Geometry Of Space Research Articles

Geometry of universal embedding spaces for almost complex manifolds

Geometry of universal embedding spaces for almost complex manifolds

Green Heat and Wind Factors in Sustainable Urban Development of Mountain-Basin Relief

The article studies mountain basin relief’s influence on the heat and wind regime organization when developing it for sustainable urban planning purposes. Each time a unique relief is identified, it requires an individual design approach taking into account the identification of the aesthetic expressiveness of green settlement development. Depending on the slope, there may be different options for using the site. The article presents the features of the geometry of the mountain-basin space associated with the relative depth of the mountain basin. The thermal field of the mountain-basin space, which is formed due to the energy of incoming solar radiation and the internal heat of insolated urban development, is analyzed. The city’s mountain-basin terrain requires an individual design approach, taking into account the identification of the aesthetic expressiveness of green settlement development. Based on the generalization and analysis of studies of air flows’ structure, their direction, velocity, and temperature fields, as well as the characteristics of the microclimate in the development of the mountain-basin space, an aerodynamic picture of air movement arising under the influence of natural thermal forces was obtained. The thermal field of the mountain-basin space has been established, which is formed due to the energy of incoming solar radiation and the internal heat of insolated urban development. The proposed methodology for calculating the heat and wind conditions of development is the basis for preliminary forecasting, assessment, and regulation of heat and wind parameters, as well as optimization of the sustainable development planning of mountain-basin relief.

Geometry of canonical genus 4 curves

AbstractWe apply the machinery of Bridgeland stability conditions on derived categories of coherent sheaves to describe the geometry of classical moduli spaces associated with canonical genus 4 space curves via an effective control over its wall‐crossing. This article provides the first description of a moduli space of Pandharipande–Thomas stable pairs that is used as an intermediate step toward the description of the associated Hilbert scheme, which in turn is the first example where the components of a classical moduli space were completely determined via wall‐crossing. We give a full list of irreducible components of the space of stable pairs, along with a birational description of each component, and a partial list for the Hilbert scheme. There are several long standing open problems regarding classical sheaf theoretic moduli spaces, and the present work will shed light on further studies of such moduli spaces such as Hilbert schemes of curves and moduli of stable pairs that are very hard to tackle without the wall‐crossing techniques.

Variational Estimation for Mechanical Systems on Lie Groups based on Geometric Mechanics

Geometric mechanics analyzes mechanical systems in the framework of variational mechanics, while accounting for the geometry of the configuration space. From the late 1970s, developments in this area produced several schemes for geometric control of mechanical systems in continuous time and discrete time. In the mid to late 2000s, geometric mechanics was first applied to state estimation of mechanical systems, particularly systems evolving on Lie groups as configuration manifolds, like rigid body systems. Much of the existing work on geometric mechanics-based estimation has been in continuous time, using deterministic, semi-stochastic and stochastic approaches. While the body of existing literature on discrete-time estimation schemes on Lie groups is not as extensive, the literature on this topic is contemporaneous with continuous-time schemes. This work describes some recent and ongoing research on geometric mechanics-based estimation schemes in continuous and discrete time from the mid-2010s, which were developed using the Lagrange-d’Alembert principle applied to rigid body systems. This approach gives (deterministic) observer designs with strong stability and robustness properties. This work concludes with potential extensions of this approach to mechanical systems with principal fiber bundles as configuration manifolds.

Animatable Virtual Humans: Learning Pose-Dependent Human Representations in UV Space for Interactive Performance Synthesis.

We propose a novel representation of virtual humans for highly realistic real-time animation and rendering in 3D applications. We learn pose dependent appearance and geometry from highly accurate dynamic mesh sequences obtained from state-of-the-art multiview-video reconstruction. Learning pose-dependent appearance and geometry from mesh sequences poses significant challenges, as it requires the network to learn the intricate shape and articulated motion of a human body. However, statistical body models like SMPL provide valuable a-priori knowledge which we leverage in order to constrain the dimension of the search space, enabling more efficient and targeted learning and to define pose-dependency. Instead of directly learning absolute pose-dependent geometry, we learn the difference between the observed geometry and the fitted SMPL model. This allows us to encode both pose-dependent appearance and geometry in the consistent UV space of the SMPL model. This approach not only ensures a high level of realism but also facilitates streamlined processing and rendering of virtual humans in real-time scenarios.

Concepts of an ideal city in the history of philosophy.

