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https://en.wikipedia.org/wiki/Presidential%20Award%20for%20Excellence%20in%20Mathematics%20and%20Science%20Teaching | The Presidential Award for Excellence in Mathematics and Science Teaching (PAEMST) is the highest recognition that a kindergarten through 12th-grade mathematics or science teacher may receive for outstanding teaching in the United States. Authorized by the Education for Economic Security Act in 1984, this program authorizes the President to bestow up to 108 awards each year. The National Science Foundation (NSF) administers PAEMST on behalf of The White House Office of Science and Technology Policy.
Eligibility
The following are the eligibility criteria for nominees. They must:
Be highly qualified teachers, as deemed by their states, districts, or schools; teachers in private/independent schools should meet the spirit of the principles and provisions of the No Child Left Behind Act, Public Law 107-110.
Hold a degree or appropriate credentials in the category for which they are applying.
Teach in one of the 50 states or four U.S. jurisdictions. The jurisdictions are Washington, D.C., Puerto Rico, Department of Defense Schools, and the U.S. territories as a group (American Samoa, Guam, the Commonwealth of the Northern Mariana Islands, and the U.S. Virgin Islands).
Be full-time employees of the school or school district.
Have at least 5 years of mathematics or science teaching experience prior to application.
Teach mathematics or science at the appropriate grade level in a public or private school. The award is made to teachers of kindergarten through 6th grades and 7th through 12 grades in alternate years.
Not have received the national PAEMST award in any prior competition or category.
See also
Awards and decorations of the United States government
List of mathematics awards
References
External links
Official site
Teacher awards
National Science Foundation
Mathematics education awards
Science education in the United States
Science and technology in the United States
Recurring events established in 1983
1983 establishments in Washington, D.C.
American education awards
Civil awards and decorations of the United States |
https://en.wikipedia.org/wiki/Law%20of%20the%20unconscious%20statistician | In probability theory and statistics, the law of the unconscious statistician, or LOTUS, is a theorem which expresses the expected value of a function of a random variable in terms of and the probability distribution of .
The form of the law depends on the type of random variable in question. If the distribution of is discrete and one knows its probability mass function , then the expected value of is
where the sum is over all possible values of . If instead the distribution of is continuous with probability density function , then the expected value of is
Both of these special cases can be expressed in terms of the cumulative probability distribution function of , with the expected value of now given by the Lebesgue–Stieltjes integral
In even greater generality, could be a random element in any measurable space, in which case the law is given in terms of measure theory and the Lebesgue integral. In this setting, there is no need to restrict the context to probability measures, and the law becomes a general theorem of mathematical analysis on Lebesgue integration relative to a pushforward measure.
Etymology
This proposition is (sometimes) known as the law of the unconscious statistician because of a purported tendency to think of the identity as the very definition of the expected value, rather than (more formally) as a consequence of its true definition. The naming is sometimes attributed to Sheldon Ross' textbook Introduction to Probability Models, although he removed the reference in later editions. Many statistics textbooks do present the result as the definition of expected value.
Joint distributions
A similar property holds for joint distributions, or equivalently, for random vectors. For discrete random variables X and Y, a function of two variables g, and joint probability mass function :
In the absolutely continuous case, with being the joint probability density function,
Special cases
A number of special cases are given here. In the simplest case, where the random variable takes on countably many values (so that its distribution is discrete), the proof is particularly simple, and holds without modification if is a discrete random vector or even a discrete random element.
The case of a continuous random variable is more subtle, since the proof in generality requires subtle forms of the change-of-variables formula for integration. However, in the framework of measure theory, the discrete case generalizes straightforwardly to general (not necessarily discrete) random elements, and the case of a continuous random variable is then a special case by making use of the Radon–Nikodym theorem.
Discrete case
Suppose that is a random variable which takes on only finitely or countably many different values , with probabilities . Then for any function of these values, the random variable has values , although some of these may coincide with each other. For example, this is the case if can take on both values |
https://en.wikipedia.org/wiki/International%20Space%20Olympics | The International Space Olympics (ISO) is an annual two-week competition for teenagers aged from 14 to 18, held in Korolyov, Russia. The competition includes examinations in Mathematics, Physics, Computer Science, and English Literature, in addition to presentation of a space related research project.
On days when participants are not competing, they are given tours of some of Russia's top space facilities and areas of cultural significance, and even have a chance to videochat with astronauts and cosmonauts on the ISS. Participants come from a wide range of countries, with each country represented as a team. In previous years, teams have attended from Germany, Greece, Israel, United Kingdom, the United States, Kazakhstan, Australia, Spain and Russia. Overall, in 2012 there were 130 participating students, in 2013 and 2014 - over 200. Over the years, the International Space Olympics has been attended by over 1,850 Russian students, as well as more than 800 students from other countries.
The Olympics are held on the initiative of S. P. Korolyov Energia Space and Rocket Corporation and has an international status. The Opening Ceremony is usually started by the heads of Korolyov City, the Moscow region, municipal heads of education, pilots, cosmonauts and representatives from RSC Energia and MCC (mission control center in Korolyov).
Sometimes, in the media, the Space Olympics is referred to as an "international space olympics camp" for students or "an intellectual marathon". When in Russia, the competitors usually stay at a hostel, and at the end of the week they are taken to the country to sum up the whole event, to hold the award ceremony, to celebrate, and to have a "good-bye" party.
In 2014 China and Venezuela voiced their desire to participate in Space Olympic Games for the first time. Moreover, it is this year that freshmen and sophomores were considered for participation.
In the 2011 competition, Georgia Jones won 1st place in Literature.
In the 2012 competition, Alex Gagliano won 1st place in Astrophysics presentations and Austin Chung won 1st place in Space History and Policy.
In the 2013 competition, Josh Ting won 1st place in Astrophysics and Hema Narlapati won 1st place in Space Policy.
References
Links
High Tech Seminars in Korolev
Science competitions |
https://en.wikipedia.org/wiki/Non-sampling%20error | In statistics, non-sampling error is a catch-all term for the deviations of estimates from their true values that are not a function of the sample chosen, including various systematic errors and random errors that are not due to sampling. Non-sampling errors are much harder to quantify than sampling errors.
Non-sampling errors in survey estimates can arise from:
Coverage errors, such as failure to accurately represent all population units in the sample, or the inability to obtain information about all sample cases;
Response errors by respondents due for example to definitional differences, misunderstandings, or deliberate misreporting;
Mistakes in recording the data or coding it to standard classifications;
Pseudo-opinions given by respondents when they have no opinion, but do not wish to say so
Other errors of collection, nonresponse, processing, or imputation of values for missing or inconsistent data.
An excellent discussion of issues pertaining to non-sampling error can be found in several sources such as Kalton (1983) and Salant and Dillman (1995),
See also
Errors and residuals in statistics
Sampling error
References
Survey methodology
Errors and residuals
Auditing terms |
https://en.wikipedia.org/wiki/Discrete%20Mathematics%20%28journal%29 | Discrete Mathematics is a biweekly peer-reviewed scientific journal in the broad area of discrete mathematics, combinatorics, graph theory, and their applications. It was established in 1971 and is published by North-Holland Publishing Company. It publishes both short notes, full length contributions, as well as survey articles. In addition, the journal publishes a number of special issues each year dedicated to a particular topic. Although originally it published articles in French and German, it now allows only English language articles. The editor-in-chief is Douglas West (University of Illinois, Urbana).
History
The journal was established in 1971. The very first article it published was written by Paul Erdős, who went on to publish a total of 84 papers in the journal.
Abstracting and indexing
The journal is abstracted and indexed in:
According to the Journal Citation Reports, the journal has a 2020 impact factor of 0.87.
Notable publications
The 1972 paper by László Lovász on the study of perfect graphs ()
The 1973 short note "Acyclic orientations of graphs" by Richard Stanley on the study of the chromatic polynomial and its generalizations ()
Václav Chvátal introduced graph toughness in 1973 ()
The 1975 paper by László Lovász on the linear programming relaxation for the set cover problem.
The 1980 paper by Philippe Flajolet on the combinatorics of continued fractions. ()
The 1985 paper by Bressoud and Zeilberger proved Andrews's q-Dyson conjecture ()
References
External links
Combinatorics journals
English-language journals
Discrete mathematics
Academic journals established in 1971
Elsevier academic journals
Semi-monthly journals |
https://en.wikipedia.org/wiki/Expected%20loss | Expected loss is the sum of the values of all possible losses, each multiplied by the probability of that loss occurring.
In bank lending (homes, autos, credit cards, commercial lending, etc.) the expected loss on a loan varies over time for a number of reasons. Most loans are repaid over time and therefore have a declining outstanding amount to be repaid. Additionally, loans are typically backed up by pledged collateral whose value changes differently over time vs. the outstanding loan value.
Three factors are relevant in analyzing expected loss:
Probability of default (PD)
Exposure at default (EAD)
Loss given default (LGD)
Simple example
Original home value $100, loan to value 80%, loan amount $80
outstanding loan $75
current home value $70
liquidation cost $10
Loss given default = Magnitude of likely loss on the exposure / Exposure at default
-$75 loan receivable write off Exposure at default
+$70 house sold
-$10 liquidation cost paid =
-$15 Loss
Express as a %
-15/75 =
20% Loss given default
Probability of default
Since there is negative equity 50 homeowners out of 100 will "toss the keys to the bank and walk away", therefore:
50% probability of default
Expected loss
In %
20% x 50% =10%
In currency
currency loss x probability
$15 * .5 = $7.5
check
loss given default * probability of default * Exposure at default
20% * 50% * $75 = $7.5
Recalculating expected loss
Expected loss is not time-invariant, but rather needs to be recalculated when circumstances change. Sometimes both the probability of default and the loss given default can both rise, giving two reasons that the expected loss increases.
For example, over a 20-year period only 5% of a certain class of homeowners default. However, when a systemic crisis hits, and home values drop 30% for a long period, that same class of borrowers changes their default behavior. Instead of 5% defaulting, say 10% default, largely due to the fact the LGD has catastrophically risen.
To accommodate for that type of situation a much larger expected loss needs to be calculated. This is the subject to considerable research at the national and global levels as it has a large impact on the understanding and mitigation of systemic risk.
See also
Systemic risk
Loss function
Potential future exposure
References
Financial risk
Loans
Credit risk |
https://en.wikipedia.org/wiki/Balance%20equation | In probability theory, a balance equation is an equation that describes the probability flux associated with a Markov chain in and out of states or set of states.
Global balance
The global balance equations (also known as full balance equations) are a set of equations that characterize the equilibrium distribution (or any stationary distribution) of a Markov chain, when such a distribution exists.
For a continuous time Markov chain with state space , transition rate from state to given by and equilibrium distribution given by , the global balance equations are given by
or equivalently
for all . Here represents the probability flux from state to state . So the left-hand side represents the total flow from out of state i into states other than i, while the right-hand side represents the total flow out of all states into state . In general it is computationally intractable to solve this system of equations for most queueing models.
Detailed balance
For a continuous time Markov chain (CTMC) with transition rate matrix , if can be found such that for every pair of states and
holds, then by summing over , the global balance equations are satisfied and is the stationary distribution of the process. If such a solution can be found the resulting equations are usually much easier than directly solving the global balance equations.
A CTMC is reversible if and only if the detailed balance conditions are satisfied for every pair of states and .
A discrete time Markov chain (DTMC) with transition matrix and equilibrium distribution is said to be in detailed balance if for all pairs and ,
When a solution can be found, as in the case of a CTMC, the computation is usually much quicker than directly solving the global balance equations.
Local balance
In some situations, terms on either side of the global balance equations cancel. The global balance equations can then be partitioned to give a set of local balance equations (also known as partial balance equations, independent balance equations or individual balance equations). These balance equations were first considered by Peter Whittle. The resulting equations are somewhere between detailed balance and global balance equations. Any solution to the local balance equations is always a solution to the global balance equations (we can recover the global balance equations by summing the relevant local balance equations), but the converse is not always true. Often, constructing local balance equations is equivalent to removing the outer summations in the global balance equations for certain terms.
During the 1980s it was thought local balance was a requirement for a product-form equilibrium distribution, but Gelenbe's G-network model showed this not to be the case.
Notes
Queueing theory |
https://en.wikipedia.org/wiki/Relatively%20hyperbolic%20group | In mathematics, the concept of a relatively hyperbolic group is an important generalization of the geometric group theory concept of a hyperbolic group. The motivating examples of relatively hyperbolic groups are the fundamental groups of complete noncompact hyperbolic manifolds of finite volume.
Intuitive definition
A group G is relatively hyperbolic with respect to a subgroup H if, after contracting the Cayley graph of G along H-cosets, the resulting graph equipped with the usual graph metric becomes a δ-hyperbolic space and, moreover, it satisfies a technical condition which implies that quasi-geodesics with common endpoints travel through approximately the same collection of cosets and enter and exit these cosets in approximately the same place.
Formal definition
Given a finitely generated group G with Cayley graph Γ(G) equipped with the path metric and a subgroup H of G, one can construct the coned off Cayley graph as follows: For each left coset gH, add a vertex v(gH) to the Cayley graph Γ(G) and for each element x of gH, add an edge e(x) of length 1/2 from x to the vertex v(gH). This results in a metric space that may not be proper (i.e. closed balls need not be compact).
The definition of a relatively hyperbolic group, as formulated by Bowditch goes as follows. A group G is said to be hyperbolic relative to a subgroup H if the coned off Cayley graph has the properties:
It is δ-hyperbolic and
it is fine: for each integer L, every edge belongs to only finitely many simple cycles of length L.
If only the first condition holds then the group G is said to be weakly relatively hyperbolic with respect to H.
The definition of the coned off Cayley graph can be generalized to the case of a collection of subgroups and yields the corresponding notion of relative hyperbolicity. A group G which contains no collection of subgroups with respect to which it is relatively hyperbolic is said to be a non relatively hyperbolic group.
Properties
If a group G is relatively hyperbolic with respect to a hyperbolic group H, then G itself is hyperbolic.
If a group G is relatively hyperbolic with respect to a group H then it acts as a geometrically finite convergence group on a compact space, its Bowditch boundary
If a group G is relatively hyperbolic with respect to a group H that has solvable word problem, then G has solvable word problem (Farb), and if H has solvable conjugacy problem, then G has solvable conjugacy problem (Bumagin)
If a group G is torsion-free relatively hyperbolic with respect to a group H, and H has a finite classifying space, then so does G (Dahmani)
If a group G is relatively hyperbolic with respect to a group H that satisfies the Farrell-Jones conjecture, then G satisfies the Farrell-jones conjecture (Bartels).
More generally, in many cases (but not all, and not easily or systematically), a property satisfied by all hyperbolic groups and byH can be suspected to be satisfied by G
The isomorphism problem fo |
https://en.wikipedia.org/wiki/Angolans%20in%20Portugal | Angolans in Portugal form the country's second-largest group of African migrants, after Cape Verdeans. In 2006, official statistics showed 28,854 legal Angolan residents in Portugal. However, this number is likely an underestimate of the true size of the community, as it does not count people of Angolan origin who hold Portuguese citizenship. In 2022 INE counted 31,614 Angolans living in Portugal.
Migration history
Large-scale migratory flow from Angola to Portugal began in the 1970s, around the time of Angolan independence. However, this early flow consisted largely of retornados, white Portuguese born in Angola. The bulk of mixed-race or black African migrants came later. After the 2002 peace agreement which ended the Angolan Civil War, many Angolan migrants in Portugal returned to Angola. By 2003, statistics of the Angolan embassy in Portugal showed that between 8,000 and 10,000 had already returned, and that 400 people a week were flying from Portugal to the Angolan capital Luanda. However, statistics of the Instituto Nacional de Estatística showed that the population of Angolan legal residents did not decrease from 2001 to 2003, but instead grew by 12.6% (from 22,751 to 25,616 people).
Demographics of Migrants
The most intense period for migration was in the 1990s. Most Angolans who came to Portugal would come under a tourist visa and overstay, converting into residents and citizens. Others would seek asylum. A sizeable majority of those who did not migrate to Portugal instead migrated to South Africa. Of the migrants, many were either young adults or teenagers. 69.5% of Angolans who migrated to Portugal were male. 76.4% of migrants had some sort of occupation at the time of their asylum interviews. 68.8% of incoming migrants had completed a basic level of education. Migrants typically came from middle to low-middle socioeconomic backgrounds. Emigration to Portugal from Angola was a result of war, economic instability, academic aspirations, and new opportunities. Existing networks created linkages by which made immigration and transitioning to Portuguese-living easier. Linkages to the Angolan homeland remained strong despite physical detachment, these linkages are especially true when speaking in regards to economic obligations.
The Process for Family Migration
In 1998, Law 244/98 in Lisbon was passed allowing for families to reconnect in Portugal. The condition was that the individual that is already residing in Lisbon is supposed to have lived there for at least one year. The individual within Portugal can also petition for his or her family member that lives outside of Portugal. The process to residency for the petitioned family member begins with temporary living status, that can be renewed once it expires. After appealing for additional years, they may be authorized for residency, not depending on the status of the already resident individual who petitioned. This can all take place under the prerequisite that the migrant can provid |
https://en.wikipedia.org/wiki/PPSMI | Pengajaran dan Pembelajaran Sains dan Matematik Dalam Bahasa Inggeris (PPSMI) (the teaching and learning of science and mathematics in English) is a government policy aimed at improving the command of the English language among pupils at primary and secondary schools in Malaysia. In accordance to this policy, the Science and Mathematics subjects are taught in the English medium as opposed to the Malay medium used before. This policy was introduced in 2003 by the then-Prime Minister of Malaysia, Mahathir Mohamad. PPSMI has been the subject of debate among academics, politicians and the public alike, which culminated to the announcement of the policy's reversal in 2012 by the Deputy Prime Minister, Muhyiddin Yassin.
History
PPSMI's inception as a Malaysian Government policy was the result of the Cabinet meeting on 19 July 2002 under the administration of the fourth prime minister, Mahathir Mohamad. According to the Malaysian Ministry of Education, the policy would be run in stages, starting with the 2003 school session, pioneered by the all students of Year 1 in primary education level, and Form 1 of the secondary education level. PPSMI was then fully implemented to all secondary students in 2007, and to all primary students in 2008.
Objective
According to the statement regarding PPSMI in the Ministry of Education's website:
When proposing the policy, Mahathir Mohamad was in the opinion that Malaysia's progress is declining in the age of globalisation, and he had hoped that this policy gives a competitive edge to the nation, following the footsteps of Singapore and India which are moving forward because of their utilisation of the English language.
Implementation
PPSMI was implemented for the 2003 school session students enrolling in Year 1 and Form 1 in primary and secondary schools respectively. Students of other grades are not affected, and continued to study Mathematics and Science in the mother tongue. PPSMI learning materials were offered in the form of packages consisting of these components:
Textbooks: Given to students as the basic source for learning on concepts and skills in Science and Mathematics.
Activity Books: Given as supplementary material for students to practice their understanding of the concepts learned from textbooks. These "Buku Latihan dan Aktiviti" or "BLA" were provided for students of Year 1 only.
Teacher's Guide: Material prepared for teachers as reference and guide to plan and implement effective teaching of the Science and Mathematics.
MyCD or "Pupil's CD-ROM": "BLA" in the form of multimedia presentations recorded in compacts discs and included with every Activity Book. The contents consists of an interactive games and simulators as well as electronic tests.
Teacher's CD-ROM: This material was meant to help teachers to plan and implement effective teaching of the Science and Mathematics. Among the contents are questions banks, additional activities as well as URLs to websites with relevance to the subject. |
https://en.wikipedia.org/wiki/Lim%20Jong-eun | Lim Jong-Eun (; born 18 June 1990) is a South Korean footballer who currently plays for Ulsan Hyundai.
Club career statistics
Honours
Club
Ulsan Hyundai
K League 1: 2022
References
External links
FIFA Player Statistics
1990 births
Living people
Men's association football defenders
South Korean men's footballers
Ulsan Hyundai FC players
Seongnam FC players
Jeonnam Dragons players
Jeonbuk Hyundai Motors players
K League 1 players |
https://en.wikipedia.org/wiki/RV%20coefficient | In statistics, the RV coefficient
is a multivariate generalization of the squared Pearson correlation coefficient (because the RV coefficient takes values between 0 and 1).
It measures the closeness of two set of points that may each be represented in a matrix.
The major approaches within statistical multivariate data analysis can all be brought into a common framework in which the RV coefficient is maximised subject to relevant constraints. Specifically, these statistical methodologies include:
principal component analysis
canonical correlation analysis
multivariate regression
statistical classification (linear discrimination).
One application of the RV coefficient is in functional neuroimaging where it can measure
the similarity between two subjects' series of brain scans
or between different scans of a same subject.
Definitions
The definition of the RV-coefficient makes use of ideas
concerning the definition of scalar-valued quantities which are called the "variance" and "covariance" of vector-valued random variables. Note that standard usage is to have matrices for the variances and covariances of vector random variables.
Given these innovative definitions, the RV-coefficient is then just the correlation coefficient defined in the usual way.
Suppose that X and Y are matrices of centered random vectors (column vectors) with covariance matrix given by
then the scalar-valued covariance (denoted by COVV) is defined by
The scalar-valued variance is defined correspondingly:
With these definitions, the variance and covariance have certain additive properties in relation to the formation of new vector quantities by extending an existing vector with the elements of another.
Then the RV-coefficient is defined by
Shortcoming of the coefficient and adjusted version
Even though the coefficient takes values between 0 and 1 by construction, it seldom attains values close to 1 as the denominator is often too large with respect to the maximal attainable value of the denominator.
