blow up and global existence for a nonlocal degenerate parabolic system abstract this paper investigates the blow up and global existence of nonnegative solutions of the system u t u m a v p v t v n b u q x t 0 t with homogeneous dirichlet boundary data where r n is a bounded domain with smooth boundary m n 1 1 p q a b 0 and dx it is proved that if pq mn there exist both global and blow up nonnegative solutions when pq mn we show that if the domain is sufficiently small every nonnegative solution is global while if the domain is large enough that is if it contains a sufficiently large ball there exists no global solution in particular when p n q m we show that every positive solution exists globally iff x dx 1 ab where x is the unique positive solution of the linear elliptic problem x 1 x x 0 x

the blow up rate for a system of heat equations with nonlinear boundary conditions this paper deals with the blow up rate estimates of positive solutions for systems of heat equations with nonlinear boundary conditions the upper and lower bounds of blow up rate are obtained

parabolic evolution equations and nonlinear boundary conditions on etudie le probleme aux valeurs limites et initiales u 7b a t u f t u 7b b t u g t u 0 t t u 0 u 0 qui peut etre considere comme une contrepartie abstraite a des problemes paraboliques semilineaires

the effect of the number of simulations on the exponents obtained by finite size scaling relations for the seven dimensional ising model on the creutz cellular automaton abstract the seven dimensional ising model is simulated on the creutz cellular automaton using the finite size lattice with the linear dimension 4 l 8 the exponents in the finite size scaling relations for the magnetic susceptibility the order parameter and the specific heat at the infinite lattice critical temperature are computed to be 3 544 24 3 505 55 3 501 52 1 78 64 1 77 38 1 76 46 and 0 02 11 0 01 17 0 01 16 using 4 l 8 for 7 14 and 21 independent simulations respectively as the number of independent simulations increase exponents are in very good agreement with the theoretical predictions of 7 2 7 4 and 0 respectively the values for the critical temperature of infinite lattice 12 870 12 12 870 10 12 869 3 and 12 871 15 12 871 5 12 871 5 are obtained from the straight line fit of the magnetic susceptibility maxima and the specific heat using 4 l 8 for 7 14 and 21 independent simulations respectively as the number of independent simulations increases the obtained results are in very good agreement with the more precise value of 12 86902 33

is a complex valued stepsize advantageous in complex valued gradient learning algorithms complex gradient methods have been widely used in learning theory and typically aim to optimize real valued functions of complex variables the stepsize of complex gradient learning methods cglms is a positive number and little is known about how a complex stepsize would affect the learning process to this end we undertake a comprehensive analysis of cglms with a complex stepsize including the search space convergence properties and the dynamics near critical points furthermore several adaptive stepsizes are derived by extending the barzilai borwein method to the complex domain in order to show that the complex stepsize is superior to the corresponding real one in approximating the information in the hessian a numerical example is presented to support the analysis

an extension of the fletcher reeves method to linear equality constrained optimization problem in this paper combining the feasible direction method with the conjugate gradient method we propose a feasible fletcher reeves conjugate gradient method for solving linear equality constrained optimization problem the directions generated by the method are feasible and descent for the objective function under mild conditions we show that the proposed method with exact line search is globally convergent numerical experiments are also given to show the efficiency of the method

a limited memory steepest descent method the possibilities inherent in steepest descent methods have been considerably amplified by the introduction of the barzilai borwein choice of step size and other related ideas these methods have proved to be competitive with conjugate gradient methods for the minimization of large dimension unconstrained minimization problems this paper suggests a method which is able to take advantage of the availability of a few additional long vectors of storage to achieve a significant improvement in performance both for quadratic and non quadratic objective functions it makes use of certain ritz values related to the lanczos process lanczos in j res nat bur stand 45 255 282 1950 some underlying theory is provided and numerical evidence is set out showing that the new method provides a competitive and more simple alternative to the state of the art l bfgs limited memory method