This article reviews the primary ideas that have been proposed about the «ideal city» from the time of ancient civilization to the present, and it draws key findings that are important for comprehending the details of contemporary urban development. The article pays a lot of attention to the social essence of the dynamics of the city, ideas about the ideal organization of urban space and its infrastructure. The comparison of modern models of the ideal city with the views of Plato, Aristotle, Al-Farabi and other thinkers and scientists on the ideal city testifies to the importance of these ideas. The authors consider the evolution of ideas about the geometry of urban space, conditioned by certain socio-ethical, aesthetic, mythological and religious priorities, basic scientific concepts and prevailing worldviews, as well as practical tasks of ensuring the safety of the city. Projects of urbanization and social reconstruction, where the ideas of spiritual and moral transformation of man and society prevail, for all their utopianism, cannot sink into the past. They encourage thinking about the possibility of building models of urban and social reconstruction that meet millennial aspirations for security, justice, the common good, and comfort. The article draws attention to the growing negative consequences of the urbanization process, the need for a serious justification of the policy of formation and development of megacities, its implementation taking into account the principle of multifactoriality.

Квантовая география: задачи типизации, классификации и районирования

The possibility of using the mathematical apparatus of the quantum theory of computer science for modeling geographical phenomena in solving problems of typification, classification and zoning to explain the reasons for the integrity of territorial objects is discussed. The methods of trivial and principal bundles of differential geometry and functional analysis of virtual internal spaces of geosystems (geomers and geochores) are used. The advantage of the quantum description lies in the linear simplicity of equations and formulas reflecting both the typical functions of discrete quantum states of geosystems and the superposition of these states realized in the observed spatial patterns in nature and society. This approach is the result of the ascent from a simple understanding of geographical objects as systems of continuous connection of elements to the allocation of independent discrete layers of independent functioning (geomers), their polysystem equivalence in geochores and holysystem integrity in region. The integrity of objects is described by general equations that do not take into account the influence of the geographical environment, therefore they allow creating models of different geosystems in the same way. For quantitative analysis of cartographic data, methods of reliability theory are used, based on the identification of frequency distributions of geomer areas by height, calculation of reliability and safety indicators. The application of the ideas of quantum geoinformatics is demonstrated by the example of assessing the factorial influence on the habitus of landscapes based on the signs of territorial manifestation of facies of different factorial series (facies classes) of the localities of the Maly Khamar-Daban ridge and the altitude distribution of the geomes of the Transbaikalia.

Colloid Transport and Retention in Constricted Tube Pore Spaces With Diverse Geometries and Orientations

AbstractColloidal contaminants infiltrate and can be attached onto grain surfaces of soils and aquifers, where they may persist. In this study, Lagrangian particle tracking is used to investigate particle trajectories and attachment in pore and fracture spaces modeled as three‐dimensional constricted tubes with diverse geometries and orientations relative to gravity. A comprehensive force balance arising from hydrodynamic drag and lift, gravitational settling, Brownian motion, and attractive DLVO interactions is simulated. Results show that the collection efficiency η is primarily governed by the dimensionless settling number 𝑆, representing the relative dominance of gravitational over hydrodynamic forces experienced by the particles. High‐𝑆 scenarios have larger η and are more sensitive to pore orientation, while low‐𝑆 scenarios are more sensitive to pore geometry. For all scenarios but especially low‐S scenarios, the majority of colloid attachment occurs near pore extremities, where fluid velocities are low, such that mechanical remobilization of particles attached is improbable. In low‐𝑆 scenarios, particles may spread and become immobilized at greater distances from the contamination source owing to lower η, are harder to mechanically remobilize as they attach more disproportionately at pore extremities, and have trajectories more sensitive to minor forces, rendering their environmental fates complex. Characterizing the collection efficiency and deposition morphology for various pore space geometries and orientations is crucial in understanding particle fate and developing continuum‐scale models of colloid transport in real soils, where pore spaces are heterogeneous and advection paths are tortuous.

Influence of the structure of the pore space of rocks on the wettability and adsorbed fluids of reservoir rocks

Theoretical and experimental studies of the effect of the geometry of the pore space on the wettability of collectors have been carried out. The special structure of the pore space of the Talinskaya area of the Krasnoleninsky deposit, due to the heterogeneity of the composition of rocks, leads to heterogeneity in the size and geometry of filtration channels and affects wettability. The specificity of the adsorption of polar components of oil causes changes in wettability and affects the amount of adsorbed oil. Keywords: pore space geometry; adsorbed fluids; wettability.

The Layout of the Statistical Forecasts Regarding the Fight Against of the Chaos’s Vectors from the Space Geometry of the Illegal Drugs

The Layout of the Statistical Forecasts Regarding the Fight Against of the Chaos’s Vectors from the Space Geometry of the Illegal Drugs

When Protein Structure Embedding Meets Large Language Models.

