Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Evaluating a limit with variable in the exponent For
$$\lim_{x \to \infty} \left(1- \frac{2}{x}\right)^{\dfrac{x}{2}}$$
I have to use the L'Hospital"s rule, right? So I get:
$$\lim_{x \to \infty}\frac{x}{2} \log\left(1- \frac{2}{x}\right)$$
And what now? I need to take the derivative of the log, is it:
$\dfrac{1}{1-\d... | Recall the limit:
$$\lim_{y \to \infty} \left(1+\dfrac{a}y\right)^y = e^a$$
I trust you can finish it from here, by an appropriate choice of $a$ and $y$.
| {
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"timestamp": "2023-03-29T00:00:00",
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"question_score": "1",
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Ball-counting problem (Combinatorics) I would like some help on this problem, I just can't figure it out.
In a box there are 5 identical white balls, 7 identical green balls and 10 red balls (the red balls are numbered from 1 to 10). A man picks 12 balls from the box. How many are the possibilities, in which:
a) exactl... | Hints: (a) How many ways can we choose $5$ numbers from $1,2,...,9,10$? (This will tell you how many different collections of $5$ red balls he may draw.) How many distinguishable collections of $7$ balls can he draw so that each of the seven is either green or white? Note that the answers to those two questions do not ... | {
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"timestamp": "2023-03-29T00:00:00",
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Evaluating the integral: $\lim_{R \to \infty} \int_0^R \frac{dx}{x^2+x+2}$ Please help me in this integral:
$$\lim_{R \to \infty} \int_0^R \frac{dx}{x^2+x+2}$$
I've tried as usually, but it seems tricky. Do You have an idea?
Thanks in advance!
| $$\dfrac1{x^2+x+2} = \dfrac1{\left(x+\dfrac12 \right)^2 + \left(\dfrac{\sqrt{7}}2 \right)^2}$$
Recall that
$$\int_a^b \dfrac{dx}{(x+c)^2 + d^2} = \dfrac1d \left.\left(\arctan\left(\dfrac{x+c}d\right)\right)\right \vert_{a}^b$$
I trust you can finish it from here. You will also need to use the fact that
$$\lim_{y \to \i... | {
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Proof of: If $x_0\in \mathbb R^n$ is a local minimum of $f$, then $\nabla f(x_0) = 0$.
Let $f \colon \mathbb R^n\to\mathbb R$ be a differentiable function.
If $x_0\in \mathbb R^n$ is a local minimum of $f$, then $\nabla f(x_0) = 0$.
Where can I find a proof for this theorem? This is a theorem for max/min in calculus ... | Do you know the proof for $n=1$? Can you try to mimic it for more variables, say $n=2$? Since $\nabla f(t)$ is a vector what you want to prove is that $\frac{\partial f}{\partial x_i}(t)=0$ for each $i$. That is why you need to mimic the $n=1$ proof, mostly.
Recall that for the $n=1$, we prove that $$f'(t)\leq 0$$ and ... | {
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How to find the limit for the quotient of the least number $K_n$ such that the partial sum of the harmonic series $\geq n$ Let $$S_n=1+1/2+\cdots+1/n.$$ Denote by $K_n$ the least subscript $k$ such that $S_k\geq n$. Find the limit $$\lim_{n\to\infty}\frac{K_{n+1}}{K_n}\quad ?$$
| We know that $H_n=\ln n + \gamma +\epsilon(n)$, where $\epsilon(n)\approx \frac{1}{2n}$ and in any case $\epsilon(n)\rightarrow 0$ as $n\rightarrow \infty$. If $m=H_n$ we may as a first approximation solve as $n=e^{m-\gamma}$. Hence the desired limit is $$\lim_{m\rightarrow \infty} \frac{e^{m+1-\gamma}}{e^{m-\gamma}}... | {
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Proof regarding unitary self-adjoin linear operators I'm suck on how to do the following Linear Algebra proof:
Let $T$ be a self-adjoint operator on a finite-dimensional inner product space $V$.
Prove that for all $x \in V$, $||T(x)\pm ix||^2=||T(x)||^2+||x||^2.$
Deduce that $T-iI$ is invertible and that $[(T-iI)^{-1}... | we have $$(Tx+ix,Tx+ix)=(Tx,Tx)+(ix, Tx)+(Tx, ix)+(ix,ix)=|Tx|^{2}+i(x,Tx)-i(x,Tx)+|x|^{2}$$where I assume you define the inner product to be Hermitian. I think for $Tx-ix$ it should be similar. The rest should be leave as an exercise for you; they are not that difficult.
To solve the last one, notice we have $(T-iI)^... | {
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positive Integer value of $n$ for which $2005$ divides $n^2+n+1$
How Can I calculate positive Integer value of $n$ for which $2005$
divides $n^2+n+1$
My try:: $2005 = 5 \times 401$
means $n^2+n+1$ must be a multiple of $5$ or multiple of $401$
because $2005 = 5 \times 401$
now $n^2+n+1 = n(n+1)+1$
now $n(n+1)+1$ co... | A number of the form $n^2+n+1$ has divisors of the form 3, or any number of $6n+1$, and has a three-place period in base n.
On the other hand, there are values where 2005 divides some $n^2+n-1$, for which the divisors are of the form n, 10n+1, 10n+9. This happens when n is 512 or 1492 mod 2005.
| {
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Do these definitions of congruences on categories have the same result in this context? Let $\mathcal{D}$ be a small category and let $A=A\left(\mathcal{D}\right)$
be its set of arrows. Define $P$ on $A$ by: $fPg\Leftrightarrow\left[f\text{ and }g\text{ are parallel}\right]$
and let $R\subseteq P$.
Now have a look at ... | They are identical. I will suppress the composition symbol for brevity and convenience.
Suppose first that $C \in \mathcal C_w$, and that $f C g$. Since $h C h$ and $k C k$, we have $f C g$ implies $hf C hg$, which in turn implies $hfk C hgk$. Thus $C \in \mathcal C_s$.
Suppose now that $C \in \mathcal C_s$, and that $... | {
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Power series of $\frac{\sqrt{1-\cos x}}{\sin x}$ When I'm trying to find the limit of $\frac{\sqrt{1-\cos x}}{\sin x}$ when x approaches 0, using power series with "epsilon function" notation, it goes :
$\dfrac{\sqrt{1-\cos x}}{\sin x} = \dfrac{\sqrt{\frac{x^2}{2}+x^2\epsilon_1(x)}}{x+x\epsilon_2(x)} = \dfrac{\sqr... | It is the same as in the "$\epsilon$" notation. For numerator, we want $\sqrt{x^2\left(\frac{1}{2}+o(1)\right)}$, which is $|x|\sqrt{\frac{1}{2}+o(1)}$. In the denominator, we have $x(1+o(1))$.
Remark: Note that the limit as $x\to 0$ does not exist, though the limit as $x$ approaches $0$ from the left does, and the li... | {
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How to prove the existence of infinitely many $n$ in $\mathbb{N}$,such that $(n^2+k)|n!$
Show there exist infinitely many $n$ $\in \mathbb{N}$,such that
$(n^2+k)|n!$ and $k\in N$
I have a similar problem:
Show that there are infinitely many $n \in \mathbb{N}$,such that
$$(n^2+1)|n!$$
Solution: We consider t... | Similar to your solution of $k=1$.
Consider the pell's equation $n^2 + k = (k^2+k) y^2$. This has solution $(n,y) = (k,1)$, hence has infinitely many solutions. Note that $k^2 + k = k(k+1) $ is never a square for $k\geq 2$, hence is a Pell's Equation of the form $n^2 - (k^2+k) y^2 = -k$.
Then, $2y = 2\sqrt{ \frac{ n^2+... | {
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Prove that $3^n>n^4$ if $n\geq8$ Proving that $3^n>n^4$ if $n\geq8$
I tried mathematical induction start from $n=8$ as the base case, but I'm stuck when I have to use the fact that the statement is true for $n=k$ to prove $n=k+1$. Any ideas?
Thanks!
| You want to show $3^n>n^4$. This i.e. to showing $e^{n\ln3}>e^{4\ln n}$. This means you want to show $n\ln 3>4\ln n$. It suffices to show $\frac{n}{\ln n }>\frac{4}{\ln 3}$. Since $\frac{8}{\ln 8}>\frac{4}{\ln 3}$ and since $f(x)=\frac{x}{\ln x}$ has a positive first derivative for $x\geq 8$, the result follows.
| {
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"timestamp": "2023-03-29T00:00:00",
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Parent and childs of a full d-node tree i have a full d-node tree (by that mean a tree that each node has exactly d nodes as kids).
My question is, if i get a random k node of this tree, in which position do i get his kids and his parent?
For example, if i have a full binary tree, the positions that i can find the par... | It looks like you're starting numbering at 1 for the root, and numbering "left to right" on each level/depth.
If the root has depth $0$, then there are $d^{t}$ nodes with depth $t$ from the root in a full $d$-dimensional tree. Also, the depth of node $k$ is $\ell_{k}=\lceil\log_{d}(k-1)\rceil$.
