robust h sub spl infin and mixed h sub 2 h sub spl infin filters for equalization designs of nonlinear communication systems fuzzy interpolation approach generally it is difficult to design equalizers for signal reconstruction of nonlinear communication channels with uncertain noises in this paper we propose the h sub spl infin and mixed h sub 2 h sub spl infin filters for equalization detection of nonlinear channels using fuzzy interpolation and linear matrix inequality lmi techniques first the signal transmission system is described as a state space model and the input signal is embedded in the state vector such that the signal reconstruction estimation design becomes a nonlinear state estimation problem then the takagi sugeno fuzzy linear model is applied to interpolate the nonlinear channel at different operation points through membership functions since the statistics of noises are unknown the fuzzy h sub spl infin equalizer is proposed to treat the state estimation problem from the worst case robust point of view when the statistics of noises are uncertain but with some nominal or average information available the mixed h sub 2 h sub spl infin equalizer is employed to take the advantage of both h sub 2 optimal performance with nominal statistics of noises and the h sub spl infin robustness performance against the statistical uncertainty of noises using the lmi approach the fuzzy h sub 2 h sub spl infin equalizer detector design problem is characterized as an eigenvalue problem evp the evp can be solved efficiently with convex optimization techniques

an interlaced extended kalman filter an estimation algorithm for a class of discrete time nonlinear systems is proposed the system structure we deal with is partitionable into in subsystems each affine w r t the corresponding part of the state vector the algorithm consists of a bank of m interlaced kalman filters and each of them estimates a part of the state considering the remaining parts as known time varying parameters whose values are evaluated by the other filters at the previous step the procedure neglects the subsystem coupling terms in the covariance matrix of the state estimation error and counteracts the errors so introduced by suitably increasing the noise covariance matrices comparisons through numerical simulations with the extended kalman filter and its modified versions proposed in the literature illustrate the good trade off provided by the algorithm between the reduction of the computational load and the estimation accuracy

katoʼs residue homomorphisms and reciprocity laws on arithmetic surfaces abstract we explicitly study kato s residue homomorphisms in milnor k theory which are closely related to contou carrere symbols as applications we establish several reciprocity laws for certain locally defined maps on k groups that are associated to arithmetic surfaces

the derivative of the maximal function in this note we show that the local hardy littlewood maximal operator is bounded in the sobolev space thus the maximal function often has partial derivatives we also show that the maximal operator preserves the zero boundary values in sobolev s sense

regularity of the centered fractional maximal function on radial functions abstract we study the regularity properties of the centered fractional maximal function m more precisely we prove that the map f m f is bounded and continuous from w 1 1 r d to l q r d in the endpoint case q d d if f is a radial function for d 1 the radiality assumption can be removed this corresponds to the counterparts of known results for the non centered fractional maximal function the main new idea consists in relating the centered and non centered fractional maximal function at the derivative level

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

endpoint sobolev and bv continuity for maximal operators abstract in this paper we investigate some questions related to the continuity of maximal operators in w 1 1 and bv spaces complementing some well known boundedness results letting m be the one dimensional uncentered hardy littlewood maximal operator we prove that the map f m f is continuous from w 1 1 r to l 1 r in the discrete setting we prove that m b v z b v z is also continuous for the one dimensional fractional hardy littlewood maximal operator we prove by means of counterexamples that the corresponding continuity statements do not hold both in the continuous and discrete settings and for the centered and uncentered versions

modified extended kalman filtering in this paper we compare the performance of the extended kalman filter ekf with that of a proposed filter called modified extended kalman filter mekf in terms of the actual state and the estimated state as a function of time and the integral squared error as a function of noise intensity both for dynamics and measurement noise all the simulation results indicate that the performance of the mekf is better than ekf for illustration numerical examples are presented using both ekf and mekf

nonlinear filtering of systems governed by ito differential equations with jump parameters abstract in this paper we present a system of coupled stochastic infinite dimensional differential equations of zakai type which describes the evolution of the unnormalized conditional density of a conditional markov process this generalizes the nonlinear filtering equation of zakai

