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In physics, angular velocity refers to how fast an object rotates or revolves relative to another point, i.e. how fast the angular position or orientation of an object changes with time. There are two types of angular velocity: spin angular velocity. Spin angular velocity refers to how fast a rigid body rotates with respect to its centre of rotation. Orbital angular velocity refers to how fast a point object revolves about a fixed origin, i.e. the time rate of change of its angular position relative to the origin. Spin angular velocity is independent of the choice of origin, in contrast to orbital angular velocity which depends on the choice of origin. In general, angular velocity is measured in angle per e.g. radians per second. The SI unit of angular velocity is expressed as radians/sec with the radian having a dimensionless value of unity, thus the SI units of angular velocity are listed as 1/sec. Angular velocity is represented by the symbol omega. By convention, positive angular velocity indicates counter-clockwise rotation, while negative is clockwise.

For example, a geostationary satellite completes one orbit per day above the equator, or 360 degrees per 24 hours, has angular velocity ω = 360 / 24 = 15 degrees per hour, or 2π / 24 ≈ 0.26 radians per hour. If angle is measured in radians, the linear velocity is the radius times the angular velocity, v = r ω. With orbital radius 42,000 km from the earth's center, the satellite's speed through space is thus v = 42,000 × 0.26 ≈ 11,000 km/hr. The angular velocity is positive since the satellite travels eastward with the Earth's rotation In three dimensions, angular velocity is a pseudovector, with its magnitude measuring the rate at which an object rotates or revolves, its direction pointing perpendicular to the instantaneous plane of rotation or angular displacement; the orientation of angular velocity is conventionally specified by the right-hand rule. In the simplest case of circular motion at radius r, with position given by the angular displacement ϕ from the x-axis, the orbital angular velocity is the rate of change of angle with respect to time: ω = d ϕ d t.

If ϕ is measured in radians, the distance from the x-axis around the circle to the particle is ℓ = r ϕ, the linear velocity is v = d ℓ d t = r ω, so that ω = v r. In the general case of a particle moving in the plane, the orbital angular velocity is the rate at which the position vector relative to a chosen origin "sweeps out" angle; the diagram shows the position vector r from the origin O to a particle P, with its polar coordinates. The particle has linear velocity splitting as v = v ‖ + v ⊥, with the radial component v ‖ parallel to the radius, the cross-radial component v ⊥ perpendicular to the radius; when there is no radial component, the particle moves around the origin in a circle. Since radial motion leaves the angle unchanged, only the cross-radial component of linear velocity contributes to angular velocity; the angular velocity ω is the rate of change of angular position with respect to time, which can be computed from the cross-radial velocity as: ω = d ϕ d t = v ⊥ r. Here the cross-radial speed v ⊥ is the signed magnitude of v ⊥, positive for counter-clockwise motion, negative for clockwise.

Taking polar coordinates for the linear velocity v gives magnitude v and angle θ relative to the radius vector. These formulas may be derived from r =, v

Virginia Ann Patton is an American retired businesswoman and former actress. After appearing in several films in the early 1940s, she was cast in her most well-known role as Ruth Dakin Bailey in Frank Capra's It's a Wonderful Life. In 1949, Patton retired with her final film credit being The Lucky Stiff. Patton was born in Ohio to Marie and Donald Patton, she was raised in her father's hometown of Portland, where her family relocated when she was an infant. She is a niece of General George S. Patton. Patton graduated from Jefferson High School in Portland, relocated to Los Angeles, where she attended the University of Southern California. While a student at the University of Southern California, Patton began to audition for acting parts, she collaborated in plays with screenwriter William C. deMille while in college. She had several insignificant film appearances before being cast in Capra's It's a Wonderful Life as Ruth Dakin Bailey, the wife of George Bailey's younger brother Harry. Although Capra did not know Patton she read the role for him and he signed her to a contract.

