Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
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Containment of Unitary Representations
Definition (Weak Containment): Let $G$ be a locally compact group, and let $\pi, \rho$ be unitary representations of $G$ into Hilbert spaces $\mathcal{H}$ and $\mathcal{K}$, respectively. Then $\pi$ is weakly contained in $\rho$ if for every $x \in \mathcal{H}$, for every compact... | To say that $\pi$ is contained in $\rho$ is the same as saying that $\pi$ is a subrepresentation of $\rho$, i.e., there is an isometry $V:\mathcal H\to\mathcal K$ such that $V\pi(g)V^*=\rho(g)$ for all $g\in G$.
| {
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"timestamp": "2023-03-29T00:00:00",
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Proof that if the limit of a function exists then the function is bounded in a neighborhood. The question is: Let $f:D\to\mathbb{R}$ and let $c$ be an accumulation point of $D$. Suppose that $f$ has a limit at $c$. Prove that $f$ is bounded on a neighborhood of $c$. That is, prove that there exists a neighborhood $U$ ... | Assume f is unbound in every epsilon neighborhood of c. Then it's absolute value doesn't tend toward a limit at c. Therefore, f doesn't tend toward a limit at c.
| {
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Greatest common divisor of $(x+1)^{4n+3} + x^{2n}$ and $x^3-1$. I have to find the greatest common divisor of
$$(x+1)^{4n+3} + x^{2n}$$
and
$$x^3-1$$
I know I can express the second polynomial as:
$$x^3-1 = (x-1)(x^2+x+1)$$
So I would have to check if the first polynomial is divisible by $(x^3-1)$, $(x^2+x+1)$ or $(x-... | Hint $\,\ x\!-\!1\nmid f(x)\,$ by $\,f(1)\neq 0,\,$ but $\ x^2\!+\!x\!+\!1\mid f(x)\,$ by
$\!\!\!\begin{align}\bmod\, \color{#0a0}{x^2\!+\!x\!+\!1}\!:\,\ f(x)\,\equiv\ &x^{\large 2n}+(\color{#0a0}{x\!+\!1})^{\large 4n+3}\\[.2em]
\equiv\ &x^{\large 2n}+({\color{#0a0}{-x^{\large 2}}})^{\large 4n+3}\ \ {\rm thus\ reduci... | {
"language": "en",
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Prove that $\left\lfloor{\frac{n}{2}}\right\rfloor+\left\lfloor\frac{\left\lceil\frac{n}{2}\right\rceil}{2}\right\rfloor+\cdots=n-1$.
Prove that, for $n\in \Bbb{Z}^+$,
$$\left\lfloor{\frac{n}{2}}\right\rfloor+\left\lfloor\frac{\left\lceil\frac{n}{2}\right\rceil}{2}\right\rfloor+\left\lfloor\frac{\left\lceil\frac{\left... | Any positive integer $n$ satisfies the following equation:
$$
n=\sum_{i=0}^{\left\lfloor\log_{2}{n}\right\rfloor}{\left(a_{i}2^{i}\right)}
$$
Substitute it to your equation to obtain:
$$
\begin{aligned}
<your\ equation>&=\sum_{i=0}^{\left\lfloor\log_{2}{n}\right\rfloor}{\left(a_{i}\left(2^{0}+\sum_{j=0}^{i-1}{2^{j}}\ri... | {
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Decompose Poisson random variable as sum of Poisson random variables If $X,Y$ are independent Poisson random variables with parameter $\lambda_1, \lambda_2$, then $X+Y$ is Poisson random variable with parameter $\lambda_1+\lambda_2$. I am wondering whether the converse if true, given a poisson random variable on a prob... | Suppose $W\sim\operatorname{Poisson}(\lambda).$
Suppose $0<\lambda_1 <\lambda,$ and let $\lambda_2 = \lambda - \lambda_1.$
Let $p = \dfrac{\lambda_1}\lambda = \dfrac{\lambda_1}{\lambda_1+\lambda_2}.$
Let $X\mid W \sim\operatorname{Binomial}(W,p),$ i.e. this is the number of successes in $W$ independent trials with prob... | {
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Finding a primitive element in a field with 27 elements. I am trying to construct a field with 27 elements, and find a primitive element in that field. I considered the irreducible polynomial $f(x)=x^3+2x+1$ over $\mathbb{Z}_3[x]$. Then I considered
$$\mathbb{Z}_3[x]/\langle f\rangle.$$
This is a field with $3^{\deg f... | From Arthurs answer we know that just guessing an element $at^2+bt+c$, it will likely be primitive. We have to choose at least one of $a$ and $b$ non-zero, so trying $t$ itself first is a good start. I wanted to add how the computation reduces to taking powers of matrices, i.e., linear algebra.
Identifying a polynomial... | {
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Probability of choosing envelopes
Suppose that you have 20 different letters and 10 distinctly addressed envelopes. The 20 letters consists of 10 pairs, where each pair belongs inside one of the 10 envelopes. Suppose that you place the 20 letters inside the 10 envelopes, two per envelope, but at random.
What is the pr... | Let $S_i$ be the arrangements where envelope $i$ has both of its intended letters. The number of intersections of $k$ of the $S_i$ is
$$
N_k=\overbrace{\ \ \binom{10}{k}\ \ }^{\substack{\text{number of ways}\\\text{to choose the}\\\text{$k$ envelopes}}}\ \ \overbrace{\frac{(20-2k)!}{2^{10-k}}\vphantom{\binom{10}{k}}}^{... | {
"language": "en",
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"question_score": "3",
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How to compute $\int_0^\infty \frac{\log(2+x^2)}{4+x^2}\,\mathrm dx$ Evaluate the integral
$$\int_0^\infty \frac{\log(2+x^2)}{4+x^2}dx$$
--
I started by stating that the integral from 0 to infinity should be the same as half the integral from $-\infty$ to $\infty$, that is:
$$\int_0^\infty \frac{\log(2+x^2)}{4+x^2}dx =... | Without residues.
$$ \frac{\log(2+x^2)}{4+x^2}=\frac{\log(x+i\sqrt2)+\log(x-i\sqrt2)}{(x+2i)(x-2i)}$$
$$\frac{1}{(x+2i)(x-2i)}=\frac{i}{4}\left(\frac{1}{x+2 i}-\frac{1}{x-2 i}\right)$$ So, we face four integrals looking like
$$\int \frac{\log(x+a)}{x+b}\,dx=\text{Li}_2\left(\frac{x+a}{a-b}\right)+\log (x+a) \log \left(... | {
"language": "en",
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How many number of permutations Here’s a question I’m struggling with:
There are 10 book consisting of 4 biographies and 6 novels. A person stacks four of the books together. In the stack of four books, at least 2 books must be biographies. How many possible permutations are there for stacking the four books?
I thou... | The first method is miscounting. Consider your first method of counting, and suppose you have all four biographies in the stack: $A,B,C,D$.
You are choosing $A,B$ from the biographies, then choosing $C,D$ from the remaining eight books, then permuting them. This is the same as choosing $C,D$ from the biographies, then ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Net in $\mathcal{B}^*$ converging to unbounded functional Let $\varphi$ be an unbounded functional on a Banach space $\mathcal{B}$. Can we always find a net of bounded functionals (i.e. in $\mathcal{B}^*$) converging to $\varphi$ in w*-topology?
Any proof or counterexample?
| Let $V$ be the directed set of finite dimensional subspaces of $\mathcal B$. For each $v\in V$ choose a continuous projection onto $v$ and denote it with $P_v$ (this works with Hahn-Banach and uses that $v$ is finite dimensional). Now define a functional $\varphi_v := \varphi\lvert_{v}\circ P_v$. Note that $P_v$ is con... | {
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Can someone explain why this is true? $\int f(x) g'(x)\,dx = \int f(x)\,dg$ I'm a software engineer (have been for 20+ years) and have over the past few years taken quite an interest in math. I would appreciate it if anyone is able to help me out with my question.
I understand the following:
$$\int g’(x)\,dx = \int dg$... | As Yves Daoust writes, this is from the simple fact:
$$\frac{dg}{dt} \cdot dt = /\text{ cancel dt }/=dg$$
This is independent of integration.
| {
"language": "en",
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In a function formula, can the independant variable $x$ ( ranging over $N$) be used as a simple figure inside a number? E.g. $ f(x)= 4,5x975$ Suppose $x$ ranges over {0,1,2,3,4,5,6,7,8,9}.
Is $f(x) = 4,5x975$ a valid function formula ?
What about the case where $x$ ranges over the set of natural numbers.
In that cas... | yes as it can be expressed as f(x)= 450975+1000*x when x $\in$ {0,1,..9}
Over the set of natural number, we still can have the formula
f(x)= $45*1000*10^{(⌊log10x⌋+1)}+975+1000*x$
Proof: How many digits does a number have? $\lfloor \log_{10} n \rfloor +1$
| {
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How many Steiner Symmetrizations does it take to make an arbitrary set convex? I have not seen this question investigated before but I might be wrong:
*
*Can any subset of $\mathbb{R}^d$ be turned into a convex set by finitely many steiner symmetrizations?
