sharp l p bounds on spectral clusters for lipschitz metrics we establish l p bounds on l 2 normalized spectral clusters for self adjoint elliptic dirichlet forms with lipschitz coefficients in two dimensions we obtain best possible bounds for all 2 le p le infty up to logarithmic losses for 6 p le 8 in higher dimensions we obtain best possible bounds for a limited range of p

stable bundles and integrable systems on considere la geometrie symplectique des fibres cotangents aux espaces de modules de fibres vectoriels stables sur une surface de riemann on montre que ce sont des systemes dynamiques hamiltoniens algebriquement completement integrables

stochastic determination of well capture zones conditioned on regular grids of transmissivity measurements a stochastic approach is adopted for characterizing the uncertainty in time related well capture zones in heterogeneous formations transmissivity is modeled as a random space function conditioned on regular grids of evenly distributed transmissivity measurements monte carlo analysis is used to infer the probability distribution of the resulting stochastic capture zones three measures of conditioning performance are defined and evaluated for several combinations of conditioning density transmissivity variance and transmissivity integral scale furthermore a connection is made with previous work on transport theory simulation results show that conditioning performance increases with an increasing conditioning density an increasing transmissivity integral scale and a decreasing transmissivity variance however there is a threshold combination of conditioning density and integral scale beyond which conditioning performance stagnates the practical implications of the results in general and the threshold in particular are discussed

nonstationary stochastic analysis in well capture zone design using first order taylor s series approximation 1 nonstationarity of flow fields due to pumping wells and its impact on advective transport is of particular interest in well capture zone design and wellhead protection however techniques based on monte carlo methods to characterize the associated capture zone uncertainty are time consuming and cumbersome this paper introduces an alternative approach the mean and covariance of system state variables i e head pore water velocity and particle trajectory are approximated using a first order taylor s series with sensitivity coefficients estimated from the adjoint operator for a system of discrete equations the approach allows nonstationarity due to several sources e g transmissivity pumping boundary conditions to be treated by employing numerical solution methods it is able to handle irregular geometry varying boundary conditions complicated sink source terms and different covariance functions all of which are important factors for real world applications a comparison of results for the taylor s series approximation with those from monte carlo analysis showed in general good agreement for most of the tested particles particle trajectory variance calculated using taylor s series approximation is then used to predict well capture zone probabilities under the assumption of normality of the mass transport s state variables verification of this assumption showed that not all particle trajectories depending on their starting location are normally or log normally distributed however the risk of using the first order method to delineate the confidence interval of a well capture zone is minimal since it marginally overestimates the 2 5 probability contour furthermore this should be balanced against its greater computation efficiency over the monte carlo approach

carleman inequalities for the dirac and laplace operators and unique continuation on demontre les meilleures inegalites de type carleman possibles dans le cas de l operateur de dirac on obtient la propriete de prolongement unique pour d v ou v l loc r n 3n 2 2 n 3

on nonlinear functionals of random spherical eigenfunctions we prove central limit theorems and stein like bounds for the asymptotic behaviour of nonlinear functionals of spherical gaussian eigenfunctions our investigation combines asymptotic analysis of higher order moments for legendre polynomials and in addition recent results on malliavin calculus and total variation bounds for gaussian subordinated fields we discuss applications to geometric functionals like the defect and invariant statistics e g polyspectra of isotropic spherical random fields both of these have relevance for applications especially in an astrophysical environment

random data cauchy problem for the navier stokes equations on t3 in this paper we study the random data initial value problem of the navier stokes equations in l2 t3 by using the randomization approach recently developed by n burq and n tzvetkov we prove that for almost all where is the sample space of a probability space a p for the randomized initial data f h l 2 t3 the navier stokes equations have a global solution u c 0 l 2 t3 or more precisely u et f h c 0 h 12 t3

inversion of the borden tracer experiment data investigation of stochastic moment models inversion of dagan s two and three dimensional stochastic models using freyberg s 1986 rajaram and gelhar s 1988 1991 and barry and sposito s 1990 moment data from the borden experiment is carried out to examine 1 the validity of the two dimensional dagan 1982 and three dimensional dagan 1988 models and 2 the reduction in uncertainty of the spatial moments over nonconditioned estimates a direct application of bayesian statistical inference in conjunction with monte carlo integration is used to produce posterior probability density functions for the parameters the parameter ranges from all methods show horizontal integral scales between 1 85 and 4 04 m vertical integral scales z between 0 144 and 0 459 m and log hydraulic conductivities in k between 0 120 and 0 197 these results compare well to the earlier estimates of woodbury and sudicky 1991 and robin et al 1991 we show that moment prediction uncertainty is substantially reduced when both tracer moment data and prior estimates of the in k geostatistical parameters are incorporated into theoretical formulae based on stochastic dispersion theory

randomization and the gross pitaevskii hierarchy we study the gross pitaevskii hierarchy on the spatial domain mathbb t 3 by using an appropriate randomization of the fourier coefficients in the collision operator we prove an averaged form of the main estimate which is used in order to contract the duhamel terms that occur in the study of the hierarchy in the averaged estimate we do not need to integrate in the time variable an averaged spacetime estimate for this range of regularity exponents then follows as a direct corollary the range of regularity exponents that we obtain is alpha frac 3 4 it was shown in our previous joint work with gressman j funct anal 266 7 4705 4764 2014 that the range alpha 1 is sharp in the corresponding deterministic spacetime estimate this is in contrast to the non periodic setting which was studied by klainerman and machedon commun math phys 279 1 169 185 2008 where the spacetime estimate is known to hold whenever alpha geq 1 the goal of our paper is to extend the range of in this class of estimates in a probabilistic sense we use the new estimate and the ideas from its proof in order to study randomized forms of the gross pitaevskii hierarchy more precisely we consider hierarchies similar to the gross pitaevskii hierarchy but in which the collision operator has been randomized for these hierarchies we show convergence to zero in low regularity sobolev spaces of duhamel expansions of fixed deterministic density matrices we believe that the study of the randomized collision operators could be the first step in the understanding of a nonlinear form of randomization

