one dimensional behavior of singular n dimensional solutions of semilinear heat equations we consider u x t a solution of u t u u p 1 u that blows up at time t where u n 0 t p 1 n 2 p 0 in particular if in addition p 3 then the solution is very close to a superposition of one dimensional solutions as functions of the distance to the blow up set finally we prove that the mere hypothesis that the blow up set is continuous implies that it is c 1 1 2 for any 0

profile for the imaginary part of a blowup solution for a complex valued semilinear heat equation abstract in this paper we consider the following complex valued semilinear heat equation t u u u p u c in the whole space r n where p n p 2 we aim at constructing for this equation a complex solution u u 1 i u 2 which blows up in finite time t and only at one blowup point a with the following estimates for the final profile u x t p 1 2 x a 2 8 p ln x a 1 p 1 u 2 x t 2 p p 1 2 p 1 2 x a 2 8 p ln x a 1 p 1 1 ln x a as x a note that the imaginary part is non zero and that it blows up also at point a our method relies on two main arguments the reduction of the problem to a finite dimensional one and a topological argument based on the index theory to get the conclusion up to our knowledge this is the first time where the blowup behavior of the imaginary part is derived in multi dimension

stable weights that balance covariates for estimation with incomplete outcome data weighting methods that adjust for observed covariates such as inverse probability weighting are widely used for causal inference and estimation with incomplete outcome data part of the appeal of such methods is that one set of weights can be used to estimate a range of treatment effects based on different outcomes or a variety of population means for several variables however this appeal can be diminished in practice by the instability of the estimated weights and by the difficulty of adequately adjusting for observed covariates in some settings to address these limitations this article presents a new weighting method that finds the weights of minimum variance that adjust or balance the empirical distribution of the observed covariates up to levels prespecified by the researcher this method allows the researcher to balance very precisely the means of the observed covariates and other features of their marginal and joint distributions such as variances and correlations and also for example the

optimal matching with minimal deviation from fine balance in a study of obesity and surgical outcomes in multivariate matching fine balance constrains the marginal distributions of a nominal variable in treated and matched control groups to be identical without constraining who is matched to whom in this way a fine balance constraint can balance a nominal variable with many levels while focusing efforts on other more important variables when pairing individuals to minimize the total covariate distance within pairs fine balance is not always possible that is it is a constraint on an optimization problem but the constraint is not always feasible we propose a new algorithm which returns a minimum distance finely balanced match when one is feasible and otherwise minimizes the total distance among all matched samples that minimize the deviation from fine balance perhaps we can come very close to fine balance when fine balance is not attainable moreover in any event because our algorithm is guaranteed to come as close as possible to fine balance the investigator may perform one match and on that basis judge whether the best attainable balance is adequate or not we also show how to incorporate an additional constraint the algorithm is implemented in two similar ways first as an optimal assignment problem with an augmented distance matrix second as a minimum cost flow problem in a network the case of knee surgery in the obesity and surgical outcomes study motivated the development of this algorithm and is used as an illustration in that example two of 47 hospitals had too few nonobese patients to permit fine balance for the nominal variable with 47 levels representing the hospital but our new algorithm came very close to fine balance moreover in that example there was a shortage of nonobese diabetic patients and incorporation of an additional constraint forced the match to include all of these nonobese diabetic patients thereby coming as close as possible to balance for this important but recalcitrant covariate

improved conditions for single point blow up in reaction diffusion systems abstract we study positive blowing up solutions of the system u t u v p v t v u q as well as of some more general systems for any p q 1 we prove single point blow up for any radially decreasing positive and classical solution in a ball this improves on previously known results in 3 directions i no type i blow up assumption is made and it is known that this property may fail ii no equidiffusivity is assumed i e any 0 is allowed iii a large class of nonlinearities f u v g u v can be handled which need not follow a precise power behavior as a side result we also obtain lower pointwise estimates for the final blow up profiles

inference for imputation estimators we derive an estimator of the asymptotic variance of both single and multiple imputation estimators we assume a parametric imputation model but allow for non and semiparametric analysis models our variance estimator in contrast to the estimator proposed by rubin 1987 is consistent even when the imputation and analysis models are misspecified and incompatible with one another

complete blow up after tmax for the solution of a semilinear heat equation soit un sous ensemble ouvert borne de r n a frontiere lisse on considere le probleme u t u f u dans o t u 0 sur o t u x o u 0 x x ou f r r est localement de lipschitz non decroissante et f o 0

some existence and nonexistence theorems for solutions of degenerate parabolic equations abstract the initial and boundary value problem for the degenerate parabolic equation v t v f v in the cylinder 0 r n bounded for a certain class of point functions satisfying v 0 e g v v m sign v is considered in the case that f v sign v c 1 v the equation has a global time solution the same is true for 1 provided the measure of is sufficiently small in the case that f v v is nondecreasing a condition is given on the initial state v x 0 which implies that the solution must blow up in finite time the existence of such initial states is discussed

