In this paper, the authors analyze the developed PreFirewall network application for the Floodlight software defined network (SDN) controller. This application filters rules, which are added into the firewall module of the Floodlight SDN controller in order to prevent the occurrence of anomalies among them. The rule filtering method is based on determining whether the addition of a new rule will not cause any anomalies with already added ones. If an anomaly was detected while adding the new rule, PreFirewall application should be able to resolve it and must report the detection of the anomaly. The developed network application PreFirewall passed a number of tests. As a result of the stress testing, it was found that the time of adding a new rule, when using PreFirewall, substantially increases with increase in the number of previously processed rules. Analysis of the network application PreFirewall showed that while adding a rule (the most frequent operation), in the worst case it is necessary to compare it with all existing rules, which are stored as a two-dimensional array. Thus, the operation of adding a new rule is the most time-consuming and has the greatest impact on the performance of the network application, which leads to an increase in response time. A possible way to of solving this problem is to select a data structure used to store the rules, in which the operation of adding a new rule would be simple. After analyzing the structure of the policy rules for the Floodlight SDN controller, the authors noted that a tree is the most adequate data structure for its storage. It provides optimization of memory used for storing the rules and, more important, it allows to achieve the constant complexity of the operation of adding a new rule and, consequently, solving the performance problem of the network application PreFirewall. The article is published in the authors’ wording.

We consider environmental-economical models of optimal harvesting, given by the differential equations with impulse action, which depend on random parameters. We assume, that lengths of intervals θ_k between the moments of impulses τ_k are random variables and the sizes of impulse influence depend on random parameters v_k, k = 1, 2, . . . One example of such objects is an equation with impulses, modelling dynamics of the population subject to harvesting. In the absence of harvesting, the population development is described by the differential equation ˙x = g(x) and in time moments τk some random share of resource v_k, k = 1, 2, . . . is taken from population. We can control gathering process so that to stop harvesting when its share will appear big enough to keep possible biggest the rest of a resource to increase the size of the following gathering. Let the equation ˙x = g(x) have an asymptotic stable solution ϕ(t) ≡ K and the interval (K₁, K₂) is the attraction area of the given solution (here 0 ≤ K₁ < K < K₂). We construct the control u = (u₁, . . . , u_k, . . .), limiting a share of harvesting resource at each moment of time τk, so that the quantity of the remained resource, since some moment τk0 , would be not less than the given value x ∈ (K₁, K). For any x ∈ (K₁, K) the estimations of average time profit, valid with probability one, are received. It is shown, that there is a unique x^∗ ∈ (K₁, K), at which the lower estimation reaches the greatest value. Thus, we described the way of population control at which the value of average time profit can be lower estimated with probability 1 by the greatest number whenever possible.

The problem of selection by the patch population in the absence of information on the utility of the patch, that is, the volume of its energy resources, is considered. This problem relates to the theory of optimal foraging. U. Dieckman proposed an approach to modeling the population patch distribution. The approach is based on a utility function that takes into account the amount of resources in a patch, the population − patch distance, and the measure of information certainty on patch utility. In this case, the Boltzmann distribution is used to describe the population patch distribution. And U. Dieckman considered a static problem that does not take into account the change in the position of the population with time. In this paper, we propose a dynamic system that describes the population patch distribution, which depends on the utility of the patch. In addition the utility varies with time as a result of distance variations. The Boltzmann distribution is a particular solution of the proposed system of differential equations. The Lyapunov stability condition for the Boltzmann distribution is obtained.The utility functions of the patches, which depend on the population − patch distance and on the measure of the information certainty, are introduced. As a result, in the two-dimensional case, a space R² is divided into areas of preferred utility. Such a partition is a generalization of the Voronoi diagram.

In the paper, the analysis of the stability of the McEliece-type cryptosystem on induced codes for key attacks is examined. In particular, a model is considered when the automorphism group is trivial for the base code C, on the basis of which the induced code F^l_q ⊗ C is constructed. In this case, as shown by N. Sendrier in 2000, there exists such a mapping, called a complete discriminant, by means of which a secret permutation that is part of the secret key of a McEliece-type cryptosystem can be effectively found. The automorphism group of the code F^l_q ⊗ C is nontrivial, therefore there is no complete discriminant for this code. This suggests a potentially high resistance of the McEliece-type cryptosystem on the code F^l_q ⊗ C. The algorithm for splitting the support for the code F^l_q ⊗ C is constructed and the efficiency of this algorithm is compared with the existing attack on the key of the McElice type cryptosystem based on the code F^l_q ⊗ C.

