Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userOn the support of a non-autocorrelated function on a hyperbolic surface
65
0
0.0
(
0
)
Authors
Konstantin Golubev
Ask ChatGPT about the research
No Arabic abstract
Let $f$ be a non-negative square-integrable function on a finite volume hyperbolic surface $Gammabackslashmathbb{H}$, and assume that $f$ is non-autocorrelated, that is, perpendicular to its image under the operator of averaging over the circle of a fixed radius $r$. We show that in this case the support of $f$ is small, namely, it satisfies $mu(supp{f}) leq (r+1)e^{-frac{r}{2}} mu(Gammabackslashmathbb{H})$. As a corollary, we prove a lower bound for the measurable chromatic number of the graph, whose vertices are the points of $Gammabackslashmathbb{H}$, and two points are connected by an edge if there is a geodesic of length $r$ between them. We show that for any finite covolume $Gamma$ the measurable chromatic number is at least $e^{frac{r}{2}}(r+1)^{-1}$.
rate research
Read More
We introduce a function model for the Teichmuller space of a closed hyperbolic Riemann surface. Then we introduce a new metric by using the maximum norm on the function space on the Teichmuller space. We prove that the identity map from the Teichmuller space equipped with the usual Teichmuller metric to the Teichmuller space equipped with this new metric is uniformly continuous. Furthermore, we also prove that the inverse of the identity, that is, the identity map from the Teichmuller space equipped with this new metric to the Teichmuller space equipped with the usual Teichmuller metric, is continuous. Therefore, the topology induced by the new metric is just the same as the topology induced by the usual Teichmuller metric on the Teichmuller space. We give a remark about the pressure metric and the Weil-Petersson metric.
In this paper, we first consider a class of expanding flows of closed, smooth, star-shaped hypersurface in Euclidean space $mathbb{R}^{n+1}$ with speed $u^alpha f^{-beta}$, where $u$ is the support function of the hypersurface, $f$ is a smooth, symmetric, homogenous of degree one, positive function of the principal curvatures of the hypersurface on a convex cone. For $alpha le 0<betale 1-alpha$, we prove that the flow has a unique smooth solution for all time, and converges smoothly after normalization, to a sphere centered at the origin. In particular, the results of Gerhardt cite{GC3} and Urbas cite{UJ2} can be recovered by putting $alpha=0$ and $beta=1$ in our first result. If the initial hypersurface is convex, this is our previous work cite{DL}. If $alpha le 0<beta< 1-alpha$ and the ambient space is hyperbolic space $mathbb{H}^{n+1}$, we prove that the flow $frac{partial X}{partial t}=(u^alpha f^{-beta}-eta u) u$ has a longtime existence and smooth convergence to a coordinate slice. The flow in $mathbb{H}^{n+1}$ is equivalent (up to an isomorphism) to a re-parametrization of the original flow in $mathbb{R}^{n+1}$ case. Finally, we find a family of monotone quantities along the flows in $mathbb{R}^{n+1}$. As applications, we give a new proof of a family of inequalities involving the weighted integral of $k$th elementary symmetric function for $k$-convex, star-shaped hypersurfaces, which is an extension of the quermassintegral inequalities in cite{GL2}.
We present the structure theorem for the positive support of the cube of the Grover transition matrix of the discrete-time quantum walk (the Grover walk) on a general graph $G$ under same condition. Thus, we introduce a zeta function on the positive support of the cube of the Grover transition matrix of $G$, and present its Euler product and its determinant expression. As a corollary, we give the characteristic polynomial for the positive support of the cube of the Grover transition matrix of a regular graph, and so obtain its spectra. Finally, we present the poles and the radius of the convergence of this zeta function.
In this paper, we propose an algebraic approach to determine whether two non-isomorphic caterpillar trees can have the same symmetric function generalization of the chromatic polynomial. On the set of all composition on integers, we introduce: An operation, which we call composition product; and a combinatorial polynomial, which we call the composition-lattice polynomial or L-polynomial, that mimics the weighted graph polynomial of Noble and Welsh. We prove a unique irreducible factorization theorem and establish a connection between the L-polynomial of a composition and its irreducible factorization, namely that reversing irreducible factors does not change L, and conjecture that is the only way of generating such compositions. Finally, we find a sufficient condition for two caterpillars have a different symmetric function generalization of the chromatic polynomial, and use this condition to show that if our conjecture were to hold, then the symmetric function generalization of the chromatic polynomial distinguishes among a large class of caterpillars.
Let $K$ be a discretely-valued field. Let $Xrightarrow Spec K$ be a surface with trivial canonical bundle. In this paper we construct a weak Neron model of the schemes $Hilb^n(X)$ over the ring of integers $Rsubseteq K$. We exploit this construction in order to compute the Motivic Zeta Function of $Hilb^n(X)$ in terms of $Z_X$. We determine the poles of $Z_{Hilb^n(X)}$ and study its monodromy property, showing that if the monodromy conjecture holds for $X$ then it holds for $Hilb^n(X)$ too. Sit $K$ corpus cum absoluto ualore discreto. Sit $ Xrightarrow Spec K$ leuigata superficies cum canonico fasce congruenti $mathcal{O}_X$. In hoc scripto defecta Neroniensia paradigmata $Hilb^n(X)$ schematum super annulo integrorum in $K$ corpo, $R subset K$, constituimus. Ex hoc, Functionem Zetam Motiuicam $Z_{Hilb^n(X)}$, dato $Z_X$, computamus. Suos polos statuimus et suam monodromicam proprietatem studemus, coniectura monodromica, quae super $X$ ualet, ualere super $Hilb^n(X)$ quoque demostrando.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
