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The parameter-free part $\text{PA}_2^\ast$ of $\text{PA}_2$, the 2nd order Peano arithmetic, is considered. We make use of a product/iterated Sacks forcing to define an $\omega$-model of $\text{PA}_2^\ast + \text{CA}(\Sigma^1_2)$, in which an example of the full Comprehension schema $\text{CA}$ fails. Using Cohen's forcing, we also define an $\omega$-model of $\text{PA}_2^\ast$, in which not every set has its complement, and hence the full $\text{CA}$ fails in a rather elementary way., Comment: 13 pages

Additional file 4:. The contents of the lossgainRSL program configuration file used to identify mouse genes lost in mammals with a long lifespan.

We make use of a finite support product of the Jensen minimal forcing to define a model of set theory in which the separation theorem fails for projective classes $\mathbf\Sigma^1_n$ and $\mathbf\Pi^1_n$, for a given $n\ge3$., Comment: arXiv admin note: text overlap with arXiv:1702.03566

Additional file 1. Additional information on some long-lived mammals.

Additional file 3:. Additional information on some predicted loss genes.

The paper is the second of our series of notes aimed to bring back in circulation some bright ideas of early modern set theory, mainly due to Harrington and Sami, which have never been adequately presented in set theoretic publications. We prove that if a real $a$ is random over a model $M$ and $x\in M[a]$ is another real then either (1) $x\in M$, or (2) $M[x]=M[a]$, or (3) $M[x]$ is a random extension of $M$ and $M[a]$ is a random extension of $M[x]$. This is a less-known result of old set theoretic folklore, and, as far as we know, has never been published. As a corollary, we prove that $\Sigma^1_n$-Reduction holds for all $n\ge3$, in a model extending the constructible universe $L$ by $\aleph_1$-many random reals., Comment: 11 pages

A canonization scheme for smooth equivalence relations on $\mathbb R^\omega$ modulo restriction to infinite perfect products is proposed. It shows that given a pair of Borel smooth equivalence relations $\mathsf E,\mathsf F$ on $\mathbb R^\omega$, there is an infinite perfect product $P\subseteq\mathbb R^\omega$ such that either ${\mathsf F}\subseteq{\mathsf E}$ on $P$, or, for some $j, Comment: A revised version

In a handwtitten note of 1975, Leo Harrington sketched a construction of a model of ZFC (no large cardinals or anything beyond ZFC!) in which $\mathbf\Pi^1_3$-Separation holds but $\mathbf\Sigma^1_3$-Reduction fails. The result has never appeared in a journal or book publication except for a few of old references., Comment: Revised version contains more consistent description on some results of Sami

Part 1. Plots of the mean length (calculated on the base of our data) of leader genes as a function of their distance from the structural gene for the following taxonomic groups. Figure S1.1. Actinobacteria. Figure S1.2. Bacteroides. Figure S1.3. Cyanobacteria; Part 2. Diagrams plotting the number of leader genes as a function of their distance from the structural gene for the following taxonomic groups. Figure S2.1. All Actinobacteria; Figure S2.2. All Actinobacteria (one genome per genus); Figure S2.3. Corynebacterineae; Figure S2.4. Corynebacterium; Figure S2.5. Bifidobacteriales; Figure S2.6. Bifidobacterium; Figure S2.7. Streptomyces; Figure S2.8. Mycobacterium; Figure S2.9. Other Actinobacteria; Figure S2.10. All Proteobacteria; Figure S2.11. All Proteobacteria (one genome per genus); Figure S2.12. Alpha-proteobacteria (one genome per genus); Figure S2.13. Beta-proteobacteria (one genome per genus); Figure S2.14. Delta-proteobacteria (one genome per genus); Figure S2.15. Epsilon-proteobacteria (one genome per genus); Figure S2.16. Gamma-proteobacteria (one genome per genus); Figure S2.17. All Firmicutes; Figure S2.18. All Firmicutes (one genome per genus); Figure S2.19. All Bacteroides; Figure S2.20. All Bacteroides (one genome per genus); Figure S2.21. All Spirochaetales; Figure S2.22. Acidobacteria; Figure S2.23. Aquificaceae; Figure S2.24. Chlamydiae; Figure S2.25. Chlorobia; Figure S2.26. Chloroflexi; Figure S2.27. Cyanobacteria; Figure S2.28. Deinococcusâ Thermus group; Figure S2.29. Fusobacteria; Figure S2.30. Planctomycetes; Figure S2.31. Tenericutes; Figure S2.32. Thermotogae. (PDF 288 kb)

1 (Standard and extra operations allowed to transform initial structures and the joint graph); #2 (Figurs to the algorithm of transforming a joint graph into the final form: the case of different operation weights and all operations.); #3 (Reconstruction of chromosome structures in plastids of rhodophytic branch along the tree of their evolution shown in Fig. 6 , Tables S3 a and S3 b .). (DOCX 679 kb)

Contains the details of step 4 of Algorithm 1 including auxiliary algorithms of semilocal sequence alignment and optimal key extension. (PDF 343 kb)

We prove that Solovay's set $��$ is generic over the ground model via a forcing notion whose order relation $\subseteq$-extends the given order relation., 4 pages

Part 1. Frequency plots for the distance between two adjacent structural genes in Escherichia coli K-12, Bacillus subtilis 168 and Synechocystis sp. PCC 6803. Part 2. Frequency plots for the distance between two adjacent structural genes in Actinobacteria, Cyanobacteria, Firmicutes, Proteobacteria, and Spirochaetales. (PDF 126 kb)

A generic extension $L[x,y]$ of $L$ by reals $x,y$ is defined, in which the union of $\mathsf E_0$-classes of $x$ and $y$ is a $\Pi^1_2$ set, but neither of these two $\mathsf E_0$-classes is separately ordinal-definable.

Part 1. RNA secondary structures of 5'-untranslated regions of proteins with the PF00270 and PF00271 domains in Corynebacterium diphtheria, C. glutamicum, and Bifidobacterium animalis. Figure S1.1. RNA duplex in the region from the stop codon of the leader gene to the start codon of the structural gene encoding helicase in C. diphtheria; Figure S1.2. RNA hairpin overlapping the Shine-Dalgarno sequence in the helicase in C. glutamicum; Figure S1.3. RNA hairpin overlapping two nucleotides of the helicase start codon in B. animalis and Streptomyces griseus. Part 2. Frequency of the leader-structural gene pairs as a function of the leader gene stop codon in Spirochaetales, Acidobacteria, Deinococcus-Thermus group, and Planctomycetes. Figure S2.1. Acidobacteria; Figure S2.2. Deinococcusâ Thermus group; Figure S2.3. Planctomycetes; Figure S2.4. Spirochaetales; Figure S2.5. Actinobacteria. Part 3. Sequence logo of the 30-nt 5'-leader regions of all structural genes in Actinobacteria. (PDF 365 kb)

We make use of a finite support product of the Jensen minimal $\varPi^1_2$ singleton forcing to define a model in which $\varPi^1_2$ Uniformization fails for a set with countable cross-sections. We also define appropriate submodels of the same model in which Separation fails for $\varPi^1_3$., Comment: arXiv admin note: text overlap with arXiv:1408.3901

A generic extension $L[x]$ of $L$ by a real $x$ is defined, in which the $\mathsf E_0$-class of $x$ is a lightface $\Pi^1_2$ set containing no ordinal-definable reals., Comment: arXiv admin note: substantial text overlap with arXiv:1408.3901