covariate association eliminating weights a unified weighting framework for causal effect estimation weighting methods offer an approach to estimating causal treatment effects in observational studies however if weights are estimated by maximum likelihood misspecification of the treatment assignment model can lead to weighted estimators with substantial bias and variance in this paper we propose a unified framework for constructing weights such that a set of measured pretreatment covariates is unassociated with treatment assignment after weighting we derive conditions for weight estimation by eliminating the associations between these covariates and treatment assignment characterized in a chosen treatment assignment model after weighting the moment conditions in covariate balancing weight methods for binary categorical and continuous treatments in cross sectional settings are special cases of the conditions in our framework which extends to longitudinal settings simulation shows that our method gives treatment effect estimates with smaller biases and variances than the maximum likelihood approach under treatment assignment model misspecification we illustrate our method with an application to systemic lupus erythematosus data

kernel based covariate functional balancing for observational studies covariate balance is often advocated for objective causal inference since it mimics randomization in observational data unlike methods that balance specific moments of covariates our proposal attains uniform approximate balance for covariate functions in a reproducing kernel hilbert space the corresponding infinite dimensional optimization problem is shown to have a finite dimensional representation in terms of an eigenvalue optimization problem large sample results are studied and numerical examples show that the proposed method achieves better balance with smaller sampling variability than existing methods

using dynamic programming for solving variational problems in vision dynamic programming is discussed as an approach to solving variational problems in vision dynamic programming ensures global optimality of the solution is numerically stable and allows for hard constraints to be enforced on the behavior of the solution within a natural and straightforward structure as a specific example of the approach s efficacy applying dynamic programming to the energy minimizing active contours is described the optimization problem is set up as a discrete multistage decision process and is solved by a time delayed discrete dynamic programming algorithm a parallel procedure for decreasing computational costs is discussed

refined asymptotics for the blowup of ut δu up this work is concerned with positive blowing up solutions of the semilinear heat equation ut u up in rn our main contribution is a sort of center manifold analysis for the equation in similarity variables leading to refined asymptotics for u in a backward space time parabola near any blowup point we also explore a connection between the asymptotics of u and the local geometry of the blowup set

openness of the set of non characteristic points and regularity of the blow up curve for the 1 d semilinear wave equation we consider here the 1 d semilinear wave equation with a power nonlinearity and with no restriction on initial data we first prove a liouville theorem for that equation then we consider a blow up solution its blow up curve x mapsto t x and i 0 subset mathbb r the set of non characteristic points we show that i0 is open and that t x is c1 on i0 all these results fundamentally use our previous result in 19 showing the convergence in selfsimilar variables for x in i 0

using mixed integer programming for matching in an observational study of kidney failure after surgery this article presents a new method for optimal matching in observational studies based on mixed integer programming unlike widely used matching methods based on network algorithms which attempt to achieve covariate balance by minimizing the total sum of distances between treated units and matched controls this new method achieves covariate balance directly either by minimizing both the total sum of distances and a weighted sum of specific measures of covariate imbalance or by minimizing the total sum of distances while constraining the measures of imbalance to be less than or equal to certain tolerances the inclusion of these extra terms in the objective function or the use of these additional constraints explicitly optimizes or constrains the criteria that will be used to evaluate the quality of the match for example the method minimizes or constrains differences in univariate moments such as means variances and skewness differences in multivariate moments such as correlations between covari

snakes active contour models a snake is an energy minimizing spline guided by external constraint forces and influenced by image forces that pull it toward features such as lines and edges snakes are active contour models they lock onto nearby edges localizing them accurately scale space continuation can be used to enlarge the capture region surrounding a feature snakes provide a unified account of a number of visual problems including detection of edges lines and subjective contours motion tracking and stereo matching we have used snakes successfully for interactive interpretation in which user imposed constraint forces guide the snake near features of interest

bounded efficient and doubly robust estimation with inverse weighting consider estimating the mean of an outcome in the presence of missing data or estimating population average treatment effects in causal inference a doubly robust estimator remains consistent if an outcome regression model or a propensity score model is correctly specified we build on a previous nonparametric likelihood approach and propose new doubly robust estimators which have desirable properties in efficiency if the propensity score model is correctly specified and in boundedness even if the inverse probability weights are highly variable we compare the new and existing estimators in a simulation study and find that the robustified likelihood estimators yield overall the smallest mean squared errors

properties of non simultaneous blow up in heat equations coupled via different localized sources abstract this paper deals with u t u u m x t e pv 0 t v t v u q 0 t e nv x t subject to homogeneous dirichlet boundary conditions the complete classification on non simultaneous and simultaneous blow up is obtained by four sufficient and necessary conditions it is interesting that in some exponent region large initial data u 0 v 0 leads to the blow up of u v and in some betweenness simultaneous blow up occurs for all of the nonnegative exponents we find that u v blows up only at a single point if m 1 n 0 while u v blows up everywhere for 0 m 1 n 0 moreover blow up rates are considered for both non simultaneous and simultaneous blow up solutions