Given known diagonal blocks and of dimensions and respectively, assuming that without loss of generality, it has been proved that the maximal attainable numerator is
where (resp. ) denotes the diagonal matrix of the eigenvalues of (resp. ) sorted decreasingly from the upper leftmost corner to the lower rightmost corner and is the matrix .
In light of this, Mordant and Segers proposed an adjusted version of the RV coefficient in which the denominator is the maximal value attainable by the numerator. It reads
The impact of this adjustment is clearly visible in practice.
See also
Congruence coefficient
Distance correlation
References
Covariance and correlation |
https://en.wikipedia.org/wiki/H.%20W.%20Lloyd%20Tanner | Henry William Lloyd Tanner (generally known as H. W. Lloyd Tanner) (17 January 1851 – 6 March 1915) was Professor of Mathematics at the University College of South Wales and Monmouthshire from 1883 to 1909.
Life
Tanner was born on 17 January 1851 at Burham, Kent and was educated at Bristol Grammar School and Jesus College, Oxford, where he was taught by John Griffiths. He was appointed Professor of Mathematics and Astronomy at the University College of South Wales and Monmouthshire (now Cardiff University) in 1883, and held the post until 1909. He was a Fellow of the Royal Society and a Fellow of the Royal Astronomical Society. Tanner published various papers on differential equations and other subjects in mathematics.
The President of the Royal Society, Sir William Crookes, said in his anniversary address in November 1915 that Tanner's death meant that mathematical science had lost "one of its most distinguished exponents", one who published "many important investigations in mathematics" that were "distinguished by great ingenuity and originality". Crookes also said that the university was deeply indebted to Tanner's educational and administrative talents.
References
1851 births
1915 deaths
19th-century British mathematicians
20th-century British mathematicians
People educated at Bristol Grammar School
Alumni of Jesus College, Oxford
Academics of Cardiff University
Fellows of the Royal Society
People from Tonbridge and Malling (district) |
https://en.wikipedia.org/wiki/List%20of%20mosques%20in%20Iran | In 2015 it was estimated, as per official statistics, that there are 47,291 Shiite mosques and 10,344 Sunni mosques in Iran.
List of mosques in Iran
This is a list of mosques in Iran.
Ardabil Province
Jome mosque
Jameh Mosque of Germi
Jameh Mosque of Namin
East Azerbaijan Province
Jameh Mosque of Ahar
Jameh Mosque of Tabriz
Jameh Mosque of Sarab
Hajj Safar Ali Mosque
Saheb-ol-Amr Mosque
Jameh Mosque of Marand
Jameh Mosque of Mehrabad
Blue Mosque, Tabriz
Stone Tark Mosque
Mirpanj Mosque
Gilan Province
Hajj Samad Khan Mosque
Chahar Padshahan
Golestan Province
Jameh Mosque of Gorgan
Fars Province
Jameh Mosque of Atigh
Vakil Mosque
Nasir-ol-molk Mosque
Jameh Mosque of Lar
Jameh Mosque of Kabir Neyriz
Jameh Mosque of Jahrom
Jameh Mosque of Darab
Jameh Mosque of Arsanjan
Hamadan Province
Jameh Mosque of Sarabi
Hormozgan Province
Malek bin Abbas Mosque
Jameh Mosque of Bastak
Jameh Mosque of Bandar Abbas
Jameh Mosque of Qiblah
Jameh Mosque of Qeshm
Isfahan Province
Agha Bozorg Mosque
Jameh Mosque of Ashtarjan
Jameh Mosque of Isfahan
Jameh Mosque of Khansar
Jameh Mosque of Khozan
Jameh Mosque of Zavareh
Jameh Mosque of Golpayegan
Jameh Mosque of Nain
Jameh Mosque of Natanz
Jameh Mosque of Nushabad
Jarchi Mosque
Agha Nour Mosque
Ali Gholi Agha Mosque
Barsian mosque and minaret
Darvazeh No Mosque
Dashti Mosque
Rahim Khan Mosque
Gar mosque and minaret
Ilchi Mosque
Roknolmolk Mosque
Kaj Mosque
Maghsoudbeyk Mosque
Meydan Mosque, Kashan
Mohammad Jafar Abadei Mosque
Hakim Mosque, Isfahan
Seyyed Mosque (Isfahan)
Shah Mosque (Isfahan)
Sheikh Lotfollah Mosque
Safa Mosque
Tabriziha Mosque
Lonban Mosque
Mesri Mosque
Hafshuye Mosque
Kerman Province
Jameh Mosque of Kerman
Malek Mosque
Hajj Agha Ali Mosque
Pamenar Mosque, Kerman
Kermanshah Province
Jameh Mosque of Kermanshah
Jameh Mosque of Shafei
Emad o dolah Mosque
Abdullah ibn Umar Mosque
Hajj Shahbazkhan Mosque
Khuzestan Province
Jameh Mosque of Dezful
Jameh Mosque of Shushtar
Rangooniha Mosque
Jameh Mosque of Khorramshahr
Kurdistan Province
Dar ul-Ihsan Mosque
Hajar Khatoon Mosque
Lorestan Province
Jameh Mosque of Borujerd
Soltani Mosque of Borujerd
Markazi Province
Jameh Mosque of Arak
Agha Zia ol Din Mosque
Jameh Mosque of Saveh
Mazandaran Province
Farahabad Mosque
Jameh Mosque of Amol
Jameh Mosque of Babol
Jameh Mosque of Sari
Qazvin Province
Jameh Mosque of Qazvin
Al-Nabi Mosque, Qazvin
Heidarieh Mosque, Qazvin
Jameh Mosque of Qerveh
Qom Province
Jameh Mosque of Qom
Jameh Mosque of Pachian
Imam Hasan al-Askari Mosque
Azam mosque of Qom
Chehel Akhtaran Mosque
Jamkaran Mosque
Razavi Khorasan Province
Sunni Mosque of Lotfabad
Gonbad Kabud Mosque
Goharshad Mosque
Haji Jalal Mosque
Howz-e Ma'jardar Mosque
Jameh Mosque of Gonabad
Jameh Mosque of Kashmar
Jameh Mosque of Marandiz
Jameh Mosque of Nishapur
Jameh Mosque of Radkan
Jameh Mosque of Sabzevar
Khosro |
https://en.wikipedia.org/wiki/Artur%20Tlisov | Artur Ruslanovich Tlisov (; born 10 June 1982) is a Russian former football player. He made his debut in the Russian Premier League in 2001 for FC Chernomorets Novorossiysk.
Career statistics
Honours
Russian Premier League champion: 2003.
External links
Profile on the FC Kuban Krasnodar site
1982 births
Living people
People from Cherkessk
Russian men's footballers
Russia men's under-21 international footballers
Men's association football midfielders
FC Chernomorets Novorossiysk players
PFC CSKA Moscow players
FC Kuban Krasnodar players
Russian Premier League players
Sportspeople from Karachay-Cherkessia
FC Nart Cherkessk players |
https://en.wikipedia.org/wiki/Math%2055 | Math 55 is a two-semester long freshman undergraduate mathematics course at Harvard University founded by Lynn Loomis and Shlomo Sternberg. The official titles of the course are Studies in Algebra and Group Theory (Math 55a) and Studies in Real and Complex Analysis (Math 55b). Previously, the official title was Honors Advanced Calculus and Linear Algebra.
Description
In the past, Harvard University's Department of Mathematics had described Math 55 as "probably the most difficult undergraduate math class in the country." But Math 55 lecturer for 2022 Professor Denis Auroux clarified that "if you’re reasonably good at math, you love it, and you have lots of time to devote to it, then Math 55 is completely fine for you."
Formerly, students would begin the year in Math 25 (which was created in 1983 as a lower-level Math 55) and, after three weeks of point-set topology and special topics (for instance, in 1994, p-adic analysis was taught by Wilfried Schmid), students would take a quiz. As of 2012, students may choose to enroll in either Math 25 or Math 55 but are advised to "shop" both courses and have five weeks to decide on one.
Depending on the professor teaching the class, the diagnostic exam may still be given after three weeks to help students with their decision. In 1994, 89 students took the diagnostic exam: students scoring more than 50% on the quiz could enroll in Schmid's Math 55 (15 students), students scoring between 10 and 50% could enroll in Benedict Gross's Math 25: Theoretical Linear Algebra and Real Analysis (55 students), and students scoring less than 10% were advised to enroll in a course such as Math 21: Multivariable Calculus (19 students).
In the past, problem sets were expected to take from 24 to 60 hours per week to complete, although some claim that it is closer to 20 hours. In 2022, on average, students spend a total of 20 to 30 hours per week on this class, including homework. In addition to visiting their professor during office hours, students are encouraged to work together on homework assignments. Many have spent much time together working in the "war room" (a place in the Grays Basement) and have developed long-lasting friendships. Taking many other challenging courses and extracurricular activities in the same semester is ill-advised.
Students typically typeset their homework in LaTeX and essentially write their own textbook for this class, which ends with a take-home final exam.
Historical retention rate
In 1970, Math 55 covered almost four years worth of department coursework in two semesters, and subsequently, it drew only the most diligent of undergraduates. Of the 75 students who enrolled in the 1970 offering, by course end, only 20 remained due to the advanced nature of the material and time-constraints under which students were given to work. David Harbater, a mathematics professor at the University of Pennsylvania and student of the 1974 Math 55 section at Harvard, recalled of his experience, "Seventy |
https://en.wikipedia.org/wiki/ICTCM%20Award | The ICTCM Award is presented each year at the International Conference on Technology in Collegiate Mathematics sponsored by Pearson Addison–Wesley & Pearson Prentice Hall publishers. This award, now in its twelfth year, was established by Pearson Education to recognize an individual or group for excellence and innovation in using technology to enhance the teaching and learning of mathematics. Electronic conference proceedings are available beginning with ICTCM 7. List of free electronic journals in mathematics
ICTCM Award recipients
1997 Chicago, ICTCM-10
To: Paul Velleman, Cornell University
For: ActiveStats
1998 New Orleans, ICTCM-11
To: Laurie Hopkins, Columbia College, SC and Amelia Kinard, Columbia College, SC
For: An Update on the Impact of handheld CAS Systems on Developmental Algebra
To: Deborah Hughes Hallett, University of Arizona; Eric Connally, Wellesley College; Rajini Jesudason, Wellesley College (Currently: Benjamin Banneker Charter Public School); Ralph Teixeira, Harvard University; and Graeme Bird, Harvard University
For: Information, Data and Decisions
To: John C. Miller, The City College of CUNY
For: xyAlgebra: Algebra Courseware with Intelligent Help at Every Step
1999 San Francisco, ICTCM-12
To: Michael E. Gage (et al.), University of Rochester and Arnold K. Pizer, University of Rochester
For: WeBWork
To: Christopher Weaver, New Mexico State University
For: Mathematics Accessible to Visually Impaired Students
2000 Atlanta, ICTCM-13
To: Deborah Hughes Hallett, University of Arizona and Richard Thompson, University of Arizona
For: Computer Texts for Business Mathematics
2001 Baltimore, ICTCM-14
To: Bob Richardson, Massey University, New Zealand and Brian Felkel, Appalachian State University, Boone, NC
For: Networked Business Mathematics
2002 Orlando, ICTCM-15
To: Robert L. Devaney, Boston University
For: The Dynamical Systems and Technology Project
2003 Chicago, ICTCM-16
To: James H. Curry, University of Colorado and Anne Dougherty, University of Colorado,
For: Mathematics Visualization Toolkit
2004 New Orleans, ICTCM-17
To: Mike Martin, Johnson County Community College, Overland Park, KS and Steven J. Wilson, Johnson County Community College, Overland Park, KS
For: Dynamic Web Tools for Undergraduate Mathematics
2006 Orlando, ICTCM-18
To: Sarah L. Mabrouk, Framingham State College, Framingham, MA
For: 'Interactive MS Excel Workbooks
2007 Boston, ICTCM-19
To: Mark H. Holmes, Rensselaer Polytechnic Institute
For: Integrating the Learning of Mathematics and Science Using Interactive Teaching and Learning Strategies
2008 San Antonio, ICTCM-20
To: Douglas B. Meade, University of South Carolina and Philip B. Yasskin, Texas A&M University
For: Maplets for Calculus, Tutoring without the Tutor
2009 New Orleans, ICTCM-21
To: Douglas Ensley, Shippensburg University and Barbara Kaskosz, University of Rhode Island
For: Flash and Math Applets: Learn by Example
2010 Chicago, ICTCM-22
To: Paul See |
https://en.wikipedia.org/wiki/International%20Conference%20on%20Technology%20in%20Collegiate%20Mathematics | The International Conference on Technology in Collegiate Mathematics (ICTCM) is an annual conference sponsored by Pearson Addison-Wesley & Pearson Prentice Hall publishers. Electronic proceedings have been available for many years and are included in the List of free electronic journals in mathematics.
Since ICTCM 10, the conference has awarded an annual ICTCM Award to recognize an individual or group for excellence and innovation in using technology to enhance the teaching and learning of mathematics.
References
Mathematics conferences |
https://en.wikipedia.org/wiki/Size%20theory | In mathematics, size theory studies the properties of topological spaces endowed with -valued functions, with respect to the change of these functions. More formally, the subject of size theory is the study of the natural pseudodistance between size pairs.
A survey of size theory can be found in
.
History and applications
The beginning of size theory is rooted in the concept of size function, introduced by Frosini. Size functions have been initially used as a mathematical tool for shape comparison in computer vision and pattern recognition.
An extension of the concept of size function to algebraic topology was made in the 1999 Frosini and Mulazzani paper
where size homotopy groups were introduced, together with the natural pseudodistance for -valued functions.
An extension to homology theory (the size functor) was introduced in 2001.
The size homotopy group and the size functor are strictly related to the concept of persistent homology group
studied in persistent homology. It is worth to point out that the size function is the rank of the -th persistent homology group, while the relation between the persistent homology group
and the size homotopy group is analogous to the one existing between homology groups and homotopy groups.
In size theory, size functions and size homotopy groups are seen as tools to compute lower bounds for the natural pseudodistance.
Actually, the following link exists between the values taken by the size functions , and the natural pseudodistance
between the size pairs
,
An analogous result holds for size homotopy group.
The attempt to generalize size theory and the concept of natural pseudodistance to norms that are different from the supremum norm has led to the study of other reparametrization invariant norms.
See also
Size function
Natural pseudodistance
Size functor
Size homotopy group
Size pair
Matching distance
References
Topology
Algebraic topology |
https://en.wikipedia.org/wiki/Palazzo%20Grimani%20di%20Santa%20Maria%20Formosa | {
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The Palazzo Grimani of Santa Maria Formosa is a State museum, located in Venice in the Castello district, near Campo Santa Maria Formosa.
History
The palace can be reached by land from Ruga Giuffa (map). The water entry, very used in ancient times, is located on the San Severo canal. The Palazzo constitutes for the city of Venice a particularly precious novelty for the originality of the architecture, for the decorations and for its history.
The original medieval building was built at the confluence of the canals of San Severo and Santa Maria Formosa, and purchased later by Antonio Grimani, who became a doge in 1521, and subsequently passed on as a legacy, in the third decade of the 16th century, to the grandsons Vettore Grimani, Procurator de Supra for the Venetian Republic, and Giovanni Grimani, Patriarch of Aquileia, who refurbished the old structure inspired by architectural models taken from classicism. The two brothers wanted to give "modern" forms to the building and had it decorated with fresco cycles and elegant stucco. In 1558, at the death of Vettore, Giovanni became the sole owner of the building: he added an extension of the palace. Decorating the rooms were many artists including Federico Zuccari, architect of the monumental staircase, and Camillo Mantovano. The patriarch Giovanni Grimani set up his refined collection of antiques, including sculptures, marbles, vases, bronzes and gems, in the rooms of the palace. In 1587 he donated the collection of sculptures and gems to the Serenissima: after his death the first ones were placed in the anti-room of the Marciana Library and today they are the founding nucleus of the National Archaeological Museum of Venice.
Until 1865, it was property of the Santa Maria Formosa branch of the Grimani family; but the palace had deteriorated and passed through several owners, until in 1981 it was acquired by the Superintendence for Architectural and Environmental Heritage of the city of Venice and became a state property. Open to the public on December 20, 2008, after a long restoration, it is currently a museum belonging to the Veneto Museum Pole
The long restoration by the Superintendence included the interior decorations, including: the Camerino di Callisto, with stucco by Giovanni da Udine; the Camerino di Apollo, with frescoes by Francesco Salviati and Giovanni da Udine; the Sala del Doge Antonio, decorated with stucco and polychrome marbles; the Sala a Fogliami by Camillo Mantovano, with the ceiling entirely covered with fruit trees, flowers and animals; and the Tribune that once housed more than a hundred pieces of the archaeological collection. Here, the sculpture depicting the Kidnapping of Ganimede is suspended in the center of the |
https://en.wikipedia.org/wiki/Symmetry%20%28disambiguation%29 | Symmetry may refer to:
Generally:
Symmetry, the broad concept
In mathematics, science and technology:
Symmetry (geometry), of shapes in a metric space such as the plane
Symmetry in mathematics, of mathematical structures in general
Symmetry (physics), a physical or mathematical feature of the system (observed or intrinsic) that is "preserved" under some change
Symmetry in biology, the balanced distribution of duplicate body parts or shapes
Molecular symmetry in chemistry
Symmetry (Sequent Computer Systems), a line of SMP computers by Sequent Computer Systems
Symmetry Magazine, a Fermilab/SLAC publication covering advanced physics
In arts and entertainment:
Symmetry (band), American instrumental duo
Symmetry (film), a Polish film
"Symmetry" (Dead Zone), an episode of the television series Dead Zone
Symmetric scale, in music
Symmetry (Saga album), a studio album by Canadian rock band Saga
"Symmetry", a song by Little Boots on the album Hands
Other uses:
Facial symmetry, a component of attractiveness
"Symmetry", street name of salvinorin B ethoxymethyl ether, a dissociative drug
Symmetry (horse)
Symmetry (journal) |
https://en.wikipedia.org/wiki/Cornel%20Pavlovici | Cornel Pavlovici (2 April 1942 – 8 January 2013) was a Romanian footballer who played as a striker.
Death
Pavlovici died on 8 January 2013.
Career statistics
Total matches played in Romanian First League: 134 matches – 57 goals.
Topscorer of Romanian First League: 1964.
Under-23 team: 8 matches – 0 goals
International goals
Notes
The 1966–67 Second League appearances made for ASA Târgu Mureş are unavailable.
Honours
Steaua București
Cupa României: 1966
References
External links
1942 births
2013 deaths
Footballers from Bucharest
Romanian men's footballers
Romania men's international footballers
Romanian people of Serbian descent
Liga I players
Liga II players
CSM Jiul Petroșani players
Faur București players
FC Rapid București players
FC Olimpia Satu Mare players
FC Steaua București players
FC Argeș Pitești players
FC Petrolul Ploiești players
FC Progresul București players
FC Brașov (1936) players
FC Drobeta-Turnu Severin players
FCM Târgoviște players
ASA Târgu Mureș (1962) players
Footballers at the 1964 Summer Olympics
Olympic footballers for Romania
Men's association football forwards |
https://en.wikipedia.org/wiki/Ren%C3%A9%20Gartler | René Gartler (born 21 October 1985) is an Austrian football coach and a former player. He is an assistant coach with LASK.
Career statistics
References
External links
René Gartler Interview
Rauswurf und Maulkorb für Rene Gartler
Gartler unterschrieb beim Lask
1985 births
Living people
Footballers from Vienna
Austrian men's footballers
Austria men's youth international footballers
Austrian expatriate men's footballers
Austrian Football Bundesliga players
2. Liga (Austria) players
Austrian Regionalliga players
2. Bundesliga players
SK Rapid Wien players
SKN St. Pölten players
FC Lustenau 07 players
Kapfenberger SV players
SV Sandhausen players
LASK players
FC Juniors OÖ players
Men's association football forwards
Austrian expatriate sportspeople in Germany
Expatriate men's footballers in Germany
Austrian football managers |
https://en.wikipedia.org/wiki/Toshikazu%20Sunada | is a Japanese mathematician and author of many books and essays on mathematics and mathematical sciences. He is professor emeritus of both Meiji University and Tohoku University. He is also distinguished professor of emeritus at Meiji in recognition of achievement over the course of an academic career. Before he joined Meiji University in 2003, he was professor of mathematics at Nagoya University (1988–1991), at the University of Tokyo (1991–1993), and at Tohoku University (1993–2003). Sunada was involved in the creation of the School of Interdisciplinary Mathematical Sciences at Meiji University and is its first dean (2013–2017). Since 2019, he is President of Mathematics Education Society of Japan.
Main work
Sunada's work covers complex analytic geometry, spectral geometry, dynamical systems, probability, graph theory, discrete geometric analysis, and mathematical crystallography. Among his numerous contributions, the most famous one is a general construction of isospectral manifolds (1985), which is based on his geometric model of number theory, and is considered to be a breakthrough in the problem proposed by Mark Kac in "Can one hear the shape of a drum?" (see Hearing the shape of a drum). Sunada's idea was taken up by Carolyn S. Gordon, David Webb, and Scott A. Wolpert when they constructed a counterexample for Kac's problem. For this work, Sunada was awarded the Iyanaga Prize of the Mathematical Society of Japan (MSJ) in 1987. He was also awarded Publication Prize of MSJ in 2013, the Hiroshi Fujiwara Prize for Mathematical Sciences in 2017, the Prize for Science and Technology (the Commendation for Science and Technology by the Minister of Education, Culture, Sports, Science and Technology) in 2018, and the 1st Kodaira Kunihiko Prize in 2019.
