Unified Formulae Research Articles

Moments of Distance from a Vertex to a Uniformly Distributed Random Point within Arbitrary Triangles

We study the cumulative distribution function (CDF), probability density function (PDF), and moments of distance between a given vertex and a uniformly distributed random point within a triangle in this work. Based on a computational technique that helps us provide unified formulae of the CDF and PDF for this random distance then we compute its moments of arbitrary orders, based on which the variance and standard deviation can be easily derived. We conduct Monte Carlo simulations under various conditions to check the validity of our theoretical derivations. Our method can be adapted to study the random distances sampled from arbitrary polygons by decomposing them into triangles.

Unified parameter of in-plane and out-of-plane constraint effects and its correlation with brittle fracture toughness of steel

In this work, the equivalent plastic strain $$\varepsilon _{p}$$ distributions ahead of crack tips for the experimental specimens with combined in-plane and out-of-plane constraints under brittle fracture condition in the literature were calculated by three-dimensional finite element. The constraint parameter $$A_{p}$$ based on the areas surrounded by $$\varepsilon _{p}$$ isolines ahead of crack tips has been comparatively analyzed with several constraint parameters ( $$T$$ -stress, $$A_{2}$$ , $$Q$$ and stress triaxiality $$h$$ ) based on the crack-tip stress fields, and the capability of parameter $$A_{p}$$ for characterizing in-plane and out-of-plane crack-tip constraint effects for brittle fracture has been identified. The results show that the parameter $$A_{p}$$ has a good correlation with brittle fracture toughness $$K_{Jc}$$ and $$J_{c}$$ of various specimens with different constraint levels, and it is a unified measure parameter of in-plane and out-of-plane constraint for brittle fracture. The unified correlation lines and formulae of the normalized brittle fracture toughness $$K_{Jc}/K_{ref}$$ and $$J_{Ic}/J_{ref}$$ with $$\sqrt{A_p}$$ have been obtained for the steel, and they may be used to determine constraint-dependent or structurally relevant fracture toughness of specimens or cracked components with any constraint levels. The application methodology of the constraint parameter $$A_{p}$$ for structural integrity assessments needs to be further investigated by numerical calculations and experiments.

A Computer Aided Kineto-Static Analysis Method for 3-RRRT Parallel Manipulator Based on Vector Bond Graph

For improving the reliability and efficiency of the kineto-static analysis of complex robot systems, the corre- sponding vector bond graph procedure is proposed. By the kinematic constraint condition, the vector bond graph model of universal joint are made. Based on this, the vector bond model of 3-RRRT parallel Manipulator is built. For solving the algebraic difficulties brought by differential causality in system automatic kineto-static analysis, the effective bond graph augment method is proposed. From the algebraic relations of input and output vectors in the basic fields, junction struc- ture and Euler-junction structure of system vector bond graph model, the unified formulae of driving moment (or force) and constraint forces at joints are derived, which are easily derived on a computer in a complete form. As a result, the automatic kineto-static analysis of 3-RRRT parallel Manipulator is realized, the validity of the method presented here is illustrated.

The Dynamic Modelling and Simulation of Spatial Multibody Systems with Prismatic Joint Based on Vector Bond Graph

For the modelling and simulation of complex spatial multibody systems, the vector bond graph method is proposed. By the kinematic constraint condation, spatial prismatic joint can be modeled by vector bond graph. For the algebraic difficulties brought by differential causality in system automatic modeling and simulation, the effective decoupling method is proposed and the differential causalities in system vector bond graph model can be eliminated. In the case of considering EJS, the unified formulae of system state space equations and constraint forces at joints are derived, which are easily derived on a computer and very suitable for spatial multibody systems. As a result, the unified modelling and simulation for complex spatial multibody systems are realized, its validity is illustrated by a practical example.

