Farthest neighbors, maximum spanning trees and related problems in higher dimensions

2005 ◽  
pp. 105-116 ◽  
Author(s):  
Pankaj K. Agarwal ◽  
Jiří Matoušek ◽  
Subhash Suri
Keyword(s):  
Spanning Trees ◽  
Higher Dimensions

scholarly journals Farthest neighbors, maximum spanning trees and related problems in higher dimensions

1992 ◽  
Vol 1 (4) ◽  
pp. 189-201 ◽  
Author(s):  
Pankaj K. Agarwal ◽  
Jiří Matoušek ◽  
Subhash Suri
Keyword(s):  
Spanning Trees ◽  
Higher Dimensions

scholarly journals TREE-WEIGHTED NEIGHBORS AND GEOMETRIC k SMALLEST SPANNING TREES

1994 ◽  
Vol 04 (02) ◽  
pp. 229-238 ◽  
Author(s):  
DAVID EPPSTEIN
Keyword(s):  
Euclidean Distance ◽  
Spanning Trees ◽  
Nearest Neighbors ◽  
Higher Dimensions ◽  
Four Dimensions ◽  
Euclidean Metric ◽  
Rectilinear Metric ◽  
Point Set ◽  
Tree Path ◽  
Path Lengths

We compute the k smallest spanning trees of a point set in the planar Euclidean metric in time O(n log n log k + k min (k, n)1/2 log (k/n)), and in the rectilinear metric in time O(n log n + n log log n log k + k ( min {k, n})1/2 log (k/n)). In three or four dimensions our time bound is O(n4/3 + ∊ + k( min {k, n})1/2 log (k/n)), and in higher dimensions the bound is O(n2−2/(⌈d/2⌉+1)+∊ + kn1/2 log n). Our technique involves a method of computing nearest neighbors using a modified set of distances formed by subtracting tree path lengths from the Euclidean distance.

Hedgehogs in higher dimensions and their applications

Astérisque ◽  
2020 ◽  
Vol 416 ◽  
pp. 213-251
Author(s):  
Mikhail LYUBICH ◽  
Remus RADU ◽  
Raluca TANASE
Keyword(s):  
Higher Dimensions

Hedgehogs in higher dimensions and their applications

Astérisque ◽  
2020 ◽  
Vol 416 ◽  
pp. 213-251
Author(s):  
Mikhail LYUBICH ◽  
Remus RADU ◽  
Raluca TANASE
Keyword(s):  
Higher Dimensions

Existence of magnetic monopoles in higher dimensions is analogous to unipolar gravity

2020 ◽  
Author(s):  
Deep Bhattacharjee
Keyword(s):  
Flux Density ◽  
Higher Dimensions ◽  
Closed Surface ◽  
Magnetic Monopoles ◽  
Surface Integral ◽  
Flux Tubes ◽  
Spatial Dimensions ◽  
Net Flux ◽  
Field Lines ◽  
The Universe

