monotonicity properties of the neumann heat kernel in the ball abstract a well known conjecture of r laugesen and c morpurgo asserts that the diagonal of the neumann heat kernel of the unit ball u r n is a strictly increasing radial function in this paper we use probabilistic arguments to settle this conjecture and to prove some inequalities for the neumann heat kernel in the ball

galerkin s method for ordinary differential equations subject to generalized nonlinear boundary conditions where xk belongs to tc a finite dimensional subspace of cc l 11 r il ii and i qj is a projection mapping cc l 11 r i i x r onto a subspace of c 1 11 r 11 ii having the same dimension as rck there is a vast collection of literature concerning the application of galerkin s method to differential equations 1 3 12 16231 we make a special reference to the papers of cesari 4 5 and urabe 22 23 our results provide an extension and a unification of previous results of urabe that were conderned with the determination of periodic solutions to systems of ordinary differential equations 22 and with the solvability of 112 0022 0396192 3 00

projection methods for nonlinear boundary value problems where l d l c x y is a linear map n x y is nonlinear and x and y are banach spaces recently cesari and mckenna s proved that if the underlying spaces possess a schauder basis or more generally they have what the authors call property l it is theoretically possible to establish a finite dimensional alternative problem for 1 3 their paper unities previous results 12 131 that established conditions under which 1 3 could be reduced to finite 282 0022 0396 85 3 00

minimizing neumann fundamental tones of triangles an optimal poincare inequality abstract the first nonzero eigenvalue of the neumann laplacian is shown to be minimal for the degenerate acute isosceles triangle among all triangles of given diameter hence an optimal poincare inequality for triangles is derived the proof relies on symmetry of the neumann fundamental mode for isosceles triangles with aperture less than 3 antisymmetry is proved for apertures greater than 3

hot spots conjecture for a class of acute triangles we show that the hot spots conjecture of j rauch holds for acute triangles if one of the angles is not larger than pi 6 more precisely we show that the second neumann eigenfunction on those acute triangles has no maximum or minimum inside the domain we first simplify the problem by showing that absence of critical points on two sides implies no critical points inside a triangle this result applies to any acute triangle and might help prove the conjecture for arbitrary acute triangles then we show that there are no critical points on two sides assuming one small angle we also establish simplicity for the smallest positive neumann eigenvalue for all non equilateral acute triangles this result was already known for obtuse triangles and it fails for the equilateral case

a counterexample to the hot spots conjecture we construct a counterexample to the hot spots conjecture there exists a bounded connected planar domain with two holes such that the second eigenvalue of the laplacian in that domain with neumann boundary conditions is simple and such that the corresponding eigenfunction attains its strict maximum at an interior point of that domain

eigenvalue inequalities for the laplacian with mixed boundary conditions abstract inequalities for the eigenvalues of the negative laplacian subject to mixed boundary conditions on polyhedral and more general bounded domains are established the eigenvalues subject to a dirichlet boundary condition on a part of the boundary and a neumann boundary condition on the remainder of the boundary are estimated in terms of either dirichlet or neumann eigenvalues the results complement several classical inequalities between dirichlet and neumann eigenvalues due to polya payne levine and weinberger friedlander and others

grand antiprism and quaternions vertices of the four dimensional 4d semi regular polytope the grand antiprism and its symmetry group of order 400 are represented in terms of quaternions with unit norm it follows from the icosian representation of the e8 root system which decomposes into two copies of the root system of h4 the symmetry of the grand antiprism is a maximal subgroup of the coxeter group w h4 it is the group aut h2 h 2 which is constructed in terms of 20 quaternionic roots of the coxeter diagram h2 h 2 the root system of h4 represented by the binary icosahedral group i of order 120 constitutes the regular 4d polytope 600 cell when its 20 quaternionic vertices corresponding to the roots of the diagram h2 h 2 are removed from the vertices of the 600 cell the remaining 100 quaternions constitute the vertices of the grand antiprism we give a detailed analysis of the construction of the cells of the grand antiprism in terms of quaternions the dual polytope of the grand antiprism has also been constructed

noncrystallographic coxeter group h4 in e8 the e8 lattice is constructed in terms of icosians by matching two sets of f4 lattices described by quaternions embedding the noncrystallographic group h4 into the weyl group w e8 has been described using matrix generators with an emphasis on the relevant coxeter elements the conjugacy classes of h4 in terms of quaternions and the characters of the two four dimensional irreducible representations are explicitly calculated

