corrected confidence bands for functional data using principal components functional principal components fpc analysis is widely used to decompose and express functional observations curve estimates implicitly condition on basis functions and other quantities derived from fpc decompositions however these objects are unknown in practice in this article we propose a method for obtaining correct curve estimates by accounting for uncertainty in fpc decompositions additionally pointwise and simultaneous confidence intervals that account for both model and decomposition based variability are constructed standard mixed model representations of functional expansions are used to construct curve estimates and variances conditional on a specific decomposition iterated expectation and variance formulas combine model based conditional estimates across the distribution of decompositions a bootstrap procedure is implemented to understand the uncertainty in principal component decomposition quantities our method compares favorably to competing approaches in simulation studies that include both densely and sparsely observed functions we apply our method to sparse observations of cd4 cell counts and to dense white matter tract profiles code for the analyses and simulations is publicly available and our method is implemented in the r package refund on cran

exact thresholds for dpll on random xor sat and np complete extensions of xor sat this paper discusses a model of constraint satisfaction problems known as uniquely extendible constraint satisfaction problems this model includes and generalizes xor sat and the model includes an np complete problem that appears to share many of the threshold characteristics of random sat in this paper we find an exact threshold in the behavior of two versions of dpll on random instances of this problem one version uses the unit clause heuristic and the other uses the generalized unit clause heuristic specifically for dpll with the unit clause heuristic we prove that there is a clause density c smaller than the satisfiability threshold such that for random instances with density smaller than this threshold dpll with unit clause will find a satisfying assignment in linear time with uniformly positive probability however for random instances with density larger than this threshold dpll with unit clause will require exponential time with uniformly positive probability to find a satisfying assignment we then find the equivalent threshold density for dpll with the generalized unit clause heuristic we also prove the analog of the 2 p sat conjecture for this class of problems

random constraint satisfaction easy generation of hard satisfiable instances in this paper we show that the models of random csp instances proposed by xu and li k xu w li exact phase transitions in random constraint satisfaction problems journal of artificial intelligence research 12 2000 93 103 k xu w li many hard examples in exact phase transitions with application to generating hard satisfiable instances technical report corr report cs cc 0302001 revised version in theoretical computer science 355 2006 291 302 are of theoretical and practical interest indeed these models called rb and rd present several nice features first it is quite easy to generate random instances of any arity since no particular structure has to be integrated or property enforced in such instances then the existence of an asymptotic phase transition can be guaranteed while applying a limited restriction on domain size and on constraint tightness in that case a threshold point can be precisely located and all instances have the guarantee to be hard at the threshold i e to have an exponential tree resolution complexity next a formal analysis shows that it is possible to generate forced satisfiable instances whose hardness is similar to unforced satisfiable ones this analysis is supported by some representative results taken from an intensive experimentation that we have carried out using complete and incomplete search methods

the evolution fractional p laplacian equation in rn fundamental solution and asymptotic behaviour abstract we consider the natural time dependent fractional p laplacian equation posed in the whole euclidean space with parameters p 2 and s 0 1 fractional exponent we show that the cauchy problem for data in the lebesgue l q spaces is well posed and show that the solutions form a family of non expansive semigroups with regularity and other interesting properties as main results we construct the self similar fundamental solution for every mass value m and prove that general finite mass solutions converge towards that fundamental solution having the same mass and convergence holds in all l q spaces a number of additional properties and estimates complete the picture

the h infty calculus and sums of closed operators we develop a very general operator valued functional calculus for operators with an h infty calculus we then apply this to the joint functional calculus of two sectorial operators when one has an h infty calculus using this we prove theorem of dore venni type on sums of sectorial operators and apply our results to the problem of l p maximal regularity our main assumption is the r boundedness of certain sets of operators and therefore methods from the geometry of banach spaces are essential here in the final section we exploit the special banach space structure of l 1 spaces and c k spaces to obtain some more detailed results in this setting

the h calculus and sums of closed operators we develop a very general operator valued functional calcu lus for operators with an h 1 calculus we then apply this to the joint functional calculus of two commuting sectorial operators when one has an h 1 calculus using this we prove theorem of dore venni type on sums of commuting sectorial operators and apply our results to the problem of lp maximal regularity our main assumption is the r boundedness of certain sets of operators and therefore methods from the geometry of ba nach spaces are essential here in the final section we exploit the special banach space structure of l1 spaces and c k spaces to obtain some more detailed results in this setting

a dynamically consistent method to solve nonlinear multidimensional advection reaction equations with fractional diffusion abstract this work is motivated by an extension of both the burgers fisher and the burgers huxley equations in multiple dimensions considering riesz fractional diffusion initial boundary conditions which are positive and bounded are imposed on a closed and bounded set and a finite difference method is proposed to approximate the solutions of the fractional model the methodology is a linear and implicit technique which is based on fractional centered differences we show in this manuscript that the method can be expressed in vector form using a minkowski matrix under suitable conditions the main properties of minkowski matrices are used then to establish the existence and the uniqueness of the solutions of the finite difference method as well as the capability of the technique to preserve the positivity and the boundedness additionally we show that the method is a consistent technique which is stable and convergent with first order of convergence in time and second order in space some illustrative simulations show that the scheme is capable of preserving the positivity and the boundedness of the numerical approximations

