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We consider a hyperbolic equation with variable leading part and nonlinearity in the lower-order term. The coefficients of the equation are smooth functions constant beyond some compact domain in the three-dimensional space. A plane wave with direction falls to the heterogeneity from the exterior of this domain. A solution to the corresponding Cauchy problem for the original equation is measured at boundary points of the domain for a time interval including the moment of arrival of the wave at these points. The unit vector is assumed to be a parameter of the problem and can run through all possible values sequentially. We study the inverse problem of determining the coefficient of the nonlinearity on using this information about solutions. We describe the structure of a solution to the direct problem and demonstrate that the inverse problem reduces to an integral geometry problem. The latter problem consists of constructing the desired function on using given integrals of the product of this function and a weight function. The integrals are taken along the geodesic lines of the Riemannian metric associated with the leading part of the differential equation. We analyze this new problem and find some stability estimate for its solution, which yields a stability estimate for solutions to the inverse problem. [ABSTRACT FROM AUTHOR]

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We prove solenoidal injectivity for the geodesic X-ray transform of tensor fields on simple Riemannian manifolds with C 1 , 1 metrics and non-positive sectional curvature. The proof of the result rests on Pestov energy estimates for a transport equation on the non-smooth unit sphere bundle of the manifold. Our low regularity setting requires keeping track of regularity and making use of many functions on the sphere bundle having more vertical than horizontal regularity. Some of the methods, such as boundary determination up to gauge and regularity estimates for the integral function, have to be changed substantially from the smooth proof. The natural differential operators such as covariant derivatives are not smooth. [ABSTRACT FROM AUTHOR]

An inverse problem for a second-order hyperbolic equation containing two nonlinear terms is studied. The problem is to reconstruct the coefficients of the nonlinearities. The Cauchy problem with a point source located at a point is considered. This point is a parameter of the problem and successively runs over a spherical surface . It is assumed that the desired coefficients are nonzero only in a domain lying inside . The trace of the solution of the Cauchy problem on is specified for all possible values of and for times close to the arrival of the wave from the source to the points on the surface ; this allows reducing the inverse problem under consideration to two successively solved problems of integral geometry. Solution stability estimates are found for these two problems. [ABSTRACT FROM AUTHOR]

Some questions concerning the inversion of the classical and generalized integral Radon transforms are discussed. The main issue is to determine information about the integrand if the values of some integrals are known. A feature of this work is that a function is integrated over hyperplanes in a finite-dimensional Euclidean space and the integrands depend not only on the variables of integration, but also on some of the variables characterizing the hyperplanes. The independent variables describing the known integrals are fewer than those in the unknown integrand. We consider discontinuous integrands defined on specifically introduced pseudoconvex sets. A Stefan-type problem of finding discontinuity surfaces of the integrand is posed. Formulas for solving the problem under study are derived by applying special integro-differential operators to known data. [ABSTRACT FROM AUTHOR]

We show that the Holmes-Thompson area of every Finsler disk of radius r whose interior geodesics are length-minimizing is at least .... Furthermore, we construct examples showing that the inequality is sharp and observe that equality is attained by a non-rotationally-symmetric metric. This contrasts with Berger's conjecture in the Riemannian case, which asserts that the round hemisphere is extremal. To prove our theorem we discretize the Finsler metric using random geodesics. As an auxiliary result, we include a proof of the integral geometry formulas of Blaschke and Santaló for Finsler manifolds with almost no trapped geodesics. [ABSTRACT FROM AUTHOR]

Sickle cell disease causes the deformation of erythrocytes into sickle cells. The study of this process using digital images of peripheral blood smears can help specialists to quantify the number of deformed cells in order to gauge the severity of the illness. A new method for classifying red blood cells into three categories: healthy, sickle cell disease, and other deformations is proposed. This method does not require obtaining the contour of each cell but instead utilizes information obtained from a small number of points, obtained through appropriate geometric sampling and the use of stereological formulas. The parameters utilized for classification are the bending energy times length (E) and the circular shape factor (F). In normal cells, which exhibit an almost circular shape, these parameters typically have values close to (1,1). To assess the effectiveness of classification using the parameters (E,F), a synthetic curve dataset and a dataset of red blood cells are employed, applying various supervised and unsupervised classification methods. [ABSTRACT FROM AUTHOR]

Copyright of Proceedings of the YSU A: Physical & Mathematical Sciences is the property of Publishing House of Yerevan State University and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

