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ՀԱՅԿԱԿԱՆ ՄԱԹԵՄԱՏԻԿԱԿԱՆ ՄԻՈՒԹՅՈՒՆ

Տարեկան նստաշրջան, 18-19 Հոկտեմբեր, 2019

ՄԻՋԱԶԳԱՅԻՆ ԳԻՏԱԺՈՂՈՎ

ՆՎԻՐՎԱԾ ԵՐԵՎԱՆԻ ՊԵՏԱԿԱՆ ՀԱՄԱԼՍԱՐԱՆԻ

100-ԱՄՅԱԿԻՆ

Թեզիսներ

Abstracts

ARMENIAN MATHEMATICAL UNION

Annual Session, 18-19 October, 2019

INTERNATIONAL CONFERENCE

DEDICATED TO THE 100th ANNIVERSARY OF

YEREVAN STATE UNIVERSITY

Երևան 2019 Yerevan

Կազմակերպչական կոմիտե՝ Organizing Committee:

Բարխուդարյան Ռ. Barkhudaryan R.

Դավիդով Ս. Davidov S.

Մովսիսյան Յու. Movsisyan Yu.

Ռաֆայելյան Ս. Rafayelyan S.

Սահակյան Ա. Sahakyan A.

Օհանյան Վ. Ohanyan V.

Գիտական կոմիտե՝ Scientific Committee:

Բարխուդարյան Ռ. Barkhudaryan R.

Բարսեղյան Գ. Barsegian G

Բեկլարյան Լ. (Ռուսաստան) Beklaryan L. (Russia)

Բուդաղյան Լ. (Նորվեգիա) Budaghyan L. (Norway)

Գևորգյան Պ. (Ռուսաստան) Gevorgyan P. (Russia)

Գրիգորյան Մ. Grigoryan M.

Դալլաքյան Ռ. Dallakyan R.

Կարապետյանց Ա. (Ռուսաստան) Karapetyants A. (Russia)

Հարությունյան Տ. Harutyunyan T.

Մելիքյան Հ. (ԱՄՆ) Melikyan H. (USA)

Մովսիսյան Յու. (ղեկավար) Movsisyan Yu. (Chairman)

Պողոսյան Է. Pogossian E.

Պողոսյան Լ. (գիտական

քարտուղար)

Poghosyan L. (Scientific

Secretary)

Սահակյան Ա. Sahakyan A.

Սերգեև Ա. (Ռուսաստան) Sergeev A. (Russia)

Փամբուկչյան Վ. (ԱՄՆ) Pambuccian V. (USA)

Օհանյան Վ. Ohanyan V.

Բովանդակություն / Content

ABRAHAMYAN L. R. 𝑴-isotopic semirings ........................................................................ 5

AHARONYAN N. G. Distribution and moments of two random points in a convex

domain .......................................................................................................................... 7

ALAVERDYAN Y. On Some Method of Generating Balanced Bent Functions ................. 9

AVETISYAN K. Two-sided estimates for reproducing kernels and related operators ..... 10

BABAYAN A. H. The Dirichlet Problem for in-homogeneous hyperbolic equation ......... 11

BARSEGIAN G. Some new principles (in real analysis, geometry and complex analysis)

with consequent two trends in equations .................................................................... 13

BEKLARYAN L. A. New approach in the question of existence of periodic and

bounded solutions for the functional-differential equations of pointwice type ....... 14

DALLAKYAN R. On the analogue of one Werbitsky's theorem for Djrbashyan

products ......................................................................................................................... 20

GALOYAN L. N. , GRIGORYAN M. G. On the uniform convergence of negative order

Cesaro means of Fourier and Fourier-Walsh series ..................................................... 21

GASPARYAN K. V. The stochastic calculus in nonstandard probability spaces ............... 22

GEVORGYAN L. On richardson iterative method of solution of operator equations ...... 23

GEVORGYAN P. Local and global equivariant fibrations................................................... 24

GEVORKYAN A. Formation of bell states by random mappings of the

two-dimensional fock state ........................................................................................... 25

GEVORKYAN A., ALEKSANYAN A. On a mapping problem between 𝟑𝑫 Euclidean

and 𝟑𝑫 Riemannian spaces .......................................................................................... 27

GRIGORYAN M. On the structure of universal functions with respect

to the classical systems ................................................................................................. 29

HARUTYUNYAN H. O. Calculation of a geometric probability for a triangle ................... 31

KECHEJIAN H., OHANYAN V.K., BARDAKHCHYAN V.G. On Poission Mixture of

Lognormal Distirbutions ................................................................................................ 33

KHACHATRYAN R. Gradient projection method and continuous selection

of multi-valued mappings ............................................................................................ 35

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MELIKYAN H. Maxiamal Subalgebras in Restricted Simple Cartan Type Lie Algebras

and Superalgebras ........................................................................................................ 36

MINASYAN A. Assessing excess risk in robust estimation of gaussian mean with

parameter contamination ............................................................................................ 37

MIRZOYAN V. A. NALBANDYAN G. A. Geometry of а class of semisymmetric

submanifolds of Codimention two with a unity index of regularity

in Euclidean spaces ........................................................................................................ 38

MOVSISYAN YU. M. Hyperidentities and related second order formulae ....................... 40

NAZARYAN A., HARUTYUNYAN D., PAMBUCCIAN V. Characterization of interior

points of a triangle in a weak absolute geometry ........................................................ 41

NIGIYAN S. A. Programming is based on the non-classical theory of computability ...... 42

OHANYAN V. K. Tomography problems in stochastic geometry ..................................... 44

PETROSYAN G. Some questions about the comparative analysis possibilities of

modeling classical petri nets and colored petri nets .................................................... 46

POGOSSIAN E. On Consequences of Constructiveness of Models of Cognizing ............... 48

SARGSYAN A., GRIGORYAN M. Universal functions for classes 𝑳𝒑[𝟎, 𝟏)𝟐, 𝒑 ∈ (𝟎, 𝟏)

with respect to the double Walsh system .................................................................... 50

SARGSYAN S., GRIGORYAN M. On the L1-convergence and behavior of coefficients of

Fourier-Vilenkin series ................................................................................................... 51

SIMONYAN L. S. On the representation of functions by walsh double system

in weighted 𝑳𝝁𝒑[𝟎, 𝟏)𝟐 – spaces ..................................................................................... 52

4

M -ISOTOPIC SEMIRINGS

L.R. Abrahamyan

Artsakh State University, Stepanakert

E-mail: liana [email protected]

A semiring is a setR equipped with two binary operations + and ·, called additionand multiplication, such that

1. (R,+) is a commutative monoid with the identity element 0:

(a+ b) + c = a+ (b+ c)

0 + a = a+ 0 = a

a+ b = b+ a

2. (R, ·) is a monoid with the identity element 1:

(a · b) · c = a · (b · c)1 · a = a · 1 = a

3. Multiplication left and right distributes over addition:

a · (b+ c) = (a · b) + (a · c)(a+ b) · c = (a · c) + (b · c)

4. Multiplication by 0 annihilates R:

0 · a = a · 0 = 0

Two rings Q(+, ·) and Q(+, ) are called k-isotopic if there exist bijective map-pings α, β, γ : Q→ Q such that:

1) α(x · y) = β(x) γ(y),

2) α, β, γ ∈ Aut[Q(+)].

Theorem 1 (Albert [1, 2], Kurosh [3]). If a ring with an identity element is k-isotopic to an associative ring, then they are isomorphic.

We introduce the following general concept of isotopy.Two semirings Q(+1, ·1) and Q(+2, ·2) are called K-isotopic, if there exist bijec-

tive mappings α, β, γ : Q→ Q such that

AMU Annual Math Session, 2019_________________________________________

5

1) α(x·1y) = β(x)2γ(y),

2) α, β, γ : Q(+1) → Q′(+2) isomorphic mappings.

Theorem 2. If a ring with an identity element is K-isotopic to an associative ring,then they are isomorphic.

Theorem 3. K-isotopic semirings are isomorphic.

In the books [4, 5] isotopy of algebras is defined as follows.Two algebras (Q,Ω) and (Q,Ω′) with binary operations are called M -isotopic,

if there exist bijective mappings α, β, γ : Q→ Q′, ψ : Ω → Ω′, such that ψ : Ω → Ω′

preserves the arity of operations and

αA(x, y) = (ψA)(βx, βy)

for all A ∈ Ω.

Theorem 4. Two M -isotopic semirings are isomorphic.

References

[1] Albert A.A. Quasigroup I. Trans. Amer. Math. Soc., 54, 1943, 507–519.

[2] Albert A.A. Quasigroup II. Trans. Amer. Math. Soc., 55, 1944, 401–419.

[3] Kurosh A.G. Lectures in General Algebra. Nauka, Moscow, 1973.

[4] Movsisyan Yu.M. Introduction to the theory of algebras with hyperidentities.Yerevan State University Press, Yerevan, 1986.

[5] Movsisyan Yu.M. Hyperidentities and hypervarieties in algebras. Yerevan StateUniversity Press, Yerevan, 1990.

6

DISTRIBUTION AND MOMENTS OF TWO RANDOM POINTS IN ACONVEX DOMAIN

N. G. Aharonyan

Yerevan State University, Yerevan, Armenia

E-mail: [email protected]

Among more popular applications of Stochastic Geometry are Stereology andTomography (see [3]). In the last century German mathematician W. Blaschke for-mulated the problem of investigation of bounded convex domains in the plane usingprobabilistic methods. In particular, the problem of recognition of bounded convexdomains D by chord length distribution. Let G be the space of all lines g in theEuclidean plane. Random lines generate chords of random length in convex domainD. The corresponding distribution function is called the chord length distributionfunction

FD(x) =1

|∂D|µg ∈ G : χ(g) = g ∩ D ≤ x

where |∂D| is the perimeter of D, and µ is invariant measure with respect to the groupof Euclidean motions (translations and rotations). The chord length distributionfunction FD(y) are independent of the positions of the domains in the plane, thusit coincides for congruent domains. Let P1 and P2 be two points chosen at random,independently and with uniform distribution in D. We are going to find the densityfunction of the distance ρ(P1, P2) between P1 and P2. Firstly, we find the distributionfunction Fρ(x) of ρ(P1, P2). By definition,

Fρ(x) =1

∥D∥2

∫∫(P1,P2):ρ(P1,P2)≤x

dP1 dP2, (1)

where dPi, i = 1, 2 is the Lebesgue measure in the plane R2.In the paper [5], a formula for the density function fρ(x) = F ′

ρ(x) of the distanceρ(P1, P2):

fρ(x) =1

∥D∥2

[2π x ∥D∥ − 2x2 |∂D|+ 2x |∂D|

∫ x

0FD(u) du

], (2)

where FD(·) is the chord length distribution function for the domain D. The ob-tained formula permits to calculate the density function by means of the chordlength distribution function of D. Therefore if we know the explicit form of thechord length distribution function for a domain, using (2) we can calculate densityfunction fρ(x) of the distance between two random points in D. In [4] the explicit


7

form of the chord length distribution function is given for any regular polygon. Con-sequently, density fρ(x) can be calculated for any regular polygon by applying theresult of [4] (see also [2]) and formula (5).