backward estimates for nonnegative solutions to a class of singular parabolic equations abstract in this paper we improve a recent result of the same authors recalde and vespri 2015 by using the same approach in a more tricky way more precisely we study a class of quasilinear singular parabolic equations with l coefficients whose prototypes are the p laplacian 2 n n 1 p 2 equations let us consider nonnegative solutions defined in r r n we recall that in recalde and vespri 2015 starting from the value attained in a point x 0 t 0 by the solution we proved sharp estimates in the stripe 1 e t 0 r n in this note we show that quite surprisingly sharp estimates hold also for the remote past i e also for the stripe 0 t 0 r n in the last section we briefly show how these results can be adapted to equations of porous medium type in the fast diffusion range i e n 2 n m 1

nonequilibrium fluctuations for a tagged particle in mean zero one dimensional zero range processes we prove a non equilibrium functional central limit theorem for the position of a tagged particle in mean zero one dimensional zero range process the asymptotic behavior of the tagged particle is described by a stochastic differential equation governed by the solution of the hydrodynamic equation

weak solutions of the porous medium equation in a cylinder we show that if d r n is a bounded domain with smooth boundary and u l m d e t u 0 solves u t u m m 1 in the sense of distributions on d 0 t and vanishes on d 0 t in a suitable weak sense then u is holder continuous in d 0 t

density and tails of unimodal convolution semigroups abstract we give sharp bounds for the isotropic unimodal probability convolution semigroups when their levy khintchine exponent has matuszewska indices strictly between 0 and 2

combinatorial sharpness criterion and phase transition classification for random csps we investigate the nature of the phase transition sharp or coarse for random constraint satisfaction problems we first give a sharp threshold criterion specified for csps which is derived from friedgut bourgain s one thus we get a complete and precise classification of the nature of the threshold for symmetric boolean csps in particular we show that it is governed by two local properties strongly related to the problems 1 sat and 2 xor sat

asymptotic behavior for the heat equation in nonhomogeneous media with critical density abstract we study the long time behavior of solutions to the heat equation in nonhomogeneous media with critical singular density x 2 t u u in r n 0 in dimensions n 3 the asymptotic behavior proves to have some interesting and quite striking properties we show that there are two completely different asymptotic profiles depending on whether the initial data u 0 vanishes at x 0 or not moreover in the former the results are true only for radially symmetric solutions and we provide counterexamples to convergence to symmetric profiles in the general case

theory of extended solutions for fast diffusion equations in optimal classes of data radiation from singularities this paper is devoted to constructing a general theory of nonnegative solutions for the equation

on properties of functional principal components analysis summary functional data analysis is intrinsically infinite dimensional functional principal component analysis reduces dimension to a finite level and points to the most significant components of the data however although this technique is often discussed its properties are not as well understood as they might be we show how the properties of functional principal component analysis can be elucidated through stochastic expansions and related results our approach quantifies the errors that arise through statistical approximation in successive terms of orders n 1 2 n 1 n 3 2 where n denotes sample size the expansions show how spacings among eigenvalues impact on statistical performance the term of size n 1 2 illustrates first order properties and leads directly to limit theory which describes the dominant effect of spacings thus for example spacings are seen to have an immediate first order effect on properties of eigenfunction estimators but only a second order effect on eigenvalue estimators our results can be used to explore properties of existing methods and also to suggest new techniques in particular we suggest bootstrap methods for constructing simultaneous confidence regions for an infinite number of eigenvalues and also for individual eigenvalues and eigenvectors

optimal existence and uniqueness theory for the fractional heat equation abstract we construct a theory of existence uniqueness and regularity of solutions for the fractional heat equation t u s u 0 0 s 1 posed in the whole space r n with data in a class of locally bounded radon measures that are allowed to grow at infinity with an optimal growth rate we consider a class of nonnegative weak solutions and prove that there is an equivalence between nonnegative data and solutions which is given in one direction by the representation formula in the other one by the initial trace we review many of the typical properties of the solutions in particular we prove optimal pointwise estimates and new harnack inequalities

an algorithm for random signed 3 sat with intervals interval k sat k isat is a generalization of classical k sat where the variables can take values in 0 1 instead of 0 1 and the literals are of the form x i for intervals i 0 1 it falls within the class of signed satisfiability problems we propose an algorithm for 3 isat and analyze it on uniformly random formulas the algorithm follows the unit clause paradigm enhanced by a very limited backtracking option using wormald s ode method we prove that if m n 2 3 with high probability our algorithm succeeds in finding an assignment of values to the variables satisfying the formula

fractional nonlinear degenerate diffusion equations on bounded domains part i existence uniqueness and upper bounds abstract we investigate quantitative properties of nonnegative solutions u t x 0 to the nonlinear fractional diffusion equation t u l f u 0 posed in a bounded domain x r n with appropriate homogeneous dirichlet boundary conditions as l we can use a quite general class of linear operators that includes the two most common versions of the fractional laplacian s 0 s 1 in a bounded domain with zero dirichlet boundary conditions but it also includes many other examples since our theory only needs some basic properties that are typical of linear heat semigroups the nonlinearity f is assumed to be increasing and is allowed to be degenerate the prototype is the power case f u u m 1 u with m 1 in this paper we propose a suitable class of solutions of the equation and cover the basic theory we prove existence uniqueness of such solutions and we establish upper bounds of two forms absolute bounds and smoothing effects as well as weighted l 1 estimates the class of solutions is very well suited for that work the standard laplacian case s 1 is included and the linear case m 1 can be recovered in the limit in a companion paper bonforte and vazquez in preparation we will complete the study with more advanced estimates like the upper and lower boundary behaviour and harnack inequalities for which the results of this paper are needed

