Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Solve for $a,b,c,d \in \Bbb R$, given that $a^2+b^2+c^2+d^2-ab-bc-cd-d+\frac 25 =0$ Today, I came across an equation in practice mock-test of my coaching institute, aiming for engineering entrance examination (The course for the test wasn't topic-specific, it was a test of complete high school mathematics). It was havi... | Multiply by $2$ and rearrange to \begin{align*}(a-b)^2 + (b-c)^2 + (c-d)^2 + (d-a)^2 + 2ad - 2d + \frac{4}{5} = 0. \tag{$\star$}\end{align*} For fixed $a$ and $d$, the minimum value of $(a-b)^2 + (b-c)^2 + (c-d)^2$ is $\frac{(d - a)^2}{3}$, with equality if and only if $a, b, c, d$ is an arithmetic progression by the ... | {
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Two different points of a metric space are contained in disjoint open balls To prove: Let $(X,d)$ be a metric space and $x,y \in X$ two different points. Show that there exist $r_x, r_y \gt 0$ such that $B(x, r_x) \cap B(y,r_y) = \varnothing$.
My solution: Let $r_x, r_y = {d(x,y)\over 2}$. Suppose that $B(x, r_x) \cap ... | Your proof is fine.
You could slightly reword your proof to make it appear in natural language to not be a proof by contradiction. I don't think this fundamentally changes anything, but it might make the proof read a little more nicely.
Let $z$ be arbitary, and let us show that $z \notin B(x,r_x) \cap B(y,r_y)$. Witho... | {
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Riemannian metric tensor defined on (0,2)-tensors I'm interested in the intuition for the following calculation:
$|Ric_g|_g^2 = g^{ia}g^{jb}R_{ij}R_{ab}$,
where $R_{ij}$ are the components of the Ricci curvature tensor.
Here's my thought process: Locally, we have
$|Ric_g|_g^2 = g(Ric_g,Ric_g) = R_{ij}R_{ab} \hspace{... | Once we've fixed a metric $g$, we can write this as a total contraction
$$(A, B) \mapsto \operatorname{contr}(g^{-1} \otimes g^{-1} \otimes A \otimes B) ,$$ (here, $\operatorname{contr}$ just denotes the appropriate composition of trace operators) or in abstract index notation,
$$(A_{ab}, B_{cd}) \mapsto g^{ac} g^{bd} ... | {
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How to build smooth families of functions satisfying a power unity? The equation $$f_1(t)^2+f_2(t)^2 = 1$$
Has the very famous solution $\cases{f_1(t) = \sin(kt)\\f_2(t) = \cos(kt)}$
Sometimes called the trigonometric unity or the triangle union. Sin and cos are also functions which are famous for being very well-behav... | Easily.
Choose arbitrary functions (as well-behaved as one likes) $g_i$ with $|g_i| \le 1$ for $i < k$. Define $$f_1 = g_1\\f_2 = g_2\sqrt{1 - f_1^2}\\f_3 = g_3\sqrt{1 - f_1^2 - f_2^2}\\\vdots$$
Finally, choose $f_k$ so that $f_k^2 = 1 - f_1^2 - f_2^2 - ... - f_{k-1}^2$. Obviously if the functions are to be continuous... | {
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Finite cardinality modules over $R=\Lambda(\Gamma)\cong\mathbb Z_p[[T]]$ Let $Q$ be a finite cardinality module over $R$. Let $\Gamma\cong\mathbb Z_p$, ($\Gamma$ is written multiplicatively, and in $\mathbb Z_p[[T]]$ corresponds to $(1+T)^{\mathbb Z_p}$). We have the following obvious inclusions among invariant modules... | The action of $R$ on $Q$ gives a homomorphism from $\mathbb{Z}_p$ to the group $A$ of automorphisms of $Q$. Since $Q$ is finite, $A$ is finite, so this homomorphism factors through the quotient $\mathbb{Z}_p\to\mathbb{Z}/p^n$ for some $n$. For that value of $n$, then, $Q^{\Gamma^{p^n}}=Q$.
(Note that the fact that th... | {
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Example about about function has no upper and lower bound I want to find a function $f:[-1,1]$ to $\mathbb{R}$ which has no upper bound and lower bound.
Does the linear function $f(x) = \tan(x)$ work, and if so how?
Appreaciate any help with that.
| Consider for example $x\mapsto \frac{x}{(x-1)(x+1)}$ (you may set an arbitrary value at $1$ and $-1$ so that it's well-defined).
If you want to stick with $\tan$, consider $g:x\mapsto \frac{\pi}2 x$. Then $\tan \circ g$ with some additional values at $x=-1$ and $x=1$ fits the bill.
| {
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Primitive Recursion - A function which grabs an arbitrary argument I am studying partial recursive functions, and while I think I understand most elements of how to prove a given function is primitive recursive, there is one particular pattern that I can't come up with a good explanation for.
Specifically, how would I ... | As per @wet's comment on the post, $f$ may be defined by:
$f(x_1, \dots, x_n, i) = \sum_{j=0}^{i} \chi_{=}(i, j) \cdot x_j$
Where $\chi_{=}(x, y)$ is the characteristic function for the predicate $x = y$.
Then, since summation, multiplication and $\chi_{=}$ are all primitive recursive (by the usual proofs for each), an... | {
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Group Theory Example What is an example of a group $G$ and a subgroup $H$ such that $|G : H|$ is infinite? I am unsure of how to approach it as it was given as an open ended exercise. Would the group $\mathbb{Z}$ and any subgroup work?
| No, most subgroups of $\mathbb{Z}$ are of the form $n\mathbb{Z}$, which have index $n$. There is one subgroup of $\mathbb{Z}$ which has infinite index, the trivial group. It will have infinite index in any infinite group.
For other options, note that the group $\mathbb{Q}$ is not finitely generated, which means that $\... | {
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Proof verification for $\lim_{x\rightarrow 0} \left(\frac{9}{x} - 9\cot(x)\right) = 0$ About 15 minutes ago I came across a question on MSE asking about
$$\lim_{x\rightarrow 0} \left(\frac{9}{x} - 9\cot(x)\right)$$
Four people instantly answered it - two of the solutions used L'H$\hat{\mathrm{o}}$pital's rule, and the ... | Alright this is embarrassing but I just figured it out as I made the post -
I need to know the limit exists before I can make such an assertion:
$$f(x) = \sin(1/x)$$
is an odd function, but it clearly doesn't approach any limit as $x$ tends to $0$. If I can show that the limit exists, then it must be zero. There are pl... | {
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Prove that if $n$ is a positive odd integer then $1947\mid (46^n+296\cdot 13^n)$ This is an exercise from The Kürschák Mathematics competition from the year 1947:
Prove that if $n$ is a positive odd integer then $1947\mid (46^n+296\cdot 13^n)$.
I have the solutions in the back of the book but I would like to tackle t... | Because for $n=1$ it's true and for all odd $n\geq3$ we obtain:
$$46^n+296\cdot13^n=46\cdot46^{n-1}-46\cdot13^{n-1}+(46+296\cdot13)13^{n-1}=$$
$$=46\cdot(46^2-13^2)\left(46^{\frac{n-1}{2}-1}+...+13^{\frac{n-1}{2}-1}\right)+2\cdot1947\cdot13^{n-1}=$$
$$=1947\left(46\left(46^{\frac{n-1}{2}-1}+...+13^{\frac{n-1}{2}-1}\rig... | {
"language": "en",
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Representation of Dirac function over Sobolve space Let $1<p<\infty$ and $q=\frac{p}{p-1}$. We know that $l\in (W^{k,p}(\Omega))'$ ($l$ is a continuous functional over $W^{k,p}$(\Omega)) if and only if there exits $\{f_\alpha\}\subset L^q(\Omega)$ such that
$$
l(v)=\sum_{|\alpha|\leq k}\int_{\Omega} f_\alpha(x) \parti... | Let us take $f_1=\begin{cases} x+1,\,\, x<0\\ 0,\,\, x\geq 0\end{cases}\,\, f_0=\begin{cases} 1,\,\, x<0\\ 0,\,\, x\geq 0\end{cases}$. Then we can obtain by integration by parts that
$$
v(0)=\int_{-1}^1 f_0(x) v(x)+f_1(x) v'(x)dx\,\,\, \forall v\in H^1(-1,1)
$$
| {
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Non-Traditional Definition of Dual Basis in $\mathbb{C}^N$ From an undergraduate book on Harmonic Analysis:
Question 1: What is the relationship between this description of a "dual basis" and this more traditional one?
Question 2: To check my understanding, let $N = 2$ and consider the basis $\{(2, 0), (0, 2i) \... | It is the same, under the canonical identification of $\mathbb C^N $ with its dual. The duality consists of writing every functional as $v\longmapsto \langle v,w\rangle $.
So from the formula you have $$\langle w_k,v_j\rangle=\delta_{k,j}, $$ which makes $w_1,\ldots, w_N $ the dual basis.
To your second question: yes.... | {
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What is the derivative of a polynomial at $\infty$? Let $f$ be a polynomial defined on the Riemann sphere. I'm struggling to understand in what sense such a map can be said to be "holomorphic" at $\infty$. What is the derivative of $f$ at $\infty$?
