Search Results (680)

We propose a Ramsey approach to the dimensional analysis of physical systems, which complements the seminal Buckingham theorem. Dimensionless constants describing a given physical system are represented as vertices of a graph, referred to as a dimensions graph. Two vertices are connected by an aqua-colored edge if they share at least one common dimensional physical quantity and by a brown edge if they do not. In this way, a bi-colored complete Ramsey graph is obtained. The relations introduced between the vertices of the dimensions graph are non-transitive. According to the Ramsey theorem, a monochromatic triangle must necessarily appear in a dimensions graph constructed from six vertices, regardless of the order of the vertices. Mantel–Turán analysis is applied to study these graphs. The proposed Ramsey approach is extended to graphs constructed from fundamental physical constants. A physical interpretation of the Ramsey analysis of dimensions graphs is suggested. A generalization of the proposed Ramsey scheme to multi-colored Ramsey graphs is also discussed, along with an extension to infinite sets of dimensionless constants. The introduced dimensions graphs are invariant under rotations of reference frames, but they are sensitive to Galilean and Lorentz transformations. The coloring of the dimensions graph is independent of the chosen system of units. The number of vertices in a dimensions graph is relativistically invariant and independent of the system of units. Full article

Shunting locomotives exhibit a wide and unpredictable range of power profiles. This unpredictability makes it impossible to rely on offline optimizations or predictive methods combined with online optimization. To maintain optimal performance across this broad range of operating conditions, the online control strategy must be robust. This article proposes a robust method to determine the optimal parameter combinations for an online energy management strategy of a hybrid fuel cell battery shunting locomotive, ensuring optimality across all scenario conditions. The first step involves extracting a statistically representative subspace for simulation, both in terms of parameter combinations and scenario conditions. A response surface model (numerical twin) is then constructed to extrapolate results across the entire space based on this simulated subspace. Using this model, the optimal solution is identified through metaheuristic algorithms (minimization search). Finally, the proposed solution is validated against a set of expert-defined scenarios. The result of the methodology ensures robust optimization across an infinite number of scenarios by minimizing the impact on both the fuel cell and the battery, without increasing mission costs. Full article

This paper considers a heterogeneous queuing system with an unlimited number of servers, where the parameters are determined by a random environment. A distinctive feature is that the parameters of the exponential distribution of the request processing time do not change their values until the end of service. Thus, the devices in the system under consideration are heterogeneous. For the study, a method of asymptotic analysis is proposed under the condition of extremely rare changes in the states of the random environment. We consider the following problem. Cloud node accepts requests of one type that have a similar intensity of arrival and duration of processing. Sometimes an input scheduler switches to accept requests of another type with other intensity and duration of processing. We model the system as an infinite-server queue in a random environment, which influences the arrival intensity and service time of new requests. The random environment is modeled by a Markov chain with a finite number of states. Arrivals are modeled as a Poisson process with intensity dependent on the state of the random environment. Service times are exponentially distributed with rates also dependent on the state of the random environment at the time moment when the request arrived. When the environment changes its state, requests that are already in the system do not change their service times. So, we have requests of different types (serviced with different rates) present in the system at the same time. For the study, we consider a situation where changes of the random environment are made rarely. The method of asymptotic analysis is used for the study. The asymptotic condition of a rarely changing random environment (entries of the generator of the corresponding Markov chain tend to zero) is used. A multi-dimensional joint steady-state probability distribution of the number of requests of different types present in the system is obtained. Several numerical examples illustrate the comparisons of asymptotic results to simulations. Full article

