Keldysh Institute of Applied Mathematics

With students of Physics chiefly in mind, we have collected the material on special functions that is most important in mathematical physics and quan­ tum mechanics. We have not attempted to provide the most extensive collec­ tion possible of information about special functions, but have set ourselves the task of finding an exposition which, based on a unified approach, ensures the possibility of applying the theory in other natural sciences, since it pro­ vides a simple and effective method for the independent solution of problems that arise in practice in physics, engineering and mathematics. For the American edition we have been able to improve a number of proofs; in particular, we have given a new proof of the basic theorem (§3). This is the fundamental theorem of the book; it has now been extended to cover difference equations of hypergeometric type (§§12, 13). Several sections have been simplified and contain new material. We believe that this is the first time that the theory of classical or­ thogonal polynomials of a discrete variable on both uniform and nonuniform lattices has been given such a coherent presentation, together with its various applications in physics.

Progress in the area of MHD stability and disruptions, since the publication of the 1999 ITER Physics Basis document (1999 Nucl. Fusion 39 2137-2664), is reviewed. Recent theoretical and experimental research has made important advances in both understanding and control of MHD stability in tokamak plasmas. Sawteeth are anticipated in the ITER baseline ELMy H-mode scenario, but the tools exist to avoid or control them through localized current drive or fast ion generation. Active control of other MHD instabilities will most likely be also required in ITER. Extrapolation from existing experiments indicates that stabilization of neoclassical tearing modes by highly localized feedback-controlled current drive should be possible in ITER. Resistive wall modes are a key issue for advanced scenarios, but again, existing experiments indicate that these modes can be stabilized by a combination of plasma rotation and direct feedback control with non-axisymmetric coils. Reduction of error fields is a requirement for avoiding non-rotating magnetic island formation and for maintaining plasma rotation to help stabilize resistive wall modes. Recent experiments have shown the feasibility of reducing error fields to an acceptable level by means of non-axisymmetric coils, possibly controlled by feedback. The MHD stability limits associated with advanced scenarios are becoming well understood theoretically, and can be extended by tailoring of the pressure and current density profiles as well as by other techniques mentioned here. There have been significant advances also in the control of disruptions, most notably by injection of massive quantities of gas, leading to reduced halo current fractions and a larger fraction of the total thermal and magnetic energy dissipated by radiation. These advances in disruption control are supported by the development of means to predict impending disruption, most notably using neural networks. In addition to these advances in means to control or ameliorate the consequences of MHD instabilities, there has been significant progress in improving physics understanding and modelling. This progress has been in areas including the mechanisms governing NTM growth and seeding, in understanding the damping controlling RWM stability and in modelling RWM feedback schemes. For disruptions there has been continued progress on the instability mechanisms that underlie various classes of disruption, on the detailed modelling of halo currents and forces and in refining predictions of quench rates and disruption power loads. Overall the studies reviewed in this chapter demonstrate that MHD instabilities can be controlled, avoided or ameliorated to the extent that they should not compromise ITER operation, though they will necessarily impose a range of constraints.

Guiding of relativistically intense laser pulses with peak power of 0.85 PW over 15 diffraction lengths was demonstrated by increasing the focusing strength of a capillary discharge waveguide using laser inverse bremsstrahlung heating. This allowed for the production of electron beams with quasimonoenergetic peaks up to 7.8 GeV, double the energy that was previously demonstrated. Charge was 5 pC at 7.8 GeV and up to 62 pC in 6 GeV peaks, and typical beam divergence was 0.2 mrad.

Soil organisms, including earthworms, are a key component of terrestrial ecosystems. However, little is known about their diversity, their distribution, and the threats affecting them. We compiled a global dataset of sampled earthworm communities from 6928 sites in 57 countries as a basis for predicting patterns in earthworm diversity, abundance, and biomass. We found that local species richness and abundance typically peaked at higher latitudes, displaying patterns opposite to those observed in aboveground organisms. However, high species dissimilarity across tropical locations may cause diversity across the entirety of the tropics to be higher than elsewhere. Climate variables were found to be more important in shaping earthworm communities than soil properties or habitat cover. These findings suggest that climate change may have serious implications for earthworm communities and for the functions they provide.

