Bundle Projection Research Articles

Derivations of forms along a map: the framework for time-dependent second-order equations

A comprehensive theory is presented concerning derivations of scalar and vector-valued forms along the projection π : R × TM → R × M. It is the continuation of previous work on derivations of forms along the tangent bundle projection and is prompted by the need for a scheme which is adapted to the study of time-dependent second-order equations. The overall structure of the theory closely follows the pattern of this preceding work, but there are many features which are certainly not trivial transcripts of the time-independent situation. As before, a crucial ingredient in the classification of derivations is a non-linear connection on the bundle π. In the presence of a given second-order system, such a connection is canonically defined and gives rise to two important operations: the dynamical covariant derivative, which is a derivation of degree 0, and the Jacobi endomorphism, which is a type (1, 1) tensor field along π. The theory is developed in such a way that all results readily apply to the more general situation of a bundle π : J 1 E → E, where E is fibred over R, but need not be the trivial fibration R × M → R.

A complete set of Bianchi identities for tensor fields along the tangent bundle projection

The various derivations defined along the tangent bundle projection tau in a series of papers by Martinez, Carinena and Sarlet (1992) are expressed as components of a single linear connection Del on E, the tangent bundle of the evolution space E=R*T M. This connection is equivalent to a system of second-order ordinary differential equations (SODE) on M. Using the linear connection, we calculate the torsion and curvature of (E, Del ), the components of which are expressed in terms of the tensors along tau defined by Martinez et. al. From these, the full set of Bianchi identities are calculated. We also show that the generalized Jacobi equation, defined by several authors, is precisely the horizontal component of the conventional Jacobi equation along geodesics of (E, Del ). Finally, we use this to show that if a Jacobi field of the lift of a SODE solution is a certain lift, then it can be extended to a symmetry of the SODE.

Myelination of a key relay zone in the hippocampal formation occurs in the human brain during childhood, adolescence, and adulthood.

A previous study demonstrated that myelination of the superior medullary lamina along the surface of the parahippocampal gyrus is occurring in human brain during adolescence. To further investigate whether postnatal increases of myelination may continue during the second decade and possibly even longer, the extent of myelination in this region has been analyzed in 164 psychiatrically normal individuals aged newborn to 76 years. Cross sections of the hippocampal formation with adjoining hippocampal gyrus were analyzed on a blinded basis using either a global rating scale or measurements of the area of myelin staining. A curvilinear increase in the extent of myelination between the first and sixth decades of life (r = .71 and r = .67, respectively) was observed. When the area of myelination was expressed relative to brain weight, there was a twofold increase between the first and second decades and an additional increase of 60% between the fourth and sixth decades. Female subjects showed a significantly greater degree of myelin staining than did male subjects during the interval of ages 6 to 29 years; however, after the third decade, there were no gender differences in the area of myelin staining. The increased staining of myelin during the first and second decades principally occurred in the subicular region and adjacent portions of the presubiculum. During the fourth through sixth decades, however, it extended to progressively more lateral locations along the surface of the presubiculum. The precise origin(s) of the axons showing progressive myelination is unknown; however, the axons in the subiculum may include some perforant path fibers, while those found in the presubiculum may include cingulum bundle projections. Overall, our data are consistent with the idea that both early and late postnatal increases of myelination occur in a key corticolimbic relay area of the human brain and underscore the importance of applying a neurodevelopmental perspective to the study of psychopathology during childhood, adolescence, and even adulthood.

Development of the hamster serotoninergic system: Cell groups and diencephalic projections

