Fractals with Positive Length and Zero Buffon Needle Probability

2003 ◽  
Vol 110 (4) ◽  
pp. 314 ◽  
Author(s):  
Yuval Peres ◽  
Karoly Simon ◽  
Boris Solomyak
Keyword(s):  
Positive Length

Maximal Areas of Reuleaux Polygons

1970 ◽  
Vol 13 (2) ◽  
pp. 175-179 ◽  
Author(s):  
G. T. Sallee
Keyword(s):  
Finite Number ◽  
Convex Set ◽  
Constant Width ◽  
Plane Convex ◽  
Circular Arcs ◽  
Positive Length ◽  
Reuleaux Polygon ◽  
Reuleaux Polygons

In this paper we provide new proofs of some interesting results of Firey [2] on isoperimetric ratios of Reuleaux polygons. Recall that a Reuleaux polygon is a plane convex set of constant width whose boundary consists of a finite (odd) number of circular arcs. Equivalently, it is the intersection of a finite number of suitably chosen congruent discs. For more details, see [1, p. 128].If a Reuleaux polygon has n sides (arcs) of positive length (where n is odd and ≥ 3), we will refer to it as a Reuleaux n-gon, or sometimes just as an n-gon. If all of the sides are equal, it is termed a regular n-gon.

Multi-clustered high-energy solutions for a phase transition problem

2005 ◽  
Vol 135 (4) ◽  
pp. 731-765
Author(s):  
Patricio L. Felmer ◽  
Salomé Martínez ◽  
Kazunaga Tanaka
Keyword(s):  
Phase Transition ◽  
Singular Perturbation ◽  
Energy Function ◽  
High Energy ◽  
Number Of Layers ◽  
Transition Problem ◽  
Phase Transition Problem ◽  
Positive Length ◽  
Limit Energy

We study the balanced Allen–Cahn problem in a singular perturbation setting. We are interested in the behaviour of clusters of layers, i.e. a family of solutions uε(x) with an increasing number of layers as ε → 0. In particular, we give a characterization of cluster of layers with asymptotically positive length by means of a limit energy function and, conversely, for a given admissible pattern, i.e. for a given a limit energy function, we construct a family of solutions with the corresponding behaviour.

scholarly journals The functional organization of area V2, I: Specialization across stripes and layers

2002 ◽  
Vol 19 (2) ◽  
pp. 187-210 ◽  
Author(s):  
STEWART SHIPP ◽  
SEMIR ZEKI
Keyword(s):  
Spectral Sensitivity ◽  
Functional Organization ◽  
Cortical Layer ◽  
Orientation Sensitivity ◽  
Area V2 ◽  
Layer Analysis ◽  
Positive Length ◽  
Varied Size ◽  
Direction Sensitivity ◽  
Marginal Zones

We used qualitative tests to assess the sensitivity of 1043 V2 neurons (predominantly multiunits) in anesthetised macaque monkeys to direction, length, orientation, and color of moving bar stimuli. Spectral sensitivity was additionally tested by noting ON or OFF responses to flashed stimuli of varied size and color. The location of 649 units was identified with respect to cycles of cytochrome oxidase stripes (thick-inter-thin-inter) and cortical layer. We used an initial 8-way stripe classification (4 stripes, and 4 “marginal” zones at interstripes boundaries), and a 9-way layer classification (5 standard layers (2–6), and 4 “marginal” strata at layer boundaries). These classes were collapsed differently for particular analyses of functional distribution; the main stripe-by-layer analysis was performed on 18 compartments (3 stripes × 6 layers). We found direction sensitivity only within thick stripes, orientation sensitivity mainly in thick stripes and interstripes, and spectral sensitivity mainly in thin stripes. Positive length summation was relatively more frequent in thick stripes and interstripes, and negative length/size summation in thin stripes. All these “majority” characteristics of stripes were most prominent in layers 3A and 3B. By contrast, “minority” characteristics (e.g. spectral sensitivity in thick stripes; positive size summation in thin stripes) tended to be most frequent in the outer layers, that is, layers 2 and 6. In consequence, going by the four functions tested, the distinctions between stripes were maximal in layer 3, moderate in layer 2, and minimal in layer 6.

scholarly journals Path-set induced closure operators on graphs

Filomat ◽  
2016 ◽  
Vol 30 (3) ◽  
pp. 863-871 ◽  
Author(s):  
Josef Slapal
Keyword(s):  
Image Analysis ◽  
Digital Image ◽  
Digital Image Analysis ◽  
Closure Operator ◽  
Simple Graph ◽  
Digital Topology ◽  
Closure Operators ◽  
Vertex Set ◽  
Positive Length ◽  
Path Connectedness

Given a simple graph, we associate with every set of paths of the same positive length a closure operator on the (vertex set of the) graph. These closure operators are then studied. In particular, it is shown that the connectedness with respect to them is a certain kind of path connectedness. Closure operators associated with sets of paths in some graphs with the vertex set Z2 are discussed which include the well known Marcus-Wyse and Khalimsky topologies used in digital topology. This demonstrates possible applications of the closure operators investigated in digital image analysis.

On cardinality questions concerning automaton mappings

2003 ◽  
Vol 40 (1-2) ◽  
pp. 71-82
Author(s):  
A. Ádám ◽  
M. Laczkovich
Keyword(s):  
The Family ◽  
Finite Set ◽  
Positive Length ◽  
Automaton Mapping

Let F+(X) be the set of words of positive length over a finite set X. By an automaton mapping (over (X,Y)) we understand a mapping of F+(X) into a finite set Y where |Y|?1). The family of all mappings over (X,Y) may be considered as an infinite automaton U having 2? states. U has at most 2^{2?} subautomata and at most 2? countable subautomata. We show that these bounds are actually attained.

