scholarly journals On the Mean-Width of Isotropic Convex Bodies and their Associated Lp-Centroid Bodies

2014 ◽  
Author(s):  
E. Milman
Keyword(s):  
Convex Bodies ◽  
Mean Width ◽  
The Mean ◽  
Centroid Bodies

scholarly journals Sections of Convex Bodies in John’s and Minimal Surface Area Position

2021 ◽  
Author(s):  
David Alonso-Gutiérrez ◽  
Silouanos Brazitikos
Keyword(s):  
Minimal Surface ◽  
Surface Area ◽  
Convex Bodies ◽  
Regular Simplex ◽  
Symmetric Convex ◽  
Polar Bodies ◽  
Mean Width ◽  
Centrally Symmetric ◽  
The Mean ◽  
Minimal Surface Area

Abstract We prove several estimates for the volume, the mean width, and the value of the Wills functional of sections of convex bodies in John’s position, as well as for their polar bodies. These estimates extend some well-known results for convex bodies in John’s position to the case of lower-dimensional sections, which had mainly been studied for the cube and the regular simplex. Some estimates for centrally symmetric convex bodies in minimal surface area position are also obtained.

scholarly journals Interaction of Poisson hyperplane processes and convex bodies

2019 ◽  
Vol 56 (4) ◽  
pp. 1020-1032 ◽  
Author(s):  
Rolf Schneider
Keyword(s):  
Convex Body ◽  
Convex Bodies ◽  
Random Polytope ◽  
Mean Width ◽  
The Mean

AbstractGiven a stationary and isotropic Poisson hyperplane process and a convex body K in ${\mathbb R}^d$ , we consider the random polytope defined by the intersection of all closed half-spaces containing K that are bounded by hyperplanes of the process not intersecting K. We investigate how well the expected mean width of this random polytope approximates the mean width of K if the intensity of the hyperplane process tends to infinity.

On characterizations of sausages via inequalities and roots of Steiner polynomials

2017 ◽  
Vol 17 (4) ◽  
Author(s):  
Jesús Yepes Nicolás
Keyword(s):  
Isoperimetric Inequality ◽  
Convex Bodies ◽  
Extremal Sets ◽  
Mean Width ◽  
The Family ◽  
Algebraic Properties ◽  
The Mean

AbstractWe prove that sausages are the family of ‘extremal sets’ in relation to certain linear improvements of Minkowski’s first inequality when working with projection/sections assumptions. In particular they characterize the equality cases of the corresponding linear refinements of both the isoperimetric inequality and Urysohn’s inequality. We also characterize sausages by algebraic properties of the roots of Steiner polynomials, in which other functionals of convex bodies such as the inradius, the mean width or the diameter are involved.

Excess entropy of the systems of molecules differing in size and shape

1983 ◽  
Vol 48 (1) ◽  
pp. 192-198 ◽  
Author(s):  
Tomáš Boublík
Keyword(s):  
Equation Of State ◽  
Hard Spheres ◽  
Excess Entropy ◽  
Convex Bodies ◽  
Pure Substance ◽  
Liquid Equilibrium ◽  
Simple Rules ◽  
Different Shapes ◽  
The Mean

The excess entropy of mixing of mixtures of hard spheres and spherocylinders is determined from an equation of state of hard convex bodies. The obtained dependence of excess entropy on composition was used to find the accuracy of determining ΔSE from relations employed for the correlation and prediction of vapour-liquid equilibrium. Simple rules were proposed for establishing the mean parameter of nonsphericity for mixtures of hard bodies of different shapes allowing to describe the P-V-T behaviour of solutions in terms of the equation of state fo pure substance. The determination of ΔSE by means of these rules is discussed.

scholarly journals Ethnic variation in medial orbital wall anatomy and its implications for decompression surgery

2021 ◽  
Vol 21 (1) ◽  
Author(s):  
Minhui Amy Chan ◽  
Farah Ibrahim ◽  
Arjunan Kumaran ◽  
Kailing Yong ◽  
Anita Sook Yee Chan ◽  
...  
Keyword(s):  
Relative Position ◽  
Ethmoid Sinus ◽  
Decompression Surgery ◽  
Ethnic Variation ◽  
Sinus Wall ◽  
Orbital Wall ◽  
Mean Width ◽  
Significant Difference ◽  
The Mean ◽  
Medial Orbital Wall

