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Constructing the Asymptotics of Solutions to Differential Sturm–Liouville Equations in Classes of Oscillating Coefficients Moscow Univ. Math. Bull. Pub Date : 2023-12-19 N. F. Valeev, E. A. Nazirova, Ya. T. Sultanaev
Abstract The article is focused on the development of a method allowing one to construct asymptotics for solutions to ODEs of arbitrary order with oscillating coefficients on the semiaxis. The idea of the method is presented on the example of studying the asymptotics of the Sturm–Liouville equation.
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Modeling the Degenerate Singularities of Integrable Billiard Systems by Billiard Books Moscow Univ. Math. Bull. Pub Date : 2023-12-19 A. A. Kuznetsova
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Spectrum of Schrödinger Operator in Covering of Elliptic Ring Moscow Univ. Math. Bull. Pub Date : 2023-12-19 M. A. Nikulin
Abstract The stationary Schrödinger equation is studied in a domain bounded by two confocal ellipses and in its coverings. The order of dependence of the Laplace operator eigenvalues on sufficiently small distance between the foci is obtained. Coefficients of the power series expansion of said eigenvalues are calculated up to and including the square of half the focal distance.
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On Testing the Symmetry of Innovation Distribution in Autoregression Schemes Moscow Univ. Math. Bull. Pub Date : 2023-12-19 M. V. Boldin, A. R. Shabakaeva
Abstract We consider a stationary linear \(AR(p)\) model with zero mean. The autoregression parameters, as well as the distribution function (d.f.) \(G(x)\) of innovations, are unknown. We test symmetry of innovations with respect to zero in two situations. In the first case the observations are a sample from a stationary solution of \(AR(p)\). We estimate parameters and find residuals. Based on them
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A Lower Bound on Complexity of a Locator Cellular Automaton Solution for the Closest Neighbor Search Problem Moscow Univ. Math. Bull. Pub Date : 2023-12-19 D. I. Vasilev, E. E. Gasanov
Abstract The paper considers the application of the locator cellular automaton model to the closest neighbor search problem. The locator cellular automaton model assumes the possibility for each cell to translate a signal through any distance using the ether. It was proven earlier that the ether model allows solving the problem with logarithmic time. In this paper we have derived a logarithmic lower
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On the Computation Complexity of the Systems of Finite Abelian Group Elements Moscow Univ. Math. Bull. Pub Date : 2023-11-05 V. V. Kochergin
Abstract The computation complexity of the systems of the finite Abelian group elements is studied in the paper. The complexity of computation means the minimal number of group operations required to calculate elements of the system over the basis elements, all results of intermediate calculations may be used multiple times. We define the Shannon function \(L(n,m)\) as the maximal complexity of \(m\)-elements
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Jordan–Kronecker Invariants of Singular Pencils for Six-Dimensional Real Nilpotent Lie Algebras Moscow Univ. Math. Bull. Pub Date : 2023-11-05 F. I. Lobzin
Abstract In this paper, we calculate the Jordan–Kronecker invariants of singular pencils for six-dimensional nilpotent Lie algebras.
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Orthorecursive Expansion with Respect to Modified Faber–Schauder System Moscow Univ. Math. Bull. Pub Date : 2023-11-05 P. S. Stepanyants, A. K. Paunov
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Classes of Linear Approximations Providing Different Stability or Instability Types of Differential Systems Moscow Univ. Math. Bull. Pub Date : 2023-11-05 I. N. Sergeev
Abstract The paper studies relationships (inclusions, coincidences, and noncoincidences) between classes of linear approximations that provide various properties of Lyapunov, Perron, and upper-limit stability or instability (from global to particular) of the zero solution to a differential system of arbitrary order. A complete set of noncoinciding stability classes is presented and some considerations
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Boundedness of the Set of Solutions to a Linear Homogeneous System Uniform Along the Initial Segment Moscow Univ. Math. Bull. Pub Date : 2023-11-05 N. L. Margolina, K. E. Shiryaev
Abstract The paper contains definitions of some properties of solutions to linear systems of ordinary differential equations and proof of the fact that these properties are not the same for unbounded systems.
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Reconstruction of the Schrödinger Operator with a Singular Potential on Half-Line by Its Prescribed Essential Spectrum Moscow Univ. Math. Bull. Pub Date : 2023-11-05 G. A. Agafonkin
Abstract Singular Schrödinger operators on \(L^{2}([0,+\infty))\) with the potential of the form \(\sum_{k=1}^{+\infty}a_{k}\delta_{x_{k}}\), where \(x_{k}\,{>}\,0\) and \(a_{k}\,{\in}\,\mathbb{R}\), are considered. It is constructively proved that every closed semibounded set \(S\subset\mathbb{R}\) can be the essential spectrum of such operator.
