We prove versions of various standard inequalities in which the dependence of the constant on the metric is explicit. 1 Technical results Definition. Let (M, g) be a smooth Riemannian n-manifold and let x ∈ M . Given Q > 1, k ∈ N, and p > n, the (Q, k, p)-harmonic radius at x, rH(Q, k, p)(x), is the supremum of reals r such that, on the geodesic ball Bx(r) of center x and radius r, there is a h...

A (k; n)-arc in PG(2; q) is usually deened to be a set K of k points in the plane such that some line meets K in n points but such that no line meets K in more than n points. There is an extensive literature on the topic of (k; n)-arcs. Here we keep the same deenition but allow K to be a multiset, that is, permit K to contain multiple points. The case k = q 2 + q + 2 is of interest because it i...

Usually creative telescoping is used to derive recurrences for sums. In this article we show that the non-existence of a creative telescoping solution, and more generally, of a parameterized telescoping solution, proves algebraic independence of certain types of sums. Combining this fact with summation-theory shows transcendence of whole classes of sums. Parameterized telescoping Given f1(k), ....

A set of points, S ⊆ PG(r, q), is said to be %-saturating if, for any point x ∈ PG(r, q), there exist %+ 1 points in S that generate a subspace in which x lies. The cardinality of a smallest possible set S with this property is denoted by k(r, q, %). We give a short survey of what is known about k(r, q, 1) and present new results for k(r, q, 2) for small values of r and q. One construction pres...

We study the combinatorial function L(k, q), the maximum number of nonzero weights a linear code of dimension k over Fq can have. We determine it completely for q = 2, and for k = 2, and provide upper and lower bounds in the general case when both k and q are ≥ 3. A refinement L(n, k, q), as well as nonlinear analogues N(M, q) and N(n,M, q), are also introduced and studied.

We consider the Grassmann graph of k-dimensional subspaces an n-dimensional vector space over q-element field and its subgraph Γ(n,k)q which consists all non-degenerate linear [n,k]q codes. assume that 1<k<n−1. It is well-known every automorphism induced by a semilinear corresponding or isomorphism to dual second possibility realized only for n=2k. Our result following: if q≥3 k≠2, then can be ...

The purpose of this paper is to construct q-Euler numbers and polynomials by using p-adic q-integral equations on Z p. Finally, we will give some interesting formula related to these q-Euler numbers and polynomials. The usual Bernoulli numbers are defined by ∞ k=0 B k t k k! = t e t − 1 , which can be written symbolically as e Bt = t e t −1 , interpreted to mean B k must be replaced by B k when...

(0, 1) [628]. (0, 2) [962]. (0, 2, t) [637]. (0, α) [696, 844]. (1,−1) [518]. (1, 2) [1269]. (17, 9) [351]. (17q, 17, 2) [364]. (2) [1198]. (2, 2) [1179]. (2, 2) [1431]. (2, 7) [1432]. (2, 8) [667, 1133]. (2, n) [904]. (2, q) [1432]. (2, q) [1231]. (255, k) [657]. (25q, 25, 3) [364]. (28, 12, 11) [117]. (2) [1270]. (2) [1452]. (3, 4) [1788, 1872]. (3, 5∗, v) [578]. (3, 8) [667]. (3, p) [635]. (...

The modulus of quasipositivity q(K) of a knot K was introduced as a tool in the knot theory of complex plane curves, and can be applied to Legendrian knot theory in symplectic topology. It has also, however, a straightforward characterization in ordinary knot theory: q(K) is the supremum of the integers f such that the framed knot (K, f) embeds non-trivially on a fiber surface of a positive tor...