In a joint work with Atsushi Katsuda, Sunada also established a geometric analogue of Dirichlet's theorem on arithmetic progressions in the context of dynamical systems (1988). One can see, in this work as well as the one above, how the concepts and ideas in totally different fields (geometry, dynamical systems, and number theory) are put together to formulate problems and to produce new results.
His study of discrete geometric analysis includes a graph-theoretic interpretation of Ihara zeta functions, a discrete analogue of periodic magnetic Schrödinger operators as well as the large time asymptotic behaviors of random walk on crystal lattices. The study of random walk led him to the discovery of a "mathematical twin" of the diamond crystal out of an infinite universe of hypothetical crystals (2005). He named it the K4 crystal due to its mathematical relevance (see the linked article). What was noticed by him is that the K4 crystal has the "strong isotropy property", meaning that for any two vertices x and y of the crystal net, and for any ordering of the edges adjacent to x and any ordering of the edges adjacent to y, there is a net-preserving congruence taking x to y and each x-edge to the s |
https://en.wikipedia.org/wiki/Potato%20peeling | In computational geometry, the potato peeling or convex skull problem is a problem of finding the convex polygon of the largest possible area that lies within a given non-convex simple polygon. It was posed independently by Goodman and Woo, and solved in polynomial time by Chang and Yap. The exponent of the polynomial time bound is high, but the same problem can also be accurately approximated in near-linear time.
References
Convex hulls
Computational geometry |
https://en.wikipedia.org/wiki/Markov%20model | In probability theory, a Markov model is a stochastic model used to model pseudo-randomly changing systems. It is assumed that future states depend only on the current state, not on the events that occurred before it (that is, it assumes the Markov property). Generally, this assumption enables reasoning and computation with the model that would otherwise be intractable. For this reason, in the fields of predictive modelling and probabilistic forecasting, it is desirable for a given model to exhibit the Markov property.
Introduction
There are four common Markov models used in different situations, depending on whether every sequential state is observable or not, and whether the system is to be adjusted on the basis of observations made:
Markov chain
The simplest Markov model is the Markov chain. It models the state of a system with a random variable that changes through time. In this context, the Markov property suggests that the distribution for this variable depends only on the distribution of a previous state. An example use of a Markov chain is Markov chain Monte Carlo, which uses the Markov property to prove that a particular method for performing a random walk will sample from the joint distribution.
Hidden Markov model
A hidden Markov model is a Markov chain for which the state is only partially observable or noisily observable. In other words, observations are related to the state of the system, but they are typically insufficient to precisely determine the state. Several well-known algorithms for hidden Markov models exist. For example, given a sequence of observations, the Viterbi algorithm will compute the most-likely corresponding sequence of states, the forward algorithm will compute the probability of the sequence of observations, and the Baum–Welch algorithm will estimate the starting probabilities, the transition function, and the observation function of a hidden Markov model.
One common use is for speech recognition, where the observed data is the speech audio waveform and the hidden state is the spoken text. In this example, the Viterbi algorithm finds the most likely sequence of spoken words given the speech audio.
Markov decision process
A Markov decision process is a Markov chain in which state transitions depend on the current state and an action vector that is applied to the system. Typically, a Markov decision process is used to compute a policy of actions that will maximize some utility with respect to expected rewards.
Partially observable Markov decision process
A partially observable Markov decision process (POMDP) is a Markov decision process in which the state of the system is only partially observed. POMDPs are known to be NP complete, but recent approximation techniques have made them useful for a variety of applications, such as controlling simple agents or robots.
Markov random field
A Markov random field, or Markov network, may be considered to be a generalization of a Markov chain in multiple dimensi |
https://en.wikipedia.org/wiki/Boolean%20model%20%28probability%20theory%29 | For statistics in probability theory, the Boolean-Poisson model or simply Boolean model for a random subset of the plane (or higher dimensions, analogously) is one of the simplest and most tractable models in stochastic geometry. Take a Poisson point process of rate in the plane and make each point be the center of a random set; the resulting union of overlapping sets is a realization of the Boolean model . More precisely, the parameters are and a probability distribution on compact sets; for each point of the Poisson point process we pick a set from the distribution, and then define as the union
of translated sets.
To illustrate tractability with one simple formula, the mean density of equals where denotes the area of and The classical theory of stochastic geometry develops many further formulae.
As related topics, the case of constant-sized discs is the basic model of continuum percolation
and the low-density Boolean models serve as a first-order approximations in the
study of extremes in many models.
References
Spatial processes |
https://en.wikipedia.org/wiki/Tevian%20Dray | Tevian Dray (born March 17, 1956) is an American mathematician
who has worked in general relativity, mathematical physics,
geometry, and both science and mathematics education. He was elected a Fellow of the American Physical Society in 2010.
He has primarily worked in the area of classical general relativity. His
research results include confirmation of the existence of solutions of
Einstein's equation containing gravitational radiation, the use of
computer algebra to classify exact solutions of Einstein's equation, an
analysis of a class of gravitational shock waves (including one of the few
known exact 2-body solutions in general relativity), and
the study of signature change, a possible model for the
Big Bang. More recently, his work has focused on applications of the
octonions to the theory of fundamental particles.
He was a graduate student under Rainer K. Sachs at
Berkeley, where he received his Ph.D.
in 1981, although much of his dissertation research was done in collaboration with
Abhay Ashtekar. The context of his dissertation, titled The Asymptotic Structure of a Family of Einstein-Maxwell Solutions focused on families of spacetimes which describe accelerating black holes, and which contain gravitational radiation. This demonstrated the existence of exact radiating solutions to the Einstein field equations.
He is currently a professor of mathematics at
Oregon State University. In addition to his ongoing work in mathematical physics,
he has made significant contributions in science education, where he directs
the Vector Calculus Bridge Project,
an attempt to teach vector
calculus the way it is used by scientists and engineers, and is part of the
development team of the Paradigms Project,
a complete
restructuring of the undergraduate physics major around several core
"paradigms". He has written a book
on special relativity and a sequel on general relativity using differential forms.
, and is coauthor of The Geometry of the Octonions released in 2015.
Bibliography
(2012) Tevian Dray, The Geometry of Special Relativity (A K Peters/CRC Press)
(2014) Tevian Dray, Differential Forms and The Geometry of General Relativity (A K Peters/CRC Press)
(2015) Tevian Dray and Corinne A. Manogue, The Geometry of the Octonions (World Scientific)
References
External links
Tevian Dray's home page
Vector Calculus Bridge Project
Paradigms Project
1956 births
Living people
20th-century American mathematicians
21st-century American mathematicians
American relativity theorists
Oregon State University faculty
Fellows of the American Physical Society
Massachusetts Institute of Technology alumni |
https://en.wikipedia.org/wiki/Stochastic%20geometry | In mathematics, stochastic geometry is the study of random spatial patterns. At the heart of the subject lies the study of random point patterns. This leads to the theory of spatial point processes, hence notions of Palm conditioning, which extend to the more abstract setting of random measures.
Models
There are various models for point processes, typically based on but going beyond the classic homogeneous Poisson point process (the basic model for complete spatial randomness) to find expressive models which allow effective statistical methods.
The point pattern theory provides a major building block for generation of random object processes, allowing construction of elaborate random spatial patterns. The simplest version, the Boolean model, places a random compact object at each point of a Poisson point process. More complex versions allow interactions based in various ways on the geometry of objects. Different directions of application include: the production of models for random images either as set-union of objects, or as patterns of overlapping objects; also the generation of geometrically inspired models for the underlying point process
(for example, the point pattern distribution may be biased by an exponential factor involving the area of the union of the objects; this is related to the Widom–Rowlinson model of statistical mechanics).
Random object
What is meant by a random object? A complete answer to this question requires the theory of random closed sets, which makes contact with advanced concepts from measure theory. The key idea is to focus on the probabilities of the given random closed set hitting specified test sets. There arise questions of inference (for example, estimate the set which encloses a given point pattern) and theories of generalizations of means etc. to apply to random sets. Connections are now being made between this latter work and recent developments in geometric mathematical analysis concerning general metric spaces and their geometry. Good parametrizations of specific random sets can allow us to refer random object processes to the theory of marked point processes; object-point pairs are viewed as points in a larger product space formed as the product of the original space and the space of parametrization.
Line and hyper-flat processes
Suppose we are concerned no longer with compact objects, but with objects which are spatially extended: lines on the plane or flats in 3-space. This leads to consideration of line processes, and of processes of flats or hyper-flats. There can no longer be a preferred spatial location for each object; however the theory may be mapped back into point process theory by representing each object by a point in a suitable representation space. For example, in the case of directed lines in the plane one may take the representation space to be a cylinder. A complication is that the Euclidean motion symmetries will then be expressed on the representation space in a somewhat unu |
https://en.wikipedia.org/wiki/Least-upper-bound%20property | In mathematics, the least-upper-bound property (sometimes called completeness or supremum property or l.u.b. property) is a fundamental property of the real numbers. More generally, a partially ordered set has the least-upper-bound property if every non-empty subset of with an upper bound has a least upper bound (supremum) in . Not every (partially) ordered set has the least upper bound property. For example, the set of all rational numbers with its natural order does not have the least upper bound property.
The least-upper-bound property is one form of the completeness axiom for the real numbers, and is sometimes referred to as Dedekind completeness. It can be used to prove many of the fundamental results of real analysis, such as the intermediate value theorem, the Bolzano–Weierstrass theorem, the extreme value theorem, and the Heine–Borel theorem. It is usually taken as an axiom in synthetic constructions of the real numbers, and it is also intimately related to the construction of the real numbers using Dedekind cuts.
In order theory, this property can be generalized to a notion of completeness for any partially ordered set. A linearly ordered set that is dense and has the least upper bound property is called a linear continuum.
Statement of the property
Statement for real numbers
Let be a non-empty set of real numbers.
A real number is called an upper bound for if for all .
A real number is the least upper bound (or supremum) for if is an upper bound for and for every upper bound of .
The least-upper-bound property states that any non-empty set of real numbers that has an upper bound must have a least upper bound in real numbers.
Generalization to ordered sets
More generally, one may define upper bound and least upper bound for any subset of a partially ordered set , with “real number” replaced by “element of ”. In this case, we say that has the least-upper-bound property if every non-empty subset of with an upper bound has a least upper bound in .
For example, the set of rational numbers does not have the least-upper-bound property under the usual order. For instance, the set
has an upper bound in , but does not have a least upper bound in (since the square root of two is irrational). The construction of the real numbers using Dedekind cuts takes advantage of this failure by defining the irrational numbers as the least upper bounds of certain subsets of the rationals.
Proof
Logical status
The least-upper-bound property is equivalent to other forms of the completeness axiom, such as the convergence of Cauchy sequences or the nested intervals theorem. The logical status of the property depends on the construction of the real numbers used: in the synthetic approach, the property is usually taken as an axiom for the real numbers (see least upper bound axiom); in a constructive approach, the property must be proved as a theorem, either directly from the construction or as a consequence of some other form |
https://en.wikipedia.org/wiki/Tur%C3%A1n%27s%20brick%20factory%20problem | In the mathematics of graph drawing, Turán's brick factory problem asks for the minimum number of crossings in a drawing of a complete bipartite graph. The problem is named after Pál Turán, who formulated it while being forced to work in a brick factory during World War II.
A drawing method found by Kazimierz Zarankiewicz has been conjectured to give the correct answer for every complete bipartite graph, and the statement that this is true has come to be known as the Zarankiewicz crossing number conjecture. The conjecture remains open, with only some special cases solved.
Origin and formulation
During World War II, Hungarian mathematician Pál Turán was forced to work in a brick factory, pushing wagon loads of bricks from kilns to storage sites. The factory had tracks from each kiln to each storage site, and the wagons were harder to push at the points where tracks crossed each other. Turán was inspired by this situation to ask how the factory might be redesigned to minimize the number of crossings between these tracks.
Mathematically, this problem can be formalized as asking for a graph drawing of a complete bipartite graph, whose vertices represent kilns and storage sites, and whose edges represent the tracks from each kiln to each storage site.
The graph should be drawn in the plane with each vertex as a point, each edge as a curve connecting its two endpoints, and no vertex placed on an edge that it is not incident to. A crossing is counted whenever two edges that are disjoint in the graph have a nonempty intersection in the plane. The question is then, what is the minimum number of crossings in such a drawing?
Turán's formulation of this problem is often recognized as one of the first studies of the crossing numbers of graphs.
(Another independent formulation of the same concept occurred in sociology, in methods for drawing sociograms, and a much older puzzle, the three utilities problem, can be seen as a special case of the brick factory problem with three kilns and three storage facilities.)
Crossing numbers have since gained greater importance, as a central object of study in graph drawing
and as an important tool in VLSI design
and discrete geometry.
Upper bound
Both Zarankiewicz and Kazimierz Urbanik saw Turán speak about the brick factory problem in different talks in Poland in 1952,
and independently published attempted solutions of the problem, with equivalent formulas for the number of crossings.
As both of them showed, it is always possible to draw the complete bipartite graph (a graph with vertices on one side, vertices on the other side, and edges connecting the two sides) with a number of crossings equal to
The construction is simple: place vertices on the -axis of the plane, avoiding the origin, with equal or nearly-equal numbers of points to the left and right of the -axis.
Similarly, place vertices on the -axis of the plane, avoiding the origin, with equal or nearly-equal numbers of points above and below the -axis |
https://en.wikipedia.org/wiki/Panos%20Papasoglu | Panos Papasoglu (; original name is also transliterated in English as Panagiotis Papazoglou) is a Greek mathematician, Lecturer of Mathematics at the Mathematics Department of the University of Oxford. His main research interests are group theory and geometric group theory.
He got his doctorate under Hyman Bass in Columbia University in 1993. Papasoglou taught at University of Paris XI until he returned to Greece in the early 2000s. He taught at the University of Athens until the late 2000s.
Notes
External links
Personal web page
Panagiotis Papasoglu in Mathematics Genealogy Project
arxiv.org list of papers
Year of birth missing (living people)
Living people
Academic staff of the University of Paris
Group theorists
Columbia Graduate School of Arts and Sciences alumni |
https://en.wikipedia.org/wiki/Random%20binary%20tree | In computer science and probability theory, a random binary tree is a binary tree selected at random from some probability distribution on binary trees. Two different distributions are commonly used: binary trees formed by inserting nodes one at a time according to a random permutation, and binary trees chosen from a uniform discrete distribution in which all distinct trees are equally likely. It is also possible to form other distributions, for instance by repeated splitting. Adding and removing nodes directly in a random binary tree will in general disrupt its random structure, but the treap and related randomized binary search tree data structures use the principle of binary trees formed from a random permutation in order to maintain a balanced binary search tree dynamically as nodes are inserted and deleted.
For random trees that are not necessarily binary, see random tree.
Binary trees from random permutations
For any set of numbers (or, more generally, values from some total order), one may form a binary search tree in which each number is inserted in sequence as a leaf of the tree, without changing the structure of the previously inserted numbers. The position into which each number should be inserted is uniquely determined by a binary search in the tree formed by the previous numbers. For instance, if the three numbers (1,3,2) are inserted into a tree in that sequence, the number 1 will sit at the root of the tree, the number 3 will be placed as its right child, and the number 2 as the left child of the number 3. There are six different permutations of the numbers (1,2,3), but only five trees may be constructed from them. That is because the permutations (2,1,3) and (2,3,1) form the same tree.
Expected depth of a node
For any fixed choice of a value in a given set of numbers, if one randomly permutes the numbers and forms a binary tree from them as described above, the expected value of the length of the path from the root of the tree to is at most , where "" denotes the natural logarithm function and the introduces big O notation. For, the expected number of ancestors of is by linearity of expectation equal to the sum, over all other values in the set, of the probability that is an ancestor of . And a value is an ancestor of exactly when is the first element to be inserted from the elements in the interval . Thus, the values that are adjacent to in the sorted sequence of values have probability of being an ancestor of , the values one step away have probability , etc. Adding these probabilities for all positions in the sorted sequence gives twice a Harmonic number, leading to the bound above. A bound of this form holds also for the expected search length of a path to a fixed value that is not part of the given set.
To understand it by using min-max records. The number in a random permutation is the min (max) record means it is the min (max) value from the first position to its position. Consider a simple example = (2, 4 |
https://en.wikipedia.org/wiki/Pierce%E2%80%93Birkhoff%20conjecture | In abstract algebra, the Pierce–Birkhoff conjecture asserts that any piecewise-polynomial function can be expressed as a maximum of finite minima of finite collections of polynomials. It was first stated, albeit in non-rigorous and vague wording, in the 1956 paper of Garrett Birkhoff and Richard S. Pierce in which they first introduced f-rings. The modern, rigorous statement of the conjecture was formulated by Melvin Henriksen and John R. Isbell, who worked on the problem in the early 1960s in connection with their work on f-rings. Their formulation is as follows:
For every real piecewise-polynomial function , there exists a finite set of polynomials such that .
Isbell is likely the source of the name Pierce–Birkhoff conjecture, and popularized the problem in the 1980s by discussing it with several mathematicians interested in real algebraic geometry.
The conjecture was proved true for n = 1 and 2 by Louis Mahé.
Local Pierce–Birkhoff conjecture
In 1989, James J. Madden provided an equivalent statement that is in terms of the real spectrum of and the novel concepts of local polynomial representatives and separating ideals.
Denoting the real spectrum of A by , the separating ideal of α and β in is the ideal of A generated by all polynomials that change sign on and , i.e., and . Any finite covering of closed, semi-algebraic sets induces a corresponding covering , so, in particular, when f is piecewise polynomial, there is a polynomial for every such that and . This is termed the local polynomial representative of f at .
Madden's so-called local Pierce–Birkhoff conjecture at and , which is equivalent to the Pierce–Birkhoff conjecture, is as follows:
Let , be in and f be piecewise-polynomial. It is conjectured that for every local representative of f at , , and local representative of f at , , is in the separating ideal of and .
References
Further reading
Conjectures
Real algebraic geometry
Unsolved problems in geometry |
https://en.wikipedia.org/wiki/List%20of%20universities%20in%20Australia%20by%20enrollment | This is a comprehensive list of all universities in Australia by total university enrolment. The data is gathered from the Department of Education and Training Higher Education statistics from 2016. For accuracy of comparison, all data is measured in Equivalent Full-Time Student Load (EFTSL) except for "Total Students".
National universities
Australian Capital Territory
New South Wales
Northern Territory
Queensland
South Australia
Tasmania
Victoria
Western Australia
Largest universities
By all students
By EFTSL
By enrolments
By undergraduate students
By postgraduate students
References
External links
Australian University Rankings
Australia education-related lists
Australia |
https://en.wikipedia.org/wiki/Tourism%20in%20Bolivia | Tourism in Bolivia is one of the economic sectors of the country. According to data from the National Institute of Statistics of Bolivia (INE), there were over 1.24 million tourists that visited the country in 2020, making Bolivia the ninth most visited country in South America. the Bolivia is a country with great tourism potential, with many attractions, due to its diverse culture, geographic regions, rich history and food. In particular, the salt flats at Uyuni are a major attraction.
World Heritage Site
In the country there are six World Heritages declared by the UNESCO:
The ruins of the city of Tiwanaku, capital of the 6th-century empire that ruled the southern Andes
The city of Potosí, historic city known for its religious and civic monuments and Cerro Rico
The Amazon, a large rainforest and sanctuary for wildlife.
Noel Kempff Mercado National Park, representative place of the Amazon and its immense biodiversity, located on a large plateau, covered by vast forests and magnificent waterfalls.
Madidi National Park, the most diverse place in Bolivia, declared by National Geographic to be one of the 20 best places to visit in the world.
Toro Toro National Park, where found paleontological wealth (thousands of dinosaurs footprints), caves, waterfalls, rock paintings and other places of interest are.
The Jesuit Missions of Chiquitos, the only active missions of all of South America.
The Fort Samaipata, the big rock carved by the Incas in the foothills of the Andes as the limit of his empire.
The Carnival of Oruro,a festival in which Catholicism is mixed with paganism.
Destination
Lake Titicaca, the world's highest navigable lake.
The Isla del Sol, the sacred place of the Incas and birthplace of the founders of the Inca Empire, Manco Cápac and Mama Ocllo
The Isla de la Luna, another sacred place of the Incas near the Isla del Sol.
Copacabana, a small town on the shores of Titicaca, home to the Virgin of Copacabana, crowned queen of Bolivia.
The Andes, the longest mountain range in the world, spanning the entire continent, and has exceptionally attractive regions:
The ski slope containing the highest restaurant in the world, called Chacaltaya.
The highest mountain in the country: Nevado Sajama, with the highest forest in the world.
The salt flats of Uyuni and Coipasa, the largest salt flats in the world.
Bolivia also is the only country in the world in having the only hotel totally fabricated of salt, found in the Uyuni.
The lakes Green lake and Red Lagoon, the sanctuary of the Andean flamingos with one of the largest active volcanoes in the world, the Licancabur.
The historic cities of:
Potosí with its Cerro Rico, formerly the largest deposit of silver in the world.
Sucre, the constitutional capital city of Bolivia, and The City of Four Names, which is home to one of the oldest universities in the Americas.
Cal Orcko is a paleontological site, found in the quarry of a cement factory, in the Department of Chuquisaca.
Cas |
https://en.wikipedia.org/wiki/P-adic%20L-function | In mathematics, a p-adic zeta function, or more generally a p-adic L-function, is a function analogous to the Riemann zeta function, or more general L-functions, but whose domain and target are p-adic (where p is a prime number). For example, the domain could be the p-adic integers Zp, a profinite p-group, or a p-adic family of Galois representations, and the image could be the p-adic numbers Qp or its algebraic closure.
