positive radial solutions and non radial bifurcation for semilinear elliptic equations in annular domains abstract we discuss the existence and multiplicity of positive radial solutions and the non radial bifurcation of u f u 0 in and u 0 on where is an annular domain of r n n 2 we prove that if f u 0 for u 0 and lim u f u u then there exists 0 such that there are at least two positive radial solutions for each 0 at least one for and none for if f 0 0 lim u 0 f u u 1 and uf u 1 e f u for u 0 e 0 then there exists a variational solution for 0 1 where 1 is the least eigenvalue of if f 0 0 lim u 0 f u u 0 and lim u f u u then there exists at least one positive radial solution for any 0 we obtain some precise multiplicity results for narrow annulus and show that the non radial bifurcation occurs if the growth of f u is rapid enough as u

perturbation of doubly periodic solution branches with applications to the cahn hilliard equation abstract in this paper we prove the existence of doubly periodic solutions of certain nonlinear elliptic problems on r 2 and study the geometry of their nodal domains in particular we will show that if we perturb a nonlinear elliptic equation exhibiting a small amplitude doubly periodic solution whose nodal domains form a checkerboard pattern then the perturbed equation will have a unique nearby solution which is still doubly periodic but for which the nodal line structure breaks up moreover we indicate what can happen if we start with a large amplitude doubly periodic solution whose nodal domains form a checkerboard pattern and we relate these solutions to the cahn hilliard equation and spinodal decomposition

symmetry breaking for δu 2δe u 0 on a disk with general boundary conditions we discuss the radially symmetric solutions and the symmetry breaking of the equation u 2 e u 0 in d and u b u n 0 on d where d is the unit disk in 2 0 and b is a constant we prove that for any b 0 such that there are exactly two radially symmetric solutions for 0 one for and none for b for where m is a positive integer there are b k 1 m such that the equation has symmetry breaking at k b on the lower branch of radially symmetric solutions

covariance controllers for linear continuous time systems abstract necessary and sufficient conditions are given for the existence of a controller of specified order n c which will assign a specified matrix value x to the covariance of the closed loop system if a solution exists the entire set of stabilizing controllers that assign x are parameterized by a certain skew symmetric matrix

on the number of positive solutions of weakly non linear elliptic equations when a parameter is large on etudie les solutions positives du probleme u f u dans u 0 sur ou est un domaine borne lisse de r n f r r est c 1 et est grand

on the zero set of the wave function in superconductivity we consider the ginzburg landau functional in a multiply connected planar domain with enclosed magnetic flux particular attention is given to the zero set of the order parameter we show that there exist applied fields for which the zero set is of codimension 1

a nash game approach to mixed h sub 2 h sub spl infin control the established theory of nonzero sum games is used to solve a mixed h sub 2 h sub spl infin control problem our idea is to use the two pay off functions associated with a two player nash game to represent the h sub 2 and h sub spl infin criteria separately we treat the state feedback problem and we find necessary and sufficient conditions for the existence of a solution both the finite and infinite time problems are considered in the infinite horizon case we present a full stability analysis the resulting controller is a constant state feedback law characterized by the solution to a pair of cross coupled riccati equations which may be solved using a standard numerical integration procedure we begin our development by considering strategy sets containing linear controllers only at the end of the paper we broaden the strategy sets to include a class of nonlinear controls it turns out that this extension has no effect on the necessary and sufficient conditions for the existence of a solution or on the nature of the controllers

a bifurcation diagram of solutions to an elliptic equation with exponential nonlinearity in higher dimensions we consider the following semilinear elliptic equation where b 1 is the unit ball in d d 3 0 and p 0 firstly following merle and peletier we show that there exists an eigenvalue p such that has a solution p w p satisfying lim x 0 w p x secondly we study a bifurcation diagram of regular solutions to it follows from the result of dancer that has an unbounded bifurcation branch of regular solutions that emanates from u 0 0 here using the singular solution we show that the bifurcation branch has infinitely many turning points around p when 3 d 9 we also investigate the morse index of the singular solution in the d 11 case

lqg control with an h sup infinity performance bound a riccati equation approach an lqg linear quadratic gaussian control design problem involving a constraint on h sup infinity disturbance attenuation is considered the h sup infinity performance constraint is embedded within the optimization process by replacing the covariance lyapunov equation by a riccati equation whose solution leads to an upper bound on l sub 2 performance in contrast to the pair of separated riccati equations of standard lqg theory the h sup infinity constrained gains are given by a coupled system of three modified riccati equations the coupling illustrates the breakdown of the separation principle for the h sup infinity constrained problem both full and reduced order design problems are considered with an h sup infinity attenuation constraint involving both state and control variables an algorithm is developed for the full order design problem and illustrative numerical results are given