Nuclei of the circadian visual system are extensively innervated by serotoninergic neurons and rhythmicity is modulated by the serotoninergic system. This study investigated the temporal relationships between prenatal origins of serotoninergic cell groups and perinatal innervation of structures in the hamster circadian visual system as well as in the remaining diencephalon. Serotonin-immunoreactive (5-HT-IR) neurons of the B4-B9 complex were first seen on embryonic day 8 (E8). The number of neurons increases sharply by E10 when the first 5-HT-IR cells are evident in the medulla (B1-B3 complex). The distribution of serotoninergic neurons in the hamster brainstem is generally adult-like by E14. Thick 5-HT-IR fibers arch around the mesencephalic flexure at E10 and reach more rostral mesencephalic areas at E11. A branch of the medial forebrain bundle (MFB) projects ventrally toward the retrochiasmatic area; a second branch ascends along the fasciculus retroflexus. Fibers cross the midline in the supraoptic commissure by E12, other arrive in the lateral geniculate region, and a branch of the MFB extends toward the mammillary area. At E13, a periventricular medial thalamic branch of the MFB is seen, axons appear in the supramammillary commissure, and a fine fasciculus between the medial thalamus and intergeniculate leaflet is visible. Lateral, paraventricular, and retrochiasmatic hypothalamic areas and centro- and dorsomedial thalamus are densely innervated at E14. The mammillary area and lateral geniculate body are moderately innervated, and the first fibers appear in the deep laminae of the superior colliculus. The innervation of the suprachiasmatic nuclei, periventricular hypothalamus, and superficial layers of the superior colliculus occurs postnatally. The results are consistent with serotoninergic system development in other species.

On isotropic submanifolds and evolution of quasicaustics

We study classification problems for generic isotropic submanifolds. The classification list of simple and unimodal singularities is obtained and the generic evolutions of quasicaustics in small dimension are classified. Examples encountered in geometric optics are presented. 0. Introduction and preliminaries. Let X be a manifold, and ω be a 2-form on X. The pair (X, ω) is called a symplectic manifold if ω is closed, i.e. dω = 0 and nondegenerate [AM]. The representative model of a symplectic manifold is a cotangent bundle T*M, endowed with the canonical 2-form CUM = dϋu, where the 1-form $M on T*M (Liouville form) is defined by (u, ϋM) = (TπM(u), τ r M (u)) , for each u e TT*M. The mapping TUM is the tangent mapping of %: T*M -» M and ττ*M: TT*M —• T*M is the tangent bundle projection. If (#,•) are local coordinates introduced in M, and (p*, q{) are corresponding local coordinates in T*M then O>M has the normal (Darboux) form coM = EU<tPi*dqi [We].

Derivations of differential forms along the tangent bundle projection II

The study of the calculus of forms along the tangent bundle projection τ, initiated in a previous paper with the same title, is continued. The idea is to complete the basic ingredients of the theory up to a point where enough tools will be available for developing new applications in the study of second-order dynamical systems. A list of commutators of important derivations is worked out and special attention is paid to degree zero derivations having a Leibnitz-type duality property. Various ways of associating tensor fields along τ to corresponding objects on TM are investigated. When the connection coming from a given second-order system is used in this process, two important concepts present themselves: one is a degree zero derivation called the dynamical covariant derivative; the other one is a type (1, 1) tensor field along τ, called the Jacobi endomorphism. It is illustrated how these concepts play a crucial role in describing many of the interesting geometrical features of a given dynamical system, which have been dealt with in the literature.

Derivations of differential forms along the tangent bundle projection

We study derivations of the algebra of differential forms along the tangent bundle projection τ : TM [rarr] M and of the module of vector-valued forms along τ. It is shown that a satisfactory classification and characterization of such derivations requires the extra availability of a connection on TM. By composing the first prolongation of forms and fields along τ with the section of T 2M representing a given second-order vector field Г, one obtains a corresponding calculus of forms on TM associated to this particular Г, which completely explains and generalizes earlier work on this subject.

A geometric approach to Noether's Second Theorem in time-dependent lagrangian mechanics

We analyse Noether's Second Theorem from a geometric viewpoint using the concepts of vector fields and forms along tangent bundle projections.

Vertically of (-1)-Lines in Scrolls Over Smooth Surfaces

AbstractLet S be a smooth surface contained as an ample divisor in a smooth complex projective threefold X, which is a P1 -bundle, and assume that induces OP1 (1) on the fibres of X. The following fact is proven. The restriction to S of the bundle projection of X is exactly the reduction morphism of the pair provided that this one is not a conic bundle. The proof is very simple and does not involve any consideration on the nefness of the adjoint bundle Some applications of the proof are given.

Pharmacotherapy for Akinesia Following Anterior Communicating Artery Aneurysm Hemorrhage.

