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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 deal with integrals of invariant measures of pairs of planes in euclidean space $\mathbb{E}^3$ as considered by Hug and Schneider. In this paper we express some of these integrals in terms of functions of the visual angle of a convex set. As a consequence of our results we evaluate the deficit in a Crofton-type inequality due to Blashcke., 16 pages

The replication of any European contingent claim by a static continuous portfolio of calls and puts, formally proven by Carr and Madan (1998), extends to multi-asset claims with absolutely homogeneous payoff. Using sophisticated tools from integral geometry, we show how such claims may be replicated with a continuum of vanilla basket calls and derive closed-form solutions to replicate two-asset best-of and worst-of options. We also derive a novel mathematical formula to invert the Radon transform which we apply to obtain a tractable expression of the joint implied distribution. Consequently, a large class of multi-asset options admit a model-free price enforced by arbitrage, just as single-asset European claims do.

Abstract The technique of numerical analysis of three-dimensional tomographic images of the pore space of soil objects has been used in this paper. It applies methods of integral geometry, topology and morphological analysis. To characterize quantitatively the transformation of the pore space structure, tomographic images of four undisturbed soils were analyzed, i.e., heavy loamy agro-gray soil (Retic Phaeozem), agromineral (Sapric Rheic Mineralic Histosols), and hypnum (brown moss Sapric Rheic Histosols) peat soils in dry and wet conditions. For samples of the subplow horizon in agro-gray soil, a decrease in both Betty numbers was observed on wetting, where the zero number (b0) stands for the amount of topologically simple closed pores, and the first number (b1) indicates a decrease in pore connectivity, which varies in a narrower range of pore sizes as compared to b0. When a sample of agromineral peat soil is moistened, the Euler–Poincaré characteristic is negative ​in the pore range of 0.1–0.16 mm, which points to the predominating complicated branched structure of the pore space and high pore connectivity. When hypnum moss is saturated, a lot of tunnel pores get narrower (“collapse”), and the connectivity decreases due to the structural specifics of long-stemmed plant residues. The number of pores and connections between them in peat soils is an order of magnitude higher than those in the subplow horizon A of the agro-gray soil. The provided quantitative changes in the considered parameters of tomographic images of the soil pore space confirm the possibility of applying them for estimating the transformation of the pore space in soils.

For a one-parameter family of simple metrics of constant curvature ($4\kappa$ for $\kappa\in (-1,1)$) on the unit disk $M$, we first make explicit the Pestov-Uhlmann range characterization of the geodesic X-ray transform, by constructing a basis of functions making up its range and co-kernel. Such a range characterization also translates into moment conditions {\it a la} Helgason-Ludwig or Gel'fand-Graev. We then derive an explicit Singular Value Decomposition for the geodesic X-ray transform. Computations dictate a specific choice of weighted $L^2-L^2$ setting which is equivalent to the $L^2(M, dVol_\kappa)\to L^2(\partial_+ SM, d\Sigma^2)$ one for any $\kappa\in (-1,1)$.

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 1+1 and 2 + 2 2+2 tensors fields. We show how the anisotropic perturbations of averaged isotopic travel-times of q S qS -polarized elastic waves provide partial information about the mixed ray transform of 2 + 2 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 \text{Dirichlet-to-Neumann} map of an isotropic elastic wave equation on a smooth and bounded domain in three dimensional Euclidean space.

We use geometric microlocal methods to compute an asymptotic expansion of mean first arrival time for Brownian particles on Riemannian manifolds. This approach provides a robust way to treat this problem, which has thus far been limited to very special geometries. This paper can be seen as the Riemannian 3-manifold version of the planar result of [1] and thus enable us to see the full effect of the local extrinsic boundary geometry on the mean arrival time of the Brownian particles. Our approach also connects this question to some of the recent progress on boundary rigidity and integral geometry [21] and [18] .

—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.

In this note, we study the Radon transform and its dual on the discrete matrices by defining hyperplanes as being infinite sets of solutions of linear Diophantine equations. We then give an inversion formula and a support theorem.

We consider the operator of generalized exponential ray transform whose values are the initial data in the emission computer tomography problem. We establish connections between exponential ray transforms of different orders and obtain the differential equations satisfied by the exponential ray transforms. We prove the uniqueness theorems for boundary value and initial-boundary value problems for the corresponding equations We also discuss close connections of generalized exponential ray transforms with integral geometry of tensor fields and tomography problems.

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.

