Integration of combinatorial decompositions in the presence of collinearities

Many results in Combinatorial Integral Geometry are derived by integration of the combinatorial decompositions associated with finite point sets {P _i } given in the plane ?². However, most previous cases of integration of the decompositions in question were carried out for the point sets {P _i } containing no triads of collinear points, where the familiar algorithm sometimes called the “Four indicator formula” can be used. The present paper is to demonstrate that the complete combinatorial algorithm valid for sets {P _i } not subject to the mentioned restriction opens the path to various results, including the field of Stochastic Geometry. In the paper the complete algorithm is applied first in an integration procedure in a study of the perforated convex domains, i.e convex domains containing a finite array of non-overlapping convex holes. The second application is in the study of random colorings of the plane that are Euclidean motions invariant in distribution, basing on the theory of random polygonal windows from the so-called Independent Angles (IA) class. The method is a direct averaging of the complete combinatorial decompositions written for colorings observed in polygonal windows from the IA class. The approach seems to be quite general, but promises to be especially effective for the random coloring generated by random Poisson polygon process governed by the Haar measure on the group of Euclidean motions of the plane, assuming that a point P ∈ ?² is colored J if P is covered by exactly J polygons of the Poisson process. A general theorem clearing the way for Laplace transform treatment of the random colorings induced on line segments is formulated.     

Uniform Convexity and Associate Spaces

We prove that the associate space of a generalized Orlicz space L^?(·) is given by the conjugate modular ?* even without the assumption that simple functions belong to the space. Second, we show that every weakly doubling Φ-function is equivalent to a doubling Φ-function. As a consequence, we conclude that L^?(·) is uniformly convex if ? and ?* are weakly doubling.     

Über eine kombinatorisch-geometrische Frage von Hadwiger und Debrunner

The problem of the partition-numbersJ _?(p, q), considered by Hadwiger and Debrunner for the family ?=C ⁿ of convex bodies, is extended to simplicial complexes and arbitrary families assuming only the validity of Helly’s theorem. We obtain results similar to those of Hadwiger and Debrunner. Further we show the existence of all partition-numbers for the family? = H ⁿC of homothets of a convex body and we get new informations on the partition-numbers for the family of parallel rectangles.     

Chord calculus and stochastic geometry

Chord calculus is a collection of integration procedures applied to to the combinatorial decompositions that give the solution of the Buffon-Sylvester problem for n needles in a plane or the similar problem in IR ³. It is a source of various integral geometry identities, some of which find their application in Stochastic geometry. In the present paper these applications are focused on random convex polygons and polyhedrons, where we define certain classes where rather simple tomography analysis is possible. The choice of these classes (the Independent Angles class and the Independent Orientations class) is due to the nature of the results of the Chord calculus. The last section points at an application of the convex polygons from the Independent Angles class to Boolean sets in the plane (Boolean models) whose probability distibutions are invariant with respect to the group of Euclidean motions of the plane.     

Mappings connected with the gradient of conformal radius

In this paper we prove the following conformity criterion for the gradient of conformal radius ?R(D, z) of a convex domain D: the boundary ?D has to be a circumference. We calculate coefficients K(r) for K(r)-quasiconformal mappings ?R(D(r), z), D(r) ? D, 0 < r < 1, and complete the results obtained by F. G. Avkhadiev and K.-J. Wirths for the structure of boundary elements of quasiconformal mappings of the domain D.     

Every graph is (2,3)-choosable

A total weighting of a graph G is a mapping ? that assigns to each element z ∈ V (G)∪E(G) a weight ?(z). A total weighting ? is proper if for any two adjacent vertices u and v, ∑ _e∈E(u) ?(e)+?(u)≠∑ _e∈E(v) ?(e)+?(v). This paper proves that if each edge e is given a set L(e) of 3 permissible weights, and each vertex v is given a set L(v) of 2 permissible weights, then G has a proper total weighting ? with ?(z) ∈ L(z) for each element z ∈ V (G)∪E(G).     

Convex hull of Brownian motion ind-dimensions

We supposeK(w) to be the boundary of the closed convex hull of a sample path ofZ _t(w), 0 ≦t ≦ 1 of Brownian motion ind-dimensions. A combinatorial result of Baxter and Borndorff Neilson on the convex hull of a random walk, and a limiting process utilizing results of P. Levy on the continuity properties ofZ _t(w) are used to show that the curvature ofK(w) is concentrated on a metrically small set.     

