existence and uniqueness of multi bump bound states of nonlinear schrodinger equations with electromagnetic fields abstract in this paper we are concerned with the existence and uniqueness of multi bump bound states of the nonlinear schrodinger equations with electromagnetic potential i t i a x 2 v x p 2 x r n for sufficiently small 0 where i is the imaginary unit 2 p 2 n n 2 for n 3 and 2 p for n 1 2 v x is a bounded real function on r n and a x a 1 x a 2 x a n x is such that a j x is a bounded real function on r n for j 1 2 n for any finite collection of a 1 a 2 a k of non degenerate critical points of v x we show that there exists a solution x t exp iet u x which is a small perturbation of a sum of one bump solutions concentrated at a 1 a 2 a k and u x is unique up to rotations for small 0

structured condition numbers and small sample condition estimation of symmetric algebraic riccati equations abstract this paper is devoted to a structured perturbation analysis of the symmetric algebraic riccati equations by exploiting the symmetry structure based on the analysis the upper bounds for the structured normwise mixed and componentwise condition numbers are derived due to the exploitation of the symmetry structure our results are improvements of the previous work on the perturbation analysis and condition numbers of the symmetric algebraic riccati equations our preliminary numerical experiments demonstrate that our condition numbers provide accurate estimates for the change in the solution caused by the perturbations on the data moreover by applying the small sample condition estimation method we propose a statistical algorithm for practically estimating the condition numbers of the symmetric algebraic riccati equations

radial solutions for the brezis nirenberg problem involving large nonlinearities abstract let us consider the problem 0 1 u a x u u p in b 1 u 0 in b 1 u 0 on b 1 where b 1 is the unit ball in r n n 3 and a x 0 is a smooth radial function under some suitable assumptions on the regular part of the green function of the operator u n 1 r u a r u we prove the existence of a radial solution to 0 1 for p large enough

a multiplicity result for singular nls equations with magnetic potentials abstract in this paper we obtain multiple solutions u r n c of the singular nonlinear schrodinger equation with an external electromagnetic field 1 i a x 2 u v x e w x u u p 2 u x r n where n 2 2 p 2 a is a c 1 real valued vector magnetic potential v x is a real continuous lower bounded electric potential and e w x is a perturbing potential depending on which may be singular our result covers the meaningful physical cases w x 1 x w x 1 x 2 we prove that the number of solutions to the eq 1 is bounded below by some number which depends on the category of the set of global minima of v for sufficiently small

a comparison of sample size methods for the logrank statistic several methods are available for sample size calculation for clinical trials when survival curves are to be compared using the logrank statistic we discuss advantages and disadvantages of some of these methods and present simulation results under exponential proportional hazards and non proportional hazard situations

multiplicity results for some nonlinear schrodinger equations with potentials multiple semiclassical standing waves for a class of nonlinear schrodinger equations with potentials are found by means of a perturbative variational method

on the existence and profile of nodal solutions for a two dimensional elliptic problem with large exponent in nonlinearity we study the existence of nodal solutions to the boundary value problem u u p 1u in a bounded smooth domain in ir with homogeneous dirichlet boundary condition when p is a large exponent we prove that for p large enough there exist at least two pairs of solutions which change sign exactly once and whose nodal lines intersect the boundary of

multi peak solutions for a singularly perturbed semilinear elliptic problem this paper is concerned with the existence of multi peak solutions of the following problem formula where is a smooth bounded domain in rn n 2 e 0 is a small positive number and p 2 2nn 2 for n 3 p 2 for n 2 using a variational method we construct positive solutions concentrating near given saddle points of the distance function dist an explicit expression for the dominant parts of the solutions is also obtained

separation of branches of o n 1 invariant solutions for a semilinear elliptic equation abstract we consider the problem u u u p in a u 0 in a u 0 on a where a is an annulus in r n n 2 p 1 and 0 recent results ensure that there exists a sequence p k of exponents p k at which a nonradial bifurcation from the radial solution occurs exploiting the properties of o n 1 invariant spherical harmonics we introduce two suitable cones k 1 and k 2 of o n 1 invariant functions that allow to separate the branches of bifurcating solutions getting their unboundedness

condition numbers of algebraic riccati equations in the frobenius norm consider the continuous time algebraic riccati equation care and the discrete time algebraic riccati equation dare which arise in linear control and system theory it is known that appropriate assumptions on the coefficient matrices guarantee the existence and uniqueness of hermitian positive semidefinite stabilizing solutions in this note we apply the theory of condition developed by rice to define condition numbers of the care and dare in the frobenius norm and derive explicit expressions of the condition numbers in a uniform manner both the complex case and real case are considered and connections to certain existing condition numbers of the care and dare are discussed

