1.
Group theory
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In mathematics and abstract algebra, group theory studies the algebraic structures known as groups. Groups recur throughout mathematics, and the methods of group theory have influenced many parts of algebra, linear algebraic groups and Lie groups are two branches of group theory that have experienced advances and have become subject areas in their own right. Various physical systems, such as crystals and the hydrogen atom, thus group theory and the closely related representation theory have many important applications in physics, chemistry, and materials science. Group theory is central to public key cryptography. The first class of groups to undergo a systematic study was permutation groups, given any set X and a collection G of bijections of X into itself that is closed under compositions and inverses, G is a group acting on X. If X consists of n elements and G consists of all permutations, G is the symmetric group Sn, in general, an early construction due to Cayley exhibited any group as a permutation group, acting on itself by means of the left regular representation. In many cases, the structure of a group can be studied using the properties of its action on the corresponding set. For example, in this way one proves that for n ≥5 and this fact plays a key role in the impossibility of solving a general algebraic equation of degree n ≥5 in radicals. The next important class of groups is given by matrix groups, here G is a set consisting of invertible matrices of given order n over a field K that is closed under the products and inverses. Such a group acts on the vector space Kn by linear transformations. In the case of groups, X is a set, for matrix groups. The concept of a group is closely related with the concept of a symmetry group. The theory of groups forms a bridge connecting group theory with differential geometry. A long line of research, originating with Lie and Klein, the groups themselves may be discrete or continuous. Most groups considered in the first stage of the development of group theory were concrete, having been realized through numbers, permutations, or matrices. It was not until the nineteenth century that the idea of an abstract group as a set with operations satisfying a certain system of axioms began to take hold. A typical way of specifying an abstract group is through a presentation by generators and relations, a significant source of abstract groups is given by the construction of a factor group, or quotient group, G/H, of a group G by a normal subgroup H. Class groups of algebraic number fields were among the earliest examples of factor groups, of much interest in number theory

2.
Density of air
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The density of air is the mass per unit volume of Earths atmosphere. Air density, like air pressure, decreases with increasing altitude and it also changes with variation in temperature or humidity. At sea level and at 15 °C air has a density of approximately 1.225 kg/m3 according to ISA.058 J/ in SI units and this quantity may vary slightly depending on the molecular composition of air at a particular location. Therefore, At IUPAC standard temperature and pressure, dry air has a density of 1.2754 kg/m3, at 20 °C and 101.325 kPa, dry air has a density of 1.2041 kg/m3. At 70 °F and 14.696 psi, dry air has a density of 0.074887 lb/ft3 and this occurs because the molar mass of water is less than the molar mass of dry air. For any gas, at a temperature and pressure, the number of molecules present is constant for a particular volume. So when water molecules are added to a volume of air. Hence the mass per volume of the gas decreases. The density of air may be calculated as a mixture of ideal gases. In this case, the pressure of water vapor is known as the vapor pressure. One formula used to find the saturation pressure is, p s a t =6.1078 ×107.5 T T +237.3 where T = is in degrees C. Despite minor differences to define all formulations the predicted mass of dry air. Importantly, some of the examples are not normalized so that the composition is equal to unity, air Density Atmosphere of Earth International Standard Atmosphere U. S