Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Is $\frac{d}{dx} \ln(\phi(x))$ decreasing where $\phi$ is the Gaussian cumulative distribution function? I was plotting the function $x \mapsto \frac{d}{dx} \ln(\phi(x))$ and it seams like this is a decreasing function. But somehow I cannot prove this. Here, $\phi$ is the Gaussian cumulative distribution function, i.e.... | $(ln (\phi(x))'=\frac {\gamma (x)} {\phi(x)}$. In view of smoothness this function is decreasing iff its derivative is $\leq 0$. So we require $\phi \gamma' \leq \gamma^{2}$ or $\phi \leq \frac {\gamma^{2}} {\gamma'}$. Look at the behavior at $\infty$ to see that this is false.
| {
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Why do Bourbaki define the characteristic of a ring the way they do? Here is the definition of the characteristic of a ring $R$ that is common in everyday usage (for example in Lang's Algebra and Wikipedia): take the unique homomorphism
$$
\mathbb{Z} \to R
$$
and define $\operatorname{char}(R)$ to be the smallest nonne... | I can't speak to their motivation, but it seems that a little later on they remark that a quotient of a ring has the same characteristic as the original ring, which would be false if they allowed rings not containing fields.
| {
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If $R_1$ and $R_2$ have the same cardinality, then $R_1 = R_2$ In proving one direction (the part at most) of this statement from Wikipedia
I re-phrase it as below theorem. Could you please leave me some hints (not involved modular arithmetic) to prove it.
Let
*
*$0<p<q < N$ be natural numbers.
*$R_1$ be the se... | HINT.
Let $a=\gcd (p,N).$ Let $p=p'a$ and $N=N'a.$ Then $\gcd(p',N')=1.$
Now $p'ax=px=Ny+r=N'ay+r$ with $r\in R_1$ iff $r=r'a$ for some $ r'\in R'_1$, where $R'_1$ is the set of remainders when multiples of $p'$ are divided by $N'.$
So $R_1=\{ar': r'\in R'_1\}.$
Use $\gcd(p',N')=1$ to obtain $R'_1=\{0,..., N'-1\},$ wh... | {
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Parametrization the curve of intersection of sphere $x^2+y^2+z^2=5$ and cylinder $x^2+\left(y-\frac{1}{2}\right)^2=\left(\frac{1}{2}\right)^2$. I've seen similar posts here but none of the answers helped me. I am trying to parametrize a curve of intersection of a (top half $z>0$) sphere $x^2+y^2+z^2=5$ and cylinder $x^... | A third alternative is using Stokes' theorem, parameterize the surface enclosed by the curve and integrate... And at first it seems like you are going to end up with a nicer integral
\begin{eqnarray}
&& \int_0^\pi d\phi \int_0^{\arcsin\left[\frac{1}{\sqrt{5}}\sin(\phi)\right]} d\theta \, \left[5 \sin(\theta)\right] (\n... | {
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Small Span Theorem & Understanding Span It seems I'm a bit confused about the concept of span. The definition of span makes complete sense to me (i.e. set of all linear combinations of vectors), but I'm confused how this works in the context of the Small Span Theorem:
Let $f$ be continuous on $\left [a, b \right]$. Gi... | What I'm understanding by "span" in this context seems to be the range of $f$ on a particular interval of a partition. Basically what the theorem is saying is that for a continuous function, you can always partition it in such a way that the maximum deviation of $f$ within any given partition is at most $\epsilon$
This... | {
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$1996$ Austrian-Polish Number theory problem
Let $k \ge 1$ be a positive integer. Prove that there exist exactly $3^{k-1}$ natural numbers $n$ with the following properties:
(i) $n$ has exactly $k$ digits (in decimal representation),
(ii) all the digits of $n$ are odd,
(iii) $n$ is divisible by $5$,
(iv) the number $... | As you found out yourself (with the exception of the $+1$ mentioned by user3052655, coming from dividing $a_k=5$ by $5$), we have
$$\begin{eqnarray}
m=n/5 & = & 2(10^{k-2}a_1+10^{k-3}a_2+...+a_{k-1})+1 = \\
& = & 10^{k-2}(2a_1)+10^{k-3}(2a_2)+\ldots+10^1(2a_{k-2})+(2a_{k-1}+1).
\end{eqnarray}$$
If you look at th... | {
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For $x^a = e^2x$. Determine the solutions as a power of$ e$ For $x^a = e^2x$. Determine the solutions as a power of $e$
One obvious solution is $0$ but I fail to find the second solution.
$x^a - e^2x = 0$
$x(x^{a-1} - e^2) = 0$
I am not sure how to find the other solution please help
| Let $x$=$e^k$
Then, $$\therefore e^{ka}=e^{k+2}$$$$\Rightarrow ka=k+2$$$$\Rightarrow k= \frac{2}{a-1}$$$$\therefore x=e^{\frac {2}{a-1}}$$
| {
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Prove that there exist infinitely many primes $p$ such that $13 \mid p^3+1$ $\textbf{Question:}$Prove that there exist infinitely many primes $p$ such that $13 \mid p^3+1$
I could easily see that the given is equivalent to showing that there are infinitely many primes $p$ such that, $p \equiv \{4,10,12\} \pmod{13}$
T... | Yes, there is an elementary proof which requires nothing more than quadratic reciprocity.
Thanks to @lhf for the pointer to Murty’s classical paper on this. I had heard about it but never seen the proof before, and the result of Schur cited in the paper was enlightening. Murty’s paper is already sufficient to answer ... | {
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$\det(I+A)=1+\operatorname{Tr}(A)$ if $\operatorname{rank}(A)=1$
Let $A$ be a complex matrix of rank $1$. Show that $$\det (I+A) = 1 + \operatorname{Tr}(A)$$ where $\det(X)$ denotes the determinant of $X$ and $\operatorname{Tr}(X)$ denotes the trace of $X$.
Any hint, please. I do not get how to combine the ideas of r... | Assume $A$ is diagonal. As its rank is $1$, it has only one non-zero eigenvalue $\lambda$. Then
$$ \det(I+A) = 1+\lambda = 1 + \mathrm{tr}\, A. $$
Assume that $A$ is diagonalisable, so that $A = PDP^{-1}$. Then
$$\det(I + A) = \det(I + PDP^{-1}) = \det(P(I + D)P^{-1} ) = \det(I + D)$$
Similarly
$$1 + \mathrm{tr}\,A = ... | {
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Closest matrix that achieves positive semidefinite condition Suppose we have two symmetric positive semidefinite $n$ dimensional matrices $A$ and $B$. We use the notation $X\leq Y$ means that $Y-X$ is positive semidefinite.
Suppose $A \not\leq B$ i.e. $B-A$ has at least one negative eigenvalue. We are interested in per... | Some thoughts on the problem:
As a further simplification, I suggest that we say that $\tilde A$ does not only satisfy $\tilde A \leq B$, but also has a rank of $1$. If your hypothesis is correct, then this assumption should not change our answer. Write
$$
A = \alpha xx^T, \quad \tilde A = \beta yy^T
$$
for some scala... | {
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Hasse's Theorem: min and max values with or without point of infinity? I have to calculate the min and max values of a field.
Min: $\lfloor{q+1-2 \sqrt{q}}\rfloor$
Max: $\lfloor{q+1+2 \sqrt{q}}\rfloor$
According to Hasse. BUT the exercise says that min and max should be found together with the point of infinity. So sho... | Hasse's theorem is usually stated as
$$ | \# E(\mathbb{F}_q) - (q + 1) | \leq 2\sqrt{q}$$
When we talk about the points on an elliptic curve $E/K$ where $K$ is a field, we are always talking about the points on the projective curve (that is, including the point at infinity). Thus if $E/\mathbb{F}_q$ is given by
$$f(x,y... | {
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Evaluate using Residues $\int_0^{2\pi}\frac{d\theta}{1+a\sin\theta}$ I need to evaluate the following using residues: $\int_0^{2\pi}\frac{d\theta}{1+a\sin\theta}$ where $-1<\theta<1$.
I suppose the $a$ in front of $\sin\theta$ is throwing me off. I was thinking I could let $z=e^{i\theta}$ and so $\sin\theta=\frac{z-z^... | I guess nothing prevents you from exploiting some symmetry before switching to the computation of residues.
$$\int_{0}^{2\pi}\frac{d\theta}{1+a\sin\theta}=\int_{0}^{\pi}\frac{d\theta}{1+a\sin\theta}+\int_{0}^{\pi}\frac{d\theta}{1-a\sin\theta}=\int_{0}^{\pi}\frac{2\,d\theta}{1-a^2\sin^2\theta}$$
equals
$$ 4\int_{0}^{\pi... | {
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A non-empty subset of integers bounded above has a maximum Suppose the set $\mathrm A$ $\neq$ $\emptyset$ , $\mathrm A$ $\subseteq$ $\Bbb Z$ is bounded above. Then since $\Bbb Z$ $\subseteq$ $\Bbb R$, I know that by the completeness axiom there exists a $supremum$ for the set $\mathrm A$, say $s$ $=$ $sup$($\mathrm A $... | If $A$ is your subset of $\mathbb{Z}$, consider the identity function, $i:A\rightarrow \mathbb{R}$. $f(A)$ is closed since $i$ is continuous and $A$ is closed, and $f(A)$ is bounded since there is an $M$ such that each term of the sequence is bounded, so $|x_n| \le |i(x_n)| \le M$ for all $n$. Therefore, $f(A)$ is com... | {
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Find the number of functions $4f^3(x)=13f(x)+6$
$f(x)$ is a function satisfying
$$4f^3(x)=13f(x)+6$$
$\forall\ x\in [-3,3]$ and is discontinuous only and only at all integers in $[-3,3]$. If $N$ denotes the unit digit of the number of all such functions, find $N$.
