necessarily in 0 1 2 3 are used in the base 4 expansion of integers and when bad branches are allowed in the spectral labeling these new conditions yield new examples of spectra and in particular lead to a surprizing example which shows that a maximal set of orthogonal exponentials is not necessarily an orthonormal basis

the compound poisson limit ruling periodic extreme behaviour of non uniformly hyperbolic dynamics we prove that the distributional limit of the normalised number of returns to small neighbourhoods of periodic points of certain non uniformly hyperbolic dynamical systems is compound poisson the returns to small balls around a fixed point in the phase space correspond to the occurrence of rare events or exceedances of high thresholds so that there is a connection between the laws of return times statistics and extreme value laws the fact that the fixed point in the phase space is a repelling periodic point implies that there is a tendency for the exceedances to appear in clusters whose average sizes is given by the extremal index which depends on the expansion of the system at the periodic point we recall that for generic points the exceedances in the limit are singular and occur at poisson times however around periodic points the picture is different the respective point processes of exceedances converge to a compound poisson process so instead of single exceedances we have entire clusters of exceedances occurring at poisson times with a geometric distribution ruling its multiplicity the systems to which our results apply include general piecewise expanding maps of the interval rychlik maps maps with indifferent fixed points manneville pomeau maps and benedicks carleson quadratic maps

convergent sequences of dense graphs i subgraph frequencies metric properties and testing we consider sequences of graphs gn and define various notions of convergence related to these sequences left convergence defined in terms of the densities of homomorphisms from small graphs into gn right convergence defined in terms of the densities of homomorphisms from gn into small graphs and convergence in a suitably defined metric in part i of this series we show that left convergence is equivalent to convergence in metric both for simple graphs gn and for graphs gn with nodeweights and edgeweights one of the main steps here is the introduction of a cut distance comparing graphs not necessarily of the same size we also show how these notions of convergence provide natural

non uniform dependence on initial data of solutions to the euler equations of hydrodynamics we show that continuous dependence on initial data of solutions to the euler equations of incompressible hydrodynamics is optimal more precisely we prove that the data to solution map is not uniformly continuous in sobolev h s topology for any s in mathbb r if the domain is the flat torus mathbb t n mathbb r n 2 pi mathbb z n and for any s 0 if the domain is the whole space mathbb r n

spectral gaps for sets and measures if x is a closed subset of the real line denote by gx the supremum of the size of the gap in the fourier spectrum of a measure taken over all non trivial finite complex measures supported on x in this paper we attempt to find gx in terms of x

on the rotation two component camassa holm system modelling the equatorial water waves abstract in this paper a modified two component camassa holm system with the effect of the coriolis force in the rotating fluid is derived which is a model in the equatorial water waves the effects of the coriolis force caused by the earth s rotation and nonlocal nonlinearities on blow up criteria and wave breaking phenomena are then investigated our refined analysis relies on the method of characteristics and conserved quantities and is proceeded with the riccati type differential inequality finally conditions which guarantee the permanent waves are obtained by using a method of the lyapunov function

ground states for exponential random graphs we propose a perturbative method to estimate the normalization constant in exponential random graph models as the weighting parameters approach infinity as an application we give evidence of discontinuity in natural parametrization along the critical directions of the edge triangle model

applications of stein s method for concentration inequalities stein s method for concentration inequalities was introduced to prove concentration of measure in problems involving complex dependencies such as random permutations and gibbs measures in this paper we provide some extensions of the theory and three applications 1 we obtain a concentration inequality for the magnetization in the curie weiss model at critical temperature where it obeys a nonstandard normalization and super gaussian concentration 2 we derive exact large deviation asymptotics for the number of triangles in the erd s renyi random graph g n p when p 0 31 similar results are derived also for general subgraph counts 3 we obtain some interesting concentration inequalities for the ising model on lattices that hold at all temperatures

