Coin flipping from a cosmic source: On error correction of truly random bits

We study a problem related to coin flipping, coding theory, and noise sensitivity. Consider a source of truly random bits x \in \bits^n, and $k$ parties, who have noisy versions of the source bits y^i \in \bits^n, where for all $i$ and $j$, it holds that \Pr[y^i_j = x_j] = 1 - \eps, independently for all $i$ and $j$. That is, each party sees each bit correctly with probability $1-\epsilon$, and incorrectly (flipped) with probability $\epsilon$, independently for all bits and all parties. The parties, who cannot communicate, wish to agree beforehand on {\em balanced} functions f_i : \bits^n \to \bits such that $\Pr[f_1(y^1) = ... = f_k(y^k)]$ is maximized. In other words, each party wants to toss a fair coin so that the probability that all parties have the same coin is maximized. The functions $f_i$ may be thought of as an error correcting procedure for the source $x$. When $k=2,3$ no error correction is possible, as the optimal protocol is given by $f_i(x^i) = y^i_1$. On the other hand, for large values of $k$, better protocols exist. We study general properties of the optimal protocols and the asymptotic behavior of the problem with respect to $k$, $n$ and \eps. Our analysis uses tools from probability, discrete Fourier analysis, convexity and discrete symmetrization

The Fermat-Torricelli problem in normed planes and spaces

We investigate the Fermat-Torricelli problem in d-dimensional real normed spaces or Minkowski spaces, mainly for d=2. Our approach is to study the Fermat-Torricelli locus in a geometric way. We present many new results, as well as give an exposition of known results that are scattered in various sources, with proofs for some of them. Together, these results can be considered to be a minitheory of the Fermat-Torricelli problem in Minkowski spaces and especially in Minkowski planes. This demonstrates that substantial results about locational problems valid for all norms can be found using a geometric approach

Geometric Dilation and Halving Distance

Let us consider the network of streets of a city represented by a geometric graph G in the plane. The vertices of G represent the crossroads and the edges represent the streets. The latter do not have to be straight line segments, they may be curved. If one wants to drive from a place p to some other place q, normally the length of the shortest path along streets, d_G(p,q), is bigger than the airline distance (Euclidean distance) |pq|. The (relative) DETOUR is defined as delta_G(p,q) := d_G(p,q)/|pq|. The supremum of all these ratios is called the GEOMETRIC DILATION of G. It measures the quality of the network. A small dilation value guarantees that there is no bigger detour between any two points. Given a finite point set S, we would like to know the smallest possible dilation of any graph that contains the given points on its edges. We call this infimum the DILATION of S and denote it by delta(S). The main results of this thesis are - a general upper bound to the dilation of any finite point set S, delta(S) - a lower bound for a specific set P, delta(P)>(1+10^(-11))pi/2, which approximately equals 1.571 In order to achieve these results, we first consider closed curves. Their dilation depends on the HALVING PAIRS, pairs of points which divide the closed curve in two parts of equal length. In particular the distance between the two points is essential, the HALVING DISTANCE. A transformation technique based on halving pairs, the HALVING PAIR TRANSFORMATION, and the curve formed by the midpoints of the halving pairs, the MIDPOINT CURVE, help us to derive lower bounds to dilation. For constructing graphs of small dilation, we use ZINDLER CURVES. These are closed curves of constant halving distance. To give a structured overview, the mathematical apparatus for deriving the main results of this thesis includes - upper bound: * the construction of certain Zindler curves to generate a periodic graph of small dilation * an embedding argument based on a number theoretical result by Dirichlet - lower bound: * the formulation and analysis of the halving pair transformation * a stability result for the dilation of closed curves based on this transformation and the midpoint curve * the application of a disk-packing result In addition, this thesis contains - a detailed analysis of the dilation of closed curves - a collection of inequalities, which relate halving distance to other important quantities from convex geometry, and their proofs; including four new inequalities - the rediscovery of Zindler curves and a compact presentation of their properties - a proof of the applied disk packing result.Geometrische Dilation und Halbierungsabstand Man kann das von den Straßen einer Stadt gebildete Netzwerk durch einen geometrischen Graphen in der Ebene darstellen. Die Knoten dieses Graphen repräsentieren die Kreuzungen und die Kanten sind die Straßen. Letztere müssen nicht geradlinig sein, sondern können beliebig gekrümmt sein. Wenn man nun von einem Ort p zu einem anderen Ort q fahren möchte, dann ist normalerweise die Länge des kürzesten Pfades über Straßen, d_G(p,q), länger als der Luftlinienabstand (euklidischer Abstand) |pq|. Der (relative) UMWEG (DETOUR) ist definiert als delta_G(p,q) := d_G(p,q)/|pq|. Das Supremum all dieser Brüche wird GEOMETRISCHE DILATION (GEOMETRIC DILATION) von G genannt. Es ist ein Maß für die Qualität des Straßennetzes. Ein kleiner Dilationswert garantiert, dass es keinen größeren Umweg zwischen beliebigen zwei Punkten gibt. Für eine gegebene endliche Punktmenge S würden wir nun gerne bestimmen, was der kleinste Dilationswert ist, den wir mit einem Graphen erreichen können, der die gegebenen Punkte auf seinen Kanten enthält. Dieses Infimum nennen wir die DILATION von S und schreiben kurz delta(S). Die Haupt-Ergebnisse dieser Arbeit sind - eine allgemeine obere Schranke für die Dilation jeder beliebigen endlichen Punktmenge S: delta(S) - eine untere Schranke für eine bestimmte Menge P: delta(P)>(1+10^(-11))pi/2, was ungefähr der Zahl 1.571 entspricht Um diese Ergebnisse zu erreichen, betrachten wir zunächst geschlossene Kurven. Ihre Dilation hängt von sogenannten HALBIERUNGSPAAREN (HALVING PAIRS) ab. Das sind Punktpaare, die die geschlossene Kurve in zwei Teile gleicher Länge teilen. Besonders der Abstand der beiden Punkte ist von Bedeutung, der HALBIERUNGSABSTAND (HALVING DISTANCE). Eine auf den Halbierungspaaren aufbauende Transformation, die HALBIERUNGSPAARTRANSFORMATION (HALVING PAIR TRANSFORMATION), und die von den Mittelpunkten der Halbierungspaare gebildete Kurve, die MITTELPUNKTKURVE (MIDPOINT CURVE), helfen uns untere Dilationsschranken herzuleiten. Zur Konstruktion von Graphen mit kleiner Dilation benutzen wir ZINDLERKURVEN (ZINDLER CURVES). Dies sind geschlossene Kurven mit konstantem Halbierungspaarabstand. Die mathematischen Hilfsmittel, mit deren Hilfe wir schließlich die Hauptresultate beweisen, sind unter anderem - obere Schranke: * die Konstruktion von bestimmten Zindlerkurven, mit denen periodische Graphen kleiner Dilation gebildet werden können * ein Einbettungsargument, das einen zahlentheoretischen Satz von Dirichlet benutzt - untere Schranke: * die Definition und Analyse der Halbierungspaartransformation * ein Stabilitätsresultat für die Dilation geschlossener Kurven, das auf dieser Transformation und der Mittelpunktkurve basiert * die Anwendung eines Kreispackungssatzes Zusätzlich enthält diese Dissertation - eine detaillierte Analyse der Dilation geschlossener Kurven - eine Sammlung von Ungleichungen, die den Halbierungsabstand zu anderen wichtigen Größen der Konvexgeometrie in Beziehung setzen, und ihre Beweise; inklusive vier neuer Ungleichungen - die Wiederentdeckung von Zindlerkurven und eine kompakte Darstellung ihrer Eigenschaften - einen Beweis des angewendeten Kreispackungssatzes

