Foundations, Volume 2, Issue 3 (September 2022) – 21 articles

In this work, we use similarly separable vectors in separable Hilbert spaces to provide generalized integral results related to majorization, Niezgoda, and Ćebysév type inequalities. Next, we furnish some refinements of these inequalities. Theorems obtained in this work extend and improve several known results in the literature. An important aspect of our work is that these inequalities are directly related to Arithmetic, Geometric, Harmonic, and Power means. These means have played an important role in many branches of arts and sciences since the last 2600 years. Full article

The work is devoted to the results of processing electromagnetic radiation signals obtained during laboratory loading of marble and diabase samples using a technique for determining the parameters of microcracks, developed and published by the author earlier. As a result of such processing, certain patterns were found in the nature of the evolution of the oscillatory process ensemble of microcracks. For example, solitary non-linear waves almost always preceded a sequence of High Frequency traces. Equations for straight lines approximating High Frequency traces in logarithmic coordinates, close to the equation of the Gutenberg–Richter law. Due to the similarity of seismic processes at different scale levels, the results of modeling at the microscale level can be used to describe seismic processes at the macroscale level, for example, to study the processes occurring immediately before destruction and at the time of destruction in order to search for repeatability and regularities. The regularities obtained can be used in the development of a predictive criterion that makes it possible to predict the time of one or another geophysical (seismic) event. Full article

Polyynes are alternations of single and triple bonds between carbon atoms, while cumulenes are successions of double bonds. Since the triple bond is the strongest bond between two carbon atoms, recent preoccupations included synthesizing and condensing cyclic polyynes and cumulenes and their clusters. Density functional theory calculations predicted stable monocyclic rings formation for a number of C atoms equal to or higher than 16. Alternative to the series of Carbon atoms are alternations of Boron and Nitrogen. Large rings (such as those of 24 atoms) can be crossed and thus small clusters can be formed. Patterns of three crosses seem to further stabilize the atomic ensemble. Clusters of $4C_{24}$ and $4B_{12}{N}_{12}$ (96 atoms) as well as $4C_{26}$ (104 atoms) have been designed, and their conformation has been studied here. Full article

In order to explore the evolution characteristics of multi-scale rock-like material failure, we studied the acoustic emission (AE) and electromagnetic radiation (EMR) characteristics of different scale rock-like materials by using the AE-EMR experimental system of coal and rock failure, and the AE and EMR response law of rockburst in mining sites was analyzed. The results show that under uniaxial loading, the stress–strain curve of the specimen has a compaction stage, linear elastic stage, elastic–plastic stage and failure stage. The cumulative AE count, AE energy and stress level of the specimen have an exponential relationship during loading and compression. The cumulative EMR counts of loading and unloading showed a trend of first decreasing and then increasing with the increase in stress level. Electromagnetic radiation and microseismic hypocentral distance show an abnormal change trend when rockburst occurs, and this abnormal phenomenon can be used as a precursor feature signal for rockburst monitoring and early warning. Full article

The accurate monitoring and early warning of coal and rock dynamic disasters become challenging in complex geological environments. Mostly, the signal information contains interferences, which misguides the technician, and thus leads to inaccurate monitoring results. To reduce the influence of interference signals, the synchronous response of the acoustic emission (AE) and electromagnetic emission (EME) signals before the failure of coal specimens during uniaxial loading was investigated in this study. Additionally, the coupling relationship between M value, AE energy/AE ringing count per unit time, and the damage of coal is established, and the early warning index of AE and EME (R value) was computed and verified through the field investigations. The results show that a strong synchronization of the acoustoelectric signals occurs only after the specimen enters the strain strengthening area. The analysis of the obtained results showed that the M value of the AE-EME synchronous response signal represents a strong degree of damage occurring in the coal body, however, this is still subject to false alarms. In contrast, the analysis of the R value accurately helped in determining the damage evaluation, thus, it can be regarded as one of the precursors of the imminent failure of coal. With R > 1, the specimen is closed to the failure state, thereby dangerous regions are identified with a dense concentration of R > 1 events. The obtained R value index through on-site AE and EME monitoring corresponds closely with the stress distribution cloud map of the roadway. It is inferred that the anti-interference ability and the reliability of the R value index are stronger than the routine early warning indicators of the single-AE or EME energy. Full article