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Invariant Sums of Products of Differentials Moscow Univ. Math. Bull. Pub Date : 2023-11-05 F. M. Malyshev
Abstract Based on the method proposed for solving the so-called \((r,s)\)-systems of linear equations, it is proved that the orders of homogeneous invariant differential operators \(n\) of smooth real functions of one variable take values from \(n\) to \(\dfrac{n(n+1)}{2}\), and the dimension of the space of all such operators does not exceed \(n!\). A classification of invariant differential operators
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Existence of Infinite, Everywhere Discontinuous Spectra of Upper Exponents of Oscillation of Signs, Zeros, and Roots of Third-Order Differential Equations Moscow Univ. Math. Bull. Pub Date : 2023-10-01
Abstract Examples of two linear homogeneous differential equations of the third order are constructed, the spectra of the upper strong exponents of oscillation of signs, zeros, and roots of one of which coincide with the set of rational numbers of the segment \([0,1]\) and those of the other equation coincide with the set of irrational numbers of the segment \([0,1]\) augmented with the number zero
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Two Theorems on Minimal Generalized Computable Numberings Moscow Univ. Math. Bull. Pub Date : 2023-08-17 M. Kh. Faizrahmanov
Abstract The paper proves that for any set \(A\) that computes a noncomputable computably enumerable set, any infinite \(A\)-computable family has an infinite number of pairwise nonequivalent minimal \(A\)-computable numberings. It is established that an arbitrary set \(A\leqslant_{T}\emptyset^{\prime}\) is low if and only if any infinite \(A\)-computable family with the greatest set under inclusion
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The Gromov–Hausdorff Distance between Vertex Sets of Regular Polygons Inscribed in a Single Circle Moscow Univ. Math. Bull. Pub Date : 2023-08-17 T. K. Talipov
Abstract We calculate the Gromov–Hausdorff distance between vertex sets of regular polygons endowed with the round metric. We give a full answer for the case of \(n\)- and \(m\)-gons with \(m\) divisible by \(n\). We also calculate all distances to \(2\)-gons and \(3\)-gons.
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Estimating the Smallest Positive Zero of Sine Series of a Harmonic Function in a Disk Moscow Univ. Math. Bull. Pub Date : 2023-08-17 T. Yu. Semenova
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Unsolvability of Some Problems about Piecewise-Polynomial Functions Moscow Univ. Math. Bull. Pub Date : 2023-08-17 S. B. Gashkov
Abstract The algorithmic unsolvability is proved for some problems concerning piecewise polynomials of one variable with infinite number of nodes.
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On Complexification of Max-Stable Distributions Moscow Univ. Math. Bull. Pub Date : 2023-08-17 A. V. Lebedev
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On the Method of Lyapunov Functionals for a Linear First-Order Volterra Integrodifferential Equation with Delay on the Semiaxis Moscow Univ. Math. Bull. Pub Date : 2023-08-17 S. Iskandarov, A. Khalilov
Abstract Sufficient conditions are established to ensure the estimation, boundedness, power-law absolute integrability on the semiaxis, the tendency to zero under the tendency to infinity of the independent variable of all solutions of the linear Volterra integrodifferential equation of the first order with delay. For this purpose, a generalized Lyapunov functional is constructed. An illustrative example
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Maximal PI-Exponents of Finite-Dimensional Algebras Moscow Univ. Math. Bull. Pub Date : 2023-08-17 M. V. Zaicev
Abstract We construct a series of examples of finite-dimensional algebras such that their PI-exponent coincides with the dimension. All these algebras are not simple.
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The Set of Limiting Realizable Values of Topological Entropy of Continuous Mappings of a Cantor Set Moscow Univ. Math. Bull. Pub Date : 2023-08-17 A. N. Vetokhin
Abstract It is established that in any neighborhood of each continuous mapping of a Cantor perfect set there is a mapping with a given topological entropy.
