Abstract. Let p > 3 be a prime, and let m be an integer with p ∤ m. In the paper we solve some conjectures of Z.W. Sun concerning Pp−1 k=0 2k k 3 /mk (mod p2), Pp−1 k=0 2k k 4k 2k /mk (mod p) and Pp−1 k=0 2k k 2 4k 2k /mk (mod p2). In particular, we show that P p−1 2 k=0 2k k 3 ≡ 0 (mod p2) for p ≡ 3, 5, 6 (mod 7). Let {Pn(x)} be the Legendre polynomials. In the paper we also show that P[ p 4 ]...

Let R ∗ = R2k+1 \ {~0} (k ≥ 1) and π: R ∗ → S2k be the map sending ~r ∈ R ∗ to ~r |~r| ∈ S 2k . Denote by P → R ∗ the pullback by π of the canonical principal SO(2k)-bundle SO(2k + 1) → S2k . Let E] → R ∗ be the associated co-adjoint bundle and E] → T ∗R ∗ be the pullback bundle under projection map T ∗R ∗ → R ∗ . The canonical connection on SO(2k + 1) → S2k turns E] into a Poisson manifold. Th...

and Applied Analysis 3 Thus, the origin of system (13) is an element critical point. It could be investigated using the classical integral factor method. Now, we consider the following system: dx dt = y + A30x3n + A21x2ny + A12xny2 + A03y3, dy dt = −x2n−1 + xn−1 (B30x3n + B21x2ny + B12xny2 + B03y3) . (14) When n = 2k + 1, by those transformations, system (14) is changed into dx dt = −y − √2k + ...

Gyarfas, A., Graphs with k odd cycle lengths, Discrete Mathematics 103 (1992) 41-48. 41 If G is a graph with k ~ 1 odd cycle lengths then each block of G is either K2k+Z or contains a vertex of degree at most 2k. As a consequence, the chromatic number of G is at most 2k + 2. For a graph G let L(G) denote the set of odd cycle lengths of G, i.e., L( G) = {2i + 1: G contains a cycle of length 2i +...

We prove an incidence theorem for points and curves in the complex plane. Given a set of m points in R and a set of n curves with k degrees of freedom, Pach and Sharir proved that the number of point-curve incidences is O ( m k 2k−1n 2k−2 2k−1 +m+n ) . We establish the slightly weaker bound Oε ( m k 2k−1 n 2k−2 2k−1 +m+n ) on the number of incidences between m points and n (complex) algebraic c...

An L(d,1)-labeling of a graph G is an assignment of nonnegative integers to the vertices such that adjacent vertices receive labels that differ by at least d and those at a distance of two receive labels that differ by at least one, where d 1. Let d1 (G) denote the least such that G admits an L(d,1)-labeling using labels from {0, 1, . . . , }. We prove that (i) if d 1, k 2 and m0, . . . , mk−1 ...

and Applied Analysis 3 Proposition 8 (see [33]). The Bernoulli’s polynomial Bj(x) satisfies the following properties: (i) B 0 (x) = 1, B 1 (x) = x − 1/2, B k (x) = ∑ k j=0 C j k B j x , (ii) B 2k+1 (1) = B 2k+1 (0) = B 2k+1 = 0, (iii) B 2k (1) = B 2k (0) = B 2k , (iv) B k (x) = (1/(k + 1))B 󸀠 k+1 (x), k = 1, 2, . . . . 3.2. The Euler-Maclaurin Method and Discretization. Let q > 0 be an integer,...

Let T be a hamiltonian tournament with n vertices and a hamiltonian cycle of T. For a cycle C k of length k in T we denote I (C k) = jA() \ A(C k)j, the number of arcs that and C k have in common. Let f(n; k; T;) = maxfI (C k)jC k Tg and f(n; k) = minff(n; k; T;)jT is a hamiltonian tournament with n vertices, and a hamiltonian cycle of Tg. In a previous paper 3] we studied the case of n 2k ? 4 ...

We first give a proof of Lemma 2.1 which bounds the error of the (t + 1)-st iterate (ψ(X t+1)) in terms of the error incurred by the t-th iterate and the optimal solution. Proof of Lemma 2.1 Recall that ψ(X) = 1 2 ∥A(X) − b∥ 2 2. Since ψ(·) is a quadratic function, we have ψ(X t+1) − ψ(X t) = ⟨∇ψ(X t), X t+1 − X t ⟩ + 1 2 ∥A(X t+1 − X t)∥ 2 2 ≤ ⟨A T (A(X t) − b), X t+1 − X t ⟩ + 1 2 · (1 + δ 2k...