text:
# Math Help - Strategy and theory differences between two forms of poker

1. ## Strategy and theory differences between two forms of poker

I was reflecting recently on a cruise I took with my family last Spring. They had a casino of course with a variety of games, but I was really only interested in poker. My dad and I signed up for a nine-man, winner take all tournament of Texas hold'em and I was truly baffled with the way the rest of the table was playing. Several older gentlemen were playing every single hand no matter what! So I thought I'd give some tips as well as slightly talk about the theory differences between these styles of the same game.

My first question is what is the difference between a tournament chip and a cash chip when you're sitting at the table?

A cash chip (for a cash game) is simply worth it's value. Nothing more, nothing less.

What about a tournament chip though? Let's say you enter a nine-main tournament (buy-in of $110) where the top three finishers get paid, and the casino starts you off with 1,500 chips. How much money would you say is sitting in front of you? What is the total value of your chips once the game starts? Well the simple answer is$0. Sorry to tell you. Why?

Until you make it to at least 3rd place, you haven't made any money. The 4,5,6,7,8,9th finishes all walk away with nothing. So, until you can claim at least 3rd place in this particular tournament, you don't have any prize money value to attribute to your chips.

-------------------------
Example: (I'm going to assume if you're reading this you understand the game of Texas hold'em, how blinds work, etc)

Say you get dealt AA and are in middle position with $1000 in cash chips. The blinds are$5/$10 and someone in front of you raises it to$55. You re-raise to $200. It folds back around to him and he shoves all-in, you call, and your hand holds up. What was your profit here?$1000 my friend. Nice hand.

Now say you get the same hand and the same action follows except you are now in a tournament just like I described above, with top 3 getting paid. You each have $1000 in tournament chips and after the hand is over, you get his$1000. How much money or equity did you gain?

I said before the simple answer is \$0, but the actual answer is it's not zero. In the poker community, the way to find out your equity in a given tournament is called an ICM calculation.

My next post will be all about ICM and if for some crazy chance this sparks some interest, I'll write a third post about how this affects optimal play at the table.

format: plain


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Prove that every real number x in [0,1] has a decimal expansion.

1. Mar 21, 2013

robertjordan

1. The problem statement, all variables and given/known data

Prove that every real number x in [0,1] has a decimal expansion.

2. Relevant equations

Let $x\in{[0,1]}$, then the decimal expansion for x is an infinite sequence $(k_{i})^{\infty}_{i=1}$ such that for all i, $k_i$ is an integer between 0 and 9 and such that $x\in\left[\frac{k_1}{10}+\frac{k_2}{10^2}+\cdots +\frac{k_n}{10^n},\frac{k_1}{10} +\frac{k_2}{10^2}+ \cdots +\frac{k_{n}+1}{10^n}\right]$.

We call that interval above $I_{k_1,k_2,\ldots,k_n}$

3. The attempt at a solution

Assume BWOC that there exists a real number $t\in[0,1]$ with no decimal expansion. That means there exists an $N\in{\mathbb{Z}}$ such that for all sequences $(k_{i})^{N}_{i=1}$, $t{\notin}\left[\frac{k_1}{10}+\frac{k_2}{10^2}+\cdots +\frac{k_N}{10^N},\frac{k_1}{10}+ \frac{k_2}{10^2}+ \cdots +\frac{k_{N}+1}{10^N}\right]$.

But $I_{0,0,\ldots,0}\cup I_{0,0,\ldots,0,1} \cup I_{0,0,\ldots,0,2} \cup \cdots \cup I_{9,9,9,\ldots,9} = [0,1]$
(That big string of unions is supposed to denote breaking up [0,1] into the union of intervals of size 10-N, but I didn't know how exactly to write it... you get the idea though.)

So $t{\notin}I_{0,0,\ldots,0}\wedge t{\notin}I_{0,0,\ldots,0,1} \wedge t{\notin}I_{0,0,\ldots,0,2} \wedge \cdots \wedge t{\notin}I_{9,9,9,\ldots,9}$ implies $t{\notin}[0,1]$, which is a contradiction.

Last edited by a moderator: Mar 22, 2013
2. Mar 22, 2013

Staff: Mentor

You have to be careful with the negation. "For all x exists a sequence such that for all i [...]" negated is "There is a t without a sequence such that for all i [...]", or "There is a t such that for all sequences, there is an i where [...] is wrong."
The problem here: Those intervals for different i are not independent. If you consider some specific i, it is possible to find an appropriate interval, sure. But that restricts the intervals for other i. The proof is still possible, but you have to take care of this. You could consider the first i where t is not in the interval, for example - if there are i with that property, there is a first one as well.


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# Klein 4 group is the only proper normal subgroup in $A_4$

How do you show that the only nontrivial normal subgroup of $A_4$, which is also not the whole group is the Klein 4 group, denoted by $V$ (or isomorphic to the Klein 4 group)?

I've shown before that $V$ is a normal subgroup of $S_4$, and that $V \subset A_4$. Is there a way to use those facts?

-
If $A$ is normal in $B$ and is contained in $C$ which is a subgroup of $B$, then $A$ is normal in $C$. Can you prove this? – Gerry Myerson Dec 3 '12 at 4:22
Yes, that's pretty clear to me. But how would that force $V$ to be the only normal subgroup that's nontrivial and not the whole group? – chubbycantorset Dec 3 '12 at 4:25
It doesn't. It just shows $V$ is normal in $A_4$. Other methods are needed to show it's the only such. See the answers. – Gerry Myerson Dec 3 '12 at 4:33

1) A subgroup of a group is normal in it iff it is a union of conjugacy classes

2) Two permutations in $\,S_n\,$ are conjugate iff they have the same cycle decomposition (i.e., the same lengths of cycles and the same ammount of cycles of each length)

3) A conjugacy class of an even permutation in $\,S_n\,$ remains exactly the same class in $\,A_n\,$ unless all the disjoint cycles in the cyclic decomposition of the permutation are of odd and different lengths, in which case the equivalence class splits in two classes in $\,A_n\,$

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It's possible that the question has arisen before conjugacy classes have been discussed. – Gerry Myerson Dec 3 '12 at 4:32
Perhaps, but it is my personal experience, both as student and lecturer, that permutation groups usually appear after conjugacy is learnt. – DonAntonio Dec 3 '12 at 4:33
I'm not too familiar with applications of conjugacy classes, but would the Sylow theorems help here? We do know that the order of $A_4$ = 12 = 3*2^2, and the order of $V$ = 2^2. – chubbycantorset Dec 3 '12 at 4:35
Well, yes: if you do what Gerry suggested and write down all the subgroups, then you'll realize $\,V\,$ is the only Sylow 2-subgroup of $\,A_4\,$. Checking the other subgroups will be easy to realize that is the only non-trivial one. One more hint, which follows from this of course: $\,A_4\,$ has no subgroup of order $\,6\,$ – DonAntonio Dec 3 '12 at 4:42
Thanks, really helped to know that $A_4$ doesn't have a subgroup of order 6. – chubbycantorset Dec 3 '12 at 4:50

First, you find all the subgroups of $A_4$. It's not that hard --- there aren't all that many of them. Then you look at each in turn and work out whether or not it is normal. If you have any trouble along the way, write back.

-
I actually haven't been taught how to find all subgroups of a given group. Is there a method for doing it, or do you just have to brute force it? – chubbycantorset Dec 3 '12 at 4:37
In this case it is worth to brutal force it. – Bombyx mori Dec 3 '12 at 4:42
Well, it's an educated brute force. Show that if a subgroup contains two $3$-cycles $a$ and $b$ with $a\ne b^2$ (for example, $a=(1\ 2\ 3),b=(1\ 2\ 4)$) then it's the whole group; show that if a subgroup contains a $3$-cycle and a product of transpositions (for example, $a=(1\ 2\ 3),b=(1\ 2)(3\ 4)$) then it's the whole group; since every group element is of the type $(1\ 2\ 3)$ or $(1\ 2)(3\ 4)$ (or is the identity), that doesn't leave much else to try, as far as subgroups go. – Gerry Myerson Dec 3 '12 at 9:50


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# Happy GeoHacking

I just got done teaching a short workshop on GIS in Python. Something may be odd to those of you who use FOSS packages to do GIS: I didn’t use any GeoPandas.

This wasn’t a constraint I wanted. But, just getting pure pip-installable packages like Shapely and PySAL was difficult enough. The admins didn’t want to install one of the big scientific Python distributions and would only let something with minimal impact get used.

So, in composing the workshop, I focused on Pandas, PySAL, and Shapely. In it, I had the students build an analogue to the GeoPandas dataframe by constructing dataframes with series containing a Shapely object. This was remarkably easy, but helped the attendees understand how to do spatial operations on Shapely objects.

Overall, though, I was struck with how easy it was to replicate the core class features of GeoPandas, notably without the use of a heavy IO library like Fiona. Every few months, the PySAL team tends to discuss whether or not to migrate to using Fiona on the backend, instead of our currently existing, pure Python solutions. I’ve been working on fixing up these solutions to get ready for a release of PySAL in Python 3, so moving to something like Fiona looked like it’d save a ton of work. But, the costs in speed and install difficulty for adding a dependency on OGR has never been a popular option.

So, just now, I pushed up a pseudo-fork of GeoPandas that incorporates the main functionalities of GeoPandas while trying to strip any reference or call to Fiona or OGR. The idea here would be that this fits in any space PySAL fits in, and works as an alternative frontend to PySAL’s current heavy leverage of raw numpy matrices and vectors.

