Singular normal form for the Painlev\'e equation P1

We show that there exists a rational change of coordinates of Painlev\'e's P1 equation $y''=6y^2+x$ and of the elliptic equation $y''=6y^2$ after which these two equations become analytically equivalent in a region in the complex phase space where $y$ and $y'$ are unbounded. The region of equivalence comprises all singularities of solutions of P1 (i.e. outside the region of equivalence, solutions are analytic). The Painlev\'e property of P1 (that the only movable singularities are poles) follows as a corollary. Conversely, we argue that the Painlev\'e property is crucial in reducing P1, in a singular regime, to an equation integrable by quadratures

Effects of boundary conditions on irreversible dynamics

We present a simple one-dimensional Ising-type spin system on which we define a completely asymmetric Markovian single spin-flip dynamics. We study the system at a very low, yet non-zero, temperature and we show that for empty boundary conditions the Gibbs measure is stationary for such dynamics, while introducing in a single site a $+$ condition the stationary measure changes drastically, with macroscopical effects. We achieve this result defining an absolutely convergent series expansion of the stationary measure around the zero temperature system. Interesting combinatorial identities are involved in the proofs

Gaussian Fluctuation in Random Matrices

Let $N(L)$ be the number of eigenvalues, in an interval of length $L$, of a matrix chosen at random from the Gaussian Orthogonal, Unitary or Symplectic ensembles of ${\cal N}$ by ${\cal N}$ matrices, in the limit ${\cal N}\rightarrow\infty$. We prove that $[N(L) - \langle N(L)\rangle]/\sqrt{\log L}$ has a Gaussian distribution when $L\rightarrow\infty$. This theorem, which requires control of all the higher moments of the distribution, elucidates numerical and exact results on chaotic quantum systems and on the statistics of zeros of the Riemann zeta function. \noindent PACS nos. 05.45.+b, 03.65.-wComment: 13 page

Star Unfolding Convex Polyhedra via Quasigeodesic Loops

We extend the notion of star unfolding to be based on a quasigeodesic loop Q rather than on a point. This gives a new general method to unfold the surface of any convex polyhedron P to a simple (non-overlapping), planar polygon: cut along one shortest path from each vertex of P to Q, and cut all but one segment of Q.Comment: 10 pages, 7 figures. v2 improves the description of cut locus, and adds references. v3 improves two figures and their captions. New version v4 offers a completely different proof of non-overlap in the quasigeodesic loop case, and contains several other substantive improvements. This version is 23 pages long, with 15 figure

Cohomological BRST aspects of the massless tensor field with the mixed symmetry (k,k)

The main BRST cohomological properties of a free, massless tensor field that transforms in an irreducible representation of GL(D,R), corresponding to a rectangular, two-column Young diagram with k>2 rows are studied in detail. In particular, it is shown that any non-trivial co-cycle from the local BRST cohomology group H(s|d) can be taken to stop either at antighost number (k+1) or k, its last component belonging to the cohomology of the exterior longitudinal derivative H(gamma) and containing non-trivial elements from the (invariant) characteristic cohomology H^{inv}(delta|d).Comment: Latex, 50 pages, uses amssym

Gauge-Invariant and Gauge-Fixed D-Brane Actions

The first part of this paper presents actions for Dirichlet p-branes embedded in a flat 10-dimensional space-time. The fields of the (p+1)-dimensional world-volume theories are the 10d space-time coordinates $X^m$, a pair of Majorana-Weyl spinors $\theta_1$ and $\theta_2$, and a U(1) gauge field $A_{\mu}$. The N = 2A or 2B super-Poincare group in ten dimensions is realized as a global symmetry. In addition, the theories have local symmetries consisting of general coordinate invariance of the world volume, a local fermionic symmetry (called ``kappa''), and U(1) gauge invariance. A detailed proof of the kappa symmetry is given that applies to all cases (p = 0,1, . . ., 9). The second part of the paper presents gauge-fixed versions of these theories. The fields of the 10d (p = 9) gauge-fixed theory are a single Majorana-Weyl spinor $\lambda$ and the U(1) gauge field $A_{\mu}$. This theory, whose action turns out to be surprisingly simple, is a supersymmetric extension of 10d Born-Infeld theory. It has two global supersymmetries: one represents an unbroken symmetry, and the second corresponds to a broken symmetry for which $\lambda$ is the Goldstone fermion. The gauge-fixed supersymmetric D-brane theories with $p<9$ can be obtained from the 10d one by dimensional reduction.Comment: 33 pages, latex, no figures; revised as requested by refere

Transmission Properties of the oscillating delta-function potential

We derive an exact expression for the transmission amplitude of a particle moving through a harmonically driven delta-function potential by using the method of continued-fractions within the framework of Floquet theory. We prove that the transmission through this potential as a function of the incident energy presents at most two real zeros, that its poles occur at energies $n\hbar\omega+\varepsilon^*$ ($0<Re(\varepsilon^*)<\hbar\omega$), and that the poles and zeros in the transmission amplitude come in pairs with the distance between the zeros and the poles (and their residue) decreasing with increasing energy of the incident particle. We also show the existence of non-resonant "bands" in the transmission amplitude as a function of the strength of the potential and the driving frequency.Comment: 21 pages, 12 figures, 1 tabl

Finite N Fluctuation Formulas for Random Matrices

For the Gaussian and Laguerre random matrix ensembles, the probability density function (p.d.f.) for the linear statistic $\sum_{j=1}^N (x_j - )$ is computed exactly and shown to satisfy a central limit theorem as $N \to \infty$. For the circular random matrix ensemble the p.d.f.'s for the linear statistics ${1 \over 2} \sum_{j=1}^N (\theta_j - \pi)$ and $- \sum_{j=1}^N \log 2|\sin \theta_j/2|$ are calculated exactly by using a constant term identity from the theory of the Selberg integral, and are also shown to satisfy a central limit theorem as $N \to \infty$.Comment: LaTeX 2.09, 11 pages + 3 eps figs (needs epsf.sty