How large is the k-th moment of the Euclidean distance ρk(D) between thesetwo points? In other words, we need to calculate the quantity

ρk(D) =1

[S(D)]2

∫D

∫D∥Q1 −Q2∥k dQ1 dQ2, k = 1, 2, 3, . . .

where S(D) is the area of D, and ∥Q1 − Q2∥ is the Euclidean distance betweenpoints Q1 and Q2. dQi, i = 1, 2 is an element of Lebesgue measure in the plane.The present problem was stated in [6] (see also [7]). We can rewrite ρk(D) to thefollowing form:

ρk(D) =2 |∂D|

(k + 2)(k + 3) [S(D)]2

∫ ∞

0xk+3 fD(x) dx, k = 1, 2, 3 . . . ,

where fD(y) is the density function of FD(y). Therefore, if we know the explicit formof the length chord density function we can calculate the k-th moment of the distancebetween two random points in D. It is not difficult to calculate ρk(D) for a disc,regular triangle, a rectangle, a rhombus, a regular pentagon and regular hexagon.This formula allows to find an explicit form of k-th moment of the distance for thoseD for which the chord lenght distribution is known (see [1], [2] and [4]).

References

[1] Aharonyan N.G., Ohanyan V.K. Calculation of geometric probabilities usingCovariogram of convex bodies. Journal of Contemporary Mathematical Analysis(Armenian Academy of Sciences), 53 (2), 2018, 112–120.

[2] Aharonyan N.G., Ohanyan V.K. Moments of the distance between two randompoints. Modeling of Artificial Intelligence, (2), 2016, 20–29.

[3] Gardner R. J. Geometric Tomography. Cambridge University Press, New York,2006.

[4] Harutyunyan H. S. and Ohanyan V.K. Chord length distribution function forregular polygons. Advances in Applied Probability, (41), 2009, 358–366.

[5] N. G. Aharonyan and V. K. Ohanyan,“Kinematic measure of intervals lying indomains”, Journal of Contemporary Mathematical Analysis (Armenian Academyof sciences), 46 (5), 280 - 288, 2011.

[6] Santalo L.A. Integral Geometry and Geometric Probability. Addison-Wesley,Reading, Mass, 2004.

[7] Burgstaller B. and Pillichshammer F. The average distance between two points.Bull. Aust. Math. Soc., (80), 2009, 353–359.

8

ON SOME METHOD OF GENERATING BALANCED BENT FUNCTIONS

Yeghisabet Alaverdyan

EKENG CJSC, Yerevan, Armenia


Boolean functions applicable in cryptography must be of high nonlinearity toprevent the cryptosystem from linear attacks and correlation attacks. For an evennumber of variables, Boolean functions endowed with maximum nonlinearity arecalled bent functions, and with respect to their algebraic, combinatorial, and cryp-tographic properties, bent functions are becoming permanently analyzed and gen-eralized.

It has been proven that any linear combination of the output functions of S-boxes is balanced. However, no conclusive approaches have been presented yet toconstruct all S-boxes so that they satisfy the property that any linear combinationof the outputs is also bent. For this reason, further investigation of the properties ofbent functions as well as of the methods to generate them is still an actual problem.

Bent functions with their good cryptographic immunity, are not balanced, andthis very fact imposes serious restrictions in their usage in specific ciphers due totheir inherit vulnerability against differential cryptanalysis.

Construction of a class of bent functions over balanced finite groups can standfor an alternative to introduce cryptographically oriented Boolean functions withoptimum algebraic degree, optimum algebraic immunity and a required high non-linearity.


9

TWO-SIDED ESTIMATES FOR REPRODUCING KERNELS AND RELATEDOPERATORS

Karen Avetisyan



Over the unit ball B in Rn (n ≥ 2), we study Bergman type projection operatorsTβ (β > 0) with a special Poisson-Bergman type kernel Pβ(x, y). For a partic-ular choice of parameters, the operators continuously project mixed norm spacesL(p, q, α) onto their harmonic subspaces. Further studies of operators Tβ requiresome lower estimates for the kernels Pβ(x, y). In this talk, two-sided estimates forthe mixed norm of the kernels Pβ(x, y) in L(p, q, α) are established.


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THE DIRICHLET PROBLEM FOR IN-HOMOGENEOUS HYPERBOLICEQUATION

A.H. Babayan

National Polytechnic University of Armenia, Yerevan


Introduction. We consider the Dirichlet problem for a second order hyperbolicequation in a unit disc. The solution is obtained in the form of series by Chebysheffpolynomials.

Formulation of the problem and obtained result. Let D be the unit disk ofthe complex plane and Γ = ∂D its boundary. We consider in D the second orderdifferential equation

A2Uxx(x, y) +A1Uxy(x, y) +A0Uyy(x, y) = 0, (x, y) ∈ D, (1)

where Ak are such real constants (A0 = 0) that the roots λj (j = 1, 2) of character-istic equation A0+A1λ+A2λ

2 = 0, are real and different. We consider the Dirichletproblem in the classical formulation, that is, we seek the solution of (1) in the classC2(D)

∩C(α)(D) satisfying Dirichlet conditions on the boundary Γ.

U |Γ = fj(x, y) (x, y) ∈ Γ. (2)

Here f ∈ C(α)(Γ) is a given function. The equation (1) is hyperbolic, thereforethe problem (1), (2) is not correct (see [1]). But in many cases this problem maybe successfully investigated. In [2] this problem for equation uxx − uyy = 0 in therectangle was investigated, and the parameters of rectangle, where the solution ofthe corresponding Dirichlet problem exists and unique where found. Then F. Johnin [3] investigated the problem for equation uxy = 0 in arbitrary domain. In thework of R.A. Alexandryan [4] the homogeneous problem (1), (2) for arbitrary hy-perbolic equation in the unit disc was investigated. In these works the existence anduniqueness of the solution of the problem was reduced to the investigation of theautomorphisms of the boundary of the domain. In recent work [5] the Dirichlet prob-lem for hyperbolic system of the first order differential equations in two dimensionaldomains was considered.

In the present talk we present new way of the solution of the problem (1), (2),which is applicable to arbitrary type of equation (1) (not only hyperbolic). It wasproved, that if λ1 and λ2 are the roots of the characteristic equation A2 + A1λ +


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A0λ2 = 0, (λj = λj = 0), and αj such angles, that

cosαj =1√

1 + λ2j

, sinαj =λj√1 + λ2

j

,

then the problem (1), (2) has a unique solution if and only if the condition

sinn(α1 − α2) = 0, n = 1, 2, . . .

holds. The solution of the problem is represented in the form

U(r, φ) =∞∑k=0

BkTk(rcos(φ− α1)) +∞∑k=0

CkTk(rcos(φ− α2)),

where Tk - first kind Chebysheff polynomials of order k, and constants Bk and Ck

are uniquely determined by the boundary function f .

References

[1] Courant R., Hilbert D. Methods of Mathematical Physics. Vol. 2 Wiley IP. NewYork, 1989.

[2] Bourgin D.G., Duffin R. The Dirichlet problem for the vibrating string equation.Bull. AMS, 45, 1939, 851–859.

[3] John F. The Dirichlet Problem for a Hyperbolic Equation. American J. of Math.,63 (1), 1941, 141–155.

[4] Alexandryan R.A. Spectral properties of operators arising from systems of dif-ferential equations of Sobolev type. Tr. Mosk. Mat. Obs., 9, 1960, 455–505.

[5] Zhura N.A., Soldatov A.P. A boundary-value problem for a first-order hyperbolicsystems in a two-dimensional domain. Izv. RAN, Ser. Mat., 81 (3), 2017, 83–108.

12

SOME NEW PRINCIPLES (IN REAL ANALYSIS, GEOMETRY ANDCOMPLEX ANALYSIS) WITH CONSEQUENT TWO TRENDS IN

EQUATIONS

G. Barsegian

Institute of mathematics of National Academy of Sciences of Armenia


In this talk we present some new principles related to the basic concepts inmathematics such as arbitrary smooth real functions, arbitrary smooth curves inthe plane, arbitrary meromorphic (particularly analytic) functions in an arbitrarydomain. Among them:

principle of zeros of real functions permitting to give bounds for the number ofzeros of the functions;

principle of angles for the plane curves permitting to compare rotations of thecurves around a given center and tangential rotations;

principle of logarithmic derivatives of meromorphic functions permitting to com-pare different integral including the logarithmic derivatives.

All these principles are brand new, i.e. have no predecessors.

Then we will show how the principles can be applied to some basic equations.This leads to two brand new trends respectively in real and complex differentialequations.


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NEW APPROACH IN THE QUESTION OF EXISTENCE OF PERIODIC ANDBOUNDED SOLUTIONS FOR THE FUNCTIONAL-DIFFERENTIAL

EQUATIONS OF POINTWICE TYPE ∗

L.A. Beklaryan

Central Economics and Mathematics Institute RAS,Nachimovsky prospect 47, 117418 Moscow, Russia

E-mail: [email protected], [email protected]

The report is devoted to the functional-differential equation of pointwise type

x(t) = f(t, x(q1(t), . . . , x(qs(t))), t ∈ BR, (1)

where: f : R × Rns −→ Rn is a map of the class C(0); qj(.), j = 1, . . . , s are thediffeomorphisms of the real line of the class C(1), preserving orientation; BR is theclose interval [t0, t1], the close half-line [t0,+∞), or the real line R.

