In this paper, we establish a supercongruence:$$\sum\limits_{k=1}^{n-1}\frac{x^k}{[k]_q}≡[n]_q\sum\limits_{k=1}^{n−1}\frac{x^k}{[k]_q}H_{k−1}(q)+\frac{1−(x;q)_n+q^{\Bigg( \begin{matrix} n \\ 2 \end{matrix} \Bigg)}(−x)^n}{[n]_q}\\({\rm mod} \ Φ_n(q)^2),$$where $n$ is a positive integer and $x$ is a variable. We also get some interesting $q$-congruences involving $q$-harmonic numbers.
The generalized Gabor transform (GGT) associated with the Jacobi-Cherednik operator is a novel addition to the class of Gabor transforms. Knowing the fact that the study of the time-frequency analysis is both theoretically interesting and practically useful, the aim of this article is to explore the main theorems of harmonic analysis for this novel transformation. The second aim is to study some quantitative uncertainty principles associated with the proposed transform.
This paper is concerned with a generalized Hilbert tensor which is defined by a generating vector. The necessary and sufficient conditions are derived for the positive semi-definiteness/definiteness of the even $m$-order $n$-dimensional generalized Hilbert tensors. Furthermore, the conditions for the positive definiteness of the $k$-th differential of multivariate homogeneous polynomial for Hilbert tensor are presented.
Let $\mathcal{G}$ be a stratified Lie group and $\Delta = -\sum_{j=1}^{n_1} X_j^2$ be the sub-Laplacian of $\mathcal{G}$, where $\{X_j\}_{1 \leq j \leq n_1}$ is a basis of the left-invariant vector fields of degree one. Given that $0 < \alpha < 1$, this paper studied the operators $X_j \Delta^{-\frac{1+\alpha}{2}}$ for $j=1,\dots,n_1$ and established their uniform endpoint estimates from $L^\infty(\mathcal{G})$ to $\mathrm{BMO}_\alpha(\mathcal{G})$, from $L^{\frac{Q}{\alpha}}(\mathcal{G})$ to $\mathrm{BMO}(\mathcal{G})$ and from $L_{\alpha}^{\frac{Q}{\alpha}}(\mathcal{G})$ to $\mathrm{BMO}_\alpha(\mathcal{G})$. This paper also considered the uniform endpoint estimates for commutators $[b, X_j \Delta^{-\frac{1+\alpha}{2}}]$, $j=1,\dots,n_1$ from $L^{\frac{Q}{\beta}}(\mathcal{G})$ to $\mathrm{BMO}_\alpha(\mathcal{G})$ if $b \in \mathrm{BMO}_\beta(\mathcal{G})$ and $0 < \alpha < \beta < 1$ as well as their characterisation of the uniform endpoint estimates $H^{\frac{Q}{Q+\alpha}}(\mathcal{G}) \to L^{\frac{Q}{Q-\alpha}}(\mathcal{G})$ and $L^{\frac{Q}{\alpha}}(\mathcal{G}) \to \mathrm{BMO}_\alpha(\mathcal{G})$. As a consequence of our results, the corresponding endpoint estimates of Riesz transforms $X_j \Delta^{-\frac{1}{2}}$, $j=1,\dots,n_1$ on $\mathcal{G}$ can be recovered as $\alpha \to 0$.
We classify all ruled surfaces in Euclidean space that are critical points of the Dirichlet energy, obtaining explicit parametrizations of these surfaces.
We prove that the tangent cone at the first blow-up time of the mean curvature flow of a closed symplectic surface in a compact Ka & uml;hler-Einstein surface consists of a finite union of planes in R4. Furthermore, when the flow develops a Type I* singularity at (X0,T), then the tangent cone is a holomorphic cone.
We establish global existence of the pluriclosed flow with arbitrary initial data on Oeljeklaus-Toma manifolds, and Gromov-Hausdorff convergence of blowdown limits to a torus under natural conjectural bounds on the flow at infinity. In the case of generalized Kähler-Ricci flow we prove refined a priori estimates in support of these conjectural bounds.
We review recent progress concerning the analysis of Lagrangians on immersions into Rd depending on the first and second fundamental forms and their covariant derivatives.
