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https://en.wikipedia.org/wiki/Gambler%27s%20ruin | In statistics, gambler's ruin is the fact that a gambler playing a game with negative expected value will eventually go broke, regardless of their betting system.
The concept was initially stated: A persistent gambler who raises their bet to a fixed fraction of the gambler's bankroll after a win, but does not reduce it after a loss, will eventually and inevitably go broke, even if each bet has a positive expected value.
Another statement of the concept is that a persistent gambler with finite wealth, playing a fair game (that is, each bet has expected value of zero to both sides) will eventually and inevitably go broke against an opponent with infinite wealth. Such a situation can be modeled by a random walk on the real number line. In that context, it is probable that the gambler will, with virtual certainty, return to their point of origin, which means going broke, and is ruined an infinite number of times if the random walk continues forever. This is a corollary of a general theorem by Christiaan Huygens, which is also known as gambler's ruin. That theorem shows how to compute the probability of each player winning a series of bets that continues until one's entire initial stake is lost, given the initial stakes of the two players and the constant probability of winning. This is the oldest mathematical idea that goes by the name gambler's ruin, but not the first idea to which the name was applied. The term's common usage today is another corollary to Huygens's result.
The concept has specific relevance for gamblers. However it also leads to mathematical theorems with wide application and many related results in probability and statistics. Huygens's result in particular led to important advances in the mathematical theory of probability.
History
The earliest known mention of the gambler's ruin problem is a letter from Blaise Pascal to Pierre Fermat in 1656 (two years after the more famous correspondence on the problem of points). Pascal's version was summarize |
https://en.wikipedia.org/wiki/Parametrization%20%28geometry%29 | In mathematics, and more specifically in geometry, parametrization (or parameterization; also parameterisation, parametrisation) is the process of finding parametric equations of a curve, a surface, or, more generally, a manifold or a variety, defined by an implicit equation. The inverse process is called implicitization. "To parameterize" by itself means "to express in terms of parameters".
Parametrization is a mathematical process consisting of expressing the state of a system, process or model as a function of some independent quantities called parameters. The state of the system is generally determined by a finite set of coordinates, and the parametrization thus consists of one function of several real variables for each coordinate. The number of parameters is the number of degrees of freedom of the system.
For example, the position of a point that moves on a curve in three-dimensional space is determined by the time needed to reach the point when starting from a fixed origin. If are the coordinates of the point, the movement is thus described by a parametric equation
where is the parameter and denotes the time. Such a parametric equation completely determines the curve, without the need of any interpretation of as time, and is thus called a parametric equation of the curve (this is sometimes abbreviated by saying that one has a parametric curve). One similarly gets the parametric equation of a surface by considering functions of two parameters and .
Non-uniqueness
Parametrizations are not generally unique. The ordinary three-dimensional object can be parametrized (or "coordinatized") equally efficiently with Cartesian coordinates (x, y, z), cylindrical polar coordinates (ρ, φ, z), spherical coordinates (r, φ, θ) or other coordinate systems.
Similarly, the color space of human trichromatic color vision can be parametrized in terms of the three colors red, green and blue, RGB, or with cyan, magenta, yellow and black, CMYK.
Dimensionality
Generally, |
https://en.wikipedia.org/wiki/Glossary%20of%20order%20theory | This is a glossary of some terms used in various branches of mathematics that are related to the fields of order, lattice, and domain theory. Note that there is a structured list of order topics available as well. Other helpful resources might be the following overview articles:
completeness properties of partial orders
distributivity laws of order theory
preservation properties of functions between posets.
In the following, partial orders will usually just be denoted by their carrier sets. As long as the intended meaning is clear from the context, will suffice to denote the corresponding relational symbol, even without prior introduction. Furthermore, < will denote the strict order induced by
A
Acyclic. A binary relation is acyclic if it contains no "cycles": equivalently, its transitive closure is antisymmetric.
Adjoint. See Galois connection.
Alexandrov topology. For a preordered set P, any upper set O is Alexandrov-open. Inversely, a topology is Alexandrov if any intersection of open sets is open.
Algebraic poset. A poset is algebraic if it has a base of compact elements.
Antichain. An antichain is a poset in which no two elements are comparable, i.e., there are no two distinct elements x and y such that x ≤ y. In other words, the order relation of an antichain is just the identity relation.
Approximates relation. See way-below relation.
Antisymmetric relation. A homogeneous relation R on a set X is antisymmetric, if x R y and y R x implies x = y, for all elements x, y in X.
Antitone. An antitone function f between posets P and Q is a function for which, for all elements x, y of P, x ≤ y (in P) implies f(y) ≤ f(x) (in Q). Another name for this property is order-reversing. In analysis, in the presence of total orders, such functions are often called monotonically decreasing, but this is not a very convenient description when dealing with non-total orders. The dual notion is called monotone or order-preserving.
Asymmetric relation. A homogeneo |
https://en.wikipedia.org/wiki/Tagged%20union | In computer science, a tagged union, also called a variant, variant record, choice type, discriminated union, disjoint union, sum type or coproduct, is a data structure used to hold a value that could take on several different, but fixed, types. Only one of the types can be in use at any one time, and a tag field explicitly indicates which one is in use. It can be thought of as a type that has several "cases", each of which should be handled correctly when that type is manipulated. This is critical in defining recursive datatypes, in which some component of a value may have the same type as that value, for example in defining a type for representing trees, where it is necessary to distinguish multi-node subtrees and leaves. Like ordinary unions, tagged unions can save storage by overlapping storage areas for each type, since only one is in use at a time.
Description
Tagged unions are most important in functional programming languages such as ML and Haskell, where they are called datatypes (see algebraic data type) and the compiler is able to verify that all cases of a tagged union are always handled, avoiding many types of errors. Compile-time checked sum types are also extensively used in Rust, where they are called enum. They can, however, be constructed in nearly any programming language, and are much safer than untagged unions, often simply called unions, which are similar but do not explicitly track which member of a union is in use currently.
Tagged unions are often accompanied by the concept of a type constructor, which is similar but not the same as a constructor for a class. Type constructors produce a tagged union type, given the initial tag type and the corresponding type.
Mathematically, tagged unions correspond to disjoint or discriminated unions, usually written using +. Given an element of a disjoint union A + B, it is possible to determine whether it came from A or B. If an element lies in both, there will be two effectively distinct copies of th |
https://en.wikipedia.org/wiki/OpenAFS | OpenAFS is an open-source implementation of the Andrew distributed file system (AFS). AFS was originally developed at Carnegie Mellon University, and developed as a commercial product by the Transarc Corporation, which was subsequently acquired by IBM. At LinuxWorld on 15 August 2000, IBM announced their plans to release a version of their commercial AFS product under the IBM Public License. This became OpenAFS. Today, OpenAFS is actively developed for a wide range of operating system families including: AIX, Mac OS X, Darwin, HP-UX, Irix, Solaris, Linux, Microsoft Windows, FreeBSD, NetBSD.
Foundation
The OpenAFS Foundation was established on May 20, 2013 as a non-profit organization dedicated to fostering the stability and growth of OpenAFS.
Governance
Governance of the project is split between the board of elders who consider issues of strategic direction, and the gatekeepers who control the source repository.
Licensing
Although there is no legal entity that owns the OpenAFS source code, copyright on many files is attributed to IBM. Most of the source is covered by the IPL, however several files in the tree are covered by university vanity licenses. All applicable licenses are listed in a file in the source repository called openafs/doc/LICENSE.
Development
The contributors over the last five years have made significant improvements to both the implementation and the AFS3 protocol without breaking interoperability with the IBM/Transarc releases. Since that announcement was written, several large development projects have been integrated, such as: 64-bit MS-Windows support, MS-Windows 7 support, Apple-Mac OS X v10.4-v10.9 support, and the demand attach fileserver.
Many development projects are at various stages of completion. The following are several prominent examples:
Fileserver back-end utilizing object storage
rxtcp
rxgk
rxk5
Instrumentation framework
Byte-range locking support
Deployment
The existing user base includes small single server ce |
https://en.wikipedia.org/wiki/Clairaut%27s%20equation | In mathematical analysis, Clairaut's equation (or the Clairaut equation) is a differential equation of the form
where is continuously differentiable. It is a particular case of the Lagrange differential equation. It is named after the French mathematician Alexis Clairaut, who introduced it in 1734.
Solution
To solve Clairaut's equation, one differentiates with respect to , yielding
so
Hence, either
or
In the former case, for some constant . Substituting this into the Clairaut's equation, one obtains the family of straight line functions given by
the so-called general solution of Clairaut's equation.
The latter case,
defines only one solution , the so-called singular solution, whose graph is the envelope of the graphs of the general solutions. The singular solution is usually represented using parametric notation, as , where .
The parametric description of the singular solution has the form
where is a parameter.
Examples
The following curves represent the solutions to two Clairaut's equations:
In each case, the general solutions are depicted in black while the singular solution is in violet.
Extension
By extension, a first-order partial differential equation of the form
is also known as Clairaut's equation.
See also
D'Alembert's equation
Chrystal's equation
Legendre transformation
Notes
References
.
.
.
Ordinary differential equations |
https://en.wikipedia.org/wiki/Strangeness | In particle physics, strangeness ("S") is a property of particles, expressed as a quantum number, for describing decay of particles in strong and electromagnetic interactions which occur in a short period of time. The strangeness of a particle is defined as:
where n represents the number of strange quarks () and n represents the number of strange antiquarks (). Evaluation of strangeness production has become an important tool in search, discovery, observation and interpretation of quark–gluon plasma (QGP). Strangeness is an excited state of matter and its decay is governed by CKM mixing.
The terms strange and strangeness predate the discovery of the quark, and were adopted after its discovery in order to preserve the continuity of the phrase: strangeness of particles as −1 and anti-particles as +1, per the original definition. For all the quark flavour quantum numbers (strangeness, charm, topness and bottomness) the convention is that the flavour charge and the electric charge of a quark have the same sign. With this, any flavour carried by a charged meson has the same sign as its charge.
Conservation
Strangeness was introduced by Murray Gell-Mann, Abraham Pais, Tadao Nakano and Kazuhiko Nishijima to explain the fact that certain particles, such as the kaons or the hyperons and , were created easily in particle collisions, yet decayed much more slowly than expected for their large masses and large production cross sections. Noting that collisions seemed to always produce pairs of these particles, it was postulated that a new conserved quantity, dubbed "strangeness", was preserved during their creation, but not conserved in their decay.
In our modern understanding, strangeness is conserved during the strong and the electromagnetic interactions, but not during the weak interactions. Consequently, the lightest particles containing a strange quark cannot decay by the strong interaction, and must instead decay via the much slower weak interaction. In most cases |
https://en.wikipedia.org/wiki/127%20%28number%29 | 127 (one hundred [and] twenty-seven') is the natural number following 126 and preceding 128. It is also a prime number.
In mathematics
As a Mersenne prime, 127 is related to the perfect number 8128. 127 is also the largest known Mersenne prime exponent for a Mersenne number, , which is also a Mersenne prime. It was discovered by Édouard Lucas in 1876 and held the record for the largest known prime for 75 years.
is the largest prime ever discovered by hand calculations as well as the largest known double Mersenne prime.
Furthermore, 127 is equal to , and 7 is equal to , and 3 is the smallest Mersenne prime, making 7 the smallest double Mersenne prime and 127 the smallest triple Mersenne prime.
There are a total of 127 prime numbers between 2,000 and 3,000.
127 is also a cuban prime of the form , . The next prime is 131, with which it comprises a cousin prime. Because the next odd number, 129, is a semiprime, 127 is a Chen prime. 127 is greater than the arithmetic mean of its two neighboring primes; thus, it is a strong prime.
127 is a centered hexagonal number.
It is the seventh Motzkin number.
127 is a palindromic prime in nonary and binary.
127 is the first Friedman prime in decimal. It is also the first nice Friedman number in decimal, since , as well as binary since .
127 is the sum of the sums of the divisors of the first twelve positive integers.
127 is the smallest prime that can be written as the sum of the first two or more odd primes: .
127 is the smallest odd number that cannot be written in the form , for is a prime number, and is an integer, since and are all composite numbers.
127 is an isolated prime where neither nor is prime.
127 is the smallest digitally delicate prime in binary.
127 is the 31st prime number and therefore it is the smallest Mersenne prime with a Mersenne prime index.
127 is the largest number with the property where is the th prime number. There are only two numbers with that property; the other one is 43.
127 is |
https://en.wikipedia.org/wiki/Mitchell%27s%20embedding%20theorem | Mitchell's embedding theorem, also known as the Freyd–Mitchell theorem or the full embedding theorem, is a result about abelian categories; it essentially states that these categories, while rather abstractly defined, are in fact concrete categories of modules. This allows one to use element-wise diagram chasing proofs in these categories. The theorem is named after Barry Mitchell and Peter Freyd.
Details
The precise statement is as follows: if A is a small abelian category, then there exists a ring R (with 1, not necessarily commutative) and a full, faithful and exact functor F: A → R-Mod (where the latter denotes the category of all left R-modules).
The functor F yields an equivalence between A and a full subcategory of R-Mod in such a way that kernels and cokernels computed in A correspond to the ordinary kernels and cokernels computed in R-Mod. Such an equivalence is necessarily additive.
The theorem thus essentially says that the objects of A can be thought of as R-modules, and the morphisms as R-linear maps, with kernels, cokernels, exact sequences and sums of morphisms being determined as in the case of modules. However, projective and injective objects in A do not necessarily correspond to projective and injective R-modules.
Sketch of the proof
Let be the category of left exact functors from the abelian category to the category of abelian groups . First we construct a contravariant embedding by for all , where is the covariant hom-functor, . The Yoneda Lemma states that is fully faithful and we also get the left exactness of very easily because is already left exact. The proof of the right exactness of is harder and can be read in Swan, Lecture Notes in Mathematics 76.
After that we prove that is an abelian category by using localization theory (also Swan). This is the hard part of the proof.
It is easy to check that the abelian category is an AB5 category with a generator
.
In other words it is a Grothendieck category and therefore has an |
https://en.wikipedia.org/wiki/Thoralf%20Skolem | Thoralf Albert Skolem (; 23 May 1887 – 23 March 1963) was a Norwegian mathematician who worked in mathematical logic and set theory.
Life
Although Skolem's father was a primary school teacher, most of his extended family were farmers. Skolem attended secondary school in Kristiania (later renamed Oslo), passing the university entrance examinations in 1905. He then entered Det Kongelige Frederiks Universitet to study mathematics, also taking courses in physics, chemistry, zoology and botany.
In 1909, he began working as an assistant to the physicist Kristian Birkeland, known for bombarding magnetized spheres with electrons and obtaining aurora-like effects; thus Skolem's first publications were physics papers written jointly with Birkeland. In 1913, Skolem passed the state examinations with distinction, and completed a dissertation titled Investigations on the Algebra of Logic. He also traveled with Birkeland to the Sudan to observe the zodiacal light. He spent the winter semester of 1915 at the University of Göttingen, at the time the leading research center in mathematical logic, metamathematics, and abstract algebra, fields in which Skolem eventually excelled. In 1916 he was appointed a research fellow at Det Kongelige Frederiks Universitet. In 1918, he became a Docent in Mathematics and was elected to the Norwegian Academy of Science and Letters.
Skolem did not at first formally enroll as a Ph.D. candidate, believing that the Ph.D. was unnecessary in Norway. He later changed his mind and submitted a thesis in 1926, titled Some theorems about integral solutions to certain algebraic equations and inequalities. His notional thesis advisor was Axel Thue, even though Thue had died in 1922.
In 1927, he married Edith Wilhelmine Hasvold.
Skolem continued to teach at Det kongelige Frederiks Universitet (renamed the University of Oslo in 1939) until 1930 when he became a Research Associate in Chr. Michelsen Institute in Bergen. This senior post allowed Skolem to condu |
https://en.wikipedia.org/wiki/Exact%20functor | In mathematics, particularly homological algebra, an exact functor is a functor that preserves short exact sequences. Exact functors are convenient for algebraic calculations because they can be directly applied to presentations of objects. Much of the work in homological algebra is designed to cope with functors that fail to be exact, but in ways that can still be controlled.
Definitions
Let P and Q be abelian categories, and let be a covariant additive functor (so that, in particular, F(0) = 0). We say that F is an exact functor if whenever
is a short exact sequence in P then
is a short exact sequence in Q. (The maps are often omitted and implied, and one says: "if 0→A→B→C→0 is exact, then 0→F(A)→F(B)→F(C)→0 is also exact".)
Further, we say that F is
left-exact if whenever 0→A→B→C→0 is exact then 0→F(A)→F(B)→F(C) is exact;
right-exact if whenever 0→A→B→C→0 is exact then F(A)→F(B)→F(C)→0 is exact;
half-exact if whenever 0→A→B→C→0 is exact then F(A)→F(B)→F(C) is exact. This is distinct from the notion of a topological half-exact functor.
If G is a contravariant additive functor from P to Q, we similarly define G to be
exact if whenever 0→A→B→C→0 is exact then 0→G(C)→G(B)→G(A)→0 is exact;
left-exact if whenever 0→A→B→C→0 is exact then 0→G(C)→G(B)→G(A) is exact;
right-exact if whenever 0→A→B→C→0 is exact then G(C)→G(B)→G(A)→0 is exact;
half-exact if whenever 0→A→B→C→0 is exact then G(C)→G(B)→G(A) is exact.
It is not always necessary to start with an entire short exact sequence 0→A→B→C→0 to have some exactness preserved. The following definitions are equivalent to the ones given above:
F is exact if and only if A→B→C exact implies F(A)→F(B)→F(C) exact;
F is left-exact if and only if 0→A→B→C exact implies 0→F(A)→F(B)→F(C) exact (i.e. if "F turns kernels into kernels");
F is right-exact if and only if A→B→C→0 exact implies F(A)→F(B)→F(C)→0 exact (i.e. if "F turns cokernels into cokernels");
G is left-exact if and only if A→B→C→0 exact implies 0→G |
https://en.wikipedia.org/wiki/Integral%20test%20for%20convergence | In mathematics, the integral test for convergence is a method used to test infinite series of monotonous terms for convergence. It was developed by Colin Maclaurin and Augustin-Louis Cauchy and is sometimes known as the Maclaurin–Cauchy test.
Statement of the test
Consider an integer and a function defined on the unbounded interval , on which it is monotone decreasing. Then the infinite series
converges to a real number if and only if the improper integral
is finite. In particular, if the integral diverges, then the series diverges as well.
Remark
If the improper integral is finite, then the proof also gives the lower and upper bounds
for the infinite series.
Note that if the function is increasing, then the function is decreasing and the above theorem applies.
Proof
The proof basically uses the comparison test, comparing the term with the integral of over the intervals
and , respectively.
The monotonous function is continuous almost everywhere. To show this, let . For every , there exists by the density of a so that . Note that this set contains an open non-empty interval precisely if is discontinuous at . We can uniquely identify as the rational number that has the least index in an enumeration and satisfies the above property. Since is monotone, this defines an injective mapping and thus is countable. It follows that is continuous almost everywhere. This is sufficient for Riemann integrability.
Since is a monotone decreasing function, we know that
and
Hence, for every integer ,
and, for every integer ,
By summation over all from to some larger integer , we get from ()
and from ()
Combining these two estimates yields
Letting tend to infinity, the bounds in () and the result follow.