The paper considers measures in the space $$\mathbb{E}$$ of planes in $$\mathbb{R}^{3}$$ , and combinatorial decompositions for their values on ‘‘Buffon sets’’ in $$\mathbb{E}$$ . 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 $$\Psi_{F}$$ is defined that depends on ‘‘general’’ continuous additive wedge function $$F(W)$$ . The question is: when $$\Psi_{F}$$ becomes a measure in the space $$\mathbb{E}$$ ? Theorem 2 points at special ‘‘tetrahedral inequalities’’, the analogues of triangular inequalities of the planar theory. If $$\Psi_{F}$$ satisfies these ‘‘tetrahedral inequalities’’, then $$\Psi_{F}$$ becomes a measure and the corresponding $$F(W)$$ is called a ‘‘wedge metric’’ (to stress the connection of the paper’s topic with Hilbert’s Fourth Problem).

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. National Science Foundation [1751498] Published version Funding NA was supported by the National Science Foundation under grant 1751498.

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.

Under consideration are the operators of angular moments which map the values of generalized attenuated ray transforms (ART) into the set of symmetric $$p $$ -tensor fields. The differential relations between the values of ARTs of various orders, acting on stationary or dynamic source distributions $$f $$ , serve as the basis for establishing the differential connections between the tensor fields of angular moments of various orders $$k $$ and ranks $$p $$ . The particular cases are indicated allowing us to obtain some previously known results. Connections of the ARTs of order $$k $$ with the problems of tomography and integral geometry as well as the established properties and connections between ARTs and angular moments can be useful as additional information when constructing the iterative algorithms for solving the problems of dynamic refraction tensor tomography.

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.

Considered one of the founding fathers of integral geometry, Luis Santal\'o has contributed to various areas of mathematics. His work has applications in number theory, in the theory of differential equations, in stochastic geometry, in functional analysis, and also in theoretical physics. Between the 1950's and the 1970's, he wrote a series of papers on general relativity and on the attempts at generalizing Einstein's theory to formulate a unified field theory. His main contribution in this subject was to provide a classification theorem for the plethora of tensors that were populating Einstein's generalized theory. This paper, which is the transcript of the conferences delivered by the authors in occasion of the celebration of the 100th anniversary of Luis Santal\'o's birth, revisits his work on theoretical physics., Comment: 10 pages. This paper is the transcript of the conferences delivered by the authors in occasion of the celebration of the 100th anniversary of Luis Santal\'o's birth. v2 minor error corrected; version accepted for publication in EPJH

In this paper we deal with a general type of integral formulas of the visual angle, among them those of Crofton, Hurwitz and Masotti, from the point of view of Integral Geometry. The purpose is twofold: to provide an interpretation of these formulas in terms of integrals of densities with respect to the canonical measure in the space of pairs of lines and to give new simpler proofs of them.

We study the problem of recovering a block-sparse signal from under-sampled observations. The non-zero values of such signals appear in few blocks, and their recovery is often accomplished using a $\ell_{1,2}$ optimization problem. In applications such as DNA micro-arrays, some prior information about the block support, i.e., blocks containing non-zero elements, is available. A typical way to consider the extra information in recovery procedures is to solve a weighted $\ell_{1,2}$ problem. In this paper, we consider a block sparse model, where the block support has intersection with some given subsets of blocks with known probabilities. Our goal in this work is to minimize the number of required linear Gaussian measurements for perfect recovery of the signal by tuning the weights of a weighted $\ell_{1,2}$ problem. For this goal, we apply tools from conic integral geometry and derive closed-form expressions for the optimal weights. We show through precise analysis and simulations that the weighted $\ell_{1,2}$ problem with optimal weights significantly outperforms the regular $\ell_{1,2}$ problem. We further examine the sensitivity of the optimal weights to the mismatch of block probabilities, and conclude stability under small probability deviations., Comment: arXiv admin note: text overlap with arXiv:1704.05397

International audience; The work reported in this paper focuses on the description and characterization of liquid flows in a gaseous environment such as those undergoing an atomization process. The objective here is to describe the complex forms encountered in order to study their dynamics on the basis of new indicators. Completing a work presented at ILASS-Europe 2019 (Paris), the concept of parallel surfaces is convened and extended to all scales of liquid systems. This approach leads to the identification of specific scales. By way of illustrations, this multi-scale description is applied to synthetic systems as well as to the case of the Rayleigh-Plateau instability.

