Abstract:
We discover that some unstable vacua have long memory. By that we mean that even in the theories containing only massive particles, there are correllators and expectation values which grow with time. We examine the cases of instabilities caused by the constant electric fields, expanding and contracting universes and, most importantly, the global de Sitter space. In the last case the interaction leads to a remarkable UV/IR mixing and to a large back reaction. This gives reasons to believe that the cosmological constant problem could be resolved by the infrared physics.

Abstract:
Necessary optimality conditions and numerical methods for solving an optimal control problem for a linear continuous-time dynanical system with controlled coefficients and quadratic goal functional are discussed.

Abstract:
Treebanks, such as the Penn Treebank (PTB), offer a simple approach to obtaining a broad coverage grammar: one can simply read the grammar off the parse trees in the treebank. While such a grammar is easy to obtain, a square-root rate of growth of the rule set with corpus size suggests that the derived grammar is far from complete and that much more treebanked text would be required to obtain a complete grammar, if one exists at some limit. However, we offer an alternative explanation in terms of the underspecification of structures within the treebank. This hypothesis is explored by applying an algorithm to compact the derived grammar by eliminating redundant rules -- rules whose right hand sides can be parsed by other rules. The size of the resulting compacted grammar, which is significantly less than that of the full treebank grammar, is shown to approach a limit. However, such a compacted grammar does not yield very good performance figures. A version of the compaction algorithm taking rule probabilities into account is proposed, which is argued to be more linguistically motivated. Combined with simple thresholding, this method can be used to give a 58% reduction in grammar size without significant change in parsing performance, and can produce a 69% reduction with some gain in recall, but a loss in precision.

Abstract:
In this article the opportunity is given to the proof of application the discriminant analysis of bearing ability of the wheel pair of the car. In particular, we offer to classify parameter of bearing ability of the wheel-axle assembly of the car. This parameter is the area of sliding zones of the connection elements, using the calculation results of the deflected mode of the wheel-axle assembly. The method allows to use mathematical models for studying the parameters of the wheel-axle assembly reliability of the wheel with an axis of the wheel pair of the car.

Abstract:
Application of the method of the principal components at the analysis of bearing ability of the wheel pair of the car. In this article it is given proof of the method of the principal components to the analysis of calculation of the stress-deformed condition of the wheel pair of the freight car. The statistical result estimation method allows to get mathematical models for researching parameters of the reliability of the wheel pair at changing of loading parameters.

Abstract:
For every $n = 2^k > 8$ there exist exactly $[(k+1)/2]$ mutually nonequivalent $Z_4$-linear extended perfect codes with distance 4. All these codes have different ranks.

Abstract:
If $N=2^k > 8$ then there exist exactly $[(k-1)/2]$ pairwise nonequivalent $Z_4$-linear Hadamard $(N,2N,N/2)$-codes and $[(k+1)/2]$ pairwise nonequivalent $Z_4$-linear extended perfect $(N,2^N/2N,4)$-codes. A recurrent construction of $Z_4$-linear Hadamard codes is given.

Abstract:
A vertex coloring of a graph is called "perfect" if for any two colors $a$ and $b$, the number of the color-$b$ neighbors of a color-$a$ vertex $x$ does not depend on the choice of $x$, that is, depends only on $a$ and $b$ (the corresponding partition of the vertex set is known as "equitable"). A set of vertices is called "completely regular" if the coloring according to the distance from this set is perfect. By the "weight distribution" of some coloring with respect to some set we mean the information about the number of vertices of every color at every distance from the set. We study the weight distribution of a perfect coloring (equitable partition) of a graph with respect to a completely regular set (in particular, with respect to a vertex if the graph is distance-regular). We show how to compute this distribution by the knowledge of the color composition over the set. For some partial cases of completely regular sets, we derive explicit formulas of weight distributions. Since any (other) completely regular set itself generates a perfect coloring, this gives universal formulas for calculating the weight distribution of any completely regular set from its parameters. In the case of Hamming graphs, we prove a very simple formula for the weight enumerator of an arbitrary perfect coloring. Codewords: completely regular code; equitable partition; partition design; perfect coloring; perfect structure; regular partition; weight distribution; weight enumerator.

Abstract:
We show that any binary $(n=2^m-3, 2^{n-m}, 3)$ code $C_1$ is a part of an equitable partition (perfect coloring) $\{C_1,C_2,C_3,C_4\}$ of the $n$-cube with the parameters $((0,1,n-1,0)(1,0,n-1,0)(1,1,n-4,2)(0,0,n-1,1))$. Now the possibility to lengthen the code $C_1$ to a 1-perfect code of length $n+2$ is equivalent to the possibility to split the part $C_4$ into two distance-3 codes or, equivalently, to the biparticity of the graph of distances 1 and 2 of $C_4$. In any case, $C_1$ is uniquely embeddable in a twofold 1-perfect code of length $n+2$ with some structural restrictions, where by a twofold 1-perfect code we mean that any vertex of the space is within radius 1 from exactly two codewords.