Non-commutative counting invariants and curve complexes

In our previous paper, viewing $D^b(K(l))$ as a non-commutative curve, where $K(l)$ is the Kronecker quiver with $l$-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 $D^b(K(l))$. 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 $D^b(Q)$ for two affine quivers, for $A_n$ and $D_4$ we derive formulas for counting of the subcategories of type $D^b(A_k)$ in $D^b(A_n)$, whereas for the two affine quivers and for $D_4$ we determine and count all generated by an exceptional collection subcategories. Estimating the numbers counting non-commutative curves in $D^b({\mathbb P}^2)$ 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 $D^b(P^2)$. 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 $\mathcal J$ in the definition of $C_{\mathcal J, P}(\mathcal T)$ 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 $Q$, $Q'$ (i.e. quivers of extended Dynkin type) there are only finitely many full triangulated subctegories in $D^b(Rep_{\mathbb K}(Q))$, which are equivalent to $D^b(Rep_{\mathbb K}(Q'))$, where ${\mathbb K}$ 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