Empirical Distributions and Uniform-in-Dimension Estimations of Scatter in the Problem of Existence of Typical Distributions

A connection between empirical measures and dimension-invariant estimations in the problem of existence of typical distributions for finite-dimensional spaces of random variables is established. Bibliography: 3 titles.     

Multivariate hypothesis testing and its application to quality control

A test is proposed for multivariate hypotheses about the quantile of a continuous distribution. A multivariate normal distribution is examined. Some examples of applying the results to quality control by a quantitative index are presented.Translated from Statisticheskie Metody, pp. 169–188, 1980.     

Using the binomial and the negative binomial distributions for interval estimation in exponential problems

The article solves the interval estimation problem for the availability index of various systems with an exponential reliability function.Translated from Statisticheskie Metody, pp. 142–152, 1982.     

A generalization of Turán's theorem

In this paper, we obtain an asymptotic generalization of Turán's theorem. We prove that if all the non‐trivial eigenvalues of a d‐regular graph G on n vertices are sufficiently small, then the largest K_t‐free subgraph of G contains approximately (t ? 2)/(t ? 1)‐fraction of its edges. Turán's theorem corresponds to the case d = n ? 1. © 2005 Wiley Periodicals, Inc. J Graph Theory     

Verification of the standard model of shear stress transport and its modified version that takes into account the streamline curvature and estimation of the applicability of the Menter combined boundary conditions in calculating the ultralow profile drag for an optimally configured cylinder–coaxial disk arrangement

A modification of the popular model of shear stress transport aimed at calculating the separation flow of an incompressible viscous liquid is justified. The modification eliminates the nonphysical pumping of the vortex viscosity in the cores of large-scale vortices. It has been verified with regard to the influence of the streamline curvature on the vortex viscosity by introducing a reciprocal linear function of the turbulent Richardson number with the Isaev–Kharchenko–Usachov constant equal to 0.02.Verification is based on solving the test problem an axisymmetric steady flow about a disk–cylinder tandem with an optimally configured nose, which has an ultralow profile drag for a Reynolds number of 5 × 10⁵. It has been shown that the Menter combined boundary conditions are valid if y⁺y of the wall does not exceed two.     

On the densities of cliques and independent sets in graphs

Let r; s≥2 be integers. Suppose that the number of blue r-cliques in a red/blue coloring of the edges of the complete graph K _n is known and fixed. What is the largest possible number of red s-cliques under this assumption? The well known Kruskal-Katona theorem answers this question for r = 2 or s = 2. Using the shifting technique from extremal set theory together with some analytical arguments, we resolve this problem in general and prove that in the extremal coloring either the blue edges or the red edges form a clique.     

Judicious partitions of directed graphs

The area of judicious partitioning considers the general family of partitioning problems in which one seeks to optimize several parameters simultaneously, and these problems have been widely studied in various combinatorial contexts. In this paper, we study essentially the most fundamental judicious partitioning problem for directed graphs, which naturally extends the classical Max Cut problem to this setting: we seek bipartitions in which many edges cross in each direction. It is easy to see that a minimum outdegree condition is required in order for the problem to be nontrivial, and we prove that every directed graph with m edges and minimum outdegree at least two admits a bipartition in which at least edges cross in each direction. We also prove that if the minimum outdegree is at least three, then the constant can be increased to . If the minimum outdegree tends to infinity with n, then the constant increases to . All of these constants are best‐possible, and provide asymptotic answers to a question of Alex Scott. © 2014 Wiley Periodicals, Inc. Random Struct. Alg., 48, 147–170, 2016     

Compatible Hamilton cycles in random graphs

A graph is Hamiltonian if it contains a cycle passing through every vertex. One of the cornerstone results in the theory of random graphs asserts that for edge probability , the random graph G(n, p) is asymptotically almost surely Hamiltonian. We obtain the following strengthening of this result. Given a graph , an incompatibility system over G is a family where for every , the set F_v is a set of unordered pairs . An incompatibility system is Δ‐bounded if for every vertex v and an edge e incident to v, there are at most Δ pairs in F_v containing e. We say that a cycle C in G is compatible with if every pair of incident edges of C satisfies . This notion is partly motivated by a concept of transition systems defined by Kotzig in 1968, and can be used as a quantitative measure of robustness of graph properties. We prove that there is a constant such that the random graph with is asymptotically almost surely such that for any μnp‐bounded incompatibility system over G, there is a Hamilton cycle in G compatible with . We also prove that for larger edge probabilities , the parameter μ can be taken to be any constant smaller than . These results imply in particular that typically in G(n, p) for , for any edge‐coloring in which each color appears at most μnp times at each vertex, there exists a properly colored Hamilton cycle. Furthermore, our proof can be easily modified to show that for any edge‐coloring of such a random graph in which each color appears on at most μnp edges, there exists a Hamilton cycle in which all edges have distinct colors (i.e., a rainbow Hamilton cycle). © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 49, 533–557, 2016     

Large almost monochromatic subsets in hypergraphs

We show that for all ℓ and ε > 0 there is a constant c = c(ℓ, ε) > 0 such that every ℓ-coloring of the triples of an N-element set contains a subset S of size $ c\sqrt {\log N} $ c\sqrt {\log N} such that at least 1 − ε fraction of the triples of S have the same color. This result is tight up to the constant c and answers an open question of Erdős and Hajnal from 1989 on discrepancy in hypergraphs. For ℓ ≥ 4 colors, it is known that there is an ℓ-coloring of the triples of an N-element set whose largest monochromatic subset has cardinality only Θ(log log N). Thus, our result demonstrates that the maximum almost monochromatic subset that an ℓ-coloring of the triples must contain is much larger than the corresponding monochromatic subset. This is in striking contrast with graphs, where these two quantities have the same order of magnitude. To prove our result, we obtain a new upper bound on the ℓ-color Ramsey numbers of complete multipartite 3-uniform hypergraphs, which answers another open question of Erdős and Hajnal.     

Matrix methods for the calculation of stability diagrams in quadrupole mass spectrometry

The theory of the computer calculation of the stability of ion motion in periodic quadrupole fields is considered. A matrix approach for the numerical solution of the Hill equation and examples of calculations of stability diagrams are described. The advantage of this method is that it can be used for any periodic waveform. The stability diagrams with periodic rectangular waveform voltages are calculated with this approach. Calculations of the conventional stability diagram of the 3-D ion trap and the first six regions of stability of a mass filter with this method are presented. The stability of the ion motion for the case of a trapping voltage with two or more frequencies is also discussed. It is shown that quadrupole excitation with the rational angular frequency omega = Nomega/P (where N, P are integers and omega is the angular frequency of the trapping field) leads to splitting of the stability diagram along iso-beta lines. Each stable region of the unperturbed diagram splits into P stable bands. The widths of the unstable resonance lines depend on the amplitude of the auxiliary voltage and the frequency. With a low auxiliary frequency splitting of the stability diagram is greater near the boundaries of the unperturbed diagram. It is also shown that amplitude modulation of the trapping RF voltage by an auxiliary signal is equivalent to quadrupole excitation with three frequencies. The effect of modulation by a rational frequency is similar to the case of quadrupole excitation, although splitting of the stability diagram differs to some extent. The methods and results of these calculations will be useful for studies of higher stability regions, resonant excitation, and non-sinusoidal trapping voltages.