Journal of Mathematical Study (JMS) is a comprehensive mathematical journal published jointly by Global Science Press and Xiamen University. It publishes original research and survey papers, in English, of high scientific value in all major fields of mathematics, including pure mathematics, applied mathematics, operational research, and computational mathematics. JMS is edited by a group of leading researchers and aims to become a premier mathematical journal.
A Note on the Global Existence for the Strong Solution of the Stochastic Three-Dimensional Axially Symmetric Euler Equations
This paper establishes the global existence of the pathwise solutions for the stochastic 3-D incompressible Euler equations with the axially symmetric initial data and axially symmetric additive noise.
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Generalized Estimation of Numerical Radius for the off-Diagonal 2×2 Operator Matrices and Its Application
We give some generalized upper bounds for the numerical radius of off-diagonal 2×2 operator matrices. These inequalities are mainly based on the extension Buzano inequality and the generalized Young inequality. And our bounds refine and generalize the existing related upper bounds. Moreover, the conclusion is applied to the non-monic operator polynomials and gives a new bound for the eigenvalues of these operator polynomials.
A Regularity Result for the Incompressible Elastodynamics with a Free Interface
We consider the incompressible inviscid elastodynamics equations with a free surface and establish the regularity of solutions for elastic system. Compared with the previous result on this free boundary problem [ Gu X and Wang F, Well-posedness of the free boundary problem in incompressible elastodynamics under the mixed type stability condition, J. Math. Anal. Appl., 2020, 482(1): 123529] in space $H^3,$ we are able to establish the regularity in space $H^{2.5+δ}.$ It is achieved by reformulating the system into the Lagrangian formulation, presenting the uniform estimates for the pressure, the tangential estimates for the system, as well as the divergence and curl estimates.
On a Class of Degenerate Nonlinear Elliptic Equations in Weighted Sobolev Spaces
In this paper we are interested in the existence of solutions for Dirichlet problem associated to the degenerate nonlinear elliptic equations in the setting of the weighted Sobolev spaces $W^{1,p}_ 0 (Ω,ω).$
Common Fixed Point Results for a Pair of Generalized Nonlinear Mappings in Complete Metric Spaces
In this paper, under the asymptotically regular condition, we investigate the existence of a unique common fixed point for a pair of generalized nonlinear mappings in the framework of complete metric spaces. Additionally, we extend our analysis to the case involving two control functions. Our work generalizes some results in recent papers.
Some $p$-Adic Hardy Operators and Their Commutators
In this paper, we will study the sharp estimates of $p$-adic weighted Hardy operator on central and noncentral weighted $p$-adic Morrey spaces. Moreover, we can obtain the sharp bound of generalized $p$-adic Hardy operator. In addition, the commutator that is generated by the generalized $p$-adic Hardy operator and the central BMO function is also bounded on $p$-adic Morrey spaces.
Non-Traveling Wave Solutions of (3+1)-Dimensional Variable Coefficients BLMP Equation with a Compound Technology
We investigate non-traveling wave solutions of the (3+1)-dimensional Boiti-Leon-Manna-Pempinelli equation with time-dependent coefficients, which describes the propagation of waves in incompressible fluids. We creatively combine the extended three-wave method with the generalized variable separation method and successfully obtain sixty exact non-traveling solutions including kink-like solutions, singular solitary wave-like solutions, periodic solitary wave-like solutions, periodic kink-like solutions, periodic cross-kink-like waves, homoclinic breather wave-like solutions and so on. The variable coefficients and arbitrary functions in the obtained solutions are easy to exhibit abundant soliton structures, which may be of great significance for explaining some practical physical phenomena. By contour plots, 2D plots, and 3D plots, we analyze the dynamic characteristics of periodic cross-kink-like solution, singular solitary wave-like solution, homoclinic breather wave-like solution. Additionally, we show changes of solutions under different tails to illustrate the influence of tails on solutions.
