the analytical solution of the additive constant problem ifd is a measure of dissimilarity on a finite set withn elements the smallest positive constantc such thatd c has an euclidean representation for allc c is shown to be the largest eigen value of a matrix of size 2n

some uniqueness and exact multiplicity results for a predator prey model in this paper we consider positive solutions of a predator prey model with diffusion and under homogeneous dirichlet boundary conditions it turns out that a certain parameter m in this model plays a very important role a good understanding of the existence stability and number of positive solutions is gained when m is large in particular we obtain various results on the exact number of positive solutions our results for large m reveal interesting contrast with that for the well studied casem 0 i e the classical lotka volterra predator prey model

s shaped bifurcation curve for a nonlocal boundary value problem abstract in this paper by using tools from bifurcation theory particularly the crandall rabinowitz bifurcation theorem the implicit function theorem and the persistence results of degenerate solutions due to e n dancer a result will be proved for a s shaped bifurcation curve of a nonlocal boundary value problem the main results of this paper partially extend the main results in 11 12 to the nonlocal boundary value problems

non negative solutions for a class of non positone problems in the recent past many results have been established on non negative solutions to boundary value problems of the form where 0 f 0 0 positone problems in this paper we consider the impact on the non negative solutions when f 0 f u to be convex to guarantee uniqueness of positive solutions and f u to be appropriately concave for multiple positive solutions this is in contrast to the case of positone problems where the roles of convexity and concavity were interchanged to obtain similar results we further establish the existence of non negative solutions with interior zeros which did not exist in positone problems

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

a theorem on s shaped bifurcation curve for a positone problem with convex concave nonlinearity and its applications to the perturbed gelfand problem abstract we study the bifurcation curve and exact multiplicity of positive solutions of the positone problem u x f u 0 1 x 1 u 1 u 1 0 where 0 is a bifurcation parameter f c 2 0 satisfies f 0 0 and f u 0 for u 0 and f is convex concave on 0 under a mild condition we prove that the bifurcation curve is s shaped on the u plane we give an application to the perturbed gelfand problem u x exp a u a u 0 1 x 1 u 1 u 1 0 where a 0 is the activation energy parameter we prove that if a a 4 166 the bifurcation curve is s shaped on the u plane our results improve those in s h wang on s shaped bifurcation curves nonlinear anal 22 1994 1475 1485 and p korman y li on the exactness of an s shaped bifurcation curve proc amer math soc 127 1999 1011 1020

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

existence of a double s shaped bifurcation curve with six positive solutions for a combustion problem abstract we study the bifurcation curve of positive solutions of the combustion problem with nonlinear boundary conditions given by u x exp u u 0 x 1 u 0 0 u 1 u 1 1 u 1 1 u 1 u 1 1 u 1 0 where 0 is called the frank kamenetskii parameter or ignition parameter 0 is the activation energy parameter u x is the dimensionless temperature and the reaction term exp u u is the temperature dependence obeying the simple arrhenius reaction rate law we prove rigorously that for 1 6 459 for some constant 1 the bifurcation curve is double s shaped on the u plane and the problem has at least six positive solutions for a certain range of positive we give rigorous proofs of some computational results of goddard ii shivaji and lee j goddard ii r shivaji e k lee a double s shaped bifurcation curve for a reaction diffusion model with nonlinear boundary conditions bound value probl 2010 art id 357542 23 pp

s shaped bifurcation curves for nonlinear two parameter problems abstract we consider the nonlinear eigenvalue problem u t 1 u t u t 2 u t p u t 0 t i 1 2 1 2 u 1 2 u 1 2 0 where p 2 is a constant 0 and 0 are parameters this equation was introduced by crandall and rabinowitz 1973 as the typical model which develops the s shaped bifurcation curve when p 3 and 0 1 it is known that if p 2 and 0 1 then there exists a unique solution u r c 2 i with u where 0 is a given constant furthermore is s shaped in this paper we establish the asymptotic formulas for as 0 1 2 when 0 1 where 1 and 2 are two turning points of and is the unique positive solution to the algebraic equation 1 x x 2 x p 0 we also characterize the asymptotic shape of u t as by establishing the asymptotic behavior of u q as when 0 1 is a fixed constant

curves of positive solutions of boundary value problems on time scales abstract let t r be a time scale with a inf t b sup t we consider the nonlinear boundary value problem 2 p t u t q t u t f t u t on t 4 u a u b 0 where r 0 and f t r r satisfies the conditions f t 0 t t r f t f t t t r we prove a strong maximum principle for the linear operator defined by the left hand side of 1 and use this to show that for every solution u of 1 2 u is positive on t a b in addition we show that there exists max 0 possibly max such that if 0 max then 1 2 has a unique solution u while if max then 1 2 has no solution the value of max is characterised as the principal eigenvalue of an associated weighted eigenvalue problem in this regard we prove a general existence result for such eigenvalues for problems with general nonnegative weights

