Search Results (82)

For the correct execution of the preliminary design of a transport ship, among other things, approximate formulas enabling the calculation of the weight of the unladen ship and the location of the center of gravity are necessary. The aim of the conducted research was to develop approximate formulas for calculating the weight and center of gravity of an empty container ship with a size ranging from 270 TEU to 3100 TEU, depending on the basic design parameters: ship speed V, deadweight DWT, and number of TEU containers. Since the weight of an unladen container ship has a very large impact on the ship’s operating parameters, an additional aim was to obtain regression formulas with greater accuracy than similar formulas published in the literature. Simple and multiple regression methods were used to develop regression formulas. The obtained results were verified on the basis of experimentally measured parameters obtained from built ships. The regression formulas presented in this article are characterized by high accuracy, greater than that of similar formulas published in the literature, and were developed for container ships currently under construction. A novelty of this study is the development of regression formulas for weight classes, which make up the total weight of an unladen ship. Full article

One of the main motivations of this paper is to construct generating functions for generalization of the Touchard polynomials (or generalization exponential functions) and certain special numbers. Many novel formulas and relations for these polynomials are found by using the Euler derivative operator and functional equations of these functions. Some novel relations among these polynomials, beta polynomials, Bernstein polynomials, related to Binomial distribution from discrete probability distribution classes, are given. Full article

By applying the “coefficient extraction method” to the symmetric transformation of hypergeometric series due to Chu and Zhang (2014), an overview is presented systematically for a large class of infinite series of convergence rate “$-1/4$” concerning harmonic numbers. Numerous closed formulae in terms of mathematical constants (such as $\pi$, $ln2$ and the Riemann zeta values) are established. They may serve as a reference source for readers in their further investigations. Full article

By means of the coefficient extraction method, we examine a transformation of a classical hypergeometric series. Three classes of infinite series (of convergence rate “$1/4$”) with harmonic numbers in numerators and cubic central binomial coefficients in denominators are expressed in terms of odd Euler sums. Several new closed formulae are established. Full article

Heusler compounds and alloys constitute a burgeoning class of materials with exceptional properties, holding immense promise for advanced technologies. Electronic band structure calculations are instrumental in driving research in this field. Nowotny–Juza compounds are similar to Semi-Heusler compounds containing one instead of two transition metal atoms in their chemical formula. Recently, they have been widely referred to as “$p^0$-d or $d^0$-d Semi-Heusler compounds”. Building upon our previous studies on $p^0$-d or $d^0$-d Semi-Heusler compounds featuring Li or K, we now explore a new class of $d^0$-d compounds incorporating alkaline earth metals and more specifically Mg which is well-known to occupy all possible sites in Heusler compounds. These compounds, with the general formula MgZ(Ga, Ge, or As), where Z is a transition metal, are investigated for their structural, electronic, and magnetic properties, specifically within the context of the three possible $C{1}_{\mathrm{b}}$ structures including also the effect of tetragonalization which is shown not to affect the equilibrium cubic type. Our findings demonstrate that a significant number of these compounds exhibit magnetic behavior, with several displaying half-metallicity, making them highly attractive for spintronic applications. This research provides a crucial foundation for future experimental investigations into these promising materials. Full article

The notion of “randomness” in the mathematical theory of composites has typically been used abstractly within measure theory, making practical applications difficult. In contrast, engineering sciences often discuss randomness too loosely, lacking a theoretical foundation. This paper aims to bridge the gap between theory and applications, focusing on the effective properties of two-dimensional conducting composites with non-overlapping circular inclusions. It is shown that there is no universal minimum number of inclusions per cell in simulations of random composites. Even minor changes to Random Sequential Addition algorithms lead to different formulas for the effective constants. Application of the analytical representative volume element (aRVE) theory methodologically and practically addresses the diversity issue of random composites based on homogenization principles. In particular, it examines how the spatial arrangement of inclusions impacts the overall composite properties. The proposed method can be applied to a large number of inclusions and to symbolically given geometric and physical parameters relevant to optimal design problems. The method leverages structural sums and enables a more refined classification of different classes of composites, which was unattainable using previous approaches. The obtained results demonstrate a diversity of apparently similar composites. This paper outlines the investigation strategy and provides a detailed description of each step. Full article

