Spectral Measures: Riemann-Lebesgue Problem

Given a Hilbert space the Lebesgue measure.

Consider a selfadjoint Hamiltonian:
$$H:\mathcal{D}\to\mathcal{H}$$

Denote its associated Borel spectral measure by:
$$E:\mathcal{B}(\mathbb{C})\to\mathcal{B}(\mathcal{H})$$
its corresponding evaluated measures by:
$$\mu_{\varphi\psi}(A):=\langle\varphi,E(A)\psi\rangle$$
$$\nu_\varphi(A):=\|E(A)\varphi\|^2$$
and its related functional calculus by:
$$\varphi\in\mathcal{D}f(H):\quad\langle f(H)\varphi,\eta\rangle:=\int fd\mu_{\varphi\eta}\quad(\eta\in\mathcal{H})$$

Then it follows by Riemann-Lebesgue:
$$e^{itH}\varphi\stackrel{|t|\to\infty}{\rightharpoonup}0\quad(\varphi\in\mathcal{H}_{ac})$$

How to prove this from scratch?

Caution

Be warned that the above does not hold strongly:
$$\|e^{itH}\varphi\|\equiv\|\varphi\|\nrightarrow0$$
(The problem lies in missing pointwise convergence.)

Application

For compact operators this turns into a strong version:
$$C\in\mathcal{C}(\mathcal{H}):\quad Ce^{itH}\varphi\stackrel{|t|\to\infty}{\to}0\quad(\varphi\in\mathcal{H})$$
(This applies for example in scattering theory.)

Remember that $E(A)^{2}=E(A)=E(A)^{\star}$. So
$$
            \mu_{x,x}=(E(A)x,x)=(E(A)^{2}x,x)=(E(A)x,E(A)^{\star}x)=(E(A),E(A)x)=\nu_{x}.
$$
It is interesting that $\mu_{x,y} << \mu_{x,x}$ and $\mu_{x,y} << \mu_{y,y}$ because
$$
         |\mu_{x,y}(A)| \le \mu_{x,x}(A)^{1/2}\mu_{y,y}(A)^{1/2}.
$$
Radon-Nikodym gives you something interesting, and it can help define and study Fourier transforms in an abstract way. Pick a vector $x$ and stick to it. Then $y \mapsto f_{x,y}$ has some nice properties, where $d\mu_{y,x}=f_{x,y}d\mu_{x,x}$. This gives a representation of the cyclic subspace spanned by $x$ under the operators $T=\int \lambda dE(\lambda)$ and $T^{\star}$ as an $L^{2}_{\mu_{x}}$ space. On this subspace $T$ becomes a multiplication operator by the spectral parameter $\lambda$, which is what classical Fourier transforms do. This offers a type of generalized Fourier transform, and is a subset of the GNS theory of $C^{\star}$ algebra representation. Quantum Mechanics benefits from this point-of-view.

The probability measures are absolutely continuous:
$$\varphi\in\mathcal{H}_{ac}\iff\nu_\varphi\ll\lambda$$

It holds the estimate:
$$|\mu_{\varphi\eta}(A)|=|\langle E(A)\varphi,\eta\rangle|\leq\|E(A)\varphi\|\cdot\|\eta\|=\sqrt{\nu_\varphi(A)}\cdot\|\eta\|$$

So one has absolute continuity:
$$\nu_\varphi\ll\lambda\implies|\mu_{\varphi\eta}|\ll\lambda\implies\mu_{\varphi\eta}\ll\lambda\implies\mu_{\varphi\eta}\ll\lambda$$

One obtains an integrable derivative:
$$h\in\mathcal{L}(\lambda):\quad|\mu|=\int h\mathrm{d}\lambda$$

Arriving at the operator analogue of Riemann Lebesgue:
$$\langle e^{itH}\varphi,\eta\rangle=\int_{-\infty}^{+\infty}e^{itx}\mathrm{d}\mu_{\varphi\eta}(x)=\int_{-\infty}^{+\infty}e^{itx}u(x)\mathrm{d}|\mu_{\varphi\eta}|(x)=\int_{-\infty}^{+\infty}e^{itx}u(x)h(x)\mathrm{d}\lambda(x)\stackrel{|t|\to\infty}{\longrightarrow}0$$


An equation with angles of triangle satisfies 60 degrees triad, can we say that the triangle is equilateral? Question:
In a triangle ABC, the equation $2\cos A \sin C=\sin B $ holds true. What type of triangle is it?
Choices available are: a) isosceles b) equilateral
Solution:
$$2\cos A \sin C=\sin B $$
$$\Rightarrow \sin(A+C) -\sin (A-C)=\sin B$$
$$\Rightarrow \sin(A+C) -\sin (A-C)=\sin(180º - (A+C))$$
$$\Rightarrow \sin(A+C) -\sin (A-C)=\sin (A+C)$$
$$\Rightarrow \sin (A-C),  A = C$$
Hence isosceles.
Confusion:
But, using $A=B=C=\frac{\pi}{3}$ in the original equation, we get $2*\frac{1}{2}*\frac{\sqrt3}{2}=\frac{\sqrt3}{2}$ which is true.
I tried to obtain a proof for both "the triangle is equilateral" and "the triangle is not equilateral", but I end in deadlock situation, so could not obtain any proof for any of the two claims.
So, just because the equation satisfies, $A=B=C=\frac{\pi}{3}$, can we say that the triangle is equilateral?
Textbook has "isosceles" as the answer.

You have successfully proved that $A=C$. However, you cannot show that $A=B=C$. So the triangle cannot be shown to be equilateral. However, every equilateral triangle, also has both sides equal, so equilateral triangles also satisfy your condition. So what we have is,
Every equilateral triangle satisfies your equation, but not every triangle that satisfies your equation is equilateral, but it is definitely isosceles. So the correct answer would be isosceles.

Your solution is right.
Your example is also right because an equilateral triangle is also an isosceles triangle.
Formally you can say that the triangle isosceles or equilateral, as for example, you can say that the triangle is isosceles or right-angled isosceles triangle with $\beta=90^{\circ},$ but it's better to say that the answer it's: The triangle is an isosceles triangle.


Is there a way to prove that a theorem has no elementary proof? Or to prove that something may have no proof? Recently I was trying to prove something, more or less elementarily, but eventually started going in circles. My prof said that the proof involves mathematical tools that I've not seen yet, and that it's unlikely there is an elementary proof.

Which got me wondering: would there be a way to prove that there is no elementary proof to a particular theorem, or even a particular kind of theorem? (other than trivial cases such as theorems already involving non-elementary maths)

So, I guess, to rephrase: a way to prove that theorem $X$ can be proved using mathematical tools that are not higher than those needed to state $X$ in the first place?

Or how about theorems that are true but maybe have no proof; can this be known? Can we prove that a theorem may or may not be true, but would have no proof if it were?

I guess this is more metamathematics than mathematics, as it would involve theorems about the properties of theorems.... is there a branch of math that studies this sort of thing? Is this what "proof theory" is?

If there is such a subject: any book recommendations that would be accessible to someone in his 2nd undergrad year? This subject is interesting to me.

I don't know of any way to prove that a theorem can or cannot be proven by "elementary" methods, and without giving a rigorous definition to what "elementary" means I doubt one would be able to proceed in this direction.

What I do know, is that you should look up Gödel's incompleteness theorems. The branch of math that these sorts of things are studied under is logic. An excellent book on this topic is Gödel, Escher, Bach. It is appropriate for an undergrad, indeed it was written as a popular book and assumes very little mathematical knowledge, although the more math you understand the deeper some parts of the book will seem.

I caution you, the book will seem fairly abstruse at times, and indeed it is a tome (upwards of 700 pages). The part you are interested in is about half way through. I would recommend taking your time. Once you realize how interwoven the content really is with the presentation, you will realize (perhaps) how dense the book is and how much you may have missed. I Think Surely Most Everything Truly Abstracts.*

As a gentler introduction, I would recommend listening to this podcast of Radiolab on loops. It's a great show in general, but if you want to just hit the part about Gödel it is 38 minutes in.

Unprovability, in mathematics, is called independence. One method commonly used to prove independence is called forcing. From the wikipedia article, I see that it is covered in Set Theory: An Introduction to Independence Proofs, by Kenneth Kunen. I have not read this book, but from the online preview and the comments on the wikipedia page I would have to say that it requires a strong mathematical background.

Yes, you are right. As far as I know the works concerning your answer is on metalogic. I think books about Godel's Incompleteness theorem. and Logicians like Russell could give you a little hint. I have scanned some books about this but it's not as easy as it seems. Definitions are mandatory to make the discussion about "provability " in a certain space S of a certain theorem X. category theory, set theory ( foundations of Mathematics).. One good venture in your kind of question.


Linear combinations of nilpotent elements This is meant to be a trivial question in a book that I'm reading, but after banging my head against it for some time and searching online I haven't found a solution.

Question: given a ring $R$ and two nilpotent elements $x$ and $y$, prove that for any $a,b \in R$ the element $ax + by$ is nilpotent. Hint: use the binomial expansion.

I'm having no trouble showing that $x+y$ is nilpotent, but when including the "coefficients" I'm having some trouble. I started by letting $n$ and $m$ be such that $x^n=0$ and $y^m=0$, and letting $k=\max\{n,m\}$. Then applying the binomial theorem gives:

$$ (ax+by)^k = \sum_{i=0}^k \binom{k}{i} \;(ax)^i \;(by)^{k-i} $$

If multiplication were commutative then this would be easy, as multiplication isn't necessarily commutative in a ring, I don't really know what to do. Both suggestions and complete answers are appreciated.

This community wiki solution is intended to clear the question from the unanswered queue.

As discovered in the comments, the user simply overlooked the requirement of commutativity, and in other comments learned how things can go wrong.

I am also going to point out that if the coefficients are distracting, then one can just prove independently that $xr$ and $x+y$ are nilpotent if $x,y$ are, and then the claim about linear combinations follows.

In general, if R is semicommutative then the set nil(R) of all nilpotent elements of R forms an ideal, so the result follows.


What is the value of $\frac{\sin x}x$ at $x=0$? On plotting graph for $\frac{\sin x}{x}$ using Wolfram|Alpha and Google, got that :

also, I can get the value of $\lim_{x\rightarrow 0} \frac{\sin x}{x} = 1$ using squeeze theorem and as illustrated on sources such as MIT.

But I'm not able to understand how the function is defined at $x=0$ and its value came out to be $1$ at that point. As promised in the plot for the function.

Using limits I got the idea about the behavior of the function in the neighborhood of that point but not its value exactly at $x=0$.

A function  $f(x)=\frac{\sin x}{x}$ cannot be  defined for $x=0$ since division by $0$ is not defined in any numerical field. So this function is not continuous in $x=0$  but, since  ( as noted in OP)
$$
\lim_{x \to 0}\frac{\sin x}{x}=1
$$
the discontinuity is removable and the function defined as :
$$
f: \mathbb{R}\to \mathbb{R} \quad f(x)= \begin {cases}\frac{\sin x}{x} &\text{for }x \ne 0 \\ 1 &\text{for }x=0\end {cases}
$$

is continuous in $(-\infty, +\infty)$.

Consider the Maclaurin expansion of $\frac{sin(x)}{x}$

$$sin(x)=x-\frac{x^3}{3!}+\frac{x^5}{5!}-...$$

Dividing by $x$ you get a series that has a value of $1$ at $x=0$


How to prove that $\{(x,y) \in \mathbb{R}^2 \mid x \neq y\}$ is open? Prove that the subset $S := \{(x,y) \in \mathbb{R}^2 \mid x \neq y\}$ is open, with respect to a topology induced by the usual Euclidean distance.

I'm not even sure how to begin. Can anybody help me prove this?

Hint: What is its complement $S^c$? Do you know how to show that this one is closed?

HINT: Choose any point in $\mathbb{R}^2$ and show there's a bigger point using the triangle inequality for all points in $\mathbb{R}^2$.


Maximum volume of a box given perimeter and surface area What would be maximum volume of a rectangular box with a given perimeter $P$ and surface area $S$?

I tried to solve following equations, where $l$ is length, $b$ is base, $h$ is height, $P$ is the perimeter, $V$ is volume, and $S$ is the surface area.

$$2(l + b + h) = P  $$
$$2(lb + bh + hl) = S$$

Need to maximize $$V=lbh$$.

Thanks.

Hint

From the perimeter, you can extract $l$. From the surface, you can extract $b$. So, now, $V$ is only a function of $h$ and you want to maximize $V$. Then, ...

I am sure that you can take from here.

The maximum value of $ab$ with condition $a+b=\text{constant}$ is given by $a=b=\text{constant}/2$...

Similarly in this case if we have $l=b=h=\text{perimeter}/6$, we will get the maximum volume for the same perimeter.


