Related papers: Oscillatory Integrals and Fractal Dimension
We discuss a number of techniques for determining the Minkowski dimension of bounded subsets of some Euclidean space of any dimension, including: the box-counting dimension and equivalent definitions based on various box-counting functions;…
We introduce a general purpose algorithm for rapidly computing certain types of oscillatory integrals which frequently arise in problems connected to wave propagation and general hyperbolic equations. The problem is to evaluate numerically…
We consider a generalization of the notion of spaces of homogeneous type, inspired by recent work of Street [21] on the multi-parameter Carnot-Caratheodory geometry, which imbues such spaces with differentiability structure. The setting…
We study two classes of radial integrals involving a product of bound and continuum one-electron states. Using a representation of the continuum part with an expansion on complex Gaussian Type Orbitals, such integrals can be performed…
We present a simple and efficient method to incorporate anharmonic effects in the vibrational \textcolor{black}{analyses} of molecules within density functional theory (DFT) calculations. This approach is closely related to the traditional…
The fractal dimension of a surface allows its degree of roughness to be characterized quantitatively. However, limited effort is attempted to calculate the fractal dimension of surfaces computed from precisely known atomic coordinates from…
In this paper, we delve into the fascinating realm of fractal calculus applied to fractal sets and fractal curves. Our study includes an exploration of the method analogues of the separable method and the integrating factor technique for…
With extensive variational simulations, dissipative quantum phase transitions in the sub-Ohmic spin-boson model are numerically studied in a dense limit of environmental modes. By employing a generalized trial wave function composed of…
By means of analytical and numerical methods we analyze the phase diagram of polaritons in one-dimensional coupled cavities. We locate the phase boundary, discuss the behavior of the polariton compressibility and visibility fringes across…
We discuss a variation of Takagi curves based, however, more on algebraic than geometric principles. Namely, we construct functions of loops in a special binary representation. The graph of these functions usually has chaotic and fractal…
We study certain families of oscillatory integrals $I_\varphi(a)$, parametrised by phase functions $\varphi$ and amplitude functions $a$ globally defined on $\mathbb{R}^d$, which give rise to tempered distributions, avoiding the standard…
In this paper, we consider the uniform estimate for the oscillatory integral with stationary phase, which was previously studied by Alazard-Burq-Zuily. We significantly reduce the order of required regularity condition on the phase and…
We construct spectral triples and, in particular, Dirac operators, for the algebra of continuous functions on certain compact metric spaces. The triples are countable sums of triples where each summand is based on a curve in the space.…
We propose a simple scheme to measure squeezing and phase properties of a harmonic oscillator. We treat in particular the case of a the field, but the scheme may be easily realized in ion traps. It is based on integral transforms of…
We present a closed-form, exact analytical solution, valid at finite times, to a class of multiple integrals with highly oscillatory kernels. Our approach leverages the intimate connection between these integrals and the minimal…
We review the theoretical framework that establishes a crucial bridge between the general Steiner-type formula of Hug, Last, and Weil and the theory of complex (fractal) dimensions of Lapidus et all. Two novel families of geometric…
In this paper, we discuss the Higuchi algorithm which serves as a widely used estimator for the box-counting dimension of the graph of a bounded function $f : [0,1] \to \R$. We formulate the method in a mathematically precise way and show…
The possibility of production of pulses with scale-invariant properties at presence of fluctuations of parameters of self-oscillatory system with three-dimensional phase space is shown. The system of equations of inertial nonlinearity…
The manner in which probability amplitudes of paths sum up to form wave functions of a harmonic oscillator, as well as other, simple 1-dimensional problems, is described. Using known, closed-form, path-based propagators for each problem, an…
Introduction Path Integrals - Introduction - Propagator - Free Particle - Path Integral Representation of Quantum Mechanics - Particle on a Ring - Particle in a Box - Driven Harmonic Oscillator - Semiclassical Approximation - Imaginary Time…