c1 α isometric embeddings of polar caps abstract we study isometric embeddings of c 2 riemannian manifolds in the euclidean space and we establish that the holder space c 1 1 2 is critical in a suitable sense in particular we prove that for 1 2 the levi civita connection of any isometric immersion is induced by the euclidean connection whereas for any 1 2 we construct c 1 isometric embeddings of portions of the standard 2 dimensional sphere for which such property fails

remarks on the helicity of the 3 d incompressible euler equations in this note we obtain a sufficient condition on the regularity of the weak solutions to guarantee the conservation of helicity for the 3 d incompressible euler equations as a corollary we obtain a lower bound of the vorticities for a weak solution of the euler equations

on nonperiodic euler flows with holder regularity in isett regularity in time along the coarse scale flow for the euler equations 2013 the first author proposed a strengthening of onsager s conjecture on the failure of energy conservation for incompressible euler flows with holder regularity not exceeding 1 3 this stronger form of the conjecture implies that anomalous dissipation will fail for a generic euler flow with regularity below the onsager critical space l t infty b 3 infty 1 3 due to low regularity of the energy profile this paper is the first and main paper in a series of two the results of which may be viewed as first steps towards establishing the conjectured failure of energy regularity for generic solutions with holder exponent less than 1 5 the main result of the present paper shows that any given smooth euler flow can be perturbed in c 1 5 epsilon t x on any pre compact subset of mathbb r times mathbb r 3 to violate energy conservation furthermore the perturbed solution is no smoother than c 1 5 epsilon t x as a corollary of this theorem we show the existence of nonzero c 1 5 epsilon t x solutions to euler with compact space time support generalizing previous work of the first author isett holder continuous euler flows in three dimensions with compact support in time 2012 to the nonperiodic setting

regularity of inviscid shell models of turbulence in this paper we continue the analytical study of the sabra shell model of energy turbulent cascade we prove the global existence of weak solutions of the inviscid sabra shell model and show that these solutions are unique for some short interval of time in addition we prove that the solutions conserve energy provided that the components of the solution satisfy ensuremath mid u n ensuremath mid ensuremath le c k n ensuremath 1 3 sqrt n phantom rule 0 2em 0ex mathrm log n 1 ensuremath 1 for some positive absolute constant c which is the analog of the onsager s conjecture for the euler s equations moreover we give a beal kato majda type criterion for the blow up of solutions of the inviscid sabra shell model and show the global regularity of the solutions in the two dimensional parameters regime

a new statistic for influence in linear regression since the seminal article by cook the usual way to measure the influence of an observation in a statistical model is to delete the observation from the sample and compute a convenient norm of the change in the parameters or in the vector of forecasts in this article we define a new way to measure the influence of an observation based on how this observation is being influenced by the rest of the data more precisely the new statistic we propose is defined as the squared norm of the vector of changes of the forecast of one observation when each of the sample points are deleted one by one we prove that this new statistic has asymptotically a normal distribution and is able to detect a group of high leverage similar outliers that will be undetected by cook s statistic we show in several examples that the proposed statistic is useful for detecting heterogeneity in regression models in large high dimensional datasets

optimal besov differentiability for entropy solutions of the eikonal equation in this paper we study the eikonal equation in a bounded planar domain we prove the equivalence among optimal besov regularity the finiteness of every entropy production and the validity of a kinetic formulation

diagnostics for use with regression recursive residuals conventional regression case diagnostics use the fit of the regression model to all cases to assess the deviation of individual cases from the full sample unfortunately as few as two bad cases can make these diagnostics completely unreliable an alternative approach is that based on recursive fitting in which the sample is stripped away case by case and the regression refitted after each deletion until there are only as many cases as parameters recursive residuals are residuals obtained from these successive recursive fits they are particularly effective for diagnosis when the assumptions of the regression model do not hold for the full data set but do hold after some bad cases are stripped away conventional regression case diagnostics provide not only residuals but also measures of leverage and influence and in this article i propose analogous single case diagnostics for use with recursive fitting furthermore since recursive fitting focuses on the compatibility of various subregressions with

the standardized influence matrix and its applications abstract in this article we introduce the standardized influence matrix sim of parameter estimators as a conjugate of the sample covariance matrix of standardized influence functions sifs evaluated at the data points we propose principal component analysis for the sim and its complement sim analysis as a diagnostic tool for regression or other statistical inference and provide theoretical insight to the local influence defined by cook we show that sim analysis reveals the multivariate structure of outlying and or influential points specifically sim analysis uncovers hidden structures of influence such as clustering that cannot be identified by the lengths of the standardized influences finally examples in linear regression show that a diagnostic method using sim is more effective if robust parameter estimates are used in calculating the sample sim

inviscid dyadic model of turbulence the fixed point and onsager s conjecture properties of an infinite system of nonlinearly coupled ordinary differential equations are discussed this system models some properties present in the equations of motion for an inviscid fluid such as the skew symmetry and the three dimensional scaling of the quadratic nonlinearity it is proved that the system with forcing has a unique equilibrium and that every solution blows up in finite time in h5 6 norm onsager s nuovo cimento 6 279 287 1949 conjecture is confirmed for the model system

