sharp estimates for solutions of multi bubbles in compact riemann surfaces in this paper we consider a sequence of multibubble solutions u k of the equation 0 1 0 u k k he uk f m he u k d 1 0 in m where h is a c 2 positive function in a compact riemann surface m and k is a constant satisfying lim k k 8m for some positive integer m 1 we prove among other things that k 8m 2 m m j 1h 1 p k j 0 log h p k j 8m 2k p k j k j e k j o e k j where p k j are centers of the bubbles of u k and k j are the local maxima of u k after adding a constant this yields a uniform bound of solutions as k converges to 8m from below provided that 0 log h p k j 8m 2k p k j 0 it generalizes a previous result due to ding jost li and wang 18 and nolasco and tarantello 31 which says that any sequence of minimizers u k is uniformly bounded if k 0 for any maximum point p of the sum of 2 log h and the regular part of the green function where k is the gaussian curvature of m the analytic work of this paper is the first step toward computing the topological degree of 0 1 which was initiated by li 24

mixed type solutions of the su 3 models on a torus in this paper we study mixed type solutions of su 3 chern simons system see 1 4 below on a two dimensional flat torus nolasco and tarantello commun math phys 213 599 639 2000 among other things nolasco and tarantello obtained solutions of 1 4 as minimizers of several functionals closely related to 1 4 and showed that if n 1 n 2 1 then one of those minimizers turns out to be a mixed type solution that is one component tends to ln frac 1 2 pointwise a e and the other component converges to a solution of a mean field equation we call these kinds of solutions mixed type i solutions in this paper we prove two main results i the asymptotic analysis of mixed type i solutions with arbitrary configuration of vortex points and ii the existence of mixed type i solutions under a non degenerate condition this non degenerate condition also ensures some uniqueness result in particular our results imply that when n 1 n 2 1 there are only two mixed type i solutions of 1 4

existence of bubbling solutions for chern simons model on a torus we study the existence of bubbling solutions for the the following chern simons higgs equation delta u frac1 varepsilon 2 rm e u 1 rm e u 4 pi sum i 1 2k delta p i quad text in omega where is a torus if k 1 for any critical point q of the associated sum of the green functions we introduce a quantity d q see 1 11 below we show that for any non degenerate critical point q with d q 0 the above problem has a solution ue satisfying that e 0 ue blows up at q the calculations in this paper also show that if a sequence of solutions ue blows up at q as e 0 then q must be a critical point of the associated sum of the green functions and d q leqq 0 so the condition d q 0 is almost necessary to obtain our result we also construct solutions with k bubbles for k geqq 2

a singular sphere covering inequality uniqueness and symmetry of solutions to singular liouville type equations we derive a singular version of the sphere covering inequality which was recently introduced in gui and moradifam invent math https doi org 10 1007 s00222 018 0820 2 2018 suitable for treating singular liouville type problems with superharmonic weights as an application we deduce new uniqueness results for solutions of the singular mean field equation both on spheres and on bounded domains as well as new self contained proofs of previously known results such as the uniqueness of spherical convex polytopes first established in luo and tian proc am math soc 116 4 1119 1129 1992 furthermore we derive new symmetry results for the spherical onsager vortex equation

robust estimation of the variance in moment methods for extra binomial and extra poisson variation when faced with data in the form of overdispersed counts or proportions moment methods allow consistent parameter estimation when only the form of the mean and variance is specified if the variance form is misspecified these methods still yield consistent parameter estimates though with lower efficiency and the variances of the estimates will be inconsistent a variance correction is available that yields consistent variance estimates in these circumstances the asymptotic and small sample efficiencies of this correction are calculated and its performance under variance misspecification is studied a group randomized breast self examination prevention study that is now underway serves as a focal point for the study of these properties the use of the variance correction in modelling is illustrated on a teratology data set

mean field equation of liouville type with singular data topological degree we consider the following mean field equation gv h ev mh ev 1 4 j 1n j qj 1 on m where m is a compact riemann surface with volume 1 h is a positive c1 function on m and and j are constants satisfying j 1 in this paper we derive the topological degree counting formula for noncritical values of we also give several applications of this formula including existence of the curvature 1 metric with conic singularities doubly periodic solutions of electroweak theory and a special case of self gravitating strings 2015 wiley periodicals inc