*If yes, is the number of symmetrizations necessarily bound... | Koch's snow flake needs an infinite number of Steiner symmetrizations.
| {
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How to do a quick estimation if $x_2 \ll x_1$ holds for the roots of a quadratic equation - to apply quick and easy root-finding formula? Wikipedia provides an interesting method of (approximately) solving a quadratic equation:
Vieta's formulas provide a useful method for finding the roots of a
quadratic in the case w... | One root is much smaller than the other when $|ac| \ll b^2$ because then the square root in the quadratic formula is very close to $b$. The approximation given comes from replacing the square root by $b$ and taking the minus sign so the two terms add. This is also the time that the calculation of the other root suffe... | {
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Circumcircle of a square and an arbitrary point inside it; prove: $|A_1B_1|\cdot|C_1D_1|=|A_1D_1|\cdot|B_1C_1|$
Point $T$ is inside the square $ABCD$. Let $A_1,B_1,C_1,D_1$ the other intersection point of the lines $AT,BT,CT,DT$ respectively and the circumcircle of the square $ABCD$. Prove:
$$|A_1B_1|\cdot|C_1D_1|=|... | Because $$\frac{A_1D_1\cdot B_1C_1}{A_1B_1\cdot C_1D_1}=\frac{\frac{A_1D_1}{AD}\cdot\frac{B_1C_1}{BC}}{\frac{A_1B_1}{AB}\cdot\frac{C_1D_1}{CD}}=\frac{\frac{A_1T}{DT}\cdot\frac{C_1T}{BT}}{\frac{A_1T}{BT}\cdot\frac{C_1T}{DT}}=1$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3578871",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Prove that $(ay-bx)^2+(az-cx)^2\ge (bz-cy)^2$ Let be $a,b,c,x,y,z>0$ such that $ax\ge \sqrt{(b^2+c^2)(y^2+z^2)}$. Prove that
$$(ay-bx)^2+(az-cx)^2\ge (bz-cy)^2$$
I tried to expand
$$a^2(y^2+z^2)+x^2(b^2+c^2)+2bcyz\ge b^2z^2+c^2y^2+2abxy+2acxz$$
Here my idea was to use the condition after the means inequality:
$$a^2(y^2... | Using Cauchy-Schwarz:
$$
\begin{aligned} \left[\left(\frac{c}{a}\right)^2+\left(\frac{b}{a}\right)^2\right]\cdot \left[(ay-bx)^2+(cx-az)^2\right]&\geq \left[\frac{c}{a}(ay-bx)+\frac{b}{a}(cx-az)\right]^2\\
&=(cy-bz)^2\\
\end{aligned}
$$
and similarly
$$
\begin{aligned} \left[\left(\frac{z}{x}\right)^2+\left(\frac{y}{x}... | {
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How many integer solutions are there for the equation $c_1 + c_2 + c_3 + c_4 = 25$, where $c_i \ge 0$ for all $1 \le i \le 4$ Question Statement:
How many integer solutions are there for the equation $c_1 + c_2 + c_3 + c_4 = 25$, where $c_i \ge 0$ for all $1 \le i \le 4$.
I would like to solve this problem using combi... | Generating Function Method
Associate to each variable the polynomial $p(x) = \sum_{i=0}^{25} x^i$. Then the product
$$ \left(p(x)\right)^4 = 1 + 4 x + 10 x^2 + \cdots + 3276 x^{25} + \cdots $$
exhibits the fact that there are $3276$ solutions to the equation. It also exhibits the number of solutions of \begin{align... | {
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"timestamp": "2023-03-29T00:00:00",
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Prove $x^6-6x^4+12x^2-11$ is irreducible over $\mathbb{Q}$ Extracted from Pinter's Abstract Algebra, Chapter 27, Exercise B1:
Let $p(x) = x^6-6x^4+12x^2-11$, which we can transform into a polynomial in $\Bbb{Z}_3[x]$:
\begin{align*}
x^6+1
\end{align*}
Since none of the three elements $0,1,2$ in $\Bbb{Z}_3$ is a ro... | Update: The answer is wrong but see my comment!
I think the reasoning should be like the following. As $p(x)$ has integer coefficients and is monic every zero of $p$ that lies in $\mathbb{Q}$ is also integer. But every integer zero of $p$ must divide the absolute term which is 11. Therefore, it could only be $\pm1$ or ... | {
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"timestamp": "2023-03-29T00:00:00",
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Show that distribution of function of LRT statistic for normal mean hypothesis testing is normally distributed Suppose $X_1 ... X_n$ ~$^{iid}$ N($\mu, \sigma$), with $\sigma$ known. What is the distribution of $-2ln(\lambda)$ where $\lambda$ is the LRT statistic for testing $H_0:\mu = \mu_0, H_1:\mu \neq \mu_0$?
So we ... | For unknown $\sigma^2$
$$\lambda=\left(\frac{\sum(X_i-\bar{X})^2}{\sum(X_i-\mu_0)^2}\right) =\left(\frac{\sum(X_i-\bar{X})^2}{\sum(X_i-\bar{X})^2+n(\bar{X}-\mu_0)^2}\right)$$
$$=\left(\frac{1}{1+\frac{n(\bar{X}-\mu_0)^2}{\sum(X_i-\bar{X})^2}}\right)$$
$$=\left(\frac{1}{1+\frac{n(\bar{X}-\mu_0)^2}{(n-1)\frac{1}{n-1}\sum... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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"question_score": "1",
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In a set of 5 bottles,1 has a fracture.If you select a pair of bottles, probability that the fractured bottle is chosen is? Its also mentioned that this question is an example of Sampling without replacement
My question is , by the general method of how I do these kind of problems , I would assume that there are two po... | Here is another viewpoint.
Suppose you don't know there is a flawed bottle. You choose two of the five bottles. You are then told that one of the five bottles contains a crack. It is equally likely to be any of the five bottles, so each bottle has a probability of $\frac{1}{5}$ of being the cracked one.
You hold two bo... | {
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"timestamp": "2023-03-29T00:00:00",
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"question_score": "2",
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Identity related to $\sum_{k=0}^{n}\frac{x^k}{\binom{n}{k}}$ How it can be shown that:
$$\sum_{k=0}^{n}\frac{x^k}{\binom{n}{k}}=\left(n+1\right)\left(\frac{x}{x+1}\right)^{n+1}\sum_{k=1}^{n+1}\frac{1+x^k}{\left(1+x\right)k}\left(\frac{1+x}{x}\right)^{k}$$
for $x \ne-1$
I tried Additive Forms of Reciprocal Pascal’s Id... | Your result is a kind of generalization of Newton's binomial identity with binomial coefficients replaced by their inverses.
You will find in page 2 of this reference by Toufik Mansour, University of Haifa) the more general expression :
$$\sum_{k=0}^{n}\frac{a^kb^{n-k}}{\binom{n}{k}}=\frac{n+1}{(a+b)\left(\tfrac{1}{a}... | {
"language": "en",
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Given $x_{n} \to x_{0}$ as $n \to \infty$, and $e^{x}=\sum_{k=0}^{\infty}\frac{x^{k}}{k!}$, prove that $\lim_{n \to \infty}e^{x_{n}} = e^{x_{0}}$ Problem: Given a convergent sequence $x_{n} \to x_{0}$ as $n \to \infty$, and that e is defined as $e^{x}=\sum_{k=0}^{\infty}\frac{x^{k}}{k!}$, prove that $\lim_{n \to \infty... | Suppose we can prove that $x_k\to 0\Rightarrow e^{x_k}\to 1.$ Then, if $x_k\to x_0,\ y_k:=x_k-x_0\to 0$ and then $e^{y_k}=e^{x_k-x_0}\to 1$ and this implies that $e^{x_k}\to e^{x_0}$.
So, it suffices to prove the result for $x_0= 0.$ But this is easy: choose $K$ large enough so that $k>K\Rightarrow |x_k|<\epsilon<1.$... | {
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Proving a function(with two variables) is continuous I am having a difficulty solving this problem. First of all, I'm sorry that the problem isn't well written but I am not very good with typing out math problems, due to the fact that I am new to this, so I hope it's at least understandable. Next, I want to say that I'... | In these problems with roots, a typical strategy is the one of “rationalize” the fraction:
If you write:
$$\frac{5-\sqrt{25-x^2-y^2}}{7-\sqrt{49-x^2-y^2}}\cdot \frac{5+\sqrt{25-x^2-y^2}}{5+\sqrt{25-x^2-y^2}}\cdot \frac{7+\sqrt{49-x^2-y^2}}{7+\sqrt{49-x^2-y^2}}= \frac{x^2+y^2}{x^2+y^2}\cdot\frac {7+\sqrt{49-x^2-y^2}} {5... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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I get a contradiction in the theory of free abelian groups. What am I doing wrong? Hi: The definition I'll use is this: Let $F$ be an abelian group and $X$ a subset of $F$. Then $F$ is a free abelian group on $X$ if for every abelian group $G$ and every function $f$ from $X$ to $G$ there is a homomorphism $\phi$ from $... | You are using $\langle X \rangle$ to mean two different things, and conflating them:
*
*You are using it to mean the free abelian group on $X$.
*You are using it to mean the subgroup of $G$ generated by $X$.
These are not the same thing, but you assume that they are.
| {
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"timestamp": "2023-03-29T00:00:00",
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Unclear Answer Book on Calculus by Michael Spivak (3rd edition) Question 11-26. The question goes as follows
Suppose that $f'(x)\geq M>0$ $\forall x\in [0,1]$. Show that there is an interval of length $\frac{1}{4}$ on which $|f|\geq M/4$.
and the answer book states
Note that $f$ is increasing. If $f(1/2)\geq 0$, the... | Essentially, you can "integrate" the expression $f'(x)\geq M$ to deduce for $x>a$ $$f(x) \geq M \cdot (x-a) + f(a) \space \space [*].$$ To prove this, use the mean value theorem: assuming the usual conditions are met if $x>a$ then $\exists c $ with $a \leq c \leq x$ such that $$\frac {f(x)-f(a)}{x-a} = f'(c)$$ which r... | {
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Permutation representation contains trivial representation Let $G$ be a finite group and $H \vartriangleleft G$ a normal subgroup.
