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Bologna Process

The Bologna Process is a series of ministerial meetings and agreements between European countries to ensure comparability in the standards and quality of higher-education qualifications. The process has created the European Higher Education Area under the Lisbon Recognition Convention. It is named after the University of Bologna, where the Bologna declaration was signed by education ministers from 29 European countries in 1999. The process was opened to other countries in the European Cultural Convention of the Council of Europe, and government meetings have been held in Prague (2001), Berlin (2003), Bergen (2005), London (2007), Leuven (2009), Budapest-Vienna (2010), Bucharest (2012), Yerevan (2015), Paris (2018), and Rome (2020).

Before the signing of the Bologna declaration, the Magna Charta Universitatum was issued at a meeting of university rectors celebrating the 900th anniversary of the University of Bologna (and European universities) in 1988. One year before the declaration, education ministers Claude Allègre (France), Jürgen Rüttgers (Germany), Luigi Berlinguer (Italy) and Baroness Blackstone (UK) signed the Sorbonne declaration in Paris in 1998, committing themselves to "harmonising the architecture of the European Higher Education system". The Bologna Process has 49 participating countries.

Signatories
Signatories of the Bologna Accord, members of the European Higher Education Area, are: All member states of the EU are participating in the process, with the European Commission also a signatory. Monaco and San Marino are the only members of the Council of Europe which did not adopt the process.
 * 1999: Austria, Belgium (Flemish Community and French Community of Belgium separately), Bulgaria, the Czech Republic, Denmark, Estonia, Finland, France, Germany, Greece, Hungary, Iceland, Ireland, Italy, Latvia, Lithuania, Luxembourg, Malta, Netherlands, Norway, Poland, Portugal, Romania, Slovakia, Slovenia, Spain, Sweden, Switzerland, United Kingdom.
 * 2001: Croatia, Cyprus, Liechtenstein, Turkey, European Commission
 * 2003: Albania, Andorra, Bosnia and Herzegovina, North Macedonia, Russia, Serbia, Vatican City
 * 2005: Armenia, Azerbaijan, Georgia, Moldova and Ukraine
 * May 2007: Montenegro
 * 2010: Kazakhstan
 * May 2015: Belarus

The ESU, EUA, EURASHE, EI, ENQA, UNICE, the Council of Europe and UNESCO are part of the process' follow-up. Other groups at this level are ENIC, NARIC and EURODOC.

Rejected countries
Four countries, Israel, Kyrgyzstan, and the unrecognized Northern Cyprus and Kosovo, have applied to join but did not meet the membership criteria.

Israel is not a party to the European Cultural Convention of the Council of Europe, although it has observer status. Although Israel is not geographically part of Europe, it is part of the UNESCO European Region (although not a UNESCO member). Israel has also ratified the Lisbon Recognition Convention but, under the criteria of the 2003 Berlin Communiqué, it is ineligible for the Bologna Process.

Kosovo is not a party to the European Cultural Convention of the Council of Europe. Although Serbia is a party, Kosovo declared independence from it and has theoretically been a part of the Bologna Process since the Kosovo War. It was suggested that Kosovo could be associated with the process in a category appropriate to its situation, such as guest or special-observer status.

Kyrgyzstan is not a party to the European Cultural Convention of the Council of Europe, although has also ratified the Lisbon Recognition Convention.

Northern Cyprus is not a party to the European Cultural Convention of the Council of Europe and not recognized as an independent political entity by any member of the Bologna Process except Turkey.

Qualifications framework
The basic framework is three cycles of higher-education qualifications. The framework adopted by the ministers at their meeting in Bergen in 2005 defines the qualifications in terms of learning outcomes, which are statements of what students know and can do on completing their degrees. In describing the cycles, the framework uses the European Credit Transfer and Accumulation System (ECTS):

 * First cycle: typically 180–240 ECTS credits (a minimum of 60 credits per academic year), usually awarding a bachelor's degree. The European Higher Education Area did not introduce a bachelor-with-honours programme, which allows graduates to receive a "BA hons." degree (used, for example, in the UK, Australia, New Zealand, and Canada) and sometimes enables graduates to begin doctoral studies without first obtaining a master's degree.
 * Second cycle: typically 60–120 ECTS credits (a minimum of 60 credits per academic year), usually awarding a master's degree.
 * Third cycle (doctoral degree): There is no concrete ECTS range, since the disciplines vary in length and comprehensiveness. However, some countries have minimum credit weight requirements on doctoral degrees. Those country-level requirements typically require 120–420 ECTS of study.

In most cases, it would take three to four years to earn a bachelor's degree and another one or two years for a master's degree. Doctoral degrees usually require another two to four years of specialization, primarily individual research under a mentor. Degree names may vary by country. One academic year normally corresponds to 60 ECTS credits, equivalent to 1,500–1,800 hours of study.

Effects by state
The process, an intergovernmental agreement between EU and non-EU countries, does not have the status of EU legislation. Since the Bologna Declaration is not a treaty or convention, there are no legal obligations for the signatory states; participation and cooperation are voluntary.

Although the declaration was created without a formal affiliation with EU institutions, the European Commission (which has supported European projects such as the Tuning and TEEP projects) plays an important role in implementing the process. Most countries do not fit the framework, using their traditional systems. The process, which will result in bilateral agreements between countries and institutions which recognise each other's degrees, is moving from strict convergence in time spent on qualifications towards a competency-based system which will have an undergraduate and postgraduate division (with a bachelor's degree in the former and a master's and doctorate in the latter).

In mainland Europe, five-year-plus first degrees are common. Many do not complete their studies, and many countries are introducing bachelor-level qualifications. The situation is evolving as the Bologna Process is implemented.

Some countries introduced the European Credit Transfer and Accumulation System (ECTS) and discussed their degree structures, qualifications, financing and management of higher education and mobility programmes. At the institutional level, the reform involved higher-education institutions, their faculties or departments, student and staff representatives and other factors. Priorities varied by country and institution.

Andorra
In Andorra, degrees are awarded by the state in all three cycles (bachelor's, master's and doctoral). The University of Andorra has adapted its classroom studies to the European Higher Education Area in accordance with the Bologna Agreement. The degree workload is counted in European credits, with a European equivalent of 180 credits (three years) for bachelor's degrees and 120 credits (two years) for master's degrees.

Austria
Austria's situation is similar to Germany's, with the lowest undergraduate degrees the Magister (FH) and Diplom (FH) (designed to take three or four years). The lowest graduate degrees are Magister and Diplom, which typically fulfill a thesis requirement (including final examination and thesis defence) and can be obtained after four to six years of study. In 2000 many curricula began to be converted into bachelor's degrees (Bakkalaureat; the term was replaced by "bachelor's" in most curricula by 2007) and master's (Magisterstudium) programmes, with nominal durations of six semesters (three years) and three to four semesters (18 months to two years) respectively.

Enrollment in a doctoral programme generally requires a master's degree in a related field. Although the nominal duration of doctoral programmes is two or three years, the time to graduate varies considerably and is generally longer.

Armenia
Armenia ratified the Bologna Process in 2005 and is a member of the European Higher Education Area.

Azerbaijan
Azerbaijan is a full member of the Bologna Process since 2005.

Belarus
Belarus became a member of the European Higher Education Area at a conference in Yerevan, Armenia, in May 2015.

Croatia
In Croatia, implementation of the Bologna Process began during the 2005–2006 academic year. Diploma degree became baccalaureate (bachelor's degree, prvostupnik), and the programmes were shortened from four to about three years. Magisterij (master's degree) is achieved after two additional years of post-graduate study. The doktorat degree (doctorate) may be received after three more years (eight years total).

The typical length of study is three years for a bachelor's degree (baccalaureus), two years for a master's degree (magistar) and three years for a doctor of science (doktor znanosti). A local distinction is made between vocational and academic degrees at the baccalaureate level, and between engineering and other programs at levels below the doctoral.

There are several exceptions. The first degree in economics still takes four years, and the master's degree is obtained after an additional year at the University of Zagreb's Faculty of Economics and Zagreb School of Economics and Management. The four-plus-one-year system also applies to fine arts and music. Medical and related studies replace the bachelor's degree with six-year first professional degrees and graduate Doctor of Medicine (doktor medicine) degrees.

The old degrees are translated as follows:
 * Diploma holders hold master's degrees (magistar inženjer for engineers and magistar for others).
 * The old master's degree is grandfathered into magistar znanosti (Master of Science), an intermediate title between the new master's degree and a doctorate for local use.
 * Doctoral degrees remain the same.

In May 2008, about 5,000 students protested weak funding, imprecisely defined new rules and the poor results of the Bologna reform.

Denmark
Denmark introduced the 3+2+3 system in 1971 with an education-management working group of the Society of Danish Engineers and a 1984–85 group of the Federation of Danish Industries, both headed by Hans Bruno Lund. Before the adoption of international standards, the lowest degree normally awarded at universities in Denmark was equivalent to a master's degree (Kandidat/cand.mag). Although bachelor's degrees have been obtained after three years of study, most students continue the additional two years required for a master's degree. Mid-length (two-to four-year) professional degrees have been adapted as professional bachelor's degrees ($3 1/2$ years).

Finland
In the Finnish pre-Bologna system, higher education was divided between universities and polytechnics. In universities, degrees were divided in most fields into a three-year bachelor's degree (kandidaatti) and a two-year master's degree (maisteri). In these fields, the Bologna Process resulted in no change.

In engineering, universities only offered a $5 1/2$-year master's program (diplomi-insinööri). This has been replaced by a three-year bachelor's degree (tekniikan kandidaatti) and a two-year master's degree (diplomi-insinööri), for which the English names are Bachelor of Science (Technology) and Master of Science (Technology). A corresponding change has been made in military higher education, where the officer's degree was divided between bachelor's and master's programmes. Finnish Universities of Applied Sciences, which have offered bachelor's-equivalent engineering programmes, began offering master's-degree programs in 2005. Some Master of Engineering (insinööri (ylempi AMK)) programmes are taught in English.

Only medicine and dentistry retain their non-standard degree structure, where the Licentiate (higher than a master's degree, but less extensive than Doctor of Medicine or Dentistry degrees) is the basic degree. A six-year program of at least 360 ECTS credits leads to the Licentiate of Medicine (lääketieteen lisensiaatti) degree. There is an intermediate title (but not an academic degree) of lääketieteen kandidaatti, and no master's degree.

Polytechnic degrees are considered bachelor's degrees in international use. In domestic use, bachelors transferring from polytechnics to universities may be required to amass a maximum of 60 ECTS credits of additional studies before beginning master's-level studies. In conjunction with the Bologna Process, polytechnics have the right to award master's degrees.

France
In France the baccalauréat, awarded at the end of secondary education, allows students to enter university. Before the LMD reform amid the 2000s which implemented the Bologna Process, it was followed by a two-year Diplôme d'études universitaires générales (DEUG) and a third-year Licence (the equivalent of a UK bachelor's degree).

Students could then pursue a Maîtrise, a one-year research degree which could be followed by a one-year vocational degree (the Diplôme d'études supérieures spécialisées or DESS) or research degree (the Diplôme d'études approfondies, or DEA). The DEA, preparation for a doctorate, was equivalent to the M. Phil. Students could then pursue a doctorat (PhD), which took at least three years.

