Topic

# Antisymmetric relation

About: Antisymmetric relation is a(n) research topic. Over the lifetime, 3322 publication(s) have been published within this topic receiving 64365 citation(s). The topic is also known as: antisymmetric property & anti-symmetric property.

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TL;DR: In this article, the authors define a set of generalized density matrices for the Hermitean density matrix of order $k, which is further antisymmetric in each set of these indices.

Abstract: In order to calculate the average value of a physical quantity containing also many-particle interactions in a system of $N$ antisymmetric particles, a set of generalized density matrices are defined. In order to permit the investigation of the same physical situation in two complementary spaces, the Hermitean density matrix of order $k$ has two sets of indices of each $k$ variables, and it is further antisymmetric in each set of these indices.Every normalizable antisymmetric wave function may be expanded in a series of determinants of order $N$ over all ordered configurations formed from a basic complete set of one-particle functions ${\ensuremath{\psi}}_{k}$, which gives a representation of the wave function and its density matrices also in the discrete $k$-space. The coefficients in an expansion of an eigenfunction to a particular operator may be determined by the variation principle, leading to the ordinary secular equation of the method of configurational interaction. It is shown that the first-order density matrix may be brought to diagonal form, which defines the "natural spin-orbitals" associated with the system. The situation is then partly characterized by the corresponding occupation numbers, which are shown to lie between 0 and 1 and to assume the value 1, only if the corresponding spin-orbital occurs in all configurations necessary for describing the situation. If the system has exactly $N$ spin-orbitals which are fully occupied, the total wave function may be reduced to a single Slater determinant. However, due to the mutual interaction between the particles, this limiting case is never physically realized, but the introduction of natural spin-orbitals leads then instead to a configurational expansion of most rapid convergence.In case the basic set is of finite order $M$, the best choice of this set is determined by a form of extended Hartree-Fock equations. It is shown that, in this case, the natural spin-orbitals approximately fulfill some equations previously proposed by Slater.

2,590 citations

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01 Jan 1997

TL;DR: In this paper, the authors present a one-dimensional analysis of fiber-reinforced composite materials and their properties, including the properties of the components of a Lamina and their relationship with other components.

Abstract: Introduction and Mathematical Preliminaries Fiber-Reinforced Composite Materials. Vectors and Tensors. Matrices. Transformation of Vector and Tensor Components. Integral Relations. Equations of Anisotropic Elasticity Classification of Equations. Kinematics. Kinetics. Constitutive Equations. Equations of Thermoelasticity and Electroelasticity. Summary. Virtual Work Principles and Variational Methods Virtual Work. The Variational Operator and Functionals. Extrema of Functionals. Virtual Work Principles. Variational Methods. Summary. Introduction to Composite Materials Basic Concepts and Terminology. Constitutive Equations of a Lamina. Transformation of Stresses and Strains. Plane Stress Constitutive Relations. Classical and First-Order Theories of Laminated Composite Plates Introduction. An Overview of ESL Laminate Theories. The Classical Laminated Plate Theory. The First-Order Laminated Plate Theory. Stiffness Characteristics for Selected Laminates. One-Dimensional Analysis of Laminated Plates Introduction. Analysis of Laminated Beams Using CLPT. Analysis of Laminated Beams Using FSDT. Cylindrical Bending Using CLPT. Cylindrical Bending Using FSDT. Closing Remarks. Analysis of Specially Orthotropic Plates Using CLPT Introduction. Bending of Simply Supported Plates. Bending of Plates with Two Opposite Edges Simply Supported. Bending of Rectangular Plates with Various Boundary Conditions. Buckling of Simply Supported Plates Under Compressive Loads. Buckling of Rectangular Plates Under Inplane Shear Load. Vibration of Simply Supported Plates. Buckling and Vibration of Plates with Two Parallel Edges Simply Supported. Closure. Analytical Solutions of Rectangular Laminates Using CLPT Governing Equations in Terms of Displacements. Admissible Boundary Conditions for the Navier Solutions. Navier Solutions of Antisymmetric Cross-Ply Laminates. The Navier Solutions of Antisymmetric Angle-Ply Laminates. The LTvy Solutions. Analysis of Midplane Symmetric Laminates. Transient Analysis. Summary. Analytical Solutions of Rectangular Laminates Using FSDT Introduction. Simply Supported Antisymmetric Cross-Ply Laminates. Simply Supported Antisymmetric Angle-Ply Laminates. Antisymmetric Cross-Ply Laminates with Two Opposite Edges Simply Supported. Antisymmetric Angle-Ply Laminates with Two Opposite Edges Simply Supported. Transient Solutions. Summary. Finite Element Analysis of Composite Laminates Introduction. Laminated Beams and Plate Strips by CLPT. Timoshenko Beam/Plate Theory. Numerical Results for Beams and Plate Strips. Finite Element Models of Laminated Plates (CLPT). Finite Element Models of Laminated Plates (FSDT). Summary. Refined Theories of Laminated Composite Plates Introduction. A Third-Order Plate Theory. Higher-Order Laminate Stiffness Characteristics. The Navier Solutions. LTvy Solutions of Cross-Ply Laminates. Displacement Finite Element Model. Layerwise Theories and Variable Kinematic Models In troduction. Development of the Theory. Finite Element Model. Variable Kinematic Formulations. Nonlinear Analysis of Composite Laminates Introduction. Nonlinear Stiffness Coefficients. Solution Methods for Nonlinear Algebraic Equations. Computational Aspects and Numerical Examples. Closure. Index Most chapters include Exercises and References for Additional Reading

