Let Ps(d) be the probability that a random 0/1-matrix of size d× d is singular, and let E(d) be the expected number of 0/1-vectors in the linear subspace spanned by d − 1 random independent 0/1-vectors. (So E(d) is the expected number of cube vertices on a random affine hyperplane spanned by vertices of the cube.) We prove that bounds on Ps(d) are equivalent to bounds on E(d): Ps(d) =

Ziel: Dreidimensionale MRT-Sequenzen erlauben sowohl eine verbesserte räumliche Auflösung bei guten Signalund Kontrasteigenschaften als auch multiplanare Rekonstruktionen. Wir verglichen Cube, eine 3-D-FLAIR-Sequenz, mit einer Standard-2D-FLAIR-Sequenz bei der Bildgebung der multiplen Sklerose (MS). Material und Methoden: Die Untersuchungenwurden an einem 3-Tesla-Scanner in der klinischen Routi...

Let Ps(d) be the probability that a random 0/1-matrix of size d× d is singular, and let E(d) be the expected number of 0/1-vectors in the linear subspace spanned by d − 1 random independent 0/1-vectors. (So E(d) is the expected number of cube vertices on a random affine hyperplane spanned by vertices of the cube.) We prove that bounds on Ps(d) are equivalent to bounds on E(d): Ps(d) =

PURPOSE Three-dimensional (3 D) MRI sequences allow improved spatial resolution with good signal and contrast properties as well as multiplanar reconstruction. We sought to compare Cube, a 3 D FLAIR sequence, to a standard 2 D FLAIR sequence in multiple sclerosis (MS) imaging. MATERIALS AND METHODS Examinations were performed in the clinical routine on a 3.0 Tesla scanner. 12 patients with de...

Consider n sensors placed randomly and independently with the uniform distribution in a d−dimensional unit cube (d ≥ 2). The sensors have identical sensing range equal to r, for some r > 0. We are interested in moving the sensors from their initial positions to new positions so as to ensure that the d−dimensional unit cube is completely covered, i.e., every point in the d−dimensional cube is wi...

A collection H of integers is called an affine d-cube if there exist d+ 1 positive integers x0, x1, . . . , xd so that H = { x0 + ∑ i∈I xi : I ⊆ {1, 2, . . . , d} } . We address both density and Ramsey-type questions for affine d-cubes. Regarding density results, upper bounds are found for the size of the largest subset of {1, 2, . . . , n} not containing an affine d-cube. In 1892 Hilbert publi...

A collection H of integers is called an aane d-cube if there exist d + 1 positive integers x 0 ; x 1) : We address both density and Ramsey-type questions for aane d-cubes. Regarding density results, upper bounds are found for the size of the largest subset of f1; 2; : : : ; ng not containing an aane d-cube. In 1892 Hilbert published the rst Ramsey-type result for aane d-cubes by showing that fo...

The design of underground spaces is mainly carried out using empirical, analytical, and numerical methods. The convergence confinement method (CCM) is an analytical technique that is widely utilized in analyzing the stability of underground spaces. However, the main challenge in the stability analysis is the selection of an accurate constitutive model for rock mass, and particularly, its post-f...

Triangulating the regular d-cube I = [0, 1] in a “simple” way has many applications, like solving differential equations by finite element methods or calculating fixed points. See, for example, [7]. Determining the smallest number of simplices needed has brought special attention both from a theoretical point of view and from an applied one (see [6, Section 14.5.2] for a recent survey). Before ...

We consider tilings and packings of R by integral translates of cubes [0, 2[, which are 4Z-periodic. Such cube packings can be described by cliques of an associated graph, which allow us to classify them in dimension d ≤ 4. For higher dimension, we use random methods for generating some examples. Such a cube packing is called non-extendible if we cannot insert a cube in the complement of the pa...