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In mathematics, a polynomial is an expression consisting of variables and coefficients, that involves only the operations of addition, subtraction and non-negative integer exponents of variables. An example of a polynomial of a single indeterminate, x, is x2 − 4x + 7. An example in three variables is x3 + 2xyz2 − yz + 1. Polynomials appear in many areas of science. For example, they are used to form polynomial equations, which encode a wide range of problems, from elementary word problems to complicated scientific problems. In advanced mathematics, polynomials are used to construct polynomial rings and algebraic varieties, central concepts in algebra and algebraic geometry; the word polynomial joins two diverse roots: the Greek poly, meaning "many," and the Latin nomen, or name. It was derived from the term binomial by replacing the Latin root bi- with the Greek poly-; the word polynomial was first used in the 17th century. The x occurring in a polynomial is called either a variable or an indeterminate.

When the polynomial is considered as an expression, x is a fixed symbol which does not have any value. However, when one considers the function defined by the polynomial x represents the argument of the function, is therefore called a "variable". Many authors use these two words interchangeably, it is common to use uppercase letters for indeterminates and corresponding lowercase letters for the variables of the associated function. A polynomial P in the indeterminate x is denoted either as P or as P. Formally, the name of the polynomial is P, not P, but the use of the functional notation P date from the time where the distinction between a polynomial and the associated function was unclear. Moreover, the functional notation is useful for specifying, in a single phrase, a polynomial and its indeterminate. For example, "let P be a polynomial" is a shorthand for "let P be a polynomial in the indeterminate x". On the other hand, when it is not necessary to emphasize the name of the indeterminate, many formulas are much simpler and easier to read if the name of the indeterminate do not appear at each occurrence of the polynomial.

The ambiguity of having two notations for a single mathematical object may be formally resolved by considering the general meaning of the functional notation for polynomials. If a denotes a number, a variable, another polynomial, or, more any expression P denotes, by convention, the result of substituting a for x in P. Thus, the polynomial P defines the function a ↦ P, the polynomial function associated to P. Frequently, when using this notation, one supposes that a is a number; however one may use it over any domain where multiplication are defined. In particular, if a is a polynomial P is a polynomial. More when a is the indeterminate x the image of x by this function is the polynomial P itself. In other words, P = P, which justifies formally the existence of two notations for the same polynomial. A polynomial is an expression that can be built from constants and symbols called indeterminates or variables by means of addition and exponentiation to a non-negative integer power. Two such expressions that may be transformed, one to the other, by applying the usual properties of commutativity and distributivity of addition and multiplication are considered as defining the same polynomial.

A polynomial in a single indeterminate x can always be written in the form a n x n + a n − 1 x n − 1 + ⋯ + a 2 x 2 + a 1 x + a 0, where a 0, …, a n are constants and x is the indeterminate. The word "indeterminate" means that x represents no particular value, although any value may be substituted for it; the mapping that associates the result of this substitution to the substituted value is a function, called a polynomial function. This can be expressed more concisely by using summation notation: ∑ k = 0 n a k x k That is, a polynomial can either be zero or can be written as the sum of a finite number of non-zero terms; each term consists of the product of a number – called the coefficient of the term – and a finite number of indeterminates, raised to nonnegative integer powers. The exponent on an indeterminate in a term is called the degree of that indeterminate in that term; because x = x1, the degree of an indeterminate without a written exponent is one. A term with

Ahmed Abdallah

Ahmed Abdallah Abderemane was a Comorian politician. He was a member of the French Senate from 1959 to 1973, President of the Comoros from 25 October 1978 until his death. Abdallah was born on the island of Anjouan, he began participating in the government in the 1940s. He was the President of the general council from 1949 until 1953, was the chairman of the chamber of deputies during the 1970s. In 1972, now leader of his political party, the Comoros Democratic Union, became president of the government council and Chief Minister of the Comoros. Jaffar, in turn, would be overthrown by Ali Soilih in 1976. Abdallah staged a coup against Soilih in 1978 with the help of mercenary Bob Denard. After Said Atthoumani had served as "Chairman of the Politico-Military Directorate" for ten days and Mohamed Ahmed assumed the titles of "Co-Chairmen of the Politico-Military Directorate." On 22 July, their titles were changed to "Co-Chairmen of the Directorate," and on 3 October, Abdallah became the lone chair.

