ERROR ANALYSIS OF THE MINIMUM DISTANCE ENERGY OF A POLYGONAL KNOT AND THE MÖBIUS ENERGY OF AN APPROXIMATING CURVE

2010 ◽  
Vol 19 (08) ◽  
pp. 975-1000 ◽  
Author(s):  
ERIC J. RAWDON ◽  
JOSEPH WORTHINGTON
Keyword(s):  
Error Analysis ◽  
Upper Bound ◽  
Minimum Distance ◽  
Energy Functionals ◽  
Explicit Bound ◽  
Knot Energy ◽  
The Given

Energy minimizing smooth knot configurations have long been approximated by finding knotted polygons that minimize discretized versions of the given energy. However, for most knot energy functionals, the question remains open on whether the minimum polygonal energies are "close" to the minimum smooth energies. In this paper, we determine an explicit bound between the Minimum-Distance Energy of a polygon and the Möbius Energy of a piecewise-C2 knot inscribed in the polygon. This bound is written in terms of the ropelength and the number of edges and can be used to determine an upper bound for the minimum Möbius Energy for different knot types.

scholarly journals ADAPTIVE ALGORITHM FOR DETERMINING THE PARAMETERS OF AN INACCESSIBLE POINT OF AN OBJECT

2021 ◽  
pp. 58-70
Author(s):  
Aleksandr Brailov ◽  
Vitaliy Panchenko
Keyword(s):  
Adaptive Algorithm ◽  
Minimum Distance ◽  
Three Dimensional ◽  
Geometrical Model ◽  
Multiple Variables ◽  
The Common ◽  
Relative Errors ◽  
The Given ◽  
Real Experimental Data

In the present research the optimizing approach to the determination of the parameters of an inaccessible point of an object is developed. The common issues are revealed and essential steps of their resolution are identified. The essence of the problem is an objective contradiction between a requirement for the location of points A and B of the centers of the sighting tubes of optical devices in the same horizontal plane P1 and the lack of a real possibility to perform such to achieve this an identical one-level arrangement without error. The aim of the study is to develop strategies for determining the position of an inaccessible point of an object in the minimum domain between intersecting sighting rays as well as an adaptive algorithm for determining the values of the parameters of an inaccessible point under the given absolute and relative errors. To achieve this aim, the following problems are formulated and solved in the paper: 1. Develop strategies for determining the position of the inaccessible point of the object in the minimum domain between the intersecting sighting rays. 2. Develop an adaptive algorithm for determining the values of the parameters of an inaccessible point based on the specified absolute and relative errors. In the proposed optimizing approach, the three-dimensional geometrical model with crossed directional rays for the determination of coordinates of the inaccessible point of an object is developed. It is discussed that points С and C', coordinated of which to be determined, locates in domain [CDM, CEM], [C'D'M, C'E'M] of the minimum distance ρmin between crossed directional rays. The optimizing problem of the determination of coordinates of an inaccessible point of an object in space is reduced to a problem of the determination of the minimum distance between two crossed directional rays. It’s known from the theory of function of multiple variables that function ρ = f (tC'D', tC'E') reaches its extremum ρmin when its partial derivatives by each variable are equal to zero. Three strategies for selecting the position of the inaccessible point C (xC, yC, zC) in the found minimum region [CDM, CEM] are proposed. The required point C' (xC', yC', zC') can be located, for example, in the middle of the minimum segment [C'D'M, C'E'M]. The essence of the adaptive algorithm is in optimizing the variation of the initial values of data α, α', β, γ, γ', AB, at which the absolute and relative errors of the coordinates of the inaccessible point satisfy the error values set by the customer (0.0001-1.2%) The proposed approach is verified using real experimental data.

scholarly journals Convergence analysis of time-discretisation schemes for rate-independent systems

2019 ◽  
Vol 25 ◽  
pp. 65 ◽  
Author(s):  
Dorothee Knees
Keyword(s):  
Convergence Analysis ◽  
Numerical Schemes ◽  
Solution Concepts ◽  
Continuous Solutions ◽  
Energy Functionals ◽  
Convergence Of Solutions ◽  
Time Discretisation ◽  
Energetic Solutions ◽  
Nonconvex Energy ◽  
The Given

It is well known that rate-independent systems involving nonconvex energy functionals in general do not allow for time-continuous solutions even if the given data are smooth. In the last years, several solution concepts were proposed that include discontinuities in the notion of solution, among them the class of global energetic solutions and the class of BV-solutions. In general, these solution concepts are not equivalent and numerical schemes are needed that reliably approximate that type of solutions one is interested in. In this paper, we analyse the convergence of solutions of three time-discretisation schemes, namely an approach based on local minimisation, a relaxed version of it and an alternate minimisation scheme. For all three cases, we show that under suitable conditions on the discretisation parameters discrete solutions converge to limit functions that belong to the class of BV-solutions. The proofs rely on a reparametrisation argument. We illustrate the different schemes with a toy example.

