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Anomaly in kernel/safe_typing.ml
It seems I have a knack for breaking things. The following was an attempt at defining a "type whose elements can be seen as elements of another structure":
From HB Require Import structures.
HB.mixin Record Setoid_of_Type A :=
{ eqv : A -> A -> Prop ; foo : True }.
HB.structure Definition Setoid := { A of Setoid_of_Type A }.
Infix "≡" := eqv (at level 79).
Section Sub.
Variable (A : Type) (X : Setoid.type).
HB.mixin Record Sub_for_Type := { elem : A -> Setoid.sort X }.
(* Error: Anomaly "File "kernel/safe_typing.ml", line 413, characters 2-8: Assertion failed." *)
Not sure that's the most reasonable thing to do, but getting anomalies from kernel/safe_typing.ml probably isn't what an HB command is supposed to do.
Hum, I guess I really need to improve this error ;-)
The problem is that HB.mixin starts a module, but you are inside a section.
I've just pushed a commit to (coq-elpi) master, it now says: "Error: This elpi code cannot be run within a section since it opens a module"
I see. So while one can define parametric instances within sections, one really has to list the parameters for parametric mixins.
Well, replacing everything following the Infix command with:
HB.mixin Record Refl_of_Type (X : Setoid.type) (A : Type) :=
{ elem_of : A -> Setoid.sort X; reflP (a : A) : (elem_of a) ≡ (elem_of a) }.
I get:
ty-deps: BUG: could not get the parameters and the dependencies of
(forall (X : Setoid.type) (A : Type) (elem_of : A -> X), (forall a : A, elem_of a ≡ elem_of a) -> axioms_ X A)
This is probably undocumented.
You should "flag" the type A with something like (A : indexed Type) where indexed is an identity function defined on top of the structures.v file
I'm not sure to follow. If I replace (A : Type) with (A : indexed Type), then the error persists unchanged. In particular, the error message does not mention indexed.
Yes, the error message sucks, but it should now show forall (X : setoid.type) (A : indexed Type)... (and you said unchanged, so I want to be sure...). Maybe there is a bug, but the code does look for
indexed in order to determine parameters.
From HB Require Import structures.
HB.mixin Record Setoid_of_Type A :=
{ eqv : A -> A -> Prop ; foo : True }.
HB.structure Definition Setoid := { A of Setoid_of_Type A }.
Infix "≡" := eqv (at level 79).
HB.mixin Record Refl_of_Type (X : Setoid.type) (A : indexed Type) :=
{ elem_of : A -> Setoid.sort X; reflP (a : A) : (elem_of a) ≡ (elem_of a) }.
still yields:
ty-deps: BUG: could not get the parameters and the dependencies of
(forall (X : Setoid.type) (A : Type) (elem_of : A -> X), (forall a : A, elem_of a ≡ elem_of a) -> axioms_ X A)
Ah, and while we're on the subject of HB.mixin. Is there a deeper reason why HB.mixin cannot take a pre-existing Record, along the lines of HB.instance. I was a bit sad when I noticed that HB.mixin
does not support notations in records.
Accepting an existing record as a mixin is not trivial.
For the notation things, I looked at it, and my understanding is that the notations are (in Coq) lost just after the record declaration. In particular the type and number of arguments of the "fields"
changes and the notation are not adjusted. This is why I did not try to support them :-/
The bug, well, it is there. I won't be able to look into it before wednesday
Enrico Tassi said:
For the notation things, I looked at it, and my understanding is that the notations are (in Coq) lost just after the record declaration. In particular the type and number of arguments of the
"fields" changes and the notation are not adjusted. This is why I did not try to support them :-/
Well, if there were notations in HB.mixin records, I would probably remove the Ops structure from my development. That's the one that also caused the "criss-cross" inheritance. But writing down the
15 axioms without notations would be pretty much unreadable.
That being said, I managed to port the graph theory library to HB. :tada:
The only two things that are still not as nice as I would like is this test thing we talked about and the smashed notations ...
Last updated: Oct 13 2024 at 01:02 UTC | {"url":"https://coq.gitlab.io/zulip-archive/stream/237868-Hierarchy-Builder-devs-.26-users/topic/Anomaly.20in.20kernel.2Fsafe_typing.2Eml.html","timestamp":"2024-11-04T21:28:18Z","content_type":"text/html","content_length":"18441","record_id":"<urn:uuid:66de1bb8-57c2-413a-80df-a579e1ae186f>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.16/warc/CC-MAIN-20241104194528-20241104224528-00763.warc.gz"} |
Trading Scenario: Margin Call Level at 100% and No Separate Stop Out Level - TradeCenterFXTrading Scenario: Margin Call Level at 100% and No Separate Stop Out Level
Trading Scenario: Margin Call Level at 100% and No Separate Stop Out Level
Let’s now take all the margin jargon you’ve learned from the previous lessons and apply them by looking at trading scenarios with different Margin Call and Stop Out Levels.
Different retail forex brokers and CFD providers have different margin call policies. Some only operate only with Margin Calls, while others define separate Margin Call and Stop Out Levels.
In this lesson, we will go through a real-life trading scenario where you are using a broker that only operates with a Margin Call.
The broker defines its Margin Call Level at 100% and has no separate Stop Out Level.
What happens to your margin account when you’re in a trade that goes terribly wrong?
Let’s go find out!
Step 1: Deposit Funds Into Trading Account
Let’s say you have an account balance of $1,000.
This is how it’d look in your trading account:
Long / Short FX Pair Position Size Entry Price Current Price Margin Level Equity Used Margin Free Margin Balance Floating P/L
– $1,000 – $1,000 $1,000 –
Step 2: Calculate Required Margin
You want to go long EUR/USD at 1.15000 and want to open a 1 mini lot (10,000 units) position. The Margin Requirement is 2%.
How much margin (Required Margin) will you need to open the position?
Since EUR is the base currency. this mini lot is 10,000 euros, which means the position’s Notional Value is €10,000.
Since our trading account is denominated in USD, we need to convert the value of the EUR to USD to determine the Notional Value of the trade.
$1.15 = €1
$11,500 = €10,000
The Notional Value is $11,500.
Now we can calculate the Required Margin:
Required Margin = Notional Value x Margin Requirement
$230 = $11,500 x .02
Assuming your trading account is denominated in USD, since the Margin Requirement is 2%, the Required Margin will be $230.
Step 3: Calculate Used Margin
Aside from the trade we just entered, there aren’t any other trades open.
Since we just have a SINGLE position open, the Used Margin will be the same as Required Margin.
Step 4: Calculate Equity
Let’s assume that the price has moved slightly in your favor and your position is now trading at breakeven.
This means that your Floating P/L is $0.
Let’s calculate your Equity:
Equity = Balance + Floating Profits (or Losses)
$1,000 = $1,000 + $0
The Equity in your account is now $1,000.
Step 5: Calculate Free Margin
Now that we know the Equity, we can now calculate the Free Margin:
Free Margin = Equity - Used Margin
$770 = $1,000 - $230
Step 6: Calculate Margin Level
Now that we know the Equity, we can now calculate the Margin Level:
Margin Level = (Equity / Used Margin) x 100%
435% = ($1,000 / $230) x 100%
At this point, this is how your account metrics would look in your trading platform:
Long / Short FX Pair Position Size Entry Price Current Price Margin Level Equity Used Margin Free Margin Balance Floating P/L
– $1,000 – – $1,000 –
Long EUR/USD 10,000 1.15000 1.15000 435% $1,000 $230 $770 $1,000 $0
EUR/USD drops 500 pips!
There are reports of a zombie outbreak in Paris.
EUR/USD falls 500 pips and is now trading at 1.10000.
Let’s see how your account is affected.
Used Margin
You’ll notice that the Used Margin has changed.
Because the exchange rate has changed, the Notional Value of the position has changed.
This requires recalculating the Required Margin.
Whenever there’s a change in the price for EUR/USD, the Required Margin changes.
With EUR/USD now trading at 1.1000 (instead of 1.15000), let’s see how much Required Margin is needed to keep the position open.
Since our trading account is denominated in USD, we need to convert the value of the EUR to USD to determine the Notional Value of the trade.
$1.10 = €1
$11,000 = €10,000
The Notional Value is $11,000.
Previously, the Notional Value was $11,500. Since EUR/USD has fallen, this means that EUR has weakened. And since your account is denominated in USD, this causes the position’s Notional Value to
Now we can calculate the Required Margin:
Required Margin = Notional Value x Margin Requirement
$220 = $11,000 x .02
Notice that because the Notional Value has decreased, so has the Required Margin.
Since the Margin Requirement is 2%, the Required Margin will be $220.
Previously, the Required Margin was $230 (when EUR/USD was trading at 1.15000).
The Used Margin is updated to reflect changes in Required Margin for every position open.
In this example, since you only have one position open, the Used Margin will be equal to the new Required Margin.
Floating P/L
EUR/USD has fallen from 1.15000 to 1.10000, a difference of 500 pips.
Since you’re trading 1 mini lot, a 1 pip move equals $1.
This means that you have a Floating Loss of $500.
Floating P/L = (Current Price - Entry Price) x 10,000 x $X/pip
-$500 = (1.1000 - 1.15000) x 10,000 x $1/pip
Your Equity is now $500.
Equity = Balance + Floating P/L
$500 = $1,000 + (-$500)
Free Margin
Your Free Margin is now $280.
Free Margin = Equity - Used Margin
$280 = $500 - $220
Margin Level
Your Margin Level has decreased to 227%.
Margin Level = (Equity / Used Margin) x 100%
227% = ($500 / $220) x 100%
Your Margin Level is still above 100% so all is still well.
Account Metrics
This is how your account metrics would look in your trading platform:
Long / Short FX Pair Position Size Entry Price Current Price Margin Level Equity Used Margin Free Margin Balance Floating P/L
– $1,000 – $1,000 $1,000 –
Long EUR/USD 10,000 1.15000 1.15000 435% $1,000 $230 $770 $1,000 $0
Long EUR/USD 10,000 1.15000 1.10000 227% $500 $220 $280 $1,000 -$500
EUR/USD drops another 288 pips!
EUR/USD falls another 288 pips and is now trading at 1.07120.
Used Margin
With EUR/USD now trading at 1.07120 (instead of 1.10000), let’s see how much Required Margin is needed to keep the position open.
Since our trading account is denominated in USD, we need to convert the value of the EUR to USD to determine the Notional Value of the trade.
$1.07120 = €1
$10,712 = €10,000
The Notional Value is $10,712.
Now we can calculate the Required Margin:
Required Margin = Notional Value x Margin Requirement
$214 = $10,712 x .02
Notice that because the Notional Value has decreased, so has the Required Margin.
Since the Margin Requirement is 2%, the Required Margin will be $214.
Previously, the Required Margin was $220 (when EUR/USD was trading at 1.10000).
The Used Margin is updated to reflect changes in Required Margin for every position open.
In this example, since you only have one position open, the Used Margin will be equal to the new Required Margin.
Floating P/L
EUR/USD has now fallen from 1.15000 to 1.07120, a difference of 788 pips.
Since you’re trading 1 mini lot, a 1 pip move equals $1.
This means that you have a Floating Loss of $788.
Floating P/L = (Current Price - Entry Price) x 10,000 x $X/pip
-$788 = (1.07120 - 1.15000) x 10,000 x $1/pip
Your Equity is now $212.
Equity = Balance + Floating P/L
$212 = $1,000 + (-$788)
Free Margin
Your Free Margin is now –$2.
Free Margin = Equity - Used Margin
-$2 = $212 - $214
Margin Level
Your Margin Level has decreased to 99%.
Margin Level = (Equity / Used Margin) x 100%
99% = ($212 / $214) x 100%
At this point, your Margin Level is now below the Margin Call Level!
Account Metrics
This is how your account metrics would look in your trading platform:
Long / Short FX Pair Position Size Entry Price Current Price Margin Level Equity Used Margin Free Margin Balance Floating P/L
– $1,000 – $1,000 $1,000 –
Long EUR/USD 10,000 1.15000 1.15000 435% $1,000 $230 $770 $1,000 $0
Long EUR/USD 10,000 1.15000 1.10000 227% $500 $220 $280 $1,000 -$500
Long EUR/USD 10,000 1.15000 1.07120 99% $212 $214 -$2 $1,000 -$788
Your trading platform will automatically close out your trade!
Two things will happen when your trade is closed:
1. Your Used Margin will be “released”.
2. Your Floating Loss will be “realized”.
Your Balance will be updated to reflect the Realized Loss.
Now that your account has no open positions and is “flat”, your Free Margin, Equity, and Balance will be the same.
There is no Margin Level or Floating P/L because there are no open positions.
Let’s see how your trading account changed from start to finish.
Long / Short FX Pair Position Size Entry Price Current Price Margin Level Equity Used Margin Free Margin Balance Floating P/L
– $1,000 – $1,000 $1,000 –
Long EUR/USD 10,000 1.15000 1.15000 435% $1,000 $230 $770 $1,000 $0
Long EUR/USD 10,000 1.15000 1.10000 227% $500 $220 $280 $1,000 -$500
Long EUR/USD 10,000 1.15000 1.07120 99% $212 $214 -$2 $1,000 -$788
– $212 – $212 $212 –
Before the trade, you had $1,000 in cash. Now you’re left with $212!
You’ve lost 79% of your capital.
% Gain/Loss = ((Ending Balance - Starting Balance) / Starting Balance) x 100%
-79% = (($212 - $1,000) / $1,000) x 100%
Some traders suffer a terrible side effect when finding out their trade has been automatically liquidated.
In the next lesson, we provide a different trading scenario where your broker has a separate Margin Call AND Stop Out Level.
Let’s see the difference between happens there versus what happened here. | {"url":"https://tradecenterfx.com/learn-forex/trading-scenario-margin-call-level-at-100-and-no-separate-stop-out-level/","timestamp":"2024-11-11T01:51:22Z","content_type":"text/html","content_length":"102834","record_id":"<urn:uuid:c7439a65-bb9b-4c78-b38f-088b5104b50c>","cc-path":"CC-MAIN-2024-46/segments/1730477028202.29/warc/CC-MAIN-20241110233206-20241111023206-00436.warc.gz"} |
What is a parallelogram?What is a parallelogram?What is a parallelogram?
What is a parallelogram?
Let’s begin with defining some of the properties of a parallelogram.
1. A parallelogram is a four sided polygon.
2. The opposite sides of a parallelogram are parallel. So if you look at the picture below, sides AB and CD are parallel and AD and BC are parallel
3. The opposite angles of a parallelogram are congruent. So angles B and D are congruent and angles A and C are congruent. ( See picture above) For example if angle B is 120 degrees then angle D will
also be 120 degrees. The same rule applies for angles D and B
4. The consecutive angles in a parallelogram are supplementary which means they add to 180 degrees. Therefore, if angle D measures 120 degrees, angle A will measure 60 degrees because they are
consecutive angles and add up to 180 degrees.
5. In a parallelogram any two consecutive angles will be supplementary to each other. In the picture, A and B,B and C, C and D, and D and A are all consecutive angles and therefore, are
Let’s look at a practice problem. Given a parallelogram in which angle C is has a measure of 80 degrees. What are the other three angles?
1. Angle A will also be 80 degrees because it is opposite angle C.
2. If angle A is 80 degrees then angle B will be 100 degrees because they are consecutive angles.
3. Angle D will be 100 degrees because it is opposite angle B
For a visual explanation please watch the video
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LibGuides: OWL - Online Writing Lab: Quotations
How to incorporate a direct quote within the text
Ask yourself these questions if you are considering using the words of another instead of summarizing the information in your own words.
1. Are the author's words so impressive or so unique that I couldn't express those ideas as well?
2. Is the author's language so succinct that it would take me twice as many words to explain the same thought?
3. Is the terminology so precise that I could not explain the meaning?
If you answer yes to any of these questions, then include the quotation into your text.
IQI--Introduce, Quote, Interpret
Now that you've decided to include the words of another in your paper, make sure that you incorporate them in such a way that they enhance your ideas and are understood by the reader.
1. Introduce the quoted material by telling the reader some information about the writer: name (the first time an author is referred to use first and last names. After that, use only last the name).
Add some perinent information about the author to give credibility: medieval mystic....
2. Include the quotation. Be sure to include the words within double quotation marks. Always include the page number where the quotation is located.
Chapter 5 in Kate Turabian’s A Manual for Writers explains the use of quotations in great detail but there are some general guidelines. If you're using footnotes or endnotes, the superscript
number (I) goes outside the period and quotation marks (i.e. ." I). If you utilize the parenthetical style of documentation, the page number is included after the quotation mark, inside the
parenthesis, and within the sentence (i.e. "(3).)
Pages 95-98 in the Publication Manual of the American Psychological Association contain the AP guidelines for including quotations. In general, include the page number inside the parenthesis
within the sentence (i.e. (p. 102) ). If you have not given the author's name within the text, include the author's last name, year, and page number inside the parenthesis (i.e. Mapes, 1902, p.
3. Always include a "coming-away" observation after the quoted material to interpret those ideas. This serves three essential functions: to explain the meaning of the quoted words; to restore your
authority; and to reestablish your voice. Never assume the reader will understand the quotation or how those words relate to your points. Your words and your ideas are what are important-not some
else's thoughts.
Example from a paper:
In Numbers 27:7 the Lord says, " What Zelophehad's daughters are saying is right. You must certainly give them property as an inheritance." God's brief ruling in favor of the daughters should remind
us today that in the "Court of Justice" the disenfranchised deserve a hearing. | {"url":"https://lpts.libguides.com/c.php?g=144546&p=946083","timestamp":"2024-11-11T03:29:30Z","content_type":"text/html","content_length":"29839","record_id":"<urn:uuid:a464294d-504d-4bfd-92b9-1ef636cf6f6d>","cc-path":"CC-MAIN-2024-46/segments/1730477028216.19/warc/CC-MAIN-20241111024756-20241111054756-00410.warc.gz"} |
Gerry Brady - Department of Computer Science
Contact Info
Crerar 398-A
Studied mathematics and physics as an undergraduate at the University of Chicago, graduating with a degree in mathematics with honors in the College. In the Honors Program at the University of
Chicago in both mathematics and physics. Began study of probability theory, statistics, numerical linear algebra, and numerical analysis as a result of interest in physics. Number theory was a
mathematical interest. Pursued graduate study at the University of Chicago and received a doctorate in logic from the University of Oslo, Norway, in 1997. Expanded version of doctoral thesis
published as a book in 2000 by Elsevier, in Studies in History and Philosophy of Mathematics series. Was a member of a pioneering team that computerized editing, production, and publication of the
Astrophysical Journal and the Astronomical Journal for the American Astronomical Society at the University of Chicago Press in the 1990s.
How well do you feel MPCS has kept up with the demands graduates face in the workplace?
“The courses I teach in the MPCS all emphasize problem solving and help students improve their problem-solving skills. Problem solving is important in the workplace, and in recent years many of our
graduates have been hired at leading firms, in part on the strength of the problem-solving skills acquired in their MPCS courses.”
What do you see as the most important advantage of receiving a master’s degree from the University of Chicago MPCS?
“Graduates of the MPCS are perceived to be intelligent, creative, and able to learn new subjects quickly. The University of Chicago’s superb academic reputation carries over to the MPCS.”
Ph.D. thesis published as book, From Peirce to Skolem, in 2000.
Several writings in mathematical and categorical logic.
• Geraldine Brady and Todd H. Trimble. The topology of relational calculus. Submitted to Advances in Mathematics.
• Geraldine Brady. From Peirce to Skolem: A Neglected Chapter in the History of Mathematical Logic. Elsevier Science: North-Holland, 2000.
Mathematical Reviews: MR 1834718
• Geraldine Brady and Todd H. Trimble. A categorical interpretation of C. S. Peirce’s System Alpha. Journal of Pure and Applied Algebra, 149: 213-239, 2000. Mathematical Reviews: MR 17627665
• Geraldine Brady and Todd H. Trimble. A string diagram calculus for predicate logic. Preprint, November 1998.
• Geraldine Brady. The Contributions of Peirce, Schroeder, Loewenheim, and Skolem to the Development of First-Order Logic. Doctoral Dissertation, Universitetet i Oslo, 1997.
• Geraldine Brady. From the algebra of relations to the logic of quantifiers. Studies in the Logic of Charles Sanders Peirce, Indiana University Press, 1997.
• Stuart A. Kurtz and Geraldine Brady. Existential Graphs: I. January 1997.
• Ph.D. University of Oslo, Norway, Mathematical Logic.
• M.A. University of Chicago, Mathematical Logic.
• B.A. University of Chicago with honors in Mathematics. | {"url":"https://computerscience.uchicago.edu/people/gerry-brady/","timestamp":"2024-11-07T10:03:14Z","content_type":"text/html","content_length":"115870","record_id":"<urn:uuid:c82c6d01-13c4-4606-930f-a566b561d5b9>","cc-path":"CC-MAIN-2024-46/segments/1730477027987.79/warc/CC-MAIN-20241107083707-20241107113707-00706.warc.gz"} |
Limits of Functions of Three Variables
Limits of Functions of Three Variables
We have just looked at Limits of Functions of Two Variables. Recall that for a two variable real-valued function, $z = f(x, y)$, then $\lim_{(x, y) \to (a,b)} f(x, y) = L$ if $\forall \epsilon > 0$ $
\exists \delta > 0$ such that if $x \in D(f)$ and $0 < \sqrt{(x - a)^2 + (y - b)^2} < \delta$ then $\mid f(x,y) - L \mid < \epsilon$. We are now going to extend this concept further to functions of
three variables.
Definition: Let $w = f(x, y, z)$ be a three variable real-valued function. Then the Limit of $f(x, y, z)$ as $(x, y, z)$ Approaches $(a,b,c)$ is $L$ denoted $\lim_{(x, y, z) \to (a,b,c)} f(x, y, z) =
L$ if $\forall \epsilon > 0$ $\exists \delta > 0$ such that if $(x, y, z) \in D(f)$ and $0 < \sqrt{(x - a)^2 + (y - b)^2 + (z - c)^2} < \delta$ then $\mid f(x, y, z) - L \mid < \epsilon$.
Once again, it is important to note that $\sqrt{(x - a)^2 + (y - b)^2 + (z - c)^2}$ represents the distance between the points $(x, y, z)$ and $(a, b, c)$. Thus, we can reformulate the definition
above as follows. The limit as $(x, y, z)$ approaches $(a, b, c)$ is the real number $L$ if for all $\epsilon > 0$ there exists a $\delta > 0$ such that if the distance between the points $(x, y, z)$
and $(a, b, c)$ is less than $\delta$ but not $0$, then the distance between $f(x, y, z)$ and $L$ is less than $\epsilon$.
We can also look at limits of functions of more than $3$ variables:
Definition: Let $z = f(x_1, x_2, ..., x_n)$ be an $n$ variable real-valued function. Then the Limit of $f(x_1, x_2, ..., x_n)$ as $(x_1, x_2, ..., x_n)$ Approaches $(a_1, a_2, ..., a_n)$ is $L$
denoted $\lim_{(x_1, x_2, ..., x_n) \to (a_1, a_2, ..., a_n)} f(x_1, x_2, ..., x_n) = L$ if $\forall \epsilon > 0$ $\exists \delta > 0$ such that if $(x_1, x_2, ..., x_n) \in D(f)$ and $0 < \sqrt
{(x_1 - a_1)^2 + (x_2 - a_2)^2 + ... + (x_n - a_n)^2} < \delta$ then $\mid f(x_1, x_2, ..., x_n) - L \mid < \epsilon$. | {"url":"http://mathonline.wikidot.com/limits-of-functions-of-three-variables","timestamp":"2024-11-06T07:55:36Z","content_type":"application/xhtml+xml","content_length":"16617","record_id":"<urn:uuid:b29f9820-16ea-482b-8cf4-a5bbba5e1a76>","cc-path":"CC-MAIN-2024-46/segments/1730477027910.12/warc/CC-MAIN-20241106065928-20241106095928-00454.warc.gz"} |
Arithmetic Trinomial Cube
Welcome to A2Z Montessori Australia
Montessori learning equipment supplies and imaginative, inspiring and creative educational toys.
Arithmetic Trinomial Cube
Montessori Arithmetic Trinomial Cube
Arithmetic Trinomial Cube
Arithmetic Trinomial Cube
This Arithmetic Trinomial Cube is designed to support the child in their deeper exploration of algebra. The Arithmetic Trinomial Cube is a physical representation of the equation (a + b + c)³. Each
of the component pieces are color-coded to represent hierarchical values from 1 unit to 1 million. This makes it possible for the child to equate an arithmetical value to each of the cubes and
The cube is contained in quality timber box with lid and hinged sides.
Package Size and Weight: 14cm x 14cm x 11cm, 1.39kg
Write a review / your experience using Arithmetic Trinomial Cube :
A2Z Montessori Australia, 5-6/50 Gateway Drive, Noosaville QLD 4566
Contact Hani on info@a2zmontessori.com.au
phone: 1300-723-725 or 07 5449 9825
Used expressions: Arithmetic Trinomial Cube Montessori Arithmetic Trinomial Cube
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Mathematics Websites | TEAM
The sites found on this page have been compiled by those students enrolled in EDU 4303, EDU 5223, and EDU 5903 at Tarleton State University since January 2000. The students have provided the
evaluations of each site.
Teacher Math Websites
A Homepage For New Math Teachers
This is a site for new math teachers and it has ideas for classroom management, favorite math problems, and sites to go for a wide variety of activities.
American Mathematical Society
The one I found has places to go and look at journal articles, places to look at lesson plans, and a lot of other cool info. about math.
Bamdad’s Math Comics
These comics strips are very funny and relevant to the classroom and real-life math problems that teachers and students alike will love, and there’s tons of them on this site on about every math
topic imaginable.
The Bureau of Labor Statistics
Lists information about occupations for which varying levels of mathematical background are required. I think you’ll find it quite interesting. (To give an idea of the comprehensive nature of the
list, “Fast-food counter workers” are included in the list as an occupation for which arithmetic is important).
California Mathematics Council
This site has online resources, issues for parents and teachers, and links to other math websites.
Educational Reference for Business and Money Math
This is a great site to use when teaching children money managing and financing.
Enchanted Mind
There is much interesting data about the brain and creativity here. There are also some cool java puzzles to enhance creativity here including Rubik’s Cube, Tangrams, and Pentominoes. This is a fun
site and also a good one for kids to use to play around with puzzles like this.
Fibonacci Numbers and the Golden Section
This is the home page for the Fibonacci numbers, the Golden String, and the Golden Section. Over 200 pages of material, but clicking on the first link takes you to an immediate explanation and a list
of sites that demonstrate Fibonacci numbers in nature. Lots of math links. Also great puzzles utilizing Fibonacci numbers.
FunBrain.Com Math Baseball
A fun math baseball game that you can play by yourself or with a friend. It also gives you a choice of math problems and the the level you want to practice.
Helping Your Child Learn Math
This great, little site provides some extremely useful activities for either school or home. The activities were rated by grade level and focused on math or science. Activities included measurement
in the kitchen, math in the grocery store, and a neat exercise in geometry focused around the recycling of cans and paper boxes. Great visualization exercises!
This is a great resource for students preparing for college life, and beyond. It focuses on improving math skills, and helping students who are preparing for the assessment test.
Key Curriculum Press
Innovators in Mathematics curriculum. Connections to technology in the mathematics classroom.
The Math Forum
Interesting site filled with problems, puzzles, and lesson plans for math.
Math Solutions – MathWay
Fantastic website dedicated to helping students solve problems from Basic Math to Linear Algebra, even Chemistry problems! This website offers examples as well as step by step procedures with problem
This is a neat website. This website had a tab for homework help, practice problems, ask an expert, calculator & tools (this was so neat because it had a calculator for everything-you could even plot
charts on a x and y axis. It also had math games and a store to buy manipulatives. I wish I knew this site when I was taking algebra!
MathType by Wiris
New MathType for Windows–creating math notation for word processing & Web pages.
Mudd Math Fun Facts
Mudd Math Fun Facts presents engaging math tidbits and challenges that can be used to start daily math classes.
National Council of Teachers of Mathematics
Online: Standards 2000 and relationships between research and the NCTM Standards and much more!
National STEM Centre
A wealth of information regarding the teaching of science, technology, engineering and science.
Teachers Lab
The Annenberg/CPB Math and Science Project
Elementary Math Websites
This user-friendly site contains math information for every elementary level. This is an interactive site that can be used by teachers, parents, and students. It is organized by topic and grade
level. It has games, puzzles, and riddles. There are also links to other useful web sites.
Games, flash cards, and homework help for elementary students
This site contains a lesson on fractals for elementary and middle school students. It explains what fractals are, tells you how to make them and has several links to other information, including
teacher resources.
This site contains information on fractions – adding, subtracting, multiplying, and dividing. It also has quizzes to take after each discussion.
Funbrain – Shape Save Game
This site is a geometry game that explores area and perimeter. It is rewarded by giving you an archaeology picture and information making it interdisciplinary with geography and history. There are 4
levels of difficulty and is good for grades 3-5.
FunBrain.Com Math Baseball
A fun math baseball game that you can play by yourself or with a friend. It also gives you a choice of math problems and the the level you want to practice.
Helping Your Child Learn Math
This great, little site provides some extremely useful activities for either school or home. The activities were rated by grade level and focused on math or science. Activities included measurement
in the kitchen,
This site is very user-friendly and has a large variety of activities to pursue including math games and music. The elementary math games include mostly counting activities and addition by counting
up dots on the screen. The intermediate math game includes addition, subtraction, multiplication, division, fractions, decimals, measurements, and graphs. The student may choose which level they are
on and after they finish the computer gives them a percentile score.
Math Flash Cards
A complete set of enter active flash cards for addition, subtraction, multiplication, division, and addition and subtraction.
Math for Kids
This site has a medieval adventure game involving problem solving. It was designed by fourth graders for upper elementary students. Math word problems have a medieval theme.
The Math Magic Resources for K-4 Teachers
Here’s a teacher’s website that has many links to online games! Some of the games need help but many are excellent! She also links to many sites that offer games not listed on the teacher’s website!
Really good links here!
Math Stories.com!
It has tons of story problems categorized in lots of different ways. Great for kids who are having trouble with story problems!
Mighty Math Club
This site has a lot of interesting activities for children to get involved in and it is very easy to get around in.
Saxon Math
This site belongs to a textbook publisher. There are several exceptional math activities for K-6. Additionally, information is offered over a broad spectrum.
School House Rocks
Do you remember this famous cartoon aired on Saturday mornings on ABC? This site has lots of cool information for science, math, social studies, and grammar. It has information that has been
previously aired. It has neat songs to remember important content area information. Have fun reliving some of your favorite memories!
Silver Burdett Ginn Mathematics
Mathematics lesson plans and activities for grades K-6. Options include Teacher Tools, Home Connection and other useful links.
Middle School Websites
This site contains a lesson on fractals for elementary and middle school students. It explains what fractals are, tells you how to make them and has several links to other information, including
teacher resources.
This is a neat website. This website had a tab for homework help, practice problems, ask an expert, calculator & tools (this was so neat because it had a calculator for everything-you could even plot
charts on a x and y axis. It also had math games and a store to buy manipulatives. I wish I knew this site when I was taking algebra!
Unfolding Polyhedra
This is a really cool website for studying polyhedrons.
IXL – 6th Grade
This website helps students with most skills that they will use in 6^th grade. There is a list of different subjects that they can practice.
IXL – 7th Grade
This website is for students in the 7^th grade. It has helpful games and practices.
IXL – 8th Grade
This website is for students in the 8^th grade. It has helpful games and practices.
Gives examples on solving equivalent forms of rational numbers.
Utah Education Network
Shows how to find equivalent forms for positive rational numbers.
The Rational Number Project
This website is helpful for teachers when they are about to teach rational numbers.
This is a game that helps students with adding and subtracting fractions.
Jamit Fractions
This is a game that helps students with adding and subtracting fractions.
High School Math Websites
Algebra OnlineSM
This description was taken directly from Algebra OnlineSM. Algebra OnlineSM is a free service designed to allow students, parents, and educators throughout the world to communicate. This includes
free private tutoring, live chat, and a message board, among many other features. Questions and discussions relating to all levels of mathematics (not just Algebra) are welcome. Algebra OnlineSM is
the future in education!
Holt, Rinehart and Winston
See Algebra 1 Interactions–motivating activities, math connections and real world application and integrated technology.
Key Curriculum Press
Innovators in Mathematics curriculum. Connections to technology in the mathematics classroom.
Math In Daily Life
This web site gives great examples of how math affects everyday decisions. Real life examples are given of probability, savings and credit, home decorating, and population growth. Great for answering
the question “Why do we need to know this?”.
The Math Forum
Interesting site filled with problems, puzzles, and lesson plans for math.
This is a neat website. This website had a tab for homework help, practice problems, ask an expert, calculator & tools (this was so neat because it had a calculator for everything-you could even plot
charts on a x and y axis. It also had math games and a store to buy manipulatives. I wish I knew this site when I was taking algebra!
Pascal’s Triangle and Its Patterns
This web site is devoted to Pascal’s Triangle. Gives information on how the triangle is constructed, Sums of Rows, Prime Numbers, Hockey Stick Patterns, Sierpinski’s Triangle and much more! Great Web
Totally Tessellated
This visually appealing math website that includes a large galley of tessellations, or the repeated patterns of interlocking shapes. There are also illustrated and animated tutorials, information on
the artist M.C. Escher, an interactive gallery of art, hands-on activities, and printable timetables.
Zona Land
This is a great site that has both mathematics and science. You can find educational and entertaining items pertaining to physics, to the mathematical sciences, and to mathematics in general. I did
my paper over measurement. I found a lot of neat ways to teach measurement in the third grade. These ways consist of finding the tallest third grade, measuring the teacher, and cooking. These are all
fun ways to teach.
All Level Math Websites
This is a really good website, because it offers a variety of math related material such as math tools and math games that integrate technology with math curriculum. For example, the online flashcard
game and the metric converter game that it has for students to play. Further more, it gives links to more fun math games that the students can enjoy. It also, offers great links to sites such as Kid
References, Teacher References, Teacher Resources, and Math Job listings. It offers you math articles that deal with anything from the history of math to women in math. The thing that really makes
this site stand out is that it is very well organized and easy to scroll through. I usually find myself surfing through websites and ending up nowhere close to where I first began. I would definitely
recommend this site to everyone.
Center for Excellence in Education, Rice University
This site contains a goldmine of math lessons for grades K-12. There are math problems to solve online and the opportunity to submit answers and find out if you have worked the problem correctly.
This site makes math a fun and encouraging experience — especially for those who experience difficulty. (Algebra fun with calendars is great fun for the whole family!)
This site is “an amusement park of mathematics and more!” It has a wide variety of mathematical activities for teachers, parents, and kids from 3-100! It includes fractals, brain benders, geometry
and other topics as well as links to other math sites. This site also has a great science section. This is terrific for increasing mathematical abilities in kids of all ages.
This web site demonstrates how math is used in the real world of building construction. Has connections between real world situations and classroom math. Has lessons for students or for teachers
broken down into K-2, 3-5, 6-7 & high school.
Enchanted Mind
There is much interesting data about the brain and creativity here. There are also some cool java puzzles to enhance creativity here including Rubik’s Cube, Tangrams, and Pentominoes. This is a fun
site and also a good one for kids to use to play around with puzzles like this.
Escape from Knab
This is a great site. You have landed on the planet Knab. You are given only a short time to “earn” enough money for a ticket back to earth. The games include choosing a job, finding a place to live,
monthly expenses, investments, writing a check, and checking and savings accounts. It’s really a great way to teach “real life” while playing a game.
This web site has lots of math and science lesson plans. It is very easy to use because you can click whether you are interested in math or science. Then select the topic you need. It will give you a
whole list of lesson ideas to look through.
Frazier Park School – Mathematics Page
This address will connect you and your students to math games in “Fun Brain, “Kid Klok” and much more. The site has a number of different activities for all grade levels in different areas in math.
The Math Forum
Interesting site filled with problems, puzzles, and lesson plans for math.
Tutor Pace
This address will connect you to live math help one-on-one on an interactive whiteboard with experienced university professors. The online tutoring program is research based which helps when needing
additional help with homework or preparing for a test.
Mathematics Lessons That Are Fun, Fun, Fun!
All the math activities you might ever need to teach kindergarten through 12th.
Math World
This web site provides many ideas about interactive things for math. This page gives many ideas about what can be used in teaching math for students in Pre-K through 12th grade.
Mrs. Glosser’s Math Goodies
You will find interactive math lessons that have real-world problems. This award winning site includes lessons for integers, understanding percent, number theory,circumference and area of circle, and
perimeter and area of polygons.
Interesting folded-paper, hands-on activities to teach kids about geometry, symmetry, etc.
Summit Learning
This is a web-site for math of any level. All you have to do is put in the grade level that you want and specific area
in math that you are looking for.
Woodka Web
This site was created by Donna Woodka, focuses on connecting girls with math, science and technology. It helps parents and teachers encourage girls interest in math, science and technology. This site
also contains connections to other sites relating to math, science, and technology. | {"url":"https://www.tarleton.edu/team/resources/mathematics-websites/","timestamp":"2024-11-06T11:05:59Z","content_type":"text/html","content_length":"89271","record_id":"<urn:uuid:a601eb8c-8b6b-4f78-b39a-15b6214f0e21>","cc-path":"CC-MAIN-2024-46/segments/1730477027928.77/warc/CC-MAIN-20241106100950-20241106130950-00531.warc.gz"} |
Constraints on the original ejection velocity fields of asteroid families
Direito de acesso
Acesso aberto
Asteroid families form as a result of large-scale collisions among main belt asteroids. The orbital distribution of fragments after a family-forming impact could inform us about their ejection
velocities. Unfortunately, however, orbits dynamically evolve by a number of effects, including the Yarkovsky drift, chaotic diffusion, and gravitational encounters with massive asteroids, such that
it is difficult to infer the ejection velocities eons after each family's formation. Here, we analyse the inclination distribution of asteroid families, because proper inclination can remain constant
over long time intervals, and could help us to understand the distribution of the component of the ejection velocity that is perpendicular to the orbital plane (υW). From modelling the initial break
up, we find that the distribution of υW of the fragments, which manage to escape the parent body's gravity, should be more peaked than a Gaussian distribution (i.e. be leptokurtic) even if the
initial distribution was Gaussian. We surveyed known asteroid families for signs of a peaked distribution of υW using a statistical measure of the distribution peakedness or flatness known as
kurtosis. We identified eight families whose υW distribution is significantly leptokurtic. These cases (e.g. the Koronis family) are located in dynamically quiet regions of themain belt, where,
presumably, the initial distribution of υW was not modified by subsequent orbital evolution. We suggest that, in these cases, the inclination distribution can be used to obtain interesting
information about the original ejection velocity field.
Como citar
Monthly Notices of the Royal Astronomical Society, v. 457, n. 2, p. 1332-1338, 2016.
Itens relacionados | {"url":"https://repositorio.unesp.br/items/ec2f76a1-e550-4030-925a-a02afffcb879","timestamp":"2024-11-07T22:46:12Z","content_type":"text/html","content_length":"433579","record_id":"<urn:uuid:af571dc9-c91c-43d8-8e9d-ceb1a81b941a>","cc-path":"CC-MAIN-2024-46/segments/1730477028017.48/warc/CC-MAIN-20241107212632-20241108002632-00261.warc.gz"} |
Mortgage Discharge Penalties
This is the most difficult topic related to mortgages and it will continue to be confusing until the laws in Canada are changed to require consistency on how lending institutions charge their
Most lenders charge an early payoff penalty on closed mortgages if the debt is paid prior to the maturity of the term. The lending institution must describe the penalty they could charge on the
mortgage document.
The most common penalty is:
The greater of three months interest penalty OR the interest rate differential. In other words, whichever amount is the larger of these two figures will be your penalty. Other kinds of penalties are
listed below.
Three Months Interest Penalty
If you are paying off your mortgage before the maturity date, most lending institutions charge three months interest penalty (or an interest differential penalty).
Your present mortgage balance is multiplied by your current interest rate and multiplied by three.
Interest Rate Differential
This usually means the difference between the interest rate on your mortgage contract compared to the rate at which the lending institution can re-lend the money.
For example:
If your mortgage has a balance of $125,000 at 9.25%, you have 2 years left to go and the current 2 year mortgage rate is 6.25%.
Then the lending institution will probably charge you – $125,000 X 24 months X 3% (9.25 – 6.25) = $7,266.21
However, just to further confuse the issue, the penalty above has not been present valued. This is when a lender charges a lower penalty because you are paying all of the ‘extra’ interest (in the
example 3%) now, not over the remaining term. Some lenders present value, other lenders do not.
Other Penalty Calculations
Methods of calculating penalties are as varied as the lenders’ imaginations! The following outline describes some penalties charged by lenders.
Some examples:
1. Greater of three months interest penalty OR the interest rate differential.
2. CMHC mortgages registered prior to July 1999 – during the first three years, the penalty is the greater of 3 months interest OR interest rate differential. After three years of payments made on a
4 or 5 year term (or longer) the penalty is three months interest.
3. CMHC mortgages registered after July 1999 – CMHC mortgages will now have the same penalty clause as the institution lending you the mortgage funds.
4. Two months penalty interest (based on the floating rate in effect at the time of payout) calculated on the outstanding balance during the first three years of the term and no penalty charged at
all for the remaining years of the term.
5. The mortgage can not be paid out unless there is an arm’s length sale – then the penalty is 3% of the outstanding mortgage balance.
6. The mortgage can not be paid out unless there is an arm’s length sale – then the penalty is the greater of three months interest OR 3% of the outstanding balance.
7. Same as above, but not more than three months interest in years 4 and 5 of a five year term.
8. For non-arm’s length sales – it is the greater of three months interest OR interest rate differential to the bond rate for the remaining term.
9. For arm’s length sales – it is the greater of three months interest OR interest rate differential to the current posted mortgage rate for the remaining term.
Here’s More Confusion
• Do not assume the same lender charges penalties the same way for each type of mortgage. Examples 1, 4 & 5 above are all charged by the same lender on different products.
• Do not assume the penalty charges you agreed to with the original mortgage document are the same when you renew with the same lender. Their policies concerning penalty charges are always
• Do not assume the same wording means the same calculation with different lenders. For example the term ‘interest rate differential’ means very different penalty policies with different lenders.
The terminology is not used consistently.
• Do not assume your legal representative or real estate agent is familiar with all the different ‘twists and turns’ of penalty charges.
There have recently been class action law suits against at least two Canadian lending institutions over their practices regarding the calculation of penalty charges. The law is still evolving
regarding acceptable practices for calculating penalties.
Click here to fill out a short application and get a no-obligation mortgage quote, or contact me directly. | {"url":"https://www.delcomortgages.ca/mortgage-articles/mortgage-discharge-penalties/","timestamp":"2024-11-12T15:45:16Z","content_type":"text/html","content_length":"49539","record_id":"<urn:uuid:bfdc7725-ae5b-4af1-a8d0-03c05bc3a9e2>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.63/warc/CC-MAIN-20241112145015-20241112175015-00509.warc.gz"} |
estimated standard deviation
A special case of an estimate of the variability of data is the estimated standard deviation. For example, if we know that data follows the Normal distribution (say people's heights), we might work
out the standard deviation of the population for a sample, and then use this as an estimate of the true standard deviation (s.d., σ).
Used on page 84 | {"url":"https://alandix.com/glossary/hcistats/estimated%20standard%20deviation","timestamp":"2024-11-06T18:52:45Z","content_type":"application/xhtml+xml","content_length":"9881","record_id":"<urn:uuid:959854b4-e67f-4724-b32c-e71c97830b3f>","cc-path":"CC-MAIN-2024-46/segments/1730477027933.5/warc/CC-MAIN-20241106163535-20241106193535-00472.warc.gz"} |
Custom Rivet
So I have a pretty good understanding of how matrices work in Maya, but that's a very different thing from being able to apply that knowledge. So I decided to try and create a matrix based rivet.
So attaching a locator to a surface is easy if you know about the four-by-four matrix node. Just create a point on surface info node and plug the position, normal vector, and one of the tangent
vectors into the matrix and then turn the matrix into tranform attributes using a decompose matrix node.
But that's been done before, so I wanted to try for something more complicated. I wanted to implement a twist and normal distance attribute on the rivet. The end result was this.
The difference between this system and a simple matrix rivet constraint is that instead of directly plugging in the position, I scale the normal vector and add it to the position vector. I also do
some trigonometry with the normal vector and the surface tangent vector to create the twist (that's what the expression node is there for).
In retrospect using a compose matrix node and a multiply matrix node would have been a far easier way to implement twist, but it was nice to practice applying trig to Maya (and refresh my expression
syntax knowledge).
If I ever decide to revamp my surface based facial rigging script then this would be a much better method of turning vectors into transformations (originally I used aim constraints).
This was also great practice for understanding the math behind the UVN deformer using matrices. | {"url":"https://www.jonah-reinhart.com/single-post/2018/02/18/custom-rivet","timestamp":"2024-11-14T08:36:21Z","content_type":"text/html","content_length":"1050465","record_id":"<urn:uuid:5ea63d54-1bf5-49dc-8c8c-7e9c6d4872e0>","cc-path":"CC-MAIN-2024-46/segments/1730477028545.2/warc/CC-MAIN-20241114062951-20241114092951-00836.warc.gz"} |
The functions in InformationValue package are broadly divided in following categories:
1. Diagnostics of predicted probability scores
2. Performance analysis
3. Functions that aid accuracy improvement
First, lets define the meaning of the various terms used in this document.
How to install
install.packages("InformationValue") # For stable CRAN version
devtools::install_github("InformationValue") # For latest dev version.
Definitions of functions
Sensitivity, a.k.a True Positive Rate is the proportion of the events (ones) that a model predicted correctly as events, for a given prediction probability cut-off.
Specificity, a.k.a * 1 - False Positive Rate* is the proportion of the non-events (zeros) that a model predicted correctly as non-events, for a given prediction probability cut-off.
False Positive Rate is the proportion of non-events (zeros) that were predicted as events (ones)
False Negative Rate is the proportion of events (ones) that were predicted as non-events (zeros)
Mis-classification error is the proportion of observations (both events and non-event) that were not predicted correctly.
Concordance is the percentage of all-possible-pairs-of-predicted Ones and Zeros where the scores of actual ones are greater than the scores of actual zeros. It represents the predictive power of a
binary classification model.
Weights of Evidence (WOE) provides a method of recoding the categorical x variable to continuous variables. For each category of a categorical variable, the WOE is calculated as:
$$WOE = ln\left(\frac{perc\ good\ of\ all\ goods}{perc\ bad\ of\ all\ bads} \right)$$
In above formula, goods is synonymous with ones, events, positives or responders and bads is synonymous with zeros, non-events, negatives or non-responders.
Information Value (IV) is a measure of the predictive capability of a categorical x variable to accurately predict the goods and bads. For each category of x, information value is computed as:
IV=(perc good of all goods−perc bad of all bads)*WOE
The total IV of a variable is the sum of IV’s of its categories. Here is what the values of IV mean according to Siddiqi (2006):
• Less than 0.02, then the predictor is not useful for modeling (separating the Goods from the Bads)
• 0.02 to 0.1, then the predictor has only a weak relationship.
• 0.1 to 0.3, then the predictor has a medium strength relationship.
• 0.3 or higher, then the predictor has a strong relationship.
Here is a sample MS Excel file that shows how to calculate WOE and Information Value.
KS Statistic or Kolmogorov-Smirnov statistic is the maximum difference between the cumulative true positive and cumulative false positive rate. It is often used as the deciding metric to judge the
efficacy of models in credit scoring. The higher the ks_stat, the more efficient is the model at capturing the responders (Ones). This should not be confused with the ks.test function.
1.1 plotROC
The plotROC uses the ggplot2 framework to create the ROC curve and prints the AUROC inside. It comes with an option to display the change points of the prediction probability scores on the graph if
you set the Show.labels = T.
plotROC(actuals=ActualsAndScores$Actuals, predictedScores=ActualsAndScores$PredictedScores)
You can also get the sensitivity matrix used to make the plot by turning on returnSensitivityMat = TRUE.
sensMat <- plotROC(actuals=ActualsAndScores$Actuals, predictedScores=ActualsAndScores$PredictedScores, returnSensitivityMat = TRUE)
1.2. sensitivity or recall
Sensitivity, also considered as the ‘True Positive Rate’ or ‘recall’ is the proportion of ‘Events’ (or ‘Ones’) correctly predicted by the model, for a given prediction probability cutoff score. The
default cutoff score for the specificity function unless specified by the threshold argument is taken as 0.5.
sensitivity(actuals = ActualsAndScores$Actuals, predictedScores = ActualsAndScores$PredictedScores)
#> [1] 1
If the objective of your problem is to maximise the ability of your model to detect the ‘Events’ (or ‘Ones’), even at the cost of wrongly predicting the non-events (‘Zeros’) as an event (‘One’), then
you could set the threshold as determined by the optimalCutoff() with optimiseFor='Ones'.
NOTE: This may not be the best example, because we are able to achieve the maximum sensitivity of 1 with the default cutoff of 0.5. However, I am showing this example just to understand how this
could be implemented in real projects.
max_sens_cutoff <- optimalCutoff(actuals=ActualsAndScores$Actuals, predictedScores = ActualsAndScores$PredictedScores, optimiseFor='Ones') # determine cutoff to maximise sensitivity.
print(max_sens_cutoff) # This would be cut-off score that achieved maximum sensitivity.
#> [1] 0.5531893
sensitivity(actuals = ActualsAndScores$Actuals, predictedScores = ActualsAndScores$PredictedScores, threshold=max_sens_cutoff)
#> [1] 1
1.3. specificity
For a given probability score cutoff (threshold), specificity computes what proportion of the total non-events (zeros) were predicted accurately. It can alo be computed as 1 - False Positive Rate. If
unless specified, the default threshold value is set as 0.5, which means, the values of ActualsAndScores$PredictedScores above 0.5 is considered as events (Ones).
specificity(actuals=ActualsAndScores$Actuals, predictedScores=ActualsAndScores$PredictedScores)
#> [1] 0.1411765
If you wish to know what proportion of non-events could be detected by lowering the threshold:
specificity(actuals=ActualsAndScores$Actuals, predictedScores=ActualsAndScores$PredictedScores, threshold = 0.35)
#> [1] 0.01176471
1.4. precision
For a given probability score cutoff (threshold), precision or ‘positive predictive value’ computes the proportion of the total events (ones) out of the total that were predicted to be events (ones).
precision(actuals=ActualsAndScores$Actuals, predictedScores=ActualsAndScores$PredictedScores)
#> [1] 0.5379747
1.5. npv
For a given probability score cutoff (threshold), npv or ‘negative predictive value’ computes the proportion of the total non-events (zeros) out of the total that were predicted to be non-events
npv(actuals=ActualsAndScores$Actuals, predictedScores=ActualsAndScores$PredictedScores)
#> [1] 1
1.6. youdensIndex
Youden’s J Index (Youden 1950), calculated as
represents the proportions of correctly predicted observations for both the events (Ones) and nonevents (Zeros). It is particularly useful if you want a single measure that accounts for both
false-positive and false-negative rates
youdensIndex(actuals=ActualsAndScores$Actuals, predictedScores=ActualsAndScores$PredictedScores)
#> [1] 0.1411765
2.1. misClassError
Mis-Classification Error is the proportion of all events that were incorrectly classified, for a given probability cutoff score.
misClassError(actuals=ActualsAndScores$Actuals, predictedScores=ActualsAndScores$PredictedScores, threshold=0.5)
#> [1] 0.4294
2.2. Concordance
Concordance is the percentage of predicted probability scores where the scores of actual positive’s are greater than the scores of actual negative’s. It is calculated by taking into account the
scores of all possible pairs of Ones and Zeros. If the concordance of a model is 100%, it means that, by tweaking the prediction probability cutoff, we could accurately predict all of the events and
Concordance(actuals=ActualsAndScores$Actuals, predictedScores=ActualsAndScores$PredictedScores)
#> $Concordance
#> [1] 0.8730796
#> $Discordance
#> [1] 0.1269204
#> $Tied
#> [1] 0
#> $Pairs
#> [1] 7225
2.3. somersD
somersD computes how many more concordant than discordant pairs exist divided by the total number of pairs. Larger the Somers D value, better model’s predictive ability.
$$Somers D = \frac{Concordant Pairs - Discordant Pairs}{Total Pairs} $$
somersD(actuals=ActualsAndScores$Actuals, predictedScores=ActualsAndScores$PredictedScores)
#> [1] 0.7461592
2.4. ks_stat
ks_stat computes the kolmogorov-smirnov statistic that is widely used in credit scoring to determine the efficacy of binary classification models. The higher the ks_stat more effective is the model
at capturing the responders.
ks_stat(actuals=ActualsAndScores$Actuals, predictedScores=ActualsAndScores$PredictedScores)
#> [1] 0.6118
2.5. ks_plot
ks_plot plots the lift is capturing the responders (Ones) against the the random case where we don’t use the model. The more curvier (higher) the model curve, the better is your model.
> ks_plot(actuals=ActualsAndScores$Actuals, predictedScores=ActualsAndScores$PredictedScores)
How to interpret this plot?
This plot aims to answer the question: Which part of the population should I target for my marketing campaign and what conversion rate can I expect? Now, lets understand how it is computed and what
those numbers mean.
After computing the prediction probability scores from a given logit model, the datapoints are sorted in descending order of prediction probability scores. This set of data points is split into 10
groups (or ranks) as marked on the X-axis, such that, the group ranked 1 contains the top 10% of datapoints with highest prediction probability scores, group ranked 2 containing the next 10% and so
The ‘random’ line in above chart corresponds to the case of capturing the responders (‘Ones’) by random selection, i.e., when you don’t have any model (generated probability scores) at disposal.
While the ‘model’ line represents the case of capturing the responders if you go by the model generated probability scores, where you begin by targeting datapoint with highest probability scores. In
simpler terms, it represents the proportion of total responders you can expect to capture as you keep targeting the data point starting from the bucket rank 1 through 10, in that order. So, now you
know which part of the population to target and what is the expected conversion rate.
For example, from the above chart for instance, by targeting first 40% of the population, the model will be able to capture 70.59% of total responders(Ones), while without the model, you can expect
to capture only 40% of responders by random targeting.
3.1. optimalCutoff
optimalCutoff determines the optimal threshold for prediction probability score based on your specific problem objectives. By adjusting the argument optimiseFor as follows, you can find the optimal
cutoff that: 1. Ones: maximizes detection of events or ones 2. Zeros: maximizes detection of non-events or zeros 3. Both: control both false positive rate and false negative rate by maximizing the
Youden’s J Index. 4. misclasserror: minimizes misclassification error (default)
> optimalCutoff(actuals = ActualsAndScores$Actuals, predictedScores = ActualsAndScores$PredictedScores) # returns cutoff that gives minimum misclassification error.
> optimalCutoff(actuals = ActualsAndScores$Actuals, predictedScores = ActualsAndScores$PredictedScores,
+ optimiseFor = "Both") # returns cutoff that gives maximum of Youden's J Index
> # > [1] 0.6431893
By setting the returnDiagnostics=TRUE you can get the sensitivityTable that shows the FPR, TPR, YOUDENSINDEX, SPECIFICITY, MISCLASSERROR for various values of cutoff.
> sens_table <- optimalCutoff(actuals = ActualsAndScores$Actuals, predictedScores = ActualsAndScores$PredictedScores,
+ optimiseFor = "Both", returnDiagnostics = TRUE)$sensitivityTable
3.2. WOE
Computes the Weights Of Evidence (WOE) for each group of a given categorical X and binary response Y.
WOE(X=SimData$X.Cat, Y=SimData$Y.Binary)
3.3. WOETable
Generates the WOE table showing the percentage goods, bads, WOE and IV for each category of X. WOE for a given category of X is computed as:
$$WOE = ln(\frac{perc.Good}{perc.Bad}) $$
options(scipen = 999, digits = 2)
WOETable(X=SimData$X.Cat, Y=SimData$Y.Binary)
CAT GOODS BADS TOTAL PCT_G PCT_B WOE IV
Group1 179 1500 1679 0.0246488571 0.0659108885 -0.9835731 0.0405842251
Group2 346 525 871 0.0476452768 0.0230688110 0.7253020 0.0178253591
Group3 560 1354 1914 0.0771137428 0.0594955620 0.2593798 0.0045698000
Group4 6 6 6 0.0008262187 0.0002636436 1.1422615 0.0006426079
Group5 4595 16369 20964 0.6327458001 0.7192635557 -0.1281591 0.0110880366
Group6 577 461 1038 0.0794546957 0.0202566131 1.3667057 0.0809063559
Group7 658 1670 2328 0.0906086478 0.0733807892 0.2108875 0.0036331398
Group8 327 859 1186 0.0450289177 0.0377449688 0.1764527 0.0012852725
Group9 14 14 14 0.0019278436 0.0006151683 1.1422615 0.0014994184
3.4. IV
Compute the information value of a given categorical X (Factor) and binary Y (numeric) response. The information value of a category of X is calculated as:
The IV of the categorical variables is the sum of information value of its individual categories.
options(scipen = 999, digits = 4)
IV(X=SimData$X.Cat, Y=SimData$Y.Binary)
#> 1] 0.162
#> attr(,“howgood”)
#> [1] “Highly Predictive”
“He who gives up code safety for code speed deserves neither.”
For more information and examples, visit rstatistics.net | {"url":"http://r-statistics.co/Information-Value-With-R.html","timestamp":"2024-11-09T23:27:44Z","content_type":"text/html","content_length":"50111","record_id":"<urn:uuid:da544d1f-25f5-4b85-8e9a-301a19bd083e>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.10/warc/CC-MAIN-20241109214337-20241110004337-00149.warc.gz"} |
Newton's Second Law of Motion
Bidang, Iban people. Sarawak 20th century, 50 x 101 cm. Lintah motif. From the Teo Family collection, Kuching. Photograph by D Dunlop.
Consider a composite atom P described by a repetitive chain of historically ordered events
$\Psi \left( \bar{r}, t \right) ^{\sf{P}} = \left( \sf{\Omega}_{1}, \sf{\Omega}_{2} \ldots \, \sf{\Omega}_{\it{i}} \, \ldots \, \sf{\Omega}_{\it{f}} \, \ldots \right)$
where each event $\sf{\Omega}$ is characterized by its momentum $\bar{p}$ and time of occurrence $t$. Let P interact with a particle called $\sf{X}$ between initial and final events so that there is
some variation in P's motion described by
$\Delta \bar{p} = \bar{p}_{\it{f}} - \bar{p}_{\it{i}}$
$\Delta t = t_{\it{f}} - t_{\it{i}}$
According to the usual narrative of Newtonian mechanics, the particle
impresses a
like a push or a pull that causes the change in P's motion.
Sir Isaac Newton
says that
"A change in motion is proportional to the motive force impressed and takes place along the straight line in which that force is impressed."^1
This relationship is called Newton's second law of motion. It can be mathematically expressed by defining an algebraic vector called the force as
$ \overline{F} \equiv \frac{ \Delta \overline{p} }{ \Delta t } $
If P is a Newtonian particle in dynamic equilibrium, then its momentum is related to its velocity by $\bar{p} = m \overline{\sf{v}}$ where $m$ is its mass. Also for many sorts of interactions the
mass of a Newtonian particle can be considered constant, then $\Delta \bar{p} = m \, \Delta\overline{\sf{v}}$ and the force can be written in terms of the acceleration, $\, \overline{a} \,$, as
$ \overline{F} = m \frac{ \Delta\overline{\sf{v}} }{ \Delta t } = m \overline{a} $
Momentum is conserved. So all particles that have momentum can cause changes in the momentum of another particle if they are absorbed or emitted. Then X, the particle that is absorbed or emitted, is
often called a force-carrying particle. Force-carrying particles that are material or charged are usually easy to detect in the immediate vicinity of an interaction. So the forces they impart are
called contact forces. Phenomena like automobile collisions and gunshot wounds can be understood using contact forces. X is much less conspicuous if it is ethereal and neutral. Such particles can be
difficult to detect, and the forces that they carry may seem to come from far away. So they are often referred to as exchange particles to suggest a remote origin, and their effects are called
action-at-a-distance forces. If X is imaginary then its force may seem like some random background fluctuation coming from nowhere specific. Many of these force-carrying particles are difficult to
characterize and distinguish as individuals, so it is often more convenient to group lots of them together and refer to them collectively as force fields. For example we may vaguely refer to a set of
photons as an electromagnetic field, or a collection of gravitons as a gravitational field.
Here is a link to the most recent version of this content, including the full text.
Newton's Second Law of Motion
page revision: 192, last edited: 02 Aug 2022 00:15 | {"url":"http://wikimechanics.org/force","timestamp":"2024-11-08T06:00:35Z","content_type":"application/xhtml+xml","content_length":"21302","record_id":"<urn:uuid:0c6253e0-0de6-407a-9a5d-274c8f1f7054>","cc-path":"CC-MAIN-2024-46/segments/1730477028025.14/warc/CC-MAIN-20241108035242-20241108065242-00568.warc.gz"} |
Texas Sharpshooter
Quick Note
I’m trying something new! This blog post is available in two places, both here and on a Jupyter notebook. Over there, you can tweak and execute my source code, using it as a sandbox for your own
explorations. Over here, it’s just a boring ol’ webpage without any fancy features, albeit one that’s easier to read on the go. Choose your own adventure!
Oh also, CONTENT WARNING: I’ll briefly be discussing sexual assault statistics from the USA at the start, in an abstract sense.
[5:08] Now this might seem pedantic to those not interested in athletics, but in the athletic world one percent is absolutely massive. Just take for example the 2016 Olympics. The difference
between first and second place in the men’s 100-meter sprint was 0.8%.
I’ve covered this argument from Rationality Rules before, but time has made me realise my original presentation had a problem.
His name is Steven Pinker.
%matplotlib inline
import matplotlib.pyplot as plt
import pandas as pd
assault = pd.read_csv('https://gitlab.com/hjhornbeck/texas_sharpshooter/raw/master/data/pinker_rape_usa.tsv',sep='\t')
plt.rcParams['figure.dpi'] = 96
plt.rcParams['figure.figsize'] = [9.5, 6]
plt.plot( assault['# Year'], assault['Rate'], 'b' )
plt.title('Forcible Rape, USA, Police reports')
plt.ylabel('Rate per 100,000')
He looks at that graph, and sees a decline in violence. I look at that chart, and see an increase in violence. How can two people look at the same data, and come to contradictory conclusions?
Simple, we’ve got at least two separate mental models.
import emcee
import numpy as np
import os
import scipy.optimize as spop
# Some of this code is based on the following examples:
# https://emcee.readthedocs.io/en/v2.2.1/user/line/
# https://jakevdp.github.io/blog/2014/06/14/frequentism-and-bayesianism-4-bayesian-in-python/
def lnLinear( theta, x, y ):
intercept, slope, lnStdDev = theta
prior = -1.5*np.log1p(slope*slope) - lnStdDev
model = slope*x + intercept
inv_sig2 = 1. / (model**2 * np.exp(2*lnStdDev))
return prior - .5*( np.sum( ((y-model)**2) * inv_sig2 - np.log(inv_sig2) ) )
# Model 1: What's happened over the last two decades?
negLnLin = lambda *args: -lnLinear(*args)
max_year = np.max( assault['# Year'] )
start = 1991
end = max_year
mask = assault['# Year'] > start
print("Finding the maximal likelihood, please wait ...", end='')
intercept_1, slope_1, error_1 = spop.minimize( negLnLin, [1000,-1,5],
args=(assault['# Year'][mask], assault['Rate'][mask]) )['x']
print(" done.")
# Model 2: Where are the extremes?
min_year = np.min( assault['# Year'] )
slope_2 = (float(assault['Rate'][assault['# Year']==max_year]) - float(assault['Rate'][assault['# Year']==min_year])) / \
(max_year - min_year)
intercept_2 = float(assault['Rate'][assault['# Year']==min_year] - slope_2*min_year)
# Model 3: What trendlines fit the data?
ndim, nwalkers, nsamples, keep = 3, 64, 300, 1
seed = [np.array([intercept_2,slope_2,3.]) + 1e-4*np.random.randn(ndim) for i in range(nwalkers)]
sampler = emcee.EnsembleSampler(nwalkers, ndim, lnLinear, threads=os.cpu_count(),
args=[ np.array(assault['# Year']), np.array(assault['Rate']) ] )
print("Running an MCMC sampler, please wait ...", end='')
sampler.run_mcmc(seed, nsamples)
print(" done.")
model_3 = sampler.chain[:, -keep:, :].reshape((-1, ndim))
print("Charting the results, please wait ...")
plt.plot( assault['# Year'], assault['Rate'], 'k' )
plt.ylabel('Rate per 100,000')
plt.plot( assault['# Year'], slope_1*assault['# Year'] + intercept_1, 'r' )
plt.plot( assault['# Year'], slope_2*assault['# Year'] + intercept_2, 'g' )
for entry in model_3:
plt.plot( assault['# Year'], entry[1]*assault['# Year'] + entry[0], 'b', alpha=0.05 )
plt.legend( ['Original Data', 'Model 1 (Pinker)', 'Model 2 (Mine)', 'Model 3 (Mine)'])
Finding the maximal likelihood, please wait ... done.
Running an MCMC sampler, please wait ... done.
Charting the results, please wait ...
All Pinker cares about is short-term trends here, as he’s focused on “The Great Decline” in crime since the 1990’s. His mental model looks at the general trend over the last two decades of data, and
discards the rest of the datapoints. It’s the model I’ve put in red.
I used two seperate models in my blog post. The first is quite crude: is the last datapoint better than the first? This model is quite intuitive, as it amounts to “leave the place in better shape
than when you arrived,” and it’s dead easy to calculate. It discards all but two datapoints, though, which is worse than Pinker’s model. I’ve put this one in green.
The best model, in my opinion, wouldn’t discard any datapoints. It would also incorporate as much uncertainty as possible about the system. Unsurprisingly, given my blogging history, I consider
Bayesian statistics to be the best way to represent uncertainty. A linear model is the best choice for general trends, so I went with a three-parameter likelihood and prior:
This third model encompasses all possible trendlines you could draw on the graph, but it doesn’t hold them all to be equally likely. Since time is short, I used an MCMC sampler to randomly sample the
resulting probability distribution, and charted that sample in blue. As you can imagine this requires a lot more calculation than the second model, but I can’t think of anything superior.
Which model is best depends on the context. If you were arguing just over the rate of police-reported sexual assault from 1992 to 2012, Pinker’s model would be pretty good if incomplete. However, his
whole schtick is that long-term trends show a decrease in violence, and when it comes to sexual violence in particular he’s the only one who dares to talk about this. He’s not being self-consistent,
which is easier to see when you make your implicit mental models explicit.
Pointing at Variance Isn’t Enough
Let’s return to Rationality Rules’ latest transphobic video. In the citations, he explicitly references the men’s 100m sprint at the 2016 Olympics. That’s a terribly narrow window to view athletic
performance through, so I tracked down the racetimes of all eight finalists on the IAAF’s website and tossed them into a spreadsheet.
dataset = pd.read_csv('https://gitlab.com/hjhornbeck/texas_sharpshooter/raw/master/data/100_metre.tsv',delimiter='\t',parse_dates=[1],dayfirst=True,dtype={'Result':np.float64,'Wind':np.float64})
olympics_2016 = dataset['Competition'] == "Rio de Janeiro Olympic Games"
finals = dataset['Race'] == "F"
fastest_time = min(dataset['Result'][olympics_2016 & finals])
table = {"Athlete":dataset['# Name'][olympics_2016 & finals],
"Result":dataset['Result'][olympics_2016 & finals],
"Delta":dataset['Result'][olympics_2016 & finals] - fastest_time
print("Rio de Janeiro Olympic Games, finals")
print( pd.DataFrame(table).sort_values("Result").to_string(index=False) )
Rio de Janeiro Olympic Games, finals
Athlete Result Delta
bolt 9.81 0.00
gatlin 9.89 0.08
de grasse 9.91 0.10
blake 9.93 0.12
simbine 9.94 0.13
meite 9.96 0.15
vicaut 10.04 0.23
bromell 10.06 0.25
Here, we see exactly what Rationality Rules sees: Usain Bolt, the current world record holder, earned himself another Olympic gold medal in the 100m sprint. First and third place are separated by a
tenth of a second, and the slowest person in the finals was a mere quarter of a second behind the fastest. That’s a small fraction of the time it takes to complete the event.
mask_2016 = (dataset['Date'] > '2016-01-01') & (dataset['Date'] < '2017-01-01')
names_2016 = pd.Categorical( dataset['# Name'] ).categories
all_2016 = list()
for name in names_2016:
temp = np.array( dataset['Result'][mask_2016 & (dataset['# Name'] == name)] )
all_2016.append( temp )
all_career = list()
for name in names_2016:
all_career.append( np.max( dataset['Date'][dataset['# Name'] == name] ) - np.min( dataset['Date'][dataset['# Name'] == name] ) )
all_races = list()
for name in names_2016:
all_races.append( len( dataset['Result'][dataset['# Name'] == name] ) )
mean_time = sorted( np.linspace(0,len(all_2016)-1,len(all_2016),dtype=int), key=lambda x:np.mean(all_2016[x]))
median_time = sorted( np.linspace(0,len(all_2016)-1,len(all_2016),dtype=int), key=lambda x:np.median(all_2016[x]))
min_time = sorted( np.linspace(0,len(all_2016)-1,len(all_2016),dtype=int), key=lambda x:np.min(all_2016[x]))
fastest_time = np.min([ np.min(data) for data in all_2016 ])
print('Race times in 2016, sorted by fastest time')
print("{0:16} {1:16} {2:16} {3:16} {4:16}".format('Name','Min time', 'Mean', 'Median', 'Personal max-min'))
print("{}".format( '-' * (6*16 + 5) ))
for i in min_time:
print("{0:16} {1:16} {2:12.2f} {3:12.2f} {4:12.2f}".format( names_2016[i], np.min(all_2016[i]), np.mean(all_2016[i]),
np.median(all_2016[i]), np.max(all_2016[i]) - np.min(all_2016[i]) ))
Race times in 2016, sorted by fastest time
Name Min time Mean Median Personal max-min
gatlin 9.8 9.95 9.94 0.39
bolt 9.81 9.98 10.01 0.34
bromell 9.84 10.00 10.01 0.30
vicaut 9.86 10.01 10.02 0.33
simbine 9.89 10.10 10.08 0.43
de grasse 9.91 10.07 10.04 0.41
blake 9.93 10.04 9.98 0.33
meite 9.95 10.10 10.05 0.44
Here, we see what I see: the person who won Olympic gold that year didn’t have the fastest time. That honour goes to Justin Gatlin, who squeaked ahead of Bolt by a hundredth of a second.
Come to think of it, isn’t the fastest time a poor judge of how good an athlete is? Picture one sprinter with a faster average time than another, and a second with a faster minimum time. The first
athlete will win more races than the second. By that metric, Gatlin’s lead grows to three hundredths of a second.
The mean, alas, is easily tugged around by outliers. If someone had an exceptionally good or bad race, they could easily shift their overall mean a decent ways from where the mean of every other
result lies. The median is a lot more resistant to the extremes, and thus a fairer measure of overall performance. By that metric, Bolt is now tied for third with Trayvon Bromell.
We could also judge how good an athlete is by how consistent they were in the given calendar year. By this metric, Bolt falls into fourth place behind Bromell, Jimmy Vicaut, and Yohan Blake. Even if
you don’t agree to this metric, notice how everyone’s race times in 2016 varies between three and four tenths of a second. It’s hard to argue that a performance edge of a tenth of a second matters
when even at the elite level sprinters’ times will vary by significantly more.
But let’s put on our Steven Pinker glasses. We don’t judge races by medians, we go by the fastest time. We don’t award records for the lowest average or most consistent performance, we go by the
fastest time. Yes, Bolt didn’t have the fastest 100m time in 2016, but now we’re down to hundredths of a second; if anything, we’ve dug up more evidence that itty-bitty performance differences
matter. If I’d just left things at that last paragraph, which is about as far as I progressed the argument last time, a Steven Pinker would likely have walked away even more convinced that
Rationality Rules got it right.
I don’t have to leave things there, though. This time around, I’ll make my mental model as explicit as possible. Hopefully by fully arguing the case, instead of dumping out data and hoping you and I
share the same mental model, I could manage to sway even a diehard skeptic. To further seal the deal, the Jupyter notebook will allow you to audit my thinking or even create your own model. No need
to take my word.
I’m laying everything out in clear sight. I hope you’ll give it all a look before dismissing me.
Model Behaviour
Our choice of model will be guided by the assumptions we make about how athletes perform in the 100 metre sprint. If we’re going to do this properly, we have to lay out those assumptions as clearly
as possible.
1. The Best Athlete Is the One Who Wins the Most. Our first problem is to decide what we mean by “best,” when it comes to the 100 metre sprint. Rather than use any metric like the lowest possible
time or the best overall performance, I’m going to settle on something I think we’ll both agree to: the athlete who wins the most races is the best. We’ll be pitting our models against each other
as many times as possible via virtual races, and see who comes out on top.
2. Pobody’s Nerfect. There is always going to be a spanner in the works. Maybe one athlete has a touch of the flu, maybe another is going through a bad breakup, maybe a third got a rock in their
shoe. Even if we can control for all that, human beings are complex machines with many moving parts. Our performance will vary. This means we can’t use point estimates for our model, like the
minimum or median race time, and instead must use a continuous statistical distribution.This assumption might seem like begging the question, as variance is central to my counter-argument, but
note that I’m only asserting there’s some variance. I’m not saying how much variance there is. It could easily be so small as to be inconsequential, in the process creating strong evidence that
Rationality Rules was right.
3. Physics Always Wins. No human being can run at the speed of light. For that matter, nobody is going to break the sound barrier during the 100 metre sprint. This assumption places a hard
constraint on our model, that there is a minimum time anyone could run the 100m. It rules out a number of potential candidates, like the Gaussian distribution, which allow negative times.
4. It’s Easier To Move Slow Than To Move Fast. This is kind of related to the last one, but it’s worth stating explicitly. Kinetic energy is proportional to the square of the velocity, so building
up speed requires dumping an ever-increasing amount of energy into the system. Thus our model should have a bias towards slower times, giving it a lopsided look.
Based on all the above, I propose the Gamma distribution would make a suitable model.
(Be careful not to confuse the distribution with the function. I may need the Gamma function to calculate the Gamma distribution, but the Gamma function isn’t a valid probability distribution.)
import scipy.stats as spst
x = np.linspace(0,10,1023)
print("Three versions of the Gamma Distribution")
plt.subplot( 131 )
plt.plot( x, spst.gamma.pdf(x, 5, scale=.7) )
plt.subplot( 132 )
plt.plot( x, spst.gamma.pdf(x, 1, scale=2) )
plt.subplot( 133 )
plt.plot( x, spst.gamma.pdf(x, 200, scale=.025) )
Three versions of the Gamma Distribution
It’s a remarkably flexible distribution, capable of duplicating both the Exponential and Gaussian distributions. That’s handy, as if one of our above assumptions is wrong the fitting process could
still come up with a good fit. Note that the Gamma distribution has a finite bound at zero, which is equivalent to stating that negative values are impossible. The variance can be expanded or
contracted arbitrarily, so it isn’t implicitly supporting my arguments. Best of all, we’re not restricted to anchor the distribution at zero. With a little tweak …
… we can shift that zero mark wherever we wish. The $b$ parameter sets the minimum value our model predicts, while α controls the underlying shape and β controls the scale or rate associated with
this distribution. α < 1 nets you the Exponential, and large values of α lead to something very Gaussian. Conveniently for me, SciPy already supports this three-parameter tweak.
My intuition is that the Gamma distribution on the left, with α > 1 but not too big, is the best model for athlete performance. That implies an athlete’s performance will hover around a specific
value, and while they’re capable of faster times those are more difficult to pull off. The Exponential distribution, with α < 1, is most favourable to Rationality Rules, as it asserts the race time
we’re most likely to observe is also the fastest time an athlete can do. We’ll never actually see that time, but what we observe will cluster around that minimum.
Running the Numbers
Enough chatter, let’s fit some models! For this one, my prior will be
which is pretty light and only exists to filter out garbage values.
import sys
def lnprob( theta, data ):
alpha, beta, b = theta
if (alpha <= 0) or (beta <= 0) or (b <= 0):
return -np.inf
return np.sum( spst.gamma.logpdf( data, alpha, scale=beta, loc=b ) )
ndim, nwalkers, nsamples, keep = 3, 64, 300, 5
models = [[] for x in all_2016]
summaries = [[] for x in all_2016]
print("Generating some models for 2016 race times (a few seconds each) ...")
print( "{:16}\t{:16}\t{:16}\t{:16}".format("# name","α","β","b") )
for loc,idx in enumerate(min_time):
data = all_2016[idx]
mean = np.mean( data ) - fastest_time # adjust for the location offset
seed = list()
i = 0
while i < nwalkers: beta = np.random.rand()*1.5 + .5 b = fastest_time - np.random.rand()*.3 if lnprob( [mean*beta, beta, b], data ) > -np.inf:
seed.append( [mean*beta, beta, b] )
i += 1
sampler = emcee.EnsembleSampler(nwalkers, ndim, lnprob, args=[data], threads=os.cpu_count())
sampler.run_mcmc(seed, nsamples)
samples = sampler.chain[:, -keep:, :].reshape((-1, ndim))
alpha_mcmc, beta_mcmc, b_mcmc = map(lambda v: (v[1], v[2]-v[1], v[1]-v[0]),
zip(*np.percentile(samples, [16, 50, 84], axis=0)))
print("{:16}".format(names_2016[idx]), end='')
print("\t{:.3f} (+{:.3f} -{:.3f})".format(*alpha_mcmc), end='')
print("\t{:.3f} (+{:.3f} -{:.3f})".format(*beta_mcmc), end='')
print("\t{:.3f} (+{:.3f} -{:.3f})".format(*b_mcmc))
models[idx] = samples
summaries[idx] = [*alpha_mcmc, *beta_mcmc, *b_mcmc]
print("... done.")
Generating some models for 2016 race times (a few seconds each) ...
# name α β b
gatlin 0.288 (+0.112 -0.075) 1.973 (+0.765 -0.511) 9.798 (+0.002 -0.016)
bolt 0.310 (+0.107 -0.083) 1.723 (+0.596 -0.459) 9.802 (+0.008 -0.025)
bromell 0.339 (+0.115 -0.082) 1.677 (+0.570 -0.404) 9.836 (+0.004 -0.032)
vicaut 0.332 (+0.066 -0.084) 1.576 (+0.315 -0.400) 9.856 (+0.004 -0.013)
simbine 0.401 (+0.077 -0.068) 1.327 (+0.256 -0.226) 9.887 (+0.003 -0.018)
de grasse 0.357 (+0.073 -0.082) 1.340 (+0.274 -0.307) 9.907 (+0.003 -0.022)
blake 0.289 (+0.103 -0.085) 1.223 (+0.437 -0.361) 9.929 (+0.001 -0.008)
meite 0.328 (+0.089 -0.067) 1.090 (+0.295 -0.222) 9.949 (+0.000 -0.003)
... done.
This text can’t change based on the results of the code, so this is only a guess, but I’m pretty sure you’re seeing a lot of α values less than one. That really had me worried when I first ran this
model, as I was already conceding ground to Rationality Rules by focusing only on the 100 metre sprint, where even I think that physiology plays a significant role. I did a few trial runs with a
prior that forced α > 1, but the resulting models would hug that threshold as tightly as possible. Comparing likelihoods, the α < 1 versions were always more likely than the α > 1 ones.
