Arbitrary Mathematical Functions Research Articles

Theoretical Analysis of the RX J0209.6−7427 X-Ray Spectrum during a Giant Outburst

We model the spectral formation occurring in the binary X-ray pulsar (XRP) RX J0209.6−7427 during the 2019 super-Eddington outburst. Using a theoretical model previously developed by the authors, we are able to produce spectra that closely resemble the phase-averaged X-ray spectra observed using the Nuclear Spectroscopic Telescope Array and Insight-HXMT during low- and high-luminosity states of the outburst, respectively. The theoretical model simulates the accretion of fully ionized gas in a dipole magnetic field and includes a complete description of the radiation hydrodynamics, matter distribution, and spectral formation. Type II X-ray outbursts provide an opportunity to study accretion over a large range of luminosities for the same neutron star. The analysis performed here represents the first time both the outburst low and high states of an accretion-powered XRP are modeled using a physics-based model rather than standard phenomenological fitting with arbitrary mathematical functions. We find that the outer polar cap radius remains constant and the column is more fully filled with increasing luminosity, Comptonized bremsstrahlung dominates the formation of the phase-averaged X-ray spectrum, and a negative correlation exists between cyclotron centroid energy and luminosity, as expected. The super-Eddington nature of the outburst is rendered possible owing to the low scattering cross section for photons propagating parallel to the magnetic field. We also find that emission through the column top dominates in both the low and high states, implying that the pulse profiles should have a roughly sinusoidal shape, which agrees with observed properties of ultraluminous XRPs.

Utilizing optimal physics-informed neural networks for dynamical analysis of nanocomposite one-variable edge plates

This study presents an innovative Physics-Informed Neural Network (PINN) approach designed to predict the dynamic responses of the One-Variable Edge Plate (OVEP), a unique plate structure characterized by an edge defined by arbitrary mathematical functions. The OVEP is constructed from a nanocomposite material reinforced with graphene nanoplatelets. Utilizing an optimized PINN pipeline, this research successfully predicts the vibration characteristics of the OVEP, encompassing linear, nonlinear vibrations, and beating phenomena. The study also demonstrates that the optimal PINN outperforms conventional neural network in terms of stability, accuracy (achieving above 99% accuracy), and efficiency in predicting long-duration vibrations. Additionally, the computational time required for generating testing results is notably diminished compared to traditional partial differential equation (PDE) solvers (reduced by about 3 to 12 times). To demonstrate model robustness, synthetic noise is intentionally introduced into the training data. The results not only enhance our understanding of the complex dynamics of the OVEP but also highlight the effectiveness of the proposed PINN framework in capturing and forecasting the dynamic behaviors of advanced plate structures. This research presents a promising potential for addressing dynamic problems in the fields of aerospace, civil, and mechanical engineering.

Free vibration analysis of functionally graded magneto-electro-elastic plates with in-plane material heterogeneity

A three dimensional elasticity analysis for the transverse free vibration characteristics of functionally graded magneto-electro-elastic plates based on the scaled boundary finite element method (SBFEM) incorporated with the precise integration algorithm (PIA) is presented. The material properties of magneto-electro-elastic plates are changing along the in-plane direction with arbitrary mathematical functions. In the proposed methodology, the strategies of only discretizing the in-plane surface and utilizing the two-dimensional spectral elements to construct diagonal coefficient matrices are adopted, which contributes to decreasing the calculation effort. The derivation process begins with the three dimensional governing equations of magneto-electro-elastic materials. Neither the plate mechanical kinematics nor invoking assumptions on the spatial distributions of electric and magnetic quantities are adopted. Built upon the introduced scaled boundary coordinates, the principle of virtual work and the technique of dual vectors, a first order ordinary differential SBFEM matrix equation for the in-plane functionally graded magneto-electro-elastic plates is obtained. Its general solution is analytically denoted as the matrix exponent. To improve the computation accuracy of the matrix exponent, the PIA is utilized to form the stiffness matrix. By virtue of the kinetic energy technique, it is convenient to construct the mass matrix of the in-plane functionally graded magneto-electro-elastic plates based on the SBFEM for the first time. Finally, comparisons of flexural frequency parameters with those from exact solutions and other numerical methods are provided. The accuracy, effectiveness, and versatility of the employed technique are validated. Moreover, additional numerical exercises are conducted to exhibit the influences of boundary conditions, material gradation functions, and aspect ratios on the free vibration behaviors of in-plane functionally graded magneto-electro-elastic rectangular, circular, and perforated plates.

