Sets In Euclidean Space Research Articles

Killed Brownian motion and the Brunn–Minkowski inequalities

AbstractWe construct triplets of killed Brownian motions to obtain the Brunn–Minkowski inequalities concerning the solutions of the equation (1/2)Δψ − h ψ = g on three interrelated compact sets in Euclidean space. These, in particular, include inequalities relating to the solutions of the Schrödinger equation and the Poisson equation on the three compact convex sets and an inequality relating to harmonic functions.

Endpoint estimates for first-order Riesz transforms associated to the Ornstein–Uhlenbeck operator

In the setting of Euclidean space with the Gaussian measure \gamma , we consider all first-order Riesz transforms associated to the infinitesimal generator of the Ornstein–Uhlenbeck semigroup. These operators are known to be bounded on L^p(\gamma) , for 1< p< \infty . We determine which of them are bounded from H^1(\gamma) to L^1(\gamma) and from L^\infty(v) to BMO( \gamma ). Here H^1(\gamma) and BMO( \gamma ) are the spaces introduced in this setting by the first two authors. Surprisingly, we find that the results depend on the dimension of the ambient space.

On Euclidean Designs and Potential Energy

We study Euclidean designs from the viewpoint of the potential energy. For a finite set in Euclidean space, we formulate a linear programming bound for the potential energy by applying harmonic analysis on a sphere. We also introduce the concept of strong Euclidean designs from the viewpoint of the linear programming bound, and we give a Fisher type inequality for strong Euclidean designs. A finite set on Euclidean space is called a Euclidean $a$-code if any distinct two points in the set are separated at least by $a$. As a corollary of the linear programming bound, we give a method to determine an upper bound on the cardinalities of Euclidean $a$-codes on concentric spheres of given radii. Similarly we also give a method to determine a lower bound on the cardinalities of Euclidean $t$-designs as an analogue of the linear programming bound.

Comparison results for capacity

E* Abstract. We obtain in this paper bounds for the capacity of a compact set K with smooth boundary. If K is contained in an (n + 1)-dimensional Cartan- Hadamard manifold and the principal curvatures of @K are larger than or equal to H0 > 0, then Cap(K) > (n 1)H0 vol(@K). When K is contained in an (n+1)- dimensional manifold with non-negative Ricci curvature and the mean curvature of @K is smaller than or equal to H0, we prove the inequality Cap(K) 6 (n 1)H0 vol(@K). In both cases we are able to characterize the equality case. Finally, if K is a convex set in Euclidean space R n+1 which admits a supporting sphere of radius H −1 0 at any boundary point, then we prove Cap(K) > (n 1)H0 vol(@K) and that equality holds for the round sphere of radius H −1 0 .

Densities and liftings for derived algebras

Motivated by the construction of the algebra of Jordan measurable sets from the algebra of measurable sets in Euclidean spaces, we determine a natural class of structures (X,S,A,I), where (X,S) is a topological space, A is an algebra of subsets of X, and I is an ideal of A, so that the derived structure (X,S,∂A,∂I) given by ∂A:={E∈A:∂E∈I} and ∂I:=∂A∩I belongs to the same class. We provide a characterization of the derived structures whose ideal contains no nonempty open sets and derive from it that each such structure has a natural strong density and (assuming completeness) a strong lifting.

The family of bi-Lipschitz classes of delone sets in euclidean space has the cardinality of the continuum

It is proved that the family of bi-Lipschitz classes of Delone sets in Euclidean space of dimension at least 2 has the cardinality of the continuum.

$d$-representability of simplicial complexes of fixed dimension

Let K be a simplicial complex with vertex set V = v 1 ,…, v n . The complex K is d -representable if there is a collection { C 1 ,…, C n } of convex sets in R d such that a subcollection { C i 1 ,…, C i j } has a nonempty intersection if and only if { v i 1 ,…, v i j } is a face of K . In 1967 Wegner proved that every simplicial complex of dimension d is (2 d +1)-representable. He also conjectured that his bound is the best possible, i.e., that there are d -dimensional simplicial complexes which are not 2 d -representable. However, he was not able to prove his conjecture. We prove that his suggestion was indeed right. Thus we add another piece to the puzzle of intersection patterns of convex sets in Euclidean space.

An approximation algorithm for solving a problem of cluster analysis

The authors provide some 2-approximation algorithm for an intractable problem to which one can reduce the problem of partitioning a vector set in Euclidean space into the two subsets (clusters) having the minimum sum of distance squares.

A Hybrid Statistical Technique for Modeling Recurrent Tracks in a Compact Set

In this technical note we present a hybrid statistical approach for modeling a vehicle's behavior as it traverses a compact set in Euclidean space. We use Symbolic Transfer Functions (STF), developed by the authors for modeling stochastic input/output systems whose inputs and outputs are both purely symbolic. We apply STF to our problem by assuming that the input symbols represent regions of space through which a track is passing while the output represents specific linear functions that more precisely model the behavior of the track. A target's behavior is modeled at two levels of precision: The symbolic model provides a probability distribution on the next region of space and behavior (linear function) that a vehicle will execute, while the continuous model predicts the position of the vehicle using classical statistical methods. The following results are presented: (i) An algorithm that parsimoniously partitions the space of the vehicle and models the behavior in the partitions with linear functions. (ii) A demonstration of our approach using real-world ship track data.