I have a chart $z\to\frac1z$ mapping $\infty$ to $0$ and vice versa. So... | To avoid confusion, it's convenient to use different variables for different coordinate functions.
Let $z$ be the standard (affine) coordinate, and define $w = 1/z$. Your question is whether $f(z)$ is differentiable at $z = \infty$, which is the same thing as whether $f(1/w)$ is differentiable at $w = 0$.
To allay worr... | {
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Check that $a_n = \sum_{k=1}^n \frac{1}{n+k}$ is bounded from above by $\frac 34$ Any ideas on how to check that
$
a_n = \sum_{k=1}^n\frac{1}{n+k} \le \frac 34\,?
$
It's pretty easy to see that the sequence is monotonic and has an almost trivial upper bound of $1$ (and a trivial lower bound of $1/2$).
I solved it sever... | If you already proved that $H_{2n}-H_n$ is increasing with respect to $n$ you are essentially done.
Here $H_n$ is the $n$-th harmonic number, $\sum_{k=1}^{n}\frac{1}{k}$, and by Riemann sums
$$\lim_{n\to +\infty}\left(H_{2n}-H_n\right)=\lim_{n\to +\infty} \sum_{k=1}^{n}\frac{1}{k+n}=\lim_{n\to +\infty}\frac{1}{n}\sum_{... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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If $e$ (Euler's constant) $= (1+\frac{1}{n})^n$ as $n$ approaches infinity, why is $e^x$ not equal to $e$? if x equals any number, real or not real. If $e$ (Euler's constant) $= (1+\frac{1}{n})^n$ as $n$ approaches infinity, why is $e^x$ not equal to $e$ if x equals any number, real or not real.
Let r = any real number... | Just as $\lim \frac{2x}{x} \neq \lim \frac{x}{x},$ despite both fractions being formally $\infty/\infty.$ It’s not enough to notice that numerator and denominator are both $\infty$. You need to know they go to infinity at the same rate, to conclude that the ratio is one.
Similarly in the limit $\displaystyle \lim_{n\t... | {
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Transpose formula to find a value Can someone help with this please?
Ive differentiated a formula to get a value, now I need to find the positive value for t for when $\frac{dR}{dt} = 0$
So:
$0 = (27t^{0.5} e^{-3t}) + (-54t^{1.5} e^{-3t})$
How would go about finding t here?
| You can divide out $27t^{0.5}e^{-3t}$ and be left with a linear equation.
| {
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How to show that $\frac{\pi}{3}\le \iint_D \left(x^2+(y-2)^2\right)^{-1/2}\,dx\,dy\le \pi$ where $D$ is the unit disc.
How to show that $$\frac{\pi}{3}\le \iint_D \frac{dx\ dy}{\sqrt{x^2+(y-2)^2}}\le \pi$$ where $D$ is the unit disc centered at the origin?
I was trying to integrate it using polar coordinates but got ... | We have $D=\{(x,y)\ |\ x^2+y^2\leq 1\}$. Clearly $x^2+(y-2)^2 \geq 1$. So we get:
\begin{align}
\iint_D \frac{1}{\sqrt[]{x^2+(y-2)^2}} dxdy \leq \iint_D dx dy = \pi
\end{align}
For the other inequality we note that the function $f(x,y)=x^2 + (y-2)^2$ can only have its maximum on the boundary of $D$ (why?). Let $\phi(... | {
"language": "en",
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Completing squares with three variables. I want to complete the squares for this polynomial
$2x^2+2y^2-z^2+2xy+3xz-4yz$
Is there any kind of easy and non-confusing way to solve it?
I’ve done this up until now:
$$2x^2+2y^2-z^2+2xy+3xz-4yz$$
$$2x^2+2xy+3xz-4yz+2y^2-z^2$$
$$2(x^2+xy+\frac{3}{2}xz)-4yz+2y^2-z^2$$
$$2(x^2+... | We can do it simply using $(a \pm b )^2 = a^2 \pm 2ab + b^2$
$$2x^2 + 2y^2 -z^2 + 2xy +3xz - 4yz $$
$$ x^2 + y^2 + 2xy + x^2 +2(x)(\frac{3}{2}z)+(\frac{3}{2}z)^2 -(\frac{3}{2}z)^2+y^2 -2(y)(2z)+(2z)^2 -(2z)^2-z^2$$
$$(x+y)^2 +(x+\frac{3}{2}z)^2 +(y-2z)^2 - (\frac{\sqrt{29}}{2}z)^2 $$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2514408",
"timestamp": "2023-03-29T00:00:00",
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Showing the Metric Function Satisfies the Triangle Inequality Let $d(X,Y)=E\left[ \frac{|X-Y|}{1+|X-Y|}\right]$. How can we show
$$d(X,Z)\leq d(X,Y)+d(Y,Z)?$$
| You basically only need two things here:
*
*the first as Eric points out is the linearity of the expectation
*the second is the easy to verify inequality $\frac{\lvert a+b\rvert}{1+\lvert a+b\rvert}\leq \frac{ \lvert a\rvert+\lvert b\rvert}{1+\lvert a\rvert+\lvert b\rvert} $ (you can verify this by cross multiplyi... | {
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Simple algebra derivation I am reading a paper and came across, what the author claims, is simple algebra. I made a few attempts, but have struggled. The equivalence claim is
$$\frac{y+2}{n+4} = \left(\frac{n}{n+4}\right)\frac{y}{n} + \left(1-\frac{n}{n+4}\right)\frac{1}{2}$$
| $$\frac{y+2}{n+4} =\frac{y}{n+4}+\frac{2}{n+4}$$
$$ =\frac{y}{n+4}.\frac{n}{n}+\frac{2+\frac{n}{2}-\frac{n}{2}}{n+4} $$
$$=\left(\frac{n}{n+4}\right)\frac{y}{n} +\left(\frac{2+\frac{n}{2}-\frac{n}{2}}{n+4}\right) .\frac{2}{2}$$
$$=\left(\frac{n}{n+4}\right)\frac{y}{n} +\left(\frac{n+4-n}{n+4}\right).\frac{1}{2}$$
$$=\l... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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one to one correspondance between points on the number axis and real numbers My question is arose by the three statements:
*
*An interval could be thought of as a line segment on the number axis
according to this book.
*I think it is true that every line segment has two end points.
*The Cantor-Dedekind axiom:
Th... | The closed ray $[a,+\infty) = \{ x \ge a \}$ has one more additional point than the open ray $(a,+\infty) = \{ x \gt a \}$, but both are defined using the real number $a$ as an 'endpoint'.
Just for fun, let's define the subset $P \subset \mathbb R$ by
$\quad P = \{ y \in \mathbb R \,| \, y = x^2 \text{ for some } x \in... | {
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Extract determinant and trace from structure constants of basis of matrix algebra Let $K$ be a field, $n$ a natural number. Suppose that $(e_i)_{i=1}^{n^2}$ is a $K$-basis of the matrix algebra $\mathbb{M}_n(K)$ (on which multiplication is defined in the usual manner). One does not know what these elements $e_i$ look l... | The determinant of an element $A\in M_n(K)$ can be computed by looking at $A$ in isolation. To compute this we don't even use the fact that $M_n(K)$ is a $K$-vector space, much less the algebra structure. So a different algebra structure will have no influence on any determinant values: however the property $\det(AB) ... | {
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Book reference for Double/ triple integrals Can someone please suggest me a Calculus book that includes Double integrals, triple integrals, volume bounded between two curves, line integrals and surface inetgrals?
I am looking for a book with plenty of examples and with geometrical approach.
Thanks
| Other than J. Callahan's book (as suggest by @Harto Saarinen)which is a great book, there are two books that i find it "complete" and those are focusing on both theoretical aspect and practical purpose. V. Zorich - "Mathematical Analysis Vol. I and II" and Moskowitz and Paliogiannis - Function of Several Variables. The... | {
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Hint to find angle $\hat{C}$ excuse me ! I put right picture ...sorry $\hat {D}=150$ my typing was wrong $105$
I need some hint to find the angle $\hat{C}$
All we know is that $$AB=DA=DC\\\hat{D}=150$$
I get stuck to find $CB$ or angle $\hat{C}$
| Drawing the diagonal $BD$, we find that $BD=a\sqrt{2}$ and $CDB=105^{\circ}$. We can use the Cosine Rule to find $BC$:
$$BC^2=a^2+(a\sqrt{2})^2-2a^2\sqrt{2}\cos105$$
Thus, $BC=a\sqrt{2+\sqrt{3}}$.
We can now use the Cosine Rule again to find $\hat{C}$:
$$\hat{C}=\cos^{-1}\left(\frac{a^2+a^2(2+\sqrt{3})-2a^2}{2a^2\sqrt{... | {
"language": "en",
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$\alpha\le \beta \iff \exists !\gamma(\alpha+\gamma=\beta)$ prove that $\alpha\le \beta \iff \exists !\gamma(\alpha+\gamma=\beta)$ where $\alpha,\gamma,\beta$ are ordinals. My first attempt at a proof is as follows
$(\rightarrow)$ suppose that $\alpha\le \beta$ then we have two cases one when $\alpha<\beta$ and one wh... | If you've already proved that ordinal addition is increasing in the right argument (that is, that $\alpha+\beta<\alpha+\gamma$ whenever $\beta<\gamma$), then your second half is just fine. However, you haven't actually proved what you're supposed to prove for the non-equal case of the first half.
