Rafik Aramyan 's Publications (Full profile)

Paper

R. M. Mnatsakanov, R. H. Aramyan, Recovery of bivariate functions from the values of its Radon transform using Laplace inversion, Journal of Computational and Applied Mathematics, 395, pp. 113557 , (2021)

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Abstract:
The problems of recovering a multivariate function f from the scaled values of its Laplace and Radon transforms are studied, and two novel methods for approximating and estimating the unknown function are proposed. Moreover, using the empirical counterparts of the Laplace transform of the underlying function, a new estimate of the Radon transform itself is obtained. Under smoothed conditions on the underlying function, the uniform convergence of the proposed constructions is established, and their accuracy is illustrated graphically with several simple examples.

DOI:
https://doi.org/10.1016/j.cam.2021.113557

Full text:
cam113557 (1).pdf
R. Aramyan, V. Mnatsakanyan , Conditional moments for a d-dimensional convex body, J. of Contemp. Math. Analysis , 56, (3), pp. 3-9, (2021)

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Abstract:
For a d-dimensional convex body, we define new integral geometric concepts: conditional moments of the random chord length and conditional moments of the distance of two independent uniformly distributed points in the body. Also in this article, the relations between the concepts are found.

DOI:
https://www.springer.com/journal/11957

Full text:
ContMath2103002AramyanKOR (1).pdf
Rafik H. Aramyan, Robert M. Mnatsakanov , To recovering the moments from the spherical mean Radon transform, Journal of Mathematical Analysis and Applications , 490, (2), pp. 413-422, (2020)

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Abstract:
This article deals with characterizations of a function in terms of its circular mean Radon transform. We present a new approach (the consistency method) showing how to describe the class of real-valued, planar functions fwhich have the given circular mean Radon transform Mfover circles centered on the unit circle. Also, expressions are derived for the geometric moments of an unknown function in terms of its circular mean Radon transform.

DOI:
https://doi.org/10.1016/j.jmaa.2020.124334

Full text:
1-s2.0-S0022247X20304960-main (1) (1).pdf
R.H. Aramyan, A description of linearly additive metrics on , Transactions of the Moscow Math. Soc. , 81, (1), pp. 115-121 , (2020)

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Abstract:
There is an integral-geometric approach, proposed by Busemann, for building linearly additive metrics on Rn (it uses hyperplanes). Hilbert's Fourth Problem was solved with the help of this construction. In this article, we present a new description (using straight lines) of linearly additive metrics on Rn, generated by a norm. There is a link between this description and the sine transform.

DOI:
https://doi.org/10.1090/mosc/304

Full text:
S0077-1554-2021-00304-7.pdf
R. Aramyan and D. Yeranyan, Chord length distribution and the distance between two random points in a convex body in Rn, General Letters in Mathematics, 9, (2), pp. 74-79, (2020)

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Abstract:
In this article for n-dimensional convex body D the relation between the chord length distribution function and the distribution function of the distance between two random points in D was found. Also the relation between their moments was found. Keywords

DOI:
http://doi.org/10.31559/glm2020.9.2.3

Full text:
GLM-9-2-3.pdf
Kazaryan Maretta; Simonyan Arsen; Simavoryan Simon; Ulitina Elena; Aramyan Rafik, Waste disposal facilities monitoring based on high-resolution information features of space images , E3S Web of Conferences, 157, (2020)

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DOI:
https://doi.org/10.1051/e3sconf/202015702029
R Aramyan, V Mnatsakanyan, Conditional moments of the distance distribution two random points in a convex domain in R2, Proceedings of the YSU, Physical and Mathematical Sciences , 54, (1), pp. 1-7, (2020)

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Abstract:
In this article we deno two new integral geometric concepts: conditional moments of the chord length distribution of a convex domain and conditional moments of the distribution of the distance of two independent uniformly dis- tributed points in a convex domain. Also in this article the relation between the two concepts are found.
R.H. Aramyan, A Flag representation for n-dimensional convex body, The Journal of Geometric Analysis, 29, (3), pp. 2998-3009, (2019)

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Abstract:
The cosine representation of the support function of a centrally symmetric convex body plays a fundamental role in integral geometry. In this article, one new so-called flag representation for the support function of an origin symmetric n-dimensional convex body in terms of surface curvature functions of the convex body is found. Using the representation, we propose a sufficient condition for an origin symmetric n-dimensional convex body to be a zonoid. The condition has a local equatorial description

