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ML Foundations Development 33

Can someone explain to me how there can be different kinds of infinities? I was reading "The man who loved only numbers" by Paul Hoffman and came across the concept of countable and uncountable infinities, but they're only words to me. Any help would be appreciated.

mathfactor is one I listen to. Does anyone else have a recommendation?

I have read a few proofs that $\sqrt{2}$ is irrational. I have never, however, been able to really grasp what they were talking about. Is there a simplified proof that $\sqrt{2}$ is irrational?

I'm looking for a nice, quick online graphing tool. The ability to link to, or embed the output would be handy, too.

I was reading up on the Fibonacci Sequence, $1,1,2,3,5,8,13,\ldots $ when I noticed some were able to calculate specific numbers. So far I've only figured out creating an array and counting to the value, which is incredibly simple, but I reckon I can't find any formula for calculating a Fibonacci number based on it's position. Is there a way to do this? If so, how are we able to apply these formulas to arrays?

I'm told by smart people that $$0.999999999\ldots=1$$ and I believe them, but is there a proof that explains why this is?

Given the semi-major axis and a flattening factor, is it possible to calculate the semi-minor axis?

In mathematics, there seem to be a lot of different types of numbers. What exactly are: Real numbers Integers Rational numbers Decimals Complex numbers Natural numbers Cardinals Ordinals And as workmad3 points out, some more advanced types of numbers (I'd never heard of) Hyper-reals Quaternions Imaginary numbers Are there any other types of classifications of a number I missed?

By matrix-defined, I mean $$\left<a,b,c\right>\times\left<d,e,f\right> = \left| \begin{array}{ccc} i & j & k\\ a & b & c\\ d & e & f \end{array} \right|$$ ...instead of the definition of the product of the magnitudes multiplied by the sign of their angle, in the direction orthogonal) If I try cross producting two vectors with no $k$ component, I get one with only $k$, which is expected. But why? As has been pointed out, I am asking why the algebraic definition lines up with the geometric definition.

I'm looking for an online or software calculator that can show me the history of items I typed in, much like an expensive Ti calculator. Can you recommend any?

What length of rope should be used to tie a cow to an exterior fence post of a circular field so that the cow can only graze half of the grass within that field? updated: To be clear: the cow should be tied to a post on the exterior of the field, not a post at the center of the field.

I've learned that the dot product is just one of many possible inner product spaces. Can someone explain this concept? When is it useful to define it as something other than the dot product?

This is simply not clicking for me. I'm currently learning math during the summer vacation and I'm on the chapter for relations and functions. There are five properties for a relation: Reflexive - $R \rightarrow R$ Symmetrical - $R \rightarrow S$ ; $S \rightarrow R$ Antisymmetrical - $R \rightarrow S$ && ($R \rightarrow R$|| $S \rightarrow S$) Asymmetrical -$R \rightarrow S$ && !($R \rightarrow R$|| $S \rightarrow S$) Transitive - if $R \rightarrow S$ && $S \rightarrow T$, then $R \rightarrow T$ If that's not what you call the properties in English, then please let me know- I have to study it in German, unfortunately, and these are my rough translations. Continuing on, I just don't know what to do with this information practically. The examples of the book are horrible: "Is the same age as" is apparently reflexive, symmetrical and transitive. "Is related to" is also apparently reflexive, symmetrical and transitive. "Is older than" is asymmetric, antisymmetric and transitive. There are more useless examples like this. I have no idea how it comes to these conclusions because we're talking about a literal statement. I was hoping perhaps for some real mathematical examples, but the book falls short on those. I would greatly appreciate it if somebody could explain the above example and perhaps give me a better use for Relations other than... that. Also, how can a relation be asymmetrical and antisymmetrical at the same time? Don't they cancel each other out?

I recently discovered Benford's Law. I find it very fascinating. I'm wondering what are some of the real life uses of Benford's law. Specific examples would be great.

Given two points $p_1$ , $p_2$ around the origin $(0,0)$ in $2D$ space, how would you calculate the angle from $p_1$ to $p_2$? How would this change in $3D$ space?

I hear a lot about Elliptic Curve Cryptography these days, but I'm still not quite sure what they are or how they relate to crypto...

I've recently started reading about Quaternions, and I keep reading that for example they're used in computer graphics and mechanics calculations to calculate movement and rotation, but without real explanations of the benefits of using them. I'm wondering what exactly can be done with Quaternions that can't be done as easily (or easier) using more tradition approaches, such as with Vectors?

Let's say I know $X$ is a Gaussian variable. Moreover, I know $Y$ is a Gaussian variable and $Y=X+Z$. Let $X$ and $Z$ be independent. How can I prove that $Y$ is a Gaussian random variable if and only if $Z$ is a Gaussian R.V.? It's easy to show the other way around ($X$ and $Z$ are orthogonal and normal, hence create a Gaussian vector hence any linear combination of the two is a Gaussian variable). Thanks

Rings, groups, and fields all feel similar. What are the differences between them, both in definition and in how they are used?

What purposes do the Dot and Cross products serve? Do you have any clear examples of when you would use them?

I have the one or other interesting Math blog in my feedreader that I follow. It would be interesting to compile a list of Math blogs that are interesting to read, and do not require research-level math skills. I'll start with my entries: Division By Zero Tanya Khovanova’s Math Blog

I'm talking in the range of 10-12 years old, but this question isn't limited to only that range. Do you have any advice on cool things to show kids that might spark their interest in spending more time with math? The difficulty for some to learn math can be pretty overwhelming. Do you have any teaching techniques that you find valuable?