Protein structure analysis is essential in various bioinformatics domains such as drug discovery, disease diagnosis, and evolutionary studies. Within structural biology, the classification of protein structures is pivotal, employing machine learning algorithms to categorize structures based on data from databases like the Protein Data Bank (PDB). To predict protein functions, embeddings based on protein sequences have been employed. Creating numerical embeddings that preserve vital information while considering protein structure and sequence presents several challenges. The existing literature lacks a comprehensive and effective approach that combines structural and sequence-based features to achieve efficient protein classification. While large language models (LLMs) have exhibited promising outcomes for protein function prediction, their focus primarily lies on protein sequences, disregarding the 3D structures of proteins. The quality of embeddings heavily relies on how well the geometry of the embedding space aligns with the underlying data structure, posing a critical research question. Traditionally, Euclidean space has served as a widely utilized framework for embeddings. In this study, we propose a novel method for designing numerical embeddings in Euclidean space for proteins by leveraging 3D structure information, specifically employing the concept of contact maps. These embeddings are synergistically combined with features extracted from LLMs and traditional feature engineering techniques to enhance the performance of embeddings in supervised protein analysis. Experimental results on benchmark datasets, including PDB Bind and STCRDAB, demonstrate the superior performance of the proposed method for protein function prediction.

SO(3)-Irreducible Geometry in Complex Dimension Five and Ternary Generalization of Pauli Exclusion Principle

Motivated by a ternary generalization of the Pauli exclusion principle proposed by R. Kerner, we propose a notion of a Z3-skew-symmetric covariant SO(3)-tensor of the third order, consider it as a 3-dimensional matrix, and study the geometry of the 10-dimensional complex space of these tensors. We split this 10-dimensional space into a direct sum of two 5-dimensional subspaces by means of a primitive third-order root of unity q, and in each subspace, there is an irreducible representation of the rotation group SO(3)↪SU(5). We find two SO(3)-invariants of Z3-skew-symmetric tensors: one is the canonical Hermitian metric in five-dimensional complex vector space and the other is a quadratic form denoted by K(z,z). We study the invariant properties of K(z,z) and find its stabilizer. Making use of these invariant properties, we define an SO(3)-irreducible geometric structure on a five-dimensional complex Hermitian manifold. We study a connection on a five-dimensional complex Hermitian manifold with an SO(3)-irreducible geometric structure and find its curvature and torsion.

Evolution of hierarchy and irreversibility in theoretical cell differentiation model.

The process of cell differentiation in multicellular organisms is characterized by hierarchy and irreversibility in many cases. However, the conditions and selection pressures that give rise to these characteristics remain poorly understood. By using a mathematical model, here we show that the network of differentiation potency (differentiation diagram) becomes necessarily hierarchical and irreversible by increasing the number of terminally differentiated states under certain conditions. The mechanisms generating these characteristics are clarified using geometry in the cell state space. The results demonstrate that the hierarchical organization and irreversibility can manifest independently of direct selection pressures associated with these characteristics, instead they appear to evolve as byproducts of selective forces favoring a diversity of differentiated cell types. The study also provides a new perspective on the structure of gene regulatory networks that produce hierarchical and irreversible differentiation diagrams. These results indicate some constraints on cell differentiation, which are expected to provide a starting point for theoretical discussion of the implicit limits and directions of evolution in multicellular organisms.

Analysis of potassium ion diffusion from neurons to capillaries: Effects of astrocyte endfeet geometry.

Neurovascular coupling (NVC) refers to a local increase in cerebral blood flow in response to increased neuronal activity. Mechanisms of communication between neurons and blood vessels remain unclear. Astrocyte endfeet almost completely cover cerebral capillaries, suggesting that astrocytes play a role in NVC by releasing vasoactive substances near capillaries. An alternative hypothesis is that direct diffusion through the extracellular space of potassium ions (K+ ) released by neurons contributes to NVC. Here, the goal is to determine whether astrocyte endfeet present a barrier to K+ diffusion from neurons to capillaries. Two simplified 2D geometries of extracellular space, clefts between endfeet, and perivascular space are used: (i) a source 1μm from a capillary; (ii) a neuron 15 μm from a capillary. K+ release is modelled as a step increase in [K+ ] at the outer boundary of the extracellular space. The time-dependent diffusion equation is solved numerically. In the first geometry, perivascular [K+ ] approaches its final value within 0.05 s. Decreasing endfeet cleft width or increasing perivascular space width slows the rise in [K+ ]. In the second geometry, the increase in perivascular [K+ ] occurs within 0.5s and is insensitive to changes in cleft width or perivascular space width. Predicted levels of perivascular [K+ ] are sufficient to cause vasodilation, and the rise time is within the time for flow increase in NVC. These results suggest that direct diffusion of K+ through the extracellular space is a possible NVC signalling mechanism.