The fitting process was telling me my intuition was wrong, and the best model here is the one that most favours Rationality Rules. Look at the b values, too. There’s no way I could have sorted the
models based on that parameter before I fit them; instead, I sorted them by each athlete’s minimum time. Sure enough, the model is hugging the fastest time each athlete posted that year, rather than
a hypothetical minimum time they could achieve.
athlete = 0
x = np.linspace(9.5,11,300)
for i,row in enumerate(models[athlete]):
if i < 100:
plt.plot( x, spst.gamma.pdf(x, row[0], scale=row[1], loc=row[2]), alpha=0.05, color='k' )
plt.title("100 models of {}'s 2016 race times".format(names_2016[athlete]))
Charting some of the models in the posterior drives this home. I’ve looked at a few by tweaking the “player” variable, as well as the output of multiple sample runs, and they all are dominated by
Exponential distributions.
Dang, we’ve tilted the playing field quite a ways in Rationality Rules’ favour.
Still, let’s simulate some races. For each race, I’ll pick a random trio of parameters from each model’s posterior and feet that into SciPy’s random number routines to generate a race time for each
sprinter. Fastest time wins, and we tally up those wins to estimate the odds of any one sprinter coming in first.
Before running those simulations, though, we should make some predictions. Rationality Rules’ view is that (emphasis mine) …
[9:18] You see, I absolutely understand why we have and still do categorize sports based upon sex, as it’s simply the case that the vast majority of males have significant athletic advantages
over females, but strictly speaking it’s not due to their sex. It’s due to factors that heavily correlate with their sex, such as height, width, heart size, lung size, bone density, muscle mass,
muscle fiber type, hemoglobin, and so on. Or, in other words, sports are not segregated due to chromosomes, they’re segregated due to morphology.
[16:48] Which is to say that the attributes granted from male puberty that play a vital role in explosive events – such as height, width, limb length, and fast twitch muscle fibers – have not
been shown to be sufficiently mitigated by HRT in trans women.
[19:07] In some events – such as long-distance running, in which hemoglobin and slow-twitch muscle fibers are vital – I think there’s a strong argument to say no, [transgender women who
transitioned after puberty] don’t have an unfair advantage, as the primary attributes are sufficiently mitigated. But in most events, and especially those in which height, width, hip size, limb
length, muscle mass, and muscle fiber type are the primary attributes – such as weightlifting, sprinting, hammer throw, javelin, netball, boxing, karate, basketball, rugby, judo, rowing, hockey,
and many more – my answer is yes, most do have an unfair advantage.
… human morphology due to puberty is the primary determinant of race performance. Since our bodies change little after puberty, that implies your race performance should be both constant and
consistent. The most extreme version of this argument states that the fastest person should win 100% of the time. I doubt Rationality Rules holds that view, but I am pretty confident he’d place the
odds of the fastest person winning quite high.
The opposite view is that the winner is due to chance. Since there are eight athletes competing here, each would have a 12.5% chance of winning. I certainly don’t hold that view, but I do argue that
chance plays a significant role in who wins. I thus want the odds of the fastest person winning to be somewhere above 12.8%, but not too much higher.
simulations = 15000
wins = [0] * len(all_2016)
print("Simulating {} races, please wait ...".format(simulations), end='')
for sim in range(simulations):
times = list()
for athlete,_ in enumerate(all_2016):
choice = int( np.random.rand()*len(models[athlete]) )
times.append( models[athlete][choice][2] + np.random.gamma( models[athlete][choice][0], models[athlete][choice][1] ) )
wins[ np.argmin(times) ] += 1
print(" done.")
by_wins = sorted( np.linspace(0,len(all_2016)-1,len(all_2016),dtype=int), key=lambda x: wins[x], reverse=True)
print("Number of wins during simulation")
for i,athlete in enumerate(by_wins):
print( "{:24} {:8} ({:.2f}%)".format(names_2016[athlete], wins[athlete], wins[athlete]*100./simulations) )
Simulating 15000 races, please wait ... done.
Number of wins during simulation
gatlin 5174 (34.49%)
bolt 4611 (30.74%)
bromell 2286 (15.24%)
vicaut 1491 (9.94%)
simbine 530 (3.53%)
de grasse 513 (3.42%)
blake 278 (1.85%)
meite 117 (0.78%)
Whew! The fastest 100 metre sprinter of 2016 only had a one in three chance of winning Olympic gold. Of the eight athletes, three had odds better than chance of winning. Even with the field tilted in
favor of Rationality Rules, this strongly hints that other factors are more determinative of performance than fixed physiology.
But let’s put our Steven Pinker glasses back on for a moment. Yes, the odds of the fastest 100 metre sprinter winning the 2016 Olympics are surprisingly low, but look at the spread between first and
last place. What’s on my screen tells me that Gatlin is 40-50 times more likely to win Olympic gold than Ben Youssef Meite, which is a pretty substantial gap. Maybe we can rescue Rationality Rules?
In order for Meite to win, though, he didn’t just have to beat Gatlin. He had to also beat six other sprinters. If p[M] represents the geometric mean of Meite beating one sprinter, then his odds of
beating seven are p[M]^7. The same rationale applies to Gatlin, of course, but because the geometric mean of him beating seven other racers is higher than p[M], repeatedly multiplying it by itself
results in a much greater number. With a little math, we can use the number of wins above to estimate how well the first-place finisher would fare against the last-place finisher in a one-on-one
win_ratio = float(np.max(wins)) / float(np.min(wins))
prob_head2head = np.power( win_ratio, 1./7. ) / (1 + np.power( win_ratio, 1./7. ))
print("In the above simulation, {} was {:.1f} times more likely to win Olympic gold than {}.".format(
names_2016[by_wins[0]], win_ratio, names_2016[by_wins[-1]] ))
print("But we estimate that if they were racing head-to-head,", end='')
print(" {} would win only {:.1f}% of the time.".format( names_2016[by_wins[0]], prob_head2head*100. ))
difference = (np.min(all_2016[by_wins[-1]]) - np.min(all_2016[by_wins[0]])) / np.min(all_2016[by_wins[0]])
print(" (For reference, their best race times in 2016 differed by {:.2f}%.)".format( difference * 100. ))
In the above simulation, gatlin was 39.5 times more likely to win Olympic gold than meite.
But we estimate that if they were racing head-to-head, gatlin would win only 62.8% of the time.
(For reference, their best race times in 2016 differed by 1.53%.)
For comparison, FiveThirtyEight gave roughly those odds for Hilary Clinton becoming the president of the USA in 2016. That’s not all that high, given how “massive” the difference is in their best
race times that year.
This is just an estimate, though. Maybe if we pitted our models head-to-head, we’d get different results?
headCount = max( simulations >> 3, 100 )
maxFound = 0
print("Wins when racing head to head ({} simulations each)".format( headCount ))
print("{:10}".format("LOSER->"), end='')
for _,idx in enumerate(min_time):
print("{:>10}".format(names_2016[idx]), end='')
for x,x_ind in enumerate(min_time):
print("{:10}".format(names_2016[x_ind]), end='')
for y in range(len(min_time)):
if y <= x:
print("{:10}".format(""), end='')
wins = 0
for rand in range(headCount):
choice = int( np.random.rand()*len(models[x_ind]) )
x_time = models[x_ind][choice][1] + np.random.gamma( models[x_ind][choice][0], models[x_ind][choice][2] )
choice = int( np.random.rand()*len(models[min_time[y]]) )
y_time = models[min_time[y]][choice][1] + np.random.gamma( models[min_time[y]][choice][0], models[min_time[y]][choice][2] )
if y_time < x_time:
wins += 1
temp = wins*100./headCount
if temp < 50: temp = 100 - temp if temp > maxFound:
maxFound = temp
print("{:9.1f}%".format(wins*100./headCount), end='')
print("The best winning percentage was {:.1f}% (therefore the worst losing percent was {:.1f}%).".format(
maxFound, 100-maxFound ))
Wins when racing head to head (1875 simulations each)
LOSER-> gatlin bolt bromell vicaut simbine de grasse blake meite
gatlin 48.9% 52.1% 55.8% 56.4% 59.5% 63.5% 61.9%
bolt 52.2% 57.9% 55.8% 57.9% 65.8% 60.2%
bromell 52.4% 55.3% 55.0% 65.2% 59.0%
vicaut 51.7% 52.2% 59.8% 59.3%
simbine 52.3% 57.7% 57.1%
de grasse 57.0% 54.7%
blake 47.2%
The best winning percentage was 65.8% (therefore the worst losing percent was 34.2%).
Nope, it’s pretty much bang on! The columns of this chart represents the loser of the head-to-head, while the rows represent the winner. That number in the upper-right, then, represents the odds of
Gatlin coming in first against Meite. When I run the numbers, I usually get a percentage that’s less than 5 percentage points off. Since the odds of one person losing is the odds of the other person
winning, you can flip around who won and lost by subtracting the odds from 100%. That explains why I only calculated less than half of the match-ups.
I don’t know what’s on your screen, but I typically get one or two match-ups that are below 50%. I’m again organizing the calculations by each athlete’s fastest time in 2016, so if an athlete’s win
ratio was purely determined by that then every single value in this table would be equal to or above 50%. That’s usually the case, thanks to each model favouring the Exponential distribution, but
sometimes one sprinter still winds up with a better average time than a second’s fastest time. As pointed out earlier, that translates into more wins for the first athlete.
Getting Physical
Even at this elite level, you can see the odds of someone winning a head-to-head race are not terribly high. A layperson can create that much bias in a coin toss, yet we still both outcomes of that
toss to be equally likely.
This doesn’t really contradict Rationality Rules’ claim that fractions of a percent in performance matter, though. Each of these athletes differ in physiology, and while that may not have as much
effect as we thought it still has some effect. What we really need is a way to substract out the effects due to morphology.
If you read that old blog post, you know what’s coming next.
[16:48] Which is to say that the attributes granted from male puberty that play a vital role in explosive events – such as height, width, limb length, and fast twitch muscle fibers – have not
been shown to be sufficiently mitigated by HRT in trans women.
According to Rationality Rules, the physical traits that determine track performance are all set in place by puberty. Since puberty finishes roughly around age 15, and human beings can easily live to
75, that implies those traits are fixed for most of our lifespan. In practice that’s not quite true, as (for instance) human beings lose a bit of height in old age, but here we’re only dealing with
athletes in the prime of their career. Every attribute Rationality Rules lists is effectively constant.
So to truly put RR’s claim to the test, we need to fit our model to different parts of the same athlete’s career, and compare those head-to-head results with the ones where we raced athletes against
each other.
table = {"Athlete":[], "First Result":[], "Latest Result":[]}
for name in names_2016:
mask_name = dataset['# Name'] == name
dates = dataset["Date"][mask_name]
table['Athlete'].append( name )
table['First Result'].append( dates.min() )
table['Latest Result'].append( dates.max() )
print( pd.DataFrame(table) )
Athlete First Result Latest Result
0 blake 2005-07-13 2019-06-21
1 bolt 2007-07-18 2017-08-05
2 bromell 2012-04-06 2019-06-08
3 de grasse 2012-06-08 2019-06-20
4 gatlin 2000-05-13 2019-07-05
5 meite 2003-07-11 2018-06-16
6 simbine 2010-03-13 2019-06-20
7 vicaut 2008-07-05 2019-07-02
That dataset contains official IAAF times going back nearly two decades, in some cases, for those eight athletes. In the case of Bolt and Meite, those span their entire sprinting career.
Which athlete should we focus on? It’s tempting to go with Bolt, but he’s an outlier who broke the mathmatical models used to predict sprint times. Gatlin would have been my second choice, but
between his unusually long career and history of doping there’s a decent argument that he too is an outlier. Bromell seems free of any issue, so I’ll go with him. Don’t agree? I made changing the
athlete as simple as altering one variable, so you can pick whoever you like.
I’ll divide up these athlete’s careers by year, as their performance should be pretty constant over that timespan, and for this sport there’s usually enough datapoints within the year to get a decent
athlete = 2 # look at the indicies on the previous table
min_races = 3 # minimum number of races per year; filters out thin data
print("{0} vs. {0}, model building ...".format( names_2016[athlete] ))
mask_ath = dataset['# Name'] == names_2016[athlete]
min_year = np.min( dataset['Date'][ mask_ath ] ).year
max_year = np.max( dataset['Date'][ mask_ath ] ).year
years = list()
models_ath = list()
summaries_ath = list()
for year in range(min_year, max_year+1):
mask_year = (dataset['Date'] > '{}-01-01'.format(year)) & (dataset['Date'] < '{}-01-01'.format(year+1)) data = dataset['Result'][ mask_ath & mask_year ] if len(data) >= min_races:
years.append( year )
mean = np.mean( data ) - fastest_time # adjust for the bation offset
seed = list()
i = 0
while i < nwalkers: beta = np.random.rand()*1.5 + .5 b = fastest_time - np.random.rand()*.3 if lnprob( [mean*beta, beta, b], data ) > -np.inf:
seed.append( [mean*beta, beta, b] )
i += 1
sampler = emcee.EnsembleSampler(nwalkers, ndim, lnprob, args=[data], threads=os.cpu_count())
sampler.run_mcmc(seed, nsamples)
samples = sampler.chain[:, -keep:, :].reshape((-1, ndim))
alpha_mcmc, beta_mcmc, b_mcmc = map(lambda v: (v[1], v[2]-v[1], v[1]-v[0]),
zip(*np.percentile(samples, [16, 50, 84], axis=0)))
print("{}".format(year), end='')
print("\t{:.3f} (+{:.3f} -{:.3f})".format(*alpha_mcmc), end='')
print("\t{:.3f} (+{:.3f} -{:.3f})".format(*beta_mcmc), end='')
print("\t{:.3f} (+{:.3f} -{:.3f})".format(*b_mcmc))
models_ath.append( samples )
summaries_ath.append( [*alpha_mcmc, *beta_mcmc, *b_mcmc] )
print("... done.")
print("{0} vs. {0}, head to head ({1} simulations)".format( names_2016[athlete], headCount ))
print("{:7}".format("LOSER->"), end='')
for year in years:
print("{:>7}".format(year), end='')
maxFound = 0
for x_ind,x in enumerate( years ):
print("{:7}".format(x), end='')
for y_ind,y in enumerate( years ):
if y <= x:
print("{:7}".format(""), end='')
wins = 0
for rand in range(headCount):
choice = int( np.random.rand()*len(models_ath[x_ind]) )
x_time = models_ath[x_ind][choice][2] + np.random.gamma( models_ath[x_ind][choice][0], models_ath[x_ind][choice][1] )
choice = int( np.random.rand()*len(models_ath[y_ind]) )
y_time = models_ath[y_ind][choice][2] + np.random.gamma( models_ath[y_ind][choice][0], models_ath[y_ind][choice][1] )
if y_time < x_time:
wins += 1
temp = wins*100./headCount
if temp < 50: temp = 100 - temp if temp > maxFound:
maxFound = temp
print("{:6.1f}%".format(wins*100./headCount), end='')
print("The best winning percentage was {:.1f}% (therefore the worst losing percent was {:.1f}%).".format(
maxFound, 100-maxFound ))
bromell vs. bromell, model building ...
year α β b
2012 0.639 (+0.317 -0.219) 0.817 (+0.406 -0.280) 10.370 (+0.028 -0.415)
2013 0.662 (+0.157 -0.118) 1.090 (+0.258 -0.195) 9.970 (+0.018 -0.070)
2014 0.457 (+0.118 -0.070) 1.556 (+0.403 -0.238) 9.762 (+0.007 -0.035)
2015 0.312 (+0.069 -0.064) 2.082 (+0.459 -0.423) 9.758 (+0.002 -0.016)
2016 0.356 (+0.092 -0.104) 1.761 (+0.457 -0.513) 9.835 (+0.005 -0.037)
... done.
bromell vs. bromell, head to head (1875 simulations)
LOSER-> 2012 2013 2014 2015 2016
2012 61.3% 67.4% 74.3% 71.0%
2013 65.1% 70.7% 66.9%
2014 57.7% 48.7%
2015 40.2%
The best winning percentage was 74.3% (therefore the worst losing percent was 25.7%).
Again, I have no idea what you’re seeing, but I’ve looked at a number of Bromell vs. Bromell runs, and every one I’ve done shows at least as much variation, if not more, than runs that pit Bromell
against other athletes. Bromell vs. Bromell shows even more variation in success than the coin flip benchmark, giving us justification for saying Bromell has a significant advantage over Bromell.
I’ve also changed that variable myself, and seen the same pattern in other athletes. Worried about a lack of datapoints causing the model to “fuzz out” and cover a wide range of values? I thought of
that and restricted the code to filter out years with less than three races. Honestly, I think it puts my conclusion on firmer ground.
Texas Sharpshooter Fallacy: Ignoring the difference while focusing on the similarities, thus coming to an inaccurate conclusion. Similar to the gambler’s fallacy, this is an example of inserting
meaning into randomness.
Rationality Rules loves to point to sporting records and the outcome of single races, as on the surface these seem to justify his assertion that differences in performance of fractions of a percent
matter. In reality, he’s painting a bullseye around a very small subset of the data and ignoring the rest. When you include all the data, you find Rationality Rules has badly missed the mark.
Physiology cannot be as determinative as Rationality Rules claims, other factors must be important enough to sometimes overrule it.
And, at long last, I can call bullshit on this (emphasis mine):
[17:50] It’s important to stress, by the way, that these are just my views. I’m not a biologist, physiologist, or statistician, though I have had people check this video who are.
Either Rationality Rules found a statistician who has no idea of variance, which is like finding a computer scientist who doesn’t know boolean logic, or he never actually consulted a statistician.
Chalk up yet another lie in his column. | {"url":"https://freethoughtblogs.com/reprobate/2019/08/01/texas-sharpshooter/","timestamp":"2024-11-10T01:24:52Z","content_type":"text/html","content_length":"99106","record_id":"<urn:uuid:ea3488e0-e05f-4d1b-a416-661961cc9633>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.3/warc/CC-MAIN-20241110005602-20241110035602-00622.warc.gz"} |
Barodiffusion in a rotating binary mixture over an infinite rotating disk
Barodiffusion (diffusion of species brought about by pressure-gradients) in an isothermal, incompressible, Newtonian binary mixture in a steady, laminar, axisymmetric, rotating motion over an
infinite, rotating, impermeable disk is considered. In view of some potential applications the simplifying features of isotopic mixtures are taken into consideration in the formulation of the
problem. An exact solution for the barodiffusion problem analogous to the von Karman flow solutions is given. Results clearly identifying the small but significant separative action of the
pressure-gradients in this configuration are presented for Schmidt numbers of order unity (typical of gaseous mixtures).
Zeitschrift Angewandte Mathematik und Physik
Pub Date:
May 1975
□ Binary Mixtures;
□ Diffusion Theory;
□ Pressure Gradients;
□ Rotating Disks;
□ Rotating Fluids;
□ Axisymmetric Flow;
□ Gaseous Diffusion;
□ Laminar Flow;
□ Pressure Effects;
□ Schmidt Number;
□ Space Commercialization;
□ Steady Flow;
□ Von Karman Equation;
□ Fluid Mechanics and Heat Transfer;
□ Exact Solution;
□ Potential Application;
□ Gaseous Mixture;
□ Mathematical Method;
□ Binary Mixture | {"url":"https://ui.adsabs.harvard.edu/abs/1975ZaMP...26..337S/abstract","timestamp":"2024-11-13T21:16:15Z","content_type":"text/html","content_length":"37908","record_id":"<urn:uuid:b9808a1c-5813-4816-890a-be2d0efc1a04>","cc-path":"CC-MAIN-2024-46/segments/1730477028402.57/warc/CC-MAIN-20241113203454-20241113233454-00371.warc.gz"} |
ncl_ngritd: Given the coordinates of a point, this routine performs a rotation - Linux Manuals (3)
ncl_ngritd (3) - Linux Manuals
ncl_ngritd: Given the coordinates of a point, this routine performs a rotation
NGRITD - Given the coordinates of a point, this routine performs a rotation of that point about a specified axis by a specified angle.
CALL NGRITD (IAXS,ANGL,UCRD,VCRD,WCRD)
#include <ncarg/ncargC.h>
void c_ngritd(int iaxs, float angl, float *ucrd, float *ycrd,
float *wcrd)
IAXS (an input expression of type INTEGER) specifies the axis about which rotation is to be done (1 for the U axis, 2 for the V axis, and 3 for the W axis).
ANGL (an input expression of type REAL) specifies the magnitude, in degrees, of the rotation angle.
UCRD (an input/output variable of type REAL) specifies the U coordinate of the point being rotated.
VCRD (an input/output variable of type REAL) specifies the V coordinate of the point being rotated.
WCRD (an input/output variable of type REAL) specifies the W coordinate of the point being rotated.
The C binding argument descriptions are the same as the FORTRAN argument descriptions.
This routine is used by NGGCOG and NGGSOG to effect the rotations that are used to generate an object of a specified shape at a specified point on the surface of the globe.
NGRITD assumes that the UVW coordinate system is right-handed. Positive values of ANGL give counter-clockwise rotations and negative values of ANGL give clockwise rotations. (When IAXS = 1, ANGL = 90
carries the positive V axis into the positive W axis; when IAXS = 2, ANGL = 90 carries the positive W axis into the positive U axis; when IAXS = 3, ANGL = 90 carries the positive U axis into the
positive V axis.)
Use the ncargex command to see the following relevant example: cpex10.
To use NGRITD or c_ngritd, load the NCAR Graphics libraries ncarg, ncarg_gks, and ncarg_c, preferably in that order.
Copyright (C) 1987-2009
University Corporation for Atmospheric Research
The use of this Software is governed by a License Agreement. | {"url":"https://www.systutorials.com/docs/linux/man/3-ncl_ngritd/","timestamp":"2024-11-14T15:43:52Z","content_type":"text/html","content_length":"9537","record_id":"<urn:uuid:98197d2a-5a28-43a9-835d-1ad450b6c1f0>","cc-path":"CC-MAIN-2024-46/segments/1730477028657.76/warc/CC-MAIN-20241114130448-20241114160448-00418.warc.gz"} |
3 Digit By 3 Digit Multiplication Worksheets On Grid Paper
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Three digit Multiplication Practice Worksheet 03
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How do you multiply 3 digit numbers by 3 digits Line up both your numbers with one on top of the other Make sure that the places match up for the ones tens and hundreds Multiply the top number by the
last digit of the second number the ones unit Write down the answer to this beneath the line
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Encouraging constant method, offering support, and creating a favorable knowing environment are advantageous steps. | {"url":"https://crown-darts.com/en/3-digit-by-3-digit-multiplication-worksheets-on-grid-paper.html","timestamp":"2024-11-12T07:21:51Z","content_type":"text/html","content_length":"29607","record_id":"<urn:uuid:2b3dfd75-b392-4bdc-a52b-6720e6fc798d>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.58/warc/CC-MAIN-20241112045844-20241112075844-00605.warc.gz"} |
The "rule of transposition" is not a commonly used term in mathematics, but it might be referring to the concept of transposition in the context of equations. Transposition is a fundamental technique
used in algebra to manipulate equations and move terms from one side of an equation to the other side while preserving equality.
Here's how the rule of transposition works:-
Suppose you have an equation of the form ax + b = c, and you want to isolate the variable x. To do this, you can use the rule of transposition, which involves the following steps:
1. Start with the original equation:-ax + b = c.
2. To isolate x, move the term containing b to the other side of the equation:- Subtract b from both sides of the equation. This step is sometimes called "transposing" or "moving terms."Original
equation: ax + b = c . After transposition: ax = c − b
3. Now, the variable x is isolated on the left side of the equation:- To find the value of x\, divide both sides of the equation by the coefficient \(a:x = (c − b)/a. The result is an equation
where x is isolated on one side, and you can easily solve for the value of x. So, in summary, the rule of transposition involves changing the position of terms in an equation while maintaining
equality, and it's commonly used when solving linear equations to isolate variables.
The rule of transposition, also known as "moving terms" or "changing sides," is a fundamental concept in algebra that allows you to change the position of terms in an equation while maintaining
equality. This is particularly useful when isolating variables in equations. Let's look at an example:-
Suppose you have the following equation and you want to isolate the variable x:
3x + 7 = 22
To do this, you can use the rule of transposition, which involves changing the position of the terms to isolate x:
1. Start with the original equation: 3x + 7 = 22
2. Move the constant term (7) to the other side of the equation by subtracting it from both sides. This step is known as transposition:
3x = 22 − 7
1. Now, the variable x is isolated on the left side of the equation:- 3x = 15
2. To solve for x, divide both sides by the coefficient 3: 3x = 15
3. Simplify the right side: x = 5
So, the solution to the equation is x = 5. You've isolated the variable x by using the rule of transposition to move the constant term to the other side of the equation. | {"url":"https://www.math-edu-guide.com/CLASS-6-Algebra-Rules-Of-Transposition.html","timestamp":"2024-11-06T07:17:46Z","content_type":"text/html","content_length":"21053","record_id":"<urn:uuid:3eab8ab0-2b91-4b76-bfec-944dce6f1586>","cc-path":"CC-MAIN-2024-46/segments/1730477027910.12/warc/CC-MAIN-20241106065928-20241106095928-00354.warc.gz"} |
--- title: "Contextualizing tree distances" author: "[Martin R. Smith](https://smithlabdurham.github.io/)" output: rmarkdown::html_vignette bibliography: ../inst/REFERENCES.bib csl: ../inst/
apa-old-doi-prefix.csl vignette: > %\VignetteIndexEntry{Contextualizing tree distances} %\VignetteEngine{knitr::rmarkdown} %\VignetteEncoding{UTF-8} --- Once you understand [how to use "TreeDist"]
(Using-TreeDist.html) to calculate tree distances, the next step is to provide some context for the calculated distances. ## Normalizing The maximum value of most tree distance metrics scales with
the size of the trees being compared. Typically, the resolution of the trees also impacts the range of possible values. As such, it can be difficult to interpret the tree distance value without
suitable context. Normalizing a distance metric is one way to render its meaning more obvious. Selecting an appropriate normalizing constant may require careful consideration of the purpose to which
a tree distance metric is being put. The default normalization behaviour of each function when `normalize = TRUE` is listed in the [function reference](../reference/index.html), or can be viewed by
typing `?FunctionName` in the R terminal. ### Nye _et al._ tree similarity Let's work through a simple example using the Nye _et al_. [-@Nye2006] similarity metric to compare two imperfectly-resolved
trees. ```{r, fig.width=6, out.width="90%", fig.align="center"} library("TreeDist") tree1 <- ape::read.tree(text = '(A, ((B, ((C, D), (E, F))), (G, (H, (I, J, K)))));') tree2 <- ape::read.tree(text =
'(A, (B, (C, D, E, (J, K)), (F, (G, H, I))));') VisualizeMatching(NyeSimilarity, tree1, tree2, Plot = TreeDistPlot, matchZeros = FALSE) ``` This is a nice metric to start with, because the maximum
similarity between each pair of splits is defined as one. (Astute readers might worry that the minimum similarity is greater than zero -- that's a harder problem to overcome.) As such, the maximum
similarity possible between two 11-leaf trees is [`NSplits(11)`](https://ms609.github.io/TreeTools/reference/NSplits.html) = `r suppressMessages(library("TreeTools")); NSplits(11)`. Normalizing
against this value tells us how similar the two trees are, compared to two identical eleven-leaf binary trees. ```{r} NyeSimilarity(tree1, tree2, normalize = FALSE) / 8 NyeSimilarity(tree1, tree2,
normalize = 8) ``` This approach will result in a similarity score less than one if two trees are identical, but not fully resolved (i.e. binary). As such, we might prefer to compare the tree
similarity to the maximum score possible for two trees of the specified resolution. This value is given by the number of splits in the least resolved of the two trees: ```{r} NyeSimilarity(tree1,
tree2, normalize = min(TreeTools::NSplits(list(tree1, tree2)))) ``` More concisely, we can provide a normalizing function: ```{r} NyeSimilarity(tree1, tree2, normalize = min) ``` This approach will
produce a similarity of one if one tree is a less-resolved version of another (and thus not identical). If we are comparing lists of trees, this best value will depend on the number of splits in each
pair of trees. We can use the function `pmin()` to select the less resolved of each pair of trees: ```{r} NyeSimilarity(list(tree1, tree2), list(tree1, tree2), normalize = pmin) ``` To avoid these
limitations, we may instead opt to normalize against the average number of splits in the two trees. This is the default normalization method for [`NyeSimilarity()`](../reference/NyeSimilarity.html):
```{r} NyeSimilarity(tree1, tree2, normalize = TRUE) ``` Finally, if `tree1` is a "target" tree -- perhaps one that has been used to simulate data from, or which is independently known to be true or
virtuous -- we may wish to normalize against the best possible match to that tree. In that case, the best possible score is ```{r} TreeTools::NSplits(tree1) ``` and our normalized score will be
```{r} NyeSimilarity(tree1, tree2, normalize = TreeTools::NSplits(tree1)) ``` ### Normalizing to random similarity The diameter (maximum possible distance) of the Nye _et al_. tree similarity metric
is easy to calculate, but this is not the case for all metrics. For example, the clustering information distance metric [@SmithDist] ranges in principle from zero to the total clustering entropy
present in a pair of trees. But with even a modest number of leaves, no pairs of trees exist in which every split in one tree is perfectly contradicted by every other split in the other; as such, any
pair of trees will necessarily have some degree of similarity. In such a context, it can be relevant to normalize tree similarity against the _expected_ distance between a pair of random trees,
rather than a maximum value [see @Vinh2010]. On this measure, distances greater than one denote trees that are more different than expected by chance, whereas a distance of zero denotes identity.
With the quartet divergence, the expected tree distance is readily calculated: any given quartet has a one in three chance of matching by chance. ```{r} library("Quartet", exclude = "RobinsonFoulds")
expectedQD <- 2 / 3 normalizedQD <- QuartetDivergence(QuartetStatus(tree1, tree2), similarity = FALSE) / expectedQD ``` The expected distance is more difficult to calculate for other metrics, but can
be approximated by sampling random pairs of trees. Measured distances between 10 000 pairs of random bifurcating trees with up to 200 leaves are available in the data package '[TreeDistData](https://
github.com/ms609/TreeDistData/)'. We can view (normalized) distances for a selection of methods: ```{r, fig.width=7, fig.height=4, message=FALSE} if (requireNamespace("TreeDistData", quietly = TRUE))
{ library("TreeDistData", exclude = "PairwiseDistances") data("randomTreeDistances", package = "TreeDistData") methods <- c("pid", "cid", "nye", "qd") methodCol <- c(pid = "#e15659", cid = "#58a14e",
nye = "#edc949", qd = "#af7aa1") oldPar <- par(cex = 0.7, mar = c(5, 5, 0.01, 0.01)) nLeaves <- as.integer(dimnames(randomTreeDistances)[[3]]) plot(nLeaves, type = "n", randomTreeDistances["pid",
"mean", ], ylim = c(0.54, 1), xlab = "Number of leaves", ylab = "Normalized distance between random tree pairs") for (method in methods) { dat <- randomTreeDistances[method, , ] lines(nLeaves, dat
["50%", ], pch = 1, col = methodCol[method]) polygon(c(nLeaves, rev(nLeaves)), c(dat["25%", ], rev(dat["75%", ])), border = NA, col = paste0(methodCol[method], "55")) } text(202, randomTreeDistances
[methods, "50%", "200"] + 0.02, c("Different phylogenetic information", "Clustering information distance", expression(paste(plain("Nye "), italic("et al."))), "Quartet divergence" ), col = methodCol
[methods], pos = 2) par(oldPar) } ``` or use these calculated values to normalize our tree distance: ```{r, eval = FALSE} expectedCID <- randomTreeDistances["cid", "mean", "9"] ClusteringInfoDistance
(tree1, tree2, normalize = TRUE) / expectedCID ``` ## Testing similarity to a known tree Similarity has two components: precision and accuracy [@Smith2019]. A tree can be 80% similar to a target tree
because it contains 80% of the splits in the target tree, and no incorrect splits -- or because it is a binary tree in which 10% of the splits present are resolved incorrectly and are thus positively
misleading. In such a comparison, of course, it is more sensible to talk about split _information_ than just the number of splits: an even split may contain more information than two very uneven
splits, so the absence of two information-poor splits may be preferable to the absence of one information-rich split. As such, it is most instructive to think of the proportion of information that
has been correctly resolved: the goal is to find a tree that is as informative as possible about the true tree. Ternary diagrams allow us to visualise the quality of a reconstructed tree with
reference to a known "true" tree: ```{r fig.align="center", fig.height=1.8, fig.width=6, out.width="80%"} testTrees <- list( trueTree = ape::read.tree(text = '(a, (b, (c, (d, (e, (f, (g, h)))))));'),
lackRes = ape::read.tree(text = '(a, (b, c, (d, e, (f, g, h))));'), smallErr = ape::read.tree(text = '(a, (c, (b, (d, (f, (e, (g, h)))))));'), bigErr = ape::read.tree(text = '(a, (c, (((b, d), (f,
h)), (e, g))));') ) VisualizeMatching(MutualClusteringInfo, testTrees$trueTree, testTrees$lackRes) points(4, 7.5, pch = 2, cex = 3, col = "#E69F00", xpd = NA) VisualizeMatching(MutualClusteringInfo,
testTrees$trueTree, testTrees$smallErr) points(4, 7.5, pch = 3, cex = 3, col = "#56B4E9", xpd = NA) VisualizeMatching(MutualClusteringInfo, testTrees$trueTree, testTrees$bigErr) points(4, 7.5, pch =
4, cex = 3, col = "#009E73", xpd = NA) ``` Better trees plot vertically towards the "100% shared information" vertex. Resolution of trees increases towards the right; trees that are more resolved may
be no better than less-resolved trees if the addition of resolution introduces error. ```{r, fig.width=4, fig.align="center", fig.asp=1, out.width="50%"} if (requireNamespace("Ternary", quietly =
TRUE)) { library("Ternary") oldPar <- par(mar = rep(0.1, 4)) TernaryPlot(alab = "Absent information", blab = "Shared information", clab = "Misinformation", lab.cex = 0.8, lab.offset = 0.18, point =
"left", clockwise = FALSE, grid.col = "#dedede", grid.minor.lines = 0, axis.labels = 0:10 / 10, axis.col = "#aaaaaa") HorizontalGrid() correct <- MutualClusteringInfo(testTrees$trueTree, testTrees)
resolved <- ClusteringEntropy(testTrees) unresolved <- resolved["trueTree"] - resolved incorrect <- resolved - correct TernaryPoints(cbind(unresolved, correct, incorrect), pch = 1:4, cex = 2, col =
Ternary::cbPalette8[1:4]) par(oldPar) } ``` ### Example Here's a noddy real-world example applying this to a simulation-style study. First, let's generate a starting tree, which will represent our
reference topology: ```{r} set.seed(0) trueTree <- TreeTools::RandomTree(20, root = TRUE) ``` Then, let's generate 200 degraded trees. We'll move away from the true tree by making a TBR move, then
reduce resolution by taking the consensus of this tree and three trees from its immediate neighbourhood (one NNI move away). ```{r} treeSearchInstalled <- requireNamespace("TreeSearch", quietly =
TRUE) if (treeSearchInstalled) { library("TreeSearch", quietly = TRUE) # for TBR, NNI oneAway <- structure(lapply(seq_len(200), function(x) { tbrTree <- TBR(trueTree) ape::consensus(list(tbrTree, NNI
(tbrTree), NNI(tbrTree), NNI(tbrTree))) }), class = "multiPhylo") } else { message("Install \"TreeSearch\" to run this example") } ``` And let's generate 200 more trees that are even more degraded.
This time we'll move further (three TBR moves) from the true tree, and reduce resolution by taking a consensus with three trees from its wider neighbourhood (each two NNI moves away). ```{r} if
(treeSearchInstalled) { threeAway <- structure(lapply(seq_len(200), function(x) { tbrTree <- TBR(TBR(TBR(trueTree))) ape::consensus(list(tbrTree, NNI(NNI(tbrTree)), NNI(NNI(tbrTree)), NNI(NNI
(tbrTree)))) }), class = "multiPhylo") } ``` Now let's calculate their tree similarity scores. We need to calculate the amount of information each tree has in common with the true tree: ```{r} if
(treeSearchInstalled) { correct1 <- MutualClusteringInfo(trueTree, oneAway) correct3 <- MutualClusteringInfo(trueTree, threeAway) } ``` The amount of information in each degraded tree: ```{r} if
(treeSearchInstalled) { infoInTree1 <- ClusteringEntropy(oneAway) infoInTree3 <- ClusteringEntropy(threeAway) } ``` The amount of information that could have been resolved, but was not: ```{r} if
(treeSearchInstalled) { unresolved1 <- ClusteringEntropy(trueTree) - infoInTree1 unresolved3 <- ClusteringEntropy(trueTree) - infoInTree3 } ``` And the amount of information incorrectly resolved:
```{r} if (treeSearchInstalled) { incorrect1 <- infoInTree1 - correct1 incorrect3 <- infoInTree3 - correct3 } ``` In preparation for our plot, let's colour our one-away trees orange , and our
three-away trees blue : ```{r, collapse=TRUE} col1 <- hcl(200, alpha = 0.9) col3 <- hcl(40, alpha = 0.9) spec1 <- matrix(col2rgb(col1, alpha = TRUE), nrow = 4, ncol = 181) spec3 <- matrix(col2rgb
(col3, alpha = TRUE), nrow = 4, ncol = 181) spec1[4, ] <- spec3[4, ] <- 0:180 ColToHex <- function(x) rgb(x[1], x[2], x[3], x[4], maxColorValue = 255) spec1 <- apply(spec1, 2, ColToHex) spec3 <-
apply(spec3, 2, ColToHex) ``` Now we can plot this information on a ternary diagram. ```{r, fig.width=7, fig.align="center", fig.asp=5/7, out.width="70%"} if (treeSearchInstalled && requireNamespace
("Ternary", quietly = TRUE)) { layout(matrix(c(1, 2), ncol = 2), widths = c(5, 2)) oldPar <- par(mar = rep(0, 4)) TernaryPlot(alab = "Information absent in degraded tree", blab = "\n\nCorrect
information in degraded tree", clab = "Misinformation in degraded tree", point = "left", clockwise = FALSE, grid.minor.lines = 0, axis.labels = 0:10 / 10) HorizontalGrid() coords1 <- cbind
(unresolved1, correct1, incorrect1) coords3 <- cbind(unresolved3, correct3, incorrect3) ColourTernary(TernaryDensity(coords1, resolution = 20), spectrum = spec1) ColourTernary(TernaryDensity(coords3,
resolution = 20), spectrum = spec3) TernaryDensityContour(coords3, col = col3, nlevels = 4) TernaryDensityContour(coords1, col = col1, nlevels = 4) if (requireNamespace("kdensity", quietly = TRUE)) {
library("kdensity") HorizontalKDE <- function(dat, col, add = FALSE) { lty <- 1 lwd <- 2 kde <- kdensity(dat) kdeRange <- kdensity:::get_range(kde) if (add) { lines(kde(kdeRange), kdeRange, col =
col, lty = lty, lwd = lwd) } else { plot(kde(kdeRange), kdeRange, col = col, lty = lty, lwd = lwd, ylim = c(0, 1), main = "", axes = FALSE, type = "l") } # abline(h = 0:10 / 10) # Useful for
confirming alignment } par(mar = c(1.8, 0, 1.8, 0)) # align plot limits with ternary plot HorizontalKDE(correct1 / infoInTree1, col1, add = FALSE) HorizontalKDE(correct3 / infoInTree3, col3, add =
TRUE) mtext("\u2192 Normalized tree quality \u2192", 2) } par(oldPar) } else { message("Install \"TreeSearch\" and \"Ternary\" to generate this plot") } ``` In the ternary plot, the vertical
direction corresponds to the normalized tree quality, as depicted in the accompanying histogram. ## What next? You may wish to: - Explore the [Ternary package](https://ms609.github.io/Ternary/) -
[Interpret tree distance metrics](https://ms609.github.io/TreeDistData/articles/09-expected-similarity.html) - Compare trees with [different tips](different-leaves.html) - Review [available distance
measures](https://ms609.github.io/TreeDist/index.html) and the corresponding [TreeDist functions](https://ms609.github.io/TreeDist/reference/index.html#section-tree-distance-measures) - Construct
[tree spaces](treespace.html) to visualize landscapes of phylogenetic trees ## References | {"url":"https://cran.ma.ic.ac.uk/web/packages/TreeDist/vignettes/using-distances.Rmd","timestamp":"2024-11-13T05:18:26Z","content_type":"text/plain","content_length":"16103","record_id":"<urn:uuid:63a85107-81cc-4525-95de-cfc7e6adbb2b>","cc-path":"CC-MAIN-2024-46/segments/1730477028326.66/warc/CC-MAIN-20241113040054-20241113070054-00765.warc.gz"} |
Computation of sensitivity coefficients in fixed source simulations with SERPENT2
Aurelio Muttoni, Alain Nussbaumer, Xhemsi Malja
For the dimensioning and assessment of structures, it is common practice to compare action effects with sectional resistances. Extensive studies have been performed to quantify the model uncertainty
on the resistance side. However, for statically indetermi ...
Ernst & Sohn | {"url":"https://graphsearch.epfl.ch/en/publication/309839","timestamp":"2024-11-03T00:54:04Z","content_type":"text/html","content_length":"100652","record_id":"<urn:uuid:50d63ff8-cff9-47a5-af5a-d83470f6f048>","cc-path":"CC-MAIN-2024-46/segments/1730477027768.43/warc/CC-MAIN-20241102231001-20241103021001-00078.warc.gz"} |
FIR Filter Implementation in Verilog HDL
1) I need a simple to code in Verilog HDL for implementing a FIR Filter.
Filter equation is as under :
y[n] = 1/3(x[n] + x [n-1] + x[n-2]
2) Is there any book availaible for implementing DSPs in Verilog HDL..
Start a New Thread
The easiest (although maybe not the most efficient) way to implement this filter would be something like:
/** Verilog code */
module fir (
input signed [SAMPLE_WIDTH] new_sample, //incoming sample
input clk, //sample clock
output reg signed [SAMPLE_WIDTH] y); //filtered output
//use signed registers to perform signed arithmetic
always @ (posedge clk) begin
y <= (x_n + x_nm1 + x_nm2) / 3; //y[n]
x_nm2 <= x_nm1; //x[n-2]
x_nm1 <= x_n; //x[n-1]
x_n <= new_sample; //x[n]
/** End of Verilog code */
The biggest problem with this code is that it uses a divide. As you probably know, division is very slow and expensive in hardware, and so you're much better off trying to massage your filter
coefficients into powers of two in order to be able to perform division using a bit shift (shifting left by n corresponds to multiplication by 2^n, shifting right by n corresponds to division by 2^
n). If you're working with a large filter with many taps, and can't make them all into powers of 2, you might be better off using a system clock that runs (much) faster than the sample clock, and
sharing one multiplier between all filter taps.
For information on Verilog DSP work, I suggest you start by looking at some of the design examples given at
Hope this helps.
1) I need a simple to code in Verilog HDL for implementing a FIR Filter.
>Filter equation is as under :
>y[n] = 1/3(x[n] + x [n-1] + x[n-2]
>2) Is there any book availaible for implementing DSPs in Verilog HDL..
Start a New Thread
> 1) I need a simple to code in Verilog HDL for implementing a FIR Filter.
> >Filter equation is as under :
> >y[n] = 1/3(x[n] + x [n-1] + x[n-2]
> >
> >2) Is there any book availaible for implementing DSPs in Verilog HDL..
There's some Verilog code in Uwe Meyer-Baese's book on DSP with FPGAs. (I
have this book but haven't actually tried to use it yet, so can't give it a
thumbs-up or thumbs down.)
-- john, KE5FX
Start a New Thread
1) I need a simple to code in Verilog HDL for implementing a FIR Filter.
>Filter equation is as under :
>y[n] = 1/3(x[n] + x [n-1] + x[n-2]
The difference equation that was posted is simply a moving average. Couple things, look at the filter requirements (cutoff freq, position nulls, etc). If you change the order of the moving average to
4 you will not need the divide. There are resource efficient algorithms for moving average. A recursive structure (similar to a CIC filter) only requires adders.
I would also suggest using something like MyHDL (www.myhdl.org) to design, implement, and simulate any DSP HDL. It is much easier to integrate signal processing simulation with the HDL simulation.
Makes analyzing and verifying DSP blocks much easier.
>2) Is there any book availaible for implementing DSPs in Verilog HDL..
The Uwe Meyer-Baese book is decent. As previously mentioned it has Verilog and VHDL examples. A general DSP book like Lyons' "Understanding Digital Signal Processing" would also be a good book.
Couple other VSLI DSP books out there as well, Parhi and Wanhammar. The later books are a more in depth but good. I don't think the Parhi has HDL (Verilog/VHDL) examples but the others do.
Start a New Thread | {"url":"https://www.dsprelated.com/showthread/fpgadsp/46-1.php","timestamp":"2024-11-12T20:03:43Z","content_type":"text/html","content_length":"60936","record_id":"<urn:uuid:883522ac-6d72-441e-951e-a8accfb8bb4a>","cc-path":"CC-MAIN-2024-46/segments/1730477028279.73/warc/CC-MAIN-20241112180608-20241112210608-00399.warc.gz"} |
Dissecting the Pragmatic Speeder's Rationale
Drivers typically speed for one of two reasons: they derive pleasure from going fast, or they want to arrive at their destination sooner. In this post I analyze how effective speeding is at "making
up time".
There's no question that driving faster will shorten your trip time, assuming you don't end up in an accident or stopped by a cop to be handed a speeding ticket because of it. Well, how much time do
you actually save?
Let's say you're traveling a distance
down a highway (no stops along the way) where the posted speed limit is
. We'll call the duration of the trip at the speed limit
, which would be calculated from Equation 1.
Equation 1: Duration of trip if driven at the posted speed limit
Now you're thinking to yourself that you're not actually going to drive the speed limit, but some other speed
. We'll bring in another quantity, your speed ratio, alpha, which is simply
divided by
. If you're speeding, alpha > 1. If you're driving below the speed limit (say due to traffic or weather), alpha < 1.
Equation 2: Ratio of actual speed to posted speed limit
So the actual amount time your trip will take at velocity
, calculated from Equation 3 below.
Equation 3: Actual duration of trip
From here we can calculate how deviating from the posted speed limit will affect trip duration, whether you're speeding in an effort to make up time or trying to figure out how much extra time you'll
need when you're driving through a snowstorm.
Equation 4: Difference between actual duration and minimum legal duration
So if you want to cut your trip duration by just 10%, you need to drive 11.1% faster than the posted speed limit. A 10% cut in trip duration is only a difference of 6 minutes per hour. In a lot of
places you can get away with 15 to 20% over the limit without drawing the attention of police, but even at 20% over you only cut the trip duration by 16.7% (10 minutes per hour). Not really an
appreciable amount of time for typical trips. Say you left the house for work late and are trying to make up 10 minutes on a 30 minute commute. In that case you'd need to increase your average speed
by 50%! Good way to draw the attention of police, not to mention the several-fold increase to your risk of involvement in a fatal collision. Here's a graph and table to show you how your speed ratio
affects your trip duration.
To summarize, speeding a little bit saves you an insignificant amount of time, but in order to make up an appreciable amount of time you need to drive like a maniac. If you consider city driving
instead of highway driving, the effect of speeding is even less significant because you spend so much time stopped at red lights. So, while it is technically true that speeding shortens your trip
duration, the savings are hardly worth the added risks or costs associated with speeding, such as significantly increased likelihood of involvement in a fatal collision, wasted fuel, speeding fines,
and higher insurance premiums. | {"url":"http://alohonyai.blogspot.com/2015/11/dissecting-pragmatic-speeders-rationale.html","timestamp":"2024-11-05T03:31:25Z","content_type":"text/html","content_length":"87229","record_id":"<urn:uuid:71670bb9-4d26-4b63-b626-c0e250fc4e86>","cc-path":"CC-MAIN-2024-46/segments/1730477027870.7/warc/CC-MAIN-20241105021014-20241105051014-00502.warc.gz"} |
Семинары: Victor-Emmanuel Brunel, On the estimation of convex polytopes
Аннотация: Set estimation started to interest scientists since the 1960's, even earlier, when stochastic geometry was developing much, dealing with many questions related to optimization,
approximation, imaging, economics, ... Although it was studied from a probabilistic and geometric prospective, it really arose as a statistical question in the early 1990's, when boundary fragments
and convex sets were investigated in the minimax framework. I will give a brief historical summary of set estimation, and I will focus on the estimation of convex polytopes. In particular I will
propose new estimators that are nearly optimal in the minimax setup, and I will talk about adaptation with respect to the number of vertices of the unknown polytope. | {"url":"https://m.mathnet.ru/php/seminars.phtml?option_lang=rus&presentid=6966","timestamp":"2024-11-11T14:02:15Z","content_type":"text/html","content_length":"6880","record_id":"<urn:uuid:1514f4dc-8e97-490e-8aa7-2598ead2f2ef>","cc-path":"CC-MAIN-2024-46/segments/1730477028230.68/warc/CC-MAIN-20241111123424-20241111153424-00300.warc.gz"} |
Riemannian Projection-free Online Learning
Riemannian Projection-free Online Learning
Zihao Hu · Guanghui Wang · Jacob Abernethy
Great Hall & Hall B1+B2 (level 1) #505
Abstract: The projection operation is a critical component in a wide range of optimization algorithms, such as online gradient descent (OGD), for enforcing constraints and achieving optimal regret
bounds. However, it suffers from computational complexity limitations in high-dimensional settings or when dealing with ill-conditioned constraint sets. Projection-free algorithms address this issue
by replacing the projection oracle with more efficient optimization subroutines. But to date, these methods have been developed primarily in the Euclidean setting, and while there has been growing
interest in optimization on Riemannian manifolds, there has been essentially no work in trying to utilize projection-free tools here. An apparent issue is that non-trivial affine functions are
generally non-convex in such domains. In this paper, we present methods for obtaining sub-linear regret guarantees in online geodesically convex optimization on curved spaces for two scenarios: when
we have access to (a) a separation oracle or (b) a linear optimization oracle. For geodesically convex losses, and when a separation oracle is available, our algorithms achieve $O(T^{\frac{1}{2}})$,
$O(T^{\frac{3}{4}})$ and $O(T^{\frac{1}{2}})$ adaptive regret guarantees in the full information setting, the bandit setting with one-point feedback and the bandit setting with two-point feedback,
respectively. When a linear optimization oracle is available, we obtain regret rates of $O(T^{\frac{3}{4}})$ for geodesically convex losses and $O(T^{\frac{2}{3}}\log T)$ for strongly geodesically
convex losses. | {"url":"https://nips.cc/virtual/2023/poster/70236","timestamp":"2024-11-10T09:34:47Z","content_type":"text/html","content_length":"47770","record_id":"<urn:uuid:ce124c74-ada7-4ec8-83d5-a61beaf2c9ad>","cc-path":"CC-MAIN-2024-46/segments/1730477028179.55/warc/CC-MAIN-20241110072033-20241110102033-00275.warc.gz"} |
Maximum mutational robustness in genotype-phenotype maps follows a self-similar blancmange-like curve
Data from: Maximum mutational robustness in genotype-phenotype maps follows a self-similar blancmange-like curve
Data files
Jul 02, 2023 version files 1.62 MB
Phenotype robustness, defined as the average mutational robustness of all the genotypes that map to a given phenotype, plays a key role in facilitating neutral exploration of novel phenotypic
variation by an evolving population. By applying results from coding theory, we prove that the maximum phenotype robustness occurs when genotypes are organised as bricklayer’s graphs, so called
because they resemble the way in which a bricklayer would fill in a Hamming graph. The value of the maximal robustness is given by a fractal continuous everywhere but differentiable nowhere
sums-of-digits function from number theory. Interestingly, genotype-phenotype (GP) maps for RNA secondary structure and the HP model for protein folding can exhibit phenotype robustness that exactly
attains this upper bound. By exploiting properties of the sums-of-digits function, we prove a lower bound on the deviation of the maximum robustness of phenotypes with multiple neutral components
from the bricklayer’s graph bound, and show that RNA secondary structure phenotypes obey this bound. Finally, we show how robustness changes when phenotypes are coarse-grained and derive a formula
and associated bounds for the transition probabilities between such phenotypes.
This data contains the data and code required to generate presented in "Maximum Mutational Robustness in Genotype-Phenotype Maps Follows a Self-similar Blancmange-like Curve" by Mohanty et al.,
published in Journal of the Royal Society Interface.
The exact maximum robustness curve corresponding to the robustness of the bricklayer's graphs, as well as the interpolated curve, can be generated from the RoBound Calculator, available free of
charge and open source on GitHub (https://github.com/vaibhav-mohanty/RoBound-Calculator).
All bounds (e.g. Figure 1, 9, 10, and 11 as well as the bounds shown in Figure 3 or 7) can be calculated using the RoBound Calculator (https://github.com/vaibhav-mohanty/RoBound-Calculator).
In Figure 3, the RNA and HP model neutral component sizes and robustness values are provided in the files hp5x5_components.csv, hp24_components.csv, rna12_components.csv, and rna15_components.csv.
These results obtainedd from Greenbury et al., "Genetic Correlations Greatly Increase Mutational Robustness and Can Both Reduce and Enhance Evolvability," PLOS Computational Biology, 2016.
In Figure 4, we show the deviation of RNA12 and RNA15 neutral networks from the maximum (bricklayer's) robustness. This data is obtained from rna12.csv (alternatively rna12.mat) and rna15.csv
(alternatively rna15.mat), which were also results obtained from Greenbury et al., "Genetic Correlations Greatly Increase Mutational Robustness and Can Both Reduce and Enhance Evolvability," PLOS
Computational Biology, 2016. The code to produce the figure is found in nc_err_corr.m
In Figure 7, the bounds can be calculated using the RoBound Calculator (https://github.com/vaibhav-mohanty/RoBound-Calculator). The raw data is obtained from using ViennaRNA (https://
www.tbi.univie.ac.at/RNA/) to calculate the dot-bracket structures. These structures are then fed into the RNASHAPES software (https://bibiserv.cebitec.uni-bielefeld.de/rnashapes) to generate the
coarse-grained data. The data files rna12abstract.mat and rna15abstract.mat contain the frequency and robustness values for the neutral networks at various levels of coarse-graining.
In Figure 8, the RNA12 transition probabilites between phenotypes can be obtained from rna12_theta.csv, which provides the phi_pq matrix when each column's sum is normalized to 1. The script
phi_critical_ranges.m produces Figure 8.
Figures 2, 5, and 6 are schematics and have no associated data.
Usage notes
CSV files can be opened easily. MATLAB (or associated python packages) can open the .mat files. MATLAB or Octave can be used to load the .m scripts.
Works referencing this dataset | {"url":"https://datadryad.org:443/stash/dataset/doi:10.5061/dryad.sj3tx969f","timestamp":"2024-11-11T10:53:34Z","content_type":"text/html","content_length":"52337","record_id":"<urn:uuid:9e79ca9a-4055-4591-9521-53193067cd92>","cc-path":"CC-MAIN-2024-46/segments/1730477028228.41/warc/CC-MAIN-20241111091854-20241111121854-00859.warc.gz"} |
What Wins Ball Games?
By: Tom Seifert
In Major League Baseball, a wide array of traditional and advanced statistics can be used to evaluate players, teams, or the league as a whole. With such a diverse set of metrics to evaluate
performance, there is no consensus among players, executives, or fans about which of these metrics measure performance most successfully. While advanced statistics like wRC+ (weighted runs created
plus) attempt to quantify a player or team's performance with a single value, the foundations for these metrics always begin with traditional statistics such as hits, strikeouts, and walks.
Similarly, this article will consider purely traditional team metrics while applying statistical regression methods to identify what metrics correlate to wins and to what degree these correlative
metrics affect a team's win count.
The data set I will analyze contains season data for all MLB teams for 2019, 2021, and 2022 (2020 is omitted because of the shortened season due to COVID-19), a total of 90 observations. My outcome
variable will be the number of games won by a team in a particular season, and I will attempt to create a multiple linear regression model to assess the significance of the linear relationship
between a team's number of wins and several of a team's offensive, defensive, and miscellaneous team statistics from a given season.Â
The offensive predictors include a team's number of hits, number of home runs, number of walks, and number of strikeouts by batters in a season. The defensive predictors are number of hits allowed,
number of home runs allowed, number of walks allowed, number of strikeouts by pitchers, and fielding percentage (% of plays where an error is not made) in a season. The last predictor will be the
number of fans who attended a team's games in the season. I will utilize statistical testing, assumption verification, and model selection to identify which combination of these ten predictors most
effectively predicts an MLB team's number of wins in a season.
Here is a description of each variable used in the regression analysis:
Summary Statistics
Looking at the summary statistics for all of the variables, it can be seen that there is a variety of distributions among the variables, all of which are numerical. Numbers like attendance can go as
high as almost four million with a 12-digit variance, while fielding percentage can only reach one at maximum and has a near zero variance.
I will now construct a multiple linear regression model and perform a t-test for slopes to test whether each individual predictor is significant in predicting a team's number of wins in a season.Â
Model Summary and T-Tests
Using a significance level of 0.05, the t-test results show that six of our predictors have p-values below 0.05 and are thus significant, while attendance, FP, SO, and SOA have p-values above 0.05,
rendering them insignificant. This indicates that the estimated slopes of the significant variables (H, HR, BB, HA, HRA, and BBA) differ significantly from 0, and they thus have a significant effect
on wins, the outcome variable in our model. Conversely, FP, SO, and SOA do not have a significant effect on the outcome variable because their estimated slopes do not significantly differ from 0.Â
The model resulted in an R-squared value of 0.897 and a 68.8 F-statistic with a p-value of 2.2x10^-16^ for the F-test. These results communicate that the model is highly likely to have at least one
significant predictor, and the predictor variables explain 89.7% of the variation in wins.Â
Verifying Assumptions
In order to further assess the model's goodness of fit, the assumptions for multiple linear regression must be verified. The assumptions that must be verified include linearity between the response
variable and the predictors, normality of the error terms, constant variance of the error terms, and no multicollinearity between predictors. It is also worthwhile to check for any outliers or
influential points using standardized residuals, leverage, and Cook's Distance.
Here are four diagnostic plots that will help check the model assumptions:
To satisfy the assumption of linearity, the residuals must be randomly scattered about the line y = 0 in the residuals vs. fitted values plot. This holds true, so the linearity assumption is met by
the model.
In order to verify the assumption of normality of the error terms, the normal Q-Q plot must form an approximately straight line. This holds true for the normal Q-Q plot shown, verifying the normality
of the error terms.
Constant Variance
The assumption of constant variance of the error terms can be checked in the residuals vs. fitted plot. In this plot, the spread of the residuals around the line y = 0 is fairly consistent and does
not significantly change with the fitted values, so the constant variance assumption is met.
To check for multicollinearity, the Variance Inflation Factor (VIF) for each variable in the model can be calculated. If none of the VIF values are greater than 5, then the correlation between
predictors does not disrupt the coefficient estimates. The VIF values for each predictor are shown below.
None of these values exceed 5, so the assumption of no multicollinearity between predictors is satisfied.
Outliers and Influential Points
Using standardized residuals, leverage, and Cook's Distance, outliers and influential points can be identified. By looking at the standardized residuals vs. leverage plot, there is only one with
standardized residuals above 2 or below -2, making it an outlier. There are twenty-two high leverage points, which are categorized by having leverage values higher than 2(p+1)/n (p = number of
predictors, n = number of observations), or higher than two times the average leverage value in the model, (p+1)/n. These leverage points have a significant effect on the model, but are not
necessarily outliers, as a good leverage point will have a high influence on the model while still following the regression pattern. There are eight points with Cook's Distances greater than 4/(n -
2), making them outliers.Â
Despite the outliers and influential points present in the model, none of the influential points are outliers, so they do not disrupt model estimates very much. In addition, these points do not cause
any of the model assumptions to be violated. Therefore, it will be not necessary to transform this model to account for outliers.Â
After verifying the assumptions, the model can be improved using model selection to ensure the best combination of predictor variables is included in the model. This method may also help reduce the
number of outliers and influential points in the model.
Variable Selection
The model is shown to meet the assumptions of multiple linear regression. Now, model selection can be used to find the best combination of predictors in predicting the outcome variable, wins. The
forward regression model selection method searches for the best-fitting model by starting with no predictors and continuously adding the most significant predictor not yet in the model, eventually
resulting in the original full model. The best of the ten models can then be identified by looking for the highest adjusted R-squared, lowest CP, and lowest BIC between the models. The best-fitting
models will have the best predicting power without introducing too many variables, as this complexity could cause over-fitting, worsening the model's ability to generalize the model to new
observations accurately.
Here are graphical comparisons between the ten models based on their adjusted R-squared, CP, and BIC values.
Using forward selection, the BIC and CP of the models stop significantly decreasing after the model with six variables is considered. R-squared stops significantly increasing after the model with six
variables. The model with seven variables has a slightly lower value of CP and a slightly higher value of R-squared in comparison to the six-variable model.Â
In order to choose between the six and seven-variable models, I will conduct a partial F-test to analyze the significance of the predictor that is present in the seven-variable model and absent in
the six-variable model. The null and alternative hypotheses for the test are as follows:
H-Null: The reduced model is more significant in predicting wins than the full model
H-1: The full model is a more significant predictor of wins than the reduced model
As shown, the p-value for this test is greater than the significance level of 0.05, so we fail to reject the null hypothesis that the reduced model outperforms the model using FP as a predictor,
meaning that FP it is an insignificant predictor in our model and the reduced six variable model will be chosen as the final model. It is also worth noting that the six-variable model has only two
high leverage points, a large decrease from the twenty-two high leverage points present in the original model. The final model summary is shown below:
After investigating the relationship between the number of games an MLB team wins in a season and ten predictor variables with a goal of creating a multiple linear regression model to fit the
relationship, six predictors were chosen for the final wins model: hits, home runs, walks, hits allowed, home runs allowed, and walks allowed.
These results suggest that the number of hits, home runs, and walks are important metrics of baseball that teams should try to maximize on offense and limit for opposing teams on defense in order to
increase win potential.Â
However, the model does not fully explain the variation in team wins; only 89.23%. The outliers and influential points in the data set could be accounted for more effectively by utilizing the
weighted least squares method, which could improve the model's goodness of fit. If given more time, further investigation of other predictor variables and more observations of data could yield an
improved model.
Further application of the model could involve introducing new data to test the model on in order to evaluate how well it predicts a team's number of wins. Assuming that the model predicts team wins
fairly accurately, further investigation of how to maximize and minimize metrics positively and negatively correlated with wins respectively could help MLB teams train and prepare their players and
rosters to give them the best chance to succeed. | {"url":"https://www.bruinsportsanalytics.com/post/what-wins-ball-games","timestamp":"2024-11-10T22:26:31Z","content_type":"text/html","content_length":"1050490","record_id":"<urn:uuid:6607052a-833c-45f3-9258-62ff2435a914>","cc-path":"CC-MAIN-2024-46/segments/1730477028191.83/warc/CC-MAIN-20241110201420-20241110231420-00783.warc.gz"} |
Logarithmic inequalities under a symmetric polynomial dominance order
We consider a dominance order on positive vectors induced by the elementary symmetric polynomials. Under this dominance order we provide conditions that yield simple proofs of several monotonicity
questions. Notably, our approach yields a quick (4 line) proof of the so-called “sum-of-squared-logarithms” inequality conjectured in (Bîrsan, Neff, and Lankeit, J. Inequalities and Applications
(2013); P. Neff, Y. Nakatsukasa, and A. Fischle; SIMAX, 35, 2014). This inequality has been the subject of several recent articles, and only recently it received a full proof, albeit via a more
elaborate complex-analytic approach. We provide an elementary proof, which, moreover, extends to yield simple proofs of both old and new inequalities for Rényi entropy, subentropy, and quantum Rényi
Dive into the research topics of 'Logarithmic inequalities under a symmetric polynomial dominance order'. Together they form a unique fingerprint. | {"url":"https://portal.fis.tum.de/en/publications/logarithmic-inequalities-under-a-symmetric-polynomial-dominance-o","timestamp":"2024-11-04T21:39:26Z","content_type":"text/html","content_length":"49138","record_id":"<urn:uuid:85a23eba-5196-4b13-bdb7-7cb65a3b5db8>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.16/warc/CC-MAIN-20241104194528-20241104224528-00718.warc.gz"} |
groupoid object in an (oo,1)-category
I have added to groupoid object in an (infinity,1)-category a section Equivalent characterizations with some of those equivalent characterizations.
Or rather, for the moment I have mostly concentrated on adding a little remark on how to translate from the “cone-style” conditions as they appear in HTT to the equivalent
“powering-style”-conditions, as they appear in “I2CATGC”.
In fact, even for the ambient $\infty$-category being $\infty Grpd$, a more thorough model theory presentation of the theory of groupoid objects would be desireable. The two articles mentioned above
focus mostly on models for the actual objects of that $\infty$-category. It’s section 3 of the second article that gives a genuine model for groupoid objects in $\infty Grpd$ (“invertible Segal
spaces”). It would for instance be nice to have a Quillen equivalence form there to a model structure for effective epimorphisms in $\infty Grpd$. Things like that. I am wondering if this has been
done in citable form somewhere, or if it still needs to be written out.
I have added to the References at groupoid object in an (∞,1)-category the items
• Adding inverses to diagrams encoding algebraic structures, Homology, Homotopy and Applications 10 (2008), no. 2, 149–174. (arXiv:0610291)
Adding inverses to diagrams II: Invertible homotopy theories are spaces, Homology, Homotopy and Applications, Vol. 10 (2008), No. 2, pp.175-193. (web, arXiv:0710.2254)
wherein model category presentations of the $\infty$-categories of groupoid objects in $\infty Grpd$ are discussed.
It seems sort of straightforward to generalize this to model categories presenting $\infty$-categories of groupoid objects in more general presentable $\infty$-categories / $\infty$-toposes. The
groupoidal version of Segal space objects in model structures of simplicial (pre)sheaves.
But: has it been written out? Is anyone aware of something?
finally added the central theorem about delooping in an $\infty$-topos: to a new section Delooping
added to groupoid object in an (infinity,1)-category a subsecton with a remark on the notion of $(\infty,1)$-quotients / homotopy quotients.
polished and expanded somewhat the entry groupoid object in an (infinity,1)-category
There is a certain ambiguity of denoting the (oo,1)-categorial equivalence from pointed connected object in a Grothendieck (oo,1)-topos to group objects in that topos by $\Omega$, namely see
proposition 7 at groupoid object in an (oo,1)-category. This is because, for X a pointed connected object, the notation could mean either a pointed object internal to the said topos or a group object
internal to the said topos.
I was troubled by this ambiguity when writing up a proof at suspension object that, internal to a Grothendieck (oo,1)-topos, suspending is equivalent smashing with the classfying space of the
integers. There, for G a group object, I use the notation $\Omega {\mathbf{B}} G$ to denote a pointed object. Taking the adjunction $(\Omega \vdash {\mathbf{B}})$ in the current notation per se, this
notation $\Omega {\mathbf{B}} G$ ought to (by abstract nonsense) refer to a group object equivalent to $G$.
One suggestion, if we follow Lurie and take the “complete Segal-space style” presentation of a group object as a simplicial object satisfying some conditions, then is to write the categorial
equivalence as $\check{C}(*\to X):{\mathrm{PointedConn}}\to {\mathrm{Grp}}$, which sends a pointed connected object $X$ to the (underlying simplicial set) of the Cech nerve of the based map $*\to X$
from the terminal object.
This is about whether to leave the forgetful functor from groups to pointed objects notationally implicit.
By the way, over at suspension object you are using notation in both ways. First you say that $\Omega \mathbf{B}$ lands in pointed objects, but in the first proof you use $\Omega$ as landing in group
My suggestion is: say locally, eg in the entry on suspension, ecplicitly what the conventions are. There won’t ever be consistent conventions across all nLab entries.
Indeed, since you write F for the free group functor, it would be most ntural to call the forgetful functor just U, as usual, and not Omega B.
Urs: the link in #8 does not work as you made a typo. suspension object
Thanks Urs, for pointing out the inconsistencies I made.
There are really two right adjoint functors from group objects in question here. One is, following your suggestion, the functor $U:{\mathrm{Grp}}\to {\mathrm{Pointed}}$ which sends a group $G$
(regarded as a simplicial object) to $G_1$ (note that $G_1$ could possibly not be 0-connected). Its left adjoint is $F:{\mathrm{Pointed}}\to {\mathrm{Grp}}$ which sends a pointed object $X$ to the
(underlying simplicial object) of the Cech nerve of $*\to \Sigma X$. The other is the functor ${\mathbf{B}}:{\mathrm{Grp}}\to{\mathrm{PointedConnected}}$ which sends a group $G$ to the colimit of $G$
, regarded as a diagram over the simplex category. (The Lab calls this functor ${\mathbf{B}}$; In the 0-truncated case, this functor is usually called the nerve; Lurie calls it geometric realization
$|-|$.) Its left adjoint ${\mathrm{PointedConnected}}\to{\mathrm{Grp}}$ sends a pointed connected object $X$ to the (underlying simplicial object) of the Cech nerve of $*\to X$. It is really the
interplay of these two functors that proves the concretization of the suspension functor.
Yes, and I am really fond of wring $\mathbf{B}$ and not writing “nerve” or “geometric realization” because the latter two are conceptually misleading or at best highly ambiguous, no matter how
standard they may be.
Is there a reason for your use of the letter $\mathbf{B}$?
Sure, for $G$ a topological group regarded as an infinity-group via its underlying homotopy type, then the super-traditional classifying space construction $B G$ is the delooping of $G$ in the $\
infty$-topos $\infty Grpd$. The boldface $\mathbf{B}$ is to denote delooping in any other $\infty$-topos. The boldface is to be suggestive of “delooping remembering additional structure” (fat
structure). More precisely, if $\mathbf{H}$ is a cohesive $\infty$-topos then under suitable conditions the functor $\Pi : \mathbf{H}\longrightarrow \infty Grpd$ takes $\mathbf{B}G$ to $B G$. | {"url":"https://nforum.ncatlab.org/discussion/1369/groupoid-object-in-an-oo1category/","timestamp":"2024-11-11T18:34:02Z","content_type":"application/xhtml+xml","content_length":"73623","record_id":"<urn:uuid:7a1728d0-5196-4266-bc75-e0fe7d5e85a7>","cc-path":"CC-MAIN-2024-46/segments/1730477028235.99/warc/CC-MAIN-20241111155008-20241111185008-00598.warc.gz"} |
[Solved] It is possible to investigate the thermo | SolutionInn
It is possible to investigate the thermo chemical properties of hydrocarbons with molecular modeling methods. (a) Use
It is possible to investigate the thermo chemical properties of hydrocarbons with molecular modeling methods.
(a) Use electronic structure software to predict ∆cHo values for the alkane’s methane through pentane. To calculate ∆cHo values, estimate the standard enthalpy of formation of CnH2 (n+l) (g) by
performing semi-empirical calculations (for example, AMI or PM3 methods) and use experimental standard enthalpy of formation values for CO2 (g) and H20 (I).
(b) Compare your estimated values with the experimental values of ∆cHo (Table 2.5) and comment on the reliability of the molecular modeling method.
(c) Test the extent to which the relation ∆cHo= k {(M/ (g mol-1)} n holds and find the numerical values for k and n.
Fantastic news! We've Found the answer you've been seeking! | {"url":"https://www.solutioninn.com/it-is-possible-to-investigate","timestamp":"2024-11-02T12:51:31Z","content_type":"text/html","content_length":"82259","record_id":"<urn:uuid:7dde4478-e17a-4eac-8b43-7e074f6e965b>","cc-path":"CC-MAIN-2024-46/segments/1730477027710.33/warc/CC-MAIN-20241102102832-20241102132832-00197.warc.gz"} |
Volume of a Prism - Formula, Derivation, Definition, Examples - Grade Potential Houston, TX
Volume of a Prism - Formula, Derivation, Definition, Examples
A prism is a vital shape in geometry. The figure’s name is derived from the fact that it is created by taking a polygonal base and expanding its sides as far as it cross the opposing base.
This article post will talk about what a prism is, its definition, different kinds, and the formulas for surface areas and volumes. We will also offer instances of how to utilize the information
What Is a Prism?
A prism is a three-dimensional geometric figure with two congruent and parallel faces, called bases, which take the shape of a plane figure. The other faces are rectangles, and their count depends on
how many sides the similar base has. For example, if the bases are triangular, the prism would have three sides. If the bases are pentagons, there would be five sides.
The characteristics of a prism are astonishing. The base and top both have an edge in common with the additional two sides, creating them congruent to one another as well! This states that all three
dimensions - length and width in front and depth to the back - can be broken down into these four parts:
1. A lateral face (meaning both height AND depth)
2. Two parallel planes which make up each base
3. An fictitious line standing upright across any provided point on any side of this shape's core/midline—known collectively as an axis of symmetry
4. Two vertices (the plural of vertex) where any three planes join
Types of Prisms
There are three major types of prisms:
• Rectangular prism
• Triangular prism
• Pentagonal prism
The rectangular prism is a common kind of prism. It has six faces that are all rectangles. It looks like a box.
The triangular prism has two triangular bases and three rectangular faces.
The pentagonal prism consists of two pentagonal bases and five rectangular faces. It seems almost like a triangular prism, but the pentagonal shape of the base sets it apart.
The Formula for the Volume of a Prism
Volume is a measure of the total amount of space that an thing occupies. As an crucial figure in geometry, the volume of a prism is very important for your studies.
The formula for the volume of a rectangular prism is V=B*h, assuming,
V = Volume
B = Base area
h= Height
Consequently, since bases can have all sorts of figures, you are required to retain few formulas to calculate the surface area of the base. However, we will go through that later.
The Derivation of the Formula
To extract the formula for the volume of a rectangular prism, we are required to look at a cube. A cube is a three-dimensional object with six sides that are all squares. The formula for the volume
of a cube is V=s^3, where,
V = Volume
s = Side length
Now, we will have a slice out of our cube that is h units thick. This slice will make a rectangular prism. The volume of this rectangular prism is B*h. The B in the formula implies the base area of
the rectangle. The h in the formula refers to height, that is how dense our slice was.
Now that we have a formula for the volume of a rectangular prism, we can generalize it to any kind of prism.
Examples of How to Utilize the Formula
Since we understand the formulas for the volume of a pentagonal prism, triangular prism, and rectangular prism, let’s utilize these now.
First, let’s figure out the volume of a rectangular prism with a base area of 36 square inches and a height of 12 inches.
V=432 square inches
Now, let’s try another problem, let’s figure out the volume of a triangular prism with a base area of 30 square inches and a height of 15 inches.
V=450 cubic inches
Considering that you possess the surface area and height, you will work out the volume without any issue.
The Surface Area of a Prism
Now, let’s talk regarding the surface area. The surface area of an object is the measurement of the total area that the object’s surface occupies. It is an crucial part of the formula; therefore, we
must know how to find it.
There are a several varied ways to find the surface area of a prism. To figure out the surface area of a rectangular prism, you can use this: A=2(lb + bh + lh), assuming,
l = Length of the rectangular prism
b = Breadth of the rectangular prism
h = Height of the rectangular prism
To compute the surface area of a triangular prism, we will utilize this formula:
b = The bottom edge of the base triangle,
h = height of said triangle,
l = length of the prism
S1, S2, and S3 = The three sides of the base triangle
bh = the total area of the two triangles, or [2 × (1/2 × bh)] = bh
We can also use SA = (Perimeter of the base × Length of the prism) + (2 × Base area)
Example for Computing the Surface Area of a Rectangular Prism
First, we will figure out the total surface area of a rectangular prism with the ensuing dimensions.
l=8 in
b=5 in
h=7 in
To solve this, we will replace these values into the respective formula as follows:
SA = 2(lb + bh + lh)
SA = 2(8*5 + 5*7 + 8*7)
SA = 2(40 + 35 + 56)
SA = 2 × 131
SA = 262 square inches
Example for Finding the Surface Area of a Triangular Prism
To find the surface area of a triangular prism, we will figure out the total surface area by following same steps as before.
This prism consists of a base area of 60 square inches, a base perimeter of 40 inches, and a length of 7 inches. Thus,
SA=(Perimeter of the base × Length of the prism) + (2 × Base Area)
SA = (40*7) + (2*60)
SA = 400 square inches
With this knowledge, you will be able to compute any prism’s volume and surface area. Check out for yourself and see how easy it is!
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Physically based rendering: informal introduction
In this post I will give you an informal introduction (and my personal understanding) about Physically based rendering.
Physically Based Rendering (PBR) is one of the latest, and most exciting trend in computer graphics. PBR is "everywhere" in computer graphics. But wait, what is it PBR
Before giving an answer and try to give a detail definition of PBR we need to understand well some important concepts.
What is light?
Light is a form of electromagnetic radiation. Specifically, it is a small subset of the entire electromagnetic radiation spectrum with wavelength between 400 nm and 700 nm. The set of studies and
techniques that try to describe and measure how the electromagnetic radiation of light is propagated, reflected and transmitted is called radiometry. What are the fundamental quantities described by
radiometry? The first one is the called flux, it describes the amount of radiant energy emitted, reflected or transmitted from a surface per unit time. The radiant energy is the energy of an
electromagnetic radiation. The unit measure of flux is joules per seconds $\frac{J}{s}$, and it is usually reported with the Greek letter $\phi$.
Other two important quantities of radiometry are irradiance and radiant exitance. The first one described flux arriving at a surface per unit area. The second one describe flux leaving a surface per
unit area (Pharr et al., 2010 [1]). Formally irradiance is described with the following equation:
$E = \frac{d\phi}{dA}$
where the differential flux $d\phi$ is computed over the differential area $dA$. It is measured as units of watt per square meter.
Before proceeding to the last radiometry quantity definition, it is useful to give the definition of solid angle. A solid angle is an extension of a 2D angle in 3D on a unit sphere. It is the total
area projected by an object on a unit sphere centered at a point $p$. It is measured in steradians. The entire unit sphere corresponds to a solid angle of $4\pi$ (the surface area of the unit
sphere). A solid angle is usually indicated as $\Omega$, but it is possible also to represent it with $\omega$, that is the set of all direction vectors anchored at $p$ that point toward the area on
the unit sphere and the object (Pharr et al., 2010 [1]). Now it is possible to give the definition of radiance, that is flux density per unit solid angle per unit area:
$L=\frac{d\phi}{d\omega \ dA^{\perp}}$
In this case $dA^{\perp}$ is the projected area $dA$ on a surface perpendicular to $\omega$. So radiance describe the limit of measurement of incident light at the surface as a cone of incident
directions of interest ${d\omega}$ becomes very small, and as the local area of interest on the surface $dA$ also becomes very small (Pharr et al., 2010 [1]). It is useful to make a distinction
between radiance arriving at a point, usually called incident radiance and indicated with $L_{i}(p,\omega)$, and radiance leaving a point called exitant radiance and indicated with $L_{o}(p,\omega)$.
This distinction will be used in the equations described below. It is important also to note another useful property, that connect the two types of radiance:
$L_{i}(p,\omega) eq L_{o}(p,\omega)$
The rendering equation
The rendering equation was introduced by James Kajiya in 1986 [2]. Sometimes it is also called the LTE, Light Transport Equation. It is the equation that describes the equilibrium distribution of
radiance in a scene (Pharr et al., 2010 [3]) . It gives the total reflected radiance at a point as a sum of emitted and reflected light from a surface. This is the formula of the rendering equation:
$L_{o}(p,\omega) = L_{e}(p,\omega) + \int_{\Omega}f_{r}(p,\omega_{i},\omega_{0})L_{i}(p,\omega)\cos\theta_{i}d\omega_ {i}$
In this formula the meaning of each symbol are:
• $p$ is a point on a surface in the scene
• $\omega_{o}$ is the outgoing light direction
• $\omega_{i}$ is the incident light direction
• $L_{o}(p,\omega)$ is the exitant radiance at a point $p$
• $L_{e}(p,\omega)$ is the emitted radiance at a point $p$
• $\Omega$ is the unit hemisphere centered around the normal at point $p$
• $\int_{\Omega}...d\omega_{i}$ is the integral over the unit hemisphere
• $f_{r}(p,\omega_{i},\omega_{0})$ is the Bidirectional Reflectance Distribution Function and we will talk about it in a few moments
• $L_{i}(p,\omega)$ is the incident radiance arriving at a point $p$
• $\cos\theta_{i}$ is given by the dot product between 𝜔: and the normal at point $p$, and is the attenuation factor of the irradiance due to incident angle
One of the main component of the rendering equation previously described is the Bidirectional Reflectance Distribution Function (BRDF). This function describes how light is reflected from a surface.