Applications
The harmonic series
diverges because, using the natural logarithm, its antiderivative, and the fundamental theorem of calculus, we get
On the other hand, the series
(cf. Riemann zeta function)
converges for every , because by the pow |
https://en.wikipedia.org/wiki/Monocyte | Monocytes are a type of leukocyte or white blood cell. They are the largest type of leukocyte in blood and can differentiate into macrophages and monocyte-derived dendritic cells. As a part of the vertebrate innate immune system monocytes also influence adaptive immune responses and exert tissue repair functions. There are at least three subclasses of monocytes in human blood based on their phenotypic receptors.
Structure
Monocytes are amoeboid in appearance, and have nongranulated cytoplasm. Thus they are classified as agranulocytes, although they might occasionally display some azurophil granules and/or vacuoles. With a diameter of 15–22 μm, monocytes are the largest cell type in peripheral blood. Monocytes are mononuclear cells and the ellipsoidal nucleus is often lobulated/indented, causing a bean-shaped or kidney-shaped appearance. Monocytes compose 2% to 10% of all leukocytes in the human body.
Development
Monocytes are produced by the bone marrow from precursors called monoblasts, bipotent cells that differentiated from hematopoietic stem cells. Monocytes circulate in the bloodstream for about one to three days and then typically migrate into tissues throughout the body where they differentiate into macrophages and dendritic cells.
Subpopulations
In humans
The first clear description of monocyte subsets by flow cytometry dates back to the late 1980s, when a population of CD16-positive monocytes was described. Today, three types of monocytes are recognized in human blood:
The classical monocyte is characterized by high level expression of the CD14 cell surface receptor (CD14++ CD16− monocyte)
The non-classical monocyte shows low level expression of CD14 and additional co-expression of the CD16 receptor (CD14+CD16++ monocyte).
The intermediate monocyte expresses high levels of CD14 and low levels of CD16 (CD14++CD16+ monocytes).
While in humans the level of CD14 expression can be used to differentiate non-classical and intermediate monocytes, the sla |
https://en.wikipedia.org/wiki/Abstraction%20%28mathematics%29 | Abstraction in mathematics is the process of extracting the underlying structures, patterns or properties of a mathematical concept, removing any dependence on real world objects with which it might originally have been connected, and generalizing it so that it has wider applications or matching among other abstract descriptions of equivalent phenomena. Two of the most highly abstract areas of modern mathematics are category theory and model theory.
Description
Many areas of mathematics began with the study of real world problems, before the underlying rules and concepts were identified and defined as abstract structures. For example, geometry has its origins in the calculation of distances and areas in the real world, and algebra started with methods of solving problems in arithmetic.
Abstraction is an ongoing process in mathematics and the historical development of many mathematical topics exhibits a progression from the concrete to the abstract. For example, the first steps in the abstraction of geometry were historically made by the ancient Greeks, with Euclid's Elements being the earliest extant documentation of the axioms of plane geometry—though Proclus tells of an earlier axiomatisation by Hippocrates of Chios. In the 17th century, Descartes introduced Cartesian co-ordinates which allowed the development of analytic geometry. Further steps in abstraction were taken by Lobachevsky, Bolyai, Riemann and Gauss, who generalised the concepts of geometry to develop non-Euclidean geometries. Later in the 19th century, mathematicians generalised geometry even further, developing such areas as geometry in n dimensions, projective geometry, affine geometry and finite geometry. Finally Felix Klein's "Erlangen program" identified the underlying theme of all of these geometries, defining each of them as the study of properties invariant under a given group of symmetries. This level of abstraction revealed connections between geometry and abstract algebra.
In mathemati |
https://en.wikipedia.org/wiki/Analytical%20mechanics | In theoretical physics and mathematical physics, analytical mechanics, or theoretical mechanics is a collection of closely related alternative formulations of classical mechanics. It was developed by many scientists and mathematicians during the 18th century and onward, after Newtonian mechanics. Since Newtonian mechanics considers vector quantities of motion, particularly accelerations, momenta, forces, of the constituents of the system, an alternative name for the mechanics governed by Newton's laws and Euler's laws is vectorial mechanics.
By contrast, analytical mechanics uses scalar properties of motion representing the system as a whole—usually its total kinetic energy and potential energy—not Newton's vectorial forces of individual particles. A scalar is a quantity, whereas a vector is represented by quantity and direction. The equations of motion are derived from the scalar quantity by some underlying principle about the scalar's variation.
Analytical mechanics takes advantage of a system's constraints to solve problems. The constraints limit the degrees of freedom the system can have, and can be used to reduce the number of coordinates needed to solve for the motion. The formalism is well suited to arbitrary choices of coordinates, known in the context as generalized coordinates. The kinetic and potential energies of the system are expressed using these generalized coordinates or momenta, and the equations of motion can be readily set up, thus analytical mechanics allows numerous mechanical problems to be solved with greater efficiency than fully vectorial methods. It does not always work for non-conservative forces or dissipative forces like friction, in which case one may revert to Newtonian mechanics.
Two dominant branches of analytical mechanics are Lagrangian mechanics (using generalized coordinates and corresponding generalized velocities in configuration space) and Hamiltonian mechanics (using coordinates and corresponding momenta in phase space). |
https://en.wikipedia.org/wiki/Windows%209x | Windows 9x is a generic term referring to a series of Microsoft Windows computer operating systems produced from 1995 to 2000, which were based on the Windows 95 kernel and its underlying foundation of MS-DOS, both of which were updated in subsequent versions. The first version in the 9x series was Windows 95, which was succeeded by Windows 98 and then Windows Me, which was the third and last version of Windows on the 9x line, until the series was superseded by Windows XP.
Windows 9x is predominantly known for its use in home desktops. In 1998, Windows made up 82% of operating system market share.
Internal release versions for versions of Windows 9x are 4.x. The internal versions for Windows 95, 98, and Me are 4.0, 4.1, and 4.9, respectively. Previous MS-DOS-based versions of Windows used version numbers of 3.2 or lower. Windows NT, which was aimed at professional users such as networks and businesses, used a similar but separate version number between 3.1 and 4.0. All versions of Windows from Windows XP onwards are based on the Windows NT codebase.
History
Windows prior to 95
The first independent version of Microsoft Windows, version 1.0, released on November 20, 1985, achieved little popularity. Its name was initially "Interface Manager", but Rowland Hanson, the head of marketing at Microsoft, convinced the company that the name Windows would be more appealing to consumers. Windows 1.0 was not a complete operating system, but rather an "operating environment" that extended MS-DOS. Consequently, it shared the inherent flaws and problems of MS-DOS.
The second installment of Microsoft Windows, version 2.0, was released on December 9, 1987, and used the real-mode memory model, which confined it to a maximum of 1 megabyte of memory. In such a configuration, it could run under another multitasking system like DESQview, which used the 286 Protected Mode.
Microsoft Windows scored a significant success with Windows 3.0, released in 1990. In addition to improved capa |
https://en.wikipedia.org/wiki/Nine%20lemma | In mathematics, the nine lemma (or 3×3 lemma) is a statement about commutative diagrams and exact sequences valid in the category of groups and any abelian category. It states: if the diagram to the right is a commutative diagram and all columns as well as the two bottom rows are exact, then the top row is exact as well. Likewise, if all columns as well as the two top rows are exact, then the bottom row is exact as well. Similarly, because the diagram is symmetric about its diagonal, rows and columns may be interchanged in the above as well.
The nine lemma can be proved by direct diagram chasing, or by applying the snake lemma (to the two bottom rows in the first case, and to the two top rows in the second case).
Linderholm (p. 201) offers a satirical view of the nine lemma:
"Draw a noughts-and-crosses board... Do not fill it in with noughts and crosses... Instead, use curved arrows... Wave your hands about in complicated patterns over this board. Make some noughts, but not in the squares; put them at both ends of the horizontal and vertical lines. Make faces. You have now proved:
(a) the Nine Lemma
(b) the Sixteen Lemma
(c) the Twenty-five Lemma..."
There are two variants of nine lemma: sharp nine lemma and symmetric nine lemma (see Lemmas 3.3, 3.4 in Chapter XII of ).
References
Homological algebra
Lemmas in category theory |
https://en.wikipedia.org/wiki/Alienators%3A%20Evolution%20Continues | Alienators: Evolution Continues (also known as Evolution: The Animated Series in some regions) is an animated comic science fiction television series. It is a continuation of the 2001 Ivan Reitman science fiction film, Evolution. 26 episodes were produced.
Created by Louis Gassin, the series is produced by DIC Entertainment, L.P. in association with The Montecito Picture Company, Columbia TriStar Television and Dentsu Inc., the latter of which handled distribution of the series in Asian territories. Although the Evolution intellectual property was owned by DreamWorks (and is now the property of Paramount Pictures which own the studio's pre-2010 live-action back catalog), the series itself is now owned by WildBrain (formerly DHX Media).
The series was the basis for a Game Boy Advance game, that was developed by Digital Eclipse and published by Activision.
Premise
As with the film, the premise of the series is that a meteor carrying organisms that evolve at a very quick rate crashes into the Arizona Desert; single-celled alien organisms on this meteor quickly evolve into monstrous creatures, dubbed the "Genus". A team of scientists, the Alienators, must eliminate all these creatures before they destroy all life on Earth. The characters, Ira Kane, Harry Block, Lucy Mai, and Wayne Green must study all these extraterrestrial lifeforms and find a way to defeat them all. The "Genus" is led by a "humanoid manifestation" named Scopes.
Characters
Scientist Ira Kane – his expertise, along with his excitement and passion about science, helps him lead the team in eliminating the Genus.
Scientist Harry Block – a coach who always keeps up on game scores, etc., and loves coaching the women's team. His strategies help the team to victory.
Lt. Lucy Mai – a tough and aggressive but benevolent lieutenant trained by a Special Forces team called The Blue Berets. She is very talented with martial arts and acrobatic. Lucy uses many kinds of gymnastics (such as handsprings, cartwh |
https://en.wikipedia.org/wiki/Extreme%20point | In mathematics, an extreme point of a convex set in a real or complex vector space is a point in that does not lie in any open line segment joining two points of In linear programming problems, an extreme point is also called vertex or corner point of
Definition
Throughout, it is assumed that is a real or complex vector space.
For any say that and if and there exists a such that
If is a subset of and then is called an of if it does not lie between any two points of That is, if there does exist and such that and The set of all extreme points of is denoted by
Generalizations
If is a subset of a vector space then a linear sub-variety (that is, an affine subspace) of the vector space is called a if meets (that is, is not empty) and every open segment whose interior meets is necessarily a subset of A 0-dimensional support variety is called an extreme point of
Characterizations
The of two elements and in a vector space is the vector
For any elements and in a vector space, the set is called the or between and The or between and is when while it is when The points and are called the of these interval. An interval is said to be a or a if its endpoints are distinct. The is the midpoint of its endpoints.
The closed interval is equal to the convex hull of if (and only if) So if is convex and then
If is a nonempty subset of and is a nonempty subset of then is called a of if whenever a point lies between two points of then those two points necessarily belong to
Examples
If are two real numbers then and are extreme points of the interval However, the open interval has no extreme points.
Any open interval in has no extreme points while any non-degenerate closed interval not equal to does have extreme points (that is, the closed interval's endpoint(s)). More generally, any open subset of finite-dimensional Euclidean space has no extreme points.
The extreme points of the closed unit |
https://en.wikipedia.org/wiki/160%20%28number%29 | 160 (one hundred [and] sixty) is the natural number following 159 and preceding 161.
In mathematics
160 is the sum of the first 11 primes, as well as the sum of the cubes of the first three primes.
Given 160, the Mertens function returns 0. 160 is the smallest number n with exactly 12 solutions to the equation φ(x) = n.
In telecommunications
The number of characters permitted in a standard short message service
The number for Dial-a-Disc (1966–1991), a telephone number operated by the General Post Office in the United Kingdom, which enabled callers to hear the latest chart hits
See also
160s
List of highways numbered 160
United Nations Security Council Resolution 160
United States Supreme Court cases, Volume 160
Article 160 of the Constitution of Malaysia
Norris School District 160, Lancaster County, Nebraska
References
External links
Number Facts and Trivia: 160
Integers |
https://en.wikipedia.org/wiki/170%20%28number%29 | 170 (one hundred [and] seventy) is the natural number following 169 and preceding 171.
In mathematics
170 is the smallest n for which φ(n) and σ(n) are both square (64 and 324 respectively). But 170 is never a solution for φ(x), making it a nontotient. Nor is it ever a solution to x - φ(x), making it a noncototient.
170 is a repdigit in base 4 (2222) and base 16 (AA), as well as in bases 33, 84, and 169. It is also a sphenic number.
170 is the largest integer for which its factorial can be stored in IEEE 754 double-precision floating-point format. This is probably why it is also the largest factorial that Google's built-in calculator will calculate, returning the answer as 170! = 7.25741562 × 10306.
There are 170 different cyclic Gilbreath permutations on 12 elements, and therefore there are 170 different real periodic points of order 12 on the Mandelbrot set.
See also
170s
E170 (disambiguation)
F170 (disambiguation)
List of highways numbered 170
United States Supreme Court cases, Volume 170
United Nations Security Council Resolution 170
Pennsylvania House of Representatives, District 170
References
External links
The Number 170
Number Facts and Trivia: 170
The Positive Integer 170
Prime curiosities: 170
Integers |
https://en.wikipedia.org/wiki/180%20%28number%29 | 180 (one hundred [and] eighty) is the natural number following 179 and preceding 181.
In mathematics
180 is an abundant number, with its proper divisors summing up to 366. 180 is also a highly composite number, a positive integer with more divisors than any smaller positive integer. One of the consequences of 180 having so many divisors is that it is a practical number, meaning that any positive number smaller than 180 that is not a divisor of 180 can be expressed as the sum of some of 180's divisors. 180 is a Harshad number and a refactorable number.
180 is the sum of two square numbers: 122 + 62. It can be expressed as either the sum of six consecutive prime numbers: 19 + 23 + 29 + 31 + 37 + 41, or the sum of eight consecutive prime numbers: 11 + 13 + 17 + 19 + 23 + 29 + 31 + 37. 180 is an Ulam number, which can be expressed as a sum of earlier terms in the Ulam sequence only as 177 + 3.
180 is a 61-gonal number, while 61 is the 18th prime number.
Half a circle has 180 degrees, and thus a U-turn is also referred to as a 180.
Summing Euler's totient function φ(x) over the first + 24 integers gives 180.
In binary it is a digitally balanced number, since its binary representation has the same number of zeros as ones (10110100).
A triangle has three interior angles that collectively total 180 degrees. In general, the interior angles of an -sided polygon add to degrees.
In religion
The Book of Genesis says that Isaac died at the age of 180.
Other
180 is the highest score possible with three darts.
See also
List of highways numbered 180
United Nations Security Council Resolution 180
United States Supreme Court cases, Volume 180
Pennsylvania House of Representatives, District 180
References
External links
Integers |
https://en.wikipedia.org/wiki/Network%20Based%20Application%20Recognition | Network Based Application Recognition (NBAR) is the mechanism used by some Cisco routers and switches to recognize a dataflow by inspecting some packets sent.
The networking equipment which uses NBAR does a deep packet inspection on some of the packets in a dataflow, to determine which traffic category the flow belongs to. Used in conjunction with other features, it may then program the internal application-specific integrated circuits (ASICs) to handle this flow appropriately. The categorization may be done with Open Systems Interconnection (OSI) layer 4 info, packet content, signaling, and so on but some new applications have made it difficult on purpose to cling to this kind of tagging.
The NBAR approach is useful in dealing with malicious software using known ports to fake being "priority traffic", as well as non-standard applications using dynamic ports. That's why NBAR is also known as OSI layer 7 categorization.
On Cisco routers, NBAR is mainly used for quality of service and network security purposes.
References
External links
Network Based Application Recognition: RTP Payload Classification, Cisco.
Block P2P Traffic on a Cisco IOS Router using NBAR Configuration Example, Cisco.
Computer network security |
https://en.wikipedia.org/wiki/190%20%28number%29 | 190 (one hundred [and] ninety) is the natural number following 189 and preceding 191.
In mathematics
190 is a triangular number, a hexagonal number, and a centered nonagonal number, the fourth figurate number (after 1, 28, and 91) with that combination of properties. It is also a truncated square pyramid number.
Integers from 191 to 199
191
191 is a prime number.
192
192 = 26 × 3 is a 3-smooth number, the smallest number with 14 divisors.
193
193 is a prime number.
194
194 = 2 × 97 is a Markov number, the smallest number written as the sum of three squares in five ways, and the number of irreducible representations of the Monster group.
195
195 = 3 × 5 × 13 is the smallest number expressed as a sum of distinct squares in 16 different ways.
196
196 = 22 × 72 is a square number.
197
197 is a prime number and a Schröder–Hipparchus number.
198
198 = 2 × 32 × 11 is the smallest number written as the sum of four squares in ten ways.
No integer factorial ever ends in exactly 198 zeroes in base 10 or in base 12.
There are 198 ridges on a U.S. dollar coin.
199
199 is a prime number and a centered triangular number.
In other fields
190 is the telephonic number of the 27 Brazilian Military Polices.
See also
190 (disambiguation)
References
Integers |
https://en.wikipedia.org/wiki/Deep%20packet%20inspection | Deep packet inspection (DPI) is a type of data processing that inspects in detail the data being sent over a computer network, and may take actions such as alerting, blocking, re-routing, or logging it accordingly. Deep packet inspection is often used for baselining application behavior, analyzing network usage, troubleshooting network performance, ensuring that data is in the correct format, checking for malicious code, eavesdropping, and internet censorship, among other purposes. There are multiple headers for IP packets; network equipment only needs to use the first of these (the IP header) for normal operation, but use of the second header (such as TCP or UDP) is normally considered to be shallow packet inspection (usually called stateful packet inspection) despite this definition.
There are multiple ways to acquire packets for deep packet inspection. Using port mirroring (sometimes called Span Port) is a very common way, as well as physically inserting a network tap which duplicates and sends the data stream to an analyzer tool for inspection.
Deep Packet Inspection (and filtering) enables advanced network management, user service, and security functions as well as internet data mining, eavesdropping, and internet censorship. Although DPI has been used for Internet management for many years, some advocates of net neutrality fear that the technique may be used anticompetitively or to reduce the openness of the Internet.
DPI is used in a wide range of applications, at the so-called "enterprise" level (corporations and larger institutions), in telecommunications service providers, and in governments.
Background
DPI technology boasts a long and technologically advanced history, starting in the 1990s, before the technology entered what is seen today as common, mainstream deployments. The technology traces its roots back over 30 years, when many of the pioneers contributed their inventions for use among industry participants, such as through common standards and |
https://en.wikipedia.org/wiki/Data%20engineering | Data engineering refers to the building of systems to enable the collection and usage of data. This data is usually used to enable subsequent analysis and data science; which often involves machine learning. Making the data usable usually involves substantial compute and storage, as well as data processing
History
Around the 1970s/1980s the term information engineering methodology (IEM) was created to describe database design and the use of software for data analysis and processing. These techniques were intended to be used by database administrators (DBAs) and by systems analysts based upon an understanding of the operational processing needs of organizations for the 1980s. In particular, these techniques were meant to help bridge the gap between strategic business planning and information systems. A key early contributor (often called the "father" of information engineering methodology) was the Australian Clive Finkelstein, who wrote several articles about it between 1976 and 1980, and also co-authored an influential Savant Institute report on it with James Martin. Over the next few years, Finkelstein continued work in a more business-driven direction, which was intended to address a rapidly changing business environment; Martin continued work in a more data processing-driven direction. From 1983 to 1987, Charles M. Richter, guided by Clive Finkelstein, played a significant role in revamping IEM as well as helping to design the IEM software product (user data), which helped automate IEM.
