Proof of Dirac Delta Sifting Property With Volume Integral

The Dirac delta function possess the sifting property which states,

$ \int _{a}^{b} f'( x) \delta ( x-x’) dx’=\begin{cases} f( x) & a< x< b\ 0 & otherwise \end{cases} $

I suspect by analogy, it also posses a similar property for volume integrals,

$ \int _{V} f'(\mathbf{r}) \delta (\mathbf{r} -\mathbf{r} ‘) d^{3}\mathbf{r} ‘=\begin{cases} f(\mathbf{r}) & \mathbf{r\ } \in V\ 0 & otherwise \end{cases} .$

Can anyone confirm this property – how would I go about proving it?

Uniform inequality of the form $\text{Proba}(\sup_{v \in [-M,M]^k}|p^Tv-\hat{p}_n^Tv| \le \epsilon_n) \ge 1 – \delta$

Let $ M > 0$ , $ k$ be a positive integer, and $ \mathcal V:=[-M,M]^k$ . Finally, let $ p \in \Delta_k$ , (where $ \Delta_k$ is the $ (k-1)$ -dimensional probability simplex) and let $ \hat{p}_n$ be an empirical version of $ p$ based on an iid sample of size $ n$ . Given $ \delta \in (0, 1)$ , my objective to obtain a uniform-bound of the form

[Objective] $ \text{Proba}(\sup_{v \in \mathcal V}|p^Tv-\hat{p}_n^Tv| \le \epsilon_n) \ge 1 – \delta $ , for some $ \epsilon_n >0 $ (the smaller the better).

Idea using covering argument

Presumably, for each $ v \in \mathcal V$ , I can use Bernstein’s inequality to control $ |p^Tv-\hat{p}_n^Tv|$ . For example,

$ $ \text{Proba}\left(|p^Tv-\hat{p}_n^Tv| \le \left(\operatorname{Var}_p(v)\frac{\log(2/\delta)}{n}\right)^{1/2} + \frac{2M\log(2/\delta)}{3n}\right) \ge 1 -\delta. $ $

On the other hand,

The mapping $ G:v \mapsto |p^Tv-\hat{p}_n^Tv|$ is $ 2$ -Lipschitz w.r.t the $ \ell_\infty$ -norm on $ \mathbb R^k$ .

Indeed, for all $ v’,v \in \mathcal V$ , one has $ $ \begin{split} |G(v’)-G(v)| &:= ||p^Tv’-\hat{p}_n^Tv’|-|p^Tv-\hat{p}_n^Tv|| \le |p^Tv’-\hat{p}_n^Tv’-(p^Tv-\hat{p}_n^Tv)|\ &= |p^T(v’-v)-\hat{p}_n^T(v’-v)| \le |p^T(v’-v)|+|\hat{p}_n^T(v’-v)| \ &\le (\|p\|_1+\|\hat{p}_n\|_1)\|v’-v\|_\infty = 2 \|v’-v\|_\infty, \end{split} $ $ where the first and second inequalities are triangle inequalities, the third inequality is a Cauchy-Schwarz inequality, and the last inequality is because $ p,\hat{p}_n \in \Delta_k$ are probability distributions.

Also, the sup-norm covering number of $ \mathcal V$ is $ \mathcal N_\infty(\mathcal V;\varepsilon)\le(2M/\varepsilon)^k$ .

By using the fact that $ \|v\|_\infty \le M$ for all $ v \in \mathcal V$ , I can replace the variance term in the above Bernstein bound (i.e we’d use a Hoeffding inequality instead) to get $ \operatorname{Var}_p(v) \le M^2$ for all $ v \in \mathcal V$ , and then use covering arguments (e.g see https://mathoverflow.net/a/322161/78539) to get an inequality of the sough-for form [Objective] above. However, such an inequality is presumably “blurred”.

Question

How can these ramblings be pieced together to obtain a strong uniform inequality of the form [objective] ? Of course, I’m more than happy to learn other tricks for obtain such a results, which might not use any of the ideas I’ve discussed above.

Delta TSA-Precheck status removed

For the longest time: Delta had me running the gauntlet as Precheck. Today we were demoted to the standard inspection. The airline representative indicated that the decision was driven by TSA and has been applied to a number of passengers.

Given that TSA has incentive (they are resource constrained) to expedite travelers as Pre-Check, what possible cause would they have to revert Recheck passengers to standard status?

How the side ($\Delta L$) opposite to the angle ($\alpha$) change by a small change in the angle ($\Delta \alpha$)

How do you proof?

In an arbitrary triangle ABC if the sides b and c remain fixed in length, but the angle $ \alpha$ changes by a small amount $ \Delta \alpha$ . Then the opposite side to $ \alpha$ changes by an amount

$ $ \Delta a = \frac{bc}{a} \sin{\alpha} \Delta \alpha $ $ .

Hodge theory: $\Delta \alpha = 0$ iff $d\alpha = d^* \alpha = 0$ on a noncompact manifold?

Let $ M$ be a Riemannian manifold (connected, oriented).

One can define the co-differential $ d^* : \Omega^k(M, \mathbb{R}) \to \Omega^{k-1}(M, \mathbb{R})$ even if $ M$ is not compact (for example use the definition with the Hodge star, there is also a definition with the covariant derivative). $ d$ and $ d^*$ are formally self-adjoint, in the sense that $ \langle d \alpha, \beta \rangle_{L^2} = \langle \alpha, d^* \beta \rangle_{L^2}$ if $ \alpha$ or $ \beta$ has compact support.

One can then define the Hodge Laplacian $ \Delta = d d^* + d^* d$ . If $ M$ is compact, then \begin{equation} \Delta \alpha = 0\quad \Leftrightarrow \quad d\alpha = 0 \text{ and } d^* \alpha = 0~. \end{equation}

What if $ M$ is not assumed compact? The direction $ \Leftarrow$ still obviously holds.

What about $ \Rightarrow$ ?

Thoughts: In the compact case, it is easy to show $ \Rightarrow$ : one just writes $ \langle \Delta \alpha, \alpha \rangle_{L^2} = \langle d \alpha, d\alpha \rangle_{L^2} + \langle d^* \alpha, d^* \rangle_{L^2}$ and quickly concludes. I still the argument might still work in the noncompact case by evaluating $ \langle \Delta \alpha, \varphi \alpha \rangle_{L^2} $ for a wisely chosen bump function $ \varphi$ , but I can’t quite make it work. Otherwise maybe it can be proven by direct computation somehow, after all $ \Delta \alpha$ , $ d \alpha$ , and $ d^* \alpha$ are all local quantities.

CQRS, microservices and delta replication

We have a micro-service that has a Domain Model and have an analytical service for the domain which has its own Query Model. The domain model and the query model are stored in separate persistencies.

Currently our Query Model uses a sub-set of the attributes from the Domain model. However going forward we have requirement to add additional attributes from the Domain Model to Query Model, in this case are there any recommendations with regards to the best approach that can be used to populate the ‘delta’ part of the Query Model with what is available in Domain Model?

As an aside, this also seems to be a weakness of the CQRS any enhancements to the Query Model would require some sort of reload of the data to populate the enhanced part of the Query Model. Or are we doing something wrong here?

Magento 2 migration Tool: does PRODUCTS model support DELTA (incremental)?

when browsing and reading guides it looks like Customers + Orders support incremental data migration (delta: the newly added items) …. bu I also read that this is supported for products : but is this really so?

Somehow it seems it was the plan to support Products Delta migration …. but it never made it through?

Using Magento 2.3.1 : does the 2.3.1 Data Migration Tool support incremental migration for Products?

thx