It represents a constant of proportionality between the differential exitant radiance and the differential irradiance at a point $p$ (Pharr et al., 2010 [1]). The parameter of this function are: the
incident light direction, the outgoing light direction and a point on the surface. The formula for this function in terms of radiometric quantities is the following:
$f_{r}(p,\omega_{i},\omega_{o}) = \frac{dL_{o}(p,\omega_{o})}{dE(p,\omega_{I})}$
The BRDF has two important properties:
• it is a symmetric function, so for all pair of directions $f_{r}(p,\omega_{i},\omega_{o}) = f_{r}(p,\omega_ {o},\omega_{i})$
• it satisfies the energy conservation principle: the light reflected is less than or equal to the incident light.
A lot of models has been developed to describe the BRDF of different surfaces. In particular, in the last years the microfacet models have gained attention. In these kind of models the surface is
represented as composed by infinitely small microfacets that model in a more realistic way the vast majority of surfaces in the real world. Each one of these microfacets has is geometric definition
(in particular its normal).
Some specific material surfaces, for example glass, reflect and transmit light at the same time. So a fraction of light goes through the material. For this reason, there’s another function, the
Bidirectional Transmittance Distribution Function, BTDF, defined in the same way as the BRDF, but with the directions $\omega_{i}$ and $\omega_{o}$ placed in the opposite hemisphere around $p$ (Pharr
et al., 2010 [1]). It is usually indicated as $f_{t}(p,\omega_{i},\omega_{o})$. The Fresnel equations tries to define the behaviour of light between different surfaces. They also help us to get the
balance between different kind of reflections changes based on the angle at which you view the surface.
Physically Based Rendering
So let's go back to our original question: What is PBR? PBR is a model that enclose a set of techniques that try to simulate how the light behaves in the real world. Taking an extraction from the
Wikipedia definition:
PBR is often characterized by an approximation of a real, radiometric bidirectional reflectance distribution function (BRDF) to govern the essential reflections of light, the use of reflection
constants such as specular intensity, gloss, and metallicity derived from measurements of real-world sources, accurate modeling of global illumination in which light bounces and/or is emitted
from objects other than the primary light sources, conservation of energy which balances the intensity of specular highlights with dark areas of an object, Fresnel conditions that reflect light
at the sides of objects perpendicular to the viewer, and accurate modeling of roughness resulting from microsurfaces.
You can see from the definition that PBR is a model that uses all the concepts we saw previously in this article to try to get the most accurate results in terms of realism in a computer graphics
applications. PBR engines and asset pipelines let the artist define materials in terms of more realistic components, instead of tweaking ad-hoc parameters based on the type of the surface. Usually in
these kind of engine/assets pipeline the main parameter used to specify a surface features are:
• albedo/diffuse: this component controls the base color/reflectivity of the surface
• metallic: this component specifies the is the surface is metallic or not
• roughness: this component specifies how rough a surface is on a per texel basis
• normal: this component is a classical normal map of the surface
What results can you achieve suing PBR? These are two example images: the first one is taken from my physically based spectral path tracing engine Spectral Clara Lux Tracer and the second one is
taken from PBRT, the physically based engine described in the book "Physically based rendering: from theory to implementation" by M. Pharr, W. Jakob, G. Humphreys .
Some PBR scenes generated using PBRT and Spectral Clara Lux Tracer
How cool are these images????
[1] M. Pharr and G. Humphreys, “Color and radiometry,” in Physically based rendering: from theory to implementation, 2nd Edition ed., Burlington, Massachusetts: Morgan Kaufmann, 2010, ch. 5, pp.
[2] J. T. Kajiya, “The Rendering Equation,” in SIGGRAPH '86, Dallas, 1986, pp. 143-150.
[3] M. Pharr and G. Humphreys, “Light transport I: surface reflection,” in Physically based rendering: from theory to implementation, 2nd ed., Burlington, Morgan Kaufmann, 2010, ch. 15, pp. 760-770. | {"url":"https://www.fabrizioduroni.it/2017/12/07/physically-base-rendering-introduction/","timestamp":"2024-11-04T21:59:47Z","content_type":"text/html","content_length":"329708","record_id":"<urn:uuid:5d0c121f-935c-40b8-b8ca-443f687dd7ff>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.16/warc/CC-MAIN-20241104194528-20241104224528-00268.warc.gz"} |
Simulation of Lumbar Spinal Stenosis Using the Finite Element Method
Computers, Materials & Continua
Simulation of Lumbar Spinal Stenosis Using the Finite Element Method
1Department of Mathematics, Faculty of Science, Khon Kaen University, Khon Kaen, 40002, Thailand
2Department of Mathematics, Faculty of Science, Mahasarakham University, Mahasarakham, 44150, Thailand
*Corresponding Author: Kamonchat Trachoo. Email: kamonchat.t@msu.ac.th
Received: 02 March 2021; Accepted: 04 May 2021
Abstract: Lumbar spine stenosis (LSS) is a narrowing of the spinal canal that results in pressure on the spinal nerves. This orthopedic disorder can cause severe pain and dysfunction. LSS is a common
disabling problem amongst elderly people. In this paper, we developed a finite element model (FEM) to study the forces and the von Mises stress acting on the spine when people bend down. An
artificial lumbar spine (L3) was generated from CT data by using the FEM, which is a powerful tool to study biomechanics. The proposed model is able to predict the effect of forces which apply to the
lumbar spine. In addition, FEM allows us to investigate the tests into the lumbar spine instead of applying the tests to the real spine in humans. The proposed model is highly accurate and provides
precise information about the lumbar spine (L3). We investigate the behavior of humans in daily life which effects to the lumbar spine in a normal person and a patient with LSS. The computational
results revealed high displacement levels around the spinal canal and lower displacement levels in the spinal body when bending down. The total displacement of the axial load in a normal person was
higher when compared with patients with LSS. Higher degree bends resulted in a lower total displacement when compared with lower degree bends, while the von Mises stress decreased as the bending
degree increased.
Keywords: Lumbar spinal stenosis; finite element method; mathematical model; von Mises stress
As the population is aging, the incidence of orthopedic problems among elderly people such as osteoporosis, osteonecrosis, primary and secondary bone tumors, scoliosis, low bone density,
osteoarthritis, Paget’s disease, and gout is increasing. These orthopedic disorders can cause severe pain and dysfunction, particularly when affecting the spine. The spine or backbone is an important
part of the human body because it supports the body structure and connects the nervous system. The spine is composed of the cervical, thoracic, lumbar, sacrum, and coccyx. The lumbar spine consists
of five spinal columns (L1-L5) and supports most of the upper part of the body while also protecting the spinal cord and nerves from injury. Lumbar spinal stenosis (LSS) is a common disease found in
the elderly population all around the world [1,2]. This disease was first described in the 1950s [3]. LSS occurs as a result of narrowing of the spinal canal, which results in pressure on the spine
and the spinal nerve root. The pressure causes pain in the back, buttocks, and legs [4]. It may also cause loss of sensation and weakness in the feet and legs, as well as sexual dysfunction.
Therefore, in order to reduce the incidence of LSS and to develop appropriate therapeutic interventions, there is a need to understand the biomechanics of LSS.
Finite element simulation models (FEM) are now increasingly used to explore the biomechanical properties of the spine and to guide surgical interventions [5–9]. Xu et al. [10] utilized this method to
develop five FEMs of the lumbar spines (L1-L5). They showed that the models confirmed that the computational results were consistent with the experimental results. Finley et al. [11] developed an
open-access FEM of the human lumbar spine for both healthy and degenerating lumbar spine. These models could be used to study the biomechanics of the lumbar spine. Gupta et al. [12] used finite
element analysis to model the internal stress and strain in the craniovertebral junction (CVJ) region caused by different implants. On the other hand, Chung et al. studied the effect of implanting an
artificial disc on L4 and L5 using the FEM [13], and Zhong et al. [14] also used this model to evaluate the impact of a new cage as a space holder on the lumbar spine. A number of research studied
about the von Mises stress and total displacement of spine [15–18]. The von Mises stress is often used to analyze the risk of developing burst fracture in bones of various grades [19]. It is also
used to interpret the six stress components acting on the materials [20]. However, to our knowledge, no FEM has yet been developed evaluating the stress and strain in LSS.
This study aimed to develop a FEM to compare the effect of the total displacement and von Mises stress in a normal person and in a patient with LSS while bending down using an artificial lumbar spine
by using a lumbar vertebra model reconstructed from a computed tomography (CT) scan.
2.1 Construction of the Lumbar Vertebra Model
A two-dimensional and three-dimensional model of the third lumbar (L3) vertebra was constructed using the CT data of a human lumbar spine. The CT data were taken from a healthy person and a patient
with LSS. The complete geometry of a healthy lumbar spine is illustrated in Fig. 1, and the geometry of a patient with LSS is illustrated in Fig. 2. The finite element representation of the lumbar
spine models was obtained by subdividing the solids into a mesh of triangular elements. The bone dimensions were 7.02 cm × 7.70 cm × 4.5 cm. The mesh of the normal lumbar spine geometry consisted of
22,630 elements, and the mesh of lumbar spinal with stenosis consisted of 19,008 elements, as shown in Fig. 3.
The lumbar spine was assumed to consist of von Mises elastoplastic material. According to the principles of continuum mechanics, the displacement, stress fields, and stress equilibrium in the lumbar
spine can be defined using the following equations;
ξij(u)=12(ui,j+uj,i) (1)
where σ is the stress tensor, ξ is the strain tensor, u is the displacement, fi is the body force, and Cijkl is the tensor-elastic constant.
The parameters used during the numerical simulation are shown in Tab. 1. For the domain, shown in Fig. 3, we imposed three boundary conditions based on the axial load on a person with a normal spine
and a person with LSS while bending down. The outer boundaries of the lumbar spine for both normal person and LSS patients were fixed to prevent translation and rotation of the domain. The inner
boundaries of the domains were not fixed because this study investigated the effect of the force on the spinal structure.
2.3 Effect of Forces Applied on the Human Lumbar Spine
The effect of forces applied on the human lumbar spine was simulated based on an average woman’s weight of 58.58 Kg. The forces on the human body when a person bends down at 30-degrees (F30),
45-degrees (F45), and 60-degrees (F60) that apply to the lumbar spine in the X-axis and Y-axis can be described as follows;
(I) F30: Fx=−298.27 N, Fy=−483.14 N
(II) F45: Fx=256.88 N, Fy=416.08 N
(III) F60: Fx=−92.02 N, Fy=−149.05 N.
The FEM was used to find the numerical solution of the boundary value by multiplying Eq. (1) with the weighting function v(x). The total weighted residual error was then set to zero, and the
following equations were used to derive the model.
∫Ωσij,jvidΩ+∫ΩfividΩ=0. (2)
From symmetry of σij,j, we obtained
σij,jvi=(σij,jvi)j−σijvi,j. (3)
Substituting Eq. (3) into Eq. (2), we obtained
∫Ω[(σij,jvi)j−σijvi,j]dΩ+∫ΩfividΩ=0. (4)
The divergence theorem was then applied as follows
∫Ω(σij,jvi)jdΩ=∫∂ΩσijvinidS. (5)
Eq. (5) was then substituted into Eq. (4) to obtain
∫Ω−σijvi,jdΩ+∫ΩfividΩ+∫∂ΩσijvinidS=0. (6)
The surface fraction boundary condition was explained by the equation
{FxFy}=[σxxτxyτxyσyy]{nxny} (7)
which is equivalent to
Fi=σijnj. (8)
From Eq. (8) and Eq. (6), we obtained
∫Ωσijvi,jdΩ=∫ΩfividΩ+∫∂ΩviFidS. (9)
we arranged Eq. (9) to
∫Ωσijξij(v)dΩ=∫ΩvifidΩ+∫∂ΩviFdS. (10)
The third equation was substituted in the system (1) into Eq. (10) to obtain the equation
∫ΩCijklξklξijdΩ=∫ΩvifidΩ+∫∂ΩviFdS. (11)
We then assumed that
f=[fxfy]andD=[∂∂x00∂∂y∂∂y∂∂x]. (12)
Eq. (11) was subsequently rearranged to obtain
∫ΩCξ(Dv)dΩ=∫ΩvfdΩ+∫∂ΩvFdS, ∫ΩC(Du)(Dv)dΩ=∫ΩvfdΩ+∫∂ΩvFdS, ∫Ω(Dv)TC(Du)dΩ=∫ΩvTfdΩ+∫∂ΩvTFdS.
Hence, the variational statement for the boundary value problem was finally stated as follows:
Find u∈V such that
a(u,v)=L(v)∀v∈V, (13)
a(u,v)=∫Ω(Dv)TC(Du)dΩ, L(v)=∫ΩvTfdΩ+∫∂ΩvTFdS, V={v∈[H1(Ω)]2∣v=0 on ∂Ω}.
In order to find the numerical solution of this variational boundary value problem, we imposed this problem in an N-dimensional subspace by using the basic function {ϕi}i=1N with an approximate u and
v as follows;
u=∑i=1NΦiuiandv=∑i=1NΦivi (14)
Φj=[ϕj00ϕj],uj=[uxjuyi]andvj=[vxjvyi]. (15)
Eq. (14) was substituted into Eq. (13). Since vi was an arbitrary value, we then obtained the following equation;
a(Φj,Φi)uj=L(Φi),(i,j=1,2,…,N) (16)
which represented a system of 2N equations in terms of unknowns {(uxj,uyj)} for j=1,2,…,N.
Finally, this problem was solved using the quasi-Newton method. The computational analysis was performed using the COMSOL multiphysics (COMSOL Inc., MA, USA).
The effects of the total displacement and von Mises stress on a patient with a normal lumbar spine and a patient with LSS while bending down are illustrated in Figs. 2 and 3. Figs. 4 to 6 show the
total displacement of the axial load of the LSS patient and for a person without disease while bending down at 30, 45, and 60 degrees, respectively. The findings of this study indicate that the
highest displacement occurs around the spinal canal and the lowest displacement occurs around the external part of the lumbar spine. This means that when a human bends down, there are some parts of
the spine perturbing, especially in the areas around the spinal canal.
The total displacement was then compared using the cross-section line of the domain of the lumbar spine, as shown in Fig. 7. We then transformed the length of the cross-section line into a unit
length. The findings of this analysis indicate that the total displacement is higher in a person with a normal spine when compared with a person with LSS, as illustrated in Fig. 8. Moreover, as the
bending degree increased, the total displacement decreased, as shown in Fig. 9. This means that a smaller degree bend resulted in high perturbation on parts of the spine and affected the displacement
of the lumbar spine.
Figs. 10–12 show the von Mises stress of the axial load in the lumbar spine in a normal person and a patient with LSS when both people bend down at 30, 45, and 60-degrees. The results indicate a high
level of stress closer to the spine canal. Fig. 13 shows the von Mises stress of the lumbar spine in a normal person and a patient with LSS at the cross-section lines described in Fig. 7. The results
indicate higher von Mises stress levels on the axial load of the lumbar spine in a normal person when compared with a patient with LSS. In Fig. 13, the cross-sectional line was transformed into a
unit length. The cross-sectional von Mises stress analysis indicated that the left half of the lumbar spine has higher stress levels when compared with the right side. Moreover, the von Mises stress
in both normal and diseased spines increased as the bending degree decreased, as shown in Fig. 14.
A mathematical model of the lumbar spine has been developed to study the total displacement and von Mises stress between a normal person and a patient with LSS by using the finite element method.
Numerical simulations were carried out to evaluate the effect of the forces on the lumbar spine when people bend down. The results showed that high displacement levels occurred around the spinal
canal, while a lower displacement was observed around the periphery of the human spine. The total displacement of the axial load in a normal person was higher when compared with a patient with LSS.
Higher degree bends resulted in a lower total displacement when compared with lower degree bends, while the von Mises stress decreased as the bending degree increased.
Acknowledgement: The authors are thankful to the reviewers. This research was supported by the Basic Research Fund of Khon Kaen University. Moreover, this research was also financially supported by
Mahasarakham University. The authors also would like to thank the Department of Civil Engineering, Faculty of Engineering, Khon Kaen University for providing the COMSOL Multiphysics package.
Funding Statement: This research was supported by the Basic Research Fund of Khon Kaen University. This research was also financially supported by Mahasarakham University.
Conflicts of Interest: The authors declare that they have no conflicts of interest to report regarding the present study.
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This work is licensed under a Creative Commons Attribution 4.0 International License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is
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qubolite: A Toolbox for Working with QUBO » Lamarr-Blog
© Thore Gerlach & Lamarr-Institut
Quantum Computing (QC) has ushered in a new era of computation, promising to solve problems that are practically infeasible for classical computers. One of the most exciting applications of quantum
computing is its ability of solving combinatorial optimization problems, such as Quadratic Unconstrained Binary Optimization (QUBO). This problem class has regained significant attention with the
advent of Quantum Computing. These hard-to-solve combinatorial problems appear in many different domains, including finance, logistics, Machine Learning and Data Mining. For more details on QC and
optimization in Machine Learning, we refer to our blog entries “Quantum computers: new potential for machine learning (in german)” and “Optimization in Machine Learning (in german)”.
To harness the power of Quantum Computing for QUBO, we introduce qubolite, a Python package comprising utilities for creating, analyzing, and solving QUBO instances, which incorporates current
research algorithms developed by scientists at the Lamarr Institute.
Understanding QUBO: Optimization of Binary Variables
Before we dive into qubolite, let us understand what QUBO is. As the name already indicates, we are concerned with the problem of finding binary values that optimize a quadratic objective function.
Mathematically, this problem can be expressed as:
where x is a binary vector and Q is a symmetric matrix, encoding the problem’s parameters.
Through their discrete nature, instances of QUBO arise mainly in the domain of combinatorial optimization, that is, where we want to find an optimal configuration of variables among a finite number
of possibilities. However, such problems are often hard or even intractable to solve with classical computers, since the number of possible solutions grows exponentially with the problem dimension,
i.e., the number of variables. Quantum Computing can potentially offer significant speedup in solving QUBO problems, thanks to algorithms like Quantum Approximate Optimization Algorithm (QAOA) and
Quantum Annealing. However, since today’s quantum hardware is very prone to errors and limited in computing power, one has to be very careful in designing suitable problem formulations which can be
solved with the quantum resources available to us. Integrated control errors are a prominent example of these limitations, which describe the physical errors appearing when implementing a given QUBO
problem on the hardware. During this process, parameters can be altered, which then leads the quantum annealer to solve a different problem, obtaining sub-optimal solutions of the original problem.
This effect is visualized in Fig. 1.
Fig. 1: QUBO solvers have a limited parameter resolution, leading to perturbations that may result in false optima. © Sascha Mücke
Unlocking QUBO: Introducing qubolite
To make working with QUBO more accessible, scientists at the Lamarr Institute have developed qubolite, a comprehensive Python package which provides a range of utilities:
• QUBO utilities: Many utilities which are useful when working with QUBO are integrated, such as easy instantiation, conversion to and from the Ising model, computing the parameters’ dynamic range,
sampling, or partial assignments of certain variables.
• Problem solvers: The package provides state-of-the-art solvers such as simulated annealing, but also a brute-force solver, with an efficient, parallelized implementation in C/C++.
• Pre-processing: To get the most out of the available solvers, methods for pre-processing are implemented. They can either identify certain properties of solutions and reduce the size of the
search space, or optimally condition QUBO instances for use on real quantum computers.
• QUBO embeddings: Different real-life problems formulated as QUBO are provided, ranging from Machine Learning problems, such as clustering, to general combinatorial problems.
Getting started with qubolite
To get started with qubolite, you can install the package using pip:
We use all optional dependencies here, relying on multiple open-source python packages, such as sklearn and igraph. The default version does only depend on the numpy package. After the installation
we can instantiate QUBO problems. To this end, we consider a simple clustering problem, which is generated using the Machine Learning package sklearn:
This data set is visualized in Fig. 2 (left). The goal now is to separate this data set into two clusters. We can import the clustering embedding and obtain a QUBO instance as follows:
The solution of this QUBO problem represents a perfect clustering of the dataset. Since our problem size here is rather small (30), we can use a brute-force approach, which rates the quality of every
possible solution and returns the best one.
The obtained solution is visualized in Fig. 2 (right), indicating the cluster assignments with different colors. Of course, with an increasing size of our data set at hand, a brute-force approach is
not possible anymore, since the number of solutions grows exponentially with the problem dimension.
Fig. 2: Exemplary data set (left). Corresponding clustering using a QUBO formulation where the binary variables with value 0 are blue and the variables with value 1 are red (right). © Thore Gerlach
Preprocessing for Quantum Computers
Since quantum computers utilize the quantum-mechanical phenomenon of superposition, they inherit the ability to search efficiently for an optimal solution in exponentially large spaces. Nowadays,
quantum hardware is still in its infancy since it possesses a limited size of qubits and is prone to errors.
How strong these errors impact the solution quality is largely dependent on the dynamic range of the given QUBO matrix parameters. The dynamic range corresponds to the number of bits required to
encode the QUBO parameters faithfully, considering the covered value range and small gradations. Scientists at the Lamarr Institute developed an algorithm for reducing the dynamic range of a given
QUBO problem, while maintaining the optimal solution. This method is implemented in qubolite and can be used in the following way:
Not only the performance of real quantum annealers is increased by reducing the dynamic range, but also the performance of classical hardware solvers which have a limited parameter bit-precision. One
such exemplary hardware solver is our IAIS Evo Annealer — more details on this technology can be found in our blog-entry “Preparing for the age of quantum with the IAIS Evo Annealer”.
Bridging the Gap: Cutting-Edge Quantum Research and Real-World Impact
qubolite opens up new possibilities for solving complex optimization problems with Quantum Computing (QC), since state-of-the-art research conducted at the Lamarr Institute is integrated in
real-time. The Python package provides a user-friendly and versatile toolbox for harnessing the potential of QC. It is not only used in our pure research projects but also in various industry
projects at Fraunhofer IAIS, one of the four Lamarr partner institutions. As quantum hardware continues to advance, qubolite contributes to making quantum optimization accessible and impactful in a
wide range of industries, being a freely usable software package.
More information on qubolite:
Optimum-Preserving QUBO Parameter Compression
Mücke, Sascha et al., 2023, arXiv preprint, pdf | {"url":"https://lamarr-institute.org/blog/qubolite-qubo/","timestamp":"2024-11-05T21:43:25Z","content_type":"text/html","content_length":"175255","record_id":"<urn:uuid:69e7412d-94d7-451d-b988-66901a23f2f0>","cc-path":"CC-MAIN-2024-46/segments/1730477027895.64/warc/CC-MAIN-20241105212423-20241106002423-00429.warc.gz"} |
TEACHER’S GUIDE: ( download http://www.stick.com.ar/guide.pdf )
This activity enables students to measure the circumference of Earth. Groups of students at two distant schools will take data and then collaborate, in essentially the same way that Eratosthenes
measured Earth’s circumference millennia ago in Egypt.
• Describe the geometry of how sunlight strikes Earth at different latitudes
• Describe how the circumference of Earth was first measured millennia ago
• Describe how to determine local noon
• Measure the angle of the sun at local noon
• Collaborate with another school some distance away to determine the circumference of Earth.
Your class will need measurements of the shortest shadow length and stick length to be taken at some other school located a considerable distance away, either north or south.
You will enter your school's information on the Stick Project website, <http://stick.com.ar/planilla.html/>, at the same time as the other school's information. Once the partner schools have shared
the measurements, the circumference of the earth is automatically calculated.
National Standards Addressed
This activity addresses the following Benchmarks and National Science Education Standards (NSES):
Benchmarks, K-12: The Mathematical World
Symbolic Relationships
When a relationship is represented in symbols, numbers can be substituted for all but one of the symbols and the possible values of the remaining symbol computed. . . .
Distances and angles that are inconvenient to measure directly can be found from measurable distances and angles using scale drawings or formulas.
The basic idea of mathematical modeling is to find a mathematical relationship that behaves in the same way as the objects or processes under investigation. A mathematical model may give insight
about how something really works or may fit observations very well without any intuitive meaning.
Benchmarks, K-12: The Nature of Science
The Scientific Enterprise
The early Egyptian, Greek, Chinese, Hindu, and Arabic cultures are responsible for many scientific and mathematical ideas and technological inventions.
NSES, K-12: History and Nature of Science
Science as a Human Endeavor
Individuals and teams have contributed and will continue to contribute to the scientific enterprise.
Historical Perspectives
In history, diverse cultures have contributed scientific knowledge and technologic inventions.
NSES, K-12: Unifying Concepts and Processes
Developing Student Understanding
• Evidence, models, and explanation
• Constancy, change, and measurement
NSES, K-12: Science as Inquiry
Abilities necessary to do scientific inquiry
• Design and conduct scientific investigations
• Use technology and mathematics to improve investigations and communications
• Formulate and revise scientific explanations and models using logic and evidence
For each student
• A copy of the Student Activity Guide
For each group
• stick or dowel, about 60 centimeters (cm) long
• stiff cardboard (to provide a smooth, flat surface)
• measuring tape
• pieces of blank paper
• pencil
To be shared by several groups
• carpenter’s level
• mason's plumb line
In this activity, you will work together with students at another school to measure the circumference of Earth. You will use the same methods and principles that Eratosthenes used more than two
thousand years ago.
Eratosthenes was a Greek living in Alexandria, Egypt, in the third century, BC. He knew that on a certain day at noon in Syene, a town a considerable distance to the south, the sun shone straight
down a deep well. This observation meant that the sun was then directly overhead in Syene, as shown in Figure 1.
Figure 1:
Light rays shining straight down a well in Syene at noon, when the sun is directly overhead. No shadow is cast.
Eratosthenes also knew that when the sun was directly overhead in Syene, it was not directly overhead in Alexandria, as shown in Figure 2. Notice that in both drawings, the sun’s rays are shown as
Figure 2:
Sunlight shining down a well in Alexandria at noon, on the same day as the observation shown in Figure 1. The sun is not directly overhead. The gray bar at the bottom left shows the shadow cast by
the side of the well. The angle of the sun’s rays and the size of the shadow are exaggerated.
In Figure 2, the side of the well casts a shadow on the bottom.
Eratosthenes used a shadow like this to find the circumference of Earth. When the sun was directly overhead in Syene, he measured the shadow of an object in Alexandria at noon. From the length of the
shadow, the height of the object, and the distance between Syene and Alexandria, he calculated the circumference of Earth. His value agreed quite well with the modern one.
How Eratosthenes Found the Circumference of Earth
Figure 3:
The sun is directly overhead at noon at Syene (S). Alexandria is at point “A.”
How did he do it, more than two thousand years ago? Take a look at Figure 3. Syene is represented by point “S,” and Alexandria by point “A”. In Figure 3, the arc length between S and A is d, and the
angle corresponding to the arc SA is ?. The radius of Earth is R. As suggested above, let’s assume that the sun’s rays are parallel. Since the ray that strikes Syene, at point S, is perpendicular to
the surface of Earth, the sun is directly overhead there.When the sun was directly overhead in Syene, Eratosthenes measured the shadow of a tower in Alexandria at noon,1 shown in Figure 4. Since both
the tower at A, which is perpendicular to Earth’s surface, and the ray of sunlight at point S both point to the center of Earth, and the rays of sunlight are parallel, the angle between the sunlight
and the tower is equal to ? (Alternate interior angles are equal.)
Figure 4:
The geometry of Eratosthenes’ measurement. He measured the length of the tower and its shadow at noon at Alexandria. Then he determined the angle of sunlight with the vertical, which is the same as
the angle subtended by Syene (S) and Alexandria
(A) at the center of Earth.
The tower and its shadow form two sides of a right triangle, as shown in Figure 4. Although trigonometry hadn’t yet been invented, Eratosthenes’ procedure can be expressed in the language of trig as
follows: The length of the shadow, the height of the tower (which he knew), and the angle ?, given here in degrees, are related by
Inverting tan? gives the value of ?. Using ratio and proportion, the arc length d is the same fraction of Earth’s circumference C as ? is of 360 degrees.
Rearranging for the circumference C,
How You Can Find the Radius of Earth
Rather than find Earth’s circumference, we suggest you find its radius, so that you can more easily compare your measured value with the accepted value.
Eratosthenes was lucky, because he knew of a place where the sun was directly overhead at noon on a certain day. Can you do the experiment even without that information?
Fortunately you can, as shown in Figures 5 and 6. You will make the shadow measurements at your school and share results with a class from a school at another location.
You can then do a subtraction to find the angle you need. Be sure to plan ahead—ideally, both schools make the measurement on the same day.
Figure 5:
The geometry for measuring the radius of Earth using the data of two collaborating schools separated by a north-south distance. Each will measure the angle of the sun, ?[A] at one location and ?[B]
at the other, at local noon.
We need two points, A and B, separated by a north-south distance d, shown on Figure 5. The experiment will work best if d is as large as possible.Take a look at Figure 6. Your school and the
collaborating school are represented by the points A and B, and the angles ?[A] and ?[B] correspond to points A and B.
Figure 6:
The relationship among the direction of sunlight, the sticks, and
the two angles ?[A] and ?[B].
At point A,
and likewise at point B.
Figures 5 and 6 show that the angle corresponding to the arc AB is just the difference ?[B] - ?[A] We can find the radius of Earth in the same way that Eratosthenes found the circumference in
Rearranging and simplifying,
Making the Measurement at Local Noon
On any day, local noon is the instant when the sun reaches its highest point in the sky. To determine it, plant the stick in the ground, making sure the stick is vertical using a plumb bob or a
carpenter’s level. In the late morning, measure the shadow’s length at regular time intervals. The shadow will get shorter as noon approaches, and then get longer again once noon has passed. The
shortest length is what you will substitute into equation above to find the value of ?[B] or ?[A] for your location.
Below are three additional activities:
• How Shadows Change During the Day
• Shadows on Earth
• Latitude
How Shadows Change During the Day
If your students have not thought much about shadows,they might benefit by starting with this preliminary activity.Give each group a 5cm straw piece, a sheet of 21cm x 29cm paper, and some tape. Ask
them to tape the straw so it stands in the center of the paper, as shown in Figure 7, and also to indicate the direction of north on one of the long sides. If you provide them with a compass, they
can orient the paper.
Figure 7:
Five-centimeter straw mounted vertically on a piece of paper.
Students predict and then measure the shadow of this straw at different times.
Their challenge is to imagine that this paper is set on level ground in sunlight and to predict the location and length of the shadow of the stick on the hour during the day. Discuss these
predictions to bring out their thinking. Then have them do the experiment and compare their predictions and results.
Shadows on Earth
Materials for each group: five 4-cm straw pieces, tape, and a piece of 21cm x 29cm paper.
Explain to students that they will make a model of shadows at different points on Earth. Have them draw a straight line across a piece of 21cm x 29cm paper and tape the five straw pieces, equally
spaced, along this line, so the straws stand straight up. Ask how the paper and straws could be a model of sticks placed at different locations on Earth (curve the paper, with the straws on the
convex side).
Explain that to avoid damaging their eyes, they should never look directly at the sun. In sunlight, ask students how the paper and straws can model the Eratosthenes experiment. (Facing the sun, hold
the paper at the ends of the long sides and curve it so the straws point out. Turn the paper to make the shadow of one straw disappear. Make this straw point directly at the sun. The straw without a
shadow models the well at Syene.) Have students describe what happens to the shadows of the other straws and relate the shadow of each one to its position. Ask students to relate these shadows to the
shadows Eratosthenes used to measure Earth’s circumference. See Figure 8.
Figure 8:
Model of shadows of sticks at noon, at different
latitudes and the same longitude.
Extension Activity: Latitude
Once experimentation is complete and the results reported, you can have students relate the measurements they have made to the definition of latitude.
a. Ask them to define latitude (the length of arc, or angle from the center of Earth, measured north or south from the equator).
b. Referring them to Figure 3, ask them to assume that point S is on the equator. Ask on what day the sun would be directly overhead at noon at S.
c. If your students made their shadow measurement on the vernal or autumnal equinox, the resulting angle would be equal to the latitude, as shown in Figure 3 (remember point S is on the equator). If
possible, have them try to do this by measuring the shadow of a stick on or near March 21 or September 21.
d. In the Eratosthenes experiment, the angle (?[B] - ?[A]) is the same as the difference in latitude of the two schools, so students could determine this difference immediately by subtracting the two
latitudes of the collaborating schools. Of course, we want students to make measurements and compare, rather than look up the answer in an atlas. If students point this out, you can remind them that
they are reenacting an historical experiment.
Notes on Introduction
In the discussion of Figures 1 and 2, which show sunlight in wells at two different locations, we assume the following:
• The rays of sunlight are parallel (see next section for more detail)
• The sides of the well are vertical.
Notes on How Eratosthenes Found the Circumference of Earth
Here are Eratosthenes’ assumptions:
Earth is a sphere. In fact, Earth bulges by about .3% at the equator, but we can safely neglect this difference.
The sun is very far away, so sunlight can be represented by parallel rays. The sun is indeed very far away, but it is not a point source, since its diameter is about 1/100 of the Earth sun distance.
Figure 9:
Notice the penumbra—the partially illuminated region—at the
end of the stick’s shadow (drawing not to scale).
As shown in Figure 9, there is a penumbral region at the end of the shadow, a region only partially illuminated by the sun. If the stick is 1 meter (m) long, the penumbral region will be more than 1
centimeter (cm) wide, which limits the accuracy of the measurement of the shadow length. The penumbra size scales up with the length of the stick, so using a longer stick does not increase the
accuracy of the measurement.
Alexandria is directly north of Syene: This is only approximately true. Find an atlas and compare the location of Alexandria and Aswan (built on the site of Syene).
You might ask students to comment on Eratosthenes’ assumptions, considering that he was working more tan 2,000 years ago.
Notes on How You Can Find the Radius of Earth
Ask students to discuss the distance d between the two schools. Is it better for d to be large or small? (large) Why? (The larger d, the larger the value of the angle ?[B] - ?[A] The larger this
angle, the smaller the percentage error in its measurement). Find the location of the two schools and ask students how to find the distance between them (use the scale on the map). Ask how well the
two schools line up north-tosouth.
Discuss how an east-west displacement might affect the outcome of the experiment. (Convert the difference in longitudes of the two schools into a time difference, using ratio and proportion and the
fact that 360° of longitude corresponds to 24 hours. Then compare the difference with the uncertainty in identifying local noon.)
Note that near local noon, the shadow length does not change much with time. For this reason, missing local noon by a few minutes is not important. Suggest that students practice measuring the shadow
length at noon in advance. If you need to look up the time of local noon, see the last reference.
Ask students how to select a suitable date for both schools to make measurements. Should they check the weather forecast first?
Try making the measurement of the shadow length yourself. If possible, drive the stick into the ground, and check with the level, or a plumb line, to be sure that the stick is vertical.
Tape copy paper on top of the sheet of cardboard, and check with the level to make sure that this surface is horizontal. Be sure your stick is short enough so the shadow doesn’t extend off the paper.
Ask students to describe what height they will measure (the distance from the top of the stick to the surface where they see the shadow).
The experience is summarized in the following video. In this video the subtitles in English can be activated.
Find out the latitude of my city
Solar noon in my city
What Is Solar Noon?
One of the things we do is "measure with a tape measure or a ruler" So we are going to remember how a measurement is read.
Videos with simple explanations
Carl Sagan - Cosmos – Eratosthenes https://www.youtube.com/watch?v=G8cbIWMv0rI
ERATOSTHENES https://www.youtube.com/watch?v=wPR3XhIDP9w
ERATÓSTENES Y LA MEDIDA DE LA TIERRA https://www.youtube.com/watch?v=2tmiWjLSMcA
In this video let's see how the shadow moves. The shadow will begin to shorten its length until it reaches a pussycat and then it will begin to be longer. The shorter shadow is what we need.
PROYECTO STICK (Original)
Project Stick (Subtitled)
Web pages to know the weather, (surely you know a better one)
Students may have difficulty understanding how the method presented here, with two schools along the same north-south line (the same meridian) working together, permits them to measure Earth’s radius
on any day of the year. If you have a globe, mount two Straw pieces on the same meridian. Place the globe in the beam of an overhead projector. Ask a student to rotate the globe so the location of
one straw is at local noon.
Ask what time it is at the other straw (local noon also).
How can they tell? (The shadows have mínimum length.) Have a student change the orientation of the globe axis and repeat.
A different way to visualize the geometry is to imagine the plane containing the center of Earth and points A and B. As Earth turns on its axis, this plane sweeps through all of space. When this
plane is oriented so the sun lies within it, then the shadow of each stick lies within the plane as well, so it is local noon at the location of each stick.
There are several ways to assess students’ performance in the Stick project.
• Students can choose one or more of the first three objectives and write or present the description that is specified in those objectives.
• Students can present a portfolio of student work, explanations, and drawings to show how they measured the radius of Earth.
• Students can prepare a written or oral presentation to a younger student on Eratosthenes’ measurement of the circumference of Earth. | {"url":"http://www.stick.com.ar/guide1.html","timestamp":"2024-11-02T04:25:40Z","content_type":"text/html","content_length":"60790","record_id":"<urn:uuid:08a580ce-8100-4846-9bd5-19fea7db37a9>","cc-path":"CC-MAIN-2024-46/segments/1730477027677.11/warc/CC-MAIN-20241102040949-20241102070949-00121.warc.gz"} |
Two supplementary angles are in the ratio \
Hint: Here we assume both the angles as separate variables. Then using the information in the question we form an equation of sum of angles. Also, using the concept of ratio, we write the angles in
the ratio form and equate it to the given ratio.
* Two angles are said to be supplementary to each other if they have the sum of angles as \[{180^ \circ }\]
* Ratio \[m:n\]can be written in the form of fraction as \[\dfrac{m}{n}\].
Complete answer:
Let us assume one angle as ‘x’ and another angle as ‘y’.
Since we know both the angles are supplementary angles, then their sum must be equal to \[{180^ \circ }\]
So we can write the sum of angles x and y as \[{180^ \circ }\]
\[ \Rightarrow x + y = {180^ \circ }\] … (1)
Now we are given that the angles are in the ratio \[5:4\]
So we can write that the ratio of angles x and y is \[5:4\].
Substitute the values of angles we assumed in the beginning of the solution.
\[ \Rightarrow x:y = 5:4\] … (2)
Now since we know we can convert the ratio \[a:b\] into a form of fraction as \[\dfrac{a}{b}\].
Therefore, we can write the equation (2) as
\[ \Rightarrow \dfrac{x}{y} = \dfrac{5}{4}\]
Multiply both sides of the equation by y
\[ \Rightarrow \dfrac{x}{y} \times y = \dfrac{5}{4} \times y\]
Cancel the same terms from numerator and denominator.
\[ \Rightarrow x = \dfrac{5}{4}y\] … (3)
Substitute the value of x from equation (3) in equation (1).
\[ \Rightarrow \dfrac{{5y}}{4} + y = {180^ \circ }\]
Take LCM on the left hand side of the equation.
\[ \Rightarrow \dfrac{{5y + 4y}}{4} = {180^ \circ }\]
Calculate the sum in the numerator.
\[ \Rightarrow \dfrac{{9y}}{4} = {180^ \circ }\]
Multiply both sides by \[\dfrac{4}{9}\]
\[ \Rightarrow \dfrac{{9y}}{4} \times \dfrac{4}{9} = {180^ \circ } \times \dfrac{4}{9}\]
Cancel out the same terms from numerator and denominator.
\[ \Rightarrow y = {\left( {20 \times 4} \right)^ \circ }\]
Calculate the product.
\[ \Rightarrow y = {80^ \circ }\]
Now we substitute the value of y in equation (1) to calculate the value of x.
\[ \Rightarrow x + {80^ \circ } = {180^ \circ }\]
Shift all constants to one side of the equation.
\[ \Rightarrow x = {180^ \circ } - {80^ \circ }\]
Calculate the value on RHS.
\[ \Rightarrow x = {100^ \circ }\]
\[\therefore \]Two supplementary angles that are in the ratio \[5:4\] are \[{100^ \circ }\] and \[{80^ \circ }\]
Students many times make the equation formed by the ratio as a complex equation when they cross multiply the values to both sides and then solve. Always try to keep that value on one side of the
equation which can later be directly substituted in another equation. Also, keep in mind ratio should always be in simplest form i.e. there should not be any common factor between numerator and | {"url":"https://www.vedantu.com/question-answer/two-supplementary-angles-are-in-the-ratio-54-class-10-maths-cbse-5fd85dd38238151f0da05ea2","timestamp":"2024-11-11T05:04:17Z","content_type":"text/html","content_length":"170088","record_id":"<urn:uuid:e2cc1925-8062-4bd8-a41b-8b97318022de>","cc-path":"CC-MAIN-2024-46/segments/1730477028216.19/warc/CC-MAIN-20241111024756-20241111054756-00801.warc.gz"} |
Geometric Accuracy Prediction and Improvement for Additive Manufacturing Using Triangular Mesh Shape Data
One major impediment to wider adoption of additive manufacturing (AM) is the presence of larger-than-expected shape deviations between an actual print and the intended design. Since large shape
deviations/deformations lead to costly scrap and rework, effective learning from previous prints is critical to improve build accuracy of new products for cost reduction. However, products to be
built often differ from the past, posing a significant challenge to achieving learning efficacy. The fundamental issue is how to learn a predictive model from a small set of training shapes to
predict the accuracy of a new object. Recently an emerging body of work has attempted to generate parametric models through statistical learning to predict and compensate for shape deviations in AM.
However, generating such models for 3D freeform shapes currently requires extensive human intervention. This work takes a completely different path by establishing a random forest model through
learning from a small training set. One novelty of this approach is to extract features from training shapes/products represented by triangular meshes, as opposed to point cloud forms. This
facilitates fast generation of predictive models for 3D freeform shapes with little human intervention in model specification. A real case study for a fused deposition modeling (FDM) process is
conducted to validate model predictions. A practical compensation procedure based on the learned random forest model is also tested for a new part. The overall shape deviation is reduced by 44%,
which shows a promising prospect for improving AM print accuracy.
Issue Section:
Research Papers
1 Introduction
Additive manufacturing (AM) has moved from solely a tool for prototyping to a critical technology for the production of functional parts used in a growing number of fields such as aerospace and
medicine [1–6]. Yet, a common problem for AM is the presence of undesirable geometric shape deformations that may lead to scrap or rework [7–9]. This can aggravate existing high costs in AM and
hamper further industrial adoption. Geometric complexity of three-dimensional (3D) objects is one major issue, among others, that hinders efforts to achieve consistent shape accuracy across a large
variety of products, particularly considering the nature of one-of-a-kind manufacturing or low-product production in AM [10,11]. Unlike mass production, learning to produce one product or one family
of products in accurate geometries is insufficient for AM.
A growing body of research seeks to address this shape deformation issue through predictive modeling and compensation approaches. As summarized in Fig. 1, there are two main categories of predictive
modeling approaches reported in the literature for shape deformation control: physics-based approaches utilizing finite element modeling [12–16] and data-driven approaches based on statistical and
machine learning [7,17–22].
Physics-based modeling uses first principle to simulate the physical phenomena underlying an AM process. Results from these simulations can be effective for predicting thermal and mechanical behavior
of parts during a print. For instance, physics-based models have been applied to simulate residual stresses in produced parts, give insight into part distortion, and predict spatiotemporal
temperature of feedstock in a build envelope, among many other uses [12–15]. Challenges faced by physics-based modeling include the computational complexity of simulations and the need to account for
a wide variety of physical phenomena that affect a process [16]. Furthermore, these phenomena can often be specific to a single method of AM, i.e., results from a simulation of a selective laser
melting machine would not be useful for modeling a machine using material extrusion.
Data-driven approaches for shape deformation control utilize data either from processes or from products to establish process-oriented models or product shape-oriented models. These surrogate models
greatly reduce computational costs. Process-oriented models seek to address geometric differences caused by process variables. Empirical and statistical methods have been applied to the investigation
and modeling of AM processes [23–26]. Factors such as layer thickness and flowrate are varied to discover optimal settings for quality control. Tong et al. [17,27], for example, utilizes polynomial
regression models to predict shrinkages in spatial directions and corrects material shrinkage and kinematic errors caused by motion of the extruder by altering computer-aided design (CAD) designs.
One downside to process-oriented models is that the product shapes and their impact on shape deformations are often not considered.
Product shape-oriented models seek to account for this by using the geometry of the manufactured part to inform error predictions. A critical step in shape-oriented modeling is the mathematical
representation of shape deformation for freeform 3D objects. Three main representation approaches have been reported in the literature: point cloud representation, parametric representation, and
triangular mesh representation. Point cloud-based approaches have sought to describe geometry using coordinates of points on a product boundary. Xu et al. [28], for example, presented a framework for
establishing the optimal correspondence between points on a deformed shape and a CAD model. A compensation profile based on this correspondence is then developed and applied to prescriptively alter
the CAD model. A different point cloud-based approach is presented in Ref. [29], which sought to utilize deep learning to enable thermal distortion modeling. A part's thermal history was captured
using a thermal camera focused on the build plate of a printer employing laser-based additive manufacturing. This information was then used to train a convolutional neural network, which gave a
distortion prediction for each point. The method was demonstrated using a number of 3D printed disks. Another related study focused on the use of transfer learning between models for different AM
materials [30]. The model that was employed utilized information regarding a point's position on the disk shape that was printed to predict geometric distortion. In addition to proper shape
registration and correspondence, one challenge for this approach is that models based on point cloud representations of shape deformation can be highly shape dependent, making it hard to translate
knowledge from shape to shape. As a result, the datasets in the previous articles are highly homogeneous.
Parametric representation approaches transform the point cloud data to extract deformation patterns or reduce complexity due to shape variety. Huang et al., for example, demonstrated a modeling and
compensation method by representing the shape deformation in a polar coordinate system [7]. One advantage of this approach is that it decouples geometric shape complexity from deformation modeling
through transformation, making systematic spatial error patterns more apparent and easier to analyze. Huang et al. showed [7,31–33] that this approach was able to reduce errors in stereolithography
(SLA) printed cylinders by up to 90% and in SLA printed 2D freeform shapes by 50% or more. One disadvantage of this method is that it requires a parametric function to be fit to the surface of each
shape that is to be modeled. Unfortunately, this can become prohibitively tedious for complex 3D shapes [34].
To account for this problem and expedite model building, this article proposes a shape-oriented modeling approach based on features extracted from triangular mesh shape representations of printed
objects. This form of shape representation is an ideal candidate because of the ease with which it can describe complex 3D geometries. Furthermore, parts manufactured using AM are almost universally
handled as triangular mesh files. The STL file is the most common format for transferring 3D shapes from CAD software or databases to slicing software for a 3D printer. It stores 3D shapes in the
form of a simple triangular mesh and has maintained widespread popularity over the past several decades due to its simplicity and wide compatibility across systems. Other more recent file formats for
3D printing such as the additive manufacturing file format (AMF) and 3D manufacturing format (3MF) formats^^2 [35] incorporate functionality beyond the storage of a single triangular mesh, such as
color and texture, more naturally defined curves, and more. These formats have found support from government and industry and are growing in adoption. Because this modeling method can utilize the
same data structure that the part is produced with, its simplicity and accuracy are increased.
Other work has used triangular mesh representations of geometry in seeking to improve print accuracy often by selecting ideal orientations for printing or by focusing on geometric differences for
specific error-prone features. Chowdhury et al. [36] proposed an approach for selecting the optimal build orientation of a part using a model with orientation-based variables relevant to a part's
final geometric accuracy. These variables were derived from the part's STL file. Their method combined this model with compensation produced by a neural network trained on finite element analysis
data to reduce the overall error of the part [36,37]. Moroni et al. [38] demonstrated a means of identifying cylindrical voids in a part's shape using a triangular mesh. This approach then predicted
the dimensional error of the cylindrical voids based on their geometric properties. Moroni et al. [39] also extended this method to selecting optimal print orientation.
The method proposed in this article seeks to predict and compensate for geometric deviations across the entire surface of a given part, making it a useful tool for increasing the shape accuracy of an
AM machine. It begins by performing feature extraction from triangular mesh representations of manufactured parts. These features are used alongside deviation measurement data for the respective
parts to train a random forest machine learning model. This model can then be used to predict errors for future prints. Finally, a compensated 3D model based on these predictions can be generated and
printed, resulting in a part with reduced geometric deviations. This process is illustrated in Fig. 2. One major contribution of this approach is that it quickly facilitates modeling of freeform
surfaces that would likely be exceedingly difficult to model using parametric function-based approaches.
An experiment to validate this approach using a number of benchmarking objects produced on an fused deposition modeling (FDM) 3D printer will be presented. The experiment used a dataset of four
objects and their corresponding geometric deviations to train a machine learning model. This model was then used to make predictions for a new shape that was treated as a testing dataset. The
predicted deviations for this shape compared favorably to the actual deviations of the shape when printed, demonstrating the potential of this approach for applications in error prediction. Finally,
these predictions were utilized to generate a compensated CAD file of the shape, which was printed and evaluated. This compensated part was found to have average deviations that were 44% smaller than
those of the uncompensated original print.
2 Feature Extraction for Triangular Mesh-Based Shape Deviation Representation
Modeling complex surfaces that cannot be easily described analytically present a challenge to many existing modeling methodologies. One way to address this challenge is the use of a finite number of
predictor variables that capture certain geometric properties of a surface that are deemed relevant based on prior engineering-informed knowledge. These predictor variables can be computed for an
evenly distributed set of points across the surface of an object. These points will then function as instances in the model for which predictions can be made and to which position modifications can
be applied for the purpose of compensation. Here, a set of eight predictor variables x, corresponding to each relevant property under consideration, is constructed using feature extraction from a
triangular mesh describing the shape to be printed. For simplicity, the vertices that make up the shape's triangular mesh can be considered the instances in the model. To produce an unbiased model,
it is necessary that the triangular mesh be uniformly dense across the surface of the shape and have triangles of the consistent size. This can be achieved by remeshing an object's STL file using one
of several algorithms [40]. This article considers three broad areas of phenomena that have been shown to affect print accuracy. These include position within the print bed, orientation and curvature
of a surface, and thermal expansion effects.
2.1 Position-Related Predictors.
The first area of significance for feature extraction is the physical position of a vertex in a print bed. Several studies have demonstrated that position within a printer's print bed is
significantly correlated with the resulting accuracy of printed parts [17,19]. In the context of FDM, this location dependency can be connected to extruder positioning, while for other processes like
digital light processing, this can be connected to optical variation [41]. For the nth vertex, the first three predictor variables (x[n][,1], x[n][,2], and x[n][,3]) used in this model correspond to
the x, y, and z coordinates of each vertex. These predictors seek to capture errors related to the actual position of the printed object within the print bed. For the validation experiment that will
follow, objects were positioned in the slicing software so that the position values from the STL file were exactly each vertex's position within the 3D printer's print bed. One implication of this is
that the same object printed in different orientations or locations will have different predictor sets.
2.2 Surface Orientation and Curvature Predictors.
The next area of significance is the orientation and curvature of a surface. This will be used due to the association of properties such as surface slope with common print errors [
]. Furthermore, surface curvature in the
plane can influence how the material is deposited. The next four predictor variables are derived from the set of normal vectors corresponding to the
triangular faces adjacent to a given vertex
. Each normal vector
= 1, 2, …
, is expressed in spherical notation with radius 1, an elevation angle, and an azimuth angle. The predictor variables are calculated as follows and illustrated in Fig.
. Figure
depicts how these predictor variables would be calculated for a single vertex
(or instance) on the triangular mesh, which is shown as a black dot on the mesh, and the expanded view to the left.
The first of these predictor variables is the median value of the azimuth angles in the set. This can be interpreted as the direction that the geometric features are facing. This is a useful term for
predicting shape-related print errors. In Fig.
, the vector that represents the median azimuth and elevation angle is shown in red, both on the triangular mesh, and the expanded view. The first predictor variable is shown as
. The second of these variables is the range in azimuth angles (i.e., max
– min
). This can be interpreted as an indicator of the curvature of the surface. Changes in curvature can affect how material is deposited from the extruder in material extrusion-based processes, or how
energy is concentrated on feedstock in powder bed fusion-based processes, and so on. In Fig.
, the third variable is shown as
. The third of these variables is the median value of the elevation angles in the set. This can be interpreted as the slope of the geometric features. This is of particular interest due to the
correlation between slope and common print errors. This variable is also useful for detecting overhangs, which can be difficult to print accurately. Finally, the fourth of these variables is the
range in elevation angles (i.e., max
– min
). This can be interpreted as the degree to which the slope changes over the surface described by the triangular faces. This has relevance to shape-dependent errors.
2.3 Material Expansion/Shrinkage Predictor.
The final predictor variable proposed here is the distance from each vertex to an axis in the
-direction placed at the center of each shape (which in the case of the validation experiment intersects the point 0,0,0 on the printer's build platform):
This distance between the z-axis placed at the center of the shape and a single vertex (or instance) n on the triangular mesh is shown in Fig. 4. The feature is of significance due to the thermal
expansion effects of the printed materials [43]. If an object is formed at a high temperature, as it cools, the printed material's coefficient of linear thermal expansion dictates the degree to which
its overall size is reduced. Such temperature changes can lead to warping, residual stresses, and dimensional inaccuracies [44,45]. This is further complicated by the fact that heat can be
concentrated at different locations over short periods of time. Objects of larger size expand and contract by a greater absolute distance due to scaling. Points on the surface that are at a greater
distance from what can be considered the center of the object will therefore experience a greater degree of displacement. This necessitates a proxy for a point's distance from the rough center of
expansion to be accounted for.
Given an STL file, the set of each of these predictor variables can be quickly calculated for each vertex. They can give a good idea of the relevant geometric factors that can influence the accuracy
of a 3D print. The relative efficacy of each of these predictor variables will be briefly evaluated in Sec. 5.4.
3 Shape Deviation Measurement and Calculation
A procedure for measuring deviations across the surface of a printed object is presented here. It is important that deviation values be calculated at each vertex on an object's triangular mesh. This
then allows for deviations to be used as the response variable corresponding to each set of predictor variables.
This procedure begins by producing a dense point cloud of measurements of the surface of a 3D printed object. In the validation experiment described later, each object was scanned using a ROMER
Absolute Arm with an attached laser scanner manufactured by Hexagon Manufacturing Intelligence. According to the manufacturer, this scanner has an accuracy of 80 μm. The objects were each scanned
with several passes from different angles so as to create scans with between 5,00,000 and 1.6 million points. In comparison, each design STL file has approximately 50,000 data points.
Registration is performed according to the methodology presented in Ref. [46]. Each point cloud is first aligned against its corresponding STL file manually. Kinematic constraints are applied in this
process. For example, the scan points on the table (ground points) are used to fix the height and orientation about x and y-axes of the scan point cloud. Ground points are produced when the laser
scanner detects the surface the scanned object is resting on and are illustrated in Fig. 5. Alignment is then refined using a modified version of the ICP algorithm. In this version, translation is
only allowed along the x and y-axes, while rotation is only allowed about the z-axis so as to preserve the initial alignment according to table points.
Once registration is completed, it is necessary to calculate the distances between each vertex in the designed triangular mesh and the 3D scan point cloud. To reduce noise from outlier points in the
scan point cloud, a mesh of the scan point cloud was generated using screened Poisson surface reconstruction (SPSR) [47]. The shortest distance between each vertex on the triangular mesh v[n] and the
surface of the scanned triangular mesh was calculated. Because shortest distance deviation is used, this SPSR reduces error minimization bias caused by always selecting the noisy points in the cloud
that are closest to the designed shape. Instead, distance to the “averaged” or smoothed surface is used. The shortest distance between v[n] and the scanned mesh is returned in the form of a vector d[
n]. The magnitude of deviation in the direction normal to the triangular mesh at each vertex is then calculated as y[n] = d[n] · N[n], where N[n] is the vector (x[n,4], x[n,6], 1) expressed in
Cartesian coordinates. Signs correspond with whether the deviation represents a dimension that is too large or too small. This results in a set of response values representing deviation values that
are normal to the surface of the designed triangular mesh. These values are used as the set of response variables y[1] through y[N].
For a training dataset containing multiple printed parts, data {(x[n], y[n]), n = 1, 2, … N} is the ensemble of the total N vertices from all of the shapes. Note that each vertex may have a different
number of adjacent triangle faces. For the validation experiment conducted in Sec. 5, for example, there are four triangular mesh files that correspond to four different shapes that are all included
in the training dataset.
4 Random Forest Model to Predict Shape Deviation With Extracted Features
To learn and predict shape deviations, it is necessary to develop a predictive model based on the training data. Because triangular mesh files often contain tens of thousands of vertices, the size of
the datasets generated by this method can be cumbersome, posing a computational challenge for machine learning methods. Conversely, because of the small number of example shapes that might be
available for model training, the approach must also be flexible and generalize well under covariate shift. One computationally efficient modeling approach that can be utilized in this situation is
the random forest method. One way to quantify the computational efficiency of a machine learning algorithm is time complexity, which reflects the number of computations that must be performed to
generate a model and thus time. The random forest algorithm has a worst-case scenario time complexity on the order of $O(MKN~2logN~)$, where M is the number of trees in the random forest, K is the
number of variables drawn at each node, and Ñ is the number of data points N multiplied by 0.632, since bootstrap samples draw 63.2% of data points on average [48]. As a point of comparison, an
algorithm such as Gaussian Process regression has a worst-case scenario time complexity on the order of O(N^3) [49]. For the training sets utilized in the proof-of-concept experiments that will
follow, this is roughly three orders of magnitude more complex.
4.1 Random Forest Method.
Researchers have successfully applied machine learning to make accurate predictions in a wide range of applications related to manufacturing. One particularly popular algorithm for applications is
random forest modeling, which has been applied to predicting surface roughness of parts produced with AM, fault diagnosis of bearings, and tool wear, to name just a few use cases [50–52]. The random
forest algorithm is a means of supervised ensemble learning originally conceived by Breiman [53]. It utilizes regression or classification trees, which are a method of machine learning that
recursively segments a given dataset into increasingly small groups based on predictor variables, allowing it to produce a response value given a new set of predictors [54]. The resulting structure
of this segmentation process resembles the roots of a tree and is shown in Fig. 6. The random forest algorithm constructs an ensemble, or forest, of these trees, each trained on a subset of the
overall dataset [40]. This process is explained in further detail later and is illustrated in Fig. 7.
The goal of a regression tree is to generate a set of rules that efficiently segment the given training set using predictor variables in a way that generates accurate predictions of a response
variable. This process begins with a single node and randomly chooses a set of predictor variables to be used in dividing the dataset. Given P total predictor variables, it is generally recommended
that the number of predictor variables sampled for each node be set to P/3 in the case of regression and $P$ in the case of classification [55]. By using this subset of predictor variables, the
algorithm seeks to split the data at the node in a manner that minimizes the sum of the squares of error for each response label y[i]:
This process is then repeated for each resulting node until a predetermined condition is met. Two common conditions include a predetermined minimum number of data observations at a node or a maximum
tree depth. Once the stopping condition is met, each of the terminal nodes is labeled with the average value of the responses for the observations contained by that node. New predictions are
generated using a set of predictor variable values to navigate down the tree until arriving at a terminal node that corresponds to the predicted response value.
The random forest algorithm begins by generating subsets or “bootstrap samples” from the overall dataset. These bootstrap samples are drawn randomly from the overall dataset with replacement,
allowing for some data to be shared between samples [53]. A regression tree is then trained for each bootstrap sample.
To make predictions using a generated forest, the predictor variables are used to generate individual predictions from each tree. The average of this set of predictions is then given as the overall
output of the ensemble.
One benefit of the random forest algorithm for this application is that the addition of irrelevant data (predictor sets are highly dissimilar to those for the predicted part) does not strongly affect
predictions based on the most relevant data. In this way, the individual trees can naturally accommodate diverse data sets in training without substantial degradation in prediction quality.
4.2 Feature Selection.
To gain an understanding of the relative importance of the predictor variables used in this model, the out-of-bag permuted predictor change in error for each predictor variable was calculated during
the validation experiment [
]. For each tree in the random forest, the training set data that was not used to train the tree, also referred to as the out-of-bag observations
is used to generate a set of predictions of the response variable
. The mean squared error (MSE) of this set of predictions, i.e., out-of-bag error is defined as follows:
Then, for the first predictor variable
, each of its values in the dataset are permuted so as to randomize the values of that predictor variable's input. A new set of predictions is generated using this data, and the MSE of these
predictions is calculated. The change in prediction error is defined as the difference between the original and changed MSE values:
A large value of $ΔErrorx1$ indicates that this is a significant predictor variable, since randomizing its input causes the predictions of the regression tree to become much worse.
This process is repeated for each of the predictor variables in the dataset and for each of the regression trees in the random forest. The change in MSE is calculated for each predictor variable and
averaged over all of the trees in the random forest. Each of these results is divided by the standard deviation of the ΔError values for the entire ensemble and is outputted as the final significance
value S(
) for each predictor variable:
4.3 Measuring Covariance Shift to Determine Feasibility of Prediction.
It is important to note that for this model as with most models, generating predictions that require large degrees of extrapolation will likely result in poor predictions. Consequently, it is
necessary to ensure that the training dataset comes from a distribution that is similar to the shape one wishes to predict. A methodology for determining the similarity of shapes according to the
predictor variables generated in Sec.
is presented here. The result is a distance metric between any two triangular meshes that can be utilized to estimate whether a training dataset for modeling has adequate similarity to the shape one
wishes to predict for. Denote the distribution of the training and test sets to be
, respectively, and then
is the ideal case and the predictions can be made with confidence. In practice, however, test distribution
will differ arbitrarily from the training distribution
. Such a change is known as covariate shift [
]. This is due to the fact that we wish to predict errors for shapes that are different than the shapes that have already been printed. Sugiyama et al. [
] note that the Kullback–Leibler divergence between two distributions for datasets can be interpreted as an estimator for the level of covariate shift between them. An approach based on this is
utilized here. Jensen–Shannon divergence is utilized instead in order to gain symmetry between distance measurements, and independent distributions for each predictor variable are calculated for the
sake of computational cost. To estimate the distributions
for each feature, kernel density estimation [
] is applied to get the density estimation of features
= 1, …, 8 as follows:
for all
in the dataset for the first shape, and
for all
in the dataset of the second shape, where
(·) is the normal kernel, which is the same for both distributions:
After determining the probability distributions for both datasets, the covariate shift for each feature
can be quantified using Jensen–Shannon divergence [
) is the Kullback–Leibler divergence [
] defined as follows:
A final divergence metric between two shapes can be given as the sum of the Jensen–Shannon divergences for each predictor variable:
4.4 Prescriptive Compensation of Shape Deviation.
Once predictions are made for a part that is to be printed, it becomes necessary to leverage these predictions to improve the part's eventual quality. This method for compensating for positioning
error was utilized in Refs. [7,27,28]. The general idea is that if a portion of the object is predicted to be too large or small by a certain amount, the shape of the object can be altered in the
opposite direction by a corresponding amount before the object is printed, thus resulting in part with less error. For each vertex on the triangular mesh v[n], a new compensated vertex is generated
by translating the vertex a distance of $−y^n$ along a vector normal to the surface at that point. This vector can be calculated in spherical coordinates as (x[n,4], x[n,6], 1). This process is
illustrated in Fig. 8. It should be noted that this is not an optimal approach, like what is presented by Huang et al. [7,18], but is instead a heuristic. One implication of this is that in many
situations, the optimal compensation for a part is different from the negative value of the observed deviation.
The reason a nonoptimal approach is taken here is due to the nature of random forest modeling, as small changes in the predictor set do not yield large if any changes in the response from the
prediction function. This is because according to the regression tree algorithm, all values within a certain region of the n-dimensional predictor space will return the same value. In the validation
experiment that will follow, for instance, only 29% of the points on the compensated STL file showed different values of predicted error after compensation (assuming the compensated STL file is then
considered the ideal shape). Of those that did, the average change in predicted error was 0.0013 mm, which is well below the resolution of the 3D printer used in this study. Once each vertex is
modified, a new compensated STL file is generated for printing.
5 Validation Experiment
5.1 Test Object Design, Printing, and Measurement.
To test the efficacy of the proposed method, a case study utilizing an FDM 3D printer was constructed. The goal of the experiment was to determine whether the geometric accuracy of a previously
unseen shape could be improved using accuracy data from several other related shapes using the proposed method. In addition, the predictive accuracy of the model for the unseen shape was also
evaluated. The experiment is designed to mirror a situation that might be encountered in an industrial setting—when a manufacturer is about to print a new part, but only has accuracy data for a small
number of somewhat related shapes. The previously discussed methodologies for finding the most relevant accuracy data one possesses, leveraging that data to generate predictions, and using those
predictions to improve accuracy through compensation are all evaluated.
A dataset of 3D printed shapes was generated on an FDM printer with four shapes being used for model training and one always withheld for model testing. These included a half-ovoid, a half-teardrop,
a triangular pyramid, a half-snail shell, and a knob shape [65]. These objects are chosen to represent different geometries including varying curved and flat faces, various topologies, and edges of
differing angles. The edge length of each triangle in the mesh of each object was set to be approximately half a millimeter during remeshing. It is important to note that because this process takes
the original parts and moves them to a much higher mesh density, accuracy is preserved. This is because small triangles can express a freeform shape with greater accuracy than large triangles.
Accuracy for freeform parts would likely not be preserved in the opposite direction. Following this remeshing process, the benchmarking objects were printed on a MakerBot Replicator FDM 3D printer
using MakerBot brand Polylactic Acid filament. Each object was printed with full infill. Care was taken to ensure that the point defined as the origin in the triangular mesh file for each object was
printed at the exact center of the print bed. This ensured that the positions of each vertex in the triangular mesh directly corresponded to the positions of the printed objects within the printer's
build envelope. The printed test objects are shown in Fig. 9.
The deviation values for each of the 3D printed shapes were calculated according to the procedure in Sec. 3. These deviation values are shown in Fig. 10, which is a heatmap of deviation values across
the surface of each shape. The color at each point indicates the extent of the deviations across the surface. Red points correspond to parts of the shape that are too large, while blue points
correspond to parts of the shape that are too small.
To better understand the distribution of deviation magnitudes, a histogram showing the frequencies with which various magnitudes of deviation values occur is shown in Fig. 11. This histogram is
specifically for the half-ovoid shape, which is withheld as the testing dataset for one iteration of the experiment. None of the values from the bottom surface of this shapes is included as the
deviations are assumed to be zero based on the assumptions used during registration. It can be seen that most deviations are within 0.3 mm of the desired dimension.
5.2 Model Training Results.
To better understand the efficacy of the method for prediction, two different models were trained. For the first model, the half-teardrop, triangular pyramid, half-snail shell, and knob shape were
used as the training data, while the half-ovoid was used as the testing dataset. For the second model, the half-ovoid, triangular pyramid, half-snail shell, and knob shape were used as the training
data, while the half-teardrop was used as the testing dataset. For each model, an ensemble of regression trees was trained using the random forest method and matlab's Statistics and Machine Learning
Toolbox. The minimum number of observations at each node was set to 200, while the number of trees in the ensemble was set to 30. This ensemble size was chosen due to the fact that experimental
results indicated that further increases in the ensemble size for this dataset yield increasingly small gains in out-of-bag error, as shown in Fig. 12.
Thanks to the simplicity of the random forest algorithm, the total training time was less than 30 s, while predictions can be generated at a speed of roughly 110,000 predictions per second. The
relative significance of each predictor variable was calculated for the trained models according to the procedure described in Sec. 4.2. These values are shown in Fig. 13. These results suggest that
each of the predictor variables contributes to the overall accuracy of the model, however to differing degrees.
Table 1 compares the covariate shift metrics between each shape in the dataset. The final values are divided by the maximum covariate shift in the table to produce a normalized set. It can be seen
that the half-ovoid dataset withheld for testing in the first model is most similar to the half-teardrop and triangular pyramid shapes. Conversely, the knob shape shows a greater magnitude of
covariate shift from most of its peers, indicating that predictions made for this shape would likely be of poorer quality. If one wished to generate predictions for the knob, it would be advisable to
train the model on data more representative of its unique shape.
5.3 Model Prediction Results.
By using the testing shapes that were withheld from the training sets, a new set of predictions was generated for each random forest model. The mean absolute error (MAE) of predictions for out-of-bag
data in the training dataset, as well as the MAE of predictions for the withheld shapes, are provided in Table 2. The first error quantifies the accuracy of the model when making new predictions for
the overall shapes (but not individual datapoints) that it has already seen in training. The second error quantifies the accuracy of predictions made for a new shape that the model has not seen
during training. The predictions for deviation across the surface of the half-ovoid are graphed alongside the actual deviation values for the shape, allowing for comparison. This is illustrated in
Fig. 14 with the same coloring scheme as shown in Fig. 10.
Plots of predicted deviation values versus actual deviation values for the out-of-bag data used in model training, as well as for the testing shape, are given in Figs. 15 and 16. For reference, the
lines $y^=y+0.1mm$ and $y^=y–0.1mm$ are provided. Predictions that fall outside these bounds might be considered of low quality. It can be seen from these results that this method is capable of
producing reasonably accurate predictions for a previously unseen shape from a small training set of just four related shapes.
The predictions shown in Figs. 15 and 16 might be useful for an operator of a 3D printer seeking to determine whether a specific 3D printed shape will be within a prespecified tolerance before
beginning the print. This procedure might also be of use when determining the best orientation with which to print an object to maximize accuracy. Figure 16 also demonstrates that there is room for
improving the accuracy of the model. This would likely include expansion of the initial training set and refinement of the initial predictor variables. This article builds upon the work presented in
Ref. [66]. For an additional example of how prediction using this methodology can be implemented, see Section 5 of Ref. [66].
5.4 Compensation Results.
A compensated STL file for the half-ovoid shape was generated according to the procedure given in Sec.
using the first model's predictions. This STL file was then printed in the same manner and with the same material as the previous objects. Its dimensional accuracy was measured, and deviations are
shown alongside the noncompensated half-ovoid in Fig.
. It can be seen that error is substantially reduced using the compensation methodology. The dimensional error is quantified for the compensated and noncompensated half-ovoid as the average vertex
error according to Eq.
and given in Table
It can be seen from the results in Table 2 that the application of the presented compensation methodology results in a 44% reduction in average vertex error and a 50% reduction in root-mean-square
vertex error for the testing shape.
6 Conclusion and Future Work
This study establishes a new data-driven, nonparametric model to predict shape accuracy of 3D printed products by learning from triangular meshes of a small set of training shapes. The accuracy of a
new 3D shape can be quickly predicted through a trained random forest model with little human intervention in specifying models for complicated 3D geometries. With features extracted from triangular
meshes, the proposed modeling approach is shown to produce reasonable predictions of shape deviation for a new part based on a limited training set of previous print data. The trained model's
out-of-bag prediction error is 0.0580 mm, while its testing dataset error was 0.0713 mm. Compensation leveraging these predictions is also shown to be effective, resulting in a 44% reduction in
average vertex deviation.
One further interesting insight gained from the presented experiment was that quality of the data is a necessary condition for reasonable predictions. Table 1 presents that only two of the shapes in
the training set had low covariate shift as compared with the testing data set. This is likely toward the lower bound on what can be utilized to maintain accurate predictions. Those wishing to
utilize this methodology should therefore ensure that their training dataset contains an adequate amount of data similar to the shapes they wish to predict for. Applications where this is already
naturally the case can be found under the concept of “mass customization,” where similarly shaped products are produced with small custom differences introduced per customer specifications. These
might include the 3D printing of retainers, custom footwear, and medical implants among many other fields. The methodology for determining shape similarity based on covariate shift of the presented
predictor variables might be utilized across other shape deviation modeling methodologies for which these conditions are significant to ensure the sufficiency of training data.
Future work might focus on a number of areas. First, incorporating information regarding the overall topology to further improve the prediction accuracy might be worthwhile. Second, new predictor
variables based on local surface geometry might be added in future studies. Predictor variables for this study were developed using an empirical approach based on domain knowledge. A more rigorous
mathematical selection process might be examined in the future work. If new predictors are developed, they can be evaluated using the procedure given in Sec. 4.2, and if shown to be effective, easily
added to the methodology. Finally, modeling methodologies that incorporate spatial autocorrelation might be investigated as a means for improving accuracy.
This research was supported by the National Science Foundation (NSF) (Grant No. CMMI-1544917) and by a graduate research fellowship from the Rose Hills Foundation.
Conflict of Interest
There are no conflicts of interest.
Data Availability Statement
The datasets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request.
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A triangle has corners at (6, 4 ), ( 2, -2), and ( 5, -8). If the triangle is reflected across the x-axis, what will its new centroid be? | HIX Tutor
A triangle has corners at #(6, 4 )#, ( 2, -2)#, and #( 5, -8)#. If the triangle is reflected across the x-axis, what will its new centroid be?
Answer 1
If you reflect across the x-axis, then the x-coordinates will not change and the y-coordinates will equal $- y$
#(6,4)rarr(6,-4)# #(2,-2)rarr(2,2)# #(5,-8)rarr(5,8)#
Coordinates of centroid is [(x1 + x2 + x3)/3, (y1 + y2 + y3)/3]
centroid #=[(6+2+5)/3, (-4+2+8)/3)]#
centroid #=(13/3, 2)#
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Answer 2
To find the centroid of the reflected triangle, you first find the centroid of the original triangle, then reflect it across the x-axis. The centroid of the original triangle is the average of its
x-coordinates and the average of its y-coordinates.
Original centroid: x-coordinate = (6 + 2 + 5) / 3 = 13 / 3 ≈ 4.33 y-coordinate = (4 - 2 - 8) / 3 = -6 / 3 = -2
Reflected centroid: Reflect the y-coordinate across the x-axis: -(-2) = 2
So, the new centroid of the reflected triangle is (4.33, 2).
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Answer from HIX Tutor
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
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Subatomic particle - Gravity, Quarks, Hadrons | Britannica
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Also called:
elementary particle
The weakest, and yet the most pervasive, of the four basic forces is gravity. It acts on all forms of mass and energy and thus acts on all subatomic particles, including the gauge bosons that carry
the forces. The 17th-century English scientist Isaac Newton was the first to develop a quantitative description of the force of gravity. He argued that the force that binds the Moon in orbit around
Earth is the same force that makes apples and other objects fall to the ground, and he proposed a universal law of gravitation.
According to Newton’s law, all bodies are attracted to each other by a force that depends directly on the mass of each body and inversely on the square of the distance between them. For a pair of
masses, m[1] and m[2], a distance r apart, the strength of the force F is given by F = Gm[1]m[2]/r^2. G is called the constant of gravitation and is equal to 6.67 × 10^−11 newton-metre^2-kilogram^−2.
The constant G gives a measure of the strength of the gravitational force, and its smallness indicates that gravity is weak. Indeed, on the scale of atoms the effects of gravity are negligible
compared with the other forces at work. Although the gravitational force is weak, its effects can be extremely long-ranging. Newton’s law shows that at some distance the gravitational force between
two bodies becomes negligible but that this distance depends on the masses involved. Thus, the gravitational effects of large, massive objects can be considerable, even at distances far outside the
range of the other forces. The gravitational force of Earth, for example, keeps the Moon in orbit some 384,400 km (238,900 miles) distant.
Newton’s theory of gravity proves adequate for many applications. In 1915, however, the German-born physicist Albert Einstein developed the theory of general relativity, which incorporates the
concept of gauge symmetry and yields subtle corrections to Newtonian gravity. Despite its importance, Einstein’s general relativity remains a classical theory in the sense that it does not
incorporate the ideas of quantum mechanics. In a quantum theory of gravity, the gravitational force must be carried by a suitable messenger particle, or gauge boson. No workable quantum theory of
gravity has yet been developed, but general relativity determines some of the properties of the hypothesized “force” particle of gravity, the so-called graviton. In particular, the graviton must have
a spin quantum number of 2 and no mass, only energy.
The first proper understanding of the electromagnetic force dates to the 18th century, when a French physicist, Charles Coulomb, showed that the electrostatic force between electrically charged
objects follows a law similar to Newton’s law of gravitation. According to Coulomb’s law, the force F between one charge, q[1], and a second charge, q[2], is proportional to the product of the
charges divided by the square of the distance r between them, or F = kq[1]q[2]/r^2. Here k is the proportionality constant, equal to 1/4πε[0] (ε[0] being the permittivity of free space). An
electrostatic force can be either attractive or repulsive, because the source of the force, electric charge, exists in opposite forms: positive and negative. The force between opposite charges is
attractive, whereas bodies with the same kind of charge experience a repulsive force. Coulomb also showed that the force between magnetized bodies varies inversely as the square of the distance
between them. Again, the force can be attractive (opposite poles) or repulsive (like poles).
Magnetism and electricity are not separate phenomena; they are the related manifestations of an underlying electromagnetic force. Experiments in the early 19th century by, among others, Hans Ørsted
(in Denmark), André-Marie Ampère (in France), and Michael Faraday (in England) revealed the intimate connection between electricity and magnetism and the way the one can give rise to the other. The
results of these experiments were synthesized in the 1850s by the Scottish physicist James Clerk Maxwell in his electromagnetic theory. Maxwell’s theory predicted the existence of electromagnetic
waves—undulations in intertwined electric and magnetic fields, traveling with the velocity of light.
Max Planck’s work in Germany at the turn of the 20th century, in which he explained the spectrum of radiation from a perfect emitter (blackbody radiation), led to the concept of quantization and
photons. In the quantum picture, electromagnetic radiation has a dual nature, existing both as Maxwell’s waves and as streams of particles called photons. The quantum nature of electromagnetic
radiation is encapsulated in quantum electrodynamics, the quantum field theory of the electromagnetic force. Both Maxwell’s classical theory and the quantized version contain gauge symmetry, which
now appears to be a basic feature of the fundamental forces.
The electromagnetic force is intrinsically much stronger than the gravitational force. If the relative strength of the electromagnetic force between two protons separated by the distance within the
nucleus was set equal to one, the strength of the gravitational force would be only 10^−36. At an atomic level the electromagnetic force is almost completely in control; gravity dominates on a large
scale only because matter as a whole is electrically neutral.
The gauge boson of electromagnetism is the photon, which has zero mass and a spin quantum number of 1. Photons are exchanged whenever electrically charged subatomic particles interact. The photon has
no electric charge, so it does not experience the electromagnetic force itself; in other words, photons cannot interact directly with one another. Photons do carry energy and momentum, however, and,
in transmitting these properties between particles, they produce the effects known as electromagnetism.
In these processes energy and momentum are conserved overall (that is, the totals remain the same, in accordance with the basic laws of physics), but, at the instant one particle emits a photon and
another particle absorbs it, energy is not conserved. Quantum mechanics allows this imbalance, provided that the photon fulfills the conditions of Heisenberg’s uncertainty principle. This rule,
described in 1927 by the German scientist Werner Heisenberg, states that it is impossible, even in principle, to know all the details about a particular quantum system. For example, if the exact
position of an electron is identified, it is impossible to be certain of the electron’s momentum. This fundamental uncertainty allows a discrepancy in energy, ΔE, to exist for a time, Δt, provided
that the product of ΔE and Δt is very small—equal to the value of Planck’s constant divided by 2π, or 1.05 × 10^−34 joule seconds. The energy of the exchanged photon can thus be thought of as
“borrowed,” within the limits of the uncertainty principle (i.e., the more energy borrowed, the shorter the time of the loan). Such borrowed photons are called “virtual” photons to distinguish them
from real photons, which constitute electromagnetic radiation and can, in principle, exist forever. This concept of virtual particles in processes that fulfill the conditions of the uncertainty
principle applies to the exchange of other gauge bosons as well. | {"url":"https://www.britannica.com/science/subatomic-particle/Gravity","timestamp":"2024-11-03T20:36:06Z","content_type":"text/html","content_length":"126040","record_id":"<urn:uuid:a0e8ffd3-d9c4-4bfb-8f58-7a92ec1fb2ac>","cc-path":"CC-MAIN-2024-46/segments/1730477027782.40/warc/CC-MAIN-20241103181023-20241103211023-00874.warc.gz"} |
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Monodromies of hyperelliptic families of genus three curves. Thesis to the Mathematical Institute for the degree of Doctor of Science. March 2001. - Tohoku University, Sendai, Japan, 2001. - (Tohoku
Mathematical Publications; 21).
