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https://en.wikipedia.org/wiki/Idempotent%20%28ring%20theory%29 | In ring theory, a branch of mathematics, an idempotent element or simply idempotent of a ring is an element such that . That is, the element is idempotent under the ring's multiplication. Inductively then, one can also conclude that for any positive integer . For example, an idempotent element of a matrix ring is precisely an idempotent matrix.
For general rings, elements idempotent under multiplication are involved in decompositions of modules, and connected to homological properties of the ring. In Boolean algebra, the main objects of study are rings in which all elements are idempotent under both addition and multiplication.
Examples
Quotients of Z
One may consider the ring of integers modulo , where is squarefree. By the Chinese remainder theorem, this ring factors into the product of rings of integers modulo , where is prime. Now each of these factors is a field, so it is clear that the factors' only idempotents will be and . That is, each factor has two idempotents. So if there are factors, there will be idempotents.
We can check this for the integers , . Since has two prime factors ( and ) it should have idempotents.
From these computations, , , , and are idempotents of this ring, while and are not. This also demonstrates the decomposition properties described below: because , there is a ring decomposition . In the identity is and in the identity is .
Quotient of polynomial ring
Given a ring and an element such that , then the quotient ring
has the idempotent . For example, this could be applied to , or any polynomial .
Idempotents in split-quaternion rings
There is a hyperboloid of idempotents in the split-quaternion ring.
Types of ring idempotents
A partial list of important types of idempotents includes:
Two idempotents and are called orthogonal if . If is idempotent in the ring (with unity), then so is ; moreover, and are orthogonal.
An idempotent in is called a central idempotent if for all in , th |
https://en.wikipedia.org/wiki/Network%20Control%20Protocol%20%28ARPANET%29 | The Network Control Protocol (NCP) was a communication protocol for a computer network in the 1970s and early 1980s. It provided the transport layer of the protocol stack running on host computers of the ARPANET, the predecessor to the modern Internet.
NCP preceded the Transmission Control Protocol (TCP) as a transport layer protocol used during the early ARPANET. NCP was a simplex protocol that utilized two port numbers, establishing two connections, for two-way communications. An odd and an even port were reserved for each application layer application or protocol. The standardization of TCP and UDP reduced the need for the use of two simplex ports for each application down to one duplex port.
There is some confusion over the name, even among the engineers who worked with the ARPANET. Originally, there was no need for a name for the protocol stack as a whole, so none existed. When the development of TCP started, a name was required for its predecessor, and the pre-existing acronym 'NCP' (which originally referred to Network Control Program, the software which implemented this stack) was organically adopted for that use. Eventually, it was realized that the original expansion of that acronym was inappropriate for its new meaning, so a new quasi-backronym was created, 'Network Control Protocol' - again, organically, not via a formal decision.
History
NCP was first specified and described in the ARPANETs earliest RFC documents in 1969 after a series of meetings on the topic with engineers from UCLA, University of Utah, and SRI. It was finalized in in early 1970, and deployed to all nodes on the ARPANET in December 1970. It remained in use until the end of 1982; see Flag Day below.
NCP provided connections and flow control between processes running on different ARPANET host computers. Application services, such as remote login and the file transfer, would be built on top of NCP, using it to handle connections to other host computers.
On the ARPANET, the protocol |
https://en.wikipedia.org/wiki/Active%20noise%20control | Active noise control (ANC), also known as noise cancellation (NC), or active noise reduction (ANR), is a method for reducing unwanted sound by the addition of a second sound specifically designed to cancel the first. The concept was first developed in the late 1930s; later developmental work that began in the 1950s eventually resulted in commercial airline headsets with the technology becoming available in the late 1980s. The technology is also used in road vehicles, mobile telephones, earbuds, and headphones.
Explanation
Sound is a pressure wave, which consists of alternating periods of compression and rarefaction. A noise-cancellation speaker emits a sound wave with the same amplitude but with inverted phase (also known as antiphase) relative to the original sound. The waves combine to form a new wave, in a process called interference, and effectively cancel each other out – an effect which is called destructive interference.
Modern active noise control is generally achieved through the use of analog circuits or digital signal processing. Adaptive algorithms are designed to analyze the waveform of the background aural or nonaural noise, then based on the specific algorithm generate a signal that will either phase shift or invert the polarity of the original signal. This inverted signal (in antiphase) is then amplified and a transducer creates a sound wave directly proportional to the amplitude of the original waveform, creating destructive interference. This effectively reduces the volume of the perceivable noise.
A noise-cancellation speaker may be co-located with the sound source to be attenuated. In this case it must have the same audio power level as the source of the unwanted sound in order to cancel the noise. Alternatively, the transducer emitting the cancellation signal may be located at the location where sound attenuation is wanted (e.g. the user's ear). This requires a much lower power level for cancellation but is effective only for a single user. |
https://en.wikipedia.org/wiki/Bochs | Bochs (pronounced "box") is a portable x86-32 and x86-64 IBM PC compatible emulator and debugger mostly written in C++ and distributed as free software under the GNU Lesser General Public License. It supports emulation of the processor(s) (including protected mode), memory, disks, display, Ethernet, BIOS and common hardware peripherals of PCs.
Many guest operating systems can be run using the emulator including DOS, several versions of Microsoft Windows, BSDs, Linux, Xenix and Rhapsody OS (precursor of Mac OS X Public Beta). Bochs runs on many host operating systems, including Android OS, Linux, macOS, PlayStation 2, Windows, and Windows CE along with its derivatives.
Bochs is mostly used for operating system development (when an emulated operating system crashes, it does not crash the host operating system, so the emulated OS can be debugged) and to run other guest operating systems inside already running host operating systems. It can also be used to run older software — such as PC games — which will not run on non-compatible, or too fast computers.
History
Bochs started as a program with a commercial license, at the price of US$25, for use as-is. If a user needed to link it to other software, that user would have to negotiate a special license. That changed on 22 March 2000, when Mandrakesoft (later Mandriva) bought Bochs from lead developer Kevin Lawton and released it for Linux under the GNU Lesser General Public License. Support for hosting on Windows XP ended with version 2.6.10.
Use
Bochs emulates the hardware needed by PC operating systems, including hard drives, CD drives, and floppy drives. It doesn't utilize any host CPU virtualization features, therefore is slower than most virtualization (as opposed to emulation) software. It provides additional security by completely isolating the guest OS from the hardware. Bochs also has extensive debugging features. It is widely used for OS development, as it removes the need for constant system restarts (to |
https://en.wikipedia.org/wiki/Robinson%20projection | The Robinson projection is a map projection of a world map that shows the entire world at once. It was specifically created in an attempt to find a good compromise to the problem of readily showing the whole globe as a flat image.
The Robinson projection was devised by Arthur H. Robinson in 1963 in response to an appeal from the Rand McNally company, which has used the projection in general-purpose world maps since that time. Robinson published details of the projection's construction in 1974. The National Geographic Society (NGS) began using the Robinson projection for general-purpose world maps in 1988, replacing the Van der Grinten projection. In 1998, NGS abandoned the Robinson projection for that use in favor of the Winkel tripel projection, as the latter "reduces the distortion of land masses as they near the poles".
Strengths and weaknesses
The Robinson projection is neither equal-area nor conformal, abandoning both for a compromise. The creator felt that this produced a better overall view than could be achieved by adhering to either. The meridians curve gently, avoiding extremes, but thereby stretch the poles into long lines instead of leaving them as points.
Hence, distortion close to the poles is severe, but quickly declines to moderate levels moving away from them. The straight parallels imply severe angular distortion at the high latitudes toward the outer edges of the map – a fault inherent in any pseudocylindrical projection. However, at the time it was developed, the projection effectively met Rand McNally's goal to produce appealing depictions of the entire world.
Formulation
The projection is defined by the table:
The table is indexed by latitude at 5-degree intervals; intermediate values are calculated using interpolation. Robinson did not specify any particular interpolation method, but it is reported that others used either Aitken interpolation (with polynomials of unknown degrees) or cubic splines while analyzing area deformation on the Ro |
https://en.wikipedia.org/wiki/22%20%28number%29 | 22 (twenty-two) is the natural number following 21 and preceding 23.
In mathematics
22 is a palindromic number. It is the second Smith number, the second Erdős–Woods number, and the fourth large Schröder number. It is also a Perrin number, from a sum of 10 and 12.
22 is the sixth distinct semiprime, and the fouth of the form where is a higher prime. It is the second member of the second cluster of discrete biprimes (21, 22), where the next such cluster is (38, 39). It contains an aliquot sum of 14 (itself semiprime), within an aliquot sequence of four composite numbers (22, 14, 10, 8, 7, 1, 0) that are rooted in the prime 7-aliquot tree.
The maximum number of regions into which five intersecting circles divide the plane is 22. 22 is also the quantity of pieces in a disc that can be created with six straight cuts, which makes 22 the seventh central polygonal number.
22 is the fourth pentagonal number, the third hexagonal pyramidal number, and the third centered heptagonal number.
is a commonly used approximation of the irrational number , the ratio of the circumference of a circle to its diameter, where in particular 22 and 7 are consecutive hexagonal pyramidal numbers. Also,
from an approximate construction of the squaring of the circle by Srinivasa Ramanujan, correct to eight decimal places.
Natural logarithms of integers in binary are known to have Bailey–Borwein–Plouffe type formulae for for all integers .
22 is the number of partitions of 8, as well as the sum of the totient function over the first eight integers, with for 22 returning 10.
22 can read as "two twos", which is the only fixed point of John Conway's look-and-say function. In other words, "22" generates the infinite repeating sequence "22, 22, 22, ..."
All regular polygons with < edges can be constructed with an angle trisector, with the exception of the 11-sided hendecagon.
There is an elementary set of twenty-two single-orbit convex tilings that tessellate two-dimensional s |
https://en.wikipedia.org/wiki/24%20%28number%29 | 24 (twenty-four) is the natural number following 23 and preceding 25.
In mathematics
24 is an even composite number, with 2 and 3 as its distinct prime factors. It is the first number of the form 2q, where q is an odd prime. It is the smallest number with at least eight positive divisors: 1, 2, 3, 4, 6, 8, 12, and 24; thus, it is a highly composite number, having more divisors than any smaller number. Furthermore, it is an abundant number, since the sum of its proper divisors (36) is greater than itself, as well as a superabundant number.
In number theory and algebra
24 is the smallest 5-hemiperfect number, as it has a half-integer abundancy index:
1 + 2 + 3 + 4 + 6 + 8 + 12 + 24 = 60 = × 24
24 is a semiperfect number, since adding up all the proper divisors of 24 except 4 and 8 gives 24.
24 is a practical number, since all smaller positive integers than 24 can be represented as sums of distinct divisors of 24.
24 is a Harshad number, since it is divisible by the sum of its digits in decimal.
24 is a refactorable number, as it has exactly eight positive divisors, and 8 is one of them.
24 is a twin-prime sum, specifically the sum of the third pair of twin primes .
24 is a highly totient number, as there are 10 solutions to the equation φ(x) = 24, which is more than any integer below 24. 144 (the square of 12) and 576 (the square of 24) are also highly totient.
24 is a polite number, an amenable number, an idoneal number, and a tribonacci number.
24 forms a Ruth-Aaron pair with 25, since the sums of distinct prime factors of each are equal (5).
24 is a compositorial, as it is the product of composite numbers up to 6.
24 is a pernicious number, since its Hamming weight in its binary representation (11000) is prime (2).
24 is the third nonagonal number.
24's digits in decimal can be manipulated to form two of its factors, as 2 * 4 is 8 and 2 + 4 is 6. In turn 6 * 8 is 48, which is twice 24, and 4 + 8 is 12, which is half 24.
24 is a congruent number, as 24 is the |
https://en.wikipedia.org/wiki/23%20%28number%29 | 23 (twenty-three) is the natural number following 22 and preceding 24.
In mathematics
Twenty-three is the ninth prime number, the smallest odd prime that is not a twin prime. It is, however, a cousin prime with 19, and a sexy prime with 17 and 29; while also being the largest member of the first prime sextuplet (7, 11, 13, 17, 19, 23). Twenty-three is also the fifth factorial prime, the second Woodall prime, and a happy number in decimal. It is an Eisenstein prime with no imaginary part and real part of the form It is also the fifth Sophie Germain prime and the fourth safe prime, and the next to last member of the first Cunningham chain of the first kind to have five terms (2, 5, 11, 23, 47). Since 14! + 1 is a multiple of 23, but 23 is not one more than a multiple of 14, 23 is the first Pillai prime. 23 is the smallest odd prime to be a highly cototient number, as the solution to for the integers 95, 119, 143, and 529. The third decimal repunit prime after R2 and R19 is R23, followed by R1031.
23 is the second Smarandache–Wellin prime in base ten, as it is the concatenation of the decimal representations of the first two primes (2 and 3) and is itself also prime.
The sum of the first nine primes up to 23 is a square: and the sum of the first 23 primes is 874, which is divisible by 23, a property shared by few other numbers.
In the list of fortunate numbers, 23 occurs twice, since adding 23 to either the fifth or eighth primorial gives a prime number (namely 2333 and 9699713).
23 has the distinction of being one of two integers that cannot be expressed as the sum of fewer than 9 cubes of positive integers (the other is 239). See Waring's problem.
The twenty-third highly composite number 20,160 is one less than the last number (the 339th super-prime 20,161) that cannot be expressed as the sum of two abundant numbers.
23 is the number of trees on 8 unlabeled nodes. It is also a Wedderburn–Etherington number, which are numbers that can be used to count |
https://en.wikipedia.org/wiki/25%20%28number%29 | 25 (twenty-five) is the natural number following 24 and preceding 26.
In mathematics
It is a square number, being 52 = 5 × 5, and hence the third non-unitary square prime of the form p2.
It is one of two two-digit numbers whose square and higher powers of the number also ends in the same last two digits, e.g., 252 = 625; the other is 76.
Twenty five has an even aliquot sum of 6, which is itself the first even and perfect number root of an aliquot sequence; not ending in (1 and 0).
It is the smallest square that is also a sum of two (non-zero) squares: 25 = 32 + 42. Hence, it often appears in illustrations of the Pythagorean theorem.
25 is the sum of the five consecutive single-digit odd natural numbers 1, 3, 5, 7, and 9.
25 is a centered octagonal number, a centered square number, a centered octahedral number, and an automorphic number.
25 percent (%) is equal to .
It is the smallest decimal Friedman number as it can be expressed by its own digits: 52.
It is also a Cullen number and a vertically symmetrical number. 25 is the smallest pseudoprime satisfying the congruence 7n = 7 mod n.
25 is the smallest aspiring number — a composite non-sociable number whose aliquot sequence does not terminate.
According to the Shapiro inequality, 25 is the smallest odd integer n such that there exist x, x, ..., x such that
where x = x, x = x.
Within decimal, one can readily test for divisibility by 25 by seeing if the last two digits of the number match 00, 25, 50, or 75.
There are 25 primes under 100.
In science
The Standard Model of physics features a total of 25 elementary particles: 12 fermions (made of 6 quarks and 6 leptons) and 13 bosons (made of 12 gauge bosons and 1 scalar boson).
The atomic number of manganese.
The average percentage DNA overlap of an individual with their half-sibling, grandparent, grandchild, aunt, uncle, niece, nephew, identical twin cousin (offspring of identical twins), or double cousin.
In religion
In Ezekiel's vision of a new temp |
https://en.wikipedia.org/wiki/26%20%28number%29 | 26 (twenty-six) is the natural number following 25 and preceding 27.
In mathematics
26 is the seventh discrete semiprime () and the fifth with 2 as the lowest non-unitary factor thus of the form (2.q), where q is a higher prime.
with an aliquot sum of 16, within an aliquot sequence of five composite numbers (26,16,15,9,4,3,1,0) to the Prime in the 3-aliquot tree.
26 is the only integer that is one greater than a square (5 + 1) and one less than a cube (3 − 1).
26 is a telephone number, specifically, the number of ways of connecting 5 points with pairwise connections.
There are 26 sporadic groups.
The 26-dimensional Lorentzian unimodular lattice II25,1 plays a significant role in sphere packing problems and the classification of finite simple groups. In particular, the Leech lattice is obtained in a simple way as a subquotient.
26 is the smallest number that is both a nontotient and a noncototient number.
There are 26 faces of a rhombicuboctahedron.
When a 3 × 3 × 3 cube is made of 27 unit cubes, 26 of them are viewable as the exterior layer. Also a 26 sided polygon is called an icosihexagon.
φ(26) = φ(σ(26)).
Properties of its positional representation in certain radixes
Twenty-six is a repdigit in base three (2223) and in base 12 (2212).
In base ten, 26 is the smallest positive integer that is not a palindrome to have a square (262 = 676) that is a palindrome.
In science
The atomic number of iron.
The number of spacetime dimensions in bosonic string theory.
Astronomy
Messier object M26, a magnitude 9.5 open cluster in the constellation Scutum.
The New General Catalogue object NGC 26, a spiral galaxy in the constellation Pegasus.
In religion
26 is the gematric number, being the sum of the Hebrew characters () being the name of the god of Israel – YHWH (Yahweh).
GOD=26=G7+O15+D4 in Simple6,74 English7,74 Gematria8,74 ('The Key': A=1, B2, C3, ..., Z26).
The Greek Strongs number G26 is "Agape", which means "Love".
The expression "For His mercy endure |
https://en.wikipedia.org/wiki/29%20%28number%29 | 29 (twenty-nine) is the natural number following 28 and preceding 30.
Mathematics
29 is the tenth prime number, and the fifth primorial prime.
29 forms a twin prime pair with thirty-one, which is also a primorial prime. Twenty-nine is also the sixth Sophie Germain prime.
29 is the sum of three consecutive squares, 22 + 32 + 42.
29 is a Lucas prime, a Pell prime, and a tetranacci number.
29 is an Eisenstein prime with no imaginary part and real part of the form 3n − 1. 29 is also the 10th supersingular prime.
None of the first 29 natural numbers have more than two different prime factors. This is the longest such consecutive sequence.
29 is a Markov number, appearing in the solutions to x + y + z = 3xyz: {2, 5, 29}, {2, 29, 169}, {5, 29, 433}, {29, 169, 14701}, etc.
29 is a Perrin number, preceded in the sequence by 12, 17, 22.
29 is the smallest positive whole number that cannot be made from the numbers {1, 2, 3, 4}, using each exactly once and using only addition, subtraction, multiplication, and division.
29 is the number of pentacubes if reflections are considered distinct.
The 29th dimension is the highest dimension for compact hyperbolic Coxeter polytopes that are bounded by a fundamental polyhedron, and the highest dimension that holds arithmetic discrete groups of reflections with noncompact unbounded fundamental polyhedra.
Religion
The Bishnois community follows 29 principles. Guru Jambheshwar had laid down 29 principles to be followed by the sect in 1485 A.D. In Hindi, Bish means 20 and noi means 9; thus, Bishnoi translates as Twenty-niners.
The number of suras in the Qur'an that begin with muqatta'at.
Science and astronomy
The atomic number of copper.
Messier object M29, a magnitude 6.6 open cluster in the constellation Cygnus.
The New General Catalogue object NGC 29, a spiral galaxy in the constellation Andromeda.
Saturn requires over 29 years to orbit the Sun.
The number of days February has in leap years.
Language and literature
|
https://en.wikipedia.org/wiki/28%20%28number%29 | 28 (twenty-eight) is the natural number following 27 and preceding 29.
In mathematics
It is a composite number; a square-prime, of the form (p2,q) where q is a higher prime. It is the third of this form and of the specific form (22.q), with proper divisors being 1, 2, 4, 7, and 14.
Twenty-eight is the second perfect number - it is the sum of its proper divisors: . As a perfect number, it is related to the Mersenne prime 7, since . The next perfect number is 496, the previous being 6.
Though perfect, 28 is not the aliquot sum of any other number other than itself, and so; unusually, is not part of a multi-number aliquot sequence.
The next perfect number, 496, has the single Aliquot sum, 652 which leads to multiple aliquot sequencing.
Twenty-eight is the sum of the totient function for the first nine integers.
Since the greatest prime factor of is 157, which is more than 28 twice, 28 is a Størmer number.
Twenty-eight is a harmonic divisor number, a happy number, a triangular number, a hexagonal number, a Leyland number of the second kind and a centered nonagonal number.
It appears in the Padovan sequence, preceded by the terms 12, 16, 21 (it is the sum of the first two of these).
