On the distribution of the nodal sets of random spherical harmonics

We study the length of the nodal set of eigenfunctions of the Laplacian on the \spheredim-dimensional sphere. It is well known that the eigenspaces corresponding to \eigval=n(n+\spheredim-1) are the spaces \eigspc of spherical harmonics of degree $n$, of dimension \eigspcdim. We use the multiplicity of the eigenvalues to endow \eigspc with the Gaussian probability measure and study the distribution of the \spheredim-dimensional volume of the nodal sets of a randomly chosen function. The expected volume is proportional to \sqrt{\eigval}. One of our main results is bounding the variance of the volume to be O(\frac{\eigval}{\sqrt{\eigspcdim}}). In addition to the volume of the nodal set, we study its Leray measure. For every $n$, the expected value of the Leray measure is $\frac{1}{\sqrt{2\pi}}$. We are able to determine that the asymptotic form of the variance is \frac{const}{\eigspcdim}.Comment: 47 pages, accepted for publication in the Journal of Mathematical Physics. Lemmas 2.5, 2.11 were proved for any dimension, some other, suggested by the referee, modifications and corrections, were mad

Properties of Non-Abelian Fractional Quantum Hall States at Filling ν=kr\nu=\frac{k}{r}

We compute the physical properties of non-Abelian Fractional Quantum Hall (FQH) states described by Jack polynomials at general filling $\nu=\frac{k}{r}$. For $r=2$, these states are identical to the $Z_k$ Read-Rezayi parafermions, whereas for $r>2$ they represent new FQH states. The $r=k+1$ states, multiplied by a Vandermonde determinant, are a non-Abelian alternative construction of states at fermionic filling $2/5, 3/7, 4/9...$. We obtain the thermal Hall coefficient, the quantum dimensions, the electron scaling exponent, and show that the non-Abelian quasihole has a well-defined propagator falling off with the distance. The clustering properties of the Jack polynomials, provide a strong indication that the states with $r>2$ can be obtained as correlators of fields of \emph{non-unitary} conformal field theories, but the CFT-FQH connection fails when invoked to compute physical properties such as thermal Hall coefficient or, more importantly, the quasihole propagator. The quasihole wavefuntion, when written as a coherent state representation of Jack polynomials, has an identical structure for \emph{all} non-Abelian states at filling $\nu=\frac{k}{r}$.Comment: 2 figure

Development of tests for measurement of primary perceptual-motor performance

Tests for measuring primary perceptual-motor performance for assessing space environment effects on human performanc

Generalized Clustering Conditions of Jack Polynomials at Negative Jack Parameter α\alpha

We present several conjectures on the behavior and clustering properties of Jack polynomials at \emph{negative} parameter $\alpha=-\frac{k+1}{r-1}$, of partitions that violate the $(k,r,N)$ admissibility rule of Feigin \emph{et. al.} [\onlinecite{feigin2002}]. We find that "highest weight" Jack polynomials of specific partitions represent the minimum degree polynomials in $N$ variables that vanish when $s$ distinct clusters of $k+1$ particles are formed, with $s$ and $k$ positive integers. Explicit counting formulas are conjectured. The generalized clustering conditions are useful in a forthcoming description of fractional quantum Hall quasiparticles.Comment: 12 page

A Paraconsistent Higher Order Logic

Classical logic predicts that everything (thus nothing useful at all) follows from inconsistency. A paraconsistent logic is a logic where an inconsistency does not lead to such an explosion, and since in practice consistency is difficult to achieve there are many potential applications of paraconsistent logics in knowledge-based systems, logical semantics of natural language, etc. Higher order logics have the advantages of being expressive and with several automated theorem provers available. Also the type system can be helpful. We present a concise description of a paraconsistent higher order logic with countable infinite indeterminacy, where each basic formula can get its own indeterminate truth value (or as we prefer: truth code). The meaning of the logical operators is new and rather different from traditional many-valued logics as well as from logics based on bilattices. The adequacy of the logic is examined by a case study in the domain of medicine. Thus we try to build a bridge between the HOL and MVL communities. A sequent calculus is proposed based on recent work by Muskens.Comment: Originally in the proceedings of PCL 2002, editors Hendrik Decker, Joergen Villadsen, Toshiharu Waragai (http://floc02.diku.dk/PCL/). Correcte

SM(2,4k) fermionic characters and restricted jagged partitions

A derivation of the basis of states for the $SM(2,4k)$ superconformal minimal models is presented. It relies on a general hypothesis concerning the role of the null field of dimension $2k-1/2$. The basis is expressed solely in terms of $G_r$ modes and it takes the form of simple exclusion conditions (being thus a quasi-particle-type basis). Its elements are in correspondence with $(2k-1)$-restricted jagged partitions. The generating functions of the latter provide novel fermionic forms for the characters of the irreducible representations in both Ramond and Neveu-Schwarz sectors.Comment: 12 page

Sensitive imaging of electromagnetic fields with paramagnetic polar molecules

We propose a method for sensitive parallel detection of low-frequency electromagnetic fields based on the fine structure interactions in paramagnetic polar molecules. Compared to the recently implemented scheme employing ultracold $^{87}$Rb atoms [B{\"o}hi \textit{et al.}, Appl. Phys. Lett. \textbf{97}, 051101 (2010)], the technique based on molecules offers a 100-fold higher sensitivity, the possibility to measure both the electric and magnetic field components, and a probe of a wide range of frequencies from the dc limit to the THz regime

Interference between independent fluctuating condensates

We consider a problem of interference between two independent condensates, which lack true long range order. We show that their interference pattern contains information about correlation functions within each condensate. As an example we analyze the interference between a pair of one dimensional interacting Bose liquids. We find universal scaling of the average fringe contrast with system size and temperature that depends only on the Luttinger parameter. Moreover the full distribution of the fringe contrast, which is also equivalent to the full counting statistics of the interfering atoms, changes with interaction strength and lends information on high order correlation functions. We also demonstrate that the interference between two-dimensional condensates at finite temperature can be used as a direct probe of the Kosterlitz-Thouless transition. Finally, we discuss generalization of our results to describe the intereference of a periodic array of independent fluctuating condensates.Comment: 7 pages, 3 figures, published versio

Dynamics of Macroscopic Wave Packet Passing through Double Slits: Role of Gravity and Nonlinearity

Using the nonlinear Schroedinger equation (Gross-Pitaevskii equation), the dynamics of a macroscopic wave packet for Bose-Einstein condensates falling through double slits is analyzed. This problem is identified with a search for the fate of a soliton showing a head-on collision with a hard-walled obstacle of finite size. We explore the splitting of the wave packet and its reorganization to form an interference pattern. Particular attention is paid to the role of gravity (g) and repulsive nonlinearity (u_0) in the fringe pattern. The peak-to-peak distance in the fringe pattern and the number of interference peaks are found to be proportional to g^(-1/2) and u_0^(1/2)g^(1/4), respectively. We suggest a way of designing an experiment under controlled gravity and nonlinearity.Comment: 10 pages, 4 figures and 1 tabl