rectifiability of measures with locally uniform cube density the conjecture that radon measures in euclidean space with positive finite density are rectifiable was a central problem in geometric measure theory for fifty years this conjecture was positively resolved by preiss in 1986 using methods entirely dependent on the symmetry of the euclidean unit ball since then due to reasons of isometric immersion of metric spaces into l infty and the uncommon nature of the sup norm even in finite dimensions a popular model problem for generalising this result to non euclidean spaces has been the study of 2 uniform measures in l 3 infty the rectifiability or otherwise of these measures has been a well known question in this paper the stronger result that locally 2 uniform measures in l 3 infty are rectifiable is proved this is the first result that proves rectifiability from an initial condition about densities for general radon measures of dimension greater than 1 outside euclidean space

a linear approximation for the regular reflection of a weak shock at a wedge abstract the problem of shock reflection by a wedge in the flow dominated by the unsteady potential flow equation is an important problem in weak regular reflection the flow behind the reflected shock is immediately supersonic and becomes subsonic further downstream the reflected shock is transonic its position is a free boundary for the unsteady potential equation which is degenerate at the sonic line in self similar coordinates applying the special partial hodograph transformation used in zhouping xin huicheng yin transonic shock in a nozzle i 2 d case comm pure appl math lvii 2004 1 51 zhouping xin huicheng yin transonic shock in a nozzle ii 3 d case ims preprint 2003 we derive a nonlinear degenerate elliptic equation with nonlinear boundary conditions in a piecewise smooth domain when the angle between incident shock and wedge is small we can see the weak regular reflection as the disturbance of normal reflection as in chen shuxing linear approximation of shock reflection at a wedge with large angle comm partial differential equations 21 78 1996 1103 1118 by linearizing the resulted nonlinear equation and boundary conditions with the above viewpoint in chen shuxing linear approximation of shock reflection at a wedge with large angle comm partial differential equations 21 78 1996 1103 1118 we obtain a linear degenerate elliptic equation with mixed boundary conditions in a curved quadrilateral domain by means of elliptic regularization techniques a delicate a priori estimate and compact arguments we show that the solution of the linearized problem is smooth in the interior and lipschitz continuous up to the degenerate boundary

diffraction of a weak shock by a wedge we match formal asymptotic expansions with differently scaled variables to obtain a uniform approximation to the similarity solution of the shock wedge diffraction problem

reliability assessment of tissue classification algorithms for multi center and multi scanner data abstract background gray and white matter volume difference and change are important imaging markers of pathology and disease progression in neurology and psychiatry such measures are usually estimated from tissue segmentation maps produced by publicly available image processing pipelines however the reliability of the produced segmentations when using multi center and multi scanner data remains understudied here we assess the robustness of six publicly available tissue classification pipelines across images acquired from different mr scanners and sites methods we used 90 t1 weighted images of a single individual scanned in 73 sessions across 27 different sites to assess the robustness of the tissue classification tools variability in dice similarity index values and tissue volumes was assessed for atropos bison classify clean fast freesurfer and spm12 results bison had the highest overall dice coefficient for gm followed by spm12 and atropos while atropos had the highest overall dice coefficient for wm followed by bison and spm12 bison had the lowest overall variability in its volumetric estimates followed by freesurfer and spm12 all methods also had significant differences between some of their estimates across different scanner manufacturers e g bison had significantly higher gm estimates and correspondingly lower wm estimates for ge scans compared to philips and siemens and different signal to noise ratio snr levels e g fast and freesurfer had significantly higher wm volume estimates for high versus medium and low snr tertiles as well as correspondingly lower gm volume estimates conclusions our comparisons provide a benchmark on the reliability of the publicly used tissue classification techniques and the amount of variability that can be expected when using large multi center and multi scanner databases

low mach number flows in time dependent domains we perform a multiple time scale single space scale analysis of a compressible fluid in a time dependent domain when the time variations of the boundary are small with respect to the acoustic velocity we introduce an average operator with respect to the fast time the averaged leading order variables satisfy modified incompressible equations which are coupled to linear acoustic equations with respect to the fast time we discuss possible initial boundary data for the asymptotic equations inherited from the initial boundary data for the compressible equations

global solutions of shock reflection by large angle wedges for potential flow when a plane shock hits a wedge head on it experiences a reflectiondiffraction process and then a self similar reflected shock moves outward as the original shock moves forward in time experimental computational and asymptotic analysis has shown that various patterns of shock reflection may occur including regular and mach reflection however most of the fundamental issues for shock reflection have not been understood including the global structure stability and transition of the different patterns of shock reflection therefore it is essential to establish the global existence and structural stability of solutions of shock reflection in order to understand fully the phenomena of shock reflection on the other hand there has been no rigorous mathematical result on the global existence and structural stability of shock reflection including the case of potential flow which is widely used in aerodynamics such problems involve several challenging difficulties in the analysis of nonlinear partial differential equations such as mixed equations of elliptic hyperbolic type free boundary problems and corner singularity where an elliptic degenerate curve meets a free boundary in this paper we develop a rigorous mathematical approach to overcome these difficulties i nvolved and establish a global theory of existence and stability for shock reflection by large angle wedges for potential flow the techniques and ideas developed here will be useful for other nonlinear problems involving similar difficulties

