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Oliver Knill

Publications and source records attributed to Oliver Knill.

125 records · Page 7Linked to original sources
arXiv2012

The Lusternik-Schnirelmann theorem for graphs

We prove the discrete Lusternik-Schnirelmann theorem telling that tcat(G) less or equal to crit(G) for a general simple graph G=(V,E). It relates the minimal number tcat(G) of in G contractible graphs covering G, with crit(G), the minimal number of critical points which an injective function f on the vertex set V can have. We also prove that the cup length cup(G) is less or equal to tcat(G) which is valid also for any finite simple graph. If cat(G) is the minimal tcat(H) among all graphs H homotopic to G and cri(G) is the minimal crit(H) among all graphs H homotopic to G, we get a relation between three homotopy invariants: an algebraic quantity (cup), a topological quantity (cat) and an analytic quantity (cri).

math.AT↗
arXiv2012

Selfsimilarity in the Birkhoff sum of the cotangent function

We prove that the Birkhoff sum S(n)/n = (1/n) sum_(k=1)^(n-1) g(k A) with g(x) = cot(Pi x) and golden ratio A converges in the sense that the sequence of functions s(x) = S([ x q(2n)])/q(2n) with Fibonacci numbers q(n) converges to a self similar limiting function s(x) on [0,1]. The function s(x) can be computed analytically. This allows to determine values like S(10^100)/10^100 accurately without that it ever would be possible to add up so many terms for this random walk. The random variables added up are Cauchy distributed random variables with almost periodic correlation. While for any continuous function g, the Birkhoff limiting function is s(x)=M x by Birkhoff's ergodic theorem, we get so examples of random variables X(n), where the limiting function of S([x n])/n converges to a nontrivial selfsimilar function s(x) along subsequences for one initial point. Hardy and Littlewood have studied the Birkhoff sum for g'(x)= -Pi csc^2(Pi x) and shown that the corresponding Birkhoff sum S'(n)/n^2 stays bounded. Sinai and Ulcigrai have found a limiting distribution for g(x) if both the rotation number A and the initial point t are integrated over. We fix the golden ratio A, start with fixed t=0 and show that the rescaled random walk converges along subsequences. The analysis sheds light on the pictures seen in previous papers with Lesieutre and Tangerman on the antiderivative G(x).

math.DS↗
arXiv2012

A Multivariable Chinese Remainder Theorem

Using an adaptation of Qin Jiushao's method from the 13th century, it is possible to prove that a system of linear modular equations a(i,1) x(i) + ... + a(i,n) x(n) = b(i) mod m(i), i=1, ..., n has integer solutions if m(i)>1 are pairwise relatively prime and in each row, at least one matrix element a(i,j) is relatively prime to m(i). The Chinese remainder theorem is the special case, where A has only one column.

math.HO↗
arXiv2012

A Brouwer fixed point theorem for graph endomorphisms

We prove a Lefschetz formula for general simple graphs which equates the Lefschetz number L(T) of an endomorphism T with the sum of the degrees i(x) of simplices in G which are fixed by T. The degree i(x) of x with respect to T is defined as a graded sign of the permutation T induces on the simplex x multiplied by -1 if the dimension of x is odd. The Lefschetz number is defined as in the continuum as the super trace of T induced on cohomology. In the special case where T is the identity, the formula becomes the Euler-Poincare formula equating combinatorial and cohomological Euler characteristic. The theorem assures in general that if L(T) is nonzero, then T has a fixed clique. A special case is a discrete Brouwer fixed point theorem for graphs: if T is a graph endomorphism of a connected graph G, which is star-shaped in the sense that only the zeroth cohomology group is nontrivial, like for connected trees or triangularizations of star shaped Euclidean domains, then there is clique x which is fixed by T. Unlike in the continuum, the fixed point theorem proven here looks for fixed cliques, complete subgraphs which play now the role of "points" in the graph. Fixed points can so be vertices, edges, fixed triangles etc. If A denotes the automorphism group of a graph, we also look at the average Lefschetz number L(G) which is the average of L(T) over A. We prove that this is the Euler characteristic of the graph G/A and especially an integer. We also show that as a consequence of the Lefschetz formula, the zeta function zeta(T,z) is a product of two dynamical zeta functions and therefore has an analytic continuation as a rational function which is explicitly given by a product formula involving only the dimension and the signature of prime orbits of simplices in G.