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Seminormal Functors and Paranormality Moscow Univ. Math. Bull. Pub Date : 2023-06-27 A. A. Ivanov
Abstract It is known that if a space \(\mathcal{F}(X)\) is hereditarily paranormal for a paracompact \(p\)-space \(X\) and normal functor \(\mathcal{F}\) of degree \(\geqslant 3\) in the category \(\mathcal{P}\) of paracompact \(p\)-spaces and their perfect maps, then \(X\) is metrizable. In this paper, a generalization of this theorem is proved for seminormal functors in the category \(\mathcal{P}\)
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$$\boldsymbol{\alpha}$$ -Monotone Sequences and the Lorentz Theorem Moscow Univ. Math. Bull. Pub Date : 2023-06-27 E. D. Alferova, M. I. Dyachenko
Abstract The properties of \(\alpha\)-monotone sequences are studied. A relationship between \(\alpha\)-monotonicity and the limiting rate of change of coefficients is revealed. Operations on sequences that do not lead out of the class \(M_{\alpha}\) are discussed. An analogue of the Lorentz theorem for trigonometric series with coefficients from the classes \(M_{\alpha}\) for \(0<\alpha<1\) is proved
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Constructive Solution to Inverse Scattering Problem for Differential Systems with a Singularity Moscow Univ. Math. Bull. Pub Date : 2023-06-27 M. Yu. Ignatiev
Abstract The inverse scattering problem for differential systems with a singularity is considered. The problem is reduced to a certain linear equation, and the solvability of the equation is proved. A reconstruction formula for coefficients of the system is obtained.
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Improvement of Relations between Moduli of Smoothness Moscow Univ. Math. Bull. Pub Date : 2023-06-27 M. K. Potapov, B. V. Simonov
Abstract Improvements for some relations between mixed fractional moduli of smoothness of functions of one variable are obtained. Their sharpness is proved.
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Comparison of Pure Greedy Algorithm with Pure Greedy Algorithm in a Pair of Dictionaries Moscow Univ. Math. Bull. Pub Date : 2023-06-27 A. S. Orlova
Abstract In this paper, the standard pure greedy algorithm (PGA) is compared with its modification, PGA in a pair of dictionaries. It is shown that PGA in a pair of dictionaries converges in a finite number of steps in certain cases, while the standard PGA for each individual dictionary has nonzero remainders at each step; at the same time, in certain cases the opposite holds true. Similarly, for the
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Calculation of the Gromov–Hausdorff Distance Using the Borsuk Number Moscow Univ. Math. Bull. Pub Date : 2023-06-06 A. O. Ivanov, A. A. Tuzhilin
Abstract The aim of this paper is to demonstrate relations between Gromov–Hausdorff distance properties and the Borsuk conjecture. The Borsuk number of a given bounded metric space \(X\) is the infimum of cardinal numbers \(n\) such that \(X\) can be partitioned into \(n\) smaller parts (in the sense of diameter). An exact formula for the Gromov–Hausdorff distance between bounded metric spaces is derived
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Parabolicity of Degenerate Singularities in Axisymmetric Euler Systems with a Gyrostat Moscow Univ. Math. Bull. Pub Date : 2023-06-06 V. A. Kibkalo
Abstract We study degenerate singularities of the well-known multiparametric family of integrable Zhukovsky cases of rigid-body dynamics, i.e., Euler tops with added constant gyrostatic moment. For an axisymmetric rigid body and systems close to it, it is proved that degenerate local and semilocal singularities are parabolic and cuspidal singularities, respectively, for all values of the set of system
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New Properties of Topological Spaces Generalizing the Extreme Disconnectedness Moscow Univ. Math. Bull. Pub Date : 2023-06-06 A. Yu. Groznova, O. V. Sipacheva
Abstract New classes \(R_{1}\), \(R_{2}\), and \(R_{3}\) of topological spaces generalizing the class of \(F\)-spaces are introduced. It is proved that all homogeneous compact subspaces of spaces in these classes and of some of their products are finite. Results on the Rudin–Keisler comparability of ultrafilters along which distinct sequences converge to the same point in \(R_{2}\)- and \(R_{3}\)-spaces
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Steiner Points in $$\boldsymbol{l_{\infty}^{2}}$$ Space Moscow Univ. Math. Bull. Pub Date : 2023-06-06 B. B. Bednov
Abstract It is proved that, for a given set of pairwise distinct points \(x_{1},\dots,x_{n}\), the sum of the distances from these points to their Steiner point in \(l_{\infty}^{2}\) space is equal to the maximum of the sum of lengths of \(\left[\dfrac{n}{2}\right]-1\) separate segments and either a semi-perimeter of a triangle or another segment with vertices in this set. The case of coincident points
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An Unsuspended Description of the $$E$$ -Theory Category Moscow Univ. Math. Bull. Pub Date : 2023-06-06 G. S. Makeev
Abstract An unsuspended picture of \(E\)-theory is obtained in the paper. In this picture, morphisms are given in terms of good enough endofunctors of \(C^{*}\)-algebras for which we construct a categorical formalism.
