Around Kolmogorov complexity: basic notions and results

Algorithmic information theory studies description complexity and randomness and is now a well known field of theoretical computer science and mathematical logic. There are several textbooks and monographs devoted to this theory where one can find the detailed exposition of many difficult results as well as historical references. However, it seems that a short survey of its basic notions and main results relating these notions to each other, is missing. This report attempts to fill this gap and covers the basic notions of algorithmic information theory: Kolmogorov complexity (plain, conditional, prefix), Solomonoff universal a priori probability, notions of randomness (Martin-L\"of randomness, Mises--Church randomness), effective Hausdorff dimension. We prove their basic properties (symmetry of information, connection between a priori probability and prefix complexity, criterion of randomness in terms of complexity, complexity characterization for effective dimension) and show some applications (incompressibility method in computational complexity theory, incompleteness theorems). It is based on the lecture notes of a course at Uppsala University given by the author

Identification of a small RNA containing the trypanosome spliced leader: a donor of shared 5' sequences of trypanosomatid mRNAs?

The 35 nucleotide spliced leader (SL) sequence is found on the 5' end of numerous trypanosome mRNAs, yet the tandemly organized reiteration units encoding this leader are not detectably linked to any of these structural genes. Here we report the presence of a class of discrete small SL RNA molecules that are derived from the genomic SL reiteration units of Trypanosoma brucei, Trypanosoma cruzi, and Leptomonas collosoma. These small SL RNAs are 135, 105, and 95 nucleotides, respectively, and contain a 5'-terminal SL or SL-like sequence. S1 nuclease analyses demonstrate that these small SL RNAs are transcribed from continuous sequence within the respective SL reiteration units. With the exception of the SL sequence and a concensus donor splice site immediately following it, these small RNAs are not well conserved. We suggest that the small SL RNAs may function as a donor of the SL sequence in an intermolecular process that places the SL at the 5' terminus of many trypanosomatid mRNAs

Towards Informative Statistical Flow Inversion

This is the accepted version of 'Towards Informative Statistical Flow Inversion', archived originally at arXiv:0705.1939v1 [cs.NI] 14 May 2007.A problem which has recently attracted research attention is that of estimating the distribution of flow sizes in internet traffic. On high traffic links it is sometimes impossible to record every packet. Researchers have approached the problem of estimating flow lengths from sampled packet data in two separate ways. Firstly, different sampling methodologies can be tried to more accurately measure the desired system parameters. One such method is the sample-and-hold method where, if a packet is sampled, all subsequent packets in that flow are sampled. Secondly, statistical methods can be used to ``invert'' the sampled data and produce an estimate of flow lengths from a sample. In this paper we propose, implement and test two variants on the sample-and-hold method. In addition we show how the sample-and-hold method can be inverted to get an estimation of the genuine distribution of flow sizes. Experiments are carried out on real network traces to compare standard packet sampling with three variants of sample-and-hold. The methods are compared for their ability to reconstruct the genuine distribution of flow sizes in the traffic

Invertibility in groupoid C*-algebras

Given a second-countable, Hausdorff, \'etale, amenable groupoid G with compact unit space, we show that an element a in C*(G) is invertible if and only if \lambda_x(a) is invertible for every x in the unit space of G, where \lambda_x refers to the "regular representation" of C*(G) on l_2(G_x). We also prove that, for every a in C*(G), there exists some x in G^{(0)} such that ||a|| = ||\lambda_x(a)||.Comment: 8 page

SYM, Chern-Simons, Wess-Zumino Couplings and their higher derivative corrections in IIA Superstring theory

