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By means of the Minkowski-type nonlinear scalarization functional, in this paper, we establish some nonlinear separation theorems via relative algebraic interior and vector closure in a general real linear space, via relative topological interior and topological closure in a real topological linear space and via quasi relative interior and topological closure in a real separated locally convex topological linear space, respectively. These new separation theorems can be applied to study those vector optimization problems with the ordering cones having possibly empty topological interior and even relative topological interior or relative algebraic interior. As their applications, we give some nonlinear scalarization characterizations of the corresponding weak efficient solutions of vector optimization problems. Moreover, we also present some concrete examples to illustrate the main results in some infinite dimensional spaces.

For any finite planar point set in general position, an interior point of the set is a point of the set such that it is not on the boundary of the convex hull of the set . For any positive integer , let be the smallest integer such that every finite planar point set with no three collinear points and with at least interior points has a subset for which the interior of the convex hull of the set contains exactly or interior points of the set . In this paper, we prove that .

Simplex algorithms governed by some pivot rule and interior point algorithms are two diverging and competitive types of algorithms for solving linear programming problems. The former moves on the underlying polyhedron, from vertex to adjacent vertex, along edges until an optimal vertex is reached while the latter approaches an optimal point by moving across interior of the polyhedron. In this article, we derive an algorithm that may be regarded as a pivot as well as interior point one. It produces a sequence of interior as well as boundary points, including vertices, until reaching a pair of exact primal and dual optimal solutions. We report encouraging computational results with dense implementation of the algorithm on a set of Netlib test problems.

In this paper functoriality of the notion of categorical interior operator is discussed and as a related topic, the property of ℱ-modal interior operator is introduced.

We analyze the impact of interior symmetries on the multiplicity of the eigenvalues of the Jacobian matrix at a fully synchronous equilibrium for the coupled cell systems associated to homogeneous networks. We consider also the special cases of regular and uniform networks. It follows from our results that the interior symmetries, as well as the reverse interior symmetries and quotient interior symmetries, of the network force the existence of eigenvalues with algebraic multiplicity greater than one. The proofs are based on the special form of the adjacency matrices of the networks induced by these interior symmetries.

The aim of this work is to introduce the concept of L-fuzzy interior systems and L-fuzzy interior operators. We start by establishing a connection between L-fuzzy interior systems and L-fuzzy interior operators. It is shown that an L-fuzzy interior system is precisely the fuzzy system in opposition to the crisp system, and an L-fuzzy interior operator is a suitable interior operator that has a close relation to an L-fuzzy interior system. It is also shown that there is a Galois correspondence between the category of L-fuzzy interior system spaces and that of L-fuzzy interior spaces.

An interior point of a finite planar point set is a point of the set that is not on the boundary of the convex hull of the set. For any integer k ≥ 1, let h(k) be the smallest integer such that every point set in the plane, no three collinear, with at least h(k) interior points, has a subset with k or k + 2 interior points of P. We prove that h(3) = 8.

An H-triangle is a triangle with corners in the set of vertices of a tiling of R^2 by regular hexagons of unit edge. It is known that any H-triangle with exactly 1 interior H-point can have at most 10 H-points on its boundary. In this note we prove that any H-triangle with exactly k interior H-points can have at most 3k+7 boundary H-points. Moreover we form two conjectures dealing with H-polygons.

An interior point of a finite planar point set is a point of the set that is not on the boundary of the convex hull of the set. For any integer k≥1, let g(k) be the smallest integer such that every planar point set in general position with at least g(k) interior points has a convex subset of points with exactly k interior points of P. In this article, we prove that g(3)=9.

An interior point of a finite point set is a point of the set that is not on the boundary of the convex hull of the set. For any integer k⩾1, let g(k) be the smallest integer such that every set of points in the plane, no three collinear, containing at least g(k) interior points has a subset of points containing exactly k interior points. We prove that g(1)=1, g(2)=4, g(3)⩾8 , and g(k)⩾k+2, k⩾4 . We also give some related results.

For hydraulic computations that take flow and pressure regulators into account, a little-known variant of the Newton method is used. Successive approximations belong to the interior of a set R specified by linear inequalities. In contrast to the base variant of the method of interior points, the Jacobian matrix is square. The solution of the problem of flow distribution is found on the boundary of R. The local convergence of the method considered is analyzed.

The aim of this paper is to present an algorithm for finding a saddle point to the constrained minimax problem. The initial problem is transformed into an equivalent equality constrained problem, and then the interior point approach is used. To satisfy the original inequality constraints a logarithmic barrier function is used and special care is given to step size parameter to keep the variables within permitted boundaries. Numerical results illustrating the method are given.