Many scholars believe that the Yi Jing 易經 (the Book of Changes) and traditional Chinese medicine share common mathematical principles, which are both predicated on the ontological of qi 氣 and the cosmological of correlative between nature and human. Traditional Chinese medicine emphasizes the systemic organization of organs, meridians, qi, and blood as central components by incorporating the mathematical principles, including the theory of “Chaos-Crack”, the infinite classification methods of yinyang 陰陽, the generative and restrictive interactions of wuxing 五行, and the metaphysical significance of special numbers such as one, two, three, etc. Traditional Chinese medicine also formulates many theories and methodologies by integrating these mathematical principles with the schemata of luoshu 洛書 and jiugong 九宮, as well as the special combination numbers such as tianliu diwu 天六地五. This research tries to explain the mathematical principles and applications from the body dimension in traditional Chinese medicine. Full article

This paper establishes two novel recurrence relations for Stirling numbers of the second kind—an L recurrence and a vertical recurrence—discovered through a probabilistic analysis of Poisson higher-order origin moments. While the link between these moments and Stirling numbers is known, our derivation via a specific expectation identity provides a clear and efficient pathway to their computation, circumventing the need for infinite series. The primary theoretical contribution is the proof of these previously undocumented combinatorial recurrences, which are of independent mathematical interest. Furthermore, we demonstrate the severe practical inadequacy of high-order sample moments as estimators, highlighting the necessity of our analytical approach to obtaining reliable estimates in applied fields. Full article

In this paper, we derive integer pseudo-parametric solutions to two sets of Diophantine equations. Moreover, we describe the so-called Double Crossed Ladder (DCL) and show how these results can be used to calculate an infinite number of integer solutions of its sides. In addition, we describe the fact that these results can be used to derive some corresponding sets of integer sides of more complex geometric structures. Full article

Due to cost and variability of geotechnical test results, the number of samples for geotechnical material parameters in one engineering project is limited, resulting in a certain degree of errors in the calculation of probability distribution, mean, and variance of mechanical parameters of the geotechnical materials. To improve the reliability of geotechnical engineering design, reducing the variance of shear strength is one of the methods. Currently, the least squares method is widely used to regress the shear strength of soil; however, the regression residuals often exhibit heteroscedasticity and correlation, which undermine the validity of the variance estimates of soil shear strength parameters. This study aims to address this issue by applying the generalized least squares method to eliminate the heteroscedasticity and correlation of regression residuals. The results of triaxial consolidated drained (CD) tests on the coarse-grained soil; triaxial unconsolidated undrained(UU), CD, and consolidated undrained (CU) tests on gravelly clay; and triaxial CD tests on sand were analyzed to estimate the mean and variance of their shear strength. The results show that while the mean values of shear strength parameters remain largely unchanged, the generalized least squares method reduces the standard deviation of cohesion by an average of 30.575% and that of the internal friction angle by 14.21%. This reduction in variability enhances the precision of parameter estimation, which is critical for reliability-based design in geotechnical engineering, as it leads to more consistent safety assessments and optimized structural designs. The reliability analysis of an infinitely long slope stability shows that the reliability index of the soil slope calculated by the traditional method is either large or small. The generalized least squares method, which eliminates the heteroscedasticity and correlation of the regression residuals, should be adopted to regress the shear strength of soil. Full article

The prepositional variation of grazie di/per + complement ‘thanks for X’ is often acknowledged in Italian grammars but has not yet been adequately examined. I appeal to key tenets of Construction Grammar to analyze 3000 tokens of this construction from the Italian Web 2020 Corpus. To fully probe the conditioning of di/per selection, I pair logistic regression of the entire dataset with a descriptive statistical analysis of various levels of constructional schematicity and frequent individual complements. Results confirm previous descriptions that per is now the majority variant and reveal that significant predictors of preposition selection include complement type (nominal, simple infinitive, compound infinitive), as well as complement complexity and quantity of intervening material (both measured in number of words). However, strong lexico-constructional effects are also observed, such that the older variant di remains strongly preferred in specific micro-constructions (e.g., grazie di tutto ‘thanks for everything’, grazie di esistere ‘thanks for existing’). These findings evince a complex case of variation which requires the joint consideration of both overall patterns and fine-grained constructional distinctions. Full article