on spectral cantor measures a probability measure in rd is called a spectral measure if it has an orthonormal basis consisting of exponentials in this paper we study spectral cantor measures we establish a large class of such measures and give a necessary and sufficient condition on the spectrum of a spectral cantor measure these results extend the studies by jorgensen and pedersen j anal math 75 1998 185 228 and strichartz j d analyse math 81 2000 209 238

uniformity of measures with fourier frames abstract we examine fourier frames and more generally frame measures for different probability measures we prove that if a measure has an associated frame measure then it must have a certain uniformity in the sense that the weight is distributed quite uniformly on its support to be more precise by considering certain absolute continuity properties of the measure and its translation we recover the characterization on absolutely continuous measures g d x with fourier frames obtained in 24 moreover we prove that the frame bounds are pushed away by the essential infimum and supremum of the function g this also shows that absolutely continuous spectral measures supported on a set if they exist must be the standard lebesgue measure on up to a multiplicative constant we then investigate affine iterated function systems ifss we show that if an ifs with no overlap admits a frame measure then the probability weights are all equal moreover we also show that the aba wang conjecture 20 is true if the self similar measure is absolutely continuous finally we will present a new approach to the conjecture of liu and wang 29 about the structure of non uniform gabor orthonormal bases of the form g g j

explicit and exact solutions concerning the antarctic circumpolar current with variable density in spherical coordinates we use spherical coordinates to devise a new exact solution to the governing equations of geophysical fluid dynamics for an inviscid and incompressible fluid with a general density distribution and subjected to forcing terms the latter are of paramount importance for the modeling of realistic flows that is flows that are observed in some averaged sense in the ocean owing to the employment of spherical coordinates we do not need to resort to approximations e g of f and plane type that simplify the geometry in the governing equations our explicit solution represents a steady purely azimuthal stratified flow with a free surface that thanks to the inclusion of forcing terms and the consideration of the earth s geometry via spherical coordinates makes it suitable for describing the antarctic circumpolar current and enables an in depth analysis of the structure of this flow in line with the latter aspect we employ functional analytical techniques to prove that the free surface distortion is defined in a unique and implicit way by means of the pressure applied at the free surface we conclude our discussion by setting out relations between the monotonicity of the surface pressure and the monotonicity of the surface distortion that concur with the physical expectations

on the wave breaking phenomena and global existence for the periodic rotation two component camassa holm system abstract considered herein is the periodic rotation two component camassa holm system which can be derived from the f plane governing equations for the geophysical water waves with a constant underlying current the nonlocal nonlinearities on blow up criteria and wave breaking phenomena are established finally a sufficient condition for global solutions is obtained by using a method of the lyapunov function

limiting entry and return times distribution for arbitrary null sets we describe an approach that allows us to deduce the limiting return times distribution for arbitrary sets to be compound poisson distributed we establish a relation between the limiting return times distribution and the probability of the cluster sizes where clusters consist of the portion of points that have finite return times in the limit where random return times go to infinity in the special case of periodic points we recover the known polya aeppli distribution which is associated with geometrically distributed cluster sizes we apply this method to several examples the most important of which is synchronisation of coupled map lattices for the invariant absolutely continuous measure we establish that the returns to the diagonal is compound poisson distributed where the coefficients are given by certain integrals along the diagonal

sharp error terms and neccessary conditions for exponential hitting times in mixing processes we prove an upper bound for the error in the exponential approximation of the hitting time law of a rare event in mixing processes with exponential decay mixing processes with a summable function and for general mixing processes with a finite alphabet in the first case the bound is uniform as a function of the measure of the event in the last two cases the bound depends also on the time scale t this allows us to get further statistical properties as the ratio convergence of the expected hitting time and the expected return time a uniform bound is a consequence we present an example that shows that this bound is sharp we also prove that second moments are not necessary for having the exponential law moreover we prove a necessary condition for having the exponential limit law

