Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Finding all solutions to $3x + 4y \equiv 1 \pmod 7$ Find all solutions $\pmod 7$: $3x + 4y$ is congruent to $1 \pmod 7$.
I have tried writing out the various equations such as $3x + 4y = 1$, $3x + 4y = 8$, $3x + 4y = 15$, etc., but I do not know how to find the finite solution.
| Hint: for any $x,y $ we always have $3x+4y=3 (x+4)+4 (y-3) $.
So by induction:
$3x+4y\equiv 1 \mod 7 \implies 3 (x\pm 4k)+4 (y\mp 3k)\equiv 1 \mod 7$
So... if you know one solution, you know infinitely many.
The question is are there any not generated from your first solution? (I.e. can we make that $\implies $ into $\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2049003",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 1
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Query related to Eq. 3.471.9 of Book of Gradeshteyn (Integration tables series and products) Equation no. 3.471.9 of Integral series and products (By Gradeshteyn) is written below $$\int_0^{\infty}x^{v-1}e^{-\frac{\beta}{x}-\gamma x}dx=2\left(\frac{\beta}{\gamma}\right)^{\frac{v}{2}}K_{v}(2\sqrt{\beta \gamma})$$ althou... | The integral will converge as long as $\text{Re}(\beta) > 0$ and $\text{Re}(\gamma) > 0$, regardless of the value of $v$. Both sides are analytic as functions of $v$ for fixed $\beta, \gamma$. So the equation should work for all $v$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2049113",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
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How to divide 2n members of the club into disjoint teams of 2 members each, when teams not labelled? I am doing this and getting $\sum_{i=0}^{2n-2} \frac{(2n-i)!}{2!(2n-i-2)!}$ but the answer given is $\frac{(2n)!}{2^n n!}$ I even tried to get this by simplifying my result, but not getting the same.
| Their answer: imagine lining up everyone in a row and then pairing up adjacent people. There are $(2n)!$ ways to line everyone up, but different line-ups can produce the same teams. To account for this overcounting, divide by $n!$ for the number of ways to arrange the $n$ teams in a row, and divide by $2$ for each pair... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2049254",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 5,
"answer_id": 0
} |
What is $\tan ^{-1} (5i/3)$
What is $\tan ^{-1} (5i/3)$
My progress: Let $\tan x= \dfrac{5i}{3}= \dfrac{\sin x}{\cos x}$
I tried using $\sin x= \dfrac{e^{ix}-e^{-ix}}{2}, \cos x= \dfrac{e^{ix}+e^{-ix}}{2}$ to show that $\dfrac{e^{ix}-e^{-ix}}{e^{ix}+e^{-ix}}= \dfrac{-5}{3}$ or $e^{2ix}= \dfrac{-1}{4}$, but I'm stuck ... | You are fine. Just solve $e^{2ix}= \dfrac{-1}{4}$, taking into account that $x$ is complex. Write $x = r + i c$ to get
$e^{-2c + 2ir}= \dfrac{-1}{4}= \dfrac{1}{4} e^{i \pi}$ and identify
$r = \pi /2$ and $c = \ln 2$. So
$$
x = \pi /2 + i \ln 2
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2049360",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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"answer_id": 0
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System of two equations with 3 unknowns and parameters Is there a way to solve this for $c_1$, $c_2$, $c_3$ in terms of $a$'s and $b$'s?
$$
\begin{cases}
a_1c_1+a_2c_2+a_3c_3=0 \\
b_1c_1+b_2c_2+b_3c_3=0
\end{cases}
$$
| In general, this system (with only two equations but three unknown) will have an infinite number of solutions. You can choose one of the $c$'s as a free variable and solve the system in terms of the $a$'s, $b$'s and the free variable as a parameter.
For example, solving for $c_1$ and $c_2$ (see Wolfram|Alpha) yields:
$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2049441",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Intuition between Ito-Formula What is the intuition behind Ito formula ? It's looking comming from no where to me. I recall that if $f\in \mathcal C^2(\mathbb R)$, then,
$$f(B_t)-f(B_0)=\int_0^t f'(B_s)dB_s+\frac{1}{2}\int_0^t f''(B_s)ds.$$
| From my poor experience in stochastic calculus, I would say that if a stochastic process depends on the Brownian $B_s$, the differential stochastic equation for such a process can be intuitively derived expanding its differential in terms of the Brownian, $\textit{i.e.}$
$$df(s,B_s) =\frac{\partial f}{\partial s}ds+ \f... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2049548",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Gelfand duality for the connection between $K$ and $C(K)$ Suppose $K_1,K_2$ are compact topological spaces, let $\pi$ be a homomorphism: $C(K_2)\rightarrow C(K_1)$, then we can find a continuous map $\tau: K_1\rightarrow K_2$, s.t $\pi(f)=f\circ\tau$. Likewise, if we have $\tau: K_1\rightarrow K_2$, then the map $\pi: ... | Gelfand duality tells you, that you have a contravariant functor between the category of compact topological spaces and the category of commutative $C^*$-algebras (which is even an equivalence).
The part you stated is only the contravariance, i.e. that your functor reverses the direction of the morphisms (continuous m... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2049659",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Computation of a series. NOTATIONS.
Let $n\in\mathbb{N}$. We define the sets $\mathfrak{M}_{0}:=\emptyset$ and
\begin{align}
\mathfrak{M}_{n}&:=\left\{m=\left(m_{1},m_{2},\ldots,m_{n}\right)\in\mathbb{N}^{n}\mid1m_{1}+2m_{2}+\ldots+nm_{n}=n\right\}&\forall n\geq1
\end{align}
and we use the notations:
\begin{align}
m!&:... | Here is a solution of a related problem, followed by a recommendation for the original problem. It would be much simpler if your sum did not have the $n$ in $(n+|m|)!\,$. In that case, we could look at the related sum
$$t_n=\sum_{m\in {\mathfrak{M} }_n}\frac{|m|!}{m!}\prod_{k=1}^n(k+1)^{-m_k}.$$
The sum for the $t$'s c... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2049773",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 1,
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If $p + q = 1$ prove that for any natural $n, m$ following is true: $(1 - p^n)^m + (1 - q^m)^n \ge 1$ Let $p, q \in \mathbb R$ be positive reals for which $p + q = 1$. How to prove that for any two natural numbers $n, m$ the following inequality is true?
$(1 - p^n)^m + (1 - q^m)^n \ge 1$
I don't have a big knowledge ab... | Let there are two coins each with probability of head $p$. Let coin 1 has been tossed $n$ times independently and this whole scheme is repeated independently for $m$ times.
Similarly coin 2 is tossed $m$ times independently and this whole scheme is repeated independently for $n$ times and independently of coin 1.
$A$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2049852",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
"answer_count": 2,
"answer_id": 1
} |
Why can we say $0\leq\sin^2(x)\leq 1$? I often see instructors write:
$0\leq\sin^2(x)\leq 1$
Why is this valid?
Isn't it supposed to be between $1$ and $-1$?
| $x \in \mathbb R$ then $x^2 \ge 0$.
If $|x| \le 1$ then $x^2 = |x|^2 = |x||x| \le |x|*1 = |x| \le 1$.
So as $-1 \le \sin x \le 1$, it follows $(\sin x)^2 \le 1$. And as $(\sin x)^2 \ge 0$, $0 \le \sin^2 x \le 1$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2049940",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Everywhere defined operators must be bounded? I have read in many places that as soon as you have an everywhere defined operator (on a Banach space), it must be automatically bounded, by the Closed Graph Theorem. However, I can't prove this using the Closed Graph Theorem (i.e., I can't prove it would be closed) and I c... | You cannot prove that, as it is not true (with the axiom of choice). The statement, which is true from the closed graph theorem, is:
If $T \colon X \to Y$ is a closed operator defined on a Banach space $X$ into a Banach space $Y$, than $T$ is bounded.
Addendum: Let $X$ be an infinite dimensional Banach space, $Y \n... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2050014",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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unconditional expectation derived from conditional binomial, Poisson and exponential. X|Y~Binomial(Y,p), Y|Z~Poisson(Z), Z~exponential(b),where p and b are constants. What is the E(X) and Var (X)
My way to solve it to to find the joint pdf of (Y|Z) and Z so we can have the marginal (unconditionally)Y. Similar get the m... | From the formulas $\mathbb{E}(\mathrm{Bin}(n,p)) = np$ and $\mathbb{V}\mathrm{ar}(\mathrm{Bin}(n,p)) = np(1 - p),$ you can get that $\mathbb{E}(X\mid Y) = Yp$ and $\mathbb{V}\mathrm{ar}(X\mid Y) = Yp(1 - p).$ Henceforth, $\mathbb{E}(X) = \mathbb{E}(Y)p$ and $\mathbb{V}\mathrm{ar}(X) = \mathbb{V}\mathrm{ar}(Y)p^2(1 - p)... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2050121",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Dividing an infinite power series by another infinite power series Let's say I have two power series
$\,\mathrm{F}\left(x\right) = \sum_{n = 0}^{\infty}\,a_{n}\,x^{n}$ and
$\,\mathrm{G}\left(x\right) = \sum_{n = 0}^{\infty}\,b_{n}\,x^{n}$.
