A note on the uniformity of the constant in the Poincar\'e inequality

The classical Poincar\'e inequality establishes that for any bounded regular domain $\Omega\subset \R^N$ there exists a constant $C=C(\Omega)>0$ such that $$\int_{\Omega} |u|^2\, dx \leq C \int_{\Omega} |\nabla u|^2\, dx \ \ \forall u \in H^1(\Omega),\ \int_{\Omega} u(x) \, dx=0.$$ In this note we show that $C$ can be taken independently of $\Omega$ when $\Omega$ is in a certain class of domains. Our result generalizes previous results in this direction.Comment: 12 pages, 1 figur

Automatic supervised information extraction of structured web data

The overall purpose of this project is, in short words, to create a system able to extract vital information from product web pages just like a human would. Information like the name of the product, its description, price tag, company that produces it, and so on. At a first glimpse, this may not seem extraordinary or technically difficult, since web scraping techniques exist from long ago (like the python library Beautiful Soup for instance, an HTML parser1 released in 2004). But let us think for a second on what it actually means being able to extract desired information from any given web source: the way information is displayed can be extremely varied, not only visually, but also semantically. For instance, some hotel booking web pages display at once all prices for the different room types, while medium-sized consumer products in websites like Amazon offer the main product in detail and then more small-sized product recommendations further down the page, being the latter the preferred way of displaying assets by most retail companies. And each with its own styling and search engines. With the above said, the task of mining valuable data from the web now does not sound as easy as it first seemed. Hence the purpose of this project is to shine some light on the Automatic Supervised Information Extraction of Structured Web Data problem. It is important to think if developing such a solution is really valuable at all. Such an endeavour both in time and computing resources should lead to a useful end result, at least on paper, to justify it. The opinion of this author is that it does lead to a potentially valuable result. The targeted extraction of information of publicly available consumer-oriented content at large scale in an accurate, reliable and future proof manner could provide an incredibly useful and large amount of data. This data, if kept updated, could create endless opportunities for Business Intelligence, although exactly which ones is beyond the scope of this work. A simple metaphor explains the potential value of this work: if an oil company were to be told where are all the oil reserves in the planet, it still should need to invest in machinery, workers and time to successfully exploit them, but half of the job would have already been done2. As the reader will see in this work, the way the issue is tackled is by building a somehow complex architecture that ends in an Artificial Neural Network3. A quick overview of such architecture is as follows: first find the URLs that lead to the product pages that contain the desired data that is going to be extracted inside a given site (like URLs that lead to ”action figure” products inside the site ebay.com); second, per each URL passed, extract its HTML and make a screenshot of the page, and store this data in a suitable and scalable fashion; third, label the data that will be fed to the NN4; fourth, prepare the aforementioned data to be input in an NN; fifth, train the NN; and sixth, deploy the NN to make [hopefully accurate] predictions

Existence of ground states for a modified nonlinear Schrodinger equation

In this paper we prove existence of ground state solutions of the modified nonlinear Schrodinger equation: $$-\Delta u+V(x)u-{1/2}u \Delta u^{2}=|u|^{p-1}u, x \in \R^N, N \geq 3,$$ under some hypotheses on $V(x)$. This model has been proposed in the theory of superfluid films in plasma physics. As a main novelty with respect to some previous results, we are able to deal with exponents $p\in(1,3)$. The proof is accomplished by minimization under a convenient constraint