Longitudinal Galilean and Carrollian limits of non-relativistic strings

It is well known that one can take an infinite speed of light limit that gives rise to non-relativistic strings with a relativistic worldsheet sigma model but with a non-relativistic target space geometry. In this work we systematically explore two further limits in which the worldsheet becomes non-Lorentzian. The first gives rise to a Galilean string with a Galilean structure on the worldsheet, extending previous work on Spin Matrix-related string theory limits. The second is a completely novel limit leading to a worldsheet theory with a Carrollian structure. We find the Nambu-Goto and Polyakov formulations of both limits and explore gauge fixing choices. Furthermore, we study in detail the case of the Galilean string for a class of target space geometries that are related to Spin Matrix target space geometries, for which the Nambu-Goto action (in static gauge) is quadratic in the fields.

Cosmological model free of singularity and inflation based on the large numbers hypothesis

It is shown that the new precise formulation of the Large Number Hypothesis (LNH), relating by means of the large number N_0 = 5.73×10^60 the modern cosmological parameters (age, size, mass, average density, and minimum temperature of the universe) with the corresponding Planck units, allows to determine the time course of these cosmological parameters during the expansion. It was found that the dimensions and mass of the universe increase linearly with time from Planck time t = t_P to the present day, starting from Planck values and increasing N_0 = 5.73×10^60 times to now. The amazing result was found that for each discrete time step (beat) with a unit Planck time ∆t = t_P, the size of the universe increases by one Planck length l_P and its mass increases by one Planck mass m_P. It is shown that the average density of the universe decreases proportionally to the square of time, and starting from the Planck density ρ_P ~ 10^96 kg m-3 decreases N_0^2 = 3.28×10^121 times to 9.46×10^-27 kg m-3 in the current epoch. The minimum measurable temperature, which is equal to the Hawking temperature for the universe T_H decreases linearly with time 5.73×10^60 times, and starting from the Planck temperature T_P = 10^32 K, it falls to 1.75×10^-29 K at the present time. It is shown that the found time course of cosmological parameters and the Planck values of the size, mass, average density, and temperature of the universe at the initial moment of the expansion t = t_P follow from the requirement to preserve the Euclidean geometry of space throughout the time of the cosmological expansion. Therefore, the suggested cosmological model based on the new formulation of LNH is free of singularity because the size and density of the universe remain finite/Planckian in the initial moments of its emergence. Besides, this model conserves the flatness and homogeneity of the universe during cosmological expansion and does not need an inflationary epoch in the early universe.

Three-dimensional concentration-polarization modeling of trace-ions in reverse osmosis

Concentration polarization (CP) is mostly evaluated through one-dimensional (1D) film models considering convective and diffusive transport of solutes. However, in multicomponent ionic solutions, electromigration and chemical speciation may also play a role in the solute transport within the CP layer, thus affecting the process performance. We developed a three-dimensional (3D) numerical model for evaluating the CP in spacer-filled feed reverse osmosis (RO) channels by coupling hydrodynamics with the transport of solutes by convection, diffusion and electromigration, and chemical speciation by several ion-pairing equilibria. A realistic spacer geometry was obtained by CT-scans of commercial RO spacers. The model revealed highly non-uniform distributions of CP, mass transfer coefficients and precipitation potential on the membrane. While electromigration tends to equalize the fluxes of dominant ions, increases locally the CP of anions and decreases that of cations, it was found not to significantly affect the boundary layer thickness. However, the chemical speciation strongly affected the CP of trace ions binding with dominant ions. Finally, the 3D results were compared with those obtained from simpler 1D models. While 1D approaches help explaining qualitatively the effects of electromigration and speciation, and serve as a reasonable first approximation, they do not capture the non-uniform distribution of concentrations, mass transfer rates or saturation indices. The model will be used to study the full length of an RO membrane module and thus provide better indicators of the overall performance of the separation process.

The Geometry of Low- and High-Level Perceptual Spaces.