on the hot spots conjecture of j rauch rodrigo banuelos research partially supported by nsf grant dms 9400854 krzysztof burdzy s research partially supported by nsf grant dms 9322689

maximizing neumann fundamental tones of triangles we prove sharp isoperimetric inequalities for neumann eigenvalues of the laplacian on triangular domains the first nonzero neumann eigenvalue is shown to be maximal for the equilateral triangle among all triangles of given perimeter and hence among all triangles of given area similar results are proven for the harmonic and arithmetic means of the first two nonzero eigenvalues

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

symmetries of the octonionic root system of e8 octonionic root system of e8 is decomposed as the 9 sets of hurwitz integers each set satisfying the binary tetrahedral group structure and the 12 sets of quaternionic units each set obeying the dicylic group structure of order 12 this fact is used to prove that the automorphism group of the octonionic root system of e7 is the finite subgroup of g2 of order 12 096 an explicit 7 7 matrix realization of which is constructed possible use of the octonionic representation of the e6 root system as orbifolds and the relevance of the binary tetrahedral structures with the statistical mechanics models are suggested

division algebras with integral elements pairing two elements of a given division algebra furnished with a multiplication rule leads to an algebra of higher dimension restricted by 8 this fact is used to obtain the roots of so 4 and sp 2 from the roots or 1 of su 2 and the weights or 1 2 of its spinor representation the root lattice of so 8 described by 24 integral quaternions are obtained by pairing two sets of roots of sp 2 the root system of f4 is constructed in terms of 24 integral and 24 half integral quaternions the root lattice of e8 expressed as 240 integral octonions are obtained by pairing two sets of roots of f4 twenty four integral quaternions of so 8 forming a discrete subgroup of su 2 are shown to be the automorphism group of the root lattices of so 8 f4 and e8 the roots of maximal subgroups so 16 e7 su 2 e6 su 3 su 9 and su 5 su 5 of e8 are identified with a simple method subsets of the discrete subgroup of su 2 leaving maximal subgroups of e8 are obtained constructions of e8 root lattice with integral octonions in seven distinct ways are made magic squares of integral lattices of goddard nahm olive ruegg and schwimmer are derived possible physical applications are suggested

vertex operators for non simply laced algebras a vertex operator construction is given for the level one representations of the affine kac moody algebras associated with non simply laced finite dimensional lie algebras using free boson and interacting fermion fields the fermion fields are constructed explicitly and a detailed discussion is given of the theory of the cocycles necessary for this and other vertex operator constructions the construction is related in detail to the folding of dynkin diagrams and a generalisation of it yields freudenthal s magic square

scaling coupling of reflecting brownian motions and the hot spots problem we introduce a new type of coupling of reflecting brownian motions in smooth planar domains called scaling coupling we apply this to obtain monotonicity properties of antisymmetric second neumann eigenfunctions of convex planar domains with one line of symmetry in particular this gives the proof of the hot spots conjecture for some known types of domains and some new ones

e8 lattice with icosians and z5 symmetry a simple method for the construction of the e8 root system with icosians is suggested it is confronted with the root system of e8 obtained by integral octonions embeddings of the maximal subgroups su 5 su 5 and su 3 e6 in e8 with respective five fold and three fold symmetries are discussed

quaternionic roots of so 8 so 9 f4 and the related weyl groups the root systems of so 8 so 9 and f4 are constructed by quaternions triality manifests itself as permutations of pure quaternion units e1 e2 and e3 it is shown that the automorphism groups of the associated root systems are the finite subgroups of o 4 generated by left right actions of unit quaternions on the root systems the relevant finite groups of quaternions the binary tetrahedral and binary octahedral groups play essential roles in the construction of the weyl groups and their conjugacy classes the relations between the dynkin indices standard orthogonal vector and the quaternionic weights are obtained

g2 2 as the automorphism group of the octonionic root system of e7 a simple method is suggested for the construction of the seven dimensional representation of the adjoint chevalley group g2 2 the automorphism group of the octonionic root system of e the maximal subgroups of g2 2 preserving the octonionic root systems of the maximal subgroups of e are identified possible implications in physics are discussed dedicated to feza giirsey on the occasion of his 70th birthday

