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https://en.wikipedia.org/wiki/Erd%C5%91s%E2%80%93Mordell%20inequality | In Euclidean geometry, the Erdős–Mordell inequality states that for any triangle ABC and point P inside ABC, the sum of the distances from P to the sides is less than or equal to half of the sum of the distances from P to the vertices. It is named after Paul Erdős and Louis Mordell. posed the problem of proving the inequality; a proof was provided two years later by . This solution was however not very elementary. Subsequent simpler proofs were then found by , , and .
Barrow's inequality is a strengthened version of the Erdős–Mordell inequality in which the distances from P to the sides are replaced by the distances from P to the points where the angle bisectors of ∠APB, ∠BPC, and ∠CPA cross the sides. Although the replaced distances are longer, their sum is still less than or equal to half the sum of the distances to the vertices.
Statement
Let be an arbitrary point P inside a given triangle , and let , , and be the perpendiculars from to the sides of the triangles.
(If the triangle is obtuse, one of these perpendiculars may cross through a different side of the triangle and end on the line supporting one of the sides.) Then the inequality states that
Proof
Let the sides of ABC be a opposite A, b opposite B, and c opposite C; also let PA = p, PB = q, PC = r, dist(P;BC) = x, dist(P;CA) = y, dist(P;AB) = z. First, we prove that
This is equivalent to
The right side is the area of triangle ABC, but on the left side, r + z is at least the height of the triangle; consequently, the left side cannot be smaller than the right side. Now reflect P on the angle bisector at C. We find that cr ≥ ay + bx for P's reflection. Similarly, bq ≥ az + cx and ap ≥ bz + cy. We solve these inequalities for r, q, and p:
Adding the three up, we get
Since the sum of a positive number and its reciprocal is at least 2 by AM–GM inequality, we are finished. Equality holds only for the equilateral triangle, where P is its centroid.
Another strengthened version
Let ABC be a triangle inscribed into a circle (O) and P be a point inside of ABC. Let D, E, F be the orthogonal projections of P onto BC, CA, AB. M, N, Q be the orthogonal projections of P onto tangents to (O) at A, B, C respectively, then:
Equality hold if and only if triangle ABC is equilateral (; )
A generalization
Let be a convex polygon, and be an interior point of . Let be the distance from to the vertex , the distance from to the side , the segment of the bisector of the angle from to its intersection with the side then :
In absolute geometry
In absolute geometry the Erdős–Mordell inequality is equivalent, as proved in , to the statement
that the sum of the angles of a triangle is less than or equal to two right angles.
See also
List of triangle inequalities
References
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External links
Alexander Bogomolny, "Erdös-Mordell Inequality", from Cut-the-Knot.
Triangle inequalities |
https://en.wikipedia.org/wiki/Methanocalculus | Methanocalculus is a genus of the Methanomicrobiales, and is known to include methanogens.
The genome of Methanocalculus is somewhat different from other genera of methanogenic archaea, with less than 90% 16S ribosomal RNA similarity. The species within Methanocalculus also have a greater tolerance to salt than other microorganisms, and they can live at salt concentrations as high as 125 g/L. Some species within Methanocalculus are neutrophiles, and Methanocalculus natronophilus, discovered in 2013, is a strict alkaliphile.
Nomenclature
The name "Methanocalculus" has Latin roots: "methano" for methane and "calculus" for gravel. Overall, it means gravel-shaped organism that produces methane.
Phylogeny
The currently accepted taxonomy is based on the List of Prokaryotic names with Standing in Nomenclature (LPSN) and National Center for Biotechnology Information (NCBI).
See also
List of Archaea genera
References
Further reading
Scientific journals
Scientific books
Scientific databases
External links
Methanocalculus at BacDive - the Bacterial Diversity Metadatabase
Archaea genera
Euryarchaeota |
https://en.wikipedia.org/wiki/Imre%20Csisz%C3%A1r | Imre Csiszár () is a Hungarian mathematician with contributions to information theory
and probability theory. In 1996 he won the Claude E. Shannon Award, the highest annual
award given in the field of information theory.
He was born on 7 February 1938 in Miskolc, Hungary. He became interested in mathematics
in middle school. He was inspired by his father who was a forest engineer and was among the first to use mathematical techniques in his area.
He studied mathematics at the Eötvös Loránd University, Budapest, and received his Diploma in 1961. He got his PhD in 1967 and the scientific degree Doctor of Mathematical Science in 1977.
Later, he was influenced by Alfréd Rényi, who was very active in the area of probability theory. In 1990 he was elected Corresponding Member of the Hungarian Academy of Sciences, and in 1995 he became Full Member. Professor Csiszar has been with the Mathematical Institute of the Hungarian
Academy of Sciences since 1961. He has been Head of the Information Theory Group there since 1968, and presently he is Head of the Stochastics Department. He is also Professor of Mathematics at the L. Eotvos University, Budapest. He has held Visiting Professorships at various universities including Bielefeld University, Germany (1981), University of Maryland, College Park (several times, last in 1992), Stanford University (1982), University of Virginia (1985–86), etc. He has been Visiting Researcher at the University of Tokyo in 1988, and at NTT, Japan, in 1994. He is married and has four children.
He is a Fellow of the IEEE, and is a member of several other learned societies, including the Bernoulli Society for Mathematical Statistics and Probability. He has received several academic awards, including the Book Excellence Award of the Hungarian Academy of Sciences for his 1981 Information Theory monograph, the 1988 Paper Award of the IEEE Information Theory Society, the 2015 IEEE Richard Hamming Medal and the Academy Award for Interdisciplinary Research of the Hungarian Academy of Sciences in 1989.
Books
With János Körner: Information Theory: Coding Theorems for Discrete Memoryless Systems, Academic Press 1981, 2nd edition Cambridge University Press 2011.
With Paul C. Shields: Information Theory and Statistics: A Tutorial, Now Publishers, Inc., 2004.
External links
Announcement of IEEE 2015 Hamming Medal
Members of the Hungarian Academy of Sciences
20th-century Hungarian mathematicians
1938 births
Living people
Information theorists
Probability theorists
Academic staff of Bielefeld University
Fellow Members of the IEEE |
https://en.wikipedia.org/wiki/2007%20Australian%20Lacrosse%20League%20season | Results and statistics for the Australian Lacrosse League season of 2007.
Game 22
Saturday, 20 October 2007, Melbourne, Victoria
Goalscorers:
Vic: Nick LeGuen 4-1, Jake Egan 2, Blair Pepperell 2, Clinton Lander 0-1, John Tokarua 0-1.
SA: Anson Carter 3, Leigh Perham 1-3, Wes Green 1, Stefan Guerin 1, Brock Pettigrove 1.
Game 23
Sunday, 21 October 2007, Melbourne, Victoria
Goalscorers:
Vic: Nick LeGuen 3-2, Blair Pepperell 2, Andrew Whitbourn 2, Clinton Lander 1-2, Sam Bullock 1-1, Jake Egan 1, Adam Townley 1, Ben Waite 1, Aaron Onofretchook 0-2, Alistair Gloutnay 0-1.
SA: Kieren Lennox 2, Will Pickett 2, Leigh Perham 1-3, Chris Averay 1-1, Anson Carter 1-1, Ryan Gaspari 1, Stefan Guerin 1, Brock Pettigrove 1.
Game 24
Friday, 26 October 2007, Adelaide, South Australia
Goalscorers:
SA: Chris Averay 2-2, Anson Carter 2-1, Leigh Perham 1-1, Stefan Guerin 1, Kieren Lennox 1, Jason MacKinnon 1, Brock Pettigrove 1, Will Pickett 1, Ryan Stone 1.
WA: Wayne Curran 6, Kim Delfs 3-1, Jesse Stack 1-2, Jason Battaglia 1, Blair Coggan 1, Brad Goddard 1, James Watson-Galbraith 1, Mark Whiteman 0-1.
Game 25
Saturday, 27 October 2007, Adelaide, South Australia
Goalscorers:
SA: Anson Carter 4, Leigh Perham 3, Chris Averay 1-2, Ryan Gaspari 1, Jack Woodford 0-1.
WA: Kim Delfs 4-3, Jesse Stack 4-1, Brad Goddard 3, Wayne Curran 2-1, Adam Delfs 1, Ben Tippett 1, James Watson-Galbraith 0-2, Mark Whiteman 0-1.
Game 26
Friday, 2 November 2007, Perth, Western Australia
Goalscorers:
WA: Wayne Curran 2, Brad Goddard 1, Jesse Stack 0-1.
Vic: Sam Bullock 3, Ben Waite 3, Adam Townley 2, Jake Egan 1, Nick LeGuen 1, Aaron Onofretchook 1, Clinton Lander 0-2, Blair Pepperell 0-2.
Game 27
Saturday, 3 November 2007, Perth, Western Australia
Goalscorers:
WA: Wayne Curran 2, Kim Delfs 2, Jason Battaglia 1, Ian Berry 1, James Watson-Galbraith 1, Sam Ramsay 0-1.
Vic: Adam Townley 4, Ben Waite 3, Nick LeGuen 2, Jake Egan 1, James Lawerson 1, Aaron Onofretchook 1, Andrew Whitbourn 1, Clinton Lander 0-1, Blair Pepperell 0-1.
ALL Table 2007
Table after completion of round-robin tournament
FINAL (Game 28)
Saturday, 10 November 2007, Melbourne, Victoria
Goalscorers:
Vic: Adam Townley 4, Nick LeGuen 2-2, Clinton Lander 2-1, Aaron Onofretchook 1-1, Andrew Whitbourn 1-1, Jake Egan 1, Ben Waite 1, Blair Pepperell 0-1.
WA: Wayne Curran 1, Kim Delfs 1, James Watson-Galbraith 1, Mark Whiteman 0-1.
All-Stars
ALL 2007 Champions (Garland McHarg Trophy): Victoria
ALL 2007 Most Valuable Player (Hobbs Perpetual Trophy): Leigh Perham (SA)
ALL 2007 All-Stars: Sam Bullock, Nick LeGuen, Cameron Shepherd, Chris Tillotson, John Tokarua, Ben Waite (Vic), Wayne Curran, Kim Delfs, Brad Goddard, Tim Kennedy, Gavin Leavy (WA), Anson Carter, Leigh Perham (SA). Coach: Murray Keen (Vic). Referee: Jason Lawrence.
See also
Australian Lacrosse League
Lacrosse in Australia
External links
Australian Lacrosse League
Lacrosse Australia
Lacrosse South Australia
Lacrosse Victoria
Western Aust |
https://en.wikipedia.org/wiki/Serdar%20%C3%96zkan | Serdar Özkan (born 1 January 1987) is a Turkish professional footballer who plays as a winger.
Career statistics
References
External links
1987 births
People from Düzce
Living people
Turkish men's footballers
Turkey men's youth international footballers
Turkey men's under-21 international footballers
Turkey men's international footballers
Men's association football midfielders
Beşiktaş J.K. footballers
İstanbulspor footballers
Akçaabat Sebatspor footballers
Samsunspor footballers
Galatasaray S.K. footballers
MKE Ankaragücü footballers
Şanlıurfaspor footballers
Elazığspor footballers
Sivasspor footballers
Eskişehirspor footballers
Antalyaspor footballers
Gençlerbirliği S.K. footballers
Bursaspor footballers
Adanaspor footballers
Süper Lig players
TFF First League players |
https://en.wikipedia.org/wiki/Ostrowski%E2%80%93Hadamard%20gap%20theorem | In mathematics, the Ostrowski–Hadamard gap theorem is a result about the analytic continuation of complex power series whose non-zero terms are of orders that have a suitable "gap" between them. Such a power series is "badly behaved" in the sense that it cannot be extended to be an analytic function anywhere on the boundary of its disc of convergence. The result is named after the mathematicians Alexander Ostrowski and Jacques Hadamard.
Statement of the theorem
Let 0 < p1 < p2 < ... be a sequence of integers such that, for some λ > 1 and all j ∈ N,
Let (αj)j∈N be a sequence of complex numbers such that the power series
has radius of convergence 1. Then no point z with |z| = 1 is a regular point for f; i.e. f cannot be analytically extended from the open unit disc D to any larger open set—not even to a single point on the boundary of D.
See also
Lacunary function
Fabry gap theorem
References
External links
Mathematical series
Theorems in complex analysis |
https://en.wikipedia.org/wiki/Iitaka%20dimension | In algebraic geometry, the Iitaka dimension of a line bundle L on an algebraic variety X is the dimension of the image of the rational map to projective space determined by L. This is 1 less than the dimension of the section ring of L
The Iitaka dimension of L is always less than or equal to the dimension of X. If L is not effective, then its Iitaka dimension is usually defined to be or simply said to be negative (some early references define it to be −1). The Iitaka dimension of L is sometimes called L-dimension, while the dimension of a divisor D is called D-dimension. The Iitaka dimension was introduced by .
Big line bundles
A line bundle is big if it is of maximal Iitaka dimension, that is, if its Iitaka dimension is equal to the dimension of the underlying variety. Bigness is a birational invariant: If is a birational morphism of varieties, and if L is a big line bundle on X, then f*L is a big line bundle on Y.
All ample line bundles are big.
Big line bundles need not determine birational isomorphisms of X with its image. For example, if C is a hyperelliptic curve (such as a curve of genus two), then its canonical bundle is big, but the rational map it determines is not a birational isomorphism. Instead, it is a two-to-one cover of the canonical curve of C, which is a rational normal curve.
Kodaira dimension
The Iitaka dimension of the canonical bundle of a smooth variety is called its Kodaira dimension.
Iitaka conjecture
Consider on complex algebraic varieties in the following.
Let K be the canonical bundle on M. The dimension of H0(M,Km), holomorphic sections of Km, is denoted by Pm(M), called m-genus. Let
then N(M) becomes to be all of the positive integer with non-zero m-genus. When N(M) is not empty, for m-pluricanonical map is defined as the map
where are the bases of H0(M,Km). Then the image of , is defined as the submanifold of .
For certain let be the m-pluricanonical map where W is the complex manifold embedded into projective space PN.
In the case of surfaces with κ(M)=1 the above W is replaced by a curve C, which is an elliptic curve (κ(C)=0). We want to extend this fact to the general dimension and obtain the analytic fiber structure depicted in the upper right figure.
Given a birational map , m-pluricanonical map brings the commutative diagram depicted in the left figure, which means that , i.e. m-pluricanonical genus is birationally invariant.
It is shown by Iitaka that given n-dimensional compact complex manifold M with its Kodaira dimension κ(M) satisfying 1 ≤ κ(M) ≤ n-1, there are enough large m1,m2 such that and are birationally equivalent, which means there are the birational map . Namely, the diagram depicted in the right figure is commutative.
Furthermore, one can select that is birational with and that is birational with both and such that
is birational map, the fibers of are simply connected and the general fibers of
have Kodaira dimension 0.
The above fiber structure is calle |
https://en.wikipedia.org/wiki/List%20of%20career%20achievements%20by%20Jack%20Nicklaus | This page details statistics, records, and other achievements pertaining to championship golfer Jack Nicklaus.
Major championships
Wins (18)
1Defeated Palmer in 18-hole playoff; Nicklaus (71), Palmer (74).
2Defeated Jacobs (2nd) & Brewer (3rd) in 18-hole playoff; Nicklaus (70), Jacobs (72), Brewer (78). 1st, 2nd and 3rd prizes awarded in this playoff.
3Defeated Sanders in 18-hole playoff; Nicklaus (72), Sanders (73).
Records and trivia
In a span of 25 years, from 1962 (age 22) to 1986 (age 46), Nicklaus won 18 professional major championships. This is the most any player has won in his career.
Nicklaus held sole possession of the lead after 54 holes of a major championship on eight occasions and won each in regulation.
Nicklaus won 10 of 12 major championships when having the lead outright or tied for the lead after 54 holes and won eight times when trailing after 54 holes.
In the above-referenced 20 major championships where Nicklaus either won (18) or finished in second place (2), he was a combined 30 strokes under par in final round scoring.
In 18 professional major championship victories, Nicklaus shot 56 rounds at even par or below.
Nicklaus won two major championships in a season on five occasions (1963, 1966, 1972, 1975, and 1980).
Nicklaus won at least one major championship in four consecutive years (1970–1973).
Nicklaus is one of five players (along with Gene Sarazen, Ben Hogan, Gary Player, and Tiger Woods) to have won all four professional major championships in his career, known as the Career Grand Slam, and the second-youngest to do so in his fifth year as a professional at age 26 (Tiger Woods, fourth year at age 24).
At age 33 in 1973, Nicklaus broke Bobby Jones' record of 13 major championships (old configuration) and Walter Hagen's record of 11 professional major championships by winning his third PGA Championship; 14th and 12th win, respectively.
One of two players to achieve a "triple career slam" i.e. winning all four major championships three times in a career, the other being Tiger Woods.
Nicklaus made 39 consecutive cuts in major championships starting at the 1969 Masters and ending by being cut at the 1978 PGA Championship. In this span he won eight times, was runner-up seven times, and had 33 top-10 finishes. This record of consecutive cuts made in major championships was equaled by Tiger Woods at the 2006 Masters.
Nicklaus holds the record for most runner-up finishes in majors with 19.
Nicklaus holds the record for most top-five finishes in major championships with 56.
One of two players to finish in the top five in all four professional major championships in two different years (1971 and 1973), the other being Tiger Woods (2000 and 2005). Both players finished in the top four in all four majors once (1973 and 2005, respectively). Bobby Jones won the "Grand Slam" under the old configuration in 1930 and Ben Hogan won all three of the majors he was able to play in 1953.
Nicklaus holds the record for most top-10 fin |
https://en.wikipedia.org/wiki/Single-linkage%20clustering | In statistics, single-linkage clustering is one of several methods of hierarchical clustering. It is based on grouping clusters in bottom-up fashion (agglomerative clustering), at each step combining two clusters that contain the closest pair of elements not yet belonging to the same cluster as each other.
This method tends to produce long thin clusters in which nearby elements of the same cluster have small distances, but elements at opposite ends of a cluster may be much farther from each other than two elements of other clusters. For some classes of data, this may lead to difficulties in defining classes that could usefully subdivide the data. However, it is popular in astronomy for analyzing galaxy clusters, which may often involve long strings of matter; in this application, it is also known as the friends-of-friends algorithm.
Overview of agglomerative clustering methods
In the beginning of the agglomerative clustering process, each element is in a cluster of its own. The clusters are then sequentially combined into larger clusters, until all elements end up being in the same cluster. At each step, the two clusters separated by the shortest distance are combined. The function used to determine the distance between two clusters, known as the linkage function, is what differentiates the agglomerative clustering methods.
In single-linkage clustering, the distance between two clusters is determined by a single pair of elements: those two elements (one in each cluster) that are closest to each other. The shortest of these pairwise distances that remain at any step causes the two clusters whose elements are involved to be merged. The method is also known as nearest neighbour clustering. The result of the clustering can be visualized as a dendrogram, which shows the sequence in which clusters were merged and the distance at which each merge took place.
