Quadrisecant

In geometry, a quadrisecant or quadrisecant line of a curve is a line that passes through four points of the curve. Every knotted curve in three-dimensional Euclidean space has a quadrisecant; the number of quadrisecants of an algebraic curve in complex projective space can be computed by a formula derived by Arthur Cayley. Quadrisecants of skew lines are associated with ruled surfaces and the Schläfli double six configuration. In three-dimensional Euclidean space, every non-trivial tame knot or link has a quadrisecant. Established in the case of knotted polygons and smooth knots by Erika Pannwitz, this result was extended to knots in suitably general position and links with nonzero linking number, to all nontrivial tame knots and links. Pannwitz proved more that the number of distinct quadrisecants is lower bounded by a function of the minimum number of boundary singularities in a locally-flat open disk bounded by the knot. Morton & Mond conjectured that the number of distinct quadrisecants of a given knot is always at least n/2, where n is the crossing number of the knot.

However, counterexamples to this conjecture have since been discovered. Two-component links have quadrisecants in which the points on the quadrisecant appear in alternating order between the two components, nontrivial knots have quadrisecants in which the four points, ordered cyclically as a,b,c,d on the knot, appear in order a,c,b,d along the quadrisecant; the existence of these alternating quadrisecants can be used to derive the Fary–Milnor theorem, a lower bound on the total curvature of a nontrivial knot. Quadrisecants have been used to find lower bounds on the ropelength of knots. Arthur Cayley derived a formula for the number of quadrisecants of an algebraic curve in three-dimensional complex projective space, as a function of its degree and genus. For a curve of degree d and genus g, the number of quadrisecants is 2 12 − g 2. In three-dimensional Euclidean space, every set of four skew lines in general position either has two quadrisecants or none. Any three of the four lines determine a doubly ruled surface, in which one of the two sets of ruled lines contains the three given lines, the other ruling consists of trisecants to the given lines.

If the fourth of the given lines pierces this surface, its two points of intersection lie on the two quadrisecants. The quadrisecants of sets of lines play an important role in the construction of the Schläfli double six, a configuration of twelve lines intersecting each other in 30 crossings. If five lines ai are given in a three-dimensional space, such that all five are intersected by a common line b6 but are otherwise in general position each of the five quadruples of the lines ai has a second quadrisecant bi, the five lines bi formed in this way are all intersected by a common line a6; these twelve lines and the 30 intersection points aibj form the double six

John Milnor

John Willard Milnor is an American mathematician known for his work in differential topology, K-theory and dynamical systems. Milnor is a distinguished professor at Stony Brook University and one of the four mathematicians to have won the Fields Medal, the Wolf Prize, the Abel Prize. Milnor was born on February 1931 in Orange, New Jersey, his father was J. Willard Milnor and his mother was Emily Cox Milnor; as an undergraduate at Princeton University he was named a Putnam Fellow in 1949 and 1950 and proved the Fary–Milnor theorem. He continued on to graduate school at Princeton under the direction of Ralph Fox and submitted his dissertation, entitled "Isotopy of Links", which concerned link groups and their associated link structure, in 1954. Upon completing his doctorate he went on to work at Princeton, he was a professor at the Institute for Advanced Study from 1970 to 1990. His students have included Tadatoshi Akiba, Jon Folkman, John Mather, Laurent C. Siebenmann, Michael Spivak, his wife, Dusa McDuff, is a professor of mathematics at Barnard College.

One of his published works is his proof in 1956 of the existence of 7-dimensional spheres with nonstandard differential structure. With Michel Kervaire, he showed that the 7-sphere has 15 differentiable structures. An n-sphere with nonstandard differential structure is called an exotic sphere, a term coined by Milnor. Egbert Brieskorn found simple algebraic equations for 28 complex hypersurfaces in complex 5-space such that their intersection with a small sphere of dimension 9 around a singular point is diffeomorphic to these exotic spheres. Subsequently Milnor worked on the topology of isolated singular points of complex hypersurfaces in general, developing the theory of the Milnor fibration whose fiber has the homotopy type of a bouquet of μ spheres where μ is known as the Milnor number. Milnor's 1968 book on his theory inspired the growth of a huge and rich research area which continues to mature to this day. In 1961 Milnor disproved the Hauptvermutung by illustrating two simplicial complexes which are homeomorphic but combinatorially distinct.

