Simple Manifold Research Articles

A support theorem for the X-ray transform on manifolds with plane covers

AbstractThis paper is concerned with support theorems of the X-ray transform on non-compact manifolds with conjugate points. In particular, we prove that all simply connected 2-step nilpotent Lie groups have a support theorem. Important ingredients of the proof are the concept of plane covers and a support theorem for simple manifolds by Krishnan. We also provide examples of non-homogeneous 3-dimensional simply connected manifolds with conjugate points which have support theorems.

Integration of Differential Graded Manifolds

Abstract We consider the problem of integration of $L_\infty $-algebroids (differential non-negatively graded manifolds) to $L_\infty $-groupoids. We first construct a “big” Kan simplicial manifold (Fréchet or Banach) whose points are solutions of a (generalized) Maurer–Cartan equation. The main analytic trick in our work is an integral transformation sending the solutions of the Maurer–Cartan equation to closed differential forms. Following the ideas of Ezra Getzler, we then impose a gauge condition that cuts out a finite-dimensional simplicial submanifold. This “smaller” simplicial manifold is (the nerve of) a local Lie $\ell $-groupoid. The gauge condition can be imposed only locally in the base of the $L_\infty $-algebroid; the resulting local $\ell $-groupoids glue up to a coherent homotopy, that is, we get a homotopy coherent diagram from the nerve of a good cover of the base to the (simplicial) category of local $\ell $-groupoids. Finally, we show that a $k$-symplectic differential non-negatively graded manifold integrates to a local $k$-symplectic Lie $\ell$-groupoid; globally, these assemble to form an $A_\infty$-functor. As a particular case for $k=2$, we obtain integration of Courant algebroids.

The general linear 2-groupoid

We deal with the symmetries of a (2-term) graded vector space or bundle. Our first theorem shows that they define a (strict) Lie 2-groupoid in a natural way. Our second theorem explores the construction of nerves for Lie 2-categories, showing that it yields simplicial manifolds if the 2-cells are invertible. Finally, our third and main theorem shows that smooth pseudofunctors into our general linear 2-groupoid classify 2-term representations up to homotopy of Lie groupoids.

The Attenuated Geodesic Ray Transform on Tensors: Generic Injectivity and Stability

We consider the attenuated geodesic ray transform defined on pairs of symmetric 2-tensors and 1-forms on a simple Riemannian manifold. We prove injectivity and stability results for a class of generic simple metrics and attenuations containing real analytic ones. In fact, methods used in this paper can be modified to generalize our results for a class of non-simple manifolds similar to Stefanov and Uhlmann Am J Math 130(1):239–268 (2008).

Some characterizations of Lorentzian manifolds

Generalized Robertson–Walker (GRW) spacetime is the generalization of the Robertson–Walker (RW) spacetime and a further generalization of GRW spacetime is the twisted spacetime. In this paper, we generalize the results of the paper [C. A. Mantica, Y. J. Suh and U. C. De, A note on generalized Robertson–Walker spacetimes, Int. J. Geom. Methods Mod. Phys. 13 (2016), Article Id: 1650079, 9 pp., doi: 101142/s0219887816500791 ]. We prove that a Ricci simple Lorentzian manifold with vanishing quasi-conformal curvature tensor is a RW spacetime. Further, we prove that a Ricci simple Lorentzian manifold with harmonic quasi-conformal curvature tensor is a GRW spacetime. As a consequence, we obtain several corollaries. Finally, we have cited some examples of the obtained results.

Regiodivergent hydrosilylation, hydrogenation, [2π + 2π]-cycloaddition and C-H borylation using counterion activated earth-abundant metal catalysis.

The widespread adoption of earth-abundant metal catalysis lags behind that of the second- and third-row transition metals due to the often challenging practical requirements needed to generate the active low oxidation-state catalysts. Here we report the development of a single endogenous activation protocol across five reaction classes using both iron- and cobalt pre-catalysts. This simple catalytic manifold uses commercially available, bench-stable iron- or cobalt tetrafluoroborate salts to perform regiodivergent alkene and alkyne hydrosilylation, 1,3-diene hydrosilylation, hydrogenation, [2π + 2π]-cycloaddition and C-H borylation. The activation protocol proceeds by fluoride dissociation from the counterion, in situ formation of a hydridic activator and generation of a low oxidation-state catalyst.

Nonadiabatic scattering of NO off Au3 clusters: A simple and robust diabatic state manifold generation method for multiconfigurational wavefunctions.

