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The Ligon-Schaaf regularization (LS mapping) was introduced in 1976 and has been used several times. However, we are not aware of any direct usage of the inverse mapping, perhaps since it appears at first sight to be quite complex, involves the use of a transcendental equation (referred to as the generalized Kepler equation) that cannot be solved in closed form, and lacks smoothness near the collision point. Here, we provide some insight into the significance of this equation, along with a very simple derivation and confirmation of the inverse LS mapping. Then we use numerical methods to investigate three applications: 1) solutions of the Kepler function, 2) calculation of orbits including time-of-flight data based on the Delaunay Hamiltonian, and 3) numerical evidence for the Birkhoff conjecture for the circular restricted 3-body problem. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the notion of a left-symmetric bialgebroid as a geometric generalization of a left-symmetric bialgebra and construct a left-symmetric bialgebroid from a pseudo-Hessian manifold. We also introduce the notion of a Manin triple for left-symmetric algebroids, which is equivalent to a left-symmetric bialgebroid. The corresponding double structure is a pre-symplectic algebroid rather than a left-symmetric algebroid. In particular, we establish a relation between Maurer–Cartan type equations and Dirac structures of the pre-symplectic algebroid which is the corresponding double structure for a left-symmetric bialgebroid. [ABSTRACT FROM AUTHOR]

In this paper we provide an analysis, both historical and mathematical, of two joint papers on the theory of connections by Élie Cartan and Jan Arnoldus Schouten that were published in 1926. These papers were the result of a fertile collaboration between the two eminent geometers that flourished in the two-year period 1925–1926. We describe the birth and the development of their scientific relationship especially in the light of unpublished sources that, on the one hand, offer valuable insight into their common research interests and, on the other hand, provide a vivid picture of Cartan's and Schouten's different technical choices. While the first part of this work is preeminently of a historical character, the second part offers a modern mathematical treatment of some contents of the two contributions. [ABSTRACT FROM AUTHOR]

Special coordinate systems are constructed in a neighborhood of a point or of a curve. Taylor expansions can then be easily inferred for the metric, the connection, or the Finsler Lagrangian in terms of curvature invariants. These coordinates circumvent the difficulties of the normal and Fermi coordinates in Finsler geometry, which in general are not sufficiently differentiable. They are obtained applying the usual constructions to the pullback of a horizontally torsionless connection. The results so obtained are easily specialized to the Berwald or Chern–Rund connections and have application in the study of the equivalence principle in Finslerian extensions of general relativity. [ABSTRACT FROM AUTHOR]

In this paper, we study the hemi-slant submanifolds of cosymplectic manifolds. Necessary and sufficient conditions for distributions to be integrable are worked out. Some important results are obtained in this direction. We study the geometry of leaves of hemi-slant submanifolds. [ABSTRACT FROM AUTHOR]

We characterise sets of uniformly absolutely continuous norm in strongly symmetric spaces of τ-measurable operators. Applications are given to the study of relatively weakly compact and relatively compact sets and to compactness properties of operators dominated in the sense of complete positivity by compact or by Dunford-Pettis operators. [ABSTRACT FROM AUTHOR]

The communication processes of knowledge creation represent a particular class of human dynamics where the expertise of individuals plays a substantial role, thus offering a unique possibility to study the structure of knowledge networks from online data. Here, we use the empirical evidence from questions-and-answers in mathematics to analyse the emergence of the network of knowledge contents (or tags) as the individual experts use them in the process. After removing extra edges from the network-associated graph, we apply the methods of algebraic topology of graphs to examine the structure of higher-order combinatorial spaces in networks for four consecutive time intervals. We find that the ranking distributions of the suitably scaled topological dimensions of nodes fall into a unique curve for all time intervals and filtering levels, suggesting a robust architecture of knowledge networks. Moreover, these networks preserve the logical structure of knowledge within emergent communities of nodes, labeled according to a standard mathematical classification scheme. Further, we investigate the appearance of new contents over time and their innovative combinations, which expand the knowledge network. In each network, we identify an innovation channel as a subgraph of triangles and larger simplices to which new tags attach. Our results show that the increasing topological complexity of the innovation channels contributes to network’s architecture over different time periods, and is consistent with temporal correlations of the occurrence of new tags. The methodology applies to a wide class of data with the suitable temporal resolution and clearly identified knowledge-content units. [ABSTRACT FROM AUTHOR]