In the early 2000s, the data and data tooling was generally held by the information technology (IT) teams in most companies. Other teams then used data for their work (e.g. reporting), and there was usually little overlap in data skillset between these parts of the business.
In the early 2010s, with the rise of the internet, the massive increase in data volumes, velocity, and variety led to the term big data to describe the data itself, and data-driven tech companies like Face |
https://en.wikipedia.org/wiki/Geographical%20pole | A geographical pole or geographic pole is either of the two points on Earth where its axis of rotation intersects its surface. The North Pole lies in the Arctic Ocean while the South Pole is in Antarctica. North and South poles are also defined for other planets or satellites in the Solar System, with a North pole being on the same side of the invariable plane as Earth's North pole.
Relative to Earth's surface, the geographic poles move by a few metres over periods of a few years. This is a combination of Chandler wobble, a free oscillation with a period of about 433 days; an annual motion responding to seasonal movements of air and water masses; and an irregular drift towards the 80th west meridian. As cartography requires exact and unchanging coordinates, the averaged locations of geographical poles are taken as fixed cartographic poles and become the points where the body's great circles of longitude intersect.
See also
Earth's rotation
Polar motion
Poles of astronomical bodies
True polar wander
References
Geodesy
Polar regions of the Earth |
https://en.wikipedia.org/wiki/Mass%20storage | In computing, mass storage refers to the storage of large amounts of data in a persisting and machine-readable fashion. In general, the term is used as large in relation to contemporaneous hard disk drives, but it has been used large in relation to primary memory as for example with floppy disks on personal computers.
Devices and/or systems that have been described as mass storage include tape libraries, RAID systems, and a variety of computer drives such as hard disk drives, magnetic tape drives, magneto-optical disc drives, optical disc drives, memory cards, and solid-state drives. It also includes experimental forms like holographic memory. Mass storage includes devices with removable and non-removable media. It does not include random access memory (RAM).
There are two broad classes of mass storage: local data in devices such as smartphones or computers, and enterprise servers and data centers for the cloud. For local storage, SSDs are on the way to replacing HDDs. Considering the mobile segment from phones to notebooks, the majority of systems today is based on NAND Flash. As for Enterprise and data centers, storage tiers have established using a mix of SSD and HDD.
Definition
The notion of "large" amounts of data is of course highly dependent on the time frame and the market segment, as storage device capacity has increased by many orders of magnitude since the beginnings of computer technology in the late 1940s and continues to grow; however, in any time frame, common mass storage devices have tended to be much larger and at the same time much slower than common realizations of contemporaneous primary storage technology.
Papers at the 1966 Fall Joint Computer Conference (FJCC) used the term mass storage for devices substantially larger than contemporaneous hard disk drives. Similarly, a 1972 analysis identified mass storage systems from Ampex (Terabit Memory) using video tape, Precision Industries (Unicon 690-212) using lasers and International Video ( |
https://en.wikipedia.org/wiki/IEEE%20802.15.4 | IEEE 802.15.4 is a technical standard which defines the operation of a low-rate wireless personal area network (LR-WPAN). It specifies the physical layer and media access control for LR-WPANs, and is maintained by the IEEE 802.15 working group, which defined the standard in 2003. It is the basis for the Zigbee, ISA100.11a, WirelessHART, MiWi, 6LoWPAN, Thread, Matter and SNAP specifications, each of which further extends the standard by developing the upper layers which are not defined in IEEE 802.15.4. In particular, 6LoWPAN defines a binding for the IPv6 version of the Internet Protocol (IP) over WPANs, and is itself used by upper layers like Thread.
Overview
IEEE standard 802.15.4 intends to offer the fundamental lower network layers of a type of wireless personal area network (WPAN) which focuses on low-cost, low-speed ubiquitous communication between devices. It can be contrasted with other approaches, such as Wi-Fi, which offer more bandwidth and requires more power. The emphasis is on very low cost communication of nearby devices with little to no underlying infrastructure, intending to exploit this to lower power consumption even more.
The basic framework conceives a 10-meter communications range with line-of-sight at a transfer rate of 250 kbit/s. Bandwidth tradeoffs are possible to favor more radically embedded devices with even lower power requirements for increased battery operating time, through the definition of not one, but several physical layers. Lower transfer rates of 20 and 40 kbit/s were initially defined, with the 100 kbit/s rate being added in the current revision.
Even lower rates can be used, which results in lower power consumption. As already mentioned, the main goal of IEEE 802.15.4 regarding WPANs is the emphasis on achieving low manufacturing and operating costs through the use of relatively simple transceivers, while enabling application flexibility and adaptability.
Key 802.15.4 features include:
real-time suitability by res |
https://en.wikipedia.org/wiki/Lusser%27s%20law | Lusser's law in systems engineering is a prediction of reliability. Named after engineer Robert Lusser, and also known as Lusser's product law or the probability product law of series components, it states that the reliability of a series of components is equal to the product of the individual reliabilities of the components, if their failure modes are known to be statistically independent. For a series of N components, this is expressed as:
where Rs is the overall reliability of the system, and rn is the reliability of the nth component.
If the failure probabilities of all components are equal, then as Lusser's colleague Erich Pieruschka observed, this can be expressed simply as:
Lusser's law has been described as the idea that a series system is "weaker than its weakest link", as the product reliability of a series of components can be less than the lowest-value component.
For example, given a series system of two components with different reliabilities — one of 0.95 and the other of 0.8 — Lusser's law will predict a reliability of
which is lower than either of the individual components.
References
Engineering failures
Reliability analysis
Reliability engineering
Statistics articles needing expert attention
Survival analysis
Systems analysis |
https://en.wikipedia.org/wiki/Mini-ITX | Mini-ITX is a motherboard form factor developed by VIA Technologies in 2001. Mini-ITX motherboards have been traditionally used in small-configured computer systems. Originally, Mini-ITX was a niche standard designed for fanless cooling with a low power consumption architecture, which made them useful for home theater PC systems, where fan noise can detract from the cinema experience.
The four mounting holes in a Mini-ITX board line up with four of the holes in ATX-specification motherboards, and the locations of the backplate and expansion slot are the same (though one of the holes used was optional in earlier versions of the ATX spec). Mini-ITX boards can therefore often be used in cases designed for ATX, micro-ATX and other ATX variants if desired.
Mini-ITX motherboards have only one expansion slot. Earlier Mini-ITX motherboards had a standard 33 MHz 5V 32-bit PCI slot, whereas newer motherboards use a PCI Express slot. Many older case designs use riser cards and some even have two-slot riser cards, although the two-slot riser cards are not compatible with all boards. Some boards based around non-x86 processors have a 3.3V PCI slot, and the Mini-ITX 2.0 (2008) boards have a PCI-Express ×16 slot; these boards are not compatible with the standard PCI riser cards supplied with older ITX (Information Technology eXtended) cases.
The HiFive Unmatched RISC-V computer uses a Mini-ITX form factor.
History
In March 2001, the chipset manufacturer VIA Technologies released a reference design for an ITX motherboard, to promote the low power C3 processor they had bought from Centaur Technology, in combination with their chipsets. Designed by Robert Kuo, VIA's chief R&D expert, the 215×191 mm VT6009 ITX Reference Board was demonstrated in "Information PC" and set-top box configurations. At that point, few manufacturers took up the ITX design, but Shuttle, Jetway, etc. produced many ITX based cube computers. Other manufactures instead produced smaller boards based on the |
https://en.wikipedia.org/wiki/Von%20Neumann%20bicommutant%20theorem | In mathematics, specifically functional analysis, the von Neumann bicommutant theorem relates the closure of a set of bounded operators on a Hilbert space in certain topologies to the bicommutant of that set. In essence, it is a connection between the algebraic and topological sides of operator theory.
The formal statement of the theorem is as follows:
Von Neumann bicommutant theorem. Let be an algebra consisting of bounded operators on a Hilbert space , containing the identity operator, and closed under taking adjoints. Then the closures of in the weak operator topology and the strong operator topology are equal, and are in turn equal to the bicommutant of .
This algebra is called the von Neumann algebra generated by .
There are several other topologies on the space of bounded operators, and one can ask what are the *-algebras closed in these topologies. If is closed in the norm topology then it is a C*-algebra, but not necessarily a von Neumann algebra. One such example is the C*-algebra of compact operators (on an infinite dimensional Hilbert space). For most other common topologies the closed *-algebras containing 1 are von Neumann algebras; this applies in particular to the weak operator, strong operator, *-strong operator, ultraweak, ultrastrong, and *-ultrastrong topologies.
It is related to the Jacobson density theorem.
Proof
Let be a Hilbert space and the bounded operators on . Consider a self-adjoint unital subalgebra of (this means that contains the adjoints of its members, and the identity operator on ).
The theorem is equivalent to the combination of the following three statements:
(i)
(ii)
(iii)
where the and subscripts stand for closures in the weak and strong operator topologies, respectively.
Proof of (i)
By definition of the weak operator topology, for any and in , the map T → <Tx, y> is continuous in this topology. Therefore, for any operator (and by substituting once and once ), so is the map
Let S be any subset of |
https://en.wikipedia.org/wiki/Bicommutant | In algebra, the bicommutant of a subset S of a semigroup (such as an algebra or a group) is the commutant of the commutant of that subset. It is also known as the double commutant or second commutant and is written .
The bicommutant is particularly useful in operator theory, due to the von Neumann double commutant theorem, which relates the algebraic and analytic structures of operator algebras. Specifically, it shows that if M is a unital, self-adjoint operator algebra in the C*-algebra B(H), for some Hilbert space H, then the weak closure, strong closure and bicommutant of M are equal. This tells us that a unital C*-subalgebra M of B(H) is a von Neumann algebra if, and only if, , and that if not, the von Neumann algebra it generates is .
The bicommutant of S always contains S. So . On the other hand, . So , i.e. the commutant of the bicommutant of S is equal to the commutant of S. By induction, we have:
and
for n > 1.
It is clear that, if S1 and S2 are subsets of a semigroup,
If it is assumed that and (this is the case, for instance, for von Neumann algebras), then the above equality gives
See also
von Neumann double commutant theorem
References
J. Dixmier, Von Neumann Algebras, North-Holland, Amsterdam, 1981.
Group theory |
https://en.wikipedia.org/wiki/Gaussian%20period | In mathematics, in the area of number theory, a Gaussian period is a certain kind of sum of roots of unity. The periods permit explicit calculations in cyclotomic fields connected with Galois theory and with harmonic analysis (discrete Fourier transform). They are basic in the classical theory called cyclotomy. Closely related is the Gauss sum, a type of exponential sum which is a linear combination of periods.
History
As the name suggests, the periods were introduced by Gauss and were the basis for his theory of compass and straightedge construction. For example, the construction of the heptadecagon (a formula that furthered his reputation) depended on the algebra of such periods, of which
is an example involving the seventeenth root of unity
General definition
Given an integer n > 1, let H be any subgroup of the multiplicative group
of invertible residues modulo n, and let
A Gaussian period P is a sum of the primitive n-th roots of unity , where runs through all of the elements in a fixed coset of H in G.
The definition of P can also be stated in terms of the field trace. We have
for some subfield L of Q(ζ) and some j coprime to n. This corresponds to the previous definition by identifying G and H with the Galois groups of Q(ζ)/Q and Q(ζ)/L, respectively. The choice of j determines the choice of coset of H in G in the previous definition.
Example
The situation is simplest when n is a prime number p > 2. In that case G is cyclic of order p − 1, and has one subgroup H of order d for every factor d of p − 1. For example, we can take H of index two. In that case H consists of the quadratic residues modulo p. Corresponding to this H we have the Gaussian period
summed over (p − 1)/2 quadratic residues, and the other period P* summed over the (p − 1)/2 quadratic non-residues. It is easy to see that
since the left-hand side adds all the primitive p-th roots of 1. We also know, from the trace definition, that P lies in a quadratic extension |
https://en.wikipedia.org/wiki/List%20of%20scientific%20constants%20named%20after%20people | This is a list of physical and mathematical constants named after people.
Eponymous constants and their influence on scientific citations have been discussed in the literature.
Apéry's constant – Roger Apéry
Archimedes' constant (, pi) – Archimedes
Avogadro constant – Amedeo Avogadro
Balmer's constant – Johann Jakob Balmer
Belphegor's prime – Belphegor (demon)
Bohr magneton – Niels Bohr
Bohr radius – Niels Bohr
Boltzmann constant – Ludwig Boltzmann
Brun's constant – Viggo Brun
Cabibbo angle – Nicola Cabibbo
Chaitin's constant – Gregory Chaitin
Champernowne constant – D. G. Champernowne
Chandrasekhar limit – Subrahmanyan Chandrasekhar
Copeland–Erdős constant – Paul Erdős and Peter Borwein
Coulomb constant (electric force constant, electrostatic constant, ) – Charles-Augustin de Coulomb
Eddington number – Arthur Stanley Eddington
Dunbar's number – Robin Dunbar
Embree–Trefethen constant
Erdős–Borwein constant
Euler–Mascheroni constant () – Leonhard Euler and Lorenzo Mascheroni
Euler's number () – Leonhard Euler
Faraday constant – Michael Faraday
Feigenbaum constants – Mitchell Feigenbaum
Fermi coupling constant – Enrico Fermi
Gauss's constant – Carl Friedrich Gauss
Graham's number – Ronald Graham
Hartree energy – Douglas Hartree
Hubble constant – Edwin Hubble
Josephson constant – Brian David Josephson
Kaprekar's constant – D. R. Kaprekar
Kerr constant – John Kerr
Khinchin's constant – Aleksandr Khinchin
Landau–Ramanujan constant – Edmund Landau and Srinivasa Ramanujan
Legendre's constant (one, 1) – Adrien-Marie Legendre
Loschmidt constant – Johann Josef Loschmidt
Ludolphsche Zahl – Ludolph van Ceulen
Mean of Phidias (golden ratio, , phi) – Phidias
Meissel–Mertens constant
Moser's number
Newtonian constant of gravitation (gravitational constant, ) – Sir Isaac Newton
Planck constant () – Max Planck
Reduced Planck constant or Dirac constant (-bar, ) – Max Planck, Paul Dirac
Ramanujan–Soldner constant – Srinivasa Ramanujan and Jo |
https://en.wikipedia.org/wiki/496%20%28number%29 | 496 (four hundred [and] ninety-six) is the natural number following 495 and preceding 497.
In mathematics
496 is most notable for being a perfect number, and one of the earliest numbers to be recognized as such. As a perfect number, it is tied to the Mersenne prime 31, 25 − 1, with 24 (25 − 1) yielding 496. Also related to its being a perfect number, 496 is a harmonic divisor number, since the number of proper divisors of 496 divided by the sum of the reciprocals of its divisors, 1, 2, 4, 8, 16, 31, 62, 124, 248 and 496, (the harmonic mean), yields an integer, 5 in this case.
A triangular number and a hexagonal number, 496 is also a centered nonagonal number. Being the 31st triangular number, 496 is the smallest counterexample to the hypothesis that one more than an even triangular prime-indexed number is a prime number. It is the largest happy number less than 500.
There is no solution to the equation φ(x) = 496, making 496 a nontotient.
E8 has real dimension 496.
In physics
The number 496 is a very important number in superstring theory. In 1984, Michael Green and John H. Schwarz realized that one of the necessary conditions for a superstring theory to make sense is that the dimension of the gauge group of type I string theory must be 496. The group is therefore SO(32). Their discovery started the first superstring revolution. It was realized in 1985 that the heterotic string can admit another possible gauge group, namely E8 x E8.
Telephone numbers
The UK's Ofcom reserves telephone numbers in many dialing areas in the 496 local block for fictional purposes, such as 0114 496-1234.
See also
AD 496
References
Integers |
https://en.wikipedia.org/wiki/Crystal%20system | In crystallography, a crystal system is a set of point groups (a group of geometric symmetries with at least one fixed point). A lattice system is a set of Bravais lattices. Space groups are classified into crystal systems according to their point groups, and into lattice systems according to their Bravais lattices. Crystal systems that have space groups assigned to a common lattice system are combined into a crystal family.
The seven crystal systems are triclinic, monoclinic, orthorhombic, tetragonal, trigonal, hexagonal, and cubic. Informally, two crystals are in the same crystal system if they have similar symmetries (albeit there are many exceptions).
Classifications
Crystals can be classified in three ways: lattice systems, crystal systems and crystal families. The various classifications are often confused: in particular the trigonal crystal system is often confused with the rhombohedral lattice system, and the term "crystal system" is sometimes used to mean "lattice system" or "crystal family".
Lattice system
A lattice system is a group of lattices with the same set of lattice point groups. The 14 Bravais lattices are grouped into seven lattice systems: triclinic, monoclinic, orthorhombic, tetragonal, rhombohedral, hexagonal, and cubic.
Crystal system
A crystal system is a set of point groups in which the point groups themselves and their corresponding space groups are assigned to a lattice system. Of the 32 crystallographic point groups that exist in three dimensions, most are assigned to only one lattice system, in which case both the crystal and lattice systems have the same name. However, five point groups are assigned to two lattice systems, rhombohedral and hexagonal, because both exhibit threefold rotational symmetry. These point groups are assigned to the trigonal crystal system.
Crystal family
A crystal family is determined by lattices and point groups. It is formed by combining crystal systems that have space groups assigned to a common l |
https://en.wikipedia.org/wiki/Kryptos | Kryptos is a sculpture by the American artist Jim Sanborn located on the grounds of the Central Intelligence Agency (CIA) headquarters, the George Bush Center for Intelligence in Langley, Virginia. Since its dedication on November 3, 1990, there has been much speculation about the meaning of the four encrypted messages it bears. Of these four messages, the first three have been solved, while the fourth message remains one of the most famous unsolved codes in the world. The sculpture continues to be of interest to cryptanalysts, both amateur and professional, who are attempting to decipher the fourth passage. The artist has so far given four clues to this passage.
Description
The main part of the sculpture is located in the northwest corner of the New Headquarters Building courtyard, outside of the Agency's cafeteria. The sculpture comprises four large copper plates with other elements consisting of water, wood, plants, red and green granite, white quartz, and petrified wood. The most prominent feature is a large vertical S-shaped copper screen resembling a scroll or a piece of paper emerging from a computer printer, half of which consists of encrypted text. The characters are all found within the 26 letters of the Latin alphabet, along with question marks, and are cut out of the copper plates. The main sculpture contains four separate enigmatic messages, three of which have been deciphered.
In addition to the main part of the sculpture, Jim Sanborn also placed other pieces of art at the CIA grounds, such as several large granite slabs with sandwiched copper sheets outside the entrance to the New Headquarters Building. Several morse code messages are found on these copper sheets, and one of the stone slabs has an engraving of a compass rose pointing to a lodestone. Other elements of Sanborn's installation include a landscaped garden area, a fish pond with opposing wooden benches, a reflecting pool, and other pieces of stone including a triangle-shaped black stone |
https://en.wikipedia.org/wiki/Computer-aided%20engineering | Computer-aided engineering can be defined as the general usage of technology to aid in tasks related to engineering analysis. Any use of technology to solve or assist engineering issues falls under this umbrella.
Overview
Following alongside the consistent improvement in computer graphics and speed, computer aid assists engineers with once complicated and time consuming tasks with the input of information and a press of a button.