The presence of nonadiabatic effects during the interaction of small molecules with metals has been observed experimentally for the last decades. Specially remarkable are the effects found for NO/Au, where experiments have suggested the presence of very strong vibronic coupling during the molecular scattering. However, the accurate inclusion of the nonadiabatic effects in periodic boundary conditions (PBC) theoretical methods remain an unapproachable challenge. Here, aiming to give some theoretical insight to the strong vibronic coupling, we have adopted a pragmatic point of view, taking use of an auxiliary simplified system, NO/Au3 . We show the importance of nonadiabatic coupling, during the scattering of NO from a Au3 cluster, using a diabatic representation of 12 electronic states of the system, including a few charge-transfer states. Our diabatic representation is obtained by rotating the orbital and configuration interaction (CI) vectors of a restricted active space (RAS) wavefunction. We present a strategy for extracting the best effective manifold of states relevant to the system, below some prescribed energy, directly from the RAS CI vectors. This scheme is able to disentangle a large dense manifold of adiabatic states with strong coupling and crossings. This approach is also shown to work for multireference configuration interaction (MRCI). By performing quantum propagations, we observed an increase in vibrational redistribution with increasing initial vibrational or translational energies. We suggest that these nonadiabatic effects should also be present at smaller energies in larger clusters. © 2018 Wiley Periodicals, Inc.

An adaptive Bayesian tracking algorithm for Internet of motion parameters based on explicit formulation and integral contour model

Aiming at multi-degree of freedom of traditional sport movement contour tracing, to improve precision and efficiency of the tracking algorithm, a kind of sport movement parameter adaptive Bayes tracking algorithm based on the explicit formulation integration outline model is put forward. First, this paper has established a simple sport movement parameter explicit formulation manifold space model, and the compact and low-dimensional model form has simplified sport movement parameter outline tracking difficulty; second, this paper has established the matrix probability decomposition model and further established a coordinative recommendation trust model of matrix probability decomposition by integrating the coordinative recommendation process by utilizing the trust scoring matrix and implements model solution based on the maximal expect method; finally, by simulation experiment, validity of the algorithm proposed has been verified.

On the classification of topological quandles

We investigate the classification of topological quandles on some simple manifolds. Precisely we classify all topological Alexander quandle structures, up to isomorphism, on the real line and the unit circle. For the closed unit interval [0,1], we conjecture that there exists only one topological quandle structure on it, i.e. the trivial one. Some evidences are provided to support our conjecture.

Inverse Problems for Magnetic Schrödinger Operators in Transversally Anisotropic Geometries

We study inverse boundary problems for magnetic Schr\"odinger operators on a compact Riemannian manifold with boundary of dimension $\ge 3$. In the first part of the paper we are concerned with the case of admissible geometries, i.e. compact Riemannian manifolds with boundary which are conformally embedded in a product of the Euclidean line and a simple manifold. We show that the knowledge of the Cauchy data on the boundary of the manifold for the magnetic Schr\"odinger operator with $L^\infty$ magnetic and electric potentials, determines the magnetic field and electric potential uniquely. In the second part of the paper we address the case of more general conformally transversally anisotropic geometries, i.e. compact Riemannian manifolds with boundary which are conformally embedded in a product of the Euclidean line and a compact manifold, which need not be simple. Here, under the assumption that the geodesic ray transform on the transversal manifold is injective, we prove that the knowledge of the Cauchy data on the boundary of the manifold for a magnetic Schr\"odinger operator with continuous potentials, determines the magnetic field uniquely. Assuming that the electric potential is known, we show that the Cauchy data determines the magnetic potential up to a gauge equivalence.

Variational encoding of complex dynamics.

Often the analysis of time-dependent chemical and biophysical systems produces high-dimensional time-series data for which it can be difficult to interpret which individual features are most salient. While recent work from our group and others has demonstrated the utility of time-lagged covariate models to study such systems, linearity assumptions can limit the compression of inherently nonlinear dynamics into just a few characteristic components. Recent work in the field of deep learning has led to the development of the variational autoencoder (VAE), which is able to compress complex datasets into simpler manifolds. We present the use of a time-lagged VAE, or variational dynamics encoder (VDE), to reduce complex, nonlinear processes to a single embedding with high fidelity to the underlying dynamics. We demonstrate how the VDE is able to capture nontrivial dynamics in a variety of examples, including Brownian dynamics and atomistic protein folding. Additionally, we demonstrate a method for analyzing the VDE model, inspired by saliency mapping, to determine what features are selected by the VDE model to describe dynamics. The VDE presents an important step in applying techniques from deep learning to more accurately model and interpret complex biophysics.

