Special Basis Research Articles

Tree Normal Forms for Quiver Representations

We explore methods for constructing normal forms of indecomposable quiver representations. The first part of the paper develops homological tools for recursively constructing families of indecomposable representations from indecomposables of smaller dimension vector. This is then specialized to the situation of tree modules, where the existence of a special basis simplifies computations and gives nicer normal forms. Motivated by a conjecture of Kac, we use this to construct cells of indecomposable representations as deformations of tree modules. The second part of the paper develops geometric tools for constructing cells of indecomposable representations from torus actions on moduli spaces of representations. As an application, we combine these methods to construct families of indecomposables grouped into cells. These actually give a normal form for all indecomposables of certain roots.

Automated solution of first order factorizable systems of differential equations in one variable

We present an algorithm which allows to solve analytically linear systems of differential equations which factorize to first order. The solution is given in terms of iterated integrals over an alphabet where its structure is implied by the coefficient matrix of the differential equations. These systems appear in a large variety of higher order calculations in perturbative Quantum Field Theories. We apply this method to calculate the master integrals of the three-loop massive form factors for different currents, as an illustration, and present the results for the vector form factors in detail. Here the solution space emerging is given by the cyclotomic harmonic polylogarithms and their associated special constants. No special basis representation of the master integrals is needed. The algorithm can be applied as well to more general cases factorizing at first order, which are based on more general alphabets, iterated integrals and associated constants.

Isogeometric boundary element method for crack propagation based on Bézier extraction of NURBS

Isogeometric boundary element method (IGABEM) based on Bézier extraction of non-uniform rational B-splines (NURBS) is developed to analyze the crack propagation and the interaction between inclusions and cracks in 2D infinite isotropic medium. We use isogeometric analysis (IGA) with NURBS to describe the geometric boundaries of inclusions and cracks which can be represented exactly no matter how coarse the discretization of the studied bodies is, so there are no geometric errors. Meanwhile, the same NURBS basis functions are used to approximate the unknown boundary quantities. For the crack boundary, the collocation points are moved inside the crack elements to meet the continuity requirement, i.e., discontinuous quadratic elements are applied on the crack boundary. Then, in order to obtain accurate stress intensity factors (SIFs), special crack tip basis functions are adopted to simulate the r variation of displacements near the crack tips. Some numerical examples are given and the obtained results have been compared with the results in the references.

Gravitational waves at their own gravitational speed

Gravitational waves propagate at the speed of light in general relativity, because of their special relativistic basis. However, light propagation is linked to the electromagnetic phenomena, with the permittivity and permeability constants as the determining factors. Is there a deeper reason why waves in a geometric theory of gravity propagate at a speed determined by electromagnetic constants? What is the relation between gravity’s own constants and the speed of gravitational waves? Our attempt to answer these fundamental questions takes us far and deep into the universe.

History states of systems and operators

We discuss some fundamental properties of discrete system-time history states. Such states arise for a quantum reference clock of finite dimension and lead to a unitary evolution of system states when satisfying a static discrete Wheeler-DeWitt-type equation. We consider the general case where system-clock pairs can interact, analyzing first their different representations and showing there is always a special clock basis for which the evolution for a given initial state can be described by a constant Hamiltonian $H$. It is also shown, however, that when the evolution operators form a complete orthogonal set, the history state is maximally entangled for any initial state, as opposed to the case of a constant $H$, and can be generated with a simple two-clock setting. We then examine the quadratic system-time entanglement entropy, providing an analytic evaluation and showing it satisfies strict upper and lower bounds determined by the energy spread and the geodesic evolution connecting the initial and final states. We finally show that the unitary operator that generates the history state can itself be considered as an operator history state, whose quadratic entanglement entropy determines its entangling power. Simple measurements on the clock enable to efficiently determine overlaps between system states and also evolution operators at any two times.