semiclassical solutions for a class of schrodinger system with magnetic potentials abstract this paper is concerned with the following nonlinear schrodinger system with magnetic potentials i e a x 2 u v x u h u x u v x r n i e a x 2 v v x v h v x u v x r n where n 3 e is a small parameter a r n r n is the magnetic vector potential and v r n r is the electric potential by applying generalized linking theorems for strongly indefinite functionals we establish the existence and multiplicity of semiclassical solutions for superquadratic and subcritical nonlinearity

the two dimensional lazer mckenna conjecture for an exponential nonlinearity abstract we consider the problem of ambrosetti prodi type u e u s 1 h x in u 0 on where is a bounded smooth domain in r 2 1 is a positive first eigenfunction of the laplacian under dirichlet boundary conditions and h c 0 we prove that given k 1 this problem has at least k solutions for all sufficiently large s 0 which answers affirmatively a conjecture by lazer and mckenna a c lazer p j mckenna on the number of solutions of a nonlinear dirichlet problem j math anal appl 84 1981 282 294 for this case the solutions found exhibit multiple concentration behavior around maxima of 1 as s

the statistical analysis of failure time data preface 1 introduction 1 1 failure time data 1 2 failure time distributions 1 3 time origins censoring and truncation 1 4 estimation of the survivor function 1 5 comparison of survival curves 1 6 generalizations to accommodate delayed entry 1 7 counting process notation bibliographic notes exercises and complements 2 failure time models 2 1 introduction 2 2 some continuous parametric failure time models 2 3 regression models 2 4 discrete failure time models bibliographic notes exercises and complements 3 inference in parametric models and related topics 3 1 introduction 3 2 censoring mechanisms 3 3 censored samples from an exponential distribution 3 4 large sample likelihood theory 3 5 exponential regression 3 6 estimation in log linear regression models 3 7 illustrations in more complex data sets 3 8 discrimination among parametric models 3 9 inference with interval censoring 3 10 discussion bibliographic notes exercises and complements 4 relative risk cox regression models 4 1 introduction 4 2 estimation of beta 4 3 estimation of the baseline hazard or survivor function 4 4 inclusion of strata 4 5 illustrations 4 6 counting process formulas 4 7 related topics on the cox model 4 8 sampling from discrete models bibliographic notes exercises and complements 5 counting processes and asymptotic theory 5 1 introduction 5 2 counting processes and intensity functions 5 3 martingales 5 4 vector valued martingales 5 5 martingale central limit theorem 5 6 asymptotics associated with chapter 1 5 7 asymptotic results for the cox model 5 8 asymptotic results for parametric models 5 9 efficiency of the cox model estimator 5 10 partial likelihood filtration bibliographic notes exercises and complements 6 likelihood construction and further results 6 1 introduction 6 2 likelihood construction in parametric models 6 3 time dependent covariates and further remarks on likelihood construction 6 4 time dependence in the relative risk model 6 5 nonnested conditioning events 6 6 residuals and model checking for the cox model bibliographic notes exercises and complements 7 rank regression and the accelerated failure time model 7 1 introduction 7 2 linear rank tests 7 3 development and properties of linear rank tests 7 4 estimation in the accelerated failure time model 7 5 some related regression models bibliographic notes exercises and complements 8 competing risks and multistate models 8 1 introduction 8 2 competing risks 8 3 life history processes bibliographic notes exercises and complements 9 modeling and analysis of recurrent event data 9 1 introduction 9 2 intensity processes for recurrent events 9 3 overall intensity process modeling and estimation 9 4 mean process modeling and estimation 9 5 conditioning on aspects of the counting process history bibliographic notes exercises and complements 10 analysis of correlated failure time data 10 1 introduction 10 2 regression models for correlated failure time data 10 3 representation and estimation of the bivariate survivor function 10 4 pairwise dependency estimation 10 5 illustration australian twin data 10 6 approaches to nonparametric estimation of the bivariate survivor function 10 7 survivor function estimation in higher dimensions bibliographic notes exercises and complements 11 additional failure time data topics 11 1 introduction 11 2 stratified bivariate failure time analysis 11 3 fixed study period survival studies 11 4 cohort sampling and case control studies 11 5 missing covariate data 11 6 mismeasured covariate data 11 7 sequential testing with failure time endpoints 11 8 bayesian analysis of the proportional hazards model 11 9 some analyses of a particular data set bibliographic notes exercises and complements glossary of notation appendix a some sets of data appendix b supporting technical material bibliography author index subject index

multi peak solutions for super critical elliptic problems in domains with small holes abstract this paper deals with the slightly super critical elliptic problem u u n 2 n 2 e in u 0 in u 0 on 1 where e 0 is a small parameter and r n is a bounded domain with smooth boundary assuming that the domain exhibits k sufficiently small holes multiple solutions are constructed by gluing double spike patterns located near each of the holes