How to prove existence of identity function Let $X$ be a non-empty set.Then, how do we show that there is a identity map from $X$ to $X$ using the axioms of set theory?

What you need to understand is the relationship between the axiom of power and the foundation axiom.  In principle, a well-founded model of set theory arises from iterated applications of the power set beginning with the empty set.  Once a set appears as an element, it will occur as a singleton in the next iteration of the power set.  Once a set occurs as a subset, its subsets will occur in the next iteration.  In particular, it pairs will occur.  Since the cumulative hierarchy also uses the axiom of union, the original set elements will be in the same set as these pairs.  So, the next iteration of the power set will form ordered pairs.  Now that you have ordered pairs, another power set iteration will generate sets composed of those ordered pairs, including the relation about which you are asking.  Note that for the identity relation, what would be a pair needs to be a singleton.  But these are generated in the same application of the power set as were the pairs.

Note that the axiom of union is used.  It guarantees that the elements of this account exist.  The same is true of the axiom of pairing.  Since the Cartesian product is, itself, the full relation, it will be composed at the same level as every other relation on the set of interest.  The separation schemes merely guarantee that what you can describe grammatically will exist relative to the sets generated in the cumulative hierarchy.

When you see remarks referring to "enough regularity", one is speaking about the level of the cumulative hierarchy needed to instantiate whatever function or relation is being considered.  However, it is very important to understand how formalist set theory inverts the sense of the constructive account above.  Suppose a particular scheme is asserted.  Then one is already considering models greater than some minimal rank needed to instantiate the scheme.  Formalism is not constructive, although the axiom of foundation is how modern set theory implements something similar to Russell's types.  And, Russell's views had been more along the lines of constructive mathematics.  That is why one can think about matters in this way.

It is true that one can simply prove things in ways suggested in the comments.  However, understanding how an existential claim is witnessed can offer a better picture of how all of the axioms are interrelated.  In other words, what "is" and what "is not" is determined by the cumulative hierarchy.


Finding the possible positions of chess knight mathematically relative to a given position From this website I found the following question:

  A chess board’s 8 rows are labelled 1 to 8, and its 8 columns a to h. Each square of the board is described by the ordered pair (column letter, row number).

  (a) A knight is positioned at (d, 3). Write down its possible positions after a single move of the knight.

  (b) If R = {1, 2, ..., 8}, C = {a, b, ..., h}, and P = {coordinates of all squares on the chess board}, use set notation to express P in terms of R and C

  (c) A rook is positioned at (g, 2). If T = {2} and G = {g}, express its possible positions after one move of the rook in terms of R, C, T and G.

For (a) is there anyway mathematically to list all the possible positions of the knight? I could for example, use a chess board to determine all the possible positions of the knight.

About (C) I completely don't get it.

The answers are provided in the website, however, I'm interested in how to solve them.

Given an initial position for the knight, the knight can move either north-east, north-west, south-east or south-west.  In each of these directions, there are usually two possible positions. For example, from $(d,3)$, the knight can move north-east to either $(e,5)$ or $(f,4)$.  Thus, there are a total of 8 possible positions for the knight if the knight starts at $(d,3)$.

Since a knight move consists of jumping 3 squares in the form of an L-shape, mathematically, the 8 possible new positions for the knight  can be obtained by just adding or subtracting 1 to one of the two coordinates of $(d,3)$ and 2 to the other coordinate. For instance, adding $(1,2)$ to $(d,3)$ gives $(e, 5)$. Here, I assume that 'adding' 1 to a letter gives the next letter, and so adding 1 to $d$ gives $e$.  Or, we can add $(2,1)$ to $(d,3)$, and we get $(f,4)$.  Or, we can add $(-1,2)$ to $(d,3)$, and we get $(c,5)$.  The number of different additions we can do is the size of the set $\{ (\pm 1, \pm 2), (\pm 2, \pm 1)\}$, which is 8.

Of course, if the knight is not near the center of the board, then the knight could fall off the board if it moves in some of these 8 directions.  So, if the knight is near the edge or corner of the board, only some (not all 8) of the possible 'additions' should be carried out.

For part (b), $P= C \times R$ should be clear, where $C$ is the set of columns of the board and $R$ the set of rows.

For part (c), observe that a rook on $(g,2)$ can either move horizontally to one of the 7 squares $(a,2), (b,2), \ldots, (h,2)$ or vertically to one of the 7 squares $(g,1), (g,3), \ldots, (g,8)$.  Hence, there are 14 possible new positions for the rook if the rook moves.  To express this set of 14 elements in set-theoretic form, observe that there are 8 positions on row 2, namely the positions $(a,2), (b,2), \ldots, (h,2)$. In other words, row 2 is the set $\{(a,2), \ldots, (h,2)\} = \{a,b,\ldots,h\} \times \{2\} = C \times T$.  Similarly, column 2 is the set $\{g\} \times \{1,\ldots,8\} = G \times R$.  The set of positions in row 2 or column 2 is their union $(C \times T) \cup (G \times R)$.  Observe that row 2 consists of 8 positions, column 2 consists of 8 positions, and their union $(C \times T) \cup (G \times R)$ consists of 15 positions (they have a common element $(g,2)$).  If a rook must move, it cannot occupy its initial position $(g,2)$.  This position is the unique element in the singleton set $\{ (g,2) \} = \{g\} \times \{2 \} = G \times T$.  If you remove the element $(g,2)$ from the 15-element set, you get the expression $((C \times T) \cup (G \times R)) - (G \times T)$.

For (a), you would move conveniently label the board position as $(4,3)$, and the $8$ possible destination squares as $(4\pm2,3\pm1)$ and $(4\pm1,3\pm2)$

Note, that's in answer to your question about listing the spaces mathematically. Using the original question notation, the knight's possible location after one move is $(\{c,e\},\{1,5\})\;\cup\;(\{b,f\},\{2,4\})$

Guessing at the unclear meaning of (c), I would say that the position of the rook after one move $(R,C)$ is: either $(R=G$ and $C \ne T)$ or  $(R\ne G$ and $C = T)$

Edit: And looking at the website you linked, I see that $R$ and $C$ are the set of possible values of rows and columns respectively, in which case I'd phrase my answer to (c) differently: the position of the rook after one move is defined by either $(R-G,T)$ or $(G, C-T)$


Solve for $x$, $3^{x^2} \lt 9^x$ $3^{x^2} < 9^x$

According to my book the solution is $x>0 \land x < 2$ so I assume this is a quadratic equation.

Therefore, I need to have $variable^2+variable+ \not variable$.

But I don't know what variable  is in this case.

I have:

$3^{x^2}$

$9^x = (3^2)^x = 3^{2x}$

and so

$3^{x^2}-3^{x2}<0$

So I think that variable should be $3^x$, so that in:
$$ax^2 + bx +c,$$
$$a = 1$$
$$b = ???$$
$$c = 0$$

But I don't know what b should be in this case...

Am I thinking right? How do I solve this?

You can use that $3^x$ is an increasing function. Hence
$$3^{x^2}<3^{2x}\iff x^2<2x\iff x^2-2x=x(x-2)<0\iff 0<x<2.$$

Use the laws of exponents.

That is, the corresponding equation is  $3^{(x^2)}=3^{(2x)}$; $x^2=2x$.

The rest follows trivially.


Question on Convergence of a sequence The sequence $y_n = \frac{1}{n+1}+\frac{1}{n+2}+\frac{1}{n+3} \dots +\frac{1}{2n}$.   Is $y_n$ convergent or divergent?

What I did:

From Mathematical Induction I derived that $y_n$ is an increasing sequence.
Now, let $y_n$ is bounded above.
Let a set Y = {$y_n$: n ∈ N}. If Y is bounded above, it has to have a supremum. Hence let $M=supY$
Therefore, $y_n \le M$ for every n ∈ N.
Hence, given $\epsilon \gt 0$, there exists some $y_k \in\Bbb Y$ such that $y_k \gt M-\epsilon$
$y_k+\epsilon \gt M$
$M \lt \frac{1}{k+1}+\frac{1}{k+2}+\frac{1}{k+3} \dots +\frac{1}{2k}+\epsilon$
Let $\epsilon = \frac{1}{2(k+1)}$  Then,
$M \lt \frac{1}{k+1}+\frac{1}{k+2}+\frac{1}{k+3} \dots +\frac{1}{2k}+\frac{1}{2(k+1)}$ $\lt$ $\frac{1}{k+2}+\frac{1}{k+3} \dots +\frac{1}{2k}+\frac{1}{2(k+1)}$=$\frac{1}{(k+1)+1}+\frac{1}{(k+1)+2} \dots +\frac{1}{2(k+1)}$ =$y_{k+1}$
Hence $M \lt y_{k+1}$ for some $y_{k+1}  \in\Bbb Y$. This is a contradiction which means, Y has no supremum which means Y is not bounded above. Hence Divergent.
I don't understand where did I go wrong in this proof because clearly Y is bounded above because $\frac{1}{n+1}+\frac{1}{n+2}+\frac{1}{n+3} \dots +\frac{1}{2n}$ $\lt$ $\frac{1}{n+1}+\frac{1}{n+1}+\frac{1}{n+1} \dots +\frac{1}{n+1}$$=$$\frac{n}{n+1}$ $\lt$$1$.
Hence 1 is clearly an upper bound.So, what is wrong with that proof?

Actually I'm not good with proving and I'm trying to improve my reasoning. Help is appreciated

Let $u_n=\frac{1}{n+1}+\ldots+\frac{1}{2n}$, we have
$$u_n=\sum_{k=n+1}^{2n}{\frac{1}{k}}=\sum_{k=1}^n{\frac{1}{n+k}}=\frac{1}{n}\sum_{k=1}^n{\frac{1}{1+\frac{k}{n}}}\underset{n\rightarrow+\infty}{\longrightarrow}\int_0^1\frac{dx}{1+x}=\ln 2$$
What is wrong in your proof is that you wrote $\frac{1}{k+1}+\ldots+\frac{1}{2(k+1)}<\frac{1}{k+2}+\ldots\frac{1}{2(k+1)}$.

Hint:
$$y_n=\dfrac1n\sum_{k=1}^n\dfrac1{1+\frac{k}{n}}$$ and use Riemann sums.

You can also note that $y_n$ is increasing and bounded since $$\dfrac12=\frac{n}{2n}\le y_n\le\frac{n}{n+1}< 1$$


Is :$\sqrt{i\pi+\sqrt{i\pi+\sqrt{i\pi+\sqrt\cdots}}}$ irrational or transcendental or real number? Is there someone who can show me if :$$\sqrt{i\pi+\sqrt{i\pi+\sqrt{i\pi+\sqrt\cdots}}}$$ is irrational or real or transcendental number ?

Thank you for any help

Let’s call $a=\sqrt{i\pi+\sqrt{i\pi\dots}}$
\begin{align}
a^2 &= i\pi+a \\
a^2-a &= i\pi \\
(a^2-a)^2 &= -\pi^2 \\
(a^2-a)(a^2-a) &= -\pi^2
\end{align}
Solve for $a$.
Hope that helps!


How to calculate the time it will take to solve a jigsaw puzzle Is there a way to know how much time it would take to solve a jigsaw puzzle if I already solve this same puzzle with a different number of pieces while timing myself?

Example of a landscape puzzle: (number of pieces) time to solve mm:ss

(24) 00:50
(54) 01:58
(96) 04:51
(150) 09:07
(384) 45:16

If the same puzzle was available in a 1000 pieces format, how long would it take me?

Also, is there a minimum number of puzzle of different size I need to complete to get a formula?

The image can greatly impact the time it will take to solve it too. If I am solving a puzzle with 2/3 of the image being a gradient sky and an image of a person, the gradient sky would take almost the same time as the person in a smaller puzzle size, but would take considerably longer the more pieces there are.

As you mention in the last paragraph, the time depends on the content, so there is no univocal relation between the number of pieces and the time.

Your data points are fairly well fitted by a linear law, and very well by a quadratic one. But we have no reason to believe that the time is super-linear, as you essentially build local aggregates of pieces and the time to locate one is roughly constant (or a slowly growing function of the total counts).

If you really want to find a statistically meaningful model, with a measure of the spread, you will have to solve hundreds of puzzles, and the result will depend on the operator.

Best Guess given only the data provided (y axis in minutes, x axis in number of pieces). Red shows your actual data, and blue shows Mathematica's best guess. It is a quadratic function, if you care to know.


Diophantine equation from the Latvian Baltic Way team selection competition 2019 So here is the problem statement:

Find all integer triples $(a, b, c)$ such that

$(a-b)^3(a+b)^2 = c^2 + 2(a-b) + 1$

The only things I have so far figured out is that (-1, 0, 0) and (0, 1, 0) are solution, gcd((a-b), c) = 1 and that c must be even.

Any ideas? Maybe even a general solution to the problem?