Taylor state dynamos found by optimal control: axisymmetric examples

Earth’s magnetic field is generated in its fluid metallic core through motional induction in a process termed the geodynamo. Fluid flow is heavily influenced by a combination of rapid rotation (Coriolis forces), Lorentz forces (from the interaction of electrical currents and magnetic fields) and buoyancy; it is believed that the inertial force and the viscous force are negligible. Direct approaches to this regime are far beyond the reach of modern high-performance computing power, hence an alternative ‘reduced’ approach may be beneficial. Taylor (Proc. R. Soc. Lond. A, vol. 274 (1357), 1963, pp. 274–283) studied an inertia-free and viscosity-free model as an asymptotic limit of such a rapidly rotating system. In this theoretical limit, the velocity and the magnetic field organize themselves in a special manner, such that the Lorentz torque acting on every geostrophic cylinder is zero, a property referred to as Taylor’s constraint. Moreover, the flow is instantaneously and uniquely determined by the buoyancy and the magnetic field. In order to find solutions to this mathematical system of equations in a full sphere, we use methods of optimal control to ensure that the required conditions on the geostrophic cylinders are satisfied at all times, through a conventional time-stepping procedure that implements the constraints at the end of each time step. A derivative-based approach is used to discover the correct geostrophic flow required so that the constraints are always satisfied. We report a new quantity, termed the Taylicity, that measures the adherence to Taylor’s constraint by analysing squared Lorentz torques, normalized by the squared energy in the magnetic field, over the entire core. Neglecting buoyancy, we solve the equations in a full sphere and seek axisymmetric solutions to the equations; we invoke $\unicode[STIX]{x1D6FC}$- and $\unicode[STIX]{x1D714}$-effects in order to sidestep Cowling’s anti-dynamo theorem so that the dynamo system possesses non-trivial solutions. Our methodology draws heavily on the use of fully spectral expansions for all divergenceless vector fields. We employ five special Galerkin polynomial bases in radius such that the boundary conditions are honoured by each member of the basis set, whilst satisfying an orthogonality relation defined in terms of energies. We demonstrate via numerous examples that there are stable solutions to the equations that possess a rapidly decreasing spectrum and are thus well-converged. Classic distributions for the $\unicode[STIX]{x1D6FC}$- and $\unicode[STIX]{x1D714}$-effects are invoked, as well as new distributions. One such new $\unicode[STIX]{x1D6FC}$-effect model possesses oscillatory solutions for the magnetic field, rarely before seen. By comparing our Taylor state model with one that allows torsional oscillations to develop and decay, we show the equilibrium state of both configurations to be coincident. In all our models, the geostrophic flow dominates the ageostrophic flow. Our work corroborates some results previously reported by Wu & Roberts (Geophys. Astrophys. Fluid Dyn., vol. 109 (1), 2015, pp. 84–110), as well as presenting new results; it sets the stage for a three-dimensional implementation where the system is driven by, for example, thermal convection.

Hierarchy of transcriptomic specialization across human cortex captured by structural neuroimaging topography.

Hierarchy provides a unifying principle for the macroscale organization of anatomical and functional properties across primate cortex, yet microscale bases of specialization across human cortex are poorly understood. Anatomical hierarchy is conventionally informed by invasive tract-tracing measurements, creating a need for a principled proxy measure in humans. Moreover, cortex exhibits marked interareal variation in gene expression, yet organizing principles of cortical transcription remain unclear. We hypothesized that specialization of cortical microcircuitry involves hierarchical gradients of gene expression. We found that a noninvasive neuroimaging measure—MRI-derived T1w/T2w mapping—reliably indexes anatomical hierarchy, and captures the dominant pattern of transcriptional variation across human cortex. We found hierarchical gradients in expression profiles of genes related to microcircuit function, consistent with monkey microanatomy, and implicated in neuropsychiatric disorders. Our findings identify a hierarchical axis linking cortical transcription and anatomy, along which gradients of microscale properties may contribute to the macroscale specialization of cortical function.

Identifying genomic hotspots of differentiation and candidate genes involved in the adaptive divergence of pea aphid host races.

