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We consider solution operators of linear ordinary boundary problems with "too many" boundary conditions, which are not always solvable. These generalized Green's operators are a certain kind of generalized inverses of differential…
The two-time Green function method in quantum electrodynamics of high-Z few-electron atoms is described in detail. This method provides a simple procedure for deriving formulas for the energy shift of a single level and for the energies and…
Deep Convolutional Neural Networks~(CNNs) offer remarkable performance of classifications and regressions in many high-dimensional problems and have been widely utilized in real-word cognitive applications. However, high computational cost…
Spectral neural operators, particularly Fourier Neural Operators (FNO), are a powerful framework for learning solution operators of partial differential equations (PDEs) due to their efficient global mixing in the frequency domain. However,…
In this letter, we investigate the tradeoff between energy efficiency (EE) and spectral efficiency (SE) in device-to-device (D2D) communications underlaying cellular networks with uplink channel reuse. The resource allocation problem is…
A new scheme has been proposed to solve the B.E. condenstates in terms of Green's function approach. It has been shown that the radial wave function of two interacting atoms, moving in a common harmonic oscillator potential modified by an…
The Muon optimizer, a matrix-structured algorithm that leverages spectral orthogonalization of gradients, is a milestone in the pretraining of large language models. However, the underlying mechanisms of Muon -- particularly the role of…
This paper introduces a hierarchical interpolative decomposition butterfly-LU factorization (H-IDBF-LU) preconditioner for solving two-dimensional electric-field integral equations (EFIEs) in electromagnetic scattering problems of perfect…
A new low-order discretization scheme for the identity operator in the magnetic field integral equation (MFIE) is discussed. Its concept is derived from the weak-form representation of combined sources which are discretized with…
We introduce basic aspects of new operator method, which is very suitable for practical solving differential equations of various types. The main advantage of the method is revealed in opportunity to find compact exact operator solutions of…
We introduce three types of partial fractional operators of variable order. An integration by parts formula for partial fractional integrals of variable order and an extension of Green's theorem are proved. These results allow us to obtain…
The localized operator partitioning method [Y. Khan and P. Brumer, J. Chem. Phys. 137, 194112 (2012)] rigorously defines the electronic energy on any subsystem within a molecule and gives a precise meaning to the subsystem ground and…
Using the framework of operator or Calder\'on preconditioning, uniform preconditioners are constructed for elliptic operators discretized with continuous finite (or boundary) elements. The preconditioners are constructed as the composition…
The method-of-moments implementation of the electric-field integral equation (EFIE) yields many code-verification challenges due to the various sources of numerical error and their possible interactions. Matters are further complicated by…
The recently introduced concept of a spectral shift operator is applied in several instances. Explicit applications include Krein's trace formula for pairs of self-adjoint operators, the Birman-Solomyak spectral averaging formula and its…
In this paper, we propose a data-driven framework for constructing efficient approximate inverse preconditioners for elliptic partial differential equations (PDEs) by learning the Green's function of the underlying operator with neural…
The mode selection and resource allocation in fog radio access networks (F-RANs) have been advocated as key techniques to improve spectral and energy efficiency. In this paper, we investigate the joint optimization of mode selection and…
Dyadic Green's function is an important tool of computational photonics, giving deeper insights into light-matter interaction. We present an operator approach to the derivation of the dyadic Green's function of a generic anisotropic…
The task to construct parametrices of elliptic differential operators on a manifold with edges requires a calculus of operators with a two-component principal symbolic hierarchy, consisting of (edge-degenerate) interior and…
The Fast Multipole Method (FMM) is an efficient numerical algorithm for computation of long-ranged forces in $N$-body problems within gravitational and electrostatic fields. This method utilizes multipole expansions of the Green's function…