Given the amount of knowledge and data accruing in the neurosciences, is it time to formulate a general principle for neuronal dynamics that holds at evolutionary, developmental, and perceptual timescales? In this paper, we propose that the brain (and other self-organised biological systems) can be characterised via the mathematical apparatus of a gauge theory. The picture that emerges from this approach suggests that any biological system (from a neuron to an organism) can be cast as resolving uncertainty about its external milieu, either by changing its internal states or its relationship to the environment. Using formal arguments, we show that a gauge theory for neuronal dynamics—based on approximate Bayesian inference—has the potential to shed new light on phenomena that have thus far eluded a formal description, such as attention and the link between action and perception. [ABSTRACT FROM AUTHOR]

In this paper, we consider the problem of fast and efficient indexing techniques for sequences evolving in non-Euclidean spaces. This problem has several applications in the areas of human activity analysis, where there is a need to perform fast search, and recognition in very high dimensional spaces. The problem is made more challenging when representations such as landmarks, contours, and human skeletons etc. are naturally studied in a non-Euclidean setting where even simple operations are much more computationally intensive than their Euclidean counterparts. We propose a geometry and data adaptive symbolic framework that is shown to enable the deployment of fast and accurate algorithms for activity recognition, dynamic texture recognition, motif discovery. Toward this end, we present generalizations of key concepts of piece-wise aggregation and symbolic approximation for the case of non-Euclidean manifolds. We show that one can replace expensive geodesic computations with much faster symbolic computations with little loss of accuracy in activity recognition and discovery applications. The framework is general enough to work across both Euclidean and non-Euclidean spaces, depending on appropriate feature representations without compromising on the ultra-low bandwidth, high speed and high accuracy. The proposed methods are ideally suited for real-time systems and low complexity scenarios. [ABSTRACT FROM AUTHOR]

The delayed feedback stabilization of rigid spacecraft attitude dynamics in the presence of an unknown time-varying delay in the measurement is addressed. The attitude representation is parameterized using minimal attitude coordinates. The time-varying delay and its derivative are assumed to be bounded. By employing a linear state feedback controller via a Lyapunov–Krasovskii functional, a general delay-dependent stability condition is characterized for the closed-loop parameterized system in terms of a linear matrix inequality (LMI) whose solution gives the suitable controller gains. An estimate of the region of attraction of the controlled system is also obtained, inside which the asymptotic stability of parameterized system is guaranteed. [ABSTRACT FROM AUTHOR]

For the pants graph, there is little known about the behaviour of geodesics, as opposed to quasi-geodesics. Brock-Masur- Minsky showed that geodesics or geodesic segments connecting endpoints satisfying a bounded combinatorics condition, such as the stable/ unstable laminations of a pseudo-Anosov, all have bounded combinatorics, outside of annuli. In this paper it is shown that there exist geodesics that also have bounded combinatorics within annuli. These geodesics are shown to have finiteness properties analogous to those of tight geodesics in the complex of curves, from which rationality of stable lengths of pseudo-Anosovs acting on the pants graph then follows from the arguments of Bowditch for the curve complex. [ABSTRACT FROM AUTHOR]