It includes finite element analysis (FEA), computational fluid dynamics (CFD), multibody dynamics (MBD), durability and optimization. It is included with computer-aided design (CAD) and computer-aided manufacturing (CAM) in the collective abbreviation "CAx".
The term CAE has been used to describe the use of computer technology within engineering in a broader sense than just engineering analysis. It was in this context that the term was coined by Jason Lemon, founder of SDRC in the late 1970s. This definition is however better known today by the terms CAx and PLM.
CAE systems are individually considered a single node on a total information network and each node may interact with other nodes on the network.
CAE fields and phases
CAE areas covered include:
Stress analysis on components and assemblies using finite element analysis (FEA);
Thermal and fluid flow analysis computational fluid dynamics (CFD);
Multibody dynamics (MBD) and kinematics;
Analysis tools for process simulation for operations such as casting, molding, and die press forming;
Optimization of the product or process.
In general, there are three phases in any computer-aided engineering task:
Pre-processing – defining the model and environmental factors to be applied to it (typically a finite element model, but facet, voxel, and thin sheet methods are also used);
Analysis solver (usually performed on high powered computers);
Post-processing of results (using visualization tools).
This cycle is iterated, often many times, either manually or with the use of commercial opt |
https://en.wikipedia.org/wiki/Duff%27s%20device | In the C programming language, Duff's device is a way of manually implementing loop unrolling by interleaving two syntactic constructs of C: the - loop and a switch statement. Its discovery is credited to Tom Duff in November 1983, when Duff was working for Lucasfilm and used it to speed up a real-time animation program.
Loop unrolling attempts to reduce the overhead of conditional branching needed to check whether a loop is done, by executing a batch of loop bodies per iteration. To handle cases where the number of iterations is not divisible by the unrolled-loop increments, a common technique among assembly language programmers is to jump directly into the middle of the unrolled loop body to handle the remainder.
Duff implemented this technique in C by using C's case label fall-through feature to jump into the unrolled body.
Original version
Duff's problem was to copy 16-bit unsigned integers ("shorts" in most C implementations) from an array into a memory-mapped output register, denoted in C by a pointer. His original code, in C, looked as follows:
send(to, from, count)
register short *to, *from;
register count;
{
do { /* count > 0 assumed */
*to = *from++;
} while (--count > 0);
}
This code assumes that initial . Since the output location is a memory-mapped register, the pointer is not incremented as would be required for a memory-to-memory copy.
If were always divisible by eight, unrolling this loop eight-fold would produce the following:
send(to, from, count)
register short *to, *from;
register count;
{
register n = count / 8;
do {
*to = *from++;
*to = *from++;
*to = *from++;
*to = *from++;
*to = *from++;
*to = *from++;
*to = *from++;
*to = *from++;
} while (--n > 0);
}
Duff realized that to handle cases where is not divisible by eight, the assembly programmer's technique of jumping into the loop body could be implemented by interlacin |
https://en.wikipedia.org/wiki/Bruck%E2%80%93Ryser%E2%80%93Chowla%20theorem | The Bruck–Ryser–Chowla theorem is a result on the combinatorics of block designs that implies nonexistence of certain kinds of design. It states that if a (v, b, r, k, λ)-design exists with v = b (a symmetric block design), then:
if v is even, then k − λ is a square;
if v is odd, then the following Diophantine equation has a nontrivial solution:
x2 − (k − λ)y2 − (−1)(v−1)/2 λ z2 = 0.
The theorem was proved in the case of projective planes by . It was extended to symmetric designs by .
Projective planes
In the special case of a symmetric design with λ = 1, that is, a projective plane, the theorem (which in this case is referred to as the Bruck–Ryser theorem) can be stated as follows: If a finite projective plane of order q exists and q is congruent to 1 or 2 (mod 4), then q must be the sum of two squares. Note that for a projective plane, the design parameters are v = b = q2 + q + 1, r = k = q + 1, λ = 1. Thus, v is always odd in this case.
The theorem, for example, rules out the existence of projective planes of orders 6 and 14 but allows the existence of planes of orders 10 and 12. Since a projective plane of order 10 has been shown not to exist using a combination of coding theory and large-scale computer search, the condition of the theorem is evidently not sufficient for the existence of a design. However, no stronger general non-existence criterion is known.
Connection with incidence matrices
The existence of a symmetric (v, b, r, k, λ)-design is equivalent to the existence of a v × v incidence matrix R with elements 0 and 1 satisfying
where is the v × v identity matrix and J is the v × v all-1 matrix. In essence, the Bruck–Ryser–Chowla theorem is a statement of the necessary conditions for the existence of a rational v × v matrix R satisfying this equation. In fact, the conditions stated in the Bruck–Ryser–Chowla theorem are not merely necessary, but also sufficient for the existence of such a rational matrix R. They can be derived from t |
https://en.wikipedia.org/wiki/C%20syntax | The syntax of the C programming language is the set of rules governing writing of software in C. It is designed to allow for programs that are extremely terse, have a close relationship with the resulting object code, and yet provide relatively high-level data abstraction. C was the first widely successful high-level language for portable operating-system development.
C syntax makes use of the maximal munch principle.
Data structures
Primitive data types
The C programming language represents numbers in three forms: integral, real and complex. This distinction reflects similar distinctions in the instruction set architecture of most central processing units. Integral data types store numbers in the set of integers, while real and complex numbers represent numbers (or pair of numbers) in the set of real numbers in floating point form.
All C integer types have and variants. If or is not specified explicitly, in most circumstances, is assumed. However, for historic reasons, plain is a type distinct from both and . It may be a signed type or an unsigned type, depending on the compiler and the character set (C guarantees that members of the C basic character set have positive values). Also, bit field types specified as plain may be signed or unsigned, depending on the compiler.
Integer types
C's integer types come in different fixed sizes, capable of representing various ranges of numbers. The type occupies exactly one byte (the smallest addressable storage unit), which is typically 8 bits wide. (Although can represent any of C's "basic" characters, a wider type may be required for international character sets.) Most integer types have both signed and unsigned varieties, designated by the and keywords. Signed integer types may use a two's complement, ones' complement, or sign-and-magnitude representation. In many cases, there are multiple equivalent ways to designate the type; for example, and are synonymous.
The representation of some types may includ |
https://en.wikipedia.org/wiki/Alexandre-Th%C3%A9ophile%20Vandermonde | Alexandre-Théophile Vandermonde (28 February 1735 – 1 January 1796) was a French mathematician, musician, and chemist who worked with Bézout and Lavoisier; his name is now principally associated with determinant theory in mathematics. He was born in Paris, and died there.
Biography
Vandermonde was a violinist, and became engaged with mathematics only around 1770. In Mémoire sur la résolution des équations (1771) he reported on symmetric functions and solution of cyclotomic polynomials; this paper anticipated later Galois theory (see also abstract algebra for the role of Vandermonde in the genesis of group theory). In Remarques sur des problèmes de situation (1771) he studied knight's tours, and presaged the development of knot theory by explicitly noting the importance of topological features when discussing the properties of knots:
"Whatever the twists and turns of a system of threads in space, one can always obtain an expression for the calculation of its dimensions, but this expression will be of little use in practice. The craftsman who fashions a braid, a net, or some knots will be concerned, not with questions of measurement, but with those of position: what he sees there is the manner in which the theads are interlaced"
The same year he was elected to the French Academy of Sciences. Mémoire sur des irrationnelles de différents ordres avec une application au cercle (1772) was on combinatorics, and Mémoire sur l'élimination (1772) on the foundations of determinant theory. These papers were presented to the Académie des Sciences, and constitute all his published mathematical work. The Vandermonde determinant does not make an explicit appearance.
He was professor at the École Normale Supérieure, member of the Conservatoire national des arts et métiers and examiner at the École polytechnique.
Honors
A special class of matrices, the Vandermonde matrices are named after him, as is an elementary fact of combinatorics, Vandermonde's identity.
Vandermonde is th |
https://en.wikipedia.org/wiki/System%206 | System 6 (or System Software 6) is a graphical user interface-based operating system for Macintosh computers, made by Apple Computer. It was released in 1988, and is part of the classic Mac OS series. It is a monolithic operating system, with cooperative multitasking based on an improved MultiFinder. The boxed version cost , and it was included with all new Macintosh computers until 1991, when it was succeeded by System 7.
Overview
MacroMaker
The MacroMaker utility was introduced in System 6. It records mouse and keyboard input as macros, and has a unique user interface intended to look and act like a tape recorder. MacroMaker was criticized for its lack of features when compared to Microsoft's AutoMac III, which was already available commercially. As MacroMaker records only the locations of mouse-clicks inside windows and not what is being clicked on or exactly when, it can not be used to automate actions in more sophisticated programs. The pre-recorded clicks miss buttons if the buttons had moved since the recording, or if they failed to appear upon playback. It records the start and end locations of mouse movements, but does not track the precise path of a movement or support pauses. MacroMaker is not compatible with System 7, in which it is succeeded by AppleScript.
Multitasking
Macintosh gained cooperative multitasking in March 1985 with Andy Hertzfeld's Switcher, which can switch between multiple full-screen applications. It was not integrated, and was only sold separately by Apple. Not many programs and features function correctly with Switcher, and it does not share the screen between applications simultaneously. Systems 5 and 6 have MultiFinder instead, which is much more mature and widely used in System 6. With MultiFinder, the Finder does not quit to free resources, and the system behaves as in the still-familiar multitasking fashion, with the desktop and other applications' windows in the background.
Hardware support
System 6 includes support for the |
https://en.wikipedia.org/wiki/Pure%20mathematics | Pure mathematics is the study of mathematical concepts independently of any application outside mathematics. These concepts may originate in real-world concerns, and the results obtained may later turn out to be useful for practical applications, but pure mathematicians are not primarily motivated by such applications. Instead, the appeal is attributed to the intellectual challenge and aesthetic beauty of working out the logical consequences of basic principles.
While pure mathematics has existed as an activity since at least ancient Greece, the concept was elaborated upon around the year 1900, after the introduction of theories with counter-intuitive properties (such as non-Euclidean geometries and Cantor's theory of infinite sets), and the discovery of apparent paradoxes (such as continuous functions that are nowhere differentiable, and Russell's paradox). This introduced the need to renew the concept of mathematical rigor and rewrite all mathematics accordingly, with a systematic use of axiomatic methods. This led many mathematicians to focus on mathematics for its own sake, that is, pure mathematics.
Nevertheless, almost all mathematical theories remained motivated by problems coming from the real world or from less abstract mathematical theories. Also, many mathematical theories, which had seemed to be totally pure mathematics, were eventually used in applied areas, mainly physics and computer science. A famous early example is Isaac Newton's demonstration that his law of universal gravitation implied that planets move in orbits that are conic sections, geometrical curves that had been studied in antiquity by Apollonius. Another example is the problem of factoring large integers, which is the basis of the RSA cryptosystem, widely used to secure internet communications.
It follows that, presently, the distinction between pure and applied mathematics is more a philosophical point of view or a mathematician's preference rather than a rigid subdivision of mathem |
https://en.wikipedia.org/wiki/Block%20matrix | In mathematics, a block matrix or a partitioned matrix is a matrix that is interpreted as having been broken into sections called blocks or submatrices. Intuitively, a matrix interpreted as a block matrix can be visualized as the original matrix with a collection of horizontal and vertical lines, which break it up, or partition it, into a collection of smaller matrices. Any matrix may be interpreted as a block matrix in one or more ways, with each interpretation defined by how its rows and columns are partitioned.
This notion can be made more precise for an by matrix by partitioning into a collection , and then partitioning into a collection . The original matrix is then considered as the "total" of these groups, in the sense that the entry of the original matrix corresponds in a 1-to-1 way with some offset entry of some , where and .
Block matrix algebra arises in general from biproducts in categories of matrices.
Example
The matrix
can be partitioned into four 2×2 blocks
The partitioned matrix can then be written as
Block matrix multiplication
It is possible to use a block partitioned matrix product that involves only algebra on submatrices of the factors. The partitioning of the factors is not arbitrary, however, and requires "conformable partitions" between two matrices and such that all submatrix products that will be used are defined. Given an matrix with row partitions and column partitions
and a matrix with row partitions and column partitions
that are compatible with the partitions of , the matrix product
can be performed blockwise, yielding as an matrix with row partitions and column partitions. The matrices in the resulting matrix are calculated by multiplying:
Or, using the Einstein notation that implicitly sums over repeated indices:
Block matrix inversion
If a matrix is partitioned into four blocks, it can be inverted blockwise as follows:
where A and D are square blocks of arbitrary size, and B and C are conformabl |
https://en.wikipedia.org/wiki/Bi-quinary%20coded%20decimal | Bi-quinary coded decimal is a numeral encoding scheme used in many abacuses and in some early computers, including the Colossus. The term bi-quinary indicates that the code comprises both a two-state (bi) and a five-state (quinary) component. The encoding resembles that used by many abacuses, with four beads indicating the five values either from 0 through 4 or from 5 through 9 and another bead indicating which of those ranges (which can alternatively be thought of as +5).
Several human languages, most notably Fula and Wolof also use biquinary systems. For example, the Fula word for 6, jowi e go'o, literally means five [plus] one. Roman numerals use a symbolic, rather than positional, bi-quinary base, even though Latin is completely decimal.
The Korean finger counting system Chisanbop uses a bi-quinary system, where each finger represents a one and a thumb represents a five, allowing one to count from 0 to 99 with two hands.
One advantage of one bi-quinary encoding scheme on digital computers is that it must have 2 bits set (one in the binary field and one in the quinary field), providing a built in checksum to verify if the number is valid or not. (Stuck bits happened frequently with computers using mechanical relays.)
Examples
Several different representations of bi-quinary coded decimal have been used by different machines. The two-state component is encoded as one or two bits, and the five-state component is encoded using three to five bits. Some examples are:
Roman and Chinese abacuses
Stibitz relay calculators at Bell Labs from Model II onwards
FACOM 128 relay calculators at Fujitsu
IBM 650 – seven bits
Two bi bits: 0 5 and five quinary bits: 0 1 2 3 4, with error checking.
Exactly one bi bit and one quinary bit is set in a valid digit. In the pictures of the front panel below and in close-up, the bi-quinary encoding of the internal workings of the machine are evident in the arrangement of the lights – the bi bits form the top of a T for each digit |
https://en.wikipedia.org/wiki/Excess-3 | Excess-3, 3-excess or 10-excess-3 binary code (often abbreviated as XS-3, 3XS or X3), shifted binary or Stibitz code (after George Stibitz, who built a relay-based adding machine in 1937) is a self-complementary binary-coded decimal (BCD) code and numeral system. It is a biased representation. Excess-3 code was used on some older computers as well as in cash registers and hand-held portable electronic calculators of the 1970s, among other uses.
Representation
Biased codes are a way to represent values with a balanced number of positive and negative numbers using a pre-specified number N as a biasing value. Biased codes (and Gray codes) are non-weighted codes. In excess-3 code, numbers are represented as decimal digits, and each digit is represented by four bits as the digit value plus 3 (the "excess" amount):
The smallest binary number represents the smallest value ().
The greatest binary number represents the largest value ().
To encode a number such as 127, one simply encodes each of the decimal digits as above, giving (0100, 0101, 1010).
Excess-3 arithmetic uses different algorithms than normal non-biased BCD or binary positional system numbers. After adding two excess-3 digits, the raw sum is excess-6. For instance, after adding 1 (0100 in excess-3) and 2 (0101 in excess-3), the sum looks like 6 (1001 in excess-3) instead of 3 (0110 in excess-3). To correct this problem, after adding two digits, it is necessary to remove the extra bias by subtracting binary 0011 (decimal 3 in unbiased binary) if the resulting digit is less than decimal 10, or subtracting binary 1101 (decimal 13 in unbiased binary) if an overflow (carry) has occurred. (In 4-bit binary, subtracting binary 1101 is equivalent to adding 0011 and vice versa.)
Motivation
The primary advantage of excess-3 coding over non-biased coding is that a decimal number can be nines' complemented (for subtraction) as easily as a binary number can be ones' complemented: just by inverting all bits. Also, when |
https://en.wikipedia.org/wiki/Cyberchase | Cyberchase is an animated sci-fi children's television series that airs on PBS Kids. The series centers around three children from Earth: Jackie, Matt and Inez, who are brought into Cyberspace, a digital universe, in order to protect the world from the villainous Hacker (Christopher Lloyd). They are able to prevent Hacker from taking over Cyberspace by means of problem-solving skills in conjunction with basic math, environmental science and wellness. In Cyberspace, they meet Digit (Gilbert Gottfried (seasons 1-13), Ron Pardo (season 14)), a "cybird" who helps them on their missions.
Cyberchase was created by WNET New York and premiered on PBS Kids on January 21, 2002. In 2010, after the season 8 finale, Cyberchase went on hiatus, but it returned in 2013 for a ninth season, followed by a tenth season in 2015. The eleventh season premiered on October 23, 2017, and the twelfth season premiered on April 19, 2019.
A thirteenth season was announced on October 19, 2020, and premiered on February 25, 2022. A fourteenth season premiered on April 21, 2023.
Plot
Motherboard is the "brain of the giant computer system that oversees all of Cyberspace". Her technician computer scientist, Dr. Marbles, kept her functioning properly until his assistant, the Hacker, turned against them. Dr. Marbles drained Hacker's battery and banished him to the Northern Frontier, where he formulated a plan to launch a virus that would attack Motherboard.
When Jackie, Matt and Inez interact with a library map in the real world, they accidentally allow Hacker access to Motherboard and she becomes infected with the virus. The kids are brought into Cyberspace and join forces with Digit, a creation of the Hacker who escaped his control. Together they protect the world from the Hacker and his clumsy, accident-prone assistants, Buzz and Delete, until they can recover the Encryptor Chip, a device stolen by Hacker that can nullify the virus and bring Motherboard back to full strength.
Cyberspace consist |
https://en.wikipedia.org/wiki/153%20%28number%29 | 153 (one hundred [and] fifty-three) is the natural number following 152 and preceding 154.
In mathematics
The number 153 is associated with the geometric shape known as the Vesica Piscis or Mandorla. Archimedes, in his Measurement of a Circle, referred to this ratio (153/265), as constituting the "measure of the fish", this ratio being an imperfect representation of .
As a triangular number, 153 is the sum of the first 17 integers, and is also the sum of the first five positive factorials:.
The number 153 is also a hexagonal number, and a truncated triangle number, meaning that 1, 15, and 153 are all triangle numbers.
The distinct prime factors of 153 add up to 20, and so do the ones of 154, hence the two form a Ruth-Aaron pair.
Since , it is a 3-narcissistic number, and it is also the smallest three-digit number which can be expressed as the sum of cubes of its digits. Only five other numbers can be expressed as the sum of the cubes of their digits: 0, 1, 370, 371 and 407. It is also a Friedman number, since 153 = 3 × 51.
The Biggs–Smith graph is a symmetric graph with 153 edges, all equivalent.
Another feature of the number 153 is that it is the limit of the following algorithm:
Take a random positive integer, divisible by three
Split that number into its base 10 digits
Take the sum of their cubes
Go back to the second step
An example, starting with the number 84:
There are 153 uniform polypeta that are generated from four different fundamental Coxeter groups in six-dimensional space.
In the Bible
The Gospel of John (chapter 21:1–14) includes the narrative of the miraculous catch of 153 fish as the third appearance of Jesus after his resurrection.