large systems of interacting particles and the porous medium equation abstract we study the empirical processes of systems of interacting particles whose time evolution is given by coupled ordinary differential equations in particular as the population size tends to infinity we derive different versions of the porous medium equation as limit dynamics

rigorous results for random 2 p sat in recent years there has been significant interest in the study of random k sat formulae for a given set of n boolean variables let bk denote the set of all possible disjunctions of k distinct non complementary literals from its variables k clauses a random k sat formula fk n m is formed by selectinguniformly and independently m clauses from bk and takingtheir conjunction motivated by insights from statistical mechanics that suggest a possible relationship between the order of phase transitions and computational complexity monasson and zecchina phys rev e 56 2 1997 1357 proposed the random 2 p sat model for a given p 0 1 a random 2 p sat formula f2 p n m has m randomly chosen clauses over n variables where pm clauses are chosen from b3 and 1 p m from b2 usingthe heuristic replica method of statistical mechanics monasson and zecchina gave a number of non rigorous predictions on the behavior of random 2 p sat formulae in this paper we give the 1rst rigorous results for random 2 p sat including the following surprising fact for p 2 5 with probability 1 o 1 a random 2 p sat formula is satisfiable if its 2 sat subformula is satisfiable that is for p 6 2 5 random 2 p sat behaves like random 2 sat

uniqueness of solutions of degenerate diffusion equations with measures as initial conditions on considere le probleme de cauchy u t u m u p sur r n 0 t u x 0 x sur r n ou m 1 p 1 n 1 0 1 p m 2 n et une mesure finie

a priori estimates for fractional nonlinear degenerate diffusion equations on bounded domains we investigate quantitative properties of the nonnegative solutions u t x geq 0 to the nonlinear fractional diffusion equation partial t u mathcal l u m 0 posed in a bounded domain x in omega subset mathbb r n with m 1 for t 0 as mathcal l we use one of the most common definitions of the fractional laplacian delta s 0 s 1 in a bounded domain with zero dirichlet boundary conditions we consider a general class of very weak solutions of the equation and obtain a priori estimates in the form of smoothing effects absolute upper bounds lower bounds and harnack inequalities we also investigate the boundary behaviour and we obtain sharp estimates from above and below in addition we obtain similar estimates for fractional semilinear elliptic equations either the standard laplacian case s 1 or the linear case m 1 are recovered as limits the method is quite general suitable to be applied to a number of similar problems

functional data analysis for sparse longitudinal data we propose a nonparametric method to perform functional principal components analysis for the case of sparse longitudinal data the method aims at irregularly spaced longitudinal data where the number of repeated measurements available per subject is small in contrast classical functional data analysis requires a large number of regularly spaced measurements per subject we assume that the repeated measurements are located randomly with a random number of repetitions for each subject and are determined by an underlying smooth random subject specific trajectory plus measurement errors basic elements of our approach are the parsimonious estimation of the covariance structure and mean function of the trajectories and the estimation of the variance of the measurement errors the eigenfunction basis is estimated from the data and functional principal components score estimates are obtained by a conditioning step this conditional estimation method is conceptually simple and straightforward to implement

structured functional additive regression in reproducing kernel hilbert spaces summary functional additive models provide a flexible yet simple framework for regressions involving functional predictors the utilization of a data driven basis in an additive rather than linear structure naturally extends the classical functional linear model however the critical issue of selecting non linear additive components has been less studied in this work we propose a new regularization framework for structure estimation in the context of reproducing kernel hilbert spaces the approach proposed takes advantage of functional principal components which greatly facilitates implementation and theoretical analysis the selection and estimation are achieved by penalized least squares using a penalty which encourages the sparse structure of the additive components theoretical properties such as the rate of convergence are investigated the empirical performance is demonstrated through simulation studies and a real data application

sharp short and long time l bounds for solutions to porous media equations with homogeneous neumann boundary conditions abstract we study a class of nonlinear diffusion equations whose model is the classical porous media equation on domains r n n 3 with homogeneous neumann boundary conditions we improve the known results in such model case proving sharp l q 0 l regularizing properties of the evolution for short time and sharp long time l bounds for convergence of solutions to their mean value the generality of the discussion allows to consider almost at the same time weighted versions of the above equation provided an appropriate weighted sobolev inequality holds in fact we show that the validity of such weighted sobolev inequality is equivalent to the validity of a suitable l q 0 l bound for solutions to the associated weighted porous media equation the long time asymptotic analysis relies on the assumed weighted sobolev inequality only and allows to prove uniform convergence to the mean value with the rate predicted by linearization in such generality

generalized linear models with functional predictors summary we present a technique for extending generalized linear models to the situation where some of the predictor variables are observations from a curve or function the technique is particularly useful when only fragments of each curve have been observed we demonstrate on both simulated and real data sets how this approach can be used to perform linear logistic and censored regression with functional predictors in addition we show how functional principal components can be used to gain insight into the relationship between the response and functional predictors finally we extend the methodology to apply generalized linear models and principal components to standard missing data problems

a general fractional porous medium equation we develop a theory of existence and uniqueness for the following porous medium equation with fractional diffusion u t 2 u m 1 u 0 x r n t 0 u x 0 f x x r n we consider data f l 1 r n and all exponents 0 0 existence and uniqueness of a weak solution is established for m m n n giving rise to an l 1 contraction semigroup in addition we obtain the main qualitative properties of these solutions in the lower range 0 m m existence and uniqueness of solutions with good properties happen under some restrictions and the properties are different from the case above m we also study the dependence of solutions on f m and moreover we consider the above questions for the problem posed in a bounded domain

