178 research outputs found
Boundaries of Amplituhedra and NMHV Symbol Alphabets at Two Loops
In this sequel to arXiv:1711.11507 we classify the boundaries of amplituhedra
relevant for determining the branch points of general two-loop amplitudes in
planar super-Yang-Mills theory. We explain the connection to
on-shell diagrams, which serves as a useful cross-check. We determine the
branch points of all two-loop NMHV amplitudes by solving the Landau equations
for the relevant configurations and are led thereby to a conjecture for the
symbol alphabets of all such amplitudes.Comment: 42 pages, 6 figures, 8 tables; v2: minor corrections and improvement
Deformations of colored sl(N) link homologies via foams
We generalize results of Lee, Gornik and Wu on the structure of deformed
colored sl(N) link homologies to the case of non-generic deformations. To this
end, we use foam technology to give a completely combinatorial construction of
Wu's deformed colored sl(N) link homologies. By studying the underlying
deformed higher representation theoretic structures and generalizing the
Karoubi envelope approach of Bar-Natan and Morrison we explicitly compute the
deformed invariants in terms of undeformed type A link homologies of lower rank
and color.Comment: 64 pages, many figure
Extensions and reductions of square-free words
A word is square-free if it does not contain a nonempty word of the form
as a factor. A famous 1906 result of Thue asserts that there exist arbitrarily
long square-free words over a -letter alphabet. We study square-free words
with additional properties involving single-letter deletions and extensions of
words.
A square-free word is steady if it remains square-free after deletion of any
single letter. We prove that there exist infinitely many steady words over a
-letter alphabet. We also demonstrate that one may construct steady words of
any length by picking letters from arbitrary alphabets of size assigned to
the positions of the constructed word. We conjecture that both bounds can be
lowered to , which is best possible.
In the opposite direction, we consider square-free words that remain
square-free after insertion of a single (suitably chosen) letter at every
possible position in the word. We call them bifurcate. We prove a somewhat
surprising fact, that over a fixed alphabet with at least three letters, every
steady word is bifurcate. We also consider families of bifurcate words
possessing a natural tree structure. In particular, we prove that there exists
an infinite tree of doubly infinite bifurcate words over alphabet of size .Comment: 11 pages, 1 figur
Local Graph Coloring and Index Coding
We present a novel upper bound for the optimal index coding rate. Our bound
uses a graph theoretic quantity called the local chromatic number. We show how
a good local coloring can be used to create a good index code. The local
coloring is used as an alignment guide to assign index coding vectors from a
general position MDS code. We further show that a natural LP relaxation yields
an even stronger index code. Our bounds provably outperform the state of the
art on index coding but at most by a constant factor.Comment: 14 Pages, 3 Figures; A conference version submitted to ISIT 2013;
typos correcte
Diszkrét matematika = Discrete mathematics
A pályázat résztvevői igen aktívak voltak a 2006-2008 években. Nemcsak sok eredményt értek el, miket több mint 150 cikkben publikáltak, eredményesen népszerűsítették azokat. Több mint 100 konferencián vettek részt és adtak elő, felerészben meghívott, vagy plenáris előadóként. Hagyományos gráfelmélet Több extremális gráfproblémát oldottunk meg. Új eredményeket kaptunk Ramsey számokról, globális és lokális kromatikus számokról, Hamiltonkörök létezéséséről. a crossig numberről, gráf kapacitásokról és kizárt részgráfokról. Véletlen gráfok, nagy gráfok, regularitási lemma Nagy gráfok "hasonlóságait" vizsgáltuk. Különféle metrikák ekvivalensek. Űj eredeményeink: Hereditary Property Testing, Inverse Counting Lemma and the Uniqueness of Hypergraph Limit. Hipergráfok, egyéb kombinatorika Új Sperner tipusú tételekte kaptunk, aszimptotikusan meghatározva a halmazok max számát bizonyos kizárt struktőrák esetén. Több esetre megoldottuk a kizárt hipergráf problémát is. Elméleti számítástudomány Új ujjlenyomat kódokat és bioinformatikai eredményeket kaptunk. | The participants of the project were scientifically very active during the years 2006-2008. They did not only obtain many results, which are contained in their more than 150 papers appeared in strong journals, but effectively disseminated them in the scientific community. They participated and gave lectures in more than 100 conferences (with multiplicity), half of them were plenary or invited talks. Traditional graph theory Several extremal problems for graphs were solved. We obtained new results for certain Ramsey numbers, (local and global) chromatic numbers, existence of Hamiltonian cycles crossing numbers, graph capacities, and excluded subgraphs. Random graphs, large graphs, regularity lemma The "similarities" of large graphs were studied. We show that several different definitions of the metrics (and convergence) are equivalent. Several new results like the Hereditary Property Testing, Inverse Counting Lemma and the Uniqueness of Hypergraph Limit were proved Hypergraphs, other combinatorics New Sperner type theorems were obtained, asymptotically determining the maximum number of sets in a family of subsets with certain excluded configurations. Several cases of the excluded hypergraph problem were solved. Theoretical computer science New fingerprint codes and results in bioinformatics were found
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