The dopamine system may be involved in three situations: the nigral projection to the basal ganglia, the mesocortical projection to the anterior cingulate gyrus, or the medial forebrain bundle projection to cortical and limbic sites. Because of the close association of dopamine systems with the known neurological syndromes of akinesia, we elected to treat a patient with akinesia due to rupture of anterior communicating artery (ACA) aneurysm with the dopamine agonist, bromocriptine. This case has important implications for the understanding of brain/behavior relationships as well as for the development of new therapies for patients who have sustained neurological injury.

The Organization of Corticopontine Fibres in Man

Patients with lesions located in the frontal and temporal lobes, in the parieto-temporo-occipital border zone, and in the anterior limb of the internal capsule, did not present ataxia or other cerebellar signs. On the other hand, patients with the ataxic hemiparesis (AH) syndrome had lesions located in the posterior limb of the internal capsule, in the corona radiata and in the central region of the cerebral cortex. These findings in man do not confirm the existence of large frontal (Arnold's bundle) and temporal (Türck's bundle) projections to the pontine nuclei and indicate that the main bulk of corticopontine fibres originates from the central region of the cerebral hemisphere and courses in the posterior limb of the internal capsule. In man, the anatomical organization of corticopontine fibres is therefore similar to that recently demonstrated in animals.

Modular elements of lattices and topological fibration

An l-arrangement is a finite collection of (I 1 )-dimensional vector subspaces of an I-dimensional vector space V over a field K. We restrict our attention to the case that K =C for a while. The complement M of the hyperplanes is an open manifold. In [3], Falk and Randell studied a libration of M: M is called strictly linearly jibered if after a suitable linear change of coordinates the restriction of the projection to the first (I 1) coordinates is a fiber bundle projection with base N the complement of an arrangement in C? ’ and fiber a complex plane C with finitely many points removed. The arrangement is called fiber-type when it allows a successive strictly linearly fibrations finally reaching C* as base. In this case they investigated several topological properties of M. Here we are interested in a combinatorial characterization of a strictly linear fibration in lattice theoretic terms. For this, the concept of modular elements defined by Stanley [S] is crucial: For example, concerning a geometric lattice arising from an essential (= the intersection of all the elements is the origin) arrangement defined over C, the existence of l-dimensional modular element is equivalent to a strictly linear libration of the complement M (2.15). This is a corollary of Theorem (2.9). As another corollary we have a characterization (2.17) that the arrangement is fiber-type if and only if it gives rise to an SS (= supersolvable) lattice, which is defined also by Stanley [9] to be a geometric lattice, which is defined also by Stanley [9] to be a geometric lattice with an ascending maximal chain consisting of modular elements. These are done in Section 2. In Section 3, we state Theorem (3.8), asserting that the algebra A associated with a finite geometric lattice studied in [S, 63 factors as a tensor product of graded subspaces by a modular element, which is Theorem (3.8). This has a topological interpretation (3.9) when the lattice originates from an arrangement defined over C: the factorization of the cohomological ring of M into that of the base and that of the fiber. Since 135 OoOl-8708/86 $7.50

6-Hydroxydopamine lesion of the ventral noradrenergic bundle blocks the effect of amphetamine on hippocampal acetylcholine

Rats treated with amphetamine exhibited an increase in hippocampal acetylcholine turnover, as determined by a mass fragmentographic technique. However, administration of amphetamine to rats which had received stereotaxically placed bilateral injections of 6-hydroxydopamine in the ventral noradrenergic bundle 10 days previously did not increase hippocampal acetylcholine turnover. Because the ventral noradrenergic bundle projects to the septal area, it is suggested that amphetamine increases acetylcholine turnover in the hippocampus by an action on noradrenergic neurons in this pathway.