the role of the green s function in a non linear elliptic equation involving the critical sobolev exponent this paper is concerned with non linear elliptic problems of the type pe u u n 2 n 2 eu u 0 on u 0 on where is a smooth and bounded domain in rn n 4 and e 0 we show that if the ue are solutions of pe which concentrate around a point as e 0 then this point cannot be on the boundary of and is a critical point of the regular part of the green s function conversely we show that for n 5 and any non degenerate critical point x0 of the regular part of the green s function there exist solutions of pe concentrating around x0 as e 0

blow up phenomena for the liouville equation with a singular source of integer multiplicity abstract we are concerned with the existence of blowing up solutions to the following boundary value problem u a x e u 4 n 0 in u 0 on where is a smooth and bounded domain in r 2 such that 0 a x is a positive smooth function n is a positive integer and 0 is a small parameter here 0 defines the dirac measure with pole at 0 we find conditions on the function a and on the domain under which there exists a solution u blowing up at 0 and satisfying a x e u 8 n 1 as 0

multi peak solutions for nonlinear schrodinger systems with magnetic potentials in r3 abstract we study the following nonlinear schrodinger system with magnetic potentials in r 3 i a x 2 u p x u 1 u 2 u v 2 u x r 3 i a x 2 v q x v 2 v 2 v u 2 v x r 3 where 0 is a small parameter 1 2 0 and 0 is a coupling constant a x p x and q x are potential functions applying the finite reduction method we prove that the nonlinear schrodinger system has multi peak solutions under some suitable conditions which are given in section 1

sample size calculation for cluster randomization trials with a time to event endpoint cluster randomization trials randomize groups called clusters of subjects called subunits between intervention arms and observations are collected from each subject in this case subunits within each cluster share common frailties so that the observations from subunits of each cluster tend to be correlated oftentimes the outcome of a cluster randomization trial is a time to event endpoint with censoring in this article we propose a closed form sample size formula for weighted rank tests to compare the marginal survival distributions between intervention arms under cluster randomization with possibly variable cluster sizes extensive simulation studies are conducted to evaluate the performance of our sample size formula under various design settings real study examples are taken to demonstrate our method

integration by parts formula for regional fractional laplacian we obtain the integration by parts formula for the regional fractional laplacian which are generators of symmetric stable processes on a subset of mathbb r n 0 2 in this formula a local operator appears on the boundary connected with the regional fractional laplacian on domain hence this formula can be understood as the green formula for the regional fractional laplacian the similar integration by parts formula is also given for more general nonlocal operators the reflected stable like processes are studied

schur complements and statistics abstract in this paper we discuss various properties of matrices of the type s h ge 1 f which we call the schur complement of e in a e f g h the matrix e is assumed to be nonsingular when e is singular or rectangular we consider the generalized schur complements s h ge f where e is a generalized inverse of e a comprehensive account of results pertaining to the determinant the rank the inverse and generalized inverses of partitioned matrices and the inertia of a matrix is given both for schur complements and for generalized schur complements we survey the known results in a historical perspective and obtain several extensions numerous applications in numerical analysis and statistics are included the paper ends with an exhaustive bibliography of books and articles related to schur complements

sample sizes based on the log rank statistic in complex clinical trials the log rank test is frequently used to compare survival curves while sample size estimation for comparison of binomial proportions has been adapted to typical clinical trial conditions such as noncompliance lag time and staggered entry the estimation of sample size when the log rank statistic is to be used has not been generalized to these types of clinical trial conditions this paper presents a method of estimating sample sizes for the comparison of survival curves by the log rank statistic in the presence of unrestricted rates of noncompliance lag time and so forth the method applies to stratified trials in which the above conditions may vary across the different strata and does not assume proportional hazards power and duration as well as sample sizes can be estimated the method also produces estimates for binomial proportions and the tarone ware class of statistics

new concentration phenomena for su 3 toda system in this paper we consider the following su 3 toda system 0 1 u1 2 2h1eu1 2h2eu2 0in b1 0 u2 2 2h2eu2 2h1eu1 0in b1 0 u1 u2 0on b1 0 where is a positive parameter which will go to zero h1 x h2 x h x is a positive smooth function and b1 0 is a unit ball in r2 in 29 the authors construct a family of solutions u1 u2 of 0 1 for h 1 such that the solutions blow up at the origin with mass 8 4 after some scaling with the limiting profile wi x i 2ewi in r2 r2 x i 2ewi where i 23 i for i 1 2 in that paper the authors ask whether there exist blow up solutions of mass 8 4 with the limiting profile w 2ew 0in r2 r2ew such that u1 is the sum of two bubbles and u2 has one bubble in this paper we prove that such solution exists