We find the entire form of the amplitude of two fermion strings (with different chirality), a massless scalar field and one closed string Ramond-Ramond (RR) in IIA superstring theory which is different from its IIB one. We make use of a very particular gauge fixing and explore several new couplings in IIA. All infinite $u$- channel scalar poles and $t,s$- channel fermion poles are also constructed. We find new form of higher derivative corrections to two fermion two scalar couplings and show that the first simple $(s+t+u)-$ channel scalar pole for $p+2=n$ case can be obtained by having new higher derivative corrections to SYM couplings at third order of $\alpha'$. We find that the general structure and the coefficients of higher derivative corrections to two fermion two scalar couplings are completely different from the derived $\alpha'$ higher derivative corrections of type IIB.Comment: 29 pages, no figure,Latex file,published version in EPJ

The Hardness of Embedding Grids and Walls

The dichotomy conjecture for the parameterized embedding problem states that the problem of deciding whether a given graph $G$ from some class $K$ of "pattern graphs" can be embedded into a given graph $H$ (that is, is isomorphic to a subgraph of $H$) is fixed-parameter tractable if $K$ is a class of graphs of bounded tree width and $W[1]$-complete otherwise. Towards this conjecture, we prove that the embedding problem is $W[1]$-complete if $K$ is the class of all grids or the class of all walls

Trypanosome mRNAs share a common 5' spliced leader sequence.

A 5'-terminal leader sequence of 35 nucleotides was found to be present on multiple trypanosome RNAs. Based on its representation in cDNA libraries, we estimate that many, if not all, trypanosome mRNAs contain this leader. This same leader was originally identified on mRNAs encoding the molecules responsible for antigenic variation, variant surface glycoproteins. Studies of selected cDNAs containing this leader sequence revealed that leader-containing transcripts can be stage-specific, stage-regulated, or constitutive. They can be abundant or rare, and transcribed from single or multigene families. No linkage between the genomic leader sequences and the structural gene exons was observed. Possible mechanisms by which the leader sequences are added to trypanosome mRNAs are discussed

The polarized expression of Na+,K+-ATPase in epithelia depends on the association between beta-subunits located in neighboring cells

The polarized distribution of Na+,K+-ATPase plays a paramount physiological role, because either directly or through coupling with co- and countertransporters, it is responsible for the net movement of, for example, glucose, amino acids, Ca2+, K+, Cl-, and CO3H- across the whole epithelium. We report here that the beta-subunit is a key factor in the polarized distribution of this enzyme. 1) Madin-Darby canine kidney (MDCK) cells (epithelial from dog kidney) express the Na+,K+-ATPase over the lateral side, but not on the basal and apical domains, as if the contact with a neighboring cell were crucial for the specific membrane location of this enzyme. 2) MDCK cells cocultured with other epithelial types (derived from human, cat, dog, pig, monkey, rabbit, mouse, hamster, and rat) express the enzyme in all (100%) homotypic MDCK/MDCK borders but rarely in heterotypic ones. 3) Although MDCK cells never express Na+,K+-ATPase at contacts with Chinese hamster ovary (CHO) cells, they do when CHO cells are transfected with beta(1)-subunit from the dog kidney (CHO-beta). 4) This may be attributed to the adhesive property of the beta(1)-subunit, because an aggregation assay using CHO (mock-transfected) and CHO-beta cells shows that the expression of dog beta(1)-subunit in the plasma membrane does increase adhesiveness. 5) This adhesiveness does not involve adherens or tight junctions. 6) Transfection of beta(1)-subunit forces CHO-beta cells to coexpress endogenous a-subunit. Together, our results indicate that MDCK cells express Na+,K+-ATPase at a given border provided the contacting cell expresses the dog P,-subunit. The cell-cell interaction thus established would suffice to account for the polarized expression and positioning of Na+,K+-ATPase in epithelial cells

Degree spectra for transcendence in fields

We show that for both the unary relation of transcendence and the finitary relation of algebraic independence on a field, the degree spectra of these relations may consist of any single computably enumerable Turing degree, or of those c.e. degrees above an arbitrary fixed $\Delta^0_2$ degree. In other cases, these spectra may be characterized by the ability to enumerate an arbitrary $\Sigma^0_2$ set. This is the first proof that a computable field can fail to have a computable copy with a computable transcendence basis