Related papers: Jacobian determinants for (nonlinear) gradient of …
Given an arbitrary planar $\infty$-harmonic function $u$, for each $\alpha>0$ we establish a quantitative local $W^{1,2}$-estimate of $|Du|^\alpha $, which is sharp as $\alpha\to0$. We also show that the distributional determinant of $u$ is…
We introduce a weak notion of $2\times 2$-minors of gradients of a suitable subclass of $BV$ functions. In the case of maps in $BV(\mathbb{R}^2;\mathbb{R}^2)$ such a notion extends the standard definition of Jacobian determinant to…
We develop a deformation-based framework for analyzing static billiard systems through the Jacobian determinant computed in noncanonical angular coordinates. Although these systems are conservative, the determinant is not identically equal…
This article presents a family of nonlinear differential identities for the spatially periodic function $u_s(x)$, which is essentially the Jacobian elliptic function $\cn^2(z;m(s))$ with one non-trivial parameter $s$. More precisely, we…
We study orthogonal polynomials and Hankel determinants generated by a symmetric semi-classical Jacobi weight. By using the ladder operator technique, we derive the second-order nonlinear difference equations satisfied by the recurrence…
Strong asymptotics on the whole complex plane of a sequence of monic Jacobi polynomials $P_n^{(\alpha_n, \beta_n)}$ is studied, assuming that $$ \lim_{n\to\infty} \frac{\alpha_n}{n}=A, \qquad \lim_{n\to\infty} \frac{\beta _n}{n}=B, $$ with…
We discuss $C^1$ regularity and developability of isometric immersions of flat domains into $\mathbb R^3$ enjoying a local fractional Sobolev $W^{1+s, \frac2s}$ regularity for $2/3 \le s< 1 $, generalizing the known results on Sobolev and…
We present two analytical formulae for estimating the sensitivity -- namely, the gradient or Jacobian -- at given realizations of an arbitrary-dimensional random vector with respect to its distributional parameters. The first formula…
This work presents a general principle, in the spirit of convex integration, leading to a method for the characterization of Young measures generated by gradients of maps in $W^{1,p}$ with $p$ less than the space dimension, whose Jacobian…
We consider properties of determinants of some random symmetric matrices issued from multivariate statistics: Wishart/Laguerre ensemble (sample covariance matrices), Uniform Gram ensemble (sample correlation matrices) and Jacobi ensemble…
For a long time it has been a challenging goal to identify all orthogonal polynomial systems that occur as eigenfunctions of a linear differential equation. One of the widest classes of such eigenfunctions known so far, is given by…
We establish some exact asymptotic results for a matching problem with respect to a family of beta distributions. Let $X_1, \ldots, X_n$ be independent random variables with common distribution the symmetric Jacobi measure $d\mu (x) = C_d…
We give a probabilistic representation for the gradient of a 2nd order linear parabolic PDE $\partial_{t}u(t,x)=(1/2)a^{ij}\partial_{ij}u(t,x)+b^{i}\partial_{i}u(t,x)$ with Cauchy initial condition $u(0,x)=f(x)$ and Neumann boundary…
We prove weak and strong versions of the coarea formula and the chain rule for distributional Jacobian determinants $Ju$ for functions $u$ in fractional Sobolev spaces $W^{s,p}(\Omega)$, where $\Omega$ is a bounded domain in $\mathbb{R}^n$…
In this short note, we compute, for large n the determinant of a class of n x n Hankel matrices, which arise from a smooth perturbation of the Jacobi weight. For this purpose, we employ the same idea used in previous papers, where the…
Geometrization of dynamics using (non)-affine parametization of arc length with time is investigated. The two archetypes of such parametrizations, the Eisenhart and the Jacobi metrics, are applied to a system of linear harmonic oscillators.…
For a homeomorphism with $p$-integrable distortion, we obtain the optimal global degree of integrability for the reciprocal of its Jacobian determinant. As an application, we strengthen the result of Dole\v{z}alov\'a, Hencl and Mal\'y…
We study long-horizon deployment of a frozen predictor under dynamic covariate shift. A time-domain Poincare inequality first reduces temporal risk volatility to derivative energy. A Jacobian-velocity theorem then supplies the corresponding…
We consider $\sigma$-harmonic mappings, that is mappings $U$ whose components $u_i$ solve a divergence structure elliptic equation ${\rm div} (\sigma \nabla u_i)=0$, for $i=1,\ldots,n $. We investigate whether, with suitably prescribed…
We propose a new starting point for comparing dose distributions in therapeutic radiation physics using a Jacobian-based measure. The measure is normalization independent, free of tunable parameters, bounded and converges to a unique value…