potential theory for regular and mach reflection of a shock at a wedge if a plane shock hits a wedge a self similar pattern of reflected shocks travels outward as the shock moves forward in time the nature of the pattern is explored for weak incident shocks strength b and small wedge angles 2 w through potential theory a number of different scalings some study of mixed equations and matching asymptotics for the different scalings the self similar equations are of mixed type a linearization gives a linear mixed flow valid away from a sonic curve near the sonic curve a shock solution is constructed in another scaling except near the zone of interaction between the incident shock and the wall where a special scaling is used the parameter c1 2w 1 b ranges from 0 to here is the polytropic constant and c1 is the sound speed behind the incident shock for 2 regular reflection weak or strong can occur and the whole pattern is reconstructed to lowest order in shock strength for 1 2 mach reflection occurs and the flow behind the reflection is subsonic and can be constructed in principle with an open elliptic problem and matched the case 0 can be solved for 1 2 2 or even larger the flow behind a mach reflection may be transonic and further investigation must be made to determine what happens the basic pattern of reflection is an almost semi circular shock issuing for regular reflection from the reflection point on the wedge and for mach reflection matched with a local interaction flow assuming their nature choosing the least entropy generation the weak regular reflection will occur for sufficiently large von neumann paradox an accumulation point of vorticity occurs on the wedge above the leading point 1994 john wiley sons inc

partial regularity of almost minimizing rectifiable g chains in hilbert space we adapt to an infinite dimensional ambient space e r reifenberg s epiperimetric inequality and a quantitative version of d preiss second moments computations to establish that the set of regular points of an almost mass minimizing rectifiable g chain in l is dense in its support whenever the group g of coefficients is so that g g g is discrete and closed

regularity of solutions to regular shock reflection for potential flow the shock reflection problem is one of the most important problems in mathematical fluid dynamics since this problem not only arises in many important physical situations but also is fundamental for the mathematical theory of multidimensional conservation laws that is still largely incomplete however most of the fundamental issues for shock reflection have not been understood including the regularity and transition of different patterns of shock reflection configurations therefore it is important to establish the regularity of solutions to shock reflection in order to understand fully the phenomena of shock reflection on the other hand for a regular reflection configuration the potential flow governs the exact behavior of the solution in c 1 1 across the pseudo sonic circle even starting from the full euler flow that is both of the nonlinear systems are actually the same in a physically significant region near the pseudo sonic circle thus it becomes essential to understand the optimal regularity of solutions for the potential flow across the pseudo sonic circle the transonic boundary from the elliptic to hyperbolic region and at the point where the pseudo sonic circle the degenerate elliptic curve meets the reflected shock a free boundary connecting the elliptic to hyperbolic region in this paper we study the regularity of solutions to regular shock reflection for potential flow in particular we prove that the c 1 1 regularity is optimal for the solution across the pseudo sonic circle and at the point where the pseudo sonic circle meets the reflected shock we also obtain the c 2 regularity of the solution up to the pseudo sonic circle in the pseudo subsonic region the problem involves two types of transonic flow one is a continuous transition through the pseudo sonic circle from the pseudo supersonic region to the pseudo subsonic region the other a jump transition through the transonic shock as a free boundary from another pseudo supersonic region to the pseudo subsonic region the techniques and ideas developed in this paper will be useful to other regularity problems for nonlinear degenerate equations involving similar difficulties

segmentation of brain mr images through a hidden markov random field model and the expectation maximization algorithm the finite mixture fm model is the most commonly used model for statistical segmentation of brain magnetic resonance mr images because of its simple mathematical form and the piecewise constant nature of ideal brain mr images however being a histogram based model the fm has an intrinsic limitation no spatial information is taken into account this causes the fm model to work only on well defined images with low levels of noise unfortunately this is often not the the case due to artifacts such as partial volume effect and bias field distortion under these conditions fm model based methods produce unreliable results here the authors propose a novel hidden markov random field hmrf model which is a stochastic process generated by a mrf whose state sequence cannot be observed directly but which can be indirectly estimated through observations mathematically it can be shown that the fm model is a degenerate version of the hmrf model the advantage of the hmrf model derives from the way in which the spatial information is encoded through the mutual influences of neighboring sites although mrf modeling has been employed in mr image segmentation by other researchers most reported methods are limited to using mrf as a general prior in an fm model based approach to fit the hmrf model an em algorithm is used the authors show that by incorporating both the hmrf model and the em algorithm into a hmrf em framework an accurate and robust segmentation can be achieved more importantly the hmrf em framework can easily be combined with other techniques as an example the authors show how the bias field correction algorithm of guillemaud and brady 1997 can be incorporated into this framework to achieve a three dimensional fully automated approach for brain mr image segmentation

