902 research outputs found
A Categorification of the Burau representation at prime roots of unity
We construct a p-DG structure on an algebra Koszul dual to a zigzag algebra
used by Khovanov and Seidel to construct a categorical braid group action. We
show there is a braid group action in this p-DG setting.Comment: 33 pages, many PSTricks figures, color viewing not essential, v2
contains minor corrections, comments welcom
Counting Value Sets: Algorithm and Complexity
Let be a prime. Given a polynomial in \F_{p^m}[x] of degree over
the finite field \F_{p^m}, one can view it as a map from \F_{p^m} to
\F_{p^m}, and examine the image of this map, also known as the value set. In
this paper, we present the first non-trivial algorithm and the first complexity
result on computing the cardinality of this value set. We show an elementary
connection between this cardinality and the number of points on a family of
varieties in affine space. We then apply Lauder and Wan's -adic
point-counting algorithm to count these points, resulting in a non-trivial
algorithm for calculating the cardinality of the value set. The running time of
our algorithm is . In particular, this is a polynomial time
algorithm for fixed if is reasonably small. We also show that the
problem is #P-hard when the polynomial is given in a sparse representation,
, and is allowed to vary, or when the polynomial is given as a
straight-line program, and is allowed to vary. Additionally, we prove
that it is NP-hard to decide whether a polynomial represented by a
straight-line program has a root in a prime-order finite field, thus resolving
an open problem proposed by Kaltofen and Koiran in
\cite{Kaltofen03,KaltofenKo05}
On some -differential graded link homologies
We show that the triply graded Khovanov-Rozansky homology of knots and links
over a field of positive odd characteristic descends to an invariant in the
homotopy category finite-dimensional -complexes.
A -extended differential on the triply graded homology discovered by
Cautis is compatible with the -DG structure. As a consequence we get a
categorification of the Jones polynomial evaluated at an odd prime root of
unityComment: 52 pages. Comments welcom
Strategy evolution on dynamic networks
Models of strategy evolution on static networks help us understand how
population structure can promote the spread of traits like cooperation. One key
mechanism is the formation of altruistic spatial clusters, where neighbors of a
cooperative individual are likely to reciprocate, which protects prosocial
traits from exploitation. But most real-world interactions are ephemeral and
subject to exogenous restructuring, so that social networks change over time.
Strategic behavior on dynamic networks is difficult to study, and much less is
known about the resulting evolutionary dynamics. Here, we provide an analytical
treatment of cooperation on dynamic networks, allowing for arbitrary spatial
and temporal heterogeneity. We show that transitions among a large class of
network structures can favor the spread of cooperation, even if each individual
social network would inhibit cooperation when static. Furthermore, we show that
spatial heterogeneity tends to inhibit cooperation, whereas temporal
heterogeneity tends to promote it. Dynamic networks can have profound effects
on the evolution of prosocial traits, even when individuals have no agency over
network structures.Comment: 45 pages; final versio
Draft Genome Sequence of Streptomyces sp. Strain JV178, a Producer of Clifednamide-Type Polycyclic Tetramate Macrolactams
Here, we report the draft genome sequence of Streptomyces sp. JV178, a strain originating from Connecticut (USA) garden soil. This strain produces the polycyclic tetramate macrolactam compounds clifednamides A and B. The draft genome contains 10.65 Mb, 9,045 predicted protein coding sequences, and several natural product biosynthetic loci
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