bifurcation from stability to instability for a free boundary problem arising in a tumor model we consider a time dependent free boundary problem with radially symmetric initial data t 0 if open image in new window and r 0 0 r in r then the stationary solution is unstable

asymptotic behavior of solutions of a free boundary problem modeling tumor spheroid with gibbs thomson relation abstract in this paper we study a free boundary problem modeling the growth of solid tumor spheroid it consists of two elliptic equations describing nutrient diffusion and pressure distribution within tumor respectively the new feature is that nutrient concentration on the boundary is less than external supply due to a gibbs thomson relation and the problem has two radial stationary solutions which differs from widely studied tumor spheroid model with surface tension effect we first establish local well posedness by using a functional approach based on fourier multiplier method and analytic semigroup theory then we investigate stability of each radial stationary solution by employing a generalized principle of linearized stability we prove that the radial stationary solution with a smaller radius is always unstable and there exists a positive threshold value of cell to cell adhesiveness such that the radial stationary solution with a larger radius is asymptotically stable for and unstable for 0

analysis of mathematical models for the growth of tumors with time delays in cell proliferation we study two mathematical models for the growth of tumors with time delays in cell proliferation one for nonnecrotic tumors in the presence of inhibitors and the other for necrotic tumors mathematical formulations of these models are retarded differential equations by using a comparison method we make rigorous analysis of these models the results show that dynamical behavior of solutions of these models are similar to that of solutions for corresponding nonretarded problems and the tumor will tend to a dormant state as time goes to infinity in favorable conditions nutrient sufficient inhibitor less while it will finally disappear in unfavorable conditions nutrient insufficient inhibitor much

convergence of line search methods for unconstrained optimization line search methods are traditional and successful methods for solving unconstrained optimization problems its convergence has attracted more attention in recent years in this paper we analyze the general results on convergence of line search methods with seven line search rules it is clarified that the search direction plays a main role in these methods and that step size guarantees the global convergence in some cases it is also proved that many line search methods have same convergence property these convergence results can enable us to design powerful effective and stable algorithms in practice finally a class of special line search methods is investigated

bisection is optimal we seek an approximation to a zero of a continuous functionf a b such thatf a 0 andf b 0 it is known that the bisection algorithm makes optimal use ofn function evaluations i e yields the minimal error which is b a 2 n 1 see e g kung 2 traub and wozniakowski 5 proposed using more general information onf by permitting the adaptive evaluations ofn arbitrary linear functionals they conjectured 5 p 170 that the bisection algorithm remains optimal even if these general evaluations are permitted this paper affirmatively proves this conjecture in fact we prove optimality of the bisection algorithm even assuming thatf is infinitely many times differentiable on a b and has exactly one simple zero

stationary solutions of a free boundary problem modeling the growth of tumors with gibbs thomson relation abstract in this paper we study a free boundary problem modeling tumor growth the model consists of two elliptic equations describing nutrient diffusion and pressure distribution within tumors respectively and a first order partial differential equation governing the free boundary on which a gibbs thomson relation is taken into account we first show that the problem may have none one or two radial stationary solutions depending on model parameters then by bifurcation analysis we show that there exist infinite many branches of non radial stationary solutions bifurcating from given radial stationary solution the result implies that cell to cell adhesiveness is the key parameter which plays a crucial role on tumor invasion

asymptotic near optimality of the bisection method we seek a approximation to a zero of an infinitely differentiable functionf 0 1 such thatf 0 0 andf 1 0 it is known that the error of the bisection method usingn function evaluations is 2 n 1 if the information used are function values then it is known that bisection information and the bisection algorithm are optimal traub and wo niakowski conjectured in 5 that the bisection information and algorithm are optimal even if far more general information is permitted they permit adaptive sequential evaluations of arbitrary linear functionals and arbitrary transformations of this information as algorithms this conjecture was established in 2 that is forn fixed the bisection information and algorithm are optimal in the worst case setting thus nothing is lost by restricting oneself to function values one may then ask whether bisection is nearly optimal in theasymptotic worst case sense that is possesses asymptotically nearly the best rate of convergence methods converging fast asymptotically like newton or secant type are of course widely used in scientific computation we prove that the answer to this question is positive for the classf of functions having zeros ofinfinite multiplicity and information consisting of evaluations of continuous linear functionals assuming that everyf inf has zeroes withbounded multiplicity there are known hybrid methods which have at least quadratic rate of convergence asn tends to infinity see e g brent 1 traub 4 and sect 1

