Explicit formulas for general order multivariate Padé approximants of pseudo-multivariate functions are constructed on specific index sets. Examples include the multivariate forms of the exponential function E (x) = ∞ ∑ j1,j2,...,jm=0 x1 1 x j2 2 · · ·x jm m (j1 + j2 + · · ·+ jm)! , the logarithm function L(x) = ∑ j1+j2+···+jm≥1 x1 1 x j2 2 · · ·x jm m j1 + j2 + · · ·+ jm , the Lauricella funct...

This work derives the linearized equations of motion, the Lagrangian density, the Hamiltonian density, and the canonical angular momentum density for general perturbations [∝ exp(imφ) with m = 0,±1, ..] of a geometrically thin self-gravitating, homentropic fluid disk including the pressure. The theory is applied to “eccentric,” m = ±1 perturbations of a geometrically thin Keplerian disk. We fin...

We discuss the calculation of semi-classical wormhole vertex operators from wave functions which satisfy the Wheeler-deWitt equation and momentum constraints, together with certain ‘wormhole boundary conditions’. We consider a massless minimally coupled scalar field, initially in the spherically symmetric ‘mini-superspace’ approximation, and then in the ‘midi-superspace’ approximation, where no...

We consider the effects of varying dispersion and nonlinearity on the stability of periodic traveling wave solutions of nonlinear PDEs of KdV type, including generalized KdV and Benjamin–Ono equations. In this investigation, we consider the spectral stability of such solutions that arise as small perturbations of an equilibrium state. A key feature of our analysis is the development of a nonloc...

We study the negative mass Schwarzschild spacetime, which has a naked singularity, and show that it is perturbatively unstable. This is achieved by first introducing a modification of the well known Regge Wheeler Zerilli approach to black hole perturbations to allow for the presence of a “kinematic” singularity that arises for negative masses, and then exhibiting exact exponentially growing sol...

Recently a covariant approach to cold matter universes in the zero-shear hypersurfaces (or longitudinal) gauge has been developed. This approach reveals the existence of an integrability condition, which does not appear in standard non-covariant treatments. A simple derivation and generalization of the integrability condition is given, based on showing that the quasi-Newtonian models are a sub-...

We consider nonlinear Schrödinger equations in R. Assume that the linear Hamiltonians have two bound states. For certain finite codimension subset in the space of initial data, we construct solutions converging to the excited states in both non-resonant and resonant cases. In the resonant case, the linearized operators around the excited states are non-self adjoint perturbations to some linear ...

We deal with Dirichlet’s problem for second order quasilinear non-divergence form elliptic equations with discontinuous coefficients. First we state suitable structure, growth, and regularity conditions ensuring solvability of the problem under consideration. Then we fix a solution u0 such that the linearized in u0 problem is non-degenerate, and we apply the Implicit Function Theorem: For all s...

Rotating spiral waves are a form of self-organization observed in spatially extended systems of physical, chemical, and biological natures. A small perturbation causes gradual change in spatial location of spiral's rotation center and frequency, i.e., drift. The response functions (RFs) of a spiral wave are the eigenfunctions of the adjoint linearized operator corresponding to the critical eige...

In this lecture we outline main results of our investigations of certain field-theoretic systems which have V-shaped field potential. After presenting physical examples of such systems we show that in static problems the exact ground state value of the field is achieved on a finite distance – there are no exponential tails. This applies in particular to soliton-like object called the topologica...