But, having just recently re-read the ESR column How to be a Hacker, I wondered how productive this conversion/strip job would actually make people. In the essay, Eric Raymond recommends that no problem should be solved twice.

And, in this case, I think I’ve certainly re-solved the problem of reading in spatial data into a GeoDataframe. But, notably, I’ve done it using a backend with much less generality than the other solutions!

So, why did I decide to do this? I can certainly handle an OGR install. Anyone who wants to use PySAL on a GeoPandas-like dataframe should themselves be able to string together simple calls to numpy.array to cast their dataframes down to vectors that PySAL can use. In fact, I’m still working on solving a core issue of GeoPandas/PySAL interoperability: construction of a spatial weights object inline with the GeoDataframe.

I guess I’m happy with the fact that this was bourne out of a realization had during that workshop. And, this was a great exercise in (yet more) io hacking with PySAL. Where some might not be able to handle an OGR install and thus not be able to grab Fiona or GeoPandas, this geodf contrib module could work. But, I think, in the future, I’ll be focused more on linking GeoDataframe instances to PySAL analytics methods (using Patsy & a fast shapely->PySAL weights constructor) rather than simply replicating an io framework that Fiona makes more general and more easy to use.

In full, swapping PySAL.core.IOHandlers in as the main engine driving IO has convinced me that an internal GeoJSON representation fully abstracted from whether the underlying data is geojson, shapefile, coverage, PostGIS, or whatever, as is provided by Fiona, is still probably the right way to do geospatial FileIO.

imported from: yetanothergeographer


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# Closed Extension Topology is not Hausdorff

## Theorem

Let $T = \left({S, \tau}\right)$ be a topological space.

Let $T^*_p = \left({S^*_p, \tau^*_p}\right)$ be the closed extension space of $T$.

Then $T^*_p$ is not a $T_2$ (Hausdorf) space.

## Proof

Aiming for a contradiction, suppose $T^*_p$ is not a $T_2$ (Hausdorf) space.

From $T_2$ Space is $T_1$ Space, $T^*_p$ is a $T_1$ (Fréchet) space.

But this contradicts Closed Extension Topology is not $T_1$

Hence by Proof by Contradiction $T^*_p$ can not be a $T_2$ (Hausdorf) space.

$\blacksquare$


text:
# Creepy, Spooky, Kooky, Ooky – The Ghost Sonata in workshop at the Actor’s Gang

Some art forms and authors are known better through satire than in the actual. Few who’ve seen Bugs Bunny can ever take opera seriously. 1999’s Resa Fantastiskt Mystisk by the Burglars of Hamm lampooned Strindberg and his like in unforgettable fashion. Sam Shepard took it in the shorts from the Future Stars of Hollywood and their Go True West. A problem arises when one unfamiliar with such an artistic legend takes in a production of a well-known work to make up for large gaps in one’s knowledge. By one, I mean me.

Brian Finney, an actor capable of great depth, tries his hand at directing Strindberg’s Ghost Sonata at The Actor’s Gang Ivy Substation space in Culver City. It isn’t until the curtain speech that we’re alerted to this being a workshop production, not reviewed, and asked to tell our friends if we happened to like it.

This is an acceptably uneven experiment best enjoyed by those who’ve seen a lot of Strindberg, who understand why he’s a big deal, and who are interested in theatrical experiments. Those people need to move quickly, it closes in a week. Finney and his large, uneven cast apply the Gang’s well-known Style and a lot of effects including a fully lipsynched first scene to a stark and rather depressing work. It is hard for the uninitiated to understand why this play is so highly regarded. There are several flashes of promise when a pairs and trios of characters get in gear and the production begins to take wing. And then it abruptly porpoises and crashes when the broadly-played commedia yanks the air out from under it.

It is the rule in technology that prototypes and breadboards need a lot of extra components and structure to provide flexibility and diagnostics during development. Further, that these are progressively and ruthlessly discarded as an idea becomes a product. It will be interesting to see this play again after the team considers it fully ready – to understand what they considered vital and what were detours. I’m not confident I’ll like the answers, though.

The Ghost Sonata
Wed-Sat 8pm until 26 April 2014
The Actor’s Gang at the Ivy Substation
9070 Venice Blvd. Culver City, CA 90232
$34.99/$30 Seniors and students
Purchase tickets online


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# Author highlighting with biblatex-apa

I'm trying to implement the author highlighting provided here:

Make one author's name bold every time it shows up in the bibliography

The solution works fine until i activate the apa style for biblatex. Heres my code:

\documentclass{article}
\usepackage[english,ngerman]{babel}
\usepackage[utf8]{inputenc}

\usepackage[
babel,
german=quotes
]{csquotes}

\usepackage[
backend=biber,
%style=apa, %<===========uncomment me!!
sortcites=true,
sorting=nyt,
hyperref=true,
backref=true,
alldates=iso8601
]{biblatex}

\begin{filecontents}{\jobname.bib}
@article{zongker_chicken_2006,
title = {Chicken chicken chicken: Chicken chicken},
volume = {12},
url = {http://www.superfrink.net/athenaeum/dougz.pdf},
shorttitle = {Chicken chicken chicken},
pages = {16--21},
number = {5},
journaltitle = {Annals of Improbable Research},
author = {Zongker, Doug},
urldate = {2013-04-05},
date = {2006}
}

@article{zongker_chicken_2007,
title = {Chicken chicken chicken: Chicken chicken},
volume = {12},
url = {http://www.superfrink.net/athenaeum/dougz.pdf},
shorttitle = {Chicken chicken chicken},
pages = {16--21},
number = {5},
journaltitle = {Annals of Improbable Research},
author = {OhterZongker, Doug},
urldate = {2013-04-05},
date = {2007}
}
\end{filecontents}
\DeclareLanguageMapping{ngerman}{ngerman-apa}

% https://tex.stackexchange.com/a/211821/48182
\newcommand{\makeauthorbold}[1]{%
\DeclareNameFormat{author}{%
\edef\tempname{{#1}}%
\ifnumequal{\value{listcount}}{1}
{\ifnumequal{\value{liststop}}{1}
{\ifnumless{\value{listcount}}{\value{liststop}}
}%
}%
}
\makeauthorbold{Zongker}

\begin{document}
\fullcite{zongker_chicken_2006}

\fullcite{zongker_chicken_2007}
\end{document}

Heres the different output without style=apa

with style=apa

Why doesn't my command overwrite the apa style ?

This post is now updated to work with biblatex >= 3.3 and biblatex-apa >= 6.8. The only change necessary is that the macro that was called name:apa:last-first in versions < 3.3 is now called name:apa:family-given. The patching has to be adapted to that name change. The old version can be found in the history of this answer.

The code you used does not work because biblatex-apa does not use the author format, but uses its own apaauthor.

I suggest you take Audrey's approach to Make specific author bold using biblatex with hashes as explained in Highlight an author in bibliography using biblatex allowing bibliography style to format it

The only thing we need to consider here is that the macro you need to patch is now called name:apa:family-given.

MWE (of course the hash 086a2ad072f9903168e35b73cd394263 is specific to the MWE, read about how to find out the hash in the answer linked to above)

\documentclass{article}
\usepackage[english,ngerman]{babel}
\usepackage[utf8]{inputenc}
\usepackage{xpatch}
\usepackage[
babel,
german=quotes
]{csquotes}

\usepackage[
backend=biber,
style=apa,
sorting=nyt,
backref=true,
alldates=iso
]{biblatex}

\newcommand*{\boldnames}{}

\newcommand*{\doboldhashes}[1]{%
\iffieldequalstr{hash}{#1}
{\bfseries\listbreak}
{}}%

\newbibmacro*{name:bold}{%
\forlistloop{\doboldhashes}{\boldnames}%
}

\xpretobibmacro{name:apa:family-given}{\begingroup\usebibmacro{name:bold}}{}{}
\xpretobibmacro{name:delim}{\begingroup\normalfont}{}{}

\xapptobibmacro{name:apa:family-given}{\endgroup}{}{}
\xapptobibmacro{name:delim}{\endgroup}{}{}

\renewcommand*{\boldnames}{}

\usepackage{filecontents}
\begin{filecontents}{\jobname.bib}
@article{zongker_chicken_2006,
title = {Chicken chicken chicken: Chicken chicken},
volume = {12},
url = {http://www.superfrink.net/athenaeum/dougz.pdf},
shorttitle = {Chicken chicken chicken},
pages = {16--21},
number = {5},
journaltitle = {Annals of Improbable Research},
author = {Zongker, Doug},
urldate = {2013-04-05},
date = {2006}
}

@article{zongker_chicken_2007,
title = {Chicken chicken chicken: Chicken chicken},
volume = {12},
url = {http://www.superfrink.net/athenaeum/dougz.pdf},
shorttitle = {Chicken chicken chicken},
pages = {16--21},
number = {5},
journaltitle = {Annals of Improbable Research},
author = {OhterZongker, Doug},
urldate = {2013-04-05},
date = {2007}
}
\end{filecontents}


text:
Logical Implication

• May 23rd 2010, 04:38 AM
skamoni
Logical Implication
Let $\theta$ be $\forall{x}\forall{y}\exists{z}((R(x,z)\wedge {R(y,z)})\wedge \forall{w}((R(x,w)\wedge {R(y,w)}) \rightarrow R(z,w)))$ and let $\phi$ be $\forall{x}\forall{y}\exists{z}((R(z,x)\wedge {R(z,y)})\wedge \forall{w}((R(w,x)\wedge {R(w,y)}) \rightarrow R(w,z)))$

Where R is a binary predicate symbol.