exact free surface equatorial flows with general stratification in spherical coordinates this paper is concerned with the construction of a new exact solution to the geophysical fluid dynamics governing equations for inviscid and incompressible fluid in the equatorial region this solution represents a steady purely azimuthal flow with a free surface the novel aspect of the solution we derive is that the flow it prescribes accommodates a general fluid stratification the density may vary both with depth and with latitude the solution is presented in the terms of spherical coordinates hence at no stage do we invoke approximations by way of simplifying the geometry in the governing equations following the construction of our explicit solution we employ functional analytic considerations to prove that the pressure at the free surface defines implicitly the shape of the free surface distortion in a unique way exhibiting also the expected monotonicity properties finally using a short wavelength stability analysis we prove that certain flows defined by our exact solution are stable for a specific choice of the density distribution

averaging in dynamical systems and large deviations the paper treats ordinary differential equations of the form frac dz varepsilon left t right dt varepsilon b z varepsilon t f t y wheref t is a hyperbolic flow large deviations bounds for the averaging principle are obtained here in the form appeared previously in f1 f2 for the case when the flowf t is replaced by a markov process

analyticity of the cauchy problem for an integrable evolution equation in this paper we consider the periodic cauchy problem for the camassa holm equation with analytic initial data and prove that its solutions are analytic in both variables globally in space and locally in time

tree structure of spectra of spectral self affine measures abstract for an integral self affine spectral measure if the zeros of its fourier transform are all integral vectors it is proven that any its spectrum has a tree structure for any subset with such tree structure a sufficient condition and a necessary condition for the subset to be a spectrum are given respectively applications are given to some known results as special cases

universality of the mean field for the potts model we consider the potts model with q colors on a sequence of weighted graphs with adjacency matrices a n allowing for both positive and negative weights under a mild regularity condition on a n we show that the mean field prediction for the log partition function is asymptotically correct whenever mathrm tr a n 2 o n in particular our results are applicable for the ising and the potts models on any sequence of graphs with average degree going to infty using this we establish the universality of the limiting log partition function of the ferromagnetic potts model for a sequence of asymptotically regular graphs and that of the ising model for bi regular bipartite graphs in both ferromagnetic and anti ferromagnetic domain we also derive a large deviation principle for the empirical measure of the colors for the potts model on asymptotically regular graphs

free surface purely azimuthal equatorial flows in spherical coordinates with stratification abstract in this paper we derive an exact solution to the governing equations for geophysical fluid dynamics in spherical coordinates which incorporates fluid stratification this solution represents a steady purely azimuthal equatorial flow with an associated free surface following the derivation of the solution we demonstrate that there is a well defined relationship between the imposed pressure at the free surface and the resulting distortion of the surface s shape finally the solution for stratified fluid flow is subjected to a short wavelength stability analysis

nonlinear large deviations abstract we present a general technique for computing large deviations of nonlinear functions of independent bernoulli random variables the method is applied to compute the large deviation rate functions for subgraph counts in sparse random graphs previous technology based on szemeredi s regularity lemma works only for dense graphs applications are also made to exponential random graphs and three term arithmetic progressions in random sets of integers

on the cauchy problem for the shallow water model with the coriolis effect abstract in this paper we are concerned with an asymptotic model for wave propagation in shallow water with the effect of the coriolis force we first establish the local well posedness in a range of the besov spaces as well as the local well posedness in the critical space then we study the gevrey regularity of the shallow water model by using a generalized cauchy kovalevsky theorem which implies that the shallow water model admits analytical solutions locally in time and globally in space moreover we obtain a precise lower bound of the lifespan and the continuity of the solution finally working with moderate weight functions that are commonly used in time frequency analysis some persistence results to the shallow water model are illustrated