The automatic detection and tracking of human eyes and, in particular, the precise localization of their centers (pupils), is a widely debated topic in the international scientific community. In fact, the extracted information can be effectively used in a large number of applications ranging from advanced interfaces to biometrics and including also the estimation of the gaze direction, the control of human attention and the early screening of neurological pathologies. Independently of the application domain, the detection and tracking of the eye centers are, currently, performed mainly using invasive devices. Cheaper and more versatile systems have been only recently introduced: they make use of image processing techniques working on periocular patches which can be specifically acquired or preliminarily cropped from facial images. In the latter cases the involved algorithms must work even in cases of non-ideal acquiring conditions (e.g in presence of noise, low spatial resolution, non-uniform lighting conditions, etc.) and without user's awareness (thus with possible variations of the eye in scale, rotation and/or translation). Getting satisfying results in pupils' localization in such a challenging operating conditions is still an open scientific topic in Computer Vision. Actually, the most performing solutions in the literature are, unfortunately, based on supervised machine learning algorithms which require initial sessions to set the working parameters and to train the embedded learning models of the eye: this way, experienced operators have to work on the system each time it is moved from an operational context to another. It follows that the use of unsupervised approaches is more and more desirable but, unfortunately, their performances are not still satisfactory and more investigations are required. To this end, this paper proposes a new unsupervised approach to automatically detect the center of the eye: its algorithmic core is a representation of the eye's shape that is obtained through a differential analysis of image intensities and the subsequent combination with the local variability of the appearance represented by self-similarity coefficients. The experimental evidence of the effectiveness of the method was demonstrated on challenging databases containing facial images. Moreover, its capabilities to accurately detect the centers of the eyes were also favourably compared with those of the leading state-of-the-art methods. [ABSTRACT FROM AUTHOR]

In this paper, we propose a novel method for parcellating the human brain into 193 anatomical structures based on diffusion tensor images (DTIs). This was accomplished in the setting of multi-contrast diffeomorphic likelihood fusion using multiple DTI atlases. DTI images are modeled as high dimensional fields, with each voxel exhibiting a vector valued feature comprising of mean diffusivity (MD), fractional anisotropy (FA), and fiber angle. For each structure, the probability distribution of each element in the feature vector is modeled as a mixture of Gaussians, the parameters of which are estimated from the labeled atlases. The structure-specific feature vector is then used to parcellate the test image. For each atlas, a likelihood is iteratively computed based on the structure-specific vector feature. The likelihoods from multiple atlases are then fused. The updating and fusing of the likelihoods is achieved based on the expectation-maximization (EM) algorithm for maximum a posteriori (MAP) estimation problems. We first demonstrate the performance of the algorithm by examining the parcellation accuracy of 18 structures from 25 subjects with a varying degree of structural abnormality. Dice values ranging 0.8–0.9 were obtained. In addition, strong correlation was found between the volume size of the automated and the manual parcellation. Then, we present scan-rescan reproducibility based on another dataset of 16 DTI images – an average of 3.73%, 1.91%, and 1.79% for volume, mean FA, and mean MD respectively. Finally, the range of anatomical variability in the normal population was quantified for each structure. [ABSTRACT FROM AUTHOR]

In this paper we investigate some subclasses of strongly regular congruences on an $E$-inversive semigroup $S$. We describe the minimum and the maximum strongly orthodox congruences on $S$ whose characteristic trace coincides with the characteristic trace of given congruences and, in each case, we present an alternative characterization for them. A description of all strongly orthodox congruences on $S$ with characteristic trace $\tau $ is given. Further, we investigate the kernel relation of strongly orthodox congruences on an $E$-inversive semigroup and give the least and the greatest element in the class of the same kernel with a given congruence. [ABSTRACT FROM AUTHOR]

Let $M$ be a complete hyperbolic 3-manifold homotopy equivalent to a compact surface $\Sigma $. Let $\Phi $ be a proper subsurface of $\Sigma $, whose boundary is sufficiently short in $M$. We show that the union of all Margulis tubes and cusps homotopic into $\Phi $ lifts to a uniformly quasiconvex subset of hyperbolic 3-space. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we investigate the small time heat kernel asymptotics on the cut locus on a class of surfaces of revolution, which are the simplest two-dimensional Riemannian manifolds different from the sphere with non-trivial cut-conjugate locus. We determine the degeneracy of the exponential map near a cut-conjugate point and present the consequences of this result to the small time heat kernel asymptotics at this point. These results give a first example where the minimal degeneration of the asymptotic expansion at the cut locus is attained. [Copyright &y& Elsevier]

