pdes differential geometry and robust statistics

... Equation (1) implies that x and x– lie on the same ray from the origin, and x–i xi = ; r– r– and squaring and adding gives 12 r– That is, project it to the circle, and invert the distance from ... fields X and Y a SMOOTH scalar (field) “X, Y‘, satisfying the following properties Symmetry: “X, Y‘ = “Y, X‘ for all X and Y, Bilinearity: “åX, ∫Y‘ = å∫“X, Y‘ for all X and Y, and scalars å and ∫ ... Equations and Derivation of Newton's Law 114 15 The Schwarzschild Metric and Event Horizons 124 16 White Dwarfs, Neutron Stars and Black Holes, by Gregory C Levine131 References and Further...

Ngày tải lên: 17/03/2014, 14:28

... fields X and Y a scalar (field) “X, Y‘, satisfying the following properties Symmetry: “X, Y‘ = “Y, X‘ for all X and Y, Bilinearity: “åX, ∫Y‘ = å∫“X, Y‘ for all X and Y, and scalars å and ∫ “X, ... Field Equations and Derivation of Newton's Law 15 The Schwarzschild Metric and Event Horizons 16 White Dwarfs, Neutron Stars and Black Holes, by Gregory C Levine Preliminaries Distance and Open Sets ... all—see any text on differential topology or differential geometry (or look at Rund's appendix) i More terminology: We shall sometimes refer to the x as the local coordinates, and to the yj as the...

Ngày tải lên: 27/03/2014, 11:52

... Equation (1) implies that x and x– lie on the same ray from the origin, and x–i xi = ; r– r– and squaring and adding gives 12 r– That is, project it to the circle, and invert the distance from ... fields X and Y a SMOOTH scalar (field) “X, Y‘, satisfying the following properties Symmetry: “X, Y‘ = “Y, X‘ for all X and Y, Bilinearity: “åX, ∫Y‘ = å∫“X, Y‘ for all X and Y, and scalars å and ∫ ... Equations and Derivation of Newton's Law 114 15 The Schwarzschild Metric and Event Horizons 124 16 White Dwarfs, Neutron Stars and Black Holes, by Gregory C Levine131 References and Further...

Ngày tải lên: 24/04/2014, 17:08

... fields X and Y a scalar (field) “X, Y‘, satisfying the following properties Symmetry: “X, Y‘ = “Y, X‘ for all X and Y, Bilinearity: “åX, ∫Y‘ = å∫“X, Y‘ for all X and Y, and scalars å and ∫ “X, ... be contravariant, and let C and D be covariant Then: ∂x–i k ∂x–j l ∂x–i ∂x–j k l i j W = k lV W, V— W— = k V ∂x ∂xl ∂x ∂x and similarly, ∂x–i ∂xl k i V— C—j = k j V Cl , ∂x ∂x– and ∂xk ∂xl C—iD—j ... presently Definition 4.9 If V and W are contravariant (or covariant) vector fields on M, and if å is a real number, we can define new fields V+W and åV by (V + W)i = Vi + Wi and (åV)i = åVi It is easily...

Ngày tải lên: 12/08/2014, 16:20

... path, and define a new function s (arc length) by s(t) = L(a, t) = length of path from t = a to t Then s is an invertible function of t, and, using s as a parameter, ||dxi/ds||2 is constant, and ... some coordinate neighborhood, and define the length of S by b ⌠  L(a, b) =  ⌡ a 42 dxidxj ± gij dt, dt dt where the sign ±1 is chosen as +1 if the curve is space-like and -1 if it is time-like ... scalar field, so is ||X|| is a scalar field, if it exists, and satisfies ||˙X|| = |˙|·||X|| for every contravariant vector field X and every scalar field ˙ The expected inequality ||X + Y|| ≤...