building a stronger instrument in an observational study of perinatal care for premature infants an instrument is a random nudge toward acceptance of a treatment that affects outcomes only to the extent that it affects acceptance of the treatment nonetheless in settings in which treatment assignment is mostly deliberate and not random there may exist some essentially random nudges to accept treatment so that use of an instrument might extract bits of random treatment assignment from a setting that is otherwise quite biased in its treatment assignments an instrument is weak if the random nudges barely influence treatment assignment or strong if the nudges are often decisive in influencing treatment assignment although ideally an ostensibly random instrument is perfectly random and not biased it is not possible to be certain of this thus a typical concern is that even the instrument might be biased to some degree it is known from theoretical arguments that weak instruments are invariably sensitive to extremely small biases for this reason strong instruments are preferred the strength of an

on the computation for blow up solutions of the nonlinear wave equation we consider in this paper the 1 dim nonlinear wave equation frac partial 2 u partial t 2 t x frac partial 2 u partial x 2 t x u 1 alpha t x alpha 0 and its finite difference analogue it is known that the solution of the current equation may become unbounded in finite time a phenomenon which is known as blow up moreover since the nonlinear wave equation enjoys finite speed of propagation even if the solution has become unbounded at certain points the solution continues to exist blows up at later times and forms the so called blow up curve up to the present however numerical approximation for blow up problems gave only the convergence up to the blow up time of vert u t cdot vert infty in fact adaptive time mesh strategy was used to numerically reproduce the phenomenon of finite time blow up nevertheless computation using such schemes can not be extended beyond the blow up time of vert u t cdot vert infty this is a fatal problem when computing blow up solutions for the nonlinear wave equation as a consequence we reconsider a finite difference scheme whose temporal grid size is given uniformly and propose an algorithm with rigorous convergence analysis to numerically reconstruct the blow up curve for the nonlinear wave equation

stable blow up patterns abstract for the semilinear heat equation ut u eu in a convex domain r n given any b we show the existence of solutions which blow up in finite time exactly at b and whose final profile has the form u t x 2 ln x b ln ln x b ln 8 t being the blow up time using a suitable set of rescaled coordinates this asymptotic behavior is proved to be stable with respect to small perturbations of the initial conditions

nonexistence of global solutions to semilinear wave equations in high dimensions on considere le probleme de cauchy pour l equation d onde semi lineaire 2 t 2 u x t u x t p p 1 u x o f x u t x o g x avec f g c o r n

using canny s criteria to derive a recursively implemented optimal edge detector a highly efficient recursive algorithm for edge detection is presented using canny s design 1 we show that a solution to his precise formulation of detection and localization for an infinite extent filter leads to an optimal operator in one dimension which can be efficiently implemented by two recursive filters moving in opposite directions in addition to the noise truncature immunity which results the recursive nature of the filtering operations leads with sequential machines to a substantial saving in computational effort five multiplications and five additions for one pixel independent of the size of the neighborhood the extension to the two dimensional case is considered and the resulting filtering structures are implemented as two dimensional recursive filters hence the filter size can be varied by simply changing the value of one parameter without affecting the time execution of the algorithm performance measures of this new edge detector are given and compared to canny s filters various experimental results are shown

singular standing ring solutions of nonlinear partial differential equations abstract we present a general framework for constructing singular solutions of nonlinear evolution equations that become singular on a d dimensional sphere where d 1 the asymptotic profile and blowup rate of these solutions are the same as those of solutions of the corresponding one dimensional equation that become singular at a point we provide a detailed numerical investigation of these new singular solutions for the following equations the nonlinear schrodinger equation i t t x 2 0 with 2 the biharmonic nonlinear schrodinger equation i t t x 2 2 0 with 4 the nonlinear heat equation t t x 2 0 with 0 and the nonlinear biharmonic heat equation t t x 2 2 0 with 0

optimal probability weights for inference with constrained precision abstractprobability weights are used in many areas of research including complex survey designs missing data analysis and adjustment for confounding factors they are useful analytic tools but can lead to statistical inefficiencies when they contain outlying values this issue is frequently tackled by replacing large weights with smaller ones or by normalizing them through smoothing functions while these approaches are practical they are also prone to yield biased inferences this article introduces a method for obtaining optimal weights defined as those with smallest euclidean distance from target weights among all sets of weights that satisfy a constraint on the variance of the resulting weighted estimator the optimal weights yield minimum bias estimators among all estimators with specified precision the method is based on solving a constrained nonlinear optimization problem whose lagrange multipliers and objective function can help assess the trade off between bias and precision of the resulting

