arXiv subjects
Publications and source records attributed to Zipeng Wang.
GRChombo is an open-source code for performing Numerical Relativity time evolutions, built on top of the publicly available Chombo software for the solution of PDEs. Whilst GRChombo uses standard techniques in NR, it focusses on applications in theoretical physics where adaptability, both in terms of grid structure, and in terms of code modification, are key drivers.
Suppose that $ω$ is a radial weight on the unit disk that satisfies both forward and reverse doubling conditions. Using Carleson measures and $T1$-type conditions, we obtain necessary and sufficient conditions of the positive Borel measure $μ$ such that the Toeplitz operator $T_{μ,ω}:L^p_a(ω)\to L_a^1(ω)$ is bounded and compact for $0<p\leq 1$. In addition, we obtain a bump condition for the bounded Toeplitz operators with $L^1(ω)$ symbol on $L^1_a(ω)$. This generalizes a result of Zhu in \cite{zhu1989}.
Let $Ω\subset \mathbb{C}$ be an arbitrary domain in the one-dimensional complex plane equipped with a positive Radon measure $μ$. For any $1\le p< \infty$, it is shown that the weighted Bergman space $A^p(Ω, μ)$ of holomorphic functions is a Banach space if and only if $A^p(Ω, μ)$ has locally uniformly bounded point evaluations. In particular, in the case $p =2$, any complete Bergman space $A^2(Ω, μ)$ is automatically a reproducing kernel Hilbert space.
We analyze the collision of three identical spin-polarized fermions at zero collision energy, assuming arbitrary finite-range potentials, and define the corresponding three-body scattering hypervolume $D_F$. The scattering hypervolume $D$ was first defined for identical bosons in 2008 by one of us. It is the three-body analog of the two-body scattering length. We solve the three-body Schrödinger equation asymptotically when the three fermions are far apart or one pair and the third fermion are far apart, deriving two asymptotic expansions of the wave function. Unlike the case of bosons for which $D$ has the dimension of length to the fourth power, here the $D_F$ we define has the dimension of length to the eighth power. We then analyze the interaction energy of three such fermions with momenta $\hbar\mathbf{k}_1$, $\hbar\mathbf{k}_2$ and $\hbar\mathbf{k}_3$ in a large periodic cubic box. The energy shift due to $D_F$ is proportional to $D_F/Ω^2$, where $Ω$ is the volume of the box. We also calculate the shifts of energy and pressure of spin-polarized Fermi gases due to a nonzero $D_F$ and the three-body recombination rate of spin-polarized ultracold atomic Fermi gases at finite temperatures.
We analyze the collision of three particles with arbitrary mass ratio at zero collision energy, assuming arbitrary short-range potentials, and generalize the three-body scattering hypervolume $D$ first defined for identical bosons in 2008. We solve the three-body Schrödinger equation asymptotically when the three particles are far apart or one pair and a third particle are far apart, deriving two asymptotic expansions of the wave function, and the parameter $D$ appears at the order $1/B^4$, where $B$ is the overall size of the triangle formed by the particles. We then analyze the ground state energy of three such particles with vanishing or negligible two-body scattering lengths in a large periodic volume of side length $L$, where the three-body parameter contributes a term of the order $D/L^6$. From this result we derive some properties of a two-component Bose gas with negligible two-body scattering lengths: its energy density at zero temperature, the corresponding generalized Gross-Pitaevskii equation, the conditions for the stability of the two-component mixture against collapse or phase separation, and the decay rates of particle densities due to three-body recombination.
In this paper, we study a family of fractional integral operators whose kernel carrying a critical index has singularity on the light-cone in R^n+1.
Let $\mathcal{T}$ be a rooted tree endowed with the natural partial order $\preceq$. Let $(Z(v))_{v\in \mathcal{T}}$ be a sequence of independent standard Gaussian random variables and let $α= (α_k)_{k=1}^\infty$ be a sequence of real numbers with $\sum_{k=1}^\infty α_k^2<\infty$. Set $α_0 =0$ and define a Gaussian process on $\mathcal{T}$ in the following way: \[ G(\mathcal{T}, α; v): = \sum_{u\preceq v} α_{|u|} Z(u), \quad v \in \mathcal{T}, \] where $|u|$ denotes the graph distance between the vertex $u$ and the root vertex. Under mild assumptions on $\mathcal{T}$, we obtain a necessary and sufficient condition for the almost sure boundedness of the above Gaussian process. Our condition is also necessary and sufficient for the almost sure uniform convergence of the Gaussian process $G(\mathcal{T}, α; v)$ along all rooted geodesic rays in $\mathcal{T}$.
Let $\mathcal{D}$ be the class of radial weights on the unit disk which satisfy both forward and reverse doubling conditions. Let $g$ be an analytic function on the unit disk $\mathbb{D}$. We characterize bounded and compact Volterra type integration operators \[ J_{g}(f)(z)=\int_{0}^{z}f(λ)g'(λ)dλ\] between weighted Bergman spaces $L_{a}^{p}(ω)$ induced by $\mathcal{D}$ weights and Hardy spaces $H^{q}$ for $0<p,q<\infty$.
