where D = (−1, 1)× (−1, 1) , f(x, y) is a given Hölder continuous function in D, and φ(x, y) is an unknown function. The equation (1) has applications in the theory of aeroelasticity [1]. Note that the equation without logarithmic singularities was many times considered in different classes of functions. In the literature the solutions of the equation (1) in bounded domains [2, 5, 6, 9] as well...

We investigate the relative logarithmic connections on a holomorphic vector bundle over complex analytic family. give sufficient condition for existence of connection singular simple normal crossing divisor. define residue and express Chern classes in terms residues.

The eeect of coupling sensitivity of chaos is known as logarithmic singular behavior of the Lyapunov exponents of coupled chaotic systems at small values of the coupling parameter. In order to study it analytically, we use a continuous-time stochastic model which can be treated by means of the Fokker-Planck equation. One main result is that the singularity depends on the uctuations of the nite-...

Low Reynolds number uid ow past a cylindrical body of arbitrary shape in an unbounded, two-dimensional domain is a singular perturbation problem involving an innnite logarithmic expansion in the small parameter ", representing the Reynolds number. We apply a hybrid asymptotic-numerical method to compute the drag coeecient, C D , and lift coeecient, C L , to within all logarithmic terms. The hyb...

Let M be a compact, connected, Riemannian manifold of dimension d, let fPt : t > 0g denote the Markov semigroups on C (M) determined by 1 2 , and let pt (x; y) denote the kernel (with respect to the Riemannian volume measure) for the operator Pt. (The existence of this kernel as a positive, smooth function is well-known, see e.g. D].) Bismut's celebrated formula, presented in B], equates r log ...

We demonstrate the occurrence of regimes with singular continuous (fractal) Fourier spectra in autonomous dissipative dynamical systems. The particular example is an ODE system at the accumulation points of bifurcation sequences associated to the creation of complicated homoclinic orbits. Two diierent mechanisms responsible for the appearance of such spectra are proposed. In the rst case when t...

In this paper, we characterize the logarithmic singularities arising in method of moments from Green’s function integrals over test domain, and use two approaches for designing geometrically symmetric quadrature rules to integrate these singular integrands. These exhibit better convergence properties than polynomials and, general, lead accuracy with a lower number points. We demonstrate their e...

Wave catastrophes are characterized by logarithmic phase singularities. Examples are light at the horizon of a black hole, sound in transsonic fluids, waves in accelerated frames, light in singular dielectrics and slow light close to a zero of the group velocity. We show that the wave amplitude grows with a half-integer power for monodirectional and symmetric wave catastrophes.

In this short note, we are concerned with the evaluation of certain convolution integrals with singular kernels. The problem is as follows. Given a kernel function K(t) which is very often singular at the origin and a density function σ(t), evaluate C(t) = ∫ t 0 K(t − τ)σ(τ)dτ for t = ∆t, 2∆t, · · · , T = N∆t. We will oftern write C(tk) = C(k∆t) = Ck in short, and similarly for σ. Direct comput...

Singular behavior and the formation of plateaus in the probability distribution in a nonadiabatically driven system are investigated numerically in the weak noise limit. A simple extension of the recently introduced logarithmic susceptibility theory is proposed to construct an approximation of the nonequilibrium potential that is valid throughout whole of the phase space.