So I'll start off:

**1)** http://linuxfreak87.googlepages.com/

**1)** Covers a lot of stuff, Maths and some physics.

**2)** http://mathworld.wolfram.com/

**2)** Amazing maths resource, lot of advanced stuff.

**3)** Basic and advanced maths here. Good tutorials.

**4)** Again more good tutorias and weekly challanges.

**5)** http://mathforum.org/dr.math/

**5)** LOTS of question solutions here, examples too. This one has helped me a lot in the past and still does

**6)** http://www.ics.uci.edu/~eppstein/junkyard/

**6)** Lots of fun geometry, useful stuff and interesting stuff here.

**7)** http://en.wikipedia.org/wiki/Category:Mathematics

**7)** As always Wikipedia is a great resource for one and all.

**8)** http://www.research.att.com/~njas/sequences/Seis.html

**8)** If your interest is number sequences this is the place to go. Useful for research.

**9)** Equations, equations and yes you guessed - MORE equations. Very useful resorce for reference.

**10)** http://home.att.net/~numericana/

**10)** Lots of interesting stuff and some other useful links too.

**11)** http://www.mcs.surrey.ac.uk/Personal/R.Knott/

**11)** Lots of interesting stuff, the mysteries of the Fibonacci Numbers etc.

**12)** http://integrals.wolfram.com/

**12)** Very useful too, online integral solver!

**13)** Maths in music, what next?

If you have more to add pease share them

Cheers,

Ryan Jones

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This is my final work on the Prime product problem.

I know it is just x^2 * y^2 = PNP^2

However the terms would just cancel out. Instead I have decided to let x^2 equal a pattern of x and PNP. So I just substituted the equation which is more complex and will not equal the right side of the equation for x^2.

In calculus where you have a complex derivative where you let du/dx equal a portion of the derivative so you can understand and simplify the manipulation of the integral. I am instead taking a more complex pattern and leaving it so x^2 does not cancel x squared. By doing this I hope it solves the pattern.

So if you could solve this polynomial equation you would solve the factorization problem.

If you couldn’t solve the polynomial? Well you could just write an algorithm that plugged in Prime numbers from smallest to largest. And because the polynomial is set up to find PNP you would get a feel for the range x was in. I mean, that this time when you try a number how far away the computed value is from PNP is significant.

So if this works it is faster than using division to factor.

But of course I await any disagreeing opinions. I now this problem gets that. But it was my final attempt before moving on to a different problem to pursue.

(((((x^2*PNP^4 + 2*PNP^2 * x^5) + x^8)/ PNP^4 ) - ((1 - x^2/(2*PNP)))) * ((PNP^2/x^2 ))) == PNP^2 Above is the pattern of x^2 * y^2 = PNP^2 It is not to be simplified yet x put and tested in that place. It is faster than division since the equation approaches PNP as the proper x is used. Smallest to largest Prime numbers are to be used. PNP = 85 x = 5 (((((x^2*PNP^4 + 2*PNP^2 * x^5) + x^8)/ PNP^4 ) - ((1 - x^2/(2*PNP)))) * ((PNP^2/x^2 ))) 85 5 4179323/578 N[4179323/578, 14] Sqrt[7230.66262975778546712802768166089965397924`14.] 85.033303062728 ((((x^2*PNP^4 + 2*PNP^2 * x^5) + x^8)/ PNP^4 ) - ((1 - x^2/(2*PNP)))) 4179323/167042 N[4179323/167042, 14] 25.019593874594 ((PNP^2/x^2)) 289 Questions to ask: Is it unique to factors of PNP or does it just give a decimal to all real numbers? Does it work for other values of PNP and x? Is it faster than factoring (recursion)?. Is it just x = x and as so not a useful pattern? Can the error be programmed? Verify then post. That is what I need to do. But I actually believe there is a pattern here. The question is does it work for all PNP and x values. I have many patterns. Some answer some of the questions. But I haven't found a polynomial I can solve after these questions have been answered. For example if this worked, I would need to solve the given equation. And this proves to be challenging. PNP = 85 x = 3 (((((x^2*PNP^4 + 2*PNP^2 * x^5) + x^8)/ PNP^4 ) - ((1 - x^2/(2*PNP)))) * ((PNP^2/x^2 ))) 85 3 847772947/130050 N[847772947/130050, 14] 6518.8231218762 Sqrt[6518.8231218762] 80.739229138481]]>

I have my own methods but I want to hear others before I share my own.