Identifying the genomic bases of adaptation to novel environments is a long-term objective in evolutionary biology. Because genetic differentiation is expected to increase between locally adapted populations at the genes targeted by selection, scanning the genome for elevated levels of differentiation is a first step towards deciphering the genomic architecture underlying adaptive divergence. The pea aphid Acyrthosiphon pisum is a model of choice to address this question, as it forms a large complex of plant-specialized races and cryptic species, resulting from recent adaptive radiation. Here, we characterized genomewide polymorphisms in three pea aphid races specialized on alfalfa, clover and pea crops, respectively, which we sequenced in pools (poolseq). Using a model-based approach that explicitly accounts for selection, we identified 392 genomic hotspots of differentiation spanning 47.3Mb and 2,484 genes (respectively, 9.12% of the genome size and 8.10% of its genes). Most of these highly differentiated regions were located on the autosomes, and overall differentiation was weaker on the X chromosome. Within these hotspots, high levels of absolute divergence between races suggest that these regions experienced less gene flow than the rest of the genome, most likely by contributing to reproductive isolation. Moreover, population-specific analyses showed evidence of selection in every host race, depending on the hotspot considered. These hotspots were significantly enriched for candidate gene categories that control host-plant selection and use. These genes encode 48 salivary proteins, 14 gustatory receptors, 10 odorant receptors, five P450 cytochromes and one chemosensory protein, which represent promising candidates for the genetic basis of host-plant specialization and ecological isolation in the pea aphid complex. Altogether, our findings open new research directions towards functional studies, for validating the role of these genes on adaptive phenotypes.

Decomposing parasite fitness reveals the basis of specialization in a two-host, two-parasite system.

The ecological specialization of parasites–whether they can obtain high fitness on very few or very many different host species–is a determining feature of their ecology. In order to properly assess specialization, it is imperative to measure parasite fitness across host species; to understand its origins, fitness must be decomposed into the underlying traits. Despite the omnipresence of parasites with multiple hosts, very few studies assess and decompose their specialization in this way. To bridge this gap, we quantified the infectivity, virulence, and transmission rate of two parasites, the horizontally transmitted microsporidians Anostracospora rigaudi and Enterocytospora artemiae, in their natural hosts, the brine shrimp Artemia parthenogenetica and Artemia franciscana. Our results demonstrate that each parasite performs well on one of the two host species (A. rigaudi on A. parthenogenetica, and E. artemiae on A. franciscana), and poorly on the other. This partial specialization is driven by high infectivity and transmission rates in the preferred host, and is associated with maladaptive virulence and large costs of resistance in the other. Our study represents a rare empirical contribution to the study of parasite evolution in multihost systems, highlighting the negative effects of under‐ and overexploitation when adapting to multiple hosts.

AN ADJUSTING APPROACH TO THE DETERMINATION OF THE PERMEABILITY FUNCTIONAL DEPENDENCIES OF ANISOTROPIC COLD-ROLLED ELECTROTECHNICAL STEELS

Purpose. To develop an effective approach for the determination of mathematical functional relationships for the high-precision description of the nonlinear dependences of the permeability on the magnetic flux density, taking into account the anisotropy of modern cold-rolled electrotechnical steel, which are used in the production of new series of power transformers.Methodology. The researches were carried out using interpolation methods, approximation, regression analysis, mathematical physics, electromagnetic field theory.Findings. The methods of functional conception of nonlinear characteristics of magnetization of ferromagnetic materials, including cold-rolled anisotropic electrotechnical steels, are researched. An adjustment approach is proposed to increase the dimension of the initial data, based on the spline interpolation method on the Hermite polynomial basis for the digital data of the magnetization characteristics. This excludes an increase of the errors in determining the permeabilities in the sections between the nodal values of the magnetic flux density. For a mathematical description of the dependences of the permeability on the magnetic flux density, a special functional basis with Gaussian functions and additional error functions has been developed, that ensures the continuity of derivatives of various orders at fixed angular values between the rolling direction of anisotropic electrical steel and magnetic flux. The method of nonlinear regression in the structure of function package of the Mathcad software for the high-precision determination of regression coefficient vectors in functional descriptions of permeabilities is applied. With the help of verification and validation of the results, which adjusting nonlinear regression with computational data using the generic method and with experimental data, a significant decrease in the relative errors that do not exceed 1.62% for fixed angles between the direction of rolling of anisotropic electrical steel and magnetic flux is obtained. The application of the new regression equations substantially improves the conditions for the field simulation of electromagnetic fields in the open-circuited mode of the power transformer in the Comsol Multiphysics software, and ensures the stability of iterative computing processes.Originality. A new approach to increasing the dimension of input arrays, based on spline interpolation of digital data arrays of magnetization characteristics for fixed angles between the directions of rolling of the electrical steel and magnetic flux is implemented. For nonlinear regression equations, a new functional basis with Gaussian functions and additional error functions for mathematical descriptions with continuous derivatives for the dependences of the permeabilities of anisotropic cold-rolled electrotechnical steels on the magnetic flux density with high accuracy in the range of angles variation between the rolling directions and magnetic flux from 0o to 90o.Practical value. The approaches and techniques for the basis of Gaussian functions and additional error functions proposed in this paper make it possible to significantly improve the accuracy of determining the nonlinear characteristics of anisotropic electrotechnical steels. Also to reduce the relative errors to 1.62% when the angles between directions of rolling of the electrotechnical steels and magnetic flux vary from 0° to 90°.