energy and cross helicity conservation for the three dimensional ideal mhd equations in a bounded domain abstract in this paper we prove the energy and cross helicity conservation of weak solutions to the three dimensional ideal mhd equations in a bounded domain under the interior besov regularity conditions which are exactly same as the three dimensional periodic domain case in 3 and the boundedness and the besov type continuity conditions for both the velocity and magnetic fields near the boundary which seem crucial for the bounded domain case due to the boundary effect note that our besov type continuity conditions near the boundary are consistent with the interior besov regularity conditions

outlier detection using nonconvex penalized regression this article studies the outlier detection problem from the standpoint of penalized regression in the regression model we add one mean shift parameter for each of the n data points we then apply a regularization favoring a sparse vector of mean shift parameters the usual l1 penalty yields a convex criterion but fails to deliver a robust estimator the l1 penalty corresponds to soft thresholding we introduce a thresholding denoted by based iterative procedure for outlier detection ipod a version based on hard thresholding correctly identifies outliers on some hard test problems we describe the connection between ipod and m estimators our proposed method has one tuning parameter with which to both identify outliers and estimate regression coefficients a data dependent choice can be made based on the bayes information criterion the tuned ipod shows outstanding performance in identifying outliers in various situations compared with other existing approaches in addition ipod is much

locality of turbulent cascades in this paper we establish sufficient conditions for locality of turbulent cascades by an exact analysis of the fluid equations no assumptions of homogeneity or isotropy are invoked and the results apply to individual realizations with no resort to any statistical averaging the only requirement is suitable regularity of the turbulent solutions of the fluid equations in the high reynolds number limit corresponding to holder continuity but non differentiability in space we use a smooth filtering approach to resolve the turbulent fields both in space and in scale we discuss several physical cascades to exemplify and clarify the analysis including joint energy and helicity cascades in three space dimensions and the dual inverse energy cascade and direct enstrophy cascade in two dimensions

regularity and energy conservation for the compressible euler equations we give sufficient conditions on the regularity of solutions to the inhomogeneous incompressible euler and the compressible isentropic euler systems in order for the energy to be conserved our strategy relies on commutator estimates similar to those employed by constantin et al for the homogeneous incompressible euler equations

owa operators in regression problems we consider an application of fuzzy logic connectives to statistical regression we replace the standard least squares least absolute deviation and maximum likelihood criteria with an ordered weighted averaging owa function of the residuals depending on the choice of the weights we obtain the standard regression problems high breakdown robust methods least median least trimmed squares and trimmed likelihood methods as well as new formulations we present various approaches to numerical solution of such regression problems owa based regression is particularly useful in the presence of outliers and we illustrate the performance of the new methods on several instances of linear regression problems with multiple outliers

onsager s conjecture on the energy conservation for solutions of euler s equation we give a simple proof of a result conjectured by onsager 1 on energy conservation for weak solutions of euler s equation

fast very robust methods for the detection of multiple outliers abstract a few repeats of a simple forward search from a random starting point are shown to provide sufficiently robust parameter estimates to reveal masked multiple outliers the stability of the patterns obtained is exhibited by the stalactite plot the robust estimators used are least median of squares for regression and the minimum volume ellipsoid for multivariate outliers the forward search also has potential as an algorithm for calculation of these parameter estimates for large problems parallel computing provides appreciable reduction in computational time

dissipative continuous euler flows we show the existence of continuous periodic solutions of the 3d incompressible euler equations which dissipate the total kinetic energy

rigidity of a non elliptic differential inclusion related to the aviles giga conjecture in this paper we prove sharp regularity for a differential inclusion into a set k subset mathbb r 2 times 2 that arises in connection with the aviles giga functional the set k is not elliptic and in that sense our main result goes beyond sverak s regularity theorem on elliptic differential inclusions it can also be reformulated as a sharp regularity result for a critical nonlinear beltrami equation in terms of the aviles giga energy our main result implies that zero energy states coincide modulo a canonical transformation with solutions of the differential inclusion into k this opens new perspectives towards understanding energy concentration properties for aviles giga quantitative estimates for the stability of zero energy states can now be approached from the point of view of stability estimates for differential inclusions all these reformulations of our results are strong improvements upon a recent work by the last two authors lorent and peng where the link between the differential inclusion into k and the aviles giga functional was first observed and used our proof relies moreover on new observations concerning the algebraic structure of entropies

inviscid limit for damped and driven incompressible navier stokes equations in mathbb r 2 we consider the zero viscosity limit of long time averages of solutions of damped and driven navier stokes equations in mathbb r 2 we prove that the rate of dissipation of enstrophy vanishes stationary statistical solutions of the damped and driven navier stokes equations converge to renormalized stationary statistical solutions of the damped and driven euler equations these solutions obey the enstrophy balance

on the connection between kolmogorov microscales and friction in pipe flows of viscoplastic fluids abstract the present work extends kolmogorov s micro scales to a large family of viscoplastic fluids the new micro scales combined with gioia and chakaborty s 2006 friction phenomenology theory lead to a unified framework for the description of the friction coefficient in turbulent flows a resulting blasius type friction equation is tested against some available experimental data and shows good agreement over a significant range of hedstrom and reynolds numbers the work also comments on the role of the new expression as a possible benchmark test for the convergence of dns simulations the formula also provides limits for the maximum drag reduction of viscoplastic flows