Consider the equation already found:-
$$(a-b)\left((a^2-b^2)^2-2\right)=c^2+1,$$
where $c^2+1$ is odd or singly even.

If $a-b$ is even then the LHS would be divisible by 4. Therefore $a-b$ is odd and $c$ is even.

Then $(a^2-b^2)^2-2$ is of the form $4k-1$ and it is therefore either $-1$ or has a prime divisor of the form $4k-1$. However, $c^2+1$ cannot have a prime factor of the form $4k-1$ and so $$(a^2-b^2)^2-2=-1.$$

Therefore $a^2-b^2$ is $-1$ or $1$ and then one of $a$ and $b$ is $0$. We also have $b-a=c^2+1$ and so the solutions for $(a,b,c)$ are $(0,1,0)$ or $(-1,0,0).$


Under what conditions does $f(x)$ equal its Fourier Series My question has 3 (similar) parts, and is as follows:

Under what conditions does $f(x)$ equal its Fourier Series for all $x$, $-L \leq x \leq L$?
Under what conditions does $f(x)$ equal its Fourier sine Series for all $x$, $0 \leq x \leq L$?
Under what conditions does $f(x)$ equal its Fourier cosine Series for all $x$, $0 \leq x \leq L$?

The Fourier series of a periodic continuous function of bounded variation converges pointwise to the function.  On the other hand, the Fourier series of a periodic continuous function can diverge at infinitely many points.  See e.g. R.E. Edwards, "Fourier Series - A Modern Introduction", section 10.3.1.

For your first part, any function which will be periodic with period 2L can be decomposed into fourier series and will take exact value for all the point's, given it is equal at -L,L and has no discontinuances in between. If it has discontinouities, it will take the average value between the two ends(Dirichlet theorem).

Sine has the same conditions as above with the additional condition that the function should be odd.In cosine, the function should be even.


Find the sum of roots of the equation. Could anyone check if it is correct? $\cos(12x)=5\sin(3x)+9\tan^2(x)+\frac 1 {\tan^2(x)},0\le x\le 2\pi$

Find the sum of roots of the above equation.

The answer is $\frac 7 6 \pi+\frac {11} 6\pi =3\pi$ given by WolframAlpha.

My attempt

I decomposed $\cos(12x)$ into $1-8\sin^2(3x)+8\sin^4(3x)$ but it didn't work.

Thanks for any help.

EDIT

I found a plausible solution. Could anyone check if it is correct?

Rewrite the equation as

$\cos(12x)-5\sin(3x)=9\tan^2(x)+\frac 1 {\tan^2(x)},0\le x\le 2\pi$.

1. Since R.H.S.$=9t+\frac 1 t\ge2\sqrt{9t \cdot \frac 1 t}=6$, where $t=\tan^2x$.

$\therefore \min(\text{R.H.S.})=6$.

2. And L.H.S.$=\cos(12x)-5\sin(3x)\le1-5(-1)=6.$

$\therefore \max(\text{L.H.S.})=6$.

$\therefore \text{L.H.S.}=6=\text{R.H.S.}$.

Solve R.H.S.$=6$, $x=\frac 7 6 \pi$ or $\frac {11} 6\pi$, reject  $x=\frac 1 6 \pi$ or $\frac 5 6\pi.$

$\therefore$ The final solution is $3\pi$.

Thanks.

In the image i have done most of the part and i reduced the expression by applying the AM >= GM


vectors in graph theory I am reading from a book (Combinatorial optimization by Schrijver) and at one point I am not clear as to how his arguments follow. Consider the following:

Let $G(V,E)$ be an undirected graph and let $w:E\rightarrow R_+$. For any subset $F$ of $E$, denote $w(F)=\sum_{e\in F}w(e)$. Denote the incidence vector of $F$ in $\mathbb{R}^E$ by $\chi^F$, i.e. for any edge $e$ in $E$, $\chi^F(e)=1$ if $e\in F$ and $\chi^F(e)=0$ otherwise.  Considering $w$ as a vector in in $\mathbb{R}^E$ we have $w(F)=w^T\chi^F$.

My doubts are as follows:

Is $E$ a vector space and $w,\chi^F$ linear transformations from $E$ to $\mathbb{R}$? Why are they referred to as vectors? Aren't they functions?
What does the term incidence vector mean?
How does it mathematically follow that $w(F)=w^T\chi^F$ ?

1) Functions over finite sets may be equivalently thought of as vectors in the usual sense. Fix a finite set $E$, ordered arbitrarily as $E = \{ e_1, e_2, \ldots, e_m \}$. Then the set of functions $\mathcal F = \{ f : E \to \mathbb R \}$ is a real vector space of dimension $|E| = m$. In fact, this has the standard basis $\{ \delta_e \colon E \to \mathbb R \}_{e \in E}$ given by
$$
\delta_e(e') = \begin{cases}
1, &e' = e,
\\ 0, &e' \ne e.
\end{cases}
$$

Based on this idea alone, we are justified in calling a function $f : E \to \mathbb R$ a vector. After all, what are members of a vector space called, if not vectors? However, to cement the idea even more, we go a bit further. Imagine stacking up the values of the function $f$ as a column vector:
$$
f = \left( \begin{array}{c}
f(e_1) \\ f(e_2) \\ f(e_3) \\ \vdots \\ f(e_m)
\end{array} \right).
$$
Then what we get is a correspondence (an isomorphism, in fact) between the function space $\mathcal F$ and the more familiar space of $m$-dimensional column vectors. This "vector notation" is often preferred over the function representation, since it is more compact. Moreover, this works seamlessly in conjunction with matrices (and matrices are obviously important in linear algebra).

2) An incidence vector $\chi^F$ is essentially the characteristic function of the set $F \subseteq E$ -- seen as a vector. The characteristic function $\chi^F$ is defined as
$$
\chi^F(e) = \begin{cases}
1, &e \in F,
\\ 0, &e \notin F.
\end{cases}
$$ Correspondingly, we can write $\chi^F$ as the $m$-vector with $1$ in the $i^{th}$ position if $e_i \in F$, and $0$ otherwise.

3) Given two vectors $a$ and $b$, you might have seen the inner product between them,  defined as
$$
a^\mathrm T b = \sum_{i=1}^m a_i b_i.
$$
Unfolding this to the function representation, we can define the inner product of two functions $f, g : E \to \mathbb R$ to be
$$
\langle f, g \rangle = f^\mathrm T g = \sum_{e \in E} f(e) g(e).
$$
In the special case when $f = w$ and $g = \chi^F$, we get
$$
\langle w, \chi^F \rangle = w^\mathrm T \chi^F = \sum_{e \in E} w(e) \chi^F(e) = \sum_{e \in F} w(e),
$$
which is $w(F)$ by definition.

Let vi = (#i,yi), • • • ,vn = (xn,yn) be n two-dimensional vectors, where
each Xi and each y¿ is an integer whose absolute valué does not exceed
2™/2/(i00v
/ñ). Show that there are two disjoint sets I,J C {1,2,..., n}
such that
! > = $ > ; •
iei jeJ


Need help understanding the derivation of Hydrostatic Equilibrium in a star First and foremost I am sorry for this; in order to make my questions clear I must first upload the lecture notes from my institution $^\zeta$ for the derivation of Hydrostatic Equilibrium:

$^\zeta$ Lecture notes courtesy of Imperial College London, Astrophysics dept, 2017-2018 edition.

Firstly, in equation $(3.1)$ why is there no negative sign for $F_g$? Last time I checked, Newton's law of gravitation told us that

$$F=-\frac{GMm}{r^2}$$

Nextly, I have a conceptual problem with equation $(3.2)$; when they use the term 'pressure' I am going to assume that they are referring to the radiation pressure that is released due to nuclear fusion reactions taking place in the sun. In that formula it must be the case that $P(r) \gt P(r+\delta r)$ if this is the case then $(3.2)$ should read $$F_p=-\left(\frac{dP}{dr}\right)\delta r\,\delta A $$

Correct me if I'm wrong but Hydrogen nuclei are fused together to generate the more stable Helium and hence lots of radiation pressure acting outwards from the centre of the star. If this is correct then $P(r+\delta r)=0$, since the outside face of that cylindrical volume element in Figure $3.1$ will not 'feel' any of this radiation pressure because this pressure only acts on the inner face. What am I not understanding here?

Moving on to equation $(3.3)$, let's suppose the sign of the gravitational force in $(3.1)$ really is neglected. Does this mean that the negative sign in $(3.3)$ is due to $P(r+\delta r) \lt P(r)$ and hence $$\frac{GM(r)\rho(r)}{r^2}=-\left(\frac{dP}{dr}\right)\,?$$ I know this is the same equation as in $(3.3)$ but I have written it with the negative sign next to the pressure differential as I would like to know where this minus sign originates from. Does anyone know?

My final concern is the sign in $(3.4)$ as last time I checked $$g=-\frac{GM}{r^2}$$ and not $$g=\frac{GM}{r^2}$$

Hence, equation $(3.4)$ should read
$$\left(\frac{dP}{dr}\right)=\rho(r)\,g\,?$$

I forgot to mention that I have done some research and this is the closest derivation I could find, which, needless to say is no less helpful than the notes I have.

The pressure they are talking about is not radiation pressure, it is just the ambient pressure in the star.  The equations here are the same as hydrostatic equilibrium in the earth's atmosphere.  The outward pressure has to equal the sum of the inward pressure and the gravitational force on the cylinder.  That gives a relationship between the fall of pressure and the mass density of the gas.

In the star, light does not travel far before being scattered.  Its pressure acts in all directions, not just outward.


String Theory: What to do? This is going to be a relatively broad/open-ended question, so I apologize before hand if it is the wrong place to ask this.

Anyways, I'm currently a 3rd year undergraduate starting to more seriously research possible grad schools.  I find myself in somewhat of a weird spot as my primary interests lie in physics, but I usually can't stand the imprecision with which most physicists do physics.  I eventually would like to do work relevant to the quest of finding a theory of everything, but because I do not like the lack of rigor in physics, I have decided not to go to graduate school in physics.  However, when looking at the research interests of faculty members, I've found that most institutions have zero, one, or occasionally two (mathematics) faculty members working in this area.  Am I just not looking in the right places?  Where I am to go if I am looking to get into a field like String Theory from a mathematician's perspective?

As a separate but related question, I've found the prerequites for string theory to be quite daunting.  At this point, I feel as if it will be at least another year or two before I can even start learning the fundamentals of the theory (I won't even be taking a course in QFT until next year).  To be honest, I am starting to feel a little scared that I won't have enough time to do my thesis work in a field related to string theory.  Compared to algebraic topology or something, which I took last year, this year and next I could be learning more advanced aspects of the field so that by the time I got to grad school I could immediately jump in and start tackling a problem, whereas with string theory I feel as if I won't be able to really do this until my third year of grad school or so.  Is this something I should actually be worried about, or am I worrying about nothing?

Also, if any of you have studied string theory, I would be interested in knowing what subjects I should study to prepare myself and textboks that you recommend for studying from.  I would prefer textbooks about physics written by a mathematician or at least a great deal of mathematical rigor, although I am willling to compromise.

Thanks before hand for all the help/suggestions.  I am interested to hear mathematicians' take on this.

EDIT:  A comment made me think that I should point out that I am ultimately interested in a theory of everything, not string theory per se.  At this point, because I have so much to learn, I think that if I head in the direction of string theory (learn things like QFT, GR, Conformal Field Theory, Supersymmetry, etc.) I can't go wrong.  It won't be for awhile until I have to really make a choice between candidates for a TOE.

UPDATE+: Nature's article G. Ellis & J. Silk - Scientific Method: Defend the Integrity of Physics led to the conference in Münich "Why Trust a Theory?" about the dangers of non-empirical confirmation, defended by some popular string theorists basically to mend the scientific method so that their "theory" can be justified without direct evidence, with philosophers and physicists presenting arguments and raising the concern that there is A REAL PROBLEM in the community. The talks by David Gross, Massimo Pigliucci and Carlo Rovelli were specially insightful. I personally endorse Carlo Rovelli's philosophy of science emphasizing the need of conceptual understanding to advance as the great thinkers of the past: A critical look at strings. A good skeptic, and now well-known, blog with up-to-date criticism about string 'theory' and some of its theorists' attitudes is P. Woit's Not Even Wrong. Other phenomenological quantum gravity skeptic blog is Sabine Hossenfelder's Backreaction.

String "theory" has produced an enormous mathematical framework of insights, hidden inter-connections and tools but it has failed so far to produce any empirical physical theory that actually solves the problems it was claimed to be able to solve. The question is not whether string theory is interesting mathematically, but whether it is a framework worth funding disproportionately with hype in the media by many important professionals selling it as a tested final theory (or even calling it a physics theory since there is still no known way to get a 4-dimensional limit without supersymmetry at low energy including the standard model of particle physics and non-perturbative general relativity with a positive cosmological constant). This slide by Rovelli in his Münich talk is a good summary:

Original answer (years ago before the LHC results)

I am going to try to answer your question by talking about my particular experience since I had interests and intentions similar to yours. At the end I give some book recommendations.