the g generalized hitchin systems and prym varieties in this article we consider the generalized hitchin systems introduced by bottacin and markman for an arbitrary reductive complex group g they have the structure of lagrangian fibrations pr u by generalized prym varieties over sets u parametrizing families of weyl invariant curves the g generalized hitchin systems satisfy a rank two condition and one can find invariant varieties x which distinguish these systems it is then shown that there is a correspondence between these integrable systems and the varieties x k d h which are equipped with an appropriate two form with values in the cartan subalgebra h

role of the calibration process in reducing model predictive error 1 an equation is derived through which the variance of predictive error of a calibrated model can be calculated this equation has two terms the first term represents the contribution to predictive error variance that results from an inability of the calibration process to capture all of the parameterization detail necessary for the making of an accurate prediction if a model is uncalibrated with parameter values being supplied solely through outside information this is the only term required the second term represents the contribution to predictive error variance arising from measurement noise in an overdetermined system such as that which may be obtained through parameter lumping e g through the introduction of a spatial zonation scheme this is the only term required it is shown however that parameter lumping is a form of implicit regularization and that ignoring the implied first term of the predictive error variance equation can potentially lead to underestimation of predictive error variance a model s role as a predictor of environmental behavior can be enhanced if it is calibrated in such a way as to reduce the variance of those predictions which it is required to make it is shown that in some circumstances this can be accomplished through overfitting against historical field data it can also be accomplished by giving greater weight to those measurements which carry the greatest information content with respect to a required prediction this suggests that a departure may be necessary from the custom of using a single calibrated model for the making of many different predictions instead model calibration may need to be repeated many times so that in each case the calibration process is optimized for the making of a specific model prediction

on the lp norm of spectral clusters for compact manifolds with boundary we use microlocal and paradifferential techniques to obtain l8 norm bounds for spectral clusters associated with elliptic second order operators on two dimensional manifolds with boundary the result leads to optimal lq bounds in the range 2 q for l2 normalized spectral clusters on bounded domains in the plane and more generally for two dimensional compact manifolds with boundary we also establish new sharp lq estimates in higher dimensions for a range of exponents qn q

l 2 restriction bounds for eigenfunctions along curves in the quantum completely integrable case we show that for a quantum completely integrable system in two dimensions the l2 normalized joint eigenfunctions of the commuting semiclassical pseudodifferential operators satisfy restriction bounds of the form int gamma varphi j hbar 2 ds mathcal o log hbar for generic curves on the surface we also prove that the maximal restriction bounds of burq gerard tzvetkov bgt are generically attained for certain exceptional subsequences of eigenfunctions

stochastic capture zone delineation within the generalized likelihood uncertainty estimation methodology conditioning on head observations a stochastic methodology to evaluate the predictive uncertainty in well capture zones in heterogeneous aquifers with uncertain parameters is presented the approach is based on the generalized likelihood uncertainty estimation methodology the hydraulic conductivity is modeled as a random space function allowing for the uncertainty that stems from the imperfect knowledge of the parameters of the correlation structure parameters are sampled from prior distributions and are used for the generation of a large number of hydraulic conductivity fields which are subsequently used to solve the groundwater flow equation a likelihood is calculated for every simulation based on some goodness of fit measure between simulated heads and available observations using inverse particle tracking a capture zone is determined which is assigned the likelihood calculated for that particular simulation statistical analysis of the ensemble of all simulations enables the predictive uncertainty of the well capture zones to be defined results are presented for a hypothetical test case and different likelihood definitions used in the conditioning process the results show that the delineated capture zones are most sensitive to the mean hydraulic conductivity and the variance whereas the integral scale of the variogram is the parameter with the smallest influence for all likelihood measures the prior uncertainty is reduced considerably by introducing the observation heads but the reduction is most effective for the very selective likelihood definition the method presented can be used in real applications to quantify the uncertainty in the location and extent of well capture zones when little or no information is available about the hydraulic properties through the conditioning on head observations