efficient bounded influence regression estimation abstract the least squares estimator for in the classical linear regression model is strongly efficient under certain conditions however in the presence of heavy tailed errors and or anomalous data the least squares efficiency can be markedly reduced in this article we propose an estimator that limits the influence of any small subset of the data and show that it satisfies a first order condition for strong efficiency subject to the constraint we then show that the estimator is asymptotically normal the article concludes with an outline of an algorithm for computing a bounded influence regression estimator and with an example comparing least squares robust regression as developed by huber and the estimator proposed in this article

onsager s conjecture in bounded domains for the conservation of entropy and other companion laws we show that weak solutions of general conservation laws in bounded domains conserve their generalized entropy and other respective companion laws if they possess a certain fractional differentiability of order one third in the interior of the domain and if the normal component of the corresponding fluxes tend to zero as one approaches the boundary this extends various recent results of the authors

unmasking outliers and leverage points a confirmation abstract identification of multiple outliers and leverage points is difficult because of the masking effect recently rousseeuw and van zomeren suggested using high breakdown robust estimation methods the least median of squares and minimum volume ellipsoid for unmasking these observations these methods tend to declare too many observations as extreme however a stepwise analysis is proposed here for confirmation of outliers and leverage points detected using the robust methods diagnostic measures are constructed for observations added back to the reduced sample they are shown graphically the complementary use of robust and diagnostic methods gives satisfactory results in analyzing two data sets one data set consists often bad and four good leverage points four or 10 using a different cutoff extreme observations of the other data set of size 28 are identified using the robust methods but the stepwise analysis confirms only one the limitations of atkinson s confirmatory approach are discusse

dissipative euler flows with onsager critical spatial regularity for any 0 we show the existence of continuous periodic weak solutions v of the euler equations that do not conserve the kinetic energy and belong to the space lt1 cx1 3 e namely x v x t is e holder continuous in space at a e time t and the integral t 1 3 edt is finite a well known open conjecture of l onsager claims that such solutions exist even in the class lt cx1 3 e 2016 wiley periodicals inc

on the energy of inviscid singular flows abstract it is known that the energy of a weak solution to the euler equation is conserved if it is slightly more regular than the besov space b 3 1 3 when the singular set of the solution is or belongs to a smooth manifold we derive various l p space regularity criteria dimensionally equivalent to the critical one in particular if the singular set is a hypersurface the energy of u is conserved provided the one sided non tangential limits to the surface exist and the non tangential maximal function is l 3 integrable while the maximal function of the pressure is l 3 2 integrable the results directly apply to prove energy conservation of the classical vortex sheets in both 2d and 3d at least in those cases where the energy is finite

onsager s conjecture almost everywhere in time in recent works by isett holder continuous euler flows in three dimensions with compact support in time pp 1 173 2012 and later by buckmaster et al ann math 2015 iterative schemes were presented for constructing solutions belonging to the holder class c 1 5 e of the 3d incompressible euler equations which do not conserve the total kinetic energy the cited work is partially motivated by a conjecture of lars onsager in 1949 relating to the existence of c 1 3 e solutions to the euler equations which dissipate energy in this note we show how the later scheme can be adapted in order to prove the existence of non trivial holder continuous solutions which for almost every time belong to the critical onsager holder regularity c 1 3 e and have compact temporal support

location of multiple outliers in two way tables using tetrads using residuals to isolate multiple outhers in a two way table may lead to very poor results a new diagnostic is proposed the median of the tetrads associated with a cell this procedure can efficiently identify multiple outliers in a single step the use of a half normal plot provides a good indicator of the actual number of outhers in the table

exact results on stationary turbulence in 2d consequences of vorticity conservation we establish a series of exact results for a model of stationary turbulence in two dimensions the incompressible navier stokes equation in a periodic domain with stochastic force white noise in time essentially all of our conclusions follow from the simple consideration of the simultaneous conservation of energy and enstrophy by the inertial dynamics our main results are as follows 1 we show the blow up of mean energy as l02e for 0 when there is no ir dissipation at the large length scale l0 2 with an additional ir dissipation we establish the validity of the traditional cascade directions and magnitudes of flux of energy and enstrophy for 0 assuming finite mean energy in the limit 3 we rigorously establish the balance equations for the energy and vorticity invariants in the 2d steady state and the forward cascade of the higher order vorticity invariants assuming finite mean values 4 we derive exact inequalities for scaling exponents in the 2d enstrophy range as follows if l p l p then 2 23 and p 0 for p 3 if the minimum holder exponent of the vorticity hmin 0 then we establish a 2d analogue of the refined similarity hypothesis which improves these bounds the most novel and interesting conclusion of this work is the connection established between intermittency in 2d and negative holder singularities of the vorticity we show that the latter are necessary for deviations from the 1967 kraichnan scaling to occur

onsager s conjecture with physical boundaries and an application to the vanishing viscosity limit author s bardos claude titi edriss s wiedemann emil abstract we consider the incompressible euler equations in a bounded domain in three space dimensions recently the first two authors proved onsager s conjecture for bounded domains i e that the energy of a solution to these equations is conserved provided the solution is h quot older continuous with exponent greater than 1 3 uniformly up to the boundary in this contribution we relax this assumption requiring only interior h quot older regularity and continuity of the normal component of the energy flux near the boundary the significance of this improvement is given by the fact that our new condition is consistent with the possible formation of a prandtl type boundary layer in the vanishing viscosity limit

