Las trans-S-variedades y la clasificación de Gray-Hervella.

Recently, trans-S manifolds have been defined as a natural generalization of f-Kenmotsu, S-manifolds and C-manifolds. This uses f-structure techniques to extend results for almost contact manifolds, where trans-Sasakian are a generalization of Kenmotsu, Sasakian and cosymplectic manifolds. The defintion (sic) of trans-S is formulated using the covariant derivative of the tensor f. Although this formulation coincides with the characterization of trans-Sasakian when s = 1, the latter type of manifolds were not initially defined in this way. Trans-Sasakian manifolds were defined in 1985 using the Gray-Hervella’s classification of almost Hermitian manifolds. Therefore, one could ask whether trans-S manifolds could be defined using this classification and how they relate with almost Hermitian manifolds. The objective of this project is to study how trans-S-manifolds and almost trans-S-manifolds relates with the Gray-Hervella’s classification. We will find both similarities and differences with the trans-Sasakian case (s = 1).Las trans-S-variedades han sido definidas recientemente como una generalización natural de las f-Kenmotsu, S-variedades y C-variedades. Estos resultados representan una versión para f-structuras de algunos resulados ya conocidos en variedades casi contacto, donde las trans-Sasakianas generalizan variedades como las Kenmotsu, las Sasakianas y las cosimpléticas. La definición de variedad trans-S ha sido formulada en términos de la derivada covariante del tensor f. Aunque esta definición coincide con la caracterización de las trans-Sasakianas cuando s = 1, este último tipo de variedad no fue definida de esta manera en un principio. Las variedades trans-Sasakianas fueron definidas en 1985 usando la clasificación de Gray y Hervella para variedades casi Hermitianas. Por tanto, una pregunta interesante sobre variedades trans-S es si podrían definirse de la misma manera y cómo se relacionan con la clase de variedades Hermitianas. El objetivo de este proyecto es estudiar como se relacionan las variedades trans-S y casi trans-S con la clasificación de Gray y Hervella. Descubriremos algunas similitudes con el caso trans-Sasakiano (s = 1) pero también algunas diferencias.Universidad de Sevilla. Máster Universitario en Matemática

Persistence Partial Matchings Induced by Morphisms between Persistence Modules

The notion of persistence partial matching, as a generalization of partial matchings between persistence modules, is introduced. We study how to obtain a persistence partial matching Gf , and a partial matching Mf , induced by a morphism f between persistence modules, both being linear with respect to direct sums of morphisms. Some of their properties are also provided, including their stability after a perturbation of the morphism f, and their relationship with other induced partial matchings already de ned in TDA.Ministerio de Ciencia e Innovación PID2019-107339GB-I0

On the stability of persistent entropy and new summary functions for Topological Data Analysis

Persistent entropy of persistence barcodes, which is based on the Shannon entropy, has been recently defined and successfully applied to different scenarios: characterization of the idiotypic immune network, detection of the transition between the preictal and ictal states in EEG signals, or the classification problem of real long-length noisy signals of DC electrical motors, to name a few. In this paper, we study properties of persistent entropy and prove its stability under small perturbations in the given input data. From this concept, we define three summary functions and show how to use them to detect patterns and topological features

Characterising epithelial tissues using persistent entropy

In this paper, we apply persistent entropy, a novel topological statis- tic, for characterization of images of epithelial tissues. We have found out that persistent entropy is able to summarize topological and geomet- ric information encoded by -complexes and persistent homology. After using some statistical tests, we can guarantee the existence of signi cant di erences in the studied tissues.Ministerio de Economía y Competitividad MTM2015-67072-

Persistent entropy: a scale-invariant topological statistic for analyzing cell arrangements

In this work, we develop a method for detecting differences in the topological distribution of cells forming epithelial tissues. In particular, we extract topological information from their images using persistent homology and a summary statistic called persistent entropy. This method is scale invariant, robust to noise and sensitive to global topological features of the tissue. We have found significant differences between chick neuroepithelium and epithelium of Drosophila wing discs in both, larva and prepupal stages. Besides, we have tested our method, with good results, with images of mathematical tesselations that model biological tissues

Stable topological summaries for analyzing the organization of cells in a packed tissue

We use topological data analysis tools for studying the inner organization of cells in segmented images of epithelial tissues. More specifically, for each segmented image, we compute different persistence barcodes, which codify the lifetime of homology classes (persistent homology) along different filtrations (increasing nested sequences of simplicial complexes) that are built from the regions representing the cells in the tissue. We use a complete and well-grounded set of numerical variables over those persistence barcodes, also known as topological summaries. A novel combination of normalization methods for both the set of input segmented images and the produced barcodes allows for the proven stability results for those variables with respect to small changes in the input, as well as invariance to image scale. Our study provides new insights to this problem, such as a possible novel indicator for the development of the drosophila wing disc tissue or the importance of centroids’ distribution to differentiate some tissues from their CVT-path counterpart (a mathematical model of epithelia based on Voronoi diagrams). We also show how the use of topological summaries may improve the classification accuracy of epithelial images using a Random Forest algorithm.Ministerio de Ciencia e Innovación PID2019-107339GB-I0

