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Expository article on the problem of determining the maximum number of equiangular lines with a fixed angle, and the associated problem of second eigenvalue multiplicity in graphs.
It is shown that Bell's counterfactuals admit joint quasiprobability distributions (i.e. joint distributions exist, but may not be non-negative). A necessary and sufficient condition for the existence among them of a true probability…
We consider eigenvalue problems in quantum mechanics in one dimension. Hamiltonians contain a class of double well potential terms, x^6 + \alpha x^2, for example . The space coordinate is continued to a complex plane and the connection…
We consider inverse scattering problems for the three-dimensional Hartree equation. We prove that if the unknown interaction potential $V(x)$ of the equation satisfies some rapid decay condition, then we can uniquely determine the exact…
The double-layer potential plays an important r$\hat{\rm o}$le in solving boundary value problems of elliptic equations. Here, in this paper, we aim at introducing and investigating double layer potentials for a generalized bi-axially…
A two dimensional eigenvalue problem (2DEVP) of a Hermitian matrix pair $(A, C)$ is introduced in this paper. The 2DEVP can be viewed as a linear algebraic formulation of the well-known eigenvalue optimization problem of the parameter…
We introduce a notion of strong periodicity of a module over a finite-dimensional algebra over a field. We prove that the existence of such modules over certain idempotent algebras is both a necessary and sufficient condition for the…
In this article we find the equivalent conditions to assure the existence and uniqueness of positive solutions to semilinear elliptic equations wih double power nonlinearities. As a bonus, we give a simpler proof of our former result that…
We study conditions of reduction of multidimensional wave equations - a system of d'Alembert and Hamilton equations. Necessary conditions for compatibility of such reduction conditions are proved. Possible types of the reduced equations and…
One of the most widely problem studied in quantum mechanics is of an infinite square-well potential. In a minimal-length scenario its study requires additional care because the boundary conditions at the walls of the well are not well…
We study an eigenvalue problem in the framework of double phase variational integrals and we introduce a sequence of nonlinear eigenvalues by a minimax procedure. We establish a continuity result for the nonlinear eigenvalues with respect…
First, a new sufficient condition for uniqueness of weak solutions is proved for the system of 2D viscous Primitive Equations. Second, global existence and uniqueness are established for several classes of weak solutions with partial…
The Bethe equations for the periodic XXX and XXZ spin chains admit singular solutions, for which the corresponding eigenvalues and eigenvectors are ill-defined. We use a twist regularization to derive conditions for such singular solutions…
We prove general necessary optimality conditions for delta-nabla isoperimetric problems of the calculus of variations.
Sufficient conditions for the well-posedness of the initial value problem for the scalar wave equation are obtained in space-times with hypersurface singularities
The Bellman equation and its continuous-time counterpart, the Hamilton-Jacobi-Bellman (HJB) equation, serve as necessary conditions for optimality in reinforcement learning and optimal control. While the value function is known to be the…
The inverse spectral problem for the second-order differential pencil with quadratic dependence on the spectral parameter is studied. We obtain sufficient conditions for the global solvability of the inverse problem, prove its local…
The inverse problem for the differential operator pencil with complex periodic potential and discontinuous coefficients on the axis is studied. Main characteristics of the fundamental solutions are investigated, the spectrum of the operator…
We establish the unique solvability of a coupling problem for entire functions which arises in inverse spectral theory for singular second order ordinary differential equations/two-dimensional first order systems and is also of relevance…
We deduce a sufficient condition of the exponential (integral) turnpike property for infinite dimensional generalized linear-quadratic optimal control problems in terms of structural properties of the control system, such as exponential…