Mathematically, the linkage function – the distance D(X,Y) between clusters X and Y – is described by the expression
where X and Y are any two sets of elements considered as clusters, and d(x,y) denotes the distance between the two elements x and y.
Naive algorithm
The following algorithm is an agglomerative scheme that erases rows and columns in a proximity matrix as old clusters are merged into new ones. The proximity matrix contains all distances . The clusterings are assigned sequence numbers and is the level of the -th clustering. A cluster with sequence number m is denoted (m) and the proximity between clusters and is denoted .
The single linkage algorithm is composed of the following steps:
Begin with the disjoint clustering having level and sequence number .
Find the most similar pair of clusters in the current clustering, say pair , according to where the minimum is over all pairs of clusters in the current clustering.
Increment the sequence number: . Merge clusters and into a single cluster to form the next clustering . Set the level of this clustering to
Update th |
https://en.wikipedia.org/wiki/Hellinger%20distance | In probability and statistics, the Hellinger distance (closely related to, although different from, the Bhattacharyya distance) is used to quantify the similarity between two probability distributions. It is a type of f-divergence. The Hellinger distance is defined in terms of the Hellinger integral, which was introduced by Ernst Hellinger in 1909.
It is sometimes called the Jeffreys distance.
Definition
Measure theory
To define the Hellinger distance in terms of measure theory, let and denote two probability measures on a measure space that are absolutely continuous with respect to an auxiliary measure . Such a measure always exists, e.g . The square of the Hellinger distance between and is defined as the quantity
Here, and , i.e. and are the Radon–Nikodym derivatives of P and Q respectively with respect to . This definition does not depend on , i.e. the Hellinger distance between P and Q does not change if is replaced with a different probability measure with respect to which both P and Q are absolutely continuous. For compactness, the above formula is often written as
Probability theory using Lebesgue measure
To define the Hellinger distance in terms of elementary probability theory, we take λ to be the Lebesgue measure, so that dP / dλ and dQ / dλ are simply probability density functions. If we denote the densities as f and g, respectively, the squared Hellinger distance can be expressed as a standard calculus integral
where the second form can be obtained by expanding the square and using the fact that the integral of a probability density over its domain equals 1.
The Hellinger distance H(P, Q) satisfies the property (derivable from the Cauchy–Schwarz inequality)
Discrete distributions
For two discrete probability distributions and ,
their Hellinger distance is defined as
which is directly related to the Euclidean norm of the difference of the square root vectors, i.e.
Also,
Properties
The Hellinger distance forms a bounded metric on the space of probability distributions over a given probability space.
The maximum distance 1 is achieved when P assigns probability zero to every set to which Q assigns a positive probability, and vice versa.
Sometimes the factor in front of the integral is omitted, in which case the Hellinger distance ranges from zero to the square root of two.
The Hellinger distance is related to the Bhattacharyya coefficient as it can be defined as
Hellinger distances are used in the theory of sequential and asymptotic statistics.
The squared Hellinger distance between two normal distributions and is:
The squared Hellinger distance between two multivariate normal distributions and is
The squared Hellinger distance between two exponential distributions and is:
The squared Hellinger distance between two Weibull distributions and (where is a common shape parameter and are the scale parameters respectively):
The squared Hellinger distance between two Poisson di |
https://en.wikipedia.org/wiki/List%20of%20Bradford%20Bulls%20records%20and%20statistics | This is a list of all the records and statistics of rugby league side Bradford Bulls. It concentrates on the records of the team and the performances of the players who have played for this team. Since the re-brand in 1996 the Bulls have gone on to win many honours and awards. Under the re-brand the Bulls played their first game against Batley Bulldogs in the 1996 Challenge Cup on 4 February 1996, Bradford won the match 60–18. As of 7 October 2021 the Bulls have played 788 games.
Team records
Team wins, losses, ties and draws
Matches played
Results summary
Highest scores
Lowest scores
Biggest wins
Biggest losses
Individual records
Most matches as captain
Most career appearances
Most career points
Most career tries
Most career goals
Most career drop goals
Most points in a season
Most tries in a season
Most goals in a season
Most drop goals in a season
Most points in a match
Most tries in a match
Most goals in a match
Most drop goals in a match
Attendance records
Season average attendance
Highest match attendance
Coaching
Coaching records
References
Statistics |
https://en.wikipedia.org/wiki/List%20of%20Tamworth%20F.C.%20managers | This is a list of managers of Tamworth Football Club.
Statistics
''Information correct as of 3 March 2019. Only competitive matches are counted. Wins, losses and draws are results at the final whistle; the results of penalty shoot-outs are not counted.
References
General
Tamworth F.C. Official Website
Tamworth F.C. Heritage Website
Specific
Managers
Tamworth |
https://en.wikipedia.org/wiki/Clinical%20prediction%20rule | A clinical prediction rule or clinical probability assessment specifies how to use medical signs, symptoms, and other findings to estimate the probability of a specific disease or clinical outcome.
Physicians have difficulty in estimated risks of diseases; frequently erring towards overestimation, perhaps due to cognitive biases such as base rate fallacy in which the risk of an adverse outcome is exaggerated.
Methods
In a prediction rule study, investigators identify a consecutive group of patients who are suspected of having a specific disease or outcome. The investigators then obtain a standard set of clinical observations on each patient and a test or clinical follow-up to define the true state of the patient. They then use statistical methods to identify the best clinical predictors of the patient's true state. The probability of disease will depend on the patient's key clinical predictors. Published methodological standards specify good practices for developing a clinical prediction rule.
A survey of methods concluded "the majority of prediction studies in high impact journals do not follow current methodological recommendations, limiting their reliability and applicability", confirming earlier findings from the diabetic literature
Effect on health outcomes
Few prediction rules have had the consequences of their usage by physicians quantified.
When studied, the impact of providing the information alone (for example, providing the calculated probability of disease) has been negative.
However, when the prediction rule is implemented as part of a critical pathway, so that a hospital or clinic has procedures and policies established for how to manage patients identified as high or low risk of disease, the prediction rule has more impact on clinical outcomes.
The more intensively the prediction rule is implemented the more benefit will occur.
Examples of prediction rules
Apache II
CHADS2 for risk of stroke with atrial fibrillation
CURB-65
Model for End-Stage Liver Disease
Ranson criteria
Centor criteria
Pneumonia severity index
Wells score (disambiguation)
Orthopaedics
Abbreviated Injury Scale
Harris Hip Score
Injury Severity Score
Kocher criteria
Mirel's Score
NACA score
Ottawa ankle rules
Ottawa knee rules
Pittsburgh knee rules
Revised Trauma Score
References
External links
Clinical Prediction Website
Clinical prediction rules online calculators
Health informatics
Evidence-based medicine
Medical scoring system |
https://en.wikipedia.org/wiki/Harry%20V.%20Roberts | Harry V. Roberts (1923–2004), American statistician, was a distinguished teacher and a pioneer in looking at the applications of Bayesian statistics to business decision making and in Total Quality Management.
Roberts began teaching at the University of Chicago Graduate School of Business in 1949 as an instructor of statistics. He was promoted to assistant professor in 1951. He earned his Ph.D. from the University of Chicago in 1955, and was appointed associate professor. He was made full professor in 1959, and was named Sigmund E. Edelstone Professor of Statistics and Quality Management in 1991. In 1997, Roberts was awarded the Norman Maclean Faculty Award from the University of Chicago for his contributions to teaching and to the student experience on campus. In recognition of his career achievements, the Chicago chapter of the American Statistical Association created the Harry V. Roberts Statistical Advocate Award, first given in January 2002.
His varied research interests also included interactive computing; time series analysis; the relation between statistical theory and practical decision making; survey methodology and practice; and productivity and quality improvement.
Roberts was the co-author of many influential publications, including two groundbreaking books: Basic Methods of Marketing Research (with James Lorie) and the textbook, Statistics: A New Approach (with W. Allen Wallis). He also co-authored an early work on the random walk hypothesis of stock market prices, “Differencing of Random Walks and Near Random Walks,” with Nicholas Gonedes, published in the Journal of Econometrics in 1977.
Roberts was an early computer enthusiast, and was especially interested in developing computer methods for statistical analysis. In the late 1960s, Roberts, in collaboration with his wife June and Robert Ling, developed a statistics package called Interactive Data Analysis (IDA), used for statistical instruction at a number of top business schools.
Toward the end of his career, Roberts helped develop a curriculum in Total Quality Management at Chicago GSB and co-authored Quality is Personal: A Foundation for Total Quality Management (with B. F. Sergesketter). This was a novel application of TQM methods to self-improvement, reminiscent of self-experimentation.
External links
Univ. of Chicago Obituary
Chicago Chapter of Amer. Statistical Association
1923 births
2004 deaths
University of Chicago alumni
American statisticians
Fellows of the American Statistical Association |
https://en.wikipedia.org/wiki/NUTS%20statistical%20regions%20of%20Austria | The Nomenclature of Territorial Units for Statistics (NUTS) is a geocode standard for referencing the subdivisions of Austria for statistical purposes. The standard is developed and regulated by the European Union. The NUTS standard is instrumental in delivering the European Union's Structural Funds. The NUTS code for Austria is AT and a hierarchy of three levels is established by Eurostat. Below these is a further levels of geographic organisation - the local administrative unit (LAU). In Austria, the LAU 2 is municipalities.
Overall
NUTS Levels
Local administrative units
Below the NUTS levels, the two LAU (Local Administrative Units) levels are:
The LAU codes of Austria can be downloaded here: NUTS codes
NUTS 3 definitions
AT1 Eastern Austria (Ostösterreich)
AT11 Burgenland
AT111 Mittelburgenland (Bezirk Oberpullendorf)
AT112 Nordburgenland (Eisenstadt, Rust, Bezirk Eisenstadt-Umgebung, Bezirk Mattersburg, Bezirk Neusiedl am See)
AT113 Südburgenland (Bezirk Güssing, Bezirk Jennersdorf, Bezirk Oberwart)
AT12 Lower Austria (Niederösterreich)
AT121 Mostviertel-Eisenwurzen (Waidhofen an der Ybbs, Bezirk Amstetten, Bezirk Melk, Bezirk Scheibbs)
AT122 Niederösterreich-Süd (Wiener Neustadt, Bezirk Lilienfeld, Bezirk Neunkirchen, Bezirk Wiener Neustadt-Land)
AT123 Sankt Pölten (Sankt Pölten, Bezirk Sankt Pölten-Land)
AT124 Waldviertel (Krems, Bezirk Gmünd, Bezirk Horn, Bezirk Krems-Land, Bezirk Waidhofen an der Thaya, Bezirk Zwettl)
AT125 Weinviertel (Bezirk Hollabrunn, part of Bezirk Mistelbach, part of Bezirk Gänserndorf)
AT126 Wiener Umland/Nordteil (Bezirk Korneuburg, Bezirk Tulln, part of Bezirken Gänserndorf, part of Bezirk Mistelbach, part of Bezirken Wien-Umgebung)
AT127 Wiener Umland/Südteil (Bezirk Bruck an der Leitha, Bezirk Baden, Bezirk Mödling, part of Bezirk Wien-Umgebung)
AT13 Vienna (Wien)
AT130 Wien
AT2 Southern Austria (Südösterreich)
AT21 Carinthia (Kärnten)
AT211 Klagenfurt-Villach (Klagenfurt, Villach, Bezirk Klagenfurt-Land, Bezirk Villach-Land)
AT212 Oberkärnten (Bezirk Feldkirchen, Bezirk Hermagor, Bezirk Spittal an der Drau)
AT213 Unterkärnten (Bezirk Sankt Veit an der Glan, Bezirk Völkermarkt, Bezirk Wolfsberg)
AT22 Styria (Steiermark)
AT221 Graz (Graz, Bezirk Graz-Umgebung)
AT222 Liezen (Bezirk Liezen)
AT223 Östliche Obersteiermark (Bezirk Bruck an der Mur, Bezirk Leoben, Bezirk Mürzzuschlag)
AT224 Oststeiermark (Bezirk Feldbach, Bezirk Fürstenfeld, Bezirk Hartberg, Bezirk Radkersburg, Bezirk Weiz)
AT225 West- und Südsteiermark (Bezirk Deutschlandsberg, Bezirk Leibnitz, Bezirk Voitsberg)
AT226 Westliche Obersteiermark (Bezirk Judenburg, Bezirk Knittelfeld, Bezirk Murau)
AT3 Western Austria (Westösterreich)
AT31 Upper Austria (Oberösterreich)
AT311 Innviertel (Bezirk Braunau am Inn, Bezirk Grieskirchen, Bezirk Ried im Innkreis, Bezirk Schärding)
AT312 Linz-Wels (Linz, Wels, Bezirk Linz-Land, Bezirk Wels-Land, Bezirk Eferding, part of Bezirk Urfahr-Umgebung)
AT313 Mühlviertel (Bezirk |
https://en.wikipedia.org/wiki/Vantieghems%20theorem | In number theory, Vantieghems theorem is a primality criterion. It states that a natural number n≥3 is prime if and only if
Similarly, n is prime, if and only if the following congruence for polynomials in X holds:
or:
Example
Let n=7 forming the product 1*3*7*15*31*63 = 615195. 615195 = 7 mod 127 and so 7 is prime
Let n=9 forming the product 1*3*7*15*31*63*127*255 = 19923090075. 19923090075 = 301 mod 511 and so 9 is composite
References
. An article with proof and generalizations.
Factorial and binomial topics
Modular arithmetic
Theorems about prime numbers |
https://en.wikipedia.org/wiki/Hugo%20Machado | Hugo Miguel Alves Machado (born 4 July 1982 in Lisbon) is a Portuguese footballer who plays for Clube Oriental de Lisboa as an attacking midfielder.
Club statistics
References
External links
Barreirense official profile
Persian League stats
1982 births
Living people
Portuguese men's footballers
Footballers from Lisbon
Men's association football midfielders
Primeira Liga players
Liga Portugal 2 players
Segunda Divisão players
Sporting CP B players
C.F. Estrela da Amadora players
F.C. Barreirense players
Associação Naval 1º de Maio players
Clube Oriental de Lisboa players
Real S.C. players
GS Loures players
C.D. Cova da Piedade players
Cypriot First Division players
Apollon Limassol FC players
Olympiakos Nicosia players
Alki Larnaca FC players
Azerbaijan Premier League players
FK Standard Sumgayit players
Persian Gulf Pro League players
Zob Ahan Esfahan F.C. players
Sanat Naft Abadan F.C. players
I-League players
Churchill Brothers FC Goa players
Football League (Greece) players
Athens Kallithea F.C. players
OFI Crete F.C. players
Portuguese expatriate men's footballers
Expatriate men's footballers in Cyprus
Expatriate men's footballers in Azerbaijan
Expatriate men's footballers in Iran
Expatriate men's footballers in India
Expatriate men's footballers in Greece
Portuguese expatriate sportspeople in Cyprus
Portuguese expatriate sportspeople in Azerbaijan
Portuguese expatriate sportspeople in Iran
Portuguese expatriate sportspeople in India
Portuguese expatriate sportspeople in Greece |
https://en.wikipedia.org/wiki/Number%20Theory%20Foundation | The Number Theory Foundation (NTF) is a non-profit organization based in the United States which supports research and conferences in the field of number theory, with a particular focus on computational aspects and explicit methods.
The NTF funds the Selfridge prize awarded at each Algorithmic Number Theory Symposium (ANTS) and is a regular supporter of several conferences and organizations in number theory, including the Canadian Number Theory Association (CNTA), Women in Numbers (WIN), and the West Coast Number Theory (WCNT) conference.
History
The NTF was created in 1999 via a grant from John Selfridge with an initial board of directors including Paul Bateman, John Brillhart, Richard Blecksmith, Brian Conrey, Ronald Graham, Richard Guy, Carl Pomerance, John Selfridge, Sam Wagstaff, and Hugh Williams. Carl Pomerance served as President of the foundation for its first two decades and was succeeded by Andrew Sutherland in 2019.
References
External links
Number theory
Foundations based in the United States |
https://en.wikipedia.org/wiki/Kenneth%20P.%20Williams | Kenneth Powers Williams (August 25, 1887 – September 25, 1958) was a professor of mathematics at Indiana University, a distinguished soldier, and a Reserve Officers' Training Corps commander. He was known as the "Father of ROTC" at Indiana University.
Early life and education
Kenneth Powers Williams was born in Urbana, Ohio on August 25, 1887, to John H. and Eva Augusta (Powers) Williams. He attended Clark College from 1905 to 1906. Williams then enrolled at Indiana University where he received his A.B. in 1908 and his A.M. degree in 1909. In 1911, he went to Princeton University where he earned a PhD. in 1913.
Academic career
In 1909, Williams was named an instructor of mathematics in the Department of Mathematics at Indiana University. He was granted a leave from there in 1911 to attend Princeton University. In 1914, Williams returned and resumed his teaching career in the Department of Mathematics.
During his academic career at I.U., Williams served as an Assistant Professor from 1914-1919, an Associate Professor from 1919-1924, and as a Full Professor from 1924-1937. He was awarded the Chair of the Department of Mathematics in 1938 which he served as until 1944. Williams retired in 1957 as a Distinguished Service Professor Emeritus.
Williams made many noteworthy achievements in the fields of mathematics, astronomy, and history over his career. He is best known for his five-volume book, Lincoln Finds a General, published from 1949–1957. Williams had intended to write seven books, but became ill and died just before Volume 5 was published. Volume 1 was reprinted by Indiana University Press, Volumes 3 and 4 were reprinted by University of Kansas Press as "Grant Rises in the West." Another publisher, Macmilian Press, published the five-volume work. (See: Bibliography of Ulysses S. Grant)
Military career
In addition to his career in education, Williams also had a successful military career. He first served as a first lieutenant with the Indiana National Guard near the Mexican border in 1916 during the Mexican Border Expedition. On April 17, 1917, Indiana University established a military training program in Bloomington called the Student Army Training Corps with Williams in command. In 1919, the Student Army Training Corps was renamed the Reserve Officers Training Corps. From 1917 to 1919, he served in the Rainbow Division as a U.S. Army captain of field artillery with the American Expeditionary Force. From 1921 to 1931, he served as an officer in the Indiana National Guard Field Artillery. From 1931 to 1939, he served as a colonel and the chief of staff of the 38th division of the Indiana National Guard before commanding the 113th Quartermaster Regiment, and eventually becoming quartermaster of the 38th division in national service.
Marriage
Williams married Mrs. Ellen (Laughlin) Scott on August 20, 1920. They had no children.
Death
After several months of illness, Williams died on September 25, 1958.
