Abstract For a Galois extension $K/F$ with $\text {char}(K)\neq 2$ and $\mathrm {Gal}(K/F) \simeq \mathbb {Z}/2\mathbb {Z}\oplus {Z}$ , we determine the $\mathbb {F}_{2}[\mathrm {Gal}(K/F)]$ -module structure of $K^{\times }/K^{\times 2}$ . Although there are an infinite number (pairwise nonisomorphic) indecomposable {F}_{2}[\mathbb {Z}]$ -modules, our decomposition includes at most nine types....

and an abelian sheaf F of torsion abelian groups on X, the natural base change map gRf∗F ∼ −→ Rf ′ ∗(g F) is an isomorphism. By limit arguments and considerations with geometric stalks as discussed last time, we can arrange that F is a Z/nZ-sheaf for some n > 0 and it suffices to treat the case where S = Spec (A) for a strictly henselian local ring and S′ = Spec (k′) for a separably closed fiel...

We show that for a given exact category, there exists bijection between semibricks (pairwise Hom-orthogonal set of bricks) and length wide subcategories (exact extension-closed abelian subcategories). In particular, we category is if only simple objects form semibrick, is, the Schur's lemma holds.

In this paper, using the exact expression for the pairwise error probability derived in terms of the message symbol distance between two message vectors rather than the codeword symbol distance between two transmitted codeword matrices, the exact closed form expressions for the symbol error probability of any linear orthogonal space-time block codes in slow Rayleigh fading channel are derived f...

The purpose of this paper is twofold. First we derive theoretically, using appropriate transformation on x(n), the closed-form solution of the nonlinear difference equation x(n+1) = 1/(±1 + x(n)), n ∈ N_0. The form of solution of this equation, however, was first obtained in [10] but through induction principle. Then, with the solution of the above equation at hand, we prove a case ...

Let F denote a family of pairwise disjoint convex sets in the plane. F is said to be in convex position, if none of its members is contained in the convex hull of the union of the others. For any xed k 5, we give a linear upper bound on P k (n), the maximum size of a family F with the property that any k members of F are in convex position, but no n are.

Let F=∪i=1kSdi be the union of pairwise vertex-disjoint k stars order d1+1,…,dk+1, respectively, where k≥2 and d1≥⋯≥dk≥1. In this paper, we present two sharp upper bounds for spectral radius F-free (bipartite) graphs characterize all corresponding extremal graphs. Moreover, minimum least eigenvalue adjacency matrix an graph are obtained.

1.1 Closed sets Let k be an algebraically closed field. Let A = Ak = {a = (a1, . . . , an) : ai ∈ k} = k be affine space. So A is a point, and A = k. Let An = k[T1, . . . , Tn] be the k-algebra of polynomials. If f ∈ An, a ∈ A, then f(a) ∈ k. Suppose S ⊆ An be a subset. Then Z(S) = {a ∈ A : f(a) = 0,∀f ∈ S} ⊆ A. A subset Z ⊆ A is called closed if Z = Z(S) for some S ⊆ An. Example 1.1. If n = 1,...

New and old results on closed polynomials, i.e., such polynomials f ∈ k[x1, . . . , xn]\k that the subalgebra k[f ] is integrally closed in k[x1, . . . , xn], are collected in the paper. Using some properties of closed polynomials we prove the following factorization theorem: Let f ∈ k[x1, . . . , xn] \ k, where k is algebraically closed. Then for all but finite number μ ∈ k the polynomial f + ...