arXiv subjects

Ho Lee

Publications and source records attributed to Ho Lee.

32 records · Page 2Linked to original sources

arXiv2016

Late-time behaviour of Israel particles in a FLRW spacetime with $Λ>0$

In this paper we study the relativistic Boltzmann equation in a spatially flat FLRW spacetime. We consider Israel particles, which are the relativistic counterpart of the Maxwellian particles, and obtain global-in-time existence and the asymptotic behaviour of solutions. The main argument of the paper is to use the energy method of Guo, and we observe that the method can be applied to study small solutions in a cosmological case. It is the first result of this type where a physically well-motivated scattering kernel is considered for the general relativistic Boltzmann equation.

math-ph↗

arXiv2016

Asymptotic behaviour of the Boltzmann equation as a cosmological model

As a Newtonian cosmological model the Vlasov-Poisson-Boltzmann system is considered, and a slightly modified Boltzmann equation, which describes the stability of an expanding universe, is derived. Asymptotic behaviour of solutions turns out to depend on the expansion of the universe, and in this paper we consider the soft potential case and will obtain asymptotic behaviour.

math-ph↗

arXiv2016

Future global existence and asymptotic behaviour of solutions to the Einstein-Boltzmann system with Bianchi I symmetry

In this paper we study the Einstein-Boltzmann system with Bianchi I symmetry. Isotropization of Bianchi I spacetimes was proved in [13] in the Vlasov case, and asymptotic behaviour of the relativistic Boltzmann equation was studied in [9] in the isotropic case. These results are generalized in the present paper to the non-isotropic and Boltzmann case. We assume that the initial data are small in a suitable sense, and show that solutions of the Einstein-Boltzmann system are future geodesically complete, that they isotropize and have a dust-like behaviour at late times. Detailed information about the metric and the matter terms is obtained. In short, the solution is asymptotically the Einstein-de Sitter solution.

gr-qc↗

arXiv2015

Low dimensional Born-Infeld equations coupled with a collisionless matter model

We consider the Born-Infeld nonlinear electromagnetic field equations and study its Cauchy problem in the case that the Vlasov equation is considered as a matter model. In the present paper, the Vlasov equation is considered on the so-called one and one-half dimensional phase space, and in consequence the Born-Infeld equations are reduced to a quasilinear hyperbolic system with two unknowns. A transformation is introduced in order to make the field equations easy to handle, and suitable assumptions are made on initial data so that the nonlinearity of the field is controlled.

math-ph↗

arXiv2014

Quantum knots and the number of knot mosaics

Lomonaco and Kauffman developed a knot mosaic system to introduce a precise and workable definition of a quantum knot system. This definition is intended to represent an actual physical quantum system. A knot (m,n)-mosaic is an $m \times n$ matrix of mosaic tiles ($T_0$ through $T_{10}$ depicted in the introduction) representing a knot or a link by adjoining properly that is called suitably connected. $D^{(m,n)}$ is the total number of all knot (m,n)-mosaics. This value indicates the dimension of the Hilbert space of these quantum knot system. $D^{(m,n)}$ is already found for $m,n \leq 6$ by the authors. In this paper, we construct an algorithm producing the precise value of $D^{(m,n)}$ for $m,n \geq 2$ that uses recurrence relations of state matrices that turn out to be remarkably efficient to count knot mosaics. $$ D^{(m,n)} = 2 \, \| (X_{m-2}+O_{m-2})^{n-2} \| $$ where $2^{m-2} \times 2^{m-2}$ matrices $X_{m-2}$ and $O_{m-2}$ are defined by $$ X_{k+1} = \begin{bmatrix} X_k & O_k \\ O_k & X_k \end{bmatrix} \ \mbox{and } \ O_{k+1} = \begin{bmatrix} O_k & X_k \\ X_k & 4 \, O_k \end{bmatrix} $$ for $k=0,1, \cdots, m-3$, with $1 \times 1$ matrices $X_0 = \begin{bmatrix} 1 \end{bmatrix}$ and $O_0 = \begin{bmatrix} 1 \end{bmatrix}$. Here $\|N\|$ denotes the sum of all entries of a matrix $N$. For $n=2$, $(X_{m-2}+O_{m-2})^0$ means the identity matrix of size $2^{m-2} \times 2^{m-2}$.