It is also a Keith number, because it recurs in a Fibonacci-like sequence started from its decimal digits: 2, 8, 10, 18, 28...
Twenty-eight is the ninth and last number in early Indian magic square of order 3.
There are twenty-eight convex uniform honeycombs.
Twenty-eight is the only positive integer that has a unique Kayles nim-value.
Twenty-eight is the only known number that can be expressed as a sum of the first nonnegative (or positive) integers (), a sum of the first primes () and a sum of the first nonprimes (), and it is unlikely that any other number has this property.
There are twenty-eight oriented diffeomorphism classes of manifolds homeomorphic to the 7-sphere.
There are 28 elements of the cuboid: 8 vertices, 12 edges, 6 faces, 2 3-dimensional elements (interior a |
https://en.wikipedia.org/wiki/Separable%20polynomial | In mathematics, a polynomial P(X) over a given field K is separable if its roots are distinct in an algebraic closure of K, that is, the number of distinct roots is equal to the degree of the polynomial.
This concept is closely related to square-free polynomial. If K is a perfect field then the two concepts coincide. In general, P(X) is separable if and only if it is square-free over any field that contains K,
which holds if and only if P(X) is coprime to its formal derivative D P(X).
Older definition
In an older definition, P(X) was considered separable if each of its irreducible factors in K[X] is separable in the modern definition. In this definition, separability depended on the field K; for example, any polynomial over a perfect field would have been considered separable. This definition, although it can be convenient for Galois theory, is no longer in use.
Separable field extensions
Separable polynomials are used to define separable extensions: A field extension is a separable extension if and only if for every in which is algebraic over , the minimal polynomial of over is a separable polynomial.
Inseparable extensions (that is, extensions which are not separable) may occur only in positive characteristic.
The criterion above leads to the quick conclusion that if P is irreducible and not separable, then D P(X) = 0.
Thus we must have
P(X) = Q(X p)
for some polynomial Q over K, where the prime number p is the characteristic.
With this clue we can construct an example:
P(X) = X p − T
with K the field of rational functions in the indeterminate T over the finite field with p elements. Here one can prove directly that P(X) is irreducible and not separable. This is actually a typical example of why inseparability matters; in geometric terms P represents the mapping on the projective line over the finite field, taking co-ordinates to their pth power. Such mappings are fundamental to the algebraic geometry of finite fields. Put another way, |
https://en.wikipedia.org/wiki/Antoni%20Zygmund | Antoni Zygmund (December 25, 1900 – May 30, 1992) was a Polish mathematician. He worked mostly in the area of mathematical analysis, including especially harmonic analysis, and he is considered one of the greatest analysts of the 20th century. Zygmund was responsible for creating the Chicago school of mathematical analysis together with his doctoral student Alberto Calderón, for which he was awarded the National Medal of Science in 1986.
Biography
Born in Warsaw, Zygmund obtained his Ph.D. from the University of Warsaw (1923) and was a professor at Stefan Batory University at Wilno from 1930 to 1939, when World War II broke out and Poland was occupied. In 1940 he managed to emigrate to the United States, where he became a professor at Mount Holyoke College in South Hadley, Massachusetts. In 1945–1947 he was a professor at the University of Pennsylvania, and from 1947, until his retirement, at the University of Chicago.
He was a member of several scientific societies. From 1930 until 1952 he was a member of the Warsaw Scientific Society (TNW), from 1946 of the Polish Academy of Learning (PAU), from 1959 of the Polish Academy of Sciences (PAN), and from 1961 of the National Academy of Sciences in the United States. In 1986 he received the National Medal of Science.
In 1935 Zygmund published in Polish the original edition of what has become, in its English translation, the two-volume Trigonometric Series. It was described by Robert A. Fefferman as "one of the most influential books in the history of mathematical analysis" and "an extraordinarily comprehensive and masterful presentation of a ... vast field". Jean-Pierre Kahane called the book "The Bible" of a harmonic analyst. The theory of trigonometric series had remained the largest component of Zygmund's mathematical investigations.
His work has had a pervasive influence in many fields of mathematics, mostly in mathematical analysis, and particularly in harmonic analysis. Among the most significant were the resu |
https://en.wikipedia.org/wiki/Optimal%20control | Optimal control theory is a branch of control theory that deals with finding a control for a dynamical system over a period of time such that an objective function is optimized. It has numerous applications in science, engineering and operations research. For example, the dynamical system might be a spacecraft with controls corresponding to rocket thrusters, and the objective might be to reach the Moon with minimum fuel expenditure. Or the dynamical system could be a nation's economy, with the objective to minimize unemployment; the controls in this case could be fiscal and monetary policy. A dynamical system may also be introduced to embed operations research problems within the framework of optimal control theory.
Optimal control is an extension of the calculus of variations, and is a mathematical optimization method for deriving control policies. The method is largely due to the work of Lev Pontryagin and Richard Bellman in the 1950s, after contributions to calculus of variations by Edward J. McShane. Optimal control can be seen as a control strategy in control theory.
General method
Optimal control deals with the problem of finding a control law for a given system such that a certain optimality criterion is achieved. A control problem includes a cost functional that is a function of state and control variables. An optimal control is a set of differential equations describing the paths of the control variables that minimize the cost function. The optimal control can be derived using Pontryagin's maximum principle (a necessary condition also known as Pontryagin's minimum principle or simply Pontryagin's principle), or by solving the Hamilton–Jacobi–Bellman equation (a sufficient condition).
We begin with a simple example. Consider a car traveling in a straight line on a hilly road. The question is, how should the driver press the accelerator pedal in order to minimize the total traveling time? In this example, the term control law refers specifically to the way |
https://en.wikipedia.org/wiki/Indium%20tin%20oxide | Indium tin oxide (ITO) is a ternary composition of indium, tin and oxygen in varying proportions. Depending on the oxygen content, it can be described as either a ceramic or an alloy. Indium tin oxide is typically encountered as an oxygen-saturated composition with a formulation of 74% In, 8% Sn, and 18% O by weight. Oxygen-saturated compositions are so typical that unsaturated compositions are termed oxygen-deficient ITO. It is transparent and colorless in thin layers, while in bulk form it is yellowish to gray. In the infrared region of the spectrum it acts as a metal-like mirror.
Indium tin oxide is one of the most widely used transparent conducting oxides because of its electrical conductivity and optical transparency, the ease with which it can be deposited as a thin film, and its chemical resistance to moisture. As with all transparent conducting films, a compromise must be made between conductivity and transparency, since increasing the thickness and increasing the concentration of charge carriers increases the film's conductivity, but decreases its transparency.
Thin films of indium tin oxide are most commonly deposited on surfaces by physical vapor deposition. Often used is electron beam evaporation, or a range of sputter deposition techniques.
Material and properties
ITO is a mixed oxide of indium and tin with a melting point in the range 1526–1926 °C (1800–2200 K, 2800–3500 °F), depending on composition. The most commonly used material is an oxide of a composition of ca. In4Sn. The material is a n-type semiconductor with a large bandgap of around 4 eV. ITO is both transparent to visible light and relatively conductive. It has a low electrical resistivity of ~10−4 Ω·cm, and a thin film can have an optical transmittance of greater than 80%. These properties are utilized to great advantage in touch-screen applications such as mobile phones.
Common uses
Indium tin oxide (ITO) is an optoelectronic material that is applied widely in both research and indu |
https://en.wikipedia.org/wiki/Ludolph%20van%20Ceulen | Ludolph van Ceulen (, ; 28 January 1540 – 31 December 1610) was a German-Dutch mathematician from Hildesheim. He emigrated to the Netherlands.
Biography
Van Ceulen moved to Delft most likely in 1576 to teach fencing and mathematics and in 1594 opened a fencing school in Leiden. In 1600 he was appointed the first professor of mathematics at the Engineering School, Duytsche Mathematique, established by Maurice, Prince of Orange, at the relatively new Leiden University. He shared this professorial level at the school with the surveyor and cartographer, , which shows that the intention was to promote practical, rather than theoretical instruction.
The curriculum for the new Engineering School was devised by Simon Stevin who continued to act as the personal advisor to the Prince. At first the professors at Leiden refused to accept the status of Van Ceulen and Van Merwen, especially as they taught in Dutch rather than Latin. Theological professors generally believed that practical courses were not acceptable studies for a university, but they were not willing to reject the School outright since it was founded by Prince Maurice.
Leiden University governors heard in April 1600 that Adriaan Metius, a fortification advisor to Prince Maurice and the States General, had been recruited and raised to the level of a full professor to teach mathematics at the rival Franeker University. The Leiden governors' main problem was to match Franeker University, without raising the status too much of Duytsche Mathematique. So they quickly recruited mathematician Rudolf Snellius to the university—as distinct from the Engineering School—but then relegated him to the Faculty of Arts.
When the first degrees were to be conferred on Engineering School graduates in 1602 (under protest from the University) the governors and University's senate refused to award them except via an examination conducted by the Universities' own mathematics professor, Rudolf Snellius—ensuring that Van Ceulen an |
https://en.wikipedia.org/wiki/Stanis%C5%82aw%20Saks | Stanisław Saks (30 December 1897 – 23 November 1942) was a Polish mathematician and university tutor, a member of the Lwów School of Mathematics, known primarily for his membership in the Scottish Café circle, an extensive monograph on the theory of integrals, his works on measure theory and the Vitali–Hahn–Saks theorem.
Life and work
Stanisław Saks was born on 30 December 1897 in Kalisz, Congress Poland, to an assimilated Polish-Jewish family. In 1915 he graduated from a local gymnasium and joined the newly recreated Warsaw University. In 1922 he received a doctorate of his alma mater with a prestigious distinction maxima cum laude. Soon afterwards he also passed his habilitation and received the Rockefeller fellowship, which allowed him to travel to the United States. Around that time he started publishing articles in various mathematical journals, mostly the Fundamenta Mathematicae, but also in the Transactions of the American Mathematical Society. He participated in the Silesian Uprisings and was awarded the Cross of the Valorous and the Medal of Independence for his bravery. Following the end of the uprising he returned to Warsaw and resumed his academic career.
For most of it he studied the theories of functions and functionals in particular. In 1930 he published his most notable book, the Zarys teorii całki (Sketch on the Theory of the Integral), which later got expanded and translated into several languages, including English (Theory of the Integral), French (Théorie de l'Intégrale) and Russian (Teoriya Integrala). Despite his successes, Saks was never awarded the title of professor and remained an ordinary tutor, initially at his alma mater and the Warsaw University of Technology, and later at the Lwów University and Wilno University. He was also an active socialist and a journalist at the Robotnik weekly (1919–1926) and later a collaborator of the Association of Socialist Youth.
Saks wrote a mathematics book with Antoni Zygmund, Analytic Functions, in |
https://en.wikipedia.org/wiki/Negative%20temperature | Certain systems can achieve negative thermodynamic temperature; that is, their temperature can be expressed as a negative quantity on the Kelvin or Rankine scales. This should be distinguished from temperatures expressed as negative numbers on non-thermodynamic Celsius or Fahrenheit scales, which are nevertheless higher than absolute zero. A system with a truly negative temperature on the Kelvin scale is hotter than any system with a positive temperature. If a negative-temperature system and a positive-temperature system come in contact, heat will flow from the negative- to the positive-temperature system. A standard example of such a system is population inversion in laser physics.
Thermodynamic systems with unbounded phase space cannot achieve negative temperatures: adding heat always increases their entropy. The possibility of a decrease in entropy as energy increases requires the system to "saturate" in entropy. This is only possible if the number of high energy states is limited. For a system of ordinary (quantum or classical) particles such as atoms or dust, the number of high energy states is unlimited (particle momenta can in principle be increased indefinitely). Some systems, however (see the examples below), have a maximum amount of energy that they can hold, and as they approach that maximum energy their entropy actually begins to decrease.
History
The possibility of negative temperatures was first predicted by Lars Onsager in 1949.
Onsager was investigating 2D vortices confined within a finite area, and realized that since their positions are not independent degrees of freedom from their momenta, the resulting phase space must also be bounded by the finite area. Bounded phase space is the essential property that allows for negative temperatures, and can occur in both classical and quantum systems. As shown by Onsager, a system with bounded phase space necessarily has a peak in the entropy as energy is increased. For energies exceeding the value wher |
https://en.wikipedia.org/wiki/Diacetyl | Diacetyl ( ; IUPAC systematic name: butanedione or butane-2,3-dione) is an organic compound with the chemical formula (CH3CO)2. It is a yellow liquid with an intensely buttery flavor. It is a vicinal diketone (two C=O groups, side-by-side). Diacetyl occurs naturally in alcoholic beverages and is added as a flavoring to some foods to impart its buttery flavor.
Chemical structure
A distinctive feature of diacetyl (and other vicinal diketones) is the long C–C bond linking the carbonyl centers. This bond distance is about 1.54 Å, compared to 1.45 Å for the corresponding C–C bond in 1,3-butadiene. The elongation is attributed to repulsion between the polarized carbonyl carbon centers.
Occurrence and biosynthesis
Diacetyl arises naturally as a byproduct of fermentation. In some fermentative bacteria, it is formed via the thiamine pyrophosphate-mediated condensation of pyruvate and acetyl CoA. Sour (cultured) cream, cultured buttermilk, and cultured butter are produced by inoculating pasteurized cream or milk with a lactic starter culture, churning (agitating) and holding the milk until a desired pH drop (or increase in acidity) is attained. Cultured cream, cultured butter, and cultured buttermilk owe their tart flavour to lactic acid bacteria and their buttery aroma and taste to diacetyl. Malic acid can be converted to lactic acid to make diacetyl.
Production
Diacetyl is produced industrially by dehydrogenation of 2,3-butanediol. Acetoin is an intermediate.
Applications
In food products
Diacetyl and acetoin are two compounds that give butter its characteristic taste. Because of this, manufacturers of artificial butter flavoring, margarines or similar oil-based products typically add diacetyl and acetoin (along with beta-carotene for the yellow color) to make the final product butter-flavored, because it would otherwise be relatively tasteless.
Electronic cigarettes
Diacetyl is used as a flavoring agent in some liquids used in electronic cigarettes. People nearby ma |
https://en.wikipedia.org/wiki/NetWare | NetWare is a discontinued computer network operating system developed by Novell, Inc. It initially used cooperative multitasking to run various services on a personal computer, using the IPX network protocol.
The original NetWare product in 1983 supported clients running both CP/M and MS-DOS, ran over a proprietary star network topology and was based on a Novell-built file server using the Motorola 68000 processor. The company soon moved away from building its own hardware, and NetWare became hardware-independent, running on any suitable Intel-based IBM PC compatible system, and able to utilize a wide range of network cards. From the beginning NetWare implemented a number of features inspired by mainframe and minicomputer systems that were not available in its competitors' products.
In 1991, Novell introduced cheaper peer-to-peer networking products for DOS and Windows, unrelated to their server-centric NetWare. These are NetWare Lite 1.0 (NWL), and later Personal NetWare 1.0 (PNW) in 1993.
In 1993, the main NetWare product line took a dramatic turn when version 4 introduced NetWare Directory Services (NDS, later renamed eDirectory), a global directory service based on ISO X.500 concepts (six years later, Microsoft released Active Directory). The directory service, along with a new e-mail system (GroupWise), application configuration suite (ZENworks), and security product (BorderManager) were all targeted at the needs of large enterprises.
By 2000, however, Microsoft was taking more of Novell's customer base and Novell increasingly looked to a future based on a Linux kernel. The successor to NetWare, Open Enterprise Server (OES), released in March 2005, offers all the services previously hosted by NetWare 6.5, but on a SUSE Linux Enterprise Server; the NetWare kernel remained an option until OES 11 in late 2011.
The final update release was version 6.5SP8 of May 2009; NetWare is no longer on Novell's product list. NetWare 6.5SP8 General Support ended in 2010; E |
https://en.wikipedia.org/wiki/38%20%28number%29 | 38 (thirty-eight) is the natural number following 37 and preceding 39.
In mathematics
specifically, the 11th discrete Semiprime, it being the 7th of the form (2.q).
the first member of the third cluster of two discrete semiprimes 38, 39 the next such cluster is 57, 58.
with an aliquot sum of 22 in an aliquot sequence of five composite numbers (38,22,14,10,8,7,1,0) to the Prime in the 7-aliquot tree. 34 is the first semiprime within a chain of 4 semiprimes in its aliquot sequence (38,22,14,10). The next semiprime with a four semiprime chain is 166.
38! − 1 yields which is the 16th factorial prime.
There is no answer to the equation φ(x) = 38, making 38 a nontotient.
38 is the sum of the squares of the first three primes.
37 and 38 are the first pair of consecutive positive integers not divisible by any of their digits.
38 is the largest even number which cannot be written as the sum of two odd composite numbers.
The sum of each row of the only non-trivial (order 3) magic hexagon is 38.
In science
The atomic number of strontium
Astronomy
The Messier object M38, a magnitude 7.0 open cluster in the constellation Auriga
The New General Catalogue object NGC 38, a spiral galaxy in the constellation Pisces
In other fields
Thirty-eight is also:
The 38th parallel north is the pre-Korean War boundary between North Korea and South Korea.
The number of slots on an American roulette wheel (0, 00, and 1 through 36; European roulette does not use the 00 slot and has only 37 slots)
The Ishihara test is a color vision test consisting of 38 pseudoisochromatic plates.
A "38" is often the name for a snub nose .38 caliber revolver.
The 38 class is the most famous class of steam locomotive used in New South Wales
Gerald Ford, 38th President of the United States
Arnold Schwarzenegger, 38th Governor of California, most recent Republican governor of California, and the second governor to be born outside of the United States
Cats have a total of 38 chromosomes in their genom |
https://en.wikipedia.org/wiki/Substitution%20matrix | In bioinformatics and evolutionary biology, a substitution matrix describes the frequency at which a character in a nucleotide sequence or a protein sequence changes to other character states over evolutionary time. The information is often in the form of log odds of finding two specific character states aligned and depends on the assumed number of evolutionary changes or sequence dissimilarity between compared sequences. It is an application of a stochastic matrix. Substitution matrices are usually seen in the context of amino acid or DNA sequence alignments, where they are used to calculate similarity scores between the aligned sequences.
Background
In the process of evolution, from one generation to the next the amino acid sequences of an organism's proteins are gradually altered through the action of DNA mutations. For example, the sequence
ALEIRYLRD
could mutate into the sequence
ALEINYLRD
in one step, and possibly
AQEINYQRD
over a longer period of evolutionary time. Each amino acid is more or less likely to mutate into various other amino acids. For instance, a hydrophilic residue such as arginine is more likely to be replaced by another hydrophilic residue such as glutamine, than it is to be mutated into a hydrophobic residue such as leucine. (Here, a residue refers to an amino acid stripped of a hydrogen and/or a hydroxyl group and inserted in the polymeric chain of a protein.) This is primarily due to redundancy in the genetic code, which translates similar codons into similar amino acids. Furthermore, mutating an amino acid to a residue with significantly different properties could affect the folding and/or activity of the protein. This type of disruptive substitution is likely to be removed from populations by the action of purifying selection because the substitution has a higher likelihood of rendering a protein nonfunctional.
If we have two amino acid sequences in front of us, we should be able to say something about how likely they are t |
https://en.wikipedia.org/wiki/Homogeneous%20space | In mathematics, a homogeneous space is, very informally, a space that looks the same everywhere, as you move through it, with movement given by the action of a group. Homogeneous spaces occur in the theories of Lie groups, algebraic groups and topological groups. More precisely, a homogeneous space for a group G is a non-empty manifold or topological space X on which G acts transitively. The elements of G are called the symmetries of X. A special case of this is when the group G in question is the automorphism group of the space X – here "automorphism group" can mean isometry group, diffeomorphism group, or homeomorphism group. In this case, X is homogeneous if intuitively X looks locally the same at each point, either in the sense of isometry (rigid geometry), diffeomorphism (differential geometry), or homeomorphism (topology). Some authors insist that the action of G be faithful (non-identity elements act non-trivially), although the present article does not. Thus there is a group action of G on X which can be thought of as preserving some "geometric structure" on X, and making X into a single G-orbit.
Formal definition
Let X be a non-empty set and G a group. Then X is called a G-space if it is equipped with an action of G on X. Note that automatically G acts by automorphisms (bijections) on the set. If X in addition belongs to some category, then the elements of G are assumed to act as automorphisms in the same category. That is, the maps on X coming from elements of G preserve the structure associated with the category (for example, if X is an object in Diff then the action is required to be by diffeomorphisms). A homogeneous space is a G-space on which G acts transitively.