Estimates for the Torsion Function and Sobolev Constants

Let Ω be an open set in Euclidean space, and let u : Ω → ℝ + be the expected lifetime of Brownian motion in Ω. It is shown that if u ∈ L p (Ω) for some p ∈ [1, ∞ ) then (i) u ∈ L q (Ω) for all q ∈ [p, ∞ ], and (ii) ${trace}\left(e^{t\Delta_{\Omega}}\right)<\infty$ for all t > 0, where − ΔΩ is the Dirichlet Laplacian acting in L 2(Ω). Pointwise bounds are obtained for u in terms of the first Dirichlet eigenfunction for Ω, assuming that the spectrum of − ΔΩ is discrete. It is shown that if Ω is open, bounded and connected in the plane and $\partial\Omega$ has an interior wedge with opening angle α at vertex v then the first Dirichlet eigenfunction and u are comparable near v if and only if α ≥ π/2. Two sided estimates are obtained for the Sobolev constant $$ C_p(\Omega):= \inf\left\{\Vert \nabla u \Vert_2^2: u \in C_0^{\infty}(\Omega),\ \Vert u\Vert_p = 1\right\}, $$ where 0 < p < 2, provided − ΔΩ satisfies a strong Hardy inequality, and the distance to the boundary function δ ∈ L 2p/(2 − p)(Ω).

SEMILOCAL CONVERGENCE OF NEWTON'S METHOD FOR SINGULAR SYSTEMS WITH CONSTANT RANK DERIVATIVES

We provide a semilocal convergence result for approximating a solution of a singular system with constant rank derivatives, using Newton's method in an Euclidean space setting. Our approach uses more precise estimates and a combination of two Lipschitz-type conditions leading to the following advantages over earlier works [13], [16], [17], [29]: tighter bounds on the distances involved, and a more precise information on the location of the solution. Numerical examples are also provided in this study.

Energy Minimization, Periodic Sets and Spherical Designs

We study energy minimization for pair potentials among periodic sets in Euclidean spaces. We derive some sufficient conditions under which a point lattice locally minimizes the energy associated to a large class of potential functions. This allows in particular to prove a local version of Cohn and Kumar's conjecture that $\mathsf{A}_2$, $\mathsf{D}_4$, $\mathsf{E}_8$ and the Leech lattice are globally universally optimal, regarding energy minimization, and among periodic sets of fixed point density.

On the visibility in well-behaved random sets in Euclidean space

In Calka et al. (2009), the decay of the probability of reaching distance at least r in some direction from a given point without hitting any ball, in the Poisson Boolean model of continuum percolation, was studied. The methods used in Calka et al. (2009) include coverage techniques, and the most precise results were obtained in dimension 2 . In this note, we strengthen some of the results obtained in Calka et al. (2009) to dimension 3 and higher and at the same time extend them to more general random sets.

Smoothness of sets in Euclidean spaces

We study some properties of smooth sets in the sense defined by Hungerford. We prove a sharp form of Hungerford's theorem on the Hausdorff dimension of their boundaries on Euclidean spaces and show the invariance of the definition under a class of automorphisms of the ambient space.

Mass problems associated with effectively closed sets

The study of mass problems and Muchnik degrees was originally motivated by Kolmogorov's non-rigorous 1932 interpretation of intuitionism as a calculus of problems. The purpose of this paper is to summarize recent investigations into the lattice of Muchnik degrees of nonempty effectively closed sets in Euclidean space. Let $\mathcal{E}_\mathrm{w}$ be this lattice. We show that $\mathcal{E}_\mathrm{w}$ provides an elegant and useful framework for the classification of certain foundationally interesting problems which are algorithmically unsolvable. We exhibit some specific degrees in $\mathcal{E}_\mathrm{w}$ which are associated with such problems. In addition, we present some structural results concerning the lattice $\mathcal{E}_\mathrm{w}$. One of these results answers a question which arises naturally from the Kolmogorov interpretation. Finally, we show how $\mathcal{E}_\mathrm{w}$ can be applied in symbolic dynamics, toward the classification of tiling problems and $\boldsymbol{Z}^d$-subshifts of finite type.

