Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
prove inequation a,b,c,d $\in \mathbb{R}$ $a,b,c,d \gt 0$ and $ c^2 +d^2=(a^2 +b^2)^3$ prove that $$ \frac{a^3}{c} + \frac{b^3}{d} \ge 1$$
If I rewrite the inequation like $ \frac{a^3}{c} + \frac{b^3}{d} \ge \frac{c^2 +d^2}{(a^2 +b^2)^3}$ and manage to simplfy it brings me nowhere. I try with
Cauchy-Schwarz Inequa... | Using Titu's Lemma, we have
$$ \dfrac{a^3}{c} + \dfrac{b^3}{d} \ge \dfrac{(a^2+b^2)^2}{ac+bd}$$
So, we are left to prove that
$$ (a^2+b^2)^2 \geq (ac+bd)\tag{1} $$
Using Cauchy-Schwarz inequality, we have
$$(a^2+b^2)(c^2+d^2) \geq (ac+bd)^2\tag{2}$$
Using the given proposition, $$c^2+d^2 =(a^2+b^2)^3$$ in $(2)$ and t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1836753",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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How to determine which of the following matrices are similar? If we have the following three matrices:
$$
A=\begin{bmatrix}
7 &1 \\
-5 &3
\end{bmatrix},\;\;
B=\begin{bmatrix}
5 &-1 \\
1 &5
\end{bmatrix},\;\;
C=\begin{bmatrix}
5 &1 \\
1 &5
\end{bmatrix}.
$$
What is the right procedure to determine if matrices are... | $A$ and $B$ cannot be similar to $C$, since $\det(A)=\det(B)=26\neq \det(C)=24$.
On the other hand $A$ and $B$ are similar with
$$
P=\begin{pmatrix} 2 & 2 \cr 6 & 2 \end{pmatrix}.
$$
Here we have $B=PAP^{-1}$. I do not need eigenvalues for this computation. I start with an invertible matrix $P=\begin{pmatrix} s_1 & s_... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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For matrices $C, D$, show that $(CD)^{100} \neq C^{100} D^{100}$ The question is to prove this is false:
$(CD)^{100} = C^{100}\cdot D^{100}$, where $C$ and $D$ are matrices.
I looked through my textbook and could not find a proof for this.
| A matrix corresponds to some (linear) operation on a vector. Let's take 2-component vectors and actions $C$ "make first component zero" and $D$ "swap components". Then $(CD)^{100}$ is "swap components, then make first component zero, repeat $100$ times" (which gives you zero vector, obviously) while $C^{100}D^{100}$ is... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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In an $\Bbb{N}$-graded domain $A$, units are homogeneous
Let $A$ be a graded domain, with additive subgroups $A_n,\,\forall\,n\geq 0$, s.t. ${A_n\cdot A_m}\subseteq A_{n+m}\,\forall\,n,m\geq 0$, and $A=\bigoplus_{n=0}^\infty\, A_n$ as abelian groups. I wish to show that if $a, a^{-1}\in A$, then $a\in A_0$.
What I've... | Since $1 \in A_0$, every $\delta_n$ with $n \ge 1$ must be zero. The highest degree $\delta_i$ will be the product of the highest degree $a_i$ and the highest degree $b_i$, and since $A$ is a domain this is non-zero. Therefore since the degrees add the highest $a_i$ and $b_i$ must both be in degree $0$, i.e. $a$ and ... | {
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Question about simplification in summation I am reading a book where the following example is shown.
$$= \sum_{0\le n-j\le n} (a+b(n-j)) $$
$$= \sum_{0\le j\le n} (a+bn-bj) $$
Why is n-j being simplified to j? I don't understand why this is possible? To specify my question, what rule of simplification is being used for... | It should be clear that $a + b(n-j) = a +bn - bj$, so it is really just a question of how
$$
\sum_{0\leq n-j \leq n} = \sum_{0\leq j\leq n}
$$
First: I assume that $n$ is a fixed number and the the index variable is $j$. So you just have to convince yourself that when $j$ ranges over an interval such that $0\leq n-j \l... | {
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"timestamp": "2023-03-29T00:00:00",
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What is the meaning of $\Bbb{Q}[x]/f(x)$? I am very confused with the meaning of $\Bbb{Q}(x)/f(x)$. Does it mean the set of all polynomials modulo $f(x)$?
If it does then how can we say that $\Bbb{R}[x]/(x^2+1)$ is isomorphic to set of complex numbers?
| Yes that's what it means. Take the isomorphism $\phi: \mathbb{R}[x]/(x^2+1)\rightarrow \mathbb{C}$ by $\phi: (a+bx) \mapsto (a+bi)$. I'll leave it to you to show that it is an isomorphism.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Is it possible to convert the polar equation $\ r = k \cos (\theta n) + 2$ into cartesian form?
Is it possible to convert the polaer equation
$$\ r = k \cos (\theta n) + 2$$
into cartesian form? Here, $k$ is some constant and $n$ is any positive whole number greater than $2$.
The farthest that I managed to get... | If you want an expression that holds for all $n \geq 2$ then it might be a tad ugly in Cartesian, because:
$$\cos(n\theta) = \sum_{\text{even }k} (-1)^{k/2}{n \choose k}\cos^{n-k} \theta \sin^k \theta \\ = (x^2+y^2)^{-n/2}\sum_{\text{even }k} (-1)^{k/2}{n \choose k} x^{n-k}y^k.$$
But this will get you there.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Cardinality of subsets with finite intersections Let $\ F_0 $ be a family of disjoint subsets of $ C$. $\ |C|= \aleph_0$.
Prove that $\ (*) |F_0|\leq\aleph_0 $.
This part was relatively simple, in the presence of choice an injection can be defined from each set to one of its elements in C. By Cantor the conclusion i... | Hint (for both the $n=1$ case and the inductive step): Suppose $F_n$ is uncountable, and let $F_n(x)=\{A\in F_n:x\in A\}$ for each $x\in C$. Show that $F_n(x)$ must be uncountable for some $x\in C$, and then apply the induction hypothesis to $\{A\setminus\{x\}:A\in F_n(x)\}$.
| {
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"timestamp": "2023-03-29T00:00:00",
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Intermediate Value Property Justification I am working on showing the Intermediate Value Property holds for a certain function. I noticed the one page on here talks about this idea, but I can not follow with what they are trying to say. [The problem is posted below - Ed.] . Any guidance would be appreciated. I just am ... | So if you believe the intermediate value theorem, then you know that if $x_1, x_2 > 0$ or $x_1, x_2 < 0$, the property follows. So suppose $x_1 \leq 0 < x_2$. It's enough to show that there exists $c \in (0, x_1)$. To do this, consider the sequence $$(t_n)_{n \in \mathbb{n}} = \left( \frac{1}{ 2 \pi n + \arcsin (k) } \... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1837513",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Is the following integral identity true or not? Is the following statement true or not?$$\int_{-\infty}^\infty xf(x)\,dx = \left. {d\over{dt}} \int_{-\infty}^\infty e^{tx}f(x)\,dx\right|_{t = 0}$$
| The statement is if Leibnitz's rule can be applied to $e^{tx}f(x)$ around $t=0$. Specifically in this case you need an integrable function $\theta$ such that $|xe^{tx}f(x)|\leq \theta(x)$ for $t$ near $0$.
| {
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"timestamp": "2023-03-29T00:00:00",
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Regarding element-wise derivative of matrices
Let $X=(x_{ij})$ be a real n-by-n matrix where $x_{ij}$ are in a range such that $X$ is insvertible. Let $Y=X^{-1}=(y_{pq})$, and regard $y_{pq}=y_{pq} (x_{11},x_{12},\ldots,x_{nn})$ as a function of $x_{ij}$. Prove that $\frac{\partial y_{pq}}{\partial . x_{ij}}=-y_{pi}y... | I guess the argument is cleaner if you use:
$$
0=\frac{\partial (XY)}{\partial x_{ij}} =\frac{\partial X}{\partial x_{ij}}Y+ X\frac{\partial Y}{\partial x_{ij}}
$$
So
$$
\frac{\partial Y}{\partial x_{ij}}=-X^{-1}\frac{\partial X}{\partial x_{ij}}Y=-Y\frac{\partial X}{\partial x_{ij}}Y
$$
So taking the $(l,k)$ matrix en... | {
"language": "en",
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Well-ordering principle and theorem Could somebody clearly explain the difference between the well-ordering principle and the well-ordering theorem? Is one of these related to the Principle of Mathematical Induction, and the other to the Axiom of Choice? Thanks.
| The "well-ordering theorem" is the statement that for any set $X$, there is a relation $<$ on $X$ which is a well-ordering. This statement is equivalent to the axiom of choice.
