Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Suspension of Cuntz algebra is traceless I saw a conclusion in a reference book: the suspension of the Cuntz algebra $C_0((0,1))\otimes \mathcal O_2$ has no tracial states.
My thought: there are many tracial states on $C_0((0,1))$. We take a tracial state $\tau$ on $C_0((0,1))$, then we can define a tracial state $\til... | Your $\tilde\tau$ is not well-defined. You have
$$
\tilde\tau(x\otimes 1)=\tau(x)=\tilde\tau(x\otimes (-1))=-\tilde\tau(x\otimes 1)=-\tau(x),
$$
a contradiction unless $\tau=0$.
If you had a trace $\varphi$ on $A\otimes \mathcal O_2$, it induces a trace $\psi$ on $\mathcal O_2$ by
$$
\psi(x)=\varphi(1\otimes x).
$$... | {
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Proving $\lim_{n\to\infty} \left( 1+\frac{x}{n} \right)^n = \lim_{n\to\infty} \left( 1+\frac{1}{n} \right)^{nx}$ with a certain method. I saw in another post on the website a simple proof that $$\lim_{n\to\infty} \left( 1+\frac{x}{n} \right)^n = \lim_{m\to\infty} \left( 1+\frac{1}{m} \right)^{mx}$$
which consists of su... | For $x$ negative, we must substitute $n$ with $-mx$ with $m$ positive. We can use the fact that we know what the limit $(1-\frac{1}{m})^{-m}$ is, and equate this with the usual definition of $e$.
| {
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A mother has two children. What is the probability, given this information that both are girls? The Problem
You know that your new neighbors have two children. One day you
see the mother taking a walk with a girl. What is the probability that the other child is also a girl?
a) Given that the mother chooses the younger ... | In $b$, you are interested in $P(D^c | G^*)$, not in $P(D^c \cap G^*)$ - we already know that $G^*$ happened. It's standard applying of Bayes rule: $P(D^c | G^*) = \frac{P(G^* | D^c) \cdot P(D^c)}{P(G^*)}$.
For numerator, $P(G^* | D^c) = \frac{1}{2}$ (if children have the same gender, then $G^*$ is equal to both of the... | {
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Mistake in solving $-\int \frac{1}{x\sqrt{x^2-1}}$ I have this function $$f:(-\infty ,-1)\rightarrow \mathbb{R}, f(x)=\frac{1}{x\sqrt{x^{2}-1}}$$ and I need to find the primitives of $f(x)$.So because $x<-1$ I need to calculate $-\int \frac{1}{x\sqrt{x^2-1}}\,dx=-\arctan(\sqrt{x^2-1})+C$ but in my book the correct answ... | It is known that $$\arctan(x)+\arctan(1/x)=C$$ where $C$ is a constant. So your solution can be written as $$\arctan\left(\frac1{\sqrt{x^2-1}}\right):=u$$Then using some trig identities, $$\frac1{\sqrt{x^2-1}}=\tan u\\\sec^2 u=1+\tan^2 u=\frac{x^2}{x^2-1}\\\sin^2 u=1-\cos^2 u=1-\left(\frac{x^2-1}{x^2}\right)=\frac1{x^2... | {
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Minimal polynomial of extension of degree 2 over a finite field with characteristic 2 I'm struggling to solve the following question.
Let $F$ be a finite field with characteristic 2 and $L/F$ be a finite extension with $[L:F]=2$. Prove that there exists $\alpha\in L$ such that $L = F(\alpha) $ and the minimal polynom... | Consider the polynomials $p_a(x)=x^2-x-a$ for $a\in F$.
Observe that for $a\neq b$, both in $F$, the roots of $p_a$ and $p_b$ are disjoint. This is because a common root $r$ implies $r^2-r-a=0=r^2-r-b$, from where $a=b$ follows.
Also, for a given $a\in F$, the two roots, $r_1$ and $r_2$, of $p_a$ satisfy $r_1+r_2=1$. I... | {
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Prove that $a^x - b^y = 0$ where $x = \sqrt{\log_a b}$ & $y = \sqrt{\log_b a}$ , $a > 0$, $b > 0$ & $a, b \ne1$ I'm solving logarithm questions. I got stuck in this question.
Prove that $a^x - b^y = 0$ where $x = \sqrt{\log_a b}$ & $y = \sqrt{\log_b a}$ , $a > 0$, $b > 0$ & $a, b \ne1$
I've tried to solve it. I'm unab... | Hint: $$y = \frac{1}{x},$$
and $$a^x = b^{1/x} \iff a^{x^2}=b$$
| {
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Find the CDF and PDF of $W$ where $W =.7 - b(.7 - y)^2, 0
A random variable $Y \sim U[0,1]$. Let $W = \frac7{10} - b\cdot(\frac7{10} - y)^2, 0<b<1$.Completely specify the CDF and PDF of $W$.Also show that the PDF of W integrates to $1$.
So I have worked through most of this problem. Importantly, the transformation is $... | That is not quite what I get...
$$\begin{align}F_W(w) &=\mathsf P(0.7 -b(0.7-Y)^2\leqslant w)
\\[1ex]&=\mathsf P(Y\leq 0.7-\sqrt{\tfrac{0.7-w}b})+\mathsf P (Y\geq 0.7+\sqrt{\tfrac{0.7-w}b})
\\[1ex]&=(0.7-\sqrt{\tfrac{0.7-w}b})\mathbf 1_{0\leq (0.7-\sqrt{\tfrac{0.7-w}b})\leq 1}+(0.3-\sqrt{\tfrac{0.7-w}b})\mathbf 1_{0\le... | {
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Prove that $inf(S_1)=sup(S_2)$
$f:I \to \Bbb R$, $I=[0,1]$ continuous and differentiable on $(0,1)$ s.t $f(0)<0<f(1)$ and $f'(x)\neq 0$ $\forall x \in (0,1)$. Let $S_1=\{x \in I | f(x) >0\}$ and $S_2=\{x \in I | f(x) <0\}$. Prove that $inf(S_1)=sup(S_2)$.
My attempt:
If $\exists x\in I$ s.t $f(x)<f(0)$ then $\exists ... | Since $f'$ has IVP and $f'(x) \neq 0$ for all $x$ it follows that $f'(x) >0$ for all $x$ or $f'(x) <0$ for all $x$. In other words, $f$ is strictly increasing or strictly decreasing. But $f(0)<f(1)$ so the second possibility is ruled out and $f$ must be strictly increasing.
Now let $a$ be the infimum of $S_1$ and $b$ b... | {
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Is the set of non invertible matrices simply connected? What are their homotopy and homology groups? It is fairly easy to see that the set of non invertible matrices is path connected. Are they simply connected? If not what is their fundamental group? What are their homotopy and homology groups. I'm looking for the ans... | The space of non-invertible matrices (with real or complex coefficients) is contractible. There is an explicit homotopy between the identity map and the constant map equal to the zero matrix, simply given by:
$$H(A,t) = tA.$$
In particular it is simply connected, and all its higher homotopy and homology groups vanish.
| {
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Prove $A^2 = A$ where $A = I_n − \alpha\alpha^T$ and $\alpha$ is an $n × 1$ vector with $\alpha^T\alpha=1$ Consider a $n × n$ matrix $A = I_n − \alpha\alpha^T$, where $I_n$ is the $n × n$
identity matrix and $α$ is an $n × 1$ column vector such that $\alpha^T\alpha = 1$. Show that $A^2 = A$.
My proof:
$\alpha\alpha^T$... | $\alpha^{T}$ is a vector. Inverse of a vector does not make sense. Just calculate $A^{2}$: $A^{2}=I-2\alpha \alpha^{T}I +\alpha \alpha^{T}\alpha \alpha^{T}I$. Since $\alpha (\alpha^{T}\alpha) \alpha^{T}=\alpha \alpha^{T}$ by hypothesis we get $A^{2}=A$.
| {
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If $P$ is an invertible linear operator on a finite dimensional vector space over a finite field, then there is some $n>0$ such that $P^n=I$ From Berkeley problems in Mathematics:
problem 7.4.21: let $P$ be a linear operator on a finite dimensional vector space over a finite field. show that if $P$ is invertible, then... | I'll make a try:
Take $P,P^2,P^3,...$. Then these can't be all different because the have entries from a finite field. Then $P^t=P^s\Rightarrow P^n=I$ since $P$ is invertible
| {
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Why are $q$ and $p+1$ relatively prime if $q$ divides $p-1$? Looking at the solution of an exercise on Sylow theorems, I see that, if $q$ and $p$ are odd primes such that $q|(p-1),$ then $q$ and $p(p+1)$ are relatively prime.
I understand that $q$ and $p$ are relatively prime, but is there a reason (theorem) why $q$ an... | Consider $q = 3$, $p = 7$. Obviously in this instance $p - 1$ is a multiple of $q$, let's say $mq$, where $m \geq 2$. Then $p(p + 1)$$ = p^2 + p = (mq + 1)^2 + mq + 1$. And $(mq + 1)^2 + mq + 1 = m^2 q^2 + 2mq + 1 + mq + 1$$ = m^2 q^2$$ + 3mq + 2.$
Since $q$ is at least $3$, and $m^2 q^2 + 3mq$ is a multiple of $q$, an... | {
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What is a non-logical concrete example of a choiceless elementary topos? What is a non-logical concrete example of a choiceless elementary topos?
