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Complying with Rigging and Material Handling Safety
Published in Frank R. Spellman, Surviving an OSHA Audit, 2020
The parallelogram law provides that if two concurrent forces are laid out vectorially, with either forces pointing toward or both away from their point of intersection, the parallelogram represents the resultant of the force. The concurrent forces must have both direction and magnitude if their resultant is to be determined.
Ancient sizing rules and limit analysis of masonry arches
Published in Pere Roca, Paulo B. Lourenço, Angelo Gaetani, Historic Construction and Conservation, 2019
Pere Roca, Paulo B. Lourenço, Angelo Gaetani
The use of vectors and parallelogram law are so intuitive that their origin is unknown, and probably, they date back to Aristotle (384–322 BC) or Heron of Alexandria (1st century AD). One of the first scholars who scientifically derived the parallelogram rule for calculating the resultant force was Stevin (1548–1620) in his book Beghinselen des Waterwichts (Principles on the Weight of Water, 1634). Subsequently, Varignon (1654–1722) introduced the funicular polygon and the polygon of forces in his work Nouvelle Mécanique ou Statique, published posthumously in 1725. Eventually, Culmann (1821–1881) discovered in 1864–1865 the structural relationship between funicular polygon and polygon of forces through a planar correlation of the projective geometry.
Rigging and Material Handling Safety
Published in Frank R. Spellman, Kathern Welsh, Safe Work Practices for Wastewater Treatment Plants, 2018
Frank R. Spellman, Kathern Welsh
Frequently, two or more forces act together to produce the effect of a single force, called a resultant. This resolution of forces can be explained by either the triangle law or the parallelogram law. The triangle law provides that if two concurrent forces are laid out vectorially with the beginning of the second force at the end of the first, the vector connecting the beginning and the end of the forces represents the resultant of the two forces (see Figure 16.2A). The parallelogram law provides that if two concurrent forces are laid out vectorially, with either forces pointing toward or both away from their point of intersection, a parallelogram represents the resultant of the force. The concurrent forces must have both direction and magnitude if their resultant is to be determined (see Figure 16.2B). If the individual forces are known or if one of the individual forces and the resultant are known, the resultant force may be simply calculated by either the trigonometric method (sines, cosines, and tangents) or the graphic method (which involves laying out the known force, or forces, at an exact scale and in the exact directions in either a parallelogram or triangle and then measuring the unknown to the same scale).
Visualizing Laguerre polynomials as a complete orthonormal set for the inner product space ℙ n
Published in International Journal of Mathematical Education in Science and Technology, 2023
Named after the French mathematician Edmond Laguerre (1834–1886), the set forms an orthogonal set of polynomials with the associated integral inner product . In quantum mechanics, they appear in eigenfunctions satisfying the radial Schrödinger equation for the Hydrogen atom. This note illustrates the integral inner product in the vector space of polynomials of degree with real coefficients – the mapping defined via for – in a DGS/MATLAB-facilitated learning environment. Also explored are the defining properties (symmetry, linearity, positive definiteness) of the integral inner product along with other notions inherent in the inner product space, such as norm, distance, orthogonal projection, Cauchy–Schwarz Inequality, Triangle Inequality, Pythagorean Theorem, Parallelogram Law, orthogonality and orthonormality, orthonormal basis, and coordinates relative to an orthonormal basis. The article also demonstrates the diversity of ways through which Laguerre polynomials can be used as an orthonormal basis for the inner product space in a technology-assisted learning environment. More details on Laguerre Polynomials can be found in Arfken (1985), Hochstrasser (1972), Szegö (1975).
A new algorithm for the minimax location problem with the closest distance
Published in Optimization, 2022
S. Nobakhtian, A. Raeisi Dehkordi
Suppose that is a minimizing sequence for the problem (1), i.e. Next suppose that and are two arbitrary subsequences of . From the parallelogram law, we have Multiplying the above inequalities by , we get If we set , then By taking the maximum of the above inequalities, we have Since , it follows that By taking the limit, we have Therefore there exists such that Moreover, To prove the uniqueness of , suppose that are two distinct optimal solutions of problem (1). From (6), we obtain Indeed, .
Visualizing the inner product space ℝm×n in a MATLAB-assisted linear algebra classroom
Published in International Journal of Mathematical Education in Science and Technology, 2018
Prelude: This classroom note considers Frobenius inner product – the mapping defined via for real m × n matrices A and B – in a MATLAB-facilitated learning environment. The examples illustrate the defining properties (symmetry, linearity, positive definiteness) of the inner product along with other notions inherent in the inner product space , such as Frobenius norm (length), distance between vectors, angle between vectors, orthogonal projection, Cauchy–Schwarz inequality, triangle inequality, Pythagorean theorem, parallelogram law, orthogonality and orthonormality, orthonormal basis, completeness relation, coordinates relative to an orthonormal basis and Parseval's identity.