The "well-ordering principle" has (at least) two different meanings. The first meaning is just another name for the well-ordering theorem. T... | {
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Inequality : $\sum_{k=1}^n x_k\cdot \sum_{k=1}^n \frac{1}{x_k} \geq n^2$ I have to show the inequality of $$\left(\sum_{i=1}^n x_i\right)*\left(\sum_{i=1}^n \frac{1}{x_i}\right) \geq n^2.$$For $x_1, ... x_n \in \mathbb{R_{>0}}$ and $ n \geq 1$. I wanted to show this inequality by induction. The basis is clear, but I am... | Using Induction on $n$, $\sum_{i=1}^{n+1}x_i$.$\sum_{i=1}^{n+1}1/x_i$$\geq n^2+x_{n+1}$$(\frac{1}{x_1}+$$\frac{1}{x_2}+....+$$\frac{1}{x_n}$)$+$$\frac{1}{x_{n+1}}$$(x_1+x_2+......+x_n)$
$=n^2+1+(\frac{x_{n+1}}{x_1}+\frac{x_1}{x_{n+1}})+....+ (\frac{x_{n+1}}{x_n}+\frac{x_n}{x_{n+1}})$$\geq n^2+1+2n=(n+1)^2$
*
*Not... | {
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"timestamp": "2023-03-29T00:00:00",
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Limit of $\frac{\pi^h-1}{h}$ as h approaches zero Can someone help me find this limit here. I only know how to use L'Hospital's rule but I want to be able to evaluate this limit without using differentiation.
$$\lim \limits_{h \to 0} \frac{\pi^h-1}{h}$$
The reason I want this limit is because just like $e$ can be expre... | $$\lim_{x\to 0} \frac{e^{hln\pi}-1}{h} = (H) = $$
$$\lim_{x\to 0} \frac{ln{\pi} * lne^{hln\pi}}{1} = ln \pi $$
| {
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"timestamp": "2023-03-29T00:00:00",
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Prove by induction $3+3 \cdot 5+ \cdots +3 \cdot 5^n = \frac{3(5^{n+1} -1)}{4}$ My question is:
Prove by induction that $$3+3 \cdot 5+ 3 \cdot 5^2+ \cdots +3 \cdot 5^n = \frac{3(5^{n+1} -1)}{4}$$ whenever $n$ is a nonnegative integer.
I'm stuck at the basis step.
If I started with $1$. I get the right hand side is $1... | Helping out with the problem.
I'm stuck at the basis step.
$p(n)$: $3+3 \cdot 5+ 3 \cdot 5^2+ \cdots +3 \cdot 5^n = \dfrac{3(5^{n+1} -1)}{4}$. Where $n \in \{0, 1, 2, \dots \}$.
We can rewrite the predicate. $p(n)$: $\sum_0^n3\cdot5^n = \dfrac{3(5^{n+1} -1)}{4}$
Base case:
Here you need to start at 0 because we onl... | {
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Finding the possible positions of chess knight mathematically relative to a given position From this website I found the following question:
A chess board’s 8 rows are labelled 1 to 8, and its 8 columns a to h. Each square of the board is described by the ordered pair (column letter, row number).
(a) A knight is posi... | Given an initial position for the knight, the knight can move either north-east, north-west, south-east or south-west. In each of these directions, there are usually two possible positions. For example, from $(d,3)$, the knight can move north-east to either $(e,5)$ or $(f,4)$. Thus, there are a total of 8 possible po... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Area of polygon of hyperbolic disc Let us consider hyperbolic disc. I use uniform tessellation {5,4}.Here 5 stands for pentagon, 4 for number of polygons sharing the same vertex.
{hyperbolic disc}
There exists formula which defines area of convex hyperbolic polygon:
$$A_{m} = \{ \pi(m-2) - (a(1)+...+a(m))\} \frac{1}{-... | First I have to point you at a misconception you have, and I fear it will be confusing.
(I only understand just about half of it myself)
The disk model you use is the Poincare disk model (see https://en.wikipedia.org/wiki/Poincar%C3%A9_disk_model ) and has a fixed absolute distance and a fixed curvature of -1 (this cu... | {
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Let $P(x)=ax^{2014}-bx^{2015}+1$ and $Q(x)=x^2-2x+1$ such that $Q(x)|P(x)$, find $a+b$
Let $P(x)=ax^{2014}-bx^{2015}+1$ and $Q(x)=x^2-2x+1$ be the polynomials where $a$ and $b$ are real numbers. If polynomial $P$ is divisible by $Q$, what is the value of $a+b$.
This is what I have tried so far: Since $Q(x)|P(x)$ we h... | Since $Q$ divides $P$, we have that $x-1$ divides $P$ twice. Hence $1$ is a root of $P$ and of its derivative $P'(x)=2014ax^{2013}$. So $a-b^{2015}+1=0$ and $2014a=0$. Now we can solve to get $a=0$, but in that case $Q$ cannot divide $P$, since $P$ has degree $0$.
If we actually have $P(x)=ax^{2015}-bx^{2014}+1$ then i... | {
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"timestamp": "2023-03-29T00:00:00",
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Feasible point of a system of linear inequalities Let $P$ denote $(x,y,z)\in \mathbb R^3$, which satisfies the inequalities:
$$-2x+y+z\leq 4$$ $$x \geq 1$$ $$y\geq2$$ $$ z \geq 3 $$ $$x-2y+z \leq 1$$ $$ 2x+2y-z \leq 5$$
How do I find an interior point in $P$?
Is there a specific method, or should I just try some rando... | Clever guessing is ok, random guessing isn't advised. Since $x\geq 1$, $y\geq 2$ and $z\geq 3$. You should try some values that are close to the boundary, since then they are less likely to break the other conditions. So an $x$ value slightly greater than $1$, a $y$ value that is slightly greater than $2$, and a $z$ va... | {
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Arrange black and white balls so that each pair of white balls is separated by at least two black balls I am trying to solve the following question:
How many linear arrangements of $m$ white balls and $(n-m)$ black balls are possible such that each pair of white balls is separated by at least two black balls? The whit... | Let $x_0,x_1,\dots,x_m$ represent the number of black balls between the respective white balls. Specifically, $x_0$ is the number of black balls to the left of the first white ball. $x_1$ is the number of black balls between the first and second white ball, etc...
$\underbrace{\bullet}_{x_0}\circ\underbrace{\bullet\b... | {
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Confused about notation ":=" versus plain old "=" Relating to sets, I find the following in a text book:
"...the set S := {1, 2, 3}".
The book has an extensive notation appendix, but the":=" notation is not included.
What exactly does ":=" mean, and how is it different from just "=", and how is it read? Many thanks f... | The notation $A=B$ means $A$ is equal to $B$. The notation $A:=B$ means "Let $A=B$." It means you're saying what you will mean when you write $A$.
I suspect the $\text{“}{:=}\text{''}$ notation hasn't existed for more than about a half a century, so it's brand-new.
| {
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Is $\frac {x^2 + 5x}{x} = x+5$? We are graphing functions in class and the function $f(x) = \frac {x^2 + 5x}{x}$, came up and our teacher simplified it to $x+5$ and graphed that with a hole in the function at $x=0$.
I started wondering, how in algebra can we say that $\frac {x^2 + 5x}{x} = x+5$, when the graphs of each... | The key here is that the domain of the function $f$ is not all real numbers. It's $\Bbb R \setminus \{0\}$ (meaning all real numbers except $0$). So once you know that it should be clear that $\frac{x^2+5x}{x}$ is exactly equal to $x+5$ on that domain.
If we wanted to specify that, we'd write $f: \Bbb R \setminus \{0... | {
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Find a six digit integer Find an integer with six different digits such that the six digit integer is divisible by each of its digits.
For example, find ABCDEF such that A, B, C, D, E and F all can divide the number ABCDEF.
Show your answer along with mathematical reasoning.
| $4\\
24\\
624\\
3624\\
183624$
2,4,8 are good to work with, because once you find three digits at the end, you can put whatever you want in the front.
3,6,9 are good to work with too, because if you find something that divides by 3 or 9, you can shuffle the digits and it still divides by 3 (or 9).
and 1 is just a no-br... | {
"language": "en",
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Operation of permutations on functions Let $P$ be the additive group of mappings from $\mathbf{Z}^n$ to $\mathbf{Z}$. For $f \in P$ and $\sigma \in \mathfrak{S}_n$ (the symmetric group of degree $n$) let $\sigma f$ be the element of $P$ defined by
$$\sigma f(z_1,\dots,z_n) = f(z_{\sigma(1)},\dots,z_{\sigma(n)}).$$
Is ... | One way to dissolve the confusion here is to think of it this way. A point $(z_1,...,z_n)$ is a function $z:\{1,...,n\}\to Z$. Hence, the point $(z_{\sigma(1)},...,z_{\sigma(n)})$, is the composition $z\circ \sigma$. Now we can think of the domain of $f$ as the set of functions
$$Z^n\simeq \{z:\{1,...,n\}\to Z\}\;,$$
a... | {
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Given $f(x,y)$ is a continuous function, Do these integrals equal? Given range $\{ 0 \le x \le 1, 0 \le y \le 1\}$
Do these integrals equal?