If I define the function
$\displaystyle{\,\mathrm{H}\left(x\right) =
\frac{\math... | The standard way
(in other words,
there is nothing original
in what I am doing here)
to get $H(x)$
is to write
$H(x)G(x) = F(x)$
and get an iteration
for the $c_n$.
$\begin{array}\\
H(x)G(x)
&=\sum_{i=0}^{\infty} c_{i} x^{i} \sum_{j=0}^{\infty} b_{j} x^{j}\\
&=\sum_{i=0}^{\infty} \sum_{j=0}^{\infty} c_{i}b_{j} x^{i+j}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2050216",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "17",
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For the following language, prove, without using Rice's Theorem, whether it is in D, SD but not D, or not in SD. The following language is: $L = \{<M>|¬L(M)∈D \}$
Let's say there is a TM called regTM.
regTM = $\{<M>|L(M) $ is regular$ \}$
I know that regTM is undecidable, therefore I am led to believe any TM for L woul... | Um why think about regular languages?
Hint
Can $L$ be enumerated? If it can, can you construct a TM that decides whether a program $P$ halts on an input $X$ or not? You can easily simulate $P$ on $X$ and if it halts you would observe it, so can you given $P$ and $X$ computably construct a program $Q$ that accepts an un... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Spectrum for a bounded linear operator and its adjoint on a Banach space are same.
I have to show that spectrum for a bounded linear operator and its adjoint on a Banach space are the same. Spectrum is defined as $$ \sigma(T)=\{\lambda\in \mathbb{K}\ :\ T-\lambda I \ \text{is invertible}.\} $$
I have to show $\sigma(... | $T-\lambda I$ is invertible if and only if $(T-\lambda I)^*=T^*-\lambda I$ is invertible: Since for every linear operator $A$ invertibility of $A$ and of $A^*$ are equivalent, which follows by taking the adjoints of, e.g., $AA^{-1}=I$ and $A^{-1}A=I$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2050473",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
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Must an automorphism on the group of real numbers under multiplication maintain sign? Suppose we have an automorphism $\phi$ under the group $(\mathbb{R}^{\#},\,\cdot)$. I need to show that $\phi$ preserves the sign of the numbers, or that $\phi(\mathbb{R}^+)=\mathbb{R}^+$ and $\phi(\mathbb{R}^-)=\mathbb{R}^-$.
I've h... | Can you think of a property, in terms of multiplication only, that positive numbers have but negative ones don't? (HINT: think about $x^2$ . . .)
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2050566",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Composition of functions is continuous?
If $f$ any $g$ be two functions defined from $[0,1]$ to $[0,1]$ with $f$ strictly increasing. Then
*
*if $g$ is continuous, is $f\circ g$ continuous?
*if $f$ is continuous, is $f\circ g$ continuous?
*if $f$ and $f\circ g$ are continuous, is $g$ continuous?
Here, $f\circ g... | In plain English:
*
*if $g$ is continuous, is $f\circ g$ continuous?
Not necessarily. We know that $f$ is strictly increasing, but that does not imply that it is continuous.
Counter-example: Define $f$ as any strictly increasing, non-continuous function.
In other words, stating that the input to $f$ "changes smoo... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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I need help with understanding expected values Consider a discrete random variable $X$ that takes on the values $x_1,...,x_n$ and for every $x_i$ there's a probability $p(x_i)=p_i$. I am attempting to understand why the expectation $E[X]$ is defined the way it is.
$$E[X]=x_1p_1+...+x_np_n$$
I cannot figure out why it ... | I will try to give you some more insight:
suppose you have a random variable $X$ which takes two values (say 4 and 8) with a chance of 50% each.
$P(X=4) = P(X=8) =.5 = p_1 = p_2$.
You can imagine that $X$ is linked to a fair coin and every Head corresponds to 4 and tail corresponds to 8.
The expected value is a nu... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Transform expression from $2^{-n}$ to $x \times 10^{-m}$ I'm trying the understand the concept of how floating number are stored in the computer. While trying to understand it I ran into following kind of transformations:
$2^{-n} \approx x \times 10^{-m}$
For example:
$2^{-1074} \approx 5 \times 10^{-324}$
Can you plea... | You want to convert $a\times 2^b$ to $c\times 10^d$. Obviously, you noticed that $d=\lfloor \log (a\times 2^b)\rfloor$. So, now you need $c$ which is given by $c=a\times 10^{-d}\times 2^{b}$ what you get easily using logarithms. Hope it helps. ( Note: The floor function used to find the exponent "d" gives us the intege... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Find the minimum-variance unbiased estimator for given $\tau(\theta)$
Let $X = (X_1, \dots, X_n)$ - a sample from the distribution $U (0,\theta)$. Prove that $T(X) = X_{(n)}$ is complete and sufficient estimation for $\theta$ and find the minimum-variance unbiased estimator $T^*(X)$ for a differentiable function $\tau... | You have $$E_{\theta}[g(T)]=\int_0^\theta g(x)\frac{nx^{n-1}}{\theta^n}\,dx,\quad\theta>0,$$ for every measurable function $g:(0,\infty)\rightarrow\mathbb{R}$ such that $g(x)x^{n-1}$ is Lebesgue integrable on $(0,\theta)$ for all $\theta>0$. By the Fundamental Theorem of Calculus for the Lebesgue integral, $$\frac{d}{d... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2051015",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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How to draw arrowheads? I am trying draw an arrow from $\begin{bmatrix}x_1\\y_1\end{bmatrix}$ to $\begin{bmatrix}x_2\\y_2\end{bmatrix}$. Here is my work.
If I draw an arrow rotating, then I can draw arrow pointing at any direction. Here is the Java code (full code). This section runs in infinite loop.
x1 = 200; y1=... | Most likely your atan2() function isn't returning what you expect in that region. You may need to add $\pi$ to the result for those values (i.e., when $x_2<x_1$) to get the branch of $\tan^{-1}$ that you desire.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2051149",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 2
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Is $R^n$ projective as a $M_n(R)$-module? It is clear that $R^n$ is not free over $M_n(R)$. But is it projective?
I suspect that it should be projective because we can probably come up with a projective basis, but I'm not sure how to find the basis.
Moreover, ideally if $R^n$ were projective over $M_n(R)$, we would ge... | Yes: $R^n$ is a finitely generated projective generator of $\textrm{Mod-}R$, so it is a finitely generated projective generator as a module over its endomorphism ring, which is the ring of matrices $M_n(R)$.
This is quite easy in general. Let $P_R$ be a finitely generated projective generator of $\mathrm{Mod\text{-}}R$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2051237",
"timestamp": "2023-03-29T00:00:00",
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$A$ is $n\times n$ complex matrix with $A^2=A$. then need to show that Rank$A$=Trace$A$ So, I have that $A^n=A$ for every natural number $n$ which gives me that determinant of $A$ is either $0$ or the $(n-1)^{th}$ roots of unity, but I don't know how to take it from here.What do I do now? Thanks in advance!
| For $x\in\Bbb R^n$ we have $x=(x-Ax)+Ax$ and since $x-Ax\in\ker A$ and $Ax\in Im (A)$ and by the rank-nullity theorem we have
$$\Bbb R^n=Im(A)\oplus \ker(A)$$
Let $p=rank(A)$ and let $\mathcal B=(e_1,\ldots,e_p)$ a basis of $Im(A)$ and $(e_{p+1},\ldots,e_n)$ a basis of $\ker A$ so $(e_1,\ldots,e_n)$ is a basis of $\Bbb... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
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Graphical interpretation of composition of functions We know function and its inverse are mirror image about the line y=x and also that their composition is identity function (y=x again)
So I was wondering if there is a link? I tried to look up for graphical interpretation of composition of functions but I couldn't fun... | If $(x,f(x))$ is a point of the graph of the invertible function $y=f(x)$, than tha point $(f(x),f^{-1}(f(x)))$ is a point of the graph of its inverse function $f^{-1}$, so, since $f^{-1}(f(x))=x$, any point $(x,f(x))$ has its symmetric point $(f(x),x)$, with respect the line $y=x$ on the graph of the inverse function... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2051672",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Difference between conditional probability and Bayes rule I know the Bayes rule is derived from the conditional probability. But intuitively, what the difference? The equation looks the same to me. The nominator is the joint probability and the denominator is the probability of the given outcome.
This is the conditiona... | Baye's thereom uses inverse or posterior probability and also it uses the total probability of an event. You are considering two events here, maybe considering a more general case would shed light on the difference
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2051826",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Alternative way of calculating the multiplicative inverse Suppose I want to find the inverse of $7^{-1} \equiv 1 \mod 19 $, is there a quick way to do instead of extended euclidean algorithm? (Assume the numbers are not large).
Thank you
| Yeah, the way most people do it in competitive programming is as follows:
Suppose you have a number $k$ that is coprime to $n$ and you want to find the inverse of $k$ mod $n$. Then it is just $k^{\varphi(n)-1}$.
