This article constructs examples of associative submanifolds in $G_2$-manifolds obtained by resolving $G_2$-orbifolds using Joyce's generalised Kummer construction. As the approach $G_2$-orbifolds, volume tends to zero. partially verifies a prediction due Halverson and Morrison.

In [1], the multiple Frobenius-Euler numbers and polynomials were constructed. In this paper we give some interesting formulae which are related to the multiple Frobenius-Euler polynomials. The main purpose of this paper is to give the Kummer type congruences for the multiple Frobenius-Euler numbers. §

It is shown that a known asymptotic expansion of the Kummer function U(a, b, z) as a tends to infinity is valid for z on the full Riemann surface of the logarithm. A corresponding result is also proved in a more general setting considered by Olver (1956).

Let {φk}k=0, n < m, be a family of polynomials orthogonal with respect to the positive semi-definite bilinear form

We prove that the complex cobordism class of any hyper-K\{a}hler manifold dimension $2n$ is a unique combination with rational coefficients classes products punctual Hilbert schemes $K3$ surfaces. also similar result using generalized Kummer varieties instead schemes. As key step, we establish closed formula for top Chern character their tangent bundles.

In this paper, we study the discrete logarithm problem in the finite fields Fqn where n|q−1. The field is called a Kummer field or a Kummer extension of Fq. It plays an important role in improving the AKS primality proving algorithm. It is known that we can efficiently construct an element g with order greater than 2 in the fields. Let Sq(•) be the function from integers to the sum of digits in...