Suppose have arbitrary ellipse with center (x,y) and its radius (a,b) . I want obtain rectangle that sides tangent of peripheral ellipse. the below image describe issue .
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I could go on and on, but I'll go onto explain some of the basic syntax of LaTeX.
Syntax
Functions & General Syntax
Basically put, if you want to write a math equation in LaTeX, you just write it. If you wanted f(x) = 3, then bung that between to math tags and you're done, producing [math]f(x)=3[/math]. Don't worry about extra spaces or carriage returns, because in general LaTeX will ignore them. It does get a little more complex than this, but don't worry about that for now. Remember that any letters you type in will be presumed to be some kind of variable and hence will be italicised.
We also have functions to display more complex things like matrices and fractions, and they have the syntax of having a \ before them, usually followed by some kind of argument. For example, \sin will produce the function sin and \frac{num}{denom} will produce a fraction with a specified numerator and denominator. More on these later.
Also remember that LaTeX is case sensitive, so \sigma is NOT the same as \Sigma.
Subscripts and Superscripts
This is perhaps one of the easiest things to do in LaTeX, and one of the most useful. Let's, for the sake of argument, say you wanted to write x^{2}. Then you'd write x^{2}, producing [math]x^2[/math]. Notice that you don't necessarily need the { and } in cases where you only have 1 thing in the index, for example x^2. But it does care if you want to write something like [math]x^{3x+2}[/math]. Subscripts are done similarly, but you use the _ operator instead of ^. If you want both subscript and superscript, then use the syntax x^{2}_{1} - which is equivalent to x_{1}^{2}.
Fractions and functions
As I've mentioned, fractions are generated by using the function \frac{num}{denom}. For example:
[math]\frac{1}{3}[/math]
[math]\frac{7}{x^2}[/math]
If you want smaller fractions, you can use \tfrac, to produce things like [math]\tfrac{1}{2}[/math] which will fit into a line nicely without having to seperate it.
LaTeX has some nice in-built functions like \sin, \cos, etc. I'm not going to write them all down here, but I'll point you to a website at the end of the document that contains them. Likewise, you can write symbols (such as infinity by using \infty) and Greek letters (e.g. \phi, \Sigma, \sigma, etc)
Bracketing
You can get all your usual brackets just by typing them straight in; for instance, (, |, [, etc. However, sometimes they won't be the right size, especially if you want to write something like (1/2)^{n}. You can get around this by using the \left and \right commands, and then placing your favourite brackets after them. For instance, to write (1/2)^{n}, we have:
[math]\left( \frac{1}{2} \right)^{n}[/math]
Integrals, Summations and Limits
Integrals can be produced by using \int, summations by \sum and limits by \lim. You can put limits on them all in the right places by using the normal subscript/superscript commands. For instance:
[math]\int_a^b x^2 \,dx[/math]
[math]\lim_{n\to\infty} \frac{1}{n} = 0[/math]
[math]\sum_{n=1}^{\infty}\frac{1}{n^2} = \frac{\pi^2}{6}.[/math]
Summary
There's a lot more things you can do with LaTeX, and I'll try to add to this as time goes by. Have a look at:
http://www.maths.tcd.ie/~dwilkins/LaTeXPrimer/'>http://www.maths.tcd...ns/LaTeXPrimer/ - the LaTeX primer
http://omega.albany.edu:8008/Symbols.html'>http://omega.albany....08/Symbols.html - some symbols that you might find useful.
If you have any questions about the system, send me a PM and I'll try to help
Cheers.
Figure 1: A series of 30x7 matrices of natural numbers such that their prime factors are distributed periodically throughout the matrix. The slot on the left represents the factors of prime number 5, the second shows the factors of 7, the third shows the factors of 11 and the fourth shows the factors of 13. Each frame of the animation shows a progression of levels in the matrix.
Note that since 5 and 7 are both factors of the primorial 210, they never move or they behave like standing waves while all larger prime factor patterns each propagate at a different rate or a different “frequency over time”.
I plugged it into Wolfram Alpha, but solving for x is still a challenge.
x^2 + (2 x^5)/N^2
alternate form
(x^2 * (N^2 + 2 * x^3) / N^2
Plug in the known value N. Still impossible to solve. But if you plug in both x and N it proves true.
Do you believe me now?
