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To get the answer $\frac{7}{10}\cdot\frac{1}{10}$, the new question can be: "What is the probability that an irregular student scores an A?".
In this case you would have these calculations
$$P(\text{Grade A and No 100%})=P(\text{No 100%})\cdot P(\text{Grade A}|\text{No 100%})=\frac{7}{10}\cdot\frac{1}{10}$$
("$\te... |
For a group $G$, let $G'$ denote the commutator of the group $G$, and if $H \leq G$ the left cosets will be denoted as $gH$. Now, I understand the fact that $[SL_2(\mathbb{Z}):GL_2(\mathbb{Z})'] = 2.$
My question is:
>Does there exist an element of order $4$ in $GL_2(\mathbb{Z})/GL_2(\mathbb{Z})'$?
What I ha... |
Does there exist an element of order $4$ in $GL_2(\mathbb{Z})/GL_2(\mathbb{Z})'$? |
True, high entropy means high information, or high uncertainty.
> How do we reconcile these two notions?
Think of $X$ as the data, or observed value. Then, to say
that **before knowing $X$** we had much uncertainty
is equivalent to say
that **after knowing $X$** we have gained much information.
By contras... |
Here is a different approach. Rewrite the equation as $$|\cos x| = -\sin 3x \implies \cos^2 x = \sin^23x$$ so that $$\sin^2x = 1-(3\sin x - 4\sin^3 x)^2$$ ie $$16t^3-24t^2+10t-1=0 \iff (2t-1)(8t^2-8t+1)=0\iff (t-\sin^2(\pi/4))(t-\sin^2(\pi/8))(t-\sin^2(3\pi/8))=0$$ where $t:= \sin^2 x$ and since $\sin^2(\pi/8)+\sin^2(3... |
Convergence of Pólya urns: what does it mean "expanding real polynomials in a basis"? |
I'm reading these [notes][1], and in the proof of this **Proposition 2**, which states:
> Let $d\ge 2$ and $S\ge 1$ be integers. Let also $(\alpha_1,\dots,\alpha_d)\in\mathbb{N}^d \setminus\{ 0 \}$. Let $(P_n)_{n\ge0}$ be the $d$-color Pólya urn random process having $S\cdot Id$ as replacement matrix and $(\alpha_... |
Here is a different approach. Rewrite the equation as $$|\cos x| = -\sin 3x \implies \cos^2 x = \sin^23x$$ so that $$\sin^2x = 1-(3\sin x - 4\sin^3 x)^2$$ ie $$16t^3-24t^2+10t-1=0 \iff (2t-1)(8t^2-8t+1)=0\iff (t-\sin^2(\pi/4))(t-\sin^2(\pi/8))(t-\sin^2(3\pi/8))=0$$ where $t:= \sin^2 x$ and since $\sin^2(\pi/8)+\sin^2(3... |
By your logic, the matrix $[1]$ is not invertible, since $[1]$ is the product of the matrix $[1, 0]$ with the matrix $\begin{bmatrix}1\\0\end{bmatrix}$.
The statement you are trying to use is:
> If $A$ and $B$ are inverible square matrices, then $C=A\cdot B$ is
> invertible, and $C^{-1} = B^{-1}\cdot A^{-1}$.... |
Setting
$$S=a+b, \ \ P=ab$$
with the classical inequality (Arith. Mean vs. Geom. Mean)
$$S \ge 2\sqrt{P}\tag{1}$$
your relationship, which can be written :
$$8(1-ab)=2^{a+b+2ab}(a+b)$$
becomes :
$$8(1-P)=2^{S+2P}S$$
(1) implies :
$$8(1-P) \ge 2^{2\sqrt{P}+2P}2 \sqrt{P}$$
$$\underbrace{4(1... |
a, b and c are 3 Independent Vectors. We can generate Orthonormal basis using those 3 vectors using Gram-Schmidt method. Lets say those 3 orthogonal basis generated from a, b and c are $A$, $B$ and $C$. We can generate orthonormal basis by dividing those orthogonal basis with their length.
I am trying to figure out ... |
Are Orthogonal Basis perpendicular to Original Basis in Gram-Schmidt? |
how to simplify this expression so that I get an expression only?
