This paper investigates the conditions for the equivalence of regular surfaces with respect to the action of a certain subgroup of linear transformations. This subgroup is pseudo-orthogonal and preserves a metric structure defined by a matrix with specific sign properties. The study focuses on elementary surfaces, which are considered as mappings from the square of the parameter domain (0,1)×(0,1) into an n-dimensional real vector space. The regularity of a surface is determined by the non-vanishing determinant of a special matrix composed of its partial derivatives. The paper also introduces the concept of surface equivalence. The main theorem establishes necessary and sufficient conditions for the equivalence of regular surfaces under the action of the pseudo-orthogonal group. These conditions are expressed through equalities between products of matrices constructed from the partial derivatives of the surfaces and the pseudo-orthogonal matrix. The obtained results provide a theoretical foundation for understanding the relationships between regular surfaces under the action of the pseudo-orthogonal group and contribute to the further study of their geometric properties and transformations.
Decision making uses analytical tools all around. Business decision making applies these tools in form of business analytics (BA). Information technology (IT) too has been pivotal in management decisions and functions. BA and IT are recent advancement combined for efficient decision making. Supply chain management (SCM) utilizes BA and IT as competitive tool. The studies of technological, managerial, strategic and economic factors and implementation process with its results on performances are going on. Effectiveness and success of BA is unconvinced up to its measurement is done with proper methods and tools. Present study focuses on method of construct development. The methodology follows the process of investigation and measurement development by Churchill for describing and organizing information about the influence of analytics and information technology in supply chain (SC) decision making and functions on supply chain performances in SCOR areas. The study has been carried out with pilot data from Indian plastics manufactures. Reliability and validity of measures was done methodologically. The results of work provide measures which may be used as a tool for future studies in area of business analytics, SC decision making and SC performances.
This paper introduces a model-agnostic approach to study statistical synergy, a form of emergence in which patterns at large scales are not traceable from lower scales. Our framework leverages various multivariate extensions of Shannon's mutual information, and introduces the O-information as a metric that is capable of characterizing synergy- and redundancy-dominated systems. The O-information is a symmetric quantity, and can assess intrinsic properties of a system without dividing its parts into "predictors" and "targets." We develop key analytical properties of the O-information, and study how it relates to other metrics of high-order interactions from the statistical mechanics and neuroscience literature. Finally, as a proof of concept, we present an exploration on the relevance of statistical synergy in Baroque music scores.
The solution and diffusion of interstitial non-metallic solutes (INSs) like H, He, O, C, N, P, and S is common in refractory high-entropy alloys (RHEAs) and essentially controls the RHEAs properties. However, the disorder local chemical environments of RHEAs hinder the quantitative prediction of the stability and diffusivity of INSs and the understanding of the underlying mechanism. Based on the tight-binding models, we propose an analytic model for determining the stability and diffusivity of INSs in RHEAs, by approximating the bonding length between INSs and their neighbors with the atomic radius of the neighbors in elemental states. This predictive model identifies that the energetics of INSs depends linearly on the d-band width of their neighbors, with the slope determined by the valence of INSs. Our scheme provides an electronic-level understanding of INSs in RHEAs and explains key experimental observations, which can serve as an effective tool for designing advanced RHEAs.
This article explores the theory of Riemann double integration for functions whose values are intervals in the framework of time scale calculus. We define the Riemann double ∆-integral and Riemann double ∇-integral for interval valued functions, namely interval Riemann ∆∆-integral and interval Riemann ∇∇integral. Some key theorems in the article discuss the uniqueness of the integral, the equality of the interval Riemann double integral to the Riemann double integral when function is degenerate, necessary and sufficient conditions for integrability, proving integrability of a function without knowing the actual value of the integral. Additionally the relationship between the interval Riemann double integral and Riemann double integral for two interval-valued functions is estableshed via Hausdorff-Pompeiu distance. Elementary properties of the integral such as linearity property, subset property and others are established. Using the concept of generalized Hukuhara difference, alternate definitions of the interval Riemann ∆∆-integral and interval Riemann ∇∇-integral are formulated and theorems proving the equivalence of the integrals defined in both approaches are established. Theorems proving the equivalence of interval Riemann ∆- and ∇integrals previously defined in both approaches are also shown.
Anshul S. Tomar, Shaede Perzanowski, Ricardo Mejia-Alvarez
et al.
