Related papers: Up-to-constants bounds on the two-point Green's fu…
It is commonly believed that the normalized gaps between consecutive ordinates $t_n$ of the zeros of the Riemann zeta function on the critical line can be arbitrarily large. In particular, drawing on analogies with random matrix theory, it…
By introducing multipe-site correlation functions, we propose a hierarchical Green function approach, and apply it to study the characteristic properties of a 2D square lattice Hubbard model by solving the equation of motions of a…
In general, for higher order elliptic equations and boundary value problems like the biharmonic equation and the linear clamped plate boundary value problem neither a maximum principle nor a comparison principle or -- equivalently -- a…
New bounds for the $k$-th order derivatives of the solutions of the normal and multivariate normal Stein equations are obtained. Our general order bounds involve fewer derivatives of the test function than those in the existing literature.…
Using a five dimensional (5D) warped model with two branes along the extra dimension, we study the Green's functions for gauge bosons with a mass gap $m_g = \rho/2$ and a continuum for $s > m_g^2$. We find that the Green's functions exhibit…
This is a continuation and completion of the program (initiated in \cite{GrN1,GrN2}) to derive pointwise estimates on the Green function and sharp bounds on the semigroup of linearized Navier-Stokes around a generic stationary boundary…
We consider global 2-SLE$_{\kappa}$ $(\eta_1, \eta_2)$ in a topological rectangle with $\kappa\in (4,8)$. We derive the law of a random hitting point of the curves and show that, conditional on this random hitting point, the pair of two…
We establish a local Harnack inequality in a neighborhood of an indecomposable singular point of a stationary integral varifold. Extending the method of Gr\"uter and Widman \cite{gruter1982green}, we construct the Green function on a…
By using the Lyapunov-Schmidt reduction method without perturbation, we consider existence results for the conformal scalar curvature on S^n (n greater or equal to 3) when the prescribed function (after being projected to R^n) has two close…
The logarithmic coefficients $\gamma_n$ of an analytic and univalent function $f$ in the unit disk $\mathbb{D}=\{z\in\mathbb{C}:|z|<1\}$ with the normalization $f(0)=0=f'(0)-1$ are defined by $\log \frac{f(z)}{z}= 2\sum_{n=1}^{\infty}…
Around 2008, Schramm conjectured that the critical probabilities for Bernoulli bond percolation satisfy the following continuity property: If $(G_n)_{n\geq 1}$ is a sequence of transitive graphs converging locally to a transitive graph $G$…
Two-particle Green's functions and the vertex functions play a critical role in theoretical frameworks for describing strongly correlated electron systems. However, numerical calculations at two-particle level often suffer from large…
An inverse problem for the two-dimensional Schrodinger equation with $L^p_{com}$-potential, $p>1$, is considered. Using the $\overline{\partial}$-method, the potential is recovered from the Dirichlet-to-Neumann map on the boundary of a…
Lower and upper bounds for a given function are important in many mathematical and engineering contexts, where they often serve as a base for both analysis and application. In this short paper, we derive piecewise linear and quadratic…
We consider shape, size and regularity of the hulls of the chordal Schramm-Loewner evolution driven by a symmetric alpha-stable process. We obtain derivative estimates, show that the complements of the hulls are Hoelder domains, prove that…
We provide larger step-size restrictions for which gradient descent based algorithms (almost surely) avoid strict saddle points. In particular, consider a twice differentiable (non-convex) objective function whose gradient has Lipschitz…
We prove that any disjoint union of finitely many simple curves in the upper half-plane can be generated in a unique way by the chordal multiple-slit Loewner equation with constant weights.
We review some recently completed research that establishes the scaling limit of Fomin's identity for loop-erased random walk on Z^2 in terms of the chordal Schramm-Loewner evolution (SLE) with parameter 2. In the case of two paths, we…
Let $\lambda_2(G)$ and $\kappa'(G)$ be the second largest eigenvalue and the edge-connectivity of a graph $G$, respectively. Let $d$ be a positive integer at least 3. For $t=1$ or 2, Cioaba proved sharp upper bounds for $\lambda_2(G)$ in a…
We provide explicit formulas for the Green function of an elliptic PDE in the infinite strip and the half-plane. They are expressed in elementary and special functions. Proofs of uniqueness and existence are also given.