identifying multiple outliers in multivariate data summary we propose a procedure for the detection of multiple outliers in multivariate data let xbe an n x p data matrix representing n observations onp variates we first order the n observations using an appropriately chosen robust measure of outlyingness then divide the data set into two initial subsets a basic subset which containsp 1 good observations and a nonbasic subset which contains the remaining n p 1 observations second we compute the relative distance from each point in the data set to the centre of the basic subset relative to the possibly singular covariance matrix of the basic subset third we rearrange the n observations in ascending order accordingly then divide the data set into two subsets a basic subset which contains the first p 2 observations and a non basic subset which contains the remaining n p 2 observations this process is repeated until an appropriately chosen stopping criterion is met the final non basic subset of observations is declared an outlying subset the procedure proposed is illustrated and compared with existing methods by using several data sets the procedure is simple computationally inexpensive suitable for automation computable with widely available software packages effective in dealing with masking and swamping problems and most importantly successful in identifying multivariate outliers

the energy balance relation for weak solutions of the density dependent navier stokes equations abstract we consider the incompressible inhomogeneous navier stokes equations with constant viscosity coefficient and density which is bounded and bounded away from zero we show that the energy balance relation for this system holds for weak solutions if the velocity density and pressure belong to a range of besov spaces of smoothness 1 3 a density dependent version of the classical karman howarth monin relation is derived

on the euler equations of incompressible fluids euler equations of incompressible fluids use and enrich many branches of mathematics from integrable systems to geometric analysis they present important open physical and mathematical problems examples include the stable statistical behavior of ill posed free surface problems such as the rayleightaylor and kelvin helmholtz instabilities the paper describes some of the open problems related to the incompressible euler equations with emphasis on the blowup problem the inviscid limit and anomalous dissipation some of the recent results on the quasigeostrophic model are also mentioned

analysing data from hormone receptor assays assay results for hormone receptors in human breast cancer are generally plotted in the form of a scatchard plot usually a line is fitted to the data points by least squares regression and thence the concentration of binding sites present in the cancer is calculated the subsequent treatment of the patient with hormones depends on this value so a good estimate of it is necessary in this study three methods of representing the data namely the scatchard plot the reciprocal plot and the woolf plot were investigated and three ways of fitting lines to the data points of each graph namely least squares regression an unweighted robust procedure and a weighted robust procedure were examined when the data were well behaved all plots gave similar answers for the concentration of binding sites irrespective of the regression technique used however when there were up to three outlying points the scatchard plot particularly with least squares regression analysis performed poorly the robust regression analyses yielded more consistent results on all plots in particular the hitherto mistrusted reciprocal scale seemed to perform well under robust regression but other evidence indicated instability in the plot it is concluded that the most reliable way of representing binding data in these experiments is by a woolf plot and that the subsequent line fitting is made resistant by an unweighted robust regression analysis

procedures for the identification of multiple outliers in linear models abstract we consider the problem of identifying and testing multiple outliers in linear models the available outlier identification methods often do not succeed in detecting multiple outliers because they are affected by the observations they are supposed to identify we introduce two test procedures for the detection of multiple outliers that appear to be less sensitive to this problem both procedures attempt to separate the data into a set of clean data points and a set of points that contain the potential outliers the potential outliers are then tested to see how extreme they are relative to the clean subset using an appropriately scaled version of the prediction error the procedures are illustrated and compared to various existing methods using several data sets known to contain multiple outliers also the performances of both procedures are investigated by a monte carlo study the data sets and the monte carlo indicate that both procedures are effective in the detection of multiple outliers

the identification of influential subsets in regression using a fuzzy clustering strategy a fuzzy clustering strategy is used to identify subsets of influential observations in regression as part of the fuzzy clustering strategy the analyst can explore the uniqueness of selected subsets and the degree of belonging of observations to selected subsets this is accomplished by either varying a fuzzy parameter or the number of clusters once the observations or subsets have been identified the analyst can then compute regression diagnostics to confirm their degree of influence in regression bootstrapping and high breakdown procedures were used to confirm the influence of the previously identified subsets this fuzzy clustering strategy is applied to the modified data on wood specific gravity and an augmented production dataset both datasets have been previously presented in the literature

influence analysis based on the case sensitivity function the case sensitivity function approach to influence analysis is introduced as a natural smooth extension of influence curve methodology in which both the insights of geometry and the power of convex analysis are available in it perturbation is defined as movement between probability vectors defining weighted empirical distributions a euclidean geometry is proposed giving such perturbations both size and direction the notion of the salience of a perturbation is emphasized this approach has several benefits a general probability case weight analysis results answers to a number of outstanding questions follow directly rescaled versions of the three usual finite sample influence curve measures seen now to be required for comparability across different sized subsets of cases are readily available these new diagnostics directly measure the salience of the infinitesimal perturbations involved their essential unity both within and between subsets is evident geometrically finally it is shown how a relaxation strategy in which a high dimensional 0 ncm discrete problem is replaced by a low dimensional o n continuous problem can combine with convex optimization results to deliver better performance in challenging multiple case influence problems further developments are briefly indicated

assessment of local influence summary statistical models usually involve some degree of approximation and therefore are nearly always wrong because of this inexactness an assessment of the influence of minor perturbations of the model is important we discuss a method for carrying out such an assessment the method is not restricted to linear regression models and it seems to provide a relatively simple unified approach for handling a variety of problems