The DESS was created in 1975 for students who completed four-year degrees. Intended as a doctorate with a more practical approach than research, it included the production of a 120-page paper which was defended to a jury of three international specialists in the field. The mini-thesis was kept in the libraries of the university issuing the DESS, unlike a PhD dissertation (distributed by its author to every French university library).

Higher education in France is also provided by non-university institutions dedicated to specific subjects. The Diplôme d'ingénieur (engineering diploma) is awarded to students after five years of study in state-recognized Écoles d'ingénieurs, particularly the Grandes Écoles such as Mines, Centrale and ENAC.

Although the baccalauréat and doctorat are unchanged in the Bologna system (known in France as LMD reform), the DEUG and licence have been merged into a three-year Licence. The Maîtrise, DESS and DEA have been combined into a two-year master's degree, which can be work- (master professionnel) or research-oriented (master recherche). The Diplôme d'ingénieur degree is still separate from a university degree, but holders may legally claim a master's degree as well.

Strikes occurred in 2002 and 2003 and 2007 protesting LMD reform, focusing more on under-funding of French universities since May 1968 than on the Bologna Process. Although the two major student organisations object to some aspects of its application to the French system, they generally welcome the European process.

Georgia
Although Georgia joined the Bologna Process in 2005 at the Bergen summit, steps towards the establishment of the European Higher Education Area were completed earlier. Since the end of the 1990s, many Georgian universities (mostly private) have introduced limited educational programs allowing students to graduate with a bachelor's degree (four years) and earn a master's degree (one to two years) while preserving the old five-to-six-year scheme. During the Soviet era, the only degree was the discontinued Specialist.

Cycles of higher education are divided into first (bachelor's degree with 240 credits), second (master's degree, 120 credits) and third (doctorate, 180 credits). Human and veterinary medicine and dentistry (300–360 credits) are integrated programs with a qualification equal to a master's degree.

Greece
Greece joined the Bologna Process in 1999. Since 2007, more-intensive steps towards the establishment of the European Higher Education Area were completed.

Hungary
In Hungary, the Bologna system applies to those who began their university education in or after September 2006. One hundred and eight majors were available for selection (compared with over 400 in 2005), of which six are exempt from the bachelor's-master's division: law, human and veterinary medicine, dentistry, pharmacy and architecture.

According to an online poll by the National Tertiary Education Information Centre, 65 percent of respondents thought it unnecessary to adopt the system. The new system provides less of a guarantee that students will obtain a master's degree, because many will complete their education after the three-year bachelor's degree. Students are expected to study more unrelated subjects during the first three years, due to the smaller number of majors.

Iceland
In Iceland, bachelor's degrees are usually three years in duration; master's degrees are two years, and doctoral degrees range from three to six years.

Ireland
In Ireland, bachelor's degrees are commonly three to four years in duration; master's and doctoral degrees are basically similar to those in the UK. Bachelor's degrees are first-cycle qualifications. Except for the MA at the Trinity College Dublin, a master's degree is always a postgraduate degree (teaching or research). The generic outcomes for Irish degrees are laid out in the 2003 National Framework of Qualifications. In 2006, Ireland was the first country to verify the compatibility of its national framework with that of the EHEA.

Italy
Italy fits the framework since its 1999 adoption of the 3+2 system. The first degree is the Laurea triennale, which may be obtained after three years of study. Selected students may then complete their studies with two additional years of specialization leading to the Laurea Magistrale.

The Laurea corresponds to a bachelor's degree; the Laurea Magistrale, corresponding to a master's degree, grants access to third-cycle programmes (post-MA degrees, doctorates or specialized schools) lasting two to five years (completing a PhD usually takes three years). A five-year degree, Laurea Magistrale a Ciclo Unico (Single-cycle Master's Degree) is awarded in law (Giurisprudenza). A five-year degree, Laurea Magistrale a Ciclo Unico (Single-cycle Master's Degree) is awarded in Primary teacher education (Scienze della Formazione Primaria), in Architecture (Architettura), in Pharmacy (Farmacia) and Pharmaceutical Sciences (Chimica e Tecnologia Farmaceutiche), as well as in visual arts (Accademia di Belle Arti) and music (Conservatorio di Musica). A six-year degree, Laurea Magistrale a Ciclo Unico (Single-cycle Master's Degree) is awarded in medicine (Medicina) and dentistry (Odontoiatria). The title for BA and BS undergraduate students is Dottore and for MA, MFA, MSc, MD and MEd graduate students Dottore magistrale (abbreviated Dott., Dott.ssa or Dr.). This should not be confused with PhD and post-MA graduates, whose title is Dottore di Ricerca (Research Doctor).

The Italian system has two types of postgraduate degree (called "Master", not to be confused with the master's degree). Laurea Magistrale (120 ECTS) allows access to third-cycle programmes, and Master universitario (at least 60 ECTS) may be divided into first- (second cycle) and second-level master's degrees (third cycle). A first-level master's degree is accessible by a first-cycle degree and "does not allow access to PhD and to 3rd cycle programmes, since this type of course does not belong to the general requirements established at national level, but it is offered under the autonomous responsibility of each university".

Kazakhstan
Kazakhstan has been a full member of the Bologna Process and European Higher Education Area since 2010.

Netherlands
The Netherlands differentiates between HBO (university of applied sciences) and WO (scientific education or research universities). WO and HBO have adopted the bachelor's-master's system. It generally requires three (WO) or four (HBO) years of education to obtain a bachelor's degree; graduates may then apply for a master's program at a university, which generally require one to two years to complete. An HBO bachelor graduate may have to pass one year of pre-master's education to bridge the gap between their HBO study and (research-oriented) WO study to be admitted to a WO Master's programme, which may grant degrees such as MA, MSc and LLM. There are also HBO master's studies, and HBOs are the only institutions offering associate degrees. A masters degree is the default (legal) requirement to enter a PhD program in the Netherlands. Dispensation can be granted on an individual base, judged by a special committee. With an HBO master's degree, it is also possible to follow a professional doctorate at a university of applied sciences.

Portugal
Due to the Bologna Process, in 2005 new licenciatura (licentiate) degrees were organized at university and polytechnic institutions of Portugal. Previously a four- to six-year programme, equivalent to 300 ECTS, it is now a three-year first cycle and the only requirement for the two-year second cycle which awards a master's degree. Some Bologna courses are integrated five- or six-year programmes awarding a joint master's degree, a common practice in medicine. In engineering, despite the use of two cycles, an engineer may be licensed only after obtaining a master's degree. Master's degrees attained after five or six years of study correspond to the old undergraduate degrees known as licenciatura. The new licenciatura, obtained after three years of study, corresponds to the discontinued bacharelato awarded by polytechnics from the 1970s to the early 2000s (roughly equivalent to an extended associate degree). Old and new master's degrees are the first graduate degree before a doctorate, and the old and new licenciatura are undergraduate degrees.

The licenciatura degree (a four- to six-year course) was required for applicants who wished to undertake the old master's and doctoral programmes, but admission was reserved for those with a licenciatura degree with a grade above 14 (out of 20). After the changes introduced by the Bologna Process, the master's degree is conferred at the end of a programme roughly equivalent in time to many old licenciatura programmes. The process was developed to improve the education system to one based on the development of competency rather than the transmission of knowledge. Its goal was the development of a system of easily comparable degrees to simplify the comparison of qualifications across Europe. Its flexibility and transparency is intended to enable wider recognition of student qualifications, facilitating movement around a European Higher Education Area based on two main cycles (undergraduate and graduate) and providing third-cycle degrees for doctoral candidates.

Russia
The Russian higher education framework was basically incompatible with the process. The generic, lowest degree in all universities since the Soviet era is the Specialist, which can be obtained after five to six years of study. Since the mid-1990s, many universities have introduced limited programmes allowing students to graduate with a bachelor's degree in four years and a master's degree in an additional one to two years while preserving the old system.

After accession to the Bologna process happened in 2003, in October 2007, Russia moved to two-tier education in line with the Bologna Process. Universities inserted a BSc diploma in the middle of their standard specialist programmes, but the transition to MS qualification has not been completed.

Although Specialists and masters are eligible for doctoral programmes (Aspirantura), bachelors are not; the Specialist degree is being discontinued. In most universities bachelor's- and master's-degree education is free of charge, although some state and private universities charge for one or both. The labour market does not yet understand BSc diplomas, but some universities made the program similar to classical education and the MS stage remains mandatory for most graduates.

Sweden
A bill proposing new regulations in the field of Higher Education was presented to Parliament in 2005, and the new system came into force in July 2007. The new system of degrees will have two degrees, of different lengths, in each cycle.

Students may not always be offered all the combinations above to obtain a degree. The högskoleexamen is usually not offered, and many schools require students to obtain the kandidatexamen before obtaining a magisterexamen or masterexamen. Most third-cycle programmes require at least a magisterexamen, although the legal minimal requirement is either a degree from the second cycle or four years of finished courses, where at least one year is second cycle.

In July 2007 a new system of credits, compatible with the European Credit Transfer and Accumulation System, was introduced in which one credit (högskolepoäng) in the new system corresponds to one ECTS credit and two-thirds of a credit in the old system (poäng).

Some Swedish universities have introduced the ECTS standard grading scale for all students, and others will use it only for international students. Since criterion-referenced grading is used instead of relative grading in the Swedish educational system, the 10-, 25-, 30-, 25- and 10-percent distribution of students among A, B, C, D and E will not be done. Some universities only give Fail or Pass grades (F or P) for certain courses (such as internship and thesis projects) or assignments, such as laboratory exercises.

Seminars
Several Bologna Process seminars have been held. The first devoted to a single academic discipline, Chemistry Studies in the European Higher Education Area (which approved Eurobachelor), was held in June 2004 in Dresden.

title: Bologna Process


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Separable extension

In field theory, a branch of algebra, an algebraic field extension $$E/F$$ is called a separable extension if for every $$\alpha\in E$$, the minimal polynomial of $$\alpha$$ over $F$ is a separable polynomial (i.e., its formal derivative is not the zero polynomial, or equivalently it has no repeated roots in any extension field). There is also a more general definition that applies when $E$ is not necessarily algebraic over $F$. An extension that is not separable is said to be inseparable.

Every algebraic extension of a field of characteristic zero is separable, and every algebraic extension of a finite field is separable. It follows that most extensions that are considered in mathematics are separable. Nevertheless, the concept of separability is important, as the existence of inseparable extensions is the main obstacle for extending many theorems proved in characteristic zero to non-zero characteristic. For example, the fundamental theorem of Galois theory is a theorem about normal extensions, which remains true in non-zero characteristic only if the extensions are also assumed to be separable.

The opposite concept, a purely inseparable extension, also occurs naturally, as every algebraic extension may be decomposed uniquely as a purely inseparable extension of a separable extension. An algebraic extension $$E/F$$ of fields of non-zero characteristic $p$ is a purely inseparable extension if and only if for every $$\alpha\in E\setminus F$$, the minimal polynomial of $$\alpha$$ over $F$ is not a separable polynomial, or, equivalently, for every element $x$ of $E$, there is a positive integer $k$ such that $$x^{p^k} \in F$$.

