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https://en.wikipedia.org/wiki/Horizontal%20line%20test | In mathematics, the horizontal line test is a test used to determine whether a function is injective (i.e., one-to-one).
In calculus
A horizontal line is a straight, flat line that goes from left to right. Given a function (i.e. from the real numbers to the real numbers), we can decide if it is injective by looking at horizontal lines that intersect the function's graph. If any horizontal line intersects the graph in more than one point, the function is not injective. To see this, note that the points of intersection have the same y-value (because they lie on the line ) but different x values, which by definition means the function cannot be injective.
Variations of the horizontal line test can be used to determine whether a function is surjective or bijective:
The function f is surjective (i.e., onto) if and only if its graph intersects any horizontal line at least once.
f is bijective if and only if any horizontal line will intersect the graph exactly once.
In set theory
Consider a function with its corresponding graph as a subset of the Cartesian product . Consider the horizontal lines in :. The function f is injective if and only if each horizontal line intersects the graph at most once. In this case the graph is said to pass the horizontal line test. If any horizontal line intersects the graph more than once, the function fails the horizontal line test and is not injective.
See also
Vertical line test
Inverse function
Monotonic function
References
Basic concepts in set theory |
https://en.wikipedia.org/wiki/Muscat | Muscat (, ) is the capital and most populated city in Oman. It is the seat of the Governorate of Muscat. According to the National Centre for Statistics and Information (NCSI), the total population of Muscat Governorate was 1.72 million as of September 2022. The metropolitan area spans approximately and includes six provinces called , making it the largest city in the Arabian Peninsula by area. Known since the early 1st century AD as an important trading port between the west and the east, Muscat was ruled by various indigenous tribes as well as foreign powers such as the Persians, the Portuguese Empire and the Ottoman Empire at various points in its history. A regional military power in the 18th century, Muscat's influence extended as far as East Africa and Zanzibar. As an important port-town in the Gulf of Oman, Muscat attracted foreign tradesmen and settlers such as the Persians, Balochs and Sindhis. Since the accession of Qaboos bin Said as Sultan of Oman in 1970, Muscat has experienced rapid infrastructural development that has led to the growth of a vibrant economy and a multi-ethnic society. Muscat is termed as a Beta - Global City by the Globalization and World Cities Research Network.
The Hajar Mountains dominate the landscape of Muscat. The city lies on the Arabian Sea along the Gulf of Oman and is in the proximity of the strategic Straits of Hormuz. Low-lying white buildings typify most of Muscat's urban landscape, while the port-district of Muttrah, with its corniche and harbour, form the north-eastern periphery of the city. Muscat's economy is dominated by trade, petroleum, liquified natural gas and porting.
Toponymy
Ptolemy's Map of Arabia identifies the territories of Cryptus Portus and Moscha Portus. Scholars are divided in opinion on which of the two is related to the city of Muscat. Similarly, Arrianus references Omana and Moscha in Voyage of Nearchus. Interpretations of Arrianus' work by William Vincent and Jean Baptiste Bourguignon d'Anville conclude that Omana was a reference to Oman, while Moscha referred to Muscat. Similarly, other scholars identify Pliny the Elder's reference to Amithoscuta to be Muscat.
The origin of the word Muscat is disputed. Some authors claim that the word has Arabic origins – from moscha, meaning an inflated hide or skin. Other authors claim that the name Muscat means anchorage or the place of "letting fall the anchor". Other derivations include muscat from Old Persian, meaning strong-scented, or from Arabic, meaning falling-place, or hidden. Cryptus Portus is synonymous with Oman ("hidden land"). But "Ov-man" (Omman), and the old Sumerian name Magan (Maa-kan), means sea-people in Arabic. An inhabitant is a Muscatter, Muscatian, Muscatite or Muscatan. In 1793 AD the capital was transferred from Rustaq to Muscat.
History
Evidence of communal activity in the area around Muscat dates back to the 6th millennium BC in Ras al-Hamra, where burial sites of fishermen have been found. The graves appear |
https://en.wikipedia.org/wiki/Multiplication%20table | In mathematics, a multiplication table (sometimes, less formally, a times table) is a mathematical table used to define a multiplication operation for an algebraic system.
The decimal multiplication table was traditionally taught as an essential part of elementary arithmetic around the world, as it lays the foundation for arithmetic operations with base-ten numbers. Many educators believe it is necessary to memorize the table up to 9 × 9.
History
Pre-modern times
The oldest known multiplication tables were used by the Babylonians about 4000 years ago. However, they used a base of 60. The oldest known tables using a base of 10 are the Chinese decimal multiplication table on bamboo strips dating to about 305 BC, during China's Warring States period.
The multiplication table is sometimes attributed to the ancient Greek mathematician Pythagoras (570–495 BC). It is also called the Table of Pythagoras in many languages (for example French, Italian and Russian), sometimes in English. The Greco-Roman mathematician Nichomachus (60–120 AD), a follower of Neopythagoreanism, included a multiplication table in his Introduction to Arithmetic, whereas the oldest surviving Greek multiplication table is on a wax tablet dated to the 1st century AD and currently housed in the British Museum.
In 493 AD, Victorius of Aquitaine wrote a 98-column multiplication table which gave (in Roman numerals) the product of every number from 2 to 50 times and the rows were "a list of numbers starting with one thousand, descending by hundreds to one hundred, then descending by tens to ten, then by ones to one, and then the fractions down to 1/144."
Modern times
In his 1820 book The Philosophy of Arithmetic, mathematician John Leslie published a multiplication table up to 99 × 99, which allows numbers to be multiplied in pairs of digits at a time. Leslie also recommended that young pupils memorize the multiplication table up to 50 × 50.
The illustration below shows a table up to 12 × 12, which is a size commonly used nowadays in English-world schools.
Because multiplication of integers is commutative, many schools use a smaller table as below. Some schools even remove the first column since 1 is the multiplicative identity.
The traditional rote learning of multiplication was based on memorization of columns in the table, in a form like
1 × 10 = 10
2 × 10 = 20
3 × 10 = 30
4 × 10 = 40
5 × 10 = 50
6 × 10 = 60
7 × 10 = 70
8 × 10 = 80
9 × 10 = 90
This form of writing the multiplication table in columns with complete number sentences is still used in some countries, such as Bosnia and Herzegovina, instead of the modern grids above.
Patterns in the tables
There is a pattern in the multiplication table that can help people to memorize the table more easily. It uses the figures below:
Figure 1 is used for multiples of 1, 3, 7, and 9. Figure 2 is used for the multiples of 2, 4, 6, and 8. These patterns can be used to memorize the multiples of any nu |
https://en.wikipedia.org/wiki/De%20Moivre%27s%20formula | In mathematics, de Moivre's formula (also known as de Moivre's theorem and de Moivre's identity) states that for any real number and integer it holds that
where is the imaginary unit (). The formula is named after Abraham de Moivre, although he never stated it in his works. The expression is sometimes abbreviated to .
The formula is important because it connects complex numbers and trigonometry. By expanding the left hand side and then comparing the real and imaginary parts under the assumption that is real, it is possible to derive useful expressions for and in terms of and .
As written, the formula is not valid for non-integer powers . However, there are generalizations of this formula valid for other exponents. These can be used to give explicit expressions for the th roots of unity, that is, complex numbers such that .
Example
For and , de Moivre's formula asserts that
or equivalently that
In this example, it is easy to check the validity of the equation by multiplying out the left side.
Relation to Euler's formula
De Moivre's formula is a precursor to Euler's formula
which establishes the fundamental relationship between the trigonometric functions and the complex exponential function.
One can derive de Moivre's formula using Euler's formula and the exponential law for integer powers
since Euler's formula implies that the left side is equal to while the right side is equal to
Proof by induction
The truth of de Moivre's theorem can be established by using mathematical induction for natural numbers, and extended to all integers from there. For an integer , call the following statement :
For , we proceed by mathematical induction. is clearly true. For our hypothesis, we assume is true for some natural . That is, we assume
Now, considering :
See angle sum and difference identities.
We deduce that implies . By the principle of mathematical induction it follows that the result is true for all natural numbers. Now, is clearly true since . Finally, for the negative integer cases, we consider an exponent of for natural .
The equation (*) is a result of the identity
for . Hence, holds for all integers .
Formulae for cosine and sine individually
For an equality of complex numbers, one necessarily has equality both of the real parts and of the imaginary parts of both members of the equation. If , and therefore also and , are real numbers, then the identity of these parts can be written using binomial coefficients. This formula was given by 16th century French mathematician François Viète:
In each of these two equations, the final trigonometric function equals one or minus one or zero, thus removing half the entries in each of the sums. These equations are in fact valid even for complex values of , because both sides are entire (that is, holomorphic on the whole complex plane) functions of , and two such functions that coincide on the real axis necessarily coincide everywhere. Here are the concrete instances of t |
https://en.wikipedia.org/wiki/James%20M.%20Buchanan | James McGill Buchanan Jr. (; October 3, 1919 – January 9, 2013) was an American economist known for his work on public choice theory originally outlined in his most famous work, The Calculus of Consent, co-authored with Gordon Tullock in 1962. He continued to develop the theory, eventually receiving the Nobel Memorial Prize in Economic Sciences in 1986. Buchanan's work initiated research on how politicians' and bureaucrats' self-interest, utility maximization, and other non-wealth-maximizing considerations affect their decision-making. He was a member of the Board of Advisors of The Independent Institute as well as of the Institute of Economic Affairs, a member of the Mont Pelerin Society (MPS) and MPS president from 1984 to 1986, a Distinguished Senior Fellow of the Cato Institute, and professor at George Mason University.
Early life
Buchanan was born in Murfreesboro, Tennessee, the eldest of the three children of James and Lila (Scott) Buchanan. His paternal grandfather, John P. Buchanan, was governor of Tennessee from 1891 to 1893. According to Buchanan's 1992 memoir, when his father, James Buchanan, Sr., married in 1918 and began his family he borrowed heavily to mechanize and improve the farm, including the acquisition of a herd of Jersey cattle. The Buchanan farm suffered during the 1920sby the time James Buchanan, Jr. was old enough to work on the farm, all the work was done either manually or with mules and horses. Buchanan described his life on the farm as "genteel poverty" with neither indoor plumbing nor electricity. The house did contain his grandfather's library of books on politics. Unlike in other farm families where children regularly stayed home to help as farm labor, his mother insisted he never miss a day of school. While completing his first university degree in 1940 at Middle Tennessee State Teachers College he continued to live at home and work on the farm. In 1941 he completed his M.S. at the University of Tennessee.
World War II
Buchanan attended the United States Naval Reserve Midshipmen's School in New York starting in September 1941. He was assigned to Honolulu, Hawaii in March 1942, where he served as an officer on Admiral Chester W. Nimitz's operations planning staff in the United States Navy.
Buchanan attributed his dislike of what he considered "Eastern elites" to his six months of Navy officer training in New York in 1941. He believed that there was overt discrimination against young men from the South or West in favor of those who had attended what he called establishment universities in the Northeast. In a 2011 interview, Buchanan said that out of twenty "boys from the establishment universities, 12 or 13 were picked against a background of a total of 600 [men]." He was completing his last month of training in New York during the December 7, 1941 attack on Pearl Harbor. Buchanan said that "in balance" his work in operations planning during the war was "easy." He was discharged in November 1945.
Education
Wit |
https://en.wikipedia.org/wiki/Jacobson%20radical | In mathematics, more specifically ring theory, the Jacobson radical of a ring R is the ideal consisting of those elements in R that annihilate all simple right R-modules. It happens that substituting "left" in place of "right" in the definition yields the same ideal, and so the notion is left-right symmetric. The Jacobson radical of a ring is frequently denoted by J(R) or rad(R); the former notation will be preferred in this article, because it avoids confusion with other radicals of a ring. The Jacobson radical is named after Nathan Jacobson, who was the first to study it for arbitrary rings in .
The Jacobson radical of a ring has numerous internal characterizations, including a few definitions that successfully extend the notion to rings without unity. The radical of a module extends the definition of the Jacobson radical to include modules. The Jacobson radical plays a prominent role in many ring and module theoretic results, such as Nakayama's lemma.
Definitions
There are multiple equivalent definitions and characterizations of the Jacobson radical, but it is useful to consider the definitions based on if the ring is commutative or not.
Commutative case
In the commutative case, the Jacobson radical of a commutative ring R is defined as the intersection of all maximal ideals . If we denote as the set of all maximal ideals in R thenThis definition can be used for explicit calculations in a number of simple cases, such as for local rings , which have a unique maximal ideal, Artin rings, and products thereof. See the examples section for explicit computations.
Noncommutative/general case
For a general ring with unity R, the Jacobson radical J(R) is defined as the ideal of all elements such that whenever M is a simple R-module. That is,
This is equivalent to the definition in the commutative case for a commutative ring R because the simple modules over a commutative ring are of the form for some maximal ideal , and the only annihilators of in R are in , i.e. .
Motivation
Understanding the Jacobson radical lies in a few different cases: namely its applications and the resulting geometric interpretations, and its algebraic interpretations.
Geometric applications
Although Jacobson originally introduced his radical as a technique for building a theory of radicals for arbitrary rings, one of the motivating reasons for why the Jacobson radical is considered in the commutative case is because of its appearance in Nakayama's lemma. This lemma is a technical tool for studying finitely generated modules over commutative rings which has an easy geometric interpretation: If we have a vector bundle over a topological space , and pick a point , then any basis of can be extended to a basis of sections of for some neighborhood .
Another application is in the case of finitely generated commutative rings, meaning R is of the form
for some base ring k (such as a field, or the ring of integers). In this case the nilradical and the Jacobson radi |
https://en.wikipedia.org/wiki/Stephen%20Baxter%20%28author%29 | Stephen Baxter (born 13 November 1957) is an English hard science fiction author. He has degrees in mathematics and engineering.
Writing style
Strongly influenced by SF pioneer H. G. Wells, Baxter has been vice-president of the international H. G. Wells Society since 2006. His fiction falls into three main categories of original work plus a fourth category, extending other authors' writing; each has a different basis, style, and tone.
Baxter's "Future History" mode is based on research into hard science. It encompasses the Xeelee Sequence, which consists of nine novels (including the Destiny's Children trilogy and Vengeance/Redemption duology that is set in alternate timeline), plus three volumes collecting the 52 short pieces (short stories and novellas) in the series, all of which fit into a single timeline stretching from the Big Bang singularity of the past to his Timelike Infinity singularity of the future. These stories begin in the present day and end when the Milky Way galaxy collides with Andromeda five billion years in the future. The central narrative is that of humanity rising and evolving to become the second most powerful race in the universe, next to the god-like Xeelee. Character development tends to be secondary to the depiction of advanced theories and ideas, such as the true nature of the Great Attractor, naked singularities and the great battle between baryonic and dark matter lifeforms. The Manifold Trilogy is another example of Baxter's future history mode, even more conceptual than the Xeelee sequence. Each novel is focused on a potential explanation of the Fermi paradox. The two-part disaster series Flood and Ark (followed by three additional stories, "Earth III," "Earth II," and "Earth I") which also fits into this category, where catastrophic events unfold in the near future and humanity must adapt to survive in three radically different planetary environments. In 2013, Baxter released his short story collection Universes which featured stories set in Flood/Ark, Jones & Bennet and Anti-Ice universes. Baxter signed a contract for two new books, Proxima and Ultima, both of which are names of planets, and they were released in 2013 and 2014, respectively.
A second category in Baxter's work is based on readings in evolutionary biology and human/animal behaviour. Elements of this appear in his future histories (especially later works like the Destiny's Children series and Flood/Ark), but here it is the focus. The major work in this category is Evolution, which imagines the evolution of humanity in the Earth's past and future. The Mammoth Trilogy, written for young adults, shares similar themes and concerns as it explores the present, past, and future of a small herd of mammoths found surviving on an island in the Arctic Ocean.
A third category of Baxter's fiction is alternate history, based on research into history. These stories are more human, with characters portrayed with greater depth and care. This includes his NA |
https://en.wikipedia.org/wiki/Grover%27s%20algorithm | In quantum computing, Grover's algorithm, also known as the quantum search algorithm, is a quantum algorithm for unstructured search that finds with high probability the unique input to a black box function that produces a particular output value, using just evaluations of the function, where is the size of the function's domain. It was devised by Lov Grover in 1996.
The analogous problem in classical computation cannot be solved in fewer than evaluations (because, on average, one has to check half of the domain to get a 50% chance of finding the right input). Charles H. Bennett, Ethan Bernstein, Gilles Brassard, and Umesh Vazirani proved that any quantum solution to the problem needs to evaluate the function times, so Grover's algorithm is asymptotically optimal. Since classical algorithms for NP-complete problems require exponentially many steps, and Grover's algorithm provides at most a quadratic speedup over the classical solution for unstructured search, this suggests that Grover's algorithm by itself will not provide polynomial-time solutions for NP-complete problems (as the square root of an exponential function is an exponential, not polynomial, function).
Unlike other quantum algorithms, which may provide exponential speedup over their classical counterparts, Grover's algorithm provides only a quadratic speedup. However, even quadratic speedup is considerable when is large, and Grover's algorithm can be applied to speed up broad classes of algorithms. Grover's algorithm could brute-force a 128-bit symmetric cryptographic key in roughly 264 iterations, or a 256-bit key in roughly 2128 iterations. It may not be the case that Grover's algorithm poses a significantly increased risk to encryption over existing classical algorithms, however.
Applications and limitations
Grover's algorithm, along with variants like amplitude amplification, can be used to speed up a broad range of algorithms. In particular, algorithms for NP-complete problems generally contain exhaustive search as a subroutine, which can be sped up by Grover's algorithm. The current best algorithm for 3SAT is one such example. Generic constraint satisfaction problems also see quadratic speedups with Grover. These algorithms do not require that the input be given in the form of an oracle, since Grover's algorithm is being applied with an explicit function, e.g. the function checking that a set of bits satisfies a 3SAT instance.
Grover's algorithm can also give provable speedups for black-box problems in quantum query complexity, including element distinctness and the collision problem (solved with the Brassard–Høyer–Tapp algorithm). In these types of problems, one treats the oracle function f as a database, and the goal is to use the quantum query to this function as few times as possible.
Cryptography
Grover's algorithm essentially solves the task of function inversion. Roughly speaking, if we have a function that can be evaluated on a quantum computer, Grover's alg |
https://en.wikipedia.org/wiki/Finitely%20generated%20abelian%20group | In abstract algebra, an abelian group is called finitely generated if there exist finitely many elements in such that every in can be written in the form for some integers . In this case, we say that the set is a generating set of or that generate .
Every finite abelian group is finitely generated. The finitely generated abelian groups can be completely classified.
Examples
The integers, , are a finitely generated abelian group.
The integers modulo , , are a finite (hence finitely generated) abelian group.
Any direct sum of finitely many finitely generated abelian groups is again a finitely generated abelian group.
Every lattice forms a finitely generated free abelian group.
There are no other examples (up to isomorphism). In particular, the group of rational numbers is not finitely generated: if are rational numbers, pick a natural number coprime to all the denominators; then cannot be generated by . The group of non-zero rational numbers is also not finitely generated. The groups of real numbers under addition and non-zero real numbers under multiplication are also not finitely generated.
Classification
The fundamental theorem of finitely generated abelian groups can be stated two ways, generalizing the two forms of the fundamental theorem of finite abelian groups. The theorem, in both forms, in turn generalizes to the structure theorem for finitely generated modules over a principal ideal domain, which in turn admits further generalizations.
Primary decomposition
The primary decomposition formulation states that every finitely generated abelian group G is isomorphic to a direct sum of primary cyclic groups and infinite cyclic groups. A primary cyclic group is one whose order is a power of a prime. That is, every finitely generated abelian group is isomorphic to a group of the form
where n ≥ 0 is the rank, and the numbers q1, ..., qt are powers of (not necessarily distinct) prime numbers. In particular, G is finite if and only if n = 0. The values of n, q1, ..., qt are (up to rearranging the indices) uniquely determined by G, that is, there is one and only one way to represent G as such a decomposition.
The proof of this statement uses the basis theorem for finite abelian group: every finite abelian group is a direct sum of primary cyclic groups. Denote the torsion subgroup of G as tG. Then, G/tG is a torsion-free abelian group and thus it is free abelian. tG is a direct summand of G, which means there exists a subgroup F of G s.t. , where . Then, F is also free abelian. Since tG is finitely generated and each element of tG has finite order, tG is finite. By the basis theorem for finite abelian group, tG can be written as direct sum of primary cyclic groups.
Invariant factor decomposition
We can also write any finitely generated abelian group G as a direct sum of the form
where k1 divides k2, which divides k3 and so on up to ku. Again, the rank n and the invariant factors k1, ..., ku are uniquely determined by G ( |
https://en.wikipedia.org/wiki/Non-Euclidean%20geometry | In mathematics, non-Euclidean geometry consists of two geometries based on axioms closely related to those that specify Euclidean geometry. As Euclidean geometry lies at the intersection of metric geometry and affine geometry, non-Euclidean geometry arises by either replacing the parallel postulate with an alternative, or relaxing the metric requirement. In the former case, one obtains hyperbolic geometry and elliptic geometry, the traditional non-Euclidean geometries. When the metric requirement is relaxed, then there are affine planes associated with the planar algebras, which give rise to kinematic geometries that have also been called non-Euclidean geometry.
Principles
The essential difference between the metric geometries is the nature of parallel lines. Euclid's fifth postulate, the parallel postulate, is equivalent to Playfair's postulate, which states that, within a two-dimensional plane, for any given line and a point A, which is not on , there is exactly one line through A that does not intersect . In hyperbolic geometry, by contrast, there are infinitely many lines through A not intersecting , while in elliptic geometry, any line through A intersects .
Another way to describe the differences between these geometries is to consider two straight lines indefinitely extended in a two-dimensional plane that are both perpendicular to a third line (in the same plane):
In Euclidean geometry, the lines remain at a constant distance from each other (meaning that a line drawn perpendicular to one line at any point will intersect the other line and the length of the line segment joining the points of intersection remains constant) and are known as parallels.
In hyperbolic geometry, they "curve away" from each other, increasing in distance as one moves further from the points of intersection with the common perpendicular; these lines are often called ultraparallels.
In elliptic geometry, the lines "curve toward" each other and intersect.
History
Background
Euclidean geometry, named after the Greek mathematician Euclid, includes some of the oldest known mathematics, and geometries that deviated from this were not widely accepted as legitimate until the 19th century.
The debate that eventually led to the discovery of the non-Euclidean geometries began almost as soon as Euclid wrote Elements. In the Elements, Euclid begins with a limited number of assumptions (23 definitions, five common notions, and five postulates) and seeks to prove all the other results (propositions) in the work. The most notorious of the postulates is often referred to as "Euclid's Fifth Postulate", or simply the parallel postulate, which in Euclid's original formulation is:
If a straight line falls on two straight lines in such a manner that the interior angles on the same side are together less than two right angles, then the straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles.
Other mathematicians have de |
https://en.wikipedia.org/wiki/Direct%20sum%20of%20modules | In abstract algebra, the direct sum is a construction which combines several modules into a new, larger module. The direct sum of modules is the smallest module which contains the given modules as submodules with no "unnecessary" constraints, making it an example of a coproduct. Contrast with the direct product, which is the dual notion.
The most familiar examples of this construction occur when considering vector spaces (modules over a field) and abelian groups (modules over the ring Z of integers). The construction may also be extended to cover Banach spaces and Hilbert spaces.
See the article decomposition of a module for a way to write a module as a direct sum of submodules.
Construction for vector spaces and abelian groups
We give the construction first in these two cases, under the assumption that we have only two objects. Then we generalize to an arbitrary family of arbitrary modules. The key elements of the general construction are more clearly identified by considering these two cases in depth.
Construction for two vector spaces
Suppose V and W are vector spaces over the field K. The cartesian product V × W can be given the structure of a vector space over K by defining the operations componentwise:
(v1, w1) + (v2, w2) = (v1 + v2, w1 + w2)
α (v, w) = (α v, α w)
for v, v1, v2 ∈ V, w, w1, w2 ∈ W, and α ∈ K.
The resulting vector space is called the direct sum of V and W and is usually denoted by a plus symbol inside a circle:
It is customary to write the elements of an ordered sum not as ordered pairs (v, w), but as a sum v + w.
The subspace V × {0} of V ⊕ W is isomorphic to V and is often identified with V; similarly for {0} × W and W. (See internal direct sum below.) With this identification, every element of V ⊕ W can be written in one and only one way as the sum of an element of V and an element of W. The dimension of V ⊕ W is equal to the sum of the dimensions of V and W. One elementary use is the reconstruction
of a finite vector space from any subspace W and its orthogonal complement:
This construction readily generalizes to any finite number of vector spaces.
Construction for two abelian groups
For abelian groups G and H which are written additively, the direct product of G and H is also called a direct sum . Thus the Cartesian product G × H is equipped with the structure of an abelian group by defining the operations componentwise:
(g1, h1) + (g2, h2) = (g1 + g2, h1 + h2)
for g1, g2 in G, and h1, h2 in H.
Integral multiples are similarly defined componentwise by
n(g, h) = (ng, nh)
for g in G, h in H, and n an integer. This parallels the extension of the scalar product of vector spaces to the direct sum above.
The resulting abelian group is called the direct sum of G and H and is usually denoted by a plus symbol inside a circle:
It is customary to write the elements of an ordered sum not as ordered pairs (g, h), but as a sum g + h.
The subgroup G × {0} of G ⊕ H is isomorphic to G and is often identifie |
https://en.wikipedia.org/wiki/Quadratic%20formula | In elementary algebra, the quadratic formula is a formula that provides the two solutions, or roots, to a quadratic equation. There are other ways of solving a quadratic equation instead of using the quadratic formula, such as completing the square.
Given a general quadratic equation of the form
with representing an unknown, with , and representing constants, and with , the quadratic formula is:
where the plus–minus symbol "±" indicates that the quadratic equation has two solutions. Written separately, they become:
Each of these two solutions is also called a root (or zero) of the quadratic equation. Geometrically, these roots represent the -values at which any parabola, explicitly given as , crosses the -axis.
As well as being a formula that yields the zeros of any parabola, the quadratic formula can also be used to identify the axis of symmetry of the parabola, and the number of real zeros the quadratic equation contains.
The expression is known as the discriminant. If , , and are real numbers and then
When , there are two distinct real roots or solutions to the equation .
When , there is one repeated real solution.
When , there are two distinct complex solutions, which are complex conjugates of each other.
Equivalent formulations
The quadratic formula may also be written as
or
Because these formulas allow re-use of intermediately calculated values, these may be easier to use when computing with a calculator or by hand. When the discriminant is negative, complex roots are involved and the quadratic formula can be written as:
Muller's method
A lesser known quadratic formula, also named "citardauq", which is used in Muller's method and which can be found from Vieta's formulas, provides (assuming ) the same roots via the equation:
For positive , the subtraction causes cancellation in the standard formula (respectively negative and addition), resulting in poor accuracy. In this case, switching to Muller's formula with the opposite sign is a good workaround.
Derivations of the formula
Many different methods to derive the quadratic formula are available in the literature. The standard one is a simple application of the completing the square technique. Alternative methods are sometimes simpler than completing the square, and may offer interesting insight into other areas of mathematics.
By completing the square
Divide the quadratic equation by , which is allowed because is non-zero:
Subtract from both sides of the equation, yielding:
The quadratic equation is now in a form to which the method of completing the square is applicable. In fact, by adding a constant to both sides of the equation such that the left hand side becomes a complete square, the quadratic equation becomes:
which produces:
Accordingly, after rearranging the terms on the right hand side to have a common denominator, we obtain:
The square has thus been completed. If the discriminant is positive, we can take the square root of both sides, yielding the |
https://en.wikipedia.org/wiki/Base%20%28topology%29 | In mathematics, a base (or basis; : bases) for the topology of a topological space is a family of open subsets of such that every open set of the topology is equal to the union of some sub-family of . For example, the set of all open intervals in the real number line is a basis for the Euclidean topology on because every open interval is an open set, and also every open subset of can be written as a union of some family of open intervals.