It seems (though I cou... | {
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Example of nonseparable partial Lipschitz continuous gradient Let $F \colon \mathbb{R}^{n} \times \mathbb{R}^{m} \to \mathbb{R}$ with partial Lipschitz continuous gradient, that is:
*
*for any fixed $y \in \mathbb{R}^{m}$, $\nabla_{x} F \left( x , y \right)$ is Lipschitz continuous with Lipschitz constant $L_{1} \l... | What about $F:\mathbb{R}\times\mathbb{R}\to\mathbb{R}:(x,y)\mapsto\sin(x+y)$? It satisfies the hypothesis, but it isn't of the form you suggest.
| {
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A region in the plane that has to intersect unit circle If a region (meaning open, connected and non-empty subset) of the plane intersects the unit circle, does it mean it has to contain points both inside and outside of the unit circle?
I wanted to make sure that there is no weird construction that would provide a cou... | Yes, it would have to contain a point both inside and outside the circle.
Let $P$ be a point of the circle and $U_\epsilon$ an epsilon neighborhood of $P$ centered at $P$ and contained in the region.
Then the ray from the origin containing $P$ contains points of the region at distances $1-\frac{\epsilon}{2}$ and $1+\fr... | {
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Questions concerning Theorem 2.30 of Baby Rudin First question I have is the following:
Is [0, 1] open relative to [0, 1]?
It seems open to me because for x in (0, 1), x is definitely an interior point of [0, 1] and for x=0, 1 there is a neighborhood centered at each point which is completely contained in [0, 1] bec... | The answer to your first question is yes.
As for the second question, the definition is correct as stated. Just because $Y$ is the intersection of $Y$ and a non-open subset of $X$ ($Y$ in this case) does not mean there isn't an open subset $G$ of $X$ such that $Y$ is the intersection of $G$ and $Y$ (any open superset o... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2516120",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
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Why is the limit of $\frac{1}{n}$ is $0$ however the series $\sum_{n=1}^{\infty} \frac{1}{n}$ is divergent? As I recall, according to the test for divergence, if you have a series $\sum_{n=1}^{\infty}a_{n}$ and if the limit of $a_{n}$ is $0$, then the series is convergent. the limit of $\frac{1}{n}$ is $0$. However, if... | the reason is that $\frac1x$ doesnt converge to the limit fast enough.
yes $\lim\limits_{x\to\infty}\frac1x=0$ but notice that, for example, that also $\lim\limits_{x\to\infty}\frac1{x^2}=0$, what is the difference between the $2$?
$x^{-1}>x^{-2}$ for $x>1$. what does it means? well $x^{-2}$ goes to $0$ as $x$ goes to ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2516186",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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Let $V$ be a finite dimensional vector space over a field $F$ and $T$ a linear operator on $V$ such that $T^2 = I_V$.
Let $V$ be a finite dimensional vector space over a field $\mathbb{F}$
and $T$ a linear operator on $V$ such that $T^2 = I_V$. If $\mathbb{F}
= \mathbb{R}$ or $\mathbb{C}$, show that $T$ is diagonal... | It suffices to show that every generalised eigenvector of $T$ is an eigenvector of $T.$ Let $v \neq 0$ be a generalised eigenvector of $T$ with corresponding eigenvalue $c.$ By induction, we only need consider the case where $(T-cI)^2(v)=0.$ (why?) Clearly $c \neq 0$ since if $c=0$ then we'd have $0 = T^2v =v,$ contra... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2516301",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 3
} |
Exercise on equivalent norms I need to show that 2 norms are equivalent in this space: $X=C^{1}([a,b])$ .
The first norm is the standard $ ||f||_{1} $. The second norm is $ ||f||_{x} = |f(a)| + ||f^{'}||_{\infty} $.
I can do one inequality, namely $ ||f||_{1} < C||f||_{x} $, could you please help me to find the oppo... | Consider the sequence $(f_n)_n$ in $C^1([0,\pi])$ defined by
$$
f_n(x) = \sin(2nx).
$$
Note that
$$
\|f_n\|_1 = \int_0^\pi \lvert \sin(2nx) \rvert dx = \frac{1}{2n} \int_0^{2n\pi } \lvert \sin(x) \rvert dx
$$
Because $\lvert \sin(x) \rvert = \lvert \sin(x+\pi) \rvert$, we have
$$
\|f_n\|_1 = \int_0^\pi \sin(x) dx = -\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2516444",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
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Solve $A^2=B$ where $B$ is the $3\times3$ matrix whose only nonzero entry is the top right entry Find all the matrices $A$ such that $$A^2= \left( \begin {array}{ccc} 0&0&1\\ 0&0&0 \\ 0&0&0\end {array} \right) $$ where $A$ is a $3\times 3$ matrix.
$A= \left( \begin {array}{ccc} 0&1&1\\ 0&0&1 \\ 0&0&0\end {array} \ri... | Note that $A^4=0$. Thus all eigenvalues of $A$ must be $0$ thus its Jordan normal form has one of the following forms
$$
A_1=\left( \begin {array}{ccc} 0&0&0\\ 0&0&0 \\ 0&0&0\end {array} \right) \text{ or }
A_2=\left( \begin {array}{ccc} 0&1&0\\ 0&0&0 \\ 0&0&0\end {array} \right)
\text{ or }
A_3=\left( \begin {array}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2516554",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 1
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Improper Integrals and general continuous functions $\int^1_0t^{-\frac{1}{2}}f(t)dt$ Let $f:[0,1]\to\mathbb{R}$ be continuous, then i want to show that the integral $$\int^1_0t^{-\frac{1}{2}}f(t)dt$$ is convergent.
I know I need to use the improper integral $$\lim_{a\to0}\int^1_at^{-\frac{1}{2}}f(t)dt$$
I have been wor... | If $f$ is continuous on $[0,1]$, then $f$ has a maximum in the interval. Let $M=\max_{x\in[0,1]}f$. So you have that $$\int_a^1 t^{-\frac{1}{2}}f(t)\mathrm{d}t\leq M\int_a^1t^{-\frac{1}{2}}\mathrm{d}t=2M[t^{\frac{1}{2}}]_{t=a}^{t=1}=2M(1-\sqrt{a})$$
By the monotonicity of the integral. Taking the limit yields the resul... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2516661",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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If numbers of heads and tails are independent, then number of tosses $N \sim \mathrm{Poisson}$
A fair coin is tossed a random number $N$ of times, giving a total $X$ of heads and $Y=N-X$ tails. Show that if $X$ and $Y$ are independent and the generating function $G_N(s)$ of $N$ exists for $s$ in a neighbourhood of $s=... | I should start by saying that I don't have much experience with generating functions, but it looks to me like what you have there is simply a functional equation to solve: $G_n(s) = (G_N(1/2+s/2))^2$. The hint from the book is to introduce a new function to simplify that: let $H(s) = G_N(1-s)$ and our equation becomes ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2516774",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
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Baire’s Category Theorem Example I wish to find an example showing that Baire’s Category Theorem may not work when applied to uncountable collections of open and dense subsets. My first thought was the rationals bu the rationals is not considered an open set right?
| Consider $X=\Bbb R$ with the usual topology. It's a complete metric space.
Define $U_a=\Bbb R-\{a\}$ for each $a\in \Bbb R$. Dense and open. But
$\bigcap_{a\in\Bbb R}U_a=\emptyset$ is certainly not dense.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2516870",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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"answer_id": 0
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Functions and Algebra question. Given the function $g(x) = 8 − 2x$
*
*Find $g(2x-3)$
The answer to the question is $14-4x$
I have no idea how the lecturer worked it out and I just jotted down the answer, while trying to do it I just can't seem to work it out. If anyone could give me the steps to tackle this prob... | Given that $g(x)=8-2x$, let $Y=2x-3$.
Then, $g(Y)=8-2Y=8-2(2x-3)=8-4x+6=14-4x$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2517003",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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A silly question with unprovability By Gödel's incompleteness theorems, we can get a true but unprovable sentance $\psi$. However, we know it is true since its falsity implies contradiction. Then, why couldn't we accept this "proof" since it shows that $\psi$ must be true? Does it mean that the law of excluded middle c... | We start with an appropriate theory $T$ and make the Gödel sentence $G_T$. When we argue that $G_T$ is true, we do not make that argument within $T$. We generally need to assume something beyond $T$ - such as "$T$ is consistent" - in order to show that $G_T$ is true. But, if $T$ is an appropriate theory, the incomplet... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2517199",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
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Direct sums in projective modules Let $P$ be a projective module and $P=P_1+N$, where $P_1$ is a direct summand of $P$ and $N$ is a submodule. Show that there is $P_2\subseteq N$ such that $P=P_1\oplus P_2$.
