Let k,N ∈ N. We view members of {0, 1, . . . , k − 1} as words of length N on the alphabet {0, 1, . . . , k − 1}. A variable word is a word w1w2 · · ·wN on the alphabet {0, 1, . . . , k−1, x} in which the letter x (the variable) occurs. Indices i for which wi = x will be called wildcards, and {i : wi = x} will be called the wildcard set. We denote variable words by w(x), e.g. w(x) = 02x1x3210x ...

To each symmetric n × n matrix W with non-zero complex entries, we associate a vector space N , consisting of certain symmetric n × n matrices. If W satisfies n ∑ x=1 Wa,x Wb,x = nδa,b (a, b = 1, . . . , n), then N becomes a commutative algebra under both ordinary matrix product and Hadamard product (entry-wise product), so that N is the Bose-Mesner algebra of some association scheme. If W sati...

In this paper we continue our investigation on “Extremal problems under dimension constraint” introduced in [2]. Let E(n, k) be the set of (0,1)-vectors in Rn with k one’s. Given 1 ≤ m, w ≤ n let X ⊂ E(n, m) satisfy span(X)∩ E(n, w) = ∅. How big can |X | be? This is the main problem studied in this paper. We solve this problem for all parameters 1 ≤ m, w ≤ n and n > n0(m, w).

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where a and W satisfy (a1) a(t) is a continuous almost periodic function of t with a(t) ≥ a0 > 0 for all t ∈ R. (W1) There is a ξ ∈ R\{0} such that W ∈ C(R\{ξ},R). (W2) limx→ξW (x) = −∞. (W3) There is a neighborhood N of ξ and U ∈ C(N\{ξ},R) such that |U(x)| → ∞ as x→ ξ and |U ′(x)|2 ≤ −W (x) for x ∈ N\{ξ}, (W4) W (x) < W (0) = 0 if x 6= 0 and W ′′(0) is negative definite. (W5) There is a const...

Let X be a smooth projective variety of dimension n over an algebraically closed field k with char(k) = p > 0 and F : X → X1 be the relative Frobenius morphism. For any vector bundle W on X , we prove that instability of F∗W is bounded by instability of W ⊗ T(Ω X ) (0 ≤ l ≤ n(p − 1))(Corollary 4.8). When X is a smooth projective curve of genus g ≥ 2, it implies F∗W being stable whenever W is st...

family of (0,1) sequences with desired autocorrelation and cross-correlation properties providing asynchronous multi-access communications with easy synchronization and good performance in OCDMA communication networks [1,3]. In this section we review a few of the most important algorithms in generating OOCs. An optical orthogonal code) , , , (c a w n λ λ is a family C of (0,1) sequences of leng...

We discuss the asymptotic behavior (as n → ∞) of the entropic integrals En = − ∫ 1 −1 log ( p n (x) ) p n (x)w(x) dx , and Fn = − ∫ 1 −1 log ( p n (x)w(x) ) p n (x)w(x) dx, when w is the symmetric Pollaczek weight on [−1, 1] with main parameter λ ≥ 1, and pn is the corresponding orthonormal polynomial of degree n. It is well known that w does not belong to the Szegő class, which implies in part...