The Stokes conjecture for waves with vorticity

We study stagnation points of two-dimensional steady gravity free-surface water waves with vorticity. We obtain for example that, in the case where the free surface is an injective curve, the asymptotics at any stagnation point is given either by the “Stokes corner flow” where the free surface has a corner of 120°, or the free surface ends in a horizontal cusp, or the free surface is horizontally flat at the stagnation point. The cusp case is a new feature in the case with vorticity, and it is not possible in the absence of vorticity. In a second main result we exclude horizontally flat singularities in the case that the vorticity is 0 on the free surface. Here the vorticity may have infinitely many sign changes accumulating at the free surface, which makes this case particularly difficult and explains why it has been almost untouched by research so far. Our results are based on calculations in the original variables and do not rely on structural assumptions needed in previous results such as isolated singularities, symmetry and monotonicity.

The Gelfand-Kirillov conjecture and Gelfand-Tsetlin modules for finite W-algebras

We address two problems with the structure and representation theory of finite W-algebras associated with general linear Lie algebras. Finite W-algebras can be defined using either Kostant`s Whittaker modules or a quantum Hamiltonian reduction. Our first main result is a proof of the Gelfand-Kirillov conjecture for the skew fields of fractions of finite W-algebras. The second main result is a parameterization of finite families of irreducible Gelfand-Tsetlin modules using Gelfand-Tsetlin subalgebra. As a corollary, we obtain a complete classification of generic irreducible Gelfand-Tsetlin modules for finite W-algebras. (C) 2009 Elsevier Inc. All rights reserved.

The "refoldability" of selected proteins in ionic liquids as a stabilization criterion, leading to a conjecture on biogenesis

The folding of proteins is usually studied in dilute aqueous solutions of controlled pH, but it has recently been demonstrated that reversible unfolding can occur in other media. Particular stability is conferred on the protein (folded or unfolded) when the process occurs in ‘protic ionic liquids’ (pILs) of controlled proton activity. This activity (‘effective pH’) is determined by the acid and base components of the pIL and is characterized in the present study by the proton chemical shift of the N–H proton. Here we propose a ‘refoldability’ or ‘refolding index’ (RFI) metric for assessing the stability of folded biomolecules in different solvent media, and demarcate high RFI zones in hydrated pIL media using ribonuclease A and hen egg white lysozyme as examples. Then we show that, unexpectedly, the same high RFIs can be obtained in pIL media that are 90% inorganic in character (simple ammonium salts). This leads us to a conjecture related to the objections that have been raised to ‘primordial soup’ theories for biogenesis, objections that are based on the observation that all the bonds involved in biomacromolecule formation are hydrolyzed in ordinary aqueous solutions unless specifically protected. The ingredients for primitive ionic liquids (NH3, CO, HCN, CO2, and water) were abundant in the early earth atmosphere, and many experiments have shown how amino acids could form from them also. Cyclical concentration in evaporating inland seas could easily produce the type of ambient-temperature, non-hydrolyzing, media that we have demonstrated here may be hospitable to biomolecules, and that may be actually encouraging of biopolymer assembly. Thus a plausible variant of the conventional ‘primordial soup’ model of biogenesis is suggested.

Towards a worldsheet derivation of the Maldacena conjecture

A U(2,2 vertical bar 4)-invariant A-model constructed from fermionic superfields has recently been proposed as a sigma model for the superstring on AdS(5) X S(5). After explaining the relation of this A-model with the pure spinor formalism, the A-model action is expressed as a gauged linear sigma model. In the zero radius limit, the Coulomb branch of this sigma model is interpreted as D-brane holes which are related to gauge-invariant N = 4 d=4 super-Yang-Mills operators. As in the worldsheet derivation of open-closed duality for Chem-Simons theory, this construction may lead to a worldsheet derivation of the Maldacena conjecture. Intriguing connections to the twistorial formulation of N = 4 Yang-Mills are also noted. (Republished with permission of JHEP from JHEP 0803:031, 2008.)

Challenging the weak cosmic censorship conjecture with charged quantum particles

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

Verifying Harder's Conjecture for Classical and Siegel Modular Forms

A conjecture by Harder shows a surprising congruence between the coefficients of “classical” modular forms and the Hecke eigenvalues of corresponding Siegel modular forms, contigent upon “large primes” dividing the critical values of the given classical modular form. Harder’s Conjecture has already been verified for one-dimensional spaces of classical and Siegel modular forms (along with some two-dimensional cases), and for primes p 37. We verify the conjecture for higher-dimensional spaces, and up to a comparable prime p.

Numerical test of the Cardy-Jacobsen conjecture in the site-diluted Potts model in three dimensions

We present a microcanonical Monte Carlo simulation of the site-diluted Potts model in three dimensions with eight internal states, partly carried out on the citizen supercomputer Ibercivis. Upon dilution, the pure model’s first-order transition becomes of the second order at a tricritical point. We compute accurately the critical exponents at the tricritical point. As expected from the Cardy-Jacobsen conjecture, they are compatible with their random field Ising model counterpart. The conclusion is further reinforced by comparison with older data for the Potts model with four states.

On a conjecture of Lapidus and van Frankenhuysen

In this paper it is shown that a conjecture of Lapidus and van Frankenhuysen of 2003 on the existence of a vertical line such that the density of the complex dimensions of nonlattice fractal strings with M scaling ratios off this line vanishes in the limit as M→∞, fails on the class of nonlattice self-similar fractal strings.

Three new Mersenne primes, and a conjecture /

Bibliography: leaf 6.