A Vector Bond Graph Method of Kineto-Static Analysis for Spatial Multibody Systems

In order to increase the reliability and efficiency of the kineto-static analysis of complex multibody systems, the corresponding vector bond graph procedure is proposed. By the kinematic constraint condition, spatial multibody systems can be modeled by vector bond graph. For the algebraic difficulties brought by differential causality in system automatic kineto-static analysis, the effective decoupling method is proposed, thus the differential causalities in system vector bond graph model can be eliminated. In the case of considering EJS, the unified formulae of driving moment and constraint forces at joints are derived based on vector bond graph, which are easily derived on a computer in a complete form and very suitable for spatial multibody systems. As a result, the automatic kineto-static analysis of spatial multibody system on a computer is realized, its validity is illustrated by the spatial multibody system with three degrees of freedom.

Statics and Stiffness Model of Serial-Parallel Manipulator Formed by k Parallel Manipulators Connected in Series

Abstract The statics and stiffness model of serial-parallel manipulators (S-PMs) formed by k parallel manipulators (PMs) connected in series is established in this paper. The S-PMs can provide features of both serial manipulators (SMs) and PMs. First, the unified formulae for solving the statics and stiffness of S-PMs are derived. Second, a k(PS + RPS + SPS) S-PM is analyzed to illustrate this model. Finally, an analytic solved example for 5(PS + RPS + SPS) S-PM is given. The established model can offer an essential theoretical basis for S-PMs.

A Vector Bond Graph Method of Kineto-Static Analysis for Complex Planar Linkage

For the kineto-static analysis of complex planar linkage, the procedure based on vector bond graph is proposed. The constraint force vectors at joints can be considered as unknown effort source vectors and added to the corresponding 0-junctions of the system vector bond graph model, most of the differential causalities in system vector bond graph model can be eliminated . In the case of mixed causality, the unified formulae of driving moment and constraint forces at joints are derived based on vector bond graph, which are easily derived on a computer in a complete form. As a result, the very difficult algebraic problem caused by differential causality and nonlinear junction structure can be overcome, and the automatic kineto-static analysis of complex planar linkage on a computer is realized. By a practical example, the validity of this procedure is illustrated.

Unified analytical formulae for the effective properties of periodic fibrous composites

Two-phase fibre-reinforced elastic composites with transversely isotropic constituents and two types of fibres periodic distribution (square and hexagonal arrays) with perfect contact conditions at the interface are considered. Unified simple analytical formulae for their effective properties are given. Such formulae involve two orders of approximation of semi-analytical expressions obtained via asymptotic homogenization. The limit case of empty fibres is also presented. The influence of each approximation is illustrated. Theoretical and experimental comparisons are included. The objective is to provide information at hand which could be useful for applications to bone mechanics and ultrasonic transducers, and for validation of numerical codes.

A Gyrobondgraph Method of Dynamic and Static Force Analysis for Spacial Multibody Systems

In order to increase the efficiency and reliability of dynamic and static force analysis for spatial multibody systems containing the coupling of multi-energy domains, a method based on gyrobondgraph is introduced. The procedure of modeling spacial multibody systems by bond graphs and its dynamic principle are described. The unified formulae of driving moment and constraint forces at joints are derived based on gyrobondgraph, they are easily generated on a computer in a complete form. As a result, the very difficult algebraic problem caused by differential causality and nonlinear junction structure can be overcome, and the automatic dynamic analysis of multibody systems on a computer is realized. By a practical example, the validity of this procedure is illustrated.

The Analysis of Elements of Human Resources Development

The aim of development has no longer nowadays only achieving an increase in the national income, but exceeded to interest in other aspects within the scope of economic development, almost the factors affiliated with the development of human resources and to provide the major needs. On this basis, new indicators and referents have been widely used to express these capabilities. The importance of this research started from the magnificent role of the human development of evaluating the better life and achieving higher levels of social and economic welfare. The objective of research is to explain the scope of development achieved in Arab countries and the scope of human resources development and the differences among these countries. The problem of the current research is to find a unified formulae among the world countries that include all primary components of the human development. This can be seen through the human resources measurement in the world countries. It hypothesized that the Arab countries have realized variously a very advanced rate in the human recourses. The quantities method has been used on the basis of allowed data analysis. The research concluded that Arab countries have profoundly achieved an acute progress in human resources development. So, the national rate of human resources increased from 0.518 in 1975 to 0.684 in 2002, the compound annual growth 1.03%. Arab countries have been able to reduce the gap considerably, where the gap has noticeably expanded inside the countries separately. .

