Elementary symmetric polynomialIn mathematics, specifically in commutative algebra, the elementary symmetric polynomials are one type of basic building block for symmetric polynomials, in the sense that any symmetric polynomial can be expressed as a polynomial in elementary symmetric polynomials. That is, any symmetric polynomial P is given by an expression involving only additions and multiplication of constants and elementary symmetric polynomials.
Tangent half-angle formulaIn trigonometry, tangent half-angle formulas relate the tangent of half of an angle to trigonometric functions of the entire angle. The tangent of half an angle is the stereographic projection of the circle onto a line. Among these formulas are the following: From these one can derive identities expressing the sine, cosine, and tangent as functions of tangents of half-angles: Using double-angle formulae and the Pythagorean identity gives Taking the quotient of the formulae for sine and cosine yields Combining the Pythagorean identity with the double-angle formula for the cosine, rearranging, and taking the square roots yields and which, upon division gives Alternatively, It turns out that the absolute value signs in these last two formulas may be dropped, regardless of which quadrant α is in.
Logarithmic derivativeIn mathematics, specifically in calculus and complex analysis, the logarithmic derivative of a function f is defined by the formula where is the derivative of f. Intuitively, this is the infinitesimal relative change in f; that is, the infinitesimal absolute change in f, namely scaled by the current value of f. When f is a function f(x) of a real variable x, and takes real, strictly positive values, this is equal to the derivative of ln(f), or the natural logarithm of f.
Wirtinger derivativesIn complex analysis of one and several complex variables, Wirtinger derivatives (sometimes also called Wirtinger operators), named after Wilhelm Wirtinger who introduced them in 1927 in the course of his studies on the theory of functions of several complex variables, are partial differential operators of the first order which behave in a very similar manner to the ordinary derivatives with respect to one real variable, when applied to holomorphic functions, antiholomorphic functions or simply differentiabl
BelgiumBelgium, officially the Kingdom of Belgium, is a country in Northwestern Europe. The country is bordered by the Netherlands to the north, Germany to the east, Luxembourg to the southeast, France to the southwest, and the North Sea to the northwest. It covers an area of and has a population of more than 11.5 million, making it the 22nd most densely populated country in the world and the 6th most densely populated country in Europe, with a density of .
European UnionThe European Union (EU) is a supranational political and economic union of member states that are located primarily in Europe. The union has a total area of and an estimated total population of over 448 million. The EU has often been described as a sui generis political entity (without precedent or comparison) combining the characteristics of both a federation and a confederation. Containing 5.
AphidAphids are small sap-sucking insects and members of the superfamily Aphidoidea. Common names include greenfly and blackfly, although individuals within a species can vary widely in color. The group includes the fluffy white woolly aphids. A typical life cycle involves flightless females giving live birth to female nymphs—who may also be already pregnant, an adaptation scientists call telescoping generations—without the involvement of males. Maturing rapidly, females breed profusely so that the number of these insects multiplies quickly.
BrexitBrexit (ˈbrɛksᵻt,_ˈbrɛgzᵻt; portmanteau of "British exit") was the withdrawal of the United Kingdom (UK) from the European Union (EU) at 23:00 GMT on 31 January 2020 (00:00 1 February 2020 CET). The UK is the only sovereign country to have left the EU. The UK had been a member state of the EU or its predecessor the European Communities (EC), sometimes both at the same time, since 1 January 1973. Following Brexit, EU law and the Court of Justice of the European Union no longer have primacy over British laws.
De factoDe facto (deɪ_ˈfæktoʊ,_di-,_də- ; de facto deː ˈfaktoː, in fact) describes practices that exist in reality, regardless of whether they are officially recognized by laws or other formal norms. It is commonly used to refer to what happens in practice, in contrast with de jure ("by law"), which refers to things that happen according to official law, regardless of whether the practice exists in reality. In jurisprudence, it mainly means "practiced, but not necessarily defined by law" or "practiced or is valid, but not officially established".
International System of UnitsThe International System of Units, internationally known by the abbreviation SI (for Système International), is the modern form of the metric system and the world's most widely used system of measurement. Established and maintained by the General Conference on Weights and Measures (CGPM), it is the only system of measurement with an official status in nearly every country in the world, employed in science, technology, industry, and everyday commerce.
00 (zero) is a number representing an empty quantity. As a number, 0 fulfills a central role in mathematics as the additive identity of the integers, real numbers, and other algebraic structures. In place-value notation such as decimal, 0 also serves as a numerical digit to indicate that that position's power of 10 is not multiplied by anything or added to the resulting number. This concept appears to have been difficult to discover. Common names for the number 0 in English are zero, nought, naught (nɔːt), nil.
Silver nitrateSilver nitrate is an inorganic compound with chemical formula AgNO3. It is a versatile precursor to many other silver compounds, such as those used in photography. It is far less sensitive to light than the halides. It was once called lunar caustic because silver was called luna by ancient alchemists who associated silver with the moon. In solid silver nitrate, the silver ions are three-coordinated in a trigonal planar arrangement. Albertus Magnus, in the 13th century, documented the ability of nitric acid to separate gold and silver by dissolving the silver.