matching measure benjamini schramm convergence and the monomer dimer free energy we define the matching measure of a lattice l as the spectral measure of the tree of self avoiding walks in l we connect this invariant to the monomer dimer partition function of a sequence of finite graphs converging to l this allows us to express the monomer dimer free energy of l in terms of the matching measure exploiting an analytic advantage of the matching measure over the mayer series then leads to new rigorous bounds on the monomer dimer free energies of various euclidean lattices while our estimates use only the computational data given in previous papers they improve the known bounds significantly

canonical measures on metric graphs and a kazhdan s theorem we extend the notion of canonical measures to all possibly non compact metric graphs this will allow us to introduce a notion of hyperbolic measures on universal covers of metric graphs kazhdan s theorem for riemann surfaces describes the limiting behavior of canonical arakelov measures on finite covers in relation to the hyperbolic measure we will prove a generalized version of this theorem for metric graphs allowing any infinite galois cover to replace the universal cover we will show all such limiting measures satisfy a version of gauss bonnet formula which using the theory of von neumann dimensions can be interpreted as a trace formula in the special case where the infinite cover is the universal cover we will provide explicit methods to compute the corresponding limiting hyperbolic measure our ideas are motivated by non archimedean analytic and tropical geometry

on the variation of maximal operators of convolution type abstract in this paper we study the regularity properties of two maximal operators of convolution type the heat flow maximal operator associated to the gauss kernel and the poisson maximal operator associated to the poisson kernel in dimension d 1 we prove that these maximal operators do not increase the l p variation of a function for any p 1 while in dimensions d 1 we obtain the corresponding results for the l 2 variation similar results are proved for the discrete versions of these operators

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

boundedness and unboundedness results for some maximal operators on functions of bounded variation abstract we characterize the space bv i of functions of bounded variation on an arbitrary interval i r in terms of a uniform boundedness condition satisfied by the local uncentered maximal operator m r from bv i into the sobolev space w 1 1 i by restriction the corresponding characterization holds for w 1 1 i we also show that if u is open in r d d 1 then boundedness from bv u into w 1 1 u fails for the local directional maximal operator m t v the local strong maximal operator m t s and the iterated local directional maximal operator m t d m t 1 nevertheless if u satisfies a cone condition then m t s bv u l 1 u boundedly and the same happens with m t v m t d m t 1 and m r

interpolation between holder and lebesgue spaces with applications abstract classical interpolation inequality of the type u x c u y u z 1 is well known in the case when x y z are lebesgue spaces in this paper we show that this result may be extended by replacing norms y or x by suitable holder semi norm we shall even prove sharper version involving weak lorentz norm we apply this result to prove the gagliardo nirenberg inequality for a wider scale of parameters

sobolev regularity of polar fractional maximal functions abstract we study the sobolev regularity on the sphere s d of the uncentered fractional hardy littlewood maximal operator m at the endpoint p 1 when acting on polar data we first prove that if q d d 0 d and f is a polar w 1 1 s d function we have m f q d f 1 we then prove that the map f m f is continuous from w 1 1 s d to l q s d when restricted to polar data our methods allow us to give a new proof of the continuity of the map f m f from w rad 1 1 r d to l q r d moreover we prove that a conjectural local boundedness for the centered fractional hardy littlewood maximal operator m implies the continuity of the map f m f from w 1 1 to l q in the context of polar functions on s d and radial functions on r d

functions of bounded variation the derivative of the one dimensional maximal function and applications to inequalities we prove that if f i r r is of bounded variation then the uncentered maximal function mf is absolutely continuous and its derivative satisfies the sharp inequality dmf l 1 i df i this allows us to obtain under less regularity versions of classical inequalities involving derivatives

