On the Minimum Polynomial and Applications

In this note we study the computation of the minimum polynomial of a matrix $A$ and how we can use it for the computation of the matrix $A^n$. We also describe the form of the elements of the matrix $A^{-n}$ and we will see that it is closely related with the computation of the Drazin generalized inverse of $A$. Next we study the computation of the exponential matrix and finally we give a simple proof of the Leverrier - Faddeev algorithm for the computation of the characteristic polynomial.

A simple proof of Linas’s theorem on Riemann zeta function

Linas Vepštas gives rapidly converging infinite representatives for values of Riemann zeta function at \left(4m-1 \right), where m is a natural number. In this paper, we give a new simple proof. Also, we obtain two equation of values of Bernoulli numbers’ generating function by applying a corollary given in this paper.

Big Building Data 2.0 - a Big Data Platform for Smart Buildings

The modern built environment is now connected. Multiple software and protocols are used in buildings of many kinds, thus creating a fascinating and heterogeneous environment. Within this context, applied research can be complicated and would benefit from a single data location across projects and users. The first version of BBData tried to solve this problem, BBData v2.0 is an update with a better-defined scope and a new codebase. The solution has been open sourced and simplified with a full software rewrite. Its components are now state-of-the-art and proven to be stable in industrial settings. The achieved performances have been thoroughly tested. Together with its new architecture, BBData v2.0 now accommodates the needs of modern experiments; efficient for simple proof of concepts while keeping the possibility to scale up to city-level projects. This flexibility makes BBData a good candidate for research while being able to scale in production settings.

On the Green function of the killed fractional Laplacian on the periodic domain

We give a very simple proof of the positivity and unimodality of the Green function for the killed fractional Laplacian on the periodic domain. The argument relies on the Jacobi triple product and a probabilistic representation of the Green function. We also show by a contour integration that the Green function is completely monotone on the positive part of the periodic domain.

A simple proof of the generalized Leibniz rule on bounded Euclidean domains

This paper is devoted to a simple proof of the generalized Leibniz rule in bounded domains. The operators under consideration are the so-called spectral Laplacian and the restricted Laplacian. Equations involving such operators have lately been considered by Constantin and Ignatova in the framework of the SQG equation [P. Constantin and M. Ignatova, Critical SQG in bounded domains, Ann. PDE 2 2016, 2, Article ID 8] in bounded domains, and by two of the authors [Q.-H. Nguyen and J. L. Vázquez, Porous medium equation with nonlocal pressure in a bounded domain, Comm. Partial Differential Equations 43 2018, 10, 1502–1539] in the framework of the porous medium with nonlocal pressure in bounded domains. We will use the estimates in this work in a forthcoming paper on the study of porous medium equations with pressure given by Riesz-type potentials.

A Consecutive Lehmer Code for Parabolic Quotients of the Symmetric Group

In this article we define an encoding for parabolic permutations that distinguishes between parabolic $231$-avoiding permutations. We prove that the componentwise order on these codes realizes the parabolic Tamari lattice, and conclude a direct and simple proof that the parabolic Tamari lattice is isomorphic to a certain $\nu$-Tamari lattice, with an explicit bijection. Furthermore, we prove that this bijection is closely related to the map $\Theta$ used when the lattice isomorphism was first proved in (Ceballos, Fang and Mühle, 2020), settling an open problem therein.