the fractional laplacian in power weighted l p spaces integration by parts formulas and self adjointness abstract we consider the fractional laplacian operator s let s 0 1 on euclidean space and investigate the validity of the classical integration by parts formula that connects the l 2 r d scalar product between a function and its fractional laplacian to the nonlocal norm of the fractional sobolev space h s r d more precisely we focus on functions belonging to some weighted l 2 space whose fractional laplacian belongs to another weighted l 2 space we prove and disprove the validity of the integration by parts formula depending on the behaviour of the weight x at infinity the latter is assumed to be like a power both near the origin and at infinity the two powers being possibly different our results have direct consequences for the self adjointness of the linear operator formally given by 1 s the generality of the techniques developed allows us to deal with weighted l p spaces as well

heat kernel estimates for the fractional laplacian with dirichlet conditions we give sharp estimates for the heat kernel of the fractional laplacian with dirichlet condition for a general class of domains including lipschitz domains

representation of solutions and large time behavior for fully nonlocal diffusion equations abstract we study the cauchy problem for a nonlocal heat equation which is of fractional order both in space and time we prove four main theorems i a representation formula for classical solutions ii a quantitative decay rate at which the solution tends to the fundamental solution iii optimal l 2 decay of mild solutions in all dimensions iv l 2 decay of weak solutions via energy methods the first result relies on a delicate analysis of the definition of classical solutions after proving the representation formula we carefully analyze the integral representation to obtain the quantitative decay rates of ii next we use fourier analysis techniques to obtain the optimal decay rate for mild solutions here we encounter the critical dimension phenomenon where the decay rate attains the decay rate of that in a bounded domain for large enough dimensions consequently the decay rate does not anymore improve when the dimension increases the theory is markedly different from that of the standard caloric functions and this substantially complicates the analysis finally we use energy estimates and a comparison principle to prove a quantitative decay rate for weak solutions defined via a variational formulation our main idea is to show that the l 2 norm is actually a subsolution to a purely time fractional problem which allows us to use the known theory to obtain the result

non local dirichlet forms and symmetric jump processes we consider the non local symmetric dirichlet form e f given by with f the closure with respect to e 1 of the set of c 1 functions on r d with compact support where e 1 f f e f f f rd f x 2 dx and where the jump kernel j satisfies for 0 2 x y 1 this assumption allows the corresponding jump process to have jump intensities whose sizes depend on the position of the process and the direction of the jump we prove upper and lower estimates on the heat kernel we construct a strong markov process corresponding to e f we prove a parabolic harnack inequality for non negative functions that solve the heat equation with respect to e finally we construct an example where the corresponding harmonic functions need not be continuous

nonlocal equations with measure data we develop an existence regularity and potential theory for nonlinear integrodifferential equations involving measure data the nonlocal elliptic operators considered are possibly degenerate and cover the case of the fractional p laplacean operator with measurable coefficients we introduce a natural function class where we solve the dirichlet problem and prove basic and optimal nonlinear wolff potential estimates for solutions these are the exact analogs of the results valid in the case of local quasilinear degenerate equations established by boccardo and gallouet j funct anal 87 149 169 1989 partial differ equ 17 641 655 1992 and kilpelainen and mal ann scuola norm sup pisa cl sci iv 19 591 613 1992 acta math 172 137 161 1994 as a consequence we establish a number of results that can be considered as basic building blocks for a nonlocal nonlinear potential theory fine properties of solutions calderon zygmund estimates continuity and boundedness criteria are established via wolff potentials a main tool is the introduction of a global excess functional that allows us to prove a nonlocal analog of the classical theory due to campanato ann mat pura appl iv 69 321 381 1965 our results cover the case of linear nonlocal equations with measurable coefficients and the one of the fractional laplacean and are new already in such cases

restricted likelihood ratio tests for functional effects in the functional linear model the goal of our article is to provide a transparent robust and computationally feasible statistical approach for testing in the context of scalar on function linear regression models assuming linearity between response and predictors we are interested in testing for the necessity of functional effects our methods are motivated by and applied to a large longitudinal study involving diffusion tensor imaging of intracranial white matter tracts in a susceptible cohort in the context of this study we conduct hypothesis tests that are motivated by anatomical knowledge and support recent findings regarding the relationship between cognitive impairment and white matter demyelination r code and data are in the examples of refund rlrt pfr supplementary materials for this article are available online

models for random constraint satisfaction problems we introduce a class of models for random constraint satisfaction problems this class includes and generalizes many previously studied models we characterize those models from our class which are asymptotically interesting in the sense that the limiting probability of satisfiability changes significantly as the number of constraints increases we also discuss models which exhibit a sharp threshold for satisfiability in the sense that the limiting probability jumps from 0 to 1 suddenly as the number of constraints increases