In this study, we formulate and analyze a susceptible–infected–susceptible (SIS) dynamic on metapopulation networks, where each node has a finite carrying capacity and the motion of individuals is modulated by vacant space at the destination. We obtain that the vacancy-dependent mobility pattern results in various asymptotic population distributions on heterogeneous metapopulation networks. The resulting population distributions have remarkable impact on the behavior of SIS dynamics. We show that, for the given total number of individuals, higher heterogeneity in population distributions facilitates epidemic spreading in terms of both a smaller epidemic threshold and larger macroscopic incidence. Moreover, we analytically obtain a sufficient condition that the disease-free equilibrium becomes unstable and an endemic state arises. Contrary to the absence of an epidemic threshold in the standard diffusion case without excluded-volume effects, the finite carrying capacity induces a nonzero epidemic threshold under certain conditions in the limit of infinite network sizes with an unbounded maximum degree. Our analytical results agree well with numerical simulations. Full article

Spot melting is an emerging alternative to traditional line melting in electron beam powder bed fusion, dividing a layer into thousands of individual spots. This method allows for an almost infinite number of spot arrangements and spot melting sequences to tailor material and part properties. To enhance the productivity of spot melting, the number of spots can be reduced by increasing the beam diameter. However, this results in rough surfaces due to the staircase effect. The classical approach to counteract these effects is to melt a contour that surrounds the infill area. Creating effective contours is challenging because the melted area ought to cover the artifacts from the staircase effect and avoid porosity in the transition area between the infill and contour, all while minimizing additional energy and melt time. In this work, we propose an algorithm for generating a spot melting sequence for contour lines surrounding the infill area. Additionally, we compare three different approaches for combining the spot melting of infill and contour areas, each utilizing a combination of large infill spots and small contour spots. The quality of the contours is evaluated based on optical inspection as well as the porosity between infill and contour using electron optical images, balanced against the additional energy input. The most suitable approach is used to build a complex brake caliper. Full article

The wide application of light emitting diodes (LEDs) in lighting systems has necessitated the inclusion of spectral tunability by using multi-color LED chips. Since the lighting requirement depends on the specific application, it is very important to have flexibility in terms of the driving conditions. While many applications use single or rather white color, some recent applications require multi-spectral lighting systems especially for agricultural or human-medical treatment applications. These systems are underexplored and pose specific challenges. In this paper, a mixture of red, green, blue, white (RGB-W) LED chips was used to develop a compact light engine specifically for agricultural applications. A computational study was performed to understand the optical distribution. Later, attention was turned into development of prototype light engines followed by experimental validation for both the thermal and optical characteristics. Each LED string was driven separately at different current levels enabling an option for obtaining an infinite number of colors for numerous applications. Each LED string on the developed light engine was driven at 300 mA, 500 mA, 700 mA, and 900 mA current levels, and the optical and thermal parameters were recorded simultaneously. A set of computational models and an experimental study were performed to understand the optical and thermal characteristics simultaneously. Full article

The traditional process for kinematic synthesis of planar mechanisms involves setting a few prescribed positions, then solving a set of equations to identify a vector chain that exactly reproduces those positions. In evaluating these equations, designers often must sift through multiple “infinities” of solutions corresponding to some number of free-choice variables that each have an infinite number of possible values. In this vast solution space, some combination of those variables will produce the most optimal solution, but finding that optimal solution is not trivial. There are two extremes for addressing the impossibility of sifting through infinite possible values. First, one could use analytical techniques to make educated estimates of the optimal values. Or, alternatively, a designer could completely remove their perspective from the process, passing the problem into a computer and programming it to sift through millions (or orders of magnitude more) possible solutions. The present work proposes a novel intermediate step in the analytical synthesis process that functions as a middle ground between these extremes. Optimizing solution density involves a designer manually manipulating the problem definition to increase the percentage of solutions that have pivots in acceptable locations. This is accomplished by changing the values of δ_j and α_j (prescribed translation and rotation of the moving plane, respectively) to manipulate the position of the poles. A physical example, designing a 7-bar parallel-motion generator, shows that applying this method yields more passing solutions when comparing over the same search depth. Specifically, 0.008% of solutions pass the design criteria without applying the method, and 3.154% pass after optimizing. This approach can reduce the computational load placed on a computer running a search script, as designers can use larger increments on the free choices without skipping over a family of solutions. Full article

