This paper addresses the estimation problem of an unknown drift parameter matrix for a fractional Ornstein-Uhlenbeck process in a multi-dimensional setting. To tackle this problem, we propose a novel approach based on rough path theory that allows us to construct pathwise rough path estimators from both continuous and discrete observations of a single path. Our approach is particularly suitable for high-frequency data. To formulate the parameter estimators, we introduce a theory of pathwise Itô integrals with respect to fractional Brownian motion. By establishing the regularity of fractional Ornstein-Uhlenbeck processes and analyzing the long-term behavior of the associated Lévy area processes, we demonstrate that our estimators are strongly consistent and pathwise stable. Our findings offer a new perspective on estimating the drift parameter matrix for fractional Ornstein-Uhlenbeck processes in multi-dimensional settings, and may have practical implications for fields including finance, economics, and engineering.

This paper is the sequel to our study of heat kernel on Ricci shrinkers [29]. In this paper, we improve many estimates in [29] and extend the recent progress of Bamler [2]. In particular, we drop the compactness and curvature boundedness assumptions and show that the theory of $\mathbb{F}$-convergence holds naturally on any Ricci flows induced by Ricci shrinkers.

For a general normed vector space, a special optimal value function called a maximal time function is considered. This covers the farthest distance function as a special case, and has a close relationship with the smallest enclosing ball problem. Some properties of the maximal time function are proven, including the convexity, the lower semicontinuity, and the exact characterizations of its subdifferential formulas.

In this paper, by characterizing Carleson measures, we investigate a class of bounded Toeplitz operator between weighted Bergman spaces with Békollé weights over the half-plane for all index choices.

This paper investigates the relative Kolmogorov $n$-widths of $2\pi$-periodic smooth classes in $\widetilde{L}_{q}$. We estimate the relative widths of $\widetilde{W}^{r} H_{p}^{\omega}$ and its generalized class $K_{p}H^{\omega}(P_{r})$, where $K_{p}H^{\omega}(P_{r})$ is defined by a self-conjugate differential operator $P_{r}(D)$ induced by $P_{r}(t):= t^{\sigma} \Pi_{j=1}^{l}(t^{2}- t_{j}^{2}),~t_{j} > 0,~j=1, 2 ,\cdots, l,~l \geq 1,~\sigma \geq 1,~r=2l+\sigma .$ Also, the modulus of continuity of the $r$-th derivative, or $r$-th self-conjugate differential, does not exceed a given modulus of continuity $\omega$. Then we obtain the asymptotic results, especially for the case $p=\infty , 1\leq q \leq \infty$.

A cautious projection BFGS method is proposed for solving nonconvex unconstrained optimization problems. The global convergence of this method as well as a stronger general convergence result can be proven without a gradient Lipschitz continuity assumption, which is more in line with the actual problems than the existing modified BFGS methods and the traditional BFGS method. Under some additional conditions, the method presented has a superlinear convergence rate, which can be regarded as an extension and supplement of BFGS-type methods with the projection technique. Finally, the effectiveness and application prospects of the proposed method are verified by numerical experiments.

We study the global unique solutions to the 2-D inhomogeneous incompressible MHD equations, with the initial data $(u_{0},B_{0})$ being located in the critical Besov space $\dot{B}_{p,1}^{-1+\frac{2}{p}}(\mathbb{R}^{2}) \,\, (1<p<2)$ and the initial density $\rho_{0}$ being close to a positive constant. By using weighted global estimates, maximal regularity estimates in the Lorentz space for the Stokes system, and the Lagrangian approach, we show that the 2-D MHD equations have a unique global solution.

This paper is devoted to the Cauchy problem for the generalized damped Boussinesq equation with a nonlinear source term in the natural energy space. With the help of linear time-space estimates, we establish the local existence and uniqueness of solutions by means of the contraction mapping principle. The global existence and blow-up of the solutions at both subcritical and critical initial energy levels are obtained. Moreover, we construct the sufficient conditions of finite time blow-up of the solutions with arbitrary positive initial energy.

In this paper, we reconstruct strongly-decaying block sparse signals by the block generalized orthogonal matching pursuit (BgOMP) algorithm in the $l_2$-bounded noise case. Under some restraints on the minimum magnitude of the nonzero elements of the strongly-decaying block sparse signal, if the sensing matrix satisfies the the block restricted isometry property (block-RIP), then arbitrary strongly-decaying block sparse signals can be accurately and steadily reconstructed by the BgOMP algorithm in iterations. Furthermore, we conjecture that this condition is sharp.

Evolution and interaction of plane waves of the multidimensional zero-pressure gas dynamics system leads to the study of the corresponding one dimensional system. In this paper, we study the initial value problem for one dimensional zero-pressure gas dynamics system. Here the first equation is the Burgers equation and the second one is the continuity equation. We consider the solution with initial data in the space of bounded Borel measures. First we prove a general existence result in the algebra of generalized functions of Colombeau. Then we study in detail special solutions with $\delta$-measures as initial data. We study interaction of waves originating from initial data concentrated on two point sources and interaction with classical shock/rarefaction waves. This gives an understanding of plane-wave interactions in the multidimensional case. We use the vanishing viscosity method in our analysis as this gives the physical solution.