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Link to the article:
https://research.ijcaonline.org/volume113/number3/pxc3901586.pdf
My question is: Why can’t I combine the Law of Sines and the property of similar triangles to solve 2 different similar triangles of different sizes?
And secondly why can’t I say the proportions in one triangle are not proportional to the same proportion in a similar triangle? I mean if all sides and angles are known in one and the other similar triangle all the angles are known, there has to be a method to solve the segments of the unknown, second similar triangle. Is there a method I do not know of?
For example, you know the lengths and angles of triangle one; and you know the angles of a smaller, similar triangle angle two.
I know, if you new one side of angle 2 you could solve with similar triangles no problem. If it were that easy the problem wouldn’t be a one-way-function.
I just wanted to get advice on the possibility and if anyone sees this as a worthy problem.
To be specific I do not know the lengths of the segments on angle 2, but I do know the equations that make up the lengths. Does anyone believe it would be possible to solve angle 2 with only equations as the known for angle 2?
I have put much thought into this. Because being able to break a one-way-function is the result. I know that there are infinite similar angles. And yes, I know trigonometry. This has just been bugging me. Why can’t I do this? Or it must be possible? But are there any higher geometry or higher math’s that solve such a problem? I’m sure someone has encountered this problem before.
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I discovered reverse Fibonacci sequence. Can you comment me what you think and what math value have this discovery?
The article is in attachment.
Thanks.
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I am trying to work my holiday (vacation) entitlement out. Can somebody please tell me if I am right?
I get 27 days holiday plus the bank holidays (public holidays). My work year is 1^{st} of April to 31^{st} of March. Due to Easter moving about that means this year I get 7 days as bank holidays. In total this year I get 34 days holiday entitlement
If I worked full time I would do 7.5 hours per day and work 5 days per week = 37.5 hours per week. 34 holidays days x 7.5 hours a day = 255 hours holiday entitlement. If I was full time every time I take holiday day I would lose 7.5 hours/1 day from my holiday entitlement.
I don’t work full time. I do 18.45 hours per week which is half a week’s work. I don’t work 2.5 days though. I work 3 days per week and do 6.45 hours a day.
Based on that information am I correct in saying
This year I get 127.5 hours holiday entitlement (half of a full time worker). Every time I take a holiday day I take off 6.15 hours from my holiday entitlement
Or should it simply be
I get 34 days holiday entitlement. Every time I take a holiday day I just take a day off my holiday entitlement.
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i have a toolbox with a padlock that opens via a 4-digit sequence. so, for example 8734. and the padlock opens.
so when i lock the toolbox sometimes im a bit lazy and i just move the 4 slot around a bit, without touching the 873. one of the other guys reckons its easier for someone to work out the code to open the lock if i have only moved 1, instead of all 4 of them.
is he right? or does the law of averages mean there are still the same chances someone will guess it regardless of how many or few i move.
]]>y=x1 cos(f1+ teta) + x2 cos(2*f1);
The peak values are straightforward for cases where the frequencies are equal or when they are non-multiples. I am having trouble to quantify the maximum and minimum values of 'y'.
Note: teta, x1 and x2 all these vary from time to time. I am therefore looking for a generalised expression (accuracy <5%).
Thanks in advance.
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Or so you'd think.
However, recently one thing came to mind. Suppose it was 6 hours before or after noon during a fall or spring equinox (for our purposes it wouldn't matter) at the equator. Since it was "halfway" between sun-over-the-horizon and sun-overhead, I presume the solar angle would be 45 degrees, right?
Now suppose at the same time someone else was, let's say, at 45 degrees latitude; (north or south for our purposes wouldn't matter) at the same time. How would one determine, then, what the solar angle there would be? Is there some sort of angular equivalent of the "vector components" used in physics and in linear algebra? If so, what would these angular equivalents be, and how would you add them to determine the combined effects of time of day and degrees of latitude on solar angle?
]]>So I'm reviewing my rules of radicals prior to teaching it to students, and found out I'm a little rusty on them.
Suppose you hit an answer that ends with a prime number as your radicand. Provided you used mathematically valid reasoning to get there, does this prime-number radicand now suggest that you arrived at the most simplified form, or are there "dead ends" distinct from the right answer?
]]>Much as before, the idea is that in your post, you surround equations with special characters, and MathJax will convert the contained text into an equation for you. There's two types of equation that you can typeset:
\( y=x^2 \)
. Note that we do not support $ signs as most LaTeX users would be familiar with, since this occurs too frequently in text.
\[ y = \int f(x) dx \]
, which we note is exactly what one would type in a usual LaTeX document.
