Perpendicular

Summary

In elementary geometry, two geometric objects are perpendicular if their intersection forms right angles (angles that are 90 degrees or π/2 radians wide) at the point of intersection called a foot. The condition of perpendicularity may be represented graphically using the perpendicular symbol, ⟂. Perpendicular intersections can happen between two lines (or two line segments), between a line and a plane, and between two planes. Perpendicularity is one particular instance of the more general mathematical concept of orthogonality; perpendicularity is the orthogonality of classical geometric objects. Thus, in advanced mathematics, the word "perpendicular" is sometimes used to describe much more complicated geometric orthogonality conditions, such as that between a surface and its normal vector. A line is said to be perpendicular to another line if the two lines intersect at a right angle. Explicitly, a first line is perpendicular to a second line if (1) the two lines meet; and (2) at

Official source

https://en.wikipedia.org/wiki/Perpendicular

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Let parallel to.parallel to be a norm in R-d whose unit ball is B. Assume that V subset of B is a finite set of cardinality n, with Sigma(v is an element of V) v = 0. We show that for every integer k with 0

Elsevier Science Inc2016

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Helmuth Berger, Philippe Bugnon, Arnaud Magrez, Manish Sharma

Electrodeposited bismuth telluride (Bi2Te3) nanowires using anodic alumina (AAO) templates show anisotropic thermal properties in a direction parallel and perpendicular to the nanowire/nanochannel axis. The present study reports thermal diffusivity measurement of templated Bi2Te3 nanowires in a direction perpendicular to the nanowire axis, using an active infrared thermography based noncontact technique. Measurements were performed on empty AAO templates of varying pore dimensions, AAO/Bi2Te3 nanocomposites and a single crystal Bi2Te3 sample. A first order lower bound model estimation showed nearly five-fold reduction of thermal conductivity in 200 nm Bi2Te3 nanowires as compared to the bulk values. (C) 2013 Elsevier B.V. All rights reserved.

Elsevier Science Bv2014

Uniaxial 2D Superlattice of Fe-4 Molecular Magnets on Graphene

Harald Brune, Luca Gragnaniello, Fabian Paschke, Stefano Rusponi

We demonstrate that electrospray deposition enables the fabrication of highly periodic self-assembled arrays of Fe4H single molecule magnets on graphene/Ir(111). The energetic positions of molecular states are probed by means of scanning tunneling spectroscopy, showing pronounced long- and short-ranged spatial modulations, indicating the presence of both locally varying intermolecular as well as adsorption-site dependent molecule substrate interactions. From the magnetic field dependence of the X-ray magnetic circular dichroism signal, we infer that the magnetic easy axis of each Fe4H molecule is oriented perpendicular to the sample surface and that after the deposition the value of the uniaxial anisotropy is identical to the one in bulk. Our findings therefore suggest that the observed interaction of the molecules with their surrounding does not modify the molecular magnetism, resulting in magnets that retain their bulk magnetic properties.

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