In this paper, conjugate $k$-holomorphic functions and generalized $k$-holomorphic functions are defined in the two-dimensional complex space, and the corresponding Riemann boundary value problems and the inverse problems are discussed on generalized bicylinders. By the characteristics of the corresponding functions and boundary properties of the Cauchy type singular integral operators with conjugate $k$-holomorphic kernels, the general solutions and special solutions of the corresponding boundary value problems are studied in a detailed fashion, and the integral expressions of the solutions are obtained.

For any $s\in(0,1)$, let the nonlocal Sobolev space $X^s(\mathbb{R} ^N)$ be the linear space of Lebesgue measure functions from $\mathbb{R} ^N$ to $\mathbb{R} $ such that any function $u$ in $X^s(\mathbb{R} ^N)$ belongs to $L^2(\mathbb{R} ^N)$ and the function $(x,y)\longmapsto\big(u(x)-u(y)\big)\sqrt{K(x-y)}$ is in $L^2(\mathbb{R} ^N,\mathbb{R} ^N)$. First, we show, for a coercive function $V(x)$, the subspace $E:=\bigg\{u\in X^s(\mathbb{R} ^N):\int_{\mathbb{R} ^N}V(x)u^2{\rm d}x<+\infty\bigg\}$ of $X^s(\mathbb{R} ^N)$ is embedded compactly into $L^p(\mathbb{R}^N)$ for $p\in[2,2_s^*)$, where $2_s^*$ is the fractional Sobolev critical exponent. In terms of applications, the existence of a least energy sign-changing solution and infinitely many sign-changing solutions of the nonlocal Schrödinger equation $-{\mathcal{L}_K}u+V(x)u=f(x,u),\ x\in\ \mathbb{R} ^N$ are obtained, where $-{\mathcal{L}_K}$ is an integro-differential operator and $V$ is coercive at infinity.

This paper is concerned with the Cauchy problem for a 3D fluid-particle interaction model in the so-called flowing regime in $\mathbb{R}^{3}$. Under the smallness assumption on both the external potential and the initial perturbation of the stationary solution in some Sobolev spaces, the existence and uniqueness of global smooth solutions in $H^{3}$ of the system are established by using the careful energy method.

In the present paper, we prove the existence, non-existence and multiplicity of positive normalized solutions $(\lambda_c, u_c)\in \mathbb{R}\times H^1(\mathbb{R}^N)$ to the general Kirchhoff problem $-M\left(\int_{\mathbb{R} ^N}|\nabla u|^2 {\rm d}x\right)\Delta u +\lambda u=g(u) \hbox{in} \mathbb{R} ^N, u\in H^1(\mathbb{R} ^N),N\geq 1,$ satisfying the normalization constraint $ \int_{\mathbb{R}^N}u^2{\rm d}x=c, $ where $M\in C([0,\infty))$ is a given function satisfying some suitable assumptions. Our argument is not by the classical variational method, but by a global branch approach developed by Jeanjean \textit{et al}. [J Math Pures Appl, 2024, 183: 44-75] and a direct correspondence, so we can handle in a unified way the nonlinearities $g(s)$, which are either mass subcritical, mass critical or mass supercritical.

We study equations in divergence form with piecewise $C^{\alpha }$ coefficients. The domains contain corners and the discontinuity surfaces are attached to the edges of the corners. We obtain piecewise $C^{1,\alpha }$ estimates across the discontinuity surfaces and provide an example to illustrate the issue regarding the regularity at the corners.

Let $\mu$ be a positive Borel measure on the interval $[0,1)$. The Hankel matrix $\mathcal{H}_{\mu}=(\mu_{n,k})_{n,k\geq 0}$ with entries $\mu_{n,k}=\mu_{n+k}$, where $\mu_{n}=\int_{[0,1)}t^n{\rm d}\mu(t)$, induces, formally, the operator $\mathcal{DH}_\mu(f)(z)=\sum\limits_{n=0}^\infty\left(\sum\limits_{k=0}^\infty \mu_{n,k}a_k\right)(n+1)z^n , z\in \mathbb{D},$ where $f(z)=\sum\limits_{n=0}^\infty a_nz^n$ is an analytic function in $\mathbb{D}$. We characterize the measures $\mu$ for which $\mathcal{DH}_\mu$ is bounded (resp., compact) operator from the logarithmic Bloch space $\mathscr{B}_{L^{\alpha}}$ into the Bergman space $\mathcal{A}^p$, where $0\leq\alpha<\infty,0<p<\infty$. We also characterize the measures $\mu$ for which $\mathcal{DH}_\mu$ is bounded (resp., compact) operator from the logarithmic Bloch space $\mathscr{B}_{L^{\alpha}}$ into the classical Bloch space $\mathscr{B}$.