the blow up problem for a semilinear parabolic equation with a potential abstract let be a bounded smooth domain in r n we consider the problem u t u v x u p in 0 t with dirichlet boundary conditions u 0 on 0 t and initial datum u x 0 m x where m 0 is positive and compatible with the boundary condition we give estimates for the blow up time of solutions for large values of m as a consequence of these estimates we find that for m large the blow up set concentrates near the points where p 1 v attains its maximum

blow up behavior of positive solutions for a chemical fuel ignition device model blow up behavior of positive solutions of a semi linear parabolic system arising from thermal explosion which subject to the homogenous dirichlet boundary conditions is investigated in particular sufficient conditions for the solutions to blow up are obtained

adjusting treatment effect estimates by post stratification in randomized experiments summary experimenters often use post stratification to adjust estimates post stratification is akin to blocking except that the number of treated units in each stratum is a random variable because stratification occurs after treatment assignment we analyse both post stratification and blocking under the neyman rubin model and compare the efficiency of these designs we derive the variances for a post stratified estimator and a simple difference in means estimator under different randomization schemes post stratification is nearly as efficient as blocking the difference in their variances is of the order of 1 n2 with a constant depending on treatment proportion post stratification is therefore a reasonable alternative to blocking when blocking is not feasible however in finite samples post stratification can increase variance if the number of strata is large and the strata are poorly chosen to examine why the estimators variances are different we extend our results by conditioning on the observed number of treated units in each stratum conditioning also provides more accurate variance estimates because it takes into account how close or far a realized random sample is from a comparable blocked experiment we then show that the practical substance of our results remains under an infinite population sampling model finally we provide an analysis of an actual experiment to illustrate our analytical results

optimal probability weights for estimating causal effects of time varying treatments with marginal structural cox models marginal structural cox models have been used to estimate the causal effect of a time varying treatment on a survival outcome in the presence of time dependent confounders these methods rely on the positivity assumption which states that the propensity scores are bounded away from zero and one practical violations of this assumption are common in longitudinal studies resulting in extreme weights that may yield erroneous inferences truncation which consists of replacing outlying weights with less extreme ones is the most common approach to control for extreme weights to date while truncation reduces the variability in the weights and the consequent sampling variability of the estimator it can also introduce bias instead of truncated weights we propose using optimal probability weights defined as those that have a specified variance and the smallest euclidean distance from the original untruncated weights the set of optimal weights is obtained by solving a constrained quadratic optimization problem the proposed weights are evaluated in a simulation study and applied to the assessment of the effect of treatment on time to death among people in sweden who live with human immunodeficiency virus and inject drugs

multiple blow up rates in a coupled heat system with mixed type nonlinearities abstract this paper studies heat equations with inner absorptions and coupled boundary fluxes of mixed type nonlinearities at first the critical exponent is obtained and simply described via a characteristic algebraic system introduced by us then as the main results of the paper three blow up rates are established under different dominations of nonlinearities for the one dimensional case and represented in another characteristic algebraic system in particular it is observed that unlike those in previous literature on parabolic models with absorptions two of the multiple blow up rates obtained here do depend on the absorption exponents in the known works the absorptions affect the blow up criteria the blow up time as well as the initial data required for the blow up of solutions all without changing the blow up rates to our knowledge this is the first example of absorption dependent blow up rates exploiting the significant interactions among diffusions inner absorptions and nonlinear boundary fluxes in the coupled system it is also proved that the blow up of solutions in the model occurs on the boundary only

global existence for a vector valued nonlinear heat equation in this paper we are concerned with the global existence the uniqueness the nonnegativity of classical solutions of a semilinear parabolic coupled system

existence and nonexistence of global solutions for a semilinear reaction diffusion system abstract this paper is concerned with the blow up and global existence of nonnegative solutions to the following cauchy problem u t u v p t 0 x r n v t v a x u q t 0 x r n u x 0 u 0 x v x 0 v 0 x x r n where the constants p q 0 and a x 0 is on the order x m as x m r the fujita critical exponent is determined when m 0 and some results of global existence of solution under some assumptions when m 0 are also obtained the results extend those in escobedo and herrero 1991 9 and indicate that m affects the fujita critical exponent

the central role of the propensity score in observational studies for causal effects abstract the results of observational studies are often disputed because of nonrandom treatment assignment for example patients at greater risk may be overrepresented in some treatment group this paper discusses the central role of propensity scores and balancing scores in the analysis of observational studies the propensity score is the estimated conditional probability of assignment to a particular treatment given a vector of observed covariates both large and small sample theory show that adjustment for the scalar propensity score is sufficient to remove bias due to all observed covariates applications include matched sampling on the univariate propensity score which is equal percent bias reducing under more general conditions than required for discriminant matching multivariate adjustment by subclassification on balancing scores where the same subclasses are used to estimate treatment effects for all outcome variables and in all subpopulations and visual representation of multivariate adjustment by a two dimensional plot author

