Dichotomy for generic supercuspidal representations of G2G_2

The local Langlands conjectures imply that to every generic supercuspidal irreducible representation of $G_2$ over a $p$-adic field, one can associate a generic supercuspidal irreducible representation of either $PGSp_6$ or$PGL_3$. We prove this conjectural dichotomy, demonstrating a precise correspondence between certain representations of $G_2$ and other representations of $PGSp_6$ and $PGL_3$. This correspondence arises from theta correspondences in $E_6$ and $E_7$, analysis of Shalika functionals, and spin L-functions. Our main result reduces the conjectural Langlands parameterization of generic supercuspidal irreducible representations of $G_2$ to a single conjecture about the parameterization for $PGSp_6$.Comment: Version 2: Mistakes in Prop 3.2 and 3.5 corrected. Results strengthened in case p=2. Changes made throughout for consistency with stronger results and reformulatio

Osmotic force resisting chain insertion in a colloidal suspension

We consider the problem of inserting a stiff chain into a colloidal suspension of particles that interact with it through excluded volume forces. The free energy of insertion is associated with the work of creating a cavity devoid of colloid and sufficiently large to accomodate the chain. The corresponding work per unit length is the force that resists the entry of the chain into the colloidal suspension. In the case of a hard sphere fluid, this work can be calculated straightforwardly within the scaled particle theory; for solutions of flexible polymers, on the other hand, we employ simple scaling arguments. The forces computed in these ways are shown, for nanometer chain and colloid diameters, to be of the order of tens of pN for solution volume fraction for biophysical processes such as the ejection of DNA from viral capsids into the cell cytoplasm.Comment: 16 pages,3 figures. Accepted for publication in European Physical Journal

Electrostatic complexation of spheres and chains under elastic stress

We consider the complexation of highly charged semiflexible polyelectrolytes with oppositely charged macroions. On the basis of scaling arguments we discuss how the resulting complexes depend on the persistence length of the polyelectrolyte, the salt concentration, and the sizes and charges of the chain and the macroions. We study first the case of complexation with a single sphere and calculate the wrapping length of the chain. We then extend our considerations to complexes involving many wrapped spheres and study cooperative effects. The mechanical properties of such a complex under an external deformation are evaluated.Comment: 16 pages, submitted to J. Chem. Phy

Hybrid bounds for twisted L-functions

The aim of this paper is to derive bounds on the critical line Rs 1/2 for L- functions attached to twists f circle times chi of a primitive cusp form f of level N and a primitive character modulo q that break convexity simultaneously in the s and q aspects. If f has trivial nebentypus, it is shown that L(f circle times chi, s) << (N vertical bar s vertical bar q)(epsilon) N-4/5(vertical bar s vertical bar q)(1/2-1/40), where the implied constant depends only on epsilon > 0 and the archimedean parameter of f. To this end, two independent methods are employed to show L(f circle times chi, s) << (N vertical bar s vertical bar q)(epsilon) N-1/2 vertical bar S vertical bar(1/2)q(3/8) and L(g,s) << D-2/3 vertical bar S vertical bar(5/12) for any primitive cusp form g of level D and arbitrary nebentypus (not necessarily a twist f circle times chi of level D vertical bar Nq(2))

Organized condensation of worm-like chains

We present results relevant to the equilibrium organization of DNA strands of arbitrary length interacting with a spherical organizing center, suggestive of DNA-histone complexation in nucleosomes. We obtain a rich phase diagram in which a wrapping state is transformed into a complex multi-leafed, rosette structure as the adhesion energy is reduced. The statistical mechanics of the "melting" of a rosette can be mapped into an exactly soluble one-dimensional many-body problem.Comment: 15 pages, 2 figures in a pdf fil

Bounding sup-norms of cusp forms of large level

Let f be an $L^2$-normalized weight zero Hecke-Maass cusp form of square-free level N, character $\chi$ and Laplacian eigenvalue $\lambda\geq 1/4$. It is shown that $\| f \|_{\infty} \ll_{\lambda} N^{-1/37}$, from which the hybrid bound $\|f \|_{\infty} \ll \lambda^{1/4} (N\lambda)^{-\delta}$ (for some $\delta > 0$) is derived. The first bound holds also for $f = y^{k/2}F$ where F is a holomorphic cusp form of weight k with the implied constant now depending on k.Comment: version 3: substantially revised versio

What do emulsification failure and Bose-Einstein condensation have in common?

Ideal bosons and classical ring polymers formed via self-assembly, are known to have the same partition function, and so analogous phase transitions. In ring polymers, the analogue of Bose-Einstein condensation occurs when a ring polymer of macroscopic size appears. We show that a transition of the same general form occurs within a whole class of systems with self-assembly, and illustrate it with the emulsification failure of a microemulsion phase of water, oil and surfactant. As with Bose-Einstein condensation, the transition occurs even in the absence of interactions.Comment: 7 pages, 1 figure, typeset with EUROTeX, uses epsfi

Distinguished non-Archimedean representations

For a symmetric space (G,H), one is interested in understanding the vector space of H-invariant linear forms on a representation \pi of G. In particular an important question is whether or not the dimension of this space is bounded by one. We cover the known results for the pair (G=R_{E/F}GL(n),H=GL(n)), and then discuss the corresponding SL(n) case. In this paper, we show that (G=R_{E/F}SL(n),H=SL(n)) is a Gelfand pair when n is odd. When $n$ is even, the space of H-invariant forms on \pi can have dimension more than one even when \pi is supercuspidal. The latter work is joint with Dipendra Prasad

Uniqueness of Bessel models: the archimedean case

In the archimedean case, we prove uniqueness of Bessel models for general linear groups, unitary groups and orthogonal groups.Comment: 22 page