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

exact multiplicity of positive solutions for a class of semilinear problem ii we consider the positive solutions to the semilinear problem u f u 0 in b n u 0 on b n where bn is the unit ball in rn n 1 and is a positive parameter it is well known that if is a smooth function then any positive solution to the equation is radially symmetric and all solutions can be parameterized by their maximum values we develop a unified approach to obtain the exact multiplicity of the positive solutions for a wide class of nonlinear functions u and the precise shape of the global bifurcation diagrams are rigorously proved our technique combines the bifurcation analysis stability analysis and topological methods we show that the shape of the bifurcation curve depends on the shape of the graph of function u u as well as the growth rale of

on the classification and evolution of bifurcation curves for a one dimensional prescribed curvature problem with nonlinearity exp aua u abstract we study the classification and evolution of bifurcation curves of positive solutions u for the one dimensional prescribed curvature problem u x 1 u x 2 exp a u a u l x l u l u l 0 where 0 is a bifurcation parameter and l a 0 are two evolution parameters we prove that on u plane for 0 a 36 17 2 118 the bifurcation curve is shaped while for a 36 17 the bifurcation curve is shaped or reversed e like shaped in particular for a a 4 107 the bifurcation curve is i shaped if l 0 small enough and ii reversed e like shaped if l is large enough

the three solutions theorem for p laplacian boundary value problems abstract the three solutions theorem is established for p laplacian boundary value problems in the presence of two pairs of lower and upper solutions which are well ordered by the relationship between the lower and upper solutions and degree theory in addition an application of the three solutions theorem is given

multiple solutions of some semilinear elliptic equations in slender cylindrical domains we analyze the semilinear problem ul ul 0 in 0 2l ul 0 at 0 2l where rn 1 is a bounded smooth domain and is a given nonlinearity in the limit l the solutions of that problem are related to those of the n 1 dimensional problem v v 0 in v 0 at in particular necessary and sufficient conditions on a given solution of the latter v are given for the former to have a solution ul such that ul x1 x2 xn v x2 xn as l in most part of the cylinder 0 2l related bifurcation problems and some generalizations are also considered

proof of a conjecture for the one dimensional perturbed gelfand problem from combustion theory we study global bifurcation curves and the exact multiplicity of positive solutions for the two point boundary value problem arising in combustion theory left begin array l u prime prime x lambda exp left frac au a u right 0 quad 1 x 1 u 1 u 1 0 end array right where lambda 0 is the frank kamenetskii parameter and a 0 is the activation energy parameter we prove that there exists a critical bifurcation value a0 approx 4 069 such that on the lambda u infty plane the bifurcation curve is s shaped for a a 0 and is monotone increasing for 0 a leqq a 0 that is we prove the long standing conjecture for the one dimensional perturbed gelfand problem we also study in the a lambda left vert u right vert infty space the shape and structure of the bifurcation surface

partial distance correlation with methods for dissimilarities distance covariance and distance correlation are scalar coefficients that characterize independence of random vectors in arbitrary dimension properties extensions and applications of distance correlation have been discussed in the recent literature but the problem of defining the partial distance correlation has remained an open question of considerable interest the problem of partial distance correlation is more complex than partial correlation partly because the squared distance covariance is not an inner product in the usual linear space for the definition of partial distance correlation we introduce a new hilbert space where the squared distance covariance is the inner product we define the partial distance correlation statistics with the help of this hilbert space and develop and implement a test for zero partial distance correlation our intermediate results provide an unbiased estimator of squared distance covariance and a neat solution to the problem of distance correlation for dissimilarities rather than distances