]]>Hello to everyone,

Is it possible to write a mathematical formula that will give us the size of an object at various distances?

For example, let's say we have a large cube that is 60 feet in size, and we wanted to know how many feet the cube would appear to be when we looked at it at these distances:

.5 of a mile

1 mile

2 miles

Is there a formula that can accurately calculate what the size of the cube would be when looking at it at these distances?

Thank you,

Chris

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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.

]]>
Quote

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?

]]>Dimension of the object for reference camera tilt angle are given as well as contour area of this object on image plane. Parameters of camera (fields of view, focal length) are also known.

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 .

]]>On scienceforums.net, we've implemented a small LaTeX system to allow you to typeset equations (in other words, cut out all the x^2 stuff and make things easier to read for everyone). The basic principle behind it is this: you have a LaTeX string, and you surround it by [math][/math] tags. I'll come to the syntax of the actual string in a moment.

For those who can already use LaTeX (and indeed, those who can't), a few things to note. In the system we've implemented, a tex file is created, surrounding the string you input with a \begin{display} environment so there is no need for $, $, \[ etc. Also note that we've included the standard AMS files for you; if anyone wants any special characters, I'm sure we can probably accommodate your needs.

The images are clickable, so you can see the code that was used to make them by clicking.

Now that's all out of the way, onto some examples

[math]x^2_1[/math] - Indexes (both subscript and superscript) on variables

[math]f(x) = \sin(x)[/math] - A simple function.

[math]\frac{dy}{dxx} = \frac{1}{1+x^2}[/math] - Example of fractions - you can create small fractions by using \tfrac.

[math]\int_{-\infty}^{\infty} e^{-x^2} = \sqrt{\pi}[/math] - A nice integral.

[math]\mathcal{F}_{x} [\sin(2\pi k_0 x)](k) = \int_{-\infty}^{\infty} e^{-2\pi ikx} \left( \frac{e^{2\pi ik_{0}x} - e^{-2\pi ik_{0}x}}{2i} \right)\, dx[/math] - a Fourier Transformation, which is rather large.

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.

Adding Time to a 2D Prime Factor Harmonic Matrix to demonstrate the “behavior” of standing vs moving prime factor “waves”

Previously I had introduced the Prime Factor Harmonic Matrix which showed that prime factors behaved like waves or specifically 1 dimensional waves that either behave like moving waves or standing harmonic waves within a 2 dimensional matrix of natural numbers.

A PFHM is simply any matrix of natural numbers that is dimensioned according to a primorial.

When this is done, the pattern of prime factors that make up the primorial behave like standing waves while prime factors that are not part of the primorial would behave like moving waves.

In order to better demonstrate this behavior, I added the dimension of time to a 2D 30x7 PFHM which is orthoganal to the plane.

Another words I created an animated gif which shows a progression of matrices level by level. i.e.

210 = 2x3x5x7

- 1-210
- 211-420
- 421-630
- ...

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?

]]>
Link to the article:

https://research.ijcaonline.org/volume113/number3/pxc3901586.pdf

Discussion about continuity of line, how continuity is related to uncountability and the continuum hypothesis.

The real line is made of real numbers which are points. Points are discrete objects, but lines are continuous objects. How does continuity arise out of discreteness when points make line? The idea of uncountability solves this problem. Rational numbers are countable, the line they make contains holes. Real numbers are uncountable, the line they make is continuous. So, continuity must be created by the uncountability of the points of a continuous line. One can imagine that uncountable points are so numerous on the real line that real numbers are squeezed together.

