For a pseudo-convex $CR$-structure on an odd dimensional manifold, we introduce a family of canonical torsion-free linear connections. Every connection in this family is uniquely determined by an almost contact structure associated with the given pseudo convex $CR$-structure. We study the change of the connections in this family under the gauge transformations and, accordingly, the corresponding change of the gauge tensor fields. The Bochner type curvature tensor field we get is invariant under gauge transformations.

In the paper first order neutral differential equations with “maxima” are considered. Sufficient conditions for oscillation of all solutions are obtained. The asymptotic behavior of the nonoscillatory solutions is investigated.

Let $C$ be the Kloosterman code over finite fields. In this paper, we give some bound for the weights of codewords in $C$ and show that the weights of the code $C$ are uniformly distributed with respect to the Sato-Tate measure by using the result of Katz.

The purpose of this paper is to determine the set of non-isomorphic indecomposable $RG$-lattices, where $R$ is a certain ring of algebraic integers, and $G$ is a cyclic group of prime order.

Orthogonal 24-run $2^5$ factorial designs derivable from saturated two-symbol orthogonal arrays of strength 2, size 24 and index 6 by selecting five columns are classified computationally into 63 isomorphic classes with respect to the permutation of factors and levels within factors. Specific features of those 63 representative designs are considered.

An initial value problem for an impulsive system of differential-difference equations is considered. By means of piecewise continuous auxiliary functions which are modifications of classical Lyapunov’s functions, some sufficient conditions for global stability of the zero solution of such problems are presented. The discontinuity of these auxiliary functions corresponds to the fact that solutions of the systems under consideration are piecewise continuous functions.

Suppose a dynamical system $dx/dt=\mathbf{F}(\text{x};\mu )$, $\text{x}\in {\mathbf{R}}^s$, $\mu \in {\mathbf{R}}^m$, has a hyperbolic saddle at $\text{x}=0$ with a homoclinic loop, for $\mu ={\mu }^0$. When $\mu$ varies from ${\mu }^0$, the loop will be destroyed in general. For $s=2$, Perko proved that, if $\mu$ varies on an $(m-1)$ dimensional hypersurface, then the system remains to admit homoclinic orbit. We consider here the same problem for $s=3$. The result is: if $\mu$ varies on an $(m-2)$ hypersurface, then the system remains to admit homoclinic orbit.

For a Frobenius-Sandwich of regular semi-local rings $R\subset R^{\prime }\subset R^p,$ $R/R^{\prime }$ has a $p$-basis if and only if $R$ has constant rank as $R^{\prime }$-module.