You can calculate this really fast using exponentiation by squaring.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2052094",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
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In given figure, prove In the given figure, two chords $AB$ and $CD$ intersect at right angles at $X$. Prove that $\text{arc } AD - \text{arc } CA =\text{arc }BD - \text{arc } BC$
My Attempt:
Join $AC$ and $AD$
$1. \quad \angle CAB+\angle ACD=\angle AXD$
$2. \quad \angle DAB +\angle ADC=\angle AXC$.
$3. \quad \angle A... | $$\angle AOD=2\angle ACD$$
$$\angle AOC=2\angle ADC$$
$$\angle BOD=2\angle BAD$$
$$\angle BOC=2\angle BAC$$
$$\angle ACD+\angle BAC=\angle BAD+\angle ADC=90^{\circ}$$
$$\therefore \quad \angle ACD-\angle ADC=\angle BAD-\angle BAC$$
$$\therefore \quad \angle AOD-\angle AOC=\angle BOD-\angle BOC$$
$$\therefore \quad \ope... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2052243",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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An interesting differential equations problem Suppose we have four ants, initially at rest, at the four corners of a square centered at the origin. They start walking clockwise, each ant walking directly toward the one in front of him. Suppose also that each ant walks with unit velocity, derive a differential equation ... | here is my try. let us look for a symmetric solution. so i will assume the ants at $re^{i\theta}= z, iz, -z, -iz$
the differential equation satisfied by the ant at $z$ is $$ \frac{d}{dt}\left(re^{i\theta}\right) = e^{i (\theta + 3\pi/4)} .$$ this can be written as
$$\frac{d r}{dt} = -\frac 1 {\sqrt 2}, \ \frac{d \t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2052361",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 6,
"answer_id": 0
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Derivative of a definite integral. I'm studying about Fourier Transform and the author wrote this:
$f(b) = \displaystyle \int_{0}^{\infty} e^{-ax^2}cos(bx)dx$
(b is the variable here)
Therefore:
$f'(b) = \displaystyle - \int_{0}^{\infty} xe^{-ax^2}sin(bx)dx$
Well, I cannot see this straightforward, and I am having tro... | Use Leibniz's formula for derivation under integral sign:
$$\frac{\partial f(a,b)}{\partial b}=\frac{\partial }{\partial b}\int_{0}^{\infty}\exp{(-ax^2)}\cos{(bx)}\,dx=\int_{0}^{\infty}\exp{(-ax^2)}\frac{\partial \cos{(bx)}}{\partial b}\,dx$$
The rest is obvious.
Hope it helps
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Limit of a sequence including infinite product. $\lim\limits_{n \to\infty}\prod_{k=1}^n \left(1+\frac{k}{n^2}\right)$ I need to find the limit of the following sequence:
$$\lim\limits_{n \to\infty}\prod_{k=1}^n \left(1+\frac{k}{n^2}\right)$$
|
PRIMER:
In THIS ANSWER, I showed using only the limit definition of the exponential function and Bernoulli's Inequality that the logarithm function satisfies the inequalities
$$\bbox[5px,border:2px solid #C0A000]{\frac{x-1}{x}\le \log(x)\le x-1} \tag 1$$
for $x>0$.
Note that we have
$$\begin{align}
\log\left(\prod_... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2052624",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
"answer_count": 4,
"answer_id": 3
} |
Isomorphisms between quotient groups Is it true that if $\mathbb{Z} \cong A/B$ with $A, B$ abelian groups then $A \cong \mathbb{Z} \times B$? I think it must be true, but can't show it.
| If $A/B\simeq \mathbf Z$, it is isomorphic to a direct summand of $A$. Indeed, consider the commutative triangle:
\begin{align}
p: A \longrightarrow & A/B\\
s\nwarrow\;&\enspace\downarrow \varphi \\
&\enspace\,\mathbf Z
\end{align}
where $p$ is the canonical map, $\varphi$ the given isomorphism and $ s$ is the homomorp... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2052751",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Prove using mean value theorem. $|\arctan(\frac{a}{4})^{4}-\arctan(\frac{b}{4})^{4}|\leq \pi^{3}\cdot|a-b|$
I started with $f(x)=\arctan(\frac{x}{4})^{4}$.
The function is continuous on $[a,b]$ and differentiable in (a,b)
So there exists a $c\in (a,b)$ such that $f'(c)=\frac{\arctan(\frac{b}{4})^{4}-\arctan(\frac{a}{... | Let $f(x) = x^4$, and suppose $|x| \le M$, then $|f'(x)| \le 4 M^3$ for $|x| \le M$.
Using the mean value theorem
we have $|f(x)-f(y)| \le 4 M^3 |x-y|$.
Let $M = {\pi \over 2}$ and note that $|\arctan x| \le M$ and
$|\arctan' x| \le 1$, hence
$|(\arctan x)^4 - (\arctan y)^4| \le {1 \over 2} \pi^3 |\arctan x -\arctan y ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2052865",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 1
} |
Prove that $ (\frac{1+\sqrt5}{2})^{n}+(\frac{1+\sqrt5}{2})^{n-1} = (\frac{1+\sqrt5}{2})^{n+1}$. I have to get from $ (\frac{1+\sqrt5}{2})^{n}+(\frac{1+\sqrt5}{2})^{n-1}$ to $(\frac{1+\sqrt5}{2})^{n+1}$ however I do not know how to get there since i do not know what to do with the exponents.
(not sure if I used the rig... | HINT: Let $\varphi=\frac12\left(1+\sqrt5\right)$; you have to show that $\varphi^n+\varphi^{n-1}=\varphi^{n+1}$. Divide through by $\varphi^{n-1}$ to see that all you really need to show is that $\varphi+1=\varphi^2$. That’s a matter of fairly straightforward arithmetic.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2052999",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
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Showing that a matrix is invertible by factorisation Let $P_5$ be the vector space of polynomials of degree $\leq$ 5 over $Q$.
$P_5 = {a_0 + a_1x + a_2x^2 + a_3x^3 + a_4x^4 + a_5x^5: a_i \in Q}$ and let $D: P_5 \longrightarrow P_5$ be the linear map $D(\alpha) = \frac{d \alpha}{dx}$. By factorising the expression $D^6... | In a ring $R$, $u \in R$ is invertible if $u v = v u = 1$, for some $v \in R$.
Here $R$ is the ring (under composition) of $\mathbb{Q}$-linear operators on $P_5$, $u = D^4 + D^2$, and $1_R = Id$. Can you see what $D^6$ does to $P_5$? If so, you will see that you have found the $v$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2053108",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
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$n \cdot \text{lcm}(a,b,2016) = ab-2016$
Find the largest even number $n$ such that there exist positive integers $a,b$ with $n \cdot \text{lcm}(a,b,2016) = ab-2016$.
I tried using the fact that $\text{lcm}(a,b,2016) \geq a,\text{lcm}(a,b,2016) \geq b,$ and $\text{lcm}(a,b,2016) \geq 2016$, but didn't see how to use ... | Clearly 2016 divides $ab$. Let $m=\mathrm{lcm}(a,b,2016)$; then
$m\ge2016$
$\implies\ ab-2016=nm\ge2016n$
$\implies\ n\le\dfrac{ab}{2016}-1$
Hence $n=\dfrac{ab}{2016}-1$ to be as large as possible. Then
$nm=ab-2016$ $\implies$ $m=2016$
So $a,b$ divide $m$ and $a,b\le m=2016$ $\implies$ $\dfrac{ab}{2016}\le2016$.
Also $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2053221",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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deriving relation for circle in complex plane Prove that $|z-z_1|²+|z-z_2|²=k$ will represent a circle if $|z_1-z_2|²\leq2k$
I tried using the concept of family of circles, but it didn't help me
| Expressing z as $x+yi, z_1=a_1+b_1i, z_2=a_2+b_2i$ so re writing our equation we get $(x-a_1)^2+(y-b_1)^2+(x-a_2)^2+(y-b_2)^2=k$ through simplifying we get that this a circle
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2053345",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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How to recognize that a Poisson Process that is split has two separate Poisson Process through viewing it as a filtration? I read in a paper that if we have a Poisson Process with rate $\lambda$, and that each arrival is separated into two separate categories, $A$ and $B$ with probability $p$ and $1-p$, then each of th... | Comment: It seems you are asking for intuitive arguments. Here are three relevant ones:
1) Suppose a radioactive source emits particles into a counter according
to a Poisson process with rate $\lambda.$ Now a piece of lead foil is placed
between the source and the counter that randomly absorbs half of the
particles. Do... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2053435",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
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Can an integral domain have an element that has no square root but has a square root in the field of fractions? A lemma states:
Let $R$ be a UFD and $F=\operatorname{Frac}(R)$. Let $d\in R$, then equation $a^2=d$ has a root in $R$ iff it has a root in $F$.