Bases are ubiquitous throughout topology. The sets in a base for a topology, which are called , are often easier to describe and use than arbitrary open sets. Many important topological definitions such as continuity and convergence can be checked using only basic open sets instead of arbitrary open sets. Some topologies have a base of open sets with specific useful properties that may make checking such topological definitions easier.
Not all families of subsets of a set form a base for a topology on . Under some conditions detailed below, a family of subsets will form a base for a (unique) topology on , obtained by taking all possible unions of subfamilies. Such families of sets are very frequently used to define topologies. A weaker notion related to bases is that of a subbase for a topology. Bases for topologies are also closely related to neighborhood bases.
Definition and basic properties
Given a topological space , a base (or basis) for the topology (also called a base for if the topology is understood) is a family of open sets such that every open set of the topology can be represented as the union of some subfamily of . The elements of are called basic open sets.
Equivalently, a family of subsets of is a base for the topology if and only if and for every open set in and point there is some basic open set such that .
For example, the collection of all open intervals in the real line forms a base for the standard topology on the real numbers. More generally, in a metric space the collection of all open balls about points of forms a base for the topology.
In general, a topological space can have many bases. The whole topology is always a base for itself (that is, is a base for ). For the real line, the collection of all open intervals is a base for the topology. So is the collection of all open intervals with rational endpoints, or the collection of all open intervals with irrational endpoints, for example. Note that two different bases need not have any basic open set in common. One of the topological properties of a space is the minimum cardinality of a base for its topology, called the weight of and denoted . From the examples above, the real line has countable weight.
If is a base for the topology of a space , it satisfies the following properties:
(B1) The elements of cover , i.e., every point belongs to some element of .
(B2) For every and every point , there exists some such that .
Property (B1) corresponds to the fact that is an open set; property |
https://en.wikipedia.org/wiki/Initial%20and%20terminal%20objects | In category theory, a branch of mathematics, an initial object of a category is an object in such that for every object in , there exists precisely one morphism .
The dual notion is that of a terminal object (also called terminal element): is terminal if for every object in there exists exactly one morphism . Initial objects are also called coterminal or universal, and terminal objects are also called final.
If an object is both initial and terminal, it is called a zero object or null object. A pointed category is one with a zero object.
A strict initial object is one for which every morphism into is an isomorphism.
Examples
The empty set is the unique initial object in Set, the category of sets. Every one-element set (singleton) is a terminal object in this category; there are no zero objects. Similarly, the empty space is the unique initial object in Top, the category of topological spaces and every one-point space is a terminal object in this category.
In the category Rel of sets and relations, the empty set is the unique initial object, the unique terminal object, and hence the unique zero object.
In the category of pointed sets (whose objects are non-empty sets together with a distinguished element; a morphism from to being a function with ), every singleton is a zero object. Similarly, in the category of pointed topological spaces, every singleton is a zero object.
In Grp, the category of groups, any trivial group is a zero object. The trivial object is also a zero object in Ab, the category of abelian groups, Rng the category of pseudo-rings, R-Mod, the category of modules over a ring, and K-Vect, the category of vector spaces over a field. See Zero object (algebra) for details. This is the origin of the term "zero object".
In Ring, the category of rings with unity and unity-preserving morphisms, the ring of integers Z is an initial object. The zero ring consisting only of a single element 0 = 1 is a terminal object.
In Rig, the category of rigs with unity and unity-preserving morphisms, the rig of natural numbers N is an initial object. The zero rig, which is the zero ring, consisting only of a single element 0 = 1 is a terminal object.
In Field, the category of fields, there are no initial or terminal objects. However, in the subcategory of fields of fixed characteristic, the prime field is an initial object.
Any partially ordered set can be interpreted as a category: the objects are the elements of , and there is a single morphism from to if and only if . This category has an initial object if and only if has a least element; it has a terminal object if and only if has a greatest element.
Cat, the category of small categories with functors as morphisms has the empty category, 0 (with no objects and no morphisms), as initial object and the terminal category, 1 (with a single object with a single identity morphism), as terminal object.
In the category of schemes, Spec(Z), the prime spectrum of the ring o |
https://en.wikipedia.org/wiki/Preadditive%20category | In mathematics, specifically in category theory, a preadditive category is
another name for an Ab-category, i.e., a category that is enriched over the category of abelian groups, Ab.
That is, an Ab-category C is a category such that
every hom-set Hom(A,B) in C has the structure of an abelian group, and composition of morphisms is bilinear, in the sense that composition of morphisms distributes over the group operation.
In formulas:
and
where + is the group operation.
Some authors have used the term additive category for preadditive categories, but here we follow the current trend of reserving this term for certain special preadditive categories (see below).
Examples
The most obvious example of a preadditive category is the category Ab itself. More precisely, Ab is a closed monoidal category. Note that commutativity is crucial here; it ensures that the sum of two group homomorphisms is again a homomorphism. In contrast, the category of all groups is not closed. See Medial category.
Other common examples:
The category of (left) modules over a ring R, in particular:
the category of vector spaces over a field K.
The algebra of matrices over a ring, thought of as a category as described in the article Additive category.
Any ring, thought of as a category with only one object, is a preadditive category. Here composition of morphisms is just ring multiplication and the unique hom-set is the underlying abelian group.
These will give you an idea of what to think of; for more examples, follow the links to below.
Elementary properties
Because every hom-set Hom(A,B) is an abelian group, it has a zero element 0. This is the zero morphism from A to B. Because composition of morphisms is bilinear, the composition of a zero morphism and any other morphism (on either side) must be another zero morphism. If you think of composition as analogous to multiplication, then this says that multiplication by zero always results in a product of zero, which is a familiar intuition. Extending this analogy, the fact that composition is bilinear in general becomes the distributivity of multiplication over addition.
Focusing on a single object A in a preadditive category, these facts say that the endomorphism hom-set Hom(A,A) is a ring, if we define multiplication in the ring to be composition. This ring is the endomorphism ring of A. Conversely, every ring (with identity) is the endomorphism ring of some object in some preadditive category. Indeed, given a ring R, we can define a preadditive category R to have a single object A, let Hom(A,A) be R, and let composition be ring multiplication. Since R is an abelian group and multiplication in a ring is bilinear (distributive), this makes R a preadditive category. Category theorists will often think of the ring R and the category R as two different representations of the same thing, so that a particularly perverse category theorist might define a ring as a preadditive category with exactly one object (in the s |
https://en.wikipedia.org/wiki/Monomorphism | In the context of abstract algebra or universal algebra, a monomorphism is an injective homomorphism. A monomorphism from to is often denoted with the notation .
In the more general setting of category theory, a monomorphism (also called a monic morphism or a mono) is a left-cancellative morphism. That is, an arrow such that for all objects and all morphisms ,
Monomorphisms are a categorical generalization of injective functions (also called "one-to-one functions"); in some categories the notions coincide, but monomorphisms are more general, as in the examples below.
In the setting of posets intersections are idempotent: the intersection of anything with itself is itself. Monomorphisms generalize this property to arbitrary categories. A morphism is a monomorphism if it is idempotent with respect to pullbacks.
The categorical dual of a monomorphism is an epimorphism, that is, a monomorphism in a category C is an epimorphism in the dual category Cop. Every section is a monomorphism, and every retraction is an epimorphism.
Relation to invertibility
Left-invertible morphisms are necessarily monic: if l is a left inverse for f (meaning l is a morphism and ), then f is monic, as
A left-invertible morphism is called a split mono or a section.
However, a monomorphism need not be left-invertible. For example, in the category Group of all groups and group homomorphisms among them, if H is a subgroup of G then the inclusion is always a monomorphism; but f has a left inverse in the category if and only if H has a normal complement in G.
A morphism is monic if and only if the induced map , defined by for all morphisms , is injective for all objects Z.
Examples
Every morphism in a concrete category whose underlying function is injective is a monomorphism; in other words, if morphisms are actually functions between sets, then any morphism which is a one-to-one function will necessarily be a monomorphism in the categorical sense. In the category of sets the converse also holds, so the monomorphisms are exactly the injective morphisms. The converse also holds in most naturally occurring categories of algebras because of the existence of a free object on one generator. In particular, it is true in the categories of all groups, of all rings, and in any abelian category.
It is not true in general, however, that all monomorphisms must be injective in other categories; that is, there are settings in which the morphisms are functions between sets, but one can have a function that is not injective and yet is a monomorphism in the categorical sense. For example, in the category Div of divisible (abelian) groups and group homomorphisms between them there are monomorphisms that are not injective: consider, for example, the quotient map , where Q is the rationals under addition, Z the integers (also considered a group under addition), and Q/Z is the corresponding quotient group. This is not an injective map, as for example every integer is mapped |
https://en.wikipedia.org/wiki/Figure-eight%20knot%20%28mathematics%29 | In knot theory, a figure-eight knot (also called Listing's knot) is the unique knot with a crossing number of four. This makes it the knot with the third-smallest possible crossing number, after the unknot and the
trefoil knot. The figure-eight knot is a prime knot.
Origin of name
The name is given because tying a normal figure-eight knot in a rope and then joining the ends together, in the most natural way, gives a model of the mathematical knot.
Description
A simple parametric representation of the figure-eight knot is as the set of all points (x,y,z) where
for t varying over the real numbers (see 2D visual realization at bottom right).
The figure-eight knot is prime, alternating, rational with an associated value
of 5/3, and is achiral. The figure-eight knot is also a fibered knot. This follows from other, less simple (but very interesting) representations of the knot:
(1) It is a homogeneous closed braid (namely, the closure of the 3-string braid σ1σ2−1σ1σ2−1), and a theorem of John Stallings shows that any closed homogeneous braid is fibered.
(2) It is the link at (0,0,0,0) of an isolated critical point of a real-polynomial map F: R4→R2, so (according to a theorem of John Milnor) the Milnor map of F is actually a fibration. Bernard Perron found the first such F for this knot, namely,
where
Mathematical properties
The figure-eight knot has played an important role historically (and continues to do so) in the theory of 3-manifolds. Sometime in the mid-to-late 1970s, William Thurston showed that the figure-eight was hyperbolic, by decomposing its complement into two ideal hyperbolic tetrahedra. (Robert Riley and Troels Jørgensen, working independently of each other, had earlier shown that the figure-eight knot was hyperbolic by other means.) This construction, new at the time, led him to many powerful results and methods. For example, he was able to show that all but ten Dehn surgeries on the figure-eight knot resulted in non-Haken, non-Seifert-fibered irreducible 3-manifolds; these were the first such examples. Many more have been discovered by generalizing Thurston's construction to other knots and links.
The figure-eight knot is also the hyperbolic knot whose complement has the smallest possible volume, , where is the Lobachevsky function. From this perspective, the figure-eight knot can be considered the simplest hyperbolic knot. The figure eight knot complement is a double-cover of the Gieseking manifold, which has the smallest volume among non-compact hyperbolic 3-manifolds.
The figure-eight knot and the (−2,3,7) pretzel knot are the only two hyperbolic knots known to have more than 6 exceptional surgeries, Dehn surgeries resulting in a non-hyperbolic 3-manifold; they have 10 and 7, respectively. A theorem of Lackenby and Meyerhoff, whose proof relies on the geometrization conjecture and computer assistance, holds that 10 is the largest possible number of exceptional surgeries of any hyperbolic knot. However |
https://en.wikipedia.org/wiki/Biproduct | In category theory and its applications to mathematics, a biproduct of a finite collection of objects, in a category with zero objects, is both a product and a coproduct. In a preadditive category the notions of product and coproduct coincide for finite collections of objects. The biproduct is a generalization of finite direct sums of modules.
Definition
Let C be a category with zero morphisms. Given a finite (possibly empty) collection of objects A1, ..., An in C, their biproduct is an object in C together with morphisms
in C (the projection morphisms)
(the embedding morphisms)
satisfying
, the identity morphism of and
, the zero morphism for
and such that
is a product for the and
is a coproduct for the
If C is preadditive and the first two conditions hold, then each of the last two conditions is equivalent to when n > 0. An empty, or nullary, product is always a terminal object in the category, and the empty coproduct is always an initial object in the category. Thus an empty, or nullary, biproduct is always a zero object.
Examples
In the category of abelian groups, biproducts always exist and are given by the direct sum. The zero object is the trivial group.
Similarly, biproducts exist in the category of vector spaces over a field. The biproduct is again the direct sum, and the zero object is the trivial vector space.
More generally, biproducts exist in the category of modules over a ring.
On the other hand, biproducts do not exist in the category of groups. Here, the product is the direct product, but the coproduct is the free product.
Also, biproducts do not exist in the category of sets. For, the product is given by the Cartesian product, whereas the coproduct is given by the disjoint union. This category does not have a zero object.
Block matrix algebra relies upon biproducts in categories of matrices.
Properties
If the biproduct exists for all pairs of objects A and B in the category C, and C has a zero object, then all finite biproducts exist, making C both a Cartesian monoidal category and a co-Cartesian monoidal category.
If the product and coproduct both exist for some pair of objects A1, A2 then there is a unique morphism such that
for
It follows that the biproduct exists if and only if f is an isomorphism.
If C is a preadditive category, then every finite product is a biproduct, and every finite coproduct is a biproduct. For example, if exists, then there are unique morphisms such that
for
To see that is now also a coproduct, and hence a biproduct, suppose we have morphisms for some object . Define Then is a morphism from to , and for .
In this case we always have
An additive category is a preadditive category in which all finite biproducts exist. In particular, biproducts always exist in abelian categories.
References
Additive categories
Limits (category theory) |
https://en.wikipedia.org/wiki/Endomorphism%20ring | In mathematics, the endomorphisms of an abelian group X form a ring. This ring is called the endomorphism ring of X, denoted by End(X); the set of all homomorphisms of X into itself. Addition of endomorphisms arises naturally in a pointwise manner and multiplication via endomorphism composition. Using these operations, the set of endomorphisms of an abelian group forms a (unital) ring, with the zero map as additive identity and the identity map as multiplicative identity.
The functions involved are restricted to what is defined as a homomorphism in the context, which depends upon the category of the object under consideration. The endomorphism ring consequently encodes several internal properties of the object. As the resulting object is often an algebra over some ring R, this may also be called the endomorphism algebra.
An abelian group is the same thing as a module over the ring of integers, which is the initial object in the category of rings. In a similar fashion, if R is any commutative ring, the endomorphisms of an R-module form an algebra over R by the same axioms and derivation. In particular, if R is a field, its modules M are vector spaces and the endomorphism ring of each is an algebra over the field R.
Description
Let be an abelian group and we consider the group homomorphisms from A into A. Then addition of two such homomorphisms may be defined pointwise to produce another group homomorphism. Explicitly, given two such homomorphisms f and g, the sum of f and g is the homomorphism . Under this operation End(A) is an abelian group. With the additional operation of composition of homomorphisms, End(A) is a ring with multiplicative identity. This composition is explicitly . The multiplicative identity is the identity homomorphism on A.
If the set A does not form an abelian group, then the above construction is not necessarily additive, as then the sum of two homomorphisms need not be a homomorphism. This set of endomorphisms is a canonical example of a near-ring that is not a ring.
Properties
Endomorphism rings always have additive and multiplicative identities, respectively the zero map and identity map.
Endomorphism rings are associative, but typically non-commutative.
If a module is simple, then its endomorphism ring is a division ring (this is sometimes called Schur's lemma).
A module is indecomposable if and only if its endomorphism ring does not contain any non-trivial idempotent elements. If the module is an injective module, then indecomposability is equivalent to the endomorphism ring being a local ring.
For a semisimple module, the endomorphism ring is a von Neumann regular ring.
The endomorphism ring of a nonzero right uniserial module has either one or two maximal right ideals. If the module is Artinian, Noetherian, projective or injective, then the endomorphism ring has a unique maximal ideal, so that it is a local ring.
The endomorphism ring of an Artinian uniform module is a local ring.
The endomorphi |
https://en.wikipedia.org/wiki/Diagonalization | In logic and mathematics, diagonalization may refer to:
Matrix diagonalization, a construction of a diagonal matrix (with nonzero entries only on the main diagonal) that is similar to a given matrix
Diagonal argument (disambiguation), various closely related proof techniques, including:
Cantor's diagonal argument, used to prove that the set of real numbers is not countable
Diagonal lemma, used to create self-referential sentences in formal logic
Table diagonalization, a form of data reduction used to make interpretation of tables and charts easier. |
https://en.wikipedia.org/wiki/Identity%20matrix | In linear algebra, the identity matrix of size is the square matrix with ones on the main diagonal and zeros elsewhere. It has unique properties, for example when the identity matrix represents a geometric transformation, the object remains unchanged by the transformation. In other contexts, it is analogous to multiplying by the number 1.
Terminology and notation
The identity matrix is often denoted by , or simply by if the size is immaterial or can be trivially determined by the context.
The term unit matrix has also been widely used, but the term identity matrix is now standard. The term unit matrix is ambiguous, because it is also used for a matrix of ones and for any unit of the ring of all matrices.
In some fields, such as group theory or quantum mechanics, the identity matrix is sometimes denoted by a boldface one, , or called "id" (short for identity). Less frequently, some mathematics books use or to represent the identity matrix, standing for "unit matrix" and the German word respectively.
In terms of a notation that is sometimes used to concisely describe diagonal matrices, the identity matrix can be written as
The identity matrix can also be written using the Kronecker delta notation:
Properties
When is an matrix, it is a property of matrix multiplication that
In particular, the identity matrix serves as the multiplicative identity of the matrix ring of all matrices, and as the identity element of the general linear group , which consists of all invertible matrices under the matrix multiplication operation. In particular, the identity matrix is invertible. It is an involutory matrix, equal to its own inverse. In this group, two square matrices have the identity matrix as their product exactly when they are the inverses of each other.
When matrices are used to represent linear transformations from an -dimensional vector space to itself, the identity matrix represents the identity function, for whatever basis was used in this representation.
The th column of an identity matrix is the unit vector , a vector whose th entry is 1 and 0 elsewhere. The determinant of the identity matrix is 1, and its trace is .
The identity matrix is the only idempotent matrix with non-zero determinant. That is, it is the only matrix such that:
When multiplied by itself, the result is itself
All of its rows and columns are linearly independent.
The principal square root of an identity matrix is itself, and this is its only positive-definite square root. However, every identity matrix with at least two rows and columns has an infinitude of symmetric square roots.
The rank of an identity matrix equals the size , i.e.:
See also
Binary matrix (zero-one matrix)
Elementary matrix
Exchange matrix
Matrix of ones
Pauli matrices (the identity matrix is the zeroth Pauli matrix)
Householder transformation (the Householder matrix is built through the identity matrix)
Square root of a 2 by 2 identity matrix
Unitary matrix
Zero matrix |
https://en.wikipedia.org/wiki/Hexagon | In geometry, a hexagon (from Greek , , meaning "six", and , , meaning "corner, angle") is a six-sided polygon. The total of the internal angles of any simple (non-self-intersecting) hexagon is 720°.
Regular hexagon
A regular hexagon has Schläfli symbol {6} and can also be constructed as a truncated equilateral triangle, t{3}, which alternates two types of edges.
A regular hexagon is defined as a hexagon that is both equilateral and equiangular. It is bicentric, meaning that it is both cyclic (has a circumscribed circle) and tangential (has an inscribed circle).
The common length of the sides equals the radius of the circumscribed circle or circumcircle, which equals times the apothem (radius of the inscribed circle). All internal angles are 120 degrees. A regular hexagon has six rotational symmetries (rotational symmetry of order six) and six reflection symmetries (six lines of symmetry), making up the dihedral group D6. The longest diagonals of a regular hexagon, connecting diametrically opposite vertices, are twice the length of one side. From this it can be seen that a triangle with a vertex at the center of the regular hexagon and sharing one side with the hexagon is equilateral, and that the regular hexagon can be partitioned into six equilateral triangles.
Like squares and equilateral triangles, regular hexagons fit together without any gaps to tile the plane (three hexagons meeting at every vertex), and so are useful for constructing tessellations. The cells of a beehive honeycomb are hexagonal for this reason and because the shape makes efficient use of space and building materials. The Voronoi diagram of a regular triangular lattice is the honeycomb tessellation of hexagons. It is not usually considered a triambus, although it is equilateral.
Parameters
The maximal diameter (which corresponds to the long diagonal of the hexagon), D, is twice the maximal radius or circumradius, R, which equals the side length, t. The minimal diameter or the diameter of the inscribed circle (separation of parallel sides, flat-to-flat distance, short diagonal or height when resting on a flat base), d, is twice the minimal radius or inradius, r. The maxima and minima are related by the same factor:
and, similarly,
The area of a regular hexagon
For any regular polygon, the area can also be expressed in terms of the apothem a and the perimeter p. For the regular hexagon these are given by a = r, and p, so
The regular hexagon fills the fraction of its circumscribed circle.
If a regular hexagon has successive vertices A, B, C, D, E, F and if P is any point on the circumcircle between B and C, then .
It follows from the ratio of circumradius to inradius that the height-to-width ratio of a regular hexagon is 1:1.1547005; that is, a hexagon with a long diagonal of 1.0000000 will have a distance of 0.8660254 between parallel sides.
Point in plane
For an arbitrary point in the plane of a regular hexagon with circumradius , whose distances to the c |
https://en.wikipedia.org/wiki/Free%20group | In mathematics, the free group FS over a given set S consists of all words that can be built from members of S, considering two words to be different unless their equality follows from the group axioms (e.g. st = suu−1t but s ≠ t−1 for s,t,u ∈ S). The members of S are called generators of FS, and the number of generators is the rank of the free group.
An arbitrary group G is called free if it is isomorphic to FS for some subset S of G, that is, if there is a subset S of G such that every element of G can be written in exactly one way as a product of finitely many elements of S and their inverses (disregarding trivial variations such as st = suu−1t).
A related but different notion is a free abelian group; both notions are particular instances of a free object from universal algebra. As such, free groups are defined by their universal property.
History
Free groups first arose in the study of hyperbolic geometry, as examples of Fuchsian groups (discrete groups acting by isometries on the hyperbolic plane). In an 1882 paper, Walther von Dyck pointed out that these groups have the simplest possible presentations. The algebraic study of free groups was initiated by Jakob Nielsen in 1924, who gave them their name and established many of their basic properties. Max Dehn realized the connection with topology, and obtained the first proof of the full Nielsen–Schreier theorem. Otto Schreier published an algebraic proof of this result in 1927, and Kurt Reidemeister included a comprehensive treatment of free groups in his 1932 book on combinatorial topology. Later on in the 1930s, Wilhelm Magnus discovered the connection between the lower central series of free groups and free Lie algebras.
Examples
The group (Z,+) of integers is free of rank 1; a generating set is S = {1}. The integers are also a free abelian group, although all free groups of rank are non-abelian. A free group on a two-element set S occurs in the proof of the Banach–Tarski paradox and is described there.
On the other hand, any nontrivial finite group cannot be free, since the elements of a free generating set of a free group have infinite order.
In algebraic topology, the fundamental group of a bouquet of k circles (a set of k loops having only one point in common) is the free group on a set of k elements.
Construction
The free group FS with free generating set S can be constructed as follows. S is a set of symbols, and we suppose for every s in S there is a corresponding "inverse" symbol, s−1, in a set S−1. Let T = S ∪ S−1, and define a word in S to be any written product of elements of T. That is, a word in S is an element of the monoid generated by T. The empty word is the word with no symbols at all. For example, if S = {a, b, c}, then T = {a, a−1, b, b−1, c, c−1}, and
is a word in S.
If an element of S lies immediately next to its inverse, the word may be simplified by omitting the c, c−1 pair:
A word that cannot be simplified further is called reduced.
The free group F |
https://en.wikipedia.org/wiki/Power%20series | In mathematics, a power series (in one variable) is an infinite series of the form
where an represents the coefficient of the nth term and c is a constant. Power series are useful in mathematical analysis, where they arise as Taylor series of infinitely differentiable functions. In fact, Borel's theorem implies that every power series is the Taylor series of some smooth function.
In many situations, c (the center of the series) is equal to zero, for instance when considering a Maclaurin series. In such cases, the power series takes the simpler form
Beyond their role in mathematical analysis, power series also occur in combinatorics as generating functions (a kind of formal power series) and in electronic engineering (under the name of the Z-transform). The familiar decimal notation for real numbers can also be viewed as an example of a power series, with integer coefficients, but with the argument x fixed at . In number theory, the concept of p-adic numbers is also closely related to that of a power series.
Examples
Polynomial
Any polynomial can be easily expressed as a power series around any center c, although all but finitely many of the coefficients will be zero since a power series has infinitely many terms by definition. For instance, the polynomial can be written as a power series around the center as
or around the center as
This is because of the Taylor series expansion of f(x) around is
as and the non-zero derivatives are , so and , a constant.
Or indeed the expansion is possible around any other center c. One can view power series as being like "polynomials of infinite degree," although power series are not polynomials.
Geometric series, exponential function and sine
The geometric series formula
which is valid for , is one of the most important examples of a power series, as are the exponential function formula
and the sine formula
valid for all real x.
These power series are also examples of Taylor series.
On the set of exponents
Negative powers are not permitted in a power series; for instance, is not considered a power series (although it is a Laurent series). Similarly, fractional powers such as are not permitted (but see Puiseux series). The coefficients are not allowed to depend on thus for instance:
is not a power series.
Radius of convergence
A power series is convergent for some values of the variable , which will always include (as usual, evaluates as and the sum of the series is thus for ). The series may diverge for other values of . If is not the only point of convergence, then there is always a number with such that the series converges whenever and diverges whenever . The number is called the radius of convergence of the power series; in general it is given as
or, equivalently,
(this is the Cauchy–Hadamard theorem; see limit superior and limit inferior for an explanation of the notation). The relation
is also satisfied, if this limit exists.
The set of the complex numbers |
https://en.wikipedia.org/wiki/Roman%20surface | In mathematics, the Roman surface or Steiner surface is a self-intersecting mapping of the real projective plane into three-dimensional space, with an unusually high degree of symmetry. This mapping is not an immersion of the projective plane; however, the figure resulting from removing six singular points is one. Its name arises because it was discovered by Jakob Steiner when he was in Rome in 1844.
The simplest construction is as the image of a sphere centered at the origin under the map This gives an implicit formula of
Also, taking a parametrization of the sphere in terms of longitude () and latitude (), gives parametric equations for the Roman surface as follows:
The origin is a triple point, and each of the -, -, and -planes are tangential to the surface there. The other places of self-intersection are double points, defining segments along each coordinate axis which terminate in six pinch points. The entire surface has tetrahedral symmetry. It is a particular type (called type 1) of Steiner surface, that is, a 3-dimensional linear projection of the Veronese surface.
Derivation of implicit formula
For simplicity we consider only the case r = 1. Given the sphere defined by the points (x, y, z) such that
we apply to these points the transformation T defined by say.
But then we have
and so as desired.
Conversely, suppose we are given (U, V, W) satisfying
(*)
We prove that there exists (x,y,z) such that
(**)
for which
with one exception: In case 3.b. below, we show this cannot be proved.
1. In the case where none of U, V, W is 0, we can set
(Note that (*) guarantees that either all three of U, V, W are positive, or else exactly two are negative. So these square roots are of positive numbers.)
It is easy to use (*) to confirm that (**) holds for x, y, z defined this way.
2. Suppose that W is 0. From (*) this implies
and hence at least one of U, V must be 0 also. This shows that is it impossible for exactly one of U, V, W to be 0.