A Note on the Determinant of a Special Class of $Q$-Walk Matrices
For a graph $G$ of order $n,$ its $Q$-walk matrix is defined by $W_Q(G) = [e,Qe,···,Q^{n−1}e],$ where $Q$ is the signless Laplacian matrix of $G$ and $e$ denotes the all-one column vector. Let $G \circ P_k$ represent the rooted product graph of $G$ and a path $P_k.$ In this note, we establish the relationship between determinants of $W_Q(G)$ and $W_Q(G \circ P_k )$ for $k=2,3.$
Characterizations of Umbilical Hypersurfaces by Partially Overdetermined Problems in Space Forms
In this paper, we characterize the rigidity of umbilical hypersurfaces by a Serrin-type partially overdetermined problem in space forms, which generalizes the similar results in Euclidean half-space and Euclidean half-ball. Guo-Xia first obtained these rigidity results when the Robin boundary condition on the support hypersurface is homogeneous, at this time the target umbilical hypersurface has orthogonal contact angle with the support. However, in this paper we can obtain any contact angle $θ ∈ (0,π)$ by changing the Robin boundary condition to be inhomogeneous.
An Alternative Proof for the Upper Bound of Curvature Integral on Manifolds with Lower Sectional Curvature Bound
Petrunin proved that the integral of scalar curvature in a unit ball is bounded from above in terms of the dimension of the manifold and the lower bound of the sectional curvature. In this paper, we give an alternative proof for this result. The main difference between this proof and Petrunin’s original proof is that we construct a stratified finite covering and apply it directly to the given manifold, rather than arguing by contradiction for a sequence of manifolds, which satisfy some technical lifting properties.
On Conical Asymptotically Flat Manifolds
We prove a conjecture of Petrunin and Tuschmann on the non-existence of asymptotically flat 4-manifolds asymptotic to the half plane. We also survey recent progress and questions concerning gravitational instantons, which serve as our motivation for studying this question.
Moduli Spaces of Metrics with Lower or Pinched Curvature Bounds and Their Compactifications
We present a metric approach to the study of moduli spaces of metrics with certain curvature bounds and suitable other geometric constraints, and their compactifications. This is accompanied by a deeper discussion of related results, questions and problems in the realm of positive and positively pinched sectional as well as Ricci curvature. Regarding the latter two topics, we place special emphasis on corresponding works and contributions of Rong X and his collaborators.
Rigidity for Einstein Manifolds under Bounded Covering Geometry
In this note, we prove three rigidity results for Einstein manifolds with bounded covering geometry. (1) An almost flat manifold $(M,g)$ must be flat if it is Einstein, i.e. ${\rm Ric}_g =λg$ for some real number $λ.$ (2) A compact Einstein manifold with a non-vanishing and almost maximal volume entropy is hyperbolic. (3) A compact Einstein manifold admitting a uniform local rewinding almost maximal volume is isometric to a space form.
Pair-of-Pants Decompositions of 4-Manifolds Diffeomorphic to General Type Hypersurfaces
In this paper, we show that a smooth 4-manifold diffeomorphic to a complex hypersurface in $\mathbb{CP}^3$ of degree $d≥5$ can be decomposed as the union of $d(d−4)^2$ copies of 4-dimensional pair-of-pants and certain subsets of ${\rm K3}$ surfaces.
Volume Gap Theorems for Ricci Nonnegative Metrics and Einstein Metrics
In this note, we prove some gap theorems of asymptotic volume ratio for Ricci nonnegative metrics, and gap theorems of volume for Einstein metrics.
Affine Isoperimetric Type Inequalities for Static Convex Domains in Hyperbolic Space
In this paper, the notion of hyperbolic ellipsoids in hyperbolic space is introduced. Using a natural orthogonal projection from hyperbolic space to Euclidean space, we establish affine isoperimetric type inequalities for static convex domains in hyperbolic space. Moreover, equality of such inequalities is characterized by these hyperbolic ellipsoids.
Monotone Sequences of Metric Spaces with Compact Limits
In this paper, we consider a fixed metric space (possibly an oriented Riemannian manifold with boundary) with an increasing sequence of distance functions and a uniform upper bound on diameter. When the fixed space endowed with the point-wise limit of these distances is compact, then there is uniform and Gromov-Hausdorff (GH) convergence to this space. When the fixed metric space also has an integral current structure of uniformly bounded total mass (as is true for an oriented Riemannian manifold with boundary that has a uniform bound on total volume), we prove volume preserving intrinsic flat convergence to a subset of the GH limit whose closure is the whole GH limit. We provide a review of all notions and have a list of open questions at the end.