Development of a Quiz – Implementation of a (Self-) Assessment Tool and its Integration in Moodle

Technology Enhanced Learning has become more popular in recent times and many organizations and universities use it as a key instrument in various teaching and training scenarios. At the University of Technology of Graz, some courses require randomized quizzes where question variables can be assigned by arbitrary mathematical functions and this feature is missing in the current solutions. This article describes the development of a quiz application that can be integrated into Moodle by utilization of the Learning Tools Interoperability protocol (LTI). The PHP application is built to support programmable questions that can contain JavaScript and HTML code. Teachers are able to build interactive, randomized quizzes, in which the random variables can be assigned with complex mathematical functions. Furthermore, the application provides a programmable grading mechanism. With this mechanism, it is possible for students to self-assess their performance, as well as for teachers to formally assess their students' learning success automatically and send the results back to Moodle (or other LTI compatible consumer applications).

Общее решение второй основной задачи для упругого кольца при нулевых главных векторах сил, приложенных к его границам

For the first time formulated and solved the problem for the ring with the set on the border of the radial displacement, in the form of Fourier series. The technique differed from used in the classic monograph N.I. Muskhelishvili has been developed and applied to the solution of the first boundary value problem for elastic ring. The solution demonstrates the flexibility of the method of analytic functions, allowing analytically solve the boundary problems which have almost arbitrary complexity. In addition, the convenience of application of is computer algebra system Mathematica was demonstrated. It allows to build a stress distribution with help of complex analytic functions, without prior separation of the real and imaginary parts. A comparison of the stress state of the ring at the given variable radial displacement of borders based on the results of calculations in Mathematica and ANSYS 10 ED was fulfilled. It is found that the results coincide with a high accuracy both qualitatively and quantitatively. For the first time the possibility of program application of the piecewise constant approximation of the variables static loads applied to the body was demonstrated. It was shown that arbitrary mathematical functions can be used for boundary conditions.

Ordered, random, monotonic and non-monotonic digital nanodot gradients.

Cell navigation is directed by inhomogeneous distributions of extracellular cues. It is well known that noise plays a key role in biology and is present in naturally occurring gradients at the micro- and nanoscale, yet it has not been studied with gradients in vitro. Here, we introduce novel algorithms to produce ordered and random gradients of discrete nanodots – called digital nanodot gradients (DNGs) – according to monotonic and non-monotonic density functions. The algorithms generate continuous DNGs, with dot spacing changing in two dimensions along the gradient direction according to arbitrary mathematical functions, with densities ranging from 0.02% to 44.44%. The random gradient algorithm compensates for random nanodot overlap, and the randomness and spatial homogeneity of the DNGs were confirmed with Ripley's K function. An array of 100 DNGs, each 400×400 µm2, comprising a total of 57 million 200×200 nm2 dots was designed and patterned into silicon using electron-beam lithography, then patterned as fluorescently labeled IgGs on glass using lift-off nanocontact printing. DNGs will facilitate the study of the effects of noise and randomness at the micro- and nanoscales on cell migration and growth.

Study on Scroll Profile Design Using Involute of Elliptic for Scroll Compressor

The scroll compressor has the conjugate motion of meshing by the fixed scroll and the planet scroll. Several kinds of geometrical curves can be adopted to form the scroll compressor profile such as involute of circle with variable and constant radius, algebraic spiral, and so on, to improve the performance of scroll compressor. This study proposes a new scroll profile design using elliptic involute and two-envelope method to create a new conjugate meshing scroll of variable cross section. The curve of the scroll section on both sides can be specified as an arbitrary mathematical function. First, the base scroll swept out by the central base circle. Next, the base circle translation path envelope of the planet scroll is moved by the basic involute curve. The base circle translation path envelope is the first envelope. The second envelope generated by the planet scroll conjugate profile is the fixed scroll. By using two-envelope method, the variety of conjugate scroll can be produce with the cross section and symmetrical variables for parametric design.

