On Sufficient Conditions for Convergence of Two-Dimensional Continued Fractions

By the method of majorant fractions and equivalent transformations, the analogies of leszyski–Pringsheim criteria for two-dimensional continued fractions are obtained.     

Analogs of the Śleszyński–Pringsheim Criteria for Two-Dimensional Continued Fractions

Using the majorant fraction method and equivalent transformations, we obtain analogues of the leszyski–Pringsheim convergence for two-dimensional continued fractions.     

Locally nilpotent groups with the min−∞−n condition

The study of locally nilpotent groups with the weak minimality condition for normal subgroups, the min^––n condition, is continued. The following results are obtained.THEOREM 1. A locally nilpotent group with the min^––n condition is countable.THEOREM 2. If G is a locally nilpotent group with the min^––n condition whose periodic part is nilpotent and the orders of the elements of the periodic part are bounded in the aggregate, then G=t(G)A, where the subgroup A is minimax.THEOREM 3. Suppose G is a locally nilpotent group with the min^––n condition and T is its periodic part. If T is nilpotent and G/T is Abelian, then G=TA, where the subgroup A is minimax.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 3, pp. 340–346, March, 1990.     

Lebesgue constants of multiple de la Vallée-Poussin sums over polyhedra

One considers linear summation methods for the multiple Fourier series the multidimensional analogues of the de la Vallé-Poussin sums. The summation of the Fourier series is carried out over the homotheties of an m-dimensional starshaped polyhedron . It is shown that if has rational vertices, then the Lebesgue constants of the considered methods, with the accuracy of O((p+1)^–1. log^m–1 (n+2)) are equal to where is the Fourier transform of the function . The exact value of the principal term of the Lebesgue constant is computed in two particular cases: 1) is obtained from an m-dimensional cube by means of a linear nonsingular transformation; 2) =0. is an m-dimensional simplex.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 125, pp. 154–165, 1983.     

On Differences of Two kth Powers of Integers

The arithmetic function rk–(n) counts the number of ways to write a natural number n as the difference of two kth powers (k 3 fixed). The investigation of the asymptotic behavior of the Dirichlet summatory function of rk–(n) leads in a natural way to a certain error term Pk–(t). In this article we establish a mean-square upper bound and an -estimate for Pk–(t).     

Characterization of the least concave majorant of brownian motion,conditional on a vertex point,with application to construction

The characterization of the least concave majorant of brownian motion by Pitman (1983,Seminar on Stochastic Processes, 1982 (eds. E. Cinlar, K. L. Chung and R. K. Getoor), 219–228, Birkhäuser, Boston) is tweaked, conditional on a vertex point. The joint distribution of this vertex point is derived and is shown to be generated with extreme ease. A procedure is then outlined by which one can construct the least concave majorant of a standard Brownian motion path over any finite, closed subinterval of (0, ∞). This construction is exact in distribution. One can also construct a linearly interpolated version of the Brownian motion path (i.e. we construct the Brownian motion path over a grid of points and linearly interpolate) corresponding to this least concave majorant over the same finite interval. A discussion of how to translate the aforementioned construction to the least concave majorant of a Brownian bridge is also presented.     

Wavelet and Weyl Transforms Associated with the Spherical Mean Operator

The admissible wavelets associated with spherical mean operator and corresponding Weyl transforms are defined. The admissible condition is given in the generalized Fourier transforms, and Plancherel formula, Parseval formula, Reproducing formula and Reproducing kernel are studied. The criteria of boundedness of the Weyl transform on the L^p– spaces is given in term of the symbol function .     

An incompressibility condition for a certain class of integral functionals. I

One considers the integral functional (y), depending on the mapping of the domain R^m into R^m, on the set of mappings y, subjected to the incompressibility condition: det y=1. One computes its first and second variations. The obtained results are compared with the formulas arising from the formal application of the method of the undetermined Lagrange multipliers. One gives an application to problems of elasticity theory.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 115, pp. 203–214, 1982.     

The Riemann–Hilbert Problem for Analytic Description of the DM Solitons

A simple exact formula is derived for the profile of the optical pulse propagating over a DM fiber with zero mean dispersion. The dissipation is neglected, and dispersion is assumed to be constant along the adjacent legs of the waveguide, thus providing the applicability of the integrable NLS models within each leg. The formula describes a class of solutions called dispersion-managed solitons (DM solitons), which are periodic along the waveguide and exponentially localized in time. The DM solitons are parameterized by a certain class of spectral data, specified from numerical simulations. Using a related Riemann–Hilbert problem, we reconstruct a profile of the DM soliton from the given spectral data. For sufficiently long legs, the leading term of DM soliton is found in explicit form by asymptotic undressing of the Riemann–Hilbert problem. The analytic results are compared with numerical simulations.     

Metrische Zusammenhänge und Torsion bei einfachen p-Vektoren

In the foundations of differential geometry for simple p-vectors, that is for the bundle of the p(n–p)+1 dimensional Grassmann cone over an n-dimensional manifold, as shown in [8] the (nonlinear) connection can be split into a linear combination of elements consisting of p–1 vectors and one connection component _h ⁱ .In the present paper these components are transformed into metric ones, and the representative element of the covariant components is improved. The p- and (n–p)-torsion elements are defined, and these are then shown to admit to a splitting theorem for the respective torsion tensors. Finally a decomposition of the connection components is given such that the minimal condition for the complementary space GK^n-p, associated to GK^n–p, does not contain the volume function.     