We study the problem of the integral geometry, in which the functions are integrated over hyperplanes in the n-dimensional Euclidean space, n = 2 ⁢ m + 1 . The integrand is the product of a function of n variables called the density and weight function depending on 2 ⁢ n variables. Such an integration is called here the weighted Radon transform, which coincides with the classical one if the weight function is equal to one. It is proved the uniqueness for the problem of determination of the surface on which the integrand is discontinuous. [ABSTRACT FROM AUTHOR]

Poisson processes in the space of k -dimensional totally geodesic subspaces (k -flats) in a d -dimensional standard space of constant curvature κ ∈ { − 1 , 0 , 1 } are studied, whose distributions are invariant under the isometries of the space. We consider the intersection processes of order m together with their (d − m (d − k)) -dimensional Hausdorff measure within a geodesic ball of radius r. Asymptotic normality for fixed r is shown as the intensity of the underlying Poisson process tends to infinity for all m satisfying d − m (d − k) ≥ 0. For κ ∈ { − 1 , 0 } the problem is also approached in the set-up where the intensity is fixed and r tends to infinity. Again, if 2 k ≤ d + 1 a central limit theorem is shown for all possible values of m. However, while for κ = 0 asymptotic normality still holds if 2 k > d + 1 , we prove for κ = − 1 convergence to a non-Gaussian infinitely divisible limit distribution in the special case m = 1. The proof of asymptotic normality is based on the analysis of variances and general bounds available from the Malliavin–Stein method. We also show for general κ ∈ { − 1 , 0 , 1 } that, roughly speaking, the variances within a general observation window W are maximal if and only if W is a geodesic ball having the same volume as W. Along the way we derive a new integral-geometric formula of Blaschke–Petkantschin type in a standard space of constant curvature. [ABSTRACT FROM AUTHOR]

Sickle cell disease causes the deformation of erythrocytes into sickle cells. The study of this process using digital images of peripheral blood smears can help specialists to quantify the number of deformed cells in order to gauge the severity of the illness. A new method for classifying red blood cells into three categories: healthy, sickle cell disease, and other deformations is proposed. This method does not require obtaining the contour of each cell but instead utilizes information obtained from a small number of points, obtained through appropriate geometric sampling and the use of stereological formulas. The parameters utilized for classification are the bending energy times length (E) and the circular shape factor (F). In normal cells, which exhibit an almost circular shape, these parameters typically have values close to (1,1). To assess the effectiveness of classification using the parameters (E,F), a synthetic curve dataset and a dataset of red blood cells are employed, applying various supervised and unsupervised classification methods.

In this article, we consider the weighted conical Radon transform—the transform is motivated by Compton camera imaging as well as optical tomography. Our contribution involves introducing new inversion formulas for the weighted conical Radon transform, including explicit formulas and properties associated with convolution frames. Furthermore, we propose reconstruction formulas that solve for variety weighted parameters in the two-dimensional space.

We study the problem of proper discretizing and sampling issues related to geodesic X-ray transforms on simple surfaces, and illustrate the theory on simple geodesic disks of constant curvature. Given a notion of band limit on a function, we provide the minimal sampling rates of its X-ray transform for a faithful reconstruction. In Cartesian sampling, we quantify the quality of a sampling scheme depending on geometric parameters of the surface (e.g., curvature and boundary curvature), and the coordinate system used to represent the space of geodesics. When aliasing happens, we explain how to predict the location, orientation, and frequency of the artifacts. [ABSTRACT FROM AUTHOR]

Let be the -dimensional Euclidean space and be a bounded convex body . Consider a family of support planes for which is an envelope. How can we obtain information about from the support planes? Conditions under which a given convex body is the envelope of a family of planes are obtained. Therefore the distances of these planes from the origin will be the support functions of this body D. In particular, we have cited expressions for the surface area and the volume of body in terms of support planes. [ABSTRACT FROM AUTHOR]

We show that the geodesic ray transform is injective on scalar functions on spherically symmetric reversible Finsler manifolds where the Finsler norm satisfies a Herglotz condition. We use angular Fourier series to reduce the injectivity problem to the invertibility of generalized Abel transforms and by Taylor expansions of geodesics we show that these Abel transforms are injective. Our result has applications in linearized boundary rigidity problem on Finsler manifolds and especially in linearized elastic travel time tomography. [ABSTRACT FROM AUTHOR]

