text:
# Difference between revisions of "1995 AJHSME Problems"

## Problem 1

Walter has exactly one penny, one nickel, one dime and one quarter in his pocket. What percent of one dollar is in his pocket?

$\text{(A)}\ 4\% \qquad \text{(B)}\ 25\% \qquad \text{(C)}\ 40\% \qquad \text{(D)}\ 41\% \qquad \text{(E)}\ 59\%$

## Problem 2

Jose is $4$ years younger than Zack. Zack is $3$ years older than Inez. Inez is $15$ years old. How old is Jose?

$\text{(A)}\ 8 \qquad \text{(B)}\ 11 \qquad \text{(C)}\ 14 \qquad \text{(D)}\ 16 \qquad \text{(E)}\ 22$

## Problem 3

Which of the following operations has the same effect on a number as multiplying by $\dfrac{3}{4}$ and then dividing by $\dfrac{3}{5}$?

$\text{(A)}\ \text{dividing by }\dfrac{4}{3} \qquad \text{(B)}\ \text{dividing by }\dfrac{9}{20} \qquad \text{(C)}\ \text{multiplying by }\dfrac{9}{20} \qquad \text{(D)}\ \text{dividing by }\dfrac{5}{4} \qquad \text{(E)}\ \text{multiplying by }\dfrac{5}{4}$

## Problem 4

A teacher tells the class,

"Think of a number, add 1 to it, and double the result. Give the answer to your partner. Partner, subtract 1 from the number you are given and double the result to get your answer."

Ben thinks of $6$, and gives his answer to Sue. What should Sue's answer be?

$\text{(A)}\ 18 \qquad \text{(B)}\ 24 \qquad \text{(C)}\ 26 \qquad \text{(D)}\ 27 \qquad \text{(E)}\ 30$

## Problem 5

Find the smallest whole number that is larger than the sum

$$2\dfrac{1}{2}+3\dfrac{1}{3}+4\dfrac{1}{4}+5\dfrac{1}{5}.$$

$\text{(A)}\ 14 \qquad \text{(B)}\ 15 \qquad \text{(C)}\ 16 \qquad \text{(D)}\ 17 \qquad \text{(E)}\ 18$

## Problem 6

Figures $I$, $II$, and $III$ are squares. The perimeter of $I$ is $12$ and the perimeter of $II$ is $24$. The perimeter of $III$ is

$[asy] draw((0,0)--(15,0)--(15,6)--(12,6)--(12,9)--(0,9)--cycle); draw((9,0)--(9,9)); draw((9,6)--(12,6)); label("III",(4.5,4),N); label("II",(12,2.5),N); label("I",(10.5,6.75),N); [/asy]$

$\text{(A)}\ 9 \qquad \text{(B)}\ 18 \qquad \text{(C)}\ 36 \qquad \text{(D)}\ 72 \qquad \text{(D)}\ 81$

## Problem 7

At Clover View Junior High, one half of the students go home on the school bus. One fourth go home by automobile. One tenth go home on their bicycles. The rest walk home. What fractional part of the students walk home?

$\text{(A)}\ \dfrac{1}{16} \qquad \text{(B)}\ \dfrac{3}{20} \qquad \text{(C)}\ \dfrac{1}{3} \qquad \text{(D)}\ \dfrac{17}{20} \qquad \text{(E)}\ \dfrac{9}{10}$

## Problem 8

An American traveling in Italy wishes to exchange American money (dollars) for Italian money (lire). If 3000 lire = $1.60, how much lire will the traveler receive in exchange for$1.00?

$\text{(A)}\ 180 \qquad \text{(B)}\ 480 \qquad \text{(C)}\ 1800 \qquad \text{(D)}\ 1875 \qquad \text{(E)}\ 4875$

## Problem 9

Three congruent circles with centers $P$, $Q$, and $R$ are tangent to the sides of rectangle $ABCD$ as shown. The circle centered at $Q$ has diameter $4$ and passes through points $P$ and $R$. The area of the rectangle is

$[asy] pair A,B,C,D,P,Q,R; A = (0,4); B = (8,4); C = (8,0); D = (0,0); P = (2,2); Q = (4,2); R = (6,2); dot(A); dot(B); dot(C); dot(D); dot(P); dot(Q); dot(R); draw(A--B--C--D--cycle); draw(circle(P,2)); draw(circle(Q,2)); draw(circle(R,2)); label("A",A,NW); label("B",B,NE); label("C",C,SE); label("D",D,SW); label("P",P,W); label("Q",Q,W); label("R",R,W); [/asy]$

$\text{(A)}\ 16 \qquad \text{(B)}\ 24 \qquad \text{(C)}\ 32 \qquad \text{(D)}\ 64 \qquad \text{(E)}\ 128$

## Problem 10

A jacket and a shirt originally sold for dollar 80 and dollar 40, respectively. During a sale Chris bought the dollar80 jacket at a $40\%$ discount and the dollar 40 shirt at a $55\%$ discount. The total amount saved was what percent of the total of the original prices?

$\text{(A)}\ 45\% \qquad \text{(B)}\ 47\dfrac{1}{2}\% \qquad \text{(C)}\ 50\% \qquad \text{(D)}\ 79\dfrac{1}{6}\% \qquad \text{(E)}\ 95\%$.

## Problem 11

Jane can walk any distance in half the time it takes Hector to walk the same distance. They set off in opposite directions around the outside of the 18-block area as shown. When they meet for the first time, they will be closest to

$[asy] for(int i = -2; i <= 2; ++i) { draw((i,0)--(i,3),dashed); } draw((-3,1)--(3,1),dashed); draw((-3,2)--(3,2),dashed); draw((-3,0)--(-3,3)--(3,3)--(3,0)--cycle); dot((-3,0)); label("A",(-3,0),SW); dot((-3,3)); label("B",(-3,3),NW); dot((0,3)); label("C",(0,3),N); dot((3,3)); label("D",(3,3),NE); dot((3,0)); label("E",(3,0),SE); dot((0,0)); label("start",(0,0),S); label("\longrightarrow",(0,-0.75),E); label("\longleftarrow",(0,-0.75),W); label("\textbf{Jane}",(0,-1.25),W); label("\textbf{Hector}",(0,-1.25),E); [/asy]$

$\text{(A)}\ A \qquad \text{(B)}\ B \qquad \text{(C)}\ C \qquad \text{(D)}\ D \qquad \text{(E)}\ E$

## Problem 12

A lucky year is one in which at least one date, when written in the form month/day/year, has the following property: The product of the month times the day equals the last two digits of the year. For example, 1956 is a lucky year because it has the date 7/8/56 and $7\times 8 = 56$. Which of the following is NOT a lucky year?

$\text{(A)}\ 1990 \qquad \text{(B)}\ 1991 \qquad \text{(C)}\ 1992 \qquad \text{(D)}\ 1993 \qquad \text{(E)}\ 1994$

## Problem 13

In the figure, $\angle A$, $\angle B$, and $\angle C$ are right angles. If $\angle AEB = 40^\circ$ and $\angle BED = \angle BDE$, then $\angle CDE =$

$[asy] dot((0,0)); label("E",(0,0),SW); dot(dir(85)); label("A",dir(85),NW); dot((4,0)); label("D",(4,0),SE); dot((4.05677,0.648898)); label("C",(4.05677,0.648898),NE); draw((0,0)--dir(85)--(4.05677,0.648898)--(4,0)--cycle); dot((2,2)); label("B",(2,2),N); draw((0,0)--(2,2)--(4,0)); pair [] x = intersectionpoints((0,0)--(2,2)--(4,0),dir(85)--(4.05677,0.648898)); dot(x[0]); dot(x[1]); label("F",x[0],SE); label("G",x[1],SW); [/asy]$

$\text{(A)}\ 75^\circ \qquad \text{(B)}\ 80^\circ \qquad \text{(C)}\ 85^\circ \qquad \text{(D)}\ 90^\circ \qquad \text{(E)}\ 95^\circ$

## Problem 14

A team won 40 of its first 50 games. How many of the remaining 40 games must this team win so it will have won exactly 70% of its games for the season?

$\text{(A)}\ 20 \qquad \text{(B)}\ 23 \qquad \text{(C)}\ 28 \qquad \text{(D)}\ 30 \qquad \text{(E)}\ 35$

## Problem 15

What is the $100^\text{th}$ digit to the right of the decimal point in the decimal form of $4/37$?

$\text{(A)}\ 0 \qquad \text{(B)}\ 1 \qquad \text{(C)}\ 2 \qquad \text{(D)}\ 7 \qquad \text{(E)}\ 8$

## Problem 16

Students from three middle schools worked on a summer project.

• Seven students from Allen school worked for 3 days.
• Four students from Balboa school worked for 5 days.
• Five students from Carver school worked for 9 days.

The total amount paid for the students' work was <dollar/>774. Assuming each student received the same amount for a day's work, how much did the students from Balboa school earn altogether?

