Related papers: The Prolongation Problem for the Heavenly Equation
The Bessel point process is a rigid point process on the positive real line and its conditional measure on a bounded interval $[0,R]$ is almost surely an orthogonal polynomial ensemble. In this article, we show that if $R$ tends to…
Consider an operator equation $F(u)=0$ in a real Hilbert space. The problem of solving this equation is ill-posed if the operator $F'(u)$ is not boundedly invertible, and well-posed otherwise. A general method, dynamical systems method…
Explicit solutions of differential equations of complex fractional orders with respect to functions and with continuous variable coefficients are established. The representations of solutions are given in terms of some convergent infinite…
A nonlinear operator equation $F(x)=0$, $F:H\to H,$ in a Hilbert space is considered. Continuous Newton's-type procedures based on a construction of a dynamical system with the trajectory starting at some initial point $x_0$ and becoming…
We generalize the Benney lattice and show that the new system of equations can be reduced to a generalized Chaplygin gas as well as the heavenly equation. We construct two infinite sets of conserved charges and show that one of the sets can…
Sufficient conditions are determined on the parameters such that the generalized and normalized Bessel function of the first kind and other related functions belong to subclasses of starlike and convex functions defined in the unit disk…
We obtain approximation results for general positive linear operators satisfying mild conditions, when acting on discontinuous functions and absolutely continuous functions having discontinuous derivatives. The upper bounds, given in terms…
We obtain an explicit H\"older regularity result for viscosity solutions of a class of second order fully nonlinear equations leaded by operator that are neither convex/concave nor uniformly elliptic.
We discuss the cosmological constant problem in the context of higher codimension brane world scenarios with infinite-volume extra dimensions.
In this paper, we briefly explain the spectral expansion problem for differential operators defined on the entire real line, generated by a differential expression with periodic, complex-valued coefficients.
We discuss a general combinatorial framework for operator ordering problems by applying it to the normal ordering of the powers and exponential of the boson number operator. The solution of the problem is given in terms of Bell and Stirling…
In an Oberbeck-Boussinesq model, rigorously derived, which includes compressibility, one could expect that the onset of convection for the B\'enard problem occurs at a higher critical Rayleigh number. Since of the difficulties related to…
On a constraint manifold we give an explicit formula for the Hessian matrix of a cost function that involves the Hessian matrix of a prolonged function and the Hessian matrices of the constraint functions. We give an explicit formula for…
The Cauchy problem for fractional derivatives linear systems of ordinary differential equations with constant coefficients is considered, where at first the analytic expressions are given through the matrix exponent of its corresponding…
In this paper, we consider a resolvent problem arising from the $Q$-tensor model for liquid crystal flows in the half-space. Our purpose is to show the $\mathcal{R}$-boundedness for the solution operator families of the resolvent problem…
We show that, under certain regularity assumptions, there exists a linear extension operator.
We solve the probability continuity equation within the Madelung-Bohm framework, assuming a separable phase expressed as $S(x,t) = Q(x)\dot{\nu}(t) + \mu(t)$. Using operator methods, we reformulate the wave function's amplitude into a form…
The Operator axioms have produced new real numbers with new operators. New operators naturally produce new equations and thus extend the traditional mathematical models which are selected to describe various scientific rules. So new…
We introduce a method of obtaining a higher order extension problem, \'a la Caffarelli-Silvestre, utilizing ideas from renormalization. Moreover, we give an alternative perspective of the recently developed extension problem for the…
In this article, we use a Berndtsson-Andersson operator and the Bergman metric in order to solve the $\bar\partial$ equation on convex domains of finite type for forms satisifying a Carleson condition and get norm estimates of the solution…