instruction stringlengths 12 30k |
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Question:
> In isosceles $\triangle ABC,\ AB=AC.$ $H$ is a point on $AC$, take points $E$ and $F$ in
> turn on the extension line of $BC$, and take point $BD$ on the extension
> line of $CB$, so that $EF=DB$, make $EG // AC$ across point $E$ to cross the
> extension line of $DH$ at point $G$, connect $AF$, and ... |
Question:
> In isosceles $\triangle ABC,\ AB=AC.$ $H$ is a point on $AC$, take points $E$ and $F$ in
> turn on the extension line of $BC$, and take point $BD$ on the extension
> line of $CB$, so that $EF=DB$, make $EG // AC$ across point $E$ to cross the
> extension line of $DH$ at point $G$, connect $AF$, and ... |
Consider an operator \(T_a\) acting on elements of $L^2(\mathbb{R})$, defined to symmetrically translate a function by plus and minus $a$. This operator is mathematically expressed as:
$$
(T_a f)(x) = \frac{1}{2}[f(x-a) + f(x+a)]
$$
which translates a function symmetrically by plus and minus, and its action in the ... |
Assume that we have a function:
$$
y=f\left(x,z(x)\right)
$$e take the derivative of this function with respect to $x,$ we
get:
$$
\frac{\partial y}{\partial x}=\frac{\partial f}{\partial x}+\frac{\partial f}{\partial z}\times\frac{\partial z}{\partial x}
$$
My question is this: while $f\left(.\right)$ is an un... |
Question:
> In isosceles $\triangle ABC,\ AB=AC.$ $H$ is a point on $AC$, take points $E$ and $F$ in
> turn on the extension line of $BC$, and take point $BD$ on the extension
> line of $CB$, so that $EF=DB$, make $EG // AC$ across point $E$ to cross the
> extension line of $DH$ at point $G$, connect $AF$, and ... |
Question:
> In isosceles $\triangle ABC,\ AB=AC.$ $H$ is a point on $AC$, take points $E$ and $F$ in
> turn on the extension line of $BC$, and take point $BD$ on the extension
> line of $CB$, so that $EF=DB$, make $EG // AC$ across point $E$ to cross the
> extension line of $DH$ at point $G$, connect $AF$, and ... |
The question is about $\sigma$-algebras and to prove or give a counter-example to the statement that for $X$ a random variable, $\sigma(X) \subseteq \sigma(X^2)$.
I think the statement is false. Suppose $X(\omega) = \omega$ (e.g. $N(0, 1)$). Then $(-\infty, a) \in \sigma(X)$, $(-\infty, b)$ is a Borel subset but $(-... |
I have a silly question about the pullback of $ 1 $-forms. Take two smooth manifolds $ M $ and $ N $ and some smooth map $ F\colon M\to N $ between them. Take some $ 1 $-form $ \eta $ on $ N $. Suppose that with respect to some coordinates $ (y^1,\dots,y^m) $ on an open subset $ V $ of $ N $ we can write
$$
\eta{\res... |
Question:
> In isosceles $\triangle ABC,\ AB=AC.$ $H$ is a point on $AC$, take points $E$ and $F$ in
> turn on the extension line of $BC$, and take point $BD$ on the extension
> line of $CB$, so that $EF=DB$, make $EG // AC$ across point $E$ to cross the
> extension line of $DH$ at point $G$, connect $AF$, and ... |
Question:
> In isosceles $\triangle ABC,\ AB=AC.$ $H$ is a point on $AC$, take points $E$ and $F$ in
> turn on the extension line of $BC$, and take point $BD$ on the extension
> line of $CB$, so that $EF=DB$, make $EG // AC$ across point $E$ to cross the
> extension line of $DH$ at point $G$, connect $AF$, and ... |
Hi everyone I'm doing an exercise of logic from the Science of programming book of David Gries, one of the exercises says:
> Prove that from $\neg q$ follows q $\Rightarrow (p \land $ $\neg p)$
I proved that in the following way:
From $\neg q$ infer $q \Rightarrow (p \land $ $\neg p)$
1. $\neg q$ Pr... |
Are you told that events $B$ and $C$ are "independent", or are they "conditionally independent given $A$"?
