Let Q: ℝn→ℝ be a quadratic form given by Q(x) = xTAx, for some symmetric matrix A. If B is an orthonormal basis for ℝn, and P is the transition matrix from B-coordinates to standard coordinates, then Q is expressed using B-coordinates as Q(x)=[x]BTK[x]B, where...
In transforming vectors in three-dimensional space, rotation matrices are often encountered. Rotation matrices are used in two senses: they can be used to rotate a vector into a new position or they can be used to rotate a coordinate basis (or coordinate system) into a new one. In this ca...
At first, compositional data need to be expressed in coordinates of an orthonormal basis on the simplex (with respect to the Aitchison geometry). The mathematical interpretation of the orthonormal coordinates is derived from the procedure by which they are constructed (called sequential binary ...
specify the localx,y, andzaxes in rectangular form with respect to the global coordinate system. Alternatively, you can specifylocalAxesas a single 3-by-3 matrix. If you do so,localAxesexpands to a 3-by-3-by-Narray with identical 3-by-3 matrices. The default value is the identity ...
Letbe a closed manifold and letbe a self-adjoint Laplace type operator on, which is of the form (1.1) for some Laplace type operatoron. The Laplace–Beltrami operator with respect to the cone metric onis the simplest operator of this type, but we also allow more general Laplace type opera...
The international date line roughly corresponds to the 180-degree meridian, although it deviates with respect to national borders. Because the Earth is divided into 24 time zones, each plus or minus a given number of hours from Greenwich, there arises a line on the Earth where the plus and ...
The basis of analytical geometry is the cartesian coordinate system, which aids in the depiction of lines, curves, and geometric forms in the n-dimensional plane. Let’s find out more about the terminologies used in the cartesian system. One can indicate the position of a particular point in...
" theIntervaltype is parameterized on two Enums:BasisandOpenness. Again, runtime introspection of enum value and runtime branching is required inside generic functions, a disadvantage compared to the D version. It is possible this could be written with type specialized generic functions, although ...
since they define convenient linear basis functions over simplices. More generally, many applied fields such as computational physics and mechanics rely heavily on barycentric coordinates since it corresponds to linear basis functions in the finite ...
Network dynamics have been proposed as a mechanistic substrate for the information transfer across cortical and hippocampal circuits. However, little is known about the mechanisms that synchronize and coordinate these processes across widespread brain regions during offline states. Here we address the hypot...