Integral Geometry Methods Research Articles

Complete Barrett-Crane model and its causal structure

The causal structure is a quintessential element of continuum spacetime physics and needs to be properly encoded in a theory of Lorentzian quantum gravity. Established spin foam (and tensorial group field theory (TGFT)) models mostly work with relatively special classes of Lorentzian triangulations (e.g. built from spacelike tetrahedra only), obscuring the explicit implementation of the local causal structure at the microscopic level. We overcome this limitation and construct a full-fledged model for Lorentzian quantum geometry the building blocks of which include spacelike, lightlike and timelike tetrahedra. We realize this within the context of the Barrett-Crane TGFT model. Following an explicit characterization of the amplitudes via methods of integral geometry, and the ensuing clear identification of local causal structure, we analyze the model's amplitudes with respect to its (space)time-orientation properties and provide also a more detailed comparison with the framework of causal dynamical triangulations (CDT).

Analysis of Tomographic Images of the Soil Pore Space Structure by Integral Geometry Methods

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.

Quantitative Analysis of Samples of Natural Hydrocarbon Reservoirs by the Methods of Integral Geometry and Topology

Quantitative Analysis of Samples of Natural Hydrocarbon Reservoirs by the Methods of Integral Geometry and Topology

Investigation of Beltrami fields by methods of integral geometry

Investigation of Beltrami fields by methods of integral geometry

Quantitative Analysis of Pore Space Structure in Dry and Wet Soil by Integral Geometry Methods

We present a methodology for a numerical analysis of three-dimensional tomographic images in this paper. The methodology is based on integral geometry, topology, and morphological analysis methods. It involves calculating cumulative and non-cumulative pore size distributions of Minkowski functionals and Betti numbers. We investigated 13 samples in dry and wet (saturated beyond the field capacity) conditions within different horizons of the Phaeozem albic. For samples of the arable horizon, an increase in the Euler characteristic was observed in the process of wetting. For samples from the A2, AB and B2 horizons, the Euler-Poincare characteristic decreased during wetting. It has been proven that both Betti numbers (number of isolated pores and number of “tunnels”) decrease with swelling of the AB and B2 horizons at a depth of 20–90 cm. For samples from the arable horizon, another dependence was observed: A Betti number of zero increased first but decreased during wetting. Based on the change in topological characteristics, two methods of changing the topology of the void space of the soil were demonstrated. The above-described quantitative changes of proposed parameters of pore space tomographic images prove the possibility and progressiveness of their usage for the pore space transformation estimate.

Cycles Cohomology and Geometrical Correspondences of Derived Categories to Field Equations

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.

РАЗВИТИЕ У СТУДЕНТОВ ПРИКЛАДНОГО МАТЕМАТИЧЕСКОГО МЫШЛЕНИЯ ПРИ ОБУЧЕНИИ ОБРАТНЫМ И НЕКОРРЕКТНЫМ ЗАДАЧАМ

In the article the author draws the reader’s attention to the fact that, while mastering the process of learning the theory and methodology of inverse and ill-posed problems, students not only form the fundamental knowledge in the field of inverse and ill-posed problems, applied and computational mathematics, mathematical modeling of processes and phenomena, but also develop one of the most important component of mathematical ability creative and applied mathematical thinking.It is emphasized that the search for solutions to inverse and ill-posed problems, students acquire profound knowledge in such scientific fields as seismology, gravimetry, magnetometry, Geophysics, astrophysics, imaging, electrodynamics, atmospheric optics, quantum scattering theory and other scientific fields. When teaching inverse and ill-posed problems, students also learn the mathematical methods, which are not included in the content of traditional mathematics applied and computational mathematics, and can only be purchased in the teaching of special courses. Among them, spectral analysis, the method of Volterra operator equations, Sobolev method, method of scales of Banach spaces of analytic functions, the method of integral geometry, the method of tensor analysis, methods of computational mathematics and other mathematical methods.

Erythrocyte shape classification using integral-geometry-based methods.

Erythrocyte shape deformations are related to different important illnesses. In this paper, we focus on one of the most important: the Sickle cell disease. This disease causes the hardening or polymerization of the hemoglobin that contains the erythrocytes. The study of this process using digital images of peripheral blood smears can offer useful results in the clinical diagnosis of these illnesses. In particular, it would be very valuable to find a rapid and reproducible automatic classification method to quantify the number of deformed cells and so gauge the severity of the illness. In this paper, we show the good results obtained in the automatic classification of erythrocytes in normal cells, sickle cells, and cells with other deformations, when we use a set of functions based on integral-geometry methods, an active contour-based segmentation method, and a k-NN classification algorithm. Blood specimens were obtained from patients with Sickle cell disease. Seventeen peripheral blood smears were obtained for the study, and 45 images of different fields were obtained. A specialist selected the cells to use, determining those cells which were normal, elongated, and with other deformations present in the images. A process of automatic classification, with cross-validation of errors with the proposed descriptors and with other two functions used in previous studies, was realized.