1,331 citations

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TL;DR: In this paper, the classical configuration space of a system of identical particles is examined and the effect of particle spin in the present formalism is discussed. But this is only the case in which the particles move in three- or higher-dimensional space.

Abstract: The classical configuration space of a system of identical particles is examined. Due to the identification of points which are related by permutations of particle indices, it is essentially different, globally, from the Cartesian product of the one-particle spaces. This fact is explicity taken into account in a quantization of the theory. As a consequence, no symmetry constraints on the wave functions and the observables need to be postulated. The two possibilities, corresponding to symmetric and antisymmetric wave functions, appear in a natural way in the formalism. But this is only the case in which the particles move in three- or higher-dimensional space. In one and two dimensions a continuum of possible intermediate cases connects the boson and fermion cases. The effect of particle spin in the present formalism is discussed.

1,028 citations

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TL;DR: In this article, the full development and analysis of four models for the transversely vibrating uniform beam are presented, including the Euler-Bernoulli, Rayleigh, shear and Timoshenko models.

Abstract: In this paper, the full development and analysis of four models for the transversely vibrating uniform beam are presented. The four theories are the Euler–Bernoulli, Rayleigh, shear and Timoshenko. First, a brief history of the development of each beam model is presented. Second, the equation of motion for each model, and the expressions for boundary conditions are obtained using Hamilton's variational principle. Third, the frequency equations are obtained for four sets of end conditions: free–free, clamped–clamped, hinged–hinged and clamped–free. The roots of the frequency equations are presented in terms of normalized wave numbers. The normalized wave numbers for the other six sets of end conditions are obtained using the analysis of symmetric and antisymmetric modes. Fourth, the orthogonality conditions of the eigenfunctions or mode shape and the procedure to obtain the forced response using the method of eigenfunction expansion is presented. Finally, a numerical example is shown for a non-slender beam to signify the differences among the four beam models.

742 citations

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TL;DR: In this article, the Green-Schwarz mechanism can take a generalized form, resulting from the combined action of all the antisymmetric tensors, and some of their implications for the low-energy effective field theory are pointed out.

Abstract: An interesting feature of some open superstring models in D < 10 is the simultaneous presence, in the spectrum, of gauge fields and of a number of antisymmetric tensor fields. In these cases the Green-Schwarz mechanism can (and does) take a generalized form, resulting from the combined action of all the antisymmetric tensors. These novelties are illustrated referring to some simple rational models in six dimensions, and some of their implications for the low-energy effective field theory are pointed out

605 citations