The elementary topoi made from choiceless models of ZF, or stuff like the realizability topos, are (I believe) choiceless, but is there a concrete example from other branches... | Assuming you mean choice in the "every epimorphism has a section" sense, counterexamples among categories of presheaves are plentiful.
For a very quick and silly example, there's the category $\mathbf{Set}^G$ for a non-trivial group $G$. In this case, the sole representable functor in the category is the transitive act... | {
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Construct $\varphi (z)$ such that $\int_{|z|=1} \frac{\varphi (z)}{z-w} dz =0$ I have this problem to complex analysis.
Construct $\varphi (z)$ a continuous function nonzero in $S^{1}$ such that $$\int_{|z|=1} \frac{\varphi (z)}{z-w} dz =0$$ for $|w|<1$.
I have the idea to take $\varphi (z) = (z-w)f(z)$ with $f(z)$ ana... | Take $\varphi(z)=\frac1z$. Then, if $w=0$,$$\int_{\lvert z\rvert=1}\frac{\varphi(z)}{z-w}\,\mathrm dz=\int_{\lvert z\rvert=1}\frac1{z^2}\,\mathrm dz=0.$$And, if $w\neq0$,\begin{align}\int_{\lvert z\rvert=1}\frac{\varphi(z)}{z-w}\,\mathrm dz&=\frac1w\int_{\lvert z\rvert=1}\frac1{z-w}-\frac1z\,\mathrm dz\\&=0,\end{align}... | {
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Calculate $\mathbb{P}[Y=y|X=x]$ where $X$=# claims reported diring firs year, $Y$=# total of claims that will eventually be reported A property-casualty insurance company issues automobile policies
on a calendar year basis only. Let $X$ be a random variable
representing the number of accident claims reported during cal... | $Y$ represents the total random claim count on year 2005 issued policies. $X$ represents the subset of those claims that were reported in the same calendar year of issue. For a sufficiently large number of claims made, you would expect that $X \approx Y \cdot d$. Another way to say this is that given that $Y$ claims... | {
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if $f’(x)/g’(x) = f(x)/g(x)$, what can we say about the relationship? Suppose for all $x$,
$$f’(x)/g’(x) = f(x)/g(x),$$
Then what can we say about the relationship between $f(x)$ and $g(x)$?
I think the only solution is $f(x) = cg(x)$ for some non-zero constant $c$. Is there any other possible relationship?
| Then the numerator of quotient rule for derivative of $f/g$ vanishes and so $f/g$ constant. Need to do more work on it for proof...
| {
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What is all N that make $2^N + 1$ divisible by 3? When $N = 1$, $2^N + 1 = 3$ which is divisible by $3$.
When $N = 7$, $2^N + 1 = 129$ which is also divisible by $3$.
But when $N = 2$, $2^N + 1 = 5$ which is not divisible by $3$.
A quick lookout can determine that when $N$ is an odd number, $2^N + 1$ is divisible by $3... | $((-1)+3))^N+1= $
$\sum_{k=0}^{N}\binom{N}{k}(-1)^{N-k}3^k+1=$
$(-1)^N +1+\sum_{k=1}^{N}(-1)^{N-k}3^k.$
The expression is divisible by $3$ $\iff$
$(-1)^N+1=0.$
Hence?
| {
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Volume of Viviani’s Window I’m taking Real Analysis and I have an assignment to do on calculating the volume of Viviani’s window or dome.
I have to solve this for the sphere $$x^2+y^2+z^2=4$$ and the cylinder $$(x-1)^2+y^2=1$$
Here's a visual representation of the problem:
Here's a figure obtained from the intersecti... | Here is a detailed expalantion of @DanielWainfleet's idea. Let $\mathcal{V}$ denote the region. Then the the intersection of $\mathcal{V}$ and the plane $x = x_0$ is described by
$$ \mathcal{I}(x_0) \quad : \quad y^2 + z^2 \leq 4 - x_0^2 \quad \text{and} \quad y^2 \leq 1 - (1 - x_0)^2.$$
The area of $\mathcal{I}(x_0)$... | {
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What does $d \mathbb{P}(\omega)$ mean in expectation of r.v. $f$? What does $d \mathbb{P}(\omega)$ mean in expectation of r.v. $f$?
$$\mathbb{E} f = \int_{\Omega} f(\omega) d \mathbb{P}(\omega)$$
Yes sure it's some "infinitesimal", but should this mean that $\mathbb{P}(\omega)$ is a variable of $f$? Since in elementary... | Part of what might be confusing to you already shows up in discrete probability. Here is an example: we roll a fair die, and let $X$ be the resulting number. We (that is, math teachers) say things like, the expected value of $X$ is $EX=\sum_{i=1}^6 i (1/6)$ and the expected value of $X^2$ is $EX^2=\sum_{i=1}^6 i^2 (1/... | {
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Matrix similarity by changing basis
Let $$A= \pmatrix{0&1&1\\1&0&0\\2&1&0}$$ and $$B = \pmatrix{0&1&0\\1&0&1\\1&2&0}.$$ Show that $A$ and $B$ are similar in $\mathbb R$.
We can do this by showing that $A$ and $B$ are similar to the same diagonal matrix: they have the same characteristic polynomial, i.e. $$\chi = X³ -... | Observe that one matrix is obtained from the other by swapping the first two columns and then the first two rows.
This means that the permutation matrix $P=\pmatrix{0&1&0\\1&0&0\\0&0&1}$ is the change of basis that you are after, in other words: $$A=PBP^{-1}$$
where moreover $P^{-1}=P$. Multiplication by $P$ on the ri... | {
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What's the remainder when $x^{7} + x^{27} + x^{47} +x^{67} + x^{87}$ is divided by $x ^ 3 - x$ What's the remainder when $x^{7} + x^{27} + x^{47} +x^{67} + x^{87}$ is divided by $x ^ 3 - x$ in terms of $x$?I tried factoring $x$ from both polynomials but I don't know what to do next since there'd be a $1$ in the second ... | $xf(x^2)\,\bmod\, x(x^2\!-\!1)\, =\, x\,(\overbrace{f(\color{#c00}{x^2})\,\bmod\, x^2\!-\!1}^{\color{#c00}{\Large x^2\ \equiv\,\ 1}})\, =\, xf(\color{#c00}{ 1})\, =\, 5x$
| {
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Regular expression that represents the words s.t. start with $a$ and have an odd quantity of $a$'s Find a regular expression that represents the language "The words over the alphabet $\{a,b,c\}$ such that end with a pair of letters, or start with $a$ and have in total an odd quantity of $a$'s".
The null word is repres... | The answer is:
$a((b+c)^*a(b+c)^*a(b+c)^*)^*$.
| {
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What formula could generate this sequence related to the Collatz conjecture The collatz conjecture states that every number eventually reaches $1$ under the repeated iteration of
$$ f_0(n) =
\begin{cases}
n/2, & \text{if $n$ even} \\
3n+1, & \text{else}
\end{cases}$$
As a number is guaranteed to be even after the $3n... | This is not an exact answer to your question, but an simple observation.
Looks like a fractal or reccurence sequence, similar to fibonacci, or one think it might have recurrent relationships. Most likely this is a sequence which can not be shortcut, i.e. one might have to compute every step along the way to get the num... | {
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Determining the convergence and divergence So I have a problem of
$$a_n = \left(1 + \frac{2}{n}\right)^n$$
I need to determine whether it is diverging or converging and find the limit if it is converging I found an answer on symbol lab of $e^2$ but I do not know how they got that type of answer
| Well, once you know the series converges, that is actually one of the definitions of $e^2.$ Here is another way.
Consider $\ln(a_n)=n\ln\left(1+\frac{2}{n}\right).$
Now we look at the form
$$\frac{\ln(1+2/n)}{1/n}=\ln(a_n).$$
We see that
$$\frac{(\ln(1+2/n))'}{(1/n)}=\frac{\frac{1}{1+2/n}}{-1/n^2}\cdot\frac{-2}{n^2}=... | {
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Inclusion Exclusion Application If $A$, $B$, and $C$ are finite sets then, the number of elements in EXACTLY ONE (i.e. at most one) of the sets $A$,$B$,$C$:$$n(A)+n(B)+n(C)-2 \times n(A \cap B)-2 \times n(A \cap C)-2 \times n(C \cap B) + 3 \times n(A \cap B \cap C)$$
I can derive the above through inclusion-exclusion, ... | Suppose we have $n$ sets $A_1,A_2,\dots,A_n$. For $k=1,2,\dots,n$ define
$$S_k=\sum|A_{n_1}\cap A_{n_2}\cap\cdots \cap A_{n_k}|,$$ where the sum is taken over all $k$-subsets
$\{n_1,n_2,\dots,n_k\}\subseteq \{1,2,\dots,n\}$
I claim that the number of elements that occur in exactly one of the sets $A_1,A_2,\dots A_n$... | {
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A finite group is isomorphic to the direct product of two normal subsets with trivial intersection I have to show the following:
Let $G$ be a finite group and $N_1 , N_2$ normal in $G$.
