Open Half-spaces Research Articles

The e-support function of an e-convex set and conjugacy for e-convex functions

A subset of a locally convex space is called e-convex if it is the intersection of a family of open halfspaces. An extended real-valued function on such a space is called e-convex if its epigraph is e-convex. In this paper we introduce a suitable support function for e-convex sets as well as a conjugation scheme for e-convex functions.

Asymptotic direction in random walks in random environment revisited

Recently Simenhaus in [Sim07] proved that for any elliptic random walk in random environment, transience in the neighborhood of a given direction is equivalent to the a.s. existence of a deterministic asymptotic direction and to transience in any direction in the open half space defined by this asymptotic direction. Here we prove an improved version of this result and review some open problems.

A representation of convex semilinear sets

If F is an ordered field, a subset of n-space over F is said to be semilinear just in case it is a finite Boolean combination of translates of closed halfspaces, where a closed halfspace is the set of all points obeying a homogeneous weak linear inequality with coefficients from F. Andradas, Rubio, and Velez have shown that closed (open) convex semilinear sets are finite intersections of translates of closed (open) halfspaces (an open halfspace is defined as before, but with a strict inequality). This paper represents arbitrary convex semilinear sets in a manner analogous to that of Andradas, Rubio, and Velez.

Radial solutions of a polyharmonic equation with power nonlinearity

We classify the regular radial solutions of the equation Δ m u = u | u | p − 1 with m a positive integer and p ∈ ( 1 , ∞ ) . Any such solution u is uniquely determined by the center values of u and its iterated Laplacians up to order m − 1 , that is, by the “center point” ( u ( 0 ) , Δ u ( 0 ) , … , Δ m − 1 u ( 0 ) ) . Most solutions have finite exit radius (that is, they are unbounded and defined on open balls of finite radius), are eventually monotonic with respect to the radial variable, and diverge to ∞ or − ∞ . The center points of these solutions form two open half-spaces in R m , separated by an ( m − 1 ) -dimensional manifold M . Solutions with center points on M are either global, or they have finite exit radius and oscillate about 0 with unbounded amplitude. Our results yield precise information regarding the existence, multiplicity, and nature of large radial solutions of Δ m u = u | u | p − 1 on a ball of given radius.

Moebius transformations and the Poincare distance in the quaternionic setting

In the space ℍ of quaternions, we investigate the natural, invariant geometry of the open, unit disc Δ ℍ and of the open half-space ℍ + . These two domains are diffeomorphic via a Cayley-type transformation. We first study the geometrical structure of the groups of Mobius transformations of Δ ℍ and ℍ + and identify original ways of representing them in terms of two (isomorphic) groups of matrices with quaternionic entries. We then define the cross-ratio of four quaternions, prove that, when real, it is invariant under the action of the Mobius transformations, and use it to define the analog of the Poincare distances and differential metrics on Δ ℍ and ℍ + .

Adelic amoebas disjoint from open halfspaces

We show that a conjecture of Einsiedler, Kapranov, and Lind on adelic amoebas of subvarieties of tori and their intersections with open halfspaces of complementary dimension is false for subvarieties of codimension greater than one that have degenerate projections to smaller dimensional tori. We prove a suitably modified version of the conjecture using algebraic methods, functoriality of tropicalization, and a theorem of Zhang on torsion points in subvarieties of tori.

Analyzing linear systems containing strict inequalities via evenly convex hulls

The evenly convex hull of a given set is the intersection of all the open halfspaces which contain such set (hence the convex hull is contained in the evenly convex hull). This paper deals with finite dimensional linear systems containing strict inequalities and (possibly) weak inequalities as well as equalities. The number of inequalities and equalities in these systems is arbitrary (possibly infinite). For such kind of systems a consistency theorem is provided and those strict inequalities (weak inequalities, equalities) which are satisfied for every solution of a given system are characterized. Such results are formulated in terms of the evenly convex hull of certain sets which depend on the coefficients of the system.

Dual Characterizations of Set Containments with Strict Convex Inequalities

Characterizations of the containment of a convex set either in an arbitrary convex set or in the complement of a finite union of convex sets (i.e., the set, described by reverse-convex inequalities) are given. These characterizations provide ways of verifying the containments either by comparing their corresponding dual cones or by checking the consistency of suitable associated systems. The convex sets considered in this paper are the solution sets of an arbitrary number of convex inequalities, which can be either weak or strict inequalities. Particular cases of dual characterizations of set containments have played key roles in solving large scale knowledge-based data classification problems where they are used to describe the containments as inequality constraints in optimization problems. The idea of evenly convex set (intersection of open half spaces), which was introduced by W. Fenchel in 1952, is used to derive the dual conditions, characterizing the set containments.