The simplest nontrivial example of a (purely) inseparable extension is $$E=\mathbb{F}_p(x) \supseteq F=\mathbb{F}_p(x^p)$$, fields of rational functions in the indeterminate x with coefficients in the finite field $$\mathbb{F}_p=\mathbb{Z}/(p)$$. The element $$x\in E$$ has minimal polynomial $$f(X)=X^p -x^p \in F[X]$$, having $$f'(X) = 0$$ and a p-fold multiple root, as $$f(X)=(X-x)^p\in E[X]$$. This is a simple algebraic extension of degree p, as $$E = F[x]$$, but it is not a normal extension since the Galois group $$\text{Gal}(E/F)$$ is trivial.

Informal discussion
An arbitrary polynomial $f$ with coefficients in some field $F$ is said to have distinct roots or to be square-free if it has $deg f$ roots in some extension field $$E\supseteq F$$. For instance, the polynomial $g(X) = X^{ 2} − 1$ has precisely $deg g = 2$ roots in the complex plane; namely $1$ and $−1$, and hence does have distinct roots. On the other hand, the polynomial $h(X) = (X − 2)^{2}$, which is the square of a non-constant polynomial does not have distinct roots, as its degree is two, and $2$ is its only root.

Every polynomial may be factored in linear factors over an algebraic closure of the field of its coefficients. Therefore, the polynomial does not have distinct roots if and only if it is divisible by the square of a polynomial of positive degree. This is the case if and only if the greatest common divisor of the polynomial and its derivative is not a constant. Thus for testing if a polynomial is square-free, it is not necessary to consider explicitly any field extension nor to compute the roots.

In this context, the case of irreducible polynomials requires some care. A priori, it may seem that being divisible by a square is impossible for an irreducible polynomial, which has no non-constant divisor except itself. However, irreducibility depends on the ambient field, and a polynomial may be irreducible over $F$ and reducible over some extension of $F$. Similarly, divisibility by a square depends on the ambient field. If an irreducible polynomial $f$ over $F$ is divisible by a square over some field extension, then (by the discussion above) the greatest common divisor of $f$ and its derivative $f$ is not constant. Note that the coefficients of $f$ belong to the same field as those of $f$, and the greatest common divisor of two polynomials is independent of the ambient field, so the greatest common divisor of $f$ and $f$ has coefficients in $F$. Since $f$ is irreducible in $F$, this greatest common divisor is necessarily $f$ itself. Because the degree of $f$ is strictly less than the degree of $f$, it follows that the derivative of $f$ is zero, which implies that the characteristic of the field is a prime number $p$, and $f$ may be written
 * $$f(x)= \sum_{i=0}^ka_ix^{pi}.$$

A polynomial such as this one, whose formal derivative is zero, is said to be inseparable. Polynomials that are not inseparable are said to be separable. A separable extension is an extension that may be generated by separable elements, that is elements whose minimal polynomials are separable.

Separable and inseparable polynomials
An irreducible polynomial $f$ in $F[X]$ is separable if and only if it has distinct roots in any extension of $F$ (that is if it may be factored in distinct linear factors over an algebraic closure of $F)$. Let $f$ in $F[X]$ be an irreducible polynomial and $f '$ its formal derivative. Then the following are equivalent conditions for the irreducible polynomial $f$ to be separable:
 * If $E$ is an extension of $F$ in which $f$ is a product of linear factors then no square of these factors divides $f$ in $E[X]$ (that is $f$ is square-free over $E$).
 * There exists an extension $E$ of $F$ such that $f$ has $deg(f)$ pairwise distinct roots in $E$.
 * The constant $1$ is a polynomial greatest common divisor of $f$ and $f '$.
 * The formal derivative $f '$ of $f$ is not the zero polynomial.
 * Either the characteristic of $F$ is zero, or the characteristic is $p$, and $f$ is not of the form $$\textstyle\sum_{i=0}^k a_iX^{pi}.$$

Since the formal derivative of a positive degree polynomial can be zero only if the field has prime characteristic, for an irreducible polynomial to not be separable, its coefficients must lie in a field of prime characteristic. More generally, an irreducible (non-zero) polynomial $f$ in $F[X]$ is not separable, if and only if the characteristic of $F$ is a (non-zero) prime number $p$, and $f(X)=g(X^{p}$) for some irreducible polynomial $g$ in $F[X]$. By repeated application of this property, it follows that in fact, $$f(X)=g(X^{p^n})$$ for a non-negative integer $n$ and some separable irreducible polynomial $g$ in $F[X]$ (where $F$ is assumed to have prime characteristic p).

If the Frobenius endomorphism $$x\mapsto x^p$$ of $F$ is not surjective, there is an element $$a\in F$$ that is not a $p$th power of an element of $F$. In this case, the polynomial $$X^p-a$$ is irreducible and inseparable. Conversely, if there exists an inseparable irreducible (non-zero) polynomial $$\textstyle f(X)=\sum a_iX^{ip}$$ in $F[X]$, then the Frobenius endomorphism of $F$ cannot be an automorphism, since, otherwise, we would have $$a_i=b_i^p$$ for some $$b_i$$, and the polynomial $f$ would factor as $$\textstyle \sum a_iX^{ip}=\left(\sum b_iX^{i}\right)^p.$$

If $K$ is a finite field of prime characteristic p, and if $X$ is an indeterminate, then the field of rational functions over $K$, $K(X)$, is necessarily imperfect, and the polynomial $f(Y)=Y^{p}−X$ is inseparable (its formal derivative in Y is 0). More generally, if F is any field of (non-zero) prime characteristic for which the Frobenius endomorphism is not an automorphism, F possesses an inseparable algebraic extension.

A field F is perfect if and only if all irreducible polynomials are separable. It follows that $F$ is perfect if and only if either $F$ has characteristic zero, or $F$ has (non-zero) prime characteristic $p$ and the Frobenius endomorphism of $F$ is an automorphism. This includes every finite field.

Separable elements and separable extensions
Let $$E\supseteq F$$ be a field extension. An element $$\alpha\in E$$ is separable over $F$ if it is algebraic over $F$, and its minimal polynomial is separable (the minimal polynomial of an element is necessarily irreducible).

If $$\alpha,\beta\in E$$ are separable over $F$, then $$\alpha+\beta$$, $$\alpha\beta$$ and $$1/\alpha$$ are separable over F.

Thus the set of all elements in $E$ separable over $F$ forms a subfield of $E$, called the separable closure of $F$ in $E$.

The separable closure of $F$ in an algebraic closure of $F$ is simply called the separable closure of $F$. Like the algebraic closure, it is unique up to an isomorphism, and in general, this isomorphism is not unique.

A field extension $$E\supseteq F$$ is separable, if $E$ is the separable closure of $F$ in $E$. This is the case if and only if $E$ is generated over $F$ by separable elements.

If $$E\supseteq L\supseteq F$$ are field extensions, then $E$ is separable over $F$ if and only if $E$ is separable over $L$ and $L$ is separable over $F$.

If $$E\supseteq F$$ is a finite extension (that is $E$ is a $F$-vector space of finite dimension), then the following are equivalent. The equivalence of 3. and 1. is known as the primitive element theorem or Artin's theorem on primitive elements. Properties 4. and 5. are the basis of Galois theory, and, in particular, of the fundamental theorem of Galois theory.
 * 1) $E$ is separable over $F$.
 * 2) $$E = F(a_1, \ldots, a_r)$$ where $$a_1, \ldots, a_r$$ are separable elements of $E$.
 * 3) $$E = F(a)$$ where $a$ is a separable element of $E$.
 * 4) If $K$ is an algebraic closure of $F$, then there are exactly $$[E : F]$$ field homomorphisms of $E$ into $K$ that fix $F$.
 * 5) For any normal extension $K$  of $F$ that contains $E$, then there are exactly $$[E : F]$$ field homomorphisms of $E$ into $K$ that fix $F$.

Separable extensions within algebraic extensions
Let $$E \supseteq F$$ be an algebraic extension of fields of characteristic $p$. The separable closure of $F$ in $E$ is $$S=\{\alpha\in E \mid \alpha \text{ is separable over } F\}.$$ For every element $$x\in E\setminus S$$ there exists a positive integer $k$ such that $$x^{p^k}\in S,$$ and thus $E$ is a purely inseparable extension of $S$. It follows that $S$ is the unique intermediate field that is separable over $F$ and over which $E$ is purely inseparable.

If $$E \supseteq F$$ is a finite extension, its degree $[E : F]$ is the product of the degrees $[S : F]$ and  $[E : S]$. The former, often denoted $[E : F]_{sep}$, is referred to as the separable part of $[E : F]$,  or as the ' of $E/F$; the latter is referred to as the inseparable part of the degree or the '. The inseparable degree is 1 in characteristic zero and a power of $p$ in characteristic $p > 0$.

On the other hand, an arbitrary algebraic extension $$E\supseteq F$$ may not possess an intermediate extension $K$ that is purely inseparable over $F$ and over which $E$ is separable. However, such an intermediate extension may exist if, for example, $$E\supseteq F$$ is a finite degree normal extension (in this case, $K$ is the fixed field of the Galois group of $E$ over $F$). Suppose that such an intermediate extension does exist, and $[E : F]$ is finite, then $[S : F] = [E : K]$, where $S$ is the separable closure of $F$ in $E$. The known proofs of this equality use the fact that if $$K\supseteq F$$ is a purely inseparable extension, and if $f$ is a separable irreducible polynomial in $F[X]$, then $f$ remains irreducible in K[X] ). This equality implies that, if $[E : F]$ is finite, and $U$ is an intermediate field between $F$ and $E$, then $[E : F]_{sep} = [E : U]_{sep}⋅[U : F]_{sep}$.

The separable closure $F^{sep}$ of a field $F$ is the separable closure of $F$ in an algebraic closure of $F$. It is the maximal Galois extension of $F$. By definition, $F$ is perfect if and only if its separable and algebraic closures coincide.

Separability of transcendental extensions
Separability problems may arise when dealing with transcendental extensions. This is typically the case for algebraic geometry over a field of prime characteristic, where the function field of an algebraic variety has a transcendence degree over the ground field that is equal to the dimension of the variety.

For defining the separability of a transcendental extension, it is natural to use the fact that every field extension is an algebraic extension of a purely transcendental extension. This leads to the following definition.

A separating transcendence basis of an extension $$E\supseteq F$$ is a transcendence basis $T$ of $E$ such that $E$ is a separable algebraic extension of $F(T)$. A finitely generated field extension is separable if and only it has a separating transcendence basis; an extension that is not finitely generated is called separable if every finitely generated subextension has a separating transcendence basis.

Let $$E\supseteq F$$ be a field extension of characteristic exponent $p$ (that is $p = 1$ in characteristic zero and, otherwise, $p$ is the characteristic). The following properties are equivalent: where $$\otimes_F$$ denotes the tensor product of fields, $$F^p$$ is the field of the $E$th powers of the elements of $F$ (for any field $F$), and $$F^{1/p}$$ is the field obtained by adjoining to $L$ the $E$th root of all its elements (see Separable algebra for details).
 * $p$ is a separable extension of $F$,
 * $$E^p$$ and $F$ are linearly disjoint over $$F^p,$$
 * $$F^{1/p} \otimes_F E$$ is reduced,
 * $$L \otimes_F E$$ is reduced for every field extension $F$ of $p$,

Differential criteria
Separability can be studied with the aid of derivations. Let $E$ be a finitely generated field extension of a field $F$. Denoting $$\operatorname{Der}_F(E,E)$$ the $E$-vector space of the $F$-linear derivations of $E$, one has
 * $$\dim_E \operatorname{Der}_F(E,E) \ge \operatorname{tr.deg}_F E,$$

and the equality holds if and only if E is separable over F (here "tr.deg" denotes the transcendence degree).

