Spherical Space Research Articles

Ultra-strong diffusion-weighted MRI reveals cerebellar grey matter abnormalities in movement disorders

Structural brain MRI has proven invaluable in understanding movement disorder pathophysiology. However, most work has focused on grey/white matter volumetric (macrostructural) and white matter microstructural effects, limiting understanding of frequently implicated grey matter microstructural differences. Using ultra-strong spherical tensor encoding diffusion-weighted MRI, a persistent MRI signal was seen in healthy cerebellar grey matter even at high diffusion-weightings (b ​≥ 10,000 s/mm2). Quantifying the proportion of this signal (denoted fs), previously ascertained to originate from inside small spherical spaces, provides a potential proxy for cell body density. In this work, this approach was applied for the first time to a clinical cohort, including patients with diagnosed movement disorders in which the cerebellum has been implicated in symptom pathophysiology. Five control participants (control group 1, median age 24.5 years (20–39 years), imaged at two timepoints, demonstrated consistency in measurement of all three measures - MD (Mean Diffusivity) fs, and Ds (dot diffusivity)- with intraclass correlation coefficients (ICC) of 0.98, 0.86 and 0.76, respectively. Comparison with an older control group (control group 2 (n = 5), median age 51 years (43–58 years)) found no significant differences, neither with morphometric nor microstructural (MD (p = 0.36), fs (p = 0.17) and Ds (p = 0.22)) measures. The movement disorder cohort (Parkinson’s Disease, n = 5, dystonia, n = 5. Spinocerebellar Ataxia 6, n = 5) when compared to the age-matched control cohort (Control Group 2) identified significantly lower MD (p < 0.0001 and p < 0.0001) and higher fs values (p < 0.0001 and p < 0.0001) in SCA6 and dystonia cohorts respectively. Lobar division of the cerebellum found these same differences in the superior and inferior posterior lobes, while no differences were seen in either the anterior lobes or with Ds measurements. In contrast to more conventional measures from diffusion tensor imaging, this framework provides enhanced specificity to differences in restricted spherical spaces in grey matter (including small cells) by eliminating signals from cerebrospinal fluid and axons. In the context of human and animal histopathology studies, these findings potentially implicate the cerebellar Purkinje and granule cells as contributors to the observed signal differences, with both cell types having been implicated in several neurological disorders through both postmortem and animal model studies. This novel microstructural imaging approach shows promise for improving movement disorder diagnosis, prognosis, and treatment.

Отчуждение: кризис самоидентичности

The appeal to the problem of alienation of personality in the conditions of modern reality has relevance due to the fact that it allows analyzing the status of such a phenomenon as the crisis of self-identity. The ways of the constitution of the self by a person in the conditions of the emerging digital reality acquire the meaning of “answers” to the problem of self-alienation of the individual. The purpose of the study is to consider the newly emerging aspects of the process of alienation associated with the formation of a reality that goes beyond the empirical space of the main spheres of public life that has become familiar to the broad masses of people. The objectives of this work are: firstly, a review of the main methodological approaches to the study of alienation in the history of philosophical thought; secondly, an analysis of new forms of alienation emerging in the digital space. The most adequate to the set goals and objectives is the comparative approach, which is a transdisciplinary methodology that allows avoiding one-sided interpretation of the phenomena under study. The article examines the ideas about the nature of alienation in classical and non-classical philosophy, due to various factors of reality that generate the “discontinuity” of human existence. It is shown that the category of “alienation” is a product of understanding the contradictions between the laws of nature and society with human nature. New forms of alienation are subjected to philosophical reflection. The paper draws attention to the fact that the crisis of personal identity is largely due to the increasing role and importance of a person in the modern world. The consequence of his active life position is not only socio-cultural transformations, but also the emergence of new problems that indicate an identity crisis and require deep philosophical reflection, the search for answers to the problem of human self-alienation. The novelty of the work lies in determining the significance of the fact of constructing a virtual self-identity by a person as a way to solve the problem of self-alienation. The article is of an installation nature, its conclusions should be considered as theoretical prerequisites for the analysis of such problems as the peculiarities of modern man’s being, his identity and polyidentity, as well as the problems of determining the nature of the emerging universal hyperreality.