on l p resolvent estimates and the density of eigenvalues for compact riemannian manifolds we address an interesting question raised by dos santos ferreira kenig and salo forum math 2014 about regions mathcal r g subset mathbb c for which there can be uniform l frac 2n n 2 to l frac 2n n 2 resolvent estimates for delta g zeta zeta in mathcal r g where delta g is the laplace beltrami operator with metric g on a given compact boundaryless riemannian manifold of dimension n geq 3 this is related to earlier work of kenig ruiz and the third author duke math j 55 329 347 1987 for the euclidean laplacian in which case the region is the entire complex plane minus any disc centered at the origin presently we show that for the round metric on the sphere s n the resolvent estimates in dos santos ferreira et al in forum math 2014 involving a much smaller region are essentially optimal we do this by establishing sharp bounds based on the distance from zeta to the spectrum of delta s n in the other direction we also show that the bounds in dos santos ferreira et al in forum math 2014 can be sharpened logarithmically for manifolds with nonpositive curvature and by powers in the case of the torus mathbb t n mathbb r n mathbb z n with the flat metric the latter improves earlier bounds of shen int math res not 1 1 31 2001 the work of dos santos ferreira et al in forum math 2014 and shen in int math res not 1 1 31 2001 was based on hadamard parametrices for delta g zeta 1 ours is based on the related hadamard parametrices for cos t sqrt delta g and it follows ideas in sogge in ann math 126 439 447 1987 of proving l p multiplier estimates using small time wave equation parametrices and the spectral projection estimates from sogge in j funct anal 77 123 138 1988 this approach allows us to adapt arguments in berard math z 155 249 276 1977 and hlawka monatsh math 54 1 36 1950 to obtain the aforementioned improvements over dos santos ferreira et al in forum math 2014 and shen in int math res not 1 1 31 2001 further improvements for the torus are obtained using recent techniques of the first author bourgain in israel j math 193 1 441 458 2013 and his work with guth bourgain and guth in geom funct anal 21 1239 1295 2011 based on the multilinear estimates of bennett carbery and tao math z 2 261 302 2006 our approach also allows us to give a natural necessary condition for favorable resolvent estimates that is based on a measurement of the density of the spectrum of sqrt delta g and moreover a necessary and sufficient condition based on natural improved spectral projection estimates for shrinking intervals as opposed to those in sogge in j funct anal 77 123 138 1988 for unit length intervals we show that the resolvent estimates are sensitive to clustering within the spectrum which is not surprising given sommerfeld s original conjecture sommerfeld in physikal zeitschr 11 1057 1066 1910 about these operators

bayesian geostatistical design task driven optimal site investigation when the geostatistical model is uncertain 1 geostatistical optimal design optimizes subsurface exploration for maximum information toward task specific prediction goals until recently most geostatistical design studies have assumed that the geostatistical description i e the mean trends covariance models and their parameters is given a priori this contradicts as emphasized by rubin and dagan 1987a the fact that only few or even no data at all offer support for such assumptions prior to the bulk of exploration effort we believe that geostatistical design should 1 avoid unjustified a priori assumptions on the geostatistical description 2 instead reduce geostatistical model uncertainty as secondary design objective 3 rate this secondary objective optimal for the overall prediction goal and 4 be robust even under inaccurate geostatistical assumptions bayesian geostatistical design follows these guidelines by considering uncertain covariance model parameters we transfer this concept from kriging like applications to geostatistical inverse problems we also deem it inappropriate to consider parametric uncertainty only within a single covariance model the matern family of covariance functions has an additional shape parameter controlling model shape by a parameter converts covariance model selection to parameter identification and resembles bayesian model averaging over a continuous spectrum of covariance models this is appealing since it generalizes bayesian model averaging from a finite number to an infinite number of models we illustrate how our approach fulfills the above four guidelines in a series of synthetic test cases the underlying scenarios are to minimize the prediction variance of 1 contaminant concentration or 2 arrival time at an ecologically sensitive location by optimal placement of hydraulic head and log conductivity measurements results highlight how both the impact of geostatistical model uncertainty and the sampling network design vary according to the choice of objective function

a full bayesian approach to parameter inference from tracer travel time moments and investigation of scale effects at the cape cod experimental site a method for inverting the travel time moments of solutes in heterogeneous aquifers is presented and is based on peak concentration arrival times as measured at various samplers in an aquifer the approach combines a lagrangian rubin and dagan 1992 solute transport framework with full bayesian hydrogeological parameter inference in the full bayesian approach the noise values in the observed data are treated as hyperparameters and their effects are removed by marginalization the prior probability density functions pdfs for the model parameters horizontal integral scale velocity and log k variance and noise values are represented by prior pdfs developed from minimum relative entropy considerations analysis of the cape cod massachusetts field experiment is presented inverse results for the hydraulic parameters indicate an expected value for the velocity variance of log hydraulic conductivity and horizontal integral scale of 0 42 m d 0 26 and 3 0 m respectively while these results are consistent with various direct field determinations the importance of the findings is in the reduction of confidence range about the various expected values on selected control planes we compare observed travel time frequency histograms with the theoretical pdf conditioned on the observed travel time moments we observe a positive skew in the travel time pdf which tends to decrease as the travel time distance grows we also test the hypothesis that there is no scale dependence of the integral scale with the scale of the experiment at cape cod we adopt two strategies the first strategy is to use subsets of the full data set and then to see if the resulting parameter fits are different as we use different data from control planes at expanding distances from the source the second approach is from the viewpoint of entropy concentration no increase in integral scale with distance is inferred from either approach over the range of the cape cod tracer experiment

oscillatory integrals and unique continuation for second order elliptic differential equations a well known counterexample see 11 indicates that the condition v e lpn2 is in the best possible nature this counterexample however has to do loc with unique continuation from a point rather than from open sets as in theorem 1 1 we hope to treat unique continuation from a point in a subsequent paper results of this type for constant coefficient operators were obtained by sawyer 14 for low dimensions and jerison and kenig 11 for general n there was a lot of previous work and we refer the reader to 11 14 for further references also jerison and kenig proved the stronger result that there is unique continuation from a point for constant coefficient operators for variable coefficient operators our results improve one of h6rmander 8 which says that there is unique continuation from open sets for solutions to 1 1