On 25 October, Abdallah assumed the title of president and remained in office until his death, despite three separate coup attempts against him. In 1982, Abdallah had the UDC and all other parties abolished, a new party, the Comorian Union for Progress, was set up. Comoros became a one-party state, with the UCP being the only legal party. Abdallah was re-elected unopposed in 1984. On 26 November 1989, he was shot dead in his Moroni office during a coup led by Ali Soilih's half-brother, Said Mohamed Djohar. Djohar took control of the country the next day. List of heads of state of the Comoros

Ministry of Foreign Affairs (Saudi Arabia)

The Ministry of Foreign Affairs is the ministry responsible for handling the Kingdom of Saudi Arabia's external relations. The ministry oversees "political and financial international relations" and monitors the Kingdom's diplomatic relations, it was created in 1930 by a royal decree issued by King Abdulaziz Al Saud, being the first ministerial body created by the King. While consolidating the newly formed Kingdom of Nejd and Hejaz, King Abdulaziz, established foreign diplomatic relations by sending representatives and receiving delegations from various states. In 1926, he established the directorate general for foreign affairs in Mecca; the first director general of foreign affairs was Abdullah Beg Al Damluji, ruler of Mecca at that time. In 1930, a royal decree was issued to elevate the directorate general to the ministry of foreign affairs. King Abdulaziz appointed Prince Faisal, as the first foreign minister; the Kingdom of Saudi Arabia was formally established by King Abdulaziz in 1932. The ministry was made up of five departments, namely the private office and the departments of oriental affairs, administrative affairs, political affairs and consular affairs.

The ministry began establishing diplomatic missions abroad. The first one was opened in Cairo in 1926 followed by another in London 1930; the number of missions expanded further after that. Aside from a brief interjection, Prince Faisal continued to serve after he succeeded the throne as King. After his assassination in 1975, Faisal was succeeded as foreign minister by Prince Saud. Saud was the longest-serving foreign minister of any country in current political times, The ministry launched a magazine, The Diplomat, in 2007, it was rumored in 2010 that the next foreign minister would be Prince Turki Al Faisal, Saud's younger brother, after Saud retired, which however did not occur. The senior officials in the ministry are as follows: The following is the list of foreign ministers since its foundation: The ministers of state for foreign affairs served are as follows: Omar Al Saqqaf Nizar Madani Adel al-Jubeir The building of the ministry is in Riyadh and was designed by Henning Larsen, it blends both monumental styles of Islamic architecture.

Larsen received the Aga Khan Award for Architecture in 1989 for his work on the building. Built in 1984, building consists of meeting and prayer rooms, a library and a banquet hall. Externally, the building appears as a fortress, carved out of a single piece of stone. Foreign relations of Saudi Arabia Politics of Saudi Arabia

Murder, She Baked

Murder, She Baked is an American/Canadian television film series, based on the cozy mystery novels written by Joanne Fluke. The television films centered around small-town baker Hannah Swensen, portrayed by Alison Sweeney, Detective Mike Kingston, played by Cameron Mathison. Five films for the Hallmark Movies & Mysteries channel were created based on Fluke's book series, with the final to air on March 26, 2017; the films are set in the fictional town of Eden Lake in Minnesota. Alison Sweeney as Hannah Swensen, a local baker who owns the Cookie Jar Cameron Mathison as Detective Mike Kingston, a widowed homicide detective who moves to town from the city Lisa Durupt as Andrea Todd, Hannah's sister Toby Levins as Deputy Bill Todd, a local detective, married to Andrea Gabriel Hogan as Doctor Norman Rhodes, a dentist and Hannah's close friend Barbara Niven as Delores Swensen and Andrea's mother Juliana Wimbles as Lisa Herman, a helper at Hannah's bakery Garry Chalk as Mayor Bascomb, the town's Mayor Murder, She Baked: A Chocolate Chip Cookie Mystery on IMDb Murder, She Baked: A Plum Pudding Murder Mystery on IMDb Murder, She Baked: A Peach Cobbler Mystery on IMDb Murder, She Baked: A Deadly Recipe on IMDb Murder, She Baked: Just Desserts on IMDb

