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https://en.wikipedia.org/wiki/Affine%20differential%20geometry | Affine differential geometry is a type of differential geometry which studies invariants of volume-preserving affine transformations. The name affine differential geometry follows from Klein's Erlangen program. The basic difference between affine and Riemannian differential geometry is that affine differential geometry studies manifolds equipped with a volume form rather than a metric.
Preliminaries
Here we consider the simplest case, i.e. manifolds of codimension one. Let be an n-dimensional manifold, and let ξ be a vector field on transverse to such that for all where ⊕ denotes the direct sum and Span the linear span.
For a smooth manifold, say N, let Ψ(N) denote the module of smooth vector fields over N. Let be the standard covariant derivative on Rn+1 where
We can decompose DXY into a component tangent to M and a transverse component, parallel to ξ. This gives the equation of Gauss: where is the induced connexion on M and is a bilinear form. Notice that ∇ and h depend upon the choice of transverse vector field ξ. We consider only those hypersurfaces for which h is non-degenerate. This is a property of the hypersurface M and does not depend upon the choice of transverse vector field ξ. If h is non-degenerate then we say that M is non-degenerate. In the case of curves in the plane, the non-degenerate curves are those without inflexions. In the case of surfaces in 3-space, the non-degenerate surfaces are those without parabolic points.
We may also consider the derivative of ξ in some tangent direction, say X. This quantity, DXξ, can be decomposed into a component tangent to M and a transverse component, parallel to ξ. This gives the Weingarten equation: The type-(1,1)-tensor is called the affine shape operator, the differential one-form is called the transverse connexion form. Again, both S and τ depend upon the choice of transverse vector field ξ.
The first induced volume form
Let be a volume form defined on Rn+1. We can induce a volume form on M given by given by This is a natural definition: in Euclidean differential geometry where ξ is the Euclidean unit normal then the standard Euclidean volume spanned by X1,...,Xn is always equal to ω(X1,...,Xn). Notice that ω depends on the choice of transverse vector field ξ.
The second induced volume form
For tangent vectors X1,...,Xn let be the given by We define a second volume form on M given by where Again, this is a natural definition to make. If M = Rn and h is the Euclidean scalar product then ν(X1,...,Xn) is always the standard Euclidean volume spanned by the vectors X1,...,Xn.
Since h depends on the choice of transverse vector field ξ it follows that ν does too.
Two natural conditions
We impose two natural conditions. The first is that the induced connexion ∇ and the induced volume form ω be compatible, i.e. ∇ω ≡ 0. This means that for all In other words, if we parallel transport the vectors X1,...,Xn along some curve in M, with respect to the connexion ∇, then t |
https://en.wikipedia.org/wiki/Chemical%20graph%20theory | Chemical graph theory is the topology branch of mathematical chemistry which applies graph theory to mathematical modelling of chemical phenomena.
The pioneers of chemical graph theory are Alexandru Balaban, Ante Graovac, Iván Gutman, Haruo Hosoya, Milan Randić and Nenad Trinajstić (also Harry Wiener and others).
In 1988, it was reported that several hundred researchers worked in this area, producing about 500 articles annually. A number of monographs have been written in the area, including the two-volume comprehensive text by Trinajstić, Chemical Graph Theory, that summarized the field up to mid-1980s.
The adherents of the theory maintain that the properties of a chemical graph (i.e., a graph-theoretical representation of a molecule) give valuable insights into the chemical phenomena. Others contend that graphs play only a fringe role in chemical research. One variant of the theory is the representation of materials as infinite Euclidean graphs, particularly crystals by periodic graphs.
See also
Chemical graph generator
Molecule mining
MATH/CHEM/COMP
Topological index
References
Theoretical chemistry
Mathematical chemistry
Application-specific graphs |
https://en.wikipedia.org/wiki/Q-function | In statistics, the Q-function is the tail distribution function of the standard normal distribution. In other words, is the probability that a normal (Gaussian) random variable will obtain a value larger than standard deviations. Equivalently, is the probability that a standard normal random variable takes a value larger than .
If is a Gaussian random variable with mean and variance , then is standard normal and
where .
Other definitions of the Q-function, all of which are simple transformations of the normal cumulative distribution function, are also used occasionally.
Because of its relation to the cumulative distribution function of the normal distribution, the Q-function can also be expressed in terms of the error function, which is an important function in applied mathematics and physics.
Definition and basic properties
Formally, the Q-function is defined as
Thus,
where is the cumulative distribution function of the standard normal Gaussian distribution.
The Q-function can be expressed in terms of the error function, or the complementary error function, as
An alternative form of the Q-function known as Craig's formula, after its discoverer, is expressed as:
This expression is valid only for positive values of x, but it can be used in conjunction with Q(x) = 1 − Q(−x) to obtain Q(x) for negative values. This form is advantageous in that the range of integration is fixed and finite.
Craig's formula was later extended by Behnad (2020) for the Q-function of the sum of two non-negative variables, as follows:
Bounds and approximations
The Q-function is not an elementary function. However, it can be upper and lower bounded as,
where is the density function of the standard normal distribution, and the bounds become increasingly tight for large x.
Using the substitution v =u2/2, the upper bound is derived as follows:
Similarly, using and the quotient rule,
Solving for Q(x) provides the lower bound.
The geometric mean of the upper and lower bound gives a suitable approximation for :
Tighter bounds and approximations of can also be obtained by optimizing the following expression
For , the best upper bound is given by and with maximum absolute relative error of 0.44%. Likewise, the best approximation is given by and with maximum absolute relative error of 0.27%. Finally, the best lower bound is given by and with maximum absolute relative error of 1.17%.
The Chernoff bound of the Q-function is
Improved exponential bounds and a pure exponential approximation are
The above were generalized by Tanash & Riihonen (2020), who showed that can be accurately approximated or bounded by
In particular, they presented a systematic methodology to solve the numerical coefficients that yield a minimax approximation or bound: , , or for . With the example coefficients tabulated in the paper for , the relative and absolute approximation errors are less than and , respectively. The coefficients for many variations of |
https://en.wikipedia.org/wiki/Philomaths | The Philomaths, or Philomath Society ( or Towarzystwo Filomatów; from the Greek φιλομαθεῖς "lovers of knowledge"), was a secret student organization that existed from 1817 to 1823 at the Imperial University of Vilnius.
History
The society was created on 1 October 1817 in Vilna, Vilna Governorate, Russian Empire, which had acquired those territories in the Partitions of Poland in 1794. The society was composed of students and alumni of the Imperial University of Vilna.
Notable members included Józef Jeżowski (co-founder and president), Jan Czeczot (co-founder), Józef Kowalewski (co-founder), Onufry Pietraszkiewicz (co-founder), Tomasz Zan (co-founder), Adam Mickiewicz (co-founder), Antoni Edward Odyniec, Ignacy Domejko, Teodor Łoziński, Franciszek Malewski, , Aleksander Chodźko, Michał Kulesza. Most of them were students, but some members and supported included faculty and former alumni.
Its structure was a cross between freemason organization and a learned society. It was divided into two chapters – scientific-mathematic and literary. The members of the latter discussed literary works, and the organization aims were self-educational and didactic; however, around 1819-1820, the members became split on whether the organizations should concentrate on self-education (Jeżowski) or take a more active role in restoring Poland's independence (Mickiewicz), eventually the second faction gained dominance and new social and political goals emerged.
The discussions increasingly turned toward romanticist ideas that were banned by the Russian Empire for their pro-independence currents; history of the Polish–Lithuanian Commonwealth was studied, pro-independence works written and circulated. The organizations inspired the creation of many similar youth organizations across the former Grand Duchy of Lithuania, and it established ties with similar clandestine pro-Polish organizations in Congress Poland and the rest of partitioned lands, such as the Patriotic Society (Towarzystwo Patriotyczne) of Walerian Łukasiński, and even Russian organizations such as the Decembrists.
Two closely related groups were formed:
The Radiant Association (Towarzystwo Promienistych, from "," the "Radiant Ones"), (1820) a legal organization created by Tomasz Zan, and disbanded under pressure from University authorities, in May 1820;
The Filaret Association (Zgromadzenie Filaretów, Filaretai, Towarzystwo Przyjaciół Pożytecznej Zabawy, filareci, from the Greek "philáretos," "Lovers of Virtue), (1834) a secret organization created by Zan within the Philomaths after the dissolution of the Radiants. It continued the traditions of the Radiants, but with a much clearer pro-independence goal, and was dedicated to the study of Polish and Lithuania patriotic literature. It was disbanded in 1823 after the arrests of the Philomaths.
Ignacy Domeyko described the spirit prevailing in the Philomaths and the Filaret Association as: "purely national, patriotic, Polish – but free from plots and |
https://en.wikipedia.org/wiki/Nicholas%20Mann%20%28occult%20writer%29 | Nicholas R. Mann (born 1952) is the author of books on geomancy, mythology, the Celtic tradition, sacred geometry and, most recently, archaeoastronomy. Glastonbury, England, Avebury, England, Sedona, Arizona (USA) and Washington, DC (USA) are all locations which feature in his work. His book Druid Magic: The Practice of Celtic Wisdom, co-written with Maya Sutton, PhD, has been described by the British Druid Order as"One of the best practical guides available..." He is also an illustrator, producing the images for the Silver Branch Cards, a Celtic divination deck of his own design. He was born in Sussex, England. He lives in Somerset, England with his partner Philippa Glasson, with whom he co-authored The Star Temple of Avalon: Glastonbury's Ancient Observatory Revealed.
Bibliography
Avalon’s Red and White Springs: The Healing Waters of Glastonbury with Dr. Philippa Glasson (2005) Green Magic
The Dark God: A Personal Journey Through the Underworld (1996) Llewellyn Publications ,
Druid Magic: The Practice of Celtic Wisdom with Maya Sutton, PhD (2001) Llewellyn ,
Energy Secrets of Glastonbury Tor (2004) Green Magic
The Giants of Gaia with Marcia Sutton, PhD (1995) Brotherhood of Life Books ,
Glastonbury Tor: A Guide to the History and Legends (Pamphlet) (1993) Triskele Publications
His Story: Masculinity in the Post-Patriarchal World (1995) Llewellyn Publications ,
The Isle of Avalon: Sacred Mysteries of Arthur and Glastonbury (2001) Green Magic ,
The Keltic Power Symbols: Native Traditions, the Keltic Goddess and God - The Serpent, the Power Animals and the Pictish Symbol Stones (1987) Triskele
The Cauldron and the Grail (1986) Triskele
Reclaiming the Gods: Magic, Sex, Death and Football (2002) Green Magic ,
The Sacred Geometry of Washington, D.C.: The Integrity and Power of the Original Design (2006) Green Magic ,
Sedona: Sacred Earth: A Guide to Geomantic Applications in the Red Rock Country (1989) Zivah Pub ,
Sedona: Sacred Earth - A Guide to the Red Rock Country (2005) Light Technology Publishing ,
The Silver Branch Cards: Divination using Druid Celtic Symbolism & Mythology (2000) Druidways
The Star Temple of Avalon: Glastonbury's Ancient Observatory Revealed with Philippa Glasson (2007)
Avebury Cosmos: The Neolithic World of Avebury henge,Silbury Hill, West Kennet long barrow,the Sanctuary & the Longstones Cove (2011) O-Books
Notes
1952 births
English occult writers
Alumni of the University of London
Living people |
https://en.wikipedia.org/wiki/New%20York%20Knicks%20all-time%20roster | This is a list of players, both past and current, who appeared at least in one game for the New York Knicks NBA franchise.
Players
Note: Statistics are correct through the end of the season.
A to B
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|align="left"| || align="center"|G || align="left"|LIU Brooklyn || align="center"|1 || align="center"| || 28 || 220 || 15 || 23 || 43 || 7.9 || 0.5 || 0.8 || 1.5 || align=center|
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|align="left"| || align="center"|F/C || align="left"|Baylor || align="center"|1 || align="center"| || 68 || 1,287 || 301 || 68 || 398 || 18.9 || 4.4 || 1.0 || 5.9 || align=center|
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|align="left"| || align="center"|G/F || align="left"|UCLA || align="center"|1 || align="center"| || 71 || 2,371 || 266 || 144 || 909 || 33.4 || 3.7 || 2.0 || 12.8 || align=center|
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|align="left"| || align="center"|F/C || align="left"|Morehead State || align="center"|1 || align="center"| || 50 || 453 || 120 || 25 || 192 || 9.1 || 2.4 || 0.5 || 3.8 || align=center|
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|align="left"| || align="center"|C || align="left"|Kansas || align="center"|2 || align="center"|– || 107 || 1,306 || 467 || 89 || 430 || 12.2 || 4.4 || 0.8 || 4.0 || align=center|
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|align="left"| || align="center"|G || align="left"|Arizona || align="center"|2 || align="center"|–|| 29 || 533 || 60 || 97 || 239 || 18.4 || 2.1 || 3.3 || 8.2 || align=center|
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|align="left"| || align="center"|F/C || align="left"|UNLV || align="center"|2 || align="center"|– || 70 || 1,062 || 297 || 77 || 300 || 15.2 || 4.2 || 1.1 || 4.3 || align=center|
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|align="left"| || align="center"|G/F || align="left"|Michigan State || align="center"|1 || align="center"| || 33 || 373 || 69 || 29 || 133 || 11.3 || 2.1 || 0.9 || 4.0 || align=center|
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|align="left"| || align="center"|F || align="left"|Indiana || align="center"|2 || align="center"|– || 27 || 83 || 31 || 5 || 42 || 3.1 || 1.1 || 0.2 || 1.6 || align=center|
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|align="left"| || align="center"|G/F || align="left"|Georgia || align="center"|4 || align="center"|– || 245 || 5,321 || 726 || 285 || 1,733 || 21.7 || 3.0 || 1.2 || 7.1 || align=center|
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|align="left"| || align="center"|G/F || align="left"|Georgia || align="center"|1 || align="center"| || 27 || 496 || 60 || 48 || 136 || 18.4 || 2.2 || 1.8 || 5.0 || align=center|
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|align="left"| || align="center"|F || align="left"|Missouri State || align="center"|1 || align="center"| || 1 || 10 || 2 || 0 || 3 || 10.0 || 2.0 || 0.0 || 3.0 || align=center|
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|align="left"| || align="center"|F || align="left"|Greece || align="center"|1 || align="center"| || 2 || 6 || 1 || 0 || 6 || 3.0 || 0.5 || 0.0 || 3.0 || align=center|
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|align="left" bgcolor="#FFCC00"|+ || align="center"|F || align="left"|Syracuse || align="center"|7 || align="center"|– || 412 || 14,813 || 2,865 || 1,328 || 10,186 || 36.0 || 7.0 || 3.2 || 24.7 || align=center|
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|align="left"| || align="center"|G || align="left"|UNLV || align="center"|4 || align="center"|– || 293 || 6,146 || 559 || 1,237 || 1,906 || 21.0 || 1.9 || 4.2 || 6.5 || align=center|
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https://en.wikipedia.org/wiki/Hundred-dollar%2C%20Hundred-digit%20Challenge%20problems | The Hundred-dollar, Hundred-digit Challenge problems are 10 problems in numerical mathematics published in 2002 by . A $100 prize was offered to whoever produced the most accurate solutions, measured up to 10 significant digits. The deadline for the contest was May 20, 2002. In the end, 20 teams solved all of the problems perfectly within the required precision, and an anonymous donor aided in producing the required prize monies. The challenge and its solutions were described in detail in the book .
The problems
From :
A photon moving at speed 1 in the xy-plane starts at t = 0 at (x, y) = (0.5, 0.1) heading due east. Around every integer lattice point (i, j) in the plane, a circular mirror of radius 1/3 has been erected. How far from the origin is the photon at t = 10?
The infinite matrix A with entries is a bounded operator on . What is ?
What is the global minimum of the function
Let , where is the gamma function, and let be the cubic polynomial that best approximates on the unit disk in the supremum norm . What is ?
A flea starts at on the infinite 2D integer lattice and executes a biased random walk: At each step it hops north or south with probability , east with probability , and west with probability . The probability that the flea returns to (0, 0) sometime during its wanderings is . What is ?
Let A be the 20000×20000 matrix whose entries are zero everywhere except for the primes 2, 3, 5, 7, ..., 224737 along the main diagonal and the number 1 in all the positions with . What is the (1, 1) entry of ?
A square plate is at temperature . At time , the temperature is increased to along one of the four sides while being held at along the other three sides, and heat then flows into the plate according to . When does the temperature reach at the center of the plate?
The integral depends on the parameter α. What is the value of α in [0, 5] at which I(α) achieves its maximum?
A particle at the center of a 10×1 rectangle undergoes Brownian motion (i.e., 2D random walk with infinitesimal step lengths) till it hits the boundary. What is the probability that it hits at one of the ends rather than at one of the sides?
Solutions
0.3233674316
0.9952629194
1.274224152
−3.306868647
0.2143352345
0.06191395447
0.7250783462
0.4240113870
0.7859336743
3.837587979 × 10−7
These answers have been assigned the identifiers , , , , , , , , , and in the On-Line Encyclopedia of Integer Sequences.
References
Review (June 2005) from Bulletin of the American Mathematical Society.
Numerical analysis
Recreational mathematics
Mathematics competitions |
https://en.wikipedia.org/wiki/Lee%20Kang-jo | Lee Kang-jo (Hangul: 이강조; Hanja: 李康助; born October 27, 1954) is a South Korean football manager.
Club career statistics
Coach & manager career
1985–1986: Yukong Elephants Trainer
1987–1989: Gangneung Jeil High School Manager
1990–2002: Sangmu FC Manager
2003–present: Gwangju Sangmu FC Manager
International goals
Results list South Korea's goal tally first.
External links
1954 births
Living people
Men's association football midfielders
South Korean men's footballers
South Korea men's international footballers
South Korean football managers
K League 1 players
1980 AFC Asian Cup players
1984 AFC Asian Cup players
Gimcheon Sangmu FC managers
jeju United FC managers
Korea University alumni
Asian Games gold medalists for South Korea
Medalists at the 1978 Asian Games
Asian Games medalists in football
Footballers at the 1978 Asian Games |
https://en.wikipedia.org/wiki/Muconic%20acid | Muconic acid is a dicarboxylic acid. There are three isomeric forms designated trans,trans-muconic acid, cis,trans-muconic acid, and cis,cis-muconic acid which differ by the geometry around the double bonds. Its name is derived from mucic acid.
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trans,trans-Muconic acid is a metabolite of benzene in humans. The determination of its concentration in urine is therefore used as a biomarker of occupational or environmental exposure to benzene. Synthetically, trans,trans-muconic acid can be prepared from adipic acid.
cis,cis-Muconic acid is produced by some bacteria by the enzymatic degradation of various aromatic chemical compounds.
The bioproduction of muconic acid is of interest because of its potential use as a platform chemical for the production of several valuable consumer bioplastics including nylon-6,6, polyurethane, and polyethylene terephthalate.
See also
Dicarboxylic acid
2-Aminomuconic acid
Notes
Dicarboxylic acids |
https://en.wikipedia.org/wiki/Kleene%E2%80%93Rosser%20paradox | In mathematics, the Kleene–Rosser paradox is a paradox that shows that certain systems of formal logic are inconsistent, in particular the version of Haskell Curry's combinatory logic introduced in 1930, and Alonzo Church's original lambda calculus, introduced in 1932–1933, both originally intended as systems of formal logic. The paradox was exhibited by Stephen Kleene and J. B. Rosser in 1935.
The paradox
Kleene and Rosser were able to show that both systems are able to characterize and enumerate their provably total, definable number-theoretic functions, which enabled them to construct a term that essentially replicates Richard's paradox in formal language.
Curry later managed to identify the crucial ingredients of the calculi that allowed the construction of this paradox, and used this to construct a much simpler paradox, now known as Curry's paradox.
See also
List of paradoxes
References
Andrea Cantini, "The inconsistency of certain formal logics", in the Paradoxes and Contemporary Logic entry of Stanford Encyclopedia of Philosophy (2007).
Lambda calculus
Mathematical paradoxes
Self-referential paradoxes |
https://en.wikipedia.org/wiki/Studentized%20range%20distribution | In probability and statistics, studentized range distribution is the continuous probability distribution of the studentized range of an i.i.d. sample from a normally distributed population.
Suppose that we take a sample of size n from each of k populations with the same normal distribution N(μ, σ2) and suppose that is the smallest of these sample means and is the largest of these sample means, and suppose s² is the pooled sample variance from these samples. Then the following statistic has a Studentized range distribution.
Definition
Probability density function
Differentiating the cumulative distribution function with respect to q gives the probability density function.
Note that in the outer part of the integral, the equation
was used to replace an exponential factor.
Cumulative distribution function
The cumulative distribution function is given by
Special cases
If k is 2 or 3, the studentized range probability distribution function can be directly evaluated, where is the standard normal probability density function and is the standard normal cumulative distribution function.
When the degrees of freedom approaches infinity the studentized range cumulative distribution can be calculated for any k using the standard normal distribution.
Applications
Critical values of the studentized range distribution are used in Tukey's range test.
The studentized range is used to calculate significance levels for results obtained by data mining, where one selectively seeks extreme differences in sample data, rather than only sampling randomly.
The Studentized range distribution has applications to hypothesis testing and multiple comparisons procedures. For example, Tukey's range test and Duncan's new multiple range test (MRT), in which the sample x1, ..., xn is a sample of means and q is the basic test-statistic, can be used as post-hoc analysis to test between which two groups means there is a significant difference (pairwise comparisons) after rejecting the null hypothesis that all groups are from the same population (i.e. all means are equal) by the standard analysis of variance.
Related distributions
When only the equality of the two groups means is in question (i.e. whether μ1 = μ2), the studentized range distribution is similar to the Student's t distribution, differing only in that the first takes into account the number of means under consideration, and the critical value is adjusted accordingly. The more means under consideration, the larger the critical value is. This makes sense since the more means there are, the greater the probability that at least some differences between pairs of means will be significantly large due to chance alone.
Derivation
The studentized range distribution function arises from re-scaling the sample range R by the sample standard deviation s, since the studentized range is customarily tabulated in units of standard deviations, with the variable . The derivation begins with a perfectly general form of the |
https://en.wikipedia.org/wiki/Spacetime%20topology | Spacetime topology is the topological structure of spacetime, a topic studied primarily in general relativity. This physical theory models gravitation as the curvature of a four dimensional Lorentzian manifold (a spacetime) and the concepts of topology thus become important in analysing local as well as global aspects of spacetime. The study of spacetime topology is especially important in physical cosmology.