Social Media, the Public Sphere, and Public Space in Authoritarian Settings: A Case Study of Egypt’s Tahrir Square and 18 Years of User Generated Content

Taking Egypt as a case study, this article examines the relationship between social media and Habermas’ public sphere and public space in less democratic settings. Using a survey of Tahrir Square protesters and process tracing to examine 18 years of social media use, the article proposes a theoretical model highlighting the role of social media in carving out an eventful public sphere, despite the lack of public space to promote communicative action in authoritarian settings. Unlike the process described by Habermas, which takes place in democratic societies, the narratives presented in this study reveal a different pattern of communication which centers around an event through which larger issues are discussed within a defiant space. This happens through initiating conversation and drawing attention to the event, engaging counterpublics, establishing links of affinity, and possibly, calling for action. Factors influencing the success of an eventful public sphere under state control include the nature of the event, its effect on the publics, the public’s affinity to the event, potential social or cultural resistance, the amount of risk involved, and the potential scope of participation.

Effects of plasma expansion plumes in view of pulses laser irradiating centimeter-scale spherical space debris

The objective of this article was to investigate the dynamic evolution behaviors of plasma expansion plumes by pulses laser irradiating centimeter-scale spherical space debris. A calculated model of centimeter-scale spherical space debris irradiated by pulses laser was firstly deduced based on FEM (finite element method)/COMSOL, and the action rules of plasma expansion plumes by pulses laser-generated irradiating the debris were simulated for different laser powers and action times. The results showed that the velocity of plasma expansion plumes was increased with the increase of laser powers and action times. Especially, when the laser power was 700 kW and the action time was close to 25 μs, the maximum velocity of plasma expansion plumes approached 1.91 km/s, and the diffusion radius of plasma expansion plumes was increased by about 2.5 mm. Further, the diffusion radius was about twice that of 400 kW when the action time reached about 48 μs. As a result, by simulating the transient flow process of nanosecond pulses laser irradiating small spherical space debris, the flow field evolution information and plasma plumes evolution characteristics of centimeter-scale space debris at nanosecond time resolution were revealed.

ԲԵՄԱԿԱՆ ԳՈՐԾՈՂՈՒԹՅՈՒՆ-ՀԱՆԴԻՍԱՏԵՍ ՀԱՐԱԲԵՐՈՒԹՅԱՆ ՀԻՄՆԱԽՆԴԻՐԸ 1960-1980-ԱԿԱՆ ԹԹ. ՀԱՅ ԹԱՏՐՈՆՈՒՄ ԱՐԵՎՄՏՅԱՆ ԹԱՏՐՈՆԻ ՈՐՈՆՈՒՄՆԵՐԻ ՀԱՄԱՏԵՔՍՏՈՒՄ

In the 20th century, the stage space was understood as one of the most important theatrical categories. Culturologist, philosopher E. Surio distinguishes two types of stage space: a cube and a sphere. Surio highlights the main characteristics of these two types of spaces: isolation, the unchanging point of view of the audience, the specific material environment characteristic of the cube, and the open nature of the spherical space, involving the audience in its radius, making him a participant of the performance.

HDSS-Net: A Novel Hierarchically Designed Network With Spherical Space Classifier for Ship Recognition in SAR Images

HDSS-Net: A Novel Hierarchically Designed Network With Spherical Space Classifier for Ship Recognition in SAR Images

Analysis and assessment of life-cycle carbon emissions of space frame structures

The construction industry takes up a significant part of global energy consumption and carbon emissions, and thus accurate calculation and evaluation of full life-cycle energy consumption and carbon emissions of different types of building structures are important for low carbon-oriented structural design. Large-span structures usually consume more materials, need more complex construction processes, and thus have a larger environmental impact. In this study, the life cycle assessment of the carbon emission of space frame structures which is a classical type of large-span structure is conducted. Space frame structures with bolted spherical joints and space frame structures with welded hollow spherical joints are considered. The life cycle of a space frame structure is divided into four phases including component production, construction, maintenance, and demolition. The tiered hybrid method that combines process-based calculation and input-output analysis is adopted to evaluate the carbon emission of space frame structures. Parametrical studies on the height-span ratio and grid number are carried out using the proposed assessment scheme. The carbon emissions of the space frame structures with different types of joints are investigated and compared. It is found that lower material consumption does not always result in lower carbon emission, and thus it is necessary to evaluate the life-cycle carbon emission during lower carbon-oriented design of space frame structures. An equation for approximately and quickly estimating the carbon emission of space frame structures is fitted from the material consumption to carbon emission datum obtained from the parametrical studies. This study provides a useful reference for the carbon emission assessment and low-carbon oriented design of space frame structures.