Abstract: The operator involved in quasiconvex functions is and this also arises as the governing operator in a worst case tug-of-war (Kohn and Serfaty (2006) [7]) and principal curvature of a surface. In Barron et al. (2012) [4] a comparison principle for was proved. A new and much simpler proof is presented in this paper based on Barles and Busca (2001) [3] and Lu and Wang (2008) [8]. Since is the minimal principal curvature of a surface, we show by example that does not have a unique solution, even if . Finally, we complete the identification of first order evolution problems giving the convex envelope of a given function. [Copyright &y& Elsevier]

In this paper, using the Lie group analysis method, we study the invariance properties of the time fractional fifth-order KdV equation. A systematic research to derive Lie point symmetries to time fractional fifth-order KdV equation is performed. In the sense of point symmetry, all of the vector fields and the symmetry reductions of the fractional fifth-order KdV equation are obtained. At last, by virtue of the sub-equation method, some exact solutions to the fractional fifth-order KdV equation are provided. [ABSTRACT FROM AUTHOR]

Given a at connection α on a manifoldM with values in a filtered L∞-algebra g, we construct a morphism holα∞ : C·(M) → BU∞(g), which generalizes the holonomy map associated to a at connection with values in a Lie algebra. The construction is based on Gugenheim's A∞-version of de Rham's theorem, which in turn is based on Chen's iterated integrals. Finally, we discuss examples related to the geometry of configuration spaces of points in Euclidean space Rd, and to generalizations of the holonomy representations of braid groups. [ABSTRACT FROM AUTHOR]

Dislocated symmetric spaces are introduced, and implications and nonimplications among various kinds of convergence axioms are derived. [ABSTRACT FROM AUTHOR]

We have used high-energy x-ray scattering to map the strain fields around crack tips in fracture specimens of a bulk metallic glass under load at room temperature and below. From the measured strain fields we can calculate the components of the stress tensor as a function of position and determine the size and shape of the plastic process zone around the crack tip. Specimens tested at room temperature develop substantial plastic zones and achieve high stress intensities () prior to fracture. Specimens tested at cryogenic temperatures fail at reduced but still substantial stress intensities () and show only limited evidence of crack-tip plasticity. We propose that the difference in behavior is associated with changes in the flow stress and elastic constants, which influence the number density of shear bands in the plastic zone and thus the strain required to initiate fracture on an individual band. A secondary effect is a change in the triaxial state of stress around the crack tip due to the temperature dependence of Poisson's ratio. It is likely that this ability to map elastic strains on the microscale will be useful in other contexts, although interpreting shifts in the position of the scattering peaks in amorphous materials in terms of elastic strains must be done with caution. [ABSTRACT FROM AUTHOR]

Biological systems fundamentally exist out of equilibrium in order to preserve organized structures and processes. Many changing cellular conditions can be represented as transitions between nonequilibrium steady states, and organisms have an interest in optimizing such transitions. Using the Hatano-Sasa Y-value, we extend a recently developed geometrical framework for determining optimal protocols so that it can be applied to systems driven from nonequilibrium steady states. We calculate and numerically verify optimal protocols for a colloidal particle dragged through solution by a translating optical trap with two controllable parameters. We offer experimental predictions, specifically that optimal protocols are significantly less costly than naive ones. Optimal protocols similar to these may ultimately point to design principles for biological energy transduction systems and guide the design of artificial molecular machines. [ABSTRACT FROM AUTHOR]

The present paper deals with a study of ø-pseudo symmetric and ø-pseudo Ricci symmetric (LCS)n-manifolds. It is shown that every ø-pseudo symmetric (LCS)n- manifold and ø-pseudo Ricci symmetric (LCS)n,-manifold are η-Einstein manifold. [ABSTRACT FROM AUTHOR]

