THE SIGNATURE OF PARTIALLY DEFINED LOCAL COEFFICIENT SYSTEMS

2008 ◽  
Vol 17 (12) ◽  
pp. 1455-1481 ◽  
Author(s):  
MARKUS BANAGL
Keyword(s):  
Local System ◽  
Fiber Bundles ◽  
Characteristic Class ◽  
Intersection Homology ◽  
Entire Space ◽  
Singular Spaces ◽  
Codimension Two ◽  
Stratified Space ◽  
Stratified Pseudomanifold ◽  
Definition Of

Intersection homology enables the definition of a twisted signature of a stratified pseudomanifold with coefficients in a local system that is typically only given on the top stratum. If the local system extends to the entire space, then the twisted signature can be computed by a version of a characteristic class formula first observed by Atiyah for smooth fiber bundles. In the first part of this paper, we construct examples of singular spaces, equipped with local systems that do not extend, such that the above characteristic class formula fails. In the second part, we consider smooth codimension two embeddings of manifolds. We view the target as a stratified space with bottom stratum the image of the embedding and top stratum the complement. When the embedded manifold is a sphere, we establish various formulae that compute the twisted signature even when the local system does not extend from the top stratum to the entire space.

scholarly journals Refined intersection homology on non-Witt spaces

2014 ◽  
Vol 07 (01) ◽  
pp. 105-133 ◽  
Author(s):  
Pierre Albin ◽  
Markus Banagl ◽  
Eric Leichtnam ◽  
Rafe Mazzeo ◽  
Paolo Piazza
Keyword(s):  
Homology Theory ◽  
Cohomology Theory ◽  
The Self ◽  
Intersection Cohomology ◽  
Intersection Homology ◽  
Stratified Spaces ◽  
Constructible Sheaves ◽  
De Rham Theory ◽  
De Rham ◽  
Definition Of

We investigate a generalization to non-Witt stratified spaces of the intersection homology theory of Goresky–MacPherson. The second-named author has described the self-dual sheaves compatible with intersection homology, and the other authors have described a generalization of Cheeger's L2 de Rham cohomology. In this paper we first extend both of these cohomology theories by describing all sheaf complexes in the derived category of constructible sheaves that are compatible with middle perversity intersection cohomology, though not necessarily self-dual. Our main result is that this refined intersection cohomology theory coincides with the analytic de Rham theory on Thom–Mather stratified spaces. The word "refined" is motivated by the fact that the definition of this cohomology theory depends on the choice of an additional structure (mezzo-perversity) which is automatically zero in the case of a Witt space.

The Regional Innovation Capacity Model

2012 ◽  
pp. 243-256
Author(s):  
Antonio Lerro ◽  
Giovanni Schiuma
Keyword(s):  
Local System ◽  
Point Of View ◽  
Future Research ◽  
Regional Innovation ◽  
Knowledge Based ◽  
Future Research Agenda ◽  
Capacity Model ◽  
Innovation Dynamics ◽  
Innovation Capacity ◽  
Definition Of

This chapter aims to present a conceptual model aimed to understand the Intellectual Capital-based (IC) characteristics of the regional innovation capacity. The proposed Regional Innovation Capacity Model (RICM) can be used for interpretative and normative purposes to analyse the innovation dynamics taking place at regional and territorial level. From an interpretative point of view, the model identifies the pillars grounding the innovation capacity of a local system. While, from a normative perspective, the model can inspire the definition of guidelines driving the design and the implementation of actions, projects and programmes aimed to stimulate and sustain regional development dynamics. The RICM adopts a knowledge-based perspective assuming that IC, in the forms of regional knowledge assets, and knowledge dynamics, in the form of knowledge transfer and learning processes, are the drivers of innovative processes and outputs. The chapter concludes proposing a future research agenda.

