15,861 research outputs found
Relationships Between Generalized Bernoulli Numbers and Polynomials and Generalized Euler Numbers and Polynomials
In this paper, concepts of the generalized Bernoulli and Euler numbers and polynomials are introduced, and some relationships between them are established
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
The Dominant Eigenvalue of an Essentially Nonnegative Tensor
It is well known that the dominant eigenvalue of a real essentially
nonnegative matrix is a convex function of its diagonal entries. This convexity
is of practical importance in population biology, graph theory, demography,
analytic hierarchy process and so on. In this paper, the concept of essentially
nonnegativity is extended from matrices to higher order tensors, and the
convexity and log convexity of dominant eigenvalues for such a class of tensors
are established. Particularly, for any nonnegative tensor, the spectral radius
turns out to be the dominant eigenvalue and hence possesses these convexities.
Finally, an algorithm is given to calculate the dominant eigenvalue, and
numerical results are reported to show the effectiveness of the proposed
algorithm
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