Honors and awards
Williams received m |
https://en.wikipedia.org/wiki/Vietoris%E2%80%93Rips%20complex | In topology, the Vietoris–Rips complex, also called the Vietoris complex or Rips complex, is a way of forming a topological space from distances in a set of points. It is an abstract simplicial complex that can be defined from any metric space M and distance δ by forming a simplex for every finite set of points that has diameter at most δ. That is, it is a family of finite subsets of M, in which we think of a subset of k points as forming a (k − 1)-dimensional simplex (an edge for two points, a triangle for three points, a tetrahedron for four points, etc.); if a finite set S has the property that the distance between every pair of points in S is at most δ, then we include S as a simplex in the complex.
History
The Vietoris–Rips complex was originally called the Vietoris complex, for Leopold Vietoris, who introduced it as a means of extending homology theory from simplicial complexes to metric spaces. After Eliyahu Rips applied the same complex to the study of hyperbolic groups, its use was popularized by , who called it the Rips complex. The name "Vietoris–Rips complex" is due to .
Relation to Čech complex
The Vietoris–Rips complex is closely related to the Čech complex (or nerve) of a set of balls, which has a simplex for every finite subset of balls with nonempty intersection. In a geodesically convex space Y, the Vietoris–Rips complex of any subspace X ⊂ Y for distance δ has the same points and edges as the Čech complex of the set of balls of radius δ/2 in Y that are centered at the points of X. However, unlike the Čech complex, the Vietoris–Rips complex of X depends only on the intrinsic geometry of X, and not on any embedding of X into some larger space.
As an example, consider the uniform metric space M3 consisting of three points, each at unit distance from each other. The Vietoris–Rips complex of M3, for δ = 1, includes a simplex for every subset of points in M3, including a triangle for M3 itself. If we embed M3 as an equilateral triangle in the Euclidean plane, then the Čech complex of the radius-1/2 balls centered at the points of M3 would contain all other simplexes of the Vietoris–Rips complex but would not contain this triangle, as there is no point of the plane contained in all three balls. However, if M3 is instead embedded into a metric space that contains a fourth point at distance 1/2 from each of the three points of M3, the Čech complex of the radius-1/2 balls in this space would contain the triangle. Thus, the Čech complex of fixed-radius balls centered at M3 differs depending on which larger space M3 might be embedded into, while the Vietoris–Rips complex remains unchanged.
If any metric space X is embedded in an injective metric space Y, the Vietoris–Rips complex for distance δ and X coincides with the Čech complex of the balls of radius δ/2 centered at the points of X in Y. Thus, the Vietoris–Rips complex of any metric space M equals the Čech complex of a system of balls in the tight span of M.
Relation to unit disk |
https://en.wikipedia.org/wiki/Point%20class | Point class may refer to
Pointclass sets in mathematics
Point-class sealift ship
Point-class cutter |
https://en.wikipedia.org/wiki/Householder%27s%20method | In mathematics, and more specifically in numerical analysis, Householder's methods are a class of root-finding algorithms that are used for functions of one real variable with continuous derivatives up to some order . Each of these methods is characterized by the number , which is known as the order of the method. The algorithm is iterative and has a rate of convergence of .
These methods are named after the American mathematician Alston Scott Householder.
Method
Householder's method is a numerical algorithm for solving the equation . In this case, the function has to be a function of one real variable. The method consists of a sequence of iterations
beginning with an initial guess .
If is a times continuously differentiable function and is a zero of but not of its derivative, then, in a neighborhood of , the iterates satisfy:
, for some
This means that the iterates converge to the zero if the initial guess is sufficiently close, and that the convergence has order or better. Furthermore, when close enough to , it commonly is the case that for some . In particular,
if is even and then convergence to will be from the right;
if is even and then convergence to will be from the left;
if is odd and then convergence to will be from the side where it starts; and
if is odd and then convergence to will alternate sides.
Despite their order of convergence, these methods are not widely used because the gain in precision is not commensurate with the rise in effort for large . The Ostrowski index expresses the error reduction in the number of function evaluations instead of the iteration count.
For polynomials, the evaluation of the first derivatives of at using the Horner method has an effort of polynomial evaluations. Since evaluations over iterations give an error exponent of , the exponent for one function evaluation is , numerically , , , for , and falling after that. By this criterion, the case (Halley's method) is the optimal value of .
For general functions the derivative evaluation using the Taylor arithmetic of automatic differentiation requires the equivalent of function evaluations. One function evaluation thus reduces the error by an exponent of , which is for Newton's method, for Halley's method and falling towards 1 or linear convergence for the higher order methods.
Motivation
First approach
Suppose is analytic in a neighborhood of and . Then has a Taylor series at and its constant term is zero. Because this constant term is zero, the function will have a Taylor series at and, when , its constant term will not be zero. Because that constant term is not zero, it follows that the reciprocal has a Taylor series at , which we will write as and its constant term will not be zero. Using that Taylor series we can write
When we compute its -th derivative, we note that the terms for conveniently vanish:
using big O notation. We thus get that the ratio
If is the zero of that is clos |
https://en.wikipedia.org/wiki/Drazin%20inverse | In mathematics, the Drazin inverse, named after Michael P. Drazin, is a kind of generalized inverse of a matrix.
Let A be a square matrix. The index of A is the least nonnegative integer k such that rank(Ak+1) = rank(Ak). The Drazin inverse of A is the unique matrix AD that satisfies
It's not a generalized inverse in the classical sense, since in general.
If A is invertible with inverse , then .
If A is a block diagonal matrix
where is invertible with inverse and is a nilpotent matrix, then
Drazin inversion is invariant under conjugation. If is the Drazin inverse of , then is the Drazin inverse of .
The Drazin inverse of a matrix of index 0 or 1 is called the group inverse or {1,2,5}-inverse and denoted A#. The group inverse can be defined, equivalently, by the properties AA#A = A, A#AA# = A#, and AA# = A#A.
A projection matrix P, defined as a matrix such that P2 = P, has index 1 (or 0) and has Drazin inverse PD = P.
If A is a nilpotent matrix (for example a shift matrix), then
The hyper-power sequence is
for convergence notice that
For or any regular with chosen such that the sequence tends to its Drazin inverse,
Jordan normal form and Jordan-Chevalley decomposition
As the definition of the Drazin inverse is invariant under matrix conjugations, writing , where J is in Jordan normal form, implies that . The Drazin inverse is then the operation that maps invertible Jordan blocks to their inverses, and nilpotent Jordan blocks to zero.
More generally, we may define the Drazin inverse over any perfect field, by using the Jordan-Chevalley decomposition where is semisimple and is nilpotent and both operators commute. The two terms can be block diagonalized with blocks corresponding to the kernel and cokernel of . The Drazin inverse in the same basis is then defined to be zero on the kernel of , and equal to the inverse of on the cokernel of .
See also
Constrained generalized inverse
Inverse element
Moore–Penrose inverse
Jordan normal form
Generalized eigenvector
References
External links
Drazin inverse on Planet Math
Group inverse on Planet Math
Matrices
de:Pseudoinverse#Ausgewählte weitere Versionen von verallgemeinerten Inversen |
https://en.wikipedia.org/wiki/Group%20inverse | In mathematics, group inverse may refer to:
the inverse element in a group or in a subgroup of another, not necessarily group structure, e.g. in a subgroup of a semigroup
the Drazin inverse |
https://en.wikipedia.org/wiki/Raghu%20Raj%20Bahadur | Raghu Raj Bahadur (30 April 1924 – 7 June 1997) was an Indian statistician considered by peers to be "one of the architects of the modern theory of mathematical statistics".
Biography
Bahadur was born in Delhi, India, and received his BA (1943) and MA (1945) in mathematics from St. Stephen’s College, University of Delhi . He received his doctorate from the University of North Carolina under Herbert Robbins in 1950 after which he joined University of Chicago. He worked as a research statistician at the Indian Statistical Institute in Calcutta from 1956 to 1961. He spent the remainder of his academic career in the University of Chicago. He is a cousin to Madhur Jaffrey.
Contributions
He published numerous papers and is best known for the concepts of "Bahadur efficiency" and the Bahadur–Ghosh–Kiefer representation (with J. K. Ghosh and Jack Kiefer).
He also framed the Anderson–Bahadur algorithm along with Theodore Wilbur Anderson which is used in statistics and engineering for solving binary classification problems when the underlying data have multivariate normal distributions with different covariance matrices.
Legacy
He held the John Simon Guggenheim Fellowship (1968–69) and was the 1974 Wald Lecturer of the IMS. He was the President of the Institute of Mathematical Statistics during 1974–75 and was elected a Fellow of the American Academy of Arts and Sciences in 1986.
References
External links
1924 births
1997 deaths
Indian statisticians
20th-century Indian mathematicians
American statisticians
20th-century American mathematicians
Fellows of the American Academy of Arts and Sciences
Presidents of the Institute of Mathematical Statistics
University of Chicago faculty
American academics of Indian descent
Scientists from Delhi
Indian emigrants to the United States
Mathematical statisticians |
https://en.wikipedia.org/wiki/Parallel%20mesh%20generation | Parallel mesh generation in numerical analysis is a new research area between the boundaries of two scientific computing disciplines: computational geometry and parallel computing. Parallel mesh generation methods decompose the original mesh generation problem into smaller subproblems which are solved (meshed) in parallel using multiple processors or threads. The existing parallel mesh generation methods can be classified in terms of two basic attributes:
the sequential technique used for meshing the individual subproblems and
the degree of coupling between the subproblems.
One of the challenges in parallel mesh generation is to develop parallel meshing software using off-the-shelf sequential meshing codes.
Overview
Parallel mesh generation procedures in general decompose the original 2-dimensional (2D) or 3-dimensional (3D) mesh generation problem into N smaller subproblems which are solved (i.e., meshed) concurrently using P processors or threads. The subproblems can be formulated to be either tightly coupled, partially coupled or even decoupled. The coupling of the subproblems determines the intensity of the communication and the amount/type of synchronization required between the subproblems.
The challenges in parallel mesh generation methods are: to maintain stability of the parallel mesher (i.e., retain the quality of finite elements generated by state-of-the-art sequential codes) and at the same time achieve 100% code re-use (i.e., leverage the continuously evolving and fully functional off-the-shelf sequential meshers) without substantial deterioration of the scalability of the parallel mesher.
There is a difference between parallel mesh generation and parallel triangulation. In parallel triangulation a pre-defined set of points is used to generate in parallel triangles that cover the convex hull of the set of points. A very efficient algorithm for parallel Delaunay triangulations appears in Blelloch et al. This algorithm is extended in Clemens and Walkington for parallel mesh generation.
Parallel mesh generation software
While many solvers have been ported to parallel machines, grid generators have left behind. Still the preprocessing step of mesh generation remains a sequential bottleneck in the simulation cycle. That is why the need for developing of stable 3D parallel grid generator is well-justified.
A parallel version of the MeshSim mesh generator by Simmetrix Inc., is available for both research and commercial use. It includes parallel implementations of surface, volume and boundary layer mesh generation as well as parallel mesh adaptivity. The algorithms it uses are based on those in reference and are scalable (both in the parallel sense and in the sense that they give speedup compared to the serial implementation) and stable. For multicore or multiprocessor systems, there is also a multithreaded version of these algorithms that are available in the base MeshSim product
Another parallel mesh generator is D3D, was |
https://en.wikipedia.org/wiki/Error-correcting%20codes%20with%20feedback | In mathematics, computer science, telecommunication, information theory, and searching theory, error-correcting codes with feedback are error correcting codes designed to work in the presence of feedback from the receiver to the sender.
Problem
Alice (the sender) wishes to send a value x to Bob (the receiver). The communication channel between Alice and Bob is imperfect, and can introduce errors.
Solution
An error-correcting code is a way of encoding x as a message such that Bob will successfully understand the value x as intended by Alice, even if the message Alice sends and the message Bob receives differ. In an error-correcting code with feedback, the channel is two-way: Bob can send feedback to Alice about the message he received.
Noisy feedback
In an error-correcting code without noisy feedback, the feedback received by the sender is always free of errors. In an error-correcting code with noisy feedback, errors can occur in the feedback, as well as in the message.
An error-correcting code with noiseless feedback is equivalent to an adaptive search strategy with errors.
History
In 1956, Claude Shannon introduced the discrete memoryless channel with noiseless feedback. In 1961, Alfréd Rényi introduced the Bar-Kochba game (also known as Twenty questions), with a given percentage of wrong answers, and calculated the minimum number of randomly chosen questions to determine the answer.
In his 1964 dissertation, Elwyn Berlekamp considered error correcting codes with noiseless feedback. In Berlekamp's scenario, the receiver chose a subset of possible messages and asked the sender whether the given message was in this subset, a 'yes' or 'no' answer. Based on this answer, the receiver then chose a new subset and repeated the process. The game is further complicated due to noise; some of the answers will be wrong.
Sources
.
.
References
See also
Noisy channel coding theorem
Error detection and correction |
https://en.wikipedia.org/wiki/Reversible%20charge%20injection%20limit | For an electrode in a solution with a particular size and geometry, the reversible charge injection limit is the amount of charge that can move from the electrode to the surroundings without causing a chemical reaction that is irreversible.
References
Electrochemistry |
https://en.wikipedia.org/wiki/Anton%20Lichkov | Anton Lichkov (: born 5 August 1980 in Petrich) is a Bulgarian former footballer, who played as a defender.
External links
2007-08 Statistics, 2006-07 Statistics & 2005-06 Statistics at PFL.bg
1980 births
Living people
Bulgarian men's footballers
First Professional Football League (Bulgaria) players
PFC Slavia Sofia players
PFC Beroe Stara Zagora players
OFC Belasitsa Petrich players
FC Montana players
People from Petrich
Men's association football defenders
Footballers from Blagoevgrad Province |
https://en.wikipedia.org/wiki/Zhelyo%20Zhelev | Zhelyo Zhelev (born 24 February 1987 in Stara Zagora) is a Bulgarian footballer currently playing for Vereya Stara Zagora as a midfielder.
External links
2007-08 Statistics
Bulgarian men's footballers
1987 births
Living people
First Professional Football League (Bulgaria) players
PFC Beroe Stara Zagora players
Men's association football midfielders
Footballers from Stara Zagora |
https://en.wikipedia.org/wiki/Jouni%20Loponen | Jouni Loponen (born 1 July 1971) is a Finnish ice hockey defender. He played for the Spokane Chiefs in 1988–89, he is a 4-times Finnish champion and in 2004, he won Elitserien.
Career statistics
References
External links
Eurohockey.net profile
Finnish ice hockey defencemen
Spokane Chiefs players
HC TPS players
Living people
1971 births |
https://en.wikipedia.org/wiki/Lambda-mu%20calculus | In mathematical logic and computer science, the lambda-mu calculus is an extension of the lambda calculus introduced by Michel Parigot. It introduces two new operators: the μ operator (which is completely different both from the μ operator found in computability theory and from the μ operator of modal μ-calculus) and the bracket operator. Proof-theoretically, it provides a well-behaved formulation of classical natural deduction.
One of the main goals of this extended calculus is to be able to describe expressions corresponding to theorems in classical logic. According to the Curry–Howard isomorphism, lambda calculus on its own can express theorems in intuitionistic logic only, and several classical logical theorems can't be written at all. However with these new operators one is able to write terms that have the type of, for example, Peirce's law.
Semantically these operators correspond to continuations, found in some functional programming languages.
Formal definition
We can augment the definition of a lambda expression to gain one in the context of lambda-mu calculus. The three main expressions found in lambda calculus are as follows:
, a variable, where V is any identifier.
, an abstraction, where V is any identifier and E is any lambda expression.
, an application, where E and E'''; are any lambda expressions.
For details, see the corresponding article.
In addition to the traditional λ-variables, the lambda-mu calculus includes a distinct set of μ-variables. These μ-variables can be used to name or freeze arbitrary subterms, allowing us to later abstract on those names. The set of terms contains unnamed (all traditional lambda expressions are of this kind) and named terms. The terms that are added by the lambda-mu calculus are of the form:
is a named term, where α is a μ-variable and t is an unnamed term.
is an unnamed term, where α is a μ-variable and E'' is a named term.
Reduction
The basic reduction rules used in the lambda-mu calculus are the following:
logical reduction
structural reduction
renaming
the equivalent of η-reduction
, for α not freely occurring in u
These rules cause the calculus to be confluent. Further reduction rules could be added to provide us with a stronger notion of normal form, though this would be at the expense of confluence.
See also
Classical pure type systems for typed generalizations of lambda calculi with control
References
External links
Lambda-mu relevant discussion on Lambda the Ultimate.
Lambda calculus
Proof theory |
https://en.wikipedia.org/wiki/Mean%20curvature%20flow | In the field of differential geometry in mathematics, mean curvature flow is an example of a geometric flow of hypersurfaces in a Riemannian manifold (for example, smooth surfaces in 3-dimensional Euclidean space). Intuitively, a family of surfaces evolves under mean curvature flow if the normal component of the velocity of which a point on the surface moves is given by the mean curvature of the surface. For example, a round sphere evolves under mean curvature flow by shrinking inward uniformly (since the mean curvature vector of a sphere points inward). Except in special cases, the mean curvature flow develops singularities.
Under the constraint that volume enclosed is constant, this is called surface tension flow.
It is a parabolic partial differential equation, and can be interpreted as "smoothing".
Existence and uniqueness
The following was shown by Michael Gage and Richard S. Hamilton as an application of Hamilton's general existence theorem for parabolic geometric flows.
Let be a compact smooth manifold, let be a complete smooth Riemannian manifold, and let be a smooth immersion. Then there is a positive number , which could be infinite, and a map with the following properties:
is a smooth immersion for any
as one has in
for any , the derivative of the curve at is equal to the mean curvature vector of at .
if is any other map with the four properties above, then and for any
Necessarily, the restriction of to is .
One refers to as the (maximally extended) mean curvature flow with initial data .
Convergence theorems
Following Hamilton's epochal 1982 work on the Ricci flow, in 1984 Gerhard Huisken employed the same methods for the mean curvature flow to produce the following analogous result:
If is the Euclidean space , where denotes the dimension of , then is necessarily finite. If the second fundamental form of the 'initial immersion' is strictly positive, then the second fundamental form of the immersion is also strictly positive for every , and furthermore if one choose the function such that the volume of the Riemannian manifold is independent of , then as the immersions smoothly converge to an immersion whose image in is a round sphere.
Note that if and is a smooth hypersurface immersion whose second fundamental form is positive, then the Gauss map is a diffeomorphism, and so one knows from the start that is diffeomorphic to and, from elementary differential topology, that all immersions considered above are embeddings.
