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result(s) for
"Dimensions"
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How big is big and how small is small : the sizes of everything and why
This book is about how big is the universe and how small are quarks, and what are the sizes of dozens of things between these two extremes. It describes the sizes of atoms and planets, quarks and galaxies, cells and sequoias. It is a romp through forty-five orders of magnitude from the smallest sub-nuclear particles we have measured, to the edge of the observed universe. It also looks at time, from the epic age of the cosmos to the fleeting lifetimes of ethereal particles. It is a narrative that trips its way from stellar magnitudes to the clocks on GPS satellites, from the nearly logarithmic scales of a piano keyboard through a system of numbers invented by Archimedes and on to the measurement of the size of an atom.
Book
Generalised intermediate dimensions
We introduce a family of dimensions, which we call the
Φ
-intermediate dimensions, that lie between the Hausdorff and box dimensions and generalise the intermediate dimensions introduced by Falconer, Fraser and Kempton. This is done by restricting the relative sizes of the covering sets in a way that allows for greater refinement than in the definition of the intermediate dimensions. We also extend the theory from Euclidean space to a wider class of metric spaces. We prove that these dimensions can be used to ‘recover the interpolation’ between the Hausdorff and box dimensions of compact subsets for which the intermediate dimensions are discontinuous at
θ
=
0
, thus providing finer geometric information about such sets. We prove continuity-like results involving the Assouad and lower dimensions, which give a sharp general lower bound for the intermediate dimensions that is positive for all
θ
∈
(
0
,
1
]
for sets with positive box dimension. We also prove Hölder distortion estimates, a mass distribution principle, and a Frostman type lemma, which we use to study dimensions of product sets.
Journal Article
Mutant X : the complete collection
Alex Summers has been thrown into a parallel world! Displacing his counterpart, Havok finds himself leading a mutant team with familiar faces. The Six are the Brute, the Fallen, Ice-Man, Bloodstorm, and Marvel Woman--better known as Havok's wife, Madelyne Pryor. Life is suddenly a lot more complicated for Havok, especially because his teammates don't believe his claims about being in the wrong universe. But things get really messy when one of the Six puts plans in motion to take over their world, stoking the fires of anti-mutant sentiment. Witness this dimension's version of Wolverine, Elektra, Nick Fury, Gambit, Magneto, the Fantastic Four, and more. But where are the X-Men? And can Havok find a way home before this dangerous dimension becomes the death of him?
Book
E 8 instantons on type-A ALE spaces and supersymmetric field theories
Abstract We consider the 6d superconformal field theory realized on M5-branes probing the E 8 end-of-the-world brane on the deformed and resolved ℂ 2/ℤ k singularity. We give an explicit algorithm which determines, for arbitrary holonomy at infinity, the 6d quiver gauge theory on the tensor branch, the type-A class S description of the T 2 compactification, and the star-shaped quiver obtained as the mirror of the T 3 compactification.
Journal Article
Do institutional dimensions matter at different stages of the entrepreneurial process? A multi-country study
Entrepreneurs should navigate through different stages from the conception of an idea until the business is operational. According to these stages, we expected that the context has a different impact on an individual’s decisions. This paper analyses the role of institutional dimensions (regulative, normative, and cultural-cognitive) in the entrepreneurial process (potential, nascent, and new entrepreneurship), using data from the Global Entrepreneurship Monitor and Heritage Foundation, with a sample of 99 countries for the period 2001–2017. Through panel data, the main findings show that (a) regulations regarding new business creation have a stronger influence on new entrepreneurship, (b) social norms have more influence on potential entrepreneurs and individual perceptions regarding their self-capacity and experience to start a new business, and (c) the cultural-cognitive dimension has a stronger influence on nascent entrepreneurship. Policymakers could consider these results to promote and generate target group policies that effectively encourage entrepreneurial activity, which is also distinguished by the level of development among countries.
Journal Article
Applications of dimension interpolation to orthogonal projections
Dimension interpolation is a novel programme of research which attempts to unify the study of fractal dimension by considering various spectra which live in between well-studied notions of dimension such as Hausdorff, box, Assouad and Fourier dimension. These spectra often reveal novel features not witnessed by the individual notions and this information has applications in many directions. In this survey article, we discuss dimension interpolation broadly and then focus on applications to the dimension theory of orthogonal projections. We focus on three distinct applications coming from three different dimension spectra, namely, the Fourier spectrum, the intermediate dimensions, and the Assouad spectrum. The celebrated Marstrand–Mattila projection theorem gives the Hausdorff dimension of the orthogonal projection of a Borel set in Euclidean space for almost all orthogonal projections. This result has inspired much further research on the dimension theory of projections including the consideration of dimensions other than the Hausdorff dimension, and the study of the exceptional set in the Marstrand–Mattila theorem.
