on the equivalence of constrained and compound optimal designs abstract constrained and compound optimal designs represent two well known methods for dealing with multiple objectives in optimal design as reflected by two functionals 1 and 2 on the space of information matrices a constrained optimal design is constructed by optimizing 2 subject to a constraint on 1 and a compound design is found by optimizing a weighted average of the functionals 1 1 2 0 1 we show that these two approaches to handling multiple objectives are equivalent

the use of a canonical form in the construction of locally optimal designs for non linear problems optimal experimental designs for non linear problems depend on the values of the underlying undknown parameters in the model for various reasons there is interest in providing explicit formulae for the optimal designs as a function of the unknown parameters this paper shows that for a certain class of generalized linear models the problem can be reduced to a canonical form this simplifies the underlying problem and designs are constructed for several contexts with a single variable using geometric and other arguments

global structure of self similar solutions in a semilinear parabolic equation abstract the aim of this paper is to investigate the structure of radial solutions for a semilinear elliptic equation u y u 2 u p 1 u p 1u 0 y rn which is related to forward self similar solutions of a semilinear parabolic equation vt v v p 1v t x 0 rn we study the existence uniqueness and parameter dependency of rapidly decaying solutions with a prescribed number of zeros

designs for generalized linear models with several variables and model uncertainty standard factorial designs sometimes may be inadequate for experiments that aim to estimate a generalized linear model for example for describing a binary response in terms of several variables a method is proposed for finding exact designs for such experiments that uses a criterion allowing for uncertainty in the link function the linear predictor or the model parameters together with a design search designs are assessed and compared by simulation of the distribution of efficiencies relative to locally optimal designs over a space of possible models exact designs are investigated for two applications and their advantages over factorial and central composite designs are demonstrated

fast computation of designs robust to parameter uncertainty for nonlinear settings experimental design in nonlinear settings is complicated by the fact that the efficiency of a design depends on the unknown parameter values thus good designs need to be efficient over a range of likely parameter values bayesian design criteria provide a natural framework for achieving such robustness by averaging local design criteria over a prior distribution on the parameters a major drawback to the use of such criteria is the heavy computational burden that they impose we present a clever quadrature scheme that greatly improves the feasibility of using bayesian design criteria we illustrate the method on some designed experiments

structure of positive radial solutions to scalar field equations with harmonic potential abstract we study a nonlinear elliptic equation with a harmonic potential u x 2 u u p 1 u 0 x r n e which is related to standing waves for a nonlinear schrodinger equation the existence of positive solution for e in some energy space can be shown by the standard variational method under n 1 n and 1 p n 2 n 2 however the uniqueness is a delicate problem in this paper noting that all positive solutions of e in the energy space are radially symmetric about the origin we study the structure of positive radial solutions to e and show the uniqueness of positive solution with finite energy when n 3

the role of critical exponents in blow up theorems the sequel abstract in 27 fujita showed that for positive solutions the initial value problem in r n for u t u u p with p 1 exhibited the following behavior if p p c 1 2 n then the initial value problem does not have any nontrivial non negative solution existing on r n 0 a global solution whereas if p p c there exist global small data positive solutions as well as solutions which are non global we call such a result a blow up theorem of fujita type in 50 levine discussed the various theorems of this type that had appeared in the literature prior to 1990 in this paper we revisit the literature since 1990

the routing open shop problem on a network complexity and approximation in the routing open shop problem jobs are located at nodes of an undirected transportation network and the machines travel on the network to execute jobs in the open shop environment the machines are initially located at the same node depot and must return to the depot after completing all the jobs it is required to find a nonpreemptive schedule that minimizes the makespan we prove that the problem is np hard even on a two node network with two machines and even on a two node network with two jobs and m machines we develop polynomial time approximation heuristics and obtain bounds on their approximation performance

structural stability for time periodic one dimensional parabolic equations denote q w x 0 l x r the nonlinear term f is assumed to satisfy the following conditions f l f q r is of class c f 2 f t l x u f t x u for all te w xe o l and uer define a self adjoint operator a in x l2 0 l by a t bxx for e d a h2 0 l a ha o l for each c1 e r let a be the fractional power of a and x d a be the fractional power space equipped with the graph norm ii 11 xn 1111 11x iia i ii x throughout this section we fix a c1 1 by the above assumptions f l f 2 for any u0 e x there exists a unique local in time classical solution u t x u t x of

critical exponents for the blowup of solutions with sign changes in a semilinear parabolic equation ii abstract the blowup of solutions of the cauchy problem formula is studied let k be the set of functions on r which change sign k times it is shown that for p k 1 2 k 1 k 0 1 2 any solution with u 0 k blows up in finite time if 1 p p k whereas a global solution with u 0 k exists if p p k this is an extension of our previous result 17 in which a fast decay condition was imposed on initial data it is also shown in this paper that if u0 decays more slowly than x 2 p 1 as x then the solution blows up in finite time regardless of the number of sign changes