Answer: $6$.
Solving the cubic for $f(x)$ gives us
$$... | The following are the properties that fully characterize your function $f$:
*
*For every $x$, $f(x)\in \{2,-1/2,-3/2\}$
*$f$ must be constant on every open interval $(n,n+1)$ for $n=-3,-2,-1,0,1,2$
*among $f(n-1,n)$, $f(n)$, $f(n,n+1)$ there must be at least two different values for $n=-2,-1,0,1,2$
*$f(3) \ne f(2,... | {
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Proving that this formula all over the positive integer gives us this sequence Firstly, we have this sequence : $1,1,1,1,1,1,1,1,1,1,2,2,2,...$ which is the sequence of the number of digits in decimal expansion of $n$.
Secondly, we have this formula : $$a_n=\Bigl\lceil\log_{10}(n+1)\Bigr\rceil-\Bigl\lceil\frac{n}{n+1}\... | Firstly, for $n \ne 0$, $n$ has $d$ digits if and only if $10^{d - 1} \le n < 10^d$. One way to find $d$ from here is to take logarithms, using $d - 1 \le \log_{10}(n) < d$, so $d = \lfloor \log_{10}(n) \rfloor$.
However, this expression doesn't work so well when $n = 0$, so we have to be a bit creative. Adding $1$, we... | {
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$\frac{x^2}{by+cz}=\frac{y^2}{cz+ax}=\frac{z^2}{ax+by}=2$ If $$\frac{x^2}{by+cz}=\frac{y^2}{cz+ax}=\frac{z^2}{ax+by}=2$$ then find the value of $$\frac{c}{2c+z}+\frac{b}{2b+y}+\frac{a}{2a+x}.$$
I think all the terms need to be manipulated in some way to get the corresponding terms from the expression whose value needs ... | Hint:
$$x(x+2a)=x^2+2ax=2(ax+by+cz)$$
$$\dfrac1{2a+x}=?$$
$$\dfrac a{2a+x}=\dfrac{ax}{2(ax+by+cz)}$$
| {
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Dimension of a vector space/ subspace with a finite basis Is the dimension of a vector space/subspace with a finite basis always the same as the number of elements in each vector and if so how can I derive that from the definition of a dimension?
| "Elements" of a vector is not something that makes sense in an abstract vector space. It does make sense when considering vectors in $\Bbb{R}^n$ (or more generally, $\Bbb{F}^n$ for some field $\Bbb{F}$), but not for more general vectors (e.g. functions).
Even if we restrict ourselves to vectors in $\Bbb{R}^n$, the numb... | {
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Continuously compounded interest with additional monthly deposits What is the equation for a continuously compounded with monthly additions of $300$ dollars for the first $10$ years and $500$ for the next $20$ with an initial investment of $0$?
I know the equation $Pe^{rt}$ is used but i don't understand how to set it ... | Calling the monthly savings $C,$ i.e. ($C=\$300$) and the monthly interest $m=r/12$ (divided by $12$ because time will be given in months, and presumably the interest rate $r$ is annual); and with the number of payments (for the first part of the problem $n=10[\text{years}]\times 12 [\text{months}]=120,$ it would seem ... | {
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Evaluate $\sqrt{a+b+\sqrt{\left(2ab+b^2\right)}}$ Evaluate $\sqrt{a+b+\sqrt{\left(2ab+b^2\right)}}$
My attempt:
Let $\sqrt{a+b+\sqrt{\left(2ab+b^2\right)}}=\sqrt{x}+\sqrt{y}$
Square both sides: $a+b+\sqrt{\left(2ab+b^2\right)}=x+2\sqrt{xy}+y$
Rearrange: $\sqrt{\left(2ab+b^2\right)}-2\sqrt{xy}=x+y-a-b$
That's where my ... | Although it is not clear what range of values is acceptable for the variables, I believe I have a sense of what you are trying to do here. This is called denesting square roots.
In particular, if $a,b,c$ are positive real numbers such that $a^2-b^2 c$ is non-negative, then it holds that
$$\sqrt{a+ b\sqrt{c}}=\sqrt{\fra... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Show $\det(F_n)=1$ for all $n$ Consider the $n\times n$ matrix $F_n= (f_{i,j})$ of binomial coefficients
$$f_{i,j}=\begin{pmatrix}i-1+j-1\\i-1\end{pmatrix}$$
Prove that $\det(F_n)=1$ for all $n$.
My current idea is to apply Leibniz formula for determinants and induction, but it seems too complicated. Any better ideas ... | Using the formula $\binom{n}{k}=\binom{n-1}{k}+\binom{n-1}{k-1}$ you get, by applying the column operations
$$
\left\{
\begin{array}{lcl}
C_n&\gets& C_n-C_{n-1} \\[1mm]
C_{n-1}&\gets& C_{n-1}-C_{n-2} \\[1mm]
&\vdots\\[1mm]
C_2&\gets& C_2-C_{1}
\end{array}
\right.
$$
that
$$\det(F_n) = \det\left(
\begin{array}{c|ccc}
1&... | {
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High probability range of chromatic number Prove that there is an absolute constant $c$, for every $n>1,$ there is an interval $I_{n}$ of at most $c \sqrt{n} /$ log $n$ consecutive integers such that the probability that the chromatic number of $G(n, 0.5)$ lies in $I_{n}$ is at least 0.99.
(From Noga Alon, Joel H. Spen... | I wrote this solution when I worked through this amazing book about a year ago!
Let $\epsilon > 0$ be arbitrary for the moment. We describe three events that each happen with probability at least $1-\epsilon$ for large $n$.
Given $G\sim G(n,1/2)$, let $u$ be the smallest positive integer such that
\begin{equation}
... | {
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Can this inductive proof that $\sum_{i=0}^n2^{2i+1}=\frac23(4^n-1)$ be simplified? The general structure of equations I've used for the inductive step for proofs with a summation is something like:
We'll prove that $\sum_{i = 0}^{n + 1} (\text{something}) = (\text{closed form expression})$
\begin{align}
\sum_{i = 0}^{n... | $$\frac{2}{3}(4^n - 1) + 2^{2n + 1}= \frac23\left(\color{red}{4^n}-1+\overbrace{\color{red}{3\cdot2^{2n}}}^{=3\cdot4^n}\right)=\frac23\left(\color{red}{4\cdot4^n}-1\right) = \frac{2}{3}(4^{n + 1} - 1)$$
| {
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Calculating The Vertex Of a Square That Circumscribed Ellipse Hello everyone how can I find the vertex of a square that circumscribed the ellipse $\frac{x^2}{9}+y^2 =1$?
I tried to mark the vertex at $(u,v),(-u,v),(u,-v),(-u,-v)$ and use the equation to calculate the tangent lines to the ellipse by the vertex points, b... | It is possible to give a very quick answer if we use this nice property of the ellipse:
the locus of the intersections of perpendicular tangents to an
ellipse is a circle called director circle,
and the square of its radius is the sum of the squares of
the ellipse semi-axes.
Tangents drawn from any point on the direc... | {
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Proving that the Average Product function attains max value at the point where the Marginal Product is equal to the Average Product. I am basically trying to prove the following excerpt from my Microeconomics textbook:
When Average Product is maximum, Marginal Product is equal to the Average Product.
The Total Produc... | This answer is to supplement the answer by Trurl and reply to the query about the significance of the sign of the derivative of marginal product.
The mathematical result you need to apply is:
Let $f$ be a differentiable function of a single variable defined on
the interval $I$. If a point $x$ in the interior of $I$ is... | {
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What Is Bigger $\frac{3}{e}$ or $\ln(3)$ Hello everyone what is bigger $\frac{3}{e}$ or $\ln(3)$?
I tried to square it at $e$ up and I got:
$e^{\frac{3}{e}} = \left(e^{e^{-1}}\right)^{3\:}$ and $3$ but I don't know how to continue I also tried to convert it to a function but I didn't find.
| Hint:
Compare the logs and use that $\ln $ is concave, hence its representative curve is below each of its tangents.
| {
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Natural deduction proof that $(P\leftrightarrow \neg P)$ is a contradiction, without first deriving $(P\vee \neg P)$ I'm looking to prove that $(P \leftrightarrow \neg P)$ is a contradiction using a natural deduction proof (which is to say, I want a proof to show $(P\leftrightarrow \neg P)\vdash Q$). In case it helps, ... | (Posted after answer accepted.) Using a form of natural deduction:
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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rigorous proof of simple phenomenon Suppose we want to prove the following simple statement rigorously -
If the number of beads in a necklace is $n$ then there exists a colored (suppose we have infinitely many colors at our disposal) necklace of minimal period $k$ (rotations) iff $k|n$.