blow up analysis for a system of heat equations with nonlinear flux which obey different laws abstract we consider a system of heat equations u t u and v t v in 0 t completely coupled by nonlinear boundary conditions u e p v u v u q e v on 0 t we prove that the solutions always blow up in finite time for non zero and non negative initial values also the blow up only occurs on with c 1 t t p 2 p q max u x t c 2 t t p 2 p q log c 3 t t q 1 2 p q max v x t log c 4 t t q 1 2 p q for p q 0 0 1 and 0 p

global existence and finite time blow up for a degenerate reaction diffusion system abstract this paper investigates the blow up and global existence of solutions of the degenerate reaction diffusion system u t u m u v p v t v n u q v x t 0 t with homogeneous dirichlet boundary data where r n is a bounded domain with smooth boundary m n 1 0 and p q 0 it is proved that if m n and pq m n every nonnegative solution is global whereas if m or n or pq m n there exist both global and blow up nonnegative solutions when m n and pq m n we show that there exists 1 which depends on the parameters p q m n such that all positive solutions are global if 1 while if 1 1 all positive solutions blow up in finite time where 1 is the first dirichlet eigenvalue for the laplacian on

spectral projected gradient method with inexact restoration for minimization with nonconvex constraints this work takes advantage of the spectral projected gradient direction within the inexact restoration framework to address nonlinear optimization problems with nonconvex constraints the proposed strategy includes a convenient handling of the constraints together with nonmonotonic features to speed up convergence the numerical performance is assessed by experiments with hard spheres problems pointing out that the inexact restoration framework provides an adequate environment for the extension of the spectral projected gradient method for general nonlinearly constrained optimization

critical exponents of the two dimensional ising model with next nearest neighbor and four spin interactions on the creutz cellular automaton abstract the two dimensional ising model with nearest neighbor next nearest neighbor and four spin interactions is simulated on the creutz cellular automaton for j 0 the values of critical exponents in the antiferromagnetic interaction region are estimated within the framework of the finite size scaling theory the calculated exponents for interaction ratios r 1 20 and j 0 are in a good agreement with their theoretical values of 0 1 8 7 4 1 4 and v 1 for the ising model

on the linear convergence of descent methods for convex essentially smooth minimization consider the problem of minimizing over a polyhedral set the composition of an affine mapping with a strictly convex essentially smooth function a general result on the linear convergence of descent methods for solving this problem is presented by applying this result the linear convergence of both the gradient projection algorithm of goldstein and levitin and polyak and a matrix splitting algorithm using regular splitting is established the results do not require that the cost function be strongly convex or that the optimal solution set be bounded the key to the analysis lies in a new error bound for estimating the distance from a feasible point to the optimal solution set

weak sharp solutions of variational inequalities xuexiang huang pointed out that the statement on line 5 of page 184 of p marcotte and d zhu siam j optim 9 1999 pp 179 189 is invalid and supplied an alternative proof of the inequality that appears on line 10 here is a simple proof of this result due to an anonymous referee the substitution starts at line 4 and ends at line 11

the barzilai and borwein gradient method for the large scale unconstrained minimization problem the barzilai and borwein gradient method for the solution of large scale unconstrained minimization problems is considered this method requires few storage locations and very inexpensive computations furthermore it does not guarantee descent in the objective function and no line search is required recently the global convergence for the convex quadratic case has been established however for the nonquadratic case the method needs to be incorporated in a globalization scheme in this work a nonmonotone line search strategy that guarantees global convergence is combined with the barzilai and borwein method this strategy is based on the nonmonotone line search technique proposed by grippo lampariello and lucidi siam j numer anal 23 1986 pp 707 716 numerical results to compare the behavior of this method with recent implementations of the conjugate gradient method are presented these results indicate that the global barzilai and borwein method may allow some significant reduction in the number of line searches and also in the number of gradient evaluations