Λ-fractional differential equations are discussed since they exhibit non-locality and accuracy. Fractional derivatives form fractional differential equations, considered as describing better various physical phenomena. Nevertheless, fractional derivatives fail to satisfy the prerequisites of differential topology for generating differentials. Hence, all the sources of generating fractional differential equations, such as fractional differential geometry, the fractional calculus of variations, and the fractional field theory, are not mathematically accurate. Nevertheless, the Λ-fractional derivative conforms to all prerequisites demanded by differential topology. Hence, the various mathematical forms, including those derivatives, do not lack the mathematical accuracy or defects of the well-known fractional derivatives. A summary of the Λ-fractional analysis is presented with its influence on the sources of differential equations, such as fractional differential geometry, field theorems, and calculus of variations. Λ-fractional ordinary and partial differential equations will be discussed. Full article

In this paper, we consider the existence of multiple positive solutions to boundary value problems of nonlinear fractional differential equation with integral boundary conditions and parameter dependence. To obtain our results, we used a functional-type cone expansion-compression fixed point theorem and the Leggett–Williams fixed point theorem. Examples are included to illustrate the main results. Full article

This article is concerned with the study of a new class of hybrid fractional q-integro-difference equations involving Caputo type q-derivatives and Riemann-Liouville q-integrals of different orders with a nonlocal q-integro-initial condition. An existence result for the given problem is obtained by means of Krasnoselskii’s fixed point theorem, whereas the uniqueness of its solutions is shown by applying the Banach contraction mapping principle. We also discuss the stability of solutions of the problem at hand and find that it depends on the nonlocal parameter in contrast to the initial position of the domain. To demonstrate the application of the obtained results, examples are constructed. Full article

In one of our previous papers, we performed a comparative analysis of the experimental and theoretical cross-sections for the excitation of atomic hydrogen by electrons. We found that the theoretical ratio of the cross-section σ_2s of the excitation of the state 2s to the cross-section σ_2p of the excitation of the state 2p was systematically higher than the corresponding experimental ratio by about 20% (far beyond the experimental error margins). We showed that this discrepancy can be due to the presence of the Second Flavor of Hydrogen Atoms (SFHA) in the experimental gas and that the share of the SFHA in the mixture, required for removing this discrepancy, was about the same as the share of the usual hydrogen atoms. The theory behind the SFHA was based on the standard quantum mechanics—on the second solution of the Dirac equation for hydrogen atoms—and on the experimental fact that the charge distribution inside the proton has the peak at the center of the proton; the term “flavor” was used by the analogy with flavors of quarks. In the present paper, we used the same guiding principles, as employed in that previous study, for the comparative analysis of the experimental and theoretical cross-sections for the excitation of molecular hydrogen by electrons. We found that presumably the most sophisticated calculations, using the convergent close-coupling method involving 491 states, very significantly underestimate the corresponding experimental cross-sections for the two lowest stable triplet states. We showed that if in some hydrogen molecules one or both atoms would be the SFHA, then the above very significant discrepancy could be eliminated. We estimated that it would take such unusual hydrogen molecules to be represented in the experimental gas by the share of about 0.26. This is just by about 40% smaller than the share 0.45 of the SFHA deduced in our previous analysis of the experiment on the electron impact excitation of hydrogen atoms (rather than hydrogen molecules). It should be emphasized that from the theoretical point of view, the share of the unusual hydrogen molecules in any experimental gas and the share of the unusual hydrogen atoms (SFHA) in any experimental gas should not be expected to coincide (it would be the comparison of “apples to oranges”, rather than “apples to apples”). In addition, given the roughness of the above estimates, we can state that the results of the present paper reinforce the main conclusion of our previous papers of the very significant share of the SFHA in the experimental hydrogen gases. Thus, the experiments on the electron impact excitation of hydrogen molecules are the fourth type of atomic experiments that proved the existence of the SFHA. Full article

In the present research, single and multi-valued (k,ψ)-Hilfer type fractional boundary value problems of order in (1,2] involving nonlocal integral boundary conditions were studied. In the single-valued case, the Banach and Krasnosel’skiĭ fixed point theorems as well as the Leray–Schauder nonlinear alternative were used to establish the existence and uniqueness results. In the multi-valued case, when the right-hand side of the inclusion has convex values, we established an existence result via the Leray–Schauder nonlinear alternative method for multi-valued maps, while the second existence result, dealing with the non-convex valued right-hand side of the inclusion, was obtained by applying Covitz-Nadler fixed point theorem for multi-valued contractions. The obtained theoretical results are well illustrated by the numerical examples provided. Full article