Partial Order Embeddings with Convex Range

A careful study is made of embeddings of posets which have a convex range. We observe that such embeddings share nice properties with the homomorphisms of more restrictive categories; for example, we show that every order embedding between two lattices with convex range is a continuous lattice homomorphism. A number of posets are considered; for one of the simplest examples, we prove that every product order embedding σ : ?^? → ?^? with convex range is of the form

$$ \sigma(x)(n)=\left( (x\circ g_{\sigma})+y_{\sigma}\right)(n) ~~~~\text{if}~ n\in K_{\sigma}, $$

(1)

and σ(x)(n) = y _σ (n) otherwise, for all x ∈ ?^?, where K _σ ? ?, g _σ : K _σ → ? is a bijection and y _σ ∈ ?^?. The most complex poset examined here is the quotient of the lattice of Baire measurable functions, with codomain of the form ? ^I for some index set I, modulo equality on a comeager subset of the domain, with its ‘natural’ ordering.

     

When does the zero-one <Emphasis Type="Italic">k</Emphasis>-law fail?

The limit probabilities of the first-order properties of a random graph in the Erd?s–Rényi model G(n, n^?α), α ∈ (0, 1), are studied. A random graph G(n, n^?α) is said to obey the zero-one k-law if, given any property expressed by a formula of quantifier depth at most k, the probability of this property tends to either 0 or 1. As is known, for α = 1? 1/(2^k?1 + a/b), where a > 2^k?1, the zero-one k-law holds. Moreover, this law does not hold for b = 1 and a ≤ 2^k?1 ? 2. It is proved that the k-law also fails for b > 1 and a ≤ 2^k?1 ? (b + 1)².     

Estimates for Integral Means of Hyperbolically Convex Functions

We prove the Mejia-Pommerenke conjecture that the Taylor coefficients of hyperbolically convex functions in the disk behave like O(log^?2(n)/n) as n → ∞ assuming that the image of the unit disk under such functions is a domain of bounded boundary rotation. Moreover, we obtain some asymptotically sharp estimates for the integral means of the derivatives of such functions and consider an example of a hyperbolically convex function that maps the unit disk onto a domain of infinite boundary rotation.     

A characterization of the normal distribution using stationary max-stable processes

Consider a max-stable process of the form \(\eta (t) = \max _{i\in \mathbb {N}} U_{i} \mathrm {e}^{\langle X_{i}, t\rangle - \kappa (t)}\), \(t\in \mathbb {R}^{d}\), where \(\{U_{i}, i\in \mathbb {N}\}\) are points of the Poisson process with intensity u ^?2du on (0,∞), X _i , \(i\in \mathbb {N}\), are independent copies of a random d-variate vector X (that are independent of the Poisson process), and \(\kappa :\mathbb {R}^{d} \to \mathbb {R}\) is a function. We show that the process η is stationary if and only if X has multivariate normal distribution and κ(t)?κ(0) is the cumulant generating function of X. In this case, η is a max-stable process introduced by R. L. Smith.     

Holomorphic injective extensions of functions in <Emphasis Type="Italic">P</Emphasis>(<Emphasis Type="Italic">K</Emphasis>) and algebra generators

We present necessary and sufficient conditions on planar compacta K and continuous functions f on K in order that f generate the algebras P(K), R(K), A(K) or C(K). We also unveil quite surprisingly simple examples of non-polynomial convex compacta K ? C and f ∈ P(K) with the property that f ∈ P(K) is a homeomorphism of K onto its image, but for which f ^?1 ? P(f(K)). As a consequence, such functions do not admit injective holomorphic extensions to the interior of the polynomial convex hull \(\widehat K\). On the other hand, it is shown that the restriction f*|_G of the Gelfand-transform f* of an injective function f ∈ P(K) is injective on every regular, bounded complementary component G of K. A necessary and sufficient condition in terms of the behaviour of f on the outer boundary of K is given in order that f admit a holomorphic injective extension to \(\widehat K\). We also include some results on the existence of continuous logarithms on punctured compacta containing the origin in their boundary.     

Some Poisson structures and Lax equations associated with the Toeplitz lattice and the Schur lattice

The Toeplitz lattice is a Hamiltonian system whose Poisson structure is known. In this paper, we unveil the origins of this Poisson structure and derive from it the associated Lax equations for this lattice. We first construct a Poisson subvariety H _n of GL _n (C), which we view as a real or complex Poisson–Lie group whose Poisson structure comes from a quadratic R-bracket on gl _n (C) for a fixed R-matrix. The existence of Hamiltonians, associated to the Toeplitz lattice for the Poisson structure on H _n , combined with the properties of the quadratic R-bracket allow us to give explicit formulas for the Lax equation. Then we derive from it the integrability in the sense of Liouville of the Toeplitz lattice. When we view the lattice as being defined over R, we can construct a Poisson subvariety H _n ^τ of U _n which is itself a Poisson–Dirac subvariety of GL _n ^R (C). We then construct a Hamiltonian for the Poisson structure induced on H _n ^τ , corresponding to another system which derives from the Toeplitz lattice the modified Schur lattice. Thanks to the properties of Poisson–Dirac subvarieties, we give an explicit Lax equation for the new system and derive from it a Lax equation for the Schur lattice. We also deduce the integrability in the sense of Liouville of the modified Schur lattice.     

