Abstract:
This article is devoted to experimental investigation of a novel application of a clustering technique introduced by the authors recently in order to use robust and stable consensus functions in information security, where it is often necessary to process large data sets and monitor outcomes in real time, as it is required, for example, for intrusion detection. Here we concentrate on a particular case of application to profiling of phishing websites. First, we apply several independent clustering algorithms to a randomized sample of data to obtain independent initial clusterings. Silhouette index is used to determine the number of clusters. Second, rank correlation is used to select a subset of features for dimensionality reduction. We investigate the effectiveness of the Pearson Linear Correlation Coefficient, the Spearman Rank Correlation Coefficient and the Goodman--Kruskal Correlation Coefficient in this application. Third, we use a consensus function to combine independent initial clusterings into one consensus clustering. Fourth, we train fast supervised classification algorithms on the resulting consensus clustering in order to enable them to process the whole large data set as well as new data. The precision and recall of classifiers at the final stage of this scheme are critical for the effectiveness of the whole procedure. We investigated various combinations of several correlation coefficients, consensus functions, and a variety of supervised classification algorithms.

Abstract:
Large outliers break down linear and nonlinear regression models. Robust regression methods allow one to filter out the outliers when building a model. By replacing the traditional least squares criterion with the least trimmed squares criterion, in which half of data is treated as potential outliers, one can fit accurate regression models to strongly contaminated data. High-breakdown methods have become very well established in linear regression, but have started being applied for non-linear regression only recently. In this work, we examine the problem of fitting artificial neural networks to contaminated data using least trimmed squares criterion. We introduce a penalized least trimmed squares criterion which prevents unnecessary removal of valid data. Training of ANNs leads to a challenging non-smooth global optimization problem. We compare the efficiency of several derivative-free optimization methods in solving it, and show that our approach identifies the outliers correctly when ANNs are used for nonlinear regression.

Abstract:
This paper establishes connections between the structure of a semigroup and the minimum spans of distance labellings of its Cayley graphs. We show that certain general restrictions on the minimum spans are equivalent to the semigroup being combinatorial, and that other restrictions are equivalent to the semigroup being a right zero band. We obtain a description of the structure of all semigroups $S$ and their subsets $C$ such that $\Cay(S,C)$ is a disjoint union of complete graphs, and show that this description is also equivalent to several restrictions on the minimum span of $\Cay(S,C)$. We then describe all graphs with minimum spans satisfying the same restrictions, and give examples to show that a fairly straightforward upper bound for the minimum spans of the underlying undirected graphs of Cayley graphs turns out to be sharp even for the class of combinatorial semigroups.

Abstract:
The main theorem of this paper gives a formula forthe largest minimum distance of error-correcting codesconsidered as ideals in incidence rings defined bydirected graphs.

Abstract:
Parkinson’s disease is the second most common neu- rodegenerative disorder after Alzheimer disease affecting 1% - 2% in people >60 years old and 3% - 4% in people >80.

Gravitation is one of
the central forces playing an important role
in formation of natural systems like
galaxies and planets. Gravitational forces between particles of a
gaseous cloud transform the cloud into spherical shells and disks of higher
density during gravitational contraction. The density can reach that of a solid
body. The theoretical model was tested to model the formation of a spiral
galaxy and Saturn. The formations of a spiral galaxy and Saturn and its disk
are simulated using a novel N-body self-gravitational model. It is
demonstrated that the formation of the spirals of the galaxy and disk of the
planet is the result of gravitational contraction of a slowly rotated
particle cloud that has a shape of slightly deformed sphere for Saturn and
ellipsoid for the spiral galaxy. For Saturn, the sphere was flattened by a
coefficient of 0.8 along the axis of rotation. During the gravitational contraction, the major part of the cloud transformed into a planet and a minor part transformed into a disk. The
thin structured disk is a result of the electromagnetic
interaction in which the magnetic forces acting on charged particles of
the cloud originate from the core of the planet.

Abstract:
If the wave functions of matter expanded with time dilation for an outside observer in the same way as photons do in gravitational redshift; with some modifications the general relativity might alone explain dark matter, galaxy rotation curves, and part of the energy released in supernova explosions. Also, the event horizons of black holes couldn’t be formed when packing matter more and more densely together. Essentially, if the time dilation increases enough, the particles turn less localized to outside observers and the mass distribution of the same particles would expand into larger volume of space. Small particles deep inside a black hole might seem like dark matter instead by their gravitational influence if the time dilation alters their size enough for outside observers. At the same time, the surface particles of the black hole would be less dispersed, creating the Newtonian gravitational potential we see closer to black holes. The following research doesn’t attempt to reformulate the general relativity itself, but only proposes the idea while approximating the Milky way gravity profile to compare the hypothesis with measurements. Therefore, actually proving the hypothesis is still far off while the idea is sound at its core.

We establish the uniqueness and local existence of weak solutions for a
system of partial differential equations which describes non-linear motions of
viscous stratified fluid in a homogeneous gravity field. Due to the presence of
the stratification equation for the density, the model and the problem are new
and thus different from the classical Navier-Stokes equations.

Abstract:
In this work, the classical Borsuk conjecture is discussed, which states that any set of diameter 1 in the Euclidean space $ {\mathbb R}^d $ can be divided into $ d+1 $ parts of smaller diameter. During the last two decades, many counterexamples to the conjecture have been proposed in high dimensions. However, all of them are sets of diameter 1 that lie on spheres whose radii are close to the value $ {1}{\sqrt{2}} $. The main result of this paper is as follows: {\it for any $ r > {1}{2} $, there exists a $ d_0 $ such that for all $ d \ge d_0 $, a counterexample to Borsuk's conjecture can be found on a sphere $ S_r^{d-1} \subset {\mathbb R}^d $.

Abstract:
We use a special space of integrable functions for studying theCauchy problem for linear functional-differential equations withnonintegrable singularities. We use the ideas developed byAzbelev and his students (1995). We show that bychoosing the function ψ generating the space, one canguarantee resolubility and certain behavior of the solution nearthe point of singularity.