on the mean field type bubbling solutions for chern simons higgs equation abstract this paper is the second part of our comprehensive study on the structure of the solutions for the following chern simons higgs equation 0 1 u 1 e 2 e u 1 e u 4 j 1 n p j in u is doubly periodic on where is a parallelogram in r 2 and e 0 is a small parameter in part 1 29 we proved the non coexistence of different bubbles in the bubbling solutions and obtained an existence result for the chern simons type bubbling solutions under some nearly necessary conditions mean field type bubbling solutions for 0 1 have been constructed in 27 in this paper we shall study two other important issues for the mean field type bubbling solutions the necessary conditions for the existence and the local uniqueness the results in this paper lay the foundation to find the exact number of solutions for 0 1

singular mean field equations on compact riemann surfaces for a general class of elliptic pde s in mean field form on compact riemann surfaces with exponential nonlinearity we address the question of the existence of solutions with concentrated nonlinear term which in view of the applications are physically of definite interest in the model we also include the possible presence of singular sources in the form of dirac masses which makes the problem more difficult to attack

an improved geometric inequality via vanishing moments with applications to singular liouville equations we consider a class of singular liouville equations on compact surfaces motivated by the study of electroweak and self dual chern simons theories the gaussian curvature prescription with conical singularities and onsager s description of turbulence we analyse the problem of existence variationally and show how the angular distribution of the conformal volume near the singularities may lead to improvements in the moser trudinger inequality and in turn to lower bounds on the euler lagrange functional we then discuss existence and non existence results

linear estimation of boundary value stochastic processes part i the role and construction of complementary models this paper presents a substantial extension of the method of complementary models for minimum variance linear estimation introduced by weinert and desai in their important paper 1 specifically the method of complementary models is extended to solve estimation problems for both discrete and continuous parameter linear boundary value stochastic processes in one and higher dimensions a major contribution of this paper is an application of green s identity in deriving a differential operator representation of the estimator to clarify the development and to illustrate the range of applications of our approach two brief examples are provided one is a 1 d discrete two point boundary value process and the other is a 2 d process governed by poisson s equation on the unit disk

elliptic functions green functions and the mean field equations on tori we show that the green functions on flat tori can have either 3 or 5 critical points only there does not seem to be any direct method to attack this problem instead we have to employ sophisticated non linear partial differential equations to study it we also study the distribution of number of critical points over the moduli space of flat tori through deformations the functional equations of special theta values provide important inequalities which lead to a solution for all rhombus tori the general picture is also emerged though some of the necessary technicality is still to de developed 1 introduction and statement of results the study of geometric or analytic problems on two dimensional tori is the same as the study of problems on r 2 with doubly periodic data such situations occur naturally in sciences and mathematics since early days the mathematical foundation of elliptic functions was subsequently devel oped in the 19th century it turns out that these special functions are rather deep objects by themselves tori of different shape may result in very dif ferent behavior of the elliptic functions and their associated objects arith metic on elliptic curves is perhaps the eldest and the most vivid example in this paper we show that this is also the case for certain non linear par tial differential equations indeed researches on doubly periodic problems in mathematical physics or differential equations often restrict the study to rectangular tori for simplicity this leaves the impression that the theory for general tori may resemble much the same way as for the rectangular case however this turns out to be false we will show that the solvability of themean field equation dependson the shape of the green function which in turn depends on the geometry of the tori in an essential way recall that the green function g z w on a flat torus t c z 1 z 2 is the unique function on t t which satisfies zg z w w z 1 t

recursive filtering and smoothing for reciprocal gaussian processes with dirichlet boundary conditions the minimum mean square error mmse estimation problem for a discrete index reciprocal gaussian process impaired by additive white gaussian noise is completely solved in the general case of noisily observed dirichlet random boundary conditions finite sets of recursive equations are obtained for the computation of the filtered sequence and of the fixed point fixed interval and fixed lag smoothed sequences recursive expressions are also given for the mmse performance of the presented estimators simulation results confirm the validity of the performance analysis

a modified pseudolikelihood approach for analysis of longitudinal data we consider the analysis of longitudinal data when the covariance function is modeled by additional parameters to the mean parameters in general inconsistent estimators of the covariance variance correlation parameters will be produced when the working correlation matrix is misspecified which may result in great loss of efficiency of the mean parameter estimators albeit the consistency is preserved we consider using different working correlation models for the variance and the mean parameters in particular we find that an independence working model should be used for estimating the variance parameters to ensure their consistency in case the correlation structure is misspecified the designated working correlation matrices should be used for estimating the mean and the correlation parameters to attain high efficiency for estimating the mean parameters simulation studies indicate that the proposed algorithm performs very well we also applied different estimation procedures to a data set from a clinical trial for illustration