localized lp estimates of eigenfunctions a note on an article of hezari and riviere abstract we use a straightforward variation on a recent argument of hezari and riviere 8 to obtain localized l p estimates for all exponents larger than or equal to the critical exponent p c 2 n 1 n 1 we are able to do this directly by just using the l p bounds for spectral projection operators from our much earlier work 12 the localized bounds we obtain here imply for instance that for a density one sequence of eigenvalues on a manifold whose geodesic flow is ergodic all of the l p 2 p bounds of the corresponding eigenfunctions are relatively small compared to the general ones in 12 which are saturated on round spheres the connection with quantum ergodicity was established for exponents 2 p p c in the recent results of the author 13 and blair and the author 3 however the article of hezari and riviere 8 was the first one to make this connection in the case of negatively curved manifolds for the critical exponent p c as is well known and we indicate here bounds for the critical exponent p c imply ones for all the other exponents 2 p the localized estimates involve l 2 norms over small geodesic balls b r of radius r and we shall go over what happens for these in certain model cases on the sphere and on manifolds of nonpositive curvature we shall also state a problem of determining when one can improve on the trivial o r 1 2 estimates for these l 2 b r bounds if r 1 one can improve on the trivial estimates if one has improved l p c m bounds just by using holder s inequality however obtaining improved bounds for r 1 seems to be subtle

modulation spaces wiener amalgam spaces and brownian motions abstract we study the local in time regularity of the brownian motion with respect to localized variants of modulation spaces m s p q and wiener amalgam spaces w s p q we show that the periodic brownian motion belongs locally in time to m s p q t and w s p q t for s 1 q 1 and the condition on the indices is optimal moreover with the wiener measure on t we show that m s p q t and w s p q t form abstract wiener spaces for the same range of indices yielding large deviation estimates we also establish the endpoint regularity of the periodic brownian motion with respect to a besov type space b p s t specifically we prove that the brownian motion belongs to b p s t for s 1 p 1 and it obeys a large deviation estimate finally we revisit the regularity of brownian motion on usual local besov spaces b p q s and indicate the endpoint large deviation estimates

a comparison of seven geostatistically based inverse approaches to estimate transmissivities for modeling advective transport by groundwater flow this paper describes the first major attempt to compare seven different inverse approaches for identifying aquifer transmissivity the ultimate objective was to determine which of several geostatistical inverse techniques is better suited for making probabilistic forecasts of the potential transport of solutes in an aquifer where spatial variability and uncertainty in hydrogeologic properties are significant seven geostatistical methods fast fourier transform ff fractal simulation fs linearized cokriging lc linearized semianalytical ls maximum likelihood ml pilot point pp and sequential self calibration ss were compared on four synthetic data sets each data set had specific features meeting or not classical assumptions about stationarity amenability to a geostatistical description etc the comparison of the outcome of the methods is based on the prediction of travel times and travel paths taken by conservative solutes migrating in the aquifer for a distance of 5 km four of the methods ls ml pp and ss were identified as being approximately equivalent for the specific problems considered the magnitude of the variance of the transmissivity fields which went as high as 10 times the generally accepted range for linearized approaches was not a problem for the linearized methods when applied to stationary fields that is their inverse solutions and travel time predictions were as accurate as those of the nonlinear methods nonstationarity of the true transmissivity field or the presence of anomalies such as high permeability fracture zones was however more of a problem for the linearized methods the importance of the proper selection of the semivariogram of the log10 t field or the ability of the method to optimize this variogram iteratively was found to have a significant impact on the accuracy and precision of the travel time predictions use of additional transient information from pumping tests did not result in major changes in the outcome while the methods differ in their underlying theory and the codes developed to implement the theories were limited to varying degrees the most important factor for achieving a successful solution was the time and experience devoted by the user of the method

concerning the lp norm of spectral clusters for second order elliptic operators on compact manifolds in this paper we study the lp l2 mapping properties of a spectral projection operator for riemannian manifolds this operator is a generalization of the harmonic projection operator for spherical harmonics on s see c d sogge duke math j 53 1986 43 65 among other things we generalize the l2 restriction theorems of c fefferman e m stein and p tomas bull amer math sot 81 1975 477478 for the fourier transform in r to the setting of riemannian manifolds we obtain these results for the spectral projection operator as a corollary of a certain sobolev inequality involving d r2 for large r this sobolev inequality generalizes certain results for r of c kenig a ruiz and the author duke math j 55 1987 329 347 the main tools in the proof of the sobolev inequalities for riemannian manifolds are the hadamard parametrix cf l hbrmander acta mad 88 1968 341 370 and the analysis of linear partial differential equations vol iii springer verlag new york 1985 and oscillatory integral theorems of l carleson and p sjolin studia math 44 19x 287 299 and stein arm math stud 112 1986 307 357 0 1988 academic pnss iw

conditional estimation of solute travel time in heterogeneous formations impact of transmissivity measurements solute transport through heterogeneous formations is modeled by the travel time approach i e the time it takes a solute particle to travel from the source to a control plane due to uncertainty is a random variable characterized by its cumulative probability density function g this function is used by regulatory agencies in pollution site assessment the main aim of the present study is to examine the impact of data namely measured transmissivities upon g and reduction of uncertainty this is achieved for two dimensional steady flow of average uniform head gradient by using a lagrangian approach developed in the past dagan 1982 1984 1989 rubin 1990 1991 the impact of data is seen in conditioning on measured values and in estimation of parameters characterizing flow and formation properties the approach is illustrated by a simulation based on synthetic data the results show how uncertainty is reduced by increasing the density of measurements for a control plane sufficiently far from the source