A new class of smoothing complementarity functions over symmetric cones

A popular approach to solving the complementarity problem is to reformulate it as an equivalent system of smooth equations via a smoothing complementarity function. In this paper, first we propose a new class of smoothing complementarity functions, which contains the natural residual smoothing function and the Fischer–Burmeister smoothing function for symmetric cone complementarity problems. Then we give some unified formulae of the Fréchet derivatives associated with Jordan product. Finally, the derivative of the new proposed class of smoothing complementarity functions is deduced over symmetric cones.

Variational bounds for anisotropic elastic multiphase composites with different shapes of inclusions

In the present work, unified formulae for the overall elastic bounds for multiphase transversely isotropic composites with different geometrical types of inclusions embedded in a matrix are calculated, including the spherical and long or short continuous cylindrical fiber cases. The influence of the different geometrical configurations of the inclusions on the composites is studied. The transversely isotropic effective bounds are obtained by applying the variational formulation for anisotropic composites developed by Willis, which relies on expressions for the static transversely isotropic Green’s function. Some numerical calculations and comparisons with the effective coefficients derived from the self-consistent approach, asymptotic homogenization method, and finite element method (FEM) are shown for different aspect ratio values, exhibiting good agreement.

Improving the arithmetic of elliptic curves in the Jacobi model

The use of elliptic curve cryptosystems on embedded systems has been becoming widespread for some years. Therefore the resistance of such cryptosystems to side-channel attacks is becoming crucial. Several techniques have recently been developed. One of these consists in finding a representation of the elliptic curve such that formulae for doubling and addition are the same. Until now, one of the best results has been obtained by using the Jacobi model. In this Letter, we improve the arithmetic of elliptic curves in the Jacobi model and we relax some conditions required to work efficiently on this model. We thus obtained the fastest unified addition formulae for elliptic curve cryptography (assuming that the curve has a 2-torsion point).

Mobility and equilibrium stability analysis of pin-jointed mechanisms with equilibrium matrix SVD

Under certain load pattern, the geometrically indeterminate pin-jointed mechanisms will present certain shapes to keep static equalization. This paper proposes a matrix-based method to determine the mobility and equilibrium stability of mechanisms according to the effects of the external loads. The first and second variations of the potential energy function of mechanisms under conservative force field are analyzed. Based on the singular value decomposition (SVD) method, a new criterion for the mobility and equilibrium stability of mechanisms can be concluded by analyzing the equilibrium matrix. The mobility and stability of mechanisms can be classified by unified matrix formulae. A number of examples are given to demonstrate the proposed criterion. In the end, criteria are summarized in a table.

Two unified formulae

Abstract The trace identity, the quadratic-form identity and the tri-trace identity, which all contain the constant γ , are all simple and powerful tools for producing Hamiltonian structure. The first formula obtained in the Letter is a unified expression of the above three formulas. Its role is to open an approach for setting up similar formulas in other mathematical or physical applications. The second formula given in the Letter is a unified computational one on the constant γ which appears in the above three formulae. This succeeds in overcoming the problem on the computing the constant γ , proposed by Tu Guizhang and left open for long time.

On the asymptotic variance in nonparametric regression with fractional time-series errors

This paper focuses on deriving explicit formulae of the asymptotic variance in nonparametric regression with fractional time-series errors. Unified formulae are obtained for fixed design models with long-memory, short-memory and antipersistent errors. It is also found that in strongly antipersistent case the Uniform kernel is no longer the minimum variance one and that for a fourth-order kernel the constant in the asymptotic variance for long-memory errors may be clearly smaller than that for i.i.d. errors. The results are applied to improving an existing data-driven algorithm. Practical performance of the proposed algorithm is illustrated with simulated and real data examples.

