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Ring Lemma
In the geometry of circle packings in the Euclidean plane, the ring lemma gives a lower bound on the sizes of adjacent circles in a circle packing. Statement The lemma states: Let n be any integer greater than or equal to three. Suppose that the unit circle is surrounded by a ring of n interior-disjoint circles, all tangent to it, with consecutive circles in the ring tangent to each other. Then the minimum radius of any circle in the ring is at least the unit fraction \frac where F_i is the ith Fibonacci number. The sequence of minimum radii, from n=3, begins Generalizations to three-dimensional space are also known. Construction An infinite sequence of circles can be constructed, containing rings for each n that exactly meet the bound of the ring lemma, showing that it is tight. The construction allows halfplanes to be considered as degenerate circles with infinite radius, and includes additional tangencies between the circles beyond those required in the statement of the lem ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ring Lemma
In the geometry of circle packings in the Euclidean plane, the ring lemma gives a lower bound on the sizes of adjacent circles in a circle packing. Statement The lemma states: Let n be any integer greater than or equal to three. Suppose that the unit circle is surrounded by a ring of n interior-disjoint circles, all tangent to it, with consecutive circles in the ring tangent to each other. Then the minimum radius of any circle in the ring is at least the unit fraction \frac where F_i is the ith Fibonacci number. The sequence of minimum radii, from n=3, begins Generalizations to three-dimensional space are also known. Construction An infinite sequence of circles can be constructed, containing rings for each n that exactly meet the bound of the ring lemma, showing that it is tight. The construction allows halfplanes to be considered as degenerate circles with infinite radius, and includes additional tangencies between the circles beyond those required in the statement of the lem ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dennis Sullivan
Dennis Parnell Sullivan (born February 12, 1941) is an American mathematician known for his work in algebraic topology, geometric topology, and dynamical systems. He holds the Albert Einstein Chair at the City University of New York Graduate Center and is a distinguished professor at Stony Brook University. Sullivan was awarded the Wolf Prize in Mathematics in 2010 and the Abel Prize in 2022. Early life and education Sullivan was born in Port Huron, Michigan, on February 12, 1941.. His family moved to Houston soon afterwards. He entered Rice University to study chemical engineering but switched his major to mathematics in his second year after encountering a particularly motivating mathematical theorem. The change was prompted by a special case of the uniformization theorem, according to which, in his own words: He received his Bachelor of Arts degree from Rice in 1963. He obtained his Doctor of Philosophy from Princeton University in 1966 with his thesis, ''Triangulating h ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lemmas , part of a neuron
{{Disambiguation ...
Lemma may refer to: Language and linguistics * Lemma (morphology), the canonical, dictionary or citation form of a word * Lemma (psycholinguistics), a mental abstraction of a word about to be uttered Science and mathematics * Lemma (botany), a part of a grass plant * Lemma (mathematics), a type of proposition Other uses * ''Lemma'' (album), by John Zorn (2013) * Lemma (logic), an informal contention See also *Analemma, a diagram showing the variation of the position of the Sun in the sky *Dilemma *Lema (other) * Lemmatisation *Neurolemma Neurilemma (also known as neurolemma, sheath of Schwann, or Schwann's sheath) is the outermost nucleated cytoplasmic layer of Schwann cells (also called neurilemmocytes) that surrounds the axon of the neuron. It forms the outermost layer of the ne ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Circle Packing
In geometry, circle packing is the study of the arrangement of circles (of equal or varying sizes) on a given surface such that no overlapping occurs and so that no circle can be enlarged without creating an overlap. The associated ''packing density'', ''η'', of an arrangement is the proportion of the surface covered by the circles. Generalisations can be made to higher dimensions – this is called ''sphere packing'', which usually deals only with identical spheres. The branch of mathematics generally known as "circle packing" is concerned with the geometry and combinatorics of packings of arbitrarily-sized circles: these give rise to discrete analogs of conformal mapping, Riemann surfaces and the like. Densest packing In the two-dimensional Euclidean plane, Joseph Louis Lagrange proved in 1773 that the highest-density lattice packing of circles is the hexagonal packing arrangement, in which the centres of the circles are arranged in a hexagonal lattice (staggered rows, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
18th International Symposium, GD 2010, Konstanz, Germany, September 21-24, 2010, Revised Selected Papers
18 (eighteen) is the natural number following 17 and preceding 19. In mathematics * Eighteen is a composite number, its divisors being 1, 2, 3, 6 and 9. Three of these divisors (3, 6 and 9) add up to 18, hence 18 is a semiperfect number. Eighteen is the first inverted square-prime of the form ''p''·''q''2. * In base ten, it is a Harshad number. * It is an abundant number, as the sum of its proper divisors is greater than itself (1+2+3+6+9 = 21). It is known to be a solitary number, despite not being coprime to this sum. * It is the number of one-sided pentominoes. * It is the only number where the sum of its written digits in base 10 (1+8 = 9) is equal to half of itself (18/2 = 9). * It is a Fine number. In science Chemistry * Eighteen is the atomic number of argon. * Group 18 of the periodic table is called the noble gases. * The 18-electron rule is a rule of thumb in transition metal chemistry for characterising and predicting the stability of metal complexes. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Slope Number
In graph drawing and geometric graph theory, the slope number of a graph is the minimum possible number of distinct slopes of edges in a drawing of the graph in which vertices are represented as points in the Euclidean plane and edges are represented as line segments that do not pass through any non-incident vertex. Complete graphs Although closely related problems in discrete geometry had been studied earlier, e.g. by and , the problem of determining the slope number of a graph was introduced by , who showed that the slope number of an -vertex complete graph is exactly . A drawing with this slope number may be formed by placing the vertices of the graph on a regular polygon. Relation to degree The slope number of a graph of maximum degree is clearly at least \lceil d/2\rceil, because at most two of the incident edges at a degree- vertex can share a slope. More precisely, the slope number is at least equal to the linear arboricity of the graph, since the edges of a singl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Planar Graph