This paper examines the multiple atlas random diffeomorphic orbit model in Computational Anatomy (CA) for parameter estimation and segmentation of subcortical and ventricular neuroanatomy in magnetic resonance imagery. We assume that there exist multiple magnetic resonance image (MRI) atlases, each atlas containing a collection of locally-defined charts in the brain generated via manual delineation of the structures of interest. We focus on maximum a posteriori estimation of high dimensional segmentations of MR within the class of generative models representing the observed MRI as a conditionally Gaussian random field, conditioned on the atlas charts and the diffeomorphic change of coordinates of each chart that generates it. The charts and their diffeomorphic correspondences are unknown and viewed as latent or hidden variables. We demonstrate that the expectation-maximization (EM) algorithm arises naturally, yielding the likelihood-fusion equation which the a posteriori estimator of the segmentation labels maximizes. The likelihoods being fused are modeled as conditionally Gaussian random fields with mean fields a function of each atlas chart under its diffeomorphic change of coordinates onto the target. The conditional-mean in the EM algorithm specifies the convex weights with which the chart-specific likelihoods are fused. The multiple atlases with the associated convex weights imply that the posterior distribution is a multi-modal representation of the measured MRI. Segmentation results for subcortical and ventricular structures of subjects, within populations of demented subjects, are demonstrated, including the use of multiple atlases across multiple diseased groups. [ABSTRACT FROM AUTHOR]

In this paper we study the geometry of distributions of semi-slant submanifolds of (α,β) trans-Sasakian manifolds, some problems concerning the stability of slant submanifolds of generalized Sasakian space-forms and we also investigate the first normal Chern class for integral submanifolds of (α,β) trans-Sasakian generalized Sasakian space-forms. [ABSTRACT FROM AUTHOR]

By partitioning the hydrostatically balanced flow into a nonlinearly balanced primary-flow part and a remaining secondary-flow part and then truncating the secondary-flow vorticity advection and stretching-tilting terms in the vector vorticity equation, the previous semibalance model (SBM) in pseudoheight coordinates is rederived in terrain-following pressure coordinates, called η coordinates. The involved truncation is topologically the same as that in pseudoheight coordinates but the truncated terms in η coordinates are not equivalent to those in pseudoheight coordinates. Because its potential vorticity (PV) is conserved and invertible, the rederived SBM is suitable for studying balanced dynamics via 'PV thinking' in real weather events, such as slowly varying vortices and curved fronts in which the primary-flow velocity and secondary-flow vorticity are nearly parallel in η coordinates. [ABSTRACT FROM AUTHOR]

The article discusses several unequal tangent problems. The author examines whether there exist a pair of simple closed curves and asserts that when dealing with a smooth curve, a chord determines the tangent directions at its end-points. It states that a binormal chord automatically gives a bitangent circle with its center concurring with the midpoint of the chord, which gives a point at infinity on the equitangent locus.

Let X be a smooth curve on a ruled surface π : S → C. In this paper, we deal with the questions on the gonality and the Clifford index of X and on the composedness of line bundles on X with the covering morphism π∣X. The main theorem shows that if a smooth curve X ∼ aCo + bf satisfies some conditions on the degree of b, then a line bundle L on X with Cliff(L) ⩽ ag(C) - 1 is composed with π∣X. This implies that a part of the gonality sequence of X is computed by the gonality sequence of C as follows: dr(X) = adr(C) for r ⩽ L, where L is the length of the gonality sequence of C. [ABSTRACT FROM AUTHOR]

We show geometrical properties of a submanifold of a (LCS)n-manifold. The properties of the induced structures on such a submanifold are also studied. [ABSTRACT FROM AUTHOR]

We study semi-slant warped product submanifolds of a Kenmotsu manifold. We obtain a characterization for warped product submanifolds in terms of warping function and shape operator and finally proved an inequality for squared norm of second fundamental form. [ABSTRACT FROM AUTHOR]

Bianchi-type transformations are constructed for two-dimensional surfaces with constant negative intrinsic curvature in the space S × ℝ. [ABSTRACT FROM AUTHOR]

Abstract: Let be a four-dimensional complete noncompact Bach-flat Riemannian manifold with positive Yamabe constant. In this paper, we show that has a constant curvature if it has a nonnegative constant scalar curvature and sufficiently small -norm of trace-free Riemannian curvature tensor. Moreover, we get a gap theorem for with positive scalar curvature. [Copyright &y& Elsevier]