The number of nodes at d... | {
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Construct a linear programming problem for which both the primal and the dual problem has no feasible solution Construct (that is, find its coefficients) a linear programming problem with at most two variables and two restrictions, for which both the primal and the dual problem has no feasible solution.
For a linear p... | Consider an LP:
$$
\begin{aligned}
\min \; & x_{1}+2 x_{2} \\
\text { s.t. } & x_{1}+x_{2}=1 \\
& 2 x_{1}+2 x_{2}=3
\end{aligned}
$$
and its dual:
$$
\begin{aligned}
\max\; & y_{1}+3 y_{2} \\
\text { s.t. } & y_{1}+2 y_{2}=1 \\
& y_{1}+2 y_{2}=2
\end{aligned}
$$
They are both infeasible.
| {
"language": "en",
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notation question (bilinear form) So I have to proof the following:
for a given Isomorphism $\phi : V\rightarrow V^*$ where $V^*$ is the dual space of $V$ show that $s_{\phi}(v,w)=\phi(v)(w)$ defines a non degenerate bilinear form.
My question : Does $\phi(v)(w)$ denote the map from $v$ to a linear function $w$? (in t... | Note that $\phi$ is a map from $V$ to $V^\ast$. So for each $v \in V$, we get an element $\phi(v) \in V^\ast$. Now $V^\ast$ is the space of linear functionals on $V$, i.e.
$$V^\ast = \{\alpha: V \longrightarrow \Bbb R \mid \alpha \text{ is linear}\}.$$
So each element of $V^\ast$ is a function from $V$ to $\Bbb R$. The... | {
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Moment generating function of a stochastic integral Let $(B_t)_{t\geq 0}$ be a Brownian motion and $f(t)$ a square integrable deterministic function. Then:
$$
\mathbb{E}\left[e^{\int_0^tf(s) \, dB_s}\right] = \mathbb{E}\left[e^{\frac{1}{2}\int_0^t f^2(s) \, ds}\right]
$$
Now assume $(X_t)_{t\geq 0}$ is such that $\left... | If $X$ and $B$ are independent, yes (use the first result to compute the expectation conditional on $X$, then take the expectation).
Otherwise, no. For a counterexample, consider $X=B$ and use Itô's formula $\mathrm d (B^2)=2B\mathrm dB+\mathrm dt$.
| {
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Matrix $BA\neq$$I_{3}$ If $\text{A}$ is a $2\times3$ matrix and $\text{B}$ is a $3\times2$ matrix, prove that $\text{BA}=I_{3}$ is impossible.
So I've been thinking about this, and so far I'm thinking that a homogenous system is going to be involved in this proof. Maybe something about one of the later steps being that... | Consider the possible dimension of the columnspace of the matrix $BA$. In particular, since $A$ has at most a two-dimensional columnspace, $BA$ has at most a two-dimensional columnspace. Stated more formally, if $A$ has rank $r_a$ and $B$ has rank $r_b$, then $BA$ has rank at most $\min\{ r_a, r_b \}$.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Are the graphs of these two functions equal to each other? The functions are: $y=\frac{x^2-4}{x+2}$ and $(x+2)y=x^2-4$.
I've seen this problem some time ago, and the official answer was that they are not.
My question is: Is that really true?
The functions obviously misbehave when $x = -2$, but aren't both of them inde... | $(1)$The first function is undefined at $x = -2$,
$(2)$ the second equation is defined at $x = -2$:
$$(x + 2) y = x^2 - 4 \iff xy + 2y = x^2 - 4\tag{2}$$ It's graph includes the entire line $x = -2$. At $x = -2$, all values of y are defined, so every point lying on the line $x = -2$: each of the form $(-2, y)$ are inc... | {
"language": "en",
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Working with subsets, as opposed to elements. Especially in algebraic contexts, we can often work with subsets, as opposed to elements. For instance, in a ring we can define
$$A+B = \{a+b\mid a \in A, b \in B\},\quad -A = \{-a\mid a \in A\}$$
$$AB = \{ab\mid a \in A, b \in B\}$$
and under these definitions, singletons ... | Since we're talking about ordered rings, maybe the ordering could be applicable to each comparison, too. i.e.,
$$ a_n \le b_n \forall a \in A,b \in B$$
if you were to apply this to the sets of even integers (greater than 0) and odd integers it might look like 1<2, 3<4, etc. Of course, cardinality comes into play since... | {
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Is there a name for this given type of matrix? Given a finite set of symbols, say $\Omega=\{1,\ldots,n\}$, is there a name for an $n\times m$ matrix $A$ such that every column of $A$ contains each elements of $\Omega$?
(The motivation for this question comes from looking at $p\times p$ matrices such that every column c... | A sensible definition for this matrix would be a column-Latin rectangle, since the transpose is known as a row-Latin rectangle. Example:
A. Drisko, Transversals in Row-Latin Rectangles, JCTA 81 (1998), 181-195.
The $m=n$ case is referred to as a column-Latin square in the literature (this is in widespread use).
I fo... | {
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How to prove these two ways give the same numbers? How to prove these two ways give the same numbers?
Way 1:
Step 1 : 73 + 1 = 74. Get the odd part of 74, which is 37
Step 2 : 73 + 37 = 110. Get the odd part of 110, which is 55
Step 3 : 73 + 55 = 128. Get the odd part of 128, which is 1
Continuing this operati... | Let $M=37$ (or any odd prime for that matter).
To formalize your first "way":
You start with an odd number $a_1$ with $1\le a_1<M$ (here specifically: $a_1=1$) and then recursively let $a_{n+1}=u$, where $u$ is the unique odd number such that $M+a_n=2^lu$ with $l\in\mathbb N_0$.
By induction, one finds that $a_n$ is an... | {
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equality between the index between field with $p^{n}$ elements and $ \mathbb{F}_{p}$ and n? can someone explain this?
$ \left[\mathbb{F}_{p^{n}}:\mathbb{F}_{p}\right]=n $
| $$|\Bbb F_p|=p\;,\;\;|\Bbb F_n^n|=p^n$$
amd since any element in the latter is a unique linear combination of some elements of it and scalars from the former, it must be that those some elements are exactly $\,n\,$ in number.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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General Solution of Diophantine equation Having the equation:
$$35x+91y = 21$$
I need to find its general solution.
I know gcf $(35,91) = 7$, so I can solve $35x+917 = 7$ to find $x = -5, y = 2$. Hence a solution to $35x+91y = 21$ is $x = -15, y = 2$.
From here, however, how do I move on to finding the set of general... | Hint: If $35x + 91y = 21$ and $35x^* + 91y^* = 21$ for for some $(x,y)$ and $(x^*, y^*)$, we can subtract the two equalities and get $5(x-x^*) + 13(y-y^*) = 0$. What does this tell us about the relation between any two solutions?
Now, $5$ and $13$ share no common factor and we're dealing with integers, $13$ must divide... | {
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Comparing $\sqrt{1001}+\sqrt{999}\ , \ 2\sqrt{1000}$ Without the use of a calculator, how can we tell which of these are larger (higher in numerical value)?
$$\sqrt{1001}+\sqrt{999}\ , \ 2\sqrt{1000}$$
Using the calculator I can see that the first one is 63.2455453 and the second one is 63.2455532, but can we tell with... | You can tell without calculation if you can visualize the graph of the square-root function; specifically, you need to know that the graph is concave (i.e., it opens downward). Imagine the part of the graph of $y=\sqrt x$ where $x$ ranges from $999$ to $1001$. $\sqrt{1000}$ is the $y$-coordinate of the point on the g... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Limit of $\lim_{x \to 0}\left (x\cdot \sin\left(\frac{1}{x}\right)\right)$ is $0$ or $1$? WolframAlpha says $\lim_{x \to 0} x\sin\left(\dfrac{1}{x}\right)=0$ but I've found it $1$ as below:
$$
\lim_{x \to 0} \left(x\sin\left(\dfrac{1}{x}\right)\right) = \lim_{x \to 0} \left(\dfrac{1}{x}x\dfrac{\sin\left(\dfrac{1}{x}\... | $$\lim_{x \to 0} \left(x\cdot \sin\left(\dfrac{1}{x}\right)\right) = \lim_{\large\color{blue}{\bf x\to 0}}
\left(\frac{\sin\left(\dfrac{1}{x}\right)}{\frac 1x}\right) =
\lim_{\large\color{blue}{\bf x \to \pm\infty}} \left(\frac{\sin x}{x}\right) = 0 \neq 1$$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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"answer_id": 1
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Smooth maps on a manifold lie group $$
\operatorname{GL}_n(\mathbb R) = \{ A \in M_{n\times n} | \det A \ne 0 \} \\
\begin{align}
&n = 1, \operatorname{GL}_n(\mathbb R) = \mathbb R - \{0\} \\
&n = 2, \operatorname{GL}_n(\mathbb R) = \left\{\begin{bmatrix}a&b\\c&d\end{bmatrix}\Bigg| ad-bc \ne 0\right\}
\end{align}
$$
$(... | Well first you have to decide what exactly the "topology" on matrices is. Suppose we considered matrices just as vectors in $\mathbb{R}^{n^2}$, with the usual metric topology. Matrix multiplication by say $A$ take a matrix $B$ to another $\mathbb{R}^{n^2}$ vector where the entries are polynomials in the components of $... | {
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Why is boundary information so significant? -- Stokes's theorem Why is it that there are so many instances in analysis, both real and complex, in which the values of a function on the interior of some domain are completely determined by the values which it takes on the boundary?
