scholarly journals On the cohomology of loop spaces of compact Lie groups

1985 ◽  
Vol 26 (1) ◽  
pp. 91-99 ◽  
Author(s):  
Howard Hiller
Keyword(s):  
Homogeneous Spaces ◽  
Cell Decomposition ◽  
Point Of View ◽  
Loop Spaces ◽  
Bruhat Decomposition ◽  
Infinite Dimensional ◽  
Dimensional Group ◽  
Analogous Question ◽  
A Cell ◽  
Simply Connected

Let G be a compact, simply-connected Lie group. The cohomology of the loop space ΏG has been described by Bott, both in terms of a cell decomposition [1] and certain homogeneous spaces called generating varieties [2]. It is possible to view ΏG as an infinite dimensional “Grassmannian” associated to an appropriate infinite dimensional group, cf. [3], [7]. From this point of view the above cell-decomposition of Bott arises from a Bruhat decomposition of the associated group. We choose a generator H ∈ H2(ΏG, ℤ) and call it the hyperplane class. For a finite-dimensional Grassmannian the highest power of H carries geometric information about the variety, namely, its degree. An analogous question for ΏG is: What is the largest integer Nk = Nk(G) which divides Hk ∈ H2k(ΏG, ℤ)?Of course, if G = SU(2) = S3, one knows Nk = h!. In general, the deviation of Nk from k! measures the failure of H to generate a divided polynomial algebra in H*(ΏG, ℤ).

scholarly journals Contact metric manifolds with large automorphism group and (κ, µ)-spaces

2019 ◽  
Vol 6 (1) ◽  
pp. 294-302 ◽  
Author(s):  
Antonio Lotta
Keyword(s):  
Automorphism Group ◽  
Symmetric Operator ◽  
Homogeneous Spaces ◽  
Point Of View ◽  
Maximum Dimension ◽  
Contact Metric Manifold ◽  
Contact Metric Manifolds ◽  
Large Automorphism Group ◽  
Dimension 2 ◽  
Simply Connected

AbstractWe discuss the classifiation of simply connected, complete (κ, µ)-spaces from the point of view of homogeneous spaces. In particular, we exhibit new models of (κ, µ)-spaces having Boeckx invariant -1. Finally, we prove that the number ${{(n + 1)(n + 2)} \over 2}$ is the maximum dimension of the automorphism group of a contact metric manifold of dimension 2n +1, n ≥ 2, whose symmetric operator h has rank at least 3 at some point; if this dimension is attained, and the dimension of the manifold is not 7, it must be a (κ, µ)-space. The same conclusion holds also in dimension 7 provided the almost CR structure of the contact metric manifold under consideration is integrable.

PROPER ACTIONS ON SOME EXPONENTIAL SOLVABLE HOMOGENEOUS SPACES

2005 ◽  
Vol 16 (09) ◽  
pp. 941-955 ◽  
Author(s):  
ALI BAKLOUTI ◽  
FATMA KHLIF
Keyword(s):  
Maximal Subgroup ◽  
Homogeneous Space ◽  
Lie Group ◽  
Homogeneous Spaces ◽  
Intersection Property ◽  
Nilpotent Lie Group ◽  
Proper Actions ◽  
Simply Connected

Let G be a connected, simply connected nilpotent Lie group, H and K be connected subgroups of G. We show in this paper that the action of K on X = G/H is proper if and only if the triple (G,H,K) has the compact intersection property in both cases where G is at most three-step and where G is special, extending then earlier cases. The result is also proved for exponential homogeneous space on which acts a maximal subgroup.

Artificial Planning Experience by Means of a Heuristic Cell-Space Model: Simulating International Migration in the Urban Process

1995 ◽  
Vol 27 (10) ◽  
pp. 1647-1665 ◽  
Author(s):  
J Portugali ◽  
I Benenson
Keyword(s):  
Urban Policy ◽  
Initial Conditions ◽  
Point Of View ◽  
Town Planning ◽  
Space Model ◽  
Self Organized ◽  
A Cell ◽  
Policy And Planning ◽  
The City ◽  
Theory And Model

We suggest considering the city as a complex, open, and thus self-organized system, and describing it by means of a cell-space model. A central property of self-organizing systems is that they are not controllable—not by individuals, nor by economic, political, and planning institutions. The city, in this respect, is complex and untamable. Inability to recognize and accept this property is one of the reasons for the difficulties and problems of modernist town planning. The theory and model we present are built to describe the urban process as a historical one in which, given identical initial conditions, each simulation run is unique and never fully repeats itself. From the point of view of urban policy and planning, our heuristic model can guide decisionmakers by answering the following question: ‘given the initial conditions of an inflow of new immigrants (that is, from the ex-USSR), what possible urban scenarios can result, and what are their global structural properties?’.

