The chromatic class and the location of a graph on a closed surface

1970 ◽  
Vol 7 (6) ◽  
pp. 405-411 ◽  
Author(s):  
L. S. Mel'nikov
Keyword(s):  
Closed Surface ◽  
Chromatic Class

Dualism QM and GR

2019 ◽  
Author(s):  
Vitaly Kuyukov
Keyword(s):  
Quantum Mechanics ◽  
General Theory ◽  
Noncommutative Geometry ◽  
Quantum Tunneling ◽  
Closed Surface ◽  
Theory Of Relativity ◽  
General Theory Of Relativity ◽  
Surface Integral ◽  
Definition Of

Quantum tunneling of noncommutative geometry gives the definition of time in the form of holography, that is, in the form of a closed surface integral. Ultimately, the holography of time shows the dualism between quantum mechanics and the general theory of relativity.

Holographic part integral and exemal closed surface

2019 ◽  
Author(s):  
Vitaly Kuyukov
Keyword(s):  
Closed Surface

Holographic part integral and exemal closed surface

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.

AUTOMORPHISMS OF BRAID GROUPS ON S2, T2, P2 AND THE KLEIN BOTTLE K

2008 ◽  
Vol 17 (01) ◽  
pp. 47-53 ◽  
Author(s):  
PING ZHANG
Keyword(s):  
Braid Group ◽  
Mapping Class Group ◽  
Class Group ◽  
Klein Bottle ◽  
Closed Surface ◽  
Group Extension ◽  
Braid Groups ◽  
Mapping Class ◽  
Group B ◽  
Central Automorphisms

It is shown that for the braid group Bn(M) on a closed surface M of nonnegative Euler characteristic, Out (Bn(M)) is isomorphic to a group extension of the group of central automorphisms of Bn(M) by the extended mapping class group of M, with an explicit and complete description of Aut (Bn(S2)), Aut (Bn(P2)), Out (Bn(S2)) and Out (Bn(P2)).

An existence theorem for Robin's equation

1972 ◽  
Vol 72 (3) ◽  
pp. 489-498 ◽  
Author(s):  
R. Cade
Keyword(s):  
Integral Equation ◽  
Existence Theorem ◽  
Electric Charge ◽  
Gaussian Curvature ◽  
Closed Surface ◽  
Arithmetic Mean ◽  
Main Argument

AbstractAn existence theorem is proved for Robin's integral equation for the density of electric charge on a closed surface, under the assumptions that the surface is convex, smooth and twice continuously differentiable. The technique is essentially Neumann's method of the arithmetic mean, used by Robin himself to show that the solution, assumed to exist, can be successively approximated by a sequence. In order to facilitate the main argument of the proof, it is assumed initially that the Gaussian curvature is everywhere positive, but this restriction is subsequently removed.

scholarly journals Closed-Surface Multifunctional Antireflective Coating Made from SiO2 with TiO2 Nanocomposites

Materials ◽  
2021 ◽  
Vol 14 (6) ◽  
pp. 1367
Author(s):  
Zhiqiu Guo ◽  
Ze Zhu ◽  
Ya Liu ◽  
Changjun Wu ◽  
Hao Tu ◽  
...  
Keyword(s):  
Electron Microscopy ◽  
Short Circuit ◽  
Sol Gel ◽  
Short Circuit Current ◽  
Closed Surface ◽  
Chain Structure ◽  
High Photocatalytic Activity ◽  
Current Gain ◽  
Antireflective Coating ◽  
Reliability And Robustness

