relating equivalence and reducibility to sparse sets for various polynomial time reducibilities r this paper asks whether being r reducible to a sparse set is a broader notion than being r equivalent to a sparse set although distinguishing equivalence and reducibility to sparse sets for many one or 1 truth table reductions would imply that p ne np this paper shows that for k truth table reductions k geq 2 equivalence and reducibility to sparse sets provably differ though gavalda and watanabe have shown that for any polynomial time computable unbounded function f cdot some sets f n truth table reducible to sparse sets are not even turing equivalent to sparse sets this paper shows that extending their result to the 2 truth table case would provide a proof that p ne np additionally this paper studies the relative power of different notions of reducibility and proves that disjunctive and conjunctive truth table reductions to sparse sets are surprisingly powerful refuting a conjecture of ko

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

the structure of logarithmic advice complexity classes a nonuniform class called here full p log due to ko is studied it corresponds to polynomial time with logarithmically long advice its importance lies in the structural properties it enjoys more interesting than those of the alternative class p log specifically its introduction was motivated by the need of a logarithmic advice class closed under polynomial time deterministic reductions several characterizations of full p log are shown formulated in terms of various sorts of tally sets with very small information content a study of its inner structure is presented by considering the most usual reducibilities and looking for the relationships among the corresponding reduction and equivalence classes defined from these special tally sets

defying upward and downward separation upward and downward separation results link the collapse of small and large classes and are a standard tool in complexity theory we study the limitations of upward and downward separation pp we show that the exponential time limited nondeterminism hierarchy does not robustly possess downward separation we show that probabilistic classes do not robustly possess hartmanis immerman sewelson upward separation though np is known to robustly possess hartmanis immerman sewelson upward separation we show that np does not robustly possess hartmanis immerman sewelson upward separation with respect to strong immunity separation on the other hand we provide a structural sufficient condition for upward separation

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

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

existence and localization of solution for second order elliptic bvp in presence of lower and upper solutions without any order abstract we prove an existence and localization result for a solution of a second order elliptic equations with dirichlet and regular oblique derivative boundary conditions the main result is obtained assuming the existence of a lower and an upper solution without particular order and a positively homogeneous asymptotic behaviour of the right hand member of the equation

classical and weak solutions of a singular semilinear elliptic problem the singular semilinear elliptic equation u p x f u 0 is shown to have a unique positive classical solution inrnthat decays to zero at ifp x is simply a nontrivial nonnegative continuous function satisfying 0 t max x t p x dt providedfis a non increasing continuously differentiable function on 0 it is also shown that the equation has a unique weakh10 solution on a bounded domain provided e0 f s ds andp x l2

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

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

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

a three solution theorem for singular nonlinear elliptic boundary value problems abstract we establish a three solution theorem for singular elliptic boundary value problems of the form u f u u in u 0 on where is a bounded domain in r n n 1 with a smooth boundary here f 0 0 is a c 1 function in 0 with f 0 0 and 0 1 in particular if there exist two pairs of sub supersolutions 1 1 and 2 2 where 1 2 1 1 2 1 with 2 2 and 2 2 are strict sub and supersolutions then we establish existence of three solutions u 1 u 2 and u 3 for the above boundary value problem such that u 1 1 2 u 2 2 1 and u 3 1 1 1 2 2 1 our results extend the work in 1 and also in 17 where such multiplicity results were discussed in the non singular case 0 further our results strengthen the multiplicity results in 12 and 13 for such singular problems 0 namely establish an additional third solution we also establish a sub super solution theorem for infinite semipositone problems i e the case when f 0 0

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