All along my undergraduate studies in theoretical physics I was aiming at getting into a Ph.D. program on String Theory.... but at the same time I was getting my own personal opinions about string theory because of the knowledge about the status of physics research nowadays and my own deep understanding of theoretical physics.
Some years ago I came to the conclusion that string theory is not something worth pursuing right now for me.

(THIS IS ONLY MY OPINION BASED ON MY EXPERIENCE AND KNOWLEDGE OF PHYSICS)

Why is that so? First of all, the particle physics community now is expecting to get new physics from CERN's LHC and other projects related to dark matter, dark energy and cosmology; the theoretical physics advances for the next decade will be around fitting phenomenological models to the new data, none of them is realistically derived from string theory. Secondly and personally I have found string theory to be a very naive attempt of "final theory": it will not be a final theory nor a theory of everything because it is not even a theory yet. The string models are a TREMENDOUS framework of structures connected to each other and to almost every aspect of mathematics but are not rigorous enough because even Interacting Quantum Field Theory is not rigorous mathematically. At the present time there is no solid mathematical foundation for the Feynman Path Integral and other continuum problems of field theory which are the cornerstone of relativistic quantum theories nowadays (they are likely to be a continuum formal asymptotic approximation to some fundamental discrete physics like in Loop Quantum Gravity, Causal Dynamical Triangulations...). Thirdly, the amount of background needed to study and tackle any problem of string theory is too much for a person to devote his/her career to: you pretty much have to be a multidisciplinary mathematician able to handle with easy branches from differential and algebraic topology, differential and algebraic geometry, complex analysis and even number theory; this besides the amount of physics needed, since ordinary quantum field theory per se is a whole world hard to master for a single person. Much of the progress of the last 30 years has been done more on the mathematical side (topological quantum field theories, mirror symmetry and lots of small advancements in pure math thanks to physical insights...). At this moment an amount of theorists are shifting their string-theoretic methods and knowledge to condensed matter physics (graphene...) and such... because the original objectives are kind of stalled (landscape multiverses, lack of contact with experiment, no clues on how to get a nonperturbative formulation of the theory and deduce the standard model of particle physics...).

Lastly, and most importantly, I personally do not like the attitude of SOME of the string theory community...... Just as Richard Feynman thought of string theory in the 80s, they are behaving VERY UNSCIENTIFICALLY..... The "theory" might be right in the end but they are not been modest or cautious at all........ There is too much propaganda regarding string theory focused on attracting young students and more funding.... This is unprecedented in physics..... and many great scientists of the past would agree I think. Too many books and documentaries talking about string theory like it were already the final theory, the final answer even though it has not made a single real prediction... A true scientist must be wary of this kind of declarations since despite how much promising and rich a theory is, string theory is not even wrong, as for the time being cannot be tested or falsified, and therefore is not a "theory" in the same respect as the theory of relativity or quantum theory. I find that kind of naming vey misleading.

Other very disturbing attitude related to this is the narrow-minded behavior of SOME researchers and departments, to the point of not allowing deviations from string theory. At the present time there is no compelling empirical reason for string theory to be better than loop quantum gravity, emergent gravity or other models of quantum gravity which are pursued at the same time. Nevertheless very few people work on them and are not encouraged for graduate students at most universities.... so more and more people end up working in a very tiny small corner of string theory, just trying to go on with their careers and get settled somewhere...... without caring too much that their work may be completely wrong or that other alternatives are as much valid as theirs.
I recommend you read the article"A dialog on quantum gravity" by Carlo Rovelli.

I must say that even though I am being very critic to string theory, I was very interested in it from the very beginning and talked to many professors working on it. Besides, I acknowledge its tremendous usefulness and impact on mathematics.

But as I learned more and more physics I have found Non-Commutative Geometry or Loop Quantum Gravity and particularly its Spin Foam version, much more attractive than string theory in spite of the fact of being still in a very early stage (remember, the greatest theorists for the past 30 years going to string theory...). Their attitude towards science is more conservative and they do not attempt a theory of everything but working out first a quantum gravity theory compatible with known relativity and quantum mechanics. Besides it is a better background independent, relational model. For example I get the feeling that the discrete Spin Foam formalism is in the good direction due to its finite sums of a discrete geometry dynamics as the final substrate of the Feynman Path Integral, therefore automatically regularizing the mathematical nonrigorous difficulties of ordinary quantum field theory (and string theory). Maybe it is just another theory or attempt, maybe both models are different sides of the story...

...

So I decided eventually to shift my career towards Pure Mathematics. Pursuing graduate studies in math is a more secure path right now. Some professor I had who was a string theorist advised me that as well. Now, I can relax and study any field of mathematics I know will still be there after a decade or two, letting me fulfill a career as a researcher and/or teacher without any regrets. Thus, I see too risky a plan to study string theory right now. Nevertheless you can pay attention to advances and news about theoretical physics at the same time.
I have been told by many Math professors that a mathematician can later become a string theorist (and publish in physics journals) but it is harder for a theoretical physicist to become a pure mathematician (and publish in math journals).

If you are interested in a Theory of Everything... that is far from being discovered in the near future... In my view the only Theory of Everything is MATHEMATICS..... as the final structure of Nature and everything we will be ever able to say about it will be mathematical. The problem of the "Physical Theory of Everything" is therefore to find out which is the particular structure of our Universe/Nature, at least as far as how much can be said from within. If you carry on with physics you will be looking for the last kind of theory of everything: the final mathematical model which embodies all the laws of Nature as we know it. If you choose pure mathematics you will be looking for the first kind of theory of everything: all that can be said about any logical structure conceivable in Nature.

Maybe you will find a similar view interesting in the kind of speculative and philosophical (but fun and enlightening!) papers on the structure of a final theory and the mathematical universe by Max Tegmark:

EDIT: in response to your comments and edit addition to your question, be sure that learning General Relativity, Quantum Field Theory, Conformal Field Theory, SuperSymmetry, Topological Field Theory and the like WILL BE VERY USEFUL AND INTERESTING WHATEVER PATH YOU TAKE AFTERWARDS. I must emphasize here that I learnt many of those subjects already and before deciding to switch to Pure Mathematics, since they are part of an undergraduate education in my country. I decided to forget about String Theory ONLY AFTER I knew enough physics to check out whether I liked/understood the theory or not. So my last advice would be to learn as much as possible and when the moment comes make up your mind depending on your personal tastes and philosophy. Therefore focusing right now on string theory is irrelevant since you will have to learn in the meantime as much physics and mathematics as needed for any other specialization in theoretical physics or pure mathematics.

If you wish to understand string theory there are many books available but you have to master a lot of mathematics as well.... as it is mentioned here. My personal advice would be to become a graduate student in mathematics with enough physics background (see below) and then read books like Deligne et al. - "Quantum Fields and Strings: A Course for Mathematicians", maybe after a softer (physical) introduction with Becker/Becker/Schwartz - "String Theory and M-Theory: A Modern Introduction", Johnson - "D-Branes" and Polchinski - "String Theory vol I, II".

A quick overview for physicists of the mathematics needed (for example to understand the book by Becker) is developed in the books by Nakahara - "Geometry, Topology and Physics" and Eschrig - "Topology and Geometry for Physics" but they are not at all enough for the others. Applications such as mirror symmetry are very well explained in Hori et al. - "Mirror Symmetry", Bridgeland et al. - "Dirichlet Branes and Mirror Symmetry" and Cox - "Mirror Symmetry and Algebraic Geometry".

You will need a strong background in general relativity which may be studied with Grøn/Hervik - "Einstein's General Theory of Relativity" and Ortin - "Gravity and Strings"; and also a deep knowledge of quantum field theory which can be studied in a mathematical style with Ticciati - "Quantum Field Theory for Mathematicians" and de Faria/de Melo - "Mathematical Aspects of Quantum Field Theory". The best references for mathematical Gauge theory are Naber - "Topology, Geometry And Gauge Fields: Foundations" and Naber - "Topology, Geometry And Gauge Fields: Interactions". Also, the huge tomes by Zeidler - "Quantum field theory I. Basics in Mathematics and Physics" and Zeidler - "Quantum Field Theory II. Quantum Electrodynamics" and their future volumes seem to me the BEST mathematical approach to modern theoretical physics. Some supersymmetry will be needed like in Wess/Bagger - "Supersymmetry and Supergravity" and Terning - "Modern Supersymmetry, dynamics and duality".

For other approaches to quantum gravity the book by Rovelli - "Quantum Gravity" is the best introduction to loop quantum gravity, and can be followed by the very mathematical but understandable book by Thiemann - "Modern Canonical Quantum General Relativity" which includes hundreds of pages of mathematical appendices with the needed formalisms. A good review of most approaches is Kiefer - "Quantum Gravity" with a conceptual overview by Oriti et al. - "Approaches to Quantum Gravity".

It would be culpably stupid to plan a career in String Theory at this late stage. Don't you read the blogs? String Theory has been dying for ten years or more, and is only kept alive by the intransigence of its supporters. Try Loop Quantum Gravity if you want to make your mark.


This is a standard application of Taylor series, hence Community Wiki.
We have $\cos x=1-\frac12x^2+\frac1{24}x^4-\cdots$ and $\sin x=x-\frac16x^3+\cdots$, so
$$
\begin{aligned}
\cos (\sin x)&=1-\frac12\sin^2 x+\frac1{24}\sin^4x+\cdots\\
&=1-(\frac12x^2-\frac16x^4+O(x^6))+\frac1{24}(x^4+O(x^6))\\
&=1-\frac12x^2+\frac5{24}x^4+O(x^6).
\end{aligned}
$$
Similarly (use the binomial series) the Taylor series of
$$
1/\sqrt{1+x^2}=1-\frac12x^2+\frac9{24}x^4+O(x^6).
$$
Therefore
$$
\cos(\sin x)-\frac1{\sqrt{1+x^2}}=-\frac16x^4+
O(x^6).
$$
From this it follows immediately that the limit is equal to $-1/6$.
Everywhere $O(x^6)$ stands for terms of degree six or higher.

The problem with the attempt in the original post is that in the first step some terms of degree four were neglected. In the end we cancel a factor of $x^4$, so this was a mistake.

Write  $\cos(\sin x)-(1+x^2)^{1/2}$ as $\cos(\sin x)-\cos x+\cos x-(1+x^2)^{1/2}$
Then,
$$\cos(\sin x)-\cos x=2\sin\dfrac{x-\sin x}2\sin\dfrac{x+\sin x}2$$
Now use L'Hospital's or  Are all limits solvable without L'Hôpital Rule or Series Expansion to find $$\lim_{x\to0}\dfrac{x-\sin x}{x^3}=\dfrac16$$  so that $$\lim_{x\to0}\dfrac{\cos(\sin x)-\cos x}{x^4}=\dfrac24\lim_{x\to0}\dfrac{\sin\dfrac{x-\sin x}2}{\dfrac{x-\sin x}2}\lim_{x\to0}\dfrac{\sin\dfrac{x+\sin x}2}{\dfrac{x+\sin x}2}\lim_{x\to0}\dfrac{x-\sin x}{x^3}\lim_{x\to0}\dfrac{x+\sin x}x$$
Finally $$\cos x-(1+x^2)^{-1/2}$$
$$=1-\dfrac{x^2}{2!}+\dfrac{x^4}{4!}-\cdots-\left(1-\dfrac{x^2}2+x^4\cdot\dfrac{\left(-\dfrac12\right)\left(-\dfrac12-1\right)}{2!}+\cdots\right)$$
$$=x^4\cdot\left(\dfrac1{4!}-\dfrac38\right)+\text{terms containing higher exponents of }x$$


How to calculate integral $\int \frac{xdy-ydx}{x^2+y^2}$ where the figure is bounded by $l: (x-a)^2+(y-b)^2=1$ I tried to rewrite the function under integral in polar coordinates, but the issue is that I don't know how to deal with these separate $dy$, $dx$. Here is my try: $$\int \frac{r\cos\phi d\phi - r\sin\phi dr}{r}$$

The domain of the integrand constitutes those pairs $(x,y)$ such that $x\neq 0$ or $y\neq 0$. On its domain, the integrand is closed, because
$$\frac{\partial}{\partial x}\frac{x}{x^2+y^2}=\frac{\partial}{\partial y}\left(-\frac{y}{x^2+y^2}\right)=\frac{y^2-x^2}{(x^2+y^2)^2}\text{.}$$
Consequently, it is locally exact: on each contractible subset of its domain, it may be written as an exact differential. For example, if $x>0$ or if $x<0$, then

$$\frac{x\mathrm{d}y-y\mathrm{d}x}{x^2+y^2}=\mathrm{d}\arctan\tfrac{y}{x}$$
whereas if $y>0$ or if $y<0$,
$$\frac{x\mathrm{d}y-y\mathrm{d}x}{x^2+y^2}=-\mathrm{d}\arctan\tfrac{x}{y}$$

The behavior of the integral depends on whether the value of $a^2+b^2$ is less than or greater than $1$, corresponding to the circular path enclosing or excluding the origin, respectively. (If $a^2+b^2=1$, then the path goes through the origin, so we ignore this case as the integrand is not defined there.)