Then $G\cong N_1\times N_2$ if and only if $N_1\cap N_2 = \{e\}$.
I have no idea on either direction, so I would be grateful for any l... | This is not true unless $N_1,N_2$ generates $G$, in this case let $p_i:G\rightarrow G/N_i$ the quotient map, show that $p_1$ iduces an isomorphism $N_2\rightarrow G/N_1$. Consider $f:G\rightarrow N_1\times N_2$ defined $f(x)=(p_1(x),p_2(x))$ show that it is an isomorphism.
| {
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Apply Newton-Raphson method to find the solutions to Apply Newton-Raphson method to find the solutions to the equation $x^3-5x=0$ starting with an initial guess of $x_0 = 1$.
While using Newton Raphson method, the value doesn't converge to a specific number. Rather, every iteration either gives $1$ or $-1$. Why does th... | For the graph of $f(x)=x^3-5x$, we have $f'(x)=3x^2-5$
$$f(1)=-4$$
$$f'(-1)=4$$
$$f'(1)=-2=f'(-1)$$
The tangent line at $x=1$ is $y+4=-2(x-1)$ which is $y=-2x-2$.
The tangent line at $x=-1$ is $y-4=-2(x+1)$ which is $y=-2x+2$
Geometrically, what has happened is you are trapped in the following cycle.
Starting from $(1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3213274",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
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Counting the Number of Real Roots of A Polynomial I am interested in solving problems which involve finding the number of real roots of any polynomial.
Suppose I take a function $$f(x)=x^6+x^5+x^4+x^3+x^2+x+1$$ This does not have any real roots but I am trying to figure out if there is some analytical way that does not... | While this method isn't guaranteed to work on all polynomials, it is surprisingly effective sometimes: notice that
\begin{align*}
[x^6+x^5+x^4+x^3+x^2+x+1]
&= x^4\left(x+\tfrac{1}{2}\right)^2+\big[\tfrac{3}{4}x^4+x^3+x^2+x+1\big] \\
&= x^4\left(x+\tfrac{1}{2}\right)^2+\tfrac34x^2\left(x+\tfrac23\right)^2+\big[\tfrac{2}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3213397",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
"answer_count": 7,
"answer_id": 1
} |
The dual space of the Sobolev space $W^{k,p}(\Omega)$. Let $\Omega$ be a nice domain in $\Bbb R^n$. It is known that any element $T\in\left( W^{k,p}(\Omega)\right)^*$ admits a (possibly non-unique) representation of the form
$$
Tu = \sum_{|a|\le k} \int_\Omega f_\alpha D^\alpha u\ dx, \tag{0}
$$
where $f_\alpha \in L^... | I think that the problem is that functionals in $(1)$ are not necessarily in $(W^{k,p})^*$ rather than non-uniqueness.
Take for example the simple setting $\Omega=(0,1)$, $W^{k,p}=H^1$.
Then $f=x^{-\frac{1}{3}} \in L^2$ so $$T=-\partial_x x^{-\frac{1}{3}}=\frac{1}{3}x^{-\frac{4}{3}}$$
satisfies $(1)$.
But it is not ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3213481",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Galois group of $\overline{F}/F$
Let $K$ be the algebraic closure of $F$ where $F$ is a finite field. Show that $Gal(K/F) \simeq \hat{\mathbb{Z}}$.
I know that $\hat{\mathbb{Z}} = \varprojlim \mathbb{Z}_{n}$, so its enough to show that $Gal(K/F) \simeq \varprojlim \mathbb{Z}_{n}$. I can solve the problem if $|F|=p$, ... | More generally, Gal$(\bar F/F)$ is the inverse limit of its finite quotients, so it suffices to pick up, if possible, a convenient inductive system of finite extensions of $F$. If $F$ is a finite field (its characteristic is irrelevant), we know that every finite extension of $F$ is cyclic , with Galois group generated... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3213583",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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Can't solve complicated second order differential equation (Poisson-Boltzmann equation) Is there anyone able to solve this second order differential equation? It is the Poisson-Boltzmann equation (found in the field of electrostatics) solved on cylindrical coordinates just on the radial direction.
$$
(\varphi'+r\cdot\v... | Let $r=e^s$ ,
Then $s=\ln r$
$\dfrac{d\varphi}{dr}=\dfrac{d\varphi}{ds}\dfrac{ds}{dr}=\dfrac{1}{r}\dfrac{d\varphi}{ds}=e^{-s}\dfrac{d\varphi}{ds}$
$\dfrac{d^2\varphi}{dr^2}=\dfrac{d}{dr}\left(e^{-s}\dfrac{d\varphi}{ds}\right)=\dfrac{d}{ds}\left(e^{-s}\dfrac{d\varphi}{ds}\right)\dfrac{ds}{dr}=\left(e^{-s}\dfrac{d^2\varp... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3213733",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Computation of the laplacian of an isometric immersion Say that $X:M \to \mathbb R^3$ is an isometric immersion of an oriented riemannian surface (oriented $2$ dimensional riemannian manifold). I understand there holds a vector-valued equation, namely
$$
\Delta X = 2HN,
$$
where $H$ is the mean curvature (half the trac... | Based on my study of @Ernie060's answer, I believe I've worked out a solution.
Fix an isometric immersion $X: M \to \mathbb R^3$. For some arbitrary point $p \in X(M)$, consider an orthonormal basis $e_1, e_2$ of $T_p X(M)$. Using existence theory for systems of linear ODE, in a small $X(M)$-neighborhood of $p$ let us ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3213841",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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$\Delta \mathbf n = -2 \mathbf n$ on the Euclidean sphere Let us consider the Euclidean two-sphere, defined by the embedding in the three dimensional Euclidean space as
$$
\mathbf n \cdot \mathbf n = 1\,,
$$
where $\cdot$ denotes the standard scalar product. The metric on the sphere, in some coordinates $x^i$, is expre... | A way to prove the above is the following (this is probably a special case of the more general answer given by @Ernie060, but I still need to fill in a few details).
In three-dimensional Euclidean space $\mathbb R^3$, the metric in Cartesian coordinates is $\delta_{IJ}=\mathrm{diag}(1,1,1)$ and in spherical coordinate... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3213979",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 1
} |
Find the period of $f(x)$ with $f(x)f(y)=f(x+y)+f(x-y)$. For any real $x$, $\;\;y$
$f(x)f(y)=f(x+y)+f(x-y)$ with $f(1)=1$
Find the period of $f(x)$.
| To summary, I am going to show the following result.
Main Theorem: Let $f$ be the function mentioned in OP. If the peroid of $f$ exists, then the
peroid of $f$ only has one of the two forms: $\frac{6}{6n+1}$ and
$\frac{6}{6n+5}$ for some $n\in \mathbb{N}$.
@Micah have shown the period $f$ "has" the peroids of $2... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3214058",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
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Help with proof of $ \mathbb{C}[X] \simeq R $ where $R$ is a $ \mathbb{C}$-algebra without nilpotents I am trying to understand the proof of the following proposition:
Let $X \subset \mathbb{A}^n$ be closed. Let $ R $ be a finitely generated $ \mathbb{C}$-algebra without nilpotents. There exists an affine variety $ X ... | Well, let $f$ be a polynomial with $f^n\in I$, i.e., $\phi(f^n)=0$. Then $\phi(f)^n=\phi(f^n)=0$ and so $\phi(f)$ is nilpotent in $R$. By hypothesis $\phi(f)=0$ and so $f\in I$. Hence, $I$ is radical.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3214233",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Computing $\int_{|z-i|=\frac{3}{2}}\frac{e^{\frac{1}{z^2}}}{z^2+1}$
Compute the integral using residues: $\int_{|z-i|=\frac{3}{2}}\frac{e^{\frac{1}{z^2}}}{z^2+1}$
Inside the circumference there are the following singular points $-i$ which is a pole of order 1 and $0$ which is essential.
So:
$\int_{|z-i|=\frac{3}{2}}... | Here is a way to calculate the integral without Laurent series to find the residue at $z = 0$. It uses that the sum of all residues is $0$ including the residue at infinity:
*
*$f(z)= \frac{e^{\frac{1}{z^2}}}{z^2+1} \Rightarrow$
$$ \operatorname{Res}_{z=0}f(z) + \operatorname{Res}_{z=i}f(z) = - \left( \operatornam... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3214371",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
How to perform U substitution when there is no direct dx to du mapping? I have been asked to solve this problem using a change of variable and a power series expansion of the exponential term:
$\frac{1}{\sqrt(2\pi)} \int_0^\sqrt2 e^{-\frac{x^2}{2}} dx$
I expanded the exponential term to the Taylor Series:
$e^{-\frac{x^... | You decided to use
$$x^2=u \implies x=\sqrt u\implies dx=\frac{du}{2 \sqrt{u}}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3214511",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Find (if it exists) random variable $X$ so it is vaild: $\mathbb{E}(X)= 3, \mathbb{E}(X^2) = 8$. Find (if it exists) random variable $X$ so it is vaild: $\mathbb{E}(X)= 3, \mathbb{E}(X^2) = 8$.
