An accurate analytical model for squirt flow in anisotropic porous rocks - Part 1: Classical geometry

Seismic wave propagation in porous rocks that are saturated with a liquid exhibits significant dispersion and attenuation due to fluid flow at the pore scale, so-called squirt flow. This phenomenon takes place in compliant flat pores such as microcracks and grain contacts that are connected to stiffer isometric pores. Accurate quantitative description is crucial for inverting rock and fluid properties from seismic attributes such as attenuation. Up to now, many analytical models for squirt flow were proposed based on simplified geometries of the pore space. These models were either not compared with a numerical solution or showed poor accuracy. We present a new analytical model for squirt flow which is validated against a three-dimensional numerical solution for a simple pore geometry that has been classically used to explain squirt flow; that is why we refer to it as classical geometry. The pore space is represented by a flat cylindrical (penny-shaped) pore whose curved edge is fully connected to a toroidal (stiff) pore. Compared with correct numerical solutions, our analytical model provides very accurate predictions for the attenuation and dispersion across the whole frequency range. This includes correct low- and high-frequency limits of the stiffness modulus, the characteristic frequency, and the shape of the dispersion and attenuation curves. In a companion paper (Part 2), we extend our analytical model to more complex pore geometries. We provide as supplementary material Matlab and symbolic Maple routines to reproduce our main results.

Formal method for planning the machining of ship machine parts

При разработке технологии изготовления судовых машин, узлов и деталей используются типовые процессы-аналоги, либо описание опыта исполнителей. Известные зарубежные подходы также используют вариативный подход, основанный на типовых решениях, либо генеративный, который предусматривает как формализацию процесса проектирования, так и использование искусственного интеллекта. Как показано в статье одной из основных проблем формализации проектирования технологических процессов является невозможность однозначного представления геометрической конфигурации реальных деталей средствами современной классической геометрии. Для решения этой проблемы предлагается использовать геометрию неидеальных объектов, базисом которой является шестимерное пространство, объединяющее три линейных и три угловых измерения. Использование указанной геометрии позволило определить формальные связи между конструкцией детали и технологическим процессом её изготовления. В статье излагается разработанный авторами метод формального проектирования процессов обработки деталей на станках. Выявленные закономерности порождения геометрических конфигураций позволили разработать алгоритмы генерирования множества методов формообразования элементарных поверхностей. Также определены условия обеспечения сходимости алгоритма формирования комплектов технологических баз и последовательности их выполнения. Изложение материалов подтверждено рассмотрением процесса проектирования технологии обработки на примере реальной детали. When developing a technology for the manufacture of ship machines, units and parts, group processes are used, or a description of the experience of the performers. Well-known foreign approaches also use a variable approach based on standard solutions, or a generative one, which provides for both the formalization of the design process and the use of artificial intelligence. Main problem of formalizing the design of technological processes is the impossibility of an unambiguous representation of the geometric configuration of real parts by means of modern classical geometry. As a solution to this problem, it is proposed to use the geometry of non-ideal objects, the basis of which is a six-dimensional space that combines three linear and three angular dimensions. This geometry made it possible to determine the formal relationships between the design of the part and the technological process of its manufacture. Article describes the method developed by the authors for the formal design of the processing of parts on machine tools. Revealed patterns generation configurations of geometric made it possible to develop algorithms for formation a variety of methods for shaping elementary surfaces. Conditions for ensuring the convergence of algorithm for formation of sets of technological bases and the sequence of their implementation are determined. Presentation materials is confirmed by considering generating of technology using the example of a real part.

Space from entanglement: An information-geometric perspective

In this paper, we study the construction of classical geometry from the quantum entanglement structure by using information geometry. In the information geometry of classical spacetime, the Fisher information metric is related to a blurred metric of a classical physical space. We first show that a local information metric can be obtained from the entanglement contour in a local subregion. This local information metric measures the fine structure of entanglement spectra inside the subregion, which suggests a quantum origin of the information-geometric blurred space. We study both the continuous and the classical limits of the quantum-originated blurred space by using the techniques from the statistical sampling algorithms, the sampling theory of spacetime and the projective limit. A scheme for going from a blurred space with quantum features to a classical geometry is also explored.

Spinors, Twistors and Classical Geometry

The paper studies explicitly the Hitchin system restricted to the Higgs fields on a fixed very stable rank 2 bundle in genus 2 and 3. The associated families of quadrics relate to both the geometry of Penrose's twistor spaces and several classical results.

Triangle conics and cubics

This is a paper about triangle cubics and conics in classical geometry with elements of projective geometry. In recent years, N.J. Wildberger has actively dealt with this topic using an algebraic perspective. Triangle conics were also studied in detail by H.M. Cundy and C.F. Parry recently. The main task of the article was to develop an algorithm for creating curves, which pass through triangle centers. During the research, it was noticed that some different triangle centers in distinct triangles coincide. The simplest example: an incenter in a base triangle is an orthocenter in an excentral triangle. This was the key for creating an algorithm. Indeed, we can match points belonging to one curve (base curve) with other points of another triangle. Therefore, we get a new intersting geometrical object. During the research were derived number of new triangle conics and cubics, were considered their properties in Euclidian space. In addition, was discussed corollaries of the obtained theorems in projective geometry, what proves that all of the descovered results could be transfered to the projeticve plane.