If $a^2+b^2>1$, then the path excludes the origin and is the boundary of a disk $D$ on which the integrand is defined. Green's theorem allows us to write
$$\int_{l}\frac{x\mathrm{d}y-y\mathrm{d}x }{x^2+y^2}
=\int_D\left(\frac{\partial}{\partial x}\frac{x}{x^2+y^2}-\frac{\partial}{\partial y}\left(-\frac{y}{x^2+y^2}\right)\right)\mathrm{d}x\,\mathrm{d}y =0\text{.}$$

If $a^2+b^2<1$, then the path includes the origin and passes through all four quadrants. Let $l_i$ be the portion of the path lying in quadrant $i$. On each quadrant we have an expression for the integrand as an exact differential in terms of the arctangent function,so that
$$\int_{l_i}\frac{x\mathrm{d}y-y\mathrm{d}x}{x^2+y^2}=\frac{\pi}{2}$$
$$\int_{l}\frac{x\mathrm{d}y-y\mathrm{d}x}{x^2+y^2}=4\left(\frac{\pi}{2}\right)=2\pi\text{.}$$

We have $x=a+ \cos( \phi)$ and $y=b +\sin( \phi)$ with $ \phi \in [0,2 \pi].$ Then

$\int_l \frac{xdy-ydx}{x^2+y^2}= \int_0^{ 2 \pi} \frac{(a+ \cos( \phi)) \ \cos ( \phi)-(b + \sin( \phi)) (- \sin ( \phi))}{1} d \phi=\int_0^{ 2 \pi}( a \cos ( \phi)+b \sin( \phi)+1) d \phi.$

Your turn !


Find the number of real roots for $x+\sqrt{a^2-x^2}=b$, $a>0$, $b>0$, as a function of $a$ and $b$ Given: (1) $x+\sqrt{a^2-x^2}=b$, $(a,b,x)\subset \mathbb R$, $a>0$, $b>0$.
Find: number of roots for (1), given possible values for $a$ and $b$.

This is a question from a book for the preparation for math contests.
It states as final answer: (a) 1 root if $b<a$; and (b) 2 roots if $a<b<a\sqrt{2}$.
I'm having difficulties on finding this answer. I don't know whether it is correct or perhaps I'm not finding the right approach.
I started moving $x$ in (1) to the left, to get
$$\sqrt{a^2-x^2}=b-x$$
Before proceeding with squaring both sides, I saved 2 needed conditions for checking the final solution (c1) $a^2-x^2\ge 0$ and (c2) $b-x\ge 0$. Then squaring both sides, we get:
$$a^2-x^2=b^2+x^2-2bx\Leftrightarrow 2x^2-2bx+(b^2-a^2)=0$$
with discriminant $\triangle$ defined by:
$$\triangle=4(2a^2-b^2)$$
From this it is easy to see that a condition for 2 roots is (c3) $\sqrt{2}a>b,$ and for 1 root is (c4) $\sqrt{2}a=b,$ as $a>0$ and $b>0$. Then I find the roots as $$x=\frac{2b\pm \sqrt{\triangle}}{4}=\frac{b\pm \sqrt{2a^2-b^2}}{2}$$
From this point, I can't see a way to reach the stated answer, if it is right.
Full solutions or helpful hints are welcome. Sorry if it is a duplicate.

As $a>0$, posing $x=as$ and $b=\beta a$, the problem is equivalent to discuss the real roots of
$$s+\sqrt{1-s^2}=\beta$$ with $\beta>0$. The function has real values for $\left| s\right|<1$. Moreover, $f(s)=s+\sqrt{1-s^2}$ has a single maximum at $s=1/\sqrt{2}$ with $f(1/\sqrt{2})=\sqrt{2}$, $f(0)=f(1)=1$ and finally $f(-1)<0$. Thus, there are no real root for $\beta>\sqrt{2}$, 2 real roots for $1<\beta<\sqrt{2}$, a single real root for $0<\beta<1$ and a double root at $\beta=\sqrt{2}$.

it must be $$a\geq |x|$$ and $$b\geq x$$
after squaring we get the equation
$$0=2x^2-2bx+b^2-a^2=0$$
solving this equation we get
$$x_1=\frac{b}{2}+\frac{1}{2}\sqrt{2a^2-b^2}$$
$$x_2=\frac{b}{2}-\frac{1}{2}\sqrt{2a^2-b^2}$$
so we get the following solution set
$$x=b$$ and $a=b$
$$x=\frac{1}{2}(b\pm i\sqrt{b^2-2a^2}$$ and $$0<a<\frac{b}{\sqrt{2}}$$
$$x=\frac{b}{2}$$ and $$a=\frac{b}{\sqrt{2}}$$
$$x=\frac{1}{2}(b\pm \sqrt{2a^2-b^2})$$ and $$\frac{b}{\sqrt{2}}<a<b$$
$$x=\frac{1}{2}(b-\sqrt{2a^2-b^2})$$ and $$a\geq b$$
Observe that i need time nto type all these formulas


mean and standard derivation about battery The batteries produced in a factory are tested before packing: 1.5% of batteries are found to be faulty, and are scrapped. Whether or not a battery is faulty is independent of each other. Experience suggested that a ‘good’ battery could last for 36 to 45 hours when used, and that all times within this range are equally likely.

D: The total time that a box of 100 good batteries lasts when used.

(c) State the distribution, including the values of any parameters, that may be used to approximate the distribution of D, the total time that a box of 100 good batteries lasts. Also, Justify the use of this distribution briefly.

(d) Use the distribution that you have suggested in part c to calculate an approximate value for the probability that a box of 100 good batteries will last for less than 4000 hours.

i can't find out the mean and standard derivation of it, moreover, can i use normal distribution to approximate the probability, can someone help me?

Background. I suppose the $n = 100$ batteries are used sequentially, so that
the total time the box lasts is $T = X_1 + X_2 + \cdots +  X_{100},$
where $X_i \sim \mathsf{Unif}(35.45).$

By the Central Limit Theorem, the sum of such a large number of
independent uniformly distributed random variables is very
nearly normally distributed. (In fact, only $n=10$ would be large enough to get a nearly normal total. You may want to look at Wikipedia on the 'Irwin-Hall Dist'n'.)

In order to work the problem, you need to find $E(X_i) = 40.5.$ and also $Var(X_i).$ Then use these results to find $E(T)$ and $Var(T).$
Then the final answers use a normal distribution, with that mean and variance.

With that start, I will leave the details to you. This seems to be
a homework problem. Soon, there may be a similar problem on an
exam, and I want you to know how to do that. I will simulate the
answers, using R statistical software, so you will have something to check against when you've
finished working this problem.

Notes: (1) A simulation based on a million boxes
will give pretty good approximate answers. (2) If you are using printed tables of the standard normal CDF, that may involve some rounding, so your probability answer may
differ slightly from mine.

Simulation. Total times for $m = 10^6$ boxes, each with $n = 100$ batteries are simulated using R statistical software.

set.seed(2020)
t = replicate(10^6, sum(runif(100,36,45)))
mean(t);  sd(t)
[1] 4050.054     # aprx E(T) = 4050
[1] 25.98528     # aprx SD(T) = 25.98
mean(t < 4000)
[1] 0.026872     # aprx P(T < 4000) = 0.027
pnorm(4000, 4050, 25.98)
[1] 0.02714238   # P(T < 4000)

Here is a histogram of the lifetimes $T$ of a million boxes along
with the density curve of the approximate normal distribution of $T.$ The area under the curve to the left of the vertical broken
line represents $P(T < 4000).$

hist(t, prob=T, col="skyblue2",
   main="Simulated Dist'n of Box Lifetime")
 curve(dnorm(x, 4050, 25.98), add=T, lwd=2)
 abline(v = 4000, col="red", lwd=2, lty="dotted")

(c) We will use the continuous uniform distribution to approximate the distribution of D. The total time that a 100 good batteries can last is in the interval of [36 * 100, 45 * 100] = [3600, 4500]. Because every good battery is equally likely to last from 36 to 45, the total time of 100 batteries is also equally likely to be in the range of 3600 to 4500. That is why we use the continuous uniform distribution.

X - total lasting time of a 100 good batteries

probability density function: f(x) = 1/(b - a) , a <= x <= b

a = 3600
b = 4500

mean = E(X) = (a + b)/2

variance = V(X) = (b - a)^2 / 12

(d) The cumulative distribution function is: $F(x) = \int_{a}^{x} 1 /(b - a)du = x/(b - a) - a/(b - a)$

now we only substitute for our values

F(4000) = 4000/(4500 - 3600) - 3600/(4500 - 3600) = 4000/900 - 3600/900 = 4.(4) - 4 = 0.(4)


Help solving this recurrence relation I wanted to resolve the determinant of the next (nxn) matrix via recurrence relations:

$$
\begin{vmatrix}
a & 1 & 0 & 0 & 0 & 0 &.... 0 & 0 & 0 & 0 & 0\\
1 & a & 1 & 0 & 0 & 0 &.... 0 & 0 & 0 & 0 & 0 \\
0 & 1 & a & 1 & 0 & 0 &.... 0 & 0 & 0 & 0 & 0\\
0 & 0 & 1 & a & 1 & 0 &.... 0 & 0 & 0 & 0 & 0\\
.. & .. & .. & .. & .. & .. &..... & .. & .. & ..\\
0 & 0 & 0 & 0 & 0 & 0 & .... 0 & 1 & a & 1 & 0\\
0 & 0 & 0 & 0 & 0 & 0 & .... 0 & 0 & 1 & a & 1\\
0 & 0 & 0 & 0 & 0 & 0 & .... 0 & 0 & 0 & 1 & a\\
\notag
\end{vmatrix}
$$

After analyzing the matrix I found the recurrence relation:
$$
D_{n}-a*D_{n-1}  +D_{n-2}=0
$$

So the polynomial that describes this recurrence is:
$$
P(\lambda) = \lambda^2 - a * \lambda + 1
$$
The roots will be:
$$
\lambda_1 = \frac{a}{2} + \frac{\sqrt{a^2-4}}{2}\\
\lambda_2 = \frac{a}{2} - \frac{\sqrt{a^2-4}}{2}
$$
To resolve the recurrence I need 2 constants (C1 & C2) that satisfy:
$$
D_n = C_1*(\frac{a}{2} + \frac{\sqrt{a^2-4}}{2})^n + C_2*(\frac{a}{2} - \frac{\sqrt{a^2-4}}{2})^n
$$
With the initial conditions
$$
D_1 = a\\
D_2 = a^2 - 1
$$
The problem is I don't know how to resolve the equation system generated by substituting the initial conditions on the function.

Any type of help is appreciated.