math.GT↗

arXiv2014

Mosaic number of knots

Lomonaco and Kauffman developed knot mosaics to give a definition of a quantum knot system. This definition is intended to represent an actual physical quantum system. A knot $n$-mosaic is an $n \times n$ matrix of 11 kinds of specific mosaic tiles representing a knot or a link. The mosaic number $m(K)$ of a knot $K$ is the smallest integer $n$ for which $K$ is representable as a knot $n$-mosaic. In this paper we establish an upper bound on the mosaic number of a knot or a link $K$ in terms of the crossing number $c(K)$. Let $K$ be a nontrivial knot or a non-split link except the Hopf link. Then $m(K) \leq c(K) + 1$. Moreover if $K$ is prime and non-alternating except $6^3_3$ link, then $m(K) \leq c(K) - 1$.

math.GT↗

arXiv2014

Upper bound on the total number of knot $n$-mosaics

Lomonaco and Kauffman introduced a knot mosaic system to give a definition of a quantum knot system which can be viewed as a blueprint for the construction of an actual physical quantum system. A knot $n$-mosaic is an $n \times n$ matrix of 11 kinds of specific mosaic tiles representing a knot or a link by adjoining properly that is called suitably connected. $D_n$ denotes the total number of all knot $n$-mosaics. Already known is that $D_1=1$, $D_2=2$, and $D_3=22$. In this paper we establish the lower and upper bounds on $D_n$ $$\frac{2}{275}(9 \cdot 6^{n-2} + 1)^2 \cdot 2^{(n-3)^2} \ \leq \ D_n \ \leq \ \frac{2}{275}(9 \cdot 6^{n-2} + 1)^2 \cdot (4.4)^{(n-3)^2}.$$ and find the exact number of $D_4 = 2594$.

math.GT↗

arXiv2014

Small knot mosaics and partition matrices

Lomonaco and Kauffman introduced knot mosaic system to give a definition of quantum knot system. This definition is intended to represent an actual physical quantum system. A knot $(m,n)$-mosaic is an $m \times n$ matrix of mosaic tiles which are $T_0$ through $T_{10}$ depicted as below, representing a knot or a link by adjoining properly that is called suitably connected. An interesting question in studying mosaic theory is how many knot $(m,n)$-mosaics are there. $D_{m,n}$ denotes the total number of all knot $(m,n)$-mosaics. This counting is very important because the total number of knot mosaics is indeed the dimension of the Hilbert space of these quantum knot mosaics. In this paper, we find a table of the precise values of $D_{m,n}$ for $4 \leq m \leq n \leq 6$ as below. Mainly we use a partition matrix argument which turns out to be remarkably efficient to count small knot mosaics. \begin{center} \begin{tabular}{|c|r|r|r|} \hline $D_{m,n}$ & $n=4$ & $n=5$ & $n=6$ \\ \hline $m=4$ & $2594$ & $54,226$ & $1,144,526$ \\ \hline $m=5$ & & $4,183,954$ & $331,745,962$ \\ \hline $m=6$ & & & $101,393,411,126$ \\ \hline \end{tabular} \end{center}

math.GT↗

arXiv2013

Asymptotic behaviour of the relativistic Boltzmann equation in the Robertson-Walker spacetime

In this paper, we study the relativistic Boltzmann equation in the spatially flat Robertson-Walker spacetime. For a certain class of scattering kernels, global existence of classical solutions is proved. We use the standard method of Illner and Shinbrot for the global existence and apply the splitting technique of Guo and Strain for the regularity of solutions. The main interest of this paper is to study the evolution of matter distribution, rather than the evolution of spacetime. We obtain the asymptotic behaviour of solutions and will understand how the expansion of the universe affects the evolution of matter distribution.

math-ph↗

arXiv2013

The spatially homogeneous relativistic Boltzmann equation with a hard potential

In this paper, we study spatially homogeneous solutions of the Boltzmann equation in special relativity and in Robertson-Walker spacetimes. We obtain an analogue of the Povzner inequality in the relativistic case and use it to prove global existence theorems. We show that global solutions exist for a certain class of collision cross sections of the hard potential type in Minkowski space and in spatially flat Robertson-Walker spacetimes.

