New answers tagged

0

Here is what I am thinking so far, although I need help in the final step of seeing whether or not the Yoneda lemma is applicable in this setup. Consider that any monoid can be represented as a category with a single object, representing the monoid itself, and the morphisms of the category are the elements of the monoid, the associativity of their ...


1

Gödel/Scott themselves use both Ax. 2 and Ax. 3. The only difference is in Axiom 1 where you omit the box operator inside the scope of the allquantifier. Here is why I think this is a troublesome assumption. It might well be that it is accidentally the case that all individuals in our World that have a certain positive property A also have a Property B. For ...


6

Strictly speaking "will this notation I like ever catch on?" Is inappropriate for this site. But we can still weigh the pros and cons of the suggestion and mention any variants we know. Cons: Introduces yet another symbol to memorize weak improvement over the obvious alternative Rarely do you talk enough about irrationals collectively to warrant special ...


1

Disbelieving the theorem is a good way to start, I find. Every time a statement is made, try to break it. Ask "what if $x=-1$?", or whatever seems likely to make an equation or an inference break down. After all, the whole point of a proof is that the thing wasn't obvious to start with, which is why it needed proving. So put the proof to the proof. Once ...


1

The trick to memorizing/remembering proofs of theorems is to understand them. If you understand what a theorem is about and you remember the strategy for proving it, it is often not hard to fill in the details on the go. Some theorems do require some "tricks" that can be useful to remember. I think the worst thing you can do is to just mindlessly memorize ...


1

One of the best classics is Calculus Made Easy by Silvanus Phillips Thompson. It is a book on infinitesimal calculus originally published in 1910.


0

There is such a thing called orthogonal matrix in linear algebra. An orthogonal matrix is a square matrix with real entries whose columns and rows are orthogonal unit vectors: $$AA^T=I$$ Each entry is a dot product of a column of $A$ with an other column of $A$. Orthogonal matrices preserve dot products, the length of vectors as well as the angles between ...


1

The Frobenius inner product of matrices $A, B \in \mathbb R^{m \times n}$ is defined by $$\langle A, B\rangle := \mbox{tr} (A^T B) = \mbox{tr} (B A^T) = \mbox{tr} (A B^T) = \mbox{tr} (B^T A)$$ For example, the standard basis "vectors" for $\mathbb R^{2 \times 2}$ are $$M_{11} := \begin{bmatrix} 1 & 0\\ 0 & 0\end{bmatrix} \qquad \quad M_{12} := \...


4

I think it is much clearer to stipulate that $U$ is an open set. Expressing it as an element of the topology does not increase the clarity of what you are saying.


0

A set is called open if it is in the topology. It's consequently implied, by virtue of $U$ being called "open", that it's in the collection $\frak{T}$. While technically correct, I think you'd be wasting time by writing it, better to just declare it "open."


2

I think it might raise eyebrows for those who are new to set theory and topology, but that is precisely why I like this notation. One better get used to certain sets having elements which are sets containing further elements themselves. That said, if you want to “translate” Let $U$ be an open set that contains $x$ a better solution in this case ...


0

I tried to find the number of ways in which a number can be expressed in term of sum of two numbers and I ended up learning Partitions which showed me how everything can be expressed mathematically....


0

You asked: What was the first bit of mathematics that made you realize that math is beautiful? For me, it was when I was 3 years old (possibly 4), contemplating my hands and fingers. I had the sudden epiphany that 5+5 absolutely had to equal 10 every time that you added them together -- not merely that they had done so repeatedly, mind you, but that ...


3

There is a closed form. $\mbox{diag}(x)=I_n\circ (xu)$ where $\circ$ is the Hadamard product, $I_n$ the identity matrix and $u=[1,\cdots,1]$.


0

I found the following resources: http://www.educatorstechnology.com/2012/10/8-great-youtube-channels-for-math.html http://www.avatargeneration.com/2012/10/learning-math-with-youtube/ http://www.freetech4teachers.com/2012/04/seven-youtube-channels-not-named-khan.html#.V2pEdbh9600 TV show called Dara O'Braian's School of Hard Sums (highly recommended) http://...


1

Let $\delta_{ijk}$ be the tensor such that $$\delta_{ijk}=\begin{cases}1&\text{if}\;i=j=k\\ 0&\text{otherwise}\\ \end{cases}$$ (in the particular basis you're working in). Then $$\mathrm{diag}(x)_{ij}=\sum_k\delta_{ijk}x_k$$ or in the summation convention just $\delta_{ijk}x_k$.


3

Using tensor or Kronecker product notation, if $e_i = \begin{pmatrix} 0 \\ \vdots \\ 1 \\ \vdots \\ 0 \end{pmatrix}$ is the standard basis for $\mathbb{R}^{n \times 1}$ and $e^i = (0, \dots, 1, \dots, 0)$ is the standard basis for $\mathbb{R}^{1 \times n}$ then we can represent $\operatorname{diag}(x_1, \dots, x_n)$ as $$ \operatorname{diag}(x_1, \dots, ...