s shaped global bifurcation curve and hopf bifurcation of positive solutions to a predator prey model where d is a bounded domain in r n 1 with smooth boundary d and a b c d m are positive constants problem 1 0 is known as the predator prey model with holling tanner type interactions where u and v represent the densities of the prey and predator respectively hence we are only interested in positive solutions by which we mean that both u and v are positive in d it is easy to see that a 1 is necessary for the existence of positive solutions to 1 0 and we shall also assume that d 1 the case d 1 is rather different and will not be discussed here we are mainly interested in studying the number of positive solutions of 1 0 and the stability of these solutions in particular we shall show that when b d c and m fall into certain range the solution set u v a of 1 0 forms a s shaped smooth curve and that hopf bifurcation occurs along this curve this not only confirms rigorously similar numerical observations on 1 0 made in 4 but also shows that the corresponding parabolic system

reduced dimensional gaussian process emulators of parametrized partial differential equations based on isomap in this paper isomap and kernel isomap are used to dramatically reduce the dimensionality of the output space to efficiently construct a gaussian process emulator of parametrized partial differential equations the output space consists of spatial or spatio temporal fields that are functions of multiple input variables for such problems standard multi output gaussian process emulation strategies are computationally impractical and or make restrictive assumptions regarding the correlation structure the method we develop can be applied without modification to any problem involving vector valued targets and vector valued inputs it also extends a method based on linear dimensionality reduction to response surfaces that cannot be described accurately by a linear subspace of the high dimensional output space comparisons to the linear method are made through examples that clearly demonstrate the advantages of nonlinear dimensionality reduction

remarks on an s shaped bifurcation curve abstract we study the non negative solutions of the boundary value problem u exp u u x u 0 x where 0 0 r n is bounded with smooth boundary this problem arises in the theory of combustion we study the estimates on the supremum norm of the solutions and estimates on the critical values of

asymptotic behavior of solutions for finite difference equations of reaction diffusion abstract the method of upper lower solutions and the associated existence comparison theorem for continuous reaction diffusion equations are extended to a finite difference system which is a discrete version of the continuous problem of either parabolic type or elliptic type under appropriate boundary condition it is shown by using the existence comparison theorem for the discrete system that for a certain class of initial functions the corresponding time dependent solution converges monotonically to a steady state solution this monotone convergence leads to the stability property of the steady state solution as well as an estimate of the stability region in the framework of the finite difference system numerical results are given for an enzyme substrate reaction model where multiple steady state solutions exist for a certain range of physical parameters in this situation the time dependent solution converges either to the maximal solution or to the minimal solution depending on the corresponding initial function

uniqueness of positive solutions for a class of elliptic boundary value problems on demontre l unicite des solutions non negatives du probleme aux valeurs limites u f u pour x u 0 pour x ou 0 et est un domaine borne de r n a frontiere lisse

the liouville bratu gelfand problem for radial operators abstract we determine precise existence and multiplicity results for radial solutions of the liouville bratu gelfand problem associated with a class of quasilinear radial operators which includes perturbations of k hessian and p laplace operators

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

persistence and bifurcation of degenerate solutions abstract we consider a nonlinear equation f e u 0 where f is a differentiable mapping from r r x to y and x y are banach spaces when e varies from a fixed e e 0 bifurcation occurs to the solution curve s u s we study the degenerate solutions of the equation and we obtain several bifurcation theorems on the degenerate solutions which can be applied in many nonlinear problems to obtain precise global bifurcation diagrams

new exact multiplicity results with an application to a population model we obtain some new exact multiplicity results for the dirichlet boundary value problem on a unit ball b n in r n we consider several classes of nonlinearities f u including both positive and sign changing cases a crucial part of the proof is to establish positivity of solutions for the corresponding linearized problem as an application we obtain exact multiplicity results for the holling tanner population model

