Related papers: Green functions for positive-depth Deligne--Luszti…
We provide quantitative inductive estimates for Green's functions of matrices with (sub)expoentially decaying off diagonal entries in higher dimensions. Together with Cartan's estimates and discrepancy estimates, we establish explicit…
Thermodynamic functions of ionic systems were evaluated analytically using the Green's Function for Body Centered Cubic Lattices. The free energy density, chemical potential, pressure, spinodals, and coulomb ionic potentials are expressed…
Let $G(q)$ be a finite group of Lie type over a field with $q$ elements, where $q$ is a prime power. The Green functions of $G(q)$, as defined by Deligne and Lusztig, are known in \textit{almost} all cases by work of Beynon--Spaltenstein,…
We construct Green's function for second order elliptic operators of the form $Lu=-\nabla \cdot (\mathbf{A} \nabla u + \boldsymbol{b} u)+ \boldsymbol c \cdot \nabla u+ du$ in a domain and obtain pointwise bounds, as well as Lorentz space…
Several results about positive solutions -in a Lipschitz domain- of a nonlinear elliptic equation in a general form $ \Delta u(x)-g(x,u(x))=0$ are proved, extending thus some known facts in the case of $ g(x,t)=t^q$, $q>1$, and a smooth…
It has been recently established by the first and third author that on uniformly rectifiable sets the Green function is almost affine in the weak sense, and moreover, in some scenarios such Green function estimates are equivalent to the…
We derive new integral presentations for central derivative values of $L$-functions of elliptic curves defined over the rationals, basechanged to a real quadratic field $K$, twisted by ring class characters of $K$ in terms of sums along…
In this paper we study higher level Deligne--Lusztig representations of reductive groups over discrete valuation rings, with finite residue field $\mathbb{F}_q$. In previous work we proved that, at even levels, these geometrically…
The Green functions were first introduced by Green to compute the character table of GLn(q) in 1955. They were later generalized by Deligne and Lusztig for an arbitrary finite group of Lie type G(q) using l-adic cohomological methods…
We study the $Q$-Kostka polynomials $L_{\lambda\mu}(t)$ by the vertex operator realization of the $Q$-Hall-Littlewood functions $G_{\lambda}(x;t)$ and derive new formulae for $L_{\lambda\mu}(t)$. In particular, we have established stability…
Given input-output pairs of an elliptic partial differential equation (PDE) in three dimensions, we derive the first theoretically-rigorous scheme for learning the associated Green's function $G$. By exploiting the hierarchical low-rank…
The main results of this article provide asymptotics at infinity of the Green's functions near and at the spectral gap edges for "generic" periodic second-order elliptic operators on noncompact Riemannian co-compact coverings with abelian…
We construct the fundamental solution or Green function for a divergence form elliptic system in two dimensions with bounded and measurable coefficients. We consider the elliptic system in a Lipschitz domain with mixed boundary conditions.…
In this note we establish the positivity of Green's functions for a class of elliptic differential operators on closed, Riemannian manifolds.
Gross and Zagier defined certain `higher Green's functions' on products of modular curves and conjectured that the value of these functions at complex multiplication points should be logarithms of algebraic numbers. This is now a theorem of…
Recent work on the quantization of Maxwell theory has used a non-covariant class of gauge-averaging functionals which include explicitly the effects of the extrinsic-curvature tensor of the boundary, or covariant gauges which, unlike the…
We look for algebraic certificates of positivity for functions which are not necessarily polynomial functions. Similar questions were examined earlier by Lasserre and Putinar and by Putinar. We explain how these results can be understood as…
In this paper it is shown how the generating functional for Green's functions in relativistic quantum field theory and in thermal field theory can be evaluated in terms of a standard quantum mechanical path integral. With this calculational…
In this paper, we study the properties of integral functionals induced on $L^1_E (S,\mu)$ by closed convex functions on a Euclidean space $E$. We give sufficient conditions for such integral functions to be strongly rotund (well-posed). We…
Green's function characterizes a partial differential equation (PDE) and maps its solution in the entire domain as integrals. Finding the analytical form of Green's function is a non-trivial exercise, especially for a PDE defined on a…