green function painleve vi equation and eisenstein series of weight one we study the problem how many singular points of a solution t to the painleve vi equation with parameter 8 1 8 1 8 3 8 might have in c 0 1 here t0 c 0 1 is called a singular point of t if t0 0 1 t0 based on hitchin s formula we explore the connection of this problem with green function and the eisenstein series of weight one among other things we prove i there are only three solutions which have no singular points in c 0 1 ii for a special type of solutions called real solutions here any branch of a solution has at most two singular points in particular at most one pole in c 0 1 iii any riccati solution has singular points in c 0 1 iv for each n 5 and n 6 6 we calculate the number of the real j values of zeros of the eisenstein series en 1 k1 k2 of weight one where k1 k2 runs over 0 n 1 2 with gcd k1 k2 n 1 the geometry of the critical points of the green function on a flat torus e as varies in the modulim1 plays a fundamental role in our analysis of the painleve iv equation in particular the conjectures raised in 22 on the shape of the domain 5 m1 which consists of tori whose green function has extra pair of critical points are completely solved here

bayesian semiparametric analysis of developmental toxicology data modeling of developmental toxicity studies often requires simple parametric analyses of the dose response relationship between exposure and probability of a birth defect but poses challenges because of nonstandard distributions of birth defects for a fixed level of exposure this article is motivated by two such experiments in which the distribution of the outcome variable is challenging to both the standard logistic model with binomial response and its parametric multistage elaborations we approach our analysis using a bayesian semiparametric model that we tailored specifically to developmental toxicology studies it combines parametric dose response relationships with a flexible nonparametric specification of the distribution of the response obtained via a product of dirichlet process mixtures approach pdpm our formulation achieves three goals 1 the distribution of the response is modeled in a general way 2 the degree to which the distribution of the response adapts nonparametrically to the observations is driven by the data and 3 the marginal posterior distribution of the parameters of interest is available in closed form the logistic regression model as well as many of its extensions such as the beta binomial model and finite mixture models are special cases in the context of the two motivating examples and a simulated example we provide model comparisons illustrate overdispersion diagnostics that can assist model specification show how to derive posterior distributions of the effective dose parameters and predictive distributions of response and discuss the sensitivity of the results to the choice of the prior distribution

a karhunen loeve like expansion for 1 f processes via wavelets while so called 1 f or scaling processes emerge regularly in modeling a wide range of natural phenomena as yet no entirely satisfactory framework has been described for the analysis of such processes orthonormal wavelet bases are used to provide a new construction for nearly 1 f processes from a set of uncorrelated random variables

existence and uniqueness for mean field equations on multiply connected domains at the critical parameter we consider the mean field equation begin aligned left begin array l quad l vardelta u rho frac e u int varomega e u 0 hbox in varomega u 0 hbox on partial varomega end array right end aligned 1 where varomega subset mathbb r 2 is an open and bounded domain of class c 1 in his 1992 paper suzuki proved that if varomega is a simply connected domain then eq 1 admits a unique solution for rho in 0 8 pi this result for varomega a simply connected domain has been extended to the case rho 8 pi by chang chen and the second author however the uniqueness result for varomega a multiply connected domain has remained a long standing open problem which we solve positively here for rho in 0 8 pi to obtain this result we need a new version of the classical bol s inequality suitable to be applied on multiply connected domains our second main concern is the existence of solutions for 1 when rho 8 pi we obtain a necessary and sufficient condition for the solvability of the mean field equation at rho 8 pi which is expressed in terms of the robin s function gamma for varomega for example if eq 1 has no solution at rho 8 pi then gamma has a unique nondegenerate maximum point as a by product of our results we solve the long standing open problem of the equivalence of canonical and microcanonical ensembles in the onsager s statistical description of two dimensional turbulence on multiply connected domains