Gage and Hamilton extended Huisken's result to the case . Matthew Grayson (1987) showed that if is any smooth embedding, then the mean curvature flow with initial data eventually consists exclusively of embeddings with strictly positive curvature, at which point Gage and Hamilton's result applies. In summary:
If is a smooth embedding, then consider the mean curvature flow with initial data . Then is a smooth embedding for every and there exists such that has posit |
https://en.wikipedia.org/wiki/Madeiran%20land%20snail | Madeiran land snail is a common name which has been given to several different species of terrestrial gastropods, air-breathing land snails:
Caseolus calculus
Discula lyelliana, found only on the Desertas Islands in the Madeira Archipelago; critically endangered
Geomitra grabhami, critically endangered; remnant populations present on Desertas Islands of Madeira
Geomitra moniziana
Leiostyla abbreviata, extinct
Leiostyla cassida
Leiostyla corneocostata
Leiostyla gibba, extinct
Animal common name disambiguation pages |
https://en.wikipedia.org/wiki/Michael%20P.%20Drazin | Michael Peter Drazin (born 1929) is an American mathematician of British background, working in noncommutative algebra.
Background
The Drazins (Дразин) were a Russian Jewish family who moved to the United Kingdom in the years before World War I. Isaac Drazin founded in 1927 a well-known electrical goods shop in Heath Street, Hampstead, which existed for over 50 years.
Isaac Drazin married Leah Wexler, and had three sons, of whom Michael was the eldest, and Philip Drazin, also a mathematician, was the youngest, the middle son being David; and died 1 January 1993.
Life
Michael Drazin was born in London on 5 June 1929. His younger brother Philip was educated as a boarder at St Christopher School, Letchworth during World War II. The self-published memoirs of Roger Atkinson, a school friend of Michael (Mike), indicate that Michael attended King Alfred School, London, located in Hampstead, retaining contacts at the school when it was evacuated in wartime to Royston, Hertfordshire; Atkinson was a boarder at St Christopher School, Letchworth from September 1942. In 1946 Atkinson and Drazin visited Paris together.
Drazin was a student at the University of Cambridge, graduating B.A. in 1950 and M.A. in 1953. He was awarded a Ph.D. in 1953 for a dissertation Contributions to Abstract Algebra written with advisers Robert Rankin and David Rees. He was a Fellow of Trinity College, Cambridge from 1952 to 1956, during that period emigrating to the United States.
In the academic year 1957–8 Drazin was Visiting Lecturer at Northwestern University. In 1958 he began a period at RIAS Inc. (the Research Institute for Advanced Studies) in Baltimore as senior scientist, after which he took a position as associate professor at Purdue University in 1962.
Works
Drazin gave his name to a type of generalized inverse in ring theory and semigroup theory he introduced in 1958, now known as the Drazin inverse. It was later extended to contexts in operator theory.
While at RIAS, Drazin worked with Emilie Virginia Haynsworth, then at the National Bureau of Standards, within its numerical analysis program. He also worked with the metallurgist Henry Martin Otte of RIAS, and they published a book of crystallographic tables.
See also
*-regular semigroup
References
External links
Home page at Purdue
List of publications at www.math.purdue.edu
Living people
20th-century American mathematicians
21st-century American mathematicians
1929 births
Purdue University faculty |
https://en.wikipedia.org/wiki/Statistics%20Finland | Statistics Finland (, ) is the national statistical institution in Finland, established in 1865 to serve as an information service and to provide statistics and expertise in the statistical sciences. The institution employs more than 800 experts from varying fields.
The institution is led by Director General Markus Sovala.
References
External links
1865 establishments in Finland
Finland
Demographics of Finland
Government of Finland |
https://en.wikipedia.org/wiki/Yuri%20Nesterenko%20%28mathematician%29 | Yuri Valentinovich Nesterenko (; born 5 December 1946 in Kharkov, USSR, now Ukraine) is a Soviet and Russian mathematician who has written papers in algebraic independence theory and transcendental number theory.
In 1997, he was awarded the Ostrowski Prize for his proof that the numbers π and eπ are algebraically independent. In fact, he proved the stronger result:
the numbers π, eπ, and Γ(1/4) are algebraically independent over Q.
the numbers π, , and Γ(1/3) are algebraically independent over Q.
for all positive integers n, the numbers π, are algebraically independent over Q.
He is a professor at Moscow State University, where he completed the mechanical-mathematical program in 1969, then the doctorate program (Soviet habilitation) in 1973, became a professor of the Number Theory Department in 1992.
He studied under Andrei Borisovich Shidlovskii. Nesterenko's students have included Wadim Zudilin.
Publications
References
External links
A picture
Web page at Moscow State University ; switch to Windows-1251 encoding if your browser does not render correctly.
Russian mathematicians
Corresponding Members of the Russian Academy of Sciences
Living people
Year of birth missing (living people) |
https://en.wikipedia.org/wiki/Australian%20cricket%20team%20in%202007%E2%80%9308 | This article contains information, results and statistics regarding the Australian national cricket team in the 2007–08 season. Statisticians class the 2007–08 season as those matches played on tours that started between September 2007 and April 2008.
Player contracts
The 2007–08 list of contracted players was announced on 1 May 2007. Note that uncontracted players still are available for selection for the national cricket team.
Match summary
M = Matches Played, W = Won, L = Lost, D = Drawn, T = Tied, NR = No Result
Last updated 8 March 2008
Tournament Summary
Australia made the semi-final stage of the 2007 Twenty20 World Championship
Australia won the Future Cup ODI series against India 4–2
Australia won the Warne–Muralitharan Trophy against Sri Lanka 2–0
Australia won the Chappell–Hadlee Trophy against New Zealand 2–0
Australia won the Border-Gavaskar Trophy against India 2–1
Australia lost the Commonwealth Bank Series finals against India 2–0
Twenty20 World Championship
Australia's 2007–08 season began with the 2007 Twenty20 World Championship in South Africa. Australia were placed in Group B and their first official match saw them take on Zimbabwe on 12 September 2007 in Cape Town.
To prepare for the tournament, Australia played two warm-up games against New Zealand and South Africa.
Unofficial Warm-Up: v New Zealand, 8 September, Benoni
Australia were sent in to bowl by New Zealand and early wickets by Ben Hilfenhaus and Brett Lee restricted New Zealand to 4/61 off 9.2 overs. Despite this, a strong partnership of 67 between Ross Taylor and Craig McMillan saw the Kiwis claw their way back. Nathan Bracken claimed the vital wicket of Taylor for 53 and Mitchell Johnson dismissed McMillan for 60 on the last ball of the innings restricting New Zealand to a chaseable 182. Hilfenhaus was the pick of the bowlers for Australia, taking 3/11 off his 3 overs. Nathan Bracken also played a vital role taking 2/34 off his 4 overs.
Australia's innings began poorly with Shane Bond taking the wickets of Adam Gilchrist and Matthew Hayden within 4 deliveries. Brad Hodge was dismissed in the second over leaving Australia reeling at 3–15. A very strong partnership between Andrew Symonds and Michael Hussey rescued Australia, as they put on 113 for the fourth wicket. Despite their dismissals, this partnership was enough to see Australia cruise to victory with ten balls to spare. Hussey top scored for Australia scoring 72 off 44 balls, however Symonds was not far behind scoring 70 off 43 balls.
Unofficial Warm-Up: v South Africa, 9 September, Centurion
Australia were sent in to bat by South Africa and were on the back foot early. In the third over of Australia's innings, Adam Gilchrist and Brad Haddin fell in successive balls to South African pace-bowler Shaun Pollock. Brad Hodge fell in the fourth and Australia were in trouble at 3/21 after 3.4 overs. Just as he did in the warm-up against New Zealand, Andrew Symonds steadied the ship scoring a str |
https://en.wikipedia.org/wiki/Eleandro%20Pema | Eleandro Pema (born 9 February 1985) is an Albanian retired football striker. He last played for KS Dinamo
Club career
He has previously played for Samsunspor in Turkey.
Career statistics
References
External links
1985 births
Living people
Footballers from Tirana
Albanian men's footballers
Men's association football midfielders
Albania men's youth international footballers
Albania men's under-21 international footballers
Samsunspor footballers
KF Tirana players
FC Dinamo City players
Flamurtari FC players
KF Elbasani players
KS Kastrioti players
Shuvalan FK players
FC Kamza players
Kategoria Superiore players
Kategoria e Parë players
Albanian expatriate men's footballers
Expatriate men's footballers in Turkey
Albanian expatriate sportspeople in Turkey
Expatriate men's footballers in Azerbaijan
Albanian expatriate sportspeople in Azerbaijan |
https://en.wikipedia.org/wiki/Peter%20Gavin%20Hall | Peter Gavin Hall (20 November 1951 – 9 January 2016) was an Australian researcher in probability theory and mathematical statistics. The American Statistical Association described him as one of the most influential and prolific theoretical statisticians in the history of the field.
The School of Mathematics and Statistics Building at The University of Melbourne was renamed the Peter Hall building in his honour on 9 December 2016.
Education
Hall attended Sydney Technical High School in Bexley, NSW during the years 1964–1969. He placed consistently high in examination results and in his final year, was among the top achievers in his form, and the winner of Old Boys' Union Mathematics prize.
Hall earned his Doctor of Philosophy degree at the University of Oxford in 1976 for research supervised by John Kingman.
Career and research
Hall was an author in probability and statistics. MathSciNet lists him with 606 publications as of January 2016. Google Scholar lists his h-index as 113. He made contributions to nonparametric statistics, in particular for curve estimation and resampling: the bootstrap method, smoothing, density estimation, and bandwidth selection. He worked on numerous applications across fields of economics, engineering, physical science and biological science. Hall also made contributions to surface roughness measurement using fractals. In probability theory he made many contributions to limit theory, spatial processes and stochastic geometry. His paper "Theoretical comparison of bootstrap confidence intervals" (Annals of Statistics, 1988) has been reprinted in the Breakthroughs in Statistics collection.
He was an Australian Research Council (ARC) Laureate Fellow at the School of Mathematics and Statistics, University of Melbourne, and also had a joint appointment at University of California Davis. He previously held a professorship at the Centre for Mathematics and its Applications at the Australian National University.
He was an ISI Highly Cited Researcher.
He is one of only three researchers based outside of North America to win the prestigious COPSS presidents' Award.
Honours and awards
His awards and honours included:
2015 Fellow of the Academy of Social Sciences in Australia
2013 Foreign Associate, National Academy of Sciences
2013 Officer of the Order of Australia
2012 Wilks Memorial Award
2011 Australian Laureate Fellowship
2011 Guy Medal in Silver
2010 George Szekeres Medal
2009 Honorary Doctor of Science degree from The University of Sydney
2007 Matthew Flinders Medal and Lecture
2000 Elected a Fellow of the Royal Society (FRS)
1998 Invited Speaker of the International Congress of Mathematicians
1996 Fellow of the American Statistical Association
1994 Hannan Medal of the Australian Academy of Science
1990 Pitman Medal from the Statistical Society of Australia
1989 Committee of presidents of Statistical Societies Award
1987 Fellow of the Australian Academy of Science
1986 Rollo Davidson Prize, Universit |
https://en.wikipedia.org/wiki/Jean-Claude%20Pagal | Jean-Claude Pagal (born September 15, 1964 in Yaoundé, Cameroon) is a former Cameroonian footballer.
Career statistics
International goals
References
External links
Profile
1964 births
Living people
Footballers from Yaoundé
Cameroonian men's footballers
Cameroon men's international footballers
Cameroonian expatriate men's footballers
Expatriate men's footballers in China
Expatriate men's footballers in England
Expatriate men's footballers in France
Expatriate men's footballers in Malta
Expatriate men's footballers in Mexico
Cameroonian expatriate sportspeople in China
Cameroonian expatriate sportspeople in England
Cameroonian expatriate sportspeople in France
Cameroonian expatriate sportspeople in Malta
Cameroonian expatriate sportspeople in Mexico
RC Lens players
La Roche VF players
AS Saint-Étienne players
FC Martigues players
Ligue 1 players
Club América footballers
Liga MX players
Carlisle United F.C. players
Sliema Wanderers F.C. players
1990 FIFA World Cup players
1990 African Cup of Nations players
1992 African Cup of Nations players
Men's association football midfielders |
https://en.wikipedia.org/wiki/Tychonoff%20axiom | In mathematics, a Tychnoff axiom may be:
the T3½ axiom that defines Tychonoff spaces; or
any of the Tychonoff separation axioms. |
https://en.wikipedia.org/wiki/Goran%20Grani%C4%87%20%28footballer%29 | Goran Granić (born 9 July 1975) is a Bosnian and Croatian professional football manager and former player.
Managerial statistics
References
External links
1975 births
Living people
Sportspeople from Livno
Men's association football defenders
Bosnia and Herzegovina men's footballers
Croatian men's footballers
NK Neretva players
NK Zagreb players
NK Troglav 1918 Livno players
NK Rudar Velenje players
NK Olimpija Ljubljana (1945–2005) players
NK Varaždin (1931–2015) players
HNK Hajduk Split players
NK Pomorac 1921 players
FC Dinamo City players
NK Dugopolje players
Croatian Football League players
Slovenian PrvaLiga players
First Football League (Croatia) players
Kategoria Superiore players
Croatian expatriate men's footballers
Expatriate men's footballers in Slovenia
Croatian expatriate sportspeople in Slovenia
Expatriate men's footballers in Albania
Croatian expatriate sportspeople in Albania
Bosnia and Herzegovina football managers
Croatian football managers
Premier League of Bosnia and Herzegovina managers
HŠK Posušje managers |
https://en.wikipedia.org/wiki/Kalle%20Kulbok | Kalle Kulbok (born 3 April 1956 in Tallinn) is an Estonian politician.
Education
Kulbok attended the Nõo Gymnasium and achieved good results on national olympiads of mathematics. By his skills, he could have qualified for the all-Union olympiad of mathematics; however, due to his displayed lack of respect for Soviet politics, he was never selected for that round.
Politics
In 1992–1995, Kulbok belonged to Riigikogu as a member of the Independent Royalist Party of Estonia. In 1995, Kulbok was not allowed to run for Riigikogu because he refused to swear allegiance to Republic of Estonia.
In later years, his political activity has mainly involved Euroskepticism. He's earned notoriety by several interesting actions, most notably by having himself pilloried for failure to derail Estonia's entry to European Union, and his declaration of revocation of his Estonian citizenship. As of 2007, Kulbok is the only natural-born citizen of Estonia known to have voluntarily requested revocation of his citizenship without naturalising in another country.
On 19 September 2003, Kulbok petitioned the Supreme Court of Estonia to declare the referendum for Estonia's joining European Union unconstitutional. The Supreme Court refused to consider the matter, arguing that only President of Estonia can rightfully present such a petition.
Career
For many years, Kulbok worked as a specialist at EENet.
References
External links
Postimees 17 October 2004: Nimetagem usuõpetus eetikaks ning vesi ongi sogatud
1956 births
Living people
Estonian humorists
Estonian Royalist Party politicians
Members of the Riigikogu, 1992–1995
Politicians from Tallinn
21st-century Estonian politicians
20th-century Estonian politicians |
https://en.wikipedia.org/wiki/Daniyar%20Mukanov | Daniyar Mukanov () (born 16 September 1978) is a retired Kazakhstani football defender.
Career statistics
International
References
External links
Living people
1978 births
Kazakhstani men's footballers
Men's association football defenders
Kazakhstan men's international footballers
Kazakhstan Premier League players
FC Yelimay players
FC Zhetysu players
FC Vostok players
FC Tobol players
FC Aktobe players
FC Atyrau players |
https://en.wikipedia.org/wiki/Hassan%20Al-Otaibi | Hassan Al-Otaibi () (born on August 6, 1976) is a Saudi Arabian former football goalkeeper who played for Al-Hilal and Al-Qadisiya.
Statistics
Honours
International
Saudi Arabia
Islamic Solidarity Games: 2005
References
Profile at Weltfussball
Living people
1976 births
Saudi Arabian men's footballers
Al-Dera'a FC players
Al Hilal SFC players
Al Qadsiah FC players
Saudi Pro League players
Men's association football goalkeepers |
https://en.wikipedia.org/wiki/Breather%20surface | In differential geometry, a breather surface is a one-parameter family of mathematical surfaces which correspond to breather solutions of the sine-Gordon equation, a differential equation appearing in theoretical physics. The surfaces have the remarkable property that they have constant curvature , where the curvature is well-defined. This makes them examples of generalized pseudospheres.
Mathematical background
There is a correspondence between embedded surfaces of constant curvature -1, known as pseudospheres, and solutions to the sine-Gordon equation. This correspondence can be built starting with the simplest example of a pseudosphere, the tractroid. In a special set of coordinates, known as asymptotic coordinates, the Gauss–Codazzi equations, which are consistency equations dictating when a surface of prescribed first and second fundamental form can be embedded into three-dimensional space with the flat metric, reduce to the sine-Gordon equation.
In the correspondence, the tractroid corresponds to the static 1-soliton solution of the sine-Gordon solution. Due to the Lorentz invariance of sine-Gordon, a one-parameter family of Lorentz boosts can be applied to the static solution to obtain new solutions: on the pseudosphere side, these are known as Lie transformations, which deform the tractroid to the one-parameter family of surfaces known as Dini's surfaces.
The method of Bäcklund transformation allows the construction of a large number of distinct solutions to the sine-Gordon equation, the multi-soliton solutions. For example, the 2-soliton corresponds to the Kuen surface. However, while this generates an infinite family of solutions, the breather solutions are not among them.
Breather solutions are instead derived from the inverse scattering method for the sine-Gordon equation. They are localized in space but oscillate in time.
Each solution to the sine-Gordon equation gives a first and second fundamental form which satisfy the Gauss-Codazzi equations. The fundamental theorem of surface theory then guarantees that there is a parameterized surface which recovers the prescribed first and second fundamental forms. Locally the parameterization is well-behaved, but extended arbitrarily the resulting surface may have self-intersections and cusps. Indeed, a theorem of Hilbert says that any pseudosphere cannot be embedded regularly (roughly, meaning without cusps) into .
Parameterization
The parameterization with parameter is given by
References
External links
Xah Lee Web - Surface Gallery
Breather surface in Virtual Math Museum
Surfaces
Mathematics articles needing expert attention
Differential equations |
https://en.wikipedia.org/wiki/List%20of%20countries%20by%20natural%20gas%20production | This is a list of countries by natural gas production based on statistics from The World Factbook, and OECD members natural gas production by International Energy Agency (down)
Countries by natural gas production
The data in the following table comes from The World Factbook.
OECD Members natural gas production by International Energy Agency
As of 2019:
See also
List of countries by natural gas proven reserves
List of countries by natural gas consumption
Natural gas by country
World energy supply and consumption
List of countries by oil production
References
Energy-related lists by country
List |
https://en.wikipedia.org/wiki/Regular%20map | Regular map may refer to:
a regular map (algebraic geometry), in algebraic geometry, an everywhere-defined, polynomial function of algebraic varieties
a regular map (graph theory), a symmetric 2-cell embedding of a graph into a closed surface |
https://en.wikipedia.org/wiki/Montejo%20de%20Ar%C3%A9valo | Montejo de Arévalo is a municipality located in the province of Segovia, Castile and León, Spain. According to the Spanish National Statistics Institute, in 2022, the municipality has a population of 171 inhabitants.