$\sum _{t=1}^{\infty }\:\frac{\left(1+g\right)^{ \lceil \frac{t}{2}\rceil}}{\left(1+k\right)^t}$
I know that for t = 1, $\lceil \frac{t}{2}\rceil$ = 1
for t = 2, $\lceil \frac{2}{2}\rceil$ = 1
for t = 3, $\lceil \frac{3}{2}\rceil$ = 2
... |
How to simplify this expression with a ceil symbol and summation? |
We consider the uniparametric differential equation
$$\frac{dx}{dt} = \left(\lambda - x^2 + 2x\right)\left(\lambda - x^2\right).$$
I want to determine the equilibrium points as well as their stability depending on $\lambda$.
I know that (correct me if I'm wrong) if $\lambda < -1$ then there don't exist equilibrium... |
How could I determine the stability of these equilibrium points if $\lambda > 0$? |
The question Equation 1 (is repeated here to easier reference) is:
$$
\left(2 n \right)^{\log 2} = \left( 5 n \right)^{\log 5}
\tag{Eq. 1}$$
Then,
$$\begin{align*}
\left(2 \right)^{\log 2}\left(n \right)^{\log 2} &= \left( 5 \right)^{\log 5} \left( n \right)^{\log 5}
\tag{Eq. 2}\\
\frac{\left(n \right)^{\log ... |
Let us take notations
$$S=a+b, \ \ P=ab$$
keeping in mind the classical inequality (Arith. Mean vs. Geom. Mean)
$$S \ge 2\sqrt{P}\tag{1}$$
With these notations, your relationship, which can be written :
$$8(1-ab)=2^{a+b+2ab}(a+b)$$
becomes :
$$8(1-P)=2^{S+2P}S$$
(1) implies :
$$8(1-P) \ge 2^... |
Could someone please tell what results are known about the zeros of L function, $L(s,\chi)$ on the 1/2 line, where $\chi$ is a character mod $q$?
Is there an upper bound for this count when we count zeros with $|\Im s|\leq T?$
In particular, what is known about zeta function? I know there are some explicit versi... |
Say that I have 4 people and 5 seats. How many unique ways are there of arranging these 4 people into the five seats?
In questions like this I would use this formula:
$$\frac{n!}{(n-k)!}$$
However this would give me -1! in the denominator. How do I interpret this?
$$\frac{4!}{(-1)!}$$
|
Please help me prove the following equation:
$$\lim_{n\to \infty} \int_{0}^{1} \frac {\cos(x^n)}{1+x^n}\,dx=1.$$
I think I need to prove that the integrand uniformly converges to 1 (besides the endpoint) and then use a theorem but I don't know how.
Any ideas? |
I have two functions $f_a(x) = y_a$ and $f_t(x) = y_t$, where the dimension of the domain is larger than range.
Finally, I want to compute the linearised mapping from $y_t$ to $y_a$ via ${}^t J_a = \frac{\partial y_a}{\partial y_t}$.
I can not compute this directly but can compute the Jacobians $J_a= \frac{\par... |
how to simplify this expression so that I get an expression only?
$$\sum _{t=1}^{\infty }\:\frac{\left(1+g\right)^{ \lceil t/2\rceil}}{\left(1+k\right)^t}$$
I know that for t = 1, $\lceil \frac{t}{2}\rceil$ = 1
for t = 2, $\lceil \frac{2}{2}\rceil$ = 1
for t = 3, $\lceil \frac{3}{2}\rceil$ = 2
for t ... |
Let $[X_1, X_2, ..., X_r]$ be a set of independent $d_1 \times d_2$ dimensional random matrices with $\mathbb{E}(X_i) = 0$ and $\|X_i\| \leq B$ (bounded operator norm).
Introduce the sum of random matrices, $Z = \sum_{i=1}^r X_i$
Define matrix variance proxy:
$$\sigma^2 = \max \left( \| \mathbb{E}(ZZ^T)\|, \|\... |
Let $M$ be a smooth manifold (without boundary). Suppose $F:M \times I \rightarrow M$ is a smooth isotopy between two diffeomorphisms $f_0$ and $f_1$ of M. Then $F(\cdot,t):M\rightarrow M$ is an embedding. My question is can we find a new isotopy $\tilde{F}$ between $f_0$ and $f_1$ such that $\forall t \in I,\tilde{F}(... |
Can an isotopy between diffeomorphisms be through diffeomorphism? |
The answer is no, and we give two counter-examples below: one in $2$ and one in $3$ and more dimensions. The second example also appears in a slightly different form in [these old lecture notes of M. Wolf][1] (Example 8.1). However, in my opinion, this specific result is rather difficult to find if you're not already f... |
I am trying to find a "manageable" sufficient condition for a function $f\colon \mathbb{R} \to \mathbb{R}$ to satisfy
$$
|f^{(n)}(x)| \le \frac{Mn!}{(x-x_0)^{n+1}}\tag{1}
$$
for some $M \ge 0$, $x_0 \in \mathbb{R}$ and all $n \ge 0$, $x > x_0$.