3. Suppose that exactly two of U, V, W are 0. Without loss of generality we assume
(***)
It follows that
(since implies that and hence contradicting (***).)
a. In the subcase where
if we determine x and y by
and
this ensures that (*) holds. It is easy to verify that
and hence choosing the signs of x and y appropriately will guarantee
Since also
this shows that this subcase leads to the desired converse.
b. In this remaining subcase of the case 3., we have
Since
it is easy to check that
and thus in this case, where
there is no (x, y, z) satisfying
Hence the solutions (U, 0, 0) of the equation (*) with
and likewise, (0, V, 0) with
and (0, 0, W) with
(each of which is a noncompact portion of a coordinate axis, in two pieces) do not correspond to any point on the Roman surface.
4. If (U, V, W) is the point (0, 0, 0), then if any two of x, y, z are zero and the third one has absolute value 1, clearly as desired.
This covers all possible cases.
Derivation |
https://en.wikipedia.org/wiki/Formal%20power%20series | In mathematics, a formal series is an infinite sum that is considered independently from any notion of convergence, and can be manipulated with the usual algebraic operations on series (addition, subtraction, multiplication, division, partial sums, etc.).
A formal power series is a special kind of formal series, whose terms are of the form where is the th power of a variable ( is a non-negative integer), and is called the coefficient. Hence, power series can be viewed as a generalization of polynomials, where the number of terms is allowed to be infinite, with no requirements of convergence. Thus, the series may no longer represent a function of its variable, merely a formal sequence of coefficients, in contrast to a power series, which defines a function by taking numerical values for the variable within a radius of convergence. In a formal power series, the are used only as position-holders for the coefficients, so that the coefficient of is the fifth term in the sequence. In combinatorics, the method of generating functions uses formal power series to represent numerical sequences and multisets, for instance allowing concise expressions for recursively defined sequences regardless of whether the recursion can be explicitly solved. More generally, formal power series can include series with any finite (or countable) number of variables, and with coefficients in an arbitrary ring.
Rings of formal power series are complete local rings, and this allows using calculus-like methods in the purely algebraic framework of algebraic geometry and commutative algebra. They are analogous in many ways to -adic integers, which can be defined as formal series of the powers of .
Introduction
A formal power series can be loosely thought of as an object that is like a polynomial, but with infinitely many terms. Alternatively, for those familiar with power series (or Taylor series), one may think of a formal power series as a power series in which we ignore questions of convergence by not assuming that the variable X denotes any numerical value (not even an unknown value). For example, consider the series
If we studied this as a power series, its properties would include, for example, that its radius of convergence is 1. However, as a formal power series, we may ignore this completely; all that is relevant is the sequence of coefficients [1, −3, 5, −7, 9, −11, ...]. In other words, a formal power series is an object that just records a sequence of coefficients. It is perfectly acceptable to consider a formal power series with the factorials [1, 1, 2, 6, 24, 120, 720, 5040, ... ] as coefficients, even though the corresponding power series diverges for any nonzero value of X.
Arithmetic on formal power series is carried out by simply pretending that the series are polynomials. For example, if
then we add A and B term by term:
We can multiply formal power series, again just by treating them as polynomials (see in particular Cauchy product):
Notice that |
https://en.wikipedia.org/wiki/Topological%20ring | In mathematics, a topological ring is a ring that is also a topological space such that both the addition and the multiplication are continuous as maps:
where carries the product topology. That means is an additive topological group and a multiplicative topological semigroup.
Topological rings are fundamentally related to topological fields and arise naturally while studying them, since for example completion of a topological field may be a topological ring which is not a field.
General comments
The group of units of a topological ring is a topological group when endowed with the topology coming from the embedding of into the product as However, if the unit group is endowed with the subspace topology as a subspace of it may not be a topological group, because inversion on need not be continuous with respect to the subspace topology. An example of this situation is the adele ring of a global field; its unit group, called the idele group, is not a topological group in the subspace topology. If inversion on is continuous in the subspace topology of then these two topologies on are the same.
If one does not require a ring to have a unit, then one has to add the requirement of continuity of the additive inverse, or equivalently, to define the topological ring as a ring that is a topological group (for ) in which multiplication is continuous, too.
Examples
Topological rings occur in mathematical analysis, for example as rings of continuous real-valued functions on some topological space (where the topology is given by pointwise convergence), or as rings of continuous linear operators on some normed vector space; all Banach algebras are topological rings. The rational, real, complex and -adic numbers are also topological rings (even topological fields, see below) with their standard topologies. In the plane, split-complex numbers and dual numbers form alternative topological rings. See hypercomplex numbers for other low-dimensional examples.
In algebra, the following construction is common: one starts with a commutative ring containing an ideal and then considers the -adic topology on : a subset of is open if and only if for every there exists a natural number such that This turns into a topological ring. The -adic topology is Hausdorff if and only if the intersection of all powers of is the zero ideal
The -adic topology on the integers is an example of an -adic topology (with ).
Completion
Every topological ring is a topological group (with respect to addition) and hence a uniform space in a natural manner. One can thus ask whether a given topological ring is complete. If it is not, then it can be completed: one can find an essentially unique complete topological ring that contains as a dense subring such that the given topology on equals the subspace topology arising from
If the starting ring is metric, the ring can be constructed as a set of equivalence classes of Cauchy sequences in this equivalence relation |
https://en.wikipedia.org/wiki/I-adic%20topology | In commutative algebra, the mathematical study of commutative rings, adic topologies are a family of topologies on the underlying set of a module, generalizing the -adic topologies on the integers.
Definition
Let be a commutative ring and an -module. Then each ideal of determines a topology on called the -adic topology, characterized by the pseudometric The family is a basis for this topology.
Properties
With respect to the topology, the module operations of addition and scalar multiplication are continuous, so that becomes a topological module. However, need not be Hausdorff; it is Hausdorff if and only ifso that becomes a genuine metric. Related to the usual terminology in topology, where a Hausdorff space is also called separated, in that case, the -adic topology is called separated.
By Krull's intersection theorem, if is a Noetherian ring which is an integral domain or a local ring, it holds that for any proper ideal of . Thus under these conditions, for any proper ideal of and any -module , the -adic topology on is separated.
For a submodule of , the canonical homomorphism to induces a quotient topology which coincides with the -adic topology. The analogous result is not necessarily true for the submodule itself: the subspace topology need not be the -adic topology. However, the two topologies coincide when is Noetherian and finitely generated. This follows from the Artin-Rees lemma.
Completion
When is Hausdorff, can be completed as a metric space; the resulting space is denoted by and has the module structure obtained by extending the module operations by continuity. It is also the same as (or canonically isomorphic to): where the right-hand side is an inverse limit of quotient modules under natural projection.
For example, let be a polynomial ring over a field and the (unique) homogeneous maximal ideal. Then , the formal power series ring over in variables.
Closed submodules
As a consequence of the above, the -adic closure of a submodule is This closure coincides with whenever is -adically complete and is finitely generated.
is called Zariski with respect to if every ideal in is -adically closed. There is a characterization:
is Zariski with respect to if and only if is contained in the Jacobson radical of .
In particular a Noetherian local ring is Zariski with respect to the maximal ideal.
References
Sources
Commutative algebra
Topology |
https://en.wikipedia.org/wiki/ARITH-MATIC | You may have been looking for arithmetic, a branch of mathematics.
ARITH-MATIC is an extension of Grace Hopper's A-2 programming language, developed around 1955. ARITH-MATIC was originally known as A-3, but was renamed by the marketing department of Remington Rand UNIVAC.
Some ARITH-MATIC subroutines
See also
A-0 System
References
External links
Website at Boise via Internet Archive
Numerical programming languages |
https://en.wikipedia.org/wiki/ABC%20ALGOL | ABC ALGOL is an extension of the programming language ALGOL 60 with arbitrary data structures and user-defined operators, intended for computer algebra (symbolic mathematics). Despite its advances, it was never used as widely as Algol proper.
References
External links
ALGOL 60 dialect |
https://en.wikipedia.org/wiki/Unique%20factorization%20domain | In mathematics, a unique factorization domain (UFD) (also sometimes called a factorial ring following the terminology of Bourbaki) is a ring in which a statement analogous to the fundamental theorem of arithmetic holds. Specifically, a UFD is an integral domain (a nontrivial commutative ring in which the product of any two non-zero elements is non-zero) in which every non-zero non-unit element can be written as a product of irreducible elements, uniquely up to order and units.
Important examples of UFDs are the integers and polynomial rings in one or more variables with coefficients coming from the integers or from a field.
Unique factorization domains appear in the following chain of class inclusions:
Definition
Formally, a unique factorization domain is defined to be an integral domain R in which every non-zero element x of R can be written as a product (an empty product if x is a unit) of irreducible elements pi of R and a unit u:
x = u p1 p2 ⋅⋅⋅ pn with n ≥ 0
and this representation is unique in the following sense:
If q1, ..., qm are irreducible elements of R and w is a unit such that
x = w q1 q2 ⋅⋅⋅ qm with m ≥ 0,
then m = n, and there exists a bijective map φ : {1, ..., n} → {1, ..., m} such that pi is associated to qφ(i) for i ∈ {1, ..., n}.
Examples
Most rings familiar from elementary mathematics are UFDs:
All principal ideal domains, hence all Euclidean domains, are UFDs. In particular, the integers (also see fundamental theorem of arithmetic), the Gaussian integers and the Eisenstein integers are UFDs.
If R is a UFD, then so is R[X], the ring of polynomials with coefficients in R. Unless R is a field, R[X] is not a principal ideal domain. By induction, a polynomial ring in any number of variables over any UFD (and in particular over a field or over the integers) is a UFD.
The formal power series ring K[[X1,...,Xn]] over a field K (or more generally over a regular UFD such as a PID) is a UFD. On the other hand, the formal power series ring over a UFD need not be a UFD, even if the UFD is local. For example, if R is the localization of k[x,y,z]/(x2 + y3 + z7) at the prime ideal (x,y,z) then R is a local ring that is a UFD, but the formal power series ring R[[X]] over R is not a UFD.
The Auslander–Buchsbaum theorem states that every regular local ring is a UFD.
is a UFD for all integers 1 ≤ n ≤ 22, but not for n = 23.
Mori showed that if the completion of a Zariski ring, such as a Noetherian local ring, is a UFD, then the ring is a UFD. The converse of this is not true: there are Noetherian local rings that are UFDs but whose completions are not. The question of when this happens is rather subtle: for example, for the localization of k[x,y,z]/(x2 + y3 + z5) at the prime ideal (x,y,z), both the local ring and its completion are UFDs, but in the apparently similar example of the localization of k[x,y,z]/(x2 + y3 + z7) at the prime ideal (x,y,z) the local ring is a UFD but its completion is not.
Let be a field of any charac |
https://en.wikipedia.org/wiki/UFD | UFD may refer to:
Union of the Democratic Forces (France), a defunct electoral coalition ()
Unique factorization domain, in abstract algebra
United Front Department, a North Korean government body
Ural Federal District, Russia
USB flash drive, in computing
Ultra faint dwarf, a type of dwarf spheroidal galaxy |
https://en.wikipedia.org/wiki/Prime%20element | In mathematics, specifically in abstract algebra, a prime element of a commutative ring is an object satisfying certain properties similar to the prime numbers in the integers and to irreducible polynomials. Care should be taken to distinguish prime elements from irreducible elements, a concept which is the same in UFDs but not the same in general.
Definition
An element of a commutative ring is said to be prime if it is not the zero element or a unit and whenever divides for some and in , then divides or divides . With this definition, Euclid's lemma is the assertion that prime numbers are prime elements in the ring of integers. Equivalently, an element is prime if, and only if, the principal ideal generated by is a nonzero prime ideal. (Note that in an integral domain, the ideal is a prime ideal, but is an exception in the definition of 'prime element'.)
Interest in prime elements comes from the fundamental theorem of arithmetic, which asserts that each nonzero integer can be written in essentially only one way as 1 or −1 multiplied by a product of positive prime numbers. This led to the study of unique factorization domains, which generalize what was just illustrated in the integers.
Being prime is relative to which ring an element is considered to be in; for example, 2 is a prime element in but it is not in , the ring of Gaussian integers, since and 2 does not divide any factor on the right.
Connection with prime ideals
An ideal in the ring (with unity) is prime if the factor ring is an integral domain.
In an integral domain, a nonzero principal ideal is prime if and only if it is generated by a prime element.
Irreducible elements
Prime elements should not be confused with irreducible elements. In an integral domain, every prime is irreducible but the converse is not true in general. However, in unique factorization domains, or more generally in GCD domains, primes and irreducibles are the same.
Examples
The following are examples of prime elements in rings:
The integers , , , , , ... in the ring of integers
the complex numbers , , and in the ring of Gaussian integers
the polynomials and in , the ring of polynomials over .
2 in the quotient ring
is prime but not irreducible in the ring
In the ring of pairs of integers, is prime but not irreducible (one has ).
In the ring of algebraic integers the element is irreducible but not prime (as 3 divides and 3 does not divide any factor on the right).
References
Notes
Sources
Section III.3 of
Ring theory |
https://en.wikipedia.org/wiki/Irreducible%20element | In algebra, an irreducible element of an integral domain is a non-zero element that is not invertible (that is, is not a unit), and is not the product of two non-invertible elements.
The irreducible elements are the terminal elements of a factorization process; that is, they are the factors that cannot be further factorized. The irreducible factors of an element are uniquely defined, up to the multiplication by a unit, if the integral domain is a unique factorization domain. It was discovered in the 19th century that the rings of integers of some number fields are not unique factorization domains, and, therefore, that some irreducible elements can appear in some factorization of an element and not in other factorizations of the same element. The ignorance of this fact is the main error in many of the wrong proofs of Fermat's Last Theorem that were given during the three centuries between Fermat's statement and Wiles's proof of Fermat's Last Theorem.
The definition can be, and usually is, extended verbatim to the elements of an arbitrary commutative ring. For a general ring , an element of is called irreducible if it is neither left-invertible nor right-invertible, and if there exists no left-invertible element together with a right-invertible element such that .
If is an integral domain, then is an irreducible element of if and only if for all , the equation implies that the ideal generated by is equal to the ideal generated by or equal to the ideal generated by . This equivalence does not hold for general commutative rings, which is why the assumption of the ring having no zero divisors is commonly made in the definition of irreducible elements.
Relationship with prime elements
Irreducible elements should not be confused with prime elements. (A non-zero non-unit element in a commutative ring is called prime if, whenever for some and in then or ) In an integral domain, every prime element is irreducible, but the converse is not true in general. The converse is true for unique factorization domains (or, more generally, GCD domains).
Moreover, while an ideal generated by a prime element is a prime ideal, it is not true in general that an ideal generated by an irreducible element is an irreducible ideal. However, if is a GCD domain and is an irreducible element of , then as noted above is prime, and so the ideal generated by is a prime (hence irreducible) ideal of .
Example
In the quadratic integer ring it can be shown using norm arguments that the number 3 is irreducible. However, it is not a prime element in this ring since, for example,
but 3 does not divide either of the two factors.
See also
Irreducible polynomial
References
Ring theory
Algebraic properties of elements |
https://en.wikipedia.org/wiki/Pseudosphere | In geometry, a pseudosphere is a surface with constant negative Gaussian curvature.
A pseudosphere of radius is a surface in having curvature in each point. Its name comes from the analogy with the sphere of radius , which is a surface of curvature . The term was introduced by Eugenio Beltrami in his 1868 paper on models of hyperbolic geometry.
Tractroid
The same surface can be also described as the result of revolving a tractrix about its asymptote.
For this reason the pseudosphere is also called tractroid. As an example, the (half) pseudosphere (with radius 1) is the surface of revolution of the tractrix parametrized by
It is a singular space (the equator is a singularity), but away from the singularities, it has constant negative Gaussian curvature and therefore is locally isometric to a hyperbolic plane.
The name "pseudosphere" comes about because it has a two-dimensional surface of constant negative Gaussian curvature, just as a sphere has a surface with constant positive Gaussian curvature.
Just as the sphere has at every point a positively curved geometry of a dome the whole pseudosphere has at every point the negatively curved geometry of a saddle.
As early as 1693 Christiaan Huygens found that the volume and the surface area of the pseudosphere are finite, despite the infinite extent of the shape along the axis of rotation. For a given edge radius , the area is just as it is for the sphere, while the volume is and therefore half that of a sphere of that radius.
Universal covering space
The half pseudosphere of curvature −1 is covered by the interior of a horocycle. In the Poincaré half-plane model one convenient choice is the portion of the half-plane with . Then the covering map is periodic in the direction of period 2, and takes the horocycles to the meridians of the pseudosphere and the vertical geodesics to the tractrices that generate the pseudosphere. This mapping is a local isometry, and thus exhibits the portion of the upper half-plane as the universal covering space of the pseudosphere. The precise mapping is
where
is the parametrization of the tractrix above.
Hyperboloid
In some sources that use the hyperboloid model of the hyperbolic plane, the hyperboloid is referred to as a pseudosphere.
This usage of the word is because the hyperboloid can be thought of as a sphere of imaginary radius, embedded in a Minkowski space.
Pseudospherical surfaces
A pseudospherical surface is a generalization of the pseudosphere. A surface which is piecewise smoothly immersed in with constant negative curvature is a pseudospherical surface. The tractroid is the simplest example. Other examples include the Dini's surfaces, breather surfaces, and the Kuen surface.
Relation to solutions to the Sine-Gordon equation
Pseudospherical surfaces can be constructed from solutions to the Sine-Gordon equation. A sketch proof starts with reparametrizing the tractroid with coordinates in which the Gauss–Codazzi equations can be |
https://en.wikipedia.org/wiki/Curvature | In mathematics, curvature is any of several strongly related concepts in geometry. Intuitively, the curvature is the amount by which a curve deviates from being a straight line, or a surface deviates from being a plane.
For curves, the canonical example is that of a circle, which has a curvature equal to the reciprocal of its radius. Smaller circles bend more sharply, and hence have higher curvature. The curvature at a point of a differentiable curve is the curvature of its osculating circle, that is the circle that best approximates the curve near this point. The curvature of a straight line is zero. In contrast to the tangent, which is a vector quantity, the curvature at a point is typically a scalar quantity, that is, it is expressed by a single real number.
For surfaces (and, more generally for higher-dimensional manifolds), that are embedded in a Euclidean space, the concept of curvature is more complex, as it depends on the choice of a direction on the surface or manifold. This leads to the concepts of maximal curvature, minimal curvature, and mean curvature.
For Riemannian manifolds (of dimension at least two) that are not necessarily embedded in a Euclidean space, one can define the curvature intrinsically, that is without referring to an external space. See Curvature of Riemannian manifolds for the definition, which is done in terms of lengths of curves traced on the manifold, and expressed, using linear algebra, by the Riemann curvature tensor.
History
In Tractatus de configurationibus qualitatum et motuum, the 14th-century philosopher and mathematician Nicole Oresme introduces the concept of curvature as a measure of departure from straightness; for circles he has the curvature as being inversely proportional to the radius; and he attempts to extend this idea to other curves as a continuously varying magnitude.
The curvature of a differentiable curve was originally defined through osculating circles. In this setting, Augustin-Louis Cauchy showed that the center of curvature is the intersection point of two infinitely close normal lines to the curve.
Plane curves
Intuitively, the curvature describes for any part of a curve how much the curve direction changes over a small distance travelled (e.g. angle in ), so it is a measure of the instantaneous rate of change of direction of a point that moves on the curve: the larger the curvature, the larger this rate of change. In other words, the curvature measures how fast the unit tangent vector to the curve rotates (fast in terms of curve position). In fact, it can be proved that this instantaneous rate of change is exactly the curvature. More precisely, suppose that the point is moving on the curve at a constant speed of one unit, that is, the position of the point is a function of the parameter , which may be thought as the time or as the arc length from a given origin. Let be a unit tangent vector of the curve at , which is also the derivative of with respect to . Then, the deriv |
https://en.wikipedia.org/wiki/Additive%20category | In mathematics, specifically in category theory, an additive category is a preadditive category C admitting all finitary biproducts.
Definition
There are two equivalent definitions of an additive category: One as a category equipped with additional structure, and another as a category equipped with no extra structure but whose objects and morphisms satisfy certain equations.
Via preadditive categories
A category C is preadditive if all its hom-sets are abelian groups and composition of morphisms is bilinear; in other words, C is enriched over the monoidal category of abelian groups.
In a preadditive category, every finitary product (including the empty product, i.e., a final object) is necessarily a coproduct (or initial object in the case of an empty diagram), and hence a biproduct, and conversely every finitary coproduct is necessarily a product (this is a consequence of the definition, not a part of it).
Thus an additive category is equivalently described as a preadditive category admitting all finitary products, or a preadditive category admitting all finitary coproducts.
Via semiadditive categories
We give an alternative definition.
Define a semiadditive category to be a category (note: not a preadditive category) which admits a zero object and all binary biproducts. It is then a remarkable theorem that the Hom sets naturally admit an abelian monoid structure. A proof of this fact is given below.
An additive category may then be defined as a semiadditive category in which every morphism has an additive inverse. This then gives the Hom sets an abelian group structure instead of merely an abelian monoid structure.
Generalization
More generally, one also considers additive -linear categories for a commutative ring . These are categories enriched over the monoidal category of -modules and admitting all finitary biproducts.
Examples
The original example of an additive category is the category of abelian groups Ab. The zero object is the trivial group, the addition of morphisms is given pointwise, and biproducts are given by direct sums.
More generally, every module category over a ring is additive, and so in particular, the category of vector spaces over a field is additive.
The algebra of matrices over a ring, thought of as a category as described below, is also additive.
Internal characterisation of the addition law
Let C be a semiadditive category, so a category having all finitary biproducts. Then every hom-set has an addition, endowing it with the structure of an abelian monoid, and such that the composition of morphisms is bilinear.
Moreover, if C is additive, then the two additions on hom-sets must agree. In particular, a semiadditive category is additive if and only if every morphism has an additive inverse.
This shows that the addition law for an additive category is internal to that category.
To define the addition law, we will use the convention that for a biproduct, pk will denote the projection morphisms, an |
https://en.wikipedia.org/wiki/Markov%20chain | A Markov chain or Markov process is a stochastic model describing a sequence of possible events in which the probability of each event depends only on the state attained in the previous event. Informally, this may be thought of as, "What happens next depends only on the state of affairs now." A countably infinite sequence, in which the chain moves state at discrete time steps, gives a discrete-time Markov chain (DTMC). A continuous-time process is called a continuous-time Markov chain (CTMC). It is named after the Russian mathematician Andrey Markov.
Markov chains have many applications as statistical models of real-world processes, such as studying cruise control systems in motor vehicles, queues or lines of customers arriving at an airport, currency exchange rates and animal population dynamics.
Markov processes are the basis for general stochastic simulation methods known as Markov chain Monte Carlo, which are used for simulating sampling from complex probability distributions, and have found application in Bayesian statistics, thermodynamics, statistical mechanics, physics, chemistry, economics, finance, signal processing, information theory and speech processing.
The adjectives Markovian and Markov are used to describe something that is related to a Markov process.
Principles
Definition
A Markov process is a stochastic process that satisfies the Markov property (sometimes characterized as "memorylessness"). In simpler terms, it is a process for which predictions can be made regarding future outcomes based solely on its present state and—most importantly—such predictions are just as good as the ones that could be made knowing the process's full history. In other words, conditional on the present state of the system, its future and past states are independent.
A Markov chain is a type of Markov process that has either a discrete state space or a discrete index set (often representing time), but the precise definition of a Markov chain varies. For example, it is common to define a Markov chain as a Markov process in either discrete or continuous time with a countable state space (thus regardless of the nature of time), but it is also common to define a Markov chain as having discrete time in either countable or continuous state space (thus regardless of the state space).
Types of Markov chains
The system's state space and time parameter index need to be specified. The following table gives an overview of the different instances of Markov processes for different levels of state space generality and for discrete time v. continuous time:
Note that there is no definitive agreement in the literature on the use of some of the terms that signify special cases of Markov processes. Usually the term "Markov chain" is reserved for a process with a discrete set of times, that is, a discrete-time Markov chain (DTMC), but a few authors use the term "Markov process" to refer to a continuous-time Markov chain (CTMC) without explicit mention. In addit |
https://en.wikipedia.org/wiki/Cauchy%27s%20integral%20theorem | In mathematics, the Cauchy integral theorem (also known as the Cauchy–Goursat theorem) in complex analysis, named after Augustin-Louis Cauchy (and Édouard Goursat), is an important statement about line integrals for holomorphic functions in the complex plane. Essentially, it says that if is holomorphic in a simply connected domain Ω, then for any simply closed contour in Ω, that contour integral is zero.
Statement
Fundamental theorem for complex line integrals
If is a holomorphic function on an open region , and is a curve in from to then,
Also, when has a single-valued antiderivative in an open region , then the path integral is path independent for all paths in .
Formulation on simply connected regions
Let be a simply connected open set, and let be a holomorphic function. Let be a smooth closed curve. Then:
(The condition that be simply connected means that has no "holes", or in other words, that the fundamental group of is trivial.)
General formulation
Let be an open set, and let be a holomorphic function. Let be a smooth closed curve. If is homotopic to a constant curve, then:
(Recall that a curve is homotopic to a constant curve if there exists a smooth homotopy (within ) from the curve to the constant curve. Intuitively, this means that one can shrink the curve into a point without exiting the space.) The first version is a special case of this because on a simply connected set, every closed curve is homotopic to a constant curve.
Main example
In both cases, it is important to remember that the curve does not surround any "holes" in the domain, or else the theorem does not apply. A famous example is the following curve:
which traces out the unit circle. Here the following integral:
is nonzero. The Cauchy integral theorem does not apply here since is not defined at . Intuitively, surrounds a "hole" in the domain of , so cannot be shrunk to a point without exiting the space. Thus, the theorem does not apply.
Discussion
As Édouard Goursat showed, Cauchy's integral theorem can be proven assuming only that the complex derivative exists everywhere in . This is significant because one can then prove Cauchy's integral formula for these functions, and from that deduce these functions are infinitely differentiable.
The condition that be simply connected means that has no "holes" or, in homotopy terms, that the fundamental group of is trivial; for instance, every open disk , for , qualifies. The condition is crucial; consider
which traces out the unit circle, and then the path integral
is nonzero; the Cauchy integral theorem does not apply here since is not defined (and is certainly not holomorphic) at .
One important consequence of the theorem is that path integrals of holomorphic functions on simply connected domains can be computed in a manner familiar from the fundamental theorem of calculus: let be a simply connected open subset of , let be a holomorphic function, and let be a piecewise conti |
https://en.wikipedia.org/wiki/Laurent%20series | In mathematics, the Laurent series of a complex function is a representation of that function as a power series which includes terms of negative degree. It may be used to express complex functions in cases where a Taylor series expansion cannot be applied. The Laurent series was named after and first published by Pierre Alphonse Laurent in 1843. Karl Weierstrass may have discovered it first in a paper written in 1841, but it was not published until after his death.
Definition
The Laurent series for a complex function about a point is given by
where and are constants, with defined by a contour integral that generalizes Cauchy's integral formula:
The path of integration is counterclockwise around a Jordan curve enclosing and lying in an annulus in which is holomorphic (analytic). The expansion for will then be valid anywhere inside the annulus. The annulus is shown in red in the figure on the right, along with an example of a suitable path of integration labeled . If we take to be a circle , where , this just amounts
to computing the complex Fourier coefficients of the restriction of to . The fact that these integrals are unchanged by a deformation of the contour is an immediate consequence of Green's theorem.
One may also obtain the Laurent series for a complex function at . However, this is the same as when (see the example below).
In practice, the above integral formula may not offer the most practical method for computing the coefficients
for a given function ; instead, one often pieces together the Laurent series by combining known Taylor expansions. Because the Laurent expansion of a function is unique whenever
it exists, any expression of this form that equals the given function
in some annulus must actually be the Laurent expansion of .
Convergent Laurent series
Laurent series with complex coefficients are an important tool in complex analysis, especially to investigate the behavior of functions near singularities.