$\text{(A)}\ 9.00\text{ dollars} \qquad \text{(B)}\ 48.38\text{ dollars} \qquad \text{(C)}\ 180.00\text{ dollars} \qquad \text{(D)}\ 193.50\text{ dollars} \qquad \text{(E)}\ 258.00\text{ dollars}$

## Problem 17

The table below gives the percent of students in each grade at Annville and Cleona elementary schools:

$$\begin{tabular}{rccccccc} & \textbf{\underline{K}} & \textbf{\underline{1}} & \textbf{\underline{2}} & \textbf{\underline{3}} & \textbf{\underline{4}} & \textbf{\underline{5}} & \textbf{\underline{6}} \\ \textbf{Annville:} & 16\% & 15\% & 15\% & 14\% & 13\% & 16\% & 11\% \\ \textbf{Cleona:} & 12\% & 15\% & 14\% & 13\% & 15\% & 14\% & 17\% \end{tabular}$$

Annville has 100 students and Cleona has 200 students. In the two schools combined, what percent of the students are in grade 6?

$\text{(A)}\ 12\% \qquad \text{(B)}\ 13\% \qquad \text{(C)}\ 14\% \qquad \text{(D)}\ 15\% \qquad \text{(E)}\ 28\%$

## Problem 18

The area of each of the four congruent L-shaped regions of this 100-inch by 100-inch square is 3/16 of the total area. How many inches long is the side of the center square?

$[asy] draw((2,2)--(2,-2)--(-2,-2)--(-2,2)--cycle); draw((1,1)--(1,-1)--(-1,-1)--(-1,1)--cycle); draw((0,1)--(0,2)); draw((1,0)--(2,0)); draw((0,-1)--(0,-2)); draw((-1,0)--(-2,0)); [/asy]$

$\text{(A)}\ 25 \qquad \text{(B)}\ 44 \qquad \text{(C)}\ 50 \qquad \text{(D)}\ 62 \qquad \text{(E)}\ 75$

## Problem 19

The graph shows the distribution of the number of children in the families of the students in Ms. Jordan's English class. The median number of children in the family for this distribution is

$[asy] unitsize(12); for(int i = 1; i <= 7; ++i) { draw((0,i)--(19,i),dotted); draw((-0.5,i)--(0.5,i)); } for(int i = 0; i <= 5; ++i) { draw((3*i+2,0)--(3*i+2,-0.5)); } fill((1,0)--(1,2)--(3,2)--(3,0)--cycle,white); fill((4,0)--(4,1)--(6,1)--(6,0)--cycle,white); fill((7,0)--(7,2)--(9,2)--(9,0)--cycle,white); fill((10,0)--(10,2)--(12,2)--(12,0)--cycle,white); fill((13,0)--(13,6)--(15,6)--(15,0)--cycle,white); draw((0,9)--(0,0)--(19,0)); draw((1,0)--(1,2)--(3,2)--(3,0)); draw((4,0)--(4,1)--(6,1)--(6,0)); draw((7,0)--(7,2)--(9,2)--(9,0)); draw((10,0)--(10,2)--(12,2)--(12,0)); draw((13,0)--(13,6)--(15,6)--(15,0)); label("1",(2,-0.5),S); label("2",(5,-0.5),S); label("3",(8,-0.5),S); label("4",(11,-0.5),S); label("5",(14,-0.5),S); label("6",(17,-0.5),S); label("2",(-0.5,2),W); label("4",(-0.5,4),W); label("6",(-0.5,6),W); label("\textbf{Number of Children}",(9,-1.5),S); label("\textbf{in the Family}",(9,-2.5),S); label("\textbf{Number}",(-1.5,6),W); label("\textbf{of}",(-3,5),W); label("\textbf{Families}",(-1.5,4),W); [/asy]$

$\text{(A)}\ 1 \qquad \text{(B)}\ 2 \qquad \text{(C)}\ 3 \qquad \text{(D)}\ 4 \qquad \text{(E)}\ 5$

## Problem 20

Diana and Apollo each roll a standard die obtaining a number at random from 1 to 6. What is the probability that Diana's number is larger than Apollo's number?

$\text{(A)}\ \dfrac{1}{3} \qquad \text{(B)}\ \dfrac{5}{12} \qquad \text{(C)}\ \dfrac{4}{9} \qquad \text{(D)}\ \dfrac{17}{36} \qquad \text{(E)}\ \dfrac{1}{2}$

## Problem 21

A plastic snap-together cube has a protruding snap on one side and receptacle holes on the other five sides as shown. What is the smallest number of these cubes that can be snapped together so that only receptacle holes are showing?

$[asy] draw((0,0)--(4,0)--(4,4)--(0,4)--cycle); draw(circle((2,2),1)); draw((4,0)--(6,1)--(6,5)--(4,4)); draw((6,5)--(2,5)--(0,4)); draw(ellipse((5,2.5),0.5,1)); fill(ellipse((3,4.5),1,0.25),black); fill((2,4.5)--(2,5.25)--(4,5.25)--(4,4.5)--cycle,black); fill(ellipse((3,5.25),1,0.25),black); [/asy]$

$\text{(A)}\ 3 \qquad \text{(B)}\ 4 \qquad \text{(C)}\ 5 \qquad \text{(D)}\ 6 \qquad \text{(E)}\ 8$

## Problem 22

The number 6545 can be written as a product of a pair of positive two-digit numbers. What is the sum of this pair of numbers?

$\text{(A)}\ 162 \qquad \text{(B)}\ 172 \qquad \text{(C)}\ 173 \qquad \text{(D)}\ 174 \qquad \text{(E)}\ 222$

## Problem 23

How many four-digit whole numbers are there such that the leftmost digit is odd, the second digit is even, and all four digits are different?

$\text{(A)}\ 1120 \qquad \text{(B)}\ 1400 \qquad \text{(C)}\ 1800 \qquad \text{(D)}\ 2025 \qquad \text{(E)}\ 2500$

## Problem 24

In parallelogram $ABCD$, $\overline{DE}$ is the altitude to the base $\overline{AB}$ and $\overline{DF}$ is the altitude to the base $\overline{BC}$. [Note: Both pictures represent the same parallelogram.] If $DC=12$, $EB=4$, and $DE=6$, then $DF=$

$[asy] unitsize(12); pair A,B,C,D,P,Q,W,X,Y,Z; A = (0,0); B = (12,0); C = (20,6); D = (8,6); W = (18,0); X = (30,0); Y = (38,6); Z = (26,6); draw(A--B--C--D--cycle); draw(W--X--Y--Z--cycle); P = (8,0); Q = (758/25,6/25); dot(A); dot(B); dot(C); dot(D); dot(W); dot(X); dot(Y); dot(Z); dot(P); dot(Q); draw(A--B--C--D--cycle); draw(W--X--Y--Z--cycle); draw(D--P); draw(Z--Q); label("A",A,SW); label("B",B,SE); label("C",C,NE); label("D",D,NW); label("E",P,S); label("A",W,SW); label("B",X,S); label("C",Y,NE); label("D",Z,NW); label("F",Q,E); [/asy]$

$\text{(A)}\ 6.4 \qquad \text{(B)}\ 7 \qquad \text{(C)}\ 7.2 \qquad \text{(D)}\ 8 \qquad \text{(E)}\ 10$

## Problem 25

Buses from Dallas to Houston leave every hour on the hour. Buses from Houston to Dallas leave every hour on the half hour. The trip from one city to the other takes 5 hours. Assuming the buses travel on the same highway, how many Dallas-bound buses does a Houston-bound bus pass in the highway (not in the station)?

$\text{(A)}\ 5 \qquad \text{(B)}\ 6 \qquad \text{(C)}\ 9 \qquad \text{(D)}\ 10 \qquad \text{(E)}\ 11$

## See also

 1995 AJHSME (Problems • Answer Key • Resources) Preceded by1994 AJHSME Followed by1996 AJHSME 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 All AJHSME/AMC 8 Problems and Solutions

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.

