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We have $m$ people and a set of $n$ elements. Each person chooses $l$ elements out of $n$, independently from each other. What is the probability that each element is chosen at least $k$ times?
Let $X_1, \dots X_n$ be the random variables counting how many times each element is chosen. I want to determine the densit... |
$m$ people choosing $l$ elements out of $n$ independently, what is the probability of each element being chosen at least $k$ times? |
<p>I am interested in the following question:</p>
<blockquote>
<p>A bounded real plane algebraic curve <span class="math-container">$f(x,y)=0$</span> has 3-fold rotational symmetry, then <span class="math-container">$d:=\deg f$</span> is even.</p>
</blockquote>
<p>I think when converting $f$ to polar coordinates it... |
The original question follows:
<h2>
Find the value of b for which the differential equation below is exact.
$$(ye^{7xy}+x)dx+(bxe^{7xy})dy=0$$
Then solve it using that value of b.
An implicit solution is fine.
<h2>
My attempt uses
$$M=ye^{7xy}+x\text{ and } N= bxe^{7xy}$$
$ \frac{M_y-N_x}{M}= -\frac{7y}{... |
By definition the Euler's totient function counts the positive integers up to a given integer $n\in \mathbb{N}$ that are relatively prime to $n$. Consequently, if $A = \{1,\cdots, n \}$, then $\phi (n) = | \{ a\in A:(a<n)\land (a\not| n) \} |$. What elements of the set $A$ divide $n$? If $ n = \prod_{i=1}^r p_i^{\alpha... |
Suppose that $\Omega$ is a set, $(\Omega, \mathscr{G})$ is a measure space, and $Z: \Omega \to \mathbb{R}$ is a given mapping. Then Z is $\mathscr{G}$ measurable iff $$Z =\displaystyle\sum_{i=1}^\infty \lambda_i I_{A_i} \tag{1}$$ for some {$\lambda$} $\subset \mathbb{R}$ and {$A_i$} $\subset \mathscr{G}$. From(1) ded... |
<p>I am interested in the following question:</p>
<blockquote>
<p>A bounded real plane algebraic curve <span class="math-container">$f(x,y)=0$</span> has 3-fold rotational symmetry, then <span class="math-container">$d:=\deg f$</span> is even.</p>
</blockquote>
<p>I think when converting $f$ to polar coordinates it... |
I am currently working with circular data that exhibits multiple modes, and I've chosen to model this data using the Multimodal Generalized von Mises (MGvM) distribution. The MGvM distribution is an extension of the von Mises distribution, designed to better accommodate the multimodality and asymmetry in circular data ... |
The original question follows:
<h2>
Find the value of b for which the differential equation below is exact.
$$(ye^{7xy}+x)dx+(bxe^{7xy})dy=0$$
Then solve it using that value of b.
An implicit solution is fine.
<h2>
My attempt uses
$$M=ye^{7xy}+x\text{ and } N= bxe^{7xy}$$
$ \frac{M_y-N_x}{M}= -\frac{7y}{... |
I am a mathematical beginner. As we all know, the graph of a one-dimensional function is a curve, and the graph of a two-dimensional function is a surface. What is the graph of a three-dimensional function? For example, $f(x,y,z)=x^2+y^3+5xyz$. Someone said that we cannot draw the graph of a three-dimensional function.... |
Why we cannot draw the graph of a three-dimensional function? |
The original question follows:
<h2>
Find the value of b for which the differential equation below is exact.
$$(ye^{7xy}+x)dx+(bxe^{7xy})dy=0$$
Then solve it using that value of b.
An implicit solution is fine.