Nishikawa, Seiki (ed)
Proceedings of the fifth Pacific Rim geometry conference, held at Tohoku University, Sendai, Japan from July 25 to July 28, 2000. In Tohoku Mathematical Publications number 20, september 2001. -
Tohoku University, Sendai, Japan, 2001. - (Tohoku Mathematical Publications; 20).
Kikuchi, Tetsuya
Studies on commuting difference sytems arising from solvable lattice models. Thesis to the Mathematical Institute for the degree of Doctor of Science. March 2000. - Tohoku University, Sendai, Japan,
2000. - (Tohoku Mathematical Publications; 19).
Watabe, Daishi
Dirichlet problem at infinity for harmonic maps. Thesis to the Mathematical Institute for the degree of Doctor of Science. March 2000. - Tohoku University, Sendai, Japan, 2000. - (Tohoku Mathematical
Publications; 18).
Yamazaki, Takeshi
Model-theoretic studies on subsystems of second order arithmetic. Thesis to the Mathematical Institute for the degree of Doctor of Science. March 2000. - Tohoku University, Sendai, Japan, 2000. -
(Tohoku Mathematical Publications; 17).
Shimoda, Taishi
Hypoellipticity of second order differential operators with sign-changing principal symbols. Thesis presented for degree of Doctor of science. March 2000.. - Tohoku University, Sendai, Japan, 2000. -
(Tohoku Mathematical Publications; 15).
Jang, Youngho
Non-archimedean quantum mechanics. A thesis presented to the mathematical institute for the degree of doctor of science. - Tohoku University, Sendai, Japan, 1998. - (Tohoku Mathematical Publications;
Izumi, Hideaki
Non-commutative Lp-spaces constructed by the complex interpolation method. Thesis to the Mathematical Institute for the degree of Doctor of Science. January 1998. - Tohoku University, Sendai, Japan,
1998. - (Tohoku Mathematical Publications; 9).
Nishiura, Y. (ed) & Takagi, I. (ed) & Yanagida, E. (ed)
Proceedings of the international conference on asymptotics in nonlinear diffusive systems towards the understanding of singularities in dissipative structures, held at mathematical institute Tohoku
university july 28-august 1, 1997. - Tohoku University, 1998. - (Tohoku Mathematical Publications; 8).
Tanigaki, Miho
Saturation of the approximation by spectral decompositions associated with the Schrödinger operator. Thesis to the Mathematical Institute for the degree of Doctor of Science. January 1998. - Tohoku
University, Sendai, Japan, 1998. - (Tohoku Mathematical Publications; 7).
Fujiié, Setsuro
Solutions ramifiées des problèmes de Cauchy caractéristiques et fonctions hypergéométriques à deux variables. Thèse présentée en novembre 1994 à l'université Tohoku, Sendai, Japon. - Tohoku
University, Sendai, Japan, 1997. - (Tohoku Mathematical Publications; 6).
Ikai, Hisatoshi
Some prehomogeneous representations defined by cubic forms. Thesis to the Mathematical Institute for the degree of Doctor of Science. March 1997. - Tohoku University, Sendai, Japan, 1997. - (Tohoku
Mathematical Publications; 5).
Fujimori, Masami
Integral and rational points on algebraic curves of certain types and their Jacobian varieties over number fields. Thesis to the Mathematical Institute for the degree of Doctor of Science. November
1996. - Tohoku University, Sendai, Japan, 1997. - (Tohoku Mathematical Publications; 4).
Ikeda, Takeshi
Coset constructions of conformal blocks. Thesis to the Mathematical Institute for degree of Doctor of Science. Tohoku University, march 1996. - Tohoku University, 1995. - (Tohoku Mathematical
Publications; 3).
Takahashi, Tomokuni
Certain algebraic surfaces of general type with irregularity one and their canonical mappings. Thesis to the Mathematical Institute for the degree of Doctor of Science. Tohoku Univ. december 1995. -
Tohoku University, 1995. - (Tohoku Mathematical Publications; 2).
Furuhata, Hitoshi
Isometric pluriharmonic immersions of Kähler manifolds into semi-Euclidean spaces. Thesis to the Mathematical Institute for the degree of Doctor of Science. December 1995. - Tohoku University,
Sendai, Japan, 1995. - (Tohoku Mathematical Publications; 1). | {"url":"https://bmi.math.u-bordeaux.fr/cgi-bin/ouvrages?collection=TOHO","timestamp":"2024-11-09T04:14:34Z","content_type":"text/html","content_length":"11907","record_id":"<urn:uuid:4f8029cc-1460-403c-8a79-6315b7f4e44b>","cc-path":"CC-MAIN-2024-46/segments/1730477028115.85/warc/CC-MAIN-20241109022607-20241109052607-00509.warc.gz"} |
Signed graph
An undirected graph is signed if each edge has a positive or negative sign. A signed graph is called balanced if the product of all signs around every cycle is positive.
Given a signed graph, can you tell if it is balanced or not?
Input consists of several cases, each one with the number of vertices n, followed by the number of edges m, followed by m triples x y s to indicate an edge between x and y with sign s ∈ { −1, 1 }.
Assume 1 ≤ n ≤ 10^5, 0 ≤ m ≤ 5n, that vertices are numbered between 0 and n−1, x ≠ y, and that there is no more than one edge between x and y.
For every graph, print “yes” if it is balanced; otherwise print “no”. | {"url":"https://jutge.org/problems/P38073_en","timestamp":"2024-11-10T12:04:39Z","content_type":"text/html","content_length":"23390","record_id":"<urn:uuid:ac11f6a7-f123-4b3b-8898-362eecc1b67b>","cc-path":"CC-MAIN-2024-46/segments/1730477028186.38/warc/CC-MAIN-20241110103354-20241110133354-00792.warc.gz"} |
Mathematical Biology & Bioinformatics | Volume 15 Issue 2 Year 2020
Correlation of the Brain Compartments in the Attention Deficit and Hyperactivity Disorder Calculated by the Method of Virtual Electrodes from Magnetic Encephalography Data Ustinin M.N., Rykunov S.D.,
Boyko A.I.
Keldysh Institute of Applied Mathematics RAS, Moscow, Russia
Abstract. New method to study the correlation of the human brain compartments based on the magnetic encephalography data analysis was proposed. The time series for the correlation analysis are
generated by the method of virtual electrodes. First, the multichannel time series of the subject with confirmed attention deficit and hyperactivity disorder are transformed into the functional
tomogram - spatial distribution of the magnetic field sources structure on the discrete grid. This structure is provided by the inverse problem solution for all elementary oscillations, found by the
Fourier transform. Each frequency produces the elementary current dipole located in the node of the 3D grid. The virtual electrode includes the part of space, producing the activity under study. The
time series for this activity is obtained by the summation of the spectral power of all sources, covered by the virtual electrode. To test the method, in this article we selected ten basic
compartments of the brain, including frontal lobe, parietal lobe, occipital lobe and others. Each compartment was included in the virtual electrode, obtained from the subjects' MRI. We studied the
correlation between compartments in the frequency bands, corresponding to four brain rhythms: theta, alpha, beta, and gamma. The time series for each electrode were calculated for the period of 300
seconds. The correlation coefficient between power series was calculated on the 1 second epoch and then averaged. The results were represented as matrices. The method can be used to study
correlations of the arbitrary parts of the brain in any spectral band.
Key words: magnetic encephalography, spectral analysis, inverse problem solution, functional tomogram, virtual electrode, correlation, human brain rhythms. | {"url":"https://matbio.org/article.php?journ_id=37&id=468&lang=eng","timestamp":"2024-11-09T19:23:22Z","content_type":"text/html","content_length":"11696","record_id":"<urn:uuid:eea9df56-3be5-4c8d-857e-642453eb41a2>","cc-path":"CC-MAIN-2024-46/segments/1730477028142.18/warc/CC-MAIN-20241109182954-20241109212954-00330.warc.gz"} |
An ‘exploratory’ way to learn math | Colorado Academy News
Jane Doherty ’19 is really enjoying college, but she has a confession—she misses math at Colorado Academy. A Freshman at Georgetown University majoring in neurobiology, Doherty will have plenty of
opportunity to practice the math she learned at CA in collegiate statistics and calculus courses. But she feels prepared for any future challenge because of the years she spent at CA in what she
calls a “numerical playground,” where she gained “a deep love for math.”
“The way we learned math reflected the fact that the subject is not just a collection of equations to be memorized,” she says. “Rather, CA allowed me to appreciate math for the awesome thing that it
is—an entire system that leverages small known patterns to solve for large unknowns.”
Doherty describes CA’s approach to teaching math as “exploratory” and “investigative.” And that approach, she readily admits, occasionally led to moments of frustration.
“When new material was introduced in class, we weren’t just handed equations, which would’ve required much less time and brain power,” she says. “Instead, we were expected to manipulate mathematical
patterns we already knew to solve unfamiliar problems, which takes much longer and is a lot harder.”
What Doherty may not have fully realized during her years at CA was that the frustration she felt and the occasional struggles she had with math were not accidental. They were a deliberative part of
the curriculum for Upper School students, where Chair of the Mathematics Department Peter Horsch says teachers look for a balance between struggle and success as students develop their math skills.
“I like hearing students say, ‘I’m not sure, but here’s what I am thinking,’” Horsch says. “We want students to take on problems they are not sure they know how to do, perhaps struggle, perhaps not
get to a final result, but they persevere. And ultimately they see even the struggle is rewarding.”
Senior Andersen Dodge celebrates a moment of discovery in Pete Horsch’s class.
Math in Ninth Grade
Step into Horsch’s Math 1 class, a combination of algebra and geometry, and you may feel as though you have taken a wrong turn. The students are seated in small groups of four, talking quietly to
each other, as if they might be discussing the themes of a classic novel. Instead, they are talking math.
Horsch floats from group to group offering feedback. You never hear him say the words “that’s right” or “that’s wrong.” Instead, you hear this:
“What mathematical evidence supports your conclusion?”
“Your answer is good, but how do we know it’s right?”
“What was your process here?”
Students take new problems and migrate with their small groups to whiteboard walls, where they can fill the perimeter of the classroom with numbers, arrows, fractions, and different approaches to
solving a problem. At this point, the class becomes an aerobic exercise for Horsch as he bounces on cross-trainers around the room from group to group, coaching their mathematical thinking, smiling
all the time. “I love watching and listening to students doing math,” he says. “I know that it’s a tool that will allow them to understand their world, whether they go into business, engineering,
design, medicine, scientific research, art, or even teaching.”
Some groups of students join others to view their thinking. No one gives up. Horsch moves around the room, reviewing their whiteboard work and asking questions.
“I’m not convinced, are you?”
“That looks like a hard strategy. I wonder if there is another approach?”
“That’s excellent. But does it really work?”
Suddenly, Ninth Grader Dori Beck is jumping up and down at the whiteboard. “Five works! Five works! I can tell you why! Can we have more problems?” It is a classic “Aha!” moment.
Beck explains what prompted the celebration. “There are different ways to approach a problem, and Mr. Horsch is interested in how we as a group worked together, using different viewpoints, to get to
an answer,” she says. “You start to learn why things don’t work; there is a lot of trial and error involved, and when you get it, it’s rewarding.”
For Ninth Grader Hudson Parks, new to CA in his Freshman year, this approach to learning math is a departure from middle school. “I was used to having a lecture, taking notes, practicing the problem
for homework, then taking the test,” he says. Parks describes the approach at CA as more “interactive,” with communication among students and between a student and the teacher. “It’s actually easier
for me to learn,” he says. “It’s just more helpful to know why and how you got the answer, because you can build on that mathematical thinking.”
By the time the students hit the third round of problems in the class, they look like athletes who have spent the first period of the game warming up and now are ready to play. They come to their
final challenge with speed and focus. Horsch keeps up the questions and concludes with affirmation.
“This was really good work today. We were not just doing the math, but getting better at providing mathematical evidence. We were thinking about the process and finding ways of simplifying it so
something that took seven steps might only take two steps.”
Pete Horsch teaching Math 1
The method of teaching
Horsch likes to use an analogy to explain how his approach to teaching math has evolved over the years. Decades ago, he would show how a problem was solved, ask the students to practice it, and then
perform what they learned on a test. It was, he says, a lot like a music recital or a play, where students practice and then perform. But with time and teaching experience, he came to believe that
math was better viewed as a theatrical improvisation, inventing a play on the fly or as sight reading, playing music you have never seen before.
“What that may mean is that your math may not be as polished as a recital or play for which you have practiced,” he says. “You may miss a note or a line, and students have to understand that’s okay.”
Using this approach means that students are more actively involved in building their own understanding of math concepts. The peer groups they work in are chosen randomly, with no teacher’s hand
manipulating who works with whom. “The students learn from each other and they know I don’t control the group,” Horsch says. “The groups are not labeled by ability, which emphasizes that everyone has
something to contribute.” What all the students know is that they cannot look to Horsch for the answer. In response to a question, they are likely to get another question: “How did you get there?
Will this work again? Why do you think that?”
“From teaching a concept and then practicing the math, we have flipped it,” Horsch says. “We introduce the problem first and ask students to discover the underlying concept.”
Upper School Principal Dr. Jon Vogels compares this approach to the way science classes have been taught for many years. “You might start with a hypothesis in your science lab and do an experiment,”
he says. “When your lab doesn’t work, you would have to analyze why and try again.”
For individual students who might learn differently, Horsch adjusts his methods. Also, some math lessons are not driven by discovery. Instead, he teaches in a more explicit way, engaging in targeted
deliberate practice. Again, Horsch likes an analogy to explain why a mix of discovery and practice helps a student learn. “If you play tennis, hitting the ball fed by a machine can be really helpful,
but if that’s all I do, I won’t be prepared for a tennis match,” he says. “In real play, a ball can come at a different angle, height, or spin.” In short, students need both experiences—practice and
discovery—the ball machine and the competition—to best learn the game.
The value of struggle
For some students, the challenge of solving a problem they have never seen before and explaining their process—the equivalent of improvising or sight reading—may feel uncomfortable. Horsch likes to
point out that it’s not unlike the struggles competitive athletes go through at any level. Increasingly, education writers are addressing the concept of struggle, particularly in math. Jo Boaler, the
Nomellini-Olivier Professor of Education at Stanford, wrote about academic struggle in a recent article, excerpted here:
“As parents and teachers, we do just about everything we can to make sure that children don’t struggle. It turns out we are making a terrible mistake. Research shows that struggling is absolutely
critical to mastery and that the highest achieving people in the world are those who have struggled the most.
“Neuroscientists have found that mistakes are helpful for brain growth and connectivity and if we are not struggling, we are not learning. Not only is struggle good for our brains but people who know
about the value of struggle improve their learning potential. This knowledge would not be earth shattering if it was not for the fact that we in the Western world are trained to jump in and prevent
learners from experiencing struggle.
“When I was teaching middle schoolers in a research math camp a few years ago one girl stood out to me; she was nearly always wrong in her thinking, but she was always engaged, arguing her case,
pushing to understand better. An observer of the class would have described her as a low achiever, but she improved more than any other of the 84 students we taught that Summer. Her standardized test
score in mathematics improved by 450 percent after 18 struggle filled lessons. Our messages to the students—that struggle would be valued and mistakes are productive—had helped her feel good about
struggle and embrace it.
“Millions of students start the school year each year excited for all they will learn, but as soon as they struggle or see someone solve a problem with ease, they start to doubt themselves and
mentally shut down. This starts a less productive learning pathway for them. Instead they should value the time of struggle and know that they are on their way to being better, wiser and equipped
with a stronger brain.”
In Paul Tough’s book The Years That Matter Most: How College Makes or Breaks Us, he profiles Uri Treisman, PhD, who received a MacArthur Fellowship recognizing his work in math education. On his
first day in Freshman calculus at the University of Texas-Austin, Treisman tells his students, “Everybody in this class will struggle. No matter who you are, questions are going to be flying at you
that you can’t answer. And when that happens, you’re going to experience stress…. But in fact, that stress is an indicator that your understanding is deepening. It’s not a sign that you’re not
learning. It’s a sign that you are learning.”
Erin Gray teaching Math 2
Twelfth Grade calculus
The Twelfth Grade calculus class co-taught by Horsch and Erin Gray is an example of learning in action. The two teachers had a number of their Senior students when they were Freshmen in Math 1. They
have seen the students grow both in their mathematical knowledge and their view of themselves as creators of mathematical knowledge. “Their willingness to engage with difficult tasks and the success
they have is one of the most rewarding parts for me of teaching this class,” says Gray. “The continuity that we, as teachers, can maintain from Freshman year to Senior year speaks to the advantages
of being at a small school like CA.”
Senior Andersen Dodge describes math as “probably not my strongest subject,” and yet she really likes coming to class because of the one-on-one time she receives from teachers. Her classmate, Senior
Seb Parra, also admits that math is probably not his “favorite” subject, but he chose to take calculus because he wants to major in business in college and he believes that being able to take
derivatives of functions will help him as a young entrepreneur.
Like the Math 1 class, Horsch and Gray circulate from one randomly assigned group to the next, coaching and asking questions. It’s a very different approach to calculus than Gray remembers from her
days as a student. “My teacher had an overhead projector and he would work a problem or two,” she says. “We sat in rows, took notes, and then we practiced the problems.” But, she says, if you looked
at the topics in calculus that she studied, her students are learning the same content, except the learning is driven by student discovery. “This is a shared experience driven by student work,” she
says. “And the students have to be effective communicators about their mathematical work. That’s what prepares them for careers where they will need to work on diverse teams.”
“There’s no doubt this is a challenging way to teach,” Vogels says. “It demands both preparation and flexibility.”
Gray agrees. “To a casual observer, it may look like we are winging it, but we have come up with problems that point students in a particular direction, and there is always an element of surprise
that requires flexibility, creativity, and communication among students and teachers.”
She is convinced this approach serves CA students well, because it helps students build a deeper and longer lasting understanding of math. “Students retain the things they figure out better than the
things they are told,” she says. “You don’t want to have to memorize a formula for everything you do in math class, because memorizing doesn’t guarantee storage. A deep flexible understanding of a
concept—that’s the bigger win.”
In the calculus class, you hear those wins from students as they work problems.
“Wow, that’s crazy!” exclaims one student. Another chimes in, “That feels like magic!”
“No, not magic,” Horsch says. “It’s magical.” Later he explains what he means.
“Working through a math problem isn’t magic, like something behind the curtain you don’t understand,” he says. “It’s magical, because it’s fantastic how it all fits together.”
For Jane Doherty, the magical experience of CA math continues in college. “This way of learning ultimately increased my confidence so much, because I learned how to be mathematically self-reliant,
resilient, and cognitively flexible,” she says. “I learned that one’s abilities as a mathematician are measured best by how they can puzzle through new challenges by using prior foundational
information and patterns. I’m so thankful for the Math Department at CA.” | {"url":"https://news.coloradoacademy.org/an-exploratory-way-to-learn-math/","timestamp":"2024-11-11T04:13:50Z","content_type":"text/html","content_length":"116257","record_id":"<urn:uuid:e415c0c7-c658-4494-8c85-12ee3d11ed54>","cc-path":"CC-MAIN-2024-46/segments/1730477028216.19/warc/CC-MAIN-20241111024756-20241111054756-00517.warc.gz"} |
Lesson 18
Expressions with Rational Numbers
Lesson Narrative
As students start to gain fluency with rational number arithmetic, they encounter complicated numerical expressions, and algebraic expressions with variables, and there is a danger that they might
lose the connection between those expressions and numbers on the number line. The purpose of this lesson is to help students make sense of expressions, and reason about their position on the number
line, for example whether the number is positive or negative, which of two numbers is larger, or whether two expressions represent the same number. They work through common misconceptions that can
arise about expressions involving variables, for example the misconception that \(\text-x\) must always be a negative number. (It is positive if \(x\) is negative.) In the last activity they reason
about expressions in \(a\) and \(b\) given the positions of \(a\) and \(b\) on a number line without a given scale, in order to develop the idea that you can always think of the letters in an
algebraic expression as numbers and deduce, for example, that \(\frac14a\) is a quarter of the way from 0 to \(a\) on the number line, even if you don't know the value of \(a\).
When students look at a numerical expression and see without calculation that it must be positive because it is a product of two negative numbers, they are making use of structure (MP7).
Learning Goals
Teacher Facing
• Evaluate an expression for given values of the variable, including negative values, and compare (orally) the resulting values of the expression.
• Generalize (orally) about the relationship between additive inverses and about the relationship between multiplicative inverses.
• Identify numerical expressions that are equal, and justify (orally) that they are equal.
Student Facing
Let’s develop our signed number sense.
Required Preparation
Print and cut up slips from the Card Sort: The Same But Different blackline master. Prepare 1 copy for every 2 students. If possible, copy each complete set on a different color of paper, so that a
stray slip can quickly be put back.
Student Facing
• I can add, subtract, multiply, and divide rational numbers.
• I can evaluate expressions that involve rational numbers.
Glossary Entries
• rational number
A rational number is a fraction or the opposite of a fraction.
For example, 8 and -8 are rational numbers because they can be written as \(\frac81\) and \(\text-\frac81\).
Also, 0.75 and -0.75 are rational numbers because they can be written as \(\frac{75}{100}\) and \(\text-\frac{75}{100}\).
Print Formatted Materials
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To have and have not
The previous sections on Semantic Storage raised questions from readers about why the probabilistic characteristics of relationships between entities were not mentioned. The main reason for this is
that probabilities, when used, are numeric values, and according to the accepted architecture, numeric data is stored in Data Storage. Separation Data Storage from the Semantic one is technically
convenient since the set of attributes (including the probability) is different for different entities. The numerical values change regardless of the relationship - as a result of measurements and/or
calculations. So using probabilistic characteristics is certainly possible, but they are stored separately, and there is no rational reason to assign the value of probability to each relation.
However, since probabilities and the degree of confidence in many approaches are used as the significant or essential element of decision-making logic, it makes sense to analyze this aspect
Since decision-making is based on accumulated information, it is natural to start the analysis with the structuring of this information. The accumulated information consists mainly of facts of a
different nature, primarily the facts that certain events occurred in a specific sequence, the measured values had corresponding values, and so on. In addition, there are facts based on predetermined
natural-scientific laws of the macro-world and mathematical laws (the relationship between speed and acceleration, between the radius of a circle and its area). With this factual information, it
makes no sense to talk about probability, and if, nevertheless, it is required, then no other value other than precisely one is suitable.
The accumulated numerical data on facts can be processed to obtain statistical data, including the frequency of certain events in certain situations. That is, estimates of probabilities appear along
with the aggregation of facts, in other words, when forming statistical models. The corresponding frequencies can be considered estimates of probabilities, intended to be used in many decision-making
algorithms. At the same time, in essence, estimates of probability are hypotheses that correspond to reality to a greater or lesser extent - not a fact, although they are based on facts. For example,
suppose data is collected about whether a person was burned after taking the first sip of liquid from a cup. In that case, the total number of incidents and the number of burns are facts, but the
estimate of the likelihood of such an event in the future is a hypothesis and may change as data accumulates.
The essential aspect in the macro-world is that the probabilistic nature of events reflects, in most cases, not the fundamental features of the environment or the process but the consequence that
facts corresponding to different situations are collected in one set. In our case (a burn from the first sip), information about what kind of liquid they drink significantly changes the situation. If
it is beer or sparkling water, then there will be no burns at all. Adding the result of measuring the temperature of the liquid to the situation description turns the process into a completely
deterministic one.
Thus, the more detailed the description of the situation, the less random the results are; the more detailed the classification of situations, the less random the consequences will be, and the less
reasonable is the use of a probabilistic model (probabilistic hypothesis). Theoretically, using a model that considers all the factors influencing the predicted events, the situation becomes
completely deterministic. In reality, however, situations regularly occur when the factors influencing the consequences, in principle, cannot be known (measured, detected). It is the main reason for
using a probabilistic approach to decision making. However, the "obvious" conclusion "if the situation is non-deterministic, then it is useful to use a probabilistic approach" is not as logical as it
might seem at first glance.
First, when a situation arises that did not occur before, it is necessary to decide. However, at the same time, there is no data for constructing an estimate of the probability. So using the
probabilistic approach, it is necessary to have a "spare" variant of the decision-making method that does not involves probabilities of possible consequences.
Secondly, if the number of past cases, similar to the current one, is small, statistical estimates of probabilities will be deliberately inadequate. It is required to set a certain minimum amount of
statistical data that must be reached before switching to probability-based decision-making.
The third, no less important, aspect is that a reasonable approach to decision-making is based on the assumption that the more detailed the situation is, the more grounds for making an optimal
decision. However, the more factors are used to define the situation, the more different combinations of factors describing the situation. Each factor combination requires its estimates of the
probabilities. For example, if there are only eight factors, each of which has one of three possible values, the number of possible situations exceeds six thousand. When statistical data for the
current situation are insufficient to obtain adequate probability estimation, cases will not be rare cases rather frequent ones. Accordingly, the "spare" decision-making method in natural conditions
becomes frequently used.
Finally, there is the last aspect, the most logically complex and least apparent. The use of probability estimates for decision-making is based on the implicit assumption that as experience and
relevant statistical data are accumulated, the probability estimates will approach specific actual values, providing an opportunity to make optimal decisions. However, a more thorough analysis leads
to the conclusion that this assumption is erroneous. The reason for this is that the source data for assessing the probabilities are the collected statistical data on the consequences of
decision-making. The choice of actions depends on the probability estimates, the probability estimates depend on the collected statistics, and the statistics depend on the decisions made; there is a
logical loop.
A naive iterative process in which current estimates of probabilities are used to make decisions, and the consequences of a decision update statistics and, accordingly, estimates of probability is
essentially a process of solving a system of equations. If “f” is the decision-making function, “u” is the selected action, “P” is the vector of the estimates of probabilities, and “g” is the
function of updating the probabilities under the outcome of the execution of the action u, we obtain the following set of relations for a certain situation:
u = f( P )
P = g( u )
It is nothing more than a system of equations with unknowns “u” and “P”. The mathematical aspect is that the iterative process of recalculating the probability estimates will not necessarily bring
the estimates closer to the actual values. It is a consequence that the decisions aim to avoid undesirable consequences, which reduces the frequency of choosing the actions that lead to them (up to
zero in some cases), and statistics for these actions are updated less often (or never). As a result, the initial "classification" into desirable and undesirable actions, formed at the initial stage,
tends to be preserved. This formation itself is based on statistical data with a small number of samples of the initial stages.
The repository https://github.com/mrabchevskiy/probability contains a Python script that demonstrates the above features with numerical simulations. There are two possible actions, one more
beneficial but also riskier, and vice versa. If the probability of the undesirable consequences of a risky action exceeds a predetermined threshold, the less risky action is selected. The first-time
action (there are no statistics and therefore no estimates of the probability) is chosen randomly; a series of tests are carried out. Results:
Threshold: 0.100
Number of tests: 600
Steps per test: 500
Cautious action:
True probability of the unwanted consequences 0.010;
Range of probability estimation: 0.000 .. 0.026 .
Risky action:
True probability of the unwanted consequences 0.200;
Range of probability estimation: 0.100 .. 1.000 .
As we can see, the estimates of probabilities can radically differ from the actual values, while there are no obvious ways of detecting such a situation.
The only reliable way to avoid developing a stably non-optimal behavior is to make decisions not based on an assessment of the consequences of a decision but based on the need to accumulate a
sufficient amount of statistical data during a sufficiently large number of decision-making acts for each specific situation. It eliminates the mutual dependence of decisions and estimates of
probabilities and means switching to the mode of experimental study of the situation to form an adequate statistical model. Since there are many different situations, and the formation of an adequate
statistical model for each of the situations is time-consuming, such a self-learning process, which guarantees the development of optimal behavior, becomes complex and resource-hungry. Obviously, in
many cases, such an approach is undesirable or unacceptable due to the high risks.
It means that at least there is no a priori superiority of approaches based on the use of probability estimates compared to other approaches (and there are such approaches). Especially if we take
into account that in many cases, the optimal solution does not depend on the estimates of probabilities but is determined only by the nonzero probability of one or another consequence, that is, by
the facts of the possibility of one or another outcome, regardless of their probability. For example, a hungry predator may decide to catch prey regardless of a specific probability of success — just
the possibility of success is sufficient. Furthermore, his potential victim decides to run away regardless of the likelihood of success (subjective assessments of which, for an apparent reason, will
always be 100%).
Decision-making methods without using probability estimates are the subject of one of the following chapters.
• The described architecture allows the use of a probabilistic approach to decision making.
• It is irrational to endow each ratio with an attribute of probability since, for a significant fraction of them, the probability is equal to one.
• The use of decision-making algorithms based on estimates of probability also requires an alternative "spare" method.
• The naive option of refining the probability estimates as these estimates are used for decision-making can lead to the development of stable estimates that are significantly different from the
actual values, leading to non-optimal behavior.
• The optimal solution does not always depend on the probability of consequences.
• There are approaches to decision-making that do not use estimates of the likelihood of consequences. | {"url":"https://agieng.substack.com/p/probability-in-decision-making","timestamp":"2024-11-05T00:31:08Z","content_type":"text/html","content_length":"165285","record_id":"<urn:uuid:53fc4950-9391-4d74-87fd-ea0ea2b89f86>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.84/warc/CC-MAIN-20241104225856-20241105015856-00876.warc.gz"} |
Quantitative Aptitude Quiz For ESIC- UDC, Steno, MTS Prelims 2022-10th January
Q1. The ratio of numerical value of rate of interest and time period at a sum invested is 5 : 2 and men get 22.5% of invested sum as interest. Find at the same rate how much interest man will get on
the sum of Rs. 1600 for two year ?
(a) 240 Rs.
(b) 180 Rs.
(c) 160 Rs.
(d) 120 Rs.
(e) 224 Rs.
Q2. Deepak and Ayush invested Rs. 12600 together in a business. Deepak invested for 8 months and Ayush invested for 15 months, if Deepak and Ayush get profit in the ratio of 75 : 128, then find the
amount invested by Ayush ?
(a) 2400 Rs.
(b) 3000 Rs.
(c) 3600 Rs.
(d) 4800 Rs.
(e) 6400 Rs.
Q3. Mohit takes half the time to do a work as Ayush takes to complete the same work, while Veer takes same time as Ayush & Mohit takes to complete the work together. Find in how many days Veer will
complete work together, if all three complete the work in 6 days ?
(a) 8 days
(b) 12 days
(c) 6 days
(d) 9 days
(e) 18 days
Q4. A vessel contains 96ℓ of mixture of milk and water in the ratio of 5 : 3 respectively. If 24ℓ of mixture taken out and some quantity of milk and water added in the ratio of 3 : 5 so the new ratio
of milk and water becomes 15 : 13. Find quantity of water added ?
(a) 12.5 ℓ
(b) 15 ℓ
(c) 25 ℓ
(d) 7.5 ℓ
(e) 30 ℓ
Q5. The ratio between length of two trains is 2 : 3 and speed of these two trains is 108 km/hr and 96 km/hr respectively. if smaller train cross a 360 m long platform in 16 sec, then find the time
taken by both trains to cross each other running in same direction ?
(a) 90 sec
(b) 84 sec
(c) 96 sec
(d) 108 sec
(e) 112 sec
Q6. The ratio between length of a rectangle and side of a square is 8 : 9. If breadth of rectangle is 10 cm and perimeter of square is 20 cm more than that of perimeter of rectangle, then find
difference between area of square & that of rectangle?
(a) 164 cm²
(b) 172 cm²
(c) 174 cm²
(d) 156 cm²
(e) 14\]
Q7. Average age of A , B & C four years hence is 24 years and the ratio between age of B & C is 6 : 5. If age of A is 4 years less than that of C, then find average age of A & B two years hence will
(a) 17 years
(b) 19 years
(c) 21 years
(d) 20 years
(e) 22 years
Q8. A bag contains ‘a’ red balls, 5 green balls. One ball taken out at random, then probability of being red ball is 3/8 . If two balls taken out at random from bag, then find probability of being
both balls either red or green?
(a) 15/28
(b) 13/28
(c) 9/28
(d) 11/28
(e) 9/28
Q9. If an article is sold at 25% discount at mark-price then loss percent is 10%. If the article is sold at marked price then what will be profit or loss percent ?
(a) 50%
(b) 40%
(c) 30%
(d) 20%
(e) 25%
Q11. Ramesh and Suresh can do a work in 18 days and 12 days respectively. Both are working together for x days and the Suresh left the work and Ramesh can do rest of the work in 8 days. Find the
value of x is ?
(a) 6 days
(b) 4 days
(c) 8 days
(d) 12 days
(e) 10 days
Q12. Anurag and Vijay can finish a work in 8 days. If Anurag increased his efficiency by 100% of his initial efficiency and Vijay decreased his efficiency by 66⅔% of his initial efficiency then they
together finish the same work in 6 days. In how many days Vijay alone can finish the same work?
(a) 25 days
(b) 40 days
(c) 20 days
(d) 35 days
(e) 22 days
Q14. There are 10 pens and 6 pencils in a box in which three are defective. Find the probability that one of them is pencil and two are pen ?
(a) 1/2
(b) None of these
(c) 27/57
(d) 27/56
(e) 3/5
Q15. How many 3 digit number can be formed by 0, 1, 2, 5, 7, 6, 9. Which is divisible by 5 and none of digit is repeated ?
(a) 55
(b) 58
(c) 68
(d) 54
(e) 52
Click Here to Register for Bank Exams 2022 Preparation Material | {"url":"https://www.bankersadda.com/quantitative-aptitude-quiz-for-esic-udc-steno-mts-prelims-2022-10th-january/","timestamp":"2024-11-04T16:57:57Z","content_type":"text/html","content_length":"610507","record_id":"<urn:uuid:398b3b01-fc19-4d58-8495-f79d3f973e05>","cc-path":"CC-MAIN-2024-46/segments/1730477027838.15/warc/CC-MAIN-20241104163253-20241104193253-00016.warc.gz"} |
ACT vs SAT Math
The SAT and ACT are both accepted standardized tests for admission to every college in the US, so we highly recommend that you figure out which test is best for you! One way to do that is to compare
the math sections and figure out which one complements your strengths the most. You could take practice tests from each test and try to figure it out on your own. Or you can read on to find out some
of the major differences between the SAT math and ACT math!
The ACT Math is 60 minutes long and contains 60 multiple choice questions. This gives you a minute per question. The test is arranged from easy to hard, so you may spend more or less time per
question. You can use an ACT approved calculator for the entirety of the section. The maximum score is 36, and it counts for 25% of your composite score.
The SAT Math has 2 parts. The first is a no calculator section. It has 15 multiple choice questions and 5 grid-in questions to be completed in 25 minutes. This gives you about 1 minute 15 seconds per
question. The second part allows an SAT approved calculator. You get 55 minutes to complete 30 multiple choice and 8 grid-in questions. This gives you about 1 min 26 seconds per question. Like the
ACT, both SAT Math sections are arranged from easy to hard, so you may spend more or less time per question. The maximum score for the entire Math section is 800, which means it counts for 50% of
your composite score.
Reference sheet
There is no reference sheet on the ACT, so you’re responsible for knowing all formulas. The SAT provides a reference sheet with 12 commonly used geometry formulas. This might be something to consider
if you haven’t taken geometry in a while or if you struggle to remember formulas.
The SAT and ACT math covers many of the same topics. Having a solid foundation in pre-algebra, algebra 1, algebra 2, geometry, and some trigonometry is crucial for success on both tests. However, the
ACT includes graphs from trigonometry, logarithms, and matrices while the SAT does not. Geometry questions make up about 30-45% of the ACT, while the SAT contains about 10% geometry.
SAT Math Topics:
• Heart of Algebra: linear equations, inequalities, systems of linear equations/inequalities,
• Problem Solving and Data Analysis: ratios, proportions, percentages, graphs/charts, statistics, and probability.
• Passport to Advanced Math: quadratic/exponential equations and graphs.
• Additional Topics in Math: geometry and trigonometry.
ACT Math Topics
• Preparing for Higher Math (57-60%)
• Number and Quantity: real/complex numbers, integers/rational exponents, vectors, and matrices.
• Algebra: linear, polynomial, radical, and exponential equations
• Functions: linear, radical, piecewise, polynomial, and logarithmic functions.
• Geometry: shapes and solids, trigonometric ratios, trig graphs
• Statistics and Probability: spread of distributions, data collection methods, and calculating probabilities.
• Integrating Essential Skills (40-43%): rates/percentages, proportions, area and volume, mean and median, and equivalent expressions; problems that combine Preparing for Higher Math content
Question Types
Both the ACT and SAT contain problems similar to those in a textbook as well as some that require critical thinking. In general, the ACT tends to have more straightforward questions that test whether
you know the math concepts and when to apply them. The SAT usually has more word problems and data analysis questions.
The ACT and SAT math sections cover similar content in different ways. By considering the formatting, time limits, and question types, you can choose the test that shows off your math skills the | {"url":"https://blog.brainsandbrawnllc.com/act-vs-sat-math","timestamp":"2024-11-11T07:17:46Z","content_type":"text/html","content_length":"72665","record_id":"<urn:uuid:b3b7f362-a7db-4177-ab3b-4b32301ffd96>","cc-path":"CC-MAIN-2024-46/segments/1730477028220.42/warc/CC-MAIN-20241111060327-20241111090327-00325.warc.gz"} |
Formal system explained
A formal system is an abstract structure and formalization of an axiomatic system used for inferring theorems from axioms by a set of inference rules.^[1]
In 1921, David Hilbert proposed to use formal systems as the foundation of knowledge in mathematics.^[2]
The term formalism is sometimes a rough synonym for formal system, but it also refers to a given style of notation, for example, Paul Dirac's bra–ket notation.
A formal system has the following:^[3] ^[4] ^[5]
A formal system is said to be recursive (i.e. effective) or recursively enumerable if the set of axioms and the set of inference rules are decidable sets or semidecidable sets, respectively.
Formal language
See main article: Formal language, Formal grammar, Syntax (logic) and Logical form.
A formal language is a language that is defined by a formal system. Like languages in linguistics, formal languages generally have two aspects:
• the syntax is what the language looks like (more formally: the set of possible expressions that are valid utterances in the language)
• the semantics are what the utterances of the language mean (which is formalized in various ways, depending on the type of language in question)
Usually only the syntax of a formal language is considered via the notion of a formal grammar. The two main categories of formal grammar are that of generative grammars, which are sets of rules for
how strings in a language can be written, and that of analytic grammars (or reductive grammar^[6] ^[7]), which are sets of rules for how a string can be analyzed to determine whether it is a member
of the language.
Deductive system
See main article: Inference, Logical consequence and Deductive reasoning. A deductive system, also called a deductive apparatus,^[8] consists of the axioms (or axiom schemata) and rules of inference
that can be used to derive theorems of the system.^[9]
Such deductive systems preserve deductive qualities in the formulas that are expressed in the system. Usually the quality we are concerned with is truth as opposed to falsehood. However, other
modalities, such as justification or belief may be preserved instead.
In order to sustain its deductive integrity, a deductive apparatus must be definable without reference to any intended interpretation of the language. The aim is to ensure that each line of a
derivation is merely a logical consequence of the lines that precede it. There should be no element of any interpretation of the language that gets involved with the deductive nature of the system.
The logical consequence (or entailment) of the system by its logical foundation is what distinguishes a formal system from others which may have some basis in an abstract model. Often the formal
system will be the basis for or even identified with a larger theory or field (e.g. Euclidean geometry) consistent with the usage in modern mathematics such as model theory.
An example of a deductive system would be the rules of inference and axioms regarding equality used in first order logic.
The two main types of deductive systems are proof systems and formal semantics.
Proof system
See main article: Proof system and Formal proof. Formal proofs are sequences of well-formed formulas (or WFF for short) that might either be an axiom or be the product of applying an inference rule
on previous WFFs in the proof sequence. The last WFF in the sequence is recognized as a theorem.
Once a formal system is given, one can define the set of theorems which can be proved inside the formal system. This set consists of all WFFs for which there is a proof. Thus all axioms are
considered theorems. Unlike the grammar for WFFs, there is no guarantee that there will be a decision procedure for deciding whether a given WFF is a theorem or not.
The point of view that generating formal proofs is all there is to mathematics is often called formalism. David Hilbert founded metamathematics as a discipline for discussing formal systems. Any
language that one uses to talk about a formal system is called a metalanguage. The metalanguage may be a natural language, or it may be partially formalized itself, but it is generally less
completely formalized than the formal language component of the formal system under examination, which is then called the object language, that is, the object of the discussion in question. The
notion of theorem just defined should not be confused with theorems about the formal system, which, in order to avoid confusion, are usually called metatheorems.
See main article: Semantics of logic, Interpretation (logic) and Model theory.
A logical system is a deductive system (most commonly first order logic) together with additional non-logical axioms. According to model theory, a logical system may be given interpretations which
describe whether a given structure - the mapping of formulas to a particular meaning - satisfies a well-formed formula. A structure that satisfies all the axioms of the formal system is known as a
model of the logical system.
A logical system is:
• Sound, if each well-formed formula that can be inferred from the axioms is satisfied by every model of the logical system.
• Semantically complete, if each well-formed formula that is satisfied by every model of the logical system can be inferred from the axioms.
An example of a logical system is Peano arithmetic. The standard model of arithmetic sets the domain of discourse to be the nonnegative integers and gives the symbols their usual meaning.^[10] There
are also non-standard models of arithmetic.
See main article: Formalism (philosophy of mathematics).
Early logic systems includes Indian logic of Pāṇini, syllogistic logic of Aristotle, propositional logic of Stoicism, and Chinese logic of Gongsun Long (c. 325–250 BCE) . In more recent times,
contributors include George Boole, Augustus De Morgan, and Gottlob Frege. Mathematical logic was developed in 19th century Europe.
David Hilbert instigated a formalist movement called Hilbert’s program as a proposed solution to the foundational crisis of mathematics, that was eventually tempered by Gödel's incompleteness
theorems. The QED manifesto represented a subsequent, as yet unsuccessful, effort at formalization of known mathematics.
Further reading
• Raymond M. Smullyan, 1961. Theory of Formal Systems: Annals of Mathematics Studies, Princeton University Press (April 1, 1961) 156 pages
• Stephen Cole Kleene, 1967. Mathematical Logic Reprinted by Dover, 2002.
• Douglas Hofstadter, 1979. . 777 pages.
External links
• Daniel Richardson, Formal systems, logic and semantics
• William J. Rapaport, Syntax & Semantics of Formal Systems
• PlanetMath, Formal System
• Pr∞fWiki, Definition:Formal System
• Pr∞fWiki, Definition:Deductive Apparatus
• Encyclopedia of Mathematics, Formal system
• Peter Suber, Formal Systems and Machines: An Isomorphism, 1997.
• Ray Taol, Formal Systems
Some quotes from John Haugeland's `Artificial Intelligence: The Very Idea' (1985), pp. 48–64.
Notes and References
1. Web site: Formal system Logic, Symbols & Axioms Britannica . 2023-10-10 . www.britannica.com . en.
2. Book: Hilbert's Program, Stanford Encyclopedia of Philosophy . 31 July 2003. https://plato.stanford.edu/archives/spr2016/entries/hilbert-program. Zach. Richard. Hilbert's Program. Metaphysics
Research Lab, Stanford University.
3. Web site: formal system . 2023-10-10 . planetmath.org.
4. Web site: Rapaport . William J. . 25 March 2010 . Syntax & Semantics of Formal Systems . University of Buffalo.
5. Web site: Definition:Formal System - ProofWiki . 2023-10-16 . proofwiki.org.
6. Reductive grammar: (computer science) A set of syntactic rules for the analysis of strings to determine whether the strings exist in a language. Web site: Sci-Tech Dictionary McGraw-Hill
Dictionary of Scientific and Technical Terms . McGraw-Hill . 6th. About the AuthorCompiled by The Editors of the McGraw-Hill Encyclopedia of Science & Technology (New York, NY) an in-house staff
who represents the cutting-edge of skill, knowledge, and innovation in science publishing. https://www.amazon.com/McGraw-Hill-Dictionary-Scientific-Technical-Terms/dp/007042313X#
7. "There are two classes of formal-language definition compiler-writing schemes. The productive grammar approach is the most common. A productive grammar consists primarrly of a set of rules that
describe a method of generating all possible strings of the language. The reductive or analytical grammar technique states a set of rules that describe a method of analyzing any string of
characters and deciding whether that string is in the language."Web site: The TREE-META Compiler-Compiler System: A Meta Compiler System for the Univac 1108 and General Electric 645, University
of Utah Technical Report RADC-TR-69-83. C. Stephen Carr, David A. Luther, Sherian Erdmann . 5 January 2015.
8. Web site: Definition:Deductive Apparatus - ProofWiki . 2023-10-10 . proofwiki.org.
9. Hunter, Geoffrey, Metalogic: An Introduction to the Metatheory of Standard First-Order Logic, University of California Press, 1971 | {"url":"http://everything.explained.today/Formal_system/","timestamp":"2024-11-11T20:46:26Z","content_type":"text/html","content_length":"29380","record_id":"<urn:uuid:c25a6f48-43ec-4d10-b8b3-d62446a849bc>","cc-path":"CC-MAIN-2024-46/segments/1730477028239.20/warc/CC-MAIN-20241111190758-20241111220758-00568.warc.gz"} |
Tutorial 14: Addition in VHDL
Created on: 19 February 2013
Adding two unsigned 4-bit numbers in VHDL using the VHDL addition operator (+) – a 4-bit binary adder is written in VHDL and implemented on a CPLD.
There are many examples on the Internet that show how to create a 4-bit adder in VHDL out of logic gates (which boils down to using logical operators in VHDL). A full adder adds only two bits and a
carry in bit. It also has a sum bit and a carry out bit. The idea is that a number of these 1-bit adders are linked together to form an adder of the desired length.
It may not be necessary to implement an adder in this way, as VHDL can use the addition operator (+) to add two numbers. We will look at adding two STD_LOGIC_VECTOR data types together.
The design is implemented on a CPLD on the home built CPLD board. Half of the switch bank on the board (4 switches) is used to input one of the values to be added, the other half of the switch bank
is used to input the second value to be added. The result (sum) of the addition of the two 4-bit numbers is displayed on 5 LEDs.
Designing the Adder
From the previous tutorial, we know that we can add two numbers together that are of the STD_LOGIC_VECTOR data type if we use the STD_LOGIC_SIGNED or the STD_LOGIC_UNSIGNED package from the IEEE
library, e.g.:
library IEEE;
use IEEE.STD_LOGIC_UNSIGNED.ALL;
This enables us to use inputs connected to switches and outputs connected to LEDs of the STD_LOGIC_VECTOR type defined in the Ports section of the ENTITY part of the VHDL.
Adding two numbers in the ARCHITECTURE part of the VHDL is as simple as this:
SUM <= NUMBER1 + NUMBER2;
Vector Sizes
We will be using two 4-bit inputs connected to 4 switches each as our numbers that will be added together. So the size of these numbers will be 4 bits each (3 downto 0).
The biggest output number (sum) generated by adding two 4-bit numbers together will be 1111b + 1111b = 11110b. So the vector used for our sum needs to be 5 bits in length (4 downto 0).
The only problem with adding two 4 bit numbers in VHDL and then putting the result in a 5 bit vector is that the highest bit (or carry) will be truncated. In other words, even if the size of the sum
vector is 5 bits, a 4 bit sum will be placed in it.
We will now look at two ways of implementing the same 4-bit binary adder and solve the problem of adding two 4-bit numbers and displaying the 5-bit result.
4-Bit Adder, Example 1
The first way of solving the problem of adding two 4-bit numbers and displaying the 5-bit result is to make the two 4-bit input numbers 5 bits in length, but keep the MSB of each number at 0.
This code shows how to implement the adder in VHDL:
library IEEE;
use IEEE.STD_LOGIC_1164.ALL;
use IEEE.STD_LOGIC_UNSIGNED.ALL;
entity binary_4_bit_adder_top is
Port ( NUM1 : in STD_LOGIC_VECTOR (4 downto 0) := "00000";
NUM2 : in STD_LOGIC_VECTOR (4 downto 0) := "00000";
SUM : out STD_LOGIC_VECTOR (4 downto 0));
end binary_4_bit_adder_top;
architecture Behavioral of binary_4_bit_adder_top is
SUM <= NUM1 + NUM2;
end Behavioral;
The code makes all the vectors to be 5 bits wide. The two input vectors are initialized with a value of 00000b so that the MSB is always 0.
To implement the above code on the home built CPLD board, we need to compensate for the inverting inputs and outputs connected to the switches and LEDs by inverting all inputs and outputs:
library IEEE;
use IEEE.STD_LOGIC_1164.ALL;
use IEEE.STD_LOGIC_UNSIGNED.ALL;
entity binary_4_bit_adder2 is
Port ( NUM1 : in STD_LOGIC_VECTOR (4 downto 0) := "11111";
NUM2 : in STD_LOGIC_VECTOR (4 downto 0) := "11111";
SUM : out STD_LOGIC_VECTOR (4 downto 0));
end binary_4_bit_adder2;
architecture Behavioral of binary_4_bit_adder2 is
signal A : STD_LOGIC_VECTOR (4 downto 0);
signal B : STD_LOGIC_VECTOR (4 downto 0);
signal X : STD_LOGIC_VECTOR (4 downto 0);
X <= A + B;
-- compensate for inverting inputs and outputs
A <= not NUM1;
B <= not NUM2;
SUM <= not X;
end Behavioral;
In the above code, the inputs are initialized to 11111b because they will be inverted to 00000b by the code at the bottom of the ARCHITECTURE implementation.
The inputs from the switches (NUM1 and NUM2) are inverted before doing the addition. The result of the addition is inverted before displaying it on the LEDs (SUM).
4-Bit Adder, Example 2
This is a better solution to the problem of adding two 4-bit numbers and displaying the result in a 5-bit vector. The VHDL concatenation operator (&) is used to put a 0 in front of each of the the
two 4-bit numbers before adding them.
The concatenation operator can be used to join extra data to the left or the right of the vector:
RESULT <= ('0' & NUMBER); -- join a 0 to the beginning of NUMBER and put into RESULT
RESULT <= (NUMBER & '0'); -- join a 0 to the end of NUMBER and put into RESULT
VHDL code for the second adder example:
library IEEE;
use IEEE.STD_LOGIC_1164.ALL;
use IEEE.STD_LOGIC_UNSIGNED.ALL;
entity binary_4_bit_adder3_top is
Port ( NUM1 : in STD_LOGIC_VECTOR (3 downto 0); -- 4-bit number
NUM2 : in STD_LOGIC_VECTOR (3 downto 0); -- 4-bit number
SUM : out STD_LOGIC_VECTOR (4 downto 0)); -- 5 bit result
end binary_4_bit_adder3_top;
architecture Behavioral of binary_4_bit_adder3_top is
SUM <= ('0' & NUM1) + ('0' & NUM2);
end Behavioral;
Again, we need to compensate for the inverting inputs and outputs of the home built CPLD board. The VHDL code below works the same way as the code above, but inverts all inputs and outputs:
library IEEE;
use IEEE.STD_LOGIC_1164.ALL;
use IEEE.STD_LOGIC_UNSIGNED.ALL;
entity binary_4_bit_adder4_top is
Port ( NUM1 : in STD_LOGIC_VECTOR (3 downto 0);
NUM2 : in STD_LOGIC_VECTOR (3 downto 0);
SUM : out STD_LOGIC_VECTOR (4 downto 0));
end binary_4_bit_adder4_top;
architecture Behavioral of binary_4_bit_adder4_top is
signal A : STD_LOGIC_VECTOR (3 downto 0);
signal B : STD_LOGIC_VECTOR (3 downto 0);
signal X : STD_LOGIC_VECTOR (4 downto 0);
X <= ('0' & A) + ('0' & B);
-- compensate for the inverting inputs and outputs
A <= not NUM1;
B <= not NUM2;
SUM <= not X;
end Behavioral; | {"url":"https://startingelectronics.org/software/VHDL-CPLD-course/tut14-VHDL-adder/","timestamp":"2024-11-10T01:43:05Z","content_type":"text/html","content_length":"17024","record_id":"<urn:uuid:f8884a1e-057c-45cb-b7b9-c5d988dfba7a>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.3/warc/CC-MAIN-20241110005602-20241110035602-00872.warc.gz"} |
How to Create a Prediction Interval in R | Online Tutorials Library List | Tutoraspire.com
How to Create a Prediction Interval in R
by Tutor Aspire
A linear regression model can be useful for two things:
(1) Quantifying the relationship between one or more predictor variables and a response variable.
(2)Â Using the model to predict future values.
In regards to (2), when we use a regression model to predict future values, we are often interested in predicting both an exact value as well as an interval that contains a range of likely
values. This interval is known as a prediction interval.
For example, suppose we fit a simple linear regression model using hours studied as a predictor variable and exam score as the response variable. Using this model, we might predict that a student
who studies for 6 hours will receive an exam score of 91.
However, because there is uncertainty around this prediction, we might create a prediction interval that says there is a 95% chance that a student who studies for 6 hours will receive an exam score
between 85 and 97. This range of values is known as a 95% prediction interval and it’s often more useful to us than just knowing the exact predicted value.
How to Create a Prediction Interval in R
To illustrate how to create a prediction interval in R, we will use the built-in mtcars dataset, which contains information about characteristics of several different cars:
#view first six rows of mtcars
# mpg cyl disp hp drat wt qsec vs am gear carb
#Mazda RX4 21.0 6 160 110 3.90 2.620 16.46 0 1 4 4
#Mazda RX4 Wag 21.0 6 160 110 3.90 2.875 17.02 0 1 4 4
#Datsun 710 22.8 4 108 93 3.85 2.320 18.61 1 1 4 1
#Hornet 4 Drive 21.4 6 258 110 3.08 3.215 19.44 1 0 3 1
#Hornet Sportabout 18.7 8 360 175 3.15 3.440 17.02 0 0 3 2
#Valiant 18.1 6 225 105 2.76 3.460 20.22 1 0 3 1
First, we’ll fit a simple linear regression model using disp as the predictor variable and mpg as the response variable.
#fit simple linear regression model
model #view summary of fitted model
#lm(formula = mpg ~ disp, data = mtcars)
# Min 1Q Median 3Q Max
#-4.8922 -2.2022 -0.9631 1.6272 7.2305
# Estimate Std. Error t value Pr(>|t|)
#(Intercept) 29.599855 1.229720 24.070
Then, we’ll use the fitted regression model to predict the value of mpg based on three new values for disp.Â
#create data frame with three new values for disp
#use the fitted model to predict the value for mpg based on the three new values
#for disp
predict(model, newdata = new_disp)
# 1 2 3
#23.41759 21.35683 19.29607
The way to interpret these values is as follows:
• For a new car with a disp of 150, we predict that it will have a mpg of 23.41759.Â
• For a new car with a disp of 200, we predict that it will have a mpg of 21.35683 .Â
• For a new car with a disp of 250, we predict that it will have a mpg of 19.29607.Â
Next, we’ll use the fitted regression model to make prediction intervals around these predicted values:
#create prediction intervals around the predicted values
predict(model, newdata = new_disp, interval = "predict")
# fit lwr upr
#1 23.41759 16.62968 30.20549
#2 21.35683 14.60704 28.10662
#3 19.29607 12.55021 26.04194
The way to interpret these values is as follows:
• The 95% prediction interval of the mpg for a car with a disp of 150 is between 16.62968 and 30.20549.
• The 95% prediction interval of the mpg for a car with a disp of 200 is between 14.60704 and 28.10662.
• The 95% prediction interval of the mpg for a car with a disp of 250 is between 12.55021 and 26.04194.
By default, R uses a 95% prediction interval. However, we can change this to whatever we’d like using the level command. For example, the following code illustrates how to create 99% prediction
#create 99% prediction intervals around the predicted values
predict(model, newdata = new_disp, interval = "predict", level = 0.99)
# fit lwr upr
#1 23.41759 14.27742 32.55775
#2 21.35683 12.26799 30.44567
#3 19.29607 10.21252 28.37963
Note that the 99% prediction intervals are wider than the 95% prediction intervals. This makes sense because the wider the interval, the higher the likelihood that it will contain the predicted
How to Visualize a Prediction Interval in R
The following code illustrates how to create a chart with the following features:
• A scatterplot of the data points for disp and mpg
• A blue line for the fitted regression line
• Gray confidence bands
• Red prediction bands
#define dataset
data #create simple linear regression model
model #use model to create prediction intervals
predictions predict")
#create dataset that contains original data along with prediction intervals
all_data #load ggplot2 library
#create plot
ggplot(all_data, aes(x = disp, y = mpg)) + #define x and y axis variables
geom_point() + #add scatterplot points
stat_smooth(method = lm) + #confidence bands
geom_line(aes(y = lwr), col = "coral2", linetype = "dashed") + #lwr pred interval
geom_line(aes(y = upr), col = "coral2", linetype = "dashed") #upr pred interval
When to Use a Confidence Interval vs. a Prediction Interval
A prediction interval captures the uncertainty around a single value. A confidence interval captures the uncertainty around the mean predicted values. Thus, a prediction interval will always be
wider than a confidence interval for the same value.
You should use a prediction interval when you are interested in specific individual predictions because a confidence interval will produce too narrow of a range of values, resulting in a greater
chance that the interval will not contain the true value.
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C++: Euclidean distance variations
So how about this new std::hypot?
float dx = x1 - x2;
float dy = y1 - y2;
return std::hypot(dx, dy);
Time: 12139783 ticks
How about the old one std::sqrt?
float dx = x1 - x2;
float dy = y1 - y2;
return std::sqrt(dx*dx + dy*dy);
Time: 7026125 ticks (1.7 times faster)
Oh, the std::hypot performs an overflow check, so if you don't care about that just use std::sqrt
And of course the very old
float dx = x1 - x2;
float dy = y1 - y2;
return dx*dx + dy*dy;
Time: 5667812 ticks (2.1 and 1.2 times faster)
Wait, can I just do
return (x1 - x2)*(x1 - x2) + (y1 - y2)*(y1 - y2);
and let the optimizer fix it. Lets check
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How do you solve x+3y=y+110 for y? | HIX Tutor
How do you solve #x+3y=y+110# for y?
Answer 1
Solve #x+3y=y+110# for #y#.
Subract #x# from both sides of the equation.
Subtract #y# from both sides.
Simplify #3y-y# to #2y#.
Divide both sides by #2#.
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Answer 2
To solve ( x + 3y = y + 110 ) for ( y ), you can start by isolating the variable ( y ) on one side of the equation. Subtract ( y ) from both sides of the equation:
( x + 3y - y = 110 )
This simplifies to:
( x + 2y = 110 )
Then, subtract ( x ) from both sides of the equation:
( x - x + 2y = 110 - x )
This simplifies to:
( 2y = 110 - x )
Finally, divide both sides of the equation by 2 to solve for ( y ):
( \frac{2y}{2} = \frac{110 - x}{2} )
This gives:
( y = \frac{110 - x}{2} )
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Answer from HIX Tutor
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
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Let the line L be the projection of the line (x-1)/(2)=(y-3)/(1)=(z-4)
Let the line L be the projection of the line x−12=y−31=z−42 In the plane x - 2y-z = 3. If d is the distance of the point (0, 0, 6) from L, then d2 is equal to_____________
Updated on:21/07/2023
Knowledge Check
• The equation of the projection line of the line x+12=y+1−1=z+34 on the plane x+2y+z=6 is
• The projection of the line x+1−1=y2=z−13 on the plane x−2y+z=6 is the line of intersection of this plane with the plane a. 2x+y+2=0 b. 3x+y−z=2 c. 2x−3y+8z=3 d. none of these
• The point of intersection of the line x−13=y+24=z−3−2 and the plane 2x−y+3z−1=0, is | {"url":"https://www.doubtnut.com/qna/649503913","timestamp":"2024-11-06T23:27:52Z","content_type":"text/html","content_length":"407990","record_id":"<urn:uuid:3a745696-2bb2-4476-b336-a782741f15c5>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.54/warc/CC-MAIN-20241106230027-20241107020027-00471.warc.gz"} |
Examples of online calculations with the calculator - Solumaths
• Arithmetic operations The online scientific calculator can use usual arithmetic operators to perform arithmetic operations as an online calculator. The notation used is as follows:
□ + For addition
□ - For subtraction
□ * For multiplication
□ / For division
□ ^ To raise to a power
A sample calculation that can add the numbers 34 and 32. See example of arithmetic calculation: 34+32
• Operations on fractions
The scientific calculator online offers the ability to simplify a fraction , to reduce a fraction and performing all arithmetic operations on fractions using the usual operators.
The following series of examples shows how to calculate fractions.
In addition to the result, the calculator retourne également returns the different stages of the fraction calculation.
• Euclidean division
The web scientific calculator allows to do an Euclidean division, the Euclidean division can be done with integers or polynomials. The following examples show how to do a Euclidean division using
the online calculator.
See example of euclidean division between two integers: `19:4`
See example of the euclidean division between two polynomials : `x^2-4` et `x+2`
• Factoring an expression
The internet calculator allows to factor an expression specifying the calculation steps. The following examples show how to factor an expression using the online calculator.
Show example factorization of an expression : `x^2-9`
Show example factorization of an expression : `x^2-4`
• Expand an expression
The calculator scientific allows to develop an expression.
The following example shows how to expand an expression using the online calculator.
Show example of expansion of an expression : `(1+x)^2`
• Solving a linear equation
The calculator allows solving a linear equation quickly and easily, the results are given in exact form and details of the calculations are displayed. Simply enter the equation and validate. The
following example shows how to solve a linear equation using the online equality calculator .
See example of solving an equation : `x/3+3=x`
• Solving a quadratic equation
The online calculator allows solving a quadratic equation very quickly and very simply, the results are given in exact form and the steps of the calculations are displayed.
how to solve a quadratic equation using the online equation calculator .
Show example of solving an equation : `x^2+x=1`
• Solving absolute value equation
Online calculator allows solving absolute value equation quickly and easily, the results are given in exact form and the steps of the calculations are displayed.
The following example shows how to solve an absolute value equation using the online equation calculator .
Show example of solving an absolute value equation : `|x-2|=1`
• Solving exponential equation
Online calculator allows solving exponential equation quickly and easily, the results are given in exact form and the steps of the calculations are displayed.
The following example shows how to solve an exponential equation using the online equations calculator .
Show example of solving an exponential equation : `exp(x-2)=1`
• Solving logarithmic equation
The online equation calculator allows to solving logarithmic equation quickly and easily, the results are given in exact form and the steps of the calculations are displayed.
The following example shows how to solve an logarithmic equation using the online equation calculator.
See example of solving an equation : `ln(x-2)=0`
• Solving an inequation of the first degree
The online calculator allows linear inequality resolution very quickly and very simply, the exact results are given, with steps of calculation.
The following example shows how to solve an inequality with the inequalities calculator .
See example of solving an inequality : x+3>2
• Solving a quadratic inequality
The calculator allows solving a quadratic inequality very simply and very quickly, the exact results are given, with steps of calculation.
The following example shows how to solve a quadratic inequality using the online calculator.
Show example of solving an inequality : `x^2-3>2`
• Computation of the discriminant of a quadratic polynomial
The web calculator allows to calculate the discriminant of a quadratic trinomial very quickly and very simply, exact results are given with the different calculation stages.
The following example shows how to calculate discriminant using the internet calculator.
Show example of discriminant calculation : `x^2-3x+2`
• Solving a system of equations
The online solver allows solving a system of equations , the solutions are given in exact form.
The following example shows how to solve a system of equations using the online system solver.
See example of solving a system of equations : solve_system([x+y=18;3*y+2*x=46];[x;y])
• Find a linear equation from two points
The online calculator allows you to find a linear equation from two points .
The following example shows how to calculate a linear equation from two points using online calculator equation straight line . See example of calculating a linear equation which passes through
two points.
• Symbolic calculation: reduce expression
The computer algebra software (CAS) that offers this site reduces expression that is to simplify an algebraic expression. The following example shows how to simplify an expression using the
symplify calculator.
See example of simplification of an expression : b+2a+3a+4b
• Calculating a square root
The square root calculator online can be used on numeric expressions but also on symbolic expressions, it provides results in exact form.
The following example shows how to calculate the square root of a number using the square root calculator online. See example square root calculation : `sqrt(12)`
• Calculating a derivative
The online derivative calculator allows for all standard functions to calculate a derivative.
The following example shows how to calculate the derivative of a function using the online derivative calculator. Show sample calculation of derivative : differentiate(4x+4)
• Calculate a limit
The online limit calculator allows for some functions to La calculatrice de limite en ligne permet pour certaines fonctions usuelles de faire le calculate limit explaining the calculations
which make it possible to obtain the result.
The following example shows how to calculate the limit of a function thanks to limit calculator .
Show sample calculation of limit : limit(`sin(x)/x`)
• Calculation of the terms of a numerical sequence
The following example shows how to calculate the terms of a numeric sequence when it is explicitly defined . See example of calculation of the terms of the sequence (`n^2`) whose indices are
between 1 and 4.
The following example shows how to calculate the terms of a numeric sequence when it is defined by recurrence . See example of calculation of the terms of the sequence defined by recurrence `u_
(n+1)=u_n+3, u_0=2` whose indices are between 1 and 4.
• Find the equation of the tangent at a given abscissa point
The tangent line calculator allows to find an equation of the tangent at a given point of abscissa .
The following example shows how to calculate the tangent at one point using the online tangent line calculator . See calculation example of the tangent to the graph of the function `f: x-> x^2+3`
at the point of abscissa 1
• Calculating a cube root
The online cube root calculator can be used on numeric expressions, it provides exact results.
The following example shows how to calculate the cube root of a number with the cube root calculator online. See sample cube root calculation : cube_root(64)
• Calculation with vectors
The internet calculator is used with vectors , it allows to do many operations that involve vector calculation. Calculations are done in exact form and can involve literal expressions,
calculation steps are specified.
The following example shows how to calculate the norm of a vector using the online vector norm calculator . See example of calculating the norm of a coordinate vector `[1;2]`
The following example shows how to calculate the norm of a vector with coordinates that contain letters. See example of calculating the norm of a coordinate vector `[(a),(2*a),(a/2)]`
The following example shows how to compute the coordinates of a vector from the coordinates of two points. See example of calculation of the coordinates of a vector from points of coordinates [1;
2] and [5;6]
The following example shows how to compute the cross product of two vectors from the coordinates. See example of calculation of the cross product of the vectors [1;1;1] and [5;5;6]
The following example shows How to compute the dot product of two vectors from the coordinates. See example of calculation of the dot product of the vectors [1;1;1] et [5;5;6].
The following example shows how to compute the scalar triple product of three vectors from the coordinates. See example of calculation of the scalar triple product of the vectors `[(1),(1),(1)]`;
• Calculation of an antiderivative
The computer algebgra software allows for many functions to get the calculation of an antiderivative .
The following example shows how to calculate an antiderivative function with the antiderivative calculator online. Show sample calculation of antiderivative: antiderivative(4x+4) .
• Operations on complex numbers
The computer algebra software (CAS) allows for all arithmetic operations on complex numbers using the usual operators to do calculation. The symbol used is i.
1. The following example shows how to calculate a complex expression using the online calculator. See example of a complex number calculation : `5+i+3*i^2`
2. The following example shows how to simplify a complex number using the online calculator. See example of simplification of a complex number : `1/(1+i)`
• Solve a differential equation of the first order
The online calculator is able to solve a differential equation of the first order.
The following example shows how to solve a differential equation using the online calculator. See example of solving a differential equation : `solve(y'+y=0;x)`
• Statistical calculations
The online calculator allows for many statistical operations as average , variance or standard deviation calculations. The exact results are given with the calculation steps. The calculator can
manipulate numbers but also letters.
The following example shows how to calculate the average of a series using the average calculator online. See example of average calculation `([[12;14;15;4];[3;5;3;2]])`
The following example shows how to calculate the variance of a series See example of variance calculation `([12;14;15;4])`
The following example shows how to calculate the standard deviation of a series. See example of standard deviation calculation `([12;14;15;4])`
• Matrices calculation
The online matrix calculator is used with matrices , it allows for many operations involving the matrix calculation. Calculations are done in exact form and may involve literal expressions. The
calculation steps are specified.
□ Calculate the determinant of a matrix or vectors.
The calculator allows to Calculate the determinant of matrix .
The following example shows how to calculate the determinant of a matrix using the online calculator. See example calculation of the determinant of the matrix `((3,1,0),(3,2,1),(4,1,7))`.
□ Calculate the inverse of a square matrix.
The web calculator online allows to calculate the inverse of a square matrix .
The following example shows how to calculate the inverse of a square matrix using the web calculator. See example of calculating the inverse of the matrix `((3,1,0),(3,2,1),(4,1,7))`.
□ Calculate the sum of matrices.
The matrix calculator online allows to calculate the sum of two matrices .
The following example shows how to calculate the sum of two matrices using the matrix calculator online. See example of calculation of the sum of the matrices `((3,3,4),(1,2,0),(-5,1,1))+
□ Calculate the difference of matrices.
The web matrix calculator allows to calculate the difference between two matrices .
The following example shows how to calculate the difference between two matrices using the matrix calculator online. See example of calculation of the difference of the matrices `((3,3,4),
□ Calculate the product of two matrices.
The internet matrix calculator allows to calculate the product of two matrices .
The following example shows how to calculate the product of two matrices using the matrix calculator online. See example of calculation of the product of the matrices `((3,3,4),(1,2,0),
□ Calculate the trace of a matrix.
The web algebra calculator allows to calculate the trace of a matrix .
The following example shows how to calculate the trace of a matrix using the calculator online. See example of calculation of the trace of the matrix `((3,3,4),(1,2,0),(-5,1,1))`.
□ Calculate the transpose of a matrix.
The algebra calculator allows to calculate the transpose of a matrix .
The following example shows how to calculate the transpose of a matrix with the web calculator. See example of calculation of the transpose of the matrix `((3,3,4),(1,2,0),(-5,1,1))`.
• Calculate taylor series
Taylor series calculator is able to calculate taylor expansion .
The following example shows how to calculate taylor series with the taylor series calculator online See example of taylor expansion of cos(x).
• Calculate maclaurin series
Maclaurin series calculator is able to calculate maclaurin expansion .
The following example shows how to calculate maclaurin series with the Maclaurin series calculator online See example of Maclaurin expansion of sin(x).
• Transform a decimal number into a percentage
The percentage calculator is able to convert a decimal number to a percentage .
The following example shows how to transform a decimal number into a percentage with the calculator. See transformation example of 0.25 into a percentage.
• Transform a fraction into a percentage
The percentage calculator is able to convert a fraction to a percentage .
The following example shows how to transform a fraction into a percentage with the online calculator. See transformation example of `1/2` into a percentage.
• Calculate a perimeter
The perimeter calculator is able to calculate usual perimeters .
The following example shows How to calculate the perimeter of a rectangle with the online perimeter calculator. See example of rectangle perimeter calculation .
The following example shows How to calculate the perimeter of a circle or How to calculate the perimeter of a square with the perimeter calculator. See example of circle perimeter calculation and
square perimeter calculation
• Calculate the monthly payment of a loan.
Whether it is home loan or consumer loan, the following example shows how to calculate the monthly payment of a loan knowing its rate and its duration. See example of calculation of a loan of
$100000 over 20 years at the rate of 2%.
• Calculate the insurance of a credit.
The following example shows how to calculate the monthly amount of credit insurance knowing its rate and duration. See example calculation of the insurance at the rate of 0.3% on a credit of $
100000 over 20 years. | {"url":"https://www.solumaths.com/en/calculator/calculator-examples","timestamp":"2024-11-02T09:28:09Z","content_type":"text/html","content_length":"47340","record_id":"<urn:uuid:7b00fc3b-7500-4011-aee6-07ec94c1fc4a>","cc-path":"CC-MAIN-2024-46/segments/1730477027709.8/warc/CC-MAIN-20241102071948-20241102101948-00568.warc.gz"} |
CSET and CBEST
Testing and Credential Information
In preparation for applying to credential programs, students will need to demonstrate:
1. Knowledge of the subject you want to teach
2. Knowledge of basic educational skills.
To demonstrate your "Subject Matter Knowledge" students can either successfully pass the CSET (California Subject Exam for Teachers) or successfully complete a Subject Matter Waiver Program. For more
information about the UCLA Math Department's Subject Matter Waiver Program, please look under "Teach Math > Math Department Offerings" in the links at the top of the page. UCLA science departments
do not have waiver programs, and therefore science credential students must pass the CSET.
To demonstrate "Knowledge of Basic Education Skills", students must either 1) successfully pass the CBEST exam or 2) meet other minimum scores on SAT, ACT or select AP tests. Please see the
California Commission on Teaching Credentialing for the current scores necessary to satisfy the requirement, or see their CBEST page under "What are the Basic Skills Testing Options for California?".
Reimbursement for CSET and CBEST for UCLA Cal Teach Students
Contingent upon continued funding, the UCLA Cal Teach program will reimburse UCLA Cal Teach students who successfully pass the CSET exams. To be reimbursed, you must have taken the exam while
enrolled as a UCLA student, be a math or science major, and have taken at least one UCLA Cal Teach course or internship (i.e. Science Education 1SL, 10SL, 100SL, Math 71SL, 72SL, 103, 105, or summer
internship). We reimburse for CSET Single Subject exams in math or science. We will potentially reimburse for CSET Multiple Subjects under limited circumstances for math or science majors. Please
contact the Academic Coordinator at cateach@chem.ucla.edu for more information. To be reimbursed, you will need to email us a copy of your test result showing that you passed, and a copy of the
receipt showing that you paid the associated fees. Please note that we cannot reimburse for late fees as per University policy.
We will also reimburse for CBEST exams, according to the same criteria above. However, please note that taking the CBEST may not be necessary if you meet the minimum scores on the SAT, ACT, or select
AP tests. Please see the California Commission on Teaching Credentialing for the current scores necessary to satisfy the requirement, or their CBEST page under "What are the Basic Skills Testing
Options for California?".
The California Basic Educational Skills Test (CBEST)
Note that taking the CBEST may not be necessary if you meet the minimum scores on the SAT, ACT, or select AP tests. Please see the California Commission on Teaching Credentialing document, or their
CBEST page for the current scores necessary to satisfy the requirement.
The CBEST consists of three sections: Reading, Mathematics, and Writing, and includes 50 multiple choice questions in the Reading Section, 50 multiple choice questions in the Mathematics sections,
and 2 written essays.
How to Prepare?
A practice test is available on the CBEST website: https://www.ctcexams.nesinc.com/PM_CBEST.asp. Most students find reviewing this material adequate preparation. Numerous additional preparation books
are sold in bookstores and on commercial websites including Amazon.com
The California Subject Examinations for Teachers (CSET)
There are various versions of the CSET. Please check directly with the CSET website and credential programs to confirm the appropriate tests and subtests for your situation.
1) For students wanting to teach Elementary School, the CSET: Multiple Subject is required.
For the above credential, students are required to take the following CSET subtests:
#101 (which covers Reading, Language, Literature, History and Social Science)
#214 (which covers Science and Mathematics)
#103 (which covers Physical Education, Human Development and Visual and Performing Arts)
How to prepare?
UC Irvine has created a great website with self-paced preparation guides. We highly recommend this resource! http://ocw.uci.edu/collections/
2) For students wanting to teach Science, two options exist.
a) Foundation-Level General Science - A Single Subject Teaching Credential in Foundation-Level General Science authorizes teaching only in general, introductory, and integrated science (integrated
science through Grade 8 only).
For the above credential, students are required to take the following CSET subtest:
Science Subtest I: General Science (215)
b) Science with an area of concentration - A credential in this area authorizes teaching general and integrated science AND the area of concentration in high school. (Most students choose this option
as it seems to have the best appeal to the majority of schools during the hiring process)
For the above credential, students are required to take the following CSET subtests:
Science Subtest I: General Science (215)
And ONE of the following subtests:
Science Subtest II: Life Sciences (217)
Science Subtest II: Chemistry (218)
Science Subtest II: Earth and Space Sciences (219)
Science Subtest II: Physics (220)
How to prepare?
Check the CTC site for sample questions and test structure and content: https://www.ctcexams.nesinc.com/PM_CSET.asp?t=217
UC Irvine has created a website with self-paced preparation guides. http://ocw.uci.edu/collections/california_subject_examination_for_teachers__preparation_resources.html
Cal State Fresno offers free workshops on CSET preparation. http://fresnostate.edu/kremen/about/centers-projects/teachmathscience/in...
Free study guides are available to be checked out from our office at 1039 Young Hall by currently enrolled UCLA science or math students who have taken at least one Cal teach course or internship
(i.e. Science Education 1SL, 10SL, 100SL, Math 71SL, 72SL, 103, 105, or summer internship).
3) For students wanting to teach Math, various options exist
a) Foundational-Level Mathematics - A Single Subject Teaching credential in Foundational-Level Mathematics authorizes teaching only in limited mathematical content areas: general mathematics,
algebra, geometry, probability and statistics, and consumer mathematics.
For the above credential, students are required to take the following CSET subtests:
Foundational-Level Mathematics Subtest I (211)
Foundational-Level Mathematics Subtest II (212)
b) Mathematics - A Single Subject Teaching Credential in Mathematics authorizes teaching all mathematics coursework.
For the above credential, students are required to take the following CSET subtests:
Mathematics Subtest I (211)
Mathematics Subtest II (212)
Mathematics Subtest III (213)
How to prepare?
Check the CTC site for sample questions and test structure and content: https://www.ctcexams.nesinc.com/PM_CSET.asp?t=211
UC Irvine has created a website with self-paced preparation guides. http://ocw.uci.edu/collections/california_subject_examination_for_teachers__preparation_resources.html
Cal State Fresno offers free workshops on CSET preparation. http://fresnostate.edu/kremen/about/centers-projects/teachmathscience/in...
Free study guides are available to be checked out from our office at 1039 Young Hall by currently enrolled UCLA science or math students who have taken at least one Cal teach course or internship
(i.e. Science Education 1SL, 10SL, 100SL, Math 71SL, 72SL, 103, 105, or summer internship).
Additional Teaching related websites include:
Credentialing Information
California Commission on Teacher Credentialing
Teacher Resources
California Department of Education
Content Standards
Curriculum Frameworks and Instructional Materials | {"url":"http://cateach.ucla.edu/?q=content/cset-and-cbest","timestamp":"2024-11-12T00:21:12Z","content_type":"text/html","content_length":"35823","record_id":"<urn:uuid:e7005f53-3a76-41a8-9534-89386f8166bd>","cc-path":"CC-MAIN-2024-46/segments/1730477028240.82/warc/CC-MAIN-20241111222353-20241112012353-00308.warc.gz"} |
THE EQUATION OF ANGLE BISECTOR - edsmathscholar.com
In this article, we’ll learn how to determine the equation of the bisector of the angle between two lines whose equations are given. For example, suppose that the equations of lines g and h are
Figure 1
In the figure, the green and black lines represent the lines g and h, respectively.
An angle bisector is the line or line segment that divides the angle into two equal parts. In Figure 1, the red line is an angle bisector. It divides the angle formed by the green and black lines.
What is the equation of the angle bisector? This article addresses the question.
Let’s first derive the formula …
1. line g with equation A[1]x + B[1]y + C[1] = 0,
2. line h with equation A[2]x + B[2]y + C[2] = 0,
3. line k, which is a bisector of the angle between g and h
To be sought: the equation of k
Let P[0](x[0],y[0]) be any point that lies on k. Let A be on line g such that [0] to line g and [0] to line h. (See Figure 2.)
Figure 2
In Figure 2, O is the point of intersection of the lines g and h. Note that [0] is of an equal length as [0] and ∠AOP[0] = ∠BOP[0]. As a consequence, ΔOAP[0] [0]. Therefore,
Since P[0](x[0],y[0]) is any point in k, the last equation above holds for all points on the line k. As a result, the equation of k can be expressed as:
Note that from (*) it seems that there are actually two possible bisectors.