axisymmetric surface diffusion dynamics and stability of self similar pinchoff the dynamics of surface diffusion describes the motion of a surface with its normal velocity given by the surface laplacian of its mean curvature this flow conserves the volume enclosed inside the surface while minimizing its surface area we review the axisymmetric equilibria the cylinder sphere and the delaunay unduloid the sphere is stable while the cylinder is long wave unstable a subcritical bifurcation from the cylinder produces a continuous family of unduloid solutions we present computations that suggest that the stable manifold of the unduloid forms a separatrix between states that relax to the cylinder in infinite time and those that tend toward finite time pinchoff we examine the structure of the pinchoff showing it has self similar structure using asymptotic numerical and analytical methods in addition to a previously known similarity solution we find a countable set of similarity solutions each with a different asymptotic cone angle we develop a stability theory in similarity variables that selects the original similarity solution as the only linearly stable one and consequently the only observable solution we also consider similarity solutions describing the dynamics after the topological transition

automatic detection of the boundary of the calcaneus from ultrasound parametric images using an active contour model clinical assessment presents a computerized method for automated detection of the boundary of the os calcis on in vivo ultrasound parametric images using an active dynamic contour model the initial contour defined without user interaction is an iso contour extracted from the textural feature space the contour is deformed through the action of internal and external forces until stability is reached the external forces which characterize image features are a combination of gray level information and second order textural features arising from local cooccurrence matrices the broadband ultrasound attenuation bua value is then averaged within the contour obtained the method was applied to 381 clinical images the contour was correctly detected in the great majority of the cases for the short term reproducibility study the mean coefficient of variation was equal to 1 81 for bua values and 4 95 for areas in the detected region women with osteoporosis had a lower bua than age matched controls p 0 0005 in healthy women the age related decline was 0 45 db mhz yr in the group of healthy post menopausal women years since menopause weight and age were significant predictors of bua these results are comparable to those obtained when averaging bua values in a small region of interest

a symptotic behaviour for a numerical approximation of a parabolic problem with blowing up solutions abstract in this paper we study the asymptotic behaviour of a semidiscrete numerical approximation for u t u xx u p in a bounded interval 0 1 with dirichlet boundary conditions we focus in the behaviour of blowing up solutions we find that the blow up rate for the numerical scheme is the same as for the continuous problem also we find the blow up set for the numerical approximations and prove that it is contained in a neighbourhood of the blow up set of the continuous problem when the mesh parameter is small enough

covariate adjustment using propensity scores for dependent censoring problems in the accelerated failure time model in many medical studies estimation of the association between treatment and outcome of interest is often of primary scientific interest standard methods for its evaluation in survival analysis typically require the assumption of independent censoring this assumption might be invalid in many medical studies where the presence of dependent censoring leads to difficulties in analyzing covariate effects on disease outcomes this data structure is called semicompeting risks data for which many authors have proposed an artificial censoring technique however confounders with large variability may lead to excessive artificial censoring which subsequently results in numerically unstable estimation in this paper we propose a strategy for weighted estimation of the associations in the accelerated failure time model weights are based on propensity score modeling of the treatment conditional on confounder variables this novel application of propensity scores avoids excess artificial censoring caused by the confounders and simplifies computation monte carlo simulation studies and application to aids and cancer research are used to illustrate the methodology

blow up properties for heat equations coupled via different nonlinearities this paper deals with heat equations coupled via exponential and power nonlinearities subject to null dirichlet boundary conditions the complete and optimal classification on non simultaneous and simultaneous blow ups is proposed by four sufficient and necessary conditions we find out that in some exponent region the blow up properties of the solutions depend much on the choosing of initial data moreover all kinds of non simultaneous and simultaneous blow up rates are obtained

total versus single point blow up of solutions of a semilinear parabolic equation with localized reaction abstract in this paper we study the initial boundary value problem of a semilinear parabolic equation with localized reaction p u t u u p u q x t x t b 0 t u x t 0 x t b 0 t u x 0 u 0 x x b where b is a unit ball in r n x b and p q 0 for the case x 0 we completely classify blow up solutions of p into total blow up cases and single point blow up cases according to the values p and q moreover we give the blow up rates of solutions near the blow up time for the other case x 0 we show total blow up and single point blow up of solutions of p in some cases depending on p and q

existence results of steady states of semilinear reaction diffusion equations and their applications partially supported by the applied mathematical sciences subprogram of the offlice of energy research u s department of energy under contract w 31 109 eng 38 while a visit ing scientist at argonne national laboratory during the summer of 1987 and m part by the board of regents of the state of louisiana under grant leqsf 86 89 rd a 11 partially supported by the applied mathematical sciences subprogram of the office of energy research u s department of energy under contract w 31 109 eng 38 this author thanks the department of mathematics university of southwestern louisiana for an invitation to spend a semester there in the fall of 1986 during which the research reported in this paper was started

