On k-stable functions

We prove that a k-continuous or a k-stable function cannot depend on more than k4^k?1 variables and related facts.     

A Convenient Subcategory of Tych

A map f:XY between Hausdorff topological spaces is k-continuous if its restriction f| _K to every compact subspace K of X is continuous. X is called a k _R -space if every k-continuous function from X to a Tychonoff space is continuous. In this paper we investigate the category of Tychonoff k _R -spaces, and show that it is Cartesian closed (thus convenient in the sense of Wyler).     

Remarks on Banaschewski-Fomin-Shanin extensions

The notion ofB ^*-continuous andB _c ^* -continuous maps is introduced. The problem of epireflection of Banaschewski-Fomin-Shanin extension for a general Hausdorff space is investigated with the help of_p B ^* and_P B _c ^* -continuous maps.     

Results on <Emphasis Type="Italic">F</Emphasis>-continuous graphs

For any nontrivial connected graph F and any graph G, the F-degree of a vertex v in G is the number of copies of F in G containing v. G is called F-continuous if and only if the F-degrees of any two adjacent vertices in G differ by at most 1; G is F-regular if the F-degrees of all vertices in G are the same. This paper classifies all P ₄-continuous graphs with girth greater than 3. We show that for any nontrivial connected graph F other than the star K _1,k , k ⩾ 1, there exists a regular graph that is not F-continuous. If F is 2-connected, then there exists a regular F-continuous graph that is not F-regular.      

Determination of an unknown coefficient in a nonlinear heat equation

The problem of determining an unknown term k(u) in the equation k(u)u_t=(k(u)u_x)_x is considered in this paper. Applying Tikhonov's regularization approach, we develop a procedure to find an approximate stable solution to the unknown coefficient from the overspecified data.     

Computing Proximal Points of Convex Functions with Inexact Subgradients

Locating proximal points is a component of numerous minimization algorithms. This work focuses on developing a method to find the proximal point of a convex function at a point, given an inexact oracle. Our method assumes that exact function values are at hand, but exact subgradients are either not available or not useful. We use approximate subgradients to build a model of the objective function, and prove that the method converges to the true prox-point within acceptable tolerance. The subgradient g _k used at each step k is such that the distance from g _k to the true subdifferential of the objective function at the current iteration point is bounded by some fixed ε > 0. The algorithm includes a novel tilt-correct step applied to the approximate subgradient.     

Shift invariant preduals of ℓ 1(ℤ)

The Banach space ? ₁(?) admits many non-isomorphic preduals, for example, C(K) for any compact countable space K, along with many more exotic Banach spaces. In this paper, we impose an extra condition: the predual must make the bilateral shift on ? ₁(?) weak*-continuous. This is equivalent to making the natural convolution multiplication on ? ₁(?) separately weak*-continuous and so turning ? ₁(?) into a dual Banach algebra. We call such preduals shift-invariant. It is known that the only shift-invariant predual arising from the standard duality between C ₀(K) (for countable locally compact K) and ? ₁(?) is c ₀(?). We provide an explicit construction of an uncountable family of distinct preduals which do make the bilateral shift weak*-continuous. Using Szlenk index arguments, we show that merely as Banach spaces, these are all isomorphic to c ₀. We then build some theory to study such preduals, showing that they arise from certain semigroup compactifications of ?. This allows us to produce a large number of other examples, including non-isometric preduals, and preduals which are not Banach space isomorphic to c ₀.     

Computation of the canonical polynomials and applications to some optimal control problems

The Tau method is a numerical technique that consists in constructing polynomial approximate solutions for ordinary differential equations. This method has two approaches: operational and recursive. The former converts the differential problem to a matrix problem and produces approximations in terms of a prescribed orthogonal polynomials basis. In the recursive approach, we construct approximate solutions in terms of a special set of polynomials {Q _k (t); k?=?0, 1, 2...} called canonical polynomials basis. In some cases, the Q _k ??s can be obtained explicitly through a recursive formula. But no analogous formulae are reported in the literature for the general cases. In this paper, utilizing the operational Tau method, we develop an algorithm that allows to generate those canonical polynomials iteratively and explicitly. In addition, we demonstrate the capability of the operational Tau method in treating quadratic optimal control problems governed by ordinary differential equations.     

An accelerated subspace minimization three-term conjugate gradient algorithm for unconstrained optimization

A three-term conjugate gradient algorithm for large-scale unconstrained optimization using subspace minimizing technique is presented. In this algorithm the search directions are computed by minimizing the quadratic approximation of the objective function in a subspace spanned by the vectors: ?g _k+1, s _k and y _k . The search direction is considered as: d _k+1 = ?g _k+1 + a _k s _k + b _k y _k , where the scalars a _k and b _k are determined by minimization the affine quadratic approximate of the objective function. The step-lengths are determined by the Wolfe line search conditions. We prove that the search directions are descent and satisfy the Dai-Liao conjugacy condition. The suggested algorithm is of three-term conjugate gradient type, for which both the descent and the conjugacy conditions are guaranteed. It is shown that, for uniformly convex functions, the directions generated by the algorithm are bounded above, i.e. the algorithm is convergent. The numerical experiments, for a set of 750 unconstrained optimization test problems, show that this new algorithm substantially outperforms the known Hestenes and Stiefel, Dai and Liao, Dai and Yuan and Polak, Ribiére and Poliak conjugate gradient algorithms, as well as the limited memory quasi-Newton method L-BFGS and the discrete truncated-Newton method TN.     

An efficient gradient method with approximate optimal stepsize for large-scale unconstrained optimization

In this paper, we introduce a new concept of approximate optimal stepsize for gradient method, use it to interpret the Barzilai-Borwein (BB) method, and present an efficient gradient method with approximate optimal stepsize for large unconstrained optimization. If the objective function f is not close to a quadratic on a line segment between the current iterate x _k and the latest iterate x _k?1, we construct a conic model to generate the approximate optimal stepsize for gradient method if the conic model is suitable to be used. Otherwise, we construct a new quadratic model or two other new approximation models to generate the approximate optimal stepsize for gradient method. We analyze the convergence of the proposed method under some suitable conditions. Numerical results show the proposed method is very promising.