By Widder's theorem, it follows that for $x_0 = 0$ the above conditi... |
Is a morphism $f:A\rightarrow B$ uniquely determined by the compositions $f\circ g$ for all $g:T\rightarrow A$? |
If you ever need to consult a knight or knave (call them $K$) to figure out whether something $P$ is the case or not, you can ask them the following:
"Would you say that $P$ is true?"
It turns out that the answer to this question will be "yes" if $P$ is true, and "no" if $P$ is false, no matter whether $K$ is a k... |
Book:
https://www.amazon.com/Vector-Calculus-Dover-Books-Mathematics/dp/0486466205
We have this problem here on page 89.
[![pic1][1]][1]
I think I was able to solve a) and c).
I know that b) is an ellipse. And I think we can also assume that the parameters $a,b$ are positive.
But for b) I ... |
What are $r(s)$ and $\theta(s)$ for an ellipse? |
In an economic model I am working with I would like to establish that the difference of two special convex functions is convex. I know that in general the difference of convex functions need not be convex, see [here][1], but I was hoping someone can point out if the properties of my functions suffice or can provide a ... |
Is the difference of this two special convex functions convex? |
In 1899 E. Almansi proved that any polyharmonic function of order $p$, i.e. in the kernel of the differential operator $\Delta^p$, defined on a shar shaped domain $\Omega\subset\mathbb{R}^n$, can be written uniquely as a sum of $p$ harmonic function, i.e. if $\Delta^p u=0$, then $\exists!$ $f_0,...,f_{p-1}$ such that
... |
Book:
https://www.amazon.com/Vector-Calculus-Dover-Books-Mathematics/dp/0486466205
We have this problem here on page 89.
[![pic1][1]][1]
I think I was able to solve a) and c).
I know that b) is an ellipse. And I think we can also assume that the parameters $a,b$ are positive.
But for b) I ... |
In 1899 E. Almansi proved that any polyharmonic function of order $p$, i.e. in the kernel of the differential operator $\Delta^p$, defined on a shar shaped domain $\Omega\subset\mathbb{R}^n$, can be written uniquely as a sum of $p$ harmonic function, i.e. if $\Delta^p u=0$, then $\exists!$ $f_0,...,f_{p-1}$ such that
... |
Let $K$ be a compact set in $\mathbb{C}$. Is it true that there exists an element of maximum modulus in $K$ and that element belongs to the boundary of $K$. If yes, how? |
I warmly welcome an approach on differencing between two continuous fractions without solving for the fraction. I cannot find a way without completing the fraction into a single rational part, for example, given two prime integers whereby $\gcd(17,11)=1, \gcd(17,3)=1$
$$\frac{17}{11}=1+\frac{1}{1+\frac{1}{1+\frac{1}{5... |
I am learning about connections using Lee's Intro to Riemannian Manifolds, and I am confused regarding the coefficients of the pullback connection. For the setting, let $(M,g) $ be a Riemannian manifold and let $\nabla$ be a connection on $M$. From Prop 4.7 in Lee's book, if we have two local frames $(E_i)$ and $(\tild... |
Book:
https://www.amazon.com/Vector-Calculus-Dover-Books-Mathematics/dp/0486466205
We have this problem here on page 89.
[![pic1][1]][1]
I think I was able to solve a) and c).
I know that b) is an ellipse. And I think we can also assume that the parameters $a,b$ are positive.
But for b) I ... |
I am learning about connections using Lee's Intro to Riemannian Manifolds, and I am confused regarding the coefficients of the pullback connection. For the setting, let $(M,g) $ be a Riemannian manifold and let $\nabla$ be a connection on $M$. From Prop 4.7 in Lee's book, if we have two local frames $(E_i)$ and $(\tild... |
Book:
https://www.amazon.com/Vector-Calculus-Dover-Books-Mathematics/dp/0486466205
We have this problem here on page 89.
[![pic1][1]][1]
I think I was able to solve a) and c).
I know that b) is an ellipse. And I think we can also assume that the parameters $a,b$ are positive.