Types of topology
There are two main types of topology for a spacetime M.
Manifold topology
As with any manifold, a spacetime possesses a natural manifold topology. Here the open sets are the image of open sets in .
Path or Zeeman topology
Definition: The topology in which a subset is open if for every timelike curve there is a set in the manifold topology such that .
It is the finest topology which induces the same topology as does on timelike curves.
Properties
Strictly finer than the manifold topology. It is therefore Hausdorff, separable but not locally compact.
A base for the topology is sets of the form for some point and some convex normal neighbourhood .
( denote the chronological past and future).
Alexandrov topology
The Alexandrov topology on spacetime, is the coarsest topology such that both and are open for all subsets .
Here the base of open sets for the topology are sets of the form for some points .
This topology coincides with the manifold topology if and only if the manifold is strongly causal but it is coarser in general.
Note that in mathematics, an Alexandrov topology on a partial order is usually taken to be the coarsest topology in which only the upper sets are required to be open. This topology goes back to Pavel Alexandrov.
Nowadays, the correct mathematical term for the Alexandrov topology on spacetime (which goes back to Alexandr D. Alexandrov) would be the interval topology, but when Kronheimer and Penrose introduced the term this difference in nomenclature was not as clear, and in physics the term Alexandrov topology remains in use.
Planar spacetime
Events connected by light have zero separation. The plenum of spacetime in the plane is split into four quadrants, each of which has the topology of R2. The dividing lines are the trajectory of inbound and outbound photons at (0,0). The planar-cosmology topological segmentation is the future F, the past P, space left L, and space right D. The homeomorphism of F with R2 amounts to polar decomposition of split-complex numbers:
so that
is the split-complex logarithm and the required homeomorphism F → R2, Note that b is the rapidity parameter for relative motion in F.
F is in bijective correspondence with each of P, L, and D under the mappings z → –z, z → jz, and z → – j z, so each acquires the same topology. The union U = F ∪ P ∪ L ∪ D then has a topology nearly covering the plane, leaving out only the null cone on (0,0). Hyperbolic rotation of the plane does not mingle the quadrants, in fact, each one is an invariant set under the uni |
https://en.wikipedia.org/wiki/Chester%20Ittner%20Bliss | Chester Ittner Bliss (February 1, 1899 – March 14, 1979) was primarily a biologist, who is best known for his contributions to statistics. He was born in Springfield, Ohio in 1899 and died in 1979. He was the first secretary of the International Biometric Society.
Academic qualifications
Bachelor of Arts in Entomology from Ohio State University, 1921
Master of Arts from Columbia University, 1922
PhD from Columbia University, 1926
Remarkably, his statistical knowledge was largely self-taught and developed according to the problems he wanted to solve (Cochran & Finney 1979).
Nevertheless, in 1942 he was elected as a Fellow of the American Statistical Association.
Major contributions
The idea of the probit function was published by Bliss in a 1934 article in Science on how to treat data such as the percentage of a pest killed by a pesticide. Bliss proposed transforming the percentage killed into a "probability unit" (or "probit").
Arguably his most important contribution was the development, with Ronald Fisher, of an iterative approach to finding maximum likelihood estimates in the probit method of bioassay. Additional contributions in biological assay were work on the analysis of time-mortality data and of slope-ratio assays (Cochran & Finney 1979).
Bliss introduced the word rankit, meaning an expected normal order statistic.
References
Citations
Sources
C. I. Bliss (1935) The calculation of the dosage-mortality curve, Annals of Applied Biology 22, 134–167. (includes appendix by Fisher.)
W. G. Cochran, D. J. Finney. 1979 Chester Ittner Bliss, 1899–1979, Biometrics; 35(4): 715–717. pdf
D. J. Finney. 1980 Chester Ittner Bliss, 1899–1979, Journal of the Royal Statistical Society, Series A, 143(1): 92–93.
T. R. Holford & C. White (2005) Bliss, Chester Ittner, Encyclopedia of Biostatistics.
External links
University of Adelaide: Correspondence between C. I. Bliss and R. A. Fisher
University of Adelaide: Fisher's appendix to Bliss (1935)
Chester Ittner Bliss papers (MS 1165). Manuscripts and Archives, Yale University Library.
1899 births
1979 deaths
American statisticians
Fellows of the American Statistical Association
Columbia Graduate School of Arts and Sciences alumni |
https://en.wikipedia.org/wiki/List%20of%20Grand%20Slam%20girls%27%20doubles%20champions | List of Girls' Doubles Junior Grand Slam tournaments tennis champions:
Champions by year
Statistics
Most Grand Slam doubles titles
Note: when a tie, the person to reach the mark first is listed first.
Three titles in a single season
Surface Slam
Players who won Grand Slam titles on clay, grass and hard courts in a calendar year.
Channel Slam
Players who won the French Open-Wimbledon double.
Sources
ITF Australian Open
ITF Roland Garros
ITF Wimbledon
ITF US Open
References
See also
List of Grand Slam girls' singles champions
List of Grand Slam boys' singles champions
List of Grand Slam boys' doubles champions
girls
Grand Slam
Girls |
https://en.wikipedia.org/wiki/Gabriel%20Paternain | Gabriel Pedro Paternain is a Uruguayan mathematician. He is Professor of Mathematics at the University of Washington. Previously he was a professor in DPMMS at the University of Cambridge, and a fellow of Trinity College. He obtained his Licenciatura from Universidad de la Republica in Uruguay in 1987, and his PhD from the State University of New York at Stony Brook in 1991. He has lectured several undergraduate and graduate courses and has gained widespread popularity due to his entertaining and informal lecturing style, which has been recognised by the university in the past for its high calibre. He was managing editor of the mathematical journal Mathematical Proceedings of the Cambridge Philosophical Society for the period 2006–2011.
He is known for his work on dynamical and geometrical aspects of Hamiltonian systems, in particular magnetic and geodesic flows. His recent
research focuses on geometric inverse problems and his collaboration with Mikko Salo and Gunther Uhlmann yielded solutions to several inverse problems in two dimensions, including the tensor tomography problem and the proof of spectral rigidity of an Anosov surface.
In his spare time he partakes in a wide variety of sports, notably football.
References
External links
http://www.dpmms.cam.ac.uk/~gpp24/
Year of birth missing (living people)
Living people
20th-century Uruguayan mathematicians
Stony Brook University alumni
Fellows of Trinity College, Cambridge
Cambridge mathematicians
21st-century Uruguayan mathematicians |
https://en.wikipedia.org/wiki/Exceptional%20divisor | In mathematics, specifically algebraic geometry, an exceptional divisor for a regular map
of varieties is a kind of 'large' subvariety of which is 'crushed' by , in a certain definite sense. More strictly, f has an associated exceptional locus which describes how it identifies nearby points in codimension one, and the exceptional divisor is an appropriate algebraic construction whose support is the exceptional locus. The same ideas can be found in the theory of holomorphic mappings of complex manifolds.
More precisely, suppose that
is a regular map of varieties which is birational (that is, it is an isomorphism between open subsets of and ). A codimension-1 subvariety is said to be exceptional if has codimension at least 2 as a subvariety of . One may then define the exceptional divisor of to be
where the sum is over all exceptional subvarieties of , and is an element of the group of Weil divisors on .
Consideration of exceptional divisors is crucial in birational geometry: an elementary result (see for instance Shafarevich, II.4.4) shows (under suitable assumptions) that any birational regular map that is not an isomorphism has an exceptional divisor. A particularly important example is the blowup
of a subvariety
:
in this case the exceptional divisor is exactly the preimage of .
References
Algebraic geometry
Birational geometry |
https://en.wikipedia.org/wiki/Jeff%20Gill%20%28academic%29 | Jefferson Morris Gill (born December 22, 1960) is Distinguished Professor of Government, and of Mathematics & Statistics, the Director of the Center for Data Science, the Editor of Political Analysis, and a member of the Center for Behavioral Neuroscience at American University as of the Fall of 2017.
He was a Professor of Political Science at Washington University in St. Louis and the Director of the Center for Applied Statistics. He was also President of the Society for Political Methodology, and is an inaugural fellow of the Society for Political Methodology. Major areas of research and interest include: Political Methodology, American Politics, Statistical Computing, Research Methods, and Public Administration. Current research is focused on projects on work in the development of Bayesian hierarchical models, nonparametric Bayesian models, elicited prior development from expert interviews, as well in fundamental issues in statistical inference. He has extensive expertise in statistical computing, Markov chain Monte Carlo (MCMC) tools in particular. Most sophisticated Bayesian models for the social or medical sciences require complex, compute-intensive tools such as MCMC to efficiently estimate parameters of interest. Gill is an expert in these statistical and computational techniques and uses them to contribute to empirical knowledge in the biomedical and social sciences. Current theoretical work builds logically on Gill's prior applied work and adds opportunities to develop new hybrid algorithms for statistical estimation with multilevel specifications and complex time-series and spatial relationships.
Current applied work includes: energetics and cancer, long-term mental health outcomes from children's exposure to war, pediatric head trauma, analysis of mouse models, and molecular models of sickle cell disease. He also contributes to gene-wide associate studies (GWAS) that seek to discover correlated cancer genes related to obesity, diet, and exercise, as well as consult on computational genetics analysis. Other work includes Bayesian hierarchical models, Markov chain Monte Carlo theory, bureaucratic behavior in national security agencies, and issues in political epidemiology. His best known works include Essential Mathematics for Political and Social Research, with Cambridge University Press, and the third edition of Bayesian Methods for the Social and Behavioral Sciences (Chapman & Hall/CRC), which is the leading Bayesian text for these disciplines. He is the author of seven other books. His journal work has appeared in the Quarterly Journal of Political Science, Journal of the Royal Statistical Society, Journal of Politics, Electoral Studies, Statistical Science, Political Research Quarterly, Sociological Methods & Research, Public Administration Review, Journal of Public Administration Research and Theory, Canadian Journal of Political Science, Journal of Statistical Software, Political Analysis, Lancet Neurology, American Jo |
https://en.wikipedia.org/wiki/Cut%20locus%20%28Riemannian%20manifold%29 | In Riemannian geometry, the cut locus of a point in a manifold is roughly the set of all other points for which there are multiple minimizing geodesics connecting them from , but it may contain additional points where the minimizing geodesic is unique, under certain circumstances. The distance function from p is a smooth function except at the point p itself and the cut locus.
Definition
Fix a point in a complete Riemannian manifold , and consider the tangent space . It is a standard result that for sufficiently small in , the curve defined by the Riemannian exponential map, for belonging to the interval is a minimizing geodesic, and is the unique minimizing geodesic connecting the two endpoints. Here denotes the exponential map from . The cut locus of in the tangent space is defined to be the set of all vectors in such that is a minimizing geodesic for but fails to be minimizing for for any . The cut locus of in is defined to be image of the
cut locus of in the tangent space under the exponential map at . Thus, we may interpret the cut locus of in as the points in the manifold where the geodesics starting at stop being minimizing.
The least distance from p to the cut locus is the injectivity radius at p. On the open ball of this radius, the exponential map at p is a diffeomorphism from the tangent space to the manifold, and this is the largest such radius. The global injectivity radius is defined to be the infimum of the injectivity radius at p, over all points of the manifold.
Characterization
Suppose is in the cut locus of in . A standard result is that either (1) there is more than one minimizing geodesic joining to , or (2) and are conjugate along some geodesic
which joins them. It is possible for both (1) and (2) to hold.
Examples
On the standard round n-sphere, the cut locus of a point consists of the single point opposite of it (i.e., the antipodal point). On
an infinitely long cylinder, the cut locus of a point consists of the line opposite the point.
Applications
The significance of the cut locus is that the distance function from a point is smooth, except on the cut locus of and itself. In particular, it makes sense to take the gradient and Hessian of the distance function away from the cut locus and . This idea is used in the local Laplacian comparison theorem and the local Hessian comparison theorem. These are used in the proof of the local version of the Toponogov theorem, and many other important theorems in Riemannian geometry.
Cut locus of a subset
One can similarly define the cut locus of a submanifold of the Riemannian manifold, in terms of its normal exponential map.
See also
Caustic (mathematics)
References
Riemannian geometry |
https://en.wikipedia.org/wiki/List%20of%20Grand%20Slam%20boys%27%20doubles%20champions | List of Boys' Doubles Junior Grand Slam tournaments tennis champions:
Champions by year
Statistics
Most Grand Slam doubles titles
Note: when a tie, the person to reach the mark first is listed first.
Career Grand Slam
Players who won all four Grand Slam titles over the course of their careers.
The event at which the Career Grand Slam was completed indicated in bold
Three titles in a single season
Surface Slam
Players who won Grand Slam titles on clay, grass and hard courts in a calendar year.
Channel Slam
Players who won the French Open-Wimbledon double.
Sources
ITF Australian Open
ITF Roland Garros
ITF Wimbledon
ITF US Open
See also
List of Grand Slam boys' singles champions
List of Grand Slam girls' singles champions
List of Grand Slam girls' doubles champions
Tennis statistics
Boys
Grand Slam Men's Singles champions
Grand Slam
Boys |
https://en.wikipedia.org/wiki/Nonhypotenuse%20number | In mathematics, a nonhypotenuse number is a natural number whose square cannot be written as the sum of two nonzero squares. The name stems from the fact that an edge of length equal to a nonhypotenuse number cannot form the hypotenuse of a right angle triangle with integer sides.
The numbers 1, 2, 3 and 4 are all nonhypotenuse numbers. The number 5, however, is not a nonhypotenuse number as 52 equals 32 + 42.
The first fifty nonhypotenuse numbers are:
1, 2, 3, 4, 6, 7, 8, 9, 11, 12, 14, 16, 18, 19, 21, 22, 23, 24, 27, 28, 31, 32, 33, 36, 38, 42, 43, 44, 46, 47, 48, 49, 54, 56, 57, 59, 62, 63, 64, 66, 67, 69, 71, 72, 76, 77, 79, 81, 83, 84
Although nonhypotenuse numbers are common among small integers, they become more-and-more sparse for larger numbers. Yet, there are infinitely many nonhypotenuse numbers, and the number of nonhypotenuse numbers not exceeding a value x scales asymptotically with x/.
The nonhypotenuse numbers are those numbers that have no prime factors of the form 4k+1. Equivalently, they are the number that cannot be expressed in the form where K, m, and n are all positive integers. A number whose prime factors are not of the form 4k+1 cannot be the hypotenuse of a primitive integer right triangle (one for which the sides do not have a nontrivial common divisor), but may still be the hypotenuse of a non-primitive triangle.
The nonhypotenuse numbers have been applied to prove the existence of addition chains that compute the first square numbers using only additions.
See also
Pythagorean theorem
Landau-Ramanujan constant
Fermat's theorem on sums of two squares
References
External links
Integer sequences |
https://en.wikipedia.org/wiki/Super-logarithm | In mathematics, the super-logarithm is one of the two inverse functions of tetration. Just as exponentiation has two inverse functions, roots and logarithms, tetration has two inverse functions, super-roots and super-logarithms. There are several ways of interpreting super-logarithms:
As the Abel function of exponential functions,
As the inverse function of tetration with respect to the height,
As a generalization of Robert Munafo's large number class system,
For positive integer values, the super-logarithm with base-e is equivalent to the number of times a logarithm must be iterated to get to 1 (the Iterated logarithm). However, this is not true for negative values and so cannot be considered a full definition.
The precise definition of the super-logarithm depends on a precise definition of non-integer tetration (that is, for y not an integer). There is no clear consensus on the definition of non-integer tetration and so there is likewise no clear consensus on the super-logarithm for non-integer inputs.
Definitions
The super-logarithm, written is defined implicitly by
and
This definition implies that the super-logarithm can only have integer outputs, and that it is only defined for inputs of the form and so on. In order to extend the domain of the super-logarithm from this sparse set to the real numbers, several approaches have been pursued. These usually include a third requirement in addition to those listed above, which vary from author to author. These approaches are as follows:
The linear approximation approach by Rubstov and Romerio,
The quadratic approximation approach by Andrew Robbins,
The regular Abel function approach by George Szekeres,
The iterative functional approach by Peter Walker, and
The natural matrix approach by Peter Walker, and later generalized by Andrew Robbins.
Approximations
Usually, the special functions are defined not only for the real values of argument(s), but to complex plane, and differential and/or integral representation, as well as expansions in convergent and asymptotic series. Yet, no such representations are available for the slog function. Nevertheless, the simple approximations below are suggested.
Linear approximation
The linear approximation to the super-logarithm is:
which is a piecewise-defined function with a linear "critical piece". This function has the property that it is continuous for all real z ( continuous). The first authors to recognize this approximation were Rubstov and Romerio, although it is not in their paper, it can be found in their algorithm that is used in their software prototype. The linear approximation to tetration, on the other hand, had been known before, for example by Ioannis Galidakis. This is a natural inverse of the linear approximation to tetration.
Authors like Holmes recognize that the super-logarithm would be a great use to the next evolution of computer floating-point arithmetic, but for this purpose, the function need not be infinitely differ |
https://en.wikipedia.org/wiki/Political%20methodology | Political methodology is a subfield of political science that studies the quantitative and qualitative methods used to study politics. Quantitative methods combine statistics, mathematics, and formal theory. Political methodology is often used for positive research, in contrast to normative research. Psephology, a skill or technique within political methodology, is the "quantitative analysis of elections and balloting".
Journals
Political methodology is often published in the "top 3" journals (American Political Science Review, American Journal of Political Science, and Journal of Politics), in sub-field journals, and in methods-focused journals.
Political Analysis
Political Science Research and Methods
Notable researchers
Gary King
Rob Franzese
Jeff Gill
Phil Schrodt
Jan Box-Steffensmeier
Simon Jackman
Jonathan Nagler
Jim Stimson
Larry Bartels
Donald Green
References
External links
The Society for Political Methodology's homepage
US News Rankings for Political Methodology
Political science |
https://en.wikipedia.org/wiki/Characteristic%20number%20%28disambiguation%29 | Characteristic number may mean:
Characteristic number (mathematics)
Characteristic number (physics)
Characteristic number (fluid dynamics) |
https://en.wikipedia.org/wiki/Marc%20M%C3%B8ller | Marc Møller (born 7 June 1986) is a retired Danish professional football defender.
External links
Lyngby BK profile
Official Danish Superliga player statistics at danskfodbold.com
1986 births
Living people
Danish men's footballers
FC Midtjylland players
Lyngby Boldklub players
Danish Superliga players
Ikast FC players
Men's association football defenders |
https://en.wikipedia.org/wiki/Morten%20Christiansen%20%28footballer%29 | Morten Christiansen (born 4 January 1978) is a Danish professional football midfielder, who currently plays for FC Royal.
External links
Lyngby BK profile
Danish Superliga player statistics at danskfodbold.com
1978 births
Living people
Danish men's footballers
AC Horsens players
Lyngby Boldklub players
Danish Superliga players
Footballers from Aarhus
Men's association football midfielders
VSK Aarhus players |
https://en.wikipedia.org/wiki/Nicolai%20Melchiorsen | Nicolai Melchiorsen (born 9 March 1984) is a Danish professional football midfielder.
External links
Nicolai Melchiorsen official Danish Superliga statistics at danskfodbold.com
1984 births
Living people
Danish men's footballers
Akademisk Boldklub players
Lyngby Boldklub players
Viborg FF players
Danish Superliga players
Men's association football midfielders |
https://en.wikipedia.org/wiki/Biplot | Biplots are a type of exploratory graph used in statistics, a generalization of the simple two-variable scatterplot.
A biplot overlays a score plot with a loading plot.
A biplot allows information on both samples and variables of a data matrix to be displayed graphically. Samples are displayed as points while variables are displayed either as vectors, linear axes or nonlinear trajectories. In the case of categorical variables, category level points may be used to represent the levels of a categorical variable. A generalised biplot displays information on both continuous and categorical variables.
Introduction and history
The biplot was introduced by K. Ruben Gabriel (1971). Gower and Hand (1996) wrote a monograph on biplots. Yan and Kang (2003) described various methods which can be used in order to visualize and interpret a biplot. The book by Greenacre (2010) is a practical user-oriented guide to biplots, along with scripts in the open-source R programming language, to generate biplots associated with principal component analysis (PCA), multidimensional scaling (MDS), log-ratio analysis (LRA)—also known as spectral mapping—discriminant analysis (DA) and various forms of correspondence analysis: simple correspondence analysis (CA), multiple correspondence analysis (MCA) and canonical correspondence analysis (CCA) (Greenacre 2016). The book by Gower, Lubbe and le Roux (2011) aims to popularize biplots as a useful and reliable method for the visualization of multivariate data when researchers want to consider, for example, principal component analysis (PCA), canonical variates analysis (CVA) or various types of correspondence analysis.
Construction
A biplot is constructed by using the singular value decomposition (SVD) to obtain a low-rank approximation to a transformed version of the data matrix X, whose n rows are the samples (also called the cases, or objects), and whose p columns are the variables. The transformed data matrix Y is obtained from the original matrix X by centering and optionally standardizing the columns (the variables). Using the SVD, we can write Y = Σk=1,...pdkukvkT;, where the uk are n-dimensional column vectors, the vk are p-dimensional column vectors, and the dk are a non-increasing sequence of non-negative scalars. The biplot is formed from two scatterplots that share a common set of axes and have a between-set scalar product interpretation. The first scatterplot is formed from the points (d1αu1i, d2αu2i), for i = 1,...,n. The second plot is formed from the points (d11−αv1j, d21−αv2j), for j = 1,...,p. This is the biplot formed by the dominant two terms of the SVD, which can then be represented in a two-dimensional display. Typical choices of α are 1 (to give a distance interpretation to the row display) and 0 (to give a distance interpretation to the column display), and in some rare cases α=1/2 to obtain a symmetrically scaled biplot (which gives no distance interpretation to the rows or the columns, but only th |
https://en.wikipedia.org/wiki/Monte%20Carlo%20method%20for%20photon%20transport | Modeling photon propagation with Monte Carlo methods is a flexible yet rigorous approach to simulate photon transport. In the method, local rules of photon transport are expressed as probability distributions which describe the step size of photon movement between sites of photon-matter interaction and the angles of deflection in a photon's trajectory when a scattering event occurs. This is equivalent to modeling photon transport analytically by the radiative transfer equation (RTE), which describes the motion of photons using a differential equation. However, closed-form solutions of the RTE are often not possible; for some geometries, the diffusion approximation can be used to simplify the RTE, although this, in turn, introduces many inaccuracies, especially near sources and boundaries. In contrast, Monte Carlo simulations can be made arbitrarily accurate by increasing the number of photons traced. For example, see the movie, where a Monte Carlo simulation of a pencil beam incident on a semi-infinite medium models both the initial ballistic photon flow and the later diffuse propagation.
