SUMMARY / RELATED TOPICS

Ellipsoid

An ellipsoid is a surface that may be obtained from a sphere by deforming it by means of directional scalings, or more of an affine transformation. An ellipsoid is a quadric surface. Among quadric surfaces, an ellipsoid is characterized by either of the two following properties; every planar cross section is empty, or is reduced to a single point. It is bounded. An ellipsoid has three pairwise perpendicular axes of symmetry which intersect at a center of symmetry, called the center of the ellipsoid; the line segments that are delimited on the axes of symmetry by the ellipsoid are called the principal axes, or axes of the ellipsoid. If the three axes have different lengths, the ellipsoid is said to be tri-axial or scalene, the axes are uniquely defined. If two of the axes have the same length the ellipsoid is an ellipsoid of revolution called a spheroid. In this case, the ellipsoid is invariant under a rotation around the third axis, there are thus infinitely many ways of choosing the two perpendicular axes of the same length.

If the third axis is shorter, the ellipsoid is an oblate spheroid. If the three axes have the same length, the ellipsoid is a sphere. Using a Cartesian coordinate system in which the origin is the center of the ellipsoid and the coordinate axes are axes of the ellipsoid, the implicit equation of the ellipsoid has the standard form x 2 a 2 + y 2 b 2 + z 2 c 2 = 1, where a, b, c are positive real numbers; the points, lie on the surface. The line segments from the origin to these points are called the principal semi-axes of the ellipsoid, because a, b, c are half the length of the principal axes, they correspond to semi-minor axis of an ellipse. If a = b > c, one has an oblate spheroid. The ellipsoid may be parameterized in several ways, which are simpler to express when the ellipsoid axes coincide with coordinate axes. A common choice is x = a sin ⁡ cos ⁡, y = b sin ⁡ sin ⁡, z = c cos ⁡, where 0 ≤ θ ≤ π, 0 ≤ φ < 2 π. These parameters may be interpreted as spherical coordinates, where θ is the polar angle and φ is the azimuth angle of the point of the ellipsoid.

The volume bounded by the ellipsoid is V = 4 3 π a b c. Alternatively expressed, where A, B and C are the lengths of the principal axes: V = π 6 A B C ≈ 0.523 A B C. Note that this equation reduces to that of the volume of a sphere when all three elliptic radii are equal, to that of an oblate or prolate spheroid when two of them are equal; the volume of an ellipsoid is 2 3 the volume of a circumscribed elliptic cylinder, π 6 the volume of the circumscribed box. The volumes of the inscribed and circumscribed boxes are respectively: V inscribed = 8 3 3 a b c, V circumscribed = 8 a b c; the surface area of a general ellipsoid is S = 2 π c 2 + 2 π a b sin ⁡ ( E sin 2 ⁡ + F

Alessandro Zamperini

Alessandro Zamperini is an Italian footballer, in the role of defender. Alessandro Zamperini started his footballing career at Serie A giants Lazio, the club he has supported as a child, before signing his first professional contract, at fierce rivals Roma moving to English First Division club Portsmouth appearing for them 26 times and scoring 2 goals against Gillingham and Crystal Palace. Zamperini moved back to Italy with Serie A team Modena, though he did not make an appearance for them, he had spells with several Serie C1 clubs, including Acireale and Ternana Serie B before moving to side Cisco Roma in 2007, Valle del Giovenco in 2008. In 2009, he moved abroad again, joined Latvian champions FK Ventspils, scoring a goal in the 2009–10 UEFA Europa League in a surprising 1–1 away draw against Sporting Clube de Portugal. On 2 August 2013 Zamperini was suspended for 2 years due to 2011 match-fixing scandal. Alessandro Zamperini at Soccerbase Alessandro Zamperini at Soccerway Alessandro Zamperini at TuttoCalciatori.net

Kruskal's tree theorem

In mathematics, Kruskal's tree theorem states that the set of finite trees over a well-quasi-ordered set of labels is itself well-quasi-ordered under homeomorphic embedding. The theorem was proved by Joseph Kruskal, it has since become a prominent example in reverse mathematics as a statement that cannot be proved within ATR0, a finitary application of the theorem gives the existence of the notoriously fast-growing TREE function. In 2004, the result was generalized from trees to graphs as the Robertson–Seymour theorem, a result that has proved important in reverse mathematics and leads to the even-faster-growing SSCG function. We give. All trees we consider are finite. Given a tree T with a root, given vertices v, w, call w a successor of v if the unique path from the root to w contains v, call w an immediate successor of v if additionally the path from to v to w contains no other vertex. Take X to be a ordered set. If T 1, T 2 are rooted trees with vertices labeled in X, we say that T 1 is inf-embeddable in T 2 and write T 1 ≤ T 2 if there is an injective map F from the vertices of T 1 to the vertices of T 2 such that For all vertices v of T 1, the label of v precedes the label of F, If w is any successor of v in T 1 F is a successor of F, If w 1, w 2 are any two distinct immediate successors of v the path from F to F in T 2 contains F. Kruskal's tree theorem states: If X is well-quasi-ordered the set of rooted trees with labels in X is well-quasi-ordered under the inf-embeddable order defined above.

Define tree, the weak tree function, as the length of the longest sequence of 1-labelled trees such that: The tree at position k in the sequence has no more than k + n vertices, for all k. No tree is homeomorphically embeddable into any tree following it in the sequence, it is known that tree = 2, tree = 5, tree > 844424930131960, but TREE is larger than treetreetreetreetree8. For a countable label set X, Kruskal's tree theorem can be expressed and proven using second-order arithmetic. However, like Goodstein's theorem or the Paris–Harrington theorem, some special cases and variants of the theorem can be expressed in subsystems of second-order arithmetic much weaker than the subsystems where they can be proved; this was first observed by Harvey Friedman in the early 1980s, an early success of the then-nascent field of reverse mathematics. In the case where the trees above are taken to be unlabeled, Friedman found that the result was unprovable in ATR0, thus giving the first example of a predicative result with a provably impredicative proof.

This case of the theorem is still provable in Π11-CA0, but by adding a "gap condition" to the definition of the order on trees above, he found a natural variation of the theorem unprovable in this system. Much the Robertson-Seymour theorem would give another theorem unprovable inside Π11-CA0. Ordinal analysis confirms the strength of Kruskal's theorem, with the proof-theoretic ordinal of the theorem equaling the small Veblen ordinal. Suppose that P is the statement: There is some m such that if T1... Tm is a finite sequence of unlabeled rooted trees where Tk has n+k vertices Ti ≤ Tj for some i < j. All t