In graph theory, a planar graph is a graph that can be embedded in the plane, i.e., it can be drawn on the plane in such a way that its edges intersect only at their endpoints. In other words, it can be drawn in such a way that no edges cross each other. Such a drawing is called a plane graph or planar embedding of the graph. A plane graph can be defined as a planar graph with a mapping from every node to a point on a plane, and from every edge to a plane curve on that plane, such that the extreme points of each curve are the points mapped from its end nodes, and all curves are disjoint except on their extreme points. Every graph that can be drawn on a plane can be drawn on the sphere as well, and vice versa, by means of stereographic projection. Plane graphs can be encoded by combinatorial maps or rotation systems. An equivalence class of topologically equivalent drawings on the sphere, usually with additional assumptions such as the absence of isthmuses, is called a pl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Closed-form Expression
In mathematics, a closed-form expression is a mathematical expression that uses a finite number of standard operations. It may contain constants, variables, certain well-known operations (e.g., + − × ÷), and functions (e.g., ''n''th root, exponent, logarithm, trigonometric functions, and inverse hyperbolic functions), but usually no limit, differentiation, or integration. The set of operations and functions may vary with author and context. Example: roots of polynomials The solutions of any quadratic equation with complex coefficients can be expressed in closed form in terms of addition, subtraction, multiplication, division, and square root extraction, each of which is an elementary function. For example, the quadratic equation :ax^2+bx+c=0, is tractable since its solutions can be expressed as a closed-form expression, i.e. in terms of elementary functions: :x=\frac. Similarly, solutions of cubic and quartic (third and fourth degree) equations can be expresse ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Recurrence Relation
In mathematics, a recurrence relation is an equation according to which the nth term of a sequence of numbers is equal to some combination of the previous terms. Often, only k previous terms of the sequence appear in the equation, for a parameter k that is independent of n; this number k is called the ''order'' of the relation. If the values of the first k numbers in the sequence have been given, the rest of the sequence can be calculated by repeatedly applying the equation. In ''linear recurrences'', the th term is equated to a linear function of the k previous terms. A famous example is the recurrence for the Fibonacci numbers, F_n=F_+F_ where the order k is two and the linear function merely adds the two previous terms. This example is a linear recurrence with constant coefficients, because the coefficients of the linear function (1 and 1) are constants that do not depend on n. For these recurrences, one can express the general term of the sequence as a closed-form expression o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Conformal Map
In mathematics, a conformal map is a function that locally preserves angles, but not necessarily lengths. More formally, let U and V be open subsets of \mathbb^n. A function f:U\to V is called conformal (or angle-preserving) at a point u_0\in U if it preserves angles between directed curves through u_0, as well as preserving orientation. Conformal maps preserve both angles and the shapes of infinitesimally small figures, but not necessarily their size or curvature. The conformal property may be described in terms of the Jacobian derivative matrix of a coordinate transformation. The transformation is conformal whenever the Jacobian at each point is a positive scalar times a rotation matrix (orthogonal with determinant one). Some authors define conformality to include orientation-reversing mappings whose Jacobians can be written as any scalar times any orthogonal matrix. For mappings in two dimensions, the (orientation-preserving) conformal mappings are precisely the locally i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
William Thurston
William Paul Thurston (October 30, 1946August 21, 2012) was an American mathematician. He was a pioneer in the field of low-dimensional topology and was awarded the Fields Medal in 1982 for his contributions to the study of 3-manifolds. Thurston was a professor of mathematics at Princeton University, University of California, Davis, and Cornell University. He was also a director of the Mathematical Sciences Research Institute. Early life and education William Thurston was born in Washington, D.C. to Margaret Thurston (), a seamstress, and Paul Thurston, an aeronautical engineer. William Thurston suffered from congenital strabismus as a child, causing issues with depth perception. His mother worked with him as a toddler to reconstruct three-dimensional images from two-dimensional ones. He received his bachelor's degree from New College in 1967 as part of its inaugural class. For his undergraduate thesis, he developed an intuitionist foundation for topology. Following this, he r ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Burton Rodin
Burton Rodin is an American mathematician known for his research in conformal mappings and Riemann surfaces. He is a professor emeritus at the University of California, San Diego. Education Rodin received a Ph.D. at the University of California, Los Angeles in 1961. His thesis, titled ''Reproducing Formulas on Riemann Surfaces'', was written under the supervision of Leo Sario. Career He was a professor at the University of California, San Diego from 1970 to 1994. He was chair of the Mathematics Department from 1977 to 1981, and became professor emeritus in June 1994. Research Rodin's 1968 work on extremal length of Riemann surfaces, together with an observation of Mikhail Katz, yielded the first systolic geometry inequality for surfaces independent of their genus. In 1980, Rodin and Stefan E. Warschawski solved the Visser–Ostrowski problem for derivatives of conformal mappings at the boundary. In 1987 he proved the circle packing theorem#History, Thurston conjecture for circl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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