From the perspectives of the thermodynamics of irreversible processes and the theory of complex systems, a characterization of longevity and aging and their relationships with the emergence and evolution of cancer was carried out. It was found that: (1) the rate of entropy production could be used as an index of the robustness, plasticity, and aggressiveness of cancer, as well as a measure of biological age; (2) the aging process, as well as the evolution of cancer, goes through what we call a “biological phase transition”; (3) the process of metastasis, which occurs during the epithelial–mesenchymal transition (EMT), appears to be a phase transition that is far from thermodynamic equilibrium and exhibits Shilnikov chaos-like dynamic behavior, which guarantees the robustness of the process and, in turn, its unpredictability; (4) as the ferroptosis process progresses, the complexity of the dynamics that are associated with the emergence and evolution of cancer decreases. The theoretical framework that was developed in this study could contribute to a better understanding of the biophysical and chemical phenomena of longevity and aging and their relationships with cancer. Full article

The ground-based monitoring of the lower ionosphere by studying the perturbations of the subionospheric propagation of very-low-frequency/low-frequency (VLF/LF) signals is important in the research of a wide variety of geophysical and Sun/space extreme phenomena. Such perturbations are identified as anomalies in the signal received from the VLF/LF transmitters operating worldwide for military purposes, time code broadcasting, etc. Especially for the study of local ionosphere-influencing phenomena, such as earthquakes, volcanoes, typhoons, etc., the monitoring of several subionospheric propagation paths is necessary. However, it is very difficult to find in the market (or reproduce) hardware (HW) for wide-band VLF/LF receivers that could receive many different transmitters, while the involved software (SW) is mainly proprietary. Aiming to provide a low-cost and easy-to-build alternative for the scientists involved in this research field, we suggest a VLF/LF receiver setup based on amateur radio open-source HW and SW. Its key components are the so-called “mini-whip” active antenna and the freeware “SpectrumLab” and “GPS2Time”. The full HW schematics and all settings of the employed SW configuration for the proposed VLF/LF receiver setup are provided in the article. To check the reliability of the proposed receiver setup, two almost identical VLF/LF radio receivers were installed in the prefecture of Attica in Greece, in June and September of 2021, respectively. Examples of ionospheric perturbations due to different phenomena (solar flares, earthquakes, and a magnetic storm) are provided to show the ability of the proposed receiver setup to provide reliable data for ionosphere-related research. Full article

The Traub iterates generate a sequence that converges to a solution of a nonlinear equation given certain conditions. The order of convergence has been shown provided that the fifth Fréchet-derivative exists. Notice that this derivative does not appear on the Traub method. Therefore, according to the earlier results, there is no guarantee that the Traub method converges if the operator is not five times Fréchet-differentiable or more. However, the Traub method can converge, since these assumptions are only sufficient. The novelty of our new technique is the fact that only the Fréchet-derivative on the method is assumed to exist to prove convergence. Moreover, the new results does not depend on the Traub method. Consequently, the same technique can be applied on other methods. The dynamics of this method are also studied. Examples further explain the theoretical results. Full article

The acoustic activity in beam-shaped specimens made of cement is studied, assuming that the beams are loaded in three-point bending under a step-wise loading scheme. Attention is focused to the attenuation of the acoustic activity during the constant-load stage of each specific loading step. The experimental data are analyzed in terms of the interevent time intervals between any two successive acoustic hits (using the F-function concept) and, further, in terms of the power of the acoustic hits (in terms of the recently introduced P-function). It is indicated that while the mechanical load is kept constant, the acoustic activity attenuates steadily, and during the early steps of this attenuation phase, the temporal evolution of both the F- and P-functions is excellently described by an exponential law. Moreover, it is proven that for both the F- and P-functions, the relaxation exponents decrease monotonically with increasing load. This decrease becomes quite abrupt for loads exceeding about 80% of the fracture load, providing an interesting and promising pre-failure indicator, i.e., a warning signal that the specimen is entering into the stage of impending macroscopic fracture. The specific conclusions are in very satisfactory agreement, both qualitatively and quantitatively, with similar ones drawn by considering the temporal evolution of the respective b-value. Full article

In the first part of this paper, we considered several theoretical models, a static and four dynamic models without rebounds, of the throw of a fair coin landing on its edge, to demonstrate that the probability of heads or tails is less than 50%, depending on the initial toss conditions, the coin geometry and conditions of the coin and landing surfaces. For the dynamic model with rebounds that is the subject of this second part of the paper, it is found that the probability that a 50 Euro cent coin thrown from a normal height with common initial velocity conditions and appropriate surface conditions will end up on its edge is in the order of one against several thousand. Full article