a fast procedure for outlier diagnostics in large regression problems abstract we propose a procedure for computing a fast approximation to regression estimates based on the minimization of a robust scale the procedure can be applied with a large number of independent variables where the usual algorithms require an unfeasible or extremely costly computer time also it can be incorporated in any high breakdown estimation method and may improve it with just little additional computer time the procedure minimizes the robust scale over a set of tentative parameter vectors estimated by least squares after eliminating a set of possible outliers which are obtained as follows we represent each observation by the vector of changes of the least squares forecasts of the observation when each of the data points is deleted then we obtain the sets of possible outliers as the extreme points in the principal components of these vectors or as the set of points with large residuals the good performance of the procedure allows identification of multiple outliers avoiding masking effe

non uniqueness of weak solutions to 2d hypoviscous navier stokes equations abstract through an adaption of the convex integration scheme in the two dimensional case a result of the h principle type is presented for the two dimensional hypoviscous incompressible navier stokes equations it is shown that the c t 0 l x 2 weak solutions can possess compact temporal supports and thus are not unique in general

on the conserved quantities for the weak solutions of the euler equations and the quasi geostrophic equations in this paper we obtain sufficient conditions on the regularity of the weak solutions to guarantee conservation of the energy and the helicity for the incompressible euler equations the regularity of the weak solutions are measured in terms of the triebel lizorkin type of norms dot mathcal f s p q and the besov norms dot mathcal b s p q in particular in the besov space case our results refine the previous ones due to constantin e titi energy and the author of this paper helicity where the regularity is measured by a special class of the besov space norm dot mathcal b s p infty dot mathcal n s p which is the nikolskii space we also obtain a sufficient regularity condition for the conservation of the lp norm of the temperature function in the weak solutions of the quasi geostrophic equation

bounded influence rank regression when e i y i x i it is known that minimizing e i e j yields an estimate of regression that attains a bounded influence of the residual with 95 efficiency for the normal fistribution we show that introducing weights b ij e i e j achieves bounded total influence with positive breakdown mallows weights in particular are optimally efficient under a predefined bound on the gross error sensitivity a generalization of mallows weights allows additional local stability against high leverage points two numerical examples illustrate the behaviour of the estimate

calculation of complex singular solutions to the 3d incompressible euler equations abstract this paper presents numerical computations of complex singular solutions to the 3d incompressible euler equations the euler solutions found here consist of a complex valued velocity field u that contains all positive wavenumbers u satisfies the usual euler equations but with complex initial data the real valued velocity u u u where u u is an approximate singular solution to the euler equations under moore s approximation the method for computing singular solutions is an extension of that in caflisch 1993 25 for axisymmetric flow with swirl but with several improvements that prevent the extreme magnification of round off error which affected previous computations this enables the first clean analysis of the singular surface in three dimensional complex space we find singularities in the velocity field of the form u 1 for near 3 2 and u log where 0 denotes the singularity surface the logarithmic singular surface is related to the double exponential growth of vorticity observed in recent numerical simulations

holder continuous solutions of boussinesq equation with compact support abstract we show the existence of holder continuous solution of boussinesq equations in whole space which has compact support both in space and time

unmasking multivariate outliers and leverage points abstract detecting outliers in a multivariate point cloud is not trivial especially when there are several outliers the classical identification method does not always find them because it is based on the sample mean and covariance matrix which are themselves affected by the outliers that is how the outliers get masked to avoid the masking effect we propose to compute distances based on very robust estimates of location and covariance these robust distances are better suited to expose the outliers in the case of regression data the classical least squares approach masks outliers in a similar way also here the outliers may be unmasked by using a highly robust regression method finally a new display is proposed in which the robust regression residuals are plotted versus the robust distances this plot classifies the data into regular observations vertical outliers good leverage points and bad leverage points several examples are discussed

h principles for the incompressible euler equations in dissipative euler flows and onsager s conjecture arxiv 1205 3626 preprint 2012 de lellis and szekelyhidi construct holder continuous dissipative weak solutions to the incompressible euler equations in the torus mathbb t 3 the construction consists of adding fast oscillations to the trivial solution we extend this result by establishing optimal h principles in two and three space dimensions specifically we identify all subsolutions defined in a suitable sense which can be approximated in the h 1 norm by exact solutions furthermore we prove that the flows thus constructed on mathbb t 3 are genuinely three dimensional and are not trivially obtained from solutions on mathbb t 2

the energy measure for the euler and navier stokes equations the potential failure of energy equality for a solution u of the euler or navier stokes equations can be quantified using a so called energy measure the weak limit of the measures u t 2 rm d x as t approaches the first possible blowup time we show that membership of u in certain weak or strong l q l p classes gives a uniform lower bound on the lower local dimension of mathcal e more precisely it implies uniform boundedness of a certain upper s density of mathcal e we also define and give lower bounds on the concentration dimension associated to mathcal e which is the hausdorff dimension of the smallest set on which energy can concentrate both the lower local dimension and the concentration dimension of mathcal e measure the departure from energy equality as an application of our estimates we prove that any solution to the 3 dimensional navier stokes equations which is type i in time must satisfy the energy equality at the first blowup time