So I want to ask, is there a counterexample for this if $R$ ... | You already have a very good example, but since you mentioned ring of integers, and to put this in a more general context:
A (full) ring of algebraic integers (a maximal order) can never work as a counter-example. The point is that those are (basically by definition) integrally closed.
This means that every element o... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2053584",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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General solution to $(\sqrt{3}-1)\cos x+(\sqrt{3}+1)\sin x=2$ $(\sqrt{3}-1)\cos x+(\sqrt{3}+1)\sin x=2$ is said to have a general solution of $x=2n\pi\pm\frac{\pi}{4}+\frac{\pi}{12}$.
My Approach:
Considering the equation as
$$
a\cos x+b\sin x=\sqrt{a^2+b^2}\Big(\frac{a}{\sqrt{a^2+b^2}}\cos x+\frac{b}{\sqrt{a^2+b^2}}\... | Our hint is: $a\cos \theta +b\sin \theta =c$.
Given: $(\sqrt{3}-1)\cos \theta +(\sqrt{3}+1)\sin \theta =2$.
Let $(\sqrt{3}-1) = r\cos \alpha$ and $(\sqrt{3}+1) =r\sin \alpha$.
Then $r\cos \alpha \cos \theta + r\sin \alpha \sin \theta =2 \Rightarrow r\cos(\theta-\alpha) =2 \Rightarrow \cos(\theta-\alpha) =\frac{2}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2053720",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 0
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Nonsingularity and inverse of a matrix. I am having some trouble solving this question:
Let $A,B,C,D$ be $n\times n$ matrices. Suppose $C$ and $A-BC^{-1}B^T$ are nonsingular.
Show that the matrix mentioned is nonsingular and find its inverse
\begin{bmatrix}A&B\\B^T&C\end{bmatrix}
For nonsingularity $\det(M)$ is not $... | Suppose $A, B, C$, and $D$ are matrices of dimension $n × n$, $n × m$, $m × n$, and $m × m$, respectively.
In general, when $D$ is invertible, a similar identity with ${\displaystyle \det(D)}$ factored out can be derived:
$${\displaystyle \det {\begin{pmatrix}A&B\\C&D\end{pmatrix}}=\det(D)\det(A-BD^{-1}C).}$$
See mo... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2053932",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Limit of recursive sequence defined by $a_0=0$, $a_{n+1}=\frac12\left(a_n+\sqrt{a_n^2+\frac{1}{4^n}}\right)$ Given the following sequence: $a_0=0$, $$a_{n+1}=\frac12\left(a_n+\sqrt{a_n^2+\frac{1}{4^n}}\right),\ \forall n\ge 0.$$
Find $\lim\limits_{n\to\infty}a_n$.
| Hint:Take $a_n = \dfrac{\tan \theta_n}{2^n}$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2054026",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 2,
"answer_id": 1
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Which version of Rolle's theorem is correct? #According to my textbook:
Rolle's theorem states that if a function $f$ is continuous on the closed interval $[a, b]$ and differentiable on the open interval $(a, b)$ such that $f(a) = f(b)$, then $f′(x) = 0$ for some $x$ with $a ≤ x ≤ b$.
#According to Wikipedia:
If a real... | These are theorems, not definitions, and both of them are correct. Notice that if Wikipedia is correct, then your textbook is automatically correct as well: if there exists $c\in (a,b)$ such that $f'(c)=0$, then there also exists $x\in [a,b]$ such that $f'(x)=0$, since you can take $x=c$ (since $(a,b)$ is a subset of ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2054174",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
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$u_n \in L^1$ $u_n \to u$ in $L^1$ and $\int_0^1 u_n \ dt = 1$ $\implies$ $\int_0^1 u \ dt = 1$ Consider a sequence $u_n \in L^1([0,1])$. Suppose that $u_n \to u$ in $L^1([0,1])$.
If $$\int_0^1 u_n \ dt = 1$$ $\forall n \in \mathbb{N}$ then why $$\int_0^1 u \ dt = 1?$$ I'm sure this follows from a very basic result, bu... | Note that $u_n \to u$ means that $(u_n - u) \to 0$ in $L^1$, which is to say that $\int|u_n - u| \to 0$. We want to show that $\int u_n - \int u \to 0$. Note, however, that
$$
\left| \int u_n - \int u\right| =
\left| \int (u_n - u)\right| \leq \int |u_n - u|
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2054272",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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9 points in a unit square with pairwise distances greater than $\frac12$
Consider a square with sides of length 1. Can we find 9 points in it such that the distance between every pair of points is strictly greater than $\frac12$?
We allow the points to be on the boundary of the square. The question arose in a convers... | Assume there is a way for picking such $9$ points $P_1,\ldots,P_9$. Consider nine circles $\Gamma_i$ centered at $P_i$, with radius $\frac{1}{4}$. By our assumptions, these circles are disjoint and their union is contained in a square with side length $\frac{3}{2}=1+\frac{1}{4}+\frac{1}{4}$. The sum of the areas of our... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2054357",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
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Proof that any plane perpendicular to x,y plane intersection with the bivariate normal distribution has the shape of a normal distribution. The bivariate normal distribution is given by the equation:
$$f(x,y)=\frac{\exp\left(-\frac{1}{2(1-\rho)^2}\left[\left(\frac{x-\mu_1}{\sigma_1}\right)^2-2\rho\left(\frac{x-\mu_1}{\... | As you can see from another answer, it is possible to carry through
your initial idea to get a proof.
The one detail that I questioned was how we establish that the
coefficient of $x^2$ in
$$
\frac{1}{(1-\rho)^2}\left[\left(\frac{x-\mu_1}{\sigma_1}\right)^2-2\rho\left(\frac{x-\mu_1}{\sigma_1}\right)\left(\frac{mx+b-\mu... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2054498",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
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$AB − BA = A$, $A$ is a nonzero matrix then $B$ is not nilpotent. Let $A,B$ $n\times n$ matrices such that $$AB-BA=A$$ and $A$ is a nonzero matrix. Prove that $B$ is not nilpotent.
I know why $A$ is nilpotent, but how can I prove $B$ is not nilpotent?
| Here's one way to see it: define the linear transformation
$$
\Phi_B(X) = XB - BX
$$
clearly, $A$ is an eigenvector of this transformation associated with $\lambda = 1$. Now, the eigenvalues of $\Phi_B$ are necessarily of the form $\lambda - \mu$ where $\lambda,\mu$ are eigenvalues of $B$ (can be seen via vectorizatio... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2054600",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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Negative Binomial (pascal) Distribution A PC has two color options, white and gray. A customer demands the white PC with a probability of 0.3. A seller of these PCs has three of each color in stock, although this is not known to the customers. Customers arrive and independently order these PCs.
Find the probability th... | Let's distinguish some cases.
*
*First case: The white PC's are bought in the first 3 sales. The probability for this to happen is:
$$p_1 = 0.3^3 = 0.027.$$
*Second case: The white PC's are bought in the first 4 sales. Notice that the last bought PC is white (otherwise, the "experiment" stops at the first 3 sales).... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2054703",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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68-95-99.7 Rule and Normal Distribution Question. I need help solving this question. It is on my final exam review. I was given the answer but i do not know how to solve it.
1) Not everyone pays the same price for the same model of a car. Suppose the price of a car is normally distributed. The mean price of a particul... | In addition to my comment above, you could standardize by obtaining the Z-Score and find the probability that way(where Mean = 1, Standard Deviation = 0)
$$Z = \frac{X-\mu }{\sigma }$$
$$Z =\frac{22750-22000}{750}$$
$$Z=1$$
$$P(Z\geq1) = .1587$$
$$and$$
$$NormalCDF(22750,1000000,22000,750) = .1587$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2054795",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
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On solvable octic trinomials like $x^8-5x-5=0$ Solvable quintic trinomials
$$x^5+ax+b=0$$
have been completely parameterized. Finding $6$th-deg versions is relatively easy to do such as,
$$x^6+3x+3=0$$
which factors over $\sqrt{-3}$. No $7$th-deg are known, but surprisingly there are octic ones, such as the simple,
$$x... | A result of Harris [1] is that every monic palindromic polynomial of degree-8 can be factored into two monic palindromic polynomials of degree-4.
$$
\begin{align}
f(x) & = x^8 + ax^7 + bx^6 + cx^5 + dx^4 + cx^3 + bx^2 + ax + 1 \\
& = (x^4 + px^3 + qx^2 + px + 1)(x^4 + rx^3 + sx^2 + rx + 1) \\
& = x^8+x^7 (p+r... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2054945",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "14",
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A question about jointly continuous random variables. I have this question and I get the answer for it. But I really want to know how E(Y)=E(E(Y|X))
Here is the question and answer.
Question:
An observation X is taken uniformly from (0, 1). Then, let Y be an observation taken uniformly on (X, 1). Find E[Y].