solutions with multiple peaks for nonlinear elliptic equations solutions with peaks near the critical points of q x are constructed for the problem delta u lambda 2 u q x u p 2 u x is an element of r n u is an element of h 1 and 2 p 2n n 2

the dirichlet elliptic problem involving regional fractional laplacian in this paper we study the solutions of elliptic equations involving regional fractional laplacian e u f in a bounded regular domain in rn n 2 with c2 boundary subject to dirichlet boundary g on where 12 1 and the operator denotes the regional fractional laplacian we prove that when g 0 problem e admits a unique weak solution under the hypotheses that f l2 f l1 dx and f m where x dist x 2 1 and m is a space of all radon measures satisfying d finally we provide an integration by parts formula for the classical solution of e with boundary data g

on mixed and componentwise condition numbers for moore penrose inverse and linear least squares problems classical condition numbers are normwise they measure the size of both input perturbations and output errors using some norms to take into account the relative of each data component and in particular a possible data sparseness componentwise condition numbers have been increasingly considered these are mostly of two kinds mixed and componentwise in this paper we give explicit expressions computable from the data for the mixed and componentwise condition numbers for the computation of the moore penrose inverse as well as for the computation of solutions and residues of linear least squares problems in both cases the data matrices have full column row rank

uniqueness of positive multi lump bound states of nonlinear schrodinger equations in this paper we are concerned with multi lump bound states of the nonlinear schrodinger equation

solutions and perturbation estimates for the matrix equations x a x n a q in this paper we investigate the equations x a x na q for the existence of positive definite solutions and perturbation bounds for these solutions are derived a sufficient condition for uniqueness of a unique positive definite solution of the equation x a x na q is given the results are illustrated by using numerical examples

concentrating solutions for an exponential nonlinearity with robin boundary condition abstract we study the following robin boundary value problem 0 1 u 2 u p 1 e u p 0 u 0 in u n u 0 on where is a bounded domain in r 2 with smooth boundary as 0 and simultaneously 0 p 2 by using a lyapunov schmidt reduction procedure we extend for the whole range of exponents 0 p 2 the results of davila and topp 2012 who constructed bubbling solutions to problem 0 1 for p 1

bubbling solutions for an exponential nonlinearity in r2 abstract we study the following boundary value problem 0 1 u u p 1 e u p 0 u 0 in u 0 on where is a bounded domain in r 2 with smooth boundary 0 is a small parameter and 0 p 2 we construct bubbling solutions to problem 0 1 using a lyapunov schmidt reduction procedure

boundary towers of layers for some supercritical problems we consider the supercritical problem u u p 1uin d u 0on d where d is a bounded smooth domain in rn and p is smaller than the th critical sobolev exponent 2n n 2n 2 with 1 n 3 we show that in some suitable torus like domains d there exists an arbitrary large number of sign changing solutions with alternate positive and negative layers which concentrate at different rates along a dimensional submanifold of d as p approaches 2n from below

boundary concentrating solutions for an anisotropic planar nonlinear neumann problem with large exponent abstract let be a bounded domain in r 2 with smooth boundary we study the following anisotropic elliptic problem a x u a x u 0 in u 0 in u u p on where denotes the outer unit normal vector to p 1 is a large exponent and a x is a positive smooth function we construct solutions of this problem which exhibit the accumulation of arbitrarily many boundary peaks at any isolated local maximum point of a x on the boundary

on the nonlinear matrix equation x a f x a q solutions and perturbation theory abstract in this paper the nonlinear matrix equation x a f x a q is discussed sufficient conditions for the existence and uniqueness of a positive semidefinite solution are derived also conditions are given under which the solution depends continuously on the matrices a and q

existence and multiplicity of solutions for nonlinear schrodinger equations with magnetic field and hartree type nonlinearities abstract we study the existence and multiplicity of solutions of the following nonlinear schrodinger equation in the presence of magnetic field i e a x 2 u v x u k x u p u p 2 u u h 1 r n where v r n r is the external potential k r n r is the response function and a a 1 a n r n r n is the magnetic vector potential associated to an external magnetic field b that is b c u r l a under suitable assumptions on the functions v x k x and a x if the parameter e is small enough we can prove the multiplicity of bound states for the equation by variational methods

a class of rank test procedures for censored survival data summary a class of linear rank statistics is proposed for the k sample problem with rightcensored survival data the class contains as special cases the log rank test mantel 1966 cox 1972 and a test essentially equivalent to peto peto s 1972 generalization of the wilcoxon test martingale theory is used to establish asymptotic normality of test statistics under the null hypotheses considered and to derive expressions for asymptotic relative efficiencies under contiguous sequences of alternative hypotheses a class of distributions is presented which corresponds to the class of rank statistics in the sense that for each distribution there is a statistic with some optimal properties for detecting location alternatives from that distribution some monte carlo results are displayed which present small sample behaviour