Let $(V,\rho)$ be the permutation representation (over $\mathbb{C})$ of $G$ acting on the set $G/H$ (we think of the quotient group as a set) in the natural way, i.e. for $s,t \in G$: $s \c... | Another way to approach this is with Frobenius Reciprocity and induction of characters. The permutation character equals $(1_H)^G$ and $[(1_H)^G, 1_G]=[1_H,1_H]=1$.
| {
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the upper Riemann integral is equal to the upper Riemann sum (Analysis 1 by Tao) Proposition 11.3.12. (in Analysis 1 by Tao) Let $f: I \to \mathbb{R}$ be a bounded function on a bounded interval $I$. Then
$$\overline{\int}_If = \inf\{U(f, P): \text{$P$ is a partition of $I$}\}.$$
I know from the previous exercise that ... | Let $P$ be a partition of $I$ and let $U(f, P) = \sum_{J \in P; J \neq \emptyset} \: (\sup_{x \in J} f(x)) \cdot|J|$. Define a piecewise constant function $g$ with respect to the same partition $P$ where for each non-empty $J \in P$, $g(x) = \sup_{x \in J} f(x)$ for all $x \in J$. Clearly, $g$ majorizes $f$ and so $$\... | {
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"timestamp": "2023-03-29T00:00:00",
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Why does an orthogonal matrix have to be square? I understand intuitively why this has to be the case (otherwise you could lose a dimension / gain a dimension which changes the length), but what is the formal proof that an orthogonal matrix has to be square?
| Just to sum up the comments, your book says a linear transformation $T:\mathbb R^n\to\mathbb R^n$ is orthogonal if it preserves the length of vectors. The matrix of a transformation from a vector space to a vector space of the same dimension is necessarily square, so this is baked into the definition of an orthogonal m... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3580819",
"timestamp": "2023-03-29T00:00:00",
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What is the inverse of the *divergence* operator? The inverse of derivation is integral.
But what is the inverse of the divergence operator ? Doest it exist ?
| The answer by Keith is close, except note that the divergence operator is not invertible, just like the derivative. It's "inverse" would also have some degrees of freedom.
In particular, when inverting the derivative $F'=f$, we have $F(y)=\int_{x=0}^{y} f(x) dx +C$.
If instead, we want to solve $\nabla \cdot \boldsymb... | {
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First-order Logic with infinite conjunction I have an infinite set of variables X and I want to state that the property that there is a unique variable in X with value 2.
For a finite set, I would write the first-order logic formula:
$$
(x_0 = 2 \wedge x_1 \neq 2 \wedge ... \wedge x_n\neq2)
\vee
(x_0 \neq 2 \wedge x_1 ... | The name you are looking for is infinitary logic.
And yes, this is a simple example to show that first order logic cannot do everything.
To makes things a bit more formal: in logic "$X$" is a set of "constant symbols", or "0-ary function symbol", each element of $X$ is an element of your language, and not some set/num... | {
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Joint Probability Density Function with Function Bounds I have a question about joint CDFs. My understanding was given a joint PDF, the joint CDF was the integral of the joint PDF from -inf to +inf for all the random variables defined. This joint CDF should be equal to 1. However, in the question below I see a contradi... | You just missed the factor of $4$ in the second calculation. The area of the region is not the value of the integral of $f_{x,y}$ since the value of $f_{x,y}$ is $4$ in the region. You have to multiply the area by $4$.
| {
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What is the conditional probability that the second card is a Spade given that the second-to-last card is a Spade? What is the conditional probability that the second card is a Spade given that the second-to-last card is a Spade? Cards are dealt without replacement.
I know conditional probability is $$P(A \mid B) = \fr... | Let $A=$ {2nd card is a spade} and $B=$ {penultimate card is a spade}.
All the condition tells you is that for each of the other 51 positions, you have one fewer spade that it could be. So $P(A)=P(B)=\frac{13}{52}=\frac{1}{4}$, but $P(A|B)=P(B|A)=\frac{12}{51}=\frac{4}{17}$.
From there, it's easy to get $P(A \cap B)$ u... | {
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Show a sequence $(x_n)^{\infty}_{n=1}$ converges to a point of S if and only if it is eventually constant Suppose a set $S$ is given the discrete metric $d_0$. Show a sequence $(x_n)^{\infty}_{n=1}$ converges to a point of $S$ if and only if it is eventually constant; there exists $N \in \mathbb{N}$ such that $x_n=x_N$... | The discrete metric is defined by
$$d(x, y) = \begin{cases} 0 & \text{if } x = y \\ 1 & \text{otherwise.}\end{cases}$$
So, if $d(x, y) < 1/2$, then $d(x, y) \neq 1$, and hence $x = y$.
| {
"language": "en",
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Partial sum $\{\frac{s_m}{s_n}\}$ converge to $1$ implies series converge? Let $s_n$ and $s_m$ be the partial sum of the series $\sum\limits_{k=0}^\infty a_k$ with $m<n$ and $a_k > 0$ for all k. If $\{\frac{s_m}{s_n}\}$ converge to $1$, does it imply that the series converges?
| If $\frac {s_m} {s_n} \to 1$ as $n,m \to \infty$ then $\ln s_m -\ln s_n \to 0$ which means $(\ln s_n)$ is a Cauchy sequence. Hence it converges to some number $c$. It follows that $s_n \to e^{c}$ so the series is convergent.
| {
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Two Polynomials with a Common Quadratic Factor Let $f(x)=x^3-ax^2-bx-3a$ and $g(x)=x^3+(a-2)x^2-bx-3b$. If they have a common quadratic factor, then find the value of $a$ and $b$.
My Attempt
Let $h(x)$ be the common quadratic factor. Then $h(x)$ also the factor of $g(x)-f(x)$, that is
$$(2a-2)x^2+(3a-3b)$$
Since $h(x)$... | Since $(2a-2)x^2+(3a-3b)$ is a common factor and the common factor is said to be a quadratic, the (monic) common factor must look like $h(x) = x^2+\dfrac 32 \cdot \dfrac{a-b}{a-1}$.
This is ugly. So let's try and avoid the direction that this is taking us. We can assume that $h(x) = x^2 - \alpha$ where $\alpha = -\dfra... | {
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Surprising fact about a certain number-theoretic function Ante suggested the following function :
For natural number $n$ we can observe the $n$ remainders $b_1,...,b_n$ by writing $n$ as $n=a_k \cdot k+b_k$ for $1 \leq k \leq n$
Because of the familiar division-with-remainder-theorem we have $0 \leq b_k <n$
Now we can ... | We have:
$$r(b)=r(b+1)$$
$$\sum_{k=1}^{\lfloor \frac{b-1}{2} \rfloor} (b \bmod{k}) =\sum_{k=1}^{\lfloor \frac{b}{2} \rfloor} ((b+1) \bmod{k}) $$
since $n \equiv b_k \pmod{k}$. Now, we take two cases:
Case $1$ : When $b$ is odd
We have:
$$\sum_{k=1}^{\frac{b-1}{2}} (b \bmod{k}) =\sum_{k=1}^{\frac{b-1}{2}} ((b+1) \bmod{k... | {
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Bound on finite dimensional space Let $V$ be a finite dimensional normed space over $\mathbb{R}$, with norm $||.||$
Show that there exists $C>0$ such that for all $x\in V$, $\sum_{i=1}^n|x_i|\leq C||x||$.
My attempt:
Suppose $dimV=n$. Let $\{$ $e_1,e_2...,e_n$ $\}$ be a basis for $V$. Consider the unit ball,
$K=\{$ $x... | In the following by $x$, I mean $\sum_k x_k e_k$.
Note that $\|x\| \le \sum_k |x_k| \|e_k\| \le K_1 \|x\|_1$ where $K_1 = \max_k \|e_k\|$.
This is the 'easy' direction.
In particular, $\|\cdot\|$ is continuous with respect to $\|\cdot\|_1$.
Let $K_2 = \min_{\|x\|_1 = 1} \|x\|$. Since the $\| \cdot\|_1$ sphere is compac... | {
"language": "en",
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An example to show that convergence of Cesaro sum to $0$ does not imply the original sequence converges to $0$. I'd like construct a non-negative sequence $\{a_{k}\}_{k=0}^{\infty}$ with $a_{0}=0$ such that the Cesaro sum $\frac{1}{n}\sum_{k=0}^{n-1}a_{k}\longrightarrow 0$ but $a_{n}$ does not converge to $0$.
I have ... | How about $a_k=(-1)^k$ C-sum $\to 0$, but $a_k$ does not converge.
| {
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Is this data skewed or symmetrical? So I have a piece of data, however, I am having a disagreement with others whether it is symmetric or skewed.
The mean of the data is 430, and the median is 433.
The data would be skewed if the mean > median, or mean < median. However the data would be symmetric if mean ≈ median.
Bec... | The data itself is definitely skewed, by the definition you give, albeit only slightly.
However if you introduce the idea that the data graphed is only a sample from a larger population, and ask whether the sample indicates the population as a whole is skewed, this is a different question.
