b'Oliver Knill'

research

∙ 01/13/2023

The Sphere Formula

The sphere formula states that in an arbitrary finite abstract simplicia...

Oliver Knill, et al. ∙

research

∙ 05/26/2022

The Babylonian Graph

The Babylonian graph B has the positive integers as vertices and connect...

Oliver Knill, et al. ∙

research

∙ 05/23/2022

The Tree-Forest Ratio

The number of rooted spanning forests divided by the number of spanning ...

Oliver Knill, et al. ∙

research

∙ 05/23/2022

Eigenvalue bounds of the Kirchhoff Laplacian

We prove that each eigenvalue l(k) of the Kirchhoff Laplacian K of a gra...

Oliver Knill, et al. ∙

research

∙ 01/24/2022

Analytic torsion for graphs

Analytic torsion is a functional on graphs which only needs linear algeb...

Oliver Knill, et al. ∙

research

∙ 08/11/2021

Shannon capacity, Chess, DNA and Umbrellas

A vexing open problem in information theory is to find the Shannon capac...

Oliver Knill, et al. ∙

research

∙ 06/18/2021

Remarks about the Arithmetic of Graphs

The arithmetic of N, Z, Q, R can be extended to a graph arithmetic where...

Oliver Knill, et al. ∙

research

∙ 01/18/2021

Graph complements of circular graphs

Graph complements G(n) of cyclic graphs are circulant, vertex-transitive...

Oliver Knill, et al. ∙

research

∙ 12/14/2020

Complexes, Graphs, Homotopy, Products and Shannon Capacity

A finite abstract simplicial complex G defines the Barycentric refinemen...

Oliver Knill, et al. ∙

research

∙ 10/19/2020

Green functions of Energized complexes

If h is a ring-valued function on a simplicial complex G we can define t...

Oliver Knill, et al. ∙

research

∙ 12/24/2019

Constant index expectation curvature for graphs or Riemannian manifolds

An integral geometric curvature is defined as the index expectation K(x)...

Oliver Knill, et al. ∙

research

∙ 12/02/2019

More on Poincare-Hopf and Gauss-Bonnet

We illustrate connections between differential geometry on finite simple...

Oliver Knill, et al. ∙

research

∙ 11/11/2019

Poincare Hopf for vector fields on graphs

We generalize the Poincare-Hopf theorem sum_v i(v) = X(G) to vector fiel...

Oliver Knill, et al. ∙

research

∙ 10/07/2019

A simple sphere theorem for graphs

A finite simple graph G is declared to have positive curvature if every ...

Oliver Knill, et al. ∙

research

∙ 08/19/2019

Energized simplicial complexes

For a simplicial complex with n sets, let W^-(x) be the set of sets in G...

Oliver Knill, et al. ∙

research

∙ 07/07/2019

The energy of a simplicial complex

A finite abstract simplicial complex G defines a matrix L, where L(x,y)=...

Oliver Knill, et al. ∙

research

∙ 06/15/2019

A parametrized Poincare-Hopf Theorem and Clique Cardinalities of graphs

Given a locally injective real function g on the vertex set V of a finit...

Oliver Knill, et al. ∙

research

∙ 05/31/2019

More on Numbers and Graphs

In this note we revisit a "ring of graphs" Q in which the set of finite ...

Oliver Knill, et al. ∙

research

∙ 05/13/2019

Dehn-Sommerville from Gauss-Bonnet

We give a zero curvature proof of Dehn-Sommerville for finite simple gra...

Oliver Knill, et al. ∙

research

∙ 05/06/2019

The average simplex cardinality of a finite abstract simplicial complex

We study the average simplex cardinality Dim^+(G) = sum_x |x|/(|G|+1) of...

Oliver Knill, et al. ∙

research

∙ 03/25/2019

A Reeb sphere theorem in graph theory

We prove a Reeb sphere theorem for finite simple graphs. The result brid...

Oliver Knill, et al. ∙

research

∙ 11/26/2018

Cartan's Magic Formula for Simplicial Complexes

Cartan's magic formula L_X = i_X d + d i_X = (d+i_X)^2=D_X^2 relates the...

Oliver Knill, et al. ∙

research

∙ 08/22/2018

Eulerian edge refinements, geodesics, billiards and sphere coloring

A finite simple graph is called a 2-graph if all of its unit spheres S(x...

Oliver Knill, et al. ∙

research

∙ 06/17/2018

Combinatorial manifolds are Hamiltonian

Extending a theorem of Whitney of 1931 we prove that all connected d-gra...

Oliver Knill, et al. ∙

research

∙ 03/19/2018

The Cohomology for Wu Characteristics

While Euler characteristic X(G)=sum_x w(x) super counts simplices, Wu ch...

Oliver Knill, et al. ∙

research

∙ 03/05/2018

The hydrogen identity for Laplacians

For any finite simple graph G, the hydrogen identity H=L-L^(-1) holds, w...

Oliver Knill, et al. ∙

research

∙ 02/05/2018

Listening to the cohomology of graphs

We prove that the spectrum of the Kirchhoff Laplacian H0 of a finite sim...

Oliver Knill, et al. ∙

research

∙ 01/15/2018

An Elementary Dyadic Riemann Hypothesis

The connection zeta function of a finite abstract simplicial complex G i...

Oliver Knill, et al. ∙

research

∙ 11/27/2017

One can hear the Euler characteristic of a simplicial complex

We prove that that the number p of positive eigenvalues of the connectio...

Oliver Knill, et al. ∙

research

∙ 08/21/2017

On Atiyah-Singer and Atiyah-Bott for finite abstract simplicial complexes

A linear or multi-linear valuation on a finite abstract simplicial compl...

Oliver Knill, et al. ∙

research

∙ 08/05/2017

The strong ring of simplicial complexes

We define a ring R of geometric objects G generated by finite abstract s...

Oliver Knill, et al. ∙

research

∙ 05/30/2017

On a Dehn-Sommerville functional for simplicial complexes

Assume G is a finite abstract simplicial complex with f-vector (v0,v1, ....

Oliver Knill, et al. ∙

research

∙ 08/18/2007

A structure from motion inequality

We state an elementary inequality for the structure from motion problem ...

Oliver Knill, et al. ∙

research

∙ 08/17/2007

Space and camera path reconstruction for omni-directional vision

In this paper, we address the inverse problem of reconstructing a scene ...

Oliver Knill, et al. ∙

research

∙ 08/17/2007

On Ullman's theorem in computer vision

Both in the plane and in space, we invert the nonlinear Ullman transform...

Oliver Knill, et al. ∙

Oliver Knill

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