Importance of the equations of the considered type is defined by the fact that thetheory of solutions of such equations is closely connected with the theory of solitonicsolutions for the infinite-dimensional ordinary differential equations [1].

The approach, suggested for such equations, is based on formalism where theimportant constructions use the group

Q =< q1, . . . , qs >

of real line diffeomorphisms. The functions

[qj(t)− t], j = 1, . . . , s (2)

are called the deviations of argument. Using replacement of time, we can fulfill theconditions

h = maxi∈1,...,s

hqj < +∞, hqj = supt∈R

|qj(t)− t|, j = 1, . . . , s (3)

for deviating argument. Obviously, that such replacement of time can change acharacter of growth of right-hand side of equation with respect to time.

Main purpose at studying such differential equations is the study of the initial-boundary value problem

x(t) = f(t, x(q1(t), . . . , x(qs(t))), t ∈ BR, (4)

x(t) = φ(t), t ∈ R\BR, φ(.) ∈ L∞(R,Rn), (5)

x(t) = x, t ∈ R, x ∈ Rn, (6)

∗Work is supported by the Russian Foundation For Basic Research (grant N 19-01-00147)


14

which we will call the main initial-boundary value problem.Other important class of tasks is connected with studying of periodic and bounded

solutions for the original functional-differential equation of pointwise type definedon the real line.

Let’s define the Banach space of functions x(·) with weights

LnµC

(k)(R)=x(·) : x(·) ∈ C(k) (R,Rn) , max

0≤r≤ksupt∈R

∥x(r)(t)µ|t|∥Rn<+∞, µ ∈ (0, 1),

and norm∥x(·)∥(k)µ = max

0≤r≤ksupt∈R

∥x(r)(t)µ|t|∥Rn .

Let’s formulate the system of restrictions for the right-hand side of the functional-differential equations of pointwise type:

(a) f(.) ∈ C(0)(R×Rn×s,Rn) (in the condition (a) the function f(.) on the variablet can be put piecewise continuous with breaks of the first sort in points of thediscrete set);

(b) condition of quasilinear growth: for any t, zj , zj , j = 1, . . . , s

∥f (t, z1, . . . , zs) ∥Rn ≤ M0(t) +M1

s∑j=1

∥zj∥Rn , M0(.) ∈ C(0)(R,R)

and Lipshchits’s condition

∥f (t, z1, . . . , zs)− f (t, z1, . . . , zs) ∥Rn ≤ Lf

s∑j=1

∥zj − zj∥Rn

(actually M1 ≤ Lf , but the constant of M1 and Lf can be taken equal);

(c) there is µ∗ ∈ R+ such that expression

supi∈Z

M0(t+ i) (µ∗)|i|

for any t ∈ R has final value and as function of argument t is continuous.

(d) for µ∗ from (c) the family functions

fq,z1,...,zs(t) = f(q(t), z1, . . . , zs)(µ⋆)hq , q ∈ Q, z1, . . . , zs ∈ Rn,

on any finite interval is uniformly continuously.

Right part f(.) of the functional-differential equation of pointwise type we willconsider as the element of Banach space Vµ∗(R× Rns,Rn)

Vµ∗(R× Rns,Rn) =f(.) : f(.) satisfy the conditions (a)-(d)

,

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∥f(.)∥Lip = supt∈R

∥f(t, 0, . . . , 0)(µ∗)|t|∥Rn+

+ sup(t,z1,...,zs,z1,...,zs)∈R1+2ns

∥f(t, z1, . . . , zs)− f(t, z1, . . . , zs)∥Rn∑sj=1 ∥zj − zj∥Rn

,

where parameter µ∗ ∈ R+ coincides with corresponding constant from the condition(c). It is obvious that for function f(.) ∈ Vµ∗(R×Rns,Rn) the smallest value of theconstant Lf from the condition Lipschitz (condition (b)) witch coincides with valuethe second term in definition of norm f(.) . Further, speaking about Lipschitz’scondition, we will understand for the constant Lf its the smallest value.

Theorem 1 ( [2]). Let map f(·) satisfies to the conditions (a) - (d). If for someµ ∈ ( 0, µ∗) ∩ ( 0, 1) inequality

Lf

s∑j=1

µ−|hqj | < lnµ−1 (7)

is satisfied, that for any fixed initial and boundary conditions

x ∈ Rn, φ(.) ∈ L∞(R,Rn)

there is the solution (absolutely continuous)

x(.) ∈ LnµC

(0)(R)

of the main initial-boundary value problem (4)-(6). Such solution is unique andas the element of space Ln

µC(0)(R) it continuously depends on initial and boundary

conditions φ(.) ∈ L∞(R,Rn),x ∈ Rn and the right part f(.) ∈ Vµ∗(R× Rns,Rn) of equation.

Condition (7) from theorem 1 is exact and not improved. The solutions of thefunctional-differential equation from theorem 1 belonging to space Ln

µC(0)(R), have

all remarkable properties of the ordinary differential equations.For the equation

x(t) = f (t, x(t+ n1), . . . , x(t+ ns)) , t ∈ R. (8)

periodic solutions will be studied. Right part such equation is periodic on time withperiod ω.

Without loss of generality we will believe that n1 ≤ . . . ≤ ns ≤ ω. For obvious-ness at the statement we will be limited to the case when magnitude of deviationsof the argument n1, . . . , ns and period ω of the right member of equation on thevariable of time are commensurable. In this case, without loss of generality, we canconsider that n1, . . . , ns, ω are integer. The last property it is possible to achieve byof replacement of time of dilation type.

Among the main methods of studying of bounded solutions of the differentialequations it should be noted the method of directrix functions [3], [4],

16

[5], method of the integral equations [6], [8], variation methods, etc. The existenceconditions ω - periodic solution for the ordinary differential equation, based on theresearch of properties of the operator of shift are given in works of many researchers,in particular, in works [4], [7]. In them, for the ordinary differential equations suchconditions are formulated in the form of property for the right part of equation onany sphere (geometrical property for the phase portrait) and is formulated in theform of restriction: for all t ∈ [0, ω], ∥x∥Rn = r

(x, g(t, x)) ≤ 0 (9)

at any fixed radius r > 0. For the equation with delay

x(t) = f(t, x(t), x(t− 1)) (10)

the condition similar to condition (9) will take the form: for all t ∈ [0, ω], ∥x∥Rn = r,y ∈ Rn

(x, f(t, x, y)) ≤ 0, (11)

but the class of the equations with such property is very narrow.The approach for studying of periodic solutions based on the accounting of

asymptotic properties of solutions which was used for studying of solutions boththe ordinary differential equations (ODE), and wider class functional-differentialequations (FDU) of pointwise type is presented in work [9].

Let’s enter designations

M0∞µ(t) = supi∈Z

M0(t+ i)µ|i|, Af =(µ−ω − 1)

lnµ−1, Bf =

Lf∑s

j=1 µ−|nj |[

lnµ−1 − Lf∑s

j=1 µ−|nj |

] ,Cf (r)=

[Lfsr+ inf

ξ∈[0,ω]sup

τ∈[ξ,ξ+1]M0∞µ(τ)

], Cf (r)=

[Lfsr+sup

t∈R∥f(t, 0, . . . , 0)∥Rn

].

(12)

It is not difficult to notice that the condition Cf (r) ≤ Cf (r) takes place. Let’sformulate the theorem of existence of the periodic solution in terms of mean on theperiod.

Theorem 2 ( [9]). Let map f(·) satisfies to the conditions (a) - (d) and it is ω-periodical function on time, where ω ∈ Z+. If for given µ ∈ (0, µ∗) ∩ (0, 1), r > 0and all x ∈ Rn, ∥x∥Rn = r the conditions

Lf∑s

j=1 µ−|nj | < lnµ−1, (13)(

x∥x∥Rn

,∫ ω0 f(τ, x, . . . , x)dτ

)<−AfBfCf . (14)

are fulfill, then for the functional-differential equation (8) there is the ω-periodicalsolution x(·). For such solution the condition ∥x(0)∥Rn ≤ r is fulfill and it belong toball of space Rn with radius µ−ωR, where

R = r +Cf (r)[

lnµ−1 − Lf∑s

j=1 µ−|nj |

] . (15)

17

Conditions such type are new even for the ordinary differential equations. Condi-tion (13) coincide with the condition from theorem 1 and guarantees the correctnessof the existence and uniqueness theorem of the initial value problem for the func-tional differentsial equation (8). Conditions (13) - (14) from theorem 2 are exactand not improved.

Let’s formulate the theorem of existence of the bounded solution in terms ofmean on some period.

Theorem 3 ( [10]). Let map f(·) satisfies to the conditions (a) - () and functionf(t, 0, ..., 0), t ∈ R is uniformly bounded. If for given µ ∈ (0, µ∗) ∩ (0, 1), r > 0,ω ∈ Z+ and all x ∈ Rn, ∥x∥Rn = r, t ∈ R the conditions

Lf

s∑j=1

µ−|nj | < lnµ−1, (16)

supt∈R

x

∥x∥Rn,

t+ω∫t

f(τ, x, . . . , x)dτ

<− AfBf Cf (r)−1

2r

[Af (Bf + 1)Cf (r)

]2(17)

are fulfill then for initial functional-differential equation of pointwise type (8) thereis the bounded solution x(·) witch belong to ball of space Rn with radius µ−ωR, where

R = r +Cf (r)[

lnµ−1 − Lf∑s

j=1 µ−|nj |

] . (18)

Moreover, length of the maximum open intervals of the open set

P = R \ t : t ∈ R, ∥x(t)∥Rn ≤ r

are less than ω.

Conditions such type are new even for the ordinary differential equations. Condi-tion (16) coincide with the condition from theorem 1 and guarantees the correctnessof the existence and uniqueness theorem of the initial value problem for the func-tional differential equation (8). Conditions (16) - (17) from theorem 3 are exact andnot improved.

References

[1] Beklaryan L.A. Introduction to the theory of functional differential equations.Group approach. Moscow: Factorial Press, 2007, –286.

[2] Beklaryan L.A. A method for the regularization of boundary value problems fordifferential equations with deviating argument. Soviet Math. Dokl., 43 (2), 1991,567–571.

