I have my own methods but I want to hear others before I share my own.
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1: Rational thinking and the self of rational thinking
If and only if we know the definition of self in human thinking, can we know the structure of human thinking. I will start by discussing the basis of rational thinking-formal logic and use logical paradox as testing.
There are three basic laws of formal logic.
1: The law of identity
2: The law of non-contradiction
3: The law of excluded middle
The law of identity states that something is what it is. Expressed as symbols: A = A.
The law of non-contradiction says that a statement cannot be both true or false at the same time and in the same way. Expressed as symbols: A ≠ -A.
The law of excluded middle says that a statement is either true or false. Expressed as symbols: = A or = -A.
When thinking rationally, there must be a "self" existing in the process of thinking. The three laws of formal logic only use opposing concepts A and -A. To define the self of rational thinking, in addition to A and -A, we must introduce a third presence. This is in conflict with formal logic. Formal logic proves that the self of rational thinking cannot exist outside of A and –A. As it is not A or -A, the self of rational thinking can be defined as below.
1: ≠ A and ≠ -A.
2: inseparable with both A and -A.
2: Irrational thinking and the self of irrational thinking
By putting all three laws of formal logic through reversion, I derive the laws of anti-logic. The way to achieve reversion is to reverse the signs into their opposites: the “=” is now “≠”; the “≠” is now “=”; the “A” is now “-A” and the “or” is now “and”. Here I only do the odd-numbered transformation because the even-numbered transformation does not produce results of anti-logic.
The law of identity becomes two laws of difference. 1: A ≠ A; 2: A = -A.
The law of non-contradiction becomes two laws of contradiction. 1: A = -A; 2: A ≠ A.
The law of excluded middle is more complex, it can offer a variety of results. It produces two laws of middle. 1: ≠ A and ≠ -A; 2: = A and = -A.
The law of excluded middle can also derive: ≠ A or = -A, = A or ≠ -A, etc. These are not anti-logic, therefore not required.
It is easy to tell that the law of difference 1 and the law of contradiction 2 are the same; the law of difference 2, the law of contradiction 1 and the law of middle 2 are same. Removing duplicates, we get three laws of anti-logic. The law of difference: A ≠ A; The law of contradiction: A = -A; The law of middle: ≠ A and ≠ -A. The law of contradiction and the law of middle are in conflict. We separate these three anti-logic laws into two groups.
Group 1: The law of different: A ≠ A; the law of contradiction: A = -A.
Group 2: The law of different: A ≠ A; the law of middle: ≠ A and ≠ -A.
Each group represents one type of irrational thinking.
Group 1 is the feature of Chinese philosophy.
Group 2 is the feature of Indian philosophy.
Rationality, as everyone knows, is the feature of Greek philosophy.
The laws of anti-logic also only use opposing concepts A and -A, therefore the self of irrational thinking has same definition as the self of rational thinking. This guarantees that the self of thinking could freely switch between rational and irrational thinking. It’s easy to find that the self of thinking's definition include the law of middle.
Below I will discuss the relationship between the two types of irrational thinking. This is related with logical paradox.
3: Conversion between two type of irrational thinking and Logical paradox
Conversion 1: When the law of contradiction exists, applying the law of non-contradiction (cannot be both true), Any of A and -A is denied, it will result in the law of middle. Because A = -A. Denying A will also deny -A, denying -A will also deny A.
Conversion 2: When the law of middle exists, applying the law of excluded middle (cannot be both false), Any of A or -A is affirmed, it will result in the law of contradiction. Because the law of middle logically requires that "A is affirmed" will result in "-A is affirmed",
"-A is affirmed" will result in "A is affirmed".
If a self-referential proposition "I am A" is -A. The symbol is expressed as: "I = A" = -A, can infer A = -A. This proposition contains contradictory laws. If "I am A" is a self-denying judgment, then at least one of A and -A is denied. According to the conversion 1, this "I" logically satisfies the intermediate law, and this "I" is the same as the self of thinking. According to the conversion 2, the intermediate law leads to the law of contradiction. That’s the reason we have logical paradox. According to the above discussion, a proposition leading to paradox must satisfy two conditions, and 1) is a direct or indirect self-referential judgment. 2) This judgment is actually self-denying and introduces a set of contradictory propositions.
]]>In mathematics , an ellipse is a curve in a plane surrounding two focal points such that the sum of the distances to the two focal points is constant for every point on the curve.
Wikipedia starts with this definition - but that looks to me like it would define an oval shape, ie one that is symmetrical. Elsewhere I have encountered the same kind of definition - taking two pegs with a length of string longer than the distance between and scribing a line with a third peg whilst pulling the string taut makes an ellipse shape. To me this looks like it describes only very specific sort of ellipse and is not a universal description of all ellipses.
Rather than a cross section through a cone shape, this looks to me like it describes a cross section through a round cylinder. Am I missing something obvious here?
]]>Is it possible to calculate new changed contour area on image plane when camera tilt angle is changed (due to changing object distance or/and camera height)?
I would be grateful if someone could point me to methods of calculations applicable here.
I have found such method as camera transform using pinhole camera model to represent object in 3d space on 2d plane, but seems that it is not what I am looking for.
]]>Now, for every square in the grid through which the circle runs, i want to calculate the proportion of the square covered by the circle. So one of the squares close up:
I thought this would be relatively straightforward but i've ended up with a horribly convoluted way involving working at some angles where the circle crosses the squares, using that to find the area of the circle, then subtracting that from the area of a rectangle that contains that circle segment, and so on for other squares. Only need to do this for three squares as symmetry saves me a lot of work, but it's still a meandering method. Just seems to me there should be a much more simple method, but i can't figure it out.
Does anybody know of a more elegant way of solving this problem?
]]>I have a set of 11 random variables and one constant (that I don't know the value of). Every time I run an experiment all 11 variables assume a value of either 0 or 1 (all together). The question is simple: how to calculate the total number of experiments that I need to do in order for the average value of those 11 variables (from all experiments) to approach the mean of 0.5 to an accuracy of, let's say, 99.9%. After that I should be able to find out the value of my constant.
Thanks for the help!
]]>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.