I know that there is a submodule $P'$ of $P$ such that $P=P_1\oplus P'$. I wanted to consider the projection fr... | The condition $P = P_1 + N$ implies that the natural map $N \to P/P_1$ is surjective. As $P$ is projective this means the quotient map $P \to P/P_1$ factors through $N$, so there is a homomorphism $P \to N$ such that the composition $P \to N \to P/P_1$ is the quotient map. Since the quotient map gives an isomorphism ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2517469",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 2,
"answer_id": 1
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Find probability given moment generating function? So, I have a question that asks me to find the probability $P(Y<2)$ if $Y$ has a moment generating function $$M_Y(t) = (1-p+pe^t)^5$$
Is this a special distribution? Is there a trick I'm missing? Solving it algebraically/ with calculus gets really messy
| Method 1: Genarally analysing MGF
we have
$$M_Y(t)= (1-p+pe^t)^5$$
if we put 1-p=q or supposing p+q=1 we will get'
$$M_Y(t)=(q+pe^t)^5$$
we know that binomial random variable have MGF $$M_Y(t)=(q+pe^t)^n$$ after matching the corresponding terms with our give MGF we get n=5 hence
$$Y\sim Bin (5,p)$$
so
$$P(Y<2)=P(Y=0)+P... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2517587",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
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A tangent line to $y=\frac1{x^2}$ cuts the $x$-axis at $A$ and $y$ at $B$, minimize $AB$. The problem:
A tangent line to $y= \frac{1}{x^2}$ intersects the x-axis at the point A and the y-axis at the point B.
What is the length of the shortest such line segment AB?
I know that the graph of $y= \frac{1}{x^2}$ looks lik... | The derivative of $y=1/x^2$ with respect to $x$ is $-2/x^3$ so the equation of the tangent-line at $(x_0,y_0)$ is $$(y-y_0)=(-2/x_0^3)(x-x_0).$$ In this equation, when $x=0$ we have $(y-1/x_0^2)=(y-y_0)=(-2/x_0^3)(0-x_0)=2/x_0^2 .$ Hence, $y=3/x_0^2.$ So the tangent-line at $(x_0,y_0)$ meets the $y$-axis at $$P=(0,3/... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2517748",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 2
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Probability density function for the radius half the length of two equal intersecting circles Considering this diagram, assuming a uniform distribution in the area of UQWD, it is still not clear how the probability density function of r becomes $l(r)/S$. Where S is the area of UQWD. What is the proof for the pdf in th... | You can parametrize the area UQWD in polar coordinate as
$$
{\cal A} = \{(r, \theta)| r\in [0, r_m], \theta\in [\pi-l(r)/(2 r), \pi+l(r)/(2 r)]\}
$$
For a given $r_0\in [0, r_m]$, one has
$$
P(r\le r_0) = \frac{1}{S}\int_0^{r_0} dr\int_{\pi-l(r)/(2r)}^{\pi+l(r)/(2r)} r d\theta = \frac{1}{S}\int_0^{r_0} l(r) d r
$$
Henc... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2517888",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Correlation between min and max of two uniform variables Let $X$ and $Y$ be two i.i.d uniform random variables drawn from $(0,1)$. Let $A$ be $\min(X,Y)$ and $B$ be $\max(X,Y)$, what’s the correlation between $A$ and $B $ ?
| $$\overline A=\int_0^1\int_0^1\min(x,y)\,dx\,dy=\int_0^1\left[\int_0^y x\,dx+\int_y^1y\,dx\right]dy=\int_0^1\left[\frac{y^2}2+y(1-y)\right]dy\\
=\frac13.$$
$$\overline B=\int_0^1\int_0^1\max(x,y)\,dx\,dy=\int_0^1\left[\int_0^y y\,dx+\int_y^1x\,dx\right]dy=\int_0^1\left[y^2+\frac{1-y^2}2\right]dy\\
=\frac23.$$
$$\overli... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2517997",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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The max of the modulus of difference of a continuous function Let $I=[a,b]$ be a closed real interval
Let $f: I \to \mathbb{C}$ be a continuous function such that $|f(x)|$ is strictly decreasing
I would like to know if is it true that
$$
\max_{x,y \in I}
|f(x)-f(y)|
=
|f(a)-f(b)|
$$
| This is not true, consider the spiral
$$f(t):=(1-t)e^{4\pi it}$$
with $t\in[0,1/2]$. The modulus is of course decreasing and we have
$$
|f(1/2)-f(0)|=1/2.
$$
However,
$$
|f(1/4)-f(0)|=7/4.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2518153",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
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MIT Integration Bee 2017 problem:$\int_0^{\pi/2}\frac 1 {1+\tan^{2017} x} \, dx$ : Need hints This is a problem from MIT integration bee 2017.
$$\int_0^{\pi/2} \frac 1 {1+\tan^{2017} x} \, dx$$
I have tried substitution method, multiplying numerator and denominator with $\sec^2x$, breaking the numerator in terms of lin... | Setting the change of variable: $u=\frac\pi2-x $ and since, $\tan x =\cot(\frac\pi2 -x)$ we have,
\begin{align}
& \int_0^{\frac\pi2}\frac{1}{1+\tan^{2017} x} \, dx = \int_0^{\frac\pi2}\frac{1}{1+\tan^{2017} (\frac\pi2-u) } \, du \\[10pt]
= {} & \int_0^{\frac\pi2}\frac{1}{1+\cot^{2017}u} \, du = \int_0^{\frac\pi2}\frac... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2518305",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "13",
"answer_count": 2,
"answer_id": 0
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Complex analysis query regarding annulus We say that annulus is given by say $1<│z│<2$. Is it possible to have an annulus inside an annulus? Like in a domain $0\leqslant|z|\leqslant5$ can we have an annular region like $1<│z│<2$ and $3<│z│<4$? Will it still be an annulus?
| Yes it still is an annulus. As when you take the annulus $3<│z│<4$ from the given domain ,there is still the region of $0\leq|z|\leq3$ and $4\leq|z|\leq5$. From here you can again remove $1<│z│<2$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2518379",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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What is really the purpose of $i$? We started talking about imaginary numbers again this year and asked this question in class, but nobody could really give a straight answer. So if anyone could tell me the real reason we have imaginary numbers that would be great! :)
| The real reason? That sounds like you want a one-liner that you might not have heard before.
Take out the graph paper.
What happens if you keep applying $i$ to $1$? i.e, $\;i \times 1$, $\;i \times (i \times 1)$, $\;i \times (i \times (i \times 1))$, etc.
Apply $i$ to $1$ and you get $i$.
Apply $i$ to $i$ and you get $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2518491",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "12",
"answer_count": 8,
"answer_id": 1
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Alternative proof that the union of $A$ and $A'$ is closed for any set $A$ I want to see, if this alternative proof that $A \cup A'$ is closed for any set $A$ is correct. Standard references I checked after doing this problem contained a completely different proof, so I am not sure if what I have done is correct.
$A'$ ... | The proof is correct, but somewhat overly complicated. It's easy to see that $A\subseteq A'$ (consider constant sequences), so $A\cup A'=A'$ and you use in your proof that that is closed.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2518572",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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Difficulty finding the slope of the tangent line with two variables inside the equation This was a question posted in my lecture that me and my friends are unable to solve. The professor said this should be done and learned in high school, but here I am in university unable to complete this question. It might have some... | The statement "calculate the slope of the line tangent to $f(x)=(x^2+1)^q$ when $q = 3$ and $x = -1$" means that you first need to replace $q$ in the function expression with the one given to you as part of the problem to obtain the actual function and then find the slope of the line tangent to the graph of this functi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2518672",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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evaluate the triple integral $z/( x^2 + z^2)$
i set this up in the order $dxdzdy$ and i tried factoring out the z, but I'm not sure how to integrate the denominator. I tried raising it to the -1 power and putting it to the top, and using u substitution, but there is no extra x. Can someone show me detailed steps on ho... | $$\begin{aligned}\int_1^4\int_1^z\int_0^z \frac{z}{x^2+z^2}\,dx\,dy\,dz &= \int_1^4\int_1^z \left[\arctan(x/z)\right]_0^z \,dy\,dz\\\
&=\int_1^4\int_1^z \frac{\pi}{4}\,dy\,dz\\
&=\int_1^4 \frac{\pi}{4}(z-1)\,dz\\
&=\frac{\pi}{4}\left[\frac{z^2}{2}-z\right]_1^4\\
&=\frac{9\pi}{8}
\end{aligned}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2518797",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Solving matrices with unknown coefficients Thanks for reading. I've gone through the other thread on this topic but the answer is quite different to the one I've got for the following question and I need some help in checking if my answer is correct - any help is greatly appreciated :)
The question asks to solve for "k... | The matrix row reduces to the following (without doing any 'division' by terms involving $k$:
\begin{bmatrix} 1 & 1 & k & 6\\ 0 & k-1 & 1-k & -3 \\ 0 & 0 & (k-1)(k+2) & 4-6k \end{bmatrix}
When doing row reduction, it is best not to divide by a term involving $k$ (if you divide by $k-1$ as you have above for example, ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2518923",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
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Algebra of Propositional Logic How can I rewrite the following propositions in their simplest equivalent forms
i.e. Least atomic propositions
*
*$(p \land \neg p) \Rightarrow \neg p$
*$\neg ((p \land\neg p) \Rightarrow \neg q) $
*$\neg ((p \land q) \Rightarrow r)$
Thanks
| As $p\land\neg p = \bot$:
$$\bot \to \neg p == \top\lor\neg p == \top$$
*
*$\top$ (simple closed form)
*$\bot$
*$\neg((p\land q)\Rightarrow r)$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2519063",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
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Determinant construction by exterior algebra and vector space orientations A standard established way to construct the determinant is to first construct $\Lambda^p(V)$ and then observe that an endomorphism $A: V \rightarrow V$ induces $v_1 \wedge \ldots \wedge v_n \mapsto A(v_1) \wedge \ldots A(v_n)$ on $\Lambda^n(V)$ ... | I believe they first appeared in the works of Grassmann. The idea is that linear subspaces of $\Lambda^p(V)$ correspond to $p$ dimensional subspaces of $V$ thus allowing one to turn the set of $p$ dimensional subspaces into a variety.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2519188",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
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Variant of boy or girl paradox Here is an interesting question my friend brought up:
You are invited to a family party. A Boy opens the door for you. There
are two children there. What is the probability that a boy opens door for
you next time?
