Abstract:
We obtain definite integrals for products of associated Legendre functions with Bessel functions, associated Legendre functions, and Chebyshev polynomials of the first kind using orthogonality and integral transforms.

Abstract:
We obtain definite integrals for products of associated Legendre functions with Bessel functions, associated Legendre functions, and Chebyshev polynomials of the first kind using orthogonality and integral transforms.

Abstract:
A truncated trigonometric, operator-valued moment problem in section 3 of this note is solved. Let be a finite sequence of bounded operators, with arbitrary, acting on a finite dimensional Hilbert space H. A necessary and sufficient condition on the positivity of an operator kernel for the existence of an atomic, positive, operator-valued measure , with the property that for every with , the moment of coincides with the term of the sequence, is given. The connection between some positive definite operator-valued kernels and the Riesz-Herglotz integral representation of the analytic on the unit disc, operator-valued functions with positive real part in the class of operators in Section 4 of the note is studied.

Abstract:
In this article it is proven the existence of integration of indefinite integrals as infinite derivative's series expansion. This also opens a new way to integrate a definite integral.

Abstract:
It is shown that any primitive integral Apollonian circle packing captures a fraction of the prime numbers. Basically the method consists in applying the circle method, considering the curvatures produced by a well-chosen family of binary quadratic forms.

Abstract:
For an integral convex polytope $\Pc \subset \RR^N$ of dimension $d$, we call $\delta(\Pc)=(\delta_0, \delta_1,..., \delta_d)$ the $\delta$-vector of $\Pc$ and $\vol(\Pc)=\sum_{i=0}^d\delta_i$ its normalized volume. In this paper, we will establish the new equalities and inequalities on $\delta$-vectors for integral simplices whose normalized volumes are prime. Moreover, by using those, we will classify all the possible $\delta$-vectors of integral simplices with normalized volume 5 and 7.

Abstract:
We develop the affine sieve in the context of orbits of congruence subgroups of semi-simple groups acting linearly on affine space. In particular we give effective bounds for the saturation numbers for points on such orbits at which the value of a given polynomial has few prime factors. In many cases these bounds are of the same quality as what is known in the classical case of a polynomial in one variable and the orbit is the integers. When the orbit is the set of integral matrices of a fixed determinant we obtain a sharp result for the saturation number, and thus establish the Zariski density of matrices all of whose entries are prime numbers. Among the key tools used are explicit approximations to the generalized Ramanujan conjectures for such groups and sharp and uniform counting of points on such orbits when ordered by various norms.

Abstract:
We give non-torsion counterexamples against the integral Tate conjecture for finite fields. We extend the result due to Pirutka and Yagita for prime numbers 2,3,5 to all prime numbers.