Abstract Aim The EUNIS Habitat Classification is a widely used reference framework for European habitat types (habitats), but it lacks formal definitions of individual habitats that would enable their unequivocal identification. Our goal was to develop a tool for assigning vegetation‐plot records to the habitats of the EUNIS system, use it to classify a European vegetation‐plot database, and compile statistically‐derived characteristic species combinations and distribution maps for these habitats. Location Europe. Methods We developed the classification expert system EUNIS‐ESy, which contains definitions of individual EUNIS habitats based on their species composition and geographic location. Each habitat was formally defined as a formula in a computer language combining algebraic and set‐theoretic concepts with formal logical operators. We applied this expert system to classify 1,261,373 vegetation plots from the European Vegetation Archive (EVA) and other databases. Then we determined diagnostic, constant and dominant species for each habitat by calculating species‐to‐habitat fidelity and constancy (occurrence frequency) in the classified data set. Finally, we mapped the plot locations for each habitat. Results Formal definitions were developed for 199 habitats at Level 3 of the EUNIS hierarchy, including 25 coastal, 18 wetland, 55 grassland, 43 shrubland, 46 forest and 12 man‐made habitats. The expert system classified 1,125,121 vegetation plots to these habitat groups and 73,188 to other habitats, while 63,064 plots remained unclassified or were classified to more than one habitat. Data on each habitat were summarized in factsheets containing habitat description, distribution map, corresponding syntaxa and characteristic species combination. Conclusions EUNIS habitats were characterized for the first time in terms of their species composition and distribution, based on a classification of a European database of vegetation plots using the newly developed electronic expert system EUNIS‐ESy. The data provided and the expert system have considerable potential for future use in European nature conservation planning, monitoring and assessment.

We present a detailed comparison of fundamental dark matter halo properties retrieved by a substantial number of different halo finders. These codes span a wide range of techniques including friends-of-friends, spherical-overdensity and phase-space-based algorithms. We

Exact Solutions and Invariant Subspaces of Nonlinear Partial Differential Equations in Mechanics and Physics is the first book to provide a systematic construction of exact solutions via linear invariant subspaces for nonlinear differential operators. Acting as a guide to nonlinear evolution equations and models from physics and mechanics, the book

The physics of the "hot spots" on stellar surfaces and the associated variability of accreting magnetized rotating stars is investigated for the first time using fully three-dimensional magnetohydrodynamic simulations. The magnetic moment of the star is inclined relative to its rotation axis by an angle Theta. A sequence of misalignment angles was investigated, between Theta=0 and 90 degrees.Typically at small Theta the spots are observed to have the shape of a bow which is curved around the magnetic axis, while at largest Theta the spots have a shape of a bar, crossing the magnetic pole. The physical parameters (density, temperature, etc.) increase toward the central regions of the spots. At relatively low density and temperature, the spots occupy approximately 10-20 % of the stellar surface, while at the highest values of these parameters this area may be less than 1 % of the area of the star. The light curves were calculated for different Theta and inclination angles of the disk i. They show a range of variability patterns, including one maximum-per-period curves (at most of angles Theta and i), and two maximum-per-period curves (at large Theta and i). At small Theta, the funnel streams may rotate faster/slower than the star, and this may lead to quasi-periodic variability of the star. The results are of interest for understanding the variability and quasi-variability of Classical T Tauri Stars, millisecond pulsars and cataclysmic variables.