Not sure if this is the place for it, but there are similar posts for podcasts and blogs, so I'll post this one. I'd be interested in seeing a list of online resources for mathematics learning. As someone doing a non-maths degree in college I'd be interested in finding some resources for learning more maths online, most resources I know of tend to either assume a working knowledge of maths beyond secondary school level, or only provide a brief summary of the topic at hand. I'll start off by posting MIT Open Courseware, which is a large collection of lecture notes, assignments and multimedia for the MIT mathematics courses, although in many places it's quite incomplete.

I am looking for an accurate algorithm to calculate the error function $$\operatorname{erf}(x)=\frac{2}{\sqrt{\pi}}\int_0^x e^{-t^2}\ dt$$ I have tried using the following formula, (Handbook of Mathematical Functions, formula 7.1.26), but the results are not accurate enough for the application.

I am looking for a reference (book or article) that poses a problem that seems to be a classic, in that I've heard it posed many times, but that I've never seen written anywhere: that of the possibility of a man in a circular pen with a lion, each with some maximum speed, avoiding capture by that lion. References to pursuit problems in general would also be appreciated, and the original source of this problem.

This is in relation to the Euler Problem $13$ from http://www.ProjectEuler.net. Work out the first ten digits of the sum of the following one-hundred $50$-digit numbers. $37107287533902102798797998220837590246510135740250$ Now, this was my thinking: I can freely discard the last fourty digits and leave the last ten. $0135740250$ And then simply sum those. This would be large enough to be stored in a $64$-bit data-type and a lot easier to compute. However, my answer isn't being accepted, so I'm forced to question my logic. However, I don't see a problem. The last fourty digits will never make a difference because they are at least a magnitude of $10$ larger than the preceding values and therefore never carry backwards into smaller areas. Is this not correct?

It is $10$ o'clock, and I have a box. Inside the box is a ball marked $1$. At $10$:$30$, I will remove the ball marked $1$, and add two balls, labeled $2$ and $3$. At $10$:$45$, I will remove the balls labeled $2$ and $3$, and add $4$ balls, marked $4$, $5$, $6$, and $7$. $7.5$ minutes before $11$, I will remove the balls labeled $4$, $5$, and $6$, and add $8$ balls, labeled $8$, $9$, $10$, $11$, $12$, $13$, $14$, and $15$. This pattern continues. Each time I reach the halfway point between my previous action and $11$ o'clock, I add some balls, and remove some other balls. Each time I remove one more ball than I removed last time, but add twice as many balls as I added last time. The result is that as it gets closer and closer to $11$, the number of balls in the box continues to increase. Yet every ball that I put in was eventually removed. So just how many balls will be in the box when the clock strikes $11$? $0$, or infinitely many? What's going on here?

I keep seeing this symbol $\nabla$ around and I know enough to understand that it represents the term "gradient." But what is a gradient? When would I want to use one mathematically?

If you know it, also try to include the precise reason why the proof is fallacious. To start this off, let me post the one that most people know already: Let $a = b$. Then $a^2 = ab$ $a^2 - b^2 = ab - b^2$ Factor to $(a-b)(a+b) = b(a-b)$ Then divide out $(a-b)$ to get $a+b = b$ Since $a = b$, then $b+b = b$ Therefore $2b = b$ Reduce to $2 = 1$ As @jan-gorzny pointed out, in this case, line 5 is wrong since $a = b$ implies $a-b = 0$, and so you can't divide out $(a-b)$.

I keep hearing about this weird type of math called calculus. I only have experience with geometry and algebra. Can you try to explain what it is to me?

Why is $1$ not considered a prime number? Or, why is the definition of prime numbers given for integers greater than $1$?

I covered hyperbolic trigonometric functions in a recent maths course. However I was never presented with any reasons as to why (or even if) they are useful. Is there any good examples of their uses outside academia?

The Weyl equidistribution theorem states that the sequence of fractional parts ${n \xi}$, $n = 0, 1, 2, \dots$ is uniformly distributed for $\xi$ irrational. This can be proved using a bit of ergodic theory, specifically the fact that an irrational rotation is uniquely ergodic with respect to Lebesgue measure. It can also be proved by simply playing with trigonometric polynomials (i.e., polynomials in $e^{2\pi i k x}$ for $k$ an integer) and using the fact they are dense in the space of all continuous functions with period 1. In particular, one shows that if $f(x)$ is a continuous function with period 1, then for any $t$, $\int_0^1 f(x) dx = \lim \frac{1}{N} \sum_{i=0}^{N-1} f(t+i \xi)$. One shows this by checking this (directly) for trigonometric polynomials via the geometric series. This is a very elementary and nice proof. The general form of Weyl's theorem states that if $p$ is a monic integer-valued polynomial, then the sequence ${p(n \xi)}$ for $\xi$ irrational is uniformly distributed modulo 1. I believe this can be proved using extensions of these ergodic theory techniques -- it's an exercise in Katok and Hasselblatt. I'd like to see an elementary proof. Can the general form of Weyl's theorem be proved using the same elementary techniques as in the basic version?

Why is any number (other than zero) to the power of zero equal to one? Please include in your answer an explanation of why $0^0$ should be undefined.

