Pure Σ2\Sigma_2-Elementarity beyond the Core

We display the entire structure ${\cal R}_2$ coding $\Sigma_1$- and $\Sigma_2$-elementarity on the ordinals. This will enable the analysis of pure $\Sigma_3$-elementary substructures.Comment: Extensive rewrite of the introduction. Mathematical content of sections 2 and 3 unchanged, extended introduction to section 2. Removed section 4. Theorem 4.3 to appear elsewhere with corrected proo

Hidden Symmetries of Large N QCD

The local SUSY symmetry of the loop dynamics of QCD is found. The remarkable thing is, there is no einbein-gravitino on this theory, which makes it a 1D topological supergravity, or locally SUSY quantum mechanics. Using this symmetry, we derive the large $N_c$ loop equation in momentum superloop space. Introducing as before the position operator \X{\mu} we argue that the superloop equation is equivalent to invariance of correlation functions of products of these operators with respect to certain quadrilinear transformation. The applications to meson and glueball sectors as well as the chiral symmetry breaking are discussed. The 1D field theory with Quark propagating around the loop in superspace is constructed.Comment: 40 pages, 1 Postscript figure, LaTe

Classical Set Theory: Theory of Sets and Classes

This is a short introductory course to Set Theory, based on axioms of von Neumann--Bernays--G\"odel (briefly NBG). The text can be used as a base for a lecture course in Foundations of Mathematics, and contains a reasonable minimum which a good (post-graduate) student in Mathematics should know about foundations of this science.Comment: 162 page

High-throughput analyses and Bayesian network modeling highlight novel epigenetic Adverse Outcome Pathway networks of DNA methyltransferase inhibitor mediated transgenerational effects

A number of epigenetic modulating chemicals are known to affect multiple generations of a population from a single ancestral exposure, thus posing transgenerational hazards. The present study aimed to establish a high-throughput (HT) analytical workflow for cost-efficient concentration-response analysis of epigenetic and phenotypic effects, and to support the development of novel Adverse Outcome Pathway (AOP) networks for DNA methyltransferase (DNMT) inhibitor-mediated transgenerational effects on aquatic organisms. The model DNMT inhibitor 5-azacytidine (5AC) and the model freshwater crustacean Daphnia magna were used to generate new experimental data and served as prototypes to construct AOPs for aquatic organisms. Targeted HT bioassays (DNMT ELISA, MS-HRM and qPCR) in combination with multigenerational ecotoxicity tests revealed concentration-dependent transgenerational (F0-F3) effects of 5AC on total DNMT activity, DNA promoter methylation, gene body methylation, gene transcription and reproduction. Top sensitive toxicity pathways related to 5AC exposure, such as apoptosis and DNA damage responses were identified in both F0 and F3 using Gaussian Bayesian network modeling. Two novel epigenetic AOP networks on DNMT inhibitor mediated one-generational and transgenerational effects were developed for aquatic organisms and assessed for the weight of evidence. The new HT analytical workflow and AOPs can facilitate future ecological hazard assessment of epigenetic modulating chemicals

Discreteness of asymptotic tensor ranks

Tensor parameters that are amortized or regularized over large tensor powers, often called "asymptotic" tensor parameters, play a central role in several areas including algebraic complexity theory (constructing fast matrix multiplication algorithms), quantum information (entanglement cost and distillable entanglement), and additive combinatorics (bounds on cap sets, sunflower-free sets, etc.). Examples are the asymptotic tensor rank, asymptotic slice rank and asymptotic subrank. Recent works (Costa-Dalai, Blatter-Draisma-Rupniewski, Christandl-Gesmundo-Zuiddam) have investigated notions of discreteness (no accumulation points) or "gaps" in the values of such tensor parameters. We prove a general discreteness theorem for asymptotic tensor parameters of order-three tensors and use this to prove that (1) over any finite field, the asymptotic subrank and the asymptotic slice rank have no accumulation points, and (2) over the complex numbers, the asymptotic slice rank has no accumulation points. Central to our approach are two new general lower bounds on the asymptotic subrank of tensors, which measures how much a tensor can be diagonalized. The first lower bound says that the asymptotic subrank of any concise three-tensor is at least the cube-root of the smallest dimension. The second lower bound says that any three-tensor that is "narrow enough" (has one dimension much smaller than the other two) has maximal asymptotic subrank. Our proofs rely on new lower bounds on the maximum rank in matrix subspaces that are obtained by slicing a three-tensor in the three different directions. We prove that for any concise tensor the product of any two such maximum ranks must be large, and as a consequence there are always two distinct directions with large max-rank