We consider an approach to the learning by precedents problem based on the minimization of the required function model complexity. Such notions as function model, function class template, function model complexity, function complexity are introduced. The described method is used to solve the problem of pseudo-Boolean function reconstruction by precedents.
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By a stochastic Leontief type equation we mean a special class of stochastic differential equations in the Ito form, in which both in the left-hand and right-hand sides there are rectangular real matrices that form a singular pencil. Besides, in the right-hand side there are a deterministic summand, depending only on time, and impulse action. It is supposed that the diffusion coefficient of the system is given by a matrix depending only on time.
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Let $G$ be a finite unitary reflection group acting on the $n$-dimensional unitary space $U^n.$ Then $G$ acts on the polynomial ring $R = C[x_1,\dots,x_n]$ in a natural manner and there exists $n$-tuple $m_1 \ge m_2 \ge \dots \ge m_n$ of positive integers, such that the algebra $I^G$ of all $G$-invariant polynomials is generated by $n$ algebraically independent homogeneous polynomials $f_1(x_1,\dots, x_n),\dots, f_n(x_1,\dots,x_n) \in I^G$ with deg $f_i = m_i$ (a system of basic invariants of group $G$) [1].
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Let $m$ be the usual Lebesgue measure on $\mathbb{R}_+ = [0,+\infty)$. Dealing with symmetric (rearrangement invariant) spaces $\mathbf{E}$ on the standard measure space $(\mathbb{R}_+,m)$, we treat the following embeddings:
$$
\mathbf{L}_1\cap\mathbf{L}_\infty \subseteq \mathbf{\Lambda}^0_{\widetilde{V}}\subseteq \mathbf{E}^0\subseteq \mathbf{E}\subseteq \mathbf{E}^{11}\subseteq \mathbf{M}_{V_*} \subseteq \mathbf{L}_1+\mathbf{L}_\infty ,
$$
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Let $A$ be a linear dissipative operator with dense domain $\mathfrak{D}(A)$ in Hilbert space $\mathfrak{H}$ and $−i \in \rho(A)$. We consider the self-adjoint operators $B = iR−iR^\ast−2R^\ast R, \widetilde{B} = iR−iR^\ast−2RR^\ast,$ where $R = (A + iI)^{−1}.$ Let $Q = \sqrt{B}, \widetilde{Q} = \sqrt{\widetilde{B}}, \mathfrak{H}_1 = \overline{Q\mathfrak{H}}, \mathfrak{H}_1 = \overline{\widetilde{Q}\mathfrak{H}}.$
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Let $G_1$ and $G_2$ be two joined bodies with masses $m_1$ and $m_2$. Each of them has a cavity partially filled with homogeneous incompressible viscous fluids situated in domains $\Omega_1$ and $\Omega_2$ with free boundaries $\Gamma_1(t)$, $\Gamma_2(t)$ and rigit parts $S_1$, $S_2$. Let $\rho_1$, $\rho_2$ be densities of fluids. We suppose that the system oscillates (with friction) near the points $O_1$, $O_2$ which are spherical hinges.
We use the vectors of small angular displacement
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The solution finding of the linear programming problem with incomplete information and reconstruction of model parameters is investigated.
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In the paper, we study the Cauchy problem for the differential second-order equation in Hilbert space $\mathscr{H} = H_1 \oplus H_2$ of the following form
$$\mathscr{C}\frac{d^2u}{dt^2} + \mathscr{B}_0u = f(t),\quad u(0) = u^0,\quad u^\prime(0) = u^1,$$
$$
\mathscr{C} =
\left(
\begin{matrix}
I_1& 0\\
0& A
\end{matrix},
\right),\quad
\mathscr{B}_0 =
\left[
\left(
\begin{matrix}
0& 0\\
0& N
\end{matrix}
\right)
+
\left(
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Let a rigid immovable vessel be partially filled with an ideal incompressible stratified fluid.
We assume that in an equilibrium state the density of a fluid is a function of the vertical variable
$x_3$, i.e., $\rho_0 = \rho_0(x_3)$. In this case the gravitational field with constant acceleration $\overrightarrow{g} = −g\overrightarrow{e_3}$ acts
on the fluid, here $g > 0$ and $\overrightarrow{e_3}$ is unit vector of the vertical axis $Ox_3$, which is directed opposite
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In this paper, consider the n-person normal form game under uncertainty. Many coalition related concepts of equilibrium or solutions have been introduced for n-person normal form games. The main initial motivation for the inception of this direction of research is to overcome one of the drawbacks of Nash equilibrium (NE), namely, NE is not immune against coalition deviation. A coalition may improve the payoff of all its members by collectively deviating from NE. In this article, the Coalitional Equilibrium (CE), is introduced for normal form games under uncertainty.
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The sufficient conditions of the existence of Berge equilibrium situation in noncooperative game of many persons in normal form are established. On the basis of these conditions the existence of Berge equilibrium situation in mixed strategies (by compact sets of strategies of players and continuity of their payoff functions) is proved. Let us consider the history of the appearance of the Berge equilibrium notion.
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Necessary familiar concepts from fuzzy numbers theory and interval systems are presented in the article, namely, concepts of: a fuzzy number; the carrier and the membershi function of fuzzy number; a discrete and continual fuzzy number; an interval matrix; low and upper bounds of an interval matrix; a middle matrix; a matrix of radiuses; an interval vector; low and upper bounds of an interval vector; a middle vector; the vector of radiuses.