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Sets in $$\boldsymbol{\mathbb{R}^{n}}$$ Monotone Path-Connected with Respect to Some Norm Moscow Univ. Math. Bull. Pub Date : 2023-06-06 E. A. Savinova
Abstract Conditions on a path-connected set \(M\) in \(\mathbb{R}^{n}\) that are necessary and sufficient for \(M\) to be monotone path-connected with respect to some norm are obtained.
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Angular Boundary Limits for Normal Subharmonic Functions Moscow Univ. Math. Bull. Pub Date : 2023-06-06 S. L. Berberyan, R. V. Dallakyan
Abstract The paper continues the study of boundary properties of normal subharmonic functions defined in the unit disk \(D\). Theorems are obtained on the existence of angular boundary limits for normal subharmonic functions almost everywhere on the arc of the unit disk \(D\).
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Criterion for Lyapunov Reducibility of a Linear Autonomous Differential System to a Linear Autonomous Equation Moscow Univ. Math. Bull. Pub Date : 2023-06-06 I. N. Sergeev, K. V. Umansky
Abstract We establish a unified criterion for the reducibility of a linear homogeneous differential system with constant coefficients to a linear homogeneous differential equation with constant coefficients by means of both Lyapunov and periodic transformations. The resulting necessary and sufficient condition on a system is formulated in terms of properties of the Jordan normal form of its matrix
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Topology of Isoenergy 5-Surfaces of a Three-Axial Ellipsoid with a Hooke Potential Moscow Univ. Math. Bull. Pub Date : 2023-03-12 G. V. Belozerov
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Weight Systems of Framed Chord Diagrams Corresponding to Lie Algebras Moscow Univ. Math. Bull. Pub Date : 2023-03-12 D. P. Ilyutko, I. M. Nikonov
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Multivariate Records of Particle Scores in Supercritical Branching Processes with Continuous Time Moscow Univ. Math. Bull. Pub Date : 2023-03-12 A. V. Nazmutdinova
Abstract Bivariate records of particle scores in immortal supercritical branching processes with continuous time are studied. The limiting intensity of records for one score and the limiting intensity of records for both scores or at least one score are found. In the case of independent scores, mean numbers of joint records for all time are calculated. The results are illustrated by examples.
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Bounds on Orders of Linear Automata Moscow Univ. Math. Bull. Pub Date : 2023-03-12 N. V. Muravev
Abstract If input and output alphabets of a Mealy automaton coincide, then one can study the order problem with respect to the superposition operation. The paper provides exact upper bounds on orders of linear automata over finite fields and rationals.
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Recurrence Legendre Polynomials Moscow Univ. Math. Bull. Pub Date : 2023-03-12 V. N. Sorokin
Abstract The recurrence polynomials partly orthogonal with respect to Lebesgue measure on the segment symmetric with respect to the unit circle are studied. The limiting distribution of their zeros is obtained in terms of a meromorphic function on compact Riemann surface. The interpretation of the limiting measure is obtained in terms of equilibrium problems in the logarithmic potential theory.
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Implementation of Postprocessing Procedure of a Rapid Algorithm of Geometric Coding of Digital Images Using CUDA Architecture Moscow Univ. Math. Bull. Pub Date : 2023-03-12 G. V. Nosovskii, A. Yu Chekunov
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On the Attribute of Uniform Convergence of Fourier Series of the Vilenkin System in the Case of Unbounded $$\boldsymbol{p}_{\boldsymbol{k}}$$ Moscow Univ. Math. Bull. Pub Date : 2023-01-10 S. M. Voronov
Abstract Series with respect to a system of characters of a zero-dimensional compact commutative group are considered. Generalization of the test of convergence of Fourier series of the Vilenkin system in the case of unbounded quasimonotone \(p_{k}\) for functions having a generalized bounded \(\Phi\)-fluctuation, which was earlier obtained in the case of bounded sequences \({p_{k}}\), is proved.
