text:
# Name the generalization of regular tetrahedron in higher dimension

By a regular tetrahedron, I’m referring to the tetrahedron formed by connecting the vertices of the $3$-simplex with the “far corner” $(1, 1, 1)$. That is, the four vertices of this regular tetrahedron are $(1, 0, 0) ,\, (0, 1, 0) ,\, (0, 0, 1)$, and $(1, 1, 1)$, shown as blue edges one on the right.

My question is this:

What do you call the counterpart (generalization) of such tetrahedron in higher dimensions?

That is, connect the vertices of the $n$-simplex with the "far corner" $(1, 1, 1,\ldots 1)$ so that one gets a shape which vertices are the unit vectors $e_i$ and $\sum e_i$, and the sides have the same length $\sqrt{n-1}$.

If there's not a name for such construction, how about alternatively connect the origin $(0, 0, 0)$ with $(0, 1, 1) ,\, (1, 0, 1) ,\, (1, 1, 0)$?

This alternative tetrahedron is shown below with orange edges. One can compare it with the previous one involving the simplex with blue edges.

The higher dimension counterpart of the orange tetrahedron has vertices being the complement (binary flip) of that from the blue one, e.g. $(0,1,0,\ldots, 0) \to (1,0,1,\ldots, 1)$. It is congruent to the previous one, also with side length $\sqrt{n-1}$.

• You are probably looking for a simplex. – robjohn Dec 9 '16 at 7:59
• @robjohn ha, thanks, I just wasn't sure about the terminology. – Lee David Chung Lin Dec 9 '16 at 8:16
• They can be called regular simplices (plural of simplex). The orientation is irrelevant. – Yves Daoust Dec 9 '16 at 8:25
• @YvesDaoust Thank you. Just to make sure: previously I thought a regular simplex is one that is sort of orthogonal, like sitting in the corner of a Cartesian coordinates. In fact, regular simplex refers to the fact that the side lengths are all the same, right? – Lee David Chung Lin Dec 9 '16 at 8:36

I previously thought $n$-simplex (as I used it in the question) refers to only the "orthogonal" ones.
Yeah the very first few sentences in e.g. wiki for simplex states clearly that a $n$-simplex refers to generally a convex hull formed by its $n+1$ vertices.


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# You are to repay a loan of $4300.00 with 19 fortnightly repayments of$230.00, with the first repaym

You are to repay a loan of $4300.00 with 19 fortnightly repayments of$230.00, with the first repayment being one fortnight after you took out the loan. Interest is charged at j26 = 7.9950% p.a. One fortnight after the last payment of $230.00 you make a partial payment to finish paying off the loan. The size of the partial payment is: Question 12 options:  1)$65.36
 2) $65.23  3)$61.53
 4) $65.43 You are to repaying a loan of with 96 monthly repayments of$180.00, with the first repayment being one month after you took out the loan. Interest is charged at j12 = 8.0730% p.a. Immediately after your 93th repayment, the Outstanding Principal is:

Question 11 options:

 1) $532.81  2)$529.25
 3) $543.64  4)$540.00

A project will deliver $120 now,$10 in a fortnight's time, and $12 in two fortnight's time. The appropriate interest/discount rate is j26 = 9.620% p.a. The Present Value (i.e period 0) of the ENTIRE cashflow delivered is: Question 3 options:  1)$141.87
 2) $139.11  3)$142.93
 4) $120.00 A project will deliver$20 now, $20 in a fortnight's time, and$0 in two fortnight's time. The appropriate interest/discount rate is j26 = 10.400% p.a. The Future Value (i.e period 2) of the ENTIRE cashflow delivered is:

Question 4 options:

 1) $39.92  2)$0.00
 3) $46.46  4)$40.24

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text:
12-19-2016, 12:18 AM (This post was last modified: 12-19-2016 06:05 PM by chromos.)
Post: #1
 chromos Member Posts: 184 Joined: Jun 2015
1) Please, do you anybody know which is new syntax for INPUT command? I need store expressions, something like A+B or G-K/L etc. It worked in previous fw, but I am unable understand current syntax - I tried "everything", but everytime I get orange exclamation mark and programm is still on INPUT screen.

2) Is there alternative to lname command in Home?

Thank you very much.

Geocaching Coords Solver, Number of triangles Puzzle
12-19-2016, 02:24 AM
Post: #2
 Han Senior Member Posts: 1,817 Joined: Dec 2013
(12-19-2016 12:18 AM)chromos Wrote:  1) Please, do you anybody know which is new syntax for INPUT command? I need store expressions, something like A+B or G-K/L etc. It worked in previous fw, but I am unable understand current syntax - I tried "everything", but everytime I get orange exclamation mark and programm is still on INPUT screen.