radial symmetry of self similar solutions for semilinear heat equations abstract the symmetry properties of positive solutions of the equation u 1 2 x u 1 p 1 u u p 0 in r n where n 2 p n 2 n was studied it was proved that u must be radially symmetric about the origin provided u x o x 2 p 1 as x and that there exist non radial solutions u satisfying lim sup x x 2 p 1 u x 0

a 65 approximation algorithm for the two machine routing open shop problem on a two node network abstract in the routing open shop problem jobs are located at nodes of a transportation network and the machines travel on the network to execute jobs for the makespan minimizing two machine routing open shop problem on a two node network which is np hard we prove that the optimum objective value of any instance lies within the interval c 6 5 c and this interval is tight c is a trivial lower bound on the optimum objective value based on this result we obtain a linear time approximation algorithm for this problem with approximation ratio not greater than 6 5 from the tightness of the interval above it follows that this algorithm gives the best possible approximation in terms of c the problem is equivalent to a variant of the two machine open shop problem with batch setup times with two batches

sign changing stationary solutions and blowup for the nonlinear heat equation in a ball in this paper we prove that if phi is a radially symmetric sign changing stationary solution of the nonlinear heat equation u t delta u u alpha u quad quad quad quad quad in the unit ball of mathbb r n n 3 with dirichlet boundary conditions then the solution of nlh with initial value lambda phi blows up in finite time if 1 0 is sufficiently small and if is subcritical and sufficiently close to 4 n 2

slow decay of solutions in a semilinear dissipative parabolic equation this paper is concerned with a cauchy problem p u t u x x u p 1 u in r x 0 u x 0 u 0 x in r where p 1 and u0 e l r a solution u of p is said to decay fast as t if limt tl p 1u x t 0 uniformly in r and to decay slowly as t otherwise we prove that if u0 x does not decay faster than x q with some q 0

on a l functional derivative estimate relating to the cauchy problem for scalar semi linear parabolic partial differential equations with general continuous nonlinearity in this paper we consider a l infty functional derivative estimate for the first spatial derivative of bounded classical solutions u mathbb r times 0 t to mathbb r to the cauchy problem for scalar semi linear parabolic partial differential equations with a continuous nonlinearity f mathbb r to mathbb r and initial data u 0 mathbb r to mathbb r of the form sup x in mathbb r u x x t leq mathcal f t f u 0 u forall t in 0 t here mathcal f t mathcal a t to mathbb r is a functional as defined in textsection 1 we establish that the functional derivative estimate is non trivially sharp by constructing a sequence f n 0 u n where for each n in mathbb n u n mathbb r times 0 t to mathbb r is a solution to the cauchy problem with zero initial data and nonlinearity f n mathbb r to mathbb r and for which sup x in mathbb r u x n x t geq alpha 0 with lim n to infty left inf t in 0 t left sup x in mathbb r u x n cdot t mathcal f t f n 0 u n right right 0

routing open shop and flow shop scheduling problems we consider a generalization of the classical open shop and flow shop scheduling problems where the jobs are located at the vertices of an undirected graph and the machines initially located at the same vertex have to travel along the graph to process the jobs the objective is to minimize the makespan in the tour version the makespan means the time by which each machine has processed all jobs and returned to the initial location while in the path version the makespan represents the maximum completion time of the jobs we present improved approximation algorithms for various cases of the open shop problem on a general graph and the tour version of the two machine flow shop problem on a tree also we prove that both versions of the latter problem are np hard which answers an open question posed in the literature

on a singular initial boundary value problem for a reaction diffusion equation arising from a simple model of isothermal chemical autocatalysis in this paper we examine the evolution that occurs when a localized input of an autocatalyst b is introduced into an expanse of a reactant a the reaction is autocatalytic of order p so a rightarrow b at rate k a b p with rate constant k we examine the case when 0 p 1 with p geq 1 having been examined by needham merkin phil trans r soc lond a 337 261 274 1991 in particular we show that the fully reacted state is not achieved as t rightarrow infty via the propagation of a travelling wavefront as for p geq 1 but is approached uniformly in space as t rightarrow infty

robust designs for poisson regression models we consider the problem of how to construct robust designs for poisson regression models an analytical expression is derived for robust designs for first order poisson regression models where uncertainty exists in the prior parameter estimates given certain constraints in the methodology it may be necessary to extend the robust designs for implementation in practical experiments with these extensions our methodology constructs designs that perform similarly in terms of estimation to current techniques and offers the solution in a more timely manner we further apply this analytic result to cases where uncertainty exists in the linear predictor the application of this methodology to practical design problems such as screening experiments is explored given the minimal prior knowledge that is usually available when conducting such experiments it is recommended to derive designs robust across a variety of systems however incorporating such uncertainty into the design process can be a computational

non uniqueness of solutions to the cauchy problem for semilinear heat equations with singular initial data the cauchy problem for semilinear heat equations with singular initial data is studied where n 2 0 is a parameter and a 0 a 0 we show that when p n 2 n and n 2 p n 2 there exists a positive constant such that the problem has two positive self similar solutions and with if and no positive self similar solutions if furthermore for each fixed and in l rn as 0 where w0 is a non unique solution to the problem with zero initial data which is constructed by haraux and weissler