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The Estimation of Traffic Intensity Parameter for a Single-Channel Queueing System with Regenerative Input Flow Moscow Univ. Math. Bull. Pub Date : 2023-01-10 G. A. Krylova
Abstract A single-channel queueing system with regenerative input flow and an unreliable device is considered. A statistical estimation of the traffic intensity parameter \(\rho\) is proposed, its consistency and asymptotic normality are proved. An algorithm for testing the hypothesis \(\rho=\rho_{0}\) against various alternatives is presented.
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Topological Models of Propositional Logic of Problems and Propositions Moscow Univ. Math. Bull. Pub Date : 2023-01-10 A. A. Onoprienko
Abstract The propositional fragment HC of the joint logic of problems and propositions introduced by Melikhov is considered. Topological models of this logic are constructed and the completeness of the logic HC with respect to this type of models is shown. Topological models of the logic H4 introduced by Artemov and Protopopescu are also constructed.
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Convergence of a Weak Greedy Algorithm When One Vector Is Added to the Orthogonal Dictionary Moscow Univ. Math. Bull. Pub Date : 2023-01-10 A. S. Orlova
Abstract Convergence of weak greedy algorithms (WGAs) and weak orthogonal greedy algorithms (WOGAs) is studied for the subspace \(\ell_{1}\subset\ell_{2}\) and dictionaries obtained from the standard orthogonal basis by adding one vector. It is shown that the condition on a weakening sequence sufficient for convergence of WOGA in the case of the orthogonal dictionary and an approximated element from
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Solution to a Linearized System of Two-Dimensional Dynamics of Viscous Gas Moscow Univ. Math. Bull. Pub Date : 2023-01-10 A. A. Kornev, V. S. Nazarov
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Short Complete Diagnostic Tests for Logic Circuits in One Infinite Basis Moscow Univ. Math. Bull. Pub Date : 2023-01-10 K. A. Popkov
Abstract It is proved that each Boolean function can be modeled by a logic circuit with one additional input in a basis consisting of conjunctions of an arbitrary number of variables, two-input disjunction, and negation, allowing a complete diagnostic test with the length no more than \(n+1\) relative to constant faults of type \(1\) at outputs of logic gates.
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Extremes of Homogeneous Two-Parametric Gaussian Fields at Discretization of Parameters Moscow Univ. Math. Bull. Pub Date : 2023-01-10 I. A. Kozik
Abstract Gaussian homogeneous fields on two-dimensional Euclidean space are considered, whose correlation functions behave at zero in a power-law manner along each of the coordinates. Exact asymptotics are evaluated for the exceedances probabilities above infinitely growing levels on lattices with different densities along each coordinate and with infinitely decreased lattice density. Relations between
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The Continuity of Inverse in Groups Moscow Univ. Math. Bull. Pub Date : 2022-11-03 E. A. Reznichenko
Abstract A wide class of spaces, \(\Delta\)-Baire spaces, are defined. If a paratopological group \(G\) is \(\Delta\)-Baire space, then \(G\) is a topological group. Locally pseudocompact spaces, Baire \(p\)-spaces, Baire \(\Sigma\)-spaces, products of Čech-complete spaces are \(\Delta\)-Baire spaces.
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Approximate Integration of Canonical Second-Order Ordinary Differential Equations by the Chebyshev Series Method with an Error Estimation of the Solution and Its Derivative Moscow Univ. Math. Bull. Pub Date : 2022-11-03 O. B. Arushanyan, S. F. Zaletkin
Abstract An approximate method of solving the Cauchy problem for canonical second-order ordinary differential equations is considered. This method is based on using the shifted Chebyshev series and a Markov quadrature formula. A number of procedures are discussed to estimate the error of the approximate solution and its derivative expressed by partial sums of shifted Chebyshev series of a certain order
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Description of All Classes of Superfunctions Consisting of Disjunctions Moscow Univ. Math. Bull. Pub Date : 2022-11-03 I. I. Maslova
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Dual Problem of Robust Utility Maximization Moscow Univ. Math. Bull. Pub Date : 2022-11-03 A. A. Farvazova
Abstract The robust utility maximization problem with a random endowment in an abstract financial market model is considered. The utility function is assumed finite on the half-line, and the dual characterization of this problem is derived.
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Kolyada Inequality between Mixed Moduli of Smoothness in Metrics $$\boldsymbol{L}_{\boldsymbol{p}}$$ and $$\boldsymbol{L}_{\boldsymbol{\infty}}$$ Moscow Univ. Math. Bull. Pub Date : 2022-11-03 M. K. Potapov†, B. V. Simonov
Abstract Interrelations between mixed fractional moduli of smoothness considered in the \(L_{p}\) and \(L_{\infty}\) metrics are clarified in the paper.
