Geometric Analogue Research Articles

Thick embeddings of graphs into symmetric spaces via coarse geometry

We prove estimates for the optimal volume of thick embeddings of finite graphs into symmetric spaces, generalising results of Kolmogorov-Barzdin and Gromov-Guth for embeddings into Euclidean spaces. We distinguish two very different behaviours depending on the rank of the non-compact factor. For rank at least 2, we construct thick embeddings of N N -vertex graphs with volume C N ln ⁡ ( 1 + N ) CN\ln (1+N) and prove that this is optimal. For rank at most 1 1 we prove lower bounds of the form c N a cN^a for some (explicit) a > 1 a>1 which depends on the dimension of the Euclidean factor and the conformal dimension of the boundary of the non-compact factor. The main tool is a coarse geometric analogue of a thick embedding called a coarse wiring, with the key property that the minimal volume of a thick embedding is comparable to the “minimal volume” of a coarse wiring for symmetric spaces of dimension at least 3 3 . In the appendix it is proved that for each k ≥ 3 k\geq 3 every bounded degree graph admits a coarse wiring into R k \mathbb {R}^k with volume at most C N 1 + 1 k − 1 CN^{1+\frac {1}{k-1}} . As a corollary, the same upper bound holds for real hyperbolic space of dimension k + 1 k+1 and in both cases this result is optimal.

A simple model of a gravitational lens from geometric optics

We propose a simple geometric optics analog of a gravitational lens with a refractive index equal to one at large distances and scaling like n(r)2=1+C2/r2, where C is a constant. We obtain the equation for ray trajectories from Fermat's principle of least time and the Euler equation. Our model yields a very simple ray trajectory equation. The optical rays bending, reflecting, and looping around the lens are all described by a single trigonometric function in polar coordinates. Optical rays experiencing fatal attraction are described by a hyperbolic function. We use our model to illustrate the formation of Einstein rings and multiple images.

The bismar scale and elastic collisions: a geometrical analogy

Throughout history, scales have served as instrumental tools for quantifying the weight of objects, relying on a comparative assessment against a specified reference weight. Scales featuring uneven arms, such as the bismar scale, have proven particularly adept at gauging masses within a specific range relative to a predetermined reference mass. On the other hand, the kinematics of elastic collisions hinge on the inertial masses of the colliding entities. By observing the aftermath of a collision between a known reference mass and an object of unknown mass, one can deduce the latter’s mass. In this contribution, we highlight a fascinating and clear analogy between these two methodologies. We do so by adapting a geometric approach, initially applicable to the bismar scale, to both non-relativistic and relativistic elastic collisions, encompassing phenomena such as Compton scattering.

Short Remark on (p1,p2,p3)-Complex Numbers

Movements on surfaces of centered Euclidean spheres and changes between those with different radii mean complex multiplication in R3. Here, the Euclidean norm, which generates the spheres, is replaced with an inhomogeneous functional and a product is introduced in a geometric analogy. Because a change in the radius now leads to a change in the shape of the sphere, a three-dimensional dynamic complex structure is created. Statements about invariant probability densities, generalized uniform distributions on generalized spheres, geometric measure representations, and dynamic ball numbers associated with this structure are also presented.

Counting conjectures and e-local structures in finite reductive groups

We prove new results in generalised Harish-Chandra theory providing a description of the so-called Brauer–Lusztig blocks in terms of information encoded in the ℓ-adic cohomology of Deligne–Lusztig varieties. Then, we propose new conjectures for finite reductive groups by considering geometric analogues of the ℓ-local structures that lie at the heart of the local-global counting conjectures. For large primes, our conjectures coincide with the counting conjectures thanks to a connection established by Broué, Fong and Srinivasan between ℓ-structures and their geometric counterpart. Finally, using the description of Brauer–Lusztig blocks mentioned above, we reduce our conjectures to the verification of Clifford theoretic properties expected from certain parametrisations of generalised Harish-Chandra series.

Random walks on mapping class groups

This survey is concerned with random walks on mapping class groups. We illustrate how the actions of mapping class groups on Teichmüller spaces or curve complexes reveal the nature of random walks and vice versa. Our emphasis is on the analogues of classical theorems such as laws of large numbers and central limit theorems and the properties of harmonic measures under optimal moment conditions. We also explain the geometric analogy between Gromov hyperbolic spaces and Teichmüller spaces that has been used to copy the properties of random walks from one to the other.

Dimensional Methods Used in the Additive Manufacturing Process.