I tried by definition of expectation, but how can I know that, when squared, some values won't be same (which implies that new... | Note that
$$
\text{Var}(X)=E(X-EX)^2=EX^2-(EX)^2=8-9=-1<0
$$
which is impossible. Hence such a random variable does not exist.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3214712",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Common point of solutions of $(\varepsilon-x)y=y'(-x+y^2-2x^2)$ Let there be the same equation as here.
$(\varepsilon-x)y=y'(-x+y^2-2x^2)$
@JJacquelin found the integrating factor
$$\boxed{\mu=\frac{1}{(x+2\epsilon x-y^2)(\epsilon +2\epsilon x-y^2)\:y}}\tag 2$$
The implicit answer is
$$\boxed{2\epsilon\ln\left(|x+2\ep... | The conditions of the existence and uniqueness theorem do not hold when $-x + y^2 -2 x^2 = 0$. Substituting this into the ODE gives
$$(x, y) \in
\left\{ (-1/2, 0), (0, 0),
\left( \epsilon, -\sqrt {\smash[b] {\epsilon (1 + 2 \epsilon)}} \right),
\left( \epsilon, \sqrt {\smash[b] {\epsilon (1 + 2 \epsilon)}} \right) \... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3214859",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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} |
Why to choose to work with a functional instead of a function? Why to choose to work with a functional instead of a function?
Notice in a function you evaluate points and in functional you evaluate functions.
and why a linear functional is important in general ?
A functional $\phi(f)$ is linear if the domain of its exi... | You pose a dichotomy that is not so. One does not look at functionals as opposed to functions.
*
*First of all, a functional is a function.
*Second, you say that "functions evaluate points". That's not a good point of view. Typical calculus functions evaluate on numbers, as they usually evaluate as formulas. But ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3215029",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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What is the name of the operation where you find the "closeness" of 2 values from 0 to 1? This type of function comes up every now and then during my projects. Often and in performance critical parts enough that I'd like to learn more about it and see how others have implemented it.
Essentially, the purpose is to find ... | I don't think that there's a name for this in mathematics -- we tend to give names to distance functions rather than closeness functions, for instance.
The underlying distance function ($0$ when two points are the same, $1$ when they're as far apart as possible) is twice the ordinary distance on a circle of circumfer... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3215170",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
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The Second Derivative Test and the Mean Value Theorem I have been studying for the upcoming Introductory Real Analysis exam and got stuck upon the proof of the second derivative test. Here is a verbatim text of the theorem and the proof from the book:
THEOREM:
Let $I$ be an open interval containing the point $x_0$ and ... | First, apply your MVT to the interval $I=(x_0,x_0+\delta)$. Take an $x\in I$. The MVT says that there is a $c, x_0<c<x$ such that:
$f'(c)=\frac{f(x)-f(x_0)}{x-x_0} $.
Since $f'(c) >0$ (it's one of your inequalities), then $f(x)>f(x_0)$. A similar argument holds for the interval $I'=(x_0-\delta,x_0)$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3215343",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Algebraic probability There are only purple and orange marbles in a bag. There are three more purple marbles than orange marbles in the bag. Roxanne is going to take at random two marbles from the bag. The probability that Roxanne will take two marbles of the same colour is 41/81. Work out the number of orange marbles ... | As you suspected, the marbles are being replaced. This means the probability for the second marble is the same as the first marble.
Letting $x$ be the number of purple marbles:
$$\left( \frac{x}{2x+3} \right)^2 + \left( \frac{x+3}{2x+3} \right)^2 = \frac{41}{81}$$
$$\frac{2x^2+6x+9}{4x^2+12x+9} = \frac{41}{81}$$
$$162x... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3215511",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Order of an element in general linear group Let $A=GL(n,2)$ be the general linear group of size $n$ over the finite field
$\mathbb{F}_2=\{0,1\}$. For small $n=3,4,5$ etc., I observed any matrix in $A$ has of the order less than or equal to $2^n-1$. Is there any proof of this result?
| Consider $g\in GL(n,2)$ and the sequence $\{g^k\}_{k=0}^\infty$. The order of $g$ is the first index $k$ such that there is some $h<k$ such that $g^h=g^k$. Therefore the set $\{g^n\,:\, n\in\Bbb N\}$ contains exactly $\operatorname{ord}g$ distinct matrices. These matrices are all non-zero elements of the $\Bbb F_2$-alg... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3215645",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Find maximum of function $A=\sum _{cyc}\frac{1}{a^2+2}$ Let $a,b,c\in R^+$ such that $ab+bc+ca=1$. Find the maximum value of $$A=\frac{1}{a^2+2}+\frac{1}{b^2+2}+\frac{1}{c^2+2}$$
I will prove $A\le \dfrac{9}{7}$ and the equality occurs when $a=b=c=\dfrac{1}{\sqrt3 }$
$$\frac{\sum _{cyc}\left(b^2+2\right)\left(c^2+2\ri... | $$a=\tan \left(\frac {\alpha}{2}\right), b=\tan \left(\frac {\beta}{2}\right), c=\tan \left(\frac {\gamma}{2}\right)\ \ \ \ (\alpha+\beta+\gamma= \pi)$$
$$A=\dfrac { \cos^2 \left(\frac {\alpha}{2}\right) }{1+ \cos^2 \left(\frac {\alpha}{2}\right) }+ \dfrac { \cos^2 \left(\frac {\beta}{2}\right) }{1+ \cos^2 \left(... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3215787",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Degree of the extension $k(x)/k(q(x))$ for a rational function $q\in k(x)$ Let $k$ be an algebraically closed field, and $q\in k(x)$ a nonzero rational function, expressible as $q(x)=r(x)/s(x)$ for $r$ and $s$ coprime, and $d=\deg r\ge\deg s$. Then will $[k(x):k(q(x))]=d$? It is easy to see that $d$ is an upper bound, ... | Set $L:=k(x)$, and its subfield $K:=k(q)$, where $q=r/s$ for coprime polynomials $r,s\in k[x]$. The claim is that $L\cong K[t]/(r(t)-qs(t))$, and it is enough to show that $m:=r(t)-qs(t)$ is an irreducible polynomial in $K[t]$. (This is where you are making a mistake: we don't want to ask about this polynomial in $L[t]... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3215886",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Wrong answer for this integral: $\int \frac{x^2+1}{x^4-x^2+1}dx$
$$\int \frac{x^2+1}{x^4-x^2+1}dx$$
Dividing both sections by $x^2$ and adding $1$ to the denominator I get: $$\int \frac{1+\frac{1}{x^2}}{(x-\frac{1}{x})^2+1}dx$$
Setting $x-\frac{1}{x}$ as $t$ I get: $$dt=1+\frac{1}{x^2}dx$$
Then I have written it in t... | The substitution $t=x-\frac{1}{x}$ is indeed a good starting point, but I'm not sure I understand your reasoning after that. The substitution rewrites your indefinite integral as $\int\frac{dt}{t^2+1}$ (as others have noted), which you seem to have rewritten as $t+\int\frac{dt}{t^2}$. That suggests to you me you confus... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3215994",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
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What could be an example where multiplying two power series gives a bigger radius that is not infinity I'm trying to think of an example of $\sum_{n=0}^{\infty}a_nx^n, \sum_{n=0}^{\infty}b_nx^n$ with radius of convergence $R_1,R_2$ such that when we multiply them we get $\sum_{n=0}^{\infty}c_nx^n, c_n = \sum_{i+j=n}a_... | Consider the Taylor series $\sum_{k\geq0} a_k\,x^k$ of the functions
$$f(x):={2-x\over(1-x)(3-x)},\qquad g(x):={1-x\over 2-x}\ .$$
Then $R_f=1$, $\>R_g=2$, and $R_{fg}=3$, because the radius of convergence for the Taylor series at $0$ is equal to the distance from $0$ to the nearest singularity.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3216154",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
can't understand derviatives from searching all over the internet I have tried to follow a lot of tutorials out there for explaining derviatives and show (understandable)examples but i couldn't understand any. Can anyone link me a useful and easy to follow tutorial or explain derviatives for me?
| https://www.youtube.com/watch?v=N2PpRnFqnqY
Khan Academy is a great resource if you want to learn math online. It also has a wide selection of calculus tutorials that are all narrated and explained very well.
I will try to explain what a derivative is in a nutshell: it is the slope of a line at a point. For example, th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3216484",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
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Euler-Lagrange Equation for Kantorovich Dual Problem Given two probability measures $\mu$ and $\nu$, the Kantorovich Dual problem for quadratic cost is to
$$
\text{minimize} \quad \int \phi(x)d\mu + \int \psi(y)d\nu
$$
over pairs $(\phi,\psi)\in L^1(d\mu)\times L^1(d\nu)$ such that $xy \leq \phi(x) + \psi(y)$. In Vill... | It is possible to weaken the assumptions on $\mu$ and $\nu$ significantly, but it does not seem to be true without some sort of technical assumption on the measures. See Theorem 1.22 in Santambrogio's book, which he describes as "the sharpest result in the unbounded case" - there it is assumed that both measures have f... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3216619",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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Calculating $\mathbb{P}(Y \leq 1)$ given the moment generating function
Given the moment generating function $$M_Y(t)
=\frac{4-3t}{2(t-2)(t-1)}$$ with $t<1$ find $\mathbb{P}(Y \leq 1)$.