symmetry breaking bifurcation of analytic solutions to free boundary problems an application to a model of tumor growth in this paper we develop a general technique for establishing analyticity of solutions of partial differential equations which depend on a parameter e the technique is worked out primarily for a free boundary problem describing a model of a stationary tumor we prove the existence of infinitely many branches of symmetry breaking solutions which bifurcate from any given radially symmetric steady state these asymmetric solutions are analytic jointly in the spatial variables and in e 1 the model and main result in this paper we present a general technique for establishing analyticity of solutions of systems of partial differential equations which depend analytically on a parameter e the method works not only for boundary value problems but also for free boundary problems in this latter context it can be used to establish long time existence of transient solutions and also to study the existence of spatially asymmetric steady solutions since free boundary problems are typically more challenging than their boundary value counterparts we shall concentrate here on a free boundary problem from developmental biology namely a model of tumor growth to further exemplify the generality of our approach an instance of a boundary value problem in a fixed domain is presented in the last section of the paper a variety of other problems are amenable to the same analysis including in particular the hele shaw model of fluid flow 11 within the last several decades a number of mathematical models have been developed that aimed at describing the evolution of carcinomas see e g 1 5 6 8 12 13 and the references cited there the main objective of these models has been to qualitatively describe under various simplifying assumptions the growth and stability of tumor tissue analysis and simulations of such models are helping to assess the relative importance of various mechanisms affecting tumor growth as well as the efficacy of certain cancer treatments on the other hand the description of the stationary dormant configurations that arise from the models has only been addressed in the case of spherical tumors but otherwise it remains largely unexplored in this paper we develop a method for establishing analyticity of received by the editors august 17 1999 1991 mathematics subject classification primary 35b32 35r35 secondary 35b30 35b60 35c10 35j85 35q80 92c15 95c15

bifurcation for a free boundary problem with surface tension modeling the growth of multi layer tumors this paper is devoted to the study of the bifurcation of a free boundary problem modeling the growth of tumors with the effect of surface tension being considered the existence of infinitely many branches of bifurcation solutions is proved the method of analysis is based on reducing the problem to an operator equation in certain holder space with a nonlinear fredholm operator of index 0 the desired result then follows from the crandall rabinowitz bifurcation theorem

bifurcation from stability to instability for a free boundary problem modeling tumor growth by stokes equation we consider a free boundary problem modeling tumor growth in fluid like tissue the model equations include a diffusion equation for the nutrient concentration and the stokes equation with a source which represents the proliferation of tumor cells the proliferation rate and the cell to cell adhesiveness which keeps the tumor intact are two parameters which characterize the aggressiveness of the tumor for any positive radius r there exists a unique radially symmetric stationary solution with radius r r for a sequence mn r there exist symmetry breaking bifurcation branches of solutions with free boundary r r eyn 0 o e 2 n even 2 for small e w hereyn 0 is the spherical harmonic of mode n 0 furthermore the smallest mn r s aymn r is such that n n r as r in this paper we prove that the radially symmetric stationary solution with r rs is linearly stable if n rs w heren rs mn rs and we prove that strict inequality holds if is small or if is large the biological implications of these results are discussed at the end of the paper

some globally convergent iterative method based on the bisection iteration for solving nonlinear scalar equations abstract we present a globally convergent iterative method for solving nonlinear scalar equations the proposed method is a sort of acceleration technique for the bisection iteration and is always guaranteed to converge under the same condition as that for convergence of the bisection iteration the rate of convergence of the method is proved to be superlinear if the approximation is sufficiently close to the solution

bisection is not optimal on the average we seek the zero of a continuous increasing functionf 0 1 1 1 such thatf 0 1 andf 1 1 it is known that the bisection method makes optimal use ofn function evaluations within a worst case analysis in this paper we study the average error with respect to the natural measure of graf et al 1986 we prove that the bisection method is not optimal on the average actually the average error of the bisection method is about 1 2 n while the average error of the optimal method is less than n with some 1 2