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Spectral Properties of a Differential Operator with Involution Moscow Univ. Math. Bull. Pub Date : 2022-11-03 Ya. A. Granilshchikova, A. A. Shkalikov
Abstract The article defines a class of regular first-order differential operators the main part of which contains the involution operator and nonconstant coefficient functions. A scheme for proving the unconditional basis property of the eigenfunctions and associated functions of regular differential operators of this type is provided under some additional conditions. Examples of operators for which
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On the Complexity of Implementation of Characteristic Functions of the Spheres by Circuits of Functional Elements Moscow Univ. Math. Bull. Pub Date : 2022-08-31 N. P. Red’kin
Abstract For characteristic functions of spheres, an asymptotics for the complexity of their implementation by circuits of functional elements in the basis \(\{\&,\vee,-\}\) is established; the characteristic function of a sphere with the center at the vertex \(\tilde{\sigma}=(\sigma_{1},\ldots,\sigma_{n})\), \(\sigma_{1},\ldots,\sigma_{n}\in\{0,1\}\), is the Boolean function equal to one on all those
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The Bounds on the Number of Partitions of the Space $$\mathbf{F}_{\mathbf{2}}^{\boldsymbol{m}}$$ into $$\boldsymbol{k}$$ -Dimensional Affine Subspaces Moscow Univ. Math. Bull. Pub Date : 2022-08-31 I. P. Baksova, Yu. V. Tarannikov
Abstract The bounds on the number of partitions of the space \(\mathbf{F}_{2}^{m}\) into affine subspaces of dimension \(k\) are presented in the paper. Apart from their immediate interest, these bounds are important for estimating the number of bent functions generated by some constructions.
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Comparing the Computational Complexity of Monomials and Elements of Finite Abelian Groups Moscow Univ. Math. Bull. Pub Date : 2022-08-31 V. V. Kochergin
Abstract The computational complexity of the element \(a_{1}^{k_{1}}a_{2}^{k_{2}}\ldots a_{q}^{k_{q}}\) of the Abelian group \(\langle a_{1}\rangle_{u_{1}}\times\langle a_{2}\rangle_{u_{2}}\times\ldots\) \(\ldots\times\langle a_{q}\rangle_{u_{q}}\) (it is supposed that \(k_{i}
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On the Classes of Autodual Functions Implicitly Maximal in $$\boldsymbol{P}_{\boldsymbol{k}}$$ Moscow Univ. Math. Bull. Pub Date : 2022-08-31 M. V. Starostin
Abstract The problem of implicit expressibility in many-valued logics is considered. Necessary and sufficient conditions for a class of autodual functions to be implicitly maximal are obtained.
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On the Implementation of Monotone Boolean Functions by Memoryless Programs Moscow Univ. Math. Bull. Pub Date : 2022-08-31 A. V. Chashkin
Abstract The average-case complexity of computing monotone Boolean functions by straight-line programs without memory with a conditional stop in the basis of all Boolean functions of at most two variables is considered. For the set of all monotone Boolean functions of \(n\) variables, Shannon-type upper and lower bounds for the average-case complexity are established for \(n\to\infty\).
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Refined Bounds on Shannon’s Function for Complexity of Circuits of Functional Elements Moscow Univ. Math. Bull. Pub Date : 2022-08-31 S. A. Lozhkin
Abstract Earlier, the author proposed rather general approaches and methods for obtaining high accuracy and close to high accuracy asymptotic bounds on Shannon’s function for complexity in various classes of circuits. Most of the results obtained with their aid were published in a number of papers, except perhaps for the close to the high accuracy asymptotic bounds on Shannon’s function for the complexity
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On the Cardinality of Interval Int(Pol $${}_{\boldsymbol{k}}$$ ) in Partial $$\boldsymbol{k}$$ -Valued Logic Moscow Univ. Math. Bull. Pub Date : 2022-08-31 V. B. Alekseev
Abstract Let \({Pol}_{k}\) be the set of all functions of \(k\)-valued logic representable by a polynomial modulo \(k\), and let \({Int}({Pol}_{k})\) be the family of all closed classes (with respect to superposition) in the partial \(k\)-valued logic containing \({Pol}_{k}\) and consisting only of functions extendable to some function from \({Pol}_{k}\). In this paper, we prove that if \(k\) is divisible