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Let $G$ be a finite unitary reflection group acting on the $n$-dimensional unitary
space $U^n$. The algebra $I^G$ of $G$-invariant polynomials is generated by n algebraically independent
homogeneous polynomials $f_1(x_1, . . . , x_n), . . . , f_n(x_1, . . . , x_n)$ of degrees $m_1 \leqslant m_2 \leqslant · · · \leqslant m_n$ (a
system of basic invariants of group $G$) [1]. According to [4] (cf. [2]) a system ${f_1, . . . , f_n}$ of
basic invariants is said to be canonical if it satisfies the following system of partial differential
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Problem of exploration finite graphs by three agents is considered. Constructed an algorithm for exploration undirected graphs with $O (n^2)$ (n is amount of nodes of graph) time and space complexities. Two agents (which move on graph) needs two different colors (in total three colors) for graph exploration. An algorithm is based on depth-first traversal method.
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In this paper dynamical system with fixed switching points is considered. Existence conditions for a solution of a structural and parametric optimization problem in a class of structural controls and sufficient conditions for optimality are proved. A numerical method for structural optimization of linear system with quadratic functional is created.
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Considered the mathematical model of the flow of the river Tisza in view the maximum cost flows. Determined of the value of maximum flows by the method of bandwidth analysis. Provided recommendations on further use of this method.
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Formula of the evaluating of multiple integrals with using Lagrange polynomial interlineation was submitted. Cubature formula is investigated in the case when information about function is set of values on lines in $G = [-1, 1]^2$ on the class of functions with condition $|f^{(p_1, p_2)} (x, y)| \le M$. The estimation of error of approaching of the cubature formula is presented.
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We introduce strategies for parallelizing a sequential local elimination algorithm for sparse discrete optimization problems. The local elimination procedure exploits the structure of the underlying problem graph using extended elimination tree. We propose to use hybrid Master-Worker scheme where worker processors (GPUs) solve concurrently subproblems corresponding to super-nodes of extended elimination tree that are generated by a single master process (CPU).
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Authors propose an algorithm of computing block-tree structure for sparse matrices. Finkelshtein’s algorithm for constructing quasiblock structures is modified and implemented. Preliminary benchmarking with test problems was done. Quality of computed quasiblock structures is investigated.
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The concept of Steiner tree minimal with respect to inclusion is under consideration. The algorithm of constructing all Steiner trees minimal with respect to inclusion and its substantiation are given. The Steiner tree minimal with respect to inclusion which has minimal weight is considered as the solution of the Steiner tree problem.
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The problem of the ecological safety of production on regional level is considered in the article. The model of functioning of a region is built taking into account the technogenic loading on an environment and the rational use of natural resources
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In the paper, approach is expounded to the analysis of the linear optimization models built on precedent initial learning information. In supposition, that all numerical parameters are rational and limited by the bit net of computer, estimations of Kolmogorov’s complexity and nonrandomness of extraction of model as empirical regularity are got from the sample.
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The formulation of the optimization problem with precedent initial information is proposed. The main problems and tasks of the constructing reliable schemes for solving such optimization problems are highlighted. The approach for solving such problems based on the loss function is described. The example based on the metric classifiers is considered. The new class of the collective learning by precedent is introduced.
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We consider the travelling wave solutions of a nonlinear parabolic equation of the
second order, namely the equation of the Kolmogorov — Petrovsky — Piskunov type with the heat
release function on the right–hand side being analytical. We found a new analytic representation
for such a solution or, more accurately, for its inverse function which is represented as a sum
of an explicitly calculated summand and an auxiliary function defined on the unit interval. An
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Let $G_1$ and $G_2$ be two joined bodies with masses $m_1$ and $m_2$. Each of them has
a cavity partially filled with homogeneous incompressible ideal fluids situated in domains ${\Omega}_1$
и ${\Omega}_2$ with free boundaries ${\Gamma}_1(t)$, ${\Gamma}_2(t)$ and rigid parts $S_1$, $S_2$. Let ${\rho}_1$, ${\rho}_2$ be densities of fluids.
We suppose that the system oscillates (with friction) near the points $O_1$, $O_2$ which are spherical
hinges.
We use the vectors of small angular displacement
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The paper deals with the essentially infinite-dimensional elliptic operator $(Lu)(x) = j(u''(x))$ (of the Laplace-L´evy type), proposed by Yu.V. Bogdansky in 1977, for
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In this paper the problem of deposit allocation between the assets is considered. The sum of the individual deposit between the three assets (unrisky and two risky) increased within a year one can offer in the form
$$f(x, y) = r(1 − x_1 − x_2) + x_1y_1 + x_2y_2,$$
where r – profitability of unrisky assets; $y_i$ – profitability of risky assets i; action $x = (x_1, x_2)$ and $x_i (i = 1, 2)$ – part of investing funds in assets i.
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In this paper, the problem of minimal matrix correction of a system of linear algebraic inequalities was consider with the use of minimax criteria. The initial problem (a system of linear inequalities without a given property)
$$Ax ≤ b, A ∈ \mathbb{R}^{m×n}, b ∈ \mathbb{R}^m$$
there is assigned the class of corrected problems describing the independent variation of initial data
$$(A + H)x ≤ b + h.$$
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The paper considers one of the well-known mathematical models in economics, called "Aggregate Demand - Aggregate Supply model"(AD-AS). Originally this model takes the form of the following ordinary differential equation
\begin{equation*}
\dot{p} = D(p) − S(p),
\end{equation*}
where $p = p(t)$ – price, $D(p)$ – demand, $S(p)$ – supply. This functions are considered with the natural assumptions for macroeconomics.
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The notion of «Berge equilibrium» (BE) appeared in Russia in 1994-1995 in the thesis written by Vaisman (post-graduate student of Zhukovskiy V.I. at that time; he died at the