The size of any sample is div... | {
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If $\sum (a_n)^2$ converges and $\sum (b_n)^2$ converges, does $\sum (a_n+b_n)/n$ converge? Could someone help me to solve this or at least give me a hint? I've tried a few criterions and still can't really prove this, and I don't know what should I try. Any help would be appreciated
| Applying Cauchy-Schwarz twice, you get
\begin{align*}
\Big(\sum_n \frac{a_n+b_n}n\Big)^2 &\leqslant \Big(\sum_n {a_n}^2+2\sum_{n}a_nb_n+\sum_n{b_n}^2\Big)\sum_n\frac 1{n^2}\\[5pt]
&\leqslant\Big(\sum_n {a_n}^2+2\sqrt{\sum_n{a_n}^2\sum_n{b_n}^2}+\sum_n{b_n}^2\Big)\sum_n\frac 1{n^2},
\end{align*}
where each sum clearly c... | {
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If 7 dice are thrown simultaneously, then what does the probability that all six digit appears on the upper face equal to? I've approached the problem the following way :
Out of the 7 dice, I select any 6 which will have distinct numbers : 7C6.
In the 6 dice, there can be 6! ways in which distinct numbers appear.
And ... | You probably noticed that your answer differs from the correct answer by a factor 2, so apparently you count everything twice.
Suppose your dice are labeled A, B, C, D, E, F, G and you throw:
A:1
B:2
C: 3
D:4
E: 5
F: 6
G: 1
Then you count this throw twice: one time with ABCDEF as the 'special' dice showing 6 different ... | {
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Sections of the exceptional divisor of a blowup Let $C$ be a smooth curve in a smooth threefold $X$. Denote by $Y$ the blowup of $X$ along $C$ with exceptional divisor $E$. Then $E \rightarrow C$ is a $\mathbb{P}^1$-bundle over $C$.
Is it true that sections of $E \rightarrow C$ correspond to smooth surfaces $S \subse... | No. For example, let $C \subset \mathbb{P}^3$ be a twisted cubic curve. Then
$$
N_{C/X} \cong \mathcal{O}_C(5) \oplus \mathcal{O}_C(5),
$$
and a surface $S$ smooth along $C$ corresponds to a section of the sheaf $I_C(d)$ such that the composition
$$
\mathcal{O}_{\mathbb{P}^3} \to I_C(d) \to I_C/I_C^2 \otimes \mathcal{O... | {
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If you equip two isomorphic groups with homeomorphic topologies, are they isomorphic as topological groups? I'm wondering if anyone has any insight regarding the truth of the above statement. Intuitively, if I have two topological groups in which their algebraic group structures are the same up to relabelling, and topl... | If $G$ is a finite topological group and $N$ is the connected component of the identity, then $N$ is normal and the coset space $G/N$ forms a basis for the topology. Conversely, one can create any finite topological group given a choice of finite group $G$ and normal subgroup $N$ to be the connected component. (See thi... | {
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How to obtain a formula for $f(z)$ given this recurrence I am trying to figure out how to derive a formula for $f(z)$ that is a function of $z$ and maybe $k \in \mathbb{N}$:
$$
f(z) = 1+z f \bigg(\frac{z}{1+z} \bigg)
$$
As an attempt, I tried a change of variable $z=\frac{1}{x}$, and I get:
$$
f\bigg(\frac{1}{x}\bigg) ... | If we plug in $z=0$ to the original functional equation we get $f(0)=1$. Then we set $g(x)=f(1/x)$. We have, as you showed, that $$g(x)=1+\frac1{x}g(x+1).$$
Thus for integer $m>0$ we have
$$g(x)=g(x+m+1)\prod_{r=0}^{m}\frac1{x+r}+\sum_{k=0}^{m-1}\prod_{j=0}^{k}\frac1{x+j}.$$
Taking the limit as $m\to\infty$ on both sid... | {
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Integral of $\int \sin^2x\cos^4xdx$ $$\int \sin^2x\cos^4xdx$$
I tried
$$I = \int (1-\cos^2x)\cos^4xdx = \int \frac{\sec^2x-1}{\sec^6x}dx = \int \frac{\tan^2x}{\sec^6x}dx$$
Take $\tan x = t \implies \sec^2xdx = dt$
$$I = \int \frac{t^2}{(t^2+1)^4}dt$$
And I could not proceed further from here.
| $$I=\int \sin^2 x \cos^4 x dx =\frac{1}{8} \int \sin^2 2x (1+\cos 2x) dx=\frac{1}{8}\int \sin ^2 2x dx+\frac{1}{8}\int (t^2/2) dt$$
Here $\sin 2x=t$
$$\implies I=\frac{1}{16}\int (1-\cos 4x) dx +\frac{(\sin 2x)^3}{48} =\frac{1}{16} x-\frac{1}{64} \sin 4x+\frac{(\sin 2x)^3}{48} $$
| {
"language": "en",
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Calculating $\mathbb E$ and $\mathbb V$ of a random variable. $\begin{pmatrix}&&&&\mathrm{payout}\\\mathrm{age,sex}&&1&&2&&4\\68, female&&1&&-&&1\\67,male&&-&&2&&-\end{pmatrix}$
I don't know how to properly format matrices, so let's explain it:
We got two $68$ year old females. If female_1 dies we have to pay $1$. If ... | Assuming the deaths are independent, you have
$$\mathbb E(S^2)=q_{68,f}\cdot 1^2+q_{68,f}\cdot 4^2+q_{67,m}2^2+q_{67,m}2^2=q_{68,f}\cdot 17+q_{67,m}8$$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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$\overline{U}\cap V\subseteq\overline{U\cap V}$ In particular if the equality is generally false, is it true if $V$ is open?
Could someone help, me please?
| In general this is false:
Consider $U := [0,1)$ and $V:= [1,2]$. The we have $\overline{U} = [0,1]$ and thus $\overline{U} \cap V = \{1\}$ but $U \cap V = \emptyset$ and thus their closure is empty as well.
However, if $V$ is open, this is true.
| {
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If $m_n + n$ is true, then prove that $m_{n+1} + n + 1$ is true. Algebraic breakdown help I'm a 46 year old Discrete Math student, and with all of the gaps in my math education, remembering the algebra to do the last step of my Induction proofs have been the hardest part for me. How do I deal with the subscript and whe... | So if I understand correctly you want to see the truth of the biconditional
$$P \leftrightarrow p_{n+1}$$ where $P$ itself is a complex chain of smaller statements with arrows between them. The beauty of the situation is that you discovered the a different way of describing the truth or falsehood of $P$: $P$ is true if... | {
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Generalized Union and Intersection by Induction Our teacher, told us to prove,
$$\left( \bigcap_{i=1}^n A_i\right)^c = \bigcup_{i=1}^n (A_i^c) $$
By induction. He told us that it has something to do with DeMorgan.
So my question is on knowing what's on the sets. I think that the left one has all the numbers to n except... | \begin{align}
\left( \bigcap_{i=1}^{n+1} A_i\right)^c & = \left(\left( \bigcap_{i=1}^n A_i \right) \cap A_{n+1} \right)^c \\[8pt]
& = (B\cap A_{n+1})^c \\[8pt]
& = B^c \cup A_{n+1}^c & & \text{de Morgan} \\[8pt]
& = \left( \bigcap_{i=1}^n A_i \right)^c \cup A_{n+1}^c \\[8pt]
& = \left( \bigcup_{i=1}^n (A_i^c) \right) \... | {
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Compounding more frequently seems to decrease total amount when using APYs? Interest rates are often given in terms of annual growth, even when compounding happens more often than once a year. To account for this, I read that we can use the following transformation to get the periodic compounding rate, $r$.
$$ r = (1 +... | The reason is that with less compounding the money is in the account longer. Let us take a one year term and compare annual vs. semiannual compounding at $10\%$. If you deposit $1$ at the start of the year, annual compounding gets you $1.1$ while semiannual gets you $1.1025$ as you would expect. But if you contribut... | {
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Are these fields equal?
Let $\zeta_3$ be the third root of unity.
1) Does it hold that: $\mathbb{Q}(\sqrt{2},\zeta_3)=\mathbb{Q}(\sqrt{2}+\zeta_3)$ ?
2) Does it hold that $\mathbb{Q}(\sqrt[3]{2},\zeta_3)=\mathbb{Q}(\sqrt[3]{2}\zeta_3)$?
My attempt for 1) is to compute the minimal polynomial of $\sqrt{2}+\zeta_3$ as ... | $\zeta_3$ is a root of $x^2 + x + 1$ since $x^3 - 1 = (x - 1)(x^2 + x + 1)$. So its degree is $2$ not $3$.
For 2) you can do the same kind of thing. The minimal polynomial of $\sqrt[3]2 \zeta_3$ is $x^3 - 2$ so $\mathbb{Q}(\sqrt[3]2 \zeta_3) \ne \mathbb{Q}(\sqrt[3]2, \zeta_3)$. It's a 3-dimensional subspace of a 6-dime... | {
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Show $AA^+$ is symmetric Can somebody show me how $AA^+$ is symmetric if $A^+$ is the pseudoinverse of $A$?
All I can muster is:
$(AA^+)^T => (A^+)^TA^T$
I know: $(A^+)^T = (A^T)^+$ but that doesn't really seem like it gets us anywhere.
