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

approximate gcd and its application to ill conditioned algebraic equations abstract we describe two algorithms of approximate gcd greatest common divisor for polynomials with coefficients of floating point numbers one is for univariate polynomials and the other for multivariate polynomials the algorithms are careful extensions of the conventional euclidean algorithm and they are applied to solving ill conditioned algebraic equations ill conditioned univariate algebraic equations have multiple or close roots and we can ill conditioned system of algebraic equations which have an approprimate common divisor conventional numeric root finding methods such as newton s method give no satisfactory result for such ill conditioned systems by using an approximate gcd algorithm for multivariate polynomials the system is transformed into a well conditioned problems by a combination of algebraic and numeric methods we call such algorithms hybrid algorithms this paper shows the importance and possible fruitfulness of the hybrid algorithm

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

a non linear structure preserving matrix method for the low rank approximation of the sylvester resultant matrix a non linear structure preserving matrix method for the computation of a structured low rank approximation s f g of the sylvester resultant matrix s f g of two inexact polynomials f f y and g g y is considered in this paper it is shown that considerably improved results are obtained when f y and g y are processed prior to the computation of s f g and that these preprocessing operations introduce two parameters these parameters can either be held constant during the computation of s f g which leads to a linear structure preserving matrix method or they can be incremented during the computation of s f g which leads to a non linear structure preserving matrix method it is shown that the non linear method yields a better structured low rank approximation of s f g and that the assignment of f y and g y is important because s f g may be a good structured low rank approximation of s f g but s g f may be a poor structured low rank approximation of s g f because its numerical rank is not defined examples that illustrate the differences between the linear and non linear structure preserving matrix methods and the importance of the assignment of f y and g y are shown

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

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

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

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

on the diagonalization of quantum birkhoff gustavson normal form an application to quantum mechanics of one of classical perturbation theory methods the birkhoff gustavson normal form bgnf is described in the quantum case it results in the van vleck perturbation theory performed upon wick normal ordered operators algebraic aspects of this procedure and formal construction of invariants integrals of motion for a perturbed system are considered it turned out that a larger set of such operators existed in the quantum mechanics rather than in the classical one it is demonstrated that according to general results of the quantum mechanical perturbation theory the quantum bgnf may always be diagonalized and two formal processes for such diagonalization are constructed in the opposite case the classical bgnf is in general nondiagonalizable this reflects the fact that the classical perturbation theory cannot handle a system with two or more resonances possible reasons for such different behavior of two very close in spirit perturbation procedures are discussed results of the described procedure entirely performed upon the wick normal ordered operators are equivalent to those of rayleigh schrodinger perturbation expansion

common factor detection and estimation the paper introduces a noniterative maximum likelihood ml based approach for the detection and estimation of the greatest common divisor of two given polynomials the coefficients of which are assumed to have a normal distribution with a consistently estimable covariance matrix it then goes on to show how this approach can be used to derive a noniterative ml method for estimating the orders and parameters of difference equation models with autoregressive residuals numerical examples are included to illustrate the performance of the proposed methodology

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

an improved non linear method for the computation of a structured low rank approximation of the sylvester resultant matrix this paper reports on improvements to recent work on the computation of a structured low rank approximation of the sylvester resultant matrix s f g of two inexact polynomials f f y and g g y specifically it has been shown in previous work that these polynomials must be processed before a structured low rank approximation of s f g is computed the existing algorithm may still however yield a structured low rank approximation of s f g but not a structured low rank approximation of s g f which is unsatisfactory moreover a structured low rank approximation of s f g must be equal to apart from permutations of its columns a structured low rank approximation of s g f but the existing algorithm does not guarantee the satisfaction of this condition this paper addresses these issues by modifying the existing algorithm such that these deficiencies are overcome examples that illustrate these improvements are shown

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

birkhoff gustavson normal form in classical and quantum mechanics a quantum analogue of the classical transformations to the birkhoff gustavson normal form is derived with unitary transformations taking over the role of canonical transformations the author finds a striking similarity to lie transformations in the classical case the author suggests that the quantum normal form is identical to the rayleigh schrodinger perturbation series and thus usually only asymptotic to the eigenvalues the convergence question of both classical and quantum normal forms is discussed in some detail

on operators commutative with all invariants for a harmonic oscillator with commensurable frequencies for the d dimensional quantum mechanical harmonic oscillator with r commensurability relation between frequencies d independent operators commutative with the oscillator hamiltonian and all other commutative with hamiltonian operators are constructed these operators form a basis in the centre of the algebra of invariants integrals of motion for the quantum mechanical oscillator with commensurable frequencies not all of these d operators have classical analogues a classical harmonic oscillator only has d r commutative with all other invariants these classical integrals first appeared in the gustavson work on the birkhoff normalization operators with such properties are of interest for the perturbation theory since any of them may be at least formally continued to become the invariant of the perturbed hamiltonian

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

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

qr factoring to compute the gcd of univariate approximate polynomials we present a stable and practical algorithm that uses qr factors of the sylvester matrix to compute the greatest common divisor gcd of univariate approximate polynomials over spl ropf x or spl copf x an approximate polynomial is a polynomial with coefficients that are not known with certainty the algorithm of this paper improves over previously published algorithms by handling the case when common roots are near to or outside the unit circle by splitting and reversal if necessary the algorithm has been tested on thousands of examples including pairs of polynomials of up to degree 1000 and is now distributed as the program qrgcd in the snap package of maple 9

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

the calculation of the degree of an approximate greatest common divisor of two polynomials the calculation of the degree of an approximate greatest common divisor agcd of two inexact polynomials f y and g y is a non trivial computation because it reduces to the estimation of the rank loss of a resultant matrix r f g this computation is usually performed by placing a threshold on the small singular values of r f g but this method suffers from disadvantages because the numerical rank of r f g may not be defined or the noise level imposed on the coefficients of f y and g y may not be known or it may only be known approximately this paper addresses this problem by considering two methods for estimating the degree of an agcd of f y and g y such that knowledge of the noise level is not required the first method involves the calculation of the smallest angle between two subspaces that are apparent from the structure of the sylvester resultant matrix s f g and the second method uses the theory of subresultant matrices which are derived from s f g by the deletion of some of its rows and columns the two methods are compared computationally on non trivial polynomials