h 2 optimal control with an h constraint the state feedback case in this paper we consider a mixed hz optimal control problem it is assumed that the plant as well as the feedback controller are finite dimensional and linear time invariant and that the plant state is available for feedback more specifically among all the state feedback controllers that minimize the he norm of a closed loop transfer matrix we give necessary and sufficient conditions for the existence of a controller that also satisfies a prescribed h norm bound on some other closed loop transfer matrix when these conditions are met the solution to the above problem is also a global solution to the contrained optimization problem of minimizing an he norm performance measure subject to an h norm constraint we also give state space formulae for computing the solutions some easily checkable sufficient conditions for the existence of solutions are given finally we give an example in which all solutions to the constrained optimization problem are necessarily dynamic i e there is no static gain solution even though plant state is available for feedback

uniqueness and nonuniqueness for positive radial solutions of δu f u r 0 on considere le probleme de l unicite pour les solutions positives du probleme de dirichlet non lineaire u f u x 0 dans u 0 est une boule ou un anneau de r n et f 0 est superlineaire en u et satisfaisait f 0 x 0

a class of invariant procedures for assessing multivariate normality summary distribution theory pertaining to a class of invariant procedures for assessing multivariate normality is described a cramer von mises type statistic belonging to this class is investigated in greater detail its asymptotic distribution is tabulated and compared with a monte carlo simulation of its finite sample distribution and a set of quadrivariate data is examined with it

global branches of non radially symmetric solutions to a semilinear neumann problem in a disk let d r 2 be a disk and let f c 3 we assume that there is a r such that f a 0a ndf a 0 in this article we are concerned with the neumann problem u f u 0i nd u 0o n d we show the following there are unbounded continuums consisting of non radially symmetric solutions emanating from the second and third eigenvalues if f u u u u p 1 a 1 or if f is of bistable type then the unbounded branches emanating from non principal eigenvalues are unbounded in the positive direction of the branch emanating from the second eigenvalue is unique near the bifurcation point up to rotation the main tool of this article is the zero level set the nodal curve of u and ux

a test to determine the multivariate normality of a data set a test is described for multivariate normality that is useful in pattern recognition the test is based on the friedman rafsky 1979 multivariate extension of the wald wolfowitz runs test the test data are combined with a multivariate swarm of points following the normal distribution generated with mean vector and covariance matrix estimated from the test data the minimal spanning tree of this resultant ensemble of points is computed and the count of the interpopulation edges in the minimal spanning tree is used as a test statistic the simulation studied both the null case of the test and one simple deviation from normality two conclusions are made from this study first the test can be conservatively applied by using the asymptotic normality of the test statistic even for small sample sizes second the power of the test appears reasonable especially in high dimensions monte carlo experiments were performed to determine if the test is reliable in high dimensions with moderate sample size the method is compared to other such tests available in the literature

generalized riccati equations for the full and reduced order mixed norm h 2 h abstract this paper considers the mixed norm h 2 h standard problem specifically an lqg control design problem involving a constraint on h disturbance attenuation is addressed it is shown that the h 2 h dynamic compensator gains are completely characterized via coupled riccati lyapunov equations the principal result involves sufficient conditions for characterizing full and reduced order controlleers that satisfy bounds on both h 2 and h performance costs as a special case of this unified result we obtain the full order h solution to the standard control problem the pure reduced order h solution with no h 2 contribution further extensions include nonstrictly proper dynamics a direct transmission term from disturbances to h performance variables cross weighting and sensor noise plant disturbance correlation and a treatment of the pure reduced order h control problem

global bifurcations of solutions of emden fowler type equation δu x λf u x on an annulus in rn n 3 abstract in this article we study global and symmetry breaking bifurcations of solutions of emden fowler type elliptic differential equations as the main tool we use a topological invariant which is suitable for study of global bifurcations and continuations of critical points of equivariant functionals namely we will apply degree for s 1 equivariant orthogonal maps defined in rybicki 1994 nonlinear anal tma 23 83 102

adaptive smoothing and density based tests of multivariate normality abstract methods of adaptive smoothing of density estimates where the amount of smoothing applied varies according to local features of the underlying density are investigated the difficulties of applying taylor series arguments in this context are explored simple properties of the estimates are investigated by numerical integration and compared with the fixed kernel approach optimal smoothing strategies based on the multivariate normal distribution are derived as an application of these techniques two tests of multivariate normality one based on integrated squared error and one on entropy are developed and some power calculations are carried out

covariance controllers a new parametrization of the class of all stabilizing controllers abstract the class of all stabilizing controllers is parametrized in terms of a system matrix called the one at a time covariance and an arbitrary skew hermitian matrix the new parametrization is in closed form for any specified controller order and provides a deterministic version of the stochastic problem of covariance assignment