In 1984 Milnor introduced a definition of attractor. The objects generalize standard attractors, include so-called unstable attractors and are now known as Milnor attractors. Milnor's current interest is dynamics holomorphic dynamics, his work in dynamics is summarized by Peter Makienko in his review of Topological Methods in Modern Mathematics: It is evident now that low-dimensional dynamics, to a large extent initiated by Milnor's work, is a fundamental part of general dynamical systems theory. Milnor cast his eye on dynamical systems theory in the mid-1970s. By that time the Smale program in dynamics had been completed. Milnor's approach was to start over from the beginning, looking at the simplest nontrivial families of maps; the first choice, one-dimensional dynamics, became the subject of his joint paper with Thurston. The case of a unimodal map, that is, one with a single critical point, turns out to be rich; this work may be compared with Poincaré's work on circle diffeomorphisms, which 100 years before had inaugurated the qualitative theory of dynamical systems.

Milnor's work has opened several new directions in this field, has given us many basic concepts, challenging problems and nice theorems. He was an editor of the Annals of Mathematics for a number of years after 1962, he has written a number of books. In 1962 Milnor was awarded the Fields Medal for his work in differential topology, he went on to win the National Medal of Science, the Lester R. Ford Award in 1970 and again in 1984, the Leroy P Steele Prize for "Seminal Contribution to Research", the Wolf Prize in Mathematics, the Leroy P Steele Prize for Mathematical Exposition, the Leroy P Steele Prize for Lifetime Achievement "... for a paper of fundamental and lasting importance, On manifolds homeomorphic to the 7-sphere, Annals of Mathematics 64, 399–405". In 1991 a symposium was held at Stony Brook University in celebration of his 60th birthday. Milnor was awarded the 2011 Abel Prize, for his "pioneering discoveries in topology and algebra." Reacting to the award, Milnor told the New Scientist "It feels good," adding that "ne is always surprised by a call at 6 o'clock in the morning."

In 2013 he became a fellow of the American Mathematical Society, for "contributions to differential topology, geometric topology, algebraic topology and dynamical systems". Milnor, John W.. Morse theory. Annals of Mathematics Studies, No. 51. Notes by M. Spivak and R. Wells. Princeton, NJ: Princeton University Press. ISBN 0-691-08008-9. ——. Lectures on the h-cobordism theorem. Notes by L. Siebenmann and J. Sondow. Princeton, NJ: Princeton University Press. ISBN 0-691-07996-X. OCLC 58324. ——. Singular points of complex hypersurfaces. Annals of Mathematics Studies, No. 61. Princeton, NJ: Princeton University Press. ISBN 0-691-08065-8. ——. Introduction to algebraic K-theory. Annals of Mathematics Studies, No. 72. Princeton, NJ: Princeton University Press. ISBN 978-0-691-08101-4. Husemoller, Dale. Symmetric bilinear forms. New York, NY: Springer-Verlag. ISBN 978-0-387-06009-5. Milnor, John W.. Characteristic classes. Annals of Mathematics Studies, No. 76. Princeton, NJ: Princeton University Press. ISBN 0-691-08122-0. Milnor, John W..

Topology from the differentiable viewpoint. Princeton Landmarks in Mathematics. Princeton, NJ: Princeton University Press. ISBN 0-691-04833-9. —— (

Necessity and sufficiency

In logic and sufficiency are terms used to describe a conditional or implicational relationship between statements. For example, in the conditional statement "If P Q", we say that "Q is necessary for P" because P cannot be true unless Q is true. We say that "P is sufficient for Q" because P being true always implies that Q is true, but P not being true does not always imply that Q is not true; the assertion that a statement is a "necessary and sufficient" condition of another means that the former statement is true if and only if the latter is true. That is, the two statements must be either true or false. In ordinary English, "necessary" and "sufficient" indicate relations between conditions or states of affairs, not statements. Being a male sibling is a necessary and sufficient condition for being a brother. In the conditional statement, "if S N", the expression represented by S is called the antecedent and the expression represented by N is called the consequent; this conditional statement may be written in many equivalent ways, for instance, "N if S", "S only if N", "S implies N", "N is implied by S", S → N, S ⇒ N, or "N whenever S".