I know that this has something to do wit... | This is because many phenomena in nature can be described by well-defined fields. When we look at the boundaries of some surface enclosing the fields, it tells us everything we need to know.
Take a conservative field, such as an electric or gravitational field. If we want to know how much energy needs to be added or r... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/394797",
"timestamp": "2023-03-29T00:00:00",
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"question_score": "21",
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Counting Problem - N unique balls in K unique buckets w/o duplication $\mid$ at least one bucket remains empty and all balls are used I am trying to figure out how many ways one can distribute $N$ unique balls in $K$ unique buckets without duplication such that all of the balls are used and at least one bucket remains ... | A slightly different approach using the twelvefold way.
If $K>N$ then it doesn't matter how you distribute the balls since at least one bucket will always be empty. In this case we are simply counting functions from a $N$ element set to a $K$ element set. Therefore the number of distributions is $K^N$.
If $K=N$ then th... | {
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Finding an area of the portion of a plane? I need help with a problem I got in class today any help would be appreciated!
Find the area of the portion of the portion of the plane $6x+4y+3z=12$ that passes through the first octant where $x, y$, and $z$ are all positive..
I graphed this plane and got all the vertices but... | If you have the three vertices, you can calculate the length of the three sides and use Heron's formula
| {
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Linear Algebra determinant and rank relation True or False?
If the determinant of a $4 \times 4$ matrix $A$ is $4$
then its rank must be $4$.
Is it false or true?
My guess is true, because the matrix $A$ is invertible.
But there is any counter-example?
Please help me.
| You're absolutely correct. The point of mathematical proof is that you don't need to go looking for counterexamples once you've found the proof. Beforehand that's very reasonable, but once you're done you're done.
Determinant 4 is nonzero $\implies$ invertible $\implies$ full rank.
Each of these is a standard propositi... | {
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Dimensions of vector subspaces in a direct sum are additive $V = U_1\oplus U_2~\oplus~...~ \oplus~ U_n~(\dim V < ∞)$ $\implies \dim V = \dim U_1 + \dim U_2 + ... + \dim U_n.$ [Using the result if $B_i$ is a basis of $U_i$ then $\cup_{i=1}^n B_i$ is a basis of $V$]
Then it suffices to show $U_i\cap U_j-\{0\}=\emptyset$ ... | Yes, you're correct.
Were you second guessing yourself? If so, no need to:
You're argument is "spot on".
If you'd like to save yourself a little space, and work, you can write your sum as:
$$ \dim V = \sum_{i = 1}^n \dim U_i$$
"...If not, let $v\in U_i\cap U_j-\{0\}.$ Then
$$v= v(\in U_i) + \sum_{\large 1\leq j\leq n... | {
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About the induced vector measure of a Pettis integrable function(part 2) Notations: In what follows, $X$ stands for a Hausdorff LCTVS and $X'$ its topological dual. Let $(T,\mathcal{M},\mu)$ be a finite measure space, i.e., $T$ is a nonempty set, $\mathcal{M}$ a $\sigma$-algebra of subsets of $T$ and $\mu$ is a nonnega... | Let $\mu(E)=0$. Take arbitrary $x'\in X'$, then
$$
x'(m_f(E))=\int_E (x'\circ f)d\mu=0
$$
then. Since $x'\in X'$ is arbitrary by corollary of Hahn-Banach theorem $m_f(E)=0$. Thus, $m_f\ll\mu$
| {
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Finding the Taylor series of $f(z)=\frac{1}{z}$ around $z=z_{0}$ I was asked the following (homework) question:
For each (different but constant) $z_{0}\in G:=\{z\in\mathbb{C}:\,
z\neq0$} find a power series $\sum_{n=0}^{\infty}a_{n}(z-z_{0})^{n}$
whose sum is equal to $f(z)$ on some subset of $G$. Please specify
... | I think you were on the right track:
$$\frac1z=\frac1{z_0+(z-z_0)}=\frac1{z_0}\cdot\frac1{1+\frac{z-z_0}{z_0}}=\frac1z_0\left(1-\frac{z-z_0}{z_0}+\frac{(z-z_0)^2}{z_{0}^2}-\ldots\right)=$$
$$=\frac1z_0-\frac{z-z_0}{z_0^2}+\frac{(z-z_0)^2}{z_0^3}+\ldots+\frac{(-1)^n(z-z_0)^n}{z_0^{n+1}}+\ldots$$
As you can see, this is ... | {
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prove that $\sum_{n=0}^{\infty} {\frac{n^2 2^n}{n^2 + 1}x^n} $ does not converge uniformly in its' convergence radius In calculus:
Given $\displaystyle \sum_{n=0}^{\infty} {\frac{n^2 2^n}{n^2 + 1}x^n} $, prove that it converges for $-\frac{1}{2} < x < \frac{1}{2} $, and that it does not converge uniformly in the area ... | According to Hagen von Eitzen theorem, you know that it will uniformly (and even absolutly) converge for intervals included in ]-1/2;1/2[. So the problem must be there.
The definition of uniform convergence is :
$ \forall \epsilon>0,\exists N_ \varepsilon \in N,\forall n \in N,\quad [ n \ge N_ \varepsilon \Rightarrow ... | {
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Sequence $(a_n)$ s.t $\sum\sqrt{a_na_{n+1}}<\infty$ but $\sum a_n=\infty$ I am looking for a positive sequence $(a_n)_{n=1}^{\infty}$ such that $\sum_{n=1}^{\infty}\sqrt{a_na_{n+1}}<\infty$ but $\sum_{n=1}^{\infty} a_n>\infty$.
Thank you very much.
| The simplest example I can think of is $\{1,0,1,0,...\}$. If you want your elements to be strictly positive, use some fast-converging sequence such as $n^{-4}$ in place of the zeroes.
| {
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A limit on binomial coefficients Let $$x_n=\frac{1}{n^2}\sum_{k=0}^n \ln\left(n\atop k\right).$$ Find the limit of $x_n$.
What I can do is just use Stolz formula. But I could not proceed.
| $x_n=\frac{1}{n^2}\sum_{k=0}^{n}\ln{n\choose k}=\frac{1}{n^2}\ln(\prod {n\choose k})=\frac{1}{n^2}\ln\left(\frac{n!^n}{n!^2.(n-1)!^2(n-2)!^2...0!^2}\right)$ since ${n\choose k}=\frac{n!}{k!(n-k)!}$
$e^{n^2x_n}=\left(\frac{n^n(n-1)!}{n!^2}\right)e^{(n-1)^2x_{n-1}}=\left(\frac{n^{n-1}}{n!}\right)e^{(n-1)^2x_{n-1}}$
By St... | {
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sum of monotonic increasing and monotonic decreasing functions I have a question regarding sum of monotinic increasing and decreasing functions. Would appreciate very much any help/direction:
Consider an interval $x \in [x_0,x_1]$. Assume there are two functions $f(x)$ and $g(x)$ with $f'(x)\geq 0$ and $g'(x)\leq 0$. W... | Alas, the answer is no.
$$f(x)=\begin{cases}-4& x\in[0,2]\\ -2& x\in [2,4]\\0& x\in[4,6]\end{cases}$$
$$g(x)=\begin{cases}5 & x\in [0,1]\\3& x\in[1,3]\\1& x\in[3,5]\\ 0 & x\in[5,6]\end{cases}$$
$$q(x)=\begin{cases} 1 & x\in [0,1]\\ -1& x\in[1,2]\\ 1 & x\in[2,3]\\ -1 & x\in[3,4]\\ 1& x\in[4,5]\\ 0 & x\in[5,6]\end{cases}... | {
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What is the Broader Name for the Fibonacci Sequence and the Sequence of Lucas Numbers? Fibonacci and Lucas sequences are very similar in their definition. However, I could just as easily make another series with a similar definition; an example would be:
$$x_0 = 53$$
$$x_1 = 62$$
$$x_n = x_{n - 1} + x_{n - 2}$$
What I ... | Occasionally (as in the link posted by vadim123) you see "Fibonacci integer sequence".
Lucas sequences (of which the Lucas sequence is but one example) are a slight generalization. Sometimes the term Horadam sequence is used instead.
