The optimal sequence compression

This paper presents the optimal compression for sequences with undefined values. Let we have $(N-m)$ undefined and $m$ defined positions in the boolean sequence $vv V$ of length $N$. The sequence code length can\u27t be less then $m$ in general case, otherwise at least two sequences will have the same code. We present the coding algorithm which generates codes of almost $m$ length, i.e. almost equal to the lower bound. The paper presents the decoding circuit too. The circuit has low complexity which depends from the inverse density of defined values $D(vv V) = frac{N}{m}$. The decoding circuit includes RAM and random logic. It performs sequential decoding. The total RAM size is proportional to the $$logleft(D(vv V) ight) ,$$ the number of random logic cells is proportional to $$log logleft(D(vv V) ight) * left(log log logleft(D(vv V) ight) ight)^2 .$$ So the decoding circuit will be small enough even for the very low density sequences. The decoder complexity doesn\u27t depend of the sequence length at all

Complexity of Nondeterministic Functions

The complexity of a nondeterministic function is the minimum possible complexity of its determinisation. The entropy of a nondeterministic function, F, is minus the logarithm of the ratio between the number of determinisations of F and the number of all deterministic functions. We obtain an upper bound on the complexity of a nondeterministic function with restricted entropy for the worst case. These bounds have strong applications in the problem of algorithm derandomization. A lot of randomized algorithms can be converted to deterministic ones if we have an effective hitting set with certain parameters (a set is hitting for a set system if it has a nonempty intersection with any set from the system). Linial, Luby, Saks and Zuckerman (1993) constructed the best effective hitting set for the system of k-value, n-dimensional rectangles. The set size is polynomial in k log n / epsilon. Our bounds of nondeterministic functions complexity offer a possibility to construct an effective hitting set for this system with almost linear size in k log n / epsilon

Very Large Cliques are Easy to Detect

It is known that, for every constant $kgeq 3$, the presence of a $k$-clique (a complete subgraph on $k$ vertices) in an $n$-vertex graph cannot be detected by a monotone boolean circuit using fewer than $Omega((n/log n)^k)$ gates. We show that, for every constant $k$, the presence of an $(n-k)$-clique in an $n$-vertex graph can be detected by a monotone circuit using only $O(n^2log n)$ gates. Moreover, if we allow unbounded fanin, then $O(log n)$ gates are enough

A Decision Algorithm for Linear Isomorphism of Types with Complexity Cn(log2(n))

It is known that ordinary isomorphisms (associativity and commutativityof "times", isomorphisms for "times" unit and currying)provide a complete axiomatisation for linear isomorphism of types.One of the reasons to consider linear isomorphism of types instead ofordinary isomorphism was that better complexity could be expected.Meanwhile, no upper bounds reasonably close to linear were obtained.We describe an algorithm deciding if two types are linearly isomorphicwith complexity Cn(log2(n))

Exact microscopic analysis of a thermal Brownian motor

We study a genuine Brownian motor by hard disk molecular dynamics and calculate analytically its properties, including its drift speed and thermal conductivity, from microscopic theory.Comment: 4 pages, 5 figure

Low-energy quasiparticle states near extended scatterers in d-wave superconductors and their connection with SUSY quantum mechanics

Low-energy quasiparticle states, arising from scattering by single-particle potentials in d-wave superconductors, are addressed. Via a natural extension of the Andreev approximation, the idea that sign-variations in the superconducting pair-potential lead to such states is extended beyond its original setting of boundary scattering to the broader context of scattering by general single-particle potentials, such as those due to impurities. The index-theoretic origin of these states is exhibited via a simple connection with Witten's supersymmetric quantum-mechanical model.Comment: 5 page

Oxygen and glucose deprivation induces widespread alterations in mRNA translation within 20 minutes

Background: Oxygen and glucose metabolism play pivotal roles in many (patho) physiological conditions. In particular, oxygen and glucose deprivation (OGD) during ischemia and stroke results in extensive tissue injury and cell death. Results: Using time-resolved ribosome profiling, we assess gene expression levels in a neural cell line, PC12, during the first hour of OGD. The most substantial alterations are seen to occur within the first 20 minutes of OGD. While transcription of only 100 genes is significantly altered during one hour of OGD, the translation response affects approximately 3,000 genes. This response involves reprogramming of initiation and elongation rates, as well as the stringency of start codon recognition. Genes involved in oxidative phosphorylation are most affected. Detailed analysis of ribosome profiles reveals salient alterations of ribosome densities on individual mRNAs. The mRNA-specific alterations include increased translation of upstream open reading frames, site-specific ribosome pauses, and production of alternative protein isoforms with amino-terminal extensions. Detailed analysis of ribosomal profiles also reveals six mRNAs with translated ORFs occurring downstream of annotated coding regions and two examples of dual coding mRNAs, where two protein products are translated from the same long segment of mRNA, but in two different frames. Conclusions: These findings uncover novel regulatory mechanisms of translational response to OGD in mammalian cells that are different from the classical pathways such as hypoxia inducible factor (HIF) signaling, while also revealing sophisticated organization of protein coding information in certain genes

Translation initiation downstream from annotated start codons in human mRNAs coevolves with the Kozak context

Eukaryotic translation initiation involves preinitiation ribosomal complex 5′ -to-3′ directional probing of mRNA for codons suitable for starting protein synthesis. The recognition of codons as starts depends on the codon identity and on its immediate nucleotide context known as Kozak context. When the context is weak (i.e., nonoptimal), leaky scanning takes place during which a fraction of ribosomes continues the mRNA probing. We explored the relationship between the context of AUG codons annotated as starts of protein-coding sequences and the next AUG codon occurrence. We found that AUG codons downstream from weak starts occur in the same frame more frequently than downstream from strong starts. We suggest that evolutionary selection on in-frame AUGs downstream from weak start codons is driven by the advantage of the reduction of wasteful out-of-frame product synthesis and also by the advantage of producing multiple proteoforms from certain mRNAs. We confirmed translation initiation downstream from weak start codons using ribosome profiling data. We also tested translation of alternative start codons in 10 specific human genes using reporter constructs. In all tested cases, initiation at downstream start codons was more productive than at the annotated ones. In most cases, optimization of Kozak context did not completely abolish downstream initiation, and in the specific example of CMPK1 mRNA, the optimized start remained unproductive. Collectively, our work reveals previously uncharacterized forces shaping the evolution of protein-coding genes and points to the plurality of translation initiation and the existence of sequence features influencing start codon selection, other than Kozak context.Russian Science Foundation (RSF) 20-14-00121Science Foundation Ireland 210692/Z/18/ZScience Foundation Ireland 12/RC/2276_P2Erasmus+ ProgrammePlan Propio de Investigacion 2019 de la Universidad de GranadaMinistry of Economy of Spain DPI2017-84439-REuropean Union (EU) DPI2017-84439-