on a regularization of the compressible euler equations for an isothermal gas abstract we consider a leray type regularization of the compressible euler equations for an isothermal gas the regularized system depends on a small parameter 0 using riemann invariants we prove the existence of smooth solutions for the regularized system for every 0 the regularization mechanism is a non linear bending of characteristics that prevents their finite time crossing we prove that in the 0 limit the regularized solutions converge strongly however based on our analysis and numerical simulations the limit is not the unique entropy solution of the euler equations the numerical method used to support this claim is derived from the riemann invariants for the regularized system this method is guaranteed to preserve the monotonicity of characteristics

weakly nonlinear high frequency waves abstract in this paper we derive a method for finding small amplitude high frequency solutions to hyperbolic systems of quasilinear partial differential equations our solution is the sum of two parts i a superposition of small amplitude high frequency waves ii a slowly varying mean solution each high frequency wave displays nonlinear distortion of the wave profile and shocks may form shock conditions are derived for conservative systems different high frequency waves do not interact provided the frequencies and wave numbers of two waves are not linearly related to those of a third the mean solution is found by solving a linear partial differential equation this method generalizes whitham s nonlinearization technique 9 for single waves to problems where many waves are present we obtain these results by generalizing a scheme first proposed by choquet bruhat 1 which employs the method of multiple scales author

on the self similar diffraction of a weak shock into an expansion wavefront we study an asymptotic problem that describes the diffraction of a weak self similar shock near a point where its shock strength approaches zero and the shock turns continuously into an expansion wavefront an example arises in the reflection of a weak shock off a semi infinite screen the asymptotic problem consists of the unsteady transonic small disturbance equation with suitable matching conditions we obtain numerical solutions of this problem which show that the shock diffracts nonlinearly into the expansion region we also solve numerically a related half space problem with a soft boundary which shows a complex reflection pattern similar to one that occurs in the guderley mach reflection of weak shocks

trilinear resonant interactions of semilinear hyperbolic waves an important question in the study of nonlinear equations is the understanding of the asymptotic behavior of families of solutions or approximate solutions one of the difficulty is known to be the nonlinear interaction of oscillations this phenomenon has been studied for special classes of solutions such as oscillatory functions with phases and amplitudes this is the domain of geometric optics in some cases general families of bounded solutions have also been considered the present paper enters the later category but is highly inspired by the results of nonlinear geometric optics geometric optics provides a precise description of the asymptotic structure of solutions to both linear e g wh and nonlinear j hk mr hmr jmr 1 jmr 2 s hyperbolic pdes whose initial data u 0 x u0 x eu1 x 0 x e have rapid oscillations with prescribed phases 0 and a single prescribed scale 1e such solutions are important in both theory and applications l kk in the lowest approximation oscillations having different spatial scales interact only when m equals zero since otherwise the amplitudes of higher frequency oscillations become negligible relative to those of lower frequencies taking m 0 requires l estimates systems that are well posed in l occur mostly in one spatial dimension for such systems arbitrary uniformly bounded sequences of initial data may be considered but comparatively little is known about the asymptotic structure of the resulting solutions since the oscillations in such solutions may be extremely complex a natural problem is to determine their non oscillatory part which may be identified with the weak limit of the sequence of solutions the highly oscillatory nature of the solutions manifests itself in the absence of strong convergence in general of course some information about those oscillations will generally be needed in order to determine the weak limit the asymptotic structure of uniformly bounded solutions depends strongly on the nature of the nonlinearity for entropy solutions of a genuinely nonlinear conservation law t 1 or of a pair of such equations in one spatial dimension dp oscillations of amplitude o 1 do not persist for positive times so a sequence of solutions converges strongly to the solution whose initial data is the weak limit of the initial data although oscillations do persist in linear systems the weak limit of a sequence of solutions to such

justification of multidimensional single phase semilinear geometric optics for semilinear strictly hyperbolic systems lu f x u we construct and justify high frequency nonlinear asymptotic expansions of the form u e x e j u j x x e lu e f x u e 0 the study of the principal term of such expansions is called nonlinear geometric optics in the applied literature we show 1 formal expansions with periodic profiles u j can be computed to all orders ii the equations for the profiles from i are solvable and iii there are solutions of the exact equations which have the formal series as high frequency asymptotic expansion