In this survey paper, we discuss various examples of Ricci solitons and their constructions. Some open questions related to the rigidity and classification of Ricci solitons will be also discussed through those examples.
Ricci Curvature and Fundamental Groups of Effective Regular Sets
For a Gromov-Hausdorff convergent sequence of closed manifolds $M^n_i\xrightarrow{GH} X$ with ${\rm Ric}\ge -(n-1)$, ${\rm diam}(M_i) ≤ D,$ and ${\rm vol}(M_i) ≥ v > 0,$ we study the relation between $π_1(M_i)$ and $X.$ It was known before that there is a surjective homomorphism $ϕ_i : π_1(M_i) → π_1(X)$ by the work of Pan-Wei. In this paper, we construct a surjective homomorphism from the interior of the effective regular set in $X$ back to $M_i,$ that is, $ψ_i :π_1(\mathcal{R}^◦_{ϵ,δ} )→π_1(M_i).$ These surjective homomorphisms $ϕ_i$ and $ψ_i$ are natural in the sense that their composition $ϕ_i◦ψ_i$ is exactly the homomorphism induced by the inclusion map $R^◦_{ ϵ,δ}\hookrightarrow X.$
Quasi-Convex Subsets and the Farthest Direction in Alexandrov Spaces with Lower Curvature Bound
Let $F$ be a closed subset in a finite dimensional Alexandrov space $X$ with lower curvature bound. This paper shows that $F$ is quasi-convex if and only if, for any two distinct points $p,r∈F,$ if there is a direction at $p$ which is more than $\frac{π}{2}$ away from $⇑^r_p$ (the set of all directions from $p$ to $r$), then the farthest direction to $⇑^r_p$ at $p$ is tangent to $F.$ This implies that $F$ is quasi-convex if and only if the gradient curve starting from $r$ of the distance function to $p$ lies in $F.$ As an application, we obtain that the fixed point set of an isometry on $X$ is quasi-convex.
$L^4$-Bound of the Transverse Ricci Curvature under the Sasaki-Ricci Flow
In this paper, we show that the uniform $L^4$-bound of the transverse Ricci curvature along the Sasaki-Ricci flow on a compact quasi-regular transverse Fano Sasakian $(2n+1)$-manifold $M.$ Then we are able to study the structure of the limit space. As consequences, when $M$ is of dimension five and the space of leaves of the characteristic foliation is of type I, any solution of the Sasaki-Ricci flow converges in the Cheeger-Gromov sense to the unique singular orbifold Sasaki-Ricci soliton and is trivial one if $M$ is transverse $K$-stable. Note that when the characteristic foliation is of type II, the same estimates hold along the conic Sasaki-Ricci flow.
Preface: Special Issues in Honor of Professors Xiaochun Rong
A Brief Introduction on Prof. Rong’s Mathematical Work
Xiaochun Rong was born on June 17th, 1954. He graduated from Beijing Normal College (now Capital Normal University) in 1982, and received his Ph.D. from Stony Brook University in 1990. Rong was a Sloan Research Fellow in 1996-1998, and an NSF research grant awardee in 1994-2013. He has been a distinguished professor at Rutgers University in New Brunswick since 2008. Rong was a 2002 ICM invited speaker, and a 2017 Fellow of the American Mathematical Society.
Analytic Insights into an Adapted Algorithm for the Score-Based Secretary Problem
In this paper, we study some basic analytic properties of a sequence of functions $\{S^{\mu,σ}_n\}$ that is directly derived in an adaptive algorithm originating from the classical score-based secretary problem. More specifically, we show that: 1. the uniqueness of maximum points of the function sequence $\{S^{\mu,σ}_n\};$ 2. the maximum point sequence of $\{S^{\mu,σ}_n\}$ monotone increases to infinity as $n$ tends to infinity. All of the proofs are elementary but nontrivial.