COMPUTING TWISTED SIGNATURES AND L-CLASSES OF NON-WITT SPACES

2006 ◽  
Vol 92 (2) ◽  
pp. 428-470 ◽  
Author(s):  
MARKUS BANAGL
Keyword(s):  
Characteristic Class ◽  
Joint Work ◽  
Stratified Space

In previous joint work with Cappell and Shaneson, we have established an Atiyah–Lusztig–Meyer-type multiplicative characteristic class formula for the twisted signature and, more generally, the twisted $L$-class, of a stratified Witt space. The present paper shows that these formulae hold even when the stratified space does not satisfy the Witt condition. It constitutes one of the first applications of signature homology.

scholarly journals New Bifurcation Critical Criterion of Flip-Neimark-Sacker Bifurcations for Two-Parameterized Family ofn-Dimensional Discrete Systems

2012 ◽  
Vol 2012 ◽  
pp. 1-12 ◽  
Author(s):  
Shengji Yao
Keyword(s):  
Jacobian Matrix ◽  
Polynomial Equation ◽  
Discrete Systems ◽  
Feasible Region ◽  
General Sense ◽  
Codimension Two ◽  
Parameterized Family ◽  
Critical Criterion ◽  
Definition Of ◽  
Codimension Two Bifurcation

A new bifurcation critical criterion of flip-Neimark-Sacker bifurcation is proposed for detecting or anticontrolling this type of codimension-two bifurcation of discrete systems in a general sense. The criterion is built on the properties of coefficients of characteristic equations instead of the properties of eigenvalues of Jacobian matrix of nonlinear system, which is formulated using a set of simple equalities and inequalities consisting of the coefficients of characteristic polynomial equation. The inequality conditions enable us to easily pick off the fake parameter domain whereas the equality conditions are used to accurately locate the critical bifurcation point. In particular, after the bifurcation parameter piont is determined, the inequality conditions can be used to figure out the feasible region of other system parameters. Thus, the criterion is suitable for two-parameterized family ofn-dimensional discrete systems. As compared with the classical critical criterion (or definition) of flip-Neimark-Sacker bifurcation stated in terms of the properties of eigenvalues, the proposed criterion is preferable in anticontrolling or detecting the existence of flip-Neimark-Sacker bifurcation in high-dimension nonlinear systems, due to its explicit parameter mechanism of the bifurcation.

Singular Spaces, Characteristic Classes, and Intersection Homology

1991 ◽  
Vol 134 (2) ◽  
pp. 325 ◽  
Author(s):  
Sylvain E. Cappell ◽  
Julius L. Shaneson
Keyword(s):  
Characteristic Classes ◽  
Intersection Homology ◽  
Singular Spaces

scholarly journals Globalist potential of the territorial community: current issues of definition, understanding and possible implementation

2020 ◽  
pp. 39-46
Author(s):  
Denis Bobrovnyk
Keyword(s):  
Social Life ◽  
Local Governments ◽  
Social Practice ◽  
Local System ◽  
Collective Mentality ◽  
The Face ◽  
State And Local ◽  
Definition Of ◽  
International Experts ◽  
Final Consumer

The article is devoted to the study of topical issues of definition, understanding and possible realization of the globalist potentialof the territorial community.It is argued that local self-government and territorial community, firstly, are the direct subject of globalization, because they areinfluenced by its factors and species blocks (economic, political, legal, etc.), and secondly, they act as the final consumer of its benefits– the beneficiaries, and the direct or indirect object of its negative phenomena (global, regional and domestic economic crisis dueto the pandemic COVID-19, and related unemployment, closure of enterprises, delays in payment of wages salaries, pensions and othersocial benefits, sequestration of social articles of the state and local budgets, etc.); in addition, thirdly, when applied directly to locallife, they are the relevant user, producer and reproducer of its achievements, actively using them to organize their life cycle or relyingon the influence of global factors on their daily lives.It is proved that the definition of a complex system-complex phenomenology of the globalist potential of the territorial communityis due to the fact that, firstly, it is simultaneously related to domestic – municipal and constitutional law, as well as systems of generalinternational law and international relations; secondly, it has not yet been properly reflected in the science of municipal and internationallaw, although some domestic and foreign scholars and international experts have touched on this issue in an attempt to focuson this phenomenon (phenomenon), which is mostly evaluative. nature.It is established that determining the global potential of the territorial community, it should be understood that it is based on theknowledge, skills and abilities of its members, as well as formed by local governments (representative and executive), including allother bodies and subjects of the local system. self-government – to protect the existential guidelines for the existence, functioning anddevelopment of territorial community in the face of global influences and changes, taking into account its own individual and groupand collective cooperation opportunities both at the level of its own state and abroad, historical and geographical identity, guidelinesfor individual and collective mentality, appropriate sustainable forms of social life and social practice, based on the relevant moral andethical values of community.

