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SAYRE EQUATION, TANGENT FORMULA AND SAYTAN
"... V the volume of a unit cell r realspace positional vector defined within a unit cell ρ (r) electron density function with r as argument h reciprocallattice vector, which corresponds to the diffraction index hkl Fh the structure factor with h as argument; the Fourier transform of ρ (r) Fsq h the Fo ..."
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V the volume of a unit cell r realspace positional vector defined within a unit cell ρ (r) electron density function with r as argument h reciprocallattice vector, which corresponds to the diffraction index hkl Fh the structure factor with h as argument; the Fourier transform of ρ (r) Fsq h the Fourier transform of ρ2(r) fj the scattering factor of the jth atom in the unit cell f sq j the scattering factor of the jth ‘squared atom ’ in the unit cell Fh the magnitude of Fh ϕh the phase of Fh Φ3 = ϕ − h+ϕ h’+ϕ h − h ’ , the threephase structure invariant Eh the normalised structure factor corresponding to Fh Eh the magnitude of Eh σn = Σj (Zj)n, Zj is the atomic number of the jth atom in the unit cell, n is an integer 2. Sayre’s equation The Sayre equation [1] is an exact equation linking structure factors. It holds under the following conditions: i) positivity; ii) atomicity; iii) equalatom structure. Given a crystal structure represented by ρ (r), we can construct a ‘squared structure’ expressed as 2ρ2(r) = ρ (r) × ρ (r) . (1) According to the convolution theorem, the Fourier transform of (1) yields F F Fh h h h h sq V
GIAMBELLITYPE FORMULA FOR SUBBUNDLES OF THE TANGENT BUNDLE
, 1996
"... Abstract. Consider a generic ndimensional subbundle V of the tangent bundle TM on some given manifold M. Given V one can define different degeneracy loci Σr(V), r = (r1 ≤ r2 ≤ r3 ≤ · · · ≤ rk) on M consisting of all points x ∈ M for which the dimension of the subspace V j (x) ⊂ TM(x) spanned b ..."
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by all length ≤ j commutators of vector fields tangent to V at x is less than or equal to rj. Under a certain transversality assumption we ’explicitly ’ calculate the Z2cohomology classes of M dual to Σr(V) using determinantal formulas due to W. Fulton and the expression for the Chern classes
Tangent Line and Tangent Plane Approximations of Definite Integrals
"... Abstract. Oftentimes, it becomes necessary to find approximate values for definite integrals, since the majority cannot be solved through direct computation. The methods of tangent line and tangent plane approximation can be derived as methods of integral approximation in two and threedimensional ..."
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Abstract. Oftentimes, it becomes necessary to find approximate values for definite integrals, since the majority cannot be solved through direct computation. The methods of tangent line and tangent plane approximation can be derived as methods of integral approximation in two and three
ON THE EVALUATION OF THE ELASTOPLASTIC TANGENT STIFFNESS
"... Summary. A new method for the evaluation of the elastoplastic algorithmic tangent stiffness is presented. The formula is based on a simple geometric argument and does not involve matrix inversions. For regular yield surfaces we provide an explicit expression which is sufficiently approximate and com ..."
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Summary. A new method for the evaluation of the elastoplastic algorithmic tangent stiffness is presented. The formula is based on a simple geometric argument and does not involve matrix inversions. For regular yield surfaces we provide an explicit expression which is sufficiently approximate
1.2 Tangent Vectors and Tangent Spaces......................... 3
, 2005
"... It has been realised for several decades now, probably since Efron’s paper introducing the concept of statistical curvature [Efr75], that most of the main concepts and methods of differential geometry are of substantial interest in connection with the theory of statistical inference. This report de ..."
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It has been realised for several decades now, probably since Efron’s paper introducing the concept of statistical curvature [Efr75], that most of the main concepts and methods of differential geometry are of substantial interest in connection with the theory of statistical inference. This report describes in simple cases the links existing between the two theories. It is based on an article introducing the topic, by R. Kass [Kas89]. The focus is on parametric statistical
The method of sweeping tangents
"... Sweeping tangents What is the area of the shaded region between the tyre tracks of a moving bicycle such as that depicted in Figure 1? If the tracks are specified, and equations for them are known, the area can be calculated using integral calculus. Surprisingly, the area ..."
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Sweeping tangents What is the area of the shaded region between the tyre tracks of a moving bicycle such as that depicted in Figure 1? If the tracks are specified, and equations for them are known, the area can be calculated using integral calculus. Surprisingly, the area
ON RIEMANNIAN TANGENT BUNDLES
"... Abstract. We study the geometry of manifolds whose tangent bundle is endowed with a Riemannian metric. The LeviCivita connection, SchoutenVan Kampen connection and Vrănceanu connection are the main tools for this study. We obtain characterizations of special classes of vertical foliations and co ..."
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Abstract. We study the geometry of manifolds whose tangent bundle is endowed with a Riemannian metric. The LeviCivita connection, SchoutenVan Kampen connection and Vrănceanu connection are the main tools for this study. We obtain characterizations of special classes of vertical foliations
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