optimal classification for blow up phenomena in heat equations coupled via exponential sources abstract this paper deals with simultaneous and non simultaneous blow up solutions for heat equations coupled via exponential sources subject to null dirichlet boundary conditions the main results complete the previously known results on the optimal classification for simultaneous and non simultaneous blow up solutions by covering the whole ranges of exponents moreover all kinds of simultaneous and non simultaneous blow up rates are obtained

causal inference using potential outcomes design modeling decisions causal effects are defined as comparisons of potential outcomes under different treatments on a common set of units observed values of the potential outcomes are revealed by the assignment mechanism a probabilistic model for the treatment each unit receives as a function of covariates and potential outcomes fisher made tremendous contributions to causal inference through his work on the design of randomized experiments but the potential outcomes perspective applies to other complex experiments and nonrandomized studies as well as noted by kempthorne in his 1976 discussion of savage s fisher lecture fisher never bridged his work on experimental design and his work on parametric modeling a bridge that appears nearly automatic with an appropriate view of the potential outcomes framework where the potential outcomes and covariates are given a bayesian distribution to complete the model specification also this framework crisply separates scientific inference for causal effects and decisions based on s

blow up behavior for a nonlinear heat equation with a localized source in a ball in this paper we consider the initial boundary value problem of a semilinear parabolic equation with local and non local localized reactions in a ball ut u up uq x t in b r where p q 0 b r x rn x 1 there exist blow up solutions of this problem for large initial data we treat the radially symmetric and one peak non negative solution u x t u r t r x of this problem we give the complete classification of total blow up phenomena and single point blow up phenomena according to p and q i if q 1 or p q 2 then single point blow up occurs whenever solutions blow up ii if 1 p q both phenomena total blow up and single point blow up occur depending on the initial data iii if p 1 q total blow up occurs whenever solutions blow up iv if max p q 1 every solution exists globally in time

evaluation of subset matching methods and forms of covariate balance this paper conducts a monte carlo simulation study to evaluate the performance of multivariate matching methods that select a subset of treatment and control observations the matching methods studied are the widely used nearest neighbor matching with propensity score calipers and the more recently proposed methods optimal matching of an optimally chosen subset and optimal cardinality matching the main findings are i covariate balance as measured by differences in means variance ratios kolmogorov smirnov distances and cross match test statistics is better with cardinality matching because by construction it satisfies balance requirements ii for given levels of covariate balance the matched samples are larger with cardinality matching than with the other methods iii in terms of covariate distances optimal subset matching performs best iv treatment effect estimates from cardinality matching have lower root mean square errors provided strong requirements for balance specifically fine balance or strength k balance plus close mean balance in standard practice a matched sample is considered to be balanced if the absolute differences in means of the covariates across treatment groups are smaller than 0 1 standard deviations however the simulation results suggest that stronger forms of balance should be pursued in order to remove systematic biases due to observed covariates when a difference in means treatment effect estimator is used in particular if the true outcome model is additive then marginal distributions should be balanced and if the true outcome model is additive with interactions then low dimensional joints should be balanced copyright 2016 john wiley sons ltd

non simultaneous blow up in coupled heat equations with multi nonlinearities abstract this paper studies coupled heat equations with multi nonlinearities of six nonlinear parameters the critical blow up exponent is established via a complete classification for all the six nonlinear parameters where a precise analysis on the geometry of and the absorption coefficients is given for the balanced interaction situation among the multi nonlinearities the main attention is contributed to non simultaneous phenomena in the model to determine the necessary and sufficient conditions of non simultaneous blow up with suitable initial data as well as the conditions under which any blow up must be non simultaneous finally the model of the paper is characterized by a comparison with the known results for other models

dealing with limited overlap in estimation of average treatment effects summary estimation of average treatment effects under unconfounded or ignorable treatment assignment is often hampered by lack of overlap in the covariate distributions between treatment groups this lack of overlap can lead to imprecise estimates and can make commonly used estimators sensitive to the choice of specification in such cases researchers have often used ad hoc methods for trimming the sample we develop a systematic approach to addressing lack of overlap we characterize optimal subsamples for which the average treatment effect can be estimated most precisely under some conditions the optimal selection rules depend solely on the propensity score for a wide range of distributions a good approximation to the optimal rule is provided by the simple rule of thumb to discard all units with estimated propensity scores outside the range 0 1 0 9

doubly robust inference for targeted minimum loss based estimation in randomized trials with missing outcome data missing outcome data is a crucial threat to the validity of treatment effect estimates from randomized trials the outcome distributions of participants with missing and observed data are often different which increases bias causal inference methods may aid in reducing the bias and improving efficiency by incorporating baseline variables into the analysis in particular doubly robust estimators incorporate 2 nuisance parameters the outcome regression and the missingness mechanism ie the probability of missingness conditional on treatment assignment and baseline variables to adjust for differences in the observed and unobserved groups that can be explained by observed covariates to consistently estimate the treatment effect one of these nuisance parameters must be consistently estimated traditionally nuisance parameters are estimated using parametric models which often precludes consistency particularly in moderate to high dimensions recent research on missing data has focused on data adaptive estimation to help achieve consistency but the large sample properties of such methods are poorly understood in this article we discuss a doubly robust estimator that is consistent and asymptotically normal under data adaptive estimation of the nuisance parameters we provide a formula for an asymptotically exact confidence interval under minimal assumptions we show that our proposed estimator has smaller finite sample bias compared to standard doubly robust estimators we present a simulation study demonstrating the enhanced performance of our estimators in terms of bias efficiency and coverage of the confidence intervals we present the results of an illustrative example a randomized double blind phase 2 3 trial of antiretroviral therapy in hiv infected persons

