Borromean rings Mathematical properties[edit] Although the typical picture of the Borromean rings (above right picture) may lead one to think the link can be formed from geometrically ideal circles, they cannot be. Freedman and Skora (1987) prove that a certain class of links, including the Borromean links, cannot be exactly circular. Alternatively, this can be seen from considering the link diagram: if one assumes that circles 1 and 2 touch at their two crossing points, then they either lie in a plane or a sphere. A realization of the Borromean rings as ellipses 3D image of Borromean Rings Linking[edit] In knot theory, the Borromean rings are a simple example of a Brunnian link: although each pair of rings is unlinked, the whole link cannot be unlinked. Another way is that the cohomology of the complement supports a non-trivial Massey product, which is not the case for the unlink. Hyperbolic geometry[edit] Connection with braids[edit] The standard 3-strand braid corresponds to the Borromean rings.
Online Algorithms in High-frequency Trading Jacob Loveless, Sasha Stoikov, and Rolf Waeber HFT (high-frequency trading) has emerged as a powerful force in modern financial markets. Only 20 years ago, most of the trading volume occurred in exchanges such as the New York Stock Exchange, where humans dressed in brightly colored outfits would gesticulate and scream their trading intentions. Nowadays, trading occurs mostly in electronic servers in data centers, where computers communicate their trading intentions through network messages. Although the look of the venue and its participants has dramatically changed, the goal of all traders, whether electronic or human, remains the same: to buy an asset from one location/trader and sell it to another location/trader for a higher price. • They receive large amounts of data every microsecond. • They must be able to act extremely fast on the received data, as the profitability of the signals they are observing decays very quickly. Three problems can arise in HFT algorithms. . . Y = Xβ + ε .
Poincaré conjecture By contrast, neither of the two colored loops on this torus can be continuously tightened to a point. A torus is not homeomorphic to a sphere. Every simply connected, closed 3-manifold is homeomorphic to the 3-sphere. An equivalent form of the conjecture involves a coarser form of equivalence than homeomorphism called homotopy equivalence: if a 3-manifold is homotopy equivalent to the 3-sphere, then it is necessarily homeomorphic to it. The Poincaré conjecture, before being proven, was one of the most important open questions in topology. History[edit] Poincaré's question[edit] At the beginning of the 20th century, Henri Poincaré was working on the foundations of topology—what would later be called combinatorial topology and then algebraic topology. In the same paper, Poincaré wondered whether a 3-manifold with the homology of a 3-sphere and also trivial fundamental group had to be a 3-sphere. The original phrasing was as follows: Attempted solutions[edit] Dimensions[edit] Solution[edit]
Automatic Statistician Holonomy Parallel transport on a sphere depends on the path. Transporting from A → N → B → A yields a vector different from the initial vector. This failure to return to the initial vector is measured by the holonomy of the connection. The study of Riemannian holonomy has led to a number of important developments. Definitions[edit] Holonomy of a connection in a vector bundle[edit] The restricted holonomy group based at x is the subgroup coming from contractible loops γ. If M is connected then the holonomy group depends on the basepoint x only up to conjugation in GL(k, R). Choosing different identifications of Ex with Rk also gives conjugate subgroups. Some important properties of the holonomy group include: Holonomy of a connection in a principal bundle[edit] such that . , will not generally be p but rather some other point p·g in the fiber over x. The holonomy group of ω based at p is then defined as In particular, Moreover, if p ~ q then Holp(ω) = Holq(ω). Holonomy bundles[edit] Monodromy[edit]
Riemannian manifold In differential geometry, a (smooth) Riemannian manifold or (smooth) Riemannian space (M,g) is a real smooth manifold M equipped with an inner product on the tangent space at each point that varies smoothly from point to point in the sense that if X and Y are vector fields on M, then is a smooth function. of inner products is called a Riemannian metric (tensor). A Riemannian metric (tensor) makes it possible to define various geometric notions on a Riemannian manifold, such as angles, lengths of curves, areas (or volumes), curvature, gradients of functions and divergence of vector fields. Introduction[edit] In 1828, Carl Friedrich Gauss proved his Theorema Egregium (remarkable theorem in Latin), establishing an important property of surfaces. Overview[edit] Smoothness of α(t) for t in [0, 1] guarantees that the integral L(α) exists and the length of this curve is defined. Riemannian manifolds as metric spaces[edit] Properties[edit] Riemannian metrics[edit] defines a smooth function M → R.
Ricci flow Several stages of Ricci flow on a 2D manifold. In differential geometry, the Ricci flow (/ˈriːtʃi/) is an intrinsic geometric flow. It is a process that deforms the metric of a Riemannian manifold in a way formally analogous to the diffusion of heat, smoothing out irregularities in the metric. Mathematical definition[edit] Given a Riemannian manifold with metric tensor , we can compute the Ricci tensor The normalized Ricci flow makes sense for compact manifolds and is given by the equation where is the average (mean) of the scalar curvature (which is obtained from the Ricci tensor by taking the trace) and is the dimension of the manifold. The factor of −2 is of little significance, since it can be changed to any nonzero real number by rescaling t. Informally, the Ricci flow tends to expand negatively curved regions of the manifold, and contract positively curved regions. Examples[edit] If the manifold is Euclidean space, or more generally Ricci-flat, then Ricci flow leaves the metric unchanged.
Hadamard space In an Hadamard space, a triangle is hyperbolic; that is, the middle one in the picture. In fact, any complete metric space where a triangle is hyperbolic is an Hadamard space. In geometry, an Hadamard space, named after Jacques Hadamard, is a non-linear generalization of a Hilbert space. It is defined to be a nonempty[1] complete metric space where, given any points x, y, there exists a point m such that for every point z, The point m is then the midpoint of x and y: In a Hilbert space, the above inequality is equality (with ), and in general an Hadamard space is said to be flat if the above inequality is equality. fixes the circumcenter of B. The basic result for a non-positively curved manifold is the Cartan–Hadamard theorem. See also[edit] References[edit] Jump up ^ The assumption on "nonempty" has meaning: a fixed point theorem often states the set of fixed point is an Hadamard space.
Villarceau circles Conceptual animation showing how a slant cut torus reveals a pair of circles, known as Villarceau circles In geometry, Villarceau circles /viːlɑrˈsoʊ/ are a pair of circles produced by cutting a torus diagonally through the center at the correct angle. Given an arbitrary point on a torus, four circles can be drawn through it. One is in the plane (containing the point) parallel to the equatorial plane of the torus. Another is perpendicular to it. Example[edit] For example, let the torus be given implicitly as the set of points on circles of radius three around points on a circle of radius five in the xy plane Slicing with the z = 0 plane produces two concentric circles, x2 + y2 = 22 and x2 + y2 = 82. Two example Villarceau circles can be produced by slicing with the plane 3x = 4z. and The slicing plane is chosen to be tangent to the torus while passing through its center. Existence and equations[edit] The cross-section of the swept surface in the xz plane now includes a second circle.