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In mathematics, a linear combination is an expression constructed from a set of terms by multiplying each term by a constant and adding the results. The concept of linear combinations is central to related fields of mathematics. Most of this article deals with linear combinations in the context of a vector space over a field, with some generalizations given at the end of the article. Suppose that K is a field and V is a vector space over K; as usual, we call call elements of K scalars. If v1...vn are vectors and a1...an are scalars the linear combination of those vectors with those scalars as coefficients is a 1 v → 1 + a 2 v → 2 + a 3 v → 3 + ⋯ + a n v → n. There is some ambiguity in the use of the term "linear combination" as to whether it refers to the expression or to its value. In most cases the value is emphasized, like in the assertion "the set of all linear combinations of v1...vn always forms a subspace". However, one could say "two different linear combinations can have the same value" in which case the expression must have been meant.

The subtle difference between these uses is the essence of the notion of linear dependence: a family F of vectors is linearly independent if any linear combination of the vectors in F is uniquely so. In any case when viewed as expressions, all that matters about a linear combination is the coefficient of each vi. In a given situation, K and V may be specified explicitly. In that case, we speak of a linear combination of the vectors v1...vn, with the coefficients unspecified. Or, if S is a subset of V, we may speak of a linear combination of vectors in S, where both the coefficients and the vectors are unspecified, except that the vectors must belong to the set S. Finally, we may speak of a linear combination, where nothing is specified. Note that by definition, a linear combination involves only finitely many vectors. However, the set S that the vectors are taken from can still be infinite. There is no reason that n cannot be zero. Let the field K be the set R of real numbers, let the vector space V be the Euclidean space R3.

Consider the vectors e1 =, e2 = and e3 =. Any vector in R3 is a linear combination of e1, e2 and e3. To see that this is so, take an arbitrary vector in R3, write: = + + = a 1 + a 2 + a 3 = a 1 e 1 + a 2 e 2 + a 3 e 3. Let K be the set C of all complex numbers, let V be the set CC of all continuous functions from the real line R to the complex plane C. Consider the vectors f and g defined by f:= eit and g:= e−it; some linear combinations of f and g are: On the other hand, the constant function 3 is not a linear combination of f and g. To see this, suppose that 3 could b

In physics, interference is a phenomenon in which two waves superpose to form a resultant wave of greater, lower, or the same amplitude. Constructive and destructive interference result from the interaction of waves that are correlated or coherent with each other, either because they come from the same source or because they have the same or nearly the same frequency. Interference effects can be observed with all types of waves, for example, radio, surface water waves, gravity waves, or matter waves; the resulting images or graphs are called interferograms. The principle of superposition of waves states that when two or more propagating waves of same type are incident on the same point, the resultant amplitude at that point is equal to the vector sum of the amplitudes of the individual waves. If a crest of a wave meets a crest of another wave of the same frequency at the same point the amplitude is the sum of the individual amplitudes—this is constructive interference. If a crest of one wave meets a trough of another wave the amplitude is equal to the difference in the individual amplitudes—this is known as destructive interference.

Constructive interference occurs when the phase difference between the waves is an multiple of π, whereas destructive interference occurs when the difference is an odd multiple of π. If the difference between the phases is intermediate between these two extremes the magnitude of the displacement of the summed waves lies between the minimum and maximum values. Consider, for example, what happens when two identical stones are dropped into a still pool of water at different locations; each stone generates a circular wave propagating outwards from the point where the stone was dropped. When the two waves overlap, the net displacement at a particular point is the sum of the displacements of the individual waves. At some points, these will be in phase, will produce a maximum displacement. In other places, the waves will be in anti-phase, there will be no net displacement at these points. Thus, parts of the surface will be stationary—these are seen in the figure above and to the right as stationary blue-green lines radiating from the centre.