Bernoulli pads generate locally large wall shear stresses on workpieces, which can be used for cleaning, but may also damage delicate surfaces. This work presents direct measurements of the wall shear stress using constant-temperature anemometry for the first time. A hot-film sensor was calibrated in the laminar and turbulent flow regimes using a purpose-built water flow channel. The calibrated sensor was then flush mounted onto a smooth surface and a Bernoulli pad was traversed over the sensor and wall shear stress data were acquired. Numerical simulations of the flow field were also performed; they accurately predicted the maximum shear stress near the jet corner but over-predicted at large radii.
This work presents a study on the characteristics of the textbooks used in the upper courses of Physics careers. The study consists of an analysis of the hamiltonian in Analytic Mechanics which is carried out in two stages: in the first one an example was characterized; in the second one, several interviews with professors were performed. The importance
of this study lies in the fact that textbooks are one of the main instructional materials used by senior undergraduate
students so that if those texts are not easy to understand, students will make an enormous effort to learn from them and this cognitive effort might not contribute to a good performance. On the basis of this analysis, it is concluded that the example does not result in an easy reading for students. Moreover, the interviews show that the characteristics of the example make the reading difficult for those who try to learn from it.
Johanna Müller, Sophie Hermann, Florian Sammüller
et al.
We identify a recently proposed shifting operation on classical phase space as a gauge transformation for statistical mechanical microstates. The infinitesimal generators of the continuous gauge group form a non-commutative Lie algebra, which induces exact sum rules when thermally averaged. Gauge invariance with respect to finite shifting is demonstrated via Monte Carlo simulation in the transformed phase space which generates identical equilibrium averages. Our results point towards a deeper basis of statistical mechanics than previously known and they offer avenues for systematic construction of exact identities and of sampling algorithms.
Inhomogeneous flows and shear banding are of interest for a range of applications but have been eluding a comprehensive theoretical understanding, mostly due to the lack of a framework comparable to equilibrium statistical mechanics. Here we revisit models of fluids that reach a stationary state obeying mechanical equilibrium. Starting from a non-local constitutive relation, we apply the idea of a "mechanical phase transition" and map the constitutive relation onto a dynamical system through an integrating factor. We illustrate this framework for two applications: shear banding in strongly thinning complex fluids and the coexistence of a solid with its sheared melt. Our results contribute to the growing body of work following a mechanical route to describe inhomogeneous systems away from thermal equilibrium.
This paper sets out to investigate the vortex flow of spinnaker yacht sails, which are low-aspect-ratio highly cambered wings used to sail downwind. We tested three model-scale sails with the same sections but different twists over a range of angles of attack in a water tunnel at a Reynolds number of 21 000. We measured the forces with a balance and the velocity field with particle image velocimetry. The sails experience massively separated three-dimensional flow and leading-edge vortices convect at half of the free-stream velocity in a turbulent shear layer. Despite the massive flow separation, the twist of the sail does not change the lift curve slope, in agreement with strip theory. As the angle of attack and the twist vary, flow reattachment might occur in the time-average sense, but this does not necessarily result in a higher lift to drag ratio as the vorticity field is marginally affected. Finally, we investigated the effect of secondary vorticity, vortex stretching and diffusion on the vorticity fluxes. Overall, these results provide new insights into the vortex flow and associated force generation mechanism of wings with massively separated flow.
This paper proposes a theory that bridges classical analytical mechanics and nonequilibrium thermodynamics. Its intent is to derive the evolution equations of a system from a stationarity principle for a suitably augmented Lagrangian action. This aim is attained for homogeneous systems, described by a finite number of state variables depending on time only. In particular, it is shown that away from equilibrium free energy and entropy are independent constitutive functions.
The paper investigates integro-differential equations in Banach spaces with operators, which are a composition of convolution and differentiation operators. Depending on the order of action of these two operators, we talk about integro-differential operators of the Riemann—Liouville type, when the convolution operator acts first, and integro-differential operators of the Gerasimov type otherwise. Special cases of the operators under consideration are the fractional derivatives of Riemann—Liouville and Gerasimov, respectively. The classes of integro-differential operators under study also include those in which the convolution has an integral kernel without singularities. The conditions of the unique solvability of the Cauchy type problem for a linear integro-differential equation of the Riemann—Liouville type and the Cauchy problem for a linear integrodifferential equation of the Gerasimov type with a bounded operator at the unknown function are obtained. These results are used in the study of similar equations with a degenerate operator at an integro-differential operator under the condition of relative boundedness of the pair of operators from the equation. Abstract results are applied to the study of initial boundary value problems for partial differential equations with an integro-differential operator, the convolution in which is given by the Mittag-Leffler function multiplied by a power function.