Low-level features are typically continuous (e.g., the gamut between two colors), but semantic information is often categorical (there is no corresponding gradient between dog and turtle) and hierarchical (animals live in land, water, or air). To determine the impact of these differences on cognitive representations, we characterized the geometry of perceptual spaces of five domains: a domain dominated by semantic information (animal names presented as words), a domain dominated by low-level features (colored textures), and three intermediate domains (animal images, lightly texturized animal images that were easy to recognize, and heavily texturized animal images that were difficult to recognize). Each domain had 37 stimuli derived from the same animal names. From 13 participants (9F), we gathered similarity judgments in each domain via an efficient psychophysical ranking paradigm. We then built geometric models of each domain for each participant, in which distances between stimuli accounted for participants' similarity judgments and intrinsic uncertainty. Remarkably, the five domains had similar global properties: each required 5-7 dimensions, and a modest amount of spherical curvature provided the best fit. However, the arrangement of the stimuli within these embeddings depended on the level of semantic information: dendrograms derived from semantic domains (word, image, and lightly texturized images) were more "tree-like" than those from feature-dominated domains (heavily texturized images and textures). Thus, the perceptual spaces of domains along this feature-dominated to semantic-dominated gradient shift to a tree-like organization when semantic information dominates, while retaining a similar global geometry.

Geometry of Hermitian symmetric spaces under the action of a maximal unipotent group

Let [Formula: see text] be a non-compact irreducible Hermitian symmetric space of rank [Formula: see text] and let [Formula: see text] be an Iwasawa decomposition of [Formula: see text]. The group [Formula: see text] acts on [Formula: see text] by biholomorphisms and the real [Formula: see text]-dimensional submanifold [Formula: see text] intersects every [Formula: see text]-orbit transversally in a single point. Moreover [Formula: see text] is contained in a complex [Formula: see text]-dimensional submanifold of [Formula: see text] biholomorphic to [Formula: see text], the product of [Formula: see text] copies of the upper half-plane in [Formula: see text]. This fact leads to a one-to-one correspondence between [Formula: see text]-invariant domains in [Formula: see text] and tube domains in [Formula: see text]. In this setting we prove an analogue of Bochner’s tube theorem. Namely, an [Formula: see text]-invariant domain [Formula: see text] in [Formula: see text] is Stein if and only if the base [Formula: see text] of the associated tube domain is convex and “cone invariant”. We also prove the univalence of [Formula: see text]-invariant holomorphically separable Riemann domains over [Formula: see text]. This yields a precise description of the envelope of holomorphy of an arbitrary [Formula: see text]-invariant domain in [Formula: see text]. Finally, we obtain a characterization of several classes of [Formula: see text]-invariant plurisubharmonic functions on [Formula: see text] in terms of the corresponding classes of convex functions on [Formula: see text]. As an application we present an explicit Lie group theoretical description of all [Formula: see text]-invariant potentials of the Killing metric on [Formula: see text] and of the associated moment maps.

NUMERICAL MODELLING TOXIC SUBSTANCE TRANSPORT IN MINE WATER FLOWS

The purpose of this work is to develop and test a methodology for modelling the migration of toxic substances left after mining in mine water flows in a system of hydraulically connected mine workings of various sizes and elevations depending on the drainage parameters. Methods of research include the analysis of factors that influence the formation of the hydraulic regime in flooded mines, the accumulation and transport of toxic substances. Parameters of moving toxic substances in mine waters are calculated using hydraulic flow equations written for mining workings, with flooded workings being considered as pipelines with distributed recharge. Modelling of non-stationary 1D transport in water from local sources of toxic substances was performed using the finite difference method. Results. Flow rates and velocities in flooded mine workings on two hydraulically connected horizons of different elevations were calculated. For different locations of sources of toxic substances on the example of polychlorinated biphenyls, their concentrations in mine waters along the migration path and at the water hoisting at different flow rates were calculated. The influence of increased water withdrawal, dilution with additional recharge along the migration path and the position of potential sources of substances in the flooded mine were investigated. It wasshown that for the considered example, increasing the water outflow rate by two times accelerates the stabilisation of mass transport with an increase in the daily removal of substances by 3.4–6.4 times, which is more active from the upper horizon. Scientific novelty. For the first time, the transport of substances in flooded workings was simulated by combining a hydraulic flow model with a numerical transport model. Unlike 2D and 3D numerical transport models based on solid mechanics, which average the concentration in the grid blocks, the proposed approach allows the reproduction of the geometry of the mined-out space more accurately. It provides a more realistic distribution of flow velocities and concentrations depending on the parameters of the mine water withdrawal, depth of mine workings, and the mine water level. Practical significance. Using the tested methodology will make it possible to perform engineering predictions of the quality of groundwater and surface water near closed mines for different periods at different levels of flooding, water withdrawal rates and depths of the pumps for mine water drainage. In addition, the proposed technique can be used to justify the conceptual scheme of the numerical hydrogeochemical model of the post-mining areas. Keywords. Close mines, toxic substances, flooded workings, hydraulic flow, water hoisting, transport model.