A New Upper Bound on the Minimum Distance of Turbo Codes

2004 ◽  
Vol 50 (12) ◽  
pp. 2985-2997 ◽  
Author(s):  
A. Perotti ◽  
S. Benedetto
Keyword(s):  
Upper Bound ◽  
Turbo Codes ◽  
Minimum Distance

scholarly journals TANGENT-POINT SELF-AVOIDANCE ENERGIES FOR CURVES

2012 ◽  
Vol 21 (05) ◽  
pp. 1250044 ◽  
Author(s):  
PAWEŁ STRZELECKI ◽  
HEIKO VON DER MOSEL
Keyword(s):  
Upper Bound ◽  
Closed Interval ◽  
Local Energy ◽  
Triple Junctions ◽  
Tangent Point ◽  
Double Integral ◽  
Energy Formula ◽  
Knot Energy ◽  
Rectifiable Curves ◽  
Infinite Energy

We study a two-point self-avoidance energy [Formula: see text] which is defined for all rectifiable curves in ℝn as the double integral along the curve of 1/rq. Here r stands for the radius of the (smallest) circle that the is tangent to the curve at one point and passes through another point on the curve, with obvious natural modifications of this definition in the exceptional, non-generic cases. It turns out that finiteness of [Formula: see text] for q ≥ 2 guarantees that γ has no self-intersections or triple junctions and therefore must be homeomorphic to the unit circle 𝕊1 or to a closed interval I. For q > 2 the energy [Formula: see text] evaluated on curves in ℝ3 turns out to be a knot energy separating different knot types by infinite energy barriers and bounding the number of knot types below a given energy value. We also establish an explicit upper bound on the Hausdorff-distance of two curves in ℝ3 with finite [Formula: see text]-energy that guarantees that these curves are ambient isotopic. This bound depends only on q and the energy values of the curves. Moreover, for all q that are larger than the critical exponent q crit = 2, the arclength parametrization of γ is of class C1, 1-2/q, with Hölder norm of the unit tangent depending only on q, the length of γ, and the local energy. The exponent 1 - 2/q is optimal.

scholarly journals Minimum Symbol Error Probability MIMO Design under the Per-Antenna Power Constraint

2012 ◽  
Vol 2012 ◽  
pp. 1-9
Author(s):  
Enoch Lu ◽  
I.-Tai Lu
Keyword(s):  
Upper Bound ◽  
Error Probability ◽  
Minimum Distance ◽  
Necessary Conditions ◽  
Mimo Systems ◽  
Symbol Error Probability ◽  
Power Constraint ◽  
Rank One ◽  
Single User ◽  
Numerical Approaches

Approximate minimum symbol error probability transceiver design of single user MIMO systems under the practical per-antenna power constraint is considered. The upper bound of a lower bound on the minimum distance between the symbol hypotheses is established. Necessary conditions and structures of the transmit covariance matrix for reaching the upper bound are discussed. Three numerical approaches (rank zero, rank one, and permutation) for obtaining the optimum precoder are proposed. When the upper bound is reached, the resulting design is optimum. When the upper bound is not reached, a numerical fix is used. The approach is very simple and can be of practical use.

ON DAVENPORT'S CONSTANT

2008 ◽  
Vol 04 (01) ◽  
pp. 107-115 ◽  
Author(s):  
P. RATH ◽  
K. SRILAKSHMI ◽  
R. THANGADURAI
Keyword(s):  
Abelian Group ◽  
Upper Bound ◽  
Davenport Constant ◽  
The Given

In this paper, using the idea of Alford, Granville and Pomerance in [1] (or van Emde Boas and Kruyswijk [6]), we obtain an upper bound for the Davenport Constant of an Abelian group G in terms of the number of repetitions of the group elements in any given sequence. In particular, our result implies, [Formula: see text] where n is the exponent of G and k ≥ 0 denotes the number of distinct elements of G that are repeated at least twice in the given sequence.

scholarly journals A new upper bound for binary codes with minimum distance four

1998 ◽  
Vol 187 (1-3) ◽  
pp. 291-295
Author(s):  
Jun Kyo Kim ◽  
Sang Geun Hahn
Keyword(s):  
Upper Bound ◽  
Minimum Distance ◽  
Binary Codes

scholarly journals Parameterized Codes over Cycles

2013 ◽  
Vol 21 (3) ◽  
pp. 241-256 ◽  
Author(s):  
Manuel González Sarabia ◽  
Joel Nava Lara ◽  
Carlos Rentería Marquez ◽  
Eliseo Sarmíento Rosales
Keyword(s):  
Upper Bound ◽  
Minimum Distance ◽  
Upper Bounds ◽  
Lower And Upper Bounds ◽  
Evaluation Codes ◽  
Singleton Bound ◽  
Odd Cycles

AbstractIn this paper we will compute the main parameters of the parameterized codes arising from cycles. In the case of odd cycles the corresponding codes are the evaluation codes associated to the projective torus and the results are well known. In the case of even cycles we will compute the length and the dimension of the corresponding codes and also we will find lower and upper bounds for the minimum distance of this kind of codes. In many cases our upper bound is sharper than the Singleton bound.

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