convergence of quasi newton method with new inexact line search quasi newton method is a well known effective method for solving optimization problems since it is a line search method which needs a line search procedure after determining a search direction at each iteration we must decide a line search rule to choose a step size along a search direction in this paper we propose a new inexact line search rule for quasi newton method and establish some global convergent results of this method these results are useful in designing new quasi newton methods moreover we analyze the convergence rate of quasi newton method with the new line search rule

a center manifold analysis for the mullins sekerka model abstract the mullins sekerka model is a nonlocal evolution model for hypersurfaces which arises as a singular limit for the cahn hilliard equation we show that classical solutions exist globally and tend to spheres exponentially fast provided that they are close to a sphere initially our analysis is based on center manifold theory and on maximal regularity

a free boundary problem for a singular system of differential equations an application to a model of tumor growth first published in transactions of the american mathematical society in volume 355 issue 9 published by the american mathematical society

locating and computing in parallel all the simple roots of special functions using pvm an algorithm is proposed for locating and computing in parallel and with certainty all the simple roots of any twice continuously di0erentiable function in any speci1c interval to compute with certainty all the roots the proposed method is heavily based on the knowledge of the total number of roots within the given interval to obtain this information we use results from topological degree theory and in particular the kronecker picard approach this theory gives a formula for the computation of the total number of roots of a system of equations within a given region which can be computed in parallel with this tool in hand we construct a parallel procedure for the localization and isolation of all the roots by dividing the given region successively and applying the above formula to these subregions until the 1nal domains contain at the most one root the subregions with no roots are discarded while for the rest a modi1cation of the well known bisection method is employed for the computation of the contained root the new aspect of the present contribution is that the computation of the total number of zeros using the kronecker picard integral as well as the localization and computation of all the roots is performed in parallel using the parallel virtual machine pvm pvm is an integrated set of software tools and libraries that emulates a general purpose 7exible heterogeneous concurrent computing framework on interconnected computers of varied architectures the proposed algorithm has large granularity and low synchronization and is robust it has been implemented and tested and our experience is that it can massively compute with certainty all the roots in a certain interval performance information from massive computations related to a recently proposed conjecture due to elbert this issue j comput appl math 133 2001 65 83 is reported c 2001 elsevier science b v all rights reserved

stability and instability of liapunov schmidt and hopf bifurcation for a free boundary problem arising in a tumor model first published in transactions of the american mathematical society in volume 360 issue 10 published by the american mathematical society

can adaption help on the average we study adaptive information for approximation of linear problems in a separable hilbert space equipped with a probability measure it is known that adaption does not help in the worst case for linear problems we prove that adaption also doesnot help on the average that is there exists nonadaptive information which is as powerful as adaptive information this result holds for orthogonally invariant measures we provide necessary and sufficient conditions for a measure to be orthogonally invariant examples of orthogonally invariant measures include gaussian measures and in the finite dimensional case weighted lebesgue measures

bifurcation for a free boundary problem modeling the growth of tumors with a drug induced nonlinear proliferation rate abstract in this paper we study a free boundary model describing growth of tumors under action of drugs to our knowledge in theoretical discussion for free boundary problems the proliferation rate in tumor models discussed in previous bifurcation results is a linear function of nutrients and inhibitors whereas in this paper we consider the net proliferation rate as a nonlinear function depending on both nutrients and drugs first the existence and the uniqueness of radially symmetric stationary solutions are obtained second we prove that symmetry breaking solutions bifurcate from the radially symmetric stationary solutions when the concentration of drug on the boundary of tumor is less than one in the rescaled model

a new unconstrained optimization method for imprecise function and gradient values a new algorithm for unconstrained optimization is presented which is based on a modified one dimensional bisection method the algorithm actually uses only the signs of function and gradient values thus it can be applied to problems with imprecise function and gradient values it converges in one iteration on quadratic functions of n variables it rapidly minimizes general functions and it does not require evaluation or estimation of the matrix of second partial derivatives the algorithm has been implemented and tested performance information for wellknown test functions is reported q 1996 academic press inc