My work:
$... | For jointly continuous random variables , $X,Y$ with joint, marginal, and conditional density functions: $f_{X,Y}(x,y), f_X(x), f_Y(y), f_{Y\mid X}(y\mid x), f_{X\mid Y}(x\mid y)$ .
$$\begin{align}\mathsf E(\mathsf E(Y\mid X)) &= \int_\Bbb R f_X(x)\left(\int_\Bbb R y~f_{Y\mid X}(y\mid x)\operatorname d y\right)\operato... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2055023",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
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Finding a maximum likelihood estimator when derivative of log-likelihood is invalid I need to find the maximum likelihood for $\theta$ given the following:
$X_1, ..., X_n$ are sampled i.i.d from a population with the following density:
$$
f(x | \theta) = \begin{cases} e^{-(x-\theta)} & x \geq \theta \\ 0 & \text{otherw... | As you say, your expressions for the likelihood and log-likelihood are only valid when $\theta$ is less than or equal to all the observed $x_i$; otherwise the likelihood is $0$ and the log-likelihood $-\infty$
Meanwhile, as your derivative suggests, your expressions for the likelihood and log-likelihood are strictly in... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2055248",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Find all $Hom(\mathbb Z/m ,A) $, A- finite Abelian group My attempt:
I want to prove that $Hom(\mathbb Z/m, A) \simeq A$
Let's build map $k$: $f \to f(1)$. It's a group homomorphism:
$fg \to g(f(1)) = g(a) = g(1+1+..+1) = g(1)a=ba=f(1)g(1) $
$k$ is injective and surjective hence bijective.
| A less confusing way to put the question is as follows:
*
*Question: Let $m \in \Bbb Z$ and $C_m$ the cyclic group of order $m$, $A$ a finite abelian group, describe the group of homomorphisms $f : C_m \rightarrow A$ with group composition defined by $(f*g)(x) = f(x)+g(x)$.
*Answer: Let $a \in C_m$ be a generator ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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Why not learn the multi-variate chain rule in Calculus I? I am wondering why we don't learn the multi-variate chain rule in Calculus I? I know the name implies it is more suitable for multi-variable Calculus, but after learning it, I've found it very useful. Notably, one does not need to remember product rule or quot... | I used to think this, too, until I taught Calculus I.
If you, as a math student and enthusiast, like to see the product rule, etc., as special cases of the multivariate chain rule, then that is good for you and deepens your understanding.
However, my experience has been that reasoning from the general to the specific d... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2055481",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "19",
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Let $a,b,c$ be the length of sides of a triangle then prove that $a^2b(a-b)+b^2c(b-c)+c^2a(c-a)\ge0$
Let $a,b,c$ be the length of sides of a triangle then prove that:
$a^2b(a-b)+b^2c(b-c)+c^2a(c-a)\ge0$
Please help me!!!
| Let $c=\max\{a,b,c\}$, $a=x+u$, $b=x+v$ and $c=x+u+v$, where $x>0$ and $u\geq0$, $v\geq0$.
Hence, $\sum\limits_{cyc}(a^3b-a^2b^2)=(u^2-uv+v^2)x^2+(u^3+2u^2v-uv^2+v^3)x+2u^3v\geq0$.
Done!
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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"answer_id": 2
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Use Mathematical Induction to prove equation? Use mathematical induction to prove each of the following statements.
let $$g(n) = 1^3 + 2^3 + 3^3 + ... + n^3$$
Show that the function $$g(n)= \frac{n^2(n+1)^2}{4}$$ for all n in N
so the base case is just g(1) right? so the answer for the base case is 1, because 4/4 = 1
t... | First, show that this is true for $n=1$:
$\sum\limits_{k=1}^{1}k^3=\frac{1^2(1+1)^2}{4}$
Second, assume that this is true for $n$:
$\sum\limits_{k=1}^{n}k^3=\frac{n^2(n+1)^2}{4}$
Third, prove that this is true for $n+1$:
$\sum\limits_{k=1}^{n+1}k^3=$
$\color\red{\sum\limits_{k=1}^{n}k^3}+(n+1)^3=$
$\color\red{\frac{n^2... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2055695",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 6,
"answer_id": 0
} |
How can I show that $4^{1536} - 9^{4824}$ can be divided by $35$ without remainder? How can I show that $4^{1536} - 9^{4824}$ can be divided by $35$ without remainder?
I'm not even sure how to begin solving this, any hints are welcomed!
$$(4^{1536} - 9^{4824}) \pmod{35} = 0$$
| Euler's theorem implies, since $\varphi(35)=24$, that
$$4^{24}\equiv 9^{24}\equiv 1\pmod {35}$$
Since $1536$ and $4824$ are multiples of $24$, the conclusion follows:
$$4^{1536}-9^{4824}=(4^{24})^{64}-(9^{24})^{201}\equiv1^{64}-1^{201}\equiv 0\pmod{35}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2055833",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 5,
"answer_id": 1
} |
Convergence and absolute convergence of $\sum_{n=1}^{\infty} = {(-1)^n \over n + (-1)^{n-1}}$ I am trying to conclude about the convergence and absolute convergence of
$$\sum_{n=1}^{\infty} = {(-1)^n \over n + (-1)^{n-1}}$$
For absolute convergence, we can note that
$$\lvert a_n \rvert = {1 \over n + (-1)^{n-1}}$$
$$\s... | If we just study the $2N$-th partial sum
$$\sum_{n=1}^{2N}\frac{(-1)^n}{n+(-1)^{n-1}} = \sum_{k=1}^{N}\frac{1}{2k-1}-\sum_{k=1}^{N}\frac{1}{2k}=\sum_{k=1}^{N}\frac{1}{2k(2k-1)} $$
we trivially have that our series is conditionally convergent and
$$ \sum_{n\geq 1}\frac{(-1)^n}{n+(-1)^{n-1}} = \color{blue}{\log 2}.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2056011",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
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Why does $A^2=0$ imply that the column space is a subset of the null space So we have a matrix n by n matrix $A$ such that $A^2 =0$. This means that $A^2 = [Aa_1 \space Aa_2 \space \dots Aa_n] = 0$, so $Aa_1 = \dots = Aa_n = 0$.
But why does this imply that col(A) $\subset$ null(A)? The column space is the space spann... | Here's an intuitive explanation that perhaps you can make precise.
A matrix is a linear function on a vector space, and $A^2$ represents composing that function with itself. Now the column space of this matrix is essentially the image of the function, as it is the span of vectors you can get out. Now we apply $A$ once ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2056071",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
Curious limits with tanh and sin These two limits can be easily solved by using De l'Hopital Rule multiple times (I think), but I suspect that there could be an easier way... Is there?
\begin{gather}
\lim_{x\to 0} \frac{\tanh^2 x - \sin^2 x}{x^4} \\
\lim_{x\to 0} \frac{\sinh^2 x - \tan^2 x}{x^4}
\end{gather}
Thanks... | From the standard Taylor series expansions, as $x \to 0$,
$$
\begin{align}
\sin x&=x-\frac{x^3}{6}+\frac{x^5}{120}+O(x^6)
\\\tanh x&=x-\frac{x^3}{3}+\frac{2 x^5}{15}+O(x^6)
\end{align}
$$ ones gets
$$
\begin{align}
\left(\sin x\right)^2&=x^2-\frac{x^4}{3}+O(x^6)
\\\left(\tanh x\right)^2&=x^2-\frac{2 x^4}{3}+O(x^6)
\end... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2056180",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 5,
"answer_id": 2
} |
$u$ upper semicontinuous iff for all $K \subseteq U$ compact and $g \in C(K)$, $u - g$ attains its maximum on $K$? As the question title suggests, how do I see that $u$ is upper semicontinuous if and only if, for all $K \subseteq U$ compact and $g \in C(K)$, the difference $u - g$ attains its maximum on $K$?
| Here's the first half to get you started...