In particular, if $$E/F$$ is an algebraic extension, then $$\operatorname{Der}_F(E, E) = 0$$ if and only if $$E/F$$ is separable.

Let $$D_1, \ldots, D_m$$ be a basis of $$\operatorname{Der}_F(E,E)$$ and $$a_1, \ldots, a_m \in E$$. Then $$E$$ is separable algebraic over $$F(a_1, \ldots, a_m)$$ if and only if the matrix $$D_i(a_j)$$ is invertible. In particular, when $$m = \operatorname{tr.deg}_F E$$, this matrix is invertible if and only if $$\{ a_1, \ldots, a_m \}$$ is a separating transcendence basis.

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: Separable extension


text:
Smashmouth (indie rock band)

Smashmouth was a band formed by guitarist Steve Pedersen, who was also in another band called Slowdown Virginia from Omaha, Nebraska at the time. He recruited bassist Bart Volkmer and drummer Clint Schnase. It was a trio that had grown out of a four-piece called Gravy Train that had originally included Ted Stevens, who had left the band when he moved to Lincoln, Nebraska to attend University of Nebraska–Lincoln (and form Polecat). Smashmouth was the first band to feature Pedersen's songwriting skills.

In 1995, both Smashmouth and Slowdown Virginia disbanded and formed what is now called Cursive. Pedersen left the band in 1998, to form The White Octave and finally, Criteria.

Band members

 * Steve Pedersen
 * Bart Volkmer
 * Clint Schnase

Discography

 * Some of You Will Be Hermits (1995 · Lumberjack Records)

title: Smashmouth (indie rock band)


text:
Smash Mouth

Smash Mouth is an American rock band from San Jose, California. The band was formed in 1994 and was originally composed of Steve Harwell (lead vocals), Kevin Coleman (drums), Greg Camp (guitar), and Paul De Lisle (bass). With Harwell's departure in 2021, De Lisle is now the only original member. They are known for their songs "Walkin' on the Sun" (1997), "All Star" (1999), and "Then The Morning Comes" (1999), as well as a cover of The Monkees' "I'm a Believer" (2001).

The band adopted retro styles covering several decades of popular music. They have also performed numerous covers of popular songs, including War's "Why Can't We Be Friends?", Rick Astley’s “Never Gonna Give You Up”, Simple Minds' "Don't You (Forget About Me)", ? & the Mysterians' "Can't Get Enough of You Baby", The Beatles' "Getting Better", Queen's "Under Pressure", House of Pain's "Jump Around", and "I Wan'na Be Like You" from The Jungle Book. They also composed two songs for the South Korean animated film Pororo, The Racing Adventure: "Beside Myself" and "Everything Just Crazy".

1990–1997: formation and Fush Yu Mang
Smash Mouth's roots trace back to 1990 when Steve Harwell and Kevin Coleman met. Smash Mouth was then formed in 1994 by Harwell, who had formerly played in a rap group called F.O.S. (Freedom of Speech). Coleman, who was Harwell's manager, knew guitarist Greg Camp and bassist Paul De Lisle, who had both played in a local punk band, and introduced the three musicians to each other. They began rehearsing together along with Coleman, who played as the drummer. They soon developed into a band, and named themselves Smashmouth, an American football term. During their early years, the band played largely rock music.

The band's first publicity came when a demo of the song "Nervous in the Alley" was played by a San Jose radio station, KOME. Soon after, Interscope Records signed the band, and the group's debut album, Fush Yu Mang, was released in 1997, featuring another member: the keyboardist Michael Klooster. Also, upon signing to Interscope Records, the band changed their name from Smashmouth to Smash Mouth. The album eventually went double platinum led by the band's first major single "Walkin' on the Sun". The singles "The Fonz" and a cover of "Why Can't We Be Friends" from the 1998 Kevin Bacon film Wild Things were also subsequently released.

1998–2004: rise to fame, Astro Lounge, Smash Mouth, and Get the Picture?
The band's second album, Astro Lounge, was released in 1999 and marked a change in direction, as it had less of the previous ska influence and more of a pop sound. It led to more publicity for the band, and ended up being one of the most critically acclaimed albums from the group. Supported by the hit singles "All Star" (which was featured in several film soundtracks, most notably in Shrek (2001), although the official music video references Mystery Men) and "Then the Morning Comes", Astro Lounge was eventually certified as triple platinum.

Also in 1999, The East Bay Sessions was released as a collection of early songs. Shortly after the release of the album, drummer Kevin Coleman left the band due to back problems. He was initially replaced by Michael Urbano, who was quickly replaced by Mitch Marine for the tour supporting Astro Lounge, who was subsequently replaced by Urbano after the tour.

In 2001, Smash Mouth covered the Monkees' hit song "I'm a Believer". It was featured on both the soundtrack for Shrek (along with "All Star") and their self-titled album. The album sold fewer copies than the band's earlier works, eventually being certified gold. Also in 2001, the group appeared as themselves in the climactic scene of the film Rat Race.

In 2003, Get the Picture? was released, featuring the singles "You Are My Number One", "Hang On" and "Always Gets Her Way". Smash Mouth was dropped from Interscope shortly after the release of Get the Picture?. That same year, the band performed a cover of the Sherman Brothers song "I Wanna Be Like You" for the animated film The Jungle Book 2.

2005–2011: new label, fluctuating lineup, Summer Girl, and Magic
Smash Mouth released the greatest hits compilation All Star Smash Hits in 2005. The album contains some more popular songs from previous Smash Mouth albums, as well as songs from soundtrack albums that were not on the band's releases. On certain networks and time slots, the album was advertised as having 18 tracks, including an edited version of "Flo". Smash Mouth played at Gumby's Birthday Celebration in August 2005.

In December 2005, the band released a Christmas album Gift of Rock. It featured covers of Christmas songs by many artists, such as the Kinks and the Ramones, and one original song, "Baggage Claim".

Smash Mouth's fifth studio album, originally to be titled Old Habits, was recorded in 2005 and expected to be released in early 2006. The band had said that the album was much more like the ska punk featured on Fush Yu Mang and The East Bay Sessions. In September 2005, the band performed what was tentatively going to be the album's first single, "Getaway Car", on Last Call with Carson Daly. The album was delayed many times, in the hope of gaining publicity with Harwell's appearance on the reality show The Surreal Life. Smash Mouth returned to the studio intent on improving the record. Old Habits was shelved, replaced by Summer Girl, which included some remixed Old Habits tracks as well as new songs. After being delayed in much the same way Old Habits was for several months, the album was released on September 19, 2006. Smash Mouth let Sony Pictures use much of their music from Summer Girl and other songs for the movie Zoom, whose opening titles credit the film's music to the band.

Before the release of Summer Girl, drummer Michael Urbano left the band without warning on February 14, 2006 due to creative differences. He was initially replaced by former drummer Mitch Marine, and then by Jason Sutter, best known for his work with American Hi-Fi and the Rembrandts. The band released their new album, Summer Girl, later that year. In early 2007, one year after joining the band, Sutter left Smash Mouth to play drums for former Soundgarden and Audioslave frontman Chris Cornell; fill-in drummer Marine returned to Smash Mouth.

Band members changed over the course of several years. Greg Camp left the band in the summer of 2008. Smash Mouth recruited Leroy Miller to play guitar. Leroy left in 2009 and Camp returned to the band, but in 2011 Camp left once again and this time the band recruited Sean Hurwitz. Hurwitz stayed until 2012, and was replaced by Mike Krompass. Later in 2012, Hurwitz returned. In 2009 Mitch Marine left once again and was replaced by Urbano, who left again after only one year in 2010, and was replaced by Marine once again. Marine left yet again after a brief spell in 2010 and was replaced by Randy Cooke. Cooke was briefly replaced by Jason Sutter in 2011, then Charlie Paxson.

In June 2011, a writer at Something Awful offered $20 if the band's lead singer, Steve Harwell, would eat 24 eggs. Others on Twitter began offering additional sums, eventually targeted to various charities. In July 2011, Harwell accepted the challenge if fans could gather pledges of $10,000 for St. Jude Children's Hospital. The fundraising goal was reached in less than a week. A self-styled "reality TV fan," Harwell requested that his friend celebrity chef Guy Fieri prepare the eggs. The event was held at Johnny Garlic's restaurant, in Dublin, California, on October 11, 2011. With about 150 people attending, Harwell was able to finish the eggs with the help of audience members as well as the San Jose Sharks mascot, Sharkie. $15,000 was raised for charity.

2012–2019: Magic and live album
After parting ways with Universal Records and signing with 429 Records, Smash Mouth released their sixth studio album titled Magic, on September 4, 2012. The album was primarily produced by new band member Mike Krompass. The first single, also entitled "Magic", debuted on the Billboard Adult Contemporary chart at No. 22. The band spent the rest of 2012 touring behind the new project as well as promoting the release of their musical book of food recipes and things of the like — Recipes from the Road. Cooke temporarily left again toward the end of 2012, replaced by Paxson. In 2013 the band took part in the "Under the Sun" with Gin Blossoms and Sugar Ray. Paxson left in July during the tour, and was initially going to be replaced by a returning Cooke; however, he was replaced by Sutter once again. Tod Burr, former drum tech of Def Leppard and drummer of Merle Jagger, came on board in 2012 as drum and keyboard tech of Smash Mouth.

On February 1, 2013, Smash Mouth headlined the AutoNation Coast to Coast rebranding event (which combined all AutoNation dealerships into a single brand) at Wayne Huizenga Park in Fort Lauderdale, Florida with Michela Paige from Season 3 of The Voice. The event also served as a final round in the AutoNation Culture of Caring Contest.

On October 6, 2014, Smash Mouth composed two songs for the English version of the South Korean animated film Pororo, The Racing Adventure, including "Beside Myself", and "Everything Just Crazy" which plays at the end credits.

In 2014, Smash Mouth was part of the "Under the Sun" tour with Uncle Kracker, Blues Traveler, and Sugar Ray.

On June 14, 2015, Smash Mouth performed at the Taste of Fort Collins food festival in Fort Collins, Colorado. Before the encore, in an incident that lasted about three minutes, singer Steve Harwell became angry, yelled profanities at the audience, and left the stage after bread that had been distributed by the festival was thrown in the air and a few pieces were reportedly thrown onto the stage. Harwell later apologized for the incident in an interview with The Herald-Mail. Event organizer Jason Ornstein explained that he asked Harwell if he wanted him to have the DJs make an announcement instructing the crowd not to throw bread but according to Ornstein, Harwell "stormed on [stage] and took matters into his own hands". He continued "It wasn't like anyone was going to be getting hurt by throwing bread up in the air ... We just had to laugh at it, because he just really made a fool of himself." Fort Collins Police Services stated laws prohibiting disorderly conduct, harassment, and 'throwing of "missiles,"' i.e. objects, apply to event attendees.