$\int_0^1(\int_0^y f(x,y)dx)dy = \int_0^1(\int_0^x f(x,y)dy)dx$
Well, the answer is no. It seems like the triangulars are different in LHS, RHS.
I don't understand $ (D_2)$. Here:... | "It seems like the rectangles are different in LHS, RHS."
In fact, they are not rectangles! $\int_0^1\int_0^x f(x, y)dydx$ takes $x$ to be from $x= 0 $to $x= 1$ and, for each $x, y$ from $y= 0$ to $y= x%. That is a **triangle** with vertices at %(0, 0), (1, 0), and (1, 1).$ $\int_0^1\int_0^y f(x, y)dxdy$ takes $y$ t... | {
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definition of derivative Definition:
A mapping $f:U\to \mathbb{R}^n$ from an open set $U\subset \mathbb{R}^m$ into $\mathbb{R}^n$ is differentiable at a point $a\in U$ if there is a linear mapping $A:\mathbb{R}^m\to \mathbb{R}^n$ described by an $n\times m$ matrix $A$ such that for all $h$ in an $\underline{\textbf{ope... | The idea is, if $f$ is differentiable at $a$, then "$f$ is approximately affine (linear plus a constant) close to $a$". The intuitive notion of "close to" generally translates as "in some open set". Note, incidentally, that the limit condition
$$
\lim_{h \to 0} \frac{\|\epsilon(h)\|}{\|h\|} = 0
$$
implicitly assumes $\... | {
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"url": "https://math.stackexchange.com/questions/1839204",
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How to find operator with Fibonacci eigenvalues? How can I find the operator that satisfies this equation?
$$F_nx^n=Dx^n$$
Summing over $n$ we can rewrite this as $$\frac1{1-x-x^2}=D\frac1{1-x}$$
I am unsure whether this can be solved.
I am trying to solve it in a similar way to something like this:
$$nx^n=Dx^n$$ has ... | Note: There are (at least) two ways of indexing the Fibonacci series. Below I work with the $F_0 = F_1 = 1$ convention rather than the $F_0 = 0, F_1 = F_2 = 1$ convention.
I assume you want $D$ to be a differential operator, given the tags. Then we have
$$D = \sum_{k = 0}^\infty f_k(x) \frac{d^k}{dx^k}$$
for some fu... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1839290",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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$\binom{n}{k}$ is a "binomial coefficient;" $n \; P \; k$ is a "__________." If I want to search for information concerning $\binom{n}{k}$, I can't Google that symbol directly, nor can I search for something like "n C k" and get anything relevant, but because the term "binomial coefficient" exists it's possible to sear... | This is a falling factorial:
$$ (n)_k = n^{\underline k} = \underbrace{n\cdot(n-1)\cdot(n-2)\cdots(n-k+1)}_{k\text{ factors}} = \frac{n!}{(n-k)!} $$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1839398",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Prove that if $f(x) = \int_{0}^x f(t)\,dt$, then $f(x) = 0$
Prove that if $f(x) = \displaystyle\int_{0}^x f(t)\, dt$ for all $x$, then $f(x) = 0$.
I first differentiated to get $f'(x) = f(x) - f(0)$. Then by the mean value theorem there exists a $c$ in $(0,x)$ such that $f'(c)=\dfrac{f(x)-f(0)}{x}$. Thus, $f'(x) = xf... | Suppose $f$ is continuous. The fundamental theorem of calculus tells us that it is actually differentiable, and moreover by differentiating both sides we get
$$f'(x) = f(x)$$
So $f(x)$ is a function which is its own derivative, and hence is of the form $f(x) = ce^{x}$. What is the constant $c$? Simply compute
$$f(0) = ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1839457",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 5,
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Evaluate the integral $\int_{-\infty}^{\infty}{\left| 2t\cdot\text{sinc}^2(2t)\right|^2}\,dt$ I have a question in solving the integral
$$\int_{-\infty}^{\infty}{\left| 2t\cdot\text{sinc}^2(2t)\right|^2}\,dt.$$
I know that you can use Parseval's Theorem to prove that $\int_{-\infty}^{\infty}\text{sinc}^4(kt) = \frac{2\... | Integration by parts is enough. Our integral equals
$$ \int_{-\infty}^{+\infty}\frac{\sin^4(2x)}{4x^2}\,dx=\frac{1}{2}\int_{-\infty}^{+\infty}\frac{\sin^4(u)}{u^2}\,du=\int_{-\infty}^{+\infty}\frac{2\cos(u)\sin^3(u)}{u}\,du $$
but:
$$ 2\cos(u)\sin^3(u)=\frac{1}{2}\sin(2u)-\frac{1}{4}\sin(4u)\,du $$
hence:
$$ \int_{-\in... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1839521",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
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Formalization of an intuitive idea to construct a surjection Let $A$ be an arbitrary set and $B$ be any non-empty set. Furthermore, suppose that there is no injection from $A$ to $B$. I want to prove that it follows that there is a surjection from $A$ to $B$.
I have an intuitive argument in mind to prove this, but I do... | Transfinite induction allows you to iterate the process you describe, and when combined with the axiom of choice, can be used to make something much like your argument work. (You will have to be a bit careful with how you handle limit ordinals, though.)
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
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Integral equation of the form $\int_{-\infty}^{\infty} e^{-a t^4} g(x,t) dt = e^{-b x^4}$ How to solve an integral equation of the following form
\begin{align}
\int_{-\infty}^{\infty} e^{-a t^4} g(x,t) dt = e^{-b x^4}
\end{align}
where $a$ and $b$ are some positive constants.
I am not very familiar with this subject... | Hah! This is actually a specific example of something in my research! (My work attacks a more general set of integral equations, in some sense.) Let's go for something nontrivial (unlike previous answers/comments).
If you consider what I like to call a diagonal kernel, i.e. $g(x,t) = f(xt)$ for some $f$ and assume $g$ ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Property of fractions Given two fractions $\frac{h}{k}$ and $\frac{h^{'}}{k^{'}}$ both in reduced form. I am unable to find a case when $\frac{h+h^{'}}{k+k^{'}}$ does not lie in the interval $\big[ \frac{h}{k},\frac{h^{'}}{k^{'}} \big]$. Is there such a case ?
PS: I was able two prove no such case exists for consecutiv... | Let's prove that, for positive $a,b,c,d$, with $\frac{a}{b}\le\frac{c}{d}$, it holds
$$
\frac{a}{b}\le\frac{a+c}{b+d}\le\frac{c}{d}
$$
The inequality on the left is equivalent to
$$
ab+ad\le ab+bc
$$
that is, $ad\le bc$, which is true.
The inequality on the right is equivalent to
$$
ad+cd\le bc+cd
$$
that is, $ad\le bc... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Prove $\frac{a+b+c}{abc} \leq \frac{1}{a^2} + \frac{1}{b^2} + \frac{1}{c^2}$. So I have to prove
$$ \frac{a+b+c}{abc} \leq \frac{1}{a^2} + \frac{1}{b^2} + \frac{1}{c^2}.$$
I rearranged it
$$ a^2bc + ab^2c + abc^2 \leq b^2c^2 + a^2c^2 + a^2b^2 .$$
My idea from there is somehow using the AM-GM inequality. Not sure how t... | Because $$\sum_{cyc}\left(\frac{1}{a^2}-\frac{1}{ab}\right)=\frac{1}{2}\sum_{cyc}\left(\frac{1}{a^2}-\frac{2}{ab}+\frac{1}{b^2}\right)=\frac{1}{2}\sum_{cyc}\left(\frac{1}{a}-\frac{1}{b}\right)^2\geq0.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1840148",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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$a\equiv b\pmod{n}\iff a/x\equiv b/x\pmod{n/\gcd(x,n)}$ for integers $a,b,x~(x\neq 0)$ and $n\in\Bbb Z^+$? I'm trying to prove/disprove the following:
If $a,b,x$ be three integers (where $x\neq 0$) such that $x\mid a,b$ and $n$ be a positive integer, then the following congruence holds:
$$a\equiv b\pmod{n}\iff a/x\equ... | More simply, write $\ \bar a = a/x,\ \bar b = b/x\ $ and $y = \bar a - \bar b.\ $ Then we get a $1$-line proof:
$$\,n\mid a\!-\!b=xy\iff n\mid xy,ny\color{#0a0}\iff n\mid (xy,ny)\!\color{#c00}{\overset{\rm D}{=}}\!(x,n)y \iff n/(x,n)\mid y$$
We employed the universal property $\color{#0a0}\iff$ along with the $\,\co... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 1
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What is the area of the triangle having $z_1$, $z_2$ and $z_3$ as vertices in Argand plane? What is the area of the triangle having $z_1$, $z_2$ and $z_3$ as vertices in Argand plane?
Is it
$$\frac{-1}{4i}[z_1(z_2^* - z_3^*)-z_1^*(z_2-z_3)+{z_2(z_3^*)-z_3(z_2^*)}]$$
where $w^*$ denotes the complex conjugate?
| I assume you mean the oriented area of that triangle (since the number changes sign when you swap $z_2$ and $z_3$).