on the solvability of nonlinear sturm liouville problems abstract the focus of this paper is the study of nonlinear differential equations subject to general non local boundary conditions to establish sufficient conditions for the existence of solutions we use properties of the nonlinearities and their relationship with the eigenvalues of an associated linear sturm liouville problem

blow up problems for a semilinear heat equation with large diffusion abstract we consider the blow up problem of a semilinear heat equation u t d u u p in 0 t d u x t 0 on 0 t d u x 0 x 0 in where is a bounded smooth domain in r n t d 0 d 0 and p 1 we study the blow up time the location of the blow up set and the blow up profile of the blow up solution for sufficiently large d in particular we prove that for almost all initial data if d is sufficiently large then the solution blows up only near the maximum points of the orthogonal projection of the initial data from l 2 onto the second neumann eigenspace

location of hot spots in thin curved strips abstract the maxima and minima of neumann eigenfunctions of thin tubular neighbourhoods of curves on surfaces are located in terms of the maxima and minima of neumann eigenfunctions of the underlying curves in particular the hot spots conjecture for a new large class of domains possibly non convex and non euclidean is proved

scalar discrete nonlinear multipoint boundary value problems abstract in this paper we provide sufficient conditions for the existence of solutions to scalar discrete nonlinear multipoint boundary value problems by allowing more general boundary conditions and by imposing less restrictions on the nonlinearities we obtain results that extend previous work in the area of discrete boundary value problems debra l etheridge jesus rodriguez periodic solutions of nonlinear discrete time systems appl anal 62 1996 119 137 debra l etheridge jesus rodriguez scalar discrete nonlinear two point boundary value problems j difference equ appl 4 1998 127 144

nonlocal boundary value problems for discrete systems abstract our goal in this paper is to provide sufficient conditions for the existence of solutions to discrete nonlinear systems subject to multipoint boundary conditions the criteria we present depends on the size of the nonlinearity and the set of solutions to the corresponding linear homogeneous boundary value problems our analysis is based on the lyapunov schmidt procedure and brouwer s fixed point theorem the results presented extend the previous work of d etheridge and j rodriguez 1996 1998 5 6 and j rodriguez and p taylor 2007 18 19

on neumann eigenfunctions in lip domains atar s research partially supported by the fund for the promotion of research at the technion burdzy gratefull acknowledges the hospitality and financial support of technion israel and institut mittag leffler sweden this research was partially supported by nsf grant dms 0071486 and isf grant 12 98

an alternative method for boundary value problems with large nonlinearities where g r r is continuously differentiable and b and d are nonsingular 12 x n matrices historically lyapunov schmidt and the alternative method are able to reduce the problem to a finite dimensional one provided the norm of the nonlinear term is sufficiently small in the region of interest that is the weakly nonlinear case recently the finite dimensional reduction for ordinary differential equations with periodic boundary conditions was accomplished 161 in this paper a constructive technique is developed that allows us to reduce 1 1 to a finite dimensional problem

on nonlinear discrete boundary value problems mo x 1 x n 0 1 4 where n is a fixed positive integer 8 represent n by n constant matrices a t is an n by n matrix for each te 0 l h t belongs to ir for each nonnegative integer t u is in r 1 is a real parameter f and g are continuously differentiable f r i ir and g r i r for systems 1 1 1 2 we prove that if the linear system x t l a l x t h t 1 5 subject to

the hot spots conjecture for domains with two axes of symmetry consider a convex planar domain with two axes of symmetry we show that the maximum and minimum of a neumann eigenfunction with lowest nonzero eigenvalue occur at points on the boundary only we deduce j rauch s hot spots conjecture in the following form if the initial temperature distribution is not orthogonal to the first nonzero eigenspace then the point at which the temperature achieves its maximum tends to the boundary in fact the maximum point reaches the boundary in finite time if the boundary has positive curvature results of this type have already been proved by bafiuelos and burdzy bb using the heat equation and probabilistic methods to deform initial conditions to eigenfunctions we introduce here a new technique based on deformation of the domain an advantage of our method is that it works even in the case of multiple eigenvalues on the way toward our results we prove monotonicity properties for neumann eigenfunctions for symmetric domains that need not be convex and deduce a sharp comparison of eigenvalues with the dirichlet problem of independent interest