In May 2016, Smash Mouth released their first live album titled Playlist: The Very Best of Smash Mouth through Sony Music. The recordings were harvested from shows in Rapid City, South Dakota, and Manila.

2016 saw the band gain a new member in Sam Eigen and the return of Cooke. Eigen is a long-time friend of both Steve Harwell and Paul De Lisle, and has played with Alanis Morissette, Janet Jackson, and John Fogerty. He can also be heard playing guitar on several of Harwell's solo recordings over the years.

Greg Camp rejoined Smash Mouth in early 2018. An acoustic re-recording of Smash Mouth's first album, Fush Yu Mang, funded through PledgeMusic was released in 2018 for the album's 20th anniversary.

On November 1, 2018, Smash Mouth released the single "Unity". It features hip-hop artists Kool Keith and Darryl McDaniels of Run-DMC. They then toured Australia in the same year during November to sell-out crowds around the country, performing a mix of theater, festival, and pub shows. During the Arlie Beach Festival of Music in Queensland, Steve Harwell became ill and had to leave the stage. However, Nicky Bomba, Frankie J Holden, Wilbur Wilde from Melbourne Ska Orchestra, and the remainder of Smash Mouth continued the set without him. Harwell recovered the next day and the Australian tour continued without any further incidents.

2020–present: Harwell's departure and death and Missile Toes
On August 8, 2020, the band performed at the 2020 Sturgis Rally and vocalist Harwell said things such as "fuck that COVID shit" during the performance. A report from the IZA Institute of Labor Economics later cited the event as being a superspreading event.

In October 2021, the band performed at The Big Sip beer and wine festival in Bethel, New York. Harwell appeared to be intoxicated, threatening audience members and performing what looked like a Nazi salute. Following the performance, Harwell announced his retirement due to ongoing health issues. Prior to Harwell's last show, the band was already performing with a substitute singer ostensibly filling in while Harwell dealt with his health issues.

In March 2022, it was announced upon Steve Harwell's departure, that the band has recruited new lead singer Zach Goode (formerly of Ghoulspoon, Divided By Zero and The Secret Seven). On March 2, 2022, the band released a cover of Rick Astley's song "Never Gonna Give You Up". Smash Mouth released a new single, "4th of July", on July 1, 2022.

In July 28 2023, the band released a music video for "Underground Sun", their first music video in over a decade. The video was created using A.I.

On September 3, 2023, it was announced that Harwell was receiving hospice care and was expected to have only days left to live. Harwell died the next day at his home in Boise, Idaho, at the age of 56 from liver failure.

On November 6, 2023, the band announced their second Christmas album, "Missile Toes", which released on November 17. The album consists of 10 songs, six of which are covers of classic Christmas songs, and four of which are originals. Featured artists Susanna Hoffs and Sophie Grey make appearances.

On April 12, 2024, Smash Mouth released the single and video "Ride On", and on July 12, 2024 the band released their newest single and lyric video, a cover of the iconic Brady Bunch song, "Sunshine Day" featuring Greg Brady himself, Barry Williams.

As of Summer 2024, Smash Mouth are also currently embarking on the Ride On Tour with close to 100 shows, touring everywhere from Alaska to Hawaii with major festival stops in Mexico, Chile, Peru, and Malaysia along the way.

Musical style
Earlier in their career, Smash Mouth was part of the ska punk scene. After the success of their single, "Walking on the Sun", the band began including more elements of psychedelic pop and other retro styles of the 1960s in their compositions. They would ultimately come to be known as a pop rock, power pop and alternative rock band, despite still retaining some elements of their ska and reggae roots.

Band members
Current members
 * Paul De Lisle – bass, backing vocals (1994–present), lead vocals (touring 2016)
 * Michael Klooster – keyboards, programming, backing vocals (touring and session 1997–2008; 2008–present)
 * Randy Cooke – drums, percussion, backing vocals (2010–2011, 2011–2012, 2013, 2016–2018, 2018–present)
 * Sean Hurwitz – guitar, backing vocals (2011–2012, 2012–2016, 2019–present)
 * Zach Goode – lead vocals (2022–present)

Touring members
 * Mark Cervantes – percussion, theremin, backing vocals (1999–2008, 2014, 2018)

Touring substitutes
 * Rob Schwartz – guitar, backing vocals (2012)
 * Kristian Attard – bass, backing vocals (2017)
 * Danny Richardson – guitar (2018)
 * Steve Carey – drums, percussion (2020)

Former members
 * Steve Harwell – lead vocals, piano, occasional keyboards (1994–2021; died 2023)
 * Kevin Coleman – drums, percussion (1994–1999)
 * Greg Camp – guitar, backing vocals; occasional keyboards and turntables (studio) (1994–2008, 2009–2011, 2018–2019)
 * Michael Urbano – drums, percussion (1999, 2000–2006, 2009–2010, 2014, 2018)
 * Mitch Marine – drums, percussion (1999–2000, 2006, 2007–2009, 2010)
 * Jason Sutter – drums, percussion, backing vocals (2006–2007, 2011, 2013–2016)
 * Leroy Miller – guitar, backing vocals (2008–2009)
 * Charlie Paxson – drums, percussion, backing vocals (2011, 2012–2013)
 * Mike Krompass – guitar, backing vocals, additional keyboards (2012)
 * Sam Eigen – guitar, backing vocals (2016–2018)

Touring guests
 * Adam Young – lead vocals, guitar (2011)
 * Miles Zuniga – guitar (2013)

Timeline

Discography

 * Fush Yu Mang (1997)
 * Astro Lounge (1999)
 * Smash Mouth (2001)
 * Get the Picture? (2003)
 * The Gift of Rock (2005)
 * Summer Girl (2006)
 * Magic (2012)
 * Missile Toes (2023)

title: Smash Mouth


text:
Wei Tao-ming

Wei Tao-ming (October 28, 1899 – May 18, 1978) was a Chinese diplomat and public servant. He was the Republic of China's Ambassador to the United States during the Second World War and foreign minister during the years when the People's Republic of China sought to oust the ROC from the United Nations. He was also civilian Governor of Taiwan Province (1947–1949), replacing Governor General Chen Yi.

Wei enlisted USA help to support Taiwan and oppose the Chinese communists.

Early life
Wei Tao-ming was born in Jiujiang, Jiangxi province in 1899. His father, Wei Tiao-yuan, was an educator and member of Sun Yat-sen's revolutionary movement. Wei Tao-ming's early schooling was at a missionary school, though he graduated from Kiangsi (Jiangxi) First Middle School in 1918. He then studied French in Beijing for a year before moving to France in 1919. He obtained his doctorate in law from the University of Paris in 1926 and returned to China to pursue a legal career in Shanghai.

He was married to Zheng Yuxiu, the first female lawyer and judge in Chinese history. She earned her doctoral degree in law at the Sorbonne in France and was the first Chinese person to practice law at the French extraterritorial courts in Shanghai. Zheng was also one of the revolutionaries involved in the attempted assassination of military official and politician Yuan Shikai, commonly reviled in Chinese history for taking advantage of both the Qing imperial court and the Republicans. She advocated women having their own voices and choices in marriage, and wrote it into the Republic of China's law. Her autobiography, My Revolutionary Years (1944), was published while her husband was Ambassador to the United States, and is revered as one of the best first hand accounts of modern Chinese history.

Political career
He became involved with the Kuomintang. At the age of 29, Wei became president of the Judicial Yuan. From 1930 to 1931, Wei served as mayor of special municipality of Nanjing, then capital of the Republic of China.

Wei was among the Nationalist government insiders implicated in corruption during the 1942-1943 American Dollar Bond scandal. After the 1941 Japanese declaration of war against the United States and the United Kingdom, the two allies sought to support China in a concrete way despite logistical limitations following the loss of British Burma. The two countries loaned significant amounts of money to the Nationalist government. The Nationalist government decided to use USD$200 million to absorb excess fabi in an effort to curb inflation. In theory, Chinese purchasers would use fabi to buy bonds at the official exchange rate and be paid in dollars when the bonds were redeemed following victory over Japan. The American Dollar Bonds were issued on March 24, 1942. The public response was poor, with few bond sales. In October 1943, H.H. Kung sent a secret memorandum to Chiang Kai-shek asking that the bond sales end. Subscriptions were closed on October 15, 1943 and a central bank official falsely announced that all bonds had been sold. Secretly, insiders then purchased the remaining bonds using currency acquired on the black market. The result was a windfall for Nationalist government insiders including Wei, Kung, Long Yun, members of the Soong family, and others.

As Ambassador to the United States from September 1942 to 1946, Wei helped secure American material and military support for the Republic of China as it resisted Japanese invasion and Communist insurgency. His public declarations were covered by the New York Times, and he accompanied Soong Mei-ling during her visits to Washington, D.C., and New York. He resigned from his post in October 1945, citing personal reasons, and was succeeded by Wellington Koo, then Ambassador to the Court of St. James.

Wei's wife Zheng Yuxiu christened the SS China Victory in 1944. The Ceremony had both an ancient Chinese invocation to the sea gods and the traditional American tradition of a bottle of champagne breaking. The matron of honor at the launching was Mrs. T. K. Chang, wife of the Chinese consul at Los Angeles. SS China Victory was the first of a long line of Victory ships to leave the Calship building.

During his tenure as Governor of Taiwan Province, Wei created the Departments of Civil Affairs (民政廳), Finance (財政廳), Construction (建設廳), and Education (教育廳). He also employed thirteen members on the provincial board (省政委員), including those who were Taiwanese-born. He became the minister of foreign affairs after being the Governor (position succeeded by Chen Tsyr-shiou).

After the Chinese Communist Revolution, Wei spent some time in Hong Kong, then made his way back to Taiwan.

Wei served as foreign minister of the Republic of China during the 1960s and was active in maintaining U.S. support for Taipei. He also maintained a coalition in the United Nations General Assembly to reject membership for the People's Republic of China. He resigned for health reasons in 1971 as Beijing's campaign to oust the ROC from the United Nations was on the verge of succeeding.

He died in Taipei on May 18, 1978, at the age 79.

title: Wei Tao-ming


text:
Chen Cheng

Chen Cheng (January 4, 1898 – March 5, 1965), courtesy name Tsi-siou, was a Chinese political and military leader, and one of the main commanders of the National Revolutionary Army during the Second Sino-Japanese War and the Chinese Civil War.

After moving to Taiwan at the end of the civil war, he served as the Governor of Taiwan Province, Vice President, and Premier of the Republic of China (ROC). He represented the ROC in visits to the United States and helped to initiate land reforms and tax reduction programs that caused communism to become unattractive in Taiwan since peasants were able to own land.

Early life
Chen Cheng was born in Qingtian County, Zhejiang, graduated from Baoding Military Academy in 1922, and entered Whampoa Military Academy two years later. It was there that he first met Chiang Kai-shek, Commandant of the Academy. Later, Chen joined National Revolutionary Army to participate in the Northern Expedition.

Rise in military
During the Northern Expedition, Chen displayed good leadership ability. Within a year, he was promoted from commanding battalions to divisions already.

After the expedition, Chen became active in the wars against the warlords. His successes in the battles allowed him to be promoted again, to the commander of the 18th Army.