numerical trajectory calculations for the efficient inversion of transient flow and tracer observations 1 we introduce a trajectory based method for the inversion of flow and transport observations the approach operates on the output of a standard numerical simulator and is applicable under very general conditions using only head and concentration histories we efficiently construct the model parameter sensitivities required for solving the inverse problem a single simulation or forward calculation is required in order to take a step in the inversion our formulation based on asymptotic solutions for flow and transport partitions the inverse problem into an arrival time and an amplitude matching problem because the model parameter sensitivities are defined along trajectories both inverse problems scale very well with respect to model size at the migration experimental site of the grimsel rock laboratory in switzerland we utilized transient head and tracer arrival times to constrain the hydraulic conductivity between a suite of boreholes the squared data misfit is reduced by over an order of magnitude in 30 iterations of our inversion algorithm each iteration requires two runs of our numerical simulator tough2 one for each experiment to construct the trajectories and associated model parameter sensitivities we find a high conductivity channel curving in a roughly east west direction in agreement with two previous studies estimates of model parameter resolution indicate that we can image the large scale conductivity variations within the array of boreholes

comparable upper and lower bounds for boundary values of neumann eigenfunctions and tight inclusion of eigenvalues for smooth bounded domains in rn we prove upper and lower l2 bounds on the boundary data of neumann eigenfunctions and we prove quasiorthogonality of this boundary data in a spectral window the bounds are tight in the sense that both are independent of the eigenvalues this is achieved by working with an appropriate norm for boundary functions which includes a spectral weight that is a function of the boundary laplacian this spectral weight is chosen to cancel concentration at the boundary that can happen for whispering gallery type eigenfunctions these bounds are closely related to wave equation estimates due to tataru using this we bound the distance from an arbitrary helmholtz parameter e 0 to the nearest neumann eigenvalue in terms of boundary normal derivative data of a trial function u solving the helmholtz equation e u 0 this inclusion bound improves over previously known bounds by a factor of e5 6 analogously to a recently improved inclusion bound in the dirichlet case due to the first two authors finally we apply our theory to present an improved numerical implementation of the method of particular solutions for computation of neumann eigenpairs on smooth planar domains we show that the new inclusion bound improves the relative accuracy in a computed neumann eigenvalue around the 42000th from nine to fourteen digits with negligible extra numerical effort

semi classical analysis of a random walk on a manifold we prove a sharp rate of convergence to stationarity for a natural random walk on a compact riemannian manifold m g the proof includes a detailed study of the spectral theory of the associated operator

bayesian methodology to stochastic capture zone determination conditioning on transmissivity measurements 1 a methodology to determine the uncertainty associated with the delineation of well capture zones in heterogeneous aquifers is presented the log transmissivity field is modeled as a random space function and the bayesian paradigm accounts for the uncertainty that stems from the imperfect knowledge about the parameters of the stochastic model unknown parameters are treated as random quantities and characterized by a prior probability distribution log transmissivity measurements are incorporated into bayes theorem updating the prior distribution and yielding posterior estimates of the mean value and the covariance parameters of the log transmissivity conditional simulations of the log transmissivity field are generated using samples from the posterior distribution of the parameters yielding samples from the predictive distribution of the log transmissivity field the uncertainty in the model parameters is propagated to the predictive uncertainty of the capture zone by solving numerically the groundwater flow equation followed by a semianalytical particle tracking algorithm the method is applied to a set of hypothetical flow fields for various sampling densities and assuming different levels of parameter uncertainty simulation results for all the sampling densities show no univocal relation between the predictive uncertainty of the capture zones and the level of parameter uncertainty however in general the predictive uncertainty increases when parameter uncertainty is taken into account

unique continuation and absence of positive eigenvalues for schrodinger operators on considere l operateur de laplace sur r n et une fonction v x sur un sous ensemble connexe ouvert de r n on montre que si n 3 une propriete de prolongement unique est vraie pour v l loc n 2 r n dans l espace de sobolev h loc 2 q r n avec q 2n n 2

a fast fourier transform stochastic analysis of the contaminant transport problem a three dimensional stochastic analysis of the contaminant transport problem is developed in the spirit of naff 1990 the new derivation is more general and simpler than previous analysis the fast fourier transformation is used extensively to obtain numerical estimates of the mean concentration and various spatial moments data from both the borden and cape cod experiments are used to test the methodology results are comparable to results obtained by other methods and to the experiments themselves

prediction of reservoir variables based on seismic data and well observations offshore petroleum reservoirs are usually explored by acquisition of vast amounts of seismic reflection data and observations along a small number of wells drilled through the reservoirs the seismic data are indirect measurements of reservoir characteristics and have good spatial coverage but low precision the well observations are sparse but precise information from these sources of data are integrated to characterize the reservoir variables of interest reservoir porosity in the current study critical parameters in the acquisition procedure are unknown and are simultaneously estimated this process termed seismic inversion constitutes a spatial multivariate ill posed inverse problem the problem is cast in a bayesian framework with a focus on sampling from the posterior model high dimensionality nonlinear components and unknown parameters in the likelihood model along with a complex design of the conditioning data complicate the problem a fast sequential sampling algorithm containing severa

spectral cluster bounds for orthonormal systems and oscillatory integral operators in schatten spaces we generalize the l p spectral cluster bounds of sogge for the laplace beltrami operator on compact riemannian manifolds to systems of orthonormal functions the optimality of these new bounds is also discussed these spectral cluster bounds follow from schatten type bounds on oscillatory integral operators