This is my solution:
$$P(\text{boy second time | boy first time}) = \... | A boy opened the door the first time around, but we are told there are two children (and we are assuming the boy is one of the two). Assuming there is an equal chance the other child is a boy or girl, that means there is also an equal chance for the household to be a two-boy household or a boy-girl household. So, the s... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2519269",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
"answer_count": 7,
"answer_id": 0
} |
Finding a Second Partial Derivative in the Distributional Sense I am told to find $f_{xy}$ in the distributional sense, where:
$$f(x,y) = \begin{cases}
1 & y\geq x^3\\
0 & y<x^3
\end{cases}$$
Now I know that the pointwise derivatives $f_{xy}$ and $f_{yx}$ are zero everywhere. Calculating the second derivative in the di... | The solution is correct up to $$-\int_{-\infty}^\infty \phi_x(x,x^3) dx = \phi(x,x^3)\bigg|_{-\infty}^\infty$$
which is false because
$$
\phi_x(x,x^3) \ne \frac{d}{dx}(\phi(x,x^3))
$$
On the left, we first take the derivative and then plug $x^3=y$. On the right, we plug $y=x^3$ and then take the derivative.
The exp... | {
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"url": "https://math.stackexchange.com/questions/2519402",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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"answer_id": 0
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How to solve the differential equation $\cos^2(x) \frac{d^2 y}{d x^2} -2 y = -\cos(x)$.
Solve the following differential equation:
$$\cos^2(x) \frac{d^2 y}{d x^2} -2 y = -\cos(x).$$
We were asked not to solve this by the method of variation of parameters, so except that method we have tried to reduce the equation a... | Continuing from Robert Z's answer, every homongeneous solution takes the form
$$y_\text{hom}(x)=a\,\tan(x)+b\,\Big(1+x\,\tan(x)\Big)\,$$
where $a$ and $b$ are constants.
This can be done by observing that $y=\tan(x)$ is a homogenous solution. With the assumption that the general homogenous solution takes the form $y=z... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2519517",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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$X$ and $Y$ are independent rv having pdf $f(t)=\frac{1}{\pi} \frac{1}{1+t^2}$ determine pdf of $Z=\frac{X+Y}{3}$
Suppose $X$ and $Y$ are independent random variables on $\mathbb{R}$ having pdf $f(t)=\frac{1}{\pi} \frac{1}{1+t^2}$. Define $Z=\frac{X+Y}{3}$ determine the pdf of $Z$.
So the pdf of $(X,Y)$ is $f(x,y)=\f... | $X$ and $Y$ are standard Cauchy hence $\frac{X+Y}{2}$ is standard Cauchy (what easily can be seen using characteristic functions)
So the pdf of $\frac{3}{2}Z$ is also $$f(t) = \frac{1}{\pi}\frac{1}{1+t^2}$$
Hence the distribution of $Z$ is: $$f(z) = \frac{d}{dz}P(Z \le z) = \frac{d}{dz}F\left(\frac{3}{2}Z \le \frac{3}{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2519667",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Convergence of power function is hypothesis testing Let $\widehat{\theta}$ be the MLE of a parameter $\theta$ and let $\widehat{\text{se}}=\{nI(\widehat{\theta})\}^{-\frac12}$ where $I(\theta)$ is the Fisher information. Consider testing$$
H_0:\theta=\theta_0\,\,\text{versus}\,\,H_1:\theta\neq \theta_0.
$$
Consider the... | In the meantime, with the guidance of Ceph in the comment section, I believe I found an answer.
Let $H_1:\theta=\theta_1$. Under $H_1$, we define $W=(\widehat{\theta}-\theta_1)/\widehat{\text{se}} \rightsquigarrow N(0,1)$. Hence,\begin{align*}
\beta(\theta_1)&=\mathbb{P}_{\theta_1}(|Z|>z_{\alpha/2})\\
&=\mathbb{P}_{\th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2519827",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Show that the series $\sum\limits_{n=2}^{\infty} \frac {(n^3+1)^{1/3}-n}{\log n}$ converges
Show that the series $$\sum\limits_{n=2}^{\infty} \frac {(n^3+1)^{1/3}-n}{\log n}$$ converges.
I showed it using Abel's theorem and limit comparison test. Any other simpler method?
| $$\lim_{n \to \infty}\left(\frac{(n^3+1)^\frac13-n}{\ln n}\right)n^2=\lim_{n\rightarrow+\infty}\frac{n^2}{\left(\sqrt[3]{(n^3+1)^2}+n\sqrt[3]{n^3+1}+n^2\right)\ln{n}}=0,$$
then for $n$ large enough we have
$$\frac{(n^3+1)^\frac13-n}{\ln n}\le \frac{1}{ n^2}$$
the result follows by comparison test. which says that it c... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Showing that if $-1$ and $2$ are not squares in $\mathbb{Z}_p$, then $-2$ is a square I would like to prove that if $-1$ and $2$ are not squares in $\mathbb{Z}_p$, then $-2$ is a square. I have searched for some hints on this, but the answers that I find all involve cosets and quadratic reciprocity. I have also heard t... | We don't need to know that $\mathbb{Z}_p^\times$ is cyclic, which I think is a much more advanced fact. And we certainly don't need quadratic reciprocity.
In fact, all we need is that (for $p$ an odd prime) the number of squares and non-squares modulo $p$ is the same: $(p-1)/2$. I'll prove this below, and leave the c... | {
"language": "en",
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"source": "stackexchange",
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Determinant of a sum of matrices I would like to know if the following formula is well known and get some references for it.
I don't know yet how to prove it (and I am working on it), but I am quite sure of its validity, after having performed a few symbolic computations with Maple.
Given $n$ square matrices $A_1,\ldot... | Given integers $n > m > 0$, let $[n]$ be a short hand for the set $\{1,\ldots,n\}$.
For any $t \in \mathbb{R}$ and $x_1, \ldots, x_n \in \mathbb{C}$, we have the identity
$$\prod_{k=1}^n (1 - e^{tx_k}) = \sum_{P \subset [n]} (-1)^{|P|} e^{t\sum_{k\in P} x_k}$$
Treat both sides as function of $t$. Expand against $t$, on... | {
"language": "en",
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Prove that : $a_n>0$, Then, $\sum_{n=1}^\infty a_n$ converges iff $\sum_{n=1}^\infty \sin(a_n)$ converges. I am given the problem: Let $a_n>0$, prove $\sum_{n=1}^\infty a_n$ converges if and only if $\sum_{n=1}^\infty \sin(a_n)$ converges.
We have this problem in a homework, and I don't believe it can be true. The stat... | The statement as given is not true since the series $\sum \pi = \infty$, while $\sum \sin\pi = 0$, but one direction is true.
First note that $0<\sin x< x$ for each $0<x<\pi$. If $\sum a_n$ converges, then $a_n\to 0$ as $n\to\infty$. Hence there is some $N\in\Bbb N$ such that for all $n\ge N$, we have $0<a_n < \pi$. H... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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If a probability is strictly positive, is it discrete? Let $(\Omega, \mathcal{F}, P)$ be a probability space.
Call $P$ strictly positive if $P(F)>0$ for all $F \in \mathcal{F} \setminus \{\emptyset\}$.
Call $F \in \mathcal{F}$ an atom if $P(F)>0$ and $P(A)=0$ for all strict, measurable subsets $A$ of $F$.
Note that if ... | First, note that any two atoms are either equal or disjoint. For if $F_1, F_2$ are atoms with $F_1 \cap F_2 \ne \emptyset$, then $F_1 \cap F_2^c$ is a strict measurable subset of $F_1$, hence empty, meaning $F_1 \subseteq F_2$, and the reverse inclusion by symmetry.
Next, note that the number of atoms is at most count... | {
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Probably that an $80\%$-truthful person actually rolled a $6$ A person, $A$, speaks the truth $4$ out of $5$ times. The person throws a die and reports that he obtained a $6$. What is the probability that he actually rolled a $6$?