a family of hager zhang conjugate gradient methods for system of monotone nonlinear equations abstract this paper presents two modified hager zhang hz conjugate gradient methods for solving large scale system of monotone nonlinear equations the methods were developed by combining modified forms of the one parameter method by hager and zhang 2006 and the hyperplane projection technique global convergence and numerical results of the methods are established preliminary numerical results show that the proposed methods are promising and more efficient compared to the methods presented by mushtak and keyvan 2018 and sun et al 2017

augmented lagrangian methods under the constant positive linear dependence constraint qualification two augmented lagrangian algorithms for solving kkt systems are introduced the algorithms differ in the way in which penalty parameters are updated possibly infeasible accumulation points are characterized it is proved that feasible limit points that satisfy the constant positive linear dependence constraint qualification are kkt solutions boundedness of the penalty parameters is proved under suitable assumptions numerical experiments are presented

a multivariate spectral projected gradient method for bound constrained optimization in this paper we consider a multivariate spectral projected gradient mspg method for bound constrained optimization combined with a quasi newton property the multivariate spectral projected gradient method allows an individual adaptive step size along each coordinate direction on the basis of nonmonotone line search global convergence is established a numerical comparison with the traditional spg method shows that the method is promising

robust orthogonal complement principal component analysis abstractrecently the robustification of principal component analysis pca has attracted lots of attention from statisticians engineers and computer scientists in this work we study the type of outliers that are not necessarily apparent in the original observation space but can seriously affect the principal subspace estimation based on a mathematical formulation of such transformed outliers a novel robust orthogonal complement principal component analysis roc pca is proposed the framework combines the popular sparsity enforcing and low rank regularization techniques to deal with row wise outliers as well as element wise outliers a nonasymptotic oracle inequality guarantees the accuracy and high breakdown performance of roc pca in finite samples to tackle the computational challenges an efficient algorithm is developed on the basis of stiefel manifold optimization and iterative thresholding furthermore a batch variant is proposed to significantly reduce the cost in ultra high dimensions th

weak sharp minima in mathematical programming the notion of a sharp or strongly unique minimum is extended to include the possibility of a nonunique solution set these minima will be called weak sharp minima conditions necessary for the solution set of a minimization problem to be a set of weak sharp minima are developed in both the unconstrained and constrained cases these conditions are also shown to be sufficient under the appropriate convexity hypotheses the existence of weak sharp minima is characterized in the cases of linear and quadratic convex programming and for the linear complementarity problem in particular a result of mangasarian and meyer is reproduced that shows that the solution set of a linear program is always a set of weak sharp minima whenever it is nonempty consequences for the convergence theory of algorithms are also examined especially conditions yielding finite termination

global convergence properties of conjugate gradient methods for optimization this paper explores the convergence of nonlinear conjugate gradient methods without restarts and with practical line searches the analysis covers two classes of methods that are globally convergent on smooth nonconvex functions some properties of the fletcher reeves method play an important role in the first family whereas the second family shares an important property with the polak ribiere method numerical experiments are presented

estimation of the optical constants and the thickness of thin films using unconstrained optimization the problem of estimating the thickness and the optical constants of thin films using transmission data only is very challenging from the mathematical point of view and has a technological and an economic importance in many cases it represents a very ill conditioned inverse problem with many local nonglobal solutions in a recent publication we proposed nonlinear programming models for solving this problem well known software for linearly constrained optimization was used with success for this purpose in this paper we introduce an unconstrained formulation of the nonlinear programming model and we solve the estimation problem using a method based on repeated calls to a recently introduced unconstrained minimization algorithm numerical experiments on computer generated films show that the new procedure is reliable

qn like variable storage conjugate gradients both conjugate gradient and quasi newton methods are quite successful at minimizing smooth nonlinear functions of several variables and each has its advantages in particular conjugate gradient methods require much less storage to implement than a quasi newton code and therefore find application when storage limitations occur they are however slower so there have recently been attempts to combine cg and qn algorithms so as to obtain an algorithm with good convergence properties and low storage requirements one such method is the code conmin due to shanno and phua it has proven quite successful but it has one limitation it has no middle ground in that it either operates as a quasi newton code using o n 2 storage locations or as a conjugate gradient code using 7n locations but it cannot take advantage of the not unusual situation where more than 7n locations are available but a quasi newton code requires an excessive amount of storage