I have been asked to find an L structure consisting of 3 elements such that $\theta \not \models \phi$

My Professor hasn't really given us a method to do this, any help would be appreciated.
• May 25th 2010, 01:33 AM
clic-clac
Hi

If I understand, you want a 3 elements structure where $\theta$ is true but $\phi$ isn't. In these sentences, precising a binary relation symbol properties, you can try to see $R$ as an order relation (perhaps the most common binary predicates are order or equivalence relations).
In such context, $\theta$ says that for any $x,y$, there is a $\sup(x,y)$ (for $R$), and $\phi$ that for any $x,y$, there is a $\inf(x,y)$.

Then I guess you can quite easily define a partial order on three elements such that any two of them have a sup, but there are two with no inf (and in fact more: with no lower bound).


text:
Home > Windows 10 > Windows 7 Error Reporting Dialog

# Windows 7 Error Reporting Dialog

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Notify me if there are elevated command prompt. error an answer now requires 10 reputation on this site (the association bonus does not count).Disabling Error Reporting allows the experienced user to save have a different version of Windows 7 (mine is Windows 7 64-bit Home Premium!

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working, and any operations that triggered the failure is sent to Microsoft.


text:
## Thomas' Calculus 13th Edition

Published by Pearson

# Chapter 3: Derivatives - Section 3.6 - The Chain Rule - Exercises 3.6 - Page 149: 53

#### Answer

$\frac{8 \sin 2t}{(1+cos2t)^5}$

#### Work Step by Step

y=$(1+cos2t)^{-4}$ Apply the chain rule: $\frac{dy}{dt}$$=-4(1+cos2t)^{-5}.\frac{d}{dt}(1+cos2t)$ =$-4(1+cos2t)^{-5}(-sin2t).\frac{d}{dt}(2t)$ =$\frac{8 \sin 2t}{(1+cos2t)^5}$

After you claim an answer you’ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide feedback.


text:
# Historical interpretation of symmetries required by Principle of Relativity

Einstein's 1905 paper discussed electromagnetism and the laws of mechanics. But did not discuss gravity, and of course did not discuss the strong or weak nuclear forces.

• Principle of Relativity - (from translation of Einstein's 1905 paper) "the same laws of electrodynamics and optics will be valid for all frames of reference for which the equations of mechanics hold good"

1) At what point did physicists interpret this to mean all physical laws should have these symmetries?

It seems like physicists were trying to get a relativistic version of gravity fairly soon after this, so maybe immediately?

But I'm not sure when the symmetry point of view took hold. For example, would physicists back then interpret conservation of energy and momentum as a consequence of special relativity (translation invariance), or would they still consider that a consequence of the individual physical laws? As a particular example, consider the initial confusion around decays involving neutrinos. When it appeared to be a possible violation of energy conservation, did they also view this as a potential violation of relativity? Or did they not think of it in that context?

2) Since the "equations of mechanics hold good" under parity transformations, did physicists historically believe special relativity predicted parity symmetry?

For example, if parity violation was discovered in 1906, would this have been considered a blow to special relativity?

• At what point did physicists interpret this to mean all physical laws should have these symmetries? In 1905, mechanics plus optics constituted all physical laws. By extending relativity from mechanics to optics, Einstein was proposing exactly that all physical laws (as known in 1905) should have the corresponding symmetries. It is also possible to think of the symmetry we're talking about as a symmetry of spacetime itself, and this was done by Ignatowsky in 1911, en.wikisource.org/wiki/… . – Ben Crowell Jan 23 '16 at 20:47
• @BenCrowell there was also gravity at the time, and around this time was the initial steps towards quantum theories, and yes later they learned about more fundamental forces. Did physicists immediately take special relativity to require any new physical theories to have these symmetries? Or did this understanding develop over time? – BuddyJohn Jan 24 '16 at 0:50
• @BuddyJohn such a change in outlook can't have been instantaneous. – vonbrand Feb 4 '16 at 23:35


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# isosceles trapezoid area calculator

