The inverse matrix for the square matrix A of order n
with coefficients of some field exists, as it is known
then and only then, when its determinant is not equal to
zero. If the matrix A has a certain type (certain
structure), then an inverse matrix A-1 should not have
exactly the same structure. Therefore, it is interesting
to describe such square matrices A, which have an
inverse matrix A-1, having the same structure as the
matrix A, under certain conditions. For example, a
subdiagonal matrix with nonzero elements on the main
diagonal has an inverse matrix over a field of
characteristic zero, having also the form of subdiagonal
matrix. Similarly, an inverse matrix towards
symmetrical or skew-symmetric matrix is also
symmetric or skew-symmetric accordingly. Also, the
matrix inverse to non-degenerate (nonsingular)
circulant will be a circulant itself, and finally, the
matrix inverse to nonsingular quasdiagonal matrix D
will be quasdiagonal itself, and will have the same
partitioned structure as D. Thus, there is a problem of
determining these types of nonsingular matrices that
have an inverse matrix of the same type as a given
matrix. In line with this problem in the present study it
is determined such type of matrices for which an
inverse matrix has the same type, at that the conditions
are identified in explicit form, ensuring the
nonsingularity of the matrix. The matrices of three
orders are shown in detail. These results allow
determining the characteristics of fields over which
there are inverse matrices of the considered types
The problems of finding of characteristic polynomials and spectra prefractal graphs with the priming cycles are investigated, the contiguity of old edges in the trajectory isn't broken. The recurrent formula is received
The results of the research of stability of the model of neutrophilomonocytegenesis are shown in the article. With the criterion of Routh-Hurwitz it's calculated that the system of the differential equations of cells growing is asymptotically steady. Threshold values of parameters of model at which the system becomes unstable are defined
The general scheme of modern statistical science is
just like this. Mathematical Statistics is a part of
mathematics that studies the statistical structure (it
itself does not give recipes analysis of statistical
data, however, it is developing methods that are
useful for use in theoretical statistics). Theoretical
Statistics - the science dedicated to the models and
methods of analysis of concrete statistical data.
Applied Statistics (in the narrow sense) is devoted to
the statistical techniques of data collection and
processing (it includes the methodology of statistical
methods, the organization of sample surveys, the
development of statistical techniques, the creation
and use of statistical software). Applications of
statistical methods in concrete fields (in economics
and management - Econometrics, in biology -
Biometrics, in chemistry - Chemometrics, in
technical research - Technometric, in geology,
demography, sociology, medicine, history, etc.).
Often positions 2 and 3 together are called Applied
Statistics. Sometimes position 1 is called Theoretical
Statistics. These terminological differences are
related to the fact that the above-described
development of the considered scientific and applied
field not once, not completely and not always
adequately reflected in the minds of experts.
Meanwhile, there are still textbooks of appropriate
level of representation of the mid-twentieth century.
The article analyzes the post-war development of
the national statistics. We have identified five
"growth points": nonparametrics, robustness,
bootstrap, statistics of interval data, and statistics of
non-numeric data. We have discussed content,
development and the basic ideas of statistics of nonnumeric
data. We have given a number of
unresolved problems of theoretical and applied
statistics
The probabilistic model of grouping data (including multidimensional data) is described. We have also generalized Euler-Maclaurin’s formulas. With its help Sheppard’s corrections and corrections on
grouping for correlation coefficient are received. We have found and studied asymptotical corrections on grouping data generally. Accuracy of approach has been estimated
Currently, the majority of scientific, technical and economic studies use statistical methods developed mainly in the first third of the XX century. They constitute the content of common textbooks. However, mathematical statistics are rapidly developing in the next 60 years. In some situations there is a need of the transition from classical to modern methods. As an example, we discuss the problem of testing the homogeneity of two independent samples. We have considered the conditions of applicability of the traditional method of testing the homogeneity based on the use of Student's t-statistic, as well as more up-to-date methods. We describe a probabilistic model of generation of statistical data in the problem of testing the homogeneity of two independent samples. In terms of this model the concept of "homogeneity" ("no difference"), can be formalized in different ways. High degree of homogeneity is achieved if the two samples are taken from one and the same population (absolute homogeneity). In some cases it is advisable to testing the coincidence of some characteristics of the elements of the sample - mathematical expectations, medians, variances, coefficients of variation, and others (testing the homogeneity of characteristics). To test the homogeneity of mathematical expectations is often recommended classic t-test. It is believed that the samples taken from a normal distributions with equal variances. It is shown that for scientific, technical and economic data the preconditions of two-sample t-test usually are not performed. To test the homogeneity of mathematical expectations instead of t-test we have offered to use the Cramer-Welch test. We have considered the consistent nonparametric Smirnov and Lehmann-Rosenblatt tests for absolute homogeneity
We have given a critical analysis of statistical
models and methods for processing text information
in historical records to establish the times when
there were certain events, ie, to build science-based
chronology. There are three main kinds of sources