I was playing around with the squares and saw an interesting pattern in their differences. $0^2 = 0$ + 1 $1^2 = 1$ + 3 $2^2 = 4$ + 5 $3^2 = 9$ + 7 $4^2 = 16$ + 9 $5^2 = 25$ + 11 $6^2 = 36$ etc. (Also, in a very related question, which major Math Research Journal should I contact to publish my groundbreaking find in?)

I have a square that's $10\mathrm{m} \times 10\mathrm{m}$. I want to cut it in half so that I have a square with half the area. But if I cut it from top to bottom or left to right, I don't get a square, I get a rectangle! I know the area of the small square is supposed to be $50\mathrm{m}^{2}$, so I can use my calculator to find out how long a side should be: it's $7.07106781\mathrm{m}$. But my teacher said I should be able to do this without a calculator. How am I supposed to get that number by hand?

It seems like functions that are continuous always seem to be differentiable, to me. I can't imagine one that is not. Are there any examples of functions that are continuous, yet not differentiable? The other way around seems a bit simpler -- a differentiable function is obviously always going to be continuous. But are there any that do not satisfy this?

Complex numbers involve the square root of negative one, and most non-mathematicians find it hard to accept that such a number is meaningful. In contrast, they feel that real numbers have an obvious and intuitive meaning. What's the best way to explain to a non-mathematician that complex numbers are necessary and meaningful, in the same way that real numbers are? This is not a Platonic question about the reality of mathematics, or whether abstractions are as real as physical entities, but an attempt to bridge a comprehension gap that many people experience when encountering complex numbers for the first time. The wording, although provocative, is deliberately designed to match the way that many people actually ask this question.

In the past, I've come across statements along the lines of "function $f(x)$ has no closed form integral", which I assume means that there is no combination of the operations: addition/subtraction multiplication/division raising to powers and roots trigonometric functions exponential functions logarithmic functions which when differentiated gives the function $f(x)$. I've heard this said about the function $f(x) = x^x$, for example. What sort of techniques are used to prove statements like this? What is this branch of mathematics called? Merged with "How to prove that some functions don't have a primitive" by Ismael: Sometimes we are told that some functions like $\dfrac{\sin(x)}{x}$ don't have an indefinite integral, or that it can't be expressed in term of other simple functions. I wonder how we can prove that kind of assertion?

I've heard of some other paradoxes involving sets (i.e., "the set of all sets that do not contain themselves") and I understand how paradoxes arise from them. But this one I do not understand. Why is "the set of all sets" a paradox? It seems like it would be fine, to me. There is nothing paradoxical about a set containing itself. Is it something that arises from the "rules of sets" that are involved in more rigorous set theory?

I learned that the volume of a sphere is $\frac{4}{3}\pi r^3$, but why? The $\pi$ kind of makes sense because its round like a circle, and the $r^3$ because it's 3-D, but $\frac{4}{3}$ is so random! How could somebody guess something like this for the formula?

Consider the task of generating random points uniformly distributed within a circle of a given radius $r$ that is centered at the origin. Assume that we are given a random number generator $R$ that generates a floating point number uniformly distributed in the range $[0, 1)$. Consider the following procedure: Generate a random point $p = (x, y)$ within a square of side $2r$ centered at the origin. This can be easily achieved by: a. Using the random number generator $R$ to generate two random numbers $x$ and $y$, where $x, y \in [0, 1)$, and then transforming $x$ and $y$ to the range $[0, r)$ (by multiplying each by $r$). b. Flipping a fair coin to decide whether to reflect $p$ around the $x$-axis. c. Flipping another fair coin to decide whether to reflect $p$ around the $y$-axis. Now, if $p$ happens to fall outside the given circle, discard $p$ and generate another point. Repeat the procedure until $p$ falls within the circle. Is the previous procedure correct? That is, are the random points generated by it uniformly distributed within the given circle? How can one formally [dis]prove it? Background Info The task was actually given in Ruby Quiz - Random Points within a Circle (#234). If you're interested, you can check my solution in which I've implemented the procedure described above. I would like to know whether the procedure is mathematically correct or not, but I couldn't figure out how to formally [dis]prove it. Note that the actual task was to generate random points uniformly distributed within a circle of a given radius and position, but I intentionally left that out in the question because the generated points can be easily translated to their correct positions relative to the given center.

I have a collection of 3D points in the standard $x$, $y$, $z$ vector space. Now I pick one of the points $p$ as a new origin and two other points $a$ and $b$ such that $a - p$ and $b - p$ form two vectors of a new vector space. The third vector of the space I will call $x$ and calculate that as the cross product of the first two vectors. Now I would like to recast or reevaluate each of the points in my collection in terms of the new vector space. How do I do that? (Also, if 'recasting' not the right term here, please correct me.)

Basic definitions: a tiling of d-dimensional Euclidean space is a decomposition of that space into polyhedra such that there is no overlap between their interiors, and every point in the space is contained in some one of the polyhedra. A vertex-uniform tiling is a tiling such that each vertex figure is the same: each vertex is contained in the same number of k-faces, etc: the view of the tiling is the same from every vertex. A vertex-transitive tiling is one such that for every two vertices in the tiling, there exists an element of the symmetry group taking one to the other. Clearly all vertex-transitive tilings are vertex-uniform. For n=2, these notions coincide. However, Grunbaum, in his book on tilings, mentions but does not explain that for n >= 3, there exist vertex uniform tilings that are not vertex transitive. Can someone provide an example of such a tiling, or a reference that explains this?

Do primes become more or less frequent as you go further out on the number line? That is, are there more or fewer primes between $1$ and $1{,}000{,}000$ than between $1{,}000{,}000$ and $2{,}000{,}000$? A proof or pointer to a proof would be appreciated.