Anticommunist campaigns
In 1931, Chen was assigned the task of suppressing the Red Army. In various campaigns searching for the main force of Red Army, Chen's units experienced heavy casualties. In the fifth campaign against the Communists, he finally managed to defeat them, which forced the Red Army to launch its Long March.

Campaigns against the Red Army came to an end after the Xi'an Incident in which Chiang and his staff were forced to agree to co-operate with the communists against the invading Imperial Japanese Army.

War against Japan
During the Battle of Shanghai, he was one of the top military assistants of Chiang Kai Shek. It was his idea to seek a decisive action in Southern China, rather than confronting the Japanese in Northern China, where Nationalist troops were in poor condition and lacked transporting vehicles. After the fall of Shanghai and of Nanjing, Chen moved to Hubei to command the Battle of Wuhan in 1938. Wuhan was the provisional headquarters of the Chinese Army. The Japanese, however, managed, despite heavy losses to defeat the Chinese and captured Wuhan on October 25, 1938.

In the last years of the war, Chen went on to command in the Battle of Changsha, the Battle of Yichang, and the Battle of West Hubei. In 1943, he was appointed the commander of the Chinese Expeditionary Force in the Burma Theater until he was replaced by Wei Li-huang because of illness.

Chinese Civil War
After the Second Sino-Japanese War, Chen became the Chief of the General Staff and commander-in-chief of the navy. He followed Chiang's orders and began to raid the "liberated" areas of the Red Army, which launched the Chinese Civil War.

In August 1947, Chiang appointed Chen as director of the Northeastern Headquarters to command the Nationalist forces against the Communists in that area. He made the crucial mistake of dissolving the local security regiments because they had served in the Japanese-collaborationist Manchukuo Imperial Army, which made the total Nationalist strength in Manchuria fall from 1.3 million to less than 480,000. He also dismissed some of the most capable Nationalist commanders, such as Du Yuming, Sun Li-jen, Zheng Dongguo, and Chen Mingren. As a result, he suffered a series of major defeats, and Chiang recalled Chen to Nanjing and sent Wei Lihuang to replace him in Shenyang as Commander-in-Chief of the Northeast and Fan Hanjie as deputy commander-in-chief and director of Jinzhou forward command center. Chen took a sick leave in Taiwan to treat his chronic stomach ailment.

In Taiwan
Chiang appointed Chen as the Governor of Taiwan Province in 1949 to plan the development of Taiwan as a Nationalist stronghold. After the Nationalist force retreated to Taiwan, Chen went on to hold key civilian government positions such as Vice-Executive of the Kuomintang, Vice President, and Premier of the Republic of China. He was the youngest premier since promulgation of the 1947 constitution to take office.

In his years on Taiwan, he introduced various land and economic reforms and carried out the reconstruction of Taiwan. Chen's 37.5% Arable Rent Reduction initiative was credited with stopping the spread of communism in Taiwan. The policy capped the rent farmers paid to landlords at 37.5% of the harvest. Prior to the directive, landlords had often sought more than half of the crop as payment.

He was also credited with launching several construction projects. One was the Shimen Reservoir, in Taoyuan County, which reduced flooding and increased the rice crop production.

On May 19, 1949, Chen promulgated the Order of Martial Law to announce the imposition of martial law throughout Taiwan to expel communist infiltration and to buffer defense capabilities.

Death
Chen died of hepatic tumors in 1965. His cremated remains were moved to Fo Guang Shan, Kaohsiung County (now part of Kaohsiung City), in August 1995.

Honors

 * Key to the City
 * Arms of the Seal of Manila, Philippines.svg Manila, Philippines (March 20, 1960)
 * Pasay City, Philippines (March 20, 1960)

title: Chen Cheng


text:
Lyapunov stability

Various types of stability may be discussed for the solutions of differential equations or difference equations describing dynamical systems. The most important type is that concerning the stability of solutions near to a point of equilibrium. This may be discussed by the theory of Aleksandr Lyapunov. In simple terms, if the solutions that start out near an equilibrium point $$x_e$$ stay near $$x_e$$ forever, then $$x_e$$ is Lyapunov stable. More strongly, if $$x_e$$ is Lyapunov stable and all solutions that start out near $$x_e$$ converge to $$x_e$$, then $$x_e$$ is said to be asymptotically stable (see asymptotic analysis). The notion of exponential stability guarantees a minimal rate of decay, i.e., an estimate of how quickly the solutions converge. The idea of Lyapunov stability can be extended to infinite-dimensional manifolds, where it is known as structural stability, which concerns the behavior of different but "nearby" solutions to differential equations. Input-to-state stability (ISS) applies Lyapunov notions to systems with inputs.

History
Lyapunov stability is named after Aleksandr Mikhailovich Lyapunov, a Russian mathematician who defended the thesis The General Problem of Stability of Motion at Kharkov University in 1892. A. M. Lyapunov was a pioneer in successful endeavors to develop a global approach to the analysis of the stability of nonlinear dynamical systems by comparison with the widely spread local method of linearizing them about points of equilibrium. His work, initially published in Russian and then translated to French, received little attention for many years. The mathematical theory of stability of motion, founded by A. M. Lyapunov, considerably anticipated the time for its implementation in science and technology. Moreover Lyapunov did not himself make application in this field, his own interest being in the stability of rotating fluid masses with astronomical application. He did not have doctoral students who followed the research in the field of stability and his own destiny was terribly tragic because of his suicide in 1918. For several decades the theory of stability sank into complete oblivion. The Russian-Soviet mathematician and mechanician Nikolay Gur'yevich Chetaev working at the Kazan Aviation Institute in the 1930s was the first who realized the incredible magnitude of the discovery made by A. M. Lyapunov. The contribution to the theory made by N. G. Chetaev was so significant that many mathematicians, physicists and engineers consider him Lyapunov's direct successor and the next-in-line scientific descendant in the creation and development of the mathematical theory of stability.

The interest in it suddenly skyrocketed during the Cold War period when the so-called "Second Method of Lyapunov" (see below) was found to be applicable to the stability of aerospace guidance systems which typically contain strong nonlinearities not treatable by other methods. A large number of publications appeared then and since in the control and systems literature. More recently the concept of the Lyapunov exponent (related to Lyapunov's First Method of discussing stability) has received wide interest in connection with chaos theory. Lyapunov stability methods have also been applied to finding equilibrium solutions in traffic assignment problems.

Definition for continuous-time systems
Consider an autonomous nonlinear dynamical system

 * $$\dot{x} = f(x(t)), \;\;\;\; x(0) = x_0$$,

where $$x(t) \in \mathcal{D} \subseteq \mathbb{R}^n$$ denotes the system state vector, $$\mathcal{D}$$ an open set containing the origin, and $$f: \mathcal{D} \rightarrow \mathbb{R}^n$$ is a continuous vector field on $$\mathcal{D}$$. Suppose $$f$$ has an equilibrium at $$x_e$$ so that $$ f(x_e)=0 $$ then

 * 1) This equilibrium is said to be Lyapunov stable if for every $$\epsilon > 0$$ there exists a $$\delta > 0$$ such that if $$\|x(0)-x_e\| < \delta$$ then for every $$t \geq 0$$ we have $$\|x(t)-x_e\| < \epsilon$$.
 * 2) The equilibrium of the above system is said to be asymptotically stable if it is Lyapunov stable and there exists $$\delta > 0$$ such that if $$\|x(0)-x_e \|< \delta$$ then $$\lim_{t \rightarrow \infty} \|x(t)-x_e\| = 0$$.
 * 3) The equilibrium of the above system is said to be exponentially stable if it is asymptotically stable and there exist $$\alpha >0, ~\beta >0, ~\delta >0$$ such that if $$\|x(0)-x_e\| < \delta$$ then $$\|x(t)-x_e\| \leq \alpha\|x(0)-x_e\|e^{-\beta t}$$ for all $$t \geq 0$$.

Conceptually, the meanings of the above terms are the following:
 * 1) Lyapunov stability of an equilibrium means that solutions starting "close enough" to the equilibrium (within a distance $$\delta$$ from it) remain "close enough" forever (within a distance $$\epsilon$$ from it). Note that this must be true for any $$\epsilon$$ that one may want to choose.
 * 2) Asymptotic stability means that solutions that start close enough not only remain close enough but also eventually converge to the equilibrium.
 * 3) Exponential stability means that solutions not only converge, but in fact converge faster than or at least as fast as a particular known rate $$\alpha\|x(0)-x_e\|e^{-\beta t}$$.

The trajectory $$x(t) = \phi(t)$$ is (locally) attractive if

 * $$\|x(t)-\phi(t)\| \rightarrow 0 $$ as  $$ t \rightarrow \infty$$

for all trajectories $$x(t) $$ that start close enough to $$\phi(t) $$, and globally attractive if this property holds for all trajectories.

That is, if x belongs to the interior of its stable manifold, it is asymptotically stable if it is both attractive and stable. (There are examples showing that attractivity does not imply asymptotic stability.  Such examples are easy to create using homoclinic connections.)

If the Jacobian of the dynamical system at an equilibrium happens to be a stability matrix (i.e., if the real part of each eigenvalue is strictly negative), then the equilibrium is asymptotically stable.

System of deviations
Instead of considering stability only near an equilibrium point (a constant solution $$x(t)=x_e$$), one can formulate similar definitions of stability near an arbitrary solution $$x(t) = \phi(t)$$. However, one can reduce the more general case to that of an equilibrium by a change of variables called a "system of deviations". Define $$y = x - \phi(t)$$, obeying the differential equation:

 * $$\dot{y} = f(t, y + \phi(t)) - \dot{\phi}(t) = g(t, y)$$.

This is no longer an autonomous system, but it has a guaranteed equilibrium point at $$y=0$$ whose stability is equivalent to the stability of the original solution $$x(t) = \phi(t)$$.

Lyapunov's second method for stability
Lyapunov, in his original 1892 work, proposed two methods for demonstrating stability. The first method developed the solution in a series which was then proved convergent within limits. The second method, which is now referred to as the Lyapunov stability criterion or the Direct Method, makes use of a Lyapunov function V(x) which has an analogy to the potential function of classical dynamics. It is introduced as follows for a system $$ \dot{x} = f(x)$$ having a point of equilibrium at $$x=0$$. Consider a function $$V : \mathbb{R}^n \rightarrow \mathbb{R} $$ such that Then V(x) is called a Lyapunov function and the system is stable in the sense of Lyapunov. (Note that $$V(0)=0$$ is required; otherwise for example $$V(x) = 1/(1+|x|)$$ would "prove" that $$\dot x(t) = x$$ is locally stable.) An additional condition called "properness" or "radial unboundedness" is required in order to conclude global stability. Global asymptotic stability (GAS) follows similarly.
 * $$V(x)=0$$ if and only if $$x=0$$
 * $$V(x)>0$$ if and only if $$x \ne 0$$
 * $$ \dot{V}(x) = \frac{d}{dt}V(x) = \sum_{i=1}^n\frac{\partial V}{\partial x_i}f_i(x) = \nabla V \cdot f(x) \le 0$$ for all values of $$x\ne 0$$ . Note: for asymptotic stability, $$ \dot{V}(x)<0 $$ for $$x \ne 0$$ is required.