multiscale recursive estimation data fusion and regularization we describe a framework for modeling stochastic phenomena at multiple scales and for their efficient estimation or reconstruction given partial and or noisy measurements which may also be at several scales in particular multiscale signal representations lead naturally to pyramidal or tree like data structures in which each level in the tree corresponds to a particular scale of representation a class of multiscale dynamic models evolving on dyadic trees is introduced the main focus of this paper is on the description analysis and application of an extremely efficient optimal estimation algorithm for this class of models this algorithm consists of a fine to coarse filtering sweep followed by a coarse to fine smoothing step corresponding to the dyadic tree generalization of kalman filtering and rauch tung striebel smoothing the kalman filtering sweep consists of the recursive application of three steps a measurement update step a fine to coarse prediction step and a fusion step we illustrate the use of our methodology for the fusion of multiresolution data and for the efficient solution of fractal regularizations of ill posed signal and image processing problems encountered

estimation of fractal signals from noisy measurements using wavelets the role of the wavelet transformation as a whitening filter for 1 f processes is exploited to address problems of parameter and signal estimations for 1 f processes embedded in white background noise robust computationally efficient and consistent iterative parameter estimation algorithms are derived based on the method of maximum likelihood and cramer rao bounds are obtained included among these algorithms are optimal fractal dimension estimators for noisy data algorithms for obtaining bayesian minimum mean square signal estimates are also derived together with an explicit formula for the resulting error these smoothing algorithms find application in signal enhancement and restoration the parameter estimation algorithms find application in signal enhancement and restoration the parameter estimation algorithms in addition to solving the spectrum estimation problem and to providing parameters for the smoothing process are useful in problems of signal detection and classification results from simulations are presented to demonstrated the viability of the algorithms

convergence rate location and z2 condition for fully bubbling solutions to su n 1 toda systems abstract it is well known that the study of su n 1 toda systems is important not only to chern simons models in physics but also to the understanding of holomorphic curves harmonic sequences or harmonic maps from riemann surfaces to c p n one major goal in the study of su n 1 toda system on riemann surfaces is to completely understand the asymptotic behavior of fully bubbling solutions in this article we use a unified approach to study fully bubbling solutions to general su n 1 toda systems and we prove three major sharp estimates important for constructing bubbling solutions the closeness of blowup solutions to entire solutions the location of blowup points and a z 2 condition

non degeneracy and uniqueness of solutions to singular mean field equations on bounded domains abstract the aim of this paper is to complete the program initiated in 51 23 and then carried out by several authors concerning non degeneracy and uniqueness of solutions to mean field equations in particular we consider mean field equations with general singular data on non smooth domains the argument is based on the alexandrov bol inequality and on the eigenvalues analysis of linearized singular liouville type problems

green s function for the laplace beltrami operator on a toroidal surface green s function for the laplacebeltrami operator on the surface of a three dimensional ring torus is constructed an integral ingredient of our approach is the stereographic projection of the toru

uniqueness of bubbling solutions with collapsing singularities abstract the seminal work 7 by brezis and merle showed that the bubbling solutions of the mean field equation have the property of mass concentration recently lin and tarantello in 30 found that the bubbling implies mass concentration phenomena might not hold if there is a collapse of singularities furthermore a sharp estimate 23 for the bubbling solutions has been obtained in this paper we prove that there exists at most one sequence of bubbling solutions if the collapsing singularity occurs the main difficulty comes from that after re scaling the difference of two solutions locally converges to an element in the kernel space of the linearized operator it is well known that the kernel space is three dimensional so the main technical ingredient of the proof is to show that the limit after re scaling is orthogonal to the kernel space

image analysis using multigrid relaxation methods image analysis problems posed mathematically as variational principles or as partial differential equations are amenable to numerical solution by relaxation algorithms that are local iterative and often parallel although they are well suited structurally for implementation on massively parallel locally interconnected computational architectures such distributed algorithms are seriously handi capped by an inherent inefficiency at propagating constraints between widely separated processing elements hence they converge extremely slowly when confronted by the large representations of early vision application of multigrid methods can overcome this drawback as we showed in previous work on 3 d surface reconstruction in this paper we develop multiresolution iterative algorithms for computing lightness shape from shading and optical flow and we examine the efficiency of these algorithms using synthetic image inputs the multigrid methodology that we describe is broadly applicable in early vision notably it is an appealing strategy to use in conjunction with regularization analysis for the efficient solution of a wide range of ill posed image analysis problems

correlation structure of the discrete wavelet coefficients of fractional brownian motion it is shown that the discrete wavelet coefficients of fractional brownian motion at different scales are correlated and that their auto and cross correlation functions decay hyperbolically fast at a rate much faster than that of the autocorrelation of the fractional brownian motion itself the rate of decay of the correlation function in the wavelet domain is primarily determined by the number of vanishing moments of the analyzing wavelet