accelerated scaled memoryless bfgs preconditioned conjugate gradient algorithm for unconstrained optimization abstract an accelerated scaled memoryless bfgs preconditioned conjugate gradient algorithm for solving unconstrained optimization problems is presented the basic idea is to combine the scaled memoryless bfgs method and the preconditioning technique in the frame of the conjugate gradient method the preconditioner which is also a scaled memoryless bfgs matrix is reset when the beale powell restart criterion holds the parameter scaling the gradient is selected as a spectral gradient for the steplength computation the method has the advantage that in conjugate gradient algorithms the step lengths may differ from 1 by two order of magnitude and tend to vary unpredictably thus we suggest an acceleration scheme able to improve the efficiency of the algorithm under common assumptions the method is proved to be globally convergent it is shown that for uniformly convex functions the convergence of the accelerated algorithm is still linear but the reduction in the function values is significantly improved in mild conditions the algorithm is globally convergent for strongly convex functions computational results for a set consisting of 750 unconstrained optimization test problems show that this new accelerated scaled conjugate gradient algorithm substantially outperforms known conjugate gradient methods scalcg 3 4 5 6 conmin by shanno and phua 1976 1978 42 43 hestenes and stiefel 1952 25 polak ribiere polyak 1969 32 33 dai and yuan 2001 17 dai and liao 2001 t 1 14 conjugate gradient with sufficient descent condition 7 hybrid dai and yuan 2001 17 hybrid dai and yuan zero 2001 17 cg descent by hager and zhang 2005 2006 22 23 as well as quasi newton lbfgs method 26 and truncated newton method by nash 1985 27

critical behavior of the two dimensional ising model with next nearest neighbor antiferromagnetic interaction on the creutz cellular automaton the two dimensional ising model with nearest and next nearest neighbor antiferromagnetic interactions of ratio r is simulated on the creutz cellular automaton the values of the critical temperature and the critical exponents for o r are estimated within the framework of the finite size scaling theory the dependence of tc on r agrees with the crossover scaling theory for 1 75 the results for 0 r are compatible with ising critical behavior

a barzilai and borwein scaling conjugate gradient method for unconstrained optimization problems in this paper we combine the conjugate gradient method with the barzilai and borwein gradient method and propose a barzilai and borwein scaling conjugate gradient method for nonlinear unconstrained optimization problems the new method does not require to compute and store matrices associated with hessian of the objective functions and has an advantage of less computational efforts moreover the descent direction property and the global convergence are established when the line search fulfills the wolfe conditions the limited numerical experiments and comparisons show that the proposed algorithm is potentially efficient

blow up phenomena for a system of semilinear heat equations with nonlinear boundary flux abstract this paper deals with the blow up for a system of semilinear r laplace heat equations with nonlinear boundary flux it is shown that under certain conditions on the nonlinearities and data blow up will occur at some finite time and when blow up does occur upper and lower bounds for the blow up time are obtained

a new nonlinear filter constructed from the newton method and epr in image restoration the edge preserving regularization method epr is effective to remove gaussian noise and impulse noise in image restorations but till now we could not say that the problem has been solved very efficiently by state of the art optimization method in our paper we will construct a new nonlinear filter from the edge preserving model and the newton method the parameter estimation model in the nonlinear filter will be established and the solution existence of the model will be proved at the same time numerical results demonstrate that the nonlinear filter removes the gaussian noise from an image as effectively as the edge preserving regularization method does and the time for the nonlinear filter to restore an image is less than half time for the edge preserving regularization method