existence and uniqueness of the global conservative weak solutions to the rotation camassa holm equation abstract we study herein the camassa holm type equation which can be considered as a model in the shallow water for the long crested waves propagating near the equator with effect of the coriolis force due to the earth s rotation this quasi linear equation is nonlocal with higher order nonlinearities compared to the classical camassa holm equation we establish the global existence and uniqueness of the energy conservative weak solutions in the energy space h 1 to this model equation

global conservative solutions of the camassa holm equation this paper develops a new approach in the analysis of the camassa holm equation by introducing a new set of independent and dependent variables the equation is transformed into a semilinear system whose solutions are obtained as fixed points of a contractive transformation these new variables resolve all singularities due to possible wave breaking returning to the original variables we obtain a semigroup of global solutions depending continuously on the initial data our solutions are conservative in the sense that the total energy equals a constant for almost every time

on the existence of solutions and the pressure function related to the antarctic circumpolar current abstract this paper is concerned with the analysis of an incompressible inviscid fluid moving only in the azimuthal direction with no variations in this direction but allowing for a depth dependent variation as such this permits the consideration of the antarctic circumpolar current the most significant current in earth s oceans and the only current that encircles the polar axis more precisely working in spherical coordinates accounting thus for the geometry of the earth we derive an implicit equation for the function describing the free surface which is related to the pressure function we use this equation to prove by means of the implicit function theorem that any small deviation from the pressure required to maintain the free surface undisturbed gives rise to a unique veritable wave solution our approach comprises also a body force which is shown to have significant influence on the monotonicity properties of the pressure function and the height function describing the free surface

fourier frequencies in affine iterated function systems abstract we examine two questions regarding fourier frequencies for a class of iterated function systems ifs these are iteration limits arising from a fixed finite families of affine and contractive mappings in r d and the ifs refers to such a finite system of transformations or functions the iteration limits are pairs x where x is a compact subset of r d the support of and the measure is a probability measure determined uniquely by the initial ifs mappings and a certain strong invariance axiom the two questions we study are 1 existence of an orthogonal fourier basis in the hilbert space l 2 x and 2 explicit constructions of fourier bases from the given data defining the ifs

estimation of moments of sums of independent real random variables for the sum s sum x i of a sequence x i of independent symmetric or nonnegative random variables we give lower and upper estimates of moments of s the estimates are exact up to some universal constants and extend the previous results for particular types of variables x i

exponential profiles producing genuine three dimensional nonlinear flows relevant for equatorial ocean dynamics abstract a nonlinear three dimensional model for equatorial oceanic flows that are moving slowly in the azimuthal direction is investigated the consequences of including exponential terms in the azimuthal flow profile are determined some examples show that the model is able to capture simultaneously depth dependent currents with flow reversal poleward flow upwelling downwelling processes

convergent sequences of dense graphs ii multiway cuts and statistical physics we consider sequences of graphs gn and dene various notions of convergence related to these sequences including left convergence dened in terms of the densities of homomorphisms from small graphs into gn and right convergence dened in terms of the densities of homomorphisms from gn into small graphs we show that right convergence is equivalent to left convergence both for simple graphs gn and for graphs gn with nontrivial nodeweights and edgeweights other equivalent conditions for convergence are given in terms of fundamental notions from combinatorics such as maximum cuts and szemer edi partitions and fundamental notions from statistical physics like energies and free energies we thereby relate local and global properties of graph sequences quantitative forms of these results express the relationships among dierent measures of similarity of large graphs

limit theorems for partially hyperbolic systems we consider a large class of partially hyperbolic systems containing among others affine maps frame flows on negatively curved manifolds and mostly contracting diffeomorphisms if the rate of mixing is sufficiently high the system satisfies many classical limit theorems of probability theory

