On index expectation and curvature for networks

نویسنده

Oliver Knill

چکیده

We prove that the expectation value of the index function if (x) over a probability space of injective function f on any finite simple graph G = (V,E) is equal to the curvature K(x) at the vertex x. This result complements and links Gauss-Bonnet ∑ x∈V K(x) = χ(G) and Poincaré-Hopf ∑ x∈V if(x) = χ(G) which both hold for arbitrary finite simple graphs.

برای دانلود رایگان متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید

ثبت نام

اگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید

منابع مشابه

Estimation of Source Location Using Curvature Analysis

A quadratic surface can be fitted to potential-field data within 3×3 windows, which allow us to calculate curvature attributes from its coefficients. Phillips (2007) derived an equation depending on the most negative curvature to obtain the depth and structural index of isolated sources from peak values of special functions. They divided the special functions into two categories: Model-specific...

متن کامل

Estimation of coal swelling index based on chemical properties of coal using artificial neural networks

Free swelling index (FSI) is an important parameter for cokeability and combustion of coals. In this research, the effects of chemical properties of coals on the coal free swelling index were studied by artificial neural network methods. The artificial neural networks (ANNs) method was used for 200 datasets to estimate the free swelling index value. In this investigation, ten input parameters ...

متن کامل

Ductility Evaluation of Prestressed Concrete Members

In this paper, moment-curvature behavior and ductility of prestressed concrete members based on nonlinear analysis is investigated. Influence of various design parameters on ductility of flexural members is determined according to appropriate materials models for stress-strain curves of concrete, prestressing steel and conventional reinforcements. Parameters studied include reinforcing index (ω...

متن کامل

Curvature from Graph Colorings

Given a finite simple graphG = (V,E) with chromatic number c and chromatic polynomial C(x). Every vertex graph coloring f of G defines an index if (x) satisfying the Poincaré-Hopf theorem [17] ∑ x if (x) = χ(G). As a variant to the index expectation result [19] we prove that E[if (x)] is equal to curvature K(x) satisfying Gauss-Bonnet ∑ xK(x) = χ(G) [16], where the expectation is the average ov...

متن کامل