The geodesic ray transform, the mixed ray transform and the transverse ray transform of a tensor field on a surface can all be seen as what we call mixing ray transforms, compositions of the geodesic ray transform and an invertible linear map on tensor fields. We provide an approach that uses a unifying concept of symmetry to merge various earlier transforms (including mixed, transverse, and light ray transforms) into a single family of integral transforms with similar kernels. [ABSTRACT FROM AUTHOR]

In this article, we work with a generalized Saint Venant operator introduced by Vladimir Sharafutdinov [Inverse and ill-posed problems series, VSP, Utrecht, 1994] to describe the kernel of the integral moment transforms over symmetric m-tensor fields in n-dimensional Euclidean space. We also provide an equivalence between the injectivity question for the integral moment transforms and generalized Saint Venant operator over symmetric tensor fields of Schwartz class. [ABSTRACT FROM AUTHOR]

Understanding pore-scale morphology and distribution of remaining oil in pore space are of great importance to carry out in-depth tapping of oil potential. Taking two water-wet cores from a typical clastic reservoir in China as an example, X-ray CT imaging is conducted at different experimental stages of water flooding and polymer-surfactant (P-S) flooding by using a high-resolution X-ray microtomography. Based on X-ray micro-CT image processing, 3D visualization of rock microstructure and fluid distribution at the pore scale is achieved. The integral geometry newly developed is further introduced to characterize pore-scale morphology and distribution of remaining oil in pore space. The underlying mechanism of oil recovery by P-S flooding is further explored. The results show that the average diameter of oil droplets gradually decreases, and the topological connectivity becomes worse after water flooding and P-S flooding. Due to the synergistic effect of "1 + 1 > 2" between the strong sweep efficiency of surfactant and the enlarged swept volume of the polymer, oil droplets with a diameter larger than 124.58 μm can be gradually stripped out by the polymer-surfactant system, causing a more scattered distribution of oil droplets in pore spaces of the cores. The network-like oil clusters are still dominant when water flooding is continued to 98% of water cut, but the dominant pore-scale oil morphology has evolved from network-like to porous-type and isolated-type after P-S flooding, which can provide strong support for further oil recovery in the later stage of chemical flooding. [ABSTRACT FROM AUTHOR]

We introduce and study a class of determinantal probability measures generalising the class of discrete determinantal point processes. These measures live on the Grassmannian of a real, complex, or quaternionic inner product space that is split into pairwise orthogonal finite-dimensional subspaces. They are determined by a positive self-adjoint contraction of the inner product space, in a way that is equivariant under the action of the group of isometries that preserve the splitting. [ABSTRACT FROM AUTHOR]

The ray statement of the inverse problem of determining three unknown variable coefficients in the linear acoustic equation is studied. These coefficients are assumed to differ from given constants only inside some bounded domain. There are point pulse sources and acoustic receivers on the boundary of this domain. Acoustic signals are measured by a receiver near the moment of time at which the signal from a source arrives at the receiver. It is shown that this information makes it possible to uniquely determine all the three desired coefficients. Algorithmically, the original inverse problem splits into three subproblems solved successively. One of them is a well-known inverse kinematic problem (of determining the speed of sound), while the other two lead to the same integral geometry problem for a family of geodesic lines determined by the speed of sound. [ABSTRACT FROM AUTHOR]

We study the problem of contextual search, a multidimensional generalization of binary search that captures many problems in contextual decision-making. In contextual search, a learner is trying to learn the value of a hidden vector v∈[0,1]d. Every round the learner is provided an adversarially chosen context ut∈\Rd, submits a guess pt for the value of ⟨ut,v⟩, learns whether pt<⟨ut,v⟩, and incurs loss ℓ(⟨ut,v⟩,pt) (for some loss function ℓ). The learner's goal is to minimize their total loss over the course of T rounds. We present an algorithm for the contextual search problem for the symmetric loss function ℓ(θ,p)=|θ-p| that achieves Od(1) total loss. We present a new algorithm for the dynamic pricing problem (which can be realized as a special case of the contextual search problem) that achieves Od(loglogT) total loss, improving on the previous best known upper bounds of Od(logT) and matching the known lower bounds (up to a polynomial dependence on d). Both algorithms make significant use of ideas from the field of integral geometry, most notably the notion of intrinsic volumes of a convex set. To the best of our knowledge, this is the first application of intrinsic volumes to algorithm design. [ABSTRACT FROM AUTHOR]