Intuitively this seems obvious, ... | If $k\not|n$ and the period is $k$, then take every other $k^{th}$ beads. After several turns, you will get $\dfrac n{\gcd(k,n)}$ different positions, spaced by $\dfrac{k}{\gcd(k,n)}$, holding beads of the same color. Hence the period is $\gcd(k,n)<k$, a contradiction.
E.g. $k=6,n=15\to\gcd(k,n)=3$.
The beads $0,6,12,3... | {
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Help in understanding integrating a function with an absolute value My math is rusty, and although I initially thought I understand the solution, upon further examination I think I don't:
That's the original function:
$$ \Psi(x,t) = A \mathrm{e}^{-\lambda|x|} \mathrm{e}^{-\mathrm{i} \omega t} $$
\begin{align*} \lang... | Your reasoning seems correct to me. There's some dot/inner product of functions defined there (in your particular context).
https://mathworld.wolfram.com/InnerProduct.html
They just use this particular definition of the inner product, and calculate $\langle x,x\rangle$ i.e. $\langle x^2\rangle$
The rest of your reason... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3739379",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Showing a function is convex/concave given another function is concave Assuming $f(x)$ is not a linear function, given a concave decreasing function $f(x)$, I want to find whether $g(x)=f(x)\cdot x$ is convex or concave for strictly positive $x$.
However, I'm having a trouble proving it mathematically.
Since $f(x)$ is ... | If $f$ is twice differentiable and concave then $f''(x)\leq 0$. And $f$ decreasing implies $f'(x)\leq 0$. Setting $g(x)=xf(x)$, we have
\begin{align*}
g''(x)=2f'(x)+xf''(x)\leq 0
\end{align*}
for all $x>0$. It follows that $g$ is concave for all $x>0$.
If the assumption of $f$ twice differentiable is too stringent for ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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What are some questions that seem easy but aren't? I've only just started to study a degree in Mathematics and I find it extremely satisfying to have a seemingly easy question that is incredibly difficulty or tricky to work out. I've looked into heaps of competition exams (although they aren't really what I'm looking f... | There are some questions that are easy to comprehend, but for which we have no solution yet, period. I don't know if that is what you're looking for, or if you only want problems that do have a solution, but that hard to find, but here are some problems of the first kind:
Goldbach Conjecture: Every even number greater ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3739653",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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the simplest non-trivial line bundle over Riemann sphere We define Riemann sphere as $S=\mathbb{C}^2-\{0\}/\sim$. Given a point $p$ over $S$, I have seen somewhere there exists a line bundle $L_p$ associated to $p$, and $L_p$ has a non-zero holomorphic section with only one zero at $p$.
I think this construction is wel... | For simplicity let's take $P = [0: 1]$.
Cover $S$ with the two opens $U_0 = S \setminus [1: 0]$ and $U_\infty = S \setminus [0:1]$.
Now take the disjoint union of two copies of the trivial bundle:
$M = (U_0 \times \mathbb{C}) \sqcup(U_\infty \times \mathbb{C})$
We impose an equivalence relation on $M$ by the rule $([a:... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3739811",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Obtaining eigenvalues of a complex matrix Let
$$A = \begin{bmatrix} 0 & i+1 & 0 \\ i & 0 &2 -i \\ 2-i & 0 & i \end{bmatrix}$$
and eigenvalues, eigenvectors are asked.
*
*by the frequently used method ,we get the characteristic equation as $$\lambda^{3} - i (1 + \lambda )\lambda + \lambda - 8 = 0$$ no way to solve th... | The characteristic polynomial is given by
$$
\chi(t)=t^3 - it^2 + ( 1 - i)t - 8
$$
and the roots $z_1,z_2,z_3$ satisfy
$$
z_1+z_2+z_3=i,\; z_1z_2z_3=8.
$$
So we have $z_3=i-z_1-z_2$ and $z_1z_2(i-z_1-z_2)=8$.
This gives a quadratic equation.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3739959",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Show that if $X$ is absolutely continuous and $g$ is absolutely continuous on bounded intervals, then $g(X)$ is absolutely continuous. I'm taking a course on probability, and the following question showed up in the reference textbook:
Show that if $X$ is absolutely continuous with pdf $f_X(\cdot)$ and $g$ is absolutely... | Maybe a hint: Since $g$ is invertible under these assumptions,
$
F_Y(t) = P( Y \le t) = P(X \le g^{-1}(t)).
$
Calculate the derivative and in the chain rule apply the inverse function theorem of calculus.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3740077",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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Suppose every element of $\mathcal F$ is a subset of every element of $\mathcal G$. Prove that $\bigcup \mathcal F\subseteq \bigcap\mathcal G$. Not a duplicate of
Prove that if F and G are nonempty families of sets, then $\bigcup \mathcal F \subseteq \bigcap \mathcal G$
Validity of this proof: Prove that $\cup \mathcal... | Your proof is correct! Also I like it how you justify everything. You can also check this alternative approach:
Proof. Take $A \in \mathcal F$. Then, as $A \subseteq B$ for all $B \in \mathcal G$,
$$A \subseteq \bigcap_{B \in \mathcal G} B = \bigcap \mathcal G.$$
As $A$ was arbitrary, it follows that $$\bigcup \mathca... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3740222",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
exp(A+B) = exp(A)exp(B) for matrices proof In this thread, On the proof: $\exp(A)\exp(B)=\exp(A+B)$ , where uses the hypothesis $AB=BA$?, it was mentioned that absolute convergence is required for swapping sums. What theorem is used precisely?
| I think actually absolute convergence is only mentioned in the original question that is linked in the question you linked: https://math.stackexchange.com/a/356763/688699
Anyway if you're talking about rearranging an infinite sum then this is just the Riemann Series Theorem.
In particular (quoting from Wikipedia) let $... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Smith normal form and basis for the image of a module homomorphism Let's say
$$\varphi: \mathbb{Z}^n \to \mathbb{Z}^m$$
is a $\mathbb{Z}$-linear mapping and $A$ is the transformation matrix of $\varphi$ and
$$SAT = Q$$
where $Q$ is the Smith normal form of $A$ with $S$ and $T$ both invertible over the Ring $\mathbb{Z}$... | The image of $\varphi$ consists of the vectors $Av$ for $v\in\Bbb Z^n$. As $T$ is invertible over $\Bbb Z$, the $ATw$ for $w\in\Bbb Z^n$ are the same as the $Av$
for $v\in\Bbb Z^n$. So this image is the $\Bbb Z$-span of the columns of $AT$
(as well as the $\Bbb Z$-span of the columns of $T$).
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3740412",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Einstein notation for a sum of vector elements Perhaps this question is misguided but I am having difficulty writing a simple matrix expression in einstein notation.
In my expression I have a vector $v$ and I wish to define the scalar value, $a$, as the sum of the components of $v$, $v^i$. I know in einstein notation y... | The issue here is that the notion of the sum of the elements of a vector is not basis independent. So the way you would do this is to define a row-vector whose entries are all $1$s in the basis that you're working in, but note that this vector could look completely different in another basis.
I think $\delta_i$ is a lo... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3740765",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Expected number of tosses to get 3 heads (not necessarily in a row) Intuition tells me the expected number of tosses of a fair coin needed to get 3 TOTAL heads is 6.
I'm trying to show this using purely (1) Bernoulli random variables and (2) purely Geometric random variables
For approach (1), Let $X_i = 1$ if the i-th... | Since expectations are additive, the second approach works as well and is used, for example, in the coupon-collector-problem :
If we have $n$ possible outcomes, every outcome equally likely, then the number of trials to get each outcome at least once has expectation $$\sum_{j=1}^n \frac{n}{j}$$ This can be seen as a ch... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3740922",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
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How is $\sum_{n=0}^N\frac{a_n}{10^n}$ a Cauchy sequence? I've been studying the construction of real numbers this week and i've read about cantor's construction using the Cauchy sequence and Dedekind's construction. Now the book i'm reading (classic set theory for guided independent study) gives a new kind of construct... | The generic term of the sequence is $s_N=\sum_{n=0}^N\frac{a_n}{10^n}$ with $N\geq 0$. In order to show that $(s_N)_N$ is a Cauchy sequence, note that for $M\geq N\geq 0$,
$$0\leq s_M-s_N=\sum_{n=N+1}^M\frac{a_n}{10^n}\leq 9\sum_{n=N+1}^M\frac{1}{10^n}<9\sum_{n=N+1}^\infty\frac{1}{10^n}=\frac{1}{10^N}$$
where we used t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3741053",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
product of nonsingular polynomials In $R^n$, the polynomial $P$ is called nonsingular if $0$ is its regular value i.e. $\forall x s.t. p(x)=0,\mbox{then} \nabla p(x)\neq0$.
How to prove that the product of nonsingular polynomials has the property that its nonsingular points are dense in its zero set?