You plug your initial conditions in, so you get
$$D_1=a = C_1*(\frac{a}{2} + \frac{\sqrt{a^2-4}}{2}) + C_2*(\frac{a}{2} - \frac{\sqrt{a^2-4}}{2})\\D_2=a^2-1 = C_1*(\frac{a}{2} + \frac{\sqrt{a^2-4}}{2})^2 + C_2*(\frac{a}{2} - \frac{\sqrt{a^2-4}}{2})^2$$
These are two equations in the two unknowns $C_1,C_2$.  The usual substitution technique will work, though it will be messy.  You can use the recurrence to evaluate $D_0=1=C_1+C_2$.  Using that with the first gives $$a=(C_1+C_2)\frac a2+(C_1-C_2)\frac{\sqrt{a^2-4}}{2}\\C_1-C_2=\frac a{\sqrt{a^2-4}}\\C_1=\frac 12(1+\frac a{\sqrt{a^2-4}})\\C_2=\frac 12(1-\frac a{\sqrt{a^2-4}})$$

To solve for the constants we just evaluate the expression for $D_n$ at $n=1$ and $n=2$, as well as use the information that $D_1=a$ and $D_2=a^2-1$ to get
$$C_1\left(\frac{a}{2}+\frac{\sqrt{a^2-4}}{2}\right)+C_2\left(\frac{a}{2}-\frac{\sqrt{a^2-4}}{2}\right)=a$$
$$C_1\left(\frac{a}{2}+\frac{\sqrt{a^2-4}}{2}\right)^2+C_2\left(\frac{a}{2}-\frac{\sqrt{a^2-4}}{2}\right)^2=a^2-1$$
Now, since $a$ is just some constant, we can treat this as a linear system and write it as
$$A\vec{c}=\left[\begin{array}{cc}
\left(\frac{a}{2}+\frac{\sqrt{a^2-4}}{2}\right) & \left(\frac{a}{2}-\frac{\sqrt{a^2-4}}{2}\right) \\
\left(\frac{a}{2}+\frac{\sqrt{a^2-4}}{2}\right)^2 & \left(\frac{a}{2}-\frac{\sqrt{a^2-4}}{2}\right)^2
\end{array}\right]
\left[\begin{array}{c}
C_1 \\
C_2
\end{array}\right]=
\left[\begin{array}{c}
a \\
a^2-1
\end{array}\right]$$
Which has solution
$$
\left[\begin{array}{c}
C_1 \\
C_2
\end{array}\right]
=
\frac{\text{adj }A}{\det A}\left[\begin{array}{c}
a \\
a^2-1
\end{array}\right]
$$
Where
$$\det A=\left(\frac{a}{2}+\frac{\sqrt{a^2-4}}{2}\right)\left(\frac{a}{2}-\frac{\sqrt{a^2-4}}{2}\right)^2-\left(\frac{a}{2}-\frac{\sqrt{a^2-4}}{2}\right)\left(\frac{a}{2}+\frac{\sqrt{a^2-4}}{2}\right)^2$$
$$=\frac{2}{a-\sqrt{a^2-4}-2}$$
and
$$\text{adj }A=\left[\begin{array}{cc}
\left(\frac{a}{2}-\frac{\sqrt{a^2-4}}{2}\right)^2 & -\left(\frac{a}{2}-\frac{\sqrt{a^2-4}}{2}\right) \\
-\left(\frac{a}{2}+\frac{\sqrt{a^2-4}}{2}\right)^2 & \left(\frac{a}{2}+\frac{\sqrt{a^2-4}}{2}\right)
\end{array}\right]$$


$(\bar x , \bar y-1)$ is not a principal ideal of $A=\mathbb R[x,y]/(x^2+y^2-1)$ Let $A=\mathbb R[x,y]/(x^2+y^2-1)$ ; then $A$ is an integral domain . Consider the maximal ideal $P=(\bar x , \bar y-1)$ of $A$ . Then how to show that $P$ is not a principal ideal ?

We reason by contradiction, considering a generator $g$ of the principal ideal $(\overline{x},\overline{y} - 1)$.
Let $B = \mathbb{R}[t, \frac{1}{1 + t^2}]$, that is, the localization $S^{-1} \mathbb{R}[t]$ where $S \subset \mathbb{R}[t]$ is the multiplicative set generated by $1 + t^2$. The map $(x, y) \mapsto
(\frac{1 - t^2}{1 + t^2}, \frac{2t}{1 + t^2})$ induces a ring homomorphism $\phi: A \rightarrow B$. (This homomorphism can be checked to be injective, but we won't use that fact.) The ring $B$ is Euclidean, hence a UFD and it easily follows that $\phi(g)$ is of the form $\lambda (1 - t)(1 + t^2)^m$ for some $\lambda \in \mathbb{R} \setminus \{0\}$ and some $m \in \mathbb{Z}$. Let $C$ be the ring of functions from $\mathbb{R}$ to $\mathbb{C}$ of the form $\theta \mapsto \sum_{n = -N}^N \lambda_n e^{in\theta}$ with $\lambda_n \in \mathbb{C}$ and $N \ge 0$.Let $Q$ be the fraction field of $C$. It is immediate to see that $P(X) \mapsto (\theta \mapsto P(e^{i\theta}))$ induces a ring isomorphism from $\mathbb{C}[X^{\pm 1}]$ (resp. $\mathbb{C}(X)$) onto $C$ (resp. $Q$). The map $t \mapsto \tan(\frac{\theta}{2}) \Doteq \frac{e^{i\theta} - 1}{i(e^{i\theta} + 1)}$ induces a ring homomorphism $\psi: B \rightarrow Q$ such that $\psi \circ \phi(A) \subset C$ (check what are the images of $\overline{x}$ and $\overline{y}$ in $Q$). Since $\psi \circ \phi(g) =
2^m\lambda \frac{e^{i m \theta}((1 + i)e^{i\theta} + 1 - i)}{(e^{2i\theta} + 1)^m(e^{i\theta} + 1)} \in C$,
we have
$2^m\lambda \frac{X^m ((1 + i)X + 1 - i)}{(X^2 + 1)^m (X + 1)} \in \mathbb{C}[X^{\pm 1}]$. This is impossible because $-1$ is not a root of the numerator of the latter fraction.

we can consider $S=R[x,y]/<x,x^2+y^2-1>$is isomorphic to $R[y]/<y^2-1>$.remark $S$ is a principal ideal ring.if the ideal you give is principal.denote $<f^->=<x^-,(y-1)^->$.in S,they also equal.So we can get $f=k(y-1)+gx+h(x^2+y^2-1)$,where $g,h\in R[x,y],k\in R$.the same argument,we can get $f=k(y-1)+lx+h(x^2+y^2-1)$,where $k,l\in R$,this is a contradiction.


How to find $\sqrt[3]{8i}$ How do I find the following cube root?

$$\sqrt[3]{8i} = ?$$

I tried by adding $\sqrt[3]{i^3 + 8i + i}$ but that is where my imagination quits.

If you want a general method for finding roots of complex numbers, you may try my answer here, for instance.

In your case, it would work as follows: write your complex number in exponential form,

$$
8i = 8 e^{i\pi/2} \ .
$$

Then,

$$
\sqrt[3]{8e^{i\pi/2}} = \sqrt[3]{8} e^{i(\pi/2 + 2k\pi)/3} \qquad \text{for} \quad k=0,1,2 \ .
$$

That is,

$$
\sqrt[3]{8e^{i\pi/2}} = 2 e^{i(\pi/6 + 2k\pi/3)} \qquad \text{for} \quad k=0,1,2
$$

And this gives:

$$
2e^{i\pi/6} = 2\left(\cos\frac{\pi}{6} + i \sin\frac{\pi}{6}\right) = 2\left(\frac{\sqrt{3}}{2} + i\frac{1}{2} \right) \ ,
$$

$$
2e^{i5\pi/6} = 2\left(\cos\frac{5\pi}{6} + i \sin\frac{5\pi}{6}\right) = 2\left(\frac{\sqrt{3}}{2} - i\frac{1}{2} \right) \ ,
$$

and

$$
2e^{i3\pi/2} = 2\left(\cos\frac{3\pi}{2} + i \sin\frac{3\pi}{2}\right) = -2i \ .
$$

Look this site

http://fatosmatematicos.blogspot.com/2011/09/identidade-de-euler-e-as-raizes.html


Given a list of integers between $0$ and $99$, create a function that will fit all the integers in the list. Okay, so my friend LOVES to play the lottery. He makes bets often with bookies on the last $2$ digits, getting $90:1$ on his money, a losing bet.

He looks up 'systems' on social media on how to calculate what numbers will come up next based on previous draws. I wonder if it is possible to generate a 'system' to fit the existing data points, i.e. the previous $5$ draws. So that I could show him that just because you can fit a pattern on previous draws of randomly generated numbers doesn't mean that you can predict the next number.

Well, it depends.

A random number generated by a computer uses what is known as a Random Number Generator (RNG) that creates new 'random' numbers. To create a number using an RNG, you need a single number called a seed. This could be the system time, a manually entered number, a previously generated number, or something unpredictable and random by itself.

If the seed is a previously generated number, you need to use a good, unpredictable algorithm. Weak RNGs can be predicted over time, due to the flaws in them. To do so, you must:
1. Know what algorithm has been used.
2. Hope that it is weak, or has a programming flaw (indexes of arrays are often confused).
3. Know how to find such flaws (meaning you need to be good at maths, as well as programming).
4. Pray that you are right (since it is almost impossible to really exploit an RNG without an expert's help).

Note that once you have cracked the method, you should be able to get the number right every time, otherwise you are wrong. It is all-in or nothing, so your friend is just wasting his money (or not, since he is losing only by a small margin, he might just get lucky, and stop playing after winning once).

If the RNG is of some in-built function by a reputed brand (say, a DOS command echo %random%), just accept that it can't be cracked.

And there is no other way to predicting a randomly generated number, it is mathematically impossible.


Is every subgroup of the product of two cyclic groups is again a product of two cyclic groups? Well, this is my question.

Is every subgroup of the product of two cyclic groups is again a product of two cyclic groups (maybe one being trivial)?

Thanks!

Yes.
Assume first that $a,b$ in $C_a\oplus C_b$ are prime powers of the same prime $p$.
Then any subgroup $H$ of it is the direct sum of finitely many cyclic $p$-groups. If there are $n$ summands, then there are $p^n-1$ elements of order $p$ in $H$. As there are only $p^2-1$ elements of order $p$ in $G$, we see that there are at most two summands.

Now for the general finite case, we can split cyclic groups of composite orders into their prime power components, see that there are (at most) two summands per prime in $H$, and combine coprime summands again to finally obtain two summands for $H$ itself.

The above works only for the finite case.
To cover all cases, we should especially cover the case $G=\mathbb Z\oplus \mathbb Z$. But once we have that case, we immediately get the result also for the general case: If $G$ is abelean with at most two generators, it is a quotient of $\mathbb Z\oplus\mathbb Z$. A subgroup $H\le G$ maps to a subgroup of $\mathbb Z\oplus\mathbb Z$ under the canonical projection, hence is 8as we will see in a moment) generated by at most two elements, hence $H$ itself is generated by at most two elements (obtained from preimages of the generators in the quotient).

Let $H\le\mathbb Z\oplus\mathbb Z$.
If $H=0$ we are done.
Otherwise let $(a,b)\in H$ be any nonzero element.
Then $f\colon H\to\mathbb Z$, $(x,y)\mapsto ay-bx$ has nontrivial kernel and some $k\mathbb Z$ with $k\ge 0$ as image. If $k=0$, the homomorphism $H\to\mathbb Z$, $(x,y)\mapsto ax+by$ is injective (we have $ay-bx=0$ from $k=0$, hence $ax+by=0$ implies $(a^2+b^2)x=(a^2x+aby)+(b^2x-aby)=0$, hence $x=0$ and similary $y=0$); the $H$ is infinite cyclic or zero.
And if $k>0$, pick $(c,d)\in H$ with $f(c,d)=k$. Then $g\colon H\to\mathbb Z$, $(x,y)\mapsto (ax+by)-\frac{f(x,y)}k(ac+bd)$ is a homomoprhism with $\langle(c,d)\rangle$ as kernel and an infinite cyclic subgrup of $\mathbb Z$ as image, which shows that $H\cong \mathbb Z\oplus\mathbb Z$.

if $x$ is co-prime to its complex n-gon in real space, then yes. A proof is trivial if you examine the Taylor expansion around $x=i$ and then translated through a subtended angle of epsilon.


$6(m + 1)(n − 1), 6 + (m − 1)(n + 1)$ and $(m − 2)(n + 2)$ are simultaneously perfect cubes. Find all pairs of integers (m, n) such that the integers $6(m + 1)(n − 1), 6 +
(m − 1)(n + 1)$ and $(m − 2)(n + 2)$ are simultaneously perfect cubes.

I assumed $6(m + 1)(n − 1), 6 +
(m − 1)(n + 1)$ and $(m − 2)(n + 2)$ to be $a^3,b^3$ and $c^3$ respectively and deduced that $6(2mn+4)=a^3+6b^3,$ $12(n-m)=a^3-6b^3+36$ but
I can't solve it further. Please help.


Why a function from $\mathbb{R}^2$ to $\mathbb{R}$ is a surface while from $\mathbb{R}$ to $\mathbb{R}^2$ is a curve? This is a silly question, but I just can not see the reason.

The function from $\Bbb R$ to $\Bbb R^2$ has to be a curve (a unidimensional object) because you are encoding information about a single variable. So, even a function from $\Bbb R$ to $\Bbb R^3$ will also be a curve in 3-D.

However, the function $\Bbb R^2$ to $\Bbb R$ encodes information from two variables into one. Hence it is represented as a surface (a 2-dimensional object).


How can I determine the radius of a dodecahedron? I am making a dodecahedron that needs to fit inside of a sphere. The sphere has a diameter of 56mm. What is largest possible measurement of one segment of a pentagon side of a dodecahedron that would fit inside the sphere? How do I determine this?

The answers provided here are correct. Let me explain the formula.

