Vicious random walkers and a discretization of Gaussian random matrix ensembles

The vicious random walker problem on a one-dimensional lattice is considered. Many walkers take simultaneous steps on the lattice and the configurations in which two of them arrive at the same site are prohibited. It is known that the probability distribution of N walkers after M steps can be written in a determinant form. Using an integration technique borrowed from the theory of random matrices, we show that arbitrary kth order correlation functions of the walkers can be expressed as quaternion determinants whose elements are compactly expressed in terms of symmetric Hahn polynomials.     

Exact solutions to the angular Teukolsky equation with s ≠ 0

We first convert the angular Teukolsky equation under the special condition of τ ≠ 0, s ≠ 0, m=0 into a confluent Heun differential equation (CHDE) by taking different function transformation and variable substitution. And then according to the characteristics of both CHDE and its analytical solution expressed by a confluent Heun function (CHF), we find two linearly dependent solutions corresponding to the same eigenstate, from which we obtain a precise energy spectrum equation by constructing a Wronskian determinant. After that, we are able to localize the positions of the eigenvalues on the real axis or on the complex plane when τ is a real number, a pure imaginary number, and a complex number, respectively and we notice that the relation between the quantum number l and the spin weight quantum number s satisfies the relation l=∣s∣+ n, n=0, 1, 2···. The exact eigenvalues and the corresponding normalized eigenfunctions given by the CHF are obtained with the aid of Maple. The features of the angular probability distribution (APD) and the linearly dependent characteristics of two eigenfunctions corresponding to the same eigenstate are discussed. We find that for a real number τ, the eigenvalue is a real number and the eigenfunction is a real function, and the eigenfunction system is an orthogonal complete system, and the APD is asymmetric in the northern and southern hemispheres. For a pure imaginary number τ, the eigenvalue is still a real number and the eigenfunction is a complex function, but the APD is symmetric in the northern and southern hemispheres. When τ is a complex number, the eigenvalue is a complex number, the eigenfunction is still a complex function, and the APD in the northern and southern hemispheres is also asymmetric. Finally, an approximate expression of complex eigenvalues is obtained when n is greater than ∣s∣.     

Fredholm modules for quantum Euclidean spheres

The quantum Euclidean spheres, S_q^N−1, are (noncommutative) homogeneous spaces of quantum orthogonal groups, SO_q(N). The *-algebra A(S^N−1_q) of polynomial functions on each of these is given by generators and relations which can be expressed in terms of a self-adjoint, unipotent matrix. We explicitly construct complete sets of generators for the K-theory (by nontrivial self-adjoint idempotents and unitaries) and the K-homology (by nontrivial Fredholm modules) of the spheres S_q^N−1. We also construct the corresponding Chern characters in cyclic homology and cohomology and compute the pairing of K-theory with K-homology. On odd spheres (i.e., for N even) we exhibit unbounded Fredholm modules by means of a natural unbounded operator D which, while failing to have compact resolvent, has bounded commutators with all elements in the algebra A(S^N−1_q).     

Trace Extensions,Determinant Bundles,and Gauge Group Cocycles

We study the geometry of determinant line bundles associated with Dirac operators on compact odd-dimensional manifolds. Physically, these arise as (local) vacuum line bundles in quantum gauge theory. We give a simplified derivation of the commutator anomaly formula using a construction based on noncyclic trace extensions and associated nonmultiplicative renormalized determinants.     

N-fold Darboux Transformation for Integrable Couplings of AKNS Equations

For the integrable couplings of Ablowitz-Kaup-Newell-Segur(ICAKNS) equations, N-fold Darboux transformation(DT) TN, which is a 4 × 4 matrix, is constructed in this paper. Each element of this matrix is expressed by a ratio of the(4N + 1)-order determinant and 4N-order determinant of eigenfunctions. By making use of these formulae,the determinant expressions of N-transformed new solutions p~([N ]), q~([N ]), r~([N ])and s~([N ])are generated by this N-fold DT.Furthermore, when the reduced conditions q =-p*and s =-r*are chosen, we obtain determinant representations of N-fold DT and N-transformed solutions for the integrable couplings of nonlinear Schr?dinger(ICNLS) equations.Starting from the zero seed solutions, one-soliton solutions are explicitly given as an example.     