lower bounds for nodal sets of eigenfunctions we prove lower bounds for the hausdorff measure of nodal sets of eigenfunctions 0 introduction let m be a smooth closed riemannian manifold and the laplace operator a function u is said to be an eigenfunction with eigenvalue if u u 0 1 with our convention on the sign of the eigenvalues are non negative and go to infinity one of the most fundamental questions about eigenfunctions is to understand the sets where they vanish these sets are called nodal sets nodal sets are n 1 dimensional manifolds away from n 2 dimensional singular sets where the gradient also vanishes so it is natural to estimate the n 1 dimensional hausdorff measure hn 1 the main result of this short note is the following lower bound theorem 1 given a closed n dimensional riemannian manifold m there exists c so that hn 1 u 0 c 3 n 4 0 2 in particular theorem 1 gives a uniform lower bound in dimension n 3 in y s t yau conjectured the lower bound c in all dimensions this was proven for surfaces by bruning b and yau independently and for real analytic metrics by donnelly and fefferman in df but remains open in the smooth case the authors were partially supported by nsf grants dms 0606629 dms 0906233 and nsf frg grants dms 0854774 and dms 0853501 778 t h colding w p minicozzi ii the donnelly fefferman argument leads to exponentially decaying lower bounds in the smooth case see also hl in m mangoubi considered eigenfunctions on a ball in a manifold and proved lower bounds for the volume of the subset of the ball where the function is positive and the subset where it is negative combined with the isoperimetric inequality one can get the lower bound c 3 n 2 1 2n for the measure of the nodal set on the entire manifold recently sogge and zelditch sz proved the lower bound c 7 3n 8 their argument is completely different and is based in part on a beautiful new integral formula relating the l1 norm of u on the nodal set and the l1 norm of u on m finally note that some papers use 2 for the eigenvalue i e u 2 u with that convention our bound is c 3 n 2 1 finding good balls throughout m will be a fixed closed manifold with laplace operator and u will be an eigenfunction with eigenvalue and m u 2 1 we will always assume that 1 since our interest is in what happens when goes to infinity the first step is to fix a scale r depending on lemma 1 there exists a 0 so that u has a zero in every ball of radius a 3 this lemma is standard but we will include a proof since it is so short proof if u does not vanish on b r 3 then barta s theorem implies that the lowest dirichlet eigenvalue on this ball is at least let be a function that is identically one on b r 6 and that cuts off linearly to zero on the annulus b r 3 b r 6 it follows that 2 2 36r 2 vol b r 3 vol b r 6 c r2 1 1 where c comes from the bishop gromov volume comparison see p 275 of g and depends only on n the ricci curvature of m and an upper bound for r since this is impossible for r2 c 1 the lemma follows from now on we set r a 1 2 with a given by lemma 1 next we use a standard covering argument to decompose the manifold m into small balls of radius r if b is a ball in m then we write 2b for the ball with the same center as b and twice the radius lemma 2 there exists a collection bi of balls of radius r in m so that m i bi and each point of m is contained in at most cm cm m of the double balls 2bi s proof choose a maximal disjoint collection of balls b r 2 pi it follows immediately from maximality that the double balls bi br pi cover m suppose that p m is contained in balls b2r p1 b2r pk in particular the disjoint balls b r 2 p1 b 2 pk are all contained in b3r p so that k i 1 vol b r 2 pi vol b3r p 1 2

local existence of solutions to randomized gross pitaevskii hierarchies in this paper we study the local in time existence of solutions to randomized forms of the gross pitaevskii hierarchy on periodic domains in particular we study the independently randomized gross pitaevskii hierarchy and the dependently randomized gross pitaevskii hierarchy which were first introduced in the author s joint work with staffilani 2013 for these hierarchies we construct local in time low regularity solutions in spaces which contain a random component the constructed density matrices will solve the full randomized hierarchies thus extending the results from the author and staffilani s joint work where solutions solving arbitrarily long subhierarchies were given our analysis will be based on the truncation argument which was first used in the deterministic setting in the work of t chen and pavlovic 2013 the presence of randomization in the problem adds additional difficulties most notably to estimating the duhamel expansions that are crucial in the truncation argument these difficulties are overcome by a detailed analysis of the duhamel expansions in the independently randomized case we need to keep track of which randomization parameters appear in the duhamel terms whereas in the dependently randomized case we express the duhamel terms directly in terms of the initial data in both cases we can obtain stronger results with respect to the time variable if we assume additional regularity on the initial data

random data cauchy theory for supercritical wave equations ii a global existence result we prove that the subquartic wave equation on the three dimensional ball with dirichlet boundary conditions admits global strong solutions for a large set of random supercritical initial data in bigcap s 1 2 h s theta times h s 1 theta we obtain this result as a consequence of a general random data cauchy theory for supercritical wave equations developed in our previous work 6 and invariant measure considerations inspired by earlier works by bourgain 2 3 on the non linear schrodinger equation which allow us to obtain also precise large time dynamical informations on our solutions