Geometric phase for spherical magnetic particles

In this study, we study on the characterization of the behavior of a dynamically charged magnetic particle in spherical space and on the geometrical expressions of electromagnetic parameters resulting from this behavior in spherical space. First, we characterize the spherical-magnetic particles and spherical-magnetic fields, which we named associated with the spherical frame. Then, with the help of the Lorentzian force and the Newtonian force relationship, we obtain the electric field of these magnetic particles in terms of the particle’s charge and mass. Next, with the help of electric field and magnetic field, we obtain important electromagnetic parameters such as energy density, power flow, light intensity and power of these spherical magnetic particles. Finally, we obtain the geometric (Berry) phases formed by Heisenberg antiferromagnetic models of the power flows.

On a strengthening of the Blaschke–Leichtweiss theorem

The Blaschke–Leichtweiss theorem (Leichtweiss in Abh Math Semin Univ Hambg 75:257–284, 2005) states that the smallest area convex domain of constant width w in the 2-dimensional spherical space \({\mathbb {S}}^2\) is the spherical Reuleaux triangle for all \(0<w\le \frac{\pi }{2}\). In this paper we extend this result to the family of wide r-disk domains of \({\mathbb {S}}^2\), where \(0<r\le \frac{\pi }{2}\). Here a wide r-disk domain is an intersection of spherical disks of radius r with centers contained in their intersection. This gives a new and elementary proof of the Blaschke–Leichtweiss theorem. Furthermore, we investigate the higher dimensional analogue of wide r-disk domains called wide r-ball bodies. In particular, we determine their minimum spherical width (resp., inradius) in the spherical d-space \({\mathbb {S}}^d\) for all \(d\ge 2\). Also, it is shown that any minimum volume wide r-ball body is of constant width r in \({\mathbb {S}}^d\), \(d\ge 2\).

Enlarged Head Pressure-Dispersed Anchor Cable for Foundation Pit Engineering Purposes

The enlarged head pressure-dispersed anchor cable is a new type of anchor cable, and the enlarged head of the anchor solid is a multisegment body with variable cross section. Compared with the traditional tension type anchor cable structure, the stress mode of the anchor solid is changed from tensile stress to compressive stress, which is more reasonable. The development process of the plastic zone of the soil around the enlarged head anchor solid is verified by a simulation method. The spherical plastic zone space formed at the enlarged head anchorage end is conducive to the load transfer between the anchor solid and the surrounding soil and reduces the displacement of the anchor solid in the soil. In this pursuit, the present study examines the force mechanism of the enlarged head pressure-dispersed anchor cable. Subsequently, taking an actual project as an example, the anchoring capacity of the enlarged head pressure-dispersed anchor cable and enlarged head tension-type anchor cable was analyzed and compared using FLAC3D software. The results indicate that the support effect of the enlarged head pressure-dispersed anchor cable was better than that of the enlarged head tension anchor cable.

PRIN/SPRIN: On Extracting Point-Wise Rotation Invariant Features.

Point cloud analysis without pose priors is very challenging in real applications, as the orientations of point clouds are often unknown. In this paper, we propose a brand new point-set learning framework PRIN, namely, Point-wise Rotation Invariant Network, focusing on rotation invariant feature extraction in point clouds analysis. We construct spherical signals by Density Aware Adaptive Sampling to deal with distorted point distributions in spherical space. Spherical Voxel Convolution and Point Re-sampling are proposed to extract rotation invariant features for each point. In addition, we extend PRIN to a sparse version called SPRIN, which directly operates on sparse point clouds. Both PRIN and SPRIN can be applied to tasks ranging from object classification, part segmentation, to 3D feature matching and label alignment. Results show that, on the dataset with randomly rotated point clouds, SPRIN demonstrates better performance than state-of-the-art methods without any data augmentation. We also provide thorough theoretical proof and analysis for point-wise rotation invariance achieved by our methods. The code to reproduce our results will be made publicly available.

