Vol. 34 No. 2 (193) (2000)

This article considers a two-input productivity function that relates income, capital, and labor and satisfies certain initial conditions. For the intensive form of the productivity function, the existence of an asymptote is investigated.

This paper is a continuation of [1] for a certain class of non-self-adjoint semi-elliptic operators. In Section 1, an estimate for the eigenvalues of a class of non-self-adjoint semi-elliptic operators in a bounded domain is proved. In Section 2, the completeness of the eigenfunctions in L₂(Ω) is demonstrated; more precisely, it is proved that sp’(T)= L₂(Ω), where sp’(T) denotes the closed subspace covered by the eigenvectors of T corresponding to the non-zero eigenvalues.

Let the sequences \big \{beta_k\big \} ^\infty_{n=1}$ be fixed such that $\lim\limits_{k\rightarrow\infty}(M_{2k}-M_{2k-1})=+\infty, \beta_k>0, $\lim\limits_{k\rightarrow\infty} \beta_k=0.$ In this paper, we prove that there exists a function $f_0(x) \in L^1_{[0,1]}$ such that the Fourier series of the function $f_0(x)$ with respect to the subsystem $ \big \{W_{n_k}(x) $ \big \}_{k=1}^\infty=\big \{W_m(x):M_{2s-1}\leq m\leq M_{2s}, s= 1,2,... \big \}$ diverges in the $L^1_{[0,1]}$ metric, and the Fourier-Walsh coefficients satisfy the condition $\sum\limits_{k=1}^\infty |a_{n_k}|\beta_{n_k}<\infty.$

Let ${\xi_n}$ be a sequence of independent random variables $S_n=\xi_1+\dots+\xi_n, ~\bar{S_n}=\max\limits_{1\leq k\leq n}|S_k|, ~n\geq1.$ It is shown that the correct variation of the function $N_n(t)=\sum\limits_{k=1}^n P(|\xi_k|\geq t)$ at infinity, where $P$ is the probability sign, implies the limit relations $\lim\limits_{t\rightarrow+\infty}\frac{P(\bar{S_n}\geq t)}{N_n(t)}= \lim\limits_{t\rightarrow+\infty}\frac{P(|S_n|\geq t)}{N_n(t)}=1.$ Let $v>0$ be a function independent of ${\xi_n}$ is an integer random variable with finite mathematical expectation, $$ S_v=\xi_1+\dots+\xi_v, ~\bar{S_v}=\max\limits_{1\leq n\leq v}|S_n|.$$ Consider the following model. Let ${\delta_n}$ be a sequence of positive numbers, $\alfa\geq0, ~L(t) $ vary slowly at infinity, $\lim\limits_{t\rightarrow+\infty}\big(P(|\xi_n|\geq t) \ t^{-\alfa}L(t)\big)= \delta_n$ uniformly on $n\geq1.$ Denote $c_n=(v=n), A_n=\sum\limits_{k=1}^n \delta_n,~ n\geq1,~A=\sum\limits_{n\geq 1}c_n \cdot A_n$. It has been proven that within the framework of the model $\lim\limits_{t\rightarrow +\infty}\frac{P(\bar{S_v}\geq t)}{ t^{-\alfa}L(t)}= \lim\limits_{t\rightarrow+\infty}\frac{P(|S_v|\geq t)}{ t^{-\alfa}L(t)}=A.$ Under the above conditions with existence of asymmetry in $P(\xi_n <x),~ n\geq1,$, asymmetry in $P(S_v<x)$ was found.

It is well known that torsion-free spaces occupy a significant place in the field of research on the geometry of affinely connected spaces. This paper examines the properties of so-called Frenet spaces, which are spaces with affine connections and torsion. The concept of volume, which is preserved under parallel translations of vectors, is introduced in these spaces. Necessary and sufficient conditions are found for a torsion-free connection onto which a Frenet connection is mapped, preserving pseudoconnection, to be equiaffine. It is proved that for a mapping of two Frenet spaces onto each other to be geodesic, it is necessary and sufficient that the mean connections of these spaces coincide.

The well-known problem of a string moving between two supports at constant velocity [1, 2] is considered under various boundary conditions. The problem of transverse oscillations of a moving membrane (busbar) used to transport electric current is also investigated. Critical parameters defining the regions of divergent and flutter instability are established. The frequencies of stable oscillations are determined.

The stability of a self-gravitating incompressible ring of large radius (small curvature) at the marginal stability point of an infinite cylinder is investigated. Calculations are performed to a first approximation for a small parameter related to the ring curvature. It is shown that the ring is stable in the region under consideration.

This paper presents a method for calculating the neutron flux in a plane medium by solving the neutron transport equation using Legendre polynomial approximation. The results demonstrate the advantage of this method over the previous method.

A semigroup variety is called solid, if every identity in this variety is a hyperidentity. In this paper, necessary and sufficient conditions were found for a semigroup variety to be solid and distributive.

This article considers the question of pseudoradiality of topologically homogeneous spaces with the Baire property that are continuous images of σ-compact topological groups, in particular, free topological groups of compacta. The results obtained are applicable to a more specific and natural question: what properties do topologically homogeneous spaces with the Baire property have that are continuously and transiently acted upon by an σ-compact group?

The scalar field affects the configuration of stellar matter, locally modifying the gravitational constant through the corresponding distribution of matter within the star. Therefore, it is interesting to explore possible conditions at the center of the stellar configuration that differ from those in Einstein's theory.