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17,984
result(s) for
"Integration theory"
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Elliptic Theory for Sets with Higher Co-dimensional Boundaries
Many geometric and analytic properties of sets hinge on the properties of elliptic measure, notoriously missing for sets of higher
co-dimension. The aim of this manuscript is to develop a version of elliptic theory, associated to a linear PDE, which ultimately yields
a notion analogous to that of the harmonic measure, for sets of codimension higher than 1.
To this end, we turn to degenerate
elliptic equations. Let
In another article to appear, we will prove that when
eBook
Higher Orbifolds and Deligne-Mumford Stacks as Structured Infinity-Topoi
We develop a universal framework to study smooth higher orbifolds on the one hand and higher Deligne-Mumford stacks (as well as their
derived and spectral variants) on the other, and use this framework to obtain a completely categorical description of which stacks arise
as the functor of points of such objects. We choose to model higher orbifolds and Deligne-Mumford stacks as infinity-topoi equipped with
a structure sheaf, thus naturally generalizing the work of Lurie (2004), but our approach applies not only to different settings of
algebraic geometry such as classical algebraic geometry, derived algebraic geometry, and the algebraic geometry of commutative ring
spectra as in Lurie (2004), but also to differential topology, complex geometry, the theory of supermanifolds, derived manifolds etc.,
where it produces a theory of higher generalized orbifolds appropriate for these settings. This universal framework yields new insights
into the general theory of Deligne-Mumford stacks and orbifolds, including a representability criterion which gives a categorical
characterization of such generalized Deligne-Mumford stacks. This specializes to a new categorical description of classical
Deligne-Mumford stacks, a result sketched in Carchedi (2019), which extends to derived and spectral Deligne-Mumford stacks as well.
eBook
Degree Theory of Immersed Hypersurfaces
The authors develop a degree theory for compact immersed hypersurfaces of prescribed K-curvature immersed in a compact, orientable Riemannian manifold, where K is any elliptic curvature function. They apply this theory to count the (algebraic) number of immersed hyperspheres in various cases: where K is mean curvature, extrinsic curvature and special Lagrangian curvature and show that in all these cases, this number is equal to -\\chi(M), where \\chi(M) is the Euler characteristic of the ambient manifold M.
eBook
Nonlinear Diffusion Equations and Curvature Conditions in Metric Measure Spaces
The aim of this paper is to provide new characterizations of the curvature dimension condition in the context of metric measure spaces (X,\\mathsf d,\\mathfrak m). On the geometric side, the authors' new approach takes into account suitable weighted action functionals which provide the natural modulus of K-convexity when one investigates the convexity properties of N-dimensional entropies. On the side of diffusion semigroups and evolution variational inequalities, the authors' new approach uses the nonlinear diffusion semigroup induced by the N-dimensional entropy, in place of the heat flow. Under suitable assumptions (most notably the quadraticity of Cheeger's energy relative to the metric measure structure) both approaches are shown to be equivalent to the strong \\mathrm {CD}^{*}(K,N) condition of Bacher-Sturm.
eBook
Integration theories: A review of selected political and economic concepts
There are a number of integration theories based on different knowledge domains. Whatever their conceptual and analytical focus are, all these theories aim to find and provide a reliable explanation based on intelligent, perceivable, and reliable evidence of integration. Most concepts in social sciences do not have the commonality of understanding and can even be context- or culture-specific. As a result, there is no one universal perception of integration. Hence, the aim of the article is to revisit and systemize the knowledge of integration theories. in political and economic terms. The research methods applied in the article include literature review and comparable analysis. The article is structured into three parts. First, the political theories of integration are elaborated upon. Second, the economic integration theories are examined, focusing on the traditional and new economic integration theories, to proceed finally to the presentation of the economic integration theory for developing countries.