DISTANCE PRESERVING SUBTREES IN MINIMUM AVERAGE DISTANCE SPANNING TREES

2013 ◽  
Vol 05 (03) ◽  
pp. 1350010
Author(s):  
LAURENT LYAUDET ◽  
PAULIN MELATAGIA YONTA ◽  
MAURICE TCHUENTE ◽  
RENÉ NDOUNDAM
Keyword(s):  
Approximate Solution ◽  
Spanning Tree ◽  
Path Length ◽  
Spanning Trees ◽  
Undirected Graph ◽  
Average Distance ◽  
Edge Length ◽  
The Other ◽  
First Results ◽  
Positive Length

Given an undirected graph G = (V, E) with n vertices and a positive length w(e) on each edge e ∈ E, we consider Minimum Average Distance (MAD) spanning trees i.e., trees that minimize the path length summed over all pairs of vertices. One of the first results on this problem is due to Wong who showed in 1980 that a Distance Preserving (DP) spanning tree rooted at the median of G is a 2-approximate solution. On the other hand, Dankelmann has exhibited in 2000 a class of graphs where no MAD spanning tree is distance preserving from a vertex. We establish here a new relation between MAD and DP trees in the particular case where the lengths are integers. We show that in a MAD spanning tree of G, each subtree H′ = (V′, E′) consisting of a vertex [Formula: see text] and the union of branches of [Formula: see text] that are each of size less than or equal to [Formula: see text], where w+ is the maximum edge-length in G, is a distance preserving spanning tree of the subgraph of G induced by V′.

A PHASE PLANE ANALYSIS OF TRANSONIC SOLUTIONS FOR THE HYDRODYNAMIC SEMICONDUCTOR MODEL

1991 ◽  
Vol 01 (03) ◽  
pp. 347-376 ◽  
Author(s):  
URI M. ASCHER ◽  
PETER A. MARKOWICH ◽  
PAOLA PIETRA ◽  
CHRISTIAN SCHMEISER
Keyword(s):  
Phase Plane ◽  
Semiconductor Device ◽  
Regularization Parameter ◽  
Phase Plane Analysis ◽  
Singularly Perturbed Problems ◽  
Discontinuous Solutions ◽  
Plane Analysis ◽  
Positive Length ◽  
Artificial Diffusion ◽  
State 1

We present an analysis of transonic solutions of the steady state 1-dimensional unipolar hydrodynamic model for semiconductors in the isoentropic case. The approach is based on construction of the orbits of the system in the electron density-electric field phase plane and on representation of discontinuous solutions of the hydrodynamic boundary value problem by a union of trajectory pieces. These pieces are related by shocks obeying jump and entropy conditions. A continuation argument in the length of the semiconductor device under consideration is applied to construct a continuum of sub- and transonic solutions, which contains at least one solution for every positive length. We also present numerical results illustrating the various possible solution profiles. For this we use a regularization of the problem, adding artificial diffusion to obtain singularly perturbed problems which are then solved numerically using continuation in the regularization parameter.

SMALL ENERGY COMPACTNESS FOR APPROXIMATE HARMONIC MAPPINGS

2011 ◽  
Vol 13 (05) ◽  
pp. 741-763 ◽  
Author(s):  
JIAYU LI ◽  
XIANGRONG ZHU
Keyword(s):  
Compact Riemannian Manifold ◽  
Blow Up ◽  
Elliptic Systems ◽  
Compactness Theorem ◽  
Harmonic Mappings ◽  
Riemannian Surface ◽  
Energy Identity ◽  
Small Energy ◽  
Blowing Up ◽  
Positive Length

In this paper, we consider the elliptic systems [Formula: see text] where u ∈ W1, 2(R2, RK) and f ∈ L ln + L, and Ω belongs to L2(R2, MK(R)⊗R2) which is antisymmetric. In the first part we prove a compactness theorem for this system. As a corollary, we obtain the compactness theorem for a sequence of mappings from a Riemannian surface to a compact Riemannian manifold with tension fields bounded in L ln + L. In the second part we prove the energy identity for a sequence of mappings from a surface to a sphere with tension fields bounded in L ln + L. In the last section we construct a blow-up sequence of mappings from B1 to S2 with tension fields bounded in L ln + L but there exists a neck with positive length during blowing up.

scholarly journals The future is not always open

2019 ◽  
Vol 110 (1) ◽  
pp. 83-103 ◽  
Author(s):  
James D. E. Grant ◽  
Michael Kunzinger ◽  
Clemens Sämann ◽  
Roland Steinbauer
Keyword(s):  
Quantum Gravity ◽  
Lorentzian Geometry ◽  
Differentiable Structure ◽  
Low Regularity ◽  
Locally Lipschitz ◽  
The Future ◽  
Positive Length ◽  
Synthetic Approaches ◽  
Push Up

Abstract We demonstrate the breakdown of several fundamentals of Lorentzian causality theory in low regularity. Most notably, chronological futures (defined naturally using locally Lipschitz curves) may be non-open and may differ from the corresponding sets defined via piecewise $$C^1$$C1-curves. By refining the notion of a causal bubble from Chruściel and Grant (Class Quantum Gravity 29(14):145001, 2012), we characterize spacetimes for which such phenomena can occur, and also relate these to the possibility of deforming causal curves of positive length into timelike curves (push-up). The phenomena described here are, in particular, relevant for recent synthetic approaches to low-regularity Lorentzian geometry where, in the absence of a differentiable structure, causality has to be based on locally Lipschitz curves.

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