It is a well-known fact that in the field of modern manufacturing processes, additive manufacturing (AM) offers unexpected opportunities for creativity and rapid development. Compared with classical manufacturing technologies, AM offers the advantages of reducing weight and improving performance and offers excellent design capabilities for prototyping and rapid sample manufacture. To achieve its full potential regarding cost, durability, material consumption, and rigidity, as well as maintaining competitiveness, there are several research directions that have not been explored. One less frequently explored direction is the involvement of dimensional methods in obtaining an optimal and competitive final product. In this review, we intend to discuss the ways in which dimensional methods, such as geometric analogy, similarity theory, and dimensional analysis, are involved in addressing the problems of AM. To the best of our knowledge, it appears that this field of engineering has not fully maximized the advantages of these dimensional methods to date. In this review, we survey mainly polymer-based AM technology. We focus on the design and optimization of highly competitive products obtained using AM and also on the optimization of layer deposition, including their orientation and filling characteristics. With this contribution to the literature, we hope to suggest a fruitful direction for specialists involved in AM to explore the possibilities of modern dimensional analysis.

Geometric analogy between quantum dynamics and curved space through quantum hydrodynamics

In general relativity, the dynamics of objects is governed by the curvature of spacetime, which is caused by the presence of matter and energy. In contrast, in quantum mechanics, the dynamics is governed by the wavefunction, which completely describes the behavior of the particles. There is an ongoing effort to explore analogs of space and spacetime curvature in the context of quantum mechanics. Such analogies may reveal a deeper structure of quantum reality and its possible relations with Einstein’s theory of gravity. In this note, by coupling the non-relativistic Schrödinger equation with the heat equation and using the hydrodynamical formulation of quantum mechanics, we find that the dynamics of the particle is fully characterized by the normalized curvature of the wavefunction’s amplitude. Such a curvature obtains an analogy to the Ricci curvature of curved space in a Riemannian manifold. The proposed geometric correspondence provides a new pathway to explore quantum dynamics through the lens of differential geometry, the language of general relativity.

Legendrian links and déjà vu moments

A Legendrian link is called a déjà vu link if its components can be connected by a positive Legendrian isotopy but this isotopy cannot be embedded. This is the contact geometric analogue of a pair of events in a spacetime such that there are déjà vu moments on every future-directed timelike path between them. We construct déjà vu links in several geometrically relevant situations and discuss their basic properties.

Spectral properties of size-invariant shape transformation.

Size-invariant shape transformation is a technique of changing the shape of a domain while preserving its sizes under the Lebesgue measure. In quantum-confined systems, this transformation leads to so-called quantum shape effects in the physical properties of confined particles associated with the Dirichlet spectrum of the confining medium. Here we show that the geometric couplings between levels generated by the size-invariant shape transformations cause nonuniform scaling in the eigenspectra. In particular, the nonuniform level scaling, in the direction of increasing quantum shape effect, is characterized by two distinct spectral features: lowering of the first eigenvalue (ground-state reduction) and changing of the spectral gaps (energy level splitting or degeneracy formation depending on the symmetries). We explain the ground-state reduction by the increase in local breadth (i.e., parts of the domain becoming less confined) that is associated with the sphericity of these local portions of the domain. We accurately quantify the sphericity using two different measures: the radius of the inscribed n-sphere and the Hausdorff distance. Due to Rayleigh-Faber-Krahn inequality, the greater the sphericity, the lower the first eigenvalue. Then level splitting or degeneracy, depending on the symmetries of the initial configuration, becomes a direct consequence of size invariance dictating the eigenvalues to have the same asymptotic behavior due to Weyl law. Such level splittings may be interpreted as geometric analogs of Stark and Zeeman effects. Furthermore, we find that the ground-state reduction causes a quantum thermal avalanche which is the underlying reason for the peculiar effect of spontaneous transitions to lower entropy states in systems exhibiting the quantum shape effect. Unusual spectral characteristics of size-preserving transformations can assist in designing confinement geometries that could lead to classically inconceivable quantum thermal machines.

Computerized Process-Oriented Dynamic Testing of Children’s Ability to Reason by Analogy Using Log Data

Abstract: The study investigated the value of process data obtained from a group-administered computerized dynamic test of analogical reasoning, consisting of a pretest-training-posttest design. We sought to evaluate the effects of training on processes and performance, and the relationships between process measures and performance on the dynamic test. Participants were N = 86 primary school children ( Mage = 8.11 years, SD = 0.63). The test consisted of constructed-response geometrical analogy items, requiring several actions to construct an answer. Process data enabled scoring of the total time, the time taken for initial planning of the task, the time taken for checking the answer that was provided, and variation in solving time. Training led to improved performance compared to repeated practice, but this improvement was not reflected in task-solving processes. Almost all process measures were related to performance, but the effects of training or repeated practice on this relationship differed widely between measures. In conclusion, the findings seemed to indicate that investigating process indicators within computerized dynamic testing of analogical reasoning ability provided information about children’s learning processes, but that not all processes were affected in the same way by training.