I have seen this... | The question is wrong usually. But if we ask the polynomials are prime to each other, the question is true. And in the paper of Guth, we can ask this property by the prime factorization of the product.Then delete the power.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3741161",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Prove that $\lim_{n\to \infty} a_n=\frac{\sqrt{5}-1}{2}$ if $a_{n+1}=\sqrt{1-a_n}$ and $0If $0<a_0<1$ and $a_{n+1}=\sqrt{1-a_n}$, prove that:
$$\lim_{n\to \infty} a_n=\dfrac{\sqrt{5}-1}{2}$$
Here what I do is that when $n\to \infty$, $a_{n+1}=a_n$
$\therefore a_n=\sqrt{1-a_n}\Rightarrow a_{n}^2+a_n-1=0$
which implies t... | hint
It is easy to prove by induction that for all $ n\ge 0$, $0\le a_n\le 1$.
Let $$f(x)=\sqrt{1-x}$$
from $ [0,1)] $ to $[0,1)$.
$$f'(x)=\frac{-1}{2\sqrt{1-x}}<0$$
$ f$ is decreasing at $ [0,1) $, so the subsequences $ (a_{2n})$ and $( a_{2n+1})$ are monotonic and convergent.
I let you to prove they have the same lim... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3741423",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Show that $\frac{\partial}{\partial r}\int_{S_r}u(x,y)ds =0$ Show that for all $0<\rho\leq r$ $\frac{\partial}{\partial r}\int_{S_r}u(x,y)ds =0$.Well, the hypotesis is $u$ is harmonic ($\Delta u=0$) , $0\in \Omega$,and $B_{\rho}$ the ball with radios, $\rho>0$ $B_{\rho}\subset \Omega$ and $\frac{\partial}{\partial r}$ ... | One answer is that this is automatically zero by the spherical mean-value property of harmonic functions. However, this fact is generally used to prove the spherical mean-value property, so let's do it directly.
First for simplicity let's define $\phi(r) = \frac{1}{r^{n - 1}}\int_{S_r} u(x) dS(x)$. Here $x \in \mathbb{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3741562",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
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prove that if $E$ is connected and $E \subseteq F \subseteq \overline{E}$, then $F$ is connected.
Define a set $A$ to be disconnected iff there exist nonempty relatively open sets $U$ and $W$ in $A$ with $U\cap W = \emptyset$ and $A = U\cup W.$ Define a set $A$ to be connected iff it is not disconnected.(there are man... | Assume $E \subseteq F \subseteq \overline{E} \subseteq X$ and that we're working in the topology of $E$ relative to $X$. If $F$ is disconnnected, then $\exists U, V \subseteq X$ open (in $X$) such that $F \subseteq U \cup V$ and $U \cap V \cap F = \emptyset$.
But then $U \cap V \cap E = \emptyset$ and $E \subseteq F \... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3741882",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Converting between bound on probability measures and densities Suppose that $P$ and $Q$ are two probability measures on the same probability space with $P(A) \leq c Q(A)$ for each (measurable) set $A$.
Is it true that $dP/dQ$ is then bounded by $c$ $P$-almost surely?
| Let $f$ denote the Random-Nikodym derivative $\frac{dP}{dQ}$. Then
$\int_Af\,dQ\leq cQ(A)$ for all $A$ and so
$$\int_A (c-f)dQ\geq0\quad\text{for all} \quad A$$
From this, it should follow that $f\leq c$ $Q$-a.s. Take for instance $A=\{f>c\}$
(You ma use a well know fact that says that if $\phi\geq0$ and $\int \phi\,d\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3742050",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Prove that there is a positive number $B$ such that $|x_{n}| \geq B$ for all $n$ $[{x_n}]^{\infty}_{n=1}$ is a sequence of real numbers that converges to $x_0$ and that all $x_n$ and $x_0$ are nonzero.
I have done the following:
for all $n$ in $|x_{n}|\geq B$
$x_0 < B$
$\epsilon<B-x_o$
$\epsilon+x_o<B$
for all $n$ in $... | We have $x_n \to x_0$ as $n \to \infty$. Then, by definition, for any $\epsilon > 0$,
$$
\exists N \in \mathbb{N} : \forall n \geq N, | x_n - x_0 | < \epsilon
$$
If we choose $\epsilon = |\frac{x_0}{2}|$ we get that for all $n \geq N$
$$
|x_n - x_0| < |\frac{x_0}{2}| \Longrightarrow |x_n| \geq |\frac{x_0}{2}|
$$
If you... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3742222",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 2
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An alternate definition of Limit Let's say we were to define the concept of limit as follows:
$\displaystyle{\lim_{x \to c}}f(x)=L$ means that for every $x$ in the domain of $f$, there exists an $x_0 \neq x$ in the domain of $f$ such that:
$$|x-c|>|x_0-c|$$and$$|f(x)-L|\ge|f(x_0)-L|$$
I have two questions:
*
*Does th... | The issue is not what you can prove, but that these two definitions are different.
*
*Let us define the function:
$$
f(x)=\begin{cases}
1 & x=0\\
0 & x=1\\
x & x\ne 0 \text{ and } x\ne 1
\end{cases}
$$
Then, with the conventional definition, $\lim_{x\to 0} f(x)=0$. Take $x=1$. There is no point $x_0\ne x$ with $|f(x_... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3742377",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Inverse function on the interval $[1, 9.5]$ I am struggling to find the inverse of the following function $$f(x) = \frac{10}{3}\exp\big(-0.06x\big)\log\bigg(\frac{1}{5}\big(2x + 3\big)\bigg).$$ I noticed that this function is not one-to-one, so I restricted its domain to the interval $[1, 9.5]$. I am aware that there m... | I can almost guarantee that this will not have an elementary inverse. If your need is simply to calculate values, then I suggest a root-finding technique. You have this tagged as pre-calculus, but I'm going to bend that a bit and suggest Newton's method. Though setting up the recurrence for Newton's method requires a d... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3742513",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Regular level set theorem proof, manifolds Let $F:N\rightarrow M$ be a smooth man between manifolds of dimension $n$ and $m$ respectively. A non-empty regular level set $F^{-1}(c)$ where $c\in M$ is a submanifold of $N$ of dimension equal to $n-m$.
The proof starts of by something like this:
Choose a chart $(V,\psi)=(V... | Upgrading my comment to an answer:
We wish to show that $F^{-1}(c)$ is the common zero set of the functions $r^i \circ \psi \circ F$ for $i \in \{1, \dots, m\}$. In other words, we want
$$F^{-1}(c) = \bigcap_{i = 1}^m (r^i \circ \psi \circ F)^{-1}(0).$$
We already know that $F^{-1}(c) = (\psi \circ F)^{-1}(\mathbf{0})$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3742645",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Need help in solving a linear algebra ( System of Equations) quiz problem I am solving previous year quiz problem of my class and I am unable to solve this question in linear algebra.
Let $A \in M_{m \times n}(\Bbb{R})$ and let $b_0 \in \Bbb{R}^m$. Suppose the system of equations $Ax = b_0$ has a unique solution. Whic... | Recall that the system $Ax=b$ is solvable iff $b$ is an element of the span of the column vectors of $A$. Now if this solution is unique the column vectors of $A$ are linearly independent. Does this clear up things?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3742744",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
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Why is the Penrose triangle "impossible"? I remember seeing this shape as a kid in school and at that time it was pretty obvious to me that it was "impossible". Now I looked at it again and I can't see why it is impossible anymore.. Why can't an object like the one represented in the following picture be a subset of $\... | Start at the bottom left-hand corner, taking othonormal unit vectors $\pmb i$ horizontally, $\pmb j$ inward along the cross-member bottom left-hand edge, and $\pmb k$ upward and perpendicular to $\pmb i$ and $\pmb j$. I'll take the long edge of a member as $5$ times its (unit) width; the exact number doesn't matter. Th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3742825",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "165",
"answer_count": 6,
"answer_id": 3
} |
System of equations and recurrence relation I am trying to find the general solution for $N$ of the following system of equations
$$
\begin{cases}
(x_n - x_{n-1})^2 + (y_n - y_{n-1})^2 = \left(\frac{\theta}{N}\right)^2 \\
{x_n}^2 + {y_n}^2 = 1
\end{cases}
$$
with the initial values $x_0 = 1$ and $y_0 = 0$ and t... | The second equation expresses that the points $(x_n,y_n)$ remain on the unit circle, and the first, that the successive points form chords of constant length, subtending an angle $\alpha=2\arcsin\frac\theta{2N}$.