Monthly rainfall–runoff modelling using artificial neural networks

Rainfall–runoff models usually present good results, but parameter calibration sometimes is tedious and subjective, and in many cases it depends on additional data surveys in the field. An alternative to the conceptual models is provided by empirical models, which relate input and output by means of an arbitrary mathematical function that bears no direct relationship to the physical characteristics of the rainfall–runoff process. This category includes the artificial neural networks (ANNs), whose implementation is the main focus of this paper. This study evaluated the capacity of ANNs to model with accuracy the monthly rainfall–runoff process. The case study was performed in the Jangada River basin, Paraná, Brazil. The results of the three ANNs that produced the best results were compared to those of a conceptual model at monthly time scale, IPHMEN. The ANNs presented the best results with highest correlation coefficients and Nash-Sutcliffe statistics and the smallest difference of volume. Citation Machado, F., Mine, M., Kaviski, E. & Fill, H. (2011) Monthly rainfall–runoff modelling using artificial neural networks. Hydrol. Sci. J. 56(3), 349–361.

Optimisation of production and storage stability of the starter bacteria Streptococcus thermophilus and Lactobacillus plantarum

AbstractThe growth of Lactobacillus plantarum and Streptococcus thermophilus was modelled by fitting isothermal calorimetric traces to arbitrary mathematical functions. There was a correlation between certain features on the thermal trace and parameters derived from a Baranyi fit to viable count data. In the presence of 75 g L−1 lactose, S. thermophilus showed a slower rate of growth than in the presence of 5 g L−1 lactose when measured by both viable count and heat flow. A simple equation described the variation in glass transition temperature with water activity for all preparations. A master curve fit for all mortality data for S. thermophilus showed the importance of a transition temperature for the survival of freeze‐dried preparations of bacteria. The temperature at which survival began to be impaired was 15–20 °C below the glass transition temperature determined by differential scanning calorimetry. Copyright © 2007 Society of Chemical Industry

Magnetic charge and the charge quantization condition

Two viewpoints concerning magnetic charge are distinguished: that of Dirac, which is unsymmetrical, and the symmetrical one, which embodies invariance under charge rotation. It is pointed out that the latter is not in conflict with the empirical asymmetry between electric and magnetic charge. The discussion is based on an action principle that uses field strengths and the vector potential $A$ as independent variables; a second vector potential $B$ is defined nonlocally in terms of the field strengths. This nonlocality is described by an arbitrary vector function ${f}^{\ensuremath{\nu}}(y)$, subject only to the restriction ${\ensuremath{\partial}}_{\ensuremath{\nu}}{f}^{\ensuremath{\nu}}(y)=\ensuremath{\delta}(y)$ and the additional requirement of oddness, in the symmetrical formulation. The charge quantization conditions for a pair of idealized charges, $a$ and $b$, are inferred by examining the dependence of the action $W$ on the choice of the arbitrary mathematical function $f$, and requiring the uniqueness of $\mathrm{exp}[\mathrm{iW}]$. For the unsymmetrical viewpoint the half-integer condition of Dirac is obtained, $\frac{{e}_{a}{g}_{b}}{4\ensuremath{\pi}}=\frac{1}{2}n$, while the symmetrical formulation requires the integer condition $\frac{({e}_{a}{g}_{b}\ensuremath{-}{e}_{b}{g}_{a})}{4\ensuremath{\pi}}=n$. The Dirac injunction, "a string must never pass through a charged particle," is criticized as unnecessarily restrictive, owing to its origin in a classical action context. As simplified by a restriction to small momentum transfers, permitting the neglect of form-factor and vacuum-polarization effects, the dynamics of a realistic system of two spin- \textonehalf{} dyons is shown to involve the same interaction structure used in the idealized discussion.

Platelet production and platelet survival in polycythaemia vera with special reference to the spleen size.

Studies of platelet survival and of platelet production rate were performed in 20 patients with active polycythaemia vera (PV). 8 of them were classified as non‐splenomegalic and 12 had splenomegaly. The results were compared with a control material consisting of 16 haematologically normal subjects of similar age. Platelet recovery was significantly lower in splenomegalic as compared with non‐splenomegalic PV patients. Platelet recovery in the latter group was, however, also significantly lower than in control subjects.Platelet mean life span and platelet half‐life time were obtained by fitting the experimental data to an arbitrary mathematical function by using a digital computer. The results indicate that platelet survival is shorter in PV patients with splenomegaly as compared with PV patients wihout splenomegaly.Irrespective of the peripheral platelet count the platelet production rate in all PV patients exceeded the upper range for the control subjects. The mean platelet production rate was approximately 9 times normal in the splenomegalic group, and 5 times normal in the group without splenomegaly. There was a large individual variation for the platelet production rate, and the dispersion of values was more pronounced in the splenomegalic group.