scholarly journals A generalization of Radon's theorem II

1981 ◽  
Vol 24 (3) ◽  
pp. 321-325 ◽  
Author(s):  
H. Tverberg
Keyword(s):  
Dimensional Space ◽  
Convex Polytopes ◽  
Empty Intersection ◽  
Radon’S Theorem ◽  
Positive Integers

A new proof is given of the following result: Let m and d be positive integers, and let a set of md + m − d points be given in d-dimensional space. Then the set can be partitioned into m sets such that the m convex polytopes spanned by the sets have a non-empty intersection.

p-Adic Eigen-Eunctions for Kubert Distributions

1983 ◽  
Vol 35 (4) ◽  
pp. 674-686
Author(s):  
Neal Koblitz
Keyword(s):  
Zeta Function ◽  
Dimensional Space ◽  
Zeta Functions ◽  
Dirichlet Character ◽  
One Dimensional ◽  
Fixed Complex ◽  
Image Position ◽  
Analogous Theorem ◽  
Positive Integers

Functions onR(or onR/Z, orQ/Z, or the interval (0,1)) which satisfy the identity1.1for positive integersmand fixed complexs,appear in several branches of mathematics (see [8], p. 65-68). They have recently been studied systematically by Kubert [6] and Milnor [12]. Milnor showed that for each complexsthere is a one-dimensional space of even functions and a one-dimensional space of odd functions which satisfy (1.1). These functions can be expressed in terms of either the Hurwitz partial zeta-function or the polylogarithm functions.My purpose is to prove an analogous theorem forp-adic functions. Thep-adic analog is slightly more general; it allows for a Dirichlet characterχ0(m) in front ofms–lin (1.1). The functions satisfying (1.1) turn out to bep-adic “partial DirichletL-functions”, functions of twop-adic variables (x, s) and one character variableχ0which specialize to partial zeta-functions whenχ0is trivial and to Kubota-LeopoldtL-functions whenx= 0.

A problem of integer partitions and numerical semigroups

2018 ◽  
Vol 149 (04) ◽  
pp. 969-978
Author(s):  
J. C. Rosales ◽  
M. B. Branco
Keyword(s):  
Numerical Semigroups ◽  
Integer Partitions ◽  
Empty Intersection ◽  
Positive Integers

AbstractLet C be a set of positive integers. In this paper, we obtain an algorithm for computing all subsets A of positive integers which are minimals with the condition that if x1 + … + xn is a partition of an element in C, then at least a summand of this partition belongs to A. We use techniques of numerical semigroups to solve this problem because it is equivalent to give an algorithm that allows us to compute all the numerical semigroups which are maximals with the condition that has an empty intersection with the set C.

A Remark on Convex Polytopes

1965 ◽  
Vol 8 (6) ◽  
pp. 829-830
Author(s):  
A. S. Glass
Keyword(s):  
Dimensional Space ◽  
Convex Polytopes ◽  
Convex Polytope ◽  
Finite Family ◽  
Alternative Proof

In this note we wish to present an alternative proof for the following well-known theorem [1, Theorem 16]: every convex polytope X in Euclidean n-dimensional space Rn is the intersection of a finite family of closed half-spaces. It will be supposed that the converse of this theorem has been verified by conventional arguments, namely: every bounded intersection of a finite family of closed half-spaces in Rn is a convex polytope [cf. 1, Theorem 15].

scholarly journals THE PARALLEL 3D CONVEX HULL PROBLEM REVISITED

1992 ◽  
Vol 02 (02) ◽  
pp. 163-173 ◽  
Author(s):  
NANCY M. AMATO ◽  
FRANCO P. PREPARATA
Keyword(s):  
Convex Hull ◽  
Parallel Implementation ◽  
Dimensional Space ◽  
Convex Polytopes ◽  
Three Dimensional ◽  
Convex Polyhedra ◽  
Local Criterion ◽  
Crew Pram ◽  
Disjoint Convex

In this paper we prove the correctness of a “local” criterion for computing the convex hull of the union ( “merging”) of two disjoint convex polyhedra. This criterion is structural. Therefore it can be algorithmically tested in several ways, not necessarily involving the determination of support (tangent) planes; indeed, it can be implemented by just testing for the intersection of certain planes and lines with convex polytopes. This criterion is amenable to parallel implementation and leads to a provably correct algorithm that computes the convex hull of any n points in three-dimensional space in O( log 2 n) time using O(n) processors on a CREW PRAM.

scholarly journals The modulus of the Fourier transform on a sphere determines 3-dimensional convex polytopes