lie group action and stability analysis of stationary solutions for a free boundary problem modelling tumor growth abstract in this paper we study asymptotic behavior of solutions for a free boundary problem modelling tumor growth we first establish a general result for differential equations in banach spaces possessing a lie group action which maps a solution into new solutions we prove that a center manifold exists under certain assumptions on the spectrum of the linearized operator without assuming that the space in which the equation is defined is of either d a or d a type by using this general result and making delicate analysis of the spectrum of the linearization of the stationary free boundary problem we prove that if the surface tension coefficient is larger than a threshold value then the unique stationary solution is asymptotically stable modulo translations provided the constant c is sufficiently small whereas if then this stationary solution is unstable

locating and computing all the simple roots and extrema of a function this paper describes and analyzes two algorithms for locating and computing with certainty all the simple roots of a twice continuously differentiable function f colon a b subset r to r and all the extrema of a three times continuously differentiable function in a b the first algorithm locates and computes all the simple roots or all the extrema while the second one is more efficient in the case where both simple roots and extrema are required this paper also gives analytical estimation of the expected complexity of the algorithms based on the distribution of the roots in a b here only the case of uniform distribution is examined which is also the approach to be followed when no statistical data are available for the function at hand the algorithms have been implemented and tested performance information for a well known bessel function is reported

a parabolic hyperbolic system modeling the growth of a tumor abstract in this paper we consider a model with tumor microenvironment involving nutrient density extracellular matrix and matrix degrading enzymes which satisfy a coupled system of pdes with a free boundary for this coupled parabolic hyperbolic free boundary problem we prove that there is a unique radially symmetric solution globally in time the stationary problem involves an ode system which is transformed into a singular integro differential equation we establish a well posed theorem for such general types of equations by the shooting method the theorem is then applied to our problem for the existence of a stationary solution in addition for this highly nonlinear problem we also prove the uniqueness of the stationary solution which is a nontrivial result moreover numerical simulations indicate that the stationary solution is likely locally asymptotically stable for reasonable range of parameters

why does information based complexity use the real number model abstract we explain why information based complexity uses the real number model results in the real number model are essentially the same as in floating point arithmetic with fixed precision modulo two important assumptions namely we use only stable algorithms the approximation error is not too small compared to the product of the condition number the roundoff unit of floating point arithmetic and the accumulation constant of a stable algorithm we illustrate this by an example of solving nonlinear equations by bisection we also indicate the possible tradeoffs between complexity and stability and the need of using multiple or varying precision for ill conditioned problems

analysis of a mathematical model of the growth of necrotic tumors abstract in this paper we study a model of necrotic tumor growth the tumor comprises necrotic cells which occupy a radially symmetric core and life proliferating cells which occupy a radially symmetric shell adjacent to the core the proliferating cells receive nutrients through diffusion from the outer boundary as well as by means of blood flow through a network of capillary vessels the mathematical model describes the evolution of the nutrient concentration between the boundary of the necrotic core r t and the outer boundary of the tumor r r t within the core itself the concentration is a constant nec a level under which life cells cannot be sustained both of the surfaces r t and r r t are free boundaries which are unknown in advance under some assumptions on the parameters we prove that i there exists a stationary solution with radii r s r r s ii for any initial data near the stationary solution the time dependent model has a unique solution r t with free boundaries r t r r t and iii t s and r t r s as t

implicitly restarted arnoldi with purification for the shift invert transformation the need to determine a few eigenvalues of a large sparse generalised eigenvalue problem az bx with positive semidefinite b arises in many physical situations for example in a stability analysis of the discretised navier stokes equation a common technique is to apply arnoldi s method to the shift invert transformation but this can suffer from numerical instabilities as is illustrated by a numerical example in this paper a new method that avoids instabilities is presented which is based on applying the implicitly restarted arnoldi method with the b semi inner product and a purification step the paper contains a rounding error analysis and ends with brief comments on some extensions

asymptotic stability for a free boundary problem arising in a tumor model abstract we consider a tumor model in which all cells are proliferating at a rate and their density is proportional to the nutrient concentration the model consists of a coupled system of an elliptic equation and a parabolic equation with the tumor boundary as a free boundary it is known that for an appropriate choice of parameters there exists a unique spherically symmetric stationary solution with radius r s which is independent of it was recently proved that there is a function r s such that the spherical stationary solution is linearly stable if r s and linearly unstable if r s in this paper we prove that the spherical stationary solution is nonlinearly stable or asymptotically stable if r s