nodal sets for groundstates of schrodinger operators with zero magnetic field in non simply connected domains we investigate nodal sets of magnetic schrodinger operators with zero magnetic field acting on a non simply connected domain in 2 for the case of circulation 1 2 of the magnetic vector potential around each hole in the region we obtain a characterisation of the nodal set and use this to obtain bounds on the multiplicity of the groundstate for the case of one hole and a fixed electric potential we show that the first eigenvalue takes its highest value for circulation 1 2

eigenvalues variations for aharonov bohm operators we study how the eigenvalues of a magnetic schrodinger operator of aharonov bohm type depend on the singularities of its magnetic potential we consider a magnetic potential defined everywhere in 2 except at a finite number of singularities so that the associated magnetic field is zero on a fixed planar domain we define the corresponding magnetic hamiltonian with dirichlet boundary conditions and study its eigenvalues as functions of the singularities we prove that these functions are continuous and in some cases even analytic we sketch the connection of this eigenvalue problem to the problem of finding spectral minimal partitions of the domain

on non radially symmetric bifurcation in the annulus we discuss the radially symmetric solutions and the non radially symmetric bifurcation of the semilinear elliptic equation u 2 eu 0 in and u 0 on where x r2 a2 0 a the upper branch of radial solutions has a non radially symmetric bifurcation symmetry breaking at each k a k 1 as a 0 the radial solutions will tend to the radial solutions on the disk and 0 a 1 the critical number on the disk

steady state kalman filtering with an h error bound abstract an estimator design problem is considered which involves both l 2 least squares and h worst case frequency domain aspects specifically the goal of the problem is to minimize an l 2 state estimation error criterion subject to a prespecified h constraint on the state estimation error the h estimation error constraint is embedded within the optimization process by replacing the covariance lyapunov equation by a riccati equation whose solution leads to an upper bound on the l 2 state estimation error the principal result is a sufficient condition for characterizing fixed order i e full and reduced order estimators with bounded l 2 and h estimation error the sufficient condition involves a system of modified riccati equations coupled by an oblique projection i e idempotent matrix when the h constraint is absent the sufficient condition specializes to the l 2 state estimation result given in 2

mixed norm h 2 h and entropy covariance control a convex optimisation approach in this paper we develop a covariance control problem to address a tradeoff between h 2 performance and h disturbance attenuation in particular we formulate a mixed norm h 2 h and entropy

existence of positive nonradial solutions for nonlinear elliptic equations in annular domains we study the existence of positive nonradial solutions of equation a f u 0 in sia u 0 on d la where s x 6 w a 2 and is positive and superlinear at both 0 and oo we use a bifurcation method to show that there is a nonradial bifurcation with mode k at ak 0 1 for any positive integer k if is subcritical and for large k if is supercritical when is subcritical then a nehari type variational method can be used to prove that there exists a e 0 1 such that for any a e a 1 the equation has a nonradial solution on 0 a

global solution curves for self similar equations abstract we consider positive solutions of a semilinear dirichlet problem u f u 0 for x 1 u 0 when x 1 on a unit ball in r n for four classes of self similar equations it is possible to parameterize the entire global solution curve through the solution of a single initial value problem this allows us to derive results on the multiplicity of solutions and on their morse indices in particular we easily recover the classical results of d d joseph and t s lundgren 6 on the gelfand problem surprisingly the situation turns out to be different for the generalized gelfand problem where infinitely many turns are possible for any space dimension n 3 we also derive detailed results for the equation modeling electrostatic micro electromechanical systems mems in particular we easily recover the main result of z guo and j wei 4 and we show that the morse index of the solutions increases by one at each turn we also consider the self similar henon s equation

on multiple eigenvalues for aharonov bohm operators in planar domains abstract we study multiple eigenvalues of a magnetic aharonov bohm operator with dirichlet boundary conditions in a planar domain in particular we study the structure of the set of the couples position of the pole circulation which keep fixed the multiplicity of a double eigenvalue of the operator with the pole at the origin and half integer circulation we provide sufficient conditions for which this set is made of an isolated point the result confirms and validates a lot of numerical simulations available in preexisting literature

symmetry breaking bifurcation for the one dimensional liouville type equation abstract the two point boundary value problem for the one dimensional liouville type equation u x l e u 0 x 1 1 u 1 u 1 0 is considered where 0 and l 0 in this paper a symmetry breaking result is obtained by using the morse index the problem u x l u 1 p 0 x 1 1 u 1 u 1 0 is also considered where 0 l 0 p 1 and p 1 l 4