The function \(f\in L_p[I], \;p>0,\) is called \((k,p)\)-differentiable at a point \(x_0\in I\) if there exists an algebraic polynomial of \(\pi\) of degree no more than \(k\) for which holds \( \Vert f-\pi \Vert_{L_p[J_h]} = o(h^{k+\frac{1}{p}}), \) where \(\;J_h=[x_0-h; x_0+h]\cap I.\) At an internal point for \(k=1\) and \(p=\infty\) this is equivalent to the usual definition of the function differentiability. At an interior point for \(k=1\) and \(p=\infty\), the definition is equivalent to the usual differentiability of the function. There is a standard "hierarchy" for the existence of differentials(if \(p_1<p_2,\) then \((k,p_2)\)-differentiability should be \((k,p_1)\)-differentiability. In the works of S.N. Bernstein, A.P. Calderon and A. Zygmund were given applications of such a construction to build a description of functional spaces (\(p=\infty\)) and the study of local properties of solutions of differential equations \((1\le p\le\infty)\), respectively. This article is related to the first mentioned work. The article introduces the concept of uniform differentiability. We say that a function \(f\), \((k,p)\)-differentiable at all points of the segment \(I\), is uniformly \((k,p)\)-differentiable on \(I\) if for any number \(\varepsilon>0\) there is a number \(\delta>0\) such that for each point \(x\in I\) runs \( \Vert f-\pi\Vert_{L_p[J_h]}<\varepsilon\cdot h^{k+\frac{1}{p}} \; \) for \(0<h<\delta, \; J_h = [x\!-\!H; x\!+\!h]\cap I,\) where \(\pi\) is the polynomial of the terms of the \((k, p)\)-differentiability at the point \(x\). Based on the methods of local approximations of functions by algebraic polynomials it is shown that a uniform \((k,p)\)-differentiability of the function \(f\) at some \(1\le p\le\infty\) implies  \(f\in C^k[I].\) Therefore, in this case the differentials are "equivalent". Since every function from \(C^k[I]\) is uniformly \((k,p)\)-differentiable on the interval \(I\) at \(1\le p\le\infty,\) we obtain a certain criterion of belonging to this space. The range \(0<p<1,\) obviously, can be included into the necessary condition the membership of the function \(C^k[I]\), but the sufficiency of Taylor differentiability in this range has not yet been fully proven.

Let \(n\in{\mathbb N}\), and let \(Q_n\) be the unit cube \([0,1]^n\). By \(C(Q_n)\) we denote the space of continuous functions \(f:Q_n\to{\mathbb R}\) with the norm \(\|f\|_{C(Q_n)}:=\max\limits_{x\in Q_n}|f(x)|,\) by \(\Pi_1\left({\mathbb R}^n\right)\) --- the set of polynomials of \(n\) variables of degree \(\leq 1\) (or linear functions). Let \(x^{(j)},\) \(1\leq j\leq n+1,\) be the vertices of \(n\)-dimnsional nondegenerate simplex \(S\subset Q_n\). An interpolation projector \(P:C(Q_n)\to \Pi_1({\mathbb R}^n)\) corresponding to the simplex \(S\) is defined by equalities \(Pf\left(x^{(j)}\right)= f\left(x^{(j)}\right).\) The norm of \(P\) as an operator from \(C(Q_n)\) to \(C(Q_n)\) may be calculated by the formula \(\|P\|=\max\limits_{x\in ver(Q_n)} \sum\limits_{j=1}^{n+1} |\lambda_j(x)|.\) Here \(\lambda_j\) are the basic Lagrange polynomials with respect to \(S,\) \(ver(Q_n)\) is the set of vertices of \(Q_n\). Let us denote by \(\theta_n\) the minimal possible value of \(\|P\|.\) Earlier, the first author proved various relations and estimates for values \(\|P\|\) and \(\theta_n\), in particular, having geometric character. The equivalence \(\theta_n\asymp \sqrt{n}\) takes place. For example, the appropriate, according to dimension \(n\), inequalities may be written in the form \linebreak \(\frac{1}{4}\sqrt{n}\) \(<\theta_n\) \(<3\sqrt{n}.\) If the nodes of the projector \(P^*\) coincide with vertices of an arbitrary simplex with maximum possible volume, we have \(\|P^*\|\asymp\theta_n.\)
When an Hadamard matrix of order \(n+1\) exists, holds \(\theta_n\leq\sqrt{n+1}.\) In the paper, we give more precise upper bounds of numbers \(\theta_n\) for \(21\leq n \leq 26\). These estimates were obtained with the application of maximum volume simplices in the cube. For constructing such simplices, we utilize maximum determinants containing the elements \(\pm 1.\) Also, we systematize and comment the best nowaday upper and low estimates of numbers \(\theta_n\) for a concrete \(n.\)