First I tried to convert this to the probability generating function, because than you can easily find it, but my TA told me that it... | Hints:
*
*Try decomposing your $M_Y(t)$ into partial fractions
*The moment generating function of a mixture distribution, where $Y\sim X_1$ with probability $p$ and $Y\sim X_0$ with probability $1-p$, is $M_Y(t) = pM_{X_1}(t)+(1-p)M_{X_0}(t)$
*The moment generating function of an exponential distribution with ra... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3216823",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
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Solving Equations using logarithm Here is a system of equations for which I am having difficulty solving:
\begin{cases} a^{2x}.b^{3y}=m^5 \\ a^{3x}.b^{2y}=m^{10} \end{cases}
| HINT
Notice that $$a^{2x}.b^{3y}=m^5 \\a^{3x}.b^{2y}=m^{10}$$
$$\Rightarrow a^{3x}.b^{2y}=(a^{2x}.b^{3y})^2$$
Now solve either of for $a$ or $b$. The solution you obtain will be dependent on $m$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3216927",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
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What is the logic behind the combinations with repetitions formula This month, I'm taking combinatorics classes in my school, yesterday we learned about combinations with repetitions formula. Our teacher wrote it on the board, but she didn't really explained what is the logic behind this and why the reduction to combin... | Well, a $k$-repetition of $n$ is a word $x=x_1\ldots x_n$ of length $n$ over the alphabet ${\Bbb N}_0$ (natural numbers inluding $0$) such that $x_1+\ldots+x_n=k$.
The $2$-repetitions of $4$ are
$2000,1100,1010,1001,0200,0110,0101,0020,0011,0002$.
They count the number of ways to draw $2$ numbers from $1,2,3,4$ without... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3217041",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
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Evaluating $\lim\limits_{n\to \infty} \sin \left( (2 + \sqrt 3 )^n\pi\right)$ for $n \in \mathbb N$
Evalulate $$\lim\limits_{n\to \infty} \sin \bigl( (2 + \sqrt 3 )^n\pi \bigr) \quad \text{ for } n \in \mathbb N$$
This question appeared in my high school exam.
My first idea was as $n$ is an integer then the value mus... | Note that
$$\lim_{n\to\infty}\sin\left((2+\sqrt 3)^n \right) $$
is not the same as
$$\lim_{n\to\infty}\sin\left((2+\sqrt 3)^n \pi\right) $$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3217301",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Convergence of the series below $$\sum_{n=1}^\infty\frac{(-1)^n}{\sqrt{n}}$$
I did:
$$\lim_{n\to \infty}\Biggr\vert\frac{(-1)^n}{\sqrt{n}}\Biggr\vert$$
$$\lim_{n\to \infty}\frac{1}{n^\frac{1}{2}}=0<1$$
So diverges by the Ratio Test, right?
And have this series:
$$\sum_{n=1}^\infty \bigg(\frac{3n^3+2}{2n^4+1}\bigg)^n$$
... | As for your first series, you did not show it diverges, but you can show (compare it with $1/\sqrt{n}$) that the series of the absolute values diverges, i.e. your series is not absolutely convergent. Using Leibniz's criteria for alternating sign series you can show it converges, making it simply (or conditionally) conv... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Locus of an Equation Bouncing off of another question I had here, I wondered what would be the equation of the locus of this equation: $$0 = b + ax - dy - cxy$$ Specifically $$0=xy-6y+6x-3$$ I would like a way to derive it. Also, if it is possible, may someone please give me a name for the function.
| By itself,
$$ (x-6)(y+6) = xy -6y+6x - 36 $$ You have
$$ 0 = xy-6y+6x-3 $$
Subtract!
$$ (x-6)(y+6) = -33 $$
called a hyperbola, in the same way that $xy = 1$ is a hyperbola. Just moved a bit, stretched a bit.
Let's check a point. My version has $-3 \cdot 11 = -33,$ so we can take $x=3, y=5.$ The original
$xy-6y+6x... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Is an intersection of prime ideals equal to their product? Suppose $R$ is a commutative ring with identity, and let $P_1 , P_2, ..., P_n$ be distinct prime ideals in $R$.
Is it then true that
$$ P_1 P_2 \cdots P_n = P_1 \cap P_2 \cap \cdots \cap P_n?$$
My hunch is that this is not true.
The inclusion $P_1P_2\cdots P_n\... | A counterexample is in the ring $R=\mathbb{R}[X,Y]$. Take $P_1=XR$ and $P_2 = XR+YR$. Then $P_1\cap P_2 = P_1$. And $X\in P_1$ is not an element of $P_1 P_2$.
Now let me prove something that does work: $\sqrt{P_1\dots P_n}=P_1\cap\dots \cap P_n$. And for that, the prime $P_i$'s do not need to be distinct. For the inclu... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Fundamental group of $S^n\setminus S^k$ I'm doing review questions for grad school examinations and I came across one that's stumped me for a while:
$S^n = \{(x_1, \dots , x_{n+1} \colon \Sigma x_i^2 = 1\}$ and $S^k = \{(x_1, \dots , x_{n+1} \in S^n \colon x_{k+2} = \cdots = x_{n+1} = 0\}$ What is $\pi_1(S^n\setminus S... | $S^n \setminus S^k \simeq S^{n-k-1}$ which should answer your question.
Proof. Define
$j : S^{n-k-1} \to S^n \setminus S^k , j(y_1,\dots,y_{n-k}) = (0,\dots,0,y_1,\dots, y_{n-k}),$
$p: S^n \setminus S^k \to \mathbb R^{n-k} \setminus \{ 0\}, p(x_1,\dots,x_{n+1}) = (x_{k+2},\dots,x_{n+1}).$
$r : S^n \setminus S^k \to S^... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Show that $a^3 - b^3 = c! - 18$ does not have a solution
Let $a, b,$ and $c$ be positive integers and $c \gt 6$.
Show that the equation $$a^3 - b^3 = c! - 18$$ does not have a solution for all positive integers $a, b,$ and $c$.
What I have realized so far is that if $a^3 - b^3 = c! - 18$, then it must also be true t... | You may want to try your luck with $\bmod 7$. You should know, or easily show, that all cubes are $\in\{0,1,6\}\bmod 7$ forcing $a^3-b^3\in\{0,1,2,5,6\}$. But for $c\ge 7$ you find $c!-18$ fails to meet this qualification (I will let you figure that out). This does not generally work for $c<7$, but that was excluded... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Are there true arithmetical statements that corresponds to inconsistency of inconsistent theories? Lets take Naive set theory "NvST" which is the theory whose axioms are all instances of naive unrestricted comprehension, which is of course known to be inconsistent. So $\neg$ Con(NvST) is a TRUE statement!
Question is ... | Gödel's work shows us how to write down an arithmetical statement that corresponds to $\operatorname{Con}(T)$ or $\neg\operatorname{Con}(T)$ for any theory $T$, as long as the set of axioms of $T$ is Turing-recognizable.
This works purely syntactically, and does not depend in any way of having an interpretation of the ... | {
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Squared modulus of a complex expression
I am trying to verify a formula that was presented in a paper giving the intensity transmission of a resonator. I need to confirm that the squared modulus of
$$\frac{r-\tau\exp\left(i\varphi\right)}{1-\tau r\exp\left(i\varphi\right)} \tag{1}$$
is equal to
$$\frac{\tau^{2}-2r\tau... | With $z:=\exp(i\varphi)$ the square modulus is $$\frac{(r-\tau z)(r-\tau/z)}{(1-\tau rz)(1-\tau r/z)}=\frac{r^2+\tau^2-r\tau(z+1/z)}{1+\tau^2r^2-\tau r(z+1/z)}=\frac{r^2+\tau^2-2r\tau\cos\varphi}{1+\tau^2r^2-2\tau r\cos\varphi}.$$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How to tell when a limit diverges? I have $\underset{x \to -2} \lim \frac{x^2-1}{2x+4}$ which doesn't exist because it diverges. I spent awhile trying to remove the division by zero before plugging it into an online math calculator, and I want to know how I could have known that it diverges. I'm not skilled enough to j... | $$
\frac{x^2}{2x+4}=
\frac{x^2-4+4}{2(x+2)}=
\frac{(x-2)(x+2)}{2(x+2)}+\frac{4}{2(x+2)}=\\
\frac{x-2}{2}+2\frac{1}{x+2}.
$$
It's not difficult to see that as $x$ approaches $-2$ from the left, $\frac{1}{x+2}$ goes to negative infinity: $\lim_{x\to-2^-}\frac{1}{x+2}=-\infty$. And as $x$ approaches $-2$ from the right, $... | {
"language": "en",
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What does it mean for a system of equations to have a non trivial solution? What does it mean for a system of equations to have a non trivial solution?
