comparison of cross flow and dynamic filtrations
new topics in food engineering (food science and technology)
... the end use of meat, and to understand the nutrient profile and energy content of cooked food products in various food outlets Standard methods for assessing the nutritional profile of food materials ... structure in chocolate and is caused by a variety of factors, including poor tempering of the chocolate, incorrect cooling methods and the presence of soft fats in the centres of chocolates Warm ... D and Stapley AGF, The effect of shear rate, temperature, sugar and emulsifier on the tempering of cocoa butter J Food Eng 77:936–942 (2006) [39] Briggs JL and Wang T, Influence of shearing and...
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Intro to differential geometry and general relativity s waner
... parameterizations of the ellipsoid, and use the identity cosh2x - sinh2x = For the double cone, use y3 = cx1, and x1 as a factor of y1 and y2.] y12 y22 y32 (a) Hyperboloid of One Sheet: + - = ... and South poles excluded) (International Dateline excluded) which is an open subset U of the sphere (Think of it as the surface of the earth with the Greenwich Meridian removed.) We call x1 and ... Definition 1.10 A chart of a surface S is a pair of functions x = (x1(y1, y2, y3), x2(y1, y2, y3)) which specify each of the local coordinates (parameters) x1 and x2 as smooth functions of a general point...
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Báo cáo hóa học: "Research Article Bessel’s Differential Equation and Its Hyers-Ulam Stability" docx
... Proceedings of the National Academy of Sciences of the United States of America, vol 27, pp 222–224, 1941 [3] Th M Rassias, “On the stability of the linear mapping in Banach spaces,” Proceedings of the ... xm for x ∈ (−ρ1 ,ρ1 ), where m yh is some Bessel function and cm ’s are given by (2.2) Then by Lemmas and and their proofs (replace M1 and M3 with Kε in Lemma 1), ∞ cm xm ≤ Cε y(x) − yh (x) = ... Hyers-Ulam stability of various linear differential equations of first order (see [15–18]) and further investigated the general solution of the inhomogeneous Legendre differential equation and its Hyers-Ulam...
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Intro to Differential Geometry and General Relativity - S. Warner Episode 4 ppsx
... coordinates of m By Example 4.3(c), ∂x ∂x we can arrange for vector fields X and Y such that if i = h i X (coordinates of m) = otherwise , and if i = k i Y (coordinates of m) = ... that, if M and N are tensors of type (1, 1), then: i p (a) Mj Nq is a tensor of type (2, 2) (a) (b) (e) ∂2xl ∂xi∂xj i j (b) Mj Nq is a tensor of type (1, 1) i j (c) Mj Ni is a tensor of type (0, ... components of a tensor; and (b) are skew-symmetric in all pairs in indices (c) How many independent components does Brst have? Cross Product (a) If X and Y are contravariant vectors, then their cross- product...
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Intro to Differential Geometry and General Relativity - S. Warner Episode 5 pdf
... 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 ... where c is the speed of light, proper time for reasons we shall see below.) Proof Inverting s(t) requires s'(t) ≠ But, by the Fundamental theorem of Calculus and the definition of L(a, t), ds ... doing in place of -1 in the metric? Answer Since physical units of time are (usually) not the same as physical units of space, we would like to convert the units of x4 (the units of time) to match...
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Intro to Differential Geometry and General Relativity - S. Warner Episode 6 pdf
... the standard inner product), then D has its columns satisfying if i ≠ j 0 if i = j = [Hint: use the given property of D to write down the entries of its inverse P in terms of the entries of D.] ... pq = gir [pq, r] = gir[ gqr,p + grp,q - gpq,r ] 58 Christoffel Symbols of the First Kind Christoffel Symbols of the Second Kind Neither of these gizmos are tensors, but instead transform as follows ... in 3-space.) There is no loss of generality in taking a = 1, so let us use “1, 0, 0, -∫/c‘ Here, c is the speed of light, and ∫ is a certain constant (The meaning of ∫ will emerge in due course)...