A New Type of a Wavelet Neural Network

Wavelet transformation uses a special basis widely known for its unique properties, the most important of which are its compactness and multiresolution (wavelet functions are produced from the mother wavelet by transition and dilation). Wavelet neural networks (WNN) use wavelet functions to decompose the approximated function. However, for a standard wavelet basis with fixed transition and dilation coefficients, the decomposition may be not optimal. If no inverse transformation is needed, the values of transition and dilation coefficients may be determined during network training, and the windows corresponding to various wavelet functions may overlap. In this study, we suggest a new type of a WNN—Adaptive Window WNN (AWWNN), designed primarily for signal processing, in which window positions and wavelet levels are determined with a special iterative procedure. Two modifications of this new type of WNN are tested against linear model and multi-layer perceptron on Mackey-Glass benchmark problem.

Special bases for solutions of a generalized Maxwell system in 3‐dimensional space

The aim of this paper is to study complete polynomial systems in the kernel space of conformally invariant differential operators in higher spin theory. We investigate the kernel space of a generalized Maxwell operator in 3‐dimensional space. With the already known decomposition of its homogeneous kernel space into 2 subspaces, we investigate first projections from the homogeneous kernel space to each subspace. Then, we provide complete polynomial systems depending on the given inner product for each subspace in the decomposition. More specifically, the complete polynomial system for the homogenous kernel space is an orthogonal system wrt a given Fischer inner product. In the case of the standard inner product in L2 on the unit ball, the provided complete polynomial system for the homogeneous kernel space is a partially orthogonal system. Further, if the degree of homogeneity for the respective subspaces in the decomposed kernel spaces approaches infinity, then the angle between the 2 subspaces approaches π/2.

On quaternionic tori and their moduli space

Quaternionic tori are defined as quotients of the skew field \mathbb H of quaternions by rank-4 lattices. Using slice regular functions, these tori are endowed with natural structures of quaternionic manifolds (in fact quaternionic curves), and a fundamental region in a 12-dimensional real subspace is then constructed to classify them up to biregular diffeomorphisms. The points of the moduli space correspond to suitable special bases of rank-4 lattices, which are studied with respect to the action of the group GL (4, \mathbb Z) , and up to biregular diffeomeorphisms. All tori with a non trivial group of biregular automorphisms – and all possible groups of their biregular automorphisms – are then identified, and recognized to correspond to five different subsets of boundary points of the moduli space.

A new singular element for evaluating stress intensity factors of V-shaped notches under mixed-mode load

In this paper, a new singular element is presented to evaluate stress intensity factors of V-shaped notches subjected to mixed-mode load. The proposed element takes into account special variation of displacements in the vicinity of the notch tip. The singularity at notch tip is variable unlike the crack problem where the displacements around the crack tip have variation of square root of r. In the proposed method, special basis functions considering the singularity order at notch tip are incorporated into the shape functions of the new element, and the singularity order is determined by the included angle of the notch. With the new element, more accurate displacement and stress fields in the neighborhood of the notch tip can be obtained, thus the stress intensity factors are computed more accurately. Accurate stress intensity factors are important for the V-notched structures to develop a fracture criterion. Numerical examples have demonstrated the accuracy and efficiency of the proposed method.