It is the aim of this article to give a reasonably detailed account of a specific bundle of geometric investigations and results pertaining to arithmetic groups, the geometry of the corresponding locally symmetric space X/Γ attached to a given arithmetic subgroup Γ ⊂ G of a reductive algebraic group G and its cohomology groups H*(X/Γ,ℂ). We focus on constructing totally geodesic cycles in X/Γ which originate with reductive subgroups H ⊂ G. In many cases, it can be shown that these cycles, to be called geometric cycles, yield non-vanishing (co)homology classes. Since the cohomology of an arithmetic group Γ is strongly related to the automorphic spectrum of Γ, this geometric construction of non-vanishing classes leads to results concerning, for example, the existence of specific automorphic forms. [ABSTRACT FROM AUTHOR]

Abstract: We first compute Riemannian curvature and Ricci curvature of metrics. Then we apply these formulae to discuss a special class metrics () which have constant flag curvature. We obtain the sufficient and necessary conditions that have constant flag curvature. Then we prove that such metrics must be locally projectively flat and complete their local classification. Using the same method we find a necessary condition that flag curvature of is constant and proved that there are no non-trivial Matsumoto metrics. Furthermore, we give a negative answer whether there are non-trivial metrics () of constant flag curvature when β is closed. [Copyright &y& Elsevier]

The object of the present paper is to introduce a non-flat Riemannian manifold called hyper-generalized recurrent manifolds and study its various geometric properties along with the existence of a proper example. [ABSTRACT FROM AUTHOR]

Abstract: We construct a mod 2 analogue of the Witten genus for dimensional spin manifolds, as well as modular characteristic numbers for a class of manifolds which we call manifolds. When these manifolds are actually spin, one recovers the original Witten genus on string manifolds. These genera vanish on string and complete intersections respectively in complex projective spaces. [Copyright &y& Elsevier]

In this paper, we study the basic results on the general study of null curves in the Minkowski 4-space R14. A transversal vector bundle of a null curve in R14 is constructed using a frenet Frame consisting of two real null and two space-like vectors. The null curves are characterized by using the Frenet frame. [ABSTRACT FROM AUTHOR]

We demonstrate how to combinatorially calculate the EH-class of a compatible contact structure inthe sutured Floer homology group of a balanced sutured three manifold which is associated to an abstractpartial open book decomposition. As an application we show that every contact three manifold (closed or with convex boundary) can be obtained by gluing tight contact handlebodies whose EH-classes are nontrivial. [ABSTRACT FROM AUTHOR]

Abstract: This paper is devoted to a study on closed geodesics on Finsler and Riemannian spheres. We call a prime closed geodesic on a Finsler manifold rational, if the basic normal form decomposition (cf. [Y. Long, Bott formula of the Maslov-type index theory, Pacific J. Math. 187 (1999) 113–149]) of its linearized Poincaré map contains no rotation matrix with rotation angle which is an irrational multiple of π, or irrational otherwise. We prove that if there exists only one prime closed geodesic on a d-dimensional irreversible Finsler sphere with , it cannot be rational. Then we further prove that there exist always at least two distinct prime closed geodesics on every irreversible Finsler 3-dimensional sphere. Our method yields also at least two geometrically distinct closed geodesics on every reversible Finsler as well as Riemannian 3-dimensional sphere. We prove also such results hold for all compact simply connected 3-dimensional manifolds with irreversible or reversible Finsler as well as Riemannian metrics. [Copyright &y& Elsevier]

In this paper, we introduce and discuss minimal γ-open sets in topological spaces. We establish some basic properties of minimal γ-open sets and provide an example to illustrate that minimal γ-open sets are independent of minimal open sets introduced and discussed in [3]. We obtain some properties of pre γ-open sets using properties of minimal γ-open sets. As an application of a theory of minimal γ-open sets, we obtain a sufficient condition for a γ-locally finite space to be a pre γ-T2 space. [ABSTRACT FROM AUTHOR]

We obtain the complete group of structure equations of a locally conformally almost cosymplectic structure (an lc $$ ACy $$-structure in what follows) and compute the components of the Riemannian curvature tensor on the space of the associated G-structure. Normal lc $$ ACy $$-structures are studied in more detail. In particular, we prove that the contact analogs of A. Gray’s second and third curvature identities hold on normal lc $$ ACy $$-manifolds, while the contact analog of A. Gray’s first identity holds if and only if the manifold is cosymplectic. [ABSTRACT FROM AUTHOR]

The article presents several reprints from mathematical reviews. One discusses on integral geometry in Klein spaces which is considered as a transitive Lie group G of automorphisms. Another one explores the geometry of G-structures specifically on its basic general problems of existence. Another one also explores on characteristic forms and geometric variants.