<h2>
My attempt uses
$$M=ye^{7xy}+x\text{ and } N= bxe^{7xy}$$
$ \frac{M_y-N_x}{M}= -\frac{7y}{... |
I want to correctly prove the following statement:
\begin{equation}
\lim_{x \rightarrow \infty} \frac{1}{\ln\left(\frac{x + a}{x - a}\right)} = \frac{x}{2a} \ , \ a \in \mathbb{R}^{\ast+}
\end{equation}
This should atleast be true for $a \in \mathbb{R}^{\ast+}$, but I have no idea how to correctly prove it, as... |
I'm working with the Multimodal Generalized von Mises (MGvM) distribution, and I am interested in finding the variance of this distribution. The density function is given by:
$$
f(\theta) = G_M(\mu_1, \ldots, \mu_S, \kappa_1, \ldots, \kappa_S) \exp\left(\sum_{s=1}^{S} \kappa_s \cos(s(\theta - \mu_s))\right), \quad ... |
Let $0<\epsilon <M$ be such that $M-\epsilon$ is continuity point of $F_Y$. Then $\lim \inf P(Y_n \ge M)\le \lim \inf P(Y_n >M-\epsilon)=\lim P(Y_n >M-\epsilon)=P(Y>M-\epsilon)$.We can find a sequence $(\epsilon_i)$ decreasing to $0$ such that $M-\epsilon_i$ is continuity point of $F_Y$ for each $i$. We get $\lim \... |
Can i find or assume the existence of a sequence $f_n$ that converges the divergent sequence $a^n$ or the sequence $e^{a\cdot n} $ when they are multiplied?
I can write this problem as:
Find $f_n$ such that the sequence
$g_n = f_n\cdot a^n$
converges to $a$.
Or similarly find $f_n$ such that
$h_n = f_... |
How many unital magmas (magma with an identity element) with three elements are there (up to isomorphism)?
My approach:
- List out all of the possible 2x2 multiplication tables for the two non-identity elements. There are $3^{4} = 81$ of these.
- Extend these 81 tables by adding in the rows and columns for the... |
Are there 45 unital magmas with three elements (up to isomorphism)? |
The original question follows:
<h2>
Find the value of b for which the differential equation below is exact.
$$(ye^{7xy}+x)dx+(bxe^{7xy})dy=0$$
Then solve it using that value of b.
An implicit solution is fine.
<h2>
My attempt uses
$$M=ye^{7xy}+x\text{ and } N= bxe^{7xy}$$
$ \frac{M_y-N_x}{M}= -\frac{7y}{... |
I think you are mentioning a special case of linear bilevel programming, and this book could serve you as a starting point:
[A Gentle and Incomplete Introduction to Bilevel Optimization](https://optimization-online.org/?p=17182) by Yasmine Beck and Martin Schmidt.
Visit especially Section 6 for some algorithms design... |
The function $g\colon (0,\infty)\to \mathbb{R}$, defined by $g(x)=\frac{1}{x}$ sends an open interval to an open interval. The sine function $h\colon \mathbb{R}\to [-1,1]$, $h(x)=\operatorname{sin}x$ is an open map.
So I think that the composition $f\colon (0,\infty)\to [-1,1]$ is an open map, but I found a statemen... |
Why is the function $f(x) = sin (1/x)$ not an open map? |
You already know that $$a_n-a_{n-1}=\left(\frac23\right)^n + n-\left(\frac23\right)^{n-1} - (n-1)=-\frac12\left(\frac23\right)^n+1$$ but, as explained about an answer to a previous question of yours, this is not a recursion relation since the RHS still depends on $n$. To remedy this, introduce $$b_n=a_n-a_{n-1}$$ If on... |
In the same spirit as [this question][1], I would like to prove explicitely that if $R$ is a domain which is not a UFD, then it is not a PID.
I am interested in the case where there is an element $a\in R$ which has at least two non equivalent decompositions.
Without any loss of generality, we may assume that $a=... |
$f(x) =\sin (1/x)$. Why is the function $f\colon (0,\infty)\to [-1,1]$ not an open map? |
The function $g\colon (0,\infty)\to \mathbb{R}$, defined by $g(x)=\frac{1}{x}$ sends an open interval to an open interval. The sine function $h\colon \mathbb{R}\to [-1,1]$, $h(x)=\operatorname{sin}x$ is an open map.
So I think that the composition $f\colon (0,\infty)\to [-1,1]$, $f(x)=\operatorname{sin}(\frac{1}{x})... |
$f(x) =\sin (1/x)$. Why is the function $f\colon (0,\infty)\to [-1,1]$ not an open map? |
(Related to [this question][1].)