Journal Article
Institutional dimensions and entrepreneurial activity: an international study
The purpose of this article is to examine the influence of institutional dimensions (regulative, normative and cultural-cognitive) on the probability of becoming an entrepreneur. The main findings demonstrate, through logistic regression, that a favourable regulative dimension (fewer procedures to start a business), normative dimension (higher media attention for new business) and cultural-cognitive dimension (better entrepreneurial skills, less fear of business failure and better knowing of entrepreneurs) increase the probability of being an entrepreneur. Data were obtained from both the Global Entrepreneurship Monitor and the International Institute for Management and Development for the year 2008, considering a sample of 30 countries and 36,525 individuals. The study advances the literature by providing new information on the environmental factors that affect entrepreneurial activity in the light of institutional economics. Also, the research could be useful for designing policies to foster entrepreneurship in different environments.
Journal Article
Global priorities for conservation across multiple dimensions of mammalian diversity
Conservation priorities that are based on species distribution, endemism, and vulnerability may underrepresent biologically unique species as well as their functional roles and evolutionary histories. To ensure that priorities are biologically comprehensive, multiple dimensions of diversity must be considered. Further, understanding how the different dimensions relate to one another spatially is important for conservation prioritization, but the relationship remains poorly understood. Here, we use spatial conservation planning to (i) identify and compare priority regions for global mammal conservation across three key dimensions of biodiversity—taxonomic, phylogenetic, and traits—and (ii) determine the overlap of these regions with the locations of threatened species and existing protected areas. We show that priority areas for mammal conservation exhibit low overlap across the three dimensions, highlighting the need for an integrative approach for biodiversity conservation. Additionally, currently protected areas poorly represent the three dimensions of mammalian biodiversity. We identify areas of high conservation priority among and across the dimensions that should receive special attention for expanding the global protected area network. These highpriority areas, combined with areas of high priority for other taxonomic groups and with social, economic, and political considerations, provide a biological foundation for future conservation planning efforts.
Journal Article
Localization of scalar field on the brane-world by coupling with gravity
A
bstract
In this paper, we consider a coupling mechanism between the kinetic term and the gravity, in which a coupling function
F
(
R
) is introduced into the kinetic term of the five-dimensional scalar field. Based on this scenario, we investigate the localization of scalar fields in three specific braneworld models: the Minkowski brane, the de Sitter brane, and the Anti-de Sitter brane. The brane models considered here are regular with no singularity for scalar curvature. For the Minkowshi brane case, the zero mode can always be localized on the brane, and the massive modes can be localized or quasi-localized on the brane. For the dS
4
brane case, two forms of factor
F
(
R
) is considered. The zero mode can always be localized, and the massive modes could be quasi-localized on the brane. Besides, with the second coupling factor, the scalar zero mode could be localized on the both sides of the origin of extra dimension, while the massive modes could be quasi-localized on the origin. Lastly, for the AdS
4
brane case, the localization of the scalar zero mode requires the consideration of a coupling potential
V
(Φ,
φ
), while the massive modes can still be localized on the brane with an infinite number.
Journal Article
Five-dimensional spinor helicity for all masses and spins
We develop a spinor helicity formalism for five-dimensional scattering amplitudes of any mass and spin configuration. While five-dimensional spinor helicity variables have been previously studied in the context of $\\mathcal{N}$ = 2, 4 supersymmetric Yang-Mills scattering amplitudes with spin less than two [1], we propose an alternative viewpoint that stems from d-dimensional spinor helicity variables avoiding the decomposition of a massive momentum into the sum of two massless momenta. By enumerating all possible independent little group tensors, we systematically build the full space of five-dimensional three-point tree-level scattering amplitudes for any configuration of spins and masses. Furthermore, we provide a prescription for computing the high energy limit of scattering amplitudes written in our spinor helicity variables. We also expect that our formalism will be applicable to effective field theories with higher spin, in particular, the scattering of highly spinning black holes in five dimensions.
Journal Article