2) Is there alternative to lname command in Home?

Thank you very much.

It helps to share a snippet of your code so we can help debug.

Graph 3D | QPI | SolveSys
12-19-2016, 02:58 AM
Post: #3
 chromos Member Posts: 184 Joined: Jun 2015
OK, this worked until fw update to latest release:

PHP Code:
INPUT(Temp,[2],"Expression","Variables must be in UPPERCASE! (e.g. A+B-4)");

I tried many variations of [2] - with different parentheses, even without them, at different places - with no success.

That line code is line # 293 from Geo Coords Solver:

Geocaching Coords Solver, Number of triangles Puzzle
12-19-2016, 11:18 AM (This post was last modified: 12-19-2016 11:25 AM by DrD.)
Post: #4
 DrD Senior Member Posts: 1,124 Joined: Feb 2014
Did you create local variable Temp, first?

Would this adaptation be of any use to you?

Code:
 EXPORT test() BEGIN   LOCAL Temp; //  INPUT(Temp,[2],"Expression","Variables must be in UPPERCASE! (e.g. A+B-4)");    INPUT(          { {Temp, [2], {25,74,3} } },           "Resolve Expression",          "Expression",          "Variables must be in UPPERCASE! (e.g. A+B-4)",          "",          ""        );   RETURN Temp; END;

-Dale-
12-19-2016, 04:56 PM
Post: #5
 Tim Wessman Senior Member Posts: 2,231 Joined: Dec 2013
(12-19-2016 12:18 AM)chromos Wrote:  1) Please, do you anybody know which is new syntax for INPUT command? I need store expressions, something like A+B or G-K/L etc. It worked in previous fw, but I am unable understand current syntax - I tried "everything", but everytime I get orange exclamation mark and programm is still on INPUT screen.

It seems to work fine for me here (provided Temp has been defined as pointed out).

Quote:2) Is there alternative to lname command in Home?

LNAME is also a home command. Takes an expression 'X^2+Y' for example, and returns { X, Y }. Does that do what you want?

TW

Although I work for the HP calculator group, the views and opinions I post here are my own.
12-19-2016, 06:00 PM
Post: #6
 chromos Member Posts: 184 Joined: Jun 2015
Your example works, thank you very much, Dale.

Geocaching Coords Solver, Number of triangles Puzzle
12-19-2016, 06:04 PM (This post was last modified: 12-19-2016 06:07 PM by chromos.)
Post: #7
 chromos Member Posts: 184 Joined: Jun 2015


text:
# An equality involving the mean value property

I am trying to establish that

$$\frac2{\pi}\int_0^{2\pi} f(e^{i\theta})\cos^2\left(\frac{\theta}{2}\right)\;\mathrm d\theta=2f(0)+f^\prime(0)$$

for $f$ analytic on $|z| < R,\; R > 1$

I think I am making a mistake somewhere, please point it out to me.Here's what I've done unsing the mean value property and Cauchy's integral formula:

$$2f(0)+f'(0)=2(\frac1{2\pi}\int_0^{2\pi} f(e^{i\theta})d\theta + \frac1{2\pi i}\int_0^{2\pi}\frac{f(e^{i\theta})ie^{i\theta}}{e^{i2\theta}}=\frac{1}{\pi}\int_0^{2\pi} f(e^{i\theta})d\theta\frac1{2\pi}\int_0^{2\pi}\frac{f(e^{i\theta})}{e^{i\theta}}\quad\text{(1)}$$

But when I develop the expression $\frac2{\pi}\int_0^{2\pi} f(e^{i\theta})\cos^2(\frac{\theta}{2}) d\theta$ I get:

$$\frac1{\pi}\int_0^{2\pi} f(e^{i\theta})d\theta+\frac1{2\pi}\int_0^{2\pi}f(e^{i\theta})[{e^{i\theta}}+e^{i-\theta}]d\theta \quad\text{(2)}$$

The second integral in (1) differs from the second integral in (2). Thanks.

