Polyhedral approximations of strictly convex compacta

We consider polyhedral approximations of strictly convex compacta in finite dimensional Euclidean spaces (such compacta are also uniformly convex). We obtain the best possible estimates for errors of considered approximations in the Hausdorff metric. We also obtain new estimates of an approximate algorithm for finding the convex hulls

Noise-induced escape in an excitable system

We consider the stochastic dynamics of escape in an excitable system, the FitzHugh-Nagumo (FHN) neuronal model, for different classes of excitability. We discuss, first, the threshold structure of the FHN model as an example of a system without a saddle state. We then develop a nonlinear (nonlocal) stability approach based on the theory of large fluctuations, including a finite-noise correction, to describe noise-induced escape in the excitable regime. We show that the threshold structure is revealed via patterns of most probable (optimal) fluctuational paths. The approach allows us to estimate the escape rate and the exit location distribution. We compare the responses of a monostable resonator and monostable integrator to stochastic input signals and to a mixture of periodic and stochastic stimuli. Unlike the commonly used local analysis of the stable state, our nonlocal approach based on optimal paths yields results that are in good agreement with direct numerical simulations of the Langevin equation

Weakly convex sets and modulus of nonconvexity

We consider a definition of a weakly convex set which is a generalization of the notion of a weakly convex set in the sense of Vial and a proximally smooth set in the sense of Clarke, from the case of the Hilbert space to a class of Banach spaces with the modulus of convexity of the second order. Using the new definition of the weakly convex set with the given modulus of nonconvexity we prove a new retraction theorem and we obtain new results about continuity of the intersection of two continuous set-valued mappings (one of which has nonconvex images) and new affirmative solutions of the splitting problem for selections. We also investigate relationship between the new definition and the definition of a proximally smooth set and a smooth set

Local characterization of strongly convex sets

Strongly convex sets in Hilbert spaces are characterized by local properties. One quantity which is used for this purpose is a generalization of the modulus of convexity \delta_\Omega of a set \Omega. We also show that \lim_{\epsilon \to 0} \delta_\Omega(\epsilon)/\epsilon^2 exists whenever \Omega is closed and convex

On properties of riemannian metrics associated with B-Elliptic operators

In this paper, we consider a Riemannian metric in which the Laplace-Beltrami operator coincides with a B-elliptic operator up to a facto

Solving problems of clustering and classification of cancer diseases based on DNA methylation data

The article deals with the problem of diagnosis of oncological diseases based on the analysis of DNA methylation data using algorithms of cluster analysis and supervised learning. The groups of genes are identified, methylation patterns of which significantly change when cancer appears. High accuracy is achieved in classification of patients impacted by different cancer types and in identification if the cell taken from a certain tissue is aberrant or normal. With method of cluster analysis two cancer types are highlighted for which the hypothesis was confirmed stating that among the people affected by certain cancer types there are groups with principally different methylation pattern

Space of images of the mixed Riesz hyperbolic B-potential and analytic continuation

In this paper we prove semigroup properties for the mixed Riesz hyperbolic B-potential, find its analytic continuation and describe a space of images of mixed hyperbolic Riesz B-potentials. This problem is closely related to the problem of inversion of the weighted Radon transform on Lorentzian manifold

On the stability of stationary states in diffusion models in biology and humanities

We consider an initial-boundary value problem for the system of partial differential equations describing processes of growth and spread of substance in biology, sociology, economics and linguistic

SPECIMEN EXAMINATION FOLLOWING SURGERY FOR RECTAL CANCER: OUR TECHNIQUE

Standard surgical treatment for rectal cancer is a total mesorectumectomy (TME) and it demands performing proper pathohistological examination of the removed specimen to select patients with high risk of local recurrence development and those indicated for adjuvant therapy. Pathologists should value quality of surgical intervention, examine both distal and circular resection margins (surgically mobilized mesorectal surface), define surgical clearance (distance from the circular resection margine to the tumor itself), perform pathological staging ((у)pTpN), determine histological type and stage of tumor differentiation. In this paper we describe a route of the specimen removed after the surgery for rectal cancer and its pathohistological examination technique which is based on the guidances of the British Royal Society of Pathologists specialized to the Regional Clinic Hospital requirements