Suppose that $H$ is a finite subgroup, and $g$ is an element of a group. How large can the intersection $gH\cap Hg^{-1}$ be given that $gH\ne Hg^{-1}$? Is it possible to have, say, $\frac{1}{2}|H|<|gH\cap Hg^{-1}|<|H|$?
The abelian case is easy: if the group is commutative, then $... |
Let $f: \mathbb{R}^3 \rightarrow \mathbb{R}^3 $ be defined as
$$
(u, v, w)=f(x,y,z) = (e^{2y} + e^{2z}, e^{2x} - e^{2z}, x - y)
$$
I have shown that it has differentiable inverse in a neighborhood of every point in $\mathbb{R}^3$. I want to compute
$$
\frac{\partial x}{\partial u}, \frac{\partial x}{\partial v},... |
Let $d\in\mathbb{Z}$ be an integer which is not a square (it does not have to be squarefree, though).
**Question.** Assume that $\vert d\vert\geq 3$ to avoid special cases. Is is true that $\pi=-1+\sqrt{d}$ is irreducible in $R=\mathbb{Z}[\sqrt{d}]$ ?
**Remark.** I can show this it is true un the following cases:... |
If $\vert d\vert\geq 3$, is $-1+\sqrt{d}$ an irreducible element of $\mathbb{Z}[\sqrt{d}]$? |
Since the Completeness theorem is equivalent to the ultrafilter Lemma, which implies Hahn-Banach, the implication is clear to me. I was more just wondering if there is a nice direct proof? I saw in another thread that Completeness very nicely proves the existence and uniqueness of Algebraic closures, so I thought maybe... |
**I HAVE SOLVED THE EXERCISE. I'M LIKELY TO WRITE AN ANSWER LATER**
The original question follows:
<h2>
Find the value of b for which the differential equation below is exact.
$$(ye^{7xy}+x)dx+(bxe^{7xy})dy=0$$
Then solve it using that value of b.
An implicit solution is fine.
<h2>
My attempt uses
$$M... |
So, there is apparently a result by Łojasiewicz (in some lecture notes of his: "Ensembles semi-analytiques", published in 1965) that states the following:
> Let $X \subset M$ be a subanalytic subset of an analytic manifold. Then, $X$ is semianalytic of dimension $\leq k$ if and only if **locally** in $M$, there is ... |
>Let $P(B_1) = 0.5$, $P(B_2)=0.6$, $P(A|B_1) = 0.5$, $P(A|B_2)=0.4$.
>Then: (put '?' if can't be calculated)
>
>a) $P(A \cap B_1) = $
>
>b) $P(\Omega - A | B_2) = $
>
>c) $P(B_1|A)=$
For a) I have $P(A \cap B_1) = P(B_1)P(A|B_1) = \frac{1}{2}\cdot\frac{1}{2}=\frac{1}{4}$.
In b) I have $P(\Omega - A | B_... |
> Given natural numbers $x,y,z$ such that it satisfies $x^{2}y+y^{2}z+z^{2}x-23=xy^{2}+yz^{2}+zx^{2}-25=3xyz$. Find the possible maximum value of $x+y+z$.
For the above problem I've tried using the identity $x^{3}+y^{3}+z^{3}-3xyz=(x+y+z)(x^{2}+y^{2}+z^{2}-xy-xz-yz)$ but ultimately lead to nowhere... (same thing wit... |
I want to correctly prove the following statement:
\begin{equation}
\lim_{x \rightarrow \infty} \frac{1}{x \cdot \ln\left(\frac{x + a}{x - a}\right)} = \frac{1}{2a} \ , \ a \in \mathbb{R}^{\ast+}
\end{equation}
This should atleast be true for $a \in \mathbb{R}^{\ast+}$, but I have no idea how to correctly prov... |
Let $d\in\mathbb{Z}$ be an integer which is not a square (it does not have to be squarefree, though).
**Question.** Assume that $\vert d\vert\geq 3$ to avoid special cases. Is is true that $\pi=-1+\sqrt{d}$ is irreducible in $R=\mathbb{Z}[\sqrt{d}]$ ?
