84,838 research outputs found
Derivatives of tangent function and tangent numbers
In the paper, by induction, the Fa\`a di Bruno formula, and some techniques
in the theory of complex functions, the author finds explicit formulas for
higher order derivatives of the tangent and cotangent functions as well as
powers of the sine and cosine functions, obtains explicit formulas for two Bell
polynomials of the second kind for successive derivatives of sine and cosine
functions, presents curious identities for the sine function, discovers
explicit formulas and recurrence relations for the tangent numbers, the
Bernoulli numbers, the Genocchi numbers, special values of the Euler
polynomials at zero, and special values of the Riemann zeta function at even
numbers, and comments on five different forms of higher order derivatives for
the tangent function and on derivative polynomials of the tangent, cotangent,
secant, cosecant, hyperbolic tangent, and hyperbolic cotangent functions.Comment: 17 page
Compact K\"ahler manifolds with automorphism groups of maximal rank
For an automorphism group G on an n-dimensional (n > 2) normal projective
variety or a compact K\"ahler manifold X so that G modulo its subgroup N(G) of
null entropy elements is an abelian group of maximal rank n-1, we show that
N(G) is virtually contained in Aut_0(X), the X is a quotient of a complex torus
T and G is mostly descended from the symmetries on the torus T, provided that
both X and the pair (X, G) are minimal.Comment: Added Hypothesis (C) to Theorem 1.2. No change of the proof
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