Several nozzle concepts that promise a gain in performance over existing conventional nozzles are discussed in this paper. It is shown that signie cant performance gains result from the adaptation of the exhaust e ow to the ambient pressure. Special attention is then given to altitude-adaptive nozzle concepts, which have recently received new interest in the space industry. Current research results are presented for dual-bell nozzles and other nozzles with devices for forced e ow separation and for plug nozzles with external freestream expansion. In addition, results of former research on nozzles of dual-mode engines such as dual-throat and dual-expander engines and on expansion ‐dee ection nozzles are shown. In general, e ow adaptation induces shocks and expansion waves, which result in exit proe les that are quite different from idealized one-dimensional assumptions. Flow phenomena observed in experiments and numerical simulations during different nozzle operations are highlighted, critical design aspects and operation conditions are discussed, and performance characteristics of selected nozzles are presented. The consideration of derived performance characteristics in launcher and trajectory optimization calculations reveal signie cant payload gains at least for some of these advanced nozzle concepts.

The Russian Academy of Sciences and Federal Space Agency, together with the participation of many international organizations, worked toward the launch of the RadioAstron orbiting space observatory with its onboard 10-m reflector radio telescope from the Baikonur cosmodrome on July 18, 2011. Together with some of the largest ground-based radio telescopes and a set of stations for tracking, collecting, and reducing the data obtained, this space radio telescope forms a multi-antenna ground-space radio interferometer with extremely long baselines, making it possible for the first time to study various objects in the Universe with angular resolutions a million times better than is possible with the human eye. The project is targeted at systematic studies of compact radio-emitting sources and their dynamics. Objects to be studied include supermassive black holes, accretion disks, and relativistic jets in active galactic nuclei, stellar-mass black holes, neutron stars and hypothetical quark stars, regions of formation of stars and planetary systems in our and other galaxies, interplanetary and interstellar plasma, and the gravitational field of the Earth. The results of ground-based and inflight tests of the space radio telescope carried out in both autonomous and ground-space interferometric regimes are reported. The derived characteristics are in agreement with the main requirements of the project. The astrophysical science program has begun.

The physics of the hot spots on stellar surfaces and the associated variability of accreting magnetized rotating stars is investigated for the first time using fully three-dimensional magnetohydrodynamic simulations. The magnetic moment of the star, , is inclined relative to its rotation axis, , by an angle (we call this angle the ‘‘misalignment angle’’), while the disk’s rotation axis is parallel to . A sequence of misalignment angles between 0 and 90 was investigated. The hot spots arise on the stellar surface because of the impact on the surface of magnetically channeled accretion streams. The distribution of different parameters in the hot spots reflects those in the funnel streams near the surface of the star. Typically, at small the spots as observed are shaped like a bow curved around the magnetic axis, while at the largest values of the spots are shaped like a bar crossing the magnetic pole. The physical parameters (density, velocity, temperature, matter, energy fluxes, etc.) increase toward the central regions of the spots; thus, the size of the spots is different at different values of these parameters. At relatively low density and temperature, the spots occupy approximately 10%– 20 % of the stellar surface, while at the highest values of these parameters this area may be less than 1 % of the area of the star. The size of the spots increases with the accretion rate. Rotation of the star leads to the observed variability of brightness. The light curves were calculated for different values of and inclination angles of the

Magnetohydrodynamic (MHD) simulations have been used to study disk accretion to a rotating magnetized star with an aligned dipole moment. Quiescent initial conditions were developed in order to avoid the fast initial evolution seen in earlier studies. A set of simulations was performed for different stellar magnetic moments and rotation rates. Simulations have shown that the disk structure is significantly changed insidea radius r_{br} where magnetic braking is significant. In this region the disk is strongly inhomogeneous. Radial accretion of matter slows as it approaches the area of strong magnetic field and a dense ring and funnel flow form at the magnetospheric radius r_m where the magnetic pressure is equal to the total, kinetic plus thermal, pressure of the matter.