On Multidisciplinary Potential Applications of Gauge Theories

We review how gauge theories, initial introduced for classical mechanics and quantum led theory, seem to apply to many fields of research such as information theory, computer science, economy, biology. This non-exhaustive list raises natural question on future developments of this theory. Short Communication How gauge theories appear in information theory and economics Basically, the mathematical structure of groups is among the best adapted for describing transformations and moves. A path on a group can encode the evolution of a dynamical system, or the moves of an exterior observer with respect to a given system. In the theory of quantum fields, more general objects called principal bundles, which consist in a total space P with fibers isomorphic to a (Lie) group G over a base (simplicial complex or manifold). M For one of the most simple settings, one can see when the base is a manifold (we call them continuum gauge theories) [1-5]. When the base is a simplicial complex (or a lattice), the models are called discrete gauge theories. They stand formally as an integrated version of the previous ones which (more or less formally again). The physical study consists in solving equations on P or after quantization into fields, in minimizing so-called action functionals. These action functionals are in fact acting on connections, which are infinitesimal expressions of local slices. They can be understood also as differential operators. These delicate settings coming from physics have, in the last twenty years, found two applications to our knowledge: In so-called geometric arbitrage theory» for managing portfolios in financial markets [2,6-10], where fiber bundles or principal bundles describe, along their fibers and on the base, the external and internal parameters for the prediction of the evolution of the portfolio. Fiber bundles over a manifold described in [9] are related to principal bundles in a way described in [10]. In so-called approximate reasoning in pairwise comparisons where complete studies can be performed when * G + =  and with normalized objects, with very recent axiomatization [10-19]. However, the need of so-called non-numerical ranking appeals the use of non-abelian groups G and of mathematical tools coming from discrete gauge theories [12,18]. Gauge wraith in biology and neurology The aim of this short article, giving the opinion of the sole author, is to mention that there are some hidden aspects of neurology and biology which reveal, in some kind of matching mirror process, the presence of gauge effects. This aspect is not actually truly confirmed, it is more-or-less at a step of conceptualization, and that is the reason why we talk of gauge wraith. Other authors have already mentioned the similarities of models coming from biology and neuronal science [11,15,16], each time with very mild interpretations of classical facts. From a heuristic viewpoint, this is not so surprising since «physics of life» deal with entities interacting with each other, exactly like elementary particles of nuclear physics interact. At the microscopic scale, a simplicial structure arise naturally by linking an entity (neuron, molecule, cell) with each of its neighbours making discrete gauge theories appropriate, and at macroscopic level, a first approximation of phenomena can rise form continuum gauge theories. However, as a last remark, the mathematical theory of discretization of gauge theories has to be questioned very seriously, because the classical scheme based on leads to mathematical problems while other still non studied ways to discretize a continuum model can be proposed [17].

Inverse problems for advection diffusion equations in admissible geometries

ABSTRACTWe study inverse boundary problems for the advection diffusion equation on an admissible manifold, i.e. a compact Riemannian manifold of dimension ≥3 with boundary, which is conformally embedded in a product of the Euclidean real line and a simple manifold. We prove the unique identifiability of the advection term of class and of class from the knowledge of the associated Dirichlet–to–Neumann map on the boundary of the manifold. This seems to be the first global identifiability result for possibly discontinuous advection terms.

Calderón problem for connections

ABSTRACTIn this paper, we consider the problem of identifying a connection ∇ on a vector bundle up to gauge equivalence from the Dirichlet-to-Neumann map of the connection Laplacian ∇*∇ over conformally transversally anisotropic (CTA) manifolds. This was proved in [9] for line bundles in the case of the transversal manifold being simple—we generalize this result to the case where the transversal manifold only has an injective ray transform. Moreover, the construction of suitable Gaussian beam solutions on vector bundles is given for the case of the connection Laplacian and a potential, following the works of [11]. This in turn enables us to construct the Complex Geometrical Optics (CGO) solutions and prove our main uniqueness result. We also reduce the problem to a new non-abelian X-ray transform for the case of simple transversal manifolds and higher rank vector bundles. Finally, we prove the recovery of a flat connection in general from the DN map, up to gauge equivalence, using an argument relating the Cauchy data of the connection Laplacian and the holonomy.

Unobstructed symplectic packing by ellipsoids for tori and hyperkähler manifolds

Let M be a closed symplectic manifold of volume V. We say that the symplectic packings of M by ellipsoids are unobstructed if any collection of disjoint symplectic ellipsoids (possibly of different sizes) of total volume less than V admits a symplectic embedding to M. We show that the symplectic packings by ellipsoids are unobstructed for all even-dimensional tori equipped with Kahler symplectic forms and all closed hyperkahler manifolds of maximal holonomy, or, more generally, for closed Campana simple manifolds (that is, Kahler manifolds that are not unions of their complex subvarieties), as well as for any closed Kahler manifold which is a limit of Campana simple manifolds in a smooth deformation. The proof involves the construction of a Kahler resolution of a Kahler orbifold with isolated singularities and relies on the results of Demailly–Paun and Miyaoka on Kahler cohomology classes.