asymptotic blow up analysis for singular liouville type equations with applications abstract we generalize the pointwise estimates obtained in 2 19 and 34 concerning blow up solutions of the liouville type equation u x 2 w x e u in with r 2 open and bounded 1 and w any lipschitz continuous function which satisfies 0 a w b we focus to the case left open in 2 and 34 where the parameter n whose analysis is much more involved as we need to resolve the difficulty of a genuinely non radial behaviour of blow up solutions in the worst situation there is no chance in general to resolve the profile in the form of a solution of a liouville equation in r 2 instead we need to adopt iterated blow up arguments next we refine our blow up analysis to cover a class of planar liouville type problems see 1 27 1 28 below arising from the study of cosmic strings cfr 28 35 in this context we are able to distinguish between a single blow up radial profile and the case of multiple blow up profiles typical of non radial solutions as a consequence we obtain a radial symmetry result which is interesting in itself but also contributes towards the sharp solvability issue for the planar problem 1 27 1 28

multiscale autoregressive processes ii lattice structures for whitening and modeling for pt i see ibid vol 40 no 8 p 1915 34 1992 lattice structures for the whitening and modeling of isotropic processes on trees are developed and a result relating the stability properties of these models to the reflection coefficient sequence introduced in pt i is presented this framework allows one to obtain a detailed analysis of the wold decomposition of processes on trees one interesting aspect of this is that there is a significantly larger class of singular processes on dyadic trees than on the integers

case studies in binary dispersion it is common in biomedical studies with binary responses that variability in the observed number of events exceeds binomial variability a phenomenon known as overdispersion failure to make an adjustment to the nominal standard errors can lead to seriously misleading inference for regression analysis in this note we examine a series of examples drawn from the literature to see which of two commonly used variance formulas is more adequate for describing overdispersion in applications two methods residual analysis and formal comparison are introduced we recommend that both methods be employed in seeking an appropriate variance expression for binary responses each of the five data sets exhibits substantial overdispersion one favoring the beta binomial form another favoring a constant overdispersion factor the remaining three examples exhibit no preference

classification and nondegeneracy of su n 1 toda system with singular sources we consider the following toda system open image in new window where i 1 0 is dirac measure at 0 and the coefficients a ij form the standard tri diagonal cartan matrix in this paper i we completely classify the solutions and obtain the quantization result sum j 1 n a ij int mathbb r 2 e u j dx 4 pi 2 gamma i gamma n 1 i quad forall 1 leq i leq n this generalizes the classification result by jost and wang for i 0 forall 1 leq i leq n ii we prove that if i i 1 j for all 1 i j n then any solution u i is radially symmetric w r t 0 iii we prove that the linearized equation at any solution is non degenerate these are fundamental results in order to understand the bubbling behavior of the toda system

supercritical mean field equations on convex domains and the onsager s statistical description of two dimensional turbulence we are motivated by the study of the microcanonical variational principle within onsager s description of two dimensional turbulence in the range of energies where the equivalence of statistical ensembles fails we obtain sufficient conditions for the existence and multiplicity of solutions for the corresponding mean field equation on convex and thin enough domains in the supercritical with respect to the moser trudinger inequality regime this is a brand new achievement since existence results in the supercritical region were previously known only on multiply connected domains we then study the structure of these solutions by the analysis of their linearized problems and we also obtain a new uniqueness result for solutions of the mean field equation on thin domains whose energy is uniformly bounded from above finally we evaluate the asymptotic expansion of those solutions with respect to the thinning parameter and combining it with all the results obtained so far we solve the microcanonical variational principle in a small range of supercritical energies where the entropy is shown to be concave

sharp existence results for mean field equations with singular data abstract let be a simply connected open and bounded domain in r 2 we are concerned with the nonlinear elliptic problem 0 1 v 8 e v e v 4 j 1 m j p j in v 0 on where j 0 p j denotes the dirac mass with singular point p j and p 1 p m we provide necessary and sufficient conditions for the existence of solutions to 0 1 our result is the two dimensional version of the sharp existence nonexistence result obtained in druet 2002 13 for elliptic equations with critical exponent in dimension 3 in particular we prove that the set m is open where for a given 1 m 0 0 m p 1 p m problem 0 1 has a solution