Dwijen Mukhopadhyay

Dwijen Mukhopadhyay was an Indian composer and singer whose musical career spanned six decades. He was an accomplished performer of Rabindrasangeet, Bengali basic songs and Hindi film songs, he recorded more than 1500 songs. He directed music in Bengali feature films and composed music for popular Bengali basic songs, he died on 24 December 2018 In 1944, Mukhopadhyay made his debut as a professional singer. In 1945, he made his first recording of basic Bengali songs from Megaphone Record Company. 1946 was eventful for him, as in that year, he not only started to act as an artist of All India Radio but started recording with HMV-Colombia Recording Company. In 1956, he entertained the soldiers of the Indian Army with his songs at Ladakh. Mukhopadhyay received his training in music from eminent singers of Bengal including Shri Sushanto Lahiri, Pankaj Mullick, Santidev Ghosh, Santosh Sengupta, Anadi Ghosh Dastidar and Niharbindu Sen, he is survived by a daughter, Meenakshi and a son Deb Krishna. Mukhopadhyay was introduced to the folk music of Bengal, Hindi film music, by the eminent film-music composer Salil Chowdhury.

The friendship between Dwijen Mukhopadhyay and Salil Chowdhury started in the late'40s through their common association with IPTA. The duo gave the Bengali audience songs like "Shyamal Barani Ogo Konya", "Klanti Name Go", "Ekdin Phire Jabo Chole", "Pallabini Go Sancharini" and many others, they came up with rare and beautiful tracks. Dwijen Mukhopadhyay went to Mumbai to work with Salil Chowdhury and recorded duet songs with Lata Mangeshkar for Hindi films like'Honeymoon','Maayaa','Sapan Suhaane' and solo playback in'Madhumati', he was one of the foremost exponents of Rabindrasangeet. He has interpreted Tagore's songs with devotion, drawing listeners to this music within and outside Bengal, he has performed Rabindrasangeet in noted Bengali films like'Kshudita Pashan' – a short story by Rabindranath Tagore and'Sandhya Raag' under distinguished Music Directors Ustad Ali Akbar Khan and Pandit Ravi Shankar, respectively. His Rabindrasangeet in'Kancher Swarga', Bon Palashir Padabali,'Wheel Chair' are landmarks.

Mukhopadhyay had performed the famous devotional song'Jaago Durga' as a part of the musical play'Mahisasura Mardini', a hugely popular radio programme broadcast by All India Radio, every year on the auspicious day of'Mahalaya' marking the beginning of the famous autumn festival of'Durga Puja' and may be labelled as a signature tune of the entire Bengali speaking community in the world, with a phenomenal popularity for the last 60 years. He sang before eminent dignitaries like Marshal Josip Broz Tito, Sarvepalli Radhakrishnan, Pandit Jawaharlal Nehru, Indira Gandhi, amongst others; as a member of'Indian Cultural Delegation', he toured the Soviet Union and East European countries like Poland, Czechoslovakia and Yugoslavia. He was invited to perform in the US, the UK, Switzerland, Canada and Bangladesh. Mukhopadhyay was a member of the "Experts Committee" and the "Selections Committee" at Visva-Bharati Music Board in Kolkata, responsible to certify the purity and authenticity of Tagore songs, sung by any artiste all over the world.

