Difference equation of the colored Jones polynomial for torus knot

We prove that the N-colored Jones polynomial for the torus knot T_{s,t} satisfies the second order difference equation, which reduces to the first order difference equation for a case of T_{2,2m+1}. We show that the A-polynomial of the torus knot can be derived from this difference equation. Also constructed is a q-hypergeometric type expression of the colored Jones polynomial for T_{2,2m+1}.Comment: 7 page

Investigation of effectiveness of various methods with different unknown variables for 3-D eddy current analysis

Computer codes using the A-&#966;, A-&#966;-&#937;, A*-0&#937;-E, T-&#937;, and E-&#937; methods were developed. The effects of the volume ratio of the conductor region to the whole region, the shape of the conductor, and the ratio of the hole region to the conductor region on the computer storage, the CPU time, and the accuracy of the methods are investigated systematically using a few simple models. The effect of the conductivity of the conductor is also examined. The computer storage, the CPU time, and the error are found to increase with increase of the volume ratio of the conductor region to the whole region. The computer storage and the CPU time are affected by the shape of the conductor in some methods of analysis. The error of the A*-&#937;(E-&#937;) method is larger than that of the other methods</p

Sustainable Hydrogen from Bio-Oil - Catalytic Steam Reforming of Acetic Acid as a Model Oxygenate

Studies were conducted with acetic acid (HAc) as model oxygenate for the design of active and stable catalysts for steam reforming of bio-oil. Pt/ZrO2 catalysts were prepared by wet impregnation technique. The Pt/ZrO2 catalysts showed high activities at initial time on stream, but lost its activity for steam reforming (H2 production) rapidly. During HAc/H2O reaction over Pt/ZrO2, conversion was close to 100% and constant for 3 hr, however, yields of products changed with time. In the beginning (5 min), H2 and CO2 were the main products, CH4 and CO were observed in small quantities. During HAc/H2O reaction over ZrO2 (without Pt), HAc conversion was close to 90%. The conversion of HAc and yields of the products were constant for 3 hr. However, no steam reforming activity (H2 and CO) was observed, and only acetone and CO2 were observed as products. Both Pt/ZrO2 and ZrO2 were very active for HAc conversion. However, H2 and CO, i.e., steam reforming products, were produced only over Pt/ZrO2 and not over ZrO2. ZrO2 showed acetone yields similar to those observed over Pt/ZrO2 after 25 min time on stream. The presence of acetone in the product mixture and formation of deposits on ZrO2 indicated a role for acetone in catalyst deactivation

Seiberg-Witten Curve for the E-String Theory

We construct the Seiberg-Witten curve for the E-string theory in six-dimensions. The curve is expressed in terms of affine E_8 characters up to level 6 and is determined by using the mirror-type transformation so that it reproduces the number of holomorphic curves in the Calabi-Yau manifold and the amplitudes of N=4 U(n) Yang-Mills theory on 1/2 K3. We also show that our curve flows to known five- and four-dimensional Seiberg-Witten curves in suitable limits.Comment: 18 pages, 1 figure; appendix C adde

A generalization of heterochromatic graphs

In 2006, Suzuki, and Akbari & Alipour independently presented a necessary and sufficient condition for edge-colored graphs to have a heterochromatic spanning tree, where a heterochromatic spanning tree is a spanning tree whose edges have distinct colors. In this paper, we propose $f$-chromatic graphs as a generalization of heterochromatic graphs. An edge-colored graph is $f$-chromatic if each color $c$ appears on at most $f(c)$ edges. We also present a necessary and sufficient condition for edge-colored graphs to have an $f$-chromatic spanning forest with exactly $m$ components. Moreover, using this criterion, we show that a $g$-chromatic graph $G$ of order $n$ with $|E(G)|>\binom{n-m}{2}$ has an $f$-chromatic spanning forest with exactly $m$ ($1 \le m \le n-1$) components if $g(c) \le \frac{|E(G)|}{n-m}f(c)$ for any color $c$.Comment: 14 pages, 4 figure

On the Quantum Invariant for the Brieskorn Homology Spheres

We study an exact asymptotic behavior of the Witten-Reshetikhin-Turaev invariant for the Brieskorn homology spheres $\Sigma(p_1,p_2,p_3)$ by use of properties of the modular form following a method proposed by Lawrence and Zagier. Key observation is that the invariant coincides with a limiting value of the Eichler integral of the modular form with weight 3/2. We show that the Casson invariant is related to the number of the Eichler integrals which do not vanish in a limit $\tau\to N \in \mathbb{Z}$. Correspondingly there is a one-to-one correspondence between the non-vanishing Eichler integrals and the irreducible representation of the fundamental group, and the Chern-Simons invariant is given from the Eichler integral in this limit. It is also shown that the Ohtsuki invariant follows from a nearly modular property of the Eichler integral, and we give an explicit form in terms of the L-function.Comment: 26 pages, 2 figure

Spontaneous spin current near the interface between unconventional superconductors and ferromagnets

We study theoretically the proximity effect between ferromagnets (F) and superconductors (S) with broken time-reversal symmetry (${\cal T}$). A chiral $(p_x\pm ip_y)$-wave, and a $d_{x^2-y^2}$-wave superconductor, the latter of which can form ${\cal T}$-breaking surface state, i.e., $(d_{x^2-y^2} \pm is)$-state, are considered for the S side. The spatial variations of the superconducting order parameters and the magnetization are determined by solving the Bogoliubov de Gennes equation. In the case of a chiral $(p_x\pm ip_y)$-wave superconductor, a spontaneous spin current flows along the interface, but not in the case of a $d_{x^2-y^2}$-wave superconductor. For F/S$(p_x+ip_y)$/F trilayer system, total spin current can be finite while total charge current vanishes, if the magnetization of two F layers are antiparallel.Comment: 6 pages, 11 figures. Accepted for publication in Physical Review

Method for Investigating the Spectral Characteristics of Corroded Metal

The extent of corrosion on the world’s bridges, buildings, power line towers, and various other outdoor steel structures is a significant problem faced by today’s infrastructure. For example, in the United States there are approximately 576,000 bridges, 41% of which are deemed potentially unsafe due to their substandard condition [1]. Corrosion is the major cause of these structures being evaluated as inadequate