The Monte Carlo method is necessarily statistical and therefore requires significant computation time to achieve precision. In addition Monte Carlo simulations can keep track of multiple physical quantities simultaneously, with any desired spatial and temporal resolution. This flexibility makes Monte Carlo modeling a powerful tool. Thus, while computationally inefficient, Monte Carlo methods are often considered the standard for simulated measurements of photon transport for many biomedical applications.
Biomedical applications of Monte Carlo methods
Biomedical imaging
The optical properties of biological tissue offer an approach to biomedical imaging. There are many endogenous contrasts, including absorption from blood and melanin and scattering from nerve cells and cancer cell nuclei. In addition, fluorescent probes can be targeted to many different tissues. Microscopy techniques (including confocal, two-photon, and optical coherence tomography) have the ability to image these properties with high spatial resolution, but, since they rely on ballistic photons, their depth penetration is limited to a few millimeters. Imaging deeper into tissues, where photons have been multiply scattered, requires a deeper understanding of the statistical behavior of large numbers of photons in such an environment. Monte Carlo methods provide a flexible framework that has been used by different techniques to reconstruct optical properties deep within tissue. A brief introduction to a few of these techniques is presented here.
Photoacoustic tomography In PAT, diffuse laser light is absorbed which generates a local temperature rise. This local temperature variation in turn generates ultrasound waves via thermoelastic expansion which are detected via an ultrasonic transducer. In practice, a variety of setup parameters are varied (i.e. light wavelength, transducer |
https://en.wikipedia.org/wiki/Vertex%20arrangement | In geometry, a vertex arrangement is a set of points in space described by their relative positions. They can be described by their use in polytopes.
For example, a square vertex arrangement is understood to mean four points in a plane, equal distance and angles from a center point.
Two polytopes share the same vertex arrangement if they share the same 0-skeleton.
A group of polytopes that shares a vertex arrangement is called an army.
Vertex arrangement
The same set of vertices can be connected by edges in different ways. For example, the pentagon and pentagram have the same vertex arrangement, while the second connects alternate vertices.
A vertex arrangement is often described by the convex hull polytope which contains it. For example, the regular pentagram can be said to have a (regular) pentagonal vertex arrangement.
Infinite tilings can also share common vertex arrangements.
For example, this triangular lattice of points can be connected to form either isosceles triangles or rhombic faces.
Edge arrangement
Polyhedra can also share an edge arrangement while differing in their faces.
For example, the self-intersecting great dodecahedron shares its edge arrangement with the convex icosahedron:
A group polytopes that share both a vertex arrangement and an edge arrangement are called a regiment.
Face arrangement
4-polytopes can also have the same face arrangement which means they have similar vertex, edge, and face arrangements, but may differ in their cells.
For example, of the ten nonconvex regular Schläfli-Hess polychora, there are only 7 unique face arrangements.
For example, the grand stellated 120-cell and great stellated 120-cell, both with pentagrammic faces, appear visually indistinguishable without a representation of their cells:
Classes of similar polytopes
George Olshevsky advocates the term regiment for a set of polytopes that share an edge arrangement, and more generally n-regiment for a set of polytopes that share elements up to dimension n. Synonyms for special cases include company for a 2-regiment (sharing faces) and army for a 0-regiment (sharing vertices).
See also
n-skeleton - a set of elements of dimension n and lower in a higher polytope.
Vertex figure - A local arrangement of faces in a polyhedron (or arrangement of cells in a polychoron) around a single vertex.
External links
(Same vertex arrangement)
(Same vertex and edge arrangement)
(Same vertex, edge and face arrangement)
Polytopes |
https://en.wikipedia.org/wiki/John%20Guckenheimer | John Mark Guckenheimer (born 1945) joined the Department of Mathematics at Cornell University in 1985. He was previously at the University of California, Santa Cruz (1973-1985). He was a Guggenheim fellow in 1984, and was elected president of the Society for Industrial and Applied Mathematics (SIAM), serving from 1997 to 1998. Guckenheimer received his A.B. in 1966 from Harvard and his Ph.D. in 1970 from Berkeley, where his Ph.D. thesis advisor was Stephen Smale.
His book Nonlinear Oscillations, Dynamical Systems and Bifurcation of Vector Fields (with Philip Holmes) is an extensively cited work on dynamical systems.
Research
Dr. John Guckenheimer's research has focused on three areas — neuroscience, algorithms for periodic orbits, and dynamics in systems with multiple time scales.
Neuroscience
Guckenheimer studies dynamical models of a small neural system, the stomatogastric ganglion of crustaceans — attempting to learn more about neuromodulation, the ways in which the rhythmic output of the STG is modified by chemical and electrical inputs.
Algorithms for Periodic Orbits
Employing automatic differentiation, Guckenheimer has constructed a new family of algorithms that compute periodic orbits directly. His research in this area attempts to automatically compute bifurcations of periodic orbits as well as "generate rigorous computer proofs of the qualitative properties of numerically computed dynamical systems".
Dynamics in systems with Multiple Time Scales
Guckenheimer's research in this area is aimed at "extending the qualitative theory of dynamical systems to apply to systems with multiple time scales". Examples of systems with multiple time scales include neural systems and switching controllers.
DsTool
Guckenheimer's research has also included the development of computer methods used in studies of nonlinear systems. He has overseen the development of DsTool, an interactive software laboratory for the investigation of dynamical systems.
Awards and honors
He became a SIAM Fellow in 2009.
In 2012 he became a fellow of the American Mathematical Society. He won a Leroy P. Steele Prize in 2013 for his book (coauthored with Philip Holmes), and he gave the Moser Lecture in May 2015.
Selected publications
Nonlinear Oscillations, Dynamical Systems and Bifurcation of Vector Fields (with Philip Holmes), Springer-Verlag, 1983, 453 pp.
Phase portraits of planar vector fields: computer proofs, Journal of Experimental Mathematics 4 (1995), 153–164.
An improved parameter estimation method for Hodgkin-Huxley model (with A. R. Willms, D. J. Baro and R. M. Harris-Warrick), J. Comp. Neuroscience 6 (1999), 145–168.
Computing periodic orbits and their bifurcations with automatic differentiation (with B. Meloon), SIAM J. Sci. Stat. Comp. 22 (2000), 951–985.
The forced van der Pol equation I: the slow flow and its bifurcations (with K. Hoffman and W. Weckesser), SIAM J. App. Dyn. Sys. 2 (2002), 1–35.
Notes
References
External links
John Guckenhe |
https://en.wikipedia.org/wiki/Stretching%20field | In applied mathematics, stretching fields provide the local deformation of an infinitesimal circular fluid element over a finite time interval ∆t. The logarithm of the stretching (after first dividing by ∆t) gives the finite-time Lyapunov exponent λ for separation of nearby fluid elements at each point in a flow. For periodic two-dimensional flows, stretching fields have been shown to be closely related to the mixing of a passive scalar concentration field. Until recently, however, the extension of these ideas to systems that are non-periodic or weakly turbulent has been possible only in numerical simulations.
Dynamical systems |
https://en.wikipedia.org/wiki/Left%20inverse | A left inverse in mathematics may refer to:
A left inverse element with respect to a binary operation on a set
A left inverse function for a mapping between sets
A kind of generalized inverse
See also
Left-cancellative
Loop (algebra), an algebraic structure with identity element where every element has a unique left and right inverse
Retraction (category theory), a left inverse of some morphism
Right inverse (disambiguation) |
https://en.wikipedia.org/wiki/A-equivalence | In mathematics, -equivalence, sometimes called right-left equivalence, is an equivalence relation between map germs.
Let and be two manifolds, and let be two smooth map germs. We say that and are -equivalent if there exist diffeomorphism germs and such that
In other words, two map germs are -equivalent if one can be taken onto the other by a diffeomorphic change of co-ordinates in the source (i.e. ) and the target (i.e. ).
Let denote the space of smooth map germs Let be the group of diffeomorphism germs and
be the group of diffeomorphism germs The group acts on in the natural way: Under this action we see that the map germs are -equivalent if, and only if, lies in the orbit of , i.e. (or vice versa).
A map germ is called stable if its orbit under the action of is open relative to the Whitney topology. Since is an infinite dimensional space metric topology is no longer trivial. Whitney topology compares the differences in successive derivatives and gives a notion of proximity within the infinite dimensional space. A base for the open sets of the topology in question is given by taking -jets for every and taking open neighbourhoods in the ordinary Euclidean sense. Open sets in the topology are then unions of
these base sets.
Consider the orbit of some map germ The map germ is called simple if there are only finitely many other orbits in a neighbourhood of each of its points. Vladimir Arnold has shown that the only simple singular map germs for are the infinite sequence (), the infinite sequence (), and
See also
K-equivalence (contact equivalence)
References
M. Golubitsky and V. Guillemin, Stable Mappings and Their Singularities. Graduate Texts in Mathematics, Springer.
Functions and mappings
Singularity theory
Equivalence (mathematics) |
https://en.wikipedia.org/wiki/Plus%20Magazine | Plus Magazine is an online popular mathematics magazine run under the Millennium Mathematics Project at the University of Cambridge.
Plus contains:
feature articles on all aspects of mathematics;
reviews of popular maths books and events;
a news section;
mathematical puzzles and games;
interviews with people in maths related careers;
Plus Podcast – Maths on the Move
History
Plus was initially named PASS Maths (Public Awareness and Schools Support for Maths) in 1997, when it was a project of the Interactive Courseware Research and Development Group, based jointly at the University of Cambridge and Keele University. Plus is now part of the Millennium Mathematics Project, a long term national initiative based in Cambridge and active across the UK and internationally.
Authors of articles in Plus include Stephen Hawking and Marcus du Sautoy.
Plus won the 2001 Webby for Best Science Site on the Web, and has been described as "an excellent site put together by those with a real love for the subject". In 2006 the Millennium Mathematics Project, of which Plus is a part, won the Queen's Anniversary Prize for Higher Education.
References
External links
1997 establishments in the United Kingdom
Magazines established in 1997
Mass media in Cambridge
Mathematics education in the United Kingdom
Mathematics magazines
Mathematics websites
Online magazines published in the United Kingdom
Publications associated with the University of Cambridge
Science and technology magazines published in the United Kingdom
Science education in the United Kingdom
Science podcasts |
https://en.wikipedia.org/wiki/Plus | Plus may refer to:
Mathematics
Addition
+, the mathematical sign
Music
+ (Ed Sheeran album), (pronounced "plus"), 2011
Plus (Cannonball Adderley Quintet album), 1961
Plus (Astrud Gilberto and James Last album), 1986
Plus (Matt Nathanson EP), 2003
Plus (Martin Garrix EP), 2018
Plus (band), a Japanese pop boy band
Plus (Autechre album), 2020
Companies
Plus Communication Sh.A, a cellphone company in Albania
Plus (telecommunications Poland), a mobile phone brand
Plus (British TV channel), run by Granada Sky Broadcasting
Plus (Slovak TV channel)
Plus (interbank network), Visa's ATM and debit card network
PLUS Markets, a small stock exchange in London, UK
PLUS Expressway Berhad, concessionaire of the North-South Expressway, Malaysia
PLUS (Dutch supermarket)
Plus (German supermarket)
Plus (autonomous trucking)
Plus Development, a defunct American computer storage manufacturer
Other
, the international call prefix
PLUS Loan, a United States Federal student loan
Plus Magazine, an online mathematics magazine
Promoting Logical Unified Security, a system for rating a building's security
Plus (programming language)
Professional Liability Underwriting Society, a non-profit organization
Plus (cereal), a breakfast cereal range by Australian breakfast company Uncle Tobys
Plus (novel), 1976, by Joseph McElroy
PLUS card, an expansion card type for some Tandy 1000 computer models
PLUS, abbreviation of the Freedom, Unity and Solidarity Party of Romania
See also
+ (disambiguation)
Minus (disambiguation)
Plus-size (disambiguation)
Circled plus (disambiguation) (⊕) |
https://en.wikipedia.org/wiki/Carleman%20matrix | In mathematics, a Carleman matrix is a matrix used to convert function composition into matrix multiplication. It is often used in iteration theory to find the continuous iteration of functions which cannot be iterated by pattern recognition alone. Other uses of Carleman matrices occur in the theory of probability generating functions, and Markov chains.
Definition
The Carleman matrix of an infinitely differentiable function is defined as:
so as to satisfy the (Taylor series) equation:
For instance, the computation of by
simply amounts to the dot-product of row 1 of with a column vector .
The entries of in the next row give the 2nd power of :
and also, in order to have the zeroth power of in , we adopt the row 0 containing zeros everywhere except the first position, such that
Thus, the dot product of with the column vector yields the column vector
Generalization
A generalization of the Carleman matrix of a function can be defined around any point, such as:
or where . This allows the matrix power to be related as:
General Series
Another way to generalize it even further is think about a general series in the following way:
Let be a series approximation of , where is a basis of the space containing
We can define , therefore we have , now we can prove that , if we assume that is also a basis for and .
Let be such that where .
Now
Comparing the first and the last term, and from being a base for , and it follows that
Examples
If we set we have the Carleman matrix
If is an orthonormal basis for a Hilbert Space with a defined inner product , we can set and will be . If we have the analogous for Fourier Series, namely
Properties
Carleman matrices satisfy the fundamental relationship
which makes the Carleman matrix M a (direct) representation of . Here the term denotes the composition of functions .
Other properties include:
, where is an iterated function and
, where is the inverse function (if the Carleman matrix is invertible).
Examples
The Carleman matrix of a constant is:
The Carleman matrix of the identity function is:
The Carleman matrix of a constant addition is:
The Carleman matrix of the successor function is equivalent to the Binomial coefficient:
The Carleman matrix of the logarithm is related to the (signed) Stirling numbers of the first kind scaled by factorials:
The Carleman matrix of the logarithm is related to the (unsigned) Stirling numbers of the first kind scaled by factorials:
The Carleman matrix of the exponential function is related to the Stirling numbers of the second kind scaled by factorials:
The Carleman matrix of exponential functions is:
The Carleman matrix of a constant multiple is:
The Carleman matrix of a linear function is:
The Carleman matrix of a function is:
The Carleman matrix of a function is:
Related matrices
The Bell matrix or the Jabotinsky matrix of a function is defined as
so as to satisfy the equation
These matrices were developed in 1947 by Eri |
https://en.wikipedia.org/wiki/Boundedly%20generated%20group | In mathematics, a group is called boundedly generated if it can be expressed as a finite product of cyclic subgroups. The property of bounded generation is also closely related with the congruence subgroup problem (see ).
Definitions
A group G is called boundedly generated if there exists a finite subset S of G and a positive integer m such that every element g of G can be represented as a product of at most m powers of the elements of S:
where and are integers.
The finite set S generates G, so a boundedly generated group is finitely generated.
An equivalent definition can be given in terms of cyclic subgroups. A group G is called boundedly generated if there is a finite family C1, …, CM of not necessarily distinct cyclic subgroups such that G = C1…CM as a set.
Properties
Bounded generation is unaffected by passing to a subgroup of finite index: if H is a finite index subgroup of G then G is boundedly generated if and only if H is boundedly generated.
Bounded generation goes to extension: if a group G has a normal subgroup N such that both N and G/N are boundedly generated, then so is G itself.
Any quotient group of a boundedly generated group is also boundedly generated.
A finitely generated torsion group must be finite if it is boundedly generated; equivalently, an infinite finitely generated torsion group is not boundedly generated.
A pseudocharacter on a discrete group G is defined to be a real-valued function f on a G such that
f(gh) − f(g) − f(h) is uniformly bounded and f(gn) = n·f(g).
The vector space of pseudocharacters of a boundedly generated group G is finite-dimensional.
Examples
If n ≥ 3, the group SLn(Z) is boundedly generated by its elementary subgroups, formed by matrices differing from the identity matrix only in one off-diagonal entry. In 1984, Carter and Keller gave an elementary proof of this result, motivated by a question in algebraic .
A free group on at least two generators is not boundedly generated (see below).
The group SL2(Z) is not boundedly generated, since it contains a free subgroup with two generators of index 12.
A Gromov-hyperbolic group is boundedly generated if and only if it is virtually cyclic (or elementary), i.e. contains a cyclic subgroup of finite index.
Free groups are not boundedly generated
Several authors have stated in the mathematical literature that it is obvious that finitely generated free groups are not boundedly generated. This section contains various obvious and less obvious ways of proving this. Some of the methods, which touch on bounded cohomology, are important because they are geometric rather than algebraic, so can be applied to a wider class of groups, for example Gromov-hyperbolic groups.
Since for any n ≥ 2, the free group on 2 generators F2 contains the free group on n generators Fn as a subgroup of finite index (in fact n − 1), once one non-cyclic free group on finitely many generators is known to be not boundedly generated, this will be true fo |
https://en.wikipedia.org/wiki/Dominance-based%20rough%20set%20approach | The dominance-based rough set approach (DRSA) is an extension of rough set theory for multi-criteria decision analysis (MCDA), introduced by Greco, Matarazzo and Słowiński. The main change compared to the classical rough sets is the substitution for the indiscernibility relation by a dominance relation, which permits one to deal with inconsistencies typical to consideration of criteria and preference-ordered decision classes.
Multicriteria classification (sorting)
Multicriteria classification (sorting) is one of the problems considered within MCDA and can be stated as follows: given a set of objects evaluated by a set of criteria (attributes with preference-order domains), assign these objects to some pre-defined and preference-ordered decision classes, such that each object is assigned to exactly one class. Due to the preference ordering, improvement of evaluations of an object on the criteria should not worsen its class assignment. The sorting problem is very similar to the problem of classification, however, in the latter, the objects are evaluated by regular attributes and the decision classes are not necessarily preference ordered. The problem of multicriteria classification is also referred to as ordinal classification problem with monotonicity constraints and often appears in real-life application when ordinal and monotone properties follow from the domain knowledge about the problem.
As an illustrative example, consider the problem of evaluation in a high school. The director of the school wants to assign students (objects) to three classes: bad, medium and good (notice that class good is preferred to medium and medium is preferred to bad). Each student is described by three criteria: level in Physics, Mathematics and Literature, each taking one of three possible values bad, medium and good. Criteria are preference-ordered and improving the level from one of the subjects should not result in worse global evaluation (class).
As a more serious example, consider classification of bank clients, from the viewpoint of bankruptcy risk, into classes safe and risky. This may involve such characteristics as "return on equity (ROE)", "return on investment (ROI)" and "return on sales (ROS)". The domains of these attributes are not simply ordered but involve a preference order since, from the viewpoint of bank managers, greater values of ROE, ROI or ROS are better for clients being analysed for bankruptcy risk . Thus, these attributes are criteria. Neglecting this information in knowledge discovery may lead to wrong conclusions.
Data representation
Decision table
In DRSA, data are often presented using a particular form of decision table. Formally, a DRSA decision table is a 4-tuple , where is a finite set of objects, is a finite set of criteria, where is the domain of the criterion and is an information function such that for every . The set is divided into condition criteria (set ) and the decision criterion (class) . Notice, that is |
https://en.wikipedia.org/wiki/Chromatic%20spectral%20sequence | In mathematics, the chromatic spectral sequence is a spectral sequence, introduced by , used for calculating the initial term of the Adams spectral sequence for Brown–Peterson cohomology, which is in turn used for calculating the stable homotopy groups of spheres.
See also
Chromatic homotopy theory
Adams-Novikov spectral sequence
p-local spectrum
References
Spectral sequences |
https://en.wikipedia.org/wiki/May%20spectral%20sequence | In mathematics, the May spectral sequence is a spectral sequence, introduced by . It is used for calculating the initial term of the Adams spectral sequence, which is in turn used for calculating the stable homotopy groups of spheres. The May spectral sequence is described in detail in .
References
.
Spectral sequences |
https://en.wikipedia.org/wiki/Harley%20Flanders | Harley M. Flanders (September 13, 1925 – July 26, 2013) was an American mathematician, known for several textbooks and contributions to his fields: algebra and algebraic number theory, linear algebra, electrical networks, scientific computing.
Life
Flanders was a sophomore calculus student of Lester R. Ford at the Illinois Institute of Technology and asked for more challenging reading. Ford recommended A Course in Mathematical Analysis by Edouard Goursat, translated by Earle Hedrick, which included challenging exercises. Flanders recalled in 2001 that the final exercise required a proof of a formula for the derivatives of a composite function, generalizing the chain rule, in a form now called the Faa di Bruno formula.
Flanders received his bachelors (1946), masters (1947) and PhD (1949) at the University of Chicago on the dissertation Unification of class field theory advised by Otto Schilling and André Weil.
He held the Bateman Fellowship at Caltech. He joined the faculty at University of California at Berkeley. In 1955 Flanders heard Charles Loewner speak there on continuous groups. Notes were taken and the lectures appeared in a limited form with the expectation that Loewner would produce a book on the topic. With his death in 1968 the notes drew the attention of Murray H. Protter and Flanders. They edited Loewner's talks and in 1971 The MIT Press published Charles Loewner: Theory of Continuous Groups. The book was re-issued in 2008.
Teaching posts Flanders held included the faculty at Purdue University (1960), Tel Aviv University (1970–77), visiting professor at Georgia Tech (1977–78), visiting scholar at Florida Atlantic University (1978–85), University of Michigan, Ann Arbor (1985–97, 2000–), University of North Florida (1997–2000) and, distinguished mathematician in residence at Jacksonville University (1997–2000).
Flanders was Editor-in-Chief, American Mathematical Monthly, 1969–1973. He also wrote calculus software MicroCalc, ver 1–7 (1975–).
In 1991 Flanders was invited to the first SIAM workshop on automatic differentiation, held in Breckenridge, Colorado. Flanders' chapter in the Proceedings is titled "Automatic differentiation of composite functions". He presented an algorithm inputting two n-vectors of (higher) derivatives of F and G at a point, which used the chain rule to construct a linear transformation producing the derivative of the composite F o G. With prompting from editor Griewank, Flanders included application of the algorithm to automatic differentiation of implicit functions. Recalling his early exposure to the formula of Faa di Bruno, Flanders wrote, "I think Faa's formula is quite inefficient for the practical computation of numerical (not symbolic) derivatives."
Harley Flanders died July 26, 2013, in Ann Arbor, Michigan.