I keep looking at this picture and its driving me crazy. How can the smaller circle travel the same distance when its circumference is less than the entire wheel?

Since constructive mathematics allows us to avoid things like Russell's Paradox, then why don't they replace traditional proofs? How do we know the "regular" kind of mathematics are free of paradox without a proof construction?

Prime numbers are numbers with no factors other than one and itself. Factors of a number are always lower or equal to than a given number; so, the larger the number is, the larger the pool of "possible factors" that number might have. So the larger the number, it seems like the less likely the number is to be a prime. Surely there must be a number where, simply, every number above it has some other factors. A "critical point" where every number larger than it simply will always have some factors other than one and itself. Has there been any research as to finding this critical point, or has it been proven not to exist? That for any $n$ there is always guaranteed to be a number higher than $n$ that has no factors other than one and itself?

Often in my studies (economics) the assumption of a "well-behaved" function will be invoked. I don't exactly know what that entails (I think twice continuously differentiability is one of the requirements), nor do I know why this is necessary (though I imagine the why will depend on each case). Can someone explain it to me, and if there is an explanation of the why as well, I would be grateful. Thanks! EDIT: To give one example where the term appears, see this Wikipedia entry for utility functions, which says at one point: In order to simplify calculations, various assumptions have been made of utility functions. CES (constant elasticity of substitution, or isoelastic) utility Exponential utility Quasilinear utility Homothetic preferences Most utility functions used in modeling or theory are well-behaved. They are usually monotonic, quasi-concave, continuous and globally non-satiated. I might be wrong, but I don't think "well-behaved" means monotonic, quasi-concave, continuous and globally non-satiated. What about twice differentiable?

I am learning geometric algebra, and it is incredible how much it helps me understand other branches of mathematics. I wish I had been exposed to it earlier. Additionally I feel the same way about enumerative combinatorics. What are some less popular mathematical subjects that you think should be more popular?

Okay, so hopefully this isn't too hard or off-topic. Let's say I have a very simple lowpass filter (something that smooths out a signal), and the filter object has a position variable and a cutoff variable (between 0 and 1). In every step, a value is put into the following bit of pseudocode as "input": position = position*(1-c)+input*c, or more mathematically, f(n) = f(n-1)*(1-c)+x[n]*c. The output is the value of "position." Basically, it moves a percentage of the distance between the current position and then input value, stores this value internally, and returns it as output. It's intentionally simplistic, since the project I'm using this for is going to have way too many of these in sequence processing audio in real time. Given the filter design, how do I construct a function that takes input frequency (where 1 means a sine wave with a wavelength of 2 samples, and .5 means a sine wave with wavelength 4 samples, and 0 is a flat line), and cutoff value (between 1 and 0, as shown above) and outputs the amplitude of the resulting sine wave? Sine wave comes in, sine wave comes out, I just want to be able to figure out how much quieter it is at any input and cutoff frequency combination.

The sum of two Gaussian variables is another Gaussian. It seems natural, but I could not find a proof using Google. What's a short way to prove this? Thanks! Edit: Provided the two variables are independent.

Suppose there are two face down cards each with a positive real number and with one twice the other. Each card has value equal to its number. You are given one of the cards (with value $x$) and after you have seen it, the dealer offers you an opportunity to swap without anyone having looked at the other card. If you choose to swap, your expected value should be the same, as you still have a $50\%$ chance of getting the higher card and $50\%$ of getting the lower card. However, the other card has a $50\%$ chance of being $0.5x$ and a $50\%$ chance of being $2x$. If we keep the card, our expected value is $x$, while if we swap it, then our expected value is: $$0.5(0.5x)+0.5(2x)=1.25x$$ so it seems like it is better to swap. Can anyone explain this apparent contradiction?

Given the coordinates of a point $(x, y)$, what is a procedure for determining if it lies within a polygon whose vertices are $(x_1, y_1), (x_2, y_2), \ldots , (x_n,y_n)$?

I remember hearing several times the advice that, we should avoid using a proof by contradiction, if it is simple to convert to a direct proof or a proof by contrapositive. Could you explain the reason? Do logicians think that proofs by contradiction are somewhat weaker than direct proofs? Is there any reason that one would still continue looking for a direct proof of some theorem, although a proof by contradiction has already been found? I don't mean improvements in terms of elegance or exposition, I am asking about logical reasons. For example, in the case of the "axiom of choice", there is obviously reason to look for a proof that does not use the axiom of choice. Is there a similar case for proofs by contradiction?

The following is a quote from Surely you're joking, Mr. Feynman. The question is: are there any interesting theorems that you think would be a good example to tell Richard Feynman, as an answer to his challenge? Theorems should be totally counter-intuitive, and be easily translatable to everyday language. (Apparently the Banach-Tarski paradox was not a good example.) Then I got an idea. I challenged them: "I bet there isn't a single theorem that you can tell me - what the assumptions are and what the theorem is in terms I can understand - where I can't tell you right away whether it's true or false." It often went like this: They would explain to me, "You've got an orange, OK? Now you cut the orange into a finite number of pieces, put it back together, and it's as big as the sun. True or false?" "No holes." "Impossible! "Ha! Everybody gather around! It's So-and-so's theorem of immeasurable measure!" Just when they think they've got me, I remind them, "But you said an orange! You can't cut the orange peel any thinner than the atoms." "But we have the condition of continuity: We can keep on cutting!" "No, you said an orange, so I assumed that you meant a real orange." So I always won. If I guessed it right, great. If I guessed it wrong, there was always something I could find in their simplification that they left out.

