Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userIntegral geometry of pairs of hyperplanes or lines
70
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
Croftons formula of integral geometry evaluates the total motion invariant measure of the set of $k$-dimensional planes having nonempty intersection with a given convex body. This note deals with motion invariant measures on sets of pairs of hyperplanes or lines meeting a convex body. Particularly simple results are obtained if, and only if, the given body is of constant width in the first case, and of constant brightness in the second case.
rate research
Read More
We introduce the study of frames and equiangular lines in classical geometries over finite fields. After developing the basic theory, we give several examples and demonstrate finite field analogs of equiangular tight frames (ETFs) produced by modular difference sets, and by translation and modulation operators. Using the latter, we prove that Gerzons bound is attained in each unitary geometry of dimension $d = 2^{2l+1}$ over the field $mathbb{F}_{3^2}$. We also investigate interactions between complex ETFs and those in finite unitary geometries, and we show that every complex ETF implies the existence of ETFs with the same size over infinitely many finite fields.
The use of mathematical models in the sciences often involves the estimation of unknown parameter values from data. Sloppiness provides information about the uncertainty of this task. In this paper, we develop a precise mathematical foundation for sloppiness and define rigorously its key concepts, such as `model manifold, in relation to concepts of structural identifiability. We redefine sloppiness conceptually as a comparison between the premetric on parameter space induced by measurement noise and a reference metric. This opens up the possibility of alternative quantification of sloppiness, beyond the standard use of the Fisher Information Matrix, which assumes that parameter space is equipped with the usual Euclidean metric and the measurement error is infinitesimal. Applications include parametric statistical models, explicit time dependent models, and ordinary differential equation models.
We study the geometry of manifolds carrying symplectic pairs consisting of two closed 2-forms of constant ranks, whose kernel foliations are complementary. Using a variation of the construction of Boothby and Wang we build contact-symplectic and contact pairs from symplectic pairs.
Given a domain $G subsetneq Rn$ we study the quasihyperbolic and the distance ratio metrics of $G$ and their connection to the corresponding metrics of a subdomain $D subset G$. In each case, distances in the subdomain are always larger than in the original domain. Our goal is to show that, in several cases, one can prove a stronger domain monotonicity statement. We also show that under special hypotheses we have inequalities in the opposite direction.
Let $K$ be an isotropic symmetric convex body in ${mathbb R}^n$. We show that a subspace $Fin G_{n,n-k}$ of codimension $k=gamma n$, where $gammain (1/sqrt{n},1)$, satisfies $$Kcap Fsubseteq frac{c}{gamma }sqrt{n}L_K (B_2^ncap F)$$ with probability greater than $1-exp (-sqrt{n})$. Using a different method we study the same question for the $L_q$-centroid bodies $Z_q(mu )$ of an isotropic log-concave probability measure $mu $ on ${mathbb R}^n$. For every $1leq qleq n$ and $gammain (0,1)$ we show that a random subspace $Fin G_{n,(1-gamma )n}$ satisfies $Z_q(mu )cap Fsubseteq c_2(gamma )sqrt{q},B_2^ncap F$. We also give bounds on the diameter of random projections of $Z_q(mu )$ and using them we deduce that if $K$ is an isotropic convex body in ${mathbb R}^n$ then for a random subspace $F$ of dimension $(log n)^4$ one has that all directions in $F$ are sub-Gaussian with constant $O(log^2n)$.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