Introduction: Cephalometric analysis is an essential tool used in orthodontic diagnosis and treatment planning. Aim: The aim of this study was to compare the measurement reliabilities (repeatability and reproducibility) of the Tau and Yen angles and compare them to the results obtained for the ANB angle. Methods: Repeatability and reliability assessments for the seven points (N, A, B, S, W, M, G) used in the analysis of ANB, Yen and Tau angles were performed twice with an interval of 7 days by 22 orthodontists. The measurement results for ANB, Yen and Tau angles were assessed using the Bland–Altman formula, Dahlberg formula, intraclass correlation coefficients (ICCs), R² coefficients and R&R. In order to assess the number of individual skeletal classes of sagittal discrepancy, the Pearson chi-squared test was used. With common parameters of df = 4, p < 0001, for the ANB angle, the result was χ² = 9104; for the Tau angle, χ² = 4556; and for the Yen angle, χ² = 4207. In order to determine the inter-rater reliability based on two-way ANOVA analysis without repetitions, the ICC (2,2) was used. The ICC (2,2) index at the 95% confidence level was 0.998 for the ANB angle, 0.997 for Tau and 0.998 for Yen. High values of the ICC index close to 1 indicate the agreement of the measurements and their high reliability. Results: The orthodontists in the study measured sagittal discrepancy significantly more accurately using the ANB angle compared to the Yen and Tau angles. Using a Bland–Altman plot, the bias and range of agreement within which 95% of the differences between measurements were accounted for were determined. For the ANB angle, the mean difference between measurements was 0.07 with a confidence interval of −1.55 to +1.69; for the Tau angle, the mean difference between measurements was 0.19 with a confidence interval of −2.92 to 3.30; and for the Yen angle, the mean difference was 0.09 with a confidence interval of −2.71 to +2.89. Using regression analysis, the measurements were assessed using the R2 index, which for the ANB angle was 0.952 (p < 0.001); for the Tau angle, R2 = 0.928 (p < 0.001), and for the Yen angle, R2 = 0.942 (p < 0.001). Conclusions: The obtained results of the assessment of the ANB, Tau and Yen angles confirm the thesis of the highest reliability, including repeatability and reproducibility, in the assessment of sagittal discrepancy in orthodontic diagnostics using the ANB angle, previously considered the gold standard. One of the basic factors attributed to the poorer repeatability and reproducibility of Tau and Yen measurements is human error related to the precision of determining new anthropometric points. Further studies to assess the usefulness of using the new Tau and Yen angle measurements in orthodontic diagnostics for sagittal discrepancy should be correlated with other measurements used so far, depending on the type of defects in the vertical dimension. It is necessary to consider enlarging the study group and performing longitudinal studies. Full article

A trivially zero minor of a matrix is a minor having all its terms in the Leibniz formula equal to zero. A matrix is superregular if all of its minors that are not trivially zero are nonzero. In the area of Coding Theory, superregular matrices over finite fields are connected with codes with optimum error correcting capabilities. There are two types of superregular matrices that yield two different types of codes. One has in all of its entries a nonzero element, and these are called full superregular matrices. The second interesting class of superregular matrices is formed by lower triangular Toeplitz matrices. In contrast to full superregular matrices, all general constructions of these matrices require very large field sizes. In this work, we investigate the construction of lower triangular Toeplitz superregular matrices over small finite prime fields. Instead of computing all possible minors, we study the structure of finite fields in order to reduce the possible nonzero minors. This allows us to restrict the huge number of possibilities that one needs to check and come up with novel constructions of superregular matrices over relatively small fields. Finally, we present concrete examples of lower triangular Toeplitz superregular matrices of sizes up to 10. Full article

This study introduces an innovative approach to Mersenne-type numbers. This paper introduces a new class of numbers, which we call Gersenne numbers. The aim of this paper is to define the Gersenne sequence and to investigate some of their properties, such as the recurrence relation, the summation formula, and the generating function. Moreover, the classical identities are derived, such as the Tagiuri–Vajda, Catalan, Cassini, and d’Ocagne identities for Gersenne numbers. Full article