* If $B$ and $C$ are *independent*, then, *by definition*: $\mathsf P(B, C)=\mathsf P(B)\,\mathsf P(C)$. However you do not appear to have enough information to find this, since you cannot use the **Law of Tot... |
Question:
> In isosceles $\triangle ABC,\ AB=AC.$ $H$ is a point on $AC$, take points $E$ and $F$ in
> turn on the extension line of $BC$, and take point $BD$ on the extension
> line of $CB$, so that $EF=DB$, make $EG // AC$ across point $E$ to cross the
> extension line of $DH$ at point $G$, connect $AF$, and ... |
Question:
> In isosceles $\triangle ABC,\ AB=AC.$ $H$ is a point on $AC$, take points $E$ and $F$ in
> turn on the extension line of $BC$, and take point $BD$ on the extension
> line of $CB$, so that $EF=DB$, make $EG // AC$ across point $E$ to cross the
> extension line of $DH$ at point $G$, connect $AF$, and ... |
Question:
> As shown in the picture/figure, in isosceles $\triangle ABC,\ AB=AC.$ $H$ is a point on $AC$, take points $E$ and $F$ in
> turn on the extension line of $BC$, and take point $BD$ on the extension
> line of $CB$, so that $EF=DB$, make $EG // AC$ across point $E$ to cross the
> extension line of $DH$ ... |
Consider the configuration space $\text{Conf}_n(\mathbb R^k)$, and consider the subgroup $G=\mathbb R^k\rtimes \mathbb R\leq \mathbb R^k\rtimes \text{GL}(k,\mathbb R)=\text{Aff}(\mathbb R^k)$ of dilations and translations in affine group, then there is a natural smooth action $G\times \text{Conf}_n(\mathbb R^k)\to \tex... |
Is this quotient space a manifold? |
Question:
> As shown in the picture/figure, in isosceles $\triangle ABC,\ AB=AC.$ $H$ is a point on $AC$, take points $E$ and $F$ in turn on the extension line of $BC$, and take point $BD$ on the extension line of $CB$, so that $EF=DB$, make $EG // AC$ across point $E$ to cross the extension line of $DH$ at point ... |
Question:
> As shown in the figure/picture, in isosceles $\triangle ABC,\ AB=AC.$ $H$ is a point on $AC$, take points $E$ and $F$ in turn on the extension line of $BC$, and take point $BD$ on the extension line of $CB$, so that $EF=DB$, make $EG // AC$ across point $E$ to cross the extension line of $DH$ at point ... |
Question:
> As shown in the figure/picture, in isosceles $\triangle ABC,\ AB=AC.$ $H$ is a point on $AC$, take points $E$ and $F$ in turn on the extension line of $BC$, and take point $BD$ on the extension line of $CB$, so that $EF=DB$, make $EG // AC$ across point $E$ to cross the extension line of $DH$ at point ... |
What is the quantitative relationship between $∠BAF$ and $∠CHG$ and the line in the figure that is equal to the line segment $AF$? |
What is the quantitative relationship between $∠BAF$ and $∠CHG$ and the line segment that is equal to the $AF$? |
I know that:
* If $f$ is continuous and monotonous on $[a, b]$, then $$f([a, b]) = [\min(f(a), f(b)), \max(f(a), f(b))]$$
* If $f$ is continuous and unimodal on $[a, b]$ with an extremum at $x_0 \in [a, b]$, then $$f([a, b]) = [\min\{f(a), f(x_0), f(b)\}, \max\{f(a), f(x_0), f(b)\}]$$
* If $f=\cos$ or $f=\sin$, th... |
Under what assumptions on f can I calculate f([a, b])? |
Question:
> As shown in the figure/picture, in isosceles $\triangle ABC,\ AB=AC.$ $H$ is a point on $AC$, take points $E$ and $F$ in turn on the extension line of $BC$, and take point $BD$ on the extension line of $CB$, so that $EF=DB$, make $EG // AC$ across point $E$ to cross the extension line of $DH$ at point ... |
I am looking for information and relevant terminology for a phenomenon related to the geometry of sublevel sets of functions. I believe that the possibly related fields are Morse theory and algebraic geometry, although there are probably more elementary ways to describe this phenomenon.
The motivation for my questio... |
What about these?