By joining together the diagonals of the pentagons on the dodecahedron we build a cube. For reference use the following figure:

Let us focus in the blue pentagon. The diagonal separates the pentagon
into a triangle and a trapezoid. The other 11 diagonals (chosen here. Note
that you could form up to 5 cubes by selecting different sets of diagonals)  form a cube. The vertices of the cube lie in a sphere which is the same sphere
circumscribing the dodecahedron. So, to find the radius of that sphere is just half of the diagonal of the cube.

It is a well known fact (see here Pentagon ) that the length of a diagonal $d$ of a pentagon of side length $\ell$ is given by
\begin{equation}
 d = \phi \ell
\end{equation}

where $\phi=(1+ \sqrt{5})/2$ is the golden ratio.

Now, the diagonal of the cube is
\begin{equation}
  D = \sqrt{ d^2 + d^2 + d^2} = \sqrt{3} d = \sqrt{3} \phi \ell.
\end{equation}

So the radius of the sphere is
\begin{equation}
  r = \frac{D}{2} = \frac{\sqrt{3} \, \phi \ell}{2} = \frac{\sqrt{3} (1 + \sqrt{5}) \ell }{4}.
\end{equation}

If the measure for the Dodecahedron’s diameter is 56mm then the measure for the edge of the Dodecahedron can be found by first dividing the measure for the diameter of the Dodecahedron 56mm into the square root of 3 = 1.732050807568877 and then divide the larger part of the division into the Golden ratio of 1.6180339887555 and the result will be the edge of the Dodecahedron for the correct Sphere = 19.98mm.


Differentiability of $\mathrm{max}(x, y)$ at $x=y$ I am trying to figure out differentiability of $\mathrm{max}(x, y)$. Intuitively, it should not be differentiable at $x=y$, since it changes direction "non-smoothly" at those points.

I can not, however, find a way to prove this algebraically.

Hint: Write

$$\operatorname{max}(x,y) = \frac{x + y + |x-y|}{2}$$

$\newcommand{\+}{^{\dagger}}%
 \newcommand{\angles}[1]{\left\langle #1 \right\rangle}%
 \newcommand{\braces}[1]{\left\lbrace #1 \right\rbrace}%
 \newcommand{\bracks}[1]{\left\lbrack #1 \right\rbrack}%
 \newcommand{\ceil}[1]{\,\left\lceil #1 \right\rceil\,}%
 \newcommand{\dd}{{\rm d}}%
 \newcommand{\down}{\downarrow}%
 \newcommand{\ds}[1]{\displaystyle{#1}}%
 \newcommand{\equalby}[1]{{#1 \atop {= \atop \vphantom{\huge A}}}}%
 \newcommand{\expo}[1]{\,{\rm e}^{#1}\,}%
 \newcommand{\fermi}{\,{\rm f}}%
 \newcommand{\floor}[1]{\,\left\lfloor #1 \right\rfloor\,}%
 \newcommand{\half}{{1 \over 2}}%
 \newcommand{\ic}{{\rm i}}%
 \newcommand{\iff}{\Longleftrightarrow}
 \newcommand{\imp}{\Longrightarrow}%
 \newcommand{\isdiv}{\,\left.\right\vert\,}%
 \newcommand{\ket}[1]{\left\vert #1\right\rangle}%
 \newcommand{\ol}[1]{\overline{#1}}%
 \newcommand{\pars}[1]{\left( #1 \right)}%
 \newcommand{\partiald}[3][]{\frac{\partial^{#1} #2}{\partial #3^{#1}}}
 \newcommand{\pp}{{\cal P}}%
 \newcommand{\root}[2][]{\,\sqrt[#1]{\,#2\,}\,}%
 \newcommand{\sech}{\,{\rm sech}}%
 \newcommand{\sgn}{\,{\rm sgn}}%
 \newcommand{\totald}[3][]{\frac{{\rm d}^{#1} #2}{{\rm d} #3^{#1}}}
 \newcommand{\ul}[1]{\underline{#1}}%
 \newcommand{\verts}[1]{\left\vert\, #1 \,\right\vert}$
$$
\partiald{\max\pars{x,y}}{x} = \half\bracks{1 + \sgn\pars{x - y}}
$$


Problem with multivariable calculus: $\lim_{(x,y)\to (0,0)} \frac{x^3 + y^3}{x^2 + y}$ Anyone can help me with this limit?

$$\lim_{(x,y)\to (0,0)} \frac{x^3 + y^3}{x^2 + y}$$

I'm having trouble with proving that this limit really goes to $0$

thank you

Putting polar coordinates $x=r\sin \theta,y=r\cos \theta$
$$\lim_{r\to 0} \frac{r^3\sin^3\theta+r^3\cos^3\theta}{r^2\sin^2 \theta + r\cos \theta}=\lim_{r\to 0}\frac{r(\sin \theta + \cos \theta)}{\sin^2\theta+\frac{\cos \theta}{r}}$$
Now if $\cos \theta \not\to 0$ then $\frac{|\cos \theta|}{|r|}\to \infty$ hence the limit is $0$ if $\cos\theta \to 0$ then we have that $|\sin \theta|\to 1$ and since denominator tends to $1$ and numerator tends to $0$ again the limit is $0$ hence the limit is $0$ independent of $\theta$


Is $0$ an imaginary number? My question is due to an edit to the Wikipedia article:  Imaginary number.

The funny thing is, I couldn't find (in three of my old textbooks) a clear definition of an "imaginary number". (Though they were pretty good at defining "imaginary component", etc.)

I understand that the number zero lies on both the real and imaginary axes.
 But is $\it 0$ both a real number and an imaginary number?

We know certainly, that there are complex numbers that are neither purely real, nor purely imaginary. But I've always previously considered, that a purely imaginary number had to have a square that is a real and negative number (not just non-positive).

Clearly we can (re)define a real number as a complex number with an imaginary component that is zero (meaning that $0$ is a real number), but if one were to define an imaginary number as a complex number with real component zero, then that would also include $0$ among the pure imaginaries.

What is the complete and formal definition of an "imaginary number" (outside of the Wikipedia reference or anything derived from it)?

The Wikipedia article cites a textbook that manages to confuse the issue further:

  Purely imaginary (complex) number : A complex number $z = x + iy$ is called a purely imaginary number iff $x=0$ i.e. $R(z) = 0$.

  Imaginary number : A complex number $z = x + iy$ is said to be an imaginary number if and only if $y \ne 0$ i.e., $I(z) \ne 0$.

This is a slightly different usage of the word "imaginary", meaning "non-real": among the complex numbers, those that aren't real we call imaginary, and a further subset of those (with real part $0$) are purely imaginary. Except that by this definition, $0$ is clearly purely imaginary but not imaginary!

Anyway, anybody can write a textbook, so I think that the real test is this: does $0$ have the properties we want a (purely) imaginary number to have?

I can't (and MSE can't) think of any useful properties of purely imaginary complex numbers $z$ apart from the characterization that $|e^{z}| = 1$. But $0$ clearly has this property, so we should consider it purely imaginary.

(On the other hand, $0$ has all of the properties a real number should have, being real; so it makes some amount of sense to also say that it's purely imaginary but not imaginary at the same time.)

A complex number z=a+ib where a and b are real numbers is called :
1- purely real , if b=0 ; e.g.- 56,78 ;
2- purely imaginary,  if a=0 ,e.g.- 2i, (5/2)i ;
3- imaginary,if b≠ 0 ,e.g.- 2+3i,1-i,5i ;
0 is purely imaginary and purely real but not imaginary.


How to find the square root of a polynomial? Here, by the word polynomial, I am referring to those polynomials, who have a square root in simple algebraic expressions.
But how to get the intuition, on how to expand the polynomial, so that we can factor it, and then find its square root? Or is there any other method also?
Like $\sqrt{\frac{x^2}4+\frac1{x^2}-\frac1x+\frac x2-\frac34}$,
At first look, we get instinct to first take LCM of all and add them. But after that, the numerator becomes $x^4+2x^3-3x^2-4x+4$. Now, (I don't want the answer) how to get that instinct on factoring this polynomial? Or is there a method?
Remark: Is there any faster, non-rigorous way also, of finding the square root?

The example you gave is not a polynomial as there are negative powers involved (e.g. 1/x^2). I guess that you mean to find a polynomial which, if you take the square root, yields another polynomial.
This is only possible if all the linear factors are in even multiplicity, i.e. the polynomial has 2n values for every root. If that is the case, you can take the square root as you described.

response to comment

Ok as you do not want the answer, this is what I would do:
- Find the roots
As a hint, x = 1 is a root. Use factor division on the polynomial to reduce its degree to a cubic equation and continue to find the roots.
- Check that all roots come in even frequencies, if not, we cannot take a "nice" square root.


Power Diophantine equation involving primes: $(p+q)^q-p^q-q^q+1=n^{p-q}$ Suppose $p$ and $q$ are prime numbers, and $n>1$ is a positive integer. Find all solutions to the following Diophantine equation:$$(p+q)^q-p^q-q^q+1=n^{p-q}$$

What I have tried:

Obviously $p>q$. If $q=2$, we get one solution: $(p, q, n)=(3, 2, 13)$. From now on $p>q>2$. Letting $p-q=2^k r$, where $r$ is an odd number and working on the evaluation of $2$ in the equation gives $k=1$.

I think that $r=1$ is the only possibility. But I don't know how to prove it! I tried to show that $r$ cannot have a prime divisor, but I failed!

Looking mod $p$, $q$ and $p+q$ gives $n^{p-q} \equiv 1 \pmod {pq^2(p+q)}$.

Edit: Let's put a new restriction on the equation: Order of $n$ modulo $p$ is $p-q$. If you work on equation with the new constraint, please inform me about your results!

Notice that $n^{p-q} \equiv 1 \pmod p$, so $\gcd(n, p)=1$ and it follows from the Fermat's little theorem that $n^{p-1} \equiv 1 \pmod p$. Order of $n$ modulo $p$ is $p-q$, hence $p-q|p-1$ and consequently $p-q|q-1$.

$$(p+q)^q-p^q-q^q+1=n^{(p-q)}$$

Let $p-q=\alpha$

$$(p+q)^q-p^q-q^q=n^{\alpha}-1$$

Since q is odd, number of terms of expansion $(p+q)^q$ is even and terms $p^q$ and $q^q$ will be eliminated on LHS and the remained polynomial has a factor like $[p.q(p+q)]$ which is even.On RHS we have a factor like $n-1$ which indicates n must be odd, in fact we have:

$$p.q (p+q).P(p, q)= (n-1)(n^{\alpha -1}+n^{\alpha -2}+n^{\alpha -3}+ . . .)$$

Where $P(p, q)$ denotes a polynomial having parameters p and q. This can help us to choose the number for n to be checked. For example with $p=5 $ and $q=3$ we may write:

$$(5+3)^3-5^3-3^3=3(3\times 5)(3+5)=18\times 20=n^2-1=(n-1)(n+1)$$

So $q=19$ works.If $p=2k_1-1$ and $q=2k_2+1 $, we have:

$$C^q_r (2k_1-1)(2k_2+1)(2k_1+2k_2)=(n-1)\Sigma^{\alpha-1}_{r=1}n^{\alpha - r}$$

Now for selecting p and q we must consider that the difference between p and q must be such that we can construct a relation it's sides have close values, for example consider primes $p=47$ and $q=7$, we have:

$$(47+7)^7-47^7-7^7=n^7-1$$

Number of digits of $54^7$ is :

$N_d=7 \log 54  = 13$ so n must be such that it's 7th power has approximately 13 digits or we must have:

$\log n ≈ \frac{13}{7} $  ⇒ $n  ≈ 100$

Now on LHS we have:

$(2\times 47= 94)(26)(7)=4\times47 (7\times13=91) $

Numbers 91 and 94 are closed to 100, but n must be odd so number 95 can be selected for test.


math fallacy problem: $-1= (-1)^3 = (-1)^{6/2} = \sqrt{(-1)^6}= 1$? I know there is something wrong with this but I don't know where. It's some kind of a math fallacy and it is driving me crazy. Here it is:
$$-1= (-1)^3 = (-1)^{6/2} = \sqrt{(-1)^6}= 1?$$

The "rule" $(a^b)^c = a^{bc}$ doesn't necessarily hold when $a < 0$.

Hi this is not a fallacy but this problem overlooks some facts
i.e. there will be two square roots to every number
like.
$$
\sqrt{4} = +2 \text{ as well as} -2
$$
similarly
$$
\sqrt{(-1)^6} = +1 \text{ or } -1
$$

So I think writing $$\sqrt{(-1)^6} = 1$$in this proof is wrong

And also people who are saying that  $$(a^b)^c != a^{b*c}$$ for a <0 or something like that
please calculate this in your calculator so that it proves you worng
$$10^{2.5*log10(-1)} $$


Hint for problem on $4 \times 7$-chessboard problem related to pigeonhole principle Suppose that each square of a $4 \times 7$ chessboard is colored either black or white. Prove that with any such coloring, the board must contain a rectangle (formed by the horizontal and vertical lines of the board) whose four distinct unit corner squares are all of the same color?

