let g be a (p,q) graph. an injective map f : e(g) → {±1,±2,...,±q} is said to be an edge pair sum labeling if the induced vertex function f*: v (g) → z - {0} defined by f*(v) = σp∈ev f (e) is one-one where ev denotes the set of edges in g that are incident with a vertex v and f*(v (g)) is either of the form {±k1,±k2,...,±kp/2} or {±k1,±k2,...,±k(p-1)/2} u {±k(p+1)/2} according a...

A set of words X is called unavoidable on a given alphabet A if every infinite word on A has a factor in X. For k, q ≥ 1, let c(k, q) be the number of conjugacy classes of words of length k on q letters. An unavoidable set of words of length k on q symbols has at least c(k, q) elements. We show that for any k, q ≥ 1 there exists an unavoidable set of words of length k on q symbols having c(k, q...

We consider the function m[k](q) that counts the number of cycle permutations of a finite field Fq of fixed length k such that their permutation polynomial has the smallest possible degree. We prove the upper–bound m[k](q) ≤ (k−1)!(q(q−1))/k for char(Fq) > e(k−3)/e and the lower–bound m[k](q) ≥ φ(k)(q(q−1))/k for q ≡ 1 (mod k). This is done by establishing a connection with the Fq–solutions of ...

In the “lost notebook”, Ramanujan recorded infinite product expansions for 1 √ r − ( 1−√5 2 )√ r and 1 √ r − ( 1 + √ 5 2 )√ r, where r = r(q) is the Rogers–Ramanujan continued fraction. We shall give analogues of these results that involve Ramanujan’s function k = k(q) = r(q)r(q).

We show that the q-Kneser graph q K2k:k (the graph on the k-subspaces of a 2k-space over G F(q), where two k-spaces are adjacent when they intersect trivially), has chromatic number qk + qk−1 for k = 3 and for k < q log q − q . We obtain detailed results on maximal cocliques for k = 3.

Let (− k 2)u = −u + q(x)u − k 2 u = δ(x), x ∈ R, ∂u ∂|x| − iku → 0, |x| → ∞. Assume that the potential q(x) is real-valued and compactly supported: q(x) = q(x), q(x) = 0 for |x| ≥ 1, 1 −1 |q|dx < ∞, and that q(x) produces no bound states. Let u(−1, k) and u(1, k) ∀k > 0 be the data. Theorem.Under the above assumptions these data determine q(x) uniquely.

If f(x, k) is the Jost solution and f(k) = f(0, k), then the I-function is I(k) := f ′(0, k)/f(k). It is proved that I(k) is in one-to-one correspondence with the scattering triple S := {S(k), kj , sj , 1 ≤ j ≤ J} and with the spectral function ρ(λ) of the Sturm-Liouville operator l = −d/dx + q(x) on (0,∞) with the Dirichlet condition at x = 0 and q(x) ∈ L1,1 := {q | q = q, ∫∞ 0 (1 + x)|q(x)dx ...

If f (x, k) is the Jost solution and f (x) = f (0, k), then the I-function is I(k) := f ′ (0,k) f (k). It is proved that I(k) is in one-to-one correspondence with the scattering triple S := {S(k), k j , s j , 1 ≤ j ≤ J} and with the spectral function ρ(λ) of the Sturm-Liouville operator l = − d 2 dx 2 + q(x) on (0, ∞) with the Dirichlet condition at x = 0 and q(x) ∈ L 1,1 := {q : q = q, ∞ 0 (1 ...