In the above situation, we say that N is a necessary condition for S. In common language this is saying that if the conditional statement is a true statement the consequent N must be true if S is to be true. Phrased differently, the antecedent S cannot be true without N being true. For example, in order for someone to be called Socrates, it is necessary for that someone to be Named. In the above situation, we can say S is a sufficient condition for N. Again, consider the third column of the truth table below. If the conditional statement is true if S is true, N must be true. In common terms, "S guarantees N". Continuing the example, knowing that someone is called Socrates is sufficient to know that someone has a Name. A necessary and sufficient condition requires that both of the implications S ⇒ N ⇒ S hold. From the first of these we see that S is a sufficient condition for N, from the second that S is a necessary condition for N; this is expressed as "S is necessary and sufficient for N ", "S if and only if N ", or S ⇔ N.

The assertion that Q is necessary for P is colloquially equivalent to "P cannot be true unless Q is true" or "if Q is false P is false". By contraposition, this is the same thing as "whenever P is true, so is Q"; the logical relation between P and Q is expressed as "if P Q" and denoted "P ⇒ Q". It may be expressed as any of "P only if Q", "Q, if P", "Q whenever P", "Q when P". One finds, in mathematical prose for instance, several necessary conditions that, taken together, constitute a sufficient condition, as shown in Example 5. Example 1 For it to be true that "John is a bachelor", it is necessary that it be true that he is unmarried, adult, since to state "John is a bachelor" implies John has each of those three additional predicates. Example 2 For the whole numbers greater than two, being odd is necessary to being prime, since two is the only whole number, both and prime. Example 3 Consider thunder, the sound caused by lightning. We say. Whenever there's lightning, there's thunder; the thunder does not cause the lightning, but because lightning always comes with thunder, we say that thunder is necessary for lightning.

Example 4 Being at least 30 years old is necessary for serving in the U. S. Senate. If you are under 30 years old it is impossible for you to be a senator; that is, if you are a senator, it follows that you are at least 30 years old. Example 5 In algebra, for some set S together with an operation ⋆ to form a group, it is necessary that ⋆ be associative, it is necessary that S include a special element e such that for every x in S it is the case that e ⋆ x and x ⋆ e both equal x. It is necessary that for every x in S there exist a corresponding element x″ such that both x ⋆ x″ and x″ ⋆ x equal the special element e. None of these three necessary conditions by itself is sufficient, but the conjunction of the three is. If P is sufficient for Q knowing P to be true is adequate grounds to conclude that Q is true; the logical relation is, as before, expressed as "if P Q" or "P ⇒ Q". This can be expressed as "P only if Q", "P implies Q" or several other variants, it may be the case that several sufficient conditions, when taken together, constitute a single necessary condition, as illustrated in example 5.

Example 1 "John is a king" implies. So knowing that it is true that John is a king is sufficient to know that he is a male. Example 2 A number's being divisible by 4 is sufficient for its being but being divisible by 2 is both sufficient and necessary. Example 3 An occurrence of thunder is a sufficient condition for the occurrence of lightning in the sense

ArXiv

ArXiv is a repository of electronic preprints approved for posting after moderation, but not full peer review. It consists of scientific papers in the fields of mathematics, astronomy, electrical engineering, computer science, quantitative biology, mathematical finance and economics, which can be accessed online. In many fields of mathematics and physics all scientific papers are self-archived on the arXiv repository. Begun on August 14, 1991, arXiv.org passed the half-million-article milestone on October 3, 2008, had hit a million by the end of 2014. By October 2016 the submission rate had grown to more than 10,000 per month. ArXiv was made possible by the compact TeX file format, which allowed scientific papers to be transmitted over the Internet and rendered client-side. Around 1990, Joanne Cohn began emailing physics preprints to colleagues as TeX files, but the number of papers being sent soon filled mailboxes to capacity. Paul Ginsparg recognized the need for central storage, in August 1991 he created a central repository mailbox stored at the Los Alamos National Laboratory which could be accessed from any computer.