blow up for a semilinear parabolic equation with large diffusion on rn ii we are concerned with the cauchy problem for a semilinear heat equation p tu d u u p 1u x rn t 0 u x 0 x x rn where d 0 p 1 n 3 0 and l rn l1 rn 1 x 2dx in the paper of fujishima and ishige 2011 8 the authors of this paper studied the behavior of the blow up time and the blow up set of the solution of p as d for the case rn x dx 0 in this paper as a continuation of fujishima and ishige 2011 8 we consider the case rn x dx 0 and study the behavior of the blow up time and the blow up set of the solution of p as d the behavior in the case rn x dx 0 is completely different from the one in the case rn x dx 0

a computational approach to edge detection this paper describes a computational approach to edge detection the success of the approach depends on the definition of a comprehensive set of goals for the computation of edge points these goals must be precise enough to delimit the desired behavior of the detector while making minimal assumptions about the form of the solution we define detection and localization criteria for a class of edges and present mathematical forms for these criteria as functionals on the operator impulse response a third criterion is then added to ensure that the detector has only one response to a single edge we use the criteria in numerical optimization to derive detectors for several common image features including step edges on specializing the analysis to step edges we find that there is a natural uncertainty principle between detection and localization performance which are the two main goals with this principle we derive a single operator shape which is optimal at any scale the optimal detector has a simple approximate implementation in which edges are marked at maxima in gradient magnitude of a gaussian smoothed image we extend this simple detector using operators of several widths to cope with different signal to noise ratios in the image we present a general method called feature synthesis for the fine to coarse integration of information from operators at different scales finally we show that step edge detector performance improves considerably as the operator point spread function is extended along the edge

determination of the curvature of the blow up set and refined singular behavior for a semilinear heat equation we consider u x t a solution of ut u u p 1u which blows up at some time t 0 where u rn 0 t r p 1 and n 2 p n 2 under a nondegeneracy condition we show that the mere hypothesis that the blow up set s is continuous and n 1 dimensional implies that it is c2 in particular we compute the n 1 principal curvatures and directions of s moreover a much more refined blow up behavior is derived for the solution in terms of the newly exhibited geometric objects refined regularity for s and refined singular behavior of u near s are linked through a new mechanism of algebraic cancellations that we explain in detail

the blowup property of solutions to some diffusion equations with localized nonlinear reactions abstract in this paper we investigate the blowup property of solutions to the equation u t u u x 0 t where x0 is a fixed point in the domain we show that under certain conditions the solution blows up in finite time moreover we prove that the set of all blowup points is the whole region furthermore the growth rate of solutions near the blowup time is also derived finally the results are generalized to the following nonlocal reaction diffusion equation u t u u dx

single point gradient blow up on the boundary for diffusive hamilton jacobi equations in planar domains consider the diffusive hamilton jacobi equation u t u u p p 2 on a bounded domain with zero dirichlet boundary conditions which arises in the kpz model of growing interfaces it is known that u remains bounded and that u may blow up only on the boundary in this paper under suitable assumptions on omega subset mathbb r 2 and on the initial data we show that the gradient blow up singularity occurs only at a single point x 0 in partial omega this is the first result of this kind in the study of problems involving gradient blow up phenomena in general domains of mathbb r n we also obtain results on nondegeneracy and localization of blow up points

construction of a blow up solution for the complex ginzburg landau equation in a critical case we construct a solution for the complex ginzburg landau equation in some critical case which blows up in finite time t only at one blow up point we also give a sharp description of its profile the proof relies on the reduction of the problem to a finite dimensional one and the use of index theory to conclude the interpretation of the parameters of the finite dimension problem in terms of the blow up point and time allows to prove the stability of the constructed solution

extinction properties of semilinear heat equations with strong absorption consider the initial boundary value problem for ut u uq with 0 0 q 1 the initial data are nonnegative and the boundary data vanish it is well known that the solution becomes extinct in finite time t i e u x t becomes identically zero for t t where t is some positive number in this paper we study the profile of x u x t as t t

blowup estimates for a parabolic system in a three species cooperating model abstract blowup estimates for solutions of a parabolic system which arises in a cooperating three species food chain model with homogeneous dirichlet boundary conditions are investigated the upper bound of blowup rate for any n and the lower bound of blowup rate for n 1 are obtained