In order to verify it, allow me to write your formula as $$g(z_1,z_2,z_3)=\mathfrak{Im}\frac{\overline{z_1}(z_2-z_3)+z_3\overline{z_2}}{2}$$ It holds $$g(z_1+t,z_2+t,z_3+t)-g(z_1,z_2,z_3)= \mathfrak{Im}\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1840270",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 3
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What is embedding? I am new to this so do I need to learn topology in order to understand this? Cause I come across this which says that unlike the 2D sphere, 2d saddle surface cannot be embedded in 3D Euclidean space(source: http://www.astro.yale.edu/vdbosch/astro610_lecture2.pdf - page 16) which I have trouble unders... | In that context of that lecture, they are not talking just about the general topological concept of embedding, but instead a more special metric concept of embedding. A metric embedding is a topological embedding which preserves the metric tensor in an appropriate sense.
So yes, you do need to understand topological e... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1840373",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Integrating $\cos (x+\sin (x))$ I tried to solve $$\int\cos(x+\sin(x))\,dx$$ but it seems to be way out of my league (tried u-substitution with $u=x+\sin(x)$ and couldn't find an answer). Also, no one on the Internet seems to have tried this before and Wolfram|Alpha and Symbolab aren't helping that much. If anyone can... | The Jacobi-Anger expansion gives
$$ e^{ix\sin\theta}=\sum_{n\in\mathbb{Z}}J_n(x) e^{ni\theta}\tag{1} $$
hence:
$$ e^{i\sin\theta+i\theta}=\sum_{n\in\mathbb{Z}} J_n(1)\,e^{(n+1)i\theta}\tag{2}$$
and by considering the real part of both sides:
$$ \cos(\theta+\sin\theta) = \sum_{n\in\mathbb{Z}} J_n(1)\,\cos((n+1)\theta)\t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1840502",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Question on inverse limits
1.7. Remark. The inverse limit of an inverse system of non-empty sets might be empty as the following example
shows: Let $I:=\mathbb{N}$ and $X_n:=\mathbb{N}$ for every
$n\in\mathbb{N}$. Let $\theta_{n\leftarrow
> n+1}:\mathbb{N}\to\mathbb{N},k\mapsto k+1$. Now assume
$(x_n)_{n\in\ma... | The problem is that as the indices increase the numbers decrease by $1$. Eventually we will have to hit a negative number, but $\mathbb N$ does not contain any of those.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1840600",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
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Limit of fibonacci sequence Let $f_n$ be the $n$th Fibonacci number. Find constants $a$ and $b$ such that
$$\lim_{n\to\infty} \frac{f_n}{a\cdot b^n} = 1$$
I'm somewhat confused on how to approach this problem. I know the closed form of the Fibonnaci sequence, and I think it may have something to do with this problem,... | HINT: You know that
$$f_n=\frac{\varphi^n-\widehat\varphi^n}{\sqrt5}\;,$$
where $\varphi=\frac12\left(1+\sqrt5\right)$ and $\widehat\varphi=\frac12\left(1-\sqrt5\right)$. Note that $|\widehat\varphi|<1$, so $\widehat\varphi^n\to 0$ as $n\to\infty$. Thus, for large $n$ the Fibonacci number $f_n$ is approximately ... ?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1840670",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 2
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Eigenvectors are unique up to a scalar If $ A $ is a matrix with eigenvector $ v $ corresponding to the eigenvalue $ \lambda, $ can we prove that $ v $ is unique up to $ \lambda, $ that is if $ v $ and $ v' $ are eigenvectors corresponding to $ \lambda, $ then $ v = Cv' $ for some constant $ C. $
| No. It is quite possible for an eigenspace to have more than one dimension. As commenters above pointed out, examples include the identity matrix or the zero matrix.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1840731",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Find all continuous functions $g(x)$ satisfying $\int_{0}^{f(x)}f(t)g(t)dt = g(f(x))-1$
Given a differentiable function $f(x)$, find all continuous functions $g(x)$ satisfying $$\int_{0}^{f(x)}f(t)g(t)dt = g(f(x))-1.$$
I differentiated both sides to get $f(f(x))f'(x)g(f(x)) = g'(f(x))f'(x)$. Thus, if $f'(x) \neq 0$ w... | Let $a = \inf(\text{im } f)$ and $b = \sup(\text{im }f)$, where $\text{im }f$ is the image of $f$ (either $a$ or $b$ could be infinite). Notice the following: If $y\in (a,b)$, then the equality $$\int\limits_{0}^{z}{f(t)g(t)\text{ d}t} = g(z)-1$$ holds for all $z$ in some neighborhood of $y$. By the Fundamental Theorem... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1840813",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Differential Equation Initial Value Problem Here is a pretty standard initial value problem that I'm having a little trouble with.
$$(\ln(y))^2\frac{\mathrm{d}y}{\mathrm{d}x}=x^2y$$
Given $y(1)=e^2$, find the constant $C$.
So I separated and integrated to get $\frac{(\ln(y))^3}{3}=\frac{x^3}{3}+C$. Multiplying $3$ to b... | It may be easier to solve for $C$ at the step
$$(\ln y)^3=x^3+C$$
$$2^3=1^3+C$$.
Everything else looks good.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1840916",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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"answer_id": 0
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How to solve $x<\frac{1}{x+2}$ Need some help with:
$$x<\frac{1}{x+2}$$
This is what I have done:
$$Domain: x\neq-2$$
$$x(x+2)<1$$
$$x^2+2x-1<0$$
$$x_{1,2} = \frac{-2\pm\sqrt{4+4}}{2} = \frac{2 \pm \sqrt{8}}{2} = \frac{-2\pm2\sqrt{2}}{2}$$
What about now?
| Multiply the inequality by $\;(x+2)^2>0\;$ ( obviously, $\;x\neq-2\;$) :
$$x(x+2)^2<x+2\iff x^3+4x^2+3x-2<0\iff$$
$$\iff (x+2)(x+1-\sqrt2)(x+1+\sqrt2)<0\iff \color{red}{x<-1-\sqrt 2}\;\;\text{or}\;\color{red}{-2<x<-1+\sqrt2}$$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Show that $U(8)$ is Isomorphic to $U(12)$.
Question: Show that $U(8)$ is Isomorphic to $U(12)$
The groups are:
$U\left ( 8 \right )=\left \{ 1,3,5,7 \right \}$
$U\left ( 12 \right )=\left \{ 1,5,7,11 \right \}$
I think there is a bit of subtle point that I am not fully understanding about isomorphism which is hinderi... | The preferred approach would be to explicitly construct that isomorphism, but since the problem suggests basing the proof around the concept of "order of an element", let us do so then.
The crucial thing here is that both groups are of order $4$, and a simple theorem says that there exist only two classes of groups of ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1841225",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 2,
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inequality proof I have come across a problem:
Let $a,b$ and $c$ be real numbers where $a > b$. Prove that if $ac \leq bc$, then $c \leq 0$.
I tried using the Indirect proof
If $a > b$, and $c > 0$, then by the 4th axiom of Inequality, Multiplicativity, $ac > bc$.
However, $ac \leq bc$, therefore $c \leq 0$.
I'm not... | Your proof is correct.
Here's an alternative proof.
$bc \geq ac \Longleftrightarrow c(b-a) \geq 0 \Longleftrightarrow -c(a-b) \geq 0$
Since $a > b \Longleftrightarrow (a-b) >0$, so $c \leq 0$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1841342",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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"answer_id": 1
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Finding the locus of a point $P$ if the tangents drawn from $P$ to circle $x^2 + y^2 = a^2$ so that the tangents are perpendicular to each other? Question: Find the locus of a point $P$ if the tangents drawn from $P$ to circle $x^2 + y^2 = a^2$ so that the tangents are perpendicular to each other.
I tried solving this ... | Hint: find the hidden square in the picture below.
| {
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"url": "https://math.stackexchange.com/questions/1841430",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
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Have historians responded to Raju's critique? C. K. Raju has made some outrageous criticisms of the traditional take on Euclid in particular and Western history in general. Yet he has a book published on the subject with an apparently respectable publisher in India. Have modern historians of the classical period respon... | The articles by Raju have a conspirational flavor. The history of Indian mathematics is still an uncharted territory. There are more informative unbiased articles, for instance, there are much deeper and less biased studies I've read:
A. Seidenberg, “The Origin of Mathematics,” Archive for History of Exact Sciences 18,... | {
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"url": "https://math.stackexchange.com/questions/1841508",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 3,
"answer_id": 1
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Every length minimizing $\mathcal C^1$ curve is a geodesic. Let $(M,g)$ a manifold and $\gamma (t)$ for $t\in [a,b]$ a curve $\mathcal C^1$ the is minimizing the length. Then, if $p=\gamma (t_0)$ and $q=\gamma (t_1)$, then $\gamma $ is also minimizing the length of $p$ and $q$ for all $a\leq t_0<t_1\leq b$. In polar co... | To answer the question why the proof shows that the curve is a geodesic:
Once we prove that $y^i$'s are constant, then $\gamma$ coincides with a radial curve, namely with $t \longrightarrow (t,y^1,...)$ which is by definition of polar coordinates a geodesic.
