scholarly journals SPIKES IN COSMIC CRYSTALLOGRAPHY

2002 ◽  
Vol 11 (06) ◽  
pp. 869-891 ◽  
Author(s):  
G. I. GOMERO ◽  
A. F. F. TEIXEIRA ◽  
M. J. REBOUÇCAS ◽  
A. BERNUI
Keyword(s):  
Multiple Images ◽  
Multiply Connected ◽  
Major Consequence ◽  
Statistical Fluctuations ◽  
Covering Group ◽  
Connected Case ◽  
Pair Separation ◽  
Simply Connected ◽  
The Universe ◽  
Covering Groups

If the universe is multiply connected and small the sky shows multiple images of cosmic objects, correlated by the covering group of the 3-manifold used to model it. These correlations were originally thought to manifest as spikes in pair separation histograms (PSH) built from suitable catalogues. Using probability theory we derive an expression for the expected pair separation histogram (EPSH) in a rather general topological-geometrical-observational setting. As a major consequence we show that the spikes of topological origin in PSH's are due to translations, whereas other isometries manifest as tiny deformations of the PSH corresponding to the simply connected case. This result holds for all Robertson–Walker spacetimes and gives rise to two basic corollaries: (i) that PSH's of Euclidean manifolds that have the same translations in their covering groups exhibit identical spike spectra of topological origin, making clear that even if the universe is flat the topological spikes alone are not sufficient for determining its topology; and (ii) that PSH's of hyperbolic 3-manifolds exhibit no spikes of topological origin. These corollaries ensure that cosmic crystallography, as originally formulated, is not a conclusive method for unveiling the shape of the universe. We also present a method that reduces the statistical fluctuations in PSH's built from simulated catalogues.

The Carathéodory Reflection Principle and Osgood–Carathéodory Theorem on Riemann Surfaces

2016 ◽  
Vol 59 (4) ◽  
pp. 776-793
Author(s):  
Paul M. Gauthier ◽  
Fatemeh Sharifi
Keyword(s):  
Riemann Surfaces ◽  
Reflection Principle ◽  
Conformal Mappings ◽  
Multiply Connected Domains ◽  
Multiply Connected ◽  
Connected Case ◽  
Simply Connected ◽  
Carathéodory Theorem

AbstractThe Osgood–Carathéodory theorem asserts that conformal mappings between Jordan domains extend to homeomorphisms between their closures. For multiply-connected domains on Riemann surfaces, similar results can be reduced to the simply-connected case, but we find it simpler to deduce such results using a direct analogue of the Carathéodory reflection principle.

scholarly journals R-GROUP AND WHITTAKER SPACE OF SOME GENUINE REPRESENTATIONS

2021 ◽  
pp. 1-61
Author(s):  
Fan Gao
Keyword(s):  
Series Representation ◽  
Principal Series ◽  
Symplectic Groups ◽  
Principal Series Representation ◽  
Connected Group ◽  
Covering Group ◽  
Simply Connected

Abstract For a unitary unramified genuine principal series representation of a covering group, we study the associated R-group. We prove a formula relating the R-group to the dimension of the Whittaker space for the irreducible constituents of such a principal series representation. Moreover, for certain saturated covers of a semisimple simply connected group, we also propose a simpler conjectural formula for such dimensions. This latter conjectural formula is verified in several cases, including covers of the symplectic groups.

scholarly journals A VISIBILITY-BASED PURSUIT-EVASION PROBLEM

1999 ◽  
Vol 09 (04n05) ◽  
pp. 471-493 ◽  
Author(s):  
LEONIDAS J. GUIBAS ◽  
JEAN-CLAUDE LATOMBE ◽  
STEVEN M. LAVALLE ◽  
DAVID LIN ◽  
RAJEEV MOTWANI
Keyword(s):  
Finite Graph ◽  
Initial Position ◽  
Moving Target ◽  
Multiply Connected ◽  
Pursuit Evasion ◽  
Minimum Number ◽  
Evasion Problem ◽  
Free Spaces ◽  
Simply Connected ◽  
Motion Strategy

