Geometric inequalities on Heisenberg groups

We establish geometric inequalities in the sub-Riemannian setting of the Heisenberg group $\mathbb H^n$. Our results include a natural sub-Riemannian version of the celebrated curvature-dimension condition of Lott-Villani and Sturm and also a geodesic version of the Borell-Brascamp-Lieb inequality akin to the one obtained by Cordero-Erausquin, McCann and Schmuckenschl\"ager. The latter statement implies sub-Riemannian versions of the geodesic Pr\'ekopa-Leindler and Brunn-Minkowski inequalities. The proofs are based on optimal mass transportation and Riemannian approximation of $\mathbb H^n$ developed by Ambrosio and Rigot. These results refute a general point of view, according to which no geometric inequalities can be derived by optimal mass transportation on singular spaces.Comment: to appear in Calculus of Variations and Partial Differential Equations (42 pages, 1 figure

Explaining the elongated shape of 'Oumuamua by the Eikonal abrasion model

The photometry of the minor body with extrasolar origin (1I/2017 U1) 'Oumuamua revealed an unprecedented shape: Meech et al. (2017) reported a shape elongation b/a close to 1/10, which calls for theoretical explanation. Here we show that the abrasion of a primordial asteroid by a huge number of tiny particles ultimately leads to such elongated shape. The model (called the Eikonal equation) predicting this outcome was already suggested in Domokos et al. (2009) to play an important role in the evolution of asteroid shapes.Comment: Accepted by the Research Notes of the AA

A Universal Voltammetric Cell

In eleotroanalytical research it is desirable to wor·k with a cell which is easily used for various electrochemical measurements. The requirem~μts for such a cell are: 1) ability to function with different types of electrodes,, 2) perfectly reproducible hydrodynamic conditions, 3) ·simple replacement \u27of the sample and electrodes, and 4) construction material which is inert in respect to the sample

Jacobian determinant inequality on corank 1 Carnot groups with applications

We establish a weighted pointwise Jacobian determinant inequality on corank 1 Carnot groups related to optimal mass transportation akin to the work of Cordero-Erausquin, McCann and Schmuckenschläger. In this setting, the presence of abnormal geodesics does not allow the application of the general sub-Riemannian optimal mass transportation theory developed by Figalli and Rifford and we need to work with a weaker notion of Jacobian determinant. Nevertheless, our result achieves a transition between Euclidean and sub-Riemannian structures, corresponding to the mass transportation along abnormal and strictly normal geodesics, respectively. The weights appearing in our expression are distortion coefficients that reflect the delicate sub-Riemannian structure of our space. As applications, entropy, Brunn-Minkowski and Borell-Brascamp-Lieb inequalities are established on Carnot groups

Using neural networks to obtain indirect information about the state variables in an alcoholic fermentation process

This work provides a manual design space exploration regarding the structure, type, and inputs of a multilayer neural network (NN) to obtain indirect information about the state variables in the alcoholic fermentation process. The main benefit of our application is to help experts reduce the time needed for making the relevant measurements and to increase the lifecycles of sensors in bioreactors. The novelty of this research is the flexibility of the developed application, the use of a great number of variables, and the comparative presentation of the results obtained with different NNs (feedback vs. feed-forward) and different learning algorithms (Back-Propagation vs. Levenberg–Marquardt). The simulation results show that the feedback neural network outperformed the feed-forward neural network. The NN configuration is relatively flexible (with hidden layers and a number of nodes on each of them), but the number of input and output nodes depends on the fermentation process parameters. After laborious simulations, we determined that using pH and CO2 as inputs reduces the prediction errors of the NN. Thus, besides the most commonly used process parameters like fermentation temperature, time, the initial concentration of the substrate, the substrate concentration, and the biomass concentration, by adding pH and CO2, we obtained the optimum number of input nodes for the network. The optimal configuration in our case was obtained after 1500 iterations using a NN with one hidden layer and 12 neurons on it, seven neurons on the input layer, and one neuron as the output. If properly trained and validated, this model can be used in future research to accurately predict steady-state and dynamic alcoholic fermentation process behaviour and thereby improve process control performance

Identification of epidermal Pdx1 expression discloses different roles of Notch1 and Notch2 in murine KrasG12D-induced skin carcinogenesis in vivo

Background The Ras and Notch signaling pathways are frequently activated during development to control many diverse cellular processes and are often dysregulated during tumorigenesis. To study the role of Notch and oncogenic Kras signaling in a progenitor cell population, Pdx1-Cre mice were utilized to generate conditional oncogenic KrasG12D mice with ablation of Notch1 and/or Notch2. Methodology/Principal Findings Surprisingly, mice with activated KrasG12D and Notch1 but not Notch2 ablation developed skin papillomas progressing to squamous cell carcinoma providing evidence for Pdx1 expression in the skin. Immunostaining and lineage tracing experiments indicate that PDX1 is present predominantly in the suprabasal layers of the epidermis and rarely in the basal layer. Further analysis of keratinocytes in vitro revealed differentiation-dependent expression of PDX1 in terminally differentiated keratinocytes. PDX1 expression was also increased during wound healing. Further analysis revealed that loss of Notch1 but not Notch2 is critical for skin tumor development. Reasons for this include distinct Notch expression with Notch1 in all layers and Notch2 in the suprabasal layer as well as distinctive p21 and β-catenin signaling inhibition capabilities. Conclusions/Significance Our results provide strong evidence for epidermal expression of Pdx1 as of yet not identified function. In addition, this finding may be relevant for research using Pdx1-Cre transgenic strains. Additionally, our study confirms distinctive expression and functions of Notch1 and Notch2 in the skin supporting the importance of careful dissection of the contribution of individual Notch receptors