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Ribbon Theory
In differential geometry, a ribbon (or strip) is the combination of a smooth space curve and its corresponding normal vector. More formally, a ribbon denoted by (X,U) includes a curve X given by a three-dimensional vector X(s), depending continuously on the curve arc-length s (a\leq s \leq b), and a unit vector U(s) perpendicular to X at each point. Ribbons have seen particular application as regards DNA. Properties and implications The ribbon (X,U) is called ''simple'' if X is a simple curve (i.e. without self-intersections) and ''closed'' and if U and all its derivatives agree at a and b. For any simple closed ribbon the curves X+\varepsilon U given parametrically by X(s)+\varepsilon U(s) are, for all sufficiently small positive \varepsilon, simple closed curves disjoint from X. The ribbon concept plays an important role in the Călugăreanu-White-Fuller formula, that states that :Lk = Wr + Tw , where Lk is the asymptotic (Gauss) ''linking number'', the integer number of tu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Differential Geometry
Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and multilinear algebra. The field has its origins in the study of spherical geometry as far back as antiquity. It also relates to astronomy, the geodesy of the Earth, and later the study of hyperbolic geometry by Lobachevsky. The simplest examples of smooth spaces are the plane and space curves and surfaces in the three-dimensional Euclidean space, and the study of these shapes formed the basis for development of modern differential geometry during the 18th and 19th centuries. Since the late 19th century, differential geometry has grown into a field concerned more generally with geometric structures on differentiable manifolds. A geometric structure is one which defines some notion of size, distance, shape, volume, or other rigidifying structu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Topological Fluid Dynamics
Topological ideas are relevant to fluid dynamics (including magnetohydrodynamics) at the kinematic level, since any fluid flow involves continuous deformation of any transported scalar or vector field. Problems of stirring and mixing are particularly susceptible to topological techniques. Thus, for example, the Thurston–Nielsen classification has been fruitfully applied to the problem of stirring in two-dimensions by any number of stirrers following a time-periodic 'stirring protocol' (Boyland, Aref & Stremler 2000). Other studies are concerned with flows having chaotic particle paths, and associated exponential rates of mixing (Ottino 1989). At the dynamic level, the fact that vortex lines are transported by any flow governed by the classical Euler equations implies conservation of any vortical structure within the flow. Such structures are characterised at least in part by the helicity of certain sub-regions of the flow field, a topological invariant of the equations. Helicit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
American Journal Of Mathematics
The ''American Journal of Mathematics'' is a bimonthly mathematics journal published by the Johns Hopkins University Press. History The ''American Journal of Mathematics'' is the oldest continuously published mathematical journal in the United States, established in 1878 at the Johns Hopkins University by James Joseph Sylvester, an English-born mathematician who also served as the journal's editor-in-chief from its inception through early 1884. Initially W. E. Story was associate editor in charge; he was replaced by Thomas Craig in 1880. For volume 7 Simon Newcomb became chief editor with Craig managing until 1894. Then with volume 16 it was "Edited by Thomas Craig with the Co-operation of Simon Newcomb" until 1898. Other notable mathematicians who have served as editors or editorial associates of the journal include Frank Morley, Oscar Zariski, Lars Ahlfors, Hermann Weyl, Wei-Liang Chow, S. S. Chern, André Weil, Harish-Chandra, Jean Dieudonné, Henri Cartan, Stephen S ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
American Mathematical Society
The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings, advocacy and other programs. The society is one of the four parts of the Joint Policy Board for Mathematics and a member of the Conference Board of the Mathematical Sciences. History The AMS was founded in 1888 as the New York Mathematical Society, the brainchild of Thomas Fiske, who was impressed by the London Mathematical Society on a visit to England. John Howard Van Amringe was the first president and Fiske became secretary. The society soon decided to publish a journal, but ran into some resistance, due to concerns about competing with the American Journal of Mathematics. The result was the ''Bulletin of the American Mathematical Society'', with Fiske as editor-in-chief. The de facto journal, as intended, was influential in in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Möbius Strip
In mathematics, a Möbius strip, Möbius band, or Möbius loop is a surface that can be formed by attaching the ends of a strip of paper together with a half-twist. As a mathematical object, it was discovered by Johann Benedict Listing and August Ferdinand Möbius in 1858, but it had already appeared in Roman mosaics from the third century CE. The Möbius strip is a non-orientable surface, meaning that within it one cannot consistently distinguish clockwise from counterclockwise turns. Every non-orientable surface contains a Möbius strip. As an abstract topological space, the Möbius strip can be embedded into three-dimensional Euclidean space in many different ways: a clockwise half-twist is different from a counterclockwise half-twist, and it can also be embedded with odd numbers of twists greater than one, or with a knotted centerline. Any two embeddings with the same knot for the centerline and the same number and direction of twists are topologically equivalent. All of t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
DNA Supercoil
