scholarly journals Estimating Multidimensional Persistent Homology Through a Finite Sampling

2015 ◽  
Vol 25 (03) ◽  
pp. 187-205 ◽  
Author(s):  
Niccolò Cavazza ◽  
Massimo Ferri ◽  
Claudia Landi
Keyword(s):  
Lower Bound ◽  
Euclidean Space ◽  
Persistent Homology ◽  
Betti Numbers ◽  
Finite Sample ◽  
Combinatorial Description ◽  
Persistent Betti Numbers ◽  
Finite Sampling ◽  
The One ◽  
Union Of Balls

An exact computation of the persistent Betti numbers of a submanifold [Formula: see text] of a Euclidean space is possible only in a theoretical setting. In practical situations, only a finite sample of [Formula: see text] is available. We show that, under suitable density conditions, it is possible to estimate the multidimensional persistent Betti numbers of [Formula: see text] from the ones of a union of balls centered on the sample points; this even yields the exact value in restricted areas of the domain. Using these inequalities we improve a previous lower bound for the natural pseudodistance to assess dissimilarity between the shapes of two objects from a sampling of them. Similar inequalities are proved for the multidimensional persistent Betti numbers of the ball union and the one of a combinatorial description of it.

Success probabilities for second guessers

1980 ◽  
Vol 17 (04) ◽  
pp. 1133-1137 ◽  
Author(s):  
A. O. Pittenger
Keyword(s):  
Lower Bound ◽  
Dimensional Space ◽  
Success Probability ◽  
First Person ◽  
True Location ◽  
Rotationally Symmetric ◽  
The One ◽  
Sharp Lower Bound

Two people independently and with the same distribution guess the location of an unseen object in n-dimensional space, and the one whose guess is closer to the unseen object is declared the winner. The first person announces his guess, but the second modifies his unspoken idea by moving his guess in the direction of the first guess and as close to it as possible. It is shown that if the distribution of guesses is rotationally symmetric about the true location of the unseen object, ¾ is the sharp lower bound for the success probability of the second guesser. If the distribution is fixed and the dimension increases, then for a certain class of distributions, the success probability approaches 1.

Lower Bounds to the Eigenvalues in One-Dimensional Problems by a Shift in the Weight Function

1970 ◽  
Vol 37 (2) ◽  
pp. 267-270 ◽  
Author(s):  
D. Pnueli
Keyword(s):  
Lower Bound ◽  
Weight Function ◽  
Lower Bounds ◽  
Variational Formulation ◽  
Ritz Method ◽  
The Other ◽  
One Dimensional ◽  
Systematic Shift ◽  
The One ◽  
Detailed Computation

A method is presented to obtain both upper and lower bound to eigenvalues when a variational formulation of the problem exists. The method consists of a systematic shift in the weight function. A detailed procedure is offered for one-dimensional problems, which makes improvement of the bounds possible, and which involves the same order of detailed computation as the Rayleigh-Ritz method. The main contribution of this method is that it yields the “other bound;” i.e., the one which cannot be obtained by the Rayleigh-Ritz method.

On the infimum of the spectrum of a relativistic Schrödinger operator

2017 ◽  
Vol 29 (3) ◽  
Author(s):  
Mohammed El Aïdi
Keyword(s):  
Lower Bound ◽  
Euclidean Space ◽  
Schrödinger Operator ◽  
Schrodinger Operator ◽  
Relativistic Schrödinger Operator ◽  
Explicit Lower Bound

AbstractIn this paper, we look for an explicit lower bound of the smallest value of the spectrum for a relativistic Schrödinger operator in a domain of the Euclidean space.