Certain analogues of Von Neumann's inequality for J-contractions

Let J be a selfadjoint operator satisfying the condition J²=I. It is proved that if T is a J-contraction (i.e., T*JT–J0) and f is an inner function, analytic on the spectrum of the operator T, then f(T) is also a J-contraction. It is also proved that in the case JI and J–I only the inner functions satisfy this property. One also considers other analogues of the von Neumann inequality.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 157, pp. 165–172, 1987.I express my gratitude to V. I. Matsaev for his stimulating interest in the paper.     

Best Polynomial Approximations in <Emphasis Type="Italic">L</Emphasis><Subscript>2</Subscript> and Widths of Some Classes of Functions

We obtain the exact values of extremal characteristics of a special form that connect the best polynomial approximations of functions f(x) ∈ L ₂ ^r (r ∈ ℤ₊) and expressions containing moduli of continuity of the kth order ω_k(f(^r), t). Using these exact values, we generalize the Taikov result for inequalities that connect the best polynomial approximations and moduli of continuity of functions from L ₂. For the classes (k, r, Ψ_*) defined by ω _k(f ^(r), t) and the majorant , we determine the exact values of different widths in the space L₂.__________Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 56, No. 11, pp. 1458–1466, November, 2004.     

On a problem of Bochner and von Neumann

The following theorem is proved. If there exists an everywhere dense set such that every solution of the equation v'_t=–iA^*v satisfying the condition v(0) is defined on the entire axis and is bounded, then every compact solution of the equation u'_t=iAu is an almost-periodic function.Translated from Matematicheskie Zametki, Vol. 3, No. 5, pp. 529–538, May, 1968.     

Method of functional integration and an eikonal approximation for potential scattering amplitudes

Conclusions We have obtained an exact closed expression for the potential scattering amplitude of particles with spin o and 1/2 as a functional integral with respect to trajectories. This has made possible a relatively simple expansion of the amplitude in powers of the small parameter 1/E. The first term of the expansion is an eikonal approximation for the amplitude for scattering through any angle and in the case of dynamically small angles (pR)^–1/2 is identical with the Glauber representation.We have found the asymptotic form of the scattering amplitude for two particles that exchange virtual mesons. A similar result was obtained in [7] by functional integration with respect to the external fields of the exact Green's functions and a subsequent eikonal expansion on the mass shell. The equivalence of this more accurate method to the approximation described in the present paper (see also [8, 9]) is connected with the interesting problem of the commutativity of the operations of eikonal approximation and second quatization.If the latter do commute, the Glauber representation for the amplitude (18) in quantum field theory is a consequence of the eikonal approximation in quantum mechanics.Joint Institute of Nuclear Research, Dubna. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 4, No. 1, pp. 22–32, July, 1970.     

Exact Solutions for a Family of Variable-Coefficient “Reaction–Duffing” Equations via the Bäcklund Transformation

The homogeneous balance method is extended and applied to a class of variable-coefficient reaction–duffing equations, and a Bäcklund transformation (BT) is obtained. Based on the BT, a nonlocal symmetry and several families of exact solutions of this equation are obtained, including soliton solutions that have important physical significance. The Fitzhugh–Nagumo and Chaffee–Infante equations are also considered as special cases.     

Self-action of a point charge in classical electrodynamics

A direct calculation demonstrates that the causal Green function for classical equations of an electromagnetic field contains an additional singular term cancelling the divergence in the self-action of a point charge. Thus, the problem of mass renormalization is avoided. An exact relativistic expression for the self-action force is presented as a sum of two terms. The first one gives the radiation damping and the second one describes the electromagnetic component of the particle momentum depending on its velocity and acceleration. Accordingly, the work of the force also consists of two terms: the radiation energy and the electromagnetic component of the particle energy. To perform the calculations, we have to extend the radial spherical coordinate in the -function argument to negative values.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 105, No. 2, pp. 256–269, November, 1995.     

Extremum Problem for the Wiener–Hopf Equation

The extremum problem for the Wiener–Hopf equation obtained by replacing the condition u(x) = 0, x < 0, by the condition of the minimum of the quadratic functional of the function u(x)exp(–x), – < x < , is solved in closed form.     

A remark on the essential self-adjointness of a Schrödinger operator with a singular potential

We examine the operators=–+v, v L_{2, loe} (R ⁿ ), where S satisfies a natural additional condition of a local nature. If a condition of Titchmarsh type is fulfilled at infinity, then S is essentially self-adjoint in L₂(Rⁿ).Translated from Matematicheskie Zametki, Vol. 20, No. 4, pp. 571–580, October, 1976.     

A new algorithm for approximating the least concave majorant

The least concave majorant, \(\hat F\), of a continuous function F on a closed interval, I, is defined by

$$\hat F\left( x \right) = \inf \left\{ {G\left( x \right):G \geqslant F,Gconcave} \right\},x \in I.$$

We present an algorithm, in the spirit of the Jarvis March, to approximate the least concave majorant of a differentiable piecewise polynomial function of degree at most three on I. Given any function F ∈ C⁴(I), it can be well-approximated on I by a clamped cubic spline S. We show that \(\hat S\) is then a good approximation to \(\hat F\).

We give two examples, one to illustrate, the other to apply our algorithm.