423 research outputs found
Random Triangle Theory with Geometry and Applications
What is the probability that a random triangle is acute? We explore this old
question from a modern viewpoint, taking into account linear algebra, shape
theory, numerical analysis, random matrix theory, the Hopf fibration, and much
much more. One of the best distributions of random triangles takes all six
vertex coordinates as independent standard Gaussians. Six can be reduced to
four by translation of the center to or reformulation as a 2x2 matrix
problem.
In this note, we develop shape theory in its historical context for a wide
audience. We hope to encourage other to look again (and differently) at
triangles.
We provide a new constructive proof, using the geometry of parallelians, of a
central result of shape theory: Triangle shapes naturally fall on a hemisphere.
We give several proofs of the key random result: that triangles are uniformly
distributed when the normal distribution is transferred to the hemisphere. A
new proof connects to the distribution of random condition numbers.
Generalizing to higher dimensions, we obtain the "square root ellipticity
statistic" of random matrix theory.
Another proof connects the Hopf map to the SVD of 2 by 2 matrices. A new
theorem describes three similar triangles hidden in the hemisphere. Many
triangle properties are reformulated as matrix theorems, providing insight to
both. This paper argues for a shift of viewpoint to the modern approaches of
random matrix theory. As one example, we propose that the smallest singular
value is an effective test for uniformity. New software is developed and
applications are proposed
Groups of banded matrices with banded inverses
A product A=F[subscript 1]...F[subscript N] of invertible block-diagonal matrices will be banded with a banded
inverse. We establish this factorization with the number N controlled by the bandwidths w
and not by the matrix size n. When A is an orthogonal matrix, or a permutation, or banded
plus finite rank, the factors F[subscript i] have w=1 and generate that corresponding group. In the
case of infinite matrices, conjectures remain open
Maximum flows and minimum cuts in the plane
A continuous maximum flow problem finds the largest t such that div v = t F(x, y) is possible with a capacity constraint ||(v[subscript 1], v[subscript 2])|| ≤ c(x, y). The dual problem finds a minimum cut ∂ S which is filled to capacity by the flow through it. This model problem has found increasing application in medical imaging, and the theory continues to develop (along with new algorithms). Remaining difficulties include explicit streamlines for the maximum flow, and constraints that are analogous to a directed graph. Keywords: Maximum flow; Minimum cut; Capacity constraint; Cheege
Every unit matrix is a LULU
AbstractThe four matrices L0U0L1U1 at the end of the title are triangular with ones on their main diagonals. Their product has determinant one. Following a question and theorem of Toffoli, we show that any matrix with determinant one can be factored in this way. A transformation of the plane becomes a sequence of one-dimensional shears, with n2 — 1 free parameters
The Main Diagonal of a Permutation Matrix
By counting 1's in the "right half" of consecutive rows, we locate the
main diagonal of any doubly infinite permutation matrix with bandwidth .
Then the matrix can be correctly centered and factored into block-diagonal
permutation matrices. Part II of the paper discusses the same questions for the
much larger class of band-dominated matrices. The main diagonal is determined
by the Fredholm index of a singly infinite submatrix. Thus the main diagonal is
determined "at infinity" in general, but from only rows for banded
permutations
The Jordan forms of AB and BA∗
The relationship between the Jordan forms of the matrix products AB and BA for
some given A and B was first described by Harley Flanders in 1951. Their non-zero eigenvalues and
non-singular Jordan structures are the same, but their singular Jordan block sizes can differ by 1.
We present an elementary proof that owes its simplicity to a novel use of the Weyr characteristic
Optimal stability for trapezoidal-backward difference split-steps
The marginal stability of the trapezoidal method makes it dangerous to use for highly non-linear oscillations. Damping is provided by backward differences. The split-step combination (αΔt trapezoidal, (1 – α)Δt for BDF2) retains second-order accuracy. The ‘magic choice’ a = 2 – √2 allows the same Jacobian for both steps, when Newton's method solves these implicit difference equations. That choice is known to give the smallest error constant, and we prove that a = 2 – √2 also gives the largest region of linearized stability
Eigenvalue and Eigenvector Analysis of Stability for a Line of Traffic
Many authors have recognized that traffic under the traditional car-following model (CFM) is subject to flow instabilities. A recent model achieves stability using bilateral control (BCM)—by looking both forward and backward [1]. (Looking back may be difficult or distracting for human drivers, but is not a problem for sensors.) We analyze the underlying systems of differential equations by studying their eigenvalues and eigenvectors under various boundary conditions. Simulations further confirm that bilateral control can avoid instabilities and reduce the chance of collisions
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