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We consider various definitions of degrees of discrete functions and establish relations between the number of relevant (essential) variables and degrees of two- and three-valued functions. Keywords: relevant variable, sensitivity, degree of Boolean function.

A multidimensional nonnegative matrix is called polystochastic if the sum of its entries over each line is equal to $1$. The permanent of a multidimensional matrix is the sum of products of entries over all diagonals. We prove that if $d$ is even, then the permanent of a $d$-dimensional polystochastic matrix of order $4$ is positive, and for odd $d$, we give a complete characterization of $d$-dimensional polystochastic matrices with zero permanent., Comment: 17 pages

It is proved that 1) the indicator function of some onefold or multifold independent set in a regular graph is a perfect coloring if and only if the set attain the Delsarte--Hoffman bound; 2) each transversal in a uniform regular hypergraph is an independent set attaining the Delsarte--Hoffman bound in the vertex adjacency multigraph of this hypergraph; 3) combinatorial designs with parameters $t$-$(v,k,\lambda)$ and similar $q$-designs, difference sets, Hadamard matrices, and bent functions are equivalent to perfect colorings of special graphs and multigraphs, in particular, it is true in the cases of the Johnson graphs $J(n,k)$ for $(k-1)$-$(v,k,\lambda)$ designs and the Grassmann graphs $J_2(n,2)$ for bent functions. Keywords: perfect coloring, equitable partition, transversal of hypergraph, combinatorial design, $q$-design, difference set, bent function, Johnson graph, Grassmann graph, Delsarte--Hoffman bound, Comment: This is improved version of the paper published in Siberian Mathematical Journal. We fix some misprints and a gap in the proof of Theorem 2

Comparison is carried out of the long term variation of the year averaged solar wind speed and interplanetary scintillation index with the variations of Wolf's numbers and A_P indexes of geomagnetic activity for the data of 20-24 solar activity cycles. It is shown that the slow non-monotonous trend in the scintillation parameters at middle and high heliolatitudes exists with the typical scale of order of century cycle. Correlation between the variations of Wolf's numbers and anomalies of the air temperature is analyzed for long data series from 1610 up to the present time. Possible application of the results to the global climate problem is discussed., Comment: Published in Bulletin of the Lebedev Physics Institute and translated from russian version of paper by Yandex translator with correction scientific lexis. 4 pages, 2 figures

We present analysis of the rate of giant radio pulses (GPs) emission from the Crab pulsar (B0531+21). Results of our 9 years daily observations with the Large Phased Array radio telescope of Pushchino Radio Astronomy Observatory at 111 MHz were used. Limited sample of 8753 strong individual pulses in 2004 observational sessions was further analysed. It was shown that the observed monthly averaged rate of GPs emission was highly unstable during the entire span of observations and changes by about two orders of magnitude for high-energy pulses. Data were further analysed to search for the possible connection between pulsar glitches and the process of GP emission. We have found a significant increase in the rate of emission of high-energy GPs after MJD 58064, when the largest glitch ever observed in the Crab pulsar was happened. Although considerable changes in GPs emission rate could have been caused by the propagation effects in the nebula itself, we have found that the pulsar had demonstrated high degree of intrinsic irregularity of high-energy pulses emission over long time intervals., Comment: 21 pages, 13 figures

We have used the unique low frequency sensitivity of the Large Phased Array radio telescope of Pushchino Radio Astronomy Observatory to collect dataset of the single pulse observations of second period pulsars of the Northern hemisphere. During observational sessions of 2011 - 2017 yrs. We have collected data on 71 pulsars at 111 MHz frequency using digital pulsar receiver. We have discovered Giant Radio Pulses (GRP) from pulsars B0301+09 and B1237+25, and confirmed early reported generation of anomalously strong (probable giant) pulses from B1133+16 on statistically significant dataset. Data for these pulsars and from B0950+08, B1112+50, early reported as pulsars generating GRPs were analyzed to evaluate their behavior on long time intervals. It was found that statistical criterion (power-law spectrum of GRPs distribution on energy and peak flux density) seems not to be strict for pulsars with the low magnetic field on light cylinder. Moreover, spectra of some of these pulsars demonstrate unstable behavior with the time and have complex multicomponent shape. In the dataset of B0950+08 we have detected strongest GRP from pulsars with the low magnetic field on light cylinder ever reported having the peak flux density as strong as 16.8 kJy., Comment: 15 pages, 8 figures, Research in Astronomy and Astrophysics, in press

Here we present results of our search for Giant Pulses(GPs) from pulsars of Northern Hemisphere. Our survey was carried out at a frequency of 111 MHz using the Large Phased Array (LPA) radio telescope. Up to now we have detected regular generation of strong pulses satisfying the criteria of GPs from 2 pulsars: B1133+16, B1237+25., Comment: 3 pages, 4 figures