analysis of a tumor model free boundary problem with a nonlinear boundary condition abstract in this paper we study a free boundary problem modeling the growth of nonnecrotic tumors with angiogenesis this model differs from the other tumor models studied in existing literatures at the point that it has a nonlinear boundary value condition for the nutrient concentration we first study spherically symmetric version of this model we prove that there exists a unique spherically symmetric stationary solution which is asymptotically stable under spherically symmetric perturbation next we make rigorous analysis to the spherically asymmetric version of this model by using some abstract theory of parabolic differential equations in banach manifold we prove that this free boundary problem is locally well posed in little holder spaces and the radial stationary solution is asymptotically stable in case the surface tension coefficient is larger than a threshold value whereas unstable in case is less than this threshold value

well posedness and stability of a free boundary problem modeling the growth of multi layer tumors abstract in this paper we study well posedness and stability of a free boundary problem modeling the growth of multi layer tumors under the action of external inhibitors an important feature of this problem is that the surface tension of the free boundary is taken into account we first reduce this free boundary problem into an evolution equation in little holder space and use the well posedness theory for differential equations in banach spaces of parabolic type i e equations which are treatable by using the analytic semi group theory to prove that this free boundary problem is locally well posed for initial data belonging to a little holder space next we study flat solutions of this problem we obtain all flat stationary solutions and give a precise description of asymptotic stability of these stationary solutions under flat perturbations finally we investigate asymptotic stability of flat stationary solutions under non flat perturbations by carefully analyzing the spectrum of the linearized stationary problem and employing the theory of linearized stability for differential equations in banach spaces of parabolic type we give a complete analysis of stability and instability of all flat stationary solutions under small non flat perturbations

new sequential and parallel derivative free algorithms for unconstrained minimization this paper presents sequential and parallel derivative free algorithms for finding a local minimum of smooth and nonsmooth functions of practical interest it is proved that under mild assumptions a sufficient decrease condition holds for a nonsmooth function based on this property the algorithms explore a set of search directions and move to a point with a sufficiently lower functional value if the function is strictly differentiable at its limit points a sub sequence of points generated by the algorithm converges to a first order stationary point nabla f x 0 if the function is convex around its limit points convergence of a subsequence to a point with nonnegative directional derivatives on a set of search directions is ensured preliminary numerical results on sequential algorithms show that they compare favorably with the recently introduced pattern search methods

a class of gradient unconstrained minimization algorithms with adaptive stepsize in this paper the development convergence theory and numerical testing of a class of gradient unconstrained minimization algorithms with adaptive stepsize are presented the proposed class comprises four algorithms the first two incorporate techniques for the adaptation of a common stepsize for all coordinate directions and the other two allow an individual adaptive stepsize along each coordinate direction all the algorithms are computationally efficient and possess interesting convergence properties utilizing estimates of the lipschitz constant that are obtained without additional function or gradient evaluations the algorithms have been implemented and tested on some well known test cases as well as on real life artificial neural network applications and the results have been very satisfactory

optimal solution of nonlinear equations satisfying a lipschitz condition for a given nonnegative we seek a pointx such that f x wheref is a nonlinear transformation of the cubeb 0 1 m into or p p 1 satisfying a lipschitz condition with the constantk and having a zero inb the information operator onf consists ofn values of arbitrary linear functionals which are computed adaptively the pointx is constructed by means of an algorithm which is a mapping depending on the information operator we find an optimal algorithm i e algorithm with the smallest error which usesn function evaluations computed adaptively we also exhibit nearly optimal information operators i e the linear functionals for which the error of an optimal algorithm that uses them is almost minimal nearly optimal information operators consists ofn nonadaptive function evaluations at equispaced pointsx j in the cubeb this result exhibits the superiority of the t aird and j rice procedure zsrch imsl library 1 over sobol s approach 7 for solving nonlinear equations in our class of functions we also prove that the simple search algorithm which yields a pointx x k such that left f x k right mathop min limits 1 leqq j leqq n left f x j right is nearly optimal the complexity i e the minimal cost of solving our problem is roughly equal to k m

bifurcation for a free boundary problem modeling the growth of multi layer tumors abstract in this paper we study bifurcations for a free boundary problem modeling the growth of multi layer tumors under the action of inhibitors an important feature of this problem is that the surface tension effect of the free boundary is taken into account by reducing this problem into an abstract bifurcation equation in a banach space overcoming some technical difficulties and finally using the crandall rabinowitz bifurcation theorem we prove that this problem has infinitely many branches of bifurcation solutions bifurcating from the flat solution