maximal entropy solutions under prescribed mass and energy abstract we consider a non local elliptic equation with exponential nonlinearity closely related to the mean field liouville equation the motivation for this equation is a variational entropy maximization problem under prescribed mass and energy we provide an unconditional existence proof in case of electrostatic repulsive self interaction and conditional existence and uniqueness in dimension two in the case of gravitational attractive self interaction

semi classical analysis of schrodinger operators and compactness in the neumann problem abstract we study the asymptotic behavior in a semi classical limit of the first eigenvalues i e the groundstate energies of a class of schrodinger operators with magnetic fields and the relationship of this behavior with compactness in the neumann problem on hartogs domains in c 2

nodal sets for solutions of elliptic equations on etudie sur un domaine connexe r n l ensemble zero u 1 0 d une solution u d une equation elliptique a ij d i d j u b j d j u cu 0 ou a ij b j c sont bornes et a ij est continu on demontre que la mesure de hausdorff a n 1 dimensions de u 1 0 est finie dans un voisinage d un point x 0 ou u a un ordre fini d annulation

upper and lower bounds for an eigenvalue associated with a positive eigenvector when an eigenvector of a semibounded operator is positive we show that a remarkably simple argument allows to obtain upper and lower bounds for its associated eigenvalue this theorem is a substantial generalization of barta type inequalities and can be applied to non necessarily purely quadratic hamiltonians an application for a magnetic hamiltonian is given and the case of a discrete schrodinger operator is also discussed it is shown how this approach leads to some explicit bounds on the ground state energy of a system made of an arbitrary number of attractive coulombian particles

the hot spots conjecture for a certain class of planar convex domains we prove the hot spots conjecture of rauch five problems an introduction to the qualitative theory of partial differential equations partial differential equations and related topics program tulane univ new orleans la 1974 lecture notes in mathematics springer berlin 1975 vol 446 pp 355 369 for a certain class of planar convex domains specifically we show that an eigenfunction corresponding to the lowest nonzero eigenvalue of the neumann laplacian on attains its maximum minimum at points in one class of domains is the planar convex domain satisfying diam 2 1 378 when is a disk diam 2 1 273 hence this condition indicates that is a nearly circular planar convex domain however symmetries of the domain are not assumed we give other sufficient conditions for domains for which the conjecture holds we also give a new isoperimetric inequality

uniqueness of global positive solution branches of nonlinear elliptic problems has been studied in many publications the first global description was given in 19 nonetheless there were open questions some of which are answered in this paper we motivate some of these questions now for a suitable class of smooth fimctions f ir ir problem 1 1 has infinitely many solutions for 2 1 this was shown in 15 via the method of suband supersolutions which also gives an order of these solutions 0 0 which is not excluded in 15 there is also a global branch continuum c of solutions 2 u bifurcating from the trivial solution line 2 0 at 2 21 where 1 21 f 0 is the principal eigenvalue of the linearization

nonradial maximizers for a henon type problem and symmetry breaking bifurcations for a liouville gel fand problem with a vanishing coefficient we study the radiality and nonradiality of the maximizers of begin aligned sup left int d x alpha e u 1 dx u in h 0 1 d int d left nabla u right 2dx beta 2 right end aligned from a viewpoint of the bifurcation diagram of the euler lagrange equation begin aligned delta u beta 2 frac x alpha e u int d x alpha ue udx 0 hbox in d qquad u 0 hbox on partial d end aligned where d x in mathbb r 2 x 1 we show that the maximizers are nonradial if begin aligned beta sqrt 8 pi left alpha 2 log frac 2 alpha 4 alpha frac alpha 4 2 right end aligned and that the maximizers are radial if alpha 0 or beta 0 is small we also give the bifurcation diagrams of the liouville gel fand problem with a vanishing coefficient begin aligned delta u lambda x alpha e u 0 hbox in d qquad u 0 hbox on partial d end aligned and of the associated mean field equation the analysis is based on exact solutions and explicit calculations

compactness in the neumann problem magnetic schrodinger operators and the aharonov bohm effect abstract compactness of the neumann operator in the neumann problem is studied for weakly pseudoconvex bounded hartogs domains in two dimensions a nonsmooth domain is constructed for which condition p fails to hold yet the kohn laplacian still has compact resolvent the main result in contrast is that for smoothly bounded hartogs domains the well known sufficient condition p is equivalent to compactness the analyses of compactness and condition p boil down to the asymptotic behavior of the lowest eigenvalues of two related sequences of schrodinger operators one with a magnetic field and one without parametrized by a fourier variable resulting from the hartogs symmetry the nonsmooth example is based on the aharonov bohm phenomenon of quantum physics for smooth domains not satisfying p we prove that there always exists an exceptional sequence of fourier variables for which the aharonov bohm effect is weak and thence that compactness fails to hold this sequence can be very sparse so that the lack of compactness is due to a rather subtle effect