Consider for instance the function with . As a real function, it is infinitely differentiable everywhere; as a complex function however it is not differentiable at . By replacing with in the power series for the exponential function, we obtain its Laurent series which converges and is equal to for all complex numbers except at the singularity . The graph opposite shows in black and its Laurent approximations
for = 1, 2, 3, 4, 5, 6, 7 and 50. As , the approximation becomes exact for all (complex) numbers except at the singularity .
More generally, Laurent series can be used to express holomorphic functions defined on an annulus, much as power series are used to express holomorphic functions defined on a disc.
Suppose
is a given Laurent series with complex coefficients and a complex center . Then there exists a unique inner radius and outer radius such that:
The Laurent series converges on the open annulus . To say that the Laurent series converges, we mean that both the positive degr |
https://en.wikipedia.org/wiki/Straightedge%20and%20compass%20construction | In geometry, straightedge-and-compass construction – also known as ruler-and-compass construction, Euclidean construction, or classical construction – is the construction of lengths, angles, and other geometric figures using only an idealized ruler and a pair of compasses.
The idealized ruler, known as a straightedge, is assumed to be infinite in length, have only one edge, and no markings on it. The compass is assumed to have no maximum or minimum radius, and is assumed to "collapse" when lifted from the page, so may not be directly used to transfer distances. (This is an unimportant restriction since, using a multi-step procedure, a distance can be transferred even with a collapsing compass; see compass equivalence theorem. Note however that whilst a non-collapsing compass held against a straightedge might seem to be equivalent to marking it, the neusis construction is still impermissible and this is what unmarked really means: see Markable rulers below.) More formally, the only permissible constructions are those granted by the first three postulates of Euclid's Elements.
It turns out to be the case that every point constructible using straightedge and compass may also be constructed using compass alone, or by straightedge alone if given a single circle and its center.
Ancient Greek mathematicians first conceived straightedge-and-compass constructions, and a number of ancient problems in plane geometry impose this restriction. The ancient Greeks developed many constructions, but in some cases were unable to do so. Gauss showed that some polygons are constructible but that most are not. Some of the most famous straightedge-and-compass problems were proved impossible by Pierre Wantzel in 1837 using field theory, namely trisecting an arbitrary angle and doubling the volume of a cube (see § impossible constructions). Many of these problems are easily solvable provided that other geometric transformations are allowed; for example, neusis construction can be used to solve the former two problems.
In terms of algebra, a length is constructible if and only if it represents a constructible number, and an angle is constructible if and only if its cosine is a constructible number. A number is constructible if and only if it can be written using the four basic arithmetic operations and the extraction of square roots but of no higher-order roots.
Straightedge and compass tools
The "straightedge" and "compass" of straightedge-and-compass constructions are idealized versions of real-world rulers and compasses.
The straightedge is an infinitely long edge with no markings on it. It can only be used to draw a line segment between two points, or to extend an existing line segment.
The compass can have an arbitrarily large radius with no markings on it (unlike certain real-world compasses). Circles and circular arcs can be drawn starting from two given points: the centre and a point on the circle. The compass may or may not collapse (i.e. fold after being |
https://en.wikipedia.org/wiki/Galois%20theory | In mathematics, Galois theory, originally introduced by Évariste Galois, provides a connection between field theory and group theory. This connection, the fundamental theorem of Galois theory, allows reducing certain problems in field theory to group theory, which makes them simpler and easier to understand.
Galois introduced the subject for studying roots of polynomials. This allowed him to characterize the polynomial equations that are solvable by radicals in terms of properties of the permutation group of their roots—an equation is solvable by radicals if its roots may be expressed by a formula involving only integers, th roots, and the four basic arithmetic operations. This widely generalizes the Abel–Ruffini theorem, which asserts that a general polynomial of degree at least five cannot be solved by radicals.
Galois theory has been used to solve classic problems including showing that two problems of antiquity cannot be solved as they were stated (doubling the cube and trisecting the angle), and characterizing the regular polygons that are constructible (this characterization was previously given by Gauss, but all known proofs that this characterization is complete require Galois theory).
Galois' work was published by Joseph Liouville fourteen years after his death. The theory took longer to become popular among mathematicians and to be well understood.
Galois theory has been generalized to Galois connections and Grothendieck's Galois theory.
Application to classical problems
The birth and development of Galois theory was caused by the following question, which was one of the main open mathematical questions until the beginning of 19th century:
The Abel–Ruffini theorem provides a counterexample proving that there are polynomial equations for which such a formula cannot exist. Galois' theory provides a much more complete answer to this question, by explaining why it is possible to solve some equations, including all those of degree four or lower, in the above manner, and why it is not possible for most equations of degree five or higher. Furthermore, it provides a means of determining whether a particular equation can be solved that is both conceptually clear and easily expressed as an algorithm.
Galois' theory also gives a clear insight into questions concerning problems in compass and straightedge construction. It gives an elegant characterization of the ratios of lengths that can be constructed with this method. Using this, it becomes relatively easy to answer such classical problems of geometry as
Which regular polygons are constructible?
Why is it not possible to trisect every angle using a compass and a straightedge?
Why is doubling the cube not possible with the same method?
History
Pre-history
Galois' theory originated in the study of symmetric functions – the coefficients of a monic polynomial are (up to sign) the elementary symmetric polynomials in the roots. For instance, , where 1, and are the elementary polynomi |
https://en.wikipedia.org/wiki/Commutative%20ring | In mathematics, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring properties that are not specific to commutative rings. This distinction results from the high number of fundamental properties of commutative rings that do not extend to noncommutative rings.
Definition and first examples
Definition
A ring is a set equipped with two binary operations, i.e. operations combining any two elements of the ring to a third. They are called addition and multiplication and commonly denoted by "" and ""; e.g. and . To form a ring these two operations have to satisfy a number of properties: the ring has to be an abelian group under addition as well as a monoid under multiplication, where multiplication distributes over addition; i.e., . The identity elements for addition and multiplication are denoted and , respectively.
If the multiplication is commutative, i.e.
then the ring is called commutative. In the remainder of this article, all rings will be commutative, unless explicitly stated otherwise.
First examples
An important example, and in some sense crucial, is the ring of integers with the two operations of addition and multiplication. As the multiplication of integers is a commutative operation, this is a commutative ring. It is usually denoted as an abbreviation of the German word Zahlen (numbers).
A field is a commutative ring where and every non-zero element is invertible; i.e., has a multiplicative inverse such that . Therefore, by definition, any field is a commutative ring. The rational, real and complex numbers form fields.
If is a given commutative ring, then the set of all polynomials in the variable whose coefficients are in forms the polynomial ring, denoted . The same holds true for several variables.
If is some topological space, for example a subset of some , real- or complex-valued continuous functions on form a commutative ring. The same is true for differentiable or holomorphic functions, when the two concepts are defined, such as for a complex manifold.
Divisibility
In contrast to fields, where every nonzero element is multiplicatively invertible, the concept of divisibility for rings is richer. An element of ring is called a unit if it possesses a multiplicative inverse. Another particular type of element is the zero divisors, i.e. an element such that there exists a non-zero element of the ring such that . If possesses no non-zero zero divisors, it is called an integral domain (or domain). An element satisfying for some positive integer is called nilpotent.
Localizations
The localization of a ring is a process in which some elements are rendered invertible, i.e. multiplicative inverses are added to the ring. Concretely, if is a multiplicatively closed subset of (i.e. whenever then so is ) then the localization of at , or ring of fractions |
https://en.wikipedia.org/wiki/Laurent%20polynomial | In mathematics, a Laurent polynomial (named
after Pierre Alphonse Laurent) in one variable over a field is a linear combination of positive and negative powers of the variable with coefficients in . Laurent polynomials in X form a ring denoted . They differ from ordinary polynomials in that they may have terms of negative degree. The construction of Laurent polynomials may be iterated, leading to the ring of Laurent polynomials in several variables. Laurent polynomials are of particular importance in the study of complex variables.
Definition
A Laurent polynomial with coefficients in a field is an expression of the form
where X is a formal variable, the summation index k is an integer (not necessarily positive) and only finitely many coefficients pk are non-zero. Two Laurent polynomials are equal if their coefficients are equal. Such expressions can be added, multiplied, and brought back to the same form by reducing similar terms. Formulas for addition and multiplication are exactly the same as for the ordinary polynomials, with the only difference that both positive and negative powers of X can be present:
and
Since only finitely many coefficients ai and bj are non-zero, all sums in effect have only finitely many terms, and hence represent Laurent polynomials.
Properties
A Laurent polynomial over C may be viewed as a Laurent series in which only finitely many coefficients are non-zero.
The ring of Laurent polynomials R[X, X −1] is an extension of the polynomial ring R[X ] obtained by "inverting X ". More rigorously, it is the localization of the polynomial ring in the multiplicative set consisting of the non-negative powers of X. Many properties of the Laurent polynomial ring follow from the general properties of localization.
The ring of Laurent polynomials is a subring of the rational functions.
The ring of Laurent polynomials over a field is Noetherian (but not Artinian).
If R is an integral domain, the units of the Laurent polynomial ring R[X, X −1] have the form uX k, where u is a unit of R and k is an integer. In particular, if K is a field then the units of K[X, X −1] have the form aX k, where a is a non-zero element of K.
The Laurent polynomial ring R[X, X −1] is isomorphic to the group ring of the group Z of integers over R. More generally, the Laurent polynomial ring in n variables is isomorphic to the group ring of the free abelian group of rank n. It follows that the Laurent polynomial ring can be endowed with a structure of a commutative, cocommutative Hopf algebra.
See also
Jones polynomial
References
Lang, Serge (2002), Algebra, Graduate Texts in Mathematics, 211 (Revised third ed.), New York: Springer-Verlag, , MR 1878556
Commutative algebra
Polynomials
Ring theory |
https://en.wikipedia.org/wiki/Boy%27s%20surface | In geometry, Boy's surface is an immersion of the real projective plane in 3-dimensional space found by Werner Boy in 1901. He discovered it on assignment from David Hilbert to prove that the projective plane could not be immersed in 3-space.
Boy's surface was first parametrized explicitly by Bernard Morin in 1978. Another parametrization was discovered by Rob Kusner and Robert Bryant. Boy's surface is one of the two possible immersions of the real projective plane which have only a single triple point.
Unlike the Roman surface and the cross-cap, it has no other singularities than self-intersections (that is, it has no pinch-points).
Parametrization
Boy's surface can be parametrized in several ways. One parametrization, discovered by Rob Kusner and Robert Bryant, is the following: given a complex number w whose magnitude is less than or equal to one (), let
and then set
we then obtain the Cartesian coordinates x, y, and z of a point on the Boy's surface.
If one performs an inversion of this parametrization centered on the triple point, one obtains a complete minimal surface with three ends (that's how this parametrization was discovered naturally). This implies that the Bryant–Kusner parametrization of Boy's surfaces is "optimal" in the sense that it is the "least bent" immersion of a projective plane into three-space.
Property of Bryant–Kusner parametrization
If w is replaced by the negative reciprocal of its complex conjugate, then the functions g1, g2, and g3 of w are left unchanged.
By replacing in terms of its real and imaginary parts , and expanding resulting parameterization, one may obtain a parameterization of Boy's surface in terms of rational functions of and . This shows that Boy's surface is not only an algebraic surface, but even a rational surface. The remark of the preceding paragraph shows that the generic fiber of this parameterization consists of two points (that is that almost every point of Boy's surface may be obtained by two parameters values).
Relation to the real projective plane
Let be the Bryant–Kusner parametrization of Boy's surface. Then
This explains the condition on the parameter: if then However, things are slightly more complicated for In this case, one has This means that, if the point of the Boy's surface is obtained from two parameter values: In other words, the Boy's surface has been parametrized by a disk such that pairs of diametrically opposite points on the perimeter of the disk are equivalent. This shows that the Boy's surface is the image of the real projective plane, RP2 by a smooth map. That is, the parametrization of the Boy's surface is an immersion of the real projective plane into the Euclidean space.
Symmetries
Boy's surface has 3-fold symmetry. This means that it has an axis of discrete rotational symmetry: any 120° turn about this axis will leave the surface looking exactly the same. The Boy's surface can be cut into three mutually congruent pieces.
Application |
https://en.wikipedia.org/wiki/Radius%20of%20convergence | In mathematics, the radius of convergence of a power series is the radius of the largest disk at the center of the series in which the series converges. It is either a non-negative real number or . When it is positive, the power series converges absolutely and uniformly on compact sets inside the open disk of radius equal to the radius of convergence, and it is the Taylor series of the analytic function to which it converges. In case of multiple singularities of a function (singularities are those values of the argument for which the function is not defined), the radius of convergence is the shortest or minimum of all the respective distances (which are all non-negative numbers) calculated from the center of the disk of convergence to the respective singularities of the function.
Definition
For a power series f defined as:
where
a is a complex constant, the center of the disk of convergence,
cn is the n-th complex coefficient, and
z is a complex variable.
The radius of convergence r is a nonnegative real number or such that the series converges if
and diverges if
Some may prefer an alternative definition, as existence is obvious:
On the boundary, that is, where |z − a| = r, the behavior of the power series may be complicated, and the series may converge for some values of z and diverge for others. The radius of convergence is infinite if the series converges for all complex numbers z.
Finding the radius of convergence
Two cases arise. The first case is theoretical: when you know all the coefficients then you take certain limits and find the precise radius of convergence. The second case is practical: when you construct a power series solution of a difficult problem you typically will only know a finite number of terms in a power series, anywhere from a couple of terms to a hundred terms. In this second case, extrapolating a plot estimates the radius of convergence.
Theoretical radius
The radius of convergence can be found by applying the root test to the terms of the series. The root test uses the number
"lim sup" denotes the limit superior. The root test states that the series converges if C < 1 and diverges if C > 1. It follows that the power series converges if the distance from z to the center a is less than
and diverges if the distance exceeds that number; this statement is the Cauchy–Hadamard theorem. Note that r = 1/0 is interpreted as an infinite radius, meaning that f is an entire function.
The limit involved in the ratio test is usually easier to compute, and when that limit exists, it shows that the radius of convergence is finite.
This is shown as follows. The ratio test says the series converges if
That is equivalent to
Practical estimation of radius in the case of real coefficients
Usually, in scientific applications, only a finite number of coefficients are known. Typically, as increases, these coefficients settle into a regular behavior determined by the nearest radius-limiting singularity. |
https://en.wikipedia.org/wiki/Analytic%20function | In mathematics, an analytic function is a function that is locally given by a convergent power series. There exist both real analytic functions and complex analytic functions. Functions of each type are infinitely differentiable, but complex analytic functions exhibit properties that do not generally hold for real analytic functions. A function is analytic if and only if its Taylor series about converges to the function in some neighborhood for every in its domain. It is important to note that it's a neighborhood and not just at some point , since every differentiable function has at least a tangent line at every point, which is its Taylor series of order 1. So just having a polynomial expansion at singular points is not enough, and the Taylor series must also converge to the function on points adjacent to to be considered an analytic function. As a counterexample see the Fabius function.
Definitions
Formally, a function is real analytic on an open set in the real line if for any one can write
in which the coefficients are real numbers and the series is convergent to for in a neighborhood of .
Alternatively, a real analytic function is an infinitely differentiable function such that the Taylor series at any point in its domain
converges to for in a neighborhood of pointwise. The set of all real analytic functions on a given set is often denoted by .
A function defined on some subset of the real line is said to be real analytic at a point if there is a neighborhood of on which is real analytic.
The definition of a complex analytic function is obtained by replacing, in the definitions above, "real" with "complex" and "real line" with "complex plane". A function is complex analytic if and only if it is holomorphic i.e. it is complex differentiable. For this reason the terms "holomorphic" and "analytic" are often used interchangeably for such functions.
Examples
Typical examples of analytic functions are
The following elementary functions:
All polynomials: if a polynomial has degree n, any terms of degree larger than n in its Taylor series expansion must immediately vanish to 0, and so this series will be trivially convergent. Furthermore, every polynomial is its own Maclaurin series.
The exponential function is analytic. Any Taylor series for this function converges not only for x close enough to x0 (as in the definition) but for all values of x (real or complex).
The trigonometric functions, logarithm, and the power functions are analytic on any open set of their domain.
Most special functions (at least in some range of the complex plane):
hypergeometric functions
Bessel functions
gamma functions
Typical examples of functions that are not analytic are
The absolute value function when defined on the set of real numbers or complex numbers is not everywhere analytic because it is not differentiable at 0.
Piecewise defined functions (functions given by different formulae in different regions) are typically no |
https://en.wikipedia.org/wiki/Absolute%20convergence | In mathematics, an infinite series of numbers is said to converge absolutely (or to be absolutely convergent) if the sum of the absolute values of the summands is finite. More precisely, a real or complex series is said to converge absolutely if for some real number Similarly, an improper integral of a function, is said to converge absolutely if the integral of the absolute value of the integrand is finite—that is, if
Absolute convergence is important for the study of infinite series because its definition is strong enough to have properties of finite sums that not all convergent series possess – a convergent series that is not absolutely convergent is called conditionally convergent, while absolutely convergent series behave "nicely". For instance, rearrangements do not change the value of the sum. This is not true for conditionally convergent series: The alternating harmonic series converges to while its rearrangement (in which the repeating pattern of signs is two positive terms followed by one negative term) converges to
Background
In finite sums, the order in which terms are added is both associative and commutative, meaning that grouping and rearranging do not matter. (1 + 2) + 3 is the same as 1 + (2 + 3), and both are the same as (3 + 2) + 1. However, this is not true when adding infinitely many numbers, and wrongly assuming that it is true can lead to apparent paradoxes. One classic example is the alternating sum
whose terms alternate between +1 and −1. What is the value of S? One way to evaluate S is to group the first and second term, the third and fourth, and so on:
But another way to evaluate S is to leave the first term alone and group the second and third term, then the fourth and fifth term, and so on:
This leads to an apparent paradox: does or ?
The answer is that because S is not absolutely convergent, grouping or rearranging its terms changes the value of the sum. This means and are not equal. In fact, the series does not converge, so S does not have a value to find in the first place. A series that is absolutely convergent does not have this problem: grouping or rearranging its terms does not change the value of the sum.
Definition for real and complex numbers
A sum of real numbers or complex numbers is absolutely convergent if the sum of the absolute values of the terms converges.
Sums of more general elements
The same definition can be used for series whose terms are not numbers but rather elements of an arbitrary abelian topological group. In that case, instead of using the absolute value, the definition requires the group to have a norm, which is a positive real-valued function on an abelian group (written additively, with identity element 0) such that:
The norm of the identity element of is zero:
For every implies
For every
For every
In this case, the function induces the structure of a metric space (a type of topology) on
Then, a -valued series is absolutely converge |
https://en.wikipedia.org/wiki/Spiral | In mathematics, a spiral is a curve which emanates from a point, moving farther away as it revolves around the point. It is a subtype of whorled patterns, a broad group that also includes concentric objects.
Helices
Two major definitions of "spiral" in the American Heritage Dictionary are:
a curve on a plane that winds around a fixed center point at a continuously increasing or decreasing distance from the point.
a three-dimensional curve that turns around an axis at a constant or continuously varying distance while moving parallel to the axis; a helix.
The first definition describes a planar curve, that extends in both of the perpendicular directions within its plane; the groove on one side of a gramophone record closely approximates a plane spiral (and it is by the finite width and depth of the groove, but not by the wider spacing between than within tracks, that it falls short of being a perfect example); note that successive loops differ in diameter. In another example, the "center lines" of the arms of a spiral galaxy trace logarithmic spirals.
The second definition includes two kinds of 3-dimensional relatives of spirals:
A conical or volute spring (including the spring used to hold and make contact with the negative terminals of AA or AAA batteries in a battery box), and the vortex that is created when water is draining in a sink is often described as a spiral, or as a conical helix.
Quite explicitly, definition 2 also includes a cylindrical coil spring and a strand of DNA, both of which are quite helical, so that "helix" is a more useful description than "spiral" for each of them; in general, "spiral" is seldom applied if successive "loops" of a curve have the same diameter.
In the side picture, the black curve at the bottom is an Archimedean spiral, while the green curve is a helix. The curve shown in red is a conic helix.
Two-dimensional
A two-dimensional, or plane, spiral may be described most easily using polar coordinates, where the radius is a monotonic continuous function of angle :
The circle would be regarded as a degenerate case (the function not being strictly monotonic, but rather constant).
In --coordinates the curve has the parametric representation:
Examples
Some of the most important sorts of two-dimensional spirals include:
The Archimedean spiral:
The hyperbolic spiral:
Fermat's spiral:
The lituus:
The logarithmic spiral:
The Cornu spiral or clothoid
The Fibonacci spiral and golden spiral
The Spiral of Theodorus: an approximation of the Archimedean spiral composed of contiguous right triangles
The involute of a circle, used twice on each tooth of almost every modern gear
An Archimedean spiral is, for example, generated while coiling a carpet.
A hyperbolic spiral appears as image of a helix with a special central projection (see diagram). A hyperbolic spiral is some times called reciproke spiral, because it is the image of an Archimedean spiral with a circle-inversion (see below).
The |
https://en.wikipedia.org/wiki/John%20Milnor | John Willard Milnor (born February 20, 1931) is an American mathematician known for his work in differential topology, algebraic K-theory and low-dimensional holomorphic dynamical systems. Milnor is a distinguished professor at Stony Brook University and one of the five mathematicians to have won the Fields Medal, the Wolf Prize, and the Abel Prize (the others being Serre, Thompson, Deligne, and Margulis).
Early life and career
Milnor was born on February 20, 1931, in Orange, New Jersey. His father was J. Willard Milnor, an engineer, and his mother was Emily Cox Milnor. As an undergraduate at Princeton University he was named a Putnam Fellow in 1949 and 1950 and also proved the Fáry–Milnor theorem when he was only 19 years old. Milnor graduated with an A.B. in mathematics in 1951 after completing a senior thesis, titled "Link groups", under the supervision of Robert H. Fox. He remained at Princeton to pursue graduate studies and received his Ph.D. in mathematics in 1954 after completing a doctoral dissertation, titled "Isotopy of links", also under the supervision of Fox. His dissertation concerned link groups (a generalization of the classical knot group) and their associated link structure, classifying Brunnian links up to link-homotopy and introduced new invariants of it, called Milnor invariants. Upon completing his doctorate, he went on to work at Princeton. He was a professor at the Institute for Advanced Study from 1970 to 1990.
He was an editor of the Annals of Mathematics for a number of years after 1962. He has written a number of books which are famous for their clarity, presentation, and an inspiration for the research by many mathematicians in their areas even after many decades since their publication. He served as Vice President of the AMS in 1976–77 period.
His students have included Tadatoshi Akiba, Jon Folkman, John Mather, Laurent C. Siebenmann, Michael Spivak, and Jonathan Sondow. His wife, Dusa McDuff, is a professor of mathematics at Barnard College and is known for her work in symplectic topology.
Research
One of Milnor's best-known works is his proof in 1956 of the existence of 7-dimensional spheres with nonstandard differentiable structure, which marked the beginning of a new field – differential topology. He coined the term exotic sphere, referring to any n-sphere with nonstandard differential structure. Kervaire and Milnor initiated the systematic study of exotic spheres, showing in particular that the 7-sphere has 15 distinct differentiable structures (28 if one considers orientation).
Egbert Brieskorn found simple algebraic equations for 28 complex hypersurfaces in complex 5-space such that their intersection with a small sphere of dimension 9 around a singular point is diffeomorphic to these exotic spheres. Subsequently, Milnor worked on the topology of isolated singular points of complex hypersurfaces in general, developing the theory of the Milnor fibration whose fiber has the homotopy type of a bouquet of μ |
https://en.wikipedia.org/wiki/Venn%20diagram | A Venn diagram is a widely used diagram style that shows the logical relation between sets, popularized by John Venn (1834–1923) in the 1880s. The diagrams are used to teach elementary set theory, and to illustrate simple set relationships in probability, logic, statistics, linguistics and computer science. A Venn diagram uses simple closed curves drawn on a plane to represent sets. Very often, these curves are circles or ellipses.
Similar ideas had been proposed before Venn such as by Christian Weise in 1712 (Nucleus Logicoe Wiesianoe) and Leonhard Euler (Letters to a German Princess) in 1768. The idea was popularised by Venn in Symbolic Logic, Chapter V "Diagrammatic Representation", published in 1881.
Details
A Venn diagram may also be called a set diagram or logic diagram. It is a diagram that shows all possible logical relations between a finite collection of different sets. These diagrams depict elements as points in the plane, and sets as regions inside closed curves. A Venn diagram consists of multiple overlapping closed curves, usually circles, each representing a set. The points inside a curve labelled S represent elements of the set S, while points outside the boundary represent elements not in the set S. This lends itself to intuitive visualizations; for example, the set of all elements that are members of both sets S and T, denoted S ∩ T and read "the intersection of S and T", is represented visually by the area of overlap of the regions S and T.
In Venn diagrams, the curves are overlapped in every possible way, showing all possible relations between the sets. They are thus a special case of Euler diagrams, which do not necessarily show all relations. Venn diagrams were conceived around 1880 by John Venn. They are used to teach elementary set theory, as well as illustrate simple set relationships in probability, logic, statistics, linguistics, and computer science.
A Venn diagram in which the area of each shape is proportional to the number of elements it contains is called an area-proportional (or scaled) Venn diagram.
Example
This example involves two sets of creatures, represented here as colored circles. The orange circle represents all types of creatures that have two legs. The blue circle represents creatures that can fly. Each separate type of creature can be imagined as a point somewhere in the diagram. Living creatures that have two legs and can fly—for example, parrots—are then in both sets, so they correspond to points in the region where the blue and orange circles overlap. This overlapping region would only contain those elements (in this example, creatures) that are members of both the orange set (two-legged creatures) and the blue set (flying creatures).
Humans and penguins are bipedal, and so are in the orange circle, but since they cannot fly, they appear in the left part of the orange circle, where it does not overlap with the blue circle. Mosquitoes can fly, but have six, not two, legs, so the point fo |
https://en.wikipedia.org/wiki/Inner%20automorphism | In abstract algebra an inner automorphism is an automorphism of a group, ring, or algebra given by the conjugation action of a fixed element, called the conjugating element. They can be realized via simple operations from within the group itself, hence the adjective "inner". These inner automorphisms form a subgroup of the automorphism group, and the quotient of the automorphism group by this subgroup is defined as the outer automorphism group.
Definition
If is a group and is an element of (alternatively, if is a ring, and is a unit), then the function
is called (right) conjugation by (see also conjugacy class). This function is an endomorphism of : for all
where the second equality is given by the insertion of the identity between and Furthermore, it has a left and right inverse, namely Thus, is bijective, and so an isomorphism of with itself, i.e. an automorphism. An inner automorphism is any automorphism that arises from conjugation.
When discussing right conjugation, the expression is often denoted exponentially by This notation is used because composition of conjugations satisfies the identity: for all This shows that right conjugation gives a right action of on itself.
Inner and outer automorphism groups
The composition of two inner automorphisms is again an inner automorphism, and with this operation, the collection of all inner automorphisms of is a group, the inner automorphism group of denoted .
is a normal subgroup of the full automorphism group of . The outer automorphism group, is the quotient group
The outer automorphism group measures, in a sense, how many automorphisms of are not inner. Every non-inner automorphism yields a non-trivial element of , but different non-inner automorphisms may yield the same element of .
Saying that conjugation of by leaves unchanged is equivalent to saying that and commute:
Therefore the existence and number of inner automorphisms that are not the identity mapping is a kind of measure of the failure of the commutative law in the group (or ring).
An automorphism of a group is inner if and only if it extends to every group containing .
By associating the element with the inner automorphism in as above, one obtains an isomorphism between the quotient group (where is the center of ) and the inner automorphism group:
This is a consequence of the first isomorphism theorem, because is precisely the set of those elements of that give the identity mapping as corresponding inner automorphism (conjugation changes nothing).
