3,380 research outputs found
Non-commutative counting invariants and curve complexes
In our previous paper, viewing as a non-commutative curve, where
is the Kronecker quiver with -arrows, we introduced categorical
invariants via counting of non-commutative curves. Roughly, these invariants
are sets of subcategories in a given category and their quotients. The
non-commutative curve-counting invariants are obtained by restricting the
subcategories to be equivalent to . The general definition defines
much larger class of invariants and many of them behave properly with respect
to fully faithful functors. Here, after recalling the definition, we focus on
examples and extend our studies beyond counting. We enrich our invariants with
structures: the inclusion of subcategories makes them partially ordered sets,
and considering semi-orthogonal pairs of subcategories as edges amount to
directed graphs. In addition to computing the non-commutative curve-counting
invariants in for two affine quivers, for and we derive
formulas for counting of the subcategories of type in ,
whereas for the two affine quivers and for we determine and count all
generated by an exceptional collection subcategories. Estimating the numbers
counting non-commutative curves in modulo group action we
prove finiteness and that an exact determining of these numbers leads to
proving (or disproving) of Markov conjecture. Regarding the mentioned structure
of a partially ordered set we initiate intersection theory of non-commutative
curves. Via the structure of a directed graph we build an analogue to the
classical curve complex used in Teichmueller and Thurston theory. The paper
contains many pictures of graphs and presents an approach to Markov Conjecture
via counting of subgraphs in a graph associated with . Some of the
results proved here were announced in the previous work.Comment: In v4, 65 pages, we have reorganized the paper and removed some
inaccuracies. Sections 2 to 7 are dedicated to general theory and then follow
sections with examples. In the previous version the letter in
the definition of was a set of non-trivial
pairwise non-equivalent triangulated categories. Now we remove the
restriction of non-trivialit
More finite sets coming from non-commutative counting
In our previous papers we introduced categorical invariants, which are,
roughly speaking, sets of triangulated subcategories in a given triangulated
category and their quotients. Here is extended the list of examples, where
these sets are finite. Using results by Geigle, Lenzning, Meltzer, H\"ubner for
weighted projective lines we show that for any two affine acyclic quivers ,
(i.e. quivers of extended Dynkin type) there are only finitely many full
triangulated subctegories in , which are equivalent to
, where is an algebraically closed
field. Some of the numbers counting the elements in these finite sets are
explicitly determined.Comment: 16 pages, In v3 Corollary 5.6 does not depend on any additional
conditions, because in a private communication Professor Helmut Lenzing
confirmed that (21) is correct. The last section 6 and the introduction in
the new version are slightly extended. The reference list is also update
Dynamical systems and categories
We study questions motivated by results in the classical theory of dynamical
systems in the context of triangulated and A-infinity categories. First,
entropy is defined for exact endofunctors and computed in a variety of
examples. In particular, the classical entropy of a pseudo-Anosov map is
recovered from the induced functor on the Fukaya category. Second, the density
of the set of phases of a Bridgeland stability condition is studied and a
complete answer is given in the case of bounded derived categories of quivers.
Certain exceptional pairs in triangulated categories, which we call Kronecker
pairs, are used to construct stability conditions with density of phases. Some
open questions and further directions are outlined as well.Comment: 35 page
Coffee Queue Project
In this paper, a computer vision system for counting people standing in line is presented. In this application, common techniques such as Adaptive Background Subtraction (ABS), blob tracking with Kalman filter, and occlusion resistive techniques are used to detect and track people. Additionally, a novel method using Dual Adaptive Background Subtractors (DABS) is implemented for dynamically determining the line region in a real-world crowded scene, and also as an alternative target acquisition to regular ABS. The DABS technique acts as a temporal bandpass filter for motion, helping identify people standing in line while in the presence of other moving people. This is achieved by using two ABS with different temporal adaptiveness. Unlike other computer vision papers which perform tests in highly controlled environments, the DABS technique is tested in a crowded Starbucks© at the Cal Poly student union. For any length of people standing in line, result shows that DABS has a lower mean error by one or more people when compared to ABS. Even in challenging crowded scenes where the line can reach 19 people in length, DABS achieves a Normalized RMS Error of 43%
e-Justice as adopted in Bulgaria
Associate Professor Dr George G. Dimitrov sets out the Concept on E-Justice adopted by the Bulgarian Council of Ministers in 2012, and sets out the purpose of the Multi-annual Action Plan for the period 2009-2013 in the area of European E-Justice Index words: Bulgaria; e-Justice; judicial reform; legal effect of electronic documents and electronic signatures in the judicial system; amending substantive and procedural laws
Legal aspects of electronic signatures in Bulgaria
George G. Dimitrov examines the Bulgarian law on electronic signatures in detai
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