The general classification under which all of these fall is the linear recurrence rela... | {
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question on summation? Please, I need to know the proof that
$$\left(\sum_{k=0}^{\infty }\frac{n^{k+1}}{k+1}\frac{x^k}{k!}\right)\left(\sum_{\ell=0}^{\infty }B_\ell\frac{x^\ell}{\ell!}\right)=\sum_{k=0}^{\infty }\left(\sum_{i=0}^{k}\frac{1}{k+1-i}\binom{k}{i}B_in^{k+1-i}\right)\frac{x^k}{k!}$$
where $B_\ell$, $B_i$ ar... | $$\left(\sum_{k=0}^{\infty} \dfrac{n^{k+1}}{k+1} \dfrac{x^k}{k!} \right) \left(\sum_{l=0}^{\infty} B_l \dfrac{x^l}{l!}\right) = \sum_{k,l} \dfrac{n^{k+1}}{k+1} \dfrac{B_l}{k! l!} x^{k+l}$$
$$\sum_{k,l} \dfrac{n^{k+1}}{k+1} \dfrac{B_l}{k! l!} x^{k+l} = \sum_{m=0}^{\infty} \sum_{l=0}^{m} \dfrac{n^{m-l+1}}{m-l+1} \dfrac{B... | {
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Packing circles on a line On today's TopCoder Single-Round Match, the following question was posed (the post-contest write-up hasn't arrived yet, and their explanations often leave much to be desired anyway, so I thought I'd ask here):
Given a maximum of 8 marbles and their radii, how would you put them next to each ot... | If I understand well, the centers of the marbles are on the line.
In that case, we can fix a coordinate system such that the $x$-axis is the line, and the center $C_1$ of the first marble is the origin. Then, its lowest point is $P=(0,-r_1)$. Calculate the coordinates of the centers of the next circles:
$$C_2=(r_1+r_2,... | {
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Help me prove this inequality : How would I go about proving this?
$$ \displaystyle\sum_{r=1}^{n} \left( 1 + \dfrac{1}{2r} \right)^{2r} \leq n \displaystyle\sum_{r=0}^{n+1} \displaystyle\binom{n+1}{r} \left( \dfrac{1}{n+1} \right)^{r}$$
Thank you! I've tried so many things. I've tried finding a series I could compare... | Here is the proof that $(1+1/x)^x$ is concave for $x\ge 1$.
The second derivative of $(1+1/x)^x$ is $(1+1/x)^x$ times $$p(x)=\left(\ln\left(1+\frac{1}{x}\right)-\frac{1}{1+x}\right)^2-\frac{1}{x(1+x)^2}$$
Now for $x\ge 1$, we have $$\ln(1+1/x)-\frac{2}{1+x}\le \frac{1}{x}-\frac{2}{1+x}=\frac{1-x}{x(1+x)}\le 0$$ and $$... | {
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Baire's theorem from a point of view of measure theory According to Baire's theorem, for each countable collection of open dense subsets of $[0,1]$, their intersection $A$ is dense. Are we able to say something about the Lebegue's measure of $A$? Must it be positive? Of full measure?
Thank you for help.
| Let $q_1,q_2,\ldots$ be an enumeration of the rationals. Let $I_n^m$ be an open interval centered at $q_n$ with length at most $1/m 1/2^n$. Then $\bigcup_n I_n^m$ is an open dense set with Lebesgue measure at most $1/m$. The intersection of these open dense sets has Lebesgue measure zero.
| {
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Calculating $\sqrt{28\cdot 29 \cdot 30\cdot 31+1}$ Is it possible to calculate $\sqrt{28 \cdot 29 \cdot 30 \cdot 31 +1}$ without any kind of electronic aid?
I tried to factor it using equations like $(x+y)^2=x^2+2xy+y^2$ but it didn't work.
| If you are willing to rely on the problem setter to make sure it is a natural, it has to be close to $29.5^2=841+29+.25=870.25$ The one's digit of the stuff under the square root sign is $1$, so it is either $869$ or $871$. You can either calculate and check, or note that two of the factors are below $30$ and only o... | {
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Methods for determining the convergence of $\sum\frac{\cos n}{n}$ or $\sum\frac{\sin n}{n}$ As far as I know, the textbook approach to determining the convergence of series like $$\sum_{n=1}^\infty\frac{\cos n}{n}$$ and $$\sum_{n=1}^\infty\frac{\sin n}{n}$$ uses Dirichlet's test, which involves bounding the partial sum... | Since you specifically said "Are there any other approaches to seeing that these series are convergent" I will take the bait and give an extremely sketchy argument that isn't meant to be a proof at all.
The harmonic series $\sum_{n=1}^{\infty} \frac{1}{n}$ famously diverges, but very slowly. For large $n$ the numbers b... | {
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Does $\,x>0\,$ hint that $\,x\in\mathbb R\,$?
Does $x>0$ suggest that $x\in\mathbb R$?
For numbers not in $\,\mathbb R\,$ (e.g. in $\mathbb C\setminus \mathbb R$), their sizes can't be compared.
So can I omit "$\,x\in\mathbb R\,$" and just write $\,x>0\,$?
Thank you.
| It really depends on context. But be safe; just say $x > 0, x\in \mathbb R$.
Omitting the clarification can lead to misunderstanding it. Including the clarification takes up less than a centimeter of space. Benefits of clarifying the domain greatly outweigh the consequences of omitting the clarification.
Besides one ... | {
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Simplifying this expression $(e^u-1)(e^u-e^l)$ Is it possible to write the following
$$(e^u-1)(e^u-e^l)$$
as
$$e^{f(u,l)}-1?$$
| The first expression evaluates to $e^{2u}-e^u-e^l+1$, so if your question is whether there is some algebraic manipulation that brings this into the form of an exponential minus $1$ for general $l,u$ then the answer is no.
In fact the range of the function $x\mapsto e^x-1$ in $\Bbb R$ is $(-1,\infty)$, and the product o... | {
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Evaluating $\int_0^\infty \frac{\log (1+x)}{1+x^2}dx$ Can this integral be solved with contour integral or by some application of residue theorem?
$$\int_0^\infty \frac{\log (1+x)}{1+x^2}dx = \frac{\pi}{4}\log 2 + \text{Catalan constant}$$
It has two poles at $\pm i$ and branch point of $-1$ while the integral is to b... | Following the same approach in this answer, we have
\begin{gather*}
\int_0^\infty\frac{\ln^a(1+x)}{1+x^2}\mathrm{d}x=\int_0^\infty\frac{\ln^a\left(\frac{1+y}{y}\right)}{1+y^2}\mathrm{d}y\\
\overset{\frac{y}{1+y}=x}{=}(-1)^a\int_0^1\frac{\ln^a(x)}{x^2+(1-x)^2}\mathrm{d}x\\
\left\{\text{write $\frac{1}{x^2+(1-x)^2}=\ma... | {
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find out the value of $\dfrac {x^2}{9}+\dfrac {y^2}{25}+\dfrac {z^2}{16}$ If $(x-3)^2+(y-5)^2+(z-4)^2=0$,then find out the value of $$\dfrac {x^2}{9}+\dfrac {y^2}{25}+\dfrac {z^2}{16}$$
just give hint to start solution.
| Hint: What values does the function $x^2$ acquire(positive/negarive)? What is the solution of the equation $x^2=0$?
Can you find the solution of the equation $x^2+y^2=0$?
Now, what can you say about the equation $(x-3)^2+(y-5)^2+(z-4)^2=0
$? Can you find the values of $x,y,z?$
| {
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Solving modular equations that gives GCD = 1 I have problems with understanding modular equations that gives GCD = 1. For example:
$$3x \equiv 59 \mod 100$$
So I'm getting $GCD(3, 100) = 1$. Now:
$1 = -33*3 + 100$
That's where the first question appears - I always guess those -33 and 1 (here) numbers...is there a way t... | $$\begin{eqnarray} \text{Note} &&\ 1 &=&\ 3\, (-33)\ \ \ \, +\ \ \ \, 100\, (1) \\
\stackrel{\times\, 59\ }\Rightarrow && 59\, &=&\ 3\, (-33\cdot 59) + 100\, (59)\end{eqnarray}$$
Hence $\ 59 = 3x+100y \ $ has a particular solution $\rm\:(x,y) = (-33\cdot 59, 59).\, $ By linearity, the general solution is the sum of ... | {
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HINT for summing digits of a large power I recently started working through the Project Euler challenges, but I've got stuck on #16 (http://projecteuler.net/problem=16)
$2^{15} = 32768$ and the sum of its digits is $3 + 2 + 7 + 6 + 8 = 26$. What is the sum of the digits of the number $2^{1000}$?
(since I'm a big fan ... | In this case, I'm afraid you just have to go ahead and calculate $2^{1000}$. There are various clever ways to do this, but for numbers this small a simple algorithm is fast enough.
The very simplest is to work in base 10 from the start. Faster is to work in binary, and convert to base 10 at the end, but this conversion... | {
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Find the transform I have the paper with 3 points on it. I have also a photo of this paper. How can I determine where is the paper on the photo, if I know just the positions of these points? And are 3 points enough?
It doesn't look like a linear transform: the paper turns into trapeze on the photo. But it should be abl... | I believe a projective transformation is exactly right. But with a projective transformation you can take any four points in general position (no three collinear) to any other four. So you'd better add a fourth point to reconstruct the transformation.
In terms of the linear algebra, you want to construct a $3\times 3$... | {
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If we know the eigenvalues of a matrix $A$, and the minimal polynom $m_t(a)$, how do we find the Jordan form of $A$? We have just learned the Jordan Form of a matrix, and I have to admit that I did not understand the algorithm.