In this study, we derive the sharp bounds of certain Toeplitz determinants whose entries are the coefficients of holomorphic functions belonging to a class defined on the unit disk $\mathbb{U}$. Furthermore, these results are extended to a class of holomorphic functions on the unit ball in a complex Banach space and on the unit polydisc in $\mathbb{C}^n$. The obtained results provide the bounds of Toeplitz determinants in higher dimensions for various subclasses of normalized univalent functions.

In this paper, $X$ is a locally compact Hausdorff space and ${\mathcal A}$ is a Banach algebra. First, we study some basic features of $C_0(X,\mathcal A)$ related to $\rm BSE$ concept, which are gotten from ${\mathcal A}$. In particular, we prove that if $C_0(X,\mathcal A)$ has the $\rm BSE$ property then $\mathcal A$ has so. We also establish the converse of this result, whenever $X$ is discrete and $\mathcal A$ has the BSE-norm property. Furthermore, we prove the same result for the $\rm BSE$ property of type I. Finally, we prove that $C_0(X,{\mathcal A})$ has the BSE-norm property if and only if $\mathcal A$ has so.

In this paper, we consider pseudoharmonic heat flow with small initial horizontal energy and give the existence of pseudoharmonic maps from closed pseudo-Hermitian manifolds into closed Riemannian manifolds.

We consider the singular Dirichlet problem for the Monge-Ampère type equation ${\rm det} \ D^2 u=b(x)g(-u)(1+|\nabla u|^2)^{q/2}, \ u<0, \ x \in \Omega, \ u|_{\partial \Omega}=0,$ where $\Omega$ is a strictly convex and bounded smooth domain in $\mathbb R^n$, $q\in [0, n+1)$, $g\in C^\infty(0,\infty)$ is positive and strictly decreasing in $(0, \infty)$ with $\lim\limits_{s \rightarrow 0^+}g(s)=\infty$, and $b \in C^{\infty}(\Omega)$ is positive in $\Omega$. We obtain the existence, nonexistence and global asymptotic behavior of the convex solution to such a problem for more general $b$ and $g$. Our approach is based on the Karamata regular variation theory and the construction of suitable sub-and super-solutions.

In this paper, we mainly discuss a discrete estimation of the average differential entropy for a continuous time-stationary ergodic space-time random field. By estimating the probability value of a time-stationary random field in a small range, we give an entropy estimation and obtain the average entropy estimation formula in a certain bounded space region. It can be proven that the estimation of the average differential entropy converges to the theoretical value with a probability of 1. In addition, we also conducted numerical experiments for different parameters to verify the convergence result obtained in the theoretical proofs.

In this paper, we consider a model of compressible isentropic two-fluid magnetohydrodynamics without resistivity in a strip domain in three dimensional space. By exploiting the two-tier energy method developed in [Anal PDE, 2013, 6: 1429-1533], we prove the global well-posedness of the governing model around a uniform magnetic field which is non-parallel to the horizontal boundary. Moreover, we show that the solution converges to the steady state at an almost exponential rate as time goes to infinity. Compared to the work of Tan and Wang [SIAM J Math Anal, 2018, 50: 1432-1470], we need to overcome the difficulties caused by particles.

In this article, we consider the diffusion equation with multi-term time-fractional derivatives. We first derive, by a subordination principle for the solution, that the solution is positive when the initial value is non-negative. As an application, we prove the uniqueness of solution to an inverse problem of determination of the temporally varying source term by integral type information in a subdomain. Finally, several numerical experiments are presented to show the accuracy and efficiency of the algorithm.

Infinite matrix theory is an important branch of function analysis. Every linear operator on a complex separable infinite dimensional Hilbert space corresponds to an infinite matrix with respect a orthonormal base of the space, but not every infinite matrix corresponds to an operator. The classical Schur test provides an elegant and useful criterion for the boundedness of linear operators, which is considered a respectable mathematical accomplishment. In this paper, we prove the compact version of the Schur test. Moreover, we provide the Schur test for the Schatten class $S_{2}$. It is worth noting that our main results can be applicable to the general matrix without limitation on non-negative numbers. We finally provide the Schur test for compact operators from $l_{p}$ into $l_{q}$.

Let $\{Z_n\}_{n\geq 0 }$ be a critical or subcritical $d$-dimensional branching random walk started from a Poisson random measure whose intensity measure is the Lebesugue measure on $\mathbb{R}^d$. Denote by $R_n:=\sup\{u>0:Z_n(\{x\in\mathbb{R}^d:|x|<u\})=0\}$ the radius of the largest empty ball centered at the origin of $Z_n$. In this work, we prove that after suitable renormalization, $R_n$ converges in law to some non-degenerate distribution as $n\to\infty$. Furthermore, our work shows that the renormalization scales depend on the offspring law and the dimension of the branching random walk. This completes the results of Révész [13] for the critical binary branching Wiener process.