sample size formula for clustered survival data using weighted log rank statistics summary we present a simple sample size formula for weighted log rank statistics applied to clustered survival data with variable cluster sizes and arbitrary treatment assignments within clusters this formula is based on the asymptotic normality of weighted log rank statistics under certain local alternatives in the clustered data context we also provide consistent variance estimators the derived sample size formula reduces to schoenfeld s 1983 formula for cases of no clustering or independence within clusters simulation results verify control of the type i error and accuracy of the sample size formula use of the sample size formula in an event driven clinical trial design is illustrated using data from the early treatment diabetic retinopathy study

semiclassical limit for nonlinear schrodinger equations with electromagnetic fields abstract we study the existence of standing waves for a class of nonlinear schrodinger equations in r n with both an electric and a magnetic field under suitable non degeneracy assumptions on the critical points of an auxiliary function related to the electric field we prove the existence and the multiplicity of complex valued solutions in the semiclassical limit we show that in the semiclassical limit the presence of a magnetic field produces a phase in the complex wave but it does not influence the location of peaks of the modulus of these waves

iterative solution of two matrix equations we study iterative methods for finding the maximal hermitian positive definite solutions of the matrix equations x a x 1 a q and x a x 1 a q where q is hermitian positive definite general convergence results are given for the basic fixed point iteration for both equations newton s method and inversion free variants of the basic fixed point iteration are discussed in some detail for the first equation numerical results are reported to illustrate the convergence behaviour of various algorithms

sample size formula for the proportional hazards regression model a formula is derived for determining the number of observations necessary to test the equality of two survival distributions when concomitant information is incorporated this formula should be useful in designing clinical trials with a heterogeneous patient population schoenfeld 1981 biometrika 68 316 319 derived the asymptotic power of a class of statistics used to test the equality of two survival distributions that result is extended to the case where concomitant information is available for each individual and where the proportional hazards model holds the loss of efficiency caused by ignoring concomitant variables is also computed

on concentration of least energy solutions for magnetic critical choquard equations abstract in the present paper we consider the following magnetic nonlinear choquard equation i a x 2 u g x u u x u 2 u 2 2 u in r n u h 1 r n c where n 4 2 2 n n 2 0 n 0 0 is a parameter a x r n r n is a magnetic vector potential and g x is a real valued potential function on r n using variational methods we establish the existence of least energy solution under some suitable conditions moreover the concentration behavior of solutions is also studied as

improved methods and starting values to solve the matrix equations x a x 1a i iteratively the two matrix iterations x k 1 i a x 1 k a are known to converge linearly to a positive definite solution of the matrix equations x a x 1 a i respectively for known choices of x 0 and under certain restrictions on a the convergence for previously suggested starting matrices x 0 is generally very slow this paper explores different initial choices of x 0 in both iterations that depend on the extreme singular values of a and lead to much more rapid convergence further the paper offers a new algorithm for solving the minus sign equation and explores mixed algorithms that use newton s method in part

singularly perturbed nonlinear dirichlet problems with a general nonlinearity let be a bounded domain in r n n 3 with a boundary c 2 we consider the following singularly perturbed nonlinear elliptic problem on e 2 u u f u 0 u 0 on u 0 on where the nonlinearity f is of subcritical growth under rather strong conditions on f it has been known that for small e 0 there exists a mountain pass solution u e of above problem which exhibits a spike layer near a maximum point of the distance function d from as e 0 in this paper we construct a solution u e of above problem which exhibits a spike layer near a maximum point of the distance function under certain conditions on f which we believe to be almost optimal

infinitely many solutions for the nonlinear schrodinger equations with magnetic potentials in rn in this paper we study a nonlinear schrodinger equations with magnetic potentials in rn involving subcritical growth under some decaying and weak symmetry conditions of both electric and magnetic potentials we prove that the equation has infinitely many nonradial complex valued solutions by applying the finite reduction method

condition numbers for the nonlinear matrix equation and their statistical estimation abstract in this paper we consider four different types of condition numbers i e two normwise ones mixed and componentwise ones for the nonlinear matrix equation x a f x a q and derive their explicit expressions using the probabilistic spectral norm estimator and the statistical condition estimation method these condition numbers can be estimated with high reliability two numerical examples are provided to illustrate the difference between the normwise condition numbers and the mixed and componentwise condition numbers and the efficiency of the estimators respectively

