A universal one-dimensional metric space

1974 ◽  
pp. 248-257 ◽  
Author(s):  
Stephen Leon Lipscomb
Keyword(s):  
Metric Space ◽  
One Dimensional

scholarly journals Lipschitz Retractions in Hadamard Spaces via Gradient Flow Semigroups

2016 ◽  
Vol 59 (4) ◽  
pp. 673-681 ◽  
Author(s):  
Miroslav Bačák ◽  
Leonid V. Kovalev
Keyword(s):  
Hilbert Space ◽  
Finite Subset ◽  
Metric Space ◽  
Hausdorff Metric ◽  
Gradient Flow ◽  
The Other ◽  
Dimensional Sphere ◽  
Hadamard Space ◽  
One Dimensional ◽  
The One

AbstractLet X(n), for n ∊ ℕ, be the set of all subsets of a metric space (X, d) of cardinality at most n. The set X(n) equipped with the Hausdorff metric is called a finite subset space. In this paper we are concerned with the existence of Lipschitz retractions r: X(n)→ X(n − 1) for n ≥ 2. It is known that such retractions do not exist if X is the one-dimensional sphere. On the other hand, Kovalev has recently established their existence if X is a Hilbert space, and he also posed a question as to whether or not such Lipschitz retractions exist when X is a Hadamard space. In this paper we answer the question in the positive.

scholarly journals A one-dimensional homologically persistent skeleton of an unstructured point cloud in any metric space

2015 ◽  
Vol 34 (5) ◽  
pp. 253-262 ◽  
Author(s):  
V. Kurlin
Keyword(s):  
Metric Space ◽  
Point Cloud ◽  
One Dimensional

scholarly journals On imbedding a metric space in a product of one-dimensional spaces

1957 ◽  
Vol 33 (8) ◽  
pp. 445-449 ◽  
Author(s):  
Jun-iti Nagata
Keyword(s):  
Metric Space ◽  
One Dimensional

scholarly journals Nonlinear model reduction on metric spaces. Application to one-dimensional conservative PDEs in Wasserstein spaces

2020 ◽  
Vol 54 (6) ◽  
pp. 2159-2197
Author(s):  
Virginie Ehrlacher ◽  
Damiano Lombardi ◽  
Olga Mula ◽  
François-Xavier Vialard
Keyword(s):  
Model Reduction ◽  
Metric Space ◽  
Metric Spaces ◽  
Nonlinear Approximation ◽  
Computational Cost ◽  
Approximation Methods ◽  
Bottom Line ◽  
One Dimensional ◽  
Nonlinear Model Reduction ◽  
Wasserstein Space

We consider the problem of model reduction of parametrized PDEs where the goal is to approximate any function belonging to the set of solutions at a reduced computational cost. For this, the bottom line of most strategies has so far been based on the approximation of the solution set by linear spaces on Hilbert or Banach spaces. This approach can be expected to be successful only when the Kolmogorov width of the set decays fast. While this is the case on certain parabolic or elliptic problems, most transport-dominated problems are expected to present a slow decaying width and require to study nonlinear approximation methods. In this work, we propose to address the reduction problem from the perspective of general metric spaces with a suitably defined notion of distance. We develop and compare two different approaches, one based on barycenters and another one using tangent spaces when the metric space has an additional Riemannian structure. Since the notion of linear vectorial spaces does not exist in general metric spaces, both approaches result in nonlinear approximation methods. We give theoretical and numerical evidence of their efficiency to reduce complexity for one-dimensional conservative PDEs where the underlying metric space can be chosen to be the L2-Wasserstein space.

Completely Regular Mappings and Homogeneous, Aposyndetic Continua

1981 ◽  
Vol 33 (2) ◽  
pp. 450-453 ◽  
Author(s):  
James T. Rogers
Keyword(s):  
Metric Space ◽  
Decomposition Theorem ◽  
One Dimensional ◽  
Completely Regular ◽  
New Information

The purpose of this note is to prove an improved version of Jones' Aposyndetic Decomposition Theorem. Corollaries to the new theorem re-emphasize the importance of understanding aposyndetic, homogeneous continua.The proof is a synthesis of results about homogeneous continua with results from an unexpected source: completely regular mappings. Completely regular mappings occur naturally and often in the study of homogeneous continua, which is a surprising and pleasing phenomenon, since these mappings were invented for quite another purpose [1]. The author believes that these maps are likely to provide even more new information about homogeneous continua.A continuum is a compact, connected, nonvoid metric space. A curve is a one-dimensional continuum. A continuum M is homogeneous if for each pair of points p and q belonging to M, there exists a homeomorphism h: M → M such that h(p) = q.