In this paper we prove a realizability theorem for Quinn's mapping cylinder obstructions for stratified spaces. We prove a continuously controlled version of the s-cobordism theorem which we further use to prove the relation between the torsion of an h-cobordism and the mapping cylinder obstructions. This states that the image of the torsion of an h-cobordism is the mapping cylinder obstruction of the lower stratum of one end of the h-cobordism in the top filtration. These results are further used to prove a theorem about the realizability of end obstructions. [ABSTRACT FROM AUTHOR]

We establish two-sided bounds for the complexity of two infinite series of closed orientable three-dimensional hyperbolic manifolds, the Löbell manifolds and the Fibonacci manifolds. The manifolds of the two series are indexed by an integer n and the corresponding complexity estimates are both linear in n. [ABSTRACT FROM AUTHOR]

Abstract: We define the notion of extrinsic symplectic symmetric spaces and exhibit some of their properties. We construct large families of examples and show how they fit in the perspective of a complete classification of these manifolds. We also build a natural -quantization on a class of examples. [Copyright &y& Elsevier]

Abstract: We compute the initial ideals, with respect to certain conveniently chosen term orders, of ideals of tangent cones at torus fixed points to Schubert varieties in orthogonal Grassmannians. The initial ideals turn out to be square-free monomial ideals and therefore define Stanley–Reisner face rings of simplicial complexes. We describe these complexes. The maximal faces of these complexes encode certain sets of non-intersecting lattice paths. [Copyright &y& Elsevier]

We construct $mathcal {F}$-structures on a Bott manifold and on some other manifolds obtained by Kummer-type constructions. We also prove that if $M=E# X$, where $E$ is a fiber bundle with structure group $G$ and a fiber admitting a $G$-invariant metric of non-negative sectional curvature and $X$ admits an $mathcal {F}$-structure with one trivial covering, then one can construct on $M$ a sequence of metrics with sectional curvature uniformly bounded from below and volume tending to zero (i.e. $operatorname {Vol}_K(M)=0$). As a corollary we prove that all the elements in the Spin cobordism ring can be represented by manifolds $M$ with $operatorname {Vol}_K (M)=0$. [ABSTRACT FROM AUTHOR]

It is proved that, for Tϵ ⩽ G = G(T) ⩽ ½√T, [This equation cannot converted into ASCII.txt] + Oϵ(T1+ϵ G1/2 + T1/2+ϵ G²) with some explicitly computable constants aj (a3 > 0) where, for fixed κ ∈ N, [This equation cannot converted into ASCII.txt] The generalizations to the mean square of I1 (t+U, G)-I1 (t, G) over [T, T+H] and the estimation of the mean square of I2(t + U, G) - I2(t, g) are also discussed. [ABSTRACT FROM AUTHOR]

Abstract: Hitchin shows in [N. Hitchin, The geometry of three-forms in six and seven dimensions, J. Differ. Geom. 55 (2000) 547–576. math.DG/0010054v1; N. Hitchin, Stable forms and special metrics, Global Differential Geometry: The Mathematical Legacy of Alfred Gray, in: Contemp. Math., volume 288, American Math. Soc., 2001, pp. 70–89. math.DG/0107101v1] that half-flat -structures on a six-dimensional manifold can be lifted to parallel -structure on the product . We show that Hitchin’s approach can also be used to construct nearly parallel -structures by lifting so-called nearly half-flat structures. These -structures are described by pairs of stable 2- and 3-forms with , for some . [Copyright &y& Elsevier]