Alexander Kuznetsov

A NOTE ON A SYMPLECTIC STRUCTURE ON THE SPACE OF G-MONOPOLES

Abstract:Let G be a semisimple complex Lie group with a Borel subgroup B. Let X=G/B be the flag manifold of G. Let be the projective line. Let . The moduli space of G-monopoles of topological charge α is naturally identified with the space of based maps from (C,∞) to (X,B) of degree α. The moduli space of G-monopoles carries a natural hyperkähler structure, and hence a holomorphic symplectic structure. It was explicitly computed by R. Bielawski in case G=SLn. We propose a simple explicit formula for another natural symplectic structure on . It is tantalizingly similar to R. Bielawskis formula, but in general (rank >1) the two structures do not coincide. Let P⊃G be a parabolic subgroup. The construction of the Poisson structure on generalizes verbatim to the space of based maps . In most cases the corresponding map is not an isomorphism, i.e. splits into nontrivial symplectic leaves. These leaves are explicilty described.

Trees Associated with the Motzkin Numbers

We consider plane rooted trees onn+1 vertices without branching points on odd levels. The number of such trees in equal to the Motzkin numberMn. We give a bijective proof of this statement.

A method for estimation of lateral and vertical mobility of platycoelous vertebrae of tetrapods

A new method for the estimation of dorsoventral and lateral mobility of platycoelous vertebrae with V-shaped (radial) articular facets on the zygapophyses is developed. This vertebral pattern is observed in dinosaurs, some other fossil reptiles, and in the cervical and lumbar regions of mammals. Based on theoretical biomechanical analysis of the intervertebral discs and articulations between zygapophyses, the estimation formulas are developed and calibrated, using precise measurements of mobility between cervical vertebrae of domestic sheep. The method is applied to the presacral vertebrae of the horned dinosaur Protoceratops andrewsi. In its cervical, lumbar, and anterior thoracic regions, the differences between the calculated amplitudes of movements and the sought true values are expected to range within ±5°. As compared to the sheep, Protoceratops shows a greater lateral mobility in the presacral region and reduced vertical mobility in the cervical region.

Kinematic constituents of the extreme head turn of Strix aluco estimated by means of CT-scanning

To analyze extreme sideways turn of the head in owls, a total fresh specimen of Strix aluco was frozen in respective posture and CT-scanned. The maximum turn to one side was found to be 360°, provided that the head is drawn into the shoulders. 160° of this full turn are ensured by the neck axial rotation (this includes ~90° twist of the head relative to epistropheus and, posterior to it, less than 15° per every cervical joint), and the rest 200° are ensured by combination of dorsal and lateral flexion. The 15° limit is overcome in five joints in respect of dorsiflexion, and in six joints in respect of lateral flexion. So large a degree of lateral mobility is unusual among birds, and is appreciated as a crucial adaptation to extreme head turning.

Flight of Mammals: From Terrestrial Limbs to Wings

The evolutionary acquisition of flapping flight in mammals remains one of the unresolved questions of biology. Currently, no consensus as to the morphofunctional steps through which mammals passed to gain the ability to fly by flapping wings has been reached. Flight of Mammals: From Terrestrial Limbs to Wings is the result of several years of research aimed to fill this gap in the literature. Its conclusions are based on original data obtained by dissections of musculoskeletal system of a number of species and on a biomechanical analysis of these data. In addition to a thorough discussion of anatomy and the means through which mammals acquired flapping flight, more than two-hundred detailed line drawings and images provide a picture of the mechanisms of flight in bats and colugos unavailable in any other source. The book is of interest to a wide range of biologists, not only to those who study bats. The methods and approaches used by the authors can be also applied to other groups of mammals in order to create morphofunctional scenarios of their evolution. Authors Aleksandra A. Panyutina, Leonid P. Korzun, and Alexander N. Kuznetsov are all followers of the scientific school of functional morphology developed at Department of Vertebrate Zoology, Faculty of Biology, Lomonosov Moscow State University, Moscow, Russia