This paper presents an analytic solution for aerodynamic noise generated by an unsteady wall pressure gust interacting with a spanwise-variable trailing edge in a background steady uniform flow. Viscous and nonlinear effects are neglected. The Wiener–Hopf method is used in conjunction with a non-orthogonal coordinate transformation and separation of variables to permit analytical progress. The solution is obtained in terms of a tailored modal expansion in the spanwise coordinate; however, only finitely many modes are cut-on, therefore the far-field noise can be quickly evaluated. The solution gives insight into the potential mechanisms behind the reduction of noise for plates with serrated trailing edges compared to those with straight edges. The two mechanisms behind the noise reduction are an increased destructive interference in the far field, and a redistribution of acoustic energy from low cut-on modes to higher cut-off modes. Five different test-case trailing-edge geometries are considered. The analytic solution identifies which geometries are most effective in different frequency ranges: geometries which promote destructive interference are best at low frequencies, whilst geometries which promote a redistribution of energy are better at high frequencies.
We derive an analytic expression for the mechanical pressure of a generic one-dimensional model of confined active Brownian particles (ABPs) that is valid for all values of Peclet number Pe and all confining scenarios. Our model reproduces the known scaling of bulk pressure with Pe^2 while in strong confinement pressure scales with Pe. Our analytic results are very well reproduced by simulations of ABPs in 2D. We use the pressure formula to calculate both the work performed by an active engine and its efficiency. In particular, efficiency is maximized for work cycles with finite period and not in the limit of infinitely slow cycles as in thermodynamic engines.
In this paper, we investigated a boundary value problem with the Sturm-Liouville type conditions for a linear ordinary differential equation of fractional order with delay. The condition for the unique solvability of the problem is obtained in the form △= 0. The Green function of the problem, in terms of which the solution of the boundary value problem under study is written out, is constructed. The existence and uniqueness theorem for the solution of the problem is proved. It is also showed that in the case when the condition of unique solvability is violated, i.e △ = 0, then the solution of the boundary value problem is not unique. Using the notation of the generalized Mittag-Leffler function via the generalized Wright function, we also studied the properties of the function △ as λ → ∞ and λ → −∞. Using asymptotic formulas for the generalized Wright function, a theorem on the finiteness of the number of eigenvalues of a boundary value problem with the Sturm-Liouville type conditions is proved.
The pull rod under eccentric loading exists extensively in engineering project,the computations of its stress and strength directly affect its reliability,especially its dynamic stress computations. The elastic dynamic mechanics were used here to study the computations of the bending stress and axial stress of the eccentric loaded rod under steady force,and the analytic expressions were given,numerical example was provided to show the variations of the amplitude of the bending stress and axial stress with the disturbance frequency,the theoretical analyses and the computation results of given example show that the amplitude of the bending stress and axial stress obviously depend on the disturbance frequency,which is different from that of corresponding static problems. The analyses and conclusions given here have some references to the dynamic reliability design.
Mechanical engineering and machinery, Materials of engineering and construction. Mechanics of materials
Boltzmann's ergodic hypothesis furnishes a possible explanation for the emergence of statistical mechanics in the framework of classical physics. In quantum mechanics, the Eigenstate Thermalization Hypothesis (ETH) is instead generally considered as a possible route to thermalization. This is because the notion of ergodicity itself is vague in the quantum world and it is often simply taken as a synonym for thermalization. Here we show, in an elementary way, that when quantum ergodicity is properly defined, it is, in fact, equivalent to ETH. In turn, ergodicity is equivalent to thermalization, thus implying the equivalence of thermalization and ETH. This result previously appeared in [De Palma et al., Phys. Rev. Lett. 115, 220401 (2015)], but becomes particularly clear in the present context. We also show that it is possible to define a classical analogue of ETH which is implicitly assumed to be satisfied when constructing classical statistical mechanics. Classical and quantum statistical mechanics are built according to the familiar standard prescription. This prescription, however, is ontologically justified only in the quantum world.