sample size in cluster randomized trials with time to event as the primary endpoint in cluster randomized trials groups of individuals clusters are randomized to the treatments or interventions to be compared in many of those trials the primary objective is to compare the time for an event to occur between randomized groups and the shared frailty model well fits clustered time to event data members of the same cluster tend to be more similar than members of different clusters causing correlations as correlations affect the power of a trial to detect intervention effects the clustered design has to be considered in planning the sample size in this publication we derive a sample size formula for clustered time to event data with constant marginal baseline hazards and correlation within clusters induced by a shared frailty term the sample size formula is easy to apply and can be interpreted as an extension of the widely used schoenfeld s formula accounting for the clustered design of the trial simulations confirm the validity of the formula and its use also for non constant marginal baseline hazards findings are illustrated on a cluster randomized trial investigating methods of disseminating quality improvement to addiction treatment centers in the usa copyright 2012 john wiley sons ltd

a new type of concentration solutions for a singularly perturbed elliptic problem we prove the existence of positive solutions concentrating on some higher dimensional manifolds near the boundary of the domain for a nonlinear singularly perturbed elliptic problem

two iteration processes for computing positive definite solutions of the equation x a x na q abstract in the present paper we suggest two iteration methods for obtaining positive definite solutions of nonlinear matrix equation x a x n a q for the integer n 1 we obtain sufficient conditions for existence of the solutions for the matrix equation finally some numerical examples to illustrate the effectiveness of the algorithms and some remarks

mixed interior and boundary bubbling solutions for neumann problem in r 2 let be a bounded domain in r2 with smooth boundary we consider the following problem u u up 1eup u 0 in with neumann boundary condition u 0 on where 0 is a small parameter 0 p 2 and denotes the outer normal vector to we construct solutions of this problem with k interior bubbling points and l boundary bubbling points for any k 1 and l 1

positive solutions to x a bx 1 b abstract we study the positive semidefinite solutions to the matrix equation x a bx 1 b under the assumption that a 0 we show that a positive solution exists if and only if a certain block tridiagonal operator is positive in which case the solution is given by the shorted operator or generalized schur complement of that operator

a semilinear schrodinger equation in the presence of a magnetic field we consider the semilinear stationary schrodinger equation in a magnetic field i a 2u v x u g x u u in n where v is the scalar or electric potential and a is the vector or magnetic potential we study the existence of nontrivial solutions both in the critical and in the subcritical case respectively g x u u 2 2 and g x u c 1 u p 2 where 2 p 2 the results are obtained by variational methods for g critical we use constrained minimization and for subcritical g we employ a minimax type argument in the latter case we also study the existence of infinitely many geometrically distinct solutions

multiple blow up solutions for an anisotropic 2 dimensional nonlinear neumann problem we study the following boundary value problem begin aligned left begin array ll div a x nabla u a x u 0 u 0 quad mathrm in omega frac partial u partial nu lambda u p 1 e u p quad mathrm on partial omega end array right end aligned 0 1 where omega is a bounded domain in mathbb r 2 with smooth boundary nu is the unit outer normal vector of partial omega and a x is a positive smooth function in overline omega lambda 0 is a small parameter and 0 p 2 we construct bubbling solutions to problem 0 1 which blow up at points near a local maximum of a x on partial omega

on an electromagnetic schrodinger equation with critical growth abstract in this paper we consider the electromagnetic schrodinger problem i a 2 u v x u k x u 2 2 u in h a v 1 r n see the definition in section 1 according as whether the indefinite linear is resonant or nonresonant we structure different direct sums of h a v 1 r n then by applying the linking theorem twice to the corresponding functional we establish the existence results

the existence of non topological multivortex solutions in the relativistic self dual chern simons theory we construct a general type of multivortex solutions of the self duality equations the bogomol nyi equations of 2 1 dimensional relativistic chern simons model with the non topological boundary condition near infinity for such construction we use a perturbation argument around the explicit solutions of the liouville equation

necessary and sufficient conditions for the existence of a positive definite solution of the matrix equation x a x 1a q abstract we consider the problem of when the matrix equation x a x 1 a q has a positive definite solution here q is positive definite we study both the real and the complex case this equation plays a crucial role in solving a special case of the discrete time riccati equation we present both necessary and sufficient conditions for its solvability this result is obtained by using an analytic factorization approach moreover we present algebraic recursive algorithms to compute the largest and smallest the solution of the equation respectively finally we discuss the number of solutions

desingularization of vortices for the euler equation we study the existence of stationary classical solutions of the incompressible euler equation in the planes that approximate singular stationary solutions of this equation the construction is performed by studying the asymptotics of equation varepsilon 2 delta u varepsilon u varepsilon q frac kappa 2 pi log frac 1 varepsilon p with dirichlet boundary conditions and q a given function we also study the desingularization of pairs of vortices by minimal energy nodal solutions and the desingularization of rotating vortices