Integral Geometry Methods on Deformed Categories in Field Theory II

The integral geometry methods are applied on deformed categories to obtain correspondences in the geometrical Langlands program and construct the due equivalences between geometrical objects of the moduli stacks and algebraic objects of the corresponding categories and their L_(G-opers) characterizing the solution classes to field theory equations in the belonging cohomological context such as H^0 (g[[z] ],V_critical )=C[Op_LG (D^X)] which is natural in the framework of the integral transforms to the generalizing of the Zuckerman functors that will be useful to the obtaining of the different factors of the universal functor of derived sheaves of Harish-Chandra to the Langlands geometrical program in mirror symmetry. The cosmological problem that exists is to reduce the number of field equations that are resoluble under the same gauge field (Verma modules) and to extend the gauge solutions to other fields using the topological groups symmetries that define their interactions. This extension can be given by a global Langlands correspondence between the Hecke sheaves category H_(G^^ ) ∞ on an adequate moduli stack and the holomorphic L_(G-) bundles category with a special connection (Deligne connection). The corresponding 〖D 〗_-modules may be viewed as sheaves of conformal blocks (or co-invariants) (images under a generalized version of the Penrose transform) naturally arising in the framework of conformal field theory.

Reconstruction of measures in the space of planes

In the present paper we consider the problem of inversion of the integral equation which naturally arises in the theory of measures generated by flag densities. We solve this equation by integral geometry methods and propose an inversion formula for the density of the measure in the space of planes in terms of its flag density.

SOME NEW BONNESEN-STYLE INEQUALITIES

By evaluating the containment measure of one domain to contain another, we will derive some new Bonnesen-type inequalities (Theorem 2) via the method of integral geometry. We obtain Ren's sufficient condition for one domain to contain another domain (Theorem 4). We also obtain some new geometric inequalities. Finally we give a simplified proof of the Bottema's result.

Solution of one integral equation on a sphere by methods of integral geometry

Solution of one integral equation on a sphere by methods of integral geometry

Theoretical derivation of heliostat tracking errors distribution

The tracking error probability distribution is derived on a pure theoretical basis. Methods of integral geometry and geometrical probabilities are used to this purpose. The distribution performs reasonably well when compared with measurements from a small database. The performance of the theoretical distribution is compared with that of other (empirical) probability distributions. The practical relevance of using the present approach is also explained.

Integral Geometry on Grassmann Manifolds and Calculus of Invariant Differential Operators

In this paper, we mainly deal with two problems in integral geometry, the range characterizations and construction of inversion formulas for Radon transforms on higher rank Grassmann manifolds. The results will be described explicitly in terms of invariant differential operators. We will also study the harmonic analysis on Grassmann manifolds, using the method of integral geometry. In particular, we will give eigenvalue formulas and radial part formulas for invariant differential operators.

The second virial coefficient for a strongly Brownian suspension of arbitrary convex bodies with model interaction potentials

The methods of integral geometry are used to determine the second virial coefficient of the expression for the pressure in terms of the number density for a suspension of arbitrary convex bodies with model interaction potentials. One special case considered assumes that the interaction energy is a function of only the minimum distance between the bodies, and others assume that the interaction energy is a function of the volume, surface area, and integral of the mean curvature of the intersection of the two bodies

Liberation 1, 2, 3: Theoretical analysis of the effect of space dimension on mineral liberation by size reduction

A derivation is made, for space dimensionality of 1, 2 and 3, of mineral liberation by size reduction. The mathematical solution to the problem of mineral liberation prediction is derived through the use of mathematical morphology and integral geometry methods. The solution is worked out for real particles produced by fragmentation of ores, and for ore textures which correspond to Boolean schemes with primary Poisson polyhedra, and Poisson polyhedral textures. These textures have characteristics which are often encountered in ores. For a sparse phase it is demonstrated that the degrees of liberation in 1, 2 and 3 dimensions are related by a simple relationship: Ld+i(D) = (Ld(D))2i where Ld (D) is the degree of liberation at screen size D measured in d dimensions. The degrees of liberation follow an exponential decay: ld(D)= exp[−0.693(DD50,d)] withD50,d+i=D50,d/2i

Prediction of particle composition distribution after fragmentation of heterogeneous materials

A review of problems encountered in developing prediction models for composition distribution for particles obtained by the breakage of multiphase materials is presented. The needs to characterize texture by image analysis, to describe particle production by similar methods, to relate particle production and texture, to use integral geometry methods and stochastic geometrical processes to solve the complex probability equations in the three dimensional space, are stressed. A presentation is made of models developed by other researchers, which illustrates some of their deficiencies. A complete presentation is made of a model which incorporates the required elements. A special case of the model, combining a description of texture based on a Boolean model with primary Poisson polyhedra grains, of particle production calibrated on screen fractions extracted from ground ores, independence of texture and breakage, is presented. Applications of the model to data presented by other researchers give evidence of its value especially in predicting particle composition distribution at sizes similar or coarser than grain sizes.

Methods of integral geometry and recovering a function with compact support from its projections in unknown directions

A method for recovering a function with compact support on a plane from its Radon transform is suggested. In particular, from the given projections of a function in unknown directions, these directions may be recovered (up to a rotation or a reflection), if there are ≥7 of them. This result may be applied in tomography.

Integral geometry and separation of continuous and discrete spectra in regular representation of PSL (2, R)

The separation of continuous and discrete spectra in regular representation of PSL(2, R is constructed with methods of integral geometry.

Estimation of fish density using the line intercept method.

The method of integral geometry is developed to estimate fish density. We can estimate two dimensional measure using measurement of one dimensional quantity. In this papar, we apply this method called line intercept method. The conditions using this method are as follows: searching individuals by observer, numrous readily visible individuals, large individuals and interception of object by line. In this research, it was recognized that the use of an echo sounder as a searching observer is good for the line intercept method. There are some problems in estimating biomass, but the method is a good way to estimate number of fish schools. We think this method may be used widely in the future, though it has not yet been put sufficiently into practical use in the field of fisheries.