For every pair of intersected lines there are always two angle bisectors.
The other angle bisector is represented by the yellow line in Figure 3.
Figure 3
It was said earlier that there were two angle bisectors for each pair of intersected lines. Furthermore, it can be proved that the angle bisectors are perpendicular to each other.
Now we are ready to solve the problem at the beginning of this article.
1. line g with equation
2. line h with equation
3. line k, which is a bisector of the angle between g and h
To be sought: the equation of k
Note that the lines g and h can be expressed in the form of Ax + By + C = 0 as follows.
g ≡ 7x – 2y = 0
h ≡ x – 2y = 0
Substituting A[1] = 7, B[1] = -2, C[1] = 0, A[2] = 1, B[2] = -2, and C[2] = 0 into (*), we get:
There are two angle bisectors. Their equations are determined as follows.
First angle bisector: (See Figure 4)
Figure 4
Second angle bisector: (See Figure 5)
Figure 5 | {"url":"http://edsmathscholar.com/the-equation-of-angle-bisector/","timestamp":"2024-11-11T04:10:20Z","content_type":"text/html","content_length":"67798","record_id":"<urn:uuid:b4c398d2-0984-448b-992e-c0d8b7fbcda1>","cc-path":"CC-MAIN-2024-46/segments/1730477028216.19/warc/CC-MAIN-20241111024756-20241111054756-00142.warc.gz"} |
A Fun Little Number - Chic Over 50
Posted in Fashion
A Fun Little Number
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because regular levels aren’t established.
254. Testostertone levels in adult men decline at an average ate of 1 to 2 percent each year.
255. Testosterone replacement therapy for hypogonadal men hhas been found to
improvve libido, mood, sexual function, bone density, muscle bulk, and muscle strength, reports the study.
256. Testosterone iis used mainly to treat symptoms off sexual dysfunction in women and men and hot flashes in women.
257. Prepubertal hypogonadism is usually defined by infantile genitalia and deficiency of
virilization, while the development of hypogonadism after puberty often results in comjplaints such as diminished libido, erectile dysfunction, infertility, gynecomastia, reduced masculinization,
changes in body composition, reductions in body and facial
hair, and osteoporosis.
258. Vigen R, ‘Donnell CI, Baron AE, et al. Organization of testosterone treatment
with mortality, myocardial infarction, and stroke in men with
low testosterone levels.
259. Other disagreeable side effects may include the development of acne,
enlargement of the clitoris and disposition changes, including an increase in feelings of hostility and aggressiveness.
260. Testosterone iss responsible for normal growth and development of male
sex organs and maintenance of secdondary sex characteristics and is the primary androgenic
261. Hormone Replacement Florida Therapy is a treatment in which hormones are given to prevent or reat health conditions common in menopausal women, like osteoporosis.
262. Understanding where your testosterone shots are coming from, and many men detail crucial and
soo vital yet, so readily avoided the standards by
which they’re created.
263. Now answer ongoing questions about its safety and
effectgiveness and more research is needed to reexamine present theories about thee function of
tetosterone inn women, Wierman said.
264. The Xu meta-analysis called for 27 published, randomized, placebo-controlled trials symbolizing 2,994 largely middle aged and
elderly male participants (1,773 trated with testosterone and 1,261 treated with
placebo) who reported 180 cardiovascular-related adverse events.9 This study found that tesyosterone therapy was correlated with an increased risk of adverse cardiovascular events
(Odds Ratio OR=1.5, 95% CI: 1.1-2.1); yet, mthodological issues limit conclusions.
265. Few data show that the incidence of cardiovascular dieease increases.
266. Chiefly, it’s because testosterone replacement therapy is, in addition, rlated to sleep difficulties, lipid abnormalities
and some other diseases.
267. Few data demonstrate that testosterone replacement increases the incidence of cardiovascular disease.
268. Thhis is distinctly different from using the combination oof HGH and testosterone as aan antiageing treatment.
269. Girls start to experience menopause after a certain age and there are
symptomss for example nioght sweats, as well as decrease in sexual desire and hot flashes.
270. Additionally, there are sime testosterone treatment cardiovascular hazards These side effects may be an indication that testosterone therapy is not for you.
271. Extended periods of deficiency of functionality and creation, due tto unnaturally maintaining and regulating our testosterone levels, will eeventually cause atrophy oof those glands and
drawn-out misuse of these organs can cause irreversible side effects and permanent damage and unwanted states.
272. Women can take testosterone through a spot, as a creme oor in the type of pelllet implants,
which have the greatest consistency off delivery.
273. This promotes thhe protein synthesis hoped for and anticipated by this life trahsforming
therapy and program, all while regulating to keep the
platitude, quality oof life deteriorating side impacts far away and out of sight.
274. Guys getting testosterone replacement treatment are normally
quite satisfiied with the results they experience
in terms of lower body fat, increased youthfulness, better muscle mases and naturally, sexual drive that is outstanding!
275. Girls have a 50 per cent highsr chance than men oof receiving the erroneous initial analysis following a heart attack,
according to a brand new study byy the University oof Leeds.
276. Some earlier studies had suggested that testosterone treatment could
get meen at higher risk for cardiovascular problems for example heart attack and stroke.
277. Testosterone levels iin adult men drop at an average rate
oof 1 to 2 percent annually.
278. Irrespective of the route of administration, studiss have shown progress
inn libido and sexual function in hypogonadal men.
279. But using these techniques could keep you away from the pharmacy counter to restrain your testosterone level.
280. Although the FDA approved testosterone therapy
for the treatment of disorders involving the testes, pituitary and hypothalamus, it
hasn’t been approved for treating age-related decrease in testosterone levels.
281. If you still desire to father children or aare not done
having added offspring, testosterone therapy should
n’t be taken by you.
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283. A loww fat diet is most likely going to be full of a barbarous enemy
annd sugar tto yiur testosterone levels.
284. The indications for thee use of testosterone in psycological and cognitive deterioration are still
not clear; yet, studies of healthy older men with testosterone insufficiency have yielded fascinating results.
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286. Other symptoms of testosterone deficiency include muscle weakness and vaginal dryness.
287. Testosterone is used for women with Turner’s syndrome, premature ovarian failure, HIV infection, or long-term corticosteroid
288. Testosterone gel can cause breast tenderness and enlargement
inn both women and men.
289. Testosterone therapy is widely used to help address the effects that
low testosterone can have on mood, muscle mass and strength, bone density, metabolic function and cognition.
290. Fundamentally, what testosterone repolacement therapy does is tto set
back your testosterone level to normal.
291. In one study published in the journal PLoS One, for instance,a heightened risk of heart attack was found in men younger than 65 with a history of heart
disease, and in older guys if they didn’thave a history of the ailment.
292. Since the heart health of the menn was carefully tracked, the research is
anticipated to shed more light on the security of
testosterone therapy.
293. Earlier this year, the U.S. Food and Drug Administration required producers of all authorized testosterone products to add info on the labels
to clarify the approveed uses of the medications and include advice about possible increasedd risks of heart attacks annd strokes in patients taking testosterone.
294. Also, because ther procedures in the body cease to work as a result of
you manioulating your testosterone levels through testosterone shots,
the treatment gains begin too decline, and all the feel
great” scenarios you were experiencing come to a dead stop.
295. A blow is struck by thiss finding to the multibillion dollar business that has splrung up recently around testosterone.
296. There are several kinds of over the counter testosterone supplements
available in nutritional supply stores.
297. Girls may develop symptoms of testosterone deficiency att any
age, but this illness is most common in postmenopausal women, occurring att the time when the production of other hormones begins to fall.
298. Testosterone levels can fall nawturally as mmen age, andd sometimes these amounts can become lower than the standard range seen in young,
heathy guys.
299. Thus, due to these testosterone sidde effects, one should avoid taking testosterone
supplements or medications, particularly when the man is enduring benign prostatic hypertrophy (BPH),
bleeding disorders, high cholesterol, any type of cancer, liver
or kidney disease, heart disease, etc.
300. Other symptoms of testosterone deficiency incluide muscle weaknesss and
vaginal dryness.
301. When you have experienced symptoms of low T, it is advisable to take a blood test to ascertain if your testosterone evels are low.
302. In the second study, researchers at Aurora Health Care, a biig
community-based health care system in Wisconsin, examined
demographic and health data from 7,245 men witth low
testosterone levels from 2011-2014.
303. We’re telling you which you can lose weight without pharmaceuticals for testosterone treatment.
304. This makes sense, knowing that sympltoms and
states of low Testosterone are universal and change both genders.
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306. All of the guys in the new study generally hadd higher ratees of medical coinditions — including coronary artery disease, diabetes and previous heart attacks — than men in the general citizenry.
307. Contact your doctor immediately if you experience a sudden increase in weight
or other serious side effects while using testosterone.
308. Yet men suitably diagnosed with testosterone deficiency should contemplate
treatment after ample dialogue abbout the risks as
well as advantages individual to their particular health status.
309. The Xu meta-analysis demanded 27 released, randomized, placebo-controlled trials signifying 2,994mainly middle aged
and elderly male participamts (1,773 treated with testosterone
and 1,261 treated with placebo) who reported 180 cardiovascular-related adverse events.9 This study found that testosterone treatment was associated
with an increased risk of aadverse cardiovascular events (Odds Ratio OR=1.5, 95% CI:
1.1-2.1); however, methodological ilemmas limit conclusions.
310. Before getting started on any supplement regime, it is necessary to speak to your physician to ensure
that testosterone upplements are suitable for you.
311. Testosterone is the major androgenic hormone made by the testes in resplnse tto
luteinizing hormones from the pituitary gland.
312. Men and women in America have used testosterone treatment since thhe late 1930s,
in many cases with just uncommon undesirable effects
– for mmore than 40 years.
313. The included studies symbolized 3,236 guys (1,895 men treaated with testosterone, 1,341 men treated with placebo) who reported 51 major adverse cardiovascular events, defined as
cardiovasscular death, nonfatal myocardial infarction oor stroke,
and serious acute coronary syndromes or heart failure.10 This study
didn’t find a statistically significant increased risk of these cardiovascular evens connected with testosterone therapy.
314. Those who have normal testosterone levfel must not administer
the treatment for the sheer fun of iit or for motives other than for health.
315. When youur brain assesses and scans your body
in its attept to regulate your hormonal secretion as needed
through the day and it finds that testosterone levels are fine annd elevated
resulting from a powerful testosterone treatment, its own natural
productioon ceases in fabrication.
316. When those levels dwindle down tto 0.00 and below,
you can rewst assured you won’t be feeling the exceptional benefits and energy optimizing symptoms expected from being on a testosterone shots program.
317. From six months to three years after analysis, 7.1 percent of the men on hormone therapy had new cases oof depression, compared with 5.2 percent of the others in the
318. Some of the men I Have seen that have been on android steroids do
look a bit like thee incredible hulk.
319. This is definitely different from using testosterone
as an antiageing treatment andd the combination of
320. Hello, I agree with you, unless thiis hornone is needed by you because you’re lacking, then no manner should take it, as I
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322. There might be many reasons why your selcted testosterone augmentation regimen may not be supplying you the results thatt yyou
anticipated and were hopefu for, if you didn’t obtain your Testosterone treatment through
323. Testosterone therapy suppresses regular testicular function, and it
is therefore essential to comprehend shrinkage oof the testicles
will probably occur with lng term use aas well as cause infertility
for a guy of any age Another common consequence of testosterone therapy comprises changes to
reed blood cell , and any guy experiencing testosterone therapy should be monitoring consistently by a medical provider to
assess treatment response and manage effects of therapy.
324. Also, there are prescription-based testosterone reatments that produce better results.
325. Testosterone levels can decrease naturally as men age, and sometimes these levels can become lower than the ordinary range
seen in young, healhy men.
326. The quantity of testosterone depends upon the person?s testosterone levels annd healtth conditions in blood.
327. Side effects in women comprise acne, hepatotoxicity,
and virilization and generally only occur when testosterone iis used in supraphysiologic doses.
328. Finkle WD, Greenland S, Ridgeway GK, et al. Increased threat of non-fatal
myocardial infarction following testosterone therapy prescription in men.
329. Yoour Post Cycle Cleanse flush out any remaining estrogen wiyhin your body,
enabling you to reap full benefits of youhr Testosterone injections therapy, bring them back
in line and will reset those amounts.
330. After getting info from the electronic record systems
oof 15 hospitals and 150 practices, the researchers looked at the combined cardiovascular event rate
of heart attack, steoke and death in men with low testosterone who received testosterone therapy and in those who
331. They were about 76 years old on average, about two
years older than the typical age of the guys whoo received different treatments.
332. Aging guys may also experience symptoms and signs like decreases in energy level aand difficulties with sexial function, but it’s not certain whether thesee aare brought on by the lowered
testosterone levels or due to normal aging.
333. The standard ranges foor blood tesstosterone are: Men 300-1,200 ng/dl,
Female 30-95 ng/dl.
334. These days, testosterone iis given through injections or skin patches that absorption takes
place transdermally.
335. Testosterone treatment haas Been widely advertised as a way too help low libido improvrs and recover diminished energy,
and use of the nutritional supplements is on the increase.
336. Strange testosterone levels can increase symptoms of enlarged
prostate (benign prostatic hyperplasia, or BPH).
337. Try using a quality zknc nutritional supplement iif you suspect or know that your testosterone level is low.
338. Hey! I could have sworn I’ve been to this blog before but after reading through
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339. A Healthy Life Style, along with WALKING every day, when possible, or some form
of Diet and Exercise goes along way to keep us
from aging too Quickly.
340. Testosterone therapy must always be discussed in context of healthy living
and a multitude oof other contributions
that also interface with overall wellness, sexual
function, prostate and cardiovascular disease, glycemic control, and
bpne health, all which give to a man’s energetic quality oof life.
341. Hormone Replacement Florida Therapy is a treatment in which
hormones are given to prevent or treat health conditions
common in menopausal women, including osteoporosis.
342. More research in the area off chronic illness was finished in men than in women.
343. There are also some testosterone therapy cardiovascular hazards
These side effectss maay be a sign that testosterone treaatment is not for you.
344. The chance of increased risk of these ailments with testosterone supplementation is of gredat anxiety,
because treatments for bot conditions contain androgen suppression.
345. Soome of thhe guys I’ve sen that have been on androoid steroids ddo seem a littpe
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346. Some of the men I’ve seen that have been on android steroids do appezr a little like the incredible hulk.
347. The U.S. National Institute on Aging is also anticipated
to release the results of research on the sacety of testosterone.
348. Lengthy perriods of deficiency of generation and functionality, due too unnaturally preserving and regulating yopur testosterone levels, will eventually cause atrophy of
those glands and lengthy abuse oof these organs can cause permanent damage and irreversible side effectts and states that are unwanted.
349. Anti-aging hormones haven’t been around for longitudinal studies
to habe been performed regarding their effects.
350. They dissolve slowly overr three to four months, releasing
small amounts of testosterone into the blood stream, but speeeding
upp whenn needed by the body -during strenuous actions, for
example – and slowing down during uiet times, a characteristic no other kind of hormone
therapy can offer.
351. Doctors and atients should be vigilant of the aggressive advertising used by makers that were testosterone,
Cappola said.
352. A limited variety of studies 33, 41 have demonstrated that emotional symptoms and recollection are enhanced with the inclusion of testosterone to estrogen.
353. Chiefly, it’s because testosterone replacement therapy iis also related to sleeping problems,
lipid abnormalities and several other ailments.
354. Thee inluded studies represented 3,236 men (1,895 guyys treated
with testosterone, 1,341 men treated with placebo) who reported
51 major adverse cardiovascular events, defined as cardiovascular death, nonfatal myocardial infarction oor stroke, and serious acute coronary syndromes or heart
failure.10 This study didn’t findd a statistically significant increased risk
of these cardiovascular events associated with testosterone
355. The possibility of increased risk of these ailments with testosterone supplementation is
of great concern because treatments for both illnesses contain androgen suppression.
356. Clinical staths of the patient is the best means to follow the effectiveness of testosterone therapy
because regular levels are not well established.
357. This is normally because it didn’t consaist of
the vital supplementations demanded to ensure the benefits of testosterone therapy are given the opportunity to arise and, more too the point, to keep health, unwanted -hindering side effects at
1. The results demonstratfed that of the 12 Sexuality measurements in thhe survey,
10 were significantly enhanced ffor guys in the testosterone group.
358. Baillargeon J, Urban RJ, Kuo YF et al. Danger of myocardial infarction in elderly
men receiving testosterone treatment.
359. While it’s understood that low amounts of testosterone pos an increased cardiovascular risk, the risks
versus gains of supplementation never have been certainly identified.
360. Although long term outcome data aren’t available, prescriptions
for testosterone are getting to be more common.
361. Although, numerous benefits are spelt by testosterone replacement therapy,
it can still be dangerous if not properly executed and used.
362. The results revealed that of the 12 Sexuality measurements in the survey, 10 were significantly improved for guys in the testosterone group.
363. Leengthy use of manufactured testosterone can cause shrinking of testicles,
gynecomastia (breast growth in men), reduced or increased sex drive, reduced sperm production, clitoral enlargement, male pattern baldness, and water retention.
364. Most girls can expect to spend one third of their lives in the postmenopausal stage.
365. An allergic reaction to this drug may cause a sudden increase in weight due to swelling,
although weight gain isn’t a common side effect of testosterone supplements.
366. Before initiating testosterone replacement therapy, ensure that
the analysis of hypogonadism has been verified with lab testing.
367. The recent contradictory findings on testosterone
therapy prompted Patel annd his teeam to run a big sysdtematic literature search for
studies assessing the relationship between testosterone replacement therapy
and cardiovascular events among men.
368. Incease muscle mass and help patients fesel better, have
more enerrgy and testosterone replacement therapy is
wideky used in elderly men to normalize the hormone level.
369. Good health is promoted by high amounts of testosterone in men aand lower the
risk of heart attack and hiogh blood pressure.
370. You are no longer getting optimum outcomes from disciplined work outs, and
371. This describes obvious aging, somebody’s sudden weight gain aand
loss of energy.
372. There miight be many reasons why your chosen testosterone improvement regimen may
not be supplying you the results that yoou anticipated and were
hopeful for if you didn’t obtain your Testosterone treatment through AAI.
373. Therefore, due to these testosterone side effects, onne shouuld avoid
taking testosterone supplements or medicines, particularly when the person is enduriing
benign prostatic hypertrophy (BPH), bleeding disorders,
high cholesterol, any kind of cancer, liver or
kidney disorder, heart disease, etc.
374. Some men really have low T, bbut they would
not have any symptoms of the ailment.
375. We learn a lot in thhe news about weight lifters and athletics
ussing anabolic steroids to increase muscle mass, but this is,
in addition, distinct from the HGH and testostefone combination discussed
in this article.
376. Not only were these tests pricey but at times, they were also not reliable because testosterone amount signaled in thee
blood is challenging to interpret.
377. Those numbners will be reset by your Post Cycle Cleanse, bring them back in line and
flush out any residual estrogen in yourr body, allowing
you to, once again, reap full benefits of your Testosterone injections therapy.
378. There arre health hazard related to testosterone thherapy and those risks could outweugh the benefits of
testosterone if you are not cautious about it.
And there are many testosterone myths and misconceptions which you may
want to contemplate (as well as side effects) before you determine to
start testosterone treatment.
379. Girls begin to experience menopause after a certain agge
and there are symptoms for example decline inn sexual desire,
as well as night sweats and hot flashes.
380. When you’ve experienced symptoms of low T, it truly is advisable to take a blood test to ascertain if your testosterone levels
arre low.
381. Testosterone replacement therapy has been used in people with testosterone deficiency, whether due to disease or aging.
382. Testosterone replacement therapy is just approved for gus who have low levels of testosterone linked to particular medical conditions.
383. Additionally, based on the available evidence from published studies and specialist input from an Advisory
Committee meeting , FDA has concluded thzt there’sa potential increased cardiovascular risk connected
with testosterone use.
384. Although there’s an extensive review 3 by the Institute of Medicin summarizing what’s known about testosterone therapy in elderly men, the secufity and effectiveness of testosterone
supppementation have not
besen clearly identified.
385. I wear a bioidentical hormone patch – a low dose and it does wonders
for me. I feel so much better and it does slow the effects of aging, althought it
doesn’t entirely stop them.
386. This article was updated withh more specific information about
which cardiovascular patients would bbe well served by testosterone treatment.
387. When your brain scans aand checks your body in its effort to modulate your hormonal secretion as needed throughout the day and it discovers that testosterone
levrls elevatesd resulting from an effective
testosterone treatment and are nice, its own natural production ceases in manufacture.
388. Thee male sex hormone testosterone can doo more for your body
thn simply raise sex drive.
389. Thhe results showed thyat of the 12 Sexuality measurements in the survey, 10 were significantly
impproved forr men in the testosterone group.
390. Just like other forms of testosterone, the testisterone
patch can cause low libido, oral difficulties, headaches, tiredness, hair loss,
skin irritations and many other allergy symptoms.
1. Rejuvchip Fort Lauderdale Testosterone pellets
are bio-identical, and are made using a botanical
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392. However, before you go assessing yourself into a retirement home, you can find a trustworrthy testosterone physician to immediately maintain yoour youth.
393. Yeet even Dr. Rajat Barua, the author of the veteran study,
acknowledged that the mechanics joining testosterone levels andd
cardiovascular problems are too poorlly understood –
and the signs is overly mixed – to urge testosterone treatment ffor cardiovascular issuhes alone, much less for men with normal testosterone levels.
394. While testoosterone therapy for men who want it may provide numerous health benefits like sexual function, mood, muscle
strength aand improved energy, there’s also a long list of potential
effects of taking testosterone.
395. As the testicular function slows down with age, this phenomenon is
commonly seen in men after the age of thirty.
396. Nowadays, testosterone is given through injectios orr skin patchds that absorption takes place transdermally.
397. Guys taking testosterone cypionate may suffer headaches,
high or low sex drive, hair loss, acne eruptions annd a bitter or strange taste
in the mouth.
398. Teztosterone treatment must always be discussed
in context of healthful living and a battalion of otfher contributions
that also interface with general wellness,
sexual function, prostate and cardiovascular disease,
glycemic control, and bone health, all of which contribute
to a man’s vibrant quality of life.
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400. The risks versus benefits of supplementation never have been clearly identified, while
it’s known that low levels of testosterone present ann
increased cardiovaascular risk.
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402. Low is correlated with hip fracture and height loss in postmenolausal women.
403. For each patient, weigh the possible increased risk
off major adverse cardiovascular consequences and otyher risks of testosterone replacement therapy against the possiible benefits of treatkng hypogonadism.
404. The CPG advocates that physicians avoid prescribing testosterone to improve sexual dysfunction in womesn who do not have HSDD.
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406. Thee indicators for tthe use of testosterone in emotional and cognitive impairment
are still not clear; however, studies of healthy elderly men with testosterone deficiency have given intriguing
407. Up tto 50 percent of diabetic men have low testosterone levels, and
obesity and poor lifestyle are knoqn to be directly linked to decreased testosterone
generation for millions of guys.
408. Finkle WD, Greenland S, Ridgeway GK, et al. Increased risk of non fatal myocardial infarction following testosterone
treatmment prescription inn men.
409. A patient who is unsatisfied with the results obtained through GH-GH therapy may choose to be treated afterwards with HGH replacement therapy or
vice versa.
410. Guuys who are feeling badly and have problems about
teztosterone levels deserve clinically demanding, evvidence based, and holistic
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Category: Cell Crawling
10/12/2016 Categories
0 Comments
Goal: Compare the average volume of the major protrusion for untreated cells and CK666 treated cells.
Got all the major protrusion volumes
untreated.largest.prot = c(c14_protrusions_rv2$V3, c15_protrusions_rv$V3, c10_protrusions_rv2$V3, c12_protrusions_rv$V3)
Found the mean and the standard dev
Note: for setting all NAs to 0:
x[is.na(x)] <- 0
The average volume of the major protrusion is higher, but also has a very large standard deviation.
0 Comments
0 Comments
Goal: To see if there is a correlation between the protrusion parameters matrix and the math of the cells using linear regression.
Training matrices:
DMSO treated cells (in order): cell 14, cell 15, cell 12, cell 10
CK666 treated cells (in order): cell 43 ,cell 48 , cell 55 , cell 53
variables: Count, largest protusion volume, total protustion volume, protrusion fraction, largest protrusion length, largest protrusion angle, and total protrusion angle.
Experiment 1
Dependent value (y): derivative using savitzky-golay
• untreated: training.set
• ck666: ck666.training.matrix
To get rid of rows without Y values
I did this because the smoothing window on the sg filter leaves a lot of blank values for the derivative.
Results: all the parameters were significant except largest protrusion angle, and total angles. The largest protrusion length is only significant for the first few fits.
Experiment 2
Dependent value (y)
: curve method
• untreated: training.set.curve
• Haven't completed this yet.
Experiment 3
Dependent value (y): bi-modal with cutoff (ie, is the cell turning or not turning?)
• untreated: trainingset.curve.bimodal
• untreated:
□ glm (family=binomial): fit.curve.bimodal
□ all fits loaded into matrix: fits.matrix.curve.bimodal
□ pvalues for all fits loaded into matrix: fits.matrix.curve.bimodal.pv
To do next:
add more cells to the training set
write function to find average turning point prediction
0 Comments | {"url":"https://www.meganrielmehan.com/methods-and-notes/category/cell-crawling","timestamp":"2024-11-11T04:43:54Z","content_type":"text/html","content_length":"39303","record_id":"<urn:uuid:511c49db-e88d-4b8f-9f4a-7bbd04e1c11e>","cc-path":"CC-MAIN-2024-46/segments/1730477028216.19/warc/CC-MAIN-20241111024756-20241111054756-00778.warc.gz"} |
Evolution of prisoner's dilemma strategies
While I was in France, I couldn't spend
my time going to the beach and talking about physics. I mean, sometimes the beach had too many jellyfish. And we had no internet for a time.
Our arch-nemesis, the jellyfish
So I might have spent an hour or two programming an evolutionary simulation of Prisoner's Dilemma strategies. The inspiration was that my boyfriend described to me an awful article
on BBC news
which completely distorted
a paper
(update: corrected link) on evolutionarily stable strategies for the iterated prisoner's dilemma.
This is all based on multiple hearsay, since I have read neither the news article nor the paper. So my evolutionary simulation may be completely off the mark. It's okay, this is just for fun. Later I
will look at the paper and compare what they did to what I did.
I assume readers are familiar with the
prisoner's dilemma
. The prisoner's dilemma contains the basic problem of altruism. If we all cooperate, we all get better results. But as far as the individual is concerned, it's better not to cooperate (ie to
defect). It gets more complicated when two players play multiple games of prisoner's dilemma in a row. In this case, even a purely selfish individual may wish to cooperate, or else in later
iterations of the game, their opponent might punish them for defecting.
The iterated prisoner's dilemma can be analyzed to death with economics and philosophy. But here I am interested in analyzing it from an evolutionary perspective. We imagine that individuals, rather
than intelligently choosing whether to cooperate or defect, blindly follow a strategy determined by their genes. I parametrize each person's genes with three numbers:
x: The probability of cooperating if in the previous iteration, the opponent cooperated.
y: The probability of cooperating if in the previous iteration, the opponent defected.
z: The probability of cooperating in the first iteration
I believe everything is better with visual representation, so here is a visual representation of the genespace:
Each square represents an individual. Each individual is described by three numbers x, y, and z, which are visually represented by the position and color of the square. The major strategies of the
iterated prisoner's dilemma are associated with different quadrants of the space. In the upper right quadrant is the cooperative strategy. In the lower left quadrant is the defecting strategy. In the
lower right, is the "tit for tat" strategy, wherein you cooperate if your opponent cooperate, and defect if your opponent defects. The upper left quadrant... I don't think there's a name for it.
And of course, there are hybrid strategies. For example, between cooperation and "Tit for Tat", there's "Tit for Tat with forgiveness". In the evolutionary algorithm, the only way to get from one
strategy to another is by accumulating mutations from generation to generation, so the hybrid strategies are important.
The simulation
I tried to perform the evolutionary simulation in the simplest way possible. Start with some number of individuals (here I use 20 individuals). Have each pair of individuals play an iterated
prisoner's dilemma (here I used 10 iterations).
Afterwards, I total up all the scores, and remove the individual with the worst score while duplicating the one with the best score. Then every individual mutates by randomly making small changes in
x, y, and z.
Upon testing, I found that this simulation favors defection far too much. The trouble is that it's really a zero sum game. There is no absolute standard for what sort of scores are good or bad, it's
just a comparison between the scores of different individuals. Even if defecting doesn't benefit you directly, it's still beneficial to hurt your opponents so that you come out ahead.
I'm not really sure how to solve this problem, and I'm very interested to see how researchers solve it. I tried several things, and settled for a simple modification. Rather than playing against
every other player, each individual will play against two random individuals.
It's still beneficial to hurt your opponents, but not
beneficial since you are only hurting a few of them. It's also possible that an individual will play against itself, and by defecting hurt itself.
I believe that researchers typically use the following payoffs: mutual cooperation is 3 points, mutual defection is 1 point, and if only one player defects then they get 5 points while the
cooperating player gets 0. However, I wanted to find some interesting behavior, so I adjusted the parameters to better balance the cooperation and defection strategies. I eventually settled on 4/3/1/
0 instead of 5/3/1/0.
Interestingly, rather than there being a single preferred strategy, multiple strategies can be evolutionarily stable. Actually, this may not be so surprising, since I basically adjusted the
parameters of the simulation until I saw interesting behavior. But it's still fun to watch.
Often the population will settle into the defection strategy and not move for a thousand generations. But sometimes there is a shift to a cooperative tit-for-tat hybrid strategy. This hybrid strategy
is not very stable, and always seems to eventually fall back to the defection strategy. Here I show an example of this happening:
Why does this happen? I'm not quite sure. But I think that when everyone is defecting, there's no real disadvantage to taking a tit-for-tat strategy. So genetic drift will sometimes cause the
population to shift towards tit-for-tat. Once there is a critical mass of tit-for-tat, it becomes advantageous to cooperate, since you will get back what you paid for. Soon everyone is cooperating or
using tit-for-tat. But then, perhaps by genetic drift, we lose too many tit-for-tat individuals, and it becomes advantageous to defect again. And so begins the cycle again.
Here I also show the average scores of the population over time:
If the population is defecting, then the average score will be 1. If it is cooperating, the average score will be 3. Here you see a punctuated equilibrium behavior.
Of course, you cannot draw any real conclusions from this simulation, since I did a lot of fine-tuning of the parameters, and because I'm not really interacting with the established literature. The
conclusion is that this is an interesting topic, and now I feel motivated to read some papers. When I do, I'll report back.
1. Actually I don't have individuals really play an iterated prisoner's dilemma against each other. Instead, I calculate the average result of an iterated prisoner's dilemma between those two
players. There is no chance involved in this calculation.
2. This means that each individual plays four games on average. Two games where the opponent is chosen randomly, and two games where that individual was chosen as the opponent.
This is part of a miniseries on the evolution of Prisoner's Dilemma strategies.
1. Evolution of prisoner's dilemma strategies
Extortionate strategies in Prisoner's dilemma
Prisoner's Dilemma and evolutionary stability
A modified prisoner's dilemma simulation
1 comment:
Larry Hamelin said...
Keep us posted. This is important. | {"url":"https://skepticsplay.blogspot.com/2013/08/evolution-of-prisoners-dilemma.html","timestamp":"2024-11-11T23:23:35Z","content_type":"application/xhtml+xml","content_length":"84020","record_id":"<urn:uuid:20464640-6d31-4824-a68f-d3e0c4d3caed>","cc-path":"CC-MAIN-2024-46/segments/1730477028240.82/warc/CC-MAIN-20241111222353-20241112012353-00362.warc.gz"} |
The Python Operators
In the Python Language, variables hold data values that convey the message. These values are either for logical or arithmetic operations. Meaning on python, all the process of writing codes is done
using the standard python Operators. In a very simple and clear manner, you’ll be learning the python Operators, what is it, the features of python Operators, and also you’ll be knowing the types and
functions of Python Operators. For the practical aspect, you’ll know how to apply them to Python Language, before that you will learn how to access the python operator Values. In conclusion, you’ll
be able to update the python Operators and delete some unwanted operations.
What are Python Operators
Python Operators are standard symbols that are used for logical and arithmetic operations in Python. Operators are used with operands to return a value e.g 2+8=10, 2 and 8 are the operands,(+) is the
operator and the returned Value or output is 10. Python has seven (7) Operators and they are:
1. Arithmetic Operators
2. Comparison/Relational Operators
3. Assignment Operators
4. Logical Operators
5. Bitwise Operators
6. Membership Operators
7. Identity Operators
This page is dedicated to giving complete knowledge of these operators.
Arithmetic Operators
The Arithmetic Operators are used to carry out all arithmetic or mathematical activities in python. Mathematical operations like addition, subtraction, multiplication, and division are enabled by the
mathematical symbols: ( +, -, /, *). See the table for complete arithmetic Operators symbols
Operator Description based on operands x and y
+ The addition adds two operands x+y
– Subtraction: subtracts two operands x-y
* Multiplication: multiply two operands x*y
/ Division(float): divides two operands x/y
// Floor division: divides two operands x//y
% Modulus: gives the remainder after the operands are been divided x%y
** Exponent: gives the result of x raised to power y: x**y
Example of Arithmetic Operators
Note: all the above examples will be shown at once. Take x=9 and y=4
Python Input:
# Examples of Arithmetic Operator
x = 9
y = 4
# Addition of numbers
add = x + y
# Subtraction of numbers
sub = x - y
# Multiplication of a number
mul = x * y
# Division(float) of number
div1 = x / y
# Division(floor) of number
div2 = x // y
# Modulo of both numbers
mod = x % y
# Power
p = x ** y
# print results
The above returns explain the arithmetic Operators practically
Comparison Operators in Python
Comparison Operators in Python are used to compare values. It gives True or False according to the condition. The python comparison Operators are also called Relational Operators. Take x and y as our
two values
Operator Description based on operands x and y
> Greater than: True if the left is greater than the right x>y
< Less than: True if the left is less than the right x<y
== True if two operands are equal x==y
!= Not equal to: True if two operands are not equal x!=y
>= Greater than or equal to True if the left operand is greater than or equal to the right operand x>=y
<= Less than or equal to True if the left operand is less than or equal to the right operand x<=y
Example of comparison Operators in Python
Take x=11 and y=22
Python input:
# Examples of Relational Operators
x = 11
y = 22
# x > y is False
print(x > y)
# x < y is True
print(x < y)
# x == y is False
print(x == y)
# x != y is True
print(x != y)
# x >= y is False
print(x >= y)
# x <= y is True
print(x <= y)
The above returns a practical explanation of the python comparison Operators
Assignment Operators
As the name implies, assignment Operators are used to assigning values to variables in Python. Meaning all characters must have the assignment symbol (=). Let’s see the tablet with x and y as our
Operator Description based on operands x and y
= Assign Values at the right operand to the left v=x+y
+=ad AND Adds right operand to left operand and assigning returns to left operand v+=x is equivalent to v=v+x
-= subtract AND Subtracts right operand from left operand and assign returns to left operand v-=x is equivalent to v=v-x
*= Multiply AND Multiplies right operand with the left operand and assign returns to the left operand v*=x is equivalent to v=v*x
/= Divide AND Divides the right operand with the left operand and assigns returns to the left operand v/=x is equivalent to v=v/x
% Modulus AND It takes modulus with left and right operands and assigns the result to the left operand v%=x is equivalent to v=v%x
**= Exponent AND Calculates the exponential power value of both operands and assigns Values to left v**=x is equivalent to v=v**x
&= Bitwise AND Calculates the bitwise operation on both operators and assigns the value to the left operand v&=x is equivalent to v=v&x
|= Bitwise OR Calculates the bitwise OR and assigns value to the left operand v|=x is equivalent to v=v|x
^= Bitwise xOR Calculates bitwise xOR and assigns Values to the left operand v^=x is equivalent to v=v^x
>>= Bitwise right shift Performs bitwise right shift on operands and assigns Values to left operand v>>=x is equivalent to v=v>>x
<<= Bitwise left shift Performs bitwise left shift on operands and assigns Values to the left operand v<<=x is equivalent to v=v<<x
Some examples of assignment Operators in Python
Python Input:
# Examples of Assignment Operators
x = 20
# Assign value
y = x
# Add and assign value
y += x
# Subtract and assign value
y -= x
# multiply and assign
y *= x
# bitwise shift operator
y <<= x
The above examples show the assignment Operators practically
Python Logical Operators
Logical operators are used to joining conditional statements to give logical output. It uses the Logical AND, Logical OR, and Logical NOT operations.
See the table for all Logical Operators in Python
Operators Description based on operands x and y
and Logical AND Returns True if (x and y) are true
or Logical OR Returns True if either (x or y) is true
not Logical NOT Returns True if neither ( x or y) is true
Example of Python Logical Operators
Python input:
# Examples of Logical Operator
x = True
y = False
# Print a and b is False
print(x and y)
# Print a or b is True
print(x or y)
# Print, not a is False
print(not x)
The above expresses the Python Logical Operators in a practical order
Bitwise Operators in Python
These are the bit-by-bit binary numbers operations in Python. Some examples of the Bitwise operators and syntax are expressed in the table below
Operators Description based on operands x and y
& Binary AND Copies a bit to the results if both operands have the bit value x&y
| Binary OR Copies a bit of the result if any one of the operands has the bit value x|y
^ Binary XOR Copies the bit if it is set in either x or y (x^y)
~ Bitwise NOT It indicates a Flippo bit on one operand ~x
>> Bitwise right shift The left operand(x) is moved right to the number of bits indicated by the right operand (x>>)
<< Bitwise left shift The left operand(x) is moved left to the number of bits indicated by the right operand (x<<)
Example of Python Bitwise operators
Python Input:
# Examples of Bitwise operators
x = 10
y = 4
# Print bitwise AND operation
print(x & y)
# Print bitwise OR operation
print(x | y)
# Print bitwise NOT operation
# print bitwise XOR operation
print(x ^ y)
# print bitwise right shift operation
print(x >> y)
# print bitwise left shift operation
print(x << y)
Membership Operators
The membership Operators check whether a variable or value is included in a sequence. It uses not or not as its operators.
• it signifies true if a value or variable is found in a sequence
• not in returns false if a value or variable is not in a sequence
Example of Python Membership Operators
# Python program to illustrate
# not 'in' operator
x = 24
y = 20
list = [10, 20, 30, 40, 50]
if (x is not in the list):
print("x is NOT present in the given list")
print("x is present in the given list")
if (y in the list):
print("y is present in the given list")
print("y is NOT present in the given list
x is NOT present in the given list
y is present in the given list
Python Identify Operators
The identity operators identify and compare the memory location of two objects. The identity operators operate in two categories which are:
• is: Simply true if the Values are similar( x is y)
• is not: Signifies true if the values are not similar (x is not y)
Example Of Python Identify Operators
Python Input:
x = 10
y = 20
v = a
print(x is not y)
print(x is c)
Python Precedence Operators
Python Precedence Operators are used in an operation with more than one operator to indicate which operator should operate first. It determines operations according to priority.
The Table Below Shows all of python’s precedence from highest to lowest
Operator Description
**Exponent Raise the power
~+- complement, urinary plus, and minus
*/%// Multiplication, divide, modulus, and floor division
+- Addition and subtraction
>><< Right Bitwise shift and Left Bitwise shift
& Bitwise AND
|^ Bitwise normal OR and exclusive OR
<= < > >= Comparison operators
<> == != Equality operators
= %= /= //= -= += *= **= Assignment operators
is is not Identity operators
in, not in Membership operators
not, or, and Logical operators
Example of Python Precedence Operators
Python Input:
# Examples of Operator Precedence
# Precedence of '+' & '*'
expr = 1 + 50 * 10
# Precedence of 'or' & 'and'
name = "Simon"
age = 0
if name == "Simon" or name == "Monday" and age >= 2:
print("Hello! Tanya.")
Hello! Tanya.
From the above output, python prints according to precedence.
In the Python Language, variables hold data values that convey the message. These values are either for logical or arithmetic operations.
Python Operators are standard symbols that are used for logical and arithmetic operations in Python.
Python Precedence Operators are used in an operation with more than one operator to indicate which operator should operate first. It determines operations according to priority.
In this topic, you have been taught extensively on Python Operators and all the Python Precedence Operators. It’s all on you to give the theory some practice for maximum results. For a headstart
check out the python tutorials. Good luck Coding!
Er. Pramod Yadav
Pramod Kumar Yadav is from Janakpur Dham, Nepal. He was born on December 23, 1994, and has one elder brother and two elder sisters. He completed his education at various schools and colleges in Nepal
and completed a degree in Computer Science Engineering from MITS in Andhra Pradesh, India. Pramod has worked as the owner of RC Educational Foundation Pvt Ltd, a teacher, and an Educational
Consultant, and is currently working as an Engineer and Digital Marketer. | {"url":"https://myprogrammingschool.com/python-operators/","timestamp":"2024-11-07T08:55:16Z","content_type":"text/html","content_length":"180213","record_id":"<urn:uuid:4500434f-8ed4-4fd2-882b-26fc593e3e31>","cc-path":"CC-MAIN-2024-46/segments/1730477027987.79/warc/CC-MAIN-20241107083707-20241107113707-00258.warc.gz"} |
After you have finished doing your mathematics research, you will need to present your findings to others. There are three main ways to do this:
The following sections provide information about each of these presentation strategies.
Writing Up Your Research
Your project write-up is a chance to synthesize what you have learned about your mathematics research problem and to share it with others. Most people find that when they complete their write-up it
gives them quite a bit of satisfaction. The process of writing up research forces you to clarify your own thinking and to make sure you really have rigorous arguments. You may be surprised to
discover how much more you will learn by summarizing your research experience!
If you have ever looked at a mathematics journal to see how mathematicians write up new results, you may have found that everything seemed neat and polished. The author often poses a question and
then presents a proof that leads neatly, and sometimes elegantly, to his or her solution. Mathematicians rarely talk about the dead ends they met along the way in these formal presentations. Your
write-up will be different. We hope that you will tell your reader about your thought process. How did you start? What did you discover? Where did that lead you? What were your conjectures? Did you
disprove any of them? How did you prove the ones that were true? By answering these questions, you will provide a detailed map that will take the reader through your research experience.
This guide will give you a brief overview of the parts of a mathematics research paper. Following the guide is a sample write up so you can see how one person wrote about her research experience and
shared her results.
A formal mathematics research paper includes a number of sections. These will be appropriate for your write-up as well. The sections of the report are linked so that you can see an example of each
part in the sample write-up that follows. Note that not all mathematical research reports contain all of the sections. For example, you might not have any appendices to include or you may not do a
literature review. However, your write-up should definitely contain parts 1, 2, 4, 5, 6 and 7.
1. Title [see sample]
As with any paper you write, a title is important. It should catch the attention of the reader as well as reflect the content of your paper.
2. Abstract [see sample]
An abstract is a summary paragraph in which you explain the basic purpose of your paper, state the question(s) you answered, and tell the reader what you proved. It provides an outline of your paper
so that someone who might be interested in your paper will know what he or she will learn by reading it. Usually people write their abstract after they have finished writing the body of the report
because it summarizes what they wrote, not what they plan to write.
3. Literature Review
If you did research about your topic by consulting existing literature and then built upon what you learned by extending the ideas, then it is appropriate to include a literature review in your
paper. This section should read as an analytical essay and tell your reader about the state of the art in your research area. Provide the reader with the background and the context of your topic,
demonstrating your expertise in the process. It is appropriate to identify and discuss books, articles and other source materials that you read for your project. Everything in your literature review
should be mentioned in your reference section, but not everything in your reference section belongs in your literature review. You are merely commenting on the most valuable material you have
identified that you will need to assimilate for your project. This section should be rich in footnotes or parenthetical references to original sources.
In the literature review section you may answer questions such as What kind of research has been done before? What kind of relevant studies or techniques needed to be mastered to do your project?
How have others gone about trying to solve your problem, and how does your approach differ?
4. Statement of the Problem/Introduction [see sample]
Here you set the stage for your paper. A brief introduction should give the reader some understanding of the context in which you are working. What was the source of inspiration for your problem?
Is it a modification of some other question? Why is this problem important or interesting to you (or someone else?) Be sure to state the question or questions that you will examine in your report.
You don t want to do any proving here, just tell the reader what questions you plan to discuss in your paper.
5. Body of the Report [see sample]
Take the reader on a trip through your research project. Using your log book as your guide, start with your initial explorations and conjectures. Explain any definitions and notation that you
developed. Tell the reader what you discovered as you learned more about the problem. Provide any numeric, geometric or symbolic examples that guided you toward your conjectures. Show your results.
Explain how you proved your conjectures.
The body of your report should be a mix of English narrative and more abstract representations. Be sure to include lots of examples to help your reader understand your reasoning. If your paper is
very long, you can divide the body of your report into sections so that it easier to tackle the various aspects of your work.
6. Ideas for Further Research [see sample]
As you worked on your research project, questions that you didn t have time to answer, or that were beyond the scope of what you were doing, undoubtedly surfaced. State those questions here. Maybe
you will inspire someone else to tackle those questions. Maybe you will work on them yourself at another time. By posing them here, you participate in advancing the field of mathematics. You place
your work within a larger framework of inquiry.
7. References [see sample]
As with any research paper, you must give credit to the people whose work you used in writing your report. Include articles and books that you used. It is also appropriate to cite articles and other
material that you found on the World Wide Web. There is more than one accepted form for referencing materials; here we provide examples using the APA (American Psychological Association) style. For
more complete instructions you may find one of the APA Style guides helpful. Most libraries have a copy.
│Book with one author │Gauss, Carl Friedrich. (1966). Disquisitiones arithmeticae (English edition). New York, NY: Springer-Verlag. │
│Book with two authors │O Daffer, Phares G. & Clemens, Stanley R. (1977). Geometry: An investigative approach. Menlo Park, CA: Addison-Wesley Publishing Company. │
│Edited book │Fadiman, Clifton (Ed.). (1997). Fantasia mathematica. New York: Copernicus. │
│Book, edition other than first │Brown, Stephen I. & Walter, Marion I. (1990). The art of problem posing. (2^nd ed.) Hillsdale, NJ: Erlbaum. │
│Essay or chapter in a collection│Epp, Susanna S. (1994) The role of proof in problem solving. In. Schoenfeld, Alan H. (Ed). Mathematical thinking and problem solving. (pp. 257-269). Hillsdale, NJ: │
│or anthology │Erlbaum. │
│Journal Article │Keiser, Jane M. (2000, April). The role of definition. Mathematics Teaching in the Middle School 5, 8. 506-511. │
│Article in a magazine │Shasha, Dennis E. (2002, January). Pinpointing a Polar Bear. Scientific American, 286, p. 96. │
│Daily newspaper article (note │Aoki, Naomi (2001, December 20). Genetic mutation gives hope to those with osteoporosis. Boston Globe, pp. E1, E6. │
│multiple pages) │ │
│Abstract on CD-ROM │Brown, G. N. (1993). Borderline states: Incest and adolescence [CD-ROM]. Journal of Analytical Psychology, 38 (1), 23-25. Abstract from SilverPlatter File: Psyclit │
│ │Item 80-25636 │
│Article posted on a web site │Bridgewater State College. (1998, August 5). APA Style: Sample Bibliographic Entries (4th ed). Bridgewater, MA: Clement C. Maxwell Library. Retrieved February 12, │
│ │2002, from the World Wide Web: http://www.bridgew.edu/depts/maxwell/apa.htm │
8. Appendices
In the appendices you should include any data or material that supported your research but that was too long to include in the body of your paper. Materials in an appendix should be referenced at
some point in the body of the report.
Some examples:
If you wrote a computer program to generate more data than you could produce by hand, you should include the code and some sample output.
If you collected statistical data using a survey, include a copy of the survey.
If you have lengthy tables of numbers that you do not want to include in the body of your report, you can put them in an appendix.
Sample Write-Up
SEATING UNFRIENDLY CUSTOMERS
A Combinatorics Problem
By Lisa Honeyman
February 12, 2002
The answers to combinatorial questions concerning unfriendly customers at a morning coffee shop are found. It is proven that if there are n seats and c customers who refuse to sit next to each other
at a coffee shop counter, then there are [n-c+1]P[c] ways for them to sit. It is also proven that if there are n seats and c customers who refuse to sit next to each other at a circular table, then
there are
The Problem
In a certain coffee shop, the customers are grouchy in the early morning and none of them wishes to sit next to another at the counter.
1. Suppose there are ten seats at the counter. How many different ways can three early morning customers sit at the counter so that no one sits next to anyone else?
2. What if there are n seats at the counter?
3. What if we change the number of customers?
4. What if, instead of a counter, there was a round table and people refused to sit next to each other?
I am assuming that the order in which the people sit matters. So, if three people occupy the first, third and fifth seats, there are actually 6 (3!) different ways they can do this. I will explain
more thoroughly in the body of my report.
Body of the Report
At first there are 10 seats available for the 3 people to sit in. But once the first person sits down, that limits where the second person can sit. Not only can t he sit in the now-occupied seat, he
can t sit next to it either. What confused me at first was that if the first person sat at one of the ends, then there were 8 seats left for the second person to chose from. But if the 1^st person
sat somewhere else, there were only 7 remaining seats available for the second person. I decided to look for patterns. By starting with a smaller number of seats, I was able to count the
possibilities more easily. I was hoping to find a pattern so I could predict how many ways the 10 people could sit without actually trying to count them all. I realized that the smallest number of
seats I could have would be 5. Anything less wouldn t work because people would have to sit next to each other. So, I started with 5 seats. I called the customers A, B, and C.
With 5 seats there is only one configuration that works.
│ A │ │ B │ │ C │
As I said in my assumptions section, I thought that the order in which the people sit is important. Maybe one person prefers to sit near the coffee maker or by the door. These would be different, so
I decided to take into account the different possible ways these 3 people could occupy the 3 seats shown above. I know that ABC can be arranged in 3! = 6 ways. (ABC, ACB, BAC, BCA, CAB, CBA). So
there are 6 ways to arrange 3 people in 5 seats with spaces between them. But, there is only one configuration of seats that can be used. (The 1^st, 3^rd, and 5^th).
Next, I tried 6 seats. I used a systematic approach to show that there are 4 possible arrangements of seats. This is how my systematic approach works:
Assign person A to the 1^st seat. Put person B in the 3^rd seat, because he can t sit next to person A. Now, person C can sit in either the 5^th or 6^th positions. (see the top two rows in the chart,
below.) Next suppose that person B sits in the 4^th seat (the next possible one to the right.) That leaves only the 6^th seat free for person C. (see row 3, below.) These are all the possible ways
for the people to sit if the 1^st seat is used. Now put person A in the 2^nd seat and person B in the 4^th. There is only one place where person C can sit, and that s in the 6^th position. (see row
4, below.) There are no other ways to seat the three people if person A sits in the 2^nd seat. So, now we try putting person A in the 3^rd seat. If we do that, there are only 4 seats that can be
used, but we know that we need at least 5, so there are no more possibilities.
│ │1^st│2^nd│3^rd│4^th│5^th│6^th│
│row 1 │ A │ │ B │ │ C │ │
│row 2 │ A │ │ B │ │ │ C │
│row 3 │ A │ │ │ B │ │ C │
│row 4 │ │ A │ │ B │ │ C │
Possible seats 3 people could occupy if there are 6 seats
Once again, the order the people sit in could be ABC, BAC, etc. so there are 4 * 6 = 24 ways for the 3 customers to sit in 6 seats with spaces between them.
I continued doing this, counting how many different groups of seats could be occupied by the three people using the systematic method I explained. Then I multiplied that number by 6 to account for
the possible permutations of people in those seats. I created the following table of what I found.
│Total # of seats│# of groups of 3 possible│# ways 3 people can sit in n seats │
│ │ │ │
│ (n) │ │ │
│ 5 │ 1 │ 6 │
│ 6 │ 4 │ 24 │
│ 7 │ 10 │ 60 │
│ 8 │ 20 │ 120 │
Next I tried to come up with a formula. I decided to look for a formula using combinations or permutations. Since we are looking at 3 people, I decided to start by seeing what numbers I would get if
I used [n]C[3] and [n]P[3].
[3]C[3] = 1 [4]C[3] = 4 [5]C[3] = 10 [ 6]C[3] = 20
[3]P[3] = 6 [ 4]P[3] = 24 [5]P[3] = 60 [ 6]P[3] = 120
Surprisingly enough, these numbers matched the numbers I found in my table. However, the n in [n]P[r] and [n]C[r] seemed to be two less than the total # of seats I was investigating.
Conjecture 1:
Given n seats at a lunch counter, there are[ n-2]C[3] ways to select the three seats in which the customers will sit such that no customer sits next to another one. There are[ n-2]P[3] ways to seat
the 3 customers in such a way than none sits next to another.
After I found a pattern, I tried to figure out why[ n-2]C[3] works. (If the formula worked when order didn t matter it could be easily extended to when the order did, but the numbers are smaller and
easier to work with when looking at combinations rather than permutations.)
In order to prove Conjecture 1 convincingly, I need to show two things:
(1) Each n 2 seat choice leads to a legal n seat configuration.
(2) Each n seat choice resulted from a unique n 2 seat configuration.
To prove these two things I will show
(3) There is a function, f, such that f(n 2 non-separated seats) n separated seats
(4) There is a function, g, such that g(n separated seats) n 2 non-separated seats
And then conclude that these two procedures are both functions and therefore 1 1.
Claim (1): Each (n 2)-seat choice leads to a legal n seat configuration.
Suppose there were only n 2 seats to begin with. First we pick three of them in which to put people, without regard to whether or not they sit next to each other. But, in order to guarantee that
they don t end up next to another person, we introduce an empty chair to the right of each of the first two people. It would look like this:
We don t need a third new seat because once the person who is farthest to the right sits down, there are no more customers to seat. So, we started with n 2 chairs but added two for a total of n
chairs. Anyone entering the restaurant after this procedure had been completed wouldn t know that there had been fewer chairs before these people arrived and would just see three customers sitting at
a counter with n chairs. This procedure guarantees that two people will not end up next to each other. Thus, each (n 2)-seat choice leads to a unique, legal n seat configuration.
Using mathematical notation: Suppose s[1], s[2] and s[3] are the locations selected with all three distinct. That is, s[2] s[1] + 1 and s[3] s[2] + 1. Then the people will actually end up sitting in
seats at locations s[1]'= s[1], s[2]'= s[2] + 1 and s[3]' = s[3] + 2. Combining these formulas with the original conditions yields:
s[2]' = s[2] + 1 s[1] + 2 = s[1]' + 2
s[3]' = s[3] + 2 s[2] + 3 = s[2]' + 2
Therefore, positions s[1]' s[2]', and s[3]' are all separated by at least one vacant seat.
This is a function that maps each combination of 3 seats selected from n 2 seats onto a unique arrangement of n seats with 3 separated customers. Therefore, it is invertible.
f(n 2 non-separated seats) n separated seats
Claim (2): Each 10-seat choice has a unique 8-seat configuration.
Given a legal 10-seat configuration, each of the two left-most diners must have an open seat to his/her right. Remove it and you get a unique 8-seat arrangement. If, in the 10-seat setting, we have q
[1] > q[2], q[3]; q[3] 1 > q[2], and q[2] 1 > q[1], then the 8 seat positions are q[1]' = q[2], q[2]' = q[2] 1, and q[3]' = q[3] 2. Combining these equations with the conditions we have
q[2]' = q[2] 1 which implies q[2]' > q[1] = q[1]'
q[3]' = q[3] 2 which implies q[3]' > q[2] 1 = q[2]'
Since q[3]' > q[2]' > q[1]', these seats are distinct. If the diners are seated in locations q[1], q[2], and q[3] (where q[3] 1 > q[2] and q[2] 1 > q[1]) and we remove the two seats to the right
of q[1] and q[2], then we can see that the diners came from q[1], q[2] 1, and q[3] 2. This is a function that maps a legal 10-seat configuration to a unique 8-seat configuration.
g(n separated seats) n 2 non-separated seats.
The size of a set can be abbreviated s( ). I will use the abbreviation S to stand for n separated seats and N to stand for the n 2 non-separated seats.
s(S) s(N) because f(N) S
and s(N) s(S) because g(S) N
therefore s(N) = s(S).
Because the sets are the same size, these functions are 1 1.
Using the technique of taking away and adding empty chairs, I can extend the problem to include any number of customers. For example, if there were 4 customers and 10 seats there would be [7]C[4] =
35 different combinations of chairs to use and [7]P[4] = 840 ways for the customers to sit (including the fact that order matters). You can imagine that three of the ten seats would be introduced by
three of the customers. So, there would only be 7 to start with.
In general, given n seats and c customers, we remove c-1 chairs and select the seats for the c customers. This leads to the formula[ n-(c-1)]C[c] =[ n-c+1]C[c] for the number of arrangements.
Once the number of combinations of seats is found, it is necessary to multiply by c! to find the number of permutations. Looking at the situation of 3 customers and using a little algebraic
manipulation, we get the [n]P[3] formula shown below.
[n-2]C[3] =
If we multiply[ n-2]C[3] by 3! we get [ n-2]P[3] (also by definition)
This same algebraic manipulation works if you have c people rather than 3, resulting in[ n-c+1]P[c]
Answers to Questions
1. With 10 seats there are [8]P[3] = 336 ways to seat the 3 people.
2. My formula for n seats and 3 customers is:[ n-2]P[3].
3. My general formula for n seats and c customers, is:[ n-(c-1)]P[c] =[ n-c+1]P[c]
_________________________________________________________________ _
After I finished looking at this question as it applied to people sitting in a row of chairs at a counter, I considered the last question, which asked would happen if there were a round table with
people sitting, as before, always with at least one chair between them.
I went back to my original idea about each person dragging in an extra chair that she places to her right, barring anyone else from sitting there. There is no end seat, so even the last person needs
to bring an extra chair because he might sit to the left of someone who has already been seated. So, if there were 3 people there would be 7 seats for them to choose from and 3 extra chairs that no
one would be allowed to sit in. By this reasoning, there would be [7]C[3] = 35 possible configurations of chairs to choose and [7]P[3] = 840 ways for 3 unfriendly people to sit at a round table.
Conjecture 2: Given 3 customers and n seats there are[ n-3]C[3] possible groups of 3 chairs which can be used to seat these customers around a circular table in such a way that no one sits next to
anyone else.
My first attempt at a proof: To test this conjecture I started by listing the first few numbers generated by my formula:
When n = 6 [6-3]C[3] = [3]C[3] = 1
When n = 7 [ 7-3]C[3] = [4]C[3] = 4
When n = 8 [ 8-3]C[3] =[ 5]C[3] = 10
When n = 9 [ 9-3]C[3] = [6]C[3] = 20
Then I started to systematically count the first few numbers of groups of possible seats. I got the numbers shown in the following table. The numbers do not agree, so something is wrong probably my
│Total # of seats (n)│# of groups of 3 possible│# of possible configurations │
│ 6 │ 2 │ 12 │
│ 7 │ 7 │ 42 │
│ 8 │ 16 │ 96 │
│ 9 │ 30 │ 180 │
I looked at a circular table with 8 people and tried to figure out the reason this formula doesn t work. If we remove 3 seats (leaving 5) there are 10 ways to select 3 of the 5 remaining chairs. ([5]
The circular table at the left in the figure below shows the n 3 (in this case 5) possible chairs from which 3 will be randomly chosen. The arrows point to where the person who selects that chair
could end up. For example, if chair A is selected, that person will definitely end up in seat #1 at the table with 8 seats. If chair B is selected but chair A is not, then seat 2 will end up
occupied. However, if chair A and B are selected, then the person who chose chair B will end up in seat 3. The arrows show all the possible seats in which a person who chose a particular chair could
end. Notice that it is impossible for seat #8 to be occupied. This is why the formula [5]C[3] doesn t work. It does not allow all seats at the table of 8 to be chosen.
The difference is that in the row-of-chairs-at-a-counter problem there is a definite starting point and ending point. The first chair can be identified as the one farthest to the left, and the
last one as the one farthest to the right. These seats are unique because the starting point has no seat to the left of it and the ending point has no seat to its right. In a circle, it is not so
Using finite differences I was able to find a formula that generates the correct numbers:
tn = n^3 9n^2 + 20n). I used this formula to predict the next two numbers in the chart and confirmed them by hand.
│Total # of seats (n)│# of groups of 3 possible│# of possible configurations │
│ 10 │ 50 │ 300 │
│ 11 │ 77 │ 462 │
Conjecture 3: Given 3 customers and n seats there are n^3 9n^2 + 20n) = n(n 4)(n 5) possible configurations if order matters. Note that the factored form of this cubic function gives clues to
how the problem works.
Proof: We need to establish a starting point. This could be any of the n seats. So, we select one and seat person A in that seat. Person B cannot sit on this person s left (as he faces the table),
so we must eliminate that as a possibility. Also, remove any 2 other chairs, leaving (n 4) possible seats where the second person can sit. Select another seat and put person B in it. Now, select
any other seat from the (n 5) remaining seats and put person C in that. Finally, take the two seats that were previously removed and put one to the left of B and one to the left of C.
The following diagram should help make this procedure clear.
Finally, if we decide that the order in which the diners sit does not matter, we can divide the function f(n) = n(n 4)(n 5) by 3! so we don t count each configuration 6 times. This results in the
function: g(n) =
In a manner similar to the method I used in the row-of-chairs-at-a-counter problem, this could be proven more rigorously.
An Idea for Further Research:
Consider a grid of chairs in a classroom and a group of 3 very smelly people. No one wants to sit adjacent to anyone else. (There would be 9 empty seats around each person.) Suppose there are 16
chairs in a room with 4 rows and 4 columns. How many different ways could 3 people sit? What if there was a room with n rows and n columns? What if it had n rows and m columns?
Abrams, Joshua. Education Development Center, Newton, MA. December 2001 - February 2002. Conversations with my mathematics mentor.
Brown, Richard G. 1994. Advanced Mathematics. Evanston, Illinois. McDougal Littell Inc. pp. 578-591
The Oral Presentation
Giving an oral presentation about your mathematics research can be very exciting! You have the opportunity to share what you have learned, answer questions about your project, and engage others in
the topic you have been studying. After you finish doing your mathematics research, you may have the opportunity to present your work to a group of people such as your classmates, judges at a science
fair or other type of contest, or educators at a conference. With some advance preparation, you can give a thoughtful, engaging talk that will leave your audience informed and excited about what you
have done.
Planning for Your Oral Presentation
In most situations, you will have a time limit of between 10 and 30 minutes in which to give your presentation. Based upon that limit, you must decide what to include in your talk. Come up with some
good examples that will keep your audience engaged. Think about what vocabulary, explanations, and proofs are really necessary in order for people to understand your work. It is important to keep the
information as simple as possible while accurately representing what you ve done. It can be difficult for people to understand a lot of technical language or to follow a long proof during a talk. As
you begin to plan, you may find it helpful to create an outline of the points you want to include. Then you can decide how best to make those points clear to your audience.
You must also consider who your audience is and where the presentation will take place. If you are going to give your presentation to a single judge while standing next to your project display, your
presentation will be considerably different than if you are going to speak from the stage in an auditorium full of people! Consider the background of your audience as well. Is this a group of people
that knows something about your topic area? Or, do you need to start with some very basic information in order for people to understand your work? If you can tailor your presentation to your
audience, it will be much more satisfying for them and for you.
No matter where you are presenting your speech and for whom, the structure of your presentation is very important. There is an old bit of advice about public speaking that goes something like this:
Tell em what you re gonna tell em. Tell em. Then tell em what you told em. If you use this advice, your audience will find it very easy to follow your presentation. Get the attention of the
audience and tell them what you are going to talk about, explain your research, and then following it up with a re-cap in the conclusion.
Writing Your Introduction
Your introduction sets the stage for your entire presentation. The first 30 seconds of your speech will either capture the attention of your audience or let them know that a short nap is in order.
You want to capture their attention. There are many different ways to start your speech. Some people like to tell a joke, some quote famous people, and others tell stories.
Here are a few examples of different types of openers.
You can use a quote from a famous person that is engaging and relevant to your topic. For example:
Benjamin Disraeli once said, There are three kinds of lies: lies, damn lies, and statistics. Even though I am going to show you some statistics this morning, I promise I am not going to lie to
you! Instead, . . .
The famous mathematician, Paul Erdös, said, A Mathematician is a machine for turning coffee into theorems. Today I m here to show you a great theorem that I discovered and proved during my
mathematics research experience. And yes, I did drink a lot of coffee during the project!
According to Stephen Hawking, Equations are just the boring part of mathematics. With all due respect to Dr. Hawking, I am here to convince you that he is wrong. Today I m going to show you one
equation that is not boring at all!
Some people like to tell a short story that leads into their discussion.
Last summer I worked at a diner during the breakfast shift. There were 3 regular customers who came in between 6:00 and 6:15 every morning. If I tell you that you didn t want to talk to these folks
before they ve had their first cup of coffee, you ll get the idea of what they were like. In fact, these people never sat next to each other. That s how grouchy they were! Well, their anti-social
behavior led me to wonder, how many different ways could these three grouchy customers sit at the breakfast counter without sitting next to each other? Amazingly enough, my summer job serving coffee
and eggs to grouchy folks in Boston led me to an interesting combinatorics problem that I am going to talk to you about today.
A short joke related to your topic can be an engaging way to start your speech.
It has been said that there are three kinds of mathematicians: those who can count and those who can t.
All joking aside, my mathematics research project involves counting. I have spent the past 8 weeks working on a combinatorics problem.. . .
To find quotes to use in introductions and conclusions try: http://www.quotationspage.com/
To find some mathematical quotes, consult the Mathematical Quotation Server: http://math.furman.edu/~mwoodard/mquot.html
To find some mathematical jokes, you can look at the Profession Jokes web site: http://www.geocities.com/CapeCanaveral/4661/projoke22.htm
There is a collection of math jokes compiled by the Canadian Mathematical Society at http://camel.math.ca/Recreation/
After you have the attention of your audience, you must introduce your research more formally. You might start with a statement of the problem that you investigated and what lead you to choose that
topic. Then you might say something like this,
Today I will demonstrate how I came to the conclusion that there are n(n 4)(n 5) ways to seat 3 people at a circular table with n seats in such a way that no two people sit next to each other.
In order to do this I will first explain how I came up with this formula and then I will show you how I proved it works. Finally, I will extend this result to tables with more than 3 people sitting
at them.
By providing a brief outline of your talk at the beginning and reminding people where you are in the speech while you are talking, you will be more effective in keeping the attention of your
audience. It will also make it much easier for you to remember where you are in your speech as you are giving it.
The Middle of Your Presentation
Because you only have a limited amount of time to present your work, you need to plan carefully. Decide what is most important about your project and what you want people to know when you are
finished. Outline the steps that people need to follow in order to understand your research and then think carefully about how you will lead them through those steps. It may help to write your entire
speech out in advance. Even if you choose not to memorize it and present it word for word, the act of writing will help you clarify your ideas. Some speakers like to display an outline of their talk
throughout their entire presentation. That way, the audience always knows where they are in the presentation and the speaker can glance at it to remind him or herself what comes next.
An oral presentation must be structured differently than a written one because people can t go back and re-read a complicated section when they are at a talk. You have to be extremely clear so that
they can understand what you are saying the first time you say it. There is an acronym that some presenters like to remember as they prepare a talk: KISS. It means, Keep It Simple, Student. It
may sound silly, but it is good advice. Keep your sentences short and try not to use too many complicated words. If you need to use technical language, be sure to define it carefully. If you feel
that it is important to present a proof, remember that you need to keep things easy to understand. Rather than going through every step, discuss the main points and the conclusion. If you like, you
can write out the entire proof and include it in a handout so that folks who are interested in the details can look at them later. Give lots of examples! Not only will examples make your talk more
interesting, but they will also make it much easier for people to follow what you are saying.
It is useful to remember that when people have something to look at, it helps to hold their attention and makes it easier for them to understand what you are saying. Therefore, use lots of graphs and
other visual materials to support your work. You can do this using posters, overhead transparencies, models, or anything else that helps make your explanations clear.
Using Materials
As you plan for your presentation, consider what equipment or other materials you might want use. Find out what is available in advance so you don t spend valuable time creating materials that you
will not be able to use. Common equipment used in talks include an over-head projector, VCR, computer, or graphing calculator. Be sure you know how to operate any equipment that you plan to use. On
the day of your talk, make sure everything is ready to go (software loaded, tape at the right starting point etc.) so that you don t have technical difficulties.
Visual aides can be very useful in a presentation. (See Displaying Your Results for details about poster design.) If you are going to introduce new vocabulary, consider making a poster with the words
and their meanings to display throughout your talk. If people forget what a term means while you are speaking, they can refer to the poster you have provided. (You could also write the words and
meanings on a black/white board in advance.) If there are important equations that you would like to show, you can present them on an overhead transparency that you prepare prior to the talk.
Minimize the amount you write on the board or on an overhead transparency during your presentation. It is not very engaging for the audience to sit watching while you write things down. Prepare all
equations and materials in advance. If you don t want to reveal all of what you have written on your transparency at once, you can cover up sections of your overhead with a piece of paper and slide
it down the page as you move along in your talk. If you decide to use overhead transparencies, be sure to make the lettering large enough for your audience to read. It also helps to limit how much
you put on your transparencies so they are not cluttered. Lastly, note that you can only project approximately half of a standard 8.5" by 11" page at any one time, so limit your information to
displays of that size.
Presenters often create handouts to give to members of the audience. Handouts may include more information about the topic than the presenter has time to discuss, allowing listeners to learn more if
they are interested. Handouts may also include exercises that you would like audience members to try, copies of complicated diagrams that you will display, and a list of resources where folks might
find more information about your topic. Give your audience the handout before you begin to speak so you don t have to stop in the middle of the talk to distribute it. In a handout you might include:
A proof you would like to share, but you don t have time to present entirely.
Copies of important overhead transparencies that you use in your talk.
Diagrams that you will display, but which may be too complicated for someone to copy down accurately.
Resources that you think your audience members might find useful if they are interested in learning more about your topic.
The Conclusion
Ending your speech is also very important. Your conclusion should leave the audience feeling satisfied that the presentation was complete. One effective way to conclude a speech is to review what you
presented and then to tie back to your introduction. If you used the Disraeli quote in your introduction, you might end by saying something like,
I hope that my presentation today has convinced you that . . .
Statistical analysis backs up the claims that I have made, but more importantly, . . . .
And that s no lie!
Getting Ready
After you have written your speech and prepared your visuals, there is still work to be done.
1. Prepare your notes on cards rather than full-size sheets of paper. Note cards will be less likely to block your face when you read from them. (They don t flop around either.) Use a large font
that is easy for you to read. Write notes to yourself on your notes. Remind yourself to smile or to look up. Mark when to show a particular slide, etc.
2. Practice! Be sure you know your speech well enough that you can look up from your notes and make eye contact with your audience. Practice for other people and listen to their feedback.
3. Time your speech in advance so that you are sure it is the right length. If necessary, cut or add some material and time yourself again until your speech meets the time requirements. Do not go
over time!
4. Anticipate questions and be sure you are prepared to answer them.
5. Make a list of all materials that you will need so that you are sure you won t forget anything.
6. If you are planning to provide a handout, make a few extras.
7. If you are going to write on a whiteboard or a blackboard, do it before starting your talk.
The Delivery
How you deliver your speech is almost as important as what you say. If you are enthusiastic about your presentation, it is far more likely that your audience will be engaged. Never apologize for
yourself. If you start out by saying that your presentation isn t very good, why would anyone want to listen to it? Everything about how you present yourself will contribute to how well your
presentation is received. Dress professionally. And don t forget to smile!
Here are a few tips about delivery that you might find helpful.
1. Make direct eye contact with members of your audience. Pick a person and speak an entire phrase before shifting your gaze to another person. Don t just scan the audience. Try not to look over
their heads or at the floor. Be sure to look at all parts of the room at some point during the speech so everyone feels included.
2. Speak loudly enough for people to hear and slowly enough for them to follow what you are saying.
3. Do not read your speech directly from your note cards or your paper. Be sure you know your speech well enough to make eye contact with your audience. Similarly, don t read your talk directly off
of transparencies.
4. Avoid using distracting or repetitive hand gestures. Be careful not to wave your manuscript around as you speak.
5. Move around the front of the room if possible. On the other hand, don t pace around so much that it becomes distracting. (If you are speaking at a podium, you may not be able to move.)
6. Keep technical language to a minimum. Explain any new vocabulary carefully and provide a visual aide for people to use as a reference if necessary.
7. Be careful to avoid repetitive space-fillers and slang such as umm , er , you know , etc. If you need to pause to collect your thoughts, it is okay just to be silent for a moment. (You should
ask your practice audiences to monitor this habit and let you know how you did).
8. Leave time at the end of your speech so that the audience can ask questions.
Displaying Your Results
When you create a visual display of your work, it is important to capture and retain the attention of your audience. Entice people to come over and look at your work. Once they are there, make them
want to stay to learn about what you have to tell them. There are a number of different formats you may use in creating your visual display, but the underlying principle is always the same: your work
should be neat, well-organized, informative, and easy to read.
It is unlikely that you will be able to present your entire project on a single poster or display board. So, you will need to decide which are the most important parts to include. Don t try to cram
too much onto the poster. If you do, it may look crowded and be hard to read! The display should summarize your most important points and conclusions and allow the reader to come away with a good
understanding of what you have done.
A good display board will have a catchy title that is easy to read from a distance. Each section of your display should be easily identifiable. You can create posters such as this by using headings
and also by separating parts visually. Titles and headings can be carefully hand-lettered or created using a computer. It is very important to include lots of examples on your display. It speeds up
people s understanding and makes your presentation much more effective. The use of diagrams, charts, and graphs also makes your presentation much more interesting to view. Every diagram or chart
should be clearly labeled. If you include photographs or drawings, be sure to write captions that explain what the reader is looking at.
In order to make your presentation look more appealing, you will probably want to use some color. However, you must be careful that the color does not become distracting. Avoid florescent colors, and
avoid using so many different colors that your display looks like a patch-work quilt. You want your presentation to be eye-catching, but you also want it to look professional.
People should be able to read your work easily, so use a reasonably large font for your text. (14 point is a recommended minimum.) Avoid writing in all-capitals because that is much harder to read
than regular text. It is also a good idea to limit the number of different fonts you use on your display. Too many different fonts can make your poster look disorganized.
Notice how each section on the sample poster is defined by the use of a heading and how the various parts of the presentation are displayed on white rectangles. (Some of the rectangles are blank, but
they would also have text or graphics on them in a real presentation.) Section titles were made with pale green paper mounted on red paper to create a boarder. Color was used in the diagrams to make
them more eye-catching. This poster would be suitable for hanging on a bulletin board.
If you are planning to use a poster, such as this, as a visual aid during an oral presentation, you might consider backing your poster with foam-core board or corrugated cardboard. A strong board
will not flop around while you are trying to show it to your audience. You can also stand a stiff board on an easel or the tray of a classroom blackboard or whiteboard so that your hands will be free
during your talk. If you use a poster as a display during an oral presentation, you will need to make the text visible for your audience. You can create a hand-out or you can make overhead
transparencies of the important parts. If you use overhead transparencies, be sure to use lettering that is large enough to be read at a distance when the text is projected.
If you are preparing your display for a science fair, you will probably want to use a presentation board that can be set up on a table. You can buy a pre-made presentation board at an office supply
or art store or you can create one yourself using foam-core board. With a presentation board, you can often use the space created by the sides of the board by placing a copy of your report or other
objects that you would like people to be able to look at there. In the illustration, a black trapezoid was cut out of foam-core board and placed on the table to make the entire display look more
unified. Although the text is not shown in the various rectangles in this example, you will present your information in spaces such as these.
Don t forget to put your name on your poster or display board. And, don t forget to carefully proof-read your work. There should be no spelling, grammatical or typing mistakes on your project. If
your display is not put together well, it may make people wonder about the quality of the work you did on the rest of your project.
For more information about creating posters for science fair competitions, see
Robert Gerver s book, Writing Math Research Papers, (published by Key Curriculum Press) has an excellent section about doing oral presentations and making posters, complete with many examples.
References Used
American Psychological Association. Electronic reference formats recommended by the American Psychological Association. (2000, August 22). Washington, DC: American Psychological Association.
Retrieved October 6, 2000, from the World Wide Web: http://www.apastyle.org/elecsource.html
Bridgewater State College. (1998, August 5). APA Style: Sample Bibliographic Entries (4th ed). Bridgewater, MA: Clement C. Maxwell Library. Retrieved December 20, 2001, from the World Wide Web:http:/
Crannell, Annalisa. (1994). A Guide to Writing in Mathematics Classes. Franklin & Marshall College. Retrieved January 2, 2002, from the World Wide Web: http://www.fandm.edu/Departments/Mathematics/
Gerver, Robert. 1997. Writing Math Research Papers. Berkeley, CA: Key Curriculum Press.
Moncur, Michael. (1994-2002 ). The Quotations Page. Retrieved April 9, 2002, from the World Wide Web: http://www.quotationspage.com/
Public Speaking -- Be the Best You Can Be. (2002). Landover, Hills, MD: Advanced Public Speaking Institute. Retrieved April 9, 2002, from the World Wide Web: http://www.public-speaking.org/
Recreational Mathematics. (1988) Ottawa, Ontario, Canada: Canadian Mathematical Society. Retrieved April 9, 2002, from the World Wide Web: http://camel.math.ca/Recreation/
Shay, David. (1996). Profession Jokes Mathematicians. Retrieved April 5, 2001, from the World Wide Web: http://www.geocities.com/CapeCanaveral/4661/projoke22.htm
Sieman s Foundation. (2001). Judging Guidelines Poster. Retrieved April 9, 2002, from the World Wide Web: http://www.siemens-foundation.org/science/poster_guidelines.htm,
VanCleave, Janice. (1997). Science Fair Handbook. Discovery.com. Retrieved April 9, 2002, from the World Wide Web: http://school.discovery.com/sciencefaircentral/scifairstudio/handbook/display.html,
Woodward, Mark. (2000). The Mathematical Quotations Server. Furman University. Greenville, SC. Retrieved April 9, 2002, from the World Wide Web: http://math.furman.edu/~mwoodard/mquot.html | {"url":"https://www2.edc.org/makingmath/handbook/Teacher/PresentingYourResearch/PresentingYourResearch.asp","timestamp":"2024-11-01T20:26:37Z","content_type":"text/html","content_length":"87851","record_id":"<urn:uuid:7376dc3a-6491-403f-9e52-95f2b7cca279>","cc-path":"CC-MAIN-2024-46/segments/1730477027552.27/warc/CC-MAIN-20241101184224-20241101214224-00671.warc.gz"} |
Unitary Flow
In mathematics and physics there are some results called no-go theorems, or impossibility theorems. To name just a few: Euler's solution to the problem of the seven bridges of Königsberg, Gödel's
incompleteness theorems, Bell's theorem, Kochen-Specker theorem, Penrose and Hawking's singularity theorems.
Research is an adventurous activity - you can spend years on researching a dead end, or you can stumble by luck upon something worthy without even knowing (for example,
the discovery by Penzias and Wilson
of the
cosmic microwave background radiation
). To avoid spending years looking in the wrong places, researchers use various guiding lines. Impossibility results are some of them, which are by far the most reliable. Other guidelines are
following the trends of the moment (also dictated by the need to publish and receive citations), following the opinions of authorities in the field, reading only what they read etc. I personally
consider misguided the idea of interpreting the results and filtering what you read and research by using the eyes of the authorities, no matter who they are. But it is understandable that they may
seem the best we have, and that anyway the "mainstream" follows them, so if you want to fit in, you have to do the same.
What about the impossibility theorems, aren't they more objective than just fashion trends dictated by authority figures? Of course they are. However, they apply to specific situations, contained in
the hypothesis of the theorems. Moreover, they rely on a mathematical model of reality, and not on reality itself. While I think that the physical world is isomorphic to a mathematical model, this
doesn't mean that it is isomorphic to the models we use.