For reference, the old guide is still available and has a number of useful examples for those getting started.
Finally, please note that for legacy posts, the old [math] [/math]
tags will still continue to work and these will display equations as inline. However it's likely that older posts may look different to the way that they did before.
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Firstly, I noticed that the surface area looks like it's the derivative of volume with respect to radius... which come to think of it makes sense as the rate of change in volume at a point in time is that outer spherical shell being added times its thickness.
But secondly I also noticed that the ratio of the two is r/3. As in, as if the average particle in a sphere were only a 1/3 of the way to the outside.
More generally, V/A is in length units. Am I figuring this right? Does V/A represent average distance, root mean square distance, or whatever other measure of central tendency from center to outside?
More generally than that, how is this extrapolated to other shapes? Does V/A represent anything in particular more generally or is its dimensionality usually meaningless?
]]>Urgent!
Hackers have accessed and corrupted this project multiple times despite it being saved only locally on my iPad with myself being the only legitimate user.
I fear that they may be exploiting the method for factoring large semi-primes to crack security.
I need the help of a reputable mathematician to alert CERT about this threat as they are ignoring my attempts to contact them.
https://www.fema.gov/community-emergency-response-teams-cert-action
Figure 1: The Sieve of Erasthonenes
By noting that prime factors occur at regular intervals. ie multiples of 2 reoccur every other number, multiples of 3 reoccur at every third number, etc. we can leverage this periodicity of prime factors to identify all non-prime positions within a predifined large range of natural numbers arranged in an array.
This periodicity of prime factors means that we can apply the concepts of Standing Wave Harmonics to find all composite numbers with a given range based on these wave patterns. Therefore we will also know the relative positions of all prime numbers within the same range.
Figure 2: Patterns of Standing Wave Harmonics
These positions can be stored as a "0" for non-prime or "1" for prime rather than needing to store the entire number, thus alleviating the computational issues with large numbers.
The method
The key is to arrange numbers into rows of N numbers which is defined by the product of the first n primes.
N = P₁x P₂...x Pn
The standing wave like effect of those first prime numbers will cause their prime factors allign into their periodic columns which, in turn causes the primes to allign themselves within the remaining collumns though they will be intermingled with other composite that are defined by primes greater than Pn.
for N = 2x3 = 6
multiples of 2:
xx, 02, xx, 04, xx, 06,
xx, 08, xx, 10, xx, 12,
xx, 14, xx, 16, xx, 18,
xx, 20, xx, 22, xx, 24,
xx, 26, xx, 28, xx, 30,
xx, 32, xx, 34, xx, 36,
multiples of 3:
xx, xx, 03, xx, xx, 06,
xx, xx, 09, xx, xx, 12,
xx, xx, 15, xx, xx, 18,
xx, xx, 21, xx, xx, 24,
xx, xx, 27, xx, xx, 30,
xx, xx, 33, xx, xx, 36,
after we combine those multiples we get:
xx, 02, 03, 04, xx, 06,
xx, 08, 09, 10, xx, 12,
xx, 14, 15, 16, xx, 18,
xx, 20, 21, 22, xx, 24,
xx, 26, 27, 28, xx, 30,
xx, 32, 33, 34, xx, 36,
Therefore we can see that the prime numbers must be located within the remaining columns that are not already occupied by the composite numbers.
For prime numbers greater than Pn their prime factors form diagonal patterns which define the gaps between the prime numbers in those remaining collumns.
xx, xx, xx, xx, 05, xx,
xx, xx, xx, 10, xx, xx,
xx, xx, 15, xx, xx, xx,
xx, 20, xx, xx, xx, xx,
25, xx, xx, xx, xx, 30,
xx, xx, xx, xx, 35, xx,
or
xx, xx, xx, xx, xx, xx,
07, xx, xx, xx, xx, xx,
xx, 14, xx, xx, xx, xx,
xx, xx, 21, xx, xx, xx,
xx, xx, xx, 28, xx, xx,
xx, xx, xx, xx, 35, xx,
combining all waves we get:
xx, 02, 03, 04, 05, 06,
07, 08, 09, 10, xx, 12,
xx, 14, 15, 16, xx, 18,
xx, 20, 21, 22, xx, 24,
25, 26, 27, 28, xx, 30,
xx, 32, 33, 34, 35, 36,
note the numbers in bold are recognized as primes as well as those numbers marked as xx i.e. no prime divisors. The exception is 01 which always shows up as a prime number but you can just ignore it.