Given $A = \begin{pmatrix} 1 & 1 & 1 & -1 \\ 0 & 2 & 1 & -1 \\ 1 & -1 & 2 & -1 \\ 1 & -1 & ... | Since the minimal polynomial has degree $4$ which is the same order of the matrix, you know that $A$'s smith normal form is
$\begin{pmatrix} 1 & 0 & 0 & 0\\ 0 & 1 & 0 & 0\\ 0 & 0 & 1 & 0\\ 0 & 0 & 0 & m_A(\lambda )\end{pmatrix}$.
Therefore the elementary divisors (I'm not sure this is the correct term in english) are ... | {
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If $\lim\limits_{x \to \pm\infty}f(x)=0$, does it imply that $\lim\limits_{x \to \pm\infty}f'(x)=0$?
Suppose $f:\mathbb{R} \rightarrow \mathbb{R}$ is everywhere differentiable and
*
*$\lim_{x \to \infty}f(x)=\lim_{x \to -\infty}f(x)=0$,
*there exists $c \in \mathbb{R}$ such that $f(c) \gt 0$.
Can we say anything a... | To correct a incorrect attempt,
let $f(x) = e^{-x^2} \cos(e^{x^4})$,
so
$\begin{align}f'(x) &= e^{-x^2} 4 x^3 e^{x^4}(-\sin(e^{x^3}))
-2x e^{-x^2} \cos(e^{x^3})\\
&= -4 x^3 e^{x^4-x^2} \sin(e^{x^3})
-2x e^{-x^2} \cos(e^{x^3})\\
\end{align}
$
The $e^{x^4-x^2}$ term makes
$f'(x)$ oscillate violently
and unboundedly
as $... | {
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Property of the trace of matrices Let $A(x,t),B(x,t)$ be matrix-valued functions that are independent of $\xi=x-t$ and satisfy $$A_t-B_x+AB-BA=0$$ where $X_q\equiv \frac{\partial X}{\partial q}$.
Why does it then follow that $$\frac{d }{d \eta}\textrm{Trace}[(A-B)^n]=0$$ where $n\in \mathbb N$ and $\eta=x+t$?
Is ther... | (This may not be a neat way to prove the assertion, but it's a proof anyway.) Let $\eta=x+t$ and $\nu=x-t$. Then $x=\eta+\nu$ and $t=\eta-\nu$ are functions of $\eta$ and $\nu$, $A=A(\eta+\nu,\eta-\nu)$ and similarly for $B$. As both $A$ and $B$ are independent of $\nu$, by the total derivative formula, we get
\begin{a... | {
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Infinite Series Problem Using Residues Show that $$\sum_{n=0}^{\infty}\frac{1}{n^2+a^2}=\frac{\pi}{2a}\coth\pi a+\frac{1}{2a^2}, a>0$$
I know I must use summation theorem and I calculated the residue which is:
$$Res\left(\frac{1}{z^2+a^2}, \pm ai\right)=-\frac{\pi}{2a}\coth\pi a$$
Now my question is: how do I get the l... | The method of residues applies to sums of the form
$$\sum_{n=-\infty}^{\infty} f(n) = -\sum_k \text{res}_{z=z_k} \pi \cot{\pi z}\, f(z)$$
where $z_k$ are poles of $f$ that are not integers. So when $f$ is even in $n$, you may express as follows:
$$2 \sum_{n=1}^{\infty} f(n) + f(0)$$
For this case, $f(z)=1/(z^2+a^2)$ a... | {
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Let $f :\mathbb{R}→ \mathbb{R}$ be a function such that $f^2$ and $f^3$ are differentiable. Is $f$ differentiable? Let $f :\mathbb{R}→ \mathbb{R}$ be a function such that $f^2$ and $f^3$ are differentiable. Is $f$ differentiable?
Similarly, let $f :\mathbb{C}→ \mathbb{C}$ be a function such that $f^2$ and $f^3$ are ana... | If you do want $f^{p}$ to represent the $p^{\text{th}}$ iterate, then you can let $f$ denote the characteristic function of the irrational numbers. Then $f^2$ and $f^3$ are both identically zero, yet $f$ is nowhere continuous, let alone differentiable.
| {
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How to evaluate $\sqrt[3]{a + ib} + \sqrt[3]{a - ib}$? The answer to a question I asked earlier today hinged on the fact that
$$\sqrt[3]{35 + 18i\sqrt{3}} + \sqrt[3]{35 - 18i\sqrt{3}} = 7$$
How does one evaluate such expressions? And, is there a way to evaluate the general expression
$$\sqrt[3]{a + ib} + \sqrt[3]{a - ... | As I said in a previous answer, finding out that such a simplification occurs is exactly as hard as finding out that $35+18\sqrt{-3}$ has a cube root in $\Bbb Q(\sqrt{-3})$ (well actually, $\Bbb Z[(1+\sqrt{-3})/2]$ because $35+18\sqrt{-3}$ is an algebraic integer).
Suppose $p,q,d$ are integers. Then $p+q\sqrt d$ is an ... | {
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Compute value of $\pi$ up to 8 digits I am quite lost on how approximate the value of $\pi$ up to 8 digits with a confidence of 99% using Monte Carlo. I think this requires a large number of trials but how can I know how many trials?
I know that a 99% confidence interval is 3 standard deviations away from the mean in a... | The usual trick is to use the approximation of 355/113, repeating multiples of 113, until the denominator is 355. This version of $\pi=355/113$ is the usual implied value when eight digits is given.
Otherwise, the approach is roughly $\sqrt{n}$
| {
"language": "en",
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How to find inverse of the function $f(x)=\sin(x)\ln(x)$ My friend asked me to solve it, but I can't.
If $f(x)=\sin(x)\ln(x)$, what is $f^{-1}(x)$?
I have no idea how to find the solution. I try to find
$$\frac{dx}{dy}=\frac{1}{\frac{\sin(x)}{x}+\ln(x)\cos(x)}$$
and try to solve it for $x$ by some replacing and other ... | The function fails the horizontal line test for one, very badly in fact. One to one states that for any $x$ and $y$ in the domain of the function, that $f(x) = f(y) \Rightarrow x = y$. that is to each point in the domain there exists a unique point in the range. Many functions can be made to be one to one $(1-1)$ by re... | {
"language": "en",
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The negative square root of $-1$ as the value of $i$ I have a small point to be clarified.We all know $ i^2 = -1 $ when we define complex numbers, and we usually take the positive square root of $-1$ as the value of "$i$" , i.e, $i = (-1)^{1/2} $.
I guess it's just a convention that has been accepted in maths and the... | The square roots with the properties we know are used only for Positive Real numbers.
We say that $i$ is the square root of $-1$ but this is a convention. You cannot perform operations with the usual properties of radicals if you are dealing with complex numbers, rather than positive reals.
It is a fact that, $-1$ ha... | {
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Prove that ideal generated by.... Is a monomial ideal Similar questions have come up on the last few past exam papers and I don't know how to solve it. Any help would be greatly appreciated..
Prove that the ideal of $\mathbb{Q}[X,Y]$ generated by $X^2(1+Y^3), Y^3(1-X^2), X^4$ and $ Y^6$ is a monomial ideal.
| Hint. $(X^2(1+Y^3), Y^3(1-X^2), X^4, Y^6)=(X^2,Y^3)$.
| {
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Find the limit $ \lim_{n \to \infty}\left(\frac{a^{1/n}+b^{1/n}+c^{1/n}}{3}\right)^n$ For $a,b,c>0$, Find $$ \lim_{n \to \infty}\left(\frac{a^{1/n}+b^{1/n}+c^{1/n}}{3}\right)^n$$
how can I find the limit of sequence above?