Energy moments of scattering phase shift for partly non-local momentum dependent interaction

The phase shift sum rules recently derived by Puff are extended for a general partly non-local momentum dependent potential. In achieving this, one must use Fredholm determinant of the outgoing solution of the Schrodinger equation instead of the Jost function as was done by Puff. The constants appearing in the moment relations are explicitly defined in terms of the momentum representation of the interaction.     

Generalized Wronskian and Grammian Solutions to a Isospectral B-type Kadomtsev-Petviashvili equation

Generally speaking, the BKP hierarchy which only has Pfaffian solutions. In this paper, based on the Grammian and Wronskian derivative formulae, generalized Wronskian and Grammian determinant solutions are obtained for the isospectral BKP equation (the second member on the BKP hierarchy) in the Hirota bilinear form. Especially, with the help of the properties of the computing of Young diagram, we have first applied Young diagram proved the proposition of this paper. Moreover, by considering the different combinations of the entries in Wronskian, we obtain various types of Wronskian solutions.     

Wronskian and Grammian solutions for (3+1)-dimensional Jimbo–Miwa equation

In this paper, based on the Hirota's bilinear method, the Wronskian and Grammian technique, as well as several properties of determinant, a broad set of sufficient conditions consisting of systems of linear partial differential equations are presented, which guarantees that the Wronskian determinant and the Grammian determinant solve the (3+1)-dimensional Jimbo-Miwa equation in the bilinear form, and then some special exact Wronskian and Grammian solutions are obtained by solving the differential conditions. At last, with the aid of Maple, some of these special exact solutions are shown graphically.     

Multiple lump molecules and interaction solutions of the Kadomtsev–Petviashvili I equation

In this paper, a modified version of the solution in form of a Gramian formula is employed to investigate a new type of multiple lump molecule solution of the Kadomtsev–Petviashvili I equation. The high-order multiple lump molecules consisting of M N-lump molecules are constructed by means of the Mth-order determinant and the non-homogeneous polynomial in the degree of 2N. The interaction solutions describing P line solitons radiating P of the M N-lump molecules are constructed. The dynamic behaviors of some specific solutions are analyzed through numerical simulation. All the results will enrich our understanding of the multiple lump waves of the Kadomtsev–Petviashvili I equation.     

Wronskian Approach and One-Dimensional Schrodinger Equation with Double-Well Potential

A Wronskian determinant approach is suggested to study the energy and the wave function for onedimensional Schrodinger equation. An integral equation and its corresponding Green function are constructed. As an example, we employed this approach to study the problem of double-well potential with strong coupling. A series of expansion of ground state energy up to the second order approximation of iterative procedure is given.     

Wronskian Approach and One-Dimensional Schrödinger Equation with Double-Well Potential

A Wronskian determinant approach is suggested to study the energy and the wave function for one-dimensional Schrödinger equation. An integral equation and its corresponding Green function are constructed. As an example, we employed this approach to study the problem of double-well potential with strong coupling. A series of expansion of ground state energy up to the second order approximation of iterative procedure is given.     

Gauged Goldstone boson effective action from direct integration of bardeen anomaly

We derive the effective action which describes the anomalous Goldstone boson interactions in the presence of external flavour SU(N)_L×SU(N)_R gauge fields, by a direct integration of the Bardeen anomaly. Through comparison with other results, we are able to elucidate the relationship between anomalies based on different regularization schemes. We also give a compact representation of the anomalous part of the fermionic determinant allowing for a simpler rederivation of our results.     

Normalized Ricci Flow on Riemann Surfaces and Determinant of Laplacian

In this letter, we give a simple proof of the fact that the determinant of Laplace operator in a smooth metric over compact Riemann surfaces of an arbitrary genus g monotonously grows under the normalized Ricci flow. Together with results of Hamilton that under the action of the normalized Ricci flow a smooth metric tends asymptotically to the metric of constant curvature, this leads to a simple proof of the Osgood–Phillips–Sarnak theorem stating that within the class of smooth metrics with fixed conformal class and fixed volume the determinant of the Laplace operator is maximal on the metric of constant curvatute.Mathematical Subject Classifications (2000). 58J52, 53C44.     