1. INTRODUCTIONWe define interior algebras, which are natural generalizations of Booleanrings equipped with an interior operator and of equality algebras. We char-acterize interior algebras in terms of a distinguished subsemilattice. Fullyinterior algebras, in which the subsemilattice consists of all idempotents,are generalizations of Boolean rings and semilattices. Our main theoremsdescribe all ring and semigroup varieties consisting of fully interior algebras.2. INTERIOR ALGEBRASThe notion of a Boolean ring with an interior operation has been aroundfor some time: they are Boolean rings with identity on which an interioroperation is defined, satisfying the usual Kuratowski sorts of conditions

We use Dirichlet form methods to construct and analyze a general class of reversible Markov precesses with totally disconnected state space. We study in detail the special case of bipartite Markov chains. The latter processes have a state space consisting of an “interior” with a countable number of isolated points and a, typically uncountable, “boundary.” The equilibrium measure assigns all of its mass to the interior. When the chain is started at a state in the interior, it holds for an exponentially distributed amount of time and then jumps to the boundary. It then instantaneously re-enters the interior. There is a “local time on the boundary.” That is, the set of times the process is on the boundary is uncountable and coincides with the points of increase of a continuous additive functional. Certain processes with values in the space of trees and the space of vertices of a fixed tree provide natural examples of bipartite chains. Moreover, time-changing a bipartite chain by its local time on the boundary leads to interesting processes, including particular Levy processes on local fields (for example, the p-adic numbers) that have been considered elsewhere in the literature.

The n-interior point variant of the Erdos-Szekeres problem is to show the following: For any n, n ⩾ 1 , every point set in the plane with sufficient number of interior points contains a convex polygon containing exactly n-interior points. This has been proved only for n ⩽ 3 . In this paper, we prove it for pointsets having atmost logarithmic number of convex layers. We also show that any pointset containing atleast n interior points, there exists a 2-convex polygon that contains exactly n-interior points.

This paper establishes basic results on the existence of interior points and interior paths in a nonlinear monotone complementarity problem in ℝn under very weak interior conditions. We show that the interior paths are bounded, continuous, and all the limit points of the paths are solutions to the complementarity problem. We prove that certain sets, including the solution set to the complementary problem, form a compact convex set. We also prove the existence of generalized interior points and interior paths. These generalized paths are also continuous and contain readily available starting points from which we can follow the paths to locate the solutions to the complementarity problem. We prove our results in the context of maximal monotone operators. The result presented here can be used to develop polynomial time interior point algorithms for general monotone complementarity problems.

The non---equational notion of abstract approximation space for roughness theory is introduced, and its relationship with the equational definition of lattice with Tarski interior and closure operations is studied. Their categorical isomorphism is proved, and the role of the Tarski interior and closure with an algebraic semantic of a S4---like model of modal logic is widely investigated. A hierarchy of three particular models of this approach to roughness based on a concrete universe is described, listed from the stronger model to the weaker one: (1) the partition spaces, (2) the topological spaces by open basis, and (3) the covering spaces.

A lattice polytope is a polytope in whose vertices are all in . The volume of a lattice polytope P containing exactly k ≥ 1 points in d in its interior is bounded above by . Any lattice polytope in of volume V can after an integral unimodular transformation be contained in a lattice cube having side length at most n˙n ! V. Thus the number of equivalence classes under integer unimodular transformations of lattice poly topes of bounded volume is finite. If S is any simplex of maximum volume inside a closed bounded convex body K in having nonempty interior, then K⊆ ( n + 2)S — (n+ l)s where mS denotes a nomothetic copy of S with scale factor m, and s is the centroid of S.

An interior point of a finite planar point set is a point of the set that is not on the boundary of the convex hull of the set. For any integer k ≥ 1, let h(k) be the smallest integer such that every set of points in the plane, no three collinear, containing at least h(k) interior points has a subset of points containing k or k + 1 interior points. We proved that h(3) =3 in an earlier paper. In this paper we prove that h(4) = 7.

Let us assume that a 2-torus homeomorphism $f$ isotopic to the identity has a segment of irrational slope as its rotation set $\rho(F)$. We prove that if the chain recurrent set $R(f)$ of $f$ is not chain transitive, then $\rho(F)$ has a rational point realized by a periodic point.

We discuss some aspects of the interior geometry of a metric space X. The metric on X is denoted d, the open respectively closed metric ball about a point x ∈ X is denoted B(x,r) or B r (x) respectively \( \overline B {\rm{ }}(x,r){\rm{ or }}{\overline B _r}{\rm{(}}x{\rm{)}} \).