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Abstract This work presents a Radon transform analysis of plane wave diffraction by an impedance step, such that the planes and the step have different impedances. Application of Radon transform on diffraction analysis is of vital importance in many real-time applications i.e., medical imaging, geosciences, astronomy, optics, stress analysis etc. Radon transform offers the advantage of introducing freedom in the usage of observer angle, during reconstruction of signal using plane wave diffraction. Radon transform along with Wiener–Hopf technique and asymptotic method of integration are used for computing the far field of the diffracted plane wave. The solution of plane wave diffraction by an impedance step consists of an infinite set of linear equations, which is obtained with the help of a software. Sinograms showing the effect of varying step height and impedance on the scattered field are plotted, compared and discussed. The mathematical analysis and graphical discussion show that Radon transform enables the observer angle to be varied between 0 to 2 π. The results are comparable to previous works on plane wave diffraction by an impedance step, when the observer angle is fixed. [ABSTRACT FROM AUTHOR]

Abstract Hans J. Zassenhaus conjectured that for any unit u of finite order in the integral group ring of a finite group G there exists a unit a in the rational group algebra of G such that a − 1 ⋅ u ⋅ a = ± g for some g ∈ G. We disprove this conjecture by first proving general results that help identify counterexamples and then providing an infinite number of examples where these results apply. Our smallest example is a metabelian group of order 2 7 ⋅ 3 2 ⋅ 5 ⋅ 7 2 ⋅ 19 2 whose integral group ring contains a unit of order 7 ⋅ 19 which, in the rational group algebra, is not conjugate to any element of the form ± g. [ABSTRACT FROM AUTHOR]

Let φn,K denote the largest angle in all the triangles with vertices among the n points selected at random in a compact convex subset K of Rd with nonempty interior, where d ≥ 2. It is shown that the distribution of the random variable λd(K) n3/3! (π - φn,K)d-1, where λd(K) is a certain positive real number which depends only on the dimension d and the shape of K, converges to the standard exponential distribution as n→∞. By using the Steiner symmetrization, it is also shown that λd(K), which is referred to in the paper as the elongation of K, attains its minimum if and only if K is a ball B(d) in Rd. Finally, the asymptotics of λd(B(d)) for large d is determined. [ABSTRACT FROM AUTHOR]

We consider weighted Radon transforms RW along hyperplanes in R3 with strictly positive weights W. We construct an example of such a transform with non-trivial kernel KerRW in the space of infinitely smooth compactly supported functions and with continuous weight. Moreover, in this example the weight W is rotation invariant. In particular, by this result we continue studies of Quinto (J Math Anal Appl 91(2): 510-522, 1983), Markoe and Quinto (SIAM J Math Anal 16(5), 1114-1119, 1985), Boman (J Anal Math 61(1), 395-401, 1993) and Goncharov and Novikov (An example of non-uniqueness for the weighted Radon transforms along hyperplanes in multidimensions. arXiv:1709.04194v2, 2017). We also extend our example to the case of weighted Radon transforms along two-dimensional planes in Rd,d≥3. [ABSTRACT FROM AUTHOR]

In integral geometry, intrinsic volumes are a set of geometrical variables to characterize spatial structures, for example, distribution of fluids in two‐fluid flow in porous media. McClure et al. (2018, https://doi.org/10.1103/PhysRevFluids.3.084306) utilized this principle and proposed a geometric state function based on the intrinsic volumes. In a similar approach, we find a geometrical description for free energy of a porous system with two fluids. This is also an extension of the work by Mecke (2000, https://doi.org/10.1007/3-540-45043-2%5f6) for energy of a single fluid. Several geometrical sets of spatial objects were defined, including bulk of the two fluids, interfaces, and three‐phase contact lines. We have simplified the description of free energy by showing how the intrinsic volumes of these sets are geometrically related. We obtain a description for energy as a function of seven microscopic geometrically independent variables. In addition, using a thermodynamic approach, we find an approximation for the free energy as a function of macroscopic parameters of saturation and pressure under quasi‐static conditions. The combination of the two energy descriptions, by integral geometry and thermodynamics, completes the relation between the associated variables and enables us to find the unknown coefficients of the intrinsic volumes and to calculate the amount of dissipated energy in drainage and imbibition processes. We show that the theory is consistent with a set of experiments performed by Schlüter et al. (2016a, https://doi.org/10.1002/2015WR018254, 2017a, https://doi.org/10.1002/2016WR019815). However, in order to be more conclusive, it needs to be tested with larger data sets. Key Points: A geometric description of free energy of a porous system with two immiscible fluids was foundGeometric description of free energy in two‐fluid flow is a function seven microscopic geometrically independent variablesUsing a thermodynamic approach, the amount of dissipated energy in drainage and imbibition processes was found [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

We consider the problem of comparing the volumes of two star bodies in an even-dimensional Euclidean space R2n=Cn by comparing their cross sectional areas along complex lines (special 2-dimensional real planes) through the origin. Under mild symmetry conditions on one of the bodies a Busemann-Petty type theorem holds. Quaternionic and octonionic analogs also hold. The argument relies on integration in polar coordinates coupled with Jensen’s inequality. Along the way we provide a criterion that detects which centered bodies are circular. i.e., stabilized by multiplication by complex numbers of unit modulus. Our goal is to present a Busemann-Petty type result with a minimum of required background (in the spirit of L. K. Hua’s book on the classical domains) and, in addition, to suggest characterizations of classes of star bodies by means of integral geometric inequalities. [ABSTRACT FROM AUTHOR]