Provide me a hint or full solution.
thanks ^^
| $$\lim_{n \to \infty}\ln \left(\frac{a^{1/n}+b^{1/n}+c^{1/n}}{3}\right)^n=\lim_{n \to \infty}\frac{\ln \left(\frac{a^{1/n}+b^{1/n}+c^{1/n}}{3}\right)}{\frac{1}{n}}=\lim_{x\to 0^+}\frac{\ln \left(\frac{a^{x}+b^{x}+c^{x}}{3}\right)}{x}$$
L'Hospital or the definition of the derivative solves it. Actually since the limit i... | {
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Summation of a finite series Let
$$f(n) = \frac{1}{1} + \frac{1}{2} + \frac{1}{3}+ \frac{1}{4}+...+ \frac{1}{2^n-1} \forall \ n \ \epsilon \ \mathbb{N} $$
If it cannot be summed , are there any approximations to the series ?
| $f(n)=H_{2^n-1}$, the $(2^n-1)$-st harmonic number. There is no closed form, but link gives the excellent approximation
$$H_n\approx\ln n+\gamma+\frac1{2n}+\sum_{k\ge 1}\frac{B_{2k}}{2kn^{2k}}=\ln n+\gamma+\frac1{2n}-\frac1{12n^2}+\frac1{120n^4}-\ldots\;,$$
where $\gamma\approx 0.5772156649$ is the Euler–Mascheroni con... | {
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How to find the norm of the operator $(Ax)_n = \frac{1}{n} \sum_{k=1}^n \frac{x_k}{\sqrt{k}}$? How to find the norm of the following operator
$$
A:\ell_p\to\ell_p:(x_n)\mapsto\left(n^{-1}\sum\limits_{k=1}^n k^{-1/2} x_k\right)
$$
Any help is welcome.
| Consider diagonal operator
$$
S:\ell_p\to\ell_p:(x_n)\mapsto(n^{-1/2}x_n)
$$
It is bounded and its norm is $\Vert S\Vert=\sup\{|n^{-1/2}|:n\in\mathbb{N}\} =1$
Consider Caesaro operator
$$
T:\ell_p\to\ell_p:(x_n)\mapsto\left(n^{-1}\sum\limits_{k=1}^nx_k\right)
$$
As it was proved earlier its norm is is $\Vert T\Vert\leq... | {
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Trigonometric substitution integral Trying to work around this with trig substitution, but end up with messy powers on sines and cosines... It should be simple use of trigonometric properties, but I seem to be tripping somewhere.
$$\int x^5\sqrt{x^2+4}dx $$
Thanks.
| It comes out pretty well if you put $x=2\tan\theta$. Doing it carefully, remove a $\tan\theta\sec\theta$ and everything else can be expressed as a polynomial in $\sec\theta$, hence it's easily done by substitution.
But a more "efficient" substitution is $u=\sqrt{x^2+4}$, or $u^2=x^2+4$. Then $2u\,du = 2x\,dx$ andthe in... | {
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How to show in a clean way that $z^4 + (x^2 + y^2 - 1)(2x^2 + 3y^2-1) = 0$ is a torus? How to show in a clean way that the zero-locus of $$z^4 + (x^2 + y^2 - 1)(2x^2 + 3y^2-1) = 0$$ is a torus?
| Not an answer, but here's a picture. Cheers.
| {
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Help with modular arithmetic If$r_1,r_2,r_3,r_4,\ldots,r_{ϕ(a)}$ are the distinct positive integers less than $a$ and coprime to $a$, is there some way to easily calculate, $$\prod_{k=1}^{\phi(a)}ord_{a}(r_k)$$
| The claim is true, with the stronger condition that there is some $i$ with $e_i=1$ and all other exponents are zero. The set of $r_i$'s is called a reduced residue system.
The second (now deleted) claim is false. Let $a=7$. Then $2^13^1=6^1$, two different representations.
| {
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Soving Recurrence Relation I have this relation
$u_{n+1}=\frac{1}{3}u_{n} + 4$
and I need to express the general term $u_{n}$ in terms of $n$ and $u_{0}$.
With partial sums I found this relation
$u_{n}=\frac{1}{3^n}u_{0} + 4\sum_{n=1}^n\frac{1}{3^n-1}$
But I also need to prove by mathematical induction that my $u_{n}$ ... | $$u_{ n }=\frac { 1 }{ { 3 }^{ n } } u_{ 0 }+4\sum _{ k=0 }^{ n-1 }{ \frac { 1 }{ { 3 }^{ k } } } =\frac { 1 }{ { 3 }^{ n } } u_{ 0 }+4\left( \frac { { \left( 1/3 \right) }^{ n }-1\quad }{ 1/3-1 } \right) =\frac { 1 }{ { 3 }^{ n } } u_{ 0 }-6\left( \frac { 1 }{ { 3 }^{ n } } -1 \right) $$ $$u_{ n }=\frac { 1 }{ { 3... | {
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Searching for unbounded, non-negative function $f(x)$ with roots $x_{n}\rightarrow \infty$ as $n \rightarrow \infty$ If a function $y = f(x)$ is unbounded and non-negative for all real $x$,
then is it possible that it can have roots $x_n$ such that $x_{n}\rightarrow \infty$ as $n \rightarrow \infty$.
| The function $ y = |x \sin(x)|$ has infinitely many roots $x_n$ such that $x_{n}\rightarrow \infty$ as $n \rightarrow \infty$.
| {
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How prove this $\int_{0}^{\infty}\sin{x}\sin{\sqrt{x}}\,dx=\frac{\sqrt{\pi}}{2}\sin{\left(\frac{3\pi-1}{4}\right)}$ Prove that
$$\int_{0}^{\infty}\sin{x}\sin{\sqrt{x}}\,dx=\frac{\sqrt{\pi}}{2}\sin{\left(\frac{3\pi-1}{4}\right)}$$
I have some question. Using this, find this integral is not converge, I'm wrong?
Thank you... | First make the substitution $x=u^2$ to get:
$\displaystyle \int _{0}^{\infty }\!\sin \left( x \right) \sin \left( \sqrt {x}
\right) {dx}=\int _{0}^{\infty }\!2\,\sin \left( {u}^{2} \right) \sin
\left( u \right) u{du}$,
$\displaystyle=-\int _{0}^{\infty }\!u\cos \left( u \left( u+1 \right) \right) {du}+
\int _{0}^{\i... | {
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Understanding the (Partial) Converse to Cauchy-Riemann We have that for a function $f$ defined on some open subset $U \subset \mathbb{C}$ then the following if true:
Suppose $u=\mathrm{Re}(f), v=\mathrm{Im}(f)$ and that all partial derivatives $u_x,u_y,v_x,v_y$ exists and are continuous on $U$. Suppose further that t... | Denote $h=\pmatrix{p\\q}.$
Since $u(x,\ y)$ is differentiable at the point $(x,\ y),$ increment of $u$ can be represented as
$$u(x+p,\ y+q)-u(x,\ y)=Du\;h+o(\Vert h\Vert)=pu_x(x,\ y)+qu_y(x,\ y)+o(|p|+|q|),$$
where $Du=\left(\dfrac{\partial{u}}{\partial{x}}, \ \dfrac{\partial{u}}{\partial{y}} \right).$
| {
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Finite ultraproduct I stucked when trying to prove:
If $A_\xi$ are domains of models of first order language and $|A_\xi|\le n$ for $n \in \omega$ for all $\xi$ in index set $X$ and $\mathcal U$ is ultrafilter of $X$ then $|\prod_{\xi \in X} A_\xi / \mathcal U| \le n$.
My tries:
If $X$ is finite set then $\mathcal U$ i... | The statement you are trying to prove is a consequence of Łoś's theorem - if every factor satisfies "there are no more than $n$ elements", then the set of factors that satisfy it is $X$, which is in $\mathcal{U}$, so by Łoś's theorem the ultraproduct will satisfy that sentence as well. Note that "there are no more than... | {
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Show that no number of the form 8k + 3 or 8k + 7 can be written in the form $a^2 +5b^2$ I'm studying for a number theory exam, and have got stuck on this question.
Show that no number of the form $8k + 3$ or $8k + 7$ can be written in the form $a^2 +5b^2$
I know that there is a theorem which tells us that $p$ is exp... | $8k+3,8k+7$ can be merged into $4c+3$ where $k,c$ are integers
Now, $a^2+5b^2=4c+3\implies a^2+b^2=4c+3-4b^2=4(c-b^2)+3\equiv3\pmod 4,$
But as $(2c)^2\equiv0\pmod 4,(2d+1)^2\equiv1\pmod 4,$
$a^2+b^2\equiv0,1,2\pmod 4\not\equiv3$
Clearly, $a^2+5b^2$ in the question can be generalized $(4m+1)x^2+(4n+1)y^2$ where $m,n$ a... | {
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The graph of $x^x$ I have a question about the graph of $f(x) = x^x$. How come the graph doesn't extend into the negative domain? Because, it is not as if the graph is undefined when $x=-5$. But according to the graph, that seems to be the case. Can someone please explain this?
Thanks
| A more direct answer is the reason your graphing calculator doesn't graph when $x<0$ is because there are infinite undefined "holes" and infinite defined points in the real plane. Even when you restrict the domain to $[-2,-1]$ this will still be the case.
Note that for $x^x$ when $x<0$ if you calculate for the output ... | {
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What can I do this cos term to remove the divide by 0? I was asked to help someone with this problem, and I don't really know the answer why. But I thought I'd still try.
$$\lim_{t \to 10} \frac{t^2 - 100}{t+1} \cos\left( \frac{1}{10-t} \right)+ 100$$
The problem lies with the cos term. What can I do with the cos term ... | The cos term is irrelevant. It can only wiggle between $-1$ and $1$, and is therefore killed by the $t^2-100$ term, since that approaches $0$.
For a less cluttered version of the same phenomenon, consider the function $f(x)=x\sin\left(\frac{1}{x}\right)$ (for $x\ne 0$). The absolute value of this is always $\le |x|$,... | {
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Why is the universal quantifier $\forall x \in A : P(x)$ defined as $\forall x (x \in A \implies P(x))$ using an implication? And the same goes for the existential quantifier: $\exists x \in A : P(x) \; \Leftrightarrow \; \exists x (x \in A \wedge P(x))$. Why couldn’t it be: $\exists x \in A : P(x) \; \Leftrightarrow \... | I can't answer your first question. It's just the definition of the notation.