scholarly journals Symmetry group analysis and invariant solutions of hydrodynamic-type systems

2004 ◽  
Vol 2004 (10) ◽  
pp. 487-534 ◽  
Author(s):  
M. B. Sheftel
Keyword(s):  
Group Analysis ◽  
Type Systems ◽  
Hydrodynamic Type ◽  
Recursion Operators ◽  
Higher Symmetries ◽  
Infinite Dimensional ◽  
Dimensional Group ◽  
Component Systems ◽  
Two Component Systems ◽  
Existence Conditions

We study point and higher symmetries of systems of the hydrodynamic type with and without an explicit dependence ont,x. We consider such systems which satisfy the existence conditions for an infinite-dimensional group of hydrodynamic symmetries which implies linearizing transformations for these systems. Under additional restrictions on the systems, we obtain recursion operators for symmetries and use them to construct infinite discrete sets of exact solutions of the studied equations. We find the interrelation between higher symmetries and recursion operators. Two-component systems are studied in more detail thann-component systems. As a special case, we consider Hamiltonian and semi-Hamiltonian systems of Tsarëv.

scholarly journals The Regularized Weak Functional Matching Pursuit for linear inverse problems

2019 ◽  
Vol 27 (3) ◽  
pp. 317-340 ◽  
Author(s):  
Max Kontak ◽  
Volker Michel
Keyword(s):  
Inverse Problems ◽  
Matching Pursuit ◽  
A Priori ◽  
Computation Time ◽  
Point Of View ◽  
Computational Point ◽  
Linear Inverse Problems ◽  
Infinite Dimensional ◽  
Frame Condition ◽  
Ill Posed

Abstract In this work, we present the so-called Regularized Weak Functional Matching Pursuit (RWFMP) algorithm, which is a weak greedy algorithm for linear ill-posed inverse problems. In comparison to the Regularized Functional Matching Pursuit (RFMP), on which it is based, the RWFMP possesses an improved theoretical analysis including the guaranteed existence of the iterates, the convergence of the algorithm for inverse problems in infinite-dimensional Hilbert spaces, and a convergence rate, which is also valid for the particular case of the RFMP. Another improvement is the cancellation of the previously required and difficult to verify semi-frame condition. Furthermore, we provide an a-priori parameter choice rule for the RWFMP, which yields a convergent regularization. Finally, we will give a numerical example, which shows that the “weak” approach is also beneficial from the computational point of view. By applying an improved search strategy in the algorithm, which is motivated by the weak approach, we can save up to 90  of computation time in comparison to the RFMP, whereas the accuracy of the solution does not change as much.

scholarly journals Arithmetic purity of strong approximation for semi-simple simply connected groups

2020 ◽  
Vol 156 (12) ◽  
pp. 2628-2649
Author(s):  
Yang Cao ◽  
Zhizhong Huang
Keyword(s):  
Quadratic Form ◽  
Strong Approximation ◽  
Homogeneous Spaces ◽  
Algebraic Groups ◽  
Integral Points ◽  
Spin Group ◽  
Linear Algebraic Groups ◽  
Simply Connected ◽  
Quadratic Hypersurfaces ◽  
Almost Prime

In this article we establish the arithmetic purity of strong approximation for certain semisimple simply connected linear algebraic groups and their homogeneous spaces over a number field $k$. For instance, for any such group $G$ and for any open subset $U$ of $G$ with ${\mathrm {codim}}(G\setminus U, G)\geqslant 2$, we prove that (i) if $G$ is $k$-simple and $k$-isotropic, then $U$ satisfies strong approximation off any finite number of places; and (ii) if $G$ is the spin group of a non-degenerate quadratic form which is not compact over archimedean places, then $U$ satisfies strong approximation off all archimedean places. As a consequence, we prove that the same property holds for affine quadratic hypersurfaces. Our approach combines a fibration method with subgroup actions developed for induction on the codimension of $G\setminus U$, and an affine linear sieve which allows us to produce integral points with almost-prime polynomial values.