symmetry breaking for solutions of semilinear elliptic equations with general boundary conditions we study the bifurcation of radially symmetric solutions of f u 0 onn balls into asymmetric ones we show that ifu satisfies homogeneous neumann boundary conditions the asymmetric components in the kernel of the linearized operators can have arbitrarily high dimension for general boundary conditions we prove some theorems which give bounds on the dimensions of the set of asymmetric solutions and on the structure of the kernels of the linearized operators

on the existence of positive radial solutions for nonlinear elliptic equations in annular domains abstract we study the existence of positive radially symmetric solutions of u g x f u 0 in annulus with dirichlet dirichlet neumann boundary conditions we show that the equation has a positive radial solution on any annulus if f and f are positive and f is superlinear at 0 and

on the aharonov bohm operators with varying poles the boundary behavior of eigenvalues we consider a magnetic schrodinger operator with magnetic field concentrated at one point the pole of a domain and half integer circulation and we focus on the behavior of dirichlet eigenvalues as functions of the pole although the magnetic field vanishes almost everywhere it is well known that it affects the operator at the spectral level the aharonov bohm effect phys rev 2 115 485 491 1959 moreover the numerical computations performed in bonnaillie noel et al anal pde 7 6 1365 1395 2014 noris and terracini indiana univ math j 59 4 1361 1403 2010 show a rather complex behavior of the eigenvalues as the pole varies in a planar domain in this paper in continuation of the analysis started in bonnaillie noel et al anal pde 7 6 1365 1395 2014 noris and terracini indiana univ math j 59 4 1361 1403 2010 we analyze the relation between the variation of the eigenvalue and the nodal structure of the associated eigenfunctions we deal with planar domains with dirichlet boundary conditions and we focus on the case when the singular pole approaches the boundary of the domain then the operator loses its singular character and the k th magnetic eigenvalue converges to that of the standard laplacian we can predict both the rate of convergence and whether the convergence happens from above or from below in relation with the number of nodal lines of the k th eigenfunction of the laplacian the proof relies on the variational characterization of eigenvalues together with a detailed asymptotic analysis of the eigenfunctions based on an almgren type frequency formula for magnetic eigenfunctions and on the blow up technique

gaussian mixture modeling by exploiting the mahalanobis distance in this paper the expectation maximization em algorithm for gaussian mixture modeling is improved via three statistical tests the first test is a multivariate normality criterion based on the mahalanobis distance of a sample measurement vector from a certain gaussian component center the first test is used in order to derive a decision whether to split a component into another two or not the second test is a central tendency criterion based on the observation that multivariate kurtosis becomes large if the component to be split is a mixture of two or more underlying gaussian sources with common centers if the common center hypothesis is true the component is split into two new components and their centers are initialized by the center of the old component candidate for splitting otherwise the splitting is accomplished by a discriminant derived by the third test this test is based on marginal cumulative distribution functions experimental results are presented against seven other em variants both on artificially generated data sets and real ones the experimental results demonstrate that the proposed em variant has an increased capability to find the underlying model while maintaining a low execution time

feedback and optimal sensitivity model reference transformations multiplicative seminorms and approximate inverses in this paper the problem of sensitivity reduction by feedback is formulated as an optimization problem and separated from the problem of stabilization stable feedback schemes obtainable from a given plant are parameterized salient properties of sensitivity reducing schemes are derived and it is shown that plant uncertainty reduces the ability of feedback to reduce sensitivity the theory is developed for input output systems in a general setting of banach algebras and then specialized to a class of multivariable time invariant systems characterized by n times n matrices of h infty frequency response functions either with or without zeros in the right half plane the approach is based on the use of a weighted seminorm on the algebra of operators to measure sensitivity and on the concept of an approximate inverse approximate invertibility of the plant is shown to be a necessary and sufficient condition for sensitivity reduction an indicator of approximate invertibility called a measure of singularity is introduced the measure of singularity of a linear time invariant plant is shown to be determined by the location of its right half plane zeros in the absence of plant uncertainty the sensitivity to output disturbances can be reduced to an optimal value approaching the singularity measure in particular if there are no right half plane zeros sensitivity can be made arbitrarily small the feedback schemes used in the optimization of sensitivity resemble the lead lag networks of classical control design some of their properties and methods of constructing them in special cases are presented

existence uniqueness and nondegeneracy of positive solutions of semilinear elliptic equations we study positive solutions of the dirichlet problem a u x f u x 0 x d u x o x e ad where d is an n ball we find necessary and sufficient conditions for solutions to be nondegenerate we also give some new existence and uniqueness theorems in this paper we study positive solutions of the dirichlet problem au x f u x o x f2