Differential forms
Flanders is known for advancing an approach to multivariate calculus that is independent of coordinates through treatment of differential forms.
According to Shiing-Shen Ch |
https://en.wikipedia.org/wiki/One-way%20analysis%20of%20variance | In statistics, one-way analysis of variance (or one-way ANOVA) is a technique to compare whether two samples' means are significantly different (using the F distribution). This analysis of variance technique requires a numeric response variable "Y" and a single explanatory variable "X", hence "one-way".
The ANOVA tests the null hypothesis, which states that samples in all groups are drawn from populations with the same mean values. To do this, two estimates are made of the population variance. These estimates rely on various assumptions (see below). The ANOVA produces an F-statistic, the ratio of the variance calculated among the means to the variance within the samples. If the group means are drawn from populations with the same mean values, the variance between the group means should be lower than the variance of the samples, following the central limit theorem. A higher ratio therefore implies that the samples were drawn from populations with different mean values.
Typically, however, the one-way ANOVA is used to test for differences among at least three groups, since the two-group case can be covered by a t-test (Gosset, 1908). When there are only two means to compare, the t-test and the F-test are equivalent; the relation between ANOVA and t is given by F = t2. An extension of one-way ANOVA is two-way analysis of variance that examines the influence of two different categorical independent variables on one dependent variable.
Assumptions
The results of a one-way ANOVA can be considered reliable as long as the following assumptions are met:
Response variable residuals are normally distributed (or approximately normally distributed).
Variances of populations are equal.
Responses for a given group are independent and identically distributed normal random variables (not a simple random sample (SRS)).
If data are ordinal, a non-parametric alternative to this test should be used such as Kruskal–Wallis one-way analysis of variance. If the variances are not known to be equal, a generalization of 2-sample Welch's t-test can be used.
Departures from population normality
ANOVA is a relatively robust procedure with respect to violations of the normality assumption.
The one-way ANOVA can be generalized to the factorial and multivariate layouts, as well as to the analysis of covariance.
It is often stated in popular literature that none of these F-tests are robust when there are severe violations of the assumption that each population follows the normal distribution, particularly for small alpha levels and unbalanced layouts. Furthermore, it is also claimed that if the underlying assumption of homoscedasticity is violated, the Type I error properties degenerate much more severely.
However, this is a misconception, based on work done in the 1950s and earlier. The first comprehensive investigation of the issue by Monte Carlo simulation was Donaldson (1966). He showed that under the usual departures (positive skew, unequal variances) "the F-te |
https://en.wikipedia.org/wiki/Australian%20Mathematical%20Sciences%20Institute | The Australian Mathematical Sciences Institute (AMSI) was established in 2002 for collaboration in the mathematical sciences to strengthen mathematics and statistics, especially in universities.
The Fields Institute and the Pacific Institute for the Mathematical Sciences have influenced AMSI's structure and operations.
AMSI has a membership that includes most Australian universities, CSIRO, the Australian Bureau of Statistics, the Bureau of Meteorology and the Defence Science and Technology Organisation. AMSI is located at Monash University.
Activities
AMSI pursues its mission through its three key program areas:
School Education
Research & Higher Education
Industry, Business & Government
School Education Program
AMSI's School Education program was established in 2004 under the International Centre of Excellence for Education in Mathematics (ICE-EM). Through ICE-EM, a sequence of mathematics texts, teacher resources, and professional development for school years 5–10 were developed.
In 2009, the Department of Education, Employment and Workplace Relations provided funding for the extension of ICE-EM activities under the Improving Mathematics Education in Schools (TIMES) project. This funded an expansion of the teaching professional development program across Australia, the development of teacher resource modules for years 5–10, and Maths: Make your career count—a suite of materials to promote careers in mathematics.
In work by Frank Barrington and Peter Brown, ICE-EM collected and published data on national enrolments in mathematics at year 12 and made a careful state-by-state comparison of year 12 curricula.
Research & Higher Education Program
Research
The AMSI Research Program expands and improves the mathematical sciences research base in Australia. The program promotes collaboration between member institutions and with international researchers and gives students at member institutions networking opportunities.
AMSI provides workshop sponsorship allocated through its Scientific Advisory Committee to AMSI members. AMSI also sponsors annual AMSI Lecturers and the Australian MS Mahler Lecturer.
Funding from the Department of Education, Science and Training enabled the establishment of the AMSI Access Grid Room (AGR) network. The AGR network facilitates distributed lectures, teaching, and research. A national program of collaborative teaching of advanced mathematics at honors level at multiple remote sites is now established.
Industry, Business and Government Program
AMSI highlights the relevance of mathematics to industry through wide-ranging industry-linked activities including:
AMSI's internship program
Industry workshops
Mathematical and statistical consulting services
The activities showcase the benefits of using mathematical tools in business, industry and government.
AMSI Intern
AMSI Intern is a national program that links postgraduate students and their university supervisors across all disciplines with industry pa |
https://en.wikipedia.org/wiki/Norman%20E.%20Gibbs | Norman E. Gibbs (November 27, 1941 – April 25, 2002) was an American software engineer, scholar and educational leader.
He studied to a B.Sc. in mathematics at Ursinus College (1964) and M.Sc. (1966) and Ph.D. (1969) in Computer Science at Purdue University, advised by Robert R. Korfhage. His research area was cycle generation, an area in graph theory.
Gibbs joined the faculty at Bowdoin College in Maine, Arizona State University and College of William and Mary (mathematics) in Virginia before moving to Pittsburgh, joining Carnegie Mellon University as professor of computer science and becoming the first director of the educational program at the Software Engineering Institute (1987–97). Since then he was chief information officer at Guilford College in Greensboro and University of Connecticut, jointly serving as professor of Operations and Information management. He eventually worked for Ball State University as chair of computer science (2000–02).
Articles
A cycle generation algorithm for finite undirected linear graphs, in Jnl. of the ACM, 16(4):564-68, 1969.
Tridiagonalization by permutations, in Comm. of the ACM, 17(1):20-24, 1974 (with William G. Poole, jr.)
Basic cycle generation, in Comm. of the ACM, 18(5):275-76, 1975
An Algorithm for Reducing the Bandwidth and Profile of a Sparse Matrix, in SIAM Jnr. of Numerical Analysis, 13(2):236-250, 1976 (with W. G. Poole and Paul K. Stockmeyer
A hybrid profile reduction algorithm, ACM Trans. on Math. Softw., 2(4):378-387, 1976
An introductory computer science course for all majors, ACM SIGCSE, 9(3):34-38, 1977
A model curriculum for a liberal arts degree in computer science, Comm. of the ACM, 29(3):202-210, 1986 (with Allen B. Tucker)
A Master of Software Engineering Curriculum: Recommendations from the Software Engineering Institute, IEEE Computer, 22(9):59-71, 1989 (With Gary A. Ford)
Software Engineering and Computer Science: The Impending Split?, in Educ. & Computing. 7(1-2):111-17, 1991
Books
Principles of data structures and algorithms with Pascal (William C. Brown Publ.,1987). With Robert R. Korfhage
Software Engineering Education: The Educational Needs of the Software Community (editor, with Richard E. Farley, 1987)
References
1941 births
2002 deaths
American computer scientists
20th-century American mathematicians
21st-century American mathematicians
20th-century American educators
Carnegie Mellon University faculty
Purdue University alumni
Software engineering researchers
Ursinus College alumni
Guilford College faculty
Chief information officers
Computer science educators |
https://en.wikipedia.org/wiki/Crime%20in%20India | Crime in India has been recorded since the British Raj, with comprehensive statistics now compiled annually by the National Crime Records Bureau (NCRB), under the Ministry of Home Affairs (India).
In 2021, a total of 60,96,310 crimes, comprising 36,63,360 Indian Penal Code (IPC) crimes and 24,32,950 Special and Local Laws (SLL) crimes were registered nationwide. It is a 7.65% annual decrease from 66,01,285 crimes in 2020; the crime rate (per 100,000 people) has decreased from 487.8 in 2020 to 445.9 in 2021, but still significantly higher from 385.5 in 2019. In 2021, offences affecting the human body contributed 30%, offences against property contributed 20.8%, and miscellaneous IPC crimes contributed 29.7% of all cognizable IPC crimes. Murder rate was 2.1 per 100,000, kidnapping rate was 7.4 per 100,000, and rape rate was 4.8 per 100,000 in 2021. According to the UN, the homicide rate was 2.95 per 100,000 in 2020 with 40,651 recorded, down from a peak of 5.46 per 100,000 in 1992 and essentially unchanged since 2017, higher than most countries in Asia and Europe and lower than most in the Americas and Africa although numerically one of the highest due to the large population.
Investigation rate is calculated as all cases disposed, quashed or withdrawn by police as a percentage of total cases available for investigation. The investigation rate of IPC crimes in India was 64.9% in 2021. Charge-sheeting rate is calculated as all cases, where charges were framed against accused, as a percentage of total cases disposed after investigation. The charge-sheeting rate of IPC crimes in India was 72.3% in 2021. Conviction rate is calculated as all cases, where accused was convicted by court after completion of a trial, as a percentage of total cases where trial was completed. The conviction rate of IPC crimes in India was 57.0% in 2021. In 2021, 51,540 murders were under investigation by police, of which charges were framed in 26,382; and 46,127 rapes were under investigation by police, of which charges were framed in 26,164. In 2021, 2,48,731 murders were under trial in courts, of which conviction was given in 4,304; and 1,85,836 rapes were under trial in courts, of which conviction was given in 3,368. The murder conviction rate was 42.4 and the rape conviction rate was 28.6 in 2021.
Over time
A report published by the NCRB compared the crime rates of 1953 and 2006. The report noted that burglary (known as house-breaking in India) declined over a period of 53 years by 79.84% (from 147,379, a rate of 39.3/100,000 in 1953 to 91,666, a rate of 7.9/100,000 in 2006), murder has increased by 7.39% (from 9,803, a rate of 2.61 in 1953 to 32,481, a rate of 2.81/100,000 in 2006).
Kidnapping increased by 47.80% (from 5,261, a rate of 1.40/100,000 in 1953 to 23,991, a rate of 2.07/100,000 in 2006), robbery declined by 28.85% (from 8,407, rate of 2.24/100,000 in 1953 to 18,456, rate of 1.59/100,000 in 2006) and riots have declined by 10.58% (from 20,529, a rate of 5 |
https://en.wikipedia.org/wiki/Power%20center | Power center may refer to:
Power center (geometry), the intersection point of the three radical axes of the pairs of circles
Power center (retail), an unenclosed shopping center with to of gross leasable area
See also
Power station |
https://en.wikipedia.org/wiki/List%20of%20things%20named%20after%20Paul%20Erd%C5%91s | The following are named after Paul Erdös:
Paul Erdős Award of the World Federation of National Mathematics Competitions
Erdős Prize
Erdős Lectures
Erdős number
Erdős cardinal
Erdős–Nicolas number
Erdős conjecture — a list of numerous conjectures named after Erdős; See also List of conjectures by Paul Erdős.
Erdős–Turán conjecture on additive bases
Erdős conjecture on arithmetic progressions
Erdős discrepancy problem
Erdős distinct distances problem
Burr–Erdős conjecture
Cameron–Erdős conjecture
Erdős–Faber–Lovász conjecture
Erdős–Graham conjecture — see Erdős–Graham problem
Erdős–Hajnal conjecture
Erdós Institute
Erdős–Gyárfás conjecture
Erdős–Straus conjecture
Erdős sumset conjecture
Erdős–Szekeres conjecture
Erdős–Turán conjecture
Erdős–Turán conjecture on additive bases
Copeland–Erdős constant
Erdős–Tenenbaum–Ford constant
Erdős–Bacon number
Erdős–Borwein constant
Erdős–Diophantine graph
Erdős–Mordell inequality
Chung–Erdős inequality
Erdős–Rényi model
Erdős space
Erdős theorems
de Bruijn–Erdős theorem (graph theory)
de Bruijn–Erdős theorem (incidence geometry)
Davenport–Erdős theorem
Erdős–Anning theorem
Erdős–Beck theorem
Erdős–Dushnik–Miller theorem
Erdős–Fuchs theorem
Erdős–Gallai theorem
Erdős–Ginzburg–Ziv theorem
Erdős–Kac theorem
Erdős–Ko–Rado theorem
Erdős–Nagy theorem
Erdős–Pósa theorem
Erdős–Rado theorem
Erdős–Stone theorem
Erdős–Szekeres theorem
Erdős–Szemerédi theorem
Erdős–Tetali theorem
Erdős–Wintner theorem
Erdős–Turán inequality
Erdős–Ulam problem
Erdős–Woods number
Hsu–Robbins–Erdős theorem
Erdős arcsine law
Erdős–Moser equation
Erdős Website — Collection of Mathematical Problems.
Erdos
Paul Erdős |
https://en.wikipedia.org/wiki/Gamow%20factor | The Gamow factor, Sommerfeld factor or Gamow–Sommerfeld factor, named after its discoverer George Gamow or after Arnold Sommerfeld, is a probability factor for two nuclear particles' chance of overcoming the Coulomb barrier in order to undergo nuclear reactions, for example in nuclear fusion. By classical physics, there is almost no possibility for protons to fuse by crossing each other's Coulomb barrier at temperatures commonly observed to cause fusion, such as those found in the sun. When George Gamow instead applied quantum mechanics to the problem, he found that there was a significant chance for the fusion due to tunneling.
The probability of two nuclear particles overcoming their electrostatic barriers is given by the following equation:
where is the Gamow energy,
Here, is the reduced mass of the two particles. The constant is the fine structure constant, is the speed of light, and and are the respective atomic numbers of each particle.
While the probability of overcoming the Coulomb barrier increases rapidly with increasing particle energy, for a given temperature, the probability of a particle having such an energy falls off very fast, as described by the Maxwell–Boltzmann distribution. Gamow found that, taken together, these effects mean that for any given temperature, the particles that fuse are mostly in a temperature-dependent narrow range of energies known as the Gamow window.
Derivation
Gamow first solved the one-dimensional case of quantum tunneling using the WKB approximation. Considering a wave function of a particle of mass m, we take area 1 to be where a wave is emitted, area 2 the potential barrier which has height V and width l (at ), and area 3 its other side, where the wave is arriving, partly transmitted and partly reflected. For a wave number k and energy E we get:
where and .
This is solved for given A and α by taking the boundary conditions at the both barrier edges, at and , where both and its derivative must be equal on both sides.
For , this is easily solved by ignoring the time exponential and considering the real part alone (the imaginary part has the same behavior). We get, up to factors depending on the phases which are typically of order 1, and up to factors of the order of (assumed not very large, since V is greater than E not marginally):
Next Gamow modeled the alpha decay as a symmetric one-dimensional problem, with a standing wave between two symmetric potential barriers at and , and emitting waves at both outer sides of the barriers.
Solving this can in principle be done by taking the solution of the first problem, translating it by and gluing it to an identical solution reflected around .
Due to the symmetry of the problem, the emitting waves on both sides must have equal amplitudes (A), but their phases (α) may be different. This gives a single extra parameter; however, gluing the two solutions at requires two boundary conditions (for both the wave function and its derivative), |
https://en.wikipedia.org/wiki/Capacity%20of%20a%20set | In mathematics, the capacity of a set in Euclidean space is a measure of the "size" of that set. Unlike, say, Lebesgue measure, which measures a set's volume or physical extent, capacity is a mathematical analogue of a set's ability to hold electrical charge. More precisely, it is the capacitance of the set: the total charge a set can hold while maintaining a given potential energy. The potential energy is computed with respect to an idealized ground at infinity for the harmonic or Newtonian capacity, and with respect to a surface for the condenser capacity.
Historical note
The notion of capacity of a set and of "capacitable" set was introduced by Gustave Choquet in 1950: for a detailed account, see reference .
Definitions
Condenser capacity
Let Σ be a closed, smooth, (n − 1)-dimensional hypersurface in n-dimensional Euclidean space ℝn, n ≥ 3; K will denote the n-dimensional compact (i.e., closed and bounded) set of which Σ is the boundary. Let S be another (n − 1)-dimensional hypersurface that encloses Σ: in reference to its origins in electromagnetism, the pair (Σ, S) is known as a condenser. The condenser capacity of Σ relative to S, denoted C(Σ, S) or cap(Σ, S), is given by the surface integral
where:
u is the unique harmonic function defined on the region D between Σ and S with the boundary conditions u(x) = 1 on Σ and u(x) = 0 on S;
S′ is any intermediate surface between Σ and S;
ν is the outward unit normal field to S′ and
is the normal derivative of u across S′; and
σn = 2πn⁄2 ⁄ Γ(n ⁄ 2) is the surface area of the unit sphere in ℝn.
C(Σ, S) can be equivalently defined by the volume integral
The condenser capacity also has a variational characterization: C(Σ, S) is the infimum of the Dirichlet's energy functional
over all continuously-differentiable functions v on D with v(x) = 1 on Σ and v(x) = 0 on S.
Harmonic/Newtonian capacity
Heuristically, the harmonic capacity of K, the region bounded by Σ, can be found by taking the condenser capacity of Σ with respect to infinity. More precisely, let u be the harmonic function in the complement of K satisfying u = 1 on Σ and u(x) → 0 as x → ∞. Thus u is the Newtonian potential of the simple layer Σ. Then the harmonic capacity (also known as the Newtonian capacity) of K, denoted C(K) or cap(K), is then defined by
If S is a rectifiable hypersurface completely enclosing K, then the harmonic capacity can be equivalently rewritten as the integral over S of the outward normal derivative of u:
The harmonic capacity can also be understood as a limit of the condenser capacity. To wit, let Sr denote the sphere of radius r about the origin in ℝn. Since K is bounded, for sufficiently large r, Sr will enclose K and (Σ, Sr) will form a condenser pair. The harmonic capacity is then the limit as r tends to infinity:
The harmonic capacity is a mathematically abstract version of the electrostatic capacity of the conductor K and is always non-negative and finite: 0 ≤ C(K) < +∞.
Generali |
https://en.wikipedia.org/wiki/Apply | In mathematics and computer science, apply is a function that applies a function to arguments. It is central to programming languages derived from lambda calculus, such as LISP and Scheme, and also in functional languages. It has a role in the study of the denotational semantics of computer programs, because it is a continuous function on complete partial orders. Apply is also a continuous function in homotopy theory, and, indeed underpins the entire theory: it allows a homotopy deformation to be viewed as a continuous path in the space of functions. Likewise, valid mutations (refactorings) of computer programs can be seen as those that are "continuous" in the Scott topology.
The most general setting for apply is in category theory, where it is right adjoint to currying in closed monoidal categories. A special case of this are the Cartesian closed categories, whose internal language is simply typed lambda calculus.
Programming
In computer programming, apply applies a function to a list of arguments. Eval and apply are the two interdependent components of the eval-apply cycle, which is the essence of evaluating Lisp, described in SICP. Function application corresponds to beta reduction in lambda calculus.
Apply function
Apply is also the name of a special function in many languages, which takes a function and a list, and uses the list as the function's own argument list, as if the function were called with the elements of the list as the arguments. This is important in languages with variadic functions, because this is the only way to call a function with an indeterminate (at compile time) number of arguments.
Common Lisp and Scheme
In Common Lisp apply is a function that applies a function to a list of arguments (note here that "+" is a variadic function that takes any number of arguments):
(apply #'+ (list 1 2))
Similarly in Scheme:
(apply + (list 1 2))
C++
In C++, Bind is used either via the std namespace or via the boost namespace.
C# and Java
In C# and Java, variadic arguments are simply collected in an array. Caller can explicitly pass in an array in place of the variadic arguments. This can only be done for a variadic parameter. It is not possible to apply an array of arguments to non-variadic parameter without using reflection. An ambiguous case arises should the caller want to pass an array itself as one of the arguments rather than using the array as a list of arguments. In this case, the caller should cast the array to Object to prevent the compiler from using the apply interpretation.
variadicFunc(arrayOfArgs);With version 8 lambda expressions were introduced. Functions are implemented as objects with a functional interface, an interface with only one non-static method. The standard interface
Function<T,R>
consist of the method (plus some static utility functions):
R apply(T para)
Go
In Go, typed variadic arguments are simply collected in a slice. The caller can explicitly pass in a slice in place of the variadic arguments |
https://en.wikipedia.org/wiki/Henryk%20Ross | Henryk Ross (1 May 1910 1991) was a Polish Jewish photographer who was employed by the Department of Statistics for the Jewish Council within the Łódź Ghetto during the Holocaust in occupied Poland.
About
Ross was born in 1910. Ross was a sports photographer for a Warsaw newspaper, prior to World War II.
Starting in 1940, Ross had been employed by the Department of Statistics for the Jewish Council within the Łódź Ghetto during the Holocaust in occupied Poland. Daringly, working as staff photographer, Ross also documented Nazi atrocities (such as public hangings) while remaining officially in the good graces of the German occupational administration.
Part of his official duties was taking identity photographs. He constructed a three level stage in his studio that let him photograph up to twelve people with a single negative. While the authorities only supplied him enough film for assigned work, this trick allowed him extra film he could use for unauthorized photography.
His unofficial images covered scenes from daily life, communal celebrations, children digging for scraps of food and large groups of Jews being led to deportation and being loaded into box cars.
In 1944, the Nazis started the liquidation of the Lodz ghetto and the deportation of the remaining Jews from it to Chelmno and Auschwitz. In the fall of 1944, Ross buried his photos and negatives in a box, hoping they might survive as a historical record. 800 Jews, including Henryk Ross, were temporary left in the ghetto in order to clean it (8 mass graves were already prepared for them). But the Nazis did not manage to murder them before the liberation of the ghetto by the Red Army in January 1945. Ross was able to dig up the box with the photos. Much of his material was damaged or destroyed by water; still, about half of his 6,000 images survived.
Ross in 1950 emigrated to Israel and testified during the 1961 trial of Adolf Eichmann. He died in 1991.
Legacy
In 2021, a gift of 48 of Ross's silver gelatin prints were given to the Museum of Fine Arts, Boston (MFA), which makes it one of the few US museums to own work by Ross.
References
Sources
Links
The Lodz Ghetto Photographs of Henryk Ross
Photographers from Warsaw
1910 births
1991 deaths
Łódź Ghetto inmates
Holocaust photographers
Adolf Eichmann
Polish Holocaust survivors
Polish emigrants to Israel |
https://en.wikipedia.org/wiki/Lajos%20Tak%C3%A1cs | Lajos Takács (August 21, 1924 (Maglód) – December 4, 2015) was a Hungarian mathematician, known for his contributions to probability theory and in particular, queueing theory. He wrote over two hundred scientific papers and six books.