Journal Article
Overlapping Iterated Function Systems from the Perspective of Metric Number Theory
In this paper we develop a new approach for studying overlapping iterated function systems. This approach is inspired by a famous
result due to Khintchine from Diophantine approximation which shows that for a family of limsup sets, their Lebesgue measure is
determined by the convergence or divergence of naturally occurring volume sums. For many parameterised families of overlapping iterated
function systems, we prove that a typical member will exhibit similar Khintchine like behaviour. Families of iterated function systems
that our results apply to include those arising from Bernoulli convolutions, the
For each
Last of all, we introduce a property of an iterated function system that we call being consistently
separated with respect to a measure. We prove that this property implies that the pushforward of the measure is absolutely continuous.
We include several explicit examples of consistently separated iterated function systems.
eBook
Probabilistic Methods in Geometry, Topology and Spectral Theory
This volume contains the proceedings of the CRM Workshops on Probabilistic Methods in Spectral Geometry and PDE, held from August 22-26, 2016 and Probabilistic Methods in Topology, held from November 14-18, 2016 at the Centre de Recherches Mathématiques, Université de Montréal, Montréal, Quebec, Canada. Probabilistic methods have played an increasingly important role in many areas of mathematics, from the study of random groups and random simplicial complexes in topology, to the theory of random Schrödinger operators in mathematical physics. The workshop on Probabilistic Methods in Spectral Geometry and PDE brought together some of the leading researchers in quantum chaos, semi-classical theory, ergodic theory and dynamical systems, partial differential equations, probability, random matrix theory, mathematical physics, conformal field theory, and random graph theory. Its emphasis was on the use of ideas and methods from probability in different areas, such as quantum chaos (study of spectra and eigenstates of chaotic systems at high energy); geometry of random metrics and related problems in quantum gravity; solutions of partial differential equations with random initial conditions. The workshop Probabilistic Methods in Topology brought together researchers working on random simplicial complexes and geometry of spaces of triangulations (with connections to manifold learning); topological statistics, and geometric probability; theory of random groups and their properties; random knots; and other problems. This volume covers recent developments in several active research areas at the interface of Probability, Semiclassical Analysis, Mathematical Physics, Theory of Automorphic Forms and Graph Theory.
eBook
A Unified Approach to Structural Limits and Limits of Graphs with Bounded Tree-Depth
In this paper we introduce a general framework for the study of limits of relational structures and graphs in particular, which is
based on a combination of model theory and (functional) analysis. We show how the various approaches to graph limits fit to this
framework and that they naturally appear as “tractable cases” of a general theory. As an outcome of this, we provide extensions of known
results. We believe that this puts these into a broader context. The second part of the paper is devoted to the study of sparse
structures. First, we consider limits of structures with bounded diameter connected components and we prove that in this case the
convergence can be “almost” studied component-wise. We also propose the structure of limit objects for convergent sequences of sparse
structures. Eventually, we consider the specific case of limits of colored rooted trees with bounded height and of graphs with bounded
tree-depth, motivated by their role as “elementary bricks” these graphs play in decompositions of sparse graphs, and give an explicit
construction of a limit object in this case. This limit object is a graph built on a standard probability space with the property that
every first-order definable set of tuples is measurable. This is an example of the general concept of
eBook
Using Organismic Integration Theory to Explore the Associations Between Users’ Exercise Motivations and Fitness Technology Feature Set Use
Wearable devices and applications (apps) that offer a variety of features intended to support exercisers have flooded the marketplace. Organismic integration theory (OIT) proposes that motivations to exercise can vary along a spectrum of self-determination. To best serve exercisers and assist organizations that are developing and promoting fitness technologies, we need a better understanding of how individuals’ exercise motivations influence their fitness technology feature set use. We also need to determine the impact of fitness technology features on enhancing or undermining wellness outcomes—such as subjective vitality. Our results suggest that almost every subtype of exerciser, where the subtype is defined by OIT motivations toward exercise, has a unique use profile. Our findings also suggest that the social interaction and data management features of current fitness technologies show promise in assisting well-being outcomes, but only for the more self-determined and amotivated subtypes of exercisers. This leads us to suggest that providing every type of exerciser the motivational support that best fits their motivational profile may not be a trivial task, but it ultimately may be necessary for fitness technologies to be universally useful in supporting wellness outcomes.
Journal Article