The first and long-standing option to protect a document from forgery (it is still used today) is the so-called “live” signature (or facsimile) and a clerical seal. However, nowadays the document flow is mostly electronic and often with a very large number of documents (electronic trading, bank payment systems, transactions in cryptocurrencies, etc.). The digital signature that emerged about thirty years ago works here. As a rule, the core of a digital signature is a function whose value is easily calculated for a given argument value, and the reverse, i.e., calculating the value of an argument given the value of a function, is very difficult. The article describes an analog of a digital signature on a different fundamental basis, using image codes that define them up to affine transformations.

In this paper the main ideas underlying a discrete geometric approach to visual recognition are summarized and the results obtained are described on a meaningful level. Some considerations on the role of heuristics and evidence-based methods in the recognition of visual images follow.

The area of scientific interest of V. A. Kondratiev was extraordinarily wide.He was keen not only on ordinary and partial differential equations, but also nonlinear analysis and spectral theory.His distinguishing feature was an ability to grasp the mathematical problem and ways and methods to solve it.He introduced notions of capacity, with the help of which he studied one-valued solvability of the first boundary-value problem for elliptic equations of higher orders.His results devoted to alternating solutions of high-order differential equations became classical from the very moment of introduction.Probably, there is no mathematician who studies elliptic equations in domains with angle points and is not familiar with the Kondratiev theory of pencils and does not use his results in papers.Vladimir Alexandrovich is a scholar classic who was among the founders of modern theory of behavior of solutions of elliptic equations in a neighborhood of a singular point.The bases of this theory were grounded in his doctoral thesis.This theory was developed in the papers of Kondratiev and Oleinik.About at the same time, Kondratiev with Oleinik and other mathematicians worked on the asymptotic theory of partial differential equations in unbounded domains.We can say for sure that those results were timely and will be invaluable in the future as well.In 1988, A. V. Kondratiev was granted a State Award for the series of papers "Studies of Boundary-Value Problems for Differential Operators and Applications in Mathematical Physics."In 1998, he was granted the Petrovskii Award.Six Doctors of Science and 35 PhDs are among his disciples.He wrote more than 250 papers and several monographs.His collaboration with E. M. Landis was very fruitful both in the scientific and pedagogic sense.These famous scientists headed a seminar at the Faculty of Mechanics and Mathematics of Moscow State University.A lot of mathematicians who work in Russia and abroad grew up here.Papers of Kondratiev and Landis introduced modern methods in the study of problem of absence of solutions of nonlinear equations.Kondratiev studied nonlinear equations, both ordinary and partial differential ones, until his death.L. Veron, V. A. Galaktionov, Yu.V. Egorov, V. A. Liskevich, S. I. Pokhozaev, B. V.

We describe a new approach to the problem of visual pattern recognition. This approach is based on figure encoding. The encoding used is invariant under affine transformations of discrete sets of points (discrete figures).

Images are a finite sets of points in two-dimensional or in three-dimensional Euclidean space. Restoration of the three-dimensional image from any affine transformed projection is considered. Point-to-conformity between projections a priori is not set.

The relations between the concentration of thyroxine, triiodothyronine, and thyrotropic hormone in the peripheral blood of white rats are studied in an experiment with the administration of a thyrostatic drug (merkazolil) in an optimally allowable daily dose. A high reliability of the mathematical model developed for reproducing the level of the content of hormones of the hypophysial-thyroid system is established.

The approach to recognition of the arbitrary geometrical figures which consist of finite sets of points on plane or in space is considered.

A mathematical model simulating spatial pattern formation (positioning) of floral organs is proposed. Computer experiment with this model demonstrated the following sequence of spatial pattern formation in a typical cruciferous flower: medial sepals, carpels, lateral sepals, long stamens, petals, and short stamens. The positioning was acropetal for the perianth organs and basipetal for the stamens and carpels. Organ type specification and positioning proceed non-simultaneously in different floral parts and organ type specification goes ahead of organ spatial pattern formation. Computer simulation of flower development in several mutants demonstrated that the AG and AP2 genes determine both organ type specification and formation of the zones for future organ development. The function of the AG gene is to determine the basipetal patterning zones for the development of the reproductive organs, while the AP2 gene maintains proliferative activity of the meristem establishing the acropetal patterning zone for the development of the perianth organs.