The precision of the number of fish in this narrative has long been considered peculiar, and many scholars have argued that 153 has some deeper significance. Jerome, for example, wrote that Oppian's Halieutica listed 153 species of fish, although this could not have been the intended meaning of the Gosp |
https://en.wikipedia.org/wiki/Micropaleontology | Micropaleontology (American spelling; spelled micropalaeontology in European usage) is the branch of paleontology (palaeontology) that studies microfossils, or fossils that require the use of a microscope to see the organism, its morphology and its characteristic details.
Microfossils
Microfossils are fossils that are generally between 0.001mm and 1 mm in size, the study of which requires the use of light or electron microscopy. Fossils which can be studied by the naked eye or low-powered magnification, such as a hand lens, are referred to as macrofossils.
For example, some colonial organisms, such as Bryozoa (especially the Cheilostomata) have relatively large colonies, but are classified by fine skeletal details of the small individuals of the colony.
In another example, many fossil genera of Foraminifera, which are protists are known from shells (called "tests") that were as big as coins, such as the genus Nummulites.
Microfossils are a common feature of the geological record, from the Precambrian to the Holocene. They are most common in deposits of marine environments, but also occur in brackish water, fresh water and terrestrial sedimentary deposits. While every kingdom of life is represented in the microfossil record, the most abundant forms are protist skeletons or cysts from the Chrysophyta, Pyrrhophyta, Sarcodina, acritarchs and chitinozoans, together with pollen and spores from the vascular plants.
In 2017, fossilized microorganisms, or microfossils, were announced to have been discovered in hydrothermal vent precipitates in the Nuvvuagittuq Belt of Quebec, Canada that may be as old as 4.28 billion years old, the oldest record of life on Earth, suggesting "an almost instantaneous emergence of life" (in a geological time-scale sense), after ocean formation 4.41 billion years ago, and not long after the formation of the Earth 4.54 billion years ago. Nonetheless, life may have started even earlier, at nearly 4.5 billion years ago, as claimed by some re |
https://en.wikipedia.org/wiki/Tholin | Tholins (after the Greek () "hazy" or "muddy"; from the ancient Greek word meaning "sepia ink") are a wide variety of organic compounds formed by solar ultraviolet or cosmic ray irradiation of simple carbon-containing compounds such as carbon dioxide (), methane () or ethane (), often in combination with nitrogen () or water (). Tholins are disordered polymer-like materials made of repeating chains of linked subunits and complex combinations of functional groups, typically nitriles and hydrocarbons, and their degraded forms such as amines and phenyls. Tholins do not form naturally on modern-day Earth, but they are found in great abundance on the surfaces of icy bodies in the outer Solar System, and as reddish aerosols in the atmospheres of outer Solar System planets and moons.
In the presence of water, tholins could be raw materials for prebiotic chemistry (i.e., the non-living chemistry that forms the basic chemicals of which life is made). Their existence has implications for the origins of life on Earth and possibly on other planets. As particles in an atmosphere, tholins scatter light, and can affect habitability.
Tholins may be produced in a laboratory, and are usually studied as a heterogeneous mixture of many chemicals with many different structures and properties. Using techniques like thermogravimetric analysis, astrochemists analyze the composition of these tholin mixtures, and the exact character of the individual chemicals within them.
Overview
The term "tholin" was coined by astronomer Carl Sagan and his colleague Bishun Khare to describe the difficult-to-characterize substances they obtained in his Miller–Urey-type experiments on the methane-containing gas mixtures such as those found in Titan's atmosphere. Their paper proposing the name "tholin" said:
For the past decade we have been producing in our laboratory a variety of complex organic solids from mixtures of the cosmically abundant gases , , , , HCHO, and . The product, synthesized by ultr |
https://en.wikipedia.org/wiki/Sprite%20%28operating%20system%29 | Sprite is an experimental Unix-like distributed operating system developed at the University of California, Berkeley by John Ousterhout's research group between 1984 and 1992. Its notable features include support for single system image on computer clusters and the introduction of the log-structured filesystem. The Tcl scripting language also originated in this project.
Early work
Early work on Sprite was based on the idea of making the operating system more "network aware", and thereby at the same time make it invisible to the user. The primary area of work was the building of a new network file system which made heavy use of local client-side caching in order to improve performance. After opening the file and some initial reads, the network is only used on-demand, and most user actions occur against the cache. Similar utilities allow remote devices to be mapped into the local computer's space, allowing for network printing and similar duties.
Many of the key Unix files are based on the network, including things like the password file. All machines in a network share the root directory as well. Other common Unix utilities such as finger were re-written to make them network aware as well, listing all of the people logged on across the network. This makes a Sprite network appear as if it were a single large time-sharing system, or a single-system image.
Another key addition to Sprite is process migration, which allows programs to be moved between machines at any time. The system maintains a list of machines and their state, and automatically moves processes to idle machines to improve local performance. Processes can also be "evicted" from machines to improve their performance, causing the original starter to move it to another machine on the network, or take control of it locally again. Long tasks (like compiling the Sprite system) can appear very fast.
Further development
Work on the "early" Sprite outlined above ended around 1987, but was improved during the |
https://en.wikipedia.org/wiki/Secret%20sharing | Secret sharing (also called secret splitting) refers to methods for distributing a secret among a group, in such a way that no individual holds any intelligible information about the secret, but when a sufficient number of individuals combine their 'shares', the secret may be reconstructed. Whereas insecure secret sharing allows an attacker to gain more information with each share, secure secret sharing is 'all or nothing' (where 'all' means the necessary number of shares).
In one type of secret sharing scheme there is one dealer and n players. The dealer gives a share of the secret to the players, but only when specific conditions are fulfilled will the players be able to reconstruct the secret from their shares. The dealer accomplishes this by giving each player a share in such a way that any group of t (for threshold) or more players can together reconstruct the secret but no group of fewer than t players can. Such a system is called a -threshold scheme (sometimes it is written as an -threshold scheme).
Secret sharing was invented independently by Adi Shamir and George Blakley in 1979.
Importance
Secret sharing schemes are ideal for storing information that is highly sensitive and highly important. Examples include: encryption keys, missile launch codes, and numbered bank accounts. Each of these pieces of information must be kept highly confidential, as their exposure could be disastrous; however, it is also critical that they should not be lost. Traditional methods for encryption are ill-suited for simultaneously achieving high levels of confidentiality and reliability. This is because when storing the encryption key, one must choose between keeping a single copy of the key in one location for maximum secrecy, or keeping multiple copies of the key in different locations for greater reliability. Increasing reliability of the key by storing multiple copies lowers confidentiality by creating additional attack vectors; there are more opportunities for a co |
https://en.wikipedia.org/wiki/137%20%28number%29 | 137 (one hundred [and] thirty-seven) is the natural number following 136 and preceding 138.
Mathematics
the 33rd prime number; the next is 139, with which it comprises a twin prime, and thus 137 is a Chen prime.
an Eisenstein prime with no imaginary part and a real part of the form .
the fourth Stern prime.
a Pythagorean prime: a prime number of the form , where () or the sum of two squares .
a strong prime in the sense that it is more than the arithmetic mean of its two neighboring primes.
a strictly non-palindromic number and a primeval number.
a factor of 10001 (the other being 73) and the repdigit 11111111 (= 10001 × 1111).
using two radii to divide a circle according to the golden ratio yields sectors of approximately 137.51° (the golden angle) and 222° in degree system so 137 is the largest integer before it.
In decimal notation, 1/137 = 0.007299270072992700..., so its period value happens to be palindromic and has a period length of only 8. However, this is only special to decimal, as in pentadecimal it (1/92) has a period length of twenty-four (24) and the period value is not at all palindromic.
Physics
Since the early 1900s, physicists have postulated that the number could lie at the heart of a grand unified theory, relating theories of electromagnetism, quantum mechanics and, especially, gravity.
1/137 was once believed to be the exact value of the fine-structure constant. The fine-structure constant, a dimensionless physical constant, is approximately 1/137, and the astronomer Arthur Eddington conjectured in 1929 that its reciprocal was in fact precisely the integer 137, which he claimed could be "obtained by pure deduction". This conjecture was not widely adopted, and by the 1940s, the experimental values for the constant were clearly inconsistent with the conjecture, being roughly 1/137.036. Recent work at the Kastler Brossel Laboratory in Paris reported the most precise measurement yet taking the value of this constant to the 11th d |
https://en.wikipedia.org/wiki/Predual | In mathematics, the predual of an object D is an object P whose dual space is D.
For example, the predual of the space of bounded operators is the space of trace class operators, and the predual of the space L∞(R) of essentially bounded functions on R is the Banach space L1(R) of integrable functions.
Abstract algebra
Functional analysis |
https://en.wikipedia.org/wiki/Daemon%20%28computing%29 | In multitasking computer operating systems, a daemon ( or ) is a computer program that runs as a background process, rather than being under the direct control of an interactive user. Traditionally, the process names of a daemon end with the letter d, for clarification that the process is in fact a daemon, and for differentiation between a daemon and a normal computer program. For example, is a daemon that implements system logging facility, and is a daemon that serves incoming SSH connections.
In a Unix environment, the parent process of a daemon is often, but not always, the init process. A daemon is usually created either by a process forking a child process and then immediately exiting, thus causing init to adopt the child process, or by the init process directly launching the daemon. In addition, a daemon launched by forking and exiting typically must perform other operations, such as dissociating the process from any controlling terminal (tty). Such procedures are often implemented in various convenience routines such as daemon(3) in Unix.
Systems often start daemons at boot time that will respond to network requests, hardware activity, or other programs by performing some task. Daemons such as cron may also perform defined tasks at scheduled times.
Terminology
The term was coined by the programmers at MIT's Project MAC. According to Fernando J. Corbató, who worked on Project MAC in 1963, his team was the first to use the term daemon, inspired by Maxwell's demon, an imaginary agent in physics and thermodynamics that helped to sort molecules, stating, "We fancifully began to use the word daemon to describe background processes that worked tirelessly to perform system chores". Unix systems inherited this terminology. Maxwell's demon is consistent with Greek mythology's interpretation of a daemon as a supernatural being working in the background.
In the general sense, daemon is an older form of the word "demon", from the Greek δαίμων. In the Unix System Ad |
https://en.wikipedia.org/wiki/Common%20operator%20notation | In programming languages, scientific calculators and similar common operator notation or operator grammar is a way to define and analyse mathematical and other formal expressions. In this model a linear sequence of tokens are divided into two classes: operators and operands.
Operands are objects upon which the operators operate. These include literal numbers and other constants as well as identifiers (names) which may represent anything from simple scalar variables to complex aggregated structures and objects, depending on the complexity and capability of the language at hand as well as usage context. One special type of operand is the parenthesis group. An expression enclosed in parentheses is typically recursively evaluated to be treated as a single operand on the next evaluation level.
Each operator is given a position, precedence, and an associativity. The operator precedence is a number (from high to low or vice versa) that defines which operator takes an operand that is surrounded by two operators of different precedence (or priority). Multiplication normally has higher precedence than addition, for example, so 3+4×5 = 3+(4×5) ≠ (3+4)×5.
In terms of operator position, an operator may be prefix, postfix, or infix. A prefix operator immediately precedes its operand, as in −x. A postfix operator immediately succeeds its operand, as in x! for instance. An infix operator is positioned in between a left and a right operand, as in x+y. Some languages, most notably the C-syntax family, stretches this conventional terminology and speaks also of ternary infix operators (a?b:c). Theoretically it would even be possible (but not necessarily practical) to define parenthesization as a unary bifix operation.
Operator associativity
Operator associativity determines what happens when an operand is surrounded by operators of the same precedence, as in 1-2-3: An operator can be left-associative, right-associative, or non-associative. Left-associative operators are appl |
https://en.wikipedia.org/wiki/Operator%20associativity | In programming language theory, the associativity of an operator is a property that determines how operators of the same precedence are grouped in the absence of parentheses. If an operand is both preceded and followed by operators (for example, ^ 3 ^), and those operators have equal precedence, then the operand may be used as input to two different operations (i.e. the two operations indicated by the two operators). The choice of which operations to apply the operand to, is determined by the associativity of the operators. Operators may be associative (meaning the operations can be grouped arbitrarily), left-associative (meaning the operations are grouped from the left), right-associative (meaning the operations are grouped from the right) or non-associative (meaning operations cannot be chained, often because the output type is incompatible with the input types). The associativity and precedence of an operator is a part of the definition of the programming language; different programming languages may have different associativity and precedence for the same type of operator.
Consider the expression a ~ b ~ c. If the operator ~ has left associativity, this expression would be interpreted as (a ~ b) ~ c. If the operator has right associativity, the expression would be interpreted as a ~ (b ~ c). If the operator is non-associative, the expression might be a syntax error, or it might have some special meaning. Some mathematical operators have inherent associativity. For example, subtraction and division, as used in conventional math notation, are inherently left-associative. Addition and multiplication, by contrast, are both left and right associative. (e.g. (a * b) * c = a * (b * c)).
Many programming language manuals provide a table of operator precedence and associativity; see, for example, the table for C and C++.
The concept of notational associativity described here is related to, but different from, the mathematical associativity. An operation that is math |
https://en.wikipedia.org/wiki/List%20of%20Mac%20software | The following is a list of Mac software – notable computer applications for current macOS operating systems.
For software designed for the classic Mac OS, see List of old Macintosh software.
Audio software
Digital audio workstations
Ableton Live
Ardour
Cubase
Digital Performer
GarageBand
Logic Pro and MainStage
REAPER
Reason
Renoise
Steinberg 2019 Mac Pro
Audio editing
Audacity – digital audio editor
DJing
djay – digital music mixing software
Mixxx – DJ mix software
Notation software
Impro-Visor
Finale
LilyPond
Overture
Misc audio tools
Audio Hijack – audio recorder
baudline – signal analyzer
Cog – open source audio player, supports multiple formats
fre:ac – open source audio converter and CD ripper
ixi software – free improvisation and sketching tools
Jaikoz – music file mass tagger
Max – Cycling 74's visual programming language for MIDI, audio, video; with MSP, Jitter
ReBirth – virtual synth program simulates Roland TR-808, TB-303
Recycle – music loop editor
Discontinued audio apps
Adobe Soundbooth – music and soundtrack editing
Audion – media player
BIAS Peak – mastering
iTunes – audio/video Jukebox by Apple
Logic Express – prosumer music production by Apple
Logic Studio – music writing studio package by Apple
Apple Loops Utility – production and organisation of Apple Loops
Apple Qmaster and Qadministrator
Mainstage – program to play software synthesizers live
QuickTime Pro – pro version of QuickTime
Soundtrack Pro – post production audio editor
WaveBurner – CD mastering and production software
RiffWorks – guitar recording and online song collaboration software
Chat (text, voice, image, video)
Active
Adium – multi-protocol IM client
aMSN
ChitChat
Colloquy – freeware advanced IRC and SILC client
Discord
Fire – open source, multiprotocol IM client
FaceTime – videoconferencing between Mac, iPhone, iPad and iPod Touch
iMessage – instant messaging between Mac, and iDevices
Ircle
Irssi – IrssiX and MacIrssi
Kopete
Li |
https://en.wikipedia.org/wiki/Ramanujan%E2%80%93Soldner%20constant | In mathematics, the Ramanujan–Soldner constant (also called the Soldner constant) is a mathematical constant defined as the unique positive zero of the logarithmic integral function. It is named after Srinivasa Ramanujan and Johann Georg von Soldner.
Its value is approximately μ ≈ 1.45136923488338105028396848589202744949303228…
Since the logarithmic integral is defined by
then using we have
thus easing calculation for numbers greater than μ. Also, since the exponential integral function satisfies the equation
the only positive zero of the exponential integral occurs at the natural logarithm of the Ramanujan–Soldner constant, whose value is approximately ln(μ) ≈ 0.372507410781366634461991866…
External links
Mathematical constants
Srinivasa Ramanujan |
https://en.wikipedia.org/wiki/Email%20spam | Email spam, also referred to as junk email, spam mail, or simply spam, is unsolicited messages sent in bulk by email (spamming). The name comes from a Monty Python sketch in which the name of the canned pork product Spam is ubiquitous, unavoidable, and repetitive. Email spam has steadily grown since the early 1990s, and by 2014 was estimated to account for around 90% of total email traffic.
Since the expense of the spam is borne mostly by the recipient, it is effectively postage due advertising. Thus, it is an example of a negative externality.
The legal definition and status of spam varies from one jurisdiction to another, but nowhere have laws and lawsuits been particularly successful in stemming spam.
Most email spam messages are commercial in nature. Whether commercial or not, many are not only annoying as a form of attention theft, but also dangerous because they may contain links that lead to phishing web sites or sites that are hosting malware or include malware as file attachments.
Spammers collect email addresses from chat rooms, websites, customer lists, newsgroups, and viruses that harvest users' address books. These collected email addresses are sometimes also sold to other spammers.
Overview
At the beginning of the Internet (the ARPANET), sending of commercial email was prohibited. Gary Thuerk sent the first email spam message in 1978 to 600 people. He was reprimanded and told not to do it again. Now the ban on spam is enforced by the Terms of Service/Acceptable Use Policy (ToS/AUP) of internet service providers (ISPs) and peer pressure.
Spam is sent by both otherwise reputable organizations and lesser companies. When spam is sent by otherwise reputable companies it is sometimes referred to as Mainsleaze. Mainsleaze makes up approximately 3% of the spam sent over the internet.
Spamvertised sites
Many spam emails contain URLs to a website or websites. According to a Cyberoam report in 2014, there are an average of 54 billion spam messages sent ev |
https://en.wikipedia.org/wiki/Academy%20of%20Interactive%20Arts%20%26%20Sciences | The Academy of Interactive Arts & Sciences (AIAS) is a non-profit organization of video game industry professionals. It organizes the annual Design Innovate Communicate Entertain summit, better known as D.I.C.E., which includes the presentations of the D.I.C.E. Awards.
History
Andrew S. Zucker, an attorney in the entertainment industry, founded the Academy of Interactive Arts & Sciences in 1991 and served as its first president. AIAS co-promoted numerous events with organizations such as the Academy of Television Arts and Sciences, the Directors Guild of America and Women in Film. Their first awards show program, Cybermania '94, which was hosted by Leslie Nielsen and Jonathan Taylor Thomas, was broadcast on TBS in 1994. While a second show was run in 1995, and was the first awards program to be streamed over the Web, it drew far fewer audiences as the first.
Video game industry leaders decided that they wanted to reform AIAS as a non-profit organization for the video game industry. The effort was backed by Peter Main of Nintendo, Tom Kalinske of Sega, and Doug Lowenstein, founder of the Entertainment Software Association (ESA), and with funding support from ESA. The AIAS was formally reestablished on November 19, 1996, with Marc Teren as president, soon replaced by game developer Glenn Entis. Initially, in 1998, AIAS' role was to handle the awards, originally known as the Interactive Achievement Awards. These awards were nominated and selected by game developers that are members of the organization themselves, mimicking the means by which the Academy Awards are voted by its members.
Around 2000, the ESA pulled out of funding AIAS, leading AIAS members Richard Hilleman and Lorne Lanning to suggest that AIAS create the D.I.C.E. Summit (short for "Design Innovate Communicate Entertain"), a convention centered around the presentation of the awards as a means to providing funding for the organization. The Summit was aimed at industry executives and lead developers as |
https://en.wikipedia.org/wiki/Edinburgh%20Multiple%20Access%20System | The Edinburgh Multi-Access System (EMAS) was a mainframe computer operating system at the University of Edinburgh. The system went online in 1971.