math.DS↗
arXiv2012

An index formula for simple graphs

Gauss-Bonnet for simple graphs G assures that the sum of curvatures K(x) over the vertex set V of G is the Euler characteristic X(G). Poincare-Hopf tells that for any injective function f on V the sum of i(f,x) is X(G). We also know that averaging the indices E[i(f,x)] over all functions gives curvature K(x). We explore here the situation when G is geometric of dimension d: that is if each unit sphere S(x) is geometric of dimension d-1 and that X(S(x))=0 for even d and X(S(x))=2 for odd d. The dimension of G is inductively defined as the average of 1+dim(S(x)) over all S(x) assuming the empty graph has dimension -1. We prove that any odd dimensional geometric graph G has zero curvature. This is done with the help of an index formula j(f,x) = 1-X(S(x))/2-X(B(f,x))/2, where j(x)=[i(f,x)+i(-f,x)]/2. The graph B(f,x) is the discrete level surface {y | f(y) = f(x)} intersected with S(x). It is a subgraph of the line graph of G and geometric if G is geometric. The index formula simplifies for geometric graphs: for even d it is j(f,x) = 1-X(B(f,x))/2, where B(f,x) is a (d-2)-dimensional graph. For odd d it becomes j(f,x) =-X(B(f,x))/2, where B(f,x) is an odd dimensional graph. Because by induction with respect to d, the X(B(f,x))=0 we know now that that j(f,x) is zero for all x and so, by taking expectation over f that curvature K(x) is zero for all x. We also point out that all these results hold almost verbatim for compact Riemannian manifolds and actually are much simpler there. The same integral geometric index formula is valid if f is a Morse function, i(f,x) is the index of the gradient vector field and if S(x) is a sufficiently small geodesic sphere around x and B(f,x) which is S(x) intersected with the level surface {y | f(y)=f(x)}. Also in the continuum, the symmetric index j(f,x) is constant zero everywhere if d is odd.

math.DG↗
arXiv2012

On index expectation and curvature for networks

We prove that the expectation value of the index function i(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 sum K(x) = chi(G) and Poincare-Hopf sum i(x) = chi(G) which both hold for arbitrary finite simple graphs.

math.DG↗
arXiv2012

The theorems of Green-Stokes,Gauss-Bonnet and Poincare-Hopf in Graph Theory

By proving graph theoretical versions of Green-Stokes, Gauss-Bonnet and Poincare-Hopf, core ideas of undergraduate mathematics can be illustrated in a simple graph theoretical setting. In this pedagogical exposition we present the main proofs on a single page and add illustrations. While discrete Stokes is is old, the other two results for graphs were found only recently.

math.DG↗
arXiv2012

A graph theoretical Poincare-Hopf Theorem

We introduce the index i(v) = 1 - X(S(v)) for critical points of a locally injective function f on the vertex set V of a simple graph G=(V,E). Here S(v) = {w in E | (v,w) in E, f(w)-f(v)<0} is the subgraph of the unit sphere at v in G. It is the exit set of the gradient vector field. We prove that the sum of i(v) over V is always is equal to the Euler characteristic X(G) of the graph G. This is a discrete Poincare-Hopf theorem in a discrete Morse setting. It allows to compute X(G) for large graphs for which other methods become impractical.

math.DG↗
arXiv2011

On the Dimension and Euler characteristic of random graphs

The inductive dimension dim(G) of a finite undirected graph G=(V,E) is a rational number defined inductively as 1 plus the arithmetic mean of the dimensions of the unit spheres dim(S(x)) at vertices x primed by the requirement that the empty graph has dimension -1. We look at the distribution of the random variable "dim" on the Erdos-Renyi probability space G(n,p), where each of the n(n-1)/2 edges appears independently with probability p. We show here that the average dimension E[dim] is a computable polynomial of degree n(n-1)/2 in p. The explicit formulas allow experimentally to explore limiting laws for the dimension of large graphs. We also study the expectation E[X] of the Euler characteristic X, considered as a random variable on G(n,p). We look experimentally at the statistics of curvature K(v) and local dimension dim(v) = 1+dim(S(v)) which satisfy the Gauss-Bonnet formula X(G) = sum K(v) and by definition dim(G) = sum dim(v)/|V|. We also look at the signature functions f(p)=E[dim], g(p)=E[X] and matrix values functions A(p) = Cov[{dim(v),dim(w)], B(p) = Cov[K(v),K(w)] on the probability space G(p) of all subgraphs of a host graph G=(V,E) with the same vertex set V, where each edge is turned on with probability p. If G is the complete graph or a union of cyclic graphs with have explicit formulas for the signature polynomials f and g.

