Suppose Q is a definite quadratic form on a vector space V over some totally real field K 6= Q. Then the maximal integral ZK-lattices in (V,Q) are locally isometric everywhere and hence form a single genus. We enumerate all orthogonal spaces (V,Q) of dimension at least 3, where the corresponding genus of maximal integral lattices consists of a single isometry class. It turns out, there are 469 ...

Let G be a finite p-group which has exactly one maximal subgroup H such that |H| > p. Then we have d(G) = 2, p = 2, H is a four-group, G is abelian of order 8 and type (4, 2), G is of class 3 and the structure of G is completely determined. This solves the problem Nr. 1800 stated by Y. Berkovich in [3]. We consider here only finite p-groups and our notation is standard (see [1]). If G is a p-gr...

The properties of Na-Ca-K exchange current through the plasma membrane of intact rod outer segments (ROS) isolated from bovine retinas were studied with the optical probe neutral red. Small cellular organelles such as bovine ROS do not offer an adequate collecting area to measure Na-Ca-K exchange currents with electrophysiological techniques. This study demonstrates that Na-Ca-K exchange curren...

We present a novel algorithm called DBSC, which finds subspace clusters in numerical datasets based on the concept of “dependency”. This algorithm uses a depth-first search strategy to find out the maximal subspaces: a new dimension is added to current k-subspace and its validity as a (k 1)-subspace is evaluated. The clusters within those maximal subspaces are mined in a similar fashion as maxi...

1 Varshamov-Gilbert Lemma Lemma 1.1. Fix k ≥ 1. There exists a subset W of {0, 1} with |W | ≥ exp(k/8) such that the Hamming distance, ∆(τ, τ ′) := ∑k i=1{τi 6= τ ′ i} > k/4 for all τ, τ ′ ∈W with τ 6= τ ′. Proof. Consider a maximal subset W of {0, 1} that satisfies: ∆(τ, τ ′) ≥ k/4 for all τ, τ ′ ∈W with τ 6= τ ′. (1) The meaning of maximal here is that if one tries to expand W by adding one m...

We study necessary and sufficient conditions for a valued field K with value group G and residue field k (with char K = char k) to admit a truncation closed embedding in the field of generalized power series k((G, f)) (with factor set f). We show that this is equivalent to the existence of a family (tower of complements) of k-subspaces of K which are complements of the (possibly fractional) ide...

Let R be a commutative integral domain with quotient field K and let P be a nonzero strongly prime ideal of R. We give several characterizations of such ideals. It is shown that (P : P) is a valuation domain with the unique maximal ideal P. We also study when P^{&minus1} is a ring. In fact, it is proved that P^{&minus1} = (P : P) if and only if P is not invertible. Furthermore, if P is invertib...

If F ⊆ N N is an analytic family of pairwise eventually different functions then the following strong maximality condition fails: For any countable H ⊆ N N, no member of which is covered by finitely many functions from F , there is f ∈ F such that for all h ∈ H there are infinitely many integers k such that f (k) = h(k). However if V = L then there exists a coanalytic family of pairwise eventua...

Let T be a tree on n vertices and L(T ) be its Laplacian matrix. The eigenvalues and eigenvectors of T are respectively referred to those of L(T ). With respect to a given eigenvector Y of T , a vertex u of T is called a characteristic vertex if Y [u] = 0 and there is a vertex w adjacent to u with Y [w] 6= 0; an edge e = (u, w) of T is called a characteristic edge if Y [u]Y [w] < 0. C(T, Y ) de...

An [n, k, r]-partite graph is a graph whose vertex set, V , can be partitioned into n pairwisedisjoint independent sets, V1, . . . , Vn, each containing exactly k vertices, and the subgraph induced by Vi ∪ Vj contains exactly r independent edges, for 1 ≤ i < j ≤ n. An independent transversal in an [n, k, r]-partite graph is an independent set, T , consisting of n vertices, one from each Vi. An ...