We investigate the launching of outflows from the disc-magnetosphere boundary of slowly and rapidly rotating magnetized stars using axisymmetric and exploratory 3D magnetohydrodynamic simulations. We find long-lasting outflows in the following cases. (1) In the case of slowly rotating stars, a new type of outflow, a conical wind, is found and studied in simulations. The conical winds appear in cases where the magnetic flux of the star is bunched up by the disc into an X-type configuration. The winds have the shape of a thin conical shell with a half-opening angle 30 -40 . About 10-30 per cent of the disc matter flows from the inner disc into the conical winds. The conical winds may be responsible for episodic as well as long-lasting outflows in different types of stars. There is also a low-density, higher velocity component (a jet) in the region inside the conical wind. (2) In the case of rapidly rotating stars (the 'propeller regime'), a two-component outflow is observed. One component is similar to the conical winds. A significant fraction of the disc matter may be ejected into the winds. The second component is a high-velocity, low-density magnetically dominated axial jet where matter flows along the opened polar field lines of the star. The jet has a mass flux of about 10 per cent of that of the conical wind, but its energy flux (dominantly magnetic) can be larger than the energy flux of the conical wind. The jet's angular momentum flux (also dominantly magnetic) causes the star to spin down rapidly. Propeller-driven outflows may be responsible for the jets in protostars and for their rapid spin-down. The jet is collimated by the magnetic force while the conical winds are only weakly collimated in the simulation region. Exploratory 3D simulations show that conical winds are axisymmetric about the rotational axis (of the star and the disc), even when the dipole field of the star is significantly misaligned.

This paper contains a survey of association scheme theory (with its algebraic and analytical aspects) and of its applications to coding theory (in a wide sense). It is mainly concerned with a class of subjects that involve the central notion of the distance distribution of a code. Special emphasis is put on the linear programming method, inspired by the MacWilliams transform. This produces upper bounds for the size of a code with a given minimum distance, and lower bounds for the size of a design with a given strength. The most specific results are obtained in the case where the underlying association scheme satisfies certain well-defined "polynomial properties;" this leads one into the realm of orthogonal polynomial theory. In particular, some "universal bounds" are derived for codes and designs in polynomial type association schemes. Throughout the paper, the main concepts, methods, and results are illustrated by two examples that are of major significance in classical coding theory, namely, the Hamming scheme and the Johnson scheme. Other topics that receive special attention are spherical codes and designs, and additive codes in translation schemes, including Z/sub 4/-additive binary codes.

Digital elevation models (DEMs) provide fundamental depictions of the three-dimensional shape of the Earth’s surface and are useful to a wide range of disciplines. Ideally, DEMs record the interface between the atmosphere and the lithosphere using a discrete two-dimensional grid, with complexities introduced by the intervening hydrosphere, cryosphere, biosphere, and anthroposphere. The treatment of DEM surfaces, affected by these intervening spheres, depends on their intended use, and the characteristics of the sensors that were used to create them. DEM is a general term, and more specific terms such as digital surface model (DSM) or digital terrain model (DTM) record the treatment of the intermediate surfaces. Several global DEMs generated with optical (visible and near-infrared) sensors and synthetic aperture radar (SAR), as well as single/multi-beam sonars and products of satellite altimetry, share the common characteristic of a georectified, gridded storage structure. Nevertheless, not all DEMs share the same vertical datum, not all use the same convention for the area on the ground represented by each pixel in the DEM, and some of them have variable data spacings depending on the latitude. This paper highlights the importance of knowing, understanding and reflecting on the sensor and DEM characteristics and consolidates terminology and definitions of key concepts to facilitate a common understanding among the growing community of DEM users, who do not necessarily share the same background.