An SiO2-TiO2 closed-surface antireflective coating was fabricated by the one-dipping method. TiO2 nanoparticles were mixed with a nanocomposited silica sol, which was composed of acid-catalyzed nanosilica networks and silica hollow nanospheres (HNs). The microstructure of the sol-gel was characterized by transmission electron microscopy. The silica HNs were approximately 40–50 nm in diameter with a shell thickness of approximately 8–10 nm. The branched-chain structure resulting from acidic hydrolysis grew on these silica HNs, and TiO2 was distributed inside this network. The surface morphology of the coating was measured by field emission scanning electron microscopy and atomic force microscopy. After optimization, transmittance of up to 94.03% was obtained on photovoltaic (PV) glass with a single side coated by this antireflective coating, whose refractive index was around 1.30. The short-circuit current gain of PV module was around 2.14–2.32%, as shown by the current-voltage (IV) curve measurements and external quantum efficiency (EQE) tests. This thin film also exhibited high photocatalytic activity. Due to the lack of voids on its surface, the antireflective coating in this study possessed excellent long-term reliability and robustness in both high-moisture and high-temperature environments. Combined with its self-cleaning function, this antireflective coating has great potential to be implemented in windows and photovoltaic modules.

scholarly journals A Note On Umbilics of a Closed Surface

1959 ◽  
Vol 15 ◽  
pp. 219-223
Author(s):  
Minoru Kurita
Keyword(s):  
Riemannian Manifold ◽  
Euclidean Space ◽  
Tensor Field ◽  
Symmetric Tensor ◽  
Closed Surface ◽  
Dimension 2 ◽  
Direction Field

In this paper we investigate indices of umbilics of a closed surface in the euclidean space. Most part of the discussion is concerned with a symmetric tensor field of degree 2, or rather a direction field, on a Riemannian manifold of dimension 2.

NEW POLARIZATIONS ON THE MODULI SPACES AND THE THURSTON COMPACTIFICATION OF TEICHMÜLLER SPACE

1998 ◽  
Vol 09 (01) ◽  
pp. 1-45 ◽  
Author(s):  
JØRGEN ELLEGAARD ANDERSEN
Keyword(s):  
Moduli Space ◽  
Moduli Spaces ◽  
Teichmüller Space ◽  
Closed Surface ◽  
Teichmuller Space ◽  
Open Dense Subset ◽  
Complex Structures ◽  
Thurston Boundary ◽  
Simply Connected ◽  
Singular Metric

Given a foliation F with closed leaves and with certain kinds of singularities on an oriented closed surface Σ, we construct in this paper an isotropic foliation on ℳ(Σ), the moduli space of flat G-connections, for G any compact simple simply connected Lie-group. We describe the infinitesimal structure of this isotropic foliation in terms of the basic cohomology with twisted coefficients of F. For any pair (F, g), where g is a singular metric on Σ compatible with F, we construct a new polarization on the symplectic manifold ℳ′(Σ), the open dense subset of smooth points of ℳ(Σ). We construct a sequence of complex structures on Σ, such that the corresponding complex structures on ℳ′(Σ) converges to the polarization associated to (F, g). In particular we see that the Jeffrey–Weitzman polarization on the SU(2)-moduli space is the limit of a sequence of complex structures induced from a degenerating family of complex structures on Σ, which converges to a point in the Thurston boundary of Teichmüller space of Σ. As a corollary of the above constructions, we establish a certain discontinuiuty at the Thurston boundary of Teichmüller space for the map from Teichmüller space to the space of polarizations on ℳ′(Σ). For any reducible finite order diffeomorphism of the surface, our constuction produces an invariant polarization on the moduli space.

Super-durable closed-surface antireflection thin film by silica nanocomposites

2017 ◽  
Vol 170 ◽  
pp. 143-148 ◽  
Author(s):  
Z.Q. Guo ◽  
Y. Liu ◽  
M.Y. Tang ◽  
J.H. Wang ◽  
X.P. Su
Keyword(s):  
Thin Film ◽  
Closed Surface ◽  
Silica Nanocomposites

Maximum orders of cyclic group actions on surfaces extending over the 3-dimensional torus

2017 ◽  
Vol 26 (06) ◽  
pp. 1742005
Author(s):  
Chao Wang ◽  
Shicheng Wang ◽  
Yimu Zhang
Keyword(s):  
Cyclic Group ◽  
Group Actions ◽  
Closed Surface ◽  
Dimensional Torus ◽  
Maximum Order ◽  
3 Dimensional

We determine the maximum order of cyclic group actions on the pair [Formula: see text] among all embeddings of closed surface [Formula: see text] into the 3-dimensional torus [Formula: see text] in the orientable category.

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