Trapezoid Area Definition The Trapezoid Area Calculator will calculate the area of a trapezoid if you enter in the height, the length of the top, and the length of the top of the trapezoid. For the circle inscribed into an isosceles trapezoid, the following is TRUE: The area of an isosceles trapezoid in the case of a circle being inscribed in it and if you know middle line. You are allowed to use our calculators in any project as long as you give attribution. Geometry - Calculate Trapezoid Area, through diagonals and angle between them. Similarly, as γ + δ = 180°, δ = 180° - 125° = 55°. Chocolate roll The cube of 5 cm chocolate roll weighs 30 g. How many calories will contain the same chocolate roller of a prism shape with a length of 0.5 m whose cross section is an isosceles trapezoid with bases 25 and 13 cm and legs 10 cm. To see how to calculate trapezoid area without using formulas, click here. ¡Gracias! Also, diagonals AC and BD are perpendicular. Step #2: Enter the value of long base "b". ,      -  bases of an isosceles trapezoid, ,       -  angles of an isosceles trapezoid, - height  measured at right angles to the base, =                Digit Its bases are 10 cm and 15 cm respectively. This is an easy to use area of a trapezoid calculator online. $$a = "Bigger base"$$ 2 Similarly, as γ + δ = 180°, δ = 180° - 125° = 55°. A trapezoid, also known as a trapezium, is a 4-sided shape with two parallel bases that are different lengths. You can also select the units (if any) … It supports measurements in inches, feet, … Calculate the area of an isosceles trapezoid whose bases are in the ratio of 4:3; leg b = 13 cm and height = 12 cm. This calculator calculates any isosceles triangle specified by two of its properties. Formula Knowing the Bases and the Height If you know the length of the bases and the height, you can calculate its area using the following formula: It is not a problem to calculate an isosceles triangle, for example, from its area and perimeter Let's assume that you want to calculate the area of a certain trapezoid. (Algebra, A. Baldor) The formula for the area of a trapezoid is A = ½(b 1 +b 2)h, where b 1 and b 2 are the lengths of the bases and h is the height. By using this website, you agree to our Cookie Policy. Also, the area can be calculated in a plane of two dimensions. How to calculate interior angles of isosceles trapezoids. The angles, the height h, the area and the diagonals of a trapezoid are calculated given its 4 sides. b = √ h 2 + a 2 4 θ = t a n − 1 ( 2 h a ) S = 1 2 a h b = h 2 + a 2 4 θ = t a n − 1 ( 2 h a ) S = 1 2 a h select elements Isosceles triangle Calculate the perimeter of isosceles triangle with arm length 73 cm and base length of 48 cm. Area Of A Trapezoid,How to Calculate the Area of a Trapezoid – dummies|2020-05-10. All formulas for radius of a circumscribed circle. The trapezoid is equivalent to the British definition of the trapezium. Python Math: Calculate the area of a trapezoid Last update on February 26 2020 08:09:18 (UTC/GMT +8 hours) Python Math: Exercise-3 with Solution. Trapezoidal Prism Volume Calculator. By using this website, you agree to our Cookie Policy. The area of an isosceles trapezoid can be found in another way, if known angle at the base and the radius of the inscribed circle. Formulas of angles, height and area have been found in Solve Trapezoid Given its Bases and Legs. Calculates the other elements of an isosceles triangle from the selected elements. All rights reserved. Now to use the area formula: trapezoid area = ((sum of the bases) ÷ 2) • height trapezoid area = ((55 + 30) ÷ 2) • 12 trapezoid area = 510 cm². If you know the length of the bases and the height, you can calculate its area using the following formula: Copia y pega el siguiente código en el HTML de tu página web para mostrar ahí esta fórmula y su calculadora. select elements \) Customer Voice. The isosceles trapezoid in its bigger base has equal angles, and in its smaller base has also equal angles. Customer Voice. By using this website, you agree to our Cookie Policy. It is the space of the internal surface in a figure, it is limited for the perimeter. Calculate the remaining internal angles. Note: A trapezoid is a quadrilateral with two sides parallel. Lines AB and DC are the non-parallel sides and are called legs. $$b = "Smaller base"$$. Questionnaire. This is a trapezoid with two opposite legs of equal length. The area of an isosceles trapezoid in the case of a circle being inscribed in it. Calculate the remaining internal angles. Area of Triangle when semiperimeter is given Area Of Triangle=sqrt (Semiperimeter Of Triangle * (Semiperimeter Of Triangle -Side A)* (Semiperimeter Of Triangle -Side B)* (Semiperimeter Of Triangle -Side C)) GO Radius of the circle circumscribed about an isosceles trapezoid - center of the inscribed circle. An isosceles triangle is a triangle that has two sides of equal length. Find the area of ABCD. Use the isosceles trapezoid calculator to calculate the legs of an isosceles trapezoid for your geometry problems. Al copiarlas a tu sitio acuerdas dar attribución a "© WikiFormulas.com". BYJU’S online isosceles trapezoid calculator tool performs the calculation faster, and it displays the result in a fraction of seconds. FAQ. Perimeter of a isosceles trapezoid & Calculator. 6 Questionnaire. Ex 1,300.34 enter: 1300.34. Using the area of trapezoid calculator: an example. Legs of an Isosceles Trapezoid Calculator Leg of a Trapezoid - Either of the two sides which connect the bases of a trapezoid. Isosceles trapezoid Find the area of an isosceles trapezoid, if the bases are 12 cm and 20 cm, the length of the arm is 16 cm; Isosceles trapezoid The lengths of the bases of the isosceles trapezoid are in the ratio 5:3, the arms have a length of 5 cm and height = 4.8 cm. Isosceles triangle Calculate the area of an isosceles triangle, the base of which measures 16 cm and the arms 10 cm. Formulas  for calculating Area of a trapezoid if given radius, The area of a isosceles trapezoid if you know diagonals and angle between them, Formula for calculating Area of a isosceles trapezoid if given diagonals and angle between them (, The area of an isosceles trapezoid if you know lateral side, middle line and angle, Formula for calculating Area of a trapezoid if given lateral side, middle line and angle (, The area of an isosceles trapezoid if you know bases and height, Formula for calculating Area of a trapezoid if given bases and height (. 4 The results will be shown after a click on Calculate. In the adjacent diagram, if we write AD = a, and BC = b, and the height h is the length of a line segment between AD and BC that is perpendicular to … All the data given is: α = 30° γ = 125° h = 6 cm. Let's assume that you want to calculate the area of a certain trapezoid. The length of the side of smaller triangle is 2.4 cm. , - angles of an isosceles trapezoid. Solution: Given: a = 10 cm b = 15 cm c = 8 cm Angles are calculated … Calculate its perimeter. The figure below to the left shows a trapezoid, and an isosceles trapezoid on the right. side a: parallel a,b; side b: height h : area S . Follow below steps to get trapezoid online using our online trapezoid area calculator. P = 25 cm. The perimeter of a isosceles trapezoid can be calculated using one of the following formula: if you know the length of the bigger base, the smaller base and one of the sides, could be left or right side. The area of a trapezoid across the diagonals and the angle between them is considered the conditional division of the trapezoid into four triangles, just like the area of any arbitrary quadrangle. Free Trapezoid calculator - Calculate area, perimeter, diagonals, sides and angles for trapezoids step-by-step This website uses cookies to ensure you get the best experience. All formulas for radius of a circle inscribed, All basic formulas of trigonometric identities, Height, Bisector and Median of an isosceles triangle, Height, Bisector and Median of an equilateral triangle, Angles between diagonals of a parallelogram, Height of a parallelogram and the angle of intersection of heights, The sum of the squared diagonals of a parallelogram, The length and the properties of a bisector of a parallelogram, Lateral sides and height of a right trapezoid, The area of an isosceles trapezoid if you know, Formula for calculating Area of a trapezoid if given all sides (, Formulas for calculating Area of a trapezoid if given sides and angle (, The area of an isosceles trapezoid in the case of a circle being inscribed in it, Formulas  for calculating Area of a trapezoid if given radius, diameter or bases (. Our calculator provides the calculation of all parameters of the isosceles triangle if you enter two of its parameters, e.g. The area of a trapezoid across the diagonals and the angle between them is considered the conditional division of the trapezoid into four triangles, just like the area of any arbitrary quadrangle. Diagonal of an isosceles trapezoid is the line segment joining two non-adjacent vertices of the trapezoid is calculated using Diagonal=sqrt(Side A*Side B+Side C^2).To calculate Diagonal of an isosceles trapezoid , you need Side A (a), Side B (b) and Side C (c).With our tool, you need to enter the respective value for Side A, Side B and Side C and hit the calculate button. How To Find The Area Of A Trapezoid (Formula & Video ... Recall that the median (m) of a trapezoidis the line segment linking the midpoints of the non-parallel sides.The following trapezoid TRAP looks like an isosceles trapezoid, doesn’t it? When describing formulas in plural, it is also valid to say "formulae". Calculations at an isosceles trapezoid (or isosceles trapezium). The base angles of an isosceles trapezoid are congruent. All the data given is: α = 30° γ = 125° h = 6 cm. WikiFormulas.com is a database of embeddable formulas, equations and calculators. ABCD trapezoid, bases AD = 4 and BC = 12. Isosceles Trapezoid Base Calculator - Calculate base of an isosceles trapezoid step-by-step This website uses cookies to ensure you get the best experience. Nuestras calculadoras son gratuitas y compatibles con móviles. Calculate the area of any trapezoid if you know the length of its two bases and its height. Area of a Isosceles Trapezoid Formulas & Calculator The area of a isosceles trapezoid can be calculated using the following formula: if you know the length of the bigger base, the length of the smaller base and the height. Geometry. Area of a trapezoid Calculator . $$h = "hight value"$$ Use the point as decimal separator. Formula for calculating Area of a trapezoid if given 1. all sides 2. sides and angle 3. radius of the inscribed circle 4. diagonals and angle between them 5. bases Find the area of an isosceles trapezoid - Calculator Online FAQ. In geometry, a triangular prism is a three-sided prism; it is a polyhedron made of a triangular base, a translated copy, and 3 faces joining corresponding sides. In British English the trapezoid is called the trapezium. Vincular de regreso es opcional pero bienvenido. As α + β = 180°, β = 180° - 30 ° = 150°. Similarity coefficient The ratio of similarity of two equilateral triangles is 3.5 (ie 7:2). If the rate of the sides an isosceles triangle is 7:6:7, find the base angle correct to the nearest degree. ... Trapezoid area calculator is simple and easy to use. Calculate the lateral side of trapezoid if given area of a trapezoid, angle at the base and bases or midline ( ) : lateral side of a right trapezoid : = Digit 1 2 4 6 10 F This is an easy to use area of a trapezoid calculator online. This feature is not available right now. - radius of the inscribed circle. 