monotonicity of the first eigenvalue and the global bifurcation diagram for the branch of interior peak solutions abstract let b r n n 1 be a unit ball and d r m be a bounded domain we study the global branch consisting of interior single peak solutions of the elliptic neumann problem u u u p 0 in b u 0 on b and the monotonicity of the first eigenvalue along the branch for large where 1 p p s n 2 n 2 n 3 n 1 2 when the domain is replaced with b d using this monotonicity we show that the branch of solutions concentrating on 0 d has secondary bifurcation points for n 1 and p 2 an integer we determine the global bifurcation diagram by showing the monotonicity of the time map the period function the monotonicity of the first eigenvalue along the whole branch is also proved for n 1 and p 3

pattern formation of the stationary cahn hilliard model we investigate critical points of the free energy of the cahn hilliard model of a binary alloy under the constraint of a constant mass the domain is the unit square minimizers of the energy without interfacial energy term are given by a decomposition of the two components of the alloy but the interfaces between the components are arbitrary specic patterns are only formed if an interfacial energy term is present we select such patterns of minimizers by an approximation of se quences of conditionally critical points of the free energy when the interfacial energy term tends to zero this is what we call pattern formation of the stationary cahn hilliard model mathematically it is a singular limit process we obtain the conditionally critical points by a global bifurcation analysis of the euler lagrange equation for the free energy where the mass is the bifurcation pa rameter and where the constant homogeneous mixtures give the trivial solutions by using characteristic symmetries and monotonicities of the bifurcating solutions we show that singular limits exist for all masses in the so called spinodal region and that they are minimizers of the free energy without interfacial energy term a binary alloy in a vessel can be described by a function u ir as follows u x 2 0 1 means that the mixture contains u x 100 of one component at x2 the energy density of the alloy is modelled by w u where w is a two well potential figure 1

local structure of bifurcation curves for nonlinear sturm liouville problems abstract we consider the nonlinear bifurcation problem arising in population dynamics and nonlinear schrodinger equation u t f u t u t 0 t i 0 1 u 0 u 1 0 where 0 is a parameter we mainly treat the case where f u u u p p 1 and establish the precise asymptotic expansion formulas for the bifurcation curve near the bifurcation point 2 in l q framework together with the result of the global behavior of the bifurcation curve we understand completely the structure of the bifurcation curve we also consider the nodal solution u n of the equation u t u t u t p 1 u t with u 0 u 0 and establish an asymptotic expansion formula for with respect to the gradient norm of the solution associated with as n 2

lower bounds for the first eigenvalue of the magnetic laplacian abstract we consider a riemannian cylinder endowed with a closed potential 1 form a and study the magnetic laplacian a with magnetic neumann boundary conditions associated with those data we establish a sharp lower bound for the first eigenvalue and show that the equality characterizes the situation where the metric is a product we then look at the case of a planar domain bounded by two closed curves and obtain an explicit lower bound in terms of the geometry of the domain we finally discuss sharpness of this last estimate

resonance phenomenon for a gelfand type problem we consider positive radially symmetric solutions of u eu 1 inb u 0on b where b is the unit ball in rn n 3 and 0 is a parameter we establish infinite multiplicity of regular solutions for 3 n 9 and some and we obtain a bound for the morse index and the number of solutions when n 10

non existence of a secondary bifurcation point for a semilinear elliptic problem in the presence of symmetry abstract we give two sufficient conditions for a branch consisting of non trivial solutions of an abstract equation in a banach space not to have a secondary bifurcation point when the equation has a certain symmetry when the nonlinearity f is of allen cahn type for instance f u u u 3 we apply these results to an unbounded branch consisting of non radially symmetric solutions of the neumann problem on a disk d r 2 u f u 0 in d u 0 on d and emanating from the second eigenvalue we show that the maximal continuum containing this branch is homeomorphic to r s 1 and that its closure is homeomorphic to r 2

solutions of nonlinear planar elliptic problems with triangle symmetry abstract in this paper we continue the study of the nodal domain structure of doubly periodic solutions of certain nonlinear elliptic problems initiated in fifeet al physica d100 1997 257 278 more precisely we consider small amplitude solutions of u f u 0 in r 2whose nodal domains consist of equilateral triangles tiling the plane if this equation is suitably perturbed then for genericfwe prove the existence of unique nearby solutions with triangle symmetry and show how their nodal domain geometry breaks up furthermore we treat the non generic rectangular cases which had to be excluded in fifeet al physica d100 1997 257 278 as well as other nodal domain structures