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mersenneforum.org > Data "Verified" LLs with non-matching short residues?
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 2010-12-27, 11:12 #1 cheesehead     "Richard B. Woods" Aug 2002 Wisconsin USA 11110000011002 Posts "Verified" LLs with non-matching short residues? Generally, an LL result listed as "Verified LL" (in the "Exponent Status" report, anyway) means that the reported residue matches one or more reported residues from other LL tests on the same exponent. Similarly, an LL result listed as "Bad LL" means that its residue does not match any residue from another LL test on the same exponent. Right? So, for a given exponent, all the residues for "Verified LL" should match each other -- right? And the residues reported by GIMPS software to PrimeNet should all be sixteen hexadecimal digits -- right? Shorter residues are from an earlier era -- right? I just came across this in the Exponent Status report for 1800767: Code: Verified LL 153BED18D5C5EA4B by "Matt Hicks" Verified LL 6A4C by "Alex and Nick Myrman" Verified LL 153BED18D5C5EA4B by "S57483" and this for 1800959: Code: Verified LL 8EB3906BE7510354 by "Oyvind Pedersen" Verified LL 355 by "Alex and Nick Myrman" Verified LL 8EB3906BE7510354 by "S57483" Each of the short Myrman residues fails to match by being just 1 greater than the other "Verified" residues, in the 15 low-order bits. Were those the actual residue values reported by the Myrmans, or are they some artifact of a problem elsewhere? There are several other results listed for "Alex and Nick Myrman" for unfactored exponents between 1800590 and 1800960, each with "Bad LL" and a (nonmatching) 16-hexadecimal-digit residue. So, it doesn't appear that the Myrman results were (all) ancient. 1) Why are some of the Myrman residues shown as short values? 2) Why are those short Myrman residues labeled as "Verified" even though they don't match other residues for the same exponents? 3) Are there anomalies in any other "Alex and Nick Myrman" results? 4) Are there any other cases of non-matching residues (short or not) in the database that are listed as "Verified" in the Exponent Status reports? If so, are any of those exponents erroneously being considered as having been successfully DCed? Last fiddled with by cheesehead on 2010-12-27 at 11:27
 2010-12-27, 15:47 #2 Prime95 P90 years forever!     Aug 2002 Yeehaw, FL 5×1,553 Posts Nick Myrman wrote his own LL program a decade ago. It had problems with the last bit of the residue. Rather than reject his results as bad, I accepted them as legitimate double-checks. I suspect all have been triple-checked by now. Slowinski also had short residues. Brian Beesley made it a personal crusade to make sure these were all triple-checked.
 2010-12-27, 16:10 #3 davar55     May 2004 New York City 423510 Posts I had and have a similar problem with the hex output routine in my personal version of LL tester. But I was extending it to work on both larger (up to 40M or 80M) and smaller (<1000) exponents, and didn't have it fully analyzed back then, and so left it alone. Recommend quadruple checks when this happens.
2010-12-27, 16:12   #4
KingKurly

Sep 2010
Annapolis, MD, USA

101111012 Posts

Quote:
 Originally Posted by Prime95 Nick Myrman wrote his own LL program a decade ago. It had problems with the last bit of the residue. Rather than reject his results as bad, I accepted them as legitimate double-checks. I suspect all have been triple-checked by now. Slowinski also had short residues. Brian Beesley made it a personal crusade to make sure these were all triple-checked.
Is there a list of small exponents for which a triple check might be useful? Or is that list empty? (I wouldn't mind running some such tests, but I suppose you'd have to take my word for it that I didn't look at the published residues. I assure you I have no interest in doing that; it rather defeats the purpose of the exercise.)

For kicks, I'm running M1800767 right now. It should be done in the next 15 minutes or so. I hope this doesn't count as poaching.

Edit: The 153BED18D5C5EA4B residue for M1800767 is correct, and I was awarded a whopping 0.0977 GHz-days for my troubles. I'm going to wait until I hear back from Prime95 before I do any more of this sort of work.

Last fiddled with by KingKurly on 2010-12-27 at 16:24 Reason: the LL finished

 2010-12-27, 18:57 #5 davar55     May 2004 New York City 102138 Posts Wasn't the moral issue of poaching being discussed in another thread?
2010-12-27, 22:43   #6
Prime95
P90 years forever!

Aug 2002
Yeehaw, FL

5·1,553 Posts

Quote:
 Originally Posted by KingKurly Is there a list of small exponents for which a triple check might be useful? Or is that list empty?
I suspect the list is empty.

2010-12-27, 22:47   #7
Mini-Geek
Account Deleted

"Tim Sorbera"
Aug 2006
San Antonio, TX USA

427110 Posts

Quote:
 Originally Posted by Prime95 I suspect the list is empty.
I suspect it is not, but that only trusted party(ies) could run the triple checks for it to do any good. (the list might be formed, at least in part, by looking for suspicious activity, e.g. user B often submits "double checks" in a non-verifiable way of user A's work shortly after user A "completes" a first-time test, verifiable or not)

Last fiddled with by Mini-Geek on 2010-12-27 at 22:52

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## How is a USD-chained currency like Tether economically possible?

2

(I'll try my best to keep it relevant to Bitcoin, but feel free to close it if it seems inapplicable to Bitcoin trading. It's more about economics)

Say there's a currency whose supply is controlled (e.g GBP in 1990, or USDT) to keep the exchange rate fixed (GBP/EUR or USDT/USD)

But sometimes the exchange rate is slightly higher or lower than intended. If it's higher, one can borrow more of the valuable currency, sell it, and then pay back with interest, and profit from the difference, because there's no doubt that the currency's value will be lowered to reach the exchange rate target.

In the end, George Soros and his $15 billion budget made at least 20% profit, according to the source, and GBP was forced to become free instead of 1.2 GBP/EUR . My question is, what protects USDT from this kind of economical strategy which is guaranteed to make Bitfinex, the minter of USDT, a loss? Is that because they hold the majority of USDT? ## Answers 0 Bitfinex, the issuer of Tether, claims that all of the USDT is backed by USD and USD equivalents. As such, the intrinsic value will be close to [$1 * asset impairment], and so no market manipulation/pegging is required to maintain parity, so long as their assets actually match the tether outstanding.

Although no action is required, there is an opportunity for Bitfinex to buy USDT with USD when the price dips below 100 cents and sell when its over 100 cents if they actually are fully backed. This would tighten the trading range, and make Bitfinex a profit in the process.


text:
# How the Venus Flytrap snaps

(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)

## Contents

### How the Venus Flytrap Sanos

Yoël Forterre, Jan M. Skotheim, Jacques Dumais & L. Mahadevan

Nature 433, 421-425 (27 January 2005)

## Keywords

elastic deformation, instability

## Summary

Although people rarely think about plant movement, there are examples of rapid behavior in the plant kingdom. Some examples include dispersing seeds or pollen and capturing prey. The Venus Fly Trap is one dramatic examples. Previous researcher have searched for a purely biological basis for the rapid motion of the plant, such as "acid-induced wall loosening" or a loss of turgor pressure inside special motor cells. However, much of the behavior can be understood just in terms of elastic deformation. The leaves are curved outwards (convex) in the open state and curved inwards (concave)in the closed state, with an elastic buckling instability during the transition.

## Soft Matter Aspects

As Prof. Howard Stone often advocates for fluid mechanics problems, the best way to approach this situation is to identify the relevant parameters can create a dimensionless parameter. In this case,

$\alpha = L^4 \kappa^2 / h^2$,

where L is the length of the leaf, $\kappa$ is the observed Gaussian curvature of the open lead, and h is the leaf thickness. A larger value of $\alpha$ corresponds to more difficulty in stretching the midplane by changing the curvature of the leaf.

written by: Naveen N. Sinha


text:
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Area and Perimeter of Rectangles

%
Progress
Practice Area and Perimeter of Rectangles
Progress
%
What is the area of a rectangle?
Teacher Contributed
The formula for the area A$A$ of a rectangle is A=b×h$A= b \times h$

qid: 100311


text:
mittyri
★★

Russia,
2022-06-24 15:43
(278 d 00:22 ago)

Posting: # 23079
Views: 1,387

## T1E after 1st stage in Power2Stage [Two-Stage / GS Designs]

Dear Forum Members,

could you please guide me regarding the function to be used for T1E estimation after 1st stage of Potvin methods (Power2Stage R package)?

Kind regards,
Mittyri
Helmut
★★★

Vienna, Austria,
2022-06-24 16:33
(277 d 23:32 ago)

@ mittyri
Posting: # 23081
Views: 1,192

## T1E after 1st stage in TSDs

Hi Mittyri,

❝ could you please guide me regarding the function to be used for T1E estimation after 1st stage of Potvin methods (Power2Stage R package)?

Really in the first stage? Negligible for the combinations where you get the largest TIE overall. Anyhow, your wish is my command:

library(Power2Stage) # alpha0 = 0.05 for method = "C" and alpha = c(0.0294, 0.0294) for method = "B" # and "C" are defaults; GMR = 0.95, targetpower = 0.8, if theta0 = 0.8 or 1.25, # by default nsims = 1e6. CV  <- 0.2 n1  <- 12 # That’s an evil combination; commonly the largest # TIE with a relatively small CV and very small n1. res <- data.frame(method = c("B", "C"), alpha0 = 0.05, alpha = 0.0294,                   TIE.stage1 = NA_real_, TIE.stage2 = NA_real_) for (j in 1:nrow(res)) {   tmp         <- power.tsd(method = res$method[j], n1 = n1, CV = CV, theta0 = 1.25) res[j, 4:5] <- c(tmp$pBE_s1, tmp$pBE) } res$alpha0[1] <- "" print(res, row.names = FALSE)  method alpha0  alpha TIE.stage1 TIE.stage2       B        0.0294   0.028849   0.046273       C   0.05 0.0294   0.035777   0.051100

Belt plus suspenders:

library(PowerTOST) # I don’t recommend that but let n1 = n of a fixed sample design # i.e., high chance of passing in the first stage of a TSD. CV  <- 0.2 n1  <- sampleN.TOST(CV = CV, print = FALSE)[["Sample size"]] for (j in 1:nrow(res)) {   tmp         <- power.tsd(method = res$method[j], n1 = n1, CV = CV, theta0 = 1.25) res[j, 4:5] <- c(tmp$pBE_s1, tmp$pBE) } res$alpha0[1] <- "" print(res, row.names = FALSE)  method alpha0  alpha TIE.stage1 TIE.stage2       B        0.0294   0.029142   0.034816       C   0.05 0.0294   0.045318   0.049042

PS: I changed the category from . Though you asked for the -package Power2Stage, perhaps somebody would have the balls to roll out own code.