DOI:
http://math.sci.am/10.1007/s12220-018-00102-1

Full text:
Aramyan JGA.pdf
R.H. Aramyan, To local reconstruction from the spherical mean Radon transform, Journal of Mathematical Analysis and Applications, 470, (1), pp. 102-117, (2019)

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Abstract:
The article suggests a new approach using the so-called consistency method for the inversion of the spherical mean Radon transform in 2D with detectors on a line. We present a new iterative formula that gives an algorithm to recover an unknown function supported completely on one side of a line Lfrom its spherical means over circles centered on the line. Our reconstruction formula has the benefit of being local. Such an inversion is required in problems of thermo-and photo-acoustic tomography, ultrasound reflection tomography, radar imaging, and a few others.

DOI:
http://math.sci.am/10.1016/j.jmaa.2018.09.055
R.H. Aramyan, The sine representation of centrally symmetric convex bodies, J. Contemp. Math. Anal., Armen. Acad. Sci. , 53, (6), pp. 307-312 , (2018)

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Abstract:
The problem of the sine representation for the support function of centrally symmetric convex bodies is studied. We describe a subclass of centrally symmetric convex bodies which is dense in the class of centrally symmetric convex bodies. Also, we obtain an inversion formula for the sine-transform.

DOI:
http://math.sci.am/10.3103/S1068362318060018

Full text:
IzvestiyaNAS.pdf
R.Aramyan, Zonoids with an equatorial characterization, Applications of Mathematics, 61, (4), pp. 413-422, (2016)

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Abstract:
It is known that local equatorial characterization of zonoids does not exist. The question arises: Is there a subclass of zonoids admitting a local equatorial characterization. In this article, a sufficient condition is found for a centrally symmetric convex body to be a zonoid. The condition has a local equatorial description. Using the condition one can define a subclass of zonoids admitting a local equatorial characterization. It is also proved that a convex body whose boundary is an ellipsoid belongs to the class.

DOI:
http://math.sci.am/10.1007/s10492-016-0139-5

Full text:
AplMat_61-2016-4_4.pdf
R.Aramyan, Reconstruction of Three Dimensional Convex Bodies from the Curvatures of Their Shadows, American Journal of Computational Mathematics, 5, pp. 86-95, (2015)

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Abstract:
In this article, we study necessary and sufficient conditions for a function, defined on the space of flags to be the projection curvature radius function for a convex body. This type of inverse problems has been studied by Christoffel, Minkwoski for the case of mean and Gauss curvatures. We suggest an algorithm of reconstruction of a convex body from its projection curvature radius function by finding a representation for the support function of the body. We lead the problem to a system of differential equations of second order on the sphere and solve it applying a consistency method suggested by the author of the article.

DOI:
http://math.sci.am/10.4236/ajcm.2015.52007
R. Aramyan, A. Manucharyan, A representation for the support function of a convex body, Proceedings of the YSU, Physical and Mathematical Sciences, 3, (2015(3)), pp. 1-7, (2015)
R.Aramyan, A Solution of Generalized Cosine Equation in Hilbert’s Fourth Problem, Advances in Pure Mathematics, 4, (6), pp. 234-241, (2014)

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Abstract:
A solution to Hilbert’s fourth problem leads to the integral equation which can be called the generalized cosine equation. In the present paper, we propose an inversion formula for the solution of the generalized cosine equation using integral and stochastic geometry methods.