extreme value laws for dynamical systems with countable extremal sets we consider stationary stochastic processes arising from dynamical systems by evaluating a given observable along the orbits of the system we focus on the extremal behaviour of the process which is related to the entrance in certain regions of the phase space which correspond to neighbourhoods of the maximal set mathcal m i e the set of points where the observable is maximised the main novelty here is the fact that we consider that the set mathcal m may have a countable number of points which are associated by belonging to the orbit of a certain point and may have accumulation points in order to prove the existence of distributional limits and study the intensity of clustering given by the extremal index we generalise the conditions previously introduced in freitas adv math 231 5 2626 2665 2012 stoch process appl 125 4 1653 1687 2015

global diffeomorphism of the lagrangian flow map defining equatorially trapped water waves abstract the aim of this paper is to prove that a three dimensional lagrangian flow which defines equatorially trapped water waves is dynamically possible this is achieved by applying a mixture of analytical and topological methods to prove that the nonlinear exact solution to the geophysical governing equations derived by constantin 2012 is a global diffeomorphism from the lagrangian labelling variables to the fluid domain beneath the free surface

the trajectories of particles in stokes waves analyzing a free boundary problem for harmonic functions we show that there are no closed particle paths in an irrotational inviscid traveling wave propagating at the surface of water over a flat bed within a period each particle experiences a backward forward motion with a slight forward drift

spectrality of the planar sierpinski family abstract let be a borel probability measure with compact support in r 2 is called a spectral measure if there exists a countable set r 2 such that e e 2 i x is an orthonormal basis for l 2 in this note we prove that the integral sierpinski measure a d is a spectral measure if and only if a d is admissible this completely settles the spectrality of integral sierpinski measures in r 2

upper tails and independence polynomials in random graphs abstract the upper tail problem in the erd s renyi random graph g g n p asks to estimate the probability that the number of copies of a graph h in g exceeds its expectation by a factor 1 chatterjee and dembo showed that in the sparse regime of p 0 as n with p n for an explicit h 0 this problem reduces to a natural variational problem on weighted graphs which was thereafter asymptotically solved by two of the authors in the case where h is a clique here we extend the latter work to any fixed graph h and determine a function c h such that for p as above and any fixed 0 the upper tail probability is exp c h o 1 n 2 p log 1 p where is the maximum degree of h as it turns out the leading order constant in the large deviation rate function c h is governed by the independence polynomial of h defined as p h x i h k x k where i h k is the number of independent sets of size k in h for instance if h is a regular graph on m vertices then c h is the minimum between 1 2 2 m and the unique positive solution of p h x 1

large deviations of subgraph counts for sparse erdős renyi graphs abstract for any fixed simple graph h v e and any fixed u 0 we establish the leading order of the exponential rate function for the probability that the number of copies of h in the erd s renyi graph g n p exceeds its expectation by a factor 1 u assuming n h p 1 with h 1 2 where 1 is the maximum degree of h this improves on a previous result of chatterjee and the second author who obtained h c e for a constant c 0 moreover for the case of cycle counts we can take as large as 1 2 we additionally obtain the sharp upper tail for schatten norms of the adjacency matrix as well as the sharp lower tail for counts of graphs for which sidorenko s conjecture holds as a key step we establish quantitative versions of szemeredi s regularity lemma and the counting lemma suitable for the analysis of random graphs in the large deviations regime

the compound poisson distribution and return times in dynamical systems previously it has been shown that some classes of mixing dynamical systems have limiting return times distributions that are almost everywhere poissonian here we study the behaviour of return times at periodic points and show that the limiting distribution is a compound poissonian distribution we also derive error terms for the convergence to the limiting distribution we also prove a very general theorem that can be used to establish compound poisson distributions in many other settings

instability of internal equatorial water waves abstract in the following paper we present criteria for the hydrodynamical instability of internal equatorial water waves we show by way of the short wavelength perturbation approach that certain geophysical waves propagating above the equatorial thermocline are linearly unstable when the wave steepness exceeds a given threshold