Succinctly, if X is an object of the category C, then the structure of a G-space is a homomorphism:
into the group of automorphisms of the object X in the category C. The pair (X, ρ) defines a homogeneous space provided ρ(G) is a transitive group of symmetries of the underlying set of X.
|
https://en.wikipedia.org/wiki/Lyapunov%20stability | Various types of stability may be discussed for the solutions of differential equations or difference equations describing dynamical systems. The most important type is that concerning the stability of solutions near to a point of equilibrium. This may be discussed by the theory of Aleksandr Lyapunov. In simple terms, if the solutions that start out near an equilibrium point stay near forever, then is Lyapunov stable. More strongly, if is Lyapunov stable and all solutions that start out near converge to , then is said to be asymptotically stable (see asymptotic analysis). The notion of exponential stability guarantees a minimal rate of decay, i.e., an estimate of how quickly the solutions converge. The idea of Lyapunov stability can be extended to infinite-dimensional manifolds, where it is known as structural stability, which concerns the behavior of different but "nearby" solutions to differential equations. Input-to-state stability (ISS) applies Lyapunov notions to systems with inputs.
History
Lyapunov stability is named after Aleksandr Mikhailovich Lyapunov, a Russian mathematician who defended the thesis The General Problem of Stability of Motion at Kharkov University in 1892. A. M. Lyapunov was a pioneer in successful endeavors to develop a global approach to the analysis of the stability of nonlinear dynamical systems by comparison with the widely spread local method of linearizing them about points of equilibrium. His work, initially published in Russian and then translated to French, received little attention for many years. The mathematical theory of stability of motion, founded by A. M. Lyapunov, considerably anticipated the time for its implementation in science and technology. Moreover Lyapunov did not himself make application in this field, his own interest being in the stability of rotating fluid masses with astronomical application. He did not have doctoral students who followed the research in the field of stability and his own destiny was t |
https://en.wikipedia.org/wiki/Combinatorial%20topology | In mathematics, combinatorial topology was an older name for algebraic topology, dating from the time when topological invariants of spaces (for example the Betti numbers) were regarded as derived from combinatorial decompositions of spaces, such as decomposition into simplicial complexes. After the proof of the simplicial approximation theorem this approach provided rigour.
The change of name reflected the move to organise topological classes such as cycles-modulo-boundaries explicitly into abelian groups. This point of view is often attributed to Emmy Noether, and so the change of title may reflect her influence. The transition is also attributed to the work of Heinz Hopf, who was influenced by Noether, and to Leopold Vietoris and Walther Mayer, who independently defined homology.
A fairly precise date can be supplied in the internal notes of the Bourbaki group. While topology was still combinatorial in 1942, it had become algebraic by 1944. This corresponds also to the period where homological algebra and category theory were introduced for the study of topological spaces, and largely supplanted combinatorial methods.
Azriel Rosenfeld (1973) proposed digital topology for a type of image processing that can be considered as a new development of combinatorial topology. The digital forms of the Euler characteristic theorem and the Gauss–Bonnet theorem were obtained by Li Chen and Yongwu Rong. A 2D grid cell topology already appeared in the Alexandrov–Hopf book Topologie I (1935).
See also
Hauptvermutung
Topological combinatorics
Topological graph theory
Notes
References
Algebraic topology
Combinatorics
es:Topología combinatoria |
https://en.wikipedia.org/wiki/Principal%20homogeneous%20space | In mathematics, a principal homogeneous space, or torsor, for a group G is a homogeneous space X for G in which the stabilizer subgroup of every point is trivial. Equivalently, a principal homogeneous space for a group G is a non-empty set X on which G acts freely and transitively (meaning that, for any x, y in X, there exists a unique g in G such that , where · denotes the (right) action of G on X).
An analogous definition holds in other categories, where, for example,
G is a topological group, X is a topological space and the action is continuous,
G is a Lie group, X is a smooth manifold and the action is smooth,
G is an algebraic group, X is an algebraic variety and the action is regular.
Definition
If G is nonabelian then one must distinguish between left and right torsors according to whether the action is on the left or right. In this article, we will use right actions.
To state the definition more explicitly, X is a G-torsor or G-principal homogeneous space if X is nonempty and is equipped with a map (in the appropriate category) such that
x·1 = x
x·(gh) = (x·g)·h
for all and all and such that the map given by
is an isomorphism (of sets, or topological spaces or ..., as appropriate, i.e. in the category in question).
Note that this means that X and G are isomorphic (in the category in question; not as groups: see the following). However—and this is the essential point—there is no preferred 'identity' point in X. That is, X looks exactly like G except that which point is the identity has been forgotten. (This concept is often used in mathematics as a way of passing to a more intrinsic point of view, under the heading 'throw away the origin'.)
Since X is not a group, we cannot multiply elements; we can, however, take their "quotient". That is, there is a map that sends to the unique element such that .
The composition of the latter operation with the right group action, however, yields a ternary operation , which serves as an affine generalizati |
https://en.wikipedia.org/wiki/Serre%E2%80%93Swan%20theorem | In the mathematical fields of topology and K-theory, the Serre–Swan theorem, also called Swan's theorem, relates the geometric notion of vector bundles to the algebraic concept of projective modules and gives rise to a common intuition throughout mathematics: "projective modules over commutative rings are like vector bundles on compact spaces".
The two precise formulations of the theorems differ somewhat. The original theorem, as stated by Jean-Pierre Serre in 1955, is more algebraic in nature, and concerns vector bundles on an algebraic variety over an algebraically closed field (of any characteristic). The complementary variant stated by Richard Swan in 1962 is more analytic, and concerns (real, complex, or quaternionic) vector bundles on a smooth manifold or Hausdorff space.
Differential geometry
Suppose M is a smooth manifold (not necessarily compact), and E is a smooth vector bundle over M. Then Γ(E), the space of smooth sections of E, is a module over C∞(M) (the commutative algebra of smooth real-valued functions on M). Swan's theorem states that this module is finitely generated and projective over C∞(M). In other words, every vector bundle is a direct summand of some trivial bundle: for some k. The theorem can be proved by constructing a bundle epimorphism from a trivial bundle This can be done by, for instance, exhibiting sections s1...sk with the property that for each point p, {si(p)} span the fiber over p.
When M is connected, the converse is also true: every finitely generated projective module over C∞(M) arises in this way from some smooth vector bundle on M. Such a module can be viewed as a smooth function f on M with values in the n × n idempotent matrices for some n. The fiber of the corresponding vector bundle over x is then the range of f(x). If M is not connected, the converse does not hold unless one allows for vector bundles of non-constant rank (which means admitting manifolds of non-constant dimension). For example, if M is a zero-dime |
https://en.wikipedia.org/wiki/Universal%20enveloping%20algebra | In mathematics, the universal enveloping algebra of a Lie algebra is the unital associative algebra whose representations correspond precisely to the representations of that Lie algebra.
Universal enveloping algebras are used in the representation theory of Lie groups and Lie algebras. For example, Verma modules can be constructed as quotients of the universal enveloping algebra. In addition, the enveloping algebra gives a precise definition for the Casimir operators. Because Casimir operators commute with all elements of a Lie algebra, they can be used to classify representations. The precise definition also allows the importation of Casimir operators into other areas of mathematics, specifically, those that have a differential algebra. They also play a central role in some recent developments in mathematics. In particular, their dual provides a commutative example of the objects studied in non-commutative geometry, the quantum groups. This dual can be shown, by the Gelfand–Naimark theorem, to contain the C* algebra of the corresponding Lie group. This relationship generalizes to the idea of Tannaka–Krein duality between compact topological groups and their representations.
From an analytic viewpoint, the universal enveloping algebra of the Lie algebra of a Lie group may be identified with the algebra of left-invariant differential operators on the group.
Informal construction
The idea of the universal enveloping algebra is to embed a Lie algebra into an associative algebra with identity in such a way that the abstract bracket operation in corresponds to the commutator in and the algebra is generated by the elements of . There may be many ways to make such an embedding, but there is a unique "largest" such , called the universal enveloping algebra of .
Generators and relations
Let be a Lie algebra, assumed finite-dimensional for simplicity, with basis . Let be the structure constants for this basis, so that
Then the universal enveloping algebra is the |
https://en.wikipedia.org/wiki/Tensor%20algebra | In mathematics, the tensor algebra of a vector space V, denoted T(V) or T(V), is the algebra of tensors on V (of any rank) with multiplication being the tensor product. It is the free algebra on V, in the sense of being left adjoint to the forgetful functor from algebras to vector spaces: it is the "most general" algebra containing V, in the sense of the corresponding universal property (see below).
The tensor algebra is important because many other algebras arise as quotient algebras of T(V). These include the exterior algebra, the symmetric algebra, Clifford algebras, the Weyl algebra and universal enveloping algebras.
The tensor algebra also has two coalgebra structures; one simple one, which does not make it a bialgebra, but does lead to the concept of a cofree coalgebra, and a more complicated one, which yields a bialgebra, and can be extended by giving an antipode to create a Hopf algebra structure.
Note: In this article, all algebras are assumed to be unital and associative. The unit is explicitly required to define the coproduct.
Construction
Let V be a vector space over a field K. For any nonnegative integer k, we define the kth tensor power of V to be the tensor product of V with itself k times:
That is, TkV consists of all tensors on V of order k. By convention T0V is the ground field K (as a one-dimensional vector space over itself).
We then construct T(V) as the direct sum of TkV for k = 0,1,2,…
The multiplication in T(V) is determined by the canonical isomorphism
given by the tensor product, which is then extended by linearity to all of T(V). This multiplication rule implies that the tensor algebra T(V) is naturally a graded algebra with TkV serving as the grade-k subspace. This grading can be extended to a Z-grading by appending subspaces for negative integers k.
The construction generalizes in a straightforward manner to the tensor algebra of any module M over a commutative ring. If R is a non-commutative ring, one can still perform the co |
https://en.wikipedia.org/wiki/X.400 | X.400 is a suite of ITU-T recommendations that define the ITU-T Message Handling System (MHS).
At one time, the designers of X.400 were expecting it to be the predominant form of email, but this role has been taken by the SMTP-based Internet e-mail. Despite this, it has been widely used within organizations and was a core part of Microsoft Exchange Server until 2006; variants continue to be important in military and aviation contexts.
History
The first X.400 Recommendations were published in 1984 (Red Book), and a substantially revised version was published in 1988 (Blue Book). New features were added in 1992 (White Book) and subsequent updates. Although X.400 was originally designed to run over the OSI transport service, an adaptation to allow operation over TCP/IP, RFC 1006, has become the most popular way to run X.400.
Developed in cooperation with ISO/IEC, the X.400-series recommendations specify OSI standard protocols for exchanging and addressing electronic messages. The companion F.400-series of recommendations define Message Handling Services built on Message Handling Systems (MHS), as well as access to and from the MHS for public services. In the late 1990s, the ITU-T consolidated recommendations F.400 and X.400 and published the ITU-T F.400/X.400 (06/1999) recommendation "Message handling system and service overview".
The X.400-series recommendations define the technical aspects of the MHS: ITU-T Rec. X.402 | (ISO/IEC 10021-2) defines the overall system architecture of an MHS, ITU-T Rec. X.411 | (ISO/IEC 10021-4) defines the Message Transfer Service (MTS) and its functional component the Message Transfer Agent (MTA), and ITU-T Rec. X.413 | (ISO/IEC 10021-5) defines the Message Store. All ITU-T recommendations provide specific terms for descriptions of system entities and procedures. For example, messages (email) exchanged among people is referred to as Interpersonal Messaging (IPM); electronically structured business documents (e.g., invoices, purcha |
https://en.wikipedia.org/wiki/Stereopticon | A stereopticon is a slide projector or relatively powerful "magic lantern", which has two lenses, usually one above the other, and has mainly been used to project photographic images. These devices date back to the mid 19th century, and were a popular form of entertainment and education before the advent of moving pictures.
Magic lanterns originally used rather weak light sources, like candles or oil lamps, that produced projections that were just large and strong enough to entertain small groups of people. During the 19th century stronger light sources, like limelight, became available.
For the "dissolving views" lantern shows that were popularized by Henry Langdon Childe since the late 1830s, lanternists needed to be able to project two aligned pictures in the same spot on a screen, gradually dimming a first picture while revealing a second one. This could be done with two lanterns, but soon biunial lanterns (with two objectives placed one above the other) became common.
William and Frederick Langenheim from Philadelphia introduced a photographic glass slide technology at the Crystal Palace Exhibition in London in 1851. For circa two centuries magic lanterns had been used to project painted images from glass slides, but the Langenheim brothers seem to have been the firsts to incorporate the relatively new medium of photography (introduced in 1839). To enjoy the details of photographic slides optimally, the stronger lanterns were needed.
By 1860 Massachusetts chemist and businessman John Fallon improved a large biunial lantern, imported from England, and named it ‘stereopticon’.
For a usual fee of ten cents, people could view realistic images of nature, history, and science themes. The two lenses are used to dissolve between images when projected. This "visual storytelling" with technology directly preceded the development of the first moving pictures.
The term stereopticon has been widely misused to name a stereoscope. The stereopticon has not commonly be |
https://en.wikipedia.org/wiki/Zymology | Zymology, also known as zymurgy, is an applied science that studies the biochemical process of fermentation and its practical uses. Common topics include the selection of fermenting yeast and bacteria species and their use in brewing, wine making, fermenting milk, and the making of other fermented foods.
Fermentation
Fermentation can be simply defined, in this context, as the conversion of sugar molecules into ethanol and carbon dioxide by yeast.
Fermentation practices have led to the discovery of ample microbial and antimicrobial cultures on fermented foods and products.
History
French chemist Louis Pasteur was the first 'zymologist' when in 1857 he connected yeast to fermentation. Pasteur originally defined fermentation as "respiration without air".
Pasteur performed careful research and concluded:
The German Eduard Buchner, winner of the 1907 Nobel Prize in chemistry, later determined that fermentation was actually caused by a yeast secretion, which he termed 'zymase'.
The research efforts undertaken by the Danish Carlsberg scientists greatly accelerated understanding of yeast and brewing. The Carlsberg scientists are generally acknowledged as having jump-started the entire field of molecular biology.
Products
All alcoholic drinks including beer, cider, kombucha, kvass, mead, perry, tibicos, wine, pulque, hard liquors (brandy, rum, vodka, sake, schnapps), and soured by-products including vinegar and alegar
Yeast leavened breads including sourdough, salt-rising bread, and others
Cheese and some dairy products including kefir and yogurt
Chocolate
Dishes including fermented fish, such as garum, surströmming, and Worcestershire sauce
Some vegetables such as kimchi, some types of pickles (most are not fermented though), and sauerkraut
A wide variety of fermented edibles made from soy beans, including fermented bean paste, nattō, tempeh, and soya sauce
Notes
References
Sources
External links
Winemaking: Fundamentals of winemaking: zymology
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https://en.wikipedia.org/wiki/One-way%20function | In computer science, a one-way function is a function that is easy to compute on every input, but hard to invert given the image of a random input. Here, "easy" and "hard" are to be understood in the sense of computational complexity theory, specifically the theory of polynomial time problems. Not being one-to-one is not considered sufficient for a function to be called one-way (see Theoretical definition, below).
The existence of such one-way functions is still an open conjecture. Their existence would prove that the complexity classes P and NP are not equal, thus resolving the foremost unsolved question of theoretical computer science. The converse is not known to be true, i.e. the existence of a proof that P≠NP would not directly imply the existence of one-way functions.
In applied contexts, the terms "easy" and "hard" are usually interpreted relative to some specific computing entity; typically "cheap enough for the legitimate users" and "prohibitively expensive for any malicious agents". One-way functions, in this sense, are fundamental tools for cryptography, personal identification, authentication, and other data security applications. While the existence of one-way functions in this sense is also an open conjecture, there are several candidates that have withstood decades of intense scrutiny. Some of them are essential ingredients of most telecommunications, e-commerce, and e-banking systems around the world.
Theoretical definition
A function f : {0,1}* → {0,1}* is one-way if f can be computed by a polynomial time algorithm, but any polynomial time randomized algorithm that attempts to compute a pseudo-inverse for f succeeds with negligible probability. (The * superscript means any number of repetitions, see Kleene star.) That is, for all randomized algorithms , all positive integers c and all sufficiently large n = length(x) ,
where the probability is over the choice of x from the discrete uniform distribution on {0,1}n, and the randomness |
https://en.wikipedia.org/wiki/Metacentric%20height | The metacentric height (GM) is a measurement of the initial static stability of a floating body. It is calculated as the distance between the centre of gravity of a ship and its metacentre. A larger metacentric height implies greater initial stability against overturning. The metacentric height also influences the natural period of rolling of a hull, with very large metacentric heights being associated with shorter periods of roll which are uncomfortable for passengers. Hence, a sufficiently, but not excessively, high metacentric height is considered ideal for passenger ships.
Metacentre
When a ship heels (rolls sideways), the centre of buoyancy of the ship moves laterally. It might also move up or down with respect to the water line. The point at which a vertical line through the heeled centre of buoyancy crosses the line through the original, vertical centre of buoyancy is the metacentre. The metacentre remains directly above the centre of buoyancy by definition.
In the diagram above, the two Bs show the centres of buoyancy of a ship in the upright and heeled conditions. The metacentre, M, is considered to be fixed relative to the ship for small angles of heel; however, at larger angles the metacentre can no longer be considered fixed, and its actual location must be found to calculate the ship's stability.
It can be calculated using the formulae:
Where KB is the centre of buoyancy (height above the keel), I is the second moment of area of the waterplane around the rotation axis in metres4, and V is the volume of displacement in metres3. KM is the distance from the keel to the metacentre.
Stable floating objects have a natural rolling frequency, just like a weight on a spring, where the frequency is increased as the spring gets stiffer. In a boat, the equivalent of the spring stiffness is the distance called "GM" or "metacentric height", being the distance between two points: "G" the centre of gravity of the boat and "M", which is a point called the metacen |
https://en.wikipedia.org/wiki/RobotWar | RobotWar is a programming game written by Silas Warner. This game, along with the companion program RobotWrite, was originally developed in the TUTOR programming language on the PLATO system in the 1970s. Later the game was commercialized and adapted for the Apple II series of computers and published by Muse Software in 1981. The premise is that in the distant future of 2002, war was declared hazardous to human health, and now countries settled their differences in a battle arena full of combat robots. As the manual states, "The task set before you is: to program a robot, that no other robot can destroy!"
The main activity of the game is to write a computer program that operates a (simulated) robot. The player selects multiple robots which do battle in an arena until only one is left standing. The robots do not have direct knowledge of the location or velocity of any of the other robots; they only use radar pulses to deduce distance, and perhaps use clever programming techniques to deduce velocity. There is no way for the player to actually take part in the battle.
Robot programming
The robots' language is similar to BASIC. There are 34 registers that can be used as variables or for the robots' I/O functions. An example program from the game manual is:
SCAN
AIM + 5 TO AIM ; MOVE GUN
AIM TO RADAR ; SEND RADAR PULSE
LOOP
IF RADAR < 0 GOSUB FIRE ; TEST RADAR
GOTO SCAN
FIRE
0 - RADAR TO SHOT ; FIRE THE GUN
ENDSUB
The robot with this program sweeps its radar in a circle, firing off radar pulses, and when it detects another robot, fires a projectile set to explode at the correct distance as estimated by the radar pulse. This particular robot stands still throughout the entire battle, as it never assigns any number to its movement registers.
Reception
Harry White reviewed RobotWar in The Space Gamer No. 45. White commented that "RobotWar is worth [...] the price. And if you don't h |
https://en.wikipedia.org/wiki/Functional%20equation | In mathematics, a functional equation
is, in the broadest meaning, an equation in which one or several functions appear as unknowns. So, differential equations and integral equations are functional equations. However, a more restricted meaning is often used, where a functional equation is an equation that relates several values of the same function. For example, the logarithm functions are essentially characterized by the logarithmic functional equation
If the domain of the unknown function is supposed to be the natural numbers, the function is generally viewed as a sequence, and, in this case, a functional equation (in the narrower meaning) is called a recurrence relation. Thus the term functional equation is used mainly for real functions and complex functions. Moreover a smoothness condition is often assumed for the solutions, since without such a condition, most functional equations have very irregular solutions. For example, the gamma function is a function that satisfies the functional equation and the initial value There are many functions that satisfy these conditions, but the gamma function is the unique one that is meromorphic in the whole complex plane, and logarithmically convex for real and positive (Bohr–Mollerup theorem).