Hypothesis Emerging energy-related technologies deal with multiscale hierarchical structures, intricate surface morphology, non-axisymmetric interfaces, and complex contact lines where wetting is difficult to quantify with classical methods. We hypothesise that a universal description of wetting on multiscale surfaces can be developed by using integral geometry coupled to thermodynamic laws. The proposed approach separates the different hierarchy levels of physical description from the thermodynamic description, allowing for a universal description of wetting on multiscale surfaces. Theory and Simulations The theoretical framework is presented followed by application to limiting cases of wetting on multiscale surfaces. Limiting cases include those considered in the Wenzel, Cassie-Baxter, and wicking state models. Wetting characterisation of multiscale surfaces is explored by conducting simulations of a fluid droplet on a structurally rough surface and a chemically heterogeneous surface. Findings The underlying origin of the classical wetting models is shown to be rooted within the proposed theoretical framework. Integral geometry provides a topological-based wetting metric that is not contingent on any type of wetting state. The wetting metric is demonstrated to account for multiscale features along the common line in a scale consistent way; providing a universal description of wetting for multiscale surfaces.

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.

We study the problem of reconstructing a block-sparse signal from compressively sampled measurements. In certain applications, in addition to the inherent block-sparse structure of the signal, some prior information about the block support, i.e., blocks containing non-zero elements, might be available. Although many block-sparse recovery algorithms have been investigated in the Bayesian framework, it is still unclear how to incorporate the information about the probability of occurrence into regularization-based block-sparse recovery in an optimal sense. In this letter, we bridge between these fields by the aid of a new concept in conic integral geometry. Specifically, we solve a weighted optimization problem when the prior distribution about the block support is available. Moreover, we obtain the unique weights that minimize the expected required number of measurements. Our simulations on both synthetic and real data confirm that these weights considerably decrease the required sample complexity.

Motivated by questions in real enumerative geometry (Borcea et al., in Discrete Comput Geom 35(2):287–300, 2006; Burgisser and Lerario, in J Reine Angew Math, https://doi.org/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 $$X_1, \ldots , X_{d_{k,n}}\subset {\mathbb {R}}\text {P}^n$$, where $$d_{k,n}=(k+1)(n-k)$$, are in random position if each one of them is randomly translated by elements $$g_1, \ldots , g_{{d_{k,n}}}$$ sampled independently from the orthogonal group with the uniform distribution. Denoting by $$\tau _k(X_1, \ldots , X_{d_{k,n}})$$ the average number of k-dimensional projective subspaces (k-flats) which are simultaneously tangent to all the hypersurfaces we prove that $$\begin{aligned} \tau _k(X_1, \ldots , X_{d_{k,n}})={\delta }_{k,n} \cdot \prod _{i=1}^{d_{k,n}}\frac{|\Omega _k(X_i)|}{|\text {Sch}(k,n)|}, \end{aligned}$$where $${\delta }_{k,n}$$ is the expected degree from [6] (the average number of k-flats incident to $$d_{k,n}$$ many random $$(n-k-1)$$-flats), $$|\text {Sch}(k,n)|$$ is the volume of the Special Schubert variety of k-flats meeting a fixed $$(n-k-1)$$-flat (computed in [6]) and $$|\Omega _k(X)|$$ is the volume of the manifold $$\Omega _k(X)\subset \mathbb {G}(k,n)$$ of all k-flats tangent to X. We give a formula for the evaluation of $$|\Omega _k(X)|$$ in terms of some curvature integral of the embedding $$X\hookrightarrow {\mathbb {R}}\text {P}^n$$ and we relate it with the classical notion of intrinsic volumes of a convex set: $$\begin{aligned} \frac{|\Omega _{k}(\partial C)|}{|\text {Sch}(k, n)|}=4V_{n-k-1}(C),\quad k=0, \ldots , n-1. \end{aligned}$$As a consequence we prove the universal upper bound: $$\begin{aligned} \tau _k(X_1, \ldots , X_{d_{k,n}})\le {\delta }_{k, n}\cdot 4^{d_{k,n}}. \end{aligned}$$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 $$X_1, \ldots , X_4$$ such that the intersection $$\Omega _1(X_1)\cap \cdots \cap \Omega _1(X_4)\subset \mathbb {G}(1,3)$$ is transverse and consists of at least m lines. Finally, we present analogous results for semialgebraic hypersurfaces (not necessarily convex) satisfying some nondegeneracy assumptions.

