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Recently, an asymptotic solution of the Rayleigh equation for an empty bubble in $N$ dimensions has been obtained. Here we give the closed--from general analytical solution of this equation. We also find the general solution of the Rayleigh…
It is well known that finite-dimensional polyhedral convex sets can be generated by finitely many points and finitely many directions. Representation formulas in this spirit are obtained for convex polyhedra and generalized convex polyhedra…
We demonstrate how one can describe explicitly the present arbitrariness in solutions of relativistic wave equations in external electromagnetic fields of special form. This arbitrariness is connected to the existence of a transformation,…
In this paper, we determine all unitary solutions to the Yang-Baxter equation in dimension four. Quantum computation motivates this study. This set of solutions will assist in clarifying the relationship between quantum entanglement and…
This article simply presents several coordinate systems for 2 and 3-dimensional hyperbolic spaces, describing the general solutions of Helmholtz equation in each one of these systems.
We present a result about solvability in $W^{2}_{p}$, $p>d$, in the whole space $\bR^{d}$ of Bellman's equations with VMO ``coefficients''. Parabolic equations are touched upon as well.
Complex structures can only form in a universe that allows for bound states. While this is clearly observed in three-dimensions, added degrees of freedom in a higher-dimensional space preclude the immediate assumption that binding…
A solution of the Einstein vacuum field equations is constructed within the contex of perturbation theory. The solution possesses a graphical representation in terms of diagrams.
Complete sets of commutation relations for arbitrary pairs of quantum minors are computed, with explicit coefficients in closed form.
Relativistic mechanics on an arbitrary manifold is formulated in the terms of jets of its one-dimensional submanifolds. A generic relativistic Lagrangian is constructed. Relativistic mechanics on a pseudo-Riemannian manifold is particularly…
In this paper, the existence and uniqueness of solution of the Cauchy problem for abstract Boussinesq equation is obtained. By applying this result, the Cauchy problem for systems of Boussinesq equations of finite or infinite orders are…
We review very briefly the main mathematical structures and results in some important areas of Quantum Mechanics involving PDEs and formulate open problems.
It is shown that solutions of the Neumann problem for the Poisson equation in an arbitrary convex $n$-dimensional domain are uniformly Lipschitz. Applications of this result to some aspects of regularity of solutions to the Neumann problem…
Relativistic field equations for a gas in special and general relativity are determined from the Boltzmann equation. The constitutive equations are obtained from the Chapman-Enskog methodology applied to a relativistic model equation…
We prove extension-dimensional versions of finite dimensional selection and approximation theorems. As applications, we obtain several results on extension dimension.
We solve the Kato square root problem for general elliptic operators and systems with measurable and complex coefficients on any domain of the Euclidean space. The operators are subject to Dirichlet boundary conditions. We also allow…
We give a formula to compute the dimension of the generic component of the moduli space of an irreducible germ of curve in the complex plane.
In this work we solve the Dirac equation by constructing the exact bound state solutions for a mixing of vector and scalar generalized Hartmann potentials. This is done provided the vector potential is equal to or minus the scalar…
We derive the complete list of singularities of propagators in theories with arbitrary (complex) masses and for arbitrary diagram. We derive in a closed form differential equations for the propagator as a function of the momentum and…
We solve the classical Dirichlet problem for a general complex Hessian equation on a small ball in $\bC^n$. Then, we show that there is a continuous solution, in pluripotential theory sense, to the Dirichlet problem on compact Hermitian…