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 effects of indefinite nonlinear boundary conditions on the structure of the positive solutions set of a logistic equation abstract we investigate a semilinear elliptic equation with a logistic nonlinearity and an indefinite nonlinear boundary condition both depending on a parameter overall we analyze the effect of the indefinite nonlinear boundary condition on the structure of the positive solutions set based on variational and bifurcation techniques our main results establish the existence of three nontrivial non negative solutions for some values of as well as their asymptotic behavior these results suggest that the positive solutions set contains an s shaped component in some case as well as a combination of a c shaped and an s shaped components in another case

existence and multiplicity results for a class of semilinear elliptic equations abstract we study the existence and multiplicity of nonnegative solutions as well as the behavior of corresponding parameter dependent branches to the equation u 1 u u m u n in a bounded domain r n endowed with the zero dirichlet boundary data where 0 m 1 and n 0 when 0 the obtained solutions can be seen as steady states of the corresponding reaction diffusion equation describing a model of isothermal autocatalytic chemical reaction with termination in addition to the main new results we formulate a few relevant conjectures

classification and evolution of bifurcation curves for the one dimensional perturbed gelfand equation with mixed boundary conditions abstract we study the classification and evolution of bifurcation curves of positive solutions for the one dimensional perturbed gelfand equation with mixed boundary conditions given by u x exp a u a u 0 0 x 1 u 0 0 u 1 c 0 we prove that for positive a a 0 0 501 and c 0 the bifurcation curve is strictly increasing on the u plane and there exists a positive 0 such that the problem has no positive solution for 0 0 and exactly one positive solution for 0 while for a a 1 4 107 there exists c 1 c 1 a 1 057 such that on the u plane i when 0 c c 1 the bifurcation curve is s shaped and the problem has at least three positive solutions for some range of positive ii when c c 1 the bifurcation curve is shaped and the problem has at least two positive solutions for some range of positive

classification and evolution of bifurcation curves for the one dimensional minkowski curvature problem and its applications abstract in this paper we study the classification and evolution of bifurcation curves of positive solutions for the one dimensional minkowski curvature problem u 1 u 2 f u in l l u l u l 0 where l 0 f c 0 c 2 0 and f u 0 for u 0 furthermore we show that for sufficiently large l 0 the bifurcation curve s l may have arbitrarily many turning points finally we apply these results to obtain the global bifurcation diagrams for ambrosetti brezis cerami problem liouville bratu gelfand problem and perturbed gelfand problem with the minkowski curvature operator respectively moreover we will make two lists which show the different properties of bifurcation curves for minkowski curvature problems corresponding semilinear problems and corresponding prescribed curvature problems

a contribution to the study of the metric and euclidean structures of dissimilarities this paper is concerned with the geometric properties of dissimilarity coefficients defined on finite sets and especially with their euclidean nature we present several particular transformations which preserve euclideanarity and we complete through the study of a one parameter family the current knowledge of the metric and euclidean structure of coefficients based on binary data these results are directly deduced from two theorems which prove the positive semi definite status of some quadratic forms which play a large role in some definitions of dissimilarity commonly used

robust kernel isomap isomap is one of widely used low dimensional embedding methods where geodesic distances on a weighted graph are incorporated with the classical scaling metric multidimensional scaling in this paper we pay our attention to two critical issues that were not considered in isomap such as 1 generalization property projection property 2 topological stability then we present a robust kernel isomap method armed with such two properties we present a method which relates the isomap to mercer kernel machines so that the generalization property naturally emerges through kernel principal component analysis for topological stability we investigate the network flow in a graph providing a method for eliminating critical outliers the useful behavior of the robust kernel isomap is confirmed through numerical experiments with several data sets

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

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

evolution of bifurcation curves for semipositone problems when nonlinearities develop multiple zeroes we discuss existence uniqueness and multiplicity results on positive solutions to u x email protected u x 0 for email protected 0 1 u 0 0 u 1 where l 0 is a constant and u is a smooth function satisfying 0 0 semipositone in particular we follow a class of nonlinearities u as they evolve from having one positive zero to three positive zeros and compare the corresponding bifurcation curves we establish that the bifurcation curve splits into two connected components in the critical case when u develops the second zero

exact multiplicity of positive solutions for concave convex and convex concave nonlinearities abstract this note gives an unified treatment of the exact multiplicity results for both s shaped and reversed s shaped bifurcation for positive solutions of the two point problem u f u 0 for 1 x 1 u 1 u 1 0 depending on a positive parameter for both concave convex and convex concave nonlinearities f u

exact multiplicity and global structure of solutions for a class of semilinear elliptic equations abstract in this paper we establish an exact multiplicity result of solutions for a class of semilinear elliptic equation we also obtain a precise global bifurcation diagram of the solution set as a result an open problem presented by c h hsu and y w shih c h hsu y w shih solutions of semilinear elliptic equations with asymptotic linear nonlinearity nonlinear anal 50 2002 275 283 is completely solved our argument is mainly based on bifurcation theory and continuation method

