A singularly perturbed elliptic problem with Dirichlet boundary conditions is considered in the case of multiple roots of the degenerate equation. A complete asymptotic expansion of the solution is constructed and justified. It is qualitatively different from the known expansion in the case where the root of the degenerate equation is simple: the asymptotic expansion of the solution being in fractional powers of the small parameter, boundary-layer variables have a different scale, boundary-layer series is constructed using a non-standard algorithm, the boundary layer in the vicinity of the domain boundary consists of three zones with different behavior of the solution in different zones.

Modern numerical methods allowing to solve contrast structure problems in the most efficient way are described. These methods include explicit-implicit Rosenbrock schemes with complex coefficients and fully implicit backward optimal Runge–Kutta schemes. As an integration argument, it is recommended to choose the length of the integral curve arc. This argument provides high reliability of the calculation and sufficiently decreases the complexity of computations for low-order systems. In order to increase the efficiency, we propose an automatic step selection algorithm based on curvature of the integral curve. This algorithm is as efficient as standard algorithms and has sufficiently larger reliability. We show that along with such an automatic step selection it is possible to calculate a posteriori asymptotically precise error estimation. Standard algorithms do not provide such estimations and their actual error quite often exceeds the user-defined tolerance by several orders. The applicability limitations of numerical methods are investigated. In solving superstiff problems, they sometimes do not provide satisfactory results. In such cases, it is recommended to imply approximate analytical methods. Consequently, numerical and analytical methods are complementary.

In this paper, we consider a numerical solution of Maxwell’s curl equations for piecewise uniform dielectric medium by the example of a one-dimensional problem. For obtaining the second order accuracy, the electric field grid node is placed into the permittivity discontinuity point of the medium. If the dielectric permittivity is large, the problem becomes singularly perturbed and a contrast structure appears. We propose a piecewise quasi-uniform mesh which resolves all characteristic solution parts of the problem (regular part, boundary layer and transition zone placed between them) in detail. The features of the mesh are discussed. 

The work is devoted to the dynamic properties of the solutions of boundary value problems associated with the classical system of Fermi – Pasta – Ulam (FPU). We study this problem in infinite-dimensional case, when a countable number of roots of characteristic equations tend to an imaginary axis. Under these conditions, we built a special non-linear partial differential equation, which plays the role of a quasinormal form, i.e, it defines the dynamics of the original boundary value problem with the initial conditions in a sufficiently small neighborhood of the equilibrium state. The modified Korteweg de Vries (KdV) equation and the Korteweg de Vries Burgers (KdVB) one are quasinormal forms depending on the parameter values. Under some additional assumptions, we apply the procedure of renormalization to the obtained boundary value problems. This procedure leads to an infinite-dimensional system of ordinary differential equations. We describe a method of folding this system in the special boundary value problem, which is an analogue of the normal form. The main result is that the analytical methods of nonlinear dynamics explored the interaction of waves moving in different directions, in the problem of the FPU. It was shown that waves influence on each other is asymptotically small and does not change the shape of waves, contributing only a shift in their speed, which does not change over time.

Creating adequate mathematical models of processes in living nature is an important task of modern biophysics. Blood clotting, nerve impulse propagation, reduction of the heart muscle, the pattern-formation in nature are auto-wave processes. FitzHugh–Nagumo system of equations is used to describe the auto-wave processes in active media. Such math problems are usually solved by numerical methods. The use of resource-intensive algorithms is required in the case of auto-wave solutions with sharp gradients. Therefore, it is appropriate to use the analytical methods for this type of problems. In this paper, the asymptotic method of contrast structures theory is used to obtain an approximate solution of a singularly perturbed system of FitzHugh–Nagumo type. The method allows to reduce the non-linear system of equations to a number of problems that can be solved analytically or with a stable numerical algorithm. This study presents the asymptotic approximation of a stationary auto-wave solution of the considered system. Additionally, this paper provides a formula that specifies the location of internal transition layers. The results were compared with the numerical solution. The application of contrast structures theory to the study of active media models can be used for analytical studies of other such systems, improving existing models and increasing the efficiency of the numerical calculations.