I know there is a similar question like this
but my doubts are different from it and als... | Your error would appear to be in P(E3|E2)=0
You are missing out the fact that the user may report a 6 being throw when in fact one has not
In fact your use of E3 is not helpful as it covers two scenarios.
What you are interested in is
User throws a six AND tells the truth
User lies AND reports that they threw a 6.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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How to solve the complex root $z^\sqrt5 =1$ with square root?
Solve
$$z^\sqrt5 =1$$
for $z$ and state how many unique solutions are possible.
I tried to convert $1$ to polar form and got $z=\exp^\left((2k\pi+2\pi)i/\sqrt5\right)$. Could someone please help me out?
| Let $z=re^{i\theta}$.
Since $z^\sqrt5=r^\sqrt5e^{i\sqrt5\theta}=1=1e^{\left(0+2k\pi\right)i}$ for $k \in \mathbb{Z}$, $r=1$.
Now we have $e^{i\sqrt5\theta}=e^{2k\pi i}$, which is exactly where you have reached.
Then we have $\sqrt5\theta=2k\pi$
So according to you, you are using $0 \leq \theta \leq 2\pi$.
We know from... | {
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Prove that for all n $(-1)^1[nC1(1+rln10)/(1+ln10^n)^r1] +(-1)^2[nC2(1+rln10)/(1+ln10^n)^r]+....=0 $ Prove that for nbelongs to natural number
$$(-1)^1{n\choose1}\dfrac{(1+r\ln10)}{(1+\ln(10^n))^r} +(-1)^2{n\choose2}\dfrac{(1+r\ln10)}{(1+\ln(10^n))^r}+....=0 $$
I have proved this by induction which clearly is not the... | Consider the binomial theorem $$(1+x)^n = 1 + \binom{n}{1} x + ... \binom{n}{n} x^n$$
But $x = -1$, we obtain:
$$(-1)^0{n\choose0}+(-1)^1{n\choose1} +(-1)^2{n\choose2}+... (-1)^n{n\choose n} =0$$
Therefore your series evaluates to $-\frac{1+r\ln{10}}{(1+\ln{10})^r}$ for all $n$ (Naturals).
| {
"language": "en",
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Evaluate the limit $\lim_{x\rightarrow 0}\frac{1-\sqrt{1+x^2}\cos x}{\tan^4(x)}$ Calculate the following limit :
$$\lim_{x\rightarrow 0}\frac{1-\sqrt{1+x^2}\cos x}{\tan^4(x)}$$
This is what I have tried:
Using Maclaurin series for $ (1+x)^a $:
$$(1+x^2)^{1/2}=1+\frac{1}{2!}x^2\quad \text{(We'll stop at order 2)}$$
Usi... | Without using L'Hospital & Taylor's Expansion,
$$=\dfrac{1-(1+x^2)(1-\sin^2x)}{1+\cos x\sqrt{1+x^2}}\cdot\dfrac{\cos^4x}{\sin^4x}$$
$$=\dfrac{x^2\sin^2x+(\sin x-x)(\sin x+x)}{x^4}\cdot\dfrac{\cos^4x}{\left(\dfrac{\sin x}x\right)^4(1+\cos x\sqrt{1+x^2})}$$
Now $\dfrac{(\sin x-x)(\sin x+x)}{x^4}=\left(\dfrac{\sin x}x+1\r... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Show that if $Z\cup E$ is measurable then so is $E.$ Problem:
Let $E,Z \in\mathbb{R}^{n}$ and $Z$ be a set of measure zero. Show that if $Z\cup E$ is measurable then so is $E.$
What I have done:
Since $Z\cup E$ is measurable then there exists
$G$
such that $Z\cup E\subset G$ consequently $ E\subset G$.
Also... | Hint
I suppose you are talking about the Lebesgue measure (tell me if that is not the case). Since $Z$ is a null set, it is measurable. Define the following set:
\begin{align}
A:=[Z\cup E] \cap [Z\cap E^c]^c
\end{align}
Now where is the set $A$ equal to? And what can you say about it?
Edit: Thanks to @Bungo for poin... | {
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Prove that $\underline\int_{a}^bf\ge0$ when $f(x)\ge 0$ My question is: Suppose that the bounded function $f:[a,b]\rightarrow \Bbb R$ has the property $f(x)\ge 0$ for all $x$ in $[a,b]$. Prove that $\underline\int_{a}^bf\ge0$. I am just confused on where to start. I would think you have to use the definition of a lower... | Hint:
We know that the lower integral is defined as the supremum of the lower sums over all partitions.
So, it suffices to show that there exists a partition whose lower sum is non-negative, as the supremum of lower sums is at least as large as any particular lower sum.
So, can you pick ANY partition, (say, $\{a,b\}... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Matrix multiplication: row x column vs. column x row I am wondering if there is any inherent difference between multiplying two matrices row by column (standard way to multiply) vs. column by row.
Asking specifically relating to a question in a textbook which asks: Express each column matrix of AB as a linear combinati... | Multiplying column-by-row is the same as multiplying row-by-column in reverse order$^\ast$. So if you invent a new matrix multiplication denoted by, say, $\rtimes$, where $A\rtimes B$ is multiplication column-by-row, then $A\rtimes B=BA$, where $BA$ is the standard row-by-column multiplication.
Okay, now let us answer ... | {
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What is the distribution of real numbers with biased digits? Suppose I have an infinite sequence of biased bits where the probability of $1$ is $2/3$ and the probability of $0$ is $1/3.$ If I view these as the digits in the binary expansion of a real number, then this sequence defines a real number in the interval $[0,... | Your recursion gives a condition the cumulative distribution function satisfies (in a sense, you have fractal copies of the function in itself), but there are several functions which satisfy this.
You would not expect the cumulative distribution function to be a smooth function since for example values of the binary fo... | {
"language": "en",
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How rough can differential form, manifold and chain be for Stokes theorem to hold? When I first learned Stokes theorem, everything is assumed to be smooth to prevent any strange things happen. But to apply to more cases, I may need to use a version of Stokes theorem that holds for rougher forms, chains and manifolds. F... | Stokes' theorem holds for manifolds with corners. The following is Theorem $16.25$ on page $419$ in John M. Lee's book $\textit{Introduction to Smooth Manifolds}$.
$\textbf{Theorem}$ (Theorem $16.25$, [Lee]). Let $M$ be an oriented smooth $n$-manifold with corners, and let $\omega$ be a compactly supported smooth $(n-... | {
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Prove that the set of natural numbers (in base 10) with exactly one of the digits equal to 7 is countably infinite. Prove that the set of natural numbers (in base 10) with exactly one
of the digits equal to 7 is countably infinite.
"In base 10" means that it's the natural numbers between 0 and 9, correct?
What might t... | If $S$ is a subset of $T$, then $|S| \leq |T|$. The set in your question is a subset of the natural numbers, so it is either finite or countably infinite (and it's easy to see that it isn't finite).
| {
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Confusing Bayes Theorem Example I'm trying to find out the probability that I have a disease given that it is in my family history so $P(disease|history)$.
If the rate of having a disease is $P(disease)=\frac{1}{1000}$ and the rate that those who have the disease have a family history of the disease is $P(history|disea... | Any statement of probability depends on some background information, call it $I$. Now we have two scenarios, we can consider background information $I_1$ which is your state of knowledge not knowing whether you have a family history of this disease, or background information $I_2=I_1\land \text{history}$.
It's certainl... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Prove if $f$ is continuous at $x_0$ and $f(x_0)>\mu$, then $f(x)>\mu ,\forall x$ in some neighborhood of $x_0$
Prove if $f$ is continuous at $x_0$ and $f(x_0)>\mu$, then $f(x)>\mu ,\forall x$ in some neighborhood of $x_0$
My attempt:
$f$ is continuous implies:
$$\forall \epsilon>0, \exists \delta >0 \text{ s.t.} |f(x... | Hint:
Set $\epsilon = \frac{|f(x_0)-\mu|}{2}$
| {
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"timestamp": "2023-03-29T00:00:00",
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What are some members of $\mathcal C^1[0,1]$ Can anyone give examples of a few functions that belong to $\mathcal C^1[0,1]$ and some functions that do not belong there?
It’s the set of all continuous functions on $[0,1]$ which are continuously differentiable on $(0,1)$ where the derived functions has continuous extens... | For instance, $f(x)=x$ belongs to $C^1\bigl([0,1]\bigr)$, because $f'(x)=1$. On the other hand, $s(x)=\sqrt x$ does not belong to $C^1\bigl([0,1]\bigr)$, beacuse, when $x\in(0,1]$, $s'(x)=\frac1{2\sqrt x}$, and you cannot extend it to a continuous function of $[0,1]$, since $\lim_{x\to0^+}s'(x)$ does not exist (in $\ma... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2522923",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Plot multiple functions with log axis on Wolfram Alpha I want to plot two functions on the same graph using a log y axis on Wolfram Alpha, but I can't find a way to do this.
I've tried things like log plot 2^(3x-1), e^x, x=1..10, but this doesn't work (despite plot 2^(3x-1), e^x, x=1..10 working perfectly fine).
| Try:
logplot {2^(3x-1), e^x}, (x, 1, 10)
Here it is as a link on WA
Here it is as a Mathematica command
LogPlot[{2^(-1 + 3*x), E^x}, {x, 1, 10},
PlotLabels -> Placed["Expressions", Above], ImageSize -> Large]
Here is the output
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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What are some of the most classical rings studied in Algebraic Geometry? I am curious what are the typical rings studied in Algebraic Geometry?