Non-inner automorphisms of finite -groups
A result of Wolfgang Gaschütz says that if is a finite non-abelian -group, then has an automorphism of -power order which is not inner.
It is an open problem whether every non-abelian -group has an automorphism of order . The latter question has positive answer whenever has one of the following conditions:
is nilpotent of class 2
is a regular -group
is a powerful -group
The centralizer in |
https://en.wikipedia.org/wiki/Genus%20%28mathematics%29 | In mathematics, genus (: genera) has a few different, but closely related, meanings. Intuitively, the genus is the number of "holes" of a surface. A sphere has genus 0, while a torus has genus 1.
Topology
Orientable surfaces
The genus of a connected, orientable surface is an integer representing the maximum number of cuttings along non-intersecting closed simple curves without rendering the resultant manifold disconnected. It is equal to the number of handles on it. Alternatively, it can be defined in terms of the Euler characteristic χ, via the relationship χ = 2 − 2g for closed surfaces, where g is the genus. For surfaces with b boundary components, the equation reads χ = 2 − 2g − b. In layman's terms, it's the number of "holes" an object has ("holes" interpreted in the sense of doughnut holes; a hollow sphere would be considered as having zero holes in this sense). A torus has 1 such hole, while a sphere has 0. The green surface pictured above has 2 holes of the relevant sort.
For instance:
The sphere S2 and a disc both have genus zero.
A torus has genus one, as does the surface of a coffee mug with a handle. This is the source of the joke "topologists are people who can't tell their donut from their coffee mug."
Explicit construction of surfaces of the genus g is given in the article on the fundamental polygon.
In simpler terms, the value of an orientable surface's genus is equal to the number of "holes" it has.
Non-orientable surfaces
The non-orientable genus, demigenus, or Euler genus of a connected, non-orientable closed surface is a positive integer representing the number of cross-caps attached to a sphere. Alternatively, it can be defined for a closed surface in terms of the Euler characteristic χ, via the relationship χ = 2 − k, where k is the non-orientable genus.
For instance:
A real projective plane has a non-orientable genus 1.
A Klein bottle has non-orientable genus 2.
Knot
The genus of a knot K is defined as the minimal genus of all Seifert surfaces for K. A Seifert surface of a knot is however a manifold with boundary, the boundary being the knot, i.e.
homeomorphic to the unit circle. The genus of such a surface is defined to be the genus of the two-manifold, which is obtained by gluing the unit disk along the boundary.
Handlebody
The genus of a 3-dimensional handlebody is an integer representing the maximum number of cuttings along embedded disks without rendering the resultant manifold disconnected. It is equal to the number of handles on it.
For instance:
A ball has genus 0.
A solid torus D2 × S1 has genus 1.
Graph theory
The genus of a graph is the minimal integer n such that the graph can be drawn without crossing itself on a sphere with n handles (i.e. an oriented surface of the genus n). Thus, a planar graph has genus 0, because it can be drawn on a sphere without self-crossing.
The non-orientable genus of a graph is the minimal integer n such that the graph can be drawn without crossing itself on a sp |
https://en.wikipedia.org/wiki/Mrs.%20Miniver%27s%20problem | Mrs. Miniver's problem is a geometry problem about the area of circles. It asks how to place two circles and of given radii in such a way that the lens formed by intersecting their two interiors has equal area to the symmetric difference of and (the area contained in one but not both circles). It was named for an analogy between geometry and social dynamics enunciated by fictional character Mrs. Miniver, who "saw every relationship as a pair of intersecting circles". Its solution involves a transcendental equation.
Origin
The problem derives from "A Country House Visit", one of Jan Struther's newspaper articles appearing in the Times of London between 1937 and 1939 featuring her character Mrs. Miniver. According to the story:
She saw every relationship as a pair of intersecting circles. It would seem at first glance that the more they overlapped the better the relationship; but this is not so. Beyond a certain point the law of diminishing returns sets in, and there are not enough private resources left on either side to enrich the life that is shared. Probably perfection is reached when the area of the two outer crescents, added together, is exactly equal to that of the leaf-shaped piece in the middle. On paper there must be some neat mathematical formula for arriving at this; in life, none.
Louis A. Graham and Clifton Fadiman formalized the mathematics of the problem and popularized it among recreational mathematicians.
Solution
The problem can be solved by cutting the lune along the line segment between the two crossing points of the circles, into two circular segments, and using the formula for the area of a circular segment to relate the distance between the crossing points to the total area that the problem requires the lune to have. This gives a transcendental equation for the distance between crossing points but it can be solved numerically. There are two boundary conditions whose distances between centers can be readily solved: the farthest apart the centers can be is when the circles have equal radii, and the closest they can be is when one circle is contained completely within the other, which happens when the ratio between radii is . If the ratio of radii falls beyond these limiting cases, the circles cannot satisfy the problem's area constraint.
In the case of two circles of equal size, these equations can be simplified somewhat. The rhombus formed by the two circle centers and the two crossing points, with side lengths equal to the radius, has an angle radians at the circle centers, found by solving the equation from which it follows that the ratio of the distance between their centers to their radius is .
See also
Goat problem#Interior grazing problem, another problem of equalizing the areas of circular lunes and lenses
References
Circles
Area
Mathematical problems |
https://en.wikipedia.org/wiki/Division | Division or divider may refer to:
Mathematics
Division (mathematics), the inverse of multiplication
Division algorithm, a method for computing the result of mathematical division
Military
Division (military), a formation typically consisting of 10,000 to 25,000 troops
Divizion, a subunit in some militaries
Division (naval), a collection of warships
Science
Cell division, the process in which biological cells multiply
Continental divide, the geographical term for separation between watersheds
Division (biology), used differently in botany and zoology
Division (botany), a taxonomic rank for plants or fungi, equivalent to phylum in zoology
Division (horticulture), a method of vegetative plant propagation, or the plants created by using this method
Division, a medical/surgical operation involving cutting and separation, see ICD-10 Procedure Coding System
Technology
Beam compass, a compass with a beam and sliding sockets for drawing and dividing circles larger than those made by a regular pair of compasses
Divider caliper or compass, a caliper
Frequency divider, a circuit that divides the frequency of a clock signal
Society
Administrative division, territory into which a country is divided
Census division, an official term in Canada and the United States
Diairesis, Plato's method of definition by division
Division (business), of a business entity is a distinct part of that business but the primary business is legally responsible for all of the obligations and debts of the division
Division (political geography), a name for a subsidiary state or prefecture of a country
Division (sport), a group of teams in organised sport who compete for a divisional title
In parliamentary procedure:
Division of the assembly, a type of formally recorded vote by assembly members
Division of a question, to split a question into two or more questions
Partition (politics), the process of changing national borders or separating political entities
Police division, a large territorial unit of the British police
Places
Division station (CTA North Side Main Line), a station on the Chicago Transit Authority's North Side Main Line
Division station (CTA Blue Line), a station on the Chicago Transit Authority's 'L' system, serving the Blue Line
Division Mountain, on the Continental Divide along the Alberta - British Columbia border of Canada
Division Range, Humboldt County, Nevada
Music
Division (10 Years album), 2008
Division (The Gazette album), 2012
Divisions (album), by Starset, 2019
Division (music), a type of ornamentation or variation found in early music
Divider, as in Schenkerian music analysis, a consonant subdivision of a consonant interval
"Division", a song by Aly & AJ from Insomniatic, 2007
"Divider", a song by Scott Weiland from the album 12 Bar Blues (album), 1998
Other uses
Divider, a central reservation in Bangladesh
Division of the field, a concept in heraldry
Division (logical fallacy), when one reasons logically that something true of a thing must also |
https://en.wikipedia.org/wiki/Catalan%27s%20conjecture | Catalan's conjecture (or Mihăilescu's theorem) is a theorem in number theory that was conjectured by the mathematician Eugène Charles Catalan in 1844 and proven in 2002 by Preda Mihăilescu at Paderborn University. The integers 23 and 32 are two perfect powers (that is, powers of exponent higher than one) of natural numbers whose values (8 and 9, respectively) are consecutive. The theorem states that this is the only case of two consecutive perfect powers. That is to say, that
History
The history of the problem dates back at least to Gersonides, who proved a special case of the conjecture in 1343 where (x, y) was restricted to be (2, 3) or (3, 2). The first significant progress after Catalan made his conjecture came in 1850 when Victor-Amédée Lebesgue dealt with the case b = 2.
In 1976, Robert Tijdeman applied Baker's method in transcendence theory to establish a bound on a,b and used existing results bounding x,y in terms of a, b to give an effective upper bound for x,y,a,b. Michel Langevin computed a value of for the bound, resolving Catalan's conjecture for all but a finite number of cases.
Catalan's conjecture was proven by Preda Mihăilescu in April 2002. The proof was published in the Journal für die reine und angewandte Mathematik, 2004. It makes extensive use of the theory of cyclotomic fields and Galois modules. An exposition of the proof was given by Yuri Bilu in the Séminaire Bourbaki. In 2005, Mihăilescu published a simplified proof.
Pillai's conjecture
Pillai's conjecture concerns a general difference of perfect powers : it is an open problem initially proposed by S. S. Pillai, who conjectured that the gaps in the sequence of perfect powers tend to infinity. This is equivalent to saying that each positive integer occurs only finitely many times as a difference of perfect powers: more generally, in 1931 Pillai conjectured that for fixed positive integers A, B, C the equation has only finitely many solutions (x, y, m, n) with (m, n) ≠ (2, 2). Pillai proved that the difference for any λ less than 1, uniformly in m and n.
The general conjecture would follow from the ABC conjecture.
Paul Erdős conjectured that the ascending sequence of perfect powers satisfies for some positive constant c and all sufficiently large n.
Pillai's conjecture means that for every natural number n, there are only finitely many pairs of perfect powers with difference n. The list below shows, for n ≤ 64, all solutions for perfect powers less than 1018, as . See also for the smallest solution (> 0).
See also
Beal's conjecture
Equation xy = yx
Fermat–Catalan conjecture
Mordell curve
Ramanujan–Nagell equation
Størmer's theorem
Tijdeman's theorem
Thaine's theorem
Notes
References
Predates Mihăilescu's proof.
External links
Ivars Peterson's MathTrek
On difference of perfect powers
Jeanine Daems: A Cyclotomic Proof of Catalan's Conjecture
Conjectures
Conjectures that have been proved
Diophantine equations
Theorems in number |
https://en.wikipedia.org/wiki/J.%20B.%20S.%20Haldane | John Burdon Sanderson Haldane (; 5 November 18921 December 1964), nicknamed "Jack" or "JBS", was a British-Indian scientist who worked in physiology, genetics, evolutionary biology, and mathematics. With innovative use of statistics in biology, he was one of the founders of neo-Darwinism. He served in the Great War, and obtained the rank of captain. Despite his lack of an academic degree in the field, he taught biology at the University of Cambridge, the Royal Institution, and University College London. Renouncing his British citizenship, he became an Indian citizen in 1961 and worked at the Indian Statistical Institute for the rest of his life.
Haldane's article on abiogenesis in 1929 introduced the "primordial soup theory", which became the foundation for the concept of the chemical origin of life. He established human gene maps for haemophilia and colour blindness on the X chromosome, and codified Haldane's rule on sterility in the heterogametic sex of hybrids in species. He correctly proposed that sickle-cell disease confers some immunity to malaria. He was the first to suggest the central idea of in vitro fertilisation, as well as concepts such as hydrogen economy, cis and trans-acting regulation, coupling reaction, molecular repulsion, the darwin (as a unit of evolution), and organismal cloning.
In 1957, Haldane articulated Haldane's dilemma, a limit on the speed of beneficial evolution, an idea which is still debated today. He willed his body for medical studies, as he wanted to remain useful even in death. He is also remembered for his work in human biology, having coined "clone", "cloning", and "ectogenesis". With his sister, Naomi Mitchison, Haldane was the first to demonstrate genetic linkage in mammals. Subsequent works established a unification of Mendelian genetics and Darwinian evolution by natural selection whilst laying the groundwork for modern synthesis, and helped to create population genetics.
Haldane was a professed socialist, Marxist, atheist, and secular humanist whose political dissent led him to leave England in 1956 and live in India, becoming a naturalised Indian citizen in 1961. Arthur C. Clarke credited him as "perhaps the most brilliant science populariser of his generation". Brazilian-British biologist and Nobel laureate Peter Medawar called Haldane "the cleverest man I ever knew". According to Theodosius Dobzhansky, "Haldane was always recognized as a singular case"; Ernst Mayr described him as a "polymath"; Michael J. D. White described him as "the most erudite biologist of his generation, and perhaps of the century"; James Watson described him as "England's most clever and eccentric biologist" and Sahotra Sarkar described him as "probably the most prescient biologist of this [20th] century." According to a Cambridge student, "he seemed to be the last man who might know all there was to be known."
Biography
Early life and education
Haldane was born in Oxford in 1892. His father was John Scott Haldane, |
https://en.wikipedia.org/wiki/Displacement | Displacement may refer to:
Physical sciences
Mathematics and physics
Displacement (geometry), is the difference between the final and initial position of a point trajectory (for instance, the center of mass of a moving object). The actual path covered to reach the final position is irrelevant.
Particle displacement, a measurement of distance of the movement of a particle in a medium as it transmits a wave (represented in mathematics by the lower-case Greek letter ξ)
Displacement field (mechanics), an assignment of displacement vectors for all points in a body that is displaced from one state to another
Electric displacement field, as appears in Maxwell's equations
Wien's displacement law, a relation concerning the spectral distribution of blackbody radiation
Angular displacement, a change in orientation of a rigid body, the amount of rotation about a fixed axis.
Engineering
Engine displacement, the total volume of air/fuel mixture an engine can draw in during one complete engine cycle
Displacement (fluid), an object immersed in a fluid pushes the fluid out of the way
Positive displacement meter, a pump or flow meter which processes a definite fluid volume per revolution
Displacement has several meanings related to ships and boats
Displacement hull, where the moving hull's weight is supported by buoyancy alone and it must displace water from its path rather than planing on the water's surface
Displacement speed, a rule of thumb for non planing watercraft to estimate their theoretical maximum speed
Displacement (ship), several related measurements of a ship's weight
Insulation displacement connector, a type of electrical connector
Displacement mapping, a technique in 3D computer graphics
Chemistry
Single displacement reaction, a chemical reaction concerning the exchange of ions
Double displacement reaction, a chemical reaction concerning the exchange of ions
Radioactive displacement law of Fajans and Soddy, elements/isotopes created during radioactive decay
Geology
Earth Crustal Displacement, an aspect of the Pole shift hypothesis
Medicine
Displacement (orthopedic surgery), change in alignment of the fracture fragments
Social sciences
Displacement (linguistics), the ability of humans (and possibly some animals) to communicate ideas that are remote in time and/or space
Forced displacement, by persecution or violence
Displacement (psychology), a sub-conscious defense mechanism
Displacement (parapsychology), a statistical or qualitative correspondence between targets and responses.
Development-induced displacement, the displacement of population for economic development
Displacement may occur during gentrification.
Sport
Displacement (fencing), a movement that avoids or dodges an attack
Other
Child displacement
See also
Offset (disambiguation)
Transformation (geometry) |
https://en.wikipedia.org/wiki/Vector%20field | In vector calculus and physics, a vector field is an assignment of a vector to each point in a space, most commonly Euclidean space . A vector field on a plane can be visualized as a collection of arrows with given magnitudes and directions, each attached to a point on the plane. Vector fields are often used to model, for example, the speed and direction of a moving fluid throughout three dimensional space, such as the wind, or the strength and direction of some force, such as the magnetic or gravitational force, as it changes from one point to another point.
The elements of differential and integral calculus extend naturally to vector fields. When a vector field represents force, the line integral of a vector field represents the work done by a force moving along a path, and under this interpretation conservation of energy is exhibited as a special case of the fundamental theorem of calculus. Vector fields can usefully be thought of as representing the velocity of a moving flow in space, and this physical intuition leads to notions such as the divergence (which represents the rate of change of volume of a flow) and curl (which represents the rotation of a flow).
A vector field is a special case of a vector-valued function, whose domain's dimension has no relation to the dimension of its range; for example, the position vector of a space curve is defined only for smaller subset of the ambient space.
Likewise, n coordinates, a vector field on a domain in n-dimensional Euclidean space can be represented as a vector-valued function that associates an n-tuple of real numbers to each point of the domain. This representation of a vector field depends on the coordinate system, and there is a well-defined transformation law (covariance and contravariance of vectors) in passing from one coordinate system to the other.
Vector fields are often discussed on open subsets of Euclidean space, but also make sense on other subsets such as surfaces, where they associate an arrow tangent to the surface at each point (a tangent vector).
More generally, vector fields are defined on differentiable manifolds, which are spaces that look like Euclidean space on small scales, but may have more complicated structure on larger scales. In this setting, a vector field gives a tangent vector at each point of the manifold (that is, a section of the tangent bundle to the manifold). Vector fields are one kind of tensor field.
Definition
Vector fields on subsets of Euclidean space
Given a subset of , a vector field is represented by a vector-valued function in standard Cartesian coordinates . If each component of is continuous, then is a continuous vector field. It is common to focus on smooth vector fields, meaning that each component is a smooth function (differentiable any number of times). A vector field can be visualized as assigning a vector to individual points within an n-dimensional space.
One standard notation is to write for the unit vectors in the c |
https://en.wikipedia.org/wiki/Pre-abelian%20category | In mathematics, specifically in category theory, a pre-abelian category is an additive category that has all kernels and cokernels.
Spelled out in more detail, this means that a category C is pre-abelian if:
C is preadditive, that is enriched over the monoidal category of abelian groups (equivalently, all hom-sets in C are abelian groups and composition of morphisms is bilinear);
C has all finite products (equivalently, all finite coproducts); note that because C is also preadditive, finite products are the same as finite coproducts, making them biproducts;
given any morphism f: A → B in C, the equaliser of f and the zero morphism from A to B exists (this is by definition the kernel of f), as does the coequaliser (this is by definition the cokernel of f).
Note that the zero morphism in item 3 can be identified as the identity element of the hom-set Hom(A,B), which is an abelian group by item 1; or as the unique morphism A → 0 → B, where 0 is a zero object, guaranteed to exist by item 2.
Examples
The original example of an additive category is the category Ab of abelian groups.
Ab is preadditive because it is a closed monoidal category, the biproduct in Ab is the finite direct sum, the kernel is inclusion of the ordinary kernel from group theory and the cokernel is the quotient map onto the ordinary cokernel from group theory.
Other common examples:
The category of (left) modules over a ring R, in particular:
the category of vector spaces over a field K.
The category of (Hausdorff) abelian topological groups.
The category of Banach spaces.
The category of Fréchet spaces.
The category of (Hausdorff) bornological spaces.
These will give you an idea of what to think of; for more examples, see abelian category (every abelian category is pre-abelian).
Elementary properties
Every pre-abelian category is of course an additive category, and many basic properties of these categories are described under that subject. This article concerns itself with the properties that hold specifically because of the existence of kernels and cokernels.
Although kernels and cokernels are special kinds of equalisers and coequalisers, a pre-abelian category actually has all equalisers and coequalisers.
We simply construct the equaliser of two morphisms f and g as the kernel of their difference g − f; similarly, their coequaliser is the cokernel of their difference.
(The alternative term "difference kernel" for binary equalisers derives from this fact.)
Since pre-abelian categories have all finite products and coproducts (the biproducts) and all binary equalisers and coequalisers (as just described), then by a general theorem of category theory, they have all finite limits and colimits.
That is, pre-abelian categories are finitely complete.
The existence of both kernels and cokernels gives a notion of image and coimage.
We can define these as
im f := ker coker f;
coim f := coker ker f.
That is, the image is the kernel of the cokernel, and the coimage is the |
https://en.wikipedia.org/wiki/Complete%20category | In mathematics, a complete category is a category in which all small limits exist. That is, a category C is complete if every diagram F : J → C (where J is small) has a limit in C. Dually, a cocomplete category is one in which all small colimits exist. A bicomplete category is a category which is both complete and cocomplete.
The existence of all limits (even when J is a proper class) is too strong to be practically relevant. Any category with this property is necessarily a thin category: for any two objects there can be at most one morphism from one object to the other.
A weaker form of completeness is that of finite completeness. A category is finitely complete if all finite limits exists (i.e. limits of diagrams indexed by a finite category J). Dually, a category is finitely cocomplete if all finite colimits exist.
Theorems
It follows from the existence theorem for limits that a category is complete if and only if it has equalizers (of all pairs of morphisms) and all (small) products. Since equalizers may be constructed from pullbacks and binary products (consider the pullback of (f, g) along the diagonal Δ), a category is complete if and only if it has pullbacks and products.
Dually, a category is cocomplete if and only if it has coequalizers and all (small) coproducts, or, equivalently, pushouts and coproducts.
Finite completeness can be characterized in several ways. For a category C, the following are all equivalent:
C is finitely complete,
C has equalizers and all finite products,
C has equalizers, binary products, and a terminal object,
C has pullbacks and a terminal object.
The dual statements are also equivalent.
A small category C is complete if and only if it is cocomplete. A small complete category is necessarily thin.
A posetal category vacuously has all equalizers and coequalizers, whence it is (finitely) complete if and only if it has all (finite) products, and dually for cocompleteness. Without the finiteness restriction a posetal category with all products is automatically cocomplete, and dually, by a theorem about complete lattices.
Examples and nonexamples
The following categories are bicomplete:
Set, the category of sets
Top, the category of topological spaces
Grp, the category of groups
Ab, the category of abelian groups
Ring, the category of rings
K-Vect, the category of vector spaces over a field K
R''-Mod, the category of modules over a commutative ring R''
CmptH, the category of all compact Hausdorff spaces
Cat, the category of all small categories
Whl, the category of wheels
sSet, the category of simplicial sets
The following categories are finitely complete and finitely cocomplete but neither complete nor cocomplete:
The category of finite sets
The category of finite abelian groups
The category of finite-dimensional vector spaces
Any (pre)abelian category is finitely complete and finitely cocomplete.
The category of complete lattices is complete but not cocomplete.
The category of metric spaces, Met, is fin |
https://en.wikipedia.org/wiki/Slugging%20percentage | In baseball statistics, slugging percentage (SLG) is a measure of the batting productivity of a hitter. It is calculated as total bases divided by at-bats, through the following formula, where AB is the number of at-bats for a given player, and 1B, 2B, 3B, and HR are the number of singles, doubles, triples, and home runs, respectively:
Unlike batting average, slugging percentage gives more weight to extra-base hits such as doubles and home runs, relative to singles. Plate appearances resulting in walks, hit-by-pitches, catcher's interference, and sacrifice bunts or flies are specifically excluded from this calculation, as such an appearance is not counted as an at-bat (these are not factored into batting average either).
The name is a misnomer, as the statistic is not a percentage but an average of how many bases a player achieves per at bat. It is a scale of measure whose computed value is a number from 0 to 4. This might not be readily apparent given that a Major League Baseball player's slugging percentage is almost always less than 1 (as a majority of at bats result in either 0 or 1 base). The statistic gives a double twice the value of a single, a triple three times the value, and a home run four times. The slugging percentage would have to be divided by 4 to actually be a percentage (of bases achieved per at bat out of total bases possible). As a result, it is occasionally called slugging average, or simply slugging, instead.
A slugging percentage is always expressed as a decimal to three decimal places, and is generally spoken as if multiplied by 1000. For example, a slugging percentage of .589 would be spoken as "five eighty nine," and one of 1.127 would be spoken as "eleven twenty seven."
Facts about slugging percentage
A slugging percentage is not just for the use of measuring the productivity of a hitter. It can be applied as an evaluative tool for pitchers. It is not as common but it is referred to as slugging-percentage against.
In 2019, the mean average SLG among all teams in Major League Baseball was .435.
The maximum slugging percentage has a numerical value of 4.000. However, no player in the history of MLB has ever retired with a 4.000 slugging percentage. Four players tripled in their only at bat and therefore share the Major League record, when calculated without respect to games played or plate appearances, of a career slugging percentage of 3.000. This list includes Eric Cammack (2000 Mets); Scott Munninghoff (1980 Phillies); Eduardo Rodríguez (1973 Brewers); and Charlie Lindstrom (1958 White Sox).
Example calculation
For example, in 1920, Babe Ruth played his first season for the New York Yankees. In 458 at bats, Ruth had 172 hits, comprising 73 singles, 36 doubles, 9 triples, and 54 home runs, which brings the total base count to . His total number of bases (388) divided by his total at bats (458) is .847 which constitutes his slugging percentage for the season. This also set a record for Ruth which stood until |
https://en.wikipedia.org/wiki/Norbert%20Wiener | Norbert Wiener (November 26, 1894 – March 18, 1964) was an American mathematician, computer scientist and philosopher. He became a professor of mathematics at the Massachusetts Institute of Technology (MIT). A child prodigy, Wiener later became an early researcher in stochastic and mathematical noise processes, contributing work relevant to electronic engineering, electronic communication, and control systems.
Wiener is considered the originator of cybernetics, the science of communication as it relates to living things and machines,
with implications for engineering, systems control, computer science, biology, neuroscience, philosophy, and the organization of society. His work heavily influenced computer pioneer John von Neumann, information theorist Claude Shannon, anthropologists Margaret Mead and Gregory Bateson, and others.
Norbert Wiener is credited as being one of the first to theorize that all intelligent behavior was the result of feedback mechanisms, that could possibly be simulated by machines and was an important early step towards the development of modern artificial intelligence.
Biography
Youth
Wiener was born in Columbia, Missouri, the first child of Leo Wiener and Bertha Kahn, Jewish immigrants from Lithuania and Germany, respectively. Through his father, he was related to Maimonides, the famous rabbi, philosopher and physician from Al Andalus, as well as to Akiva Eger, chief rabbi of Posen from 1815 to 1837.
Leo had educated Norbert at home until 1903, employing teaching methods of his own invention, except for a brief interlude when Norbert was 7 years of age. Earning his living teaching German and Slavic languages, Leo read widely and accumulated a personal library from which the young Norbert benefited greatly. Leo also had ample ability in mathematics and tutored his son in the subject until he left home. In his autobiography, Norbert described his father as calm and patient, unless he (Norbert) failed to give a correct answer, at which his father would lose his temper.
In “The Theory of Ignorance”, a paper he wrote at the age of 10, he disputed “man’s presumption in declaring that his knowledge has no limits”, arguing that all human knowledge “is based on an approximation”, and acknowledging “the impossibility of being certain of anything.”
He graduated from Ayer High School in 1906 at 11 years of age, and Wiener then entered Tufts College. He was awarded a BA in mathematics in 1909 at the age of 14, whereupon he began graduate studies of zoology at Harvard. In 1910 he transferred to Cornell to study philosophy. He graduated in 1911 at 17 years of age.