Can someone give a simple explanation as to why the harmonic series $$\sum_{n=1}^\infty\frac1n=\frac 1 1 + \frac 12 + \frac 13 + \cdots $$ doesn't converge, on the other hand it grows very slowly? I'd prefer an easily comprehensible explanation rather than a rigorous proof regularly found in undergraduate textbooks.

If you could go back in time and tell yourself to read a specific book at the beginning of your career as a mathematician, which book would it be?

Is there a way of taking a number known to limited precision (e.g. $1.644934$) and finding out an "interesting" real number (e.g. $\displaystyle\frac{\pi^2}{6}$) that's close to it? I'm thinking of something like Sloane's Online Encyclopedia of Integer Sequences, only for real numbers. The intended use would be: write a program to calculate an approximation to $\displaystyle\sum_{n=1}^\infty \frac{1}{n^2}$, look up the answer ("looks close to $\displaystyle\frac{\pi^2}{6}$") and then use the likely answer to help find a proof that the sum really is $\displaystyle \frac{\pi^2}{6}$. Does such a thing exist?

I'm not a real Mathematician, just an enthusiast. I'm often in the situation where I want to learn some interesting Maths through a good book, but not through an actual Maths textbook. I'm also often trying to give people good Maths books to get them "hooked". So the question: What are good books, for laymen, which teach interesting Mathematics, but actually does it in a "real" way. For example, "Fermat's Last Enigma" doesn't count, since it doesn't actually feature any Maths, just a story, and most textbook don't count, since they don't feature a story. My favorite example of this is "Journey Through Genius", which is a brilliant combination of interesting storytelling and large amounts of actual Mathematics. It took my love of Maths to a whole other level. Edit: A few more details on what I'm looking for. The audience of "laymen" should be anyone who has the ability (and desire) to understand actual mathematics, but does not want to learn from a textbook. Obviously I'm thinking about myself here, as a programmer who loves mathematics, I love being exposed to real maths, but I'm not going to get into it seriously. That's why books that show actual maths, but give a lot more exposition (and much clearer explanations, especially of what the intuition should be) are great. When I say "real maths", I'm talking about actual proofs, formulas, or other mathematical theories. Specifically, I'm not talking about philosophy, nor am I talking about books which only talk about the history of maths (Simon Singh style), since they only talk about maths, they don't actually show anything. William Dunham's books and Paul J. Nahin's books are good examples.

Is there a consensus in the mathematical community, or some accepted authority, to determine whether zero should be classified as a natural number? It seems as though formerly $0$ was considered in the set of natural numbers, but now it seems more common to see definitions saying that the natural numbers are precisely the positive integers.

Below is a visual proof (!) that $32.5 = 31.5$. How could that be? (As noted in a comment and answer, this is known as the "Missing Square" puzzle.)

Are there any simple methods for calculating the digits of $\pi$? Computers are able to calculate billions of digits, so there must be an algorithm for computing them. Is there a simple algorithm that can be computed by hand in order to compute the first few digits?

The arithmetic hierarchy defines the $\Pi_1$ formulae of arithmetic to be formulae that are provably equivalent to a formula in prenex normal form that only has universal quantifiers, and $\Sigma_1$ if it is provably equivalent to a prenex normal form with only existential quantifiers. A formula is $\Delta_1$ if it is both $\Pi_1$ and $\Sigma_1.$ These formulae are often called recursive: why?

Any homomorphism $φ$ between the rings $\mathbb{Z}_{18}$ and $\mathbb{Z}_{15}$ is completely defined by $φ(1)$. So from $$0 = φ(0) = φ(18) = φ(18 \cdot 1) = 18 \cdot φ(1) = 15 \cdot φ(1) + 3 \cdot φ(1) = 3 \cdot φ(1)$$ we get that $φ(1)$ is either $5$ or $10$. But how can I prove or disprove that these two are valid homomorphisms?

How can I show that $(n-1)!\equiv-1 \pmod{n}$ if and only if $n$ is prime? Thanks.

In category theory, a subobject of $X$ is defined as an object $Y$ with a monomorphism, from $Y$ to $X$. If $A$ is a subobject of $B$, and $B$ a subobject of $A$, are they isomorphic? It is not true in general that having monomorphisms going both ways between two objects is sufficient for isomorphy, so it would seem the answer is no. I ask because I'm working through the exercises in Geroch's Mathematical Physics, and one of them asks you to prove that the relation "is a subobject of" is reflexive, transitive and antisymmetric. But it can't be antisymmetric if I'm right...

I've read about about higher-order logics (i.e. those that build on first-order predicate logic) but am not too clear on their applications. While they are capable of expressing a greater range of proofs (though never all, by Godel's Incompleteness theorem), they are often said to be less "well-behaved". Mathematicians generally seem to stay clear of such logics when possible, yet they are certainly necessary for prooving some more complicated concepts/theorems, as I understand. (For example, it seems the reals can only be constructed using at least $2^{\text{nd}}$ order logic.) Why is this, what makes them less well-behaved or less useful with respect to logic/proof theory/other fields?

When I tried to approximate $$\int_{0}^{1} (1-x^7)^{1/5}-(1-x^5)^{1/7}\ dx$$ I kept getting answers that were really close to $0$, so I think it might be true. But why? When I ask Mathematica, I get a bunch of symbols I don't understand!

