Krupski and V. The result applies to different finite or infinite topological dimensions of metrizable spaces. The classical Hurewicz-Menger-Tumarkin theorem in dimension theory says that connected topological n-manifolds with or without boundary are Cantor manifolds i.

The theorem was almost immediately strengthened by Mazurkiewicz who proved that regions i. The Hurewicz-Menger-Tumarkin theorem has many generalizations.

In particular, it is known that regions of homogeneous locally compact metric spaces are Cantor manifolds including their infinite-dimensional versions [5, 6]. It was proved in [3] that no weakly infinite-dimensional subset cuts the product of a countable number of nondegenerate metric continua. Moreover, our result holds true for a very general dimension function DK considered in [4] which captures the covering dimension, cohomological dimension dimG with respect to any Abelian group G as well as the extraordinary dimension dimL with respect to a given CWcomplex L, and has its counterparts in infinite dimensions including C-spaces and weakly infinite-dimensional spaces.

## Some sharp restriction theorems for homogeneous manifolds

Basic facts on Cantor manifolds and their stronger variations with respect to dimension DK or to the above-mentioned infinite dimensions. Documents: Advanced Search Include Citations. KrupskiV. Citations: 2 - 2 self. Abstract Abstract. Powered by:.Full-text: Access denied no subscription detected However, an active subscription may be available with MSP at msp.

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As an application, we prove a counterpart for signature classes of a codimension-two vanishing theorem for the index of the Dirac operator on spin manifolds the latter is due to Hanke, Pape and Schick, and is a development of well-known work of Gromov and Lawson.

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### Survey on the Generalized R. L. Moore Problem

Export citation. Export Cancel. References O AttieQuasi-isometry classification of some manifolds of bounded geometryMath. You have access to this content. You have partial access to this content. You do not have access to this content. More like this.Thanks for helping us catch any problems with articles on DeepDyve. We'll do our best to fix them. Check all that apply - Please note that only the first page is available if you have not selected a reading option after clicking "Read Article".

Include any more information that will help us locate the issue and fix it faster for you. Benedetto 1. A celebrated result due to Stein and Thomas says that if M has non-vanishing Gaussian curvature, then the estimate 1.

See [1] and Theorem 7 in Section 3 below.

However, a sharp necessary and sufficient condition is not currently available. It should also be noted that even in codimension one, the more general LP, L q. Enjoy affordable access to over 18 million articles from more than 15, peer-reviewed journals.

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DeepDyve requires Javascript to function. Please enable Javascript on your browser to continue. Some sharp restriction theorems for homogeneous manifolds Carli, Laura ; Iosevich, Alex.

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## Note di Matematica

We give an updated extended survey of results related to the celebrated unsolved generalized R. Moore problem. In particular, we address the problem of characterizing codimension one manifold factors, i. A main part of the paper is devoted to many efficient general position techniques, which have been used to solve special cases of this problem.

Let R n denote the n -dimensional Euclidean space.

The Generalized R. Moore Problem asks:. This is a classical problem that has remained unsolved for over sixty years. One major importance of this problem is its potential applications to manifold recognition problems such as the famous Busemann Conjecture and Bing-Borsuk Conjecture — see our recent survey [ 37 ]. InDaverman published a most excellent survey on the Generalized R.

Moore Problem [ 21 ]. In the present paper, we extend this discussion to significant developments since that point in time. Our survey will focus on developments afterespecially with respect to general position strategies. An early result of R. Moore states [ 46 ] :. Could this result be generalized to higher dimensions? Since no analogous characterizations for higher dimensional manifolds existed at the time, generalizations of his theorem to higher dimensions using the same approach were not possible.

One of the most general cases is addressed in the following theorem see [ 303141 ] :. Suppose G is a usc decomposition of a complete metric space X. InBing [ 7 ] constructed the Dogbone space which was realized by an upper semicontinuous cellular decomposition of R 3 but failed to be homeomorphic to R 3thereby demonstrating that the answer to the previous question is no.

Many examples of decompositions of manifolds whose elements are cellular, or even more generally cell-like, that do not generate manifolds, have now been discovered cf.

Shortly after the construction of his Dogbone space, Bing [ 8 ] discovered a very surprising result: the product of the Dogbone space with the real line is homeomorphic to R 4. Is this always the case? Or could elements be so tangled, that even within a product of R there is insufficient room to obtain the desired shrinking of elements.

Thus emerged the Generalized R. Moore Problem :.In lower regularity we provide examples of such non periodic manifolds that can be realized as proper leaves. The given non periodic examples are repetitive leaving open the non repetitive case.

These are, as far as we know, the first examples of this kind. Moreover, with some additional work, we can provide examples of manifolds that cannot be realized as leaves of any codimension one C 2 foliation on a compact manifold but can be realized as leaves of codimension two C 2 foliations, this is again the first example of this kind. We also present a hyperbolic analogue of the cut-and-project method that naturally produces examples of chaotic Delone sets.

Data: Xoves 9 de xaneiro de Hora: h. We give a description of the complete Euclidean hypersurfaces that admit non-trivial infinitesimal bendings. We also present some results concerning genuine infinitesimal bendings of submanifolds in low codimension.

That an infinitesimal bending is genuine means that it is not determined by an infinitesimal bending of a submanifold of larger dimension. We show that a strong local condition for a submanifold to be genuinely infinitesimally bendable is to be ruled and we estimate the dimension of the rulings.

Finally, we describe the situation for infinitesimal bendings of compact submanifolds in codimension 2. This is a joint work with M. This is joint work with A. In this talk, I will introduce a classification result of Homogeneous Lagrangian submanifolds obtained by the actions of connected closed subgroups of the solvable part of the Iwasawa decomposition in the complex hyperbolic space.