18

[3] Krasnosel’skii M.A. and Perov A. I. and Povolockii A. I. and Zabreiko P.P.Plane Vector Fields. New York: Academic Press, 1966, –245.

[4] Krasnosel’skii M.A. The Operator of Translation along the Trajectories of Dif-ferential Equations. Providence: American Mathematical Society, 1968.

[5] Krasnosel’skii M.A. and Zabreiko P.P. Geometrical Methods of Nonlinear Anal-ysis. New York: Springer, 1984, –512.

[6] Rozenvasser V.N. Oscillation of nonlinear systems. Moscow: Nauka, 1969.

[7] Daletskii Yu. L. and Krein M.G. Stability of solutions of differential equationsin Banach space. Providence : American Mathematical Society, 1974, –529.

[8] Perov A. I. and Kostrub I.D. Bounded solutions of nonlinear vector-matrix dif-ferential equations of n-th order. Voronezh: CPI Science Book, 2013, –227.

[9] Beklaryan L.A. New approach in the question of existence of periodic solutionfor the functional-differential equations of pointwice type. Izvestiya: Mathemat-ics 82 (6), 2018, 1077-1107.

[10] Beklaryan L.A. New approach in the question of existence of bounded solutionfor the functional-differential equations of pointwice type. Izvestiya: Mathemat-ics (in the press)

19

ON THE ANALOGUE OF ONE WERBITSKY’S THEOREM FORDJRBASHYAN PRODUCTS

Rubik Dallakyan

National Polytechnic University of Armenia, Vanadzor, Armenia


The following result is obtained in the paper

Theorem. Let −1 < α 6 0, and Bα(z; an) be the convergent infinite M.M. Djr-bashyan product. Then following assertions are equivalent:

1. The sequence an is a (WN)-sequence;

2. Bα(k) = O

(1

k

), k → ∞;

3.

∞∑k=m

∣∣∣Bα(k)∣∣∣ = O

(1

m

), m → ∞;

4. Bα(eiφ; an) ∈ Lip

(1

p, p

)for some p ∈ (1,∞);

5.1

2π

2π∫0

∣∣B′′α(re

iφ; an)∣∣ dφ 6 const

1− r.

For Blaschke products (i.e. for α = 0) this theorem was proved by Werbinsky in[1].

References

[1] Werbitsky I. E. On a Tylor coefficient and Lp-continuity moduli of the Blaschkeproduct. Zapiski nauchn. sem LOMI, 107, 1982, 27-35.

[2] Djrbashyan M.M. Integral transforms and function representation in complexdom. M., Nauka, 1966.


20

ON THE UNIFORM CONVERGENCE OF NEGATIVE ORDER CESAROMEANS OF FOURIER AND FOURIER-WALSH SERIES

L.N. Galoyan, M.G. Grigoryan



The following theorems are true

Theorem 1. There exists a function g ∈ C[−π, π] , such that for an arbitraryincreasing sequence of natural numbers nk and for each α ∈ (−1,−1

2) we have|x ∈ [−π, π], lim supk→∞σnk

(g, x)) = ∞||> 0.

Theorem 2. There exists an increasing sequence of natural numbers nk , suchthat for any positive number ε ∈ (0, 1) and for each f ∈ L1[−π, π] one can finda function g ∈ C[−π, π] with |x ∈ [0, 1],f(x) = g(x)|| ε such that for any α <0, α = −1,−2, . . . the Cesaro means σnk

(g, x) of trigonometric Fourier series offunction g(x)converges to it uniformly on [−π, π].

This work was supported by the RA MES State Committee of Science, in theframes of the research project 18T-1A148.

References

[1] Grigoryan M.G., Galoyan L.N. On the uniform convergence of negative orderCesaro means of Fourier series. Journal of Mathematical Analysis and Appli-cation, 434 (1), 2016, 554–567.

[2] Galoyan L., Grigoryan M. Application of negative order Cesaro summabilitymethods to Fourier-Walsh series of functions from L∞[0, 1). Colloquium Math-ematicum, 158 (2), 2019, 187–216.


21

THE STOCHASTIC CALCULUS IN NONSTANDARD PROBABILITY SPACES

K. V. Gasparyan



Usually, the stochastic calculus is considered for probability spaces satisfyingthe so-called “usual” conditions on the filtration given on these spaces. However,especially in financial mathematics, the case where these conditions are violated is ofinterest. The stochastic calculus for “unusual” case was constructed in [1] (see, also,[2]). The application of stochastic calculus in the theory of stochastic differentialequations and nonlinear filtering theory were considered in [3], [4].

Recently, interest of this theory in connection with its applications in financialmathematics reappeared again (see [5], [6]).

References

[1] Gal‘chouk L. Stochastic integrals with respect to optional semimartingales. The-ory of Probab. and Appl., 29 (1), 1985, 93–108.

[2] Kuhn C., Stroh M. A note on stochastic integration with respect to optionalsemimartingales. Electronic Commun. in Probab., 14, 2009, 192–201.

[3] Gasparyan K. Stochastic equations with respect to the optional semimartingales.Izvestiya VUZ, Math., 29 (12), 1985, 79–83.

[4] Abdelghani M., Melnikov A. On linear stochastic equations of optional semi-martingales and their applications. Statistics and Probab. Letters, 125, 2017,207–214.

[5] Gasparyan K. About uniform optional supermartingale decomposition in non-standard case. In: Stochastic and PDE Methods in Financial Mathematics,Yerevan, 2012, 14–16.

[6] Abdelghani M., Melnikov A. Optional decomposition of optional supermartin-gales and applications to filtering and finance. Stochastic, 91 (6), 2019, 797–816.


22

ON RICHARDSON ITERATIVE METHOD OF SOLUTION OF OPERATOREQUATIONS

Levon Gevorgyan

National Polytechnic University of Armenia, Yerevan, Armenia


Let A be a (bounded) operator, acting in a Banach space X. One of the mostefficient ways of solving the equation Ax = b is the Richardson iterations method[1], defined by the formula

xn+1 = xn + α (b−Axn) , n ∈ Z+, α ∈ C.

The starting element x0- ”guess” may be arbitrary, e.g. x0 = θ or x0 = b. If someinformation on the properties of the operator A and/or the right side b is availableit may be used to accelerate the rate of convergence. The parameter α should bechosen ([2], [3]) to assure the inequality ∥I − αA∥ < 1.

In this talk some details of this approach will be considered.

References

[1] Richardson L. F. The approximate arithmetical solution by finite differencesof physical problems involving differential equations, with an application to thestresses in a masonry dam. Phil. Trans. Roy. Soc. A.,210, 1910, 307–357.

[2] Opfer G., Schober G. Richardson’s iteration for nonsymmetric matrices. LinearAlgebra Appl., 58, 1984, 343–361.

[3] Gevorgyan L. An analysis of the convergence of the one-step iterative methodin a Krylov space. J. Contemp. Math. Anal., 44 (5), 2009, 324–334.


23

LOCAL AND GLOBAL EQUIVARIANT FIBRATIONS

P. S. Gevorgyan

Moscow Stat Pedagogical University, Russia


The well known Hurewicz Theorem in the theory of fibrations states that in theparacompact situation a local fibration is a fibration, where local is in terms of anopen cover of the base. The main result of this paper is to prove similar local toglobal theorem in the theory of equivariant fibrations. A classification of equivariantfibrations with unique path lifting property is also given.


24

FORMATION OF BELL STATES BY RANDOM MAPPINGS OF THETWO-DIMENSIONAL FOCK STATE

Ashot Gevorkyan

Institute for Informatics and Automation Problems, NAS of RA

Institute of Chemical Physics, NAS of RA E-mail: g [email protected]

One of the important directions in the development of quantum communicationsis related to the problem of creating and controlling Bell’s states. We propose anew way to create and control such states, based on quantum mechanics with afundamental environment recently developed by the author [1].

Let us consider the quantum subsystem+random environment as a joint system(JS), which is described in the framework of the stochastic differential equation ofthe Langevin-Schrodinger type:

i∂tΨstc = H(x, t; f(t)

)Ψstc, x = (x1, x2), ∂t ≡ ∂/∂t, (1)

where the evolution operator H(x, t; f(t)

)of JS is represented as two linearly

coupled 1D harmonic oscillators immersed in a random environment:

H(x, t; f(t)

)=

1

2

2∑l=1

[− ∂2

∂x2l+Ω2

(t; f(t)

)x2l

]+ ω

(t; f(t)

)x1x2, (2)

x1, x2; t ∈ (−∞,+∞). Note that Ω(t; f(t)

)and ω

(t; f(t)

)are some random

functions. As for the f(t) function, this is a random Markov process, which turnson at the moment of time t0 > −∞ and turns off at the moment of time t′0 < +∞.It is easy to show that for times t < t0 equation (1)-(2) describes 2D Fock states.

Theorem. If a complex probabilistic process Ψstc

(x, t; f(t)

)satisfies the equation

(1)-(2), and the random function f(t) =(f i(t), f r(t)

), respectively, respectively,

satisfies to the correlation relations of white noise:

⟨fυl (t)⟩ = 0, ⟨fυ

l (t)fυl (t

′)⟩ = 2ϵυl δ(t− t′), υ = (i, r), (3)

where ϵυ denotes the power of the random fluctuations, then the mathematical expec-tations of the Bell states can be constructed exactly in the form of multiple integralsand solutions of second-order partial differential equations (PDE):

Ψ∓(q1, q2, t) = E[Ψ∓

QSE

]=

1√2

|0⟩1 ⊗ |0⟩2 ∓ |1⟩1 ⊗ |1⟩2

,

Φ∓(q1, q2, t) = E[Φ∓QSE

]=

1√2

|0⟩1 ⊗ |1⟩2 ∓ |1⟩1 ⊗ |0⟩2

, (4)


25

where q1 = (x1 − x2)/√2 and q2 = (x1 + x2)/

√2, in addition, Ψ∓

QS(q1, q2, t) =

⟨Ψ∓QSE⟩Rξ and Φ∓

QS(q1, q2, t) = ⟨Φ∓QSE⟩Rξ denote Bell states, which are obtained

after averaging the corresponding expressions over the functional space Rξ. As for

the wave states |0⟩l and |1⟩l, they are calculated and have the following form:

|0⟩l = (g−l )1/2

∫ +∞

−∞

∫ +∞

0W

( 12, 12)

l (u1, u2, t) exp1

2

(iu1 − u2

)q2l

du1du2,

|1⟩l = 2(g−l )1/2ql

∫ +∞

−∞

∫ +∞

0W

( 32, 32)

l (u1, u2, t) exp1

2

(iu1 − u2

)q2l

du1du2, (5)

where g−l = (Ω−l /π)

1/2 and (Ω−l )

2 = limt→−∞[Ω2

(t; f(t)

)− (−1)lω

(t; f(t)

)]. In

addition, the function W(p,k)l is a solution of the following second-order PDE:

∂tW(p,k)l =

Ll − (pu1 + iku2)

W

(p,k)l . (6)

where the operator Ll has the form:

Ll = ϵl

(∂ 2

∂u21+ µ

∂ 2

∂u22

)+

∂

∂u1

(u21 − u22 +Ω2

0l(t))+ 2u1

∂

∂u2u2.