Additional modes of access were soon added: FTP in 1991, Gopher in 1992, the World Wide Web in 1993. The term e-print was adopted to describe the articles, it began as a physics archive, called the LANL preprint archive, but soon expanded to include astronomy, computer science, quantitative biology and, most statistics. Its original domain name was xxx.lanl.gov. Due to LANL's lack of interest in the expanding technology, in 2001 Ginsparg changed institutions to Cornell University and changed the name of the repository to arXiv.org. It is now hosted principally with eight mirrors around the world, its existence was one of the precipitating factors that led to the current movement in scientific publishing known as open access. Mathematicians and scientists upload their papers to arXiv.org for worldwide access and sometimes for reviews before they are published in peer-reviewed journals. Ginsparg was awarded a MacArthur Fellowship in 2002 for his establishment of arXiv; the annual budget for arXiv is $826,000 for 2013 to 2017, funded jointly by Cornell University Library, the Simons Foundation and annual fee income from member institutions.

This model arose in 2010, when Cornell sought to broaden the financial funding of the project by asking institutions to make annual voluntary contributions based on the amount of download usage by each institution. Each member institution pledges a five-year funding commitment to support arXiv. Based on institutional usage ranking, the annual fees are set in four tiers from $1,000 to $4,400. Cornell's goal is to raise at least $504,000 per year through membership fees generated by 220 institutions. In September 2011, Cornell University Library took overall administrative and financial responsibility for arXiv's operation and development. Ginsparg was quoted in the Chronicle of Higher Education as saying it "was supposed to be a three-hour tour, not a life sentence". However, Ginsparg remains on the arXiv Scientific Advisory Board and on the arXiv Physics Advisory Committee. Although arXiv is not peer reviewed, a collection of moderators for each area review the submissions; the lists of moderators for many sections of arXiv are publicly available, but moderators for most of the physics sections remain unlisted.

Additionally, an "endorsement" system was introduced in 2004 as part of an effort to ensure content is relevant and of interest to current research in the specified disciplines. Under the system, for categories that use it, an author must be endorsed by an established arXiv author before being allowed to submit papers to those categories. Endorsers are not asked to review the paper for errors, but to check whether the paper is appropriate for the intended subject area. New authors from recognized academic institutions receive automatic endorsement, which in practice means that they do not need to deal with the endorsement system at all. However, the endorsement system has attracted criticism for restricting scientific inquiry. A majority of the e-prints are submitted to journals for publication, but some work, including some influential papers, remain purely as e-prints and are never published in a peer-reviewed journal. A well-known example of the latter is an outline of a proof of Thurston's geometrization conjecture, including the Poincaré conjecture as a particular case, uploaded by Grigori Perelman in November 2002.

Perelman appears content to forgo the traditional peer-reviewed journal process, stating: "If anybody is interested in my way of solving the problem, it's all there – let them go and read about it". Despite this non-traditional method of publication, other mathematicians recognized this work by offering the Fields Medal and Clay Mathematics Millennium Prizes to Perelman, both of which he refused. Papers can be submitted in any of several formats, including LaTeX, PDF printed from a word processor other than TeX or LaTeX; the submission is rejected by the arXiv software if generating the final PDF file fails, if any image file is too large, or if the total size of the submission is too large. ArXiv now allows one to store and modify an incomplete submission, only finalize the submission when ready; the time stamp on the article is set. The standard access route is through one of several mirrors. Sev

Smoothness

In mathematical analysis, the smoothness of a function is a property measured by the number of derivatives it has that are continuous. A smooth function is a function. Differentiability class is a classification of functions according to the properties of their derivatives. Higher order differentiability classes correspond to the existence of more derivatives. Consider an open set on the real line and a function f defined on that set with real values. Let k be a non-negative integer; the function f is said to be of class Ck if the derivatives f′, f′′... F are continuous; the function f is said to be of class C ∞, or smooth. The function f is said to be of class Cω, or analytic, if f is smooth and if its Taylor series expansion around any point in its domain converges to the function in some neighborhood of the point. Cω is thus contained in C∞. Bump functions are examples of functions in C∞ but not in Cω. To put it differently, the class C0 consists of all continuous functions; the class C1 consists of all differentiable functions.