Hence $$(x_n,y_n)=(\cos n\alpha,\sin n\alpha).$$
By the way, the system has in fact $2^n$ distinct solutio... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3742927",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
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Understanding Serge Lang's Definition of Homotopy I have been following Serge Lang's Complex Analysis text book and today I came across a chapter on homotopy. I have trouble visualising and honestly, understanding the definition that he has given in his book. Here is the definition from his book
Could somebody explain... | By definition, $\psi(t,c)=\gamma(t)$. Since $\psi$ is continuous, if $c_1$ is slightly bigger than $c$, then $t\mapsto\psi(t,c_1)$ is a path which is close to $\gamma$. And if $c_2$ is slightly bigger than $c_1$, then $t\mapsto\psi(t,c_2)$ is a path which is close to the previous one. And so on, until you reach $d$. So... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3743039",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
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a linear map on $W$
Define $W = \{(a_1, a_2,\cdots) : a_i \in \mathbb{F}, \exists N\in\mathbb{N}, \forall n \geq N, a_n = 0\},$ where $\mathbb{F} = \mathbb{R} $ or $\mathbb{C}$ and $W$ has the standard inner product, which is given by $\langle(a_1,a_2,\cdots), (b_1,b_2,\cdots)\rangle = \sum_{i=1}^\infty a_i \overline{... | Let $e_i\in W$ satisfy $(e_i)_j=1$ if $i=j$ and $(e_i)_j=0$ otherwise. Then $$\langle Te_i,e_j\rangle=\cases{1\,\,{\rm if}\,\,j\leq i,\\0 \,\,{\rm if}\,\,j>i.}$$
Thus if $T^*$ exists: $$\langle e_i,T^*e_j\rangle=\cases{1\,\,{\rm if}\,\,j\leq i,\\0 \,\,{\rm if}\,\,j>i.}$$
So we have $(T^*e_j)_i=1$ for all $i\geq j$ whi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3743178",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Equality of the totient function of two multiples of $x$ I'm looking to solve for $x \in \mathbb{N}$ in the equation $\phi(4 x) = \phi(5 x)$. I know the totient function $\phi(y)$ just gives the number of integers less than or equal to $y$ that are coprime to $y$. I tried approaching it like a normal equation and expan... | We use the fact that the totient function is multiplicative. Let:
$$x=2^a5^by$$
where $\gcd(y,10)=1$. Then:
$$\phi(4x)=\phi(5x) \implies \phi(2^{a+2}5^bx)=\phi(2^a5^{b+1}x)$$
Using the fact that the totient function is multiplicative, we yield:
$$\phi(2^{a+2})\phi(5^b)\phi(x)=\phi(2^a)\phi(5^{b+1})\phi(x)$$
Cancelling ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3743288",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Quadratic Forms on a (finite dimensional real) vector space with same zero set are scalar multiples? Let $V$ be a finite-dimensional vector space over $\Bbb{R}$, let $g,h:V \times V \to \Bbb{R}$ be bilinear symmetric functionals, and let $Q_g,Q_h:V \to \Bbb{R}$ be the associated quadratic forms. Suppose that $Q_g, Q_h$... | Partial answer:
As can be seen in the positive definite case, it does not generally hold that two bilinear forms are scalar multiplies if they have the same zero set.
However, the statement becomes true for the case of $\dim V \leq 3$ if we extend the bilinear form to a $\Bbb C$-bilinear form over $\Bbb C$. Without lo... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3743404",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
} |
Ascoli-Arzelà example and equicontinuity My teacher did an example of Ascoli-Arzelà Theorem that I really don't understand, above all how he show that a series of functions is equicontinuous.
Let $f_n:=x^n \quad , \quad x\in [0,1] \quad , \quad f_n\subset C([0,1],\Bbb{R})$
$\forall x, \quad \exists \lim_{x\to +\infty}f... | I don't think the behavior of the derivative at the point $x=1$ proves (without much effort) that the sequence is not uniformly equi-continuous . So your teacher's argument is not complete and not a good approach for this.
If the sequence is uniformly equicontinuous then there exists $\delta >0$ such that $|f_n(x)-f_n(... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3743536",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
Is $\int_{0}^{\infty} e^{-(a^2x^2+\frac{b^2}{x^2})}dx=\int_{0}^{\infty}e^{-(b^{2}{X}^2+\frac{a^2}{X^2})}dX$ for arbitrary $a,b$ and fixed range? The problem says,
If $\int_{0}^{\infty} \mathbb{e^{-(a^2x^2+\frac{b^2}{x^2})dx}=\frac{\sqrt{\pi}}{2a}.e^{-2ab}} \longrightarrow(i)$, then prove that $\mathbb{\int_{0}^{\infty... | You can "plug i into iii" via the "integration-by-substitution"
$$
\mathscr{X} = x \\
d \mathscr{X} = dx
$$
(i.e., using the chain rule in the simplest possible way); this shows that the integral you're calling "iii" is the same as the integral you're calling "i".
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3743644",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Are coslice categories of preadditive categories preadditive? I do not see any natural way how a coslice category of a preadditive category can be preadditive (other than in some degenerate cases). However, they are given as an example in Popescu's book "Abelian Categories with Applications to Rings and Modules" (1973)... | You are right and it is easy to come up with examples. In fact, every non-trivial example does it: Suppose that $C$ has a non-initial object $X$. Then there exists an object $Y$ with a non-zero morphism $f'\colon X\to Y$. Moreover, let $f\colon X\to Y$ be the zero morphism (i.e., the unit of the abelian group $\hom(X,Y... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3743772",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Evaluating the $ \lim_{n \to \infty} \prod_{1\leq k \leq n} (1+\frac{k}{n})^{1/k}$ I am really struggling to work out the limit of the following product:
$$ \lim_{n \to \infty} \prod_{1\leq k \leq n} \left (1+\frac{k}{n} \right)^{1/k}.$$
So far, I have spent most of my time looking at the log of the above expression. I... |
I thought that it would be instructive to present an approach that does not rely on Riemann Sums, but rather makes use of the Taylor Series of $\log(1+x)$. To that end, we proceed.
The function $\log(1+x)$ can be represented by its Taylor Series, $\log(1+x)=\sum_{\ell=1}\frac{(-1)^{\ell-1}}{\ell}x^\ell$ for $-1<x... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3743837",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 3,
"answer_id": 1
} |
Biholomorphism between Riemann Surfaces. Let's consider the homeomorphism $\psi : \mathbb C \rightarrow B_{1}(0), z \rightarrow \frac {z}{|z|+1}$.
Let $Z$ be be the Riemann Surface with topological space $\mathbb C$ induced by the chart $( \mathbb C,\psi)$.
I have to show that $Z$ is biholomorphic $B_{1}(0)$ .
I think ... | One point of importance here is that $g$ need only be holomorphic as a map $B_1(0) \to Z$, not to the usual complex structure on $\mathbb{C}$. For this it suffices to check that $g$ is a diffeomorphism and $\varphi \circ g$ is holomorphic for all coordinate charts $\varphi$ in an atlas for the Riemann surface structure... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3744081",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Divergent integrals $\int_{a}^{\infty}\frac{dx}{f(x)}$ and $\int_{a}^{\infty}\frac{dx}{f(x)+b}$ for positive $f$ and $b$ Assume that $f\colon [a,\infty)\to(0,\infty)$ is continuous ($a\in\mathbb{R}$) and such that
$$\int_{a}^{\infty}\frac{dx}{f(x)}=+\infty$$
and let $b>0$. Can we claim that
$$\int_{a}^{\infty}\frac{dx}... | This is false in general. First, we may as well take $a=0$. If $f(x)$ is not positive for $[0,a)$, modify the function and consider $f(x-a)$ instead of $f(x)$. Second, note that it is sufficient to consider a positive function that is discontinuous at a countable set $S$ (but otherwise continuous) but whose limits exis... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3744216",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 0
} |
Evaluating $\int _0^1\frac{\ln \left(x^2+x+1\right)}{x\left(x+1\right)}\:dx$ How can i evaluate this integral, maybe differentiation under the integral sign? i started expressing the integral as the following,
$$\int _0^1\frac{\ln \left(x^2+x+1\right)}{x\left(x+1\right)}\:dx=\int _0^1\frac{\ln \left(x^2+x+1\right)}{x}\... | Solution using harmonic series
$$\int _0^1\frac{\ln \left(x^2+x+1\right)}{x\left(x+1\right)}\:dx=\int _0^1\frac{\ln \left(x^2+x+1\right)}{x}\:dx-\int _0^1\frac{\ln \left(x^2+x+1\right)}{x+1}\:dx\:$$
$$\int _0^1\frac{\ln \left(x^2+x+1\right)}{x}\:dx=\underbrace{\int _0^1\frac{\ln \left(1-x^3\right)}{x}\:dx}_{x^3\to x}-... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3744330",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
} |
Why is a stochastic matrix a $l^2$ contraction If $P$ is a doubly stochastic matrix i.e. $P=(p_{ij})_{1\leq i,j \leq n}$ is s.t. the row sums $\sum_j p_{ij}=1$ for all $i$ and $\sum_i p_{ij}=1$ for all $j$, then may I know why
$$||Px||\leq ||x||$$
for all $x\in \mathbb{R}^n$ where $||\cdot||$ is the $l^2$ norm?
| This is related to the more general formulation that if $P\colon L^2\to L^2$ is given by $Pf(x) = \int p(x,y)\,f(y)\,dy$, and
\begin{align*}
\sup_x\int|p(x,y)|\,dy &\le 1,\\
\sup_y\int|p(x,y)|\,dx &\le 1, \tag{$\ast$}
\end{align*}
then $\|P\|_{L^2\to L^2}\le 1$. To see this, recall that
\begin{align*}
\|P\|_{L^2\to L^2... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3744453",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Trace of map & matrices Let $1\leq m,n\in \mathbb{N}$ and let $\mathbb{K}$ be a field.
For $a\in M_m(\mathbb{K})$ we consider the map $\mu_a$ that is defined by $$\mu_a: \mathbb{K}^{m\times n}\rightarrow \mathbb{K}^{m\times n}, \ c\mapsto ac$$
I want to show that $trace(\mu_a)=n\cdot trace(a)$ .