stability and spectral convergence of fourier method for nonlinear problems on the shortcomings of the 2 3 2 3 de aliasing method the high order accuracy of fourier method makes it the method of choice in many large scale simulations we discuss here the stability of fourier method for nonlinear evolution problems focusing on the two prototypical cases of the inviscid burgers equation and the multi dimensional incompressible euler equations the fourier method for such problems with quadratic nonlinearities comes in two main flavors one is the spectral fourier method the other is the 2 3 2 3 pseudo spectral fourier method where one removes the highest 1 3 1 3 portion of the spectrum this is often the method of choice to maintain the balance of quadratic energy and avoid aliasing errors two main themes are discussed in this paper first we prove that as long as the underlying exact solution has a minimal c 1 alpha c 1 spatial regularity then both the spectral and the 2 3 2 3 pseudo spectral fourier methods are stable consequently we prove their spectral convergence for smooth solutions of the inviscid burgers equation and the incompressible euler equations on the other hand we prove that after a critical time at which the underlying solution lacks sufficient smoothness then both the spectral and the 2 3 2 3 pseudo spectral fourier methods exhibit nonlinear instabilities which are realized through spurious oscillations in particular after shock formation in inviscid burgers equation the total variation of bounded pseudo spectral fourier solutions must increase with the number of increasing modes and we stipulate the analogous situation occurs with the 3d incompressible euler equations the limiting fourier solution is shown to enforce l 2 l 2 energy conservation and the contrast with energy dissipating onsager solutions is reflected through spurious oscillations

location of several outliers in multiple regression data using elemental sets the outlying tendency of any case in a multiple regression of p predictors may be estimated by drawing all subsets of size p from the remaining cases and fitting the model each such subset yields an elemental residual for the case in question and a suitable summary statistic of them can be used as an estimate of the case s outlying tendency we propose two such summary statistics an unweighted median which is of bounded influence and a weighted median which is more efficient but less robust the computational load of the procedure is reduced by using random samples in place of the full set of subsets of size p as a byproduct the method yields useful information on the influence or leverage of cases and the mutual masking of high leverage points

robust regression using repeated medians abstract the repeated median algorithm is a robustified u statistic in which nested medians replace the single mean unlike many generalizations of the univariate median repeated median estimates maintain the high 50 breakdown value and can resist the effects of outliers even when they comprise nearly half of the data because they are calculated directly not iteratively repeated median procedures can be used as starting values for iterative robust estimation methods for bivariate linear regression with symmetric errors repeated median estimates are unbiased and fisher consistent and their efficiency under gaussian sampling can be comparable to the efficiency of the univariate median author

a sufficient condition for the kolmogorov 4 5 law for stationary martingale solutions to the 3d navier stokes equations we prove that statistically stationary martingale solutions of the 3d navier stokes equations on mathbb t 3 subjected to white in time colored in space forcing satisfy the kolmogorov 4 5 law in an averaged sense and over a suitable inertial range using only the assumption that the kinetic energy is o nu 1 as nu rightarrow 0 where is the inverse reynolds number this plays the role of a weak anomalous dissipation no energy balance or additional regularity is assumed aside from that satisfied by all martingale solutions from the energy inequality if the force is statistically homogeneous then any homogeneous martingale solution satisfies the spherically averaged 4 5 law pointwise in space an additional hypothesis of approximate isotropy in the inertial range gives the traditional version of the kolmogorov law we demonstrate a necessary condition by proving that energy balance and an additional quantitative regularity estimate as nu rightarrow 0 imply that the 4 5 law or any similar scaling law cannot hold

leave k out diagnostics for time series abstract the authors propose diagnostics for arima model fitting for time series formed by deleting observations from the data and measuring the change in the estimates of the parameters the use of leave one out diagnostics is a well established tool in regression analysis this document demonstrates the efficacy of observation deletion based diagnostics for arima models addressing issues special to the time diagnostics based on the innovations variance it is shown that the dependency aspect of time series data gives rise to a smearing effect which confounds the diagnostics for the coefficients it is also shown that the diagnostics based on the innovations variance are much clearer and more sensitive than those for the coefficients a leave k out diagnostics approach is proposed to deal with patches of outliers and problems caused by masking are handled by use of iterative deletion an overall strategy for arima model fitting is given and applied to two data sets

inconsistency of resampling algorithms for high breakdown regression estimators and a new algorithm because high breakdown estimators hbes are impractical to compute exactly in large samples approximate algorithms are used the algorithm generally produces an estimator with a lower consistency rate and breakdown value than the exact theoretical estimator this discrepancy grows with the sample size with the implication that huge computations are needed for good approximations in large high dimension samples the workhorse for hbes has been the elemental set or basic resampling algorithm this turns out to be completely ineffective in high dimensions with high levels of contamination however enriching it with a concentration step turns it into a method that can handle even high levels of contamination provided that the regression outliers are located on random cases it remains ineffective if the regression outliers are concentrated on high leverage cases we focus on the multiple regression problem but several of the broad conclusions notably those of the inadequacy of fixed numbers of

on self similar solutions to the incompressible euler equations abstract recent numerical simulations have shown the existence of multiple self similar solutions to the cauchy problem for the 2 dimensional incompressible euler equation with initial vorticity in l l o c p r 2 1 p toward a rigorous validation of these computations in this paper we analytically construct self similar solutions i on an outer domain of the form x r and ii in a neighborhood of the points where the solution exhibits a spiraling vortex singularity the outer solution is obtained as the fixed point of a contractive transformation based on the biot savart formula and integration along characteristics the inner solution is constructed using a system of adapted coordinates following the approach of v elling 2016 17