The source of a p-adic L-function tends to be one of two types. The first source—from which Tomio Kubota and Heinrich-Wolfgang Leopoldt gave the first construction of a p-adic L-function —is via the p-adic interpolation of special values of L-functions. For example, Kubota–Leopoldt used Kummer's congruences for Bernoulli numbers to construct a p-adic L-function, the p-adic Riemann zeta function ζp(s), whose values at negative odd integers are those of the Riemann zeta function at negative odd integers (up to an explicit correction factor). p-adic L-functions arising in this fashion are typically referred to as analytic p-adic L-functions. The other major source of p-adic L-functions—first discovered by Kenkichi Iwasawa—is from the arithmetic of cyclotomic fields, or more generally, certain Galois modules over towers of cyclotomic fields or even more general towers. A p-adic L-function arising in this way is typically called an arithmetic p-adic L-function as it encodes arithmetic data of the Galois module involved. The main conjecture of Iwasawa theory (now a theorem due to Barry Mazur and Andrew Wiles) is the statement that the Kubota–Leopoldt p-adic L-function and an arithmetic analogue constructed by Iwasawa theory are essentially the same. In more general situations where both analytic and arithmetic p-adic L-functions are constructed (or expected), the statement that they agree is called the main conjecture of Iwasawa theory for that situation. Such conjectures represent formal statements concerning the philosophy that special values of L-functions contain arithmetic information.
Dirichlet L-functions
The Dirichlet L-function is given by the analytic continuation of
The Dirichlet L-function at negative integers is given by
where Bn,χ is a generalized Bernoulli number defined by
for χ a Dirichlet character with conductor f.
Definition using interpolation
The Kubota–Leopoldt p-adic L-function Lp(s, χ) interpolates the Dirichlet L-function with the Euler factor at p removed.
More precisely, Lp(s, χ) is the unique continuous function of the p-adic number s such that
for positive integers n divisible by p − 1. The right hand side is just the usual Dirichlet L-function, except that the Euler factor at p is removed, otherwise it would not be p-adically continuous. The continuity of the right hand side is closely related to the Kummer congruences.
When n is not divisible by p − 1 this does not usually hold; instead
for positive integers n.
Here χ is twisted by a power of the Teichmüller character ω.
Viewed as a p-a |
https://en.wikipedia.org/wiki/Twelve%20Jewels%20of%20Islam | The Twelve Jewels of Islam in the Nation of Gods and Earths is a variant of the Supreme Alphabet and Supreme Mathematics that the group's members use to understand the meaning of the universe. All three systems comprise the Universal Language. These jewels are also shared by The Nation of Islam.
The twelve principles
Knowledge
Wisdom
Understanding
Freedom
Justice
Equality
Food
Clothing
Shelter
Love
Peace
Happiness
References
Five-Percent Nation |
https://en.wikipedia.org/wiki/Magnus%20expansion | In mathematics and physics, the Magnus expansion, named after Wilhelm Magnus (1907–1990), provides an exponential representation of the solution of a first-order homogeneous linear differential equation for a linear operator. In particular, it furnishes the fundamental matrix of a system of linear ordinary differential equations of order with varying coefficients. The exponent is aggregated as an infinite series, whose terms involve multiple integrals and nested commutators.
The deterministic case
Magnus approach and its interpretation
Given the coefficient matrix , one wishes to solve the initial-value problem associated with the linear ordinary differential equation
for the unknown -dimensional vector function .
When n = 1, the solution simply reads
This is still valid for n > 1 if the matrix satisfies for any pair of values of t, t1 and t2. In particular, this is the case if the matrix is independent of . In the general case, however, the expression above is no longer the solution of the problem.
The approach introduced by Magnus to solve the matrix initial-value problem is to express the solution by means of the exponential of a certain matrix function
:
which is subsequently constructed as a series expansion:
where, for simplicity, it is customary to write for and to take t0 = 0.
Magnus appreciated that, since , using a Poincaré−Hausdorff matrix identity, he could relate the time derivative of to the generating function of Bernoulli numbers and
the adjoint endomorphism of ,
to solve for recursively in terms of "in a continuous analog of the BCH expansion", as outlined in a subsequent section.
The equation above constitutes the Magnus expansion, or Magnus series, for the solution of matrix linear initial-value problem. The first four terms of this series read
where is the matrix commutator of A and B.
These equations may be interpreted as follows: coincides exactly with the exponent in the scalar ( = 1) case, but this equation cannot give the whole solution. If one insists in having an exponential representation (Lie group), the exponent needs to be corrected. The rest of the Magnus series provides that correction systematically: or parts of it are in the Lie algebra of the Lie group on the solution.
In applications, one can rarely sum exactly the Magnus series, and one has to truncate it to get approximate solutions. The main advantage of the Magnus proposal is that the truncated series very often shares important qualitative properties with the exact solution, at variance with other conventional perturbation theories. For instance, in classical mechanics the symplectic character of the time evolution is preserved at every order of approximation. Similarly, the unitary character of the time evolution operator in quantum mechanics is also preserved (in contrast, e.g., to the Dyson series solving the same problem).
Convergence of the expansion
From a mathematical point of view, the convergence pro |
https://en.wikipedia.org/wiki/Capelli%27s%20identity | In mathematics, Capelli's identity, named after , is an analogue of the formula det(AB) = det(A) det(B), for certain matrices with noncommuting entries, related to the representation theory of the Lie algebra . It can be used to relate an invariant ƒ to the invariant Ωƒ, where Ω is Cayley's Ω process.
Statement
Suppose that xij for i,j = 1,...,n are commuting variables. Write Eij for the polarization operator
The Capelli identity states that the following differential operators, expressed as determinants, are equal:
Both sides are differential operators. The determinant on the left has non-commuting entries, and is expanded with all terms preserving their "left to right" order. Such a determinant is often called a column-determinant, since it can be obtained by the column expansion of the determinant starting from the first column. It can be formally written as
where in the product first come the elements from the first column, then from the second and so on. The determinant on the far right is Cayley's omega process, and the one on the left is the Capelli determinant.
The operators Eij can be written in a matrix form:
where are matrices with elements Eij, xij, respectively. If all elements in these matrices would be commutative then clearly . The Capelli identity shows that despite noncommutativity there exists a "quantization" of the formula above. The only price for the noncommutativity is a small correction: on the left hand side. For generic noncommutative matrices formulas like
do not exist, and the notion of the 'determinant' itself does not make sense for generic noncommutative matrices. That is why the Capelli identity still holds some mystery, despite many proofs offered for it. A very short proof does not seem to exist. Direct verification of the statement can be given as an exercise for n = 2, but is already long for n = 3.
Relations with representation theory
Consider the following slightly more general context. Suppose that and are two integers and for , be commuting variables. Redefine by almost the same formula:
with the only difference that summation index ranges from to . One can easily see that such operators satisfy the commutation relations:
Here denotes the commutator . These are the same commutation relations which are satisfied by the matrices which have zeros everywhere except the position , where 1 stands. ( are sometimes called matrix units). Hence we conclude that the correspondence defines a representation of the Lie algebra in the vector space of polynomials of .
Case m = 1 and representation Sk Cn
It is especially instructive to consider the special case m = 1; in this case we have xi1, which is abbreviated as xi:
In particular, for the polynomials of the first degree it is seen that:
Hence the action of restricted to the space of first-order polynomials is exactly the same as the action of matrix units on vectors in . So, from the representation theory point of view, the sub |
https://en.wikipedia.org/wiki/Journal%20of%20Applied%20Mathematics%20and%20Mechanics | The Journal of Applied Mathematics and Mechanics, also known as Zeitschrift für Angewandte Mathematik und Mechanik or ZAMM is a monthly peer-reviewed scientific journal dedicated to applied mathematics. It is published by Wiley-VCH on behalf of the Gesellschaft für Angewandte Mathematik und Mechanik. The editor-in-chief is Holm Altenbach (Otto von Guericke University Magdeburg). According to the Journal Citation Reports, the journal has a 2022 impact factor of 2.3.
Publication history
The journal's first issue appeared in 1921, published by the Verein Deutscher Ingenieure and edited by Richard von Mises.
References
External links
Mathematics journals
Monthly journals
Wiley-VCH academic journals
English-language journals |
https://en.wikipedia.org/wiki/Cubic%20field | In mathematics, specifically the area of algebraic number theory, a cubic field is an algebraic number field of degree three.
Definition
If K is a field extension of the rational numbers Q of degree [K:Q] = 3, then K is called a cubic field. Any such field is isomorphic to a field of the form
where f is an irreducible cubic polynomial with coefficients in Q. If f has three real roots, then K is called a totally real cubic field and it is an example of a totally real field. If, on the other hand, f has a non-real root, then K is called a complex cubic field.
A cubic field K is called a cyclic cubic field if it contains all three roots of its generating polynomial f. Equivalently, K is a cyclic cubic field if it is a Galois extension of Q, in which case its Galois group over Q is cyclic of order three. This can only happen if K is totally real. It is a rare occurrence in the sense that if the set of cubic fields is ordered by discriminant, then the proportion of cubic fields which are cyclic approaches zero as the bound on the discriminant approaches infinity.
A cubic field is called a pure cubic field if it can be obtained by adjoining the real cube root of a cube-free positive integer n to the rational number field Q. Such fields are always complex cubic fields since each positive number has two complex non-real cube roots.
Examples
Adjoining the real cube root of 2 to the rational numbers gives the cubic field . This is an example of a pure cubic field, and hence of a complex cubic field. In fact, of all pure cubic fields, it has the smallest discriminant (in absolute value), namely −108.
The complex cubic field obtained by adjoining to Q a root of is not pure. It has the smallest discriminant (in absolute value) of all cubic fields, namely −23.
Adjoining a root of to Q yields a cyclic cubic field, and hence a totally real cubic field. It has the smallest discriminant of all totally real cubic fields, namely 49.
The field obtained by adjoining to Q a root of is an example of a totally real cubic field that is not cyclic. Its discriminant is 148, the smallest discriminant of a non-cyclic totally real cubic field.
No cyclotomic fields are cubic because the degree of a cyclotomic field is equal to φ(n), where φ is Euler's totient function, which only takes on even values except for φ(1) = φ(2) = 1.
Galois closure
A cyclic cubic field K is its own Galois closure with Galois group Gal(K/Q) isomorphic to the cyclic group of order three. However, any other cubic field K is a non-Galois extension of Q and has a field extension N of degree two as its Galois closure. The Galois group Gal(N/Q) is isomorphic to the symmetric group S3 on three letters.
Associated quadratic field
The discriminant of a cubic field K can be written uniquely as df2 where d is a fundamental discriminant. Then, K is cyclic if and only if d = 1, in which case the only subfield of K is Q itself. If d ≠ 1 then the Galois closure N of K contains a unique quadratic field k |
https://en.wikipedia.org/wiki/Scott%27s%20trick | In set theory, Scott's trick is a method for giving a definition of equivalence classes for equivalence relations on a proper class (Jech 2003:65) by referring to levels of the cumulative hierarchy.
The method relies on the axiom of regularity but not on the axiom of choice. It can be used to define representatives for ordinal numbers in ZF, Zermelo–Fraenkel set theory without the axiom of choice (Forster 2003:182). The method was introduced by .
Beyond the problem of defining set representatives for ordinal numbers, Scott's trick can be used to obtain representatives for cardinal numbers and more generally for isomorphism types, for example, order types of linearly ordered sets (Jech 2003:65). It is credited to be indispensable (even in the presence of the axiom of choice) when taking ultrapowers of proper classes in model theory. (Kanamori 1994:47)
Application to cardinalities
The use of Scott's trick for cardinal numbers shows how the method is typically employed. The initial definition of a cardinal number is an equivalence class of sets, where two sets are equivalent if there is a bijection between them. The difficulty is that almost every equivalence class of this relation is a proper class, and so the equivalence classes themselves cannot be directly manipulated in set theories, such as Zermelo–Fraenkel set theory, that only deal with sets. It is often desirable in the context of set theory to have sets that are representatives for the equivalence classes. These sets are then taken to "be" cardinal numbers, by definition.
In Zermelo–Fraenkel set theory with the axiom of choice, one way of assigning representatives to cardinal numbers is to associate each cardinal number with the least ordinal number of the same cardinality. These special ordinals are the ℵ numbers. But if the axiom of choice is not assumed, for some cardinal numbers it may not be possible to find such an ordinal number, and thus the cardinal numbers of those sets have no ordinal number as representatives.
Scott's trick assigns representatives differently, using the fact that for every set there is a least rank in the cumulative hierarchy when some set of the same cardinality as appears. Thus one may define the representative of the cardinal number of to be the set of all sets of rank that have the same cardinality as . This definition assigns a representative to every cardinal number even when not every set can be well-ordered (an assumption equivalent to the axiom of choice). It can be carried out in Zermelo–Fraenkel set theory, without using the axiom of choice, but making essential use of the axiom of regularity.
Scott's trick in general
Let be an equivalence relation of sets. Let be a set and its equivalence class with respect to . If is non-empty, we can define a set, which represents , even if is a proper class. Namely, there exists a least ordinal , such that is non-empty. This intersection is a set, so we can take it as the representative of . We |
https://en.wikipedia.org/wiki/Gerald%20B.%20Whitham | Gerald Beresford Whitham FRS (13 December 1927 – 26 January 2014) was a British–born American applied mathematician and the Charles Lee Powell Professor of Applied Mathematics (Emeritus) of Applied & Computational Mathematics at the California Institute of Technology. He received his Ph.D. from the University of Manchester in 1953 under the direction of Sir James Lighthill. He is known for his work in fluid dynamics and waves.
Academic career
Whitham was born in Halifax, West Yorkshire. He received his Ph.D. from University of Manchester in 1953. He was a Faculty Member in the Department of Mathematics at the Massachusetts Institute of Technology during 1959–1962. He left MIT to join California Institute of Technology, Pasadena, California where he was instrumental in setting up the applied mathematics program in 1962.
Honors and awards
Whitham is a Fellow of the American Academy of Arts and Sciences since 1959. In 1965, Whitham was elected a Fellow of the Royal Society.
Whitham received the Norbert Wiener Prize in Applied Mathematics in 1980, jointly awarded by the Society for Industrial and Applied Mathematics (SIAM) and the American Mathematical Society (AMS). This prize was awarded "for an outstanding contribution to applied mathematics in the highest and broadest sense." Whitham was honored "for his broad contributions to the understanding of fluid dynamical phenomena and his innovative contributions to the methodology through which that understanding can be constructed".
Selected articles
Articles
(See George C. McVittie.)
Books
G. B. Whitham, Linear and Nonlinear Waves, John Wiley & Sons (1974).
References
External links
20th-century American mathematicians
21st-century American mathematicians
Massachusetts Institute of Technology School of Science faculty
California Institute of Technology faculty
Fellows of the Royal Society
Fellows of the American Academy of Arts and Sciences
Fluid dynamicists
People from Halifax, West Yorkshire
1927 births
2014 deaths
British emigrants to the United States |
https://en.wikipedia.org/wiki/2007%E2%80%9308%20NK%20Dinamo%20Zagreb%20season | This article shows statistics of individual players for the football club Dinamo Zagreb It also lists all matches that Dinamo Zagreb played in the 2007–08 season.
Competitions
Overall
Prva HNL
Classification
Results summary
Results by round
Results by opponent
Source: Prva HNL 2007–08 article
UEFA Cup
Classification
Matches
Competitive
Goalscorers
External links
Dinamo Zagreb official website
GNK Dinamo Zagreb seasons
Dinamo Zagreb
Croatian football championship-winning seasons |
https://en.wikipedia.org/wiki/Erich%20Kamke | Erich Kamke (18 August 1890 – 28 September 1961) was a German mathematician, who specialized in the theory of differential equations. Also, his book on set theory became a standard introduction to the field.
Biography
Kamke was born in Marienburg, West Prussia, German Empire (modern Malbork, Poland).
After attending school in Stettin, Kamke studied mathematics and physics from 1909 at the University of Giessen and the University of Göttingen. He was a volunteer in the signals force in World War I. In 1919, he married Dora Heimowitch, who was the daughter of a Jewish businessman. He earned his doctorate in 1919 at the University of Göttingen under Edmund Landau with thesis Verallgemeinerungen des Waring-Hilbertschen Satzes (Generalizations of the Waring-Hilbert theorem). While teaching between 1920 and 1926, Kamke earned his habilitation at the University of Münster in 1922. He was appointed as a professor at the University of Tübingen in 1926.
Because of his marriage and his opposition to National Socialism, he was denounced by fellow mathematician Erich Schönhardt and eventually forced to retire in 1937.
Following World War II, he was reappointed as a professor at the University of Tübingen, and was instrumental in the organisation of a mathematical congress in Tübingen in autumn 1946, the first scientific congress in Germany after the war. In 1948, he re-established the German Mathematical Society, and was the chairman until 1952, when he became vice-president of the International Mathematical Union, which he remained until 1954.
He died in Rottenburg am Neckar from a heart attack.
Works
Das Lebesguesche Integral. Eine Einführung in die neuere Theorie der reellen Funktionen, B. G. Teubner, Leipzig 1925
Mengenlehre, Sammlung Göschen/Walter de Gruyter, Berlin 1928
Differentialgleichungen reeller Funktionen, Akademische Verlagsgesellschaft, Leipzig 1930; ab der 4. (überarbeiteten) Auflage 1962 in zwei Bänden:
Band 1: Gewöhnliche Differentialgleichungen
Band 2: Partielle Differentialgleichungen
Einführung in die Wahrscheinlichkeitstheorie, S. Hirzel, Leipzig 1932
Differentialgleichungen. Lösungsmethoden und Lösungen I. Gewöhnliche Differentialgleichungen, Leipzig 1942
Differentialgleichungen. Lösungsmethoden und Lösungen II. Partielle Differentialgleichungen 1. Ordnung für eine gesuchte Funktion, Leipzig 1944
Das Lebesgue-Stieltjes-Integral, B. G. Teubner, Leipzig 1956
References
1890 births
1961 deaths
20th-century German mathematicians
People from Malbork
People from West Prussia
German Army personnel of World War I
University of Giessen alumni
University of Göttingen alumni
University of Münster alumni
Academic staff of the University of Tübingen |
https://en.wikipedia.org/wiki/2008%E2%80%9309%20PFC%20CSKA%20Sofia%20season | The 2008–09 season was PFC CSKA Sofia's 61st consecutive season in A Group. This article shows player statistics and all matches (official and friendly) that the club have and will play during the 2008–09 season.
Players
Squad information
Appearances for competitive matches only
|-
|colspan="14"|Players sold or loaned out after the start of the season:
|}
As of game played start of season
Players in/out
Summer transfers
In:
Out:
Winter transfers
In:
Out:
Pre-season and friendlies
Pre-season
On-season (autumn)
Mid-season
On-season (spring)
Competitions
A Group
Table
Results summary
Results by round
Fixtures and results
Bulgarian Super Cup
Bulgarian Cup
See also
PFC CSKA Sofia
External links
Official Site
PFC CSKA Sofia seasons
Cska Sofia |
https://en.wikipedia.org/wiki/Cayley%27s%20%CE%A9%20process | In mathematics, Cayley's Ω process, introduced by , is a relatively invariant differential operator on the general linear group, that is used to construct invariants of a group action.
As a partial differential operator acting on functions of n2 variables xij, the omega operator is given by the determinant
For binary forms f in x1, y1 and g in x2, y2 the Ω operator is . The r-fold Ω process Ωr(f, g) on two forms f and g in the variables x and y is then
Convert f to a form in x1, y1 and g to a form in x2, y2
Apply the Ω operator r times to the function fg, that is, f times g in these four variables
Substitute x for x1 and x2, y for y1 and y2 in the result
The result of the r-fold Ω process Ωr(f, g) on the two forms f and g is also called the r-th transvectant and is commonly written (f, g)r.
Applications
Cayley's Ω process appears in Capelli's identity, which
used to find generators for the invariants of various classical groups acting on natural polynomial algebras.
used Cayley's Ω process in his proof of finite generation of rings of invariants of the general linear group. His use of the Ω process gives an explicit formula for the Reynolds operator of the special linear group.
Cayley's Ω process is used to define transvectants.
References
Reprinted in
Invariant theory |
https://en.wikipedia.org/wiki/Maier%27s%20theorem | In number theory, Maier's theorem is a theorem about the numbers of primes in short intervals for which Cramér's probabilistic model of primes gives a wrong answer.
The theorem states that if π is the prime-counting function and λ is greater than 1 then
does not have a limit as x tends to infinity; more precisely the limit superior is greater than 1, and the limit inferior is less than 1. The Cramér model of primes predicts incorrectly that it has limit 1 when λ≥2 (using the Borel–Cantelli lemma).
Proofs
Maier proved his theorem using Buchstab's equivalent for the counting function of quasi-primes (set of numbers without prime factors lower to bound , fixed). He also used an equivalent of the number of primes in arithmetic progressions of sufficient length due to Gallagher.
gave another proof, and also showed that most probabilistic models of primes incorrectly predict the mean square error
of one version of the prime number theorem.
See also
Maier's matrix method
References
Theorems in analytic number theory
Probabilistic models
Theorems about prime numbers |
https://en.wikipedia.org/wiki/Polyakov%20formula | In differential geometry and mathematical physics (especially string theory), the Polyakov formula expresses the conformal variation of the zeta functional determinant of a Riemannian manifold. Proposed by Alexander Markovich Polyakov this formula arose in the study of the quantum theory of strings. The corresponding density is local, and therefore is a Riemannian curvature invariant. In particular, whereas the functional determinant itself is prohibitively difficult to work with in general, its conformal variation can be written down explicitly.
