Representations of Composite Braids and Invariants for Mutant Knots and Links in Chern-Simons Field Theories

We show that any of the new knot invariants obtained from Chern-Simons theory based on an arbitrary non-abelian gauge group do not distinguish isotopically inequivalent mutant knots and links. In an attempt to distinguish these knots and links, we study Murakami (symmetrized version) $r$-strand composite braids. Salient features of the theory of such composite braids are presented. Representations of generators for these braids are obtained by exploiting properties of Hilbert spaces associated with the correlators of Wess-Zumino conformal field theories. The $r$-composite invariants for the knots are given by the sum of elementary Chern-Simons invariants associated with the irreducible representations in the product of $r$ representations (allowed by the fusion rules of the corresponding Wess-Zumino conformal field theory) placed on the $r$ individual strands of the composite braid. On the other hand, composite invariants for links are given by a weighted sum of elementary multicoloured Chern-Simons invariants. Some mutant links can be distinguished through the composite invariants, but mutant knots do not share this property. The results, though developed in detail within the framework of $SU(2)$ Chern-Simons theory are valid for any other non-abelian gauge group.Comment: Latex, 25pages + 16 diagrams available on reques

Chirality of Knots 9429_{42} and 107110_{71} and Chern-Simons Theory

Upto ten crossing number, there are two knots, $9_{42}$ and $10_{71}$ whose chirality is not detected by any of the known polynomials, namely, Jones invariants and their two variable generalisations, HOMFLY and Kauffman invariants. We show that the generalised knot invariants, obtained through $SU(2)$ Chern-Simons topological field theory, which give the known polynomials as special cases, are indeed sensitive to the chirality of these knots.Comment: 15 pages + 7 diagrams (available on request

Multiparametric and coloured extensions of the quantum group GLq(N)GL_q(N) and the Yangian algebra Y(glN)Y(gl_N) through a symmetry transformation of the Yang-Baxter equation

Inspired by Reshetikhin's twisting procedure to obtain multiparametric extensions of a Hopf algebra, a general `symmetry transformation' of the `particle conserving' $R$-matrix is found such that the resulting multiparametric $R$-matrix, with a spectral parameter as well as a colour parameter, is also a solution of the Yang-Baxter equation (YBE). The corresponding transformation of the quantum YBE reveals a new relation between the associated quantized algebra and its multiparametric deformation. As applications of this general relation to some particular cases, multiparametric and coloured extensions of the quantum group $GL_q(N)$ and the Yangian algebra $Y(gl_N)$ are investigated and their explicit realizations are also discussed. Possible interesting physical applications of such extended Yangian algebras are indicated.Comment: 21 pages, LaTeX (twice). Interesting physical applications of the work are indicated. To appear in Int. J. Mod. Phys.

Medical Diagnosis with Multimodal Image Fusion Techniques

Image Fusion is an effective approach utilized to draw out all the significant information from the source images, which supports experts in evaluation and quick decision making. Multi modal medical image fusion produces a composite fused image utilizing various sources to improve quality and extract complementary information. It is extremely challenging to gather every piece of information needed using just one imaging method. Therefore, images obtained from different modalities are fused Additional clinical information can be gleaned through the fusion of several types of medical image pairings. This study's main aim is to present a thorough review of medical image fusion techniques which also covers steps in fusion process, levels of fusion, various imaging modalities with their pros and cons, and  the major scientific difficulties encountered in the area of medical image fusion. This paper also summarizes the quality assessments fusion metrics. The various approaches used by image fusion algorithms that are presently available in the literature are classified into four broad categories i) Spatial fusion methods ii) Multiscale Decomposition based methods iii) Neural Network based methods and iv) Fuzzy Logic based methods. the benefits and pitfalls of the existing literature are explored and Future insights are suggested. Moreover, this study is anticipated to create a solid platform for the development of better fusion techniques in medical applications

Boosts, Schwarzschild Black Holes and Absorption cross-sections in M theory

$D$ dimensional neutral black strings wrapped on a circle are related to $(D-1)$ dimensional charged black holes by boosts. We show that the boost has to be performed in the covering space and the boosted coordinate has to be compactified on a circle with a Lorentz contracted radius. Using this fact we show that the transition between Schwarzschild black holes to black p-branes observed recently in M theory is the well-known black hole- black string transition viewed in a boosted frame. In a similar way the correspondence point where an excited string state goes over to a neutral black hole is mapped exactly to the correspondence point for black p-branes. In terms of the $p$ brane quantities the equation of state for an excited string state becomes identical to that of a 3+1 dimensional massless gas for all $p$. Finally, we show how boosts can be used to relate Hawking radiation rates. Using the known microscopic derivation of absorption by extremal 3-branes and near-extremal 5D holes with three large charges we provide a microscopic derivation of absorption of 0-branes by seven and five dimensional Schwarzschild black holes in a certain regime.Comment: Some references added, minor clarifications (harvmac, 16 pages

Hawking Radiation from Four-dimensional Schwarzschild Black Holes in M-theory

Recently a method has been developed for relating four dimensional Schwarzschild black holes in M-theory to near-extremal black holes in string theory with four charges, using suitably defined ``boosts'' and T-dualities. We show that this method can be extended to obtain the emission rate of low energy massless scalars for the four dimensional Schwarzschild hole from the microscopic picture of radiation from the near extremal hole

Spectral solutions for the Schr\"odinger equation with a regular singularity

We propose a modification in the Bethe-like ansatz to reproduce the hydrogen atom spectrum and the wave functions. Such a proposal provided a clue to attempt the exact quantization conditions (EQC) for the quantum periods associated with potentials V (x) which are singular at the origin. In a suitable limit of the parameters, the potential can be mapped to |x| potential. We validate our EQC proposition by numerically computing the Voros spectrum and matching it with the true spectrum for |x| potential. Thus we have given a route to obtain the spectral solution for the one dimensional Schr\"odinger equation involving potentials with regular singularity at the origin.Comment: 26 pages, 6 figures, 5 tables; A mathematica file is attached as an ancillary fil