Trivial means obviously but how can a equation have a obvious solution?
My book says if solution is non trivial than determinant of coefficient of variables is 0.
| The word is often used about systems of equations such as
$$ 3x+5y-12z = 0 \\ x-y+5z = 0 \\ 5x+y+3z=0 $$
where it is immediately obvious that setting all of the unknowns to $0$ will solve the system. The question is then whether the system has other solutions than that.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Randomly generate a sorted set with uniform distribution I have an ordered set $S = \langle S_1, S_2, .., S_M \rangle$ from which I want to draw a sample of $N$ elements in such a way that the sample is non-strictly totally ordered (as with $\leq$ and the integers), and all the possible occur with equal probability. Th... | I would just draw $3$ random numbers and only accept ones in ascending order, that way you have equal probabilities (i.e. rejection method as discussed)
Otherwise,
Sample the triples by using monte carlo. I.e. count how many different combinations are permissible, and accept each with equal probability according to di... | {
"language": "en",
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"source": "stackexchange",
"question_score": "7",
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Number Theory Puzzle: Competition Problem I have been struggling the past few hours with a problem I initially thought to be easy and simple. Nothing comes to mind besides guessing numbers and the answer. Any help or hints would be greatly appreciated!
Thank You!
Problem:
If any digit of a given 4-digit number is dele... | We have $14$ such numbers. With the numbers @RossMillikan sir have noted, we also can have $a|ab$ and $b|ab$ case. Which will give numbers like $1200,1500,2400,3600,4800$. The full list is given below: $$ 1100, 1200, 1500, 2200, 2400, 3300, 3600, 4400, 4800, 5500, 6600, 7700, 8800, 9900$$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Question about the definition of involute of a plane curve I'm studying about involute of a plane curve from here and there's a small point that is really bothering me and I can not understand it.
Assuming curve is parameterized by arc length, the involute is defined as $$\gamma(t) = \beta(t) - t \beta'(t) $$
Why is th... | The model for the involute is this: Take a circle, with scotch tape wrapped around it. Start to peel the scotch tape off and follow the point $P$ at the end of the tape. As you go counterclockwise around the circle, the tape is tangent, but the line segment from the point of contact to $P$ goes in the direction opposit... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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$SL(n,\Bbb{C})$ is a regular submanifold of $GL(n,\Bbb{C})$
Let $SL(n,\Bbb{C})$ be the group of matrices of complex entries and determinant $1$.
I want to prove that $SL(n,\Bbb{C})$ is a regular submanifold of $GL(n,\Bbb{C})$.
An idea is to use the Regular Level Set Theorem because $$SL(n,\Bbb{C})=f^{-1}(1)$$ where $... | You can avoid the calculations by using Lie group theory, because $GL(n,\mathbb{C})$ is a Lie group and $SL(n,\mathbb{C})$ a closed subgroup. Cartan's theorem then says $SL(n,\mathbb{C})$ is a regular Lie subgroup, in particular a regular submanifold. Note that this also works for $\mathbb{R}$ instead of $\mathbb{C}$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3219238",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Finiteness of $\int_0^1 \left(\sum_{n=1}^\infty \frac{n^\alpha e^{- t n^\alpha}}{1 - e^{- t n^\alpha}} \right)^{1/2} \, \mathrm{d}t$ Let $\alpha > 1$.
My question is: is $\int_0^1 \left(\sum_{n=1}^\infty \frac{n^\alpha e^{- t n^\alpha}}{1 - e^{- t n^\alpha}} \right)^{1/2} \, \mathrm{d}t$ finite ?
For $t \in ]0, 1]$, th... | As Beautiful Art pointed out, it is enough to find the behaviour about $t=0$. Defining $$f(t)=\sum_{n=1}^\infty \frac{n^\alpha}{{\rm e}^{t n^\alpha}-1}$$ you can calculate the Mellin-Transform
$${\cal M}_f(s) = \Gamma(s)\zeta(s)\zeta\left(\alpha(s-1)\right)$$ which converges whenever $\Re(s)>1+\frac{1}{\alpha}$. The in... | {
"language": "en",
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Evaluate $\sum_{n=1}^\infty \frac{(-1)^{[\sqrt{n}]}}{n^p}$ for $ 0 < p \leq 1$ How do you evaluate $$\sum_{n=1}^\infty \frac{(-1)^{[\sqrt{n}]}}{n^p}$$ for the case $0 < p \leq 1$?
Now I have successfully proved that the series converges for the case $p > 1$ and divergence of $p\leq 0$ is trivial, but I cannot deal with... | $$S(N) = \sum_{n=1}^{N}(-1)^{\lfloor\sqrt{n}\rfloor}$$
is such that $S((2M)^2)=-2M$ and $S((2M+1)^2)=2M-1$.
In particular $|S(N)|\leq\sqrt{N}$ and this inequality is sharp for any square $N$, but the sign of $S(N)$ depends on the parity of $\lfloor\sqrt{N}\rfloor$. If we assume $p\in\left(\frac{1}{2},1\right]$ and app... | {
"language": "en",
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Is $\text{abs}: [-1, 1] \to [0,1]$ a covering map? I read that for any covering space $p:C \to X$ the cardinality of the fibre is the same for every $x \in X$ if $X$ is connected.
So what is wrong with my example: $\text{abs}: [-1,1] \to [0,1]$?
Clearly the fibre has cardinality 2 everywhere except for $x=0$ where the ... | $abs$ is not a covering map. That's because no open neighbourhood of $0$ in $[-1,1]$ can be mapped homeomorphically onto its image. And covering maps are local homeomorphisms.
In particular your $U=[0,0.5)$ example, as a neighbourhood of $0$ in the codomain, is not correct. Its preimage under $abs$ is $(-0.5,0.5)$ whic... | {
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Prove by induction on $n$ that $(1)(2)+(2)(3)+...+n(n+1)={1\over 3}n(n+1)(n+2)$ So after testing myself with this question, I was unable to solve it. I was able to prove the base case $n=1$, but I was pretty lost on the induction step. I took a look at the solution and here it is: Solution to problem
I understand it up... | This is related to the distributive property which basically states that $\rm \color{red}ab+\color{red}ac=\color{red}a\cdot(b+c)$.
In your example
$$\rm \color{blue}{\frac13(k+1)(k+2)}\cdot k+ \color{blue}{\frac13(k+1)(k+2)}\cdot3=\color{blue}{\frac13(k+1)(k+2)}\cdot(k+3)$$
| {
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If an exercise says "the measure of the smallest angle in a triangle is $20$", does that mean no other angle can have measure $20$? I have a question about this exercise:
If the measure of the smallest angle in a triangle is 20, then the measure of the greatest possible angle in this triangle is .....
$$a ) 90$$
$$b... | What you're asking is a valid question: there is a difference between the minimal element and the least element.
In this question, I would assume that it's OK for the other angle to be 20. Because if not, then there is no measurement for the 2nd smallest angle. What would it be? 20.01? 20.0001? It's analogous to asking... | {
"language": "en",
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Subset of a subgroup is not closed under group actions In the following image (from "Field Arithmetic by Fried & Jarden" Page 6, Lemma 1.2.2(b)), red rectangle,
I'm trying to figure out why it's right to claim $h^{-1} \in H$.
I thought the following solved it:
$g=k_ih_i$ and $g=kh^{-1}$ $\Rightarrow k_ih_i=kh^{-1} \R... | The proof is not correct.
You have $g = k_ih_i$ with $k_i \in K, h_i \in H_i$. But then $h_i = k_i^{-1}g \in K^{-1}g$ and we conclude $H_i \cap K^{-1}g \ne \emptyset$ and not $H_i \cap g^{-1}K \ne \emptyset$. But then we can correctly show that there exists $h \in \bigcap_{i \in I} (H_i \cap K^{-1}g) = (\bigcap_{i \i... | {
"language": "en",
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Convolution of functions $f,g\in L^1([0,1])$ Problem: convolution ($f*g) $ of functions $f,g\in L^1([0,1])$, where:
$$f(x) = \frac{3}{5-4\cos{4\pi x}},$$
$$g(x) = \frac{2\cos{2\pi x}}{5-4\cos{4\pi x}},$$
and $$(f * g)(x) = \int_{0}^{1}f(x-y)g(y)dy.$$
I'm not sure how to approach this problem. This is a problem from a c... | First note a couple of translation symmetries:
$$
f(x-1/2) = \frac{3}{5-4\cos{4\pi (x-1/2)}} = \frac{3}{5-4\cos{4\pi x}} = f(x) \\
g(x+1/2) = \frac{2\cos{2\pi (x+1/2)}}{5-4\cos{4\pi (x+1/2)}} = \frac{-2\cos{2\pi x}}{5-4\cos{4\pi x}} = -g(x+1/2)
$$
Then we split the integral into two parts and translate the second one:
... | {
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Why are parallelograms defined as quadrilaterals? What term would encompass polygons with greater than two parallel pairs? It seems the definition of a parallelogram is locked to quadrilaterals for some reason. Is there a reason for this? Why couldn't a parallelogram (given the way the word seems rather than as a mathe... | I'm going to propose, out of the blue, terms like "hexaparallelogram", "octaparallelogram", and so forth.