The unified Riemann-Liouville fractional derivative formulae

In this paper, we obtain two unified fractional derivative formulae. The first involves the product of two general class of polynomials and the multivariable $H$-function. The second fractional derivative formula also involves the product of two general class of polynomials and the multivariable $H$-function and has been obtained by the application of the first fractional derivative formula twice and it has two independent variables instead of one. The polynomials and the functions involved in both the fractional derivative formulae as well as their arguments are quite general in nature and so our findings provide interesting unifications and extensions of a number of (known and new) results. For the sake of illustration, we point out that the fractional derivative formulae recently obtained by Srivastava, Chandel and Vishwakarma [11], Srivastava and Goyal [12], Gupta, Agrawal and Soni [4], Gupta and Agrawal [3] follow as particular cases of our findings. In the end, we record a new fractional derivative formula involving the product of the Konhauser biorthogonal polynomials, the Jacobi polynomials and the product of $r$ different modified Bessel functions of the second kind as a simple special case of our first formula.

Relative amplitude preserved P–P and P–SV wave seismic sections

In seismic wave propagation, the amplitude decay along travel time comprises two components: the spherical divergence and the earth absorption effect in an inhomogeneous medium. In reservoir geophysics, both the P–P and P–SV wave seismic sections are required to be relative-amplitude preserved so that they can be used for comparative interpretation and analysis. Therefore, each of the following two processing steps, divergence correction and inverse Q filtering, applied to both seismic sections should be relative-amplitude preserved. In this paper, the spherical divergence corrections on the P–P and P–SV wave sections are expressed in unified formulae in which the formula for the P–SV wave section differs from that for the P–P wave section by a factor (1 + γ)-1, where γ is the ratio of P- to S-wave velocities, applicable to both the scale and power coefficients simultaneously. Divergence-corrected seismic sections are then used to estimate the earth Q models, which in turn are used to design inverse Q filters. Only when divergence correction and inverse Q filtering, both of which are amplitude-related processing, applied to the P–P and P–SV wave sections are adequately accurate, the amplitude information contained in these seismic sections can be reliably used for lithological analysis in reservoir geophysics.

Unified radiation formulae for classical and quantum electrodynamics

There are deep conceptual differences between the classical and quantummechanical treatment of electromagnetic radiation processes. Nevertheless, it ispossible to give a formally unified description of the spectral and angulardistribution of radiation in both cases in terms of four-dimensional Fouriertransforms of currents. We present parallel derivations of the basic radiationformulae utilizing classical electrodynamics as well as spinor quantumelectrodynamics. In addition both derivations allow for the presence of amedium with an index of refraction. The practical application of thesemethods is illustrated by calculations of some specific radiation problems.

Certain fractional derivative formulae involving the product of a general class of polynomials and the multivariableH-function

In the present paper, we obtain three unified fractional derivative formulae (FDF). The first involves the product of a general class of polynomials and the multivariableH-function. The second involves the product of a general class of polynomials and two multivariableH-functions and has been obtained with the help of the generalized Leibniz rule for fractional derivatives. The last FDF also involves the product of a general class of polynomials and the multivariableH-function but it is obtained by the application of the first FDF twice and it involves two independent variables instead of one. The polynomials and the functions involved in all our fractional derivative formulae as well as their arguments which are of the typex ρ Π =1 (x t i +α i ) σ i are quite general in nature. These formulae, besides being of very general character have been put in a compact form avoiding the occurrence of infinite series and thus making them useful in applications. Our findings provide interesting unifications and extensions of a number of (new and known) results. For the sake of illustration, we give here exact references to the results (in essence) of five research papers [2, 3,10, 12, 13] that follow as particular cases of our findings. In the end, we record a new fractional derivative formula involving the product of the Hermite polynomials, the Laguerre polynomials and the product ofr different Whittaker functions as a simple special case of our first formula.