What is a unital homomorphism? Why are they important?

So you have $2$ dice and you want to get at least a $1$ or a $5$ (on the dice not added). How do you go about calculating the answer for this question. This question comes from the game farkle.

Books on Number Theory for anyone who loves Mathematics? (Beginner to Advanced & just for someone who has a basic grasp of math)

Background: Many (if not all) of the transformation matrices used in $3D$ computer graphics are $4\times 4$, including the three values for $x$, $y$ and $z$, plus an additional term which usually has a value of $1$. Given the extra computing effort required to multiply $4\times 4$ matrices instead of $3\times 3$ matrices, there must be a substantial benefit to including that extra fourth term, even though $3\times 3$ matrices should (?) be sufficient to describe points and transformations in 3D space. Question: Why is the inclusion of a fourth term beneficial? I can guess that it makes the computations easier in some manner, but I would really like to know why that is the case.

I'm looking for course notes and assignments and hopefully some example exams for Discrete Math, I'm taking a placement exam in the subject after having taken it 4 years ago.

I'm very interested in Computer Science (computational complexity, etc.). I've already finished a University course in the subject (using Sipser's "Introduction to the Theory of Computation"). I know the basics, i.e. Turing Machines, Computability (Halting problem and related reductions), Complexity classes (time and space, P/NP, L/NL, a little about BPP). Now, I'm looking for a good book to learn about some more advanced concepts. Any ideas?

I am vaguely familiar with the broad strokes of the development of group theory, first when ideas of geometric symmetries were studied in concrete settings without the abstract notion of a group available, and later as it was formalized by Cayley, Lagrange, etc (and later, infinite groups being well-developed). In any case, it's intuitively easy for me to imagine that there was substantial lay, scientific, and artistic interest in several of the concepts well-encoded by a theory of groups. I know a few of the corresponding names for who developed the abstract formulation of rings initially (Wedderburn etc.), but I'm less aware of the ideas and problems that might have given rise to interest in ring structures. Of course, now they're terribly useful in lots of math, and $\mathbb{Z}$ is a natural model for elementary properties of commutative rings, and I'll wager number theorists had an interest in developing the concept. And if I wanted noncommutative models, matrices are a good place to start looking. But I'm not even familiar with what the state of knowledge and formalization of things like matrices/linear operators was at the time rings were developed, so maybe these aren't actually good examples for how rings might have been motivated. Can anyone outline or point me to some basics on the history of the development of basic algebraic structures besides groups?

I'm looking to find out if there's any easy way to calculate the number of ways to tile a $3 \times 2n$ rectangle with dominoes. I was able to do it with the two codependent recurrences f(0) = g(0) = 1 f(n) = f(n-1) + 2g(n-1) g(n) = f(n) + g(n-1) where $f(n)$ is the actual answer and $g(n)$ is a helper function that represents the number of ways to tile a $3 \times 2n$ rectangle with two extra squares on the end (the same as a $3 \times 2n+1$ rectangle missing one square). By combining these and doing some algebra, I was able to reduce this to f(n) = 4f(n-1) - f(n-2) which shows up as sequence A001835, confirming that this is the correct recurrence. The number of ways to tile a $2 \times n$ rectangle is the Fibonacci numbers because every rectangle ends with either a verticle domino or two horizontal ones, which gives the exact recurrence that Fibonacci numbers do. My question is, is there a similar simple explanation for this recurrence for tiling a $3 \times 2n$ rectangle?

I know that the Fibonacci sequence can be described via the Binet's formula. However, I was wondering if there was a similar formula for $n!$. Is this possible? If not, why not?

What are the best books and lecture notes on category theory?

Context: Rings. Are $x \cdot 0 = 0$ and $x \cdot 1 = x$ and $-(-x) = x$ axioms? Arguably three questions in one, but since they all are properties of the multiplication, I'll try my luck...

The Fibonacci sequence is very well known, and is often explained with a story about how many rabbits there are after $n$ generations if they each produce a new pair every generation. Is there any other reason you would care about the Fibonacci sequence?

I know that the harmonic series $1 + \frac12 + \frac13 + \frac14 + \cdots$ diverges. I also know that the sum of the inverse of prime numbers $\frac12 + \frac13 + \frac15 + \frac17 + \frac1{11} + \cdots$ diverges too, even if really slowly since it's $O(\log \log n)$. But I think I read that if we consider the numbers whose decimal representation does not have a certain digit (say, 7) and sum the inverse of these numbers, the sum is finite (usually between 19 and 20, it depends from the missing digit). Does anybody know the result, and some way to prove that the sum is finite?

One of the first things ever taught in a differential calculus class: The derivative of $\sin x$ is $\cos x$. The derivative of $\cos x$ is $-\sin x$. This leads to a rather neat (and convenient?) chain of derivatives: sin(x) cos(x) -sin(x) -cos(x) sin(x) ... An analysis of the shape of their graphs confirms some points; for example, when $\sin x$ is at a maximum, $\cos x$ is zero and moving downwards; when $\cos x$ is at a maximum, $\sin x$ is zero and moving upwards. But these "matching points" only work for multiples of $\pi/4$. Let us move back towards the original definition(s) of sine and cosine: At the most basic level, $\sin x$ is defined as -- for a right triangle with internal angle $x$ -- the length of the side opposite of the angle divided by the hypotenuse of the triangle. To generalize this to the domain of all real numbers, $\sin x$ was then defined as the Y-coordinate of a point on the unit circle that is an angle $x$ from the positive X-axis. The definition of $\cos x$ was then made the same way, but with adj/hyp and the X-coordinate, as we all know. Is there anything about this basic definition that allows someone to look at these definitions, alone, and think, "Hey, the derivative of the sine function with respect to angle is the cosine function!" That is, from the unit circle definition alone. Or, even more amazingly, the right triangle definition alone. Ignoring graphical analysis of their plot. In essence, I am asking, essentially, "Intuitively why is the derivative of the sine the cosine?"