Differential geometric global smoothings of simple normal crossing complex surfaces with trivial canonical bundle

AbstractLetXXbe a simple normal crossing (SNC) compact complex surface with trivial canonical bundle which includes triple intersections. We prove that ifXXisdd-semistable, then there exists a family of smoothings in a differential geometric sense. This can be interpreted as a differential geometric analogue of the smoothability results due to Friedman, Kawamata-Namikawa, Felten-Filip-Ruddat, Chan-Leung-Ma, and others in algebraic geometry. The proof is based on an explicit construction of local smoothings around the singular locus ofXX, and the first author’s existence result of holomorphic volume forms on global smoothings ofXX. In particular, these volume forms are given as solutions of a nonlinear elliptic partial differential equation. As an application, we provide several examples ofdd-semistable SNC complex surfaces with trivial canonical bundle including double curves, which are smoothable to complex tori, primary Kodaira surfaces, andK3K3surfaces. We also provide several examples of such complex surfaces including triple points, which are smoothable toK3K3surfaces.

Структура текстовой модели стихиры «Приидите, ублажим Иосифа»

The relevance of the topic is stipulated by the intensive development of the theory of textual-musical forms in modern Russian musicology. The novelty of the research is determined by the method of analysis of archaic text models by applying the principle of symmetry. In the article the unique composition of the verbal text in the masterpieces of the early Russian monophonic chants, the origins of which go back to the Byzantine tradition of the 7th and 8th centuries AD, is examined for the first time. The repetitive identical syntactical constructions — the elements unconnected to the plot — are combined into a complex polytextual structure. The stichera Come, Let Us Thank Joseph is based on the interaction of plot and non-plot elements. The rhythm of the form in its line composition of end-to-end development is created by the periodic symmetry of threefold repetition. The group of small texts representing the direct speech of the characters: Joseph, Mary, and Simeon the God-Receiver is embedded in the “large text”: the call to prayer and the prayer of believers. The author of the article describes it with a mathematical formula, and offers a geometric analogue in the form of a gnomon — a figure suggested by Greek mathematician Heron of Alexandria in the second half of the 1st century AD. The archaic text code of the stichera has traversed from century to century and acquired a modern sound due to the expressive means corresponding to the musical stylistic system of the epoch. The polished text models provide canonical ritual chants with centuries-old functioning in the form of the variant multitude of musical compositions.

Modern Dimensional Analysis Involved in Polymers Additive Manufacturing Optimization

The paper aims to use Modern Dimensional Analysis (MDA) to study the polymers additive manufacturing optimization. The original part of the work is represented by the application of this nonconventional method in the field of polymers additive manufacturing. The laws of the model provide the complete sets of dimensionless variables, which cannot be offered by any of the classical methods (such as Geometric Analogy, Theory of Similarity, and Classical Dimensional Analysis). The validation of the method was performed experimentally. The original part of the work is represented by the application of this nonconventional method in the field of polymers additive manufacturing optimization. An application is presented and the necessary steps are analyzed one by one.

In Silico Prediction of the Metabolic Resistance of Vitamin D Analogs against CYP3A4 Metabolizing Enzyme.

The microsomal cytochrome P450 3A4 (CYP3A4) and mitochondrial cytochrome P450 24A1 (CYP24A1) hydroxylating enzymes both metabolize vitamin D and its analogs. The three-dimensional (3D) structure of the full-length native human CYP3A4 has been solved, but the respective structure of the main vitamin D hydroxylating CYP24A1 enzyme is unknown. The structures of recombinant CYP24A1 enzymes have been solved; however, from studies of the vitamin D receptor, the use of a truncated protein for docking studies of ligands led to incorrect results. As the structure of the native CYP3A4 protein is known, we performed rigid docking supported by molecular dynamic simulation using CYP3A4 to predict the metabolic conversion of analogs of 1,25-dihydroxyvitamin D2 (1,25D2). This is highly important to the design of novel vitamin D-based drug candidates of reasonable metabolic stability as CYP3A4 metabolizes ca. 50% of the drug substances. The use of the 3D structure data of human CYP3A4 has allowed us to explain the substantial differences in the metabolic conversion of the side-chain geometric analogs of 1,25D2. The calculated free enthalpy of the binding of an analog of 1,25D2 to CYP3A4 agreed with the experimentally observed conversion of the analog by CYP24A1. The metabolic conversion of an analog of 1,25D2 to the main vitamin D hydroxylating enzyme CYP24A1, of unknown 3D structure, can be explained by the binding strength of the analog to the known 3D structure of the CYP3A4 enzyme.