Gravity has been leaking in higher dimensions in the bulk. Gravity being a closed string is not attached or does not have any endpoints unlike photons to any Dirichlet (p)-Branes and therefore can travel inter-dimensional without any hindrance. In LHC, CERN, Gravitons are difficult to detect as they last for such a short span of time and in most of the cases invisible as because they can escape to higher spatial dimensions to the maximum of 10, as per 'M'-Theory. Gravity being one of the 4-Fundamental forces is weaker than all 3 (strong and weak nuclear force, electromagnetism) and therefore a famous problem has been made in particle physics called the 'hierarchy problem'. Through comprehensive analysis and research I have come to the conclusion that if dimension is 5 (or 4 if we neglect the temporal dimensions) then an old approach is there for the compactification of the dimensions as per Kaluza-Klein theory and the most important implications of this theory is that an unification of electromagnetism with gravitation occurs in the fifth dimensions, therefore we can conclude that both the charge (electric as well as magnetic and gravity) are dependent of each other in case of Dimensions greater than 4 (5 if time is added). Now, basic principles of electromagnetic theory states that the field-flux density through a closed surface like a T 2 Torus when integrated over the surface area leads to a zero flux. That means there is no flux outside this closed surface integral. However, if the surface is open then the field flux density is not zero and this preserves the concept of magnetic monopoles. However, in a paper in 1931,[1] Dirac approaches monopole theory of magnetism through a different perspectives that, if all the electrical charges of the universe is quantized[2] then there is a suitable (not yet proved though) existence of monopoles; however this are not well understood as of today's scenario. In condensed matter physics, plasma physics and magneto hydrodynamics, there are flux tubes and as the both ends of the flux tubes are independent of each other then the net flux through the cylinder is zero as the amount of field lines entering the tube on one side is equal to the amount of field lines exit from the other end. And in the sides of the cylinder or the flux tube there is no escape of field lines, hence, net flux is conserved. There also exists a type of 'Quasiparticles' that can act as a monopole.[3][4][5] Now, from the perspectives of the Guess law of electromagnetism, if there exists a magnetic monopole then the net charge or flux density over a surface is not zero rather the divergence of the flux density B is 4 [6]and an alternative approach of the 'monopole' can be achieved by increasing the spatial dimensions by a factor of 1 or more. The Gravity has no such poles and therefore can be considered as a unipolar flux density existing throughout the universe and is applicable to the inverse square law of decreasing magnitude via distance as 1/r 2. However, a magnet is always of bipolar with a north and South Pole. If a magnet can be broken then also the broken parts develop the other poles and become bipolar. However, there are tiny domains inside a magnet and if a magnet can be heated to approx. 700℃ then all the poles disappeared and if its cooled quickly, rather very quickly then the tiny domains inside the magnet would not get enough time to rearrange themselves and multipolar magnet is developed therefore to preserve the bipolar properties, the magnet should be cooled slowly allowing the time given to the tiny domains top rearrange themselves. Therefore, even multipole can be achieved quite easily but not the monopoles. So, the equation for a closed surface integral of a flux density without monopole is ∯(S) B dS = 0 or ∇ • B = 0 and that closed surface can be considered as 2 types namely (we will discuss about torus) as because in string theory compactification of higher spatial dimensions occurs in torus.

Optimal ventcel graphs, minimal cost spanning trees and asymptotic probabilities

2007 ◽  
Vol 1 (1) ◽  
pp. 265-275 ◽  
Author(s):  
Chiang Tzuu-Shuh ◽  
Chow Yunshyong
Keyword(s):  
Spanning Trees ◽  
Minimal Cost

Spectral Methods

2018 ◽  
Author(s):  
S. G. Rajeev
Keyword(s):  
Simple Model ◽  
Differential Equations ◽  
Boundary Conditions ◽  
Linear Algebra ◽  
Spectral Methods ◽  
Differential Operators ◽  
Higher Dimensions ◽  
Standard Linear ◽  
Sommerfeld Equation ◽  
Spectral Discretization

Thenumerical solution of ordinary differential equations (ODEs)with boundary conditions is studied here. Functions are approximated by polynomials in a Chebychev basis. Sections then cover spectral discretization, sampling, interpolation, differentiation, integration, and the basic ODE. Following Trefethen et al., differential operators are approximated as rectangular matrices. Boundary conditions add additional rows that turn them into square matrices. These can then be diagonalized using standard linear algebra methods. After studying various simple model problems, this method is applied to the Orr–Sommerfeld equation, deriving results originally due to Orszag. The difficulties of pushing spectral methods to higher dimensions are outlined.

scholarly journals On overspinning of black holes in higher dimensions

2021 ◽  
Vol 31 ◽  
pp. 100758
Author(s):  
Sanjar Shaymatov ◽  
Naresh Dadhich
Keyword(s):  
Black Holes ◽  
Higher Dimensions
Sign in / Sign up

Export Citation Format

Share Document