Examples
Recurrence relations can be seen as functional equations in functions over the integers or natural numbers, in which the differences between terms' indexes can be seen as an application of the shift operator. For example, the recurrence relation defining the Fibonacci numbers, , where and
, which characterizes the periodic functions
, which characterizes the even functions, and likewise , which characterizes the odd functions
, which characterizes the functional square roots of the function g
(Cauchy's functional equation), satisfied by linear maps. The equation may, contingent on the axiom of choice, also have other pathological nonlinear solutions, whose existence can be proven with a Hamel basis for the re |
https://en.wikipedia.org/wiki/Ring%20of%20symmetric%20functions | In algebra and in particular in algebraic combinatorics, the ring of symmetric functions is a specific limit of the rings of symmetric polynomials in n indeterminates, as n goes to infinity. This ring serves as universal structure in which relations between symmetric polynomials can be expressed in a way independent of the number n of indeterminates (but its elements are neither polynomials nor functions). Among other things, this ring plays an important role in the representation theory of the symmetric group.
The ring of symmetric functions can be given a coproduct and a bilinear form making it into a positive selfadjoint graded Hopf algebra that is both commutative and cocommutative.
Symmetric polynomials
The study of symmetric functions is based on that of symmetric polynomials. In a polynomial ring in some finite set of indeterminates, a polynomial is called symmetric if it stays the same whenever the indeterminates are permuted in any way. More formally, there is an action by ring automorphisms of the symmetric group Sn on the polynomial ring in n indeterminates, where a permutation acts on a polynomial by simultaneously substituting each of the indeterminates for another according to the permutation used. The invariants for this action form the subring of symmetric polynomials. If the indeterminates are X1, ..., Xn, then examples of such symmetric polynomials are
and
A somewhat more complicated example is
X13X2X3 + X1X23X3 + X1X2X33 + X13X2X4 + X1X23X4 + X1X2X43 + ...
where the summation goes on to include all products of the third power of some variable and two other variables. There are many specific kinds of symmetric polynomials, such as elementary symmetric polynomials, power sum symmetric polynomials, monomial symmetric polynomials, complete homogeneous symmetric polynomials, and Schur polynomials.
The ring of symmetric functions
Most relations between symmetric polynomials do not depend on the number n of indeterminates, other than that so |
https://en.wikipedia.org/wiki/Projective%20module | In mathematics, particularly in algebra, the class of projective modules enlarges the class of free modules (that is, modules with basis vectors) over a ring, by keeping some of the main properties of free modules. Various equivalent characterizations of these modules appear below.
Every free module is a projective module, but the converse fails to hold over some rings, such as Dedekind rings that are not principal ideal domains. However, every projective module is a free module if the ring is a principal ideal domain such as the integers, or a polynomial ring (this is the Quillen–Suslin theorem).
Projective modules were first introduced in 1956 in the influential book Homological Algebra by Henri Cartan and Samuel Eilenberg.
Definitions
Lifting property
The usual category theoretical definition is in terms of the property of lifting that carries over from free to projective modules: a module P is projective if and only if for every surjective module homomorphism and every module homomorphism , there exists a module homomorphism such that . (We don't require the lifting homomorphism h to be unique; this is not a universal property.)
The advantage of this definition of "projective" is that it can be carried out in categories more general than module categories: we don't need a notion of "free object". It can also be dualized, leading to injective modules. The lifting property may also be rephrased as every morphism from to factors through every epimorphism to . Thus, by definition, projective modules are precisely the projective objects in the category of R-modules.
Split-exact sequences
A module P is projective if and only if every short exact sequence of modules of the form
is a split exact sequence. That is, for every surjective module homomorphism there exists a section map, that is, a module homomorphism such that f h = idP . In that case, is a direct summand of B, h is an isomorphism from P to , and is a projection on the summand |
https://en.wikipedia.org/wiki/OpenMSX | openMSX is a free software emulator for the MSX architecture. It is available for multiple platforms, including Microsoft Windows and POSIX systems such as Linux
For copyright reasons, the emulator cannot be distributed with original MSX-BIOS ROM images. Instead, openMSX includes C-BIOS, a minimal implementation of the MSX BIOS, allowing some games to be played without the original ROM image. It is possible for the user to replace C-BIOS by native BIOS if they prefer.
OpenMSX emulates a large amount of MSX systems and MSX related hardware, including:
MSXturboR
Moonsound
IDE Controller by Sunrise
GFX9000
Pioneer Palcom LaserDisc
Also some computer systems similar to MSX are emulated, like the SpectraVideo SVI-318/328, ColecoVision and Sega SG-1000.
Notable features include:
Hard- and software Scalers
Debugging
Tcl Script Support
Cheat Finder (through Tcl)
Game Trainers (through Tcl)
Audio/Video recording
Reverse support (go back in emulated time to correct mistakes or debug what happened)
OpenMSX has an open communication protocol to communicate with the openMSX emulator. Utilizing this communication protocol enables to write versatile add-ons for openMSX. Projects making use of this protocol include the following applications:
openMSX Catapult (by the openMSX team)
openMSX Debugger (by the openMSX team)
openMSXControl plugin
NekoLauncher openMSX
openMSX Peashooter
openMSX Control Plugin for Gedit
Currently Catapult, a GUI developed for the emulator that is part of the project, is being redeveloped utilizing Python and the Qt toolkit.
The openMSX Debugger is also under development, written in C++, also utilizing the Qt Toolkit.
References
Sources
Project Homepage
Project Forum
C-BIOS Compatibility Page
openMSX 0.5.1 review (2005)
NekoLauncher openMSX
openMSX Peashooter
openMSX Control Plugin for Gedit
openMSX development builds for Mac, Windows, Android & Dingux
Free emulation software
Free software programmed in Tcl
Free software projects
Linux emulat |
https://en.wikipedia.org/wiki/Israel%20Gelfand | Israel Moiseevich Gelfand, also written Israïl Moyseyovich Gel'fand, or Izrail M. Gelfand (, , ; – 5 October 2009) was a prominent Soviet-American mathematician. He made significant contributions to many branches of mathematics, including group theory, representation theory and functional analysis. The recipient of many awards, including the Order of Lenin and the first Wolf Prize, he was a Foreign Fellow of the Royal Society and professor at Moscow State University and, after immigrating to the United States shortly before his 76th birthday, at Rutgers University. Gelfand is also a 1994 MacArthur Fellow.
His legacy continues through his students, who include Endre Szemerédi, Alexandre Kirillov, Edward Frenkel, Joseph Bernstein, David Kazhdan, as well as his own son, Sergei Gelfand.
Early years
A native of Kherson Governorate, Russian Empire (now, Odesa Oblast, Ukraine), Gelfand was born into a Jewish family in the small southern Ukrainian town of Okny. According to his own account, Gelfand was expelled from high school under the Soviets because his father had been a mill owner. Bypassing both high school and college, he proceeded to postgraduate study at the age of 19 at Moscow State University, where his advisor was the preeminent mathematician Andrei Kolmogorov. He received his PhD in 1935.
Gelfand immigrated to the United States in 1989.
Work
Gelfand is known for many developments including:
the book Calculus of Variations (1963), which he co-authored with Sergei Fomin;
Gelfand's formula, which expresses the spectral radius as a limit of matrix norms.
the Gelfand representation in Banach algebra theory;
the Gelfand–Mazur theorem in Banach algebra theory;
the Gelfand–Naimark theorem;
the Gelfand–Naimark–Segal construction;
Gelfand–Shilov spaces;
the Gelfand–Pettis integral;
the representation theory of the complex classical Lie groups;
contributions to the theory of Verma modules in the representation theory of semisimple Lie algebras (with I. N. |
https://en.wikipedia.org/wiki/Conformal%20field%20theory | A conformal field theory (CFT) is a quantum field theory that is invariant under conformal transformations. In two dimensions, there is an infinite-dimensional algebra of local conformal transformations, and conformal field theories can sometimes be exactly solved or classified.
Conformal field theory has important applications to condensed matter physics, statistical mechanics, quantum statistical mechanics, and string theory. Statistical and condensed matter systems are indeed often conformally invariant at their thermodynamic or quantum critical points.
Scale invariance vs conformal invariance
In quantum field theory, scale invariance is a common and natural symmetry, because any fixed point of the renormalization group is by definition scale invariant. Conformal symmetry is stronger than scale invariance, and one needs additional assumptions to argue that it should appear in nature. The basic idea behind its plausibility is that local scale invariant theories have their currents given by where is a Killing vector and is a conserved operator (the stress-tensor) of dimension exactly . For the associated symmetries to include scale but not conformal transformations, the trace has to be a non-zero total derivative implying that there is a non-conserved operator of dimension exactly .
Under some assumptions it is possible to completely rule out this type of non-renormalization and hence prove that scale invariance implies conformal invariance in a quantum field theory, for example in unitary compact conformal field theories in two dimensions.
While it is possible for a quantum field theory to be scale invariant but not conformally invariant, examples are rare. For this reason, the terms are often used interchangeably in the context of quantum field theory.
Two dimensions vs higher dimensions
The number of independent conformal transformations is infinite in two dimensions, and finite in higher dimensions. This makes conformal symmetry much more constraini |
https://en.wikipedia.org/wiki/Quotient%20module | In algebra, given a module and a submodule, one can construct their quotient module. This construction, described below, is very similar to that of a quotient vector space. It differs from analogous quotient constructions of rings and groups by the fact that in these cases, the subspace that is used for defining the quotient is not of the same nature as the ambient space (that is, a quotient ring is the quotient of a ring by an ideal, not a subring, and a quotient group is the quotient of a group by a normal subgroup, not by a general subgroup).
Given a module over a ring , and a submodule of , the quotient space is defined by the equivalence relation
if and only if
for any in . The elements of are the equivalence classes The function sending in to its equivalence class is called the quotient map or the projection map, and is a module homomorphism.
The addition operation on is defined for two equivalence classes as the equivalence class of the sum of two representatives from these classes; and scalar multiplication of elements of by elements of is defined similarly. Note that it has to be shown that these operations are well-defined. Then becomes itself an -module, called the quotient module. In symbols, for all in and in :
Examples
Consider the polynomial ring, with real coefficients, and the -module . Consider the submodule
of , that is, the submodule of all polynomials divisible by . It follows that the equivalence relation determined by this module will be
if and only if and give the same remainder when divided by .
Therefore, in the quotient module , is the same as 0; so one can view as obtained from by setting . This quotient module is isomorphic to the complex numbers, viewed as a module over the real numbers
See also
Quotient group
Quotient ring
Quotient (universal algebra)
References
Module theory
Module |
https://en.wikipedia.org/wiki/August-Wilhelm%20Scheer | August-Wilhelm Scheer (born July 27, 1941) is a German Professor of business administration and business information at Saarland University, and founder and director of IDS Scheer AG, a major IT service and software company. He is known for the development of the Architecture of Integrated Information Systems (ARIS) concept.
Biography
In 1972 Scheer received a PhD from University of Hamburg with the thesis "Kosten- und kapazitätsorientierte Ersatzpolitik bei stochastisch ausfallenden Produktionsanlagen". In 1974 he obtained his Habilitation also in Hamburg with a thesis about project control.
In 1975 Scheer took over one of the first chairs for information systems and founded the institute for information systems (IWi) at Saarland University, which he led until 2005. In 1984 he founded IDS Scheer, a Business Process Management (BPM) software company, which is widely regarded as the founder of the BPM industry. In 1997 he also founded IMC AG, a company for innovative learning technologies and spin-off of Saarland University, together with Wolfgang Kraemer, Frank Milius and Wolfgang Zimmermann.
In 2003 Scheer was awarded the Philip Morris Research Prize and the Ernst & Young Entrepreneur of the Year Award. In December 2005 he was awarded the Erich Gutenberg price and in the same month the Federal Cross of Merit first class. In 2005 he was also elected as a fellow of the Gesellschaft für Informatik. Since 2006, he has been a member of the council for innovation and growth of the Federal Government. 2007 he was honored as a HPI-Fellow by the Hasso-Plattner-Institut (HPI) für Softwaresystemtechnik and was elected President of the German Association for Information Technology, Telecommunications and New Media.
On June 4, 2010 Scheer was awarded with the Design Science Lifetime Achievement Award at University of St. Gallen. He received the prize as a recognition for his contribution to design science research.
Work
His research focuses on information and business pr |
https://en.wikipedia.org/wiki/Jet%20Set%20Willy | Jet Set Willy is a platform video game originally written by Matthew Smith for the ZX Spectrum home computer. It was published in 1984 by Software Projects and ported to most home computers of the time.
The game is a sequel to Manic Miner published in 1983, and the second game in the Miner Willy series. It spent over three months at the top of the charts and was the UK's best-selling home video game of 1984.
The player controls Miner Willy as he tidies up his mansion after a massive party to get some sleep. Players navigate Willy through 60 screens of the mansion and grounds, collecting glowing items while avoiding hazards and guardians.
The game features classical music from Beethoven, Grieg, Bach, and Mozart. Initially the game could not be completed due to various bugs, but fixes for these were released by Software Projects. Jet Set Willy included a copy protection measure in the form of a card with coloured codes, making it more difficult to duplicate. Various expanded versions and ports were released, as well as third-party editing tools that allowed players to design their own rooms and sprites.
Plot
A tired Miner Willy has to tidy up all the items left around his house after a huge party. With this done, his housekeeper Maria will let him go to bed. Willy's mansion was bought with the wealth obtained from his adventures in Manic Miner, but much of it remains unexplored and it appears to be full of strange creatures, possibly a result of the previous (missing) owner's experiments. Willy must explore the enormous mansion and its grounds (including a beach and a yacht) to fully tidy up the house so he can get some much-needed sleep.
Gameplay
Jet Set Willy is a flip-screen platform game in which the player moves the protagonist, Willy, from room to room in his mansion collecting objects. Unlike the screen-by-screen style of its prequel, the player can explore the mansion at will. Willy is controlled using only left, right and jump. He can climb stairs by w |
https://en.wikipedia.org/wiki/Encephalitis%20lethargica | Encephalitis lethargica is an atypical form of encephalitis. Also known as "sleeping sickness" or "sleepy sickness" (distinct from tsetse fly–transmitted sleeping sickness), it was first described in 1917 by neurologist Constantin von Economo and pathologist Jean-René Cruchet. The disease attacks the brain, leaving some victims in a statue-like condition, speechless and motionless. Between 1915 and 1926, an epidemic of encephalitis lethargica spread around the world. The exact number of people infected is unknown, but it is estimated that more than one million people contracted the disease during the epidemic, which directly caused more than 500,000 deaths. Most of those who survived never returned to their pre-morbid vigour.
Signs and symptoms
Encephalitis lethargica is characterized by high fever, sore throat, headache, lethargy, double vision, delayed physical and mental response, sleep inversion and catatonia. In severe cases, patients may enter a coma-like state (akinetic mutism). Patients may also experience abnormal eye movements ("oculogyric crises"), Parkinsonism, upper body weakness, muscular pains, tremors, neck rigidity, and behavioral changes including psychosis. Klazomania (a vocal tic) is sometimes present.
Cause
The causes of encephalitis lethargica are uncertain. Though it used to be believed that it was connected to the Spanish flu epidemic, modern research provides arguments against this claim. Some studies have explored its origins in an autoimmune response, and, separately or in relation to an immune response, links to pathologies of infectious disease—viral and bacterial, such as in the case of influenza, where a link with encephalitis is clear. Postencephalitic Parkinsonism was clearly documented to have followed an outbreak of encephalitis lethargica following the 1918 influenza pandemic; evidence for viral causation of the Parkinson's symptoms is circumstantial (epidemiologic, and finding influenza antigens in encephalitis lethargica pati |
https://en.wikipedia.org/wiki/Finite%20intersection%20property | In general topology, a branch of mathematics, a non-empty family A of subsets of a set is said to have the finite intersection property (FIP) if the intersection over any finite subcollection of is non-empty. It has the strong finite intersection property (SFIP) if the intersection over any finite subcollection of is infinite. Sets with the finite intersection property are also called centered systems and filter subbases.
The finite intersection property can be used to reformulate topological compactness in terms of closed sets; this is its most prominent application. Other applications include proving that certain perfect sets are uncountable, and the construction of ultrafilters.
Definition
Let be a set and a nonempty family of subsets of that is, is a subset of the power set of Then is said to have the finite intersection property if every nonempty finite subfamily has nonempty intersection; it is said to have the strong finite intersection property if that intersection is always infinite.
In symbols, has the FIP if, for any choice of a finite nonempty subset of there must exist a point Likewise, has the SFIP if, for every choice of such there are infinitely many such
In the study of filters, the common intersection of a family of sets is called a kernel, from much the same etymology as the sunflower. Families with empty kernel are called free; those with nonempty kernel, fixed.
Families of examples and non-examples
The empty set cannot belong to any collection with the finite intersection property.
A sufficient condition for the FIP intersection property is a nonempty kernel. The converse is generally false, but holds for finite families; that is, if is finite, then has the finite intersection property if and only if it is fixed.
Pairwise intersection
The finite intersection property is strictly stronger than pairwise intersection; the family has pairwise intersections, but not the FIP.
More generally, let be a positi |
https://en.wikipedia.org/wiki/Online%20skill-based%20game | Online skill-based games are online games in which the outcome of the game is determined by the player's physical skill (like fast reaction or dexterity) or mental skill (logic abilities, strategic thinking, trivia knowledge). As in off-line games of skill, the definition has legal meaning, as playing games of chance for money is an illegal act in several countries.
Categories
Most skill-based games, or skillgames, fall into five categories:
Arcade games involve quick fingers and quick thinking. These games are generally sped-up puzzle games.
Puzzle games rely on logic abilities and require the user to solve certain types of puzzles. While not as fast-paced as arcade games, these games often come with a time limit.
Word games are puzzle games using word problems, like rearranging letters to make words.
Trivia games test the user's knowledge of trivia in specific categories or in general.
Fantasy sport games rely on the participants ability to assemble the best group of players.
Card games are played with playing cards online and requires good use of probability and other mathematical tactics.
History
Around 2000, Disney invested millions in a new online skill-based game company called Skillgames.com (formerly PureSkill.com). Manhattan-based Skillgames, with endorsements by Disney-owned properties such as ESPN and ABC, was to develop skill-based games such as "Hole-In-One Golf," "Soap Opera Trivia" and others implemented as Java applets on their site. Players could win prizes up to a million dollars their first time playing. Skillgames, the brainchild of Walker Digital, also the parent company of Priceline.com, fell on hard times in 2001. Congress had begun to threaten a crack-down on Internet gambling, and although the company was confident of the distinction between games of skill and games of chance, Disney decided to withdraw its investment. Skillgames management announced a business model change in late spring of 2001 and rounds of layoffs followed. |
https://en.wikipedia.org/wiki/Systems%20programming | Systems programming, or system programming, is the activity of programming computer system software. The primary distinguishing characteristic of systems programming when compared to application programming is that application programming aims to produce software which provides services to the user directly (e.g. word processor), whereas systems programming aims to produce software and software platforms which provide services to other software, are performance constrained, or both (e.g. operating systems, computational science applications, game engines, industrial automation, and software as a service applications).
Systems programming requires a great degree of hardware awareness. Its goal is to achieve efficient use of available resources, either because the software itself is performance critical or because even small efficiency improvements directly transform into significant savings of time or money.
Overview
The following attributes characterize systems programming:
The programmer can make assumptions about the hardware and other properties of the system that the program runs on, and will often exploit those properties, for example by using an algorithm that is known to be efficient when used with specific hardware.
Usually a low-level programming language or programming language dialect is used so that:
Programs can operate in resource-constrained environments
Programs can be efficient with little runtime overhead, possibly having either a small runtime library or none at all
Programs may use direct and "raw" control over memory access and control flow
The programmer may write parts of the program directly in assembly language
Often systems programs cannot be run in a debugger. Running the program in a simulated environment can sometimes be used to reduce this problem.
Systems programming is sufficiently different from application programming that programmers tend to specialize in one or the other.
In systems programming, often limited programm |
https://en.wikipedia.org/wiki/ARMM%20%28Usenet%29 | Automated Retroactive Minimal Moderation (ARMM) was a program developed by Richard Depew in 1993 to aid in the control of Usenet abuse. Concerned by abusive posts emanating from certain anonymous-posting sites, Depew developed ARMM to allow news administrators to automatically issue cancel messages for such posts. This was a controversial act, as many news administrators and users were concerned about censorship of the netnews medium.