The physics of systems of quantum emitters in waveguide quantum electrodynamics is significantly influenced by the relation between their spatial separation and the wavelength of the emitted photons. If the distance that separates a pair of emitters meets specific resonance conditions, the photon amplitudes produced from decay may destructively interfere. In an infinite-waveguide setting, this effect gives rise to bound states in the continuum, where a photon remains confined between the emitters. In the case of a finite-length waveguide with periodic boundary conditions, there exist two such relevant distances for a given arrangement of the quantum emitters, leading to states in which a photon is confined to either the shorter or the longer path that connects the emitters. If the ratio of the shorter and the longer path is a rational number, these two kinds of resonant eigenstates are allowed to co-exist for the same Hamiltonian. In this paper, we investigate the existence of quasi-degenerate resonant doublets of a pair of identical emitters coupled to a linear waveguide mode. The states that form the doublet are searched among the ones in which a single excitation tends to remain bound to the emitters. We investigate the spectrum in a finite range around degeneracy points to check whether the doublet remains well separated from the closest eigenvalues in the spectrum. The identification of quasi-degenerate doublets opens the possibility to manipulate the emitters-waveguide system as an effectively two-level system in specific energy ranges, providing an innovative tool for quantum technology tasks. Full article

The occurrence of knots as solutions of dynamical systems has been widely studied in the literature. In particular, ways to determine families of knots as solutions of differential equations have been described in several papers. In this article, an infinite family of dynamical systems, based on torus knots, is built each of which has the property that an infinite number of cable knots from torus knots (i.e., iterated torus knots) are obtained as solutions. One such dynamical system, based on the trefoil knot, is explicitly constructed. The methodology described herein may also be applied to any torus knot, and even to any other knot as long as a parametrization is provided for the latter. An example of application of the method is presented for the case of the figure eight knot, which is not a torus knot. Also, a possible application in cryptography is sketched. Full article

The often used analytical representation of the Maxwell–Boltzmann classical speed distribution function (F) for elastic, indivisible particles assumes an infinite limit for the speed. Consequently, volume and the number of particles (n) extend to infinity: Both infinities contradict assumptions underlying this non-relativistic formulation. Finite average kinetic energy and temperature (T) result from normalization of F removing n: However, total energy (i.e., heat of the collection) remains infinite because n is infinite. This problem persists in recent adaptations. To better address real (finite) systems, wherein T depends on heat, we generalize this one-parameter distribution (F, cast in energy) by proposing a two-parameter gamma distribution function (F*) in energy which reduces to F at large n. Its expectation value of kT (k = Boltzmann’s constant) replicates F, whereas the shape factor depends on n and affects the averages, as expected for finite systems. We validate F* via a first-principle, molecular dynamics numerical model of energy and momentum conserving collisions for 26, 182, and 728 particles in three-dimensional physical space. Dimensionless calculations provide generally applicable results; a total of 10⁷ collisions suffice to represent an equilibrated collection. Our numerical results show that individual momentum conserving collisions in three-dimensions provide symmetrical speed distributions in all Cartesian directions. Thus, momentum and energy conserving collisions are the physical cause for equipartitioning of energy: Validity of this theorem for other systems depends on their specific motions. Our numerical results set upper limits on kinetic energy of individual particles; restrict the n particles to some finite volume; and lead to a formula in terms of n for conserving total energy when utilizing F* for convenience. Implications of our findings on matter under extreme conditions are briefly discussed. Full article