It was known as Montejo de la Vega de Arévalo until the beginning of the 20th century, due to its geography.
Demography
Landmarks
Santo Tomas de Aquino's church: It's a gothic-mudéjar temple.
Casas solariegas
Virgen de los Huertos' church: It was edified in the same spot where appeared the sculpture of the virgin that gives name to the church. That sculpture was moved to Santo Tomás de Aquino's church, but it continued returning to that spot until the church was built.
Chain
References
Municipalities in the Province of Segovia |
https://en.wikipedia.org/wiki/La%20P%C3%B3veda%20de%20Soria | La Póveda de Soria is a municipality located in the province of Soria, Castile and León, Spain. According to the latest 2019 data from the Spanish National Institute of Statistics (INE), the municipality has a population of 123 inhabitants.
References
Municipalities in the Province of Soria |
https://en.wikipedia.org/wiki/Anisohedral%20tiling | In geometry, a shape is said to be anisohedral if it admits a tiling, but no such tiling is isohedral (tile-transitive); that is, in any tiling by that shape there are two tiles that are not equivalent under any symmetry of the tiling. A tiling by an anisohedral tile is referred to as an anisohedral tiling.
Existence
The first part of Hilbert's eighteenth problem asked whether there exists an anisohedral polyhedron in Euclidean 3-space; Grünbaum and Shephard suggest that Hilbert was assuming that no such tile existed in the plane. Reinhardt answered Hilbert's problem in 1928 by finding examples of such polyhedra, and asserted that his proof that no such tiles exist in the plane would appear soon. However, Heesch then gave an example of an anisohedral tile in the plane in 1935.
Convex tiles
Reinhardt had previously considered the question of anisohedral convex polygons, showing that there were no anisohedral convex hexagons but being unable to show there were no such convex pentagons, while finding the five types of convex pentagon tiling the plane isohedrally. Kershner gave three types of anisohedral convex pentagon in 1968; one of these tiles using only direct isometries without reflections or glide reflections, so answering a question of Heesch.
Isohedral numbers
The problem of anisohedral tiling has been generalised by saying that the isohedral number of a tile is the lowest number orbits (equivalence classes) of tiles in any tiling of that tile under the action of the symmetry group of that tiling, and that a tile with isohedral number k is k-anisohedral. Berglund asked whether there exist k-anisohedral tiles for all k, giving examples for k ≤ 4 (examples of 2-anisohedral and 3-anisohedral tiles being previously known, while the 4-anisohedral tile given was the first such published tile). Goodman-Strauss considered this in the context of general questions about how complex the behaviour of a given tile or set of tiles can be, noting a 10-anisohedral example of Myers. Grünbaum and Shephard had previously raised a slight variation on the same question.
Socolar showed in 2007 that arbitrarily high isohedral numbers can be achieved in two dimensions if the tile is disconnected, or has coloured edges with constraints on what colours can be adjacent, and in three dimensions with a connected tile without colours, noting that in two dimensions for a connected tile without colours the highest known isohedral number is 10.
Joseph Myers has produced a collection of tiles with high isohedral numbers, particularly a polyhexagon with isohedral number 10 (occurring in 20 orbits under translation) and another with isohedral number 9 (occurring in 36 orbits under translation).
References
External links
John Berglund, Anisohedral Tilings Page
Joseph Myers, Polyomino, polyhex and polyiamond tiling
Tessellation |
https://en.wikipedia.org/wiki/Arthur%20Harold%20Stone | Arthur Harold Stone (30 September 1916 – 6 August 2000) was a British mathematician born in London, who worked at the universities of Manchester and Rochester, mostly in topology. His wife was American mathematician Dorothy Maharam.
Stone studied at Trinity College, Cambridge. His first paper dealt with squaring the square, he proved the Erdős–Stone theorem with Paul Erdős and is credited with the discovery of the first two flexagons, a trihexaflexagon and a hexahexaflexagon while he was a student at Princeton University in 1939. His Ph.D. thesis, Connectedness and Coherence, was written in 1941 under the direction of Solomon Lefschetz. He served as a referee for The American Mathematical Monthly journal in the 1980s.
The Stone metrization theorem has been named after him, and he was a member of a group of mathematicians who published pseudonymously as Blanche Descartes. He is not to be confused with American mathematician Marshall Harvey Stone.
See also
Ham sandwich theorem
References
External links
1916 births
2000 deaths
20th-century American mathematicians
20th-century English mathematicians
Alumni of Trinity College, Cambridge
British expatriate academics in the United States
English expatriates in the United States
Topologists
Mathematicians from London
Princeton University alumni |
https://en.wikipedia.org/wiki/Cogo | Cogo may refer to:
COGO, coordinate geometry software
CoGo, a bikeshare system in Columbus, Ohio
Cogo, Equatorial Guinea, town
Cogo, Tibet, village
Cogo (toy company), a Chinese company of brick toys |
https://en.wikipedia.org/wiki/Istv%C3%A1n%20F%C3%A1ry | István Fáry (30 June 1922 – 2 November 1984) was a Hungarian-born mathematician known for his work in geometry and algebraic topology. He proved Fáry's theorem that every planar graph has a straight-line embedding in 1948, and the Fáry–Milnor theorem lower-bounding the curvature of a nontrivial knot in 1949.
Biography
Fáry was born June 30, 1922, in Gyula, Hungary. After studying for a master's degree at the University of Budapest, he moved to the University of Szeged, where he earned a Ph.D. in 1947. He then studied at the Sorbonne before taking a faculty position at the University of Montreal in 1955. He moved to the University of California, Berkeley in 1958 and became a full professor in 1962. He died on November 2, 1984, in El Cerrito, California.
Selected publications
.
.
References
External links
Photos from the Oberwolfach Photo Collection
1922 births
1984 deaths
20th-century Hungarian mathematicians
University of California, Berkeley College of Letters and Science faculty
Geometers
Topologists
University of Paris alumni
Hungarian expatriates in France
Hungarian expatriates in Canada
Hungarian expatriates in the United States |
https://en.wikipedia.org/wiki/List%20of%20Major%20League%20Baseball%20career%20games%20started%20leaders | In baseball statistics, a pitcher is credited with a game started (denoted by GS) if he is the first pitcher to pitch for his team in a game.
Cy Young holds the Major League Baseball games started record with 815. Young is the only pitcher in MLB history to start more than 800 career games. Nolan Ryan (773), Don Sutton (756), Greg Maddux (740), Phil Niekro (716), Steve Carlton (709), Roger Clemens (707), and Tommy John (700) are the only other pitchers to have started 700 or more games their career.
Key
List
Stats updated as of October 1, 2023.
Notes
References
External links
Games s
Major League Baseball statistics |
https://en.wikipedia.org/wiki/Stack%20%28mathematics%29 | In mathematics a stack or 2-sheaf is, roughly speaking, a sheaf that takes values in categories rather than sets. Stacks are used to formalise some of the main constructions of descent theory, and to construct fine moduli stacks when fine moduli spaces do not exist.
Descent theory is concerned with generalisations of situations where isomorphic, compatible geometrical objects (such as vector bundles on topological spaces) can be "glued together" within a restriction of the topological basis. In a more general set-up the restrictions are replaced with pullbacks; fibred categories then make a good framework to discuss the possibility of such gluing. The intuitive meaning of a stack is that it is a fibred category such that "all possible gluings work". The specification of gluings requires a definition of coverings with regard to which the gluings can be considered. It turns out that the general language for describing these coverings is that of a Grothendieck topology. Thus a stack is formally given as a fibred category over another base category, where the base has a Grothendieck topology and where the fibred category satisfies a few axioms that ensure existence and uniqueness of certain gluings with respect to the Grothendieck topology.
Overview
Stacks are the underlying structure of algebraic stacks (also called Artin stacks) and Deligne–Mumford stacks, which generalize schemes and algebraic spaces and which are particularly useful in studying moduli spaces. There are inclusions: schemes ⊆ algebraic spaces ⊆ Deligne–Mumford stacks ⊆ algebraic stacks (Artin stacks) ⊆ stacks. and give a brief introductory accounts of stacks, , and give more detailed introductions, and describes the more advanced theory.
Motivation and history
The concept of stacks has its origin in the definition of effective descent data in .
In a 1959 letter to Serre, Grothendieck observed that a fundamental obstruction to constructing good moduli spaces is the existence of automorphisms. A major motivation for stacks is that if a moduli space for some problem does not exist because of the existence of automorphisms, it may still be possible to construct a moduli stack.
studied the Picard group of the moduli stack of elliptic curves, before stacks had been defined. Stacks were first defined by , and the term "stack" was introduced by for the original French term "champ" meaning "field". In this paper they also introduced Deligne–Mumford stacks, which they called algebraic stacks, though the term "algebraic stack" now usually refers to the more general Artin stacks introduced by .
When defining quotients of schemes by group actions, it is often impossible for the quotient to be a scheme and still satisfy desirable properties for a quotient. For example, if a few points have non-trivial stabilisers, then the categorical quotient will not exist among schemes, but it will exist as a stack.
In the same way, moduli spaces of curves, vector bundles, or other geometric ob |
https://en.wikipedia.org/wiki/Seashell%20surface | In mathematics, a seashell surface is a surface made by a circle which spirals up the z-axis while decreasing its own radius and distance from the z-axis. Not all seashell surfaces describe actual seashells found in nature.
Parametrization
The following is a parameterization of one seashell surface:
where and \\
Various authors have suggested different models for the shape of shell. David M. Raup proposed a model where there is one magnification for the x-y plane, and another for the x-z plane. Chris Illert proposed a model where the magnification is scalar, and the same for any sense or direction with an equation like
which starts with an initial generating curve and applies a rotation and exponential magnification.
See also
Helix
Seashell
Spiral
References
C. Illert (Feb. 1983), "the mathematics of Gnomonic seashells", Mathematical Biosciences 63(1): 21-56.
C. Illert (1987), "Part 1, seashell geometry", Il Nuovo Cimento 9D(7): 702-813.
C. Illert (1989), "Part 2, tubular 3D seashell surfaces", Il Nuovo Cimento 11D(5): 761-780.
C. Illert (Oct 1990),"Nipponites mirabilis, a challenge to seashell theory?", Il Nuovo Cimento 12D(10): 1405-1421.
C. Illert (Dec 1990), "elastic conoidal spires", Il Nuovo Cimento 12D(12): 1611-1632.
C. Illert & C. Pickover (May 1992), "generating irregularly oscillating fossil seashells", IEE Computer Graphics & Applications 12(3):18-22.
C. Illert (July 1995), "Australian supercomputer graphics exhibition", IEEE Computer Graphics & Applications 15(4):89-91.
C. Illert (Editor 1995), "Proceedings of the First International Conchology Conference, 2-7 Jan 1995, Tweed Shire, Australia", publ. by Hadronic Press, Florida USA. 219 pages.
C. Illert & R. Santilli (1995), "Foundations of Theoretical Conchology", publ. by Hadronic Press, Florida USA. 183 pages plus coloured plates.
Deborah R. Fowler, Hans Meinhardt, and Przemyslaw Prusinkiewicz. Modeling seashells. Proceedings of SIGGRAPH '92 (Chicago, Illinois, July 26–31, 1992), In Computer Graphics, 26, 2, (July 1992), ACM SIGGRAPH, New York, pp. 379–387.
Callum Galbraith, Przemyslaw Prusinkiewicz, and Brian Wyvill. Modeling a Murex cabritii sea shell with a structured implicit surface modeler. The Visual Computer vol. 18, pp. 70–80. http://algorithmicbotany.org/papers/murex.tvc2002.html
Surfaces |
https://en.wikipedia.org/wiki/Nonlinearity%20%28journal%29 | Nonlinearity is a peer-reviewed scientific journal published by IOP Publishing and the London Mathematical Society. The journal publishes papers on nonlinear mathematics, mathematical physics, experimental physics, theoretical physics and other areas in the sciences where nonlinear phenomena are of fundamental importance. The Editors-in-Chief are Tasso J Kaper (Boston University) for IOP Publishing and Konstantin Khanin (University of Toronto) for the London Mathematical Society.
Abstracting and indexing
The journal is abstracted and indexed in Science Citation Index, Current Contents/Physical, Chemical & Earth Sciences, Inspec, CompuMath Citation Index, Mathematical Reviews, MathSciNet, Zentralblatt MATH, and VINITI Database RAS. According to the Journal Citation Reports, the journal has a 2020 impact factor of 2.129.
See also
Journal of Physics A
Inverse Problems
London Mathematical Society
IOP Publishing
References
External links
Physics journals
IOP Publishing academic journals
Academic journals established in 1988
Mathematics journals
Monthly journals
English-language journals
London Mathematical Society |
https://en.wikipedia.org/wiki/Carlos%20B%C3%A1ez%20%28footballer%2C%20born%201982%29 | Carlos Báez Appleyard (born 12 June 1982 in Asunción, Paraguay) is a Paraguayan football defender currently playing for O'Higgins in Chile.
External links
Argentine Primera statistics
1982 births
Living people
Footballers from Asunción
Paraguayan men's footballers
Paraguayan expatriate men's footballers
Men's association football defenders
Cerro Porteño players
Club Atlético Independiente footballers
Arsenal de Sarandí footballers
Cúcuta Deportivo footballers
O'Higgins F.C. footballers
Chilean Primera División players
Argentine Primera División players
Categoría Primera A players
Expatriate men's footballers in Argentina
Expatriate men's footballers in Chile
Expatriate men's footballers in Colombia |
https://en.wikipedia.org/wiki/Meco%2C%20Spain | Meco is a municipality in the eastern part of the Autonomous Community of Madrid, (Spain). In 2006, Meco had a population of 11,094 (Spanish National Statistics Institute).
The town is located to the north of the River Henares, in the comarca of La Campiña del Henares, and also in one of the two natural sub-comarcas that make up the Comarca de Alcalá, La Campiña del Henares, characterized by vast grain fields and gentle hills. It is 8 km to the north of the metropolitan area of Alcalá de Henares and 4 km from the University of Alcalá. It provides residences for the university and also is part of the Henares Industrial Corridor.
On the east, Meco borders the municipalities of Azuqueca de Henares and Villanueva de la Torre, of Guadalajara, and also the autonomous community of Castile-La Mancha. It can be reached by the Royal Cattle Track of Galiana or Riojana, and by the last section of the Henares Canal.
Geography
Altitude: 673 m
Latitude: 40° 33′ 00″ N
Longitude: 003° 19′ 59″ E
Located at the edge of one of the terraces of the River Henares, 35 km from the capital, Madrid. The rivers Las Monjas, Villanueva, and its tributary, Valdegatos.
Meco also has the distinction of being Spain's most distant populated area from the sea.
Economy
A town with a dry-soil agricultural tradition, Meco has long been an exporter of wheat, flour and bread to Alcalá, Guadalajara, and Madrid. Also, its economy is based on salt mining.
History
There are archaeological remains from the Iron Age, as well as a Roman necropolis. The municipality may have originated as a Roman agricultural estate, and may have been a wheat staple in Muslim times. After the re-conquest of the Media Marca (Middle March) by Alfonso VI, it came to form part of the Commune of Villa and Land of Guadalajara. In the 15th century, King Juan II gave it to the Marquis of Santillana, Iñigo López de Mendoza, who passed it ti his second son, Iñigo López de Mendoza, Count of Tendilla. The son of Iñigo López de Mendoza, also of the same name, added the title of Marquis of Móndejar by buying this villa from the Catholic Monarchs. In 1801, Alcalá de Henares was separated from the administration of Toledo and became capital of the administrative region. It also separated from Guadalajara, coming to form part of the region of Alcalá de Henares.
Art
Among the buildings of the town, a magnificent parish Church of Nuestra Señora de la Asunción (Meco) stands out. It is dedicated to Our Lady of the Assumption, and is located on Plaza de la Constitución. Construction started in the middle of the 16th century in the Gothic Transitional style, and was completed in the 17th in Baroque style. The presbytery was therefore built of limestone masonry, while the sacristy and tower were brick.
A hall church with columns, its design has been attributed to Rodrigo Gil de Hontañón by some historians. It has three naves, the middle one with barrel vaults flattened at the top, and the lateral ones with pointed arches. It |
https://en.wikipedia.org/wiki/Italian%20school | The Italian School refers to several different Italian schools of thought, including:
Italian School (art)
Italian School (philosophy)
Italian school of algebraic geometry
Italian school of swordsmanship
Italian school of criminology
Italian school of engraving
Italian school of elitism
Italian school of singing |
https://en.wikipedia.org/wiki/Aleksandre%20Gogoberishvili | Aleksandr Gogoberishvili (; born 16 February 1977) is a Georgian former footballer.
Career statistics
Honors
FC Baku
Azerbaijan Premier League: 2005–06
References
External links
1977 births
Living people
Men's footballers from Georgia (country)
Georgia (country) men's international footballers
FC Dinamo Tbilisi players
FC Guria Lanchkhuti players
FC Merani Tbilisi players
FC Locomotive Tbilisi players
Expatriate men's footballers from Georgia (country)
Expatriate men's footballers in Russia
Expatriate men's footballers in Azerbaijan
Russian Premier League players
FC Anzhi Makhachkala players
FC Baku players
Qarabağ FK players
Shuvalan FK players
FC Sioni Bolnisi players
Dinamo Zugdidi players
Turan Tovuz players
Expatriate sportspeople from Georgia (country) in Azerbaijan
Men's association football midfielders
FC Shevardeni-1906 Tbilisi players
FC WIT Georgia players
Footballers from Tbilisi |
https://en.wikipedia.org/wiki/Roman%20Akhalkatsi | Roman Akhalkatsi (; born 20 February 1980) is a Georgian former footballer who played as a midfielder.
He was the first Georgian to play in the A PFG.
Azerbaijan statistics
References
External links
Profile at KLISF
Player profile
1980 births
Living people
Men's footballers from Georgia (country)
Expatriate men's footballers in Russia
Expatriate men's footballers in Azerbaijan
FC Lokomotiv Moscow players
FC Baltika Kaliningrad players
Karvan FK players
Simurq PIK players
People from Gori, Georgia
Expatriate sportspeople from Georgia (country) in Azerbaijan
FC Dila Gori players
FC Kolkheti-1913 Poti players
FC Torpedo Kutaisi players
FC Metalurgi Rustavi players
Men's association football midfielders |
https://en.wikipedia.org/wiki/Suleyman%20Camara | Souleymane Camara (born 10 April 1984) is an Ivorian football striker whose last known club was Karvan in the Azerbaijan Premier League.
