We derive a simple formula for the second-order subdifferential of the maximum of coordinates which allows us to construct this set immediately from its argument and the direction to which it is applied. This formula can be combined with a chain rule recently proved by Mordukhovich and Rockafellar [9] in order to derive a similarly simple formula for the extended partial second-order subdiffere...

We provide a simple and exact formula for the minimum Miller loop length in Atei pairing based on Brezing-Weng curves, in terms of the involved parameters, under a mild condition on the parameters. It will be also shown that almost all cryptographically useful/meaningful parameters satisfy the mild condition. Hence the simple and exact formula is valid for them. It will also turn out that the f...

Using generating functions, we derive a multiple point formula for every generic immersion f : M # N between even dimensional oriented manifolds. This produces explicit formulas for the signature and Pontrjagin numbers of the multiple point manifolds. The formula takes a particular simple form in many special cases, e.g. when f is nullhomotopic we recover Szűcs’s formula in [3]. It also include...

— We discover another one-parameter generalization of Postnikov's hook length formula for binary trees. The particularity of our formula is that the hook length h v appears as an exponent. As an application, we derive another simple hook length formula for binary trees when the underlying parameter takes the value 1/2.

A classical result of MacMahon gives a simple product formula for the generating function of major index over the symmetric group. A similar factorial-type product formula for the generating function of major index together with sign was given by Gessel and Simion. Several extensions are given in this paper, including a recurrence formula, a specialization at roots of unity and type B analogues.

Among various approximation formulas for the gamma function, Smith showed that [Formula: see text] which is a little-known but accurate and simple one. In this note, we prove that the function [Formula: see text] is strictly increasing and concave on [Formula: see text], which shows that Smith's approximation is just an upper one.

We propose and prove the Verlinde formula for the quantization of the Higgs bundle moduli spaces and stacks for any simple and simply-connected group. This generalizes the equivariant Verlinde formula for the case of SU(n) proposed in [GP]. We further establish a Verlinde formula for the quantization of parabolic Higgs bundle moduli spaces and stacks.

We present a new method for generating a [Formula: see text]-surface from a triangular network of compatible surface strips. The compatible surface strips are given by a network of polynomial curves with an associated implicitly defined surface, which fulfill certain compatibility conditions. Our construction is based on a new concept, called bubble patches, to represent the single surface patc...