scholarly journals A Note about Bezdek's Conjecture on Covering an Annulus by Strips

10.37236/894 ◽  
2008 ◽  
Vol 15 (1) ◽  
Author(s):  
Yuqin Zhang ◽  
Ren Ding
Keyword(s):  
Convex Region ◽  
Plane Region ◽  
Minimal Width ◽  
Parallel Lines ◽  
Closed Plane

A closed plane region between two parallel lines is called a strip. András Bezdek posed the following conjecture: For each convex region $K$ there is an $\varepsilon>0$ such that if $\varepsilon K$ lies in the interior of $K$ and the annulus $K\backslash \varepsilon K$ is covered by finitely many strips, then the sum of the widths of the strips must be at least the minimal width of $K$. In this paper, we consider problems which are related to the conjecture.

Applications of outwardly simple line families to plane convex sets

1977 ◽  
Vol 82 (3) ◽  
pp. 353-356 ◽  
Author(s):  
K. J. Falconer
Keyword(s):  
Convex Sets ◽  
Convex Set ◽  
Plane Convex ◽  
Minimal Width ◽  
Parallel Lines

We use the concept of outwardly simple line families (see Hammer and Sobczyk (4)) first to obtain conditions involving mid-chords that ensure that a plane convex set is centro-symmetric, and secondly to show that it is possible to inscribe a semicircle of diameter ω in any convex set of minimal width ω in at least 3 different ways. We show that a plane convex set X is centro-symmetric if every mid-chord of X (that is every chord of X mid-way between two parallel lines of support of X) bisects the area of X, or alternatively if every mid-chord of X is a diameter of X (that is a longest chord in some direction). Hammer and Smith (3) have used outwardly simple line families in a different way to show that a plane convex set X is centro-symmetric if every diameter of X bisects the area of X, or if every diameter of X bisects the perimeter of X.

Parallel Lines

2004 ◽  
Author(s):  
Nina Davenport
Keyword(s):  
Parallel Lines

On an area-preserving inverse curvature flow of convex closed plane curves

2021 ◽  
Vol 280 (8) ◽  
pp. 108931
Author(s):  
Laiyuan Gao ◽  
Shengliang Pan ◽  
Dong-Ho Tsai
Keyword(s):  
Curvature Flow ◽  
Plane Curves ◽  
Area Preserving ◽  
Closed Plane

Reflections on Parallel Lines in Sociology and Linguistics

2019 ◽  
Vol 7 (2) ◽  
pp. 137-155
Author(s):  
Deping Lu
Keyword(s):  
Parallel Lines ◽  
And Linguistics

scholarly journals On Structure Descriptors Related with Intramolecular Energy of Alkanes

2004 ◽  
Vol 59 (10) ◽  
pp. 694-698 ◽  
Author(s):  
Ivan Gutman ◽  
Boris Furtula ◽  
Biljana Arsić
Keyword(s):  
Organic Molecule ◽  
Wiener Index ◽  
Critical Value ◽  
Fixed Number ◽  
Methyl Groups ◽  
Parallel Lines ◽  
Analytical Expressions

In an earlier work it was demonstrated that the Zenkevich index U provides a measure of the intramolecular energy of an organic molecule, and that - in the case of alkanes - it is related to the Wiener index. We now show that U is much closer related to the recently introduced variable Wiener index Wλ : Within sets of isomeric alkanes, the relation between U and Wλ is linear, the (U,Wλ )-points forming several, mutually parallel, lines. Each such line pertains to a group of isomers possessing a fixed number of methyl groups. There exists a critical value of the parameter λ for which all the (U,Wλ )-lines coalesce, in which case the relation between U and Wλ becomes independent of the number of methyl groups. Approximate analytical expressions for the (U,Wλ )-dependence are deduced.