With the numbers arranged in a 2D array, we can treat it like a matrix and therefore we can ignore the value of the numbers themselves and only define the relative positions of all composite numbers within the array which of course also defines the relative positions of all of the prime numbers within the array.
Since we are only treating positions of the array as prime (1) or non-prime (0), then we can alleviate the issues with computational complexity of long numbers by only dealing with their primality and position.
Example:
By arranging numbers into rows of N numbers then you will notice that all primes will become aligned into columns that number fewer than columns of composite numbers.
e.g.
For the first 2 primes determine the positions of the first 11 primes within the first 36 numbers.
N = 2x3 = 6
0,1,1,0,1,0,
1,0,0,0,1,0,
1,0,0,0,1,0,
1,0,0,0,1,0,
0,0,0,0,1,0,
1,0,0,0,0,0,
For the first 3 primes we can define the positions of the first 29 primes within the first 100 numbers.
N = 2x3x5 = 30
0,1,1,0,1,0,1,0,0,0,1,0,1,0,0,0,1,0,1,0,0,0,1,0,0,0,0,0,1,0,
1,0,0,0,0,0,1,0,0,0,1,0,1,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,
1,0,0,0,0,0,1,0,0,0,1,0,1,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,1,0,
1,0,0,0,0,0,1,0,0,0,1,0,1,0,0,0,1...
where
1 ≈ prime number position
0 ≈ composite number position
We can scale the Node ranges to the size of the prime numbers which we are focused on by adding more primes to the composite node.
N₂ = 2x3 = 6 N² = 36
N₃ = N₂x5 = 30 N² = 900
N₄ = N₃x7 = 210 N² = 44,100
N₅ = N₄x11 = 2310 N² = 5,336,100
N₆ = N₅x13 = 30,030 N² = 901,800,900
N₇ = N₆x17 = 510,510 N² = 260,620,460,100
N₈ = N₇x19 = 9,699,690 N² = 94,083,986,096,100
N₉ = N₈x23 = 223,092,870 N² = 49,770,428,644,836,900
N₁₀ = N₉x27 = 6,023,507,490 N² = 36,282,642,482,086,100,100
N₁₁ = N₁₀x31 = 186,728,732,190 N² = 34,867,619,425,284,742,196,100
N₁₂ = N₁₁x37 = 6,908,963,091,030 N² = 47,733,770,993,214,812,066,460,900
...
Nn
Below is an excel spreadsheet that uses the first 4 prime numbers to define a node length:
N = 2x3x5x7 = 210
This node was then used to find the first 8555 prime number positions from within 88,200 natural numbers by using the wave patterns of all prime factors to eliminate all composite numbers leaving only primes behind.
I was able to validate the correctness of those prime numbers to be 100% correct compared to downloaded samples.
While this is not important for finding small primes, the method uses the same process for finding prime number positions without needing access or operate on the large numbers themselves.
All prime number positions can be predifined using prime factor wave-like patterns so that accessing large prime numbers should be usable in real time.
While the spreadsheet had over 70 sheets involved, I show sample sheets for two prime factor patterns for 11 and 101.
The third sheet shows the prime number positions are shown in red within a field of natural numbers shown in grey, although the numbers themselves are just there to provide context for the result.
The actual numbers used in formulas are simply a 0 or 1 to show primality as shown in the fourth sheet.
Figure 3: Proof of concept in using an Excel spreadsheet which uses wave patterns of prime factors shown as the patterns of "1" intersecting harmonic columns shown as the white cells in order to derive the positions for the first 8555 prime numbers within the first 88,200 natural numbers.
While creating this kind of matrix for the largest prime numbers in question would be a large undertaking the memory and speed cost can be greatly reduced relative to current methods by dealing with relative prime number positions only without needing to store or perform operations on large prime numbers directly.
It may even be possible to model prime number positions using electronic signal waves or perhaps light wave frequencies at wavelengths that correspond with the prime factor periodicity in order to determining where they intersect with prime harmonic standing waves in order to identify the prime number positions.
Ultimately, with access to large primes in near real time, it should be possible to use methods of infinite compression, by taking large strings of binary data, and converting it to small number keys using universally accessable large mersenne primes to compress and decompress the data at either end.
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