We consider a wide class of summatory functions $F\left\{f;N,p^m\right\}={\sum }_{k\le N}f\left(p^{m}k\right)$, $m\in {\mathbb{Z}}_{+}\cup \left\{0\right\}$ associated with the multiplicative arithmetic functions f of a scaled variable $k\in {\mathbb{Z}}_{+}$, where p is a prime number. Assuming an asymptotic behavior of the summatory function, $F\{f;N,1\}\stackrel{N\to \infty }{=}{G}_1\left(N\right)\left[1+\mathcal{O}\left(G_2\left(N\right)\right)\right]$, where $G_1\left(N\right)=N^{a_1}{\left(\log N\right)}^{b_1}$, $G_2\left(N\right)=N^{-a_2}{\left(\log N\right)}^{-b_2}$ and $a_1,a_2\ge 0$, $-\infty <b_1,b_2<\infty$, we calculate the renormalization function $R\left(f;N,p^m\right)$, defined as a ratio $F\left\{f;N,p^m\right\}/F\{f;N,1\}$, and find its asymptotics $R_{\infty }\left(f;p^m\right)$ when $N\to \infty$. We prove that a renormalization function is multiplicative, i.e., $R_{\infty }\left(f;{\prod }_{i=1}^n{p}_i^{m_i}\right)={\prod }_{i=1}^n{R}_{\infty }\left(f;p_i^{m_i}\right)$ with n distinct primes $p_i$. We extend these results to the other summatory functions ${\sum }_{k\le N}f\left(p^m{k}^l\right)$, $m,l,k\in {\mathbb{Z}}_{+}$ and ${\sum }_{k\le N}{\prod }_{i=1}^n{f}_i\left(k{p}^{m_i}\right)$, $f_i\ne f_j$, $m_i\ne m_j$. We apply the derived formulas to a large number of basic summatory functions including the Euler $\varphi \left(k\right)$ and Dedekind $\psi \left(k\right)$ totient functions, divisor ${\sigma }_n\left(k\right)$ and prime divisor $\beta \left(k\right)$ functions, the Ramanujan sum $C_q\left(n\right)$ and Ramanujan $\tau$ Dirichlet series, and others. Full article

In this paper, we investigate time-dependent Kantowski–Sachs spherically symmetric teleparallel $F\left(T\right)$ gravity with a scalar field source. We begin by setting the exact field equations to be solved and solve conservation laws for possible scalar field potential, $V\left(\varphi \right)$, solutions. Then, we find new non-trivial teleparallel $F\left(T\right)$ solutions by using power-law and exponential ansatz for each potential case arising from conservation laws, such as linear, quadratic, or logarithmic, to name a few. We find a general formula allowing us to compute all possible new teleparallel $F\left(T\right)$ solutions applicable for any scalar field potential and ansatz. Then, we apply this formula and find a large number of exact and approximate new teleparallel $F\left(T\right)$ solutions for several types of cases. Some new $F\left(T\right)$ solution classes may be relevant for future cosmological applications, especially concerning dark matter, dark energy quintessence, phantom energy leading to the Big Rip event, and quintom models of physical processes. Full article

A generalized scientific review with elements of additions and clarifications has been carried out on the methods of theoretical research on the electrophysical properties of crystals with ionic–molecular chemical bonds (CIMBs). The main theoretical tools adopted are the methods of quasi-classical kinetic theory as applied to ionic subsystems relaxing in layered dielectrics (natural silicates, crystal hydrates, various types of ceramics, and perovskites) in an electric field. A universal (applicable for any CIMBs class crystals) nonlinear quasi-classical kinetic equation of theoretical and practical importance has been constructed. This equation describes, in complex with the Poisson equation, the mechanism of ion-relaxation polarization and conductivity in a wide range of polarizing field parameters (0.1–1000 MV/m) and temperatures (1–1550 K). The physical model is based on a system of non-interacting ions (due to the low concentration in the crystal) moving in a one-dimensional, spatially periodic crystalline potential field, perturbed by an external electric field. The energy spectrum of ions is assumed to be continuous. Elements of quantum mechanical theory in a quasi-classical model are used to mathematically describe the influence of tunnel transitions of hydrogen ions (protons) during the interaction of proton and anion subsystems in hydrogen-bonded crystals (HBC) on the polarization of the dielectric in the region of nitrogen (50–100 K) and helium (1–10 K) temperatures. The mathematical model is based on the solution of a system of nonlinear Fokker-Planck and Poisson equations, solved by perturbation theory methods (via expanding solutions into infinite power series in a small dimensionless parameter). Theoretical frequency and temperature spectra of the dielectric loss tangent were constructed and analyzed, the molecular parameters of relaxers were calculated, and the physical nature of the maxima of the experimental temperature spectra of dielectric losses for a number of HBC crystals was discovered. The low-temperature maximum, which is caused by the quantum tunneling of protons and is absent in the experimental spectra, was theoretically calculated and investigated. The most effective areas of scientific and technical application of the theoretical results obtained were identified. The application of the equations and recurrent formulas of the constructed model to the study of nonlinear optical effects in elements of laser technologies and nonlinear radio wave effects in elements of microwave signal control systems is of the greatest interest. Full article