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

local convergence for permutations and local limits for uniform rho avoiding permutations with rho 3 we set up a new notion of local convergence for permutations and we prove a characterization in terms of proportions of consecutive pattern occurrences we also characterize random limiting objects for this new topology introducing a notion of shift invariant property corresponding to the notion of unimodularity for random graphs we then study two models in the framework of random pattern avoiding permutations we compute the local limits of uniform rho avoiding permutations for rho 3 when the size of the permutations tends to infinity the core part of the argument is the description of the asymptotics of the number of consecutive occurrences of any given pattern for this result we use bijections between rho avoiding permutations and rooted ordered trees local limit results for galton watson trees the second moment method and singularity analysis

on the continuous and discontinuous maximal operators abstract in the first part of this paper we study the regularity properties of a wide class of maximal operators these results are used to show that the spherical maximal operator is continuous w 1 p r n w 1 p r n when p n n 1 other given applications include fractional maximal operators and maximal singular integrals on the other hand we show that the restricted hardy littlewood maximal operator m where the supremum is taken over the cubes with radii greater than 0 is bounded from l p r n to w 1 p r n but discontinuous

h sub spl infin fuzzy estimation for a class of nonlinear discrete time dynamic systems this paper studies h sub spl infin fuzzy filtering design for state estimation of nonlinear discrete time systems with bounded but unknown disturbances first the takagi and sugeno 1985 fuzzy model is proposed to represent a nonlinear system next using a linear matrix inequality lmi approach the h sub spl infin fuzzy filtering problems are characterized in terms of a linear matrix inequality problem lmip the lmip can be efficiently solved using convex optimization techniques simulation examples are given to illustrate the design procedure and the applicability of the proposed method the results indicate that the proposed method is suitable for practical applications

on the growth of l 2 invariants for sequences of lattices in lie groups we study the asymptotic behaviour of betti numbers twisted torsion and other spectral invariants of sequences of locally symmetric spaces our main results are uniform versions of the degeorge wallach theorem of a theorem of delorme and various other limit multiplicity theorems a basic idea is to adapt the notion of benjamini schramm convergence bs convergence originally introduced for sequences of finite graphs of bounded degree to sequences of riemannian manifolds and analyze the possible limits we show that bs convergence of locally symmetric spaces g k implies convergence in an appropriate sense of the normalized relative plancherel measures associated to l2 g this then yields convergence of normalized multiplicities of unitary representations betti numbers and other spectral invariants on the other hand when the corresponding lie group g is simple and of real rank at least two we prove that there is only one possible bs limit i e when the volume tends to infinity locally symmetric spaces always bs converge to their universal cover g k this leads to various general uniform results when restricting to arbitrary sequences of congruence covers of a fixed arithmetic manifold we prove a strong quantitative version of bs convergence which in turn implies upper estimates on the rate of convergence of normalized betti numbers in the spirit of sarnak xue an important role in our approach is played by the notion of invariant random subgroups for higher rank simple lie groups g we exploit rigidity theory and in particular the nevo stuck zimmer theorem and kazhdan s property t to obtain a complete understanding of the space of irs s of g 2017 department of mathematics princeton university

sparse regular random graphs spectral density and eigenvectors we examine the empirical distribution of the eigenvalues and the eigenvectors of adjacency matrices of sparse regular random graphs we find that when the degree sequence of the graph slowly increases to infinity with the number of vertices the empirical spectral distribution converges to the semicircle law moreover we prove concentration estimates on the number of eigenvalues over progressively smaller intervals we also show that with high probability all the eigenvectors are delocalized

an l p theory of sparse graph convergence i limits sparse random graph models and power law distributions we introduce and develop a theory of limits for sequences of sparse graphs based on lp graphons which generalizes both the existing l1 theory of dense graph limits and its extension by bollob as and riordan to sparse graphs without dense spots in doing so we replace the no dense spots hypothesis with weaker assumptions which allow us to analyze graphs with power law degree distributions this gives the rst broadly applicable limit theory for sparse graphs with unbounded average degrees in this paper we lay the foundations of the lp theory of graphons characterize convergence and develop corresponding random graph models while we prove the equivalence of several alternative metrics in a companion paper