I have done the followi... | With a matrix representation for $\mu_a$ you may compute the trace somewhat simpler as the sum of its diagonal elements, without having to worry about diagonalising the map. To get such a representation you may use the trick of introducing a $\delta$ (over indices) and rewrite in the following index form (like a disgui... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3744627",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
Finding sign of leading coefficient of a quadratic equation In a given quadratic equation $f(x)=ax^2+bx+c$ if $f(-1)>-4, f(1)<0$ and $f(3)>5$, then how can I find the sign of $a$?
Answer in the textbook: $a>0$
| If $a=0$, then you have a straight line. Then $$f(1)=\frac{f(-1)+f(3)}2>\frac{-4+5}2=\frac 12>0$$
Then for a parabola with $a>0$ you have $$f(\frac{x+y}2)<\frac{f(x)+f(y)}2$$ and for $a<0$ you have $$f(\frac{x+y}2)>\frac{f(x)+f(y)}2$$
Since $$\frac{f(-1)+f(3)}2>0$$ and $$f(1)<0$$ then $$f(\frac{-1+3}2)<\frac{f(-1)+f(3)... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3744841",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
} |
Series with coefficients coming from beta function I recently came across the series
$$G(t):=\sum_{n=0}^\infty\frac{t^{n}}{B(n+1,\xi+1)}$$
defined for $t\in(0,1)$ and $\xi+1>0$. I am trying to get a better-looking form for this sum, something more "usable" in general, but I don't know how to handle the beta function. I... | Here is the general non-integer solution (leading to the same solution as @K.dafaoite integer solution). From the paper by E.Stade The reciprocal of the beta function we see that if $n+\xi+1>0$:
$$\frac{1}{B(n+1, \xi+1)}=\frac{n+\xi+1}{2\pi i}\int_{|u|=1}\left(1+\frac{1}{u}\right)^n(1+u)^\xi\frac{du}{u}$$
where the int... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3745150",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
Prove or disprove that $PQ = P + Q - I$ if $P$ and $Q$ are disjoint permutation matrices whose cycle lengths sum to $n.$
Prove or disprove that if the matrices $P$ and $Q$ represent disjoint permutation cycles in $S_{n}$ with sum of cycle lengths equal to $n,$ then $PQ = P+Q-I$.
MY TRY:
Let's start by an example. Let... | Not an answer but a generalisation of the above:-
Let $P_{1}, P_{2},\cdots P_{n}$ ($n\gt 2$) represent disjoint permutation cycles, we'll have $$\prod_{i=1}^{n}P_{i} = \sum_{i=1}^{n}P_{i}-(n-1)I$$
Proof :- We'll prove it by induction. It holds for $n=2$, as proved in the problem. Let it holds for $n = k$ i.e. $$\prod... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3745261",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 3,
"answer_id": 2
} |
Trader with 60% chance of gaining 50% and 40% and 40% chance of dropping 50%. Average return per day calculation. I am reading Paul Wilmott's Frequently Asked Questions in Quantitative Finance, and there is a question that states the following:
Every day a trader either makes 50% with prob- ability 0.6 or loses 50% wi... | $$e^{0.6\ln 1.5+0.4\ln 0.5}=e^{0.6\ln 1.5}\cdot e^{0.4\ln 0.5}=(1.5)^{0.6}\cdot (0.5)^{0.4}=0.966\ldots\quad(\because e^{n\ln x}=e^{\ln x^n}=x^n)$$ times the money at the beginning of the day becomes the money at the end of the day on average because the probability of getting $1.5$ times the money at the end of the da... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3745372",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
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Evaluate $\lim\limits_{x \to 1} \left( \frac{1}{x-1}-\frac{3}{1-x^3} \right)$ Evaluate $$\lim_{x \to 1} \left( \frac{1}{x-1}-\frac{3}{1-x^3} \right)$$
My attempt: $$\lim_{x \to 1} \left( \frac{1}{x-1}-\frac{3}{1-x^3} \right) = \lim_{x \to 1} \frac{x+2}{x^2+x+1}=1$$
According to the answer key, this limit does not exist... | $\lim_{x \to 1} \left( \frac{1}{x-1}-\frac{3}{1-x^3} \right)=\lim_{x\to 1} \frac{x^2 + x + 4}{x^3 - 1}=\lim_{x\to 1}\frac{p(x)}{q(x)}$, where $p(x) =x^2+x+4, q(x) =x^3-1 $
Note that, $p(x) \to 6, q(x) \to 0$ as $x\to 1$
Now let's claim that the limit $\lim_{x\to 1}\frac{p(x)}{q(x)}$ doesn't exist on set of real nos. ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3745531",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 5,
"answer_id": 3
} |
Mean and extinction probability of a Galton-Watson branching with pmf of offspring produced $P(Q=q) = (q+1)(1-r)^2r^q, 0Initial population is $X_0 = g$, ($g$ being a positive number or $0$) and the probability mass function of the number of offsprings $(q)$ produced by an individual is $P(Q=q) = (q+1)(1-r)^2r^q, 0<r<1$... | It is straightforward to show that if the offspring distribution has finite mean, that is,
$$
\mathbb E[Q] = \sum_{k=0}^\infty k\cdot\mathbb P(Q=k) :=\mu <\infty
$$
then the expected population size at time $n$, conditioned on $\{X_0=1\}$ is given by
$$
\mathbb E[X_n\mid X_0=1] = \mu^n.
$$
If $g$ is a positive integer,... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3745713",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
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Closure of a subgroup is again a subgroup Let $G$ be a topological group and $H$ a subgroup. Then $\overline{H}$ is again a subgroup.
Attempt: Let $x,y \in \overline{H}$. Choose nets $\{x_\alpha\}_{\alpha \in I}$ and $\{y_\beta\}_{\beta \in J}$ with $x_\alpha \to x, y_\beta \to y$ and these nets are in $H$. Then we get... | Seems ok, but I'd use "pre-image of open is open" directly instead of converging nets.
Let $x,y\in\overline H$.
Suppose $xy\notin\overline H$. By openness of $\overline H^\complement$, there is a neighbourhood of $U$ of $(x,y)\in G\times G$ such that $uv\notin \overline H$ for all $(u,v)\in U$. $U$ contains some $V_x\t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3745908",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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Calculating the size of uniform cabinet doors on two walls I know using this board for this problem is overkill, but I'm struggling with something that I know should be simple.
I'm building kitchen cabinets on two different walls. I'm going to buy the doors for these cabinets from a manufacturer in bulk, and all cabine... | If you measure the width in inches, a width of $w$ will give you $\frac {180}w$ doors along the $15'$ wall and $\frac {108}w$ doors along the $9'$ wall. If the fractions do not come out even, throw away any remainder. To get the most doors you want to choose $w$ as small as possible. This will make the cabinets quit... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3746033",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
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Assume both r and s are rational then $\frac{r}{s}\in\mathbb{Q}$ Currently done with showing for both r+s and r $\times s$ rational.
Now stuck with the following part; If we know that r and s are both rationals what could we say about
$\frac{r}{s}$
Assume both r and s are rationals.
Then we would have by definition r =... | Yes, it's a valid proof.
The only "error" that may be corrected is that you must assume $s\in\mathbb{Q}\backslash\{0\}$, because if $s=0$ then $r/s$ is undefined and consequently we get that $r/s\notin\mathbb{Q}$.
So, as conclussion for your proof:
$$r\in\mathbb{Q}\phantom{a}, \phantom{a}s\in\mathbb{Q}\backslash\{0\} \... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3746187",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Approximating a correlated random walk on a 2D grid I have been working on modeling the movement of ants, and a number of studies suggest the use of correlated random walks. These are biased random walks, where the direction of the next step is dependent on the direction of the previous step. The way this is usually mo... | I think one of the easier ways of doing this is calculating the individual probabilities of every angle using the probability density formula for Gaussian distribution. In the grid, the random walker can take steps only in 90$^{\circ}$ and 180$^\circ$. Diagonal steps ($45^\circ$, $135^\circ$, etc) can be added as well.... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3746444",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
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Proof of Infinitude of Primes by Euler's Product Formula is Circular? Many guides will refer to Euler's product formula as simple way to prove that the number of primes is infinite.
$$\sum_n\frac{1}{n} = \prod_p \frac{1}{1-\frac{1}{p}}$$
The argument is that if the primes were finite, the product on the right hand side... | No, the proof is not circular. If we assume there are a finite number of primes $p_1,\ldots,p_k$, then we would assume that any $n\in\mathbb{N}$ would be able to be factorized as $p_1^{\alpha_1}\ldots p_k^{\alpha_k}$. (The proof of the Fundamental Theorem of Arithmetic does not require that there be an infinite number ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3746532",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "15",
"answer_count": 3,
"answer_id": 2
} |
Geometric explanation for the convergence of $\sum_{n = 1}^{\infty} (1/a)^n$ when $1I'm studying some series and I'm trying to see them in a more geometric way, thinking of integrals or parts of something big (you will know what i mean). But I can't find a geometric way to see why those should converge!