**Remark.** I can show this it is true un the following cases:... |
$\newcommand{\Pj}{\mathbb{P}}$I am trying to show that if $k$ is an infinite field, then for any $r$-dimensional closed subset $X\subset\Pj^n_k$ there are $r$ hyperplanes (i.e., $V(f_i)$ for linear homogeneous polynomials $f_i\in k[x_0,\dots,x_n]$) such that $X\cap\bigcap_{i=1}^r V(f_i)$ consists of finitely many point... |
Let $f,g,h$ be real functions and $F$ be a probability measure. Suppose that the following holds:
$$
\lim_{t\to\infty}\frac{h(t)}{\int_x f(x,t) dF(x)} = 1,
$$
$$
\forall x: \lim_{t\to\infty}\frac{f(x,t)}{g(x,t)} = 1.
$$
Under what assumptions does it hold that
$$
\lim_{t\to\infty}\frac{h(t)}{\int_x g(x... |
When can we interchange functions inside the integral? |
The following is problem 13 [here][1].
>Consider functions in $\mathcal{C}^2[0,1]$ and $a,b>0$. In this case, if we define:
>$$\lVert f \rVert:=\lVert f \rVert_\infty+ a \lVert f' \rVert_\infty +b \lVert f'' \rVert$$
>This norm makes $\mathcal{C}^2[0,1]$ into a Banach algebra if and only if $a^2\geq b$
I hav... |
Consider this particular instance for the $0/0$ case of the Stolz-Cesaro Theorem: $$\lim_{n \to \infty} a_n = \lim_{n \to \infty} b_n = 0$$ with $a_n$ and $b_n$ two sequences of real numbers, $b_n$ strictly monotone, whereby $$\lim_{n \to \infty} \frac{a_{n+1} - a_{n}}{b_{n+1} - b_{n}} = 0$$ then also $$\lim_{n \to \i... |
Hint:
1. What happens to the matrix if you replace basis $\{v_1, v_2, \ldots, v_n\}$ with let's say $\{\alpha v_1, v_2, \ldots, v_n\}$ for $\alpha \not= 0$?
2. What happens if you re-order the basis? In particular, let's say you exchange $v_i$ and $v_j$? |
We have:
$X$~$N\left(\mu,\sigma^2\right)$
We know that $\eta\ge0$ and thus I need to find $E\left[e^{\eta X}\right]$.
So what I thought of doing:
$$E\left[e^{\eta X}\right]=E\left[e^{\left(-j\right)j\eta X}\right]=\Phi _{_X}\left(-j\eta \right)=e^{j\mu \omega -\frac{1}{2}\sigma ^2\omega ^2}|_{_{_{\omega =-... |
Finding characteristic function of normal distribution for positive numbers only $X$~$N\left(\mu,\sigma ^2\right),E\left[e^{\eta X}\right]=?$ |
The following is problem 13 [here][1].
