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[3] [4] [5] Despite being differentiable nowhere, the function is continuous: Since the terms of the infinite series which defines it are bounded by ± a n {\textstyle \pm a^{n}} and this has finite sum for 0 < a < 1 {\textstyle 0<a<1} , convergence of the sum of the terms is uniform by the Weierstrass M-test with M n = a n {\textstyle M_{n}=a ...
In the field of representation theory in mathematics, a projective representation of a group G on a vector space V over a field F is a group homomorphism from G to the projective linear group = /, where GL(V) is the general linear group of invertible linear transformations of V over F, and F ∗ is the normal subgroup consisting of nonzero scalar multiples of the identity transformation (see ...
Its value is +1 if n is the product of an even number of prime numbers, and −1 if it is the product of an odd number of primes. Explicitly, the fundamental theorem of arithmetic states that any positive integer n can be represented uniquely as a product of powers of primes: n = p 1 a 1 ⋯ p k a k , where p 1 < p 2 < ... < p k are primes and ...
u n = 3u n−1 − 2u n−2 with initial values u 0 = −1, u 1 = 0. [11] I have found another recurrence sequence that seems to possess the same property; it is the one whose general term is v n = v n−2 + v n−3 with initial values v 0 = 3, v 1 = 0, v 2 = 2.
Quartic or biquadratic reciprocity is a collection of theorems in elementary and algebraic number theory that state conditions under which the congruence x 4 ≡ p (mod q) is solvable; the word "reciprocity" comes from the form of some of these theorems, in that they relate the solvability of the congruence x 4 ≡ p (mod q) to that of x 4 ≡ q (mod p).
In mathematics, the silver ratio is a geometrical proportion close to 70/29.Its exact value is 1 + √2, the positive solution of the equation x 2 = 2x + 1.. The name silver ratio results from analogy with the golden ratio, the positive solution of the equation x 2 = x + 1.
1921: Pi Student Chapter was formed in Toronto thereby expanding Alpha Omega into an international scope; 1924: First non-US alumni chapter was founded in Toronto; 1932: (October 7), merger with Alpha Zeta Gamma completed with chartering of Alpha Kappa and Alpha Lambda. 1936: Alpha Omega establishes the Achievement Medal.