An early version of ARMM contained a bug which caused it to post follow-ups to its own messages, recursively sending posts to the news.admin.policy newsgroup. This was an early example of (unintentional) Usenet spam.
References
Usenet
Spam filtering |
https://en.wikipedia.org/wiki/Ford%20circle | In mathematics, a Ford circle is a circle in the Euclidean plane, in a family of circles that are all tangent to the -axis at rational points. For each rational number , expressed in lowest terms, there is a Ford circle whose center is at the point and whose radius is . It is tangent to the -axis at its bottom point, . The two Ford circles for rational numbers and (both in lowest terms) are tangent circles when and otherwise these two circles are disjoint.
History
Ford circles are a special case of mutually tangent circles; the base line can be thought of as a circle with infinite radius. Systems of mutually tangent circles were studied by Apollonius of Perga, after whom the problem of Apollonius and the Apollonian gasket are named. In the 17th century René Descartes discovered Descartes' theorem, a relationship between the reciprocals of the radii of mutually tangent circles.
Ford circles also appear in the Sangaku (geometrical puzzles) of Japanese mathematics. A typical problem, which is presented on an 1824 tablet in the Gunma Prefecture, covers the relationship of three touching circles with a common tangent. Given the size of the two outer large circles, what is the size of the small circle between them? The answer is equivalent to a Ford circle:
Ford circles are named after the American mathematician Lester R. Ford, Sr., who wrote about them in 1938.
Properties
The Ford circle associated with the fraction is denoted by or There is a Ford circle associated with every rational number. In addition, the line is counted as a Ford circle – it can be thought of as the Ford circle associated with infinity, which is the case
Two different Ford circles are either disjoint or tangent to one another. No two interiors of Ford circles intersect, even though there is a Ford circle tangent to the x-axis at each point on it with rational coordinates. If is between 0 and 1, the Ford circles that are tangent to can be described variously as
the circles where |
https://en.wikipedia.org/wiki/Index%20of%20biochemistry%20articles | Biochemistry is the study of the chemical processes in living organisms. It deals with the structure and function of cellular components such as proteins, carbohydrates, lipids, nucleic acids and other biomolecules.
Articles related to biochemistry include:
0–9
2-amino-5-phosphonovalerate - 3' end - 5' end
A
ABC-Transporter Genes - abl gene - acetic acid - acetyl CoA - acetylcholine - acetylcysteine - acid - acidic fibroblast growth factor - acrosin - actin - action potential - activation energy - active site - active transport - adenosine - adenosine diphosphate (ADP) - adenosine monophosphate (AMP) - adenosine triphosphate (ATP) - adenovirus - adrenergic receptor - adrenodoxin - aequorin - aerobic respiration - agonist - alanine - albumin - alcohol - alcoholic fermentation - alicyclic compound - aliphatic compound - alkali - allosteric site - allostery - allotrope - allotropy - alpha adrenergic receptor - alpha helix - alpha-1 adrenergic receptor - alpha-2 adrenergic receptor - alpha-beta T-cell antigen receptor - alpha-fetoprotein - alpha-globulin - alpha-macroglobulin - alpha-MSH - Ames test - amide - amine - amino - amino acid - amino acid receptor - amino acid sequence - amino acid sequence homology - aminobutyric acid - ammonia - AMPA receptor - amyloid - anabolism - anaerobic respiration - analytical chemistry - androgen receptor - angiotensin - angiotensin II - angiotensin receptor - ankyrin - annexin II - antibiotic - antibody - apoenzyme - apolipoprotein - apoptosis - aquaporin - archaea - arginine - argipressin - aromatic amine - aromatic compound - arrestin - Arrhenius equation - aryl hydrocarbon receptor - asparagine - aspartic acid - atom - atomic absorption spectroscopy - atomic mass - atomic mass unit - atomic nucleus - atomic number - atomic orbital - atomic radius - Atomic weight - ATP synthase - ATPase - atrial natriuretic factor - atrial natriuretic factor receptor - Avogadro constant - axon
B
B cell - bacteria - bacterial conjugation - |
https://en.wikipedia.org/wiki/Missing%20square%20puzzle | The missing square puzzle is an optical illusion used in mathematics classes to help students reason about geometrical figures; or rather to teach them not to reason using figures, but to use only textual descriptions and the axioms of geometry. It depicts two arrangements made of similar shapes in slightly different configurations. Each apparently forms a 13×5 right-angled triangle, but one has a 1×1 hole in it.
Solution
The key to the puzzle is the fact that neither of the 13×5 "triangles" is truly a triangle, nor would either truly be 13x5 if it were, because what appears to be the hypotenuse is bent. In other words, the "hypotenuse" does not maintain a consistent slope, even though it may appear that way to the human eye.
A true 13×5 triangle cannot be created from the given component parts. The four figures (the yellow, red, blue and green shapes) total 32 units of area. The apparent triangles formed from the figures are 13 units wide and 5 units tall, so it appears that the area should be S = = 32.5 units. However, the blue triangle has a ratio of 5:2 (=2.5), while the red triangle has the ratio 8:3 (≈2.667), so the apparent combined hypotenuse in each figure is actually bent. With the bent hypotenuse, the first figure actually occupies a combined 32 units, while the second figure occupies 33, including the "missing" square.
The amount of bending is approximately unit (1.245364267°), which is difficult to see on the diagram of the puzzle, and was illustrated as a graphic. Note the grid point where the red and blue triangles in the lower image meet (5 squares to the right and two units up from the lower left corner of the combined figure), and compare it to the same point on the other figure; the edge is slightly under the mark in the upper image, but goes through it in the lower. Overlaying the hypotenuses from both figures results in a very thin parallelogram (represented with the four red dots) with an area of exactly one grid square (Pick's theorem g |
https://en.wikipedia.org/wiki/Timothy%20Gowers | Sir William Timothy Gowers, (; born 20 November 1963) is a British mathematician. He is Professeur titulaire of the Combinatorics chair at the Collège de France, and director of research at the University of Cambridge and Fellow of Trinity College, Cambridge. In 1998, he received the Fields Medal for research connecting the fields of functional analysis and combinatorics.
Education
Gowers attended King's College School, Cambridge, as a choirboy in the King's College choir, and then Eton College as a King's Scholar, where he was taught mathematics by Norman Routledge. In 1981, Gowers won a gold medal at the International Mathematical Olympiad with a perfect score. He completed his PhD, with a dissertation on Symmetric Structures in Banach Spaces at Trinity College, Cambridge in 1990, supervised by Béla Bollobás.
Career and research
After his PhD, Gowers was elected to a Junior Research Fellowship at Trinity College. From 1991 until his return to Cambridge in 1995 he was lecturer at University College London. He was elected to the Rouse Ball Professorship at Cambridge in 1998. During 2000–2 he was visiting professor at Princeton University. In May 2020 it was announced that he would be assuming the title chaire de combinatoire at the College de France beginning in October 2020, though he intends to continue to reside in Cambridge and maintain a part-time affiliation at the university, as well as enjoy the privileges of his life fellowship of Trinity College.
Gowers initially worked on Banach spaces. He used combinatorial tools in proving several of Stefan Banach's conjectures in the subject, in particular constructing a Banach space with almost no symmetry, serving as a counterexample to several other conjectures. With Bernard Maurey he resolved the "unconditional basic sequence problem" in 1992, showing that not every infinite-dimensional Banach space has an infinite-dimensional subspace that admits an unconditional Schauder basis.
After this, Gowers turned to c |
https://en.wikipedia.org/wiki/DI%20unit | A DI unit (direct input or direct inject) is an electronic device typically used in recording studios and in sound reinforcement systems to connect a high output impedance unbalanced output signal to a low-impedance, microphone level, balanced input, usually via an XLR connector and XLR cable. DIs are frequently used to connect an electric guitar or electric bass to a mixing console's microphone input jack. The DI performs level matching, balancing, and either active buffering or passive impedance matching/impedance bridging. DI units are typically metal boxes with input and output jacks and, for more expensive units, “ground lift” and attenuator switches.
DI boxes are extensively used with professional and semi-professional PA systems, professional sound reinforcement systems and in sound recording studios. Manufacturers produce a wide range of units, from inexpensive, basic, passive units to expensive, sophisticated, active units. DI boxes may provide numerous features and user-controllable options (e.g., a user-selectable 0dB, 20dB or 40dB pad and/or a "ground lift" switch). They may come in different types of enclosures, usually a metal chassis that helps to protect against electrical interference. Some bass amplifiers have built-in DI units, so that the bass amp's output signal can be connected directly to a mixing board in a sound reinforcement/live show or recording context.
Terminology
DI units are also referred to as a DI box, direct box, or simply DI, with each letter pronounced, as in "Dee Eye." The term is variously claimed to stand for direct input, direct injection, direct induction or direct interface.
History
Passive direct boxes first appeared in the United States in the middle 1960s, most notably in Detroit at radio stations and recording studios like Motown, United Sound Systems, Golden World Records, Tera Shirma Studios and the Metro-Audio Capstan Roller remote recording truck. These DIs were custom made by engineers like Ed Wolfrum with his |
https://en.wikipedia.org/wiki/Banach%E2%80%93Mazur%20game | In general topology, set theory and game theory, a Banach–Mazur game is a topological game played by two players, trying to pin down elements in a set (space). The concept of a Banach–Mazur game is closely related to the concept of Baire spaces. This game was the first infinite positional game of perfect information to be studied. It was introduced by Stanisław Mazur as problem 43 in the Scottish book, and Mazur's questions about it were answered by Banach.
Definition
Let be a non-empty topological space, a fixed subset of and a family of subsets of that have the following properties:
Each member of has non-empty interior.
Each non-empty open subset of contains a member of .
Players, and alternately choose elements from to form a sequence
wins if and only if
Otherwise, wins.
This is called a general Banach–Mazur game and denoted by
Properties
has a winning strategy if and only if is of the first category in (a set is of the first category or meagre if it is the countable union of nowhere-dense sets).
If is a complete metric space, has a winning strategy if and only if is comeager in some non-empty open subset of
If has the Baire property in , then is determined.
The siftable and strongly-siftable spaces introduced by Choquet can be defined in terms of stationary strategies in suitable modifications of the game. Let denote a modification of where is the family of all non-empty open sets in and wins a play if and only if
Then is siftable if and only if has a stationary winning strategy in
A Markov winning strategy for in can be reduced to a stationary winning strategy. Furthermore, if has a winning strategy in , then has a winning strategy depending only on two preceding moves. It is still an unsettled question whether a winning strategy for can be reduced to a winning strategy that depends only on the last two moves of .
is called weakly -favorable if has a winning strategy in . Then, is a Baire space if and |
https://en.wikipedia.org/wiki/Distribution%20function%20%28physics%29 | In molecular kinetic theory in physics, a system's distribution function is a function of seven variables, , which gives the number of particles per unit volume in single-particle phase space. It is the number of particles per unit volume having approximately the velocity near the position and time . The usual normalization of the distribution function is
where N is the total number of particles and n is the number density of particles – the number of particles per unit volume, or the density divided by the mass of individual particles.
A distribution function may be specialised with respect to a particular set of dimensions. E.g. take the quantum mechanical six-dimensional phase space, and multiply by the total space volume, to give the momentum distribution, i.e. the number of particles in the momentum phase space having approximately the momentum .
Particle distribution functions are often used in plasma physics to describe wave–particle interactions and velocity-space instabilities. Distribution functions are also used in fluid mechanics, statistical mechanics and nuclear physics.
The basic distribution function uses the Boltzmann constant and temperature with the number density to modify the normal distribution:
Related distribution functions may allow bulk fluid flow, in which case the velocity origin is shifted, so that the exponent's numerator is , where is the bulk velocity of the fluid. Distribution functions may also feature non-isotropic temperatures, in which each term in the exponent is divided by a different temperature.
Plasma theories such as magnetohydrodynamics may assume the particles to be in thermodynamic equilibrium. In this case, the distribution function is Maxwellian. This distribution function allows fluid flow and different temperatures in the directions parallel to, and perpendicular to, the local magnetic field. More complex distribution functions may also be used, since plasmas are rarely in thermal equilibrium.
The mathem |
https://en.wikipedia.org/wiki/Point-to-point%20%28telecommunications%29 | In telecommunications, a point-to-point connection refers to a communications connection between two communication endpoints or nodes. An example is a telephone call, in which one telephone is connected with one other, and what is said by one caller can only be heard by the other. This is contrasted with a point-to-multipoint or broadcast connection, in which many nodes can receive information transmitted by one node. Other examples of point-to-point communications links are leased lines and microwave radio relay.
The term is also used in computer networking and computer architecture to refer to a wire or other connection that links only two computers or circuits, as opposed to other network topologies such as buses or crossbar switches which can connect many communications devices.
Point-to-point is sometimes abbreviated as P2P. This usage of P2P is distinct from P2P meaning peer-to-peer in the context of file sharing networks or other data-sharing protocols between peers.
Basic data link
A traditional point-to-point data link is a communications medium with exactly two endpoints and no data or packet formatting. The host computers at either end take full responsibility for formatting the data transmitted between them. The connection between the computer and the communications medium was generally implemented through an RS-232 or similar interface. Computers in close proximity may be connected by wires directly between their interface cards.
When connected at a distance, each endpoint would be fitted with a modem to convert analog telecommunications signals into a digital data stream. When the connection uses a telecommunications provider, the connection is called a dedicated, leased, or private line. The ARPANET used leased lines to provide point-to-point data links between its packet-switching nodes, which were called Interface Message Processors.
Modern links
With the exception of passive optical networks, modern Ethernet is exclusively point-to-point |
https://en.wikipedia.org/wiki/Point-to-multipoint%20communication | In telecommunications, point-to-multipoint communication (P2MP, PTMP or PMP) is communication which is accomplished via a distinct type of one-to-many connection, providing multiple paths from a single location to multiple locations.
Point-to-multipoint telecommunications is typically used in wireless Internet and IP telephony via gigahertz radio frequencies. P2MP systems have been designed with and without a return channel from the multiple receivers. A central antenna or antenna array broadcasts to several receiving antennas and the system uses a form of time-division multiplexing to allow for the return channel traffic.
Modern point-to-multipoint links
In contemporary usage, the term point-to-multipoint wireless communications relates to fixed wireless data communications for Internet or voice over IP via radio or microwave frequencies in the gigahertz range.
Point-to-multipoint is the most popular approach for wireless communications that have a large number of nodes, end destinations or end users. Point to Multipoint generally assumes there is a central base station to which remote subscriber units or customer premises equipment (CPE) (a term that was originally used in the wired telephone industry) are connected over the wireless medium. Connections between the base station and subscriber units can be either line-of-sight or, for lower-frequency radio systems, non-line-of-sight where link budgets permit. Generally, lower frequencies can offer non-line-of-sight connections. Various software planning tools can be used to determine feasibility of potential connections using topographic data as well as link budget simulation. Often the point to multipoint links are installed to reduce the cost of infrastructure and increase the number of CPE's and connectivity.
Point-to-multipoint wireless networks employing directional antennas are affected by the hidden node problem (also called hidden terminal) in case they employ a CSMA/CA medium access control protoco |
https://en.wikipedia.org/wiki/Germline | In biology and genetics, the germline is the population of a multicellular organism's cells that pass on their genetic material to the progeny (offspring). In other words, they are the cells that form the egg, sperm and the fertilised egg. They are usually differentiated to perform this function and segregated in a specific place away from other bodily cells.
As a rule, this passing-on happens via a process of sexual reproduction; typically it is a process that includes systematic changes to the genetic material, changes that arise during recombination, meiosis and fertilization for example. However, there are many exceptions across multicellular organisms, including processes and concepts such as various forms of apomixis, autogamy, automixis, cloning or parthenogenesis. The cells of the germline are called germ cells. For example, gametes such as a sperm and an egg are germ cells. So are the cells that divide to produce gametes, called gametocytes, the cells that produce those, called gametogonia, and all the way back to the zygote, the cell from which an individual develops.
In sexually reproducing organisms, cells that are not in the germline are called somatic cells. According to this view, mutations, recombinations and other genetic changes in the germline may be passed to offspring, but a change in a somatic cell will not be. This need not apply to somatically reproducing organisms, such as some Porifera and many plants. For example, many varieties of citrus, plants in the Rosaceae and some in the Asteraceae, such as Taraxacum, produce seeds apomictically when somatic diploid cells displace the ovule or early embryo.
In an earlier stage of genetic thinking, there was a clear distinction between germline and somatic cells. For example, August Weismann proposed and pointed out, a germline cell is immortal in the sense that it is part of a lineage that has reproduced indefinitely since the beginning of life and, barring accident, could continue doing so indef |
https://en.wikipedia.org/wiki/Rank%20%28computer%20programming%29 | In computer programming, rank with no further specifications is usually a synonym for (or refers to) "number of dimensions"; thus, a two-dimensional array has rank two, a three-dimensional array has rank three and so on.
Strictly, no formal definition can be provided which applies to every programming language, since each of them has its own concepts, semantics and terminology; the term may not even be applicable or, to the contrary, applied with a very specific meaning in the context of a given language.
In the case of APL the notion applies to every operand; and dyads ("binary functions") have a left rank and a right rank.
The box below instead shows how rank of a type and rank of an array expression could be defined (in a semi-formal style) for C++ and illustrates a simple way to calculate them at compile time.
#include <type_traits>
#include <cstddef>
/* Rank of a type
* -------------
*
* Let the rank of a type T be the number of its dimensions if
* it is an array; zero otherwise (which is the usual convention)
*/
template <typename T> struct rank
{
static const std::size_t value = 0;
};
template<typename T, std::size_t N>
struct rank<T[N]>
{
static const std::size_t value = 1 + rank<T>::value;
};
template <typename T>
constexpr auto rank_v = rank<T>::value;
/* Rank of an expression
*
* Let the rank of an expression be the rank of its type
*/
template <typename T>
using unqualified_t = std::remove_cv_t<std::remove_reference_t<T>>;
template <typename T>
auto rankof(T&& expr)
{
return rank_v<unqualified_t<T>>;
}
Given the code above the rank of a type T can be calculated at compile time by
rank<T>::value
or the shorter form
rank_v<T>
Calculating the rank of an expression can be done using
rankof(expr)
See also
Rank (linear algebra), for a definition of rank as applied to matrices
Rank (J programming language), a concept of the same name in the J programming language
Arrays
Programming language topics |
https://en.wikipedia.org/wiki/Chikungunya | Chikungunya is an infection caused by the Chikungunya virus (CHIKV). Symptoms include fever and joint pains. These typically occur two to twelve days after exposure. Other symptoms may include headache, muscle pain, joint swelling, and a rash. Symptoms usually improve within a week; however, occasionally the joint pain may last for months or years. The risk of death is around 1 in 1,000. The very young, old, and those with other health problems are at risk of more severe disease.
The virus is spread between people by two types of mosquitos: Aedes albopictus and Aedes aegypti. They mainly bite during the day. The virus may circulate within a number of animals including birds and rodents. Diagnosis is by either testing the blood for the virus's RNA or antibodies to the virus. The symptoms can be mistaken for those of dengue fever and Zika fever. It is believed most people become immune after a single infection.
The best means of prevention is overall mosquito control and the avoidance of bites in areas where the disease is common. This may be partly achieved by decreasing mosquito access to water and with the use of insect repellent and mosquito nets. There is no vaccine and no specific treatment as of 2016. Recommendations include rest, fluids, and medications to help with fever and joint pain.
While the disease typically occurs in Africa and Asia, outbreaks have been reported in Europe and the Americas since the 2000s. In 2014 more than a million suspected cases occurred. In 2014 it was occurring in Florida in the continental United States but as of 2016 there were no further locally acquired cases. The disease was first identified in 1952 in Tanzania. The term is from the Kimakonde language and means "to become contorted".
Signs and symptoms
Around 85% of people infected with Chikungunya virus experience symptoms, typically beginning with a sudden high fever above . The fever is soon followed by severe muscle and joint pain. Pain usually affects multiple joints |
https://en.wikipedia.org/wiki/Pontryagin%20duality | In mathematics, Pontryagin duality is a duality between locally compact abelian groups that allows generalizing Fourier transform to all such groups, which include the circle group (the multiplicative group of complex numbers of modulus one), the finite abelian groups (with the discrete topology), and the additive group of the integers (also with the discrete topology), the real numbers, and every finite-dimensional vector space over the reals or a -adic field.