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Intro to Differential Geometry and General Relativity - S. Warner Episode 7 pptx
... expression of the form: ‡ This is my own version of the proof There is a version in Bernard Schutz's book, but the proof there seems overly complicated and also has some gaps relating to consistncy of ... 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 tensor with ... paramaterization) is normal to the curve, and its magnitude is a measure of how fast the curve is “turning,” and so we call the derivative i of T the curvature of C If C happens to be on a manifold,...
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Intro to Differential Geometry and General Relativity - S. Warner Episode 8 ppsx
... c ≠ 1) We think of V(4) as the unit vector in the direction of time, and V(1), V(2) and V(3) as the spatial basis vectors Using parallel translation, we obtain a similar set of vectors at each ... definition of C1 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 ... the first of the above (boxed) formulas into the second, and using symmetry of the second derivatives and the metric tensor, we find (exercise set) Covariant Curvature Tensor in Terms of the Metric...
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Intro to Differential Geometry and General Relativity - S. Warner Episode 10 docx
... each of the three components of momentum instead of energy now gives us the law of conservation of momentum (3 coordinates): Ta1,1 + Ta2,2 + Ta3,3 + Ta4,4 = for a = 1, 2, Combining all of these and ... (force = rate of change of momentum) and we interpret force per unit area as pressure What about the fourth coordinate? The 4th coordinate of the 4-momentum is the energy A component of the form ... M; what's straight in the eyes of one chart is curved in the eyes of another “Geodesic,” on the other hand, does make sense, since it is independent of the choice of coordinates If we can write...
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Intro to Differential Geometry and General Relativity - S. Warner Episode 11 ppt
... 102 (Here and in what follows, we are ignoring terms of order O(˙2).) Substituting this information in (II), and using the fact that i P ,4 = ∂ i (mov ), ∂x the time-rate of change of momentum, ... and Antigravity Newton's law of gravity says that a particle of mass M exerts a force on another particle of mass m according to the formula F =- r GMmr , r r where r = “x, y, z‘, r = |r |, and ... combination of 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:...
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Intro to Differential Geometry and General Relativity - S. Warner Episode 12 pps
... 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 ... 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 = 2M Case (Extremely Dense Stars) Radius of the star, ... version and write out the Christoffel symbols: dP m0 s = ¶risPrPi d† dP m0 s = gik(grk,s + gks,r - gsr,k) PrPi d† dP m0 s = (grk,s + gks,r - gsr,k) PrPk d† But the sum of the second and third...
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Intro to Differential Geometry and General Relativity - S. Warner Episode 13 pptx
... because of the high temperatures One can think of the star as a gas of positive charge atomic nuclei and negative charge electrons Each metal nucleus is a few thousand times heavier than the set of ... This change reduces the radius R0 of the neutron star by a factor of ‡2000 (the ratio of mn to me) and R0 ‡ 10 km One of these would comfortably fit on Long Island but would produce somewhat disruptive ... known laws of physics are capable of interrupting the collapse of a neutron star In a sense, the laws of physics leave the door open for the formation of stellar black holes References and Suggested...
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Bài giảng Chapter 17 Free Energy and Thermodynamics
... exothermic and more random ∆H is negative and large and ∆S is negative but small ∆H is positive but small and ∆S is positive and large or high temperature ∆G will be positive when ∆H is + and ∆S ... Memorize the Laws of Thermodynamics • You can break a California State Law but you can NOT break a Thermodynamic Law Chemistry, Julia Burdge, 2nd e., McGraw Hill First Law of Thermodynamics • you ... enthalpy and the entropy • The enthalpy is a comparison of the bond energy of the reactants to the products bond energy = amount needed to break a bond ∆H • The entropy factors relates to the randomness/orderliness...
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Tài liệu Makingbiodiesel joshua tickell free energy biodiesel pptx
... be started and cooled down on diesel or biodiesel fuel This also requires the use of an extra fuel tank and a valve to switch between the tank of diesel or biodiesel fuel and the tank of vegetable ... methanol, and lye If you are using new vegetable oil, always use 3.5 grams of lye per liter of oil Since each batch of used cooking oil is different, the amount of lye in each batch of biodiesel ... amount of lye, make a small test batch of biodiesel in a blender before attempting a reaction in a large mixing tank For the test batch, use 100 86 milliliters of vegetable oil and 20 milliliters of...