Georg Cantor called the set of real numbers continuum, so he probably thought of creating continuity with discreteness when inventing uncountability. But, what does continuity really mean?

Please read the article at

PDF Continuity and uncountability

Continuity and uncountability

Peng Kuan 彭宽

27 September 2016

Abstract: Discussion about continuity of line, how continuity is related to uncountability and the continuum hypothesis.

What is uncountability for?

The real line is made of real numbers which are points. Points are discrete objects, but lines are continuous objects. How does continuity arise out of discreteness when points make line? The idea of uncountability solves this problem. Rational numbers are countable, the line they make contains holes. Real numbers are uncountable, the line they make is continuous. So, continuity must be created by the uncountability of the points of a continuous line. One can imagine that uncountable points are so numerous on the real line that real numbers are squeezed together.

Georg Cantor called the set of real numbers continuum, so he probably thought of creating continuity with discreteness when inventing uncountability. But, what does continuity really mean?

Cauchy’s continuity

I haven’t found existing definition for continuity of line but definitions for continuous function instead. For example, using a Cauchy’s sequence s=(x_i | x_i∈R)_(i∈N) which converges to a point a, the continuity of a function f (x) at point a is defined as follow:

lim┬(i→∞)〖x_i 〗=a⇒ lim┬(i→∞)f(x_i )=f(a) (1)

The real line is the function f=0, satisfies this definition at any real numbers and is continuous. I call this definition Cauchy’s continuity.

However, if a and xi are rational numbers, the converging sequence will be entirely in ℚ and Cauchy’s continuity will allow the set of rational numbers to be continuous, which is wrong. So, Cauchy’s continuity is inadequate to define continuity of line.

Geometric continuity

Figure 1

Line is a geometric form that represents the form of real objects, for example conductive wire, water pipe, trajectory of planets etc. To illustrate the continuity of line, imagine the lines in Figure 1 as a conductive wire interrupted between points A and B. When electric current flows in the wire and the interruption, electrons move in the conductive medium of the wire and make an electric arc through the air in the interruption. To cross the interruption an electron must quit the conductive medium from point A, pass through the air and enter the point B. Following this image, continuous line is a mathematical medium in the form of line in which a point can move without quitting. An interruption is a location where a moving point must quit the medium.

So, I propose the following definition of continuity:

A line is continuous between 2 points C and D if the space between them is zero. Equivalently, the line is continuous between C and D if a moving point can go from C to D without crossing other point else than C and D. If all points of a line satisfy this condition, then the line is everywhere continuous.

C and D are said to be in contact and adjacent to each other. In the following, this kind of continuity will be referred to as geometric continuity.

Real line

Is the real line geometrically continuous? No interruption can be found on the real line, but the condition of geometric continuity is not satisfied. Take 2 different real numbers a and b and bring them close to each other, no matter how close they are, they are always separated by infinitely many other numbers. If an imaginary electron goes from a to b, it must cross many other points else than a and b. So, the real line is interrupted between a and b but not geometrically continuous.

Also, being not in contact with other point, a is an isolated point. As a can be any real numbers, all real numbers are isolated and the set ℝ is discrete. So, ℝ is not a continuum.

Constructing continuous line

Figure 2

Why are real numbers discrete? Let us see Figure 2. The points on the right are in contact to each other and they are continuous. The distance between the centers of adjacent points is denoted by d and the width of points by w. These points are continuous because w=d.

On the left, the distance between the points is still d but the width of points is smaller, w<d, this makes them discrete. However, we can shrink the distance d on the left to make the points continuous again.

If the points were real numbers, the width of points equals zero and, however small the value of d is, the points are always separated by a distance because d>0. Therefore, that the width of points is zero is the reason that makes real numbers discrete. This also proves that uncountability is unrelated to continuity. Indeed, real numbers are uncountable and discrete at the same time.