References
Conformal geometry
Spectral theory
String theory |
https://en.wikipedia.org/wiki/Main%20conjecture%20of%20Iwasawa%20theory | In mathematics, the main conjecture of Iwasawa theory is a deep relationship between p-adic L-functions and ideal class groups of cyclotomic fields, proved by Kenkichi Iwasawa for primes satisfying the Kummer–Vandiver conjecture and proved for all primes by
. The Herbrand–Ribet theorem and the Gras conjecture are both easy consequences of the main conjecture.
There are several generalizations of the main conjecture, to totally real fields, CM fields, elliptic curves, and so on.
Motivation
was partly motivated by an analogy with Weil's description of the zeta function of an algebraic curve over a finite field in terms of eigenvalues of the Frobenius endomorphism on its Jacobian variety. In this analogy,
The action of the Frobenius corresponds to the action of the group Γ.
The Jacobian of a curve corresponds to a module X over Γ defined in terms of ideal class groups.
The zeta function of a curve over a finite field corresponds to a p-adic L-function.
Weil's theorem relating the eigenvalues of Frobenius to the zeros of the zeta function of the curve corresponds to Iwasawa's main conjecture relating the action of the Iwasawa algebra on X to zeros of the p-adic zeta function.
History
The main conjecture of Iwasawa theory was formulated as an assertion that two methods of defining p-adic L-functions (by module theory, by interpolation) should coincide, as far as that was well-defined. This was proved by for Q, and for all totally real number fields by . These proofs were modeled upon Ken Ribet's proof of the converse to Herbrand's theorem (the Herbrand–Ribet theorem).
Karl Rubin found a more elementary proof of the Mazur–Wiles theorem by using Thaine's method and Kolyvagin's Euler systems, described in and , and later proved other generalizations of the main conjecture for imaginary quadratic fields.
In 2014, Christopher Skinner and Eric Urban proved several cases of the main conjectures for a large class of modular forms. As a consequence, for a modular elliptic curve over the rational numbers, they prove that the vanishing of the Hasse–Weil L-function L(E, s) of E at s = 1 implies that the p-adic Selmer group of E is infinite. Combined with theorems of Gross-Zagier and Kolyvagin, this gave a conditional proof (on the Tate–Shafarevich conjecture) of the conjecture that E has infinitely many rational points if and only if L(E, 1) = 0, a (weak) form of the Birch–Swinnerton-Dyer conjecture. These results were used by Manjul Bhargava, Skinner, and Wei Zhang to prove that a positive proportion of elliptic curves satisfy the Birch–Swinnerton-Dyer conjecture.
Statement
p is a prime number.
Fn is the field Q(ζ) where ζ is a root of unity of order pn+1.
Γ is the largest subgroup of the absolute Galois group of F∞ isomorphic to the p-adic integers.
γ is a topological generator of Γ
Ln is the p-Hilbert class field of Fn.
Hn is the Galois group Gal(Ln/Fn), isomorphic to the subgroup of elements of the ideal class group of Fn whose order is a p |
https://en.wikipedia.org/wiki/Weierstrass%E2%80%93Erdmann%20condition | The Weierstrass–Erdmann condition is a mathematical result from the calculus of variations, which specifies sufficient conditions for broken extremals (that is, an extremal which is constrained to be smooth except at a finite number of "corners").
Conditions
The Weierstrass-Erdmann corner conditions stipulate that a broken extremal of a functional satisfies the following two continuity relations at each corner :
Applications
The condition allows one to prove that a corner exists along a given extremal. As a result, there are many applications to differential geometry. In calculations of the Weierstrass E-Function, it is often helpful to find where corners exist along the curves. Similarly, the condition allows for one to find a minimizing curve for a given integral.
References
Calculus of variations |
https://en.wikipedia.org/wiki/E.%20Brian%20Davies | Edward Brian Davies FRS (born 13 June 1944) is a former professor of Mathematics, King's College London (1981–2010), and is the author of the popular science book Science in the Looking Glass: What do Scientists Really Know. In 2010, he was awarded a Gauss Lecture by the German Mathematical Society.
Publications
Books
References
External links
Brian Davies' King's College London Home Page
Brian Davies' Personal Home Page
Full proof? Let's trust it to the black box Times Higher Education 1 September 2006
New Scientist book review of Why Beliefs Matter
London Mathematical Society Review of Why Beliefs Matter: Reflections on the Nature of Science by Colva Roney-Dougal
1944 births
Living people
Alumni of Jesus College, Oxford
Academics of King's College London
Fellows of King's College London
Fellows of the Royal Society
21st-century British mathematicians
Welsh mathematicians
Fellows of St John's College, Oxford
Academic journal editors
20th-century British mathematicians |
https://en.wikipedia.org/wiki/Motivic%20zeta%20function | In algebraic geometry, the motivic zeta function of a smooth algebraic variety is the formal power series:
Here is the -th symmetric power of , i.e., the quotient of by the action of the symmetric group , and is the class of in the ring of motives (see below).
If the ground field is finite, and one applies the counting measure to , one obtains the local zeta function of .
If the ground field is the complex numbers, and one applies Euler characteristic with compact supports to , one obtains .
Motivic measures
A motivic measure is a map from the set of finite type schemes over a field to a commutative ring , satisfying the three properties
depends only on the isomorphism class of ,
if is a closed subscheme of ,
.
For example if is a finite field and is the ring of integers, then defines a motivic measure, the counting measure.
If the ground field is the complex numbers, then Euler characteristic with compact supports defines a motivic measure with values in the integers.
The zeta function with respect to a motivic measure is the formal power series in given by
.
There is a universal motivic measure. It takes values in the K-ring of varieties, , which is the ring generated by the symbols , for all varieties , subject to the relations
if and are isomorphic,
if is a closed subvariety of ,
.
The universal motivic measure gives rise to the motivic zeta function.
Examples
Let denote the class of the affine line.
If is a smooth projective irreducible curve of genus admitting a line bundle of degree 1, and the motivic measure takes values in a field in which is invertible, then
where is a polynomial of degree . Thus, in this case, the motivic zeta function is rational. In higher dimension, the motivic zeta function is not always rational.
If is a smooth surface over an algebraically closed field of characteristic , then the generating function for the motives of the Hilbert schemes of can be expressed in terms of the motivic zeta function by Göttsche's Formula
Here is the Hilbert scheme of length subschemes of . For the affine plane this formula gives
This is essentially the partition function.
References
Functions and mappings
Algebraic geometry |
https://en.wikipedia.org/wiki/Oversampling%20and%20undersampling%20in%20data%20analysis | Within statistics, oversampling and undersampling in data analysis are techniques used to adjust the class distribution of a data set (i.e. the ratio between the different classes/categories represented). These terms are used both in statistical sampling, survey design methodology and in machine learning.
Oversampling and undersampling are opposite and roughly equivalent techniques.
There are also more complex oversampling techniques, including the creation of artificial data points with algorithms like Synthetic minority oversampling technique.
Motivation for oversampling and undersampling
Both oversampling and undersampling involve introducing a bias to select more samples from one class than from another, to compensate for an imbalance that is either already present in the data, or likely to develop if a purely random sample were taken. Data Imbalance can be of the following types:
Under-representation of a class in one or more important predictor variables. Suppose, to address the question of gender discrimination, we have survey data on salaries within a particular field, e.g., computer software. It is known women are under-represented considerably in a random sample of software engineers, which would be important when adjusting for other variables such as years employed and current level of seniority. Suppose only 20% of software engineers are women, i.e., males are 4 times as frequent as females. If we were designing a survey to gather data, we would survey 4 times as many females as males, so that in the final sample, both genders will be represented equally. (See also Stratified Sampling.)
Under-representation of one class in the outcome (dependent) variable. Suppose we want to predict, from a large clinical dataset, which patients are likely to develop a particular disease (e.g., diabetes). Assume, however, that only 10% of patients go on to develop the disease. Suppose we have a large existing dataset. We can then pick 9 times the number of patients who did not go on to develop the disease for every one patient who did.
Oversampling is generally employed more frequently than undersampling, especially when the detailed data has yet to be collected by survey, interview or otherwise. Undersampling is employed much less frequently. Overabundance of already collected data became an issue only in the "Big Data" era, and the reasons to use undersampling are mainly practical and related to resource costs. Specifically, while one needs a suitably large sample size to draw valid statistical conclusions, the data must be cleaned before it can be used. Cleansing typically involves a significant human component, and is typically specific to the dataset and the analytical problem, and therefore takes time and money. For example:
Domain experts will suggest dataset-specific means of validation involving not only intra-variable checks (permissible values, maximum and minimum possible valid values, etc.), but also inter-variable checks. For e |
https://en.wikipedia.org/wiki/Goss%20zeta%20function | In the field of mathematics, the Goss zeta function, named after David Goss, is an analogue of the Riemann zeta function for function fields. proved that it satisfies an analogue of the Riemann hypothesis. proved results for a higher-dimensional generalization of the Goss zeta function.
References
Zeta and L-functions |
https://en.wikipedia.org/wiki/Normal%20model | Normal model may refer to:
Normal distribution, a type of continuous probability distribution
A model of interpreting equality (see Interpretation (logic)#Interpreting equality) |
https://en.wikipedia.org/wiki/Levi-Civita%20field | In mathematics, the Levi-Civita field, named after Tullio Levi-Civita, is a non-Archimedean ordered field; i.e., a system of numbers containing infinite and infinitesimal quantities. Each member can be constructed as a formal series of the form
where are real numbers, is the set of rational numbers, and is to be interpreted as a fixed positive infinitesimal. The support of , i.e., the set of indices of the nonvanishing coefficients must be a left-finite set: for any member of , there are only finitely many members of the set less than it; this restriction is necessary in order to make multiplication and division well defined and unique. The ordering is defined according to the dictionary ordering of the list of coefficients, which is equivalent to the assumption that is an infinitesimal.
The real numbers are embedded in this field as series in which all of the coefficients vanish except .
Examples
is an infinitesimal that is greater than , but less than every positive real number.
is less than , and is also less than for any positive real .
differs infinitesimally from 1.
is greater than and even greater than times any positive real number, no matter how big, but is still less than every positive real number.
is greater than any real number.
is interpreted as .
is a valid member of the field, because the series is to be construed formally, without any consideration of convergence.
Definition of the field operations and positive cone
If and are two Levi-Civita series, then
their sum is the pointwise sum .
their product is the Cauchy product .
(One can check that the support of this series is left-finite and that for each of its elements , the set is finite, so the product is well defined.)
the relation holds if (i.e. has non-empty support) and the least non-zero coefficient of is strictly positive.
Equipped with those operations and order, the Levi-Civita field is indeed an ordered field extension of where the series is a positive infinitesimal.
Properties and applications
The Levi-Civita field is real-closed, meaning that it can be algebraically closed by adjoining an imaginary unit (i), or by letting the coefficients be complex. It is rich enough to allow a significant amount of analysis to be done, but its elements can still be represented on a computer in the same sense that real numbers can be represented using floating point.
It is the basis of automatic differentiation, a way to perform differentiation in cases that are intractable by symbolic differentiation or finite-difference methods.
The Levi-Civita field is also Cauchy complete, meaning that relativizing the definitions of Cauchy sequence and convergent sequence to sequences of Levi-Civita series, each Cauchy sequence in the field converges. Equivalently, it has no proper dense ordered field extension.
As an ordered field, it has a natural valuation given by the rational exponent corresponding to the first non zero coefficient of a |
https://en.wikipedia.org/wiki/Interdecile%20range | In statistics, the interdecile range is the difference between the first and the ninth deciles (10% and 90%). The interdecile range is a measure of statistical dispersion of the values in a set of data, similar to the range and the interquartile range, and can be computed from the (non-parametric) seven-number summary.
Despite its simplicity, the interdecile range of a sample drawn from a normal distribution can be divided by 2.56 to give a reasonably efficient estimator of the standard deviation of a normal distribution. This is derived from the fact that the lower (respectively upper) decile of a normal distribution with arbitrary variance is equal to the mean minus (respectively, plus) 1.28 times the standard deviation.
A more efficient estimator is given by instead taking the 7% trimmed range (the difference between the 7th and 93rd percentiles) and dividing by 3 (corresponding to 86% of the data falling within ±1.5 standard deviations of the mean in a normal distribution); this yields an estimator having about 65% efficiency. Analogous measures of location are given by the median, midhinge, and trimean (or statistics based on nearby points).
See also
Interquartile range
Robust measures of scale
Standard deviation
Statistical analysis
References
Scale statistics |
https://en.wikipedia.org/wiki/Redheffer%20matrix | In mathematics, a Redheffer matrix, often denoted as studied by , is a square (0,1) matrix whose entries aij are 1 if i divides j or if j = 1; otherwise, aij = 0. It is useful in some contexts to express Dirichlet convolution, or convolved divisors sums, in terms of matrix products involving the transpose of the Redheffer matrix.
Variants and definitions of component matrices
Since the invertibility of the Redheffer matrices are complicated by the initial column of ones in the matrix, it is often convenient to express where is defined to be the (0,1) matrix whose entries are one if and only if and . The remaining one-valued entries in then correspond to the divisibility condition reflected by the matrix , which plainly can be seen by an application of Mobius inversion is always invertible with inverse . We then have a characterization of the singularity of expressed by
If we define the function
then we can define the Redheffer (transpose) matrix to be the nxn square matrix in usual matrix notation. We will continue to make use this notation throughout the next sections.
Examples
The matrix below is the 12 × 12 Redheffer matrix. In the split sum-of-matrices notation for , the entries below corresponding to the initial column of ones in are marked in blue.
A corresponding application of the Mobius inversion formula shows that the Redheffer transpose matrix is always invertible, with inverse entries given by
where denotes the Moebius function. In this case, we have that the inverse Redheffer transpose matrix is given by
Key properties
Singularity and relations to the Mertens function and special series
Determinants
The determinant of the n × n square Redheffer matrix is given by the Mertens function M(n). In particular, the matrix is not invertible precisely when the Mertens function is zero (or is close to changing signs). As a corollary of the disproof of the Mertens conjecture, it follows that the Mertens function changes sign, and is therefore zero, infinitely many times, so the Redheffer matrix is singular at infinitely many natural numbers.
The determinants of the Redheffer matrices are immediately tied to the Riemann Hypothesis through this relation with the Mertens function, since the Hypothesis is equivalent to showing that for all (sufficiently small) .
Factorizations of sums encoded by these matrices
In a somewhat unconventional construction which reinterprets the (0,1) matrix entries to denote inclusion in some increasing sequence of indexing sets, we can see that these matrices are also related to factorizations of Lambert series. This observation is offered in so much as for a fixed arithmetic function f, the coefficients of the next Lambert series expansion over f provide a so-called inclusion mask for the indices over which we sum f to arrive at the series coefficients of these expansions. Notably, observe that
Now in the special case of these divisor sums, which we can see from the above expansion, |
https://en.wikipedia.org/wiki/Fractional%20Calculus%20and%20Applied%20Analysis | Fractional Calculus and Applied Analysis is a peer-reviewed mathematics journal published by Walter de Gruyter. It covers research on fractional calculus, special functions, integral transforms, and some closely related areas of applied analysis.
The journal is abstracted and indexed in Science Citation Index Expanded, Scopus, Current Contents/Physical, Chemical and Earth Sciences, Zentralblatt MATH, and Mathematical Reviews.
The journal's Founding Editors were Professors Eric Love, Ian Sneddon, Bogoljub Stanković, Rudolf Gorenflo, Danuta Przeworska-Rolewicz, Gary Roach, Anatoly Kilbas, and Wen Chen.
References
External links
Mathematics journals
Academic journals established in 1998
Quarterly journals
De Gruyter academic journals
English-language journals |
https://en.wikipedia.org/wiki/Winifred%20Asprey | Winifred "Tim" Alice Asprey (April 8, 1917 – October 19, 2007) was an American mathematician and computer scientist. She was one of only around 200 women to earn PhDs in mathematics from American universities during the 1940s, a period of women's underrepresentation in mathematics at this level.
She was involved in developing the close contact between Vassar College and IBM that led to the establishment of the first computer science lab at Vassar.
Family
Asprey was born in Sioux City, Iowa; her parents were Gladys Brown Asprey, Vassar class of 1905, and Peter Asprey Jr. She had two brothers, actinide and fluorine chemist Larned B. Asprey (1919–2005), a signer of the Szilárd petition, and military historian and writer Robert B. Asprey (1923–2009) who dedicated several of his books to his sister Winifred.
Education and work
Asprey attended Vassar College in Poughkeepsie, New York, where she earned her undergraduate degree in 1938. As a student there, Asprey met Grace Hopper, the "First Lady of Computing," who taught mathematics at the time. After graduating, Asprey taught at several private schools in New York City and Chicago before going on to earn her MS and PhD degrees from the University of Iowa in 1942 and 1945, respectively. Her doctoral advisor was the topologist Edward Wilson Chittenden.
Asprey returned to Vassar College as a professor. By then, Grace Hopper had moved to Philadelphia to work on UNIVAC (Universal Automatic Computer) project. Asprey became interested in computing and visited Hopper to learn about the foundations of computer architecture. Asprey believed that computers would be an essential part of a liberal arts education.
At Vassar, Asprey taught mathematics and computer science for 38 years and was the chair of the mathematics department from 1957 until her retirement in 1982.
She created the first Computer Science courses at Vassar, the first being taught in 1963, and secured funds for the college's first computer, making Vassar the second college in the nation to acquire an IBM System/360 computer in 1967.
Asprey connected with researchers at IBM and other research centers and lobbied for computer science at Vassar. In 1989, due to her contributions, the computer center she started was renamed the Asprey Advanced Computation Laboratory.
References
External links
Profile at Vassar College Innovators Gallery
Winifred Asprey Papers at Vassar College Archives and Special Collections Library
Vassar College alumni
University of Iowa alumni
Vassar College faculty
American women computer scientists
20th-century American women scientists
American women mathematicians
Computer science educators
1917 births
2007 deaths
20th-century American mathematicians
Mathematicians from Iowa
20th-century women mathematicians
20th-century American scientists
American computer scientists
21st-century American women |
https://en.wikipedia.org/wiki/Electronic%20Journal%20of%20Combinatorics | The Electronic Journal of Combinatorics is a peer-reviewed open access scientific journal covering research in combinatorial mathematics.
The journal was established in 1994 by Herbert Wilf (University of Pennsylvania) and Neil Calkin (Georgia Institute of Technology). The Electronic Journal of Combinatorics is a founding member of the Free Journal Network. According to the Journal Citation Reports, the journal had a 2017 impact factor of 0.762.
Editors-in-chief
Current
The current editors-in-chief at Electronic Journal of Combinatorics are:
Maria Axenovich, Karlsruhe Institute of Technology, Germany
Miklós Bóna, University of Florida, United States
Julia Böttcher, London School of Economics, United Kingdom
Richard A. Brualdi, University of Wisconsin, Madison, United States
Zdeněk Dvořák, Charles University, Czech Republic
Eric Fusy, CNRS/LIX, École Polytechnique, France
Catherine Greenhill, UNSW Sydney, Australia
Felix Joos, Universität Heidelberg, Germany
Brendan McKay, Australian National University, Australia
Bojan Mohar, Simon Fraser University, Canada
Marc Noy, Universitat Politècnica de Catalunya, Spain
Greta Panova, University of Southern California, United States
Alexey Pokrovskiy, University College London, United Kingdom
Gordon Royle, University of Western Australia, Australia
Bruce Sagan, Michigan State University, United States
Paco Santos, University of Cantabria, Spain
Maya Stein, University of Chile, Chile
Edwin van Dam, Tilburg University, Netherlands
Ian Wanless, Monash University, Australia
David Wood, Monash University, Australia
Qing Xiang, Southern University of Science and Technology, China
Since 2013, one of the editors-in-chief has been designated the Chief Editorial Officer. The present officer is Bruce Sagan.
Past
The following people have been editors-in-chief of the Electronic Journal of Combinatorics:
Dynamic surveys
In addition to publishing normal articles, the journal also contains a class of articles called Dynamic Surveys that are not assigned to volumes and can be repeatedly updated by the authors.
Open access
Since its inception, the journal has operated under the diamond-model open access model, charging no fees to either authors or readers. It is a founding member of the Free Journal Network.
Copyright
Since its inception, the journal has left copyright of all published material with its authors. Instead, authors provide the journal with an irrevocable licence to publish and agree that any further publication of the material acknowledges the journal. Since 2018, authors have been strongly encouraged to release their articles under a Creative Commons license.
References
External links
Combinatorics journals
Academic journals established in 1994
Open access journals
English-language journals
Online-only journals |
https://en.wikipedia.org/wiki/Theodore%20James%20Courant | Theodore James "Ted" Courant is an American mathematician who has conducted research in the fields of differential geometry and classical mechanics. In particular, he made seminal contributions to the study of Dirac manifolds, which generalize both symplectic manifolds and Poisson manifolds, and are related to the Dirac theory of constraints in physics. Some mathematical objects in this field have since been named after him, including the Courant bracket and Courant algebroid.
Education and career
Courant received his B.A. degree from Reed College, and his Ph.D. from The University of California, Berkeley, where he was a student of Alan Weinstein.
After teaching at the University of California, Santa Cruz and the University of Minnesota, Courant moved to secondary education at private schools in California including The Branson School and Wildwood School.
Personal life
Ted Courant is the grandson of Richard Courant.
