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Bobo Hua

Publications and source records attributed to Bobo Hua.

At least 55 records · Page 3Linked to original sources
arXiv2023

On the complexity of Cayley graphs on a dihedral group

In this paper, we investigate the complexity of an infinite family of Cayley graphs $\mathcal{D}_{n}=Cay(\mathbb{D}_{n}, b^{\pmβ_1},b^{\pmβ_2},\ldots,b^{\pmβ_s}, a b^{γ_1}, a b^{γ_2},\ldots, a b^{γ_t} )$ on the dihedral group $\mathbb{D}_{n}=\langle a,b| a^2=1, b^n=1,(a\,b)^2=1\rangle$ of order $2n.$ We obtain a closed formula for the number $τ(n)$ of spanning trees in $\mathcal{D}_{n}$ in terms of Chebyshev polynomials, investigate some arithmetical properties of this function, and find its asymptotics as $n\to\infty.$ Moreover, we show that the generating function $F(x)=\sum\limits_{n=1}^\inftyτ(n)x^n$ is a rational function with integer coefficients.

math.CO↗
arXiv2023

Uniqueness class of solutions to a class of linear evolution equations

In this paper, we study the wave equation on infinite graphs. On one hand, in contrast to the wave equation on manifolds, we construct an example for the non-uniqueness for the Cauchy problem of the wave equation on graphs. On the other hand, we obtain a sharp uniqueness class for the solutions of the wave equation. The result follows from the time analyticity of the solutions to the wave equation in the uniqueness class. In the last part, we extend the result to a wide class of linear evolution equations.

math.AP↗
arXiv2023

The existence of ground state solutions for nonlinear p-Laplacian equations on lattice graphs

In this paper, we study the nonlinear $p$-Laplacian equation $$-Δ_{p} u+V(x)|u|^{p-2}u=f(x,u) $$ with positive and periodic potential $V$ on the lattice graph $\mathbb{Z}^{N}$, where $Δ_{p}$ is the discrete $p$-Laplacian, $p \in (1,\infty)$. The nonlinearity $f$ is also periodic in $x$ and satisfies the growth condition $|f(x,u)| \leq a(1+|u|^{q-1})$ for some $ q>p$. We first prove the equivalence of three function spaces on $\mathbb{Z}^{N}$, which is quite different from the continuous case and allows us to remove the restriction $q>p^{*}$ in [SW10], where $p^{*}$ is the critical exponent for $ W^{1,p}(Ω) \hookrightarrow L^{q}(Ω)$ with $Ω\subset \mathbb{R}^{N}$ bounded. Then, using the method of Nehari [Neh60, Neh61], we prove the existence of ground state solutions to the above equation.

math.AP↗
arXiv2023

Liouville theorems for ancient solutions of subexponential growth to the heat equation on graphs

Mosconi proved Liouville theorems for ancient solutions of subexponential growth to the heat equation on a manifold with Ricci curvature bounded below. We extend these results to graphs with bounded geometry: for a graph with bounded geometry, any nonnegative ancient solution of subexponential growth in space and time to the heat equation is stationary, and thus is a harmonic solution.

math.DG↗
arXiv2023

A combinatorial curvature flow in spherical background geometry

In [12], the existence of ideal circle patterns in Euclidean or hyperbolic background geometry under the combinatorial conditions was proved using flow approaches. It remains as an open problem for the spherical case. In this paper, we introduce a combinatorial geodesic curvature flow in spherical background geometry, which is analogous to the combinatorial Ricci flow of Chow and Luo in [4]. We characterize the sufficient and necessary condition for the convergence of the flow. That is, the prescribed geodesic curvature satisfies certain geometric and combinatorial condition if and only if for any initial data the flow converges exponentially fast to a circle pattern with given total geodesic curvature on each circle. Our result could be regarded as a resolution of the problem in the spherical case. As far as we know, this is the first combinatorial curvature flow in spherical background geometry with fine properties, and it provides an algorithm to find the desired ideal circle pattern.

math.GT↗
arXiv2022

Hyperbolic polyhedral surfaces with regular faces

We study hyperbolic polyhedral surfaces with faces isometric to regular hyperbolic polygons satisfying that the total angles at vertices are at least $2π.$ The combinatorial information of these surfaces is shown to be identified with that of Euclidean polyhedral surfaces with negative combinatorial curvature everywhere. We prove that there is a gap between areas of non-smooth hyperbolic polyhedral surfaces and the area of smooth hyperbolic surfaces. The numerical result for the gap is obtained for hyperbolic polyhedral surfaces, homeomorphic to the double torus, whose 1-skeletons are cubic graphs.

math.MG↗
arXiv2022

Steklov flows on trees and applications

We introduce the Steklov flows on finite trees, i.e. the flows (or currents) associated with the Steklov problem. By constructing appropriate Steklov flows, we prove the monotonicity and rigidity of the first nonzero Steklov eigenvalues on trees: for finite trees $\g_1$ and $\g_2,$ the first nonzero Steklov eigenvalue of $\g_1$ is greater than or equal to that of $\g_2$, provided that $\g_1$ is a subgraph of $\g_2.$ Moreover, we give the sufficient and necessary condition in which the equality holds.