We will call the set of all positive even numbers E and the set of all positive integers N. At first glance, it seems obvious that E is smaller than N, because for E is basically N with half of its terms taken out. The size of E is the size of N divided by two. You could see this as, for every item in E, two items in N could be matched (the item x and x-1). This implies that N is twice as large as E On second glance though, it seems less obvious. Each item in N could be mapped with one item in E (the item x*2). Which is larger, then? Or are they both equal in size? Why? (My background in Set theory is quite extremely scant)

I have been learning some functional programming recently and I so I have come across monads. I understand what they are in programming terms, but I would like to understand what they are mathematically. Can anyone explain what a monad is using as little category theory as possible?

Say we have a function $\displaystyle f(z)=\sum_{n=0}^\infty a_n z^n$ with radius of convergence $R>0$. Why is the radius of convergence only $R$? Can we conclude that there must be a pole, branch cut or discontinuity for some $z_0$ with $|z_0|=R$? What does that mean for functions like $$f(z)=\begin{cases} 0 & \text{for $z=0$} \\\ e^{-\frac{1}{z^2}} & \text{for $z \neq 0$} \end{cases}$$ that have a radius of convergence $0$?

We define Lie algebras abstractly as algebras whose multiplication satisfies anti-commutativity and Jacobi's Identity. A particular instance of this is an associative algebra equipped with the commutator bracket: $[a,b]=ab-ba$. However, the notation suggests that this bracket is the one we think about. Additionally, the right adjoint to the functor I just mentioned creates the universal enveloping algebra by quotienting the tensor algebra by the tensor version of this bracket; but we could always start with some arbitrary Lie algebra with some other satisfactory bracket and apply this functor. My question is "Why the commutator bracket?" Is it purely from a historical standpoint (and if so could you explain why)? Or is there a result that says any Lie algebra is essentially one with the commutator bracket (maybe something about the faithfulness of the functor from above)? I know of (a colleague told me) a proof that the Jacobi identity is also an artifact of the right adjoint to the universal enveloping algebra. He can show that it is the necessary identity for the universal enveloping algebra to be associative (if someone knows of this in the literature I would also appreciate the link to this!) I hope this question is clear, if not, I can revise and try to make it a bit more specific.

It appears to me that only Triangles, Squares, and Pentagons are able to "tessellate" (is that the proper word in this context?) to become regular 3D convex polytopes. What property of those regular polygons themselves allow them to faces of regular convex polyhedron? Is it something in their angles? Their number of sides? Also, why are there more Triangle-based Platonic Solids (three) than Square- and Pentagon- based ones? (one each) Similarly, is this the same property that allows certain Platonic Solids to be used as "faces" of regular polychoron (4D polytopes)?

Two questions: Find a bijective function from $(0,1)$ to $[0,1]$. I haven't found the solution to this since I saw it a few days ago. It strikes me as odd--mapping a open set into a closed set. $S$ is countable. It's trivial to find a bijective function $f:\mathbb{N}\to\mathbb{N}\setminus S$ when $|\mathbb{N}| = |\mathbb{N}\setminus S|$; let $f(n)$ equal the $n^{\text{th}}$ smallest number in $\mathbb{N}\setminus S$. Are there any analogous trivial solutions to $f:\mathbb{R}\to\mathbb{R}\setminus S$?

I just got my statistics test back and I am totally confused about one of the questions! A study was done that took a simple random sample of 40 people and measured whether the subjects were right-handed or left-handed, as well as their ages. The study showed that the proportion of left-handed people and the ages had a strong negative correlation. What can we conclude? Explain your answer. I know that we can't conclude that getting older causes people to become right-handed. Something else might be causing it, not the age. If two things are correlated, we can only conclude association, not causation. So I wrote: We can conclude that many people become right-handed as they grow older, but we cannot tell why. That's exactly what association means, but my teacher marked me wrong! What mistake did I make? Is 40 too small of a sample size to make any conclusions?

I've sort of gotten a grasp on the Chain rule with one variable. If you hike up a mountain at 2 feet an hour, and the temperature decreases at 2 degrees per feet, the temperature would be decreasing for you at $2\times 2 = 4$ degrees per hour. But I'm having a bit more trouble understanding the Chain Rule as applied to multiple variables. Even the case of 2 dimensions $$z = f(x,y),$$ where $x = g(t)$ and $y = h(t)$, so $$\frac{dz}{dt} = \frac{\partial z}{dx} \frac{dx}{dt} + \frac{\partial z}{dy} \frac{dy}{dt}.$$ Now, this is easy enough to "calculate" (and figure out what goes where). My teacher taught me a neat tree-based graphical method for figuring out partial derivatives using chain rule. All-in-all, it was rather hand-wavey. However, I'm not sure exactly how this works, intuitively. Why, intuitively, is the equation above true? Why addition? Why not multiplication, like the other chain rule? Why are some multiplied and some added?