I will also mention a construction of Homogeneous Lagrangian submanifolds in Hermitian symmetric spaces of noncompact type with higher rank. We give a complete classification of the local geometries possible if the torsion is assumed parallel; this generalizes a previous result of Opozda in the torsion free setting; these geometries are all locally homogeneous. If the torsion is not parallel, we assume the underlying surface is locally homogeneous and provide a complete classification in this setting as well.

Any surface can be realized as a leaf of foliation of a compact 3-manifold, but there are strong restrictions on the topology of generic leaves.

I will speak about compact spaces foliated by hyperbolic surfaces, and discuss which surfaces can coexist as leaves of such a lamination. I will present a combinatorial construction that yields minimal laminations by hyperbolic surfaces with prescribed surfaces as leaves, with a precise control of the topology of the surfaces that appear.

These examples embed inbut some of them exhibit a combination of leaves which cannot occur in codimension one. This is joint work with J. Brum, M. Para ello, en primer lugar, estudiaremos el caso G finito. Es un trabajo conjunto con Pedro Chocano y Manuel A.

I will introduce the concepts mentioned above, discuss about the motivation to study the asymptotic Dirichlet problem for the mean curvature operator, and show some results about the conditions under which it can be solved. The talk is based on joint works with J. Casteras, I.

Holopainen and J. I'll explain a new proof of this theorem. Curso de doutoramento.

**Principles of Riemannian Geometry in Neural Networks - TDLS**

Paul A. Schweitzer S. Hiroshi Tamaru. Cecilia Herrera. Yasuo Matsushita. Juan J. Homare Tadano. Andreas Kollross.Author: David F. Snyder Journal: Trans. Abstract: Let be an upper semicontinuous decomposition used of the -manifold into subcontinua having the shape of closed orientable -manifolds. The primary objective of this investigation is to determine the local connectivity properties of the decomposition space if is -acyclic and is finite dimensional. The Leray-Grothendieck spectral sequence of the decomposition map is analyzed, which relegates the principal part of the investigation to studying the structure of the Leray sheaf of and its relation to the local cohomology of.

Let denote the subset of over which the Leray sheaf is not locally constant, the subset of over which the Leray sheaf is not locally Hausdorff, and. Then we get as our main result, which extends work of R.

Daverman and J. Walsh, and generalizes a result of D. Coram and P. Duvall as well, Theorem. Let be a - acyclic decomposition of the - manifold such thatis finite dimensional, and the set does not locally separate. Then is a generalized - manifold, if eitheror and is orientable.

References [Enhancements On Off] What's this? Ancel, The locally flat approximation of cell-like embedding relationsPh. Bonahon and L. Lecture Note Ser. Press, Cambridge,pp. MR [Br2] Glen E. BredonGeneralized manifolds, revisitedTopology of Manifolds Proc. MR [C] J. Duvall Jr. MR [CD4] D. DavermanDecompositions of manifolds into codimension one submanifoldsCompositio Math.

MR [D2] R. DavermanThe 3-dimensionality of certain codimension-3 decompositionsProc. DavermanDecompositions of manifoldsPure and Applied Mathematics, vol. MR [D4] R. DavermanDecompositions into submanifolds of fixed codimensionGeometric and algebraic topology, Banach Center Publ. MR [DH] R. Daverman and L. HuschDecompositions and approximate fibrationsMichigan Math. Daverman and John J. MR [DW2] R.We hope this content on epidemiology, disease modeling, pandemics and vaccines will help in the rapid fight against this global problem.

Click on title above or here to access this collection. The Hindmarsh--Rose model of neural action potential is revisited from the point of view of global bifurcation analysis, with the singular perturbation parameter held fixed. Of particular concern is a parameter regime where lobe-shaped regions of irregular bursting undergo a transition to stripe-shaped regions of periodic bursting.

The boundary of each stripe represents a fold bifurcation that causes a smooth spike adding transition where the number of spikes in each burst is increased by one.

It is shown via numerical path-following that the lobe-to-stripe transition is organized by a sequence of codimension-one and -two homoclinic bifurcations. Specifically, each of a sequence of homoclinic bifurcation curves in the parameter plane is found to undergo a sharp turn, due to interaction between a two-dimensional unstable manifold and the one-dimensional slow manifold that persists from the singular limit.

Implications of this mechanism for other excitable systems are discussed. Sign in Help View Cart. Article Tools. Add to my favorites. Recommend to Library. Email to a friend. Digg This. Notify Me! E-mail Alerts. RSS Feeds. SIAM J. Related Databases.

Web of Science You must be logged in with an active subscription to view this. Keywords bifurcation analysiscodimension-two homoclinic degeneraciesHindmarsh--Rose modelperiod adding. Publication Data. Publisher: Society for Industrial and Applied Mathematics. Journal of Nonlinear Science Communications in Nonlinear Science and Numerical Simulation 83 Communications in Nonlinear Science and Numerical Simulation 82 Advanced Concepts: Analysis of Nonlinear Oscillators.

Communications in Nonlinear Science and Numerical Simulation 79 The Journal of Mathematical Neuroscience 8 Acta Mathematica Sinica, English Series 34 :6, Frontiers in Computational Neuroscience Physics Letters A :6, Journal of Computational Neuroscience 41 :3, International Journal of Bifurcation and Chaos 26 Journal of Computational Neuroscience 40 :3, Nonlinear Dynamics 83 :3, Alessio Franci and Rodolphe Sepulchre.

Multiple Time Scale Dynamics, EPL Europhysics Letters :2,

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Homogeneous manifolds in codimension two revisited