Related papers: A unifying perspective on linear continuum equatio…

We consider resolvents of operators taking the form ${\bf A}=\Gamma_1{\bf B}\Gamma_1$ where $\Gamma_1({\bf k})$ is a projection that acts locally in Fourier space and ${\bf B}({\bf x})$ is an operator that acts locally in real space. Such…

Following advances in the abstract theory of composites, we develop rapidly converging series expansions about $z=\infty$ for the resolvent ${\bf R}(z)=[z{\bf I}-{\bf P}^\dagger{\bf Q}{\bf P}]^{-1}$ where ${\bf Q}$ is an orthogonal…

Enlarging on Parts I, II, and III we write more equations in the desired format of the extended abstract theory of composites. We focus on a multitude of equations involving higher order derivatives. The motivation is that results and…

This chapter deals with the notion of the resolvent of a self-adjoint operator. We pay special attention to the convergence of unbounded self-adjoint operators in several resolvent senses, and how they are related to the convergence of…

Let $q\neq \pm 1$ be a complex number of modulus one. This paper deals with the operator relation $AB=qBA$ for self-adjoint operators $A$ and $B$ on a Hilbert space. Two classes of well-behaved representations of this relation are studied…

Convergence of operators acting on a given Hilbert space is an old and well studied topic in operator theory. The idea of introducing a related notion for operators acting on arying spaces is natural. However, it seems that the first…

For a function of a type $ \left| \mathbf{r}_1{+}\ldots {+}\mathbf{r}_{_N} \right|^{-\nu} \in \mathbb{R} $ from the many-dimensional vectors $ \mathbf{r}_s $ in Euclidean space, the successive algebraic approach is the derivation of the…

The resolvent convergence of self-adjoint operators via the technique of $\Gamma$-convergence of quadratic forms is adapted to incorporate complex Hilbert spaces. As an application, we find effective operators to the Dirichlet Laplacian…

We elaborate on a recently conjectured relation of Painlev\'e transcendents and 2D CFT. General solutions of Painlev\'e VI, V and III are expressed in terms of $c=1$ conformal blocks and their irregular limits, AGT-related to instanton…

Following some past advances, we reformulate a large class of linear continuum science equations in the format of the extended abstract theory of composites so that we can apply this theory to better understand and efficiently solve those…

This paper introduces a very fast method for the computation of the resolvent of fractional powers of operators. The analysis is kept in the continuous setting of (potentially unbounded) self adjoint positive operators in Hilbert spaces.…

In this paper we introduce a class of mathematical objects called \emph{extensors} and develop some aspects of their theory with considerable detail. We give special names to several particular but important cases of extensors. The…

We consider here a spherically symmetric but inhomogeneous universe filled with a massless scalar field. The model obeys two constraints. The first one is that the gradient of the scalar field is timelike everywhere. The second constraint…

We derive global analytic representations of fundamental solutions for a class of linear parabolic systems with full coupling of first order derivative terms where coefficient may depend on space and time. Pointwise convergence of the…

Monotone operators, especially in the form of subdifferential operators, are of basic importance in optimization. It is well known since Minty, Rockafellar, and Bertsekas-Eckstein that in Hilbert space, monotone operators can be understood…

In the whole space $R^d$, $d\ge 2$, we study homogenization of a divergence form elliptic fourth-order operator $A_\varepsilon$ with measurable $\varepsilon$-periodic coefficients, where $\varepsilon$ is a small parameter. For the resolvent…

In this paper we study invertible extensions of a symmetric operator in a Hilbert space $H$. All such extensions are characterized by a parameter in the generalized Neumann's formulas. Generalized resolvents, which are generated by the…

We consider a class of linear integral operators with impulse responses varying regularly in time or space. These operators appear in a large number of applications ranging from signal/image processing to biology. Evaluating their action on…

The subject of the article is linear systems of wave equations on cosmological backgrounds with convergent asymptotics. The condition of convergence corresponds to the requirement that the second fundamental form, when suitably normalised,…

We prove that any given function can be smoothly approximated by functions lying in the kernel of a linear operator involving at least one fractional component. The setting in which we work is very general, since it takes into account…