Electromagnetic radiation (EMR) technology makes it possible to evaluate changes in the stress-strain state (SSS) in the “tunnel lining-enclosing rock mass” system at a high level of interference, and to create schemes of long-term EMR control in tunnels (geotechnical monitoring systems). The issues of the variations in EMR signals are extremely important for monitoring systems: based on anomalous deviations from the normal regime one can draw conclusions about changes in the SSS, leading to geodynamic phenomena (e.g., rock bursts). This article presents data obtained during laboratory studies on samples and field studies in transport tunnels. Also, some results of long-term geotechnical monitoring by a set of methods is presented: EMR and tensometry of the tunnel lining, both methods are in the automatic mode. The ability of an EMR control system to respond to earthquakes affecting tunnel structures is shown. An analysis of long-term EMR studies was conducted, which showed the periodic oscillation of the “tunnel lining-enclosing rock mass” system. In a stable compressed state, minima of EMR pulses are recorded; when the rock mass and lining material are stretched, charges are separated on the edges of micro-defects and EMR increases; complete separation of the edges of micro-defects leads to the termination of intense EMR. The same occurs in the opposite direction during the compression of micro-defects and micro-fractures in the rock mass and concrete lining. The periods of compression and expansion are closely related to temperature fluctuations. The results differ in detail and, therefore, in to be more confident, additional studies are needed in various host rock massifs and types of tunnel lining. Full article

Considering that a fair coin has two sides and a cylindrical edge, the probability that it would fall on its edge is calculated, yielding the probability of heads or tails of less than 50%. In this first part, the theoretical models for a static case and for five dynamic cases, without rebounds, show that there is a small probability that the coin does not fall on its head or tail, depending on the initial toss conditions, the coin geometry and conditions of the coin and landing surfaces. It is found that the probability that a 50 Eurocent coin thrown from a normal height with common initial velocity conditions and appropriate surface conditions will end up on its edge is in the order of one against several thousand. In the second part of the paper, the dynamic model with rebounds is investigated. Full article

For the excitation of the n = 2 states of hydrogen atoms due to electron impact, we compared the experimental and theoretical ratios of the cross-sections σ_2s/σ_2p. We found this theoretical ratio to be systematically higher than the corresponding experimental ratio by about 20%—far beyond the experimental error margins. We suggest that this discrepancy can be explained by the presence of the Second Flavor of Hydrogen Atoms (SFHA) in the experimental hydrogen gas. The explanation is based on the fact that, in the experiments, the cross-section σ_2s was determined by using the quenching technique—by applying an electric field that mixed the 2s and 2p states, followed by the emission of the Lyman-alpha line from the 2p state. However, the SFHA only had the s-states, so the quenching technique would not count the excitation of the SFHA in the 2s state and, thus, lead to the underestimation of the cross-section σ_2s. We estimates the share of the SFHA in the experimental hydrogen gas required for eliminating the above discrepancy and found this share to be about the same as the share of the usual hydrogen atoms. Thus, our results constitute the third proof from atomic experiments that the SFHA does exist, the first proof being related to the experimental distribution of the linear momentum in the ground state of hydrogen atoms, and the second proof being related to the experimental cross-section of charge exchange between hydrogen atoms and low-energy protons. Full article

Near-infrared spectroscopy (NIRS) allows for the rapid estimation of a wide range of wood properties. Typically, NIRS studies on wood have utilized benchtop spectrometers, but efforts to utilize NIR hyperspectral imaging to examine wood and wood products have increased. Compared to benchtop NIR systems, hyperspectral imaging has several advantages (speed, visualization of spatial variability), but the data typically have a lower signal-to-noise ratio as well as fewer wavelengths saved; thus, hyperspectral imaging systems have a larger spectral sampling interval (SSI). Furthermore, the SSI and wavelength range varies considerably among different HSI cameras. NIR-HSI systems based on indium gallium arsenide (InGaAs) detectors have a wavelength range typically from 900 to 1700 nm, while short-wave infrared hyperspectral imaging (SWIR-HSI) systems based on mercury cadmium telluride (MCT) detectors have the ‘full’ NIR wavelength range from 1000 to 2500 nm. These factors may influence the performance of wood property calibrations. We compared one NIR-HSI (900–1700 nm) and three SWIR-HSI (1000–2500 nm) commercially available cameras with an NIRS benchtop spectrometer (1100–2500 nm). The performance of specific gravity (SG) and stiffness (MOE) calibration models was compared with one-hundred Douglas-fir (Pseudotsuga menziesii) samples. The limited wavelength range of an NIR-HSI camera provided the best models for MOE, whereas the NIR-HSI and two SWIR-HSI cameras provided similar SG results. SWIR-HSI models heavily favored wavelengths greater than 1900 nm. Full article