RNA polymers are constructed from four distinct nucleotide bases. The sequence of these nucleotide bases determines both the folded conformation and the biological function of RNA. It recently has been established that disparately related functional classes of evolved RNA possess similar base composition biases despite a lack of sequence similarity, folded structure, or metabolic function. We h...

Heegaard Floer theory is a kind of topological quantum field theory, assigning graded groups to closed, connected, oriented 3-manifolds and group homomorphisms to smooth, oriented 4dimensional cobordisms. Bordered Heegaard Floer homology is an extension of Heegaard Floer homology to 3-manifolds with boundary, with extended-TQFT-type gluing properties. In this survey, we explain the formal struc...

Given a double complex X there are spectral sequences with the E2 terms being either HI (HII(X)) or HII(HI(X)). But if HI(X) = HII(X) = 0 both spectral sequences have all their terms 0. This can happen even though there is nonzero (co)homology of interest associated with X. This is frequently the case when dealing with Tate (co)homology. So in this situation the spectral sequences may not give ...

Let $R$ be a 2-torsion free semiprime ring with extended centroid $C$, $U$ the Utumi quotient ring of $R$ and $m,n>0$ are fixed integers. We show that if $R$ admits derivation $d$ such that $b[[d(x), x]_n,[y,d(y)]_m]=0$ for all $x,yin R$ where $0neq bin R$, then there exists a central idempotent element $e$ of $U$ such that $eU$ is commutative ring and $d$ induce a zero derivation on $(1-e)U$. ...

We use the theory of hyperplane arrangements to construct natural bases for the homology of partition lattices of types A, B and D. This extends and explains the " splitting basis " for the homology of the partition lattice given in [19], thus answering a question asked by R. Stanley. More explicitly, the following general technique is presented and utilized. Let A be a central and essential hy...

1.1. Khovanov’s Homology and Conjecture. In his 1999 paper [K1] in Duke Mathematical Journal, Mikhail Khovanov showed that the Jones link polynomial is the graded Euler characteristic of a bigraded homology module H (D) over Z[c], associated to a diagram D of the link. H (D) is the homology of a bigraded chain complex C(D) with a (1, 0)-bigraded differential. He also explained how each link cob...

We study the topology of matching and chessboard complexes. Our main results are as follows. 1. We prove conjectures of A. Björner, L. Lovász, S. T. Vrećica, and R. T. Živaljević on the connectivity of these complexes. 2. We show that for almost all n, the first nontrivial homology group of the matching complex on n vertices has exponent three. 3. We prove similar but weaker results on the expo...

The skein lasagna module is an extension of Khovanov–Rozansky homology to the setting a four-manifold and link in its boundary. This invariant plays role Hilbert space associated fully extended (4+ϵ)-dimensional TQFT. We give general procedure for expressing terms handle decomposition four-manifold. use this calculate few examples, show that can sometimes be locally infinite dimensional.

In this paper, we show that if we extend D-term hybrid inflation by adding a light scalar field which couples to a waterfall field, large non-Gaussianity can be generated at the end of inflation. Contribution of cosmic strings generated after D-term inflation to the Cosmic Microwave Background (CMB) angular power spectrum can be as low as 10% and this also evade the spectral index problem. cmli...