The idea of using a Reconfigurable Intelligent Surface (RIS) consisting of a large array of passive scattering elements to assist wireless communication systems has recently attracted much attention from academia and industry. A central issue with RIS is how much power they can effectively convey to the target radio nodes. Regarding this question, several power level models exist in the literature but few have been validated through experiments. In this paper, we propose a radar cross section-based received power model for an RIS-aided wireless communication system that is rooted in the physical properties of RIS. Our proposed model follows the intuition that the received power is related to the distances from the transmitter/receiver to the RIS, the angles in the TX-RIS-RX triangle, the effective area of each element, and the reflection coefficient of each element. To the best of our knowledge, this paper is the first to model the angle-dependent phase shift of the reflection coefficient, which is typically ignored in existing literature. We further measure the received power with our experimental platform in different scenarios to validate our model. The measurement results show that our model is appropriate both in near field and far field and can characterize the impact of angles well.
We introduce a 2-parameter Hedberg's method, and use it to prove a Hardy-Littlewood-Sobolev theorem for a new type of fractional integral operators, whose kernel has singularity on the light-cone.
In this note, we shall prove an explicit formula on the probability of the level line of the Gaussian Free Field (GFF) with mixed boundary condition terminating at the free boundary, which generalizes the results of GFF with Dirichlet boundary condition.
In this first part of our project, we prove a classical Hardy-Littlewood-Sobolev result for a new family of fractional integral operators whose kernel has singularity appeared on the light cone in R^n+1.
In this note, we study a class of random subsets of positive integers induced by Bernoulli random variables. We obtain sufficient conditions such that the random set is almost surely lacunary, does not have bounded gaps and contains infinitely many arithmetic progressions, respectively.
We study the regularity of Fourier integral operators, by allowing their symbols to satisfy certain multi-parameter characteristics. As a result, we give an extension of Seeger-Sogge-Stein theorem on product spaces.
We study the metric projection onto the closed convex cone in a real Hilbert space $\mathscr{H}$ generated by a sequence $\mathcal{V} = \{v_n\}_{n=0}^\infty$. The first main result of this paper provides a sufficient condition under which we can identify the closed convex cone generated by $\mathcal{V}$ with the following set: \[ \mathcal{C}[[\mathcal{V}]]: = \bigg\{\sum_{n=0}^\infty a_n v_n\Big|a_n\geq 0,\text{ the series }\sum_{n=0}^\infty a_n v_n\text{ converges in $\mathscr{H}$}\bigg\}. \] Then, by adapting classical results on general convex cones, we give a useful description of the metric projection of a vector onto $\mathcal{C}[[\mathcal{V}]]$. As applications, we obtain the best approximations of many concrete functions in $L^2([-1,1])$ by polynomials with non-negative coefficients.
For $-1<α<\infty$, let $ω_α(z)=(1+α)(1-|z|^2)^α$ be the standard weight on the unit disk. In this note, we provide descriptions of the boundedness and compactness for the Toeplitz operators $T_{μ,β}$ between distinct weighted Bergman spaces $L_{a}^{p}(ω_α)$ and $L_{a}^{q}(ω_β)$ when $0<p\leq1$, $q=1$, $-1<α,β<\infty$ and $0<p\leq 1<q<\infty, -1<β\leqα<\infty$, respectively. Our results can be viewed as extensions of Pau and Zhao's work in \cite{Pau}. Moreover, partial of main results are new even in the unweighted settings.
The generalized Lotka-Volterra (gLV) equations are a mathematical proxy for ecological dynamics. We focus on a gLV model of the gut microbiome, in which the evolution of the gut microbial state is determined in part by pairwise inter-species interaction parameters that encode environmentally-mediated resource competition between microbes. We develop an in silico method that controls the steady-state outcome of the system by adjusting these interaction parameters. This approach is confined to a bistable region of the gLV model. The two steady states of interest are idealized as either a "healthy" or "diseased" steady state of the gut microbiome. In this method, a dimensionality reduction technique called steady-state reduction (SSR) is first used to generate a two-dimensional (2D) gLV model that approximates the high-dimensional dynamics on the 2D subspace spanned by the two steady states. Then a bifurcation analysis of the 2D model analytically determines parameter modifications that drive a disease-prone initial condition to the healthy steady state. This parameter modification of the reduced 2D model guides parameter modifications of the original high-dimensional model, resulting in a change of steady-state outcome in the high-dimensional model. This control method, called SPARC (SSR-guided parameter change), bypasses the computational challenge of directly determining parameter modifications in the original high-dimensional system. SPARC could guide the development of indirect bacteriotherapies, which seek to change microbial compositions by deliberately modifying gut environmental variables such as gut acidity or macronutrient availability.
We introduce and give a more or less complete study of a family of branching-Toeplitz operators on the Hilbert space $\ell^2(T_q)$ indexed by a rooted homogeneous tree $T_q$ of degree $q\ge 2$. The finite dimensional analogues of such operators form a very natural family of structured sparse matrices called branching-Toeplitz matrices and will also be investigated. The branching-Toeplitz operators/matrices in this paper should be viewed as natural generalizations of the standard Toeplitz operators/matrices. We will apply our results to construct a family of determinantal point processes on homogeneous trees which are branching-type strong stationary stochastic processes.