10 a = 4 cm. Isosceles Trapezoid Calculator is a free online tool that displays the area of an isosceles trapezoid. Trapezoid MO-5-Z8 F, Area of a triangle - "side angle side" (SAS) method, Area of a triangle - "side and two angles" (AAS or ASA) method, Surface area of a regular truncated pyramid, All formulas for perimeter of geometric figures, All formulas for volume of geometric solids. Calculate the area of any trapezoid if you know the length of its two bases and its height. Isosceles triangle Calculator . Isosceles Trapezoid Calculator. A right triangular prism has rectangular sides, otherwise it is oblique. Lines AC (or q) and BD (or p) are called diagonals The line perpendicular to lines AD & BC is called the height or altitude. ✉ wikiformulas@gmail.com 1 The area of a isosceles trapezoid can be calculated using the following formula: if you know the length of the bigger base, the length of the smaller base and the height. EN: isosceles-trapezoid-perimeter-area-calculator description This website uses cookies to ensure you get the best experience. In Isosceles trapezium, the two parallel lines or sides are called as the bases and the two non parallel sides (other than the bases) are called as the legs. Find the area of an isosceles trapezoid if the lengths of its bases are 16 cm and 30 cm, and the diagonals are perpendicular to each other. Formula Knowing the Bases and one Side. A trapezoid is a geometric figure with 4 sides that has 2 parallel sides. Calculate the circumference and area of a trapezoid. * * * * * * * * * Trapezoids * * * * * * * * * ALL TRAPEZOIDS have the following properties: Home / Mathematics / Area; Calculates the area of a trapezoid given two parallel sides and the height. Enter the three side lengths, choose the number of decimal places and click Calculate. Calculate the perimeter and area of the larger triangle. Using the area of trapezoid calculator: an example. To calculate Diagonal of an isosceles trapezoid , you need Side A (a), Side B (b) and Side C (c). P = 25 cm. a, b and c are the sides of the trapezoid. OwlCalculator.com. Write a Python program to calculate the area of a trapezoid. © 2017 WikiFormulas.com Home / Mathematics / Triangle; Calculates the other elements of an isosceles triangle from the selected elements. To calculate the elements of the iso trapezoid; enter the two bases, a side or height. a = 4 cm. The area of an isosceles (or any) trapezoid is equal to the average of the lengths of the base and top (the parallel sides) times the height. It has 3 fields in which you need to feed value. - diameter of the inscribed circle. As α + β = 180°, β = 180° - 30 ° = 150°. If you only know the side lengths of a regular trapezoid, you can break the trapezoid into simple shapes to find the height and finish your calculation. base b and an arm a. It is a representation of a rule or a general principle using letters. Area of an Isosceles Trapezoid = [ (a+b)h]/2 square units For example, the bases of the isosceles trapezoid are 2 cm, and 5 cm and height is 5cm, then the area is: Area = [ (2+4)5]/2 Area = 30/2 Formulas for calculating Area of a trapezoid if given sides and angle ( A ) : 2. Please try again later. Trapezoid area = ((sum of the bases) ÷ 2) • height Lines BC and AD are parallel and are called bases. It is a geometric figure with 4 sides and 4 vertices, the isosceles trapezoid has 2 parallel sides and will always have diferent lengths betwen them; the other two sides are equal length. By … Solved Examples. With our tool, you need to enter the respective value for Side A, Side B and Side C and hit the calculate button. It supports measurements in inches, feet, … Diagonal of an isosceles trapezoid is the line segment joining two non-adjacent vertices of the trapezoid is calculated using Diagonal=sqrt(Side A*Side B+Side C^2). Question 1: A trapezoid has legs 8 cm each in length. Step #1: Enter the value of short base "a". , - bases of an isosceles trapezoid. Free Trapezoid calculator - Calculate area, perimeter, diagonals, sides and angles for trapezoids step-by-step This website uses cookies to ensure you get the best experience. The Area of isosceles trapezoid formula is $\large Area\;of\;Isosceles\;Trapeziod=h\left [\frac{a+b}{2}\right]$ Where, a and b are the parallel sides of trapezoid. Trapezoid on the right Calculate trapezoid area, through diagonals and angle between them of smaller triangle a., and an isosceles trapezoid calculator tool performs the calculation of all parameters of the sides isosceles... A rule or a general principle using letters places isosceles trapezoid area calculator click Calculate angle between them, you to!: parallel a, b ; side b: height h: area S a trapezoid legs... Perimeter and area of a certain trapezoid otherwise it is the space the... Base length of its two bases and its height steps to get trapezoid using... 2017 WikiFormulas.com all rights reserved its two bases and its height formulas in plural, it is quadrilateral. Arm length 73 cm and 15 cm respectively Either of the two bases, a side height! To Calculate the area of a trapezoid – dummies|2020-05-10 it displays the result in a figure, is. Trapezium ): area S website, you agree to our Cookie Policy also equal angles and BC =.. Two bases, a side or height 1: a trapezoid are congruent of two dimensions dar. Sides, otherwise it is the space of the two sides which connect the bases of a trapezoid, known. Shape with two opposite legs of an isosceles triangle is 2.4 cm base angles of an isosceles with. Equal length if you know the length of its two bases and its height two! Wikiformulas.Com all rights reserved the side of smaller triangle is 7:6:7, find the base angles of isosceles... To the left shows a trapezoid – dummies|2020-05-10 click here displays the isosceles trapezoid area calculator in a fraction seconds... With arm length 73 cm and base length of its parameters, e.g left shows a is. Base of an isosceles triangle specified by two of its parameters, e.g of any trapezoid you. Prism has rectangular sides, otherwise it is oblique that has two of. Bases and legs triangles is 3.5 ( ie 7:2 ) legs 8 cm each in length similarity coefficient ratio. And the height h: area S cm each in length enter two of two! Calculate trapezoid area, through diagonals and angle between them its bigger has! A right triangular prism has rectangular sides, otherwise it is oblique agree to our Policy... A figure, it is oblique and legs its height principle using letters a triangle that 2! Through diagonals and angle between them a fraction of seconds known as a trapezium, a! Has rectangular sides, otherwise it is the space of the sides an isosceles in... As γ + δ = 180° - 30 ° = 150° similarity coefficient the ratio of of... That has 2 parallel sides arm length 73 cm and 15 cm.! Byju ’ S online isosceles trapezoid are calculated given its 4 sides that has parallel... Or height is 3.5 ( ie 7:2 ) a figure, it also. Abcd trapezoid, how to Calculate the perimeter and area of a trapezoid calculator.. Prism has rectangular sides, otherwise it is oblique calculator tool performs calculation. Embeddable formulas, click here, b ; side b: height h, the.. Legs of equal length, find the base angles of an isosceles trapezoid in the case of a trapezoid how... Want to Calculate trapezoid area without using formulas, click here you two! Of decimal places and click Calculate area and the height h: area S copiarlas a sitio. Triangle is a 4-sided shape with two opposite legs of equal length of long base  b.! Using our online trapezoid area without using formulas, equations and calculators and it displays the in. Called the trapezium below steps to get trapezoid online using our online trapezoid area calculator simple! Byju ’ S online isosceles trapezoid for your geometry problems Calculate the area of a trapezoid the an... Best experience A. Baldor ) When describing formulas in plural, it is the space of the trapezium certain.! Trapezium ) rights reserved area S acuerdas dar attribución a  © WikiFormulas.com '' long . A side or height... trapezoid area calculator is simple and easy to use program to Calculate area! = 12, it is limited for the perimeter and area of an isosceles if! Faster isosceles trapezoid area calculator and an isosceles triangle, the area of a certain trapezoid a side or height find the angles. Parallel bases that are different lengths been found in Solve trapezoid given its 4 that. To the nearest degree two bases and its height number of decimal places and click.... In plural, it is a representation of a trapezoid calculator tool performs the faster! Assume that you want to Calculate the area of a trapezoid long base  a '' parameters e.g... The legs of an isosceles triangle from the selected elements feed value for the perimeter © WikiFormulas.com '' which... Online trapezoid area calculator 7:6:7, find the base of which measures 16 cm and base length of its.... Angles, the area of the isosceles trapezoid calculator tool performs the calculation of all parameters of trapezium. Question 1: enter the three side lengths, choose the number decimal... Online isosceles trapezoid in the case of a trapezoid is a 4-sided shape isosceles trapezoid area calculator two which! H: area S click Calculate each in length sides, otherwise it is oblique the rate of side... Get trapezoid online using our online trapezoid area without using formulas, click here the length of the surface... Certain trapezoid trapezoid step-by-step this website, you agree to our Cookie Policy the angles, the angles. To our Cookie Policy dar attribución a  © WikiFormulas.com '' found in Solve given. = 180° - 125° = 55° ( Algebra, A. Baldor ) When describing formulas plural... Bigger base has also equal angles, and an isosceles trapezoid calculator Calculate. Trapezoid ( or isosceles trapezium ) also valid to say  formulae '' shown after a click on Calculate find... Calculated given its 4 sides that has 2 parallel sides and are called legs, AD... Baldor ) When describing formulas in plural, it is the space of the side of smaller triangle is trapezoid...: a trapezoid is equivalent to the British definition of the side of triangle... # 2: enter the value of long base  b '' below to the British definition of the trapezoid... The number of decimal places and click Calculate website, you agree to Cookie! Faster, and it displays the result in a plane of two equilateral is... Uses cookies to ensure you get the best experience has 3 fields in which you need to feed.... Uses cookies to ensure you get the best experience cookies to ensure you get best! Using this website uses cookies to ensure you get the best experience a rule a... A plane of two dimensions figure, it is the space of the trapezoid. The number of decimal places and click Calculate of the internal surface a! The trapezoid is equivalent to the British definition of the isosceles trapezoid on isosceles trapezoid area calculator right known as a trapezium is. A database of embeddable formulas, equations and calculators choose the number of decimal places and Calculate. All parameters of the iso trapezoid ; enter the three side lengths, the! ✉ wikiformulas @ gmail.com © 2017 WikiFormulas.com all rights reserved faster, and in its smaller base also. Are calculated given its bases are 10 cm and base length of its parameters e.g. Of all parameters of the two sides of equal length trapezoid are calculated given bases. In which you need to feed value formulae '' figure below to the British definition of the iso ;! Calculator provides the calculation faster, and in its smaller base has also equal angles, the height:... Result in a figure, it is the space of the iso trapezoid enter! ° = 150° - 30 ° = 150° and BC = 12 enter the value of long base  ''. To say  formulae '' Either of the trapezium h: area S to... English the trapezoid is called the trapezium, a side or height in a figure, it is for! - Calculate base of an isosceles triangle from the selected elements calculators in any project as long as you attribution!, the area and the diagonals of a trapezoid has legs 8 cm each in length legs of isosceles! 2017 WikiFormulas.com all rights reserved calculator is simple and easy to use area a... Abcd trapezoid, and it displays the result in a fraction of seconds trapezoid online. Is 2.4 cm attribución a  © WikiFormulas.com '' isosceles trapezoid area calculator specified by two of its two,... Larger triangle trapezoid ( or isosceles trapezium ) ie 7:2 ) cm each length. A quadrilateral with two parallel sides: enter the two sides of equal length and angle between.! Area without using formulas, equations and calculators 8 cm each in length trapezoid area calculator ; Calculates area! Embeddable formulas, equations and calculators a tu sitio acuerdas dar attribución a ©. Online using our online trapezoid area, through diagonals and angle between them WikiFormulas.com all rights reserved is.... The nearest degree and area have been found in Solve trapezoid given parallel! That are different lengths in its smaller base has also equal angles, the base of measures. Two bases and its height ( or isosceles trapezium ) angles, height and area of the two sides equal... Base  b '' that you want to Calculate trapezoid area calculator online using our online trapezoid without. Also equal angles, height and area of a trapezoid is called the.. Geometry - Calculate trapezoid area calculator, δ = 180°, δ 180°!


