scholarly journals Minimal pairs of bounded closed convex sets as minimal representations of elements of the Minkowski–Rådström–Hörmander spaces

2009 ◽  
Author(s):  
Jerzy Grzybowski ◽  
Diethard Pallaschke ◽  
Ryszard Urbański
Keyword(s):  
Convex Sets ◽  
Minimal Pairs ◽  
Minimal Representations ◽  
Hörmander Spaces

scholarly journals Minimal pairs of compact convex sets

2004 ◽  
Author(s):  
Diethard Pallaschke ◽  
Ryszard Urbański
Keyword(s):  
Convex Sets ◽  
Compact Convex ◽  
Minimal Pairs

On the amount of minimal pairs of convex sets

2010 ◽  
Vol 25 (1) ◽  
pp. 89-96
Author(s):  
Jerzy Grzybowski ◽  
Diethard Pallaschke ◽  
Ryszard Urbański
Keyword(s):  
Convex Sets ◽  
Minimal Pairs

Minimal Pairs of Convex Sets

2002 ◽  
pp. 49-90
Author(s):  
Diethard Pallaschke ◽  
Ryszard Urbański
Keyword(s):  
Convex Sets ◽  
Minimal Pairs

On the Use of POCS for Restoring the “Missing Wedge” in Electron Tomography

1990 ◽  
Vol 48 (1) ◽  
pp. 520-521
Author(s):  
Neng-Yu Zhang ◽  
Bruce F. McEwen ◽  
Joachim Frank
Keyword(s):  
Function Space ◽  
Convex Sets ◽  
Electron Tomography ◽  
Spinal Ganglion ◽  
Fourier Space ◽  
Missing Information ◽  
Voltage Electron ◽  
High Voltage Electron Microscope ◽  
Energy Bound ◽  
Spatial Boundedness

Reconstructions of asymmetric objects computed by electron tomography are distorted due to the absence of information, usually in an angular range from 60 to 90°, which produces a “missing wedge” in Fourier space. These distortions often interfere with the interpretation of results and thus limit biological ultrastructural information which can be obtained. We have attempted to use the Method of Projections Onto Convex Sets (POCS) for restoring the missing information. In POCS, use is made of the fact that known constraints such as positivity, spatial boundedness or an upper energy bound define convex sets in function space. Enforcement of such constraints takes place by iterating a sequence of function-space projections, starting from the original reconstruction, onto the convex sets, until a function in the intersection of all sets is found. First applications of this technique in the field of electron microscopy have been promising.To test POCS on experimental data, we have artificially reduced the range of an existing projection set of a selectively stained Golgi apparatus from ±60° to ±50°, and computed the reconstruction from the reduced set (51 projections). The specimen was prepared from a bull frog spinal ganglion as described by Lindsey and Ellisman and imaged in the high-voltage electron microscope.

Final Consonant Deletion in African American Children Speaking Black English

1993 ◽  
Vol 24 (3) ◽  
pp. 161-166 ◽  
Author(s):  
Michael J. Moran
Keyword(s):  
African American ◽  
Black English ◽  
African American Children ◽  
Vowel Length ◽  
Minimal Pairs ◽  
Speech Evaluation ◽  
American Children

The purpose of this study was to determine whether African American children who delete final consonants mark the presence of those consonants in a manner that might be overlooked in a typical speech evaluation. Using elicited sentences from 10 African American children from 4 to 9 years of age, two studies were conducted. First, vowel length was determined for minimal pairs in which final consonants were deleted. Second, listeners who identified final consonant deletions in the speech of the children were provided training in narrow transcription and reviewed the elicited sentences a second time. Results indicated that the children produced longer vowels preceding "deleted" voiced final consonants, and listeners perceived fewer deletions following training in narrow transcription. The results suggest that these children had knowledge of the final consonants perceived to be deleted. Implications for assessment and intervention are discussed.

The Standard Metric

2017 ◽  
Author(s):  
Bernhard M¨uhlherr ◽  
Holger P. Petersson ◽  
Richard M. Weiss
Keyword(s):  
Affine Transformation ◽  
Convex Sets ◽  
Finite Dimension ◽  
Affine Subspace ◽  
Affine Transformations ◽  
Real Vector ◽  
Euclidean Metric ◽  
Affine Hyperplane ◽  
Tits Building ◽  
Convex Closure

This chapter presents some results about groups generated by reflections and the standard metric on a Bruhat-Tits building. It begins with definitions relating to an affine subspace, an affine hyperplane, an affine span, an affine map, and an affine transformation. It then considers a notation stating that the convex closure of a subset a of X is the intersection of all convex sets containing a and another notation that denotes by AGL(X) the group of all affine transformations of X and by Trans(X) the set of all translations of X. It also describes Euclidean spaces and assumes that the real vector space X is of finite dimension n and that d is a Euclidean metric on X. Finally, it discusses Euclidean representations and the standard metric.

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