scholarly journals Upper bounds for packings of spheres of several radii

2014 ◽  
Vol 2 ◽  
Author(s):  
DAVID DE LAAT ◽  
FERNANDO MÁRIO DE OLIVEIRA FILHO ◽  
FRANK VALLENTIN
Keyword(s):  
Linear Programming ◽  
Euclidean Space ◽  
Semidefinite Programming ◽  
Unit Sphere ◽  
Upper Bound ◽  
Convex Bodies ◽  
Classical Problem ◽  
Upper Bounds ◽  
Sphere Packings ◽  
Spherical Caps

AbstractWe give theorems that can be used to upper bound the densities of packings of different spherical caps in the unit sphere and of translates of different convex bodies in Euclidean space. These theorems extend the linear programming bounds for packings of spherical caps and of convex bodies through the use of semidefinite programming. We perform explicit computations, obtaining new bounds for packings of spherical caps of two different sizes and for binary sphere packings. We also slightly improve the bounds for the classical problem of packing identical spheres.

scholarly journals Sums of distances between points of a sphere

1982 ◽  
Vol 5 (4) ◽  
pp. 707-714 ◽  
Author(s):  
Glyn Harman
Keyword(s):  
Euclidean Space ◽  
Unit Sphere ◽  
Upper Bound ◽  
Dimensional Euclidean Space ◽  
Spherical Caps

GivenNpoints on a unit sphere ink+1dimensional Euclidean space, we obtain an upper bound for the sum of all the distances they determine which improves upon earlier work by K. B. Stolarsky whenkis even. We use his method, but derive a variant of W. M. Schmidt's results for the discrepancy of spherical caps which is more suited to the present application.

scholarly journals The Closest Packing of Spherical Caps in n Dimensions

1955 ◽  
Vol 2 (3) ◽  
pp. 139-144 ◽  
Author(s):  
R. A. Rankin
Keyword(s):  
Euclidean Space ◽  
Unit Sphere ◽  
Closest Packing ◽  
Dimensional Unit ◽  
Spherical Caps ◽  
Image Position

Let Sn denote the “surface” of an n-dimensional unit sphere in Euclidean space of n dimensions. We may suppose that the sphere is centred at the origin of coordinates O, so that the points P(x1, x2, …, xn) of Sn satisfyWe suppose that n≥2.

scholarly journals On Euclidean Designs and Potential Energy

10.37236/8 ◽  
2012 ◽  
Vol 19 (1) ◽  
Author(s):  
Tsuyoshi Miezaki ◽  
Makoto Tagami
Keyword(s):  
Linear Programming ◽  
Lower Bound ◽  
Euclidean Space ◽  
Potential Energy ◽  
Harmonic Analysis ◽  
Upper Bound ◽  
Type Inequality ◽  
Linear Programming Bound ◽  
Concentric Spheres ◽  
Finite Set

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.

scholarly journals k-Point semidefinite programming bounds for equiangular lines

2021 ◽  
Author(s):  
David de Laat ◽  
Fabrício Caluza Machado ◽  
Fernando Mário de Oliveira Filho ◽  
Frank Vallentin
Keyword(s):  
Linear Programming ◽  
Euclidean Space ◽  
Semidefinite Programming ◽  
Spherical Codes ◽  
Equiangular Lines

AbstractWe propose a hierarchy of k-point bounds extending the Delsarte–Goethals–Seidel linear programming 2-point bound and the Bachoc–Vallentin semidefinite programming 3-point bound for spherical codes. An optimized implementation of this hierarchy allows us to compute 4, 5, and 6-point bounds for the maximum number of equiangular lines in Euclidean space with a fixed common angle.

scholarly journals Isosceles Sets

10.37236/230 ◽  
2009 ◽  
Vol 16 (1) ◽  
Author(s):  
Yury J. Ionin
Keyword(s):  
Euclidean Space ◽  
Upper Bound ◽  
Isosceles Triangle ◽  
Upper Bounds ◽  
The Other ◽  
The Third ◽  
Exact Answer ◽  
Hamming Space ◽  
Erdős Problem ◽  
Set Of Points