a feasible method for optimization with orthogonality constraints minimization with orthogonality constraints e g x top x i and or spherical constraints e g vert x vert 2 1 has wide applications in polynomial optimization combinatorial optimization eigenvalue problems sparse pca p harmonic flows 1 bit compressive sensing matrix rank minimization etc these problems are difficult because the constraints are not only non convex but numerically expensive to preserve during iterations to deal with these difficulties we apply the cayley transform a crank nicolson like update scheme to preserve the constraints and based on it develop curvilinear search algorithms with lower flops compared to those based on projections and geodesics the efficiency of the proposed algorithms is demonstrated on a variety of test problems in particular for the maxcut problem it exactly solves a decomposition formulation for the sdp relaxation for polynomial optimization nearest correlation matrix estimation and extreme eigenvalue problems the proposed algorithms run very fast and return solutions no worse than those from their state of the art algorithms for the quadratic assignment problem a gap 0 842 to the best known solution on the largest problem tai256c in qaplib can be reached in 5 min on a typical laptop

a class of nonmonotone stabilization methods in unconstrained optimization this paper deals with the solution of smooth unconstrained minimization problems by newton type methods whose global convergence is enforced by means of a nonmonotone stabilization strategy in particular a stabilization scheme is analyzed which includes different kinds of relaxation of the descent requirements an extensive numerical experimentation is reported

on the steplength selection in gradient methods for unconstrained optimization abstract the seminal paper by barzilai and borwein 1988 has given rise to an extensive investigation leading to the development of effective gradient methods several steplength rules have been first designed for unconstrained quadratic problems and then extended to general nonlinear optimization problems these rules share the common idea of attempting to capture in an inexpensive way some second order information however the convergence theory of the gradient methods using the previous rules does not explain their effectiveness and a full understanding of their practical behaviour is still missing in this work we investigate the relationships between the steplengths of a variety of gradient methods and the spectrum of the hessian of the objective function providing insight into the computational effectiveness of the methods for both quadratic and general unconstrained optimization problems our study also identifies basic principles for designing effective gradient methods

re entrant phase transitions of the blume emery griffiths model for a simple cubic lattice on the cellular automaton the spin 1 ising beg model with the nearest neighbor bilinear and biquadratic interactions and single ion anisotropy is simulated on a cellular automaton which is improved from the creutz cellular automaton cca for a simple cubic lattice the simulations have been made for several sets of parameters k jk j and d jd j in the 3 d j 0 3 d j 0 and 1 k j 0 1 k j 0 parameter regions the re entrant and double re entrant phase transitions of the beg model are determined from the temperature variations of the thermodynamic quantities m q and the phase diagrams characterizing phase transitions are compared with those obtained from other methods

avoiding the maratos effect by means of a nonmonotone line search ii inequality constrained problems feasible iterates when solving inequality constrained optimization problems via sequential quadratic programming sqp it is potentially advantageous to generate iterates that all satisfy the constraints all quadratic programs encountered are then feasible and there is no need for a surrogate merit function feasibility of the successive iterates is in fact required in many contexts such as in real time applications or when the objective function is not defined outside the feasible set it has recently been shown that this is indeed possible by means of a suitable perturbation of the original sqp iteration without losing superlinear convergence in this context the well known maratos effect is compounded by the possible infeasibility of the full step of one even close to a solution these difficulties have been accommodated by making use of a suitable modification of a bending technique proposed by mayne and polak requiring evaluation of the constraints function at an auxiliary point at each iteration in part i

a globally convergent bfgs method with nonmonotone line search for non convex minimization in this paper we propose a modified bfgs broyden fletcher goldfarb shanno method with nonmonotone line search for unconstrained optimization under some mild conditions we show that the method is globally convergent without a convexity assumption on the objective function we also report some preliminary numerical results to show the efficiency of the proposed method