n waves in elasticity we study the large time behavior of solutions of the one dimensional equations of elasticity typically a nonconvex system of conservation laws we show that solutions with compact initial support converge in l1 to a superposition of n waves at algebraic rate likewise we show that the perturbation expansion of weakly nonlinear geometric optics is uniformly valid to second order our analysis uses approximate scalar laws together with l1 stability of scalar solutions and decay in total variation of the wave speed

concentrated nearly monotonic epiperimetric measures in euclidean space we characterize holder continuously differentiable m dimensional submanifolds of euclidean space among m rectifiable sets s in terms of growth conditions on the m density ratios of the hausdorff measure h m l s

the control of the false discovery rate in multiple testing under dependency benjamini and hochberg suggest that the false discovery rate may be the appropriate error rate to control in many applied multiple testing problems a simple procedure was given there as an fdr controlling procedure for independent test statistics and was shown to be much more powerful than comparable procedures which control the traditional familywise error rate we prove that this same procedure also controls the false discovery rate when the test statistics have positive regression dependency on each of the test statistics corresponding to the true null hypotheses this condition for positive dependency is general enough to cover many problems of practical interest including the comparisons of many treatments with a single control multivariate normal test statistics with positive correlation matrix and multivariate t furthermore the test statistics may be discrete and the tested hypotheses composite without posing special difficulties for all other forms of dependency a simple conservative modification of the procedure controls the false discovery rate thus the range of problems for which a procedure with proven fdr control can be offered is greatly increased 1 1 simultaneous hypotheses testing the control of the increased type i error when testing simultaneously a family of hypotheses is a central issue in the area of multiple comparisons rarely are we interested only in whether all hypotheses are jointly true or not which is the test of the intersection null hypothesis in most applications we infer about the individual hypotheses realizing that some of the tested hypotheses are usually true we hope not all and some are not we wish to decide which ones are not true indicating statistical discoveries an important such problem is that of multiple endpoints in a clinical trial a new treatment is compared with an existing one in terms of a large number of potential benefits endpoints

a numerical study of riemann problems for the two dimensional unsteady transonic small disturbance equation we study a two parameter family of riemann problems for the unsteady transonic small disturbance utsd equation also called the two dimensional burgers equation the two parameters a and b which define oblique shock initial data correspond to the slopes of the initial shock waves in the upper half plane for each a and b the three constant states in the upper half plane satisfy the rankine hugoniot conditions across the shocks this leads to a two parameter family of oblique shock interaction problems in this paper we present a numerical study of global solution behavior for the values of a and b in a previously obtained bifurcation diagram our study supplements the related theoretical results and conjectures recently obtained by s canic and b l keyfitz we employ a high resolution numerical method which reveals fine solution structures our findings confirm theoretical results and conjectures about the solution patterns and deepen the understanding of the structure of several intricate wave in

a linear approximation for the regular reflection of a weak shock at a wedge satisfying sonic condition abstract the problem of shock reflection by a wedge which the flow is dominated by the unsteady potential flow equation is a important problem in weak regular reflection the flow behind the reflected shock is immediately supersonic and becomes subsonic further downstream the reflected shock is transonic its position is a free boundary for the unsteady potential equation which is degenerate at the sonic line in self similar coordinates applying the special partial hodograph transformation used in zhouping xin huicheng yin transonic shock in a nozzle i 2 d case comm pure appl math 57 2004 1 51 zhouping xin huicheng yin transonic shock in a nozzle ii 3 d case ims preprint 2003 we derive a nonlinear degenerate elliptic equation with nonlinear boundary conditions in a piecewise smooth domain when the angle which between incident shock and wedge is small we can see that weak regular reflection as the disturbance of normal reflection as in shuxing chen linear approximation of shock reflection at a wedge with large angle comm partial differential equations 21 78 1996 1103 1118 by linearizing the resulted nonlinear equation and boundary conditions with above viewpoint we obtain a linear degenerate elliptic equation with mixed boundary conditions and a linear degenerate elliptic equation with oblique boundary conditions in a curved quadrilateral domain by means of elliptic regularization techniques delicate a priori estimate and compact arguments we show that the solution of linearized problem with oblique boundary conditions is smooth in the interior and lipschitz continuous up to the degenerate boundary

controlling the false discovery rate a practical and powerful approach to multiple testing summary the common approach to the multiplicity problem calls for controlling the familywise error rate fwer this approach though has faults and we point out a few a different approach to problems of multiple significance testing is presented it calls for controlling the expected proportion of falsely rejected hypotheses the false discovery rate this error rate is equivalent to the fwer when all hypotheses are true but is smaller otherwise therefore in problems where the control of the false discovery rate rather than that of the fwer is desired there is potential for a gain in power a simple sequential bonferronitype procedure is proved to control the false discovery rate for independent test statistics and a simulation study shows that the gain in power is substantial the use of the new procedure and the appropriateness of the criterion are illustrated with examples