Let K be a quadratic extension of Q, B a quaternion algebra over Q and A = B ⊗Q K. Let O be a maximal order in A extending an order in B. The projective norm one group PO1 is shown to be isomorphic to the spinorial kernel group O′(L), for an explicitly determined quadratic Z-lattice L of rank four, in several general situations. In other cases, only the local structures of O and L are given at ...

A nite new algorithm is proposed for clustering m given points in n-dimensional real space into k clusters by generating k planes that constitute a local solution to the nonconvex problem of minimizing the sum of squares of the 2-norm distances between each point and a nearest plane. The key to the algorithm lies in a formulation that generates a plane in n-dimensional space that minimizes the ...

A nite new algorithm is proposed for clustering m given points in n-dimensional real space into k clusters by generating k planes that constitute a local solution to the nonconvex problem of minimizing the sum of squares of the 2-norm distances between each point and a nearest plane. The key to the algorithm lies in a formulation that generates a plane in n-dimensional space that minimizes the ...

Abstract. We develop conditions under which a product ∏∞ i=0 Ti of matrices chosen from a possibly infinite set of matrices S = {Tj |j ∈ J} converges. We obtain the following conditions which are sufficient for the convergence of the product: There exists a vector norm such that all matrices in S are nonexpansive with respect to this norm and there exists a subsequence {ik}k=0 of the sequence o...

On the basis of the concept of the interval valued intuitionistic fuzzy sets introduced by K. Atanassov, the notion of interval valued intuitionistic fuzzy subsemimodule of a semimodule with respect to t-norm T and s-norm S is given and the characteristic properties are described. The homomorphic image and inverse image are investigated. In particular, by the help of the congruence relations on...

Let X0, X1, . . . , Xk with k ∈ IN ∪ {∞} be sequence spaces (finite or infinite dimensional) over C or IR with absolute norms Ni for i = 0, . . . , k, (i.e., with 1unconditional bases) such that dimX0 = k. Define an absolute norm on the cross product space (also known as the X0 1-unconditional sum) X1 × · · · ×Xk by N(x1, . . . , xk) = N0(N1(x1), . . . , Nk(xk)) for all (x1, . . . , xk) ∈ X1 × ...

w,q = supφ∈BX́ ( ∑k j=1 | φ(xj) | ) 1 q . This is a natural generalization of the concept of (p; q)-summing operators and in the last years has been studied by several authors. The infimum of the L > 0 for which the inequality holds defines a norm ‖.‖as(p;q) for the case p ≥ 1 or a p-norm for the case p < 1 on the space of (p; q)-summing homogeneous polynomials. The space of all m-homogeneous (p...

w,q = supφ∈BX́ ( ∑k j=1 | φ(xj) | ) 1 q . This is a natural generalization of the concept of (p; q)-summing operators and in the last years has been studied by several authors. The infimum of the L > 0 for which the inequality holds defines a norm ‖.‖as(p;q) for the case p ≥ 1 or a p-norm for the case p < 1 on the space of (p; q)-summing homogeneous polynomials. The space of all m-homogeneous (p...

For a metric space (K, d) the Banach space Lip(K) consists of all scalar-valued bounded Lipschitz functions on K with the norm ‖f‖L = max(‖f‖∞, L(f)), where L(f) is the Lipschitz constant of f . The closed subspace lip(K) of Lip(K) contains all elements of Lip(K) satisfying the lip-condition lim0<d(x,y)→0 |f(x) − f(y)|/d(x, y) = 0. For K = ([0, 1], | · |), 0 < α < 1, we prove that lip(K) is a p...

Let L/K be an abelian extension of number fields. In Lecture 22 we defined the norm group T m := NL/K (IL m)Rm (see Definition 22.26) that we claim is equal to the kernel of the Artin L/K K map ψm : Im → Gal(L/K), provided that the modulus m is divisible by the conductor L/K K of L (see Definition 22.23). We showed that T m contains ker ψm (Proposition 22.27), L/K L/K and in Theorem 22.28 we pr...