rational krylov sequence methods for eigenvalue computation abstract algorithms to solve large sparse eigenvalue problems are considered a new class of algorithms which is based on rational functions of the matrix is described the lanczos method the arnoldi method the spectral transformation lanczos method and rayleigh quotient iteration all are special cases but there are also new algorithms which correspond to rational functions with several poles in the simplest case a basis of a rational krylov subspace is found in which the matrix eigenvalue problem is formulated as a linear matrix pencil with a pair of hessenberg matrices

analysis of a mathematical model for tumor growth under indirect effect of inhibitors with time delay in proliferation in this paper a mathematical model for tumor growth with time delay in proliferation under indirect effect of inhibitor is studied the delay represents the time taken for cells to undergo mitosis nonnegativity of solutions is investigated the steady state analysis is presented with respect to the magnitude of the delay existence of hopf bifurcation is proved for some parameter values local and global stability of the stationary solutions are proved for other ones the analysis of the effect of inhibitor s parameters on tumor s growth is presented the results show that dynamical behavior of solutions of this model is similar to that of solutions for corresponding non retarded problems for some parameter values

bifurcation solutions of a free boundary problem modeling tumor growth with angiogenesis abstract in this paper we study bifurcation solutions from the unique radial solution of a free boundary problem modeling stationary state of tumors with angiogenesis this model comprises two elliptic equations describing the distribution of the nutrient concentration x and the inner pressure p p x unlike similar tumor models that have been intensively studied in the literature where dirichlet boundary condition for is imposed in this model the boundary condition for is a robin boundary condition existence and uniqueness of a radial solution of this model have been successfully proved in a recently published paper 20 in this paper we study existence of nonradial solutions by using the bifurcation method let k k 2 be the sequence of eigenvalues of the linearized problem we prove that there exists a positive integer k 2 such that in the two dimension case for any k k k is a bifurcation point and in the three dimension case for any even k k k is also a bifurcation point

analysis of a free boundary problem modeling the growth of multicell spheroids with angiogenesis abstract in this paper we study a free boundary problem modeling the growth of vascularized tumors the model is a modification to the byrne chaplain tumor model that has been intensively studied during the past two decades the modification is made by replacing the dirichlet boundary value condition with the robin condition which causes some new difficulties in making rigorous analysis of the model particularly on existence and uniqueness of a radial stationary solution in this paper we successfully solve this problem we prove that this free boundary problem has a unique radial stationary solution which is asymptotically stable for large surface tension coefficient whereas unstable for small surface tension coefficient tools used in this analysis are the geometric theory of abstract parabolic differential equations in banach spaces and spectral analysis of the linearized operator

restrictions on implicit filtering techniques for orthogonal projection methods abstract we consider the class of the orthogonal projection methods opm to solve iteratively large eigenvalue problems an opm is a method that projects a large eigenvalue problem on a smaller subspace in this subspace an approximation of the eigenvalue spectrum can be computed from a small eigenvalue problem using a direct method examples of opms are the arnoldi and the davidson method we show how an opm can be restarted implicitly and explicitly this restart can be used to remove a specific subset of vectors from the approximation subspace this is called explicit filtering an implicit restart can also be combined with an implicit filtering step i e the application of a polynomial or rational function on the subspace even if inaccurate arithmetic is assumed however the condition for the implicit application of a filter is that the rank of the residual matrix must be small

a dimension reducing method for unconstrained optimization abstract a new method for unconstrained optimization in r n is presented this method reduces the dimension of the problem in such a way that it can lead to an iterative approximate formula for the computation of n 1 components of the optimum while its remaining component is computed separately using the final approximations of the other components it converges quadratically to a local optimum and it requires storage of order n 1 n 1 besides it does not require a good initial guess for one component of the optimum and it does not directly perform gradient evaluations thus it can be applied to problems with imprecise gradient values moreover a procedure for transforming the matrix formed by our method into a symmetric as well as into a diagonal one is presented furthermore the proposed dimension reducing scheme using finite difference gradient and hessian is presented the methods have been implemented and tested performance information for well known test functions is reported