topology of finite graphs this paper derives from a course in group theory which i gave at berkeley in 1982 i wanted to prove the standard theorems on free groups and discovered that after a few preliminaries the notion of locally injective map or immersion of graphs was very useful this enables one to see in an effective easy algorithmic way just what happens with finitely generated free groups one can understand in this way 1 howson s theorem that if a and b are finitely generated subgroups of a free group then a b is finitely generated and 2 m hall s theorem that finitely generated subgroups of free groups are closed in the profinite topology during this course s m gersten came up with a simple proof of h neumann s inequality on the ranks in howson s theorem one of the ideas in gersten s proof was to use core graphs graphs with no trees hanging on subsequently i found that some consequences of a paper of greenberg s could be proved using core graphs and covering translations of immersions the most striking such result is that if a and b are finitely generated subgroups of a free group and if a c b is of finite index in both a and b then a b is of finite index in a v b the subgroup generated by a ub

exact multiplicity results for boundary value problems with nonlinearities generalizing cubic using techniques of bifurcation theory we present two exact multiplicity results for boundary value problems of the type the first result concerns the case when the nonlinearity is independent of x and behaves like a cubic in u the second one deals with a class of nonlinearities with explicit x dependence

remark on a sufficient condition for multiplicity in a model in chemical reactor theory we consider the model u l s u exp k l u x e 0 1 u 0 0 u 1 which arises in chemical reactor theory here l s and k are positive constants it is known that if there exists an a e 0 s such that h a email protected f s ds a2 f a 0 then there exists a range of l for which there exists at least three nonnegative solutions and using numerical evaluation it was conjectured that such values of a exist in this paper we give an analytical proof of this conjecture

testing for homogeneity of multivariate dispersions using dissimilarity measures summary testing homogeneity of dispersions may be of its own scientific interest as well as an important auxiliary step verifying assumptions of a main analysis the problem is that many biological and ecological data are highly skewed and zero inflated also the number of variables often exceeds the sample size thus data analysts often do not rely on parametric assumptions but use a particular dissimilarity measure to calculate a matrix of pairwise differences this matrix is then the basis for further statistical inference anderson 2006 proposed a distance based test of homogeneity of multivariate dispersions for a one way anova design for which a matrix of pairwise dissimilarities can be taken as an input the key idea like in levene s test is to replace each observation with its distance to an estimated group center in this paper we suggest an alternative approach that is based on the means of within group distances and does not require group centre calculations to obtain the test statistic we show that this approach can have theoretical as well as practical advantages a permutation procedure that gives type i error close to the prescribed value even in small samples is described

quotients of surface groups and homology of finite covers via quantum representations we prove that for each sufficiently complicated orientable surface s there exists an infinite image linear representation rho of pi 1 s such that if gamma in pi 1 s is freely homotopic to a simple closed curve on s then rho gamma has finite order furthermore we prove that given a sufficiently complicated orientable surface s there exists a regular finite cover s rightarrow s such that h 1 s mathbb z is not generated by lifts of simple closed curves on s and we give a lower bound estimate on the index of the subgroup generated by lifts of simple closed curves we thus answer two questions posed by looijenga and independently by kent kisin marche and mcmullen the construction of these representations and covers relies on quantum text so 3 representations of mapping class groups

a simplified proof of a conjecture for the perturbed gelfand equation from combustion theory abstract for the perturbed gelfand s equation on the unit ball in two dimensions y du and y lou 5 proved that the curve of positive solutions is exactly s shaped for sufficiently small values of the secondary parameter we present a simplified proof and some extensions this problem is prominent in combustion theory see e g the book of j bebernes and d eberly 1