DNA supercoiling refers to the amount of twist in a particular DNA strand, which determines the amount of strain on it. A given strand may be "positively supercoiled" or "negatively supercoiled" (more or less tightly wound). The amount of a strand’s supercoiling affects a number of biological processes, such as compacting DNA and regulating access to the genetic code (which strongly affects DNA metabolism and possibly gene expression). Certain enzymes, such as topoisomerases, change the amount of DNA supercoiling to facilitate functions such as DNA replication and transcription. The amount of supercoiling in a given strand is described by a mathematical formula that compares it to a reference state known as "relaxed B-form" DNA. Overview In a "relaxed" double-helical segment of B-DNA, the two strands twist around the helical axis once every 10.4–10.5 base pairs of sequence. Adding or subtracting twists, as some enzymes do, imposes strain. If a DNA segment under twist s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Knot Theory
In the mathematical field of topology, knot theory is the study of knot (mathematics), mathematical knots. While inspired by knots which appear in daily life, such as those in shoelaces and rope, a mathematical knot differs in that the ends are joined so it cannot be undone, Unknot, the simplest knot being a ring (or "unknot"). In mathematical language, a knot is an embedding of a circle in 3-dimensional Euclidean space, \mathbb^3 (in topology, a circle is not bound to the classical geometric concept, but to all of its homeomorphisms). Two mathematical knots are equivalent if one can be transformed into the other via a deformation of \mathbb^3 upon itself (known as an ambient isotopy); these transformations correspond to manipulations of a knotted string that do not involve cutting it or passing through itself. Knots can be described in various ways. Using different description methods, there may be more than one description of the same knot. For example, a common method of descr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Knots And Graphs
In mathematics, a knot is an embedding of the circle into three-dimensional Euclidean space, (also known as ). Often two knots are considered equivalent if they are ambient isotopic, that is, if there exists a continuous deformation of which takes one knot to the other. A crucial difference between the standard mathematical and conventional notions of a knot is that mathematical knots are closed — there are no ends to tie or untie on a mathematical knot. Physical properties such as friction and thickness also do not apply, although there are mathematical definitions of a knot that take such properties into account. The term ''knot'' is also applied to embeddings of in , especially in the case . The branch of mathematics that studies knots is known as knot theory and has many relations to graph theory. Formal definition A knot is an embedding of the circle () into three-dimensional Euclidean space (), or the 3-sphere (), since the 3-sphere is compact. Two knots a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bollobás–Riordan Polynomial
The Bollobás–Riordan polynomial can mean a 3-variable invariant polynomial of graphs on orientable surfaces, or a more general 4-variable invariant of ribbon graphs, generalizing the Tutte polynomial The Tutte polynomial, also called the dichromate or the Tutte–Whitney polynomial, is a graph polynomial. It is a polynomial in two variables which plays an important role in graph theory. It is defined for every undirected graph G and contai .... History These polynomials were discovered by . Formal definition The 3-variable Bollobás–Riordan polynomial of a graph G is given by :R_G(x,y,z) =\sum_F x^y^z^, where the sum runs over all the spanning subgraphs F and * v(G) is the number of vertices of G; * e(G) is the number of its edges of G; * k(G) is the number of components of G; * r(G) is the rank of G, such that r(G) = v(G)- k(G); * n(G) is the nullity of G, such that n(G) = e(G)-r(G); * bc(G) is the number of connected components of the boundary of G. See also * G ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Twist (mathematics)
In differential geometry Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and multili ..., the twist of a ''Ribbon (mathematics), ribbon'' is its rate of change (mathematics), rate of axial rotation. Let a ribbon (X,U) be composted of space curve X=X(s), where s is the arc length of X, and U=U(s) the a unit normal vector, perpendicular at each point to X. Since the ribbon (X,U) has edges X and X'=X+\varepsilon U, the twist (or ''total twist number'') Tw measures the average winding number, winding of the edge curve X' around and along the axial curve X. According to Love (1944) twist is defined by : Tw = \dfrac \int \left( U \times \dfrac \right) \cdot \dfrac ds \; , where dX/ds is the unit tangent vector to X. The total twist number Tw can be decomposed (Moffatt & Ricca 1992) into ''normalized ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Smoothness
In mathematical analysis, the smoothness of a function is a property measured by the number of continuous derivatives it has over some domain, called ''differentiability class''. At the very minimum, a function could be considered smooth if it is differentiable everywhere (hence continuous). At the other end, it might also possess derivatives of all orders in its domain, in which case it is said to be infinitely differentiable and referred to as a C-infinity function (or C^ function). Differentiability classes Differentiability class is a classification of functions according to the properties of their derivatives. It is a measure of the highest order of derivative that exists and is continuous for a function. Consider an open set U on the real line and a function f defined on U with real values. Let ''k'' be a non-negative integer. The function f is said to be of differentiability class ''C^k'' if the derivatives f',f'',\dots,f^ exist and are continuous on U. If f is k-differ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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