Recall that in the operator Ll the parameter µ = ϵil/ϵrl ∈ [0, 1] denotes a constant,

and Ω20l(t) =

[Ω2

(t; 0

)− (−1)lω

(t; 0

)]is a certain regular function of time.

References

[1] Gevorkyan A. S. Nonrelativistic quantum mechanics with fundamental envi-ronment, Theoretical Concepts of Quantum Mechanics. 8, 2012, 161–186.Ed. Prof. M. R. Pahlavani, ISBN: 978-953-51-0088-1.

26

ON A MAPPING PROBLEM BETWEEN 3D EUCLIDEAN AND 3DRIEMANNIAN SPACES

A. Gevorkyan1,2 and A. Aleksanyan1

1Institute for Informatics and Automation Problems of NAS, Yerevan, Armenia2Institute of Chemical Physics of NAS, Yerevan, Armenia

E-mail: g [email protected]

As is well known, the temporal evolution of a classical system is uniquely de-termined by the Hamilton equations, which form a system of ordinary second-orderdifferential equations. The mathematical problem is to integrate this system ofdifferential equations and find all possible functions of variable ”t” (usual time),which, after substituting into the equations, turns them into an identity. In the caseof dynamical Poincare systems, the system of equations, as a rule, cannot be fullyintegrated, since the number of integrals of motion is often less than the number ofdegrees of freedom.

In a series of papers [1, 2, 3], using the classical three-body problem as an ex-ample, it was shown that at formulation the problem on Riemannian geometry newhidden symmetries of the dynamical system are revealed. This allows us to makethe integration of the problem more complete. Note that in the formulation of thedynamical problem on a Riemannian manifold, the key role is played by the proofof the homeomorphism theorem between the 3D Euclidean subspace E3 and the 3Dconformal Euclidean space M(3).

However, in the proof of the theorem an important role is played by the auxiliarymanifold (see Fig. 1), which arises as a result of the mapping subspace E3 onto themanifold M(3).

In particular, it was shown that the corresponding system of underdetermined

Figure 1: In this diagram all spaces are homeomorphic to each other, i.e. E3 ≃S(3) ≃ M(3).


27

Figure 2: Four basic manifolds generated by the mapping of the subspace E3 ontothe manifold M(3). Three out of six holes are visible on the first manifold; however,manifolds of the fourth type also have single hole topologies.

algebraic equations generates 84 separate 3D manifolds that satisfy the four basicsymmetries (see FIG. 2). In the work, the geometric and topological featuresof these manifolds are studied in detail. In particular, as shown, it is advisable torepresent each set of similar manifolds in the form of the usual sum of sets. The studyshows that each of the combined 3D manifolds is embedded in the 9D Euclideanspace as an oriented manifold.

And finally, it is important to note that all oriented 3D manifolds, with theexception of the first type manifolds (the first figure on the left, FIG. 2) have commonsets of points using which it can be combined into a new 3D manifold.

References

[1] Gevorkyan A. S. On reduction of the general three-body Newtonian problem andthe curved geometry. Journal of Physics: Conference Series, 496, 012030, 2014.

[2] Gevorkyan A. S. On the motion of classical three-body system with considerationof quantum fluctuations. Physics of Atomic Nuclei, 80 (2), 2017, 358–365.

[3] Gevorkyan A. S. The Three-body Problem in Riemannian Geometry. HiddenIrreversibility of the Classical Dynamical System. Lob. Jour. of Mathematics,40 (8), 2019, 1058–1068.

28

ON THE STRUCTURE OF UNIVERSAL FUNCTIONS WITH RESPECT TOTHE CLASSICAL SYSTEMS∗

Martin Grigoryan



Functions’ existences, being universal in different senses,(the universality of thefunctions was manifested by the argument translations, the difference quotients, highorder derivatives and Taylor series respectively),were considered in many papers. Inthis lecture we will construct functions such that their universality is manifestedin the classical systems through their Fourier series. Let M =M [0, 1) -be the theclass of all Lebesgue measurable functions on [0 , 1 ) with almost everywhere (a.e.)convergence and let Lp [0 , 1 ], p > 0 be the space of all measurable functions f(x) on[0 , 1 ] that satisfy the condition

∫ 10 |f(x)|pdx < ∞. By |E| we denote the Lebesgue

measure of a measurable set E ⊂ [0, 1). The purpose is to describe the structureof functions that are universal for M with respect to the signs of its Fourier-Walshcoefficients.

Theorem. For every δ ∈ (0, 1) there exists a measurable set E ⊂ [0, 1] with|E| > 1−δ, such that for every almost everywhere finite measurable function on [0, 1]one can find a integrable (modified) function U ∈ L1[0, 1), with strictly decreasingFourier -Walsh coefficients ck(U)∞k=0 , and converging by L1[0, 1) norm Fourier -Walsh series, which which coincides with U (x) = f(x) on E and has the followingproperties: the function U(x) is universal both for all spaces Lp[0, 1], p ∈ (0, 1),and for the class M with respect to the Walsh system Wk(x) in sense of signsits Fourier-Walsh coefficients, that is, if for each almost everywhere finite measur-able function function g(x) one can find numbers δk = ±1, k = 1, 2, . . ., such thatthe series

∑∞k=0 δkck(U)Wk(x) converges to f in Lp[0, 1] (respectively in M almost

everywhere on [0, 1]. The purpose to describe the structure of functions that areuniversal for Lp(0, 1)-spaces, 0 ≤ p < 1,with respect to the signs of Fourier-Walshcoefficients.

References

[1] Grigoryan MG. On the universal and strong property related to Fourier-Walshseries. Banach Journal of Math. Analysis, 11 (3), 2017, 698–712.

∗This work was supported by the RA MES State Committee of Science, in the frames of theresearch project 18T-1A148.


29

[2] Grigoryan M.G., Sargsyan A.A.On the universal function for the class Lp(0, 1),0 < p < 1. Journal of Func. Anal., 270 (8), 2016, 3111–3133.

[3] Grigoryan M.G., Galoyan L.N. On the universal functions. Journal of Approx-imation Theory 225, 2018, 191–208.

[4] Grigoryan M.G. and Sargsyan A.A. The structure of universal functions forLp(0, 1)-spaces, 0 < p < 1. Sbornik: Mathematics, 209 (1), 2018, 35–55.

[5] Grigoryan M.G., Galoyan L.N. On Fourier series that are universal modulosigns Studia Mathematica, 249 (2), 2019, 215–231.

30

CALCULATION OF A GEOMETRIC PROBABILITY FOR A TRIANGLE

H. O. Harutyunyan



Let Rn (n ≥ 2) be the n-dimensional Euclidean space, D ⊂ Rn be a boundedconvex body with inner points, and Vn be the n-dimensional Lebesgue measure inRn. The function

C(D, h) = Vn(D ∩ (D+ h)), h ∈ Rn,

is called the covariogram of the body D. Here D+h = x+h, x ∈ D (see [1], [2]).Let Sn−1 denote the (n − 1)-dimensional sphere in Rn of radius 1 centered at

the origin and L(ω) be a random segment of length l > 0, which intersects D.Denote by P(L(ω) ⊂ D) probability, that random segment of length l in Rn

having a common point with body D entirely lying in body D. Let Πru⊥D bethe orthogonal projection of D onto the hyperplane u⊥ (here u⊥ stands for thehyperplane with normal u, passing through the origin). For any body D of n-dimensional space Rn we have (see [3])

P(L(ω) ⊂ D) =1

On−1

∫Sn−1

C(D,u, l)

Vn(D) + l · bD(u)du

where bD(u) = Vn−1(Πru⊥D) and On−1 = σn−1(Sn−1) the surface area of the unit

sphere in Rn. For any planar bounded convex domain we have

P(L(ω) ⊂ D) =1

πS(D) + l|∂D|

∫ π

0C(D,u, l) du.

The relationship between P (L(ω) ⊂ D) and the covariogram of any triangle weobtain in [4]. The main goal of the talk is to present the explicit form of P (L(ω) ⊂ D)for any triangle on the plane.

References

[1] Schneider R. and Weil W. Stochastic and Integral Geometry. Springer, 2008.

[2] Gasparyan A. and Ohanyan V.K. Recognition of triangles by covariogram. Jour-nal of Contemporary Mathematical Analysis (Armenian Academy of sciences),48 (3), 2013, 110–122.


31

[3] Aharonyan N.G. and Ohanyan V.K. Calculation of geometric probabilities us-ing Covariogram of convex bodies Journal of Contemporary Mathematical Anal-ysis (Armenian Academy of sciences), 46 (5), 2018, 280–288.

[4] Aharonyan N.G. and Harutyunyan H.O. Geometric Probability calculation fora triangle. Proceedings of Yerevan State University, 51 (3), 2017, 211–216.