Persistence Partial Matchings Induced by Morphisms between Persistence Modules

The notion of persistence partial matching, as a generalization of partial matchings between persistence modules, is introduced. We study how to obtain a persistence partial matching $\mathcal{G}_f$, and a partial matching $\mathcal{M}_f$, induced by a morphism $f$ between persistence modules, both being linear with respect to direct sums of morphisms. Some of their properties are also provided, including their stability after a perturbation of the morphism $f$, and their relationship with other induced partial matchings already defined in TDA

Bilateral early activation of retinal microglial cells in a mouse model of unilateral laser-induced experimental ocular hypertension

The immune system plays an important role in glaucomatous neurodegeneration. Retinal microglial reactivation associated with ganglion cell loss could reportedly contribute to the glaucoma progression. Recently we have described signs of microglia activation both in contralateral and ocular hypertension (OHT) eyes involving all retinal layers 15 days after OHT laser induction in mice. However, no works available have analyzed the microglial activation at earliest time points after OHT induction (24 h) in this experimental model. Thus, we seek to describe and quantify signs of microglia activation and differences depending on the retinal layer, 24 h after unilateral laser-induced OHT. Two groups of adult Swiss mice were used: age-matched control (naïve) and lasered. In the lasered animals, OHT eyes as well as contralateral eyes were analyzed. Retinal whole-mounts were immunostained with antibodies against Iba-1 and MHC-II. We quantified the number of microglial cells in the photoreceptor layer (OS), outer plexiform layer (OPL), and inner plexiform layer (IPL); the number of microglial vertical processes connecting the OPL and OS; the area of the retina occupied by Iba-1+ cells (Iba1-RA) in the nerve fiber layer-ganglion cell layer (NFL-GCL), the total arbor area of microglial cells in the OPL and IPL and; Iba-1+ cell body area in the OPL, IPL and NFL-GCL. In contralateral and OHT eyes the morphological features of Iba-1+ cell activation were: migration, enlargement of the cell body, higher degree of branching and reorientation of the processes, radial disposition of the soma and processes toward adjacent microglial plexuses, and presence of amoeboid cells acting as macrophages. These signs were more pronounced in OHT eyes. Most of Iba-1+ cells did not express MHC-II; rather, only dendritic and rounded cells expressed it. In comparison with naïve eyes, in OHT eyes and contralateral eyes no significant differences were found in the microglial cell number; but there was a significant increase in Iba1-RA. The total arbor area of microglial cells was significantly decreased in: i) OHT eyes with respect contralateral eyes and naïve-eyes in IPL; ii) OHT eyes with respect to naïve eyes in OPL. The number of microglial vertical processes connecting the OPL and OS were significantly increased in contralateral eyes compared with naïve-eyes and OHT eyes. In OPL, IPL and NFL-GCL, the cell body area of Iba-1+ cells was significantly greater in OHT eyes than in naïve and contralateral eyes, and greater in contralateral eyes than in naïve eyes. A non-proliferative microglial reactivation was detected both in contralateral eyes and in OHT eyes in an early time after unilateral laser-induced OHT (24 h). This fast microglial activation, which involves the contralateral eye, could be mediated by the immune system

Retinal Molecular Changes Are Associated with Neuroinflammation and Loss of RGCs in an Experimental Model of Glaucoma

Signaling mediated by cytokines and chemokines is involved in glaucoma-associated neuroinflammation and in the damage of retinal ganglion cells (RGCs). Using multiplexed immunoassay and immunohistochemical techniques in a glaucoma mouse model at different time points after ocular hypertension (OHT), we analyzed (i) the expression of pro-inflammatory cytokines, anti-inflammatory cytokines, BDNF, VEGF, and fractalkine; and (ii) the number of Brn3a+ RGCs. In OHT eyes, there was an upregulation of (i) IFN-γ at days 3, 5, and 15; (ii) IL-4 at days 1, 3, 5, and 7 and IL-10 at days 3 and 5 (coinciding with downregulation of IL1-β at days 1, 5, and 7); (iii) IL-6 at days 1, 3, and 5; (iv) fractalkine and VEGF at day 1; and (v) BDNF at days 1, 3, 7, and 15. In contralateral eyes, there were (i) an upregulation of IL-1β at days 1 and 3 and a downregulation at day 7, coinciding with the downregulation of IL4 at days 3 and 5 and the upregulation at day 7; (ii) an upregulation of IL-6 at days 1, 5, and 7 and a downregulation at 15 days; (iii) an upregulation of IL-10 at days 3 and 7; and (iv) an upregulation of IL-17 at day 15. In OHT eyes, there was a reduction in the Brn3a+ RGCs number at days 3, 5, 7, and 15. OHT changes cytokine levels in both OHT and contralateral eyes at different time points after OHT induction, confirming the immune system involvement in glaucomatous neurodegeneration