Therefore, we have shown that small segments (because we rest... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1841663",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Sum of digits equality $Z$ return sum of digits of number: $Z(15)=6$.
If for some $W\in\mathbb{N}$, $Z(W)=100;~Z(44W)=800$ find
$$
ZZ(2015!)+ZZZ(2015!)+ZZZZ(2015!)+Z(3W)
$$
I don't have experience with this kind of problem, please give a hint what should I start? What approach is right?
| The sum of digits function returns about $4.5$ times the base $10$ log of a number, because the base $10$ log gives the number of digits and the average digit is $4.5$. It also maintains the value of the number $\bmod 9$ by the classic divisibility test. $2015!$ has about $5500$ digits (you can use Stirling to get th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1841726",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Tricky detail in extreme value theorem proof I am reading Pugh's Real Mathematical Analysis, and in chapter 1, section 6, ``The Skeleton of Calculus,'' Pugh supplies a proof of the Extreme Value Theorem. I am having trouble understanding one particular point in the proof. Note that he proves the existence of maximums, ... | My guess: the idea is to prove that $f(c)=M$. He proves it by contradiction. Assume that $f(c)<M$ then by continuity we can go a bit further to $c+\delta$ and still have all functional values being under $M$. It contradicts the choice of $c$ as the supremum of $X$.
P.S. It is easy, but you have to mention also that $c\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1841865",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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What does it mean to perform calculus upon functions of complex values? Complex numbers exist in a plane. This would lead me to believe that calculus views them as multivariate, but I am not real sure. How would one define a rate of change for a complex number valued function, or the area underneath it. Could someone e... | At a very simple level, any complex number $z$ can be expressed as $x + i y.$
and $f(z) = u(x,y) + i v (x,y)$
The definition of derivative is the same definition.
$f'(z) = \lim_\limits{z\to z_0} \frac {f(z) - f(z_0)}{z-z_0}$
$\frac {\partial f}{\partial x} = \frac {\partial u}{\partial x} + i \frac{\partial u}{\partial... | {
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"timestamp": "2023-03-29T00:00:00",
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Recurrence relation solution $a_{n}=\frac{a_{n-1}}{a_{n-1}-a_{n-2}}$ I want to find the analytic form of the recurrence relation $$a_{n}=\frac{a_{n-1}}{a_{n-1}-a_{n-2}},\,a_{1}=2,\,a_{2}=1 $$ but when looking at the results they seem chaotic. Is it possible that it simply doesn't have any analytic form?
| It seems to be chaotic I calculated and plotted the first 60 points of the solution.
If there was a solution it would have to have some periodic tendancies but it seems chaotic where these tendencies are and has unpredictable spikes well above the range of a normal Sin or Cos function.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1842010",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Calc 3: Calculate Work Done on Particle I've been working on this problem for a while and I'm pretty stuck. I tried it multiple different ways, by the last time I attempted it I realized that I hadn't converted kilometers to meters the entire time.
Anyway, the last approach I tried after doing the conversion was to int... | Smart Method
The work done is equal to the change in potential energy. The potential of a gravitational field:
$$
v(\vec{r})=-\frac{GM}{\|\vec{r} \|}\ \phantom{aaaa}\dots(1)
$$
So the work done is
$$
m\Delta v=-\frac{GMm}{\|\vec{r}_1 \|}+\frac{GMm}{\|\vec{r}_2 \|}
$$
Where $\vec{r}_1$ is the initial radial vector and $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1842097",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Prove that $\sum_{n=1}^{\infty }\frac{B_{2n}}{(2n-1)!}=\frac{1}{2}-\frac{1}{(e-1)^2}$ Prove that $$\sum_{n=1}^{\infty }\frac{B_{2n}}{(2n-1)!}=\frac{1}{2}-\frac{1}{(e-1)^2}$$
My idea is to find the Taylor series of $\frac{1}{(e^x-1)^2}$, but it seems not useful.
Any helps, thanks
| An alternative approach is to use the integral representation
$$ B_{2n} = (-1)^{n}4n \int_{0}^{\infty} \frac{x^{2n-1}}{e^{2 \pi x}-1} \, dx.$$
Specifically,
$$ \begin{align}\sum_{n=1}^{\infty} \frac{B_{2n}}{(2n-1)!} &= \sum_{n=1}^{\infty} \frac{1}{(2n-1)!} (-1)^{n-1} 4n \int_{0}^{\infty} \frac{x^{2n-1}}{e^{2 \pi x}-1} ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1842183",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Show that : $\lim\limits_{h\to 0}\frac {h^5} {2h^4} \frac1{\sqrt{h^2 + h^4}} = \lim\limits_{h \to 0}\frac {h^5} {2h^5}$ In my textbook, I found the following step, but I don't understand how the author gets there.
$$\lim_{h\to 0} {{\frac {h^5} {2h^4} \over \sqrt{h^2 + h^4}}} = \lim_{h \to 0}\frac {h^5} {2h^5}$$
| The comment by Claude explains the step of that author but only for $\;h>0\;$, but the step is wrong for negative values of $\;h\;$ :
$$\frac{\frac{h^5}{2h^4}}{\sqrt{h^2+h^4}}=\frac h{2|h|\sqrt{1+h^2}}=\begin{cases} \cfrac1{2\sqrt{1+h^2}}, &h>0\xrightarrow[h\to0^-]{}-\cfrac12\\{}\\-\cfrac1{2\sqrt{1+h^2}},&h<0\xrightarr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1842301",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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product of integral of reciprocal functions let us consider the integral of the following question :
this question seems quit interesting for me and that why i have decided to think about, this, first of all i was thinking to use function like
$\frac{1}{x}$
because if $f(x)=\frac{1}{x}$
then definitely $\frac{... | Differentiating both sides we get,$$f\int\dfrac{1}{f}dx + \frac{1}{f}\int{f}dx=0 \implies f^2\int\frac{1}{f}dx+\int fdx=0$$
Differentiate both sides again to get,$$f+ 2ff'\int\frac{1}{f}dx + f=0$$ $$\implies 2ff'\int\frac{1}{f}dx=2f\implies \int\frac{1}{f}dx=\frac{1}{f'}\implies\frac{1}{f}=\frac{f''}{f'^2}$$ $$\implies... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1842377",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
} |
Calculating $\int_0^{\infty } \frac{\ln (x)}{\sqrt{x} \left(a^2+x^2\right)^2} \, \mathrm{d}x$ using contour integration I can do this integral using the keyhole contour the answer is:$$\int_0^{\infty } \frac{\ln (x)}{\sqrt{x} \left(a^2+x^2\right)^2} \, \mathrm{d}x = -\frac{\pi (-6 \ln (a)+3 \pi +4)}{8 \sqrt{2}a^{7/2}}... | This answer is based on Feynman's trick/differentiation under the integral sign.
Well, we may just compute:
$$I(\alpha)=\int_{0}^{+\infty}\frac{x^{\alpha-1/2}}{(x^2+1)^2}\,dx \tag{1}$$
through the substitution $\frac{1}{x^2+1}=u$, Euler's beta function and the $\Gamma$ reflection formula to get:
$$ I(\alpha) = \frac{\p... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1842497",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Concept of roots in Quadratic Equation $a$ , $b$, $c$ are real numbers where a is not equal to zero and the quadratic equation
\begin{align}
ax^2 + bx +c =0
\end{align}
has no real roots then prove that $c(a+ b+ c)>0$ and $a(a+ b + c) >0$
My Approach : As the equation has no real roots then its discriminant is... | For the second part, another justification is that since the discriminant is less than zero, then it must hold that $a,c$ have the same sign. Why? If $a,c$ had opposite signs then the discriminant $\Delta = b^2-4ac$ would have been positive.
Since $c,(a+b+c)$ have the same sign and $c,a$ have the same sign then $a,(a+b... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1842568",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
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The first step in the proof of the Pólya-Vinogradov Inequality. The well-known Pólya-Vinogradov Inequality states:
$$\forall m, n \in \mathbb{N}: \displaystyle \left|{\sum_{k \mathop = m}^{m+n} \left({\frac k p}\right)}\right| < \sqrt p \ \ln p,$$
where $\left({\frac k p}\right)$ is the Legendre symbol.
I would like to... | The key idea here is orthogonality: the complex number $\omega_r = e^{2\pi ir/p}$ has the property that $\omega_r^p = 1$. This makes it easy to evaluate $\sum_{a=0}^{p-1} \omega_r^a$, since
$$(1 - \omega_r)(1 + \omega_r + \omega_r^2 + \cdots + \omega_r^{p-1}) = 1 - \omega_r^p = 0.$$
As long as $\omega_r \ne 1$, we can... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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How to rigorously deduce the Laurent series of $\log\frac{z-p}{z-q}$? Of course, the logarithm here is defined on the ring region $|z|>R\ge\max\{|p|,|q|\}$ as
$$\log\frac{z-p}{z-q}=\int_{z_0}^z \left(\frac1{w-p}-\frac1{w-q}\right)\mathrm d w. $$
Here the integral is along an arbitrary curve connecting $z_0$, a fixed p... | Yes, you can compute the Laurent series directly from the integral
definition.
