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We consider the X-ray transform in a projective space over a finite field. It is well known (after Bolker) that this transform is injective. We formulate an analog of Gelfand's admissibility problem for the Radon transform, which asks for a classification of all minimal sets of lines for which the restricted Radon transform is injective. The solution involves doubly ruled quadric surfaces. [ABSTRACT FROM AUTHOR]

The chord‐length distribution function [γ′′(r)] of any bounded polyhedron has a closed analytic expression which changes in the different subdomains of the r range. In each of these, the γ′′(r) expression only involves, as transcendental contributions, inverse trigonometric functions of argument equal to R[r, Δ1], Δ1 being the square root of a second‐degree r polynomial and R[x, y] a rational function. As r approaches δ, one of the two end points of an r subdomain, the derivative of γ′′(r) can only show singularities of the forms |r − δ|−n and |r − δ|−m+1/2, with n and m appropriate positive integers. Finally, the explicit analytic expressions of the primitives are also reported. [ABSTRACT FROM AUTHOR]

Multiphase flow in porous media is strongly influenced by the pore-scale arrangement of fluids. Reservoir-scale constitutive relationships capture these effects in a phenomenological way, relying only on fluid saturation to characterize the macroscopic behavior. Working toward a more rigorous framework, we make use of the fact that the momentary state of such a system is uniquely characterized by the geometry of the pore-scale fluid distribution. We consider how fluids evolve as they undergo topological changes induced by pore-scale displacement events. Changes to the topology of an object are fundamentally discrete events. We describe how discontinuities arise, characterize the possible topological transformations and analyze the associated source terms based on geometric evolution equations. Geometric evolution is shown to be hierarchical in nature, with a topological source term that constrains how a structure can evolve with time. The challenge associated with predicting topological changes is addressed by constructing a universal geometric state function that predicts the possible states based on a non-dimensional relationship with two degrees of freedom. The approach is validated using fluid configurations from both capillary and viscous regimes in ten different porous media with porosity between 0.10 and 0.38. We show that the non-dimensional relationship is independent of both the material type and flow regime. We demonstrate that the state function can be used to predict history-dependent behavior associated with the evolution of the Euler characteristic during two-fluid flow. [ABSTRACT FROM AUTHOR]

We derive bounds relating Renegar’s condition number to quantities that govern the statistical performance of convex regularization in settings that include the $\ell _{1}$ -analysis setting. Using results from conic integral geometry, we show that the bounds can be made to depend only on a random projection, or restriction, of the analysis operator to a lower dimensional space, and can still be effective if these operators are ill-conditioned. As an application, we get new bounds for the undersampling phase transition of composite convex regularizers. Key tools in the analysis are Slepian’s inequality and the kinematic formula from integral geometry. [ABSTRACT FROM AUTHOR]

In this paper, the Orlicz version of the classical dual Cauchy-Kubota formula is given and the concept of dual affine quermassintegrals is extended to dual Orlicz mixed affine quermassintegrals in the framework of Orlicz Brunn-Minkowski theory. Some inequalities for dual Orlicz mixed affine quermassintegrals are obtained, such as dual Orlicz-Minkowski inequality and dual Orlicz-Brunn-Minkowski inequality. [ABSTRACT FROM AUTHOR]

General integral geometric invariants for convex bodies are introduced and two integral geometric inequalities for them are established. The equality cases for the inequalities are kinematic formulas, which are characterizations of integral geometric valuations. Those characterizations are analogues of Hadwiger's characterization theorem. [ABSTRACT FROM AUTHOR]