The Hodge, Tate and Mumford-Tate conjectures are proved for the fibre product of two non-isotrivial 1-parameter families of regular surfaces with geometric genus 1 under some conditions on degenerated fibres, the ranks of the N\'eron - Severi groups of generic geometric fibres and representations of Hodge groups in transcendental parts of rational cohomology.
Let \(\pi_i:X_i\to C\quad (i = 1, 2)\) be a projective non-isotrivial family (possibly with degeneracies) over a smooth projective curve \(C\). Assume that the discriminant loci \(\Delta_i=\{\delta\in C\,\,\vert\,\, Sing(X_{i\delta})\neq\varnothing\} \quad (i = 1, 2)\) are disjoint, \(h^{2,0}(X_{ks})=1,\quad h^{1,0}(X_{ks}) = 0\) for any smooth fibre \(X_{ks}\), and the following conditions hold:
\((i)\) for any point \(\delta \in \Delta_i\) and the Picard-Lefschetz transformation \( \gamma \in GL(H^2 (X_{is}, Q)) \), associated with a smooth part \(\pi'_i: X'_i\to C\setminus\Delta_i\) of the morphism \(\pi_i\) and with a loop around the point \(\delta \in C\), we have \((\log(\gamma))^2\neq0\);
\((ii)\) the variety \(X_i \, (i = 1, 2)\), the curve \(C\) and the structure morphisms \(\pi_i:X_i\to C\) are defined over a finitely generated subfield \(k \hookrightarrow C\).
If for generic geometric fibres \(X_{1s}\) \, and \, \(X_{2s}\) at least one of the following conditions holds: \((a)\) \(b_2(X_{1s})- rank NS(X_{1s})\) is an odd prime number, \(\quad\,\,\) \(b_2(X_{1s})- rank NS(X_{1s})\neq b_2(X_{2s})- rank NS(X_{2s})\); \((b)\) the ring \(End_{ Hg(X_{1s})} NS_ Q(X_{1s})^\perp\) is an imaginary quadratic field, \(\quad\,\, b_2(X_{1s})- rank NS(X_{1s})\neq 4,\) \(\quad\,\, End_{ Hg(X_{2s})} NS_ Q(X_{2s})^\perp\) is a totally real field or \(\,\, b_2(X_{1s})- rank NS(X_{1s})\,>\, b_2(X_{2s})- rank NS(X_{2s})\) ; \((c)\) \([b_2(X_{1s})- rank NS(X_{1s})\neq 4, \, End_{ Hg(X_{1s})} NS_ Q(X_{1s})^\perp= Q\); \(\quad\,\,\) \(b_2(X_{1s})- rank NS(X_{1s})\neq b_2(X_{2s})- rank NS(X_{2s})\),
then for the fibre product \(X_1 \times_C X_2\) the Hodge conjecture is true, for any smooth projective \(k\)-variety \(X_0\) with the condition \(X_1 \times_C X_2\) \(\widetilde{\rightarrow}\) \(X_0 \otimes_k C\) the Tate conjecture on algebraic cycles and the Mumford-Tate conjecture for cohomology of even degree are true.

We establish lower estimates for an integral functional
$$\int\limits_\Omega f(u(x), \nabla u(x)) \, dx ,$$
where \(\Omega\) -- a bounded domain in \(\mathbb{R}^n \; (n \geqslant 2)\), an integrand \(f(t,p) \, (t \in [0, \infty),\; p \in \mathbb{R}^n)\) -- a function that is \(B\)-measurable with respect to a variable \(t\) and is convex and even in the variable \(p\), \(\nabla u(x)\) -- a gradient (in the sense of Sobolev) of the function \(u \colon \Omega \rightarrow \mathbb{R}\). In the first and the second sections we utilize properties of permutations of differentiable functions and an isoperimetric inequality \(H^{n-1}( \partial A) \geqslant \lambda(m_n A)\), that connects \((n-1)\)-dimensional Hausdorff measure \(H^{n-1}(\partial A )\) of relative boundary \(\partial A\) of the set \(A \subset \Omega\) with its \(n\)-dimensional Lebesgue measure \(m_n A\). The integrand \(f\) is assumed to be isotropic, i.e. \(f(t,p) = f(t,q)\) if \(|p| = |q|\).
Applications of the established results to multidimensional variational problems are outlined. For functions \( u \) that vanish on the boundary of the domain \(\Omega\), the assumption of the isotropy of the integrand \( f \) can be omitted. In this case, an important role is played by the Steiner and Schwartz symmetrization operations of the integrand \( f \) and of the function \( u \). The corresponding variants of the lower estimates are discussed in the third section. What is fundamentally new here is that the symmetrization operation is applied not only to the function \(u\), but also to the integrand \(f\). The geometric basis of the results of the third section is the Brunn-Minkowski inequality, as well as the symmetrization properties of the algebraic sum of sets.