Interference of light is a common phenomenon that can be explained classically by the superposition of waves, however a deeper understanding of light interference requires knowledge of wave-particle duality of light, due to quantum mechanics. Prime examples of light interference are the famous double-slit experiment, laser speckle, anti-reflective coatings and interferometers. Traditionally the classical wave model is taught as a basis for understanding optical interference, based on the Huygens–Fresnel principle; the above can be demonstrated in one dimension by deriving the formula for the sum of two waves. The equation for the amplitude of a sinusoidal wave traveling to the right along the x-axis is W 1 = A cos ⁡ where A is the peak amplitude, k = 2 π / λ is the wavenumber and ω = 2 π f is the angular frequency of the wave. Suppose a second wave of the same frequency and amplitude but with a different phase is traveling to the right W 2 = A cos ⁡ where φ is the phase difference between the waves in radians.

The two waves will superpose and add: the sum of the two waves is W 1 + W 2 = A. Using the trigonometric identity for the sum of two cosines: cos ⁡ a + cos ⁡ b = 2 cos ⁡ cos ⁡, this can be written W 1 + W 2 = 2 A cos ⁡ cos ⁡; this represents a wave at the original frequency, traveling to the right like the components, whose amplitude is proportional to the cosine of φ / 2. Constructive interference: If the phase difference is an multiple of π: φ = …, − 4 π, − 2 π, 0, 2 π, 4 π, …

Mexico is one of the 51 founding members of the United Nations and was admitted into the organization in 1945. Since Mexico is a full member of all the UN agencies and participates within the organization and has diplomatic relations with most member states. On 26 June 1945, Mexico was represented in San Francisco by Ezequiel Padilla Peñaloza, Francisco Castillo Nájera and Manuel Tello Baurraud in the signing of the United Nations Charter; the country was formally admitted into the organization on 7 November 1945. Since the beginning, Mexico has participated in the social and economic activities of the UN's various specialized agencies and other international organizations concerned with social and economic improvement. However, due to restraints in the Mexican constitution, Mexico is prohibited from contributing troops for peacekeeping missions abroad unless Mexico has formally declared war on a country. Calls have been made from Mexican politicians to amend the constitution in order to partake in UN peacekeeping missions.

This situation started to change after President Enrique Peña Nieto's address to the General Assembly on 24 September 2014, when he stated that "Mexico has taken the decision to participate in U. N. peacekeeping missions, taking part in humanitarian tasks that benefit civil society". Mexico has been elected four times to the United Nations Security Council; the Mexican Government is vehemently opposed to adding new members to the Security Council. Mexico and eight other countries created a group called the "Uniting for Consensus", where they are opposed to new permanent members, however they would like to raise the number of more non-permanent members to 20. List of terms as an elected member to the Security Council: 1946 1980–1981 2002–2003 2009–2010 Mexico maintains permanent representation to the United Nations headquarters in New York City and to the other main UN agencies based in Geneva, Paris and Vienna. Foreign relations of Mexico League of Nations Permanent Mission of Mexico to the United Nations United Nations Official website of the Permanent Mission of Mexico to the UN Official website of the Ministry of Foreign Affairs of Mexico Official website of the United Nations

A financial intermediary is an institution or individual that serves as a middleman among diverse parties in order to facilitate financial transactions. Common types include commercial banks, investment banks, pooled investment funds, stock exchanges. Financial intermediaries reallocate otherwise uninvested capital to productive enterprises through a variety of debt, equity, or hybrid stakeholding structures. Through the process of financial intermediation, certain assets or liabilities are transformed into different assets or liabilities; as such, financial intermediaries channel funds from people who have surplus capital to those who require liquid funds to carry out a desired activity. A financial intermediary is an institution that facilitates the channeling of funds between lenders and borrowers indirectly; that is, savers give funds to an intermediary institution, that institution gives those funds to spenders. This may be in the form of mortgages. Alternatively, they may lend the money directly via the financial markets, eliminate the financial intermediary, known as financial disintermediation.

In the context of climate finance and development, financial intermediaries refer to private sector intermediaries, such as banks, private equity, venture capital funds, leasing companies and pension funds, micro-credit providers. International financial institutions provide funding via companies in the financial sector, rather than directly financing projects; the hypothesis of financial intermediaries adopted by mainstream economics offers the following three major functions they are meant to perform: Creditors provide a line of credit to qualified clients and collect the premiums of debt instruments such as loans for financing homes, auto, credit cards, small businesses, personal needs. Converting short-term liabilities to long term assets Risk transformationConverting risky investments into risk-free ones. Convenience denomination Matching small deposits with large loans and large deposits with small loans There are two essential advantages from using financial intermediaries: Cost advantage over direct lending/borrowing Market failure protection.