The discriminant $\Delta = 18abcd - 4b^3d + b^2 c^2 - 4ac^3 - 27a^2d^2$ of the cubic polynomial $ax^3 + bx^2 + cx+ d$ indicates not only if there are repeated roots when $\Delta$ vanishes, but also that there are three distinct, real roots if $\Delta > 0$, and that there is one real root and two complex roots (complex conjugates) if $\Delta < 0$. Why does $\Delta < 0$ indicate complex roots? I understand that because of the way that the discriminant is defined, it indicates that there is a repeated root if it vanishes, but why does $\Delta$ greater than $0$ or less than $0$ have special meaning, too?

This answer suggests that there are explicit polynomial equations for which the existence (or nonexistence) of integer solutions is unprovable. How can this be?

I know there must be something unmathematical in the following but I don't know where it is: \begin{align} \sqrt{-1} &= i \\\\\ \frac1{\sqrt{-1}} &= \frac1i \\\\ \frac{\sqrt1}{\sqrt{-1}} &= \frac1i \\\\ \sqrt{\frac1{-1}} &= \frac1i \\\\ \sqrt{\frac{-1}1} &= \frac1i \\\\ \sqrt{-1} &= \frac1i \\\\ i &= \frac1i \\\\ i^2 &= 1 \\\\ -1 &= 1 \quad !!? \end{align}

An equivalence relation is defined by three properties: reflexivity, symmetry and transitivity. Doesn't symmetry and transitivity implies reflexivity? Consider the following argument. For any $a$ and $b$, $a R b$ implies $b R a$ by symmetry. Using transitivity, we have $a R a$. Source: Exercise 8.46, P195 of Mathematical Proofs, 2nd (not 3rd) ed. by Chartrand et al

$\frac17 = 0.(142857)$... with the digits in the parentheses repeating. I understand that the reason it's a repeating fraction is because $7$ and $10$ are coprime. But this...cyclical nature is something that is not observed by any other reciprocal of any natural number that I know of (besides multiples of $7$). (if I am wrong, I hope that I may find others through this question) By "cyclical," I mean: 1/7 = 0.(142857)... 2/7 = 0.(285714)... 3/7 = 0.(428571)... 4/7 = 0.(571428)... 5/7 = 0.(714285)... 6/7 = 0.(857142)... Where all of the repeating digits are the same string of digits, but shifted. Not just a simple "they are all the same digits re-arranged", but the same digits in the same order, but shifted. Or perhaps more strikingly, from the wikipedia article: 1 × 142,857 = 142,857 2 × 142,857 = 285,714 3 × 142,857 = 428,571 4 × 142,857 = 571,428 5 × 142,857 = 714,285 6 × 142,857 = 857,142 What is it about the number $7$ in relation to the base $10$ (and its prime factorization $2\cdot 5$?) that allows its reciprocal to behave this way? Is it (and its multiples) unique in having this property? Wikipedia has an article on this subject, and gives a form for deriving them and constructing arbitrary ones, but does little to show the "why", and finding what numbers have cyclic inverses.

Let's say, I have 4 yellow and 5 blue balls. How do I calculate in how many different orders I can place them? And what if I also have 3 red balls?

I just came back from an intense linear algebra lecture which showed that linear transformations could be represented by transformation matrices; with more generalization, it was later shown that affine transformations (linear + translation) could be represented by matrix multiplication as well. This got me to thinking about all those other transformations I've picked up over the past years I've been studying mathematics. For example, polar transformations -- transforming $x$ and $y$ to two new variables $r$ and $\theta$. If you mapped $r$ to the $r$ axis and $\theta$ to the $y$ axis, you'd basically have a coordinate transformation. A rather warped one, at that. Is there a way to represent this using a transformation matrix? I've tried fiddling around with the numbers but everything I've tried to work with has fallen apart quite embarrassingly. More importantly, is there a way to, given a specific non-linear transformation, construct a transformation matrix from it?

I like to use physical demonstrations when teaching mathematics (putting physics in the service of mathematics, for once, instead of the other way around), and it'd be great to get some more ideas to use. I'm looking for nontrivial ideas in abstract mathematics that can be demonstrated with some contraption, construction or physical intuition. For example, one can restate Euler's proof that $\sum \frac{1}{n^2} = \frac{\pi^2}{6}$ in terms of the flow of an incompressible fluid with sources at the integer points in the plane. Or, consider the problem of showing that, for a convex polyhedron whose $i^{th}$ face has area $A_i$ and outward facing normal vector $n_i$, $\sum A_i \cdot n_i = 0$. One can intuitively show this by pretending the polyhedron is filled with gas at uniform pressure. The force the gas exerts on the $i_th$ face is proportional to $A_i \cdot n_i$, with the same proportionality for every face. But the sum of all the forces must be zero; otherwise this polyhedron (considered as a solid) could achieve perpetual motion. For an example showing less basic mathematics, consider "showing" the double cover of $SO(3)$ by $SU(2)$ by needing to rotate your hand 720 degrees to get it back to the same orientation. Anyone have more demonstrations of this kind?

Are there any interesting and natural examples of semigroups that are not monoids (that is, they don't have an identity element)? To be a bit more precise, I guess I should ask if there are any interesting examples of semigroups $(X, \ast)$ for which there is not a monoid $(X, \ast, e)$ where $e$ is in $X$. I don't consider an example like the set of real numbers greater than $10$ (considered under addition) to be a sufficiently 'natural' semigroup for my purposes; if the domain can be extended in an obvious way to include an identity element then that's not what I'm after.