Career statistics
References
External links
Player profile
1984 births
Living people
Ivorian men's footballers
Expatriate men's footballers in Azerbaijan
Ivorian expatriate sportspeople in Azerbaijan
Footballers from Abidjan
Men's association football forwards
Karvan FK players |
https://en.wikipedia.org/wiki/Ramiz%20Kerimov | Ramiz Kerimov (born 4 August 1981) is an Azerbaijani former football goalkeeper.
Career statistics
References
External links
Player profile
1981 births
Living people
Azerbaijani men's footballers
Khazar Lankaran FK players
Men's association football goalkeepers
MOIK Baku players |
https://en.wikipedia.org/wiki/Fizuli%20Mammadov | Fizuli Mammadov (born 8 September 1977) is an Azerbaijani footballer playing as a defender.
National team statistics
External links
Profile at footballzz.co.uk
1977 births
Living people
Azerbaijani men's footballers
Azerbaijani expatriate men's footballers
Simurq PIK players
Machine Sazi F.C. players
FC Spartak Ivano-Frankivsk players
Expatriate men's footballers in Iran
Expatriate men's footballers in Ukraine
Men's association football defenders
Azerbaijan men's international footballers |
https://en.wikipedia.org/wiki/Alim%20Qurbanov | Alim Qurbanov (born 5 December 1977) is a retired Azerbaijani footballer who spent most of his career playing for club Khazar Lankaran as a midfielder.
National team statistics
International goals
References
External links
1977 births
Living people
Azerbaijani men's footballers
Azerbaijan men's international footballers
Footballers from Baku
Khazar Lankaran FK players
Azerbaijan Premier League players
Men's association football midfielders |
https://en.wikipedia.org/wiki/Kostadin%20Dzhambazov | Kostadin Dzhambazov (Bulgarian Cyrillic: Костадин Джамбазов; born 6 July 1980 in Burgas) is a former Bulgarian footballer, who played as a defender.
External links
2006–07 Statistics at PFL.bg
Bulgarian men's footballers
PFC Slavia Sofia players
PFC Litex Lovech players
FC Chernomorets Burgas players
PFC Neftochimic Burgas (2009–2014) players
Khazar Lankaran FK players
OFC Nesebar players
First Professional Football League (Bulgaria) players
1980 births
Living people
Men's association football defenders
Bulgarian expatriate men's footballers
Bulgarian expatriate sportspeople in Azerbaijan
Expatriate men's footballers in Azerbaijan
Footballers from Burgas |
https://en.wikipedia.org/wiki/Derivator | In mathematics, derivators are a proposed frameworkpg 190-195 for homological algebra giving a foundation for both abelian and non-abelian homological algebra and various generalizations of it. They were introduced to address the deficiencies of derived categories (such as the non-functoriality of the cone construction) and provide at the same time a language for homotopical algebra.
Derivators were first introduced by Alexander Grothendieck in his long unpublished 1983 manuscript Pursuing Stacks. They were then further developed by him in the huge unpublished 1991 manuscript Les Dérivateurs of almost 2000 pages. Essentially the same concept was introduced (apparently independently) by Alex Heller.
The manuscript has been edited for on-line publication by Georges Maltsiniotis. The theory has been further developed by several other people, including Heller, Franke, Keller and Groth.
Motivations
One of the motivating reasons for considering derivators is the lack of functoriality with the cone construction with triangulated categories. Derivators are able to solve this problem, and solve the inclusion of general homotopy colimits, by keeping track of all possible diagrams in a category with weak equivalences and their relations between each other. Heuristically, given the diagramwhich is a category with two objects and one non-identity arrow, and a functorto a category with a class of weak-equivalences (and satisfying the right hypotheses), we should have an associated functorwhere the target object is unique up to weak equivalence in . Derivators are able to encode this kind of information and provide a diagram calculus to use in derived categories and homotopy theory.
Definition
Prederivators
Formally, a prederivator is a 2-functorfrom a suitable 2-category of indices to the category of categories. Typically such 2-functors come from considering the categories where is called the category of coefficients. For example, could be the category of small categories which are filtered, whose objects can be thought of as the indexing sets for a filtered colimit. Then, given a morphism of diagramsdenote byThis is called the inverse image functor. In the motivating example, this is just precompositition, so given a functor there is an associated functor . Note these 2-functors could be taken to bewhere is a suitable class of weak equivalences in a category .
Indexing categories
There are a number of examples of indexing categories which can be used in this construction
The 2-category of finite categories, so the objects are categories whose collection of objects are finite sets.
The ordinal category can be categorified into a two category, where the objects are categories with one object, and the functors come form the arrows in the ordinal category.
Another option is to just use the category of small categories.
In addition, associated to any topological space is a category which could be used as the indexing category.
Moreover |
https://en.wikipedia.org/wiki/Near-semiring | In mathematics, a near-semiring, also called a seminearring, is an algebraic structure more general than a near-ring or a semiring. Near-semirings arise naturally from functions on monoids.
Definition
A near-semiring is a set S with two binary operations "+" and "·", and a constant 0 such that (S, +, 0) is a monoid (not necessarily commutative), (S, ·) is a semigroup, these structures are related by a single (right or left) distributive law, and accordingly 0 is a one-sided (right or left, respectively) absorbing element.
Formally, an algebraic structure (S, +, ·, 0) is said to be a near-semiring if it satisfies the following axioms:
(S, +, 0) is a monoid,
(S, ·) is a semigroup,
(a + b) · c = a · c + b · c, for all a, b, c in S, and
0 · a = 0 for all a in S.
Near-semirings are a common abstraction of semirings and near-rings [Golan, 1999; Pilz, 1983]. The standard examples of near-semirings are typically of the form M(Г), the set of all mappings on a monoid (Г; +, 0), equipped with composition of mappings, pointwise addition of mappings, and the zero function. Subsets of M(Г) closed under the operations provide further examples of near-semirings. Another example is the ordinals under the usual operations of ordinal arithmetic (here Clause 3 should be replaced with its symmetric form c · (a + b) = c · a + c · b. Strictly speaking, the class of all ordinals is not a set, so the above example should be more appropriately called a class near-semiring. We get a near-semiring in the standard sense if we restrict to those ordinals strictly less than some multiplicatively indecomposable ordinal.
Bibliography
Golan, Jonathan S., Semirings and their applications. Updated and expanded version of The theory of semirings, with applications to mathematics and theoretical computer science (Longman Sci. Tech., Harlow, 1992, . Kluwer Academic Publishers, Dordrecht, 1999. xii+381 pp.
Krishna, K. V., Near-semirings: Theory and application, Ph.D. thesis, IIT Delhi, New Delhi, India, 2005.
Pilz, G., Near-Rings: The Theory and Its Applications, Vol. 23 of North-Holland Mathematics Studies, North-Holland Publishing Company, 1983.
The Near Ring Main Page at the Johannes Kepler Universität Linz
Willy G. van Hoorn and B. van Rootselaar, Fundamental notions in the theory of seminearrings, Compositio Mathematica v. 18, (1967), pp. 65–78.
Algebraic structures |
https://en.wikipedia.org/wiki/Separable%20algebra | In mathematics, a separable algebra is a kind of semisimple algebra. It is a generalization to associative algebras of the notion of a separable field extension.
Definition and first properties
A ring homomorphism (of unital, but not necessarily commutative rings)
is called separable if the multiplication map
admits a section
that is a homomorphism of A-A-bimodules.
If the ring is commmutative and maps into the center of , we call a separable algebra over .
It is useful to describe separability in terms of the element
The reason is that a section σ is determined by this element. The condition that σ is a section of μ is equivalent to
and the condition that σ is a homomorphism of A-A-bimodules is equivalent to the following requirement for any a in A:
Such an element p is called a separability idempotent, since regarded as an element of the algebra it satisfies .
Examples
For any commutative ring R, the (non-commutative) ring of n-by-n matrices is a separable R-algebra. For any , a separability idempotent is given by , where denotes the elementary matrix which is 0 except for the entry in position , which is 1. In particular, this shows that separability idempotents need not be unique.
Separable algebras over a field
A field extension L/K of finite degree is a separable extension if and only if L is separable as an associative K-algebra. If L/K has a primitive element with irreducible polynomial , then a separability idempotent is given by . The tensorands are dual bases for the trace map: if are the distinct K-monomorphisms of L into an algebraic closure of K, the trace mapping Tr of L into K is defined by . The trace map and its dual bases make explicit L as a Frobenius algebra over K.
More generally, separable algebras over a field K can be classified as follows: they are the same as finite products of matrix algebras over finite-dimensional division algebras whose centers are finite-dimensional separable field extensions of the field K. In particular: Every separable algebra is itself finite-dimensional. If K is a perfect field – for example a field of characteristic zero, or a finite field, or an algebraically closed field – then every extension of K is separable so that separable K-algebras are finite products of matrix algebras over finite-dimensional division algebras over field K. In other words, if K is a perfect field, there is no difference between a separable algebra over K and a finite-dimensional semisimple algebra over K.
It can be shown by a generalized theorem of Maschke that an associative K-algebra A is separable if for every field extension the algebra is semisimple.
Group rings
If K is commutative ring and G is a finite group such that the order of G is invertible in K, then the group ring K[G] is a separable K-algebra. A separability idempotent is given by .
Equivalent characterizations of separability
There are several equivalent definitions of separable algebras. A K-algebra |
https://en.wikipedia.org/wiki/Kachurovskii%27s%20theorem | In mathematics, Kachurovskii's theorem is a theorem relating the convexity of a function on a Banach space to the monotonicity of its Fréchet derivative.
Statement of the theorem
Let K be a convex subset of a Banach space V and let f : K → R ∪ {+∞} be an extended real-valued function that is Fréchet differentiable with derivative df(x) : V → R at each point x in K. (In fact, df(x) is an element of the continuous dual space V∗.) Then the following are equivalent:
f is a convex function;
for all x and y in K,
df is an (increasing) monotone operator, i.e., for all x and y in K,
References
(Proposition 7.4)
Convex analysis
Theorems in functional analysis |
https://en.wikipedia.org/wiki/K%C5%8Dmura%27s%20theorem | In mathematics, Kōmura's theorem is a result on the differentiability of absolutely continuous Banach space-valued functions, and is a substantial generalization of Lebesgue's theorem on the differentiability of the indefinite integral, which is that Φ : [0, T] → R given by
is differentiable at t for almost every 0 < t < T when φ : [0, T] → R lies in the Lp space L1([0, T]; R).
Statement
Let (X, || ||) be a reflexive Banach space and let φ : [0, T] → X be absolutely continuous. Then φ is (strongly) differentiable almost everywhere, the derivative φ′ lies in the Bochner space L1([0, T]; X), and, for all 0 ≤ t ≤ T,
References
(Theorem III.1.7)
Theorems in measure theory
Theorems in functional analysis |
https://en.wikipedia.org/wiki/Moreau%27s%20theorem | In mathematics, Moreau's theorem is a result in convex analysis named after French mathematician Jean-Jacques Moreau. It shows that sufficiently well-behaved convex functionals on Hilbert spaces are differentiable and the derivative is well-approximated by the so-called Yosida approximation, which is defined in terms of the resolvent operator.
Statement of the theorem
Let H be a Hilbert space and let φ : H → R ∪ {+∞} be a proper, convex and lower semi-continuous extended real-valued functional on H. Let A stand for ∂φ, the subderivative of φ; for α > 0 let Jα denote the resolvent:
and let Aα denote the Yosida approximation to A:
For each α > 0 and x ∈ H, let
Then
and φα is convex and Fréchet differentiable with derivative dφα = Aα. Also, for each x ∈ H (pointwise), φα(x) converges upwards to φ(x) as α → 0.
References
(Proposition IV.1.8)
Convex analysis
Theorems in functional analysis |
https://en.wikipedia.org/wiki/Annales%20Scientifiques%20de%20l%27%C3%89cole%20Normale%20Sup%C3%A9rieure | Annales Scientifiques de l'École Normale Supérieure is a French scientific journal of mathematics published by the Société Mathématique de France. It was established in 1864 by the French chemist Louis Pasteur and published articles in mathematics, physics, chemistry, biology, and geology. In 1900, it became a purely mathematical journal. It is published with help of the Centre national de la recherche scientifique. Its web site is hosted by the mathematics department of the École Normale Supérieure.
External links
Archive (1864–2013)
Mathematics journals
Publications established in 1864
Multilingual journals
Multidisciplinary scientific journals
Société Mathématique de France academic journals |
https://en.wikipedia.org/wiki/National%20Civil%20Registry%20%28Colombia%29 | The National Civil Registry () is the government agency of Colombia charged with collecting and storing the vital statistics and identifying information of all citizens, counts votes of campaigns for the Senate, presidency and the vice presidency, and to regulate the distribution and organization of identity documentation for each citizen for legal purposes.
Colombian citizens obtain their ID (), to be able to vote, and also do all the permits tramits. Their headquarters are located in Bogotá. The current manager of the office is Alexander Vega Rocha. The current legislation made it mandatory for all citizens to carry this document and be able to present the upon request by the authorities. Lack of the allows the local authorities to detain the citizen while the identity is verified in the government data base. This differs with the American legislation. The is a required document for entering and departing the country regardless of the place of residence or second nationality of a Colombian citizen. This requirement is only waved to Colombians that have renounced to the Colombian citizenship, after a lengthy process. The Colombian government made significant changes to the cedula and is requiring all citizens to change to the new national ID in preparation for the presidential elections early 2010.
References
Government of Colombia |
https://en.wikipedia.org/wiki/Berger%20inequality | In mathematics, Berger inequality may refer to
Berger's inequality for Einstein manifolds;
the Berger–Kazdan comparison theorem. |
https://en.wikipedia.org/wiki/Bochner%20identity | In mathematics — specifically, differential geometry — the Bochner identity is an identity concerning harmonic maps between Riemannian manifolds. The identity is named after the American mathematician Salomon Bochner.
Statement of the result
Let M and N be Riemannian manifolds and let u : M → N be a harmonic map. Let du denote the derivative (pushforward) of u, ∇ the gradient, Δ the Laplace–Beltrami operator, RiemN the Riemann curvature tensor on N and RicM the Ricci curvature tensor on M. Then
See also
Bochner's formula
References
External links
Differential geometry
Mathematical identities |
https://en.wikipedia.org/wiki/Furstenberg%27s%20proof%20of%20the%20infinitude%20of%20primes | In mathematics, particularly in number theory, Hillel Furstenberg's proof of the infinitude of primes is a topological proof that the integers contain infinitely many prime numbers. When examined closely, the proof is less a statement about topology than a statement about certain properties of arithmetic sequences. Unlike Euclid's classical proof, Furstenberg's proof is a proof by contradiction. The proof was published in 1955 in the American Mathematical Monthly while Furstenberg was still an undergraduate student at Yeshiva University.
Furstenberg's proof
Define a topology on the integers , called the evenly spaced integer topology, by declaring a subset U ⊆ to be an open set if and only if it is a union of arithmetic sequences S(a, b) for a ≠ 0, or is empty (which can be seen as a nullary union (empty union) of arithmetic sequences), where
Equivalently, U is open if and only if for every x in U there is some non-zero integer a such that S(a, x) ⊆ U. The axioms for a topology are easily verified:
∅ is open by definition, and is just the sequence S(1, 0), and so is open as well.
Any union of open sets is open: for any collection of open sets Ui and x in their union U, any of the numbers ai for which S(ai, x) ⊆ Ui also shows that S(ai, x) ⊆ U.
The intersection of two (and hence finitely many) open sets is open: let U1 and U2 be open sets and let x ∈ U1 ∩ U2 (with numbers a1 and a2 establishing membership). Set a to be the least common multiple of a1 and a2. Then S(a, x) ⊆ S(ai, x) ⊆ Ui.
This topology has two notable properties:
Since any non-empty open set contains an infinite sequence, a finite non-empty set cannot be open; put another way, the complement of a finite non-empty set cannot be a closed set.
The basis sets S(a, b) are both open and closed: they are open by definition, and we can write S(a, b) as the complement of an open set as follows:
The only integers that are not integer multiples of prime numbers are −1 and +1, i.e.
Now, by the first topological property, the set on the left-hand side cannot be closed. On the other hand, by the second topological property, the sets S(p, 0) are closed. So, if there were only finitely many prime numbers, then the set on the right-hand side would be a finite union of closed sets, and hence closed. This would be a contradiction, so there must be infinitely many prime numbers.
Topological properties
The evenly spaced integer topology on is the topology induced by the inclusion , where is the profinite integer ring with its profinite topology.
It is homeomorphic to the rational numbers with the subspace topology inherited from the real line, which makes it clear that any finite subset of it, such as , cannot be open.
Notes
References
Keith Conrad https://kconrad.math.uconn.edu/blurbs/ugradnumthy/primestopology.pdf
External links
Furstenberg's proof that there are infinitely many prime numbers at Everything2
Article proofs
General topology
Prime numbers |
https://en.wikipedia.org/wiki/Valley%20Junior/Senior%20High%20School | Valley Junior/Senior High School is a public school in New Kensington, Westmoreland County in the state of Pennsylvania. According to the National Center for Education Statistics, in the 2018–2019 school year, the School reported an enrollment of 792 pupils in grades 9th through 12th.
Demographics of student body
As of 2009.
Alternative education
Valley High School has an alternative education program for students with behavioral issues, those who have been chronically truant or are expelled from the traditional school programs. Students work toward graduation under the supervision of a teacher using online OdysseyWare software.
Awards and recognition
Valley High School's Junior ROTC program was named an Honor Unit with Distinction in 2006 and 2009, scoring in the 96th percentile in an inspection held once every three years.
In 1998, a team of students representing Valley High School tied for third place in an international Space Settlement Design contest sponsored by NASA, for their research project, entitled "Space Colonies, A Design Study."
Ten Commandments controversy
On March 20, 2012, the Freedom From Religion Foundation (FFRF) sent a letter of complaint about a large granite monument with 10 Commandments predominantly displayed near the main entrance to the school, citing that the school is in violation of the establishment clause of the First Amendment.
On October 13, 2012, approximately 50 people attended a rally in support of keeping the monument at Valley High School.
On September 14, 2012, the FFRF and four New Kensington residents filed suit against the school district seeking a declaration that the monument is unconstitutional, a permanent injunction directing its removal, nominal damages and costs and attorneys’ fees. The Judge, Terrance McVerry ruled on Dec. 19, 2012 allowing the plaintiffs the right to use pseudonyms. On Jan. 22, 2013 he denied the motion to dismiss, allowing the case to continue.
On July 27, 2015, the District Court ruled that the Plaintiffs did not have standing.