Surface integrals and harmonic functions

Using the notion of inferior mean due to M. Heins, we establish two inequalities for such a mean relative to a positive harmonic function defined on the open unit ball or half-space in .

Residue formulae for vector partitions and Euler–MacLaurin sums

Let V be an n-dimensional real vector space endowed with a rank-n lattice Γ . The dual lattice Γ ∗ = Hom(Γ.Z) is naturally a subset of the dual vector space V ∗. Let Φ = [β1, β2, . . . , βN ] be a sequence of not necessarily distinct elements of Γ ∗, which span V ∗ and lie entirely in an open halfspace of V ∗. In what follows, the order of elements in the sequence will not matter. The closed cone C(Φ) generated by the elements of Φ is an acute convex cone, divided into open conic chambers by the (n− 1)-dimensional cones generated by linearly independent (n−1)-tuples of elements of Φ . Denote by ZΦ the sublattice of Γ ∗ generated by Φ . Pick a vector a ∈ V ∗ in the cone C(Φ), and denote by ΠΦ(a) ⊂ R+ the convex polytope consisting of all solutions x = (x1, x2, . . . , xN) of the equation ∑Nk=1 xkβk = a in nonnegative real numbers xk . This is a closed convex polytope called the partition polytope associated to Φ and a. Conversely, any closed convex polytope can be realized as a partition polytope. If λ ∈ Γ ∗, then the vertices of the partition polytope ΠΦ(λ) have rational coordinates. We denote by ιΦ(λ) the number of points with integral coordinates in ΠΦ(λ). Thus ιΦ(λ) is the number of solutions of the equation ∑N k=1 xkβk = λ in nonnegative integers xk . The function λ → ιΦ(λ) is called the vector partition function associated to Φ . Obviously, ιΦ(λ) vanishes if λ does not belong to C(Φ) ∩ZΦ .

On the completeness of eigenmodes in a parallel plate waveguide with a perfectly matched layer termination

An explicit proof of the completeness of the eigenmodes of a grounded parallel plate waveguide, with a perfectly matched layer (PML) termination is given. The proof is based on a general theorem governing the completeness of sets of complex exponentials. The Green's function of the structure is then obtained by means of a biorthogonalization procedure. A comparison with the Green's function of a grounded halfspace indicates that a PML termination simulates the open halfspace efficiently. Index Terms—Completeness, eigenmodes, Green's function, parallel plate waveguide, perfectly matched layer (PML).

On large deviations of Markov processes with discontinuous statistics

This paper establishes a process-level large deviations principle for Markov processes in the Euclidean space with a discontinuity in the transition mechanism along a hyperplane. The transition mechanism of the process is assumed to be continuous on one closed half-space and also continuous on the complementary open half-space. Similar results were recently obtained for discrete time processes by Dupuis and Ellis and by Nagot. Our proof relies on the work of Blinovskii and Dobrushin, which in turn is based on an earlier work of Dupuis and Ellis.

Characterization of R-Evenly Quasiconvex Functions

A function defined on a locally convex space is called evenly quasiconvex if its level sets are intersections of families of open half-spaces. Furthermore, if the closures of these open halfspaces do not contain the origin, then the function is called R-evenly quasiconvex. In this note, R-evenly quasiconvex functions are characterized as those evenly-quasiconvex functions that satisfy a certain simple relation with their lower semicontinuous hulls.

Right inverses for linear, constant coefficient partial differential operators on distributions over open half spaces

Results of Hormander on evolution operators together with a character- ization of the present authors (Ann. Inst. Fourier, Grenoble 40, 619 - 655 (1990)) are used to prove the following: Let P C z1 z n and denote by Pm its principal part. If P Pm is dominated by Pm then the following assertions for the partial differential operators P D and Pm D are equivalent for N S n 1 :

Minimal 2-Fold Coverings of E d

Let N(k, d) be the smallest positive integer such that given any finite collection of open halfspaces which k-fold coversEd, there exists a subcollection of cardinality less than or equal toN(k,d) which k-fold coversEd. A well-known corollary to Helly's theorem proves N(1,d) =d+1. This provides an inductive base from which we show N(k; d) exists for all positive integers k.Our main result is \(N(2,d) = \left\lfloor {((d + 3)/2)^2 } \right\rfloor \).