Ngày tải lên: 12/08/2014, 16:20

... ||V—|| = On the other hand, we can also use G— to compuate ||V—||2, and we get ||V—||2 = , k showing that k = Hence, G— = G, and also D has the desired form This proves (b) (and also (c), by the ... multiplying…) Deduce that x–i(t) = zi + rit for some constants ri and si Finally, set t = and t = to conclude that x–i(t) = x–0i + ∆x–it, and apply (c) above.] 55 If the x–i-system is moving with a ... Lorentz Transformations of Special Relativity If two Lorentz frames x and x– have the same coordinates at (x, y, z, t) = (0, 0, 0, 0), and if the x–-frame is moving in the x-direction with a speed of...

Ngày tải lên: 12/08/2014, 16:20

... that it is a tensor If x– and y– are any two oriented coordinate systems at m and change-of-coordinate matrices D and E with respect to some inertial frame x at m, and if the coordinates of the ... another, then det(D) = +1 E3 has two orientations: one given by any left-handed system, and the other given by any right-handed system In the homework, you will see that spheres are orientable, ... definition of the “contravariant” derivative, Xa|b of Xa with respect to xb, and show that Xa|b = if and only if Xa|b = Geodesics and Local Inertial Frames Let us now apply some of this theory to curves...

Ngày tải lên: 12/08/2014, 16:20

... above) j Warning: The integrand term ¶i Xi is not constant, and must be evaluated as a function of t using the path C1 However, if the path is a small one, then the integrand is approximately equal ... A…B…C has only one nonzero coordinate: the first (b) Show that, if we consider A, B and C as 3-vectors a , b and c respectively by ignoring their first (zero) coordinate, then b c (A…B…C)1 = a ... h     If Xj is parallel along C, which has parametrization with domain [a, b] and corresponding points å and ∫ on M, then, since j h dx dX j = -¶i h Xi dt dt ……… (I) ……… (II) we can integrate...

Ngày tải lên: 12/08/2014, 16:20

... (since the lengths of the vectors a, b and c are equal to the respective areas) If this force was not zero, then there would be a resultant force F on the prism, and hence an acceleration of its material ... is volume in Euclidean 4-space, and where we take the limit as ∆V’0 But now, generalizing to 4-space is forced on us: first replace momentum by the 4momentum P, and then, noting that n∆S∆x4 is ... above formula in differential form: V n P T (dV ) = T (n dV) = dP V This describes T as a function which converts the covariant vector dV into a contravariant P field (P ), and thus suggests...

Ngày tải lên: 12/08/2014, 16:20

... of the local geometry in space-time But we are getting ahead of ourselves Finally, we generalize the (second order differential) operator Ô to some yet-to-bedetermined second order differential ... gab,ij, gab,i and gab, and must also be symmetric (since T is) Examples of such a tensors are the Ricci tensors Rab, gabR, as well as gab Let us take a linear combination as our candidate: Rab ... gravitational force from a simple distortion of the metric In other words and this is what Einstein realized—gravity is nothing more than the geometry of spacetime; it is not a mysterious “force” at all Exercise...

Ngày tải lên: 12/08/2014, 16:20

... outside a star; that is, r > rs, we find that r > 2M as well, and so 1-2M/r is positive, and never zero (If r ≤ 2M, we are inside the star, and the Schwarzschild metric no longer applies.) r rs r = ... excellent differential geometry notes, wherein we saw that we can take E = for a particle starting at rest far from the black hole In other words,  dr   = 2M/r  d† Notice that this is constant and ... refer to the last formula in Section 10, and obtain Newton's Law of Gravity To how many kilograms does one unit of M correspond? 15 The Schwarzschild Metric and Event Horizons We saw that the metric...