The Pontryagin dual of a locally compact abelian group is the locally compact abelian topological group formed by the continuous group homomorphisms from the group to the circle group with the operation of pointwise multiplication and the topology of uniform convergence on compact sets. The Pontryagin duality theorem establishes Pontryagin duality by stating that any locally compact abelian group is naturally isomorphic with its bidual (the dual of its dual). The Fourier inversion theorem is a special case of this theorem.
The subject is named after Lev Pontryagin who laid down the foundations for the theory of locally compact abelian groups and their duality during his early mathematical works in 1934. Pontryagin's treatment relied on the groups being second-countable and either compact or discrete. This was improved to cover the general locally compact abelian groups by Egbert van Kampen in 1935 and André Weil in 1940.
Introduction
Pontryagin duality places in a unified context a number of observations about functions on the real line or on finite abelian groups:
Suitably regular complex-valued periodic functions on the real line have Fourier series and these functions can be recovered from their Fourier series;
Suitably regular complex-valued functions on the real line have Fourier transforms that are also functions on the real line and, just as for periodic functions, these functions can be recovered from their Fourier transforms; and
Complex-valued functions on a finite abelian group have discrete |
https://en.wikipedia.org/wiki/Typed%20lambda%20calculus | A typed lambda calculus is a typed formalism that uses the lambda-symbol () to denote anonymous function abstraction. In this context, types are usually objects of a syntactic nature that are assigned to lambda terms; the exact nature of a type depends on the calculus considered (see kinds below). From a certain point of view, typed lambda calculi can be seen as refinements of the untyped lambda calculus, but from another point of view, they can also be considered the more fundamental theory and untyped lambda calculus a special case with only one type.
Typed lambda calculi are foundational programming languages and are the base of typed functional programming languages such as ML and Haskell and, more indirectly, typed imperative programming languages. Typed lambda calculi play an important role in the design of type systems for programming languages; here, typability usually captures desirable properties of the program (e.g., the program will not cause a memory access violation).
Typed lambda calculi are closely related to mathematical logic and proof theory via the Curry–Howard isomorphism and they can be considered as the internal language of certain classes of categories. For example, the simply typed lambda calculus is the language of Cartesian closed categories (CCCs)
Kinds of typed lambda calculi
Various typed lambda calculi have been studied. The simply typed lambda calculus has only one type constructor, the arrow , and its only types are basic types and function types . System T extends the simply typed lambda calculus with a type of natural numbers and higher order primitive recursion; in this system all functions provably recursive in Peano arithmetic are definable. System F allows polymorphism by using universal quantification over all types; from a logical perspective it can describe all functions that are provably total in second-order logic. Lambda calculi with dependent types are the base of intuitionistic type theory, the calculus of construc |
https://en.wikipedia.org/wiki/Trojan%20Room%20coffee%20pot | The Trojan Room coffee pot was a coffee machine located in the Computer Laboratory of the University of Cambridge, England. Created in 1991 by Quentin Stafford-Fraser and Paul Jardetzky, it was migrated from their laboratory network to the web in 1993, becoming the world's first webcam.
To save people working in the building the disappointment of finding the coffee machine empty after making the trip to the room, a camera was set up providing a live picture of the coffee pot to all desktop computers on the office network. After the camera was connected to the Internet a few years later, the coffee pot gained international renown as a feature of the fledgling World Wide Web, until being retired in 2001.
Development
The 128×128 px greyscale camera was connected to the laboratory's local network through a video capture card fitted on an Acorn Archimedes computer. Researcher Quentin Stafford-Fraser wrote the client software, dubbed XCoffee and employing the X Window System protocol, while his colleague Paul Jardetzky wrote the server program.
In 1993, web browsers gained the ability to display images, and it soon became clear that this would be an easier way to make the picture available to users. The camera was connected to the Internet and the live picture became available via HTTP in November of the same year, by computer scientists Daniel Gordon and Martyn Johnson. It therefore became visible worldwide and grew into a popular landmark of the early web.
Retirement
Following the laboratory's move to its current premises, the camera was eventually switched off, at 09:54 UTC on . Coverage of the shutdown included front-page mentions in The Times and The Washington Post, as well as articles in The Guardian and Wired.
The last of the four or five coffee machines seen online, a Krups, was auctioned on eBay for to the German news website Der Spiegel. The pot was later refurbished pro bono by Krups employees, and was switched on again in the magazine's editorial off |
https://en.wikipedia.org/wiki/Brickwork | Brickwork is masonry produced by a bricklayer, using bricks and mortar. Typically, rows of bricks called courses are laid on top of one another to build up a structure such as a brick wall.
Bricks may be differentiated from blocks by size. For example, in the UK a brick is defined as a unit having dimensions less than and a block is defined as a unit having one or more dimensions greater than the largest possible brick.
Brick is a popular medium for constructing buildings, and examples of brickwork are found through history as far back as the Bronze Age. The fired-brick faces of the ziggurat of ancient Dur-Kurigalzu in Iraq date from around 1400 BC, and the brick buildings of ancient Mohenjo-daro in present day Pakistan were built around 2600 BC. Much older examples of brickwork made with dried (but not fired) bricks may be found in such ancient locations as Jericho in Palestine, Çatal Höyük in Anatolia, and Mehrgarh in Pakistan. These structures have survived from the Stone Age to the present day.
Brick dimensions are expressed in construction or technical documents in two ways as co-ordinating dimensions and working dimensions.
Coordination dimensions are the actual physical dimensions of the brick with the mortar required on one header face, one stretcher face and one bed.
Working dimensions is the size of a manufactured brick. It is also called the nominal size of a brick.
Brick size may be slightly different due to shrinkage or distortion due to firing, etc.
An example of a co-ordinating metric commonly used for bricks in the UK is as follows:
Bricks of dimensions 215 mm × 102.5 mm × 65 mm;
Mortar beds (horizontal) and perpends (vertical) of a uniform 10 mm.
In this case the co-ordinating metric works because the length of a single brick (215 mm) is equal to the total of the width of a brick (102.5 mm) plus a perpend (10 mm) plus the width of a second brick (102.5 mm).
There are many other brick sizes worldwide, and many of them use this same co-ordi |
https://en.wikipedia.org/wiki/Active%20and%20passive%20transformation | Geometric transformations can be distinguished into two types: active or alibi transformations which change the physical position of a set of points relative to a fixed frame of reference or coordinate system (alibi meaning "being somewhere else at the same time"); and passive or alias transformations which leave points fixed but change the frame of reference or coordinate system relative to which they are described (alias meaning "going under a different name"). By transformation, mathematicians usually refer to active transformations, while physicists and engineers could mean either.
For instance, active transformations are useful to describe successive positions of a rigid body. On the other hand, passive transformations may be useful in human motion analysis to observe the motion of the tibia relative to the femur, that is, its motion relative to a (local) coordinate system which moves together with the femur, rather than a (global) coordinate system which is fixed to the floor.
In three-dimensional Euclidean space, any proper rigid transformation, whether active or passive, can be represented as a screw displacement, the composition of a translation along an axis and a rotation about that axis.
The terms active transformation and passive transformation were first introduced in 1957 by Valentine Bargmann for describing Lorentz transformations in special relativity.
Example
As an example, let the vector , be a vector in the plane. A rotation of the vector through an angle θ in counterclockwise direction is given by the rotation matrix:
which can be viewed either as an active transformation or a passive transformation (where the above matrix will be inverted), as described below.
Spatial transformations in the Euclidean space R3
In general a spatial transformation may consist of a translation and a linear transformation. In the following, the translation will be omitted, and the linear transformation will be represented by a 3×3 matrix .
Active transfo |
https://en.wikipedia.org/wiki/Mathematical%20puzzle | Mathematical puzzles make up an integral part of recreational mathematics. They have specific rules, but they do not usually involve competition between two or more players. Instead, to solve such a puzzle, the solver must find a solution that satisfies the given conditions. Mathematical puzzles require mathematics to solve them. Logic puzzles are a common type of mathematical puzzle.
Conway's Game of Life and fractals, as two examples, may also be considered mathematical puzzles even though the solver interacts with them only at the beginning by providing a set of initial conditions. After these conditions are set, the rules of the puzzle determine all subsequent changes and moves. Many of the puzzles are well known because they were discussed by Martin Gardner in his "Mathematical Games" column in Scientific American. Mathematical puzzles are sometimes used to motivate students in teaching elementary school math problem solving techniques. Creative thinkingor "thinking outside the box"often helps to find the solution.
List of mathematical puzzles
This list is not complete.
Numbers, arithmetic, and algebra
Cross-figures or cross number puzzles
Dyson numbers
Four fours
KenKen
Water pouring puzzle
The monkey and the coconuts
Pirate loot problem
Verbal arithmetics
24 Game
Combinatorial
Cryptograms
Fifteen Puzzle
Kakuro
Rubik's Cube and other sequential movement puzzles
Str8ts a number puzzle based on sequences
Sudoku
Sujiko
Think-a-Dot
Tower of Hanoi
Bridges Game
Analytical or differential
Ant on a rubber rope
See also: Zeno's paradoxes
Probability
Monty Hall problem
Tiling, packing, and dissection
Bedlam cube
Conway puzzle
Mutilated chessboard problem
Packing problem
Pentominoes tiling
Slothouber–Graatsma puzzle
Soma cube
T puzzle
Tangram
Involves a board
Conway's Game of Life
Mutilated chessboard problem
Peg solitaire
Sudoku
Nine dots problem
Chessboard tasks
Eight queens puzzle
Knight's Tour
No-three- |
https://en.wikipedia.org/wiki/Biomolecule | A biomolecule or biological molecule is a loosely used term for molecules present in organisms that are essential to one or more typically biological processes, such as cell division, morphogenesis, or development. Biomolecules include the primary metabolites which are large macromolecules (or polyelectrolytes) such as proteins, carbohydrates, lipids, and nucleic acids, as well as small molecules such as vitamins and hormones. A more general name for this class of material is biological materials. Biomolecules are an important element of living organisms, those biomolecules are often endogenous, produced within the organism but organisms usually need exogenous biomolecules, for example certain nutrients, to survive.
Biology and its subfields of biochemistry and molecular biology study biomolecules and their reactions. Most biomolecules are organic compounds, and just four elements—oxygen, carbon, hydrogen, and nitrogen—make up 96% of the human body's mass. But many other elements, such as the various biometals, are also present in small amounts.
The uniformity of both specific types of molecules (the biomolecules) and of certain metabolic pathways are invariant features among the wide diversity of life forms; thus these biomolecules and metabolic pathways are referred to as "biochemical universals" or "theory of material unity of the living beings", a unifying concept in biology, along with cell theory and evolution theory.
Types of biomolecules
A diverse range of biomolecules exist, including:
Small molecules:
Lipids, fatty acids, glycolipids, sterols, monosaccharides
Vitamins
Hormones, neurotransmitters
Metabolites
Monomers, oligomers and polymers:
Nucleosides and nucleotides
Nucleosides are molecules formed by attaching a nucleobase to a ribose or deoxyribose ring. Examples of these include cytidine (C), uridine (U), adenosine (A), guanosine (G), and thymidine (T).
Nucleosides can be phosphorylated by specific kinases in the cell, producing nucl |
https://en.wikipedia.org/wiki/Applied%20probability | Applied probability is the application of probability theory to statistical problems and other scientific and engineering domains.
Scope
Much research involving probability is done under the auspices of applied probability. However, while such research is motivated (to some degree) by applied problems, it is usually the mathematical aspects of the problems that are of most interest to researchers (as is typical of applied mathematics in general).
Applied probabilists are particularly concerned with the application of stochastic processes, and probability more generally, to the natural, applied and social sciences, including biology, physics (including astronomy), chemistry, medicine, computer science and information technology, and economics.
Another area of interest is in engineering: particularly in areas of uncertainty, risk management, probabilistic design, and Quality assurance.
History
Having initially been defined at a symposium of the American Mathematical Society in the later 1950s, the term "applied probability" was popularized by Maurice Bartlett through the name of a Methuen monograph series he edited, Applied Probability and Statistics. The area did not have an established outlet until 1964, when the Journal of Applied Probability came into existence through the efforts of Joe Gani.
See also
Areas of application:
Ruin theory
Statistical physics
Stoichiometry and modelling chemical reactions
Ecology, particularly population modelling
Evolutionary biology
Optimization in computer science
Telecommunications
Options pricing in economics
Ewens's sampling formula in population genetics
Operations research
Gaming mathematics
Stochastic processes:
Markov chain
Poisson process
Brownian motion and other diffusion processes
Queueing theory
Renewal theory
Additional information and resources
Applied Probability Trust
INFORMS Institute for Operations Research and the Management Sciences
References
Further reading
Baeza-Yates, R. (2005) Recent advances in a |
https://en.wikipedia.org/wiki/Dragon%20curve | A dragon curve is any member of a family of self-similar fractal curves, which can be approximated by recursive methods such as Lindenmayer systems. The dragon curve is probably most commonly thought of as the shape that is generated from repeatedly folding a strip of paper in half, although there are other curves that are called dragon curves that are generated differently.
Heighway dragon
The Heighway dragon (also known as the Harter–Heighway dragon or the Jurassic Park dragon) was first investigated by NASA physicists John Heighway, Bruce Banks, and William Harter. It was described by Martin Gardner in his Scientific American column Mathematical Games in 1967. Many of its properties were first published by Chandler Davis and Donald Knuth. It appeared on the section title pages of the Michael Crichton novel Jurassic Park.
Construction
The Heighway dragon can be constructed from a base line segment by repeatedly replacing each segment by two segments with a right angle and with a rotation of 45° alternatively to the right and to the left:
The Heighway dragon is also the limit set of the following iterated function system in the complex plane:
with the initial set of points .
Using pairs of real numbers instead, this is the same as the two functions consisting of
Folding the dragon
The Heighway dragon curve can be constructed by folding a strip of paper, which is how it was originally discovered. Take a strip of paper and fold it in half to the right. Fold it in half again to the right. If the strip was opened out now, unbending each fold to become a 90-degree turn, the turn sequence would be RRL, i.e. the second iteration of the Heighway dragon. Fold the strip in half again to the right, and the turn sequence of the unfolded strip is now RRLRRLL – the third iteration of the Heighway dragon. Continuing folding the strip in half to the right to create further iterations of the Heighway dragon (in practice, the strip becomes too thick to fold sharply after |
https://en.wikipedia.org/wiki/Breeding%20back | Breeding back is a form of artificial selection by the deliberate selective breeding of domestic (but not exclusively) animals, in an attempt to achieve an animal breed with a phenotype that resembles a wild type ancestor, usually one that has gone extinct. Breeding back is not to be confused with dedomestication.
It must be kept in mind that a breeding-back breed may be very similar to the extinct wild type in phenotype, ecological niche, and to some extent genetics, but the gene pool of that wild type was different prior to its extinction. Even the superficial authenticity of a bred-back animal depends on the particular stock used to breed the new lineage. As a result of this, some breeds, like Heck cattle, are at best a vague look-alike of the extinct wild type aurochs, according to the literature.
Background
The aim of breeding back programs is to restore the wild traits which may have been unintentionally preserved in the lineages of domesticated animals. Commonly, not only the new animal's phenotype, but also its ecological capacity, are considered in back-breeding projects, as hardy, "bred back" animals may be used in certain conservation projects. In nature, usually only individuals well suited to their natural circumstances will survive and reproduce, whereas humans select animals with additional attractive, docile or productive characteristics, protecting them from the dangers once found in their ancestral environment (predation, drought, disease, extremes of weather, lack of mating opportunities, etc.). In such cases, selection criteria in nature differ from those found in domesticated conditions. Because of this, domesticated animals often differ significantly in phenotype, behaviour and genetics from their wild forerunners. It is the hope of breeding-back programs to re-express, within a new breeding lineage, the wild, ancient traits that may have "lain buried" in the DNA of domestic animals.
In many cases, the extinct wild type ancestors of a give |
https://en.wikipedia.org/wiki/Plus%E2%80%93minus%20sign | The plus–minus sign, , is a mathematical symbol with multiple meanings:
In mathematics, it generally indicates a choice of exactly two possible values, one of which is obtained through addition and the other through subtraction.
In experimental sciences, the sign commonly indicates the confidence interval or uncertainty bounding a range of possible errors in a measurement, often the standard deviation or standard error. The sign may also represent an inclusive range of values that a reading might have.
In medicine, it means "with or without".
In engineering, the sign indicates the tolerance, which is the range of values that are considered to be acceptable or safe, or which comply with some standard or with a contract.
In botany, it is used in morphological descriptions to notate "more or less".
In chemistry, the sign is used to indicate a racemic mixture.
In chess, the sign indicates a clear advantage for the white player; the complementary minus-plus sign, , indicates the same advantage for the black player.
In electronics, this sign may indicate a dual voltage power supply, such as ±5 volts means +5 volts and −5 volts, when used with audio circuits and operational amplifiers.
In linguistics, it may indicate a distinctive feature, such as [±voiced].
In philosophy, the symbol ± or ∓ can be used to indicate a yinyang concept. Although Yin(-) and Yang(+) are in opposition, they coordinate and help each other in a unity. Yin and Yang are interdependent and coexist as two sides of the same concept.
History
A version of the sign, including also the French word ou ("or"), was used in its mathematical meaning by Albert Girard in 1626, and the sign in its modern form was used as early as 1631, in William Oughtred's Clavis Mathematicae.
Usage
In mathematics
In mathematical formulas, the symbol may be used to indicate a symbol that may be replaced by either the plus and minus signs, or , allowing the formula to represent two values or two equations.
If , one may give |
https://en.wikipedia.org/wiki/Open%20Inventor | Open Inventor, originally IRIS Inventor, is a C++ object-oriented retained mode 3D graphics toolkit designed by SGI to provide a higher layer of programming for OpenGL. Its main goals are better programmer convenience and efficiency. Open Inventor exists as both proprietary software and free and open-source software, subject to the requirements of the GNU Lesser General Public License (LGPL), version 2.1.
The primary objective was to make 3D programming accessible by introducing an object-oriented API, allowing developers to create complex scenes without the intricacies of low-level OpenGL. The toolkit incorporated features like scene graphs, pre-defined shapes, and automatic occlusion culling to streamline scene management. While Open Inventor focused on ease of use, the OpenGL Performer project, spawned from the same context, emphasized performance optimization. The two projects later converged in an attempt to strike a balance between accessibility and performance, culminating in initiatives like Cosmo 3D and OpenGL++. These projects underwent various stages of development and refinement, contributing to the evolution of 3D graphics programming paradigms.
Early history
Around 1988–1989, Wei Yen asked Rikk Carey to lead the IRIS Inventor project. Their goal was to create a toolkit that made developing 3D graphics applications easier to do. The strategy was based on the premise that people were not developing enough 3D applications with IRIS GL because it was too time-consuming to do so with the low-level interface provided by IRIS GL. If 3D programming were made easier, through the use of an object oriented API, then more people would create 3D applications and SGI would benefit. Therefore, the credo was always “ease of use” before “performance”, and soon the tagline “3D programming for humans” was being used widely.
Use
OpenGL (OGL) is a low level application programming interface that takes lists of simple polygons and renders them as quickly as possible. To |
https://en.wikipedia.org/wiki/Prey%20%28novel%29 | Prey is the thirteenth novel by Michael Crichton under his own name and the twenty-third overall. It was first published in November 2002, making it his first novel of the twenty-first century. An excerpt was first published in the January–February 2003 issue of Seed magazine. Prey brings together themes from two earlier Crichton best-selling novels, Jurassic Park and The Andromeda Strain and serves as a cautionary tale about developments in science and technology, in particular, nanotechnology, genetic engineering, and distributed artificial intelligence.
The book features relatively new advances in the computing/scientific community, such as artificial life, emergence (and by extension, complexity), genetic algorithms, and agent-based computing. Fields such as population dynamics and host-parasite coevolution are also at the heart of the novel.
Film rights to the book were purchased by 20th Century Fox.
Plot summary
The novel is narrated by protagonist Jack Forman, an unemployed software programmer, played the role of a stay-at-home husband. His wife, Julia, serves as a high ranking executive for a Nanorobotics company called Xymos and claims to be working on revolutionary imaging technology, which takes up most of her time. This lead to the husband being suspicious of her having an affair.
In the novel, Julia showed Jack a video of Xymos nanobots being injected into a human test subject one night and Jack was impressed at the same time suspicious of the new technology. The couple's baby Amanda awakes in agony, but was cured instantly after an emergency room MRI was performed that same night.
