Mathematical Economics

You can find here an example of the use of the metric entropy integral in the case of the Wiener process. We will work out this example when we examine the notion of continuity of functions between metric spaces. Furthermore, you can find here, a simple example, built upon an already examined one, that makes evident that non total boundness does not imply the existence of one element balls (as is typically the case in discrete spaces). Furthermore it implies that open balls can be non totally bo

We continued the examination of the notion of total boundness. We provided several remarks regarding hereditarity, etc, that implied that the notion coincides with that of boundness on Euclidean spaces. We briefly discussed the notion of metric entropy and described the Metric Entropy Integral Theorem of Dudley as an astonishing application of the notion of total boundness in the theory of Gaussian Processes. As previously you can find notes for total boundness here and for the aforementioned re

You can find here a generalization of the issue of metrics comparison w.r.t. boundness, here an example where the property is "somehow lost in the limit", and here a result on boundness obtained by some asymptotic comparison of balls between two metrics.

We have been occupied with further examples, and the issues of hereditarity on subsets of boundness, equivalence of metrics w.r.t. boundness, and the "invariance" of boundness w.r.t. finite products and the metrics we have defined on such products. You can find notes for the above here, here, and here.

We moved on to a refinement of the notion, obtaining the notion of total boundness, and we have initially provided with the non-equivalence of the two notions. You can find notes for the above her

You can find here the example of a metric space with a metric defined in such a way so that some of the open balls are non-convex (the carrier set is simply the reals-hence they have the algebraic structure for the notion of convexity to make sense), along with a generalization. This has emerged by a discussion with one of your collegues.

We begun studying properties of metric spaces via the definition of the open and the closed balls that the metric defines. We have shown that these cannot in any case be empty, obey some monotonicity property,  they can separate points in a metric space, while this does not generally hold for pseudo metrics, and the local information that they convey about their center, can be conveyed by a "countable description". The example of the real line endowed with the usual metric and the real line endo

We have been occupied with further examples of metrics and subsequent metric spaces, and have shown that it is possible that different metrics on the same carrier set can obey relations. We suspected that such relations might imply analogous ones between the relevant properties that each metric endows the space with, and that provides as with a motivation of further examination of such relations.

We examined the important example of the space of bounded real functions on a non-empty domain endow

After some brief discussion of the course's scope and aims, as well as of some prerequisite notions, we begun with the definition of a distance function (metric) w.r.t. a non empty set of reference as a real function defined on the product of this set with itself that satisfies positivity, separation, symmetry and triangle inequality. The example of the discrete metric showed that any such set bears at least one such function, and further examples implied that more than one metrics can exist, so

We have been occupied with Brouwer's FPT and its subsequent application for the verification of the existence of Nash equilibria in a class of finite non-co-operative games, in a restricted setting that avoids the use of its generalization to correspendences, i.e. Kakutani's FPT.  You can find notes on the above here.

We have been occupied with the development of preparatory notions for the Brouwer's FPT. You can find notes for them here. (Notice that the sketch of proof of Borsuk's Lemma is out of the scope of the lectures. Anyhow, the notes contain a correction of the sketch as presented in the class. Specifically, the isomorphism between the homology groups of the same order does not hold for retracts in general, but refers to the stronger notion of deformation retracts. Check the notes for the corrected a