bubbling solutions for relativistic abelian chern simons model on a torus we prove the existence of bubbling solutions for the following chern simons higgs equation delta u frac 1 varepsilon 2 e u 1 e u 4 pi sum j 1 n delta p j quad rm in omega where is a torus we show that if n 4 and p 1 p j j 2 n then for small e 0 the above problem has a solution u e and as e 0 u e blows up at the vertex point p 1 and satisfies frac 1 varepsilon 2 e u 1 e u to 4 pi n delta p 1 this is the first result for the existence of a solution which blows up at a vertex point

modeling and estimation of discrete time gaussian reciprocal processes discrete time gaussian reciprocal processes are characterized in terms of a second order two point boundary value nearest neighbor model driven by a locally correlated noise whose correlation is specified by the model dynamics this second order model is the analog for reciprocal processes of the standard first order state space models for markov processes the model is used to obtain a solution to the smoothing problem for reciprocal processes the resulting smoother obeys second order equations whose structure is similar to that of the kalman filter for gauss markov processes it is shown that the smoothing error is itself a reciprocal process

on the efficiency of quasi likelihood estimation summary a quasi likelihood method has been proposed by wedderburn 1974 for the estimation of parameters in regression models when there is some assumed relationship between the mean and variance of each observation but not necessarily a fully specified likelihood if the underlying distribution comes from a natural exponential family the quasi likelihood estimates maximize the likelihood and so have full asymptotic efficiency under more general distributions there is some loss of efficiency which is investigated here three types of model are discussed in detail models with constant variance models with constant coefficient of variation and models with overdispersion relative to some exponential family the asymptotic efficiency of quasi likelihood estimation is calculated under some particular distributions and then more generally via an approximation for small departures from the corresponding natural exponential family the possibility of refinement of the quasi likelihood approach to incorporate additional information about the underlying distribution is considered

on the spectrum of fractional brownian motions fractional brownian motions fbms provide useful models for a number of physical phenomena whose empirical spectra obey power laws of fractional order however due to the nonstationary nature of these processes the precise meaning of such spectra remains generally unclear two complementary approaches are proposed which are intended to clarify this point the first one based on a time frequency analysis takes into account the nonstationary nature of fbm and puts emphasis on time averaged measurements the second one based on a time scale analysis is matched to self similarity properties of fbm and reveals an underlying stationary structure relative to each time scaling

on a sharp sobolev type inequality on two dimensional compact manifolds motivated by the asymptotic analysis of double vortex condensates in the chern simons higgs theory we construct a suitable minimizing sequence for a sharp sobolev inequality a la moser for two dimensional compact manifolds as a consequence we first obtain a direct proof of the sharp character of such an inequality secondly and more interestingly we use such minimizing sequence to show that for the flat torus the corresponding extremal problem attains its infimum

bubbling solutions for the su 3 chern simons model on a torus we consider the following nonlinear system derived from the su 3 chern simons models on a torus where denotes the dirac measure at when if we look for a solution with then 0 1 is reduced to the chern simons higgs equation the existence of bubbling solutions to 0 1 has been a longstanding problem in this paper we prove the existence of such solutions such that even if 2012 wiley periodicals inc

modeling and estimation of multiresolution stochastic processes an overview is provided of the several components of a research effort aimed at the development of a theory of multiresolution stochastic modeling and associated techniques for optimal multiscale statistical signal and image processing a natural framework for developing such a theory is the study of stochastic processes indexed by nodes on lattices or trees in which different depths in the tree or lattice correspond to different spatial scales in representing a signal or image in particular it is shown how the wavelet transform directly suggests such a modeling paradigm this perspective then leads directly to the investigation of several classes of dynamic models and related notions of multiscale stationarity in which scale plays the role of a time like variable the investigation of models on homogeneous trees is emphasized the framework examined here allows for consideration in a very natural way of the fusion of data from sensors with differing resolutions also thanks to the fact that wavelet transforms do an excellent job of compressing large classes of covariance kernels it is seen that these modeling paradigms appear to have promise in a far broader context than one might expect

degree counting and shadow system for toda system of rank two one bubbling abstract we initiate the program for computing the leray schauder topological degree for toda systems of rank two this program still contains a lot of challenging problems for analysts as the first step we prove that if a sequence of solutions u 1 k u 2 k blows up then one of h j e u j k m h j e u j k d v g j 1 2 tends to a sum of dirac measures this is so called the phenomena of weak concentration our purposes in this article are i to introduce the shadow system due to the bubbling phenomena when one of parameters i crosses 4 and j 4 n where 1 i j 2 ii to show how to calculate the topological degree of toda systems by computing the topological degree of the general shadow systems iii to calculate the topological degree of the shadow system for one point blow up we believe that the degree counting formula for the shadow system would be useful in other problems