Any hints on this problem, I guess it could be solved with a clever application of the pigeonhole principle, but I have some difficulty seeing what the objects and what the bags are in which to put the objects to apply the pigeonhole principle, and then argue that such an coloring must exist. It is taken from a textbook on discrete mathematics, and in the text a version of the Monotone Subsequence Theorem by Erdös/Szekeres is proven (using the pigeonhole principle). Hence, I guess it is somehow related to this.

Let's say the chessboard has $4$ columns and $7$ rows. Each row must land in (at least) one of the following $6$ bags:

First and second squares are black.
First and third squares are black.
Second and third squares are black.
First and second squares are white.
First and third squares are white.
Second and third squares are white.

By the pigeonhole principle, two rows are in the same bag, and that gives you your rectangle with all four corner squares of the same color.

Note that you only need a $3\times7$ chessboard for this to work.

isnt there a trivial case?

"the board must contain a rectangle (formed by the horizontal and vertical lines of the board) whose four distinct unit corner squares are all of the same color?"

Each square of the board is a rectangle and every corner of the square must be the same color because it is only one square, a solid color


What are natural numbers? What are the natural numbers?

Is it a valid question at all?
My understanding is that a set satisfying Peano axioms is called "the natural numbers" and from that one builds integers, rational numbers, real numbers, etc. But without any uniqueness theorem how can we call it "the" natural numbers? If any set satisfying the axioms is called natural numbers then maybe there is no "natural numbers" as a definite object as we know it in real world, but just a class of objects satisfying certain axioms, like Noetherian rings?

Ravichandran's answer is right: the natural numbers are the numbers 0, 1, 2, 3, ... .  We can directly understand these numbers, based on our inductive definition of how to count in English or another natural language, even before we create an axiom system for them.

The standard list of axioms that we use to characterize the natural numbers was stated by Peano. Several people have brought up first-order logic in answers here, but that isn't what you want to use to establish categoricity (I will come back to that). Dedekind was the first to prove that we can axiomatize the natural numbers in a way that all models of our axioms are isomorphic, using second-order logic with second-order semantics.

  Theorem (after Dedekind). Say that we have two structures $(A, 0_A, S_A)$ and $(B, 0_B,S_B)$ each of which is a model of Peano's axioms for the natural numbers with successor, and each of which has the property that every nonempy subset of the domain has a least element. Then there is a bijection $f\colon A \to B$ such that $f(0_A) = 0_B$ and for all $a \in A$ $f(S_A(a)) = S_B(f(a))$.

The theorem is proved using the axiom of induction repeatedly, but the idea is completely transparent. Suppose that Sisyphus is given a new task. He will count out all the natural numbers in order in Greek, while a fellow tortured soul will count out all the natural numbers in order in English, at the same speed as Sisyphus. Just based on Ravichandran's assessment of what the natural numbers are, the map that sends each number spoken by Sisyphus to the number spoken by his companion at the same time is obviously an isomorphism between the "Greek natural numbers" and the "English natural numbers."

This argument cannot be captured in first-order logic, not even in first-order ZFC. But that isn't the fault of the natural numbers: first order logic can't give a categorical set of axioms for any infinite structure. Dedekind's proof can be cast in ZFC in the sense that it shows that the models of Peano's axioms in a certain model of ZFC are all isomorphic to each other in that model. Moreover, ZFC is sound in the sense that the things it proves about the natural numbers are correct. But as long as we use first-order semantics for ZFC, it can't fully capture the natural numbers.

The key point of second order semantics is that "every nonempty subset" means every nonempty subset.  If a structure $(A, 0_A, S_A)$ satisfies a certain finite set of axioms that are all true in $\mathbb{N}$, then $\mathbb{N}$ can be identified with an initial segment of $A$. In this case, if $A - \mathbb{N}$ is nonempty it will not have a least element, and so $A$ will not satisfy the second-order Peano axioms. Again, this argument cannot be completely captured in first-order logic because $\mathbb{N}$ cannot be fully captured.

Even if we think of Peano's axioms as only specifying an isomorphism class of structures, the situation is not like Noetherian rings, where there are many non-isomorphic examples. $\mathbb{N}$ is more like the finite field on two elements. The question of which model of the second-order Peano axioms is really the "natural numbers" is like the question of which isomorphic copy of $F_2$ is "really" $F_2$. It's a fine question for philosophers, but as mathematicians we have a perfectly good idea what $F_2$ is, up to isomorphism, and a good idea what $\mathbb{N}$ is, up to isomorphism. We also have axiom systems that let us prove things about these structures. $F_2$ is easier to apprehend because it's finite, but this only makes $\mathbb{N}$ more interesting.

Each natural number is a set. A possibility would be:

0 is the set that contains the empty set $\{\emptyset\}$.
1 is the set that contains the empty set and 0 $\{\emptyset,\{\emptyset\}\}$
2 is the set that contains the empty set and 1 $\{\emptyset,\{\emptyset,\{\emptyset\}\}\}$
3 is the set that contains the empty set and 2 $\{\emptyset,\{\emptyset,\{\emptyset,\{\emptyset\}\}\}$

and so on. But keep in mind that the representation is not very important, because all correct representations (like the above) you can think of, are isomorphic.


What is exactly a “Point”? I read somewhere that a line is made up of infinite points.
Between any two points on that line, there are another infinite points.
and between any two points BETWEEN those 2 points there are another infinite points.

Is this true? If so , HOW?

Also, i have read in physics that the smallest possible length is the “Planck’s Length” . Nothing can be smaller than it. So is the point even smaller than the Planck’s length?

Is a point spherical in shape? Rectangular? Square? Or it does not has any shape? If so, it must not have any length, that means a point does not exist.
Is this correct?

First of all and most importantly: Mathematical existence has nothing whatsoever to do with physical existence. Just because something does not "exist physically" does not mean it cannot exist mathematically. Mathematics was partly created to model physics, but this does not mean that mathematics cannot make simplifications (like believing there are points). But even then, applications of mathematics go way beyond "just" physics. You can also think of it this way: Mathematics is so amazing, besides the physics you are used to it may explain any kind of physics you can come up with in your head.

A mathematical point (in let's say two dimensions) is just a pair $(x_0,y_0)$ consisting of real numbers $x_0,y_0$. If you have another point $(x_1,y_1)$, you get a line through these two points. It's the set of all points $(x,y)$ with $y=ax+b$, where $a = \frac{y_0-y_1}{x_0-x_1}$ and $b = y_0 - mx_0$. Of course, there are infinitely many points $(x,y)$ on said line (take any $x$ and calculate the corresponding $y$). But even between $(x_0,y_0)$ and $(x_1,y_1)$, there are infinitely many points. Take any $\lambda\in [0,1]$, then $(\lambda_0 + (1-\lambda)x_1, \lambda y_0 + (1-\lambda) y_1)$ will be a point "between" $(x_0,y_0)$ and $(x_1,y_1)$.

The concept of a point can be generalized to abritary dimensions ($\mathbb{R}^n)$, other rings ($\mathbb{Z}^n$, $\mathbb{Q}^n$, $\mathbb{C}^n$, ...), later to points in a metric space and "finally" to points in a topological space. Then points need not actually represent anything physical. For example you could take the vertices of a graph (that is a thing with vertices and edges between those vertices) to be your points and measure distance between these vertices (how many steps do I need to get from one vertix to another vertix?). This yields a metric space, which does not have anything directly to do with physics, but which is still very interesting and important.

There are also other approaches to formalizing the concept of 'point', for example incidence geometry: It abstractly deals with points and lines, and may be used to formalize different kinds of geometries synthetically.

A point is a abstract theoretical mathematical construction. It does not exist in the real world hence it can be smaller than a Planck length.. It has no shape, zero width, zero length, zero height, etc - it is zero dimensional.

From this you should be able to conceptualize how you can have an infinite number of points between any two given points.


RATIONAL INEQUALITY - Find the values of a such that range of $f(x)=\frac {x+1}{a+x^2}$ contains $[0,1]$ Find the values of $\text{“}a\text{''}$ such that range of $f(x)=\frac{x+1}{a+x^2}$ contains $[0,1]$
 where  am i  wrong ??:

I took $2$ cases :
Case $1$ : $a+x^2 \gt 0$
, I solved the inequality $f(x) \ge 0$ , $f(x) \le 1$,

Case $2$: $a+x^2 \lt 0$
 , Then I solved inequality $f(x) \ge 0$ , $f(x) \le 1$

If i am wrong ,

HOW CAN I SOLVE THIS  PROBLEM ?

Your mistake is quite common, and it happened in the step of converting a text question into mathematical language.

Think about what the expression "the range of $f$ contains $[0,1]$" actually means. It means that $[0,1]$ is a subset of the range of $f$, or in other words, that every $y\in[0,1]$, is an element of the range of $f$.

For example, the function $f(x)=x$ satisfies the condition "the range of $f$ contains $[0,1]$", since the range of $f$ is $\mathbb R$, and $\mathbb R$ contains $[0,1]$.

On the other hand, the function $g(x)=\frac12$ does not satisfy the condition, because the range of $g$ is $\{\frac12\}$.

Another function, $h(x)=\sin^2 x$, also satisfies the condition, but "just barely", since the range of $h$ is $[0,1]$, which contains $[0,1]$ (but any smaller set does not).

Now, think about what the demand $0\leq f(x)\leq 1$ actually means. It means that for every $x$, you have $f(x)\in [0,1]$. In other words, you demand that the range of $f$ is contained in $[0,1].

Let's take a look at the previous examples:

$f(x)=x$ does not satisfy this condition, since $f(x)\leq 1$ is not true for $x=3$.

$g(x)=\frac12$ obviously satisfies this condition.

$h(x)=\sin^2 x$ satisfies this condition, but barely (both $0$ and $1$ are reached by $h$ at some point).

So, the conclusion here is that you were looking at functions that have ranges, contained in $[0,1]$, instead of looking at functions that have ranges that contain $[0,1]$.

Here $a$ can not be negative or $0\;,$ otherwise $f(x)$ is unbounded.

Now for $a>0,$ we get $f(x)<0\;\forall x<-1$

Hence $a\in \phi$


How to find out the solid angle subtended by a tetrahedron at its vertex? Given that the angles between the consecutive lateral edges AB, AC & AD meeting at the vertex A of a tetrahedron ABCD are $ α, β, γ$ (as shown in the diagram below). Is there any set-formula to find out the solid angle subtended by the tetrahedron at the same vertex?

Note: A tetrahedron is a solid having 4 triangular faces, 6 edges & 4 vertices. Three triangular faces meet together at each of four vertices & each of six edges is shared (common) by two adjacent triangular faces.

Denote the solid angle by $\omega$ and let $v_1,v_2,v_3$ be the vectors from vertex $A$ along the edges $AB, AD, AC$. Then we have (using the usual cross product, dot product, and Euclidean norm):

$$(4 \pi)\omega + \pi = \cos ^{-1} \left( \frac{ (v_1 \times v_2) \cdot (v_1 \times v_3)}{||v_1 \times v_2|| ||v_1 \times v_3||}\right) + \cos ^{-1} \left( \frac{ (v_2 \times v_1) \cdot (v_2 \times v_3)}{||v_2 \times v_1|| ||v_2 \times v_3||}\right) + \cos ^{-1} \left( \frac{ (v_3 \times v_1) \cdot (v_3 \times v_2)}{||v_3 \times v_1|| ||v_3 \times v_2||}\right)$$

The solid angle $\omega$ subtended at a vertex by any tetrahedron having (vertex) angles $\alpha, \beta$ & $\gamma$ between consecutive lateral edges meeting at the same vertex, is given by the following HCR's Generalized Formula
$$\omega=\cos^{-1}\left(\frac{\cos\alpha-\cos\beta\cos\gamma}{\sin\beta\sin\gamma}\right)-\sin^{-1}\left(\frac{\cos\beta-\cos\alpha\cos\gamma}{\sin\alpha\sin\gamma}\right)-\sin^{-1}\left(\frac{\cos\gamma-\cos\alpha\cos\beta}{\sin\alpha\sin\beta}\right)$$
OR
$$\omega=\cos^{-1}\left(\frac{\cos\alpha-\cos\beta\cos\gamma}{\sin\beta\sin\gamma}\right)+\cos^{-1}\left(\frac{\cos\beta-\cos\alpha\cos\gamma}{\sin\alpha\sin\gamma}\right)+\cos^{-1}\left(\frac{\cos\gamma-\cos\alpha\cos\beta}{\sin\alpha\sin\beta}\right)-\pi$$
It is worth noticing that the above formula has internal symmetry i.e. the vertex-angles $\alpha, \beta$ & $\gamma$ can be taken in any order/sequence but the result obtained in each case remains the same.