>Consider functions in $\mathcal{C}^2[0,1]$ and $a,b>0$. In this case, if we define:
>$$\lVert f \rVert:=\lVert f \rVert_\infty+ a \lVert f' \rVert_\infty +b \lVert f'' \rVert$$
>This norm makes $\mathcal{C}^2[0,1]$ into a Banach algebra if and only if $a^2\geq 2 b$
I h... |
I've recently become a little obsessed with all the ways limits can be composed - and the preconditions for this to take place. I took $A,B_{1},B_{2},C \subseteq \mathbb{R}$ and $f:A\to B_{1},\ g:B_{2} \to C$ and further assumed $f[A] \subseteq B_{2}$. And then I listed out all the ways to compose $f$ and $g$ with limi... |
What are the conditions to compose limits to infinity? |
Assume M is 2 state- ergodic Markov chain - i.e it's irreducible and aperiodic, with Stochastic matrix $Q \in \mathbb{R}^2 $ .<br /> Then $Q(1,1) = (1,1)$. Hence $Q(\alpha,\alpha) = (\alpha,\alpha)$. <br /> Now, since the stationary distribution (under these settings it exists and it's unique) holds that it's a probabi... |
Why $(0.5,0.5)$ isn't stationary distribution for every 2 state ergodic Markov chain? |
**Hint** $\, $ The *key idea* is that any *periodicity* of the exponential map $\,n\mapsto a^n\,$ allows us to use **modular order reduction** on *exponents* as in the results below. We can find *small* periods $\,e\,$ such that $\,a^{\large e}\equiv 1\,$ either by [Euler's totient](https://en.wikipedia.org/wiki/Euler'... |
In three dimensions, a multivector consists of a scalar, a vector, a bivector and a tri-vector. Is there a term that generalizes these names? For example, in an $n$-dimensional space, can I use the terminology $k$-vector (where $0\leq k\leq n$) to refer to the different scalar and vector types, such that a 0-vector is ... |
Is there a single term that generalizes the names of the individual vector and scalar types in a multivector? |
I have this equation, seen from a video of facebook where there is a complicate solution (I remember that there is a specific website),
$$(2n)^{\log 2}=(5n)^{\log 5}$$
I have adopted this approach:
$$(2)^{\log 2}(n)^{\log 2}=(5)^{\log 5}(n)^{\log 5} \iff 2n^{\log 2}=5n^{\log 5}$$
After
$$\frac{n^{\log 2}}... |
Let $\mathfrak{g}$ be a Lie algebra. How can I show that $H^1(\mathfrak{g})\cong \mathfrak{g}/[\mathfrak{g},\mathfrak{g}]$, the first real cohomology space of $\mathfrak{g}$?
My try: Let $c:\mathfrak{g}\to \mathbb{R}$ be a one-cocycle. Then $c$ is a linear map such that $c([X,Y])=0$ for all Lie algebra elements $X,Y... |
How to show that $H^1 (\mathfrak{g})\cong \mathfrak{g}/[\mathfrak{g},\mathfrak{g}]$ for a Lie algebra $\mathfrak{g}$? |
In three dimensions, a multivector consists of a scalar, a vector, a bivector and a tri-vector. Is there a term that generalizes these names? For example, in an $n$-dimensional space, can I use the terminology $k$-vector (where $0\leq k\leq n$) to refer to the different scalar and vector types, such that a 0-vector is ... |
I wish to consider a principal bundle which has a fibre $F$ of spatial $d$-dimensional momenta over a base manifold of $\mathbb{R}$. So, naturally the momenta $k^i$ is the generator of the structure group which is the group of translations with the Lie algebra $[k^i, k^j] = 0$. Also, the generators of this group are al... |
In the last test I took, there was a question:
>Provide an example of a discrete random variable that cannot be modeled in the classical probability model.
I know that the classical probability model means that all elementary events are equally probable. Hence, since a random variable is a function from $\Omega$ t... |
Suppose that $\Omega$ is a set, $(\Omega, \mathscr{G})$ is a measure space, and $Z: \Omega \to \mathbb{R}$ is a given mapping. Then Z is $\mathscr{G}$ measurable iff $$Z =\displaystyle\sum_{i=1}^\infty \lambda_i I_{A_i} \tag{1}$$ for some {$\lambda$} $\subset \mathbb{R}$ and {$A_i$} $\subset \mathscr{G}$. From(1) ded... |
>Cone: A subset $ S \subseteq \mathbb{R}^n$ is a cone if $\alpha \geq 0 \implies \alpha S \subseteq S.$
>Polar: A Polar $K^*$ of a cone $K$ is a closed convex cone such that
$$K^*=\{y \in \mathbb{R}^n \mid x\in K \implies (x,y) \geq 0\}$$
where $(x,y)$ denote the standard inner product of vectors.
**Problem** I... |
I wish to consider a principal bundle which has a fibre $F$ of spatial $d$-dimensional momenta over a base manifold of $\mathbb{R}$. So, naturally the momenta $k^i$ is the generator of the structure group which is the group of translations with the Lie algebra $[k^i, k^j] = 0$. Also, the generators of this group are al... |
For example, if I have a set of all permutations of the set {1, 2, 3, 4} and the permutation 1, 4, 3, 2 how do i calculate which element of the set it is without calculating all the permutations? |
Someone told me recently that the category of modules over a Frobenius algebra is a Frobenius category. Where can I find a reference for a proof of this?