For a singularly perturbed one-dimensional parabolic equation with a perturbation parameter ε multiplying the highest-order derivative in the equation, ε ∈ (0,1], an initial-boundary value Neumann problem is considered on a segment. In this problem, when the parameter ε tends to zero, boundary layers appear in neighborhoods of the lateral boundary. In the paper, convergence of the problem solution and its regular and singular components are studied. It is shown that standard finite difference schemes on uniform grids used to numerically solve this problem do not converge ε-uniformly. The error in the grid solution grows unboundedly when the parameter ε → 0. The use of a special difference scheme on Shishkin grid which is a piecewise-uniform mesh with respect to x condensing in neighborhoods of boundary layers and a uniform mesh in t, constructed by using monotone grid approximations of the differential problems, allows us to find a numerical solution of this problem convergent in the maximum norm ε-uniformly. The results of the numerical experiments confirm the theoretical results.

In this paper, for a singularly perturbed parabolic reaction-diffusion equation with a perturbation parameter <i>ε</i>², <i>ε</i> ∈ (0,1], multiplying the highest-order derivative in the equation, an initial-boundary value Dirichlet problem is considered. For this problem, a standard difference scheme constructed by using monotone grid approximations of the differential problem on uniform grids, is studied in the presence of computer perturbations. Perturbations of grid solutions are studied, which are generated by computer perturbations, i.e., the computations on a computer. The conditions imposed on admissible computer perturbations are obtained under which the accuracy of the perturbed computer solution is the same by order as the solution of an unperturbed difference scheme, i.e., a standard scheme in the absence of perturbations. The schemes of this type with controlled computer perturbations belong to computer difference schemes, also named reliable difference schemes.

We consider some singular perturbation problems in the case where a degenerate equation has intersecting roots (this case is also referred to as ‘ the exchange of stabilities’). Such problems often occur as models in chemical kinetics. There are lots of works that establish the existence and asymptotic behavior of solutions to such problems. Due to exchange of stabilities, a typical solution approaches the non-smooth (but continuous) composite root of the degenerate equation as the perturbation parameter gets smaller. In a number of problems a regular part of the perturbative term dominates the singular one, so an additional condition on the regular part is needed to improve the stability of a composite root in the vicinity of the intersection point. Inversion of that condition results in a loss or a blow-up of the solution for sufficiently small values of the perturbation parameter. We prove some results of this kind by means of the nonlinear capacity argument and discuss their role in developing numerical algorithms for the problems under consideration.

Recall the Lebesgue's singular function. We define a Lebesgue's singular function \(L(t)\) as the unique continuous solution of the functional equation
$$
L(t) = qL(2t) +pL(2t-1),
$$
where \(p,q>0\), \(q=1-p\), \(p\ne q\).
The moments of Lebesque' singular function are defined as
$$
M_n = \int_0^1t^n dL(t), \quad n = 0, 1, \dots
$$
The main result of this paper is
$$
M_n =
n^{\log_2 p} e^{-\tau(n)}\left(1 + \mathcal{O}(n^{-0.99})\right),
$$
where
$$
\tau(x) =
\frac12\ln p + \Gamma'(1)\log_2 p +\frac1{\ln 2}\frac{\partial}{\partial z}\left.Li_{z}\left(-\frac{q}{p}\right)\right|_{z=1}
%+\\
\\
+\frac1{\ln 2}\sum_{k\ne0}
\Gamma(z_k)Li_{z_k+1}\left(-\frac{q}{p}\right) x^{-z_k},
$$
$$
z_k = \frac{2\pi ik}{\ln 2}, \ \ k\ne 0.
$$
The proof is based on analytic techniques such as the poissonization and the Mellin transform.

