Fractal Analysis of Interval Series of Prime Numbers

The paper presents a fractal analysis of interval series, the members of which are consecutive deviations of natural prime numbers. Fractal analysis, which has been in full force since the end of the last century, has made it possible to identify new, unusual properties of geometric and physical objects and processes, including predicting the behavior of time and spatial series. The combination of two structural blocks - the spatial interval series of increasing power and the fractal set made it possible to apply the fractal technique to the study of the sequence of intervals. With its help, the idea of the phenomenon of intervals of prime numbers as a structure that does not contradict the nature of most natural phenomena is expanded. Using the Hurst method and scaling it is established that the appearance of intervals of prime numbers is not random. With restrictions on the available computer memory, criteria-based estimates of interval series of different capacities were carried out and it was found that they have the properties of scale invariance, multifractality and self-similarity. The performed estimates confirm that the continuum of primes at all scale levels belongs to fractal sets.

Uniqueness and numerical reconstruction for inverse problems dealing with interval size search

<p style='text-indent:20px;'>We consider a heat equation and a wave equation in one spatial dimension. This article deals with the inverse problem of determining the size of the spatial interval from some extra boundary information on the solution. Under several different circumstances, we prove uniqueness, non-uniqueness and some size estimates. Moreover, we numerically solve the inverse problems and compute accurate approximations of the size. This is illustrated with several satisfactory numerical experiments.</p>

An Accurate Approximation of the Two-Phase Stefan Problem with Coefficient Smoothing

In this work, we consider the heat transfer problems with phase change. The mathematical model is described through a two-phase Stefan problem and defined in the whole domain that contains frozen and thawed subdomains. For the numerical solution of the problem, we present three schemes based on different smoothing of the sharp phase change interface. We propose the method using smooth coefficient approximation based on the analytical smoothing of discontinuous coefficients through an error function with a given smoothing interval. The second method is based on smoothing in one spatial interval (cell) and provides a minimal length of smoothing calculated automatically for the given values of temperatures on the mesh. The third scheme is a convenient scheme using a linear approximation of the coefficient on the smoothing interval. The results of the numerical computations on a model problem with an exact solution are presented for the one-dimensional formulation. The extension of the method is presented for the solution of the two-dimensional problem with numerical results.

Sticky-reflected stochastic heat equation driven by colored noise

UDC 519.21 We prove the existence of a sticky-reflected solution to the heat equation on the spatial interval driven by colored noise. The process can be interpreted as an infinite-dimensional analog of the sticky-reflected Brownian motion on the real line, but now the solution obeys the usual stochastic heat equation except for points where it reaches zero. The solution has no noise at zero and a drift pushes it to stay positive. The proof is based on a new approach that can also be applied to other types of SPDEs with discontinuous coefficients.

Simulation of the Vibratory Behavior of Slender Shafts Subject to Transverse Loads Moving in the Axial Direction

In various machines of the manufacturing industry, and in particular in paper converting machinery, there are shafts operating under conditions similar to that of a slender beam subjected to a transverse load moving in the axial direction. This condition can lead to vibrations and consequent deterioration of the machine performance and of the product quality. The problem has been theoretically studied in the literature since the 1990s. While shaft mass and stiffness are universally considered among the most influential parameters on its vibratory behavior, less obvious and not investigated in the literature is the influence of the spatial interval between two successive loads, an aspect that should be considered in the shaft design phase. In fact, if that is less than the length of the shaft, i.e., if there is more than one transverse load on the shaft at a given time, the vibration level may decrease with respect to the single-load configuration. This work describes the development of a mathematical model of a slender shaft hinged at its ends, representing the rotor of a paper roll perforating unit, with the SW Mathematica. The effect of a load moving axially at a given speed followed by similar loads after given spatial intervals was simulated investigating the influence of speed and load interval on shaft vibrations and resonance. The results showed how reducing the load interval can lead to a reduction of the shaft vibration which is a useful indication on possible design corrective actions.

Infinitely many periodic solutions for a semilinear wave equation with x-dependent coefficients

This paper is devoted to the study of periodic (in time) solutions to an one-dimensional semilinear wave equation with x-dependent coefficients under various homogeneous boundary conditions. Such a model arises from the forced vibrations of a nonhomogeneous string and propagation of seismic waves in nonisotropic media. By combining variational methods with an approximation argument, we prove that there exist infinitely many periodic solutions whenever the period is a rational multiple of the length of the spatial interval. The proof is essentially based on the spectral properties of the wave operator with x-dependent coefficients.

Investigation of a Signal Demodulation Method based on Wavelet Transformation for OFDR to Enhance Its Distributed Sensing Performance

Optical fiber distributed sensing that is based on optical frequency domain reflectometer (OFDR) is a promising technology for achieving a highest spatial resolution downwards to several millimeters. An OFDR signal demodulation method that is based on Morlet wavelet transformation (WT) is demonstrated in detail to improve the resolution of distributed sensing physical quantity under a high spatial resolution, aiming at the trade-off between spatial and spectrum resolution. The spectrum resolution, spatial interval of the measured gauges, and spatial resolution can be manually controlled by adjusting the wavelet parameters. The experimental results that were achieved by the wavelet transformation (WT) method are compared with these by short time Fourier transformation (STFT) method and they indicate that significant improvements, such as strain resolution of 1 με, spatial resolution of 5 mm, average repeatability of 4.3 με, and stability of 7.3 με within one hour, have been achieved. The advantages of this method are high spatial and spectral resolution, robust, and applicability with current OFDR systems.

Existence of multiple periodic solutions to a semilinear wave equation with x-dependent coefficients

This paper is concerned with the periodic (in time) solutions to an one-dimensional semilinear wave equation with x-dependent coefficients. Such a model arises from the forced vibrations of a nonhomogeneous string and propagation of seismic waves in nonisotropic media. By combining variational methods with saddle point reduction technique, we obtain the existence of at least three periodic solutions whenever the period is a rational multiple of the length of the spatial interval. Our method is based on a delicate analysis for the asymptotic character of the spectrum of the wave operator with x-dependent coefficients, and the spectral properties play an essential role in the proof.

Dependence of the Oppel-Kundt illusion on configuration of the components

We studied a size estimation of spatial intervals, which were limited by two straight or curved lines. Intervals widths varied from 2.56 to 3.12 deg. Additional lines could be within intervals. The task of the observers was to estimate the separation between the distant lines. Three experiments were carried out. In the first experiment, several additional lines divided spatial intervals into equal parts. Thus, we investigated the Oppel-Kundt illusion. We founded the invariant dependence of the illusion on number of lines with respect to the interval size. In the second experiment, only two additional lines were arranged inside the interval on different distances. Maximal illusion was obtained for the equal distance between the all four lines. In the third experiment, the two curved parallel lines limited spatial interval. Two other equidistant lines could be within the intervals. Revealed here illusions were greater than illusions for the intervals bounded by straight lines. Model of modules gave a good approximation to data of the Oppel-Kund illusion.