on s shaped bifurcation curves for multiparameter positone problems abstract we study the existence of multiple positive solutions to the two point boundary value problem u x f u x o u 0 0 u 1 u 1 where 0 0 here f is a smooth function such that f 0 on 0 r for some 0 r in particular we consider the case when f is initially convex and then concave we discuss sufficient conditions for the existence of at least three positive solutions for a certain range independent of of we apply our results to the nonlinearity f u exp cu c u c 4 which arises in combustion theory and to the nonlinearity f u u exp c 1 u 0 fixed c 4 4 which arises in chemical reactor theory

on s shaped bifurcation curves where sz c r n 2 1 is bounded with smooth boundary aq readers interested in the derivation of the problem are referred to the work by bebernen l particularly section 2 relevant work on the analysis of this problem has been done by brown et al 2 shivaji 3 61 hastings and mcleod 7 sattinger 8 parks 9 and tam lo clearly u 0 is a solution of 1 l when 2 0 it follows from the contraction mapping theorem that l l always has a solution for a sufficiently small moreover let

higher order derivatives and perturbation bounds for determinants the first derivative of the determinant function is given by the well known jacobi s formula we obtain three different expressions for all higher order derivatives norms of these derivatives are then evaluated exactly

existence and multiplicity of positive solutions for classes of singular elliptic pdes we consider the boundary value problem u g u u in u 0on where rn is a bounded domain is a nonnegative function in l such that 0 on some subset of of positive measure and g 0 r is continuous we establish the existence of three positive solutions when g 0 0 positone the graph of s 1g s is roughly s shaped and 0 we also prove that there exists at least one positive solution when g 0 0 and 0 1 we employ the method of sub super solutions to prove our results

s shaped bifurcation curves in ecosystems abstract we consider the existence of multiple positive solutions to the steady state reaction diffusion equation with dirichlet boundary conditions of the form u u u 2 k c u 2 1 u 2 x u 0 x here u div u is the laplacian of u 1 is the diffusion coefficient k and c are positive constants and r n is a smooth bounded region with in c 2 this model describes the steady states of a logistic growth model with grazing in a spatially homogeneous ecosystem it also describes the dynamics of the fish population with natural predation in this paper we discuss the existence of multiple positive solutions leading to the occurrence of an s shaped bifurcation curve we prove our results by the method of sub supersolutions

a proposed upstream weight numerical method for simulating pollutant transport in groundwater an alternative upstream weighting numerical method is proposed to simulate the transport process in groundwater under a variety of conditions the discretized system of the solute transport equations is derived from the integral form that expresses the balance of mass in each element an upstream weight that depends on the local peclet number is directly added to the basis functions when convective transport dominates the dispersive transport this technique can eliminate the oscillation of solutions efficiently with only a nominal increase of computation effort over that of the general linear triangular galerkin finite element method the accuracy of the proposed numerical model is tested against two classical problems for which analytical solutions exist the agreement is nearly exact an additional numerical example is used to illustrate the applicability of this model

perturbation bounds for the operator absolute value it is shown how to estimate the norms of the derivatives of all orders of the map that takes an invertible operator to the positive part in its polar decomposition using this perturbation bounds of any order can be obtained for this map

quantum analysis non commutative differential and integral calculi a new scheme of quantum analysis namely a non commutative calculus of operator derivatives and integrals is introduced this treats differentiation of an operator valued function with respect to the relevant operator in a banach space in this new scheme operator derivatives are expressed in terms of the relevant operator and its inner derivation explicitly derivatives of hyperoperators are also defined some possible applications of the present calculus to quantum statistical physics are briefly discussed

variation of induced linear operators abstract let v be an n dimensional inner product space let be an irreducible character of the symmetric group s m and let v be the symmetry class of tensors associated with it let a be a linear operator on v and let k a be the operator it induces on v we obtain an explicit expression for the norm of the derivative of the map a k a in terms of the singular values of a two special cases of this problem antisymmetric and symmetric tensor products have been studied earlier and our results reduce to the earlier ones in these cases