For your second question, by definition of '$\rightarrow$', we have
$\exists x (x\in A \rightarrow P(x)) \leftrightarrow \exists x\neg(x\in A \wedge \neg P(x))$
I think you will agree that this is quite different from
$\exists x (x\in A \wedg... | {
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How many points can you find on $y=x^2$, for $x \geq 0$, such that each pair of points has rational distance? Open problem in Geometry/Number Theory. The real question here is:
Is there an infinite family of points on $y=x^2$, for $x \geq 0$, such that the distance between each pair is rational?
The question of "if no... | The answer to the "infinite family" question appears to be, no. Jozsef Solymosi and Frank de Zeeuw, On a question of Erdős and Ulam, Discrete Comput. Geom. 43 (2010), no. 2, 393–401, MR2579704 (2011e:52024), prove (according to the review by Liping Yuan) that no irreducible algebraic curve other than a line or a circle... | {
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What is the cardinality of $\Bbb{R}^L$? By $\Bbb{R}^L$, I mean the set that is interpreted as $\Bbb{R}$ in $L$, Godel's constructible universe. For concreteness, and to avoid definitional questions about $\Bbb{R}$, I'm looking at the set ${\cal P}(\omega)$ as a proxy. I would think it needs to be countable, since only ... | It is impossible to give a complete answer to this question just in $\sf ZFC$.
For once, it is possible that $V=L$ is true in the model you consider, so in fact $\Bbb R^L=\Bbb R$, so the cardinality is $\aleph_1$.
On the other hand it is possible that the universe is $L[A]$ where $A$ is a set of $\aleph_2$ Cohen reals,... | {
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In any tree, what is the maximum distance between a vertex of high degree and a vertex of low degree? In any undirected tree $T$, what is the maximum distance from any vertex $v$ with $\text{deg}(v) \geq 3$ to the closest (in a shortest path sense) vertex $y$ with $\text{deg}(y) \leq 2$? That is, $y$ can be leaf.
It se... | In your second question (shortest path to vertex of degree $\le 2$), the bound $\operatorname{diam}(G) / 2$ holds, simply by noticing that the ends of the longest path in a tree are leaves, and the "worst" that a graph can do is have the vertex of degree $3$ or higher right in the center. But in fact, this holds for t... | {
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Yitang Zhang: Prime Gaps Has anybody read Yitang Zhang's paper on prime gaps? Wired reports "$70$ million" at most, but I was wondering if the number was actually more specific.
*EDIT*$^1$:
Are there any experts here who can explain the proof? Is the outline in the annals the preprint or the full accepted paper?
| 70 million is exactly what is mentioned in the abstract.
It is quite likely that this bound can be reduced; the author says so in the paper:
This result is, of course, not optimal. The condition $k_0 \ge 3.5 \times 10^6$ is also crude and there are certain ways to relax it. To replace the right side of (1.5) by a valu... | {
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The integral of $\frac{1}{1+x^n}$ Motivated by this question:
Integration of $\displaystyle \int\frac{1}{1+x^8}\,dx$
I got curious about finding a general expression for the integral $\int \frac{1}{1+x^n},\,n \geq 1$. By factoring $1+x^n$, we can get an answer for any given $n$ (in terms of logarithms, arctangents, etc... | I showed in THIS ANSWER, that a general solution is given by
$$\bbox[5px,border:2px solid #C0A000]{\int\frac{1}{x^n+1}dx=-\frac1n\sum_{k=1}^n\left(\frac12 x_{kr}\log(x^2-2x_{kr}x+1)-x_{ki}\arctan\left(\frac{x-x_{kr}}{x_{ki}}\right)\right)+C'}
$$
where $x_{kr}$ and $x_{ki}$ are the real and imaginary parts of $x_k$, re... | {
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Solve the equation $\sqrt{3x-2} +2-x=0$
Solve the equation: $$\sqrt{3x-2} +2-x=0$$
I squared both equations $$(\sqrt{3x-2})^2 (+2-x)^2= 0$$
I got $$3x-2 + 4 -4x + x^2$$
I then combined like terms $x^2 -1x +2$
However, that can not be right since I get a negative radicand when I use the quadratic equation.
$x = 1/2... | $$\sqrt{3x-2} +2-x=0$$
Isolating the radical:$$\sqrt{3x-2} =-2+x$$
Squaring both sides:$$\bigg(\sqrt{3x-2}\bigg)^2 =\bigg(-2+x\bigg)^2$$
Expanding $(-2+x)^2$ and gathering like terms: $$3x-2=-2(-2+x)+x(-2+x)$$
$$3x-2=4-2x-2x+x^2$$
Set x equal to zero:$$3x-2=4-4x+x^2$$
Gather like terms:$$0=4+2-3x-4x+x^2$$
Factor the q... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/398984",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 2
} |
Riemann integral and Lebesgue integral $f:R\rightarrow [0,\infty)$ is a Lebesgue-integrable function. Show that
$$
\int_R f \ d m=\int_0^\infty m(\{f\geq t\})\ dt
$$
where $m$ is Lebesgue measure.
I know the question may be a little dump.
| We have, using Fubini and denoting by$\def\o{\mathbb 1}\def\R{\mathbb R}$ $\o_A$ the indicator function of a set $A \subseteq \R$
\begin{align*}
\int_\R f(x)\, dx &= \int_\R \int_{[0,\infty)}\o_{[0,f(x)]}(t)\, dt\,dx\\
&= \int_{[0,\infty)} \int_\R \o_{[0,f(x)]}(t)\, dx\, dt\\
&= \int_{[0,\infty)} \in... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/399058",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Improper integrals Question I was asked to define the next intergrals and I want to know if I did it right:
$$1) \int^\infty_a f(x)dx = \lim_{b \to \infty}\int^b_af(x)dx$$
$$2) \int^b_{-\infty} f(x)dx = \lim_{a \to -\infty}\int^b_af(x)dx$$
$$3) \int^\infty_{-\infty} f(x)dx = \lim_{b \to \infty}\int^b_0f(x)dx + \lim_{a ... | The first two definitions you gave are the standard definitions, for $f$ say continuous everywhere. The third is more problematical, It is quite possible that the definition in your course is
$$\lim_{a\to-\infty, b\to \infty} \int_a^b f(x)\,dx.$$
So $a\to-\infty$, $b\to\infty$ independently.
What you wrote down would ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/399119",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Solve equations $\sqrt{t +9} - \sqrt{t} = 1$ Solve equation: $\sqrt{t +9} - \sqrt{t} = 1$
I moved - √t to the left side of the equation $\sqrt{t +9} = 1 -\sqrt{t}$
I squared both sides $(\sqrt{t+9})^2 = (1)^2 (\sqrt{t})^2$
Then I got $t + 9 = 1+ t$
Can't figure it out after that point.
The answer is $16$
| An often overlooked fact is that
$$ \sqrt{t^2}=\left|t\right|$$
Call me paranoid but here's how I would solve this
$$\sqrt{t +9} - \sqrt{t} = 1$$
$$\sqrt{t +9} = \sqrt{t}+1$$
$$\left|t +9\right| = \left(\sqrt{t}+1\right)\left(\sqrt{t}+1\right)$$
$$\left|t +9\right| = \left|t\right|+2\sqrt{t}+1$$
Since the original e... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/399199",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "13",
"answer_count": 7,
"answer_id": 5
} |
What is the physical meaning of fractional calculus? What is the physical meaning of the fractional integral and fractional derivative?
And many researchers deal with the fractional boundary value problems, and what is the physical background?
What is the applications of the fractional boundary value problem?
| This may not be what your looking for but...
In my line of work I use fractional Poisson process a lot, now these arise from sets of Fractional Differential Equations and the physical meaning behind this is that the waiting times between events is no longer exponentially distributed but instead follows a Mittag-Leffler... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/399266",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 4,
"answer_id": 0
} |
Improper Integral $\int_{1/e}^1 \frac{dx}{x\sqrt{\ln{(x)}}} $ I need some advice on how to evaluate it.
$$\int\limits_\frac{1}{e}^1 \frac{dx}{x\sqrt{\ln{(x)}}} $$
Thanks!
| Here's a hint:
$$
\int_{1/e}^1 \frac{1}{\sqrt{\ln x}} {\huge(}\frac{dx}{x}{\huge)}.
$$
What that is hinting at is what you need to learn in order to understand substitutions. It's all about the chain rule. The part in the gigantic parentheses becomes $du$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/399423",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Convergence of $\sum \frac{a_n}{(a_1+\ldots+a_n)^2}$ Assume that $0 < a_n \leq 1$ and that $\sum a_n=\infty$. Is it true that
$$ \sum_{n \geq 1} \frac{a_n}{(a_1+\ldots+a_n)^2} < \infty $$
?
I think it is but I can't prove it. Of course if $a_n \geq \varepsilon$ for some $\varepsilon > 0$ this is obvious.