For a fixed k ∈ {1, …, d}, consider arbitrary random vectors X0, …, Xk ∈ ℝd such that the (k + 1)-tuples (UX0, …, UXk) have the same distribution for any rotation U. Let A be any nonsingular d × d matrix. We show that the k-dimensional volume of the convex hull of affinely transformed Xi satisfies \[|{\rm{conv}}(A{X_{\rm{0}}} \ldots ,A{X_k}){\rm{|}}\mathop {\rm{ = }}\limits^{\rm{D}} (|{P_\xi }\varepsilon |/{\kappa _k})|{\rm{conv}}\left( {{X_0}, \ldots ,{X_k}} \right)\] , where ɛ:= {x ∈ ℝd : x┬ (A┬ A)−1x ≤ 1} is an ellipsoid, Pξ denotes the orthogonal projection to a uniformly chosen random k-dimensional linear subspace ξ independent of X0, …, Xk, and κk is the volume of the unit k-dimensional ball. As an application, we derive the following integral geometry formula for ellipsoids: ck,d,p ∫Ad,k |ɛ ∩ E|p+d+1 μd,k(dE) = |ɛ|k+1 ∫Gd,k |PLɛ|pνd,k(dL), where $c_{k,d,p} = \big({\kappa_{d}^{k+1}}/{\kappa_k^{d+1}}\big) ({\kappa_{k(d+p)+k}}/{\kappa_{k(d+p)+d}})$ . Here p > −1 and Ad,k and Gd,k are the affine and the linear Grassmannians equipped with their respective Haar measures. The p = 0 case reduces to an affine version of the integral formula of Furstenberg and Tzkoni (1971).

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.

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.

Finite—difference analogue of the two-dimensional problem of integral geometry with a weight function are studied. The stability estimate for the considered problem are obtained.

In this paper, we consider the problems of integral geometry on families of parabolas. We examined the application of the integral geometry problem to seismic problems. We shall get the inverse formula for the problem of integral geometry on families of parabolas. Numerical experiments demonstrate the performance of the algorithm.

Every three-dimensional object can be computed by a two-dimensional plane with the help of integration. Inspired by Crofton formulas and the Cavalieri's Principle, this work derives a general method for the surface area of a polygonal in three-dimensional space. In fact, the surface area of a three-dimensional object can be subdivided into a finite number of small rectangle. This research represents the area of a rectangle by the number of the intersection point between the rectangle and the line passing through the rectangle in all directions. Next, this research computes the proportionality constant D of integration. Eventually, this research extends the result to a boarder discussion on the application of the obtained result to a smooth surface in R^3. Within the process of integration , the volume of a four-dimensional object in TS2 is calculated. This research jumps out from the conventional representation of the surface area using the one-dimension integral geometry. The reader will realize another technique of representing the surface area with the integration of the number of intersection points in a sub-divided parallelograms. Moreover, not only does the research extent the concept of Cavalieri's principle to a three-dimensional application, but also the solution incites a possible way using the intersection point to explore the volume or the surface area of an object in a higher dimension world.

The algebras of valuations on $S^6$ and $S^7$ invariant under the actions of $\mathrm G_2$ and $\mathrm{Spin}(7)$ are shown to be isomorphic to the algebra of translation-invariant valuations on the tangent space at a point invariant under the action of the isotropy group. This is in analogy with the cases of real and complex space forms, suggesting the possibility that the same phenomenon holds in all Riemannian isotropic spaces. Based on the description of the algebras the full array of kinematic formulas for invariant valuations and curvature measures in $S^6$ and $S^7$ is computed. A key technical point is an extension of the classical theorems of Klain and Schneider on simple valuations.

We prove that the geodesic X-ray transform is injective on scalar functions and (solenoidally) on one-forms on simple Riemannian manifolds $(M,g)$ with $g \in C^{1,1}$. In addition to a proof, we produce a redefinition of simplicity that is compatible with rough geometry. This $C^{1,1}$-regularity is optimal on the H\"older scale. The bulk of the article is devoted to setting up a calculus of differential and curvature operators on the unit sphere bundle atop this non-smooth structure., Comment: 29 pages, 2 figures

The study of the relationships between the integration invariants and the different classes of operators, as well as of functions inside the context of the integral geometry, establishes diverse homologies in the dual space of the functions. This is given in the class of cohomology of the integral operators that give solution to certain class of differential equations in field theory inside a holomorphic context. By this way, using a cohomological theory of appropriate operators that establish equivalences among cycles and cocycles of closed submanifolds, line bundles and contours can be obtained by a cohomology of general integrals, useful in the evaluation and measurement of fields, particles, and physical interactions of diverse nature that occurs in the space-time geometry and phenomena. Some of the results applied through this study are the obtaining of solutions through orbital integrals for the tensor of curvature R μν , of Einstein’s equations, and using the imbedding of cycles in a complex Riemannian manifold through the duality: line bundles with cohomological contours and closed submanifolds with cohomological functional. Concrete results also are obtained in the determination of Cauchy type integral for the reinterpretation of vector fields.

