Density of Local Maxima of the Distance Function to a Set of Points in the Plane

We investigate the circumstances under which the distance function to a closed set in a Banach space having a one-sided directional derivative equal to 1 or −1 implies the existence of nearest points. In reflexive spaces we show that at a dense set of points outside a closed set the distance function has a directional derivative equal to 1.

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Superdifferential Analysis of the Takagi-Van Der Waerden Functions

Set-Valued and Variational Analysis ◽

10.1007/s11228-021-00620-1 ◽

2021 ◽

Author(s):

Juan Ferrera ◽

Javier Gómez Gil ◽

Jesús Llorente

Keyword(s):

Hausdorff Dimension ◽

Hausdorff Measure ◽

Global Maximum ◽

Dimensional Hausdorff Measure ◽

Local Maxima ◽

Set Of Points ◽

Do So

AbstractIn this work we completely describe the superdifferential of the Takagi-Van der Waerden functions and, as a consequence, the local maxima of these functions are characterized. Regarding the set of points where the superdifferential is not empty, we calculate its Hausdorff dimension as well as its corresponding Hausdorff measure. To do so, for any even integer greater than or equal to two we determine the 1/2-dimensional Hausdorff measure of the set of points where Takagi-Van der Waerden functions attain their global maximum.

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Long-term Cyclic Variations of Prominences, Green and Red Coronae over Solar Cycles

International Astronomical Union Colloquium ◽

10.1017/s0252921100064496 ◽

2000 ◽

Vol 179 ◽

pp. 201-204

Author(s):

Vojtech Rušin ◽

Milan Minarovjech ◽

Milan Rybanský

Keyword(s):

Emission Lines ◽

High Latitudes ◽

Green Line ◽

Solar Cycles ◽

The South ◽

Coronal Emission ◽

Local Maxima ◽

The North ◽

Cyclic Variations

AbstractLong-term cyclic variations in the distribution of prominences and intensities of green (530.3 nm) and red (637.4 nm) coronal emission lines over solar cycles 18–23 are presented. Polar prominence branches will reach the poles at different epochs in cycle 23: the north branch at the beginning in 2002 and the south branch a year later (2003), respectively. The local maxima of intensities in the green line show both poleward- and equatorward-migrating branches. The poleward branches will reach the poles around cycle maxima like prominences, while the equatorward branches show a duration of 18 years and will end in cycle minima (2007). The red corona shows mostly equatorward branches. The possibility that these branches begin to develop at high latitudes in the preceding cycles cannot be excluded.

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Video Graphics and High-Voltage Electron Microscopy For Analysis of Microtubule Distributions In Fixed Cells

Proceedings, annual meeting, Electron Microscopy Society of America ◽

10.1017/s0424820100181373 ◽

1990 ◽

Vol 48 (1) ◽

pp. 524-525

Author(s):

Richard Mcintosh ◽

David Mastronarde ◽

Kent McDonald ◽

Rubai Ding

Keyword(s):

Electron Microscopy ◽

Cross Sections ◽

Cell Shape ◽

Video Camera ◽

Computer Memory ◽

High Voltage Electron Microscopy ◽

Thin Sections ◽

Voltage Electron ◽

Morphogenetic Event ◽

Set Of Points

Microtubules (MTs) are cytoplasmic polymers whose dynamics have an influence on cell shape and motility. MTs influence cell behavior both through their growth and disassembly and through the binding of enzymes to their surfaces. In either case, the positions of the MTs change over time as cells grow and develop. We are working on methods to determine where MTs are at different times during either the cell cycle or a morphogenetic event, using thin and thick sections for electron microscopy and computer graphics to model MT distributions.One approach is to track MTs through serial thin sections cut transverse to the MT axis. This work uses a video camera to digitize electron micrographs of cross sections through a MT system and create image files in computer memory. These are aligned and corrected for relative distortions by using the positions of 8 - 10 MTs on adjacent sections to define a general linear transformation that will align and warp adjacent images to an optimum fit. Two hundred MT images are then used to calculate an “average MT”, and this is cross-correlated with each micrograph in the serial set to locate points likely to correspond to MT centers. This set of points is refined through a discriminate analysis that explores each cross correlogram in the neighborhood of every point with a high correlation score.

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The Generalized Distance Function: A Classification Technique for the Biological and Social Sciences

PsycEXTRA Dataset ◽

10.1037/e596702009-001 ◽

1953 ◽

Author(s):

Richard S. Melton

Keyword(s):

Social Sciences ◽

Distance Function ◽

Classification Technique ◽

Generalized Distance

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Parallel Residues

10.23943/princeton/9780691166902.003.0021 ◽

2017 ◽

Author(s):

Bernhard M¨uhlherr ◽

Holger P. Petersson ◽

Richard M. Weiss

Keyword(s):

Distance Function ◽

Length Function ◽

Coxeter System ◽

Vertex Set ◽

Ordered Pairs

This chapter considers the notion of parallel residues in a building. It begins with the assumption that Δ‎ is a building of type Π‎, which is arbitrary except in a few places where it is explicitly assumed to be spherical. Δ‎ is not assumed to be thick. The chapter then elaborates on a hypothesis which states that S is the vertex set of Π‎, (W, S) is the corresponding Coxeter system, d is the W-distance function on the set of ordered pairs of chambers of Δ‎, and ℓ is the length function on (W, S). It also presents a notation in which the type of a residue R is denoted by Typ(R) and concludes with the condition that residues R and T of a building will be called parallel if R = projR(T) and T = projT(R).

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On Asymmetric Distances

Analysis and Geometry in Metric Spaces ◽

10.2478/agms-2013-0004 ◽

2013 ◽

Vol 1 ◽

pp. 200-231 ◽

Cited By ~ 6

Author(s):

Andrea C.G. Mennucci

Keyword(s):

Total Variation ◽

Calculus Of Variations ◽

Distance Function ◽

Metric Space ◽

Metric Spaces ◽

Geodesic Segment ◽

Intrinsic Metric ◽

Typical Application ◽

Variation Formula

Abstract In this paper we discuss asymmetric length structures and asymmetric metric spaces. A length structure induces a (semi)distance function; by using the total variation formula, a (semi)distance function induces a length. In the first part we identify a topology in the set of paths that best describes when the above operations are idempotent. As a typical application, we consider the length of paths defined by a Finslerian functional in Calculus of Variations. In the second part we generalize the setting of General metric spaces of Busemann, and discuss the newly found aspects of the theory: we identify three interesting classes of paths, and compare them; we note that a geodesic segment (as defined by Busemann) is not necessarily continuous in our setting; hence we present three different notions of intrinsic metric space.

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