symplectic structures their backlund transformations and hereditary symmetries abstract it is shown that compatible symplectic structures lead in a natural way to hereditary symmetries we recall that a hereditary symmetry is an operator valued function which immediately yields a hierarchy of evolution equations each having infinitely many commuting symmetries all generated by this hereditary symmetry furthermore this hereditary symmetry usually describes completely the soliton structure and the conservation laws of these equations this result then provide us with a method for constructing hereditary symmetries and hence exactly solvable evolution equations in addition we show how symplectic structures transform under backlund transformations this leads to a method for generating a whole class of symplectic structures from a given one several examples and applications are given illustrating the above results also the connection of our results with those of gelfand and dikii and of magri is briefly pointed out

a problem on completeness of exponentials let be a nite positive measure on the real line for a 0 denote by ea the family of exponential functions ea fe ist j s2 0 a g

non spectrality of planar self affine measures with three elements digit set abstract the self affine measure m d associated with an affine iterated function system d x m 1 x d d d is uniquely determined the problems of determining the spectrality or non spectrality of a measure m d have been received much attention in recent years one of the non spectral problem on m d is to estimate the number of orthogonal exponentials in l 2 m d and to find them in the present paper we show that for an expanding integer matrix m m 2 z and the three elements digit set d given by m a b d c and d 0 0 1 0 0 1 if a c b d 3 z then there exist at most 3 mutually orthogonal exponentials in l 2 m d and the number 3 is the best this confirms the three elements digit set conjecture on the non spectrality of self affine measures in the plane

a three dimensional autonomous nonlinear dynamical system modelling equatorial ocean flows abstract we investigate a nonlinear three dimensional model for equatorial flows finding exact solutions that capture the most relevant geophysical features depth dependent currents poleward or equatorial surface drift and a vertical mixture of upward and downward motions

on the inverse spectral problem for the camassa holm equation abstract a key basis for seeking periodic solutions of the camassa holm equation is to understand the associated spectral problem y 1 4 y my the periodic spectrum can be recovered from the norming constants and the elements of the auxiliary spectrum the potential can then be reconstructed from the periodic spectrum a necessary and sufficient condition for exponential decrease of the widths 2n 2n 1for a sequence 0 m c1 r

global existence and blow up phenomena for an integrable two component camassa holm shallow water system abstract this paper is concerned with global existence and blow up phenomena for an integrable two component camassa holm shallow water system a new global existence result and several new blow up results of strong solutions to the system are presented our obtained results for the system are sharp and improve considerably earlier results

when does a bernoulli convolution admit a spectrum abstract in this paper we solve a long standing problem on bernoulli convolutions in particular we show that the bernoulli convolution with contraction rate 0 1 admits a spectrum if and only if is the reciprocal of an even integer

non spectral problem for a class of planar self affine measures abstract the self affine measure m d corresponding to an expanding matrix m m n r and a finite subset d r n is supported on the attractor or invariant set of the iterated function system d x m 1 x d d d the spectral and non spectral problems on m d including the spectrum tiling problem implied in them have received much attention in recent years one of the non spectral problem on m d is to estimate the number of orthogonal exponentials in l 2 m d and to find them in the present paper we show that if a b c z a 1 c 1 and a c z 3 z m a b 0 c and d 0 0 1 0 0 1 then there exist at most 3 mutually orthogonal exponentials in l 2 m d and the number 3 is the best this extends several known conclusions the proof of such result depends on the characterization of the zero set of the fourier transform m d and provides a way of dealing with the non spectral problem

on the μm d orthogonal exponentials abstract let m d be a self affine measure corresponding to a given affine iterated function system d x m 1 x d d d in the present paper we will study the problem of how to determine the l 2 m d space has finite or infinite orthogonal exponentials such research is motivated by a conjecture on the non spectrality of m d we first obtain a non spectral criterion which extends the result of dutkay and jorgensen in opposition to the condition of this criterion we then obtain some conditions which imply the infinite orthogonal systems in l 2 m d these are necessary for further investigation on the spectrality of m d as an application we completely settle the corresponding problem for the generalized planar sierpinski family