EMAS was a powerful and efficient general purpose multi-user system which coped with many of the computing needs of the University of Edinburgh and the University of Kent (the only other site outside Edinburgh to adopt the operating system).
History
Originally running on the ICL System 4/75 mainframe (based on the design of the IBM 360) it was later reimplemented on the ICL 2900 series of mainframes (as EMAS 2900 or EMAS-2) where it ran in service until the mid-1980s. Near the end of its life, the refactored version was back-ported (as EMAS-3) to the Amdahl 470 mainframe clone, and thence to the IBM System/370-XA architecture (the latter with help from the University of Kent, although they never actually ran EMAS-3). The National Advanced System (NAS) VL80 IBM mainframe clone followed later. The final EMAS system (the Edinburgh VL80) was decommissioned in July 1992.
The University of Kent system went live in December 1979, and ran on the least powerful machine in the ICL 2900 range - an ICL 2960, with 2 MB of memory, executing about 290k instructions per second. Despite this, it reliably supported around 30 users. This number increased in 1983 with the addition of an additional 2 MB of memory and a second Order Code Processor (OCP) (what is normally known as a CPU) running with symmetric multiprocessing. This system was decommissioned in August 1986.
Features
EMAS was written entirely in the Edinburgh IMP programming language, with only a small number of critical functions using embedded assembler within IMP sources. It had several features that were advanced for the time, including dynamic linking, multi-level storage, an efficient scheduler, a separate user-space kernel ('director'), a user-level shell ('basic command interpreter'), a comprehensive archiving system and a memory-mapped file architecture.
Such feat |
https://en.wikipedia.org/wiki/Web%20developer | A web developer is a programmer who develops World Wide Web applications using a client–server model. The applications typically use HTML, CSS, and JavaScript in the client, and any general-purpose programming language in the server. HTTP is used for communications between client and server. A web developer may specialize in client-side applications (Front-end web development), server-side applications (back-end development), or both (full-stack development).
Prerequisite
There are no formal educational or license requirements to become a web developer. However, many colleges and trade schools offer coursework in web development. There are also many tutorials and articles which teach web development, often freely available on the web - for example, on JavaScript.
Even though there are no formal requirements, web development projects require web developers to have knowledge and skills such as:
Using HTML, CSS, and JavaScript
Programming/coding/scripting in one of the many server-side languages or frameworks
Understanding server-side/client-side architecture and communication of the kind mentioned above
Ability to utilize a database
See also
Frontend and backend
Outline of web design and web development
User interface
User experience
Website design
Web development
Software developer
References
External links
The US Department of Labor's description of Web Developers
/ World Wide Web Consortium (W3C)
Web development
Computer occupations |
https://en.wikipedia.org/wiki/Expansion%20pack | An expansion pack, expansion set, supplement, or simply expansion, is an addition to an existing role-playing game, tabletop game, video game, collectible card game or miniature wargame. These add-ons usually add new game areas, weapons, objects, characters, adventures or an extended storyline to an already-released game.
While board game expansions are typically designed by the original creator, video game developers sometimes contract out development of the expansion pack to a third-party company, it may choose to develop the expansion itself, or it may do both.
Board games and tabletop RPGs may have been marketing expansions since the 1970s, and video games have been releasing expansion packs since the 1980s, early examples being the Dragon Slayer games Xanadu Scenario II and Sorcerian. Other terms for the concept are module and, in certain games' marketing, adventure.
Characteristics
The price of an expansion pack is usually much less than that of the original game. As expansion packs consist solely of additional content, most require the original game in order to play. Games with many expansions often begin selling the original game with prior expansions, such as The Sims Deluxe Edition (The Sims with The Sims: Livin' Large). These bundles make the game more accessible to new players. When games reach the end of their lifespan, the publisher often releases a 'complete' or 'gold' collection, which includes the game and all its subsequent expansions.
Stand-alone expansion packs
Some expansion packs do not require the original game in order to use the new content, as is the case with Half-Life: Blue Shift, Uncharted: The Lost Legacy or Sonic & Knuckles. Some art, sound, and code are reused from the original game. In some cases, a stand-alone expansion such as Heroes of Might and Magic III: The Shadow of Death, or Dungeon Siege: Legends of Aranna includes the original game.
Console game expansion packs
Expansion packs are most commonly released for PC games, |
https://en.wikipedia.org/wiki/Potrzebie | Potrzebie (; dative/locative of potrzeba, "a need") is a Polish word popularized by its non sequitur use as a running gag in the early issues of Mad not long after the comic book began in 1952.
Origin
Mad editor Harvey Kurtzman spotted the word printed in the Polish language section of a multi-languaged "Instructions for Use" sheet accompanying a bottle of aspirin, and Kurtzman, who was fascinated with unusual words, decided it would make an appropriate but meaningless background gag. After cutting the word out of the instruction sheet, he made copies and used rubber cement to paste "Potrzebie" randomly into the middle of Mad satires.
Appearances
Potrzebie was first used in a story in Mad #11 (May 1954), where it was the exclamation of a character who spoke only in foreign languages and song lyrics, in "Murder the Story", a parody illustrated by Jack Davis. It was used again in Bernard Krigstein's "From Eternity Back to Here!" in Mad #12 (June 1954) on an airplane advertising banner. With the same type font, it reappeared in Jack Davis's "Book! Movie!" in Mad #13 (July 1954), pasted into a panel as the title of an abstract painting seen in the background. In the same issue the word appears as POTS-REBIE, emblazoned on a cauldron in which Robinson Crusoe is roasting a frankfurter. This piece reappeared in one of the earliest Mad paperbacks, Bedside Mad It was illustrated as a rebus in "Puzzle Pages!" in Mad #19 (January 1955). These stories, like others in Mad comics, were written by Harvey Kurtzman. Frequent repetition gave it the status of a catch phrase or in-joke among the readership which continues to the present day. In the first Mad Style Guide, edited by Bhob Stewart in 1994, the word was made available for display on T-shirts and other licensed Mad products. It also sees occasional use as a metasyntactic variable by hackers.
A typical appearance of the word is exemplified by the Mad version of Chaucer's Canterbury Tales (from Mad #43, December 1958) |
https://en.wikipedia.org/wiki/Ring%20of%20integers | In mathematics, the ring of integers of an algebraic number field is the ring of all algebraic integers contained in . An algebraic integer is a root of a monic polynomial with integer coefficients: . This ring is often denoted by or . Since any integer belongs to and is an integral element of , the ring is always a subring of .
The ring of integers is the simplest possible ring of integers. Namely, where is the field of rational numbers. And indeed, in algebraic number theory the elements of are often called the "rational integers" because of this.
The next simplest example is the ring of Gaussian integers , consisting of complex numbers whose real and imaginary parts are integers. It is the ring of integers in the number field of Gaussian rationals, consisting of complex numbers whose real and imaginary parts are rational numbers. Like the rational integers, is a Euclidean domain.
The ring of integers of an algebraic number field is the unique maximal order in the field. It is always a Dedekind domain.
Properties
The ring of integers is a finitely-generated -module. Indeed, it is a free -module, and thus has an integral basis, that is a basis of the -vector space such that each element in can be uniquely represented as
with . The rank of as a free -module is equal to the degree of over .
Examples
Computational tool
A useful tool for computing the integral closure of the ring of integers in an algebraic field is the discriminant. If is of degree over , and form a basis of over , set . Then, is a submodule of the spanned by . pg. 33 In fact, if is square-free, then forms an integral basis for . pg. 35
Cyclotomic extensions
If is a prime, is a th root of unity and is the corresponding cyclotomic field, then an integral basis of is given by .
Quadratic extensions
If is a square-free integer and is the corresponding quadratic field, then is a ring of quadratic integers and its integral basis is given by if and by if |
https://en.wikipedia.org/wiki/Equicontinuity | In mathematical analysis, a family of functions is equicontinuous if all the functions are continuous and they have equal variation over a given neighbourhood, in a precise sense described herein.
In particular, the concept applies to countable families, and thus sequences of functions.
Equicontinuity appears in the formulation of Ascoli's theorem, which states that a subset of C(X), the space of continuous functions on a compact Hausdorff space X, is compact if and only if it is closed, pointwise bounded and equicontinuous.
As a corollary, a sequence in C(X) is uniformly convergent if and only if it is equicontinuous and converges pointwise to a function (not necessarily continuous a-priori).
In particular, the limit of an equicontinuous pointwise convergent sequence of continuous functions fn on either metric space or locally compact space is continuous. If, in addition, fn are holomorphic, then the limit is also holomorphic.
The uniform boundedness principle states that a pointwise bounded family of continuous linear operators between Banach spaces is equicontinuous.
Equicontinuity between metric spaces
Let X and Y be two metric spaces, and F a family of functions from X to Y. We shall denote by d the respective metrics of these spaces.
The family F is equicontinuous at a point x0 ∈ X if for every ε > 0, there exists a δ > 0 such that d(ƒ(x0), ƒ(x)) < ε for all ƒ ∈ F and all x such that d(x0, x) < δ.
The family is pointwise equicontinuous if it is equicontinuous at each point of X.
The family F is uniformly equicontinuous if for every ε > 0, there exists a δ > 0 such that d(ƒ(x1), ƒ(x2)) < ε for all ƒ ∈ F and all x1, x2 ∈ X such that d(x1, x2) < δ.
For comparison, the statement 'all functions ƒ in F are continuous' means that for every ε > 0, every ƒ ∈ F, and every x0 ∈ X, there exists a δ > 0 such that d(ƒ(x0), ƒ(x)) < ε for all x ∈ X such that d(x0, x) < δ.
For continuity, δ may depend on ε, ƒ, and x0.
For uniform continuity, δ may depend on ε a |
https://en.wikipedia.org/wiki/Ideal%20norm | In commutative algebra, the norm of an ideal is a generalization of a norm of an element in the field extension. It is particularly important in number theory since it measures the size of an ideal of a complicated number ring in terms of an ideal in a less complicated ring. When the less complicated number ring is taken to be the ring of integers, Z, then the norm of a nonzero ideal I of a number ring R is simply the size of the finite quotient ring R/I.
Relative norm
Let A be a Dedekind domain with field of fractions K and integral closure of B in a finite separable extension L of K. (this implies that B is also a Dedekind domain.) Let and be the ideal groups of A and B, respectively (i.e., the sets of nonzero fractional ideals.) Following the technique developed by Jean-Pierre Serre, the norm map
is the unique group homomorphism that satisfies
for all nonzero prime ideals of B, where is the prime ideal of A lying below .
Alternatively, for any one can equivalently define to be the fractional ideal of A generated by the set of field norms of elements of B.
For , one has , where .
The ideal norm of a principal ideal is thus compatible with the field norm of an element:
Let be a Galois extension of number fields with rings of integers .
Then the preceding applies with , and for any we have
which is an element of .
The notation is sometimes shortened to , an abuse of notation that is compatible with also writing for the field norm, as noted above.
In the case , it is reasonable to use positive rational numbers as the range for since has trivial ideal class group and unit group , thus each nonzero fractional ideal of is generated by a uniquely determined positive rational number.
Under this convention the relative norm from down to coincides with the absolute norm defined below.
Absolute norm
Let be a number field with ring of integers , and a nonzero (integral) ideal of .
The absolute norm of is
By convention, the norm of the zer |
https://en.wikipedia.org/wiki/James%20Clark%20%28programmer%29 | James Clark (born 23 February 1964) is a software engineer and creator of various open-source software including groff, expat and several XML specifications.
Education and early life
Clark was born in London and educated at Charterhouse School and Merton College, Oxford where he studied Mathematics and Philosophy.
Career
Clark has lived in Bangkok, Thailand since 1995, and is permanent Thai resident. He owns a company called Thai Open Source Software Center, which provides him a legal framework for his open-source activities. Clark is the author and creator of groff, as well as an XML editing mode for GNU Emacs.
Work on XML
Clark served as technical lead of the working group that developed XML—notably contributing the self-closing, empty element tag syntax, and the name XML. His contributions to XML are cited in dozens of books on the subject. Clark is the author or co-author of a number of influential specifications and implementations, including:
DSSSL: An SGML transformation and styling language.
Expat: An open-source XML parser.
XSLT: XSL Transformations, a part of the XSL family. He was the editor of the XSLT 1.0 specification.
XPath: Path language for addressing XML documents; used by XSLT but also as a free-standing language. He was the editor of the XPath 1.0 specification.
TREX: Tree regular experessions for XML (TREX) is a schema language for XML. TREX has been merged with RELAX to create RELAX NG.
RELAX NG: an XML Schema language, with both an explicit XML syntax and a compact syntax. Clark was critical of the XML Schema (W3C) language (also known as XSD) and developed RELAX NG in response
Jing: An implementation of RELAX NG.
Clark Notation: A way to express an XML Name in a compact way
Ballerina: Ballerina is an open source general-purpose programming language for application programmers.
Clark is listed as a member of the working group that developed the Java Stream processing API for XML (StAX) JSR 173 at the JCP.
Software Industry Pro |
https://en.wikipedia.org/wiki/Ecotone | An ecotone is a transition area between two biological communities, where two communities meet and integrate. It may be narrow or wide, and it may be local (the zone between a field and forest) or regional (the transition between forest and grassland ecosystems). An ecotone may appear on the ground as a gradual blending of the two communities across a broad area, or it may manifest itself as a sharp boundary line.
Etymology
The word ecotone was coined by Alfred Russel Wallace, who first observed the abrupt boundary between two biomes in 1859. It is formed as a combination of ecology plus -tone, from the Greek tonos or tension – in other words, a place where ecologies are in tension.
Features
There are several distinguishing features of an ecotone. First, an ecotone can have a sharp vegetation transition, with a distinct line between two communities. For example,
a change in colors of grasses or plant life can indicate an ecotone. Second, a change in physiognomy (physical appearance of a plant species) can be a key indicator. Water bodies, such as estuaries, can also have a region of transition, and the boundary is characterized by the differences in heights of the macrophytes or plant species present in the areas because this distinguishes the two areas' accessibility to light. Scientists look at color variations and changes in plant height. Third, a change of species can signal an ecotone. There will be specific organisms on one side of an ecotone or the other.
Other factors can illustrate or obscure an ecotone, for example, migration and the establishment of new plants. These are known as spatial mass effects, which are noticeable because some organisms will not be able to form self-sustaining populations if they cross the ecotone. If different species can survive in both communities of the two biomes, then the ecotone is considered to have species richness; ecologists measure this when studying the food chain and success of organisms. Lastly, the abundance |
https://en.wikipedia.org/wiki/Backdraft | A backdraft (North American English) or backdraught (British English) is the abrupt burning of superheated gasses in a fire caused when oxygen rapidly enters a hot, oxygen-depleted environment; for example, when a window or door to an enclosed space is opened or broken. Backdrafts are typically seen as a blast of smoke and/or flame out of an opening of a building. Backdrafts present a serious threat to firefighters. There is some debate concerning whether backdrafts should be considered a type of flashover (see below).
Burning
When material is heated enough, it begins to break down into smaller compounds, including flammable or even explosive gas, typically hydrocarbons. This is called pyrolysis, and does not require oxygen. If oxygen is also provided, then the hydrocarbons can combust, starting a fire.
If material undergoing pyrolysis is later given sufficient oxygen, the hydrocarbons will ignite, and therefore, combustion takes place.
Cause
A backdraft can occur when a compartment fire has little or no ventilation. Due to this, little or no oxygen can flow into the compartment. Then, because fires reduce oxygen, the oxygen concentration decreases. When the oxygen concentration becomes too low to support combustion, some or all of the combustion switches to pyrolysis. However, the hydrocarbons and smoke (primarily particulate matter) remain at a temperature hot enough to auto-ignite. If oxygen is then re-introduced to the compartment, e.g. by opening a door or window to a closed room, while the gasses are still hot enough to auto-ignite, combustion will restart, often abruptly or even explosively, as the gasses are heated by the combustion and expand rapidly because of the rapidly increasing temperature, combined with the energy released from combustion.
The colour and movement of smoke is used by firefighters to infer fire conditions, including the risk of backdraft. Characteristic warning signs of a backdraft include yellow or brown smoke, smoke which exits |
https://en.wikipedia.org/wiki/PiHex | PiHex was a distributed computing project organized by Colin Percival to calculate specific bits of . 1,246 contributors used idle time slices on almost two thousand computers to make its calculations. The software used for the project made use of Bellard's formula, a faster version of the BBP formula.
History
To calculate the five trillionth digit (and the following seventy-six digits) took 13,500 CPU hours, using 25 computers from 6 different countries. The forty trillionth digit required 84,500 CPU hours and 126 computers from 18 different countries. The highest calculation, the one quadrillionth digit, took 1.2 million CPU hours and 1,734 computers from 56 different countries. Total resources: 1,885 computers donated 1.3 million CPU hours. The average computer that was used to calculate would have taken 148 years to complete the calculations alone.
After setting three records, calculating the five trillionth bit, the forty trillionth bit, and the quadrillionth bit, the project ended on September 11, 2000.
While the PiHex project calculated the least significant digits of ever attempted in any base, the second place is held by Peter Trueb who computed some 22+ trillion digits in 2016 and third place by houkouonchi who derived the 13.3 trillionth digit in base 10.
Algorithm
Unlike most computations of , which compute results in base 10, PiHex computed in base 2 (bits), because Bellard's formula and the BBP formula could only be used to compute in base 2 at the time.
The final bit strings for each of the three calculations resulted as such:
Binary digits of from five trillion minus three to five trillion and seventy-six (completed August 30, 1998):
0000 0111 1110 0100 0101 0111 0011 0011 1100 1100
^ Five trillionth bit of
0111 1001 0000 1011 0101 1011 0101 1001 0111 1001
Binary digits of from forty trillion minus three to forty trillion and sixty-four (February 9, 1999):
1010 0000 1111 1001 1111 1111 0011 0111 0001 1101
^ Forty trillionth bi |
https://en.wikipedia.org/wiki/Okinawa%20diet | The Okinawa diet describes the traditional dietary practices of indigenous people of the Ryukyu Islands (belonging to Japan), which were claimed to have contributed to their relative longevity over a period of study in the 20th century.
Relative longevity
As assessed over 1949 to 1998, people from the Ryukyu Islands (of which Okinawa is the largest) had a life expectancy among the highest in the world (83.8 years vs. 78.9 years in the United States), although the male life expectancy rank among Japanese prefectures plummeted in the 21st century.
Okinawa had the longest life expectancy in all prefectures of Japan for almost 30 years prior to 2000. The relative life expectancy of Okinawans has since declined, due to many factors including Westernization. In 2000, Okinawa dropped in its ranking for longevity advantage for men to 26th out of 47 within the prefectures of Japan. In 2015, Japan had the highest life expectancy of any country: 90 years for women and for men, 84 years.
Although there are myriad factors that could account for differences in life expectancy, calorie restriction and regular physical activity could be factors. People have promoted the "Okinawa diet", despite the fact that the diet alone is unlikely to solely explain high life expectancy among seniors on Okinawa in the 20th century.
Indigenous islanders' diet
The traditional diet of the islanders contained sweet potato, green-leafy or root vegetables, and soy foods, such as miso soup, tofu or other soy preparations, occasionally served with small amounts of fish, noodles, or lean meats, all cooked with herbs, spices, and oil. Although the traditional Japanese diet usually includes large quantities of rice, the traditional Okinawa diet consisted of smaller quantities of rice; instead the staple was sweet potato. The Okinawa diet had only 30% of the sugar and 15% of the grains of the average Japanese dietary intake.