generalized functional linear models we propose a generalized functional linear regression model for a regression situation where the response variable is a scalar and the predictor is a random function a linear predictor is obtained by forming the scalar product of the predictor function with a smooth parameter function and the expected value of the response is related to this linear predictor via a link function if in addition a variance function is specified this leads to a functional estimating equation which corresponds to maximizing a functional quasi likelihood this general approach includes the special cases of the functional linear model as well as functional poisson regression and functional binomial regression the latter leads to procedures for classification and discrimination of stochastic processes and functional data we also consider the situation where the link and variance functions are unknown and are estimated nonparametrically from the data using a semiparametric quasi likelihood procedure an essential step in our proposal is dimension reduction by approximating the predictor processes with a truncated karhunen loeve expansion we develop asymptotic inference for the proposed class of generalized regression models in the proposed asymptotic approach the truncation parameter increases with sample size and a martingale central limit theorem is applied to establish the resulting increasing dimension asymptotics we establish asymptotic normality for a properly scaled distance between estimated and true functions that corresponds to a suitable l 2 metric and is defined through a generalized covariance operator as a consequence we obtain asymptotic tests and simultaneous confidence bands for the parameter function that determines the model the proposed estimation inference and classification procedures and variants with unknown link and variance functions are investigated in a simulation study we find that the practical selection of the number of components works well with the aic criterion and this finding is supported by theoretical considerations we include an application to the classification of medflies regarding their remaining longevity status based on the observed initial egg laying curve for each of 534 female medflies

uniqueness and properties of distributional solutions of nonlocal equations of porous medium type abstract we study the uniqueness existence and properties of bounded distributional solutions of the initial value problem for the anomalous diffusion equation t u l u 0 here l can be any nonlocal symmetric degenerate elliptic operator including the fractional laplacian and numerical discretizations of this operator the function r r is only assumed to be continuous and nondecreasing the class of equations include nonlocal generalized porous medium equations fast diffusion equations and stefan problems in addition to very general uniqueness and existence results we obtain stability l 1 contraction and a priori estimates we also study local limits continuous dependence and properties and convergence of a numerical approximation of our equations

functional linear regression analysis for longitudinal data we propose nonparametric methods for functional linear regression which are designed for sparse longitudinal data where both the predictor and response are functions of a covariate such as time predictor and response processes have smooth random trajectories and the data consist of a small number of noisy repeated measurements made at irregular times for a sample of subjects in longitudinal studies the number of repeated measurements per subject is often small and may be modeled as a discrete random number and accordingly only a finite and asymptotically nonincreasing number of measurements are available for each subject or experimental unit we propose a functional regression approach for this situation using functional principal component analysis where we estimate the functional principal component scores through conditional expectations this allows the prediction of an unobserved response trajectory from sparse measurements of a predictor trajectory the resulting technique is flexible and allows for different patterns regarding the timing of the measurements obtained for predictor and response trajectories asymptotic properties for a sample of n subjects are investigated under mild conditions as n oo and we obtain consistent estimation for the regression function besides convergence results for the components of functional linear regression such as the regression parameter function we construct asymptotic pointwise confidence bands for the predicted trajectories a functional coefficient of determination as a measure of the variance explained by the functional regression model is introduced extending the standard r 2 to the functional case the proposed methods are illustrated with a simulation study longitudinal primary biliary liver cirrhosis data and an analysis of the longitudinal relationship between blood pressure and body mass index

transformations of self similar solutions for porous medium equations of fractional type abstract we consider four different models of nonlinear diffusion equations involving fractional laplacians and study the existence and properties of classes of self similar solutions such solutions are an important tool in developing the general theory we introduce a number of transformations that allow us to map complete classes of solutions of one equation into those of another one thus providing us with a number of new solutions as well as interesting connections special attention is paid to the property of finite propagation

positivity local smoothing and harnack inequalities for very fast diffusion equations abstract we investigate qualitative properties of local solutions u t x 0 to the fast diffusion equation t u u m m with m 1 corresponding to general nonnegative initial data our main results are quantitative positivity and boundedness estimates for locally defined solutions in domains of the form 0 t with r d they combine into forms of new harnack inequalities that are typical of fast diffusion equations such results are new for low m in the so called very fast diffusion range precisely for all m m c d 2 d the boundedness statements are true even for m 0 while the positivity ones cannot be true in that range

global positivity estimates and harnack inequalities for the fast diffusion equation abstract we investigate local and global properties of positive solutions to the fast diffusion equation u t u m in the range d 2 d m 1 corresponding to general nonnegative initial data for the cauchy problem posed in the whole euclidean space r d we prove sharp local positivity estimates weak harnack inequalities and elliptic harnack inequalities we use them to derive sharp global positivity estimates and a global harnack principle for the mixed initial and boundary value problem posed in a bounded domain of r d with homogeneous dirichlet condition we prove weak and elliptic harnack inequalities our work shows that these fast diffusion flows have regularity properties comparable and in some senses better than the linear heat flow

initial trace of positive solutions to fractional diffusion equation with absorption in this paper we prove the existence of an initial trace t u of any positive solution u of the semilinear fractional diffusion equation h t u u f t x u 0 in r times r n where n 1 where the operator with 0 1 is the fractional laplacian and f r times r n times r r is a caratheodory function satisfying f t x u u 0 for all t x u r times r n times r we define the regular set of the trace t u as an open subset of r u r n carrying a nonnegative radon measive u such that lim t 0 ru u t x x dx ru d c 2 0 r u and the singular set s u r n r u as the set points a such that lim sup t 0 b a u t x dx 0 we study the reverse problem of constructing a positive solution to h with a given initial trace s where s r n is a closed set and is a positive radon measure on r r n s and develop the case f t x u t u p where 1 and p 1

some tools for functional data analysis multivariate data analysis permits the study of observations which are finite sets of numbers but modern data collection situations can involve data or the processes giving rise to them which are functions functional data analysis involves infinite dimensional processes and or data the paper shows how the theory of l splines can support generalizations of linear modelling and principal components analysis to samples drawn from random functions spline smoothing rests on a partition of a function space into two orthogonal subspaces one of which contains the obvious or structural components of variation among a set of observed functions and the other of which contains residual components this partitioning is achieved through the use of a linear differential operator and we show how the theory of polynomial splines can be applied more generally with an arbitrary operator and associated boundary constraints these data analysis tools are illustrated by a study of variation in temperature precipitation patterns among some canadian weather stations