DOI:
http://math.sci.am/10.4236/apm.2014.46031
R. H. Aramyan, A representation for convex bodies, Armenian Journal of Mathematics, 5, (1), pp. . 69-74, (2013)
R. H. Aramyan, Convex bodies and measures in the space of planes, Journal of Contemporary Mathematical Analysis (Armenian Academy of Sciences), 47, (2), pp. 19--30, (2012)
R. H. Aramyan, Reconstruction of centrally symmetric convex bodies in R^n, Buletinul Acad. De Stiinte A R. Moldova, 65, (1), pp. 28-32, (2011)
R. H. Aramyan, The integral and stochastic geometric approach to Hilbertâ??s fourth problem, Proceedings of the 8th Seminar on probability and Stochastic processes University of Guilian, pp. 28-29, (2011)
R. H. Aramyan, Reconstruction of measures in the space of planes, Lobachevskii Journal of Mathematics, 32, (4), pp. 241 -- 246, (2011)
R. H. Aramyan, Measure of planes intersecting a convex body, 3, (1), pp. 1-7, (2010)
R. H. Aramyan, Generalized Radon transform on the sphere, Analysis (Munich) , 30, (3), pp. 271-284, (2010)
R. Aramyan, Solution of an integral equation by consistency method, Lithuanian Math. Journal, 50, (2), pp. 133 -- 139, (2010)
R. G. Aramyan, Solution of one integral equation on the sphere by methods of integral geometry, Doklady Mathematics (Russian Academy of Sciences), 79, (3), pp. 325-328, (2009)
R. H. Aramyan, A representation for the generating density of convex bodies, Lobachevskii Journal of Mathematics, 30, (2), pp. 97-101, (2009)
R. H. Aramyan, Reconstruction of centrally-symmetrical convex bodies in R^3, Vestnik fiz-mat nauk RAU, 2, pp. 10 -- 15, (2009)
R.H.Aramyan, Reconstruction of centrally-symmetrical convex bodies by projection curvature radii , Journal of Contemporary Math. Anal. (Armenian Academy of Sciences), 39, (5), pp. 50 -68, (2004)
R.H. Aramyan, Generalized Radon transform with an application in convexity theory, Journal of Contemporary Mathematical Analysis (Armenian Academy of Sciences), 38, (3), pp. 16-36, (2003)
R. H. Aramyan, Radii of projection curvature and generating density of Convex bodies in $R^n$, Izv. NAN Armenii, Matematika, . In Russian, 36, (1), pp. 48-59, (2001)

Published translation

Journal of Contemporary Math. Anal.

36

(1)

, pp. 48-59

(2001)
R. H. Aramyan, An approach to generalized Funk equations, I, Izv. NAN Armenii, Matematika, . In Russian, 36, (1), pp. 48-59, (2001)

Published translation

Journal of Contemporary Math. Anal.

36

(1)

, pp. 48-59

(2001)
R. V. Ambartzumian, A. Der Kiureghian, V. K. Oganian, H. S. Sukiasian, R. H. Aramyan, Poisson random planes model in tunnelbuilding, Izv. NAN Armenii, Matematika, In Russian, 31, (2), (1996)

Published translation

Journal of Contemporary Math. Anal.

31

(2)

(1996)
R. H. Aramyan, Measure generation in the space of planes and spherical Euler functional, Izv. NAN Armenii, Matematika, In Russian, 29, (4), (1994)

Published translation

Journal of Contemporary Math. Anal.

29

(4)

, pp. 52-74,

(1994)
R. H. Aramyan, Flag-representation of convex bodies by stochastic approximation of Delone, Izv. NAN Armenii, Matematika, In Russian, 28, (2), (1993)

Published translation

Journal Math. Anal.

28

(2)

(1993)
R. H. Aramyan, Recovering rose of directions from flag densities in R^3, Izv. NAN Armenii, Matematika, In Russian, 27, (5), (1992)

Published translation

Journal Math. Anal.

27

(5)

, pp. 22-35,

(1992)
R. H. Aramyan, Flag representations and curvature measures of convex bodies, Izv. NAN Armenii, Matematika, In Russian, 23, (1), (1988)

Published translation

Journal Math. Anal.

23

(1)

(1988)
R. H. Aramyan, About stochastic approximation of convex bodies, Izv. NAN Armenii, Matematika, In Russian, 22, (5), (1987)

Published translation

Journal Math. Anal.

22

(5)

(1987)
R. H. Aramyan, To calculation of probabilities of nonaffine shapes, Izv. NAN Armenii, Matematika, In Russian, 20, (4), (1985)

Published translation

Journal Math. Anal.

20

(4)

(1985)

Conference Paper

R. H. Aramyan, About stochastic approximation of convex bodies, Abstracts of Communications: in "5th Vilnius Conference on Probability Theory and Mathematical Statistics", Vilnius, (1989)
R. H. Aramyan, About stochastic approximation of convex bodies, Abstracts of Communications: in " International Conference on Geometry, Novosibirsk, Russia, (1987)

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