Dietary divergence is associated with increased intra-specific competition in a marine predator

Optimal foraging theory predicts that when food is plentiful all individuals should take a small range of preferred prey types, but as competition increases less preferred prey will be included in the diet. This dietary switching may not be uniform among individuals, which produces discrete dietary clusters. We tested this hypothesis for gentoo penguins at Bird Island, South Georgia, using stable isotope analysis and biologging. Competition, in the form of the density of foraging dives, increased markedly from incubation to chick-rearing owing to increased foraging effort. Birds responded behaviourally by exploiting a greater portion of the available foraging radius and increasing dive depths. Dietary niche width doubled and two discrete dietary clusters appeared; one comprising birds that consumed mostly krill and another that ate a greater proportion of demersal fish. There were no differences in morphology between the dietary classes, but birds in the fish class had a tendency to dive deeper, which suggests a behavioural basis for specialization. Our findings are consistent with the hypothesis that intra-specific competition expands the population’s dietary niche width and drives divergence in diets among individuals.

Genomewide signatures of selection in Epichloë reveal candidate genes for host specialization.

Host specialization is a key process in ecological divergence and speciation of plant-associated fungi. The underlying determinants of host specialization are generally poorly understood, especially in endophytes, which constitute one of the most abundant components of the plant microbiome. We addressed the genetic basis of host specialization in two sympatric subspecies of grass-endophytic fungi from the Epichloë typhina complex: subsp. typhina and clarkii. The life cycle of these fungi entails unrestricted dispersal of gametes and sexual reproduction before infection of a new host, implying that the host imposes a selective barrier on viability of the progeny. We aimed to detect genes under divergent selection between subspecies, experiencing restricted gene flow due to adaptation to different hosts. Using pooled whole-genome sequencing data, we combined FST and DXY population statistics in genome scans and detected 57 outlier genes showing strong differentiation between the two subspecies. Genomewide analyses of nucleotide diversity (π), Tajima's D and dN/dS ratios indicated that these genes have evolved under positive selection. Genes encoding secreted proteins were enriched among the genes showing evidence of positive selection, suggesting that molecular plant-fungus interactions are strong drivers of endophyte divergence. We focused on five genes encoding secreted proteins, which were further sequenced in 28 additional isolates collected across Europe to assess genetic variation in a larger sample size. Signature of positive selection in these isolates and putative identification of pathogenic function supports our findings that these genes represent strong candidates for host specialization determinants in Epichloë endophytes. Our results highlight the role of secreted proteins as key determinants of host specialization.

ПОВЫШЕНИЕ КАЧЕСТВА ЖИЗНИ В МОНОГОРОДАХ НА ОСНОВЕ РАЗВИТИЯ СЕЛЬСКОГО ХОЗЯЙСТВА

The article is devoted to the problem of living standard improvement in single-industry towns on the basis of regional specialization. In modern conditions, much attention is paid to the complex of factors and conditions for the formation of certain areas of economic activity. The North Caucasus Federal District (NCFD) has significant potential for agricultural development, which can contribute to solving the problems of single-industry towns with industrial specialization. These statistics indicate several positive trends in the socio-economic situation in the region: an increase in the number of inhabitants of the rural population, a steady increase in the share of food products and agricultural raw materials in the total volume of Russian imported goods. The main importance of rural areas in the reproductive concept of the North Caucasus regions is determined by the functions of the social-territorial system of communities: industrial, demographic, socio-cultural, nature protection, recreational. Research methods: systematic, formal-logical, analysis and modeling of socioeconomic processes. The scientific novelty of the proposed approach is a comprehensive assessment of agriculture in the region as the main factor in improving the socio-economic situation of single-industry towns. As a result, the research has revealed the positive impact of the industry on the living standards in the single-industry towns of the North Caucasus Federal District (NCFD).

Constructive basis for the development of the extreme zones of Siberia and the Russian arctic

The paper describes the constructive basis for building and structure designing in the difficult regions of Siberia and the Russian Arctic and development of a new approach to the design of buildings and structures in the Russian Arctic in the face of climate change. Severe climate and weather conditions of the northwestern territories of Russia are rapidly changing due to the global warming. The structural changes in development and construction, following this irreversible process, are highlighted and discussed in relation to the foundations and constructive systems. The paper provides an overview of methods and technological approaches to the extreme zone construction. The application of modern methods of design and construction with the involvement of competent "practitioners" increases the pace of construction and reduces its cost; Architectural and constructive solutions should focus on the developing areas of business and tourism. We suggest the possible protective measures against the transgression of the World Ocean and thawing of the permafrost. The term “special Arctic bases”, denoting the new type of settlements in the Russian Arctic, is introduced. The introduction of constructive solutions and the development of planning solutions for the bases will help to ensure a stable activity in the region.