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Article Text

Informed consent and participant perceptions of influenza vaccine trials in South Africa
1. K Moodley1,
2. M Pather2,
3. L Myer3
1. 1Bioethics Unit, Centre for Applied Ethics and Department of Family Medicine and Primary Care, Faculty of Health Sciences, University of Stellenbosch, South Africa
2. 2Department of Family Medicine and Primary Care, University of Stellenbosch, South Africa
3. 3School of Public Health and Family Medicine, University of Cape Town, South Africa
1. Correspondence to:
 Prof K Moodley
 Bioethics Unit—Tygerberg Division, PO Box 19063, Tygerberg 7505, South Africa; km{at}sun.ac.za

## Abstract

Background and objectives: There are few insights from sub-Saharan Africa on research participants’ experiences of the informed consent process, particularly in the context of randomised controlled trials, where issues of randomisation and the use of placebos may be confusing concepts for participants. This study investigated the knowledge and perceptions of the informed consent process among individuals participating in influenza vaccine trials in two disadvantaged communities in South Africa.

Method: Four to 12 months after completion of the trials, participants were contacted to return to participate in the informed consent study. The semistructured questionnaire administered to assess recall of trial procedures and the informed consent process covered key issues including: purpose of the study; awareness that the study was not part of routine treatment; voluntary nature of participation and freedom to withdraw; randomisation; placebos; and remuneration.

Results: A total of 334 participants (93% of the original vaccine trial sample; mean age 68 years, median level of education grade 8, 69% women) completed the questionnaire. Only 21% were able to recall that they were allocated randomly to the different treatment arms. Only 19% of those involved in the placebo controlled study had interpreted the concept of placebo as an inactive medication.

Conclusion: Although a good general recall of trial concepts was demonstrated, only a small proportion of the participants correctly interpreted and recalled the concepts of randomisation and placebos. Informed consent in this and similarly disadvantaged communities may often be inadequate and new ways to improve understanding of the research process should be explored.

• informed consent
• randomised control trials
• participant comprehension
• participant recall
• placebos
• random allocation

## Statistics from Altmetric.com

The randomised controlled trial (RCT) is the most widely accepted source of evidence for the safety, tolerability, and effectiveness of medical interventions. The main reason for this is scientific: properly conducted trials produce valid data which advance knowledge of optimal medical care. However in the process, the interests of participants may be unavoidably subjugated for the common good.1,2

## INFORMED CONSENT IN RANDOMISED CONTROLLED TRIALS

Informed consent is fundamental to the ethical conduct of randomised controlled trials and is a critical component of the research process. Defined as “an autonomous authorization by individuals of a medical intervention or of involvement in research”,3 the principle of informed consent is enshrined in all major guidelines for the ethical conduct of biomedical research.4–6 Informed consent is a process, based on verbal and written communication between participants and trial staff (or other individuals recruiting participants). The main pragmatic worry about informed consent is the different ways in which the process can fail—for example, because consent is not sought or because participants may not adequately understand the issues involved.

Written trial materials are a central component of the informed consent process that is required by most major ethical guidelines.4–6 To enhance understanding of informed consent forms and related patient information materials, it is essential that these documents are highly readable. Research in South Africa and other developing countries often uses relatively complex and detailed patient information leaflets written in academic language, sometimes imported from developed countries. This occurs despite concerns regarding the readability of informed consent documents in developed countries like the USA.7 In South Africa, there is the additional challenge of enrolling patients who are often educationally disadvantaged, as most biomedical research involves previously disadvantaged non-white population groups. In this research setting, one or more of the 11 official languages may be spoken, adding to the complexity of translation of consent documents to the consent process. Finding the most accurate words to explain research related concepts, such as placebo and randomisation, in each language is often challenging.

Conducting empirical research on the understanding of the informed consent process can be complex as a result of the confounding effect of recall in conducting a test of understanding. Differentiating between recall and understanding is a significant problem in informed consent research.8 This problem may be compounded by the timing of the test of understanding. Furthermore, there is a relation between severity of disease and retention of trial related information.9

Although the factors that may affect participants’ understanding and recall of informed consent are likely to vary across research settings, there has been surprisingly little research investigating the comprehension and/or recall of informed consent in developing countries. Two South African studies have been published to date. The first study concluded that consent was informed but not always voluntary.10 The second study found that despite the participants having eight years of schooling or more, they had “poor knowledge about the most basic details of the trial”.11 The authors concluded that subjects’ participation could not be seen as informed. Regarding the voluntary nature of the consent, the same study found that although the respondents believed that their participation was voluntary they were clearly aware of the lack of alternative sources of care.11 Data on informed consent in other developing countries have focused mainly on willingness to participate in human immunodeficiency virus (HIV) vaccine trials. A hypothetical informed consent study conducted in Uganda found that trial related procedures like randomisation, placebos, and blinding were unfamiliar.12 An informed consent study conducted in Brazil, also on HIV vaccine trials, found that only 30% of the sample understood the concept of randomisation.13

To address the lack of empirical evidence regarding the assessment of informed consent in South Africa, we investigated the recall of informed consent procedures among participants in two industry sponsored influenza vaccine trials conducted in Cape Town, South Africa. We focused, in particular, on participants’ recall of key aspects of trial design that were a focus of the informed consent process, as well as the determinants of participants’ overall perceptions.

## METHODS

### Study setting

This study is based on two trials of an experimental intranasal influenza vaccine, sponsored by a major pharmaceutical firm, each conducted over a 12 month period during 2001 and 2002. The 2001 trial used an intranasal placebo as its control arm, while the 2002 trial was unblinded and based on comparison with an existing intramuscular (IM) vaccine. Participants were drawn from two impoverished non-white communities outside Cape Town, and were recruited through community meetings and word of mouth. All trial related contacts took place at the provincial day hospitals located in the communities. Interested individuals were eligible for the trials if they were over 60 years of age, agreed to the study procedures, and were gauged mentally competent by a score of 21 or higher on a standard Mini-Mental State Examination (MMSE). This is a short (eight item) quantitative examination that is commonly used to assess cognitive impairment and is widely used both internationally and in South Africa, particularly in geriatric populations.14,15 In the USA, the median MMSE score for normal adults ranges from 29 for individuals with nine or more years of education to 22 for those with four or fewer years of education.16 As compensation for their involvement, R50.00 (approximately US$8.30 or €6.7) was given to trial participants at each of three regularly scheduled study visits, and a further R20.00 (approximately US$3.30 or €2.7) was provided to meet transport costs for unscheduled visits. These amounts are not considered a significant income supplement in this setting.17

### Informed consent procedure

All participants received three separate explanations of the trial procedures prior to agreeing to participate. Community meetings during the recruitment phase included a discussion of the risks and benefits associated with the trial. Several weeks later, groups of interested individuals then discussed their participation with study nurses, which included a step by step verbal explanation of a patient information leaflet. The design of this leaflet was based on the International Conference on Harmonization guidelines for Good Clinical Practice. Participants were given the leaflet in their home language (Afrikaans or English) to take home, to re-read, and to discuss their participation with family members. Finally, on the day of enrolment into the trial, study doctors reviewed the leaflet again with the participants before written informed consent was provided. After this, participants were randomised during their enrolment visit using a scratch card system that allowed the trial assignment to take place in front of participants.

At the time of the study the authors (who were investigators on the influenza vaccine trials) were unaware of the application of readability scores to informed consent documents. We did, however, feel that the forms were written in academic language and that the trial information would be difficult for participants to understand. Hence, the detailed informed consent process outlined above in which enhanced verbal explanations were provided at community, group, and individual levels. Despite this, study personnel felt that comprehension may have been poor after enrolment so the information materials provided to trial participants were subsequently scored for readability using the Flesch readability formula and the Fry readability scale. Analysis of the patient information leaflet revealed a Flesch score of 46, which would fall into the “difficult” category, and this corresponded to a Fry score indicating a grade 12 reading level.18–20 After this assessment, the informed consent study described here was initiated by the clinical investigators at the two vaccine trial sites.

### Assessment of participants’ perceptions of the informed consent process

Four to 12 months after completion of the trial, participants were contacted to return to participate in the informed consent study. A semistructured questionnaire was designed to assess knowledge and perceptions of trial procedures and the informed consent process. The questionnaires covered awareness that the study constituted research and not routine treatment, an understanding of the purpose of the study, voluntary nature of participation and freedom to withdraw, interpretation of randomisation, use of placebos, and participant remuneration. Questions regarding the use of placebos were only administered to participants in the 2001 placebo controlled vaccine trial. Interviews lasted approximately 20 minutes and were conducted in participants’ home language by a study nurse who was not affiliated with the trial site. Participants were assured by study staff that their involvement and responses would not affect their health care in any way.

### Data analysis

We analysed the quantitative data using SAS (version 8.02, Cary, USA). Two simple scoring systems were developed a priori to gauge participants’ overall understanding of the trial from the informed consent procedure. Firstly, a six item informed consent understanding score was constructed for all participants (from both the placebo and IM vaccine trials) using questions regarding informed consent: whether participation was part of standard medical care or a research project; what vaccine was being tested; whether participation was voluntary; whether the decision to participate would affect standard medical care; whether the participant could withdraw voluntarily from the study; and whether an effective ‘flu vaccine was available to all participants as part of routine medical care. With this system participants could receive a maximum score of six if they correctly responded to each question, and a minimum score of zero.

Secondly, a separate placebo understanding score, based on four items, was administered to only participants in the 2001 placebo controlled trial. The items included: whether the participant understood the concept of randomisation by chance; whether the participant understood the concept of “inactive medicine”; whether they correctly stated that “inactive medicine” meant no effective treatment; and whether they understood that their trial assignment was determined by chance (rather than by a study doctor, nurse, or the participant’s own decision). The range of possible scores was from four to zero.

Data were summarised using means, medians, and proportions. Throughout the analysis, the correlation coefficients (r) and Student’s t tests were used to examine the relation between understanding scores and participant characteristics, and χ2 tests were used to compare the proportions of subjects answering each question correctly. Multiple linear regression was used to estimate the effect of participant characteristics on the scores achieved for each scale. All models reported here were adjusted for age, level of education, and MMSE score, each modelled as a continuous variable. The coefficients (β) from these models can be interpreted as adjusted differences in mean scores, with a 1 unit change in the independent variable (that is, age, education, MMSE) corresponding to a β unit change in the mean score on the scale in question. Model fit was assessed using standard diagnostic procedures,21 and all statistical tests were two sided at α = 0.05.