The mathematical model imitating floral organ spatial pattern formation (positioning) was developed. Computer experiments performed on its basis demonstrated that organ spatial pattern formation in typical crucifer flower occurred in successive order: medial sepals, carpels, lateral sepals, long stamens, petals and short stamens. The positioning was advanced in two directions, acropetally in the perianth and basipetally in the stamens and carpels. The organ type specifying and positioning take place non-simultaneously in different floral areas. The organ type specifying passed ahead of organ primordial spatial pattern formation. The modeling of flower development of several mutants demonstrated that arabidopsis genes AP2 and AG in addition to specifying floral organ types also determine the particular zones in the floral meristem for futur organ development. The AG gene controls the formation of basipetal patterning zones where the reproductive organs develop, AP2 maintains the proliferative activity in the floral meristem that form acropetal patterning zones where perianth organ develop.

A system for formal description of Arabidopsis thaliana floral development is proposed. It is based on correlations between elementary modules composing the flower and activity profiles of the genes controlling determination of floral organ identity. Flower development has been formally described for wild-type plant and mutants in floral morphogenesis (leafy, apetala1, apetala2, apetala3, pistillata, agamous, and superman genes). A mathematical (automata–theoretical) model of genetic control of determination of floral organ identity has been developed. The model adequately predicts the pattern of organogenetic changes in double mutants in floral development, i.e., predicts the type of interactions between these genes.

We introduce special codes of finite sets of points on the plane which give a possibility to reconstruct, under some conditions, the initial sets. The results can be extended to the case of /i-dimensional Euclidean spaces with n > 2. A finite set A of points on the plane is called a figure. Letne N, Nn (1,2, ...,n}. Let the figure A consist of n points, then it is called an η-figure as well. We enumerate the points of the η-figure A, i.e., we assign an element of Nn to each point of the figure A in such a way that different points obtain different numbers. We denote the point numbered / by a/, i e Nni and the set {a\, ...,an} is denoted by ξΑ. Let k e Nn, {/Ί, ...,4} c Nn, k > 2, and iu Φ iv for u Φ v, u, v = l, ...,&. The set A' = {a„,..., aik} will be referred to as a fc-subset of A. If for a point a\. of a Λ-subset A there are three points alu, aiv, ais in A' such that they do not lie on one line, the indices ip if,, iv, is are pair-wise different, and the point a-{. is in the triangle on the vertices a/,, aiv, ais or on its boundary, then the point αί} is called an interior point of the fc-subset A'. The points of A' which are not interior are called contour points of A'. Let MA> be the polygon with the vertices in the contour points of A'. Let k > 2 and let 5(1 /A, be the area of the polygon Μ χ which is called the area of the /:-subset A. If the points of A' are placed on one line, then A is called collinear and we set Sil _ifc =0. If k = 2, then 5/1(2 is the distance between the points #,, and a/2. Let A' = {aM, ...,a /Jt} and A" = {fl/,,...,ο^} be two fc-subsets of A. For A' andA / ; we set A / · · · · \ ^Ί··Λ· if 5,·,...Λ * 0, and p(i\, ...,4j'i> -»Λ) is undefined if 5,,...7t = 0. The graph of the partial function p is denoted by TA. A code of an η-figure A is the pair (ξΑ, ΤΑ)\ ^-figures A and Β with the codes (ξΑ, ΤΑ) and (ξ^, ΤΒ) are called Λ-equivalent if there exists a bijection ψ: ξΑ -> ξ^ such that for any fc-subsets {a/,, ...,a/t} and {a,·,, ...,a/J of A *UDC 519.716. Originally published in Diskretnaya Matematika (1996) 8, No.4, 57-61 (in Russian). Translated by V. F. Kolchin.