L-space intervals for graph manifolds and cables

We present a graph manifold analog of the Jankins-Neumann classification of Seifert fibered spaces over $S^2$ admitting taut foliations, providing a finite recursive formula to compute the L-space Dehn-filling interval for any graph manifold with torus boundary. As an application of a generalization of this result to Floer simple manifolds, we compute the L-space interval for any cable of a Floer simple knot complement in a closed three-manifold in terms of the original L-space interval, recovering a result of Hedden and Hom as a special case.

L-space intervals for graph manifolds and cables

We present a graph manifold analog of the Jankins–Neumann classification of Seifert fibered spaces over$S^{2}$admitting taut foliations, providing a finite recursive formula to compute the L-space Dehn-filling interval for any graph manifold with torus boundary. As an application of a generalization of this result to Floer simple manifolds, we compute the L-space interval for any cable of a Floer simple knot complement in a closed three-manifold in terms of the original L-space interval, recovering a result of Hedden and Hom as a special case.

Generic injectivity and stability of inverse problems for connections

ABSTRACTWe consider the nonlinear problem of determining a connection and a Higgs field from the corresponding parallel transport along geodesics on a compact Riemannian manifold with boundary, in any dimension. The problem can be reduced to an integral geometry question of some attenuated geodesic ray transform through a pseudolinearization argument. We show injectivity (up to natural obstructions) and stability estimates for both the linear and nonlinear problems for generic simple metrics and generic connections and Higgs fields, including the real-analytic ones. We consider the problems on simple manifolds to make the exposition of the main ideas clear and concise, many results of this paper still hold under some assumptions weaker than simplicity.

Flow Lectin Affinity Chromatography−A Model with Sambucus nigra Agglutinin

The study of proteomes has become an important way to assess changes occurring in disease conditions. Nonetheless, the dominance of highly abundant serum proteins significantly complicates the discovery and analysis of low-abundant glycoproteins that may constitute potential disease biomarkers. Several techniques have been used to perform serum proteome partitioning such as affinity chromatography based on protein A and G, avian IgY, monoclonal antibodies and lectins. Lectins are particularly well-suited to bind and selectively isolate certain glycan structures present in complex samples like biological ones. The lectin reversibly captures glycoproteins with a particular glycostructure present in the sample in different abundance rates, yielding, after elution, an enriched pool of glycoproteins for glycoproteomic studies. However, lectin affinity chromatography (LAC) for biochemical applications is mainly carried out in batch conditions, which implies a long procedure time, expensive reagents or columns and special equipments or installations. This work proposes the development of a lab-made and easy to implement flow LAC procedure, using Sambucus nigra agglutinin (SNA) immobilized on agarose beads to isolate the STn glycan (a pan-carcinoma biomarker) present in glycoproteins of cancer patients serum. Under flow conditions, the proposed LAC procedure allows the automation of the process, especially important when several samples need to be run. Additionally, the flow system does not require expensive installations or experienced personnel, and it can be tailored to perform LAC with any lectin, according to the purpose of the glycoproteomic analysis. Significance: This work aims to demonstrate the feasibility and the benefits of performing lectin affinity chromatography under flow conditions and with a very simple flow manifold, when compared with the batch procedure usually carried out in Biochemistry and Molecular Biology laboratories. The easiness to implement this simple manifold, without requiring expensive equipments or installations, allows its use in any laboratory, even in those with financial limitations. Several advantages of the developed methodology are pointed out, versus the use of batch procedures or commercial kits or columns, especially when routine analyses need to be performed.

Morrison-Kawamata cone conjecture for hyperkähler manifolds

Let M be a simple hyperkahler manifold, that is, a simply connected compact holomorphically symplectic manifold of Kahler type with h 2,0 = 1. Assuming b2(M) 6 5, we prove that the group of holomorphic automorphisms of M acts on the set of faces of its Kahler cone with finitely many orbits. This statement is known as Morrison-Kawamata cone conjecture for hyperkahler manifolds. As an implication, we show that a hyperkahler manifold has only finitely many non-equivalent birational models. The proof is based on the following observation, proven with ergodic theory. Let M be a complete Riemannian orbifold of dimension at least three, constant negative curvature and finite volume, and {Si} an infinite set of complete, locally geodesic hypersurfaces. Then the union of Si is dense in M.