The defining property of an artificial physical self-replicating system, such as a self-replicating robot, is that it has the ability to make copies of itself from basic parts. Three questions that immediately arises in the study of such systems are: (1) How complex is the whole robot in comparison to each basic part? (2) How disordered can the parts be while having the robot successfully replicate? (3) What design principles can enable complex self-replicating systems to function in disordered environments generation after generation? Consequently, much of this article focuses on exploring different concepts of entropy as a measure of disorder, and how symmetries can help in reliable self replication, both at the level of assembly (by reducing the number of wrong ways that parts could be assembled), and also as a parity check when replicas manufacture parts generation after generation. The mathematics underpinning these principles that quantify artificial physical self-replicating systems are articulated here by integrating ideas from information theory, statistical mechanics, ergodic theory, group theory, and integral geometry. [ABSTRACT FROM AUTHOR]

We consider the inverse problem of determining the coefficient of the nonlinear term in an equation whose main part is the wave operator. The properties of the solution of the direct problem are studied; in particular, the existence and uniqueness of a bounded solution in a neighborhood of the characteristic cone is established, and the structure of this solution is written out. The problem of finding the unknown function is reduced to the problem of integral geometry on a family of straight lines with a weight function invariant with respect to rotations around some fixed point. The uniqueness of the solution of the inverse problem is established, and an algorithm for reconstructing the desired function is proposed. [ABSTRACT FROM AUTHOR]

This paper analyzes network disconnection due to randomly occurring natural disasters in a geographical area, which is divided into sub-areas on the basis of disaster-vulnerable (DV) levels. The DV levels against an earthquake, for example, depend on the soil-foundation-structure. The network is a communications cable network that is planned to be built between two given network nodes. To make the optical cable working between these network nodes, electric supply from a power grid to these nodes is necessary, where the power grid has already existed. In this paper, the geographical conditions are derived under which a cable route of the network maximizes the probability of connecting two given nodes for any level of disaster when the geographical route information of the power grid is given. The achievable maximum probability of connecting two given nodes or its bound is explicitly given. [ABSTRACT FROM AUTHOR]

Intersection formulae of Croton type for general geometric probes are well known in integral geometry. For the special case of cylinders with non necessarily convex direktrix, however, no equivalent formula seems to exist in the literature. We derive such formula resorting to motion invariant probability elements associated with test systems, instead of using a traditional approach. Because cylinders are seldom used as probes in stereological practice, however, this note is mainly of a theoretical nature.

In this work, we explore how the emergence of collective motion in a system of particles is influenced by the structure of their domain. Using the Vicsek model to generate flocking, we simulate two-dimensional systems that are confined based on varying obstacle arrangements. The presence of obstacles alters the topological structure of the domain where collective motion occurs, which, in turn, alters the scaling behavior. We evaluate these trends by considering the scaling exponent and critical noise threshold for the Vicsek model, as well as the associated diffusion properties of the system. We show that obstacles tend to inhibit collective motion by forcing particles to traverse the system based on curved trajectories that reflect the domain topology. Our results highlight key challenges related to the development of a more comprehensive understanding of geometric structure's influence on collective behavior.

The paper studies the problem of inverting the integral transformation of Radon, whose formula, under traditional restrictions, gives the integrand values at any point. For the case when such a function is discontinuous and depends not only on the points of 3D space, but also on the parameters characterizing the plane of integration, these integrals have been named the generalized Radon transform (GRT). For the GRT inversion problem, the matching between quantities of known variables and variables of the integrand did not allow us to fully find the desired function. In this paper, only a part of such a function was selected, namely, the discontinuity surface of the integrand for the GRT. An algorithm for solving the problem was put forward, and it was supported by a concrete example.

This paper studies the Hausdorff dimension of the intersection of isotropic projections of subsets of ℝ2n, as well as dimension of intersections of sets with isotropic planes. It is shown that if A and B are Borel subsets of ℝ2n of dimension greater than m, then for a positive measure set of isotropic m-planes, the intersection of the images of A and B under orthogonal projections onto these planes have positive Hausdorff m-measure. In addition, if A is a measurable set of Hausdorff dimension greater than m, then there is a set B ⊂ ℝ2n with dim B ⩽ m such that for all x ∈ ℝ2n\B there is a positive measure set of isotropic m-planes for which the translate by x of the orthogonal complement of each such plane, intersects A on a set of dimension dim A – m. These results are then applied to obtain analogous results on the nth Heisenberg group.