I will give just a simple example. Remember the problem of the
Seven Bridges of Königsberg
. It was solved negatively by Euler in 1735, and led to graph theory and anticipated the idea of topology. The problem is to walk through the city by crossing each bridge once and only once. Here is
a map, which is of course an idealization:
Euler reduced the problem to an even more idealized one. He denoted the shores and the islands by vertices, and the bridges by edges, and obtained probably the first graph in the history:
Euler was then able to show immediately that there is no way to walk and cross each of the bridges once and only once (without jumping like Mario or swimming in the river, or being teleported!). The
reason is that an even number of edges have to meet at the vertices which are not those where you start or end the trip. But there are no such vertices in the above graph, so all four have to be
starting or ending vertices. But at most two vertices can be used to start and end, so the problem has negative answer.
This illustrates the main point of this article. The problem has a negative answer, but this doesn't mean that in reality the answer is negative too. The mathematical model is an idealization, which
forgets one thing: that the Pregel river has a spring, a source of origin. If we add the spring to the map, we obtain a different problem:
This problem has a simple solution, which is obtained by "going back to the origin":
This is "thinking outside the box", literally, because you have to go outside the original picture box. I came up with this solution years ago, when I was in school and read about Euler's solution.
Of course, it doesn't contradict Euler's theorem, because the resulting graph is different than the one he considered, as we can see below:
Hence, Euler's theorem itself tells us how to solve the problem associated with this graph. The problem is solved by the very theorem which one considers to forbid the existence of a solution.
The main point of this simple example is that even in simple cases we don't actually know the true settings in which we apply the no-go theorems, or we ignore them to idealize the problem. We are
applying the no-go theorems in the dark, so perhaps, rather than being guidelines, they are blocking our access to the real solutions of the real problems. While most researchers try to avoid being
in contradiction with impossibility theorems, maybe it is good to reopen closed cases from time to time. | {"url":"http://www.unitaryflow.com/2014/03/","timestamp":"2024-11-08T07:13:50Z","content_type":"application/xhtml+xml","content_length":"75080","record_id":"<urn:uuid:8272ad68-b8b8-4049-9dfa-2c57b4602822>","cc-path":"CC-MAIN-2024-46/segments/1730477028032.87/warc/CC-MAIN-20241108070606-20241108100606-00852.warc.gz"} |
Determining Your Weight Throughout the Solar System
Gravitation 2
Gravity Throughout the Solar System
Reviewing some background concepts…These will require some thought:
1. Explain the difference between weight and mass as commonly defined in physics (hint: in physics, mass is NOT just the “amount of matter” in an object…think about the influence of mass on motion in
your definition.)
2. Explain the difference between the following three concepts: the acceleration DUE to gravity (9.8m/s/sec), the gravitational FIELD strength (9.8 N/Kg), and gravitational FORCE (a value in
Newtons). It might be helpful to use a specific example.
Some Calculations:
3. Determine your mass in Kg (1 Kg = 2.2 lbs).
4. Calculate the force of gravity between you and earth using Newton’s Universal Law of Gravitation. Show the entire calculation. Use the provided planetary data sheet for all necessary values.
5. Determine how much you would weigh at a distance 3X the earth’s radius above the earth’s surface. (what is your total distance in this situation?). Show the full calculation.
6. The space shuttle often orbits at a distance of about 200 miles above the earth’s surface. Convert this distance into meters (1 mi = 1609 m) and calculate the force of the earth on an 80 Kg
astronaut at this point in space. Some of your best friends believe that astronauts on the shuttle float because “there is no gravity in space”. Does your calculation here support that view? What
percentage of one’s surface weight is present 200 miles above the earth’s surface?
7. Determine how much you would weigh at the surface of the moon. (i.e. what is the force of gravity between you and the moon?). Show the full calculation.
8. Determine how much you would weigh at the surface of Jupiter. Show the full calculation.
9. Determine how much you would weigh at the surface of the sun.
10. Determine the force of gravity between the earth and the moon.
11. Imagine that the sun was to shrink without losing mass (we will discuss why some stars collapse very soon in class). What would happen to the forces of gravity on objects at the sun’s surface?
Explain your reasoning fully. What would happen to the force of gravity between the earth and sun? Explain.
12. If the sun were to shrink to the size of the earth without losing any mass, how much would a 1 kg object weigh at its surface? Show your calculation clearly.
13. If the sun were to shrink to the size of the earth without losing mass, would the force of gravity of the sun acting on the earth and other planets change? Explain.
14. Describe the difference between free fall motion, projectile motion, and orbital motion. What do each of these have in common? | {"url":"https://docest.com/doc/208189/determining-your-weight-throughout-the-solar-system","timestamp":"2024-11-06T18:41:21Z","content_type":"text/html","content_length":"24229","record_id":"<urn:uuid:07fe08e0-0303-4532-b480-84d803273d15>","cc-path":"CC-MAIN-2024-46/segments/1730477027933.5/warc/CC-MAIN-20241106163535-20241106193535-00201.warc.gz"} |
How Many Meters per second Is 363.3 Knots?
363.3 knots in meters per second
How many meters per second in 363.3 knots?
363.3 knots equals 186.898 meters per second
Unit Converter
Conversion formula
The conversion factor from knots to meters per second is 0.514444444444, which means that 1 knot is equal to 0.514444444444 meters per second:
1 kt = 0.514444444444 m/s
To convert 363.3 knots into meters per second we have to multiply 363.3 by the conversion factor in order to get the velocity amount from knots to meters per second. We can also form a simple
proportion to calculate the result:
1 kt → 0.514444444444 m/s
363.3 kt → V[(m/s)]
Solve the above proportion to obtain the velocity V in meters per second:
V[(m/s)] = 363.3 kt × 0.514444444444 m/s
V[(m/s)] = 186.89766666651 m/s
The final result is:
363.3 kt → 186.89766666651 m/s
We conclude that 363.3 knots is equivalent to 186.89766666651 meters per second:
363.3 knots = 186.89766666651 meters per second
Alternative conversion
We can also convert by utilizing the inverse value of the conversion factor. In this case 1 meter per second is equal to 0.005350521586684 × 363.3 knots.
Another way is saying that 363.3 knots is equal to 1 ÷ 0.005350521586684 meters per second.
Approximate result
For practical purposes we can round our final result to an approximate numerical value. We can say that three hundred sixty-three point three knots is approximately one hundred eighty-six point eight
nine eight meters per second:
363.3 kt ≅ 186.898 m/s
An alternative is also that one meter per second is approximately zero point zero zero five times three hundred sixty-three point three knots.
Conversion table
knots to meters per second chart
For quick reference purposes, below is the conversion table you can use to convert from knots to meters per second
knots (kt) meters per second (m/s)
364.3 knots 187.412 meters per second
365.3 knots 187.927 meters per second
366.3 knots 188.441 meters per second
367.3 knots 188.955 meters per second
368.3 knots 189.47 meters per second
369.3 knots 189.984 meters per second
370.3 knots 190.499 meters per second
371.3 knots 191.013 meters per second
372.3 knots 191.528 meters per second
373.3 knots 192.042 meters per second | {"url":"https://convertoctopus.com/363-3-knots-to-meters-per-second","timestamp":"2024-11-04T23:29:14Z","content_type":"text/html","content_length":"34251","record_id":"<urn:uuid:185ad3ed-aab1-4017-9170-dfe8487eea90>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.84/warc/CC-MAIN-20241104225856-20241105015856-00134.warc.gz"} |
Freeee ???????? loooooooooooooool ,,, don't let it free man hadii kele adigaa ku dhib mudan ,, na mean ???
Freeee ???????? loooooooooooooool ,,, don't let it free man hadii kele adigaa ku dhib mudan ,, na mean ???
Why wear such thing?? what happened to the tracksuit bottoms and shorts??!! That’s ultimate comfiness
I love the Green Jumper though..
Like what ?? ,,, coz i don't get your point
May is the language coz i'm not good at English and would be better if you can post come picture samples ,,,,,,,,,
Hi people,
I was walking home on a Friday night after a work meeting and heard this very familiar voice tone. Yep it was a fellow country man speaking on the phone; he was laud, very laud that I could hear him
from across the road. I couldn’t believe my eyes though when I saw him I really had to look at him twice and you won’t guess, he was wearing a bright maroon suite with black shirt and a tie. He was
good looking I must say with a wadaad beard but the COLOUR, I was like WHAAAAAAAAAAAYYYYYYY???? It looked like he was going to a fancy dress party.
I walked past him as he entered the Pizza Hut, Wallahi I stopped outside and had to beat the urge of following him to tell him not to wear these colours as they are inappropriate for Muslim men, but
I didn’t coz I thought he might misunderstand me.
If you were in my position would you actually tell the person about their dress sense or colour of cloths?
And more importantly why do our people where these bright colours?? Its ceeb and?
loooooooooool Rose, loool. I Don't tent bother myself with what people do or wear, but looooool, i can't believe it bothered you so much! loool
Kimiya, loool they were trying to look good for the mag! loool
Was it this type of colour, Rose?
It's not that bad you know. Of course I only stick with navy, black and grey. :cool:
Why do you think it's 'ceeb'? Are you saying it's one of those signs to show that one, err, plays for the other team?
since wen somalis learn to consider the colour of the cloths we wear,,, remember folks 80% of our fellow somalis hardly get something to wear,
Wake up guys!!!!!!!!!!!
Originally posted by NGONGE:
Was it this type of colour, Rose?
NGONG it wasn't that colour. Not Maroon, i got my colours mixed up it was Terracotta, between Orange and red, strange colour for a suit
^ UK ROSE, a better question would ask who on earth makes these suits?
Centurion, guess that means you are definitely not a trendsetter.
Originally posted by Centurion:
Terracota's a lovely colour!
You female, saaxib? This is nothing to do with you thinking it a lovely colour by the way. I'm just surprised that, as a man, you actually know what 'Terracota' is.
Maybe I'm just old.
Rose, I've just checked out the colour you're talking about. I agree with Centurion, it's a nice colour.
Here is a really cool suit (don't care about the colour either). | {"url":"https://www.somaliaonline.com/community/topic/14264-our-fashionable-somali-men/page/2/?tab=comments#comment-221762","timestamp":"2024-11-04T11:31:46Z","content_type":"text/html","content_length":"284306","record_id":"<urn:uuid:ea6cd961-f73c-4969-b429-f2e4fb58bacc>","cc-path":"CC-MAIN-2024-46/segments/1730477027821.39/warc/CC-MAIN-20241104100555-20241104130555-00866.warc.gz"} |
what is the relationship between the dynamical casimir effect and virtual particles?
Since virtual particles are disturbances in a field, and not particles in any sense, as explained here, how is it that true photons arise from them when excited with kinetic energy via the dynamical
casimir effect?
While the effect is described here, I was hoping to get an answer that touches more with the virtual particle's role in the effect, if it is of sigificance.
This post imported from StackExchange Physics at 2014-04-01 13:25 (UCT), posted by SE-user jzn
All interactions at the quantum dynamical level are subject to an analysis with Feynman diagrams, and the Casimir effect is not different in this case. Virtual pairs are internal lines in Feynman
R.L.Jaffe has a arXiv paper that goes into the detail on the Casimir force and corresponding diagrams.
the abstract:
In discussions of the cosmological constant, the Casimir effect is often invoked as decisive evidence that the zero point energies of quantum fields are "real''. On the contrary, Casimir effects
can be formulated and Casimir forces can be computed without reference to zero point energies. They are relativistic, quantum forces between charges and currents. The Casimir force (per unit
area) between parallel plates vanishes as "alpha", the fine structure constant, goes to zero, and the standard result, which appears to be independent of "alpha", corresponds to the "alpha"-->
infinity limit.
He separates the Casimir effect from the vacuum fluctuations effect.From the conclusions:
I have presented an argument that the experimental confirmat ion of the Casimir effect does not establish the reality of zero point fluctuations. Casimir forces can be calculated without
reference to the vacuum and, like any other dynamical effect in QED, vanish as α→0. The vacuum-to-vacuum graphs (See Fig. 1) that define the zero point energy do not enter the calculation of the
Casimir force, which instead only involves graphs with external lines. So the concept of zero point fluctuations is a heuristic and calculational aid in the description of the Casimir effect, but
not a necessity.
This post imported from StackExchange Physics at 2014-04-01 13:25 (UCT), posted by SE-user anna v
Your sea of virtual photons normally come in and out of existence in pairs. But with a mirror moving relativistically, those pairs separate and hence become real photons (producing light). This is
explained in any paper on the effect, so I advise doing so for better enlightenment.
This post imported from StackExchange Physics at 2014-04-01 13:25 (UCT), posted by SE-user Chris Gerig | {"url":"https://www.physicsoverflow.org/11201/relationship-between-dynamical-casimir-virtual-particles","timestamp":"2024-11-07T22:42:03Z","content_type":"text/html","content_length":"130956","record_id":"<urn:uuid:f7d9f9de-baf3-40f5-9a21-f9f6be5df095>","cc-path":"CC-MAIN-2024-46/segments/1730477028017.48/warc/CC-MAIN-20241107212632-20241108002632-00774.warc.gz"} |
Kikkoman Giveaway
This post may contain affiliate links. Read our disclosure policy.
Kikkoman is sharing its special ingredient to help homecooks Kikk-up holiday entertaining this winter – PANKO! A longtime chef’s secret, Kikkoman Panko is not just a crunchy addition to Chinese food,
but every homecook’s secret weapon to create crispy, savory and unique dishes. Today, I have a Kikkoman giveaway.
Broaden your flavor profiles and add texture to your dishes by adding Kikkoman Panko to vegetables, casseroles and stuffing! I used to to make a yummy crispy orange baked french toast. I also have
used it to make a green bean casserole.
From Panko and Kara-age to Teriyaki and Soy Sauce glazes, Kikkoman products help you prepare simple, versatile and delicious meals. Your guests will love. Whether looking for the perfect side to
compliment that turkey, preparing an appetizer that makes you the star of any party, or finding tasty variations of classic family meal staples, Kikkoman has the solution.
Kikkoman has provided two full sized products for me to giveaway to a lucky ready. To enter, tell me how you like to use panko bread crumbs. Or what recipe you would like to try using them. You
have until Sunday December 18 at midnight (MST) to enter.
Kikkoman also has a buy one get one free coupon available right now. If you want to try some recipes right away!
45 Comments
1. LOVE Panko…mostly I use for grilled veggies and all seafood like shrimp, fish, and oysters…nothing like it.
2. I first used panko bread crumbs in an oven-fried fish reicpe I did, and I was amazed! Great product!
3. Love baked fish coated in Panko.
4. I use it for homemade fish sticks and ANYTIME bread crumbs are called for in a recipe! LOVE Panko!
5. I use them on many different things but my fav way is coconut oven baked chicken tenders. Nice and crispy thanks to panko crumbs.
6. I use them to make zucchini fries.
7. Pingback: Crispy Orange Baked French Toast with Caramel Syrup | Real Mom Kitchen
8. I use them to top Mac & Cheese!
9. I’ve heard of them in a few things on recipe shows. I’d love to try them out!
10. I use panko in most recipes calling for bread crumbs. I’m not sure if they make a difference (shhhh; probably shouldn’t be admitting this on a recipe blog!) but it makes me feel all gourmet and
11. Panko crumbs are great!!!!! Our fav way is to pound out chicken breasts, dip in egg and a little water, then dredge in panko crumbs and parmesan cheese. yum.
12. have never tried them but i would try them out with baked chicken.
13. We love them in meatballs or to crust mozzarella sticks!
14. I would make shrimp tempura and crab cakes!
15. I use panko crumbs to make mozzerella sticks!
16. Fish, pork chops, chicken, cheese sticks, I just love the crunchiness
17. My family likes it on Tilapia Filets with Cafe Rio dressing drizzled on top.
18. Would try it on my baked chicken breasts!
19. I would use them as a breading for chicken nuggets.
20. I like to use panko breadcrumbs on pork chops. Taste great after baking in the oven!
21. I love to use Panko breadcrumbs in making my own cheese sticks done in the oven. Thanks for the chance to enter.
22. I use Panko crumbs on my roaster parmesan potatoes, fantastic crunch!
23. I love to use panko to bread fish Chicken or pork as well as shrimp it makes everything I bread taste crispy and light.
24. I want to try it on my oven-fried chicken breasts instead of Bisquick!
25. I like to slice up pork tenderloin and bake it rolled in panko and seasonings ヅ *Thanks* for the giveaway!
26. I made a delicious Jalpeno Popper dip with these last year.
27. we love panko bread crumbs, we use them for so much! My kids favorite is probably chicken nuggets and mozzarella sticks.
28. There’s this Japanese dish called katsu that is amazing and uses panko. You should try it!
29. I like to use panko on top of my baked mac and cheese.
30. homemade chicken strips
31. I would love to use panko to make chicken parmesan in the oven! 🙂
32. I’d like to try it on tilapia
33. I have never used panko but I would love to try.
34. I use the Panko in your recipe for Breaded Chicken with Basil Cream Sauce. It has become a family favorite. Thanks!
35. cheese sticks
36. chicken roll-ups!
37. I coat chicken in ranch dressing then dip in panko crumbs and bake – just like fried chicken but my kids love it cause of the slight taste of ranch! I like to mix it with coconut and put on
chicken tenders. I also tried it in some green bean casserole – YUMMY! May try to use it on my funeral potatoes for Christmas dinner!!
38. cheese sticks, shrimo, chicken, the list is endless!! and i so want to win!! pick me, pick me!! 😉 please and thank you!! 😉
39. That sounds really yummy on cheese sticks!!!! I need to try that out sometime soon. That would be a great appetiser!
40. I’ve used the Panko on fried chicken…..YUM!!!!! It’s nice a crispy, just the way that fried chicken should be!
41. Panko would be great for meatloaf!
42. I use Panko in meatloaf and on fish fillets and eggplant. I love it!!!!
43. I’ve never tried Panko but have been wanting to, so I would make fried, or baked chicken
44. I use panko when I make roasted broccoli. So yummy!
45. I want to make mozzarella sticks! I’ve yet to try panko! | {"url":"https://www.realmomkitchen.com/kikkoman-giveaway-2/","timestamp":"2024-11-04T15:14:48Z","content_type":"text/html","content_length":"436430","record_id":"<urn:uuid:94f3c67d-0e3c-450a-93cf-886fb87795d4>","cc-path":"CC-MAIN-2024-46/segments/1730477027829.31/warc/CC-MAIN-20241104131715-20241104161715-00621.warc.gz"} |
Eitan Adler's thoughts
I've taken a brief break from blogging about my Cormen readings but I decided to write up the answers to chapter 11. Note that the chapters and question numbers may not match up because I'm using an
older edition of the book.
Question 11.1-1:
Suppose that a dynamic set $S$ is represented by a direct address table $T$ of length $m$. Describe a procedure that finds the maximum element of $S$. What is the worst case performance of your
Assuming the addresses are sorted by key: Start at the end of the direct address table and scan downward until a non-empty slot is found. This is the maximum and if not:
1. Initialize $max$ to $-\infty$
2. Start at the first address in the table and scan downward until a used slot is found. If you reach the end goto #5
3. Compare key to $max$. If it is greater assign it to $max$
4. Goto #2
5. Return $max$
The performance of this algorithm is $\Theta(m)$. A slightly smaller bound can be found in the first case of $\Theta(m - max)$
Question 11.1-2:
Describe how to use a bit vector to represent a dynamic set of distinct elements with no satellite data. Dictionary operations should run in $O(1)$ time.
Initialize a bit vector of length $|U|$ to all $0$s. When storing key $k$ set the $k$th bit and when deleting the $k$th bit set it to zero. This is $O(1)$ even in a non-transdichotomous model
though it may be slower.
Question 11.1-3:
Suggest how to implement a direct address table in which the keys of stored elements do not need to be distinct and the elements can have satellite data. All three dictionary operations must take
$O(1)$ time.
Each element in the table should be a pointer to the head of a linked list containing the satellite data. $nul$ can be used for non-existent items.
Question 11.1-4:
We wish to implement a dictionary by using direct addressing on a large array. At the start the array entries may contain garbage, and initializing the entire array is impractical because of its
size. Describe a scheme for implementing a direct address dictionary on the array. Dictionary operations should take $O(1)$ time. Using an additional stack with size proportional to the number of
stored keys is permitted.
On insert the array address is inserted into a stack. The array element is then initialized to the value of the location in the stack.
On search the array element value is to see if it is pointing into the stack. If it is the value of the stack is checked to see if it is pointing back to the array.^[1]
On delete, the array element can be set to a value not pointing the stack but this isn't required. If the element points to the value of the stack, it is simply popped off. If it is pointing to
the middle of the stack, the top element and the key element are swapped and then the pop is performed. In addition the value which the top element was pointing to must be modified to point to
the new location
Question 11.2-1:
Suppose we have use a hash function $h$ to hash $n$ distinct keys into an array $T$ of length $m$. Assuming simple uniform hashing what is the expected number of collisions?
Since each new value is equally likely to hash to any slot we would expect $n/m$ collisions.
Question 11.2-2:
Demonstrate the insertion of the keys: $5, 28, 19, 15, 20, 33, 12, 17, 10$ into a hash table with 9 slots and $h(k) = k \mod{9}$^[2]
hash values
1 28 -> 19 -> 1
6 15 -> 33
Question 11.2-3:
If the keys were stored in sorted order how is the running time for successful searches, unsuccessful searches, insertions, and deletions affected under the assumption of simple uniform hashing?
Successful and unsuccessful searches are largely unaffected although small gains can be achieved if if the search bails out early once the search finds a key later in the sort order than the one
being searched for.
Insertions are the most affected operation. The time is changed from $\Theta(1)$ to $O(n/m)$
Deletions are unaffected. If the list was doubly linked the time remains $O(1)$. If it was singly linked the time remains $O(1 + \alpha)$
Question 11.2-4:
Suggest how storage for elements can be allocated and deallocated within the ash table by linking all unused slots into a free list. Assume one slot can store a flag and either one element or two
pointers. All dictionary operations should run in $O(1)$ expected time.
Initialize all the values to a singly linked free list (flag set to false) with a head and tail pointer. On insert, use the memory pointed to by the head pointer and set the flag to true for the
new element and increment the head pointer by one. On delete, set the flag to false and insert the newly freed memory at the tail of the linked list.
Question 11.2-5:
Show that if $|U| > nm$ with $m$ the number of slots, there is a subset of $U$ of size $n$ consisting of keys that all hash to the same slot, so that the worst case searching time for hashing
with chaining is $\Theta(n)$
Assuming the worst case of $|U|$ keys in the hash tabe assuming the optimial case of simple uniform hashing all m slots will have $|U|/m = n$ items. Removing the assumption of uniform hashing
will allow some chains to become shorter at the expense of other chains becoming longer. There are more items then the number of slots so at least one slot must have at least $n$ items by the
pigeon hole principle.
Question 11.3-1:
Suppose we wish to search a linked list of length $n$, where every element contains a key $k$ along with a hash value $h(k)$. Each key is a long character string. How might we take advantage of
the hash values when searching the list for an element of a given key?
You can use $h(k)$ to create a bloom filter of strings in the linked list. This is an $\Theta(1)$ check to determine if it is possible that a string appears in the linked list.
Additionally, you can create a hash table of pointers to elements in the linked list with that hash value. this way you only check a subset of the linked list. Alternatively, one can keep the
hash of the value stored in the linked list as well and compare the hash of the search value to the hash of each item and only do the long comparison if the hash matches.
Question 11.3-2:
Suppose that a string of length $r$ is hashed into $m$ slots by treating it as a radix-128 number and then using the division method. The number $m$ is easily represented as a 32 bit word but the
string of $r$ character treated as a radix-128 number takes many words. How can we apply the division method to compute the hash of the character string without using more than a constant number
of words outside of the string itself?
Instead of treating the word as a radix-128 number some form of combination could be used. For example you may add the values of each character together modulus 128.
Question 11.3-4:
Consider a hash table of size $m = 1000$ and a corresponding hash function $h(k) = \lfloor m (k A \mod{1})\rfloor$ for $ A = \frac{\sqrt{5} - 1}{2}$ Compute the locations to which the keys 61,
62, 63, 64, 65 are mapped.
key hash
1. This is required because it is possible that the random garbage in the array points to the stack by random chance | {"url":"https://blog.eitanadler.com/2012/05/","timestamp":"2024-11-13T06:33:21Z","content_type":"application/xhtml+xml","content_length":"56866","record_id":"<urn:uuid:ead68d6a-4994-45c1-9c1d-a3386f2056fb>","cc-path":"CC-MAIN-2024-46/segments/1730477028326.66/warc/CC-MAIN-20241113040054-20241113070054-00329.warc.gz"} |
Midpoints and the Pythagorean Theorem
Sometimes an answer seems too easy.
How can Prime Numbers and the Pythagorean Theorem be the missing truth we have all been looking for? A lot of work will need to be done for this to be regarded as true, but here’s an explanation of
why we should do that work.
At the end of a decade’s thought, the solution to how two infinite points can resonate relies on treating them both as prime numbers whose midpoint can be deduced as the result of the Pythagorean
Theorem. Of course we are not looking for the hypotnuese of a triangle, we simply want to know how two values can have a common denominator between them.
In this case it is not necessary to have a real number, only a way to understand how two infinite points derived from assumptions about the Void can interact, or resonate. How these points came to be
in this situation is another story, suffice to say they exist and are in relationship, sharing a matching value, which in this case is “1”+ to equal a Prime number.
So why does the Pythagorean Theorem work for this? Well of course this is not geometry so it is probabley wrong to call it the Pythagorean Theorum, for this we can better call it the Pythagorean
Tone. You see it is the midpoint harmonic of two spheres in resonance, each sphere is the product of a reduction algorithm which has entirely subdivided the space around a central point to account
for it all within 1/1048512, which is the diameter of the last central sphere, and the cycle begins again using that value as “1”.
More than that the starting value of the reduction algorithm was probably more than that many iterations since it began. We are not going to address this issue here. The main thing with this is to
describe the idea that there is a way to reconcile the interaction of two infinite points by squaring two prime numbers and finding their square root to act as the midpoint value of them.
In another post on this site I explain the details of this as components of a Functional Cosmology. In that post I refer to this resonance as “Relationship”, signifying that is the third state of
creation in which separation into individuated parts begins. Although this began with the reducing points interacting with original substrate value (frequency potential) of the Intrusion into the
Void, it has remained within all relationships since then. Yes, all objects, such as molecules and larger are constantly in Resonance with one another. Like interacts with like, point-to-point.
each with a unique connection to the original Awareness.
Was Pythagoras telling us about the Music of the Spheres? | {"url":"https://www.spacimetrics.com/blog/2018/03/25/midpoints-and-the-pythagorean-theorem/","timestamp":"2024-11-09T01:02:48Z","content_type":"application/xhtml+xml","content_length":"80949","record_id":"<urn:uuid:394f7d7a-c189-4fb5-83fa-7deed15ff76b>","cc-path":"CC-MAIN-2024-46/segments/1730477028106.80/warc/CC-MAIN-20241108231327-20241109021327-00273.warc.gz"} |
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Forum Jump
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Trying to find phase and group delay of transfer functions
Trying to find phase and group delay of transfer functions
I have several functions I would like to get the phase and group delay. In python I did the following
from sympy import pi, Symbol, I , atan, im, re, simplify, collect, trigsimp, sign
fr = Symbol('fr', real=True) fo = Symbol('fo', real=True) fn = Symbol('fn', real=True) f = Symbol('f',real=True) q = Symbol('q', real=True) qn = Symbol('qn', real=True)
wr = 2pifr wo = 2pifo wn = 2pifn w = 2pif s = I*w
Lp1 - s domain
H_lp1 = wr/(s+wr)
(re_part,im_part) = H_lp1.expand(complex=True).as_real_imag()
H_lp1_ph = atan(im_part/re_part)
Which returns the correct answer but when I try a more complicated function such as
H_lpn2 = ((s2+wn2)(wo/wn)2)/(s2+s(wo/q)+wo**2)
(re_part,im_part) = H_lpn2.expand(complex=True).as_real_imag()
H_lpn2_ph_expanded = atan(im_part/re_part)
H_lpn2_ph = simplify(H_lpn2_ph_expanded)
It does not return the complete solution. Any ideas? | {"url":"https://ask.sagemath.org/question/29386/trying-to-find-phase-and-group-delay-of-transfer-functions/","timestamp":"2024-11-13T17:38:14Z","content_type":"application/xhtml+xml","content_length":"47675","record_id":"<urn:uuid:104ac07f-ba36-43f0-afa6-a9e80501da3a>","cc-path":"CC-MAIN-2024-46/segments/1730477028387.69/warc/CC-MAIN-20241113171551-20241113201551-00358.warc.gz"} |
math gcd Python - Find Greatest Common Divisor with math.gcd() Function
With Python, we can calculate the greatest common divisor of two numbers with the math gcd() function.
import math
The Python math module has many powerful functions which make performing certain calculations in Python very easy.
One such calculation which is very easy to perform in Python is finding the greatest common divisor (GCD) of two numbers.
We can find the GCD of two numbers easily with the Python math module gcd() function. The math.gcd() function takes two integers and returns the GCD of those two integers.
Below are some examples of how to use math.gcd() in Python to find the GCD of two numbers.
import math
How to Get the Greatest Common Divisor of a List in Python with gcd() Function
To find the GCD of a list of numbers in Python, we need to use the fact that the GCD of a list of numbers will be the max of all pairwise GCDs in a list of numbers.
To get the GCD of a list of integers with Python, we loop over all integers in our list and find the GCD at each iteration in the loop.
Below is an example function in Python which will calculate the GCD of a list of integers using a loop and the math gcd() function.
import math
def gcd_of_list(ints):
gcd = math.gcd(ints[0],ints[1])
for i in range(2,len(ints)):
gcd = math.gcd(gcd,ints[i])
return gcd
Hopefully this article has been useful for you to understand how to use the gcd() math function in Python to find the greatest common divisors of a list of numbers. | {"url":"https://daztech.com/math-gcd-python/","timestamp":"2024-11-07T13:21:32Z","content_type":"text/html","content_length":"241144","record_id":"<urn:uuid:997ae5c0-379f-40d2-a0ba-0bb4726d522d>","cc-path":"CC-MAIN-2024-46/segments/1730477027999.92/warc/CC-MAIN-20241107114930-20241107144930-00386.warc.gz"} |
Exploring Gauge-Higgs Inflation with Extra Dimensions: U(1) Gauge Theory on a Warped Background
(1) Toshiki Kawai, Department of Physics, Hokkaido University, Sapporo 060-0810, Japan (E-mail: t-kawai@higgs3.sci.hokudai.ac.jp);
(2) Yoshiharu Kawamura, Department of Physics, Shinshu University, Matsumoto 390-8621, Japan (E-mail: haru@azusa.shinshu-u.ac.jp).
Table of Links
2 U(1) gauge theory on a warped background
3 Gauge-Higgs inflation on a warped background
4 Conclusions and discussions, Acknowledgements, and References
2 U(1) gauge theory on a warped background
2.1 Randall-Sundrum metric and action integral
The spacetime is assumed to be 5d one with the RS metric given by [8, 9]
2.2 Conjugate boundary conditions
where β is a constant called a twisted phase, the superscript C denotes a 4d charge conjugation, θC is a real number, and the asterisk means the complex conjugation. Then, the covariant derivatives
obey the relations:
2.3 Mass spectrum
Then, the action integral is rewritten as
2.4 Effective potential
Let us derive the effective potential for the Wilson line phase θ(= θ(x)). Taking the standard procedure, a d-dimensional effective potential involving one degree of freedom at the one-loop level is
given b
[3] We introduce both Mψ and cσ′ (y) in a general standpoint, and we will see that cσ′ (y) is forbidden by imposing specific boundary conditions on fields in the next subsection. | {"url":"https://companybrief.tech/exploring-gauge-higgs-inflation-with-extra-dimensions-u1-gauge-theory-on-a-warped-background","timestamp":"2024-11-11T11:50:12Z","content_type":"text/html","content_length":"17277","record_id":"<urn:uuid:ef34ecea-6176-4cbf-be23-9c4db43f1a5c>","cc-path":"CC-MAIN-2024-46/segments/1730477028228.41/warc/CC-MAIN-20241111091854-20241111121854-00000.warc.gz"} |
Data Structures
Good knowledge of common data structures is needed in computer science.
This article describes common data structures, and present cost and benefits.
A List is an abstract data type representing a finite collection of ordered elements.
Typically, a List is allocated dynamically depending on the implementation. For example, Array-based lists allocate elements in contiguous memory, and linked lists allocate elements randomly.
Arrays and Array-based Lists
An Array is a collection of elements identified by an index.
An Array provides constant time access to any element in its list. So reading and writing is O(1).
Typically Arrays are statically allocated in a contiguous space of memory, their size is fixed, and the index is zero-based.
ArrayList and Vector are Array-based lists, providing O(1) access time like Arrays. An additional benefit of these implementations is their size can grow and shrink, and the amortized time for
insertion is O(1).
Linked Lists
A Linked List is a sequence of nodes representing contiguous elements.
Typically, the Linked Lists nodes are dynamically allocated. Pointers are used to link nodes, and special pointers to head (first node) and tail (last node) are provided.
In a singly linked list, each node points to the next in the list. In a doubly linked list, each node points both to the next and the previous one.
Unlike arrays, don’t provide constant time access to an item, but you can add/remove items from head or tail in constant time.
A Stack is a list-based structure that provides restricted access to its elements. The most recent element added is the first to be removed. Hence it has LIFO (Last In, First Out) ordering.
Typically, a stack provides the following operations
• add (or push): add an item to the top of the stack
• remove (or pop): remove and return the item on the top of the stack
• peek (or top): look at the element on top of the stack without removing
• empty: test if the stack is empty
Stacks allow add and remove in O(1) time. Note that, unlike Arrays, Stacks don’t provide constant time access to a random node, but O(N), as in the worst case you need to scan all elements to find
the desired one.
A Queue is a list-based structure that provides restricted access to its elements. The first element added is the first that will be removed, hence it has FIFO (First In, First Out) ordering.
Typically, a queue provides the following operations:
• add (or enqueue): add an item to the tail of the queue
• remove (or dequeue): remove an item from the head of the queue
• peek: look at the item at the head of the queue without removing it
A Queue provides constant time add and remove elements.
A Set is an abstract data type that stores unique values without ordering.
Typically, a set provides the following operations
• add: add the element if not already contained in the set
• contains: test if the element is already contained in the set
• remove: remove one item from the set
• size: get the number of elements
The HashSet is a commonly used implementation of the Set that provides amortized and average time performance O(1) for basic operations.
A Map (or Associative Array) is a data structure mapping a key K to a value V. It is used for fast lookups or searching data. It stores data as a <K, V> pair, where a key is unique
Typically, a map provides the following operations
• put: add the pair <K,V> where the key is unique
• get: return the value V corresponding to the given key K
• remove: remove the pair <K,V> with the given key.
• contains: test if the map contains the given key
The HashMap is a commonly used implementation of the Map that provides amortized constant time performance O(1) for basic operations.
Binary trees
A Tree is a hierarchical tree structure made of nodes. Each node, typically, has a value and a list of references to child nodes. The ancestor of all nodes in a tree is called root.
A Binary Tree is a tree where each node have up to two child nodes.
A Binary Search Tree (BST) is a Binary Tree with a specific property: For each node with a value V, all the left children have values lesser than V, and the right children greater than V.
BST example
BSTs allow fast lookup, addition, and removal of items. For example, a search on a BST with N nodes takes O(log N) in time.
A Heap represents an almost complete binary tree that satisfies the heap property. There are two types of Heaps:
• max heap: for any given node the value of the parent is >= than the value of the child
• min heap: the value of a parent is <= w.r.t. the value of a child
Typically, heaps are used to represent priority queues and are implemented as arrays, where the tree structure is computed based on the indices of data in the array.
Min Heap example
A Graph is a collection of vertices with edges connecting some of the nodes. A vertex is also called a node. A graph is directed when its edges have a specified direction from source to target.
A common way of representing a graph is the adjacency list, where each vertex stores a list of adjacent vertices.
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Arrow Diameter: Is it the Key to Archery Success?
A statistical analysis of the effects of arrow diameter on score was done in order to determine if having a larger arrow diameter would help to increase one’s score. To gather the data four different
skill levels of archers were found and they shot two different arrow diameters, one ideal and one that was one size larger. Once all the data was gathered I used the Central Limit Theorem to conclude
that the mean of my data, x, was normal distributed. Considering the mean of my data was normally distributed, all of the methods described below were able to be used to complete the statistical
analysis. A table of the descriptive statistics can be found in Appendix B. With all of the descriptive statistics completed a null hypothesis and an alternative hypothesis were needed in order to
complete the hypothesis tests. //0:(//,- /Z2 ) = 0 and Ha: /ix> /z2 were used as the parameters for the hypothesis test, with an alpha of 0.05. //, was the average score of the ideal arrow diameter,
whereas /z2 was the average score of the larger diameter arrow. Using these parameters the null hypothesis was rejected for the beginning archer, with a p-value of 0.00903. Yet, the null hypothesis
was unable to be rejected for the other three categories of archers. Once the hypothesis tests were completed, an analysis of variance, or ANOVA, was conducted. This analysis gave a p-value of
0.00294. Therefore, the null hypothesis was rejected, stating that arrow diameter may affect score. It was determined that arrow diameter does seem to affect score, but it does not appear to improve
an archer’s score.
PubMed ID | {"url":"https://scholars.carroll.edu/items/6f9fdcd0-8aa3-47dc-8148-ec7a9e20ba60","timestamp":"2024-11-11T12:59:03Z","content_type":"text/html","content_length":"474374","record_id":"<urn:uuid:48256607-986c-4634-a9dc-f3abdf78085b>","cc-path":"CC-MAIN-2024-46/segments/1730477028230.68/warc/CC-MAIN-20241111123424-20241111153424-00527.warc.gz"} |
The Discrete Fourier Transform: Amplitude scaling between the time and frequency domains
1 Introduction
Consider a real valued discrete time signal
for amplitude
Is there a single-sided frequency domain representation of
The short answer is provided below. More detail and proofs are provided later on.
2 (TLDR) Amplitude scaling in the time and frequency domains
The DFT of the signal given by equation 1, calculated in the conventional way as used in Matlab and Python, is defined as
resulting in a set of other frequency variables is also possible; we’ll stick with the representation in equation 2 as it’s notationally simplest).
• We assume a rectangular window has been used to obtain the
• As we’ll see later on, implementing the DFT via equation 2 causes the complex amplitude entries to be scaled by a factor We will called this re-scaled spectrum
• Rescaling DFT amplitudes is useful where preserving the physical units of the time domain signal, e.g., sound pressure or surface velocity, will be of relevance in the frequency domain analysis.
• The spectrum structure of even or odd. Therefore, the exact scaling calculations are listed separately in the next two sub-sections.
2.2 DFT scaling computations: N is even
where we note that the entries in provided later on.
Evaluating the DFT for even
2.3 DFT scaling computations: N is odd
where we note that the entry in provided later on.
2.4 Amplitude scaling key result
With reference to the cosine signal of equation 1, implementation of equation 3 or 4, as appropriate, will give us
as long as
Therefore, for the scaled spectrum
In other words, the implied physical units of
Note: If the DFT is instead expressed in terms of cyclic frequency (Hz), this result may be written as
for entries other than at 0 Hz and Nyquist.
Or, in words, if we examine the frequency bin of
2.5 Caveats
It is important to note that equation 5 is not, in general, exactly true. In particular, it does not compensate for spectral leakage effects that cause the signal energy within
In fact, it is only strictly true under the following assumptions:
1. Periodicity: The signal 1, 5 to hold for each component).
2. Stationarity: The signal components within
3. Windowing: As noted earlier, a rectangular window has been used.
Despite these caveats, in appropriate settings, and when the number of samples
2.6 Example 1 – Single frequency cosine wave
This section provides an illustrative example where we know, in advance, the values of 1. This ensures that
Imagine that the signal
Let’s choose some specific numbers, as follows:
where we see that
Figure 1 shows a plot of 3).
Figure 1: Upper: The signal 7 plotted in the time domain. Centre: The regular double-sided DFT (via equation 2, with only the lower half of spectrum shown). Lower: The single-sided and scaled DFT
(via equation 3).
Observations from Figure 1:
• Upper: As expected, the time domain signal
• Centre: The regular DFT-computed frequency domain representation reveals a single non-zero complex amplitude, at bin index know from elsewhere that a conjugate copy of this lower half appears in
the upper half of the spectrum, with another non-zero bin entry at
• Lower: The scaled DFT-computed frequency domain representation reveals a single non-zero complex amplitude, at bin index 5 is indeed satisfied. We therefore have an appropriate way to scale our
DFT so its units match those of our time domain signal.
2.7 Example 2 – Signal with three periodic frequency components (+ Matlab code)
We can expand the previous example for a signal containing three frequency components that are periodic in 2, which starts at 0, and what is required in Matlab, which starts at 1).
N = 80; % Total number of samples
A1 = [1,3,8]; % Signal amplitude (time domain)
k1 = [6;10;17]; % Signal frequency (time domain)
nVec = 0:N-1; % Time index vector
kVec_SS = nVec((0:(N/2))+1); % Single-sided frequency index vector
x = A1*cos((2*pi*k1*nVec)./N);% Make a discrete time domain signal
x_dft = fft(x); % Basic DFT of x
x_dft_S = x_dft((0:(N/2))+1); % Single-sided DFT of x (even-length N)
x_dft_S((1:(N/2)-1)+1) = (2/N)*x_dft_S((1:(N/2)-1)+1); % Scaled single-sided DFT of x
x_dft_S(1) = (1/N)*x_dft_S(1); % Deal with DC
x_dft_S(end) = (1/N)*x_dft_S(end); % Deal with Nyquist
stem(kVec_SS, abs(x_dft_S))
xlabel('Frequency bin')
ylabel('DFT magnitude (scaled)')
The result of the code above is shown in Figure 2.
Figure 2: The figure obtained by running the code in Example 2. Three discrete frequency components are clearly present, and their scaled DFT magnitudes match those of the original discrete time
2.8 Example 3 – Cosine wave with linear ramping (i.e., a non-stationary signal)
Three important caveats about the use of DFT scaling were outlined above. In this example, we briefly illustrate the relevance of the second caveat, i.e., stationarity.
which is the product of our original cosine wave (equation 1) and a linearly increasing ramp from 0:1.
The signal defined by equation 8 is interesting, because it seems to have only a single frequency component, but an evolving amplitude. Hence, it is not stationary, and there is no sense in which it
has a single time domain amplitude. We would expect that scaling the complex amplitudes of its DFT in the frequency domain, via equation 3, will provide a similarly ambiguous result.
Figure 3: Upper: The signal 8 plotted in the time domain. Centre: The regular double-sided DFT (via equation 2, with only the lower half of spectrum shown). Lower: The single-sided and scaled DFT
(via equation 3).
Analysis of this signal, following a similar structure to that of Example 1, is illustrated in Figure 3. We observe:
• Upper: The linear ramp has an average value of 0.5.
• Centre: The spectrum no longer has a single frequency component at
• Lower: The scaled DFT has 8, we know that this does not correspond to a steady cosine wave. In real world measurements where we can’t be sure that the time domain signal is steady, care is needed
in interpreting the scaled DFT.
• Key message: Care is needed in how we interpret scaled DFT amplitudes, particularly when our time domain signal varies in amplitude over the course of the analysis window, and/or is not strictly
periodic within the analysis window.
3 (In detail) Amplitude scaling in the time and frequency domains
3.1 Introduction
Having demonstrated the basic calculations required to produce a scaled version of the DFT, such that amplitude units in the time and frequency domains match each other (with caveats), in this
section we will lay out some of the mathematical proofs that underpin this useful result.
3.2 DFT of a cosine signal that is periodic in
Let’s start by evaluating the DFT of the pure cosine signal defined in equation 1, using equation 2 and Euler’s identity, to give
Notice that in cases where either
Notice also that in cases where see here for details), we can re-express equation 9 as a closed form finite geometric series, to give
Equations 9 and 10 are very useful results, which will become clear when we consider what happens for particular values of
3.2.1 The case where
Consider 9, the exponent within
Meanwhile, the exponent within 10, and see that the numerator evaluates to 0 (since
Via a similar argument, all terms of
from which we notice that the amplitude term
As expected, pair of oppositely rotating complex phasors, and the summarised below.
3.2.2 The case where
Consider 9, the exponent in the
Meanwhile, the exponent in the 10, and see that the numerator evaluates to 0 (since
Via a similar argument, all terms of
Similar comments can be made about the meaning of those the preceding sub-section.
3.2.3 Special case where
Consider 2, a few steps of algebra reveal that
The entry at
For the signal 1, which is perfectly periodic in
In the specific case where unit step, scaled by 17 becomes
By inspection we see that the average value of
3.2.4 Special case where
Consider 2, a few steps of algebra reveal that
This is interesting since, as with
If 12 and 15 that
However, if 9 we obtain
3.3 Summary
Pulling together all of the above, for a single frequency cosine signal
At last, these expressions provide the DFT scaling condition given at the start in equation 3, in cases where
In cases where
4 Amplitude scaling for periodic signals with a constant phase shift
What happens to the results in the preceding section if we introduce a phase shift
Following the same analysis, but skipping most of the steps, it is possible to show, via a similar argument to that in equation 9, that
where 9.
Equation 27 reveals a pair of complex-valued spectral amplitudes, which we know via the Hermitian symmetry property of the DFT to be the conjugates of each other. Following a similar analysis to that
above, at
Via the multiplicative property of complex numbers (magnitude of a product is the product of the magnitudes), the magnitude is given by
since same computations as previously.
In other words, whatever the initial phase of the discrete time signal, the DFT amplitude scaling approach can be applied in the same way. However, in general it should be applied to the magnitude of
the (complex valued) spectral amplitudes.
5 Power spectra
To be added.
6 Further reading
[1] Jens Ahrens et al (2020). Tutorial on Scaling of the Discrete Fourier Transform and the Implied Physical Units of the Spectra of Time-Discrete Signals. Audio Engineering Society Convention
e-Brief 600, 2020.
[2] Mathworks: Amplitude Estimation and Zero Padding
[3] Mathworks: Webinar on Understanding Power Spectral Density and the Power Spectrum | {"url":"https://selfnoise.co.uk/resources/signals-and-dft/dft-scaling/","timestamp":"2024-11-05T15:09:22Z","content_type":"text/html","content_length":"142679","record_id":"<urn:uuid:7ece88fc-5f15-437f-aedb-4d3c016a4833>","cc-path":"CC-MAIN-2024-46/segments/1730477027884.62/warc/CC-MAIN-20241105145721-20241105175721-00889.warc.gz"} |
A Visual Guide to Self-Labelling Images
A self-supervised method to generate labels via simultaneous clustering and representation learning
In the past year, several methods for self-supervised learning of image representations have been proposed. A recent trend in the methods is using Contrastive Learning (SimCLR, PIRL, MoCo) which have
given very promising results.
However, as we had seen in our survey on self-supervised learning, there exist many other problem formulations for self-supervised learning. One promising approach is: > Combine clustering and
representation learning together to learn both features and labels simultaneously.
A paper Self-Labelling(SeLa) presented at ICLR 2020 by Asano et al. of the Visual Geometry Group(VGG), University of Oxford has a new take on this approach and achieved the state of the art results
in various benchmarks.
The most interesting part is that we can auto-generate labels for images in some new domain with this method and then use those labels independently with any model architecture and regular supervised
learning methods. Self-Labelling is a very practical idea for industries and domains with scarce labeled data. Let’s understand how it works.
Solving The Chicken and Egg Problem
At a very high level, the Self-Labelling method works as follows:
• Generate the labels and then train a model on these labels
• Generate new labels from the trained model
• Repeat the process
But, how will you generate labels for images in the first place without a trained model? This sounds like the chicken-and-egg problem where if the chicken came first, what did it hatch from and
if the egg came first, who laid the egg?
The solution to the problem is to use a randomly initialized network to bootstrap the first set of image labels. This has been shown to work empirically in the DeepCluster paper.
The authors of DeepCluster used a randomly initialized AlexNet and evaluated it on ImageNet. Since the ImageNet dataset has 1000 classes, if we randomly guessed the classes, we would get an baseline
accuracy of 1/1000 = 0.1%. But, a randomly initialized AlexNet was shown to achieve 12% accuracy on ImageNet. This means that a randomly-initialized network possesses some faint signal in its
Thus, we can use labels obtained from a randomly initialized network to kick start the process which can be refined later.
Self-Labelling Pipeline
Let’s now understand how the self-labelling pipeline works.
As seen in the figure above, we first generate labels for augmented unlabeled images using a randomly initialized model. Then, the Sinkhorn-Knopp algorithm is applied to cluster the unlabeled images
and get a new set of labels. The model is again trained on these new set of labels and optimized with cross-entropy loss. Sinkhorn-Knopp algorithm is run once in a while during the course of training
to optimize and get new set of labels. This process is repeated for a number of epochs and we get the final labels and a trained model.
Step by Step Example
Let’s see how this method is implemented in practice with a step by step example of the whole pipeline from the input data to the output labels:
1. Training Data
First of all, we get N unlabeled images \[I_1, ..., I_N\] and take batches of them from some dataset. In the paper, batches of 256 unlabeled images are prepared from the ImageNet dataset.
2. Data Augmentation
We apply augmentations to the unlabeled images so that the self-labelling function learned is transformation invariant. The paper first randomly crops the image into size 224*224. Then, the image is
converted into grayscale with a probability of 20%. Color Jitter is applied to this image. Finally, the horizontal flip is applied 50% of the time. After the transformations are applied, the image is
normalized with a mean of[0.485, 0.456, 0.406] and a standard deviation of [0.229, 0.224, 0.225].
This can be implemented in PyTorch for some image as:
import torchvision.transforms as transforms
from PIL import Image
im = Image.open('cat.png')
aug = transforms.Compose([
transforms.ColorJitter(0.4, 0.4, 0.4, 0.4),
transforms.Normalize(mean=[0.485, 0.456, 0.406],
std=[0.229, 0.224, 0.225])
aug_im = aug(im)
3. Choosing Number of Clusters(Labels)
We then need to choose the number of clusters(K) we want to group our data in. By default, ImageNet has 1000 classes so we could use 1000 clusters. The number of clusters is dependent on the data and
can be chosen either by using domain knowledge or by comparing the number of clusters against model performance. This is denoted by:
\[ y_1, ..., y_N \in {1, ..., K} \]
The paper experimented with the number of clusters ranging from 1000(1k) to 10,000(10k) and found the ImageNet performance improves till 3000 but slightly degrades when using more clusters than that.
So the papers use 3000 clusters and as a result 3000 classes for the model.
4. Model Architecture
A ConvNet architecture such as AlexNet or ResNet-50 is used as the feature extractor. This network is denoted by \(\color{#885e9c}{\phi(} I \color{#885e9c}{)}\) and maps an image I to feature vector
\(m \in R^D\) with dimension D.
Then, a classification head is used which is simply a single linear layer that converts the feature vectors into class scores. These scores are converted into probabilities using the softmax
\[ p(y=.|x_i) = softmax( \color{#3b6eb5}{h\ o\ \phi(}x_i \color{#3b6eb5}{)} ) \]
5. Initial Random Label Assignment
The above model is initialized with random weights and we do a forward pass through the model to get class predictions for each image in the batch. These predicted classes are assumed as the initial
6. Self Labelling with Optimal Transport
Using these initial labels, we want to find a better distribution of images into clusters. To do that, the paper uses a novel approach quite different than K-means clustering that was used in
DeepCluster. The authors apply the concept of optimal transport from operations research to tackle this problem.
Let’s first understand the optimal transport problem with a simple real-world example:
• Suppose a company has two warehouses A and B and each has 25 laptops in stock. Two shops in the company require 25 laptops each. You need to decide on an optimal way to transport the laptops from
the warehouse to the shops.
• There are multiple possible ways to solve this problem. We could either assign all laptops from warehouse A to shop 1 and all laptops from warehouse B to shop 2. Or we could switch the shops. Or
we could transfer 15 laptops from warehouse A and remaining 10 from warehouse B. The only constraint is that the number of laptops allocated from a warehouse cannot exceed their current limit
i.e. 25.
• But, if we know the distance from each warehouse to the shops, then we can find an optimal allocation with minimal travel. Here, we can see intuitively that the best allocation would be to
deliver all 25 laptops from warehouse B to shop 2 since the distance is less than warehouse A. And we can deliver the 25 laptops from warehouse A to shop 1. Such optimal allocation can be found
out using the Sinkhorn-Knopp algorithm.
Now, that we understand the problem, let’s see how it applies in our case of cluster allocation. The authors have formulated the problem of assigning the unlabeled images into clusters as an optimal
transport problem in this way:
1. Problem:
Generate an optimal matrix Q that allocates N unlabeled images into K clusters.
2. Constraint:
The unlabeled images should be divided equally into the K clusters. This is referred to as the equipartition condition in the paper.
3. Cost Matrix:
The cost of allocating each image to a cluster is given by the model performance when trained using these clusters as the labels. Intuitively, this means the mistake model is making when we
assign an unlabeled image to some cluster. If it is high, then that means our current label assignment is not ideal and so we should change it in the optimization step.
We find the optimal matrix Q using a fast-variant of the Sinkhorn-Knopp algorithm. This algorithm involves a single matrix-vector multiplication and scales linearly with the number of images N. In
the paper, they were able to reach convergence on ImageNet dataset within 2 minutes when using GPU to accelerate the process. For the algorithm and derivation of Sinkhorn-Knopp, please refer to the
Sinkhorn Distances paper. There is also an excellent blogpost by Michiel Stock that explains Optimal Transport ^1.
7. Representation Learning
Since we have updated labels Q, we can now take predictions of the model on the images and compare it to their corresponding cluster labels with a classification cross-entropy loss. The model is
trained for a fixed number of epochs and as the cross-entropy loss decrease, the internal representation learned improves.
\[ E(p|y_1, ..., y_N) = -\frac{1}{N} \sum_{i=1}^{N} logp(y_i \mid x_i) \]
8. Scheduling Cluster Updates
The optimization of labels at step 6 is scheduled to occur at most once an epoch. The authors experimented with not using self-labelling algorithm at all to doing the Sinkhorn-Knopp optimization once
per epoch. The best result was achieved at 80.
This shows that self-labeling is giving us a significant increase in performance compared to no self-labeling (only random-initialization and augmentation).
Label Transfer
The labels obtained for images from self-labelling can be used to train another network from scratch using standard supervised training.
In the paper, they took labels assigned by SeLa with AlexNet and retrained another AlexNet network from scratch with those labels using only 90-epochs to get the same accuracy.
They did another interesting experiment where 3000 labels obtained by applying SeLa to ResNet-50 was used to train AlexNet model from scratch. They got 48.4% accuracy which was higher than 46.5%
accuracy obtained by training AlexNet from scratch directly. This shows how labels can be transferred between architectures.
The authors have published their generated labels for the ImageNet dataset. These can be used to train a supervised model from scratch.
The author have also setup an interactive demo here to look at all the clusters found from ImageNet.
Insights and Results
1. Small Datasets: CIFAR-10/CIFAR-100/SVHN
The paper got state of the art results on CIFAR-10, CIFAR-100 and SVHN datasets beating best previous method AND. An interesting result is very small improvement(+0.8%) on SVHN, which the authors say
is because the difference between supervised baseline of 96.1 and AND’s 93.7 is already small (<3%).
The authors also evaluated it using weighted KNN and an embedding size of 128 and outperformed previous methods by 2%.
2. What happens to equipartition assumption if the dataset is imbalanced?
The paper has an assumption that images are equally distributed over classes. So, to test the impact on the algorithm when it’s trained on unbalanced datasets, the authors prepared three datasets out
of CIFAR-10:
• Full: Original CIFAR-10 dataset with 5000 images per class
• Light Imbalance: Remove 50% of images in the truck class of CIFAR-10
• Heavy Imbalance: Remove 10% of first class, 20% of second class and so on from CIFAR-10
When evaluated using linear probing and kNN classification, SK(Sinkhorn-Knopp) method beat K-means on all three conditions. In light imbalance, no method was affected much. For heavy imbalance, all
methods dropped in performance but the performance decrease was lower for self-supervised methods using k-means and self-labelling than supervised ones. The self-labelling method beat even supervised
method on CIFAR-100. Thus, this method is robust and can be applied for an imbalanced dataset as well.
Code Implementation
The official implementation of Self-Labelling in PyTorch by the paper authors is available here. They also provide pretrained weights for AlexNet and Resnet-50.
BibTeX citation:
author = {Chaudhary, Amit},
title = {A {Visual} {Guide} to {Self-Labelling} {Images}},
date = {2020-04-10},
url = {https://amitness.com/posts/self-labelling.html},
langid = {en}
For attribution, please cite this work as:
Chaudhary, Amit. 2020.
“A Visual Guide to Self-Labelling Images.”
April 10, 2020. | {"url":"https://amitness.com/posts/self-labelling","timestamp":"2024-11-07T09:10:39Z","content_type":"application/xhtml+xml","content_length":"53134","record_id":"<urn:uuid:7cc39e6d-2307-419c-babe-0a7a2a723296>","cc-path":"CC-MAIN-2024-46/segments/1730477027987.79/warc/CC-MAIN-20241107083707-20241107113707-00823.warc.gz"} |
Volume of a Cone (Formulas & Examples) | Free Lesson
Volume of a Cone Lesson
Cone Volume Formula
The formula for volume of a cone is given as:
V = ^1⁄[3]πhr^2
Where h is the height and r is the radius.
The Formula for the Volume of a Cone
Volume of a Cone Example Problem
A cone-shaped rock formation is measured to have a height of 4 meters and a radius of 2 meters. What is the volume of the rock formation in cubic meters?
1. Let's plug the given dimensions into the volume formula.
2. V = ^1⁄[3]πhr^2
3. V = ^1⁄[3]π(4)(2^2) = ^16⁄[3]π
4. The rock formation's volume is ^16⁄[3]π cubic meters. | {"url":"https://www.voovers.com/geometry/volume-of-a-cone/","timestamp":"2024-11-07T04:13:39Z","content_type":"text/html","content_length":"196081","record_id":"<urn:uuid:7b933ade-df56-4c69-aaef-3d4fa10dc854>","cc-path":"CC-MAIN-2024-46/segments/1730477027951.86/warc/CC-MAIN-20241107021136-20241107051136-00369.warc.gz"} |
Benford's Law and the 2020 Election
Before getting into the topic at hand, let me remind the forum about it's policy on discussing the 2020 election. As a remind,
rule 19
states the following:
Quote: Rule 19
In an effort to keep the focus of the forum on gambling, Vegas, and math, comments of a political, racial, religious, sexual, or otherwise controversial nature are not allowed. We recommend
taking such discussion elsewhere (Added 8/13/19).
However, since there is such significant betting on the election, that we do allow discussion of that. As a general rule, we allow mathematical and scientific discussion of anything, to include the
election and the coronavirus. Such discussion must be of an academic nature. A good rule of thumb if a post about a sensitive topic is allowable if it does not betray the personal opinion of the
With that out of the way, there is an accusation out there that the Michigan election results were faked as evidenced by allegedly not being in compliance with
Benford's Law
. Briefly, that suggests that the digits of numbers of a random nature tend to follow certain distributions. A good way to catch falsified numbers, for example on an income tax return, is if the
digits do not follow expected distributions. Human beings are notoriously bad at randomizing without the aid of computers.
What got me interested in the topic was
this post
by Evenbob at DT, alleging the vote counts by county in Michigan were he says in part, "Applied to the election in MI, Biden’s vote numbers in do not match Benford’s law at a 99.999% level." I have
surfed around a bit and the best argument I have found about this is
There is Undeniable Mathematical Evidence the Election is Being Stolen
at The Red Elephants. This articles makes a number of points, just one of them about Benford's Law. I will not endeavor to argue the other points here.
First, here is a list of the county by county election results in Michigan, which allegedly has the fake vote totals. At this time (10:47 AM Nov 9, 2020) 99% of the vote is in, so these totals can be
expected to change slightly.
County Biden Trump Total
Alcona County 2,142 4,848 6,990
Alger County 2,053 3,014 5,067
Allegan County 24,447 41,381 65,828
Alpena County 6,000 10,686 16,686
Antrim County 7,289 9,783 17,072
Arenac County 2,773 5,928 8,701
Baraga County 1,475 2,512 3,987
Barry County 11,804 23,473 35,277
Bay County 21,718 30,919 52,637
Benzie County 5,480 6,600 12,080
Berrien County 37,438 43,518 80,956
Branch County 6,161 14,066 20,227
Calhoun County 28,417 35,900 64,317
Cass County 9,122 16,686 25,808
Charlevoix County 6,939 9,841 16,780
Cheboygan County 5,435 10,171 15,606
Chippewa County 6,651 10,682 17,333
Clare County 5,199 10,861 16,060
Clinton County 21,963 25,095 47,058
Crawford County 2,612 4,955 7,567
Delta County 7,605 13,206 20,811
Dickinson County 4,569 8,469 13,038
Eaton County 31,297 31,797 63,094
Emmet County 9,662 12,135 21,797
Genesee County 120,082 99,199 219,281
Gladwin County 4,524 9,893 14,417
Gogebic County 3,573 4,600 8,173
Grand Traverse County 28,682 30,502 59,184
Gratiot County 6,693 12,104 18,797
Hillsdale County 5,883 17,037 22,920
Houghton County 7,755 10,380 18,135
Huron County 5,349 11,949 17,298
Ingham County 94,221 47,640 141,861
Ionia County 10,899 20,655 31,554
Iosco County 5,371 9,760 15,131
Iron County 2,493 4,216 6,709
Isabella County 14,072 14,815 28,887
Jackson County 32,004 47,381 79,385
Kalamazoo County 83,674 56,823 140,497
Kalkaska County 3,003 7,436 10,439
Kent County 186,753 165,318 352,071
Keweenaw County 672 862 1,534
Lake County 2,288 3,946 6,234
Lapeer County 16,368 35,480 51,848
Leelanau County 8,793 7,915 16,708
Lenawee County 20,916 31,539 52,455
Livingston County 48,218 76,980 125,198
Luce County 842 2,109 2,951
Mackinac County 2,589 4,258 6,847
Macomb County 225,561 264,535 490,096
Manistee County 6,107 8,321 14,428
Marquette County 20,465 16,287 36,752
Mason County 6,802 10,207 17,009
Mecosta County 7,373 13,265 20,638
Menominee County 4,315 8,117 12,432
Midland County 20,493 27,675 48,168
Missaukee County 1,967 6,648 8,615
Monroe County 32,975 52,710 85,685
Montcalm County 9,703 21,815 31,518
Montmorency County 1,628 4,171 5,799
Muskegon County 45,508 44,544 90,052
Newaygo County 7,874 18,864 26,738
Oakland County 433,982 325,916 759,898
Oceana County 4,944 8,892 13,836
Ogemaw County 3,475 8,253 11,728
Ontonagon County 1,391 2,358 3,749
Osceola County 3,214 8,928 12,142
Oscoda County 1,342 3,466 4,808
Otsego County 4,743 9,779 14,522
Ottawa County 64,566 100,511 165,077
Presque Isle County 2,912 5,343 8,255
Roscommon County 5,166 9,670 14,836
Saginaw County 51,068 50,784 101,852
St. Clair County 31,363 59,184 90,547
St. Joseph County 9,262 18,128 27,390
Sanilac County 5,966 16,194 22,160
Schoolcraft County 1,589 3,090 4,679
Shiawassee County 15,371 23,154 38,525
Tuscola County 8,713 20,310 29,023
Van Buren County 16,800 21,591 38,391
Washtenaw County 157,130 56,241 213,371
Wayne County 587,074 264,149 851,223
Wexford County 5,838 12,102 17,940
Michigan presidential results
at politico.com.
Next, here are frequency figures of the first and second digits of, shall we say, random large numbers.
Digit First digit expectations Second digit expectations
0 0.00% 11.97%
1 30.10% 11.39%
2 17.61% 10.88%
3 12.49% 10.43%
4 9.69% 10.03%
5 7.92% 9.67%
6 6.69% 9.34%
7 5.80% 9.04%
8 5.12% 8.76%
9 4.58% 8.50%
Total 100.00% 100.00%
The next three tables shall show the first digit frequency for total votes, Biden votes, and Trump votes, as well as the expected total by digit according to Benford's Law. I shall also state the chi
square statistic and p value. The p value is the probability that random results would be less skewed that those observed.
Before I continue, an argument could be made that a total of 83 counties is not enough to apply the chi-squared test. This argument has some merit. The rule of thumb is there should be an expected
total of each digit of at least 5. In this case, the expected total of nines is only 3.80. For the limited data, I can't think of a better test, but am all ears to ideas.
That said, here is the table for the first digit of total votes.
Digit Count Expectations
1 30 24.99
2 14 14.62
3 9 10.37
4 5 8.04
5 6 6.57
6 7 5.56
7 3 4.81
8 7 4.25
9 2 3.80
Total 83 83.00
chi-squred statitistic = 6.110748
p value = 0.634828
Here is the table for the first digit of Biden votes.
Digit Count Expectations
1 15 24.99
2 17 14.62
3 9 10.37
4 8 8.04
5 11 6.57
6 9 5.56
7 5 4.81
8 4 4.25
9 5 3.80
Total 83 83.00
chi-squred statitistic = 10.080136
p value = 0.259446
Here is the table for the first digit of Trump votes.
Digit Count Expectations
1 22 24.99
2 13 14.62
3 11 10.37
4 11 8.04
5 7 6.57
6 2 5.56
7 3 4.81
8 7 4.25
9 7 3.80
Total 83 83.00
chi-squred statitistic = 9.134425
p value = 0.331083
All three p values look very normal. So, how about the second digits.
Here is the table for the second digit of total votes.
Digit Count Expectations
0 9 9.93
1 8 9.45
2 10 9.03
3 3 8.66
4 7 8.33
5 11 8.02
6 9 7.75
7 11 7.50
8 8 7.27
9 7 7.05
Total 83 83.00
chi-squred statitistic = 7.33793
p value = 0.60198
Here is the table for the second digit of Biden votes.
Digit Count Expectations
0 7 9.93
1 12 9.45
2 8 9.03
3 8 8.66
4 10 8.33
5 8 8.02
6 8 7.75
7 8 7.50
8 9 7.27
9 5 7.05
Total 83 83.00
chi-squred statitistic = 3.11012
p value = 0.95977
Here is the table for the second digit of Trump votes.
chi-squred statitistic = 3.11012
p value = 0.95977
Digit Count Expectations
0 14 9.93
1 9 9.45
2 8 9.03
3 8 8.66
4 6 8.33
5 4 8.02
6 14 7.75
7 7 7.50
8 6 7.27
9 7 7.05
Total 83 83.00
chi-squred statitistic = 9.81754
p value = 0.36546
Bottom line is these election results look perfectly compliant with Benford's Law to me.
Of course, I welcome all comments and arguments to the contrary, so long as they are mathematical in nature and don't betray personal political opinions.
"For with much wisdom comes much sorrow." -- Ecclesiastes 1:18 (NIV)
This guy seems legit and he found
significant deviations from Benfords
Law in the election results for Biden.
"It's not called gambling if the math is on your side."
I’ve always encountered Benford’s law when there was a power law driving generation of the numbers. What’s the corresponding principle that would make the law applicable to election results?
The race is not always to the swift, nor the battle to the strong; but that is the way to bet.
• Threads: 300
• Posts: 11825
Joined: Dec 22, 2009
To apply Benfords law to this election would it not make sense to first examine results in other elections?
If I remember correctly Mondale versus Reagan 84 resulted in defeat 49:1 statewide. How does Benfords law apply there.
I'm not of the mathematicians club so forgive me if that question is a stupid one
For Whom the bus tolls; The bus tolls for thee
Quote: darkoz
To apply Benfords law to this election would it not make sense to first examine results in other elections?
If I remember correctly Mondale versus Reagan 84 resulted in defeat 49:1 statewide. How does Benfords law apply there.
I'm not of the mathematicians club so forgive me if that question is a stupid one
I think it's a mathematical formula to determine if numbers were randomly generated as they would in counting vote totals in a county
A human input a vote total for all the counties trying to use random numbers
When somebody doesn't believe me, I could care less. Some get totally bent out of shape when not believed. Weird. I believe very little on all forums
Quote: EvenBob
This guy seems legit and he found
significant deviations from Benfords
Law in the election results for Biden.
Go to the 20 minute mark. Look at the rows for 4 and 5. The results are skewed because Biden has too many 4s and too few 5s. Now recall that this is a first digit test. Ask yourself: is it GOOD to
replace 5s with 4s at the start of numbers when you're cheating and trying to win?
"So as the clock ticked and the day passed, opportunity met preparation, and luck happened." - Maurice Clarett
Quote: rdw4potus
Go to the 20 minute mark. Look at the rows for 4 and 5. The results are skewed because Biden has too many 4s and too few 5s. Now recall that this is a first digit test. Ask yourself: is it GOOD
to replace 5s with 4s at the start of numbers when you're cheating and trying to win?
Is this in a foreign language? I have
zero math training.
"It's not called gambling if the math is on your side."
Quote: EvenBob
Is this in a foreign language? I have
zero math training.
If you don't
realize that 4<5
then how can
you say that
seems legit?
Quote: sabre
If you don't
realize that 4<5
then how can
you say that
seems legit?
From the comments section of the video:
Mathematician here. I'm afraid you've made a big mistake in this analysis by not first plotting a histogram of the magnitudes of the dataset. Looking at it, it seems this is very restricted. Almost
all numbers come in at 4 or 5 digits. As a result, Benford's law would not be expected to fit well.
This problem then worsens as you look at specific areas, and renders your chi squared tests invalid.
In order to properly test against an expectation of Benford you need data which ranges nicely over several orders of magnitude.
Answer by the video author:
12 hours ago
Hi, and many thanks for a thought-provoking comment. I am aware of these applicability limitations of the Benford's law. However, there are two redeeming counterarguments one can make here. First,
all of the tests have at least 30 (and generally much more) observations for all digits, basically a textbook case for Chi-squared applicability. Some alternative tests like Kolmogorov-Smirnov or
Kuiper would be most relevant in case of smaller datasets (Chi-squared has higher power in larger samples). Second, as for orders of magnitude, the data does span seven, with most coming in 3 to 5
digits (3 is actually more frequent than 5). The more important question is whether you would have expected such data to abide by Benford's law under the null. This can be evidenced by Monte Carlo
simulations and past elections. I plan to do a second video on the topic later in the week, addressing most notable suggestions and comments, including yours (will certainly do histograms, Monte
Carlo, past elections, second-digit tests, and, if time allows, alternative goodness-of-fit tests).
"It's not called gambling if the math is on your side."
Just playing devils advocate for math reasons
Aren't there ways of cheating to avoid being caught by Benfords law
Such as raising each county of Bidens total by the same exact amount.
It still generates a random number because you are adding the exact same number of votes to an already random numbers keeping them looking random.
When somebody doesn't believe me, I could care less. Some get totally bent out of shape when not believed. Weird. I believe very little on all forums
Quote: EvenBob
This guy seems legit and he found
significant deviations from Benfords
Law in the election results for Biden.
That is a reasonable argument he makes and the guy does know how to perform statistical tests.
His smoking gun at the end of the video is Biden votes in 755 voting jurisdictions in what he calls Swing States. Here is his table.
Number Actual Expected
1 205.00 227.28
2 138.00 132.95
3 94.00 94.33
4 102.00 73.17
5 44.00 59.78
6 54.00 50.54
7 43.00 43.78
8 40.00 38.62
9 35.00 34.55
Total 755.00 755.00
I did the math and he is right, the p value is 1.97%. There is a shortage of Biden vote totals that begin with 1 or 5 and a surplus that begin with a 4. However, the results of results this skewed or
more is close to 2%. My response is 2% is not that low. You can slice and dice the data all kinds of ways and it won't be hard to find some test that has a value under 2%.
To make any accusations of voter fraud there should be evidence beyond a reasonable doubt and 2% is definitely more than enough to have reasonable doubt. At least to me.
Second, I don't necessarily buy that voting district voting populations make for a good Benford test. For example 21 out of Michigan's 83 counties have a voting total in the range of 12,080 to
18,135. That is 25.3% in that tight range. I suspect that given typical rural county sizes and population densities that there are a lot of counties that frankly look similar to each other, violating
the assumption of independence in Benford's test.
To be more convinced, I'd like to see the same test applied to the same states but different years.
In conclusion, decent video and I don't dispute the math. However, I just don't agree that the results he presents look very fishy.
"For with much wisdom comes much sorrow." -- Ecclesiastes 1:18 (NIV)
If total ballots and total trump look normal, that's a pretty strong indication that the Biden count is real but odd. Manipulating the Biden vote requires manipulating the total ballots and/or the
trump vote.
"So as the clock ticked and the day passed, opportunity met preparation, and luck happened." - Maurice Clarett
Quote: rdw4potus
If total ballots and total trump look normal, that's a pretty strong indication that the Biden count is real but odd. Manipulating the Biden vote requires manipulating the total ballots and/or
the trump vote.
Can you give some numbers to back this up, please.
"For with much wisdom comes much sorrow." -- Ecclesiastes 1:18 (NIV)
Can statistical findings based on Benford's Law be used as court evidence ? I mean any court case not specifically to US Election 2020 case.
Quote: ssho88
Can statistical findings based on Benford's Law be used as court evidence ? I mean any court case not specifically to US Election 2020 case.
It is admissible at least in cases of showing fraud in a company’s financial statements.
The race is not always to the swift, nor the battle to the strong; but that is the way to bet.
It’s still not clear to me that Benford’s law should apply to county level election results. But it does pop up in unexpected places. Would love to see past elections analyzed to see how they lined
The race is not always to the swift, nor the battle to the strong; but that is the way to bet.
Census.gov has a spreadsheet of
County Population Totals: 2010-2019
. I previously speculated county populations may not adhere to Benford's Law well because there are many counties with populations between 10,000 and 20,000, at least in Michigan.
To explore that further, I downloaded the entire USA spreadsheet and applied Benford's Law to the first and second digits of the populations of all 3,142 counties. The following table shows the
number of counties by the leading digit in the population for 2019. The expected column shows the expected total per Benford's Law.
First Digit Total Expected
1 954 945.84
2 590 553.28
3 371 392.56
4 307 304.49
5 230 248.79
6 205 210.35
7 163 182.21
8 175 160.72
9 147 143.77
Total 3142 3142.00
The result of a chi-squared test has a p value of 37.42%. So that looks very normal.
The next table does the same thing, but with the second digit in the population figures.
Second Digit Total Expected
0 377 376.03
1 328 357.84
2 355 341.92
3 322 327.80
4 307 315.17
5 302 303.76
6 324 293.38
7 284 283.89
8 274 275.15
9 269 267.06
Total 3142 3142.00
A chi-squared test of this table has a p value of 68.59%. So again, it conforms with Benford very nicely.
"For with much wisdom comes much sorrow." -- Ecclesiastes 1:18 (NIV)
I found this paper on Benford’s law as applied to elections. I haven’t had a chance to read more than the abstract so far. But wanted to link it here in case others are interested. It’s from 2017 so
shouldn’t have a politically motivated conclusion as applied to current election.
The race is not always to the swift, nor the battle to the strong; but that is the way to bet.
Quote: unJon
I found this paper on Benford’s law as applied to elections. I haven’t had a chance to read more than the abstract so far. But wanted to link it here in case others are interested. It’s from 2017
so shouldn’t have a politically motivated conclusion as applied to current election.
Here is a more
direct link
. I skimmed the paper. It basically uses Benford to analyze some elections in the Ukraine that were suspected of being fixed. I suspect similar academic papers will be written about this election.
So far, I have yet to see anything in 2020 that fails a Benford test beyond a reasonable doubt.
"For with much wisdom comes much sorrow." -- Ecclesiastes 1:18 (NIV)
I'm thinking about this more today. How does Benford work with numbers that are derived from each other? Say I start with county totals. They're random. If I split those county totals into two
subsets, is it appropriate to expect each of those sub-sets to also conform to the original county-level expected digit distribution? I don't think it is. If 30% of the original set starts with a 1,
I think it's much harder for 30% of each subset to also start with a 1.
"So as the clock ticked and the day passed, opportunity met preparation, and luck happened." - Maurice Clarett
Here's a guy who does Benford's Law for
a living and he found 'massive fraud' in
GA. You can skip the first half where he
mostly explains how this works.
"It's not called gambling if the math is on your side."
Quote: EvenBob
Here's a guy who does Benford's Law for
a living and he found 'massive fraud' in
GA. You can skip the first half where he
mostly explains how this works.
This guy just does an eyeball test of the results with statements like, "You can just tell there are a lot of anomalies in here." So I had to do a chi-squared test myself on his table of the first
digit for Georgia. Here are the same tables he has at the 9:30 point.
This first table is the first digit of the Biden vote total in Georgia's 159 counties.
Digit Count Expectations
1 47 47.86
2 40 28.00
3 16 19.87
4 21 15.41
5 7 12.59
6 9 10.64
7 9 9.22
8 5 8.13
9 5 7.28
Total 159 159.00
Here are the results of a chi-squared test:
chi-squred statitistic = 12.600898
p value = 0.126339
That p value is 1.14 standard deviations south of an 0.5 average.
Next, here is the same table for Trump votes.
Digit Count Expectations
1 43 47.86
2 36 28.00
3 12 19.87
4 15 15.41
5 9 12.59
6 17 10.64
7 9 9.22
8 9 8.13
9 9 7.28
Total 159 159.00
chi-squred statitistic = 11.230501
p value = 0.188978
That p value is 0.88 standard deviations south of an 0.5 average.
With p values of 12.6% and 18.9% for the two tables, this is a little more skewed than expectations, but nothing that screams fraud. Show me something that is more than three standard deviations from
expectations and I'll start to take fraud accusations seriously.
"For with much wisdom comes much sorrow." -- Ecclesiastes 1:18 (NIV)
• Threads: 216
• Posts: 12635
Joined: Feb 28, 2010
What things might affect distribution besides fraud? For instance, parties gerrymandering their district certainly creates artificial population distribution of the electorate.
Sanitized for Your Protection
Quote: Wizard
and I'll start to take fraud accusations seriously.
This is why statistical math is such a
joke. I have a friend in his 80's who
was a math teacher for 55 years and
still makes money tutoring on the
net. He wants nothing to do with
stat or probability math because
he says you can manipulate it to
say anything you want it to say.
Four people will come up with
four different conclusions using
the same input data. To him this
isn't real math, it's voodoo. I
understand now what he's talking
about. It's joke math.. Like the math
people who tell me you can't beat
roulette. I just smile and answer
'whatever you say' and go out and
beat it every time.
"It's not called gambling if the math is on your side."
Quote: EvenBob
This is why statistical math is such a
I, too, throw tantrums when the stats don’t reaffirm my prior beliefs.
Ding Dong the Witch is Dead
The sheer scale of the purported voter fraud is mind-boggling. It would take thousands of people in hundreds of county Boards of Election across several states. Scores of republicans, including at
least one republican secretary of state, all involved in a coordinated effort to defeat Donald Trump! And some people really think that's more likely than the less popular candidate losing a free and
fair election.
"So as the clock ticked and the day passed, opportunity met preparation, and luck happened." - Maurice Clarett
People interpret statistics however they want, not manipulate. The nation would be a lot less dumb if our educational system put students on a statistics and probability path rather than calculus
path by default.
Quote: mcallister3200
People interpret statistics however they want, not manipulate. The nation would be a lot less dumb if our educational system put students on a statistics and probability path rather than calculus
path by default.
Mark Twain said 'There are three kinds of lies:
lies, damned lies, and statistics.' He was
mainly talking about political stats. I can
guarantee that if I post 10 different sources
for voter fraud using Benford, all the math
people here will say nope, their math is
wrong, nothing to see here. So there is
no point in even discussing it.