The red blood cell deformation caused by disorders like sickle cell disease can be assessed by observing blood samples under a microscope. This manual process is cumbersome and prone to errors but can be supported by automated techniques that allow red blood cells to be classified according to the shape they present. There are proposals in the literature that use functions based on integral geometry to obtain a description of the cells’ contour before performing classification, reaching 96.16% accuracy with the use of the k-Nearest Neighbor (KNN) classifier. In those approaches, the classification-confusion cases persist mainly in the classes of most significant interest, which are those related to the detection of deformed cells. In this work, we use artificial neural networks-based classifiers, trained with the characteristics obtained from integral geometry-based functions, to classify erythrocytes into normal, sickle, and other deformations classes. Our proposal achieves accuracy of 98.40%. This result is superior to those of previous studies concerning the classes of greatest interest. Also, our approach is computationally more efficient than previous works, making it suitable for supporting medical follow-up diagnosis of sickle cell disease.

Volterra integral equations of the first kind arise naturally in various problems. Many problems in mathematical physics, engineering, and integral geometry are often reduced to first-kind Volterra integral equations. A survey of regularization methods for first kind Volterra equations is given by Lamm. This chapter uses the Legendre wavelet to solve Volterra integral equations of the first kind. The aim is to present an efficient numerical method to find a stable solution of Volterra integral equations of the first kind. The chapter commences with an introduction to Legendre wavelets and an approximation of function using Legendre wavelets. It also introduces a local regularization method and presents the numerical method to find an approximate solution to the given problem. The convergence and error analysis is considered and some numerical examples are presented to verify the accuracy of the numerical method.

Deep neural networks trained on jet images have been successful in classifying different kinds of jets. In this paper, we identify the crucial physics features that could reproduce the classification performance of the convolutional neural network in the top jet vs. QCD jet classification. We design a neural network that considers two types of substructural features: two-point energy correlations, and the IRC unsafe counting variables of a morphological analysis of jet images. The new set of IRC unsafe variables can be described by Minkowski functionals from integral geometry. To integrate these features into a single framework, we reintroduce two-point energy correlations in terms of a graph neural network and provide the other features to the network afterward. The network shows a comparable classification performance to the convolutional neural network. Since both networks are using IRC unsafe features at some level, the results based on simulations are often dependent on the event generator choice. We compare the classification results of Pythia 8 and Herwig 7, and a simple reweighting on the distribution of IRC unsafe features reduces the difference between the results from the two simulations., Comment: 43 pages, 25 figures, published in JHEP

In this chapter, a sparse linear model for joint activity detection and channel estimation in IoT networks is introduced. We present the problem formulation for both the cases of single-antenna and multiple-antenna BSs. A convex relaxation approach based on lp-norm minimization is firstly introduced, followed by a smoothed primal-dual first-order algorithm to solve it. The theoretical analysis of the convex relaxation approach based on the conic integral geometry theory is further presented. Furthermore, an iterative threshold algorithm, namely approximate message passing (AMP), is introduced, followed by the performance analysis based on the state evolution technique. Simulation results are also presented to demonstrate the performance of different algorithms.

Properties of operators of generalized attenuated ray transforms (ART) are investigated. Starting with Radon transform in the mathematical model of computer tomography, attenuated ray transform in emission tomography and longitudinal ray transform in tensor tomography, we come to the operators of ART of order k over symmetric m-tensor fields, depending on spatial and temporal variables. The operators of ART of order k over tensor fields contain complex-valued absorption, different weights, and depend on time. Connections between ART of various orders are established by means of application of linear part of transport equation. This connections lead to the inhomogeneous k-th order differential equations for the ART of order k over symmetric m-tensor field. The right hand parts of such equations are m-homogeneous polynomials containing the components of the tensor field as the coefficients. The polynomial variables are the components \(\xi ^j\) of direction vector \(\xi \) participating in differential part of transport equation. Uniqueness theorems of boundary-value and initial boundary-value problems for the obtained equations are proved, with significant application of Gauss-Ostrogradsky theorem. The connections of specified operators with integral geometry of tensor fields, emission tomography, photometry and wave optics allow to treat the problem of inversion of the ART of order k as the inverse problem of determining the right hand part of certain differential equation.

