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Ho Lee

Publications and source records attributed to Ho Lee.

At least 19 recordsLinked to original sources
arXiv2026

The massless Boltzmann equation in Minkowski spacetime

We study the spatially homogeneous, massless Boltzmann equation in Minkowski spacetime for a certain range of hard and soft interactions. For hard interactions, we derive a Povzner-type inequality for massless particles and show that solutions exist for all time into the future. For soft interactions, we employ singular weights to control singularities at $ p = 0 $, which arise from the masslessness of particles, to obtain local existence. These results, which are among rather few proofs of existence for the massless Boltzmann equation, are motivated by our earlier work on the massless Einstein--Boltzmann system in certain cosmological settings.

math.AP↗
arXiv2026

Asymptotic behavior of small solutions to the Vlasov--Klein--Gordon system in high dimensions

We study the asymptotic behavior of small solutions to the Vlasov--Klein--Gordon system in high dimensions. The standard argument of Glassey and Strauss \cite{GS87} for studying small solutions to the Vlasov--Maxwell system does not apply to the Vlasov--Klein--Gordon system due to the massiveness of the Klein--Gordon field. In this paper we use the vector field method and consider solutions in dimensions $ n \geq 4 $ with the hyperboloidal foliation of the Minkowski spacetime to obtain the asymptotic properties for the Vlasov--Klein--Gordon system.

math.AP↗
arXiv2025

Parallel-plate chambers as radiation-hard detectors for time-based beam diagnostics in carbon-ion radiotherapy

Accurate range verification of carbon ion beams is critical for the precision and safety of charged particle radiotherapy. In this study, we evaluated the feasibility of using a parallel-plate ionization chamber for real-time, time-based diagnostic monitoring of carbon ion beams. The chamber featured a 0.4 mm gas gap defined by metallic electrodes and was filled with carbon dioxide (CO$_2$), a non-polymerizing gas suitable for high-rate applications. Timing precision was assessed via self-correlation analysis, yielding a precision approaching one picosecond for one-second acquisitions under clinically relevant beam conditions. This level of timing accuracy translates to a water-equivalent range uncertainty of approximately 1 mm, which meets the recommended clinical tolerance for carbon ion therapy. Furthermore, the kinetic energy of the beam at the synchrotron extraction point was determined from the measured orbital period, with results consistently within 1 MeV/nucleon of the nominal energy. These findings demonstrate the potential of parallel-plate chambers for precise, real-time energy and range verification in clinical carbon ion beam quality assurance.

physics.med-ph↗
arXiv2024

Homogeneous solutions to the Einstein-matter equations with a magnetic field and a conformal gauge singularity

We study massless solutions to the Einstein equations coupled to different matter models with a magnetic field and a conformal gauge singularity assuming spatial homogeneity with three commuting spatial translations. We show that there are no solutions in the case that the matter model is a radiation fluid. If the matter is described via kinetic theory we obtain that there exist unique solutions to the Einstein-Vlasov system and the Einstein-Boltzmann system for a certain range of soft potentials. For both the Vlasov and the Boltzmann case we also obtain asymptotic expansions close to the initial conformal gauge singularity.

gr-qc↗
arXiv2023

Future of Bianchi I magnetic cosmologies with kinetic matter

We show under the assumption of small data that solutions to the Einstein-Vlasov system with a pure magnetic field and Bianchi I symmetry isotropise and tend to dust solutions. We also obtain the decay rates for the main variables. This generalises part of the work [V.~G.~LeBlanc, Classical Quantum Gravity 14, 2281-2301 (1997)] concerning the future behaviour of orthogonal perfect fluids with a linear equation of state in the presence of a magnetic field to the Vlasov case.

gr-qc↗
arXiv2023

Future global existence of homogeneous solutions to the Einstein-Boltzmann system with soft potentials

We study the Cauchy problem for the Einstein-Boltzmann system with soft potentials in a cosmological setting. We assume the Bianchi I symmetry to describe a spatially homogeneous, but anisotropic universe and consider a cosmological constant $ \Lambda > 0 $ to describe an accelerated expansion of the universe. For the Boltzmann equation we introduce a new weight function and apply the method of Illner and Shinbrot to obtain the future global existence of spatially homogeneous, small solutions. For the Einstein equations we assume that the initial value of the Hubble variable is close to $ ( \Lambda / 3 )^{ 1 / 2 } $. We obtain the future global existence and asymptotic behavior of spatially homogeneous solutions to the Einstein-Boltzmann system with soft potentials.

gr-qc↗
arXiv2023

Well-posedness of anisotropic and homogeneous solutions to the Einstein-Boltzmann system with a conformal-gauge singularity