Being a bit mor... | Consider the following continuous function:
$$f(s) = \frac{1}{a^s}$$
Compare this to the left-continuous step function:
$$g(s) = \frac{1}{a^{\lceil s \rceil}}$$
Note that
$$\int_0^n g(s) ds \equiv \sum_{i=1}^{n} \frac{1}{a^i} \quad \forall n \in \mathbb{N}$$
Also note that $\lceil s \rceil \geq s \implies f(s) \geq g(s... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3746741",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
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$\int_0 ^ \frac{\pi}{2}[\sin 2x (1+\cos 3x) ]dx$ . Here $[t]$ denotes the greatest integer less than or equal to $t$. $\int_0 ^ \frac{\pi}{2}[\sin 2x (1+\cos 3x) ]dx$ . Here $[t]$ denotes the greatest integer function.
Can anyone give me a hint?
| $$y=\sin(2x)(1+\cos(3x)),x\in[0,\pi/2]$$note that $0\leq 2x\leq \pi \to 0\leq \sin(2x)\leq 1\\1+\cos(3x)\geq0\\$
now find max,min of $y=\sin(2x)(1+\cos(3x)) $ on $x\in[0,\pi/2]$ it seems $0\leq y<1 $so $[y]=0$ so $\int_0^{\pi/2}[y]dx\to 0
\\$
https://www.desmos.com/calculator/inubfmxu8p
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3746875",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
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Does $\partial A$ determine $A$? Given a bounded closed set $A$ in $\mathbb R^n$, can $A$ be uniquely determined by $\partial A$, except for the boundary itself?
Or, use it differently, given two bounded closed sets $A_1, A_2$ in $\mathbb R^n$ with $\partial A_1 = \partial A_2$, $A_1 \ne\partial A_1$, and $A_2 \ne\part... | I think the other examples given so far are not correct. I think this works:
Take two disks in $\Bbb{R}^2$, and let one set be the union of one disk and the circle around the other, and let the other set be the union of the other disk with the circle around the former. Then they are closed, bounded, different than thei... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3747178",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 3,
"answer_id": 1
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Prove: $\tan{\frac{x}{2}}\sec{x}= \tan{x} - \tan{\frac{x}{2}}$ I was solving a question which required the above identity to proceed but I never found its proof anywhere. I tried to prove it but got stuck after a while.
I reached till here:
To Prove: $$\tan{\frac{x}{2}}\sec{x}= \tan{x} - \tan{\frac{x}{2}}$$
But I don'... | In terms of $t=\tan\tfrac{x}{2}$, the LHS is $t\cdot\frac{1+t^2}{1-t^2}$, while the RHS is $\frac{2t}{1-t^2}-t=t\cdot\frac{2-(1-t^2)}{1-t^2}=t\cdot\frac{1+t^2}{1-t^2}$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3747280",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 3
} |
Does remainder of 2^-1 divide by 7 exists? (mod 7) Decimal? I am confused as to how decimal plays a part compared to the multiplicative inverse.
For example, I know that $2^2\equiv 2^5\equiv 4\bmod 7$ (the pattern is 1,2,4, for every power of 3)
This then implies that $2^{-1}\equiv 4 \bmod 7 .$
However $2^{-1}$ is a de... | $2^{-1}\equiv 4 \bmod 7$ makes perfect sense because $2 \cdot 4 \equiv 1 \bmod 7$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3747405",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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Graph proving a cycle inequality I was solving some old exams from my university and I've stumbled across this one which I didn't know how tto think through, it says:
Give a graph $H$. Let $u(H)$ be the number of vertices of $H$ of degree 1. Let $C$ be a cycle in $H$. Show that $l(C) <= v(H) - u(H) $
| HINT: Can a vertex of degree $1$ belong to $C$?
| {
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"url": "https://math.stackexchange.com/questions/3747486",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Why is $ \frac{5}{64}((161+72\sqrt{5})^{-n}+(161+72\sqrt{5})^{n}-2)$ always a perfect square? I'm working on a puzzle, and the solution requires me somehow establishing that
$$ f(n):=\frac{5}{64}\Big(\big(161+72\sqrt{5}\big)^{-n}+\big(161+72\sqrt{5}\big)^{n}-2\Big)$$
is a perfect square for $n\in \mathbb{Z}_{\geq 0}$.
... | It's $$\frac{5}{64}\left((9+4\sqrt5)^{2n}+(9+4\sqrt5)^{-2n}-2\right)=\frac{5}{64}\left((9+4\sqrt5)^n-(9-4\sqrt5)^n\right)^2.$$
Can you end it now?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3747774",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 0
} |
Why rotations with two angles in $3D$ do not form a group? Let us use any parametrization of $3D$ rotations with three angles (e.g. Euler angles or yaw-pitch-roll), and throw away one of the angles (just assign it a fixed value).
Will the remaining set of transformation form a group? If yes — which? If no — why?
Follo... | Let consider some 90 degree rotations.
We have two rotation matrices that generate our group.
$P=\pmatrix{&1&\\-1&&\\&&1}$ and $R=\pmatrix{1&&\\&&1\\&-1&}$
Traditionally our third rotation matrix
$Y=\pmatrix{&&1\\&1&\\-1&&}$ has been left out.
But $PRP^{-1} = Y$
Two rotations will generate the 3rd rotation and hence $S... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3747886",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Why does $\frac{a}{b}<0$ imply $ab<0$? I'm not sure if this was asked before, but my question is: why does $\frac{a}{b}<0$ imply $ab<0$? How do you prove it both intuitively and rigorously(using math)? I think I understand it intuitively: it's becuase for $\frac{a}{b}$ to be negative, exactly one of $a$ or $b$ has to b... | If you know one, you can obtain the other by multiplying/dividing by $b^2$, which is possible since that will always be a positive quantity. Hence, it suffices to show that any one of these statements is true.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3748124",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 3
} |
A kind of Cauchy's Condensation Test? Let $f:[0,\infty) \to \mathbb{R}$ be a positive , monotone decreasing function and
$ f(x) \to 0$ as $x\to \infty$
Let $\{a_n\}$ be a bounded sequence such that $a_n \ge 0$ and $\displaystyle\sum _{n=1}^{\infty} a_n=\infty$ .
Then prove that $\sum_{n\ge 1} f(n)$ and $\sum_{n\ge 1}a_... | PARTIAL answer:
$(a_n)$ is bounded so there exists an $M > 0$ such that for all $n$ we have $0 \leq a_n \leq M$. Moreover, as you suggested, we can assume without loss of generality that $a_n > 0$ for all $n$.
Let $N_k = \min \{ n : a_1 + \dots + a_n \geq k\}$.
$N_k$ is well defined and finite for all $k$ because $\sum... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3748269",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
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A question regarding the proof of $\gcd(a^m-1, a^n-1) = a^{\gcd(m,n)}-1$ I have a problem trying to understand the proof:
Theorem $\boldsymbol{1.1.5.}$ For natural numbers $a,m,n$, $\gcd\left(a^m-1,a^n-1\right)=a^{\gcd(m,n)}-1$
Outline. Note that by the Euclidean Algorithm, we have
$$
\begin{align}
\gcd\left(a^m-1,a^... | Suppose :
$gcm(m, n)= k$
$m=m_1 k$
$n=n_1 k$
Then we have:
$a^m-1=(a^K)^{m_1}-1=(a^k-1)[(a^k)^{m_1-1}+(a^k)^{m_1-2}+ . . . $
$a^n-1=(a^K)^{n_1}-1=(a^k-1)[(a^k)^{n_1-1}+(a^k)^{n_1-2}+ . . . $
That means:
$gcd(a^m-1, a^n-1)=a^k-1=a^{gcm(m, n)}-1$
Notice that the factor $(a^k-1)$ exists for both cases where m and n are bo... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3748376",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Without the ZF axiom of regularity can any infinite sets be constructed? Update with Direct Question
Based on Asaf's comments, here is a related question:
Prove that the mapping $n \mapsto n \cup \{n\}$ on the set $\Bbb N$ is injective without the axiom of foundation.
The wikipedia $\text{ZF}$ article under axiom 7 c... | It is true, if $x=\{y\}$ and $y=\{x\}$ and $x\neq y$, then $x\cup\{x\}=\{x,y\}=y\cup\{y\}$. So without assuming the axiom of regularity the map $x\mapsto\{x\}$ is not provably injective.
But just because it is not necessarily injective on the entire universe does not mean that we cannot find a set on which it is inject... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3748489",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
What is the relation of the Infinite power series with these fraction series? Take this infinite series:
$$S = 1 + \sum_{n=1}^\infty\prod_{i=1}^n\frac{2i+1}{4i} = 1 + \frac{3}{4} + \frac{3\times5}{4\times8} + \frac{3\times5\times7}{4\times8\times12} + ....$$
We want to find the sum of this series. I didn't know how to ... | $\newcommand{\bbx}[1]{\,\bbox[15px,border:1px groove navy]{\displaystyle{#1}}\,}
\newcommand{\braces}[1]{\left\lbrace\,{#1}\,\right\rbrace}
\newcommand{\bracks}[1]{\left\lbrack\,{#1}\,\right\rbrack}
\newcommand{\dd}{\mathrm{d}}
\newcommand{\ds}[1]{\displaystyle{#1}}
\newcommand{\expo}[1]{\,\mathrm{e}^{#1}\,}
\new... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Prove that $s(z)=\frac{1}{1+e^{-z}}$ is always increasing I want to prove that $s(z)=\frac{1}{1+e^{-z}}$ is always increasing.
