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The Maharashtra State Board of Secondary and Higher Secondary Education (Abbreviation: MSBSHSE) is a statutory and autonomous body established under the "Maharashtra Secondary Boards Act" 1965 (amended in 1977). [1] Most important task of the board, among few others, is to conduct the SSC for 10th class and HSC for 12th class examinations. [2]
For some other divergent geometric series, including Grandi's series with ratio −1, and the series 1 + 2 + 4 + 8 + ⋯ with ratio 2, one can use the general solution for the sum of a geometric series with base 1 and ratio , obtaining , but this summation method fails for 1 + 1 + 1 + 1 + ⋯, producing a division by zero.
The sum is taken over all combinations of nonnegative integer indices k 1 through k m such that the sum of all k i is n. That is, for each term in the expansion, the exponents of the x i must add up to n. [1] [a] In the case m = 2, this statement reduces to that of the binomial theorem. [1]
In mathematics, the floor function is the function that takes as input a real number x, and gives as output the greatest integer less than or equal to x, denoted ⌊x⌋ or floor(x). Similarly, the ceiling function maps x to the least integer greater than or equal to x, denoted ⌈x⌉ or ceil(x). [1]
It will introduce students to the more abstract concepts in subjects of mathematics, sciences, social sciences, arts and humanities. Secondary Stage: Classes 9 to 12, covering the ages of 14–18 years. It is again subdivided into two parts: classes 9 and 10 covering the first phase while classes 11 and 12 covering the second phase.
Lefschetz proved that any normal function determined a class in H 2 (X, Z) and that the class of ν Γ is the fundamental class of Γ. Furthermore, he proved that a class in H 2 (X, Z) is the class of a normal function if and only if it lies in H 1,1. Together with Poincaré's existence theorem, this proves the theorem on (1,1)-classes.
The curl of the gradient of any continuously twice-differentiable scalar field (i.e., differentiability class) is always the zero vector: =. It can be easily proved by expressing ∇ × ( ∇ φ ) {\displaystyle \nabla \times (\nabla \varphi )} in a Cartesian coordinate system with Schwarz's theorem (also called Clairaut's theorem on equality ...
Wiles had the insight that in many cases this ring homomorphism could be a ring isomorphism (Conjecture 2.16 in Chapter 2, §3 of the 1995 paper [4]). He realised that the map between R {\displaystyle R} and T {\displaystyle \mathbf {T} } is an isomorphism if and only if two abelian groups occurring in the theory are finite and have the same ...