the simple quadratic ising model with crossing bonds abstract we investigate the ferromagnetic critical line of the simple quadratic ising model with nearest neighbour and next nearest neighbour interactions by means of monte carlo renormalisation group and analytic methods a comparison of the monte carlo results with the renormalisation group trajectories yields the asymptotic behaviour of the critical line in the strong coupling limit a number of figures of spin configurations is included to illustrate the physical behaviour of the model

convergence properties of nonmonotone spectral projected gradient methods in a recent paper a nonmonotone spectral projected gradient spg method was introduced by birgin et al for the minimization of differentiable functions on closed convex sets and extensive presented results showed that this method was very efficient in this paper we give a more comprehensive theoretical analysis of the spg method in doing so we remove various boundedness conditions that are assumed in existing results such as boundedness from below of f boundedness of xk or existence of accumulation point of xk if f is uniformly continuous we establish the convergence theory of this method and prove that the spg method forces the sequence of projected gradients to zero moreover we show under appropriate conditions that the spg method has some encouraging convergence properties such as the global convergence of the sequence of iterates generated by this method and the finite termination etc therefore these results show that the spg method is attractive in theory

a two dimensional ising model with non universal critical behaviour it is shown that the two dimensional square lattice ising model with the nearest neighbour ferromagnetic interaction j the next nearest neighbour antiferromagnetic interaction j 0 and the four body interaction jint shows non universal critical behaviour when j is small the estimations of critical temperature tc and the critical exponent of susceptibility are performed within errors of 0 003 and 1 2 respectively

minimization of locally lipschitzian functions this paper presents a globally convergent model algorithm for the minimization of a locally lipschitzian function the algorithm is built on an iteration function of two arguments and the convergence theory is developed parallel to analogous results for the problem of solving systems of locally lipschitzian equations application of the theory to a wide range of nonsmooth optimization problems is discussed these include the minimax problem the composite optimization problem the implicit programming problem and others a recently developed nonmonotone linesearch technique is shown to be applicable in this nonsmooth context and an extension to constrained problems is also presented

the effect of the increase of linear dimensions on exponents obtained by finite size scaling relations for the six dimensional ising model on the creutz cellular automaton the six dimensional ising model is simulated on the creutz cellular automaton using the finite size lattices with the linear dimension 4

a nonmonotone line search technique and its application to unconstrained optimization a new nonmonotone line search algorithm is proposed and analyzed in our scheme we require that an average of the successive function values decreases while the traditional nonmonotone approach of grippo lampariello and lucidi siam j numer anal 23 1986 pp 707 716 requires that a maximum of recent function values decreases we prove global convergence for nonconvex smooth functions and r linear convergence for strongly convex functions for the l bfgs method and the unconstrained optimization problems in the cute library the new nonmonotone line search algorithm used fewer function and gradient evaluations on average than either the monotone or the traditional nonmonotone scheme

projected newton methods for optimization problems with simple constraints we consider the problem min f x x geqq 0 and propose algorithms of the form x k 1 x k alpha k d k nabla f x k where cdot denotes projection on the positive orthant alpha k is a stepsize chosen by an armijo like rule and d k is a positive definite symmetric matrix which is partly diagonal we show that d k can be calculated simply on the basis of second derivatives of f so that the resulting newton like algorithm has a typically superlinear rate of convergence with other choices of d k convergence at a typically linear rate is obtained the algorithms are almost as simple as their unconstrained counterparts they are well suited for problems of large dimension such as those arising in optimal control while being competitive with existing methods for low dimensional problems the effectiveness of the newton like algorithm is demonstrated via computational examples involving as many as 10 000 variables extensions to general linearly constrained pr