We introduce and solve some new problems of efficient reconstruction of an unknown sequence from its versions distorted by errors of a certain type. These erroneous versions are considered as outputs of repeated transmissions over a channel, either a combinatorial channel defined by the maximum number of permissible errors of a given type, or a discrete memoryless channel. We are interested in the smallest N such that N erroneous versions always suffice to reconstruct a sequence of length n, either exactly or with a preset accuracy and/or with a given probability. We are also interested in simple reconstruction algorithms. Complete solutions for combinatorial channels with some types of errors of interest in coding theory, namely, substitutions, transpositions, deletions, and insertions of symbols are given. For these cases, simple reconstruction algorithms based on majority and threshold principles and their nontrivial combination are found. In general, for combinatorial channels the considered problem is reduced to a new problem of reconstructing a vertex of an arbitrary graph with the help of the minimum number of vertices in its metrical ball of a given radius. A certain sufficient condition for solution of this problem is presented. For a discrete memoryless channel, the asymptotic behavior of the minimum number of repeated transmissions which are sufficient to reconstruct any sequence of length n within Hamming distance d with error probability /spl epsiv/ is found when d/n and /spl epsiv/ tend to 0 as n/spl rarr//spl infin/. A similar result for the continuous channel with discrete time and additive Gaussian noise is also obtained.

Sequence variants in gene regulatory regions alter gene expression and contribute to phenotypes of individual cells and the whole organism, including disease susceptibility and progression. Single-nucleotide variants in enhancers or promoters may affect gene transcription by altering transcription factor binding sites. Differential transcription factor binding in heterozygous genomic loci provides a natural source of information on such regulatory variants. We present a novel approach to call the allele-specific transcription factor binding events at single-nucleotide variants in ChIP-Seq data, taking into account the joint contribution of aneuploidy and local copy number variation, that is estimated directly from variant calls. We have conducted a meta-analysis of more than 7 thousand ChIP-Seq experiments and assembled the database of allele-specific binding events listing more than half a million entries at nearly 270 thousand single-nucleotide polymorphisms for several hundred human transcription factors and cell types. These polymorphisms are enriched for associations with phenotypes of medical relevance and often overlap eQTLs, making candidates for causality by linking variants with molecular mechanisms. Specifically, there is a special class of switching sites, where different transcription factors preferably bind alternative alleles, thus revealing allele-specific rewiring of molecular circuitry.

Stationary magnetohydrodynamic (MHD) outflows from a rotating, conducting Keplerian accretion disk threaded by B-field are investigated numerically by time-dependent, axisymmetric (2.5D) simulations using a Godunov-type code. A large class of stationary magneto-centrifugally driven winds are found where matter is accelerated from a thermal speed at the disk to much larger velocity, greater than the fast magnetosonic speed and larger than the escape speed. The flows are approximately spherical outflows with only small collimation within the simulation region. Numerical results are shown to coincide with the theoretical predictions of ideal, axisymmetric MHD to high accuracy. Investigation of the influence of outer boundary conditions, particularly that on the toroidal component of magnetic field shows that the commonly used ``free'' boundary condition leads to artificial magnetic forces which can act to give spurious collimation. New boundary conditions are proposed which do not generate artificial forces. Artificial results may also arise for cases where the Mach cones on the outer boundaries are partially directed into the simulation region.

A new set of special functions, which has a wide range of applications from number theory to integrability of nonlinear dynamical systems, is described. We study multiple orthogonal polynomials with respect to <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p greater-than 1"> <mml:semantics> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>&gt;</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">p &gt; 1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> weights satisfying Pearson’s equation. In particular, we give a classification of multiple orthogonal polynomials with respect to classical weights, which is based on properties of the corresponding Rodrigues operators. We show that the multiple orthogonal polynomials in our classification satisfy a linear differential equation of order <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p plus 1"> <mml:semantics> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">p+1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . We also obtain explicit formulas and recurrence relations for these polynomials.

Compact, tunable, radially symmetric focusing of electrons is critical to laser-plasma accelerator (LPA) applications. Experiments are presented demonstrating the use of a discharge-capillary active plasma lens to focus 100-MeV-level LPA beams. The lens can provide tunable field gradients in excess of 3000 T/m, enabling cm-scale focal lengths for GeV-level beam energies and allowing LPA-based electron beams and light sources to maintain their compact footprint. For a range of lens strengths, excellent agreement with simulation was obtained.