Pure Mathematics

Every theorem of Classical Analysis, Functional Analysis or of the Measure Theory that states a property of sequences leads to a class of filters for which this theorem is valid. Sometimes such class of filters is trivial (say, all filters or the filters with countable base), but in several ...

We can regard operations that discard information, like specializing to a particular case or dropping the intermediate steps of a proof, as projections, and operations that reconstruct information as liftings. By working with several projections in parallel we can make sense of statements like “Set is the archetypal Cartesian Closed Category”, which means that proofs about CCCs can be done in the “archetypal language” and then lifted to proofs in the general setting. The method works even when our archetypal language is diagrammatical, has potential ambiguities, is not completely formalized, and does not have semantics for all terms. We illustrate the method with an example from hyperdoctrines and another from synthetic differential geometry.

In this paper, I show that the idea of logical determinism can be traced back from the Old Babylonian period at least. According to this idea, there are some signs (omens) which can explain the appearance of all events. These omens demonstrate the will of gods and their power realized through natural forces. As a result, each event either necessarily appears or necessarily disappears. This idea can be examined as the first version of eternalism – the philosophical belief that each temporal event (including past and future events) is actual. In divination lists in Akkadian presented as codes we can reconstruct Boolean matrices showing that the Babylonians used some logical-algebraic structures in their reasoning. The idea of logical contingency was introduced within a new mood of thinking presented by the Greek prose – historical as well as philosophical narrations. In the Jewish genre ’aggādōt, the logical determinism is supposed to be in opposition to the Greek prose.

We prove the existence and uniqueness of solution for a first-order ordinary differential equation with periodic boundary conditions admitting only the existence of a lower solution. To this aim, we prove an appropriate fixed point theorem in partially ordered sets.

We study the integrability of polynomial vector fields using Galois theory of linear differential equations when the associated foliations is reduced to a Riccati type foliation. In particular we obtain integrability results for some families of quadratic vector fields, Li\'enard equations and equations related with special functions such as Hypergeometric and Heun ones. The Poincar\'e problem for some families is also approached.

We investigate the 3-edge coloring problem, based on the idea to give an algorithm, that attempts to minimize the use of color 4 (in a 4-edge coloring), and to study the factors that force it to fail. More specific, we introduce the notion of the ”position” of crossing points (or crossing edges) and the notion of the ”connection by bicolor paths”. A first result of this approach is that we can always assign a 4-edge coloring in a connected, bridgeless and cubic graph G using the fourth color only in crossing edges. Furthermore, color 4 is needed at most in half of these pairs of crossing edges. Finally, we bound the number of times color 4 is required in terms of the distance between the crossing points in the drawing of the graph.

In 2012, R.M. Aron, D. Carando, T.W. Gamelin, S. Lassalle, and M. Maestre presented that the Cluster Value Theorem in the infinite dimensional Banach space setting holds for the Banach algebra $\mathcal{H}^\infty (B_{c_0})$. On the other hand, B.J. Cole and T.W. Gamelin showed in 1986 that $\mathcal{H}^\infty (\ell_2 \cap B_{c_0})$ is isometrically isomorphic to $\mathcal{H}^\infty (B_{c_0})$ in the sense of an algebra. Motivated by this work, we are interested in a class of open subsets $U$ of a Banach space $X$ for which $\mathcal{H}^\infty (U)$ is isometrically isomorphic to $\mathcal{H}^\infty (B_{c_0})$. We prove that there exist polydisk type domains $U$ of any infinite dimensional Banach space $X$ with a Schauder basis such that $\mathcal{H}^\infty (U)$ is isometrically isomorphic to $\mathcal{H}^\infty (B_{c_0})$, which also generalizes the result by Cole and Gamelin. We also show that the Cluster Value Theorem is true for $\mathcal{H}^\infty (U)$. As the dual space $X^*$ is...