math.SP↗
arXiv2022

Discrete Schwarz rearrangement in lattice graphs

In this paper, we prove a discrete version of the generalized Riesz inequality on $\mathbb{Z}^d$. As a consequence, we will derive the extended Hardy-Littlewood and Pólya-Szegö inequalities. We will also establish cases of equality in the latter. Our approach is totally novel and self-contained. In particular, we invented a definition for the discrete rearrangement in higher dimensions. Moreover, we show that the definition "suggested" by Pruss does not work. We solve a long-standing open question raised by Alexander Pruss in [Pru98, p494], Duke Math Journal, and discussed with him in several communications in 2009-2010, [Pru10]. Our method also provides a line of attack to prove other discrete rearrangement inequalities and opens the door to the establishment of optimizers of many important discrete functional inequalities in $\mathbb{Z}^d,$ $d\geq2$. We will also discuss some applications of our findings. To the best of our knowledge, our results are the first ones in the literature dealing with discrete rearrangement on $\mathbb{Z}^d,$ $d\geq2$.

math.AP↗
arXiv2022

Gap theorems for ends of smooth metric measure spaces

In this paper, we establish two gap theorems for ends of smooth metric measure space $(M^n, g,e^{-f}dv)$ with the Bakry-Émery Ricci tensor $\mathrm{Ric}_f\ge-(n-1)$ in a geodesic ball $B_o(R)$ with radius $R$ and center $o\in M^n$. When $\mathrm{Ric}_f\ge 0$ and $f$ has some degeneration outside $B_o(R)$, we show that there exists an $ε=ε(n,\sup_{B_o(1)}|f|)$ such that such a space has at most two ends if $R\leε$. When $\mathrm{Ric}_f\ge\frac 12$ and $f(x)\le\frac 14d^2(x,B_o(R))+c$ for some constant $c>0$ outside $B_o(R)$, we can also get the same gap conclusion.

math.DG↗
arXiv2022

Extremal functions for the second-order Sobolev inequality on groups of polynomial growth

In this paper, we prove the second-order Sobolev inequalities on Cayley graphs of groups of polynomial growth. We use the discrete Concentration-Compactness principle to prove the existence of extremal functions for best constants in supercritical cases. As applications, we get the existence of positive ground state solutions to the $p$-biharmonic equations and the Lane-Emden systems.

math.AP↗
arXiv2022

Upper bounds for the Steklov eigenvalues on trees

In this paper, we study the upper bounds for discrete Steklov eigenvalues on trees via geometric quantities. For a finite tree, we prove sharp upper bounds for the first nonzero Steklov eigenvalue by the reciprocal of the size of the boundary and the diameter respectively. We also prove similar estimates for higher order Steklov eigenvalues.

math.SP↗
arXiv2021

Graphs with positive spectrum

In this paper, we prove sharp $\ell^2$ decay estimates of nonnegative generalized subharmonic functions on graphs with positive Laplacian spectrum, which extends the result by Li and Wang on Riemannian manifolds.

math.DG↗
arXiv2021

On $L^p$ Liouville theorems for Dirichlet forms

We study harmonic functions for general Dirichlet forms. First we review consequences of Fukushima's ergodic theorem for the harmonic functions in the domain of the $ L^{p} $ generator. Secondly we prove analogues of Yau's and Karp's Liouville theorems for weakly harmonic functions. Both say that weakly harmonic functions which satisfy certain $ L^{p} $ growth criteria must be constant. As consequence we give an integral criterion for recurrence.

math.FA↗
arXiv2021

Existence of ground state solutions to some Nonlinear Schrödinger equations on lattice graphs

In this paper, we study the nonlinear Schrödinger equation $ -Δu+V(x)u=f(x,u) $on the lattice graph $ \mathbb{Z}^{N}$. Using the Nehari method, we prove that when $f$ satisfies some growth conditions and the potential function $V$ is periodic or bounded, the above equation admits a ground state solution. Moreover, we extend our results from $\mathbb{Z}^{N}$ to quasi-transitive graphs.

math.AP↗
arXiv2021

Every Salami has two ends

A salami is a connected, locally finite, weighted graph with non-negative Ollivier Ricci curvature and at least two ends of infinite volume. We show that every salami has exactly two ends and no vertices with positive curvature. We moreover show that every salami is recurrent and admits harmonic functions with constant gradient. The proofs are based on extremal Lipschitz extensions, a variational principle and the study of harmonic functions. Assuming a lower bound on the edge weight, we prove that salamis are quasi-isometric to the line, that the space of all harmonic functions has finite dimension, and that the space of subexponentially growing harmonic functions is two-dimensional. Moreover, we give a Cheng-Yau gradient estimate for harmonic functions on balls.

math.DG↗