A lectin horseradish peroxidase study of the origin of ascending fibers in the medial forebrain bundle of the rat. The lower brainstem

The origins of projections within the medial forebrain bundle from the lower brainstem were examined with the horseradish peroxidase technique. Labeled cells were found in at least 15 lower brainstem nuclei following injections of a conjugate or horseradish peroxidase and wheat germ agglutinin at various levels of the medial forebrain bundle. Dense labeling was observed in the following cell groups (from caudal to rostral): A1 (above the lateral reticular nucleus); A2 (mainly within the nucleus of the solitary tract); a distinct group of cell trailing ventrolaterally from the medial longitudinal fasciculus at the level of the rostral pole of the inferior olive; raphe magnus; nucleus incertus; dorsolateral tegmental nucleus (of Castaldi); locus coeruleus; nucleus subcoeruleus; caudal part of the dorsal (lateral) parabrachial nucleus; and raphe pontis. Distinct but light labeling was seen in raphe pallidus and obscurus, nucleus prepositus hypoglossi, nucleus gigantocellularis pars ventralis, and the ventral (medial) parabrachial nucleus. Sparse labeling was observed throughout the medullary and caudal pontine reticular formation. Several lower brainstem nuclei were found to send strong projections along the medial forebrain bundle to very anterior levels of the forebrain. They were: A1, A2, raphe magnus (rostral part), nucleus incertus, dorsolateral tegmental nucleus, raphe pontis and locus coeruleus. With the exception of the locus coeruleus, attention has only recently been directed to the ascending projections of most of the nuclei mentioned above. Evidence was reviewed indicating that fibers from lower brainstem nuclei with ascending medial forebrain bundle projections distribute to widespread regions of the forebrain. It is concluded from the present findings that several medullary cell groups are capable of exerting a direct effect on the forebrain and that the medial forebrain bundle is the major ascending link between the lower brainstem and the forebrain.

The A-B-C (Allocortex-Brainstem-Core) circuitry of endocrine-autonomic integration and regulation:: A proposed hypothesis on the anatomical-functional relationships between estradiol sites of action and peptidergic-aminergic neuronal systems

A sex steroid hormone sensitive brainstem-allocortex axis of neuronal cell groups and projections is recognized with convergent pathways of aminergic-peptidergic messenger systems, which subserves the adjustment for varying reproductive and environmental conditions and the coordination of endocrine-autonomic functions. Main stations in the A-B-C (Allocortex-Brainstem-Core) periventricular axis include the substantia gelatinosa, nucleus (n.) tractus solitarii-dorsal vagal nucleus-area postrema complex, locus ceruleus, n. parabrachialis, central gray and associated raphe nuclei, ventral tegmental area, lateral and periventricular hypothalamus, n. paraventricularis, bed nucleus of the stria terminalis, preoptic-septal nuclei and n. centralis amygdalae with associated amygdaloid nuclei, as well as the ventral and dorsal allocortex. All of these stations and their periventricular and medial forebrain bundle projections contain estradiol sites of action and represent elements of earlier defined periventricular estradiol-target neuron systems. Results from colocalization of 3H estradiol by thaw-mount autoradiography and aminergic and peptidergic messengers by immunohistochemistry or other histochemical techniques indicate direct nuclear effects of estradiol on certain noradrenalin, dopamine, gamma aminobutyric acid, somatostatin, and neurophysin neurons. Additional data about correspondence of estradiol-target neuron accumulations with neuronal sites of peptide messenger production suggest direct effects of estradiol on certain enkephalin, endorphin, corticotropin releasing hormone, adrenalin, serotonin, cholecystokinin, pancreatic polypeptide and gonadotropin releasing hormone neurons—and probably others. As documented for the pituitary, and as an approach to understand varying and dual effects, it is postulated that estradiol activation of brain messenger systems parallels the heterogenous estradiol binding in the A-B-C system. This is expressed in the concept of differential Multiple Activation of Heterogenous Systems (MAHS). Different hormonal states probably result in differential alterations of estradiol receptor numbers in neuronal groups, and account for related changes in the differential stimulation of activational and inhibitory messenger systems, perhaps, providing one explanation for a transfer from a positive to a negative “feedback” and related effects. These effects, thus, include “indirect” actions of the steroid by involving not only one messenger system, but a number of such systems.