degree counting for toda system with simple singularity one point blow up abstract in this paper we study the degree counting formula of the rank two toda system with simple singular source when 1 0 4 4 8 and 2 4 n the key step is to derive the degree formula of the shadow system which arises from the bubbling solutions as 1 tends to 4 in order to compute the topological degree of the shadow system we need to find some suitable deformation during this deformation we shall deal with new difficulty arising from the phenomenon blow up does not necessarily imply concentration of mass this phenomenon occurs due to the collapsing of singularities this is a continuation of the previous work 25

bubbling solutions for a skew symmetric chern simons system in a torus abstract we establish the existence of bubbling solutions for the following skew symmetric chern simons system u 1 1 e 2 e u 2 1 e u 1 4 i 1 n 1 p i 1 u 2 1 e 2 e u 1 1 e u 2 4 i 1 n 2 p i 2 over a parallelogram with doubly periodic boundary condition where e 0 is a coupling parameter and p denotes the dirac measure concentrated at p we obtain that if n 1 1 n 2 1 1 there exists an e 0 0 such that for any e 0 e 0 the above system admits a solution u 1 e u 2 e satisfying u 1 e and u 2 e blow up simultaneously at the point p and 1 e 2 e u j e 1 e u i e 4 n i p 1 i j 2 i j as e 0 where the location of the point p defined by 1 12 satisfies the condition 1 13

reciprocal processes on a tree modeling and estimation issues motivated by multiresolution decomposition methods such as the discrete wavelet transformation the authors introduce reciprocal processes on truncated n ary trees the authors discuss the relationship between such processes and nearest neighbor models the authors show that they can derive a recursive description of the process and that all reciprocal processes on n ary trees reduce to autoregressive processes in the case of zero valued boundary values at the bottom of the tree corresponding to truncation of the tree the authors then study the smoothing equations associated with such models

the wavelet transform time frequency localization and signal analysis two different procedures for effecting a frequency analysis of a time dependent signal locally in time are studied the first procedure is the short time or windowed fourier transform the second is the wavelet transform in which high frequency components are studied with sharper time resolution than low frequency components the similarities and the differences between these two methods are discussed for both schemes a detailed study is made of the reconstruction method and its stability as a function of the chosen time frequency density finally the notion of time frequency localization is made precise within this framework by two localization theorems

wavelet packets of nonstationary random processes contributing factors for stationarity and decorrelation the paper addresses the analysis and interpretation of second order random processes by using the wavelet packet transform it is shown that statistical properties of the wavelet packet coefficients are specific to the filtering sequences characterizing wavelet packet paths these statistical properties also depend on the wavelet order and the form of the cumulants of the input random process the analysis performed points out the wavelet packet paths for which stationarization decorrelation and higher order dependency reduction are effective among the coefficients associated with these paths this analysis also highlights the presence of singular wavelet packet paths the paths such that stationarization does not occur and those for which dependency reduction is not expected through successive decompositions the focus of the paper is on understanding the role played by the parameters that govern stationarization and dependency reduction in the wavelet packet domain this is addressed with respect to semi analytical cumulant expansions for modeling different types of nonstatonarity and correlation structures the characterization obtained eases the interpretation of random signals and time series with respect to the statistical properties of their coefficients on the different wavelet packet paths

a generalized trudinger moser inequality on a compact riemannian surface abstract let g be a compact riemannian surface let h be two smooth functions on with d v g 0 and h 0 h 0 in this paper using a method of blow up analysis we prove that the functional 1 j h u 1 2 g u 2 d v g 8 1 d v g u d v g 8 log h e u d v g is bounded from below in w 1 2 g moreover we obtain a sufficient condition under which j h attains its infimum in w 1 2 g these results generalize the main results in ding jost li wang 1997 and yang zhu 2017

some new smoother implementations for discrete time gaussian reciprocal processes abstract the smoothing problem for discrete time gaussian reciprocal processes is examined for this class of process the smoother takes the form of a second order two point boundary value nearest neighbour model this smoother is non causal but admits causal implementations which extend the two filter and rauch tung striebel smoothing formulae of gauss markov processes a multi rate implementation based on a block cyclic reduction method is also described