But for b) I ... |
Book:
https://www.amazon.com/Vector-Calculus-Dover-Books-Mathematics/dp/0486466205
We have this problem here on page 89.
[![pic1][1]][1]
I think I was able to solve a) and c).
I know that b) is an ellipse. And I think we can also assume that the parameters $a,b$ are positive.
But for b) I ... |
I'm currently working on solving a partial differential equation (PDE) using the method of characteristics and have encountered a conceptual challenge regarding the independence of first integrals (constant solutions). The PDE in question is:
$$u_x(x+z) + u_y \cdot y + u_z(z+ y^2) = 0$$
To solve this, I've derived ... |
The identity function is an injection $[0,1]\setminus X\to [0,1]$. If we can also find an injection $[0,1]\to [0,1]\setminus X$, we know that there is a bijection by the [Schroeder-Bernstein Theorem][1]. We can construct such an injection by sending $0$ to $0$ and "compressing" each interval $\left(\frac{1}{n+1},\frac{... |
Let's state a general zero-counting form for a general $L$-function. Take an $L$-function with the functional equation
$$ \Lambda(s) := L(s) Q^s \prod_{\nu = 1}^N \Gamma( \alpha_\nu + \beta_\nu )
= \omega \overline{\Lambda(1 - \overline{s})}.$$
Define the *degree* of $L$ to be
$$ d_L := 2 \sum_{\nu} \alpha_\n... |
Book:
https://www.amazon.com/Vector-Calculus-Dover-Books-Mathematics/dp/0486466205
We have this problem here on page 89.
[![pic1][1]][1]
I think I was able to solve a) and c).
I know that b) is an ellipse. And I think we can also assume that the parameters $a,b$ are positive.
But for b) I ... |
For the past few days I've been studying the topic of functional analysis. I was making an exercise in which I needed to prove the following identity
$$G\Big(\frac{\partial}{\partial b}\Big)F(b) = F\Big(\frac{\partial}{\partial x}\Big)[G(x)e^{xb}]|_{x=0},$$
in which we can assume $F$ and $G$ can be expanded in a conv... |
This is related to a question I posted before: https://math.stackexchange.com/q/4872424/1295158.
Since that question has not received a reply, here I post a simplified version with the $N$ random variables reduced to one random variable. This may help me or other people to think about that question.
Let function $f... |
Every elliptic curve over a field $K$ can be mapped to a smooth, projective genus 1 curve (also defined over $K$) with a $K$-rational point, and vice versa. As I understand it, curves *without* said $K$-rational points can *never* be mapped to elliptic curves (since clearly then the elliptic curve group law cannot be e... |
Motivated by [this question][1], I ask the question below.
Let $S\subset M$ be a closed in $M$ but not necessary compact, embedded submanifold of a complete Riemannian manifold $M.$ Assume that for each $p \in M$, there exist a **unique** closest point on $S$, i.e a unique point $\tilde s(p) \in S$ such that $d(p,\t... |
Let function $f:[0,1]^{N+1} \mapsto \mathbb{R}$ be continuously differentiable.
Let $Y_1,\cdots,Y_{N}$ be random variables with support on $[0,1]$ with continuously differentiable joint PDF.
$$g(x):= \mathbb{P}(f(x,Y_1,\cdots,Y_{N})<0).$$
There is $y^* \in (0,1)$ such that for $Y_1\gtrless y^*$,
$$
f(0,Y_1,... |
In my answer to a recent
[MSE question](https://math.stackexchange.com/q/4875434/)
I mention the elliptic curve
[LMFDB 1728.n4](https://www.lmfdb.org/EllipticCurve/Q/1728a1)
with integer points $(-1,\pm1)$. The elliptic divisibility
sequence associated with these points is defined by initial values
and recursion
... |
The formula of RSI indicator (RSI = 100 - (100 / (1 + RS), RS = Average Gain / Average Loss), given the definition of division is to split an amount into equally-sized group.