of knowledge of ancient history: ancient texts, the
remains of material culture and traditions. The
specific date of the extracted by archaeologists
objects in most cases can not be found. The group of
Academician A.T. Fomenko has developed and
applied new statistical methods for analysis of
historical texts (Chronicle), based on the intensive
use of computer technology. Two major scientific
results were: the majority of historical records that
we know now, are duplicated (in particular,
chronicles, describing the so-called "Ancient Rome"
and "Middle Ages", talking about the same events);
the known historical chronicles tell us about real
events, separated from the present time for not more
than 1000 years. It was found that chronicles
describing the history of "ancient times" and
"Middle Ages" and the chronicle of Chinese history
and the history of various European countries do not
talk about different, but about the same events. We
have the attempt of a new dating of historical events
and restoring the true history of human society
based on new data. From the standpoint of statistical
methods of historical records and images of their
fragments – they are special cases of non-numeric
objects of nature. Therefore, developed by the group
of A.T. Fomenko computer-statistical methods are
the part of non-numerical statistics. We have
considered some methods of statistical analysis of
chronicles applied by the group of A.T. Fomenko:
correlation method of maximums; dynasties method;
the method of attenuation frequency; questionnaire
method codes. New chronology allows us to
understand much of the battle of ideas in modern
science and mass consciousness. It becomes clear
the root cause of cautious attitude of the West
towards Russia
The article deals with mathematical models of
management decision-making to select the option to
protect the AU, based on sufficient statistical
information about attacks on the AU. The amount of a
priori uncertainty about the choice of protection option
in GIS was described with Boltzmann's entropy.
Introduction of the value within Shannon’s definition
of mutual information is called the context random
variables, it allows removing the uncertainty regarding
the actions of the enemy, and it enables decisionmakers
to choose protection options. The model of
decision for choosing the type of protection of the AIS
presented in the article is based on sufficient statistical
information about the attacks to the system
components. In the ideal case, for decision-making,
we use large sample statistical data that provides high
accuracy control system for protection of information.
Based on the available amount of information
available to the IPA, against the acts of SIN, it is
possible to choose a decision on the choices you make
The article is devoted to the discussion of the
organization of clinical-statistical studies and
experiments. We have considered the examples of
the application of statistical methods in scientific
medical research. Under the clinical-statistical
research we understand specially organized
collection and analysis of medical data about the
course of disease in patients, research of the
dynamics of objective and subjective indicators of
the state of reaction to these or other therapeutic
effects. We study one, two or more groups of
individuals (patients or healthy), conclusions are
drawn on the whole group, but not for each
individual patient. The purpose of research - to
transfer the conclusions reached for the sample to
the general population, i.e., clinical and statistical
study focused on the production of useful
recommendations concerning those patients who fall
into the field of view of doctors after the end of the
study. There are two main types of research -
prospective and retrospective. The first related to the
analysis of the last patients, the second - to
monitoring the course of their disease in the future.
We have considered typical mistakes in the
organization of clinical-statistical studies. When
planning a research, we usually distinguish the
experimental and control groups, which are identical
or similar in all respects except for the studied
factors (exposure). We discuss the various options
for blind methods and consider the application of
statistical models and methods in scientific medical
research. We have analyzed examples of confidence
estimation of proportion (probability) and the
homogeneity test of probabilities. For statistical
modeling we use the Poisson distribution in the case
of small probability. With its help, we analyze
statistical data on the opisthorchiasis
Fuzzy sets are the special form of objects of nonnumeric
nature. Therefore, in the processing of the
sample, the elements of which are fuzzy sets, a
variety of methods for the analysis of statistical data
of any nature can be used - the calculation of the
average, non-parametric density estimators,
construction of diagnostic rules, etc. We have told
about the development of our work on the theory of
fuzziness (1975 - 2015). In the first of our work on
fuzzy sets (1975), the theory of random sets is
regarded as a generalization of the theory of fuzzy
sets. In non-fiction series "Mathematics.
Cybernetics" (publishing house "Knowledge") in
1980 the first book by a Soviet author fuzzy sets is
published - our brochure "Optimization problems
and fuzzy variables". This book is essentially a
"squeeze" our research of 70-ies, ie, the research on
the theory of stability and in particular on the
statistics of objects of non-numeric nature, with a
bias in the methodology. The book includes the
main results of the fuzzy theory and its note to the
random set theory, as well as new results (first
publication!) of statistics of fuzzy sets. On the basis
of further experience, you can expect that the theory
of fuzzy sets will be more actively applied in
organizational and economic modeling of industry
management processes. We discuss the concept of
the average value of a fuzzy set. We have
considered a number of statements of problems of
testing statistical hypotheses on fuzzy sets. We have
also proposed and justified some algorithms for
restore relationships between fuzzy variables; we
have given the representation of various variants of
fuzzy cluster analysis of data and variables and
described some methods of collection and
description of fuzzy data