Thus, a C1 function is a function whose derivative exists and is of class C0. In general, the classes Ck can be defined recursively by declaring C0 to be the set of all continuous functions and declaring Ck for any positive integer k to be the set of all differentiable functions whose derivative is in Ck−1. In particular, Ck is contained in Ck−1 for every k, there are examples to show that this containment is strict. C∞, the class of infinitely differentiable functions, is the intersection of the sets Ck as k varies over the non-negative integers; the function f = { x if x ≥ 0, 0 if x < 0 is continuous, but not differentiable at x = 0, so it is of class C0 but not of class C1. The function g = { x 2 sin if x ≠ 0, 0 if x = 0 is differentiable, with derivative g ′ = { − cos + 2 x sin if x ≠ 0, 0 if x = 0; because cos oscillates as x → 0, g’ is not continuous at zero. Therefore, g is differentiable but not of class C1. Moreover, if one takes g = x4/3sin in this example, it can be used to show that the derivative function of a differentiable function can be unbounded on a compact set and, that a differentiable function on a compact set may not be locally Lipschitz continuous.

The functions f = | x | k + 1 where k is are continuous and k times differentiable at all x. But at x = 0 they are not times differentiable, so they are of class Ck but not of class Cj where j > k. The exponential function is analytic, so, of class Cω; the trigonometric functions are analytic wherever they are defined. The function f = { e − 1 1 − x 2 if | x | < 1, 0 otherwise is smooth, so of class C∞, but it is not analytic at x = ±1, so it is not of class Cω. The function f is an example of a smooth function with compact support. A function f: U ⊂ R n → R defined on an open set U of R n is said to be of class C k {\displayst

Baseball

Baseball is a bat-and-ball game played between two opposing teams who take turns batting and fielding. The game proceeds when a player on the fielding team, called the pitcher, throws a ball which a player on the batting team tries to hit with a bat; the objectives of the offensive team are to hit the ball into the field of play, to run the bases—having its runners advance counter-clockwise around four bases to score what are called "runs". The objective of the defensive team is to prevent batters from becoming runners, to prevent runners' advance around the bases. A run is scored when a runner advances around the bases in order and touches home plate; the team that scores the most runs by the end of the game is the winner. The first objective of the batting team is to have a player reach first base safely. A player on the batting team who reaches first base without being called "out" can attempt to advance to subsequent bases as a runner, either or during teammates' turns batting; the fielding team tries to prevent runs by getting batters or runners "out", which forces them out of the field of play.

Both the pitcher and fielders have methods of getting the batting team's players out. The opposing teams switch forth between batting and fielding. One turn batting for each team constitutes an inning. A game is composed of nine innings, the team with the greater number of runs at the end of the game wins. If scores are tied at the end of nine innings, extra innings are played. Baseball has no game clock. Baseball evolved from older bat-and-ball games being played in England by the mid-18th century; this game was brought by immigrants to North America. By the late 19th century, baseball was recognized as the national sport of the United States. Baseball is popular in North America and parts of Central and South America, the Caribbean, East Asia in Japan and South Korea. In the United States and Canada, professional Major League Baseball teams are divided into the National League and American League, each with three divisions: East and Central; the MLB champion is determined by playoffs. The top level of play is split in Japan between the Central and Pacific Leagues and in Cuba between the West League and East League.

The World Baseball Classic, organized by the World Baseball Softball Confederation, is the major international competition of the sport and attracts the top national teams from around the world. A baseball game is played between two teams, each composed of nine players, that take turns playing offense and defense. A pair of turns, one at bat and one in the field, by each team constitutes an inning. A game consists of nine innings. One team—customarily the visiting team—bats in the top, or first half, of every inning; the other team -- customarily the home team -- bats in second half, of every inning. The goal of the game is to score more points than the other team; the players on the team at bat attempt to score runs by circling or completing a tour of the four bases set at the corners of the square-shaped baseball diamond. A player bats at home plate and must proceed counterclockwise to first base, second base, third base, back home to score a run; the team in the field attempts to prevent runs from scoring and record outs, which remove opposing players from offensive action until their turn in their team's batting order comes up again.

When three outs are recorded, the teams switch roles for the next half-inning. If the score of the game is tied after nine innings, extra innings are played to resolve the contest. Many amateur games unorganized ones, involve different numbers of players and innings; the game is played on a field whose primary boundaries, the foul lines, extend forward from home plate at 45-degree angles. The 90-degree area within the foul lines is referred to as fair territory; the part of the field enclosed by the bases and several yards beyond them is the infield. In the middle of the infield is a raised pitcher's mound, with a rectangular rubber plate at its center; the outer boundary of the outfield is demarcated by a raised fence, which may be of any material and height. The fair territory between home plate and the outfield boundary is baseball's field of play, though significant events can take place in foul territory, as well. There are three basic tools of baseball: the ball, the bat, the glove or mitt: The baseball is about the size of an adult's fist, around 9 inches in circumference.