Motivated by questions in real enumerative geometry (Borcea et al., in Discrete Comput Geom 35(2):287–300, 2006; Bürgisser and Lerario, in J Reine Angew Math, 10.1515/crelle-2018-0009, 2018; Megyesi and Sottile, in Discrete Comput Geom 33(4):617–644, 2005; Megyesi et al., in Discrete Comput Geom 30(4):543–571, 2003; Sottile and Theobald, in Trans Am Math Soc 354(12):4815–4829, 2002; Proc Am Math Soc 133(10):2835–2844, 2005; in: Goodman et al., in Surveys on discrete and computational geometry. AMS, Providence, 2008) we investigate the problem of the number of flats simultaneously tangent to several convex hypersurfaces in real projective space from a probabilistic point of view (here by "convex hypersurfaces" we mean that these hypersurfaces are boundaries of convex sets). More precisely, we say that smooth convex hypersurfaces X 1 , ... , X d k , n ⊂ R P n , where d k , n = (k + 1) (n - k) , are in random position if each one of them is randomly translated by elements g 1 , ... , g d k , n sampled independently from the orthogonal group with the uniform distribution. Denoting by τ k (X 1 , ... , X d k , n ) the average number of k-dimensional projective subspaces (k-flats) which are simultaneously tangent to all the hypersurfaces we prove that τ k (X 1 , ... , X d k , n ) = δ k , n · ∏ i = 1 d k , n | Ω k (X i) | | Sch (k , n) | , where δ k , n is the expected degree from [6] (the average number of k-flats incident to d k , n many random (n - k - 1) -flats), | Sch (k , n) | is the volume of the Special Schubert variety of k-flats meeting a fixed (n - k - 1) -flat (computed in [6]) and | Ω k (X) | is the volume of the manifold Ω k (X) ⊂ G (k , n) of all k-flats tangent to X. We give a formula for the evaluation of | Ω k (X) | in terms of some curvature integral of the embedding X ↪ R P n and we relate it with the classical notion of intrinsic volumes of a convex set: | Ω k (∂ C) | | Sch (k , n) | = 4 V n - k - 1 (C) , k = 0 , ... , n - 1. As a consequence we prove the universal upper bound: τ k (X 1 , ... , X d k , n ) ≤ δ k , n · 4 d k , n . Since the right hand side of this upper bound does not depend on the specific choice of the convex hypersurfaces, this is especially interesting because already in the case k = 1 , n = 3 for every m > 0 we can provide examples of smooth convex hypersurfaces X 1 , ... , X 4 such that the intersection Ω 1 (X 1) ∩ ⋯ ∩ Ω 1 (X 4) ⊂ G (1 , 3) is transverse and consists of at least m lines. Finally, we present analogous results for semialgebraic hypersurfaces (not necessarily convex) satisfying some nondegeneracy assumptions. [ABSTRACT FROM AUTHOR]

Like his colleagues de Prony, Petit, and Poisson at the Ecole Polytechnique, Cauchy used infinitesimals in the Leibniz-Euler tradition both in his research and teaching. Cauchy applied infinitesimals in an 1826 work in differential geometry where infinitesimals are used neither as variable quantities nor as sequences but rather as numbers. He also applied infinitesimals in an 1832 article on integral geometry, similarly as numbers. We explore these and other applications of Cauchy's infinitesimals as used in his textbooks and research articles. An attentive reading of Cauchy's work challenges received views on Cauchy's role in the history of analysis and geometry. We demonstrate the viability of Cauchy's infinitesimal techniques in fields as diverse as geometric probability, differential geometry, elasticity, Dirac delta functions, continuity and convergence. [ABSTRACT FROM AUTHOR]

We consider the attenuated geodesic ray transform defined on pairs of symmetric 2-tensors and 1-forms on a simple Riemannian manifold. We prove injectivity and stability results for a class of generic simple metrics and attenuations containing real analytic ones. In fact, methods used in this paper can be modified to generalize our results for a class of non-simple manifolds similar to Stefanov and Uhlmann Am J Math 130(1):239–268 (2008). [ABSTRACT FROM AUTHOR]