Okinawan cuisine consists of smaller meal portions of green and yellow vegeta |
https://en.wikipedia.org/wiki/Overeating | Overeating occurs when an individual consumes more calories in relation to the energy that is expended via physical activity or expelled via excretion, leading to weight gain and often obesity. Overeating is the defining characteristic of binge eating disorder.
This term may also be used to refer to specific episodes of over-consumption. For example, many people overeat during festivals or while on holiday.
Overeating can be a symptom of binge eating disorder or bulimia nervosa.
In a broader sense, hyperalimentation includes excessive food administration through other means than eating, e.g. through parenteral nutrition.
Treatment
Cognitive behavioural therapy, individual therapy, and group therapy are often beneficial in helping people keep track of their eating habits and changing the way they cope with difficult situations. Often overeating and the related binge eating are related to dieting, body image issues, as well as social pressures.
There are several 12-step programs that helps overeaters, such as Overeaters Anonymous or Food Addicts in Recovery Anonymous and others.
It is quite clear through research, and various studies that overeating causes addictive behaviors.
In some instances, overeating has been linked to the use of medications known as dopamine agonists, such as pramipexole.
See also
Compulsive overeating
Counterregulatory eating
Behavioral addiction
Binge eating
Binge eating disorder
Food drunk
Gluttony
Inflammation#Systemic inflammation and overeating
Mindless Eating: Why We Eat More Than We Think (book)
Polyphagia
Fasting
References
Further reading
Kessler, David A. The End of Overeating: Taking Control of the Insatiable American Appetite (2009)
External links
Eating behaviors
Habits
Hyperalimentation |
https://en.wikipedia.org/wiki/Vanish%20at%20infinity | In mathematics, a function is said to vanish at infinity if its values approach 0 as the input grows without bounds. There are two different ways to define this with one definition applying to functions defined on normed vector spaces and the other applying to functions defined on locally compact spaces.
Aside from this difference, both of these notions correspond to the intuitive notion of adding a point at infinity, and requiring the values of the function to get arbitrarily close to zero as one approaches it. This definition can be formalized in many cases by adding an (actual) point at infinity.
Definitions
A function on a normed vector space is said to if the function approaches as the input grows without bounds (that is, as ). Or,
in the specific case of functions on the real line.
For example, the function
defined on the real line vanishes at infinity.
Alternatively, a function on a locally compact space , if given any positive number , there exists a compact subset such that
whenever the point lies outside of In other words, for each positive number the set has compact closure.
For a given locally compact space the set of such functions
valued in which is either or forms a -vector space with respect to pointwise scalar multiplication and addition, which is often denoted
As an example, the function
where and are reals greater or equal 1 and correspond to the point on vanishes at infinity.
A normed space is locally compact if and only if it is finite-dimensional so in this particular case, there are two different definitions of a function "vanishing at infinity".
The two definitions could be inconsistent with each other: if in an infinite dimensional Banach space, then vanishes at infinity by the definition, but not by the compact set definition.
Rapidly decreasing
Refining the concept, one can look more closely to the of functions at infinity. One of the basic intuitions of mathematical analysis is that the Fourier |
https://en.wikipedia.org/wiki/Standard%20basis | In mathematics, the standard basis (also called natural basis or canonical basis) of a coordinate vector space (such as or ) is the set of vectors, each of whose components are all zero, except one that equals 1. For example, in the case of the Euclidean plane formed by the pairs of real numbers, the standard basis is formed by the vectors
Similarly, the standard basis for the three-dimensional space is formed by vectors
Here the vector ex points in the x direction, the vector ey points in the y direction, and the vector ez points in the z direction. There are several common notations for standard-basis vectors, including {ex, ey, ez}, {e1, e2, e3}, {i, j, k}, and {x, y, z}. These vectors are sometimes written with a hat to emphasize their status as unit vectors (standard unit vectors).
These vectors are a basis in the sense that any other vector can be expressed uniquely as a linear combination of these. For example, every vector v in three-dimensional space can be written uniquely as
the scalars , , being the scalar components of the vector v.
In the -dimensional Euclidean space , the standard basis consists of n distinct vectors
where ei denotes the vector with a 1 in the th coordinate and 0's elsewhere.
Standard bases can be defined for other vector spaces, whose definition involves coefficients, such as polynomials and matrices. In both cases, the standard basis consists of the elements of the space such that all coefficients but one are 0 and the non-zero one is 1. For polynomials, the standard basis thus consists of the monomials and is commonly called monomial basis. For matrices , the standard basis consists of the m×n-matrices with exactly one non-zero entry, which is 1. For example, the standard basis for 2×2 matrices is formed by the 4 matrices
Properties
By definition, the standard basis is a sequence of orthogonal unit vectors. In other words, it is an ordered and orthonormal basis.
However, an ordered orthonormal basis is not necessa |
https://en.wikipedia.org/wiki/Alidade | An alidade () (archaic forms include alhidade, alhidad, alidad) or a turning board is a device that allows one to sight a distant object and use the line of sight to perform a task. This task can be, for example, to triangulate a scale map on site using a plane table drawing of intersecting lines in the direction of the object from two or more points or to measure the angle and horizontal distance to the object from some reference point's polar measurement. Angles measured can be horizontal, vertical or in any chosen plane.
The alidade sighting ruler was originally a part of many types of scientific and astronomical instrument. At one time, some alidades, particularly using circular graduations as on astrolabes, were also called diopters. With modern technology, the name is applied to complete instruments such as the 'plane table alidade'.
Origins
The word in Arabic (, , ), signifies the same device. In Greek and Latin, it is respectively called , "dioptra", and , "fiducial line".
The earliest alidades consisted of a bar, rod or similar component with a vane on each end. Each vane (also called a pinnule or pinule) has a hole, slot or other indicator through which one can view a distant object. There may also be a pointer or pointers on the alidade to indicate a position on a scale. Alidades have been made of wood, ivory, brass and other materials.
Examples of old alidade types
The figure on the left displays drawings that attempt to show the general forms of various alidades that can be found on many antique instruments. Real alidades of these types could be much more decorative, revealing the maker's artistic talents as well as his technical skills. In the terminology of the time, the edge of an alidade at which one reads a scale or draws a line is called a fiducial edge.
Alidade B in the diagram shows a straight, flat bar with a vane at either end. No pointers are used. The vanes are not centred on the bar but offset so that the sight line coincides with |
https://en.wikipedia.org/wiki/Powerful%20number | A powerful number is a positive integer m such that for every prime number p dividing m, p2 also divides m. Equivalently, a powerful number is the product of a square and a cube, that is, a number m of the form m = a2b3, where a and b are positive integers. Powerful numbers are also known as squareful, square-full, or 2-full. Paul Erdős and George Szekeres studied such numbers and Solomon W. Golomb named such numbers powerful.
The following is a list of all powerful numbers between 1 and 1000:
1, 4, 8, 9, 16, 25, 27, 32, 36, 49, 64, 72, 81, 100, 108, 121, 125, 128, 144, 169, 196, 200, 216, 225, 243, 256, 288, 289, 324, 343, 361, 392, 400, 432, 441, 484, 500, 512, 529, 576, 625, 648, 675, 676, 729, 784, 800, 841, 864, 900, 961, 968, 972, 1000, ... .
Equivalence of the two definitions
If m = a2b3, then every prime in the prime factorization of a appears in the prime factorization of m with an exponent of at least two, and every prime in the prime factorization of b appears in the prime factorization of m with an exponent of at least three; therefore, m is powerful.
In the other direction, suppose that m is powerful, with prime factorization
where each αi ≥ 2. Define γi to be three if αi is odd, and zero otherwise, and define βi = αi − γi. Then, all values βi are nonnegative even integers, and all values γi are either zero or three, so
supplies the desired representation of m as a product of a square and a cube.
Informally, given the prime factorization of m, take b to be the product of the prime factors of m that have an odd exponent (if there are none, then take b to be 1). Because m is powerful, each prime factor with an odd exponent has an exponent that is at least 3, so m/b3 is an integer. In addition, each prime factor of m/b3 has an even exponent, so m/b3 is a perfect square, so call this a2; then m = a2b3. For example:
The representation m = a2b3 calculated in this way has the property that b is squarefree, and is uniquely defined by this property.
|
https://en.wikipedia.org/wiki/Lotka%E2%80%93Volterra%20equations | The Lotka–Volterra equations, also known as the Lotka–Volterra predator–prey model, are a pair of first-order nonlinear differential equations, frequently used to describe the dynamics of biological systems in which two species interact, one as a predator and the other as prey. The populations change through time according to the pair of equations:
where
the variable is the population density of prey (for example, the number of rabbits per square kilometre);
the variable is the population density of some predator (for example, the number of foxes per square kilometre);
and represent the instantaneous growth rates of the two populations;
represents time;
The prey's parameters, and , describe, respectively, the maximum prey per capita growth rate, and the effect of the presence of predators on the prey growth rate.
The predator's parameters, , , respectively describe the predator's per capita death rate, and the effect of the presence of prey on the predator's growth rate.
All parameters are positive and real.
The solution of the differential equations is deterministic and continuous. This, in turn, implies that the generations of both the predator and prey are continually overlapping.
The Lotka–Volterra system of equations is an example of a Kolmogorov model, which is a more general framework that can model the dynamics of ecological systems with predator–prey interactions, competition, disease, and mutualism.
Biological interpretation and model assumptions
The prey are assumed to have an unlimited food supply and to reproduce exponentially, unless subject to predation; this exponential growth is represented in the equation above by the term . The rate of predation on the prey is assumed to be proportional to the rate at which the predators and the prey meet; this is represented above by . If either or is zero, then there can be no predation. With these two terms the prey equation above can be interpreted as follows: the rate of change of the prey's p |
https://en.wikipedia.org/wiki/Stenomask | A stenomask is a hand-held microphone built into a padded, sound-proof enclosure that fits over the speaker's mouth or nose and mouth. Some lightweight versions may be fitted with an elastic neck strap to hold them in place while freeing the user's hands for other tasks. The purpose of a stenomask is to allow a person to speak without being heard by other people, and to keep background noise away from the microphone.
A stenomask is useful for speech recognition applications, because it allows voice transcription in noisy environments. Perhaps more importantly, a stenomask silences the user's voice so that it does not interfere with the surrounding environment such as a court or a classroom. The user can verbally identify the speaker, indicate gestures and unspoken answers, and describe activities as they take place.
An operator of a stenomask can be trained to "re-voice" everything he or she hears into a stenomask connected to a speech recognition system, for a real-time text transcription of everything spoken. This allows a "voice writer" to produce instant text feeds within a courtroom and distribute them in plain text format immediately after a proceeding. The equipment can also interface with litigation management software.
A trained operator using a stenomask connected to a pre-trained speech recognition system can exceed 180 words per minute while at the same time exceeding 95 percent accuracy. They may also modify the pronunciation of the words they are speaking in order to improve accuracy.
In comparison to conventional approaches like Gregg shorthand and stenotype technology, the main disadvantage of stenomask technology is the distinctive visual appearance of the operator when speaking into the stenomask. As the Dallas Morning News put it, they "can channel their inner Darth Vader." In covering Wisconsin's first official voice writer, the Racine Journal Times began by explaining that the mask is not "a way to summon his minions to build the Deat |
https://en.wikipedia.org/wiki/Harmonic%20divisor%20number | In mathematics, a harmonic divisor number or Ore number is a positive integer whose divisors have a harmonic mean that is an integer. The first few harmonic divisor numbers are
1, 6, 28, 140, 270, 496, 672, 1638, 2970, 6200, 8128, 8190 .
Harmonic divisor numbers were introduced by Øystein Ore, who showed that every perfect number is a harmonic divisor number and conjectured that there are no odd harmonic divisor numbers other than 1.
Examples
The number 6 has the four divisors 1, 2, 3, and 6. Their harmonic mean is an integer:
Thus 6 is a harmonic divisor number. Similarly, the number 140 has divisors 1, 2, 4, 5, 7, 10, 14, 20, 28, 35, 70, and 140. Their harmonic mean is
Since 5 is an integer, 140 a harmonic divisor number.
Factorization of the harmonic mean
The harmonic mean of the divisors of any number can be expressed as the formula
where is the sum of th powers of the divisors of : is the number of divisors, and is the sum of divisors .
All of the terms in this formula are multiplicative, but not completely multiplicative.
Therefore, the harmonic mean is also multiplicative.
This means that, for any positive integer , the harmonic mean can be expressed as the product of the harmonic means of the prime powers in the factorization of .
For instance, we have
and
Harmonic divisor numbers and perfect numbers
For any integer M, as Ore observed, the product of the harmonic mean and arithmetic mean of its divisors equals M itself, as can be seen from the definitions. Therefore, M is harmonic, with harmonic mean of divisors k, if and only if the average of its divisors is the product of M with a unit fraction 1/k.
Ore showed that every perfect number is harmonic. To see this, observe that the sum of the divisors of a perfect number M is exactly 2M; therefore, the average of the divisors is M(2/τ(M)), where τ(M) denotes the number of divisors of M. For any M, τ(M) is odd if and only if M is a square number, for otherwise each divisor d of M can be pair |
https://en.wikipedia.org/wiki/Central%20angle | A central angle is an angle whose apex (vertex) is the center O of a circle and whose legs (sides) are radii intersecting the circle in two distinct points A and B. Central angles are subtended by an arc between those two points, and the arc length is the central angle of a circle of radius one (measured in radians). The central angle is also known as the arc's angular distance. The arc length spanned by a central angle on a sphere is called spherical distance.
The size of a central angle is or (radians). When defining or drawing a central angle, in addition to specifying the points and , one must specify whether the angle being defined is the convex angle (<180°) or the reflex angle (>180°). Equivalently, one must specify whether the movement from point to point is clockwise or counterclockwise.
Formulas
If the intersection points and of the legs of the angle with the circle form a diameter, then is a straight angle. (In radians, .)
Let be the minor arc of the circle between points and , and let be the radius of the circle.
If the central angle is subtended by , then
If the central angle is not subtended by the minor arc , then is a reflex angle and
If a tangent at and a tangent at intersect at the exterior point , then denoting the center as , the angles (convex) and are supplementary (sum to 180°).
Central angle of a regular polygon
A regular polygon with sides has a circumscribed circle upon which all its vertices lie, and the center of the circle is also the center of the polygon. The central angle of the regular polygon is formed at the center by the radii to two adjacent vertices. The measure of this angle is
See also
Chord (geometry)
Inscribed angle
Great-circle navigation
References
External links
interactive
interactive
Inscribed and Central Angles in a Circle
Angle
Circles
Elementary geometry
Angle |
https://en.wikipedia.org/wiki/Inscribed%20angle | In geometry, an inscribed angle is the angle formed in the interior of a circle when two chords intersect on the circle. It can also be defined as the angle subtended at a point on the circle by two given points on the circle.
Equivalently, an inscribed angle is defined by two chords of the circle sharing an endpoint.
The inscribed angle theorem relates the measure of an inscribed angle to that of the central angle subtending the same arc.
The inscribed angle theorem appears as Proposition 20 on Book 3 of Euclid's Elements.
Theorem
Statement
The inscribed angle theorem states that an angle θ inscribed in a circle is half of the central angle 2θ that subtends the same arc on the circle. Therefore, the angle does not change as its vertex is moved to different positions on the circle.
Proof
Inscribed angles where one chord is a diameter
Let O be the center of a circle, as in the diagram at right. Choose two points on the circle, and call them V and A. Draw line VO and extended past O so that it intersects the circle at point B which is diametrically opposite the point V. Draw an angle whose vertex is point V and whose sides pass through points A and B.
Draw line OA. Angle BOA is a central angle; call it θ. Lines OV and OA are both radii of the circle, so they have equal lengths. Therefore, triangle VOA is isosceles, so angle BVA (the inscribed angle) and angle VAO are equal; let each of them be denoted as ψ.
Angles BOA and AOV add up to 180°, since line VB passing through O is a straight line. Therefore, angle AOV measures 180° − θ.
It is known that the three angles of a triangle add up to 180°, and the three angles of triangle VOA are:
180° − θ
ψ
ψ.
Therefore,
Subtract
from both sides,
where θ is the central angle subtending arc AB and ψ is the inscribed angle subtending arc AB.
Inscribed angles with the center of the circle in their interior
Given a circle whose center is point O, choose three points V, C, and D on the circle. Draw lines |
https://en.wikipedia.org/wiki/Chinese%20wall | A Chinese wall or ethical wall is an information barrier protocol within an organization designed to prevent exchange of information or communication that could lead to conflicts of interest. For example, a Chinese wall may be established to separate people who make investments from those who are privy to confidential information that could improperly influence the investment decisions. Firms are generally required by law to safeguard insider information and ensure that improper trading does not occur.
Etymology
Bryan Garner's Dictionary of Modern Legal Usage states that the metaphor title "derives of course from the Great Wall of China", although an alternative explanation links the idea to the screen walls of Chinese internal architecture.
The term was popularized in the United States following the stock market crash of 1929, when the U.S. government legislated information separation between investment bankers and brokerage firms, in order to limit the conflict of interest between objective company analysis and the desire for successful initial public offerings. Rather than prohibiting one company from engaging in both businesses, the government permitted the implementation of Chinese-wall procedures.
A leading note on the subject published in 1980 in the University of Pennsylvania Law Review titled "The Chinese Wall Defense to Law-Firm Disqualification" perpetuated the use of the term.
Objections to the term Chinese wall
There have been disputes about the use of the term for some decades, particularly in the legal and banking sectors. The term can be seen both as culturally insensitive and an inappropriate reflection on Chinese culture and trade, which are now extensively integrated into the global market.
In Peat, Marwick, Mitchell & Co. v. Superior Court (1988), Presiding Justice Harry W. Low, a Chinese American, wrote a concurring opinion specifically in order "to express my profound objection to the use of this phrase in this context". He called the te |
https://en.wikipedia.org/wiki/144%20%28number%29 | 144 (one hundred [and] forty-four) is the natural number following 143 and preceding 145. 144 is a dozen dozens, or one gross.
In mathematics
144 is the square of 12. It is also the twelfth Fibonacci number, following 89 and preceding 233, and the only Fibonacci number (other than 0, and 1) to also be a square. 144 is the smallest number with exactly 15 divisors, but it is not highly composite since the smaller number 120 contains 16. 144 is also equal to the sum of the eighth twin prime pair, (71 + 73). It is divisible by the value of its φ function, which returns 48 in its case, and there are 21 solutions to the equation φ(x) = 144. This is more than any integer below it, which makes it a highly totient number.
As a square number in decimal notation, 144 = 12 × 12, and if each number is reversed the equation still holds: 21 × 21 = 441. 169 shares this property, 13 × 13 = 169, while 31 × 31 = 961. Also in decimal, 144 is the largest of only four sum-product numbers, and it is a Harshad number, since 1 + 4 + 4 = 9, which divides 144.
144 is the smallest number whose fifth power is a sum of four (smaller) fifth powers. This solution was found in 1966 by L. J. Lander and T. R. Parkin, and disproved Euler's sum of powers conjecture. It was famously published in a paper by both authors, whose body consisted of only two sentences:
A regular ten-sided decagon has an internal angle of 144 degrees, which is equal to four times its own central angle, and equivalently twice the central angle of a regular five-sided pentagon.
The snub 24-cell, one of three semiregular polytopes in the fourth dimension, contains a total of 144 polyhedral cells: 120 regular tetrahedra and 24 regular icosahedra.
The maximum determinant in a 9 by 9 matrix of zeroes and ones is 144.
