Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userAn elementary approach to the dimension of measures satisfying a first-order linear PDE constraint
78
0
0.0
(
0
)
Authors
Adolfo Arroyo-Rabasa
Ask ChatGPT about the research
No Arabic abstract
We give a simple criterion on the set of probability tangent measures $mathrm{Tan}(mu,x)$ of a positive Radon measure $mu$, which yields lower bounds on the Hausdorff dimension of $mu$. As an application, we give an elementary and purely algebraic proof of the sharp Hausdorff dimension lower bounds for first-order linear PDE-constrained measures; bounds for closed (measure) differential forms and normal currents are further discussed. A weak structure theorem in the spirit of [Ann. Math. 184(3) (2016), pp. 1017-1039] is also discussed for such measures.
rate research
Read More
We establish higher integrability estimates for constant-coefficient systems of linear PDEs [ mathcal A mu = sigma, ] where $mu in mathcal M(Omega;V)$ and $sigmain mathcal M(Omega;W)$ are vector measures and the polar $frac{dmu}{d |mu|}$ is uniformly close to a subspace $L$ of $V$ intersecting the wave cone of $mathcal A$ only at the origin. More precisely, we prove local compensated compactness estimates of the form [ |mu|_{L^p(Omega)} lesssim |mu|(Omega) + |sigma|(Omega), qquad Omega Subset Omega. ] Here, the exponent $p$ belongs to the (optimal) range $1 leq p < d/(d-k)$, $d$ is the dimension of $Omega$, and $k$ is the order of $mathcal A$. For canceling constant-rank operators we also obtain the limiting case $p = d/(d-k)$. We consider applications to compensated compactness as well as applications to the theory of functions of bounded variation and bounded deformation.
We study truncated objects using elementary methods. Concretely, we use universes and the resulting natural number object to define internal truncation levels and prove they behave similar to standard truncated objects. Moreover, we take an elementary approach to localizations, giving various equivalent conditions that characterize localizations and constructing a localization out of a sub-universe of local objects via an internal right Kan extension. We then use this general approach, as well as an inductive approach, to construct truncation functors. We use the resulting truncation functors to prove classical results about truncations, such as Blakers-Massey theorem, in the elementary setting. We also give examples of non-presentable $(infty, 1)$-categories where the elementary approach can be used to define and compute truncations. Finally, we turn around and use truncations to study elementary $(infty, 1)$-toposes and show how they can help us better understand subobject classifiers and universes
By an easy trick taken from caloric polynomial theory we construct a family $mathscr{B}$ of $almost regular$ domains for the caloric Dirichlet problem. $mathscr{B}$ is a basis of the Euclidean topology. This allows to build, with a basically elementary procedure, the Perron solution to the caloric Dirichlet problem on every bounded domain.
Time-varying mixture densities occur in many scenarios, for example, the distributions of keywords that appear in publications may evolve from year to year, video frame features associated with multiple targets may evolve in a sequence. Any models that realistically cater to this phenomenon must exhibit two important properties: the underlying mixture densities must have an unknown number of mixtures, and there must be some smoothness constraints in place for the adjacent mixture densities. The traditional Hierarchical Dirichlet Process (HDP) may be suited to the first property, but certainly not the second. This is due to how each random measure in the lower hierarchies is sampled independent of each other and hence does not facilitate any temporal correlations. To overcome such shortcomings, we proposed a new Smoothed Hierarchical Dirichlet Process (sHDP). The key novelty of this model is that we place a temporal constraint amongst the nearby discrete measures ${G_j}$ in the form of symmetric Kullback-Leibler (KL) Divergence with a fixed bound $B$. Although the constraint we place only involves a single scalar value, it nonetheless allows for flexibility in the corresponding successive measures. Remarkably, it also led us to infer the model within the stick-breaking process where the traditional Beta distribution used in stick-breaking is now replaced by a new constraint calculated from $B$. We present the inference algorithm and elaborate on its solutions. Our experiment using NIPS keywords has shown the desirable effect of the model.
We consider a non-linear parabolic partial differential equation (PDE) on $mathbb R^d$ with a distributional coefficient in the non-linear term. The distribution is an element of a Besov space with negative regularity and the non-linearity is of quadratic type in the gradient of the unknown. Under suitable conditions on the parameters we prove local existence and uniqueness of a mild solution to the PDE, and investigate properties like continuity with respect to the initial condition and blow-up times. We prove a global existence and uniqueness result assuming further properties on the non-linearity. To conclude we consider an application of the PDE to stochastic analysis, in particular to a class of non-linear backward stochastic differential equations with distributional drivers.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