Here, based on the definition of division, does it mean we are splitting Average Gain into equally-sized Average Loss group? I couldn't form ... |
What is the meaning of Division (Average Gain / Average Loss) in the RSI indicator? |
**I do not want to use index notation.**
I want to compute the derivative
$$
D_x (Axx^\top A) = ?
$$
where $A$ is an $n\times n$ symmetric matrix and $x$ in a vector in $\mathbb{R}^n$. I tried resources such as the matrix calculus cookbook but they don't deal with scenarios like this: Here the function $f(x) =... |
It is a problem found in Physics. From the edge of a (in distance r to the center) disk a (mathematical) pendulum is hanging. The disk spinns with a velocity w, which also stimulates the hanging pendulum.(As can be seen in the picutre below)
[![enter image description here][1]][1]
[![enter image description here][2]]... |
What is the distance between $\tau=(1,1.,..)$ and $c_0$? |
In Hatcher's Algebraic Topology on page $409$ in the third paragraph it is written
> Given a fibration $p : E \to B$ with fiber $F = p^{−1} (b_0 )$, we know that the
inclusion of $F$ into the homotopy fiber $F_p$ is a homotopy equivalence. Recall that $F_p$
consists of pairs $(e, \gamma)$ with $e \in E$ and $\gamma... |
Error in Hatcher, Algebraic Topology? |
> **Theorem.** Let $Z\sim N(0,1)$ and $Y\sim \chi^2(v)$ are two independent random variabels. Then, $$T = \frac{Z}{\sqrt{Y/v}} \sim
t-student$$ with degrees of freedom $v$.
Is it the converse always true? If $T\sim t-student$ with degrees of freedom $v$, then there exist two random variables $Z\sim N(0,1),Y\sim \c... |
In Hatcher's Algebraic Topology on page $409$ in the third paragraph it is written
> Given a fibration $p : E \to B$ with fiber $F = p^{−1} (b_0 )$, we know that the
inclusion of $F$ into the homotopy fiber $F_p$ is a homotopy equivalence. Recall that $F_p$
consists of pairs $(e, \gamma)$ with $e \in E$ and $\gamma... |
Motivated by [this question][1], I ask the question below.
Let $S\subset M$ be a closed in $M$ but not necessary compact, embedded submanifold of a complete Riemannian manifold $M.$ Assume that for each $p \in M$, there exist a **unique** closest point on $S$, i.e a unique point $\tilde s(p) \in S$ such that $d(p,\t... |
I read about the Johnson Lindenstrauss lemma, which (roughly) states that if $x \in \mathbb{R}^n$ and $A \in \mathbb{R}^{k \times n}$ is a "random matrix", then with high probability $\|x\|_2 \approx \|Ax\|_2$.
The proof goes by assuming that $A_{i,j}$ are i.i.d random variables with mean $0$ and variance $1$, and ... |
A structure of semilattice over T is the same thing than a structure of finite abelian monoid such that $\forall t \in T$, $t² = t$. \
Given a semilattice T, we get an abelian monoid by defining $a.b$ = $a \wedge b$ and conversely given an abelian monoid T such that $\forall t \in T$, $t² = t$, we get a semilattice by... |
Let
$$
f: \mathbb{R}^m \longrightarrow \mathbb{R}
$$
be a smooth, real-valued function. Then it holds that
$$
f(x)- f(0) = x\cdot \nabla f(0)+ x\ \mbox{Hess}f(0)\ x^T + \sum_{|\alpha|=3,\alpha\in\mathbb{N}^m} \frac{x^\alpha}{\alpha!} \int_{0}^{1}(1-t)^2\ {\partial^\alpha} f(tx)\ dt.
$$
Now if we assume tha... |
Motivated by [this question][1], I ask the question below.
Let $S\subset M$ be a closed in $M$ but not necessary compact, embedded submanifold of a complete Riemannian manifold $M.$ Assume that for each $p \in M$, there exist a **unique** closest point on $S$, i.e a unique point $\tilde s(p) \in S$ such that $d(p,\t... |
I have stumbled upon an interesting question whilst working on my thesis.
You are given a matrix of integers $A=(a_{i,j})$ with $1\leq i\leq k$ and $1\leq j \leq r$ and a tuple $(b_1, \dots, b_r)$ such that $\sum_{i=1}^r b_i=k$.
You have to calculate the lexicographic maximal string $(a_{i_1, j_1}, a_{i_2, j_2}, ... |
I am working on a special video editing task and I have faced some math problem
I have video that I want to speed up for example to 200%, and to reverse (or 'cancel') it I need to do 100/200 * 100 = 50%, so this is not a problem as I get back exactly where I was, the problem if I want to accelerate from 100% to 200%... |
Let $V$ be a vector space with an increasing filtration $V_j, j\in \mathbb Z$, we assume the filtration is Hausdorff $\bigcap_j V_j=0$ and exhaustive $\bigcup_j V_j=V$. Consider the associated graded space $\text{gr}V=\bigoplus_j V_j/{V_{j-1}}$, when is $\text{gr}V\cong V$ as a vector space?