We prove that higher moment maps on area measures of a euclidean vector space are injective, while the kernel of the centroid map equals the image of the first variation map. Based on this, we introduce the space of smooth dual area measures on a finite-dimensional euclidean vector space and prove that it admits a natural convolution product which encodes the local additive kinematic formulas for groups acting transitively on the unit sphere. As an application of this new integral-geometric structure, we obtain the local additive kinematic formulas in hermitian vector spaces in a very explicit way. [ABSTRACT FROM AUTHOR]

The classical Crofton formula explains how intrinsic volumes of a convex body K in n -dimensional Euclidean space can be obtained from integrating a measurement function at sections of K with invariantly moved affine flats. Motivated by stereological applications, we present variants of Crofton's formula, where the flats are constrained to contain a fixed linear subspace L 0 , but are otherwise invariantly rotated. This main result generalizes a known rotational Crofton formula, which only covers the case dim ⁡ L 0 = 0. The proof combines a suitable Blaschke–Petkantschin formula with the classical Crofton formula. We also argue that our main result is best possible, in the sense that one cannot estimate intrinsic volumes of a set, based on lower-dimensional sections, other than those given by our result. Finally, we provide a proof for a well-established variant: an integral relation for vertical sections. Our formula is stated for intrinsic volumes of a given set, complementing the classical approach for Hausdorff measures. [ABSTRACT FROM AUTHOR]

In 1977 D. G. Kendall considered diffusions of shape induced by independent Brownian motions in Euclidean space. In this paper, we consider a different class of diffusions of shape, induced by the projections of a randomly rotating, labelled ensemble. In particular, we study diffusions of shapes induced by projections of planar triangular configurations of labelled points onto a fixed straight line. That is, we consider the process in Sigma(3)(1) (the shape space of triads in R) that results from extracting the 'shape information' from the projection of a given labelled planar triangle as this evolves under the action of Brownian motion in SO(2). We term the thus-defined diffusions Radon diffusions and derive explicit stochastic differential equations and stationary distributions. The latter belong to the family of angular central Gaussian distributions. In addition, we discuss how these Radon diffusions and their limiting distributions are related to the shape of the initial triangle, and explore whether the relationship is bijective. The triangular case is then used as a basis for the study of processes in Sigma(k)(1) arising from projections of an arbitrary number, k, of labelled points on the plane. Finally, we discuss the problem of Radon diffusions in the general shape space Sigma(k)(n).

Any compact body with regular boundary in ℝN defines a two-valued function on the space of affine hyperplanes: the volumes of the two parts into which these hyperplanes cut the body. This function is never algebraic if N is even and is very rarely algebraic if N is odd: all known bodies defining algebraic volume functions are ellipsoids (and have been essentially found by Archimedes for N = 3). We demonstrate a new series of locally algebraically integrable bodies with algebraic boundaries in spaces of arbitrary dimensions, that is, of bodies such that the corresponding volume functions coincide with algebraic ones in some open domains of the space of hyperplanes intersecting the body. [ABSTRACT FROM AUTHOR]

The mechanical and use properties of metal alloys depend on several factors, including the amount and the geometry of impurities (inclusions). In this context, image analysis enables these inclusions to be studied from digital images acquired by various systems such as optical/electron microscopy or X-ray tomography. This paper therefore aims to present some geometrical and morphometrical tools of image analysis, in order to characterize inclusions in metal alloys. To achieve this quantification, many geometrical and morphometrical features are traditionally used to quantitatively describe a population of objects (inclusions). Integral geometry, via Minkowski's functionals (in 2D: area, perimeter, Euler-Poincaré number), has been particularly investigated in image analysis. Nevertheless, they are sometimes insufficient for the characterization of complex microstructures (such as aggregates/agglomerates of objects). Other quantitative parameters are then necessary in order to discriminate or group different families of objects. In particular, shape diagrams are mathematical representations in the Euclidean plane for studying the morphology (shape) of objects, regardless of their size. In addition, this representation also makes it possible to analyze the evolution from one shape to another. In conclusion, image analysis using integral geometry and shape diagrams provide efficient tools with known mathematical properties to quantitatively describe inclusions (providing separate information on size and shape). The geometrical characteristics of these inclusions could thereafter be related to the mechanical properties of the metal alloys. [ABSTRACT FROM AUTHOR]