A function $u\in C(U)$ is upper semicontinuous provided that for all $x \in U$, and for all sequences $(x_n)_{n=1}^{\infty}\subset U$ such that $x_n \rightarrow x$, $$\limsup u(x_n) \leq u(x)$$
I make the assumption that you're considering real-valued functions on a metric sp... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2056310",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
Prove that if lim f(x) = L1 and lim g(x) = L2, then lim (f(x))^(g(x)) = L1^L2 I am trying to prove that if
$$
\lim_{x \to c} (f(x)) = L_1
\\ \lim_{x \to c} (g(x)) = L_2
\\ L_1, L_2 \geq 0
$$
Then
$$
\lim_{x \to c} f(x)^{g(x)} = (L_1)^{L_2}
$$
I am doing this for fun, and my prof said that it shouldn't be too hard, but ... | It is often convenient to write $0^0=1,$ for example, in "Let $p(x)=\sum_{j=0}^n a_jx^j$ " it is assumed that $a_0x^0=a_0$ when $x=0.$
But if $L_1=L_2=0$ then $f(x)/g(x)$ can converge to any non-negative value, or fail to converge. Examples: Let $c=0:$
(1). Let $f_1(x)=1/e^{1/|x|}$ for $x\ne 0$ and $g_1(x)=|x|.$ Th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2056499",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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"answer_id": 1
} |
Factor $64x^9 - 125y^6$ I'm trying to help my daughter with Algebra 2. This is a homework assignment. I've done a fair amount of searching but I just can't figure this out. Mathway, an online tool gave the answer:
$(4x^3 - 5y^2) (16x^6 + 20x^3y^2 + 25y^4)$
I can pattern match a bit but can't figure out what's going ... | Way to do this:
$a^3-b^3=(a-b)(a^2+ab+b^2)$
$64x^9 - 125y^6 = (4x^3)^3 - (5y^2)^3$
$= (4x^3 - 5y^2) [(4x^3)^2 + 4x^3 * 5y^2 + (5y^2)^2]$
$= (4x^3 - 5y^2) (16x^6 + 20x^3y^2 + 25y^4)$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2056616",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
How to find Generator Matrix from a given Parity Check Matrix? I'm given a Parity Check Matrix
\begin{bmatrix}0&1&1&1&1&0&0\\1&0&1&1&0&1&0\\ 1&1&0&1&0&0&1\end{bmatrix}
and I have to find the Generator Matrix of it.I spent many days try to solve it but I can't
| This parity-check matrix is in the standard form $$[P^T|I_{n-k}]$$
The generator matrix is hence given by
$$G=[I_k|P]=\begin{bmatrix}
1 &0& 0& 0& 0& 1& 1\\ 0& 1& 0& 0& 1& 0& 1\\0& 0& 1& 0& 1& 1& 0\\ 0& 0& 0& 1& 1& 1& 1
\end{bmatrix}$$
You can verify that $GH^T=0$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2056750",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
Show that $a^2 \equiv a \mod (1+i)$ for all $a \in \mathbb{Z}[i]$. This is a problem statement from a section on quotient rings in Abstract Algebra, so I assume it requires the use of the FIT for rings. When looking around for similar problems, I was only able to find examples with number theory, which isn't really wha... | Note that $ 2 = (1+i)(1-i) \equiv 0 \pmod{1+i} $ and any $ a + bi \in \mathbf Z[i] $ with $ a, b \in \mathbf Z $ is congruent to an integer modulo $ 1 + i $, thus any element of $ \mathbf Z[i] $ is congruent to either $ 0 $ or $ 1 $ modulo $ 1+i $.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2056824",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 5,
"answer_id": 2
} |
$∃x¬(\varphi ∨ \psi) → ∃x(¬\varphi ∨ ¬\psi)$ and $∃y(\varphi ∧ \psi) → (∀x$ $\varphi ∧ ∀y$ $\psi)$ For each of the following formulas, indicate if is or not a first order logic theorem, whatever the formulas $\varphi$ and $\psi$. Justify, showing that exists a natural deduction of the corresponding formula or indicatin... | Hint
1st) Consider that $\lnot (\varphi \lor \psi)$ is equivalent to $\lnot \varphi \land \lnot \psi$.
2nd) Consider : "there exists a number that is $=0$ and $\ge 0$".
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2056904",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Big-O of recursive function Let $f:\mathbb{Z}_+ \to \mathbb{Z}_+$ be the function defined by $f(k)=3f(k-1)+2$ for any $k \in \mathbb{Z}_+$. Prove that $f(n)$ is $O(6^n)$.
How do I prove it with mathematical induction?
| Let $g(k) = f(k)+1$, then $g(k) = 3 g(k-1)$ is a geometric sequence, hence $g(k)=3^k g(0)$ and $f(k) = 3^k (f(0)+1)-1$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2056993",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Does there exists a finite abelian group $G$ containing exactly $60$ elements of order $2$? Suppose there exists such a group. Then Lagrange's theorem assures that the group is of even order.
But I conclude from this and this that such a group has odd number of elements of order $2$. Giving us contradiction.
Hence ther... | Clearly, the order of $G$ cannot be an odd, it's obvious. Suppose the order of $G$ is $2n$. Since order of identity is always $1$ i.e. $|e|=1$. So we have left only $2n-1$ element which is odd in number. Elements which are not their own inverses, these elements and their inverses exist in pairs.
Then it should be even ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2057111",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 2,
"answer_id": 1
} |
Limit of a series with upper bound in the summand? I have constructed a model of drug dosing, and to find the maximum quantity of drug in the body after an infinite number of doses, I believe I must compute this limit:
$$\lim_{n \to \infty} D \displaystyle\sum_{i=1}^{n} e^{-(n-i)k\Delta t}$$
where $D, k, \text{and } \D... | Your sum is equivalent to $$\lim_{n\to \infty} De^{-nk\Delta t}\sum_{i=1}^{n} (e^{k\Delta t})^i$$
Which is a geometric series.
$$\lim_{n\to \infty} De^{-nk\Delta t}\sum_{i=1}^{n} (e^{k\Delta t})^i=\lim_{n\to \infty} De^{-nk\Delta t} \frac{e^{k\Delta t}(e^{k\Delta t n}-1)}{e^{k\Delta t}-1}$$
Carry out the multiplicatio... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Alternatives to the politician theorem A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R and S are invited to a birthday party. We are told that:
*
*each pair of guests have exactly one common friend
*A and G only have one common friend: C
*A and G are friends
*I and C only have one common friend: A
How many... | I had never encountered this surprising theorem - thanks for posting.
In order to give a concrete example, the question adds some extra conditions. Under these $A$ has at least three friends: $G$ (by the second condition) and $I$ & $C$ (by the third condition). Therefore it is $A$ who is the politician. Note that this ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2057393",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 1
} |
Isomorphic groups but not isomorphic rings Provide an example of two rings that have the same characteristic, are isomorphic as groups but are not isomorphic as rings.
I'm confused with how to being. I know that having the same characteristic means that the concatenation is the same number to receive the zero element.
| The group isomorphism refers to the additive structure.
Let $R$ be any ring. We can define two ring structures on the set $R\times R$: the addition is the same, so the two additive groups are not only isomorphic, but identical:
$$
(a,b)+(c,d)=(a+c,b+d)
$$
We can define two different multiplications:
$$
(a,b)\cdot(c,d)=... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2057709",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "12",
"answer_count": 7,
"answer_id": 6
} |
Find all positive integer solutions for M Find all positive integer solutions M, where $x,y,z$ are non-negative integers from the equations,
$x + y + z = 94$
$4x + 10y + 19z = M$
Attempt:
I multiplied equation 1 by 4, and subtracted it from equation 2 to get
$6y + 15z = M-376$
I know $M-376$ has to be a multiple of 3, ... | Let $S$ be the set of all combinations of the form $2y+5z$ such that $y$ and $z$ are non-negative integers with $y+z\leq 94$.
You want the set $3S+376$.
So we just need to determine $S$.
We shall prove that $S$ contains every integer in the range $\{1,2,\dots , 94\times 5\}$ except a select few.
Which numbers that are ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2057819",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 0
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A normal subgroup with index relatively prime to its order This is an exercise from Lang's Algebra. The theorem and my work on it are below:
Let $G$ be a finite group and $N$ be a normal subgroup such that $N$ and $G/N$ has relatively prime orders. I need to show the following two:
i)Let $H$ be a subgroup of $G$ having... | For $i)$:
Since $|H|=|G/N|$, $|H|$ and $|N|$ are relatively prime by assumption which means that if $x\in H\cap N$, $x=e$ for the order of $x$ must divide both $|H|$ and $|N|$. Because
$$|HN|=\frac{|H||N|}{|H\cap N|}$$
we see that $|HN|=|H||N|=|G|$.
For $ii)$ what you've written is not correct. For one, $f_2$ isn't eve... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2057918",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
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Confirming the Existence the PDE Solutions in Sobolev Spaces So far I have only the most basic understanding of the Sobolev space, like the existence of a unique weak solution, or what it means in the PDE:
$$\Delta u = -f ,\qquad \text{where } f \text{ is a Schwartz function}$$
However, how do I think about a PDE when ... | I assume that $\Omega$ is bounded and has smooth boundary and that you have zero Dirichlet boundary conditions. If $f\in L^2(\Omega)$ (in fact one can take weaker conditions $f\in H^{-1}(\Omega)$ here) then consider the weak formulation of the Poisson equation:
$$\int \nabla u \cdot \nabla v = \int f v, \forall v\in H... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2057996",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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"answer_id": 0
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Graph $\text{Im}\left(\frac{1}{z}\right)=1$ I used the identity $z=x+iy$ which resulted in $\text{Im}\left(\frac{1}{x+iy}\right)$. Multiplying by the conjugate, I found that this was equal to $\text{Im}\left(\frac{x-iy}{x^2+y^2}\right)$, which by splitting this fraction into two terms is $\frac{-y}{x^2+y^2}=1$. Multipl... | You're almost there. We have
$$x^2+y^2=-y\implies x^2+(y+1/2)^2=\frac14$$
In the complex plane, $x=\text{Re}(z)$ and $y=\text{Im}(z)$. Therefore, this is a circle with center $(0,-1/2)$ and radius $1/2$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2058120",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
} |
Equation with matrix and determinant Given
$$A={\begin{bmatrix}
\det(A) & \det(A)+a \\
\det(A) +b & \det(A)+c
\end{bmatrix}}$$
where $a,b,c$ are given and $A$ is unknown, is it possible to use some clever tricks concerning determinants for this case? (instead of direct calculations).