Acute Sets In Euclidean Spaces

A finite set ℋ in ℝd is called an acute set if any angle determined by three points of ℋ is acute. We examine the maximal cardinality α(d) of a d-dimensional acute set. The exact value of α(d) is known only for d≤3. For each d≥4 we improve on the best known lower bound for α(d). We present different approaches. On one hand, we give a probabilistic proof that α(d)>c·1.2d. (This improves a random construction given by Erdo˝s and Furedi.) On the other hand, we give an almost exponential constructive example which outdoes the random construction in low dimension (d≤250). Both approaches use the small dimensional examples that we found partly by hand (d=4, 5) and partly by computer (6≤d≤10). We also investigate the following variant of the above problem: what is the maximal size κ(d) of a d-dimensional cubic acute set (that is, an acute set contained in the vertex set of a d-dimensional hypercube)? We give an almost exponential constructive lower bound, and we improve on the best known upper bound.

Covering planar sets

Problems connected with the classical Borsuk problem on partitioning a set in Euclidean space into subsets of smaller diameter, and also connected with the Nelson-Hadwiger problem on the chromatic number of Euclidean space, are studied. New bounds are obtained for the quantities and , where the suprema are taken over all sets of unit diameter on a plane, and where the quantities and are defined for a given bounded set as follows: Here the are subsets, is the diameter of , is the distance between the points and , and . The bounds obtained for are better than any known before; this paper is the first to consider the values .Bibliography: 19 titles.

Tube Formulas and Complex Dimensions of Self-Similar Tilings

We use the self-similar tilings constructed in (Pearse in Indiana Univ. Math J. 56(6):3151---3169, 2007) to define a generating function for the geometry of a self-similar set in Euclidean space. This tubularzeta function encodes scaling and curvature properties related to the complement of the fractal set, and the associated system of mappings. This allows one to obtain the dimensions of the self-similar tiling as the poles of the tubularzeta function and hence develop a tube formula for self-similar tilings in ?d. The resulting power series in ?is a fractal extension of Steiner's classical tube formula for convex bodies K?? d . Our sum has coefficients related to the curvatures of the tiling, and contains terms for each integer i=0,1,?,d?1, just as Steiner's does. However, our formula also contains a term for each This provides further justification for the term complex dimension. It also extends several aspects of the theory of fractal strings to higher dimensions and sheds new light on the tube formula for fractals strings obtained in (Lapidus and van Frankenhuijsen in Fractal Geometry, Complex Dimensions and Zeta Functions: Geometry and Spectra of Fractal Strings, 2006).

Information-theoretic inequalities on unimodular Lie groups

Classical inequalities used in information theory such as those of de Bruijn, Fisher, Cramér, Rao, and Kullback carry over in a natural way from Euclidean space to unimodular Lie groups. These are groups that possess an integration measure that is simultaneously invariant under left and right shifts. All commutative groups are unimodular. And even in noncommutative cases unimodular Lie groups share many of the useful features of Euclidean space. The rotation and Euclidean motion groups, which are perhaps the most relevant Lie groups to problems in geometric mechanics, are unimodular, as are the unitary groups that play important roles in quantum computing. The extension of core information theoretic inequalities defined in the setting of Euclidean space to this broad class of Lie groups is potentially relevant to a number of problems relating to information gathering in mobile robotics, satellite attitude control, tomographic image reconstruction, biomolecular structure determination, and quantum information theory. In this paper, several definitions are extended from the Euclidean setting to that of Lie groups (including entropy and the Fisher information matrix), and inequalities analogous to those in classical information theory are derived and stated in the form of fifteen small theorems. In all such inequalities, addition of random variables is replaced with the group product, and the appropriate generalization of convolution of probability densities is employed. An example from the field of robotics demonstrates how several of these results can be applied to quantify the amount of information gained by pooling different sensory inputs.

On a generalization of distance sets

A subset X in the d-dimensional Euclidean space is called a k-distance set if there are exactly k distinct distances between two distinct points in X and a subset X is called a locally k-distance set if for any point x in X, there are at most k distinct distances between x and other points in X. Delsarte, Goethals, and Seidel gave the Fisher type upper bound for the cardinalities of k-distance sets on a sphere in 1977. In the same way, we are able to give the same bound for locally k-distance sets on a sphere. In the first part of this paper, we prove that if X is a locally k-distance set attaining the Fisher type upper bound, then determining a weight function w, ( X , w ) is a tight weighted spherical 2 k-design. This result implies that locally k-distance sets attaining the Fisher type upper bound are k-distance sets. In the second part, we give a new absolute bound for the cardinalities of k-distance sets on a sphere. This upper bound is useful for k-distance sets for which the linear programming bound is not applicable. In the third part, we discuss about locally two-distance sets in Euclidean spaces. We give an upper bound for the cardinalities of locally two-distance sets in Euclidean spaces. Moreover, we prove that the existence of a spherical two-distance set in ( d − 1 ) -space which attains the Fisher type upper bound is equivalent to the existence of a locally two-distance set but not a two-distance set in d-space with more than d ( d + 1 ) / 2 points. We also classify optimal (largest possible) locally two-distance sets for dimensions less than eight. In addition, we determine the maximum cardinalities of locally two-distance sets on a sphere for dimensions less than forty.