Highly Connected Embeddings in Codimension Two

In this paper we study semilocal knots over $f$ into $\xi$, that is, embeddings of a manifold $N$ into $E(\xi )$, the total space of a $2$-disk bundle over a manifold $M$, such that the restriction of the bundle projection $p:E(\xi ) \to M$ to the submanifold $N$ is homotopic to a normal map of degree one, $f:N \to N$. We develop a new homology surgery theory which does not require homology equivalences on boundaries and, in terms of these obstruction groups, we obtain a classification (up to cobordism) of semilocal knots over $f$ into $\xi$. In the simply connected case, the following geometric consequence follows from our classification. Every semilocal knot of a simply connected manifold $M\# K$ in a bundle over $M$ is cobordant to the connected sum of the zero section of this bundle with a semilocal knot of the highly connected manifold $K$ into the trivial bundle over a sphere.

Plateau's problem for surfaces of constant mean curvature from a global point of view

This paper is concerned with the set of solutions of Plateau's problem for surfaces of constant mean curvature. By methods of global analysis this set is represented in an infinite dimensional fibre bundle over the set of boundary curves. Both the manifold structure and the Fredholm property of the bundle projection depend on the branch type of the surfaces under consideration.

Some James numbers of Stiefel manifolds

The purpose of this note is to determine some unstable James numbers of Stiefel manifolds. We denote the real numbers by R, the complex numbers by C, and the quaternions by H. Let F be one of these fields with the usual norm, and d = dimRF. Let On, k = On, k(F) be the Stiefel manifold of all orthonormal k–frames in Fn, and q: On, k → Sdn−1 the bundle projection which associates with each frame its last vector. Then the James number O{n, k} = OF{n, k} is defined as the index of q* πdn−1(On, k) in πdn−1(Sdn−1). We already know when O{n, k} is 1 (cf. (1), (2), (3), (13), (33)), and also the value of OK{n, k} (cf. (1), (13), (15), (34)). In this note we shall consider the complex and quaternionic cases. For earlier work see (11), (17), (23), (27), (29), (31) and (32). In (27) we defined the stable James number , which was a divisor of O{n, k}. Following James we shall use the notations X{n, k}, Xs{n, k}, W{n, k} and Ws{n, k} instead of OH{n, k}, , Oc{n, k} and respectively. In (27) we noticed that O{n, k} = Os{n, k} if n ≥ 2k– 1, and determined Xs{n, k} for 1 ≤ k ≤ 4, and also Ws{n, k} for 1 ≤ k ≤ 8. On the other hand Sigrist (31) calculated W{n, k} for 1 ≤ k ≤ 4. He informed the author that W{6,4} was not 4 but 8. Since Ws{6,4} = 4 (cf. § 5 below) this yields that the unstable James number does not equal the stable one in general.

Projections of the optic tectum and the mesencephalic nucleus of the trigeminal nerve in the tegu lizard (Tupinambis nigropunctatus).

Fibers undergoing Wallerian degeneration following tectal lesions were demonstrated with the Nauta and Fink-Heimer methods and traced to their termination. Four of the five distinct fiber paths originating in the optic tectum appear related to vision, while one is related to the mesencephalic nucleus of the trigeminus. The latter component of the tectal efferents distributes fibers to 1) the main sensory nucleus of the trigeminus, 2) the motor nucleus of the trigeminus, 3) the nucleus of tractus solitarius, and 4) the intermediate gray of the cervical spinal cord. The principal ascending bundle projects to the nucleus rotundus, three components of the ventral geniculate nucleus and the nucleus ventromedialis anterior ipsilaterally, before it crosses in the supraoptic commissure and terminates in the contralateral nucleus rotundus, ventral geniculate nucleus and a hitherto unnamed region dorsal to the nucleus of the posterior accessory optic tract. Fibers leaving the tectum dorso-medially terminate in the posterodorsal nucleus ipsilaterally and the stratum griseum periventriculare of the contralateral tectum. The descending fiber paths terminate in medial reticular cell groups and the rostral spinal cord contralaterally and in the torus and the lateral reticular regions ipsilaterally. The ipsilateral fascicle also issues fibers to the magnocellular nucleus isthmi.

Approximate Torus Fibrations of High Dimensional Manifolds can be Approximated by Torus Bundle Projections

Approximate Torus Fibrations of High Dimensional Manifolds can be Approximated by Torus Bundle Projections