Is my ODE a Painleve equation in disguise?

Painleve equations belong to the class y'' + a_1 {y'}^3 + 3 a_2 {y'}^2 + 3 a_3 y' + a_4 = 0, where a_i=a_i(x,y). This class of equations is invariant under the general point transformation x=Phi(X,Y), y=Psi(X,Y) and it is therefore very difficult to find out whether two equations in this class are related. We describe R. Liouville's theory of invariants that can be used to construct invariant characteristic expressions (syzygies), and in particular present such a characterization for Painleve equations I-IV.Comment: 8 pages. Based on talks presented at NEEDS 2000, Gokova, Turkey, 29 June - 7 July, 2000, and at the AMS-HKMS joint meeting 13-16 December, 2000. Submitted to J. Nonlin. Math. Phy

Onsager-Manning-Oosawa condensation phenomenon and the effect of salt

Making use of results pertaining to Painleve III type equations, we revisit the celebrated Onsager-Manning-Oosawa condensation phenomenon for charged stiff linear polymers, in the mean-field approximation with salt. We obtain analytically the associated critical line charge density, and show that it is severely affected by finite salt effects, whereas previous results focused on the no salt limit. In addition, we obtain explicit expressions for the condensate thickness and the electric potential. The case of asymmetric electrolytes is also briefly addressed.Comment: to appear in Phys. Rev. Let

Representations of an integer by some quaternary and octonary quadratic forms

In this paper we consider certain quaternary quadratic forms and octonary quadratic forms and by using the theory of modular forms, we find formulae for the number of representations of a positive integer by these quadratic forms.Comment: 20 pages, 4 tables. arXiv admin note: text overlap with arXiv:1607.0380

Generalized dilaton-Maxwell cosmic string and wall solutions

The class of static solutions found by Gibbons and Wells for dilaton-electrodynamics in flat spacetime, which describe nontopological strings and walls that trap magnetic flux, is extended to a class of dynamical solutions supporting arbitrarily large, nondissipative traveling waves, using techniques previously applied to global and local topological defects. These solutions can then be used in conjunction with S-duality to obtain more general solitonic solutions for various axidilaton-Maxwell theories. As an example, a set of dynamical solutions is found for axion, dilaton, and Maxwell fields in low energy heterotic string theory using the SL(2,R) invariance of the equations of motion.Comment: 11 pages; to appear in Phys.Lett.

Exact Quantum Solutions of Extraordinary N-body Problems

The wave functions of Boson and Fermion gases are known even when the particles have harmonic interactions. Here we generalise these results by solving exactly the N-body Schrodinger equation for potentials V that can be any function of the sum of the squares of the distances of the particles from one another in 3 dimensions. For the harmonic case that function is linear in r^2. Explicit N-body solutions are given when U(r) = -2M \hbar^{-2} V(r) = \zeta r^{-1} - \zeta_2 r^{-2}. Here M is the sum of the masses and r^2 = 1/2 M^{-2} Sigma Sigma m_I m_J ({\bf x}_I - {\bf x}_J)^2. For general U(r) the solution is given in terms of the one or two body problem with potential U(r) in 3 dimensions. The degeneracies of the levels are derived for distinguishable particles, for Bosons of spin zero and for spin 1/2 Fermions. The latter involve significant combinatorial analysis which may have application to the shell model of atomic nuclei. For large N the Fermionic ground state gives the binding energy of a degenerate white dwarf star treated as a giant atom with an N-body wave function. The N-body forces involved in these extraordinary N-body problems are not the usual sums of two body interactions, but nor are forces between quarks or molecules. Bose-Einstein condensation of particles in 3 dimensions interacting via these strange potentials can be treated by this method.Comment: 24 pages, Latex. Accepted for publication in Proceedings of the Royal Societ

Solutions of the Einstein-Dirac and Seiberg-Witten Monopole Equations

We present unique solutions of the Seiberg-Witten Monopole Equations in which the U(1) curvature is covariantly constant, the monopole Weyl spinor consists of a single constant component, and the 4-manifold is a product of two Riemann surfaces of genuses p_1 and p_2. There are p_1 -1 magnetic vortices on one surface and p_2 - 1 electric ones on the other, with p_1 + p_2 \geq 2 p_1 = p_2= 1 being excluded). When p_1 = p_2, the electromagnetic fields are self-dual and one also has a solution of the coupled euclidean Einstein-Maxwell-Dirac equations, with the monopole condensate serving as cosmological constant. The metric is decomposable and the electromagnetic fields are covariantly constant as in the Bertotti-Robinson solution. The Einstein metric can also be derived from a K\"{a}hler potential satisfying the Monge-Amp\`{e}re equations.Comment: 22 pages. Rep. no: FGI-99-

Integrability of one degree of freedom symplectic maps with polar singularities

In this paper, we treat symplectic difference equations with one degree of freedom. For such cases, we resolve the relation between that the dynamics on the two dimensional phase space is reduced to on one dimensional level sets by a conserved quantity and that the dynamics is integrable, under some assumptions. The process which we introduce is related to interval exchange transformations.Comment: 10 pages, 2 figure

Einstein metrics in projective geometry

It is well known that pseudo-Riemannian metrics in the projective class of a given torsion free affine connection can be obtained from (and are equivalent to) the solutions of a certain overdetermined projectively invariant differential equation. This equation is a special case of a so-called first BGG equation. The general theory of such equations singles out a subclass of so-called normal solutions. We prove that non-degerate normal solutions are equivalent to pseudo-Riemannian Einstein metrics in the projective class and observe that this connects to natural projective extensions of the Einstein condition.Comment: 10 pages. Adapted to published version. In addition corrected a minor sign erro