100,185 research outputs found
Vectorial Ribaucour Transformations for the Lame Equations
The vectorial extension of the Ribaucour transformation for the Lame
equations of orthogonal conjugates nets in multidimensions is given. We show
that the composition of two vectorial Ribaucour transformations with
appropriate transformation data is again a vectorial Ribaucour transformation,
from which it follows the permutability of the vectorial Ribaucour
transformations. Finally, as an example we apply the vectorial Ribaucour
transformation to the Cartesian background.Comment: 12 pages. LaTeX2e with AMSLaTeX package
Linearly dependent vectorial decomposition of clutters
This paper deals with the question of completing a monotone increasing family of
subsets of a finite set
to obtain the linearly dependent subsets of a family of
vectors of a vector space. Specifically, we demonstrate that such vectorial completions
of the family of subsets ¿ exist and, in addition, we show that the minimal
vectorial completions of the family ¿ provide a decomposition of the clutter of the
inclusion-minimal elements of ¿. The computation of such vectorial decomposition
of clutters is also discussed in some cases.Peer ReviewedPostprint (author’s final draft
Universal Vectorial and Ultrasensitive Nanomechanical Force Field Sensor
Miniaturization of force probes into nanomechanical oscillators enables
ultrasensitive investigations of forces on dimensions smaller than their
characteristic length scale. Meanwhile it also unravels the force field
vectorial character and how its topology impacts the measurement. Here we
expose an ultrasensitive method to image 2D vectorial force fields by
optomechanically following the bidimensional Brownian motion of a singly
clamped nanowire. This novel approach relies on angular and spectral tomography
of its quasi frequency-degenerated transverse mechanical polarizations:
immersing the nanoresonator in a vectorial force field does not only shift its
eigenfrequencies but also rotate eigenmodes orientation as a nano-compass. This
universal method is employed to map a tunable electrostatic force field whose
spatial gradients can even take precedence over the intrinsic nanowire
properties. Enabling vectorial force fields imaging with demonstrated
sensitivities of attonewton variations over the nanoprobe Brownian trajectory
will have strong impact on scientific exploration at the nanoscale
Two Triangles with the Same Orthocenter and a Vectorial Proof of Stevanovic's Theorem
In this article we'll emphasize on two triangles and provide a vectorial
proof of the fact that these triangles have the same orthocenter. This proof
will further allow us to develop a vectorial proof of the Stevanovic's theorem
relative to the orthocenter of the Fuhrmann's triangle.Comment: 5 pages, 2 geometrical figure
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