References
Reed College alumni
University of California, Berkeley alumni
20th-century American mathematicians
Living people
Year of birth missing (living people)
21st-century American mathematicians |
https://en.wikipedia.org/wiki/Topology%20control | Topology control is a technique used in distributed computing to alter the underlying network (modeled as a graph) to reduce the cost of distributed algorithms if run over the resulting graphs. It is a basic technique in distributed algorithms. For instance, a (minimum) spanning tree is used as a backbone to reduce the cost of broadcast from O(m) to O(n), where m and n are the number of edges and vertices in the graph, respectively.
The term "topology control" is used mostly by the wireless ad hoc and sensor networks research community. The main aim of topology control in this domain is to save energy, reduce interference between nodes and extend lifetime of the network. However, recently the term has also been gaining traction with regards to control of the network structure of electric power systems.
Topology construction and maintenance
Lately, topology control algorithms have been divided into two subproblems: topology construction, in charge of the initial reduction, and topology maintenance, in charge of the maintenance of the reduced topology so that characteristics like connectivity and coverage are preserved.
This is the first stage of a topology control protocol. Once the initial topology is deployed, specially when the location of the nodes is random, the administrator has no control over the design of the network; for example, some areas may be very dense, showing a high number of redundant nodes, which will increase the number of message collisions and will provide several copies of the same information from similarly located nodes. However, the administrator has control over some parameters of the network: transmission power of the nodes, state of the nodes (active or sleeping), role of the nodes (Clusterhead, gateway, regular), etc. By modifying these parameters, the topology of the network can change.
Upon the same time a topology is reduced and the network starts serving its purpose, the selected nodes start spending energy: Reduced topology starts losing its "optimality as soon as full network activity evolves.
After some time being active, some nodes will start to run out of energy. Especially in wireless sensor networks with multihopping, intensive packet forwarding causes nodes that are closer to the sink to spend higher amounts of energy than nodes that are farther away.
Topology control has to be executed periodically in order to preserve the desired properties such as connectivity, coverage, density.
Topology construction algorithms
There are many ways to perform topology construction:
Optimizing the node locations during the deployment phase
Change the transmission range of the nodes
Turn off nodes from the network
Create a communication backbone
Clustering
Adding new nodes to the network to preserve connectivity (Federated Wireless sensor networks)
Some examples of topology construction algorithms are:
Tx range-based
Geometry-based: Gabriel graph (GG), Relative neighborhood graph (RNG), Voronoi diagram
|
https://en.wikipedia.org/wiki/Zhongshan%20Min | {
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Zhongshan Min (), known as Cunhua () by its speakers, are three Min Chinese dialect islands in the Zhongshan region of the southern Chinese province of Guangdong. The Zhongshan Min people settled in the region from Fujian Province as early as the Northern Song dynasty period (1023–1031).
The three dialects are:
Longdu dialect, spoken mainly in Shaxi and Dachong in the west of the prefecture,
Nanlang dialect or Dongxiang dialect, spoken mainly in Nanlang and Zhangjiabian in the east, and
Sanxiang dialect, spoken in Sanxiang in the south.
According to Nicholas Bodman, the Longdu and Nanlang dialects belong to the Eastern Min group, while the Sanxiang dialect belongs to Southern Min. All three have been heavily influenced by the Shiqi dialect, the local variety of Yue Chinese.
As the dialect with the most speakers, the Longdu dialect may be taken as the representative dialect of Zhongshan Min.
Notes
References
Sources
See also
Zhongshan
Southern Min |
https://en.wikipedia.org/wiki/Semi-reflexive%20space | In the area of mathematics known as functional analysis, a semi-reflexive space is a locally convex topological vector space (TVS) X such that the canonical evaluation map from X into its bidual (which is the strong dual of the strong dual of X) is bijective.
If this map is also an isomorphism of TVSs then it is called reflexive.
Semi-reflexive spaces play an important role in the general theory of locally convex TVSs.
Since a normable TVS is semi-reflexive if and only if it is reflexive, the concept of semi-reflexivity is primarily used with TVSs that are not normable.
Definition and notation
Brief definition
Suppose that is a topological vector space (TVS) over the field (which is either the real or complex numbers) whose continuous dual space, , separates points on (i.e. for any there exists some such that ).
Let and both denote the strong dual of , which is the vector space of continuous linear functionals on endowed with the topology of uniform convergence on bounded subsets of ;
this topology is also called the strong dual topology and it is the "default" topology placed on a continuous dual space (unless another topology is specified).
If is a normed space, then the strong dual of is the continuous dual space with its usual norm topology.
The bidual of , denoted by , is the strong dual of ; that is, it is the space .
For any let be defined by , where is called the evaluation map at ;
since is necessarily continuous, it follows that .
Since separates points on , the map defined by is injective where this map is called the evaluation map or the canonical map.
This map was introduced by Hans Hahn in 1927.
We call semireflexive if is bijective (or equivalently, surjective) and we call reflexive if in addition is an isomorphism of TVSs.
If is a normed space then is a TVS-embedding as well as an isometry onto its range;
furthermore, by Goldstine's theorem (proved in 1938), the range of is a dense subset of the bidual .
A normable space is reflexive if and only if it is semi-reflexive.
A Banach space is reflexive if and only if its closed unit ball is -compact.
Detailed definition
Let be a topological vector space over a number field (of real numbers or complex numbers ).
Consider its strong dual space , which consists of all continuous linear functionals and is equipped with the strong topology , that is, the topology of uniform convergence on bounded subsets in .
The space is a topological vector space (to be more precise, a locally convex space), so one can consider its strong dual space , which is called the strong bidual space for .
It consists of all
continuous linear functionals and is equipped with the strong topology .
Each vector generates a map by the following formula:
This is a continuous linear functional on , that is, .
One obtains a map called the evaluation map or the canonical injection:
which is a linear map.
If is locally convex, from the Hahn–Banach theorem it follows that is in |
https://en.wikipedia.org/wiki/Regius%20Professor%20of%20Mathematics | The Regius Professorship of Mathematics is the name given to three chairs in mathematics at British universities, one at the University of St Andrews, founded by Charles II in 1668, the second one at the University of Warwick, founded in 2013 to commemorate the Diamond Jubilee of Elizabeth II and the third one at the University of Oxford, founded in 2016.
University of St Andrews (1668)
From 1997 to 2015 there was no Regius Professor of Mathematics. In April 2013 the post was advertised, and in 2015 Igor Rivin was appointed. He was succeeded by Kenneth Falconer in 2017.
List of Regius Professors of Mathematics
The following list may be incomplete.
1668–1674 James Gregory
1674–1688 William Sanders
1689–1690 James Fenton
1690–1707 vacant
1707–1739 Charles Gregory
1739–1765 David Gregory
1765–1807 Nicolas Vilant
1807–1809 vacant
1809–1820 Robert Haldane
1820–1858 Thomas Duncan
1857–1858 John Couch Adams
1859–1877 William L F Fischer
1877–1879 George Chrystal
1879–1921 Peter Redford Scott Lang
1921–1950 Herbert Westren Turnbull
1950–1969 Edward Thomas Copson
1970–1997 John Mackintosh Howie
1997–2015 vacant
2015–2017 Igor Rivin
2017–present Kenneth Falconer
University of Warwick (2013)
The creation of the post of the Regius Professor of Mathematics was announced in January 2013, in March 2014 Martin Hairer was appointed to the position.
University of Oxford (2016)
The creation of the post of the Regius Professor of Mathematics was announced in June 2016 and Andrew Wiles was appointed as the first holder of the chair in May 2018.
In August 2020, it was announced that the Regius Professorship in Mathematics at the University of Oxford will become a permanent fixture at Merton College.
References
Mathematics education in the United Kingdom
Mathematics Regius Professor
Professorships at the University of Oxford
Professorships in mathematics |
https://en.wikipedia.org/wiki/Vector%20multiplication | In mathematics, vector multiplication may refer to one of several operations between two (or more) vectors. It may concern any of the following articles:
Dot product – also known as the "scalar product", a binary operation that takes two vectors and returns a scalar quantity. The dot product of two vectors can be defined as the product of the magnitudes of the two vectors and the cosine of the angle between the two vectors. Alternatively, it is defined as the product of the projection of the first vector onto the second vector and the magnitude of the second vector. Thus,
Cross product – also known as the "vector product", a binary operation on two vectors that results in another vector. The cross product of two vectors in 3-space is defined as the vector perpendicular to the plane determined by the two vectors whose magnitude is the product of the magnitudes of the two vectors and the sine of the angle between the two vectors. So, if is the unit vector perpendicular to the plane determined by vectors and ,
Exterior product or wedge product – a binary operation on two vectors that results in a bivector. In Euclidean 3-space, the wedge product has the same magnitude as the cross product (the area of the parallelogram formed by sides and ) but generalizes to arbitrary affine spaces and products between more than two vectors.
Geometric product or Clifford product – for two vectors, the geometric product is a mixed quantity consisting of a scalar plus a bivector. The geometric product is well defined for any multivectors as arguments.
A bilinear product in an algebra over a field.
A Lie bracket for vectors in a Lie algebra.
Hadamard product – entrywise or elementwise product of tuples of scalar coordinates, where .
Outer product - where with results in a matrix.
Triple products – products involving three vectors.
Quadruple products – products involving four vectors.
Applications
Vector multiplication has multiple applications in regards to mathematics, but also in other studies such as physics and engineering.
Physics
The use of the cross product can help determine the moment of force, also known as torque. It can also be used to calculate the Lorentz force exerted on a charged particle moving in a magnetic field.
The dot product is used to determine the work done by a constant force.
See also
Scalar multiplication
Matrix multiplication
Vector addition
Vector algebra relations
Operations on vectors
Multiplication |
https://en.wikipedia.org/wiki/Jeong%20Jun-yeon | Jeong Jun-yeon (; born 30 April 1989) is a South Korean footballer who plays as a defender for FC Anyang.
Club career statistics
External links
1989 births
Living people
Footballers from South Jeolla Province
Men's association football defenders
South Korean men's footballers
South Korea men's under-20 international footballers
Jeonnam Dragons players
Gwangju FC players
Gimcheon Sangmu FC players
K League 1 players
K League 2 players |
https://en.wikipedia.org/wiki/Herchel%20Smith%20Professor%20of%20Pure%20Mathematics | The Herchel Smith Professorship of Pure Mathematics is a professorship in pure mathematics at the University of Cambridge. It was established in 2004 by a benefaction from Herchel Smith "of £14.315m, to be divided into five equal parts, to support the full endowment of five Professorships in the fields of Pure Mathematics, Physics, Biochemistry, Molecular Biology, and Molecular Genetics." When the position was advertised in 2004, the first holder was expected to focus on mathematical analysis.
List of Herchel Smith Professors of Pure Mathematics
2006–2013 Ben J. Green
2019–present Pierre Raphael
References
Report of the General Board on the establishment of a Herchel Smith Professorship of Pure Mathematics, Cambridge University Reporter, 3 March 2004
University left 'biggest' bequest, BBC News, 25 June 2002
Cambridge benefits from £50m Pill legacy, telegraph.co.uk, 26 June 2002
Pure Mathematics, Smith, Herchel
Faculty of Mathematics, University of Cambridge
2004 establishments in England
Pure Mathematics, Smith, Herchel
Mathematics education in the United Kingdom |
https://en.wikipedia.org/wiki/Lothar%20G%C3%B6ttsche | Lothar Göttsche (born January 21, 1961, in Sonderburg, Denmark) is a German mathematician, known for his work in algebraic geometry.
He is a research scientist at the International Centre for Theoretical Physics in Trieste, Italy. He is also editor for Geometry & Topology.
Biography
After studying mathematics at the University of Kiel, he received his Dr. rer. nat. under the direction of Friedrich Hirzebruch at the University of Bonn in 1989.
Göttsche was invited as speaker to the International Congress of Mathematicians in Beijing in 2002.
In 2012 he became a fellow of the American Mathematical Society.
Work
Göttsche received international acclaim with his formula for the generating function for the Betti numbers of the Hilbert scheme of points on an algebraic surface:
If is a smooth surface over an algebraically closed field of characteristic , then the generating function for the motives of the Hilbert schemes of can be expressed in terms of the motivic zeta function by Göttsche's formula
Here is the Hilbert scheme of length subschemes of .
Göttsche is also the author of a celebrated conjecture predicting the number of curves in certain linear systems on algebraic surfaces.
References
External links
Home page of Lothar Göttsche
20th-century German mathematicians
Algebraic geometers
People from Sønderborg Municipality
University of Kiel alumni
University of Bonn alumni
Fellows of the American Mathematical Society
1961 births
Living people
21st-century German mathematicians |
https://en.wikipedia.org/wiki/Xu-Jia%20Wang | Xu-Jia Wang (; born September 1963) is a Chinese-Australian mathematician. He is a professor of mathematics at the Australian National University and a fellow of the Australian Academy of Science.
Biography
Wang was born in Chun'an County, Zhejiang province, China. Wang obtained his B.S. in 1983 and his Ph.D. in 1990 from the Department of Mathematics of Zhejiang University (ZJU) in Hangzhou.
After completing his PhD, Wang served as lecturer and associate professor, at ZJU before departing for ANU In 1995. Wang is a Professor in the Centre for Mathematics and its Applications and Mathematical Sciences Institute of Australian National University.
Wang is well known for his work on differential equations, especially non-linear partial differential equations and their geometrical and transportational applications.
Honors and awards
Australian Mathematical Society Medal (2002)
invited speaker, 2002 International Congress of Mathematicians
Morningside Gold Medal of Mathematics, 2007
Fellow of the Australian Academy of Science (2009).
Australian Laureate Fellowship (2013)
Publications (selected)
(with Aram Karakhanyan)
(with Guji Tian)
(with Kai-Seng Chou)
(with Neil Trudinger)
(with Neil Trudinger and Xi-Nan Ma)
(with Xiaohua Zhu)
References
External links
Xu-Jia Wang's homepage at ANU
The Australian Mathematical Society Medal
浙大数学博士汪徐家当选为澳大利亚科学院院士 (in Chinese)
The Australian Academy of Science
1963 births
Living people
20th-century Chinese mathematicians
21st-century Chinese mathematicians
Academic staff of the Australian National University
Chinese emigrants to Australia
Fellows of the Australian Academy of Science
Zhejiang University alumni
Academic staff of Zhejiang University
Australian mathematicians
Scientists from Hangzhou
Mathematicians from Zhejiang
Educators from Hangzhou |
https://en.wikipedia.org/wiki/Early%20Algebra | Early Algebra is an approach to early mathematics teaching and learning. It is about teaching traditional topics in more profound ways. It is also an area of research in mathematics education.
Traditionally, algebra instruction has been postponed until adolescence. However, data of early algebra researchers shows ways to teach algebraic thinking much earlier. The National Council of Teachers of Mathematics (NCTM) integrates algebra into its Principles and Standards starting from Kindergarten.
One of the major goals of early algebra is generalizing number and set ideas. It moves from particular numbers to patterns in numbers. This includes generalizing arithmetic operations as functions, as well as engaging children in noticing and beginning to formalize properties of numbers and operations such as the commutative property, identities, and inverses.
Students historically have had a very difficult time adjusting to algebra for a number of reasons. Researchers have found that by working with students on such ideas as developing rules for the use of letters to stand in for numbers and the true meaning of the equals symbol (it is a balance point, and does not mean "put the answer next"), children are much better prepared for formal algebra instruction.
Teacher professional development in this area consists of presenting common student misconceptions and then developing lessons to move students out of faulty ways of thinking and into correct generalizations. The use of true, false, and open number sentences can go a long way toward getting students thinking about the properties of number and operations and the meaning of the equals sign.
Research areas in early algebra include use of representations, such as symbols, graphs and tables; cognitive development of students; viewing arithmetic as a part of algebraic conceptual fields
Notes
References
Blanton, M. L. Algebra and the Elementary Classroom: Transforming Thinking, Transforming Practice. (Heinemann, 2008).
J. Kaput, D. Carraher, & M. Blanton (Eds.), Algebra in the Early Grades. (Lawrence Erlbaum and Associates, 2007).
Schliemann, A.D., Carraher, D.W., & Brizuela, B. Bringing Out the Algebraic Character of Arithmetic: From Children's Ideas to Classroom Practice. (Lawrence Erlbaum Associates, 2007).
Carraher, D., Schliemann, A.D., Brizuela, B., & Earnest, D. (2006). Arithmetic and Algebra in early Mathematics Education. Journal for Research in Mathematics Education, Vol 37.
National Council of Teachers of Mathematics. Principles and Standards for School Mathematics. (Author, 2000)
External links
Tufts/TERC Early Algebra Project
Algebra education |
https://en.wikipedia.org/wiki/David%20Goss | David Mark Goss (April 20, 1952 – April 4, 2017) was a mathematician, a professor in the department of mathematics at Ohio State University, and the editor-in-chief of the Journal of Number Theory. He received his B.S. in mathematics in 1973 from University of Michigan and his Ph.D. in 1977 from Harvard University under the supervision of Barry Mazur; prior to Ohio State he held positions at Princeton University, Harvard, the University of California, Berkeley, and Brandeis University. He worked on function fields and introduced the Goss zeta function.
In 2012, he became a fellow of the American Mathematical Society.
Books
Selected papers
References
External links
Home page of David Goss
David Goss on MathSciNet
1952 births
2017 deaths
20th-century American mathematicians
21st-century American mathematicians
University of Michigan College of Literature, Science, and the Arts alumni
Harvard University alumni
Princeton University faculty
Harvard University Department of Mathematics faculty
Harvard University faculty
University of California, Berkeley faculty
Brandeis University faculty
Ohio State University faculty
Fellows of the American Mathematical Society |
https://en.wikipedia.org/wiki/Paul%20Townsend | Paul Kingsley Townsend FRS (; born 3 March 1951) is a British physicist, currently a Professor of Theoretical Physics in Cambridge University's Department of Applied Mathematics and Theoretical Physics. He is notable for his work on string theory.
Education
He received his PhD from Brandeis University in 1976 for his dissertation The 1/N expansion of scalar field theories supervised by Howard Joel Schnitzer. Since then he has over 320 publications.
Work
In 1987, , , and Paul Townsend showed that there are no superstrings in eleven dimensions (the largest number of dimensions consistent with a single graviton in supergravity theories), but supermembranes. In 1977 he was the first to formulate pure 4D N = 1 supergravity in anti-de Sitter space.
Awards and honours
He was elected a Fellow of the Royal Society in May 2000.
References
External links
(lecture by Paul Townsend)
(Workshop: Octonions and the Standard Model, Perimeter Institute, 2021)
(Workshop: Octonions and the Standard Model, Perimeter Institute, 2021)
1951 births
Living people
Brandeis University alumni
Cambridge mathematicians
British string theorists
Fellows of the Royal Society |
https://en.wikipedia.org/wiki/List%20of%20neighbourhoods%20in%20Kingston%2C%20Ontario | The City of Kingston has defined 45 distinct neighbourhoods based on census data from Statistics Canada. Different from the city's twelve electoral districts, the neighbourhoods as defined by the City all share common socio-demographic characteristics.. Detailed socio-demographic information on the city can be found in the Kingston Community Profile, 2009: A Socio-Demographic Analysis of Kingston, Ontario Canada. The profile is published by the Social Planning Council of Kingston and District (SPCKD). While some of these neighbourhoods have established their own business improvement area, others are simply a designation given by the City of Kingston in recognition of their distinct attributes and characteristics.
The divisions are arbitrary; they follow census boundaries but often chop existing, recognised neighbourhoods (including the central business district) in two.
Downtown and inner suburbs
These points (and a few others, including Polson Park, Rideau Heights and Alcan), if not part of the original city, were annexed to Kingston from Kingston Township or the former village of Portsmouth no later than the 1950s.
Kingston - West
Most (but not all) of these points were in the former Township of Kingston, annexed on January 1, 1998. The pre-amalgamation boundary is Cataraqui Creek.
Kingston - East
Most of these points were in the former Pittsburgh Township, annexed on January 1, 1998. The pre-amalgamation boundary is the Cataraqui River.
Kingston - North
Many or most of these points (except for those south of the 401) were in the former Township of Kingston, annexed on January 1, 1998.
References |
https://en.wikipedia.org/wiki/Weingarten%20function | In mathematics, Weingarten functions are rational functions indexed by partitions of integers that can be used to calculate integrals of products of matrix coefficients over classical groups. They were first studied by who found their asymptotic behavior, and named by , who evaluated them explicitly for the unitary group.
Unitary groups
Weingarten functions are used for evaluating integrals over the unitary group Ud
of products of matrix coefficients of the form
where denotes complex conjugation. Note that where is the conjugate transpose of , so one can interpret the above expression as being for the matrix element of .
This integral is equal to
where Wg is the Weingarten function, given by
where the sum is over all partitions λ of q . Here χλ is the character of Sq corresponding to the partition λ and s is the Schur polynomial of λ, so that sλd(1) is the dimension of the representation of Ud corresponding to λ.
The Weingarten functions are rational functions in d. They can have poles for small values of d, which cancel out in the formula above. There is an alternative inequivalent definition of Weingarten functions, where one only sums over partitions with at most d parts. This is no longer a rational function of d, but is finite for all positive integers d. The two sorts of Weingarten functions coincide for d larger than q, and either can be used in the formula for the integral.
Values of the Weingarten function for simple permutations
The first few Weingarten functions Wg(σ, d) are
(The trivial case where q = 0)
where permutations σ are denoted by their cycle shapes.
There exist computer algebra programs to produce these expressions.