Let $\mu$ be a positive Borel measure on the interval $[0,1)$. For $\alpha>0$, the generalized Hankel matrix $\mathcal{H}_{\mu, \alpha}=(\mu_{n, k, \alpha})_{n, k \geq 0}$ with entries $\mu_{n, k, \alpha}=\int_{[0,1)} \frac{\Gamma(n+\alpha)}{n ! \Gamma(\alpha)} t^{n+k} \mathrm{d}\mu(t)$ induces formally the operator \begin{equation*} \mathcal{H}_{\mu, \alpha}(f)(z)=\sum_{n=0}^{\infty}\left(\sum_{k=0}^{\infty} \mu_{n, k, \alpha} a_k\right) z^n \end{equation*} on the space of all analytic function $f(z)=\sum_{k=0}^{\infty} a_{k} z^{k}$ in the unit disk $\mathbb{D}$. In this paper, we characterize the measures $\mu$ for which $\mathcal{H}_{\mu, \alpha}(f)$ is well defined on the Hardy spaces $H^p(01)$ is a bounded (resp., compact) operator from the Hardy spaces $H^p(0
In this paper, we prove the existence of a solution for the exterior Dirichlet problem for Hessian equations on a non-convex ring. Moreover, the solution we obtained is smooth. This extends the result of [Bao-Li-Li, “On the exterior Dirichlet problem for Hessian equations” Trans. Amer. Math. Soc.366(2014)].
In this paper, we establish some strong laws of large numbers (SLLN) for non-independent variables under the framework of sublinear expectations. One of our main results is for blockwise m-dependent random variables and another is for orthogonal random variables, both of which are the generalization of SLLN for independent random variables in sublinear expectation spaces.
Loday introduced di-associative algebras and tri-associative algebras motivated by periodicity phenomena in algebraic K-theory. The purpose of this paper is to study the splittings of operations of di-associative algebras and tri-associative algebras. First, we introduce the notion of a quadri-dendriform algebra, which is a splitting of a di-associative algebra. We show that a relative averaging operator on dendriform algebras gives rise to a quadri-dendriform algebra. Conversely, a quadri-dendriform algebra gives rise to a dendriform algebra and a representation such that the quotient map is a relative averaging operator. Furthermore, any quadri-dendriform algebra can be embedded into an averaging dendriform algebra. Finally, we introduce the notion of six-dendriform algebras, which are a splitting of tri-associative algebras, and demonstrate that homomorphic relative averaging operators induce six-dendriform algebras.
We show that the minimal log discrepancy of any isolated Fano cone singularity is at most the dimension of the variety. This is based on its relation with dimensions of moduli spaces of orbifold rational curves. We also propose a conjectural characterization of weighted projective spaces as Fano orbifolds in terms of orbifold rational curves, which would imply that the equality holds only for smooth points.
Degenerate complex Monge-Ampère equations arise naturally in the study of geometry of singular varieties. In this paper, we prove gradient estimate and $W^{3,p}$ estimate for a class of degenerate complex Monge-Ampère equations.
Degenerate complex Monge-Ampe`re equations arise naturally in the study of geometry of singular varieties. In this paper, we prove gradient estimate and W3,p estimate for a class of degenerate complex Monge-Ampe`re equations.
We survey basic properties of the geometric flow for immersions within a Hamiltonian isotopy class and propose a definition for Type I singularities.
We show that by applying a set of existing analytical arguments, a more robust effective uniqueness result for blowups can be obtained, with multiple implications following therefrom.
In this paper, we investigate the one-dimensional parabolic equation. Using blow-up analysis, we prove that the zero set at each time slice is discrete and that the number of zeros decreases with respect to time. Moreover, it strictly declines at the singularity. This provides a new and simplified proof of the main results in existing literature, where the original proof relies on spectral theory. Besides, our results represent a localized version of those existing results with weaker conditions.
In this paper, we establish a decay estimate for the discrete four-order Schrodinger equation on the hexagonal triangulation with gamma = 0. The proof is based on the uniform estimates of oscillatory integrals, as developed by Karpushkin, along with a key result by Varchenko. Our result is to show the l(1) l(infinity) dispersive decay rate is < t >(-sigma) for any 0< sigma < 1/2. Additionally, we provide estimates for the inhomogeneous discrete fourth-order Schrodinger equation with gamma = 0.
One of the significant motivations for studying the Kähler-Ricci flow is its relation to the Analytic Minimal Model Program as initiated by Gang Tian.Carrying out this classification program requires careful analysis of the flow metric,particularly when it encounters singularities.In this note,we survey some results pertaining to the limit for the Kähler-Ricci flow.