32

ON POISSION MIXTURE OF LOGNORMAL DISTIRBUTIONS

H. Kechejian, V.K. Ohanyan, V.G. Bardakhchyan

Freepoint Commodities, Stanford, CT, USAYerevan State University, Yerevan, Armenia

E-mail: [email protected]; [email protected], [email protected]

In most stochastic processes described by jump-diffusion differential equationsone deals with reconstructing (evaluating) parameters for further usage. Mostlythis is done by assuming the process to be Levy one. However in commodity pricingframework prices has some incompatible features with this Levy processes. Mostnotable one is the existence of remarkable seasonality. Andersen [1] proposed themodel for futures prices (with intension to have non-homogeneous jumps) by thefollowing differential equation.

dF (t, T )

F (t−, T )= µ(t, T )dt+ σ(t, T )dW (t) + (em(t,T )dJ(t)− 1)

Jumps are generally taken to be non-homogenous compound Poisson point process(NCPP). Our goal is to understand by which means it is possible to do statisticalestimation of parameters. So at first we reconstruct discretization of the process for

1 interval of time. While discrete process contain e∑Nn

k=0 Yk =∏Nn

k=0 eYk , we instead

consider the sum

HSn =

∞∑k=0

eYnk

where

eYnk ∼ 1

rϕ(ln r; 0,m(n− 1, T )k)e−λλ

k

k!

We argue that this is quite good approximation if infinite number of jumps is possiblewithin small interval of time(see [3]). For this form of jump part we have shownthat

HS ∼appr LogN(µHS , σ2HS)

with

σ2HS = ln

I0(2λem(n−1,T )

)− I0

(2λe

m(n−1,T )2

)e2λe

m(n−1,T )+ 1

µ2HS = λe

m(n−1,T )2 − ln

(I0(2λe)− I0(2λ

√e)

e2λ√e

+ 1

)


33

We have also shown that Lyapunov’s central limit theorem is not applicable to thiscase. So there is no evidence for using CLT. However the method of moments cannotbe reliable as well, as examining the moments of our random variable we rigorouslyshowed that its distribution cannot be uniquely determined by its moments, asKrein’s condition (see [2]) is not satisfied in this case.

Having this we showed that moments methods applied to the process shouldcontain at least the first five initial moments, to get estimates for process parameters.MLE can be used only numerically as the final part can be approximated only withthe sum of normal and lognormal distribution which has no closed-form.

References

[1] Andersen L.Markov Models for Commodity Futures: Theory and Practice. 2008,1–45.

[2] Lin G.D. Recent developments on the moment problem. Journal of StatisticalDistributions and Applications, 4 (1), 2017.

[3] Tellambura C. and Senaratne D. Accurate computation of the MGF of the log-normal distribution and its application to sum of lognormals. IEEE Transactionson Communication, 58 (5), 2010, 1568–1577.

34

GRADIENT PROJECTION METHOD AND CONTINUOUS SELECTION OFMULTI-VALUED MAPPINGS

R. Khachatryan



Given a setM ⊆ Rm of strategies of the first player, the set of ε-optimal strategiesof the second player with loss function f(x, y) is defined to by

aε(x) = y ∈ M/f(x, y) ≤ infy∈M

f(x, y) + ε,

where ϵ > 0.

Theorem. Let E ⊆ Rn be a compact set and M ⊆ Rm be a convex and compactset. Let also f(x, y) be a continuous function satisfying the following conditions:

1) The function f(x, y) is convex with respect to y for any fixed x ∈ E;

2) The partial derivative f ′y(x, y) =

∂f(x,y)∂y exists and is continuous with respect

to x and y.

Assume also that a point (x0, y0) satisfies the condition f(x0, y0) = miny∈M

f(x0, y).

Then there exists a number K such that

yj(x0) = y0, yj(x) ∈ aϵ(x) ∀x ∈ E, j > K.

Here the sequence yj(x) is constructed by the gradient projection method:

yj+1(x) = ΠM (yj(x)− λjf′y(x, y

j(x)),

where

λj ↓ 0,

∞∑j=0

λj = ∞,

∞∑j=0

λ2j < ∞,

and ΠM (A) is the projection of the point A on the closed convex set M .

References

[1] Vasiliev F. P. Numerical methods for solving extreme problems. Nauka, Moscow,1980.

[2] Sukharev A.G., Timokhov A.V., Fedorov V.V. Optimization Methods Course.Science, Moscow, 1986.


35

MAXIAMAL SUBALGEBRAS IN RESTRICTED SIMPLE CARTAN TYPE LIEALGEBRAS AND SUPERALGEBRAS

Hayk Melikyan

Department of Mathematics and Physics, North Carolina Central University, USA


In the Lie theory, the classification problem for maximal subalgebras is a classicalone. A number of problems in geometry and algebra lead to this one. In particularthe classical problem of the classification of primitive transformations which wasposed in the 19th century by a Norwegian mathematician Sophus Lie. Nowadays,motivations for finding all maximal subalgebras of simple Lie algebras come fromboth mathematics, for example, systems of differential equations satisfying a super-position principle, and from physics, for example, symmetry breaking, complete setsof commuting operators in a quantum mechanical system, dynamical systems, etc.

This problem has been the focus of much research which produced very beautifulresults. We first recall the famous papers of E. Dynkin, from the middle of 50s oflast century, where the classification of semisimple subalgebras of complex semisim-ple Lie algebras has been obtained. Next, while studying the maximal subalgebrasin nonclassical simple modular Lie algebras over the fields of nonzero characteris-tics, new series of exceptional simple Lie algebras were discovered by H. Melikyan,nowadays it is known as Melikyan algebras.

Our goal is the characterization of maximal graded subalgebras in Cartan typemodular Lie algebras and superalgebras over the algebraically closed field of char-acteristic p > 3. The notion of an R-subalgebra and that of an S-subalgebra areintroduced for maximal subalgebras. All maximal R-subalgebras are described com-pletely. The number of conjugacy classes, representatives of conjugacy classes of allR-subalgebras are found. An invariant characterization of graded S-subalgebras isalso obtained.


36

ASSESSING EXCESS RISK IN ROBUST ESTIMATION OF GAUSSIAN MEANWITH PARAMETER CONTAMINATION

Arshak Minasyan

Yerevan State University, YerevaNN Research Lab, Yerevan, Armenia


In this work we assess the excess risk in the setup of robust (to outliers) estima-tion of Gaussian mean. The optimal minimax risk rate was established by [3] withcomputationally non-tractable estimator known as Tukey’s median [4]. In [1] and [5]computationally tractable estimators are studied with sub-optimal minimax rates.We propose an estimator whose excess risk tends to 0, when the contamination leveltends to 0. In all above-mentioned estimators the excess risk is strictly positive. Wealso establish the corresponding minimax excess risk rate of proposed estimator.

References

[1] Collier O., Dalalyan A.Minimax estimation of a p-dimensional linear functionalin sparse Gaussian models and robust estimation of the mean. arXiv:1712.05495,2018.

[2] Chen M., Gao C., Ren Z. A general decision theory for Huber’s ε-contaminationmodel. Electronic Journal of Statistics, 10, 2016, 3752 – 3774.

[3] Chen M., Gao C., Ren Z. Robust Covariance and Scatter Matrix Estimationunder Huber’s Contamination Model. Annals of Statistics, 46 (5), 2018, 1932–1960.

[4] Tukey J. W. Mathematics and the picturing of data. In Proceedings of theInternational Congress of Mathematicians, 2, 1975, 523 – 531.

[5] Cheng Y., Diakonikolas I., Ge R. High-Dimensional Robust Mean Estimationin Nearly-Linear Time. arXiv:1811.09380, 2018.


37

GEOMETRY OF CLASS OF SEMISYMMETRIC SUBMANIFOLDS OFCODIMENTION TWO WITH A UNITY INDEX OF REGULARITY IN

EUCLIDEAN SPACES

V.A. Mirzoyan, G.A. Nalbandyan

National Polytechnic University of Armenia, Yerevan, ArmeniaArtsakh State University, Stepanakert


Let M be a Riemannian manifold with a Riemannian connection ∇, a curvaturetensor R, a Ricci tensor R1 and curvature operators R(X,Y ) = ∇X∇Y −∇Y ∇X −∇[X,Y ]. If R(X,Y )R=0, then the manifold M is called semi-symmetric, while ifR(X,Y )R1 = 0, it is called Ricci-semisymmetric. The implication R(X,Y )R =0 ⇒ R(X,Y )R1 = 0 is true. The local classification of Riemannian semisymmetricmanifolds was obtained by Z. I. Szabo [1]. The basic structure theorem of Ricci-semisymmetric manifolds states that a smooth Riemannian manifold M satisfies thecondition R(X,Y )R1 = 0 if and only if it is either a two-dimensional, or an Einstein,or a semi-Einstein, or a direct product (locally) of the listed classes of manifolds [2].Some classes of Ricci-semisymmetric submanifolds in Euclidean spaces were stud-ied in [3–5]. Here we give a geometric description of a semi-Einstein submanifoldsatisfying the condition R(X,Y )R = 0.

Theorem. Suppose in an Euclidean space Em+2 the Ricci tensor of m-dimensional anormally flat ricci-semisymmetric submanifold M has only one nonzero eigenvaluewhich corresponds to only one regular principal curvature vector n of multiplicityp > 2 and let the nullity index µ and relative nullity index ν satisfy the conditionµ = ν + 1. If M is irreducible, then locally it is either a warped cone over a sphereSp(r), where the radius r is a linear (not constant) function of the coordinate ofthe curvilinear generatrix of the cone, which is a general helix with equal curvatureand torsion, located in three-dimensional space, or M it is interlacing product of a(m− p− 1)-dimensional locally Euclidean submanifold M and a warped cone over asphere Sp(r). If M is reducible, then it is an open part of one of the following directproducts:

Sp(r)× L× Lν , Sp(r)×M (0), Cp+1 × L× Lν−1, Cp+1 × Lν ,

where L is a plane curve different from the straight line, M (0) is a hypersurface ofrank 1 (hence, locally Euclidean), which is the envelope of a one-parameter family of(ν +1)-dimensional planes, Cp+1 is the hypercon over the sphere Sk(r), Cp+1 is thewarped cone over the sphere Sk(r), Lν−1 and Lν are the planes of the correspondingdimensions.


38

From the multiplicity of the vector n and the condition µ = ν +1 it follows thatthe submanifold M satisfies the semisymmetrisity condition R(X,Y )R = 0.

References

[1] Szabo Z. I. Structure theorems on Riemannian spaces satisfying R(X,Y )·R=0.I. The local version. J. Differential Geom., 17 (4), 1982, 531–582.