analysis of a mathematical model for tumor growth with gibbs thomson relation abstract in this paper we study a mathematical model for the growth of nonnecrotic solid tumor the tumor is assumed to be radially symmetric and its radius r t is an unknown function of time t as tumor growth and the model is in the form of a free boundary problem the feature of the model is that a gibbs thomson relation is taken into account which results an interesting phenomenon that there exist two stationary solutions depending on the model parameters the global existence and uniqueness of solution are established by denoting c the ratio of the diffusion time scale to the tumor doubling time scale we prove that for sufficiently small c 0 the stationary solution with the larger radius is asymptotically stable and the other smaller one is unstable

bifurcation for a free boundary tumor model with extracellular matrix and matrix degrading enzymes abstract we study a free boundary problem modeling solid tumor growth with ecm and mde interactions the production rate of mde by tumor cells is a nonlinear function depending on nutrients and mde this nonlinear term is more complicated for this model we can not give the explicit expressions or integral expressions for mde terms while we only get the existence and uniqueness at first we show the existence and the uniqueness of radially symmetric stationary solutions then the existence of symmetry breaking solutions bifurcating from the radially symmetric stationary solutions is obtained

a free boundary problem modeling the cell cycle and cell movement in multicellular tumor spheroids abstract this paper deals with a mathematical model describing the cell cycle dynamics and chemotactic driven cell movement in a multicellular tumor spheroid tumor cells consist of two types of cells proliferating cells and quiescent cells which have different chemotactic responses to an extracellular nutrient supply the model is a free boundary problem for a nonlinear system of reaction diffusion advection equations where the free boundary is the outer boundary of the spheroid the free boundary condition is quite novel due to different velocity of two types of cells the global existence and uniqueness of solutions to the model is proved the proof is based on a fixed point argument together with the l p theory for parabolic equations with the third boundary condition

well posedness and stability of a multi dimensional tumor growth model we study a moving boundary problem modeling the growth of in vitro tumors this problem consists of two elliptic equations describing the distribution of the nutrient and the internal pressure respectively and a first order partial differential equation describing the evolution of the moving boundary an important feature is that the effect of surface tension on the moving boundary is taken into account we show that this problem is locally well posed for a large class of initial data by using analytic semi group theory we also prove that if the surface tension coefficient is larger than a threshold value then the unique flat equilibrium is asymptotically stable whereas in the case this flat equilibrium is unstable

convergence of descent method without line search line search method is sometimes a useful and efficient technique for solving unconstrained optimization problems especially for solving small and middle scale problems however a line search procedure is necessary at each iteration leading to a significant amount of computation in order to reduce the evaluations of objective functions and gradients the line search procedure should be avoided in algorithm design in this paper we propose a new descent method without line search for unconstrained optimization problems the algorithm is required to estimate some parameters at each iteration we analyze theoretically the global convergence of the new algorithm under mild conditions these theoretical conclusions can help us in designing new efficient methods for optimization problems

analysis of a free boundary tumor model with angiogenesis we consider a free boundary problem for a spherically symmetric tumor with free boundary r 0 if liminft t 0 surprisingly we exhibit solutions when is not small where t 0 exponentially in t while r t exponentially in t finally we prove the global asymptotic stability of steady state when is sufficiently small

optac a portable software package for analyzing and comparing optimization methods by visualization abstract a software package for analyzing and comparing optimization methods is presented this package displays using different colors the regions of convergence to the minima of a given function for various optimization methods it displays also the rate of their convergence as well as the regions of divergence of these methods moreover this package gives quantitative information regarding the total convergence area in a specific domain for various minima using optac optimization analysis and comparisons we are able to see in a picture the advantages and disadvantages of any optimization method as well as to compare various methods in order to choose the proper method for a given class of problems the optac package is self contained and conforms to the ansi 1977 fortran standards

locating and computing arbitrarily distributed zeros the problem of locating and computing with certainty all the simple roots of a twice continuously differentiable function f colon a b subset r to r is studied when some additional information on the distribution of the roots in the interval is available the framework is the one proposed by siam j sci comput 17 1996 pp 1232 1248 where only the uniform case was examined this paper settles some of the problems posed there and generalizes some of its results by considering an arbitrary distribution of the roots in a b the theoretical results are accompanied by simulations in a number of problems of various size