On a cell decomposition of the Hilbert scheme of points in the plane

1988 ◽  
Vol 91 (2) ◽  
pp. 365-370 ◽  
Author(s):  
Geir Ellingsrud ◽  
Stein Arild Str�mme
Keyword(s):  
Hilbert Scheme ◽  
Cell Decomposition ◽  
Hilbert Scheme Of Points ◽  
A Cell

Tree Pattern Matching

1997 ◽  
Author(s):  
K. Zhang ◽  
D. Shasha
Keyword(s):  
Point Of View ◽  
Programming Approach ◽  
Suffix Trees ◽  
Dynamic Programming Approach ◽  
Ordered Trees ◽  
Single Leaf ◽  
Tree Pattern Matching ◽  
A Cell ◽  
Carry Over ◽  
Don't Cares

Most of this book is about stringology, the study of strings. So why this chapter on trees? Why not graphs or geometry or something else? First, trees generalize strings in a very direct sense: a string is simply a tree with a single leaf. This has the unsurprising consequence that many of our algorithms specialize to strings and the happy consequence that some of those algorithms are as efficient as the best string algorithms. From the point of view of “treeology”, there is the additional pragmatic advantage of this relationship between trees and strings: some techniques from strings carry over to trees, e.g., suffix trees, and others show promise though we don’t know of work that exploits it. So, treeology provides a good example area for applications of stringologic techniques. Second, some of our friends in stringology may wonder whether there is some easy reduction that can take any tree edit problem, map it to strings, solve it in the string domain and then map it back. We don’t believe there is, because, as you will see, tree editing seems inherently to have more data dependence than string editing. (Specifically, the dynamic programming approach to string editing is always a local operation depending on the left, upper, and upper left neighbor of a cell. In tree editing, the upper left neighbor is usually irrelevant — instead the relevant cell depends on the tree topology.) That is a belief not a theorem, so we would like to state right at the outset the key open problem of treeology: can all tree edit problems on ordered trees (trees where the order among the siblings matters) be reduced efficiently to string edit problems and back again?. The rest of this chapter proceeds on the assumption that this question has a negative response. In particular, we discuss the best known algorithms for tree editing and several variations having to do with subtree removal, variable length don’t cares, and alignment. We discuss both sequential and parallel algorithms.

Decomposable Free Loop Spaces

1987 ◽  
Vol 39 (4) ◽  
pp. 938-955 ◽  
Author(s):  
J. Aguadé
Keyword(s):  
Topological Group ◽  
Loop Space ◽  
Base Point ◽  
Point Of View ◽  
Homotopy Equivalence ◽  
Compact Open Topology ◽  
Loop Spaces ◽  
Continuous Maps

In this paper we study the spaces X having the property that the space of free loops on X is equivalent in some sense to the product of X by the space of based loops on X. We denote by ΛX the space of all continuous maps from S1 to X, with the compact-open topology. ΩX denotes, as usual, the loop space of X, i.e., the subspace of ΛX formed by the maps from S1 to X which map 1 to the base point of X.If G is a topological group then every loop on G can be translated to the base point of G and the space of free loops ΛG is homeomorphic to G × ΩG. More generally, any H-space has this property up to homotopy. Our purpose is to study from a homotopy point of view the spaces X for which there is a homotopy equivalence between ΛX and X × ΩX which is compatible with the inclusion ΩX ⊂ ΛX and the evaluation map ΛX → X.

scholarly journals Human Systems Biology and Metabolic Modelling: A Review—From Disease Metabolism to Precision Medicine

2019 ◽  
Vol 2019 ◽  
pp. 1-16 ◽  
Author(s):  
Claudio Angione
Keyword(s):  
Systems Biology ◽  
Precision Medicine ◽  
Model Building ◽  
Research Field ◽  
Point Of View ◽  
Patient Specific ◽  
Metabolic Modelling ◽  
Metabolic Systems ◽  
A Cell ◽  
Metabolic Models

In cell and molecular biology, metabolism is the only system that can be fully simulated at genome scale. Metabolic systems biology offers powerful abstraction tools to simulate all known metabolic reactions in a cell, therefore providing a snapshot that is close to its observable phenotype. In this review, we cover the 15 years of human metabolic modelling. We show that, although the past five years have not experienced large improvements in the size of the gene and metabolite sets in human metabolic models, their accuracy is rapidly increasing. We also describe how condition-, tissue-, and patient-specific metabolic models shed light on cell-specific changes occurring in the metabolic network, therefore predicting biomarkers of disease metabolism. We finally discuss current challenges and future promising directions for this research field, including machine/deep learning and precision medicine. In the omics era, profiling patients and biological processes from a multiomic point of view is becoming more common and less expensive. Starting from multiomic data collected from patients and N-of-1 trials where individual patients constitute different case studies, methods for model-building and data integration are being used to generate patient-specific models. Coupled with state-of-the-art machine learning methods, this will allow characterizing each patient’s disease phenotype and delivering precision medicine solutions, therefore leading to preventative medicine, reduced treatment, andin silicoclinical trials.

Sign in / Sign up

Export Citation Format

Share Document