n vortex equilibria for ideal fluids in bounded planar domains and new nodal solutions of the sinh poisson and the lane emden fowler equations we prove the existence of equilibria of the n vortex hamiltonian in a bounded domain omega subset mathbb r 2 which is not necessarily simply connected on an arbitrary bounded domain we obtain new equilibria for n 3 or n 4 if has an axial symmetry we obtain a symmetric equilibrium for each n in mathbb n we also obtain new stream functions solving the sinh poisson equation delta psi rho sinh psi in with dirichlet boundary conditions for 0 small the stream function psi rho induces a stationary velocity field v rho solving the euler equation in on an arbitrary bounded domain we obtain velocitiy fields having three or four counter rotating vortices if has an axial symmetry we obtain for each n a velocity field v rho that has a chain of n counter rotating vortices analogous to the mallier maslowe row of counter rotating vortices in the plane our methods also yield new nodal solutions for other semilinear dirichlet problems in particular for the lane emden fowler equation delta u u p 1 u in with p large

on some p laplacian equation with electromagnetic fields and critical nonlinearity in ℝn in this paper we consider the existence and multiplicity of solutions for p laplacian equation with electromagnetic fields and critical nonlinearity in n ep p au v x up 2u up 2u h x up up 2u for x n where p au x div u ia x up 2 u ia x u by using lions second concentration compactness principle and concentration compactness principle at infinity to prove that the ps c condition holds locally and by variational method we show that this equation has at least one solution provided that e e for any m it has m pairs of solutions if e em where e and em are sufficiently small positive numbers

infinitely many solutions for nonlinear schrodinger equations with electromagnetic fields abstract in this paper we study the nonlinear schrodinger equation with electromagnetic fields i a y 2 u v y u u p 1 u u r n c where the vector a r a 1 r a 2 r a n r is such that a j r j 1 2 n is a real function on r and v r is a positive function on r 1 p n 2 n 2 if n 3 and 1 p if n 2 we prove that the equation has infinitely many non radial complex valued solutions under conditions h 1 and h 2 which are given in section 1

solutions for conformally invariant fractional laplacian equations with multi bumps centered in lattices abstract in this paper we consider the following nonlinear elliptic equation involving the fractional laplacian with critical exponent s u k x u n 2 s n 2 s u 0 in r n where s 0 1 and n 2 2 s k 0 is periodic in x 1 x k with 1 k n 2 s 2 under some natural conditions on k near a critical point we prove the existence of multi bump solutions where the centers of bumps can be placed in some lattices in r k including infinite lattices on the other hand to obtain positive solution with infinite bumps such that the bumps locate in lattices in r k the restriction on 1 k n 2 s 2 is in some sense optimal since we can show that for k n 2 s 2 no such solutions exist

perturbation analysis for solutions of x a x na q abstract the nonlinear matrix equations x a x n a q are investigated new perturbation estimates for positive definite solutions of these equations are derived the new results are illustrated by numerical examples

singularly perturbed problems in partial differential equations a survey this survey paper contains a surprisingly large amount of material on singularly perturbed partial differential equations and indeed can serve as an introduction to some of the ideas and methods of the singular perturbation theory starting from prandtl s work a large amount of work has been done in the area of singular perturbations this paper limits its coverage to some standard singular perturbation models considered by various workers and the methods developed by numerous researchers after 1980 2000 in this review we have covered singularly perturbed partial differential equations about odes the survey has already been done by us see m k kadalbajoo k c patidar appl math comput 130 2002 457 510

multiple blow up solutions for an exponential nonlinearity with potential in r 2 abstract we study the following boundary value problem 0 1 u a x u p 1 e u p 0 u 0 in u 0 on where is a bounded domain in r 2 with smooth boundary 0 is a small parameter the function a x 0 is a smooth potential and the exponent p satisfies 0 p 2 we construct a family of solutions to problem 0 1 which blows up as 0 at some points of which stay outside the zero set of a x we relate the number of possible blow up points with the zero set of a x

infinitely many solutions for schrodinger maxwell equations with indefinite sign subquadratic potentials in this paper we deal with multiplicity of solutions of a class of sublinear schrodinger maxwell equations du v x u fu f x u inr 3 d f u 2 lim x f x 0 inr 3 under more relaxed assumptions on v and f we establish some existence criteria to guarantee that the above problem has at least one or infinitely many nontrivial solutions by using the genus properties in critical theory

multi bump solutions for the nonlinear schrodinger poisson system in this paper we study a kind of nonlinear schrodinger poisson system with a parameter e for any positive integer m we prove that there exists e m 0 such that for 0 e e m the equation has an m bump positive solution under some suitable conditions as a consequence the equation has more and more multi bump positive solutions as e 0

nonexistence of multi bubble solutions to some elliptic equations on convex domains abstract we prove the nonexistence of multi bubble solutions for several types of problems on smooth bounded convex domains problems we study include the liouville equation u e u in u 0 on in r 2 where 0 is a parameter and the almost critical problem u u n 2 n 2 e u 0 in u 0 on in higher dimensions where e 0