Harvard and World War I
The next year he returned to Harvard, while still continuing his philosophical studies. Back at Harvard, Wiener became influenced by Edward Vermilye Huntington, whose mathematical interests ranged from axiomatic foundations to engineering problems. Harvard awarded Wiener a PhD in June 1913, when he was only 19 years old, for a dissertation o |
https://en.wikipedia.org/wiki/SGP | SGP may refer to:
Events
Secret Garden Party, a UK music festival
Speedway Grand Prix, a series of motorcycling contests
Symposium on Geometry Processing, of European Association For Computer Graphics
Organisations
Businesses
Simmering-Graz-Pauker, an Austrian machine/vehicle manufacturer
Stockland Corporation Limited, an Australian property developer (ASX ticker: SGP)
Simply Good Production, a Russian video production agency
Political parties
Reformed Political Party (Staatkundig Gereformeerde Partij), the Netherlands
Socialist Equality Party (Sozialistische Gleichheitspartei), Germany
Scottish Green Party, Scotland
Professional associations
Sociedad de Gestión de Productores Fonográficos del Paraguay, for Paraguayan record producers
Society of General Physiologists, for biomedical scientists
Science
Simplified General Perturbations model, for orbital calculations
Social Golfer Problem, a problem in discrete mathematics
Transport
Shay Gap Airport, IATA airport code "SGP"
Schweizer SGP 1-1, an American glider
Subaru Global Platform, unibody automobile platform
Other uses
SGP, the ISO 3166-1 alpha-3 country code for Singapore
sgp, the ISO 639-3 code for the Singpho dialect
Stability and Growth Pact, the main EU fiscal agreement
SpaceGhostPurrp, American rapper and record producer |
https://en.wikipedia.org/wiki/Palermo%20Technical%20Impact%20Hazard%20Scale | The Palermo Technical Impact Hazard Scale is a logarithmic scale used by astronomers to rate the potential hazard of impact of a near-Earth object (NEO). It combines two types of data—probability of impact and estimated kinetic yield—into a single "hazard" value. A rating of 0 means the hazard is equivalent to the background hazard (defined as the average risk posed by objects of the same size or larger over the years until the date of the potential impact). A rating of +2 would indicate the hazard is 100 times as great as a random background event. Scale values less than −2 reflect events for which there are no likely consequences, while Palermo Scale values between −2 and 0 indicate situations that merit careful monitoring. A similar but less complex scale is the Torino Scale, which is used for simpler descriptions in the non-scientific media.
As of June 2023, one asteroid has a cumulative Palermo Scale value above −2: 101955 Bennu (−1.41). Seven have cumulative Palermo Scale values between −2 and −3: (29075) 1950 DA (−2.05), 1979 XB (−2.72), 2021 EU (−2.74), (−2.79), (−2.83), (−2.98), and (−2.98). Of those that have a cumulative Palermo Scale value between −3 and −4, one was discovered in 2023: 2023 DO (−3.60).
Scale
The scale compares the likelihood of the detected potential impact with the average risk posed by objects of the same size or larger over the years until the date of the potential impact. This average risk from random impacts is known as the background risk. The Palermo Scale value, P, is defined by the equation:
where
pi is the impact probability
T is the time interval over which pi is considered
fB is the background impact frequency
The background impact frequency is defined for this purpose as:
where the energy threshold E is measured in megatons, and yr is the unit of T divided by one year.
Positive rating
In 2002 the near-Earth object reached a positive rating on the scale of 0.18, indicating a higher-than-background threat. The value was subsequently lowered after more measurements were taken. is no longer considered to pose any risk and was removed from the Sentry Risk Table on 1 August 2002.
In September 2002, the highest Palermo rating was that of asteroid (29075) 1950 DA, with a value of 0.17 for a possible collision in the year 2880. By March 2022, the rating had been reduced to −2.0.
For a brief period in late December 2004, with an observation arc of 190 days, asteroid (then known only by its provisional designation ) held the record for the highest Palermo scale value, with a value of 1.10 for a possible collision in the year 2029. The 1.10 value indicated that a collision with this object was considered to be almost 12.6 times as likely as a random background event: 1 in 37 instead of 1 in 472. With further observation through 2021 there is no risk from Apophis for the next 100+ years.
See also
Asteroid impact avoidance
Asteroid impact prediction
Earth-grazing fireball
Impact event
List of asteroid |
https://en.wikipedia.org/wiki/Double%20pendulum | In physics and mathematics, in the area of dynamical systems, a double pendulum also known as a chaos pendulum is a pendulum with another pendulum attached to its end, forming a simple physical system that exhibits rich dynamic behavior with a strong sensitivity to initial conditions. The motion of a double pendulum is governed by a set of coupled ordinary differential equations and is chaotic.
Analysis and interpretation
Several variants of the double pendulum may be considered; the two limbs may be of equal or unequal lengths and masses, they may be simple pendulums or compound pendulums (also called complex pendulums) and the motion may be in three dimensions or restricted to the vertical plane. In the following analysis, the limbs are taken to be identical compound pendulums of length and mass , and the motion is restricted to two dimensions.
In a compound pendulum, the mass is distributed along its length. If the mass is evenly distributed, then the center of mass of each limb is at its midpoint, and the limb has a moment of inertia of about that point.
It is convenient to use the angles between each limb and the vertical as the generalized coordinates defining the configuration of the system. These angles are denoted and . The position of the center of mass of each rod may be written in terms of these two coordinates. If the origin of the Cartesian coordinate system is taken to be at the point of suspension of the first pendulum, then the center of mass of this pendulum is at:
and the center of mass of the second pendulum is at
This is enough information to write out the Lagrangian.
Lagrangian
The Lagrangian is
The first term is the linear kinetic energy of the center of mass of the bodies and the second term is the rotational kinetic energy around the center of mass of each rod. The last term is the potential energy of the bodies in a uniform gravitational field. The dot-notation indicates the time derivative of the variable in question.
Since (see Chain Rule and List of trigonometric identities)
and
substituting the coordinates above and rearranging the equation gives
There is only one conserved quantity (the energy), and no conserved momenta. The two generalized momenta may be written as
These expressions may be inverted to get
The remaining equations of motion are written as
These last four equations are explicit formulas for the time evolution of the system given its current state. No closed form solutions for and as functions of time are known, therefore solving the system can only be done numerically, using the Runge Kutta method or similar techniques.
Chaotic motion
The double pendulum undergoes chaotic motion, and shows a sensitive dependence on initial conditions. The image to the right shows the amount of elapsed time before the pendulum flips over, as a function of initial position when released at rest. Here, the initial value of ranges along the -direction from −3.14 to 3.14. The initial value ranges |
https://en.wikipedia.org/wiki/Moir%C3%A9%20pattern | In mathematics, physics, and art, moiré patterns ( , , ) or moiré fringes are large-scale interference patterns that can be produced when a partially opaque ruled pattern with transparent gaps is overlaid on another similar pattern. For the moiré interference pattern to appear, the two patterns must not be completely identical, but rather displaced, rotated, or have slightly different pitch.
Moiré patterns appear in many situations. In printing, the printed pattern of dots can interfere with the image. In television and digital photography, a pattern on an object being photographed can interfere with the shape of the light sensors to generate unwanted artifacts. They are also sometimes created deliberately – in micrometers they are used to amplify the effects of very small movements.
In physics, its manifestation is wave interference such as that seen in the double-slit experiment and the beat phenomenon in acoustics.
Etymology
The term originates from moire (moiré in its French adjectival form), a type of textile, traditionally made of silk but now also made of cotton or synthetic fiber, with a rippled or "watered" appearance. Moire, or "watered textile", is made by pressing two layers of the textile when wet. The similar but imperfect spacing of the threads creates a characteristic pattern which remains after the fabric dries.
In French, the noun moire is in use from the 17th century, for "watered silk". It was a loan of the English mohair (attested 1610). In French usage, the noun gave rise to the verb moirer, "to produce a watered textile by weaving or pressing", by the 18th century. The adjective moiré formed from this verb is in use from at least 1823.
Pattern formation
Moiré patterns are often an artifact of images produced by various digital imaging and computer graphics techniques, for example when scanning a halftone picture or ray tracing a checkered plane (the latter being a special case of aliasing, due to undersampling a fine regular pattern). This can be overcome in texture mapping through the use of mipmapping and anisotropic filtering.
The drawing on the upper right shows a moiré pattern. The lines could represent fibers in moiré silk, or lines drawn on paper or on a computer screen. The nonlinear interaction of the optical patterns of lines creates a real and visible pattern of roughly parallel dark and light bands, the moiré pattern, superimposed on the lines.
The moiré effect also occurs between overlapping transparent objects. For example, an invisible phase mask is made of a transparent polymer with a wavy thickness profile. As light shines through two overlaid masks of similar phase patterns, a broad moiré pattern occurs on a screen some distance away. This phase moiré effect and the classical moiré effect from opaque lines are two ends of a continuous spectrum in optics, which is called the universal moiré effect. The phase moiré effect is the basis for a type of broadband interferometer in x-ray and particle |
https://en.wikipedia.org/wiki/%21%20%28disambiguation%29 | ! is a punctuation mark, called an exclamation mark (33 in ASCII), exclamation point, ecphoneme, or bang.
! or exclamation point may also refer to:
Mathematics and computers
Factorial, a mathematical function
Derangement, a related mathematical function
Negation, in logic and some programming languages
Uniqueness quantification, in mathematics and logic
! (CONFIG.SYS directive), usage for unconditional execution of directives in FreeDOS configuration files
Music
! (The Dismemberment Plan album), released in 1995
! (Donnie Vie album), released in 2016
"!" (The Song Formerly Known As), a single on the 1997 album Unit by Regurgitator
Exclamation Mark (album), a 2011 album by Jay Chou
Exclamation Point, a 2010 LP by DA!
! (Trippie Redd album), released in 2019
! (Trippie Redd song), that album's title track
! (Cláudia Pascoal album), released in 2020
Other
ǃ, the IPA symbol for postalveolar click in speech
An indicator of a good chess move in punctuation
A dereference operator in BCPL
See also
!! (disambiguation)
!!! (disambiguation)
Interrobang, the nonstandard mix of a question mark and an exclamation mark
ḷ, not the exclamation mark, but a lower-case letter Ḷ used in Asturian |
https://en.wikipedia.org/wiki/Percentage | In mathematics, a percentage () is a number or ratio expressed as a fraction of 100. It is often denoted using the percent sign (%), although the abbreviations pct., pct, and sometimes pc are also used. A percentage is a dimensionless number (pure number), primarily used for expressing proportions, but percent is nonetheless a unit of measurement in its orthography and usage.
Examples
For example, 45% (read as "forty-five percent") is equal to the fraction , the ratio 45:55 (or 45:100 when comparing to the total rather than the other portion), or 0.45.
Percentages are often used to express a proportionate part of a total.
(Similarly, one can also express a number as a fraction of 1,000, using the term "per mille" or the symbol "".)
Example 1
If 50% of the total number of students in the class are male, that means that 50 out of every 100 students are male. If there are 500 students, then 250 of them are male.
Example 2
An increase of $0.15 on a price of $2.50 is an increase by a fraction of = 0.06. Expressed as a percentage, this is a 6% increase.
While many percentage values are between 0 and 100, there is no mathematical restriction and percentages may take on other values. For example, it is common to refer to 111% or −35%, especially for percent changes and comparisons.
History
In Ancient Rome, long before the existence of the decimal system, computations were often made in fractions in the multiples of . For example, Augustus levied a tax of on goods sold at auction known as centesima rerum venalium. Computation with these fractions was equivalent to computing percentages.
As denominations of money grew in the Middle Ages, computations with a denominator of 100 became increasingly standard, such that from the late 15th century to the early 16th century, it became common for arithmetic texts to include such computations. Many of these texts applied these methods to profit and loss, interest rates, and the Rule of Three. By the 17th century, it was standard to quote interest rates in hundredths.
Percent sign
The term "percent" is derived from the Latin per centum, meaning "hundred" or "by the hundred".
The sign for "percent" evolved by gradual contraction of the Italian term per cento, meaning "for a hundred". The "per" was often abbreviated as "p."—eventually disappeared entirely. The "cento" was contracted to two circles separated by a horizontal line, from which the modern "%" symbol is derived.
Calculations
The percent value is computed by multiplying the numeric value of the ratio by 100. For example, to find 50 apples as a percentage of 1250 apples, one first computes the ratio = 0.04, and then multiplies by 100 to obtain 4%. The percent value can also be found by multiplying first instead of later, so in this example, the 50 would be multiplied by 100 to give 5,000, and this result would be divided by 1250 to give 4%.
To calculate a percentage of a percentage, convert both percentages to fractions of 100, or to decim |
https://en.wikipedia.org/wiki/De%20Morgan%27s%20laws | In propositional logic and Boolean algebra, De Morgan's laws, also known as De Morgan's theorem, are a pair of transformation rules that are both valid rules of inference. They are named after Augustus De Morgan, a 19th-century British mathematician. The rules allow the expression of conjunctions and disjunctions purely in terms of each other via negation.
The rules can be expressed in English as:
The negation of a disjunction is the conjunction of the negations
The negation of a conjunction is the disjunction of the negations
or
The complement of the union of two sets is the same as the intersection of their complements
The complement of the intersection of two sets is the same as the union of their complements
or
not (A or B) = (not A) and (not B)
not (A and B) = (not A) or (not B)
where "A or B" is an "inclusive or" meaning at least one of A or B rather than an "exclusive or" that means exactly one of A or B.
In set theory and Boolean algebra, these are written formally as
where
and are sets,
is the complement of ,
is the intersection, and
is the union.
In formal language, the rules are written as
and
where
P and Q are propositions,
is the negation logic operator (NOT),
is the conjunction logic operator (AND),
is the disjunction logic operator (OR),
is a metalogical symbol meaning "can be replaced in a logical proof with", often read as "if and only if". For any combination of true/false values for P and Q, the left and right sides of the arrow will hold the same truth value after evaluation.
Another form of De Morgan's law is the following as seen in the right figure.
Applications of the rules include simplification of logical expressions in computer programs and digital circuit designs. De Morgan's laws are an example of a more general concept of mathematical duality.
Formal notation
The negation of conjunction rule may be written in sequent notation:
,
and
.
The negation of disjunction rule may be written as:
,
and
.
In rule form: negation of conjunction
and negation of disjunction
and expressed as a truth-functional tautology or theorem of propositional logic:
where and are propositions expressed in some formal system.
Substitution form
De Morgan's laws are normally shown in the compact form above, with the negation of the output on the left and negation of the inputs on the right. A clearer form for substitution can be stated as:
This emphasizes the need to invert both the inputs and the output, as well as change the operator when doing a substitution.
Set theory and Boolean algebra
In set theory and Boolean algebra, it is often stated as "union and intersection interchange under complementation", which can be formally expressed as:
where:
is the negation of , the overline being written above the terms to be negated,
is the intersection operator (AND),
is the union operator (OR).
Unions and intersections of any number of sets
The generalized form is
where is some, possibly countably |
https://en.wikipedia.org/wiki/Abelian | Abelian may refer to:
Mathematics
Group theory
Abelian group, a group in which the binary operation is commutative
Category of abelian groups (Ab), has abelian groups as objects and group homomorphisms as morphisms
Metabelian group, a group where the commutator subgroup is abelian
Abelianisation
Topology and number theory
Abelian variety, a complex torus that can be embedded into projective space
Abelian surface, a two-dimensional abelian variety
Abelian function, a meromorphic function on an abelian variety
Abelian integral, a function related to the indefinite integral of a differential of the first kind
Other mathematics
Abelian category, in category theory, a preabelian category in which every monomorphism is a kernel and every epimorphism is a cokernel
Abelian and Tauberian theorems, in real analysis, used in the summation of divergent series
Abelian extension, in Galois theory, a field extension for which the associated Galois group is abelian
Abelian von Neumann algebra, in functional analysis, a von Neumann algebra of operators on a Hilbert space in which all elements commute
Other uses
Abelian, in physics, a gauge theory with a commutative symmetry group
Hovhannes Abelian (1865–1936), Armenian actor
See also
Pre-abelian category, an additive category that has all kernels and cokernels
Niels Henrik Abel (1802–1829), Norwegian mathematician who gave his name to several different mathematical concepts
Abelians, a 4th-century Christian sect
Abel, a Biblical figure in the Book of Genesis |
https://en.wikipedia.org/wiki/Persi%20Diaconis | Persi Warren Diaconis (; born January 31, 1945) is an American mathematician of Greek descent and former professional magician. He is the Mary V. Sunseri Professor of Statistics and Mathematics at Stanford University.
He is particularly known for tackling mathematical problems involving randomness and randomization, such as coin flipping and shuffling playing cards.
Biography
Diaconis left home at 14 to travel with sleight-of-hand legend Dai Vernon, and was awarded a high school diploma based on grades given to him by his teachers after dropping out of George Washington High School. He returned to school at age 24 to learn math, motivated to read William Feller's famous two-volume treatise on probability theory, An Introduction to Probability Theory and Its Applications. He attended the City College of New York for his undergraduate work, graduating in 1971, and then obtained a Ph.D. in Mathematical Statistics from Harvard University in 1974, learned to read Feller, and became a mathematical probabilist.
According to Martin Gardner, at school, Diaconis supported himself by playing poker on ships between New York and South America. Gardner recalls that Diaconis had "fantastic second deal and bottom deal".
Diaconis is married to Stanford statistics professor Susan Holmes.
Career
Diaconis received a MacArthur Fellowship in 1982. In 1990, he published (with Dave Bayer) a paper entitled "Trailing the Dovetail Shuffle to Its Lair" (a term coined by magician Charles Jordan in the early 1900s) which established rigorous results on how many times a deck of playing cards must be riffle shuffled before it can be considered random according to the mathematical measure total variation distance. Diaconis is often cited for the simplified proposition that it takes seven shuffles to randomize a deck. More precisely, Diaconis showed that, in the Gilbert–Shannon–Reeds model of how likely it is that a riffle results in a particular riffle shuffle permutation, it takes 5 riffles before the total variation distance of a 52-card deck begins to drop significantly from the maximum value of 1.0, and 7 riffles before it drops below 0.5 very quickly (a threshold phenomenon), after which it is reduced by a factor of 2 every shuffle. When entropy is viewed as the probabilistic distance, riffle shuffling seems to take less time to mix, and the threshold phenomenon goes away (because the entropy function is subadditive).
Diaconis has coauthored several more recent papers expanding on his 1992 results and relating the problem of shuffling cards to other problems in mathematics. Among other things, they showed that the separation distance of an ordered blackjack deck (that is, aces on top, followed by 2's, followed by 3's, etc.) drops below .5 after 7 shuffles. Separation distance is an upper bound for variation distance.
Diaconis has been hired by casino executives to search for subtle flaws in their automatic card shuffling machines. Diaconis soon found some and t |
https://en.wikipedia.org/wiki/Newcomb%27s%20paradox | In philosophy and mathematics, Newcomb's paradox, also known as Newcomb's problem, is a thought experiment involving a game between two players, one of whom is able to predict the future.
Newcomb's paradox was created by William Newcomb of the University of California's Lawrence Livermore Laboratory. However, it was first analyzed in a philosophy paper by Robert Nozick in 1969 and appeared in the March 1973 issue of Scientific American, in Martin Gardner's "Mathematical Games". Today it is a much debated problem in the philosophical branch of decision theory.
The problem
There is a reliable predictor, another player, and two boxes designated A and B. The player is given a choice between taking only box B or taking both boxes A and B. The player knows the following:
Box A is transparent and always contains a visible $1,000.
Box B is opaque, and its content has already been set by the predictor:
If the predictor has predicted that the player will take both boxes A and B, then box B contains nothing.
If the predictor has predicted that the player will take only box B, then box B contains $1,000,000.
The player does not know what the predictor predicted or what box B contains while making the choice.
Game-theory strategies
In his 1969 article, Nozick noted that "To almost everyone, it is perfectly clear and obvious what should be done. The difficulty is that these people seem to divide almost evenly on the problem, with large numbers thinking that the opposing half is just being silly." The problem continues to divide philosophers today. In a 2020 survey, a modest plurality of professional philosophers chose to take both boxes (39.0% versus 31.2%).
Game theory offers two strategies for this game that rely on different principles: the expected utility principle and the strategic dominance principle. The problem is called a paradox because two analyses that both sound intuitively logical give conflicting answers to the question of what choice maximizes the player's payout.
Considering the expected utility when the probability of the predictor being right is certain or near-certain, the player should choose box B. This choice statistically maximizes the player's winnings, setting them at about $1,000,000 per game.
Under the dominance principle, the player should choose the strategy that is always better; choosing both boxes A and B will always yield $1,000 more than only choosing B. However, the expected utility of "always $1,000 more than B" depends on the statistical payout of the game; when the predictor's prediction is almost certain or certain, choosing both A and B sets player's winnings at about $1,000 per game.
David Wolpert and Gregory Benford point out that paradoxes arise when not all relevant details of a problem are specified, and there is more than one "intuitively obvious" way to fill in those missing details. They suggest that in the case of Newcomb's paradox, the conflict over which of the two strategies is "obviously correct |
https://en.wikipedia.org/wiki/Jacques%20Roubaud | Jacques Roubaud (; born 5 December 1932 in Caluire-et-Cuire, Rhône) is a French poet, writer and mathematician.
Life and career
Jacques Roubaud taught Mathematics at University of Paris X Nanterre and Poetry at EHESS. A member of the Oulipo group, he has published poetry, plays, novels, and translated English poetry and books into French such as Lewis Carroll's The Hunting of the Snark. French poet and novelist Raymond Queneau had Roubaud's first book, a collection of mathematically structured sonnets, published by Éditions Gallimard, and then invited Roubaud to join the Oulipo as the organization's first new member outside the founders.
Roubaud's fiction often suppresses the rigorous constraints of the Oulipo (while mentioning their suppression, thereby indicating that such constraints are indeed present), yet takes the Oulipian self-consciousness of the writing act to an extreme. This simultaneity both appears playfully, in his Hortense novels (Our Beautiful Heroine, Hortense Is Abducted and Hortense in Exile), and with gravity and reflection in The Great Fire of London, considered the pinnacle of his prose. The Great Fire of London (1989), The Loop (1993), and Mathematics (2012) are the first three volumes of a long, experimental, autobiographical work known as "the project" (or "the minimal project"), and the only volumes of "the project", at present, to have been translated into English. Seven volumes of "the project" have been completed and published in French. To compose The Loop, Roubaud began with a childhood memory of a snowy night in Carcassonne and then wrote nightly, without returning to correct his writing from previous nights. Roubaud's goals in writing The Loop were to discover, "My own memory, how does it work?", and to "destroy" his memories through writing them down.
Roubaud has participated in readings and lectures at the European Graduate School (2007), the Salon du Livre de Paris (2008), and the "Dire Poesia" series at Palazzo Leoni Montanari in Venice (2011).
He married Alix Cléo Roubaud in 1980; she died three years later.
Selected bibliography
La Belle Hortense (1985). Our Beautiful Heroine, trans. David Kornacker (Overlook Press, 1987).
Quelque chose noir (1986). Some Thing Black, trans. Rosmarie Waldrop. Photographs by Alix Cléo Roubaud (Dalkey Archive Press, 1990).
L'Enlèvement d'Hortense (1987). Hortense Is Abducted, trans. Dominic Di Bernardi (Dalkey Archive Press, 1989).
Échanges de la lumière (1990). Exchanges on Light, trans. Eleni Sikélianòs (La Presse, 2009).
Le Grand Incendie de Londres (Branch 1 of the Project) (1989). The Great Fire of London, trans. Dominic Di Bernardi (Dalkey Archive Press, 1991).
La Princesse Hoppy ou Le Conte du Labrador (1990). The Princess Hoppy, or The Tale of Labrador, trans. Bernard Hœpffner (Dalkey Archive Press, 1993).
L'Exil d'Hortense (1990). Hortense in Exile, trans. Dominic Di Bernardi (Dalkey Archive Press, 1992).
La Pluralité des mondes de Lewis (1991). The Plurality of |
https://en.wikipedia.org/wiki/Root%20mean%20square | In mathematics and its applications, the root mean square of a set of numbers (abbreviated as RMS, or rms and denoted in formulas as either or ) is defined as the square root of the mean square (the arithmetic mean of the squares) of the set.
The RMS is also known as the quadratic mean (denoted ) and is a particular case of the generalized mean. The RMS of a continuously varying function (denoted ) can be defined in terms of an integral of the squares of the instantaneous values during a cycle.
For alternating electric current, RMS is equal to the value of the constant direct current that would produce the same power dissipation in a resistive load.
In estimation theory, the root-mean-square deviation of an estimator is a measure of the imperfection of the fit of the estimator to the data.
Definition
The RMS value of a set of values (or a continuous-time waveform) is the square root of the arithmetic mean of the squares of the values, or the square of the function that defines the continuous waveform. In physics, the RMS current value can also be defined as the "value of the direct current that dissipates the same power in a resistor."
In the case of a set of n values , the RMS is
The corresponding formula for a continuous function (or waveform) f(t) defined over the interval is
and the RMS for a function over all time is
The RMS over all time of a periodic function is equal to the RMS of one period of the function. The RMS value of a continuous function or signal can be approximated by taking the RMS of a sample consisting of equally spaced observations. Additionally, the RMS value of various waveforms can also be determined without calculus, as shown by Cartwright.
In the case of the RMS statistic of a random process, the expected value is used instead of the mean.
In common waveforms
If the waveform is a pure sine wave, the relationships between amplitudes (peak-to-peak, peak) and RMS are fixed and known, as they are for any continuous periodic wave. However, this is not true for an arbitrary waveform, which may not be periodic or continuous. For a zero-mean sine wave, the relationship between RMS and peak-to-peak amplitude is:
Peak-to-peak
For other waveforms, the relationships are not the same as they are for sine waves. For example, for either a triangular or sawtooth wave
Peak-to-peak
In waveform combinations
Waveforms made by summing known simple waveforms have an RMS value that is the root of the sum of squares of the component RMS values, if the component waveforms are orthogonal (that is, if the average of the product of one simple waveform with another is zero for all pairs other than a waveform times itself).
Alternatively, for waveforms that are perfectly positively correlated, or "in phase" with each other, their RMS values sum directly.
Uses
In electrical engineering
Voltage
A special case of RMS of waveform combinations is:
where refers to the direct current (or average) component of the signal, and is the |
https://en.wikipedia.org/wiki/Frans%20van%20Schooten | Frans van Schooten Jr. also rendered as Franciscus van Schooten (15 May 1615, Leiden – 29 May 1660, Leiden) was a Dutch mathematician who is most known for popularizing the analytic geometry of René Descartes.
Life
Van Schooten's father, was a professor of mathematics at the University of Leiden, having Christiaan Huygens, Johann van Waveren Hudde, and René de Sluze as students.
Van Schooten met Descartes in 1632 and read his Géométrie (an appendix to his Discours de la méthode) while it was still unpublished. Finding it hard to understand, he went to France to study the works of other important mathematicians of his time, such as François Viète and Pierre de Fermat. When Frans van Schooten returned to his home in Leiden in 1646, he inherited his father's position and one of his most important pupils, Huygens.
The pendant marriage portraits of him and his wife Margrieta Wijnants were painted by Rembrandt and are kept in the National Gallery of Art:
Work
Van Schooten's 1649 Latin translation of and commentary on Descartes' Géométrie was valuable in that it made the work comprehensible to the broader mathematical community, and thus was responsible for the spread of analytic geometry to the world.
Over the next decade he enlisted the aid of other mathematicians of the time, de Beaune, Hudde, Heuraet, de Witt and expanded the commentaries to two volumes, published in 1659 and 1661. This edition and its extensive commentaries was far more influential than the 1649 edition. It was this edition that Gottfried Leibniz and Isaac Newton knew.
Van Schooten was one of the first to suggest, in exercises published in 1657, that these ideas be extended to three-dimensional space. Van Schooten's efforts also made Leiden the centre of the mathematical community for a short period in the middle of the seventeenth century.
In elementary geometry Van Schooten's theorem is named after him.