Magnesium aluminates (MgO)_x(Al₂O₃)_1−x belong to a class of refractory materials with important applications in glass and glass–ceramic technologies. Typically, these materials are fabricated from high-temperature molten phases. However, due to the difficulties in making measurements at very high temperatures, information on liquid-state structure and properties is limited. In this work, we employed the method of aerodynamic levitation with CO₂ laser heating at large scale facilities to study the structure of liquid magnesium aluminates in the system (MgO)_x(Al₂O₃)_1−x, with x = 0.33, 0.5, and 0.75, using X-ray and neutron diffraction. We determined the structure factors and corresponding pair distribution functions, providing detailed information on the short-range structural order in the liquid state. The local structures were similar across the range of compositions studied, with average coordination numbers of ${\overline{n}}_{\mathrm{A}\mathrm{l}}^{\mathrm{O}}\sim 4.5$ and ${\overline{n}}_{\mathrm{M}\mathrm{g}}^{\mathrm{O}}\sim 5.1$ and interatomic distances of $r_{\mathrm{AlO}}=1.76-1.78 \mathrm{\AA }$ and $r_{\mathrm{MgO}}=1.93-1.95 \mathrm{\AA }$. The results are in good agreement with previous molecular dynamics simulations. For the spinel endmember MgAl₂O₄ (x = 0.5), the average Mg-O and Al-O coordination numbers gave rise to conflicting values for the inversion coefficient χ, indicating that the structural formula used to describe the solid-state order-disorder transition is not applicable in the liquid state. Full article

This research produced five hybrid-bonded fiber-reinforced plastic (HB-FRP)-reinforced beam specimens, and different fatigue load amplitudes were used as parameter variables for the fatigue performance tests of the FRP–concrete interface. The results show that FRP sliding with the number of cycles can be roughly divided into fatigue initiation stage, fatigue development stage, and fatigue damage stage, and finally, because the load is too large and friction, pin, and bonding cannot provide a greater inhibition effect, FRP adhesive length is not enough to withstand the stripping load and failure. FRP slip increases with increasing fatigue load amplitude for the same number of fatigue cycles; for the specimens with fatigue damage, the interfacial stiffness of FRP–concrete decreases with the increase in the number of cyclic, and the rate of stiffness damage at the FRP–concrete interface accelerates with increasing fatigue load amplitude. For fatigue load magnitudes higher than 0.6, the slopes of fatigue bond–slip curves decrease with the increase in the number of cycles. When the fatigue load magnitude is lower than 0.4, the slope of the fatigue bond–slip curve increases with the number of cycles and is close to the slope of the monotonically loaded curve. Due to the difference in load class, the bond strength of HB-F-1 continues to decrease with the increase in fatigue times, decreasing by 26% after 100 fatigue cycles and decreasing to 7.33 MPa after 5000 fatigue cycles. The bond strength of the sample HB-F-2 first increased and then decreased with the increase in fatigue times. After 10,000 fatigue cycles, the bond strength decreased by 8%, and at 11,133,300 fatigue breaks, the bond strength of the sample HB-F-3 continued to increase with the increase in fatigue times. At 2 million fatigue load cycles, the bond strength increased to 7 MPa, far from reaching the peak strength. The empirical formulas for the fatigue life curve of HB-FRP-reinforced specimens under single steel fasteners are proposed. Full article

Two typical fixed-length random number generation problems in information theory are considered for general sources. One is the source resolvability problem and the other is the intrinsic randomness problem. In each of these problems, the optimum achievable rate with respect to the given approximation measure is one of our main concerns and has been characterized using two different information quantities: the information spectrum and the smooth Rényi entropy. Recently, optimum achievable rates with respect to f-divergences have been characterized using the information spectrum quantity. The f-divergence is a general non-negative measure between two probability distributions on the basis of a convex function f. The class of f-divergences includes several important measures such as the variational distance, the KL divergence, the Hellinger distance and so on. Hence, it is meaningful to consider the random number generation problems with respect to f-divergences. However, optimum achievable rates with respect to f-divergences using the smooth Rényi entropy have not been clarified yet in both problems. In this paper, we try to analyze the optimum achievable rates using the smooth Rényi entropy and to extend the class of f-divergence. To do so, we first derive general formulas of the first-order optimum achievable rates with respect to f-divergences in both problems under the same conditions as imposed by previous studies. Next, we relax the conditions on f-divergence and generalize the obtained general formulas. Then, we particularize our general formulas to several specified functions f. As a result, we reveal that it is easy to derive optimum achievable rates for several important measures from our general formulas. Furthermore, a kind of duality between the resolvability and the intrinsic randomness is revealed in terms of the smooth Rényi entropy. Second-order optimum achievable rates and optimistic achievable rates are also investigated. Full article