Patton said that she was the only girl the famous director signed in his entire career. Patton still gives interviews about It's a Wonderful Life and is today the last surviving credited member of the adult actors in the film. Patton made only four films after It's a Wonderful Life, including her first lead in the B-Western Black Eagle, she appeared in the drama The Burning Cross, a film about a World War II veteran who becomes embroiled with the Ku Klux Klan upon returning to his hometown. Patton was married to Cruse W. Moss from 1949 until his death in 2018, she gave up acting in the late 1940s to concentrate on raising a family with her husband in Ann Arbor, Michigan. She attended the University of Michigan. Berry, S. Torriano. Historical Dictionary of African American Cinema. Scarecrow Press. ISBN 978-0-81085-545-8. Virginia Patton on IMDb Interview with Virginia Patton in 2013

Macphersonite, Pb42 2, is a carbonate mineral, trimorphous with leadhillite and susannite. Macphersonite is white, colorless, or a pale amber in color and has a white streak, it crystallizes in the orthorhombic system with a space group of Pcab. It is soft mineral that has a high specific gravity. Macphersonite is named after Harry Gordon Macpherson, a keeper of minerals at the Royal Scottish Museum, it was discovered and accepted in 1984. The structure of macphersonite is represented as a sequence of three layers stacked along the; the first layer is a sulfate tetrahedra, the second is of lead and hydroxide, the third is a layer composed of lead and carbonate. Stacking of the three layers can be detailed as... BABCCBABCC... similar to leadhillite. Two C layers of lead carbonate in the BAB stacking provide a weak connection that leads to the perfect cleavage; the Leadhills macphersonite is a pale amber to colorless in color, while the Argentolle mine macphersonite is colorless to white. It has a luster of adamantine on fresh surfaces and elsewhere it is resinous.

Macphersonite is soft with a 2.5-3 on the Mohs hardness, has an uneven fracture with a high density of 6.5g/cm3. Macphersonite has a strong yellow fluorescence under both long and short wave, ultraviolet is displayed by the Leadhills specimens, the Argentolle material does not fluoresce. Macphersonite is found in the Leadhills region of southwest Scotland and in the Saint-Prix, Saône-et-Loire region of France, it is the rarest of the three polymorphs. It occurs in lead deposits associated with cerussite, caledonite, leadhillite and pyromorphite

Black Milk is the fourth album by Gallon Drunk, released in 1999 through Self Distribuzione. The album served as the soundtrack for the Greek film of the same name. Gallon DrunkJeremy Cottinghambass guitar Terry Edwardssaxophone, keyboards on "Hurricane" James Johnstonvocals, banjo, bass guitar, organ, Moog synthesizer, harmonica Ian White – drumsProduction and additional personnelGallon Drunk – engineering, recording Pauline Michailidis – backing vocals on "Theme from Black Milk", "Every Second of Time" and "At My Side" Tony Papamichael – engineering, recording

Calthorpes' House is a heritage-listed house located at 24 Mugga Way, Red Hill, Australian Capital Territory. It was built in 1927 for Dell Calthorpe. Harry Calthorpe was a partner in Calthorpe and Woodger, a successful Queanbeyan-based stock and station and real estate agency; the house was designed by Ken Oliphant of Oakley and Parkes, the architectural firm responsible for the residence of the Prime Minister of Australia, The Lodge. Sir Charles Rosenthal had been selected to design the house. Harry Calthorpe died at age 59 in 1950, Dell Calthorpe remained in the house until her death in 1979; the house and its contents had remained unchanged for many years, both the Calthorpe family and historians recognised its historical value and potential for a house museum. It was purchased by the Government of Australia in late 1984, was opened as a museum in 1986, it continues to operate as a museum today. Calthorpes' House is a snapshot of domestic life for a family of four in a late 1920s three-bedroom Canberra house, however it is not representative, given the larger than average house and garden, there have been some minor changes and additions over time.