[2] Mirzoyan V.A. Structure theorems for Riemannian Ric-semi-symmetric spaces.Izv. Vyssh. Uchebn. Zaved. Mat., (6), 1992, 80–89 (in Russian).

[3] Mirzoyan V.A. Structure theorems for Ricci-semisymmetric submanifolds andgeometric description of a class of minimal semi-Einstein submanifolds. Mat.Sb., 197 (7), 2006, 47–76 (in Russian).

[4] Mirzoyan V.A. Normally flat semi-Einstein submanifolds of Euclidean spaces.Izv. RAN. Ser. Mat., 75 (6), 2011, 47–78 (in Russian).

[5] Mirzoyan V.A. General Classification of Normally Flat Ric-SemisymmetricSubmanifolds. NAS of Armenia, Reports, 112 (1), 2012, 19–29.

39

HYPERIDENTITIES AND RELATED SECOND ORDER FORMULAE

Yu.M. Movsisyan



This talk is dedicated to:

1) the classification of hyperidentities;

2) characterization of algebras with classical hyperidentities and related secondorder formulae;

3) characterization of free algebras of varieties defined by hyperidentities ;

4) characterization of multiplicative groups and semigroups of fields (Jakobson-Maltsev-Fuchs problem);

5) characterization of multiplicative groups and semigroups of division rings (Jakobson-Maltsev-Fuchs general problem);

6) characterization of multiplicative groups and semigroups of division rings withnon-trivial center.


40

CHARACTERIZATION OF INTERIOR POINTS OF A TRIANGLE IN AWEAK ABSOLUTE GEOMETRY

Aram Nazaryan1, Davit Harutyunyan2, Victor Pambuccian3

1,2Yerevan State University, Yerevan, Armenia3Arizona State University, School of Mathematical and Natural Sciences, USA

E-mail: [email protected], [email protected], [email protected]

Operating in the real Euclidean plane M. Hajja and H. Martini derived [1, The-orem 12] that bore similar result to the propositions 20 and 21 of Book I of Euclid’sElements.

Theorem 1. Let P be a point in the plane of a triangle ABC. Then there exists apoint Q inside or on the boundary of ABC that satisfies.

AQ ≤ AP, BQ ≤ BP, CQ ≤ CP. (1)

As a critique of the methodology, which the authors describe as a “fancifulproof, using Zorn’s lemma and the Bolzano-Weierstrass theorem”, a question isposed weather the use of such “heavy machinery is indeed inevitable” [1, p. 13].Considering that the proof should be carried out in Hilbert’s absolute geometry(whose axioms are the plane axioms of incidence, order, and congruence of groups I,II, and III of Hilbert’s Grundlagen der Geometrie), we further explore the possibilityof deriving a more uninvolved proof that does not use any of the aforementionedtheorems.

As M. Hajja and H. Martini prove only the existence of an interior point Q, wealso develop a procedure(an algorithm) for constructing such a point [1, p. 14]. Wepropose a pair of theorems, while solving this problem within a weak plane absolutegeometry (whose axioms can be deduced inside Hilbert’s plane absolute geometry).

Theorem 2. For any point P inside or on the boundary of triangle ABC, there isno point Q, different from P , such that Q and P satisfy (1).

Theorem 3. For every point P outside of triangle ABC there exists a point Qinside of triangle ABC, such that Q and P satisfy (1)<.

In the proof of 3 we also provide an algorithm to construct such a point Q.

References

[1] Hajja M., Martini H., Proposition 21 of Book I of Euclid’s Elements: vari-ants, generalizations, and open questions. Mitteilungen der MathematischenGesellschaft in Hamburg, 33, 2013, 135–139.


41

PROGRAMMING IS BASED ON THE NON-CLASSICAL THEORY OFCOMPUTABILITY

S. A. Nigiyan



In 1936, the concepts of an algorithm and a computable function were formalized.In this formalization, the value of the function will be undefined if the value of atleast one of its arguments is undefined, and the algorithm will work endlessly if thevalue of the function is undefined [1], [2], [3].

The function implemented by the program can be defined for undefined values ofsome of its arguments, and if the function implemented by the program is undefined,then the program either stops with an undefined value, or works endlessly.

In this paper the definition of arithmetical functions with indeterminate valuesof arguments is given. The notions of computability, strong computability and λ-definability for such functions are introduced. It is proved that every λ-definablearithmetical function with indeterminate values of arguments is monotonic and com-putable. It is proved that every computable, naturally extended arithmetical func-tion with indeterminate values of arguments is λ-definable. It is also proved thatthere exist both λ-definable and non λ-definable strong computable, monotonic, notnaturally extended arithmetical functions with indeterminate values of arguments[4], [5]. Based on this theory, typed programming was translated into untyped pro-gramming [6], and the interpretation of typed and untyped functional programs wasinvestigated [7].

References

[1] Kleene S. Introduction to Metamathematics. D.Van Nostrand Company, Inc.,1952.

[2] Rogers H. Theory of Recursive Functions and Effective Computability. McGraw-Hill Book Company, 1967.

[3] Barendregt H. The Lambda Calculus. Its Syntax and Semantics. North-HollandPublishing Company, 1981.

[4] Nigiyan S.A. On Non-classical Theory of Computability. Proceedings of theYSU, Physical and Mathematical Sciences, (1), 2015, 52–60.


42

[5] Nigiyan S.A. On Arithmetical Functions with Indeterminate Values of Argu-ments. Emil Artin International Conference, Yerevan, May 27 - June 2, 2018,101–103.

[6] Nigiyan S.A., Khondkaryan T.V. On Translation of Typed Functional Programsinto Untyped Functional Programs. Proceedings of the YSU, Physical and Math-ematical Sciences, 51 (2), 2017, 177–186.

[7] Nigiyan S.A. On Interpretation of Typed and Untyped Functional Programs.Proceedings of the YSU, Physical and Mathematical Sciences, 52 (2), 2018,119–133.

43

TOMOGRAPHY PROBLEMS IN STOCHASTIC GEOMETRY

V. K. Ohanyan


E-mail: [email protected]; [email protected]

Let Rn (n ≥ 2) be the n-dimensional Euclidean space, D ⊂ Rn be a boundedconvex body. Random k-flats in Rn, 1 ≤ k ≤ n − 1 generate cross sections ofrandom size in convex body D. As D is a convex body, then obviously intersectionsof k-flats with D are always connected subsets of Rn for every k ∈ 1, . . . , n − 1.It is natural to require that the corresponding distribution of random size of crosssections to be invariant with respect to the group of all Euclidean motions in Rn.The determination of the distribution of size of cross sections has a long tradition ofapplication to collections of bounded convex bodies forming structures in metal andcrystallography. However, calculations of geometrical characteristics of random crosssections is often a difficult task. In a special case k = 1 we call the correspondingdistribution function as the chord length distribution function. For n = 2 the list ofknown results was expanded after 2005 when N.G. Aharonyan and V.K. Ohanyanobtained the explicit formula of the chord length distribution function for a regularpolygon (see [1]).

Proposition. Let D be a convex planar polygon which has m pairs of parallelsides (ai1 , aj1), . . . , (aim , ajm). The distances of the parallel lines which carry thesesegments are d1, . . . , dm, respectively, and πaik ∩ πajk denotes the length of theintersection of the orthogonal projections of both segments onto one of the carryinglines, k = 1, . . . ,m. Then for k ∈ 1, . . . ,m for which πaik ∩ πajk > 0, the chordlength density function has a discontinuity at dk, and the limit from above at dk isinfinite.

A computer program is created which gives values of a chord length distributionfunction in the case of a regular n-gon for every natural n ≥ 3 (see [2]).

A practical application these results in crystallography can be found in [3] and[4].

References

[1] Aharonyan N.G., Ohanyan V.K. Chord length distribution functions forpolygons. Journal of Contemporary Mathematical Analysis (Distributed bySpringer, see www.springerlink.com), 40 (4), 2005, 43–56.


44

[2] Harutyunyan H. S., Ohanyan V.K. Chord length distribution function for regu-lar polygons. Advances in Applied Probability (SGSA), 41 (2), 2009, 358–366.

[3] Aharonyan N.G., Ohanyan V.K. Calculation of geometric probabilities usingCovariogram of convex bodies. Journal of Contemporary Mathematical Analysis(Distributed by Springer, see www.springerlink.com), 53 (2), 2018, 110–120.

[4] Gille W., Aharonyan N.G. and Harutyunyan H. S. Chord length distribution ofpentagonal and hexagonal rods: relation to small-angle scattering. Journal ofAppl. Crystallography, 42, 2009, 326–328.

45

SOME QUESTIONS ABOUT THE COMPARATIVE ANALYSISPOSSIBILITIES OF MODELING CLASSICAL PETRI NETS AND COLORED

PETRI NETS

Goharik Petrosyan

ASPU and ISEC NAS RA

E-mail: petrosyan [email protected]

Petri nets are a tool for the study of systems. Petri net theory allows a system tobe modelled by a Petri net, a mathematical representation of the system. Analysis ofthe Petri net can then, hopefully, reveal important information about the structureand dynamic behaviour of the modelled system. This information can then be usedto evaluate the modelled system and suggest improvements or changes. Thus, thedevelopment of a theory of Petri nets is based on the application of Petri nets in themodelling and design of systems.

Colored Petri Net is a graphical oriented language for design, specification, sim-ulation and verification of systems (Jensen, 1992; Jensen, 1996; Ullman, 1998). Itis in particular well-suited for systems that consist of a number of processes whichcommunicate and synchronize. Typical examples of application areas are commu-nication protocols, distributed systems, automated production systems, work flowanalysis and VLSI chips (very large scale integration, from 106 to 107 transistors).

Acknowledgement. The paper deals with issues that relate to the comparativeanalysis of the Classical Petri Nets and the Colored Petri Nets, where, using illus-trative examples, it is found that the Classical Petri Nets have limited propertiesof modeling with the comparison of the Colored Petri Nets. In the future, we willconsider more complex systems that can be modeled with other extensions of Petrinets.

References

[1] Peterson J. L. Petri Net Theory and the Modeling of Systems. Prentice Hall,Upper Saddle River, 1981.