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Integrating U.S. Climate, Energy, and Transportation Policies potx
... Attapit, formerly of RAND’s Office of Congressional Relations, and Kate O’Neal of RAND’s Development Office for their assistance in contacting congressional staff and representatives of the business ... U.S Department of Energy, Of ce of Energy Efficiency and Renewable Energy Eric Smith U.S Environmental Protection Agency Eleen Trang Of ce of Representative Zoe Lofgren, U.S House of Representatives ... The mission of RAND Infrastructure, Safety, and Environment is to improve the development, operation, use, and protection of society’s essential physical assets and natural resources and to enhance...
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Partial Differential Equations and Fluid Mechanics doc
... soundness of an estimate of the number of degrees of freedom of a given flow (expressed by an estimate of its global attractor dimension) depends directly on the quality of the estimate of and a good ... L, < x2 < h(x1 )} ¯ ¯ ¯ and ∂Ω = Γ0 ∪ ΓL ∪ Γ1 , where Γ0 and Γ1 are the bottom and the top, and ΓL is the lateral part of the boundary of Ω We are interested in solutions of (1.1)–(1.2) in Ω that ... helps us to understand the influence of geometry of the flow and roughness of the boundary (as measured by h ) on the behaviour of the fluid 1.4 Time-dependent driving: dimension of the pullback attractor...
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Báo cáo khoa học: Insight into the phosphodiesterase mechanism from combined QM ⁄ MM free energy simulations pot
... specic (PDE 4, and 8), (b) cGMP specic (PDE 5, and 9) and (c) dual specicity both for cAMP and cGMP hydrolysis (PDE 1, 2, 3, 10 and 11) Although the structure of a small fragment of PDE4D was reported ... shows the changes of some of the geometries as a function of the MFEP coordinates At the transition state, the distances of rPO3Â and rOhP, the breaking and forming bonds, are 1.8 and 1.9 0.1 A, ... inversion of conguration is reected by the positive values of u1 and u2 (Fig 5B and Table 1) Although the O3Â atom of the ribosyl ring of AMP is quite far away from the phosphorus and the phosphorus...
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Báo cáo "On the asymptotic behavior of delay differential equations and its relationship with C0 - semigoup " potx
... Eq.(13) (see [7, 11, 12]) Applying the process of argument of parts 2.1 and 2.2 to the uniformlly stable and asymptotic equivalence of above model of population, we will give the following results ... Asymptotic equivalence of systems of differential equations, Results of mathematic science 40 (1985) 245 (Russian) [4] M Svec, Itegral and asymptotic equivelence of two systems of diffrential equations, ... solution x(t) ≡ of Eq.(4) is uniformly stable (2) If limt→∞ T (t) = 0, then the null solution x(t) ≡ of Eq.(4) is uniformly exponential stable Proof Throughout this paper in proof of theorems, we...
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Alice’s Adventures in WonderlandBy Lewis Caroll (1865)Download free eBooks of classic literature, books and novels at Planet eBook. Subscribe to our free eBooks blog and email newsletter..All in the Golden AfternoonAll in the golden afternoon Full ppt
... capital of Paris, and Paris is the capital of Rome, and Rome—no, that’s all wrong, I’m certain! I must have been changed for Mabel! I’ll try and say ‘How doth the little—‘ and she crossed her hands ... brown hair! And it’ll fetch things when you throw them, and it’ll sit up and beg for its dinner, and all sorts of things—I can’t remember half of them and it belongs to a farmer, you know, and he ... longed to get out of that dark hall, and wander about among those beds of bright flowers and those cool fountains, but she could not even get her head though the doorway; and even if my head...
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introduction to differential geometry and general relativity
... to parameterizations of the ellipsoid, and use the identity cosh2x - sinh2x = For the double cone, use y3 = cx1, and x1 as a factor of y1 and y2.] y2 y2 y2 (a) Hyperboloid of One Sheet: 12 + 22 ... Basic Premises of General Relativity 14 The Einstein Field Equations and Derivation of Newton's Law 15 The Schwarzschild Metric and Event Horizons 16 White Dwarfs, Neutron Stars and Black Holes, ... the portion of the sphere given by 0
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