The evaluations of determinants with Legendre symbol entries have close relation combinatorics and character sums over finite fields. Recently, Sun [9] posed some conjectures on this topic. In paper, we prove also study variants. For example, show the following result:

Background • Legendre Sequence For a prime p > 2 let (s n) be the Legendre sequence defined as s n = 1, n p = −1, 0, otherwise, n ≥ 0, where. p denotes the Legendre symbol. • Sidelnikov Sequence Let q be an odd prime power, g a primitive element of F q , and let η denote the quadratic character of F Background • Legendre Sequence For a prime p > 2 let (s n) be the Legendre sequence defined as s...

f((1− t)x1 + tx2) ≤ (1− t)f(x1) + tf(x2), x1, x2 ∈ C, 0 ≤ t ≤ 1, then f : X → R is convex. Proof. Let (x1, α1), (x2, α2) ∈ epi f and 0 ≤ t ≤ 1. The fact that the pairs (xi, αi) belong to epi f means in particular that f(xi) < ∞, and hence that xi ∈ C, as otherwise we would have f(xi) =∞. But (1− t)(x1, α1) + t(x2, α2) = ((1− t)x1 + tx2, (1− t)α1 + tα2), and, as x1, x2 ∈ C, f((1− t)x1 + tx2) ≤ (...

The general framework of Legendre transformation is extended to the case of symplectic groupoids, using an appropriate generalization of the notion of generating function (of a Lagrangian submanifold). 1 Tulczyjew triple and its generalization The general framework of Legendre transformation was introduced by Tulczyjew [1]. It consists in recognition of the following structure, which we call th...

We introduce a version of the calculus of variations on time scales, which includes as special cases the classical calculus of variations and the discrete calculus of variations. Necessary conditions for weak local minima are established, among them the Euler condition, the Legendre condition, the strengthened Legendre condition, and the Jacobi condition. AMS (MOS) Subject Classification. 39A10.