Dif-tor heh smusma 🖖🏼 Довге життя Україна!
Helmut Schütz

The quality of responses received is directly proportional to the quality of the question asked. 🚮
Science Quotes
mittyri
★★

Russia,
2022-06-24 23:12
(277 d 16:53 ago)

@ Helmut
Posting: # 23084
Views: 1,179

Hi Helmut,

tmp         <- power.tsd(method = res$method[j], n1 = n1, CV = CV,  theta0 = 1.25) do I understand it right that if the study is terminated after 1st stage (due to any reason), we are using the same function with n1 (not fixing n2 to 0) PS: I changed the category from . Though you asked for the -package Power2Stage, perhaps somebody would have the balls to roll out own code. I changed the category twice when wrote the post Kind regards, Mittyri Helmut ★★★ Vienna, Austria, 2022-06-25 16:07 (276 d 23:59 ago) @ mittyri Posting: # 23086 Views: 1,155 ## T1E after 1st stage in TSDs Hi Mittyri, Of course, ElMaestro is right. There Can Be Only One. ❝ do I understand it right that if the study is terminated after 1st stage (due to any reason), we are using the same function with n1 (not fixing n2 to 0) Not sure what you mean. For any of the simulated studies the decision scheme of the method is followed. The function returns counters for the studies passing in the 1st stage (pBE_s1) and of the pooled data in the 2nd stage (pBE). We can extract all numbers: library(Power2Stage) # using defaults of method = "B" tmp <- power.tsd(CV = 0.2, n1 = 12, theta0 = 1.25) if (tmp$theta0 == tmp$theta1 | tmp$theta0 == tmp$theta2) { txt <- "Type I Error" } else { txt <- "Power" } cat(formatC(tmp$nsims, digits = 7, big.mark = ","),     "simulated studies\nFailed in stage 1     :",     formatC(tmp$nsims - tmp$nsims * tmp$pBE_s1 - tmp$pct_s2 / 100 * tmp$nsims, digits = 6, big.mark = ","), sprintf("(%7.4f%%)", 100 * (tmp$nsims - tmp$nsims * tmp$pBE_s1 - tmp$pct_s2 / 100 * tmp$nsims) / tmp$nsims), "\nPassed in stage 1 :", formatC(tmp$pBE_s1 * tmp$nsims, digits = 6, big.mark = ","), sprintf("(%7.4f%%)", 100 * tmp$pBE_s1),     "\nProceeded to stage 2  :",     formatC(tmp$nsims * tmp$pct_s2 / 100, digits = 6, big.mark = ","),     sprintf("(%7.4f%%)", tmp$pct_s2), "\nFailed in stage 2 :", formatC(tmp$nsims * tmp$pct_s2 / 100 - (tmp$pBE -tmp$pBE_s1) * tmp$nsims,             digits = 6, big.mark = ","),     sprintf("(%7.4f%%)",     100 * (tmp$nsims * tmp$pct_s2 / 100 - (tmp$pBE -tmp$pBE_s1) * tmp$nsims) / tmp$nsims),     "\nPassed in stage 2     :",     formatC((tmp$pBE -tmp$pBE_s1) * tmp$nsims, digits = 6, big.mark = ","), sprintf("(%7.4f%%)", 100 * (tmp$pBE -tmp$pBE_s1)), "\nFailed in stage 1 or 2:", formatC((1 - tmp$pBE) * tmp$nsims, digits = 6, big.mark = ","), sprintf("(%7.4f%%)", 100 * (1 - tmp$pBE)),     "\nPassed in stage 1 or 2:",     formatC(tmp$pBE * tmp$nsims, digits = 6, big.mark = ","),     sprintf("(%7.4f%%) = empiric", 100 * tmp\$pBE), txt, "\n") 1,000,000 simulated studies Failed in stage 1     :  92,550 ( 9.2550%) Passed in stage 1     :  28,849 ( 2.8849%) Proceeded to stage 2  : 878,601 (87.8601%) Failed in stage 2     : 861,177 (86.1177%) Passed in stage 2     :  17,424 ( 1.7424%) Failed in stage 1 or 2: 953,727 (95.3727%) Passed in stage 1 or 2:  46,273 ( 4.6273%) = empiric Type I Error

And:

tmp <- power.tsd(CV = 0.2, n1 = 12, theta0 = 0.95) 100,000 simulated studies Failed in stage 1     :   2,325 ( 2.3250%) Passed in stage 1     :  41,333 (41.3330%) Proceeded to stage 2  :  56,342 (56.3420%) Failed in stage 2     :  13,409 (13.4090%) Passed in stage 2     :  42,933 (42.9330%) Failed in stage 1 or 2:  15,734 (15.7340%) Passed in stage 1 or 2:  84,266 (84.2660%) = empiric Power

❝ ❝

PS: I changed the category from . Though you asked for the -package Power2Stage, perhaps somebody would have the balls to roll out own code.

❝ I changed the category twice when wrote the post

I see. Feel free to change it to anyone you like.

Dif-tor heh smusma 🖖🏼 Довге життя Україна!
Helmut Schütz

The quality of responses received is directly proportional to the quality of the question asked. 🚮
Science Quotes
mittyri
★★

Russia,
2022-06-26 01:29
(276 d 14:36 ago)

@ Helmut
Posting: # 23087
Views: 1,136

## Potvin C consequences

Hi Helmut (and ElMaestro - sorry to post in this thread, but all your answers are great!),

❝ Of course, ElMaestro is right.

There Can Be Only One.

OK, I understand the logic here.
take a look at some (theoretical) situation: a Sponsor goes with Potvin C (don't ask me why), moreover, he proved that with the current CV and n1 T1E does not exceed 5%.
But as you showed in the lectures and in example above, Potvin C could not ensure 5% for all combinations. Thus, during stage 1 the CV was lower and the resulted T1E was out of 5%.
What would be the right (safe) way out in such situation?

Kind regards,
Mittyri
ElMaestro
★★★

Denmark,
2022-06-26 11:11
(276 d 04:54 ago)

@ mittyri
Posting: # 23088
Views: 1,134

## Potvin C consequences

Hey easy now,

❝ ❝ Of course, ElMaestro is right.

While that sounds good it is only very rarely the truth.

❝ take a look at some (theoretical) situation: a Sponsor goes with Potvin C (don't ask me why), moreover, he proved that with the current CV and n1 T1E does not exceed 5%.

❝ But as you showed in the lectures and in example above, Potvin C could not ensure 5% for all combinations. Thus, during stage 1 the CV was lower and the resulted T1E was out of 5%.

❝ What would be the right (safe) way out in such situation?

I can't follow you, in particular the parts in red.
Your type I error is not depending on your estimates and thus not depending on whether you went into stage 2 or not. They are depending on the true GMR, the (true) variability, N and the underlying stats assumptions; you may have estimates of the CV and GMR and your N is known.
Note also that when we speak of type I error it is often implied that it is the maximum type I error. By definition, it is on the acceptance borders that you get an idea of the type I errors.

Pass or fail!
ElMaestro
mittyri
★★

Russia,
2022-06-27 15:45
(275 d 00:20 ago)

@ ElMaestro
Posting: # 23093
Views: 1,058

## Potvin C consequences

Hi ElMaestro,

some more questions

❝ Your type I error is not depending on your estimates and thus not depending on whether you went into stage 2 or not.

power.tsd(method = "C", CV = 0.2, n1 = 12, theta0 = 1.25) p(BE)    = 0.0511 power.tsd(method = "C", CV = 0.3, n1 = 12, theta0 = 1.25) p(BE)    = 0.043743

So T1E (OK, maximum type I error) depends on the CV, isn't it?
Or may be I misunderstood your point and we should not verify T1E during estimation process since we are assuming that the framework was stated as acceptable in the Protocol?

Kind regards,
Mittyri
Helmut
★★★

Vienna, Austria,
2022-06-27 16:49
(274 d 23:16 ago)

@ mittyri
Posting: # 23094
Views: 1,036

## Potvin C consequences

Hi Mittyri,

power.tsd(method = "C", CV = 0.2, n1 = 12, theta0 = 1.25)

p(BE)    = 0.0511

power.tsd(method = "C", CV = 0.3, n1 = 12, theta0 = 1.25)

p(BE)    = 0.043743

❝ So T1E (OK, maximum type I error) depends on the CV, isn't it?

Yes, though not only on the CV but on n1 as well (though less so).

❝ Or may be I misunderstood your point and we should not verify T1E during estimation process since we are assuming that the framework was stated as acceptable in the Protocol?

In principle yes but I received a few deficiency letters asking for a “post hoc assessement of the Type I Error”. That’s not well-thought-out. Of course, we know n1 but the CV is just an estimate. The true one (used in the simulations of the framework) is unknown.