References
Some Contemporaries of Descartes, Fermat, Pascal and Huygens: Van Schooten, based on W. W. Rouse Ball's A Short Account of the History of Mathematics (4th edition, 1908)
External links
Mathematische Oeffeningen van Frans van Schooten
Biografisch Woordenboek van Nederlandse Wiskundigen: Frans van Schooten
Frans van Schooten, and his Ruler Constructions at Convergence
An e-textbook developed from Frans van Schooten 1646 by dbook
1615 births
1660 deaths
17th-century Dutch mathematicians
Leiden University alumni
Academic staff of Leiden University |
https://en.wikipedia.org/wiki/Hemisphere | Hemisphere may refer to:
In geometry
Hemisphere (geometry), a half of a sphere
As half of the Earth
A hemisphere of Earth
Northern Hemisphere
Southern Hemisphere
Eastern Hemisphere
Western Hemisphere
Land and water hemispheres
A half of the (geocentric) celestial sphere
Northern celestial hemisphere
Southern celestial hemisphere
A cultural hemisphere
As half of the brain
A cerebral hemisphere, a division of the cerebrum
A half of the cerebellum, a smaller part of the brain
Other
Hémisphère (Paradis), a 12-inch album by French artists Paradis
Hemispheres (magazine), an inflight publication
Hemispheres (TV series), Canadian and Australian news program
Hemispheres (Rush album), 1978
Hemispheres (Lily Afshar album), 2006
Hemispheres (Doseone album), 1998
L'Hemisfèric at the Ciutat de les Arts i les Ciències, Valencia, Spain
Hemisphere Project, a counternarcotics program between United States federal and state drug officials and AT&T
Hemispheres |
https://en.wikipedia.org/wiki/Geography%20of%20Anguilla | Anguilla is an island in the Leeward Islands. It has numerous bays, including Barnes, Little, Rendezvous, Shoal, and Road Bays.
Statistics
Location: Caribbean, island in the Caribbean Sea, east of Puerto Rico
Geographic coordinates: 18°15′ N, 63°10′ W
Map references: Central America and the Caribbean
Area:
total:
land:
water:
Area – comparative: about half the size of Washington, D.C.
Coastline: 61 km
Maritime claims:
exclusive fishing zone:
territorial sea:
Climate: tropical moderated by northeast trade winds
Terrain: flat and low-lying island of coral and limestone
Elevation extremes:
lowest point: Caribbean Sea 0 m
highest point: Crocus Hill 73 m
Natural resources: salt, fish, lobster
Land use:
arable land: 0%
permanent crops: 0%
permanent pastures: 0%
forests and woodland: 61.1%
other: 38.9% (mostly rock with some commercial salt ponds)
Natural hazards: frequent hurricanes and other tropical
storms (July to October)
Environment – current issues: supplies of potable water
sometimes cannot meet increasing demand largely because of poor distribution system.
Islands and cays
The territory of Anguilla consists of the island of Anguilla itself (by far the largest), as well as numerous other islands and cays, most of which are very small and uninhabited. These include:
Anguillita
Blowing Rock
Cove Cay
Crocus Cay
Deadman's Cay
Dog Island
East Cay
Little Island
Little Scrub Island
Mid Cay
North Cay
Prickly Pear Cays
Rabbit Island
Sandy Island, also known as Sand Island
Scilly Cay
Scrub Island
Seal Island
Sombrero, also known as Hat Island
South Cay
South Wager Island
West Cay
Districts
Anguilla is divided into fourteen districts:
Climate
Anguilla features a tropical wet and dry climate under the Köppen climate classification. The island has a rather dry climate, moderated by northeast trade winds. Temperatures vary little throughout the year. Average daily maxima range from about in December to in July. Rainfall is erratic, averaging about per year, the wettest months being September and October, and the driest February and March. Anguilla is vulnerable to hurricanes from June to November, peak season August to mid-October.
The island suffered damage in 1995 from Hurricane Luis.
Vegetation
Anguilla's coral and limestone terrain provide no subsistence possibilities for forests, woodland, pastures, crops, or arable lands. Its dry climate and thin soil hamper commercial agricultural development.
See also
References
External links
Caribbean-On-Line.com provides detailed maps of Anguilla.
Districts of Anguilla, Statoids.com
Anguilla 2001 Census, Government of Anguilla |
https://en.wikipedia.org/wiki/Kolmogorov%20space | In topology and related branches of mathematics, a topological space X is a T0 space or Kolmogorov space (named after Andrey Kolmogorov) if for every pair of distinct points of X, at least one of them has a neighborhood not containing the other. In a T0 space, all points are topologically distinguishable.
This condition, called the T0 condition, is the weakest of the separation axioms. Nearly all topological spaces normally studied in mathematics are T0 spaces. In particular, all T1 spaces, i.e., all spaces in which for every pair of distinct points, each has a neighborhood not containing the other, are T0 spaces. This includes all T2 (or Hausdorff) spaces, i.e., all topological spaces in which distinct points have disjoint neighbourhoods. In another direction, every sober space (which may not be T1) is T0; this includes the underlying topological space of any scheme. Given any topological space one can construct a T0 space by identifying topologically indistinguishable points.
T0 spaces that are not T1 spaces are exactly those spaces for which the specialization preorder is a nontrivial partial order. Such spaces naturally occur in computer science, specifically in denotational semantics.
Definition
A T0 space is a topological space in which every pair of distinct points is topologically distinguishable. That is, for any two different points x and y there is an open set that contains one of these points and not the other. More precisely the topological space X is Kolmogorov or if and only if:
If and , there exists an open set O such that either or .
Note that topologically distinguishable points are automatically distinct. On the other hand, if the singleton sets {x} and {y} are separated then the points x and y must be topologically distinguishable. That is,
separated ⇒ topologically distinguishable ⇒ distinct
The property of being topologically distinguishable is, in general, stronger than being distinct but weaker than being separated. In a T0 space, the second arrow above also reverses; points are distinct if and only if they are distinguishable. This is how the T0 axiom fits in with the rest of the separation axioms.
Examples and counter examples
Nearly all topological spaces normally studied in mathematics are T0. In particular, all Hausdorff (T2) spaces, T1 spaces and sober spaces are T0.
Spaces which are not T0
A set with more than one element, with the trivial topology. No points are distinguishable.
The set R2 where the open sets are the Cartesian product of an open set in R and R itself, i.e., the product topology of R with the usual topology and R with the trivial topology; points (a,b) and (a,c) are not distinguishable.
The space of all measurable functions f from the real line R to the complex plane C such that the Lebesgue integral . Two functions which are equal almost everywhere are indistinguishable. See also below.
Spaces which are T0 but not T1
The Zariski topology on Spec(R), the prime spectrum of a commutati |
https://en.wikipedia.org/wiki/Peter%20Shor | Peter Williston Shor (born August 14, 1959) is an American professor of applied mathematics at MIT. He is known for his work on quantum computation, in particular for devising Shor's algorithm, a quantum algorithm for factoring exponentially faster than the best currently-known algorithm running on a classical computer.
Early life and education
Shor was born in New York City to Joan Bopp Shor and S. W. Williston Shor. He grew up in Washington, D.C. and Mill Valley, California. While attending Tamalpais High School, he placed third in the 1977 USA Mathematical Olympiad. After graduation that year, he won a silver medal at the International Math Olympiad in Yugoslavia (the U.S. team achieved the most points per country that year). He received his B.S. in Mathematics in 1981 for undergraduate work at Caltech, and was a Putnam Fellow in 1978. He earned his PhD in Applied Mathematics from MIT in 1985. His doctoral advisor was F. Thomson Leighton, and his thesis was on probabilistic analysis of bin-packing algorithms.
Career
After being awarded his PhD by MIT, he spent one year as a postdoctoral researcher at the University of California, Berkeley, and then accepted a position at Bell Labs in New Providence, New Jersey. It was there he developed Shor's algorithm. This development was inspired by Simon's problem, where he first solved the discrete log problem (which relates point-finding on a hypercube to a torus) and,"Later that week, I was able to solve the factoring problem as well. There’s a strange relation between discrete log and factoring."Due to their similarity as HSP problems, Shor discovered a related factoring problem (Shor's algorithm) that same week for which he was awarded the Nevanlinna Prize at the 23rd International Congress of Mathematicians in 1998 and the Gödel Prize in 1999. In 1999, he was awarded a MacArthur Fellowship. In 2017, he received the Dirac Medal of the ICTP and for 2019 the BBVA Foundation Frontiers of Knowledge Award in Basic Sciences.
Shor began his MIT position in 2003. Currently, he is the Henry Adams Morss and Henry Adams Morss, Jr. Professor of Applied Mathematics in the Department of Mathematics at MIT. He also is affiliated with CSAIL and the MIT Center for Theoretical Physics (CTP).
He received a Distinguished Alumni Award from Caltech in 2007.
On October 1, 2011, he was inducted into the American Academy of Arts and Sciences. He was elected as an ACM Fellow in 2019 "for contributions to quantum-computing, information theory, and randomized algorithms". He was elected as a member of the National Academy of Sciences in 2002. In 2020, he was elected a member of the National Academy of Engineering for pioneering contributions to quantum computation.
In an interview published in Nature on October 30, 2020, Shor said that he considers post-quantum cryptography to be a solution to the quantum threat, although a lot of engineering effort is required to switch from vulnerable algorithms.
Along with three oth |
https://en.wikipedia.org/wiki/Singleton | Singleton may refer to:
Sciences, technology
Mathematics
Singleton (mathematics), a set with exactly one element
Singleton field, used in conformal field theory
Computing
Singleton pattern, a design pattern that allows only one instance of a class to exist
Singleton bound, used in coding theory
Singleton variable, a variable that is referenced only once
Singleton, a character encoded with one unit in variable-width encoding schemes for computer character sets
Singleton, an empty tag or self-closing tag in XHTML or XML coding
Social science
Singleton (global governance), a hypothetical world order with a single decision-making agency
Singleton, a consonant that is not a geminate in linguistics
Singleton, a person that is not a twin or other multiple birth
People
Singleton (surname), for a partial list of people with the surname "Singleton"
Places
United Kingdom
Singleton, Lancashire, England
Singleton, West Sussex, England
Singleton, Kent, England
Singleton Park, Swansea, Wales
Singleton Abbey
Singleton Hospital
Australia
Singleton, New South Wales
Singleton Council, New South Wales
Singleton, Western Australia
Other uses
Singleton (lifestyle), a self-description of individuals without romantic partners, particularly applied to women in their thirties introduced in the novel and film Bridget Jones's Diary
The Singleton (film), a 2015 British drama film
"Singleton", a short story by Greg Egan
Singleton (cards), a single card in a suit
The Singleton, a whisky made by Diageo
Catawba (grape) or Singleton |
https://en.wikipedia.org/wiki/Pseudometric%20space | In mathematics, a pseudometric space is a generalization of a metric space in which the distance between two distinct points can be zero. Pseudometric spaces were introduced by Đuro Kurepa in 1934. In the same way as every normed space is a metric space, every seminormed space is a pseudometric space. Because of this analogy the term semimetric space (which has a different meaning in topology) is sometimes used as a synonym, especially in functional analysis.
When a topology is generated using a family of pseudometrics, the space is called a gauge space.
Definition
A pseudometric space is a set together with a non-negative real-valued function called a , such that for every
Symmetry:
Subadditivity/Triangle inequality:
Unlike a metric space, points in a pseudometric space need not be distinguishable; that is, one may have for distinct values
Examples
Any metric space is a pseudometric space.
Pseudometrics arise naturally in functional analysis. Consider the space of real-valued functions together with a special point This point then induces a pseudometric on the space of functions, given by for
A seminorm induces the pseudometric . This is a convex function of an affine function of (in particular, a translation), and therefore convex in . (Likewise for .)
Conversely, a homogeneous, translation-invariant pseudometric induces a seminorm.
Pseudometrics also arise in the theory of hyperbolic complex manifolds: see Kobayashi metric.
Every measure space can be viewed as a complete pseudometric space by defining for all where the triangle denotes symmetric difference.
If is a function and d2 is a pseudometric on X2, then gives a pseudometric on X1. If d2 is a metric and f is injective, then d1 is a metric.
Topology
The is the topology generated by the open balls
which form a basis for the topology. A topological space is said to be a if the space can be given a pseudometric such that the pseudometric topology coincides with the given topology on the space.
The difference between pseudometrics and metrics is entirely topological. That is, a pseudometric is a metric if and only if the topology it generates is T0 (that is, distinct points are topologically distinguishable).
The definitions of Cauchy sequences and metric completion for metric spaces carry over to pseudometric spaces unchanged.
Metric identification
The vanishing of the pseudometric induces an equivalence relation, called the metric identification, that converts the pseudometric space into a full-fledged metric space. This is done by defining if . Let be the quotient space of by this equivalence relation and define
This is well defined because for any we have that and so and vice versa. Then is a metric on and is a well-defined metric space, called the metric space induced by the pseudometric space .
The metric identification preserves the induced topologies. That is, a subset is open (or closed) in if and only if is open (or closed) in |
https://en.wikipedia.org/wiki/Union | Union commonly refers to:
Trade union, an organization of workers
Union (set theory), in mathematics, a fundamental operation on sets
Union may also refer to:
Arts and entertainment
Music
Union (band), an American rock group
Union (Union album), 1998
Union (Chara album), 2007
Union (Toni Childs album), 1988
Union (Cuff the Duke album), 2012
Union (Paradoxical Frog album), 2011
Union, a 2001 album by Puya
Union, a 2001 album by Rasa
Union (Son Volt album), 2019
Union (The Boxer Rebellion album), 2009
Union (Yes album), 1991
"Union" (Black Eyed Peas song), 2005
Other uses in arts and entertainment
Union (Star Wars), a Dark Horse comics limited series
Union, in the fictional Alliance–Union universe of C. J. Cherryh
Union (Horse with Two Discs), a bronze sculpture by Christopher Le Brun, 1999–2000
The Union (Marvel Team), a Marvel Comics superhero team and comic series
Education
Union Academy (disambiguation), the name of several institutions
Union College (disambiguation), the name of several institutions
Union Institute & University, in Ohio, U.S.
Union Presbyterian Seminary, in Virginia, U.S.
Union Public Schools, a school district in Oklahoma, U.S.
Union School of Theology, in Wales
Union Theological College, in Northern Ireland
Union Theological Seminary (disambiguation), the name of several institutions
Union University (disambiguation), the name of several institutions
History and politics
Economic union, a type of trade bloc
Political union, a type of state which is composed of or created out of smaller states
Personal union, the combination of two or more states that have the same monarch
Poor law union, a former unit of local government in the United Kingdom
Real union, a union of two or more states, which share some state institutions
CDU/CSU, or the Union, a German political alliance
Union (American Civil War), U.S. states that were loyal to the U.S. federal government
Union (Hungarian-German trade union council)
Union (Madagascar), a political party
Places
Canada
Union, Elgin County, Ontario
Union, Leeds and Grenville United Counties, Ontario
Union, Nova Scotia
Union, Prince Edward Island
United States
Union, Alabama
Union, Connecticut
Union, Illinois
Union, Logan County, Illinois
Union, Indiana
Union, Iowa
Union, Kentucky
Union, Louisiana
Union Parish, Louisiana
Union, Maine
Union, Mississippi
Union, Missouri, in Franklin County
Union, Clark County, Missouri
Union, Nebraska
Union, New Hampshire
Union (CDP), New Jersey
Union (hamlet), New York, in Madison County
Union, New York, in Broome County
Union, Ohio, a city in Montgomery and Miami Counties
Union, Oregon
Union, Pennsylvania (disambiguation)
Union, South Carolina
Union, Texas
Union, Virginia
Union, Washington
Union, West Virginia
Union, Barbour County, West Virginia
Union, Wisconsin (disambiguation)
Union Township, New Jersey (disambiguation)
Arcata, California, first settled as Union
Mount Union ( |
https://en.wikipedia.org/wiki/Demographics%20of%20Bolivia | The demographic characteristics of the population of Bolivia are known from censuses, with the first census undertaken in 1826 and the most recent in 2012. The National Institute of Statistics of Bolivia (INE) has performed this task since 1950. The population of Bolivia in 2012 reached 10 million for the first time in history. The population density is 9.13 inhabitants per square kilometer, and the overall life expectancy in Bolivia at birth is 68.2 years. The population has steadily risen from the late 1800s to the present time. The natural growth rate of the population is positive, which has been a continuing trend since the 1950s; in 2012, Bolivia's birth rate continued to be higher than the death rate. Bolivia is in the third stage of demographic transition. In terms of age structure, the population is dominated by the 15–64 segment. The median age of the population is 23.1, and the gender ratio of the total population is 0.99 males per female.
Bolivia is inhabited mostly by Mestizo, Quechua and Aymara, while minorities include 37 indigenous groups (0.3% average per group). Spanish, Quechua, Aymara, Guarani languages, as well as 34 other native languages are the official languages of Bolivia. Spanish is the most-spoken language (60.7%) within the population. The main religions of Bolivia are the Catholic Church (81.8%), Evangelicalism (11.5%), and Protestantism (2.6%). There is a literacy rate of 91.2%. An estimated 7.6% of the country's gross domestic product (GDP) is spent on education. The average monthly household income was Bs.1,378 ($293) in 1994. In December 2013 the unemployment rate was 3.2% of the working population. The average urbanization rate in Bolivia is 67%.
Population
The first true estimate of the population of Bolivia came in 1826, in which 997,427 inhabitants were estimated. This number was calculated from the 1796 census organized by Francisco Gil de Taboada, which consisted of several Bolivian cities. The first modern census was completed in 1831, and ten have been completed since then. The organizer of Bolivia's censuses has changed throughout the years—Andrés de Santa Cruz (1831), The Bolivian Statistical Office (1835, 1854, 1882), The Bolivian Statistical Commission (1845), The National Immigration Bureau and The Statistics and Geographic Propaganda (1900), and The Department of Statistics and Censuses (1950)—with the INE conducting the census since 1976. The national census is supposed to be conducted every ten years, however, the 2012 census was late because of "climatic factors and the financing." The 2012 census was conducted on 21 November 2012, in which 10,027,254 inhabitants were in the country. The estimated cost of the census was $50 million.
With a population of 10.0 million in 2012, Bolivia ranks 87th in the world by population. Its population density is 9.13 inhabitants per square kilometer. The overall life expectancy in Bolivia is 65.4. The total fertility rate is 2.87 children per mother. Since 19 |
https://en.wikipedia.org/wiki/Euler%27s%20theorem | In number theory, Euler's theorem (also known as the Fermat–Euler theorem or Euler's totient theorem) states that, if and are coprime positive integers, and is Euler's totient function, then raised to the power is congruent to modulo ; that is
In 1736, Leonhard Euler published a proof of Fermat's little theorem (stated by Fermat without proof), which is the restriction of Euler's theorem to the case where is a prime number. Subsequently, Euler presented other proofs of the theorem, culminating with his paper of 1763, in which he proved a generalization to the case where is not prime.
The converse of Euler's theorem is also true: if the above congruence is true, then and must be coprime.
The theorem is further generalized by Carmichael's theorem.
The theorem may be used to easily reduce large powers modulo . For example, consider finding the ones place decimal digit of , i.e. . The integers 7 and 10 are coprime, and . So Euler's theorem yields , and we get .
In general, when reducing a power of modulo (where and are coprime), one needs to work modulo in the exponent of :
if , then .
Euler's theorem underlies the RSA cryptosystem, which is widely used in Internet communications. In this cryptosystem, Euler's theorem is used with being a product of two large prime numbers, and the security of the system is based on the difficulty of factoring such an integer.
Proofs
1. Euler's theorem can be proven using concepts from the theory of groups:
The residue classes modulo that are coprime to form a group under multiplication (see the article Multiplicative group of integers modulo n for details). The order of that group is . Lagrange's theorem states that the order of any subgroup of a finite group divides the order of the entire group, in this case . If is any number coprime to then is in one of these residue classes, and its powers modulo form a subgroup of the group of residue classes, with . Lagrange's theorem says must divide , i.e. there is an integer such that . This then implies,
2. There is also a direct proof: Let be a reduced residue system () and let be any integer coprime to . The proof hinges on the fundamental fact that multiplication by permutes the : in other words if then . (This law of cancellation is proved in the article Multiplicative group of integers modulo n.) That is, the sets and , considered as sets of congruence classes (), are identical (as sets—they may be listed in different orders), so the product of all the numbers in is congruent () to the product of all the numbers in :
and using the cancellation law to cancel each gives Euler's theorem:
See also
Carmichael function
Euler's criterion
Fermat's little theorem
Wilson's theorem
Notes
References
The Disquisitiones Arithmeticae has been translated from Gauss's Ciceronian Latin into English and German. The German edition includes all of his papers on number theory: all the proofs of quadratic reciprocity, the determination of |
https://en.wikipedia.org/wiki/Demographics%20of%20the%20Gambia | The demographic characteristics of the population of The Gambia are known through national censuses, conducted in ten-year intervals and analyzed by The Gambian Bureau of Statistics (GBOS) since 1963. The latest census was conducted in 2013. The population of The Gambia at the 2013 census was 1.8 million. The population density is 176.1 per square kilometer, and the overall life expectancy in The Gambia is 64.1 years. Since the first census of 1963, the population of The Gambia has increased every ten years by an average of 43.2 percent. Since 1950s, the birth rate has constantly exceeded the death rate; the natural growth rate is positive. The Gambia is in the second stage of demographic transition. In terms of age structure, The Gambia is dominated by 15- to 24-year-old segment (57.6%). The median age of the population is 19.9 years, and the gender ratio of the total population is 0.98 males per female.
Population
With a population of 1.88 million in 2013, The Gambia ranks 149th in the world by population. Its population density is . The overall life expectancy in The Gambia is 64.1 years. The total fertility rate of 3.98 is one of the highest in the world. Since 1950, the United Nations (UN) estimated the birth rate exceeds the death rate. The Gambia Bureau of Statistics (GBOS) estimates the population of The Gambia is expected to reach 3.6 million in 20 years. The population of The Gambia has increased each census, starting with 315 thousand in 1963 to 1.8 million in 2013. The GBOS predicted the reason for the increase from 2003 to 2013 was more coverage in the latter census compared to the former's.
Vital statistics
Registration of vital events in Gambia is not complete. The Population Department of the United Nations prepared the following estimates.
Fertility Rate (The Demographic Health Survey and Multiple Indicator Cluster Surveys)
Sources:
Fertility Rate TFR (Wanted Fertility Rate) and CBR (Crude Birth Rate):
Structure of the population (DHS 2013) (males 23,904, females 25,649, total 49,553) :
Population Estimates by Sex and Age Group (30.XII.2015) (Source: Integrated Household Survey (IHS) 2015/2016.):
Fertility data as of 2019-20 (DHS Program):
Life expectancy
Other demographic statistics
Demographic statistics according to the World Population Review in 2022.
One birth every 6 minutes
One death every 29 minutes
One net migrant every 180 minutes
Net gain of one person every 7 minutes
The following demographic statistics are from the CIA World Factbook.
Population
2,413,403 (2022 est.)
2,092,731 (July 2018 est.)
Religions
Muslim 96.4%, Christian 3.5%, other or none 0.1% (2019-20 est.)
Age structure
0-14 years: 35.15% (male 391,993/female 388,816)
15-24 years: 20.12% (male 221,519/female 225,414)
25-54 years: 36.39% (male 396,261/female 412,122)
55-64 years: 4.53% (male 48,032/female 52,538)
65 years and over: 3.81% (male 38,805/female 45,801) (2021 est.)
0-14 years: 36.97% (male 388,615 /female 385,172)
15-24 |
https://en.wikipedia.org/wiki/Rectangle | In Euclidean plane geometry, a rectangle is a quadrilateral with four right angles. It can also be defined as: an equiangular quadrilateral, since equiangular means that all of its angles are equal (360°/4 = 90°); or a parallelogram containing a right angle. A rectangle with four sides of equal length is a square. The term "oblong" is occasionally used to refer to a non-square rectangle. A rectangle with vertices ABCD would be denoted as .
The word rectangle comes from the Latin rectangulus, which is a combination of rectus (as an adjective, right, proper) and angulus (angle).
A crossed rectangle is a crossed (self-intersecting) quadrilateral which consists of two opposite sides of a rectangle along with the two diagonals (therefore only two sides are parallel). It is a special case of an antiparallelogram, and its angles are not right angles and not all equal, though opposite angles are equal. Other geometries, such as spherical, elliptic, and hyperbolic, have so-called rectangles with opposite sides equal in length and equal angles that are not right angles.
Rectangles are involved in many tiling problems, such as tiling the plane by rectangles or tiling a rectangle by polygons.
Characterizations
A convex quadrilateral is a rectangle if and only if it is any one of the following:
a parallelogram with at least one right angle
a parallelogram with diagonals of equal length
a parallelogram ABCD where triangles ABD and DCA are congruent
an equiangular quadrilateral
a quadrilateral with four right angles
a quadrilateral where the two diagonals are equal in length and bisect each other
a convex quadrilateral with successive sides a, b, c, d whose area is .
a convex quadrilateral with successive sides a, b, c, d whose area is
Classification
Traditional hierarchy
A rectangle is a special case of a parallelogram in which each pair of adjacent sides is perpendicular.
A parallelogram is a special case of a trapezium (known as a trapezoid in North America) in which both pairs of opposite sides are parallel and equal in length.
A trapezium is a convex quadrilateral which has at least one pair of parallel opposite sides.
A convex quadrilateral is
Simple: The boundary does not cross itself.
Star-shaped: The whole interior is visible from a single point, without crossing any edge.
Alternative hierarchy
De Villiers defines a rectangle more generally as any quadrilateral with axes of symmetry through each pair of opposite sides. This definition includes both right-angled rectangles and crossed rectangles. Each has an axis of symmetry parallel to and equidistant from a pair of opposite sides, and another which is the perpendicular bisector of those sides, but, in the case of the crossed rectangle, the first axis is not an axis of symmetry for either side that it bisects.
Quadrilaterals with two axes of symmetry, each through a pair of opposite sides, belong to the larger class of quadrilaterals with at least one axis of symmetry through a pa |
https://en.wikipedia.org/wiki/Domenico%20Maria%20Novara%20da%20Ferrara | Domenico Maria Novara (1454–1504) was an Italian scientist.
Life
Born in Ferrara, for 21 years he was professor of astronomy at the University of Bologna, and in 1500 he also lectured in mathematics at Rome. He was notable as a Platonist astronomer, and in 1496 he taught Nicolaus Copernicus astronomy. He was also an astrologer.
At Bologna, Novara was assisted by Copernicus, with whom he observed a lunar occultation of Aldebaran. Copernicus later used this observation to disprove Ptolemy's model of lunar distance.
Copernicus had started out as Novara's student and then became his assistant and co-worker. Novara in turn declared that his teacher had been the famous astronomer Regiomontanus, who was once a pupil of Georg Purbach. Novara was initially educated at the University of Florence, at the time a major center of Neoplatonism. He studied there under Luca Pacioli, a friend of Leonardo da Vinci.
Novara's writings are largely lost, except for a few astrological almanacs written for the university. But Copernicus' De revolutionibus orbium coelestium (published in 1543, long after Novara's death) records that on 9 March 1497 Novara witnessed Copernicus' first observation. Both men were described as "free minds and free souls," and Novara believed that his findings would have shaken Ptolemy's "unshakable" geocentric system.
Novara died in 1504 in Bologna.
References
A. Romer, "The welcoming of Copernicus's de revolutionibus: The commentariolus and its reception" Physics in Perspective, 1(2): 157–183, 1999.
External links
Encyclopedia of Medieval Italy
Copernicus and di Novara
Copernicus and his revolutions
di Novara's Influence on Copernicus
1454 births
1504 deaths
Scientists from Ferrara
Italian astrologers
15th-century astrologers
16th-century astrologers
15th-century Italian astronomers
15th-century Italian mathematicians
16th-century Italian mathematicians
16th-century Italian astronomers
Academic staff of the University of Bologna |
https://en.wikipedia.org/wiki/100-year%20flood | A 100-year flood is a flood event that has on average a 1 in 100 chance (1% probability) of being equaled or exceeded in any given year.
The 100-year flood is also referred to as the 1% flood. For coastal or lake flooding, the 100-year flood is generally expressed as a flood elevation or depth, and may include wave effects. For river systems, the 100-year flood is generally expressed as a flowrate. Based on the expected 100-year flood flow rate, the flood water level can be mapped as an area of inundation. The resulting floodplain map is referred to as the 100-year floodplain. Estimates of the 100-year flood flowrate and other streamflow statistics for any stream in the United States are available. In the UK, the Environment Agency publishes a comprehensive map of all areas at risk of a 1 in 100 year flood. Areas near the coast of an ocean or large lake also can be flooded by combinations of tide, storm surge, and waves. Maps of the riverine or coastal 100-year floodplain may figure importantly in building permits, environmental regulations, and flood insurance. These analyses generally represent 20th-century climate.