This paper addresses the problem of planning the motion of one or more pursuers in a polygonal environment to eventually "see" an evader that is unpredictable, has unknown initial position, and is capable of moving arbitrarily fast. This problem was first introduced by Suzuki and Yamashita. Our study of this problem is motivated in part by robotics applications, such as surveillance with a mobile robot equipped with a camera that must find a moving target in a cluttered workspace. A few bounds are introduced, and a complete algorithm is presented for computing a successful motion strategy for a single pursuer. For simply-connected free spaces, it is shown that the minimum number of pursuers required is Θ( lg  n). For multiply-connected free spaces, the bound is [Formula: see text] pursuers for a polygon that has n edges and h holes. A set of problems that are solvable by a single pursuer and require a linear number of recontaminations is shown. The complete algorithm searches a finite graph that is constructed on the basis of critical information changes. It has been implemented and computed examples are shown.

scholarly journals The Prime Function, the Fay Trisecant Identity, and the van der Pauw Method

2021 ◽  
Author(s):  
Hiroyuki Miyoshi ◽  
Darren Crowdy ◽  
Rhodri Nelson
Keyword(s):  
Applied Sciences ◽  
Science Literature ◽  
Multiply Connected ◽  
Schottky Double ◽  
Van Der Pauw ◽  
Simply Connected ◽  
Genus One ◽  
Physical Quantities ◽  
Prime Function ◽  
Van Der Pauw Method

AbstractThe van der Pauw method is a well-known experimental technique in the applied sciences for measuring physical quantities such as the electrical conductivity or the Hall coefficient of a given sample. Its popularity is attributable to its flexibility: the same method works for planar samples of any shape provided they are simply connected. Mathematically, the method is based on the cross-ratio identity. Much recent work has been done by applied scientists attempting to extend the van der Pauw method to samples with holes (“holey samples”). In this article we show the relevance of two new function theoretic ingredients to this area of application: the prime function associated with the Schottky double of a multiply connected planar domain and the Fay trisecant identity involving that prime function. We focus here on the single-hole (doubly connected, or genus one) case. Using these new theoretical ingredients we are able to prove several mathematical conjectures put forward in the applied science literature.

CONNECTIVITY PROPERTIES OF SIERPIŃSKI RELATIVES

Fractals ◽  
2011 ◽  
Vol 19 (04) ◽  
pp. 481-506 ◽  
Author(s):  
T. D. TAYLOR
Keyword(s):  
Fractal Dimension ◽  
Multiply Connected ◽  
Simply Connected ◽  
Totally Disconnected

This paper presents a study of the connectivity of the class of fractals known as the Sierpiński relatives. These fractals all have the same fractal dimension, but different topologies. Some are totally disconnected, some are disconnected but with paths, some are simply-connected, and some are multiply-connected. Conditions for these four cases are presented. Constructions of paths, including non-contractible closed paths in the case of multiply-connected relatives, are presented. Examples of specific relatives are provided to illustrate the four cases.

scholarly journals Hints for possible low redshift oscillation around the best fit ΛCDM model in the expansion history of the universe

2020 ◽  
Author(s):  
L Kazantzidis ◽  
H Koo ◽  
S Nesseris ◽  
L Perivolaropoulos ◽  
A Shafieloo
Keyword(s):  
Monte Carlo ◽  
Monte Carlo Simulations ◽  
Absolute Magnitude ◽  
Density Parameter ◽  
Statistical Fluctuations ◽  
Full Dataset ◽  
Λcdm Model ◽  
History Of ◽  
Best Fit ◽  
The Universe