Hybrid misclassification minimization

Given two finite point setsA andB in then-dimensional real spaceR n , we consider the NP-complete problem of minimizing the number of misclassified points by a plane attempting to divideR n into two halfspaces such that each open halfspace contains points mostly ofA orB. This problem is equivalent to determining a plane {x | x T w=γ} that maximizes the number of pointsx ∈A satisfying inx T w>γ, plus the number of pointsx ∈B satisfyingx T w<γ. A simple but fast algorithm is proposed that alternates between (i) minimizing the number of misclassified points by translation of the separating plane, and (ii) a rotation of the plane so that it minimizes a weighted average sum of the distances of the misclassified points to the separating plane. Existence of a global solution to an underlying hybrid minimization problem is established. Computational comparison with a parametric approach to solve the NP-complete problem indicates that our approach is considerably faster and appears to generalize better as determined by tenfold cross-validation.

On the theory of discrete TLM Green's functions in three-dimensional TLM

The response to a wave pulse incident on the boundary of a certain spatial domain may be represented by discrete TLM Green's functions. On the other hand, the response to a localized electromagnetic excitation at the boundary of a certain spatial domain may be calculated directly from Maxwell's equations and be represented by analytic TLM Green's functions. For low frequencies and small wave numbers, the analytic TLM Green's functions coincide with the discrete TLM Green's functions. Applying the analytic TLM Green's functions in the absorbing boundary condition at the boundary to the open half-space reduces the computational effort considerably when compared with the application of the discrete TLM Green's functions.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

The volume of hyperbolic Coxeter polytopes of even dimension

Let H denote hyperbolic space of dimension n, and let S be an index set for a finite collection of open half spaces H s in H n bounded by codimension one hyperplanes Hs. We assume that for all distinct s, t ∈ S either Hs ∩ Ht is not empty and the (interior) dihedral angle of H s ∩ H + t along Hs ∩ Ht has size π mst for certain integers mst = mts ≥ 2, or Hs ∩Ht is empty while H + s ∩H + t is not empty. In the latter case we put mst = mts = ∞ and we also put mss = 1. Under these assumptions the intersection C = ⋂ s H + s is not empty, and its closure D is called a hyperbolic Coxeter polytope. By abuse of notation let s ∈ S also denote the reflection of H in the hyperplane Hs. Now the group W of motions of H n generated by the reflections s ∈ S is discrete, and D is a strict fundamental domain for the action of W on H. Moreover (W,S) is a Coxeter group with Coxeter matrix M = (mst), i.e. W has a presentation with generators s ∈ S and relations (st)s,t = 1 for s, t ∈ S. Let l(w) denote the length of w ∈ W with respect to the generating set S, and let PW (t) ∈ Z[[t]] be the Poincare series of W defined by PW (t) = ∑ w t .

External Tangents and Closedness of Cone + Subspace

When X and Y are convex subsets of a topological vector space E, an external tangent of the ordered pair ( X, Y) is defined as an open ray T that issues from a point of X ∩ Y, is disjoint from X ∪ Y, and is such that X intersects each open halfspace containing T. It is shown that if E is a separable normed space, C is a closed convex cone in E, and L is a line through the origin in E, then the vector sum C + L = { c + ℓ: c ∈ C, ℓ ∈ L} is closed if and only if the pair ( C, L) does not admit any external tangent. When S is a subspace of finite dimension greater than 1, closedness of C + S is shown to be equivalent to the nonexistence of external tangents of a certain pair ( C′, L,), where L is a line through the origin and C′ is a second closed convex cone constructed from ( C, S). Questions about the closedness of sets of the form cone + subspace arise from Various optimization problems, from problems concerning the extension of positive linear functions, and from certain problems in matrix theory.

A Polynomial Time Algorithm That Learns Two Hidden Unit Nets

Let N be the class of functions realizable by feedforward linear threshold nets with n input units, two hidden units each of zero threshold, and an output unit. This class is also essentially equivalent to the class of intersections of two open half spaces that are bounded by planes through the origin. We give an algorithm that probably almost correctly (PAC) learns this class from examples and membership queries. The algorithm runs in time polynomial in n, ∊ (the accuracy parameter), and δ (the confidence parameter). If only examples are allowed, but not membership queries, we give an algorithm that learns N in polynomial time provided that the probability distribution D from which examples are chosen satisfies D(x) = D(−x) ∀x. The algorithm yields a hypothesis net with two hidden units, one linear threshold and the other quadratic threshold.