### Ethical approval

Each participant provided written consent, separate from the consent for participation in the vaccine trial, before being interviewed for this study. The University of Stellenbosch Pharmaceutical Trials Ethics Committee provided ethical approval for this study, as well as the two influenza vaccine trials on which this research was based.

## RESULTS

A total of 334 trial participants completed the informed consent questionnaire, 131 from the trial of nasal spray versus placebo and 203 from the trial of nasal spray versus IM vaccine. This sample represented 93% of subjects from the vaccine trials returning for the informed consent study. The mean age of participants was 68 years (range: 60–80) and 69% of the sample were women. The median level of education was grade 8 (range: none to grade 12) and the mean MMSE score was 26.5 (range: 21–30).

### Participants’ perceptions of informed consent

Table 1 shows the participant responses to each of the informed consent questions. A high proportion of participants knew that the trial was not part of their usual treatment, that the vaccine being tested was experimental, and that they could withdraw from the trial at any time. However, most subjects thought that their trial assignment was chosen by a study doctor or nurse, while only a fifth of subjects thought that their assignment was determined by chance only.

Table 1

Participant responses to informed consent related questions (n = 334)

The mean (SD) score on the six items which comprised the informed consent scale was 5.1 (0.8). No participant answered every question incorrectly, and a third of the participants (n = 109) responded correctly to each question. In bivariate analysis, participants’ scores on this scale were positively correlated with increasing level of education (r = 0.15; p<0.01) and MMSE score (r = 0.18; p<0.01), as well as with decreasing age (r = −0.12; p = 0.03). When entered into a multiple linear regression model, MMSE score (β = 0.05; p = 0.05) and age retained slight associations with comprehension of the informed consent process, although the association with educational level did not persist.

### Participants’ perceptions of placebo

The responses of the 131 participants in the placebo controlled trial to the placebo related items are presented in table 2. Less than half of those interviewed could recall the concept of an inactive placebo, and that their trial assignment was determined by chance. Only 19% indicated that a placebo involved receiving no active medicine, and more than a third of the participants (38%) stated that they would not have continued to participate in the trial if they had known that they had received an inactive placebo.

Table 2

Participant responses to placebo trial related questions (n = 131)

The mean score on the four item placebo comprehension scale was 1.2 (1.1). One participant answered all four items correctly, whereas 45 participants (34%) did not answer any of the items correctly. This score did not appear to be associated with participant age, but was strongly correlated with increasing level of education (r = 0.24; p<0.01) and higher MMSE scores (r = 0.28; p<0.01). In multivariate analysis adjusted for educational level and participant age, MMSE score remained correlated with placebo comprehension (β = 0.12; p = 0.03).

### Participant remuneration

Most participants received a total of R150.00 (approximately US$25 or €20) in remuneration over the course of the trial. When asked whether the remuneration they received was adequate, 281 subjects (84%) were satisfied with R50.00 (approximately US$8.30 or €6.7) per visit17 whereas 49 subjects (16%) either stated that the amount was too little or were unsure; no participant stated that the amount was too much. Participants who felt that the amount they received was too little, or were unsure of the amount, had higher recall scores on both the informed consent and placebo scales compared with the participants who felt that the amount they received was enough (mean informed consent recall score 5.4 v 5.0, respectively, p = 0.01; mean placebo recall score of 1.5 v 1.1, respectively, p = 0.14). The association between perceptions of the amount of remuneration and informed consent recall persisted after multivariate adjustment for participant age, education, and MMSE score (mean difference in informed consent score, those who felt the amount received was inadequate or unsure v those who felt the amount was enough, 0.30, p = 0.02).

## DISCUSSION

An assessment of the informed consent process requires the conduct of a test of understanding of the basic elements of consent22 as well as trial related concepts and procedures. However, it is often very difficult to draw a distinction between understanding and recall of trial information.23 An attempt has been made to distinguish a test of recall from a test of understanding in the form of the Deaconess Informed Consent Comprehension Test, but this method remains insufficiently tested to draw conclusions regarding reliability and validity.24 In the absence of a precise method to measure understanding, and given the fact that understanding and recall are interrelated in the cognitive processing of trial related information especially if the assessment is conducted months after the enrolment of study participants, we have focused on recall of trial related information in this study.25

Participants in these influenza vaccine trials demonstrated a good general recall of the informed consent process. However, the present results show low levels of recall surrounding two of the most complex—and arguably the most important—aspects of randomised controlled trial participation: randomisation and the concept of placebo.

### Trial participants’ understanding of randomisation and placebos

Only a small fraction of trial participants recognised that their treatment allocation was determined by chance alone, rather than by study personnel. This is consistent with the findings of Featherstone and Donovan, who reported that most patients interviewed found the concept of randomisation difficult to accept, suggesting that the term “random” has different meanings to lay and professional audiences.26 Similarly, Snowdon et al indicated how difficult it can be for patients to make sense of a treatment decision that is based on chance.27 This is so because patients are traditionally accustomed to treatment decisions being taken in their best interests. The scientific method that incorporates concepts like randomisation and blinding is often incompatible with “personal care”, a concept that requires that the doctor’s first obligation is solely the wellbeing of the patient.28 In fact, one of the consequences of a misunderstanding of the concept of randomisation is a failure to appreciate the distinction between the goals of clinical research and ordinary treatment—the therapeutic misconception.29,30 Under the therapeutic misconception, participants fail to appreciate their chances of not receiving any treatment at all if randomised to a placebo group or of merely receiving experimental, unproved treatment if allocated to the study arm of the trial.

Most participants in the placebo controlled trial did not recall that assignment to the placebo arm would involve receiving no active treatment. Several factors require consideration when interpreting these results. Firstly, with a mean age of 68 years, our study population was substantially older than research subjects in many studies. However, research focusing on geriatric populations is becoming increasingly common, and it is important to understand the perceptions of informed consent in this group.31–33 Secondly, our results are based on a follow up interview that took place up to 12 months after the influenza vaccine trials were completed. This relatively long delay between trial participation and informed consent assessment is a limitation of the study. However, it may mean that levels of comprehension were higher during the trial, although we found no differences in participants’ recall according to the time elapsed between trial participation and undertaking this study. A previous study on informed consent involving research participants 60 years and older had found similar results using a test of comprehension administered soon after the informed consent process.34 Schaeffer et al assessed the impact of disease severity on retention of trial related information. The most relevant finding of their study was that changes in recall and understanding do not always decrease over time.9 Finally, while the recall of informed consent was based on factual knowledge, several of the questions we used in this study involved participants’ subjective judgements regarding trial participation—in turn introducing the possibility of social desirability biases from participants who wish to please trial personnel. We made every effort to minimise this possibility by assuring participants that their responses would not affect their health care in any manner and by using interviewers who were not affiliated with the trial.

It is important to note that these results were observed despite an informed consent process that was relatively intensive for research studies conducted in developing countries. Prior to enrolment into the trial, participants had three separate contacts with study personnel, including two separate verbal explanations of the patient information leaflet. Study site coordinators read through the patient information leaflet one paragraph at a time with groups of five participants each, taking questions and providing lay explanations for each paragraph of the nine page patient information leaflet. These explanations were conducted in the participants’ home language. Investigators at all three sites also provided verbal explanations in layman’s terms of study related procedures. These efforts to help ensure an adequate explanation of the trial for all participants were more intensive than the informed consent process involved in most clinical and epidemiological research in this context, where participants may receive only a single explanation of the research study before being asked to participate. In this light, it is possible that in most research settings in South Africa and similar settings, participants’ comprehension of the informed consent process may be substantially lower than these results suggest. On the other hand, the difficult readability level of the patient information leaflet could have resulted in the lower scores on recall of trial related information and this could represent a limitation of this study.

### Future directions

Given the lack of empirical research to date investigating informed consent comprehension, coupled with the diversity of research settings across the developing world, further research on this topic is clearly needed. Simple evaluations of informed consent comprehension can be critical in designing more effective participant education materials, particularly in instances where relatively complex concepts such as randomisation and placebos are involved. For example, Rikkert et al describe a process of “experienced consent” in a group of elderly research participants who were subjected to a mock trial before being asked to provide consent for participation in the actual trial.31 Fitzgerald et al in their study on comprehension of informed consent in Haiti advise that formal assessment of a research participant’s comprehension of the consent form should be a routine step in the informed consent process in developing countries.35

### Summary

This study suggests that participants’ recall of informed consent in randomised controlled trials in South Africa and other developing countries may often be inadequate. Difficult and lengthy patient information leaflets impact negatively on understanding of trial related information. Hence, interventions to improve understanding of the research process, especially the concepts of randomisation and placebos, require greater attention. Given the limitations of this study in assessing recall rather than understanding, future research involving a test of understanding immediately after trial related information is presented to potential participants is indicated. Although this study is constrained by its ability to assess recall rather than understanding, it represents an important milestone in empirical research into the consent process in South Africa and has identified gaps in information retained by research participants.

## Acknowledgments

We would like to acknowledge I Joseph and D Gallant who assisted in data collection at the sites. Professor PJT de Villiers, Principal Investigator of the influenza vaccine study and head of the Department of Family Medicine, University of Stellenbosch, provided valuable support during this study.

View Abstract

## Footnotes

• This study was funded by a research grant from the Department of Family Medicine and Primary Care, University of Stellenbosch.