Abstract Background To describe the inter-ethnic variation in medial orbital wall anatomy between Chinese, Malay, Indian and Caucasian subjects. Methods Single-centre, retrospective, Computed Tomography (CT)-based observational study. 20 subjects of each ethnicity, were matched for gender and laterality. We excluded subjects younger than 16 years and those with orbital pathology. OsiriX version 8.5.1 (Pixmeo., Switzerland) and DICOM image viewing software CARESTREAM Vue PACS (Carestream Health Inc., USA) were used to measure the ethmoidal sinus length, width and volume, medial orbital wall and floor angle and the relative position of the posterior ethmoid sinus to the posterior maxillary wall. Statistical analyses were performed using Statistical Package for Social Sciences version 25.0 (IBM, USA). Results There were 12 males (60 %) in each group, with no significant difference in age (p = 0.334–0.994). The mean ethmoid sinus length in Chinese, Malay, Indian and Caucasian subjects, using the Chinese as reference, were 37.2, 36.9, 38.0 and 37.4mm, the mean width was 11.6, 10.5, 11.4 and 10.0mm (p = 0.020) and the mean ethmoid sinus volume were 3362, 3652, 3349 and 3898mm3 respectively. The mean medial orbital wall and floor angle was 135.0, 131.4, 131.0 and 136.8 degrees and the mean relative position of posterior ethmoid sinus to posterior maxillary wall were − 2.0, -0.2, -1.5 and 1.6mm (p = 0.003) respectively. Conclusions No inter-ethnic variation was found in decompressible ethmoid sinus volume. Caucasians had their posterior maxillary sinus wall anterior to their posterior ethmoidal walls unlike the Chinese, Malay and Indians. Awareness of ethnic variation is essential for safe orbital decompression.

scholarly journals Convex Bodies of Minimal Volume, Surface Area and Mean Width with Respect to Thin Shells

2008 ◽  
Vol 60 (1) ◽  
pp. 3-32 ◽  
Author(s):  
Károly Böröczky ◽  
Károly J. Böröczky ◽  
Carsten Schütt ◽  
Gergely Wintsche
Keyword(s):  
Unit Ball ◽  
Surface Area ◽  
Convex Bodies ◽  
Extreme Points ◽  
Thin Shells ◽  
Minimal Volume ◽  
Asymptotic Formulae ◽  
Mean Width

AbstractGiven r > 1, we consider convex bodies in En which contain a fixed unit ball, and whose extreme points are of distance at least r from the centre of the unit ball, and we investigate how well these convex bodies approximate the unit ball in terms of volume, surface area and mean width. As r tends to one, we prove asymptotic formulae for the error of the approximation, and provide good estimates on the involved constants depending on the dimension.

Cervical and High Thoracic Ligamentum Flavum Frequently Fails to Fuse in the Midline

2003 ◽  
Vol 99 (6) ◽  
pp. 1387-1390 ◽  
Author(s):  
Philipp Lirk ◽  
Christian Kolbitsch ◽  
Guenther Putz ◽  
Joshua Colvin ◽  
Hans Peter Colvin ◽  
...  
Keyword(s):  
Ligamentum Flavum ◽  
Epidural Needle ◽  
Complete Fusion ◽  
Thoracic Epidural ◽  
Loss Of Resistance ◽  
Ligamenta Flava ◽  
Mean Width ◽  
Human Cadavers ◽  
The Mean ◽  
High Thoracic Epidural

Background Cervical and high thoracic epidural anesthesia and analgesia have gained increasing importance in the treatment of painful conditions and as components of anesthetics for cardiac and breast surgery. In contrast to the hanging-drop technique, the loss-of-resistance technique is thought to rely on the penetration of the ligamentum flavum. However, the exact morphology of the ligamentum flavum at different vertebral levels remains controversial. Therefore, the aim of this study was to investigate the incidence and morphology of cervical and high thoracic ligamentum flavum mid-line gaps in embalmed cadavers. Methods Vertebral column specimens were obtained from 52 human cadavers. On each dissected level, ligamentum flavum mid-line gaps were recorded and evaluated with respect to shape and size. Results The following variations were encountered: complete fusion in the mid-line, mid-line fusion with a gap in the caudal part, mid-line gap, and mid-line gap with widened caudal end. The incidence of mid-line gaps at the following levels was: C3-C4: 66%, C4-C5: 58%, C5-C6: 74%, C6-C7: 64%, C7-T1: 51%, Th1-Th2: 21%, Th2-Th3: 11%, Th3-Th4: 4%, Th4-Th5: 2%, and Th5-Th6: 2%. The mean width of mid-line gaps was 1.0 +/- 0.3 mm. Conclusions In conclusion, the present study shows that gaps in the ligamenta flava are frequent at cervical and high thoracic levels but become rare at the T3/T4 level and below, such that one cannot always rely on the ligamentum flavum as a perceptible barrier to epidural needle placement at these levels.

scholarly journals Radiographic Measurement of Cochlear in Sudanese Population using High Resolution Computed Tomography (HRCT)

2021 ◽  
Vol 29 (2) ◽  
Author(s):  
Lubna Bushara ◽  
Mohamed Yousef ◽  
Ikhlas Abdelaziz ◽  
Mogahid Zidan ◽  
Dalia Bilal ◽  
...  
Keyword(s):  
Computed Tomography ◽  
Hearing Loss ◽  
High Resolution ◽  
Temporal Bone ◽  
Healthy Subjects ◽  
Cochlear Nerve ◽  
High Resolution Computed Tomography ◽  
Mean Width ◽  
The Mean ◽  
The Right