scholarly journals I.—Antiquarian Researches in Illyricum, III, and IV

Archaeologia ◽  
1885 ◽  
Vol 49 (01) ◽  
pp. 1-168 ◽  
Author(s):  
Arthur John Evans
Keyword(s):  
Middle Ages ◽  
Administrative District ◽  
North East ◽  
The North ◽  
East Region ◽  
Parallel Lines ◽  
North And South ◽  
Uncertain Boundary ◽  
The Middle Ages

Hitherto we have been concerned with the Dalmatian coast-cities and the great parallel lines of road that traversed the length of the Province from the borders of Pannonia and Italy to those of Epirus. From Salonæ there were, in addition to these highways to the North and South, at least two main-lines of Roman Way that traversed the interior ranges of the Dinaric Alps and led to the Mœsian and Dardanian borders that lay to the East and South-East. Milliary columns have been found at Salonæ, one recording the completion by Tiberius' Legate Dolabella of a line of road leading from the Colony of Salonæ to a mountain stronghold of the Ditiones—an Illyrian clan probably inhabiting what is now the North-East region of Bosnia; another, also of Tiberius' time, referring to the construction of a line, 156 miles in extent, from Salonæ to aCastellumof the Dæsitiates, an Illyrian clan belonging to theConventusor administrative district of Narona, and whose stronghold, according to the mileage given, must be sought somewhere on the Upper Drina, towards the Moesian and Dalmatian confines. This latter line may very well be that represented in theTabula Peutingerianaas leading from Salonæ to Argentaria, a name which seems to connect itself with the silver-bearing ranges lying on the uncertain boundary of the ancient Dalmatia and Dardania, and which, from its mineral riches, was still known in the Middle Ages asMonte Argentaro.

scholarly journals Curing Depth of Light-activated Nanofiller containing Resin Composites

2012 ◽  
Vol 3 (2) ◽  
pp. 119-125 ◽  
Author(s):  
Werner J Finger ◽  
Masafumi Kanehira ◽  
Yasuyuki Araki ◽  
Takehiko Wada ◽  
Andreas Utterodt ◽  
...  
Keyword(s):  
Crosslink Density ◽  
Mechanical Characteristics ◽  
Polynomial Regression ◽  
Knoop Hardness ◽  
Resin Composites ◽  
Depth Level ◽  
Depth Of Cure ◽  
Micro Raman Spectroscopy ◽  
Parallel Lines ◽  
Order Polynomial

ABSTRACT Objective To compare the depth of cure of nanofiller containing with conventional resin composites. Materials and methods Five nanofilled and nanohybrid composites were investigated and compared with a microfilled and a microhybrid reference, using the ISO 4049 scraping test, Knoop hardness (KHN) and DC profiling. Specimens from all materials (shade A3) were activated with the same LED light source for 20s. KHN (0.25N/15s) of three specimens each, produced in split molds was measured after 24 hours dark storage on sections perpendicular to the irradiated surface at 250 µm distance along the center line and two parallel lines, 0.5 mm apart, to a depth of 3 mm. Mean KHNs of the three neighboring indentations at each level were calculated. Degree of conversion (DC) was determined on specimens as mentioned above using micro-Raman spectroscopy at 125 µm distance with three measurements at each depth level. Results The depth of cure of all materials was >2 mm when determined according to the ISO test. KHN and DC data followed second order polynomial regression lines (r2 > 0.70; p < 0.001). At 2 mm depth, the KHN of six resin composites was ≤80% of the top KHN whereas the DC of all materials was ≥86% of the maximum DC at the irradiated surface. Conclusion The ISO scraping test overemphasizes the attainable depth of cure, when compared with 80% of top KHN as arbitrarily defined curing depth. KHNs reflect the crosslink density of the polymer, whereas DC additionally includes double bond conversions not contributing to enhancement of mechanical characteristics. How to cite this article Kanehira M, Araki Y, Finger WJ, Wada T, Utterodt A, Komatsu M. Curing Depth of Light-activated Nanofiller containing Resin Composites. World J Dent 2012;3(2): 119-125.

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