The tensorial curvature measures are tensor-valued generalizations of the curvature measures of convex bodies. On convex polytopes, there exist further generalizations some of which also have continuous extensions to arbitrary convex bodies. We prove a complete set of kinematic formulae for such (generalized) tensorial curvature measures. These formulae express the integral mean of the (generalized) tensorial curvature measure of the intersection of two given convex bodies (resp. polytopes), one of which is uniformly moved by a proper rigid motion, in terms of linear combinations of (generalized) tensorial curvature measures of the given convex bodies (resp. polytopes). We prove these results in a more direct way than in the classical proof of the principal kinematic formula for curvature measures, which uses the connection to Crofton formulae to determine the involved constants explicitly. [ABSTRACT FROM AUTHOR]

Short and transparent proofs of central limit theorems for intrinsic volumes of random polytopes in smooth convex bodies are presented. They combine different tools such as estimates for floating bodies with Stein’s method from probability theory. [ABSTRACT FROM AUTHOR]

The present paper is chiefly based on the report delivered by the author on March 16, 2016 at the Principle Seminar of the Department of Probability held at the Faculty of Mechanics and Mathematics of Moscow State University. We describe in general terms the relations of the theory of probability with integral geometry and with the phenomenon of measure concentration in multidimensional geometry. In this context, we also discuss one general observation by A. N. Kolmogorov about the Gauss normal distribution, which was brought to the author's attention by A. N. Shiryaev. [ABSTRACT FROM AUTHOR]

This paper deals with the isoperimetric deficit upper bound for the convex domain in a surface 𝕏𝜖2 of constant curvature 𝜖 by the containment measure of a convex domain to contain another convex domain in integral geometry. Some reverse Bonnesen-style inequalities are obtained. In particular, two of them strengthen Zhou’s result in 𝕏𝜖2 and Bottema’s result in the Euclidean plane 𝔼2. [ABSTRACT FROM AUTHOR]

Mixed volumes V ( K 1 , … , K d ) of convex bodies K 1 , … , K d in Euclidean space R d are of central importance in the Brunn–Minkowski theory. Representations for mixed volumes are available in special cases, for example as integrals over the unit sphere with respect to mixed area measures. More generally, in Hug–Rataj–Weil (2013) [11] a formula for V ( K [ n ] , M [ d − n ] ) , n ∈ { 1 , … , d − 1 } , as a double integral over flag manifolds was established which involved certain flag measures of the convex bodies K and M (and required a general position of the bodies). In the following, we discuss the general case V ( K 1 [ n 1 ] , … , K k [ n k ] ) , n 1 + ⋯ + n k = d , and show a corresponding result involving the flag measures Ω n 1 ( K 1 ; ⋅ ) , … , Ω n k ( K k ; ⋅ ) . For this purpose, we first establish a curvature representation of mixed volumes over the normal bundles of the bodies involved. We also obtain a corresponding flag representation for the mixed functionals from translative integral geometry and a local version, for mixed (translative) curvature measures. [ABSTRACT FROM AUTHOR]

For a compact oriented Finsler surface with smooth boundary, we consider the scalar and vector integral geometry problems over a general family of curves running between boundary points and parametrized by arc length. We impose a natural condition which results in the no conjugate points condition in the case when the curves in question are geodesic lines. Our main theorem generalizes Mukhometov’s theorem in several directions. We also consider these problems on a closed oriented Finsler surface. In this case the integral geometry problems make sense provided that sufficiently many curves in the family are periodic. To this end, we assume that the induced flow on the unit circle bundle of the surface is Anosov. Also, we study the cohomological equation of thermostats without conjugate points. [ABSTRACT FROM AUTHOR]

We present a short, elementary proof not involving Fourier transforms of the theorem of Cramér and Wold that a Borel probability measure is determined by its values on half-spaces. [ABSTRACT FROM AUTHOR]

In this work, we submite and study a newlclass of meromorphiccmultivalent functions defined by fractional differ-integral operator . We gain some geometric properties , such as, coefficient inequality, growth and distortion bounds, convolution properties, integral representation, radii of starlikeness and convexity, extreme points, weighted mean and arithmetic mean for functions belonging to the class (αβγση). [ABSTRACT FROM AUTHOR]

The problem of economic measurements is discussed. The system of economic indices must reflect the economic relations and mechanisms existing in society. An achievement of the XX century is the development of a system of national accounts and the gross domestic product index. However, the gross domestic product index, which is related to the Hamilton-Pontryagin function in extensive economic growth models, turns out to be inadequate under the conditions of structural changes. New problems of integral geometry related to production models that take into account the substitution of production factors are considered. [ABSTRACT FROM AUTHOR]

By studying the group of rigid motions, PSH(1), in the 3D-Heisenberg group H1,we define a density and a measure in the set of horizontal lines. We show that the volume of a convex domain D ⊂ H1 is equal to the integral of the length of chords of all horizontal lines intersecting D. As in classical integral geometry, we also define the kinematic density for PSH(1) and show that the measure of all segments with length l intersecting a convex domain D ⊂ H1 can be represented by the p-area of the boundary ∂D, the volume of D, and 2l. Both results show the relationship between geometric probability and the natural geometric quantity in [10] derived by using variational methods. The probability that a line segment be contained in a convex domain is obtained as an application of our results.