| Let $\text{det}A=d$. Then we are given
$$A={\begin{bmatrix}
d & d+a \\
d +b & d+c
\end{bmatrix}}$$
Using row operations, we get
$$
d=|A|=\begin{vmatrix}
d & d+a \\
d +b & d+c
\end{vmatrix} \rightarrow \begin{vmatrix}
d & d+a \\
b & c-a
\end{vmatrix}
\rightarrow \begin{vmatrix}
d & a \\
b & c-a-b
\end{vmatrix}
$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2058247",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
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Why is a line a closed subset of $\mathbb R^2$? I'm studying topology and I have a doubt in the following exercise. I'd appreciate some help. Let $m\neq0$ and $c$ be real numbers. Prove that the line $L=\{\langle x,y\rangle:y=mx+c\}$ is a closed subset of $\mathbb R^2$. I found similar questions here, but with answers ... | If you know that the real line is a closed subspace of $\Bbb{R}^2$ (indeed, it's a complete metric subspace, which is even stronger) the result follows easily because the real line can be mapped to any other line in $\Bbb{R}^2$ by a composition of a rotation and a translation. Since rotations and translations are isome... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2058367",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 5,
"answer_id": 4
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In the derivation of the formula for volume of a solid of revolution, how does $Δx$ "become" $\mathrm dx$? I am currently learning about the formula for the volume of a solid of revolution formed by the rotation about the $x$-axis through 2$\pi$ radians. I believe this is called the "disk" method.
Referring to the sec... | $\def\d{\mathrm{d}}\def\peq{\mathrel{\phantom{=}}{}}$The identity (Note that $Δx = \dfrac{b - a}{n}$)$$
\lim_{n → ∞} \sum_{k = 1}^n A(x_{n, k}) · \frac{b - a}{n} = \int_a^b A(x) \,\d x
$$
is the corollary of the definition of Riemann integral. For any $n \geqslant 1$, the notation $Δx$ means a small length, whereas the... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2058453",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 1
} |
Integration with Half space Gaussian I have a problem to solve and I have something that i don't know how to do.
The half-space Gaussian integral is given :
$$\int_{0}^\infty \exp(-ax^2)dx = \frac{1}2 \sqrt{\frac{\pi}{a}}$$
I have to calculate
$$\int_{0}^\infty \exp \left(-ax^2 - \frac{b}{x^2} \right)dx$$
a and b are ... |
Note that in THIS ANSWER, I presented a solution to a more general version of the integral of interest herein.
Let $I(a,b)$ be the integral given by
for $a>0$ and $b>0$.
$$I(a,b)=\int_0^\infty e^{-\left(ax^2+\frac{b}{x^2}\right)}\,dx \tag 1$$
Enforcing the substitution $x\to \sqrt[4]{b/a}x$ into $(1)$ reveals
$$\be... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2058562",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
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An Inequality $\left(\sum\limits_{k=1}^n a_k^{1/2}\right)^2\le\left(\sum\limits_{k=1}^n a_k^{1/3}\right)^3$
Why is $\left(\sum\limits_{k=1}^n a_k^{1/2}\right)^2\le\left(\sum\limits_{k=1}^n a_k^{1/3}\right)^3$ with $a_k$ nonnegative
Writing
$\left(\sum\limits_{k=1}^n a_k^{1/2}\right)^2=\left(\sum\limits_{k=1}^n a_ka... | For simplicity, let $n=2$. It is easy to generalize to the case of $n>2$. Let
$$ a_1=r^2b_1^4, a_2=r^2b_2^4$$
such that
$$ b_1^2+b_2^2=1,b_1,b_2\ge0. $$
Then the inequality becomes
$$ b_1^{\frac43}+b_2^{\frac43}\ge 1 $$
which is easy to prove. In fact, noting $0\le b_1,b_2\le1$, one has
$$ b_1^{\frac43}+b_2^{\frac43}\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2058667",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
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Continuity of a function with real paramter Let $f:\mathbb{R^3}$ $\rightarrow$ $\mathbb{R}$, defined as:
$$f(x,y,z)=\begin{cases} \left(x^2+y^2+z^2\right)^p \exp\left(\frac{1}{\sqrt{x^2+y^2+z^2}}\right)& ,\,\text{if }\quad(x,y,z) \ne (0,0,0)\quad \\
0 &,\,\text{o.w}
\end{cases}$$
Where $\,p\in \mathbb{R}$. Is this f... | $f$ is continuous in $\mathbb{R}^3\setminus\{(0,0,0)\}$, but not in the point $(0,0,0)$ since the limit of $f(x,y,z)$ when $(x,y,z)\to(0,0,0)$ does not exist: To see it recall that
$$
e^{t}=1+t+\frac{t^2}{2!}+\frac{t^3}{3!}+\frac{t^4}{4!}+\cdots,
$$
so for $t>0$ we have
$$
e^{t}\geq\frac{t^{2p+2}}{(2p+2)!}.
$$
Plugging... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2058939",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
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computing flux integral just given this question. compute the flux out of the unit circle, C. $$F(x,y)=\langle x+2y,3x+4y\rangle $$
i am not sure on how to solve this. Usually the flux would include Z function. please help!
| The flux of $F(x,y)$ across $C$ is given by $\int_C F(x,y)\cdot n\,ds$, where $n$ is the outward normal vector to $C$. Using the planar divergence theorem, you could also calculate this integral as:
$$\int_C F(x,y)\cdot n\,ds=\iint_D\nabla\cdot F(x,y)\,dA$$
Where $D$ is enclosed by $C$ (in this case $D$ is the unit dis... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2059010",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
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Prove that, if G is a bipartite graph with an odd number of vertices, then G is non-Hamiltonian Continuing with my studies in Introduction to Graph Theory 5th Edition by Robin J Wilson, one of the exercises asked to prove that, if $G$ is a bipartite graph with an odd number of vertices, then $G$ is non-Hamiltonian. Thi... | Yes, your proof is quite correct.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2059091",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 0
} |
Probability distribution for y=min(x,a) let's say a random x has the following pdf, then how we can find the distribution for $y=min (x,a)$, for $\theta <= a <=w $?
$(x-\theta)^k exp(-\lambda(x-\theta))$. for $\theta<=x<=w$
Thank you for helping!
| $\theta$ is just a shift parameter, either for $x$ and $a$ and $w$ and thus also for $y$, and we can get rid of it, leaving
$$
\left\{ \begin{gathered}
0 \leqslant a \leqslant w\quad p(a) = 1/w\quad \left( {\text{?}\;\text{supposed}} \right) \hfill \\
0 \leqslant x \leqslant w\quad p(x) = x^{\,k} e^{\, - \,\lambda... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2059209",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
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Do any major theorems of complex analysis that require holomorphic functions fail if the function is only holomorphic up to removable singularities? Or is "holomorphic on $\Omega$" universally (i.e., in practice and standard texts, and including for purposes of Ph.D. quals) understood to mean "at worst possessing a hol... | A bit less trivial (but still trivial) example is Picard theorem
| {
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"url": "https://math.stackexchange.com/questions/2059363",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Set of all finite subsets of the plane How could I determine the cardinality of the set of all finite subsets of the plane? I believe I am correct in saying that this set is equivalent to the power set of $\Bbb R^{2}$ minus all infinite sets in that set. Is it correct to say that I can map any set of size $k$ to $\Bbb ... | The set is
$$\mathcal{P}_f\left(\mathbb{R}^2\right)=\bigcup_{n=0}^{\infty}P_n\left(\mathbb{R}^2\right)$$
where $P_n\left(\mathbb{R}^2\right)=\{S\subset\mathbb{R}^2\,;\,|S|=n\}\subset{\left(\mathbb{R}^2\right)}^n$.
Now, clearly, the cardinality of $P_1\left(\mathbb{R}^2\right)$ is $\left|\mathbb{R}^2\right|=|\mathbb{R}|... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2059499",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
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Every cyclic subspace contains an eigenvector The question is :
Let $X$ be a non-null vector.Then there exists an eigenvector $Y$ of $A$ belonging to the span of $\{X, AX, A^{2} X, ... \}$.
I have tried to the best of my ability to solve it. But I don't find any right way to proceed.Please help me.
I want to add a so... | Why $Y\ne 0$: Let $h(t)=\prod_{i=2}^k(t-\beta_i).$ Then $\deg (h)<k$ and $h(t)$ is not identically $0$ so by the minimality of $k$ we have $0\ne h(A)(X).$ But $h(A)(X)=Y.$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2059608",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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"answer_id": 0
} |
bromwich inverse laplace of $\frac{1}{\sqrt{s+1}}$ I want to use the Bromwich integral to evaluate the inverse laplace of $\frac{1}{\sqrt{s+1}}$.