global bifurcation of steady state solutions in this paper we study the bifurcation of steady state solutions of a reaction diffusion equation in one space variable the steady state solutions satisfy the equation a f u 0 on the interval l 0 or 0 for some nonzero function c c a for the neumann problem we use entirely different techniques to prove that s is never critical for any cubic polynomial j this implies at once that the neumann problem can have at most one nonconstant solution having a given number of maxima or minima this solution is necessarily strongly nondegenerate in the sense that zero is not contained in the spectrum of the linearized operator see 2 for the dirichlet problem the situation is far more complicated and the bifurcation diagrams undergo qualitative changes depending on the positions of the roots off for example fig 1 shows the bifurcation diagram for the positive solutions for two different cubic functions of the form f u a u u b u c it is interesting to note however that in every case we study there are at most three solutions to the dirichlet problem for each

trefftz method fitting boundary conditions the paper presents various ways of fitting the boundary conditions in the t complete functions method the authors point out the distinct advantages of the orthogonal collocation in comparison to the equidistant collocation and the integral fit the convergence of the collatz error measures and the conditioning of the solution matrices are investigated in detail

an upstream weight multiple cell balance finite element method for solving three dimensional convection dispersion equations an upstream weight multiple cell balance finite element method is developed to simulate the three dimensional contaminant transport in aquifers the method combines the concepts of local mass balance and finite element approximations when convective transport dominates the dispersive transport the proposed technique effectively eliminates the oscillation of numerical solutions with only a nominal increase of formulation effort over that of the general galerkin finite element method two numerical examples for which analytical solutions exist are used to verify the proposed numerical solutions an additional example is presented to demonstrate the practical applicability of the proposed method

classes of reaction diffusion equations where a parameter influences the equation as well as the boundary condition abstract we study positive solutions to steady state reaction diffusion equations of the form u f u u u 0 where 0 is a bounded domain in r n n 1 with smooth boundary u is the outward normal derivative of u c 0 is strictly increasing such that 0 0 and f c 2 0 r 0 with 0 r 0 if r 0 we assume f c 2 0 r 0 with f r 0 0 and f s 0 for s r 0 and if r 0 we assume lim s f s 0 and lim s f s s 0 sublinear at note here that the parameter influences both the equation and the boundary condition we discuss existence nonexistence multiplicity and uniqueness results for the cases when a f 0 0 b f 0 0 and c f 0 0 we obtain existence and multiplicity results by the method of sub super solutions and uniqueness results by comparison principles and a priori estimates

exact multiplicity of solutions and s shaped bifurcation curves for the p laplacian perturbed gelfand problem in one space variable abstract we study exact multiplicity of positive solutions and the bifurcation curve of the p laplacian perturbed gelfand problem from combustion theory p u x exp a u a u 0 1 x 1 u 1 u 1 0 where p 1 p y y p 2 y p u is the one dimensional p laplacian 0 is the frank kamenetskii parameter u x is the dimensionless temperature and the reaction term f u exp a u a u is the temperature dependence obeying the arrhenius reaction rate law we find explicitly a a p 0 such that if the activation energy a a then the bifurcation curve is s shaped in the u plane more precisely there exist 0 such that the problem has exactly three positive solutions for exactly two positive solutions for and and a unique positive solution for 0 and

contractive maps on normed linear spaces and their applications to nonlinear matrix equations in this paper we will give necessary and sufficient conditions under which a map is a contraction on a certain subset of a normed linear space these conditions are already well known for maps on intervals in r using the conditions and banach s fixed point theorem we can prove a fixed point theorem for operators on a normed linear space the fixed point theorem will be applied to the matrix equation x in a f x a where f is a map on the set of positive definite matrices induced by a real valued map on 0 this will give conditions on a and f under which the equation has a unique solution in a certain set we will consider two examples of f in detail in one example the application of the fixed point theorem is the first step in proving that the equation has a unique positive definite solution under the conditions on a