Ngày tải lên: 12/08/2014, 16:20

... Press, 1986 David Lovelock and Hanno Rund, Tensors, Differential Forms, and Variational Principles (Dover, 1989) Charles E Weatherburn, An Introduction to Riemannian Geometry and the Tensor Calculus ... nearly massless and noninteracting) A sufficiently dense star is unstable against such an interaction and all electrons and protons are converted to neutrons leaving behind a chargeless and nonluminous ... collapse and its electrons will be pushed closer and closer to the nuclei At some point, a nuclear reaction begins to occur in which electrons 127 and protons combine to form neutrons (and neutrinos...

Ngày tải lên: 12/08/2014, 16:20

... Manifolds and Differentiable Maps 2.4 Pseudo-Groups and Models Spaces 2.5 Smooth Maps and Diffeomorphisms 2.6 Coverings and Discrete groups 2.6.1 Covering spaces and the ... being C r at a point and for that we have a modified version of the last definition: Let M and N be C r manifolds with corresponding maximal atlases AM and AN and modelled on Rn and Rd respectively ... x → gx and since x and gx are in the same orbit, we see that gU and U both project to same set p(U ) Now if x, y ∈ gU and p|gU (x) = p|gU (y) then y = hx for some h ∈ G But also x = ga (and y...

Ngày tải lên: 17/03/2014, 14:29

Ngày tải lên: 31/03/2014, 15:55

Ngày tải lên: 12/06/2014, 16:18

... DU, V H LINH, V MEHRMANN, AND D D THUAN a lot of attention; see, e.g., [5, 6, 12, 26, 29, 32, 37, 38] and [11] for a recent survey In contrast to this, the stability and robust stability analyses ... −1 G(λ0 )u = CH(λ0 )−1 −λ0 B1 and thus Δ = G(λ0 ) −1 Since e−λ0 τ B3 u = 0, B2 it follows that −λ0 B1 B2 e−λ0 τ B3 u = 0, and hence x = On the other hand, by (4.6) and (4.7) we have H(λ0 )x = ... 105–113 [23] D Hinrichsen and A J Pritchard, Mathematical Systems Theory I Modelling, State Space Analysis, Stability and Robustness, Springer-Verlag, New York, 2005 [24] G Hu and E J Davison, Real...

Ngày tải lên: 16/12/2017, 02:57

... connectors were uncoupled and the endface geometry was measured The average, maximum, and minimum permanent changes in fiber height are shown in Table Page Endface Geometry and Connector Reliability ... contact in the outside plant Endface Geometry and Connector Reliability in the Outside Plant Several industry standards exist that address APC connector endface geometry The purpose of these specifications ... potting will reduce fiber withdrawal and the possibility of air gaps Page Endface Geometry and Connector Reliability in the Outside Plant Ferrule Rotation and Apex Offset: We now examine the issue...

Ngày tải lên: 22/10/2013, 18:15

... dependency networks, and showed the best contemporary results Gim´ nez e and M` rquez (2004) used one-pass, left-to-right a and right-to-left combined tagging algorithm and achieved near state-of-the-art ... and showed current state-of-the-art results.3 Our individual models (generalized and domainspecific) are similar to Gim´ nez and M` rquez (2004) e a in that we use a subset of their features and ... al., 2003) and the SVMTool (Gim´ nez and M` rquez, e a 2004) Both systems are trained with the same training data and use configurations optimized for their best reported results Tables and show...

Ngày tải lên: 19/02/2014, 19:20

... “Mr and Mrs Smith”, “Mr And Mrs Smith”, and so on To deal with these aliasing forms, we transform all different forms into a standard form for the later ranking and identification The standard ... Yorker” and “紐約(pronounced as NiuYue)客(people, pronounced as Ke)” “橫濱” and “Yokohama”, “鈴木一朗” and “Ichiro Suzuki” “The Mask” and “ 摩 登 (modern) 大 (great) 聖 (saint)” “孫中山(Sun Zhong-Shan)” and “Sun ... retrieving candidates from returned snippets, combining both linguistic and statistical information to find the correct translation Our system can be split into three steps: candidate retrieving, candidate...

Ngày tải lên: 20/02/2014, 12:20