These include a lack of transparency, inadequate attention to social and environmental concerns, a failure to link directly to proven developmental impacts. According to the dominant economic view of monetary operations, the following institutions are or can act as financial intermediaries: Banks Mutual savings banks Savings banks Building societies Credit unions Financial advisers or brokers Insurance companies Collective investment schemes Pension funds cooperative societies Stock exchangesAccording to the alternative view of monetary and banking operations, banks are not intermediaries but "fundamentally money creation" institutions, while the other institutions in the category of supposed "intermediaries" are investment funds. Financial intermediaries are meant to bring together those economic agents with surplus funds who want to lend to those with a shortage of funds who want to borrow. In doing this, they offer the benefits of risk transformation. Specialist financial intermediaries are ostensibly enjoying a related advantage in offering financial services, which not only enables them to make profit, but raises the overall efficiency of the economy.

Their existence and services are explained by the "information problems" associated with financial markets. Debt Financial economics Investment Saving Financial market efficiency Keith. Finance and Financial Markets. New York: PALGRAVE MACMILLAN, 2005. Valdez, Steven. An Introduction To Global Financial Markets. Macmillan Press, 2007

Karim Qajymqanuly Massimov is a Kazakh politician who served as Prime Minister of Kazakhstan from 10 January 2007 to 24 September 2012 and again from 2 April 2014 to 8 September 2016. Massimov served as Deputy Prime Minister from 19 January 2006 to 9 January 2007 and as Minister of Economy and Budget Planning, Minister of Transport and Communications in 2001. President Nursultan Äbishuly Nazarbayev nominated Massimov to succeed Daniyal Akhmetov as Prime Minister on 9 January 2007; the Nur Otan party endorsed Massimov's candidacy and Parliament confirmed the nomination on 10 January. On 24 September 2012, Massimov's premiership ended when President Nazarbayev dismissed him as from the position, appointing him chief of staff of the presidential office in order to readjust the power balance between various factions within the government. Since September 2016, he is has been the incumbent was the head of Kazakhstan's National Security Committee of Kazakhstan. Born in Tselinograd in Kazakhstan, Karim Massimov graduated from People's Friendship University of Russia and studied in Beijing and at Wuhan University in Hubei, China.

Karim Massimov is a born citizen of Kazakhstan. Karim Massimov is considered a China expert. China has become an important strategic partner for Kazakhstan in recent years and the two countries are working together to develop Kazakhstan's energy resources. Massimov visited Beijing, China with Finance Minister Natalya Korzhova and Communications Minister Serik Akhmetov, Energy and Mineral Resources Minister Baktykozha Izmukhambetov from 16–17 November 2006. Massimov co-chaired the third meeting of the China-Kazakhstan Cooperation Committee with Chinese Vice Premier Wu Yi. Several accords between agencies of the two governments were signed. Massimov met with Chinese Prime Minister Wen Jiabao. Meanwhile, Kazakh President Nursultan Nazarbayev met with Liu Qi, secretary of the Beijing Party Committee, in Astana. Secretary Liu said; the goal of the visit is to deepen cooperation and mutual understanding between our countries." On 28 March 2002 in an article in Izvestia, Massimov announced that the Government of Kazakhstan planned to increase wheat exports to Iran from 100,000 to two million tons.

Massimov and Israeli Vice Premier Shimon Peres announced from Jerusalem on 29 October 2006 that the state-owned National Innovation Fund of Kazakhstan would begin investing in the Peace Valley project and other projects in the Middle East. Massimov said, "I came to Israel with a clear message to the nation in Zion from the president, that Kazakhstan is a moderate Muslim state, interested in being involved in the Middle East. Kazakhstan intends to found political and economic ties with Israel and its neighbors." Massimov expressed desire to create a free trade zone. Vice Premier Peres and Massimov agreed to establish an agriculture-school in each country; the NIF has given US\$10 million to Israeli VC fund Vertex. Massimov met with Israeli Prime Minister Ehud Olmert, who praised Kazakhstan for showing a "beautiful face of Islam. Contemporary, ever-developing Kazakhstan is a perfect example of both economic development and interethnic accord that should be followed by more Muslim states." President Nazarbayev nominated Massimov to succeed Daniyal Akhmetov as Prime Minister on 9 January 2007.