the self duality equations on a riemann surface in this paper we shall study a special class of solutions of the self dual yang mills equations the original self duality equations which arose in mathematical physics were defined on euclidean 4 space the physically relevant solutions were the ones with finite action the so called instantons the same equations may be dimensionally reduced to euclidean 3 space by imposing invariance under translation in one direction these equations also have physical relevance the solutions which have finite action in three dimensions are the magnetic monopoles if we take the reduction process one step further and consider solutions which are invariant under two translations we obtain a set of equations in the plane here however there is no clear physical meaning and indeed attempts to find finite action solutions have failed nevertheless these are the equations we shall consider despite the lack of interesting solutions in r2 the equations have the important property conformal invariance which allows them to be defined on manifolds modelled on r2 by conformal maps namely riemann surfaces we shall consider here solutions of the self duality equations defined on a compact riemann surface there are in fact solutions as we shall show and the moduli space of all solutions turns out to be a manifold with an extremely rich geometric structure which will be the focus of our study it brings together in a harmonious way the subjects of riemannian geometry topology algebraic geometry and symplectic geometry illuminating all these facets of the same object accounts for the length of this paper the self duality equations are equations from gauge theory geometrically they are defined in terms of connections on principal bundles while the group of the principal bundle may be chosen arbitrarily for the equations to make sense we restrict attention here to the simplest case of su 2 or so 3 there are two reasons for this the first and most obvious is that it simplifies calculations and avoids the use of inductive processes which are inherent in the consideration of a general lie group of higher rank the second reason is that solutions for su 2 have an intimate relationship with the internal structure of the riemann surface as a consequence of results we shall prove about solutions to the self duality equations we learn something about the moduli space of complex structures on the surface itself namely teichmuller space

random data cauchy theory for supercritical wave equations i local theory we study the local existence of strong solutions for the cubic nonlinear wave equation with data in hs m s 1 2 where m is a three dimensional compact riemannian manifold this problem is supercritical and can be shown to be strongly ill posed in the hadamard sense however after a suitable randomization we are able to construct local strong solution for a large set of initial data in hs m where s 1 4 in the case of a boundary less manifold and s 8 21 in the case of a manifold with boundary

new examples of probabilistic well posedness for nonlinear wave equations abstract we consider fractional wave equations with exponential or arbitrary polynomial nonlinearities we prove the global well posedness on the support of the corresponding gibbs measures we provide ill posedness constructions showing that the results are truly super critical in the considered functional setting we also present a result in the case of a general randomisation in the spirit of the work by n burq and the second author

logarithmic improvements in l p bounds for eigenfunctions at the critical exponent in the presence of nonpositive curvature we consider the problem of proving l p bounds for eigenfunctions of the laplacian in the high frequency limit in the presence of nonpositive curvature and more generally manifolds without conjugate points in particular we prove estimates at the critical exponent p c frac 2 d 1 d 1 where a spectrum of scenarios for phase space concentration must be ruled out our work establishes a gain of an inverse power of the logarithm of the frequency in the bounds relative to the classical l p bounds of the second author

a reassessment of the groundwater inverse problem this paper presents a functional formulation of the groundwaterflow inverse problem that is sufficiently general to accommodate most commonly used inverse algorithms unknown hydrogeological properties are assumed to be spatial functions that can be represented in terms of a possibly infinite basis function expansion with random coefficients the unknown parameter function is related to the measurements used for estimation by a forward operator which describes the measurement process in the particular case considered here the parameter of interest is the large scale log hydraulic conductivity the measurements are point values of log conductivity and piezometric head and the forward operator is derived from an upscaled groundwaterflow equation the inverse algorithm seeks the most probable or maximum a posteriori estimate of the unknown parameter function when the measurement errors and parameter function are gaussian and independent the maximum a posteriori estimate may be obtained by minimizing a least squares performance index which can be partitioned into goodness of fit and prior terms when the parameter is a stationary random function the prior portion of the performance index is equivalent to a regularization term which imposes a smoothness constraint on the estimate this constraint tends to make the problem well posed by limiting the range of admissible solutions the gaussian maximum a posteriori problem may be solved with variational methods using functional generalizations of gauss newton or gradient based search techniques several popular groundwater inverse algorithms are either special cases of or variants on the functional maximum a posteriori algorithm these algorithms differ primarily with respect to the way they describe spatial variability and the type of search technique they use linear versus nonlinear the accuracy of estimates produced by both linear and nonlinear inverse algorithms may be measured in terms of a bayesian extension of the cramer rao lower bound on the estimation error covariance this bound suggests how parameter identifiability can be improved by modifying the problem structure and adding new measurements

bounds for eigenfunctions of the laplacian on compact riemannian manifolds suppose that is an eigenfunction of with eigenvalue 0 it is proved that c1 n 14 2 where n is the dimension of m and c1 depends only upon a bound for the absolute value of the sectional curvature of m and a lower bound for the injectivity radius of m it is then shown that if m admits an isometric circle action and the metric is generic one has exceptional sequences of eigenfunctions satisfying the complementary bounds k c2 n 18k 2

a solute flux approach to transport in heterogeneous formations 1 the general framework it is common to represent solute tranport in heterogeneous formations in terms of the resident concentration c x t regarded as a random space function the present study investigates the alternative representation by q the solute mass flux at a point of a control plane normal to the mean flow this representation is appropriate for many field applications in which the variable of interest is the mass of solute discharged through a control surface a general framework to compute the statistical moments of q and of the associated total solute discharge q and mass m is established with x the direction of the mean flow a solute particle is crossing the control plane at y z and at the travel arrival time the associated expected solute flux value is proportional to the joint probability density function pdf g1 of and whereas the variance of q is shown to depend on the joint pdf g2 of the same variables for two particles in turn the statistical moments of and depend on those of the velocity components through a system of stochastic ordinary differential equations for a steady velocity field and neglecting the effect of pore scale dispersion a major simplification of the problem results in the independence of the random variables and as a consequence the pdf of and can be derived independently of a few approximate approaches to derive the statistical moments of and are outlined these methods will be explored in paper 2 in order to effectively derive the variances of the total solute discharge and mass while paper 3 will deal with the nonlinear effect of the velocity variance upon the moments of and

