Revisiting Ciric type nonunique fixed point theorems via interpolation

[EN] In this paper, we aim to revisit some non-unique fixed point theorems that were initiated by Ciric, first. We consider also some natural consequences of the obtained results. In addition, we provide a simple example to illustrate the validity of the main result.Karapinar, E. (2021). Revisiting Ciric type nonunique fixed point theorems via interpolation. Applied General Topology. 22(2):483-496. https://doi.org/10.4995/agt.2021.16562OJS48349622

Towards theory of C-symmetries

The concept of C-symmetry originally appeared in PT-symmetric quantum mechanics is studied within the Krein spaces framework

Hypothesis Testing For Network Data in Functional Neuroimaging

In recent years, it has become common practice in neuroscience to use networks to summarize relational information in a set of measurements, typically assumed to be reflective of either functional or structural relationships between regions of interest in the brain. One of the most basic tasks of interest in the analysis of such data is the testing of hypotheses, in answer to questions such as "Is there a difference between the networks of these two groups of subjects?" In the classical setting, where the unit of interest is a scalar or a vector, such questions are answered through the use of familiar two-sample testing strategies. Networks, however, are not Euclidean objects, and hence classical methods do not directly apply. We address this challenge by drawing on concepts and techniques from geometry, and high-dimensional statistical inference. Our work is based on a precise geometric characterization of the space of graph Laplacian matrices and a nonparametric notion of averaging due to Fr\'echet. We motivate and illustrate our resulting methodologies for testing in the context of networks derived from functional neuroimaging data on human subjects from the 1000 Functional Connectomes Project. In particular, we show that this global test is more statistical powerful, than a mass-univariate approach. In addition, we have also provided a method for visualizing the individual contribution of each edge to the overall test statistic.Comment: 34 pages. 5 figure

Diffusion in multiscale spacetimes

We study diffusion processes in anomalous spacetimes regarded as models of quantum geometry. Several types of diffusion equation and their solutions are presented and the associated stochastic processes are identified. These results are partly based on the literature in probability and percolation theory but their physical interpretation here is different since they apply to quantum spacetime itself. The case of multiscale (in particular, multifractal) spacetimes is then considered through a number of examples and the most general spectral-dimension profile of multifractional spaces is constructed.Comment: 23 pages, 5 figures. v2: discussion improved, typos corrected, references adde

Sharp nonuniqueness for the Navier-Stokes equations

In this paper, we prove a sharp nonuniqueness result for the incompressible Navier-Stokes equations in the periodic setting. In any dimension $d \geq 2$ and given any $p<2$, we show the nonuniqueness of weak solutions in the class $L^{p}_t L^\infty$, which is sharp in view of classical uniqueness results. The proof is based on the construction of a class of non-Leray-Hopf weak solutions. More specifically, for any $p0$, we construct non-Leray-Hopf weak solutions $u \in L^{p}_t L^\infty \cap L^1_t W^{1,q}$ that are locally smooth outside a singular set in time of Hausdorff dimension less than $\varepsilon$. As a byproduct, examples of anomalous dissipation in the class $L^{ {3}/{2} - \varepsilon}_t C^{ {1}/{3}}$ for the Euler equations are given