hypothesis testing in functional linear models functional data arise frequently in biomedical studies where it is often of interest to investigate the association between functional predictors and a scalar response variable while functional linear models flm are widely used to address these questions hypothesis testing for the functional association in the flm framework remains challenging a popular approach to testing the functional effects is through dimension reduction by functional principal component pc analysis however its power performance depends on the choice of the number of pcs and is not systematically studied in this article we first investigate the power performance of the wald type test with varying thresholds in selecting the number of pcs for the functional covariates and show that the power is sensitive to the choice of thresholds to circumvent the issue we propose a new method of ordering and selecting principal components to construct test statistics the proposed method takes into account both the association with the response and the variation along each eigenfunction we establish its theoretical properties and assess the finite sample properties through simulations our simulation results show that the proposed test is more robust against the choice of threshold while being as powerful as and often more powerful than the existing method we then apply the proposed method to the cerebral white matter tracts data obtained from a diffusion tensor imaging tractography study

harnack estimates at large sharp pointwise estimates for nonnegative solutions to a class of singular parabolic equations abstract in this paper we deal with quasilinear singular parabolic equations with l coefficients whose prototypes are the p laplacian 2 n n 1 p 2 equations in this range of the parameters we are in the so called fast diffusion case extending a recent result ragnedda et al 2013 we are able to prove harnack estimates at large i e starting from the value attained in a point by the solution we are able to give explicit and sharp pointwise estimates from below by using the barenblatt solutions in the last section we briefly show how these results can be adapted to equations of porous medium type in the fast diffusion range i e n 2 n m 1

nonequilibrium scaling limit for a tagged particle in the simple exclusion process with long jumps we prove an invariance principle for a tagged particle in a simple exclusion process out of equilibrium the scaling limit is a time inhomogeneous process of independent increments related to the solution of a fractional heat equation

the fisher kpp problem with doubly nonlinear diffusion abstract the famous fisher kpp reaction diffusion model combines linear diffusion with the typical kpp reaction term and appears in a number of relevant applications in biology and chemistry it is remarkable as a mathematical model since it possesses a family of travelling waves that describe the asymptotic behaviour of a large class solutions 0 u x t 1 of the problem posed in the real line the existence of propagation waves with finite speed has been confirmed in some related models and disproved in others we investigate here the corresponding theory when the linear diffusion is replaced by the slow doubly nonlinear diffusion and we find travelling waves that represent the wave propagation of more general solutions even when we extend the study to several space dimensions a similar study is performed in the critical case that we call pseudo linear i e when the operator is still nonlinear but has homogeneity one with respect to the classical model and the pseudo linear case the slow travelling waves exhibit free boundaries

measure valued solutions and the phenomenon of blow down in logarithmic diffusion abstract the existence of solutions of elliptic and parabolic equations with data a measure has always been quite important for the general theory a prominent example being the fundamental solutions of the linear theory in nonlinear equations the existence of such solutions may find special obstacles that can be either essential or otherwise they may lead to more general concepts of solution we give a particular review of results in the field of nonlinear diffusion as a new contribution we study in detail the case of logarithmic diffusion associated with ricci flow in the plane where we can prove existence of measure valued solutions the surprising thing is that these solutions become classical after a finite time in that general setting the standard concept of weak solution is not adequate but we can solve the initial value problem for the logarithmic diffusion equation in the plane with bounded nonnegative measures as initial data in a suitable class of measure solutions we prove that the problem is well posed the phenomenon of blow down in finite time is precisely described initial point masses diffuse into the medium and eventually disappear after a finite time t i m i 4

fractional cahn hilliard allen cahn and porous medium equations abstract we introduce a fractional variant of the cahn hilliard equation settled in a bounded domain r n and complemented with homogeneous dirichlet boundary conditions of solid type i e imposed in the whole of r n after setting a proper functional framework we prove existence and uniqueness of weak solutions to the related initial boundary value problem then we investigate some significant singular limits obtained as the order of either of the fractional laplacians appearing in the equation is let tend to 0 in particular we can rigorously prove that the fractional allen cahn fractional porous medium and fractional fast diffusion equations can be obtained in the limit finally in the last part of the paper we discuss existence and qualitative properties of stationary solutions of our problem and of its singular limits

uniqueness of very weak solutions for a fractional filtration equation abstract we prove existence and uniqueness of distributional bounded solutions to a fractional filtration equation in r d with regards to uniqueness it was shown even for more general equations in 22 that if two bounded solutions u w of 1 1 satisfy u w l 1 r d 0 t then u w we obtain here that this extra assumption can in fact be removed and establish uniqueness in the class of merely bounded solutions for nonnegative initial data we first show that a minimal solution exists and then that any other solution must coincide with it a similar procedure is carried out for sign changing solutions as a consequence distributional solutions have locally finite energy