Probability
A common misunderstanding is that a 100-year flood is likely to occur only once in a 100-year period. In fact, there is approximately a 63.4% chance of one or more 100-year floods occurring in any 100-year period. On the Danube River at Passau, Germany, the actual intervals between 100-year floods during 1501 to 2013 ranged from 37 to 192 years. The probability Pe that one or more floods occurring during any period will exceed a given flood threshold can be expressed, using the binomial distribution, as
where T is the threshold return period (e.g. 100-yr, 50-yr, 25-yr, and so forth), and n is the number of years in the period. The probability of exceedance Pe is also described as the natural, inherent, or hydrologic risk of failure. However, the expected value of the number of 100-year floods occurring in any 100-year period is 1.
Ten-year floods have a 10% chance of occurring in any given year (Pe =0.10); 500-year have a 0.2% chance of occurring in any given year (Pe =0.002); etc. The percent chance of an X-year flood occurring in a single year is 100/X. A similar analysis is commonly applied to coastal flooding or rainfall data. The recurrence interval of a storm is rarely identical to that of an associated riverine flood, because of rainfall timing and location variations among different drainage basins.
The field of extreme value theory was created to model rare events such as 100-year floods for the purposes of civil engineering. This theory is most commonly applied to the maximum or minimum observed stream flows of a given river. In desert areas where there are only ephemeral washes, this method is applied to the maximum observed rainfall over a given period of time (24-hours, 6-hours, or 3-hours). The extreme value analysis only considers the most extreme event observed in a given year. So, between the lar |
https://en.wikipedia.org/wiki/Bureau%20of%20Labor%20Statistics | The Bureau of Labor Statistics (BLS) is a unit of the United States Department of Labor. It is the principal fact-finding agency for the U.S. government in the broad field of labor economics and statistics and serves as a principal agency of the U.S. Federal Statistical System. The BLS collects, processes, analyzes, and disseminates essential statistical data to the American public, the U.S. Congress, other Federal agencies, State and local governments, business, and labor representatives. The BLS also serves as a statistical resource to the United States Department of Labor, and conducts research measuring the income levels families need to maintain a satisfactory quality of life.
BLS data must satisfy a number of criteria, including relevance to current social and economic issues, timeliness in reflecting today's rapidly changing economic conditions, accuracy and consistently high statistical quality, impartiality in both subject matter and presentation, and accessibility to all. To avoid the appearance of partiality, the dates of major data releases are scheduled more than a year in advance, in coordination with the Office of Management and Budget.
History
The Bureau of Labor was established within the Department of the Interior on June 27, 1884, to collect information about employment and labor. Its creation under the Bureau of Labor Act (23 Stat. 60) stemmed from the findings of U.S. Senator Henry W. Blair's "Labor and Capital Hearings", which examined labor issues and working conditions in the U.S. Statistician Carroll D. Wright became the first U.S. Commissioner of Labor in 1885, a position he held until 1905. The Bureau's placement within the federal government structure changed three times in the first 29 years following its formation. It was made an independent (sub-Cabinet) department by the Department of Labor Act (25 Stat. 182) on June 13, 1888. The Bureau was then incorporated into the Department of Commerce and Labor by the Department of Commerce Act (32 Stat. 827) on February 14, 1903. Finally, it was transferred under the Department of Labor in 1913 where it resides today. The BLS is now headquartered in the Postal Square Building near the United States Capitol and Union Station.
Since 1915, the BLS has published the Monthly Labor Review, a journal focused on the data and methodologies of labor statistics.
The BLS is headed by a commissioner who serves a four-year term from the date he or she takes office. The most recent Commissioner of Labor Statistics is William W. Beach, who was assumed office on March 28, 2019 Dr. William Beach was confirmed by the United States Senate on March 13, 2019. William Beach's Senate Confirmation.
Erica Groshen, who was confirmed by the U.S. Senate on January 2, 2013 and sworn in as the 14th Commissioner of Labor Statistics on January 29, 2013, for a term that ended on January 27, 2017. William Wiatrowski, Deputy Commissioner of the BLS, was serving as Acting Commissioner until the next co |
https://en.wikipedia.org/wiki/Thrust%20fault | A thrust fault is a break in the Earth's crust, across which older rocks are pushed above younger rocks.
Thrust geometry and nomenclature
Reverse faults
A thrust fault is a type of reverse fault that has a dip of 45 degrees or less.
If the angle of the fault plane is lower (often less than 15 degrees from the horizontal) and the displacement of the overlying block is large (often in the kilometer range) the fault is called an overthrust or overthrust fault. Erosion can remove part of the overlying block, creating a fenster (or window) – when the underlying block is exposed only in a relatively small area. When erosion removes most of the overlying block, leaving island-like remnants resting on the lower block, the remnants are called klippen (singular klippe).
Blind thrust faults
If the fault plane terminates before it reaches the Earth's surface, it is called a blind thrust fault. Because of the lack of surface evidence, blind thrust faults are difficult to detect until rupture. The destructive 1994 earthquake in Northridge, Los Angeles, California, was caused by a previously undiscovered blind thrust fault.
Because of their low dip, thrusts are also difficult to appreciate in mapping, where lithological offsets are generally subtle and stratigraphic repetition is difficult to detect, especially in peneplain areas.
Fault-bend folds
Thrust faults, particularly those involved in thin-skinned style of deformation, have a so-called ramp-flat geometry. Thrusts mainly propagate along zones of weakness within a sedimentary sequence, such as mudstones or halite layers; these parts of the thrust are called decollements. If the effectiveness of the decollement becomes reduced, the thrust will tend to cut up the section to a higher stratigraphic level until it reaches another effective decollement where it can continue as bedding parallel flat. The part of the thrust linking the two flats is known as a ramp and typically forms at an angle of about 15°–30° to the bedding. Continued displacement on a thrust over a ramp produces a characteristic fold geometry known as a ramp anticline or, more generally, as a fault-bend fold.
Fault-propagation folds
Fault-propagation folds form at the tip of a thrust fault where propagation along the decollement has ceased, but displacement on the thrust behind the fault tip continues. The formation of an asymmetric anticline-syncline fold pair accommodates the continuing displacement. As displacement continues, the thrust tip starts to propagate along the axis of the syncline. Such structures are also known as tip-line folds. Eventually, the propagating thrust tip may reach another effective decollement layer, and a composite fold structure will develop with fault-bending and fault-propagation folds' characteristics.
Thrust duplex
Duplexes occur where two decollement levels are close to each other within a sedimentary sequence, such as the top and base of a relatively strong sandstone layer bounded by two relativel |
https://en.wikipedia.org/wiki/Boundary%20%28topology%29 | In topology and mathematics in general, the boundary of a subset of a topological space is the set of points in the closure of not belonging to the interior of . An element of the boundary of is called a boundary point of . The term boundary operation refers to finding or taking the boundary of a set. Notations used for boundary of a set include and . Some authors (for example Willard, in General Topology) use the term frontier instead of boundary in an attempt to avoid confusion with a different definition used in algebraic topology and the theory of manifolds. Despite widespread acceptance of the meaning of the terms boundary and frontier, they have sometimes been used to refer to other sets. For example, Metric Spaces by E. T. Copson uses the term boundary to refer to Hausdorff's border, which is defined as the intersection of a set with its boundary. Hausdorff also introduced the term residue, which is defined as the intersection of a set with the closure of the border of its complement.
A connected component of the boundary of is called a boundary component of .
Common definitions
There are several equivalent definitions for the of a subset of a topological space which will be denoted by or simply if is understood:
It is the closure of minus the interior of in :
where denotes the closure of in and denotes the topological interior of in
It is the intersection of the closure of with the closure of its complement:
It is the set of points such that every neighborhood of contains at least one point of and at least one point not of :
A of a set refers to any element of that set's boundary. The boundary defined above is sometimes called the set's to distinguish it from other similarly named notions such as the boundary of a manifold with boundary or the boundary of a manifold with corners, to name just a few examples.
Properties
The closure of a set equals the union of the set with its boundary:
where denotes the closure of in
A set is closed if and only if it contains its boundary, and open if and only if it is disjoint from its boundary. The boundary of a set is closed; this follows from the formula which expresses as the intersection of two closed subsets of
("Trichotomy") Given any subset each point of lies in exactly one of the three sets and Said differently, and these three sets are pairwise disjoint. Consequently, if these set are not empty then they form a partition of
A point is a boundary point of a set if and only if every neighborhood of contains at least one point in the set and at least one point not in the set.
The boundary of the interior of a set as well as the boundary of the closure of a set are both contained in the boundary of the set.
Conceptual Venn diagram showing the relationships among different points of a subset of = set of limit points of set of boundary points of area shaded green = set of interior points of area shaded yellow = set of iso |
https://en.wikipedia.org/wiki/Cauchy%27s%20integral%20formula | In mathematics, Cauchy's integral formula, named after Augustin-Louis Cauchy, is a central statement in complex analysis. It expresses the fact that a holomorphic function defined on a disk is completely determined by its values on the boundary of the disk, and it provides integral formulas for all derivatives of a holomorphic function. Cauchy's formula shows that, in complex analysis, "differentiation is equivalent to integration": complex differentiation, like integration, behaves well under uniform limits – a result that does not hold in real analysis.
Theorem
Let be an open subset of the complex plane , and suppose the closed disk defined as
is completely contained in . Let be a holomorphic function, and let be the circle, oriented counterclockwise, forming the boundary of . Then for every in the interior of ,
The proof of this statement uses the Cauchy integral theorem and like that theorem, it only requires to be complex differentiable. Since can be expanded as a power series in the variable
it follows that holomorphic functions are analytic, i.e. they can be expanded as convergent power series.
In particular is actually infinitely differentiable, with
This formula is sometimes referred to as Cauchy's differentiation formula.
The theorem stated above can be generalized. The circle can be replaced by any closed rectifiable curve in which has winding number one about . Moreover, as for the Cauchy integral theorem, it is sufficient to require that be holomorphic in the open region enclosed by the path and continuous on its closure.
Note that not every continuous function on the boundary can be used to produce a function inside the boundary that fits the given boundary function. For instance, if we put the function , defined for , into the Cauchy integral formula, we get zero for all points inside the circle. In fact, giving just the real part on the boundary of a holomorphic function is enough to determine the function up to an imaginary constant — there is only one imaginary part on the boundary that corresponds to the given real part, up to addition of a constant. We can use a combination of a Möbius transformation and the Stieltjes inversion formula to construct the holomorphic function from the real part on the boundary. For example, the function has real part . On the unit circle this can be written . Using the Möbius transformation and the Stieltjes formula we construct the function inside the circle. The term makes no contribution, and we find the function . This has the correct real part on the boundary, and also gives us the corresponding imaginary part, but off by a constant, namely .
Proof sketch
By using the Cauchy integral theorem, one can show that the integral over (or the closed rectifiable curve) is equal to the same integral taken over an arbitrarily small circle around . Since is continuous, we can choose a circle small enough on which is arbitrarily close to . On the other hand, the integral
o |
https://en.wikipedia.org/wiki/B%C3%A6rum | {{Historical populations
|footnote = Source: Statistics Norway.
|shading = off
|1951|35838
|1961|57573
|1971|76580
|1981|80385
|1991|90579
|2001|101340
|2011|112789
|2012|114489
|2020|127731
|2021|128760
}}
Bærum () is a municipality in the Greater Oslo Region in Norway that forms an affluent suburb of Oslo on the west coast of the city. Bærum is Norway's fifth largest municipality with a population of 128,760 (2021). It is part of the electoral district and historical county of Akershus and of the newer Viken County. The administrative centre of the municipality is the town of Sandvika. Bærum was established as a municipality on 1 January 1838.
Bærum has the highest income per capita in Norway and the highest proportion of university-educated individuals. Bærum, particularly its eastern neighbourhoods bordering West End Oslo, is one of Norway's priciest and most fashionable residential areas, leading Bærum residents to be frequently stereotyped as snobs in Norwegian popular culture. The municipality has been voted the best Norwegian place to live in considering governance and public services to citizens.
Name
The name (Old Norse: Bergheimr) is composed of berg, which means "mountain", and heimr, which means "homestead" or "farm". It probably originally belonged to a farm located at the base of the prominent mountain of Kolsås. In Old Norse times, the municipality was often called Bergheimsherað, meaning "the herað (parish/district) of Bergheimr".
Coat-of-arms
The coat-of-arms was granted on 9 January 1976. They show an old silver-colored lime kiln on a green background. That was an important aspect of the local economy from the Middle Ages until around 1800. There are still some original ovens visible in the municipality.
History
The area known today as Bærum was a fertile agricultural area as far back as the Bronze Age, and several archeological finds stem from the Iron Age. The first mention of the name is from the saga of Sverre of Norway, from about 1200. There are ruins of stone churches from the 12th century at Haslum and Tanum.
The pilgrim road to Trondheim that was established after 1030 went through Bærum, and there is evidence that lime kilns were in use in the area in 850. There were shipping ports for the quicklime at Slependen and Sandvika. The lime kiln is the main motif for the municipality's coat of arms.
In the 17th century, iron ore was discovered in Bærum and the ironworks at Bærums Verk were founded. Industries such as paper mills, nail factories, sawmills, glassworks, and brickworks were established along the rivers Lysakerelven and Sandvikselva in the followi |
https://en.wikipedia.org/wiki/Residue%20%28complex%20analysis%29 | In mathematics, more specifically complex analysis, the residue is a complex number proportional to the contour integral of a meromorphic function along a path enclosing one of its singularities. (More generally, residues can be calculated for any function that is holomorphic except at the discrete points {ak}k, even if some of them are essential singularities.) Residues can be computed quite easily and, once known, allow the determination of general contour integrals via the residue theorem.
Definition
The residue of a meromorphic function at an isolated singularity , often denoted , , or , is the unique value such that has an analytic antiderivative in a punctured disk .
Alternatively, residues can be calculated by finding Laurent series expansions, and one can define the residue as the coefficient a−1 of a Laurent series.
The definition of a residue can be generalized to arbitrary Riemann surfaces. Suppose is a 1-form on a Riemann surface. Let be meromorphic at some point , so that we may write in local coordinates as . Then, the residue of at is defined to be the residue of at the point corresponding to .
Examples
Residue of a monomial
Computing the residue of a monomial
makes most residue computations easy to do. Since path integral computations are homotopy invariant, we will let be the circle with radius . Then, using the change of coordinates we find that
hence our integral now reads as
Application of monomial residue
As an example, consider the contour integral
where C is some simple closed curve about 0.
Let us evaluate this integral using a standard convergence result about integration by series. We can substitute the Taylor series for into the integrand. The integral then becomes
Let us bring the 1/z5 factor into the series. The contour integral of the series then writes
Since the series converges uniformly on the support of the integration path, we are allowed to exchange integration and summation.
The series of the path integrals then collapses to a much simpler form because of the previous computation. So now the integral around C of every other term not in the form cz−1 is zero, and the integral is reduced to
The value 1/4! is the residue of ez/z5 at z = 0, and is denoted
Calculating residues
Suppose a punctured disk D = {z : 0 < |z − c| < R} in the complex plane is given and f is a holomorphic function defined (at least) on D. The residue Res(f, c) of f at c is the coefficient a−1 of in the Laurent series expansion of f around c. Various methods exist for calculating this value, and the choice of which method to use depends on the function in question, and on the nature of the singularity.
According to the residue theorem, we have:
where γ traces out a circle around c in a counterclockwise manner. We may choose the path γ to be a circle of radius ε around c, where ε is as small as we desire. This may be used for calculation in cases where the integral can be calculated directly, bu |
https://en.wikipedia.org/wiki/Birthday%20problem | In probability theory, the birthday problem asks for the probability that, in a set of randomly chosen people, at least two will share a birthday. The birthday paradox refers to the counterintuitive fact that only 23 people are needed for that probability to exceed 50%.
The birthday paradox is a veridical paradox: it seems wrong at first glance but is, in fact, true. While it may seem surprising that only 23 individuals are required to reach a 50% probability of a shared birthday, this result is made more intuitive by considering that the birthday comparisons will be made between every possible pair of individuals. With 23 individuals, there are = 253 pairs to consider, far more than half the number of days in a year.
Real-world applications for the birthday problem include a cryptographic attack called the birthday attack, which uses this probabilistic model to reduce the complexity of finding a collision for a hash function, as well as calculating the approximate risk of a hash collision existing within the hashes of a given size of population.
The problem is generally attributed to Harold Davenport in about 1927, though he did not publish it at the time. Davenport did not claim to be its discoverer "because he could not believe that it had not been stated earlier". The first publication of a version of the birthday problem was by Richard von Mises in 1939.
Calculating the probability
From a permutations perspective, let the event be the probability of finding a group of 23 people without any repeated birthdays. Where the event is the probability of finding a group of 23 people with at least two people sharing same birthday, . is the ratio of the total number of birthdays, , without repetitions and order matters (e.g. for a group of 2 people, mm/dd birthday format, one possible outcome is ) divided by the total number of birthdays with repetition and order matters, , as it is the total space of outcomes from the experiment (e.g. 2 people, one possible outcome is ). Therefore and are permutations.
Another way the birthday problem can be solved is by asking for an approximate probability that in a group of people at least two have the same birthday. For simplicity, leap years, twins, selection bias, and seasonal and weekly variations in birth rates are generally disregarded, and instead it is assumed that there are 365 possible birthdays, and that each person's birthday is equally likely to be any of these days, independent of the other people in the group. For independent birthdays, the uniform distribution on birthdays is the distribution that minimizes the probability of two people with the same birthday; any unevenness increases this probability. The problem of a non-uniform number of births occurring during each day of the year was first addressed by Murray Klamkin in 1967. As it happens, the real-world distribution yields a critical size of 23 to reach 50%.
The goal is to compute , the probability that at least two people in |
https://en.wikipedia.org/wiki/Demographics%20of%20Cuba | The demographic characteristics of Cuba are known through census which have been conducted and analyzed by different bureaus since 1774. The National Office of Statistics of Cuba (ONE) since 1953. The most recent census was conducted in September 2012. The population of Cuba at the 2012 census was 12 million. The population density is 103.6 inhabitants per square kilometer, and the overall life expectancy in Cuba is 78.0 years. The population has always increased from one census to the next, with the exception of the 2012 census, when the count decreased by 10,000. Since 1740, Cuba's birth rate has surpassed its death rate; the natural growth rate of the country is positive. Cuba is in the fourth stage of demographic transition. In terms of age structure, the population is dominated (71.1%) by the 15- to 64-year-old segment. The median age of the population is 39.5, making it the oldest in the Americas, and the gender ratio of the total population is 0.99 males per female.
Population
According to the 2002 census, Cuba's population was 11,033,758, whereas the 2016 census numbered the population at 11,089,511. There was a drop between the 2002 and 2012 censuses which was the first drop in Cuba's population since Cuba's war of independence. This drop was due to low fertility and emigration, as during this time (fiscal years 2003 to 2012), 42,028 Cubans received legal permanent residence in the United States. Consequently, Cuba is also the oldest country in the Americas in terms of median age, due to a high amount of emigration by younger Cubans to the U.S. In the last few years before the end of the wet feet, dry feet policy on January 12, 2017, the number of Cubans moving to the United States significantly outnumbered the natural increase during those years.
Population by subdivisions
Vital statistics
Structure of the population
Racial groups
Ancestral origins
According to the previous censuses, the Chinese were counted as white.
The ancestry of Cubans comes from many sources:
Spanish
During the 18th, 19th and early part of the 20th century, large waves of Spanish immigrants from Canary Islands, Catalonia, Andalusia, Galicia, and Asturias emigrated to Cuba. Between 1820 and 1898, a total of 508,455 people left Spain, and more than 750,000 Spanish immigrants left for Cuba between 1899 and 1930, with many returning to Spain. There are 139,851 Spanish citizens living in Cuba as of 1 January 2018.
Canary Islanders
Catalans
Andalusians
Galicians
Asturians
The Slave trade brought Africans to Cuba during its early history:
Between 1842 and 1873, 221,000 African slaves entered Cuba.
Africans
People of the Americas:
Haitians
Jamaicans
Other European people that have contributed include:
Germans
French
Portuguese
Italians
Russians
People from Asia:
Chinese
Koreans
Filipino
Lebanese, Syrian, Egyptian, Palestinian (Arab Cubans)
Between 1842 and 1873, 124,800 Chinese arrived.
Genetics
An autosomal study from 2014 ha |
https://en.wikipedia.org/wiki/Lyapunov%20fractal | In mathematics, Lyapunov fractals (also known as Markus–Lyapunov fractals) are bifurcational fractals derived from an extension of the logistic map in which the degree of the growth of the population, r, periodically switches between two values A and B.
A Lyapunov fractal is constructed by mapping the regions of stability and chaotic behaviour (measured using the Lyapunov exponent ) in the a−b plane for given periodic sequences of a and b. In the images, yellow corresponds to (stability), and blue corresponds to (chaos).
Lyapunov fractals were discovered in the late 1980s by the Germano-Chilean physicist Mario Markus from the Max Planck Institute of Molecular Physiology. They were introduced to a large public by a science popularization article on recreational mathematics published in Scientific American in 1991.
Properties
Lyapunov fractals are generally drawn for values of A and B in the interval . For larger values, the interval [0,1] is no longer stable, and the sequence is likely to be attracted by infinity, although convergent cycles of finite values continue to exist for some parameters. For all iteration sequences, the diagonal a = b is always the same as for the standard one parameter logistic function.
The sequence is usually started at the value 0.5, which is a critical point of the iterative function. The other (even complex valued) critical points of the iterative function during one entire round are those that pass through the value 0.5 in the first round. A convergent cycle must attract at least one critical point. Therefore, all convergent cycles can be obtained by just shifting the iteration sequence, and keeping the starting value 0.5. In practice, shifting this sequence leads to changes in the fractal, as some branches get covered by others. For instance, the Lyapunov fractal for the iteration sequence AB (see top figure on the right) is not perfectly symmetric with respect to a and b.
Algorithm
The algorithm for computing Lyapunov fractals works as follows:
Choose a string of As and Bs of any nontrivial length (e.g., AABAB).
Construct the sequence formed by successive terms in the string, repeated as many times as necessary.
Choose a point .
Define the function if , and if .
Let , and compute the iterates .
Compute the Lyapunov exponent:In practice, is approximated by choosing a suitably large and dropping the first summand as for .
Color the point according to the value of obtained.
Repeat steps (3–7) for each point in the image plane.
More Iterations
More dimensions
Lyapunov fractals can be calculated in more than two dimensions. The sequence string for a n-dimensional fractal has to be built from an alphabet with n characters, e.g. "ABBBCA" for a 3D fractal, which can be visualized either as 3D object or as an animation showing a "slice" in the C direction for each animation frame, like the example given here.
Remark
Note that the term „fractal“ in this wikipedia-page is a colloquial denominati |
https://en.wikipedia.org/wiki/Sieve%20of%20Eratosthenes | In mathematics, the sieve of Eratosthenes is an ancient algorithm for finding all prime numbers up to any given limit.
It does so by iteratively marking as composite (i.e., not prime) the multiples of each prime, starting with the first prime number, 2. The multiples of a given prime are generated as a sequence of numbers starting from that prime, with constant difference between them that is equal to that prime. This is the sieve's key distinction from using trial division to sequentially test each candidate number for divisibility by each prime. Once all the multiples of each discovered prime have been marked as composites, the remaining unmarked numbers are primes.
The earliest known reference to the sieve (, kóskinon Eratosthénous) is in Nicomachus of Gerasa's Introduction to Arithmetic, an early 2nd cent. CE book which attributes it to Eratosthenes of Cyrene, a 3rd cent. BCE Greek mathematician, though describing the sieving by odd numbers instead of by primes.
One of a number of prime number sieves, it is one of the most efficient ways to find all of the smaller primes. It may be used to find primes in arithmetic progressions.
Overview
A prime number is a natural number that has exactly two distinct natural number divisors: the number 1 and itself.
To find all the prime numbers less than or equal to a given integer by Eratosthenes' method:
Create a list of consecutive integers from 2 through : .
Initially, let equal 2, the smallest prime number.
Enumerate the multiples of by counting in increments of from to , and mark them in the list (these will be ; the itself should not be marked).
Find the smallest number in the list greater than that is not marked. If there was no such number, stop. Otherwise, let now equal this new number (which is the next prime), and repeat from step 3.
When the algorithm terminates, the numbers remaining not marked in the list are all the primes below .
The main idea here is that every value given to will be prime, because if it were composite it would be marked as a multiple of some other, smaller prime. Note that some of the numbers may be marked more than once (e.g., 15 will be marked both for 3 and 5).
As a refinement, it is sufficient to mark the numbers in step 3 starting from , as all the smaller multiples of will have already been marked at that point. This means that the algorithm is allowed to terminate in step 4 when is greater than .
Another refinement is to initially list odd numbers only, , and count in increments of in step 3, thus marking only odd multiples of . This actually appears in the original algorithm. This can be generalized with wheel factorization, forming the initial list only from numbers coprime with the first few primes and not just from odds (i.e., numbers coprime with 2), and counting in the correspondingly adjusted increments so that only such multiples of are generated that are coprime with those small primes, in the first place.
Example
To find al |
https://en.wikipedia.org/wiki/Emil%20Julius%20Gumbel | Emil Julius Gumbel (18 July 1891, in Munich – 10 September 1966, in New York City) was a German mathematician and political writer.
Gumbel specialised in mathematical statistics and, along with Leonard Tippett and Ronald Fisher, was instrumental in the development of extreme value theory, which has practical applications in many fields, including engineering and finance. In 1958, Gumbel published a key book, Statistics of Extremes, in which he derived and analyzed the probability distribution that is now known as the Gumbel distribution in his honor.
In the 1920s and early 1930s, Gumbel was considered unusual and highly controversial in German academic circles for his vocal support of left-wing politics and pacifism, and his opposition to Fascism. His influential writings about the politically motivated Feme murders made the case that the Weimar Republic was corruptly anti-leftist and anti-republican. Gumbel publicly opposed the Nazi Party and, in 1932, he was one of the 33 prominent signers of the Urgent Call for Unity.
Biography
Born to a prominent Jewish family in Württemberg, Gumbel graduated in mathematics from the University of Munich, completing his doctoral thesis on the topic of population statistics shortly before the outbreak of the First World War. After a short period of military service, he was discharged in 1915 on medical grounds and he joined the University of Berlin to work with the prominent Russian statistician Ladislaus von Bortkiewicz. From this time onwards he became much more politically active. He joined the Independent Social Democrat Party in 1917, and became a prominent member of the pacifist New Fatherland League which was later renamed the German League for Human Rights. In January 1918, Gumbel took up a position with the electronics company Telefunken, researching sound transmitter waves, and he continued his political activities with the support of one of the firm's founders, Georg Count von Arco, a prominent member of the human rights movement. In 1922, Gumbel became Professor of Mathematical Statistics at the University of Heidelberg, where he soon found that combining academic work with politics was much more controversial, resulting in protests by students and faculty members, who were mostly right-wing, and strong criticism in the right-wing press.
Among the Nazis' most-hated public intellectuals, Gumbel was forced out of his position in Heidelberg in 1932. He then moved to France, where he taught at the École libre des hautes études in Paris, and in Lyon, as well as continuing his political activities and helping other refugees, until the German invasion of 1940. He then left Europe for the United States, where he taught at the New School for Social Research and Columbia University in New York City until his death in 1966.
When he died of lung cancer in 1966, Gumbel's papers were made a part of The Emil J. Gumbel Collection, Political Papers of an Anti-Nazi Scholar in Weimar and Exile. These papers inclu |
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