Algebra of differential operators associated with Young diagrams

We establish a correspondence between Young diagrams and differential operators of infinitely many variables. These operators form a commutative associative algebra isomorphic to the algebra of the conjugated classes of finite permutations of the set of natural numbers. The Schur functions form a complete system of common eigenfunctions of these differential operators, and their eigenvalues are expressed through the characters of symmetric groups. The structure constants of the algebra are expressed through the Hurwitz numbers.Comment: 11 page

New words in human mutagenesis

<p>Abstract</p> <p>Background</p> <p>The substitution rates within different nucleotide contexts are subject to varying levels of bias. The most well known example of such bias is the excess of C to T (C > T) mutations in CpG (CG) dinucleotides. The molecular mechanisms underlying this bias are important factors in human genome evolution and cancer development. The discovery of other nucleotide contexts that have profound effects on substitution rates can improve our understanding of how mutations are acquired, and why mutation hotspots exist.</p> <p>Results</p> <p>We compared rates of inherited mutations in 1-4 bp nucleotide contexts using reconstructed ancestral states of human single nucleotide polymorphisms (SNPs) from intergenic regions. Chimp and orangutan genomic sequences were used as outgroups. We uncovered 3.5 and 3.3-fold excesses of T > C mutations in the second position of ATTG and ATAG words, respectively, and a 3.4-fold excess of A > C mutations in the first position of the ACAA word.</p> <p>Conclusions</p> <p>Although all the observed biases are less pronounced than the 5.1-fold excess of C > T mutations in CG dinucleotides, the three 4 bp mutation contexts mentioned above (and their complementary contexts) are well distinguished from all other mutation contexts. This provides a challenge to discover the underlying mechanisms responsible for the observed excesses of mutations.</p

BGWM as Second Constituent of Complex Matrix Model

Earlier we explained that partition functions of various matrix models can be constructed from that of the cubic Kontsevich model, which, therefore, becomes a basic elementary building block in "M-theory" of matrix models. However, the less topical complex matrix model appeared to be an exception: its decomposition involved not only the Kontsevich tau-function but also another constituent, which we now identify as the Brezin-Gross-Witten (BGW) partition function. The BGW tau-function can be represented either as a generating function of all unitary-matrix integrals or as a Kontsevich-Penner model with potential 1/X (instead of X^3 in the cubic Kontsevich model).Comment: 42 page

Integrability properties of Hurwitz partition functions. II. Multiplication of cut-and-join operators and WDVV equations

Correlators in topological theories are given by the values of a linear form on the products of operators from a commutative associative algebra (CAA). As a corollary, partition functions of topological theory always satisfy the generalized WDVV equations. We consider the Hurwitz partition functions, associated in this way with the CAA of cut-and-join operators. The ordinary Hurwitz numbers for a given number of sheets in the covering provide trivial (sums of exponentials) solutions to the WDVV equations, with finite number of time-variables. The generalized Hurwitz numbers from arXiv:0904.4227 provide a non-trivial solution with infinite number of times. The simplest solution of this type is associated with a subring, generated by the dilatation operators tr X(d/dX).Comment: 24 page

Homothetic perfect fluid space-times

A brief summary of results on homotheties in General Relativity is given, including general information about space-times admitting an r-parameter group of homothetic transformations for r>2, as well as some specific results on perfect fluids. Attention is then focussed on inhomogeneous models, in particular on those with a homothetic group $H_4$ (acting multiply transitively) and $H_3$. A classification of all possible Lie algebra structures along with (local) coordinate expressions for the metric and homothetic vectors is then provided (irrespectively of the matter content), and some new perfect fluid solutions are given and briefly discussed.Comment: 27 pages, Latex file, Submitted to Class. Quantum Gra

Small Cofactors May Assist Protein Emergence from RNA World: Clues from RNA-Protein Complexes

It is now widely accepted that at an early stage in the evolution of life an RNA world arose, in which RNAs both served as the genetic material and catalyzed diverse biochemical reactions. Then, proteins have gradually replaced RNAs because of their superior catalytic properties in catalysis over time. Therefore, it is important to investigate how primitive functional proteins emerged from RNA world, which can shed light on the evolutionary pathway of life from RNA world to the modern world. In this work, we proposed that the emergence of most primitive functional proteins are assisted by the early primitive nucleotide cofactors, while only a minority are induced directly by RNAs based on the analysis of RNA-protein complexes. Furthermore, the present findings have significant implication for exploring the composition of primitive RNA, i.e., adenine base as principal building blocks

Cardy condition for open-closed field algebras

Let $V$ be a vertex operator algebra satisfying certain reductivity and finiteness conditions such that $\mathcal{C}_V$, the category of V-modules, is a modular tensor category. We study open-closed field algebras over V equipped with nondegenerate invariant bilinear forms for both open and closed sectors. We show that they give algebras over certain \C-extension of the Swiss-cheese partial dioperad, and we obtain Ishibashi states easily in such algebras. We formulate Cardy condition algebraically in terms of the action of the modular transformation $S: \tau \mapsto -\frac{1}{\tau}$ on the space of intertwining operators. We then derive a graphical representation of S in the modular tensor category $\mathcal{C}_V$. This result enables us to give a categorical formulation of Cardy condition and modular invariant conformal full field algebra over $V\otimes V$. Then we incorporate the modular invariance condition for genus-one closed theory, Cardy condition and the axioms for open-closed field algebra over V equipped with nondegenerate invariant bilinear forms into a tensor-categorical notion called Cardy $\mathcal{C}_V|\mathcal{C}_{V\otimes V}$-algebra. We also give a categorical construction of Cardy $\mathcal{C}_V|\mathcal{C}_{V\otimes V}$-algebra in Cardy case.Comment: 70 page, 105 figures, references are updated. less typos, to appear in Comm. Math. Phy