There is an air raid shelter behind the house, still preserved, was built in response to the risk of bombing of Canberra during the war. It was added to the Australian Capital Territory Heritage Register on 27 September 1996 and to the former Register of the National Estate on 25 March 1986

In mathematics, the main conjecture of Iwasawa theory is a deep relationship between p-adic L-functions and ideal class groups of cyclotomic fields, proved by Kenkichi Iwasawa for primes satisfying the Kummer–Vandiver conjecture and proved for all primes by Mazur and Wiles. The Herbrand–Ribet theorem and the Gras conjecture are both easy consequences of the main conjecture. There are several generalizations of the main conjecture, to real fields, CM fields, elliptic curves, so on. Iwasawa was motivated by an analogy with Weil's description of the zeta function of an algebraic curve over a finite field in terms of eigenvalues of the Frobenius endomorphism on its Jacobian. In this analogy, The action of the Frobenius corresponds to the action of the group Γ; the Jacobian of a curve corresponds to a module X over Γ defined in terms of ideal class groups. The zeta function of a curve over a finite field corresponds to a p-adic L-function. Weil's theorem relating the eigenvalues of Frobenius to the zeros of the zeta function of the curve corresponds to Iwasawa's main conjecture relating the action of the Iwasawa algebra on X to zeros of the p-adic zeta function.

The main conjecture of Iwasawa theory was formulated as an assertion that two methods of defining p-adic L-functions should coincide, as far as, well-defined. This was proved by Mazur & Wiles for Q, for all real number fields by Wiles; these proofs were modeled upon Ken Ribet's proof of the converse to Herbrand's theorem. Karl Rubin found a more elementary proof of the Mazur–Wiles theorem by using Thaine's method and Kolyvagin's Euler systems, described in Lang and Washington, proved other generalizations of the main conjecture for imaginary quadratic fields. In 2014, Christopher Skinner and Eric Urban proved several cases of the main conjectures for a large class of modular forms; as a consequence, for a modular elliptic curve over the rational numbers, they prove that the vanishing of the Hasse–Weil L-function L of E at s = 1 implies that the p-adic Selmer group of E is infinite. Combined with theorems of Gross-Zagier and Kolyvagin, this gave a conditional proof of the conjecture that E has infinitely many rational points if and only if L = 0, a form of the Birch–Swinnerton-Dyer conjecture.

These results were used to prove that a positive proportion of elliptic curves satisfy the Birch–Swinnerton-Dyer conjecture. P is a prime number. Fn is the field Q where ζ is a root of unity of order pn+1. Γ is the largest subgroup of the absolute Galois group of F∞ isomorphic to the p-adic integers. Γ is a topological generator of Γ Ln. Hn is the Galois group Gal, isomorphic to the subgroup of elements of the ideal class group of Fn whose order is a power of p. H∞ is the inverse limit of the Galois groups Hn. V is the vector space H∞⊗ZpQp. Ω is the Teichmüller character. Vi is the ωi eigenspace of V. hp is the characteristic polynomial of γ acting on the vector space Vi Lp is the p-adic L function with Lp = –Bk/k, where B is a generalized Bernoulli number. U is the unique p-adic number satisfying γ = ζu for all p-power roots of unity ζ Gp is the power series with Gp = LpThe main conjecture of Iwasawa theory proved by Mazur and Wiles states that if i is an odd integer not congruent to 1 mod p–1 the ideals of Zp generated by hp and Gp are equal.

The proof of Iwasawa's main conjecture by Mazur-Wiles was improved by Rubin using Kolyvagin's Euler Systems, see entry for Iwasawa theory. Coates, John. New York, pp. 203–208, MR 0255510 Iwasawa, Kenkichi, "On p-adic L-functions", Annals of Mathematics, Second Series, 89: 198–205, doi:10.2307/1970817, ISSN 0003-486X, JSTOR 1970817, MR 0269627 Manin, Yu. I.. Introduction to Modern Number Theory, Encyclopaedia of Mathematical Sciences, 49, ISBN 978-3-540-20364-3, ISSN 0938-0396, Zbl 1079.11002 Mazur, Barry.