The soul words of the Ukrainian people

The article presents the main directions of the domestic psychological science development which are devoted to the psychology of the Ukrainian person and the Ukrainian people in the context of methodolohems of their self-cognition and self-affirmation. It is noted that the changeability of the meanings of the same word forms reflects the peculiarities of human thinking of each era and therefore, through the existence of the mental, determines the meaningfulness of being which in the case of the Ukrainian understanding is characterized by both the limitlessness of the human soul cognition and the importance of comprehension the common and different in the spiritual organization of other peoples. In particular, the multi-sense mosaic of the ethno-Ukrainian self-awareness of the soul existence as life, love, dream, person, part of the body and embodied desire is argued, the inexhaustibility of the meaning of the soul as a word, term, concept and psychological category is underlined, in addition, it is substantiated the thesis that the exit of feeling and understanding beyond the limits of any scientific cognition is possible in the same way that personally present life creates itself by its existence in the infinity of its own incarnations and meanings. The reaction of the Ukrainian soul to a foreign hiding of faces under the masks of “guise” and “persona” is revealed in an adequate borrowing of the protective method of salvation from insincerity, slyness and deception in the form of the modified Latin concepts of “parsuna” and “portrait”, the dichotomy of borrowed and own is clarified, which accompanies the development of the spiritual identity of Ukrainians in the dualism of the soul-for-oneself and the psyche as a general depersonalized category-for-all that exists in parallel in soul language and psychology, without crossing in its purpose and embodiment. The attention is paid to the fact that over the centuries, an established action on the meaning of the soul remains unanswered, each time acquiring different incarnations: in soulwords as a verbal expression of the soul, expression and appeal to it; in soulsalvation as the return of the soul to itself; in spiritualizing as an active comprehensiveness of mental life in its conscious, subconscious, unconscious and superconscious dimensions. And yet, despite the wide range of forms of spiritual, a soul of a living being remains immeasurable as the soul of life, which will always be the more incomprehensible, the more specific the attempts to grasp it completely and entirely become. The statement that the whole world of a person is embodied in his soul and exists as long as a personal experience continues, as well as the life of the peoples – all those born and related by its psychosocial flow – is reflected. It has been proven that the manifestations of the peoples’ life in its past and present are covered by the concept of the soul and, in a conscious expression, are embodied in interactions on a common field – in the spherical space of communication, which acquires infinite variations of speech existence. Based on this, the need to mark the essence of the soul of everyone who belongs to one of the peoples of the world is inherent in each of these peoples, although it is realized and implemented differently, including going beyond its own borders, which involves capturing as many individuals as possible by imposing your identity on them, embodied in the monopolistic form of cosmopolitanism, chauvinism, fascism, communism, internationalism, globalism or other fluctuations of the ethnic spirit that insatiable by struggle for the primacy. Under the favorable conditions of military expansion for this spirit, it expanded beyond its own phenomenal horizons, always parasitizing on the original ethnic language base – Arabic, Greek, Persian, Latin, French, German, English, Russian and many others. The need in cognition the essence of the Ukrainian soul and its defining properties is revealed, primarily in its desire for deeper self-cognition, the fundamental research basis of which is the joint work of the Ukrainian scientists’ team of the middle of the 20th century “Ukrainian soul” that reflects the main dimensions of the ultra-complex, multi-subject object of study chosen by five scientists: “Ukrainian emotionality” – Ye. Onatskiy, “Ukrainian’s perception of the world” – O. Kulchytskiy, “Family and soul of the peoples” – B. Tsymbalisty, “Soul and song” – M. Shlemkevych, “Bibliographic review” – V. Doroshenko. It is concluded that psychology of the Ukrainian person, the Ukrainian peoples needs a creation of, an adequate for psychosophical cognition-construction, methodology of discovering unknown dimensions of their own development in which the self-movement of psycho-spiritual is possible on the path of self-reflection of its own essence in the boundless vectors of human development that must find and harmonize its own sensory-intellectual soulwords.

