scholarly journals Calibrations for minimal networks in a covering space setting

2020 ◽  
Vol 26 ◽  
pp. 40 ◽  
Author(s):  
Marcello Carioni ◽  
Alessandra Pluda
Keyword(s):  
Minimization Problem ◽  
Covering Space ◽  
Regular Hexagon ◽  
Steiner Problem ◽  
Space Setting ◽  
Regular Pentagon ◽  
Minimal Networks

In this paper, we define a notion of calibration for an approach to the classical Steiner problem in a covering space setting and we give some explicit examples. Moreover, we introduce the notion of calibration in families: the idea is to divide the set of competitors in a suitable way, defining an appropriate (and weaker) notion of calibration. Then, calibrating the candidate minimizers in each family and comparing their perimeter, it is possible to find the minimizers of the minimization problem. Thanks to this procedure we prove the minimality of the Steiner configurations spanning the vertices of a regular hexagon and of a regular pentagon.

scholarly journals On different notions of calibrations for minimal partitions and minimal networks in ℝ2

2019 ◽  
Vol 0 (0) ◽  
Author(s):  
Marcello Carioni ◽  
Alessandra Pluda
Keyword(s):  
Minimal Surfaces ◽  
Steiner Problem ◽  
Minimal Networks

Abstract Calibrations are a possible tool to validate the minimality of a certain candidate. They have been introduced in the context of minimal surfaces and adapted to the case of the Steiner problem in several variants. Our goal is to compare the different notions of calibrations for the Steiner problem and for planar minimal partitions that are already present in the literature. The paper is then complemented with remarks on the convexification of the problem, on nonexistence of calibrations and on calibrations in families.

Generalized Second Moment of Areas of Regular Polygons for Ludwick Type Material and its Application to Cantilever Column Buckling

2019 ◽  
Vol 19 (02) ◽  
pp. 1950010
Author(s):  
Joon Kyu Lee ◽  
Byoung Koo Lee
Keyword(s):  
Cross Sections ◽  
Equilateral Triangle ◽  
Regular Polygon ◽  
Regular Hexagon ◽  
Type Material ◽  
Regular Polygons ◽  
Column Buckling ◽  
Second Moment ◽  
Regular Pentagon

This study deals with the generalized second moment of area (GSMA) of regular polygon cross-sections for the Ludwick type material and its application to cantilever column buckling. In the literature, the GSMA for the Ludwick type material has only been considered for rectangular, elliptical and superellipsoidal cross-sections. This study calculates the GSMAs of regular polygon cross-sections other than those mentioned above. The GSMAs calculated by varying the mechanical constant of the Ludwick type material for the equilateral triangle, square, regular pentagon, regular hexagon and circular cross-sections are reported in tables and figures. The GSMAs obtained from this study are applied to cantilever column buckling, with results shown in tables and figures.

scholarly journals The chord length distribution function for regular polygons

2009 ◽  
Vol 41 (2) ◽  
pp. 358-366 ◽  
Author(s):  
H. S. Harutyunyan ◽  
V. K. Ohanyan
Keyword(s):  
Distribution Function ◽  
Regular Polygon ◽  
Length Distribution ◽  
Chord Length ◽  
Regular Hexagon ◽  
Regular Polygons ◽  
Chord Length Distribution ◽  
Regular Pentagon ◽  
Regular Triangle ◽  
Elementary Expression

In this paper we obtain an elementary expression for the chord length distribution function of a regular polygon. The formula is derived using δ-formalism in Pleijel identity. In the particular cases of a regular triangle, a square, a regular pentagon, and a regular hexagon, our formula coincides with the results of Sulanke (1961), Gille (1988), Aharonyan and Ohanyan (2005), and Harutyunyan (2007), respectively.