In the case of $V_j$ is ... |
When is $\text{gr}(V)\cong V$? |
We know that, geometrically, the action of a linear operator $T: \mathbb{R}^n \to \mathbb{R}^m$ is to take an element $v \in \mathbb{R}^n$, scale it by some factor $\alpha \in \mathbb{R}^+$ and rotate it through some angle $\theta$. What I'm wondering is the following -
Assuming $L$ is the matrix representation of ... |
I assume to have two random variables $X,Y$ with pdfs $p_x,p_Y$, respectively. Unfortunately, I don't have much background knowledge on probability theory. I am wondering whether assuming $p(x | y)$ is identical for all $y \in Y$ automatically implies that $X$ and $Y$ are (stochastically) independent?
My feeling is... |
I have a matrix $F\in \mathbb{R}^{n\times n^2}$ for some integer $n>1$. I know in some cases, I can decompose $F$ as
$$F = \mathbf{v}^\dagger\otimes \mathcal{F}$$
where $\mathbf{v}\in \mathbb{R}^n$ and $\mathcal{F}\in \mathbb{R}^{n\times n}$. In general, though, it's not possible to do this, and I have some counterex... |
I have found the following first part of a series:
\
$$1+x+x^2+\frac{4}{3}x^3+\frac{7}{6}x^4+\frac{6}{5}x^5+\frac{37}{30}x^6+\frac{404}{315}x^7+...$$
\
However, I just cant seem to find the pattern in the coefficients. Is any of you able to? Like, you could rewrite the denominators to $k!$, but then the numerators don... |
Which of the following collections of 3-vectors space [a,b,c] are vectors? The vectors with ab=0. |
How to solve Hoffman & Kunze exercise 8, section 10.2? |
I have stumbled upon an interesting question whilst working on my thesis.
You are given a matrix of pairwise distinct integers $A=(a_{i,j})$ with $1\leq i\leq k$ and $1\leq j \leq r$ and a tuple $(b_1, \dots, b_r)$ such that $\sum_{i=1}^r b_i=k$.
You have to calculate the lexicographic maximal string $(a_{i_1, j_... |
> Suppose we have a biased coin with probability $\frac13$ heads and
> $\frac23$ tails. If we flip this coin $n$ times, what is the
> probability we have more heads than tails?
Is this something we can solve for? My best idea is to use a tail inequality such as Markov's Inequality or Chebychev's Inequality to form... |
My interpretation of your question is "Why isn't the set of vectors from $(a, b, c) \in \mathbb{R}^3 $ for which $ab=0$ a vector space?
The answer is that it isn't closed under vector addition:
$(1, 0, 0) + (0, 1, 0) = (1, 1, 0)$
and $1 \cdot 1 = 1 \neq 0$.
All the examples in your justification are indeed... |
Could someone please tell what results are known about the zeros of L function, $L(s,\chi)$ on the 1/2 line, where $\chi$ is a character mod $q$?
Is there an upper bound for this count when we count zeros with $|\Im s|\leq T?$
In particular, what is known about zeta function? I know there are some explicit versi... |
I am learning C++ and I find `pow`, `log`, `log2`, `exp` functions provided by `cmath` extremely slow, so that I would like to implement faster versions of them.
I currently have a version using polynomial approximations, which is much faster than the ones provided by the library and very accurate, but not as accura... |
You can compute the derivative:
$$f'(x) = 4^x \ln 4 + 8^x \ln 8 + 9^x \ln 9 - 2^x \ln 2 - 6^x \ln 6 - 12^x \ln 12.$$
My instinct would be that a "nice" value of $x$ will probably make $f'(x) = 0$ (probably an integer). So, I would look for $x$ where this expression is $0$, by treating the $\ln 2$ and $\ln 3$ terms se... |
>Let be $K$ a non-empty compact subset of $\mathbb{R}^n$. Show that $S:=\mathbb{R}^n\setminus K$ is connected.
As $K$ is bounded there exists a real number $c\geq 0$ such that $\Vert x\Vert\leq \frac{c}{2}$ for all $x\in K$.