I am aware that the polynomial ring $K[x_1,\dots,x_n]$, where $K$ is a field, is of great interest.
Are there any other classical rings that are studied?
Thanks.
Update: I am a... | Surely the field of rational functions $K(X)$ for a field $K$, power-series rings, and Laurent series rings also qualify.
Based on my knowledge of the history of algebraic geometry (what little of it I have, I learned from Dieudonne's paper/lecture on the topic$^\ast$), at least the rational functions have been importa... | {
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Find the density function of T= max(X,Y) I have some problems about this question.
$X$ and $Y$ are two independent random variables: X is an exponential
random variable, Y is a uniform random variable over $[0, a]$. Given
that $EX = EY = 6$, find the density function of the random variable
$T = max(X, Y ).$
Now... | Hint: Let $F_{Z}$ be the cumulative distribution of a random variable
$Z$. If $X$ and $Y$ are independent random variables, then
\begin{multline*}
F_{T}(t)=\mathbb{P}\{T\leq t\}=\mathbb{P}\{\max\{X,Y\}\leq t\}=\mathbb{P}(\{X\leq t\}\cap\{Y\leq t\})=\mathbb{P}\{X \leq t\} \mathbb{P} \{ Y \leq t \}\\=F_{X}(t)F_{Y}(t).
\e... | {
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"timestamp": "2023-03-29T00:00:00",
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Prove using Mathematical Induction that $2^{3n}-3^n$ is divisible by $5$ for all $n≥1$. I did most of it but I stuck here I attached my working
tell me if I did correct or not thanks
My working:
EDITED: I wrote the notes as TEX
Prove using induction that $2^{3n} - 3^n \mod{5} = 0$.
Statement is true for $n = 1$:
$$2... | It's hard to read your handwriting but it looks like you have the right idea but were sloppy in your execution and made so distributive error.
Assume if $n=k$ then statement is true and
$2^{3k} - 3^k = 5P$ for some integer $P$. (Always a good idea to specify what a variable is whenever you introduce it.)
$2^{3k+1} -3... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2523392",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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linear, continuous functional, Schwartz space
Is $g:\mathbb{R}\to\mathbb{C}$ measurable such that it exists $N\in\mathbb{N}$ with $x\mapsto \frac{g(x)}{(1+x^2)^n}\in L^1(\mathbb{R},\lambda)$, then defines $g$ a continuous, linear functional $S_g:\mathcal{S}(R)\to\mathbb{C}$, where $\mathcal{S}(\mathbb{R})$ is the Schw... | Continuity was shown. You need to show that the integral will always converge though.
If $f \in \mathbb{S}$, then there exists N sufficiently large such that outside of some compact interval $[-N,N]$ we have that $|f| < \frac{1}{(1+x^2)^n}$. Thus $$\int_{-\infty}^{\infty}|f(x) g(x)| dx < \int_{-N}^{N}|f(x) g(x)| dx ... | {
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Show that $q(z) \neq 0$ on the neighborhood $N(z_0,r)$ by continuity. Suppose that $g$ is analytic and never zero on $N(z_0,r)$, and that $g$ has a zero of order $m$ at $z_0$.
By the factorization theorem, we have $g(z) = (z-z_0)^m q(z)$ where $q(z_0)\neq 0$ and $q(z)$ is analytic on $N(z_0,r)$.
How do i show that $q(z... | By mere continuity? You can’t. However, for all $z ∈ N(z_0,r)$, you have $g(z) = (z-z_0)^m q(z)$ as you say. So $q(z) = 0$ would imply $g(z) = 0$, which is impossible for $z ≠ z_0$. And on the other hand, you already have $q(z_0) ≠ 0$ by assumption.
(Remark: You can deduce that $q(z) ≠ 0$ on $N(r',z_0)$ for some real $... | {
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"url": "https://math.stackexchange.com/questions/2523666",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Notation to describe that a value is equivalent to at least one component of a vector? Suppose I have the vector $X=<1, 2, 3, 4, 5, 1>$. This notation should mean that this is a vector whose respective components are 1, 2, 3, 4, 5, and 1, in that same order. Suppose I want to show that the value 1 is equivalent to at... | Define $U_a$ as the union of the hyper planes $H_i(a)$ where $x_i=a$ $$U_a=\cup_{i=1}^{n}H_i(a)$$
Then you can say $X \in U_1$ for what you are looking for.
Unfortunately, I don’t know any pre-existing standard notation for this.
| {
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"url": "https://math.stackexchange.com/questions/2523850",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
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} |
A and B have 10 dollars each. They bet 1 dollar each time. A wins the final game iff B has no money left.
*
*For each bet, A has prob = 0.5 to win the 1 dollar from B. and prob = 0.5 to lose 1 dollar to B. How to calculate the probability that A wins the final game?
*How about for the initial state A has 20 dollars ... | By symmetry $A$ wins with probability $0.5$ for scenario (1). For (2), consider the probability that $A$ wins given that, at some point in time, $A$ has $k$ dollars and $B$ has $30 - k$ dollars. Define $p_k$ by
$$
p_k = P(A \text{ wins } | \: A\text{ has } k \text{ and } B \text{ has } 30-k).
$$
Conditioning on whether... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2524006",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Prove that $|\mathbb{R}| = |(0, 1)|$. Prove that $|\mathbb{R}| = |(0, 1)|$. (Hint: Consider the tangent function.)
This is my current thought process:
Using the hint, I map $(0, 1) \rightarrow (-\frac{\pi}{2},\frac{\pi}{2})$ by the function $f(x) = \pi x - \frac{\pi}{2}$, and
then state that since $f$ is linear, and... | Your proof is basically correct, but needs to be fleshed out just a bit. Recall that two sets have the same cardinality if there is a bijection between them. We are going to build a bijection from $(0,1)$ to $\mathbb{R}$ in two steps:
*
*Let $\varphi : (0,1) \to (-\frac{\pi}{2},\frac{\pi}{2})$ be the function $\va... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2524154",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Combinatorics : relationship between Combinations and Permutations with similar objects I have a question in regards to relationship between Combinations and Permutations with like or repeated objects.
Using an example like the the word Mississipi, basic example of using Permutations with like letters.
I have seen this... | The following are equivalent expressions for the multinomial coefficient:$${{}^{(a+b+c)}\mathrm C_{a,b,c}\\= \binom{a+b+c}{a,b,c}\\= \dfrac{(a+b+c)!}{a!~b!~c!}\\ = \binom{a+b+c}{a}\binom{b+c}b\\ \vdots \text{ and such like}}$$
The multinomial coefficient counts the ways to permute (or arrange) a multiset of $a, b,$ and... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2524296",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Is the category of class of all sets a concrete category? Since the class of all sets is not a set but is a class, is the category of class of all sets a concrete category?
(the only object is the class of all sets)
| A concrete category is a category with a faithful functor to Set, right? And faithful means that it is injective on the hom-sets for every pair of objects.
Seeing as the category with Set as the only object and the identity as the only arrow only has one hom-set, with one element, we can just map it to a singleton set ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2524418",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 1
} |
Is $1+\sqrt{5}$ a prime under the $\mathbb{Z}[{\sqrt{5}}]$ domain? The title is self-explanatory. I know it's irreducible but is it a prime? How to prove these primality and/or irreducibility of $1+\sqrt{5}$.
Can you just briefly state how a prime is defined under $\mathbb{Z}[{\sqrt{5}}]$? I know that it will only be d... | One would usually define a non-zero element $a$ of a ring $R$ to be prime if the ideal it generates is a prime ideal.
Under this definition, $1+\sqrt 5$ is not prime, since $4 =2\times2= (1+\sqrt5)(\sqrt5-1) \in \langle1+\sqrt 5\rangle$, but $2\notin\langle1+\sqrt 5\rangle$.
You should note that $\mathbb Z[\sqrt 5]$ is... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
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"answer_id": 3
} |
A coin is tossed $n$ times. What is the probability of getting odd number of heads?
A coin is tossed $n$ times. What is the probability of getting odd number of heads?
I started this chapter sometimes ago and faced in front of a tough problem. At first I started considering cases. Case-I : The probability of getting ... | First toss $n-1$ times. When you toss the $n^{th}$ time, one outcome will give an even number of heads and one outcome an odd number of heads.