UPPER BOUND OF THE DIRECTIONAL ENTROPY OF A ℤ2-ACTION

2011 ◽  
Vol 22 (07) ◽  
pp. 711-718 ◽  
Author(s):  
HASAN AKIN
Keyword(s):  
Cellular Automaton ◽  
Metric Space ◽  
Upper Bound ◽  
Natural Extension ◽  
Studia Math ◽  
Short Paper ◽  
Compact Metric Space ◽  
One Dimensional ◽  
Shift Map ◽  
Bernoulli Measures

In this short paper, we study the measure directional entropy of a ℤ2-action generated by an invertible one-dimensional linear cellular automaton and the shift map acting on the compact metric space [Formula: see text]. We obtain an upper bound of the measure directional entropy of the ℤ2-action with respect to arbitrary Bernoulli measures without making use of the natural extension previously used in the paper [M. Courbage and B. Kaminski, Studia Math. 153(3), 285–295 (2002)].

A one-dimensional self-gravitating stellar gas

1966 ◽  
Vol 25 ◽  
pp. 46-48 ◽  
Author(s):  
M. Lecar
Keyword(s):  
Gravitational Field ◽  
Phase Space ◽  
Motion Picture ◽  
Space Distribution ◽  
Two Dimensional ◽  
One Dimensional ◽  
Phase Space Distribution ◽  
Dimensional Phase Space ◽  
The Mean

“Dynamical mixing”, i.e. relaxation of a stellar phase space distribution through interaction with the mean gravitational field, is numerically investigated for a one-dimensional self-gravitating stellar gas. Qualitative results are presented in the form of a motion picture of the flow of phase points (representing homogeneous slabs of stars) in two-dimensional phase space.

Electron-Mirror Microscopic Aspects of Ferroelectric Domains of BaTiO3 And Ca2Sr (C2H5CO2)6

1970 ◽  
Vol 28 ◽  
pp. 478-479
Author(s):  
Teruo Someya ◽  
Jinzo Kobayashi
Keyword(s):  
Surface Layer ◽  
Electric Fields ◽  
Quantitative Study ◽  
Recent Progress ◽  
Ferroelectric Domain ◽  
Resolving Power ◽  
Surface Pattern ◽  
Crystal Surfaces ◽  
One Dimensional ◽  
Ferroelectric Domains

Recent progress in the electron-mirror microscopy (EMM), e.g., an improvement of its resolving power together with an increase of the magnification makes it useful for investigating the ferroelectric domain physics. English has recently observed the domain texture in the surface layer of BaTiO3. The present authors ) have developed a theory by which one can evaluate small one-dimensional electric fields and/or topographic step heights in the crystal surfaces from their EMM pictures. This theory was applied to a quantitative study of the surface pattern of BaTiO3).

Structure and function of neural networks in the retina

1988 ◽  
Vol 46 ◽  
pp. 26-27
Author(s):  
Peter Sterling
Keyword(s):  
Ganglion Cells ◽  
Three Dimensional ◽  
Structure And Function ◽  
Synaptic Connections ◽  
Visual Axis ◽  
One Dimensional ◽  
Cat Retina ◽  
Ultrathin Sections ◽  
And Function ◽  
Insight Into

The synaptic connections in cat retina that link photoreceptors to ganglion cells have been analyzed quantitatively. Our approach has been to prepare serial, ultrathin sections and photograph en montage at low magnification (˜2000X) in the electron microscope. Six series, 100-300 sections long, have been prepared over the last decade. They derive from different cats but always from the same region of retina, about one degree from the center of the visual axis. The material has been analyzed by reconstructing adjacent neurons in each array and then identifying systematically the synaptic connections between arrays. Most reconstructions were done manually by tracing the outlines of processes in successive sections onto acetate sheets aligned on a cartoonist's jig. The tracings were then digitized, stacked by computer, and printed with the hidden lines removed. The results have provided rather than the usual one-dimensional account of pathways, a three-dimensional account of circuits. From this has emerged insight into the functional architecture.

Intergrowth of modulated structures in high Tc Bi-Sr-Ca-Cu-O superconductors

1990 ◽  
Vol 48 (4) ◽  
pp. 76-77
Author(s):  
A.Q. He ◽  
G.W. Qiao ◽  
J. Zhu ◽  
H.Q. Ye
Keyword(s):  
Twin Structure ◽  
Modulated Structure ◽  
High Tc ◽  
One Dimensional ◽  
Lattice Constants ◽  
Superconducting Phases ◽  
Modulated Structures ◽  
Layer Structures

Since the first discovery of high Tc Bi-Sr-Ca-Cu-O superconductor by Maeda et al, many EM works have been done on it. The results show that the superconducting phases have a type of ordered layer structures similar to that in Y-Ba-Cu-O system formulated in Bi2Sr2Can−1CunO2n+4 (n=1,2,3) (simply called 22(n-1) phase) with lattice constants of a=0.358, b=0.382nm but the length of c being different according to the different value of n in the formulate. Unlike the twin structure observed in the Y-Ba-Cu-O system, there is an incommensurate modulated structure in the superconducting phases of Bi system superconductors. Modulated wavelengths of both 1.3 and 2.7 nm have been observed in the 2212 phase. This communication mainly presents the intergrowth of these two kinds of one-dimensional modulated structures in 2212 phase.

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