We consider the Einstein-Boltzmann system for massless particles in the Bianchi I space-time with scattering cross-sections in a certain range of soft potentials. We assume that the space-time has an initial conformal gauge singularity and show that the initial value problem is well posed with data given at the singularity. This is understood by considering conformally rescaled equations. The Einstein equations become a system of singular ordinary differential equations, for which we establish an existence theorem which requires several differentiability and eigenvalue conditions on the coefficient functions together with the Fuchsian conditions. The Boltzmann equation is regularized by a suitable choice of time coordinate, but still has singularities in momentum variables. This is resolved by considering singular weights, and the existence is obtained by exploiting singular moment estimates.

gr-qc↗
arXiv2022

Small solutions of the Einstein-Boltzmann-scalar field system with Bianchi symmetry

We show that small homogeneous solutions to the Einstein-Boltzmann-scalar field system exist globally towards the future and tend to the de Sitter solution in a suitable sense. More specifically, we assume that the spacetime is of Bianchi type I--VIII, that the matter is described by Israel particles and that there exists a scalar field with a potential which has a positive lower bound. This represents a generalization of the work [19], where a cosmological constant was considered, and a generalization of [16], where a spatially flat FLRW spacetime was considered. We obtain the global existence and asymptotic behavior of classical solutions to the Einstein-Boltzmann-scalar field system for small initial data.

gr-qc↗
arXiv2021

On almost Ehlers-Geren-Sachs theorems

We show assuming small data that massless solutions to the reflection symmetric Einstein-Vlasov system with Bianchi VII$_0$ symmetry which are not locally rotational symmetric, can be arbitrarily close to and will remain close to isotropy as regards {to} the shear. However in general the shear will not tend to zero and the Hubble normalised Weyl curvature will blow up. This generalises the work \cite{NHW,WHU}, which considered a non-tilted radiation fluid to the massless Vlasov case. This represents another example of the fact that almost Ehlers-Geren-Sachs theorems do not hold in general and that collisionless matter behaves differently than a perfect fluid.

gr-qc↗
arXiv2019

On the future of solutions to the massless Einstein-Vlasov system in a Bianchi I cosmology

We show that massless solutions to the Einstein-Vlasov system in a Bianchi I space-time with small anisotropy, i.e. small shear and small trace-free part of the spatial energy momentum tensor, tend to a radiation fluid in an Einstein-de Sitter space-time with the anisotropy $\Sigma^a_b\Sigma^b_a$ and $\tilde{w}^i_j \tilde{w}^j_i$ decaying as $O(t^{-\frac12})$.

gr-qc↗
arXiv2017

Late-time behaviour of the Einstein-Boltzmann system with a positive cosmological constant

In this paper we study the Einstein-Boltzmann system for Israel particles with a positive cosmological constant. We consider spatially homogeneous solutions of Bianchi types except IX and obtain future global existence and asymptotic behaviour of solutions to the Einstein-Boltzmann system. The result shows that the solutions converge to the de Sitter solution at late times.

gr-qc↗
arXiv2017

Bianchi I solutions of the Einstein-Boltzmann system with a positive cosmological constant

In this paper we study the future global existence and late-time behaviour of the Einstein-Boltzmann system with Bianchi I symmetry and a positive cosmological constant $Λ>0$. For the Boltzmann equation we consider the scattering kernel of Israel particles which are the relativistic counterpart of Maxwellian particles. Under a smallness assumption on initial data in a suitable norm we show that solutions exist globally in time and isotropize at late times.

math-ph↗
arXiv2017

Mutation invariance of the arc index for some Montesinos knots

For the alternating knots or links, mutations do not change the arc index. In the case of nonalternating knots, some semi-alternating knots or links have this property. We mainly focus on the problem of mutation invariance of the arc index for nonalternating knots which are not semi-alternating. In this paper, we found families of infinitely many mutant pairs/triples of Montesinos knots with the same arc index.

math.GT↗
arXiv2017

Period and toroidal knot mosaics

Knot mosaic theory was introduced by Lomonaco and Kauffman in the paper on `Quantum knots and mosaics' to give a precise and workable definition of quantum knots, intended to represent an actual physical quantum system. A knot (m,n)-mosaic is an $m \! \times \! n$ matrix whose entries are eleven mosaic tiles, representing a knot or a link by adjoining properly. In this paper we introduce two variants of knot mosaics: period knot mosaics and toroidal knot mosaics, which are common features in physics and mathematics. We present an algorithm producing the exact enumeration of period knot (m,n)-mosaics for any positive integers m and n, toroidal knot (m,n)-mosaics for co-prime integers m and n, and furthermore toroidal knot (p,p)-mosaics for a prime number p. We also analyze the asymptotics of the growth rates of their cardinality.

math.GT↗
arXiv2017

Self-similarity breaking of cosmological solutions with collisionless matter

In this paper we consider the Einstein-Vlasov system with Bianchi VII$_0$ symmetry. Under the assumption of small data we show that self-similarity breaking occurs for reflection symmetric solutions. This generalizes the previous work concerning the non-tilted fluid case \cite{WHU} to the Vlasov case, and we obtain detailed information about the late-time behaviour of metric and matter terms.

gr-qc↗