Kinematics of the shoulder girdle in bats

A well known hypothetical scheme of wing move ments in a flying bat was proposed by Hill and Smith [6]. This is probably a compilation of studies of Vaughan [7, 8, etc.] and original data of these authors. According to this scheme (Fig. 1), a significant contri bution to flapping movements of wings is made by joint action of the shoulder girdle and humerus. A key element of this model is the clavicle which is presumed to be mobile in the transverse plane, with an amplitude more than 60°, from an almost vertical position of the clavicle at the beginning of the downstroke to an almost horizontal position at its end. The scapula con nected to its distal end is presumed to slide over the surface of the thorax, circumscribing respective arch. The authors of this scheme believe that, in the upper position of the wing, the scapula lies horizontally on the dorsal surface of the thorax and, in the lower posi tion, it is located lateral to the thorax in an almost ver tical (parasagittal) position. Similar ideas were devel oped in [1, etc.].

Kinematics of Chiropteran Shoulder Girdle in Flight

New data on the mechanisms of movements of the shoulder girdle and humerus of bats are described; potential mobility is compared to the movements actually used in flight. The study was performed on the basis of morphological and functional analysis of anatomical specimens of 15 species, high speed and high definition filming of two species and X‐ray survey of Rousettus aegyptiacus flight. Our observations indicate that any excursions of the shoulder girdle in bats have relatively small input in the wing amplitude. Shoulder girdle movements resemble kinematics of a crank mechanism: clavicle plays the role of crank, and scapula—the role of connecting rod. Previously described osseous “locking mechanisms” in shoulder joint of advanced bats do not affect the movements, actually used in flight. The wing beats in bats are performed predominantly by movements of humerus relative to shoulder girdle, although these movements occupy the caudal‐most sector of available shoulder mobility. Anat Rec, 296:382–394, 2013.

Forelimb Morphology of Colugos

The chapter includes four subchapters—Wing Membrane, Skeleton, Joints and Musculature, which describe, respectively, the skin membrane, osteology, syndesmology and myology of the shoulder girdle and forelimb in both extant species of the order Dermoptera. Such detailed morphological data were never published before; our current knowledge of colugo’s anatomy is based on one description of Leche with only 37 figures for all organs, summaries of most interesting features made by Chapman and Grasse and study of the wrist osteology by Stafford and Thorington. The description is supplied with 57 original illustrations—49 black-and-white drawings of myological dissections, 7 grayscale and 1 colored photos of the skeleton.

Functional Analysis of Locomotor Apparatus of Colugos

Various aspects of biology of colugos are considered, which have an influence upon their peculiar locomotion. The osteological, syndesmological and myological features described in Chap 2, are treated in accordance with the major types of locomotion of these animals (gliding, running and clinging on thick tree trunks, and suspending under branches). The corresponding mobility of the shoulder girdle elements is analyzed. Static models of the muscular forces in the forelimb and shoulder girdle are developed for the cases of gliding and clinging onto tree trunks. The chapter includes the following subchapters: Some Biological Aspects of Colugos; Gliding; Climbing up Trunks; Climbing under Branches; Mobility of Shoulder Girdle; Static Analysis of Clinging onto Trunk; Static Analysis of Gliding. It is supplied with 4 grayscale and 11 colored illustrations, which include original drawings representing kinematic and static models being considered, and the photos of alive flying lemurs which were kindly supplied by colleagues from Singapore.

Functional Analysis of Locomotor Apparatus of Bats

Various aspects of bat locomotion are considered, including terrestrial one. Wingbeat cycle and interaction of the wing with the air are discussed in detail. A new model of the shoulder girdle mobility in flight is established, which differs significantly from the common point of view. Static model of the muscular forces in the forelimb and shoulder girdle is developed for the case of mid-downstroke in forward flight. Specific features of the forelimb musculature in bats are interpreted in functional terms. The chapter includes the following subchapters: Locomotor Features of Chiropterans; Kinematics of Chiropteran Wing; Interaction of Wing with Air; Internal Biomechanics of Wing; Static Analysis of Downstroke. It is supplied with 2 grayscale and 10 colored illustrations, which include the photos of alive bats, original drawings representing kinematic and static models being considered, and additional X-ray frame sequences of the wingbeat cycle of Rousettus aegyptiacus.