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# Probability density function of a log-normal distribution only knowing the geometric mean and geometric standard deviation

I want to plot the curve of a log-normal distribution of which I only know that the mean is $10^{-7}$ and the geometric standard deviation is $1.2$.

The probability density function of a log-normal distribution is given by:

$$f(x) = \frac{1}{x}\cdot\frac{1}{\sigma\sqrt{2\pi}}\exp\left(-\frac{(\ln x - \mu)^2}{2\sigma^2}\right)$$

However here $\sigma$ is the (arithmetic) standard deviation and $\mu$ is the arithmetic mean. The geometric mean and standard deviation are defined as

$$GM = e^{\mu_l}\quad\mathrm{where}\quad\mu_l = \frac{\sum_{i=1}^N\ln(x_i)}{N}$$

and

$$GSD = e^{\sigma_l}\quad\mathrm{where}\quad\sigma_l=\sqrt{\frac{\sum_{i=1}^N\left[\ln(x_i)-\mu_l\right]^2}{N}}$$

How can I calculate the probability density function $f(x)$ if I only know the geometric mean and standard deviation?

With the help of the answer to this question I was also able to calculate the probability density function with GM and GSD. Actually the answer is more or less in the question, but what I didn't know is that

$$\mu_l = \mu$$

and

$$\sigma_l = \sigma.$$

With that knowledge $\mu$ and $\sigma$ can easily be calculated by:

$$\mu = \ln(GM)$$

and

$$\sigma = \ln(GSD)$$

and then inserted into the probability density function, leading for e.g. $GM = 1$ and $GSD \approx 1.284$ to


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# Local-Global Principle in Graph Spectrum

The question is a bit vague, but any ideas/directions will be appreciated.

Let us fix an $n$-vertex $d$-regular graph $G=(V,E)$. As I understand it, the eigenvalues of the adjacency matrix $A$ of $G$ capture "global" information about the graph, akin to a fourier transform. For instance, the number of cycles in $G$ of length $k$ is $Tr(A^k)$ which is the $k$-th power sum of the eigenvalues.

But even this uses all the eigenvalues. In particular, take some eigenvalue $\lambda$ and its eigenvector $\vec{f}$. Does $\lambda^k$ alone have any natural correspondence with some measure on the cycles of length $k$?

Secondly, since the spectrum captures global properties of the graph, like expansion, it seems as if slightly changing the graph locally can change the spectrum and eigenvectors significantly in a global way. This can be checked empirically, though I am unclear on how to formalize this.

Does the reverse hold in any meaningful way? That is, suppose I change the graph "globally", for instance, I change every vertex/edge the same way. Would that change the spectrum and eigenvalues "locally"?

The example that illustrates my vague question is the result of Lubetzky-Peres, which says that going from the adjacency matrix $A$ to the Hashimoto matrix (of the line-digraph of $G$) changes the eigenvalues and eigenvectors locally. Each eigenvalue $\lambda$ of $A$ splits into two eigenvalues $\mu,\overline{\mu}$ which are roots of $$x^2-\lambda x+ (d-1)=0$$ and the eigenvector corresponding to $\mu$ at a "vertex" $(v,w)$ is $\mu f(w)-f(v)$. So modifying the graph "globally" as in the case of the line-digraph construction (where we add two vertices for each direction of each edge of $G$) changes the eigenvalues and eigenvectors only locally.

Is there an underlying principle here that governs the degree of locality/globality of the spectrum and eigenvectors? What are the precise class of operations on a graph that change the spectrum locally?

The procedure of going from the a graph (adjacency matrix), $X$, to the directed graph of directed of directed edges (Hashimoto operator), call it $H(X)$, is not exactly what I would call a "global change" but rather a "functorial change". Indeed, $X\mapsto H(X)$ can be extended to a functor from the category of graphs with covering maps as morphisms, call it $\mathcal{G}$, to the category of directed graphs.

Such functorial constructions are abundant. For example, let $P_k(X)$ denote the graph with the same set of vertices as $X$ and with one edge from $x$ to $y$ for every non-backtracking path $x\to y$ of length $k$. Then $X\mapsto P_k(X)$ is functorial in the sense that $P_k$ extends naturally to a functor $\mathcal{G}\to \mathcal{G}$.

For some functorial constructions $X\mapsto F(X)$ there are indeed connections between the spectrum of $X$ and that of $F(X)$, e.g. $H$ and $P_k$. I do not know of a general procedure to predict or compute them. However, such connections often have representation-theoretic interpretations, which is not to say that they cannot be proved in elementary means.

Elaborating on the representation theoretic point of view, let $T$ denote the $k$-regular tree, let $G$ denote its automorphism group (with its point-wise convergence topology), let $K$ be the stablizer of some vertex $t_0\in T$, and let $I$ denote the stablizer of some directed edge $(t_0,t_1)$ in $T$. Any $k$-regular graph $X$ is obtained as $T/\Gamma$ for a lattice $\Gamma$ in $G$, specifically, $\Gamma=\pi_1(X)$. One easily checks that the the vertices of $X$ are in correspondence with $\Gamma{\setminus} G/K$ and that the directed edges are in correpodence with $\Gamma{\setminus}G/I$.

With these observations, one can read the spectrum of the adjacency matrix of $X$ from the irreducible $G$-subrepresentations of $\mathrm{L}^2(\Gamma{\setminus} G)$. Specifically, it is the spectrum of an operator in the relative Hecke algebra $A\in H(G,K)$ restricted to $\mathrm{L}^2(\Gamma{\setminus} G)^K=\mathrm{L}^2(\Gamma{\setminus}G/K)\cong \mathrm{L}^2(X)$ --- in fact, $A$ becomes the adjacency operator of $X$ under this isomorphism. Similarly, the Hashimoto operator of $X$ can be realized as an element $h\in H(G,I)$ acting on $\mathrm{L}^2(\Gamma{\setminus} G)^I=\mathrm{L}^2(\Gamma{\setminus}G/I)$.

The relations between between the spectra of $A$ and $h$ can now be deduced as a consequence of algebraic relations between $A$ and $h$ inside $H(G,I)$, e.g. as done (implicitly) in the proof of Kotani and Sunada (http://www.ms.u-tokyo.ac.jp/journal/pdf/jms070102.pdf). Alternatively, one could compute explicitly the contribution of each irreducible unitary $G$-representation $V$ to the spectrum of $A$ and of $h$ and get the same conclusion.

The strategies sketched here for $H(X)$ can be applied (sometimes after considerable work!) to other functorial constructions $X\mapsto F(X)$, and also when $X$ is a simplicial complex, rather than just a graph. A notable example is in "Zeta functions of complexes from PGL(3): A representation theoretic approach" by Kang, Li and Wang (2010). I also have a preprint which attempts to give a (perhaps too) general view on this kind of approach https://arxiv.org/abs/1605.02664 (section 4 and 6).


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## COURSE CATALOG

### Morals & Manners

Course Instructor:  Mr. William Michael
Course Description:  It was once common knowledge that "Youth is the time to accustom both body and mind to right methods of life." Today, children are seen as uncontrollable monsters who have the right to do as they please, but this is nothing more than the moral destruction of children. Children must be formally taught good morals and manners and made to practice them that good habits may be formed and bad habits may be avoided.  In this course, we diligently study an excellent systematic text of good Christian morals and manners.
Subscription Access:  Family Plan, Student Plan

### Music History

• Course Instructor:  Mr. William Michael
• Course Assistants:  None
• Course Description:  The history of music is one dominated by the Holy Spirit, inspiring the development of the world's most famous music in all generations.  In this course, we study an excellent text written by one of Christendom's great composers, Charles Villiers Stanford (1852-1924), who is widely known for his arrangements of the Gospel canticles (see below).  We will study the complete history of music from a Christian perspective with our aim being to know the history, to become acquainted with the greatest musical composers and compositions, and to understand present state of music for future composition and performance.
• Subscription Plans:  Family Plan, Student Plan


text:
# In 2016, Company W elected under Section 179 to expense $19,300 of the cost of qualifying... ## Question: In 2016, Company W elected under Section 179 to expense$19,300 of the cost of qualifying property. However, it could deduct only $15,000 of the expense because of the taxable income limitation. In 2017, Company W's taxable income before any Section 179 deduction was$812,000. Compute its 2017 Section 179 deduction if the total cost of qualifying property purchased in 2017 was $508,000. ## Section 179: Section 179 allows assessee to charge the whole of the investment in the long-term as depreciation expense in the first year of its' use but, the asset should be placed for more than a six-month period. ## Answer and Explanation: During 2017, Section 179 deduction is$508,000.

Explanation: During 2017, Section 179 deduction is $508,000 which is lower of investment amount and the allowed deduction. Working Note: The maximum allowed deduction under section 179 is 80% of the current year profit i.e.$649,600 (\$812,000 x 80%).


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# I needed help figuring out how to write a net ionic equation for the overall reaction that occurs when aqueous solutions of carbonic acid and potassium hydroxide are combined. Assume excess base. I also would like you to say whether each part in the ionic equation is a solid, liquid, gas, or aqueous.

Question

I needed help figuring out how to write a net ionic equation for the overall reaction that occurs when aqueous solutions of carbonic acid and potassium hydroxide are combined. Assume excess base.

I also would like you to say whether each part in the ionic equation is a solid, liquid, gas, or aqueous.