topological degree for a mean field equation on riemann surfaces we consider the following mean field equations 0 1 where m is a compact riemann surface with volume 1 h is a positive continuous function on m is a real number 0 2 and where is a bounded smooth domain h is a c1 positive function on and based on our previous analytic work 14 we prove among other things that the degree counting formula for 0 1 is given by for 8m 8 m 1 2003 wiley periodicals inc

on the use of the quasi likelihood method in teratological experiments this paper concerns the impact of litter effects on the inference of dose response relationships in teratological experiments with binary response kupper et al 1986 biometrics 42 85 98 concluded that when intra litter correlations are dose dependent the use of a common intra litter correlation likelihood based on the beta binomial distribution could lead to severe bias in estimation of the parameter beta which characterizes the dose response relationship we show here that the problems of bias and coverage probability for beta could still be substantial when one uses the likelihood with heterogeneous intra litter correlations we then examine through a simulation study the performance of the quasi likelihood method wedderburn 1974 biometrika 61 439 447 and recommend that this method with a common intra litter correlation parameter be used when the number of litters is small or modest

recursive filtering and smoothing for reciprocal gaussian processes pinned boundary case the least square estimation problem for pinned to zero discrete index reciprocal gaussian processes in additive white noise is solved thus completing and extending some previous results available in the literature in particular following the innovations approach a finite set of recursive equations is obtained for the filter and for the three standard classes of smoothers fixed point fixed interval fixed lag recursive expressions for the mean square performance of the proposed estimators are also given

bubbling solutions for moser trudinger type equations on compact riemann surfaces abstract we study an elliptic equation related to the moser trudinger inequality on a compact riemann surface s g g u u e u 2 1 s s u e u 2 d v g 0 in s s u d v g 0 where 0 is a small parameter s is the area of s g is the laplace beltrami operator and d v g is the area element given any integer k 1 under general conditions on s we find a bubbling solution u which blows up at exactly k points in s as 0 when s is a flat two torus in rectangular form we find that either seven or nine families of such solutions do exist for k 2 in particular in any square flat two torus actually nine families of bubbling solutions with two bubbling points do exist if s is a riemann surface with non constant robin s function then at least two bubbling solutions with k 1 exists

longitudinal data analysis using generalized linear models summary this paper proposes an extension of generalized linear models to the analysis of longitudinal data we introduce a class of estimating equations that give consistent estimates of the regression parameters and of their variance under mild assumptions about the time dependence the estimating equations are derived without specifying the joint distribution of a subject s observations yet they reduce to the score equations for multivariate gaussian outcomes asymptotic theory is presented for the general class of estimators specific cases in which we assume independence m dependence and exchangeable correlation structures from each subject are discussed efficiency of the proposed estimators in two simple situations is considered the approach is closely related to quasi likelih ood some key ironh estimating equation generalized linear model longitudinal data quasi likelihood repeated measures

wavelet representations of stochastic processes and multiresolution stochastic models deterministic signal analysis in a multiresolution framework through the use of wavelets has been extensively studied very successfully in recent years in the context of stochastic processes the use of wavelet bases has not yet been fully investigated we use compactly supported wavelets to obtain multiresolution representations of stochastic processes with paths in l sup 2 defined in the time domain we derive the correlation structure of the discrete wavelet coefficients of a stochastic process and give new results on how and when to obtain strong decay in correlation along time as well as across scales we study the relation between the wavelet representation of a stochastic process and multiresolution stochastic models on trees proposed by basseville et al see ieee trans inform theory vol 38 p 766 784 mar 1992 we propose multiresolution stochastic models of the discrete wavelet coefficients as approximations to the original time process these models are simple due to the strong decorrelation of the wavelet transform experiments show that these models significantly improve the approximation in comparison with the often used assumption that the wavelet coefficients are completely uncorrelated

multiscale autoregressive processes i schur levinson parametrizations in many applications it is of interest to analyze and recognize phenomena occurring at different scales the recently introduced wavelet transforms provide a time and scale decomposition of signals that offers the possibility of such analysis a corresponding statistical framework to support the development of optimal multiscale statistical signal processing algorithms is described the theory of multiscale signal representation leads naturally to models of signals on trees and this provides the framework for investigation in particular the class of isotropic processes on homogeneous trees is described and a theory of autoregressive models is developed in this context this leads to generalizations of schur and levinson recursions associated properties of the resulting reflection coefficients and the initial pieces in a system theory for multiscale modeling