In 1946, Paul Erdős posed a problem of determining the largest possible cardinality of an isosceles set, i.e., a set of points in plane or in space, any three of which form an isosceles triangle. Such a question can be asked for any metric space, and an upper bound ${n+2\choose 2}$ for the Euclidean space ${\Bbb E}^{n}$ was found by Blokhuis. This upper bound is known to be sharp for $n=1$, 2, 6, and 8. We will consider Erdős' question for the binary Hamming space $H_{n}$ and obtain the following upper bounds on the cardinality of an isosceles subset $S$ of $H_{n}$: if there are at most two distinct nonzero distances between points of $S$, then $|S|\leq{n+1\choose 2}+1$; if, furthermore, $n\geq 4$, $n\ne 6$, and, as a set of vertices of the $n$-cube, $S$ is contained in a hyperplane, then $|S|\leq{n\choose 2}$; if there are more than two distinct nonzero distances between points of $S$, then $|S|\leq{n\choose 2}+1$. The first bound is sharp if and only if $n=2$ or $n=5$; the other two bounds are sharp for all relevant values of $n$, except the third bound for $n=6$, when the sharp upper bound is 12. We also give the exact answer to the Erdős problem for ${\Bbb E}^{n}$ with $n\leq 7$ and describe all isosceles sets of the largest cardinality in these dimensions.

scholarly journals L^p-norm inequality using q-moment and its applications

2019 ◽  
Author(s):  
Tomohiro Nishiyama
Keyword(s):  
Euclidean Space ◽  
Lebesgue Measure ◽  
Upper Bound ◽  
Measurable Function ◽  
Finite Measure ◽  
Upper Bounds ◽  
Norm Inequality ◽  
Lp Norm ◽  
Lp Norms ◽  
Q Moment

For a measurable function on a set which has a finite measure, an inequality holds between two Lp-norms. In this paper, we show similar inequalities for the Euclidean space and the Lebesgue measure by using a q-moment which is a moment of an escort distribution. As applications of these inequalities, we first derive upper bounds for the Renyi and the Tsallis entropies with given q-moment and derive an inequality between two Renyi entropies. Second, we derive an upper bound for the probability of a subset in the Euclidean space with given Lp-norm on the same set.

scholarly journals Some Hermite–Hadamard type integral inequalities for convex functions defined on convex bodies in ℝn\mathbb{R}^{n}

2020 ◽  
Vol 26 (1) ◽  
pp. 67-77 ◽  
Author(s):  
Silvestru Sever Dragomir
Keyword(s):  
Euclidean Space ◽  
Convex Functions ◽  
Convex Bodies ◽  
Integral Inequalities ◽  
Divergence Theorem ◽  
Several Variables ◽  
Type Integral ◽  
Bounded Convex

AbstractIn this paper, by the use of the divergence theorem, we establish some integral inequalities of Hermite–Hadamard type for convex functions of several variables defined on closed and bounded convex bodies in the Euclidean space {\mathbb{R}^{n}} for any {n\geq 2}.

Upper bounds on the energy dissipation in turbulent precession

1996 ◽  
Vol 321 ◽  
pp. 335-370 ◽  
Author(s):  
R. R. Kerswell
Keyword(s):  
Experimental Data ◽  
Energy Dissipation ◽  
Viscous Dissipation ◽  
Upper Bound ◽  
Dissipation Rate ◽  
Oblate Spheroid ◽  
Upper Bounds ◽  
Precession Rate ◽  
Long Cylinder

Rigorous upper bounds on the viscous dissipation rate are identified for two commonly studied precessing fluid-filled configurations: an oblate spheroid and a long cylinder. The latter represents an interesting new application of the upper-bounding techniques developed by Howard and Busse. A novel ‘background’ method recently introduced by Doering & Constantin is also used to deduce in both instances an upper bound which is independent of the fluid's viscosity and the forcing precession rate. Experimental data provide some evidence that the observed viscous dissipation rate mirrors this behaviour at sufficiently high precessional forcing. Implications are then discussed for the Earth's precessional response.

scholarly journals The Essential Dimension of Stacks of Parabolic Vector Bundles over Curves

2012 ◽  
Vol 10 (3) ◽  
pp. 455-488 ◽  
Author(s):  
Indranil Biswas ◽  
Ajneet Dhillon ◽  
Nicole Lemire
Keyword(s):  
Lower Bounds ◽  
Upper Bound ◽  
Vector Bundles ◽  
Upper Bounds ◽  
Essential Dimension ◽  
Parabolic Structure ◽  
Moduli Stack

AbstractWe find upper bounds on the essential dimension of the moduli stack of parabolic vector bundles over a curve. When there is no parabolic structure, we improve the known upper bound on the essential dimension of the usual moduli stack. Our calculations also give lower bounds on the essential dimension of the semistable locus inside the moduli stack of vector bundles of rank r and degree d without parabolic structure.

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