Related papers: Discrete Symbol Calculus
In the present talk we briefly demonstrate an elegant and effective technique for calculation of the trace expansion in the derivatives of background fields. One of main advantages of the technique is manifestly (super)symmetrical and gauge…
Consider an elliptic self-adjoint pseudodifferential operator $A$ acting on $m$-columns of half-densities on a closed manifold $M$, whose principal symbol is assumed to have simple eigenvalues. Relying on a basis of pseudodifferential…
This paper is devoted to conducting a comprehensive and self-contained study of the boundedness on modulation spaces of Fourier integral operators arising when solving Schr\"{o}dinger type operators. The symbols of these operators belong to…
We obtain an asymptotic formula for the counting function of the discrete spectrum for Hankel-type pseudo-differential operators with discontinuous symbols.
Wavelet based algorithms in numerical analysis are similar to other transform methods in that vectors and operators are expanded into a basis and the computations take place in this new system of coordinates. However, due to the recursive…
Symmetry plays a central role in accelerating symbolic computation involving polynomials. This chapter surveys recent developments and foundational methods that leverage the inherent symmetries of polynomial systems to reduce complexity,…
Bilinear pseudodifferential operators with symbols in the bilinear analog of all the H\"ormander classes are considered and the possibility of a symbolic calculus for the transposes of the operators in such classes is investigated. Precise…
The goal of the present work is to solve a linear dispersive equation with variable coefficient advection on an unbounded domain. In this setting, transparent boundary conditions are vital to allow waves to leave (or even re-enter) the,…
Symbolic computation, powered by modern computer algebra systems, has important applications in mathematical reasoning through exact deep computations. The efficiency of symbolic computation is largely constrained by such deep computations…
For $m\in\mathbb{R}$ we introduce the symbol classes $S^m$, $m\in\mathbb{R}$, consisting of smooth functions $\sigma$ on $\mathbb{R}^{2d}$ such that $|\partial^\alpha \sigma(z)|\leq C_\alpha (1+|z|^2)^{m/2}$, $z\in\mathbb{R}^{2d}$, and we…
An unsteady problem is considered for a space-fractional equation in a bounded domain. A first-order evolutionary equation involves the square root of an elliptic operator of second order. Finite element approximation in space is employed.…
We propose a functional calculus which allows one to apply functions to the matrix anti-commutator/commutator operator. The calculus is introduced in a straightforward manner if the operators act on symmetric matrices, and it leads to a…
This paper investigates the possibility of approximating multiple mathematical operations in latent space for expression derivation. To this end, we introduce different multi-operational representation paradigms, modelling mathematical…
We construct an abstract pseudodifferential calculus with operator-valued symbol, adapted to the treatment of Coulomb-type interactions, and we apply it to study the quantum evolution of molecules in the Born-Oppenheimer approximation, in…
In our previous work, numerical schemes for a simplified version of 3-wave kinetic equations, in which only the simple forward-cascade terms of the collision operators are kept, have been successfully designed, especially to capture the…
We obtain weighted $L^p$ inequalities for pseudo-differential operators with smooth symbols and their commutators by using a class of new weight functions which include Muckenhoupt weight functions. Our results improve essentially some…
Solving the wave equation is essential to seismic imaging and inversion. The numerical solution of the Helmholtz equation, fundamental to this process, often encounters significant computational and memory challenges. We propose an…
The numerical simulation of wave propagation in semiclassical (high-frequency) problems is well known to pose a formidable challenge. In this work, a new phase-space approach for the numerical simulation of semiclassical wave propagation,…
In this paper, we give a sharp sparse domination of pseudodifferential operators associated with symbols belonging to the H\"{o}rmander class, and fundamental solutions of dispersive equations. Furthermore, we give boundedness results of…
We prove a symbolic calculus for a class of pseudodifferential operators, and discuss its applications to $L^2$-compactness via a compact version of the $T(1)$ theorem.