Modern Method to Analyze the Heat Transfer in a Symmetric Metallic Beam with Hole

The paper aims to use Modern Dimensional Analysis to study the heat transmission through a rectangular bar with a hole. The problem is very important for monitoring a structure, made of such bars, to protect it from fire. The original part of the work is represented by the application of this nonconventional method in the field of heat transfer in bars of rectangular-tubular section. During system heating, the properties of the material change dramatically at high temperatures, which can lead to the collapse of the entire system. The Laws of the Model, further applied to the two sets of independent variables, provide the complete sets of dimensionless variables, which cannot be offered by any of the classical methods (such as Geometric Analogy, Theory of Similarity, and Classical Dimensional Analysis). The validation of the method was made experimental on both unprotected structural elements and those thermally protected with layers of intumescent paints, widely used in the field of fire protection. Finite Element Method was too applied to obtain the field of temperature in order to validate the model.

A geometric Vietoris-Begle theorem, with an application to convex subsets of topological vector lattices

We show that if L is a topological vector lattice, u:L→L is the function u(x)=x∨0, C⊂L is convex, and D=u(C) is metrizable, then D is an ANR and u|C:C→D is a homotopy equivalence, so D is contractible and thus an AR. This is proved by verifying the hypotheses of a second result: if X is a connected space that is homotopy equivalent to an ANR, Y is an ANR, and f:X→Y is a continuous surjection such that, for each y∈Y and each neighborhood V⊂Y of y, there is a neighborhood V′⊂V of y such that f−1(V′) can be contracted in f−1(V), then f is a homotopy equivalence. The latter result is a geometric analogue of the Vietoris-Begle theorem.

Secondary minibasins in orogens: Examples from the Sivas Basin (Turkey) and the sub-Alpine fold-and-thrust belt (France)

Inherited and syn-compressional salt-controlled structures in orogens are difficult to distinguish from compressional features because classical salt features (halokinetic growth strata, salt welds) can be misinterpreted as shortening-related structures (syn-kinematic growth strata, tectonic unconformities). In the Sivas basin in Turkey, syn-kinematic salt diapirism entirely shaped the architecture of the foreland and allowed the development of primary and secondary minibasins. One of the most impressive, the Tuzlagözü minibasin, formed by sinking into a salt feeder while being upturned due to compression. This minibasin has been nicknamed the “Turkish Vélodrome” because of its similarity with the well-known Vélodrome structure in the French Alps which shows growth strata and successive unconformities, commonly interpreted as foreland growth syncline features. Based on structural field descriptions and geometric analogies with the Tuzlagözü minibasin, we revisit the Vélodrome as a salt-related minibasin. We argue that the Vélodrome developed as a secondary minibasin above an Oligocene aerial salt glacier: sheet of Upper Triassic evaporites extruded during Alpine orogeny. This new interpretation enables to propose a revised evolution of the sub-Alpine fold-and-thrust belt. In line with other recent works, this study emphasises once again the important role of salt tectonics in the evolution of the sub-Alpine fold-and-thrust belt.

Sieve of Eratosthenes for Bose-Einstein condensates in optical moiré lattices

We catalog known optical moir\'e lattices and uncover exotic lattice configurations following a geometric analog of the ancient sieve of Eratosthenes algorithm for finding prime numbers. Rich dynamics of Bose-Einstein condensates loaded into these optical lattices is revealed from numerical simulations of time-of-flight interference patterns. What sets this method apart is the ability to tune the periodicity of the optical lattices without changing the wavelength of the laser, yet maintaining the local potential at the individual lattice sites. In addition, we discuss the ability to spatially translate the optical lattice through applying a structured phase only.

Various sources of distraction during analogical reasoning

Reasoning by analogy requires mapping relational correspondence between two situations to transfer information from the more familiar (source) to the less familiar situation (target). However, the presence of distractors may lead to invalid conclusions based on semantic or perceptual similarities instead of on relational correspondence. To understand the role of distraction in analogy making, we examined semantically rich four-term analogies (A:B::C:?) and scene analogies, as well as semantically lean geometric analogies and the matrix task tapping general reasoning. We examined (a) what types of lures were most distracting, (b) how the two semantically rich analogy tasks were related, and (c) how much variance in the scores could be attributed to general reasoning ability. We observed that (a) in four-term analogies the distractors semantically related to C impacted performance most strongly, as compared to the perceptual, categorical, and relational distractors, but the two latter distractor types also mattered; (b) distraction sources in four-term and scene analogies were virtually unrelated; and (c) general reasoning explained the largest part of variance in resistance to distraction. The results suggest that various sources of distraction operate at different stages of analogical reasoning and differently affect specific analogy paradigms.