a bayesian approach to outlier detection and residual analysis summary an approach to detecting outliers in a linear model is developed an outlier is defined to be an observation with a large random error generated by the linear model under consideration outliers are detected by examining the posterior distribution of the random errors an augmented residual plot is also suggested as a graphical aid in finding outliers we propose a precise definition of an outlier in a linear model which appears to lead to simple ways of exploring data for the possibility of outliers the definition is such that if the parameters of the model are known then it is also known which observations are outliers alternatively if the parameters are unknown the posterior distribution can be used to calculate the posterior probability that any observation is an outlier in a linear model with normally distributed random errors ei with mean zero and variance a 2we declare the ith observation to be an outlier if iei i ko for some choice of k the value of k can be chosen so that the prior probability of an outlier is small and thus outliers are observations which are more extreme than is usually expected realizations of normally distributed errors of more than about three standard deviations from the mean are certainly surprising and worth further investigation such outlying observations can occur under the assumed model however and this should be taken into account when deciding what to do with outliers and in choosing k note that ei is the actual realization of the random error not the usual estimated residual i the problem of outliers is studied and thoroughly reviewed by barnett lewis 1984 hawkins 1980 beckman cook 1983 and pettit smith 1985 the usual bayesian approach to outlier detection uses the definition given by freeman 1980 freeman defines an outlier to be any observation that has not been generated by the mechanism that generated the majority of observations in the data set freeman s definition therefore requires that a model for the generation of outliers be specified and is implemented by for example box tiao 1968 guttman dutter freeman 1978 and abraham box 1978 our method differs in that we define outliers as arising from the model under consideration rather than arising from a separate expanded model our approach is similar to that described by zellner moulton 1985 and is an extension of the philosophy

energy conservation for the 3d tropical climate model in bounded domains abstract in this paper we consider the energy conservation to the 3d tropical climate model in bounded domain by exploring the special structure of the nonlinear components and using the ideas introduced in 1 we can give some sufficient conditions for the weak solutions to ensure energy conservation

cauchy problem for dissipative holder solutions to the incompressible euler equations we consider solutions to the cauchy problem for the incompressible euler equations on the 3 dimensional torus which are continuous or holder continuous for any exponent theta frac 1 16 using the techniques introduced in de lellis and szekelyhidi inventiones mathematicae 9 377 407 2013 dissipative euler flows and onsager s conjecture 2012 we prove the existence of infinitely many holder continuous initial vector fields starting from which there exist infinitely many holder continuous solutions with preassigned total kinetic energy

energy conservation in two dimensional incompressible ideal fluids this note addresses the issue of energy conservation for the 2d euler system with an lp control on vorticity we provide a direct argument based on a mollification in physical space to show that the energy of a weak solution is conserved if omega nabla times u in l frac 3 2 an example of a 2d field in the class omega in l frac 3 2 epsilon for any epsilon 0 and u in b 1 3 3 infty onsager critical space see shvydkoy in discr contin dyn syst ser s 3 3 473 496 2010 is constructed with non vanishing energy flux this demonstrates sharpness of the kinematic argument which does not differentiate between 2d and 3d and requires onsager s regularity control on the solution next we show that for physically realizable solutions there is a mechanism preventing the anomalous dissipation in 2d that does not require such a control namely we prove that any solution to the euler equations produced via a vanishing viscosity limit from the navier stokes equations with omega in l p for p 1 conserves energy

statistical solutions and onsager s conjecture abstract we prove a version of onsager s conjecture on the conservation of energy for the incompressible euler equations in the context of statistical solutions as introduced recently by fjordholm et al 2017 as a byproduct we also obtain an alternative proof for the conservative direction of onsager s conjecture for weak solutions under a weaker besov type regularity assumption than previously known

least quantile regression via modern optimization we address the least quantile of squares lqs and in particular the least median of squares regression problem using modern optimization methods we propose a mixed integer optimization mio formulation of the lqs problem which allows us to find a provably global optimal solution for the lqs problem our mio framework has the appealing characteristic that if we terminate the algorithm early we obtain a solution with a guarantee on its sub optimality we also propose continuous optimization methods based on first order subdifferential methods sequential linear optimization and hybrid combinations of them to obtain near optimal solutions to the lqs problem the mio algorithm is found to benefit significantly from high quality solutions delivered by our continuous optimization based methods we further show that the mio approach leads to a an optimal solution for any dataset where the data points y i mathbf x i s are not necessarily in general position b a simple proof of the breakdown point of the lqs objective value that holds for any dataset and c an extension to situations where there are polyhedral constraints on the regression coefficient vector we report computational results with both synthetic and real world datasets showing that the mio algorithm with warm starts from the continuous optimization methods solve small n 100 and medium n 500 size problems to provable optimality in under two hours and outperform all publicly available methods for large scale n 10 000 lqs problems