The complex function $\frac{e^{st}}{\sqrt{s+1}}$ has a pole and branch point in -1. I cannot find a good contour to evaluate. Am I right when I say the countour should enclos... | $\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... | {
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If $\frac{\sqrt{31+\sqrt{31+\sqrt{31+ \cdots}}}}{\sqrt{1+\sqrt{1+ \sqrt{1+ \cdots}}}}=a-\sqrt b$, find the value of $a+b$
$\frac{\sqrt{31+\sqrt{31+\sqrt{31+ \cdots}}}}{\sqrt{1+\sqrt{1+ \sqrt{1+ \cdots}}}}=a-\sqrt b$ where $a,b$ are natural numbers. Find the value of $a+b$.
I am not able to proceed with solving this ... | √(31+√(31+√(31....))) = s
s = √(31+s)
s² = s+31
s = (5√5 + 1)/2 (by the quadratic formula)
√(1+√(1+√(1....))) = k
k = √(1+k)
k² = k+1
k = (√5 + 1)/2 (by the quadratic formula)
s/k = (5√5 + 1)/(√5 + 1)
= (5√5 + 1)(√5 - 1)/4 (multiplying the numerator and denominator both by conjugate √5 - 1)
= (24-4√5)/4
= 6-√5
Since 6 ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2059779",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Proof verification: almost linear map is injective iff the kernel only contains zero. $f:V\to W$ is a $\mathbb{R}$-$\mathbb{C}$-linear map between a $\mathbb{R}$-vector space $V$ and a $\mathbb{C}$-vector space $W$ if
$$
f(av+bw)=af(v)+bf(w):=(a+i0)f(v)+(b+i0)f(w)
$$
for any $a,b\in\mathbb{R}$ and $v,w\in V$.
Theorem:... | Your proof is fine.
Here is another reason it should work out the same: The vectors in a complex vector space also form a real vector space under vector addition and scalar multiplication by reals. Your "almost linear" maps are then linear maps to this corresponding real vector space. Since you already have this resul... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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"answer_count": 1,
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All fractions which can be written simultaneously in the forms $\frac{7k-5}{5k-3}$ and $\frac{6l-1}{4l-3}$ Find all fractions which can be written simultaneously in the forms $\frac{7k-5}{5k-3}$ and $\frac{6l-1}{4l-3}$
for some integers $k,l$.
Please check my answer and tell me is correct or not....
$$\frac{43}{31},\f... | Suppose there is integer $p$ which can be written as $\frac{6l-1}{4l-3}$ and $\frac{7k-5}{5k-3}$.
$$p= \frac{6l-1}{4l-3} =\frac{7k-5}{5k-3}$$
$$\implies kl+8k+l=6$$
$$\implies(k+1)l=(6-8k)\implies l=\frac{-2(4k-3)}{(k+1)}$$.
Which gives following integer solutions:
$(k,l)=(-15,-9),(-8,-10),(-3,-15),(-2,-22),(0,6),(1,-... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2060154",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 2
} |
Prove that $P(\bigcup_{i=1}^{\infty} A_i) = 1$ $A_i$ $(i=1,2,...)$ are independent events
$\sum_{i=1}^{\infty}P(A_i) = \infty.$
Prove that:
$P(\bigcup_{i=1}^{\infty} A_i) = 1 $
Can someone please help me out with this question?
| By virtue of the non-trivial part of Borel-Cantelli lemma, one is allowed to conclude that the event $\displaystyle \mathrm{A} = \{A_n, \mathrm{i.o.}\} = \bigcap_{k = 1}^\infty \bigcup_{n = k}^\infty A_n$ has total mass equal to one. Notice that $\mathrm{A}$ lies within the union of the $A_n$ and the exercise follows.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2060285",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 1
} |
Proving critical point of a system is a center We have
$$ \begin{cases} \dot{x} = - y - y^3 \\ \dot{y} = x \end{cases} $$
where $x,y \in \mathbb{R}$. Show that the critical point for the linear system is a $\mathbf{center}$. Prove that the type of the critical point is the same for the $\mathbf{nonlinear}$ system.
TRY:... | You can use the following Lyapunov function candidate
$$V(x,y)=1/2x^2+1/2y^2+1/4y^4,$$
which is positive definite and $V(x=0,y=0)=0$.
The derivative is given by
$$\dot{V}=x\dot{x}+y\dot{y}+y^3\dot{y}=x[-y-y^3]+yx+y^3[x]\equiv0.$$
This condition implies that the origin is a center.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2060418",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Having trouble understanding summation identity I'm having trouble understanding how the right side of this summation is equal to the left side. Any information would be helpful!
$$\sum_{k=0}^{n} {r \choose k}{s \choose n-k}={r+s \choose n}$$
| This is known as Vandermonde's Identity. It says that for each way to choose $n$ items from a set of $r+s$ items, for some $k$ we must choose $k$ from the set of $r$ and $n-k$ from the set of $s$. This translates to
$$
\binom{r+s}{n}=\sum_{k=0}^n\binom{r}{k}\binom{s}{n-k}\tag{1}
$$
Another way to look at this is to lo... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2060567",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 1
} |
Simplifying $\frac { \sqrt2 + \sqrt 6}{\sqrt2 + \sqrt3}$? Is there anything else you can do to reduce it to something "nicer" other than multiplying it by $\dfrac {\sqrt3 - \sqrt2}{\sqrt3 - \sqrt2}$ and get $\sqrt 6 -2 + \sqrt {18} - \sqrt {12}$? The reason I think there's a nicer form is because the previous problem i... | Well, you could note that
$$\sqrt6-2+\sqrt{18}-\sqrt{12}=-2+\sqrt6(1-\sqrt2+\sqrt3)$$
But beyond that, it doesn't look any better.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2060719",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Simple Dice Probability Question This problem is throwing me off because it seems extremely simple, but the answer given is not what I get. The book has the answer as 5/36, but wouldn't it be 6/36 which would reduce to 1/6? Because you could have the combinations 4/6, 5/5, 5/6, 6/4, 6/5, 6/6 which is six total combinat... | The text book must be wrong then
6 Combinations (4,6), (5,5), (5,6), (6,4), (6,5), (6,6)
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2060856",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
prove the integral function is continuous
Let $f: [a,b]\times [c,d]\to \mathbb{R}, (x,\alpha)\to f(x,\alpha)$ is a 2 variable continuous function on $[a,b]\times [c,d]$, then $h(\alpha )=\int _a^b f(x, \alpha )dx$ is a continuous function on $[c,d]$.
My attempt: I've tried to prove this using the definition of 2-var ... | Hint: the projection of a continuous function onto any coordinate is continuous, and the integral of a continuous real function is continuous.
In a general sense, we may define the space $\mathbb{R}^2$ to be the collection of ordered pairs from $\mathbb{R}$, equipped with the property that the projection of any contin... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2060964",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Calculate the derivative of $g(x)=x^2.cos(1/x^2)$. Does the derivative of this function exist at $x=0$? Calculate the derivative of $g(x)=x^2.cos(1/x^2)$. Does the derivative of this function exist at $x=0$?
I calculated the derivative to be $2.cos(1/x^2)x+2(sin(1/x^2))/x$.
I'm tempted to say that the derivative does... | $g'$ does not have limit in $0$, but it does not mean $g$ not differentiable in $0$ :
$$\frac{g(x)-g(0)}{x}=x\cos\frac{1}{x^2}\xrightarrow[x\to0]{}0$$
so $g'(0)=0$.
All you can say is that $g'$ is not continuous in $0$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2061065",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
$\sqrt{3-x}-\sqrt{-x^2+8x+10}=1$ Consider the equation
$$
\sqrt{3-x}-\sqrt{-x^2+8x+10}=1.
$$
I have solved it in a dumb way by solving the equation of degree four. So, the
only real solution is $x = -1$. Can you please suggest, maybe there are better or easier ways to solve it?
| You did it in the right way. You will end up with a $x^4$ term, and you must solve from there to get $-1.$
The above answer is right, that it gives you a $range$ of values, but if you want to solve directly, you must do it the way you have said above.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2061205",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
} |
Random placement of rooks on a chessboard $8$ rooks are placed randomly on an $8\times 8$ chess board.
What is the probability of having exactly one rook each row and each column?
I guess there is no meaningful order here?
| Hint. Eight (indistinguishable) rooks can be placed on an $8\times 8$ chess board in $\binom{64}{8}$ ways (we have to select $8$ positions among $8^2=64$).
Having exactly one rook each row and each column can be done in $8!$ ways (for each of the $8$ columns we choose e different row).
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2061325",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Is there a math symbol for $f(x) > g(x)$ except when $x=0$, in which case $f(x)=g(x)$? Title says it all, except that $x \in R^n$. For want of a better choice, I've specified the amsmath symbol $\gtrdot$ for this relation, but probably this actually means something quite different that google hasn't revealed?? I... | You could define a Boolean function: $$D^+(f, g) = \delta^{f(0)}_{g(0)} | f(x) > g(x) \forall x \neq 0 |$$ Presumably everyone reading your paper knows what the Kronecker delta and the Iverson bracket are. It's much easier to look for a prior definition of a function than it is to look for a prior use of a symbol, as y... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2061459",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 1
} |
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