sharp thresholds of graph properties and the sat problem consider g n p to be the probability space of random graphs on n vertices with edge probability p we will be considering subsets of this space defined by monotone graph properties a monotone graph property p is a property of graphs such that a p is invariant under graph automorphisims b if graph h has property p then so does any graph g having h as a subgraph a monotone symmetric family of graphs is a family defined by such a property one of the first observations made about random graphs by erdos and renyi in their seminal work on random graph theory 12 was the existence of threshold phenomena the fact that for many interesting properties p the probability of p appearing in g n p exhibits a sharp increase at a certain critical value of the parameter p bollobas and thomason proved the existence of threshold functions for all monotone set properties 6 and in 14 it is shown that this behavior is quite general and that all monotone graph properties exhibit threshold behavior i e the probability of their appearance increases from values very close to 0 to values close to 1 in a very small interval more precise analysis of the size of the threshold interval is done in 7 this threshold behavior which occurs in various settings which arise in combinatorics and computer science is an instance of the phenomenon of phase transitions which is the subject of much interest in statistical physics one of the main questions that arises in studying phase transitions is how sharp is the transition for example one of the motivations for this paper arose from the question of the sharpness of the phase transition for the property of satisfiability of a random kcnf boolean formula nati linial who introduced me to this problem suggested that although much concrete analysis was being performed on this problem the best approach would be to find general conditions for sharpness of the phase transition answering the question posed in 14 as to the relation between the length of the threshold interval and the value of the critical probability in this paper we indeed introduce a simple condition and prove it is sufficient stated roughly in the setting of random graphs the main theorem states that if a property has a coarse threshold then it can be approximated by the property of having certain given graphs as a subgraph this condition can be applied in a more

a widder s type theorem for the heat equation with nonlocal diffusion the main goal of this work is to prove that every non negative strong solutionu x t to the problem u t delta alpha 2 u 0 rm for x t in mathbb r n times 0 t 0 alpha 2 can be written as u x t int mathbb r n p t x y u y 0 dy where p t x frac 1 t n alpha p left frac x t 1 alpha right and p x int mathbb r n e i x cdot xi xi alpha d xi this result shows uniqueness in the setting of non negative solutions and extends some classical results for the heat equation by widder in 15 to the nonlocal diffusion framework

a fractional porous medium equation abstract we develop a theory of existence uniqueness and regularity for the following porous medium equation with fractional diffusion u t 1 2 u m 1 u 0 x r n t 0 u x 0 f x x r n with m m n 1 n n 1 and f l 1 r n an l 1 contraction semigroup is constructed and the continuous dependence on data and exponent is established nonnegative solutions are proved to be continuous and strictly positive for all x r n t 0

an l p a priori estimate for the tricomi equation in the upper half space we establish an lp a priori estimate for the tricomi equation our main tool is mihlin s multiplier theorem combined with well known estimates of the newtonian potential 0 introduction the purpose of this paper is to establish an lp a priori estimate for the tricomi equation in the upper half space the tricomi equation arises in transonic gas dynamics and is a typical model equation of changing type it has been extensively investigated from the various viewpoints the tricomi equation can be interpreted as an elliptic equation which degenerates on the boundary which is our viewpoint in this paper we can formulate the dirichlet boundary value problem in the upper half space as follows 02 yacuz f in r 0 2 u 0 x x on mt where r y x y 0 x e rn and x is the laplacian in the x variable an l2 a priori estimate can be obtained very easily following the presentation in 8 we suppose that u e co rn and multiply the equation 0 1 by 92u then we integrate over r r to obtain 0 3 ir dy2x 2jj v dydx rn 1 o 9y2 rn 1 o a y 2irn1l vq dx j 1 f 3 dydx where vx stands for the gradient in x e r n this yields 0 4 2u f 1 yx 4y21 l2 rn dy l2 rn l2 rn m f 1 l2 rn 11vx01il2 rn 1 7 received by the editors december 30 1996 and in revised form february 10 1998 1991 mathematics subject classification primary 35j70 35b45

continuity of the temperature in boundary heat control problems abstract motivated by the boundary heat control problems formulated in the book of duvaut and lions we study a boundary stefan problem and a boundary porous media problem we prove continuity of the solution with the appropriate modulus we also extend the results to the fractional order case and to the anomalous diffusion problems

spectrum of the fractional p laplacian in rn and decay estimate for positive solutions of a schrodinger equation abstract in this paper we prove the existence of unbounded sequence of eigenvalues for the fractional p laplacian with weight in r n we also show a nonexistence result when the weight has positive integral in addition we show some qualitative properties of the first eigenfunction including a sharp decay estimate finally we extend the decay result to the positive solutions of a schrodinger type equation

existence and uniqueness of an attractive nonlinear diffusion system we establish the existence and uniqueness of an attractive fractional coupled system such a system has applications in biological populations of cells we confirm that the fractional system under consideration admits a global solution in the sobolev space the solution is shown to be unique the technique is founded on analytic method of the fixed point theory and the fractional differential operator is scrutinized from the view of the riemann liouville differential operator finally we illustrate some entropy fractional differential inequalities regarding the solution of the system

infinite speed of propagation and regularity of solutions to the fractional porous medium equation in general domains we study the positivity and regularity of solutions to the fractional porous me dium equations ut sum 0 in 0 for m 1 and s 0 1 with dirichlet boundary data u 0 in 0 n and nonnegative initial condition u 0 u0 0 our first result is a quantitative lower bound for solutions that holds for all positive times t 0 as a consequence we find a global harnack principle stating that for any t 0 solutions are comparable to ds m where d is the distance to this is in sharp contrast with the local case s 1 where the equation has finite speed of propagation after this we study the regularity of solutions we prove that solutions are classical in the interior c in x and c1 in t and establish a sharp cxs m regularity estimate up to the boundary our methods are quite general and can be applied to wider classes of nonlocal parabolic equations of the form ut f u 0 in both in bounded and unbounded domains 2016 wiley periodicals inc