the detection of influential subsets in linear regression by using an influence matrix this paper presents a new method to identify influential subsets in linear regression problems the procedure uses the eigenstructure of an influence matrix which is defined as the matrix of uncentred covariances of the effect on the whole data set of deleting each observation normalized to include the univariate cook statistics on the diagonal it is shown that the eigenstructure of the influence matrix is useful to identify influential subsets and a procedure for detecting influential sets is proposed the method is illustrated with two examples

lack of uniqueness for weak solutions of the incompressible porous media equation in this work we consider weak solutions of the incompressible two dimensional porous media ipm equation by using the approach of de lellis szekelyhidi we prove non uniqueness for solutions in l in space and time

on the weak solutions to the maxwell landau lifshitz equations and to the hall magneto hydrodynamic equations in this paper we deal with weak solutions to the maxwell landau lifshitz equations and to the hall magneto hydrodynamic equations first we prove that these solutions satisfy some weak strong uniqueness property then we investigate the validity of energy identities in particular we give a sufficient condition on the regularity of weak solutions to rule out anomalous dissipation in the case of the hall magneto hydrodynamic equations we also give a sufficient condition to guarantee the magneto helicity identity our conditions correspond to the same heuristic scaling as the one introduced by onsager in hydrodynamic theory finally we examine the sign locally of the anomalous dissipations of weak solutions obtained by some natural approximation processes

anomalous dissipation for 1 5 holder euler flows recently the second and fourth authors developed an iterative scheme for obtaining rough solutions of the 3d incompressible euler equations in holder spaces the motivation comes from onsager s conjecture the construction involves a superposition of weakly interacting perturbed beltrami flows on infinitely many scales an obstruction to better regularity arises from the errors in the linear transport of a fast periodic flow by a slow velocity field in a recent paper the third author has improved upon the methods introducing some novel ideas on how to deal with this obstruction thereby reaching a better holder exponent albeit weaker than the one conjectured by onsager in this paper we give a shorter proof of this final result adhering more to the original scheme of the second and fourth authors and introducing some new devices more precisely we show that for any positive e there exist periodic solutions of the 3d incompressible euler equations that dissipate the total kinetic energy and belong to the holder class c 1 5 epsilon

the accuracy of elemental set approximations for regression abstract the elemental set algorithm involves performing many fits to a data set each fit made to a subsample of size just large enough to estimate the parameters in the model elemental sets have been proposed as a computational device to approximate estimators in the areas of high breakdown regression and multivariate location scale estimation where exact optimization of the criterion function is computationally intractable although elemental set algorithms are used widely and for a variety of problems the quality of the approximation they give has not been studied this article shows that they provide excellent approximations for the least median of squares least trimmed squares and ordinary least squares criteria it is suggested that the approach likely will be equally effective in the other problem areas in which exact optimization of a criterion is difficult or impossible

multiple influential point detection in high dimensional regression spaces influence diagnosis should be routinely conducted when one aims to construct a regression model despite its importance the problem of influence quantification is severely under investigated in a high dimensional setting mainly due to the difficulty of establishing a coherent theoretical framework and the lack of easily implementable procedures although some progress has been made in recent years existing approaches are ineffective in detecting multiple influential points especially due to the notorious masking and swamping effects to address this challenge we propose a new group deletion procedure referred to as mip by introducing two novel quantities named max and min statistics these two statistics have complimentary properties in that the max statistic is effective for overcoming the masking effect while the min statistic is useful for overcoming the swamping effect combining their strengths we further propose an efficient algorithm that can detect influential points with prespecified guarantees for wider applications we focus on developing the new proposal for the multiple response regression model encompassing the univariate response linear model as a special case the proposed influential point detection procedure is simple to implement efficient to run and enjoys attractive theoretical properties its effectiveness is verified empirically via extensive simulation study and data analysis

reconciling curvature and importance sampling based procedures for summarizing case influence in bayesian models abstractmethods for summarizing case influence in bayesian models take essentially two forms 1 use common divergence measures for calculating distances between the full data posterior and the case deleted posterior and 2 measure the impact of infinitesimal perturbations to the likelihood to study local case influence methods based on approach 1 lead naturally to considering the behavior of case deletion importance sampling weights the weights used to approximate samples from the case deleted posterior using samples from the full posterior methods based on approach 2 lead naturally to considering the local curvature of the kullback leibler divergence of the full posterior from a geometrically perturbed quasi posterior by examining the connections between the two approaches we establish a rationale for employing low dimensional summaries of case influence obtained entirely via the variance covariance matrix of the log importance sampling weights we illustrate the use of the proposed diagnos

markov selections for the 3d stochastic navier stokes equations we investigate the markov property and the continuity with respect to the initial conditions strong feller property for the solutions to the navier stokes equations forced by an additive noise first we prove by means of an abstract selection principle that there are markov solutions to the navier stokes equations due to the lack of continuity of solutions in the space of finite energy the markov property holds almost everywhere in time then depending on the regularity of the noise we prove that any markov solution has the strong feller property for regular initial conditions we give also a few consequences of these facts together with a new sufficient condition for well posedness

the euler equations as a differential inclusion we propose a new point of view on weak solutions of the euler equations describing the motion of an ideal incompressible fluid in r n with n 2 we give a reformulation of the euler equations as a differential inclusion and in this way we obtain transparent proofs of several celebrated results of v scheffer and a shnirelman concerning the non uniqueness of weak solutions and the existence of energy decreasing solutions our results are stronger because they work in any dimension and yield bounded velocity and pressure