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. Keywords: Cauchy--Crofton formula; center of curvature; continuity; infinitesimals; integral geometry; limite; standard part; de Prony; Poisson, Comment: 21 pages, Real Analysis Exchange

By studying the space of geodesics in $ADS_3/CFT_2$ and quantizing the geodesic motion, we relate scattering data to boundary entanglement of the CFT vacuum. The basic idea is to use a family of plane waves parametrized by coordinates of the space of geodesics i.e. kinematic space. This idea enables a simple calculation of the Berry curvature living on kinematic space. As a result we recover the Crofton form with a coefficient depending on the scattering energy. In arriving at these results the space of horocycles is used. We show that this new space used in concert with kinematic space incorporates naturally the gauge degrees of freedom responsible for an analogue of Berry's Phase. Horocycles also give a new geometric look to the strong subadditivity relation in terms of lambda lengths giving rise to shear coordinates of geodesic quadrangles. A generalization for geodesic polygons then reveals an interesting connection with $A_{n}$ cluster algebras. Here the cluster variables are the lambda lengths related to the regularized entropies of the boundary via the Ryu-Takayanagi relation. An elaboration of this idea indicates that cluster algebras might provide a natural algebraic means for encoding the gauge invariant entanglement patterns of certain boundary entangled states in the geometry of bulk geodesics. Finally using the language of integral geometry we show how certain propagators connected to the bulk, boundary and kinematic spaces are related to data of elementary scattering problems. We also present some hints how these ideas might be generalized for more general holographic scenarios.

The integral geometry methods are the techniques could be the more naturally applied to study of the characterization of the moduli stacks and solution classes (represented cohomologically) obtained under the study of the kernels of the differential operators of the corresponding field theory equations to the space-time. Then through a functorial process a classification of differential operators is obtained through of the co-cycles spaces that are generalized Verma modules to the space-time, characterizing the solutions of the field equations. This extension can be given by a global Langlands correspondence between the Hecke sheaves category on an adequate moduli stack and the holomorphic bundles category with a special connection (Deligne connection). Using the classification theorem given by geometrical Langlands correspondences are given various examples on the information that the geometrical invariants and dualities give through moduli problems and Lie groups acting.

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., 9 pages, 2 figures

We consider weighted Radon transforms $$R_W$$ along hyperplanes in $$\mathbb {R}^3$$ with strictly positive weights W. We construct an example of such a transform with non-trivial kernel $$\mathrm {Ker}R_W$$ 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 $$\mathbb {R}^d, \, d\ge 3$$ .

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.

A differential-difference analogue of a two-dimensional problem of integral geometry with a weight function is studied. A stability estimate for the problem considered is obtained.

Let $\varphi_ {n, 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 $\mathbb {R}^ d $ with nonempty interior, where $ d\ge2 $. It is shown that the distribution of the random variable $\lambda_d (K)\,\frac {n^ 3}{3!}\,(\pi-\varphi_ {n, K})^{d-1} $, where $\lambda_d (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 $n\to\infty$. By using the Steiner symmetrization, it is also shown that $\lambda_d (K)$, 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 $\mathbf {R}^d$. Finally, the asymptotics of $\lambda_d(B^{(d)})$ for large $d$ is determined.

In this article generalized attenuated ray transforms (ART) and integral angular moments are investigated. Starting from the Radon transform, the attenuated ray transform and the longitudinal ray transform, we derive the concept of ART-operators of order $k$ over functions defined on the phase space and depending on time. The ART-operators are generalized for complex-valued absorption coefficient as well as weight functions of polynomial and exponential type. Connections between ART operators of various orders are established by means of the application of the linear part of a transport equation. These connections lead to inhomogeneous differential equations of order $(k+1)$ for the ART of order $k$. Uniqueness theorems for the corresponding boundary-value and initial boundary-value problems are proved. Properties of integral angular moments of order $p$ are considered and connections between the moments of different orders are deduced. A close connection of the considered operators with mathematical models for tomography, physical optics and integral geometry allows to treat the inversion of ART of order $k$ as an inverse problem of determining the right-hand side of a corresponding differential equation., 20 pages, 1 figure

The aim of this article is to investigate the uniqueness of solution of an inverse source problem for a generalized kinetic equation on a Riemannian manifold. The problem is related with an integral geometry problem in semi-geodesic coordinates. We prove the uniqueness in a convex domain by the help of Riemannian coordinates.