uniqueness of entropy solutions to fractional conservation laws with fully infinite speed of propagation our goal is to study the uniqueness of bounded entropy solutions for a multidimensional conservation law including a non lipschitz convection term and a diffusion term of nonlocal porous medium type the nonlocality is given by a fractional power of the laplace operator for a wide class of nonlinearities the l 1 contraction principle is established despite the fact that the finite infinite speed of propagation alibaud jee 2007 cannot be exploited in our framework existence is deduced with perturbation arguments the method of proof adapted from andreianov maliki nodea 2010 requires a careful analysis of the action of the fractional laplacian on truncations of radial powers

the dirichlet problem for the fractional p laplacian evolution equation abstract we consider a model of fractional diffusion involving a natural nonlocal version of the p laplacian operator we study the dirichlet problem posed in a bounded domain of r n with zero data outside of for which the existence and uniqueness of strong nonnegative solutions is proved and a number of quantitative properties are established a main objective is proving the existence of a special separate variable solution u x t t 1 p 2 f x called the friendly giant which produces a universal upper bound and explains the large time behaviour of all nontrivial nonnegative solutions in a sharp way moreover the spatial profile f of this solution solves an interesting nonlocal elliptic problem we also prove everywhere positivity of nonnegative solutions with any nontrivial data a property that separates this equation from the standard p laplacian equation

on the cauchy problem for a general fractional porous medium equation with variable density abstract we study the well posedness of the cauchy problem for a fractional porous medium equation with a varying density 0 we establish existence of weak energy solutions uniqueness and nonuniqueness is studied as well according to the behavior of at infinity

degenerate operators of tricomi type in l p spaces and in spaces of continuous functions we study elliptic operators l with dirichlet boundary conditions on a bounded domain whose diffusion coefficients degenerate linearly at in tangential directions we compute the domain of l and establish existence uniqueness and maximal regularity of the elliptic and parabolic problems for l in lp spaces and in spaces of continuous functions moreover the analytic semigroups generated by l are consistent positive compact and exponentially stable

the cauchy problem for ut δum when 0 m 1 on considere le probleme de cauchy pour l equation de diffusion non lineaire u t u u m 1 0 sur 0 r n u 0 u 0 quand 0 m 1 on montre qu il existe une solution temporelle globale pour une fonction localement integrable u 0 on montre l unicite dans cette classe

large time behavior for a porous medium equation in a nonhomogeneous medium with critical density abstract we study the large time behavior of solutions to the cauchy problem for the porous medium equation in nonhomogeneous media with critical singular density x 2 t u u m in r n 0 where m 1 and n 3 with nonnegative initial condition u x 0 u 0 x 0 the asymptotic behavior proves to have some interesting and striking properties we show that there are different asymptotic profiles for the solutions depending on whether the continuous initial data u 0 vanishes at x 0 or not moreover when u 0 0 0 we show the convergence towards a peak type profile presenting a jump discontinuity coming from an interesting asymptotic simplification to a conservation law while when u 0 0 0 the limit profile remains continuous these phenomena illustrate the strong effect of the singularity at x 0 we improve the time scale of the convergence in sets avoiding the singularity on the way we also study the large time behavior for a porous medium equation with convection which is interesting for itself

contractive properties for the heat equation in sobolev spaces abstract consider the heat equation r u x u 0 in a cylinder 0 t r n 1 smooth lateral boundary under zero neumann or dirichlet conditions geometric conditions for are given that guarantee that for a given p x u t l p will be non increasing for any solution decay rates are also given for arbitrary and p it is shown how to construct an equivalent l p norm such that x t is non increasing in this norm

quantitative local and global a priori estimates for fractional nonlinear diffusion equations abstract we establish quantitative estimates for solutions u t x to the fractional nonlinear diffusion equation t u s u m 0 in the whole range of exponents m 0 0 s 1 the equation is posed in the whole space x r d we first obtain weighted global integral estimates that allow to establish existence of solutions for classes of large data in the core of the paper we obtain quantitative pointwise lower estimates of the positivity of the solutions depending only on the norm of the initial data in a certain ball the estimates take a different form in three exponent ranges slow diffusion good range of fast diffusion and very fast diffusion finally we show existence and uniqueness of initial traces

asymptotic expansion of solutions to the cauchy problem for doubly degenerate parabolic equations with measurable coefficients abstract we study the asymptotic behaviour of nonnegative solutions of the cauchy problem for doubly degenerate parabolic equations with variable coefficients when the initial datum has a finite mass the asymptotic expansion of the solution for a large time uniformly in whole space is established

the scaling window of the model d k csp abstract consider the model d k csp which consists of a set of n variables with the domain size d and a set of t r n ln d ln p constraints where each constraint binds k distinct variables and permits p d k tuples of values we prove that the exact phase transition of the model d k csp occurs at r c 1 as long as k ln d ln n where satisfies that 1 2 1 2 2 k 1 e 1 e 0 is a sufficiently small constant more importantly we establish the finite size scaling about this transition specifically let 0 be a given constant the probability of satisfiability of the model d k csp is greater than 1 if r 1 1 n 1 ln d is dependent on the control parameters and is less than if r 1 1 n ln d

one dimensional degenerate operators in lp spaces abstract we comprehensively study a one dimensional elliptic operator degenerating of first order at the boundary in an l p setting here the coefficient of the drift term determines the regularity and the possible boundary conditions of the problem