It wound in yarn and covered in white cowhide, with red stitching. The bat is a hitting tool, traditionally made of a solid piece of wood. Other materials are now used for nonprofessional games, it is a hard round stick, about 2.5 inches in diameter at the hitting end, tapering to a narrower handle and culminating in a knob. Bats used by adults are around 34 inches long, not longer than 42 inches; the glove or mitt is a fielding tool, made of padded leather with webbing between the fingers. As an aid in catching and holding onto the ball, it takes various shapes to meet the specific needs of differ

Curve

In mathematics, a curve is speaking, an object similar to a line but that need not be straight. Thus, a curve is a generalization of a line. Various disciplines within mathematics have given the term different meanings depending on the area of study, so the precise meaning depends on context. However, many of these meanings are special instances of the definition. A curve is a topological space, locally homeomorphic to a line. In everyday language, this means that a curve is a set of points which, near each of its points, looks like a line, up to a deformation. A simple example of a curve is the parabola, shown to the right. A large number of other curves have been studied in multiple mathematical fields. A closed curve is a curve that forms a path whose starting point is its ending point—that is, a path from any of its points to the same point. Related meanings include the graph of a function and a two-dimensional graph. Interest in curves began; this can be seen in numerous examples of their decorative use in art and on everyday objects dating back to prehistoric times.

Curves, or at least their graphical representations, are simple to create, for example by a stick in the sand on a beach. The term line was used in place of the more modern term curve. Hence the phrases straight line and right line were used to distinguish what are today called lines from curved lines. For example, in Book I of Euclid's Elements, a line is defined as a "breadthless length", while a straight line is defined as "a line that lies evenly with the points on itself". Euclid's idea of a line is clarified by the statement "The extremities of a line are points,". Commentators further classified lines according to various schemes. For example: Composite lines Incomposite lines Determinate Indeterminate The Greek geometers had studied many other kinds of curves. One reason was their interest in solving geometrical problems that could not be solved using standard compass and straightedge construction; these curves include: The conic sections studied by Apollonius of Perga The cissoid of Diocles, studied by Diocles and used as a method to double the cube.

The conchoid of Nicomedes, studied by Nicomedes as a method to both double the cube and to trisect an angle. The Archimedean spiral, studied by Archimedes as a method to trisect an angle and square the circle; the spiric sections, sections of tori studied by Perseus as sections of cones had been studied by Apollonius. A fundamental advance in the theory of curves was the advent of analytic geometry in the seventeenth century; this enabled a curve to be described using an equation rather than an elaborate geometrical construction. This not only allowed new curves to be defined and studied, but it enabled a formal distinction to be made between curves that can be defined using algebraic equations, algebraic curves, those that cannot, transcendental curves. Curves had been described as "geometrical" or "mechanical" according to how they were, or could be, generated. Conic sections were applied in astronomy by Kepler. Newton worked on an early example in the calculus of variations. Solutions to variational problems, such as the brachistochrone and tautochrone questions, introduced properties of curves in new ways.

The catenary gets its name as the solution to the problem of a hanging chain, the sort of question that became accessible by means of differential calculus. In the eighteenth century came the beginnings of the theory of plane algebraic curves, in general. Newton had studied the cubic curves, in the general description of the real points into'ovals'; the statement of Bézout's theorem showed a number of aspects which were not directly accessible to the geometry of the time, to do with singular points and complex solutions. Since the nineteenth century there has not been a separate theory of curves, but rather the appearance of curves as the one-dimensional aspect of projective geometry, differential geometry; the era of the space-filling curves provoked the modern definitions of curve. In general, a curve is defined through a continuous function γ: I → X from an interval I of the real numbers into a topological space X. Depending on the context, it is either γ or its image γ, called a curve. In general topology, when non-differentiable functions are considered, it is the map γ, called a curve, because its image may look differently from what is called a curve.

For example, the image of the Peano curve fills the square. On the other hand, when one considers curves defined by a differentiable function, this is the image of the function, called a curve; the curve is said to be simple, or a Jordan arc, if γ is injective, i.e. if for all x, y in I, we have γ = γ