Akhmetov resigned on 8 January without explanation. Analysts attributed Akhmetov's political downfall to the President's criticism of his administrative oversight of the economy; the Parliament voted overwhelmingly in favor of the nomination on 10 January with 37 out of 39 Senators and 66 of 77 Assemblymen from the Majilis voting for Massimov. Akhmetov became the Defense Minister. Massimov is fluent in Kazakh, Chinese and Arabic. At the same time, he is thought to be well connected within the Kremlin. Massimov has three kids, his hobbies are books, Muay Thai, rock climbing and golf. Massimov was the President of Federation of Amateur Muaythai of Asia in 2010; the FAMA and the Continental Federation of International Federation of Muaythai Amateurs in Asia is the first Continental Federation since 1991, supporting the work and efforts of the IFMA. He was nominated and went on to assume the position of Vice President of the IFMA, President of Muaythai federation in Kazakhstan and Vice-President of World Muaythai Council.

In 2012, Massimov was unanimously re-elected for another four-year term to head the Asian federation, recognised by the Olympic Council of Asia. He showed his support for the sport in an interview in which he stated that the sport "brings together athletes from across the world to train and compete with honour and in the spirit of cultural exchange and understanding." Government of Kazakhstan Cabinet of ministers of Kazakhstan Media related to Karim Masimov at Wikimedia Commons Official Site of Prime Minister of Kazakhstan PM Karim K. Massimov Kazakh President accepts Prime Minister's resignation Analysis: Why the world cares about Kazakhstan

Alfred "Uganda" Roberts is a conga/percussion player. As a young musician he performed with Professor Longhair, who he would continue to work with until Longhair's death, one of the many New Orleans notables Roberts has performed with. Born and raised in the historic Tremé neighborhood of New Orleans, Roberts took an interest in calypso rhythms from a young age, being introduced to the music by attending clubs frequented by the many sailors and seamen who sailed back and forth between New Orleans and the Caribbean. Beginning his career in music in his early teens, playing bongos in clubs in the French Quarter, Roberts switched to the congas in his early 20s by the encouragement of jazz drummer, Smokey Johnson. After becoming an in-demand conga/percussion player in New Orleans, Roberts established a relationship with fellow Treme resident and neighbor, record producer Allen Toussaint, leading to Roberts becoming a house percussionist in Toussaint's Sea Saint Studios, playing on recordings such as The Meters' Afrika and Hey Pocky Way, as well as Toussaint's 1972 album, Life and Faith.

In 1972, Roberts was introduced to New Orleans pianist Henry Roland Byrd, better known to the public as Professor Longhair. It was at the 2nd annual New Orleans Jazz and Heritage Festival that Jazz Fest producer/founder Quint Davis introduced the two musicians, Roberts would go on to tour and record with Professor Longhair for eight years, until Professor Longhair's death in 1980. Roberts is featured on Professor Longhair's Rock N Roll Gumbo, featuring Louisiana blues musicians Snooks Eaglin and Clarence “Gatemouth” Brown, as well as Fess' last studio album recorded, Crawfish Fiesta; the London Concert is a duo performance by Professor Roberts. Roberts toured with Willie Tee and the Wild Magnolias off and on from 1980 until 1986, when he took a hiatus from the music industry coming out of his semi-retirement to tour and record - such as on Dr. John’s albums Goin' Back to New Orleans and Dis, Dat, or D’udda. One evening in 2007, Roberts was invited by the young funk band Groovesect to join them on stage at New Orleans' Maple Leaf Bar, starting a relationship that would result in Roberts joining the band and recording on Groovesect's debut album, On The Brim.

He performed at The Bloomington Blues & Boogie Woogie Piano Festival, in Bloomington, Indiana, in 2017 and 2018. Uganda Roberts has recorded commercially with John Mooney. Uganda Roberts has performed as himself on the television series "Treme". Official site Discography Ogden Museum NOIMC Innovation Award Audio - Alfred "Uganda" Roberts And His Drums