global bifurcation and exact multiplicity of positive solutions for a positone problem with cubic nonlinearity and their applications we study the global bifurcation and exact multiplicity of positive solutions of u x lambda f epsilon u 0 1 f epsilon u epsilon u 3 sigma u 2 tau u rho where lambda epsilon 0 are two bifurcation parameters and sigma rho 0 tau 0 are constants by developing some new time map techniques we prove the global bifurcation of bifurcation curves for varying epsilon 0 more precisely we prove that for any sigma rho 0 tau 0 there exists epsilon 0 such that on the lambda parallel to u parallel to infinity plane the bifurcation curve is s shaped for 0 epsilon we also prove the global bifurcation of bifurcation curves for varying lambda 0 thus we are able to determine the exact number of positive solutions by the values of epsilon and lambda we give an application to prove a long standing conjecture for global bifurcation of positive solutions for the problem u x lambda epsilon u 3 u 2 u 1 0 1 u 1 u 1 0 which was studied by crandall and rabinowitz arch rational mech anal 52 1973 p 177 in addition we give an application to prove a conjecture of smoller and wasserman j differential equations 39 1981 p 283 lines 2 3 on the maximum number of positive solutions of a positone problem

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

uniqueness multiplicity and stability for positive solutions of a pair of reaction diffusion equations we study the number and stability of the positive solutions of a reaction diffusion equation pair when certain parameters in the equations are large the equation pair can be viewed as singular or regular perturbations of some single or essentially single equation problems for which the number and stability of their solutions can be well understood with the help of these simpler equations we are able to obtain a rather complete understanding of the number and stability of the positive solutions for the equation pair for the cases that certain parameters are large in particular we obtain a fairly satisfactory description of the positive solution set of the equation pair

applications of sturm sequences to bifurcation analysis of delay differential equation models this paper formalizes a method used by several others in the analysis of biological models involving delay differential equations in such a model the characteristic equation about a steady state is transcendental this paper shows that the analysis of the bifurcation due to the introduction of the delay term can be reduced to finding whether a related polynomial equation has simple positive real roots after this result has been established we utilize sturm sequences to determine whether a polynomial equation has positive real roots this work has extended the stability results found in previous papers and provides a novel theorem about stability switches for low degree characteristic equations

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

asymptotic length of bifurcation curves related to inverse bifurcation problems abstract we consider the nonlinear eigenvalue problem u t f u t 0 u t 0 t i 1 1 u 1 u 1 0 where f u u g u and 0 is a parameter our interest is the asymptotic length l g of the bifurcation curve g u 0 by the notion of l g we will propose a new framework of inverse bifurcation problems precisely we consider whether it is possible to characterize the unknown nonlinear term g u by l g if g u g 1 u u 2 sin u or g 2 u u 1 2 sin u then the bifurcation curves g i i 1 2 are continuous functions of 0 and g i 2 4 as furthermore they intersect the line 2 4 infinitely many times as in this paper by establishing the precise asymptotic formula for g i for 1 we obtain the precise asymptotic length l g i of g i for 1 then we consider the set g c 2 r g 2 4 as and study the existence and non existence of g whose asymptotic formula for l g is equal to those of l g i up to the second term

exact multiplicity of solutions and s shaped bifurcation curve for a class of semilinear elliptic equations abstract the set of steady state solutions to a reaction diffusion equation modeling an autocatalytic chemical reaction is completely determined when the reactor has spherical geometry and the spatial dimension is n 1 or 2 for any reaction order or n 3 for subcritical reaction order bifurcation approach and analysis of linearized problems are used to establish exact multiplicity and precise global bifurcation diagram of positive steady states

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

number of zeros of interval polynomials in this paper we develop a rigorous algorithm for counting the real interval zeros of polynomials with perturbed coefficients that lie within a given interval without computing the roots of any polynomials the result generalizes sturm s theorem for counting the roots of univariate polynomials to univariate interval polynomials

a convex matrix optimization for the additive constant problem in multidimensional scaling with application to locally linear embedding the additive constant problem has a long history in multidimensional scaling and it has recently been used to resolve the issue of indefiniteness of the geodesic distance matrix in isomap but it would lead to a large positive constant being added to all eigenvalues of the centered geodesic distance matrix often causing significant distortion of the original distances in this paper we reformulate the problem as a convex optimization of almost negative semidefinite matrix so as to achieve minimal variation of the original distances we then develop a newton cg method and further prove its quadratic convergence finally we include a novel application to the famous lle locally linear embedding in nonlinear dimensionality reduction addressing the issue when the input of lle has missing values we justify the use of the developed method to tackle this issue by establishing that the local gram matrix used in lle can be obtained through a local euclidean distance matrix the effectiveness of our method is

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