On Some Remarkable Points and Line Segments in Triangles

Let ra, rb, rc be the radii, and OA, OB, OC the centers of tangent circles at the vertices to the circumcircle of a triangle ABC and to the opposite sides. In the paper [Andrica D., Marinescu D.S. New interpolation inequalities to Euler's R≥2 // Forum Geometricorum. 2017. Vol. 17], the authors proved that 4/R £ 1/ra + 1/rb +1/rc £2/r. In the paper [Isaev I., Maltsev Yu., Monastyreva A. On some relations in geometry of a triangle // Journal of Classical Geometry. 2018. Vol. 4], it is given the following generalization of these inequalities: 1/ra + 1/rb +1/rc=2/R+1/r. In that paper, we find the area of the triangle OAOBOC (see Theorem 1). We prove some relations for the numbers R-ra, R-rb, R-rc, where R is the circumradius of a triangle ABC. Namely, we find the expressions 1/R-ra+1/R-rb + 1/R-rc и a/R-ra+b/R-rb + c/R-rc by means by the parameters p, R and r (see Theorem 2). We estimate these values (see Theorem 3). Finally, using the results of paper [Maltsev Yu., Monastyreva A. On some relations for a triangle // International Journal of Geometry. 2019. Vol. 8 (1)] and representing the expression of (1-cos(αβ))(1-cos(β-γ))(1-cos(α-γ)) by means of p, R, r, we prove new proof of the fundamental triangle inequality (see Corollary 2).

Real classical geometry with arbitrary deficit parameter(s) $$\alpha (_{I})$$ in deformed Jackiw–Teitelboim gravity

An interesting deformation of Jackiw–Teitelboim (JT) gravity has been proposed by Witten by adding a potential term $$U(\phi )$$ U ( ϕ ) as a self-coupling of the scalar dilaton field. During calculating the path integral over fields, a constraint comes from integration over $$\phi $$ ϕ as $$R(x)+2=2\alpha \delta (\vec {x}-\vec {x}')$$ R ( x ) + 2 = 2 α δ ( x → - x → ′ ) . The resulting Euclidean metric suffered from a conical singularity at $$\vec {x}=\vec {x}'$$ x → = x → ′ . A possible geometry is modeled locally in polar coordinates $$(r,\varphi )$$ ( r , φ ) by $$\mathrm{d}s^2=\mathrm{d}r^2+r^2\mathrm{d}\varphi ^2,\varphi \cong \varphi +2\pi -\alpha $$ d s 2 = d r 2 + r 2 d φ 2 , φ ≅ φ + 2 π - α . In this letter we show that there exists another family of ”exact” geometries for arbitrary values of the $$\alpha $$ α . A pair of exact solutions are found for the case of $$\alpha =0$$ α = 0 . One represents the static patch of the AdS and the other one is the non-static patch of the AdS metric. These solutions were used to construct the Green function for the inhomogeneous model with $$\alpha \ne 0$$ α ≠ 0 . We address a type of phase transition between different patches of the AdS in theory because of the discontinuity in the first derivative of the metric at $$x=x'$$ x = x ′ . We extended the study to the exact space of metrics satisfying the constraint $$R(x)+2=2\sum _{i=1}^{k}\alpha _i\delta ^{(2)}(x-x'_i)$$ R ( x ) + 2 = 2 ∑ i = 1 k α i δ ( 2 ) ( x - x i ′ ) as a modulus diffeomorphisms for an arbitrary set of deficit parameters $$(\alpha _1,\alpha _2,\ldots ,\alpha _k)$$ ( α 1 , α 2 , … , α k ) . The space is the moduli space of Riemann surfaces of genus g with k conical singularities located at $$x'_k$$ x k ′ , denoted by $$\mathcal {M}_{g,k}$$ M g , k .

«GeOrigaMetry» An Approach to the Accessibility of Geometry for Blind People

Making geometry accessible for blind people, apart from the formal aspects, can pose some difficulties, especially in terms of accessibility to figures. To deal with this problem this article focuses on paper folding where both Euclidean and origami axiomatic systems are used simultaneously. In the first case, with a ruler and compass, we can solve quadratic problems in a plane. In addition, the axioms of origami allow us to address unanswered questions with classical geometry methods, which involve cubic equations, such as the trisection of an angle. An experiment with INJA (National Institute for Blind Youth, Paris) students and other blind people will take place so that we can see the possibilities offered by this method, which brings a ludic, but rigorous approach to these complex and frequently off-putting issues. We believe that this dynamic pedagogical approach can increase interest and motivation, encourage tactile stimulation and facilitate the development of specific structures of brain plasticity. The article is written in a linear way, accessible to blind people; figures are provided to facilitate understanding for "visually impaired" people, who are not used to following a geometric concept without pictures. Finally, it should be noted that the method is particularly suitable in an inclusive education context.