Lieutenant colonel is a rank of commissioned officer in the armies, most marine forces and some air forces of the world, above a major and below a colonel. Several police forces in the United States use the rank of Lieutenant Colonel; the rank of lieutenant colonel is shortened to "colonel" in conversation and in unofficial correspondence. Sometimes, the term'half-colonel' is used in casual conversation in the British Army. A lieutenant colonel is in charge of a battalion or regiment in the army; the rank of Lieutenant Colonel is used by several police forces in the United States. The following articles deal with the rank of lieutenant colonel Lieutenant-colonel Lieutenant colonel Lieutenant colonel Lieutenant-colonel Azerbaijan – Polkovnik leytenant Afghanistan — Dagarman Arab world — Moqaddam Albania — Nënkolonel Argentina – Teniente Coronel Armenia — Pokhgndapet Austria — Oberstleutnant Belgium — Lieutenant-colonel, Luitenant-kolonel Bosnia and Herzegovina — Potpukovnik Brazil — Tenente-coronel Chile – Teniente Coronel Bulgaria — Podpolkovnik Cambodia — Lok Vorsenito Colombia — Teniente Coronel Croatia – Podpukovnik Czech Republic — Podplukovník People's Republic of China — 中校 Republic of China — 中校 Denmark — Oberstløjtnant Estonia — Kolonelleitnant Ethiopia — Lieutenant koronel Finland — Everstiluutnantti, Överstelöjtnant France — Lieutenant-colonel Germany — Oberstleutnant Nazi Germany — Obersturmbannführer Georgia – Vice-colonel Greece — Antisyntagmatarkhis Honduras — Teniente Coronel Hungary — Alezredes India ― Lieutenant Colonel Indonesia — Letnan kolonel Iran — Sarhang dovom Israel — Sgan aluf Italy — Tenente colonnello Japan — Ni sa North Korea — Jungjwa South Korea — Jungryung Latvia — Pulkvežleitnants Lithuania — Pulkininkas leitenantas Macedonia – Потполковник Malaysia – Leftenan-Kolonel Malta — Logotenent kurunell Mongolia — Дэд Хурандаа Netherlands — Luitenant-kolonel Norway — Oberstløytnant Pakistan – Lieutenant Colonel Philippines — Kalakan, Teniente Coronel Poland — Podpułkownik Portugal — Tenente-coronel Romania — Locotenent colonel Russia — Podpolkovnik Serbia — Potpukovnik Slovakia — Podplukovník Slovenia — Podpolkovnik Somalia — Gaashaanle Dhexe South Africa — Commandant/kommandant.
In mathematics, a Rota–Baxter algebra is an associative algebra, together with a particular linear map R which satisfies the Rota–Baxter identity. It appeared first in the work of the American mathematician Glen E. Baxter in the realm of probability theory. Baxter's work was further explored from different angles by Gian-Carlo Rota, Pierre Cartier, Frederic V. Atkinson, among others. Baxter’s derivation of this identity that bore his name emanated from some of the fundamental results of the famous probabilist Frank Spitzer in random walk theory. In the 1980s, the Rota-Baxter operator of weight 0 in the context of Lie algebras was rediscovered as the operator form of the classical Yang–Baxter equation, named after the well-known physicists Chen-Ning Yang and Rodney Baxter; the study of Rota–Baxter algebras experienced a renaissance this century, beginning with several developments, in the algebraic approach to renormalization of perturbative quantum field theory, dendriform algebras, associative analogue of the classical Yang–Baxter equation and mixable shuffle product constructions.
Let k be a commutative ring and let λ be given. A linear operator R on a k-algebra A is called a Rota–Baxter operator of weight λ if it satisfies the Rota–Baxter relation of weight λ: R R = R + R + λ R for all x, y ∈ A; the pair or A is called a Rota–Baxter algebra of weight λ. In some literature, θ = − λ is used in which case the above equation becomes R R + θ R = R + R, called the Rota-Baxter equation of weight θ; the terms Baxter operator algebra and Baxter algebra are used. Let R be a Rota–Baxter of weight λ. − λ I d − R is a Rota–Baxter operator of weight λ. Further, for μ in k, μ R is a Rota-Baxter operator of weight μ λ. Integration by parts Integration by parts is an example of a Rota–Baxter algebra of weight 0. Let C be the algebra of continuous functions from the real line to the real line. Let: f ∈ C be a continuous function. Define integration as the Rota–Baxter operator I = ∫ 0 x f d t. Let G = I and F = I; the formula for integration for parts can be written in terms of these variables as F G = ∫ 0 x f G d t + ∫ 0 x F g d t.
In other words I I = I + I, which shows that I is a Rota–Baxter algebra of weight 0. The Spitzer identity appeared is named after the American mathematician Frank Spitzer, it is regarded as a remarkable stepping stone in the theory of sums of independent random variables in fluctuation theory of probability. It can be understood in terms of Rota–Baxter operators. Li Guo. WHAT IS...a Rota-Baxter Algebra? Notices of the AMS, December 2009, Volume 56 Issue 11
The Board of Intermediate and Secondary Education, Multan was established on 30 March 1968. It is located near Gulgasht Colony, Multan, it is responsible to take all Matriculation and Intermediate exams of Multan Division schools and colleges as well as all the private candidates. Multan board takes the examination from 9th class to 12th Class; the Admissions are announced as follows: 9th & 10th class during November/December every year 1st & 2nd year during January/February every yearResults of these exams are announced as following schedule: Annual 10th/Combined Examination on 25 July Annual 9th Fresh Examination on 25 August Annual Inter Part-II / Combined Examination on 12 September Annual Inter Part-I Fresh Examination on 10 October Supply 10th/Combined Examination on 10 November Supply Inter Part-II / Combined Examination on 12 JanuaryAround 240,000 candidates appear for annual matric exam every year in BISE Multan and around 130,000 candidates appear for annual Intermediate exam every year.
If we include the supply exams figures in it, we can say that every year the BISE Multan manage the exam process of around 400,000 candidates. And this figure is increasing around 10% yearly. Jurisdiction of Multan Board includes Multan Division which includes following districts:- Multan Khanewal Vehari Lodhran Board of Intermediate and Secondary Education, Sahiwal Board of Intermediate and Secondary Education, Dera Ghazi Khan Board of Intermediate and Secondary Education, Bahawalpur Board of Intermediate and Secondary Education, Lahore Board of Intermediate and Secondary Education, Faisalabad Board of Intermediate and Secondary Education, Rawalpindi Board of Intermediate and Secondary Education, Gujranwala Board of Intermediate and Secondary Education, Sargodha Board of Intermediate Education, Karachi Board of Secondary Education, Karachi Board of Intermediate and Secondary Education, Hyderabad Official Website of BISE Multan