sample size calculation for the weighted rank statistics with paired survival data this paper introduces a sample size calculation method for the weighted rank test statistics with paired two sample survival data our sample size formula requires specification of joint survival and censoring distributions for modelling the distribution of paired survival variables we may use a paired exponential survival distribution that is specified by the marginal hazard rates and a measure of dependency also in most trials randomizing paired subjects the subjects of each pair are accrued and censored at the same time over an accrual period and an additional follow up period so that the paired subjects have a common censoring time under these practical settings the design parameters include type i and type ii error probabilities marginal hazard rates under the alternative hypothesis correlation coefficient accrual period or accrual rate and follow up period if pilot data are available we may estimate the survival distributions from them but we specify the censoring distribution based on the specified accrual trend and the follow up period planned for the new study through simulations the formula is shown to provide accurate sample sizes under practical settings real studies are taken to demonstrate the proposed method

concentrating solutions for a planar elliptic problem involving nonlinearities with large exponent abstract we consider the boundary value problem u u p 0 in a bounded smooth domain in r 2 with homogeneous dirichlet boundary condition and p a large exponent we find topological conditions on which ensure the existence of a positive solution u p concentrating at exactly m points as p in particular for a nonsimply connected domain such a solution exists for any given m 1

small sample statistical condition estimates for general matrix functions a new condition estimation procedure for general matrix functions is presented that accurately gauges sensitivity by measuring the effect of random perturbations at the point of evaluation in this procedure the number of extra function evaluations used to evaluate the condition estimate determines the order of the estimate that is the probability that the estimate is off by a given factor is inversely proportional to the factor raised to the order of the method the transpose free nature of this new method allows it to be applied to a broad range of problems in which the function maps between spaces of different dimensions this is in sharp contrast to the more common power method condition estimation procedure that is limited in the usual case where the frechet derivative is known only implicitly to maps between spaces of equal dimension a group of examples illustrates the flexibility of the new estimation procedure in handling a variety of problems and types of sensitivity estimates such as mix

sign changing blow up solutions for henon type elliptic equations with exponential nonlinearity abstract we study the existence of sign changing solutions with multiple concentration to the following boundary value problem u e 2 x 2 e u e u in u 0 on where 0 is a smooth bounded domain in r 2 containing the origin e 0 is a small parameter in particular we prove that if 1 then a nodal solution exists with a number of mixed positive and negative blow up points up to 4

a prescribed scalar curvature type equation almost critical manifolds and multiple solutions we present an asymptotic analysis for a perturbed prescribed scalar curvature type equation a major consequence is a non existence result in low dimension conversely we prove an existence result in higher dimensions to this aim we develop a general finite dimensional reduction procedure for perturbed variational functionals the general principle can be useful to discuss some other nonlinear elliptic pde with sobolev critical growth in bounded domains

sample size estimation for survival outcomes in cluster randomized studies with small cluster sizes summary we present a method for computing sample size for cluster randomized studies involving a large number of clusters with relatively small numbers of observations within each cluster for multivariate survival data only the marginal bivariate distribution is assumed to be known the validity of this assumption is also discussed

bubbling solutions for moser trudinger type equations on compact riemann surfaces abstract we study an elliptic equation related to the moser trudinger inequality on a compact riemann surface s g g u u e u 2 1 s s u e u 2 d v g 0 in s s u d v g 0 where 0 is a small parameter s is the area of s g is the laplace beltrami operator and d v g is the area element given any integer k 1 under general conditions on s we find a bubbling solution u which blows up at exactly k points in s as 0 when s is a flat two torus in rectangular form we find that either seven or nine families of such solutions do exist for k 2 in particular in any square flat two torus actually nine families of bubbling solutions with two bubbling points do exist if s is a riemann surface with non constant robin s function then at least two bubbling solutions with k 1 exists

statistical condition estimation for linear systems the standard approach to measuring the condition of a linear system compresses all sensitivity information into one number thus a loss of information can occur in situations in which the standard condition number with respect to inversion does not accurately reflect the actual sensitivity of a solution or particular entries of a solution it is shown that a new method for estimating the sensitivity of linear systems addresses these difficulties the new procedure measures the effects on the solution of small random changes in the input data and by properly scaling the results obtains reliable condition estimates for each entry of the computed solution moreover this approach which is referred to as small sample statistical condition estimation is no more costly than the standard 1 norm or power method 2 norm condition estimates and it has the advantage of considerable flexibility for example it easily accommodates restrictions on or structure associated with allowable perturbations the method also has a rigorous statistical theory available for the probability of accuracy of the condition estimates however it gives no estimate of an approximate null vector for nearly singular systems the theory of this approach is discussed along with several illustrative examples