Geometric Integration Theory

Krantz, Steven G.; Parks, Harold R.

doi:10.1007/978-0-8176-4679-0

Cited by 188 publications

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“…The building blocks of this space are zero sets of a Hausdorff dimension φ embedded into empty sets of a Hausdorff φ 2 which are distributed in a uniform randomness to form via an infinite number of unions and intersections a hierarchical-infinite dimensional Cantor set [68][69][70][71][72][73][74][75][76][77]. This Suslin-like operation [71] entails few fundamental equations which determine all what is required to deduce various vital invariant properties of this space which we use to mimic our own physical macro and micro quantum spacetime. Based on the above we were able to show in the last four years beyond reasonable doubt that the jewel of special relativity, i.e.…”

Section: Background Information and Preliminary Considerationsmentioning

confidence: 99%

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The Measure Concentration of Convex Geometry in a Quasi Banach Spacetime behind the Supposedly Missing Dark Energy of the Cosmos

Naschie¹

2014

AJAA

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Abstract:Based on the Banach spaces motivated theory of convex geometry in high dimensionality we give a new confirmation of the derivation of the 95.5 per cent so called dark energy density of the cosmos. The result derives directly from the purely geometric-topological phenomenon of measure-mass concentration and gives an unqualified complete confirmation of our previous results including the hidden quantum nature of Einstein's celebrated equation E = mc 2 . Thus the quantum dissection into ordinary energy of the quantum particle E(O) = mc 2 /22 and dark energy of the quantum wave E(D) = mc 2 (21/22) were validated by first a purely mathematical theorem of convex geometry and second by conservation of energy leading to E(O) + E(D) = mc 2 as in Einstein's special theory of relativity.

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Section: Background Information and Preliminary Considerationsmentioning

confidence: 99%

“…Thus although infinite dimensional, because of its hierarchal geometry reflected in its Suslin expansion [71,76] …”

Section: The Measure Concentration Of E-infinity Spacetime As An Explmentioning

confidence: 99%

The Measure Concentration of Convex Geometry in a Quasi Banach Spacetime behind the Supposedly Missing Dark Energy of the Cosmos

Naschie¹

2014

AJAA

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“…The standard reference for currents is the treatise [Federer 1969]. Other very good references are [Almgren 1986a;Burago and Zalgaller 1988;Federer and Fleming 1960;Hardt and Simon 1986;Krantz and Parks 2008;Morgan 2009;Simon 1983]. …”

Section: Crystals and Polycrystalsmentioning

confidence: 99%

B2-convexity implies strong and weak lower semicontinuity of partitions of ℝⁿ

Caraballo¹

2011

Pacific J. Math.

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We prove that B2-convexity is sufficient for lower semicontinuity of surface energy of partitions of ‫ޒ‬ n , for any n ≥ 2. We establish lower semicontinuity in the usual strong topology, assuming the regions converge in volume. We also establish lower semicontinuity in the more general situation in which we suppose integral currents associated with individual regions converge to some integral current in the weak topology of integral currents.B2-convexity, formulated by F. Morgan in 1995, is a powerful condition since it is easy to work with and since many other conditions from the literature imply it. Our results therefore imply that each of those conditions is sufficient for strong and weak lower semicontinuity of surface energy.We establish other results of independent interest, including a Lebesgue point theorem for partitions and a localization theorem, which shows that if lower semicontinuity holds locally then it holds globally.

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“…We will not discuss integral currents here in detail-see [25], [33], [29], or [7] for more details. spt [A] is the smallest closed set Q such that…”

Section: Notation and Bv Functionsmentioning

confidence: 99%

Local simplicity, topology, and sets of finite perimeter

Caraballo¹

2011

Interfaces Free Bound.

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We introduce a useful and relatively easy to check condition, local simplicity, which provides significantly more structure to sets of finite perimeter, while not being too restrictive. Local simplicity holds for minimizers in a wide variety of variational problems in materials science, biology, image processing, oncology, and other fields. We prove several regularity and structural properties of locally simple sets and their boundaries, including a vital decomposition theorem that in our setting strengthens the conclusion of theorems of H. Federer and L. Ambrosio, V. Caselles, S. Masnou, and J.-M. Morel. We establish strong connections between topology and sets of finite perimeter, so that ordinary notions of openness, closedness, and connectedness may be readily used in the finite perimeter setting.We apply these results to an image reconstruction procedure from image processing, L 1 TVminimization. The density ratio bounds computed to establish local simplicity are themselves of practical importance, as they provide concrete, easy to compute criteria to check simulations against.

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Geometric Integration Theory

Cited by 188 publications

References 0 publications

The Measure Concentration of Convex Geometry in a Quasi Banach Spacetime behind the Supposedly Missing Dark Energy of the Cosmos

The Measure Concentration of Convex Geometry in a Quasi Banach Spacetime behind the Supposedly Missing Dark Energy of the Cosmos

B2-convexity implies strong and weak lower semicontinuity of partitions of ℝⁿ

Local simplicity, topology, and sets of finite perimeter

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Geometric Integration Theory

Cited by 188 publications

References 0 publications

The Measure Concentration of Convex Geometry in a Quasi Banach Spacetime behind the Supposedly Missing Dark Energy of the Cosmos

The Measure Concentration of Convex Geometry in a Quasi Banach Spacetime behind the Supposedly Missing Dark Energy of the Cosmos

B2-convexity implies strong and weak lower semicontinuity of partitions of ℝn

Local simplicity, topology, and sets of finite perimeter

Contact Info

Product

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B2-convexity implies strong and weak lower semicontinuity of partitions of ℝⁿ