the evolution to localized and front solutions in a non lipschitz reaction diffusion cauchy problem with trivial initial data abstract in this paper we establish the existence of spatially inhomogeneous classical self similar solutions to a non lipschitz semi linear parabolic cauchy problem with trivial initial data specifically we consider bounded solutions to an associated two dimensional non lipschitz non autonomous dynamical system for which we establish the existence of a two parameter family of homoclinic connections on the origin and a heteroclinic connection between two equilibrium points additionally we obtain bounds and estimates on the rate of convergence of the homoclinic connections to the origin

asymptotic behavior and attractors for reaction diffusion equations in unbounded domains abstract in this paper we give general and flexible conditions for a reaction diffusion equation to be dissipative in an unbounded domain the functional setting is based on standard lebesgue and sobolev lebesgue spaces we show how the reaction and diffusion mechanisms have to work together to obtain the asymptotic compactness of solutions and therefore the existence of the compact attractor in particular cases our results allow us to improve some previous known results

the zero set of a solution of a parabolic equation on etudie l ensemble nul d une solution u t x de l equation u t a x t u xx b x t u x c x t u sous des hypotheses tres generales sur les coefficients a b et c

linear behaviour of solutions of a superlinear heat equation we give sufficient conditions which guarantee that solutions of a superlinear heat equation decay to zero at the same rate as the solutions of the linear heat equation with the same initial data this improves a previous result of lee and ni by showing that this behaviour holds for a significantly larger and rather tightly defined class of solutions

variational problems related to self similar solutions of the heat equation on demontre la compacite de certains plongements de sobolev sous de bonnes conditions sur les poids k x exp x c 2 r n r on considere des methodes variationnelles pour la resolution d equations elliptiques

extended weak maximum principles for parabolic partial differential inequalities on unbounded domains in this paper we establish extended maximum principles for solutions to linear parabolic partial differential inequalities on unbounded domains where the solutions satisfy a variety of growth dec

the equivalence of constrained and weighted designs in multiple objective design problems abstract several competing objectives may be relevant in the design of an experiment the competing objectives may not be easy to characterize in a single optimality criterion one approach to these design problems has been to weight each criterion and find the design that optimizes the weighted average of the criteria an alternative approach has been to optimize one criterion subject to constraints on the other criteria an equivalence theorem is presented for the bayesian constrained design problem equivalence theorems are essential in verifying optimality of proposed designs especially when as in most nonlinear design problems numerical optimization is required this theorem is used to show that the results of cook and wong on the equivalence of the weighted and constrained problems apply much more generally the results are applied to bayesian nonlinear design problems with several objectives

global similarity solutions to a class of semilinear parabolic equations existence bifurcations and asymptotics this paper examines the existence structure and bifurcation in the parameters n and p of a class of global positive similarity solutions to the semilinear parabolic equation u t n 2 u u p with p 1 and n 1 here n 2 is the n dimensional laplace operator these are related to the existence of geometric threshold in the p th order autocatalytic or chain branching reactions

on two signed solutions to a second order semi linear parabolic partial differential equation with non lipschitz nonlinearity abstract in this paper we establish the existence of a 1 parameter family of spatially inhomogeneous radially symmetric classical self similar solutions to a cauchy problem for a semi linear parabolic pde with non lipschitz nonlinearity and trivial initial data specifically we establish well posedness for an associated initial value problem for a singular two dimensional non autonomous dynamical system with non lipschitz nonlinearity additionally we establish that solutions to the initial value problem converge algebraically to the origin and oscillate as

on a decay property of solutions to the haraux weissler equation abstract we give a sufficient condition that non radial h 1 solutions to the haraux weissler equation should belong to the weighted sobolev space h 1 r n where is the weight function exp x 2 4 our result provides in some sense a connection between the solutions obtained by ode method and those by variational approach in the space h 1 r n

critical exponent for the bipolar blowup in a semilinear parabolic equation abstract the blowup of solutions of the initial boundary value problem formula is studied where is a cone inrn we say that a solutionuexhibits bipolar blowup if formula for somet 1 for the bipolar blowup in the following sense if 1 p then the bipolar blowup does not occur for any sufficiently small initial data the value ofp is expressed in terms of the dimensionnand the second dirichlet eigenvalue of the laplace beltrami operator on sn 1 in the case of r we definek polar blowup and determine critical exponents for thek polar blowup fork 1 2 3

global existence and blowup for sign changing solutions of the nonlinear heat equation abstract in this paper given 0 2 n we prove the existence of a function with the following properties the solution of the equation u t u u u on r n with the initial condition u 0 is global on the other hand the solution with the initial condition u 0 blows up in finite time if 0 is either sufficiently small or sufficiently large

robust experimental design for multivariate generalized linear models a simple heuristic is proposed for constructing robust experimental designs for multivariate generalized linear models the method is based on clustering a set of local optimal designs a method for finding local d optimal designs using available resources is also introduced clustering with its simplicity and minimal computation needs is demonstrated to outperform more complex and sophisticated methods