scholarly journals Generalized Discontinuity of Real-Valued Functions

2013 ◽  
Vol 55 (1) ◽  
pp. 1-16
Author(s):  
Rafał Zduńczyk
Keyword(s):  
Lebesgue Measure ◽  
Real Function ◽  
Baire Category ◽  
Local System ◽  
Local Systems ◽  
Basic Properties ◽  
Lebesgue Density ◽  
Definition Of ◽  
Set Of Points

Abstract We present a proof of the theorem on countability of the set of points of generalized discontinuity of an ( , ) regular real function f:X→ℝ, where is a local system in X and is a partition of X. We start with a definition of a local system in a generalized form and with basic properties of local systems. The concepts are illustrated with examples. The main result is applied both for regularities in the sense of density connected with the Lebesgue measure on ℝn (Lebesgue density) and with Baire category -density), respectively.

scholarly journals Theoretical Models of Modern Suburban Recreational Complexes

2018 ◽  
pp. 71-76
Author(s):  
Ksenia O. Bolyaeva
Keyword(s):  
Theoretical Models ◽  
Local System ◽  
Taxonomic Rank ◽  
Recreational Facilities ◽  
Open Type ◽  
Closed Type ◽  
Definition Of ◽  
Functional Profiles ◽  
Recreational Resources ◽  
Object Relative

The article discusses aspects of the formation of theoretical models of modern suburban recreational complexes (SRC). The recreational environment is defined as result of interaction, interpenetration and mutual addition of three complex factors: recreational activity, recreational resources and recreational structures. Definition of SRC as the local system of recreational suburban formations of various functional profiles consisting of recreational structures is given. SRC is distinguished by taxonomic and typological features. The taxonomic feature determines the position of the SRC in the scale of rank of the recreational formations system. The typological feature indicates qualitative differences in the recreational formations of a single taxonomic rank. The analysis of examples of completed SRC allows us to distinguish the basic theoretical models of modern SRC, depending on thelocation of the recreation object. Specific features of the formation of a specialized recreational layout of the SRC are revealed: closed, half-closed, half-open and open. SRC of the closed type contain the full list of necessary rooms for performance of recreational function and don't demand connection with recreational facilities of the region. SRC of half-closed type perform one dominating recreational function with partial use of recreational resources of the area. SRC of half-open type assumes only auxiliary recreational group and operate recreational resources of the area to perform the main recreational function. SRC of open type assume only placement and incidental service and are completely focused on recreational resources of the area. The specificity of contemporary types of SRC is the position of the main recreation object relative to the SRC itself. The object of recreation within the SRC is formed by a specialized recreational group, which sets the orientation of the SRC.

scholarly journals A Critical Exploration of Some Fundamental Terms and Terminologies in Statistics

2020 ◽  
pp. 45-50 ◽  
Keyword(s):  
Degrees Of Freedom ◽  
Grouped Data ◽  
Entire Space ◽  
Single Measure ◽  
Linearly Independent ◽  
Critical Exploration ◽  
Statistical Sense ◽  
Sample Range ◽  
Definition Of

This article focuses on the understanding of definitions of several widely used statistical terms such as degrees of freedom, locations, range, dispersion, grouped and ungrouped data. The terms have been redefined along with examples so that they stand alone to express their meaning. In this article, a new term ‘the smallest unit’ in a statistical sense has been defined and illustrated in some instances. It is also indicated how statisticians or practitioners of statistics are using it knowingly or unknowingly. We have mentioned the application of the smallest unit in the classification of data. Moreover, the concept of the smallest unit has been synced with the definition of the sample range so that the range can cover the entire space of values. Therefore, the proposed sample range can now better approximate the population range. We have shown that researchers can end up with misleading result if they treat a dataset as ungrouped data when it is truly a grouped data. This has been discussed in the computation of different percentiles. Moreover, the crux of the definition of degrees of freedom and dispersion has been pointed out which has helped repelled the confusion behind these terms. We have shown how the concept of linearly independent pieces of information is related to the definition of degrees of freedom. We have also emphasized not to mix the definition of standard deviation and/or variance with the whole concept of dispersion because the former is merely a single measure among many measures of the latter.

Sign in / Sign up

Export Citation Format

Share Document