A series of strange events begin to occur in the family and Julia begins to display strange and abusive behavior toward her family until she was injured in a car crash, leaving Jack to consult for Xymos where project manager Ricky was struggling with the nanobots.
Jack was taken to the Xymos research facility in Nevada's Basin Desert, where he was given a tour of the |
https://en.wikipedia.org/wiki/Antipodal%20point | In mathematics, two points of a sphere (or n-sphere, including a circle) are called antipodal or diametrically opposite if they are the intersections of the sphere with a diameter, a straight line passing through its center.
Given any point on a sphere, its antipodal point is the unique point at greatest distance, whether measured intrinsically (great-circle distance on the surface of the sphere) or extrinsically (chordal distance through the sphere's interior). Every great circle on a sphere passing through a point also passes through its antipodal point, and there are infinitely many great circles passing through a pair of antipodal points (unlike the situation for any non-antipodal pair of points, which have a unique great circle passing through both). Many results in spherical geometry depend on choosing non-antipodal points, and degenerate if antipodal points are allowed; for example, a spherical triangle degenerates to an underspecified lune if two of the vertices are antipodal.
The point antipodal to a given point is called its antipodes, from the Greek () meaning "opposite feet"; see . Sometimes the s is dropped, and this is rendered antipode, a back-formation.
Higher mathematics
The concept of antipodal points is generalized to spheres of any dimension: two points on the sphere are antipodal if they are opposite through the centre. Each line through the centre intersects the sphere in two points, one for each ray emanating from the centre, and these two points are antipodal.
The Borsuk–Ulam theorem is a result from algebraic topology dealing with such pairs of points. It says that any continuous function from to maps some pair of antipodal points in to the same point in Here, denotes the sphere and is real coordinate space.
The antipodal map sends every point on the sphere to its antipodal point. If points on the are represented as displacement vectors from the sphere's center in Euclidean then two antipodal points are represented by addit |
https://en.wikipedia.org/wiki/Section%20%28fiber%20bundle%29 | In the mathematical field of topology, a section (or cross section) of a fiber bundle is a continuous right inverse of the projection function . In other words, if is a fiber bundle over a base space, :
then a section of that fiber bundle is a continuous map,
such that
for all .
A section is an abstract characterization of what it means to be a graph. The graph of a function can be identified with a function taking its values in the Cartesian product , of and :
Let be the projection onto the first factor: . Then a graph is any function for which .
The language of fibre bundles allows this notion of a section to be generalized to the case when is not necessarily a Cartesian product. If is a fibre bundle, then a section is a choice of point in each of the fibres. The condition simply means that the section at a point must lie over . (See image.)
For example, when is a vector bundle a section of is an element of the vector space lying over each point . In particular, a vector field on a smooth manifold is a choice of tangent vector at each point of : this is a section of the tangent bundle of . Likewise, a 1-form on is a section of the cotangent bundle.
Sections, particularly of principal bundles and vector bundles, are also very important tools in differential geometry. In this setting, the base space is a smooth manifold , and is assumed to be a smooth fiber bundle over (i.e., is a smooth manifold and is a smooth map). In this case, one considers the space of smooth sections of over an open set , denoted . It is also useful in geometric analysis to consider spaces of sections with intermediate regularity (e.g., sections, or sections with regularity in the sense of Hölder conditions or Sobolev spaces).
Local and global sections
Fiber bundles do not in general have such global sections (consider, for example, the fiber bundle over with fiber obtained by taking the Möbius bundle and removing the zero section), so it is also usef |
https://en.wikipedia.org/wiki/Intra-species%20recognition | Intra-species recognition is the recognition by a member of a species of a conspecific (another member of the same species). In many species, such recognition is necessary for procreation.
Different species may employ different methods, but all of them are based on one or more senses (after all, this is how the organism gathers information about the environment). The recognition may happen by chemical signature (smell), by having a distinctive shape or color (sight), by emitting certain sounds (hearing), or even by behaviour patterns. Often a combination of these is used.
Among human beings, the sense of sight is usually in charge of recognizing other members of the same species, with maybe the subconscious help of smell. In particular, the human brain has a disproportionate amount of processing power dedicated to finely analyze the features of a human face. This is why we are able to distinguish almost all human beings from each other (barring look-alikes), and a human being from a similar species like some anthropomorphic ape, with only a quick glance.
Intra-species recognition systems are often subtle. For example, ornithologists have great difficulty in distinguishing the chiffchaff from the willow warbler by eye, and there is no evidence that the birds themselves can do so other than by the different songs of the male. Sometimes, intra-species recognition is fallible: in many species of frog, the males are commonly seen copulating with females of the wrong species or even with inanimate objects.
Heliconius charithonia displays intra-species recognition by roosting with conspecifics. They do this with the help of UV rhodopsins in the eye that help them distinguish between ultraviolet yellow pigments and regular yellow pigments. They have also been known to emit chemical cues in order to recognize members of their own species.
See also
Assortative mating
Sexual selection
References
Reproduction |
https://en.wikipedia.org/wiki/Group%20coded%20recording | In computer science, group coded recording or group code recording (GCR) refers to several distinct but related encoding methods for representing data on magnetic media. The first, used in bpi magnetic tape since 1973, is an error-correcting code combined with a run-length limited (RLL) encoding scheme, belonging into the group of modulation codes. The others are different mainframe hard disk as well as floppy disk encoding methods used in some microcomputers until the late 1980s. GCR is a modified form of a NRZI code, but necessarily with a higher transition density.
Magnetic tape
Group coded recording was first used for magnetic-tape data storage on 9-track reel-to-reel tape. The term was coined during the development of the IBM 3420 Model 4/6/8 Magnetic Tape Unit and the corresponding 3803 Model 2 Tape Control Unit, both introduced in 1973. IBM referred to the error correcting code itself as "group coded recording". However, GCR has come to refer to the recording format of bpi (250 bits/mm) tape as a whole, and later to formats which use similar RLL codes without the error correction code.
In order to reliably read and write to magnetic tape, several constraints on the signal to be written must be followed. The first is that two adjacent flux reversals must be separated by a certain distance on the media, defined by the magnetic properties of the media itself. The second is that there must be a reversal often enough to keep the reader's clock in phase with the written signal; that is, the signal must be self-clocking and most importantly to keep the playback output high enough as this is proportional to the density of flux transitions. Prior to bpi tapes, bpi tapes satisfied these constraints using a technique called phase encoding (PE), which was only 50% efficient. For bpi GCR tapes, a (0, 2) RLL code is used, or more specifically a (0, 2) block code sometimes also referred to as GCR (4B-5B) encoding. This code requires five bits to be written for ever |
https://en.wikipedia.org/wiki/List%20of%20biomolecules | This is a list of articles that describe particular biomolecules or types of biomolecules.
A
For substances with an A- or α- prefix such as
α-amylase, please see the parent page (in this case Amylase).
A23187 (Calcimycin, Calcium Ionophore)
Abamectine
Abietic acid
Acetic acid
Acetylcholine
Actin
Actinomycin D
Adenine
Adenosmeme
Adenosine diphosphate (ADP)
Adenosine monophosphate (AMP)
Adenosine triphosphate (ATP)
Adenylate cyclase
Adiponectin
Adonitol
Adrenaline, epinephrine
Adrenocorticotropic hormone (ACTH)
Aequorin
Aflatoxin
Agar
Alamethicin
Alanine
Albumins
Aldosterone
Aleurone
Alpha-amanitin
Alpha-MSH (Melaninocyte stimulating hormone)
Allantoin
Allethrin
α-Amanatin, see Alpha-amanitin
Amino acid
Amylase (also see α-amylase)
Anabolic steroid
Anandamide (ANA)
Androgen
Anethole
Angiotensinogen
Anisomycin
Antidiuretic hormone (ADH)
Anti-Müllerian hormone (AMH)
Arabinose
Arginine
Argonaute
Ascomycin
Ascorbic acid (vitamin C)
Asparagine
Aspartic acid
Asymmetric dimethylarginine
ATP synthase
Atrial-natriuretic peptide (ANP)
Auxin
Avidin
Azadirachtin A – C35H44O16
B
Bacteriocin
Beauvericin
beta-Hydroxy beta-methylbutyric acid
beta-Hydroxybutyric acid
Bicuculline
Bilirubin
Biopolymer
Biotin (Vitamin H)
Brefeldin A
Brassinolide
Brucine
Butyric acid
C
Cadaverine
Caffeine
Calciferol (Vitamin D)
Calcitonin
Calmodulin
Calreticulin
Camphor - (C10H16O)
Cannabinol - (C21H26O2)
Capsaicin
Carbohydrase
Carbohydrate
Carnitine
Carrageenan
Carotinoid
Casein
Caspase
Catecholamine
Cellulase
Cellulose - (C6H10O5)x
Cerulenin
Cetrimonium bromide (Cetrimide) - C19H42BrN
Chelerythrine
Chromomycin A3
Chaparonin
Chitin
α-Chloralose
Chlorophyll
Cholecystokinin (CCK)
Cholesterol
Choline
Chondroitin sulfate
Cinnamaldehyde
Citral
Citric acid
Citrinin
Citronellal
Citronellol
Citrulline
Cobalamin (vitamin B12)
Coenzyme
Coenzyme Q
Colchicine
Collagen
Coniine
Corticosteroid
Corti |
https://en.wikipedia.org/wiki/Kepler%20conjecture | The Kepler conjecture, named after the 17th-century mathematician and astronomer Johannes Kepler, is a mathematical theorem about sphere packing in three-dimensional Euclidean space. It states that no arrangement of equally sized spheres filling space has a greater average density than that of the cubic close packing (face-centered cubic) and hexagonal close packing arrangements. The density of these arrangements is around 74.05%.
In 1998, Thomas Hales, following an approach suggested by , announced that he had a proof of the Kepler conjecture. Hales' proof is a proof by exhaustion involving the checking of many individual cases using complex computer calculations. Referees said that they were "99% certain" of the correctness of Hales' proof, and the Kepler conjecture was accepted as a theorem. In 2014, the Flyspeck project team, headed by Hales, announced the completion of a formal proof of the Kepler conjecture using a combination of the Isabelle and HOL Light proof assistants. In 2017, the formal proof was accepted by the journal Forum of Mathematics, Pi.
Background
Imagine filling a large container with small equal-sized spheres: Say a porcelain gallon jug with identical marbles. The "density" of the arrangement is equal to the total volume of all the marbles, divided by the volume of the jug. To maximize the number of marbles in the jug means to create an arrangement of marbles stacked between the sides and bottom of the jug, that has the highest possible density, so that the marbles are packed together as closely as possible.
Experiment shows that dropping the marbles in randomly, with no effort to arrange them tightly, will achieve a density of around 65%. However, a higher density can be achieved by carefully arranging the marbles as follows:
For the first layer of marbles, arrange them in a hexagonal lattice (the honeycomb pattern)
Put the next layer of marbles in the lowest lying gaps you can find above and between the marbles in the first layer, r |
https://en.wikipedia.org/wiki/Alfred%20Vail | Alfred Lewis Vail (September 25, 1807 – January 18, 1859) was an American machinist and inventor. Along with Samuel Morse, Vail was central in developing and commercializing American telegraphy between 1837 and 1844.
Vail and Morse were the first two telegraph operators on Morse's first experimental line between Washington, D.C., and Baltimore, and Vail took charge of building and managing several early telegraph lines between 1845 and 1848. He was also responsible for several technical innovations of Morse's system, particularly the sending key and improved recording registers and relay magnets. Vail left the telegraph industry in 1848 because he believed that the managers of Morse's lines did not fully value his contributions.
His last assignment, superintendent of the Washington and New Orleans Telegraph Company, paid him only $900 a year, leading Vail to write to Morse,
"I have made up my mind to leave the Telegraph to take care of itself, since it cannot take care of me. I shall, in a few months, leave Washington for New Jersey, ... and bid adieu to the subject of the Telegraph for some more profitable business."
Early life
Vail's parents were Bethiah Youngs (1778–1847) and Stephen Vail (1780–1864). Vail was born in Morristown, New Jersey, where his father was an entrepreneur and industrialist who built the Speedwell Ironworks into one of the most innovative iron works of its time. Their other son George Vail, Alfred's brother, was a noted politician.
Alfred attended public schools before taking a job as a machinist at the iron works. He enrolled in New York University to study theology in 1832, where he was an active and successful student and a member of the Eucleian Society, graduating in 1836.
Involvement with Morse's telegraph
Visiting his alma mater on September 2, 1837, Vail happened to witness one of Samuel Morse's early telegraph experiments. He became fascinated by the technology and negotiated an arrangement with Morse to develop the technolog |
https://en.wikipedia.org/wiki/Sphere%20packing | In geometry, a sphere packing is an arrangement of non-overlapping spheres within a containing space. The spheres considered are usually all of identical size, and the space is usually three-dimensional Euclidean space. However, sphere packing problems can be generalised to consider unequal spheres, spaces of other dimensions (where the problem becomes circle packing in two dimensions, or hypersphere packing in higher dimensions) or to non-Euclidean spaces such as hyperbolic space.
A typical sphere packing problem is to find an arrangement in which the spheres fill as much of the space as possible. The proportion of space filled by the spheres is called the packing density of the arrangement. As the local density of a packing in an infinite space can vary depending on the volume over which it is measured, the problem is usually to maximise the average or asymptotic density, measured over a large enough volume.
For equal spheres in three dimensions, the densest packing uses approximately 74% of the volume. A random packing of equal spheres generally has a density around 63.5%.
Classification and terminology
A lattice arrangement (commonly called a regular arrangement) is one in which the centers of the spheres form a very symmetric pattern which needs only n vectors to be uniquely defined (in n-dimensional Euclidean space). Lattice arrangements are periodic. Arrangements in which the spheres do not form a lattice (often referred to as irregular) can still be periodic, but also aperiodic (properly speaking non-periodic) or random. Because of their high degree of symmetry, lattice packings are easier to classify than non-lattice ones. Periodic lattices always have well-defined densities.
Regular packing
Dense packing
In three-dimensional Euclidean space, the densest packing of equal spheres is achieved by a family of structures called close-packed structures. One method for generating such a structure is as follows. Consider a plane with a compact arrangement of |
https://en.wikipedia.org/wiki/Moment%20%28mathematics%29 | In mathematics, the moments of a function are certain quantitative measures related to the shape of the function's graph. If the function represents mass density, then the zeroth moment is the total mass, the first moment (normalized by total mass) is the center of mass, and the second moment is the moment of inertia. If the function is a probability distribution, then the first moment is the expected value, the second central moment is the variance, the third standardized moment is the skewness, and the fourth standardized moment is the kurtosis. The mathematical concept is closely related to the concept of moment in physics.
For a distribution of mass or probability on a bounded interval, the collection of all the moments (of all orders, from to ) uniquely determines the distribution (Hausdorff moment problem). The same is not true on unbounded intervals (Hamburger moment problem).
In the mid-nineteenth century, Pafnuty Chebyshev became the first person to think systematically in terms of the moments of random variables.
Significance of the moments
The -th raw moment (i.e., moment about zero) of a distribution is defined bywhereThe -th moment of a real-valued continuous function f(x) of a real variable about a value c is the integralIt is possible to define moments for random variables in a more general fashion than moments for real-valued functions — see moments in metric spaces. The moment of a function, without further explanation, usually refers to the above expression with c = 0.
For the second and higher moments, the central moment (moments about the mean, with c being the mean) are usually used rather than the moments about zero, because they provide clearer information about the distribution's shape.
Other moments may also be defined. For example, the th inverse moment about zero is and the -th logarithmic moment about zero is
The -th moment about zero of a probability density function f(x) is the expected value of and is called a raw moment or c |
https://en.wikipedia.org/wiki/Leafcutter%20ant | Leafcutter ants, a non-generic name, are any of 47 species of leaf-chewing ants belonging to the two genera Atta and Acromyrmex.
These species of tropical, fungus-growing ants are all endemic to South and Central America, Mexico, and parts of the southern United States. Leafcutter ants can carry twenty times their body weight and cut and process fresh vegetation (leaves, flowers, and grasses) to serve as the nutritional substrate for their fungal cultivates.
Acromyrmex and Atta ants have much in common anatomically; however, the two can be identified by their external differences. Atta ants have three pairs of spines and a smooth exoskeleton on the upper surface of the thorax, while Acromyrmex ants have four pairs and a rough exoskeleton. The exoskeleton itself is covered in a thin layer of mineral coating, composed of rhombohedral crystals that are generated by the ants.
Next to humans, leafcutter ants form some of the largest and most complex animal societies on Earth. In a few years, the central mound of their underground nests can grow to more than across, with smaller radiating mounds extending out to a radius of , taking up and containing eight million individuals.
The lifecycle of a leafcutter ant colony
Reproduction and colony founding
Winged females and males leave their respective nests en masse and engage in a nuptial flight known as the revoada (Portuguese) or vuelo nupcial (Spanish). Each female mates with multiple males to collect the 300 million sperm she needs to set up a colony.
Once on the ground, the female loses her wings and searches for a suitable underground lair in which to find her colony. The success rate of these young queens is very low, and only 2.5% will go on to establish a long-lived colony. To start her own fungus garden, the queen stores bits of the parental fungus garden mycelium in her infrabuccal pocket, which is located within her oral cavity. Colonies are generally founded by individual queens haplometrosis. Because |
https://en.wikipedia.org/wiki/27%20%28number%29 | 27 (twenty-seven; Roman numeral XXVII) is the natural number following 26 and preceding 28.
In mathematics
Twenty-seven is equal to the cube of three: ; also 23 (see tetration). It is divisible by the number of prime numbers below it (9).
In decimal, 27 is the first composite number not divisible by any of its digits. In base ten, it is also
It is also the first non-trivial decagonal number.
27 has a prime aliquot sum of 13 (the sixth prime number) in the aliquot sequence (27, 13, 1, 0) of only one composite number, rooted in the 13-aliquot tree.
Whereas the composite index of 27 is 17 (the cousin prime to 13), 7 is the prime index of 17; a prime reciprocal magic square based on multiples of has a magic constant of 27.
In the Collatz conjecture (i.e. the problem), a starting value of 27 requires 3 × 37 = 111 steps to reach 1, more than any smaller number.
The next two larger numbers to require more steps are 54 and 55, where the fourteenth prime number (43) requires twenty-seven steps to reach 1.
Including the null-motif, there are 27 distinct hypergraph motifs.
There are exactly twenty-seven straight lines on a smooth cubic surface, which give a basis of the fundamental representation of Lie algebra .
The unique simple formally real Jordan algebra, the exceptional Jordan algebra of self-adjoint 3 by 3 matrices of quaternions is 27-dimensional; its automorphism group is the 52-dimensional exceptional Lie algebra
There are twenty-seven sporadic groups, if the non-strict group of Lie type (with an irreducible representation that is twice that of in 104 dimensions) is included.
In Robin's theorem for the Riemann hypothesis, twenty-seven integers fail to hold for values where is the Euler–Mascheroni constant; this hypothesis is true if and only if this inequality holds for every larger
Base-specific
In base ten, if one cyclically rotates the digits of a three-digit number that is a multiple of 27, the new number is also a multiple of 27. For exa |
https://en.wikipedia.org/wiki/RAMDAC | A random-access memory digital-to-analog converter (RAMDAC) is a combination of three fast digital-to-analog converters (DACs) with a small static random-access memory (SRAM) used in computer graphics display controllers or video cards to store the color palette and to generate the analog signals (usually a voltage amplitude) to drive a color monitor. The logical color number from the display memory is fed into the address inputs of the SRAM to select a palette entry to appear on the data output of the SRAM. This entry is composed of three separate values corresponding to the three components (red, green, and blue) of the desired physical color. Each component value is fed to a separate DAC, whose analog output goes to the monitor, and ultimately to one of its three electron guns (or equivalent in non-CRT displays).
RAMDACs became obsolete as DVI, HDMI, DisplayPort and other digital interface technology became mainstream, which transfer video data digitally (via transition-minimized differential signaling or low-voltage differential signaling) and defer digital-to-analog conversion until the monitor's pixels are actuated.
The size of each DAC of the RAMDAC is 6 to 10 bits. The SRAM's word length must be at least three times as large as the size of each DAC. The SRAM acts as a color lookup table (CLUT). It usually has 256 entries (and thus an 8-bit address). If the DAC's word length is also 8 bits, we have a 256×24-bit SRAM which allows a selection of 256 out of (16.7 million) possible colors for the display. The contents of this SRAM can be altered when no pixel needs to be generated for transmission to the display, which occurs during the vertical blanking interval between every frame.