There are no problems with “Method B”. In the entire framework of CV/n1-combinations the T1E is controlled. When you explore a narrow grid (step size of CV 2% and of n1 2) you will get this (noncentral t-distribution instead of the shifted central t like in the paper):

Maximum Type I Error 0.048856 at CV 24% and n1 12.

As you already discovered, “Method C” can be problematic – though only in certain rare cases.

Maximum Type I Error 0.051250 at CV 22% and n1 12
(the yellow dot). Thick red contour at the bottom encloses T1E > 0.05.

IMHO, it was unfortunate by Potvin et al. to select the same adjusted α in both methods. Any ‘Type 2’ TSD will require more adjustment than its ‘Type 1’ counterpart.
BTW, 0.0294 in “Method B” is overly conservative; 0.0302 controls the Type I Error as well. On the other hand, for “Method C” you need 0.0282.

Consider to avoid this stuff in the future. See this post and followings for better alternatives.

Dif-tor heh smusma 🖖🏼 Довге життя Україна!
Helmut Schütz

The quality of responses received is directly proportional to the quality of the question asked. 🚮
Science Quotes
mittyri
★★

Russia,
2022-06-27 19:13
(274 d 20:52 ago)

@ Helmut
Posting: # 23096
Views: 1,009

## from Potvin C to Inverse Normal

Dear Helmut,

❝ In principle yes but I received a few deficiency letters asking for a “post hoc assessement of the Type I Error”. That’s not well-thought-out. Of course, we know n1 but the CV is just an estimate. The true one (used in the simulations of the framework) is unknown.

Fully agree! This thread was started due to deficiency letter my friend showed me some time ago. I thought I missed something in TSD, but now I see a full picture

❝ Consider to avoid this stuff in the future. See this post and followings for better alternatives.

there are zero papers I've seen by Russian experts regarding inverse normal method. I guess this is the reason why the sponsors do not want to have a risk with that invention. "Why should we care if everything works with Potvin?"

Kind regards,
Mittyri
Helmut
★★★

Vienna, Austria,
2022-06-27 20:54
(274 d 19:11 ago)

@ mittyri
Posting: # 23098
Views: 1,013

## From Potvin C to Inverse Normal: history (and outlook?)

Hi Mittyri and all,

❝ ❝ Consider to avoid this stuff in the future. See this post and followings for better alternatives.

❝ there are zero papers I've seen by Russian experts regarding inverse normal method. I guess this is the reason why the sponsors do not want to have a risk with that invention. "Why should we care if everything works with Potvin?"

Obviously it doesn’t work. Maybe it’s  time for a change.

In my studies I used only “Method C” and never received a single (‼) deficiency letter. Why? Not sure. Only because all studies stopped for success in the first stage? Or because I was ‘famous’?1 Anyhow, later I faced deficiency letters – even for “Method B”.2

Since this was an extremely unsatisfactory situation, I nudged my friends to develop an exact method (i.e., with strict Type I Error control without relying on simulations). Not an easy job, took us two years.3,4 As an aside, only the 2×2×2 crossover design is covered, but we stated:

[…] the proposed adaptive design would allow to switch from a classical two-period design to a more complex replicate design if it turns out the reference product is highly variable and the within subject-variability has to be determined as well.

Currently science fiction.
Our posters were noticed and Byron Jones started working on the stuff together will a veteran of adaptive designs, Willi Maurer. In December 2016 the method appeared in a book.5 Unfortunately the code was crude, politely speaking. In February 2018 the paper6 – containing the analytical proof and ‘improved’ code as supplementary information7 – was published. The method is extremely flexible when it comes to futility criteria (PE or CI not within certain limits, estimated total sample size above a limit), minimum stage 2 sample size protecting against dropouts, and even sample size re-estimation based on the T/R-ratio observed in the first stage (i.e., fully adaptive). Since stages are evaluated separately, not only an ANOVA can be used but also a mixed-effects model, which is recommended by the FDA and Health Canada. It took Benjamin Lang and Detlew Labes almost half a year to implement the inverse-normal combination method in Power2Stage.8 BTW, the method is acceptable for the FDA as well.9

I was told that Russian »эксперты« have altercations of the type:

“Helmut said …!” – “No, on the contrary! He said …”

and so on and so forth.
What I really said and keep saying:  Use the inverse-normal combination method!

Regrettably a parallel design is another pot of tea. Taking unequal group sizes in the first stage and heterogenicity into account is not trivial. At the very bottom of our todo list.
In the meantime use the simulation-based method10 available in Power2Stage.

1. Schütz H. Two-stage designs in bioequivalence trials. Eur J Clin Pharmacol. 2015; 71(3): 271–81. doi:10.1007/s00228-015-1806-2.
2. Last year I had to deal with a deficiency letter of the Czech agency SÚKL. “Method B” passed with flying colors in the first stage. The agency questioned the reliability of Potvin’s simulations ‘because the grid in the publication’s simulations was too sparse’. Bizarre. Of course, a narrow grid with a step size of 2 for n1 and 2% for the CV showed no inflated Type I Error as well (maximum 0.048856 for n1 12 and CV 24%). Hurray, 1.116 billion simulated studies!
3. König F, Wolfsegger M, Jaki T, Schütz H, Wassmer G. Adaptive two-stage bioequivalence trials with early stopping and sample size re-estimation. Vienna: 2014; 35th Annual Conference of the International Society for Clinical Biostatistics. Poster P1.2.88. doi:10.13140/RG.2.1.5190.0967.
4. König F, Wolfsegger M, Jaki T, Schütz H, Wassmer G. Adaptive two-stage bioequivalence trials with early stopping and sample size re-estimation. Trials. 2015; 16(Suppl 2); P218. doi:10.1186/1745-6215-16-S2-P218.
5. Patterson SD, Jones B. Bioequivalence and Statistics in Clinical Pharmacology. Boca Raton: CRC Press; 2nd edition 2017. ISBN 978-1-4665-8520-1. p. 141–187.
6. Maurer W, Jones B, Chen Y. Controlling the type 1 error rate in two-stage sequential designs when testing for average bioequivalence. Stat Med. 2018; 37(10): 1–21. doi:10.1002/sim.7614.
7. Maurer W, Jones B, Chen Y. Supplementary Information for ”Controlling the Type I error rate in two-stage sequential adaptive designs when testing for Average Bioequivalence”. Open access.
8. Labes D, Lang B. Schütz H. Power2Stage: Power and Sample-Size Distribution of 2-Stage Bioequivalence Studies. Package version 0.5-1. 2018-04-03. Current version on CRAN.
9. Lee J, Feng K, Xu M, Gong X, Sun W, Kim J, Zhang Z, Wang M, Fang L, Zhao L. Applications of Adaptive Designs in Generic Drug Development. Clin Pharm Ther. 2020; 110(1): 32–5. doi:10.1002/cpt.2050.
10. Fuglsang A. Sequential Bioequivalence Approaches for Parallel Designs. AAPS J. 2014; 16(3): 373–8. doi:10.1208/s12248-014-9571-1. Free Full text.

Seems that this is a hot topic. 210 views in 3½ days.

Dif-tor heh smusma 🖖🏼 Довге життя Україна!
Helmut Schütz

The quality of responses received is directly proportional to the quality of the question asked. 🚮
Science Quotes
ElMaestro
★★★

Denmark,
2022-06-27 19:13
(274 d 20:53 ago)

@ mittyri
Posting: # 23095
Views: 1,111

## Potvin C consequences

Hi mittyri,

❝ So T1E (OK, maximum type I error) depends on the CV, isn't it?

❝ Or may be I misunderstood your point and we should not verify T1E during estimation process since we are assuming that the framework was stated as acceptable in the Protocol?

Depends on the CV, but does not depend on the estimated CV. And that's a very significant difference. You never know anything other than your estimates, i.e. the knowledge that is generated as results of your trial do not determine the (maximum) type I error.

Pass or fail!
ElMaestro
mittyri
★★

Russia,
2022-06-27 19:22
(274 d 20:43 ago)

@ ElMaestro
Posting: # 23097
Views: 1,024

## true CV ≠ observed CV

Hi ElMaestro,

❝ Depends on the CV, but does not depend on the estimated CV. And that's a very significant difference. You never know anything other than your estimates, i.e. the knowledge that is generated as results of your trial do not determine the (maximum) type I error.

you are right, I completely agree. We cannot base TIE estimation on the observed CV.
As Helmut noted, to be on the safe side Potvin C should gone with the wind, so the framework per se should not permit such situations for given prerequisites (or even better to go with Inverse normal)

Kind regards,
Mittyri
ElMaestro
★★★

Denmark,
2022-06-24 20:25
(277 d 19:41 ago)

@ mittyri
Posting: # 23082
Views: 1,185

## T1E after 1st stage in Power2Stage

Hi Mittyri,

❝ could you please guide me regarding the function to be used for T1E estimation after 1st stage of Potvin methods (Power2Stage R package)?

That would be a misnomer, I think. The T1E is a property of the testing approach. There is an alpha at stage 1 and there is an alpha at stage 2 but there is no T1E at stage 1 (or 2). There is just a well-defined T1E for the approach.
Mixing up the terms alpha and T1E is a real danger for 2-stage approaches, since in "normal" trials they are one and the same.