bayesian assessment of the expected data impact on prediction confidence in optimal sampling design 1 incorporating hydro geo logical data such as head and tracer data into stochastic models of subsurface flow and transport helps to reduce prediction uncertainty because of financial limitations for investigation campaigns information needs toward modeling or prediction goals should be satisfied efficiently and rationally optimal design techniques find the best one among a set of investigation strategies they optimize the expected impact of data on prediction confidence or related objectives prior to data collection we introduce a new optimal design method called predia gnosis preposterior data impact assessor predia derives the relevant probability distributions and measures of data utility within a fully bayesian generalized flexible and accurate framework it extends the bootstrap filter bf and related frameworks to optimal design by marginalizing utility measures over the yet unknown data values predia is a strictly formal information processing scheme free of linearizations it works with arbitrary simulation tools provides full flexibility concerning measurement types linear nonlinear direct indirect allows for any desired task driven formulations and can account for various sources of uncertainty e g heterogeneity geostatistical assumptions boundary conditions measurement values model structure uncertainty a large class of model errors via bayesian geostatistics and model averaging existing methods fail to simultaneously provide these crucial advantages which our method buys at relatively higher computational costs we demonstrate the applicability and advantages of predia over conventional linearized methods in a synthetic example of subsurface transport in the example we show that informative data is often invisible for linearized methods that confuse zero correlation with statistical independence hence predia will often lead to substantially better sampling designs finally we extend our example to specifically highlight the consideration of conceptual model uncertainty

separating coordinates for the generalized hitchin systems and the classical r matrices we exhibit natural darboux coordinates for the generalized hitchin systems studied by bottacin and markman these systems are defined on spaces of stable pairs consisting of a vector bundle and a form valued meromorphic endomorphism of the bundle in special cases genus zero genus one the bundles are rigid and one has the rational trigonometric and elliptic gaudin systems explicit formulae are given in these cases

markov random field models for high dimensional parameters in simulations of fluid flow in porous media we give an approach for using flow information from a system of wells to characterize hydrologic properties of an aquifer in particular we consider experiments where an impulse of tracer fluid is injected along with the water at the input wells and its concentration is recorded over time at the uptake wells we focus on characterizing the spatially varying permeability field which is a key attribute of the aquifer for determining flow paths and rates for a given flow experiment as is standard for estimation from such flow data we use complicated subsurface flow code that simulates the fluid flow through the aquifer for a particular well configuration and aquifer specification in particular the permeability field over a grid the solution to this ill posed problem requires that some regularity conditions be imposed on the permeability field typically this regularity is accomplished by specifying a stationary gaussian process model for the permeability field here we use an intrinsically stationary

a bayesian approach to stochastic capture zone delineation incorporating tracer arrival times conductivity measurements and hydraulic head observations this paper presents a methodology to invert tracer arrival times and to incorporate travel time data in the delineation of well capture zones within a bayesian framework the observed arrival times are used to obtain probability based weights for each realization of the hydraulic conductivity field realizations that closely reproduce the observed arrival times are more likely to represent the true or real conductivity field than realizations yielding worse predictions and are consequently characterized by a higher conditional probability in the prediction of the capture zones the realizations are weighted by their respective probability we combine the arrival time data with conductivity measurements and head observations to delineate stochastic capture zones the conductivity measurements update the prior distributions specified for the unknown structural parameters of the conductivity field and are used as conditioning data in the generation of conditional conductivity fields the parameter values used to generate the conductivity realizations are sampled by monte carlo from the updated parameter distributions the head and travel time observations are subsequently used to assign probability based weights to the conductivity realizations by applying bayes theorem the hyperparameters of the error model are assumed unknown and their effect is accounted for by marginalization a synthetic flow setup consisting of a single extraction well in a fully confined aquifer under a regional gradient is used to illustrate the method we evaluate the relative worth of the different data types by introducing them sequentially in the inverse methodology results indicate that the different data types are complementary and that the travel time data are essential to improve the predictions of the capture zones this is confirmed by the results for the case where uncertainty in the homogeneous porosity is accounted for

l p properties for gaussian random series let c c n n n be an arbitrary sequence of l 2 n and let f c be a random series of the type f c n n g n c n e n where g n n n is a sequence of independent n c 0 1 gaussian random variables and e n n n an orthonormal basis of l 2 y the finite measure space y m being arbitrary by using the equivalence of gaussian moments and an integrability theorem due to fernique we show that a necessary and sufficient condition for f c to belong to l p y m p 2 oo for any c l 2 n almost surely is that sup n n e n l p y m oo one of the main motivations behind this result is the construction of a nontrivial gibbs measure invariant under the flow of the cubic defocusing nonlinear schrodinger equation posed on the open unit disc of r 2

on random hermite series we study integrability and continuity properties of random series of hermite functions we get optimal results which are analogues to classical results concerning fourier series like the paley zygmund or the salem zygmund theorems we also consider the case of series of radial hermite functions which are not so well behaved in this context we prove some l p bounds of radial hermite functions which are optimal when p is large