Let \(n\in {\mathbb N}\), and let \(Q_n=[0,1]^n\) be the \(n\)-dimensional
unit cube. For a nondegenerate simplex \(S\subset {\mathbb R}^n\), by
\(\sigma S\) we denote the homothetic image of \(S\)
with the center of homothety in the center of gravity of S and the
ratio of homothety \(\sigma\). We apply the following
numerical characteristics of the simplex.
Denote by \(\xi(S)\) the minimal \(\sigma>0\) with the property
\(Q_n\subset \sigma S\). By \(\alpha(S)\) we denote the minimal
\(\sigma>0\) such that \(Q_n\) is contained in a translate
of a simplex \(\sigma S\).
By \(d_i(S)\) we mean the \(i\)th axial diameter of \(S\), i.\,e.
the maximum length of a segment contained in \(S\) and parallel
to the \(i\)th coordinate axis. We apply the computational
formulae for
\(\xi(S)\), \(\alpha(S)\), \(d_i(S)\) which have been proved by the first
author. In the paper we discuss the case \(S\subset Q_n\).

Let
\(\xi_n=\min\{ \xi(S): S\subset Q_n\}. \)
Earlier the first author formulated the conjecture:
{\it if
\(\xi(S)=\xi_n\), then \(\alpha(S)=\xi(S)\).} He proved this statement
for \(n=2\) and the case when \(n+1\) is an Hadamard number, i.\,e.
there exists an Hadamard matrix of order \(n+1\). The following
conjecture is a stronger
proposition: {\it for each \(n\),
there exist \(\gamma\geq 1\), not depending on \(S\subset Q_n\), such that
\(\xi(S)-\alpha(S)\leq \gamma (\xi(S)-\xi_n).\)}
By \(\varkappa_n\) we denote the minimal
\(\gamma\) with such a property.
If \(n+1\) is an Hadamard number, then the precise value of \(\varkappa_n\)
is 1. The existence of \(\varkappa_n\) for other \(n\)
was unclear. In this paper with the use of computer methods we obtain
an equality
$$\varkappa_2 = \frac{5+2\sqrt{5}}{3}=3.1573\ldots $$
Also we prove a new estimate
$$\xi_4\leq \frac{19+5\sqrt{13}}{9}=4.1141\ldots,$$
which improves the earlier result \(\xi_4\leq \frac{13}{3}=4.33\ldots\)
Our conjecture is that \(\xi_4\) is precisely
\(\frac{19+5\sqrt{13}}{9}\). Applying this value
in numerical computations we achive the value
$$\varkappa_4 = \frac{4+\sqrt{13}}{5}=1.5211\ldots$$

Denote by \(\theta_n\) the minimal norm
of interpolation projection on the space of linear functions of \(n\)
variables as an operator from
\(C(Q_n)\)
in \(C(Q_n)\). It is known that, for each \(n\),
$$\xi_n\leq \frac{n+1}{2}\left(\theta_n-1\right)+1,$$
and for \(n=1,2,3,7\) here we have an equality.
Using computer methods we obtain the result \(\theta_4=\frac{7}{3}\).
Hence, the minimal \(n\) such that the above inequality has a strong form
is equal to 4.
%, a principal architecture of common purpose CPU and its main components are discussed, CPUs evolution is considered and drawbacks that prevent future CPU development are mentioned. Further, solutions proposed so far are addressed and new CPU architecture is introduced. The proposed architecture is based on wireless cache access that enables reliable interaction between cores in multicore CPUs using terahertz band, 0.1-10THz. The presented architecture addresses the scalability problem of existing processors and may potentially allow to scale them to tens of cores. As in-depth analysis of the applicability of suggested architecture requires accurate prediction of traffic in current and next generations of processors we then consider a set of approaches for traffic estimation in modern CPUs discussing their benefits and drawbacks. The authors identify traffic measurements using existing software tools as the most promising approach for traffic estimation, and use Intel Performance Counter Monitor for this purpose. Three types of CPU loads are considered including two artificial tests and background system load. For each load type the amount of data transmitted through the L2-L3 interface is reported for various input parameters including the number of active cores and their dependences on number of cores and operational frequency.