thermal avalanche for blowup solutions of semilinear heat equations we consider the semilinear heat equation ut u up both in n and in a bounded domain with homogeneous dirichlet boundary conditions with 1 t we consider a natural approximation by reaction problems with global solution and pass to the limit as is well known the limit solution undergoes complete blowup after it blows up at t t the continuation is identically infinite for all t t we perform here a deeper analysis of how complete blowup occurs actually the singularity set of a solution that blows up as t t propagates instantaneously at time t t to cover the whole space producing a catastrophic discontinuity between t t and t t this is called the avalanche we describe its formation as a layer appearing in the limit of the natural approximate problems after a suitable scaling the initial structure of the layer is given by the solution of a limit problem described by a simple ordinary differential equation as t proceeds past t the solutions of the approximate problems have a traveling wave behavior with a speed that we compute the situation in the inner region depends on the type of approximation a conical pattern is formed with time when we approximate the power up by a flat truncation at level n while for tangent truncations we get an exponential increase in time and a diffusive spatial pattern 2003 wiley periodicals inc

the test of the finite size scaling relations for the six dimensional ising model on the creutz cellular automaton the six dimensional ising model is simulated on the creutz cellular automaton using the finite size lattices with the linear dimension 4 l 10 the exponents in the finite size scaling relations for the magnetic susceptibility and the order parameter at the infinite lattice critical temperature are computed to be 3 00 12 and 1 47 10 using 6 l 10 respectively which are in very good agreement with the theoretical predictions of and the critical temperature for the infinite lattice is found to be 10 835 5 using 4 l 10 which is also in very good agreement with the precise results the finite size scaling relation for the magnetic susceptibility at the infinite lattice critical temperature is also valid for the maxima of the magnetic susceptibilities of the finite size lattices the finite size scaling plots of the magnetic susceptibility and the order parameter verify the finite size scaling relations about the infinite lattice critical temperature

on the worst case performance of the steepest descent algorithm for quadratic functions the existing choices for the step lengths used by the classical steepest descent algorithm for minimizing a convex quadratic function require in the worst case mathcal o c log 1 varepsilon o clog 1 iterations to achieve a precision varepsilon where c is the hessian condition number we show how to construct a sequence of step lengths with which the algorithm stops in mathcal o sqrt c log 1 varepsilon o clog 1 iterations with a bound almost exactly equal to that of the conjugate gradient method

blow up estimates for system of heat equations coupled via nonlinear boundary flux abstract this paper deals with a system of heat equations coupled via nonlinear boundary flux the precise blow up rate estimates are established together with the blow up set it is observed that there is some quantitative relationship regarding the blow up properties between the heat system with coupled nonlinear boundary flux terms and the corresponding reaction diffusion system with the same nonlinear terms as the source

blow up estimates of the positive solution of a parabolic system this paper establishes the blow up estimates for the systems ut u 0 vt v 0 in br 0 t br rn with the nonlinear boundary conditions u um1vn1 and v um2vn2 on sr 0 t where 0 m1 1 m2 and 0 n2 1 n1 we prove that c t t 2 max u x t c t t 2 and c t t 2 max v x t c t t 2 under some monotonicity assumptions on the initial values where n1 n2 1 m2 m1 1 and n1m2 1 m1 1 n2

a new family of conjugate gradient methods in this paper we develop a new class of conjugate gradient methods for unconstrained optimization problems a new nonmonotone line search technique is proposed to guarantee the global convergence of these conjugate gradient methods under some mild conditions in particular polak ribiere polyak and liu storey conjugate gradient methods are special cases of the new class of conjugate gradient methods by estimating the local lipschitz constant of the derivative of objective functions we can find an adequate step size and substantially decrease the function evaluations at each iteration numerical results show that these new conjugate gradient methods are effective in minimizing large scale non convex non quadratic functions

critical indices for the two dimensional ising model with nearest neighbor and next nearest neighbor interactions ii strip cluster approximation the spin pair correlation functions are calculated for the ising model with nearest neighbor interaction j 0 and next nearest neighbor interaction j on the square lattice in terms of three approximations of the simplified cluster variation method with strip clusters as the basic clusters the critical indices and are estimated from the obtained spin pair correlation functions with the aid of the coherent anomaly method for j j 0 5 we confirm that the system belongs to the onsager universality class in the sense of the weak universality hypothesis