model based direct adjustment abstract direct adjustment or standardization applies population weights to subclass means in an effort to estimate population quantities from a sample that is not representative of the population direct adjustment has several attractive features but when there are many subclasses it can attach large weights to small quantities of data often in a fairly erratic manner in the extreme direct adjustment can attach infinite weight to nonexistent data a noticeable inconvenience in practice this article proposes a method of model based direct adjustment that preserves the attractive features of conventional direct adjustment while stabilizing the weights attached to small subclasses the sample mean and conventional direct adjustment are both special cases of model based direct adjustment under two different extreme models for the subclass specific selection probabilities the discussion of this method provides some insights into the behavior of true and estimated propensity scores the estimated scores

blow up estimates for system of heat equations coupled via nonlinear boundary flux abstract this paper deals with a system of heat equations coupled via nonlinear boundary flux the precise blow up rate estimates are established together with the blow up set it is observed that there is some quantitative relationship regarding the blow up properties between the heat system with coupled nonlinear boundary flux terms and the corresponding reaction diffusion system with the same nonlinear terms as the source

continuous force field analysis for generalized gradient vector flow field we propose a modification of the generalized gradient vector flow field techniques based on a continuous force field analysis at every iteration the generalized gradient vector flow method obtains a new improved vector field however the numerical procedure always employs the original image to calculate the gradients used in the source term the basic idea developed in this paper is to use the resulting vector field to obtain an improved edge map and use it to calculate a new gradient based source term the improved edge map is evaluated by new continuous force field analysis techniques inspired by a preceding discrete version the approach leads to a better convergence and better segmentation accuracy as compared to several conventional gradient vector flow type methods

construction of a stable periodic solution to a semilinear heat equation with a prescribed profile we construct a periodic solution to the semilinear heat equation with power nonlinearity in one space dimension which blows up in finite time t only at one blow up point we also give a sharp description of its blow up profile the proof relies on the reduction of the problem to a finite dimensional one and the use of index theory to conclude thanks to the geometrical interpretation of the finite dimensional parameters in terms of the blow up time and blow up point we derive the stability of the constructed solution with respect to initial data

on blow up at space infinity for semilinear heat equations abstract a nonnegative blowing up solution of the semilinear heat equation u t u u p with p 1 is considered when initial data u 0 satisfies lim x u 0 m 0 u 0 m and u 0 m it is shown that the solution blows up only at space infinity and that lim x u x t is the solution of the ordinary differential equation v t v p with v 0 m

iterative approximation of solutions for semilinear parabolic equations system in this paper we present the convergence analysis of iterative schemes for solving semilinear systems resulting from multidimensional semilinear second order parabolic partial differential equations pdes defined in a domain n x and subject to cauchy boundary conditions on n using the method of integral successive iterative approximation and the principle of decay estimation we derive the bounded nonnegative local solutions and global solutions blowing up results of the solutions are also presented

estimation of regression coefficients when some regressors are not always observed abstract in applied problems it is common to specify a model for the conditional mean of a response given a set of regressors a subset of the regressors may be missing for some study subjects either by design or happenstance in this article we propose a new class of semiparametric estimators based on inverse probability weighted estimating equations that are consistent for parameter vector 0 of the conditional mean model when the data are missing at random in the sense of rubin and the missingness probabilities are either known or can be parametrically modeled we show that the asymptotic variance of the optimal estimator in our class attains the semiparametric variance bound for the model by first showing that our estimation problem is a special case of the general problem of parameter estimation in an arbitrary semiparametric model in which the data are missing at random and the probability of observing complete data is bounded away from 0 and then deriving a representation for the efficient score

single point blow up for a semilinear initial value problem abstract the initial value problem on r r is considered u t t x u xx t x u t x u t r 0 u 0 x x where 0 and is a fixed large number it is known that for some initial values the solution u t x exists only up to some finite time t and that u t as t t for the specific initial value k where 0 xx 0 r 0 k is sufficiently large it is shown that if x 0 then lim t t u t x and lim t t u x t x exist and are finite in other words blow up occurs only at the point x 0

improved double robust estimation in missing data and causal inference models recently proposed double robust estimators for a population mean from incomplete data and for a finite number of counterfactual means can have much higher efficiency than the usual double robust estimators under misspecification of the outcome model in this paper we derive a new class of double robust estimators for the parameters of regression models with incomplete cross sectional or longitudinal data and of marginal structural mean models for cross sectional data with similar efficiency properties unlike the recent proposals our estimators solve outcome regression estimating equations in a simulation study the new estimator shows improvements in variance relative to the standard double robust estimator that are in agreement with those suggested by asymptotic theory

interior imperfect ignition cannot occur on a set of positive measure for the differential inequality u t delta u geq 0 it is proved that the dead core of the solution d bigcup 0 leq t leq t d t times t with d t overline left x in omega mid u x t infty right cannot be enclosed inside a bounded domain with positive lebesgue measure