The method of direct computation of the universal (fibred) product in the category of commutative associative algebras of finite type with unity over a field is given and proven. The field of coefficients is not supposed to be algebraically closed and can be of any characteristic. Formation of fibred product of commutative associative algebras is an algebraic counterpart of gluing algebraic schemes by means of some equivalence relation in algebraic geometry. If initial algebras are finite-dimensional vector spaces, the dimension of their product obeys a Grassmann-like formula. A finite-dimensional case means geometrically the strict version of adding two collections of points containing a common part.

The method involves description of algebras by generators and relations on input and returns similar description of the product algebra. It is "ready-to-eat"\, even for computer realization. The product algebra is well-defined: taking other descriptions of the same algebras leads to isomorphic product algebra. Also it is proven that the product algebra enjoys universal property, i.e. it is indeed a fibred product. The input data are a triple of algebras and a pair of homomorphisms \(A_1\stackrel{f_1}{\to}A_0\stackrel{f_2}{\leftarrow}A_2\). Algebras and homomorphisms can be described in an arbitrary way. We prove that for computing the fibred product it is enough to restrict to the case when $f_i,i=1,2$ are surjective and describe how to reduce to the surjective case. Also the way of choosing generators and relations for input algebras is considered.

Paper is published in the author's wording.

We construct some asymptotic formulas for solutions of a certain linear second-order delay differential equation when the independent variable tends to infinity. Two features concerning the considered equation should be emphasized. First, the coefficient of this equation has an oscillatory decreasing form. Second, when the delay equals zero, this equation turns into the so-called one-dimensional Schr¨odinger equation at energy zero with Wigner–von Neumann type potential. Dynamics of the latter is well-known. The question of interest is how the behavior of solutions changes qualitatively and quantitatively when the delay is introduced in this dynamical model. This equation also attracts interest from the standpoint of the theory of oscillations of solutions of functional differential equations. We apply the method of asymptotic integration that is based on the ideas of the centre manifold theory in its presentation with respect to the systems of functional differential equations with oscillatory decreasing coefficients. The essence of the method is to construct a so-called critical manifold in the phase space of the considered dynamical system. This manifold is attractive and positively invariant, and, therefore, the dynamics of all solutions of the initial equation is determined by the dynamics of the solutions lying on the critical manifold. The system that describes the dynamics of the solutions lying on the critical manifold is a linear system of two ordinary differential equations. To construct the asymptotics for solutions of this system, we use the averaging changes of variables and transformations that diagonalize variable matrices. We reduce the system on the critical manifold to what is called the L-diagonal form. The asymptotics of the fundamental matrix of L-diagonal system may be constructed by the use of the classical Levinson’s theorem.

The article is devoted to the analysis of neural networks consisting of generalized neural elements. The first part of the article proposes a new neural network model — a modified network of generalized neural elements (MGNE-network). This network developes the model of generalized neural element, whose formal description contains some flaws. In the model of the MGNE-network these drawbacks are overcome. A neural network is introduced all at once, without preliminary description of the model of a single neural element and method of such elements interaction. The description of neural network mathematical model is simplified and makes it relatively easy to construct on its basis a simulation model to conduct numerical experiments. The model of the MGNE-network is universal, uniting properties of networks consisting of neurons-oscillators and neurons-detectors. In the second part of the article we prove the equivalence of the dynamics of the two considered neural networks: the network, consisting of classical generalized neural elements, and MGNE-network. We introduce the definition of equivalence in the functioning of the generalized neural element and the MGNE-network consisting of a single element. Then we introduce the definition of the equivalence of the dynamics of the two neural networks in general. It is determined the correlation of different parameters of the two considered neural network models. We discuss the issue of matching the initial conditions of the two considered neural network models. We prove the theorem about the equivalence of the dynamics of the two considered neural networks. This theorem allows us to apply all previously obtained results for the networks, consisting of classical generalized neural elements, to the MGNE-network.