I'm wondering whether, for more than $4$ sides, you would like your definition of hexaparallelogram to be restricted to having 3 pairs of parallel and pairwise equal sides (as in your picture - evidently these have... | {
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Further factorisation of a difference of cubes? We know that a difference of cubes can be factored using $a^3-b^3=(a-b)(a^2+ab+b^2)$. How do we know that the quadratic can't be factored further.
For example,
$$
27x^3-(x+3)^3=((3x-(x+3))(9x^2+3x(x+3)+(x+3)) \\
=(2x-3)(13x^2+15x+9)
$$
In this case we can't factor the qu... | Let $$\delta:=b^2-4ac.$$
We can factor the quadratic over $\mathbb R$ if and only if $\delta\geq 0$.
In the example above,we have
$$a = 13,b = 15,c = 9$$
hence,
$$\delta = 15^2-4\times 13\times 9=-243<0$$
so the quadratic isn't reducible over $\mathbb R$.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Intuition behind proof of Schwarz's lemma There is the very well known proof of Schwarz's lemma in complex analysis. When I read it I feel like the answers described here. I'm not sure how I would motivate and explain why one should expect the proof to work when talking to someone new to complex analysis despite the pr... | Schwarz lemma is a profound result of non-euclidean geometry as it says that conformal maps that preserve the unit disc decrease hyperbolic distance on the unit disc; there is a book by S. Dineen called precisely that (The Schwarz Lemma, Clarendon Press, 1989, reprinted in an inexpensive pb by Dover Press) that goes in... | {
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Positive-definite over $\mathbb Q$ form is positive definite over $\mathbb R$? I was reading P. Etingof's "Introduction to the representation theory" when I found this problem (and I've trouble with it): we have a quadratic form $Q(x) = \sum x_i^2 - \frac{1}{2}\sum b_{ij}x_ix_j$, $b_{ij} \in \mathbb N$ (if it can be im... | Suppose this matrix is positive semidefinite over $\mathbb{R}$, but not positive definite. Then it has a zero eigenvalue. Now show that at least one eigenvector corresponding to this eigenvalue has rational coordinates, since the form itself has rational elements. This implies that this matrix is also not positive defi... | {
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"A manifold with boundary has dimension at least 1" if it has a dimension and if it has nonempty boundary? My book is An Introduction to Manifolds by Loring W. Tu.
As can be found in the following bullet points
*
*Can a topological manifold be non-connected and each component with different dimension?
*Is $[0,1) \c... | Assuming sensible definitions, an alternative solution is to change the statement to the following:
A connected manifold with non-empty boundary has dimension at least 1
Edit: I rejected the suggested edit to change "manifold with non-empty boundary" to "manifold with boundary with non-empty boundary" because it does... | {
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"timestamp": "2023-03-29T00:00:00",
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How to solve this quadratic congruent equation by inspection I found a systematic way (c.f. How to solve this quadratic congruence equation) to solve all congruent equations of the form of $ax^2+bx+c=0\pmod{p}$, or to determine that they have no solution.
But I wonder if there is some easy way to find solutions of simp... | Servaes gave a good general method for solving quadratic equations modulo a prime.
I will elaborate on my comment about how in this particular case the answer could be found by inspection.
Adding or subtracting a multiple of $p$ to or from any of the coefficients $a,b,c$
does not materially change the equation $ax^... | {
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Why are |vertical lines| used to mark matrix determinants? This notation is sometimes used to denote the determinant:
$$ \begin{vmatrix}a & b \\ c & d\end{vmatrix} = ad-bc$$
Why? Where did this notation come from? Was there any relationship between this notation and the absolute value $|x|$ or the norm $\lVert\mathbf{x... | There is a relationship between the vertical line notation for determinant and the notation $|x|$ for absolute value and $||\mathbf{x}||$ for norm, however I do not know whether this was an intentional decision historically. The absolute value, norm, and determinant, all have at least two things in common.
*
*They a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3221128",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 1
} |
Why Does Adding the nth Derivative Increase a Function Approximation's Accuracy? I am currently taking calculus 3: sequences and series, and we've just started learning about Maclaurin and Taylor Series. I understand the concept behind them -- of these polynomials and derivatives of polynomials.
However, I do not unde... | Consider a polynomial function $P$ and let $x_0$ be a real number. Then $P(x_0)$ is the value that $P(x)$ takes at $x_0$ and, since polynomial functions are continuous, whe $x$ is close to $x_0$, then $P(x)$ is close to $P(x_0)$.
Now, consider the polynomial $P_1(x)$ of degree $1$ such that $P_1(x_0)=P(x_0)$ and that $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3221237",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
How do I test for convergence of $\Sigma_{n = 2}^{\infty} \frac{log(n)}{n \sqrt{n + 1}}$ I was trying to solve this problem.
Test $\Sigma_{n = 2}^{\infty} \frac{log(n)}{n \sqrt{n + 1}}$ for
convergence or divergence .
But I couldn't quite make a lot of progress. Here's what I tried.
*
*I tried applying integral ... | $\sum_{n = 2}^{\infty} \frac{log(n)}{n \sqrt{n + 1}}
$
The basic fact needed is that
$\dfrac{\ln(n)}{n^a}
\to 0$
as $n \to \infty$
for any $a > 0$.
Setting $a= 1/4$,
$\dfrac{\ln(n)}{n^{1/4}}
\to 0$
so that
$\dfrac{\ln(n)}{n^{1/4}}
\lt 1$
for all large enough $n$.
Therefore,
for all large enough $n$,
$\dfrac{\ln(n)}{n\s... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3221376",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Solve $\sqrt{3}\sin(x)+\cos(x)-2=0$ I need to solve the equation $$\sqrt{3}\sin(x)+\cos(x)-2=0$$
My try: I separated the radical then I squared and I noted $\cos(x)=t$ and I got a quadratic equation with $t=\frac{1}{2}$
To solve $\cos(x)=\frac{1}{2}$ I used formula $x\in\left \{ +-\arccos(\frac{1}{2})+2k\pi \right \}k\... | Your mistake was forgetting that when you square a non-$0$ equality, the result satisfies two possible equations. In particular, both $x = y$ and $x = -y$ gives that $x^2 = y^2$. From what you describe, you did the following:
$$\sqrt{3}\sin(x) = 2 - \cos(x) \tag{1}\label{eq1}$$
then square both sides to get
$$3\sin^2(x... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3221494",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
Averaging i.i.d. variables: Equal chance to be right and left of the mean? Let $\{X_i\}_{i=1}^{\infty}$ be i.i.d. random variables. Define
$$ L_n = \frac{1}{n}\sum_{i=1}^n X_i \quad \forall n \in \{1, 2, 3, …\} $$
Using the central limit theorem, it can be shown that if $E[X_i]=0$ and $0<Var(X_i)<\infty$ then:
$$ \li... | Yes it is possible for $c$ to take any value strictly between $0$ and $1$. The point is that there exist mean-zero stable distributions which are not symmetric about $0$ (of course, such a stable distribution cannot be Gaussian, and so it must have infinite variance). You may look at the Wikipedia page to see how some ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3221659",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 2,
"answer_id": 0
} |
$3$ mice in the corners of an equilateral triangle crawling at each other ODE problem At each corner of an equilateral triangle, a mouse is positioned. At time $t = 0$ the mice begin crawling towards each other. Mouse $1$ always crawls directly towards mouse $2$, mouse $2$ towards mouse $3$ and mouse $3$ towards mouse ... | Let $\omega:=e^{2\pi i/3}$. The three mice always form an equilateral triangle. Therefore we may write
$$z_k(t)=\omega^k \>z(t),\qquad z(0)=1\ .$$
According to the last sentence we have
$$\dot z_k(t)=z_{k+1}(t)-z_k(t)=(\omega-1)z_k(t)\ ,$$
so that we obtain the IVP
$$\dot z(t)=(\omega-1)\>z(t),\qquad z(0)=1$$
with solu... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3221828",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Flea on infinite chessboard jumping with irrational vector eventually changes square color Question from Engel's Problem Solving Strategies:
An infinite chessboard consists of $1 \times 1$ squares. A flea starts on a white square and makes jumps by $\alpha$ to the right and $\beta$ upwards, where $\alpha$ and $\beta$ ... | I'm not quite sure about the coordinate system you're using. If your $(0,0)$ is meant to be the bottom left corner of the first white square, then since it's on the corner, it's also on the boundary, so it could in theory be considered to be any of the $4$ squares at that corner.
To avoid any such issues, I will define... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3222010",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
$1/(51 +u^2) + 1/(51 +v^2) + 1/(51 +w^2)$ inequality I need to prove that $$1/(51 +u^2) + 1/(51 +v^2) + 1/(51 +w^2) \leq 1/20$$ given $u + v + w = 9$ and $u,v,w$ positive reals.