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Konrad Engel ◽  
Bastian Laasch
Keyword(s):  
Fourier Transform ◽  
Convex Polytopes ◽  
Convex Polytope ◽  
X Ray Diffraction ◽  
X Ray ◽  
3 Dimensional ◽  
Empty Intersection ◽  
Open Set ◽  
Ewald Sphere ◽  
The Fourier Transform

Abstract Let 𝒫 and P ′ \mathcal{P}^{\prime} be 3-dimensional convex polytopes in R 3 \mathbb{R}^{3} and S ⊆ R 3 S\subseteq\mathbb{R}^{3} be a non-empty intersection of an open set with a sphere. As a consequence of a somewhat more general result it is proved that 𝒫 and P ′ \mathcal{P}^{\prime} coincide up to translation and/or reflection in a point if | ∫ P e - i ⁢ s ⋅ x ⁢ dx | = | ∫ P ′ e - i ⁢ s ⋅ x ⁢ dx | \bigl{\lvert}\int_{\mathcal{P}}e^{-i\mathbf{s}\cdot\mathbf{x}}\,\mathbf{dx}\bigr{\rvert}=\bigl{\lvert}\int_{\mathcal{P}^{\prime}}e^{-i\mathbf{s}\cdot\mathbf{x}}\,\mathbf{dx}\bigr{\rvert} for all s ∈ S \mathbf{s}\in S . This can be applied to the field of crystallography regarding the question whether a nanoparticle modelled as a convex polytope is uniquely determined by the intensities of its X-ray diffraction pattern on the Ewald sphere.

The motion of a lunar orbiter

1966 ◽  
Vol 25 ◽  
pp. 373
Author(s):  
Y. Kozai
Keyword(s):  
Degrees Of Freedom ◽  
Dimensional Space ◽  
Artificial Satellite ◽  
Equations Of Motion ◽  
Triaxial Ellipsoid ◽  
The Moon ◽  
Secular Motion ◽  
The Earth ◽  
Close Satellite ◽  
The Mean

The motion of an artificial satellite around the Moon is much more complicated than that around the Earth, since the shape of the Moon is a triaxial ellipsoid and the effect of the Earth on the motion is very important even for a very close satellite.The differential equations of motion of the satellite are written in canonical form of three degrees of freedom with time depending Hamiltonian. By eliminating short-periodic terms depending on the mean longitude of the satellite and by assuming that the Earth is moving on the lunar equator, however, the equations are reduced to those of two degrees of freedom with an energy integral.Since the mean motion of the Earth around the Moon is more rapid than the secular motion of the argument of pericentre of the satellite by a factor of one order, the terms depending on the longitude of the Earth can be eliminated, and the degree of freedom is reduced to one.Then the motion can be discussed by drawing equi-energy curves in two-dimensional space. According to these figures satellites with high inclination have large possibilities of falling down to the lunar surface even if the initial eccentricities are very small.The principal properties of the motion are not changed even if plausible values ofJ3andJ4of the Moon are included.This paper has been published in Publ. astr. Soc.Japan15, 301, 1963.

Three Dimensional structure and organization of diploid chromosomes by optical section microscopy

1987 ◽  
Vol 45 ◽  
pp. 633-635
Author(s):  
David A. Agard ◽  
Yasushi Hiraoka ◽  
John W. Sedat
Keyword(s):  
Dimensional Space ◽  
Three Dimensional ◽  
Developmental State ◽  
Mitotic Cell ◽  
Biological Properties ◽  
Dimensional Structure ◽  
Specific Gene ◽  
Three Dimensional Structure ◽  
Complementary Techniques ◽  
Dynamic Structures

In an effort to understand the complex relationship between structure and biological function within the nucleus, we have embarked on a program to examine the three-dimensional structure and organization of Drosophila melanogaster embryonic chromosomes. Our overall goal is to determine how DNA and proteins are organized into complex and highly dynamic structures (chromosomes) and how these chromosomes are arranged in three dimensional space within the cell nucleus. Futher, we hope to be able to correlate structual data with such fundamental biological properties as stage in the mitotic cell cycle, developmental state and transcription at specific gene loci.Towards this end, we have been developing methodologies for the three-dimensional analysis of non-crystalline biological specimens using optical and electron microscopy. We feel that the combination of these two complementary techniques allows an unprecedented look at the structural organization of cellular components ranging in size from 100A to 100 microns.