He studied at the Technical University of Budapest (1943-1948), taking courses with Charles Jordan and received an M.S. for his dissertation On a Probability-theoretical Investigation of Brownian Motion (1948). From 1945-48 he was a student assistant to Professor Zoltán Bay and participated in his famous experiment of receiving microwave echoes from the Moon (1946). In 1957 he received the Academic Doctor's Degree in Mathematics for his thesis entitled "Stochastic processes arising in the theory of particle counters" (1957).
He worked as a mathematician at the Tungsram Research Laboratory (1948–55), the Research Institute for Mathematics of the Hungarian Academy of Sciences (1950–58) and was an associate professor in the Department of Mathematics of the L. Eötvös University (1953–58). He was the first to introduce semi-Markov processes in queueing theory.
He took a lecturing appointment at Imperial College in London and London School of Economics (1958), before moving to Columbia University in New York City (1959–66) and Case Western Reserve University in Cleveland (1966–87), advising over twenty Ph.D.-theses. He also held visiting appointments at Bell Labs and IBM Research, had sabbaticals at Stanford University (1966). He was a Professor of Statistics and Probability at Case Western Reserve University from 1966 until he retired as Professor Emeritus in 1987.
Takács was married to Dalma Takács, author and professor of English Literature at Notre Dame College of Ohio. He had two daughters, contemporary figurative realist artist, Judy Takács and Susan, a legal assistant.
Publications
The following is a partial list of publications
Some Investigations Concerning Recurrent Stochastic Processes of a Certain Kind, Magyar Tud. Akad. Alk. Mat.Int. Kozl. vol.3, pp. 115–128, 1954.
Investigations of Waiting Time Problems by Reduction to Markov Processes, Acta Math. Acad. Sci. Hung. vol.6, pp. 101–129, 1955.
Sojourn times for the Brownian motion, Journal of Applied Mathematics and Stochastic Analysis, vol. 11, no. 3, pp. 231–246, 1998
In memoriam: Pál Erdős (1913-1996), Journal of Applied Mathematics and Stochastic Analysis, vol. 9, no. 4, pp. 563–564, 1996
Sojourn times, Journal of Applied Mathematics and Stochastic Analysis, vol. 9, no. 4, pp. 415–426, 1996
Brownian local times, Journal of Applied Mathematics and Stochastic Analysis, vol. 8, no. 3, pp. 209–232, 1995
Limit distributions for queues and random rooted trees, Journal of Applied Mathematics and Stochastic Analysis, vol. 6, no. 3, pp. 189–216, 1993
On a probability problem connected with railway traffic, Journal of Applied Mathematics and Stochastic Analysis, vol. 4, no. 1, pp. 1–27, 1991
Conditional limit theorems for branching process |
https://en.wikipedia.org/wiki/Dirichlet%20energy | In mathematics, the Dirichlet energy is a measure of how variable a function is. More abstractly, it is a quadratic functional on the Sobolev space . The Dirichlet energy is intimately connected to Laplace's equation and is named after the German mathematician Peter Gustav Lejeune Dirichlet.
Definition
Given an open set and a function the Dirichlet energy of the function is the real number
where denotes the gradient vector field of the function .
Properties and applications
Since it is the integral of a non-negative quantity, the Dirichlet energy is itself non-negative, i.e. for every function .
Solving Laplace's equation for all , subject to appropriate boundary conditions, is equivalent to solving the variational problem of finding a function that satisfies the boundary conditions and has minimal Dirichlet energy.
Such a solution is called a harmonic function and such solutions are the topic of study in potential theory.
In a more general setting, where is replaced by any Riemannian manifold , and is replaced by for another (different) Riemannian manifold , the Dirichlet energy is given by the sigma model. The solutions to the Lagrange equations for the sigma model Lagrangian are those functions that minimize/maximize the Dirichlet energy. Restricting this general case back to the specific case of just shows that the Lagrange equations (or, equivalently, the Hamilton–Jacobi equations) provide the basic tools for obtaining extremal solutions.
See also
Dirichlet's principle
Dirichlet eigenvalue
Total variation
Oscillation
Harmonic map
References
Calculus of variations
Partial differential equations |
https://en.wikipedia.org/wiki/12th%20of%20Never | 12th of Never may refer to:
12th of Never (novel), a 2013 novel by James Patterson
"The Twelfth of Never", a song by Johnny Mathis
Twelfth of Never, an idiom of improbability |
https://en.wikipedia.org/wiki/Arthur%20J.%20Lohwater | Arthur John "Jack" Lohwater (October 20, 1922 - June 10, 1982) was an American mathematician.
He obtained a Ph.D. in mathematics at University of Rochester (1951), on the dissertation The Boundary Values of a Class of Analytic Functions, advised by Wladimir Seidel. Later he joined the faculty at University of Michigan and Case Western Reserve University. He was editor of Mathematical Reviews (1962–65). With Norman Steenrod and Sydney Gould he established important ties with Russian mathematicians, beginning with conferences in Moscow (1956, 58) and
resulting in a dictionary. Lohwater died after a long battle with lung cancer. He was married to the mathematician Marjorie White Lohwater (1925–2007).
Books
Русско-английский словарь математических терминов. (Russian-English Dictionary of the Mathematical Sciences) (American Mathematical Society, 1961). The inverse was published by Soviet Academy of Sciences (1961).
The theory of cluster sets (Cambridge University Press, 1966). With Edward Collingwood.
Global Differentiable Dynamics, Proceedings of the Conference Held at Case Western Reserve University, Cleveland, Ohio, June 2–6, 1969. With Otomar Hájek and Roger C. McCann (editors)
Translations
Publications
Mathematics in the Soviet Union, in Science 17 May 1957: 974-978
The boundary behaviour of analytic functions, in Itogi Nauki i Techniki, Mat. Anal., 10:99-259, 1973
"Introduction to Inequalities", 1982 (unpublished, reproduced with permission of Marjorie Lohwater) used in the "Introduction to Inequalities" course taught by Lohwater.
Awards
Guggenheim fellowship 1955 (mathematics)
References
20th-century American mathematicians
University of Rochester alumni
University of Michigan faculty
Case Western Reserve University faculty
1922 births
1982 deaths |
https://en.wikipedia.org/wiki/Analytic%20torsion | In mathematics, Reidemeister torsion (or R-torsion, or Reidemeister–Franz torsion) is a topological invariant of manifolds introduced by Kurt Reidemeister for 3-manifolds and generalized to higher dimensions by and .
Analytic torsion (or Ray–Singer torsion) is an invariant of Riemannian manifolds defined by as an analytic analogue of Reidemeister torsion. and proved Ray and Singer's conjecture that Reidemeister torsion and analytic torsion are the same for compact Riemannian manifolds.
Reidemeister torsion was the first invariant in algebraic topology that could distinguish between closed manifolds which are homotopy equivalent but not homeomorphic, and can thus be seen as the birth of geometric topology as a distinct field. It can be used to classify lens spaces.
Reidemeister torsion is closely related to Whitehead torsion; see . It has also given some important motivation to arithmetic topology; see . For more recent work on torsion see the books and .
Definition of analytic torsion
If M is a Riemannian manifold and E a vector bundle over M, then there is a Laplacian operator acting on the k-forms with values in E. If the eigenvalues on k-forms are λj then the zeta function ζk is defined to be
for s large, and this is extended to all complex s by analytic continuation.
The zeta regularized determinant of the Laplacian acting on k-forms is
which is formally the product of the positive eigenvalues of the laplacian acting on k-forms.
The analytic torsion T(M,E) is defined to be
Definition of Reidemeister torsion
Let be a finite connected CW-complex with fundamental group
and universal cover , and let be an orthogonal finite-dimensional -representation. Suppose that
for all n. If we fix a cellular basis for and an orthogonal -basis for , then is a contractible finite based free -chain complex. Let be any chain contraction of D*, i.e. for all . We obtain an isomorphism with , . We define the Reidemeister torsion
where A is the matrix of with respect to the given bases. The Reidemeister torsion is independent of the choice of the cellular basis for , the orthogonal basis for and the chain contraction .
Let be a compact smooth manifold, and let be a unimodular representation. has a smooth triangulation. For any choice of a volume , we get an invariant . Then we call the positive real number the Reidemeister torsion of the manifold with respect to and .
A short history of Reidemeister torsion
Reidemeister torsion was first used to combinatorially classify 3-dimensional lens spaces in by Reidemeister, and in higher-dimensional spaces by Franz. The classification includes examples of homotopy equivalent 3-dimensional manifolds which are not homeomorphic — at the time (1935) the classification was only up to PL homeomorphism, but later showed that this was in fact a classification up to homeomorphism.
J. H. C. Whitehead defined the "torsion" of a homotopy equivalence between finite complexes. This is a direct generali |
https://en.wikipedia.org/wiki/Judith%20Grabiner | Judith Victor Grabiner (born October 12, 1938) is an American mathematician and historian of mathematics, who is Flora Sanborn Pitzer Professor Emerita of Mathematics at Pitzer College, one of the Claremont Colleges. Her main interest is in mathematics in the eighteenth and nineteenth centuries.
Education
Grabiner completed a Bachelor of Science degree at the University of Chicago in 1960. She was a graduate student in the history of science at Harvard University, completing a Master of Arts in 1962 and a Ph.D. in 1966, under I. Bernard Cohen. Her PhD dissertation was on Italian mathematician Joseph-Louis Lagrange.
Career
Grabiner was an instructor at Harvard for several years, before she and her husband Sandy Grabiner moved to California. She was a professor of history at California State University, Dominguez Hills from 1972 to 1985.
Grabiner joined the mathematics department at Pitzer College in 1985, and has been the Flora Sanborn Pitzer Professor of Mathematics since 1994. Her teaching includes courses on the history of mathematics, mathematics in different cultures, and mathematics and philosophy.
Recognition
Grabiner received the Carl B. Allendoerfer Award for the best article in Mathematics Magazine in 1984, 1989, and 1996, and the Lester R. Ford Award in 1984, 1998, 2005, and 2010, for the best article in American Mathematical Monthly.
In 2003, Grabiner received one of the Mathematical Association of America's Deborah and Franklin Haimo Awards for Distinguished College or University Teaching of Mathematics. She became a fellow of the American Mathematical Society in 2012. In 2014, she was awarded the Beckenbach Book Prize.
She was the 2021 winner of the Albert Leon Whiteman Memorial Prize of the American Mathematical Society "for her outstanding contributions to the history of mathematics, in particular her works on Cauchy, Lagrange, and MacLaurin; her widely-recognized gift for expository writing; and a distinguished career of teaching, lecturing, and numerous publications promoting a better understanding of mathematics and the significant roles it plays in culture generally".
Books
References
External links
1938 births
Living people
American historians of mathematics
University of Chicago alumni
Harvard University alumni
Women mathematicians
Fellows of the American Mathematical Society |
https://en.wikipedia.org/wiki/Bayes%20linear%20statistics | Bayes linear statistics is a subjectivist statistical methodology and framework. Traditional subjective Bayesian analysis is based upon fully specified probability distributions, which are very difficult to specify at the necessary level of detail. Bayes linear analysis attempts to solve this problem by developing theory and practise for using partially specified probability models. Bayes linear in its current form has been primarily developed by Michael Goldstein. Mathematically and philosophically it extends Bruno de Finetti's Operational Subjective approach to probability and statistics.
Motivation
Consider first a traditional Bayesian Analysis where you expect to shortly know D and you would like to know more about some other observable B. In the traditional Bayesian approach it is required that every possible outcome is enumerated i.e. every possible outcome is the cross product of the partition of a set of B and D. If represented on a computer where B requires n bits and D m bits then the number of states required is . The first step to such an analysis is to determine a person's subjective probabilities e.g. by asking about their betting behaviour for each of these outcomes. When we learn D conditional probabilities for B are determined by the application of Bayes' rule.
Practitioners of subjective Bayesian statistics routinely analyse datasets where the size of this set is large enough that subjective probabilities cannot be meaningfully determined for every element of D × B. This is normally accomplished by assuming exchangeability and then the use of parameterized models with prior distributions over parameters and appealing to the de Finetti's theorem to justify that this produces valid operational subjective probabilities over D × B. The difficulty with such an approach is that the
validity of the statistical analysis requires that the subjective probabilities are a good representation of an individual's beliefs however this method results in a very precise specification over D × B and it is often difficult to articulate what it would mean to adopt these belief specifications.
In contrast to the traditional Bayesian paradigm Bayes linear statistics following de Finetti uses Prevision or subjective expectation as a primitive, probability is then defined as the expectation of an indicator variable. Instead of specifying a subjective probability for every element in the partition D × B the analyst specifies subjective expectations for just a few quantities that they are interested in or feel knowledgeable about. Then instead of conditioning an adjusted expectation is computed by a rule that is a generalization of Bayes' rule that is based upon expectation.
The use of the word linear in the title refers to de Finetti's arguments that probability theory is a linear theory (de Finetti argued against the more common measure theory approach).
Example
In Bayes linear statistics, the probability model is only partially specifi |
https://en.wikipedia.org/wiki/List%20of%20people%20by%20Erd%C5%91s%20number | Paul Erdős (1913–1996) was a Hungarian mathematician. He considered mathematics to be a social activity and often collaborated on his papers, having 511 joint authors, many of whom also have their own collaborators. The Erdős number measures the "collaborative distance" between an author and Erdős. Thus, his direct co-authors have Erdős number one, theirs have number two, and so forth. Erdős himself has Erdős number zero.
There are more than 11,000 people with an Erdős number of two. This is a partial list of authors with an Erdős number of three or less. For more complete listings of Erdős numbers, see the databases maintained by the Erdős Number Project or the collaboration distance calculators maintained by the American Mathematical Society and by zbMATH.
Zero
Paul Erdős
One
A
János Aczél
Ron Aharoni
Martin Aigner
Miklós Ajtai
Leonidas Alaoglu
Yousef Alavi
Krishnaswami Alladi
Noga Alon
Nesmith Ankeny
Joseph Arkin
Boris Aronov
David Avis
B
László Babai
Frederick Bagemihl
Leon Bankoff
Paul T. Bateman
James Earl Baumgartner
Mehdi Behzad
Richard Bellman
Vitaly Bergelson
Arie Bialostocki
Andreas Blass
Ralph P. Boas Jr
Béla Bollobás
John Adrian Bondy
Joel Lee Brenner
John Brillhart
Thomas Craig Brown
W. G. Brown
Nicolaas Govert de Bruijn
R. Creighton Buck
Stefan Burr
Steve Butler
C
Neil J. Calkin
Peter Cameron
Paul A. Catlin
Gary Chartrand
Phyllis Chinn
Sarvadaman Chowla
Fan Chung
Kai Lai Chung
Václav Chvátal
Charles Colbourn
John Horton Conway
Arthur Herbert Copeland
Imre Csiszár
D
Harold Davenport
Dominique de Caen
Jean-Marie De Koninck
Jean-Marc Deshouillers
Michel Deza
Persi Diaconis
Gabriel Andrew Dirac
Jacques Dixmier
Yael Dowker
Underwood Dudley
Aryeh Dvoretzky
E
György Elekes
Peter D. T. A. Elliott
F
Vance Faber
Siemion Fajtlowicz
Ralph Faudree
László Fejes Tóth
William Feller
Peter C. Fishburn
Géza Fodor
Aviezri Fraenkel
Péter Frankl
Gregory Freiman
Wolfgang Heinrich Johannes Fuchs
Zoltán Füredi
G
Steven Gaal
Janos Galambos
Tibor Gallai
Fred Galvin
Joseph E. Gillis
Leonard Gillman
Abraham Ginzburg
Chris Godsil
Michael Golomb
Adolph Winkler Goodman
Basil Gordon
Ronald J. Gould
Ronald Graham
Sidney Graham
Andrew Granville
Peter M. Gruber
Branko Grünbaum
Hansraj Gupta
Richard K. Guy
Michael Guy
András Gyárfás
H
András Hajnal
Gábor Halász
Haim Hanani
Frank Harary
Zdeněk Hedrlín
Hans Heilbronn
Pavol Hell
Fritz Herzog
Alan J. Hoffman
Verner Emil Hoggatt Jr.
I
Albert Ingham
J
Eri Jabotinsky
Steve Jackson
Michael Scott Jacobson
Svante Janson
Vojtěch Jarník
K
Mark Kac
Paul Chester Kainen
Shizuo Kakutani
Egbert van Kampen
Irving Kaplansky
Jovan Karamata
Ke Zhao
Paul Kelly
Péter Kiss
Murray S. Klamkin
Maria Klawe
Daniel Kleitman
Yoshiharu Kohayakawa
Jurjen Ferdinand Koksma
Péter Komjáth
János Komlós
Steven G. Krantz
Michael Krivelevich
Ewa Kubicka
Kenneth Kunen
L
Jean A. Larson
Renu C. Laskar
Jos |
https://en.wikipedia.org/wiki/Seiberg%E2%80%93Witten%20invariants | In mathematics, and especially gauge theory, Seiberg–Witten invariants are invariants of compact smooth oriented 4-manifolds introduced by , using the Seiberg–Witten theory studied by during their investigations of Seiberg–Witten gauge theory.
Seiberg–Witten invariants are similar to Donaldson invariants and can be used to prove similar (but sometimes slightly stronger) results about smooth 4-manifolds. They are technically much easier to work with than Donaldson invariants; for example, the moduli spaces of solutions of the Seiberg–Witten equations tends to be compact, so one avoids the hard problems involved in compactifying the moduli spaces in Donaldson theory.
For detailed descriptions of Seiberg–Witten invariants see , , , , . For the relation to symplectic manifolds and Gromov–Witten invariants see . For the early history see .
Spinc-structures
The Spinc group (in dimension 4) is
where the acts as a sign on both factors. The group has a natural homomorphism to SO(4) = Spin(4)/±1.
Given a compact oriented 4 manifold, choose a smooth Riemannian metric with Levi Civita connection . This reduces the structure group from the connected component GL(4)+ to SO(4) and is harmless from a homotopical point of view. A Spinc-structure or complex spin structure on M is a reduction of the structure group to Spinc, i.e. a lift of the SO(4) structure on the tangent bundle to the group Spinc. By a theorem of Hirzebruch and Hopf, every smooth oriented compact 4-manifold admits a Spinc structure. The existence of a Spinc structure is equivalent to the existence of a lift of the second Stiefel–Whitney class to a class Conversely such a lift determines the Spinc structure up to 2 torsion in A spin structure proper requires the more restrictive
A Spinc structure determines (and is determined by) a spinor bundle coming from the 2 complex dimensional positive and negative spinor representation of Spin(4) on which U(1) acts by multiplication. We have . The spinor bundle comes with a graded Clifford algebra bundle representation i.e. a map such that for each 1 form we have and . There is a unique hermitian metric on s.t. is skew Hermitian for real 1 forms . It gives an induced action of the forms by anti-symmetrising. In particular this gives an isomorphism of of the selfdual two forms with the traceless skew Hermitian endomorphisms of which are then identified.
Seiberg–Witten equations
Let be the determinant line bundle with . For every connection with on , there is a unique spinor connection on i.e. a connection such that for every 1-form and vector field . The Clifford connection then defines a Dirac operator on . The group of maps acts as a gauge group on the set of all connections on . The action of can be "gauge fixed" e.g. by the condition , leaving an effective parametrisation of the space of all such connections of with a residual gauge group action.
Write for a spinor field of positive chirality, i.e. a section o |
https://en.wikipedia.org/wiki/National%20Health%20and%20Nutrition%20Examination%20Survey | The National Health and Nutrition Examination Survey (NHANES) is a survey research program conducted by the National Center for Health Statistics (NCHS) to assess the health and nutritional status of adults and children in the United States, and to track changes over time. The survey combines interviews, physical examinations and laboratory tests.
The NHANES interview includes demographic, socioeconomic, dietary, and health-related questions. The examination component consists of medical, dental, and physiological measurements, as well as laboratory tests administered by medical personnel.
The first NHANES was conducted in 1971, and in 1999 the surveys became an annual event; the first report on the topic was published in 2001.
NHANES findings are used to determine the prevalence of major diseases and risk factors for diseases. Information is used to assess nutritional status and its association with health promotion and disease prevention. NHANES findings are also the basis for national standards for such measurements as height, weight, and blood pressure. NHANES data are used in epidemiological studies and health sciences research (including biomarkers of aging), which help develop sound public health policy, direct and design health programs and services, expand health knowledge, extend healthspan and lifespan.
Follow-up studies using NHANES data were made possible by creating linked mortality files and files based on Medicare and Medicaid data.
See also
National Archive of Computerized Data on Aging
References
External links
Official website
page for NHANES 1999-2000
DSDR page for NHANES 2001-2002
DSDR page for NHANES 2003-2004
DSDR page for NHANES 2005-2006
DSDR page for NHANES 2007-2008
Validity of U.S. Nutritional Surveillance: National Health and Nutrition Examination Survey Caloric Energy Intake Data, 1971–2010
Centers for Disease Control and Prevention
Gerontology
Health surveys |
https://en.wikipedia.org/wiki/Lester%27s%20theorem | In Euclidean plane geometry, Lester's theorem states that in any scalene triangle, the two Fermat points, the nine-point center, and the circumcenter lie on the same circle.
The result is named after June Lester, who published it in 1997, and the circle through these points was called the Lester circle by Clark Kimberling.
Lester proved the result by using the properties of complex numbers; subsequent authors have given elementary proofs, proofs using vector arithmetic, and computerized proofs.
See also
Parry circle
van Lamoen circle
References
External links
Theorems about triangles and circles |
https://en.wikipedia.org/wiki/Nokomis%2C%20Saskatchewan | Nokomis is a town in the Canadian province of Saskatchewan.
Demographics
In the 2021 Census of Population conducted by Statistics Canada, Nokomis had a population of living in of its total private dwellings, a change of from its 2016 population of . With a land area of , it had a population density of in 2021.
History
1904: The area was opened to homesteading.
1906: Florence Mary Halstead established a post office on the Halstead farm and called it "Nokomis". After the Grand Trunk Pacific Railway was built, the town requested the post office be moved into town, accepting the condition of the post-mistress that the town be renamed "Nokomis". The post office was first located in Henry's Men's Clothing Store, and moved into its own building just north of the Times Office the next year.