This study aimed to determine the measurements of the cochlea among healthy subjects and hearing deafness subjects using a High Resolution Computed Tomography (HRCT). A total of 230 temporal bone HRCT cases were retrospectively investigated in the period spanning from 2011 to 2015. Three 64-slice units were used to examine patients with clinical complaints of hearing loss conditions at three Radiology departments in Khartoum, Sudan. For the control group (A) healthy subjects, the mean width of the right and left cochlear were 5.61±0.40 mm and 5.56±0.58 mm, the height were 3.56±0.36 mm and 3.54±0.36 mm, the basal turn width were 1.87±0.19 mm and 1.88 ±0.18 mm, the width of the cochlear nerve canal were 2.02±1.23 and 1.93±0.20, cochlear nerve density was 279.41±159.02 and 306.84±336.9 HU respectively. However, for the experimental group (B), the mean width of the right and left cochlear width were 5.38±0.46 mm and 5.34±0.30 mm, the height were 3.53±0.25 mm and 3.49±0.28mm, the basal turn width were 1.76±0.13 mm, and 1.79±0.13 mm, the width of the cochlear nerve canal were 1.75±0.18mm and 1.73±0.18mm, and cochlear nerve density were 232.84±316.82 and 196.58±230.05 HU, respectively. The study found there was a significant difference in cochlea’s measurement between the two groups with a p-value < 0.05. This study had established baseline measurements for the cochlear for the healthy Sudanese population. Furthermore, it found that HRCT of the temporal bone was the best for investigation of the cochlear and could provide a guide for the clinicians to manage congenital hearing loss.

scholarly journals The Micro-Geometry Evaluation on Steel Sheets after Chemical Surface Treatment

2013 ◽  
Vol 10 (2) ◽  
pp. 22-25
Author(s):  
Dagmar Draganovská ◽  
Miroslav Tomáš ◽  
Peter Ižol
Keyword(s):  
Surface Treatment ◽  
Steel Sheet ◽  
Unit Length ◽  
Mean Deviation ◽  
Maximum Height ◽  
Chemical Surface ◽  
Steel Sheets ◽  
Mean Width ◽  
The Mean ◽  
Metal Materials

Abstract The protection of metal materials by coatings is one of the most important way to protect the surface against corrosion. Galvanisation of metal surfaces by an electric current is one of the most frequent way of surface treatment due to its wide application. The paper is oriented to the evaluation of the surface quality in terms of micro-geometry. The basic surface of the steel sheet S235JRG2 has been evaluted and compared to the degreased one, galvanized one and surface after galvanizing, chromatizing and phosphating. The microgeometry parameters such as the arithmetical mean deviation Ra, the maximum height of profile Rz, the total height of profile Rt as well as the the mean width of the profile elements RSm and the number of peaks per unit length RPc has been compared.

Processing of form and motion in area 21a of cat visual cortex

1993 ◽  
Vol 10 (1) ◽  
pp. 93-115 ◽  
Author(s):  
B. Dreher ◽  
A. Michalski ◽  
R. H. T. Ho ◽  
C. W. F. Lee ◽  
W. Burke
Keyword(s):  
Spatial Organization ◽  
Receptive Fields ◽  
Direction Selectivity ◽  
Orientation Tuning ◽  
Area 17 ◽  
Horizontal Strip ◽  
Tuning Curves ◽  
Mean Width ◽  
Area 21A ◽  
The Mean

AbstractExtracellular recordings from single neurons have been made from presumed area 21a of the cerebral cortex of the cat, anesthetized with N2O/O2/sodium pentobarbitone mixture. Area 21a contains mainly a representation of a central horizontal strip of contralateral visual field about 5 deg above and below the horizontal meridian.Excitatory discharge fields of area 21a neurons were substantially (or slightly but significantly) larger than those of neurons at corresponding eccentricities in areas 17, 19, or 18, respectively. About 95% of area 21a neurons could be activated through either eye and the input from the ipsilateral eye was commonly dominant. Over 90% and less than 10% of neurons had, respectively, C-type and S-type receptive-field organization. Virtually all neurons were orientation-selective and the mean width at half-height of the orientation tuning curves at 52.9 deg was not significantly different from that of neurons in areas 17 and 18. About 30% of area 21a neurons had preferred orientations within 15 deg of the vertical.The mean direction-selectivity index (32.8%) of area 21a neurons was substantially lower than the indices for neurons in areas 17 or 18. Only a few neurons exhibited moderately strong end-zone inhibition. Area 21a neurons responded poorly to fast-moving stimuli and the mean preferred velocity at about 12.5 deg/s was not significantly different from that for area 17 neurons.Selective pressure block of Y fibers in contralateral optic nerve resulted in a small but significant reduction in the preferred velocities of neurons activated via the Y-blocked eye. By contrast, removal of the Y input did not produce significant changes in the spatial organization of receptive fields (S or C type), the size of the discharge fields, the width of orientation tuning curves, or direction-selectivity indices.Our results are consistent with the idea that area 21a receives its principal excitatory input from area 17 and is involved mainly in form rather than motion analysis.

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