AADG: Automatic Augmentation for Domain Generalization on Retinal Image Segmentation.

Convolutional neural networks have been widely applied to medical image segmentation and have achieved considerable performance. However, the performance may be significantly affected by the domain gap between training data (source domain) and testing data (target domain). To address this issue, we propose a data manipulation based domain generalization method, called Automated Augmentation for Domain Generalization (AADG). Our AADG framework can effectively sample data augmentation policies that generate novel domains and diversify the training set from an appropriate search space. Specifically, we introduce a novel proxy task maximizing the diversity among multiple augmented novel domains as measured by the Sinkhorn distance in a unit sphere space, making automated augmentation tractable. Adversarial training and deep reinforcement learning are employed to efficiently search the objectives. Quantitative and qualitative experiments on 11 publicly-accessible fundus image datasets (four for retinal vessel segmentation, four for optic disc and cup (OD/OC) segmentation and three for retinal lesion segmentation) are comprehensively performed. Two OCTA datasets for retinal vasculature segmentation are further involved to validate cross-modality generalization. Our proposed AADG exhibits state-of-the-art generalization performance and outperforms existing approaches by considerable margins on retinal vessel, OD/OC and lesion segmentation tasks. The learned policies are empirically validated to be model-agnostic and can transfer well to other models. The source code is available at https://github.com/CRazorback/AADG.

A systematic study of a droplet breakup process in decaying homogeneous isotropic turbulence using a mesoscopic simulation approach

ABSTRACT The breakup of a spherical droplet in a decaying homogeneous isotropic turbulence is studied by solving the Cahn–Hilliard–Navier–Stokes equations. This flow provides a great opportunity to study the interactions of turbulent kinetic energy and interfacial free energy and their effects on the breakup dynamics. Three distinct stages of droplet evolution, namely, the deformation stage, the breakup stage, and the restoration stage, are identified and then analysed systematically from several perspectives: a geometric perspective, a dynamic perspective, a global energetic perspective, and a multiscale energy transfer perspective. It is found that the ending time of the breakup stage can be estimated by the Hinze criterion. The kinetic energy of the two-phase flow during the breakup stage is found to have a power-law decay with an exponent , compared to for the single-phase flow, mainly due to the enhanced viscous dissipation generated by the daughter droplets. Energy spectra of the two-phase flow show power-law decay, with a slope between and , at high wave numbers, both in the Fourier spectral space and in the spherical harmonics space.

A general construction of spherical barycentric coordinates and applications

We provide a general method for defining and efficiently computing barycentric coordinates with respect to polygons on the unit sphere. More precisely, we develop a novel explicit construction which allows us to compute the spherical barycentric coordinates from their 2D-Euclidean counterparts. In particular, we give two interesting families of spherical coordinates, one is defined for convex and non-convex spherical polygons. An interesting consequence is the possibility to construct new 3D barycentric coordinates for arbitrary polygonal meshes. Furthermore, we present an alternative construction for spherical barycentric coordinates with help of 3D barycentric coordinates for closed triangular meshes. This construction is extended to arbitrary dimensions. We show that our spherical and 3D coordinates are widely applicable to many domains. We give several examples related to spherical blending, space deformations and shape morphing in 3D.

Unleashing the Potentials of Quantum Probability Theory for Customer Experience Analytics

In information systems research, the advantages of Customer Experience (CX) and its contribution to organizations are largely recognized. The CX analytics evaluate how customers perceive products, ranging from their functional usage to their cognitive states regarding the product, such as emotions, sentiment, and satisfaction. The most recent research in psychology reveals that cognition analytics research based on Classical Probability Theory (CPT) and statistical learning, which is used to evaluate people’s cognitive states, is limited due to their reliance on rational decision-making. However, the cognitive attitudes of customers are characterized by uncertainty and entanglement, resulting in irrational decision-making bias. What is captured by traditional CPT-based data science in the context of cognition aspects of CX analytics is only a small portion of what should be captured. Current CX analytics efforts fall far short of their full potential. In this paper, we set a novel research direction for CX analytics by Quantum Probability Theory (QPT). QPT-based analytics have been introduced recently in psychology research and reveal better cognition assessment under uncertainty, with a high level of irrational behavior. Adopting recent advances in the psychology domain, this paper develops a vision and sets a research agenda for expanding the application of CX analytics by QPT to overcome CPT shortcomings, identifies research areas that contribute to the vision, and proposes elements of a future research agenda. To stimulate debate and research QPT-CX analytics, we attempt a preliminary characterization of the novel method by introducing a QPT-based rich mathematical framework for CX cognitive modeling based on quantum superposition, Bloch sphere, and Hilbert space. We demonstrate the implementation of the QPT-CX model by the use case of customers’ emotional motivator assessments while implementing quantum vector space with a set of mathematical axioms for CX analytics. Finally, we outline the key advantages of quantum CX over classical by supporting theoretical proof for each key.