Thanks!
| The comment from user759562 is correct, it is Hermitian by definition. But in the spirit of the question, lets do the computation with the definition provided here. That is, when $A$ has linearly independent columns, $A^+$ can be expressed as $A^+ = (A^*A)^{-1}A^*.$ Note that for an invertible matrix $B$, we have $(B^*... | {
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How does a vector b in the column space come from a vector in the row space? I'm working through Gilbert Strang's Introduction to Linear Algebra book and am really confused by a paragraph from chapter 4.1 titled 'Orthogonality of the Four Subspaces'. The paragraph is as follows:
Every vector goes to the column space! ... | This theorem is strange, because its not always true... It only holds when the matrix $\mathbf{A}$ has full rank. So probably context is missing here.
Anyway, to your question:
The row space that is spanned by your example matrix is NOT
$$\text{span}\left(\begin{bmatrix}1\\2\end{bmatrix}\right),$$ it is
$$\text{span}... | {
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Show convergence of $\sum \frac{z^n}{n}$ Show that the series $\displaystyle\sum \frac{z^n}{n}$ for $z=1$ diverges, but for all $z \ne 1$ with $|z|=1$ converges. Hint: Estimate $(1-z) \displaystyle\sum_{n=k}^{m} \frac{z^n}{n}$.
The first case $z=1$ is easy, this is just the harmonic series. But I am really stuck with t... | Hint:
$$\sum_{k=1}^n\frac{z^k}{k}-\sum_{k=1}^n\frac{z^{k+1}}k=\sum_{k=1}^n\frac{z^k}{k}-\sum_{k=2}^{n+1}\frac{z^k}{k-1}=z-\sum_{k=2}^{n}\frac{z^k}{k(k-1)}-\frac{z^{n+1}}n.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3585272",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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How do we find the answer of the derivative when we are not even given a point in the function? In almost all problems I came across of derivatives, we were just given the function(for ex-x3) and we were told to find the derivative of the function. As it is to my understanding, derivative is the slope of the tangent at... | When someone says to find the derivative of a function in the manner you speak of, they are wanting you to find the derivative at an arbitrary point. This ends up being another function. For example:
$f(x) = x^2$
We know that at any point $x$ the derivative is $2x$. Therefore the derivative is $f’(x) = 2x$. The evaluat... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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An inequality for the mgf using Jensen’s inequality Given non-negative random variables $X_1,X_2,...$ how to show that
$$\mathbb{E}\exp(t\max\limits_{1\leq i\leq n}X_i)\leq \sum\limits_{1\leq i\leq n}\mathbb{E}\exp(tX_i).$$
I think we should start with
$$\max\limits_{1\leq i\leq n}X_i\leq \sum\limits_{1\leq i\leq n}X... | If $t<0$ then $t \max_k X_k(\omega) \le t X_i(\omega)$ for all $i$ and if
$t \ge 0$ then $ t \max X_k(\omega) \le tX_i(\omega)$ for some $i$.
Hence $e^{t \max_k X_k(\omega) } \le \sum_k e^{t X_k(\omega)}$ and hence taking expectations
we have the desired result.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Do $\{x\mid x\in\mathbb{R},xShort question about set-builder notation.
Do
$$D=\{x \mid x \in \mathbb{R}, x < k\}$$
and
$$D=\{x \in \mathbb{R} \mid x < k\}$$
mean the same thing?
I see both of them used in different contexts and was wondering if they are interchangeable.
| They mean the same thing. I prefer $\{x \in \mathbb{R} \mid x < k \}$, because it is a clear separation between the domain ($\mathbb{R}$) and the condition ($x < k$). So I think it is easier to read, definitely when the condition gets more complicated.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Powers of roots in terms of polynomial coefficients Suppose we have a monic polynomial of degree $n$ with coefficients $c_1, c_2, c_3, \cdots, c_n$, and roots $r_1, r_2, r_3, \cdots, r_n$:
$$
x^n+c_1 x^{n-1} + c_2 x^{n-2} + c_3x^{n-3} + \cdots + c_n
$$
I'm looking to find expressions such as
$$
r_1^2 + r_2^2 + r_3^2 + ... | You can use Newton's identities.
This process would be inductive. The coefficient of $x^{n-k}$ is $(-1)^ke_k$ by the notation in the article on Newton's identities. Your desired sums are
$$p_k=r_1^k+r_2^k+\cdots+r_n^k$$
Then the formula says
$$ke_k=e_{k-1}p_1-e_{k-2}p_2-e_{k-3}p_3+\cdots+(-1)^{k-1}p_k$$
Substituting i... | {
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"source": "stackexchange",
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Disproving equality of cartesian products We are to disprove the statement $X \times Y = Y \times X \iff X = Y$ but I can't think of an example where this would be false. If $X = Y$, then wouldn't the Cartesian product be the same in either direction?
| The important point is that a Cartesian product is a set of ordered pairs. So if $X$ (say) contains an element $a$ which is not in $Y$, then in $X \times Y$ the element $a$ will appear in ordered pairs only as the first item of the pair, while in $Y \times X$ it will appear only as the second item of the pair. So no ... | {
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"timestamp": "2023-03-29T00:00:00",
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Compute $\int_0^1\frac{\ln(1-x)\ln(1+x)}{1+x}\ln\left(\frac{1+x}{2}\right)\ dx$ How to prove that
$$\int_0^1\frac{\ln(1-x)\ln(1+x)}{1+x}\ln\left(\frac{1+x}{2}\right)\ dx$$
$$=2\text{Li}_4\left(\frac12\right)-2\zeta(4)+\frac{15}8\ln(2)\zeta(3)-\frac12\ln^2(2)\zeta(2)$$
where $\text{Li}_r$ is the polylogarithm function a... | Set $x=2t-1$
$$\begin{align}
& =\int_{\frac{1}{2}}^{1}{\frac{\ln \left( t \right)\ln \left( 2t \right)}{t}\ln \left( 2-2t \right)dt} \\
& =\int_{\frac{1}{2}}^{1}{\frac{\ln \left( t \right)\ln \left( 2t \right)}{t}\left( \ln \left( 2 \right)-\sum\nolimits_{n=1}^{\infty }{\frac{{{t}^{n}}}{n}} \right)dt} \\
& =\int... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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show that $\sqrt{n+1}-\sqrt{n} \rightarrow 0$ as $n \rightarrow \infty$ show that $\sqrt{n+1}-\sqrt{n} \rightarrow 0$ as $n \rightarrow \infty$
Here is the algebric proof:
We have $a_n=\sqrt{n+1}-\sqrt{n}$, and we want to show that $\lim a_n=0$.
$$\sqrt{n+1}-\sqrt{n}=\frac{(\sqrt{n+1}-\sqrt{n})(\sqrt{n+1}+\sqrt{n})}{\... | The epsilon-delta method requires you to work out how small a $\delta$ is sufficient for a sought $\epsilon$, so you need your calculation anyway. You want to prove$$\forall\epsilon>0\exists\delta>0\left(\forall n>\frac{1}{\delta}\left(\frac{1}{\sqrt{n+1}+\sqrt{n}}<\epsilon\right)\right).$$It suffices to take $\delta=4... | {
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Use contour Integration to establish $\int\limits_{-\infty}^\infty\frac1{(x^2+a^2)(x^2+b^2)}{\rm d}x=\frac\pi{ab(a+b)}$ for $a,b>0$ Can someone help me figure this out please? This is the question along with what I have so far.
Use contour integration to establish
$$\int_{x=-\infty}^\infty\frac1{(x^2+a^2)(x^2+b^2)}... | Hint: The poles are simple, and located at $\pm ai,\pm bi$. You could use the residue theorem, if you prove the integral on the part of the contour off the real axis goes to zero.
| {
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What is the notation of a set of $n$ binary numbers where all of them are 0 except for one? From this post I found out we can define a set of $n$ binary numbers mathematically like: $\mathbb Z_2^n$.
But what if I want to further restrict this set such that all the bits must be zero except for one? For example, elements... | I don't know of any formal notation for this, but the set you are describing is precisely the powers of $2$ up to $2^{n-1}$; i.e. $\{2^a\mid a\in\mathbb Z,0\le a<n\}$. If you refer to such sets regularly, you may denote them by $A_n$ or whatever notation is convenient for you.
| {
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Trig substitution for $\sqrt{9-x^2}$ I have an integral that trig substitution could be used to simplify.
$$ \int\frac{x^3dx}{\sqrt{9-x^2}} $$
The first step is where I'm not certain I have it correct. I know that, say, $\sin \theta = \sqrt{1-cos^2 \theta}$, but is it correct in this case $3\sin \theta = \sqrt{9 - (3\c... | What you have done is absolutely correct, except where you forgot to mention that $\theta$ is in $(0, \pi)$, but you can simplify your answer further.
The book's answer might be something like $-\frac{1}{3} \sqrt{9-x^2} (x^2+18)$, which you can get by factoring out a factor of $\sqrt{9-x^2}$:
$$-9\sqrt{9-x^2} + \frac{... | {
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How to give a combinatorial proof for: If $n$ and $k$ are positive integers with $n=2k$ then $\frac{n!}{2^k}$ is an integer How can i give a combinatorial proof for if $n$ and $k$ are positive integers with $n=2k$ then $\dfrac{n!}{2^k}$ is an integer?
| If $n=2k$, $\dfrac{n!}{2^k}$ can be written as the multinomial $\dbinom{n}{2,\dots,2}$ (with $k$ $2$'s), and is therefore an integer.
| {
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How could we plot the KL divergence? I was trying to write a blogpost on information theory and I think it would be a good idea (if possible) to plot the KL divergence in a 3D-plot in order to show graphically its convexity, but I wouldn't know how to define the pdf space. How would you do it?
$$
KL(f||g)=\sum_{x \in X... | One idea would be to use the fact that a function is convex if and only if its restriction to a line is convex.