Explicit expressions for the integrals in the first cases
The explicit expressions for the integrals of first- and second-degree polynomials, obtained via the formula above, are:
Asymptotic behavior
For large d, the Weingarten function Wg has the asymptotic behavior
where the permutation σ is a product of cycles of lengths Ci, and cn = (2n)!/n!(n + 1)! is a Catalan number, and |σ| is the smallest number of transpositions that σ is a product of. There exists a diagrammatic method to systematically calculate the integrals over the unitary group as a power series in 1/d.
Orthogonal and symplectic groups
For orthogonal and symplectic groups the Weingarten functions were evaluated by . Their theory is similar to the case of the unitary group. They are parameterized by partitions such that all parts have even size.
External links
References
Random matrices
Mathematical physics |
https://en.wikipedia.org/wiki/Hilbert%20metric | In mathematics, the Hilbert metric, also known as the Hilbert projective metric, is an explicitly defined distance function on a bounded convex subset of the n-dimensional Euclidean space Rn. It was introduced by as a generalization of Cayley's formula for the distance in the Cayley–Klein model of hyperbolic geometry, where the convex set is the n-dimensional open unit ball. Hilbert's metric has been applied to Perron–Frobenius theory and to constructing Gromov hyperbolic spaces.
Definition
Let Ω be a convex open domain in a Euclidean space that does not contain a line. Given two distinct points A and B of Ω, let X and Y be the points at which the straight line AB intersects the boundary of Ω, where the order of the points is X, A, B, Y. Then the Hilbert distance d(A, B) is the logarithm of the cross-ratio of this quadruple of points:
The function d is extended to all pairs of points by letting d(A, A) = 0 and defines a metric on Ω. If one of the points A and B lies on the boundary of Ω then d can be formally defined to be +∞, corresponding to a limiting case of the above formula
when one of the denominators is zero.
A variant of this construction arises for a closed convex cone K in a Banach space V (possibly, infinite-dimensional). In addition, the cone K is assumed to be pointed, i.e. K ∩ (−K) = {0} and thus K determines a partial order on V. Given any vectors v and w in K \ {0}, one first defines
The Hilbert pseudometric on K \ {0} is then defined by the formula
It is invariant under the rescaling of v and w by positive constants and so descends to a metric on the space of rays of K, which is interpreted as the projectivization of K (in order for d to be finite, one needs to restrict to the interior of K). Moreover, if K ⊂ R × V is the cone over a convex set Ω,
then the space of rays of K is canonically isomorphic to Ω. If v and w are vectors in rays in K corresponding to the points A, B ∈ Ω then these two formulas for d yield the same value of the distance.
Examples
In the case where the domain Ω is a unit ball in Rn, the formula for d coincides with the expression for the distance between points in the Cayley–Klein model of hyperbolic geometry, up to a multiplicative constant.
If the cone K is the positive orthant in Rn then the induced metric on the projectivization of K is often called simply Hilbert's projective metric. This cone corresponds to a domain Ω which is a regular simplex of dimension n − 1.
Motivation and applications
Hilbert introduced his metric in order to construct an axiomatic metric geometry in which there exist triangles ABC whose vertices A, B, C are not collinear, yet one of the sides is equal to the sum of the other two — it follows that the shortest path connecting two points is not unique in this geometry. In particular, this happens when the convex set Ω is a Euclidean triangle and the straight line extensions of the segments AB, BC, AC do not meet the interior of one of the sides |
https://en.wikipedia.org/wiki/Chinese%20people%20in%20the%20Netherlands | Chinese people in the Netherlands form one of the largest overseas Chinese populations in continental Europe. In 2018 official statistics showed 92,644 people originating from the People's Republic of China (PRC) (including Hong Kong) and Republic of China (ROC), or people with at least one such parent. However, these statistics do not capture the whole size of the Chinese community, which since its earliest days has included not just migrants from China, but people of Chinese ethnicity drawn from among overseas Chinese communities as well.
Migration history
Early Chinese labour migration to the Netherlands was drawn primarily from two sources: peddlers from Qingtian, Zhejiang who began arriving in the country after World War I, and seamen of Guangdong origin drawn from among the British Chinese community; the latter had initially been brought in as strikebreakers in 1911. During the Great Depression, many of the seamen were laid off and also took to street peddling, especially of (peanut cakes); the Dutch referred to them as "" ("peanut man"). Their numbers dropped as a result of voluntary outmigration and deportations; by World War II, fewer than 1,000 remained.
Another group of early ethnic Chinese in the Netherlands were students; they were largely not from China, however, but were instead drawn from among Chinese communities in the Dutch East Indies. From a group of 20 in 1911, their numbers continued to increase, interrupted only by World War II; in 1957, out of the roughly 1,400 ethnic Chinese from Indonesia in the Netherlands, 1,000 were students. In 1911, these students established the Chung Hwa Hui, which was in contact with various Chinese organizations and political parties in Europe. Largely of Peranakan origin, the students tended to speak Indonesian local languages as their mother tongues, and had already done their early education at Dutch-medium schools. However, with increasing tensions in Indonesia–Netherlands relations in the late 1950s and early 1960s, the number of students dropped off sharply.
Though the number of Chinese students from Indonesia dropped off, tens of thousands of ethnic Chinese were forced to leave the country due to the violent political situation in Indonesia in 1965. Most went to China, the United States, or Australia, but those who had been educated in Dutch preferentially chose the Netherlands as their destination; there are no exact statistics, but the migrants themselves estimate that about 5,000 arrived during this period. As with the students, these migrants tended to speak no Chinese, with Indonesian languages as their mother tongues and Dutch as their academic language. In the late 1970s and early 1980s, Hong Kong also became a significant source of Chinese migrants to the Netherlands, with about 600 to 800 per year, falling off to around 300 to 400 per year by the late 1980s.
Also in the 1980s, the Netherlands began to become a popular choice for students from mainland China. Factors which i |
https://en.wikipedia.org/wiki/Annals%20of%20Statistics | The Annals of Statistics is a peer-reviewed statistics journal published by the Institute of Mathematical Statistics. It was started in 1973 as a continuation in part of the Annals of Mathematical Statistics (1930), which was split into the Annals of Statistics and the Annals of Probability.
The journal CiteScore is 5.8, and its SCImago Journal Rank is 5.877, both from 2020.
Articles older than 3 years are available on JSTOR, and all articles since 2004 are freely available on the arXiv.
Editorial board
The following persons have been editors of the journal:
Ingram Olkin (1972–1973)
I. Richard Savage (1974–1976)
Rupert G. Miller (1977–1979)
David V. Hinkley (1980–1982)
Michael D. Perlman (1983–1985)
Willem van Zwet (1986–1988)
Arthur Cohen (1988–1991)
Michael Woodroofe (1992–1994)
Larry Brown and John Rice (1995–1997)
Hans-Rudolf Künsch and James O. Berger (1998–2000)
John Marden and Jon A. Wellner (2001–2003)
Morris Eaton and Jianqing Fan (2004–2006)
Susan Murphy and Bernard Silverman (2007–2009)
Peter Bühlmann and T. Tony Cai (2010–2012)
Peter Hall and Runze Li (2013–2015)
Ed George and Tailen Hsing (2016–2018)
Richard J. Samworth and Ming Yuan (2019–2021)
Enno Mammen and Lan Wang (2022–2024)
External links
Annals of Statistics homepage
Annals of Statistics at Project Euclid
References
Statistics journals
Institute of Mathematical Statistics academic journals |
https://en.wikipedia.org/wiki/Timor-Leste%20national%20football%20team%20results | This article details the match results and statistics of the Timor-Leste national football team.
Key
Key to matches
Att. = Match attendance
(H) = Home ground
(A) = Away ground
(N) = Neutral ground
Key to record by opponent
Pld = Games played
W = Games won
D = Games drawn
L = Games lost
GF = Goals for
GA = Goals against
Results
Timor-Leste's score is shown first in each case.
Record by opponent
Notes
See also
Timor-Leste national football team records and statistics
References
External links
Timor-Leste - Fixtures & Results at FIFA.com
Results
National association football team results |
https://en.wikipedia.org/wiki/Does%20God%20Play%20Dice%3F | Does God Play Dice: The New Mathematics of Chaos is a non-fiction book about chaos theory written by British mathematician Ian Stewart. The book was initially published by Blackwell Publishing in 1989.
Summary
In this book, Stewart explains chaos theory to an audience presumably unfamiliar with it. As the book progresses the writing changes from simple explanations of chaos theory to in-depth, rigorous mathematical study. Stewart covers mathematical concepts such as differential equations, resonance, nonlinear dynamics, and probability. The book is illustrated with diagrams and graphs of mathematical concepts and equations when applicable.
The back of the book, and a summary of its content, reads, "The science of chaos is forcing scientists to rethink Einstein's fundamental assumptions regarding the way the universe behaves. Chaos theory has already shown that simple systems, obeying precise laws, can nevertheless act in a random manner. Perhaps God plays dice within a cosmic game of complete law and order. Does God Play Dice? reveals a strange universe in which nothing may be as it seems. Familiar geometric shapes such as circles and ellipses give way to infinitely complex structures known as fractals, the fluttering of a butterfly's wings can change the weather, and the gravitational attraction of a creature in a distant galaxy can change the fate of the solar system."
The title of the book is a reference to a famous quote by Albert Einstein.
Book
Ian Stewart: Does God Play Dice: The New Mathematics of Chaos, Blackwell Publishing, 1989,
References
External links
Text available at books.google.com
Books by Ian Stewart (mathematician)
Science books
1989 non-fiction books
Mathematics books |
https://en.wikipedia.org/wiki/Volume%20conjecture | In the branch of mathematics called knot theory, the volume conjecture is the following open problem that relates quantum invariants of knots to the hyperbolic geometry of knot complements.
Let O denote the unknot. For any hyperbolic knot K let be Kashaev's invariant of ; this invariant coincides with the following evaluation of the -Colored Jones Polynomial of :
Then the volume conjecture states that
where vol(K) denotes the hyperbolic volume of the complement of K in the 3-sphere.
Kashaev's Observation
observed that the asymptotic behavior of a certain state sum of knots gives the hyperbolic volume of the complement of knots and showed that it is true for the knots , , and . He conjectured that for general hyperbolic knots the formula (2) would hold. His invariant for a knot is based on the theory of quantum dilogarithms at the -th root of unity, .
Colored Jones Invariant
had firstly pointed out that Kashaev's invariant is related to the colored Jones polynomial by replacing q with the 2N-root of unity, namely, . They used an R-matrix as the discrete Fourier transform for the equivalence of these two values.
The volume conjecture is important for knot theory. In section 5 of this paper they state that:
Assuming the volume conjecture, every knot that is different from the trivial knot has at least one different Vassiliev (finite type) invariant.
Relation to Chern-Simons theory
Using complexification, rewrote the formula (1) into
where is called the Chern–Simons invariant. They showed that there is a clear relation between the complexified colored Jones polynomial and Chern–Simons theory from a mathematical point of view.
References
.
.
.
.
Knot theory
Conjectures
Unsolved problems in geometry |
https://en.wikipedia.org/wiki/Fork%E2%80%93join%20queue | In queueing theory, a discipline within the mathematical theory of probability, a fork–join queue is a queue where incoming jobs are split on arrival for service by numerous servers and joined before departure. The model is often used for parallel computations or systems where products need to be obtained simultaneously from different suppliers (in a warehouse or manufacturing setting). The key quantity of interest in this model is usually the time taken to service a complete job. The model has been described as a "key model for the performance analysis of parallel and distributed systems." Few analytical results exist for fork–join queues, but various approximations are known.
The situation where jobs arrive according to a Poisson process and service times are exponentially distributed is sometimes referred to as a Flatto–Hahn–Wright model or FHW model.
Definition
On arrival at the fork point, a job is split into N sub-jobs which are served by each of the N servers. After service, sub-job wait until all other sub-jobs have also been processed. The sub-jobs are then rejoined and leave the system.
For the fork–join queue to be stable the input rate must be strictly less than sum of the service rates at the service nodes.
Applications
Fork–join queues have been used to model zoned RAID systems, parallel computations and for modelling order fulfilment in warehouses.
Response time
The response time (or sojourn time) is the total amount of time a job spends in the system.
Distribution
Ko and Serfozo give an approximation for the response time distribution when service times are exponentially distributed and jobs arrive either according to a Poisson process or a general distribution. QIu, Pérez and Harrison give an approximation method when service times have a phase-type distribution.
Average response time
An exact formula for the average response time is only known in the case of two servers (N=2) with exponentially distributed service times (where each server is an M/M/1 queue). In this situation, the response time (total time a job spends in the system) is
where
is the utilization.
is the arrival rate of jobs to all the nodes.
is the service rate across all the nodes.
In the situation where nodes are M/M/1 queues and N > 2, Varki's modification of mean value analysis can also be used to give an approximate value for the average response time.
For general service times (where each node is an M/G/1 queue) Baccelli and Makowski give bounds for the average response time and higher moments of this quantity both in the transient and steady state situations. Kemper and Mandjes show that for some parameters these bounds are not tight and show demonstrate an approximation technique. For heterogeneous fork-join queues (fork-join queues with different service times), Alomari and Menasce propose an approximation based on harmonic numbers that can be extended to cover more general cases such as probabilistic fork, open and closed fork-join queues. |
https://en.wikipedia.org/wiki/Vital%20statistics | Vital statistics may refer to:
Vital statistics (government records), a government database recording the births and deaths of individuals within that government's jurisdiction.
Bust/waist/hip measurements, informally called vital statistics, measurements for the purpose of fitting clothes
Vital signs, measures of various physiological statistics, often taken by health professionals, in order to assess the most basic body functions
Vital Statistics (opera), a 1987 one-act opera about physiognomy, re-titled Facing Goya
See also
Vitalstatistix (disambiguation) |
https://en.wikipedia.org/wiki/Birman%E2%80%93Wenzl%20algebra | In mathematics, the Birman–Murakami–Wenzl (BMW) algebra, introduced by and , is a two-parameter family of algebras of dimension having the Hecke algebra of the symmetric group as a quotient. It is related to the Kauffman polynomial of a link. It is a deformation of the Brauer algebra in much the same way that Hecke algebras are deformations of the group algebra of the symmetric group.
Definition
For each natural number n, the BMW algebra is generated by and relations:
These relations imply the further relations:
This is the original definition given by Birman and Wenzl. However a slight change by the introduction of some minus signs is sometimes made, in accordance with Kauffman's 'Dubrovnik' version of his link invariant. In that way, the fourth relation in Birman & Wenzl's original version is changed to
(Kauffman skein relation)
Given invertibility of m, the rest of the relations in Birman & Wenzl's original version can be reduced to
(Idempotent relation)
(Braid relations)
(Tangle relations)
(Delooping relations)
Properties
The dimension of is .
The Iwahori–Hecke algebra associated with the symmetric group is a quotient of the Birman–Murakami–Wenzl algebra .
The Artin braid group embeds in the BMW algebra, .
Isomorphism between the BMW algebras and Kauffman's tangle algebras
It is proved by that the BMW algebra is isomorphic to the Kauffman's tangle algebra , the isomorphism is defined by
and
Baxterisation of Birman–Murakami–Wenzl algebra
Define the face operator as
,
where and are determined by
and
.
Then the face operator satisfies the Yang–Baxter equation.
Now with
.
In the limits , the braids can be recovered up to a scale factor.
History
In 1984, Vaughan Jones introduced a new polynomial invariant of link isotopy types which is called the Jones polynomial. The invariants are related to the traces of irreducible representations of Hecke algebras associated with the symmetric groups. showed that the Kauffman polynomial can also be interpreted as a function on a certain associative algebra. In 1989, constructed a two-parameter family of algebras with the Kauffman polynomial as trace after appropriate renormalization.
References
Representation theory
Knot theory
Diagram algebras |
https://en.wikipedia.org/wiki/Ferdinand%20Minding | Ernst Ferdinand Adolf Minding (; – ) was a German-Russian mathematician known for his contributions to differential geometry. He continued the work of Carl Friedrich Gauss concerning differential geometry of surfaces, especially its intrinsic aspects. Minding considered questions of bending of surfaces and proved the invariance of geodesic curvature. He studied ruled surfaces, developable surfaces and surfaces of revolution and determined geodesics on the pseudosphere. Minding's results on the geometry of geodesic triangles on a surface of constant curvature (1840) anticipated Beltrami's approach to the foundations of non-Euclidean geometry (1868).
Career
Minding was largely self-taught in mathematics. He attended lectures in the University of Halle and eventually graduated with a thesis "De valore intergralium duplicium quam proxime inveniendo" (1829).
Minding worked as a teacher in Elberfeld and as a university lecturer in Berlin. His work on statics drew the attention of Alexander von Humboldt. However, his 1842 bid for election to Berlin Academy, supported by Peter Gustav Lejeune Dirichlet, failed and in 1843 he relocated to the University of Dorpat, where he was a professor of mathematics for the next 40 years. In Dorpat he taught Karl Peterson and supervised his doctoral thesis that established the Gauss–Bonnet theorem and derived Gauss–Codazzi equations. Minding also worked on differential equations (Demidov prize of the St Petersburg Academy in 1861), algebraic functions, continued fractions and analytical mechanics. His list of publications consists of some 60 titles, including several books. Many of his scientific accomplishments were only recognized properly after his death.
References
External links
Differential geometers
Hyperbolic geometers
19th-century German mathematicians
Russian mathematicians
Corresponding members of the Saint Petersburg Academy of Sciences
Honorary members of the Saint Petersburg Academy of Sciences
1806 births
1885 deaths |
https://en.wikipedia.org/wiki/J%C3%A1nos%20Pintz | János Pintz (born 20 December 1950 in Budapest) is a Hungarian mathematician working in analytic number theory. He is a fellow of the Rényi Mathematical Institute and is also a member of the Hungarian Academy of Sciences. In 2014, he received the Cole Prize.
Mathematical results
Pintz is best known for proving in 2005 (with Daniel Goldston and Cem Yıldırım) that
where denotes the nth prime number. In other words, for every ε > 0, there exist infinitely many pairs of consecutive primes pn and pn+1 that are closer to each other than the average distance between consecutive primes by a factor of ε, i.e., pn+1 − pn < ε log pn. This result was originally reported in 2003 by Daniel Goldston and Cem Yıldırım but was later retracted. Pintz joined the team and completed the proof in 2005 and developed the so called GPY sieve. Later, they improved this to showing that pn+1 − pn < ε(log log n)2 occurs infinitely often. Further, if one assumes the Elliott–Halberstam conjecture, then one can also show that primes within 16 of each other occur infinitely often, which is nearly the twin prime conjecture.
Additionally,
With János Komlós and Endre Szemerédi, he disproved the Heilbronn conjecture.
With Iwaniec, he proved that for sufficiently large n there is a prime between n and n + n23/42.
Pintz gave an effective upper bound for the first number for which the Mertens conjecture fails.
He gave an O(x2/3) upper bound for the number of those numbers that are less than x and not the sum of two primes.
With Imre Z. Ruzsa, he improved a result of Linnik by showing that every sufficiently large even number is the sum of two primes and at most 8 powers of 2.
Goldston, S. W. Graham, Pintz, and Yıldırım proved that the difference between numbers which are products of exactly 2 primes is infinitely often at most 6.
See also
Prime gap
Landau's problems
Fazekas Mihály Gimnázium
Maier's theorem
References
External links
János Pintz's page at the Alfréd Rényi Institute of Mathematics
Number theorists
Institute for Advanced Study visiting scholars
Members of the Hungarian Academy of Sciences
Eötvös Loránd University alumni
Mathematicians from Budapest
20th-century Hungarian mathematicians
21st-century Hungarian mathematicians
Living people
1950 births |
https://en.wikipedia.org/wiki/Maria%20Deloria%20Knoll | Maria Deloria Knoll is an expert in the fields of epidemiology, disease surveillance, vaccine trial conduct, and bio-statistics. She currently serves as associate director of Science at the International Vaccine Access Center (IVAC), an organization dedicated to accelerating global access to life-saving vaccines, at the Johns Hopkins Bloomberg School of Public Health in Baltimore, Maryland.
Education
Maria Deloria Knoll completed her undergraduate studies in biostatistics at the University of North Carolina at Chapel Hill. She received her PhD in epidemiology from the Johns Hopkins Bloomberg School of Public Health in Baltimore, Maryland.
Research
Knoll's research interests include the epidemiology of vaccine-preventable infectious diseases and the methodologic issues pertaining to the design, conduct and analysis of epidemiologic studies and clinical trials.
Current projects include the PERCH study to determine the causes of pneumonia in settings representative of developing countries to guide the development of new pneumonia vaccines and treatment algorithms for 2015 onwards, estimating the burden and serotype distribution of pneumococcal and meningococcal disease among children and adults, measuring the impact on disease burden of pneumococcal conjugate vaccine use in Kenya and The Gambia, and evaluating the value of antigen-based diagnostic tests in blood to improve sensitivity of pneumococcal detection in pneumonia patients.
Knoll has over 20 years of experience in the design, conduct, and analysis of clinical trials and epidemiologic studies. She spent 13 years at the National Institute of Health's National Institute of Allergy and Infectious Diseases and her clinical trials experience there included close working relationships with many large and mid-sized vaccine manufacturers. Knoll also served as a Research Assistant Professor at Northwestern University's Department of Preventive Medicine in Chicago, Illinois, where she directed a course on clinical trial design, conduct and analysis in their MPH program, conducted cardiovascular and HIV-related epidemiologic studies using data from established cohorts, and designed therapeutic clinical trials for the Rehabilitation Institute of Chicago. She has written many well cited original articles and is regularly an invited speaker to various international health conferences.