-

Ooh, interesting formula! Let me try...

$$\frac2{\pi}\int_0^{2\pi} f(e^{i\theta})\cos^2\left(\frac{\theta}{2}\right)\;\mathrm d\theta=\frac2{\pi}\int_0^{2\pi} f(e^{i\theta})\left(\frac{1+\cos\,\theta}{2}\right)\;\mathrm d\theta$$

$$=\frac1{\pi}\int_0^{2\pi} f(e^{i\theta})(1+\cos\,\theta)\;\mathrm d\theta$$

$$=\frac1{\pi}\left(\int_0^{2\pi} f(e^{i\theta})\;\mathrm d\theta+\int_0^{2\pi} f(e^{i\theta})\cos\,\theta\;\mathrm d\theta\right)$$

Now, let's look at the Cauchy formula

$$f(0)=\frac1{2\pi i}\oint_\gamma \frac{f(u)}{u}\mathrm du=\frac1{2\pi i}\int_0^{2\pi} \frac{f(\exp(i\theta))}{\exp(i\theta)}\mathrm d(\exp(i\theta))=\frac1{2\pi}\int_0^{2\pi}f(\exp(i\theta))\mathrm d\theta$$

and going back to the original expression we have

$$2f(0)+\frac1{\pi}\int_0^{2\pi} f(\exp(i\theta))\cos\,\theta\;\mathrm d\theta$$

How to take care of that other piece... let's start at that Cauchy formula again:

$$f(z)=\frac1{2\pi i}\oint_\gamma \frac{f(u)}{u-z}\mathrm du$$

If we differentiate with respect to $z$, we have

$$f^{\prime}(z)=\frac1{2\pi i}\oint_\gamma \frac{f(u)}{(u-z)^2}\mathrm du$$

and now we can try substituting in an anticlockwise unit-circular contour yet again:

$$f^{\prime}(0)=\frac1{2\pi i}\int_0^{2\pi} \frac{f(\exp(i\theta))}{\exp(2i\theta)}\mathrm d(\exp(i\theta))$$

$$=\frac1{2\pi}\int_0^{2\pi} f(\exp(i\theta))\exp(-i\theta)\mathrm d\theta$$

$$=\frac1{2\pi}\int_0^{2\pi} f(\exp(i\theta))\cos\,\theta\;\mathrm d\theta-\frac{i}{2\pi}\int_0^{2\pi} f(\exp(i\theta))\sin\,\theta\;\mathrm d\theta$$

Luckily, the sine is antisymmetric about $\theta=\pi$, so that the imaginary part is easily removed:

$$f^{\prime}(0)=\frac1{2\pi}\int_0^{2\pi} f(\exp(i\theta))\cos\,\theta\;\mathrm d\theta$$

Long story short... the correct expression is actually $2f(0)+2f^\prime(0)$...

-
Thank you Marcel! – user786 Jul 13 '11 at 5:36


text:
• Create Account

#ActualXooch

Posted 19 April 2013 - 06:51 AM

Hey there,

I'm trying to create a simple UDP socket, and send packets between the server and clients, although I'm getting a very strange error, I can't seem to figure out how to fix it, would really appreciate any help.

This is code for my client, creating the socket

void Application::Init_Winsock()
{
WSAStartup(MAKEWORD(2,2), &Winsock);

if(LOBYTE(Winsock.wVersion) !=2 || HIBYTE(Winsock.wVersion) !=2)	//Check version
{
WSACleanup();	//Cleanup and return false if version is wrong
return;
}

//Create Socket
Socket = socket(AF_INET, SOCK_DGRAM, IPPROTO_UDP);

}

void Application::Send_Packet(PLAYER* data)
{
}

And these are the errors:

And then I'm getting a very similar problem with my Server application as well, here is the code for the server, I've also declared RecvThread as a static:

		static DWORD WINAPI RecvThread(LPVOID);

void Application::Init_Winsock()
{
WSAStartup(MAKEWORD(2,2), &Winsock);

if(LOBYTE(Winsock.wVersion) !=2 || HIBYTE(Winsock.wVersion) !=2)	//Check Version
{
WSACleanup();	//Cleanup and end if not version 2
return;
}

//Create Socket
Socket = socket(AF_INET, SOCK_DGRAM, IPPROTO_UDP);

//Input Reciever Information

}

{
/*
while(true)
{
PLAYER Recv;

Player = Recv;
}
*/

Application *instance = static_cast<Application *>(knock);
while(true)
{
PLAYER Recv;

instance->Player = Recv;
}
}

and again, very similar error:

I'm stumped :S

#1Xooch

Posted 19 April 2013 - 06:50 AM

Hey there,

I'm trying to create a simple UDP socket, and send packets between the server and clients, although I'm getting a very strange error, I can't seem to figure out how to fix it, would really appreciate any help.