1907: The town was named Junction City, with the hopes that it would become the largest city in western Canada
1907: The Canadian Bank of Commerce was constructed with K.W. Reikie as manager, and the Northern Crown Bank with R.S. Inkster as manager. Inkster's residence (Earl McDougall's house) was one of the first residences constructed. Others were homes of Norman Townsend and J.I. Jamieson. Ewart's hall opened above the Northern Crown Bank, and here the first schoolroom classes were held. Mabel Dobbyn, who later married K.W. Reikie of the Bank of Commerce, was the first teacher.
1908: Carloads of lumber, hardware and carpenters were arriving, and the Sash and Door Factory was kept busy. For a time the Franklin Realty Co. contemplated starting a brickyard, using the good clay of the district. Almost every train brought in new settlers, and many cars of settlers' effects. That month, the Nokomis Times building was put up on 2nd Avenue by W.C.R. Garrioch.
1908: The town was renamed Nokomis
1909: The post office was opened
1910: The Carter Land Company began purchasing land in the Nokomis district.
1912: The first coal seam was discovered south east of Nokomis in the Tate area (now known as the NSC1 Pit) (51.43935N,-104.819276W).
1914–1916: Two more mines were started, one (NCS2) and the other straight east(NCS3).
1918: Officials from Hunter Valley Coal Chain (HVCC) were sent to the Nokomis area to purchase the surface rights to as well as mineral rights
1946: After the Second World War many men returned to the area where they found work with a new oil company from the United States (ND Oil Seekers)
1947: The first well was drilled, which is known as NOW1 (Nokomis Oil Well 1). It was drilled in the formation known as the Hatfield Basin (Latitude: 51° 25' 26.117" N, Longitude: 105° 00' 47.486" W). The Hatfield Basin was mainly sweet crude oil and was extremely shallow. This made the area very popular to new oil companies.
1988: The first horizontal well was drilled in the area by the directional driller Ryan Oliver and MWD was done by Kent Ruether. This well broke many records. It was one of the fastest ever drilled, the longest ever drilled and had the b |
https://en.wikipedia.org/wiki/Imperial%2C%20Saskatchewan | Imperial is a town in the Canadian province of Saskatchewan. The town is located along Saskatchewan Highway 2.
Demographics
In the 2021 Census of Population conducted by Statistics Canada, Imperial had a population of living in of its total private dwellings, a change of from its 2016 population of . With a land area of , it had a population density of in 2021.
See also
List of communities in Saskatchewan
List of towns in Saskatchewan
Footnotes
External links
Towns in Saskatchewan
Big Arm No. 251, Saskatchewan
Division No. 11, Saskatchewan |
https://en.wikipedia.org/wiki/Even%20and%20odd%20ordinals | In mathematics, even and odd ordinals extend the concept of parity from the natural numbers to the ordinal numbers. They are useful in some transfinite induction proofs.
The literature contains a few equivalent definitions of the parity of an ordinal α:
Every limit ordinal (including 0) is even. The successor of an even ordinal is odd, and vice versa.
Let α = λ + n, where λ is a limit ordinal and n is a natural number. The parity of α is the parity of n.
Let n be the finite term of the Cantor normal form of α. The parity of α is the parity of n.
Let α = ωβ + n, where n is a natural number. The parity of α is the parity of n.
If α = 2β, then α is even. Otherwise α = 2β + 1 and α is odd.
Unlike the case of even integers, one cannot go on to characterize even ordinals as ordinal numbers of the form Ordinal multiplication is not commutative, so in general In fact, the even ordinal cannot be expressed as β + β, and the ordinal number
(ω + 3)2 = (ω + 3) + (ω + 3) = ω + (3 + ω) + 3 = ω + ω + 3 = ω2 + 3
is not even.
A simple application of ordinal parity is the idempotence law for cardinal addition (given the well-ordering theorem). Given an infinite cardinal κ, or generally any limit ordinal κ, κ is order-isomorphic to both its subset of even ordinals and its subset of odd ordinals. Hence one has the cardinal sum
References
Ordinal numbers
Parity (mathematics) |
https://en.wikipedia.org/wiki/Uniform%205-polytope | In geometry, a uniform 5-polytope is a five-dimensional uniform polytope. By definition, a uniform 5-polytope is vertex-transitive and constructed from uniform 4-polytope facets.
The complete set of convex uniform 5-polytopes has not been determined, but many can be made as Wythoff constructions from a small set of symmetry groups. These construction operations are represented by the permutations of rings of the Coxeter diagrams.
History of discovery
Regular polytopes: (convex faces)
1852: Ludwig Schläfli proved in his manuscript Theorie der vielfachen Kontinuität that there are exactly 3 regular polytopes in 5 or more dimensions.
Convex semiregular polytopes: (Various definitions before Coxeter's uniform category)
1900: Thorold Gosset enumerated the list of nonprismatic semiregular convex polytopes with regular facets (convex regular 4-polytopes) in his publication On the Regular and Semi-Regular Figures in Space of n Dimensions.
Convex uniform polytopes:
1940-1988: The search was expanded systematically by H.S.M. Coxeter in his publication Regular and Semi-Regular Polytopes I, II, and III.
1966: Norman W. Johnson completed his Ph.D. Dissertation under Coxeter, The Theory of Uniform Polytopes and Honeycombs, University of Toronto
Non-convex uniform polytopes:
1966: Johnson describes two non-convex uniform antiprisms in 5-space in his dissertation.
2000-2023: Jonathan Bowers and other researchers search for other non-convex uniform 5-polytopes, with a current count of 1297 known uniform 5-polytopes outside infinite families (convex and non-convex), excluding the prisms of the uniform 4-polytopes. The list is not proven complete.
Regular 5-polytopes
Regular 5-polytopes can be represented by the Schläfli symbol {p,q,r,s}, with s {p,q,r} 4-polytope facets around each face. There are exactly three such regular polytopes, all convex:
{3,3,3,3} - 5-simplex
{4,3,3,3} - 5-cube
{3,3,3,4} - 5-orthoplex
There are no nonconvex regular polytopes in 5 dimensions or above.
Convex uniform 5-polytopes
There are 104 known convex uniform 5-polytopes, plus a number of infinite families of duoprism prisms, and polygon-polyhedron duoprisms. All except the grand antiprism prism are based on Wythoff constructions, reflection symmetry generated with Coxeter groups.
Symmetry of uniform 5-polytopes in four dimensions
The 5-simplex is the regular form in the A5 family. The 5-cube and 5-orthoplex are the regular forms in the B5 family. The bifurcating graph of the D5 family contains the 5-orthoplex, as well as a 5-demicube which is an alternated 5-cube.
Each reflective uniform 5-polytope can be constructed in one or more reflective point group in 5 dimensions by a Wythoff construction, represented by rings around permutations of nodes in a Coxeter diagram. Mirror hyperplanes can be grouped, as seen by colored nodes, separated by even-branches. Symmetry groups of the form [a,b,b,a], have an extended symmetry, [[a,b,b,a]], like [3,3,3,3], doubling the symmetry |
https://en.wikipedia.org/wiki/Orbit%20%28control%20theory%29 | The notion of orbit of a control system used in mathematical control theory is a particular case of the notion of orbit in group theory.
Definition
Let
be a control system, where
belongs to a finite-dimensional manifold and belongs to a control set . Consider the family
and assume that every vector field in is complete.
For every and every real , denote by the flow of at time .
The orbit of the control system through a point is the subset of defined by
Remarks
The difference between orbits and attainable sets is that, whereas for attainable sets only forward-in-time motions are allowed, both forward and backward motions are permitted for orbits.
In particular, if the family is symmetric (i.e., if and only if ), then orbits and attainable sets coincide.
The hypothesis that every vector field of is complete simplifies the notations but can be dropped. In this case one has to replace flows of vector fields by local versions of them.
Orbit theorem (Nagano–Sussmann)
Each orbit is an immersed submanifold of .
The tangent space to the orbit
at a point is the linear subspace of spanned by
the vectors where denotes the pushforward of by , belongs to and is a diffeomorphism of of the form with and .
If all the vector fields of the family are analytic, then where is the evaluation at of the Lie algebra generated by with respect to the Lie bracket of vector fields.
Otherwise, the inclusion holds true.
Corollary (Rashevsky–Chow theorem)
If for every and if is connected, then each orbit is equal to the whole manifold .
See also
Frobenius theorem (differential topology)
References
Further reading
Control theory |
https://en.wikipedia.org/wiki/Binary%20cyclic%20group | In mathematics, the binary cyclic group of the n-gon is the cyclic group of order 2n, , thought of as an extension of the cyclic group by a cyclic group of order 2. Coxeter writes the binary cyclic group with angle-brackets, ⟨n⟩, and the index 2 subgroup as (n) or [n]+.
It is the binary polyhedral group corresponding to the cyclic group.
In terms of binary polyhedral groups, the binary cyclic group is the preimage of the cyclic group of rotations () under the 2:1 covering homomorphism
of the special orthogonal group by the spin group.
As a subgroup of the spin group, the binary cyclic group can be described concretely as a discrete subgroup of the unit quaternions, under the isomorphism where Sp(1) is the multiplicative group of unit quaternions. (For a description of this homomorphism see the article on quaternions and spatial rotations.)
Presentation
The binary cyclic group can be defined as:
See also
binary dihedral group, ⟨2,2,n⟩, order 4n
binary tetrahedral group, ⟨2,3,3⟩, order 24
binary octahedral group, ⟨2,3,4⟩, order 48
binary icosahedral group, ⟨2,3,5⟩, order 120
References
Cyclic |
https://en.wikipedia.org/wiki/Marshall%2C%20Saskatchewan | Marshall is a town in Saskatchewan, Canada 19 km (12 miles) from Lloydminster on the Yellowhead Highway (Highway 16).
Demographics
In the 2021 Census of Population conducted by Statistics Canada, Marshall had a population of living in of its total private dwellings, a change of from its 2016 population of . With a land area of , it had a population density of in 2021.
Notable people
Braden Holtby - NHL goaltender
See also
List of communities in Saskatchewan
List of towns in Saskatchewan
References
Wilton No. 472, Saskatchewan
Towns in Saskatchewan
Division No. 17, Saskatchewan |
https://en.wikipedia.org/wiki/Piapot%2C%20Saskatchewan | Piapot () is a hamlet within the Rural Municipality of Piapot No. 110, Saskatchewan, Canada. Listed as a designated place by Statistics Canada, the hamlet had a population of 50 in the Canada 2016 Census.
Once a thriving community, it has seen a steady decline since the 1950s and in the present day it resembles a ghost town. The hotel and saloon closed in 2006 but reopened in May 2008, embracing western heritage and culture. The Piapot Saloon and Guesthouse offers an escape from everyday life in the spirit of the original settlers as well as a gift shop and old western saloon. The only other business that is open to the public is the post office.
Demographics
In the 2021 Census of Population conducted by Statistics Canada, Piapot had a population of 40 living in 22 of its 29 total private dwellings, a change of from its 2016 population of 50. With a land area of , it had a population density of in 2021.
See also
List of communities in Saskatchewan
List of hamlets in Saskatchewan
List of place names in Canada of Indigenous origin
Piapot
References
Piapot No. 110, Saskatchewan
Former villages in Saskatchewan
Designated places in Saskatchewan
Hamlets in Saskatchewan
Division No. 4, Saskatchewan |
https://en.wikipedia.org/wiki/Eastern%20Heights%2C%20Queensland | {
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https://en.wikipedia.org/wiki/Andrew%20M.%20Bruckner | Andrew Michael Bruckner (born December 17, 1932) is an American retired mathematician, known for his contributions to real analysis.
He got his PhD in mathematics from University of California, Los Angeles (1959) on the dissertation Minimal Superadditive Extensions of Superadditive Functions advised by John Green (mathematician).
He joined the faculty at University of California, Santa Barbara.
The "Andy Award" is given annually in his name, to significant contributors to real analysis.
In 2012 he became a fellow of the American Mathematical Society.
Books
Differentiation of real functions (American Mathematical Society, 1994)
Real analysis (1997). With Judith B. Bruckner and Brian S. Thomson.
Elementary real analysis (2001). With B. Thomson and J. Bruckner.
References
20th-century American mathematicians
21st-century American mathematicians
University of California, Los Angeles alumni
University of California, Santa Barbara faculty
Fellows of the American Mathematical Society
Living people
1932 births |
https://en.wikipedia.org/wiki/Wladimir%20Seidel | Wladimir P. Seidel (December 21, 1907 – January 12, 1981) was a Russian-born German-American mathematician, and Doctor of Mathematics. He held a fellowship as a Benjamin Peirce Professor in Harvard University. During World War II, he was with the Montreal Theory group for the National Research Council of Canada.
Life
He was born in Odessa, Russian Empire on December 21, 1907.
Career
He earned his Ph.D. from Ludwig-Maximilians-Universität in München (February 26, 1930) on a dissertation entitled Über die Ränderzuordnung bei konformen Abbildungen, advised by Constantin Carathéodory.
He joined the faculty of Mathematics at Harvard University (as Benjamin Peirce Instructor, 1932–33), at University of Rochester (1941–55), at The Institute for Advanced Study in Princeton (1952–53), at University of Notre Dame (1955–63), and at Wayne State University in Detroit (since 1963).
During World War II, he was with the Montreal Theory group for the National Research Council of Canada.
The Seidel class is named after him.
He was married to Leah Lappin-Seidel (1904–1999).
Publications
References
20th-century German mathematicians
Emigrants from the Russian Empire to the United States
Emigrants from the Weimar Republic to the United States
Ludwig Maximilian University of Munich alumni
University of Rochester faculty
Harvard University Department of Mathematics faculty
Harvard University faculty
Princeton University faculty
University of Notre Dame faculty
Wayne State University faculty
Academics from Detroit
Scientists from Odesa
1907 births
1981 deaths
20th-century American mathematicians |
https://en.wikipedia.org/wiki/1973%E2%80%9374%20Winnipeg%20Jets%20season | The 1973–74 Winnipeg Jets season was their second season in the World Hockey Association (WHA).
Regular season
Season standings
Playoffs
Houston Aeros 4, Winnipeg Jets 0
Player statistics
Forwards
Note: GP= Games played; G= Goals; A= Assists; PTS = Points; PIM = Points
Defencemen
Note: GP= Games played; G= Goals; A= Assists; PTS = Points; PIM = Points
Goaltending
Note: GP= Games played; MIN= Minutes; W= Wins; L= Losses; T = Ties; SO = Shutouts; GAA = Goals against
Draft picks
Winnipeg's draft picks at the 1973 WHA Amateur Draft.
References
External links
Jets on Hockey Database
Winnipeg Jets (1972–1996) seasons
Winn
Winn |
https://en.wikipedia.org/wiki/Krak%C3%B3w%20School%20of%20Mathematics%20and%20Astrology | The Kraków School of Mathematics and Astrology () was an influential mid-to-late-15th-century group of mathematicians and astrologers at the University of Kraków (later Jagiellonian University).
Notable members
Jan of Głogów (1445–1507), author of widely recognized mathematical and astrological tracts
Marcin Biem (1470–1540), contributor to the Gregorian calendar
Marcin Bylica of Olkusz (1433–93), later court astrologer to King Matthias Corvinus of Hungary
Albert Brudzewski (1446–1495), teacher to notable scholars active at European universities
Marcin Król of Żurawica (1422–1460)
Nicolaus Copernicus (1473–1543), student at Kraków in 1491–95
See also
Kraków School of Mathematics
Polish School of Mathematics
References
History of education in Poland
Education in Kraków
History of mathematics
Jagiellonian University
Polish mathematics
Astrological organizations |
https://en.wikipedia.org/wiki/World%20Christian%20Encyclopedia | World Christian Encyclopedia is a reference work, with its third edition published by Edinburgh University Press in November 2019. The WCE is known for providing membership statistics for major world religions and Christian denominations including historical data and projections of future populations.
The data incorporated into the World Christian Encyclopedia have been made available online at the World Christian Database (WCD).
Editions
1st - 1982
The first edition, World Christian Encyclopedia: A Comparative Survey of Churches and Religions in the Modern World A.D. 1900–2000 (WCE), by David B. Barrett, was published in 1982 by Oxford University Press. Barrett was a trained aeronautical engineer who became a missionary with the Church Missionary Society (Anglican). He arrived in Nyanza Province in Western Kenya in 1957. Over the course of 14 years he traveled to 212 of 223 countries and corresponded with Christians all over the world in search of the most up-to-date statistics on Christianity and world religions. His research resulted in the first edition of the World Christian Encyclopedia in 1982.
2nd - 2001
Barrett moved to Richmond, Virginia in 1985 to work with the Southern Baptists on missionary strategy. He continued his research as an independent researcher, joined by Todd M. Johnson in 1988. With George Kurian, Barrett and Johnson produced the second edition of the World Christian Encyclopedia, in 2 volumes, in 2001 (Oxford University Press).
3rd - 2019
The third edition, written and edited by Todd M. Johnson and Gina A. Zurlo (Barrett died in 2011), was released in November 2019. Johnson and Zurlo are co-directors of the Center for the Study of Global Christianity at Gordon-Conwell Theological Seminary (South Hamilton, MA, USA).
Reception
One study found that the WCD's data was "highly correlated with other sources that offer cross-national religious composition estimates" but the database "consistently gives a higher estimate for percent Christian in comparison to other cross-national data sets". Concern has also been raised about possible bias because the World Christian Encyclopedia was originally developed as a Christian missionary tool.
Margit Warburg, a Danish researcher, has argued that the database contains numerical inaccuracies in its statistics on the Baháʼí Faith. She noted that figures given in WCE for some Western countries are highly exaggerated. For instance, the World Christian Encyclopedia reports an estimated 1,600 Baháʼís in Denmark in 1995 and 682,000 Baháʼís in the US in 1995. According to her, the Baháʼís themselves do not acknowledge such numbers; the number of registered Baháʼís in Denmark, in 1995, was about 240 and in the number in the USA was about 130,000.
References
External links
Edinburgh University Press: World Christian Encyclopedia
World Christian Database
Center for the Study of Global Christianity
1982 non-fiction books
2001 non-fiction books
Encyclopedias of religion
Religious studies bo |
https://en.wikipedia.org/wiki/Steiner%E2%80%93Lehmus%20theorem | The Steiner–Lehmus theorem, a theorem in elementary geometry, was formulated by C. L. Lehmus and subsequently proved by Jakob Steiner. It states:
Every triangle with two angle bisectors of equal lengths is isosceles.
The theorem was first mentioned in 1840 in a letter by C. L. Lehmus to C. Sturm, in which he asked for a purely geometric proof. Sturm passed the request on to other mathematicians and Steiner was among the first to provide a solution. The theorem became a rather popular topic in elementary geometry ever since with a somewhat regular publication of articles on it.
Direct proofs
The Steiner–Lehmus theorem can be proved using elementary geometry by proving the contrapositive statement: if a triangle is not isosceles, then it does not have two angle bisectors of equal length.
There is some controversy over whether a "direct" proof is possible;
allegedly "direct" proofs have been published, but not everyone agrees that these proofs are "direct."
For example, there exist simple algebraic expressions for angle bisectors in terms of the sides of the triangle. Equating two of these expressions and algebraically manipulating the equation results in a product of two factors which equal 0, but only one of them (a − b) can equal 0 and the other must be positive. Thus a = b. But this may not be considered direct as one must first argue about why the other factor cannot be 0.
John Conway
has argued that there can be no "equality-chasing" proof because the theorem (stated algebraically) does not hold over an arbitrary field, or even when negative real numbers are allowed as parameters.
A precise definition of a "direct proof" inside both classical and intuitionistic logic has been provided by Victor Pambuccian,
who proved, without presenting the direct proofs, that direct proofs must exist in both the classical logic and the intuitionistic logic setting. Ariel Kellison later gave a direct proof.
Notes
References & further reading
John Horton Conway, Alex Ryba: The Steiner-Lehmus Angle Bisector Theorem. In: Mircea Pitici (Hrsg.): The Best Writing on Mathematics 2015. Princeton University Press, 2016, , pp. 154–166
Alexander Ostermann, Gerhard Wanner: Geometry by Its History. Springer, 2012, pp. 224–225
S. Abu-Saymeh, M. Hajja, H. A. ShahAli: Another Variation on the Steiner-Lehmus Theme. Forum Geometricorum 8, 2008, pp. 131–140
External links
Paul Yiu: Euclidean Geometry Notes, Lectures Notes, Florida Atlantic University, pp. 16–17
Torsten Sillke: Steiner–Lehmus Theorem, extensive compilation of proofs on a website of the University of Bielefeld
Euclidean geometry
Theorems about special triangles |
https://en.wikipedia.org/wiki/Lucjan%20Zarzecki | Lucjan Zarzecki (1873–1925) was a Polish pedagogue and mathematician, a co-originator of national education concept. His area of study was general didactics and didactics of mathematics.
Member of the Polska Macierz Szkolna, professor and director of Pedagogics Department of the Wolna Wszechnica Polska in Warsaw.
Notable works
Charakter jako cel wychowania (1918)
Nauczanie matematyki początkowej vol. 1–3 (1919–1920)
Dydaktyka ogólna, czyli kształcenie charakteru przez nauczanie (1920)
Wstęp do pedagogiki (1922)
Wychowanie narodowe (1926)
Further reading
References
1873 births
1925 deaths
Mathematics educators
Polish educational theorists
Polish educators
19th-century Polish mathematicians
20th-century Polish mathematicians
Educators from the Russian Empire
Mathematicians from the Russian Empire |
https://en.wikipedia.org/wiki/1947%E2%80%9348%20Toronto%20Maple%20Leafs%20season | The 1947–48 Toronto Maple Leafs season involved winning the Stanley Cup.
Offseason
Regular season
Final standings
Record vs. opponents
Schedule and results
Player statistics
Regular season
Scoring
Goaltending
Playoffs
Scoring
Goaltending
Playoffs
Stanley Cup Finals
This was the debut series for Detroit's Gordie Howe, and the last for Toronto's Syl Apps who retired after the series.
Detroit Red Wings vs. Toronto Maple Leafs
Toronto wins best-of-seven series 4–0.