on assessing multivariate normality summary we discuss procedures for assessing multivariate normality based on properties of radii and angles of the multivariate normal distribution and suggest that different procedures may be efficiently combined in order to achieve omnibus tests the assumption of population multivariate normality underlies many multivariate data analytic procedures unfortunately however there are relatively few formal methods available for assess ing the validity of this underlying assumption gnanadesikan 1977 section 5 4 2 and cox and small 1978 have extensively reviewed various formal and informal procedures proposed for testing the multinormality hypothesis and it is not our purpose to survey the procedures anew rather we suggest in this note that different characteristics of multivariate normality can be examined with different procedures which may thereafter be combined to obtain omnibus tests for the multinormality hypothesis

some new tests for multivariate normality a family of statistics is presented that can be used for testing goodness of fit to a parametric family these statistics include mardia s measure of multivariate kurtosis and moore and stubblebine s test for multivariate normality the asymptotic distribution of the statistics is found under mild hypotheses on the parametric family and in the case of multivariate normality the distribution is shown to be independent of the true parameter a class of tests for multivariate normality is presented and the performance of two such tests in the bivariate case is found in simulations

mixed h sub 2 h sub infinity control a convex optimization approach the problem of finding an internally stabilizing controller that minimizes a mixed h sub 2 h sub infinity performance measure subject to an inequality constraint on the h sub infinity norm of another closed loop transfer function is considered this problem can be interpreted and motivated as a problem of optimal nominal performance subject to a robust stability constraint both the state feedback and output feedback problems are considered it is shown that in the state feedback case one can come arbitrarily close to the optimal even over full information controllers mixed h sub 2 h sub infinity performance measure using constant gain state feedback moreover the state feedback problem can be converted into a convex optimization problem over a bounded subset of n n and n q where n and q are respectively the state and input dimensions real matrices using the central h sub infinity estimator it is shown that the output feedback problem can be reduced to a state feedback problem in this case the dimension of the resulting controller does not exceed the dimension of the generalized plant

the gel fand problem for the biharmonic operator we study stable and finite morse index solutions of the equation delta 2 u e u if the equation is posed in mathbb r n we classify radial stable solutions we then construct nonradial stable solutions and we prove that unlike the corresponding second order problem no liouville type theorem holds unless additional information is available on the asymptotics of solutions at infinity thanks to this analysis we prove that stable solutions of the equation on a smoothly bounded domain supplemented with navier boundary conditions are smooth if and only if n leqq 12 we find an upper bound for the hausdorff dimension of their singular set in higher dimensions and conclude with an a priori estimate for solutions of bounded morse index provided they are controlled in a suitable morrey norm

regularity for the fractional gelfand problem up to dimension 7 abstract we study the problem s u e u in a bounded domain r n where is a positive parameter more precisely we study the regularity of the extremal solution to this problem our main result yields the boundedness of the extremal solution in dimensions n 7 for all s 0 1 whenever is for every i 1 n convex in the x i direction and symmetric with respect to x i 0 the same holds if n 8 and s 0 28206 or if n 9 and s 0 63237 these results are new even in the unit ball b 1

the zero set of the order parameter in two dimensions we survey the possible zero sets of the order parameter wavefunction in two dimensional domains certain structures are feasible including the well known abrikosov lattice the giant vortex and the line cut in multiply connected domains other proposed structures such as the vortex ring and the noninteger vortex point are shown to be impossible

nodal sets of solutions of equations involving magnetic schrodinger operator in three dimensions it is well known that the complexity of the nodal set of a function mainly comes from the singular set on which both the function and the gradient vanish the singular set of a real valued solution of a linear elliptic equation has been well investigated for a complex valued solution of a linear equation involving a magnetic schrodinger operator the structure of the nodal set has not been well investigated yet excepted in the two dimensional case in this paper we extend the arguments of garafalo and lin indiana univ math j 35 245 268 1986 and of han indiana univ math j 43 983 1002 1994 to show that the singular set of such a solution in a three dimensional domain is countably 1 rectifiable the functions considered in this paper include the order parameter in the ginzburg landau theory of superconductivity and the eigenfunctions of the magnetic schrodinger operator

proof of a conjecture for the perturbed gelfand equation from combustion theory abstract in this paper we prove that when the domain b is a two dimensional ball for all small positive e the solution curve u of the perturbed gelfand equation u eu 1 eu with u b 0 is exactly s shaped this settles a long standing conjecture on the exact number of solutions to this equation

symmetry breaking for positive solutions of semilinear elliptic equations in a recent interesting paper gidas ni and nirenberg 2 proved that positive solutions of the dirichlet problem for second order semi linear elliptic equations on balls must themselves be spherically symmetric functions here we consider the bifurcation problem for such solutions specifically we investigate the ways in which the symmetric solution can bifurcate into a nonsymmetric solution when this happens we say that the symmetry breaks to carry out this program we rely on certain results in 6 where we studied the kernel of the associated linearized operator this enables us to give some necessary conditions for symmetry to break we also find a class of functionsf where these conditions are also sufficient see equation 18 below the problem is of course to show that when zero comes into the spectrum of the linearized operator and our conditions are fulfilled bifurcation into the non radial direction actually occurs this is done by showing first that the bifurcation curve is a smooth manifold near the bifurcation point and then appealing to a theorem of vanderbauwhede 8 which gives sufficient conditions for bifurcation to occur in the presence of symmetries

