Finiteness Result for the Simplicial Branch-and-Bound Algorithm Based on ω-Subdivisions

The question of the finiteness of simplicial branch-and-bound algorithms employing only -subdivisions is considered. In Ref. 1, it was shown that this algorithm is convergent; here, it is proved that the algorithm is also finite if two assumptions are fulfilled. The first assumption requires the function values at vertices of the initial simplex to be lower than the optimal value of the problem. The second assumption requires each vertex of the initial simplex to violate at most one of the constraints defining the feasible polytope. The first assumption is mild from a theoretical point of view; the second assumption is strong, but holds always for instance when the feasible region is a hypercube.     

Topological existence and stability for stackelberg problems

The aim of this paper is to study, in a topological framework, existence and stability for the solutions to a parametrized Stackelberg problem. To this end, approximate solutions are used, more precisely, -solutions and strict -solutions. The results given are of minimal character and the standard types of constraints are considered, that is, constant constraints, constraints defined by a finite number of inequalities, and more generally constraints defined by an arbitrary multifunction.     

Some topological properties of ω-covering sets

We prove the following theorems:1. There exists an -covering with the property s ₀.2. Under cov there exists X such that is not an -covering orX \ B is not an -covering].3. Also we characterize the property of being an -covering.     

On Convergence of the Simplicial Branch-and-Bound Algorithm Based on ω-Subdivisions

The problem of minimizing a concave function over a polytope is considered. The simplicial branch-and-bound approach is presented and theoretical studies about the convergence of these algorithms are carried on. In particular, the convergence of the algorithm based on so-called -subdivisions is proved, which had been an open question for a long time.     

On Periodic Solutions of Nonlinear Functional Differential Equations

Sufficient conditions are established for the existence and uniqueness of an -periodic solution of the functional differential equation where f is a continuous operator acting from the space of n-dimensional -periodic continuous vector functions into the space of n-dimensional -periodic and summable on [0,] vector functions.     

On the Convergence of Cone Splitting Algorithms with ω-Subdivisions

We present a convergence proof of the Tuy cone splitting algorithm with a pure -subdivision strategy for the minimization of a concave function over a polytope. The key idea of the convergence proof is to associate with the current hyperplane a new hyperplane that supports the whole polytope instead of only the portion of it contained in the current cone. A branch-and-bound variant of the algorithm is also discussed.     

ε-solutions in vector minimization problems

This paper presents some properties of -solutions for vector minimization problems where the function to be optimized takes its values in the Euclidean space ^p . The results obtained generalize the classical ones for exact Pareto solutions.     

Inequality of Petrović and Giaccardi for Convex Function of Higher Order

In this paper we use Lidstone polynomials to prove further generalization of Giaccardi generalization of the well-known Petrovis inequality.     

Equicontinuity of Maps on Figure-eight Space

Let 8 be the figure-eight space and f : 8 8 be a continuous map with a periodic point. In this paper, we show that f is equicontinuous if and only if one of the following conditions holds: (1) (8) is a circle on 8 and f|_o is conjugate to a rational rotation.AMS Mathematics Subject Classification (1991) 54H20     

On <Emphasis Type="Italic">L</Emphasis><Subscript>α,ω</Subscript> complete extensions of complete theories of Boolean algebras

For a complete first order theory of Boolean algebras T which has nonisomorphic countable models, we determine the first limit ordinal = (T) such that We show that for some and for all other Ts, A nonprincipal ideal I of B is almost principal, if a is a principal ideal of B} is a maximal ideal of B. We show that the theory of Boolean algebras with an almost principal ideal has complete extensions and characterize them by invariants similar to the Tarskis invariants.Mathematics Subject Classification (2000): Primary 03C15, Secondary 03C35, 06E05Revised version: 2 February 2004     

On θ-Principal Extension and H-Closedness

A type of extensions called the -extention of topological spaces and their -equivalence and -trace systems are introduced, which ultimately characterize H-closed -extensions of a Hausdorff topological space. Also, the notion of -principal extensions is defined. A typical -principal extension consisting of certain grills on a Hausdorff space is constructed, and finally, some characterizations of H-closedness of a Hausdorff space are obtained.AMS Subject Classification (2000) 54D30 54D99     

A Simplified Convergence Proof for the Cone Partitioning Algorithm

We present a new convergence result for the cone partitioning algorithm with a pure -subdivision strategy, for the minimization of a quasiconcave function over a polytope. It is shown that the algorithm is finite when -optimal solution with > 0 are looked for, and that any cluster point of the points generated by the algorithm is an optimal solution in the case = 0. This result improves on the one given previously by the authors, its proof is simpler and relies more directly on a new class of hyperplanes and its associated simplicial lower bound.     

A closed formula for the čebyšev barycentric weights of optimal approximation in H2

The barycentric formula has several advantages over other means of evaluating the polynomial interpolating a function betweenn points in an interval. In particular, it is much more stable for sets of points clustered at the extremities of the interval, as are all the sets guaranteeing a good approximation forn sufficiently large. Also, it requires onlyO(n) operations for every function to be interpolated, once some weights, which depend only on the points, have been computed. Computing those weights usually requiresO(n²) operations; for ebyev points, however,O(n) operations suffice. We show here that all the above is also true for the optimal evaluation of functionals in H² by giving a closed formula for the corresponding weights.     

On the ω-limit Set Dichotomy of Cooperating Kolmogorov Systems

The authors study the -limit set dichotomy of the Kolmogorov systems _i=x_i f _i(x)x _i0, 1in with the cooperative and irreducible hypotheses and obtain the quasiconvergence almost everywhere when n=3, which gives an affirmative answer to the open problem by Smith [9, p.72] in the case of n=3.     

Quantitative Siegel's theorem for Galois coverings

It is known that Siegels theorem on integral points is effective for Galoiscoverings of the projective line. In this paper we obtain a quantitative version of this result, giving an explicit upper bound for the heights of S-integral K-rational points in terms of the number field K, the set of places S and the defining equation of the curve.Our main tools are Bakers theory of linear forms in logarithms and thequantitative Eisenstein theorem due to Schmidt, Dwork and van der Poorten.     

Complete stability of large order statistics

Fix an integerr1. For eachnr, letM _nr be the rth largest ofX ₁,...,X _n, where {X _n,n1} is a sequence of i.i.d. random variables. Necessary and sufficient conditions are given for the convergence of _n=r n P[|M _nr /a _n –1|<] for every >0, where {a _n} is a real sequence and –1. Moreover, it is shown that if this series converges for somer1 and some >–1, then it converges for everyr1 and every >–1.     

Asymptotic Curvature of Θ-Metrics

We give an explicit and reasonably simple expression for the curvature tensor of a -metric at boundary points, in terms of the metric tensor and invariants of the -structure. We examine the behavior of the induced metric on level sets of a defining function near the boundary and describe the asymptotic behavior of its curvature tensor. Some applications of these results are given.     

Study of Noetherian Γ-Semirings via Operator Semirings

In this paper we study Noetherian -semirings and obtain Cohens theorem for a special class of -semirings. Weak primary decomposition theorem for a particular type of -semirings is also obtained.Presently Lecturer in Mathematics, University of Burdwan, GOLAPBAG, W.B. INDIA.