The Pneumococcal Conjugate Vaccine Introduction Study (PCVIS) in The Gambia and Kenya
The GAVI Alliance and the Bill & Melinda Gates Foundation are funding two pneumococcal conjugate vaccine (PCV) impact studies in the Gambia and Kenya. In 2006, these projects were identified as critical to GAVI's objectives of generating sustainable PCV introduction. These studies, funded through the Pneumococcal Vaccines Accelerated Development and Introduction Plan (PneumoADIP), a project of IVAC at The Johns Hopkins Bloomberg School of Public Health (JHSPH), provide the earliest possible opportunity to document the health impact of |
https://en.wikipedia.org/wiki/Random%20tree | In mathematics and computer science, a random tree is a tree or arborescence that is formed by a stochastic process. Types of random trees include:
Uniform spanning tree, a spanning tree of a given graph in which each different tree is equally likely to be selected
Random minimal spanning tree, spanning trees of a graph formed by choosing random edge weights and using the minimum spanning tree for those weights
Random binary tree, binary trees with a given number of nodes, formed by inserting the nodes in a random order or by selecting all possible trees uniformly at random
Random recursive tree, increasingly labelled trees, which can be generated using a simple stochastic growth rule.
Treap or randomized binary search tree, a data structure that uses random choices to simulate a random binary tree for non-random update sequences
Rapidly exploring random tree, a fractal space-filling pattern used as a data structure for searching high-dimensional spaces
Brownian tree, a fractal tree structure created by diffusion-limited aggregation processes
Random forest, a machine-learning classifier based on choosing random subsets of variables for each tree and using the most frequent tree output as the overall classification
Branching process, a model of a population in which each individual has a random number of children
See also
Lightning tree
External links
Trees (graph theory)
Probabilistic data structures
Random graphs |
https://en.wikipedia.org/wiki/Indefinite%20sum | In discrete calculus the indefinite sum operator (also known as the antidifference operator), denoted by or , is the linear operator, inverse of the forward difference operator . It relates to the forward difference operator as the indefinite integral relates to the derivative. Thus
More explicitly, if , then
If F(x) is a solution of this functional equation for a given f(x), then so is F(x)+C(x) for any periodic function C(x) with period 1. Therefore, each indefinite sum actually represents a family of functions. However, due to the Carlson's theorem, the solution equal to its Newton series expansion is unique up to an additive constant C. This unique solution can be represented by formal power series form of the antidifference operator: .
Fundamental theorem of discrete calculus
Indefinite sums can be used to calculate definite sums with the formula:
Definitions
Laplace summation formula
where are the Cauchy numbers of the first kind, also known as the Bernoulli Numbers of the Second Kind.
Newton's formula
where is the falling factorial.
Faulhaber's formula
provided that the right-hand side of the equation converges.
Mueller's formula
If then
Euler–Maclaurin formula
Choice of the constant term
Often the constant C in indefinite sum is fixed from the following condition.
Let
Then the constant C is fixed from the condition
or
Alternatively, Ramanujan's sum can be used:
or at 1
respectively
Summation by parts
Indefinite summation by parts:
Definite summation by parts:
Period rules
If is a period of function then
If is an antiperiod of function , that is then
Alternative usage
Some authors use the phrase "indefinite sum" to describe a sum in which the numerical value of the upper limit is not given:
In this case a closed form expression F(k) for the sum is a solution of
which is called the telescoping equation. It is the inverse of the backward difference operator.
It is related to the forward antidifference operator using the fundamental theorem of discrete calculus described earlier.
List of indefinite sums
This is a list of indefinite sums of various functions. Not every function has an indefinite sum that can be expressed in terms of elementary functions.
Antidifferences of rational functions
where , the generalized to real order Bernoulli polynomials.
where is the polygamma function.
where is the digamma function.
Antidifferences of exponential functions
Particularly,
Antidifferences of logarithmic functions
Antidifferences of hyperbolic functions
where is the q-digamma function.
Antidifferences of trigonometric functions
where is the q-digamma function.
where is the normalized sinc function.
Antidifferences of inverse hyperbolic functions
Antidifferences of inverse trigonometric functions
Antidifferences of special functions
where is the incomplete gamma function.
where is the falling factorial.
(see super-exponential function)
See also
Indefinite product
Time scal |
https://en.wikipedia.org/wiki/Schur%27s%20lemma%20%28Riemannian%20geometry%29 | In Riemannian geometry, Schur's lemma is a result that says, heuristically, whenever certain curvatures are pointwise constant then they are forced to be globally constant. The proof is essentially a one-step calculation, which has only one input: the second Bianchi identity.
The Schur lemma for the Ricci tensor
Suppose is a smooth Riemannian manifold with dimension Recall that this defines for each element of :
the sectional curvature, which assigns to every 2-dimensional linear subspace of a real number
the Riemann curvature tensor, which is a multilinear map
the Ricci curvature, which is a symmetric bilinear map
the scalar curvature, which is a real number
The Schur lemma states the following:
The Schur lemma is a simple consequence of the "twice-contracted second Bianchi identity," which states that
understood as an equality of smooth 1-forms on Substituting in the given condition one finds that
Alternative formulations of the assumptions
Let be a symmetric bilinear form on an -dimensional inner product space Then
Additionally, note that if for some number then one automatically has { With these observations in mind, one can restate the Schur lemma in the following form:
Note that the dimensional restriction is important, since every two-dimensional Riemannian manifold which does not have constant curvature would be a counterexample.
The Schur lemma for the Riemann tensor
The following is an immediate corollary of the Schur lemma for the Ricci tensor.
The Schur lemma for Codazzi tensors
Let be a smooth Riemannian or pseudo-Riemannian manifold of dimension Let he a smooth symmetric (0,2)-tensor field whose covariant derivative, with respect to the Levi-Civita connection, is completely symmetric. The symmetry condition is an analogue of the Bianchi identity; continuing the analogy, one takes a trace to find that
If there is a function on such that for all in then upon substitution one finds
Hence implies that is constant on each connected component of As above, one can then state the Schur lemma in this context:
Applications
The Schur lemmas are frequently employed to prove roundness of geometric objects. A noteworthy example is to characterize the limits of convergent geometric flows.
For example, a key part of Richard Hamilton's 1982 breakthrough on the Ricci flow was his "pinching estimate" which, informally stated, says that for a Riemannian metric which appears in a 3-manifold Ricci flow with positive Ricci curvature, the eigenvalues of the Ricci tensor are close to one another relative to the size of their sum. If one normalizes the sum, then, the eigenvalues are close to one another in an absolute sense. In this sense, each of the metrics appearing in a 3-manifold Ricci flow of positive Ricci curvature "approximately" satisfies the conditions of the Schur lemma. The Schur lemma itself is not explicitly applied, but its proof is effectively carried out through Hamilton's calculations.
In the |
https://en.wikipedia.org/wiki/Indefinite%20product | In mathematics, the indefinite product operator is the inverse operator of . It is a discrete version of the geometric integral of geometric calculus, one of the non-Newtonian calculi. Some authors use term discrete multiplicative integration.
Thus
More explicitly, if , then
If F(x) is a solution of this functional equation for a given f(x), then so is CF(x) for any constant C. Therefore, each indefinite product actually represents a family of functions, differing by a multiplicative constant.
Period rule
If is a period of function then
Connection to indefinite sum
Indefinite product can be expressed in terms of indefinite sum:
Alternative usage
Some authors use the phrase "indefinite product" in a slightly different but related way to describe a product in which the numerical value of the upper limit is not given. e.g.
.
Rules
List of indefinite products
This is a list of indefinite products . Not all functions have an indefinite product which can be expressed in elementary functions.
(see K-function)
(see Barnes G-function)
(see super-exponential function)
See also
Indefinite sum
Product integral
List of derivatives and integrals in alternative calculi
Fractal derivative
References
Further reading
http://reference.wolfram.com/mathematica/ref/Product.html -Indefinite products with Mathematica
- bug in Maple V to Maple 8 handling of indefinite product
Markus Müller. How to Add a Non-Integer Number of Terms, and How to Produce Unusual Infinite Summations
Markus Mueller, Dierk Schleicher. Fractional Sums and Euler-like Identities
External links
Non-Newtonian calculus website
Mathematical analysis
Indefinite sums
Indefinite sums
Non-Newtonian calculus |
https://en.wikipedia.org/wiki/Ernst%20Specker | Ernst Paul Specker (11 February 1920, Zürich – 10 December 2011, Zürich) was a Swiss mathematician. Much of his most influential work was on Quine's New Foundations, a set theory with a universal set, but he is most famous for the Kochen–Specker theorem in quantum mechanics, showing that certain types of hidden variable theories are impossible. He also proved the ordinal partition relation ω2 → (ω2,3)2, thereby solving a problem of Erdős.
Specker received his Ph.D. in 1949 from ETH Zurich, where he remained throughout his professional career.
See also
Specker sequence
Baer–Specker group
References
External links
Biography at the University of St. Andrews
Ernst Specker (1920-2011), Martin Fürer, January 25, 2012.
Ernst Specker:Selecta, Birkhauser, 1990.
1920 births
2011 deaths
ETH Zurich alumni
Academic staff of ETH Zurich
Scientists from Zürich
Set theorists
Swiss mathematicians
20th-century Swiss mathematicians |
https://en.wikipedia.org/wiki/Nikolaus%20Hofreiter | Nikolaus Hofreiter (8 May 1904 – 23 January 1990) was an Austrian mathematician who worked mainly in number theory.
Biography
Hofreiter went to school in Linz and studied from 1923 in Vienna with Hans Hahn, Wilhelm Wirtinger, Emil Müller at the Technische Universität Wien on descriptive geometry, and Philipp Furtwängler, with whom he obtained his doctorate in 1927 on the reduction theory of quadratic forms (Eine neue Reduktionstheorie für definite quadratische Formen). In 1928 he passed the Lehramtsprüfung examination and completed the probationary year as a teacher in Vienna, but then returned to the university (first as a scientific assistant at the TU Vienna) where in 1929 he was assistant to Furtwängler and then habilitated in 1933. He was even then an excellent teacher, and gave lectures not only in Vienna but also in Graz.
His dissertation and habilitation thesis dealt with the reduction theory of quadratic forms, which Gauss, Charles Hermite and Hermann Minkowski had worked on previously. Hofreiter treated the case of four variables of a problem of Minkowski (Minkowski had solved the problem for two variables, while Robert Remak had solved it for three variables) on the product of inhomogeneous linear forms and achieved significant progress. The complete solution was only found 15 years later (and the general case is still unresolved). In 1934, he proved the existence of infinitely many real quadratic number fields without a Euclidean algorithm. In addition, he dealt with the geometry of numbers and Diophantine approximation.
In 1939, he was an associate professor and married the mathematician Margarete Dostalík (1912-2013). She was also a student of Furtwängler and did important work on algebraic equations and was working as a meteorologist in Berlin at the time. During the Second World War, he moved from Vienna and was a little later at the Hermann Goering Aviation Research Institute in Braunschweig, where his colleagues Wolfgang Gröbner from Vienna, Bernhard Baule from Graz, Ernst Peschl and Josef Laub were already working. Through his work there, together with Gröbner, he started a table of integrals. The first volume, on indefinite integrals, was published by Notdruck (Braunschweig) in 1944 and by Springer in 1949. In 1950, the second volume containing definite integrals appeared. Both parts were widely available through to the 5th 1973/75 edition. His wife, Margaret, assisted with the calculations, as well as the preparation and review of both volumes.
In addition to their work at the Aviation Research Institute, Gröbner and Hofreiter continued to give lectures and seminars at the Technical University of Braunschweig. After the war he returned to Vienna in 1946 and continued to expand on his number theory work. He also worked on linear optimization and numerical mathematics. In 1954, he became a professor, and in 1963/4 became Dean of the Faculty, and in 1965/6, he was Rector of the University of Vienna. In 1974, he retired.
In |
https://en.wikipedia.org/wiki/Partially%20ordered%20ring | In abstract algebra, a partially ordered ring is a ring (A, +, ·), together with a compatible partial order, that is, a partial order on the underlying set A that is compatible with the ring operations in the sense that it satisfies:
and
for all . Various extensions of this definition exist that constrain the ring, the partial order, or both. For example, an Archimedean partially ordered ring is a partially ordered ring where partially ordered additive group is Archimedean.
An ordered ring, also called a totally ordered ring, is a partially ordered ring where is additionally a total order.
An l-ring, or lattice-ordered ring, is a partially ordered ring where is additionally a lattice order.
Properties
The additive group of a partially ordered ring is always a partially ordered group.
The set of non-negative elements of a partially ordered ring (the set of elements for which also called the positive cone of the ring) is closed under addition and multiplication, that is, if is the set of non-negative elements of a partially ordered ring, then and Furthermore,
The mapping of the compatible partial order on a ring to the set of its non-negative elements is one-to-one; that is, the compatible partial order uniquely determines the set of non-negative elements, and a set of elements uniquely determines the compatible partial order if one exists.
If is a subset of a ring and:
then the relation where if and only if defines a compatible partial order on (that is, is a partially ordered ring).
In any l-ring, the of an element can be defined to be where denotes the maximal element. For any and
holds.
f-rings
An f-ring, or Pierce–Birkhoff ring, is a lattice-ordered ring in which and imply that for all They were first introduced by Garrett Birkhoff and Richard S. Pierce in 1956, in a paper titled "Lattice-ordered rings", in an attempt to restrict the class of l-rings so as to eliminate a number of pathological examples. For example, Birkhoff and Pierce demonstrated an l-ring with 1 in which 1 is not positive, even though it is a square. The additional hypothesis required of f-rings eliminates this possibility.
Example
Let be a Hausdorff space, and be the space of all continuous, real-valued functions on is an Archimedean f-ring with 1 under the following pointwise operations:
From an algebraic point of view the rings
are fairly rigid. For example, localisations, residue rings or limits of rings of the form are not of this form in general. A much more flexible class of f-rings containing all rings of continuous functions and resembling many of the properties of these rings is the class of real closed rings.
Properties
A direct product of f-rings is an f-ring, an l-subring of an f-ring is an f-ring, and an l-homomorphic image of an f-ring is an f-ring.
in an f-ring.
The category Arf consists of the Archimedean f-rings with 1 and the l-homomorphisms that preserve the identity.
Every ordered |
https://en.wikipedia.org/wiki/Ensemble%20learning | In statistics and machine learning, ensemble methods use multiple learning algorithms to obtain better predictive performance than could be obtained from any of the constituent learning algorithms alone.
Unlike a statistical ensemble in statistical mechanics, which is usually infinite, a machine learning ensemble consists of only a concrete finite set of alternative models, but typically allows for much more flexible structure to exist among those alternatives.
Overview
Supervised learning algorithms perform the task of searching through a hypothesis space to find a suitable hypothesis that will make good predictions with a particular problem. Even if the hypothesis space contains hypotheses that are very well-suited for a particular problem, it may be very difficult to find a good one. Ensembles combine multiple hypotheses to form a (hopefully) better hypothesis. The term ensemble is usually reserved for methods that generate multiple hypotheses using the same base learner.
The broader term of multiple classifier systems also covers hybridization of hypotheses that are not induced by the same base learner.
Evaluating the prediction of an ensemble typically requires more computation than evaluating the prediction of a single model. In one sense, ensemble learning may be thought of as a way to compensate for poor learning algorithms by performing a lot of extra computation. On the other hand, the alternative is to do a lot more learning on one non-ensemble system. An ensemble system may be more efficient at improving overall accuracy for the same increase in compute, storage, or communication resources by using that increase on two or more methods, than would have been improved by increasing resource use for a single method. Fast algorithms such as decision trees are commonly used in ensemble methods (for example, random forests), although slower algorithms can benefit from ensemble techniques as well.
By analogy, ensemble techniques have been used also in unsupervised learning scenarios, for example in consensus clustering or in anomaly detection.
Ensemble theory
Empirically, ensembles tend to yield better results when there is a significant diversity among the models. Many ensemble methods, therefore, seek to promote diversity among the models they combine. Although perhaps non-intuitive, more random algorithms (like random decision trees) can be used to produce a stronger ensemble than very deliberate algorithms (like entropy-reducing decision trees). Using a variety of strong learning algorithms, however, has been shown to be more effective than using techniques that attempt to dumb-down the models in order to promote diversity. It is possible to increase diversity in the training stage of the model using correlation for regression tasks or using information measures such as cross entropy for classification tasks.
Theoretically, one can justify the diversity concept because the lower bound of the error rate of an ensemble system can |
https://en.wikipedia.org/wiki/Segre%20class | In mathematics, the Segre class is a characteristic class used in the study of cones, a generalization of vector bundles. For vector bundles the total Segre class is inverse to the total Chern class, and thus provides equivalent information; the advantage of the Segre class is that it generalizes to more general cones, while the Chern class does not.
The Segre class was introduced in the non-singular case by Segre (1953)..
In the modern treatment of intersection theory in algebraic geometry, as developed e.g. in the definitive book of Fulton (1998), Segre classes play a fundamental role.
Definition
Suppose is a cone over , is the projection from the projective completion of to , and is the anti-tautological line bundle on . Viewing the Chern class as a group endomorphism of the Chow group of , the total Segre class of is given by:
The th Segre class is simply the th graded piece of . If is of pure dimension over then this is given by:
The reason for using rather than is that this makes the total Segre class stable under addition of the trivial bundle .
If Z is a closed subscheme of an algebraic scheme X, then denote the Segre class of the normal cone to .
Relation to Chern classes for vector bundles
For a holomorphic vector bundle over a complex manifold a total Segre class is the inverse to the total Chern class , see e.g. Fulton (1998).
Explicitly, for a total Chern class
one gets the total Segre class
where
Let be Chern roots, i.e. formal eigenvalues of where is a curvature of a connection on .
While the Chern class c(E) is written as
where is an elementary symmetric polynomial of degree in variables
the Segre for the dual bundle which has Chern roots is written as
Expanding the above expression in powers of one can see that is represented by
a complete homogeneous symmetric polynomial of
Properties
Here are some basic properties.
For any cone C (e.g., a vector bundle), .
For a cone C and a vector bundle E,
If E is a vector bundle, then
for .
is the identity operator.
for another vector bundle F.
If L is a line bundle, then , minus the first Chern class of L.
If E is a vector bundle of rank , then, for a line bundle L,
A key property of a Segre class is birational invariance: this is contained in the following. Let be a proper morphism between algebraic schemes such that is irreducible and each irreducible component of maps onto . Then, for each closed subscheme , and the restriction of ,
Similarly, if is a flat morphism of constant relative dimension between pure-dimensional algebraic schemes, then, for each closed subscheme , and the restriction of ,
A basic example of birational invariance is provided by a blow-up. Let be a blow-up along some closed subscheme Z. Since the exceptional divisor is an effective Cartier divisor and the normal cone (or normal bundle) to it is ,
where we used the notation . Thus,
where is given by .
Examples
Example 1
Let Z be a smooth cu |
https://en.wikipedia.org/wiki/Kefeng%20Liu | Kefeng Liu (; born 12 December 1965), is a Chinese-American mathematician who is known for his contributions to geometric analysis, particularly the geometry, topology and analysis of moduli spaces of Riemann surfaces and Calabi–Yau manifolds. He is a professor of mathematics at University of California, Los Angeles, as well as the executive director of the Center of Mathematical Sciences at Zhejiang University. He is best known for his collaboration with Bong Lian and Shing-Tung Yau in which they establish some enumerative geometry conjectures motivated by mirror symmetry.
Biography
Liu was born in Kaifeng, Henan province, China. In 1985, Liu received his B.A. in mathematics from the Department of Mathematics of Peking University in Beijing. In 1988, Liu obtained his M.A. from the Institute of Mathematics of the Chinese Academy of Sciences (CAS) in Beijing. Liu then went to study in the United States, obtaining a Ph.D. from Harvard University in 1993 under Shing-Tung Yau.
From 1993 to 1996, Liu was C. L. E. Moore Instructor at the Massachusetts Institute of Technology. From 1996 to 2000, Liu was an assistant professor at Stanford University. Liu joined the University of California, Los Angeles faculty in 2000, where he was promoted to full professor in 2002. In September 2003, Liu was appointed as the head of Zhejiang University's mathematics department. Liu is currently the executive director of the Center of Mathematical Sciences at Zhejiang University.
Awards and honors
Frederick E. Terman Fellow (1997-2001)
Sloan Fellowship (1998-2001)
Guggenheim Fellow (2002)
Silver Morningside Medal (1998)
Morningside Gold Medal in Mathematics (2004)
invited speaker, 2002 International Congress of Mathematicians
Editorial Work
Communications in Analysis and Geometry, Editor-in-Chief
Pure and Applied Mathematical Quarterly, Co-Editor-in-Chief.
Asian Journal of Mathematics, Editor.
Pacific Journal of Mathematics, Editor.
Notices of the International Congress of Chinese Mathematicians, Co-Editor-in-Chief.
Advanced Lectures in Mathematics, Executive Editor.
Science China Mathematics, Editor.
Mathematics and Humanities, Co-Editor-in-Chief.
References
External links
Homepage of Kefeng Liu at UCLA
Homepage of Kefeng Liu at ZJU
"A brief summary of my researches" by Kefeng Liu
The Mathematics Genealogy Project – Kefeng Liu
Knowledge, Technique, and Imagination – Kefeng Liu Hangzhou, April 2004
1965 births
Living people
20th-century American mathematicians
21st-century American mathematicians
American science writers
Educators from Henan
Harvard University alumni
Massachusetts Institute of Technology School of Science faculty
Mathematicians from Henan
Peking University alumni
People's Republic of China emigrants to the United States
Sloan Research Fellows
University of California, Los Angeles faculty
Writers from Kaifeng
Academic staff of Zhejiang University |
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