Of course, we can approach this problem by choosing two arbitrary points $a,b\not \in K$ ... |
I have the following expression given by
$$S_n := \sum_{0 \leq i \leq j \leq n} \frac{1}{i!(n-i)!j!(n-j)!} a^i b^{j-i} c^{n-j}. $$
I wanted to see if I could use the multinomial theorem to simplify it. It is evident that
$$\sum_{0 \leq i \leq j \leq n} \frac{1}{i!(n-i)!j!(n-j)!} a^i b^{j-i} c^{n-j} = \sum_{0 \leq ... |
Once I went to my old handout and encountered this problem :
There are n points which pairwise have distance at least 1. Prove that we can choose $n/7$ points of them that pairwise have distance at least $\sqrt3$.
I've attemped for a week and I've come across to $n/12$ points with drawing rings with ratio of 1 an... |
Once I went to my old handout and encountered this problem :
There are n points which pairwise have distance at least 1. Prove that we can choose $n/7$ points of them that pairwise have distance at least $\sqrt3$.
I've attemped for a week and I've come across to $n/12$ points with drawing rings with ratio of 1 an... |
Let $X$ be the set of all complex functions endowed with the uniform topology. Then the set:
$A=\{p\in X\mid p\text{ is a polynomial and } p(0)=0\}$ is discrete in $X$.
It suffices to show that any singleton is open. For a given $p\in A$ the family:
$V_{p,n} = \{f\in X\mid \forall x \in \mathbb{C}: \lvert f(x)... |
I am trying to figure out if $f: \mathbb{R}_{\geq 0}^{2n} \mapsto \mathbb{R}$ with
$f(x,y)= (\lVert x \rVert - \lVert y \rVert)^2$
is a convex function.
I would be interested if it is the case for any p-Norm, but only one of them like the eucledian norm would also be helpful.
So far I have tried:
- Usi... |
Is (\lVert x \rVert - \lVert y \rVert)^2 convex? |
I am trying to figure out if $f: \mathbb{R}_{\geq 0}^{2n} \mapsto \mathbb{R}$ with
$$f(x,y)= (\lVert x \rVert - \lVert y \rVert)^2$$
is a convex function.
I would be interested if it is the case for any p-Norm, but only one of them like the eucledian norm would also be helpful.
So far I have tried:
- Using ... |
Is $(\lVert x \rVert - \lVert y \rVert)^2$ convex? |
In my answer to a recent
[MSE question](https://math.stackexchange.com/q/4875434/)
I mention the elliptic curve
[LMFDB 1728.n4](https://www.lmfdb.org/EllipticCurve/Q/1728a1)
with integer points $(-1,\pm1)$. The
[elliptic divisibility sequence](
https://en.wikipedia.org/wiki/Elliptic_divisibility_sequence)
associ... |
Let $\phi: S^1 \rightarrow S^1$ homomorphism continouos injective. Then $\phi(x)= x$ or $\phi(x)= \overline{x}$ for all $x \in S^1$, where $\overline{x}$ is conjugate.
**Attempt.** I proved $\phi$ is bijective function then $\phi$ topological homomorphism. Then we know $\ker \left (\phi \right)={1}$, and $(-1)^2=1$ ... |
Let $\mathbb{K}$ be a finite field with $q$ elements and $V$ a $n$-dimensional $\mathbb{K}$-vectorspace.
1. How many elements are in $V$?
2. How many subsets of $V$ are a basis of $V$?
----------
For the first question I reasoned that I have $n$ dimensions that I can fill up with $q$ different elements. So ... |
Let $\mathbb{K}$ be a finite field with $q$ elements and $V$ a $n$-dimensional $\mathbb{K}$-vectorspace. How many subsets of $V$ are a basis of $V$? |
Let $A=\{a_1,a_2,a_3\}$ and $B=\{b_1,b_2,b_3\}$ be two sets, and let
\begin{gather}
\Delta(B)=\left\{\delta\in[0,1]^B\mid\sum_{b\in B}\delta(b)=1\right\}
\end{gather}
be the set of all lotteries over the set $B$. Further, let $f:A\to\Delta(B)$ and $g:A\to\Delta(B)$ be two functions satisfying $f\neq g$. Then, ... |
How to define the weighted average of two functions? |
I'm working on a problem whose solution relies on finding if an arbitrary matrix A is positive semi-definite. A is real valued, square and symmetric, and each of its entries are given by:
$A_{ii} = f(i)$
$A_{ij} = A_{ji} = f(i,j)$
The problem is that this is part of an induction proof, where the size N of the ... |
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