Let X be an open subset of ℝ 2 {\mathbb{R}^{2}}. We study the dynamic operator, 𝒜 {\mathcal{A}} , integrating over a family of level curves in X when the object changes between the measurement. We use analytic microlocal analysis to determine which singularities can be recovered by the data-set. Our results show that not all singularities can be recovered as the object moves with a speed lower than the X-ray source. We establish stability estimates and prove that the injectivity and stability are of a generic set if the dynamic operator satisfies the visibility, no conjugate points, and local Bolker conditions. We also show this results can be implemented to fan beam geometry. [ABSTRACT FROM AUTHOR]

We consider weighted ray-transforms Pw (weighted Radon transforms along oriented straight lines) in ℝd d≥2, with strictly positive weights W. We construct an example of such a transform with non-trivial kernel in the space of infinitely smooth compactly supported functions on ℝd. In addition, the constructed weight W is rotation-invariant continuous and is infinitely smooth almost everywhere on ℝd x 핊d-1. In particular, by this construction we give counterexamples to some well-known injectivity results for weighted ray transforms for the case when the regularity of W is slightly relaxed. We also give examples of continous strictly positive W such that dimker Pw ≥ n in the space of infinitely smooth compactly supported functions on ℝd for arbitrary n ∈ ℕ ∪{∞}, where W are infinitely smooth for d=2 and infinitely smooth almost everywhere for d ≥ 3. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

In the paper, the notion of dual harmonic mixed quermassintegrals in the classical Brunn-Minkowski theory is extended to that of Orlicz dual harmonic mixed quermassintegrals in the Orlicz-Brunn-Minkowski theory. The analogs of the classical dual Cauchy-Kubota formula, the dual Minkowski isoperimetric inequality and the dual Brunn-Minkowski inequality are established for this new dual Orlicz harmonic mixed quermassintegrals. [ABSTRACT FROM AUTHOR]

Abstract We present a new construction of Poisson transforms between vector bundle valued differential forms on homogeneous parabolic geometries and vector bundle valued differential forms on the corresponding Riemannian symmetric space, which can be described in terms of finite dimensional representations of reductive Lie groups. In particular, we use these operators to relate the BGG-sequences on the domain to twisted deRham sequences on the target space. Finally, we explicitly design a family of Poisson transforms between standard tractor valued differential forms for the real hyperbolic space and its boundary which are compatible with the BGG-complex. [ABSTRACT FROM AUTHOR]

Abstract In the present paper we prove some new fixed point theorems for self-mappings defined on a complete metric space with a w -distance. These results extend some previous fixed point theorems in this field to more general contractive conditions in the setting of w -distances for selfmappings which satisfy certain weaker Meir–Keeler conditions. [ABSTRACT FROM AUTHOR]

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 which gives an algorithm to recover an unknown function supported completely on one side of a line L from its spherical means over circles centered on the line L. 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 other. [ABSTRACT FROM AUTHOR]

We show that balls, circles and 2‐spheres can be identified by generalized Riesz energy among compact submanifolds of the Euclidean space that are either closed or with codimension 0, where the Riesz energy is defined as the double integral of some power of the distance between pairs of points. As a consequence, we obtain the identification by the interpoint distance distribution. [ABSTRACT FROM AUTHOR]

We prove the strong Artin conjecture for continuous, totally odd, two-dimensional representations of the absolute Galois group of a totally real field F. [ABSTRACT FROM AUTHOR]

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]

A few words on the present special topic, devoted to the multiscale modeling of granular media, and published in honor of Prof. Dominique Jeulin’s enduring contribution to the wide field of image analysis, random structures and material science.