Comparative Genomics of Smut Pathogens: Insights From Orphans and Positively Selected Genes Into Host Specialization.

Host specialization is a key evolutionary process for the diversification and emergence of new pathogens. However, the molecular determinants of host range are poorly understood. Smut fungi are biotrophic pathogens that have distinct and narrow host ranges based on largely unknown genetic determinants. Hence, we aimed to expand comparative genomics analyses of smut fungi by including more species infecting different hosts and to define orphans and positively selected genes to gain further insights into the genetics basis of host specialization. We analyzed nine lineages of smut fungi isolated from eight crop and non-crop hosts: maize, barley, sugarcane, wheat, oats, Zizania latifolia (Manchurian rice), Echinochloa colona (a wild grass), and Persicaria sp. (a wild dicot plant). We assembled two new genomes: Ustilago hordei (strain Uhor01) isolated from oats and U. tritici (strain CBS 119.19) isolated from wheat. The smut genomes were of small sizes, ranging from 18.38 to 24.63 Mb. U. hordei species experienced genome expansions due to the proliferation of transposable elements and the amount of these elements varied among the two strains. Phylogenetic analysis confirmed that Ustilago is not a monophyletic genus and, furthermore, detected misclassification of the U. tritici specimen. The comparison between smut pathogens of crop and non-crop hosts did not reveal distinct signatures, suggesting that host domestication did not play a dominant role in shaping the evolution of smuts. We found that host specialization in smut fungi likely has a complex genetic basis: different functional categories were enriched in orphans and lineage-specific selected genes. The diversification and gain/loss of effector genes are probably the most important determinants of host specificity.

Identification of beef production farms in the Pampas and Campos area that stand out in economic and environmental performance

Worldwide, native grasslands are being converted to non-native pastures and cropland. This process threatens local grassland biomes as well as the livelihoods of farm families that utilize these grasslands.In the Río de la Plata grasslands region meat production and multispecies native grasslands have coexisted for more than 400 years. Low levels of meat productivity and farm income, however, trigger replacement of native grasslands by crops and leys and threaten the survival of local beef farming systems. We studied the economic and environmental performance of beef farming in the region based on interviews and field measurements on 280 case study farms with the following aims: (a) to identify the multi-functional economic and environmental performance of beef farms across the Rio de la Plata grasslands biome; (b) to identify farms with ‘outstanding’ multi-functional performance; (c) to compare performance levels with those found in other regions; and (d) to discuss the implications of the outstanding farms for the development of new systems of meat production. The representativeness of the case study farms was ascertained by comparing them with a farm typology constructed from survey data of 15,448 beef farms situated predominantly on native grasslands in Argentina, Brazil and Uruguay. We identified seven farm types on the basis of farm size, labour, farm specialization, land use and stocking rate.We identified positive deviant farms based on Pareto-ranking and compared these with a classification based on threshold values provided by experts. Out of the 280 farms, 41 were ranked as Pareto-optimal, i.e. outperformed other farms in one or more indicators without being outperformed in other indicators. Out of these, 5 were positive deviants, achieving on average 192 kg LW ha−1 yr−1 of livestock productivity and 201 US$ ha−1 year−1 farm income, having most favourable values for fossil energy consumption, phosphorus balance, carbon footprint and having over 95% of their land under native grassland as a proxy for biodiversity conservation value. Four of these farms belonged to farm types that together represented 55% of the population, suggesting scope for widescale improvement.Compared to the values reported for the OECD countries the beef farming systems of the Río de la Plata grasslands region consume less energy and positive deviant farms demonstrated approximately average livestock productivity and carbon footprint. Increasing livestock productivity in the Rio de la Plata grasslands region resulted in a stronger decline of the carbon footprint without compromising the current negligible levels of fossil fuel energy use. Further elucidation of management practices that lead to positive deviant performance will require modelling of the interaction of pasture and herd dynamics at farm level and is needed to support targeted policy support for sustainable natural grassland-based beef production in the region.