Since the category of modules must be an abelian category with enough projectives and enough injectives, this simply amounts to showing that the projectives and i... |
I have this equation, seen from a video of facebook where there is a complicate solution (I remember that there is a specific website),
$$(2n)^{\log 2}=(5n)^{\log 5}$$
I have adopted this approach:
$$(2)^{\log 2}(n)^{\log 2}=(5)^{\log 5}(n)^{\log 5} \iff 2n^{\log 2}=5n^{\log 5}$$
After
$$\frac{n^{\log 2}}... |
Let ABCD be a convex quadrilateral, point E on BC, F and G different points on CD such that (GFC), and H is a point on AD. Prove that the lengths EG and FH intersect.
My attempt was to look at triangle GCE and line HF and to use Pasch axiom on them. To use that axiom first i need to check that line HF doesn't pass t... |
Let ABCD be a convex quadrilateral, point E on BC, F and G different points on CD such that (GFC), and H is a point on AD. Prove that the lengths EG and FH intersect.
[sketch][1]
My attempt was to look at triangle GCE and line HF and to use Pasch axiom on them. To use that axiom first i need to check that line HF... |
I have this equation from a mathematical challenge in Germany, seen from a video of facebook where there is a complicate solution (I remember that there is a specific website),
$$(2n)^{\log 2}=(5n)^{\log 5}$$
I have adopted this approach:
$$(2)^{\log 2}(n)^{\log 2}=(5)^{\log 5}(n)^{\log 5} \iff 2n^{\log 2}=5n^{... |
I'm not sure if this answers your question. If it doesn't, consider it an extended comment.
Definitions are -well- a matter of definition and thus cannot be wrong or true, but they can certainly be weird or counterintuitive.
And I want to suggest that the definitions you provided for the left/right/two-sided idea... |
A set $\Gamma$ of propositional logic formulae over $(P, C)$ is said to be finitely satisfiable if and only if every finite subset $\Gamma_{0} \subset_{\text{fin}} \Gamma$ of $\Gamma$ is satisfiable.
Let $\Gamma$ be a set of propositional logic formulae over $(P, C)$. $\Gamma$ is said to be $P$-consistent if for ev... |
Let $p$ be a prime number and let $m$ be a positive integer. We know that $\beta:=p^{-1/m}$ is an $m$-degree algebraic number (in fact, its minimal polynomial is $pX^m-1$).
I would like to prove that the algebraic number
$$
\beta^{1!}+\beta^{2!}+\beta^{3!}+\cdots+\beta^{(m-1)!}
$$
has still degree $m$. Clearly, ... |
Consider the so-called Cox-Ingersoll-Process model
\begin{equation}
dr_t=a(b-r_t)dt+\sigma \sqrt{r_t}dW_t
\end{equation}
It can be shown (Wikipedia), that the mean of this process is
\begin{equation}
\langle r_t \rangle = r_0 e^{-at }+ b(1-e^{-at})^2
\end{equation}
for some initial value $r_0$.
... |
I'm reading Miroslav Repicky's paper "A proof of the independence of the Axiom of Choice from the Boolean Prime Ideal Theorem" and I'm confused by the claim that an ideal $I$ on a Boolean algebra $B$ being prime in $OD^{V[G]}[A,f\frown h]$ implies that it must be a maximal ideal of $B$ in $OD^{V[G]}[A,f\frown h]$. For ... |
Let ABCD be a convex quadrilateral, point E on BC, F and G different points on CD such that (GFC), and H is a point on AD. Prove that the lengths EG and FH intersect.
[![sketch][1]][1]
My attempt was to look at triangle GCE and line HF and to use Pasch axiom on them. To use that axiom first i need to check that l... |
I have been trying to experiment with curves. Recently I found a curve (in $\mathbb{R}^2$) with unbounded curvature.