Diffraction contrast of dislocations in icosahedral quasicrystals

1990 ◽  
Vol 48 (4) ◽  
pp. 454-455
Author(s):  
K. Urban ◽  
Z. Zhang ◽  
M. Wollgarten ◽  
D. Gratias
Keyword(s):  
Dimensional Space ◽  
Three Dimensional ◽  
Complete Characterization ◽  
Extinction Condition ◽  
Burgers Vector ◽  
Diffraction Contrast ◽  
Lattice Geometry ◽  
Reference Space ◽  
Icosahedral Quasicrystals ◽  
The One

Recently dislocations have been observed by electron microscopy in the icosahedral quasicrystalline (IQ) phase of Al65Cu20Fe15. These dislocations exhibit diffraction contrast similar to that known for dislocations in conventional crystals. The contrast becomes extinct for certain diffraction vectors g. In the following the basis of electron diffraction contrast of dislocations in the IQ phase is described. Taking account of the six-dimensional nature of the Burgers vector a “strong” and a “weak” extinction condition are found.Dislocations in quasicrystals canot be described on the basis of simple shear or insertion of a lattice plane only. In order to achieve a complete characterization of these dislocations it is advantageous to make use of the one to one correspondence of the lattice geometry in our three-dimensional space (R3) and that in the six-dimensional reference space (R6) where full periodicity is recovered . Therefore the contrast extinction condition has to be written as gpbp + gobo = 0 (1). The diffraction vector g and the Burgers vector b decompose into two vectors gp, bp and go, bo in, respectively, the physical and the orthogonal three-dimensional sub-spaces of R6.

Flavin radicals, conformational sampling and robust design principles in interprotein electron transfer: the trimethylamine dehydrogenase-electron-transferring flavoprotein complex

2004 ◽  
Vol 71 ◽  
pp. 1-14
Author(s):  
David Leys ◽  
Jaswir Basran ◽  
François Talfournier ◽  
Kamaldeep K. Chohan ◽  
Andrew W. Munro ◽  
...  
Keyword(s):  
Electron Transfer ◽  
Dimensional Space ◽  
Three Dimensional ◽  
Kinetic Studies ◽  
Molecular Assembly ◽  
Conformational Sampling ◽  
Trimethylamine Dehydrogenase ◽  
Key Residues ◽  
Electron Transferring Flavoprotein ◽  
Electron Transfer Complex

TMADH (trimethylamine dehydrogenase) is a complex iron-sulphur flavoprotein that forms a soluble electron-transfer complex with ETF (electron-transferring flavoprotein). The mechanism of electron transfer between TMADH and ETF has been studied using stopped-flow kinetic and mutagenesis methods, and more recently by X-ray crystallography. Potentiometric methods have also been used to identify key residues involved in the stabilization of the flavin radical semiquinone species in ETF. These studies have demonstrated a key role for 'conformational sampling' in the electron-transfer complex, facilitated by two-site contact of ETF with TMADH. Exploration of three-dimensional space in the complex allows the FAD of ETF to find conformations compatible with enhanced electronic coupling with the 4Fe-4S centre of TMADH. This mechanism of electron transfer provides for a more robust and accessible design principle for interprotein electron transfer compared with simpler models that invoke the collision of redox partners followed by electron transfer. The structure of the TMADH-ETF complex confirms the role of key residues in electron transfer and molecular assembly, originally suggested from detailed kinetic studies in wild-type and mutant complexes, and from molecular modelling.

Non-Linear Dynamics of Cardiovascular Variability Signals

1994 ◽  
Vol 33 (01) ◽  
pp. 81-84 ◽  
Author(s):  
S. Cerutti ◽  
S. Guzzetti ◽  
R. Parola ◽  
M.G. Signorini
Keyword(s):  
Dimensional Space ◽  
Severe Heart Failure ◽  
Parametric Models ◽  
Deterministic Systems ◽  
Finite Dimensional ◽  
Return Maps ◽  
Pathological Conditions ◽  
Non Linear ◽  
Space State

Abstract:Long-term regulation of beat-to-beat variability involves several different kinds of controls. A linear approach performed by parametric models enhances the short-term regulation of the autonomic nervous system. Some non-linear long-term regulation can be assessed by the chaotic deterministic approach applied to the beat-to-beat variability of the discrete RR-interval series, extracted from the ECG. For chaotic deterministic systems, trajectories of the state vector describe a strange attractor characterized by a fractal of dimension D. Signals are supposed to be generated by a deterministic and finite dimensional but non-linear dynamic system with trajectories in a multi-dimensional space-state. We estimated the fractal dimension through the Grassberger and Procaccia algorithm and Self-Similarity approaches of the 24-h heart-rate variability (HRV) signal in different physiological and pathological conditions such as severe heart failure, or after heart transplantation. State-space representations through Return Maps are also obtained. Differences between physiological and pathological cases have been assessed and generally a decrease in the system complexity is correlated to pathological conditions.

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