It is easier to visualize this method of analysis by thinking of a physical system (e.g. vibrating spring and mass) and considering the energy of such a system. If the system loses energy over time and the energy is never restored then eventually the system must grind to a stop and reach some final resting state. This final state is called the attractor. However, finding a function that gives the precise energy of a physical system can be difficult, and for abstract mathematical systems, economic systems or biological systems, the concept of energy may not be applicable.

Lyapunov's realization was that stability can be proven without requiring knowledge of the true physical energy, provided a Lyapunov function can be found to satisfy the above constraints.

Definition for discrete-time systems
The definition for discrete-time systems is almost identical to that for continuous-time systems. The definition below provides this, using an alternate language commonly used in more mathematical texts.

Let (X, d) be a metric space and f : X → X a continuous function. A point x in X is said to be Lyapunov stable, if,

 * $$\forall \epsilon>0 \ \exists \delta>0 \  \forall y\in X \ \left [d(x,y)<\delta \Rightarrow \forall n \in \mathbf{N} \  d\left (f^n(x),f^n(y) \right )<\epsilon \right ].$$

We say that x is asymptotically stable if it belongs to the interior of its stable set, i.e. if,

 * $$ \exists \delta>0 \left [ d(x,y)<\delta \Rightarrow \lim_{n\to\infty} d \left(f^n(x),f^n(y) \right)=0\right ].$$

Stability for linear state space models
A linear state space model

 * $$\dot{\textbf{x}} = A\textbf{x}$$,

where $$ A$$ is a finite matrix, is asymptotically stable (in fact, exponentially stable) if all real parts of the eigenvalues of $$ A$$ are negative. This condition is equivalent to the following one:

 * $$A^\textsf{T}M + MA$$

is negative definite for some positive definite matrix $$M = M^\textsf{T}$$. (The relevant Lyapunov function is $$V(x) = x^\textsf{T}Mx$$.)

Correspondingly, a time-discrete linear state space model

 * $$\textbf{x}_{t+1} = A\textbf{x}_t$$

is asymptotically stable (in fact, exponentially stable) if all the eigenvalues of $$ A$$ have a modulus smaller than one.

This latter condition has been generalized to switched systems: a linear switched discrete time system (ruled by a set of matrices $$\{A_1, \dots, A_m\}$$)

 * $${\textbf{x}_{t+1}} = A_{i_t}\textbf{x}_t,\quad A_{i_t} \in \{A_1, \dots, A_m\}$$

is asymptotically stable (in fact, exponentially stable) if the joint spectral radius of the set $$\{A_1, \dots, A_m\}$$ is smaller than one.

Stability for systems with inputs
A system with inputs (or controls) has the form

 * $$\dot{\textbf{x}} = \textbf{f}(\textbf{x}, \textbf{u})$$

where the (generally time-dependent) input u(t) may be viewed as a control, external input, stimulus, disturbance, or forcing function. It has been shown that near to a point of equilibrium which is Lyapunov stable the system remains stable under small disturbances. For larger input disturbances the study of such systems is the subject of control theory and applied in control engineering. For systems with inputs, one must quantify the effect of inputs on the stability of the system. The main two approaches to this analysis are BIBO stability (for linear systems) and input-to-state stability (ISS) (for nonlinear systems)

Example
This example shows a system where a Lyapunov function can be used to prove Lyapunov stability but cannot show asymptotic stability. Consider the following equation, based on the Van der Pol oscillator equation with the friction term changed:

 * $$ \ddot{y} + y -\varepsilon \left( \frac{\dot{y}^{3}}{3} - \dot{y}\right) = 0.$$

Let

 * $$ x_{1} = y, x_{2} = \dot{y} $$

so that the corresponding system is

 * $$ \begin{align}

&\dot{x}_{1} = x_{2}, \\ &\dot{x}_{2} = -x_{1} + \varepsilon \left( \frac{3} - {x_{2}}\right). \end{align} $$

The origin $$ x_1= 0,\ x_2=0$$ is the only equilibrium point. Let us choose as a Lyapunov function

 * $$ V = \frac {1}{2} \left(x_{1}^{2}+x_{2}^{2} \right) $$

which is clearly positive definite. Its derivative is

\dot{V} = x_{1} \dot x_{1} + x_{2} \dot x_{2} = x_{1} x_{2} - x_{1} x_{2}+\varepsilon \frac{x_{2}^4}{3} - \varepsilon {x_{2}^2} =  \varepsilon \frac{x_{2}^4}{3} -\varepsilon {x_{2}^2}. $$

It seems that if the parameter $$ \varepsilon $$ is positive, stability is asymptotic for $$ x_{2}^{2} < 3.$$ But this is wrong, since $$ \dot{V} $$ does not depend on $$x_1$$, and will be 0 everywhere on the $$x_1$$ axis. The equilibrium is Lyapunov stable but not asymptotically stable.

Barbalat's lemma and stability of time-varying systems
It may be difficult to find a Lyapunov function with a negative definite derivative as required by the Lyapunov stability criterion, however a function $$V$$ with $$\dot{V}$$ that is only negative semi-definite may be available. In autonomous systems, the invariant set theorem can be applied to prove asymptotic stability, but this theorem is not applicable when the dynamics are a function of time.

Instead, Barbalat's lemma allows for Lyapunov-like analysis of these non-autonomous systems. The lemma is motivated by the following observations. Assuming f is a function of time only:

 * Having $$\dot{f}(t) \to 0$$ does not imply that $$f(t)$$ has a limit at $$t\to\infty$$. For example, $$f(t)=\sin(\ln(t)),\; t>0$$.
 * Having $$f(t)$$ approaching a limit as $$t \to \infty$$ does not imply that $$\dot{f}(t) \to 0$$. For example, $$f(t)=\sin\left(t^2\right)/t,\; t>0$$.
 * Having $$f(t)$$ lower bounded and decreasing ($$\dot{f}\le 0$$) implies it converges to a limit. But it does not say whether or not $$\dot{f}\to 0$$ as $$t \to \infty$$.

Barbalat's Lemma says:
 * If $$f(t)$$ has a finite limit as $$t \to \infty$$ and if $$\dot{f}$$ is uniformly continuous (a sufficient condition for uniform continuity is that $$\ddot{f}$$ is bounded), then $$\dot{f}(t) \to 0$$ as $$t \to\infty$$.

An alternative version is as follows:

 * Let $$p\in [1,\infty)$$ and $$q\in (1,\infty]$$. If $$f \in L^p(0,\infty)$$ and $${\dot f}\in L^q(0,\infty)$$, then $$f(t)\to 0$$ as $$t\to \infty.$$

In the following form the Lemma is true also in the vector valued case:

 * Let $$f(t)$$ be a uniformly continuous function with values in a Banach space $$E$$ and assume that $$\textstyle\int_0^t f(\tau)\mathrm {d}\tau$$ has a finite limit as $$t\to \infty$$. Then $$f(t)\to 0$$ as $$t\to \infty$$.

The following example is taken from page 125 of Slotine and Li's book Applied Nonlinear Control.

Consider a non-autonomous system
 * $$\dot{e}=-e + g\cdot w(t)$$
 * $$\dot{g}=-e \cdot w(t).$$

This is non-autonomous because the input $$w$$ is a function of time. Assume that the input $$w(t)$$ is bounded.

Taking $$V=e^2+g^2$$ gives $$\dot{V}=-2e^2 \le 0.$$

This says that $$V(t)\leq V(0)$$ by first two conditions and hence $$e$$ and $$g$$ are bounded. But it does not say anything about the convergence of $$e$$ to zero, as $$\dot{V}$$ is only negative semi-definite (note $$g$$ can be non-zero when $$\dot{V}$$=0) and the dynamics are non-autonomous.

Using Barbalat's lemma:

 * $$\ddot{V}= -4e(-e+g\cdot w)$$.

This is bounded because $$e$$, $$g$$ and $$w$$ are bounded. This implies $$\dot{V} \to 0$$ as $$t\to\infty$$ and hence $$e \to 0$$. This proves that the error converges.

title: Lyapunov stability


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Edwin Abbott (educator)

Edwin Abbott (12 May 1808 – 27 May 1882) was an English educator.

Abbott was born in London in 1808, the son of Edward Abbott, an oilman and Italian warehouseman descended from George Abbot, archbishop of Canterbury. In 1827 he was made headmaster of the Philological School of General Instruction, Marylebone, the school he had himself attended. As a headmaster he was known as firm but humane. He married his first cousin Jane Abbott (1806–1882) in 1831. The couple had eight children, including Edwin Abbott Abbott, author of Flatland; a daughter, Elizabeth, married John Humffreys Parry.

Abbott's social concern led him to keep up contact with the Christian Socialists. He translated the third volume of J. H. Merle D'Aubigné's history of the reformation, wrote textbooks on Latin, English grammar and arithmetic, and published a concordance to Alexander Pope.

Abbott retired as headmaster in 1872, and died at Upper Norwood in 1882. The school was renamed St Marylebone Grammar School in 1901, and one of the four senior houses was named after Abbott.

Works

 * (tr.) History of the Great Reformation, by J. H. Merle D'Aubigné, vol. 3, 1841
 * Handbook of Arithmetic and First Steps in Algebra
 * A Handbook of English Grammar, 1845
 * A Second Latin Book, 1858
 * A concordance to the works of Alexander Pope, 1875

title: Edwin Abbott (educator)


text:
Elizabeth Stanton

Elizabeth Stanton may refer to:

 * Elizabeth Stanton (television host) (born 1995), American television host
 * Elizabeth Cady Stanton (1815–1902), American social activist and abolitionist
 * Elizabeth Cady Stanton Blake (1894–1981), American painter
 * Elizabeth Stanton (Massachusetts politician), American Democratic politician

title: Elizabeth Stanton


text:
Pointe-à-Pitre International Airport

Pointe-à-Pitre International Airport or Pointe-à-Pitre Le Raizet Airport (Aérodrome de Pointe-à-Pitre Le Raizet or Aéroport Guadeloupe Pôle Caraïbes "Caribbean Hub") is an international airport serving Pointe-à-Pitre on the island of Grande-Terre in Guadeloupe, France.

The airport is located in Abymes, 2.4 km north-northeast of Pointe-à-Pitre. It is the main hub for Air Caraïbes and Air Antilles Express. Air France also has two Airbus A320 based in Pointe-à-Pitre for regional flights. It is the largest of the six airports in the archipelago. In 2019, the airport handled 2,488,782 passengers; it is the second busiest airport in the Lesser Antilles after Queen Beatrix International Airport located in Aruba, and before Grantley Adams International Airport located in Barbados.

Facilities
The airport is at an elevation of 35 ft above mean sea level. It has one paved runway designated 12/30 which measures 3125 x. Runway 12/30 is long enough to allow aircraft as large as the A380 to take off and land without difficulty. The airport was also one of the first to handle the first A380 prototype in the second week of January 2006, for two days. The same year, the airport celebrated its 40th anniversary. The airport has two interconnected terminals, Terminal 1 and Terminal 2. Terminal 1 serves both international and regional flights. Terminal 2 only serves regional flights.

The former Air Guadeloupe had its head office at the airport.