Pass or fail!
ElMaestro
mittyri
★★

Russia,
2022-06-24 23:05
(277 d 17:00 ago)

@ ElMaestro
Posting: # 23083
Views: 1,188

## T1E after 1st stage in Power2Stage

Hi ElMaestro,

❝ Mixing up the terms alpha and T1E is a real danger for 2-stage approaches, since in "normal" trials they are one and the same.

yep, I should reformulate: T1E for the adaptive study finished after 1st stage due to BE goal reached (or some other reason to stop)

Kind regards,
Mittyri
ElMaestro
★★★

Denmark,
2022-06-25 00:55
(277 d 15:10 ago)

@ mittyri
Posting: # 23085
Views: 1,136

## T1E after 1st stage in Power2Stage

Hi mittyri,

❝ yep, I should reformulate: T1E for the adaptive study finished after 1st stage due to BE goal reached (or some other reason to stop)

See my answer above. It does not change regardless of whether you stopped after stage 1, regardless of whether alpha 1 and alpha 2 are the same or different.

A relevant perspective: Imagine you do a normal trial with a 90% CI of 88.88%-99.99%. What is the type I error? The answer is 5%.
That answer -the 5% percent- does not in any way depend on the figures. Same for the two-stage approach. The type 1 error does not change with your data, it doesn't matter if you stop at stage 1 or not.

Pass or fail!
ElMaestro


text:
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This is a list of roots, suffixes, and prefixes used in medical terminology, their meanings, and their etymologies.Most of them are combining forms in New Latin and hence international scientific vocabulary.There are a few general rules about how they combine. This information should not be considered complete, up to date, and is not intended to be used in place of a visit, consultation, or advice of a legal, medical, or any other professional. Rhyming reduplication form super-duper first attested 1940. A saccade is a twitching, jerking movement, particularly of the eyeball. A surmise is a guess “sent over” available evidence, which is often not enough to prove something. They supervise, or watch “over” the schools in their respective districts. In No. Please :-). Prefixes are key morphemes in English vocabulary that begin words. super, supra & supr These ROOT-WORDS are the Prefixes SUPER & SUPR meaning OVER, ABOVE, BEYOND & GREATER IN QUALITY. Quick Summary. a prefix occurring originally in loanwords from Latin, with the basic meaning “above, beyond.”. The primary sense seems to have shifted over time from usually meaning "beyond" to usually meaning "very much," which can be contradictory. in loanwords from Latin, with the basic meaning “above, beyond.” Words formed with super-have the following general senses: “to place or be placed above or over” (superimpose), “a thing placed over another” (superstructure), “situated over” (superficial) and, more figuratively, “an individual, thing, or property that exceeds customary norms or levels” (superconductivity; superman), … There you are, on your way to having a SUPER vocabulary! Sometimes you’d like to hit those kinds of charges “over” the head! Learn vocabulary, terms, and more with flashcards, games, and other study tools. The root of a word is a unit of meaning (morpheme) and, as such, it is an abstraction, though it can usually be represented alphabetically as a word. See an example word page ». super- word-forming element meaning "above, over, beyond," from Latin super (adverb and preposition) "above, over, on the top (of), beyond, besides, in addition to," from *(s)uper-, variant form of PIE root *uper "over." Start studying super root words. You need to enable Javascript to get the best out of this site. A root word must have an affix or another root attached to it in order to form a true word. Another word for super. https://www.thefreedictionary.com/Super+(prefix), a prefix occurring orig. Supercell definition is - an unusually large storm cell; specifically : a severe storm generated by such a cell. Root word: Meanings: Origin: Examples and Definitions: a/n: ... the meaning of a word) forsaken or forfeited - completely lost; forgiven - completely given (a release of debt). Super Bowl attested from 1966; Super Glue from 1975; as a verb by 1983. The roots of “superstition,” which appeared in English in the early 15th century, are the Latin “super” (meaning, as usual, “above”) and the participle form of “stare,” which means “to stand,” giving us a basic sense of “the act of standing over or above.” The word as we first heard it was super-cadja-flawjalistic-espealedojus. And have you ever been hit with a surcharge on your cell phone bill, those sneaky little charges that go “over” what you are supposed to pay? Under definition, beneath and covered by: under a table; under a tree. in loanwords from Latin, with the basic meaning “above, beyond.” Words formed with super-have the following general senses: “to place or be placed above or over” (superimpose), “a thing placed over another” (superstructure), “situated over” (superficial) and, more figuratively, “an individual, thing, or property that exceeds customary norms or levels” (superconductivity; superman), … You say ‘hyperactive’ about someone, who is too active and cannot relax. As in superaspirin, superbug, superjacent, supernumerary, supersize, supertaster. The prefix super- and its variant sur- mean “over.”. You'll find that the roots listed below are from Greek or Latin and can't stand alone in English; they need something joined to them to make a whole word in English. That’s the original meaning of the English prefix, too, though according to the Oxford Dictionary of English Etymology, it has taken on other senses over the centuries. A vocabulary list featuring Power Prefix: super-. An easy way to remember that the prefix super-means “over” is through the comic book hero Superman, who is the man “over” all other men in terms of physical power. The prefix hyper- is a morpheme, thus, it cannot be divided. For example, it can be said that the root of the English verb form running is run , or the root of the Spanish superlative adjective amplísimo is ampli- , since those words are clearly derived from the root forms by simple suffixes that do not alter the roots in any way. Membean is an incredibly effective way to learn words and permanently remember them. aukščiausios rūšies, geriausios kokybės, aukščiausio laipsnio, Dictionary, Encyclopedia and Thesaurus - The Free Dictionary, the webmaster's page for free fun content. Learn these words beginning with the prefix super, meaning "above," "over," or "beyond." For instance, a surname is that name which is “over” a family and thereby identifies it, or the family’s last name. All text and design are copyrighted ©2010-2021 Membean, Inc. All rights reserved. super (adj.) Prefixes are key morphemes in English vocabulary that begin words. Find more ways to say super, along with related words, antonyms and example phrases at Thesaurus.com, the world's most trusted free thesaurus. SACCADE. ‘Hyperactive’ is a word that start with the prefix hyper. Want to increase your prefix power? sur This ROOT-WORD is the Prefix SUR which means OVER, ABOVE & MORE. That which is superfluous “flows over and above” what is necessary. 1 a (1) : over and above : higher in quantity, quality, or degree than : more than superhuman. Now that you have been surrounded with superlative examples of words which contain the prefixes super- and sur-, you will never again have to look those words “over” twice before knowing what they mean. The word “super” comes from the Latin word super which means "above, over, beyond". Root word definition: the form of a word after all affixes are removed | Meaning, pronunciation, translations and examples Learn more on how we help for Start a free trial of Quizlet Plus by Thanksgiving | Lock in 50% off all year Try it free As a prefix, super- originates from the Latin super, an adverb and preposition meaning above, on top of, beyond, beside. We found 605 words starting with Super. One who takes a survey of people wants to look “over” what they think. (2) : in addition : extra supertax. a prefix occurring orig. home / medterms medical dictionary a-z list / super- definition Medical Definition of Super-Medical Author: William C. Shiel Jr., MD, FACP, FACR; Super-: Prefix meaning meaning above, more than normal, or excessive. The prefix super- and its variant sur- mean “over.” Examples using this prefix include superior, supervise, surname, and surface. First, prefixes and suffixes, most of which are derived from ancient Greek or classical Latin, have a droppable -o-. Terms of Service. In English words from Old French, it appears as sur-. School. School systems love to have members of management who stand “over” all others, such as superintendents, who are in charge of entire school systems. a prefix occurring orig. This is a comprehensive words list for words that start with Super. The surface of something is etymologically the face that lies “over” what it’s covering. The adjective super is an abbreviated use of the prefix super -, which comes from the Latin super -, meaning “above,” “over,” or “beyond.” Super is another way to say "the best" — you are above the rest. Review the list below, as well as a few examples of English words that are based on these roots. If one considers the inverse function of second order tetration, called the super square root $$\text{ssrt(}x\text{)}$$, we find several other applications, mentioned below. 2, it is a belt tied over a saddle. The prefix super-and its variant sur-mean “over.” Examples using this prefix include superior, supervise, surname, and surface. That’s just supercalifragilisticexpialidocious! The prefix sub-, with its variants which all begin with su-, is a prolific part of the English language.Examples using this prefix include subway, suffer, supply, and suggest.An easy way to remember that the prefix sub-means “under” is through the word submarine, or a vehicle that travels “under” the sea. The word ingredient Memlet, shown below, is one of many ways that a word is taught in Membean. b (1) : exceeding or so as to exceed a norm superheat. super- A prefix meaning "while drunk." Speaking of superstars, the football game that stands “over” all other football games is, you got it, the Super Bowl. [9] The Oxford English Dictionary defines the word as "a nonsense word, originally used esp. Hey guys, want to get drunk and play super-football? Personal Learning, From the Latin preposition super meaning above. Definition & Meaning: Word Root Hyper Hyper- means ‘too, over, excessive, beyond’. therm - WordReference English dictionary, questions, discussion and forums. "Super"="above" is the prefix from the Greek and the root "natural" is what it sounds like so the word means "above and beyond what is considered natural". A variant of the prefix super-, which also means “above,” is the morpheme sur-. This Prefix does not indicate greatness or superiority. It was first recorded in the 1680′s, where it was used as a prefix in “superfine goods”, denoting that the goods are of the highest grade. Test Prep, Super-, sur-, mean over, above, beyond. Acri: bitter (ac… Since much of the English language is derived from Latin and Greek, there may be times when the root of a word isn't immediately recognizable because of its origin. He is the superstar in the DC Comics line of heroes, or the star that stands “over” other heroic stars, such as Batman and Wonder Woman. See more. An easy way to remember that the prefix super- means “over” is through the comic book hero Superman, who is the man “over” all other men in terms of physical power. Privacy Policy. All content on this website, including dictionary, thesaurus, literature, geography, and other reference data is for informational purposes only. The Super Bowl features the superior teams from the AFC and the NFC divisions facing off against each other, that is, the two teams that stood “over” all the rest during the football season. Words formed with super- have the following general senses: “to place or be placed above or over” … These are vocabulary words from those roots. For example, super-cooking is cooking while drunk, and super-volleyball is playing volleyball while drunk. And if you’re … The most common prefixes in English terms, and other reference data for! Beyond. ” words formed with than superhuman, beyond. ” words formed with vocabulary that words. What it ’ s covering, surname, and more with flashcards, games and. We help for Test Prep, Personal Learning, or watch “ over available. Beyond. ” Examples using this prefix include superior, supervise, or degree:. By 1983. a prefix occurring orig ’ s covering prefix super is one of many that., beyond. ” words formed with to enable Javascript to get the best of! Superjacent, supernumerary, supersize, supertaster Examples using this prefix include superior, supervise, surname, surface. Addition: extra supertax with flashcards, games, and other study tools tied. 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# The space between the table and its caption is very small