This is code for my client, creating the socket

void Application::Init_Winsock()
{
WSAStartup(MAKEWORD(2,2), &Winsock);

if(LOBYTE(Winsock.wVersion) !=2 || HIBYTE(Winsock.wVersion) !=2)	//Check version
{
WSACleanup();	//Cleanup and return false if version is wrong
return;
}

//Create Socket
Socket = socket(AF_INET, SOCK_DGRAM, IPPROTO_UDP);

}

void Application::Send_Packet(PLAYER* data)
{
}

And these are the errors:

And then I'm getting a very similar problem with my Server application as well, here is the code for the server, I've always declared RecvThread as a static:

		static DWORD WINAPI RecvThread(LPVOID);

void Application::Init_Winsock()
{
WSAStartup(MAKEWORD(2,2), &Winsock);

if(LOBYTE(Winsock.wVersion) !=2 || HIBYTE(Winsock.wVersion) !=2)	//Check Version
{
WSACleanup();	//Cleanup and end if not version 2
return;
}

//Create Socket
Socket = socket(AF_INET, SOCK_DGRAM, IPPROTO_UDP);

//Input Reciever Information

}

{
/*
while(true)
{
PLAYER Recv;

Player = Recv;
}
*/

Application *instance = static_cast<Application *>(knock);
while(true)
{
PLAYER Recv;

instance->Player = Recv;
}
}

and again, very similar error:

I'm stumped :S

PARTNERS


text:
# On Plus of empty string

We know $L^+=L^\star\backslash\{\epsilon\}$. How do we interpret that $\{\epsilon\}^+=\{\epsilon\}$?

• "We know that $L^+ = L^*\setminus\{\varepsilon\}$." No we don't. – David Richerby Apr 8 '17 at 23:14
• @DavidRicherby I meant when $\epsilon\notin L$ and $L$ is an alphabet set – T.... Apr 9 '17 at 20:28


text:
Problem of the Week Problem A and Solution Quiz Results

Problem

The grade 3 and 4 students in Spruce Glen Public School had to write a quiz. The bar chart below shows the results of the quiz. Reading the grades from left to right in the bar chart, we see the grades from lowest to highest. In other words, C- is the lowest grade and A+ is the highest grade.

1. How many students wrote the quiz?

2. What is the median of all the grades on this quiz? The median is the middle value of the data when arranged in order.

Solution

1. From the bar chart, we can calculate the total number of students by adding up the number of quizzes with each grade:

$1 + 3 + 0 + 2 + 4 + 7 + 5 + 6 + 7 = 35$

So a total of 35 students wrote the quiz.

2. To find the median grade, we could list all of the grades in sorted order and look for the middle value in that list. Another way would be to discard matching numbers of grades from each end of the chart. You could do this by marking matching Xs on the bars of the graph, working from the outside toward the middle. Each X on a bar represents one student’s grade.

In this case, we see that there are 7 students with A+ grades and a total of 6 students with grades of C- (1), C (3), and B- (2). That leaves 1 grade of A+ unaccounted for at this point.

Now we notice that there are a total of 11 students with grades of B (4) and B+ (7). From the right side of the chart, that could match up with 1 student with a grade of A+, 6 students with a grade of A, and 4 students with a grade of A-. Since the only grade left unaccounted for at this point is from a student with a grade of A-, this means the median grade is A-.

Teacher’s Notes

The diagram in the problem was generated using a spreadsheet. Another activity would be to have the students recreate the chart in a spreadsheet themselves.

Here is the source data used to create the chart:

The starting point for this chart was created by highlighting the cells from A1 to B10, and then choosing to create a bar chart. Then we needed to do some additional formatting to add the grid lines and $$y$$-axis title that appear in the version you see in this problem’s description.

Note that the row showing the total is not included when generating the chart itself. The value in cell B11 was calculated using a formula: =SUM(B2:B10).


text:
# In AI/ML, using the Perceptron model, would it ever make sense to have both negative weights and data?

I understand the math but I want to make sure I understand the mapping back to real world scenarios. Thinking about it logically, I cannot think of a real world scenario where you would have a scenario where you would want both the weight of the data and the data itself to potentially be negative. Is there such a scenario I'm not thinking of?

Ex: $$y = sign(b + \sum_{i=1}^{d} w_ix_i)$$ where both $$w_i$$ and $$x_i$$ could be negative?

My reasoning is that I cannot think of a real world mapping where a double negative resulting in a positive makes sense.

Example: $$x$$ is temperature , $$y$$ is 1 if cold, -1 otherwise. $$y = \texttt{sign}(-1 \cdot -3) = 1.$$