On the Computational Problems of Upper Convex Densities for Self-Similar Sets with the Open Set Condition
Let $E$ be a self-similar set satisfying the open set condition. Zhou and Feng posed an open problem in 2004 as follows: let $x ∈ E,$ under what conditions is there a set $U_x$ containing $x$ with $|U_x| > 0$ such that $\overline{D}^s_C(E,x) = \frac{\mathcal{H}^s(E \cap U_x)}{|U_x|^s}?$ The aim of this paper is to present a solution of this problem. Under the assumption that there exists a nonempty convex open set $V$ containing $E$ and satisfying the requirement of the open set condition, it is proved that if $x ∈ E$ and the upper convex density of $E$ at $x$ equals 1, then there exists a convex set $U_x$ containing $x$ with $|U_x| > 0$ such that $\overline{D}^s_C(E,x) =\frac{\mathcal{H}^s (E∩U_x)}{|U_x|^s}.$ Finally, as an application of this result, an equivalent condition for $E_0 = E$ is given, where $E_0 =\{x∈E|\overline{D}^s_C(E,x)=1\}.$
Normalized Solutions for a Kirchhoff Equation with Potential in $\mathbb{R}^3$
In this paper, for given mass $c>0,$ we study the existence of normalized solutions to the following nonlinear Kirchhoff equation $$\begin{cases} (a+b\int_{\mathbb{R}^3}[|\nabla u|^2+V(x)u^2]dx)[-\Delta u+V(x)u]=\lambda u+\mu|u|^{q-2}u+|u|^{p-2}u, \ \ \ {\rm in}\ \ \mathbb{R}^3, \\ \int_{\mathbb{R}^3}|u|^2dx=c^2, \end{cases}$$where $a>0, b>0, λ∈\mathbb{R},$ $5<q< p<6,$ $\mu>0$ and $V$ is a continuous non-positive function vanishing at infinity. Under some mild assumptions on $V,$ we prove the existence of a mountain pass normalized solution via the minimax principle.
Fitting Heavy-Tailed Distributions to Mortality Indexes for Longevity Risk Forecasts
Modeling and predicting mortality rates are crucial for managing and mitigating longevity risk in pension funds. To address the impacts of extreme mortality events in forecasting, researchers suggest directly fitting a heavy-tailed distribution to the residuals in modeling mortality indexes. Since the true mortality indexes are unobserved, this fitting relies on the estimated mortality indexes containing measurement errors, leading to estimation biases in standard inferences within the actuarial literature. In this paper, the empirical characteristic function (ECF) technique is employed to fit heavy-tailed distributions to the time series residuals of mortality indexes and normal distributions to the measurement errors. Through a simulation study, we empirically validate the consistency of our proposed method and demonstrate the importance and challenges associated with making inferences in the presence of measurement errors. Upon analyzing publicly available mortality datasets, we observe instances where mortality indexes may follow highly heavy-tailed distributions, even exhibiting an infinite mean. This complexity adds a layer of difficulty to the statistical inference for mortality models.
Endpoint Estimates for an Oscillatory Multiplier Associated with Wave Equations on the Torus
In this paper, we establish the endpoint estimate for an oscillatory multiplier associated with wave equations on the torus, which extends the results of Fan and Sun. In addition, we obtain a more general result for sublinear operators on compact measure spaces.
B-Statistical Uniform Integrability and $\mathfrak{L}_p$ Convergence
In this paper, we first give the concept of B-statistical uniform integrability with respect to Cesàro matrix (B-CUI), which is weaker than Cesàro uniform integrability. Then we establish some B-statistical convergence theorems for random sequence under the condition of B-CUI, which generalizes the outcomes of some known results. Finally, for some special kinds of triangular array or pairwise independent sequences of random variables, similar results are also derived without conditions of B-CUI.
The Category Types of Extensions of a Tambara-Yamagami Category
In this paper, we study the group extension of a Tambara-Yamagami category which has Frobenius-Perron dimension $2pq,$ where $p,q$ are prime numbers. We prove that there are two possible category types when $p\ne q,$ and five possible category types when $p=q.$