Awards and records
Prince of Wales Trophy
Vezina Trophy: || Turk Broda
Turk Broda, Goaltender, NHL First Team All-Star
Transactions
May 15, 1947: Traded Buck Jones and Nick Knott to the Tulsa Oilers of the USHL for cash
October 5, 1947: Traded Gordie Bell to the Washington Lions of the AHL for cash
November 2, 1947: Acquired Max Bentley and Cy Thomas from the Chicago Black Hawks for Gus Bodnar, Gaye Stewart, Bob Goldham, Bud Poile and Ernie Dickens
References
Maple Leafs on Hockey Database
Maple Leafs on Database Hockey
Stanley Cup championship seasons
Toronto Maple Leafs seasons
Toronto
Tor
1948 Stanley Cup
Toronto Maple Leafs
Toronto Maple Leafs |
https://en.wikipedia.org/wiki/Eilenberg%E2%80%93Mazur%20swindle | In mathematics, the Eilenberg–Mazur swindle, named after Samuel Eilenberg and Barry Mazur, is a method of proof that involves paradoxical properties of infinite sums. In geometric topology it was introduced by and is often called the Mazur swindle. In algebra it was introduced by Samuel Eilenberg
and is known as the Eilenberg swindle or Eilenberg telescope (see telescoping sum).
The Eilenberg–Mazur swindle is similar to the following well known joke "proof" that 1 = 0:
1 = 1 + (−1 + 1) + (−1 + 1) + ... = 1 − 1 + 1 − 1 + ... = (1 − 1) + (1 − 1) + ... = 0
This "proof" is not valid as a claim about real numbers because Grandi's series 1 − 1 + 1 − 1 + ... does not converge, but the analogous argument can be used in some contexts where there is some sort of "addition" defined on some objects for which infinite sums do make sense,
to show that if A + B = 0 then A = B = 0.
Mazur swindle
In geometric topology the addition used in the swindle is usually the connected sum of knots or manifolds.
Example : A typical application of the Mazur swindle in geometric topology is the proof that the sum of two non-trivial knots A and B is non-trivial. For knots it is possible to take infinite sums by making the knots smaller and smaller, so if A + B is trivial then
so A is trivial (and B by a similar argument). The infinite sum of knots is usually a wild knot, not a tame knot.
See for more geometric examples.
Example: The oriented n-manifolds have an addition operation given by connected sum, with 0 the n-sphere. If A + B is the n-sphere, then A + B + A + B + ... is Euclidean space so the Mazur swindle shows that the connected sum of A and Euclidean space is Euclidean space, which shows that A is the 1-point compactification of Euclidean space and therefore A is homeomorphic to the n-sphere. (This does not show in the case of smooth manifolds that A is diffeomorphic to the n-sphere, and in some dimensions, such as 7, there are examples of exotic spheres A with inverses that are not diffeomorphic to the standard n-sphere.)
Eilenberg swindle
In algebra the addition used in the swindle is usually the direct sum of modules over a ring.
Example: A typical application of the Eilenberg swindle in algebra is the proof that if A is a projective module over a ring R then there is a free module F with . To see this, choose a module B such that is free, which can be done as A is projective, and put
F = B ⊕ A ⊕ B ⊕ A ⊕ B ⊕ ⋯.
so that
A ⊕ F = A ⊕ (B ⊕ A) ⊕ (B ⊕ A) ⊕ ⋯ = (A ⊕ B) ⊕ (A ⊕ B) ⊕ ⋯ ≅ F.
Example: Finitely generated free modules over commutative rings R have a well-defined natural number as their dimension which is additive under direct sums, and are isomorphic if and only if they have the same dimension.
This is false for some noncommutative rings, and a counterexample can be constructed using the Eilenberg swindle as follows. Let X be an abelian group such that X ≅ X ⊕ X (for example the direct sum of an infinite number of copies of any nonzero |
https://en.wikipedia.org/wiki/Oleg%20Viro | Oleg Yanovich Viro () (b. 13 May 1948, Leningrad, USSR) is a Russian mathematician in the fields of topology and algebraic geometry, most notably real algebraic geometry, tropical geometry and knot theory.
Contributions
Viro developed a "patchworking" technique in algebraic geometry, which allows real algebraic varieties to be constructed by a "cut and paste" method. Using this technique, Viro completed the isotopy classification of non-singular plane projective curves of degree 7. The patchworking technique was one of the fundamental ideas which motivated the development of tropical geometry. In topology, Viro is most known for his joint work with Vladimir Turaev, in which the Turaev-Viro invariants (relatives of the Reshetikhin-Turaev invariants) and related topological quantum field theory notions were introduced.
Education and career
Viro studied at the Leningrad State University where he received his Ph.D. degree in 1974; his advisor was Vladimir Rokhlin. Viro taught from 1973 until 1991 at Leningrad State University. Since 1986 he has been a member of the Saint Petersburg Department of the Steklov Institute of Mathematics. In 1992-1997, Viro was a F. B. Jones chair professor in Topology at the University of California, Riverside.
In 1994-2003 he was a professor at Uppsala University, Sweden. On 8 February 2007, Viro and his colleague Burglind Juhl-Jöricke were forced to resign from the university. There had been a history of conflict at the Mathematics Institute, with allegations of disagreeable behavior by several parties in the conflict. A number of Swedish, European and American mathematicians protested the manner in which the two Professors of Mathematics were forced to resign. These protests include the following:
an open letter by Lennart Carleson, former president of the International Mathematical Union,
a letter by Ari Laptev, current president of the European Mathematical Society, and
a letter from M. Salah Baouendi, Arthur Jaffe, Joel Lebowitz, Elliott H. Lieb and Nicolai Reshetikhin.
As of 2009, Viro is a senior researcher at the St. Petersburg Department of the Steklov Institute of Mathematics, and a professor at Stony Brook University.
Awards and honors
Viro was an invited speaker at the International Congress of Mathematicians in 1983 (Warsaw) and the European Congress of Mathematicians in 2000 (Barcelona). He was awarded the Göran Gustafsson Prize (1997) by the Swedish government.
In 2012 he became a fellow of the American Mathematical Society.
References
External links
Oleg Viro's website
1948 births
Living people
20th-century Russian mathematicians
21st-century Russian mathematicians
Soviet mathematicians
Topologists
Algebraic geometers
University of California, Riverside faculty
Fellows of the American Mathematical Society
Stony Brook University faculty |
https://en.wikipedia.org/wiki/Julian%20Keilson | Julian Keilson (November 19, 1924 – March 8, 1999 in Rochester, New York) was an American
mathematician.
He was known for his work in probability theory. His work in survival analysis is relevant to many fields, e.g., medical research, parts supply, asset depreciation, rental pricing, etc.
He got his B.Sc. in physics from Brooklyn College,
and M.Sc. and Ph.D. from Harvard University. His Ph.D. thesis advisor was the Nobel Prize–winning professor of Physics, Julian Schwinger. Next he worked at
MIT Lincoln Laboratories and GTE Laboratories before joining the faculty at University of Rochester (1966–96) where
he started the statistics department.
He also taught at MIT Sloan School of Management (1986–92).
Books
Green's functions in probability theory (1965)
Markov chain models -- rarity and exponentiality
References
20th-century American mathematicians
Brooklyn College alumni
Harvard University alumni
University of Rochester faculty
1924 births
1999 deaths
MIT Sloan School of Management faculty
MIT Lincoln Laboratory people |
https://en.wikipedia.org/wiki/Romani%20people%20in%20Ireland | The number of Romani people in Ireland is roughly estimated, as the Central Statistics Office collects its data based on nationality and not ethnic origin. For this reason a precise demographic profile of the Romani in Ireland is not available. Some estimates of Romani in Ireland give the population at 1,700 in 2004, rising to between 2,500 and 3,000 in 2005. The majority derived from Romani populations originating in Ukraine and Hungary.
History
Origin
The Romani people originate from Northern India, presumably from the northwestern Indian states Rajasthan and Punjab.
The linguistic evidence has indisputably shown that roots of Romani language lie in India: the language has grammatical characteristics of Indian languages and shares with them a big part of the basic lexicon, for example, body parts or daily routines.
More exactly, Romani shares the basic lexicon with Hindi and Punjabi. It shares many phonetic features with Marwari, while its grammar is closest to Bengali.
Genetic findings in 2012 suggest the Romani originated in northwestern India and migrated as a group.
According to a genetic study in 2012, the ancestors of present scheduled tribes and scheduled caste populations of Northern India, traditionally referred to collectively as the Ḍoma, are the likely ancestral populations of the modern European Roma.
In February 2016, during the International Roma Conference, the Indian Minister of External Affairs stated that the people of the Roma community were children of India. The conference ended with a recommendation to the Government of India to recognize the Roma community spread across 30 countries as a part of the Indian diaspora.
Migration to Ireland
Romani have been present in Ireland since the 18th century. Traditionally, Romani arrived from Britain for seasonal work, either as farm labourers or as coppersmiths
Post-1989
After the dissolution of Eastern Bloc, thousands of Romani, among others, sought asylum in Ireland and other Western countries. Their arrival prompted contrasting editorials in the mainstream newspapers. In 1989, Romani started to arrive in Ireland, predominantly by hiding in container lorries. In the summer of 1998, several hundred Romani arrived hidden in freight containers in Rosslare Harbour, many of them illegally trafficked.
A second impetus for Romani immigration arose after the admittance of an additional 15 states to the European Union, with the populations coming to Dublin and the other major towns and cities.
References
External links
Access Ireland - Training Roma as cultural mediators
Roma Educational Needs in Ireland - Context and Challenges Lesovitch, L., June 2005 City of Dublin VEC
Ireland
Ireland
Society of Ireland |
https://en.wikipedia.org/wiki/Autonomous%20convergence%20theorem | In mathematics, an autonomous convergence theorem is one of a family of related theorems which specify conditions guaranteeing global asymptotic stability of a continuous autonomous dynamical system.
History
The Markus–Yamabe conjecture was formulated as an attempt to give conditions for global stability of continuous dynamical systems in two dimensions. However, the Markus–Yamabe conjecture does not hold for dimensions higher than two, a problem which autonomous convergence theorems attempt to address. The first autonomous convergence theorem was constructed by Russell Smith. This theorem was later refined by Michael Li and James Muldowney.
An example autonomous convergence theorem
A comparatively simple autonomous convergence theorem is as follows:
Let be a vector in some space , evolving according to an autonomous differential equation . Suppose that is convex and forward invariant under , and that there exists a fixed point such that . If there exists a logarithmic norm such that the Jacobian satisfies for all values of , then is the only fixed point, and it is globally asymptotically stable.
This autonomous convergence theorem is very closely related to the Banach fixed-point theorem.
How autonomous convergence works
Note: this is an intuitive description of how autonomous convergence theorems guarantee stability, not a strictly mathematical description.
The key point in the example theorem given above is the existence of a negative logarithmic norm, which is derived from a vector norm. The vector norm effectively measures the distance between points in the vector space on which the differential equation is defined, and the negative logarithmic norm means that distances between points, as measured by the corresponding vector norm, are decreasing with time under the action of . So long as the trajectories of all points in the phase space are bounded, all trajectories must therefore eventually converge to the same point.
The autonomous convergence theorems by Russell Smith, Michael Li and James Muldowney work in a similar manner, but they rely on showing that the area of two-dimensional shapes in phase space decrease with time. This means that no periodic orbits can exist, as all closed loops must shrink to a point. If the system is bounded, then according to Pugh's closing lemma there can be no chaotic behaviour either, so all trajectories must eventually reach an equilibrium.
Michael Li has also developed an extended autonomous convergence theorem which is applicable to dynamical systems containing an invariant manifold.
Notes
Stability theory
Fixed points (mathematics)
Theorems in dynamical systems |
https://en.wikipedia.org/wiki/Probabilistic%20causation | Probabilistic causation is a concept in a group of philosophical theories that aim to characterize the relationship between cause and effect using the tools of probability theory. The central idea behind these theories is that causes raise the probabilities of their effects, all else being equal.
Deterministic versus probabilistic theory
Interpreting causation as a deterministic relation means that if A causes B, then A must always be followed by B. In this sense, war does not cause deaths, nor does smoking cause cancer. As a result, many turn to a notion of probabilistic causation. Informally, A probabilistically causes B if As occurrence increases the probability of B. This is sometimes interpreted to reflect imperfect knowledge of a deterministic system but other times interpreted to mean that the causal system under study has an inherently indeterministic nature. (Propensity probability is an analogous idea, according to which probabilities have an objective existence and are not just limitations in a subject's knowledge).
Philosophers such as Hugh Mellor and Patrick Suppes have defined causation in terms of a cause preceding and increasing the probability of the effect. (Additionally, Mellor claims that cause and effect are both facts - not events - since even a non-event, such as the failure of a train to arrive, can cause effects such as my taking the bus. Suppes, by contrast, relies on events defined set-theoretically, and much of his discussion is informed by this terminology.)
Pearl argues that the entire enterprise of probabilistic causation has been misguided from the very beginning, because the central notion that causes "raise the probabilities" of their effects cannot be expressed in the language of probability theory. In particular, the inequality Pr(effect | cause) > Pr(effect | ~cause) which philosophers invoked to define causation, as well as its many variations and nuances, fails to capture the intuition behind "probability raising", which is inherently a manipulative or counterfactual notion.
The correct formulation, according to Pearl, should read:
Pr(effect | do(cause)) > Pr(effect | do(~cause))
where do(C) stands for an external intervention that compels the truth of C. The conditional probability Pr(E | C), in contrast, represents a probability resulting from a passive observation of C, and rarely coincides with Pr(E | do(C)). Indeed, observing the barometer falling increases the probability of a storm coming, but does not
"cause" the storm; were the act of manipulating the barometer to change the probability of storms, the falling barometer would qualify as a cause of storms. In general, formulating the notion of "probability raising" within the calculus of do-operators resolves the difficulties that probabilistic causation has encountered in the past half-century,Cartwright, N. (1989). Nature's Capacities and Their Measurement, Clarendon Press, Oxnard. among them the infamous Simpson's paradox, and clarifies pr |
https://en.wikipedia.org/wiki/Canadian%20Open%20Mathematics%20Challenge | The Canadian Open Mathematics Challenge (COMC) is an annual mathematics competition held in Canada during the month of October. This competition is run by the Canadian Mathematical Society. Students who score exceptionally well on this competition are selected to participate in the Canadian Mathematical Olympiad.
Participation
The COMC is written on a select day in October each year and is proctored by teachers across Canada. In order to participate in this competition, students must register through their school’s mathematics department and pay any fees associated with the competition. Following the day of the competition, exams are returned to a network of university partners across Canada for marking.
Eligibility
The competition is open to any student with an interest in mathematics. However, to be official participants, students must satisfy the following criteria:
must be in full-time attendance at an elementary or secondary school, or CEGEP since September of the year of the COMC; and
be less than 19 years old as of June 30 of the year of the COMC.
Students writing from outside Canada are not eligible for cash awards but compete for ranking in the International division.
Students not meeting the qualification requirements can still participate and marks as “unofficial” competitors.
Format
The COMC consists of three sections:
4 basic questions (Part A) for 4 marks each,
4 intermediary questions (Part B) for 6 marks each, and
4 advanced questions (Part C) for 10 marks each
for a total of 80 marks.
The length of the contest is 2.5 hours. Calculators are not permitted.
Awards and Prizes
There are two divisions: Canadian Awards, which is only for participants writing the exam from within Canada; and International Awards, which is only for participants writing it outside of Canada.
Canadian Award Categories
There are award categories for Best in Canada and Best in Province and Best in Region for all students as well as for students at each grade. For example:
Best in Canada Overall: all official participants (regardless of grade) in Canada compete for this, the most prestigious category,
Best in Canada, Grade XI: all Grade XI students in Canada compete for this,
Best in BC (overall): all official participants (regardless of grade) in BC,
Best in BC, Grade XII: all Grade XII students in BC,
Best in Toronto (overall): all official participants in Toronto-area schools,
Best in Toronto, Grade XII: All grade XII and Cégep students in Toronto compete for this category
etc.
The top six unique scores in any category earn awards: Gold, Silver, Bronze, Honourable Mention.
The top students in Canada also normally receive cash awards based on their ranking.
International Awards
The top official participants from outside Canada are considered for the international awards division, which is not grade-dependent. The top three unique scores are given Gold, Silver or Bronze. Additionally, any international student who achieves at least a sco |
https://en.wikipedia.org/wiki/COMC | COMC may refer to:
Canadian Open Mathematics Challenge, competition
L-2-hydroxycarboxylate dehydrogenase (NAD+), enzyme
(2R)-3-sulfolactate dehydrogenase (NADP+), enzyme |
https://en.wikipedia.org/wiki/Stars%20and%20bars%20%28combinatorics%29 | In the context of combinatorial mathematics, stars and bars (also called "sticks and stones", "balls and bars", and "dots and dividers") is a graphical aid for deriving certain combinatorial theorems. It was popularized by William Feller in his classic book on probability. It can be used to solve many simple counting problems, such as how many ways there are to put indistinguishable balls into distinguishable bins.
Statements of theorems
The stars and bars method is often introduced specifically to prove the following two theorems of elementary combinatorics concerning the number of solutions to an equation.
Theorem one
For any pair of positive integers and , the number of -tuples of positive integers whose sum is is equal to the number of -element subsets of a set with elements.
For example, if and , the theorem gives the number of solutions to (with ) as the binomial coefficient
This corresponds to compositions of an integer.
Theorem two
For any pair of positive integers and , the number of -tuples of non-negative integers whose sum is is equal to the number of multisets of cardinality taken from a set of size , or equivalently, the number of multisets of cardinality taken from a set of size .
For example, if and , the theorem gives the number of solutions to (with ) as:
This corresponds to weak compositions of an integer.
Proofs via the method of stars and bars
Theorem one proof
Suppose there are n objects (represented here by stars) to be placed into k bins, such that all bins contain at least one object. The bins are distinguishable (say they are numbered 1 to k) but the n stars are not (so configurations are only distinguished by the number of stars present in each bin). A configuration is thus represented by a k-tuple of positive integers, as in the statement of the theorem.
For example, with and , start by placing the stars in a line:
The configuration will be determined once it is known which is the first star going to the second bin, and the first star going to the third bin, etc.. This is indicated by placing bars between the stars. Because no bin is allowed to be empty (all the variables are positive), there is at most one bar between any pair of stars.
For example:
There are gaps between stars. A configuration is obtained by choosing of these gaps to contain a bar; therefore there are possible combinations.
Theorem two proof
In this case, the weakened restriction of non-negativity instead of positivity means that we can place multiple bars between stars, before the first star and after the last star.
For example, when and , the tuple (4, 0, 1, 2, 0) may be represented by the following diagram:
To see that there are possible arrangements, observe that any arrangement of stars and bars consists of a total of objects, n of which are stars and of which are bars. Thus, we only need to choose of the positions to be bars (or, equivalently, choose n of the positions to be stars).
Theorem 1 can no |
https://en.wikipedia.org/wiki/Crime%20in%20Switzerland | Crime in Switzerland is combated mainly by cantonal police. The Federal Office of Police investigates organised crime, money laundering and terrorism.
Crime statistics
In Switzerland, police registered a total of 432,000 offenses under the Criminal Code in 2019 (−0.2% compared with previous year), of which 110,140 or 25.5 percent were cases of thefts (excluding vehicles, −2.0%), and 41,944 or 9.7 percent were thefts of vehicles (including bicycles, −10.1%), 46 were killings and 161 were attempted murders. The number of cases of rape reported increased by 53 incidents or 8.5 percent over the previous year. The number of criminal pornography offenses increased by 56.1 percent to 2,837. Offenses against the Narcotics Act decreased by 0.7 percent to 75,757.
In 2014, 110,124 adults were convicted, of which 55,240 (50%) were convicted according to traffic regulation offences, 6,540 (+1.6%) for trafficking in narcotic substances, and 17,882 (−7.2%) for offenses against the Federal Act on Foreign Nationals. 83,014 or 83.4% of adult convicted people are male, and 42,289 or 42.5 percent of them Swiss citizens.
In the same year, 11,484 minors (78 percent of them male, 68 percent of them of Swiss nationality, 64.2 percent aged either 16 or 17) were convicted.
Convictions for infliction of bodily harm have steadily increased throughout the 1990s and 2000s, with 23 convictions for serious injury and 831 for light injury in 1990 as opposed to 78 and 2,342, respectively, in 2005. Convictions for rape have also slightly increased, fluctuating between 500 and 600 cases per year in the period 1985 to 1995, but between 600 and 700 cases in the period 2000 to 2005. Consistent with these trends, convictions for threats or violence directed against officials has consistently risen in the same period, from 348 in 1990 to 891 in 2003.
Types of convictions
The number of convicted persons is given in the following tables. Each class of crime references the relevant section of the Strafgesetzbuch (Criminal Code, abbreviated as StGB in German), or Betäubungsmittelgesetz (abbr. BetmG, Narcotics Act), or the Strassenverkehrsgesetz (abbr. SVG, Swiss Traffic Regulations).
2016 conviction numbers may not include convictions overturned on appeal.
Due to privacy protection laws some convictions are not included.
2016 conviction numbers may not include convictions overturned on appeal.
Due to privacy protection laws some convictions are not included.
Historic conviction rates
The historic adult conviction rates are given in the following chart:
2014 conviction numbers may not include convictions overturned on appeal.
Age at conviction
The age of the individuals at the time of their convictions is given in this chart:
2014 conviction numbers may not include convictions overturned on appeal.
Juvenile crimes
According to official statistics, there has been a total of 20,902 juvenile convictions in 2021 – 7.5 percent more than in 2020. According to a 2021 survey about |
https://en.wikipedia.org/wiki/Onno%20J.%20Boxma | Onno Johan Boxma (born 1952) is a Dutch mathematician, and Professor at the Eindhoven University of Technology, known for several contributions to queueing theory and applied probability theory.
Biography
Born in The Hague, Boxma earned his B.Sc. in Mathematics at Delft University of Technology in 1974, and his Ph.D. cum laude in Mathematic from Utrecht University in 1977 on the dissertation entitled "Analysis of Models for Tandem Queues", advised by Wim Cohen.
Boxma continued at Utrecht as faculty from 1974 to 1985, and was IBM Research postdoctoral fellow in 1978–79, before joining the faculty of Centrum Wiskunde & Informatica in Amsterdam. There he chaired the performance analysis group until 1998. He was full professor at University of Tilburg from 1987 to 1988. and since 1998 he is as full professor holding the chair of Stochastic Operations Research in the Department of Mathematics and Computer Science at Technische Universiteit Eindhoven, becoming vice dean of the department in 2009.
He was the editor-in-chief of Queueing Systems from 2004 to 2009, and scientific director of EURANDOM from 2005-2010. In 2009 he was awarded an honorary degree by the University of Haifa (Israel), and received the 2011 ACM SIGMETRICS Achievement Award in June 2011. Also, he is honorary professor in Heriot-Watt University, Edinburgh, UK (2008-2010 and 2011-2013).