In the case of KL divergence, we can pick any two pairs of distributions $(f, g)$ and $(f', g')$ and plot
$$
\mathrm{KL}(\lambda f + (1 - \lambda) f' \, || \, \lambda g + (1 - \lambda) g')
$$
as a function of... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3586784",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Physics Question Math Related: When do Physicists ever use the following expression in whatever context:
\begin{align}
\frac{d}{dt}[r_1(t)\ \cdot \ r_2(t)] = r_1 \ \cdot \ \frac{dr_2}{dt} + \frac{dr_1}{dt} \ \cdot \ r_2 \\
\frac{d}{dt}[r_1(t)\ \times \ r_2(t)] = r_1 \ \times \ \frac{dr_2}{dt} + \frac{dr_1}{dt} \ \times... | 2) Angular momentum of a solid object is defined as $\vec{L} = \vec{r} \times \vec{p}$, where $\vec{p}$ is regular momentum of that object, and $\vec{r}$ is the distance vector from the point w.r.t which the angular momentum is calculated. Thus, change in angular momentum can come either from changing the linear moment... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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For fixed hypotenuse, can the number of primitive Pythagorean triples exceed the number of non-primitive ones? For the equation,
$$a^2+b^2=c^2$$
if $c$ is fixed and the number of natural solutions for $a, b$ is greater than $1$, then can the number of primitive solutions (solutions in which $a, b, c$ are coprime) excee... | For some hypotenuse $c$, let the number of primitive solutions be greater than the number of non-primitive solutions.
Assume that $p \mid c$ for some prime $p$. Clearly, there is atleast one primitive solution $(a,b,c)$. Then, we have:
$$p^2 \mid c^2 \implies p^2 \mid (a^2+b^2)$$
It is easy to see that since $p \nmid ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3587066",
"timestamp": "2023-03-29T00:00:00",
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Upper Bound on a en exponential function I am trying to upper bound the following function and find its growth rate:
\begin{equation}
\psi(y) \stackrel{\triangle}{=} \int_{0}^{\infty}\exp\left(-\frac{(y-x)^2}{2(\sigma_0^2 + \sigma_1^2 x)}\right)f(x)\,dx,~y>0,
\end{equation}
where $f(x)>0$ satisfies $\int_{0}^{+\infty}f... | First, let us see an example in which $\lim_{y\to \infty} y\psi(y) = \infty$.
Let (log-Cauchy distribution)
$$f(x) = \frac{1}{\pi x (1 + (\ln x)^2)}, \ x > 0.$$
We have, for sufficiently large $y$,
\begin{align}
\psi(y) &= \int_{0}^{\infty}\exp\left(-\frac{(y-x)^2}{2(\sigma_0^2 + \sigma_1^2 x)}\right)
\frac{1}{\pi x ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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How many times must I apply L’Hopital? I have this limit:
$$\lim _{x\to 0}\left(\frac{e^{x^2}+2\cos \left(x\right)-3}{x\sin \left(x^3\right)}\right)=\left(\frac 00\right)=\lim _{x\to 0}\frac{\frac{d}{dx}\left(e^{x^2}+2\cos \left(x\right)-3\right)}{\frac{d}{dx}\left(x\sin \left(x^3\right)\right)}$$
$$\lim_{x\to0}\frac{2... | Let's first attack the numerator alone, repetitively differentiating until we no longer get zero. Let $N = \mathrm{e}^{x^2} + 2 \cos x - 3$.
\begin{align*}
\frac{\mathrm{d}}{\mathrm{d}x} N &= 2 x \mathrm{e}^{x^2} - 2 \sin x \xrightarrow{x \rightarrow 0} 0 \text{,} \\
\frac{\mathrm{d}^2}{\mathrm{d}x^2} N &= (4 ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Exercise and proof given in a Number Theory textbook
Prove that $(\forall m\in\mathbb N)(\exists$ $x,y \in \mathbb N)$, s.t. $x-y \geq m$ and $\sigma(x^2)=\sigma(y^2)$
$\sigma(x):=\displaystyle\sum_{i=1}^kd_i,\;d_i\mid x,\;\forall i\in\{1,\ldots,k\},\;k\le x$
Proof (in text book):
Let $n \in \mathbb{N}$ with $n > m$ ... | General framework. It's interesting to study natural numbers $\ s<t\ $ such that $\ \sigma(s)=\sigma(t).\ $ In particular, it's a tough challenge to find natural numbers $\ a<b\ $ such that $\ \sigma(a^2)=\sigma(b^2).\ $ Indeed, this last equation is the difficult part of the given exercise.
Once you have such $\ a<b\ ... | {
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"timestamp": "2023-03-29T00:00:00",
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Evaluating $\lim_{x\to\infty}\left(\frac{4^x+5^x}{4}\right)^{1/x}$ $$\lim_{x\to\infty}\left(\frac{4^x+5^x}{4}\right)^{\frac{1}{x}}=?$$
I have tried a lot but I am stuck when I solve this by using this hints. $a^x=\exp(\ln(a^x))=\exp(x\ln a)$, so then $a=\frac{4^x+5^x}{4}$. The above expression becomes
$$\lim_{x\to\inft... | Let's assume that the required limit to be calculated is $L$
And by taking the natural logarithm both sides, we would have:
$$\ln L =\lim_{ x\to \infty }\frac{\ln(\frac{4^x+5^x}{4})}{x}$$
This is an inderminate form of $\frac{\infty}{\infty}$ and can be solved by applying L' Hopital's Rule.
Applying the rule we get:
$... | {
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"timestamp": "2023-03-29T00:00:00",
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Find $a, b, c$ such that element $x=a\alpha^2+b\alpha+c \in \mathbb{Q}[x]/\langle x^3+x+11 \rangle$ I'm trying to generate a fundamental unit of the number field $K=\mathbb{Q}(\alpha)$, where $\alpha^3+\alpha+11$.
I found a non-trivial unit and I need to find $a,b,c\in\Bbb{Q}$ such that
$$\frac{(-5\alpha^2-4\alpha+8)... | A quick google search leads to the "Number field element" page of the Sage documentation, which shows that the following code does the trick in SageMath:
K.<a> = NumberField(x^3 + x + 11)
f = a.coordinates_in_terms_of_powers()
f((-5*a^2-4*a+8)*(6*a^2+6)*(8*a^2+8)^(-6)*(9*a^2+9*a)^(-11)*(10*a^2)^(-1)*(10*a^2+10*a))
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3587740",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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$\sin(x) - \sqrt3 \sin(3x) + \sin(5x) < 0$ for $0My attempt at solving this:
$\sin(x) - \sqrt3\sin(3x) + \sin(5x) < 0$
$2\sin \left(\frac{5x+x}2\right) + \cos\left(\frac{5x-x}2\right) - \sqrt 3\sin(3x) < 0$
I divide everything with 2:
$\sin(3x) + \frac12\cos(2x) - \frac {\sqrt 3}2\sin(3x) < 0$
I think I have gone the w... | $$\sin (x)+\sin (5x) - \sqrt3\sin(3x) <0\Rightarrow 2\sin(3x)\cos (2x) - \sqrt{3}\sin(3x) < 0 \\\Rightarrow \sin(3x)\left(\cos(2x) -\frac{\sqrt{3}}{2}\right)<0 $$
Case $1$:
$$ \sin(3x) < 0 \text{ and } \cos (2x) > \frac{\sqrt3}2 \implies \frac{\pi}{12}<x<\frac\pi3$$
Case $2$:
$$ \sin (3x) > 0 \text{ and } \cos (2x) < \... | {
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Let $X$ and $Y$ are topological spaces with indiscrete topologies then prove that the product topology $X\times Y$ will be indiscrete space Now $\tau_X=\{X,\emptyset\}$ and $\tau_Y=\{Y,\emptyset\}$ their product topology will be like $\tau_{X\times Y}=\{X \times Y , \emptyset \times Y , X \times \emptyset , \emptyset\}... | It turns out that $X\times\emptyset=\emptyset\times Y=\emptyset$. So, yes, the product topology on $X\times Y$ is the indiscrete topology.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Easy way to find the partial fraction I always have trouble trying to find the partial fraction, especially for complicated ones.
For example, this is what I will do to find the partial fraction of $\displaystyle \frac{8x^3+35x^2+42x+27}{x(2x+3)^3}$
*
*$\displaystyle \frac{A}{x}+\frac{B}{2x+3}+\frac{C}{(2x+3)^2}+\f... | A suggestion may be in form of $$\displaystyle \frac{8x^3+35x^2+42x+27}{x(2x+3)^3}$$
look for $(2x+3)^3=8x^3+36x^2+54x+27$ so ,we can rewrite $$\displaystyle \frac{8x^3+35x^2+42x+27}{x(2x+3)^3}=\\\frac{(2x+3)^3-x^2-12x}{x(2x+3)^3}=\\
\frac{(2x+3)^3}{x(2x+3)^3}-\frac{x(x+12)}{x(2x+3)^3}=\\
\frac{1}{x}-\frac{(x+12)}{(2x... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Impulse/Delta Function--homework help I need to solve the initial value problem:
I took the Laplace transform of both sides and this is what I have thus far:
I now need to take the inverse Laplace transform to find x(t). I can't simplify the denominator by completing the square, so I am stuck here. Is this an example ... | What you have looks correct to me. Now decompose the fraction this way:
$$G(s)=\frac 1 {(s^2+s-2)}=\frac 1 {(s+2))(s-1)}$$
$$G(s)=\frac 1 3 \left (\frac 1 {(s-1)}-\frac 1 {(s+2)} \right )$$
Then take Inverse Laplace Transform.
$$g(t)=\frac 13 (e^{t}-e^{-2t})$$
Do the same for the other fraction.
| {
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If A and B are $n \times n$ matrices where each column sums to p. Then for what values of p will the matrix AB also have all columns that sum to p? I have no idea how to approach this question. I've tried working through it with sum notation but it became jumbled. I assume there's another property of matrices that I ca... | Nice question.