However, I am not able to find one curve with unbounded torsion, for various reasons:
- It has to be in $\mathbb{R}^3$, which makes things more complicated
- Torsion is not as easy to calculate ... |
First of all, I know that there is an extremely similar question from yesterday that has been closed due to Mathematics Stack Exchange guidelines, so I can't comment and find what is incorrect in my way.
The question : find the coeffient of $(x^3*y^4*z)$ in $ (x+y+z)^5 *(1+x+y+z)^{5}$.
We know that: $(x+y+z)^5$ =... |
How to find the coeffient of $x^3*y^4*z$ in $ (x+y+z)^5 *(1+x+y+z)^{5}$? |
Someone told me recently that the category of modules over a Frobenius algebra is a Frobenius category. Where can I find a reference for a proof of this?
Since the category of modules must be an abelian category (which is exact with respect to the class of short exact sequences) with enough projectives and enough i... |
First of all, I know that there is an extremely similar question from yesterday that has been closed due to Mathematics Stack Exchange guidelines, so I can't comment and find what is incorrect in my way.
The question : find the coeffient of $(x^3*y^4*z)$ in $ (x+y+z)^5 *(1+x+y+z)^{5}$.
We know that: $(x+y+z)^5$ =... |
Let $x,y,z\in\mathbb{R}^{*}$ s.t. $xy,yz,xz\in \mathbb{Q}$. If $x^3+y^3+z^3\in \mathbb{Q}$ then $x,y,z\in \mathbb{Q}$ .
My approach:
I notice that $x^2=\frac{xy\cdot xz}{yz}\in\mathbb{Q}$.
Also $y^2, z^2\in \mathbb{Q}$.
Then $(x+y+z)(x^3+y^3+z^3)\in\mathbb{Q},$ so $x+y+z\in\mathbb{Q}$ if $x^3+y^3+z^3\neq 0$. ... |
Let $A$ be an $n\times n$ matrix which is similar to $D=diag(1,0,\ldots,0)$ and Let $P$ be the invertible matrix such that $P^{-1}AP=D$. What is the vector
$$
A
\begin{pmatrix}
1\\1\\ \vdots \\1
\end{pmatrix}=?
$$
**My attempt**: If the matrix is $2\times2$ and
$$
P=
\begin{pmatrix}
a & b\\ c & d\\
\end{p... |
First of all, I know that there is an extremely similar question from yesterday that has been closed due to Mathematics Stack Exchange guidelines, so I can't comment and find what is incorrect in my way.
The question : find the coefficient of $(x^3*y^4*z)$ in $ (x+y+z)^5 *(1+x+y+z)^{5}$.
We know that: $(x+y+z)^5$... |
How to find the coefficient of $x^3*y^4*z$ in $ (x+y+z)^5 *(1+x+y+z)^{5}$? |
For the following question, is my proof correct? Can I write it any better?
> Let $(X,d)$ be a metric space and $\gamma:[0,1] \rightarrow X$ be a continuous function such that $\gamma(0) \ne \gamma(1)$. Let $f: X \rightarrow \mathbb{R}$ be a continuous functions fuch that $f(\gamma(0)) = 1$ and $f(\gamma(1)) = 0$. P... |
Let $G$ be a group action on the set $X$, and let $f:(X,\Sigma_X)\to (Y,\Sigma_Y)$ be a measurable maximal invariant ($f$ is constant on each orbit and take different values on different orbits). Am asked to show that if $(X,\Sigma_X)$ and $(Y,\Sigma_Y)$ are standard Borel then
$$\sigma(f)=\{B\in \Sigma_X:g^{-1}(B)=... |
For the following question, is my proof correct? Can I write it any better?
> Let $(X,d)$ be a metric space and $\gamma:[0,1] \rightarrow X$ be a continuous function such that $\gamma(0) \ne \gamma(1)$. Let $f: X \rightarrow \mathbb{R}$ be a continuous function such that $f(\gamma(0)) = 1$ and $f(\gamma(1)) = 0$. Pr... |
Are you sure that is the correct question?