Work
Boxma's research focuses on the field of applied probability and stochastic operations, particularly of queueing theory and its application to the performance analysis of computer, communication and production systems.
Publications
Books, a selection:
1977. Analysis of Models for Tandem Queues. Doctoral thesis Utrecht University.
1983. Boundary Value Problems in Queueing System Analysis. Editor with Wim Cohen
Articles, a selection:
Boxma, Onno J., and Hans Daduna. "Sojourn times in queueing networks." CWI. Department of Operations Research, Statistics, and System Theory [BS] R 8916 (1989): 1-47.
Boxma, Onno J. "Workloads and waiting times in single-server systems with multiple customer classes." Queueing Systems 5.1-3 (1989): 185-214.
Cohen, Jacob Willem, and Onno J. Boxma. Boundary value problems in queueing system analysis. Elsevier, 2000.
Albrecher, Hansjörg, and Onno J. Boxma. "A ruin model with dependence between claim sizes and claim intervals." Insurance: Mathematics and Economics 35.2 (2004): 245-254.
References
1952 births
Living people
Dutch mathematicians
Delft University of Technology alumni
Utrecht University alumni
Academic staff of Utrecht University
Academic staff of Tilburg University
Academic staff of the Eindhoven University of Technology
Scientists from The Hague
Queueing theorists |
https://en.wikipedia.org/wiki/Radicial%20morphism | In algebraic geometry, a morphism of schemes
f: X → Y
is called radicial or universally injective, if, for every field K the induced map X(K) → Y(K) is injective. (EGA I, (3.5.4)) This is a generalization of the notion of a purely inseparable extension of fields (sometimes called a radicial extension, which should not be confused with a radical extension.)
It suffices to check this for K algebraically closed.
This is equivalent to the following condition: f is injective on the topological spaces and for every point x in X, the extension of the residue fields
k(f(x)) ⊂ k(x)
is radicial, i.e. purely inseparable.
It is also equivalent to every base change of f being injective on the underlying topological spaces. (Thus the term universally injective.)
Radicial morphisms are stable under composition, products and base change. If gf is radicial, so is f.
References
, section I.3.5.
, see section V.5.
Morphisms of schemes |
https://en.wikipedia.org/wiki/Logarithmic%20norm | In mathematics, the logarithmic norm is a real-valued functional on operators, and is derived from either an inner product, a vector norm, or its induced operator norm. The logarithmic norm was independently introduced by Germund Dahlquist and Sergei Lozinskiĭ in 1958, for square matrices. It has since been extended to nonlinear operators and unbounded operators as well. The logarithmic norm has a wide range of applications, in particular in matrix theory, differential equations and numerical analysis. In the finite-dimensional setting, it is also referred to as the matrix measure or the Lozinskiĭ measure.
Original definition
Let be a square matrix and be an induced matrix norm. The associated logarithmic norm of is defined
Here is the identity matrix of the same dimension as , and is a real, positive number. The limit as equals , and is in general different from the logarithmic norm , as for all matrices.
The matrix norm is always positive if , but the logarithmic norm may also take negative values, e.g. when is negative definite. Therefore, the logarithmic norm does not satisfy the axioms of a norm. The name logarithmic norm, which does not appear in the original reference, seems to originate from estimating the logarithm of the norm of solutions to the differential equation
The maximal growth rate of is . This is expressed by the differential inequality
where is the upper right Dini derivative. Using logarithmic differentiation the differential inequality can also be written
showing its direct relation to Grönwall's lemma. In fact, it can be shown that the norm of the state transition matrix associated to the differential equation is bounded by
for all .
Alternative definitions
If the vector norm is an inner product norm, as in a Hilbert space, then the logarithmic norm is the smallest number such that for all
Unlike the original definition, the latter expression also allows to be unbounded. Thus differential operators too can have logarithmic norms, allowing the use of the logarithmic norm both in algebra and in analysis. The modern, extended theory therefore prefers a definition based on inner products or duality. Both the operator norm and the logarithmic norm are then associated with extremal values of quadratic forms as follows:
Properties
Basic properties of the logarithmic norm of a matrix include:
for scalar
where is the maximal real part of the eigenvalues of
for
Example logarithmic norms
The logarithmic norm of a matrix can be calculated as follows for the three most common norms. In these formulas, represents the element on the th row and th column of a matrix .
Applications in matrix theory and spectral theory
The logarithmic norm is related to the extreme values of the Rayleigh quotient. It holds that
and both extreme values are taken for some vectors . This also means that every eigenvalue of satisfies
.
More generally, the logarithmic norm is related to the numerical ran |
https://en.wikipedia.org/wiki/Beita%2C%20Nablus | Beita (, translation: "Home") is a Palestinian town in the Nablus Governorate in the northern West Bank located southeast of Nablus. According to the Palestinian Central Bureau of Statistics, the town had a population of 11,682 in 2017. It consists of five clans which branch out to thirty families. There are many houses dating back to the Roman era. The current mayor, elected in 2004 is Arab ash-Shurafa.
The town contains four mosques and three clinics. Since 1967, under the Israeli occupation of the West Bank, more than 77 Beita villagers have been shot dead by Israeli forces, many during protests, 7 were killed between May and September 2021 during the suppression of demonstrations against the establishment of an Israeli outpost on Beita lands.
Location
Beita (including Za'tara locality) is located – south of Nablus. It is bordered by Osarin and Aqraba to the east, Awarta and Odala to the north, Huwwara and Yasuf to the west, and Yatma and Qabalan to the south.
History
There are two historical centres in Beita; Beita el-Fauqa ("The upper Beita") to the North-East and Beita et-Tahta ("The lower Beita") to the South-West. In Beita el-Fauqa, pottery sherds from the Iron Age II/Persian. Persian and Mamluk era have been found, while at Beita et-Tatha sherds from the Iron Age II, Persian, Roman/Byzantine, Byzantine, and Mamluk era have been found.
Ottoman era
Beita was incorporated into the Ottoman Empire in 1517 with all of Palestine, and both in Beita el-Fauqa and Beita et-Tatha sherds from the early Ottoman era have been found.
In 1596 Beita appeared in the tax registers as being in the Nahiya of Jabal Qubal of the Liwa of Nablus. It had a population of 50 households, all Muslim. The villagers paid taxes on wheat, barley, summer crops, olive trees, occasional revenues, goats and/or beehives, and a press for olives or grapes; a total of 8,000 Akçe.
In 1838, Edward Robinson noted Beita as a "large village", located in the El-Beitawy district, east of Nablus.
In 1882, the PEF's Survey of Western Palestine described it as "A large village, with a kind of suburb to the south, near which are ancient tombs. It is supplied by wells, and surrounded by olives. It stands upon the hills east of the Mukhnah plain, and is the capital of the district named from it."
British Mandate era
In the 1922 census of Palestine conducted by the British Mandate authorities, Beita had a population of 883, all Muslims, increasing at the time of the 1931 census to 1,194, still all Muslim, in 286 houses.
In the 1945 statistics Beita had a population of 1,580 Muslims, with 17,542 dunams of land, according to an official land and population survey. Of this, 5,666 dunams were plantations and irrigable land, 6,916 used for cereals, while 76 dunams were built-up land.
Jordanian era
In the wake of the 1948 Arab–Israeli War, and after the 1949 Armistice Agreements, Beita came under annexed Jordanian rule.
At the beginning of 1930s Shaikh Rezeq Abdelrazeq Elyan Open the |
https://en.wikipedia.org/wiki/Goldbach%E2%80%93Euler%20theorem | In mathematics, the Goldbach–Euler theorem (also known as Goldbach's theorem), states that the sum of 1/(p − 1) over the set of perfect powers p, excluding 1 and omitting repetitions, converges to 1:
This result was first published in Euler's 1737 paper "Variæ observationes circa series infinitas". Euler attributed the result to a letter (now lost) from Goldbach.
Proof
Goldbach's original proof to Euler involved assigning a constant to the harmonic series:
, which is divergent. Such a proof is not considered rigorous by modern standards. There is a strong resemblance between the method of sieving out powers employed in his proof and the method of factorization used to derive Euler's product formula for the Riemann zeta function.
Let x be given by
Since the sum of the reciprocal of every power of two is , subtracting the terms with powers of two from x gives
Repeat the process with the terms with the powers of three:
Absent from the above sum are now all terms with powers of two and three. Continue by removing terms with powers of 5, 6 and so on until the right side is exhausted to the value of 1. Eventually, we obtain the equation
which we rearrange into
where the denominators consist of all positive integers that are the non-powers minus one. By subtracting the previous equation from the definition of x given above, we obtain
where the denominators now consist only of perfect powers minus one.
While lacking mathematical rigor, Goldbach's proof provides a reasonably intuitive argument for the theorem's truth. Rigorous proofs require proper and more careful treatment of the divergent terms of the harmonic series. Other proofs make use of the fact that the sum of 1/p over the set of perfect powers p, excluding 1 but including repetitions, converges to 1 by demonstrating the equivalence:
See also
Goldbach's conjecture
List of sums of reciprocals
References
.
Theorems in analysis
Mathematical series
Articles containing proofs |
https://en.wikipedia.org/wiki/Weakly%20measurable%20function | In mathematics—specifically, in functional analysis—a weakly measurable function taking values in a Banach space is a function whose composition with any element of the dual space is a measurable function in the usual (strong) sense. For separable spaces, the notions of weak and strong measurability agree.
Definition
If is a measurable space and is a Banach space over a field (which is the real numbers or complex numbers ), then is said to be weakly measurable if, for every continuous linear functional the function
is a measurable function with respect to and the usual Borel -algebra on
A measurable function on a probability space is usually referred to as a random variable (or random vector if it takes values in a vector space such as the Banach space ).
Thus, as a special case of the above definition, if is a probability space, then a function is called a (-valued) weak random variable (or weak random vector) if, for every continuous linear functional the function
is a -valued random variable (i.e. measurable function) in the usual sense, with respect to and the usual Borel -algebra on
Properties
The relationship between measurability and weak measurability is given by the following result, known as Pettis' theorem or Pettis measurability theorem.
A function is said to be almost surely separably valued (or essentially separably valued) if there exists a subset with such that is separable.
In the case that is separable, since any subset of a separable Banach space is itself separable, one can take above to be empty, and it follows that the notions of weak and strong measurability agree when is separable.
See also
References
Functional analysis
Measure theory
Types of functions |
https://en.wikipedia.org/wiki/Arthur%20Bleksley | Arthur Edward Herbert Bleksley (1908, Matatiele – 1984) was a South African Professor of Applied Mathematics and an astronomer. Bleksley's early research involved the astrophysics and astronomy of variable stars. He encouraged science awareness in South Africa by publishing articles about science, by being on a popular radio show, and through his presentations at the Johannesburg Planetarium.
Education and career
Bleksley was born in the Eastern Cape and attended the Outeniqua High School in George. After matriculation he studied at Stellenbosch University and graduated cum laude in 1927 and went on to obtain his M.Sc. in 1929, winning the Van der Horst Prize. In 1930 he joined the Solar Research Station run by the National Geographic Society and the Smithsonian Institution at Brukkaros, the caldera of an extinct volcano in South West Africa (Namibia). American researcher William H. Hoover and his colleague Frederick Atwood Greeley, ran an observatory on the mountain from 1926 to December 1931, collecting solar radiation data so as to find a correlation with the earth's weather. To this end detailed observations were made of the Solar Constant. High-altitude observatories were set up at various locations - Mount Montezuma in Chile, initially at Mount Harqua Hala in Arizona (later moved to Table Mountain in California) and lastly Mount Brukkaros, a site selected by Charles Greeley Abbot, and later moved to Mount St. Katherine on the Sinai peninsula. The Brukkaros observatory consisted of a 10m deep tunnel in the flank of the mountain. A solar telescope or coelostat at the mouth of the tunnel passed sunlight to a spectrograph, an Ångström compensation pyrheliometer and a bolometer further in.
In 1932 Bleksley was appointed as Junior Lecturer in the Department of Applied Mathematics at the University of Witwatersrand, eventually becoming head of the department. Whilst there he worked on and completed his doctoral thesis A Statistical and Analytical Study of the Phenomenon of Long-period Stellar Variability. Bleksley developed a mathematical model of radially pulsating stars, and the observations for the Cepheid variable stars and long-period variable stars compared favourably with his model's predictions. During this period he took sabbatical leave and studied under Sir Arthur Eddington at Cambridge, Professor Hans Ludendorff of the Astrophysical Observatory at Potsdam and Ejnar Hertzsprung at Leiden. He was President of the Astronomical Society of Southern Africa in 1948/49 and one of about 100 people who attended the founding of the SA Institute of Physics on 7 July 1955.
Personal life
Bleksley showed great interest in parapsychology and in 1969 attended a conference at Saint-Paul de Vence in France. The conference dealt with creativity and its possible links to parapsychology. Other participants included Kenneth Burke, Eugenio Gaddini, the Italian psychoanalyst, Jerre Mangione, the Italian-American author, Emilio Servadio, the Italian-Indian |
https://en.wikipedia.org/wiki/Interpretation%20%28logic%29 | An interpretation is an assignment of meaning to the symbols of a formal language. Many formal languages used in mathematics, logic, and theoretical computer science are defined in solely syntactic terms, and as such do not have any meaning until they are given some interpretation. The general study of interpretations of formal languages is called formal semantics.
The most commonly studied formal logics are propositional logic, predicate logic and their modal analogs, and for these there are standard ways of presenting an interpretation. In these contexts an interpretation is a function that provides the extension of symbols and strings of symbols of an object language. For example, an interpretation function could take the predicate T (for "tall") and assign it the extension {a} (for "Abraham Lincoln"). Note that all our interpretation does is assign the extension {a} to the non-logical constant T, and does not make a claim about whether T is to stand for tall and 'a' for Abraham Lincoln. Nor does logical interpretation have anything to say about logical connectives like 'and', 'or' and 'not'. Though we may take these symbols to stand for certain things or concepts, this is not determined by the interpretation function.
An interpretation often (but not always) provides a way to determine the truth values of sentences in a language. If a given interpretation assigns the value True to a sentence or theory, the interpretation is called a model of that sentence or theory.
Formal languages
A formal language consists of a possibly infinite set of sentences (variously called words or formulas) built from a fixed set of letters or symbols. The inventory from which these letters are taken is called the alphabet over which the language is defined. To distinguish the strings of symbols that are in a formal language from arbitrary strings of symbols, the former are sometimes called well-formed formulæ (wff). The essential feature of a formal language is that its syntax can be defined without reference to interpretation. For example, we can determine that (P or Q) is a well-formed formula even without knowing whether it is true or false.
Example
A formal language can be defined with the
alphabet , and with a word being in if it begins with and is composed solely of the symbols and .
A possible interpretation of could assign the decimal digit '1' to and '0' to . Then would denote 101 under this interpretation of .
Logical constants
In the specific cases of propositional logic and predicate logic, the formal languages considered have alphabets that are divided into two sets: the logical symbols (logical constants) and the non-logical symbols. The idea behind this terminology is that logical symbols have the same meaning regardless of the subject matter being studied, while non-logical symbols change in meaning depending on the area of investigation.
Logical constants are always given the same meaning by every interpretation of the standard ki |
https://en.wikipedia.org/wiki/Kawasaki%27s%20theorem | Kawasaki's theorem or Kawasaki–Justin theorem is a theorem in the mathematics of paper folding that describes the crease patterns with a single vertex that may be folded to form a flat figure. It states that the pattern is flat-foldable if and only if alternatingly adding and subtracting the angles of consecutive folds around the vertex gives an alternating sum of zero.
Crease patterns with more than one vertex do not obey such a simple criterion, and are NP-hard to fold.
The theorem is named after one of its discoverers, Toshikazu Kawasaki. However, several others also contributed to its discovery, and it is sometimes called the Kawasaki–Justin theorem or Husimi's theorem after other contributors, Jacques Justin and Kôdi Husimi.
Statement
A one-vertex crease pattern consists of a set of rays or creases drawn on a flat sheet of paper, all emanating from the same point interior to the sheet. (This point is called the vertex of the pattern.) Each crease must be folded, but the pattern does not specify whether the folds should be mountain folds or valley folds. The goal is to determine whether it is possible to fold the paper so that every crease is folded, no folds occur elsewhere, and the whole folded sheet of paper lies flat.
To fold flat, the number of creases must be even. This follows, for instance, from Maekawa's theorem, which states that the number of mountain folds at a flat-folded vertex differs from the number of valley folds by exactly two folds. Therefore, suppose that a crease pattern consists of an even number of creases, and let be the consecutive angles between the creases around the vertex, in clockwise order, starting at any one of the angles. Then Kawasaki's theorem states that the crease pattern may be folded flat if and only if the alternating sum and difference of the angles adds to zero:
An equivalent way of stating the same condition is that, if the angles are partitioned into two alternating subsets, then the sum of the angles in either of the two subsets is exactly 180 degrees. However, this equivalent form applies only to a crease pattern on a flat piece of paper, whereas the alternating sum form of the condition remains valid for crease patterns on conical sheets of paper with nonzero defect at the vertex.
Local and global flat-foldability
Kawasaki's theorem, applied to each of the vertices of an arbitrary crease pattern, determines whether the crease pattern is locally flat-foldable, meaning that the part of the crease pattern near the vertex can be flat-folded. However, there exist crease patterns that are locally flat-foldable but that have no global flat folding that works for the whole crease pattern at once. conjectured that global flat-foldability could be tested by checking Kawasaki's theorem at each vertex of a crease pattern, and then also testing bipartiteness of an undirected graph associated with the crease pattern. However, this conjecture was disproven by , who showed that Hull's conditions are |
https://en.wikipedia.org/wiki/Sieve%20of%20Sundaram | In mathematics, the sieve of Sundaram is a variant of the sieve of Eratosthenes, a simple deterministic algorithm for finding all the prime numbers up to a specified integer. It was discovered by Indian student S. P. Sundaram in 1934.
Algorithm
Start with a list of the integers from 1 to n. From this list, remove all numbers of the form where:
The remaining numbers are doubled and incremented by one, giving a list of the odd prime numbers (i.e., all primes except 2) below .
The sieve of Sundaram sieves out the composite numbers just as the sieve of Eratosthenes does, but even numbers are not considered; the work of "crossing out" the multiples of 2 is done by the final double-and-increment step. Whenever Eratosthenes' method would cross out k different multiples of a prime , Sundaram's method crosses out for .
Correctness
If we start with integers from to , the final list contains only odd integers from to . From this final list, some odd integers have been excluded; we must show these are precisely the composite odd integers less than .
Let be an odd integer of the form . Then, is excluded if and only if is of the form , that is . Then we have:
So, an odd integer is excluded from the final list if and only if it has a factorization of the form — which is to say, if it has a non-trivial odd factor. Therefore the list must be composed of exactly the set of odd prime numbers less than or equal to .
def sieve_of_Sundaram(n):
"""The sieve of Sundaram is a simple deterministic algorithm for finding all the prime numbers up to a specified integer."""
k = (n - 2) // 2
integers_list = [True] * (k + 1)
for i in range(1, k + 1):
j = i
while i + j + 2 * i * j <= k:
integers_list[i + j + 2 * i * j] = False
j += 1
if n > 2:
print(2, end=' ')
for i in range(1, k + 1):
if integers_list[i]:
print(2 * i + 1, end=' ')
Asymptotic Complexity
The above obscure but as commonly implemented Python version of the Sieve of Sundaram hides the true complexity of the algorithm due to the following reasons:
The range for the outer i looping variable is much too large, resulting in redundant looping that can't perform any composite number representation culling; the proper range is to the array index represent odd numbers less than the square root of the range.
The code doesn't properly account for indexing of Python arrays, which are zero index based so that it ignores the values at the bottom and top of the array; this is a minor issue, but serves to show that the algorithm behind the code has not been clearly understood.
The inner culling loop (the j loop) exactly reflects the way the algorithm is formulated, but seemingly without realizing that the indexed culling starts at exactly the index representing the square of the base odd number and that the indexing using multiplication can much more easily be expressed as a simple repeated addition of the base odd |
https://en.wikipedia.org/wiki/Structural%20break | In econometrics and statistics, a structural break is an unexpected change over time in the parameters of regression models, which can lead to huge forecasting errors and unreliability of the model in general. This issue was popularised by David Hendry, who argued that lack of stability of coefficients frequently caused forecast failure, and therefore we must routinely test for structural stability. Structural stability − i.e., the time-invariance of regression coefficients − is a central issue in all applications of linear regression models.
Structural break tests
A single break in mean with a known breakpoint
For linear regression models, the Chow test is often used to test for a single break in mean at a known time period for . This test assesses whether the coefficients in a regression model are the same for periods and .
Other forms of structural breaks
Other challenges occur where there are:
Case 1: a known number of breaks in mean with unknown break points;
Case 2: an unknown number of breaks in mean with unknown break points;
Case 3: breaks in variance.
The Chow test is not applicable in these situations, since it only applies to models with a known breakpoint and where the error variance remains constant before and after the break. Bayesian methods exist to address these difficult cases via Markov chain Monte Carlo inference.
In general, the CUSUM (cumulative sum) and CUSUM-sq (CUSUM squared) tests can be used to test the constancy of the coefficients in a model. The bounds test can also be used. For cases 1 and 2, the sup-Wald (i.e., the supremum of a set of Wald statistics), sup-LM (i.e., the supremum of a set of Lagrange multiplier statistics), and sup-LR (i.e., the supremum of a set of likelihood ratio statistics) tests developed by Andrews (1993, 2003) may be used to test for parameter instability when the number and location of structural breaks are unknown. These tests were shown to be superior to the CUSUM test in terms of statistical power, and are the most commonly used tests for the detection of structural change involving an unknown number of breaks in mean with unknown break points. The sup-Wald, sup-LM, and sup-LR tests are asymptotic in general (i.e., the asymptotic critical values for these tests are applicable for sample size as ), and involve the assumption of homoskedasticity across break points for finite samples; however, an exact test with the sup-Wald statistic may be obtained for a linear regression model with a fixed number of regressors and independent and identically distributed (IID) normal errors. A method developed by Bai and Perron (2003) also allows for the detection of multiple structural breaks from data.
The MZ test developed by Maasoumi, Zaman, and Ahmed (2010) allows for the simultaneous detection of one or more breaks in both mean and variance at a known break point. The sup-MZ test developed by Ahmed, Haider, and Zaman (2016) is a generalization of the MZ test which allows for the detect |
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