Understanding pore-scale morphology and distribution of remaining oil in pore space are of great importance to carry out in-depth tapping of oil potential. Taking two water-wet cores from a typical clastic reservoir in China as an example, X-ray CT imaging is conducted at different experimental stages of water flooding and polymer-surfactant (P-S) flooding by using a high-resolution X-ray microtomography. Based on X-ray micro-CT image processing, 3D visualization of rock microstructure and fluid distribution at the pore scale is achieved. The integral geometry newly developed is further introduced to characterize pore-scale morphology and distribution of remaining oil in pore space. The underlying mechanism of oil recovery by P-S flooding is further explored. The results show that the average diameter of oil droplets gradually decreases, and the topological connectivity becomes worse after water flooding and P-S flooding. Due to the synergistic effect of “1 + 1 > 2” between the strong sweep efficiency of surfactant and the enlarged swept volume of the polymer, oil droplets with a diameter larger than 124.58 μm can be gradually stripped out by the polymer-surfactant system, causing a more scattered distribution of oil droplets in pore spaces of the cores. The network-like oil clusters are still dominant when water flooding is continued to 98% of water cut, but the dominant pore-scale oil morphology has evolved from network-like to porous-type and isolated-type after P-S flooding, which can provide strong support for further oil recovery in the later stage of chemical flooding.

We consider the mixed ray transform of tensor fields on a three-dimensional compact simple Riemannian manifold with boundary. We prove the injectivity of the transform, up to natural obstructions, and establish stability estimates for the normal operator on generic three dimensional simple manifold in the case of 1+1 and 2+2 tensors fields. We show how the anisotropic perturbations of averaged isotopic travel-times of qS-polarized elastic waves provide partial information about the mixed ray transform of 2+2 tensors fields. If in addition we include the measurement of the shear wave amplitude, the complete mixed ray transform can be recovered. We also show how one can obtain the mixed ray transform from an anisotropic perturbation of the Dirichlet-to-Neumann map of an isotropic elastic wave equation on a smooth and bounded domain in three dimensional Euclidean space. [ABSTRACT FROM AUTHOR]

Abstract—The paper addresses quantitative analysis of three-dimensional (3D) porous media (natural hydrocarbon reservoirs) based on topological invariants—the Minkowski functionals (MF) and presents the solutions of several applied problems obtained using integral geometry methods. The analysis of binarization of a 3D image of sandstone as a dynamic process is demonstrated and the stability of the Minkowski functionals to the choice of the binarization threshold is proved. The approach to solving the classification problem for samples of hydrocarbon reservoirs and finding the analog samples is proposed. Synthetic model samples of porous media and samples of real geological objects are studied. [ABSTRACT FROM AUTHOR]

For a -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. In addition, the relations between the concepts are found in this article. [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]

We relate the existence of many infinite geodesics on Alexandrov spaces to a statement about the average growth of volumes of balls. We deduce that the geodesic flow exists and preserves the Liouville measure in several important cases. The analytic tools we develop have close ties to integral geometry. [ABSTRACT FROM AUTHOR]

On simple geodesic disks of constant curvature, we derive new functional relations for the geodesic X-ray transform, involving a certain class of elliptic differential operators whose ellipticity degenerates normally at the boundary. We then use these relations to derive sharp mapping properties for the X-ray transform and its corresponding normal operator. Finally, we discuss the possibility of theoretically rigorous regularized inversions for the X-ray transform when defined on such manifolds. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

The paper considers measures in the space of planes in , and combinatorial decompositions for their values on "Buffon sets" in . These decompositions, written in terms of a "wedge function" depending on the measure, have been known since long in Combinatorial Integral Geometry, yet their explicit expressions have been well established only for "non-degenerate" Buffon sets. Theorem 1 removes this gap and presents a decomposition algorithm valid with no similar restriction. Theorem 2 presents a result in a direction converse to Theorem 1. Starting from the decomposition algorithm, a combinatorial valuation is defined that depends on "general" continuous additive wedge function . The question is: when becomes a measure in the space ? Theorem 2 points at special "tetrahedral inequalities", the analogues of triangular inequalities of the planar theory. If satisfies these "tetrahedral inequalities", then becomes a measure and the corresponding is called a "wedge metric" (to stress the connection of the paper's topic with Hilbert's Fourth Problem). [ABSTRACT FROM AUTHOR]

Intersection formulae of Crofton type for general geometric probes are well known in integral geometry. For the special case of cylinders with non necessarily convex directrix, however, no equivalent formula seems to exist in the literature. We derive this formula resorting to motion invariant probability elements associated with test systems, instead of using a traditional approach. Because cylinders are seldom used as probes in streological practice, however, this note is mainly of a theoretical nature. [ABSTRACT FROM AUTHOR]