For $\lvert w \rvert >\max\{\lvert p \rvert,\lvert q \rvert\}$,
$$
\frac{1}{w-p} - \frac{1}{w-q}
= \frac{1}{w(1-p/w)} - \frac{1}{w(1-q/w)}
= \sum_{n=1}^\infty \frac{p^n - q^n}{w^{n+1}}
$$
The terms for $n=0$ cancel, and all remaining term... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1842762",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Limit with Lambert-$W$ function I have asked a similar question about this one particular limit:
\begin{equation}
A=\lim_{c\to 1}\exp\left[ -\left(\frac{1}{1-c}\right)\left(W_{0}\left[ B\left( 1+\frac{x}{rc}\right) \right]-W_{0}[B]\right)\right]
\end{equation}
where $r>0$, $x \in \mathbb{R}$, $W_0$ is the $k=0$ branch ... | This is not an answer but it is too long for a comment.
mjqxxxx's answer contains all the required steps.
Concerning the approximation made for $B$, consider the definition $$B=\frac{(1-c)r}{1-(1-c)r}\exp\left[ \frac{(1-c)r}{1-(1-c)r} \right]$$ and let us define $a=(1-c)r$ which makes $$B=\frac a{1-a}\exp\left[ \frac{a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1842897",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Why is the complex plane shaped like it is? It's always taken for granted that the real number line is perpendicular to multiples of $i$, but why is that? Why isn't $i$ just at some non-90 degree angle to the real number line? Could someone please explain the logic or rationale behind this? It seems self-apparent to me... | Complex numbers can be constructed as couple of real numbers :
$
a+ib=(a,b)
$
with suitable definitions of the operations of sum and product ( see here).
With such definitions a complex number corresponds, in a natural way, to an element of $\mathbb{R}^2$ and we have : $1=(1,0)$ and $i=(0,1)$ and , using the usual repr... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "70",
"answer_count": 11,
"answer_id": 2
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Maximal ideal in a local artinian ring. I know that an artinian ring $A$ is the union of its units and its zero-divisors.
So every non-zero-divisor is an unit.
I also know that in a local ring every element which is out from the maximal ideal is an unit.
Can I conclude that the set of zero-divisors is the maximal ide... |
I also know that in a local ring every element which is out from the maximal ideal is an unit.
Can I conclude that the set of zero-divisors [of an Artinian local ring] is the maximal ideal of $A$?
Yes: here is an elementary way to see it.
Suppose $M$ is the unique maximal ideal of a commutative Artinian ring $A$. Sup... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1843016",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Concerning The Number of Ways of Drawing a Full House vs. Two Pair The Wikipedia entry for "Poker probability" gives the following result for the number of ways of drawing a full house:
$$ \binom{13}{1} \binom{4}{3} \binom{12}{1} \binom{4}{2} = 3, 744. $$
The logic here is conventional: From the 13 kinds you choose 1, ... | The problem with your $247,104$ is that it counts each two-pair hand two times, according to which of the pairs you mention first. But 5H-5D-7S-7H-9D is the same hand as 7S-7H-5H-5D-9D, so it gets counted both with fives first and with sevens first.
In contrast, for a full house it is unambiguous which value is the one... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1843111",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Arranging numbers around a square In how many ways numbers 1 to 12 can be arranged on a sides of squares (5 places on each sides i.e 20 places total) leaving 8 places empty?
I am getting answer as 12c5(selecting 5 numbers)*7c5(selecting another 5 numbers)*2(remaining 2)*5!(arranging the 5 numbers amongst themselves)
Bu... | I think the reason your answer is wrong is because you assumed that two sides would be filled on the square and a third side would have two numbers. For example, imagine a square with 3 numbers filled on each side with two numbers empty on each side. You did not account for such a possibility in your computation.
The s... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1843210",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
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Customer problem in poisson process (two products)
Customers arrive at a shop according to a Poisson process at rate
$\lambda$ (/minute), where they choose to buy either product $A$ (with probability p) or product $B$ (with probability $1-p$), independently. Given that during the first hour $5$ customers chose p... | The direct approach here would be to note that all customers who arrived in the first $10$ minutes buying $A$ is the same as none of them buying $B$, so we don't have to worry about the customers who bought $A$ or about the whole Poisson business at all, since we just want the probability that of $5$ customers equidist... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1843339",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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How much velocity can a canister of fuel give a spaceship? I've recently considered the issue of how much velocity a canister of fuel can provide a 'spaceship'. I assumed we could approximate a basic solution If we know the mass of the fuel $m$, the mass of the ship $M$ and the amount of energy in the fuel $E$. So the ... | The equation in the mentioned Wikipedia page is a special case of the general problem of motion equation for a system with variable mass that comes from the conservation of the momentum of the system. You can see the derivation of the pertinent equation at this page.
The pure energetic approach in OP is not correct be... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1843441",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Is there a reduction formula for $I_n=\int_{0}^{n\pi}\frac{\sin x}{1+x}\,dx$? I haven't been able to manipulate this integral. I need to find the value of $I_n$ for $n=1,2,3,4$ and arrange them in ascending order.
| Since the sine function oscillates with zeros at integer multiples of $\pi$, we introduce
$$
K_j=\int_{(j-1)\pi}^{j\pi}\frac{\sin x}{1+x}\,dx,\quad j\in\{1,2,3,4\}.
$$
Then, since $x\mapsto 1/(1+x)$ is decreasing and since $\sin(x+\pi)=-\sin(x)$ we find that
$$
K_1>-K_2>K_3>-K_4>0.
$$
The integrals we want to compare a... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Volume of the $N$-dimensional domain $\sum\limits_{k=1}^N (1 + |x_k|^a)^b\le\varepsilon$ I wish to calculate the following $N$-dimensional integral $$I = \int_0^\infty dx_1 \ldots \int_0^\infty dx_{N} \, H\left(\varepsilon - \sum_{k=1}^N (1 + x_k^a)^b\right),$$ where $a, b$ and $\varepsilon$ are positive reals with $\v... | $$I = \int\limits_0^\infty dx_1 \ldots \int\limits_0^\infty dx_{N} \, H\left(\varepsilon - \sum_{k=1}^N (1 + x_k^a)^b\right)$$
$$ = \int\limits_0^{\large\sqrt[a]{{\sqrt[b]{\varepsilon-N+1}-1}}} dx_1 \int\limits_0^{\large\sqrt[a]{{\sqrt[b]{\varepsilon-(1+x_1^a)^b-N+2}-1}}} dx_2 \ldots \int\limits_0^{\large\sqrt[a]{{\sqr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1843675",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Find point where a line of multiple vertices overlaps itself Since I'm not familiar with a lot of mathematical terminology, I will explain this problem with a little story.
Imagine you and your friend Anne have a piece of string each, and place it on a coordinate system. You decide to form a loop with your string, so t... | I think you're going to have to make some assumptions here. There can certainly be examples of lists which could represent both overlapping and non-overlapping strings, as there are lots of choices of strings to interpolate the points, and we may have that only some of them cause a self-intersection. There will be case... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Maximizing the sum of the squares of numbers whose sum is constant I wonder how one goes about to find the maximum of $\sum v_i^2$, the $v_i$'s being positive integers whose sum $\sum_i v_i$ is fixed.
| Imagine that you're aiming to cover as much of the $\sum_i v_i$ square as possible:
The bigger the largest inner square, the closer it gets to covering more of the background square. The maximum sum of squares is reached when all but one of the $v_i$ is at the specified minimum - in this case, $1$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1843982",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Let $f(x) = 5x+9$. Show that $\lim \limits_{x \to -3}f(x)=-6$
Let $f(x)=5x+9$. Show that $\lim \limits_{x \to -3}f(x)=-6$
A couple of questions about showing this and proving this. As I'm working through the problem I don't understand how I proved or showed anything as I don't understand the results I get.
I unders... | You want $\left|f(x)-(-6)\right|=5\left|x-(-3)\right|< \epsilon$, that is $\left|x-(-3)\right|\lt \epsilon/5$ and that is guaranteed if you take $\delta=\epsilon/5$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1844120",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Turing Decryption Example I know this exact same question exists but I am still having problems in understanding it. The following is given in the text:
The message m can be any integer in the set {0,1,2,…,p−1}; in particular, the message is no longer required to be a prime(p is a prime). The sender encrypts the messa... | As you said, $k\cdot k^{-1} \equiv 1 \pmod p$. Therefore, whenever we have $k\cdot k^{-1}$ in a $\pmod p$ equation, we can replace it with $1$, since they are congruent in such a system.
