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Bicyclo[1.1.0]butane is an organic compound with the formula C 4 H 6. It is a bicyclic molecule consisting of two cis-fused cyclopropane rings, and is a colorless and easily condensed gas. [1] Bicyclobutane is noted for being one of the most strained compounds that is isolatable on a large scale — its strain energy is estimated at 63.9 kcal ...
Cyclobutane-1,3-diyl is the planar four-membered carbon ring species with radical character localized at the 1 and 3 positions. The singlet cyclobutane-1,3-diyl is predicted to be the transition state for the ring inversion of bicyclobutane, proceeding via homolytic cleavage of the transannular carbon-carbon bond (Figure 3).
This image of a simple structural formula is ineligible for copyright and therefore in the public domain, because it consists entirely of information that is common property and contains no original authorship.
Since a binomial coefficient is always an integer, the nth binomial coefficient is divisible by p and hence equal to 0 in the ring. We are left with the zeroth and pth coefficients, which both equal 1, yielding the desired equation. Thus in characteristic p the freshman's dream is a valid identity.
ATS theorem (number theory) Abel's binomial theorem (combinatorics) Abel's curve theorem (mathematical analysis) Abel's theorem (mathematical analysis) Abelian and Tauberian theorems (mathematical analysis) Abel–Jacobi theorem (algebraic geometry) Abel–Ruffini theorem (theory of equations, Galois theory) Abhyankar–Moh theorem (algebraic ...
For any pair of positive integers n and k, the number of k-tuples of non-negative integers whose sum is n is equal to the number of multisets of size k − 1 taken from a set of size n + 1, or equivalently, the number of multisets of size n taken from a set of size k, and is given by
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In mathematics, in the field of abstract algebra, the structure theorem for finitely generated modules over a principal ideal domain is a generalization of the fundamental theorem of finitely generated abelian groups and roughly states that finitely generated modules over a principal ideal domain (PID) can be uniquely decomposed in much the same way that integers have a prime factorization.