2014 Crystallographic concepts are extended to quasicrystalline structures and applied to icosahedral quasicrystals. 2-dimensional N fold rotational symmetries are shown to be compatible with Bravais lattices in (at least) ~ (N) dimensions, where ~ (N) is the Euler number, while for 3-dimensional icosahedral symmetry the minimal dimension is 6. The case of icosahedral crystallography is worked ...

A necessary and sufficient condition for existence of a Banach space with a finite dimensional decomposition but without the π-property in terms of norms of compositions of projections is found. 2000 Mathematics Subject Classification. Primary 46B15; Secondary 46B07, 46B28. The problem of existence of Banach spaces with the π-property but without a finite dimensional decomposition is one of the...

Let R be an n-dimensional Euclidean space and D be an n-dimensional hyperbolic space with the Poincaré metric for n > 1. In this paper, we shall prove the following results. (i) A bijection f : D → D n is an isometry (Möbius transformation) if and only if f is triangle preserving. (ii) A bijection f : R → R is an affine transformation if and only if f is triangle preserving.

In a paper [1] the authors ask whether the Frobenius and the H norms are induced. There they claimed that the Frobenius norm is not induced, and consequently conjectured that the H-norm may not be induced. In this note it is shown that the Frobenius norm is induced on particular matrix spaces. It is then shown that the H-norm is in fact induced on a particular matrix-valued L1 space. NOTATION R...

It is shown that every fuzzy n-normed space naturally induces a locally convex topology, and that every finite dimensional fuzzy n-normed space is complete.

We study the problem of constructing a probability density in 2N -dimensional phase space which reproduces a given collection of n joint probability distributions as marginals. Only distributions authorized by quantum mechanics, i.e. depending on a (complete) commuting set of N variables, are considered. A diagrammatic or graph theoretic formulation of the problem is developed. We then exactly ...

The terminology and notation used in this paper have been introduced in the following papers: [2], [6], [10], [4], [7], [18], [8], [13], [1], [3], [5], [15], [16], [17], [21], [22], [9], [19], [20], [11], [14], and [12]. For simplicity, we use the following convention: x, y are sets, i, n are natural numbers, r, s are real numbers, and f1, f2 are n-long real-valued finite sequences. Let s be a ...

A different proof is given to the result announced in [MS2]: For each 1 ≤ k < n we give an upper bound on the minimal distance of a k-dimensional subspace of an arbitrary n-dimensional normed space to the Hilbert space of dimension k. The result is best possible up to a multiplicative universal constant. Our main result is the following extension of Dvoretzky’s theorem (from the range 1 < k < c...

The articles [2], [6], [10], [4], [7], [18], [8], [13], [1], [3], [5], [15], [16], [17], [21], [22], [9], [19], [20], [11], [14], and [12] provide the notation and terminology for this paper. Let F be a set. We introduce F is n-long as a synonym of F is empty. We introduce F is finite sequence-membered as an antonym of F is empty. We introduce F is quasi-prebasis as an antonym of F is empty. We...

In this paper, we study the existence of extremal solutions for impulsive delay fuzzy integrodifferential equations in n-dimensional fuzzy vector space, by using monotone method. We show that obtained result is an extension of the result of Rodŕıguez-López [8] to impulsive delay fuzzy integrodifferential equations in n-dimensional fuzzy vector space.