Additive functions of intervals and Hausdorff measure

1946 ◽  
Vol 42 (1) ◽  
pp. 15-23 ◽  
Author(s):  
P. A. P. Moran
Keyword(s):  
Lower Bound ◽  
Euclidean Space ◽  
Hausdorff Measure ◽  
Convex Sets ◽  
Infinite Sequence ◽  
Small Positive Number ◽  
Additive Functions ◽  
Image Position ◽  
Bounded Sets ◽  
Increasing Function

Consider bounded sets of points in a Euclidean space Rq of q dimensions. Let h(t) be a continuous increasing function, positive for t>0, and such that h(0) = 0. Then the Hausdroff measure h–mE of a set E in Rq, relative to the function h(t), is defined as follows. Let ε be a small positive number and suppose E is covered by a finite or enumerably infinite sequence of convex sets {Ui} (open or closed) of diameters di less than or equal to ε. Write h–mεE = greatest lower bound for any such sequence {Ui}. Then h–mεE is non-decreasing as ε tends to zero. We define

scholarly journals A note on bounded variation and heat semigroup on Riemannian manifolds

2007 ◽  
Vol 76 (1) ◽  
pp. 155-160 ◽  
Author(s):  
A. Carbonaro ◽  
G. Mauceri
Keyword(s):  
Lower Bound ◽  
Euclidean Space ◽  
Heat Kernel ◽  
Ricci Curvature ◽  
Bounded Variation ◽  
Riemannian Manifolds ◽  
Heat Semigroup ◽  
Functions Of Bounded Variation ◽  
Fixed Radius ◽  
Lower Bound Estimate

In a recent paper Miranda Jr., Pallara, Paronetto and Preunkert have shown that the classical De Giorgi's heat kernel characterisation of functions of bounded variation on Euclidean space extends to Riemannian manifolds with Ricci curvature bounded from below and which satisfy a uniform lower bound estimate on the volume of geodesic balls of fixed radius. We give a shorter proof of the same result assuming only the lower bound on the Ricci curvature.

scholarly journals On the lower bound of the internal energy of the one-component-plasma

2015 ◽  
Vol 22 (4) ◽  
pp. 044504 ◽  
Author(s):  
S. A. Khrapak ◽  
A. G. Khrapak
Keyword(s):  
Lower Bound ◽  
Internal Energy ◽  
Component Plasma ◽  
The One

scholarly journals ELECTRODYNAMICS IN A BACKGROUND CHIRAL FIELD

2011 ◽  
Vol 26 (38) ◽  
pp. 2879-2887
Author(s):  
F. T. BRANDT ◽  
D. G. C. MCKEON ◽  
A. PATRUSHEV
Keyword(s):  
Vector Field ◽  
Euclidean Space ◽  
Effective Action ◽  
Exact Expression ◽  
Two Dimensions ◽  
Perturbative Expansion ◽  
Dimensional Euclidean Space ◽  
Chiral Field ◽  
Axial Field ◽  
The One

We consider the one-loop effective action in four-dimensional Euclidean space for a background chiral field coupled to a spinor field. It proves possible to find an exact expression for this action if the mass m of the spinor vanishes. If m does not vanish, one can make a perturbative expansion in powers of the axial field that contributes to the chiral field, while treating the contribution of the vector field exactly when it is a constant. The analogous problem in two dimensions is also discussed.

scholarly journals SUPERSYMMETRIC YANG–MILLS THEORIES WITH EXACT SUPERSYMMETRY ON THE LATTICE

2011 ◽  
Vol 26 (30n31) ◽  
pp. 5057-5132 ◽  
Author(s):  
ANOSH JOSEPH
Keyword(s):  
Field Theory ◽  
Euclidean Space ◽  
Topological Field Theory ◽  
Strongly Coupled ◽  
Dimensional Lattice ◽  
Yang Mills ◽  
Gravitational Theories ◽  
Topological Field ◽  
Perturbative Renormalization ◽  
The One

Inspired by the ideas from topological field theory it is possible to rewrite the supersymmetric charges of certain classes of extended supersymmetric Yang–Mills (SYM) theories in such a way that they are compatible with the discretization on a Euclidean space–time lattice. Such theories are known as maximally twisted SYM theories. In this review we discuss the construction and some applications of such classes of theories. The one-loop perturbative renormalization of the four-dimensional lattice [Formula: see text] SYM is discussed in particular. The lattice theories constructed using twisted approach play an important role in investigating the thermal phases of strongly coupled SYM theories and also the thermodynamic properties of their dual gravitational theories.

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