[![enter image description here][1]][1]
[1]: https://i.stack.imgur.com/gEfis.jpg
if you look at this picture, I take $f(x)=\frac{3^n-2^n}{3^n+2^n}+\frac{5^n-3^n}{5^n+3^n} - \frac{5^n-2^n}{5^n+2^n}$ and I look for $f(x) \le 0 $ so that satisfy the $\frac{3^n-2^n}{... |
I can't speak to your professors intent but, generally it is best practice when introduced to a subject to either write completely algebraic sentences or complete prose. This is because it is often not entirely obvious when to use which one, and so switching between them can create awkward sentences that are repetitive... |
Let $R_1$, $R_2 \in \mathbb R^{N \times M}$ with $M > N$ be matrices of rank $N-1$. When can we find a symmetric matrix $P\in \mathbb R^{M\times M}$ such that the matrix equation
$$ R_1 P = R_2, $$
is satisfied? When does a solution exist? I was thinking that an approach could be rewriting the equation by vecot... |
I am working on other proofs, namely for so-called "quasi-increasing" but non-monotonic sequences.
But, my question: is the following a sufficient definition of non-monotonic?
> For all $N \in \mathbb{N}$ where $l>n>m \ge N$ there is $a_n \le a_m$ and $a_l \ge a_m$
Namely, there are always terms with preceding... |
Are you sure that is the correct question?
[![enter image description here][1]][1]
if you look at this picture, I take $f(x)=\frac{3^n-2^n}{3^n+2^n}+\frac{5^n-3^n}{5^n+3^n} - \frac{5^n-2^n}{5^n+2^n}$ and I look for $f(x) \le 0 $ so that satisfy the $\frac{3^n-2^n}{3^n+2^n}+\frac{5^n-3^n}{5^n+3^n} \leq \frac{5^n-... |
I've recently become a little obsessed with all the ways limits can be composed - and the preconditions for this to take place. I took $A,B_{1},B_{2},C \subseteq \mathbb{R}$ and $f:A\to B_{1},\ g:B_{2} \to C$ and further assumed $f[A] \subseteq B_{2}$. And then I listed out all the ways to compose $f$ and $g$ with limi... |
Is there other elementary proofs that if $m^4 + 4^n$ is prime number then $m$ is odd and $n $ is even? |
Are you sure that is the correct question?
[![enter image description here][1]][1]
if you look at this picture, I take $f(x)=\frac{3^n-2^n}{3^n+2^n}+\frac{5^n-3^n}{5^n+3^n} - \frac{5^n-2^n}{5^n+2^n}$ and I look for $f(x) \le 0 $ so that satisfy the $\frac{3^n-2^n}{3^n+2^n}+\frac{5^n-3^n}{5^n+3^n} \leq \frac{5^n-... |
There is a statement about locally convex spaces in Reed & Simon, Methods of Modern Mathematical Physics (Vol I, Section V.1) that is given without a proof.
The statement is:
Given two families of seminorms $\{\rho_\alpha\}_{\alpha \in A}$ and $\{d_\beta\}_{\beta \in B}$ over a locally convex space X, if the fami... |
I am working on other proofs, namely for so-called "quasi-increasing" but non-monotonic sequences.
But, my question: is the following a sufficient definition of non-monotonic?
> For all $N \in \mathbb{N}$ where $l>n>m \ge N$ there is $a_n \le a_m$ and $a_l \ge a_n$
Namely, there are always terms with preceding... |
Let $H\le G$ be finite and $g\in G$. How large can $gH\cap Hg^{-1}$ be if $gH\ne Hg^{-1}$? Is $\frac{1}{2}|H|<|gH\cap Hg^{-1}|<|H|$ possible? |
Another partial answer based on numerical evidence. It is easy to note that your numbers can be written as
$$
\frac{a_n}{4^n}
$$
where
$$
a_n=
1,5,22,91,370,1491,5986,23986,96028,384277,1537438,6150406,\dots\tag{*}
$$
After playing a bit with the number one finds:
$$
a_1=1;\quad a_n=4a_{n-1}+\xi_n,\tag1
$... |
Let $A$ be a abelian group and $B$ a subgroup of $A$.
Is $Aut_B(A)\to Aut(A/B)$ surjective?
Where $Aut_B(A)=\{\tau\in Aut(A)|\tau(B)=B\}$ |
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