Related papers: Wave Arithmetic: A Smooth Integral Representation …
Rational wave numbers are periodic sequences ${\mathbf \omega}={\bf A}{\bf w}(f,g)$ in which amplitude ${\bf A}$ a product of powers of trigonometric sequences and ${\bf w}(f,g)=\exp({\bf {i2}\pi ( f {\mathbf \xi} \oplus g{\bf 1})})$ is a…
We introduce a novel method for encoding integers using smooth real-valued functions whose integral properties implicitly reflect discrete quantities. In contrast to classical representations, where the integer appears as an explicit…
Wavelets are a useful basis for constructing solutions of the integral and differential equations of scattering theory. Wavelet bases efficiently represent functions with smooth structures on different scales, and the matrix representation…
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…
The arithmetic of natural numbers has a natural and simple encoding within sets, and the simplest set whose structure is not that of any natural number extends this set-theoretic representation to positive and negative integers. The…
The wave function in quantum mechanics presents an interesting challenge to our understanding of the physical world. In this paper, I show that the wave function can be understood as four intrinsic relations on physical space. My account…
The paper exploits an isomorphism between the natural numbers N and a space U of periodic sequences of the roots of unity in constructing a recursive procedure for representing and computing the prime numbers. The nth wave number ${\bf…
"Clarithmetic" is a generic name for formal number theories similar to Peano arithmetic, but based on computability logic (see http://www.cis.upenn.edu/~giorgi/cl.html) instead of the more traditional classical or intuitionistic logics.…
Spiral waves are striking self-organized coherent structures that organize spatio-temporal dynamics in dissipative, spatially extended systems. In this paper, we provide a conceptual approach to various properties of spiral waves. Rather…
Scattering of electromagnetic waves by many small particles of arbitrary shapes is reduced rigorously to solving linear algebraic system of equations bypassing the usual usage of integral equations. The matrix elements of this linear…
The choice of mathematical representation when describing physical systems is of great consequence, and this choice is usually determined by the properties of the problem at hand. Here we examine the little-known wave operator…
An integral representation of solutions of the wave equation as a superposition of other solutions of this equation is built. The solutions from a wide class can be used as building blocks for the representation. Considerations are based on…
In sight of fundamental thermal limits on further substantial performance improvements of modern digital computational processing units, wave-based analog computation is becoming an enticing alternative. A wave, as it propagates through a…
A new approach to the geometrization of the electron theory is proposed. The particle wave function is represented by a geometric entity, i.e., Clifford number, with the translation rules possessing the structure of Dirac equation for any…
Smoothed Wigner transforms have been used in signal processing, as a regularized version of the Wigner transform, and have been proposed as an alternative to it in the homogenization and / or semiclassical limits of wave equations. We…
We draw attention to various aspects of number theory emerging in the time evolution of elementary quantum systems with quadratic phases. Such model systems can be realized in actual experiments. Our analysis paves the way to a new,…
String diagrams turn algebraic equations into topological moves that have recurring shapes, involving the sliding of one diagram past another. We individuate, at the root of this fact, the dual nature of polygraphs as presentations of…
The development of wavelet theory has in recent years spawned applications in signal processing, in fast algorithms for integral transforms, and in image and function representation methods. This last application has stimulated interest in…
Wave-like spatial statistics in walking-droplet quantum analogs are typically attributed to spatial or temporal nonlocal wave effects. We show instead that such behavior arises generically from the low-dimensional nonlinear dynamics of an…
In this paper, we propose and analyze semi-implicit numerical schemes for the stochastic wave equation (SWE) with general nonlinearity and multiplicative noise. These numerical schemes, called stochastic scalar auxiliary variable (SAV)…