In particular, 144 is the sum of the divisors of 70: σ(70) = 144, where 70 is part of the only solution to the cannonball problem aside from the trivial solution, in-which the sum of the squares of the first twen |
https://en.wikipedia.org/wiki/666%20%28number%29 | 666 (six hundred [and] sixty-six) is the natural number following 665 and preceding 667.
In Christianity, 666 is referred to in (most manuscripts of) chapter 13 of the Book of Revelation of the New Testament as the "number of the beast."
In mathematics
666 is the sum of the first thirty-six natural numbers, which makes it a triangular number:
.
Since 36 is also triangular, 666 is a doubly triangular number. Also, where 15 and 21 are triangular as well, whose squares ( and ) add to 666 and have a difference of .
The number of integers which are relatively prime to 666 is also 216, ; and for an angle measured in degrees, (where here is the golden ratio).
666 is also the sum of the squares of the first seven primes , while the number of twin primes less than is 666.
A prime reciprocal magic square based on in decimal has a magic constant of 666. The twelfth pair of twin primes is (149, 151), with 151 the thirty-sixth prime number.
666 is a Smith number and Harshad number in base ten. The 27th indexed unique prime in decimal features a "666" in the middle of its sequence of digits.
The Roman numeral for 666, DCLXVI, has exactly one occurrence of all symbols whose value is less than 1000 in decreasing order (D = 500, C = 100, L = 50, X = 10, V = 5, I = 1).
In religion
Number of the beast
In the Textus Receptus manuscripts of the New Testament, the Book of Revelation () cryptically asserts 666 to be "the man's number" or "the number of a man" (depending on how the text is translated) associated with the Beast, an antagonistic creature that appears briefly about two-thirds into the apocalyptic vision. Some manuscripts of the original Koine Greek use the symbols χξϛ chi xi stigma (or χξϝ with a digamma), while other manuscripts spell out the number in words.
In modern popular culture, 666 has become one of the most widely recognized symbols for the Antichrist or, alternatively, the devil. Earnest references to the number occur both among apocalyptici |
https://en.wikipedia.org/wiki/Native%20POSIX%20Thread%20Library | The Native POSIX Thread Library (NPTL) is an implementation of the POSIX Threads specification for the Linux operating system.
History
Before the 2.6 version of the Linux kernel, processes were the schedulable entities, and there were no special facilities for threads.
However, it did have a system call — — which creates a copy of the calling process where the copy shares the address space of the caller. The LinuxThreads project used this system call to provide kernel-level threads (most of the previous thread implementations in Linux worked entirely in userland). Unfortunately, it only partially complied with POSIX, particularly in the areas of signal handling, scheduling, and inter-process synchronization primitives.
To improve upon LinuxThreads, it was clear that some kernel support and a new threading library would be required. Two competing projects were started to address the requirement: NGPT (Next Generation POSIX Threads) worked on by a team which included developers from IBM, and NPTL by developers at Red Hat. The NGPT team collaborated closely with the NPTL team and combined the best features of both implementations into NPTL. The NGPT project was subsequently abandoned in mid-2003 after merging its best features into NPTL.
NPTL was first released in Red Hat Linux 9. Old-style Linux POSIX threading is known for having trouble with threads that refuse to yield to the system occasionally, because it does not take the opportunity to preempt them when it arises, something that Windows was known to do better at the time. Red Hat claimed that NPTL fixed this problem in an article on the Java website about Java on Red Hat Linux 9.
NPTL has been part of Red Hat Enterprise Linux since version 3, and in the Linux kernel since version 2.6. It is now a fully integrated part of the GNU C Library.
There exists a tracing tool for NPTL, called POSIX Thread Trace Tool (PTT). And an Open POSIX Test Suite (OPTS) was written for testing the NPTL library against the PO |
https://en.wikipedia.org/wiki/List%20of%20IEC%20standards | This is an incomplete list of standards published by the International Electrotechnical Commission (IEC).
The numbers of older IEC standards were converted in 1997 by adding 60000; for example IEC 27 became IEC 60027.
IEC standards often have multiple sub-part documents; only the main title for the standard is listed here.
IEC 60027 Letter symbols to be used in electrical technology
IEC 60028 International standard of resistance for copper
IEC 60034 Rotating electrical machines
IEC 60038 IEC Standard Voltages
IEC 60041 Field acceptance tests to determine the hydraulic performance of hydraulic turbines, storage pumps and pump-turbines
IEC 60044 Instrument transformers
IEC 60045 Steam turbines
IEC 60050 International Electrotechnical Vocabulary
IEC 60051 Direct acting indicating analogue electrical measuring instruments and their accessories
IEC 60052 Voltage measurement by means of standard air gaps
IEC 60055 Paper-insulated metal-sheathed cables for rated voltages up to 18/30 kV (with copper or aluminium conductors and excluding gas-pressure and oil-filled cables)
IEC 60059 IEC standard current ratings
IEC 60060 High-voltage test techniques
IEC 60061 Lamp caps and holders together with gauges for the control of interchangeability and safety
IEC 60062 Marking codes for resistors and capacitors
IEC 60063 Preferred number series for resistors and capacitors
IEC 60064 Tungsten filament lamps for domestic and similar general lighting purposes – Performance requirements
IEC 60065 Audio, video and similar electronic apparatus – Safety requirements
IEC 60067 Dimensions of electronic tubes and valves
IEC 60068 Environmental testing
IEC 60071 Insulation co-ordination
IEC 60072 Dimensions and output series for rotating electrical machines
IEC 60073 Basic and safety principles for man-machine interface, marking and identification – Coding principles for indicators and actuators
IEC 60076 Power transformers
IEC 60077 Railway applications – Electric |
https://en.wikipedia.org/wiki/Conpoy | Conpoy or dried scallop is a type of Cantonese dried seafood product that is made from the adductor muscle of scallops. The smell of conpoy is marine, pungent, and reminiscent of certain salt-cured meats. Its taste is rich in umami due to its high content of various free amino acids, such as glycine, alanine, and glutamic acid. It is also rich in nucleic acids such as inosinic acid, amino acid byproducts such as taurine, and minerals, such as calcium and zinc.
Conpoy is produced by cooking raw scallops and then drying them.
Terminology
Conpoy is a loanword from the Cantonese pronunciation of 乾貝 (), which literally means "dried shell(fish)".
Usage
In Hong Kong, conpoy from two types of scallops are common. Conpoy made from Atrina pectinata or (江珧) from mainland China is small and milder in taste. Patinopecten yessoensis or (扇貝), a sea scallop imported from Japan (hotategai, 帆立貝 in Japanese), produces a conpoy that is stronger and richer in taste .
As with many dried foods, conpoy was originally made as a way to preserve seafood in times of excess. In more recent times its use in cuisine has been elevated to gourmet status. Conpoy has a strong and distinctive flavor that can be easily identified when used in rice congee, stir fries, stews, and sauces.
XO sauce, a seasoning used for frying vegetables or seafoods in Cantonese cuisine, contains significant quantities of conpoy. For example, the Lee Kum Kee formulation lists conpoy as the third ingredient on its label.
See also
References
Seafood
Food ingredients
Dried meat |
https://en.wikipedia.org/wiki/Space%20group | In mathematics, physics and chemistry, a space group is the symmetry group of a repeating pattern in space, usually in three dimensions. The elements of a space group (its symmetry operations) are the rigid transformations of the pattern that leave it unchanged. In three dimensions, space groups are classified into 219 distinct types, or 230 types if chiral copies are considered distinct. Space groups are discrete cocompact groups of isometries of an oriented Euclidean space in any number of dimensions. In dimensions other than 3, they are sometimes called Bieberbach groups.
In crystallography, space groups are also called the crystallographic or Fedorov groups, and represent a description of the symmetry of the crystal. A definitive source regarding 3-dimensional space groups is the International Tables for Crystallography .
History
Space groups in 2 dimensions are the 17 wallpaper groups which have been known for several centuries, though the proof that the list was complete was only given in 1891, after the much more difficult classification of space groups had largely been completed.
In 1879 the German mathematician Leonhard Sohncke listed the 65 space groups (called Sohncke groups) whose elements preserve the chirality. More accurately, he listed 66 groups, but both the Russian mathematician and crystallographer Evgraf Fedorov and the German mathematician Arthur Moritz Schoenflies noticed that two of them were really the same. The space groups in three dimensions were first enumerated in 1891 by Fedorov (whose list had two omissions (I3d and Fdd2) and one duplication (Fmm2)), and shortly afterwards in 1891 were independently enumerated by Schönflies (whose list had four omissions (I3d, Pc, Cc, ?) and one duplication (P21m)). The correct list of 230 space groups was found by 1892 during correspondence between Fedorov and Schönflies. later enumerated the groups with a different method, but omitted four groups (Fdd2, I2d, P21d, and P21c) even though he already |
https://en.wikipedia.org/wiki/Life%20on%20Mars | The possibility of life on Mars is a subject of interest in astrobiology due to the planet's proximity and similarities to Earth. To date, no proof of past or present life has been found on Mars. Cumulative evidence suggests that during the ancient Noachian time period, the surface environment of Mars had liquid water and may have been habitable for microorganisms, but habitable conditions do not necessarily indicate life.
Scientific searches for evidence of life began in the 19th century and continue today via telescopic investigations and deployed probes, searching for water, chemical biosignatures in the soil and rocks at the planet's surface, and biomarker gases in the atmosphere.
Mars is of particular interest for the study of the origins of life because of its similarity to the early Earth. This is especially true since Mars has a cold climate and lacks plate tectonics or continental drift, so it has remained almost unchanged since the end of the Hesperian period. At least two-thirds of Mars's surface is more than 3.5 billion years old, and it could have been habitable since 4.48 billion years ago, 500 million years before the earliest known Earth lifeforms; Mars may thus hold the best record of the prebiotic conditions leading to life, even if life does not or has never existed there.
Following the confirmation of the past existence of surface liquid water, the Curiosity, Perseverance and Opportunity rovers started searching for evidence of past life, including a past biosphere based on autotrophic, chemotrophic, or chemolithoautotrophic microorganisms, as well as ancient water, including fluvio-lacustrine environments (plains related to ancient rivers or lakes) that may have been habitable. The search for evidence of habitability, taphonomy (related to fossils), and organic compounds on Mars is now a primary objective for space agencies.
The findings of organic compounds inside sedimentary rocks and of boron on Mars are of interest as they are precursors |
https://en.wikipedia.org/wiki/500%20%28number%29 | 500 (five hundred) is the natural number following 499 and preceding 501.
Mathematical properties
500 = 22 × 53. It is an Achilles number and an Harshad number, meaning it is divisible by the sum of its digits. It is the number of planar partitions of 10.
Other fields
Five hundred is also
the number that many NASCAR races often use at the end of their race names (e.g., Daytona 500), to denote the length of the race (in miles, kilometers or laps).
the longest advertised distance (in miles) of the IndyCar Series and its premier race, the Indianapolis 500.
Slang names
Monkey (UK slang for £500; US slang for $500)
Integers from 501 to 599
500s
501
501 = 3 × 167. It is:
the sum of the first 18 primes (a term of the sequence ).
palindromic in bases 9 (6169) and 20 (15120).
502
502 = 2 × 251
vertically symmetric number
503
503 is:
a prime number.
a safe prime.
the sum of three consecutive primes (163 + 167 + 173).
the sum of the cubes of the first four primes.
a Chen prime
an Eisenstein prime with no imaginary part.
an index of a prime Lucas number.
an isolated prime
504
504 = 23 × 32 × 7. It is:
a tribonacci number.
a semi-meandric number.
a refactorable number.
a Harshad number.
is prime
the group order of the fourth smallest non-cyclic simple group A1(8) = 2G2(3)′.
the number of symmetries of the simple group PSL(2,8) that is the automorphism group of the Macbeath surface.
505
505 = 5 × 101
model number of Levi's jeans, model number of
This number is the magic constant of n×n normal magic square and n-queens problem for n = 10.
506
506 = 2 × 11 × 23. It is:
a sphenic number.
a square pyramidal number.
a pronic number.
a Harshad number.
507
507 = 3 × 132 = 232 - 23 + 1, which makes it a central polygonal number
The age Ming had before dying.
508
508 = 22 × 127, sum of four consecutive primes (113 + 127 + 131 + 137), number of graphical forest partitions of 30, since 508 = 222 + 22 + 2 it is the maximum number of regions into |
https://en.wikipedia.org/wiki/555%20%28number%29 | 555 (five hundred [and] fifty-five) is the natural number following 554 and preceding 556.
In mathematics
555 is a sphenic number. In base 10, it is a repdigit, and because it is divisible by the sum of its digits, it is a Harshad number. It is also a Harshad number in binary, base 11, base 13 and hexadecimal.
It is the sum of the first triplet of three-digit permutable primes in decimal:
.
It is the twenty-sixth number such that its Euler totient (288) is equal to the totient value of its sum-of-divisors: .
Telephone numbers
The NANP reserves telephone numbers in many dialing areas in the 555 local block for fictional purposes, such as 1-308-555-3485.
References
External links
Integers |
https://en.wikipedia.org/wiki/Y-intercept | In analytic geometry, using the common convention that the horizontal axis represents a variable x and the vertical axis represents a variable y, a y-intercept or vertical intercept is a point where the graph of a function or relation intersects the y-axis of the coordinate system. As such, these points satisfy x = 0.
Using equations
If the curve in question is given as the y-coordinate of the y-intercept is found by calculating Functions which are undefined at x = 0 have no y-intercept.
If the function is linear and is expressed in slope-intercept form as , the constant term is the y-coordinate of the y-intercept.
Multiple y-intercepts
Some 2-dimensional mathematical relationships such as circles, ellipses, and hyperbolas can have more than one y-intercept. Because functions associate x values to no more than one y value as part of their definition, they can have at most one y-intercept.
x-intercepts
Analogously, an x-intercept is a point where the graph of a function or relation intersects with the x-axis. As such, these points satisfy y=0. The zeros, or roots, of such a function or relation are the x-coordinates of these x-intercepts.
Unlike y-intercepts, functions of the form y = f(x) may contain multiple x-intercepts. The x-intercepts of functions, if any exist, are often more difficult to locate than the y-intercept, as finding the y intercept involves simply evaluating the function at x=0.
In higher dimensions
The notion may be extended for 3-dimensional space and higher dimensions, as well as for other coordinate axes, possibly with other names. For example, one may speak of the I-intercept of the current–voltage characteristic of, say, a diode. (In electrical engineering, I is the symbol used for electric current.)
See also
Regression intercept
References
Elementary mathematics
Functions and mappings |
https://en.wikipedia.org/wiki/Ad%20hoc%20network | An ad hoc network refers to technologies that allow network communications on an ad hoc basis. Associated technologies include:
Wireless ad hoc network
Mobile ad hoc network
Vehicular ad hoc network
Intelligent vehicular ad hoc network
Protocols associated with ad hoc networking
Ad hoc On-Demand Distance Vector Routing
Ad Hoc Configuration Protocol
Smart phone ad hoc network
Ad hoc wireless distribution service
References
Computer networking |
https://en.wikipedia.org/wiki/One%20half | One half (: halves) is the irreducible fraction resulting from dividing one (1) by two (2), or the fraction resulting from dividing any number by its double.
It often appears in mathematical equations, recipes, measurements, etc.
As a word
One half is one of the few fractions which are commonly expressed in natural languages by suppletion rather than regular derivation. In English, for example, compare the compound "one half" with other regular formations like "one-sixth".
A half can also be said to be one part of something divided into two equal parts. It is acceptable to write one half as a hyphenated word, one-half.
Mathematics
One half is the unique rational number that lies midway between nil and unity (which are the elementary additive and multiplicative identities) as the quotient of the first two non-zero integers, . It has two different decimal representations in base ten, the familiar and the recurring , with a similar pair of expansions in any even base; while in odd bases, one half has no terminating representation, it has only a single representation with a repeating fractional component (such as in ternary and in quinary).
Multiplication by one half is equivalent to division by two, or "halving"; conversely, division by one half is equivalent to multiplication by two, or "doubling".
A number raised to the power of one half is equal to the square root of ,
Uses
A hemiperfect number is a positive integer with a half-integer abundancy index:
where is odd, and is the sum-of-divisors function. The first three hemiperfect numbers are 2, 24, and 4320.
The area of a triangle with base and altitude is computed as,
One half figures in the formula for calculating figurate numbers, such as the -th triangular number:
and in the formula for computing magic constants for magic squares,
Successive natural numbers yield the -th metallic mean by the equation,
In the study of finite groups, alternating groups have order
By Euler, a clas |
https://en.wikipedia.org/wiki/720%20%28number%29 | 720 (seven hundred [and] twenty) is the natural number following 719 and preceding 721.
It is 6! (6 factorial) and a composite number with thirty divisors, more than any number below, making it a highly composite number. It is a Harshad number in every base from binary to decimal.
720 is expressible as the product of consecutive integers in two different ways: and .
There are 49 solutions to the equation , more than any integer below it, making 720 a highly totient number.
720 is a 241-gonal number.
In other fields
720 is:
A common vertical display resolution for HDTV (see 720p).
720° is two full rotations; the term "720" refers to a skateboarding trick.
720° is also the name of a skateboarding video game.
720 is a dual area code in the Denver Metro Area along with 303.
720° is the sum of all the defects of any polyhedron.
720 is a short form of saying Boeing 720, an airliner which is no longer in service.
For the year AD, see 720.
References
Integers |
https://en.wikipedia.org/wiki/700%20%28number%29 | 700 (seven hundred) is the natural number following 699 and preceding 701.
It is the sum of four consecutive primes (167 + 173 + 179 + 181), the perimeter of a Pythagorean triangle (75 + 308 + 317) and a Harshad number.
Integers from 701 to 799
Nearly all of the palindromic integers between 700 and 800 (i.e. nearly all numbers in this range that have both the hundreds and units digit be 7) are used as model numbers for Boeing Commercial Airplanes.
700s
701 = prime number, sum of three consecutive primes (229 + 233 + 239), Chen prime, Eisenstein prime with no imaginary part
702 = 2 × 33 × 13, pronic number, nontotient, Harshad number
703 = 19 × 37, triangular number, hexagonal number, smallest number requiring 73 fifth powers for Waring representation, Kaprekar number, area code for Northern Virginia along with 571, a number commonly found in the formula for body mass index
704 = 26 × 11, Harshad number, lazy caterer number , area code for the Charlotte, NC area.
705 = 3 × 5 × 47, sphenic number, smallest Bruckman-Lucas pseudoprime
706 = 2 × 353, nontotient, Smith number
707 = 7 × 101, sum of five consecutive primes (131 + 137 + 139 + 149 + 151), palindromic number, number of lattice paths from (0,0) to (5,5) with steps (0,1), (1,0) and, when on the diagonal, (1,1).
708 = 22 × 3 × 59, number of partitions of 28 that do not contain 1 as a part
709 = prime number; happy number. It is the seventh in the series 2, 3, 5, 11, 31, 127, 709 where each number is the nth prime with n being the number proceeding it in the series, therefore, it is a prime index number.
710s
710 = 2 × 5 × 71, sphenic number, nontotient, number of forests with 11 vertices
711 = 32 × 79, Harshad number, number of planar Berge perfect graphs on 7 nodes. Also the phone number of Telecommunications Relay Service, commonly used by the deaf and hard-of-hearing.
712 = 23 × 89, refactorable number, sum of the first twenty-one primes, totient sum for first 48 integers. It is the largest k |
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