gr-qc↗

arXiv2012

The Einstein-Boltzmann system and positivity

The Einstein-Boltzmann system is studied, with particular attention to the non-negativity of the solution of the Boltzmann equation. A new parametrization of post-collisional momenta in general relativity is introduced and then used to simplify the conditions on the collision cross-section given by Bancel and Choquet-Bruhat. The non-negativity of solutions of the Boltzmann equation on a given curved spacetime has been studied by Bichteler and by Tadmon. By examining to what extent the results of these authors apply in the framework of Bancel and Choquet-Bruhat, the non-negativity problem for the Einstein-Boltzmann system is resolved for a certain class of scattering kernels. It is emphasized that it is a challenge to extend the existing theory of the Cauchy problem for the Einstein-Boltzmann system so as to include scattering kernels which are physically well-motivated.

gr-qc↗

arXiv2004

SX Phoenicis Stars in the Globular Cluster NGC 5466

Through time-series CCD photometry of the globular cluster NGC 5466, we have detected nine SX Phoenicis stars including three new ones. All the SX Phoenicis stars are located in the blue straggler region in the color-magnitude diagram of NGC 5466. Five of them show clearly double-radial mode features, the periods of which are well consistent with the theoretical ratio of the first overtone mode to the fundamental mode (P$_{1H}$/P$_F$). Normally, it has not been easy to secure a P-L relation of the SX Phoenicis stars, because determination of the pulsational mode of the SX Phoenicis stars has been difficult. The existence of five SX Phoenicis stars in NGC5466 with double-radial modes allows us to derive reliably a P-L relation for the fundamental mode of the SX Phoenicis stars. Using seven SX Phoenicis stars including five stars with double-radial modes, we derive a P-L relation for the fundamental mode in NGC 5466, $ = - 3.25 (\pm 0.46) Log P + 14.70 (\pm 0.06), (σ= \pm0.04) $, corresponding to $ = - 3.25 (\pm0.46)Log P - 1.30 (\pm0.06) $ for an adopted distance modulus of $(m-M)_0=16.00$ and zero reddening.

astro-ph↗

arXiv2003

New SX Phoenicis Stars in the Globular Cluster M53

Through time-series CCD photometry of the metal-poor globular cluster M53, we have discovered eight new SX Phoenicis type stars (labeled from SXP1 to SXP8). All the new SX Phoenicis stars are located in the blue straggler star region of a color-magnitude diagram of M53. One of these stars (SXP2) is found to have very closely separated pulsation frequencies: $f_1/f_2 = 0.9595$ where $f_1$ and $f_2$ are primary and secondary frequencies. This may be due to excitation of non-radial modes. Six of these SX Phoenicis stars are considered to be pulsating in the fundamental mode. They show a tight linear correlation between the period and luminosity. We derive a period - luminosity relation for the fundamental mode for the period range of $-1.36 < Log P[d]< -1.15$ : $ =-3.010(\pm0.262)Log P + 15.310(\pm0.048)$ with an rms scatter of 0.038, corresponding to $ =-3.010 Log P - 1.070$ for an adopted distance modulus of $(m-M)_V=16.38$ (Harris 1996).

astro-ph↗

arXiv2001

A New SX Phe Star in the Globular Cluster M15

A new SX Phe star (labelled SXP1) found from $BV$ CCD photometry is the first to be discovered in the globular cluster M15. It is a blue straggler and is located $102\arcsec.8$ north and $285\arcsec.6$ west of the center of M15 \citep{har96}. Mean magnitudes of SXP1 are $<$$B$$>$ = 18$\fm$671 and $<$$V$$>$ = 18$\fm$445. The amplitude of variability of SXP1 is measured to be $ΔV \approx 0.15$. From multiple-frequency analysis based on the Fourier decomposition method, we detect two very closely separated pulsating frequencies: the primary frequency at $f_1=24.630$ c/d for both $B$- and $V$-bands, and the secondary frequency at $f_2=24.338$ c/d for the $B$-band and 24.343 c/d for the $V$-band. This star is the second among known SX Phe stars found to pulsate with very closely separated frequencies ($f_2/f_1\ge0.95$). These frequencies may be explained by excitation of nonradial modes; however, we have an incomplete understanding of this phenomenon in the case of SX Phe stars with relatively high amplitudes. The metallicity-period and the variability amplitude-period relations for SXP1 in M15 are found to be consistent with those for SX Phe stars in other globular clusters.

astro-ph↗