On August 9, 2016 the U.S. Third Circuit Court of Appeals unanimously ruled that Plaintiff Marie Schaub had standing and remanded the case back to Judge Terrance McVerry to be heard on its merits. On February 21, 2017, the lawsuit came to an end when the school district agreed to remove the monument from in front of the school and for the school district, through their insurance company, to pay $163,500 in legal fees, including more than $40,000 to the Freedom From Religion Foundation. In accord with the agreement, the monument was removed on March 21, 2017. It was donated to a local elementary Catholic school, Mary Queen of Apostles, where it currently resides.
Extracurriculars
New Kensington-Arnold School District offers a wide variety of clubs, activities and an extensive sports program.
Athletics
Vocational–technical education
Students in grades 10–12 may attend the Northern Westmoreland Career and Technology Center part-time, if ele |
https://en.wikipedia.org/wiki/Victor%20Guillemin | Victor William Guillemin (born 1937 in Boston) is an American mathematician. He works at the Massachusetts Institute of Technology in the field of symplectic geometry, and he has also made contributions to the fields of microlocal analysis, spectral theory, and mathematical physics.
Education and career
Guillemin obtained a B.A. at Harvard University in 1959, as well as an M. A. at the University of Chicago in 1960. He received a Ph.D. in mathematics from Harvard University in 1962; his thesis, entitled Theory of Finite G-Structures, was written under the direction of Shlomo Sternberg.
He worked at Columbia University from 1963 to 1966 and then moved to the Massachusetts Institute of Technology as assistant professor. He become associated professor in 1969 and full professor in 1973.
Awards and honors
Guillemin was awarded in 1969 a Sloan Research Fellowship, in 1988 a Guggenheim fellowship and in 1996 a Humboldt fellowship. In 1970 he was invited speaker at the International Congress of Mathematicians in Nice.
He was elected a fellow of the American Academy of Arts and Sciences in 1984 and of the United States National Academy of Sciences in 1985. In 2003, he was awarded the Leroy P. Steele Prize for Lifetime Achievement by the American Mathematical Society. In 2012 he became a fellow of the American Mathematical Society.
Research
Guillemin worked in several areas in analysis and geometry, including microlocal analysis, symplectic group actions, and spectral theory of elliptic operators on manifolds.
He is the author or co-author of numerous books and monographs, including a widely used textbook on differential topology, written jointly with Alan Pollack in 1974, and a monograph on symplectic geometry in physics, written jointly with Shlomo Sternberg in 1986.
Family
Victor Guillemins's uncle Ernst Guillemin was a Professor of Electrical Engineering and Computer Science at MIT, his younger brother Robert Charles Guillemin was a sidewalk artist, his brother-in-law Ray Raphael is an historian, and his daughter Karen Guillemin is a Professor of Biology at the University of Oregon.
Selected publications
; reprinted in 1990 as an on-line book
See also
Zoll surface
References
External links
Victor Guillemin's personal web page
Members of the United States National Academy of Sciences
20th-century American mathematicians
21st-century American mathematicians
Differential geometers
Topologists
Massachusetts Institute of Technology School of Science faculty
Harvard College alumni
University of Chicago alumni
Fellows of the American Mathematical Society
Living people
1937 births
Harvard Graduate School of Arts and Sciences alumni
Sloan Research Fellows |
https://en.wikipedia.org/wiki/Turnstile%20%28disambiguation%29 | A turnstile is a pedestrian gate.
Turnstile may also refer to:
Turnstile (symbol), symbol used in mathematics, logic, and computer science
Turnstiles (album), a 1976 studio album by Billy Joel
Turnstile antenna, set of two dipole antennas
Optical turnstile, physical security device
TURNSTILE, a codename for the UK's Central Government War Headquarters
Turnstile (band), a hardcore punk band
The Turnstile, a 1912 novel by A. E. W. Mason
in the fiction movie Tenet (film), a device that inverts entropy
See also
Turn Style, retail store
Turnstyle (band), a band from Perth, Western Australia |
https://en.wikipedia.org/wiki/MUMPS%20%28software%29 | MUMPS (MUltifrontal Massively Parallel sparse direct Solver) is a software application for the solution of large sparse systems of linear algebraic equations on distributed memory parallel computers. It was developed in European project PARASOL (1996–1999) by CERFACS, IRIT-ENSEEIHT and RAL. The software implements the multifrontal method, which is a version of Gaussian elimination for large sparse systems of equations, especially those arising from the finite element method. It is written in Fortran 90 with parallelism by MPI and it uses BLAS and ScaLAPACK kernels for dense matrix computations.
Since 1999, MUMPS has been supported by CERFACS, IRIT-ENSEEIHT, and INRIA.
The importance of MUMPS lies in the fact that it is a supported free implementation of the multifrontal method.
References
External links
WinMUMPS, files for compiling MUMPS on Windows
Free software programmed in Fortran
Numerical software
Public-domain software with source code |
https://en.wikipedia.org/wiki/B%C3%A1rbara%20M.%20Brizuela | Bárbara M. Brizuela is an American mathematics educator, and an associate professor education at Tufts University.
Education and career
Brizuela was born in the United States, though raised in Argentina and Venezuela.
She has an Ed.D from Harvard University where she studied under Eleanor Duckworth. Prior to that, she received a Master of Arts, General Studies in Education from Tufts and a Licenciada en Ciencias Pedagógicas and Licenciada en Psicopedagogía degrees from the Universidad de Belgrano. She was a Spencer Fellow at the Harvard Graduate School of Education from 1997 until 2000 and a Roy E. Larsen Fellow in 1996–1997.
She is one of the leaders of the Tufts Math, Science, Technology and Engineering Education graduate research program.
In 2008, she received a Fulbright Fellowship.
Research
Brizuela's main research focus is on mathematics education in early childhood and elementary school. She mainly studies children's learning of written mathematical representations as well as children's construction of algebraic understandings in a line of work called "Early Algebra". She is a member of the Early Algebra Project, an NSF-funded longitudinal study of the effects of introducing some algebraic concepts to children in elementary school, and was the Principal Investigator of a study created to follow up the children of the Early Algebra study into middle and high school, also funded by the NSF. She is also involved in the Noyce Teacher Fellowship Program at Tufts and in the research effort surrounding Tufts's Poincaré Institute for Mathematics Education, an NSF NSF MSP project.
Books
In 2004, her book Mathematical Development in Young Children: Exploring Notations was published. This book was later translated into Portuguese.
In 2007, she published the book Bringing Out the Algebraic Character of Arithmetic: From Children’s Ideas to Classroom Practice with her colleagues Analúcia Schliemann and David Carraher. This book was later translated into Spanish. She is also the author of Haciendo números: Las notaciones númericas vistas desde la psicología, la didáctica la historia (Editorial Paidós Mexicana, 2006)
With Brian E. Gravel she edited Show Me What You Know: Exploring Student Representations Across STEM Disciplines (Teachers College Press, 2013).
Selected journal articles
Schliemann, A.D., Carraher, D.W., & Brizuela, B. M. (2012, in press). "Algebra in Elementary School and its Impact on Middle School Learning." Recherches en Didactique des Mathématiques, Paris, France.
Caddle, M., & Brizuela, B. M. (2011). "Fifth Graders’ Additive And Multiplicative Reasoning: Establishing Connections Across Conceptual Fields Using A Graph." Journal of Mathematical Behavior, 30(3), 224–234.
Martinez, M. V., Brizuela, B. M., & Castro Superfine, A. (2011). "Integrating Algebra and Proof in High School Mathematics: An Exploratory Study". Journal of Mathematical Behavior, 30, 30–47.
Brizuela, B. M., & Alvarado, M. (2010). "First graders' w |
https://en.wikipedia.org/wiki/Bernard%20Meadows | Bernard Meadows (19 February 1915 - 12 January 2005) was a British modernist sculptor. Meadows was Henry Moore's first assistant; then part of the Geometry of Fear school, a loose-knit group of British sculptors whose prominence was established at the 1952 Venice Biennale; a Professor of Sculpture at the Royal College of Art for 20 years; and returned to assist Moore again in his last years.
Early life
Meadows was born in Norwich, and educated at the City of Norwich School, After briefly training as an accountant in 1931, he attended Norwich School of Art and then in 1936 became Henry Moore's first assistant at his studio then in Kent. He participated in the first Surrealist exhibition in London in 1936. He lived in Chalk Farm from 1937, assisting Moore in his new studio at Hampstead, and studied at the Royal College of Art (although his first application was rejected, due to his association with Moore) and at the Courtauld Institute.
In the Second World War, he initially registered as a conscientious objector, but when Nazi Germany invaded the USSR in 1941, he withdrew his objection. He was called up to the Royal Air Force and worked in air-sea rescue, serving for a time the Cocos Islands in the Indian Ocean, where he was inspired by the large crabs.
Career
He returned to Moore's studio after the war, and helped Moore with his marble sculpture Three Standing Figures 1947 and his 1949 bronze Family Group.
He went on to find acclaim. An elm figure was exhibited in the open air sculpture exhibition at Battersea Park in 1951, alongside the Festival of Britain, which went to the Tate Gallery.
He exhibited in the British Pavilion at the Venice Biennale a year later, alongside a new generation of British sculptors, including Anthony Caro, Lynn Chadwick and Eduardo Paolozzi. Their angular artworks contrasted with the more rounded styles of their seniors, Henry Moore and Barbara Hepworth, and they were dubbed by art critic Herbert Read as the "Geometry of Fear".
He held his first solo exhibition at Gimpel Fils in 1957, with four more in the decade to 1967. He also exhibited at the São Paulo Biennale in 1957, Documenta 2 in Kassel in 1959, and the 1964 Venice Biennale. He exhibited from New York City to Tokyo and produced a stream of public and private art in Britain and beyond. His edgy pieces often based on animals and seemingly carved from shrapnel could imply Cold War menace.
Meadows' work titled Public Sculpture, a controversial assembly of stone blocks and balls of dripping and dimpled metal, was commissioned for the Eastern Daily Press in 1968 at Prospect House, Norwich. It was Grade II Listed in 2018 and restored by its current owners, Alan Boswell Group, in 2022. The sculpture is on permanent display outside the building alongside an illustrated panel telling the story of Bernard Meadows and Public Sculpture.
Teaching commitments took precedence over his own work. He taught at the Royal College of Art from 1948, and was Professor of S |
https://en.wikipedia.org/wiki/Lyndon%E2%80%93Hochschild%E2%80%93Serre%20spectral%20sequence | In mathematics, especially in the fields of group cohomology, homological algebra and number theory, the Lyndon spectral sequence or Hochschild–Serre spectral sequence is a spectral sequence relating the group cohomology of a normal subgroup N and the quotient group G/N to the cohomology of the total group G. The spectral sequence is named after Roger Lyndon, Gerhard Hochschild, and Jean-Pierre Serre.
Statement
Let be a group and be a normal subgroup. The latter ensures that the quotient is a group, as well. Finally, let be a -module. Then there is a spectral sequence of cohomological type
and there is a spectral sequence of homological type
,
where the arrow '' means convergence of spectral sequences.
The same statement holds if is a profinite group, is a closed normal subgroup and denotes the continuous cohomology.
Examples
Homology of the Heisenberg group
The spectral sequence can be used to compute the homology of the Heisenberg group G with integral entries, i.e., matrices of the form
This group is a central extension
with center corresponding to the subgroup with . The spectral sequence for the group homology, together with the analysis of a differential in this spectral sequence, shows that
Cohomology of wreath products
For a group G, the wreath product is an extension
The resulting spectral sequence of group cohomology with coefficients in a field k,
is known to degenerate at the -page.
Properties
The associated five-term exact sequence is the usual inflation-restriction exact sequence:
Generalizations
The spectral sequence is an instance of the more general Grothendieck spectral sequence of the composition of two derived functors. Indeed, is the derived functor of (i.e., taking G-invariants) and the composition of the functors and is exactly .
A similar spectral sequence exists for group homology, as opposed to group cohomology, as well.
References
(paywalled)
Spectral sequences
Group theory |
https://en.wikipedia.org/wiki/Factorion | In number theory, a factorion in a given number base is a natural number that equals the sum of the factorials of its digits. The name factorion was coined by the author Clifford A. Pickover.
Definition
Let be a natural number. For a base , we define the sum of the factorials of the digits of , , to be the following:
where is the number of digits in the number in base , is the factorial of and
is the value of the th digit of the number. A natural number is a -factorion if it is a fixed point for , i.e. if . and are fixed points for all bases , and thus are trivial factorions for all , and all other factorions are nontrivial factorions.
For example, the number 145 in base is a factorion because .
For , the sum of the factorials of the digits is simply the number of digits in the base 2 representation since .
A natural number is a sociable factorion if it is a periodic point for , where for a positive integer , and forms a cycle of period . A factorion is a sociable factorion with , and a amicable factorion is a sociable factorion with .
All natural numbers are preperiodic points for , regardless of the base. This is because all natural numbers of base with digits satisfy . However, when , then for , so any will satisfy until . There are finitely many natural numbers less than , so the number is guaranteed to reach a periodic point or a fixed point less than , making it a preperiodic point. For , the number of digits for any number, once again, making it a preperiodic point. This means also that there are a finite number of factorions and cycles for any given base .
The number of iterations needed for to reach a fixed point is the function's persistence of , and undefined if it never reaches a fixed point.
Factorions for
b = (k − 1)!
Let be a positive integer and the number base . Then:
is a factorion for for all
is a factorion for for all .
b = k! − k + 1
Let be a positive integer and the number base . Then:
is a factorion for for all .
Table of factorions and cycles of
All numbers are represented in base .
See also
Arithmetic dynamics
Dudeney number
Happy number
Kaprekar's constant
Kaprekar number
Meertens number
Narcissistic number
Perfect digit-to-digit invariant
Perfect digital invariant
Sum-product number
References
External links
Factorion at Wolfram MathWorld
Arithmetic dynamics
Base-dependent integer sequences |
https://en.wikipedia.org/wiki/221%20%28disambiguation%29 | 221 may refer to:
In time:
The year:
221 AD
221 BC
In mathematics:
221 (number)
In geography:
Roads:
U.S. Route 221 in Virginia
In astronomy:
221 Eos, a main-belt asteroid
In transportation:
Aircraft:
The Boeing 221 mail plane
In weapons:
Firearms:
The .221 Remington Fireball pistol cartridge
In fiction:
221B Baker Street, the address of Sherlock Holmes
Experiment 221, the codename for Sparky, a fictional alien character in the Lilo & Stitch franchise
See also
Two to One (1978 album), album by Thelma Houston |
https://en.wikipedia.org/wiki/Parity%20of%20zero | In mathematics, zero is an even number. In other words, its parity—the quality of an integer being even or odd—is even. This can be easily verified based on the definition of "even": it is an integer multiple of 2, specifically . As a result, zero shares all the properties that characterize even numbers: for example, 0 is neighbored on both sides by odd numbers, any decimal integer has the same parity as its last digit—so, since 10 is even, 0 will be even, and if is even then has the same parity as —indeed, and always have the same parity.
Zero also fits into the patterns formed by other even numbers. The parity rules of arithmetic, such as , require 0 to be even. Zero is the additive identity element of the group of even integers, and it is the starting case from which other even natural numbers are recursively defined. Applications of this recursion from graph theory to computational geometry rely on zero being even. Not only is 0 divisible by 2, it is divisible by every power of 2, which is relevant to the binary numeral system used by computers. In this sense, 0 is the "most even" number of all.
Among the general public, the parity of zero can be a source of confusion. In reaction time experiments, most people are slower to identify 0 as even than 2, 4, 6, or 8. Some teachers —and some children in mathematics classes—think that zero is odd, or both even and odd, or neither. Researchers in mathematics education propose that these misconceptions can become learning opportunities. Studying equalities like can address students' doubts about calling 0 a number and using it in arithmetic. Class discussions can lead students to appreciate the basic principles of mathematical reasoning, such as the importance of definitions. Evaluating the parity of this exceptional number is an early example of a pervasive theme in mathematics: the abstraction of a familiar concept to an unfamiliar setting.
Why zero is even
The standard definition of "even number" can be used to directly prove that zero is even. A number is called "even" if it is an integer multiple of 2. As an example, the reason that 10 is even is that it equals . In the same way, zero is an integer multiple of 2, namely so zero is even.
It is also possible to explain why zero is even without referring to formal definitions. The following explanations make sense of the idea that zero is even in terms of fundamental number concepts. From this foundation, one can provide a rationale for the definition itself—and its applicability to zero.
Basic explanations
Given a set of objects, one uses a number to describe how many objects are in the set. Zero is the count of no objects; in more formal terms, it is the number of objects in the empty set. The concept of parity is used for making groups of two objects. If the objects in a set can be marked off into groups of two, with none left over, then the number of objects is even. If an object is left over, then the number of objects is odd. The |
https://en.wikipedia.org/wiki/Krivine%E2%80%93Stengle%20Positivstellensatz | In real algebraic geometry, Krivine–Stengle (German for "positive-locus-theorem") characterizes polynomials that are positive on a semialgebraic set, which is defined by systems of inequalities of polynomials with real coefficients, or more generally, coefficients from any real closed field.
It can be thought of as a real analogue of Hilbert's Nullstellensatz (which concern complex zeros of polynomial ideals), and this analogy is at the origin of its name. It was proved by French mathematician and then rediscovered by the Canadian .
Statement
Let be a real closed field, and = {f1, f2, ..., fm} and = {g1, g2, ..., gr} finite sets of polynomials over in variables. Let be the semialgebraic set
and define the preordering associated with as the set
where Σ2[1,...,] is the set of sum-of-squares polynomials. In other words, (, ) = + , where is the cone generated by (i.e., the subsemiring of [1,...,] generated by and arbitrary squares) and is the ideal generated by .
Let ∈ [1,...,] be a polynomial. Krivine–Stengle Positivstellensatz states that
(i) if and only if and such that .
(ii) if and only if such that .
The weak is the following variant of the . Let be a real closed field, and , , and finite subsets of [1,...,]. Let be the cone generated by , and the ideal generated by . Then
if and only if
(Unlike , the "weak" form actually includes the "strong" form as a special case, so the terminology is a misnomer.)
Variants
The Krivine–Stengle Positivstellensatz also has the following refinements under additional assumptions. It should be remarked that Schmüdgen's Positivstellensatz has a weaker assumption than Putinar's Positivstellensatz, but the conclusion is also weaker.
Schmüdgen's Positivstellensatz
Suppose that . If the semialgebraic set is compact, then each polynomial that is strictly positive on can be written as a polynomial in the defining functions of with sums-of-squares coefficients, i.e. . Here is said to be strictly positive on if for all . Note that Schmüdgen's Positivstellensatz is stated for and does not hold for arbitrary real closed fields.
Putinar's Positivstellensatz
Define the quadratic module associated with as the set
Assume there exists L > 0 such that the polynomial If for all , then ∈ (,).
See also
Positive polynomial for other positivstellensatz theorems.