I know from previous work that taking the derivative and proving that it is always greater than $0$ is one way to prove $s(z)$ is always increasing.
How do I go about proving that the derivativ... | $e^z$ is increasing, $e^{-z}$ is decreasing, $e^{-z}+1$ is decreasing and positive, $\dfrac1{e^{-z}+1}$ is increasing.
$$z_0<z_1\implies-z_0>-z_1\implies e^{-z_0}>e^{-z_1}\implies e^{-z_0}+1>e^{-z_1}+1>0
\\\implies\frac1{e^{-z_0}+1}<\frac1{e^{-z_1}+1}.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3748938",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
How find the 2001th and 2003th derivatives of $f(x)= \frac{x^5}{1+x^6}$ Lef $ f: \mathbb{R} \rightarrow \mathbb{R} $ be defined by $f(x)= \dfrac{x^5}{1+x^6}$. I want to find the 2001th and 2003th derivatives of $f$ at the point $x=0$. I tried to use the chain rule but I do not see the pattern and I do not know what the... | Note that $\frac{x^5}{1+x^6}=x^5 \sum_{k=0}^\infty (-1)^k x^{6k}=\sum_{k=0}^\infty x^{6k+5}$ by using the geometric series expansion of $\frac1{1+y}$ for $y=x^6$. Then try to write 2001 and 2003 in the form $6k+r$ with $0\leq r<6$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3749347",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 2
} |
Find a subgroup of index 3 of dihedral group $D_{12}$ Find a subgroup of index 3 in the dihedral group $D_{12}$. I know the number of elements in $D_{12}$ is 24 and also that is we have this subgroup of index 3, then we obtain that $|D_{12}:H|=8$, where $H$ is our wanted subgroup, but I don`t know how to go further.
I ... | $D_{12}$ is generated by $a,b$ with $a^2=b^{12}=1, aba=b^{-1}$. Then $b^3$ generates a subgroup $A$ of order $12/3=4$. That subgroup is normal in $D_{12}$ because $ab^3a=b^{-3}=b^9=(b^3)^3$. Then $a$ and $A$ generate a subgroup $H$ of order 8. The index of that subgroup is then 3.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3749487",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Let $G$ be a graph such that all its vertices have degree 2. Prove that $G$ is a union of pairwise disjoint cycles.
Let $G$ be a graph such that all its vertices have degree 2. Prove that
$G$ is a union of pairwise disjoint cycles.
This is Exercise 4.1.4 in the book Problem-Solving Methods in Combinatorics by Pablo S... | Hint. Consider a connected component $C$ of $G$. Then the vertices of $C$ have degree $2$. Show that $C$ is a cycle. Note that, more generally, any connected graph with all even degree vertices have an Eulerian circuit.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3749686",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Solve for $x$ in $\sin^{-1}(1-x)-2\sin^{-1}x =\frac{\pi}{2}$ Let $x=\sin y$
$$\sin^{-1}(1-\sin y)-2\sin^{-1}\sin y=\frac{\pi}{2}$$
$$\sin^{-1}(1-\sin y)-2y=\frac{\pi}{2}$$
$$1-\sin y =\sin (\frac{\pi}{2}+2y)$$
$$1-\cos2y=\sin y$$
$$\sin y(2\sin y-1)=0$$
$$x=0,~ \frac 12$$
Clearly, $x=\frac 12$ isn’t correct, because it... |
Find $x$ satisfying $\sin^{-1}(1-x)-2\sin^{-1}x =\frac{\pi}{2}$.
Common mistakes
While going step-by-step, if you go from
*
*$(\sin^{-1}a)=y\stackrel{\text{to}}{\longrightarrow}\sin(\sin^{-1}a)=a=\sin (y\pm 2n\pi)$, and from
*$\sin y=a\stackrel{\text{to}}{\longrightarrow}\sin^{-1}(\sin y)=\begin{cases}2n\pi+y&y\in... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3749783",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 5,
"answer_id": 3
} |
How to find the total distance travelled in a kinematics integration question?
I completed question a which is $\int _0^3\:t^2-6t+5 = -3$ or more specifically $3m$ to the left
Now isnt distance the absolute value of the displacement? $\left|\int _{t1}^{t2}\:v\left(t\right)\right|=\left|x1\:-x2\right|$
So for question ... | Some definitions:
*
*$$\text{average speed} = \frac{\text{distance travelled}}{\text{time
elapsed}}$$
*$$\text{speed at time }t=\left| v(t)\right|$$
*$$\text{average value of } f(t) \text{ over }
[t_1, t_2]=\frac{\int_{t_1}^{t_2}f(t)\,\mathrm{d}t}{t_2-t_1}$$
Therefore
\begin{align}\text{distance travelled ove... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3749879",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 5,
"answer_id": 4
} |
Motion down an inclined plane with leg equal to the diameter of a circle This is an exercise from Morris Kline's "Calculus: An Intuitive and Physical Approach".
An object slides down an inclined plan $OP'$ (Fig. 3-9) starting from rest at $0$. Show that the point $Q$ reached in the time $t_1$ required to fall straight... | For both paths.
Time = $\frac 12 \frac {\text {distance}}{\text {acceleration}}$
On the trip from $O$ to $Q,$ the acceleration is $g\sin A$
The distance covered is $\overline {OP} \sin A$
The alternative trip from $O$ to $P$ the accelartion is $g$ and the distance is $\overline {OP}$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3749965",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 5,
"answer_id": 4
} |
Changing a double integral into a single integral - Volterra-type integral equations I have a question regarding a calculation that i stumbled upon when proving that a Cauchy problem can be converted in a Volterra-type integral equation. Specifically, this equality:
\begin{equation*}
\int_0^t\int_0^sy(t) dt ds = \int_0... | There is an issue of confusion between the dummy variable of integration $t$ and the upper limit on the outer integral. Instead, write
$$F(t)=\int_0^t\int_0^s y(x)\,dx\,ds\tag1$$
Then, note that the region $0\le x\le s$, for $0\le s\le t$ is a triangular shaped region with vertices in the $(x,s)$-plane at $(0,0)$, $(0... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3750112",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
2D integration by parts I have the 2D integral $$\iint_{xy}\psi\textbf{u}_H\cdot\nabla_Hh\,{\rm d}x{\rm d}y,$$ with $\psi = \psi(x, y)$, $\textbf{u}_H = (u(x, y), v(x, y))$, $\nabla_H = (\partial/\partial x, \partial/\partial y)$ and $h = h(x, y)$. Is there a 2D integration by parts equivalent, so that I can remove the... | The integral seems to be of the form
$$
\iint_\Omega \mathbf{v}\cdot\nabla f \, dA
$$
The integration by parts equivalent is to use
$$
\nabla \cdot (f \mathbf{v}) = \nabla f \cdot \mathbf{v} + f (\nabla\cdot\mathbf{v})
$$
and then Stokes' theorem:
$$
\iint_\Omega \mathbf{v}\cdot\nabla f \, dA
= \iint_\Omega \left( \nab... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3750237",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
About Two Goldbach's Conjectures I've learned that
*
*"The Strong Goldbach's Conjecture" is that
'All the even natural numbers greater than 2 can be written as a sum of two prime numbers.'
And,
*"The Weak Goldbach's Conjecture" is that
'All the natural numbers greater than 5 can be written as the sum of 3 prime nu... | As stated in the Origins section of Wikipedia's "Goldbach's conjecture" article,
On $7$ June $1742$, the German mathematician Christian Goldbach wrote a letter to Leonhard Euler (letter XLIII), in which he proposed the following conjecture:
$\;\;\;\;\;$Every integer that can be written as the sum of two primes, can al... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3750593",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Solving $\frac{\cos^2\left(\frac\pi2 \cos\theta\right)}{\sin^2\theta} = 0.5$
$$\frac{\cos^2\left(\dfrac\pi2 \cos\theta\right)}{\sin^2\theta} = 0.5$$
I want to solve the above equation for $\theta$ in order to find its value, but I am stuck.
Could anyone enlighten me by a method to solve it?
| Let $x =\cos\theta$ to simplify the equation to
$$\cos(\pi x) +x^2=0$$
which has the trivial roots $\pm 1$ (excluded due to $\sin\theta \ne 0$), as well as the root that can be approximated with $\frac\pi2-\pi x+x^2 =0$, i.e.
$$x=\left(1+\sqrt{1-\frac2\pi} \right)^{-1}= 0.6239$$
(vs. the exact $ 0.6298$). Thus, the sol... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3750747",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Given a straight pyramid with a regular hexagonal base, we pass through the center of its base an alpha plane parallel to a side face. Given a straight pyramid with a regular hexagonal base, we pass through the center of its base an alpha plane parallel to a side face. Find the ratio between the area of the obtained ... |
Here is a picture of how this will look (my awkward effort to show it). S is the center of the hexagonal base and PS is perpendicular to the base. The plane in question (say Q) is parallel to ABP side face, passes through CSD and cuts CFP, FEP and DEP sides. The cross section that you see is trapezoid CXYD parallel to... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3750820",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
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