scholarly journals The chord length distribution function for regular polygons

2009 ◽  
Vol 41 (02) ◽  
pp. 358-366
Author(s):  
H. S. Harutyunyan ◽  
V. K. Ohanyan
Keyword(s):  
Distribution Function ◽  
Regular Polygon ◽  
Length Distribution ◽  
Chord Length ◽  
Regular Hexagon ◽  
Regular Polygons ◽  
Chord Length Distribution ◽  
Regular Pentagon ◽  
Regular Triangle ◽  
Elementary Expression

In this paper we obtain an elementary expression for the chord length distribution function of a regular polygon. The formula is derived using δ-formalism in Pleijel identity. In the particular cases of a regular triangle, a square, a regular pentagon, and a regular hexagon, our formula coincides with the results of Sulanke (1961), Gille (1988), Aharonyan and Ohanyan (2005), and Harutyunyan (2007), respectively.

Constrained BV functions on covering spaces for minimal networks and Plateau’s type problems

2017 ◽  
Vol 10 (1) ◽  
pp. 25-47 ◽  
Author(s):  
Stefano Amato ◽  
Giovanni Bellettini ◽  
Maurizio Paolini
Keyword(s):  
Sobolev Spaces ◽  
Minimization Problem ◽  
A Priori ◽  
Functions Of Bounded Variation ◽  
Covering Spaces ◽  
Bv Functions ◽  
Γ Convergence ◽  
Vector Valued ◽  
Minimal Networks ◽  
Natural Way

AbstractWe link covering spaces with the theory of functions of bounded variation, in order to study minimal networks in the plane and Plateau’s problem without fixing a priori the topology of solutions. We solve the minimization problem in the class of (possibly vector-valued) $\mathrm{BV}$ functions defined on a covering space of the complement of an ${(n-2)}$-dimensional compact embedded Lipschitz manifold S without boundary. This approach has several similarities with Brakke’s “soap films” covering construction. The main novelty of our method stands in the presence of a suitable constraint on the fibers, which couples together the covering sheets. In the case of networks, the constraint is defined using a suitable subset of transpositions of m elements, m being the number of points of S. The model avoids all issues concerning the presence of the boundary S, which is automatically attained. The constraint is lifted in a natural way to Sobolev spaces, allowing also an approach based on Γ-convergence.

An Augmented Iteratively Re-weighted and Refined Least Squares Method for lp Minimization Problem in Inverse Modeling of Potential Field Magnetic Data

2020 ◽  
Vol 1 (3) ◽  
Author(s):  
Maysam Abedi
Keyword(s):  
Magnetic Field ◽  
Magnetic Susceptibility ◽  
Least Squares ◽  
Minimization Problem ◽  
Potential Field ◽  
Field Data ◽  
Least Squares Method ◽  
Porphyry Copper ◽  
Magnetic Data ◽  
Magnetic Field Data

The presented work examines application of an Augmented Iteratively Re-weighted and Refined Least Squares method (AIRRLS) to construct a 3D magnetic susceptibility property from potential field magnetic anomalies. This algorithm replaces an lp minimization problem by a sequence of weighted linear systems in which the retrieved magnetic susceptibility model is successively converged to an optimum solution, while the regularization parameter is the stopping iteration numbers. To avoid the natural tendency of causative magnetic sources to concentrate at shallow depth, a prior depth weighting function is incorporated in the original formulation of the objective function. The speed of lp minimization problem is increased by inserting a pre-conditioner conjugate gradient method (PCCG) to solve the central system of equation in cases of large scale magnetic field data. It is assumed that there is no remanent magnetization since this study focuses on inversion of a geological structure with low magnetic susceptibility property. The method is applied on a multi-source noise-corrupted synthetic magnetic field data to demonstrate its suitability for 3D inversion, and then is applied to a real data pertaining to a geologically plausible porphyry copper unit.  The real case study located in  Semnan province of  Iran  consists  of  an arc-shaped  porphyry  andesite  covered  by  sedimentary  units  which  may  have  potential  of  mineral  occurrences, especially  porphyry copper. It is demonstrated that such structure extends down at depth, and consequently exploratory drilling is highly recommended for acquiring more pieces of information about its potential for ore-bearing mineralization.

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