Here's an example:
$$2x \equiv 3 \pmod{5}$$
Now, after doing some guess and check, you can find that $2\cdot 3 \equiv 1 \pmod 5$, mea... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Real Analysis, Folland Proposition 2.29 Modes of Convergence Background Information:
$f_n\rightarrow f$ in $L^1$ $\Leftrightarrow$ $\forall\epsilon > 0,\exists N$ $\forall n\geq N$ $\int |f_n - f| < \epsilon$
A sequence $\{f_n\}$ of measurable complex-valued function on $(X,M,\mu)$ converges in measure to $f$ if for e... | I believe your approach is correct. I wrote something up before realizing you had already provided a proof (more-or-less what you have, just by proving the contrapositive):
Let $E_{n,\epsilon}=\{x\colon|f_{n}(x)-f(x)|\geq\epsilon\}$. Suppose
$f_{n}$ does not converge to $f$ in measure so that there exists
an $\epsilon>... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Prove the inequalities $|e^{x}-1|\leq e^{|x|}-1\leq |x|e^{|x|}$ Prove that $|e^{x}-1|\leq e^{|x|}-1\leq |x|e^{|x|}$ for all $x\in \mathbb{C}$. I did this by Maclaurin series of $e^x$,
$$|e^{x}-1|\leq |x|+\frac{|x|^2}{2!}+\frac{|x|^3}{3!}+\mathcal{O}(|x|^{4})=e^{|x|}-1 \\
\leq |x|^2+\frac{|x|^3}{2!}+\frac{|x|^4}{3!}+\ma... | $$e^{|x|}-1\leq |x|e^{|x|}\iff 1-e^{-|x|}\le|x|\iff\int_0^{|x|}e^{-t}\,dt\le\int_0^{|x|}1\,dt$$
which is clearly true because $e^{-t}\le1$ for $t\ge0$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1844537",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Plotting an arc with no center point - a practical solution please! I need a mathematical solution to a very practical problem (laying a patio).
The attached will hopefully explain.
The center of the circle for the arc we wish to have is inaccessible (ie in the house behind walls).
There are 2 fixed points the arc mus... | If you can measure tangent angles in the diagram the following could be of use in a scaled geometrical construction:
$$ R_{old}= \dfrac{r_2^2-r_1^2}{ 2(r_2 \sin \beta - r_1 \sin \alpha )}$$
Along perpendicular bisector of given points $(1,2)$ mark old center of circle $O$ with this radius and a new center $N$ with a ne... | {
"language": "en",
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What motivated trying to express the signature of a manifold as a linear combination of pontrjagin numbers? I have tried reading a proof of the signature theorem but it is way beyond me, is there a way to motivate, in english, why anyone even started searching for such a formula? Why would one assume that the signature... | First one notes that the signature is a bordism invariant, additive, and multiplicative, hence defines $\sigma:\Omega \otimes \mathbb Q\to \mathbb Q$.
Next you note that every ring morphism from the oriented bordism ring $\Omega \to \mathbb Q$ factors through $\Omega \otimes \mathbb Q$. But we know that on the latter s... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Are cyclotomic polynomials irreducible modulo a prime? I am actually interested on cyclotomic polynomials of the form $x^n+1$ (i.e. $n$ is a power of 2, for large $n$).
Are these polynomials irreducible modulo a prime $q$? I already saw that $X^2+1$ is not irreducible modulo 5 because $Z_5$ contains roots of -1. What a... | I find this relatively general answer from the Finite Field book by Rudolf, theorem 2.47, helpful:
The cyclotomic field $K^{(n)}$ is a simple algebraic extension of $K$. If $K=\mathbb{F}_q$ with $\gcd(q,n)=1$, then the cyclotomic polynomial $Q_n$ factors into $\phi(n)/d$ distinct monic irreducible polynomials in $K[x]$... | {
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"source": "stackexchange",
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Is something wrong with this solution for $\sin 2x = \sin x$? I have this question. What are the solutions for $$
\sin 2x = \sin x; \\ 0 \le x < 2 \pi $$
My method:
$$ \sin 2x - \sin x = 0 $$
I apply the formula $$ \sin a - \sin b = 2\sin \left(\frac{a-b}{2} \right) \cos\left(\frac{a+b}{2} \right)$$
So:
$$ 2\sin\le... | $\sin A=0\implies A=n\pi$
$\cos B=0\implies B=(2m+1)\pi/2$
$m,n$ are arbitrary integers
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1845034",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 6,
"answer_id": 2
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Convergent of sequence looking like Riemann zeta function I have a question that: Given a non-negative sequence {$\epsilon_n$} ($n \in \mathbb{Z+}$) such that $\lim_{n \rightarrow +\infty}\epsilon_n = 0$.
Can we conclude that $\sum_{n \in \mathbb{Z+}}\frac{\epsilon_n}{n} < +\infty$
| Not sure what this has to do with zeta functions. The answer is no: take $\epsilon_n=1/\ln(n+1)$. Use the integral test to show the series diverges.
| {
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"timestamp": "2023-03-29T00:00:00",
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} |
How many applicants need to apply in order to meet the hiring target? Cam needs to hire $30$ new employees. Ten percent $(10\%)$ of applicants do not meet the basic business requirements for the job, $12\%$ of the remaining applicants do not pass the pre-screening assessment, $23\%$ of those remaining applicants do not... | To elaborate on John's answer: You cannot simply add $10$, $12$, $23$, and $5$ percent, because the percentages are "in tandem". After you cut away the first $10$ percent, you only cut away $12$ percent of those who remain, not of those who started the selection process.
As a simpler example, consider the following se... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1845182",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 4,
"answer_id": 1
} |
Spending and left over amount/percentages Question 1) Cam would like to spend some of his money on friends. Of the $350$ dollar budget he was given, he has to spend $75.00$ dollars on office supplies, $10.00$ dollars per person for an entrance fee to the concert that his top $3$ friends and himself will attend later th... | Question 1. He spends more on the three friends then he does on the other 9. he spends 30 dollars on them. On top we have he spends 75 dollars + 17 dollars on himself. So we have the budjet for each friend to be $((350-75-17-30)/12)=19$, but remember that 3 of his 12 friends receive an extra 10 dollars so each friend ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1845287",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
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Irreducible polynomial over $\mathbb{Q}$ implies polynomial is irreducible over $\mathbb{Z}$ Let $f(x) \in \mathbb{Z}[x]$ be a polynomial of degree $\geq 2$. Then choose correct
a) if $f(x)$ is irreducible in $ \mathbb{Z}[x] $ then it is irreducible in $ \mathbb{Q}[x] $.
b) if $f(x)$ is irreducible in $ \mathbb{Q}[x] $... | You are totally correct, (1) is true and (2) is false. The statement you quote from Wikipedia is only true, if the coefficients come from a field.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1845542",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "15",
"answer_count": 2,
"answer_id": 1
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A weight problem
I am having a hard time solving the following puzzle. Could you please me to figure it out?
A chemist has a set of five weights. She knows that it includes one 1-gram weight, and also one each 2-, 3-, 4-, and 5-gram weights, but because they are unmarked, she has no way of telling them apart except by... | Okay, this isn't a complete solution as this does seem intricate but maybe this is a good starting point.
Note there are $5! = 120$ possible ways the weights can be determined. Each weighing will have $3$ possible outcomes. The weight balances, the left is heavier, and the right is heavier. If you can figure out a w... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1845655",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
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Galois extension definition. Let $L,K$ be fields with $L/K$ a field extension. We say $L/K$ is a Galois extension if $L/K$ is normal and separable.
I don't fully understand this definition, is it saying that
1) $L$ has to be the splitting field for some polynomial in $K[x]$ and that polynomial must not have any repea... | the extension $L/K$ is galoisien is equivalently to each $x\in L$,
$x$ is algebraic over $K$ and the minimal polynomial of $x$ has
simple roots and are all in $L$. so by primitive element theorem
we get: the extension $L/K$ finite and galoisienne is equivalent
to your proposal 1) but not to 2).
but in the infinity c... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1845751",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 5,
"answer_id": 3
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Why every point of a function where differentiation exists has only one tangent? Can anyone help me out?
Why every point of a function where differentiation exists has only one tangent?
I know the slope at any point of any function is defined by differentiation at that point.But there may be another straight line which... | It's not just about a line touching. (Every line through the point touches the graph there.) It's the best linear approximation, in the sense that the error $f(x)-L(x)$ goes to $0$ faster than $x$ approaches $a$. (Here $L(x)$ is the linear function whose graph we're discussing.)
EDIT: To be more precise, if $f$ is diff... | {
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"url": "https://math.stackexchange.com/questions/1845989",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Bounds on $f(k ;a,b) =\frac{ \int_0^\infty \cos(a x) e^{-x^k} \, dx}{ \int_0^\infty \cos(b x) e^{-x^k}\, dx}$ Suppose we define a function
\begin{align}
f(k ;a,b) =\frac{ \int_0^\infty \cos(a x) e^{-x^k} \,dx}{ \int_0^\infty \cos(b x) e^{-x^k} \,dx}
\end{align}
can we show that
\begin{align}
|f(k ;a,b)| \le 1
\end{al... | Yes, the Fourier transform of $\exp(-|x|^k)$ is positive and decreasing
for all $k$ such that $0 < k \leq 2$.