Wayne Allen Root, famous Vegas oddsmaker,
says the fix was in and a few insiders made
millions by betting Biden at the right minute.
Just before Fox called AZ for Biden, Trump's
odds of winning with bookmakers went all
the way to 8 to 1, and right then millions
worldwide was bet on Biden. After Fox
called AZ the odds went back to even
money immediately. Root says a few very
rich insiders knew the election was fixed
and took full advantage of it.
Last edited by: EvenBob on Nov 11, 2020
"It's not called gambling if the math is on your side."
Quote: EvenBob
Mark Twain said 'There are three kinds of lies:
lies, damned lies, and statistics.' He was
mainly talking about political stats. I can
guarantee that if I post 10 different sources
for voter fraud using Benford, all the math
people here will say nope, their math is
wrong, nothing to see here. So there is
no point in even discussing it.
Wayne Allen Root, famous Vegas oddsmaker,
says the fix was in and a few insiders made
millions by betting Biden at the right minute.
Just before Fox called AZ for Biden, Trump's
odds of winning with bookmakers went all
the way to 8 to 1, and right then millions
worldwide was bet on Biden. After Fox
called AZ the odds went back to even
money immediately. Root says a few very
rich insiders knew the election was fixed
and took full advantage of it.
So now you're saying Fox is in on the fix? They were the only news organization to call AZ early, so I guess that's really the only way your story could possibly work. Wow, that's a very odd
"So as the clock ticked and the day passed, opportunity met preparation, and luck happened." - Maurice Clarett
Quote: EvenBob
Mark Twain said 'There are three kinds of lies:
lies, damned lies, and statistics.' He was
mainly talking about political stats. I can
guarantee that if I post 10 different sources
for voter fraud using Benford, all the math
people here will say nope, their math is
wrong, nothing to see here. So there is
no point in even discussing it.
Wayne Allen Root, famous Vegas oddsmaker,
says the fix was in and a few insiders made
millions by betting Biden at the right minute.
Just before Fox called AZ for Biden, Trump's
odds of winning with bookmakers went all
the way to 8 to 1, and right then millions
worldwide was bet on Biden. After Fox
called AZ the odds went back to even
money immediately. Root says a few very
rich insiders knew the election was fixed
and took full advantage of it.
I had Biden at -130 bet the day before election. Election night when Florida was going Trump I tried to double down on Biden at +400. Couldn’t get it down though. There was a crazy swing in the
market that I thought was an overreaction. It was.
The race is not always to the swift, nor the battle to the strong; but that is the way to bet.
Quote: EvenBob
This is why statistical math is such a
joke. I have a friend in his 80's who
was a math teacher for 55 years and
still makes money tutoring on the
net. He wants nothing to do with
stat or probability math because
he says you can manipulate it to
say anything you want it to say.
One of my favorite books is How To Lie With Statistics. It is on one of my bookshelves.
At my age, a "Life In Prison" sentence is not much of a deterrent.
I just ran across a new video on Benford's Law and the 2020 Election by one of my favorite YouTubers, Matt Parker. He mainly looks at applying Benford's Law to the Chicago election results.
"For with much wisdom comes much sorrow." -- Ecclesiastes 1:18 (NIV)
I've read about 5 articles from
people who are experts in the
field and they all say Benford's
Law used in elections is useless
because the "spread of orders of
magnitude" of the numbers
involved are not large enough.
It has to do with precinct size.
Precincts don’t have that much
size variation in them.
So that's that. Never mind.
"It's not called gambling if the math is on your side."
Quote: Wizard
I just ran across a new video on Benford's Law and the 2020 Election by one of my favorite YouTubers, Matt Parker. He mainly looks at applying Benford's Law to the Chicago election results.
I just watched this as was going to post a link. Matt makes math fun and entertaining.
“Man Babes” #AxelFabulous
Quote: EvenBob
I've read about 5 articles from
people who are experts in the
field and they all say Benford's
Law used in elections is useless
because the "spread of orders of
magnitude" of the numbers
involved are not large enough.
It has to do with precinct size.
Precincts don’t have that much
size variation in them.
So that's that. Never mind.
That is Matt's point in the video I just posted.
In my opinion a good place to start is to do a Benford test on the total population in each jurisdiction or the total voting population. Doing that in the various counties in Michigan or the combined
United States seems to pass a Benford test. These are the only two regions I've done the test on. If that seems to indicate a natural spread in population numbers, it's fair to consider a Benford
test on vote totals.
This is historic, I think it's the first time you've changed your position on something.
"For with much wisdom comes much sorrow." -- Ecclesiastes 1:18 (NIV)
Quote: Wizard
This is historic, I think it's the first time you've changed your position on something.
I never had a position to change.
I had never heard of this stupid
'law' before I posted about it. I
just assumed it was true because
it had the word 'law' attached to
it. I should have realized that in
math 'law' is just a meaningless
"It's not called gambling if the math is on your side."
Quote: EvenBob
I never had a position to change.
I had never heard of this stupid
'law' before I posted about it. I
just assumed it was true because
it had the word 'law' attached to
it. I should have realized that in
math 'law' is just a meaningless
The word "law" in mathematics seems to mean something that is always true and self-evident. For example, the Commutative Law says that a+b=b+a. It's just obvious and fundamental. Not something that I
think can be formally proven as it is part of the foundation on which math is built.
To be honest, I am now sure why Bedford's Law is a "law." It is a true statement, but it can be proven with logarithms. It seems more like a theorem to me. Perhaps one of the other math-heads on the
forum can take this further.
"For with much wisdom comes much sorrow." -- Ecclesiastes 1:18 (NIV)
Are a “property” and “law” interchangeable terms in mathematics?
Also what’s the deal with some people calling it math, and some people calling it maths, is that just like the thing where some people insist on inserting unnecessary “u’s” in inappropriate places
after an o like color/colour, behavior/behaviour etc?
Quote: Wizard
It seems more like a theorem to me.
It's a theory that says if the
numbers are just the right
kind of groupings, this
thing might work. Or not.
It's just more pretend
probability math that real
math people don't go near.
"It's not called gambling if the math is on your side."
Quote: DRich
One of my favorite books is How To Lie With Statistics. It is on one of my bookshelves.
Fascinating book. Was required reading in a stats class I took in 1989. I really should re-read. Thanks!
Quote: Wizard
I just ran across a new video on Benford's Law and the 2020 Election by one of my favorite YouTubers, Matt Parker.
Thank you for sharing. I had not heard of Matt, but that was a great video. I'll certainly start diving through more.
Two issues might cause this analysis to be suspect ... First, 83 data points is a fairly small universe to observe a genuine Benford distribution. But second, and much more importantly, it's
Michigan. and at the risk of turning this into a political discussion, you can count on the data from most counties being honest, but when it comes to Wayne County (i.e. Detroit and the surrounding
suburbs) the levels of graft and corruption there are stunning. (I know, I lived there for the first 23 years of my life) So as a test, I think it's more realistic to look at the 1165 individual
precinct reports from Wayne County rather than the overall county totals. Of course the data for each precinct can't be found anywhere on the internet (though I had no problem finding it for my home
state of NH) ...
So, at least as a check for the honesty of data coming from Wayne County, I thought I might take a look at the 2016 election ... and the results were "interesting" to say the least. First, they don't
make it easy. Transparency laws require the data be published (eventually) but the data for Detroit was scanned as image data into a PDF which required OCR software to turn back into "real" data. And
then a check to be sure the OCR didn't mess with too many data points. Finally I got this:
For President Trump the table looks like this (Wish I could show the graph, it's more informative)
Digit Count Expectations
1 252 350.7
2 137 205.2
3 164 166.5
4 153 112.9
5 145 92.3
6 89 77.9
7 94 67.6
8 59 59.6
9 50 53.4
As we can see the data looks a little wonky ... the first thing to notice is 22 precincts reported zero votes for President Trump ... virtually a statistical impossibility, even in Detroit ... but it
appears the numbers my have been manipulated .... So we begin to wonder, since President Trump won Michigan in 2016, might his campaign have been manipulating the data?
Then we have a look at the data for Secretary Clinton ... and we find the data is totally wacked .... (again, the Excel chart shows it better)
Digit Count Expectations
1 92 350.7
2 179 205.2
3 279 166.5
4 232 112.9
5 163 92.3
6 107 77.9
7 64 67.6
8 33 59.6
9 16 53.4
Benford analysis shows that the data was clearly manipulated for Secretary Clinton, and probably for President Trump ... but it really doesn't indicate how the data was changed or who was supposed to
be the beneficiary .... Thoughts?
If you cannot quantify it, it's not science.
Quote: mcallister3200
Are a “property” and “law” interchangeable terms in mathematics?
Also what’s the deal with some people calling it math, and some people calling it maths, is that just like the thing where some people insist on inserting unnecessary “u’s” in inappropriate
places after an o like color/colour, behavior/behaviour etc?
You mean English English, as opposed to 'simplified English' for American's $:o)
Psalm 25:16 Turn to me and be gracious to me, for I am lonely and afflicted. Proverbs 18:2 A fool finds no satisfaction in trying to understand, for he would rather express his own opinion.
Quote: pjt36
For President Trump the table looks like this (Wish I could show the graph, it's more informative)
Digit Count Expectations
1 252 350.7
2 137 205.2
3 164 166.5
4 153 112.9
5 145 92.3
6 89 77.9
7 94 67.6
8 59 59.6
9 50 53.4
As we can see the data looks a little wonky ... the first thing to notice is 22 precincts reported zero votes for President Trump ... virtually a statistical impossibility, even in Detroit ...
but it appears the numbers my have been manipulated .... So we begin to wonder, since President Trump won Michigan in 2016, might his campaign have been manipulating the data?
Then we have a look at the data for Secretary Clinton ... and we find the data is totally wacked .... (again, the Excel chart shows it better)
Digit Count Expectations
1 92 350.7
2 179 205.2
3 279 166.5
4 232 112.9
5 163 92.3
6 107 77.9
7 64 67.6
8 33 59.6
9 16 53.4
May I see the raw data? I'd like to to a Bedford test on the total combined votes. I suspect this is a situation like Matt Parker described in Chicago where the precincts are so small and consistent
in population that you expect first digit clumping.
"For with much wisdom comes much sorrow." -- Ecclesiastes 1:18 (NIV)
Quote: EvenBob
It's a theory that says if the
numbers are just the right
kind of groupings, this
thing might work. Or not.
It's just more pretend
probability math that real
math people don't go near.
Well not really. It works anytime there is some exponential growth or power law that generates the numbers. It’s just a question of asking whether that assumption is applicable to a given situation.
The neat thing is it pops up in unexpected places like a normal distribution does.
The race is not always to the swift, nor the battle to the strong; but that is the way to bet.
I did two very simple simulations based on the county sizes and how many people voted in 2016 (I know more people did in 2020 but it probably doesn't matter whether 39% or 50+% voted) and ran ten
elections and listed each of the county results. I had four candidates and assumed the voting was (i) close 49.5% 48.5% 1.8% 0.2% (ii) less close 51%/47%/1.8%/0.2% with a slight random variation
added. Since I don't exactly know what one is supposed to notice, other than the ratios being associated with the logarithms. I might run other simulations with a higher turnout later.
1st digit 2nd digit
0 0 7 530 0.000% 11.983%
1 18 095 6 702 28.795% 10.665%
2 11 124 6 596 17.702% 10.496%
3 8 475 6 485 13.487% 10.320%
4 6 119 6 379 9.737% 10.151%
5 5 358 6 246 8.526% 9.940%
6 4 227 5 978 6.727% 9.513%
7 3 381 5 764 5.380% 9.173%
8 3 137 5 553 4.992% 8.837%
9 2 924 5 607 4.653% 8.923%
1st digit 2nd digit
0 0 7 555 0.000% 12.023%
1 18 099 6 884 28.802% 10.955%
2 11 177 6 608 17.786% 10.516%
3 8 408 6 543 13.380% 10.412%
4 6 157 6 426 9.798% 10.226%
5 5 343 6 046 8.503% 9.621%
6 4 268 5 859 6.792% 9.324%
7 3 387 5 751 5.390% 9.152%
8 3 115 5 718 4.957% 9.099%
9 2 886 5 450 4.593% 8.673%
Quote: miplet
Here are the 2016 Wayne county results. https://www.waynecounty.com/elected/clerk/november-8-2016-general.aspx
Just eyeballing
this table of Wayne County election results in 2016
, each precinct has total votes in the three digits. It is not going to be appropriate to apply Bedford's Law to such data.
Exactly the same issue as Matt Parker discussed about Chicago.
Charlie, if you have the vote totals in a spreadsheet, can you share them please.
"For with much wisdom comes much sorrow." -- Ecclesiastes 1:18 (NIV)
Is there a way to attach a spreadsheet here? ..
miplet linked to the Wayne County data, but those are the scanned images (pita to work with)
You're almost certainly right that this data suffers the same shortcomings as Parker's analysis of Chicago
The Benford analysis for the Total vote in Wayne County shows a more reasonable Benford distribution, but you can see the spike at 2-3 from Secretary Clinton's returns easily overlayed, even on that
Digit Count Expectations
1 296 350.7
2 151 205.2
3 192 166.5
4 153 112.9
5 101 92.3
6 92 77.9
7 64 67.6
8 58 59.6
9 58 53.4
I also had a look at 2016 precinct data from Los Angeles County (4500+ datapoints) and saw similar results .... though Secretary Clinton's graphed peaked at 5 there rather than 2.... But I suspect,
again, that this data is similar to the Chicago and Detroit data. This will probably be true for any large metropolitan area as their precincts are more uniform in size. Statewide analysis at the
precinct level may work, but this data's more likely to be "honest" as it includes large areas not controlled by any one political faction.
Finally, took a look at the "last two" digits for Secretary Clinton's returns which showed a similar "random" distribution as Parker's Chicago data.
edit: Let me just add here that the expected "random" distribution for the two digit trailing test should be 11.65 for any single instance but this does show spikes above 20 for 93 and 98 ....
probably nothing :)
So since we've apparently put the "Human Generated" manipulation issue to bed (or at least failed to demonstrate it via Benford), anyone want to have a look at the Edison Research data to see if the
machines themselves have been compromised ;-P
2nd Edit: ... an interesting presentation by Dr. Shiva Ayyadurai addresses the "Machine" issue here;
MIT PhD Analysis of Michigan Votes Reveals Unfortunate Truth of US Voting Systems
.... One significant conclusion is that whatever the math, the systemic issues raised are real ... and if those aren't fixed, bookmakers are going to stop taking bets on elections.
Last edited by: pjt36 on Nov 13, 2020
If you cannot quantify it, it's not science. | {"url":"https://wizardofvegas.com/forum/questions-and-answers/all-other/35315-benfords-law-and-the-2020-election/","timestamp":"2024-11-12T22:20:22Z","content_type":"text/html","content_length":"212651","record_id":"<urn:uuid:dfbf7958-4ffc-4b3e-ab31-ca36a35f110d>","cc-path":"CC-MAIN-2024-46/segments/1730477028290.49/warc/CC-MAIN-20241112212600-20241113002600-00053.warc.gz"} |
Practical formula to calculate tension of vertical cable with hinged-fixed conditions based on vibration method
Vertical cables are widely used in the tied-arch bridges and suspension bridges as the vital components to transfer load. It is very important to accurately estimate the cable tensions in the cable
supported bridges during both construction and in-service stages. Vibration method is the most widely used method for in-situ measurement of cable tensions. But for the cables with hinged-fixed
boundary conditions, no analytical formulas can be used to describe the relationship between the frequencies and the cable tension. According to the general solution of the vibration equation and
based on its numerical computational results, practical formula to calculate tensions of vertical cables by multiple natural frequencies satisfying hinged-fixed boundary conditions is proposed in
this paper. The expression of the practical formula is the same as the solution derived from an axially loaded beam with simple supported ends and can use the first 10 order frequencies to calculate
the cable tension conveniently and accurately. Error analysis showed that when using the fundamental frequency to estimate cable force, the estimated tension errors of the cables with its
dimensionless parameter $\xi \ge$2.8 are less than 2 %. It contained nearly all of the vertical cables used in bridge engineering. In addition, with multiple natural frequencies being measured,
bending stiffness of the cable can be identified by using the formulas presented in this paper with an iterative method. At last, the practical formula in this paper is verified to have high
precision with several numerical examples, and can be conveniently applied to field test for cable-supported bridges.
1. Introduction
Vertical cables are widely used in the tied-arch bridges and suspension bridges as the vital components to transfer load from the deck to the arch ribs or towers [1]. The tension force of cables must
be measured accurately because it is very important in assessing the structural condition during construction and in-service stages [2]. Currently available techniques to estimate the cable tension
include the static methods directly measuring the tension by a load cell, and the vibration methods indirectly estimating the tension from measured natural frequencies [3]. Due to the simplicity of
application and the high accuracy reached in many cases, the identification of cable force based on the vibration method has been widely used and has become extremely popular in engineering practice
For the vertical cables, the sag-extensibility is no need to be taken into account. The existing formulas to describe the relationship between the natural frequencies and the cable tension can be
classified into several categories as shown subsequently.
The first category utilizes the flat taut string theory that neglects both sag-extensibility and bending stiffness:
where ${f}_{n}$ denotes the $n$th natural frequency in Hz, and $T$, $m$ and $l$ denote the tension, mass density and length of the cable, respectively. Given the measured frequency and the mode
order, the computation of the tension is straightforward. But this formula is restricted to long, slender cables. For the vertical cables used in tied-arch bridges and suspension bridges, the bending
stiffness are relative large and have significant effect on the natural frequencies of the system. Unacceptable errors may be introduced by ignoring the bending stiffness for the cables.
The second category formula takes the bending stiffness into account based on the vibration equation of axially loaded beam, as shown in Eq. (2) [5]:
$T=4m{l}^{2}{\left(\frac{{f}_{n}}{n}\right)}^{2}-{n}^{2}{\pi }^{2}\frac{EI}{{l}^{2}},$
where $EI$ denotes the flexural stiffness of the cable, and the other symbols are the same as Eq. (1). Though Eq. (2) considers the effect of bending stiffness, it may lead to unacceptable errors in
force prediction for short and stout cables, because it is derived from an axially-tensioned beam with hinged end boundaries rather than fixed ones.
For the cables with fixed-fixed or hinged-fixed boundary conditions which are commonly used in real bridge structures (as Figs. 1 and 2 show), there are no analytical formulas to calculate their
cable tensions from the measured frequencies. The third category formulas have been established to solve such problems. This is the so-called practical formulas.
Fig. 1The vertical cable used in arch bridge with fixed-fixed boundary conditions
Fig. 2The vertical cable used in suspension bridge with hinged-fixed boundary conditions
Zui et al. [6] presented practical formulas to estimate the cable forces from the first and second natural frequencies of cable vibration as Eq. (3)-(5) show. The formulas are based on the
approximate solutions of high accuracy for the equation of inclined cable with flexural rigidity. But for the vertical cables without sag-extensibility, these formulas can only use the fundamental
frequency for calculation.
1. In the case of using the natural frequency of first-order mode (cable with sufficiently small sag $\text{3}\le \mathrm{\Gamma }$):
$T=4m{\left({f}_{1}l\right)}^{2}\left[1-2.2\frac{C}{{f}_{1}}-0.550{\left(\frac{C}{{f}_{1}}\right)}^{2}\right],\left(17\le \xi \right),$
$T=4m{\left({f}_{1}l\right)}^{2}\left[0.865-11.6{\left(\frac{C}{{f}_{1}}\right)}^{2}\right],\left(6\le \xi \le 17\right),$
$T=4m{\left({f}_{1}l\right)}^{2}\left[0.828-10.5{\left(\frac{C}{{f}_{1}}\right)}^{2}\right],\left(0\le \xi \le 6\right).$
2. In the case of using the natural frequency of second-order mode (cable with relatively large sag $\mathrm{\Gamma }>\text{3}$):
$T=m{\left({f}_{2}l\right)}^{2}\left[1-4.40\frac{C}{{f}_{2}}-1.10{\left(\frac{C}{{f}_{2}}\right)}^{2}\right],\left(60\le \xi \right),$
$T=m{\left({f}_{2}l\right)}^{2}\left[1.03-6.33\frac{C}{{f}_{2}}-1.58{\left(\frac{C}{{f}_{2}}\right)}^{2}\right],\left(17\le \xi \le 60\right),$
$T=m{\left({f}_{2}l\right)}^{2}\left[0.882-85.0{\left(\frac{C}{{f}_{2}}\right)}^{2}\right],\left(0\le \xi \le 17\right).$
3. In the case of using the natural frequencies of high-order modes (very long cable $n\ge \text{2}$):
$T=m{\left(\frac{{f}_{n}}{n}\right)}^{2}{l}^{2}\left[1-2.20\frac{C}{{f}_{n}}\right],\left(200\le \xi \right),$
where $C=\sqrt{EI/\left(m{l}^{4}\right)}$, $\xi =\sqrt{T/\left(EI\right)}\bullet l$.
Mehrabi and Tabatabai [7] proposed a simple relationship among non-dimensional cable parameters for in-plane and out-of-plane vibration modes. The relationship accounts for both the sag-extensibility
and bending stiffness effects. The cable ends are assumed to be fixed. This relationship is most accurate when ${\lambda }^{2}$ (the sag-extensibility parameter) is less than 3.1 and $\xi$ is more
than 50. The function is shown in Eq. (6):
$\frac{{\omega }_{n}}{{\omega }_{ns}}=\alpha {\beta }_{n}-0.24\frac{\mu }{\xi }.$
In which:
$\alpha =1+0.039\mu ,{\beta }_{n}=1+\frac{2}{\xi }+\frac{\left(4+{n}^{2}{\pi }^{2}/2\right)}{{\xi }^{2}},\mathrm{}\mathrm{}\mathrm{}\mathrm{}{\omega }_{ns}=n{\omega }_{1s},$
$\mu =\left\{\begin{array}{ll}{\lambda }^{2},& \text{for}\text{\hspace{0.17em}}n=1\text{}\text{(in\hspace{0.17em}plane)}\text{,}\\ 0,& \text{for}\text{}\text{\hspace{0.17em}}n>1\text{\hspace{0.17em}}
\left(\text{in\hspace{0.17em}plane}\right),\\ 0,& \text{for\hspace{0.17em}all}\text{\hspace{0.17em}}n\text{\hspace{0.17em}}\left(\text{out\hspace{0.17em}of\hspace{0.17em}plane}\right).\end{array}\
Ren et al. [8] proposed empirical formulas to estimate cable tension based on the solutions by means of energy method and fitting the exact solutions of cable vibration equations, shown in Eqs. (7).
The cable bending stiffness effect is taken into account. The empirical formulas to estimate cable tension are based on the cable fundamental frequency only:
$T=3.432m{l}^{2}{f}^{2}-45.191\frac{EI}{{l}^{2}},\left(0\le \xi \le 18\right),$
$T=m{\left(2lf-\frac{2.363}{l}\sqrt{\frac{EI}{m}}\right)}^{2},\left(18<\xi \le 210\right),$
$T=4m{l}^{2}{f}^{2},\left(210<\xi \right).$
Gan et al. [9] proposed a practical formula of cable tension estimation based on the vibration function of Euler-Bernoulli beam with fixed-fixed boundary conditions and energy principle. The
expression is as Eq. (8) shows, it is very simple, but the precision is not very good [10]:
$T=4m{l}^{2}\frac{{f}_{n}^{2}}{{\alpha }_{n}^{2}}-{\pi }^{2}\frac{EI}{{l}^{2}}{\beta }_{n}^{2},$
where ${\alpha }_{n}=n+0.1$, ${\beta }_{n}=n+1$.
Fang et al. [11] proposed a practical formula with simple explicit form on the basis of the transverse vibration equation of a cable and by using a curve-fitting technique to replace the numerical
iterative process. The formula is shown in Eq. (9). But the cable ends are assumed to be fixed, and therefore it can’t be used in other boundary conditions:
$T=4{\pi }^{2}m{l}^{2}\frac{{f}_{n}^{2}}{{\gamma }_{n}^{2}}-\frac{EI}{{l}^{2}}{\gamma }_{n}^{2}.$
In which:
${\gamma }_{n}=n\pi +A\sqrt{\frac{EI}{m{\omega }_{n}^{2}{l}^{4}}}+B\frac{EI}{m{\omega }_{n}^{2}{l}^{4}},\mathrm{}\mathrm{}\mathrm{}\mathrm{}{A}_{n}=-18.9+26.2n+15.1{n}^{2},$
${B}_{n}=\left\{\begin{array}{ll}290,& n=1,\\ 0,& n\ge 2.\end{array}\right\$
The aforementioned practical formulas all assumed the ends of cable are both fixed, so it can’t be used to estimate the tension of cables with hinged-fixed boundary conditions. Furthermore, most of
the existing practical formulas can only use the fundamental frequency for calculation. But for in-situ test, sometimes the fundamental frequency is difficult to be measured for the reasons of
excitation methods, sensor’s sensitivity, mounting position of the sensors and so on. If the fundamental frequency of cable is not obtained, the common method is using the frequency differences
between two frequencies as the fundamental frequency. It may lead to unacceptable errors for the cables with large bending stiffness. So, new practical formulas which can solve those problems are
The objective of this paper is to propose a new practical formula that can estimate cable tension from measured natural frequencies satisfying hinged-fixed boundary conditions. To achieve this
objective, the following three tasks are performed. Firstly, from the general solution of the vibration equation of the vertical cable, practical formula of the cable tension identification
satisfying hinged-fixed boundary conditions is derived. The expression of the practical formulas is the same as the solution derived from an axially loaded beam with simply supported ends and in
which only an additional parameter ${\lambda }^{2}$ is introduced. Secondly, by using Newton iterative method, the numerical solutions of the frequency equation of the tensioned vertical cable with
hinged-fixed boundary conditions are obtained. By the least squares fitting method, and based on the theoretical results the relationship between ${\lambda }^{2}$ and the cable’s geometrical and
physical parameters is fitted. Thirdly, the feasibility and practicability of the proposed approach are examined through error analysis and by several numerical examples.
2. Basic governing equations and solutions
Vertical cables are mostly used in tied-arch bridges and suspension bridges, it have no sag, so the equation of lateral motion coincides with the equation of the motion of a beam with axial tension
as follows [12]:
$EI\frac{{\partial }^{4}w}{\partial {x}^{4}}-T\frac{{\partial }^{2}w}{\partial {x}^{2}}+m\frac{{\partial }^{2}w}{\partial {t}^{2}}=0,$
where $EI$ is the bending stiffness of the cable, $w$ is the lateral deflection due to vibration, $T$ is the axial cable force, $m$ is the mass of cable per unit length.
Using the method of separation of variables, the general solution of Eq. (10) can be obtained as follows:
$w\left(x\right)={A}_{1}\mathrm{s}\mathrm{i}\mathrm{n}\left(\delta x\right)+{A}_{2}\mathrm{c}\mathrm{o}\mathrm{s}\left(\delta x\right)+{A}_{3}\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{h}\left(\epsilon x\
right)+{A}_{4}\mathrm{c}\mathrm{o}\mathrm{s}\mathrm{h}\left(\epsilon x\right),$
$\delta =\sqrt{\sqrt{{\zeta }^{4}+{\gamma }^{4}}-{\zeta }^{2}},$
$\epsilon =\sqrt{\sqrt{{\zeta }^{4}+{\gamma }^{4}}+{\zeta }^{2}},$
${\gamma }^{4}=m{\omega }^{2}/\left(EI\right),$
${\zeta }^{2}=T/\left(2EI\right).$
$\alpha =\delta l,\beta =\epsilon l,\stackrel{^}{x}=x/l.$
Eq. (11) can be rewritten as:
$w\left(x\right)={A}_{1}\mathrm{s}\mathrm{i}\mathrm{n}\left(\alpha \stackrel{^}{x}\right)+{A}_{2}\mathrm{c}\mathrm{o}\mathrm{s}\left(\alpha \stackrel{^}{x}\right)+{A}_{3}\mathrm{s}\mathrm{i}\mathrm
{n}\mathrm{h}\left(\beta \stackrel{^}{x}\right)+{A}_{4}\mathrm{c}\mathrm{o}\mathrm{s}\mathrm{h}\left(\beta \stackrel{^}{x}\right).$
For hinged-fixed boundary conditions, the boundary conditions used to determine the constants ${A}_{1}$ through ${A}_{4}$ from Eq. (14) are:
$w\left(l\right)=0$${A}_{1}\mathrm{s}\mathrm{i}\mathrm{n}\left(\alpha \right)+{A}_{2}\mathrm{c}\mathrm{o}\mathrm{s}\left(\alpha \right)+{A}_{3}\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{h}\left(\beta \
right)+{A}_{4}\mathrm{c}\mathrm{o}\mathrm{s}\mathrm{h}\left(\beta \right)=0$
$w"\text{'}\left(0\right)=0$$-{\alpha }^{2}{A}_{2}+{\beta }^{2}{A}_{4}=0$
$w"\left(l\right)=0$${A}_{1}\alpha \mathrm{c}\mathrm{o}\mathrm{s}\left(\alpha \right)-{A}_{2}\alpha \mathrm{s}\mathrm{i}\mathrm{n}\left(\alpha \right)+{A}_{3}\beta \mathrm{c}\mathrm{o}\mathrm{s}\
mathrm{h}\left(\beta \right)+{A}_{4}\beta \mathrm{s}\mathrm{i}\mathrm{n}\mathrm{h}\left(\beta \right)=0$
The natural frequencies are obtained from the condition that the determinant of the boundary conditions is set equal to zero:
$\mathrm{d}\mathrm{e}\mathrm{t}\left({\left[\begin{array}{cccc}0& \mathrm{sin}\left(\alpha \right)& 0& \alpha \mathrm{cos}\left(\alpha \right)\\ 1& \mathrm{cos}\left(\alpha \right)& -{\alpha }^{2}& -
\alpha \mathrm{sin}\left(\alpha \right)\\ 0& \mathrm{sinh}\left(\beta \right)& 0& \beta \mathrm{cosh}\left(\beta \right)\\ 1& \mathrm{cosh}\left(\beta \right)& {\beta }^{2}& \beta \mathrm{sinh}\left
(\beta \right)\end{array}\right]}^{T}\right)=0.$
Frequency equations can be obtained from Eq. (16), as Eq. (17) shows:
$\alpha \mathrm{c}\mathrm{o}\mathrm{s}\left(\alpha \stackrel{^}{x}\right)\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{h}\left(\beta \stackrel{^}{x}\right)-\beta \mathrm{s}\mathrm{i}\mathrm{n}\left(\alpha \
stackrel{^}{x}\right)\mathrm{c}\mathrm{o}\mathrm{s}\mathrm{h}\left(\beta \stackrel{^}{x}\right)=0.$
It is a transcendental equation, so no explicit solutions can be obtained. Newton iterative method is used to calculate the value of $\beta$ versus $\alpha$, the results of $\alpha$ and $\beta$ for
the first ten order modes are shown in Fig. 3.
Fig. 3The solutions of βn versus αn for hinged-fixed boundary conditions (n= 1-10)
3. Establishment of practical formulas
From Eqs. (12) and (13) we arrive at:
${\alpha }_{n}{\beta }_{n}=\sqrt{\frac{m{\omega }_{n}^{2}{l}^{4}}{EI}},$
${\beta }_{n}^{2}-{\alpha }_{n}^{2}={l}^{2}\frac{T}{EI}.$
Combining Eqs. (18a) and (18b), $T$ can be solved as:
$T=\frac{4m{l}^{2}{\left(\frac{{f}_{n}}{n}\right)}^{2}}{{\left(\frac{{\alpha }_{n}}{n\pi }\right)}^{2}}-\frac{EI}{{l}^{2}}{\left(n\pi \right)}^{2}{\left(\frac{{\alpha }_{n}}{n\pi }\right)}^{2}.$
Defined ${\alpha }_{n}/n\pi ={\gamma }_{n}$, then:
$T=\frac{4m{l}^{2}{\left({f}_{n}/n\right)}^{2}}{{\gamma }_{n}^{2}}-\frac{EI}{{l}^{2}}{\left(n\pi \right)}^{2}{\gamma }_{n}^{2}.$
Eq. (20) is the proposed practical formula to calculate cable tension. Compared with the pre-existing formulas, this formula is derived from the general solution of the equation of motion for a cable
with axial tension $T$, so it can be used for any constraint conditions in theory.
In Eq. (20), ${\gamma }_{n}^{2}$ is an unknown parameter, for different conditions the value of ${\gamma }_{n}^{2}$ is different. If the boundary is simple supported, then ${\gamma }_{n}^{2}\equiv$1
and Eq. (20) becomes the same as Eq. (2). But for the hinged-fixed boundary conditions, ${\gamma }_{n}^{2}$ is not a constant and can’t be solved directly. From the practical point of view, fitting
formula to calculate the value of ${\gamma }_{n}^{2}$ is needed based on the theoretical results.
Fig. 4 shows the relationship between ${\gamma }_{n}^{2}$ and ${\lambda }_{n}$ for the first ten order modes of cables with hinged-fixed boundary conditions from the theoretical results. A quadratic
best fit function is obtained between ${\gamma }_{n}^{2}$ and ${\lambda }_{n}$ in the form of Eq. (21):
${\gamma }_{n}^{2}=A{\lambda }_{n}^{2}\text{+}B{\lambda }_{n}+1,$
$\left\{\begin{array}{l}{A}_{n}=-14.1{n}^{3}-22.4{n}^{2}+100.3n-42,\\ {B}_{n}=6.87n+0.7.\end{array}\right\$
Fig. 4γn2-λn relation curves a) n= 1-3, b) n= 4-10
Fig. 5 shows the comparison curves between the theoretical value and fitted value. Fig. 6 shows fitting errors of ${\gamma }_{n}^{2}$ versus $\lambda /{\lambda }_{\mathrm{m}\mathrm{a}\mathrm{x}}$.
From these figures it can be seen that: 1) Fitted values coincide well with the theoretical values, the first order frequency has the maximum fitting error and the value is no more than 1.5 %; 2)
Fitting errors decrease with the increase of the frequency order; 3) The errors of the 2-10 order frequencies are all less than 0.5 % in the whole range.
Fig. 5Comparisons of fitted value with theoretical value of γn2 versus λn: a) n= 1, b) n= 3, c) n= 5, d) n= 10
Summarize the results aforementioned, the expression of practical formulas to calculate the tension of vertical cables for hinged-fixed boundary conditions is as follows:
$T=\frac{4m{l}^{2}{\left({f}_{n}/n\right)}^{2}}{{\gamma }_{n}^{2}}-\frac{EI}{{l}^{2}}{\left(n\pi \right)}^{2}{\gamma }_{n}^{2},n=1,2,\dots ,10.$
In which:
${\gamma }_{n}^{2}=A{\lambda }_{n}^{2}\text{+}B{\lambda }_{n}+1,$
${\lambda }_{n}=\sqrt{\frac{EI}{m{\omega }_{n}^{2}{l}^{4}}},$
$\left\{\begin{array}{l}{A}_{n}=-14.1{n}^{3}-22.4{n}^{2}+100.3n-42,\\ {B}_{n}=6.87n+0.7.\end{array}\right\$
Fig. 6Curves of relative errors of γn2 versus λ/λmax
The expression of Eq. (22) is the same as Eq. (2) which is derived from the vibration equation of axially loaded beam, with the difference that an additional parameter ${\gamma }_{n}^{2}$ is needed.
For hinged-hinged boundary conditions, ${\gamma }_{n}^{2}\equiv$1. So Eq. (2) can be regarded as a special case of Eq. (3).
4. Error analysis
From the derivation process of the practical formula, it can be seen that if the frequencies are measured exactly and the structural parameters of the cable are accurate, the error of estimated cable
tension is only resulted from the error of fitted ${\gamma }_{n}^{2}$ and can be calculated theoretically.
From Eq. (23b) we obtained:
$\frac{EI}{{l}^{2}}=m{l}^{2}{\omega }_{n}^{2}{\lambda }_{n}^{2}.$
Submitting Eq. (24) into Eq. (22), then we arrive at:
$T=\frac{m{l}^{2}{\omega }_{n}^{2}}{{\left(n\pi \right)}^{2}{\gamma }_{n}^{2}}-m{l}^{2}{\omega }_{n}^{2}{\lambda }_{n}^{2}{\left(n\pi \right)}^{2}{\gamma }_{n}^{2}.$
Defining the relative error of ${\gamma }_{n}^{2}$ is $\epsilon$, so the calculated cable tension ${T}^{"}$ is:
${T}^{"}=\frac{m{l}^{2}{\omega }_{n}^{2}}{{\left(n\pi \right)}^{2}\left(1+\epsilon \right){\gamma }_{n}^{2}}-m{l}^{2}{\omega }_{n}^{2}{\lambda }_{n}^{2}{\left(n\pi \right)}^{2}\left(1+\epsilon \
right){\gamma }_{n}^{2}.$
Defined the error of $T$ is $\mathrm{\Delta }T/T$, then:
$\frac{\mathrm{\Delta }T}{T}=\frac{{T}^{"}-T}{T}=\frac{-\epsilon -{\lambda }_{n}^{2}{\left(n\pi \right)}^{4}{\gamma }_{n}^{4}\left(1+\epsilon \right)\epsilon }{1+\epsilon -{\lambda }_{n}^{2}{\left(n\
pi \right)}^{4}{\gamma }_{n}^{4}\left(1+\epsilon \right)}.$
From Fig. 6, we can get the value of $\epsilon$ for a specified $\lambda$. Then the error of $T$ can be calculated by using the Eq. (27) and the known parameters. The curves of relative errors of $T$
versus $\lambda /{\lambda }_{\mathrm{m}\mathrm{a}\mathrm{x}}$ are shown in Fig. 7.
Fig. 7Curves of relative errors of T versus λ/λmax
From Fig. 7 we can seen that: 1) The error of $T$ calculated with the fundamental frequency is larger than that calculated with other order frequencies. When $\lambda /{\lambda }_{\mathrm{m}\mathrm
{a}\mathrm{x}}<\text{0.87}$, the errors of $T$ calculated with the fundamental frequency are less than 2 %; When $\lambda /{\lambda }_{\mathrm{m}\mathrm{a}\mathrm{x}}<\text{0.93}$, the errors are
less than 5 %. 2) The errors of $T$ calculated with the 2-10 order frequencies have high precision. When $\lambda /{\lambda }_{\mathrm{m}\mathrm{a}\mathrm{x}}<\text{0.9}$, the errors of $T$
calculated with 2-10 order frequencies are all less than 1 %; When $\lambda /{\lambda }_{\mathrm{m}\mathrm{a}\mathrm{x}}<\text{0.96}$, the errors of $T$ calculated with 2-10 order frequencies are
less than 5 %; When $\lambda /{\lambda }_{\mathrm{m}\mathrm{a}\mathrm{x}}\ge \text{0.96}$, the errors of $T$ calculated with 2-10 order frequencies become larger than 5 %, and increase rapidly.
From the definition of $\lambda$ and $\xi$, we arrive at:
${\xi }_{n}=\sqrt{\frac{T}{m{\omega }_{n}^{2}{l}^{2}}}\cdot \frac{1}{{\lambda }_{n}}.$
$\xi$ is a non-dimensional parameter reflecting the effect of cable’s bending stiffness defined by Zui et al. [6]. In the common references, according to the value of $\xi$, cables can be classified
into several categories as shown subsequently. When $\xi$ is very large, the cable is called long cable and can be known as a taut string. When $\xi$ is very small, the cable is called short cable
and can be known as an axially-tensioned beam. And when $\xi$ is between the two category defined above, the cable is called medium length cable and the dynamic behaviors of them are combined the
property of axially-tensioned beam and taut string. In order to facilitate comparison with the references, the curves of relative errors of $T$ versus $\xi$ are obtained as shown in Fig. 8.
From Fig. 8 it can be seen that: 1) The errors of calculated cable tension decrease with the increase of $\xi$; 2) For the same value of $\xi$, the errors of $T$ are different when using different
order frequencies, in general, the order of frequency is higher, the error of $T$ is larger. When $\xi \ge \text{10}$, the errors of the calculated cable tension using the first ten order of
frequencies are all less than 5 %.
Table 1 lists the value of $\xi$ when the value of errors is 2 % and 5 %. From the table, if the value of $\xi$ is known, the error range of the calculated cable tensions can be pre-estimated. This
is very useful in engineering applications. In practice, 5 % are usually used as the upper limit of allowable error. So from Table 1, we can see that the practical formulas in this paper is
applicable when $\xi \ge \text{2.1}$ if the fundamental frequency is used to determine the cable force, and when $\xi \ge \text{2.8}$, the errors of the calculated cable tension using the fundamental
frequency are less than 2 %. Nearly all of the vertical cables used in bridge engineering are in this range. So the formulas in this paper have high precision in engineering application.
Fig. 8Curves of relative errors of T versus ξ
Table 1The value of ξ for specific value of errors
Error Frequency order
2 % 2.8 3.3 2.5 3 4.9 6.7 8.6 10.5 12 13.5
5 % 2.1 2.6 1.9 2.6 3.6 4.8 5.9 7.0 8.0 9.0
5. Identification of bending stiffness
The geometrical parameters of cables are very important in estimating cable tensions. Those parameters include the mass of the cable per unit length, the length of the cable, the bending stiffness of
the cable and so on. Among these parameters, the bending stiffness is the most difficult to be obtained because most of cables are using parallel steel wire or steel stranded wire and the cross
section of it no longer conform with the assumption of plane cross section. Bending stiffness has significant influence on cable force estimation especially for the short vertical cables, so
identification of bending stiffness is very important to improve the accuracy of cable tension estimation. It can be identified by using the formulas in this paper with multiple natural frequencies.
The principle of the identification is as follows:
From ${T}_{r}={T}_{s}$, we obtained:
$\frac{m{l}^{2}{\omega }_{r}^{2}}{{\left(r\pi \right)}^{2}{\gamma }_{r}^{2}}-\frac{EI}{{l}^{2}}{\left(r\pi \right)}^{2}{\gamma }_{r}^{2}=\frac{m{l}^{2}{\omega }_{s}^{2}}{{\left(s\pi \right)}^{2}{\
gamma }_{s}^{2}}-\frac{EI}{{l}^{2}}{\left(s\pi \right)}^{2}{\gamma }_{s}^{2}.$
Then $EI$ can be solved:
$EI=\frac{m{l}^{4}}{{\pi }^{4}}\frac{{\omega }_{r}^{2}/\left({r}^{2}{\gamma }_{r}^{2}\right)-{\omega }_{s}^{2}/\left({s}^{2}{\gamma }_{s}^{2}\right)}{{r}^{2}{\gamma }_{r}^{2}-{s}^{2}{\gamma }_{s}^
Because ${\gamma }_{r}$, ${\gamma }_{s}$ is a function of $EI$, Eq. (30) can’t be solved explicitly, iterative method is needed. The iterative steps are as follows:
1. The iterations start with initial value of $E{I}_{0}=\text{0}$, then substitute $E{I}_{0}=\text{0}$ into Eqs. (23a) and (23b), ${\gamma }_{r1}$, ${\gamma }_{s1}$ are calculated;
2. Substitute ${\gamma }_{r1}$, ${\gamma }_{s1}$ in to Eq. (30), $E{I}_{1}$ is obtained, if $|E{I}_{1}-E{I}_{0}|\le Tol$, then the bending stiffness is $E{I}_{0}$ and the iteration is stopped. $Tol$
is a desired tolerance. Otherwise, completing the cycle of iteration to be repeated until the desired accuracy is attained.
6. Verification examples
From the product manual of hanging-cable/tied-cable system for arch bridge pressed by OVM company which is a professional manufacturer of steel strand in China [13], several hanging-cable styles
commonly used in arch bridge are used as the verification examples. The hanging-cables are composed of 37-211 parallel wires and the length of the cable are in the range of 3 m to 70 m, which
contained nearly all of the vertical cables used in tied-arch bridges.
6.1. Estimation of cable force
Table 2 shows the comparison results of the calculated cable tension using the formulas in this paper. From this table, it can be seen that the differences between the results of Eq. (2) and the
exact value are very large and the differences are decreased with the value of $\xi$ increasing. For the cables with $\xi \ge \text{46}$, the errors will be less than 5 %. So the precisions of
calculated cable tensions using the presented formulas are improved largely. For all kinds of cables, the errors of the calculated results with all order frequencies are no more than 1.5 %.
Especially for the cables with $\xi \ge \text{46}$, the errors of the calculated cable tensions are less than 0.1 %.
Table 2Comparison of computed cable tensions
$L$ $m$ $EI$ $T$ $f$ Freq. Eq. (2) Error Error
Cable type $\xi$ method
(m) (kg/m) ($\text{kN}\text{∙}{\text{m}}^{\text{2}}$) (kN) (Hz) order ${T}_{1}$ (kN) of ${T}_{1}$ of ${T}_{2}$
${T}_{2}$ (kN)
PES(FD)7-37 3 13.6 34928 500 11 36.365 1 609 494 21.8 % 1.3 %
PES(FD)7-55 5 20.1 77195 1000 18 50.043 2 1137 997 13.7 % 0.3 %
PES(FD)7-73 10 26.6 135910 1500 33 38.185 3 1603 1497 6.9 % 0.2 %
PES(FD)7-91 15 33.5 211242 2000 46 34.522 4 2097 1997 4.9 % 0.1 %
PES(FD)7-109 20 39.3 303118 2500 57 33.274 5 2598 2498 3.9 % 0.1 %
PES(FD)7-127 30 46.4 411538 3000 81 26.444 6 3082 2999 2.7 % 0.0 %
PES(FD)7-151 40 54.1 581634 3500 98 23.055 7 3580 3500 2.3 % 0.0 %
PES(FD)7-187 50 66.9 892176 4000 106 20.31 8 4086 4001 2.2 % 0.0 %
PES(FD)7-199 60 71.0 1010133 4500 127 19.516 9 4583 4503 1.8 % 0.1 %
PES(FD)7-211 70 75.5 1135690 5000 154 19.825 10 5082 5006 1.6 % 0.1 %
6.2. Estimation of bending stiffness
If more than one order of frequencies were obtained, bending stiffness of cables can be identified by using the method presented in this paper. To verify the accuracy of bending stiffness
identification method, cable 1 and cable 10 are analyzed as examples. Table 3 shows the comparison of identified value of $EI$. Table 4 gives the solution history of $EI$ iteration when using the 1st
and 2nd order of frequencies for cable 1.
From Table 3 it can be seen that: 1) Bending stiffness of cable 1 and cable 10 are accurately identified with the errors being no more than 10 %. 2) The error of the cable with larger $\xi$ is larger
than that of the cable with smaller $\xi$. This is because the bending stiffness has little impact on the estimation results for the cable with larger $\xi$. In other words, the cable tension
identification is insensitive to the change of cable’s bending stiffness for the cable with larger $\xi$. Even the bending stiffness changed largely, the cable tension will change little. 3) The
error of identified bending stiffness using the first order frequency is the largest. It is mainly because the fitted value of ${\gamma }_{1}^{2}$ for the first order frequency is more roughly than
the others which can be seen from Fig. 5.
From Table 4 we can see that the iteration of the method presented in this paper are converged quickly, and the satisfied results are obtained after only 5 steps of iterations.
Table 3Comparison of identified EI value
Cable 1 (PES(FD)7-37) Cable 10 (PES(FD)7-211)
Identified Identified
$f$ Ture value Relative $f$ Ture value Relative
Freq. value Freq. value
order (Hz) ($\text{kN}\text{∙}{\text{m}}^{\text error order (Hz) ($\text{kN}\text{∙}{\text{m}}^{\text error
{2}}$) ($\text{kN}\text{∙}{\text{m}}^{\text {2}}$) ($\text{kN}\text{∙}{\text{m}}^{\text
{2}}$) {2}}$)
1 40.168 34928 1 1.9571 1135690
36535 4.6 % 1220137 7.4 %
2 87.863 34928 2 3.9167 1135690
3 148.02 34928 3 5.8812 1135690
34866 0.2 % 1200290 5.7 %
4 223.14 34928 4 7.8529 1135690
5 314.45 34928 5 9.8344 1135690
34776 0.4 % 1171519 3.2 %
6 422.59 34928 6 11.828 1135690
7 547.9 34928 7 13.836 1135690
34823 0.3 % 1137662 0.2 %
8 690.6 34928 8 15.861 1135690
9 850.85 34928 9 17.906 1135690
34925 0.0 % 1148930 1.2 %
10 1028.8 34928 10 19.972 1135690
Table 4Solution history of EI iteration
Iteration $E{I}_{0}$ ${\lambda }_{1}$ ${\lambda }_{2}$ ${\lambda }_{1}^{2}$ ${\lambda }_{2}^{2}$ $EI$ Difference (%)
1 0 0.000 0.000 1.00 1.00 41168 17.9 %
2 41168 0.027 0.012 1.22 1.17 36409 4.2 %
3 36409 0.025 0.011 1.20 1.16 36539 4.6 %
4 36539 0.025 0.011 1.20 1.16 36535 4.6 %
5 36535 0.025 0.011 1.20 1.16 36535 4.6 %
7. Conclusions
Practical formula for cable tension identification satisfying hinged-fixed boundary conditions is proposed based on the general solution of the transverse vibration equation of the vertical cable. It
is a unified formula with the same expression as the solution derived from an axially loaded beam with simply supported, without segmentation, and can be used to calculate tensions conveniently and
accurately. Moreover it can be used to estimate the cable tension by directly using its first ten order frequencies and taking the bending stiffness into account. The capability of this formula has
been verified through several numerical examples, which indicated that those formulas are sufficiently accurate and can be conveniently applied to in-situ measurement of vertical cables used in
bridge structures.
The formula presented in this paper, originating from the theoretical solution with few approximations and assumptions, has high precision and the errors are only resulting from the error of
coefficient fitting, and the error between the estimated results and the exact solutions can be calculated theoretically. For hinged-fixed conditions, when the cable’s dimensionless parameter $\xi \
ge \text{10}$, the errors of calculated cable tension using the first ten order of frequencies are all less than 5 %. If using the cable’s fundamental frequency to calculate, when the cable’s
dimensionless parameter $\xi \ge \text{2.8}$, the errors of calculated cable tension will be no more than 2 %. In a real bridge, most of their cables are tensioned to a relatively large internal
force with its parameter $\xi \ge \text{2.8}$ in order to make the best use of the material. So, the practical formulas in this paper can be applied to estimate nearly all of the cable tension in
real bridges conveniently and accurately.
The formula in this paper can be simultaneously used to identify the cable’s bending stiffness and force with multiple natural frequencies being measured. The iterative equation is derived and the
iterative steps are given. From the example analysis it can be seen that the bending stiffness identification method presented in this paper is converged quickly, and the satisfied results can be
obtained after only a few steps of iteration.
• Sun Y., Li H. Effect of extreme properties of vertical cable on the cable force measurement by frequency-based method. Engineering Mechanics, Vol. 30, Issue 8, 2013, p. 10-17, (in Chinese).
• Fei Q. G., Han X. L. Identification of modal parameters from structural ambient responses using wavelet analysis. Journal of Vibroengineering, Vol. 14, Issue 3, 2012, p. 1176-1186.
• Kim B. H., Park T. Estimation of cable tension force using the frequency-based system identification method. Journal of Sound and Vibration, Vol. 304, Issue 3-5, 2007, p. 1067-1072.
• Caetano E. On the identification of cable force from vibration measurements. IABSE-IASS Symposium, London, 2011.
• Humar J.L. Dynamics of structures. Prentice Hall, Upper Saddle River, NJ, 1990.
• Zui H., Shinke T., Namita Y. Practical formulas for estimation of cable tension by vibration method. Journal of Structural Engineering-ASCE, Vol. 122, Issue 6, 1996, p. 651-656.
• Mehrabi A. B., Tabatabai H. Unified finite difference formulation for free vibration of cables. Journal of Structural Engineering-ASCE, Vol. 124, Issue 11, 1998, p. 1313-1322.
• Ren W. X., Chen G., Hu W. H. Empirical formulas to estimate cable tension by cable fundamental frequency. Structural Engineering and Mechanics, Vol. 20, Issue 3, 2005, p. 363-380.
• Gan Q., Wang R. H., Rao R. Practical formula for estimation on the tensional force of cable by its measured natural frequencies. Chinese Journal of Theoretical and Applied Mechanics, Vol. 42,
Issue 5, 2010, p. 983-988, (in Chinese).
• Tang S. H., Fang Z., Yang S. Practical formula for the estimation of cable tension in frequrency method considering the effects of boundary conditions. Journal of Hunan University, Natural
Sciences, Vol. 39, Issue 8, 2012, p. 7-13, (in Chinese).
• Fang Z., Wang J. Q. Practical formula for cable tension estimation by vibration method. Journal of Bridge Engineering-ASCE, Vol. 17, Issue 1, 2012, p. 161-164.
• Clough R. W., Penzien J. Dynamics of structures. McGraw-Hill, New York, 1993.
• OVM Engineering Co., LTD. Product manuals of hanging-cable/tied-cable system for arch bridge. Liuzhou OVM Engineering Co., LTD, LiuZhou, China, 2007, (in Chinese).
About this article
25 November 2013
28 February 2014
cable supported bridge
cable tension estimation
vibration method
practical formula
bending stiffness
The authors gratefully acknowledge the financial support by Project supported by the National Natural Science Foundation of China (Grant No. 51208123; No. 51208229), the Project supported by the
Natural Science Foundation of Guangdong Province, China (Grant No. S2012040007317), and the Talent Introduction Project supported by Higher Education Department of Guangdong Province in 2012.
Copyright © 2014 JVE International Ltd.
This is an open access article distributed under the
Creative Commons Attribution License
, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. | {"url":"https://www.extrica.com/article/14873","timestamp":"2024-11-08T23:30:39Z","content_type":"text/html","content_length":"192858","record_id":"<urn:uuid:021fe55b-6075-4370-8ab0-523f5732e18c>","cc-path":"CC-MAIN-2024-46/segments/1730477028106.80/warc/CC-MAIN-20241108231327-20241109021327-00183.warc.gz"} |
Sam the Butterfly - Applying Inequalities
(Also available in Pyret)
Students discover that inequalities have an important application in video games: keeping game characters on the screen! Students apply their understanding to edit code so that it will keep Sam the
Butterfly safely in view.
Students will be able to:
Lesson Goals
• apply their understanding of inequalities to keep a game character on the screen
Student-Facing Lesson Goals • I can use what I know about inequalities to define the boundaries that will keep a character on the screen.
a mathematical description of the relationship between two variables or quantities, in which they are not necessarily equal
🔗Introducing Sam 15 minutes
Students are introduced to Sam the Butterfly, a simple activity in which they must write simple inequalities to detect when Sam has gone too far in one dimension.
• Open the Sam the Butterfly Starter File in a new tab and save a copy of your own.
• Complete Introducing Sam, clicking "Run" and using the arrow keys to investigate the program with your partner.
As students complete the worksheet and explore the program, they should notice that Sam’s coordinates are displayed at the top of the screen. When Sam is at (0,0), we only see a part of Sam’s wing
because Sam’s position is based on the center of the butterfly image. Students should observe that Sam can go up to, but not beyond, an x of -50. Students can represent this algebraically as x > -50$
\displaystyle x \gt -50$, or (for students who notice that Sam only moves in increments of 10) x ≥ -40$\displaystyle x \geq -40$.
Every time Sam moves, we want to check and see if Sam is safe.
Note: In this programming language, question marks are prounced "huh?" So safe-left? would be prounounced "safe left huh?" This can be a source of some amusement for students!
To further support students, consider asking what three functions are defined in their starter files. Then, ask students what each function should do, when working properly.
• What should our left-checking function do?
□ Check to see if x is greater than -50.
• What should our right-checking function do?
□ Check to see if x is less than 690.
• What should onscreen? do?
□ Answers may vary. Let students drive the discussion, and don’t give away the answer!
• Complete Left and Right with your partner.
• Once finished, fix the corresponding functions in your Sam the Butterfly file, and test them out.
Students will notice that fixing safe-left? keeps Sam from disappearing off the left side, but fixing safe-right? doesn’t seem to keep Sam from disappearing off the right side! When students
encounter this, encourage them to look through the code to try and figure out why.
"False" doesn’t mean "Wrong"!
A lot of students - especially confident ones - may struggle to come up with an example where safe-left? returns false:
; Students hate writing the second one! (EXAMPLE (safe-left? 189) (> 189 -50)) (EXAMPLE (safe-left? -65) (> -65 -50))
This misconception comes from confusing a statement that is "false" with a program that is "wrong". In the second example, above, the result of (safe-left? -65) is false, because "65 is greater than
-50" is a false statement. Remind your students that you want one example that’s true, and a second that’s false!
Emphasize to students that they cannot trust the behavior of a complex system! After looking closely at examples and observing that they all pass, students should suspect that the bug is elsewhere.
• Does safe-left? work correctly? How do you know?
• Does safe-right? work correctly? How do you know?
🔗Protecting Sam on Both Sides 30 minutes
Students solve a word problem involving compound inequalities, using and to compose the simpler Boundary-checking functions from the previous lesson.
Note: In this programming language, question marks are pronounced "huh?". So safe-left? would be pronounced "safe left huh?" This can be a source of some amusement for students!
Recruit three student volunteers to roleplay the functions safe-left?, safe-right?, and onscreen?. Give them 1 minute to read the contract and code, as written in the program.
Ask the volunteers what their name, Domain and Range are. Explain that you, the facilitator, will be providing a coordinate input. The functions safe-left? and safe-right? will respond with either
"true" or "false".
The function onscreen?, however, will call the safe-left? function! So the student roleplaying onscreen? should turn to safe-left? and give the input to them.
• Facilitator: "onscreen-huh 70"
• onscreen? (turns to safe-left?): "safe-left-huh 70"
• safe-left?: "true"
• onscreen? (turns back to facilitator): "true"
• Facilitator: "onscreen-huh -100"
• onscreen? (turns to safe-left?): "safe-left-huh -100"
• safe-left?: "false"
• onscreen? (turns back to facilitator): "false"
• Facilitator: "onscreen-huh 900"
• onscreen? (turns to safe-left?): "safe-left-huh 900"
• safe-left?: "true"
• onscreen? (turns back to facilitator): "true"
Hopefully your students will notice that safe-right? did not participate in this roleplay scenario at all!
• What is the problem with onscreen??
□ It’s only talking to safe-left?, it’s not checking with safe-right?
• How can onscreen? check with both?
□ It needs to talk to safe-left? AND safe-right?
• Complete Word Problem: onscreen?.
• When this function is entered into the editor, students should now see that Sam is protected on both sides of the screen.
Extension Option What if we wanted to keep Sam safe on the top and bottom edges of the screen as well? What additional functions would we need? What functions would need to change? We recommend that
students tackling this challenge define a new function is-onscreen-2.
Bring back the three new student volunteers to roleplay those functions, with the onscreen function now working properly. Make sure students provide correct answers, testing both true and false
conditions using coordinates where Sam is onscreen and offscreen.
• How did it feel when you saw Sam hit both walls?
• Are there multiple solutions for onscreen??
• Is this Top-Down or Bottom-Up design?
== Boundary Detection in the Game 10 minutes
=== Overview Students identify common patterns between two-dimensional Boundary detection and detecting whether a player is onscreen. They apply the same problem-solving and narrow mathematical
concept from the previous lesson to a more general problem.
Have students open their in-progress game file and click "Run". Invite them to analyze the movement of the danger and the target
• How are the TARGET and DANGER behaving right now?
□ They move across the screen.
• What do we want to change?
□ We want them to come back after they leave one side of the screen.
• What happens to an image’s x-coordinate when it moves off the screen?
□ An image is entirely off-screen if its x-coordinate is less than -50 and greater than 690.
• How can we make the computer understand when an image has moved off the screen?
□ We can teach the computer to compare the image’s coordinates to a boundary on the number line, just like we did with Sam the Butterfly!
Apply what you learned from Sam the Butterly to fix the safe-left?, safe-right?, and onscreen? functions in your own code.
Since the screen dimensions for their game are 640x480, just like Sam, they can use their code from Sam as a starting point.
=== Common Misconceptions
• Students will need to test their code with their images to see if the boundaries are correct for them. Students with large images may need to use slightly wider boundaries, or vice versa for
small images. In some cases, students may have to go back and rescale their images if they are too large or too small for the game.
• Students may be surprised that the same code that "traps Sam" also "resets the DANGER and TARGET ". It’s critical to explain that these functions do neither of those things! All they do is test
if a coordinate is within a certain range on the x-axis. There is other code (hidden in the teachpack) that determines what to do if the coordinate is offscreen. The ability to re-use function is
one of the most powerful features of mathematics - and programming!
• The same code that "trapped" Sam also "resets" the DANGER and the TARGET. What is actually going on?
These materials were developed partly through support of the National Science Foundation, (awards 1042210, 1535276, 1648684, and 1738598). Bootstrap by the Bootstrap Community is licensed under a
Creative Commons 4.0 Unported License. This license does not grant permission to run training or professional development. Offering training or professional development with materials substantially
derived from Bootstrap must be approved in writing by a Bootstrap Director. Permissions beyond the scope of this license, such as to run training, may be available by contacting | {"url":"https://bootstrapworld.org/materials/fall2022/en-us/lessons/inequalities3-sam-wescheme/index.shtml?pathway=algebra-wescheme","timestamp":"2024-11-07T05:55:51Z","content_type":"text/html","content_length":"36292","record_id":"<urn:uuid:00ef2103-b1c3-4456-91cf-e35396a49ec2>","cc-path":"CC-MAIN-2024-46/segments/1730477027957.23/warc/CC-MAIN-20241107052447-20241107082447-00504.warc.gz"} |
Codeforces Round #626 (Div.1, Div.2, based on Moscow Open Olympiad in Informatics, rated) - Codeforces
Right now happens the first tour of the Open Olympiad in Informatics, and tomorrow will be the second one. This contest is prepared by Moscow Olympiad Scientific Committee that you may know by Moscow
Team Olympiad, Moscow Olympiad for Young Students and Metropolises Olympiad (rounds 327, 342, 345, 376, 401, 433, 441, 466, 469, 507, 516, 541, 545, 567, 583, 594, 622).
Open Olympiad consists of the most interesting and hard problems that are proposed by a wide community of authors, so we decided to conduct a Codeforces regular round based on it, which will happen
on Mar/07/2020 12:35 (Moscow time) and will be based on both days of the Olympiad. Each division will have 6 problems and 2 hours to solve them.
We kindly ask all the community members that are going to participate in the competition to show sportsmanship by not trying to cheat in any manner, in particular, by trying to figure out problem
statements from the onsite participants. If you end up knowing some of the problems of Moscow Open Olympiad (by participating in it, from some of the onsite contestants or in any other way), please
do not participate in the round. We also ask onsite contestants to not discuss problems in public. Failure to comply with any of the rules above may result in a disqualification.
Problems of this competition were prepared by meshanya, cdkrot, wrg0ababd, vintage_Vlad_Makeev, DebNatkh, diko, voidmax, okwedook, ch_egor, V--o_o--V, Sender, grphil, mingaleg, KiKoS, Endagorion,
budalnik guided by cdkrot, ch_egor, vintage_Vlad_Makeev, GlebsHP, Zlobober and Helen Andreeva.
Problems for second division were prepared by vintage_Vlad_Makeev, ch_egor and MikeMirzayanov, to who we also want to say thanks for the Codeforces and Polygon systems.
Good luck everybody!
The scoring distribution for both divisions is standard: 500 — 1000 — 1500 — 2000 — 2500 — 3000
Due to the official competition source codes of other participants will not be available for an hour after the end of the round.
UPD2: Editorial
UPD3: Winners!
Div. 1:
Div. 2:
» 5 years ago, # |
3 years ago? wut?
• » 5 years ago, # ^ |
» +240
vintage_Vlad_Makeev Yeah, this announcement is quite vintage.
□ » 5 years ago, # ^ |
» +25
oh! so problems would be vintage as well.
Hope for no accident!
im ok meme fire!
» 5 years ago, # |
3 years ago :D
» 5 years ago, # |
» 5 years ago, # |
Please no mathforces!
5 years ago, # |
» +29
AkiLotus Blog post from 3 years ago.
Dude I thought my "5 months" were creepy enough. :D
» 5 years ago, # |
What is the onsite format and how will both days be combined in one round? If the onsite is a 5h contest every day, then joining both in a 2h one seems brutal.
• »
» The are two days, each has 4 problems for 5 hours.
• » 5 years ago, # ^ |
» ← Rev. 2 → +1
majorro Each of these problems have subtasks, so they may be given I suppose.
» 5 years ago, # |
5 years ago, # |
← Rev. 2 → -15
The announcement said that this round "will be based on both days of the Olympiad. Each division will have 6 problems and 2 hours to solve them."
What does "both" means?
A. The problems of two divisions are same problems;
B. No problem exists in both division simultaneously;
My english is poor. Could anyone help me with it?
• » 5 years ago, # ^ |
» ← Rev. 2 → -28
majorro .
• 5 years ago, # ^ |
» +33
Here "both" means that problems from two days of Olympiad will be used for Codeforces round.
Each division will have 6 problems. Some of them are common, some of them are not.
□ »
» Got it, thx :)
• » 5 years ago, # ^ |
» +61
vintage_Vlad_Makeev As you can see, preparation of this contest started a long time ago (preparing most of the problems started even before announcement writing), so some of them we prepared by
vintage Vlad Makeev and some by modern Vlad Makeev.
□ »
» preparing most of the problems started even before announcement writing
You mean more than 3 years ago? XD
5 years ago, # |
» ← Rev. 2 → +13
adamant Should I prepare my vintage Paint MS for a vintage meme about vintage_Vlad_Makeev , vintage_Vlad_Makeev , vintage_Vlad_Makeev and that other guy whose handle I'm not going to type from my
• » 5 years ago, # ^ |
» +17
rotavirus Plz make a meme about yuhao
What's the score for each problem?
• »
» The scoring distribution for both divisions is standard: 500 — 1000 — 1500 — 2000 — 2500 — 3000
» Can anyone explain how and how much rating increases or decreases depending on what factor ? I read this FAQ from codeforces Codeforces FAQ rating and div link but it is still not clear
how much it changes and on what factor?
• »
» This might be the simplest explanation: Based on your rating before a contest starts, you are given an "expected rank", ie, you are expected to perform that well. If you outperform your
expected rank then your rating increases and if you underperform, your rating decreases.
• »
» You can see the formula here. Though it is important, I don't care that much and use CF-predictor with live rating changes instead.
» 5 years ago, # |
There are 11 Candidate Master among Div.2 registration and 9 Expert among Div.1 registration. When will it be corrected?
» 5 years ago, # |
Who else checked the date? :p
Is this contest rated?
• » 5 years ago, # ^ |
» +21
vintage_Vlad_Makeev Yes, vintage contests are usually rated.
» 5 years ago, # |
5 years ago, # |
← Rev. 2 → +92
Schools are like:
adzo261 Taught in Class: Div2 A
Solved for practice: Div2 B
Homework: Div2 C
Exams: Div2 D,E,F
After solving C
Why are we not allowed to copy others code in our system to check for hacking? We are only allowed to see others code. Previously we were allowed to copy also (around 7-8 months back).
• »
» Only in edu round, you are allowed to copy.
» 5 years ago, # |
» 5 years ago, # |
How to solve Div 2 D?
» 5 years ago, # |
Problem Div1 B was Info1Cup 2017 XORsum
• » 5 years ago, # ^ |
» +12
mayank_ag Its a little difficult to understand from the editorial given. Can you simplify the approach to solve this question?
□ »
» 5 years ago, # |
How to solve Div2B
• »
» I just search for windows with dimension a*b where a*b = K. I maintained prefix array of both the given array, and searched how many window of dimension 'a*1' are there in array A and
how many window of dimension 'b*1' are there in array B. Multiply both the count and add to the final answer. This is O(sqrt(K) * (N+M)) solution.
» 5 years ago, # |
Hate SpeedForces.
How to solve div2 B??
» 5 years ago, # |
Thanks for this great problemset, I haven't seen better one on Codeforces in a while.
• » 5 years ago, # ^ |
» +50
vintage_Vlad_Makeev i already told you swistakk as the age passes contests become much and much better as they were in your young days.
» 5 years ago, # |
How to solve div 1 B. My time complexity was O(n log(n)*log(1e7)) which gave TLE on test case 4.
• »
» I tried to find for every $$$k$$$-th bit how many number are there such that $$$Bi+Bj$$$ $$$mod$$$ $$$2^{k+1}$$$ $$$>= 2^k$$$ where $$$Bi=Ai$$$ $$$mod$$$ $$$2^{k+1}$$$. I don't know if
this is correct I had some bugs in code.
□ »
» Exactly the same idea..
• »
» Same problem with me too, I implemented binary search instead of using lower_bound. It is giving TLE on test case 4. People who have used lower_bound did not get TLE. My submission: 72730048
How to solve div2 D?
What is testcase 6 in div2C?
• »
» maybe something like 6 ))()((
□ »
» What is the expected answer for that?
☆ »
» 6
I have no idea for div2 D!!!
How to solve div 2 D? Was looking at a trie based approach but how to handle carries properly?
• » 5 years ago, # ^ |
» +3
notnamed To calculate if there's a carry in n-th binary position you can modify array a, and set bit n and higher bits to 0 in every number. Then after sorting array it is easy to count pairs of
numbers with sum of 2^n or more. Pair counting step can be done in O(n), but I used binary search, which is n*log(n), and it worked fine.
Did anyone solve with idea that converting to problem for binary array and do some stuff. btw I had something to do so I couldnt continue competing.
How to solve DIV 2D?? is it based on calculating freq array of all pair less than o(n^2) if yes then how?
• »
» freq array off all pair can be calculated in o(nlogn) time using FFT but I receive Memory limit.
• »
» sorry. it can be calculated in o(10^7log(10^7)) time.
5 years ago, # |
» +51
Rewritetxdy Before the contest: I want to solve Div1 D!
After the contest: How to solve Div1 B and C ??
» 5 years ago, # |
I have never seen a shitty contest like this. The problemset(Div.2) was completely imbalanced. Disgusting!!!
5 years ago, # |
» +26
cuber_coder Div2D/1B was copied : https://oj.uz/problem/view/info1cup17_xorsum
Would still like to know how did you guys solve it ?
5 years ago, # |
← Rev. 2 → +74
My solution to B: A sum $$$s = a_i+a_j$$$ (w.l.o.g. both $$$\lt 2^{k+1}$$$) has a bit $$$2^k$$$ set iff $$$s \ge 2^k$$$ and either $$$s \lt 2^{k+1}$$$ or $$$s \ge 2^{k+1}+2^k$$$. For each
$$$k$$$, use a Fenwick tree to count them. $$$O(N \log^2)$$$ with a good constant.
» C: Consider vertices from the right half. Let's ignore vertices with no edges to the left half. Merge vertices with the same sets of left neighbours. The answer is the GCD of the total weight
and weights of all these merged vertices. Complexity: $$$O(M + N \log)$$$ with hashing.
Proof: consider the answer $$$g$$$ and its divisor $$$d$$$, which must divide the total weight. Among the merged vertices whose weights aren't divisible by $$$d$$$, find one with the smallest
degree. If we take all vertices from the left half that aren't adjacent to it, then we take all other vertices from the right half, since all other merged vertices have $$$\ge$$$ degree and
those with equal degree don't have the same set of neighbours, so their weight (total weight — weight of this not taken merged vertex) is indivisible by $$$d$$$.
D: There's a reasonably straightforward DP: for each aggressiveness $$$l$$$ and each suffix of already chosen candidates, calculate the maximum cost if we have $$$j$$$ people that fought and
crossed over to aggressiveness $$$l+1$$$. This is $$$O(N^2M)$$$, but we can notice that the number of people that can cross over to gradually higher $$$l$$$ decreases exponentially, so we can
remember the maximum $$$j$$$ that resulted in a reachable state for each $$$l$$$ and suffix. Complexity: $$$O(ok)$$$ probably.
• 5 years ago, # ^ |
← Rev. 3 → +8
Another proof of C would be the following.
For a sets $$$S$$$, $$$T$$$, define $$$\gcd(S, T)$$$ as $$$\gcd$$$ of sum of elements of $$$S$$$ and $$$T$$$. We note the following, if $$$A \subseteq B$$$ then $$$\gcd(A, B) = \gcd(A, B
- A)$$$.
» For two vertices we want $$$\gcd(N(u_1), N(u_2), N(u_1) \cup N(u_2))$$$, if $$$A = N(u_1)$$$ and $$$B = N(u_2)$$$.
$$$\gcd(A, B, A\cup B) = \gcd(A, B, A\cup B - A) = \gcd(A, B, B - A)$$$
islingr $$$= \gcd(A, B - (B - A), B - A) = \gcd(A, A \cap B, B - A)$$$
$$$= \gcd(A - A\cap B, A \cap B, B - A) = \gcd(A \cap B, A - B, B - A).$$$
Notice that all these are all the sections of the Venn Diagram of $A$ and $$$B$$$. We can easily generalize to more sets (just use induction).
We effectively have to sum all the nodes that are in the same section of the Venn Diagram. Two nodes are in the same section of the Venn Diagram if and only if, they are in the same $$$N
(u_i)$$$'s which is the same as saying they have the same left neighbours. $$$\blacksquare$$$
The dumbass I am, I didn't realize that the problem was trivial from here and managed to not solve it. :'(
5 years ago, # |
I don't know what's wrong with this code for Div2 B. It's giving WA on testcase 3. Can anyone please tell my why??
define lli long long int
define ulli unsigned long long int
pragma GCC target ("sse4.2")
using namespace std;
define fast ios_base::sync_with_stdio(0);cin.tie(NULL);cout.tie(NULL)
define time cout<<"\nTime Elapsed: " << 1.0*clock() / CLOCKS_PER_SEC << " sec\n";
int main() { fast; lli n,m,k; cin>>n>>m>>k; vector<pair<lli,lli>>vp; for(lli i=1;i<=sqrt(k);i++) { if(k%i==0) { vp.push_back({i,k/i}); if(i!=k/i) vp.push_back({k/i,i}); } } /*for(auto
i:vp) { cout<<i.first<<" "<<i.second<<"\n"; } */ vectora,b; for(lli i=0;i<n;i++) { lli x; cin>>x; a.push_back(x); } for(lli i=0;i<m;i++) { lli x; cin>>x; b.push_back(x); }
for(lli i=1;i<n;i++)
for(lli i=1;i<m;i++)
» {
Runtime_Error97 if(dpb[i-1])
/*for(lli i=0;i<dpa.size();i++)
cout<<dpa[i]<<" ";
for(lli i=0;i<dpb.size();i++)
cout<<dpb[i]<<" ";
lli las=la.size();
lli lbs=lb.size();
lli sz=vp.size();
lli total=0;
for(auto i:vp)
for(lli j=0;j<las;j++)
for(lli k=0;k<lbs;k++)
return 0;
• »
» can someone tell me how to ask question so that i will not get so many downvotes. I am new to codeforces. THis was my first contest. And i don't know what i did for getting so
many downvotes
□ »
» First important thing is to not post long code directly into your comment. Use a link instead
» 5 years ago, # |
Contest was unbalanced. Difficulty level between C and D was very wide.
» 5 years ago, # |
The gap between Div2 C and Div2D doesn't nice at all
5 years ago, # |
» +42
SPatrik For Div1 C, I made a randomized solution, which is completely wrong. I submitted it 16 times before it got Pretests passed. So if I count with 1/16 chance to pass 12 tests and with about 96
main tests (8*12), then I have about (1/16)^8≈2*10^(-10) chance to pass main tests, so about 1 in a 5 billion. I like my chances!
• »
» I really wouldn't want to be the one preparing tests for div1C. Not only is it possible to make a correct solution without realising it (see above), it's really easy to make a solution
that's almost correct and distinguishing between them reliably is hard.
□ 5 years ago, # ^ |
» +38
» Oh yeah, it was quite hard to come with idea of strong tests. However there is a test with ~7500 non-zero degrees vertices in left half and exactly one subset of size 5 of
vertices that you should handle to find proper GCD and connected graph.
You can try to find such test yourself it's a very nice problem.
□ »
» You can make a solution by simply comparing neighboring vectors by size and elements instead of hashing. But indeed building strong tests for it are so difficult to me. I prefer
» preparing problems with easy work in generating data ^o^.
• 5 years ago, # ^ |
← Rev. 6 → 0
» Mine's a randomised and a bit greedy solution as well. (greedily sorting vertices with several comparators and considering all prefix of vertices as subsets to calculate gcd).
It passes systests, submitted after the contest. 72661488
Unfortunately I tle'd systests during the contest with the same soln 72655512 thanks to using endl instead of '\n'. T-T
Greedy comparators used: Sort left vertices in order of how less connected their right neighbours are to left vertices. Sort by degree.
5 years ago, # |
» D was much easier than C, in my opinion (just a straightforward dp). Also including isolated vertices in C was pure evil, Took me around a quarter to figure that out :/
KMAASZRAA Btw is the intended solution to E parallel binary search on the value of each index and then some book-keeping of left and right pointer for each index via DSU? If so, it was a bad idea to
have it as E since not many would reach it in time (though it isn't much hard).
Nice problemset overall.
• 5 years ago, # ^ |
← Rev. 2 → +21
» For me it was the opposite: C was very straightforward, I just looked at when some $$$x$$$ divides the GCD, and it was easy to see and prove that this holds when every non-isolated
right-side vertex has value divisible by $$$x$$$ (assuming every right-side vertex has a distinct set of connections. If they do not, just join vertices with the same connections). I
mango_lassi did get one WA from the isolated vertex case though.
On the other hand, in D I had to work with this monstrosity: 72679047, which I couldn't optimise and debug in time.
» 5 years ago, # |
Is it just me or is the TL for E strict? I should have used a fast set instead of std::set.
• »
» I used multiset, set and segment tree at the same time (for different purposes), 1247 ms on pretests
□ »
» Seems like my implementation has a bad constant. My solution runs in 3sec on my machine (which is a bit faster than CF), and with fastset it runs in 1sec.
☆ »
» Ok mine is too slow too :(
• » 5 years ago, # ^ |
» +58
ksun48 n log n, TLE systests, byebye
» 5 years ago, # |
How to solve div1C?
» 5 years ago, # |
Good Balance!
» 5 years ago, # |
• »
» As the editorial, my O(NlogNlogMAX) time solution get TLE, maybe my implementation is not optimal enough. By the way, if I use int instead of int_64, I get wrong answer on the same test
case, which means within time limits.
Amazing problems, thanks for them!
» 5 years ago, # |
Btw, statement of F was TERRIBLE, it got so many stupid things I am not even gonna point them out, cause it would take me too long
For Div2, Over 2k solutions for C, and about 100 for D. Quite a shift though
» 5 years ago, # |
I Hate SpeedForces :(
» 5 years ago, # |
Good problemset, but as I suspected squishing a 2-day 4-hour(?) contest into a single 2h round is just gonna make it too difficult.
• 5 years ago, # ^ |
» +46
Yeah, it's quite sad that nobody had time for F, I like this problem very much :(
However we dropped some hard problems from original competition to make this contest more solvable, but probably 0.5-1 extra hours were probably required.
5 years ago, # |
Why is this solution to 1C getting TLE on test 72？
It seems many other people got TLE on the same testcase.
• » "Due to the official competition source codes of other participants will not be available for an hour after the end of the round."
Wait a bit or paste the code to somewhere else, if you want to get an answer.
□ »
» Probably sensible to not paste code somewhere else as that kind of defeats the purpose of making codes not available. Just wait.
☆ »
» 5 years ago, # ^ |
» -30
A rule that can be broken without repercussions is useless.
• »
» It seems I'm getting TLE due to std::cin. Why such a tight TL... Or is it intended to fail solutions with slower I/O?
□ »
» You are probably just doing it wrong. Using endl or sth like this. It's never an intention to fail solutions with iostream since it is not slower than cstdio when used properly.
☆ »
» Then how to use iostream "right"? I've done nothing but to replace std::cin>>a[i] with scanf("%lld",&a[i]) and it passed.
○ »
» ios::sync_with_stdio(0), cin.tie(0), cout.tie(0);.
» And '\n' instead of endl.
