Addendum: Linear Random Operator Equations in Mathematical Physics. III

‎A matrix LSQR algorithm for solving constrained linear operator equations

In this work‎, ‎an iterative method based on a matrix form of LSQR algorithm is constructed for solving the linear operator equation $mathcal{A}(X)=B$‎ ‎and the minimum Frobenius norm residual problem $||mathcal{A}(X)-B||_F$‎ ‎where $Xin mathcal{S}:={Xin textsf{R}^{ntimes n}~|~X=mathcal{G}(X)}$‎, ‎$mathcal{F}$ is the linear operator from $textsf{R}^{ntimes n}$ onto $textsf{R}^{rtimes s}$‎, ‎$ma...

Some Beautiful Equations of Mathematical Physics

The basic ideas and the important role of gauge principles in modern elementary particle physics are outlined. There are three theoretically consistent gauge principles in quantum field theory: the spin-1 gauge principle of electromagnetism and the standard model, the spin-2 gauge principle of general relativity, and the spin-3/2 gauge principle of supergravity. Dirac Lecture delivered at the I...

Index theory for linear self-adjoint operator equations and nontrivial solutions for asymptotically linear operator equations

We will first establish an index theory for linear self-adjoint operator equations. And then with the help of this index theory we will discuss existence and multiplicity of solutions for asymptotically linear operator equations by making use of the dual variational methods and Morse theory. Finally, some interesting examples concerning second order Hamiltonian systems, first order Hamiltonian ...

SPENCER OPERATOR AND APPLICATIONS: From Continuum Mechanics to Mathematical physics

Let us revisit briefly the foundation of n-dimensional elasticity theory as it can be found today in any textbook, restricting our study to n = 2 for simplicity. If x = (x, x) is a point in the plane and ξ = (ξ(x), ξ(x)) is the displacement vector, lowering the indices by means of the Euclidean metric, we may introduce the ”small” deformation tensor ǫ = (ǫij = ǫij = (1/2)(∂iξj + ∂jξi)) with n(n...

Hypergeometric series solutions of linear operator equations