This follows from the known case of $k=2$ (Gaussians) via an argument of
B.F.Logan cited in my 1991 paper with Odlyzko and Rush:
Noam D. Elkies, Andrew M. Odlyzko, and Jason A. Rush: On the packing densities o... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1846072",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 2,
"answer_id": 1
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If $x\in \left(0,\frac{\pi}{4}\right)$ then $\frac{\cos x}{(\sin^2 x)(\cos x-\sin x)}>8$
If $\displaystyle x\in \left(0,\frac{\pi}{4}\right)\;,$ Then prove that $\displaystyle \frac{\cos x}{\sin^2 x(\cos x-\sin x)}>8$
$\bf{My\; Try::}$ Let $$f(x) = \frac{\cos x}{\sin^2 x(\cos x-\sin x)}=\frac{\sec^2 x}{\tan^2 x(1-\ta... | We are to prove that $$\frac{\cos x}{\sin x (\cos x - \sin x)}> 8 \sin x $$
By Cauchy-Schwarz Inequality, since all quantities involved are positive
$$\bf{LHS = }\frac{1}{\cos x} + \frac{1}{\cos x-\sin x} \ge \frac{4}{\sin x + \cos x - \sin x} = \frac{4}{\sin x}$$
For the given range of x, we have $1> 2 \sin^2 x$
So $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1846161",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 6,
"answer_id": 2
} |
Properly discontinuous group actions - Hausdorffness I was told to prove the following: If an action is free and satisfies that each point has a neighborhood $U$ satisfying $U \cap gU=\emptyset$ except for finitely many $g\in G$, and moreover the space is Hausdorff, then each point has a neighborhood such that $g_1\neq... | Let $x\in X$, consider a neigborhood $U$ of $x$ such that $V_x=\{g\in G:g(U)\cap U\neq \phi\}$ is finite. Write $V_x=\{y_1,...,y_n\}$ there exists $x\in U_i, y_i\in V_i$ such that $U_i\cap V_i$ is empty since $X$ is separated, set $W=U\cap_iU_i$, $W$ is an open subset containing $x$. Remark that if $g\in G$ and $g\neq ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1846238",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Finding the principal part for the Laurent series How can I find the principal part of the Laurent series for $f(z)=\dfrac{\pi^2}{(\sin \pi z)^2}$ centered at $k$ where $k \in \mathbb{Z}$.
I think there are two ways to do it either use the formula $a_k$ for Laurent coefficients or expand manipulate $\sin^2 \pi z$ and ... | By setting $z:=k+\varepsilon$, $k \in \mathbb{Z}$, $\varepsilon \to 0$, one has by the Taylor series expansion:
$$
\sin^2 \pi z=\left(\sin (\pi k+\pi\varepsilon)\right)^2=\sin^2(\pi\varepsilon)=\pi^2 \varepsilon^2-\frac{\pi^4 \varepsilon^4}{3}+O(\varepsilon^6)
$$ giving
$$
f(z)=\frac{\pi^2}{\pi^2 \varepsilon^2-\frac{\p... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1846290",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
} |
Are there ways to solve equations with multiple variables? I am not at a high level in math, so I have a simple question a simple Google search cannot answer, and the other Stack Exchange questions does not either. I thought about this question after reading a creative math book. Here is the question I was doing, which... | Multiplying by the common denominator $7!$, we get
$$ 2520 a_2 + 840 a_3 + 210 a_4 + 42 a_5 + 7 a_6 + a_7 = 3600$$
Take this mod $7$:
$$ a_7 \equiv 2 \mod 7$$
Since $0 \le a_7 < 7$, this means $a_7 = 2$, and then (substituting this and dividing by $7$):
$$ 360 a_2 + 120 a_3 + 30 a_4 + 6 a_5 + a_6 = 514$$
Taking this mo... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1846370",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 4,
"answer_id": 2
} |
$\vDash \phi \Rightarrow\, \vDash \psi$ How can I prove the last part of the following exercise.
Show that $R\vDash \phi \Rightarrow R\vDash\psi$ for all structure $R$ implies $\vDash \phi \Rightarrow\, \vDash \psi$, but not vice versa.
I have the first part. Ans: If $\vDash \phi$ then for all structures $R$, $\phi$ ... | The statement $\vDash \phi \Rightarrow\, \vDash \psi$ tells you that if $\phi$ is true in all structures, then $\psi$ is true for all structures. But if $\phi$ is not true in all structures, it tells you nothing at all. That is, if $\phi$ is any statement that is not true in all structures, then then the implication ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1846507",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Need a solution to this Integration problem How to evaluate:$\displaystyle\int_{0}^{r}\frac{x^4}{(x^2+y^2)^{\frac{3}{2}}}dx$
I have tried substituting $x =y\tan\ A$, but failed.
| $\newcommand{\angles}[1]{\left\langle\,{#1}\,\right\rangle}
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\newcommand{\dd}{\mathrm{d}}
\newcommand{\ds}[1]{\displaystyle{#1}}
\newcommand{\expo}[1]{\,\mathrm{e}^{#1}\,}
\newcommand{\half}{{1 \ov... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1846579",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 3
} |
intuition behind having a unique regression line I understand this mathematically. we have function of 2 variables represents the sum of square errors. We have to find the $a$ and $b$ that minimize the function. there is only one minimum point.
But when I think of it, I can't see why 2 different lines would not bring t... | You have two degrees of freedom (the slope $m$ and $y$-intercept $b$ of the regression line, say), but also two constraints: The partial derivatives of the squared total error $E$ with respect to $m$ and with respect to $b$ must vanish. That means you expect only finitely many (local) minima. (As A.E. says, $E$ is stri... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1846712",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
} |
What is the expected return of this hypotetical lottery game? What is the expected return of this hypotetical lottery game?
There are 60 numbers and you must pick 6 different numbers.
Then they pick 6 numbers (that will be different from each other) at random and if your 6 numbers is equal to those 6 numbers (but dont ... | The probability of winning is 1 / total number of possible outcomes
You're looking for the number of ways to pick 6 numbers at random from 60, I am assuming without considering the order (that is, 1 3 5 7 9 51 is the same as 5 7 9 51 1 3)
This number is equal to $\binom{60}{6}$ so your probability of winning is $1/ \b... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1846808",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Limit of a sequence defined as a definite integral Given two sequences $$a_n=\int_{0}^{1}(1-x^2)^ndx$$ and $$b_n=\int_{0}^{1}(1-x^3)^ndx$$ where $n$ is a natural number, then find the value of $$\lim_{n \to \infty}(10\sqrt[n]{a_n}+5\sqrt[n]b_n)$$
I have no starts. Looks good though. Some hints please. Thanks.
| $\newcommand{\angles}[1]{\left\langle\,{#1}\,\right\rangle}
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\newcommand{\dd}{\mathrm{d}}
\newcommand{\ds}[1]{\displaystyle{#1}}
\newcommand{\expo}[1]{\,\mathrm{e}^{#1}\,}
\newcommand{\half}{{1 \ov... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1846899",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 5,
"answer_id": 3
} |
Similar Triangles--Find the measurement of the unknown side This is a question I know I got wrong on a final exam in a very easy class for teaching elementary geometry/prep for Praxis II. I actually received a 99% average in the entire course because of the single point taken off of the final. I just want to know how t... | Hint: Note that the sum of any two sides of a triangle is greater than the third side. That will be enough to eliminate $3$ of the given choices.
For example, $CD$ cannot be $5$, since $7+(6+5)$ is not greater than $18$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1847040",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
How to find $\int \frac{x^2-1}{x^3\sqrt{2x^4-2x^2+1}} dx$ How to find ?$$\int \frac{x^2-1}{x^3\sqrt{2x^4-2x^2+1}} dx$$
I tried using the substitution $x^2=z$.But that did not help much.
| By setting $x^2=z$ we are left with:
$$ \frac{1}{2}\int\frac{z-1}{z^2\sqrt{2z^2-2z+1}}\,dz=C+\frac{\sqrt{2z^2-2z+1}}{2z}.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1847140",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 5,
"answer_id": 0
} |
How can $i^2 = k^2 = j^2 = ijk = -1$ be true? I have just started to learn the basics of quaternions, but I immediately run into a wall.
Litteraly the first equation on Wikipedia is as follows
$i^2 = k^2 = j^2 = ijk = -1$
This implies
$i = \sqrt{-1}$
$j = \sqrt{-1}$
$k = \sqrt{-1}$
but now $ijk = -1$ also need to be tr... | The way to think about this is not to think of these as normal multiplication, but rather rotation. To rotate by i means to take the point at 1 and sort of move it 90 degrees up to i. Rotation by j and k is completely similar. All other numbers on the unit circles of i, j, and k for their respective multiplications fol... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1847218",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 3,
"answer_id": 2
} |
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