2,184 research outputs found
Asymptotic Behavior of Colored Jones polynomial and Turaev-Viro Invariant of figure eight knot
In this paper we investigate the asymptotic behavior of the colored Jones
polynomials and the Turaev-Viro invariants for the figure eight knot. More
precisely, we consider the -th colored Jones polynomials evaluated at
-th root of unity with a fixed limiting ratio, , of and
. We find out the asymptotic expansion formula (AEF) of the colored
Jones polynomials of the figure eight knot with close to . Nonetheless,
we show that the exponential growth rate of the colored Jones polynomials of
the figure eight knot with close to is strictly less than those with
close to . It is known that the Turaev Viro invariant of the figure
eight knot can be expressed in terms of a sum of its colored Jones polynomials.
Our results show that this sum is asymptotically equal to the sum of the terms
with close to 1. As an application of the asymptotic behavior of the
colored Jones polynomials, we obtain the asymptotic expansion formula for the
Turaev-Viro invariants of the figure eight knot. Finally, we suggest a possible
generalization of our approach so as to relate the AEF for the colored Jones
polynomials and the AEF for the Turaev-Viro invariants for general hyperbolic
knots.Comment: 40 pages, 0 figure
The Quality Control of Puerariae Lobatae Radix and Puerariae Thomsonii Radix
Puerariae Lobatae Radix (PLR) and Puerariae Thomsonii Radix (PTR) are traditional Chinese medicines used interchangeably in clinical practice, even though they possess significantly different chemical profiles. The aim of this thesis was to differentiate PLR from PTR using various analytical instruments coupled with chemometrics. Morphological results illustrate PLR possessed distinct macroscopic and microscopic features as compared to PTR. UPLC results reveal isoflavonoids were the major chemical constituents in both species, with the content of puerarin in PLR significantly greater than in PTR. PLS-DA models demonstrate both UPLC and HPTLC chromatographic fingerprints were effective in differentiating PLR from PTR. PLSR coupled with Raman spectra was able to predict the TPC and antioxidant capacities of PLR and PTR. The pharmacological results illustrate PLR possessed significantly greater anti-diabetic, cytoprotective and anti-cancer activities as compared to PTR. In summary, the results reveal the chemical fingerprints coupled with chemometrics was effective in differentiating PLR from PTR, and PLR was morphologically, chemically and pharmacologically different from PTR. This thesis provided further insight into the comprehensive nature of the quality control of two similar species and recommends changes to their descriptions in the pharmacopoeias. This will ultimately improve the quality, safety and efficacy of herbal products
Relative Reshetikhin-Turaev invariants, hyperbolic cone metrics and discrete Fourier transforms II
We prove the Volume Conjecture for the relative Reshetikhin-Turaev invariants
proposed in [29] for all pairs (M,K) such that M\K is homeomorphic to the
complement of the figure-8 knot in S^3 with almost all possible cone angles.Comment: 34 pages, 5 figures. arXiv admin note: substantial text overlap with
arXiv:2003.1005
Relative Reshetikhin-Turaev invariants, hyperbolic cone metrics and discrete Fourier transforms I
We propose the Volume Conjecture for the relative Reshetikhin-Turaev
invariants of a closed oriented -manifold with a colored framed link inside
it whose asymptotic behavior is related to the volume and the Chern-Simons
invariant of the hyperbolic cone metric on the manifold with singular locus the
link and cone angles determined by the coloring. We prove the conjecture in the
case that the ambient -manifold is obtained by doing an integral surgery
along some components of a fundamental shadow link and the complement of the
link in the ambient manifold is homeomorphic to the fundamental shadow link
complement, for sufficiently small cone angles. Together with Costantino and
Thurston's result that all compact oriented -manifolds with toroidal or
empty boundary can be obtained by doing an integral surgery along some
components of a suitable fundamental shadow link, this provides a possible
approach of solving Chen-Yang's Volume Conjecture for the Reshetikhin-Turaev
invariants of closed oriented hyperbolic -manifolds. We also introduce a
family of topological operations (the change of pair operations) that connect
all pairs of a closed oriented -manifold and a framed link inside it that
have homeomorphic complements, which correspond to doing the partial discrete
Fourier transforms to the corresponding relative Reshetikhin-Turaev invariants.
As an application, we find a Poisson Summation Formula for the discrete Fourier
transforms.Comment: 44 pages, 21 figure
Geometry of fundamental shadow link complements and applications to the 1-loop conjecture
We construct a geometric ideal triangulation for every fundamental shadow
link complement and solve the gluing equation explicitly in terms of the
holonomies of the meridians of the link for any generic character in the
distinguished component of the -character variety
of the link complement. As an application, we obtain a new formula for the
volume of a hyperideal tetrahedron in terms of its dihedral angles. Moreover,
by using the ideal triangulation, we verify the 1-loop conjecture proposed by
Dimofte and Garoufalidis for every fundamental shadow link complement. We also
prove a surgery formula for the 1-loop invariant with respect to certain nice
ideal triangulations of 3-manifolds with toroidal boundary.Comment: 52 pages, 22 figure
Structural realism and the New Cold War
The Cold War mentality such as misperception which is deep-rooted in the American and Russian government has triggered the revival of U.S.-Russian conflicts. The new Cold War would be a continuation from the past that operates with an action-and-reactions model. However, with the change of geopolitical situation, the new Cold War would be conducted in different means.
The structural constraints suggested by the structural realist would be an explanation for the emergence of new Cold War. The structural realism makes contributions of the idea of misperception,over-exaggeration, and balance of power that is a product of an anarchic international system. The concept of security dilemma would also deliver a theoretical foundation of the action-and-reaction approach between the United States and Russia.
The geopolitical struggles between the Untied States and Russia in the CIS region, Europe and Asia would be illustrated to explain the new Cold War style of confrontation. Furthermore, the geostrategic analysis emphasizes the Russian resurgence through energy to exert greater influences in a variety of regions. Last of all, the Russo-Georgian war and Ukrainian crisis would be selected as the case study to clarify the Russian revisionism in the near abroad
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