# R. V. Ambartzumian's research while affiliated with National Academy of Sciences of Armenia and other places

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## Publications (11)

Snow is an excellent water reservoir, naturally storing large quantities of water at time scales from a few days to several months. In summer-dry countries, like Armenia, runoff due to snow melt from mountain regions is highly important for a sustained water supply (irrigation, hydropower). Snow fields on Mount Aragats, Armenia's highest peak, ofte...

The paper studies random line processes Z that are translation invariant in probability distribution, and whose first and second order moment measures possess continuous densities. The purpose is to review the analytical apparatus based on the concept of horizontal or vertical windows and corresponding Palm-type probability distributions that are n...

In 1961, at A.M.S. Symposium on Convexity, P.C. Hammer proposed the following problem: how many X-ray pictures of a convex planar domain D must be taken to permit its exact reconstruction? Richard Gardner writes in his fundamental 2006 book [4] that X-rays in four different directions would do the job. The present paper points at the possibility th...

The purpose of the article is to try a probability approach to History, and to explain, on a concrete historical example,
such concepts as virtual events, historical events and miracle events. The necessary historical expertise is taken from THE SECOND WORLD WAR by Winston Churchill and THE TWO-OCEAN WAR by S.E.
Morison, the two-time winner of the...

Combinatorial integral geometry possesses some results that can be interpreted as belonging to the field of Geometric Tomography.
The main purpose of the present paper is to present a case of parallel X-ray approach to tomography of random convex polygons. However, the Introduction reviews briefly some earlier results by the author
that refer to re...

Many results in Combinatorial Integral Geometry are derived by integration of the combinatorial decompositions associated with finite point sets {P
i
} given in the plane ℝ2. However, most previous cases of integration of the decompositions in question were carried out for the point sets {P
i
} containing no triads of collinear points, where the fa...

Chord calculus is a collection of integration procedures applied to to the combinatorial decompositions that give the solution
of the Buffon-Sylvester problem for n needles in a plane or the similar problem in IR
3. It is a source of various integral geometry identities, some of which find their application in Stochastic geometry. In
the present pa...

LetH be the class of sufficiently smooth metrics defined on the Euclidean plane for which the geodesics are the usual Euclidean
liens. The general problem is to describe all metrics fromH which at each point possess the length indicatrix from a prescribed parametric class of convex figures. As a tool, a differential
equation is proposed derived fro...

Guided by analogy with Euler's spherical excess formula, we define a finite-additive functional on bounded convex polygons in ℝ 2 (the Euler functional). Under certain smoothness assumptions, we find some sufficient conditions when this functional can be extended to a planar signed measure. A dual reformulation of these conditions leads to signed m...

Guided by analogy with Euler’s spherical excess formula, we define a finite-additive functional on bounded convex polygons in ℝ 2 (the Euler functional). Under certain smoothness assumptions, we find some sufficient conditions when this functional can be extended to a planar signed measure. A dual reformulation of these conditions leads to signed m...

## Citations

... In the European Alps, glaciers are an economic factor, contributing to hydroelectric power and serving as parts of ski resorts [1]. Recently, some efforts have been undertaken to maintain snow in ski resorts [2,3] and manage meltwater production [4]. Covering glaciers with geotextiles has become widespread as an effective method to mitigate glacier ablation in Switzerland and Austria [1,5,6]. ...

... We then let f (t) to be the length of the part of the line y = t that intersects the domain. Then by the definition of t 0 we have f (t 0 ) = 0, and by the the convexity of the domain, f (t) is strictly decreasing on [0, t 0 ] (see for example Lemma 1.1 of [Am13]). We then locate a disk S of radius 1 κmax that is tangent to α at s 0 from below. ...

... In Theorem 1 we give its extension for quite general Buffon sets in IE. (The corresponding result for the space of lines in the plane have been recently announced in [10].) Theorem 2 refers to the combinatorial decompositions for the class of isotropic (but not necessarily translation invariant ) measures on IE. ...

... The corresponding area of mathematical research is called Stochastic Geometry (see [1], [2] and [10]). Among more popular applications are Stereology and Tomography (see [14], [3], [16], [21]). triangle (see [6]) and a rectangle (see [7]). ...

Reference: Tomography of bounded convex domains

... In the last section of this paper, we recall Busemann's formulation of this higher-dimensional analogue. In the paper [15], Ambartzumian and Oganian give (Theorem 1) a necessary and sufficient condition for a Finsler metric in dimension 2 to be a solution of Hilbert's problem. In the same paper, they study a parametric version of the problem, where the convex domain which is the support of the metric (which is assumed to be centrally symmetric) depends continuously on parameters. ...

Reference: On Hilbert's fourth problem