The SRAM can usually be bypassed and the DACs can be fed color directly by display data, for True color modes. In fact this has become very much the normal mode of operation of a RAMDAC since the mid-1990s, so the programmable palette is mostly retained only as a legacy feature |
https://en.wikipedia.org/wiki/Automatic%20vehicle%20location | Automatic vehicle location (AVL or ~locating; telelocating in EU) is a means for automatically determining and transmitting the geographic location of a vehicle. This vehicle location data, from one or more vehicles, may then be collected by a vehicle tracking system to manage an overview of vehicle travel. As of 2017, GPS technology has reached the point of having the transmitting device be smaller than the size of a human thumb (thus easier to conceal), able to run 6 months or more between battery charges, easy to communicate with smartphones (merely requiring a duplicate SIM card from one's mobile phone carrier in most cases) — all for less than $20 USD.
Most commonly, the location is determined using GPS and the transmission mechanism is SMS, GPRS, or a satellite or terrestrial radio from the vehicle to a radio receiver. A single antenna unit covering all the needed frequency bands can be employed. GSM and EVDO are the most common services applied, because of the low data rate needed for AVL, and the low cost and near-ubiquitous nature of these public networks. The low bandwidth requirements also allow for satellite technology to receive telemetry data at a moderately higher cost, but across a global coverage area and into very remote locations not covered well by terrestrial radio or public carriers.
Other options for determining actual location, for example in environments where GPS illumination is poor, are dead reckoning, i.e. inertial navigation, or active RFID systems or cooperative RTLS systems. These systems may be applied in combination in some cases. In addition, terrestrial radio positioning systems using a low frequency switched packet radio network have also been used as an alternative to GPS based systems.
Applications
Automatic vehicle locating is a powerful tool for managing fleets of vehicles such as service vehicles, emergency vehicles, and public transport vehicles such as buses and trains. It is also used to track mobile assets, such as |
https://en.wikipedia.org/wiki/Glossary%20of%20tensor%20theory | This is a glossary of tensor theory. For expositions of tensor theory from different points of view, see:
Tensor
Tensor (intrinsic definition)
Application of tensor theory in engineering science
For some history of the abstract theory see also multilinear algebra.
Classical notation
Ricci calculus
The earliest foundation of tensor theory – tensor index notation.
Order of a tensor
The components of a tensor with respect to a basis is an indexed array. The order of a tensor is the number of indices needed. Some texts may refer to the tensor order using the term degree or rank.
Rank of a tensor
The rank of a tensor is the minimum number of rank-one tensor that must be summed to obtain the tensor. A rank-one tensor may be defined as expressible as the outer product of the number of nonzero vectors needed to obtain the correct order.
Dyadic tensor
A dyadic tensor is a tensor of order two, and may be represented as a square matrix. In contrast, a dyad is specifically a dyadic tensor of rank one.
Einstein notation
This notation is based on the understanding that whenever a multidimensional array contains a repeated index letter, the default interpretation is that the product is summed over all permitted values of the index. For example, if aij is a matrix, then under this convention aii is its trace. The Einstein convention is widely used in physics and engineering texts, to the extent that if summation is not to be applied, it is normal to note that explicitly.
Kronecker delta
Levi-Civita symbol
Covariant tensor
Contravariant tensor
The classical interpretation is by components. For example, in the differential form aidxi the components ai are a covariant vector. That means all indices are lower; contravariant means all indices are upper.
Mixed tensor
This refers to any tensor that has both lower and upper indices.
Cartesian tensor
Cartesian tensors are widely used in various branches of continuum mechanics, such as fluid mechanics and elasticity. In cl |
https://en.wikipedia.org/wiki/Audio%20filter | An audio filter is a frequency dependent circuit, working in the audio frequency range, 0 Hz to 20 kHz. Audio filters can amplify (boost), pass or attenuate (cut) some frequency ranges. Many types of filters exist for different audio applications including hi-fi stereo systems, musical synthesizers, effects units, sound reinforcement systems, instrument amplifiers and virtual reality systems.
Types
Low-pass
Low-pass filters pass through frequencies below their cutoff frequencies, and progressively attenuates frequencies above the cutoff frequency. Low-pass filters are used in audio crossovers to remove high-frequency content from signals being sent to a low-frequency subwoofer system.
High-pass
A high-pass filter does the opposite, passing high frequencies above the cutoff frequency, and progressively attenuating frequencies below the cutoff frequency. A high-pass filter can be used in an audio crossover to remove low-frequency content from a signal being sent to a tweeter.
Band-pass
A band-pass filter passes frequencies between its two cutoff frequencies, while attenuating those outside the range. A band-reject filter attenuates frequencies between its two cutoff frequencies, while passing those outside the 'reject' range.
All-pass
An all-pass filter passes all frequencies, but affects the phase of any given sinusoidal component according to its frequency.
Applications
In some applications, such as in the design of graphic equalizers or CD players, the filters are designed according to a set of objective criteria such as passband, passband attenuation, stopband, and stopband attenuation, where the passbands are the frequency ranges for which audio is attenuated less than a specified maximum, and the stopbands are the frequency ranges for which the audio must be attenuated by a specified minimum. In more complex cases, an audio filter can provide a feedback loop, which introduces resonance (ringing) alongside attenuation. Audio filters can also be designed to |
https://en.wikipedia.org/wiki/Jedem%20das%20Seine | "" () is the literal German translation of the Latin phrase suum cuique, meaning "to each his own" or "to each what he deserves".
During World War II the phrase was cynically used by the Nazis as a motto displayed over the entrance of Buchenwald concentration camp. This has resulted in use of the phrase being considered controversial in modern Germany.
History
Jedem das Seine has been an idiomatic German expression for several centuries. For example, it is found in the works of Martin Luther and contemporaries.
It appears in the title of a cantata by Johann Sebastian Bach, Nur jedem das Seine (BWV 163), first performed at Weimar in 1715.
Some nineteenth-century comedies bear the title Jedem das Seine, including works by Johann Friedrich Rochlitz and Caroline Bernstein.
An ironic twist on the proverb, "jedem das Seine, mir das Meiste" ("to each his own, to me the most"), has been known in the reservoir of German idioms for a long time, including its inclusion in Carl Zuckmayer's 1931 play The Captain of Köpenick.
In 1937, the Nazis constructed the Buchenwald concentration camp, 7 km from Weimar, Germany. The motto Jedem das Seine was placed in the camp's main entrance gate. The gates were designed by Franz Ehrlich, a former student of the Bauhaus art school, who had been imprisoned in the camp because he was a communist.
Controversies
Several modern advertising campaigns in the German language, including ads for Nokia, REWE grocery stores, Burger King, and Merkur Bank, have been marred by controversy after using the phrase Jedem das Seine or Jedem den Seinen.
An ExxonMobil ad campaign in January 2009 touted Tchibo coffee drinks at the company's Esso stores with the slogan Jedem den Seinen!. The ads were withdrawn after protest from the Central Council of Jews in Germany, and a company spokesman said its advertising contractor had been unaware of the proverb's association with Nazism.
In March 2009, a student group associated with the Christian Democratic U |
https://en.wikipedia.org/wiki/Frequency%20domain | In mathematics, physics, electronics, control systems engineering, and statistics, the frequency domain refers to the analysis of mathematical functions or signals with respect to frequency, rather than time. Put simply, a time-domain graph shows how a signal changes over time, whereas a frequency-domain graph shows how the signal is distributed within different frequency bands over a range of frequencies. A frequency-domain representation consists of both the magnitude and the phase of a set of sinusoids (or other basis waveforms) at the frequency components of the signal. Although it is common to refer to the magnitude portion as the frequency response of a signal, the phase portion is required to uniquely define the signal.
A given function or signal can be converted between the time and frequency domains with a pair of mathematical operators called transforms. An example is the Fourier transform, which converts a time function into a complex valued sum or integral of sine waves of different frequencies, with amplitudes and phases, each of which represents a frequency component. The "spectrum" of frequency components is the frequency-domain representation of the signal. The inverse Fourier transform converts the frequency-domain function back to the time-domain function. A spectrum analyzer is a tool commonly used to visualize electronic signals in the frequency domain.
A frequency-domain representation may describe either a static function or a particular time period of a dynamic function (signal or system). The frequency transform of a dynamic function is performed over a finite time period of that function and assumes the function repeats infinitely outside of that time period. Some specialized signal processing techniques for dynamic functions use transforms that result in a joint time–frequency domain, with the instantaneous frequency response being a key link between the time domain and the frequency domain.
Advantages
One of the main reasons for us |
https://en.wikipedia.org/wiki/Mascarene%20Islands | The Mascarene Islands (, ) or Mascarenes or Mascarenhas Archipelago is a group of islands in the Indian Ocean east of Madagascar consisting of the islands belonging to the Republic of Mauritius as well as the French department of La Réunion. Their name derives from the Portuguese navigator Pedro Mascarenhas, who first visited them in April 1512. The islands share a common geologic origin in the volcanism of the Réunion hotspot beneath the Mascarene Plateau and form a distinct ecoregion with a unique flora and fauna.
Geography
The archipelago comprises three large islands, Mauritius, Réunion, and Rodrigues, plus a number of volcanic remnants in the tropics of the southwestern Indian Ocean, generally between 700 and 1500 kilometres east of Madagascar. The terrain includes a variety of reefs, atolls, and small islands. They present various topographical and edaphic regions. On the largest islands these gave rise to unusual biodiversity. The climate is oceanic and tropical.
Mauritius is located 900 km east of Madagascar. It has an area of 1865 km2. The highest point is 828 meters. Mauritius is the most populous of the Mascarene Islands, with a population of 1,252,964.
Réunion is located 150 km southwest of Mauritius. It is the largest of the islands, with an area of 2512 km2. Piton des Neiges (3069 m), an extinct volcano, is the highest peak on Réunion and in the islands. Piton de la Fournaise is an active volcano on Réunion which erupts frequently.
Rodrigues is located 574 km east of Mauritius. It has an area of 109 km2, and reaches 393 meters elevation.
St. Brandon, also known as the Cargados Carajos shoals, is a coral atoll group consisting of a barrier reef, shoals, and low islets. It is the remnant of one -or more- large volcanic islands which were submerged by rising tides. Today, the thirty or so islands that comprise St Brandon form part of the Republic of Mauritius. Around seventeen of the uninhabited islands are administered by the 'Outer Islands Developm |
https://en.wikipedia.org/wiki/Variegation | Variegation is the appearance of differently coloured zones in the leaves and sometimes the stems and fruit of plants. Species with variegated individuals are sometimes found in the understory of tropical rainforests, and this habitat is the source of a number of variegated houseplants. Variegation is caused by mutations that affect chlorophyll production or by viruses, such as mosaic viruses, which have been studied by scientists. The striking look of variegated plants is desired by many gardeners, and some have deliberately tried to induce it for aesthetic purposes. There are a number of gardening books about variegated plants, and some gardening societies specialize in them.
The term is also sometimes used to refer to colour zonation in flowers, minerals, and the skin, fur, feathers or scales of animals.
Causes
Chimeral
Plants that are chimeras contain tissues with more than one genotype. A variegated chimera contains some tissues that produce chlorophyll and other tissues which do not. Because the variegation is due to the presence of two kinds of plant tissue, propagating the plant must be by a vegetative method of propagation that preserves both types of tissue in relation to each other. Typically, stem cuttings, bud and stem grafting, and other propagation methods that results in growth from leaf axil buds will preserve variegation. Cuttings with complete variegation may be difficult, if not impossible, to propagate. Root cuttings will not usually preserve variegation, since the new stem tissue is derived from a particular tissue type within the root.
Some variegation is due to visual effects caused by reflection of light from the leaf surface. This can happen when an air layer is located just under the epidermis resulting in a white or silvery reflection. It is sometimes called blister variegation. Pilea cadierei (aluminum plant) is an example of a house plant that shows this effect. Leaves of most Cyclamen species show such patterned variegation, var |
https://en.wikipedia.org/wiki/Separation%20barrier | A separation barrier or separation wall is a barrier, wall or fence, constructed to limit the movement of people across a certain line or border, or to separate peoples or cultures. A separation barrier that runs along an internationally recognized border is known as a border barrier.
David Henley opines in The Guardian that separation barriers are being built at a record-rate around the world along borders and do not only surround dictatorships or pariah states. In 2014, The Washington Post listed notable 14 separation walls as of 2011, indicating that the total concurrent number of walls and barriers which separate countries and territories is 45.
The term "separation barrier" has been applied to structures erected in Belfast, Homs, the West Bank, São Paulo, Cyprus, and along the Greece-Turkey border and the Mexico-United States border. In 2016, Julia Sonnevend listed in her book Stories Without Borders: The Berlin Wall and the Making of a Global Iconic Event the concurrent separation barriers of Sharm el-Sheikh (Egypt), Limbang border (Brunei), the Kazakh-Uzbekistan barrier, Indian border fence with Bangladesh, United States separation barrier with Mexico, Saudi Arabian border fence with Iraq and Hungary's fence with Serbia. Several erected separation barriers are no longer active or in place, including the Berlin Wall, the Maginot Line and some barrier sections in Jerusalem.
Construction rate of separation barriers and walls
David Henley opines in The Guardian that separation barriers are being built at a record-rate around the world along borders and do not only surround dictatorships or pariah states. In 2014, The Washington Post listed notable 14 separation walls as of 2011, indicating that the total concurrent number of walls and barriers which separate countries and territories is 45.
Structures described as "separation barriers" or "separation walls"
Central Europe
Communities in the Czech Republic, Romania and Slovakia have long built Roma walls in |
https://en.wikipedia.org/wiki/L%C3%A9vy%20flight | A Lévy flight is a random walk in which the step-lengths have a stable distribution, a probability distribution that is heavy-tailed. When defined as a walk in a space of dimension greater than one, the steps made are in isotropic random directions. Later researchers have extended the use of the term "Lévy flight" to also include cases where the random walk takes place on a discrete grid rather than on a continuous space.
The term "Lévy flight" was coined by Benoît Mandelbrot, who used this for one specific definition of the distribution of step sizes. He used the term Cauchy flight for the case where the distribution of step sizes is a Cauchy distribution, and Rayleigh flight for when the distribution is a normal distribution (which is not an example of a heavy-tailed probability distribution).
The particular case for which Mandelbrot used the term "Lévy flight" is defined by the survivor function of the distribution of step-sizes, U, being
Here D is a parameter related to the fractal dimension and the distribution is a particular case of the Pareto distribution.
Properties
Lévy flights are, by construction, Markov processes. For general distributions of the step-size, satisfying the power-like condition, the distance from the origin of the random walk tends, after a large number of steps, to a stable distribution due to the generalized central limit theorem, enabling many processes to be modeled using Lévy flights.
The probability densities for particles undergoing a Levy flight can be modeled using a generalized version of the Fokker–Planck equation, which is usually used to model Brownian motion. The equation requires the use of fractional derivatives. For jump lengths which have a symmetric probability distribution, the equation takes a simple form in terms of the Riesz fractional derivative. In one dimension, the equation reads as
where γ is a constant akin to the diffusion constant, α is the stability parameter and f(x,t) is the potential. The Riesz d |
https://en.wikipedia.org/wiki/Programming%20tool | A programming tool or software development tool is a computer program that software developers use to create, debug, maintain, or otherwise support other programs and applications. The term usually refers to relatively simple programs, that can be combined to accomplish a task, much as one might use multiple hands to fix a physical object. The most basic tools are a source code editor and a compiler or interpreter, which are used ubiquitously and continuously. Other tools are used more or less depending on the language, development methodology, and individual engineer, often used for a discrete task, like a debugger or profiler. Tools may be discrete programs, executed separately – often from the command line – or may be parts of a single large program, called an integrated development environment (IDE). In many cases, particularly for simpler use, simple ad hoc techniques are used instead of a tool, such as print debugging instead of using a debugger, manual timing (of overall program or section of code) instead of a profiler, or tracking bugs in a text file or spreadsheet instead of a bug tracking system.
The distinction between tools and applications is murky. For example, developers use simple databases (such as a file containing a list of important values) all the time as tools. However a full-blown database is usually thought of as an application or software in its own right. For many years, computer-assisted software engineering (CASE) tools were sought after. Successful tools have proven elusive. In one sense, CASE tools emphasized design and architecture support, such as for UML. But the most successful of these tools are IDEs.
Uses of programming tools
Translating from human to computer language
Modern computers are very complex and in order to productively program them, various abstractions are needed. For example, rather than writing down a program's binary representation a programmer will write a program in a programming language like C, Java or Py |
https://en.wikipedia.org/wiki/Quantitative%20psychological%20research | Quantitative psychological research is psychological research that employs quantitative research methods.
Quantitative research falls under the category of empirical research.
See also
Statistics
Quantitative psychology
Quantitative research
References
Applied statistics
Experimental psychology
Quantitative research
Statistical data types
Quantitative psychology |
https://en.wikipedia.org/wiki/David%20Parnas | David Lorge Parnas (born February 10, 1941) is a Canadian early pioneer of software engineering, who developed the concept of information hiding in modular programming, which is an important element of object-oriented programming today. He is also noted for his advocacy of precise documentation.
Life
Parnas earned his PhD at Carnegie Mellon University in electrical engineering. Parnas also earned a professional engineering license in Canada and was one of the first to apply traditional engineering principles to software design.
He worked there as a professor for many years. He also taught at the University of North Carolina at Chapel Hill (U.S.), at the Department of Computer Science of the Technische Universität Darmstadt (Germany), the University of Victoria (British Columbia, Canada), Queen's University in Kingston, Ontario, McMaster University in Hamilton, Ontario, and University of Limerick (Republic of Ireland).
David Parnas received a number of awards and honors:
ACM "Best Paper" Award, 1979
Norbert Wiener Award for Social and Professional Responsibility, 1987
Two "Most Influential Paper" awards International Conference on Software Engineering, 1991 and 1995
Doctor honoris causa of the Computer Science Department, ETH Zurich, Switzerland, 1986
Fellow of the Royal Society of Canada, 1992
Fellow of the Association for Computing Machinery, 1994
Doctor honoris causa of the Louvain School of Engineering, University of Louvain (UCLouvain), Belgium, 1996
ACM SIGSOFT's "Outstanding Research" award, 1998
IEEE Computer Society's 60th Anniversary Award, 2007
Doctor honoris causa of the Faculty of Informatics, University of Lugano, Switzerland, 2008
Fellow of the Gesellschaft für Informatik, 2008
Fellow of the Institute of Electrical and Electronics Engineers (IEEE), 2009
Doctor honoris causa of the Vienna University of Technology (Dr. Tech.H.C.), Vienna Austria, 2011
Work
Modular design
In modular design, his double dictum of high cohesion within mod |
https://en.wikipedia.org/wiki/Semiring | In abstract algebra, a semiring is an algebraic structure. It is a generalization of a ring, dropping the requirement that each element must have an additive inverse. At the same time, it is a generalization of bounded distributive lattices.
The smallest semiring that is not a ring is the two-element Boolean algebra, e.g. with logical disjunction as addition. A motivating example that is neither a ring nor a lattice is the set of natural numbers under ordinary addition and multiplication, when including the number zero. Semirings are abundant, because a suitable multiplication operation arises as the function composition of endomorphism over any commutative monoid.
The theory of (associative) algebras over commutative rings can be generalized to one over commutative semirings.
Terminology
Some authors call semiring the structure without the requirement for there to be a or . This makes the analogy between ring and on the one hand and and on the other hand work more smoothly. These authors often use rig for the concept defined here. This originated as a joke, suggesting that rigs are rings without negative elements. (And this is similar to using rng to mean a ring without a multiplicative identity.)
The term dioid (for "double monoid") has been used to mean semirings or other structures. It was used by Kuntzman in 1972 to denote a semiring. (It is alternatively sometimes used for naturally ordered semirings, but the term was also used for idempotent subgroups by Baccelli et al. in 1992.)
Definition
A semiring is a set equipped with two binary operations and called addition and multiplication, such that:
is a monoid with identity element called :
is a monoid with identity element called :
Addition is commutative:
Multiplication by the additive identity annihilates :
Multiplication left- and right-distributes over addition:
Explicitly stated, is a commutative monoid.
Notation
The symbol is usually omitted from the notation; |
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