0

The function $\operatorname{diag}$ is a linear operator, so you know there is some matrix that does what you're looking for. Recall that to find a transformation matrix for a l.o. $T$ when working with the standard basis $e_1, e_2, \ldots, e_n$, we use the matrix: $$\begin{bmatrix}{T(e_1) \ \vert \ \ T(e_2) \ \vert \ \ldots \ \vert \ T(e_n) }\end{bmatrix}$...


2

Your $V$ is $$ V = \begin{pmatrix} -x_1^2 y_1 - x_1 x_2 y_2 + x_1 y_1 \\ -x_2^2 y_2 - x_1 x_2 y_1 + x_2 y_2 \\ \end{pmatrix} $$ thus a cubic polynomial for each component.


2

Most theorems have one or two tricks and the proof just unfolds from that. I would suggest just memorizing the tricks and then just figure out how the rest comes from that.


0

The underlying idea is extremely important in analysis; a simple example is that you can embed any inner product space into the space of linear functionals on its conjugate by the linear transformation $v \mapsto \langle -, v\rangle$. It's not an application of Yoneda lemma, but it's clearly the same sort of idea. In fact, there's a good analogy between ...


1

Your answers are correct and as you already stated, there is no general rule to produce counterexamples to the converse of a given result. However, it is often helpful to look at the proof of the correct result and study closely how the premises come into play. At every step where they're used, you may then try to produce an example - that doesn't satisfy ...


3

History aside, you would like all the angles that commonly occur in elementary plane geometry (and are classically constructible and are rational multiples of $\pi$) to be integer numbers of degreess. Given that we need to put $90^\circ$, $60^\circ$ (in an equilateral triangle) and $45^\circ$ (in a right isosceles triangle) and $72^\circ$ (in a regular ...


2

Sets $A$ and $B$ and a function $f$ form a counterexample to $f_*(A\cap B)\supseteq f_*(A)\cap f_*(B)$ if and only if there is some $x\in f_*(A)\cap f_*(B)$ such that $x\notin f_*(A\cap B)$. This means that there must be some $a\in A$ and $b\in B$ such that $f(a)=f(b)=x$, but there is no $c\in A\cap B$ such that $f(c)=x$. In other words, we must have $a\ne b$...


1

For high schoolers, degrees would definitely be easier than gradians. This is because $360$ is divisible by $2$, $3$, and $5$, so many angles of interest are integers in degrees while they are not in gradians: $$\frac{\pi}{3}=60^\circ=\frac{200}{3}^\text{g}$$ $$\frac{\pi}{4}=45^\circ=50^\text{g} \ \text{(OK, this works out for both.)}$$ $$\frac{\pi}{10}=18^\...


7

Resist the urge to look them up! If you are worried you are forgetting some theorem, sit down and try to re-derive it from something you do know how to prove (maybe mvt from Rolles, which maybe you remember). If you get stuck for awhile, look up a first step, then keep going yourself. "Actively" re-deriving theorems helps me make sure I understand them. It ...


0

Besides the right-handed system convention, sometimes we use clockwise sense. Of course the rotation of hour/minute/second hand of a clock. Also, whole-circle bearing is using clockwise direction (though polar coordinates take anti-clockwise as positive). Interestingly the elliptic integral $\displaystyle \int_{0}^{t} \sqrt{a^2\cos^2 t+b^2\sin^2 t} dt=aE\...


4

There is no mathematical reason for this. The reason for picking it this way (irritating people in other fields, as you can see from the comments!) is that rotating the [positive] $x$-axis onto the [positive] $y$-axis is about the simplest rotation you can think of, so we decide to call it "positive". And clearly rotating the positive $x$-axis onto the ...


1

Draw the axes with the $y$ one going downwards, while the $x$ one goes as usual. Now clockwise rotation is positive. The mathematics behind this behavior is called orientation.


6

To answer all three questions it suffices to construct a field of any given infinite cardinality, since a field is a ring which is additively a group. To that effect, let $A$ be an infinite set. Then the field $\mathbb Q(A)$ of rational functions over $\mathbb Q$ using the elements of $A$ as indeterminates is a field with the same cardinality as $A$.


6

Yes, many results of this form follow from the Löwenheim–Skolem theorem, which asserts that if a first-order theory (a particular way to write down axioms something should satisfy; this includes groups, rings, fields, and more) has an infinite model then it has a model of every infinite cardinality. The Löwenheim–Skolem theorem has much weirder ...


1

Well, I don't see how you can get $W(t)$ as a composed function of $u(t)$ as $W(t) = f\left(u(t)\right)$, but you can do the following, which may be of some use: Assuming $y(t)$ is a sufficiently-smooth function, note that $$ u(0) = 0 $$ and $$ u'(t) = y(t)\, . $$ Plug the latter expression for $y(t)$ into your integral for $W(t)$: $$ W(t) = \int_0^{\tau} d\...


0

What you want to do would fall into the field of Speech Synthesis, meaning the artificial production of human speech. Among the many approaches to speech synthesis, the one you are referring to would be Formant synthesis. Format synthesis aims to reproduce speech by adding specific individual wave sounds together. That technique is called Additive Synthesis....