■ »
» Thanks,it passed after doing this.
□ »
» Yes, I already replaced "cout<<ans<<endl" by "printf("%lld\n",ans)" and got accepted in less than one second, after failing by TLE at 72.
5 years ago, # |
» +11
sitaram After long time i will reach on specialist.
Thank you codeforces .
» 5 years ago, # |
← Rev. 3 → 0
So all the pretests in Div1B are either $$$n$$$ is even or $$$a_1\oplus a_2 \oplus a_3 \dots \oplus a_n=0$$$... lol
Pretest is weak?
» 5 years ago, # |
The most important part for Div.1 C is not to use std::endl.
• »
» Why does it matter so much? You're printing one integer.
□ »
» I thought so too during the contest. But it has multiple test cases.
☆ »
» 5 years ago, # ^ |
» -26
• »
» Even if one uses the standard ordered map for storing vectors, no problem. Just the "cout" is the problem.
My solution for Div 1 A shows Pretest passed and I did not get any points for it. Does it mean it failed system test or something else? It is weird..
» 5 years ago, # |
Will I get minus if I got wrong answer on pretest 1?
• »
» No penalties for solutions failed on (pre)test 1
□ »
» I know about NO PENALTIES. But what about getting minus?
☆ »
» You mean something like "-1" "-2"("tried" counter)?
» Failing on (pre)test 1 won't be counted in the "tried" counter
○ »
» Forget about it. I already got minus in rating.
■ »
» You mean rating change?I didn't understand it :( sorry for inconvenience
» Being a master was truly magnificent experience.
Fly_37 Au revoir!
5 years ago, # |
» +24
ZhouShang0817 I took part in Codeforces Round #626 (Div. 1, based on Moscow Open Olympiad in Informatics) just now. I passed the pretest of problem A. But the system didn't send my solution into
system test. Why does this happen？
• » 5 years ago, # ^ |
» +16
aditya_sheth Same with me :(
□ »
» Are you extra registration?
☆ »
» Yes!
○ »
» I am extra registration too. And did you use m1.codeforces or m2.codeforces.com or m3.codeforces.com?
■ »
» No i used codeforces.com, the reason seems to be extra registration.
★ »
» I suddenly accepted. It really confused me.
5 years ago, # |
← Rev. 5 → -35
My submission 72636993 got a TLE during system testing. Submitting the same exact same code does gets an AC!! Also the TLE was on test 5 wasn't it already included in the pretests??
The submissions:
The below submissions I just added another variable x, just to bypass submitting the exact same code twice. Other than that there is no difference in the codes.
Submissions image
vintage_Vlad_Makeev vintage_Vlad_Makeev ch_egor MikeMirzayanov
• » 5 years ago, # ^ |
» +12
vintage_Vlad_Makeev Your solution works quite close to time limit. There are always some fluctuations in program working time, so it's just a bad luck that your submission got TL on system tests.
□ »
» Thanks for the reply.
» 5 years ago, # |
How come my code for B fail at test case 5 ,which was already present in pretests.
• » The execution time may be different in runs.
If your code consumed almost 1000ms(e.g:996ms) in the pretests, it's probably to fail in the main tests.
□ »
» Hello, you know when the rate change?
☆ »
» in several hours? no one knows the exact time.
5 years ago, # |
Okay, I will say it. The pretest for Div1 C is fucking garbage.
I passed pretests using only 580ms. The time limit is 2s. My solution is deterministic, and runs in basically the same time given that the input size is fixed.
» It got TLE on test 72.
MofK I then changed from cout << ans << endl; to cout << ans << '\n';. It now runs in 800ms.
It means that it takes at least 1.2s to flush 500000 times, which implies there is no such test where $$$t = 500000$$$ in the pretest (I really doubt the maximum $$$t$$$ in the pretest is even
close). Now I'm pretty sure that Codeforces has the policy on problems that the pretest must satisfy i) it contains all corner cases on which some known solution fails and ii) all parameters
specified in the problem must hit their respective maximal values. With that knowledge, I put my trust in the problem setters that they have at least complied with the policy and believed that
Codeforces actually could handle 500000 flushes better than I thought. Turned out I was wrong on both, LOL.
It would be really great if anyone involved can explain the thought process (or lack thereof) behind the decision not to put such tests in the pretest.
• » 5 years ago, # ^ |
» -67
aid So you knew that flushes could be a problem, but decided to use endl anyway, hoping that jury will comply with some imaginary policy that noone except you knows? Seems like you got what you
□ » 5 years ago, # ^ |
» +99
Not including all "simple" types of max tests (and I believe $$$t=500\,000$$$ is one of them) is a questionable decision, especially considering that large $$$t$$$ might be here for a
mnbvmar reason. For example, my first idea involved factorizing a number as large as $$$10^{12}$$$, which is pretty hard when you have to do it $$$t$$$ times within 2 seconds.
□ 5 years ago, # ^ |
I did not know how long it takes for Codeforces to process that amount of flushes. I used flushes to debug my solution locally, and forgot about it. I submitted the solution, saw that it
passed within 1/3 of the time limit, and was completely convinced that it would not be that big of a problem.
» I didn't say that I deserved to pass using 500000 flushes; however, if such tests were present in the pretest, I would have known immediately what the issue was.
» For the "imaginary policy" part, feel free to enlighten yourself. Some quotes taken from the document:
MofK • Always make tests (at least) with minimum possible, maximum possible and some value in between for each variable and their combinations.
• In general, pretests should include: a test with maximum size of input (e.g. maximum n), a test against integer overflow, a few random small tests.
• There should be no warnings in polygon, except (if reasonable) "there are tests with input/output for statements [without TeX marks]". ("parameter 't' does not hit maximal value" is
one such warning).
I have involved in preparing some Codeforces rounds, I know what I am talking about. I don't pull facts out of my ass. Have a nice day.
☆ » 5 years ago, # ^ |
» -43
» First and third points are about all tests, not pretests. Second point only implies that there should be a pretest with sum of $$$n$$$ equal to $$$500000$$$. Still don't understand
why would you expect pretests to be perfect.
○ » 5 years ago, # ^ |
» +34
» FYI, here is a screenshot taken from Polygon:
As you can see, the system does warn about pretests being incomplete. In fact, if you insist on trying to nitpick things, the test with $$$t=500000$$$ and $$$n=1, m=1$$$ for each
MofK case also gives the maximum input size possible (about $$$1$$$ million more integers to read), therefore it should be included in the pretest (this is rather redundant, since the
third quote already pointed that out anyway).
■ »
» 5 years ago, # ^ |
» -74
» OK, authors are wrong. But still you relied on something that is not written in the rules (for contestants) and got punished. Seems right.
★ »
» 5 years ago, # ^ |
» +47
» Yeah I mean people also expect each other to not tell a dick joke at the funeral, even if it's not written in the rules, and will be upset (rightfully so) if one does.
» But fair point I guess. Sorry for the bad analogy.
★ »
» 5 years ago, # ^ |
» ← Rev. 3 → +26
» Ignore: in prev version Russian colypasted joke
• »
» 5 years ago, # ^ |
» 5 years ago, # |
I guess B and C got swapped :(
» so sad, this sentence "Then output k distinct integers (1≤pi≤n), indexes of the chosen elements" in div2 A, Let me misunderstanding, i think it's k distinct integers a[i], not the index
of a[i].
» 5 years ago, # |
Unfortunately, Problem Div.1 B coincides with atcoder ARC092D.. TAT But it's nice problem anyway.
There is a small typo in the input section of the problem statement of the problem Div.1 D.
» 5 years ago, # |
← Rev. 2 → +3
» 5 years ago, # |
shreya-singh Can anyone please tell me why this submission (for Div 2 B) gave run time error on TC 34, as on my compiler, for same case it gives the expected output 0 as answer? https://
• »
» in your solution problem is with this statement " i=v.size()-1 " here v.size() is unsigned integer. make sure to convert it into int before using it like this "i=(int)v.size()-1".
Can someone explain div1D because I can't get much of what the editorial is saying?
» 5 years ago, # |
Div 2 was shit first 3 very easy and other all was out of mind #fuckedup
» 5 years ago, # |
Wish Codeforces could be better and better. | {"url":"https://mirror.codeforces.com/blog/entry/52457","timestamp":"2024-11-04T07:53:23Z","content_type":"text/html","content_length":"612866","record_id":"<urn:uuid:cd4d2c38-8ec4-4812-a944-eca6d2ba6e94>","cc-path":"CC-MAIN-2024-46/segments/1730477027819.53/warc/CC-MAIN-20241104065437-20241104095437-00003.warc.gz"} |
Functional Programming Using C++ Templates (Part 1)
Template metaprogramming can initially seem baffling, but exploring its link to functional programming helps shed some light on things.
Computing can be a surprisingly deep field at times. I find that the more I learn about it, the more I'm struck by quite how many similarities there are between different areas of the subject. I was
browsing through Andrei Alexandrescu's fascinating book Modern C++ Design recently when I read about a connection which I thought was worth sharing.
As I suspect most of you will already be aware, C++ can be used for something called template metaprogramming, which makes use of C++'s template mechanism to compute things at compile time. If you
take a look at a template metaprogram, however, you'll find that it looks nothing like a 'normal' program. In fact, anything but the simplest metaprogram can start to look quite intimidating to
anyone who's unfamiliar with the idioms involved. This makes metaprogramming seem hard, and can put people off before they've even started.
Surprisingly, the key to template metaprogramming turns out to be functional programming. Normal programs are written in an imperative style: the programmer tells the computer to do things in a
certain order, and it goes away and executes them. Functional programming, by contrast, involves expanding definitions of functions until the end result can be easily computed.
Programmers who have studied computer science formally at university are likely to have already come across some form of functional programming, perhaps in a language such as Haskell, but for many
self-taught programmers the idioms of functional programming will be quite new. In this article, I hope to give a glimpse of how functional programming works, and the way it links directly to
metaprogramming in C++.
For a more detailed look at functional programming, readers may wish to take a look at [Thompson] and [Bird]. Anyone who's interested in template metaprogramming in general may also wish to take a
look at the Boost MPL library [Boost]. Finally, for a much deeper look at doing functional programming in C++, readers can take a look at [McNamara].
Compile-time lists
As a concrete example, I want to consider a simple list implementation. For those who are unfamiliar with them, Haskell lists are constructed recursively. A list is defined to be either the empty
list, [], or an element (of the appropriate type) prefixed, using the : operator, to an existing list. The example [23,9,84] = 23:[9,84] = 23:9:[84] = 23:9:84:[] shows how they work more clearly.
Working only with lists of integers (Integers in Haskell) for clarity at the moment, we can define the following functions to take the head and tail of a list:
head :: [Integer] -> Integer
head (x:xs) = x
tail :: [Integer] -> [Integer]
tail (x:xs) = xs
The head function takes a list of integers and returns an integer (namely, the first element in the list). The tail function returns the list of integers remaining after the head is removed. So far,
so mundane (at least if you're a regular functional programmer).
Now for the interesting bit. It turns out that you can do exactly the same thing in C++, using templates. (This may or may not make you think 'Aha!', depending on your temperament.) The idea (à la
Alexandrescu) is to store the list as a type. We declare lists of integers as follows:
struct NullType;
template <int x, typename xs> struct IntList;
The NullType struct represents the empty list, []; the IntList template represents non-empty lists. Using this scheme, our list [23,9,84] from above would be represented as the type IntList<23,
IntList<9, IntList<84, NullType> > >. A key point here is that neither of these structs will ever be instantiated (that's why they're just declared rather than needing to be defined): lists are
represented as types here rather than objects.
Given the above declarations, then, we can implement our head and tail functions as shown in Listing 1.
template <typename T> struct Head;
template <int x, typename xs>
struct Head<IntList<x,xs> > {
enum { value = x };
template <typename T> struct Tail;
template <int x, typename xs>
struct Tail<IntList<x,xs> > {
typedef xs result;
Listing 1
Already some important ideas are emerging here. For a start, if we ignore the fact that the C++ version of the code is far more verbose than its Haskell counterpart (largely because we're using C++
templates for a purpose for which they were never designed), the two programs are remarkably similar. We're using partial template specialization in C++ to do the job done by pattern-matching in
Haskell. Integers are being defined using enums and lists are defined using typedefs (remember once again that lists are represented as types).
Using these constructs is rather clumsy. A program outputting the head of the list [7,8], for example, currently looks like:
#include <iostream>
int main() {
std::cout << Head<IntList<7,IntList<8,
NullType> > >::value << std::endl;
return 0;
To improve this sorry state of affairs, we'll use macros (this is one of those times when the benefits of using them outweigh the disadvantages). In a manner analogous to that used for 'typelists' in
Modern C++ Design, we define the macros in Listing 2 to help with list creation.
#define NULLLIST NullType
#define INTLIST1(n1) IntList<n1, NULLLIST>
#define INTLIST2(n1,n2) IntList<n1, INTLIST1(n2)>
#define INTLIST3(n1,n2,n3) IntList<n1, INTLIST2(n2,n3)>
#define INTLIST4(n1,n2,n3,n4) IntList<n1, INTLIST3(n2,n3,n4)>
Listing 2
We also define macros for head and tail:
#define HEAD(xs) Head<xs>::value
#define TAIL(xs) Tail<xs>::result
The improvement in the readability and brevity of the code above is striking:
std::cout << HEAD(INTLIST2(7,8)) << std::endl;
From now on, we will assume that when we define a new construct, we will also define an accompanying macro to make it easier to use.
Outputting a list
Before implementing some more interesting list algorithms, it's worth briefly mentioning how to output a list. It should come as no surprise that the form of our output template differs from the
other code in this article: output is clearly done at runtime, whereas all our other list manipulations are done at compile-time. We can output lists using the code in Listing 3.
template <typename T> struct OutputList;
template <> struct OutputList<NullType> {
void operator()() {
std::cout << "Null" << std::endl;
template <int x, typename xs>
struct OutputList<IntList<x,xs> > {
void operator()() {
std::cout << x << ' ';
Listing 3
Computing the head and tail of a list constructed in a head:tail form may seem a relatively trivial example. Our next step is to try implementing something a bit more interesting: sorting. Perhaps
surprisingly, this isn't actually that difficult. The analogy between functional programming in Haskell and compile-time programming in C++ is extremely deep, to the extent that you can transform
Haskell code to C++ template code almost mechanically. For this article, we'll consider two implementations of sorting, selection sort and insertion sort (it would be just as possible, and not a
great deal harder, to implement something more efficient, like quicksort: I'll leave that as an exercise for the reader). I've confined my implementation to ordering elements using operator<, but it
can be made more generic with very little additional effort.
A simple selection sort works by finding the minimum element in a list, moving it to the head of the list and recursing on the remainder. We're thus going to need the following: a way of finding the
minimum element in a list, a way of removing the first matching element from a list and a sorting implementation to combine the two. Listing 4 shows how we'd do it in Haskell.
minElement :: [Int] -> Int
minElement [m] = m
minElement (m:ms) = if m < least then m else least
here least = minElement ms
remove :: Int -> [Int] -> [Int]
remove n (n:ms) = ms
remove n (m:ms) = m : (remove n ms)
ssort :: [Int] -> [Int]
ssort [] = []
ssort ms = minimum : ssort remainder
where minimum = minElement ms
remainder = remove minimum ms
Listing 4
We can transform this to C++ as shown in Listing 5.
// Finding the smallest element of a list
template <typename T> struct MinElement;
template <int x>
struct MinElement<IntList<x,NullType> > {
enum { value = x };
template <int x, typename xs>
struct MinElement<IntList<x,xs> > {
enum { least = MinElement<xs>::value };
enum { value = x < least ? x : least };
// Removing the first element with a given value
// from a list
template <int n, typename T> struct Remove;
template <int n, typename xs> struct Remove<n,
IntList<n,xs> > {
typedef xs result;
template <int n, int x, typename xs>
struct Remove<n, IntList<x,xs> > {
typedef IntList<x,
typename Remove<n,xs>::result> result;
// Sorting the list using selection sort
template <typename T> struct SSort;
template <> struct SSort<NullType> {
typedef NullType result; };
template <int x, typename xs>
struct SSort<IntList<x,xs> > {
enum {
minimum = MinElement<IntList<x,xs> >::value };
typedef typename Remove<minimum,
IntList<x,xs> >::result remainder;
typedef IntList<minimum,
typename SSort<remainder>::result> result;
Listing 5
The important things to note here are that each function in the Haskell code corresponds to a C++ template declaration, and each pattern-matched case in the Haskell code corresponds to a
specialization of one of the C++ templates.
Implementing insertion sort is quite interesting. The essence of the algorithm is to insert the elements one at a time into an ordered list, preserving the sorted nature of the list as an invariant.
A simple Haskell implementation of this goes as follows:
insert :: Int -> [Int] -> [Int]
insert n [] = [n]
insert n (x:xs) = if n < x
then n:x:xs else x:(insert n xs)
isort :: [Int] -> [Int]
isort [] = []
isort (x:xs) = insert x (isort xs)
Translating the insert function to C++ is not entirely trivial. The problem is that we need to generate one of two different types depending on the value of a boolean condition, which is non-obvious.
There are (at least) two solutions to this: we can either rewrite the Haskell function to avoid the situation, or we can write a special C++ template to select one of two typedefs based on a boolean
Rewriting the Haskell code could be done as follows:
insert :: Int -> [Int] -> [Int]
insert n [] = [n]
insert n (x:xs) = smaller : (insert larger xs)
where (smaller,larger) = if n < x then (n,x)
else (x,n)
This solves the problem (generating one of two different values depending on the value of a boolean condition is easy), but at the cost of a less efficient function.
The template version (using the Select template borrowed directly from Andrei's book) does a better job:
template <bool b, typename T, typename U>
struct Select {
typedef T result;
template <typename T, typename U>
struct Select<false, T, U> {
typedef U result;
This allows us to straightforwardly transform the more efficient form of the Haskell code to C++ (Listing 6).
// Inserting a value into an ordered list
template <int n, typename T> struct Insert;
template <int n> struct Insert<n, NullType> {
typedef IntList<n, NullType> result;
template <int n, int x, typename xs>
struct Insert<n, IntList<x,xs> > {
typedef IntList<n, IntList<x,xs> > before;
typedef IntList<x,
typename Insert<n,xs>::result> after;
typedef typename Select<(n < x), before,
after>::result result;
// Sorting the list using insertion sort
template <typename T> struct ISort;
template <> struct ISort<NullType> {
typedef NullType result; };
template <int x, typename xs>
struct ISort<IntList<x, xs> > {
typedef typename Insert<x,
typename ISort<xs>::result>::result result;
Listing 6
It turns out that in C++ this still isn't as efficient as it could be. The culprit is in the second specialization of Insert - by defining the before and aftertypedefs in the specialization itself,
we force them both to be instantiated even though only one is actually needed. The solution is to introduce an extra level of indirection (Listing 7).
template <int n, int x, typename xs>
struct InsertBefore {
typedef IntList<n, IntList<x,xs> > result;
template <int n, int x, typename xs>
struct InsertAfter {
typedef IntList<x,
typename Insert<n,xs>::result> result;
template <int n, int x, typename xs>
struct Insert<n, IntList<x,xs> > {
typedef InsertBefore<n,x,xs> before;
typedef InsertAfter<n,x,xs> after;
typedef typename Select<(n < x), before,
after>::result::result result;
Listing 7
This solves the problem, because now the chosen IntList template only gets instantiated if it is actually needed.
Maps and filters
One of the best things about writing in a functional language has traditionally been the ability to express complicated manipulations in a simple fashion. For example, to apply the same function f to
every element of a list xs in Haskell is as simple as writing map f xs. Similarly, filtering the list for only those elements satisfying a boolean predicate p would simply be filter p xs. A
definition of these functions in Haskell is straightforward enough:
map :: (a -> b) -> [a] -> [b]
map f [] = []
map f (x:xs) = (f x) : map f xs
filter :: (a -> Bool) -> [a] -> [a]
filter p [] = []
filter p (x:xs) = if p x then x : remainder
else remainder
where remainder = filter p xs
Achieving the same thing in C++ initially seems simple, but is actually slightly subtle. The trouble is in how to define f and p. It turns out that what we need here are template template parameters.
Both f and p are template types which yield a different result for each value of their template argument. For instance, a 'function' to multiply by two could be defined as:
template <int n> struct TimesTwo {
enum { value = n*2 };
and a predicate which only accepts even numbers could be defined as
template <int n> struct EvenPred {
enum { value = (n % 2 == 0) ? 1 : 0 };
The Map and Filter templates can then be defined as in Listing 8.
template <template <int> class f,
typename T> struct Map;
template <template <int> class f>
struct Map<f, NullType> {
typedef NullType result;
template <template <int> class f, int x,
typename xs>
struct Map<f, IntList<x, xs> > {
enum { first = f<x>::value };
typedef IntList<first,
typename Map<f,xs>::result> result;
template <template <int> class p, typename T>
struct Filter;
template <template <int> class p>
struct Filter<p, NullType> {
typedef NullType result;
template <template <int> class p, int x,
typename xs>
struct Filter<p, IntList<x,xs> > {
enum { b = p<x>::value };
typedef typename Filter<p,xs>::result remainder;
typedef typename Select<b, IntList<x,remainder>,
remainder>::result result;
Listing 8
Note that we again make use of the Select template to choose between the two different result types in Filter.
So far, we've only seen how to implement integer lists. There's a good reason for this - things like doubles, for example, can't be template parameters. All isn't entirely lost, however. It turns out
that we can make lists of anything that can be represented by integers at compile-time! The code looks something like Listing 9.
template <int n> struct Int {
typedef const int valueType;
static valueType value = n;
template <int n, int d> struct Rational {
typedef const double valueType;
static valueType value;
template <int n, int d> const double
Rational<n,d>::value = ((double)n)/d;
template <typename T> struct Head;
template <typename x, typename xs>
struct Head<List<x,xs> > {
typedef x result;
#define HEAD(xs)Head<xs>::result::value
Listing 9
The important change is in how we treat the head of the list - now we write typename x wherever we had int x before, and use the type's value field to get its actual value if we need it. The rest of
the code can be transformed to work for generic lists in a very similar fashion. There's something to be said about how we handle ordering, but that's a topic for the next article!
In this article, we've seen how template metaprogramming is intrinsically related to functional programming in languages like Haskell, and implemented compile-time lists using C++ templates. Next
time, I'll show one way of implementing ordering in generic lists, and consider how to implement compile-time binary search trees.
So what are the uses of writing code like this? One direct use of compile-time BSTs would be to implement a static table that is sorted at compile time. This can prove extremely helpful, particularly
in embedded code. There are also indirect benefits derived from learning more about template metaprogramming in general. Writing code like this can be seen as a useful stepping stone towards
understanding things like the typelists described in Andrei's book. The capabilities these provide are quite astounding and can provide us with real benefits to the brevity and structure of our code.
Till next time...
Thanks to the Overload review team for the various improvements they suggested for this article.
[Bird] Introduction to Functional Programming, Richard Bird and Philip Wadler, Prentice Hall
[Boost] http://www.boost.org/libs/mpl/doc/index.html
[McNamara] Functional Programming in C++, Brian McNamara and Yannis Smaragdakis, ICFP '00
[Thompson] Haskell: The Craft of Functional Programming, Simon Thompson, Addison Wesley
Overload Journal #81 - Oct 2007 + Programming Topics + Design of applications and programs
Browse in : All > Journals > Overload > 81 (5)
All > Topics > Programming (877)
All > Topics > Design (236)
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Browse by Authors and Editors
Number of items: 29.
Inglis, Matthew and Foster, Colin (2018) Five decades of mathematics education research. Journal for Research in Mathematics Education, 49 (4). pp. 462-500. ISSN 1945-2306
Foster, Colin, Hodgen, Jeremy and Küchemann, Dietmar (2018) Defining a rhombus. Mathematics Teaching, 260 . p. 31. ISSN 0025-5785
Foster, Colin and Inglis, Matthew (2018) How do you describe mathematics tasks? Mathematics Teaching, 260 . pp. 18-20. ISSN 0025-5785
Foster, Colin (2017) Developing mathematical fluency: comparing exercises and rich tasks. Educational Studies in Mathematics . pp. 1-21. ISSN 1573-0816
Foster, Colin (2017) Book review. Validity in educational and psychological assessment, by Paul E. Newton & Stuart D. Shaw. Research in Mathematics Education, 19 (2). pp. 108-111. ISSN 1754-0178
Foster, Colin and Inglis, Matthew (2017) Teachers’ appraisals of adjectives relating to mathematics tasks. Educational Studies in Mathematics, 95 (3). pp. 283-301. ISSN 1573-0816
Foster, Colin and Martin, David (2016) Two-dice horse race. Teaching Statistics, 38 (3). pp. 98-101. ISSN 1467-9639
Foster, Colin (2016) Confidence and ‘negative’ marking. MathematicsTeaching, 251 . pp. 11-13. ISSN 0025-5785
Foster, Colin (2016) Proof without words: integer right triangle hypotenuses without Pythagoras. College Mathematics Journal, 47 (2). p. 101. ISSN 1931-1346
Foster, Colin (2016) Confidence and competence with mathematical procedures. Educational Studies in Mathematics, 91 (2). pp. 271-288. ISSN 1573-0816
Foster, Colin and de Villiers, Michael (2015) The definition of the scalar product: an analysis and critique of a classroom episode. International Journal of Mathematical Education in Science and
Technology, 47 (5). pp. 750-761. ISSN 1464-5211
Wake, Geoffrey, Swan, Malcolm and Foster, Colin (2015) Professional learning through the collaborative design of problem-solving lessons. Journal of Mathematics Teacher Education . pp. 1-18. ISSN
Foster, Colin (2015) The Convergent–Divergent Model: an opportunity for teacher–learner development through principled task design. Educational Designer, 2 (8). pp. 1-25. ISSN 1759-1325
Foster, Colin (2015) Expression polygons. Mathematics Teacher, 109 (1). pp. 62-65. ISSN 0025-5769
Foster, Colin (2014) Closed but provocative questions: curves enclosing unit area. International Journal of Mathematical Education in Science and Technology, 46 (5). pp. 776-783. ISSN 0020-739X
Foster, Colin (2014) Confidence trick: the interpretation of confidence intervals. Canadian Journal of Science, Mathematics, and Technology Education, 4 (1). pp. 23-34. ISSN 1492-6156
Foster, Colin (2014) Exploiting unexpected situations in the mathematics classroom. International Journal of Science and Mathematics Education . ISSN 1571-0068
Foster, Colin (2014) Getting goose bumps about teaching evolution. Primary Science (131). pp. 5-7. ISSN 0269-2465
Foster, Colin (2014) Minimal interventions in the teaching of mathematics. European Journal of Science and Mathematics Education, 2 (3). pp. 147-154. ISSN 2301-251X
Foster, Colin (2013) Mathematical études: embedding opportunities for developing procedural fluency within rich mathematical contexts. International Journal of Mathematical Education in Science and
Technology, 44 (5). pp. 765-774. ISSN 0020-739X
Foster, Colin (2013) Resisting reductionism in mathematics pedagogy. The Curriculum Journal, 24 (4). pp. 563-585. ISSN 0958-5176
Foster, Colin (2012) Creationism as a misconception: socio-cognitive conflict in the teaching of evolution. International Journal of Science Education, 34 (14). pp. 2171-2180. ISSN 0950-0693
Conference or Workshop Item
Swan, Malcolm and Foster, Colin (2018) Formative assessment lessons for concept development and problem solving. In: 13th International Congress on Mathematical Education, 24 - 31 July 2016, Hamburg,
Kent, Geoff and Foster, Colin (2015) Re-conceptualising conceptual understanding in mathematics. In: Ninth Congress of European Research in Mathematics Education (CERME 9), 4-8 February 2015, Prague,
Czech Republic.
Wake, Geoffrey, Foster, Colin and Swan, Malcolm (2014) Understanding issues in mathematical problem solving and modeling: lessons from lesson study. In: 38th Conference of the International Group for
the Psychology of Mathematics Education and the 36th Conference of the North American Chapter of the Psychology of Mathematics Education, 15 - 20 July 2014, Vancouver, Canada.
Foster, Colin, Wake, Geoff and Swan, Malcolm (2014) Mathematical knowledge for teaching probem solving: lessons from lesson study. In: 38th Conference of the International Group for the Psychology of
Mathematics Education and the 36th Conference of the North American Chapter of the Psychology of Mathematics Education, 15-20 July 2014, Vancouver, Canada.
Wake, Geoffrey, Foster, Colin and Swan, Malcolm (2014) Teacher knowledge for modelling and problem solving. In: 8th British Congress of Mathematics Education, 14 - 17 April 2014, Nottingham, U.K..
Foster, Colin (2014) ‘Can’t you just tell us the rule?’: teaching procedures relationally. In: 8th British Congress of Mathematics Education, 14 - 17 April 2014, University of Nottingham.
Foster, Colin (2017) Questions pupils ask. Mathematical Association, Leicester. ISBN 9780906588918 | {"url":"https://eprints.nottingham.ac.uk/view/people/Foster=3AColin=3A=3A.html","timestamp":"2024-11-13T11:20:34Z","content_type":"application/xhtml+xml","content_length":"23795","record_id":"<urn:uuid:90db13cf-dd93-415d-83f3-431c90f54253>","cc-path":"CC-MAIN-2024-46/segments/1730477028347.28/warc/CC-MAIN-20241113103539-20241113133539-00733.warc.gz"} |
Tutte Colloquium - Shi Li | Combinatorics and Optimization
Friday, June 9, 2023 3:30 pm - 3:30 pm EDT (GMT -04:00)
Title: Online Unrelated-Machine Load Balancing and Generalized Flow with Recourse
Speaker: Shi Li
Affiliation: University at Buffalo
Location: MC 5501
Abstract: I will present the online algorithms for unrelated-machine load balancing problem with recourse. First, we shall present a (2+\epsilon)-competitive algorithm for the problem with O_\
epsilon(\log n) amortized recourse per job. This is the first O(1)-competitive algorithm for the problem with reasonable recourse, and the competitive ratio nearly matches the long-standing
best-known offline approximation guarantee. We shall also present an O(\log\log n/\log\log\log n)-competitive algorithm for the problem with O(1) amortized recourse. The best-known bounds from prior
work are O(\log\log n)-competitive algorithms with O(1) amortized recourse due to Gupta et al., for the special case of the restricted assignment model.
Along the way, we design an algorithm for the online generalized network flow problem (also known as network flow problem with gains) with recourse. In the problem, any edge uv in the network has a
gain parameter \gamma_{uv} > 0 and \theta-units of flow sent across uv from u's side becomes \gamma_{uv} \theta units of flow on the v'th side. In the online problem, there is one sink, and sources
come one by one. Upon arrival of a source, we need to send 1 unit flow from the source. A recourse occurs if we change the flow value of an edge. We give an online algorithm for the problem with
recourse at most O(1/\epsilon) times the optimum cost for the instance with capacities scaled by \frac{1}{1+\epsilon}. The (1+\epsilon)-factor improves upon the corresponding (2+\epsilon)-factor of
Gupta et al., which only works for the ordinary network flow problem. As an immediate corollary, we also give an improved algorithm for the online b-matching problem with reassignment costs.
This is based on joint work with Ravishankar Krishnaswamy and Varun Suriyanarayana. The paper will appear in STOC 2023. | {"url":"https://uwaterloo.ca/combinatorics-and-optimization/events/tutte-colloquium-shi-li","timestamp":"2024-11-10T13:24:46Z","content_type":"text/html","content_length":"78073","record_id":"<urn:uuid:bed9ad07-ef95-4b70-8de9-1453c4ae5aff>","cc-path":"CC-MAIN-2024-46/segments/1730477028186.38/warc/CC-MAIN-20241110103354-20241110133354-00013.warc.gz"} |
Cartesian product rule
Toffoli's own scheme [21] increases the dimensionality of the automaton rather than the number of generations spanned by its evolutionary rule. However, the simplest solution to the problem of
exhibiting explicitly reversible rules of evolution may well lie in increasing the number of states of the automaton, by forming their cartesian products, whose rule of evolution needs to span
but a single generation:
Thus a single cell of a
substitution of these values into Eq. 3 reproduces Eq. 1, in its turn reversible by the rule
Similar formulas would allow the formation of reversible
To understand possible complications which can arise, consider Fredkin's idea applied to a
Evolution loses the value of c, preserves d intact, yet permits the recovery of a if b were known explicitly. Supposing a third cell
from which c can now be recovered via d and f. Since d is already known, the cell
In each case evolution ``remembers'' the second component of the right cell, but the left ancestor is the one recovered by the reversed rule. Typical of neighborhoods with half integral radius,
this asymmetry seems to have no further significance. It is often seen in rules with shifting.
Harold V. McIntosh | {"url":"http://delta.cs.cinvestav.mx/~mcintosh/newweb/ra/node4.html","timestamp":"2024-11-12T18:34:28Z","content_type":"text/html","content_length":"4228","record_id":"<urn:uuid:09414e54-655e-43d5-8ab0-e8d099da079c>","cc-path":"CC-MAIN-2024-46/segments/1730477028279.73/warc/CC-MAIN-20241112180608-20241112210608-00794.warc.gz"} |
Condensed Learning Seminar
Étale Hyperdescent
Introduce the notion of hypersheaf and discuss its basic properties and equivalent characterizations. In particular, discuss the notions of homotopy and cohomological dimensions and their relation to
hyperdescent. Sketch the proof of the fact for finite dimensional qcqs schemes and analytic adic space, the corresponding Nisnevich topos has finite homotopy dimension. Explain the counterexample
[CM21, Example 4.15] in the étale case. Sketch the proof of the hypercompleteness criterion in terms of topos theoretic points (see [CM21, Theorem 4.36] and its adic analog [And23, Satz B.31]) and
use it to prove that for finite dimensional analytic adic spaces, localizing invariants satisfy étale hyperdescent after chromatic localization (see [And23, Satz 5.14]) using the algebraic analog as
input (see [CM21, Theorem 7.14]).
Date & Time
April 19, 2024 | 2:30pm – 4:30pm
Princeton University, Fine Hall 314
Event Series | {"url":"https://www.ias.edu/math/events/condensed-learning-seminar-20","timestamp":"2024-11-07T21:44:57Z","content_type":"text/html","content_length":"47599","record_id":"<urn:uuid:2a74356e-e883-4a05-996d-3664784d55a7>","cc-path":"CC-MAIN-2024-46/segments/1730477028017.48/warc/CC-MAIN-20241107212632-20241108002632-00317.warc.gz"} |
Converting Within the Metric System
Unlike other websites, YouTube videos, or all instructors based on comma shifting, this article focuses on logical converting.
Some conversion examples are detached from daily life and absurd, like converting 5.4 mm to decameters? But why? There's none of that in this article.
Before we begin ;
• You need to know approximately how long a length measurement is. For instance, when you think of 1 meter, how long does it appear in your mind? What about 1 decimeter or centimeter?
• You should be well-acquainted with the submultiples and multiples of the meter, that is, the metric sequence. Which is larger, a decimeter or a centimeter? What is the next unit of measurement
larger than a meter?
• Knowing the meanings of the names used in the metric system can facilitate your conversions. Understand well what the names signify. For instance, 'kilo' means 'thousand', and 'cent' stands for 1
• In conversions, you may see units expressed in decimal notation, so be well-acquainted with decimal representations.
Metric System
Kilo = Thousand
Hecto = Hundred
Deca = Ten
Deci = 1/10
Cent = 1/100
milim = 1/1000
Popular Units
meter - centimeter - milimeter and kilometer
Unpopuler Units
Hectometer - Decameter - Decimeter
Km < > Meter
Kilo = 1000
1 kilometer is 1000 m, so 5 km will make 5000 m
Given that 1 km is 1000 meters, 3 km is 3000 meters. What remains is 0.5 km, which can be thought of logically (like half of a thousand meters) or can be calculated.
6 km is 6000 meter
so result is $$6,75km=6750meter$$
Given that 1 km is 1000 meters, it hasn't even reached a full km. So, the result should be something like 0._ _ _.
2000 meters = 2 km, and what remains is 800 meters. How many km is 800 meters?
We replaced 'meter' with '1/1000 of a km.
Given that 1 km is 1000 meters, it hasn't even reached a full km. So, the result should be something like 0._ _ _.
meter < > centimeter
cent = 1/100
100 cent = $1
1 meter is 100 cm so , 9 m is 900 cm .
5 meters is 500 cm, and what remains is 0.7 meters.
so the result is ;
2 meters is 200 cm, and what remains is 0.78 meters.
so the result is $$2.78m=278cm$$
1 meter is 100 cm, and what remains is 8 cm .
so the result is $$108cm=1.08m$$
$$75,4 cm=....?m$$
Given that 1 m is 100 centimeters, it hasn't even reached a full m ( 100 cm ) . So, the result should be something like 0._ _ _.
$$12 cm=....?mm$$
1 cm is 10 mm so 12 cm = 120 mm
$$3.5 cm=....?mm$$
1 centimeter is 10 mm, and what remains is 0.5 cm .
so the result is $$3.5 cm=35mm$$
$$4.58 cm=....?mm$$
1 centimeter is 10 mm, 4 cm is 40 mm and what remains is 0.58 cm
so the result is $$4.58cm=45.8 mm$$
1 centimeter is 10 mm, and what remains is 5mm.
the result should be something like 1, ___ cm
so the result is $$15mm=1.5 cm$$
Other Conversions
km << >> cm
km >> m >> cm or versus .
Progressing step by step will be easier for you. First, try to convert the units you don't know into the ones you are familiar with. Make sure to read the notes at the top of the page.
What is more ? | {"url":"https://www.middleschoolmaths.com/2023/08/converting-within-metric-system.html","timestamp":"2024-11-04T04:59:40Z","content_type":"text/html","content_length":"91468","record_id":"<urn:uuid:7ae1ea03-8e78-4e31-95f7-5ef7b92fb62a>","cc-path":"CC-MAIN-2024-46/segments/1730477027812.67/warc/CC-MAIN-20241104034319-20241104064319-00729.warc.gz"} |
Step by Step Synthetic Division Tutorial: Easy Guide with Examples and Answers (Algebra 2) - Knowunity
Synthetic division is a step-by-step method for dividing polynomials by linear factors. This technique simplifies the process of polynomial long division, especially when dealing with monic linear
divisors. The transcript provides a comprehensive overview of synthetic division, including its definition, application, and examples.
Key points:
• Synthetic division is used for dividing polynomials by monic linear divisors (x - c)
• It's a shortcut method that reduces the work compared to traditional long division
• The process involves arranging coefficients and performing simple arithmetic operations
• Synthetic division is particularly useful for finding polynomial roots and factoring | {"url":"https://knowunity.com/knows/algebra-2-synthetic-divison-50207298-4193-47c7-823a-2b21253f2d77?utm_content=taxonomy","timestamp":"2024-11-12T12:10:06Z","content_type":"text/html","content_length":"424156","record_id":"<urn:uuid:1e9d7510-cbc1-4430-9201-3c346d85e805>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.45/warc/CC-MAIN-20241112113320-20241112143320-00719.warc.gz"} |
Constructing Angle Bisectors
1) Use the ANGLE BISECTOR . Further directions appear below the applet.
5) Use the PERPENDICULAR LINE tool to construct a line that passes through the incenter and is perpendicular to . 6) Use the PERPENDICULAR LINE tool to construct a line that passes through the
incenter and is perpendicular to . 7) Use the INTERSECT tool to plot the 3 points at which the lines (you constructed in steps 4 - 6) intersect the 3 sides of the triangle. 8) Go to the STEPS window.
Hide the 3 lines you constructed in steps 4 - 6. 9) Construct a circle with centered at the incenter that passes through any 1 of the points you constructed in step 7.
What do you notice? Why does this occur?
When you're done (or if you're unsure of something), feel free to check by watching the quick silent screencast below the applet. | {"url":"https://beta.geogebra.org/m/nhyqb5zc","timestamp":"2024-11-09T22:54:09Z","content_type":"text/html","content_length":"98503","record_id":"<urn:uuid:9f10f935-933a-4acb-9e36-5a9d03ebed08>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.10/warc/CC-MAIN-20241109214337-20241110004337-00321.warc.gz"} |
P(a|b) - (Actuarial Mathematics) - Vocab, Definition, Explanations | Fiveable
from class:
Actuarial Mathematics
The notation p(a|b) represents the conditional probability of event A occurring given that event B has already occurred. This concept is central to understanding how probabilities can be influenced
by prior events, highlighting the dependence of one event on another. Recognizing conditional probability allows for better decision-making and risk assessment in various contexts, as it helps
quantify uncertainty based on existing information.
congrats on reading the definition of p(a|b). now let's actually learn it.
5 Must Know Facts For Your Next Test
1. The formula for calculating conditional probability is given by the equation $$p(a|b) = \frac{p(a \text{ and } b)}{p(b)}$$, where p(a and b) is the joint probability of both events occurring.
2. Conditional probability plays a crucial role in Bayes' theorem, which allows for updating probabilities based on new evidence.
3. Understanding p(a|b) is essential for risk assessment in fields such as finance, insurance, and medicine, where decisions often depend on prior events or conditions.
4. If two events are independent, then the conditional probability simplifies to p(a|b) = p(a), meaning knowing that B has occurred gives no additional information about A.
5. Graphical representations, such as Venn diagrams or probability trees, can help visualize how conditional probabilities interact and overlap.
Review Questions
• How does p(a|b) differ from joint and marginal probabilities?
□ p(a|b) focuses on the probability of event A occurring under the condition that event B has occurred, while joint probability p(a and b) measures the likelihood of both events happening
together. Marginal probability p(a) looks at the occurrence of event A in isolation. Understanding these differences helps clarify how events influence each other and enables more accurate
probabilistic assessments.
• In what situations would it be crucial to apply conditional probability in decision-making?
□ Conditional probability is vital in scenarios where prior knowledge impacts future outcomes. For instance, in medical diagnostics, knowing a patientโ s symptoms (event B) can change the
likelihood of having a specific disease (event A). Similarly, in finance, understanding market conditions can influence investment strategies. These applications highlight how conditional
probability aids in informed decision-making by incorporating relevant background information.
• Evaluate how conditional independence affects the relationship between two events and give an example.
□ Conditional independence occurs when the occurrence of one event does not affect the probability of another event given a third event. This can be represented as p(a|b) = p(a). For example,
consider two independent diseases where knowing a patient has one disease (B) does not change the likelihood of them having the other disease (A), as long as we are aware that certain tests
(C) have been conducted. Understanding this relationship is crucial in statistics and risk analysis because it simplifies calculations and clarifies interactions between variables.
ยฉ 2024 Fiveable Inc. All rights reserved.
APยฎ and SATยฎ are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website. | {"url":"https://library.fiveable.me/key-terms/actuarial-mathematics/paorb","timestamp":"2024-11-04T05:08:47Z","content_type":"text/html","content_length":"164411","record_id":"<urn:uuid:c79eb586-5c39-4363-ab7d-fc17a7ba08f3>","cc-path":"CC-MAIN-2024-46/segments/1730477027812.67/warc/CC-MAIN-20241104034319-20241104064319-00210.warc.gz"} |
Published: 24 October 2024| Version 1 | DOI: 10.17632/ygc5mhw5jw.1
Omoshade Balogun
A Simulated panel data from a sample of N (20 and 30) across time period of T = 4 and T= 6.
Steps to reproduce
The data structure follows a normal distribution with ε_(1t )~N (0,1), where t= 1,…,T ; i = 1,…,n_k and k= 1,…,5. A total of k = 5 subjects (for instance) was studied and simulated over T = 4 and T=
6 times. Thus, a total of n = N (k×T) observations were generated. An independent variable x_(it )was considered and simulated from the Gaussian distribution; X_it~iid N(20,1). Over fixed time
interval of k = 4 and 6, the vector of parameters, β is set as β ̂=(β ̂_0it,β ̂_1it )^/= (20,1)^/.The response variable Y_(it ) was simulated from a panel date regression model Y_(it )= β_0it+ β_1it
x_1it+ ε_it. accordingly. An Unbalance time interval was infused into the data by randomly removing ω% of the total sample from the data, where ω take on values 5, 10, 15 and 20. (Balogun et al.,
University of Ilorin
Statistics, Coding in Statistics, Descriptive Statistics, Applied Statistics | {"url":"https://data.mendeley.com/datasets/ygc5mhw5jw/1","timestamp":"2024-11-04T04:57:16Z","content_type":"text/html","content_length":"99596","record_id":"<urn:uuid:37e2a87d-50f5-4cf3-8609-9741c9324b94>","cc-path":"CC-MAIN-2024-46/segments/1730477027812.67/warc/CC-MAIN-20241104034319-20241104064319-00121.warc.gz"} |
Repeated Addition Worksheet for Class 2
CBSE Worksheets for Class 2 Maths
Download free printable worksheets for CBSE Class 2 Maths with important chapter wise questions as per Latest NCERT Syllabus. These Worksheets help Grade 2 students practice Mathematics Important
Questions and exercises on various topics like Multiplication, Division, Time, Calendar, fractions, Additions, Time, Money, Subtractions, Shapes and Mental Maths and Word Problems. These free PDF
download of Class 2 maths worksheets consist of visual simulations to help your child visualize concepts being taught and reinforce their learning..
Get free Kendriya Vidyalaya Class 2 Maths Worksheets shared by teachers, parents & students to understand the concepts. All the necessary topics are covered in these 4th grade worksheets. These class
2 maths worksheets provide skills and experience necessary to ace in Exams
Latest CBSE Class 2 Maths Worksheets
Maths Topics to be covered for Class 2
• Chapter 1: What is long, what is round?
• Chapter 2: Counting in groups
• Chapter 3: How much can you carry
• Chapter 4: Counting in tens
• Chapter 5: Patterns
• Chapter 6: Footprints
• Chapter 7: Jugs and Mugs
• Chapter 8: Tens and Ones
• Chapter 9: My Funday
• Chapter 10: Add our points
• Chapter 11: Lines and lines
• Chapter 12: Give and take
• Chapter 13: The longest step
• Chapter 14: The bird comes, bird goes
• Chapter 15: How may ponytails?
For Preparation of board exams students can also check out other resource material
CBSE Class 2 Maths Question Papers
Important Questions for Class 2 Maths Chapter Wise
Maths Revision Notes for class 2
Worksheets of Other Subjects of Class 2
CBSE Worksheets of Class 2 English CBSE Worksheets of Class 2 E.V.S CBSE Worksheets of Class 2 Computer Science CBSE Worksheets of Class 2 Hindi CBSE Worksheets of Class 2 Value Education
Why do one Children need Worksheets for Practice?
It is very old saying that one can build a large building if the foundation is strong and sturdy. This holds true for studies also. Worksheets are essential and help students in the in-depth
understanding of fundamental concepts. Practicing solving a lot of worksheets, solving numerous types of questions on each topic holds the key for success. Once basic concepts and fundamentals have
been learnt, the next thing is to learn their applications by practicing problems. Practicing the problems helps us immensely to gauge how well we have understood the concepts.
There are times when students just run through any particular topic with casual awareness there by missing out on a few imperative “between the lines” concepts. Such things are the major causes of
weak fundamental understandings of students. So in such cases Worksheets act as a boon and critical helpful tool which gauges the in-depth understanding of children highlighting doubts and
misconceptions, if any.
Worksheets classifies the important aspects of any topic or chapter taught in the class in a very easy manner and increases the awareness amongst students.When students try to solve a worksheet they
get to understand what are the key important factors which needs the main focus.Sometimes it happens that due to shortage of time all the major points of any particular topic gets skipped in the
class or teacher rushes through , due to shortage of time. A worksheet thus provides a framework for the entire chapter and can help covering those important aspects which were rushed in the class
and ensure that students record and understand all key items.
In a class of its say 40 students howsoever teacher tries to be active and work towards making each student understand whatever she has to teach in the class but there are always some students who
tend to be in their own world and they wander in their thoughts.Worksheets which are provided timely to all the students, causes them to focus on the material at hand. it’s simply the difference
between passive and active learning. Worksheets of this type can be used to introduce new material, particularly material with many new definitions and terms.
Worksheets help students be focussed and attentive in the class because they know after the class is over they will be assigned a worksheet which they need to solve so if they miss or skip any point
in the class they may not be able to solve the worksheet completely and thereby lose reputation in the class.
Often students revise the chapter at home reading their respective textbooks. Thus more often than not they do miss many important points. Worksheets thus can be used intentionally to help guide
student’s to consult textbooks. Having students write out responses encourages their engagement with the textbooks, the questions chosen indicate areas on which to focus. Explicitly discussing the
worksheets and why particular questions are asked helps students reflect on what is important.
Worksheets of Other Classes | {"url":"https://www.ribblu.com/cbse/repeated-addition-worksheet-for-class-2","timestamp":"2024-11-02T04:25:06Z","content_type":"text/html","content_length":"508086","record_id":"<urn:uuid:ad2daea8-2f8d-4f78-b0ca-82a16048c348>","cc-path":"CC-MAIN-2024-46/segments/1730477027677.11/warc/CC-MAIN-20241102040949-20241102070949-00559.warc.gz"} |
how to use t variable in the bazier curve
i’m doing a practice with bazier curve, by folow a tutorial i was able to get a cruved linerenderer, but what i want to do is to move an object by a curve, and here’s the point… how to do it, is fine
to just put Time.deltatime as t variable?
this is the script
using System.Collections;
using System.Collections.Generic;
using UnityEngine;
public class Bezier : MonoBehaviour
private LineRenderer lineRenderer;
private Transform point0,point1,point2;
private Vector3[] positions = new Vector3[50];
private int numOfPoints = 50;
// Use this for initialization
void Start ()
lineRenderer.positionCount = numOfPoints;
// Update is called once per frame
void Update ()
private void DrawQuadraticBazierCurve()
for (int i = 1; i <= numOfPoints; i++)
float t = i / (float) numOfPoints;
positions[ i - 1] = CalculateQuadraticBezierPoint(t,point0.position,point1.position,point2.position);
/*private Vector3 CalculateLinearBezierPoint (float t, Vector3 p0, Vector3 p1)
return p0 + t * (p1 - p0);
// P = P0 + t ( P1 - P0 )
private Vector3 CalculateQuadraticBezierPoint (float t, Vector3 p0, Vector3 p1, Vector3 p2)
float u = 1 - t;
float tt = t * t;
float uu = u * u;
Vector3 p = uu * p0;
p += 2 * u * t * p1;
p += tt * p2;
return p;
// B(t) = (1-t)2P0 + 2(1-t)tP1 + t2P2
After you generate the Bezier curve (which you only have to do once - you don’t need to do that every frame) you have an Vector3 array with positions of points on that curve. The object you want to
move along that curve has to have access to those points and has to know how many points there are. Whether you do that by changing their accessors to public or by provinding public methods to
retrieve them is up to you.
All you need to do is make your object move towards each of these points, one by one. Have a field that stores the current point index and the position of your current target:
int currentPointIndex;
Vector3 currentTargetPosition;
In your object’s Update function you can simply MoveTowards your current target position. If you reach that position, you simply increment the index and get the next point position. You’d also want
to add a check if there are any more points, or if you’ve already gone through the entire curve:
transform.position = Vector3.MoveTowards(transform.position, currentTargetPosition, speed * Time.deltaTime);
if (transform.position == currentTargetPosition) {
if (currentPointIndex > numOfPoints - 1) {
Well i did it.
But player doesn’t rotate with curve… | {"url":"https://discussions.unity.com/t/how-to-use-t-variable-in-the-bazier-curve/215093","timestamp":"2024-11-05T06:12:17Z","content_type":"text/html","content_length":"31427","record_id":"<urn:uuid:1eb15164-44e0-4cd5-8d2c-ec5775a11bb7>","cc-path":"CC-MAIN-2024-46/segments/1730477027871.46/warc/CC-MAIN-20241105052136-20241105082136-00716.warc.gz"} |
VARX.P function (DAX) - DAX
Applies to: Calculated column Calculated table Measure Visual calculation
Returns the variance of the entire population.
VARX.P(<table>, <expression>)
Term Definition
table Any DAX expression that returns a table of data.
expression Any DAX expression that returns a single scalar value, where the expression is to be evaluated multiple times (for each row/context).
Return value
A number with the variance of the entire population.
• VARX.P evaluates <expression> for each row of <table> and returns the variance of <expression> assuming that <table> refers to the entire population.. If <table> represents a sample of the
population, then compute the variance by using VARX.S.
• VARX.P uses the following formula:
∑(x - x̃)^2/n
where x̃ is the average value of x for the entire population
and n is the population size
• Blank rows are filtered out from columnName and not considered in the calculations.
• An error is returned if columnName contains less than 2 non-blank rows
• This function is not supported for use in DirectQuery mode when used in calculated columns or row-level security (RLS) rules.
The following example shows the formula for a calculated column that calculates the variance of the unit price per product, when the formula is used in the Product table
= VARX.P(InternetSales_USD, InternetSales_USD[UnitPrice_USD] –(InternetSales_USD[DiscountAmount_USD]/InternetSales_USD[OrderQuantity])) | {"url":"https://learn.microsoft.com/en-us/dax/varx-p-function-dax","timestamp":"2024-11-14T05:29:46Z","content_type":"text/html","content_length":"44821","record_id":"<urn:uuid:7b229a17-b146-413d-87d0-f47987c53e5b>","cc-path":"CC-MAIN-2024-46/segments/1730477028526.56/warc/CC-MAIN-20241114031054-20241114061054-00810.warc.gz"} |
Favorite Theorems: Unique Witnesses
July Edition
This is the August edition of Favorite Theorems.
Perhaps you could easily find a satisfying assignment of a Boolean formula when only one solution exists, somehow using the uniqueness to cleverly guide your search. Valiant and Vazirani show that
any such procedure would give a randomized algorithm for general satisfiability and put the class NP in RP.
NP is as easy as detecting unique solutions, Leslie Valiant and Vijay Vazirani, STOC 1985, TCS 1986.
Valiant and Vazirani show there is some efficient randomized algorithm f mapping Boolean formula to Boolean formula such that for all formula φ,
• f(φ) will always have the same set of variables of φ and every satisfying assignment of f(φ) is a satisfying assignment of φ. In particular if φ is not satisfiable then f(φ) is never satisfiable.
• If φ is satisfiable then Pr(f(φ) has exactly one satisfying assignment) ≥ 1/(4n), where n is the number of variables of φ.
By repeating the process a polynomial number of times one of the formulas produced by f on a satisfiable φ will have a unique witness with exponentially small error,
Valiant and Vazirani's proof takes random half-spaces of the solution set and argues that with some reasonable probability repeating this process will yield a single solution. The techniques are
quite general and one can get similar results for nearly every known NP problem.
Mulmuley, Vazirani and Vazirani prove an "isolating lemma" which one can use to give an alternate proof of a similar theorem by putting random weights on edges and with some reasonable probability
there is a unique maximum weight clique. Mulmuley et. al. use the isolating lemma to give a simple randomized parallel algorithm for maximum matching.
Besides the immediate hardness of finding unique solutions, Valiant-Vazirani has had impact in complexity in many ways. Just a couple:
• One can use Valiant-Vazirani to find a satisfying assignment of a formula using non-adaptive queries to SAT.
• Valiant-Vazirani gives the base case of Toda's theorem.
• Similar ideas show how to create a low-degree polynomial that correctly computes the OR function at most inputs (which one can extend to AC^0 by recursion).
4 comments:
1. ECCC TR06-062
2. Isolation Lemma of [MVV87] was used to prove NL/poly = UL/poly by Reinhardt and Allender. To show that NL/poly = UL/poly it is sufficient to show that NL/poly \subseteq UL/poly, other side is
implied by definition, for which one can present a UL/poly algorithm for problem STCONN. However, to do this in UL, one have to reduce STCONN to a problem of finding a unique path in the input
graph G from s to t. This was done by a reduction used by Wigderson to show NL/poly \subseteq \Oplus L/poly. Reduction works in logspace by assigning random weights (between [1,4.|V|^2.|E|]) to
the edges of the graph, while ensuring that there will be a unique minimum weight s-t-path on weighted graph with high probability iff there is one s-t-path in original graph (i.e. a RL
reduction), and then turning the weighted graph back to a unweighted graph while preserving the reduction. In fact this weight assignment can ensure that the shortest distance between every pair
of nodes is achieved by a unique path with high probability (call it min unique graph). A weight function is then ``good'' if it converts the graph to a min unique graph. A simple counting
argument shows that in a collection of polynomially many random weight functions one is ``good'' for every possible input graphs.
Also, see some more application of isolation lemma in complexity of Matching (``Isolation, Matching, and Counting: Uniform and Nonuniform Upper Bounds'',Eric Allender and Klaus Reinhardt and S.
See Hemaspaandra and Ogihara's book for a chapter on Isolation Lemma.
Stasys Jukna's book ``Extremal Combinatorics'' has a section on Isolation.
3. What's up with ECCC TR06-062?
There seem to be much of a discussion around this report. Is the proof there valid?
4. No! Proof of ECCC TR06-062 still has some problem.
I am not sure if an incomplete result deserves much discussion - but the reason for which it was discussed was following:
Is ECCC kind of model "good"? Where good-ness was observed from various perspective, namely and most importantly can author timestamp work by ECCC submission and claim that the "proof belongs to
I rather see the purpose of submitting a paper in ECCC as a way to receive peer-review comments. I am not sure if this is an inverted way of using ECCC (people seem to have opposite view point).
For me, neither do I see it embarrassing to get my mistakes reviewed by others nor do I intend to submit so that I can claim that the "proof belongs to me"(hard to prove this claim though :), but
following explanation might serve).
I do not belong to academia, and I do not find much opportunity to get my works discussed with others or get them reviewed. I did not indented to live with that, and rather decided to post in
ECCC – as even one constructive comment will be worth for me, which will allow me to correct my work quickly. It is rather in this desperation the ``problematic paper’’ exists as TR06-062. While
I must say I am grateful to, Chris Calabro and Yann Barsamian, who commented on the earlier version, and I am working on them.
One final comment: First comment in this post:
Anonymous said...
ECCC TR06-062
9:47 PM, September 02, 2006
Is not posted by me, :) of course I would not want this to be discussed much - before I complete my duty (i.e. correct my work). | {"url":"https://blog.computationalcomplexity.org/2006/09/favorite-theorems-unique-witnesses.html?m=1","timestamp":"2024-11-09T12:33:23Z","content_type":"application/xhtml+xml","content_length":"66975","record_id":"<urn:uuid:f654392e-56a6-4777-bd11-80b1108931c6>","cc-path":"CC-MAIN-2024-46/segments/1730477028118.93/warc/CC-MAIN-20241109120425-20241109150425-00323.warc.gz"} |
The Beautiful Mathematics Behind the RSA Cryptosystem - Jon's Blog
Home Cryptography The Beautiful Mathematics Behind the RSA Cryptosystem
The Beautiful Mathematics Behind the RSA Cryptosystem
Cryptography chemejon November 10, 2021 0 Comment
What is RSA?
RSA or Rivest-Shamir-Adleman is a public-key cryptosystem that was first described in 1977 by Ron Rivest, Adi Shamir and Leonard Adleman. RSA is one of the oldest public-key cryptosystem but not the
first. The first ever publicly known system was the Diffie-Hellman key exchange published just one year prior to RSA.
Asymmetric Cryptography
Asymmetric cryptography or public-key cryptography is a type of cryptography that uses a pair of keys.
• Public key
□ Can be known by others
□ Used to encrypt data
• Private key
□ Only known by the end user
□ Used to decrypt, public key encrypted data
Asymmetric cryptography is most popularly used in SSH (secure shell) to verify identity prior to exchanging secret key to be used for symmetrical cryptographic communications.
How does RSA work?
Warning this section contains a lot of math, the math concepts are that of an engineer who has zero background of number theory and learned on the spot. Concepts are explains to the best of my
understanding and links to specific top articles are provided for people who want to dive deeper.
Trapdoor Function
A trapdoor function is a function that is easy to compute in one direction, yet very difficult to do in the backward direction. For example:
Given the equation find x and y, given that both are prime.
To solve for x and y one would need to divide 4,284,689 by several prime numbers until an answer is found, essentially using brute force. But if given 1,171 as p, one could easily divide 4,284,689 by
1,171 and get 3,659 as q.
$4,284,689\div 1,171 = 3,659$
This would be a good example of a trap door function for a human but not one for a computer who could brute force all the combinations of prime products under a second.
To prevent brute forcing, RSA uses very large prime numbers. The default length for ssh-keygen is 3072-bit, or a number that is 925 digits long.
Prime and Modulus Generation
The first step to generating a RSA key pair is to generate two prime numbers, q and p. This can be done using a primality test algorithm. The algorithm may look like this:
1. Generate a list of primes in a range of values via a prime sieve function, such as the sieve of Eratosthenes.
2. Filter the list of primes for Sophie-Germain primes candidates, this generates a list of “safe primes”.
3. The above list is checked against small known primes of less then 2^30 , those are then filtered out to provide a list of candidates.
4. Take the list of candidates and perform the Miller-Rabin primality test with a minimum of 4 trials, this generates a list of validated “safe primes”.
5. Select two primes from the validated list of “safe” prime and multiply them together, this will be the modulus. Used in the encryption and decryption key. We will call this number n. For our case
we can select p=11, n=13
$n=pq \\\\ 143=11*13 \\\\ n=143$
The above algorithm is summarized based off the the open-ssh’s implementation of ssh-keygen found here.
Find Number of Relatively Prime (coprime) Numbers
Integers are coprime when the only possible integer divisor of both of them is 1.
For example 14 and 25 are coprime because 14 and 25 share no common divisor other than 1. An easy way to thing about this is to put them into a fraction. If the fraction can be reduced then the two
numbers are not coprime to each other.
$\frac{14}{25}\text{ coprime} \\\\ \frac{16}{64} \Longrightarrow \frac{4}{17} \text{ not coprime}$
There are two functions that can be used to calculate the number of coprimes from 1 to n. Euler’s totient function and Carmichael’s totient function.
Euler’s Totient Function
Euler’s totient function counts the number of positive integers up to a given integer n that is relatively prime to n. This number is represented as the greek letter phi φ(n).
An easy way to calculate φ(n) is:
$n=14\;\;\; p=2\;\;\; q=7 \\\\ \phi(n)=(p-1)(q-1)\\\\ \phi(14)=(2-1)(7-1) \\\\ \phi(14)=6$
Carmichael’s Totient Function
Carmichael’s Totient Function is similar to Euler’s function except it is abled to be reduced by half under certain circumstances. This number is represented as the greek letter lambda λ(n). I won’t
go into detail on his exact theorem, you can click on the link to find out more. But here is an online calculator.
$n=14\\\\ \lambda(14)=6$
If needed to be calculated programmatically, Carmichael’s Totient Function is usually used and will use less resources to calculate all the prime factors, due to the smaller λ(n) in relation to φ(n)
in most cases.
Generating the Public Key
Now that we have φ(n) or λ(n) we can construct our public key. A public key is made up of a prime number e, as well as a modulus n. For our case we will use λ(n) as our coprime count.
The criteria for e is that it must fall between 1 and λ(n), which in our example is 6 and it must be coprime with n, which is 14 and λ(n). In our case the only one left is 5, so thus e is 5. We can
write our encryption formula applying our public key and a plaintext message m with the following formula:
$n=14\;\;\; e=5\;\;\; c=ciphertext \;\;\; m=plaintext\\\\ c=m^{e}\;mod\;n\\\\ c=m^{5}\;mod \;14$
Encrypting a Message
In the above example ^e mod n is the public key. Mod refers to modulus or the left over during division by a divisor. If want to encrypt a single number 4 and keep it secret. We can say that our
plaintext, m, is equal to 4. Plugging everything in we get:
$n=14\;\;\; e=5\;\;\; c=ciphertext \;\;\; m=4\\\\ c=4^{5}\;mod \;14 \\\\ c=1024\;mod \;14 \\\\ c=2$
To get the modulo I used wolframalpha, a convenient computational knowledge engine. After calculating the equation we get our encrypted value of 2. We can safely send this to the owner of the key
pair, who should be the only person that can decrypt it. When decrypted the message should resolve to our initial plaintext value of 4.
Since the public key is made public, it is not important for e to be a random number. In the default key generation for ssh-keygen this value is set at 65,537. In order to make this possible the two
prime numbers generated who’s product are n, must have 65,537 (Fermat number) coprimes or greater. This is why RSA keys have a minimum number of bits, which usually create a modulus much greater then
the Fermat number, the selected numbers are then tested to ensure that 65,537 is coprime with n and λ(n), if it is not another set is selected.
Generating the Private Key
In order to decrypt the encrypted message and turn it back into plaintext, we must create the private key. The creation of this key is pretty easy if you have all the required variables, which we
luckily calculated out in the steps above. The first step we need to do is calculate d, given the formula below, which describes the criteria for selecting d.
$\lambda=6\;\;\; e=5\;\;\; \\\\ de\;mod\; \lambda = 1 \\\\ 5d\;mod\;6 = 1 \\\\$
In plain english what we are looking for in the equation below is, what multiplied by 5 divided by 6 has a remainder of 1 (think back to your long division days.
# 1 2 3 4 5 6 7 8 9 10 11
Mod 5 4 3 2 1 0 5 4 3 2 1
We begin by finding out the numbers that are multiples by 5 that when divided by 6 give a remainder of 1, we can do this by listing out the multiples of 6 and the multiples of both 5 and 6. The 6
column is offset by 1 because we want to find the smallest multiple of 6 that can go into a multiple of 5, or else we would have no remainder..
$d=5\; or \;11 \\\\ d=11$
We can see that we get two numbers 5, and 11. We can’t pick 5 because our encryption key is 5, so that leaves 11 left for our decryption key.
Now that we have a value of 11 for d we can create our decryption formula, by substituting in our private key.
$n=14\;\;\; d=11 \;\;\; m=plaintext \;\;\; c=ciphertext$$m=c^{d}\;mod\; n \\\\ m=c^{11}\;mod\; 14 \\\\$
Decrypting A Message
Now to substitute our encrypted text value of 2.
$m=2^{11}\;mod\; 14 \\\\ m=2048\;mod\;14\\\\ m=4$
Again with the help of wolframalpha we were able to solve the equation above and get a decrypted value of 4. Which is consistent with our input plaintext value of 4.
A great online resource by Syed Umar Anis that does all the calculations can be found here. All you need to do is enter in two prime numbers.
Final Thoughts
I hope this post helped you understand RSA as much as it did for me writing it. Now you understand what happens when you generate an RSA key and how it is used by both the client and the host during
a ssh session.
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asymetrical keys carmichael's totient crypto decryption encryption euler's totient how it works modulus private key public key rsa trapdoor function
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Novel theorems and integrability for metallic structures on the frame bundle of the second order
It is well known that ‘an almost complex structure’ J that is J2 = -I
on the manifold M is called ‘an almost Hermitian manifold’ (M; J; G) if
G(JX; JY ) = G(X; Y ) and proved that (F 2M; JD; GD) is ‘an almost
Hermitian manifold’ on the frame bundle of the second order F 2M. The
term ‘an almost complex structure’ refers to the general quadratic structure J2 = pJ + qI; where p = -1; q = 0: However, this paper aims to
study the general quadratic equation J2 = pJ + qI; where p; q are positive integers, it is named as a metallic structure. The diagonal lift of
the metallic structure J on the frame bundle of the second order F 2M
is studied and shows that it is also a metallic structure. The proposed
theorem proves that the diagonal lift GD of a Riemannian metric G is a
metallic Riemannian metric on F 2M. Also, a new tensor field J~ of type
(1,1) is defined on F 2M and proves that it is a metallic structure. The
2-form and its derivative dF of a tensor field J~ are determined. Furthermore, the Nijenhuis tensor of a metallic structure J~ on the frame
bundle of the second order F 2M is calculated. Finally, a study is done
on the Nijenhuis tensor NJD of a tensor field JD of type (1,1) on F 2M
is integrable
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Identifying outliers in meta-analysis
How to identify and deal with outliers in meta-analysis
How to identify and deal with outliers in meta-analysis, using R.
One of the first steps when analysing primary data is to visualise your results. This can help identify unusual trends, outliers, and data points that have a disproportionate influence on your
results, which might not be clear by looking at summary statistics alone.
To demonstrate, let's have a look at a set of thirteen datasets from the datasaurus R package, which each contain x-values and y-values that all generate the same means and standard deviations, and
roughly the same correlation coefficients.
datasaurus_dozen %>%
group_by(dataset) %>%
mean_x = mean(x),
mean_y = mean(y),
std_dev_x = sd(x),
std_dev_y = sd(y),
corr_x_y = cor(x, y)
This will code generate the following output:
# A tibble: 13 x 6
dataset mean_x mean_y std_dev_x std_dev_y corr_x_y
<chr> <dbl> <dbl> <dbl> <dbl> <dbl>
1 away 54.3 47.8 16.8 26.9 -0.0641
2 bullseye 54.3 47.8 16.8 26.9 -0.0686
3 circle 54.3 47.8 16.8 26.9 -0.0683
4 dino 54.3 47.8 16.8 26.9 -0.0645
5 dots 54.3 47.8 16.8 26.9 -0.0603
6 h_lines 54.3 47.8 16.8 26.9 -0.0617
7 high_lines 54.3 47.8 16.8 26.9 -0.0685
8 slant_down 54.3 47.8 16.8 26.9 -0.0690
9 slant_up 54.3 47.8 16.8 26.9 -0.0686
10 star 54.3 47.8 16.8 26.9 -0.0630
11 v_lines 54.3 47.8 16.8 26.9 -0.0694
12 wide_lines 54.3 47.8 16.8 26.9 -0.0666
13 x_shape 54.3 47.8 16.8 26.9 -0.0656
Next, let's visualise these x-y relationships via a series of scatterplots:
dp <- ggplot(datasaurus_dozen, aes(x=x, y=y, colour=dataset))+
theme(legend.position = "none")+
facet_wrap(~dataset, ncol=3)
dp + scale_colour_viridis_d(option = "plasma")
Despite having similar statistical characteristics, the shape of the data was different between datasets (and now you probably understand why this package is called "datasauraus", if you look
Meta-analysis is no different. Typically, a meta-analyst will construct a forest plot to visualise effect sizes and their variances, which are used to synthesize data into a summary effect size.
Here's an example meta-analysis and forest plot and the code to construct it, using a dataset from metafor and a meta-analysis and forest plot function from the meta package.
dat <- dat.molloy2014
datcor <- metacor(ri,
data = dat,
studlab = paste(authors),
method.tau = "REML",
comb.random = TRUE,
comb.fixed = FALSE,
sm = "ZCOR")
meta::forest(datcor, print.I2 = FALSE)
## OUTPUT WITH SOME INFO OMITTED ##
Number of studies combined: k = 16
COR 95%-CI z p-value
Random effects model 0.1488 [0.0878; 0.2087] 4.75 < 0.0001
Quantifying heterogeneity:
tau^2 = 0.0081 [0.0017; 0.0378]; tau = 0.0901 [0.0412; 0.1944]
I^2 = 60.7% [32.1%; 77.2%]; H = 1.59 [1.21; 2.10]
Test of heterogeneity:
Q d.f. p-value
38.16 15 0.0009
Looking at the results of the meta-analysis, the random-effects model was statistically significant (p < .0001), with a summary effect size estimate of 0.1488. The test of heterogeneity was also
statistically significant (p < .0009).
Now let's look at our forest plot. Of course, this is very subjective, but forest plots can identify studies that are worth closer inspection.
On first glance, there aren't any studies that stand out in this meta-analysis, but it's always worth taking a more in-depth look at possible outliers and influential studies.
One approach for outlier detection is a "leave-one-out" analysis, which assesses the influence of individual studies by performing a series of meta-analyses that leave out one of the studies in the
original meta-analysis. The general concept with this analysis is that by removing one study, you can observe how much the results change, thus demonstrating the effect of the inclusion of that
particular study on your results.
It's worth noting that none of these tests within a leave-one-out analysis is a definitive test, as you're arbitrarily excluding one study per analysis that would ordinarily meet your inclusion
criteria. So if your primary meta-analysis was not statistically significant but one of your leave-one-out meta-analyses were significant, you can't make big claims regarding the significance of your
finding. Instead, consider this approach a sensitivity test.
Leave-one-out analysis can also help identify sources of heterogeneity. Perhaps there are two studies that seem to be influential—what is it about those studies that make them influential? Maybe
they include participants from the same kind of population? Sometimes these factors don't become clear until you see these studies grouped together via leave-one-out analysis.
It's straightforward to perform a leave-one-out analysis using the metafor R package. The influence()function contains a suite of leave-one-out diagnostic tests, summarised in Viechtbauer & Cheung
(2010), that you can run to help identify influential studies.
dat <- dat.molloy2014
dat <- escalc(measure="ZCOR",
ri=ri, ni=ni,
slab=paste(authors, year, sep=", "))
res <- rma(yi, vi, data=dat)
inf <- influence(res)
# OUTPUT #
rstudent dffits cook.d cov.r
Axelsson et al., 2009 0.2918 0.0485 0.0025 1.1331
Axelsson et al., 2011 0.1196 -0.0031 0.0000 1.2595
Bruce et al., 2010 1.2740 0.2595 0.0660 0.9942
Christensen et al., 1999 1.4711 0.3946 0.1439 0.9544
Christensen & Smith, 1995 0.8622 0.1838 0.0339 1.0505
Cohen et al., 2004 -0.9795 -0.2121 0.0455 1.0639
Dobbels et al., 2005 0.2177 0.0296 0.0010 1.1740
Ediger et al., 2007 -0.9774 -0.3120 0.1001 1.1215
Insel et al., 2006 0.7264 0.1392 0.0195 1.0561
Jerant et al., 2011 -1.8667 -0.5861 0.2198 0.8502
Moran et al., 1997 -1.4985 -0.2771 0.0756 1.0073
O'Cleirigh et al., 2007 1.8776 0.4918 0.2148 0.8819
Penedo et al., 2003 -1.1892 -0.2939 0.0859 1.0550
Quine et al., 2012 -0.0020 -0.0423 0.0021 1.2524
Stilley et al., 2004 0.8066 0.2126 0.0459 1.0907
Wiebe & Christensen, 1997 -0.7160 -0.1656 0.0280 1.0853
tau2.del QE.del hat weight
Axelsson et al., 2009 0.0091 37.7109 0.0568 5.6776
Axelsson et al., 2011 0.0100 36.7672 0.1054 10.5396
Bruce et al., 2010 0.0075 35.3930 0.0364 3.6432
Christensen et al., 1999 0.0068 33.5886 0.0562 5.6195
Christensen & Smith, 1995 0.0082 36.5396 0.0441 4.4069
Cohen et al., 2004 0.0084 37.1703 0.0411 4.1094
Dobbels et al., 2005 0.0094 37.6797 0.0714 7.1362
Ediger et al., 2007 0.0084 36.1484 0.0889 8.8886
Insel et al., 2006 0.0083 37.0495 0.0379 3.7886
Jerant et al., 2011 0.0047 25.0661 0.1058 10.5826
Moran et al., 1997 0.0077 35.6617 0.0369 3.6922
O'Cleirigh et al., 2007 0.0059 31.9021 0.0511 5.1150
Penedo et al., 2003 0.0080 36.3291 0.0587 5.8732
Quine et al., 2012 0.0100 37.7339 0.0998 9.9778
Stilley et al., 2004 0.0083 35.8385 0.0684 6.8403
Wiebe & Christensen, 1997 0.0087 37.7017 0.0411 4.1094
dfbs inf
Axelsson et al., 2009 0.0481
Axelsson et al., 2011 -0.0032
Bruce et al., 2010 0.2623
Christensen et al., 1999 0.3994
Christensen & Smith, 1995 0.1837
Cohen et al., 2004 -0.2112
Dobbels et al., 2005 0.0296
Ediger et al., 2007 -0.3128
Insel et al., 2006 0.1387
Jerant et al., 2011 -0.5430
Moran et al., 1997 -0.2791
O'Cleirigh et al., 2007 0.5059
Penedo et al., 2003 -0.2941
Quine et al., 2012 -0.0434
Stilley et al., 2004 0.2125
Wiebe & Christensen, 1997 -0.1642
This analysis also provides a classification for what's considered influential. Look for an asterisk in the last column of the output, called 'inf' (this particular analysis had no influential
studies, so there are no asterisks here). See the metafor documentation for what's considered influential, but here's an important point from this document worth emphasising:
Note that the chosen cut-offs are (somewhat) arbitrary. Substantively informed judgment should always be used when examining the influence of each study on the results.
You can also make the same plot (with slightly different names for the influence tests) using the dmetar package:
inf.analysis <- InfluenceAnalysis(x = datcor, random = TRUE)
plot(inf.analysis, "influence")
If you want a nice visualisation for your leave-one-out analysis, you can pair the 'meta' and 'dmetar' packages:
dat <- dat.molloy2014
datcor <- metacor(ri,
data = dat,
studlab = paste(authors),
method.tau = "REML",
comb.random = TRUE,
comb.fixed = FALSE,
sm = "ZCOR")
inf.analysis <- InfluenceAnalysis(x = datcor,
random = TRUE)
plot(inf.analysis, "es")
This plot is like a forest plot, but instead of visualising the effect size and variance for each study in each row, it visualises the summary effect sizes for meta-analyses without the study named
in each row. The original summary effect size (including all studies) is shown as a dotted vertical line, and the 95% confidence interval of the original meta-analysis is shown by the green bounds.
In this example, there isn't a big shift between the smallest correlation at the top and the largest correlation at the bottom.
Outliers just based on effect size are easier to eyeball from a forest plot, however, an outlier of heterogeneity is a little tricker to identify.
To better understand studies that are influential in terms of heterogeneity, you can also order the plot via I^2, which is one measure of heterogeneity. While a single study effect size may not have
a large influence on the summary effect size, it could have an impact on heterogeneity.
plot(inf.analysis, "I2")
Removing the Jerant et al study seems to have an appreciable effect on reducing study heterogeneity. To more closely explore this, we can construct a Baujat plot, which shows the overall
heterogeneity contribution for each study against the influence on the pooled result for each study.
plot(inf.analysis, "baujat")
Jerant et al is really sticking out now, as it has almost three times more influence on the pooled result compared to all other studies, and a comparatively high overall heterogeneity contribution.
If we look back at our leave-one-out plot ordered by correlation, removing Jerant et al leads to the largest effect size, but it's not the largest by much. This indicates that this Jerant et al is
pulling down the overall summary effect size estimate, but not by a large amount.
What do you do if your analysis identifies potential outlier studies?
I don't think that outliers shouldn't be removed from your primary analysis, unless they're implausible.
What do I mean by implausible?
To give a recent example, a man in the UK was put into the priority list for a COVID vaccine due to morbid obesity. Thinking that this may have been a mistake, as he considered his weight to be
relatively normal, he followed this up to discover that the local authorities had him registered with a BMI of 28,000, because his height of 6ft 2ins was noted as 6.2cm instead. As BMI of 40 and over
is considered 'morbidly obese', a BMI value of 28,000 is obviously implausible.
Leave-one-out analyses are better framed as robustness checks and as a way to identify sources of study heterogeneity that you may have missed otherwise. You've set your inclusion/exclusion criteria
for your meta-analysis for a reason, so if a study fulfills criteria, then it should be included.
Outlier analyses are also a good check to make sure that you have entered numbers correctly. A common reason for mistakes in meta-analysis is that standard error and standard deviation get mixed up.
Of course, this should be carefully checked for all studies, but outlier analysis is a good backstop, just in case.
It's worthwhile reporting your leave-one-out analysis, regardless of whether you discover outliers. If your result hangs on the inclusion of a single study, then your findings aren't terribly robust
and this should be made clear in the discussion of your results. | {"url":"https://www.dsquintana.blog/how-do-you-decide-which-studies-in-a-meta-analysis-are-influential-and-should-be-removed/","timestamp":"2024-11-01T20:38:29Z","content_type":"text/html","content_length":"43074","record_id":"<urn:uuid:37a4527e-0210-442d-b52a-19bf5f8a95fc>","cc-path":"CC-MAIN-2024-46/segments/1730477027552.27/warc/CC-MAIN-20241101184224-20241101214224-00527.warc.gz"} |