Any idea? Tha... | Let $A_n=\sum\limits_{k=1}^na_n$. Then
$$
\begin{align}
\sum_{n=1}^\infty\frac{A_n-A_{n-1}}{A_n^2}
&\le\frac1{a_1}+\sum_{n=2}^\infty\frac{A_n-A_{n-1}}{A_nA_{n-1}}\\
&=\frac1{a_1}+\sum_{n=2}^\infty\left(\frac1{A_{n-1}}-\frac1{A_n}\right)\\
&\le\frac2{a_1}
\end{align}
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/399565",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 0
} |
Upper bound on the difference between two elements of an eigenvector Let $W$ be the non-negative, symmetric adjacency/affinity matrix for some connected graph. If $W_{ij}$ is large, then vertex $i$ and vertex $j$ have a heavily weighted edge between them. If $W_{ij} = 0$, then no edge connects vertex $i$ to vertex $j$.... | I've seen peopl use Davis and Kahan (1970), "The Rotation of Eigenvectors by a Perturbation". It's sometimes a bit tough going, but incredibly useful for problems like this.
More info would also be useful. Is $W$ stochastic? Are there underlying latent classes that control the distribution of $W_{ij}$, e.g., with $E(W_... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/399634",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Evaluate $\int^{441}_0\frac{\pi\sin \pi \sqrt x}{\sqrt x} dx$ Evaluate this definite integral:
$$\int^{441}_0\frac{\pi\sin \pi \sqrt x}{\sqrt x} dx$$
| This integral (even the indefinite one) can be easily solved by observing:
$$\frac{\mathrm d}{\mathrm dx}\pi\sqrt x = \frac{\pi}{2\sqrt x}$$
which implies that:
$$\frac{\mathrm d}{\mathrm dx}\cos\pi\sqrt x = -\frac{\pi \sin\pi\sqrt x}{2\sqrt x}$$
Finally, we obtain:
$$\int\frac{\pi\sin\pi\sqrt x}{\sqrt x}\,\mathrm dx =... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/399840",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
} |
Function generation by input $y$ and $x$ values I wonder if there are such tools, that can output function formulas that match input conditions. Lets say I will make input like that:
$y=0, x=0$
$y=1, x=1$
$y=2, x=4$
and tool should generate for me function formula y=x^2. I am aware its is not possible to find out exa... | One of my favorite curve-fitting resources is zunzun. They have many, many possible types of curves that can fit the data you give it.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/399899",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 0
} |
Determining $\sin(15)$, $\sin(32)$, $\cos(49)$, etc. How do you in general find the trigonometric function values? I know how to find them for 30 45, and 60 using the 60-60-60 and 45-45-90 triangle but don't know for, say $\sin(15)$ or $\tan(75)$ or $\csc(50)$, etc.. I tried looking for how to do it but neither my text... | Value of $\sin{x}$ with prescribed accuracy can be calculated from Taylor's representation
$$\sin{x}=\sum\limits_{n=0}^{\infty}{\dfrac{(-1)^n x^{2n+1}}{(2n+1)!}}$$ or infinite product
$$\sin{x}=x\prod\limits_{n=1}^{\infty}{\left(1-\dfrac{x^2}{\pi^2 n^2} \right)}.$$
For some partial cases numerous trigonometric identiti... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/399948",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 0
} |
Help with combinations problem?
Initially there are $m$ balls in one bag, and $n$ in the other, where $m,n>0$. Two different operations are allowed:
a) Remove an equal number of balls from each bag;
b) Double the number of balls in one bag.
Is it always possible to empty both bags after a finite sequence of oper... | Regarding the first part...
Let $m>n$
Remove $n-1$ balls from each bag so that you have $m-n+1$ balls in one bag and $1$ ball in the other bag.
Now repeat the algorithm of doubling the balls in the bag which has $1$ ball and then taking away $1$ from each bag till you have $1$ ball in each bag. Finally remove $1$ ball ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/400026",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Books for Geometry processing Please suggest some basic books on geometry processing.
I want to learn this subject for learning algorithms in 3d mesh generation and graphics.
Please suggest me subjects or areas of mathematics i have learn in order to be understanding 3d mesh generation. I am doing self study and i am ... | See these books:
*
*Polygon Mesh Processing by Botsch et al.
*Geometry and Topology for Mesh Generation by Edelsbrunner
and these courses:
*
*Mesh Generation and Geometry Processing in Graphics, Engineering, and Modeling
*Geometry Processing Algorithms
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/400093",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
} |
How to show $x^4 - 1296 = (x^3-6x^2+36x-216)(x+6)$ How to get this result: $x^4-1296 = (x^3-6x^2+36x-216)(x+6)$?
It is part of a question about finding limits at mooculus.
| Hints: $1296=(-6)^4$ and $a^n-b^n=(a-b)(a^{n-1}+a^{n-2}b+\ldots+ab^{n-2}+b^{n-1})$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/400178",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 5,
"answer_id": 0
} |
Simple Linear Regression Question Let $Y_{i} = \beta_{0} + \beta_{1}X_{i} + \epsilon_{i}$ be a simple linear regression model with independent errors and iid normal distribution. If $X_{i}$ are fixed what is the distribution of $Y_{i}$ given $X_{i} = 10$?
I am preparing for a test with questions like these but I am ... | Let $\epsilon_i \sim N(0,\sigma^2)$. Then, we have:
$$Y_i \sim N(\beta_0 + \beta_1 X_i,\sigma^2)$$
Further clarification:
The above uses the following facts:
(a) Expectation is a linear operator,
(b) Variance of a constant is $0$,
(c) Covariance of a random variable with a constant is $0$ and finally,
(d) A linear ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/400240",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Bertrand's postulate in another point of view I was just wondering why can't we use prime number's theorem to prove Bertrand's postulate.We know that if we show that for all natural numbers $n>2, \pi(2n)-\pi(n)>0$ we are done. Why can't it be proven by just showing (By using the prime number's theorem) that for every n... | You need a precise estimate of the form $$c_1<\frac{\pi(n)\ln n}{n}<c_2.$$ With that you can derive
$\pi(2n)>c_1\frac{2n}{\ln(2n)}>2c_1\frac{n}{\ln 2+\ln n}$. If you are lucky, you can continue$2c_1\frac n{\ln2+\ln n}>c_2\frac n{\ln n}>\pi(n)$.
However, for this you better have $\frac{2c_1}{c_2} >1+\frac{\ln 2}{\ln n}$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/400308",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 5,
"answer_id": 3
} |
Prove that a cut edge is in every spanning tree of a graph
Given a simple and connected graph $G = (V,E)$, and an edge $e \in E$. Prove:
$e$ is a cut edge if and only if $e$ is in every spanning tree of $G$.
I have been thinking about this question for a long time and have made no progress.
| Hint ("only if"): Imagine you have a spanning tree in the graph which doesn't contain the cut-edge. What happens to the graph if you remove this cut edge? What happens to the spanning tree?
Hint ("if"): What happens if you remove this "indispensable" edge (the one which is in every spanning tree)? Can the resulting gra... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/400384",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
} |
Finding example of sets that satisfy conditions give examples of sets such that:
i)$A\in B$ and $A\subseteq B$
My answer : $B=\mathcal{P(A)}=\{\emptyset,\{1\},\{2\},\{1,2\}\}$ and $A=\{1,2\}$ then $A\in B$ and $A\subseteq B$
ii) $|(C\cup D)\setminus(C\cap D)|=1$
My answer is: $C=\{1,2,3\}$, $D=\{2,3\}$ then $C\cup D=\{... | Your answer is incorrect. Because $1,2\in A$, but $1,2\notin B$. Your second answer is correct.
To the last question, the answer is again correct (assuming $\sf ZF$), because $A\in B$ and $B\subseteq A$ would imply that we have $A\in A$, which is impossible due to the axiom of regularity.
To correct the first answer, c... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/400437",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Evaluating $48^{322} \pmod{25}$ How do I find $48^{322} \pmod{25}$?
| Finding the $\phi(n)$ for $25$ was easy, but what it if the $n$ was arbitrarily large?
$$48 \equiv -2( \mod 25)$$
Playing around with $-2(\mod 25)$ so as to get $1$ or $-1(\mod 25)$.
We see that $1024 \equiv -1(\mod 25)$
$$((-2)^{10})^{32} \equiv 1 (\mod 25)$$
$$(-2)^{322} \equiv 4(\mod 25)$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/400522",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 2
} |
Does Euler totient function gives exactly one value(answer) or LEAST calculated value(answer is NOT below this value)? I was studying RSA when came across Euler totient function. The definition states that- it gives the number of positive values less than $n$ which are relatively prime to $n$.
I thought I had it, until... | For some intuition about why $n$ and $m$ must be relatively prime, consider that, for $N=p_1^{a_1}p_2^{a_2}...p_n^{a_n}$,
$$\varphi(N)=N \left (1-\frac{1}{p_1}\right) \left (1-\frac{1}{p_2}\right)...\left (1-\frac{1}{p_n}\right)$$
And for $M=p_m^b$,
$$\varphi(M)=M \left (1-\frac{1}{p_m}\right)$$
If $p_m$ is not one of ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/400590",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 3
} |
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