COSMONYMIC COMPONENT OF UKRAINIAN ONYMIC IDENTITY

The article is dedicated to the study of the cosmic component of the Ukrainian onymic identity. The concept of identity is just as relevant in the movement as one of the central sources of modern Ukrainian scientific thought. The analysis of the mechanisms of proper names in the context of cultural identity is the task of our research. The purpose of the article is to consider the issue of collective identity and how it is reflected in the field of proper names, namely - cosmonyms. Cosmonyms are one of the peripheric proper names. Their functioning and place in the onymic space of the Ukrainian cultural sphere is still understudied. In our opinion, the associative aspect of the cosmonymic sphere needs further studies, where the main segment should be the concept of cosmonymic space and which segments are the main levers of influence on cultural life and onymic identity. Identity can be portrayed as an ‘individual’ sense of self, defined by a) a set of physical, psychological and interpersonal characteristics that are not fully shared with any other person, b) a range of affiliations (eg ethnicity) and social roles. Onymic identity is based on collective identity as an understanding and isolation of a group in the environment. The vision of the similarity between individual and collective identities is based on the understanding that identity is never exclusively individual. Collective identity is rooted in a number of individual identities, which in turn are chosen by the individual. Thus, a person, based on a set of internal traits and social roles, becomes part of the socio-cultural collective identity, defines it and can influence it. Onymic identity is the realization of proper names in the collective discourse, by means of which a certain group separates itself from a number of other social groups, creates its own socio-cultural identity, which is based on a general ethnic paradigm. Free associative onymic experiment is one of the reliable methodological practices of onomastic search. The experiment was conducted with Ukrainianspeaking informants, a total of 200 people (100 informants each time). The peculiarity of our study is that the experiment was conducted in two stages in 2006 and 2021. Two stimuli were chosen for this study: cosmonyms – proper names for cosmic bodies, Uranus and Pluto. The results of the experiment are summarized in quantitative terms: the total number of all reactions/the total number of different non-duplicate reactions/the total number of anonymous reactions/ anonymous non-repetitive reactions/rejections. The number of all reactions together should ideally be 100, but may be greater if some informants provided more than one reaction, or less if there were refusals. Regarding the repertoire of different types of reactions that were proposed by respondents in two stages of the experiment on the stimulus Uranus, it is marked by a certain stereotype. The most common type of reaction is, as expected, hyperonymic, due to the search for the owner of the mental lexicon of denotation, which is denoted by the stimulus. The total number of such reactions is 109 (52 in 2006 and 57 in 2021). In the second place are meronymic reactions - 23 (9 in 2006 and 14 in 2021). The third place is occupied by qualitative reactions with a total of 16 (13 in 2006 and 3 in 2021). Contiguous reactions were not numerous, but almost always present in the experimental materials - 8 reactions (2 and 6, respectively). Causative reactions took place only in 2006 - 4 times. The repertoire of types of reactions to the stimulus Pluto is very similar. Hyperonymic reactions prevail 113 (51 in 2006 and 62 in 2021), which realize the dominant function of cognition. Meronimic reactions totaling 20 (9 in 2006 and 11 in 2021) and contiguous reactions totaling 4 (2 in 2006 and 2 in 2021), as well as isolated cases of qualitative (1 in 2006) and synonymous (1 in 2021) were also identified. Thus, the analysis of the associative field of the studied stimuli cosmonyms made it possible to establish universal memoplexes as the results of collective creativity and collective experience, which is reflected in the collective memory. The obtained results testify the isomorphism of the dominant domains of the structure of associative fields of the stimuli: COSMOS - CHEMICAL ELEMENT - GOD, which occurs contrary to the theonymic origin of the cosmonyms and the cosmonymic origin of the name of the chemical element. We see further prospects for the study in expanding the repertoire of onymyc stimuli in order to investigate the memopletic nature of their associative fields.