on the gap between h 2 and entropy performance measures in h control design abstract recent papers have considered the problem of minimizing an entropy functional subject to an h performance constraint since the entropy is an upper bound for the h 2 cost there remains a gap between entropy minimization and h 2 minimization in this paper we consider a generalized cost functional involving both h 2 and entropy aspects this approach thus provides a means for optimizing h 2 performance within h control design

schrodinger operators in the twentieth century this paper reviews the past fifty years of work on spectral theory and related issues in nonrelativistic quantum mechanics

a differential method for bounding the ground state energy for a wide class of hamiltonians a novel method for obtaining lower and upper bounds for the lowest energy is presented unlike perturbative or variational techniques this method does not involve the computation of any integral a normalization factor or a matrix element it just requires the determination of the absolute minimum and maximum in the whole configuration space of the local energy associated with a normalizable trial function the calculation of the norm is not needed after a general introduction the method is applied to three non integrable systems the asymmetric annular billiard the many body spinless coulombian problem the hydrogen atom in a constant and uniform magnetic field being more sensitive than the variational methods to any local perturbation of the trial function this method can be used to systematically improve the energy bounds with a local skilled analysis an algorithm relying on this method can therefore be constructed and an explicit example for a one dimensional problem is given

existence of radial solutions for nonlinear elliptic equations with gradient terms in annular domains abstract this paper is concerned with the problem of elliptic equations in annular domains involving derivative terms using nonlinear analysis methods some results regarding existence of solutions are established more precisely at least one radial solution is obtained basing on schauder s fixed point theorem and contraction mapping theorem respectively some iterative technique is also applied which allows us to overcome difficulties that the nonlinearity is related to derivative terms

topological degree for a mean field equation on riemann surfaces we consider the following mean field equations 0 1 where m is a compact riemann surface with volume 1 h is a positive continuous function on m is a real number 0 2 and where is a bounded smooth domain h is a c1 positive function on and based on our previous analytic work 14 we prove among other things that the degree counting formula for 0 1 is given by for 8m 8 m 1 2003 wiley periodicals inc

relations between maximum entropy h control and combined h lqg control abstract the maximum entropy h control problem and a combined h lqg control problem are presented in a unified way and shown to be equivalent in the case of a full order controller and equalized h lqg weights the result links two separate approaches which have distinctly different methods of solution only two decoupled algebraic riccati equations are needed to state the controller maximum value of the entropy and minimum value of the auxiliary cost

magnetic bottles in connection with superconductivity abstract motivated by the theory of superconductivity and more precisely by the problem of the onset of superconductivity in dimension two a lot of papers devoted to the analysis in a semi classical regime of the lowest eigenvalue of the schrodinger operator with magnetic field have appeared recently here we mention the works by bernoff sternberg lu pan and del pino felmer sternberg this partially recovers questions analyzed in a different context by the authors around the question of the so called magnetic bottles our aim is to analyze the former results to treat them in a more systematic way and to improve them by giving sharper estimates of the remainder in particular we improve significantly the lower bounds and as a by product we solve a conjecture proposed by bernoff sternberg concerning the localization of the ground state inside the boundary in the case with constant magnetic fields

oscillatory patterns in the ginzburg landau model driven by the aharonov bohm potential abstract we consider the aharonov bohm magnetic potential and study the transition from normal to superconducting solutions within the ginzburg landau model of superconductivity we obtain oscillatory patterns which are consistent with the little parks effect we study also the same problem but for a regularization of the aharonov bohm potential which leads to an interesting aharonov bohm like magnetic field and we prove that the transition between superconducting and normal solutions is not monotone too our results explore a mechanism to derive the aharonov bohm magnetic potential starting from a step magnetic field thereby presenting a new aspect of magnetic steps besides their favoring of the celebrated edge states

sharp boundary behavior of eigenvalues for aharonov bohm operators with varying poles abstract in this paper we investigate the behavior of the eigenvalues of a magnetic aharonov bohm operator with half integer circulation and dirichlet boundary conditions in a bounded planar domain we establish a sharp relation between the rate of convergence of the eigenvalues as the singular pole is approaching a boundary point and the number of nodal lines of the eigenfunction of the limiting problem i e of the dirichlet laplacian ending at that point the proof relies on the construction of a limit profile depending on the direction along which the pole is moving and on an almgren type monotonicity argument for magnetic operators