The number of unit distances is almost linear for most norms

We prove that there exists a norm in the plane under which no n-point set determines more than O(n log n log log n) unit distances. Actually, most norms have this property, in the sense that their complement is a meager set in the metric space of all norms (with the metric given by the Hausdorff distance of the unit balls)

Blocking Visibility for Points in General Position

For a finite set P in the plane, let b(P) be the smallest possible size of a set Q, Q∩P=∅, such that every segment with both endpoints in P contains at least one point of Q. We raise the problem of estimating b(n), the minimum of b(P) over all n-point sets P with no three points collinear. We review results providing bounds on b(n) and mention some additional observation

Implementation and Verification of Network Interface Blocks

V rámci platformy NetCOPE se vstupní a výstupní síťové bloky používají pro odstínění návrháře síťové aplikace od problémů s implementací linkové vstvy síťového modelu ISO/OSI, zvláště pak její MAC podvrstvy. Tato bakalářská práce se zabývá návrhem, implementací a verifikací takovýchto bloků pracujících na rychlosti 10 Gb/s. Navržený vstupní síťový blok provádí kontrolu příchozích rámců a umožňuje zahazování těchto rámců na základě výsledků prováděných kontrol. Výstupní síťový blok podporuje nahrazování zdrojové MAC adresy rámce a doplnění pole FCS. Součástí obou síťových bloků jsou také různé druhy čítačů rámců. Navržené síťové bloky byly otestovány na kartách COMBO v rámci platformy NetCOPE a bylo pro ně navrženo verifikační prostředí pro jazyk SystemVerilog.Network interface blocks are basic part of the NetCOPE platform where they help to the network application designers to deal with problems of implementing the Data Link Layer of the OSI Reference Model, especially the MAC sublayer. This thesis is focused on the design and implementation of such network interface blocks operating at speed 10 Gb/s. Designed input interface block provides checking of several parts of the Ethernet frame and allows discarding of this frame based on checking results. Output interface block supports replacing frame's Source Address by a pre-set value and provides frame's CRC computation. Both network interface blocks also include a set of frames counters. Implemented network interface blocks were tested on the COMBO card. SystemVerilog verification testbench was also designed for both network interface blocks.

Ramsey-like properties for bi-Lipschitz mappings of finite metric spaces

summary:Let $(X,\rho)$, $(Y,\sigma)$ be metric spaces and $f:X\to Y$ an injective mapping. We put $\|f\|_{Lip} = \sup \{\sigma (f(x),f(y))/\rho(x,y)$; $x,y\in X$, $x\neq y\}$, and $\operatorname{dist}(f)= \|f\|_{Lip}.\|f^{-1}\|_{Lip}$ (the {\sl distortion\/} of the mapping $f$). Some Ramsey-type questions for mappings of finite metric spaces with bounded distortion are studied; e.g., the following theorem is proved: Let $X$ be a finite metric space, and let $\varepsilon>0$, $K$ be given numbers. Then there exists a finite metric space $Y$, such that for every mapping $f:Y\to Z$ ($Z$ arbitrary metric space) with $\operatorname{dist}(f)<K$ one can find a mapping $g:X\to Y$, such that both the mappings $g$ and $f|_{g(X)}$ have distortion at most $(1+\varepsilon)$. If $X$ is isometrically embeddable into a $\ell_p$ space (for some $p\in [1,\infty]$), then also $Y$ can be chosen with this property

Note on bi-Lipschitz embeddings into normed spaces

summary:Let $(X,d)$, $(Y,\rho)$ be metric spaces and $f:X\to Y$ an injective mapping. We put $\|f\|_{\operatorname{Lip}} = \sup \{\rho (f(x),f(y))/d(x,y); x,y\in X, x\neq y\}$, and $\operatorname{dist}(f)= \|f\|_{\operatorname{Lip}}.\| f^{-1}\|_{\operatorname{Lip}}$ (the {\sl distortion} of the mapping $f$). We investigate the minimum dimension $N$ such that every $n$-point metric space can be embedded into the space $\ell_{\infty }^N$ with a prescribed distortion $D$. We obtain that this is possible for $N\geq C(\log n)^2 n^{3/D}$, where $C$ is a suitable absolute constant. This improves a result of Johnson, Lindenstrauss and Schechtman [JLS87] (with a simpler proof). Related results for embeddability into $\ell_p^N$ are obtained by a similar method

Removing Degeneracy in LP-Type Problems Revisited

LP-type problems is a successful axiomatic framework for optimization problems capturing, e.g., linear programming and the smallest enclosing ball of a point set. In Matoušek and Škovroň (Theory Comput. 3:159-177, 2007), it is proved that in order to remove degeneracies of an LP-type problem, we sometimes have to increase its combinatorial dimension by a multiplicative factor of at least 1+ε with a certain small positive constant ε. The proof goes by checking the unsolvability of a system of linear inequalities, with several pages of calculations. Here by a short topological argument we prove that the dimension sometimes has to increase at least twice. We also construct 2-dimensional LP-type problems with −∞ for which removing degeneracies forces arbitrarily large dimension increas

A Combinatorial Proof of Kneser'sConjecture*

Kneser's conjecture, first proved by Lovász in 1978, states that the graph with all k-element subsets of {1, 2, . . . , n} as vertices and with edges connecting disjoint sets has chromatic number n−2k+2. We derive this result from Tucker's combinatorial lemma on labeling the vertices of special triangulations of the octahedral ball. By specializing a proof of Tucker's lemma, we obtain self-contained purely combinatorial proof of Kneser's conjectur

String graphs and separators

String graphs, that is, intersection graphs of curves in the plane, have been studied since the 1960s. We provide an expository presentation of several results, including very recent ones: some string graphs require an exponential number of crossings in every string representation; exponential number is always sufficient; string graphs have small separators; and the current best bound on the crossing number of a graph in terms of the pair-crossing number. For the existence of small separators, unwrapping the complete proof include generally useful results on approximate flow-cut dualities.Comment: Expository paper based on course note