Volumes of Polyhedra in Non-Euclidean Spaces of Constant Curvature

Computation of the volumes of polyhedra is a classical geometry problem known since ancient mathematics and retaining its importance until present time. Deriving volume formulas for 3-dimensional non-Euclidean polyhedra of a given combinatorial type is a very difficult problem. Nowadays, it is fully solved for a tetrahedron, the most simple polyhedron in the combinatorial sense. However, it is well known that for a polyhedron of a special type its volume formula becomes much simpler. This fact was noted by Lobachevskii who found the volume of the so-called ideal tetrahedron in hyperbolic space (all vertices of this tetrahedron are on the absolute). In this survey, we present main results on volumes of arbitrary non-Euclidean tetrahedra and polyhedra of special types (both tetrahedra and polyhedra of more complex combinatorial structure) in 3-dimensional spherical and hyperbolic spaces of constant curvature K = 1 and K = −1, respectively. Moreover, we consider the new method by Sabitov for computation of volumes in the hyperbolic space (described by the Poincaré model in the upper half-space). This method allows one to derive explicit volume formulas for polyhedra of arbitrary dimension in terms of coordinates of vertices. Considering main volume formulas for non-Euclidean polyhedra, we will give proofs (or sketches of proofs) for them. This will help the reader to get an idea of basic methods for computation of volumes of bodies in non-Euclidean spaces of constant curvature.

ПОЛИГАРМОНИЧЕСКОЕ УРАВНЕНИЕ НА СФЕРЕ ТРЕХМЕРНОГО ПРОСТРАНСТВА

We consider a nonhomogeneous polyharmonic equation on the unit sphere in the three-dimensional Euclidean space. Sobolev spherical spaces act as functional classes in which solutions to the spherical polyharmonic equation are sought. It is proved that for a given right-hand side of the equation, which is orthogonal to the identically-one function, the solution to the equation exists in the spherical Sobolev space and is unique there. We establish that for small variations of the right-hand side of the polyharmonic equation under consideration, its solutions change little in the corresponding norm.

Genome-wide identification and characterization of the KCS gene family in sorghum (Sorghum bicolor (L.) Moench).

The aboveground parts of plants are covered with cuticle, a hydrophobic layer composed of cutin polyester and cuticular wax that can protect plants from various environmental stresses. β-Ketoacyl-CoA synthase (KCS) is the key rate-limiting enzyme in plant wax synthesis. Although the properties of KCS family genes have been investigated in many plant species, the understanding of this gene family in sorghum is still limited. Here, a total of 25 SbKCS genes were identified in the sorghum genome, which were named from SbKCS1 to SbKCS25. Evolutionary analysis among different species divided the KCS family into five subfamilies and the SbKCSs were more closely related to maize, implying a closer evolutionary relationship between sorghum and maize. All SbKCS genes were located on chromosomes 1, 2, 3, 4, 5, 6, 9 and 10, respectively, while Chr 1 and Chr 10 contained more KCS genes than other chromosomes. The prediction results of subcellular localization showed that SbKCSs were mainly expressed in the plasma membrane and mitochondria. Gene structure analysis revealed that there was 0–1 intron in the sorghum KCS family and SbKCSs within the same subgroup were similar. Multiple cis-acting elements related to abiotic stress, light and hormone response were enriched in the promoters of SbKCS genes, which indicated the functional diversity among these genes. The three-dimensional structure analysis showed that a compact spherical space structure was formed by various secondary bonds to maintain the stability of SbKCS proteins, which was necessary for their biological activity. qRT-PCR results revealed that nine randomly selected SbKCS genes expressed differently under drought and salt treatments, among which SbKCS8 showed the greatest fold of expression difference at 12 h after drought and salt stresses, which suggested that the SbKCS genes played a potential role in abiotic stress responses. Taken together, these results provided an insight into investigating the functions of KCS family in sorghum and in response to abiotic stress.