Spaces of lower semicontinuous set-valued maps I

We introduce a lower semicontinuous analog, L ^?(X), of the well-studied space of upper semicontinuous set-valued maps with nonempty compact interval images. Because the elements of L ^?(X) contain continuous selections, the space C(X) of real-valued continuous functions on X can be used to establish properties of L ^?(X), such as the two interrelated main theorems. The first of these theorems, the Extension Theorem, is proved in this Part I. The Extension Theorem says that for binormal spaces X and Y, every bimonotone homeomorphism between C(X) and C(Y) can be extended to an ordered homeomorphism between L ^?(X) and L ^?(Y). The second main theorem, the Factorization Theorem, is proved in Part II. The Factorization Theorem says that for binormal spaces X and Y, every ordered homeomorphism between L ^?(X) and L ^?(Y) can be characterized by a unique factorization.     

A conditional limit theorem for a bivariate representation of a univariate random variable and conditional extreme values

We first consider a real random variable X represented through a random pair (R,T) and a deterministic function u as X = R?u(T). Under quite weak assumptions we prove a limit theorem for (R,T) given X>x, as x tends to infinity. The novelty of our paper is to show that this theorem for the representation of the univariate random variable X permits us to obtain in an elegant manner conditional limit theorems for random pairs (X,Y) = R?(u(T),v(T)) given that X is large. Our approach allows to deduce new results as well as to recover under considerably weaker assumptions results obtained previously in the literature. Consequently, it provides a better understanding and systematization of limit statements for the conditional extreme value models.     

Integral averaging technique for oscillation of damped half-linear oscillators

This paper is concerned with the oscillatory behavior of the damped half-linear oscillator (a(t)?_p(x′))′ + b(t)?_p(x′) + c(t)?_p(x) = 0, where ?p(x) = |x|^p?1 sgn x for x ∈ ? and p > 1. A sufficient condition is established for oscillation of all nontrivial solutions of the damped half-linear oscillator under the integral averaging conditions. The main result can be given by using a generalized Young’s inequality and the Riccati type technique. Some examples are included to illustrate the result. Especially, an example which asserts that all nontrivial solutions are oscillatory if and only if p ≠ 2 is presented.     

A singular Monge-Ampère equation on unbounded domains

In this paper, we study the Dirichlet problem for a singular Monge-Amp`ere type equation on unbounded domains. For a few special kinds of unbounded convex domains, we find the explicit formulas of the solutions to the problem. For general unbounded convex domain ?, we prove the existence for solutions to the problem in the space C∞(?) ∩ C(?). We also obtain the local C1/2-estimate up to the ?? and the estimate for the lower bound of the solutions.     

Decompositions,Partitions, and Coverings with Convex Polygons and Pseudo-Triangles

We propose a novel subdivision of the plane that consists of both convex polygons and pseudo-triangles. This pseudo-convex decomposition is significantly sparser than either convex decompositions or pseudo-triangulations for planar point sets and simple polygons. We also introduce pseudo-convex partitions and coverings. We establish some basic properties and give combinatorial bounds on their complexity. Our upper bounds depend on new Ramsey-type results concerning disjoint empty convex k-gons in point sets.     

Meromorphic Solutions for a Class of Differential Equations and Their Applications

In this note, we study the admissible meromorphic solutions for algebraic differential equation fⁿf' + P_n?1(f) = R(z)e^α(z), where P_n?1(f) is a differential polynomial in f of degree ≤ n ? 1 with small function coefficients, R is a non-vanishing small function of f, and α is an entire function. We show that this equation does not possess any meromorphic solution f(z) satisfying N(r, f) = S(r, f) unless P_n?1(f) ≡ 0. Using this result, we generalize a well-known result by Hayman.     

Unions of Fat Convex Polytopes Have Short Skeletons

The skeleton of a polyhedral set is the union of its edges and vertices. Let \(\mathcal {P}\) be a set of fat, convex polytopes in three dimensions with n vertices in total, and let f _max be the maximum complexity of any face of a polytope in \(\mathcal {P}\). We prove that the total length of the skeleton of the union of the polytopes in \(\mathcal {P}\) is at most O(α(n)?log^? n?logf _max) times the sum of the skeleton lengths of the individual polytopes.