The equivalence principle

For the example of the motion of an accelerated charge it is shown that an inertial frame of reference in which there is a homogeneous static gravitational field with strength g is physically inequivalent to a uniformly accelerated frame of reference moving with acceleration-g with respect to the inertial frame of reference. It follows from this that the equivalence principle does not hold in the usual formulation. The widely held opinion that such a principle is the basis of the general theory of relativity is not entirely correct. Einstein's theory of gravitation is based on an equivalence principle of a deeper content, which takes the form that the metric fieldg of a Riemannian space is declared to be a gravitational field. Such is the natural formulation of the equivalence principle at which Einstein subsequently arrived.Institute of High Energy Physics; Serpukhov. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 99, No. 1, pp. 121–140, April, 1994.     

The distribution of 4-full numbers

Let Q(x) denote the number of 4-full numbers not exceeding x. It is well known that $$Q(x) = \sum\limits_{j = 4}^7 {r_j x^{1/j} + R(x)}$$ where $$r_j = \mathop {res}\limits_{s = 1/j} (F(s)/s), F(s) = \mathop \prod \limits_P \left( {1 + \frac{{p^{ - 4s} }}{{1 - p^{ - s} }}} \right)$$ and R(x) is the remainder. This paper proves that $$R(x) \ll x^{3626/35461 + \varepsilon }$$ where ε is any positive number.     

Error distribution in sampled-data measurements

Joint error distributions are estimated for sampled-data measurements.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 73, pp. 66–70, 1992.     

An exponent bound on skew Hadamard abelian difference sets

A difference setD in a groupG is called a skew Hadamard difference set (or an antisymmetric difference set) if and only ifG is the disjoint union ofD, D(^–1), and {1}, whereD(^–1)={d^–1|dD}. In this note, we obtain an exponent bound for non-elementary abelian groupG which admits a skew Hadamard difference set. This improves the bound obtained previously by Johnsen, Camion and Mann.     

Transformation of coordinates with grid condensation toward the curvilinear boundary of the exterior flow region and boundary rectification

We consider a transformation of coordinates that maps the infinite region above the curve y=f(x) (f(x) 0) onto the entire upper halfplane. The original region is interpreted as an exterior flow region, and the curve described by the function y=f(x) as the flow boundary. The coordinate transformation rectifies the curvilinear boundary and can be applied for numerical solution of the equations of motion of a viscous fluid in a region with a curvilinear boundary of arbitrary shape.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 73, pp. 42–47, 1992.     

The curvatures of asymptotic lines on a domain of three-dimensional Lobachevsky space that is immersed intoE 5

It is assumed that a domain of a three-dimensional Lobachevsky space with curvature equal to – 1 is immersed into E ⁵.It is known that four asymptotic lines pass through every point on the immersed domain. It is proved that the second curvature of the asymptotic line is k ₂ =1/cos, where is the angle formed by vector ₃ from the natural frame of the asymptotic line and the normal plane of the submanifold. The third curvature k ₃ satisfies k ₃ |(d/ds) (1/k ₂ ) |and if k ₃ 0, then the fourth curvature is expressed in terms of k ₂ and k ₃.Thus, any curve in E ⁵ may not be an asymptotic line on a domain of a Lobachevsky space immersed in E ⁵.Translated from Ukrainskii Geometricheskii Sbornik, No. 35, pp. 5–10, 1992.     

Automated millimeter-wave-therapy devices of the “BAYuR” series and diagnostic and therapeutic systems based on them

The devices and systems for EHF therapy and diagnosis of the BAYuR series are examined and compared with known analogs.Scientific-Research Measuring-Systems Institute. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 37, No. 1, pp. 155–157, January, 1994.