Statistics
Annual passenger traffic (enplaned + deplaned), 2000 - 2023

10 busiest routes from Guadeloupe Pôle Caraîbes Airport in 2020

title: Pointe-à-Pitre International Airport


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Barbi Twins

Shane and Sia Barbi ( Carlson; born April 2, 1963), popularly known as The Barbi Twins, are American identical twins, cover models, co-authors, and frequently nude spokespersons as part of their animal rights advocacy.

Modeling
Shane and Sia began modeling at the age of seven when they posed for a layout in the Sears mail-order catalog. As adults, their career has included fashion and mainly pin-up modeling. The Los Angeles Times did a cover story about the Barbi Twins when their billboard went up on Sunset Boulevard in 1989 immediately catapulting them to worldwide attention. In 1989, 1990, and early 1991, they modeled for Chanel, Thierry Mugler, Jean Paul Gaultier, John Galliano and others.

Playboy
According to their 2001 E! True Hollywood Story, the billboard, and the press that came from it, caught the eye of Hugh Hefner. The twins were put on the cover of Playboy magazine in the September 1991 issue which broke records by selling out in less than two weeks. In an unprecedented move, Hefner erected a billboard to promote the Barbi Twins and their second Playboy issue (which also broke sales records).

Other print work
USA Today said that the twins career brought them "caviar dreams and champagne life styles of the rich and famous". Playboy called their covers "legendary" and credited their subsequent television appearances with "the highest ratings," remarking that the twins became Hollywood landmarks and global obsessions. The twins have also generated a degree of controversy – a Saturday morning cartoon and an MTV show were criticized for being "too sexy" or "too controversial".

Magazines and tabloids called them everything from "the sexiest twins alive" and "the best selling calendar models of the world" (Cosmopolitan UK) to referring to them as both "bizarre and bulimic" and "sexual icons and America's cartoon sweethearts" (Entertainment Weekly) They were a favorite for cover models appearing on the covers and inside not only Playboy but mainstream national and international magazines including Cosmopolitan and Redbook. Newspapers, magazines and tabloids including 'USA TODAY''.

Shane Barbi was among the many notable quotables when Newsweek's "Overheard" section (syndicated from the New York Post, Page Six) picked up a quote from her during an interview, "You can ask us about our cup size or favorite positions but please no personal questions." Stuff magazine gave the twins the title of "Sex Symbols for 1993" placing them in good company with other popular female icons such as Marilyn Monroe (1956), Farrah Fawcett (1977), Madonna (1991), Sharon Stone (1992), and Cindy Crawford (1994).

Redbook magazine took excerpts from the book "Diana's Boys" which reveals how Prince William had his first crush on the Barbi Twins.

After four years of the "best selling calendars", photographs from a photo shoot were stolen from a NY film lab and "held for ransom" but were later returned.

In 2011 were featured and voted as one of MAXIM's Hottest Famous Twins, were featured in numerous movie posters on behalf of animal causes (see Animal Activism below) and on covers and featured articles in True Cowboy Magazine.

Merchandising
Shane and Sia posed together in several internationally distributed Barbi Twins calendars from 1993 to 2007. After the success of their 1993 calendar the "Barbi Mania" continued with dozens of posters, a comic-art calendar and two comic books. Other merchandise such as promotional trading cards, postcards, gift cards, gift wrap, mugs, lighters and just about any product that could have their images (or more specifically bodies) stamped on it were for sale. The Daily Star UK ran a comic strip called "The Barbi Twins", and featuring the twin models with psychic abilities. Gift shops worldwide contained Barbi Twin products and every Spencer Gifts had dedicated a corner to Barbi Twin merchandise as E! True Hollywood Story reported.

Film and television
48 Hours did a special on the Barbi Twins in 2002 on their eating disorders, and the E! True Hollywood Story did a biography on the twins in 2001. They also had several guest appearances on talk shows for television and radio, such as The Geraldo Rivera Show and The Howard Stern Show. The twins made several appearances along with William Shatner, John Landis, and many others in the American animated series Eek! The Cat playing twin rocket scientists, "Dr. Shane and Dr. Sia". In 2007, they received an associate producer credit on Curt Johnson's animal activist documentary titled Your Mommy Kills Animals, which received highly acclaimed reviews from Variety, The Hollywood Reporter and Los Angeles Times.

In 2010 the twins again showed their support for the animal community and were featured in the movie Skin Trade.

In 2012 the twins announced their first feature documentary called I HAVE A DREAM about breed discrimination. Breed discrimination was at the root of the recent death of Lennox, a dog that was destroyed just for being a breed too similar to a pit bull.

Barbi v. Flynt
In November 2003, the Barbi Twins sued Larry Flynt and Hustler magazine for shots taken at Playboy studios that eventually found their way into the January 2004 issue of Hustler without permission. The twins had a vitamin line they were planning to be introduced in 2004 and they claimed that the pictures, allegedly taken by Shane's then-husband Ken Wahl, were never meant to be published. They sued for both the pictures and a fictional story that appeared in Hustler as the Barbi Twins claimed it ruined their health image. However, the twins' lawsuit was not successful, as Flynt possessed a valid photo release signed by the twins.

Parents
The twins are the daughters of property developer Bob Carlson (d. October 2001), a devout Christian, and Marsha Barbi, a former Miss Ohio, but their parents divorced when the twins were just two. The twins believe that their mother was always a lesbian and when she left their father, she and they traveled to West Hollywood where Marsha became an "outspoken lesbian feminist". According to the twins, Marsha had numerous lesbian lovers including some quite famous women. Then she met closeted singer Dusty Springfield in 1975 and the two women lived together for two years.

Spouses
Shane Barbi married actor Ken Wahl in 1997.

Health issues
The twins struggled with bulimia. According to their E! True Hollywood Story and a candid interview in USA TODAY their bulimia and insecurities caused them to be obsessed with crash-dieting, bingeing and purging, abusing laxatives and destructive exercise routines for up to 10 hours a day. Their disorders manifested into agoraphobia preventing them from attending the Hollywood major premieres and invitations they received, causing them to cancel events, and to turn down offers for their own television shows and movies, and even dates with their high-profile suitors.

During their recovery, the twins armed themselves with degrees in health and nutrition, and began lecturing in 2000 on "How to eat to live, not live to eat". They lectured at the Learning Annex, universities and high schools as well as on radio, television and on-line. In early 2001, they came out to the nation on the CBS television program 48HRS in a piece about eating disorders, entitled "Slim Chance". They also shared their recovery story in a chapter in a celebrity addiction book called Feeding the Fame.

The twins' first health and fitness book, Dying To Be Healthy: A Breakthrough Diet, Nutrition and Self Help Guide (1999), tells the story of their struggles with eating disorders and how they overcame it. Their second book, The Eco Anti-Diet, Plus Confession (2006), helps people understand how to be "eco and animal friendly" while being healthy and fit. 100% of the proceeds of the books go to animal charities.

Miscellaneous
By the year 2000 the twins began using their modeling career to help show support and draw focus on other issues they felt important, including publishing books on fitness and health issues, showing their support for the military with their patriotic images and words of support, and animal rights.

Early years
According to the twins, their interest in animal activism began in childhood, after experiencing, at their father's ranch, animals they'd befriended and named would end up going to slaughter. In response, they drafted a heartfelt manifesto they called "Civil Rights for Animals".

Website
The twins founded a website for animal activism called TheKittyLiberationFront.org which has now become TwinBunnies.com, and help animal charities and rescue groups worldwide, volunteer with shelters for walking and rehabilitating dogs, participate in the TNR (Trap, Neuter and Return). They made calendar featuring shelter animals, called "Pinup Pets" which was to support the hurricane animal rescue efforts (as seen in their documentary Your Mommy Kills Animals).

In a 2009 article by Fox News called the twins "the Willie Nelson of pin-up girls interviewing the twins where they spoke of how they (the twins) use their "seven and a half minutes each of fame" to speak out for animals. They participate in animal protests, such as anti-pet shops, anti-puppy mill, anti-circus, and anti-animal experimentation (vivisection) and are vegan. They held a vigil for pets lost during the pet food poisoning that occurred that same year.  They also help lobby animal bills, such as the pet food safety bill.

Wild horses
In 2009 they became very active in lobbying in favor of the ROAM act for wild horses along with Willie Nelson, and continued their support speaking openly about the horse slaughter ban, the ban on de-clawing and the fight against factory farms and animal cruelty.

Marine life
In 2010 they helped bring public awareness to the ban on seal hunting in Canada by doing a media blitz to help end the brutality of seal hunting, the dangers to the sea turtles and other endangered species affected by the BP oil spill, and helped lobby and pass HR5566, the Animal Crush Video Prohibition Act Bill (banning violent images against animals on websites).

Film and video
The Twins use their celebrity status in every way they can including offering their images up in posters displaying their "wacky humor and good looks", such as their 2010 posters to support causes such as the Anti Fur movie Skin Trade and the cruelty that happens in the dairy industry in the documentary Earthlings.=, and their 2011 covers and featured articles in True Cowboy Magazine and their continuing fight to save the wild horses (February and April 2011 issues).

Other efforts
The twins participate in animal rescue projects. In 2011, they worked the Animal Welfare Institute to support the ban on horse slaughter. and appeared in the documentary The Petition about the issue.

Finding Luka Magnotta
In 2012, the twins spearheaded a movement to track down a person posting online videos of killing kittens, believed to be Luka Magnotta. They have been searching for him since 2008. Their search ended after his capture in 2012.

Books

 * Barbi, Sia and Greg Gorman (photographer). Dying To Be Healthy: Millennium Dieting and Nutrition. Pentimento Entertainment, 1999. ISBN 1-892676-11-7
 * Reprinted as: Dying To Be Healthy: A Breakthrough Diet, Nutrition and Self-Help Guide. 2nd Edition. Triumph Books, 2001. ISBN 1-892049-42-2
 * Conte, Robert. History of the Barbi Twins. Studio Books, 1998. ISBN 1-890313-01-7
 * The Barbi Twins. The Eco Anti-Diet, plus confessions AuthorHouse, 2006 ISBN 1-4259-2722-X

Calendars

 * Barbi Twins: Shane and Sia – 1993 Calendar
 * The Barbi Twins 1995 Swimsuit Comic Art Calendar Topps Comics, 1994
 * Artists include John Byrne, Adam Hughes, Julie Bell, Miran Kim, Linser, et al.
 * Best of the Barbi Twins 1997 Calendar. H & H Global Inc.
 * Barbi Twins 2000 Health & Fitness Calendar. Pentimento Entertainment, 1999
 * "Barbi Twins" calendars for the years 2002 through 2007

Comic book

 * Conte, Robert, Peter Hsu, Matt Haley, Fastner and Larson. The Barbi Twins Adventures. Topps Comics.

title: Barbi Twins


text:
Erskine Childers

Erskine Childers may refer to:

 * Erskine Childers (author) (1870–1922), author and Irish nationalist, who served as secretary-general of the Irish delegation that negotiated the Anglo-Irish Treaty in 1921
 * Erskine Hamilton Childers (1905–1974), Fianna Fáil minister who became President of Ireland, son of the above
 * Erskine Barton Childers (1929–1996), UN civil servant, Senior Adviser to UN Director-General for Development and International Economic Co-operation, son of the above

title: Erskine Childers