On the other hand, if one puts a real number s in contact with another number r, they will occupy the same point because their widths are zero. If t is put in contact with s, the 3 numbers r, s and t will occupy the point of r. We can repeat this operation uncountably many times, we will obtain only one point, not a line. So, uncountably many points of zero width do not make continuous line.

Figure 3

Figure 4

So, to construct a geometrically continuous line the constructing points must have nonzero width, that is, w>0. What would be the value of w? Let us deconstruct the continuous line in the interval [0,1] by splitting, as shown in Figure 3 and Figure 4. The first splitting point is ½, then the resulted 2 segments are split at ¼ and ¾. And then, the 4 resulted segments are split at ⅛, ⅜, ⅝ and ⅞. The spitting goes forever and we obtain an infinite sequence of splitting points ssplit=(aiℝ )iℕ and an infinite sequence of segments.

The segments are in contact with one another, securing continuity. Their length equals the infinitesimal number ε=1/2^∞ . These segments are the constructing blocks of the original line, each one starts at its splitting point aiℕ and has the length .

Remark: The construction of geometrically continuous line proves that the controversial infinitesimal number really exist, otherwise, continuity cannot arise.

General model

Figure 5

For a general line in space such as the one shown in Figure 5, a constructing segment is determined by 6 quantities: 3 coordinates for starting position, 2 angles for direction and for length. This segment, S in Figure 5, will be referred to as infinitesimal vector-segment and is the constructing blocks for general line.

Real numbers are discrete points that are 0-dimensional objects. In the contrary, infinitesimal vector-segment has nonzero length and is a one-dimensional object. So, we have the following property:

One-dimensional geometrically continuous line is constructed only with one-dimensional objects.

Consequently, 0-dimensional points cannot construct one-dimensional line, even they are uncountably many. In general, continuous objects in higher dimension are not constructed with objects of lower dimension. For example, 2-dimensional surfaces are constructed with infinitesimal surface2 and

n-dimensional volumes with infinitesimal n-volumen.

Uncountability

How did Georg Cantor link uncountability to continuity? In fact, he constructed the continuum ℝ in two steps: 1) ℝ is uncountable; 2) Uncountability of ℝ creates continuity for the real line.

Figure 6

He concentrated himself on proving that ℝ is uncountable. The first proof he gave was based on nested intervals [a0, b0], [an, bn], as shown in Figure 6. Because anbn when n, Georg Cantor claims that the limit of an and bn is a number not included in the lists a0a and b0b, thus real numbers are uncountable.

However, does the limit of an and bn really exist? A limit is a real number which must be fully determined, that is, all the digits from 1st to th are fixed, for example . When n increases, the first m digits of an and bn get fixed and make a number that seems to converge. The first m digits of the limit may equal this number, but the limit’s last digits, from m+1st to th, will never be determined. In fact, when n increases, an and bn both vary and the points within the interval [an, bn] are all undetermined. So, the limit that Georg Cantor claims cannot exist and this proof is invalid.

In addition to this flaw which is explained in «On Cantor's first proof of uncountability», Georg Cantor’s later proofs, the power-set argument and the diagonal argument, contain also flaws, which are explained in «On the uncountability of the power set of ℕ» and «Hidden assumption of the diagonal argument». So, all 3 proofs that Georg Cantor provided fail and uncountability possibly does not exist.

About the second step Georg Cantor did nothing but simply claim that ℝ is a continuum; probably he assumed that uncountability really created continuity. But it is shown above that uncountability is not related to continuity. So, uncountability has lost its utility and becomes useless except for itself.

Continuum hypothesis

The continuum hypothesis states that there is no set whose cardinality is strictly between that of the integers which is a discrete set and the real numbers which is a continuum. The idea behind this hypothesis is that there cannot be set that is discrete and continuous at the same time. Georg Cantor tried hard to find such set; the Cantor’s ternary set is probably one of his attempts.

However, it is shown that uncountability is not proven, then the cardinality of real numbers is questionable. Anyway, ℝ is not a continuum and the continuum hypothesis makes no longer sense.

]]>

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.

]]>

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.

]]>

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?