2

Based on your needs you can choose a proper book. Usually, there are two kind of books here. Those that emphasize on the theory and those that emphasize on the application. Often, engineering students are not that much into the theory and they use books which care more about application. Books like Spivak have an emphasis on the theory. If you want to ...


2

It's just not a particularly interesting problem. Apart from the romantic (and ridiculous) idea that Fermat had a secret proof of his conjecture that was lost to history, there's not much compelling about FLT aside from the fact that it's very easy to state. What makes it interesting is that Frey proved that given a nontrivial rational point on the curve $x^...


2

For any given definition of natural numbers the existence of prime numbers is a matter of fact. Obviously the property that characterize these numbers is a consequence of the definition of the binary operation called ''multiplication''. In some sense we can consider the existence of the prime numbers a consequence of the existence of commutative rings ...


-2

This is such an interesting question! I am not an expert, I've only taken a introductory course on number theory, however what I've always thought is that the reason for prime numbers to exist, is the same as for atoms to exist. I mean, we know atoms are not the ultimate 'brick' of matter, however stick with it. The atoms bonded together form everything ...


0

Sorry this got a bit too big for a comment. Not really an answer but maybe some help to get you up and running. A quaternion $\bf q$ can be represented as a 4x4 matrix $\bf M_q$ of real values: $${\bf q} = a+b{\bf i}+c{\bf j}+d{\bf k} : {\bf M}_{\bf q}=\left[\begin{array}{rr|rr}a&b&c&d\\-b&a&-d&c\\\hline-c&d&a&-b\\-d&...


0

1). Suppose you want to build a $2$-sphere (up to homeomorphism) using convex polygons cut out of paper. It is natural to want to glue two polygons only along their edges and the edges should be glued in pairs. Further, if two edges of two polygons are to be glued, they should be of the same length so we can overlap them. Knowing that the Euler ...


1

In my opinion, it is sufficient that the objective is quasi-convex. Indeed, this ensures that all local minimizers are global minimizers and the set of global minimizers is convex. Thus, you do not have to fight against local minimizers (which are not global minimizers).


5

There is a lot of subtlety within the term "false statement". In the simplest sense, if we are studying a particular model or structure, each sentence in the appropriate formal language is true or false in that structure. There is little reason I can think to to try to pretend that a statement that is known to be false in a structure is true in that same ...


6

If you force two integers $a$ and $b$ to be equal, you get $\mathbb{Z}_{b-a}$, or modulo arithmetic. If you pretend that $x^2$ can be negative, you get complex numbers. If you say $3$ can be divided evenly by $2$, you get fractions. If you pretend that $p^n$ is large when $n$ is negative instead of positive, you get $p$-adic numbers. If you pretend there is ...


4

There is no situation where it is meaningful to assume that some false statement is true. The reason is that we would be asserting a falsehood, which means that we can derive any statement at all, not just nonsense equalities like $1 = 2$. It arises as follows: Let $P$ be a false mathematical statement, and assume that $P$ is true. Take any ...


1

Being continuous or not is often a matter of referential: origin, orientation of axes and units, or scale. When discontinuity appears, the source is often in the simplification of the model, think about reducing a moving object to its center of mass. But would physics exists without discontinuity? If an equation is continuous, and differentiable, it ...


1

This is a cleaned up version of some comments of mine on the original question. Some bad behavior is removable, some is not. The behavior that is removable in some sense "already is", from the physicists' perspective. For example, sinc can be thought of as (a multiple of) the Fourier transform of the indicator function of some interval symmetric about zero. ...


0

I would pick up a copy of Baby Rudin and apply what you learn to real life. Does the set of objects in a room belong to the countable set $\Bbb{N}$? Are there examples of a cartesian product in this very room right now? All in all, focus on what's beyond the end goal: Applying this to your topological grad school courses, your research papers, and in ...


2

For questions 1 and 2, it is impossible to answer in a sensible way without knowing what paper you are talking about. For question 3, just read more papers. After a while you'll get faster and better at discerning what parts and results of a paper interest you, and what is irrelevant for your research and therefore skippable.


0

You should find a good book or a good teacher if you want to appreciate the beauty of mathematics. If you personally experienced finding a subject beautiful and interesting then there will be no problem learning it even if you are a beginner. Note that by the terms good book and good teacher that is in accordance to your taste and therefore subjective.


2

Discontinuous functions are fairly common. What's the magnitude of the force between two point charges, or particles which can be considered point charges, $$F=\frac{kq_1 q_2}{r^2}$$ Where $q$'s are the charges, $k$ is constant, and $r$ is the distance between them. This is quite clearly discontinuous when the distance is is zero, and diverges as the ...


5

The canonical example of this is the apparent singularity that arises in spherical coordinates when you pass around the earth only to find that your longitude has gone from $0$ to $180$. Or for example the singularity that arises in the Laplacian with spherical coordinates. These are all non-physical and are a consequence of choosing a coordinate system. A ...



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