If the coin is a fair coin, this gives the result immediately. If it is not, then you need to combine the probabilities of odd/even after $n-1$ tosses with the probabilities on ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2524783",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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For what value of $a$ does the equation $|x(x-4)|=a$ have exactly 3 real solutions? I simplified the equation and got two cases where $$x(x-4)=a$$
and
$$x(x-4)=-a.$$
How do I go after that. I know how you could get $2$ solutions or $4$ solutions, but how do you get $3$ solutions?
| The expression inside the absolute value signs is a quadratic, so the graph of $y=x(x-4)$ is a parabola. Its vertex is at $(2,-4)$, so the equation $x(x-4)=-4$ has precisely $1$ solution. Meanwhile, the equation $x(x-4)=4$ has two solutions. Putting these together, you should be able to find the answer to your question... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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} |
Compute the derivative of $S(x) = \int_1^{\arcsin(x)}\frac{\sin(t)}{t}dt$ My book states the following:
FUNDAMENTAL THEOREM OF CALCULUS. Assume the function $f$ is continous in $x\in[a,b].$ Put $$S(x)=\int_a^xf(t)dt, \quad a\leq x\leq b.$$
Then, it follows that the function $S$ is differentiable in $x\in(a,b)$ with th... | HINT:
Let $y(x)=\arcsin(x)$ and use the chain rule $$\frac{dS(y(x))}{dx}=\frac{dS(y(x))}{dy(x)}\frac{dy(x)}{dx}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2525030",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 0
} |
Proving that all the real roots of Hermite polynomials are in $(-\sqrt{4n+1}, \sqrt{4n+1})$ The Hermite polynomials are given by:
$H_n(x)=(-1)^n e^{x^2} \dfrac{d^n}{dx^n}e^{-x^2}$
There is the proof that all the roots are real: https://math.stackexchange.com/a/104875/504137.
And I know the fact that they all are bounde... | It doesn't look like there are many questions on this topic on Math.SE, so just for fun let's use some tricks from matrix analysis to slightly improve the bound $\sqrt{2n-2}$ in Jack's answer.
Let $A = [a_{ij}]$ be a real $n \times n$ matrix. Define $|A| = [|a_{ij}|]$. We will write $A \geq 0$ if all $a_{ij} \geq 0$. ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2525094",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
Does sequence ${ s }_{ n }=\sum _{ k=0 }^{ n }{ \frac { 1 }{ { 1+k }^{ 2 } } } $ converge I'm reviewing some Calculus 1 convergence stuff. I want to decide wether ${ s }_{ n }:=\sum _{ k=0 }^{ n }{ \frac { 1 }{ { 1+k }^{ 2 } } } $
is convergent or not.
Since the sequence is monotnically increasing and has an upper and... | 1) Sequence is monotonically increasing .
2)Need to find an upper bound:
$\sum_{k=1}^{n} \dfrac{1}{1+k^2}\le$
$\sum_{k=1}^{n}\dfrac{1}{k^2}$.
Consider:
$\sum_{k=2}^{n}\dfrac{1}{k^2} \le$
$\sum_{k=1}^{n-1}\dfrac{1}{k(k+1)} =$
$\sum_{k=1}^{n-1}[\dfrac{1}{k} - \dfrac{1}{k+1}]=$
$1 - \dfrac{1}{n}.$
Bounded above.
Sum conve... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2525277",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 5,
"answer_id": 4
} |
Find a function that has finite values for lower norm but becomes infinite I recently came across the following question: does there exist a non-decreasing function $h : [0,1) \rightarrow \Re^+$, i.e. with non-negative range, that satisfies $\|h\|_1=1$ and $\|h\|_a\leq 1$, where $a$ is some value in $[1,2)$, i.e. for l... | Such a function cannot exist.
Assume that there exists a measurable function $f : [0,1) \to \mathbb{R}^+$ such that $\|f\|_a \le 1$, $\forall a \in [1,2)$.
Notice that $f^a \le f^b$ for any $a \le b$ with $a, b \in [1,2)$. Pick an increasing sequence $(a_n)_{n=1}^\infty$ in $[1,2)$ which converges to $2$. Then $(f^{a_n... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2525350",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
If $f$ is a entire function such that $f(z+n+im)=f(z)$ for all $z\in \mathbb{C}$ and for all $n,m \in \mathbb{Z}$, then $f$ is constant. If $f$ is a entire function such that $f(z+n+im)=f(z)$ for all $z\in \mathbb{C}$ and for all $n,m \in \mathbb{Z}$, then $f$ is constant.
I'm having trouble solving this one. Could you... | Since $f$ is continuous, $f(R)$ is a compact subset of $\mathbb C$, where $R$ is the rectangle $\{a+bi\mid a, b\in[0, 1]\}$. Hence $f(R)$ is bounded. By that $f(z+n+im)=f(z)$, it follows that $f(\mathbb C)=f(R)$. Thus $f$ is a bounded entire function. Then the Liouville theorem entails that $f$ is constant.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2525488",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Write formula for matrix in terms of Fibonacci numbers How could I express this matrix in terms of Fibonacci numbers? It seems like I'd have to use induction once I have a candidate for a formula but I'm unsure of where to start with expressing the matrix in terms of Fibonacci numbers.
Thanks in advance!
Let $T:\math... | $T(e_1)=e_2$ and $T(e_2)=(1,1)$, so the matrix is formed by columns $T(e_1)$ and $T(e_2)$:
$$T:=\begin{pmatrix} 0&1\\1&1
\end{pmatrix}$$
Note that
$$T^n=\begin{pmatrix} 0&1\\1&1
\end{pmatrix}^n=\begin{pmatrix}F_{n-1}&F_n\\F_n & F_{n+1} \end{pmatrix}.$$
Where $F_0=0$ and $F_1=1$. You can check this by induction:
$$T^{n... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2525592",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
} |
How can I approximate the Rician Distribution through the Gaussian Distribution? Which are the techniques used to approximate a distribution into another? I know that I can model a Gaussian Distribution through the parameters of mean and variance. However how can I approximate Rice Distribution through the Gaussian Dis... | You can generate a Rayleigh distribution simply as:
x_rayleigh = ( randn(1, 1e6) + 1i*randn(1, 1e6) ) / sqrt(2);
This is a complex normal distribution with zero mean and variance 1/2 per dimension. So the amplitude of x_rayleigh follows a Rayleigh distribution. The phase of x_rayleigh will be uniform.
Now if you... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2525727",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
If $f$ has a zero and $|f''|\leq M$, then $f$ is monotone on $(-h,h)$, where $h=\sqrt{2|f(0)|/3M}$
Let $f$ be twice differentiable on $\mathbb R$ and let $M$ be a bound
of $f''$, $|f''|\leq M$ on $\mathbb R$. Assume $f(0)\neq0$ and define
$h=\sqrt{\frac{2|f(0)|}{3M}}$. Prove that if $f$ has a zero in $(-h,h)$, it'... | This is not true.
Consider $ f(x) = exp (-x^2) -0.1$
It's not monotone on $(-h,h)$ for any given $h>0$, yet it's second derivative is bounded by a finite $M$ and $f(0)=1$, and it has a root.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2525846",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 1
} |
Sampling distribution of random sample I am currently working on the following question: Suppose you have a finite population $P$ of size $N$. We select a sample $S_1$ using a random sampling without replacement of size $n_1$. Then we select a sample $S_2$ from $P-S_1$ using random sampling without replacement of size ... | Yes, but more directly you could say that this probability equals:$$\binom{N}{n_1+n_2}^{-1}$$
There is no essential difference between taking two consecutive samples of sizes $n_1$, $n_2$ respectively that are afterwards joined and one sample of size $n_1+n_2$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2525944",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Cauchy Integral Formula for $\oint_{\gamma_i} \frac{z^2+1}{z(z-8)}dz~~~\gamma_i = \mathcal C(3,i), ~~i=1,4,6$ I have a question that i'd like to check my working on. Calculate the integral of $$\oint_{\gamma_i} \frac{z^2+1}{z(z-8)}dz~~~\gamma_i = \mathcal C(3,i), ~~i=1,4,6$$
(a) $\gamma_1$ is the circular contour, posi... | For c) you can decompose the fraction
as follows
$$\frac{z^2+1}{z(z-8)} =1+- \frac{1}{8z}+\frac{65}{8(z-8)}$$
Thus,
$$\oint_{\gamma_6} \frac{z^2+1}{z(z-8)}dz=\oint_{\gamma_6} dz-\frac{1}{8}\oint_{\gamma_6} \frac{dz}{z}+\frac{65}{8}\oint_{\gamma_6} \frac{dz}{(z-8)}\\-\frac{2\pi i}{8}+\frac{2\pi i*65}{8} =\color{red}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2526074",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Find the limit $\lim_{n\to\infty}\sqrt[n]{\dfrac{x^n}{2n+1}}=\text{?} \ \ \ \ :x>0$
Find the limit below:
$$\lim_{n\to\infty}\sqrt[n]{\dfrac{x^n}{2n+1}}=\text{?} \ \ \ \ :x>0$$
My Try :
$$a_n:=\sqrt[n]{\dfrac{x^n}{2n+1}}\\ \ln a_n=\dfrac{1}{n}\ln \left(\dfrac{x^n}{2n+1} \right)\\ \ln a_n=\dfrac{1}{n}\left(n\ln x-\ln(... | you are almost right the only problem is $\frac{1}{n} \ln (2n+1)$ is NOT equal to $\ln(\frac{2n+1}{n})$. But $\frac{1}{n}\ln(2n+1)\to 0$ when $n\to \infty$. Then
$$
\ln(a_n)=\ln(x) +\frac{1}{n}\ln(2n+1)\to \ln(x)
$$
Hence $a_n$ tends to $x$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2526212",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 3
} |
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