You can go for the following approach : note that if $A,B$ are $n \times n$ matrices, each having columns summing to $p$, then the sum of all entries of $A$ and $B$ are both $np$ (number of columns times sum of each column).
Now, we calculate the sum of all entries of $AB$.
$$
\sum_{i,j=1}^n (AB)_{ij} = ... | {
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If the system of inequalities $3x^2+2x-1<0$ and $(3a-2)x-a^2x+2<0$ possesses a solution, find the least natural number $a$ If the system of equations $3x^2+2x-1<0$ and $(3a-2)x-a^2x+2<0$ possesses a solution, find the least natural number $a$
My attempt is as follows:-
$$3x^2+3x-x-1<0$$
$$3x(x+1)-1(x+1)<0$$
$$x\in\left... | Inequalities can be sumized in this system:
$x<-1$ and $x>\frac{1}{3}$
$a<=\frac{3\sqrt{x}-\sqrt{x+8}}{2\sqrt{x}}$
$a>=\frac{3\sqrt{x}+\sqrt{x+8}}{2\sqrt{x}}$
The least natural number is $a=4$ for $x=\frac{1}{3}$.
Other values are:
$a=5$ for $x=\frac{1}{6}$;
$a=6$ for $x=\frac{1}{10}$;
$a=7$ for $x=\frac{1}{15}$;
$a=8$... | {
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} |
Rayleigh as exponential family - compute $\mathbb{E}(Y)$ and Var$(Y)$ where Y is a sum of independent squared Rayleigh's distribution Prove that X of Rayleigh distribution with pdf $f(x, \sigma) = \frac{x}{\sigma^2}e^{-\frac{x^2}{2\sigma^2}}\mathbb{1}_{(0, \infty)}(x)$ comes from the exponential family and then compute... | The Rayleigh distribution is a single parameter exponential family if we can write it in the form
$$ f(x: \sigma) = h(x) \exp\left( \eta(\sigma) T(x) - A(\sigma) \right)$$
Here we have $$f(x: \sigma) = x \mathbb{1}_{[0,\infty)} (x) \exp \left( \frac{-1}{2\sigma^2} x^2 - 2 \log \sigma \right)$$
so it is indeed an expone... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3589074",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Unable to prove an assertion with induction I need to prove:
$ \displaystyle \sum_{k=1}^n\frac{1}{(5k + 1) (5k + 6)} = \frac{1}{30} - \frac{1}{5(5n + 6)} $ with mathematical induction for all $n \in \mathbb{N}$.
After successfully proving it for n = 1, I try to prove it in the Induction-Step for n + 1:
$ \displaystyle... | Let $ n $ be a positive integer.
Observe that : $ \left(\forall k\in\mathbb{N}\right),\ \frac{1}{\left(5k+1\right)\left(5k+6\right)}=\frac{1}{5}\left(\frac{1}{5k+1}-\frac{1}{5k+6}\right) \cdot $
Thus, \begin{aligned} \sum\limits_{k=1}^{n}{\frac{1}{\left(5k+1\right)\left(5k+6\right)}}&=\frac{1}{5}\left(\sum\limits_{k=1}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3589245",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 3
} |
Trying to visualize a polygon in a space $X$ From Rotman's Algebraic Topology:
A polygon in a space $X$ is a $1$-chain $\pi = \sum\limits_{i=0}^k \sigma_i$ where $\sigma_i(e_1) = \sigma_i(e_0)$ for all $i$.
By a theorem proven in the book, all polygons are $1$-cycles.
From the book: a $1$-cycle is essentially "a sum ... | You may have made a transcription error when copying the definition of polygon. The definition Rotman gives (at least by the 4th corrected printing, 1998) says
A polygon in a space $X$ is a $1$-chain $\pi = \sum_{i=0}^k \sigma_i$, where $\sigma_i(e_1) = \sigma_{i+1}(e_0)$ for all $i$ (indices are read mod$(k + 1)$). ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3589612",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Would duplicates matter in cartesian product of a set? For example:
\begin{align}
A &= \{1, 1, 2\}
\\
B &= \{3, 3, 3, 2, 2, 4\}
\end{align}
Would $A$ cross $B$ equate to $\{(1,3),(1,2),(1,4),(2,3),(2,2),(2,4)\}$ without the dupes of $(1,3)$, etc.
| No, because the Cartesian product of sets is itself a set. For sets in general, we consider a set, and a set with the same entries but some duplicates, to be precisely the same.
For example, let $A=\{1,2\},B=\{3,4\},A'=\{1,1,2,2\},B'=\{3,3,4,4,4,4,4\}$. Under these conditions, $A=A',B=B',$ and in turn $A \times B = A' ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3589844",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
Show that if $(b^n-1)/(b-1)$ is a power of prime numbers, where $b,n>1$ are positive integers, then $n$ must be a prime number.
Show that if $(b^n-1)/(b-1)$ is the power of a prime number, where $b,n>1$ are positive integers, then $n$ must be a prime number.
My solution:
If $n$ is composite, then let $n=mk$, $m,k>1$,... | Let $(b^n-1)/(b-1)=p^x$ where $p$ is a prime and $x> 0$. If $n$ is composite, there are two cases.
*
*There exists a prime $q$ such that $n=q^m$ for some $m>1$. Note
$$p^x=\frac{b^n-1}{b-1}=\frac{b^{q^m}-1}{b^{q^{m-1}}-1}\cdot \frac{b^{q^{m-1}}-1}{b-1},$$
we can assume
$$\frac{b^{q^{m-1}}-1}{b-1}=p^y$$
for some $0... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3590064",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Calculate $\mathbb{E}(X-Y\mid 2X+Y).$ if $X\sim N(0,a)$ and $Y\sim N(0,b)$
Question: Given that $X$ and $Y$ are two random variables satisfying $X\sim N(0,a)$ and $Y\sim N(0,b)$ for some $a,b>0$. Assume that $X$ and $Y$ have correlation $\rho.$
Calculate
$$\mathbb{E}(X-Y \mid 2X+Y).$$
I tried to use the fact t... | The joint distribution of $(Z_1,Z_2)\equiv(X-Y,2X+Y)$ is $\mathcal{N}(0,\Sigma)$, where
$$
\Sigma=\begin{bmatrix}
a+b-2\rho\sqrt{ab} & 2a-b-\rho\sqrt{ab} \\
2a-b-\rho\sqrt{ab} & 4a+b+4\rho\sqrt{ab}
\end{bmatrix}.
$$
Then the conditional distribution of $Z_1$ given $Z_2$ is
$$
Z_1\mid Z_2=z\sim \mathcal{N}(\Sigma_{12}\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3590203",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 1
} |
Stokes Theorem application question Below is an excerpt from the book "Partial Differential Equations" by Evans. The underlined equation confuses me. Clearly it is an application of Stokes theorem, and the implication seems to be that if $f$ is any compactly supported smooth function (for simplicity say on all of $\m... | I think it's easier to go the other direction:
$$
\begin{split}
\int_{\mathbb{R^2}-B_1(0)}f_x\,dx\wedge dy &= \int_{\mathbb{R}^2-B_1(0)}d(f\,dy) = \int_{S^1}f\,dy \\
&= -\int_0^{2\pi} f\,d(\sin\theta) = -\int_0^{2\pi}f\cos\theta\,d\theta = -\int_{\partial B_1(0)} fx\, dS.
\end{split}
$$
The additional minus sign after... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3590429",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
Does the map from a set to the free object on the set have to be injective? I've seen the following definition of a free object in category theory.
Let $\mathcal C$ be a concrete category. Denote by $U\colon\mathcal C\to\mathrm{Set}$ the forgetful functor. Let $X$ be a set. Then an object $F(X)\in\mathcal C$ equipped ... | No, it does not follow from the definition. Indeed, it does not even follow from the definition in the case of modules in general. Suppose $R$ is the zero ring (the ring with one element). Then every module over $R$ has one element, and this single module is free on every possible set via every possible map.
If you ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3590541",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 1,
"answer_id": 0
} |
Hilbert Polynomial at a point? One of the review problems in my final review is the following:
Let $X\subset\mathbb P_{\mathbb C}^n$ be a hypersurface, and $P\in X$ a singular point. Let $L$ be a line not contained in $X$ that intersects $X$ at $P$.
Prove that $h_{X\cap L}(P)\geq2$, the intersection multiplicity of $X... | The following is a community wiki answer recording the discussion in the comments so that this question might be marked as answered (once this answer is upvoted or accepted).
Hm yeah I am not sure that this makes sense? Not an expert though. Could they mean like intersection number of $X\cap L$ at $P$? – user113102
Oh... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3590721",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Evaluating $\lim_{x\to 0}\frac{x\sin x-2+2\cos x}{x\ln(1+x)-x^2}$ using L'Hôpital Considering this limit, assigned to my high school students,
$$\lim_{x\to 0}\frac{x\sin x-2+2\cos x}{x\ln \left(1+x\right)-x^2}=\left(\frac00\right)=\lim_{x\to 0}\frac{\frac{d}{dx}\left(x\sin \left(x\right)-2+2\cos \left(x\right)\right)... | A quick estimate allows you to see that Taylor would yield even degree terms at the numerator (the function is even), but the constant and quadratic ones cancel each other. At the denominator, $x^2-x^2$ and cubic terms.