Notes
References
Real algebraic geometry
Algebraic varieties
German words and phrases
Theorems in algebraic geometry |
https://en.wikipedia.org/wiki/List%20of%20Ipswich%20Town%20F.C.%20seasons | Ipswich Town Football Club have played association football since their foundation in 1878. For every season in which they have played, a set of statistics exist for their results in a number of competitions, including competitions in English and European football.
Following the club's foundation, Ipswich Town played amateur football against teams from around Suffolk. During the 1880s, the club played a number of matches in the Suffolk Challenge Cup, winning it three times. Throughout the early part of the 20th century, Ipswich played in various amateur competitions including the Norfolk & Suffolk League, the South East Anglian League, the Eastern Counties League and the Southern Amateur League. Turning professional in 1936, Ipswich joined the Southern Football League before moving into the Football League by gaining entry to Division Three (South) in the 1937–38 season.
The club has won the League Championship on a single occasion, the FA Cup once, and the UEFA Cup once. This list details the club's achievements in all competitive competitions, and the top scorers for each season.
Seasons
Key
P = Played
W = Games won
D = Games drawn
L = Games lost
F = Goals for
A = Goals against
Pts = Points
Pos = Final position
N&SL = Norfolk & Suffolk League
EAL = East Anglian League
SAL = Southern Amateur League
ECL = Eastern Counties Football League
SL = Southern League
Div 1 = Football League First Division
Div 2 = Football League Second Division
Div 3(S) = Football League Third Division South
Prem = Premier League
Chmp = Championship
n/a = Not applicable
QR1 = Qualifying round 1
QR2 = Qualifying round 2
QR3 = Qualifying round 3
QR4 = Qualifying round 4
R1 = Round 1
R2 = Round 2
R3 = Round 3
R4 = Round 4
R5 = Round 5
QF = Quarter-finals
SF = Semi-finals
Top scorers shown in bold are players who were also top scorers in their division that season.
Footnotes
References
General
Specific
Seasons
Ipswich Town F.C. |
https://en.wikipedia.org/wiki/Kuratowski%20convergence | In mathematics, Kuratowski convergence or Painlevé-Kuratowski convergence is a notion of convergence for subsets of a topological space. First introduced by Paul Painlevé in lectures on mathematical analysis in 1902, the concept was popularized in texts by Felix Hausdorff and Kazimierz Kuratowski. Intuitively, the Kuratowski limit of a sequence of sets is where the sets "accumulate".
Definitions
For a given sequence of points in a space , a limit point of the sequence can be understood as any point where the sequence eventually becomes arbitrarily close to . On the other hand, a cluster point of the sequence can be thought of as a point where the sequence frequently becomes arbitrarily close to . The Kuratowski limits inferior and superior generalize this intuition of limit and cluster points to subsets of the given space .
Metric Spaces
Let be a metric space, where is a given set. For any point and any non-empty subset , define the distance between the point and the subset:
For any sequence of subsets of , the Kuratowski limit inferior (or lower closed limit) of as ; isthe Kuratowski limit superior (or upper closed limit) of as ; isIf the Kuratowski limits inferior and superior agree, then the common set is called the Kuratowski limit of and is denoted .
Topological Spaces
If is a topological space, and are a net of subsets of , the limits inferior and superior follow a similar construction. For a given point denote the collection of open neighbhorhoods of . The Kuratowski limit inferior of is the setand the Kuratowski limit superior is the setElements of are called limit points of and elements of are called cluster points of . In other words, is a limit point of if each of its neighborhoods intersects for all in a "residual" subset of , while is a cluster point of if each of its neighborhoods intersects for all in a cofinal subset of .
When these sets agree, the common set is the Kuratowski limit of , denoted .
Examples
Suppose is separable where is a perfect set, and let be an enumeration of a countable dense subset of . Then the sequence defined by has .
Given two closed subsets , defining and for each yields and .
The sequence of closed balls converges in the sense of Kuratowski when in and in , and in particular, . If , then while .
Let . Then converges in the Kuratowski sense to the entire line.
In a topological vector space, if is a sequence of cones, then so are the Kuratowski limits superior and inferior. For example, the sets converge to .
Properties
The following properties hold for the limits inferior and superior in both the metric and topological contexts, but are stated in the metric formulation for ease of reading.
Both and are closed subsets of , and always holds.
The upper and lower limits do not distinguish between sets and their closures: and .
If is a constant sequence, then .
If is a sequence of singletons, then and consist of the limit points and clu |
https://en.wikipedia.org/wiki/Lucian%20Bebchuk | Lucian Arye Bebchuk (born 1955) is a professor at Harvard Law School focusing on economics and finance.
Life and career
Bebchuk has a B.A. in mathematics and economics from the University of Haifa (1977), an LL.B. from the University of Tel Aviv (1979), an LL.M. and S.J.D. from Harvard Law School (1980 and 1984) and an M.A. and Ph.D. in economics, also from Harvard (1992 and 1993). He was a junior fellow of the Harvard Society of Fellows from 1983 to 1985. He joined the Harvard Law faculty in 1986. Bebchuck is the co-author, with Jesse Fried, of Pay without Performance: The Unfulfilled Promise of Executive Compensation.
Distinctions
Prof. Bebchuk was named one of the top 100 most influential players in corporate governance in the US by Directorship magazine. He was elected a fellow of the American Academy of Arts and Sciences in 2000. In 2004, he was awarded a Guggenheim Fellowship.
References
External links
Personal website
Faculty page
1955 births
Living people
Fellows of the American Academy of Arts and Sciences
Harvard Law School alumni
Harvard Law School faculty
Tel Aviv University alumni
Finance law scholars
University of Haifa alumni
American legal scholars |
https://en.wikipedia.org/wiki/Unistochastic%20matrix | In mathematics, a unistochastic matrix (also called unitary-stochastic) is a doubly stochastic matrix whose entries are the squares of the absolute values of the entries of some unitary matrix.
A square matrix B of size n is doubly stochastic (or bistochastic) if all its entries are non-negative real numbers and each of its rows and columns sum to 1. It is unistochastic if there exists a unitary matrix U such that
This definition is analogous to that for an orthostochastic matrix, which is a doubly stochastic matrix whose entries are the squares of the entries in some orthogonal matrix. Since all orthogonal matrices are necessarily unitary matrices, all orthostochastic matrices are also unistochastic. The converse, however, is not true. First, all 2-by-2 doubly stochastic matrices are both unistochastic and orthostochastic, but for larger n this is not the case. For example, take and consider the following doubly stochastic matrix:
This matrix is not unistochastic, since any two vectors with moduli equal to the square root of the entries of two columns (or rows) of B cannot be made orthogonal by a suitable choice of phases. For , the set of orthostochastic matrices is a proper subset of the set of unistochastic matrices.
the set of unistochastic matrices contains all permutation matrices and its convex hull is the Birkhoff polytope of all doubly stochastic matrices
for this set is not convex
for the set of triangle inequality on the moduli of the raw is a sufficient and necessary condition for the unistocasticity
for the set of unistochastic matrices takes the form of a centrosymmetric matrix and unistochasticity of any bistochastic matrix B is implied by a non-negative value of its Jarlskog invariant
for the relative volume of the set of unistochastic matrices with respect to the Birkhoff polytope of doubly stochastic matrices is
for explicit conditions for unistochasticity are not known yet, but there exists a numerical method to verify unistochasticity based on the algorithm by Haagerup
The Schur-Horn theorem is equivalent to the following "weak convexity" property of the set of unistochastic matrices: for any vector the set is the convex hull of the set of vectors obtained by all permutations of the entries of the vector (the permutation polytope generated by the vector ).
The set of unistochastic matrices has a nonempty interior. The unistochastic matrix corresponding to the unitary matrix with the entries , where and , is an interior point of .
References
.
Matrices |
https://en.wikipedia.org/wiki/Friedrich%20Kambartel | Friedrich Kambartel was a German philosopher.
Biography
Kambartel was born on 17 February 1935 in Münster, Germany. He studied physics, mathematics and philosophy at the University of Münster, where he received his PhD (in mathematics) and his “habilitation”, the postdoctoral lecture qualification (in philosophy). In 1966 he was appointed Professor of Philosophy at the University of Constance, where he took active part in making it a reform university (“Little Harvard on Lake Constance”). Kambartel had close ties to the Erlangen School of constructivist philosophy of science. He taught in Frankfurt am Main from 1993 until his retirement in 2000. He died on April 25, 2022, in Constance.
Kambartel's main research areas are the philosophy of language, the philosophy of the natural sciences, and the philosophy of mind. However, he also contributed to logic, action theory, ethics and the philosophy of economics.
His most important works are the habilitation thesis Erfahrung und Struktur (“Experience and Structure”), published by Suhrkamp in 1968, as well as the three anthologies Theorie und Begründung (1978, “Theory and Justification”), Philosophie der humanen Welt (1989, “Philosophy of the Human World”) and Philosophie und Politische Ökonomie (1998, “Philosophy and Political Economics”).
Kambartel's philosophical work is wide-ranging and manifold. Yet two major tenets are present throughout — on the one hand the primacy of practical reason (his “pragmatism”), and on the other the conception of reason as culture (“his anti-formalism”).
The first tenet shapes his contributions to the philosophies of science, mind, and action. If action and practical reason were granted primacy over thinking and theoretical reason, and if the latter were only possible on the basis of the former, then results obtained by neuroscience, for example, could never show that man is determined after all and cannot really act freely.
The second tenet does not emerge clearly until his later work, and then it also marks a distance to the constructive attempts of the Erlangen School. Reason was not to be understood exactly, e.g. to be defined as a principle or criterion. Reason was rather a culture you grow into, a social practice within which you cultivate your judgment. Conceptual judgments like Kant’s formula of man as an end in itself served as comments to parts of the “grammar” of this culture.
Bibliography
Books
Bernard Bolzano's Grundlegung der Logik. Ausgewählte Paragraphen aus der Wissenschaftslehre, Vol. 1 and 2, with supplementary summaries, an introduction and indices, edited by F. Kambartel, Hamburg, 1963, 1978².
Erfahrung und Struktur. Bausteine zu einer Kritik des Empirismus und Formalismus, Frankfurt a.M., 1968, 1976²; Span.: Buenos Aires, 1972.
Gottlob Frege: Nachgelassene Schriften, with contributions from G. Gabriel and W. Rödding, edited, introduced, and commented by H. Hermes, F. Kambartel, F. Kaulbach, Hamburg, 1969; Engl.: Oxford, 1979.
Histor |
https://en.wikipedia.org/wiki/Jurjen%20Ferdinand%20Koksma | Jurjen Ferdinand Koksma (21 April 1904, Schoterland – 17 December 1964, Amsterdam) was a Dutch mathematician who specialized in analytic number theory.
Koksma received his Ph.D. degree (cum laude) in 1930 at the University of Groningen under supervision of Johannes van der Corput, with a thesis on Systems of Diophantine Inequalities. Around the same time, aged 26, he was invited to become full professor at the Vrije Universiteit Amsterdam. He accepted and in 1930 became the first professor in mathematics at this university. Koksma is also one of the founders of the Dutch Mathematisch Centrum (today Centrum Wiskunde & Informatica).
One of Koksma's main works was the book Diophantische Approximationen, published in 1936 by Springer. He also wrote several papers with Paul Erdős.
In 1950 he became member of the Royal Netherlands Academy of Arts and Sciences.
Koksma had two brothers, Jan and Marten, who were also mathematicians.
See also
Denjoy–Koksma inequality
Koksma's equivalent classification
Koksma–Hlawka inequality
Erdős–Turán–Koksma inequality
References
Literature
Arie van Deursen: The distinctive character of the Free University in Amsterdam, 1880-2005, Eerdmans Publishing (2008).
1904 births
1964 deaths
20th-century Dutch mathematicians
Members of the Royal Netherlands Academy of Arts and Sciences
Number theorists
People from Heerenveen
University of Groningen alumni |
https://en.wikipedia.org/wiki/Baseball-Reference.com | Baseball-Reference is a website providing baseball statistics for every player in Major League Baseball history. The site is often used by major media organizations and baseball broadcasters as a source for statistics. It offers a variety of advanced baseball sabermetrics in addition to traditional baseball "counting stats".
Baseball-Reference is part of Sports Reference, LLC; according to an article in Street & Smith's Sports Business Journal, the company's sites have more than one million unique users per month.
History
Founder Sean Forman began developing the website while working on his Ph.D. dissertation in applied math and computational science at the University of Iowa. While writing his dissertation, he had also been writing articles on and blogging about sabermetrics. Forman's database was originally built from the Total Baseball series of baseball encyclopedias.
The website went online in April 2000, after first being launched in February 2000 as part of the website for the Big Bad Baseball Annual. It was originally built as a web interface to the Lahman Baseball Database, though it now employs a variety of data sources.
In 2004, Forman founded Sports Reference. Sports Reference is a website that came out of the Baseball Reference website. The company was incorporated as Sports Reference, LLC in 2007. In 2006, Forman left his job as a math professor at Saint Joseph's University in order to focus on Baseball-Reference full-time.
In February 2009, Fantasy Sports Ventures took a minority stake in Sports Reference, LLC, the parent company of Baseball-Reference, for a "low seven-figure sum".
At the end of April 2021, the site changed a number of identifying names, "discontinuing the use of nicknames that are racially or ethnically influenced" and "names based upon a player's disability", such as Chief Bender and Dummy Hoy, who are now listed as Charles Bender and Billy Hoy, respectively.
Features
The site has season, career, and minor league records (when available, back to ) for everyone who has played Major League Baseball, year-by-year team pages, all final league standings, all postseason numbers, voting results for all historic awards such as the Cy Young Award and MVP, head-to-head batter vs. pitcher career totals, individual statistical leaders for each season and all-time, managers' career records, the full results of all MLB player drafts, Negro leagues statistics (Baseball Reference added Negro League Statistics to its website in 2021), a baseball encyclopedia (the Bullpen), and box scores and game logs from every MLB game back to , among other features.
To compare ballplayers to one-another it offers "Black ink" and "Gray Ink" tests, which tally a player's dominance and overall productivity against his peers. It also offers sabremetrician Jay Jaffe's system acronymned "JAWS" for ranking players of different eras against each other by weighting their primes.
In addition, there are a number of what the website calls "Frivo |
https://en.wikipedia.org/wiki/National%20Statistics%20Office%20of%20Georgia | The National Statistics Office (GeoStat) (, sak'art'velos statistikis erovnuli samsakhuri; საქსტატი, sak'stati) is an agency in charge of national statistics and responsible for carrying out population, agricultural and other censuses in Georgia. It was established as a legal entity of public law according to the December 11, 2009 law of Georgia, succeeding the Department of Statistics of the Ministry of Economy and Sustainable Development of Georgia. The head office is located in Tbilisi.
History
The earliest references to the collection of statistics in Georgia date from the 13th century. Materials from population censuses made in the seventeenth and eighteenth centuries in various regions of the country have survived to the present day.
On November 15, 1918 a temporary Statistical Bureau was formed within the Ministry of Agriculture of the Democratic Republic of Georgia. The bureau's functions included development of materials for agricultural census, accounting of the available land and determining of norms for its distribution. On the basis of law enacted by the Constituent Assembly on July 25, 1919, a Republican Statistical Committee was formed within the same ministry. The committee was assigned to manage all types of statistical works of national importance. During the Soviet rule (1921–1991), the national statistics service was provided by the Central Statistics Division. In a newly independent Georgia, it was succeeded by the Social and Economic Information Committee established at the Parliament of Georgia (1991–1995), the State Department of Social and Economic Information (1995–1997), and the State Department of Statistics of Georgia (1997–2004). The department was subordinated to the Ministry of Economy and Sustainable Development in 2004 and made an independent agency under the current name in 2010.
Mission
GeoStat is an official authority exclusively responsible for production and dissemination of official statistics in accordance with international statistical standards and requirements. Its principal purposes are collection, editing, processing, storage, analysis and dissemination of exhaustive, up to date, reliable and com¬pa¬rable statistical data.
One of the main functions of GeoStat is provision of official statistics to the civil society, official authorities, NGOs (non-governmental organizations), the mass media, business and academic communities and other categories of users. Its information is open and accessible to all users. GeoStat regularly publishes the country's social and economic indicators on its website.
Structure
The agency is led by the executive director, who is appointed by the Prime Minister of Georgia for the term of 4 years. The Executive Director also acts as the Chairman of the GeoStat Board which consists of 8 members.
The agency consists of the central office and eleven regional branches.
The present structure of the GeoStat was put in place following a reorganisation in 2018. It consists of 1 |
https://en.wikipedia.org/wiki/Bicentric%20polygon | In geometry, a bicentric polygon is a tangential polygon (a polygon all of whose sides are tangent to an inner incircle) which is also cyclic — that is, inscribed in an outer circle that passes through each vertex of the polygon. All triangles and all regular polygons are bicentric. On the other hand, a rectangle with unequal sides is not bicentric, because no circle can be tangent to all four sides.
Triangles
Every triangle is bicentric. In a triangle, the radii r and R of the incircle and circumcircle respectively are related by the equation
where x is the distance between the centers of the circles. This is one version of Euler's triangle formula.
Bicentric quadrilaterals
Not all quadrilaterals are bicentric (having both an incircle and a circumcircle). Given two circles (one within the other) with radii R and r where , there exists a convex quadrilateral inscribed in one of them and tangent to the other if and only if their radii satisfy
where x is the distance between their centers. This condition (and analogous conditions for higher order polygons) is known as Fuss' theorem.
Polygons with n > 4
A complicated general formula is known for any number n of sides for the relation among the circumradius R, the inradius r, and the distance x between the circumcenter and the incenter. Some of these for specific n are:
where and
Regular polygons
Every regular polygon is bicentric. In a regular polygon, the incircle and the circumcircle are concentric—that is, they share a common center, which is also the center of the regular polygon, so the distance between the incenter and circumcenter is always zero. The radius of the inscribed circle is the apothem (the shortest distance from the center to the boundary of the regular polygon).
For any regular polygon, the relations between the common edge length a, the radius r of the incircle, and the radius R of the circumcircle are:
For some regular polygons which can be constructed with compass and ruler, we have the following algebraic formulas for these relations:
Thus we have the following decimal approximations:
Poncelet's porism
If two circles are the inscribed and circumscribed circles of a particular bicentric n-gon, then the same two circles are the inscribed and circumscribed circles of infinitely many bicentric n-gons. More precisely,
every tangent line to the inner of the two circles can be extended to a bicentric n-gon by placing vertices on the line at the points where it crosses the outer circle, continuing from each vertex along another tangent line, and continuing in the same way until the resulting polygonal chain closes up to an n-gon. The fact that it will always do so is implied by Poncelet's closure theorem, which more generally applies for inscribed and circumscribed conics.
Moreover, given a circumcircle and incircle, each diagonal of the variable polygon is tangent to a fixed circle.
References
External links
Elementary geometry
Types of polygons |
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