I have a table written like this:

\begin{table}
\centering
\begin{tabular}{|l||c|c|}
\hline&Column 1&Column 2\\
\hline\hline
cell1 & cell2 & cell3 \\
\hline
cell1 & cell2 & cell3 \\\hline
\end{tabular}
\caption{A test caption}
\label{tab:attempt1}
\end{table}

Now the caption is VERY NEAR the table.. Additionally, there's a big vertical white space after the caption... How can I fix that to look like any normal table with a caption?

This is the case only with tables, figures are working fine.

-
This is not the default behaviour. Which packages are you using and what \documentclass? –  Werner Jun 5 '12 at 18:25
Since the space is as expected in the article class, you should probably specify which class you're using. –  egreg Jun 5 '12 at 18:26
how about \usepackage{caption} \captionsetup[table]{skip=10pt}? –  cmhughes Jun 5 '12 at 18:26
It appears that you must have some code or package that affects this that you are not showing. Please always supply a complete document that shows the problem. article class puts a skip of size \abovecaptionskip above the caption, and sets this length to 10pt, so you should get at least that spacing. But you may be using a different class. –  David Carlisle Jun 5 '12 at 18:29
@cmhughes You should turn your comment into an answer. Also the following link might be useful to add to the answer: Why should a table caption be placed above the table? –  Alan Munn Jun 5 '12 at 18:42

## 1 Answer

You could try

\usepackage{caption}
\captionsetup[table]{skip=10pt}

in your preamble. On another note, have a look at Why should a table caption be placed above the table?

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# zbMATH — the first resource for mathematics

Fuzzy smoothness and sequences of fuzzy smooth functions. (English) Zbl 0955.26011
Summary: This paper is a continuation of the work [M. Kalina, Tatra Mt. Math. Publ. 14, 153-159 (1998; Zbl 0944.26030)]. It further investigates fuzzy smoothness of fuzzy continuous, but crisp, functions. Finally, sequences of fuzzy smooth functions and their sums are considered.

##### MSC:
 26E50 Fuzzy real analysis 26A24 Differentiation (real functions of one variable): general theory, generalized derivatives, mean value theorems 03E72 Theory of fuzzy sets, etc.
##### Keywords:
fuzzy derivative; fuzzy continuity; fuzzy smoothness
Full Text:
##### References:
 [1] Burgin, M.; Šostak, A., Fuzzification of the theory of continuous functions, Fuzzy sets and systems, 62, 71-81, (1994) · Zbl 0828.26015 [2] Dubois, D.; Prade, H., Towards fuzzy differential calculus, part 1: integration of fuzzy mappings, Fuzzy sets and systems, 8, 1-17, (1982) · Zbl 0493.28002 [3] Dubois, D.; Prade, H., Towards fuzzy differential calculus, part 2: integration on fuzzy intervals, Fuzzy sets and systems, 8, 105-116, (1982) · Zbl 0493.28003 [4] Dubois, D.; Prade, H., Towards fuzzy differential calculus, part 3: differentiation, Fuzzy sets and systems, 8, 225-233, (1982) · Zbl 0499.28009 [5] Janiš, V., Fixed points of fuzzy functions, Tatra mountains math. publ., 12, 13-19, (1997) · Zbl 0946.54036 [6] Janiš, V., Fuzzy uniformly continuous functions, Tatra mountains math. publ., 14, 177-180, (1998) · Zbl 0944.26029 [7] V. Janiš, Nearness derivatives and fuzzy differentiability, Fuzzy Sets and Systems, to appear. [8] Kalina, M., Derivatives of fuzzy functions and fuzzy derivatives, Tatra mountains math. publ., 12, 27-34, (1997) · Zbl 0951.26015 [9] Kalina, M., On fuzzy smooth functions, Tatra mountains math. publ., 14, 153-159, (1998) · Zbl 0944.26030 [10] Mesiar, R., Computation over LR-fuzzy numbers, (), 165-176
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.


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# The Static Stress Intensity Factor around the Antiplane Crack in an Infinite FGM Strip

Mathematical Problems in Engineering, Jun 2013

The problem of the static stress intensity factor around antiplane crack in an infinite strip functionally graded material was studied by using the method of integral transform-dual integral equations in this paper. The shear modulus in the two principal directions of the functionally graded material was assumed to vary proportionately as gradient model of double parameters index function. The partial differential equation was first reduced to Euler equation with Fourier cosine transform. By solving dual integral equations that were derived by applying the solution of Euler equation with the method of Copson, stress intensity factor around the crack tip was derived. And the variation curves of the dimensionless stress intensity factor with the strip height, crack length, gradient parameter, and inhomogeneous coefficient are obtained by using the numerical calculation.


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# Do all the electromagnetic radiations have dual nature i.e. particle nature & wave nature?

I have studied the dual nature of the light as particle nature & wave nature.

A photon of light energy can knock a single electron out of certain metals (usually having less forbidden-energy-gap) called photo-electric emission which shows particle nature of light.

While the phenomenon of Interference (which may be constructive or destructive depending on the wave equations) shows the wave nature of light.

The question is: Do all the electromagnetic radiations show dual nature same as that of light if "yes" then explain how? & if "no" then explain why?

• Yes, because all electromagnetic radiation can be described by photons or waves. – HDE 226868 Jul 17 '15 at 20:53
• All electromagnetic energies are described by the same equations - they all exhibit duality. – Jon Custer Jul 17 '15 at 21:02

Yes, everything shows wave-particle duality to varying degrees because "wave-particle duality" is just a name for a certain behaviour of quantum objects, and everything is believed to be a quantum object.

In particular, all electromagnetic radiation can be conceived of as being made of photons, which exhibit particle- and wave-like properties.

Yes, all electromagnetic radiation shows this 'dual nature' - which is to be expected since there is nothing really separating light from the other radiation in the spectrum apart from the arbitrary boundaries we have decided for it (i.e. visible light is just defined by what humans can see).

As you'll discover as you learn more about quantum mechanics, photons (and other particles for that matter), are not described by having a 'dual nature', rather their apparent 'dual nature' is a consequence of the quantum mechanical laws which they follow, which may seem counter-intuitive and are unlike any of the classical mechanisms you are used to.


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# All Questions

43k views

### Reverse Triangle Inequality Proof

I've seen the full proof of the Triangle Inequality \begin{equation*} |x+y|\le|x|+|y|. \end{equation*} However, I haven't seen the proof of the reverse triangle inequality: \begin{equation*} ||x|-|...
687 views

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### I have a problem understanding the proof of Rencontres numbers (Derangements)

I understand the whole concept of Rencontres numbers but I can't understand how to prove this equation $$D_{n,0}=\left[\frac{n!}{e}\right]$$ where $[\cdot]$ denotes the rounding function (i.e., $[x]$...
35k views

### Union of two vector subspaces not a subspace?

I'm having a difficult time understanding this statement. Can someone please explain with a concrete example?
6k views

### How can you prove that the square root of two is irrational?

I have read a few proofs that $\sqrt{2}$ is irrational. I have never, however, been able to really grasp what they were talking about. Is there a simplified proof that $\sqrt{2}$ is irrational?
### “Closed” form for $\sum \frac{1}{n^n}$
Earlier today, I was talking with my friend about some "cool" infinite series and the value they converge to like the Basel problem, Madhava-Leibniz formula for $\pi/4, \log 2$ and similar alternating ...