Dynamical Determinants via Dynamical Conjugacies for Postcritically Finite Polynomials

We give an analogue of Levin–Sodin–Yuditskii's study of the dynamical Ruelle determinants of hyperbolic rational maps in the case of subhyperbolic quadratic polynomials. Our main tool is to reduce to an expanding situation. We do so by applying a dynamical change of coordinates on the domains of a Markov partition constructed from the landing ray at the postcritical repelling orbit. We express the dynamical determinants as the product of an (entire) determinant with an explicit expression involving the postcritical repelling orbit, thus explaining the poles in d (z).     

非线性发展方程的Wronskian解及Young图证明

给出一般非线性发展方程构造Wronskian解的间接法. 根据Young图运算的性质给出了文中命题的证明, 并讨论了置换群特征标与Young图表达式系数间的关系. 关键词： 非线性发展方程 Wronskian解 Young图 特征标     

Characteristic polynomials of complex random matrix models

We calculate the expectation value of an arbitrary product of characteristic polynomials of complex random matrices and their Hermitian conjugates. Using the technique of orthogonal polynomials in the complex plane our result can be written in terms of a determinant containing these polynomials and their kernel. It generalizes the known expression for Hermitian matrices and it also provides a generalization of the Christoffel formula to the complex plane. The derivation we present holds for complex matrix models with a general weight function at finite-N, where N is the size of the matrix. We give some explicit examples at finite-N for specific weight functions. The characteristic polynomials in the large-N limit at weak and strong non-hermiticity follow easily and they are universal in the weak limit. We also comment on the issue of the BMN large-N limit.     

A Fredholm Determinant Representation in ASEP

In previous work (Tracy and Widom in Commun. Math. Phys. 279:815–844, 2008) the authors found integral formulas for probabilities in the asymmetric simple exclusion process (ASEP) on the integer lattice ℤ. The dynamics are uniquely determined once the initial state is specified. In this note we restrict our attention to the case of step initial condition with particles at the positive integers ℤ⁺ and consider the distribution function for the mth particle from the left. In Tracy and Widom (Commun. Math. Phys. 279:815–844, 2008) an infinite series of multiple integrals was derived for the distribution. In this note we show that the series can be summed to give a single integral whose integrand involves a Fredholm determinant. We use this determinant representation to derive (non-rigorously, at this writing) a scaling limit.     

Electromagnetically induced transparency in N-level cascade schemes

We examine electromagnetically induced transparency (EIT) in cascade schemes with N levels and N−1 fields. We show that transparency effects are present when N is odd and that destruction of EIT is present on line centre when N is even. We predict multiple dark resonances in such schemes due to multiphoton EIT effects. By examining atomic rubidium we propose methods of achieving such schemes by use of coupling rf fields into hyperfine levels.     

Critical behavior of transverse and longitudinal ac susceptibilities in a random anisotropy magnet a-Dy16Fe84

We have investigated critical lines in the H-T plane in a random anisotropy magnet (RAM) a-Dy₁₆Fe₈₄ with a small effective ratio of the anisotropy (D) to the exchange constant (J) by means of ac susceptibility (χ) in static fields H parallel and perpendicular to the ac field. We found that the transverse χ exhibits an anomaly along the irreversible line H(T_f) determined by previous magnetization measurements, while the longitudinal χ does so along a characteristic line H(T_i) in a lower temperature region. Above H(T_f) we also found an extra characteristic line H(T_c). The lines were almost independent of the measured frequency. Both the present results and previous magnetization results suggest that an equilibrium phase transition occurs, and the critical lines analogous to those in Heisenberg spin glasses are present in a weak RAM.     

The Universal Gerbe,Dixmier–Douady Class,and Gauge Theory

We clarify the relation between the Dixmier–Douady class on the space of self-adjoint Fredholm operators (universal B-field) and the curvature of determinant bundles over infinite-dimensional Grassmannians. In particular, in the case of Dirac type operators on a three dimensional compact manifold we obtain a simple and explicit expression for both forms.