a theorem on reversed s shaped bifurcation curves for a class of boundary value problems and its application abstract we study the bifurcation curves of positive solutions of the boundary value problem u x f e u x 0 1 x 1 u 1 u 1 0 where f e u g u e h u e r is a bifurcation parameter and functions g h c 0 c 2 0 satisfy five hypotheses presented herein assuming these hypotheses on fixed g and h we prove that the bifurcation curve is reverse s shaped on the e u plane that is the bifurcation curve has exactly two turning points at some points e u e and e u e such that e e and u e u e in addition we prove that e 0 thus the exact number of positive solutions can be precisely determined by the values of e and e we give an application to the two parameter bifurcation problem u x 1 u 2 e u 3 0 1 x 1 u 1 u 1 0 where e are two positive bifurcation parameters some new results are obtained

unitarily invariant norm inequalities for elementary operators involving g1 operators abstract in this paper motivated by perturbation theory of operators we present some upper bounds for f a x g b x in terms of a x b x and f a x g b x in terms of a x x b where a b are g 1 operators is a unitarily invariant norm and f g are certain analytic functions further we find some new upper bounds for the schatten 2 norm of f a x x g b several special cases are discussed as well

collocation methods and the solution of conduction convection problems the upwinded finite element method was introduced for the solution of conduction convection problems by considering the model problem u ku 0 k 0 subject to the boundary conditions u 0 1 u 1 0 in this paper we consider the solution of this problem by the collocation method using piecewise cubic hermite polynomials and a uniform grid we consider the collocation points to be arbitrary parameters which enable us to express the collocation points in terms of the upwinding parameter of the upwinded finite element method various choices of the collocation points including values which lead to complete accuracy in the nodal function values as well as the nodal values of the derivatives are discussed finally we discuss the application of these points to slightly more general problems numerical results illustrating the behaviour of these points are given

semilinear elliptic boundary value problems in chemical reactor theory abstract this paper is devoted to the study of semilinear elliptic boundary value problems arising in chemical reactor theory which obey the simple arrhenius rate law and newtonian cooling we prove that ignition and extinction phenomena occur in the stable steady temperature profile at some critical values of a dimensionless heat evolution rate

differentiation of operator functions and perturbation bounds given a smooth real function f on the positive half line consider the induced map on the set of positive hilbert space operators let f k be the kth derivative of the real function f and e5 dkf the kth frechet derivative of the operator map f we identify large classes of functions for which for k 1 2 this reduction of a noncommutative problem to a commutative one makes it easy to obtain perturbation bounds for several operator maps our techniques serve to illustrate the use of a formalism for quantum analysis that is like the one recently developed by m suzuki

matrix factorizations and their perturbations there are several ways in which a matrix can be factorized as a product of two special matrices these factorizations are often used in numerical analysis and many perturbation bounds useful in such analysis have been proved by various authors in this paper a simple method which leads to these results and several new ones is discussed in the last section related results on the lipschitz continuity of the matrix absolute value are surveyed

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

first and second order perturbation bounds for the operator absolute value norms of the first and the second derivatives of the map that takes an invertible hilbert space operator to its absolute value are estimated using this first and second order perturbation bounds for this map are obtained

multiple positive solutions of the one dimensional p laplacian in this work we investigate the existence of positive solutions of the p laplacian using the quadrature method we prove the existence of multiple solutions of the one dimensional p laplacian for 0 and determine their exact number for 0 2005 elsevier inc all rights reserved

critical behaviour of nonlinear elliptic boundary value problems suggested by exothermic reactions we consider a class of semilinear elliptic boundary value problems depending on a parameter which arise in the theory of combustion based on the results in another paper by the same author a rigorous quantitative connection is shown between the solution set of the boundary value problem and that of a simple scalar equation the semenov approximation

inequalities for the schatten p norm iv we present some inequalities for the schattenp norm of operators on a hilbert space it is shown among other things that ifa is an operator such that rea a 0 then for any operatorx ax xa p 2a x p also for any two operatorsa andb a b 2 2 a b 2 2 2 a b 2 2

derivations derivatives and chain rules for a smooth function f on the space of bounded operators in a hilbert space we obtain formulas for the nth order commutator f a x x x in terms of the frechet derivatives dmf a we illustrate the use of these formulas in obtaining bounds for norms of generalised commutators f a x xf b and their higher order analogues