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Revision Notes on Solid and Electronic Device:-

Thermoionic emission:- Thermionic emission is the phenomenon in which electrons are emitted by a metal contains free electrons which behave like the molecules of a perfect gas.
Richardson Equation:-

I = AT1/2e-b/T

Here I is the thermionic current density in amp per sq meter. T is the temperature on kelvin scale, A and b being constants.

A = ne√k/2πm, b = e?/k

Here, n is the number of electrons per unit volume, e is the charge on electron, m is the mass of electron, k is the Boltzmann’s constant and ? is the potential barrier of the metal.

There are three types of energy bands in a solid viz.

(a)Valence energy band

(b)Conduction energy band

(c)Forbidden energy gap.

Valance Energy
Band

Forbidden Energy
Band

Conduction Energy Band

In this band there are valence electrons.

No electrons are found in this band

In this band the electrons are rarely found

This band may be partially or completely filled with electrons.

This band is completely empty.

This band is either empty or partially filled with electrons.

In this band the electrons are not capable of gaining energy from external electric field.

In this band the electrons can gain energy from electric field.

The electrons in this band do not contribute to electric current.

Electrons in this band contribute in this band contribute to electric current.

In this band there are electrons of outermost orbit of atom which contribute in band formation.

In this band there are electrons which are obtained on breaking the covalent bands.

This is the band of maximum energy in which the electrons are always present.

This is the band of minimum energy which is empty.

This band can never be empty.

This band can be empty.

Energy gap or Band gap (Eg):-

(a) The minimum energy which is necessary for shifting electrons from valence band to conduction band is defined as band gap (Eg)

(b) The forbidden energy gap between the valence band and the conduction band is known as band gap (Eg). i.e. Eg = Ec – Ev

(c) As there are energy levels f electrons in an atom, similarly there are three specific energy bands for the electrons in the crystal formed by these atoms as shown in the figure,

(d) Completely filled energy bands: The energy band, in which maximum possible numbers of electrons are present according to capacity is known as completely filled bank.

(e) Partially filled energy bands: The energy band, in which number of electrons present is less than the capacity of the band, is known as partially filled energy band.

(f) Electric conduction is possible only in those solids which have empty energy band or partially filled energy band.

Various types of solids:-

On the basis of band structure of crystals, solids are divided in three categories.

(a) Insulators

(b) Semi-conductors

(c) Conductors.

Difference between Conductors, Semi-conductors and Insulators:-

S.No.

Property

Conductors

Semi-conductors

Insulators

1.

Electrical conductivity and its value

Very high

10–7 mho/m

Between those of conductors and insulators

i.e. 10–7 mho/m to 10–13 mho/m

Negligible 10–13 mho/m

2.

Resistivity and its value

Negligible Less than 10–5 W-m

Between those of conductors and insulators i.e. 10–5 W-m to 105 W-m

Very high more than 105 W-m

3.

Band structure

4.

Energy gap and its value

Zero or very small

More that in con-ductors but less than that in insu-lators e.g. in Ge, ?Eg =0.72 eV is Si, ?Eg =1.1 eV in Ga As ?Eg =1.3 eV

Very large e.g. in diamond ?Eg = 7 eV

5.

Current carriers and current flow

Due to free electrons and very high

Due to free electrons and holes more than that in insulators

Due to free electrons but negligible.

6.

Number of current carriers (electrons or holes) at ordinary temperature

Very high

very low

negligible

7.

Condition of valence band and conduction band at ordinary temperature

The valence and conduction bands are completely filled or conduction band is some what empty (e.g. in Na)

Valence band in somewhat empty and conduction band is somewhat filled

Valence band is completely filled and conduction band is completely empty.

8.

Behaviour at 0 K

Behaves like a superconductor.

Behaves like an insulator

Behaves like an insulator

9.

Temperature coefficient of resistance (a)

Positive

Negative

Negative

10.

Effects of temperature on conductivity

Conductivity decreases

Conductivity increases

Conductivity increases

11.

On increasing temperature the number of current carriers

Decreases

Increases

Increases

12.

On mixing impurities their resistance

Increases

Decreases

Remains unchanged

13.

Current flow in these takes place

Easily

Very slow

Does not take place

14.

Examples

Cu, Ag, Au, Na, Pt, Hg etc.

Ge, Si, Ga, As etc.

Wood, plastic, mica, diamond, glass etc.

The number of electrons or cotters is given by

i.e. on increasing temperature, the number of current carriers increases.

The semiconductors are of two types.

(a) Intrinsic or pure semiconductors

(b) Extrinsic or dopes semiconductors

Difference between Intrinsic and Extrinsic semiconductors:-

S.No.

Intrinsic semiconductors

Extrinsic semiconductors

1.

Pure Ge or Si is known as intrinsic semiconductor

The semiconductor, resulting from mixing impurity in it, is known as extrinsic semiconductors.

2.

Their conductivity is low (because only one electron in 109 contribute)

Their conductivity is high

3.

The number of free electrons (ni) in conduction band is equal to the number of holes (pi) in valence band.

In this case ni is not equal to pi

4.

These are not practically used

These are practically used

5.

In these the energy gap is very small

In these the energy gap is more than that in pure semiconductors.

6.

In these the Fermi energy level lies in the middle of valence band and conduction

In these the Fermi level shifts towards valence or conduction energy bands.

Child’s law:-

Ip = KVp3/2, K is the proportionality constant.

Work function:-

Work function of a metal is the amount of energy required to pull an electron from the surface of metal to a distant position.

Diode:-

It is a vacuum tube containing two electrodes, an emitter and a collector.

Diode as a rectifier:-

Rectification is the process of converting the alternating current into a unidirectional current.

Rectification can be done by making use of a diode. A diode affecting rectification is said to be acting as a rectifier.

Diode Resistance:-

(a) Static plate resistance:- (i) Rp = Vp/Ip (ii) Rp ∝ (Vp)-1/2 (iii) Rp ∝ (Ip)-1/3

(b) Dynamic plate resistance:- (i) rp = (ΔVp/ΔIp) (ii) rp∝Vp-1/2 (iii) rp∝Ip-1/2

Half-wave rectifier:-  It is the type of rectification in which only one half of the input a.c. is translated into the output.

Full-wave rectifier:- A rectifier in which current flows through the load for both the halves of input a.c. is called half-wave rectifier.

Triode:- It is a vacuum tube containing three elements namely plate, filament and grid.

Triode constants:-

(a) Plate resistance (rp):- It is defined as the ratio between changes in plate potential keeping grid potential constant to the corresponding change in plate current.

(b) Mutual Inductance (gm):- (Trans-conductance):- It is defined as the ratio between change in plate current  to the change in grid potential keeping plate potential constant required to bring about that change in current.

(c) Amplification factor (μ):- it is defined as the ratio between change in plate potential keeping grid potential constant to the change in grid potential keeping plate potential constant, in order to bring about same change in plate current.

(d) Relation between μ, rp and gm:-

(e) rp ∝Ip-1/3

(f) gm∝Ip1/3

Triode as an amplifier:- Process of increasing the amplitude of the input signal is called amplification. A triode affecting an increase in the amplitude of a signal is said to be acting as an amplifier.

Voltage gain:- It is the ratio between the output voltage (voltage across the load resistance RL) to the signal voltage.

Voltage gain = V0/eg = μRL/RL+rp = μ/[1+(rp/RL)]

This indicates that the voltage gain depend upon the load resistance. For RL=∞, voltage gain is equal to the amplification factor.

(a) Ip = (μVg/RL+rp)

(b) A =μRL/RL+rp

(c) Amax = μ

(d) μ = A [1+(rp/RL)]

(e) A = μ/2   if RL = rp

Cut off voltage:-

Vg = -(Vp/μ)

Plate current equation:-

Ip = K[Vg+(Vp/μ)]3/2

Triode as an oscillator:- A triode producing high frequency oscillating waves, of constant amplitude, is said to be acting as an oscillator.

Frequency of oscillation waves (f):- f = 1/2π√LC

Here L is the inductance and C is the capacitance.

Majority charge carrier:-

(a) For N-type semiconductor:- electron

(b) For P-type semiconductor:- hole

Electrical conductivity of semiconductors:-

Intrinsic semiconductors:-

(a) σ = e(neμe + nhμh)

Here, ne is the electron density, nh is the hole density,  is the electric mobilities and  is the hole mobilities.

(b)

Extrinsic semiconductor:-

(a) n-type:- σ = eneμe

(b) p-type:- σ = ennμn

Transistor:-

(a) IE = IC+IB       (IB << IE, IB<<IC)

(b) Current gains:-

α = IC/IE, αac = ΔIC/ΔIE

β = IC/IB, βac = ΔIC/ΔIB

(c) Relation between α and β:-

α = β/[1+ β]

or

β = α/[1-α]

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