On a fast bilateral filtering formulation using functional rearrangements

We introduce an exact reformulation of a broad class of neighborhood filters, among which the bilateral filters, in terms of two functional rearrangements: the decreasing and the relative rearrangements. Independently of the image spatial dimension (one-dimensional signal, image, volume of images, etc.), we reformulate these filters as integral operators defined in a one-dimensional space corresponding to the level sets measures. We prove the equivalence between the usual pixel-based version and the rearranged version of the filter. When restricted to the discrete setting, our reformulation of bilateral filters extends previous results for the so-called fast bilateral filtering. We, in addition, prove that the solution of the discrete setting, understood as constant-wise interpolators, converges to the solution of the continuous setting. Finally, we numerically illustrate computational aspects concerning quality approximation and execution time provided by the rearranged formulation.Comment: 29 pages, Journal of Mathematical Imaging and Vision, 2015. arXiv admin note: substantial text overlap with arXiv:1406.712

Combinatorics and Probability

For the past few decades, Combinatorics and Probability Theory have had a fruitful symbiosis, each benefitting from and influencing developments in the other. Thus to prove the existence of designs, probabilistic methods are used, algorithms to factorize integers need combinatorics and probability theory (in addition to number theory), and the study of random matrices needs combinatorics. In the workshop a great variety of topics exemplifying this interaction were considered, including problems concerning designs, Cayley graphs, additive number theory, multiplicative number theory, noise sensitivity, random graphs, extremal graphs and random matrices

A Mass Minimizing Flow for Real-Valued Flat Chains with Applications to Transport Networks

An oriented transportation network can be modeled by a 1-dimensional chain whose boundary is the difference between the demand and supply distributions, represented by weighted sums of point masses. To accommodate efficiencies of scale into the model, one uses a suitable Mα norm for transportation cost. One then finds that the minimal cost network has a branching structure since the norm favors higher multiplicity edges, representing shared transport. In this thesis, we construct a continuous flow that evolves some initial such network to reduce transport cost without altering its supply and demand distributions. Instead of limiting our scope to transport networks, we construct this M^α mass reducing flow for real-valued flat chains by finding a real current of locally finite mass with the property that its restrictions are flat chains; the slices of such a restriction dictate the flow. Keeping the boundary fixed, this flow reduces the M^α mass of the initial chain and is Lipschitz continuous under the flat-α norm. To complete the thesis, we apply this flow to transportation networks, showing that the flow indeed evolves branching transport networks to be more cost efficient

Low Eigenvalues of Laplace and Schrödinger Operators

This workshop brought together researchers interested in eigenvalue problems for Laplace and Schr¨dinger operators. The main topics o of discussions and investigations covered Dirichlet and Neumann eigenvalue problems, inequalities for the spectral gap, isoperimertic problems and sharp Lieb–Thirring type inequalities. The focus included not only the analytic and geometric sides of the problems, but also related probabilistic and computational aspects