Hence the limit will be zero, but you will need three successive applications of L'Hospital to estab... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3590839",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 1
} |
the equivalent definition of the interval in $\mathbb{R}$ Let $X$ be a subset of the real line $\mathbb{R}$. Then the following statements are equivalent.
(b) Whenever $x,y \in X$ and $x < y$, the interval $[x, y]$ is also contained in $X$.
(c) $X$ is an interval (in the sense of Definition 9.1.1).
Definition 9.1.1. ... | Hint: To show that $X$ is one of the intervals $[a,b], (a,b),[a,b),(a,b]$ (where $a =\inf X, b=\sup X$) you only have to show that $x \in X$ whenever $a <x<b$. So it makes no difference as to whether $X$ is closed or not.
Use definitions of infimum and supremum to show that $c<x<d$ for some $c,d \in X$. Then use b) to... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3591088",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Simpson's rule — where did the coefficients come from? I am reading how Simpson's Rule works for numerical integration. So I understand that given the two endpoints $x_0$ and $x_2$, and one intermediate point $x_1$, we can connect these points to make a parabolic function as an approximation to the original function wh... | The interpolating polynomial can be written as
\begin{align*}
p_2(x)= &\sum_{i=0}^2 L_i(x) f(x_i)=\sum_{i=0}^2 \frac{\prod_{j \ne i}(x-x_j)}{\prod_{j \ne i}(x_i-x_j)} f(x_i)\\
=& \frac{(x-x_1)(x-x_2)}{(x_0-x_1)(x_0-x_2)}f(x_0)+\frac{(x-x_0)(x-x_2)}{(x_1-x_0)(x_1-x_2)}f(x_1)+\frac{(x-x_0)(x-x_1)}{(x_2-x_0)(x_2-x_1)}f(x_... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3591245",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Conjecture $\frac{a}{a^r+b^r}+\frac{b}{b^r+c^r}+\frac{c}{c^r+a^r}\geq \frac{a}{a^r+c^r}+\frac{c}{c^r+b^r}+\frac{b}{b^r+a^r}$ following this kind of inequality One of my old inequality (very sharp) I propose this because I don't see it on the forum :
Let $a,b,c>0$ and $a+b+c=1$ with $r\in(\frac{1}{2},1)$ and $a\geq b ... | If $\prod\limits_{cyc}(a-b)=0$, so it's obvious.
Let $a>b>c.$
Thus, we need to prove that:
$$\sum_{cyc}\left(\frac{a}{a^r+b^r}-\frac{a}{a^r+c^r}\right)\geq0$$ or
$$\sum_{cyc}\frac{a(c^r-b^r)}{(a^r+b^r)(a^r+c^r)}\geq0$$ or
$$\sum_{cyc}a(c^r-b^r)(b^r+c^r)\geq0$$ or
$$\sum_{cyc}a(c^{2r}-b^{2r})\geq0$$ or
$$a^{2r}(b-c)+c^... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3591424",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Prove that $\frac{1}{2\pi}\int_0^{2\pi}\frac{R^2-r^2}{R^2-2Rr\cos\theta+r^2}d\theta=1$ Let $C=\{z:|z|=r|\}$ with $r<R$ oriented in + sense. calcule:
$$\int_{C}\frac{R+z}{z(R-z)}dz$$
and deduce that
$$\frac{1}{2\pi}\int_0^{2\pi}\frac{R^2-r^2}{R^2-2Rr\cos\theta+r^2}d\theta=1$$
My attempt
I proved that $$\int_{C}\frac{R+... | Define : \begin{aligned} f:\mathbb{C}\setminus\left\lbrace\frac{R}{r},\frac{r}{R}\right\rbrace&\rightarrow\mathbb{C}\\ z&\mapsto\frac{R^{2}-r^{2}}{\left(R-rz\right)\left(Rz-r\right)} \end{aligned}
Since $ r<R $, the residue theorem allows us to write : $$ \oint_{\left|z\right|=1}{f\left(z\right)\mathrm{d}z}=2\pi\,\math... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3591587",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Why is it true that if $H$ is a subgroup of a group $G$, then $1_H=1_G$? I am studying about subgroup. My definition of subgroup is that:
Let a set $G$, with a binary operation$
×:G×G→G,(a,b)↦×(a,b)=:a×b$ be a group. Then $H⊂G$ is a subgroup iff $H$ with a restriction of $×$ to $H×H$, that is,$×|_{H×H}$ is also a group... | Suppose exists $b\in G,b\ne 1_H$ such that $ab=a$ for some $a\in G.$ By multipling $a^{-1}$ on the left we have that $b=1_G$, but $1_H$ is an element of $G$ which satisies $a1_H=a$ for some $a\in G,$ (in particular, the elements of H). Therefore, $1_H=1_G$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3591734",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Converting a regular expression to its complement via automata I'm supposed to convert a regular expression $r = (\alpha\beta + \beta\alpha)^\ast$ into its complement via automata. I started out by first constructing the individual DFAs that recognize $\alpha\beta$ and $\beta\alpha$:
I then combined and closed these w... | Since typing $\alpha$ and $\beta$ is time consuming, let me take the alphabet $A = \{a, b\}$ instead.
Your language $L = (ab + ba)^*$ is the star of the prefix code $P = \{ab, ba\}$ and there is a standard algorithm to compute the minimal automaton of $P^*$ when $P$ is a finite prefix code. Here you get the automaton $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3591854",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Relation between Weierstrass $\wp$-functions
Let $\Lambda=[\lambda_1,\lambda_2]$ be a lattice with associated Weierstrass function $\wp$, and consider the Weierstrass function $\wp_2$ associated to the lattice $\Lambda_2=[\tfrac{1}{2}\lambda_1,\lambda_2]$. Prove the identities $$\wp_2(z)=\wp(z)+\wp(z+\tfrac{1}{2}\lamb... | Given a lattice $\,\Lambda,\,$ the Weierstrass $\wp$ function is characterized by being a
meromorphic doubly periodic function with period
lattice $\,\Lambda\,$ whose only poles are at points in $\,\Lambda,\,$
and whose Laurent series at the origin is
$\,\wp (z) = z^{-2} + O(z^2)$. In your first equation, note
that th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3592216",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
How to deal with binomial expansion within floor function as in $\lfloor{(a+\sqrt{b})^n\rfloor}$? In questions involving floor functions containing binomial coefficients, like example 368 in the posted image, where it asks
for $n$, a nonnegative integer, show that the integers $\lfloor{(1+\sqrt{2})^n\rfloor}$ are alt... | The point is that if you expand $(1+\sqrt 2)^n+(1-\sqrt 2)^n$ by the binomial theorem, the terms with $\sqrt 2$ raised to an odd power cancel while the ones with $\sqrt 2$ raised to an even power are equal in the two terms. The $k$ in the summation is half the power of $\sqrt 2$ in the terms we are considering. The l... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3592385",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Why are these two definite integrals equal? How can one prove that, for $0< z<1$, the two integrals
$$\int_0^\infty \frac{u^{z-1}}{1+u}du$$
and
$$\int_0^\infty \frac{u^{-z}}{1+u} du$$
are equal?
From the integral representation of the beta function
$$B(z,w)=\frac12\int_{0}^\infty \frac{u^{z-1}+u^{w-1}}{(1+u)^{z+w}} ... | If you know, say, that the result of the first integral is $\pi/\sin(\pi z)$, you can see that the second one follows from the first by a mapping $z \mapsto 1-z$, which means the result of the second one is just $\pi/\sin(\pi - \pi z) = \pi/\sin(\pi z)$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3592596",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 0
} |
Determine the validity of the argument. I have this question and was hoping I could get some help on it:
p∧q∧r → s
u →s
p∧u∧~r
∴q
I have found:
p is true
u is true
~r is true
r is false.
But I am unsure what to do to find the validity of the statement.
My thinking is that premise 1 is (~p ∨ ~q ∨ ~r) ∨ s where S is tr... | No, $q$ does not follow. You can try it with $q$ both ways and see.
As you said, the third premise means $p$ is True, $u$ is True, and $r$ is False. Then the second premise means $s$ is True.
If $q$ is True, the first premise says $T \wedge T \wedge F \to T$, which is fine.
If $q$ is False, the first premise says $T \w... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3592731",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Prove the space $L^p(X) \cap L^q(X)$ with the norm $||f||_{L^p \cap L^q}=||f||_p+||f||_q$ is a Banach space $X$ is a space with positive measure and $1\le p<q\le +\infty$. I have to prove that $L^p(X) \cap L^q(X)$ is a complete space i.e. every Cauchy's sequence converges in this space with the norm $\lVert f\rVert_{L^... | You know that both $L^p$ and $L^q$ are Banach spaces. Fix a Cauchy sequence $(f_n)_{n\geq 1}$ in $L^p\cap L^q$.
*
*Since $(f_n)_{n\geq 1}$ is a Cauchy sequence in $L^p$, it converges in $L^p$. Let $f\in L^p$ be the limit.
*Since $(f_n)_{n\geq 1}$ is a Cauchy sequence in $L^q$, it converges in $L^q$. Let $g\in L^q$ ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3592870",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
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