Theory of spinor Fermi and Bose gases in tight atom waveguides

Divergence-free pseudopotentials for spatially even and odd-wave interactions in spinor Fermi gases in tight atom waveguides are derived. The Fermi-Bose mapping method is used to relate the effectively one-dimensional fermionic many-body problem to that of a spinor Bose gas. Depending on the relative magnitudes of the even and odd-wave interactions, the N-atom ground state may have total spin S=0, S=N/2, and possibly also intermediate values, the case S=N/2 applying near a p-wave Feshbach resonance, where the N-fermion ground state is space-antisymmetric and spin-symmetric. In this case the fermionic ground state maps to the spinless bosonic Lieb-Liniger gas. An external magnetic field with a longitudinal gradient causes a Stern-Gerlach spatial separation of the corresponding trapped Fermi gas with respect to various values of $S_z$.Comment: 4+ pages, 1 figure, revtex4. Submitted to PRA. Minor corrections of typos and notatio

Method of constructing exactly solvable chaos

We present a new systematic method of constructing rational mappings as ergordic transformations with nonuniform invariant measures on the unit interval [0,1]. As a result, we obtain a two-parameter family of rational mappings that have a special property in that their invariant measures can be explicitly written in terms of algebraic functions of parameters and a dynamical variable. Furthermore, it is shown here that this family is the most generalized class of rational mappings possessing the property of exactly solvable chaos on the unit interval, including the Ulam=Neumann map y=4x(1-x). Based on the present method, we can produce a series of rational mappings resembling the asymmetric shape of the experimentally obtained first return maps of the Beloussof-Zhabotinski chemical reaction, and we can match some rational functions with other experimentally obtained first return maps in a systematic manner.Comment: 12 pages, 2 figures, REVTEX. Title was changed. Generalized Chebyshev maps including the precise form of two-parameter generalized cubic maps were added. Accepted for publication in Phys. Rev. E(1997

Solvable and/or integrable and/or linearizable N-body problems in ordinary (three-dimensional) space. I

Several N-body problems in ordinary (3-dimensional) space are introduced which are characterized by Newtonian equations of motion (``acceleration equal force;'' in most cases, the forces are velocity-dependent) and are amenable to exact treatment (``solvable'' and/or ``integrable'' and/or ``linearizable''). These equations of motion are always rotation-invariant, and sometimes translation-invariant as well. In many cases they are Hamiltonian, but the discussion of this aspect is postponed to a subsequent paper. We consider ``few-body problems'' (with, say, \textit{N}=1,2,3,4,6,8,12,16,...) as well as ``many-body problems'' (N an arbitrary positive integer). The main focus of this paper is on various techniques to uncover such N-body problems. We do not discuss the detailed behavior of the solutions of all these problems, but we do identify several models whose motions are completely periodic or multiply periodic, and we exhibit in rather explicit form the solutions in some cases

On some exceptional cases in the integrability of the three-body problem

We consider the Newtonian planar three--body problem with positive masses $m_1$, $m_2$, $m_3$. We prove that it does not have an additional first integral meromorphic in the complex neighborhood of the parabolic Lagrangian orbit besides three exceptional cases $\sum m_i m_j/(\sum m_k)^2= 1/3$, $2^3/3^3$, $2/3^2$ where the linearized equations are shown to be partially integrable. This result completes the non-integrability analysis of the three-body problem started in our previous papers and based of the Morales-Ramis-Ziglin approach.Comment: 7 page

Ground State H-Atom in Born-Infeld Theory

Within the context of Born-Infeld (BI) nonlinear electrodynamics (NED) we revisit the non-relativistic, spinless H-atom. The pair potential computed from the Born-Infeld equations is approximated by the Morse type potential with remarkable fit over the critical region where the convergence of both the short and long distance expansions slows down dramatically. The Morse potential is employed to determine both the ground state energy of the electron and the BI parameter.Comment: 4 pages, 1 figure, final version to appear in Foundation of Physic

A New Related Message Attack on RSA

Abstract. Coppersmith, Franklin, Patarin, and Reiter show that given two RSA cryptograms x e mod N and (ax + b) e mod N for known constants a, b ∈ ZN, one can compute x in O(e log 2 e) ZN-operations with some positive error probability. We show that given e cryptograms ci ≡ (aix + bi) e mod N, i = 0, 1,...e − 1, for any known constants ai, bi ∈ ZN, one can deterministically compute x in O(e) ZN-operations that depend on the cryptograms, after a pre-processing that depends only on the constants. The complexity of the pre-processing is O(e log 2 e) ZNoperations, and can be amortized over many instances. We also consider a special case where the overall cost of the attack is O(e) ZN-operations. Our tools are borrowed from numerical-analysis and adapted to handle formal polynomials over finite-rings. To the best of our knowledge their use in cryptanalysis is novel.

Exact Superpotentials from Matrix Models

Dijkgraaf and Vafa (DV) have conjectured that the exact superpotential for a large class of N=1 SUSY gauge theories can be extracted from the planar limit of a certain holomorphic matrix integral. We test their proposal against existing knowledge for a family of deformations of N=4 SUSY Yang-Mills theory involving an arbitrary polynomial superpotential for one of the three adjoint chiral superfields. Specifically, we compare the DV prediction for these models with earlier results based on the connection between SUSY gauge theories and integrable systems. We find complete agreement between the two approaches. In particular we show how the DV proposal allows the extraction of the exact eigenvalues of the adjoint scalar in the confining vacuum and hence computes all related condensates of the finite-N gauge theory. We extend these results to include Leigh-Strassler deformations of the N=4 theory.Comment: 28 pages, 1 figure, latex with JHEP.cls, replaced with typos corrected and one clarifying commen

The Phase Structure of Mass-Deformed SU(2)xSU(2) Quiver Theory

The phase structure of the finite SU(2)xSU(2) theory with N=2 supersymmetry, broken to N=1 by mass terms for the adjoint-valued chiral multiplets, is determined exactly by compactifying the theory on a circle of finite radius. The exact low-energy superpotential is constructed by identifying it as a linear combination of the Hamiltonians of a certain symplectic reduction of the spin generalized elliptic Calogero-Moser integrable system. It is shown that the theory has four confining, two Higgs and two massless Coulomb vacua which agrees with a simple analysis of the tree-level superpotential of the four-dimensional theory. In each vacuum, we calculate all the condensates of the adjoint-valued scalars.Comment: 12 pages, JHEP.cl

Uncovering Ramanujan's "Lost" Notebook: An Oral History

Here we weave together interviews conducted by the author with three prominent figures in the world of Ramanujan's mathematics, George Andrews, Bruce Berndt and Ken Ono. The article describes Andrews's discovery of the "lost" notebook, Andrews and Berndt's effort of proving and editing Ramanujan's notes, and recent breakthroughs by Ono and others carrying certain important aspects of the Indian mathematician's work into the future. Also presented are historical details related to Ramanujan and his mathematics, perspectives on the impact of his work in contemporary mathematics, and a number of interesting personal anecdotes from Andrews, Berndt and Ono

Unitarity of the Knizhnik-Zamolodchikov-Bernard connection and the Bethe Ansatz for the elliptic Hitchin systems

We work out finite-dimensional integral formulae for the scalar product of genus one states of the group $G$ Chern-Simons theory with insertions of Wilson lines. Assuming convergence of the integrals, we show that unitarity of the elliptic Knizhnik-Zamolodchikov-Bernard connection with respect to the scalar product of CS states is closely related to the Bethe Ansatz for the commuting Hamiltonians building up the connection and quantizing the quadratic Hamiltonians of the elliptic Hitchin system.Comment: 24 pages, latex fil