math.PR↗
arXiv2010

A discrete Gauss-Bonnet type theorem

We prove a prototype curvature theorem for subgraphs G of the flat triangular tesselation which play the analogue of "domains" in two dimensional Euclidean space: The Pusieux curvature K(p) = 2|S1(p)| - |S2(p)| is equal to 12 times the Euler characteristic when summed over the boundary of G. Here |S1(p)| is the arc length of the unit sphere of p and |S2(p)| is the arc length of the sphere of radius 2. This curvature 12 formula is discrete Gauss-Bonnet formula or Hopf Umlaufsatz. The curvature introduced here is motivated by analogue constructions in the continuum like the Jacobi equations for geodesic flows. For the first order curvature K1(p) = 6-|S1(p)|, Gauss-Bonnet results are much easier to show, are less "differential geometric" but generalize to rather general 2-dimensional graphs G=(V,E): The sum of the K1 curvatures over the entire graph is 6 times the Euler characteristic of G, where the Euler characteristic is defined as v-e+f where v=|V|,e=|E| and f is the number of triangles. For many abstract two dimensional graphs, the sum over all K curvatures is 60 times the Euler characteristic. Under which conditions this curvature 60 theorem holds is still under investigation. In our proof of the curvature 12 theorem, the concept of dimension for abstract graphs plays an important role: a vertex p of a finite abstract graph X=(V,E) is called 0-dimensional, if p is not connected to any other vertex. A subset G of X is called 0-dimensional if every point of G is 0-dimensional in G. A vertex p of G is called 1-dimensional if S1(p) is zero dimensional. A finite subset G of X is called 1-dimensional if any of the points in G is 1-dimensional. A point p of G is called 2 dimensional, if S1(p) is a one-dimensional graph and a subset G of the graph is called 2-dimensional, if every vertex p of G is 2-dimensional.

math.GN↗
arXiv2010

Selfsimilarity and growth in Birkhoff sums for the golden rotation

We study Birkhoff sums S(k,a) = g(a)+g(2a)+...+g(ka) at the golden mean rotation number a with periodic continued fraction approximations p(n)/q(n), where g(x) = log(2-2 cos(2 pi x). The summation of such quantities with logarithmic singularity is motivated by critical KAM phenomena. We relate the boundedness of log- averaged Birkhoff sums S(k,a)/log(k) and the convergence of S(q(n),a) with the existence of an experimentally established limit function f(x) = lim S([x q(n)])(p(n+1)/q(n+1))-S([x q(n)])(p(n)/q(n)) for n to infinity on the interval [0,1]. The function f satisfies a functional equation f(ax) + (1-a) f(x)= b(x) with a monotone function b. The limit lim S(q(n),a) for n going to infinity can be expressed in terms of the function f.

math.DS↗
arXiv2008

Analytic continuation of Dirichlet series with almost periodic coefficients

We prove that an ordinary Dirichlet series with coefficients a(n)=g(n b) has an abscissa of convergence 0 if g is an odd 1-periodic, real-analytic function and b is Diophantine. We also show that if g is odd and has bounded variation and b is of bounded Diophantine type r>1, then the abscissa of convergence is smaller or equal than 1-1/r. Using a polylogarithm expansion, we prove that if g is odd and real analytic and b is Diophantine, then the ordinary Dirichlet series has an analytic continuation to the entire complex plane.

math.CV↗
arXiv2007

A structure from motion inequality

We state an elementary inequality for the structure from motion problem for m cameras and n points. This structure from motion inequality relates space dimension, camera parameter dimension, the number of cameras and number points and global symmetry properties and provides a rigorous criterion for which reconstruction is not possible with probability 1. Mathematically the inequality is based on Frobenius theorem which is a geometric incarnation of the fundamental theorem of linear algebra. The paper also provides a general mathematical formalism for the structure from motion problem. It includes the situation the points can move while the camera takes the pictures.

cs.CV↗
arXiv2007

On Ullman's theorem in computer vision

Both in the plane and in space, we invert the nonlinear Ullman transformation for 3 points and 3 orthographic cameras. While Ullman's theorem assures a unique reconstruction modulo a reflection for 3 cameras and 4 points, we find a locally unique reconstruction for 3 cameras and 3 points. Explicit reconstruction formulas allow to decide whether picture data of three cameras seeing three points can be realized as a point-camera configuration.

cs.CV↗
arXiv2007

Space and camera path reconstruction for omni-directional vision

In this paper, we address the inverse problem of reconstructing a scene as well as the camera motion from the image sequence taken by an omni-directional camera. Our structure from motion results give sharp conditions under which the reconstruction is unique. For example, if there are three points in general position and three omni-directional cameras in general position, a unique reconstruction is possible up to a similarity. We then look at the reconstruction problem with m cameras and n points, where n and m can be large and the over-determined system is solved by least square methods. The reconstruction is robust and generalizes to the case of a dynamic environment where landmarks can move during the movie capture. Possible applications of the result are computer assisted scene reconstruction, 3D scanning, autonomous robot navigation, medical tomography and city reconstructions.

cs.CV↗
arXiv2005

The Problem of Positive Kolmogorov-Sinai entropy for the Standard map

The problem of positive Kolmogorov-Sinai entropy of the Chirikov-Standard map with respect to the invariant Lebesgue measure on the two-dimensional is open. In 1999, we believed to have a proof that the entropy can be bounded below. This approach was based on an idea of Herman to do subharmonic estimates.This document replaces an announcement I had circulated in 1999. In the present document, the incorrect parts have been deleted. The entropy conjecture is open. The references given in the text might still be helpful for people trying an operator theoretical or analytic approach to this problem in ergodic theory.

math.DS↗