Covering Points with Convex Sets of Minimum Size

2016 ◽  
pp. 166-178 ◽  
Author(s):  
Hwan-Gue Cho ◽  
William Evans ◽  
Noushin Saeedi ◽  
Chan-Su Shin
Keyword(s):  
Convex Sets ◽  
Minimum Size

Covering points with convex sets of minimum size

2018 ◽  
Vol 718 ◽  
pp. 14-23 ◽  
Author(s):  
Sang Won Bae ◽  
Hwan-Gue Cho ◽  
William Evans ◽  
Noushin Saeedi ◽  
Chan-Su Shin
Keyword(s):  
Convex Sets ◽  
Minimum Size

Preparation of integrated circuits for TEM electromigration in situ studies

1986 ◽  
Vol 44 ◽  
pp. 882-883
Author(s):  
J. V. Maskowitz ◽  
W. E. Rhoden ◽  
D. R. Kitchen ◽  
R. E. Omlor ◽  
P. F. Lloyd
Keyword(s):  
Integrated Circuits ◽  
Crystallographic Orientation ◽  
Silicon Wafers ◽  
Minimum Size ◽  
Test Vehicle ◽  
In Situ Studies ◽  
Boron Doped ◽  
Doped Silicon ◽  
Drill Holes

The fabrication of the aluminum bridge test vehicle for use in the crystallographic studies of electromigration involves several photolithographic processes, some common, while others quite unique. It is most important to start with a clean wafer of known orientation. The wafers used are 7 mil thick boron doped silicon. The diameter of the wafer is 1.5 inches with a resistivity of 10-20 ohm-cm. The crystallographic orientation is (111).Initial attempts were made to both drill and laser holes in the silicon wafers then back fill with photoresist or mounting wax. A diamond tipped dentist burr was used to successfully drill holes in the wafer. This proved unacceptable in that the perimeter of the hole was cracked and chipped. Additionally, the minimum size hole realizable was > 300 μm. The drilled holes could not be arrayed on the wafer to any extent because the wafer would not stand up to the stress of multiple drilling.

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.

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.

Transplantation of coral fragment, Acropora formosa (Scleractinia)

2014 ◽  
pp. 1
Author(s):  
Hanny Tioho ◽  
Maykel A.J Karauwan
Keyword(s):  
Fragment Size ◽  
Observation Time ◽  
Minimum Size ◽  
Average Growth Rate ◽  
Average Growth ◽  
Significant Difference ◽  
Acropora Formosa ◽  
Survival And Growth ◽  
One Way Anova ◽  
Better Than

The minimum size of coral transplants, Acropora formosa, was assessed to support their survival and growth. For this, 150 coral fragments of different sizes (5, 10, 15 cm) were transplanted close to the donor colony. Their survivorship and growth were observed for 12 months. At the end of the observation time, 90% of 15 cm-transplanted coral fragments survived, while the others (10cm and 5 cm) did 86% and 82% respectively. The average growth rate of 5 cm-coral fragments was 0.860 cm/month, while 10 and 15 cm-fragments were 0.984 cm/month and 1.108 cm/month respectively. One-way ANOVA showed that there was significant difference (p<0.05) among the three (5, 10, 15 cm) transplant initial sizes in which the longest fragment size tended to survive longer than the smaller one.  However, the smaller transplants grew better than the bigger one, 10.318 cm/year (206%) for 5 cm-transplant, 11.803 cm/year (118%) for 10 cm-transplant, and 13.299 cm/year (89%) for 15 cm-transplant, respectively. Ukuran minimal fragmen karang Acropora formosa yang ditransplantasi diduga untuk mendukung ketahanan hidup dan pertumbuhannya. Untuk itu, 150 fragmen karang ditransplantasi ke lokasi yang berdekatan dengan koloni induknya.  Ketahanan hidup dan pertumbuhan semua fragmen karang yang ditransplantasi diamati selama 12 bulan.  Pada akhir pengamatan, 90% dari fragmen karang berukuran 15 cm yang ditransplantasi dapat bertahan hidup, sedangkan yang lainnya (ukuran 10 cm dan 5 cm) masing-masing sebesar 86% dan 82%.  Rata-rata laju pertumbuhan fragmen karang dengan ukuran awal 5 cm adalah 0,860 cm/bulan, sedangkan ukuran fragmen 10 dan 15 cm masing-masing adalah 0,984 cm/bulan and 1,108 cm/bulan. ANOVA satu arah menunjukkan adanya perbedaan yang nyata (p<0.05) antara ketiga ukuran fragmen yang berbeda, di mana ukuran fragmen karang yang lebih panjang cenderung mempunyai ketahanan hidup yang lebih baik. Namun demikian, ukuran transplant yang lebih kecil memiliki pertumbuhan lebih baik dibandingkan dengan ukuran yang lebih besar, yakni10,318 cm/tahun (206%) untuk transplant berukuran 5 cm, 11,803 cm/tahun (118%) untuk 10 cm, dan 13,299 cm/tahun (89%) untuk ukuran 15 cm.

On a Two-Sided Turán Problem

10.37236/1735 ◽  
2003 ◽  
Vol 10 (1) ◽  
Author(s):  
Dhruv Mubayi ◽  
Yi Zhao
Keyword(s):  
Minimum Size ◽  
Turan Problem ◽  
Turan Problems ◽  
Turán Problem ◽  
Positive Integers

Given positive integers $n,k,t$, with $2 \le k\le n$, and $t < 2^k$, let $m(n,k,t)$ be the minimum size of a family ${\cal F}$ of nonempty subsets of $[n]$ such that every $k$-set in $[n]$ contains at least $t$ sets from ${\cal F}$, and every $(k-1)$-set in $[n]$ contains at most $t-1$ sets from ${\cal F}$. Sloan et al. determined $m(n, 3, 2)$ and Füredi et al. studied $m(n, 4, t)$ for $t=2, 3$. We consider $m(n, 3, t)$ and $m(n, 4, t)$ for all the remaining values of $t$ and obtain their exact values except for $k=4$ and $t= 6, 7, 11, 12$. For example, we prove that $ m(n, 4, 5) = {n \choose 2}-17$ for $n\ge 160$. The values of $m(n, 4, t)$ for $t=7,11,12$ are determined in terms of well-known (and open) Turán problems for graphs and hypergraphs. We also obtain bounds of $m(n, 4, 6)$ that differ by absolute constants.

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