Abstract We search for possible deviations from the expectations of the concordance ΛCDM model in the expansion history of the Universe by analysing the Pantheon Type Ia Supernovae (SnIa) compilation along with its Monte Carlo simulations using redshift binning. We demonstrate that the redshift binned best fit ΛCDM matter density parameter Ω0m and the best fit effective absolute magnitude $\cal M$ oscillate about their full dataset best fit values with considerably large amplitudes. Using the full covariance matrix of the data taking into account systematic and statistical errors, we show that at the redshifts below z ≈ 0.5 such oscillations can only occur in 4 to 5% of the Monte Carlo simulations. While statistical fluctuations can be responsible for this apparent oscillation, we might have observed a hint for some behaviour beyond the expectations of the concordance model or a possible additional systematic in the data. If this apparent oscillation is not due to statistical or systematic effects, it could be due to either the presence of coherent inhomogeneities at low z or due to oscillations of a quintessence scalar field.

scholarly journals AN INFINITE NUMBER OF CLOSED FLRW UNIVERSES FOR ANY VALUE OF THE SPATIAL CURVATURE

2012 ◽  
Vol 18 ◽  
pp. 77-80
Author(s):  
HELIO V. FAGUNDES
Keyword(s):  
Finite Volume ◽  
Infinite Number ◽  
Negative Curvature ◽  
Large Scale ◽  
Positive Curvature ◽  
Cosmological Models ◽  
Infinite Volume ◽  
Spatial Curvature ◽  
Multiply Connected ◽  
Simply Connected

The Friedman-Lemaître-Robertson-Walker cosmological models are based on the assumptions of large-scale homogeneity and isotropy of the distribution of matter and energy. They are usually taken to have spatial sections that are simply connected; they have finite volume in the positive curvature case, and infinite volume in the null and negative curvature ones. I want to call the attention to the existence of an infinite number of models, which are based on these same metrics, but have compact, finite volume, multiply connected spatial sections. Some observational implications are briefly mentioned.

scholarly journals On the fundamental group of a Lie semigroup

1992 ◽  
Vol 34 (3) ◽  
pp. 379-394 ◽  
Author(s):  
Karl-Hermann Neeb
Keyword(s):  
Fundamental Group ◽  
Lie Group ◽  
Unit Group ◽  
Universal Covering ◽  
Covering Group ◽  
Lie Semigroup ◽  
Lie Semigroups ◽  
Inclusion Mapping ◽  
Image Position ◽  
Simply Connected

The simplest type of Lie semigroups are closed convex cones in finite dimensional vector spaces. In general one defines a Lie semigroup to be a closed subsemigroup of a Lie group which is generated by one-parameter semigroups. If W is a closed convex cone in a vector space V, then W is convex and therefore simply connected. A similar statement for Lie semigroups is false in general. There exist generating Lie semigroups in simply connected Lie groups which are not simply connected (Example 1.15). To find criteria for cases when this is true, one has to consider the homomorphisminduced by the inclusion mapping i:S→G, where S is a generating Lie semigroup in the Lie group G. Our main results concern the description of the image and the kernel of this mapping. We show that the image is the fundamental group of the largest covering group of G, into which S lifts, and that the kernel is the fundamental group of the inverse image of 5 in the universal covering group G. To get these results we construct a universal covering semigroup S of S. If j: H(S): = S ∩ S-1 →S is the inclusion mapping of the unit group of S into S, then it turns out that the kernel of the induced mappingmay be identfied with the fundamental group of the unit group H(S)of S and that its image corresponds to the intersection H(S)0 ⋂π1(S), where π1(s) is identified with a central subgroup of S.

A rigidity theorem for manifolds without conjugate points

1998 ◽  
Vol 18 (4) ◽  
pp. 813-829 ◽  
Author(s):  
CHRISTOPHER B. CROKE ◽  
BRUCE KLEINER
Keyword(s):  
Companion Paper ◽  
Rigidity Theorem ◽  
Conjugate Points ◽  
Compactly Supported ◽  
Connected Case ◽  
Flat Metric ◽  
Simply Connected

In this paper we show that some nonsimply connected manifolds without conjugate points exhibit rigidity phenomena similar to that studied in [BGS, section I.5]. This is a companion paper to [Cr-Kl1] that deals with the simply connected case. In particular, we show that one cannot make a nontrivial, compactly supported, change to a complete flat metric without introducing conjugate points.

Sign in / Sign up

Export Citation Format

Share Document