Using AM-HM inequality with $(51 +u^2, 51 +v^2, 51 +w^2)$ I arrive at
$(51 +u^2 + 51 +v^2 + 51 +w^2)/3 \geq 3/((1/(51 +u^2) +1/(51 +v^2) 1/(5... | You're doing well. Note that you have $u^2+v^2+w^2$ in the denominator, and $1/x$ is a decreasing function. That means that
$$ \left( u^2+v^2+w^2 \ge \frac{(u+v+w)^2}{3} \right) \Rightarrow \left( \frac{9}{153+u^2+v^2+w^2} \le \frac{9}{153+\frac{(u+v+w)^2}{3}} \right)$$
EDIT: The answer is not correct, because the ineq... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3222160",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
Linearly independent vectors each subtracted by a linear combination of them are linearly dependent if coefficients add up to $1$ The task
I've been given the following problem:
Let $v_1, \ldots, v_n$ linearly independent vectors in an $\mathbb{F}$-vector space $V$ and $u = \lambda_1 v_1 + \ldots + \lambda_n v_n$ a li... | If you consider the matrix having as columns the coordinates of the new vectors with respect to the given linearly independent set (everything takes place in the subspace of which $\{v_1,\dots,v_n\}$ is a base):
\begin{bmatrix}
1-\lambda_1 & -\lambda_1 & \dots & -\lambda_1 \\
-\lambda_2 & 1-\lambda_2 & \dots & -\lambda... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3222596",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Sobolev imbedding theorem $H^{1,p}(\mathbb{R}^n)$ contained in $L^{{np}/(n-p)}(\mathbb{R}^n)$ (Taylor Michael)
i. I can not make sense of the following:
For (2.4) I imagine that it is the fundamental theorem of the calculation but I can not prove it formally.
Neither will it be understood how to arrive at equation 2... | Observe by the fundamental theorem of calculus, you have
\begin{align}
u(x, \mathbf{x}_{n-1})=u(x, \mathbf{x}_{n-1})-u(x_0, \mathbf{x}_{n-1}) = \int^x_{x_0} D_1u(x_1, \mathbf{x}_{n-1})\ dx_1
\end{align}
if $(x_0, \mathbf{x}_{n-1})\notin \operatorname{supp} u$ which means
\begin{align}
|u(\mathbf{x})| \leq \int^\infty_{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3222774",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Prove a property of Legendre symbol In case someone does not know the definition, I first write down the definition.
Def Let $a$ be s.t. $(a,m)=1$. Then we say $a$ is a quadratic residue modulo m if the congruence $x^2\equiv a$ (mod $m$) has a solution. If it has no solution, then we say $a$ is a quadratic nonresidue m... | Observe that both $\left(\frac{a}{p}\right)\left(\frac{b}{p}\right)$ and $\left(\frac{ab}{p}\right)$ are equal to $0,1$ or $-1$. Therefore if they are congruent modulo a prime $p>2$, they are necessarily equal.
As for the second question, if $a$ is positive you can write it as a product of primes $a=q_1\dots q_k$, and ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3222908",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Prove the following inequality involving a sum. Suppose that $m,n, q\in \mathbb{N}$ such that
$$\lambda_{n,m}=\frac{m+1/2}{(m+1/2)^2−n^2} \text{ and } \sigma_{q,m} = \sum_{k=0}^q \lambda_{k,m}.$$
Furthemore we also know that,
$$\frac{\lambda_{0,m}}{2}+\sum_{n=1}^{\infty}\lambda_{n,m}=0.$$
Then I want to show that when ... | As $\sigma_{q,m}$ is decreasing, we have $\sigma_{q,m} > \sigma_{q+1,m} > \sigma_{q+2,m} > ...$. So, as $q\to\infty$, we get $\sigma_{q,m} > \frac{\lambda_{0,m}}{2}$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3223045",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Different methods give different answers. Let A,B,C be three angles such that $ A=\frac{\pi}{4} $ and $ \tan B \tan C=p $. Let $A,B,C$ be three angles such that $ A=\frac{\pi}{4} $ and $ \tan B \tan C=p $. Find all possible values of $p$ such that $A,B$ and $C$ are angles of a triangle.
case 1- discriminant
We can re... | The reason why the 1st method is wrong :
We can rewrite the following equation
$ f(x) = x^2 - (p-1)x + p $
As we know the sum and product of $ \tan A $ and $ \tan B $
Settings discriminant greater than equal to zero.
It seems that you meant "the sum and product of $\color{red}{\tan C}$ and $\tan B$".
Considering the ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3223162",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 7,
"answer_id": 4
} |
Area of Generalized Koch Snowflake In the Koch snowflake, the zeroth iteration is an equilateral triangle, and the n-th iteration is made by adding an equilateral triangle directly in the middle of each side of the previous iteration. The area of the Koch snowflake is $8/5$ the area of the starting triangle.
If I wante... | As first conjectured by OP, for the hexagon ($n = 6$) case, the area of the generalized Koch snowflake indeed equals to $\frac{12}{5}$ of that of the seed hexagon.
This comes down to following observation. When one scale the seed hexagon to make its area two-third of that of a seed triangle, the generalized Koch snowfl... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3223258",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Sections of the projection onto the monoid of the path-conncted components Let $M$ be a topological monoid and $A:=\pi_0(M)$ be the monoid consisting of its path connected components. Assume that $A$ is countable. We have a monoid homomorphism $\pi\colon M\longrightarrow A$. I would like to check that there exists at l... | This is false. For instance, let $M=\bigcup_{n\in\mathbb{Z}}\{n\}\times[|n|,\infty)$, which is a topological monoid under coordinatewise addition. The monoid $A$ of path-components is just $\mathbb{Z}$, with $\pi:M\to A$ being the first projection. But there does not exist any nontrivial monoid homomorphism $A\to M$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3223397",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Calculate $2^{2^{2^{\cdot^{\cdot^{2}}}}} \mod 2016$ How to find $2^{2^{2^{\cdot^{\cdot^{2}}}}} \mod 2016$ where $2$ occurs $2016$ times?
My current observations:
$$2^{11} = 2048 \equiv 2048=2016 \equiv 2^5 $$
and
$$ 2^{16} \equiv 2^{11}\cdot 2^5 \equiv 2^{10} $$ and now we have $2012$ of "2" left...
| Hint: $2016 = 2^5 \cdot 3^2 \cdot 7$. Consider it separately mod $2^5$, mod $3^2$ and mod $7$, and combine using the Chinese Remainder Theorem.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3223544",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Proof involving rational functions on an elliptic curve I am working through understanding the proof I posted below, and I have questions about 3 parts. (Note, the proof assumes results developed before, but I don't have questions on those. My questions are on things in the proof that are "common knowledge" mathematics... | Q1: The behaviour of $f(x)=\frac{\alpha(x)}{\beta(x)}$ at $\infty$ can be studied as behaviour of $g(x):=f(1/x)$ at $x=0$. If $\deg\alpha>\deg\beta$, then $\alpha(1/x)$ has a higher order pole than $\beta(1/x)$, meaning that $g(x)$ has a pole at $x=0$ and $f(x)$ is note defined at $x=\infty$.
Q2: I'll give some illustr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3223670",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Why $ b =1 $ solves the contradiction in this proof? The proof is given below:
Theorem 3.3 (Pythagoras). The number $\sqrt{2}$ is irrational.
Proof. Suppose, to the contrary, that $\sqrt{2}$ is a rational number, say $\sqrt{2}=a/b$, where $a$ and $b$ are both integers with
$\gcd(a,b)=1$. Squaring, we get $a^2=2b^2$,... | If $a^2 = 1$, then $a = 1$ or $a = -1$. In the latter case, $b$ would also need to be negative for their ratio to be positive; so, let us pursue the former case without loss of generality.
If $a=1$, then we would have $1/b = \sqrt{2}$ for some positive integer $b$. But, squaring both sides would then yield that $1/b^2 ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3223801",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Determine the slope and write the Cartesian equation of the line. Write the equation of the line by the origin of coordinates that has vector director of components (1,2) determine the slope and write the Cartesian equation of the line.
My attempt:
Let $l$ a line such that pass for the origin.
Let $a$ a director vect... | I take it you want the line through the origin and $(1,2)$. The equation is $y=2x$. This is in slope-intercept form (slope $2$, $y$-intercept $0$). The slope is $m=\dfrac {\Delta y}{\Delta x}=\dfrac{2-0}{1-0}=2$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3223962",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
} |
Can the radius of convergence of a sum of two power series be an arbitrary number? Let $\sum_{n=0}^\infty a_n z^n,\sum_{n=0}^\infty b_n z^n$ be two series with the same radius convergence $R>0$. Can the radius of convergence of their sum be any positive real number which is greater than $R$?
| Yes, it can. If $\sum_{n=0}^\infty a_nz^n$ has radius of convergence $R\in(0,\infty)$, and if $r>R$, consider the series $\sum_{n=0}^\infty(-a_n+r^{-n})z^n$, whose radius of convergence is also $R$. But the radius of convergence of the sum of both series is $r$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3224131",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
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