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Let S be a semigroup. In the present paper, we determine the complex-valued solutions (f, g) of the functional equation g(xσ(y)) = g(x)g(y) − f(x)f(y) + αf(xσ(y)), x, y ∈ S, where σ : S → S is an automorphism that need not be involutive, and α ∈ C is a fixed constant. Our results generalize and extend the ones by Stetkær in The cosine addition law with an additional term. Aequat Math., no. 6, 90, 1147-1168 (2016), and also the ones by Aserrar and Elqorachi in A generalization of the cosine addition law on semigroups. Aequat Math. 97, 787–804 (2023). Some consequences of our results are presented. [ABSTRACT FROM AUTHOR]

Let S be a semigroup. Our main results are that we describe the complex-valued solutions of the following functional equations g (x σ (y)) = g (x) g (y) + f (x) f (y) , x , y ∈ S , f (x σ (y)) = f (x) g (y) + f (y) g (x) , x , y ∈ S , and f (x σ (y)) = f (x) g (y) - f (y) g (x) , x , y ∈ S , where σ : S → S is an automorphism that need not be involutive. As a consequence we show that the first two equations are equivalent to their variants. We also give some applications. [ABSTRACT FROM AUTHOR]

In the present paper reality conditions for quasi-periodic solutions of the KdV equation are determined completely. As a result, solutions in the form of non-linear waves can be plotted and investigated. The full scope of obtaining finite-gap solutions of the KdV equation is presented. It is proven that the multiply periodic ℘ 1 , 1 -function on the Jacobian variety of a hyperelliptic curve of arbitrary genus serves as the finite-gap solution, the genus coincides with the number of gaps. The subspace of the Jacobian variety where ℘ 1 , 1 , as well as other ℘-functions, are bounded and real-valued is found in any genus. This result covers every finite phase space of the KdV hierarchy, and can be extended to other completely integrable equations. A method of effective computation of this type of solutions is suggested, and illustrated in genera 2 and 3. [ABSTRACT FROM AUTHOR]

Twisted Edwards curves over finite fields have attracted great interest for their efficient and unified addition formula. In this paper, we consider twisted Edwards curves over local fields and introduce a cryptosystem based on quotient groups of twisted Edwards curves over local fields. From the study of formal groups of twisted Edwards curves and twisted Edwards curves over local fields, we give the choice of cryptographic groups. An element in these groups can be uniformly represented by two n digit p-adic numbers, whereas an element in the elliptic curves in Weierstrass form over local fields is represented by a 3 n − 2 digit p-adic number and a 4 n−3 digit p-adic number. In the cryptography on elliptic curves in Weierstrass form over local fields, five cases for different input point pairs in computing points addition have to be considered and sometimes points have to be lifted. In the cryptography on twisted Edwards curves over local fields, the addition formula is simple, unified, and complete, which is efficient, does not need lifting points, and is against the side channel analysis. Finally, a speedy point multiplication algorithm and some concrete instances are given. [ABSTRACT FROM AUTHOR]

We revisit the addition law for expectations and present a sibling law: the absolute law for expectations. We show that these two laws and their corresponding laws for probabilities can be reconciled under a single framework. As an application, we use the absolute law for expectations to calculate the mean absolute deviation. Finally, we remark on a hidden point in a related article previously published on these pages; this will help readers to avoid a potential pitfall. [ABSTRACT FROM AUTHOR]