He was a member of the'Expert Committee' of All India Radio, New Delhi that took national level audition of senior singers to promote them to'Top grade Artists' of India. Mukhopadhyay was the'External Examiner' of the Post Graduate Course in Visva Bharati University, West Bengal, India, he created a cultural organisation named'Uttarayani', India to impart training on Rabindrasangeet. Mukhopadhyay was the President of'Bani Chakra College of Music', India, he had the rare privilege of celebrating the Birth Centenary of Rabindranath Tagore in 1961 and took active part in the celebration of the 150th Birth Anniversary held in 2011. Dwijen Mukherjee in Comprehensive Website on Salil Chowdhury Dwijen Mukherjee in Sangeet Natak Academy Padma Shree-padma-award-honour Times Of India Article on Padma Shree Award The Daily Star Article


Auderath is an Ortsgemeinde – a municipality belonging to a Verbandsgemeinde, a kind of collective municipality – in the Cochem-Zell district in Rhineland-Palatinate, Germany. It belongs to the Verbandsgemeinde of Ulmen; the tourist resort of Auderath is located in the Vulkaneifel, a part of the Eifel known for its volcanic history and geological features, ongoing activity today, including gases that sometimes well up from the earth. The municipality lies some 3 km southeast of Ulmen. To Alflen and the Büchel Airbase in Büchel in the east it is 2 km. To the south lie Schmitt and Gillenbeuren, to the southwest Filz and Wollmerath. On a broader scale, Auderath lies between Koblenz. Auderath stood under the lordship of the Electorate of Trier and the Lords of Ulmen. In 1573, Trier became the overlord for good. Auderath belonged to the Electoral-Trier Amt of Ulmen. Beginning in 1794, Auderath lay under French rule. In 1815 it was assigned to the Kingdom of Prussia at the Congress of Vienna. Under Prussia, the village belonged to the Bürgermeisterei of Lutzerath and beginning in 1816 to the Cochem district.

Since 1946, it has been part of the newly founded state of Rhineland-Palatinate. The council is made up of 12 council members, who were elected by majority vote at the municipal election held on 7 June 2009, the honorary mayor as chairman. Auderath’s mayor is Helmut Krämer; the German blazon reads: Schild durch eingeschweifte gestürzte goldene Spitze, darin eine grüne Krone, gespalten. Vorn in Grün ein silberner Korb mit goldenen Broten, hinten in Grün eine silberne Henkelurne; the municipality’s arms might in English heraldic language be described thus: Tierced in mantle reversed, dexter vert a basket argent therein loaves of bread Or, sinister vert an urn of the second, in chief Or a crown azure. The crown in the gold field refers to Our Lady of the Rosary, the parish’s patron saint; the basket with the loaves of bread in the green field recalls an old custom in the municipality, that of giving out Saint Sebastian’s bread in the church. The urn stands for digs undertaken in the municipality between 1936 and 1940, which unearthed Roman burial mounds, wall remnants and potsherds in the cadastral areas of Seitert and Hinterschläf.

The tincture vert refers to the village’s founding as a country estate at the time of the clearings in the High Middle Ages, to the still rural scenery found in the municipality today. The following are listed buildings or sites in Rhineland-Palatinate’s Directory of Cultural Monuments: Catholic Church of Our Lady of the Rosary, Kirchstraße – Baroque aisleless church, 1734, expanded 1952. Hauptstraße/corner of Kirchstraße – well with pyramidal roof, 18th century. Kirchstraße/corner of Brunnenstraße – wayside cross from 1657. Auderather Mühle, west of the village – timber-frame mill solid or slated, bakehouse. Kloster St. Josef, northeast of the village – Barefoot-Carmelite convent from the 1920s and 1930s; each year at Whitsun, a forest festival is held at the Matteshütte. Volunteer fire brigade Alemannia Auderath sport club Auderath riding club Gesangverein Liederkranz Auderath Möhnenverein Catholic youth group Information about Auderath private website about Auderath