[2] Murata T. Petri Nets: Properties, Analysis and Applications. Proceedings ofthe IEEE, 77 (4), 1989, 541–580. doi:10.1109/5.24143

[3] Jensen K Coloured Petri Nets: Basic Concepts, Analysis Methods and PracticalUse. Springer-Verlag, Berlin, 1992. doi:10.1007/978-3-662-06289-0


46

[4] Jensen K. Coloured Petri Nets: Basic Concepts, Analysis Methods and PracticalUse. Volumn 1. Basic Concepts. Monographs in Theoretical Computer Science.Springer-Verlag, Berlin, 1997.

[5] Jensen K. Coloured Petri Nets: Basic Concepts, Analysis Methods and PracticalUse. Volumn 2. Analysis Methods Monographs in Theoretical Computer Science.Springer-Verlag, Berlin, 1997.

[6] Jensen K. Coloured Petri Nets: Basic Concepts, Analysis Methods and PracticalUse. Volumn 3. Practical Use. Monographs in Theoretical Computer Science.Springer-Verlag, Berlin, 1997.

[7] Jensen K. Coloured Petri Nets: A High-level Language for System Design andAnalysis. In: G. Rozenberg, Ed., Advances in Petri Nets 1990, Lecture Notesin Computer Science, 483, Springer-Verlag, Berlin, 1991, 342–416.

[8] Jensen K. Coloured Petri Nets: A High-level Language for System Design andAnalysis. In: K. Jensen and G. Rozenberg, Eds., High-Level Petri Nets. Theoryand Application, Springer-Verlag, Berlin, 1991, 44–122.

[9] Ullman J.D. Elements of ML Programming. Prentice- Hall, Upper Saddle River,1998.

[10] Bevilacquaa M., Ciarapicaa F. E., Giovannia M. Timed Coloured Petri Nets formodelling and managing processes and projects. 11th CIRP Conference on Intel-ligent Computation in Manufacturing Engineering - CIRP ICME ’17, ProcediaCIRP, 67, 2018, 58–62.

47

ON CONSEQUENCESOF CONSTRUCTIVENESS OF MODELS OF COGNIZING

Edward Pogossian

Division of Computational and Cognitive Networks, IPIA, NAS RA, Yerevan, Armenia


1.1. Artificial Intelligence, following Alonzo Church then Allan Turing [1], can beinterpreted as the branch of sciences aimed to provide adequate constructivemodels of cognizing, at least, comparable by effectiveness with mental doingsof humans. And the first such models were algorithms [1, 2].Cognizing, in turn, are mental doings on learning and organizing mental sys-tems (mss) [2] while mss are learned both by revelation of mss and by acqui-sition of mss from communities.

1.2. By definition, learning of mss at any stage assumes certain collection, thesauriof mss including certain cognizers (cogs). Other words, cognizing requiresexistence of certain starting root thesauri rTh that necessarily include certainroot cogs, rC*.

2.1. Optimistic expectations in constructing rC* comprise the assumptionAI-Ass. The highest human cognizers Cogs can construct root cognizers bothin the modes based on the models of cells, i.e., rcogsAIc, and not based onthose models, i.e., rcogsAI, that developing can attain CogsAIc and CogsAI,correspondingly, functionally, at least, equal to Cogs.

2.2. The assumption induces the following essential consequences.AI-Crl1. CogsAIc (CogsAI) being equal to Cogs will inherit from Cogs theability to reproduce themselves, i.e., to construct equal to themselves cognizers,particularly by constructing rcogsAIc (rcogsAI) developing to CogsAIc(CogsAI)equal to CogsAIc (CogsAI), thus, equal to Cogs.AI-Crl2. Constructive root cognisers rcogsntr developing themselves to cogsNtrequal to Cogs can be originated in the nature.AI-Clr3. Originated in nature cognizers cogsNtr analogously to Cogs have to beable to reproduce themselves in a variety of modes, particularly, in the cellularmode based on construction of cellular root cognizers rcogsAI.

3.1. Since algorithms can be represented as compositions of 1-/2-rels [5] in [2, 6,7] it was argued that origination of rC* can be reduced to origination of:matrices of outputs (imprints) of mental classifiers, classifiers of the types of1-/2-relationships(rels) and organization of 1-/2- rels into algorithms.


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3.2. By Shannon and Schrodinger [3] “information” is measuring negentropicity, is,apparently, inseparable from classifying and by [4] can be originated in nature.Thus, grounding origination of “information” simultaneously grounds origina-tion of the basic classifiers, i.e., the 1-/2- place rels. Therefore, significantconsequence of solution of origination of rC* states that the kernel of effectivecognition is one of universal means for being in the universe and it is not theprivilege of only cellular realities and, in general, cellular realities could beconstructed [6].

References

[1] Maltcev A. I. Algorithms and Recursive Functions. Nauka, Moscow, 1965.

[2] Pogossian E. Constructing Adequate Mental Models. Mathematical Problems ofComputer Sciences, Proc. of IIAP, Yerevan, 2018, 35–42.

[3] Shrodinger E. Mind and Matter. Cambridge, 1956.

[4] Parrondo J. M. R., Horowitz J. M., Sagawa T. Thermodynamics of information.Nature Physics, 11 (2), 2015, 131–139.

[5] Markov and Nagorni N. Theory of Algorifms. Nauka, Moscow, 1984.

[6] Pogossian E. Artificial Intelligence: Alternating the Highest Levels of Hu-man Cognizing. 12th Inter. Conf. in Comp. Sci. and Information Technologies,CSIT2019, Yerevan, 36–40.

[7] Pogossian E. Challenging the Uniqueness of Being by Cognizing. Transactionsof IIAP NAS RA, Mathematical Problems of Computer Sciences, 51, 2019,66–81.

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UNIVERSAL FUNCTIONS FOR CLASSES Lp[0, 1)2, p ∈ (0, 1), WITHRESPECT TO THE DOUBLE WALSH SYSTEM

Artsrun Sargsyan, Martin Grigoryan



We address questions on the existence and structure of universal functionsU(x, y) ∈ L1[0, 1)2 for classes Lp[0, 1)2, p ∈ (0, 1), with respect to the double Walshsystem. It is shown that there exists a measurable set E ⊂ [0, 1)2 with measurearbitrarily close to, such that, by a proper modification of any integrable functionoutside E , we can get an integrable function, which is universal for each classLp[0, 1)2, p ∈ (0, 1), with respect to the double Walsh system in the sense of signsof Fourier coefficients.


References

[1] Grigoryan M., Sargsyan A. On the structure of universal functions for classesLp[0, 1)2, p ∈ (0, 1) with respect to the double Walsh system. Mat. Sbornik(N.S.), Banach J. Math. Anal., 13 (3), 2019, 647–674.

[2] Sargsyan A., Grigoryan M. Universal functions for classes Lp[0, 1)2, p ∈ (0, 1),with respect to the double Walsh system. Positivity, November 2019, 23 (5),1261–1280.


50

ON THE L1-CONVERGENCE AND BEHAVIOR OF COEFFICIENTS OFFOURIER–VILENKIN SERIES

Stepan Sargsyan, Martin Grigoryan



In this talk, we prove the following statement that is true for both boundedand some type of unbounded Vilenkin systems: for any ε ∈ (0, 1), there exists ameasurable set E ⊂ [0, 1] whose measure |E| bigger than 1-ε such that for anyfunction f ∈ L1[0, 1), it is possible to find a function g ∈ L1[0, 1) coinciding withf on E, and Fourier series of g(x) with respect to Vilenkin system are convergentinL1[0, 1) -norm and the absolute values of non zero Fourier coefficients of g(x) aremonotonically decreasing.


References

[1] Grigoryan M.G., Sargsyan S.A. On the L1-convergence and behaviour of coef-ficients of Fourier–Vilenkin series. Positivity, 22 (3), 2018, 897–918.

[2] Grigoryan M.G., Sargsyan S.A. , On the Fourier-Vilenkin coefficients. ActaMathematica Scientia, 37B (2), 2017, 1–8.


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ON THE REPRESENTATION OF FUNCTIONS BY WALSH DOUBLESYSTEM IN WEIGHTED L

p

µ[0, 1)2-SPACES

L. S. Simonyan



In this talk I will present the following theorems:

Theorem 1. There is a measurable (weight) function µ(x; y) : 0 < µ(x; y) ≤1, (x, y) ∈ [0, 1)2 such that for each p ∈ [1,∞) and for every function f(x, y) ∈Lpµ[0, 1)2 there exist a series by double Walsh system with the following property:

limR→∞

1∫0

1∫0

∣∣∣∣∣∣∑

0≤k2+s2≤R2

bk,sφk(x)φs(y)− f(x, y)

∣∣∣∣∣∣p

µ(x; y)dxdy = 0.

This stronger theorem follows from Theorem 1:

Theorem 2. There exist a weight function µ(x; y) : 0 < µ(x; y) ≤ 1, (x, y) ∈[0, 1)2, and a series by the Walsh double system φk(x)φs(y)∞k,s=0 of the form∑∞

k,s=0 dk,sφk(x)φs(y), with∑∞

k,s=0 |dk,s|r < ∞, for all r > 2 and non-zero mem-

bers in |dk ,s |∞k,s=0 , which are in decreasing order over all rays, with the followingproperty:

for each p ∈ [1,∞) and for every function f(x, y) ∈ Lpµ[0, 1)2 one can find

δk,s = 0 or 1 such that

limR→∞

1∫0

1∫0

∣∣∣∣∣∣∑

0≤k2+s2≤R2

δk,sdk,sφk(x)φs(y)− f(x, y)

∣∣∣∣∣∣p

µ(x; y)dxdy = 0.

Remark. The proved theorem is true also for Vilenkin systems (see [3]).

References

[1] Grigoryan M.G., On the representation of functions by orthogonal series inweighted Lp spaces. Studia Math., 134 (3), 1999, 211–237.

[2] Grigoryan M.G., Grigoryan T.M., Simonyan L. S. Convergence of Fourier-Walsh double series in weighted Lp

µ[0; 1)2. Springer Proceedings in Mathematicsand Statistics, 2, Springer Nature Switzerland AG 2019, 109–137.


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[3] Simonyan L. S. On convergence of Fourier double series with respect to theVilenkin systems. Proceedings of the Yerevan State University, 52 (1), 2018,22–28.

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