>$$\int_{0}^{1} \left(\frac{5}{2} \left((x - \sqrt{x^2 - 1})^{2i} + x^4\right) - 1\right) \, dx = e^{\pi}\tag{1}$$
$$\int_{0}^{1} \left( -\frac{5}{2\left( x - \sqrt{x^2 - 1}\right)^{2i}} - \frac{5x^4}{2} + 1 \right) \, dx = e^{-\pi}\tag{2}$$
[1]: https://en.wikipedia.org/wiki/Gelfond%27s_... |
Assume that $W$ is a zero-mean Gaussian process on $[0,\infty)$ with covariance function satisfying $c(s,t):=E(W(s)W(T))=C(|t-s|)$ for a function $C:[0,\infty) \to \mathbb{R}$.
Can I exploit somehow the Ergodic theorem to claim that (in probability or almost surely)
$$
\frac{1}{\log(K_n/k_n)}\sum_{j=k_n}^{K_n} \f... |
Question:
> As shown in the figure/picture, in isosceles $\triangle ABC,\ AB=AC.$ $H$ is a point on $AC$, take points $E$ and $F$ in turn on the extension line of $BC$, and take point $BD$ on the extension line of $CB$, so that $EF=DB$, make $EG // AC$ across point $E$ to cross the extension line of $DH$ at point ... |
Question:
> As shown in the figure/picture, in isosceles $\triangle ABC,\ AB=AC.$ $H$ is a point on $AC$, take points $E$ and $F$ in turn on the extension line of $BC$, and take point $BD$ on the extension line of $CB$, so that $EF=DB$, make $EG // AC$ across point $E$ to cross the extension line of $DH$ at point ... |
Inspired by [this problem](https://math.stackexchange.com/questions/4631003/find-the-locus-of-the-point-of-intersection-of-three-mutually-perpendicular-tang), consider the paraboloid
$$ z = \dfrac{1}{4} (x^2 + y^2) $$
And the point $P(2, 3, -2)$. I want to find a set of three mutually perpendicular planes tangen... |
Question:
> As shown in the figure/picture, in isosceles $\triangle ABC,\ AB=AC.$ $H$ is a point on $AC$, take points $E$ and $F$ in turn on the extension line of $BC$, and take point $BD$ on the extension line of $CB$, so that $EF=DB$, make $EG // AC$ across point $E$ to cross the extension line of $DH$ at point ... |
Question:
> As shown in the figure/picture, in isosceles $\triangle ABC,\ AB=AC.$ $H$ is a point on $AC$, take points $E$ and $F$ in turn on the extension line of $BC$, and take point $BD$ on the extension line of $CB$, so that $EF=DB$, make $EG // AC$ across point $E$ to cross the extension line of $DH$ at point ... |
Question:
> As shown in the figure/picture, in isosceles $\triangle ABC,\ AB=AC.$ $H$ is a point on $AC$, take points $E$ and $F$ in turn on the extension line of $BC$, and take point $BD$ on the extension line of $CB$, so that $EF=DB$, make $EG // AC$ across point $E$ to cross the extension line of $DH$ at point ... |
What is the quantitative relationship between $∠BAF$ and $∠CHG$ and the line segment that is equal to $AF$? |
Question:
> As shown in the figure/picture, in isosceles $\triangle ABC,\ AB=AC.$ $H$ is a point on $AC$, take points $E$ and $F$ in turn on the extension line of $BC$, and take point $BD$ on the extension line of $CB$, so that $EF=DB$, make $EG // AC$ across point $E$ to cross the extension line of $DH$ at point ... |
$\bigcup \cal{A} := \{x:(\exists a \in \cal{A})x\in a\}$
Theorem:
$\bigcup (A \cup B) = \bigcup A \cup \bigcup B$
Attempt at proof:
$x \in \bigcup (A \cup B) \iff (\exists X \in A \cup B)x \in X $
$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \iff \exists X[(X \in A$ or $X \in B) \implies x \in X]$
... |
For some $\epsilon > 0$, suppose $\sup_{i \in \mathbb{N}}||A_i| - |B_i|| < \epsilon$ where $\mathbb{N}$ is the set of natural numbers, do we have
$|\sup_{i \in \mathbb{N}}|A_i| - \sup_{i \in \mathbb{N}}|B_i|| < \epsilon$?
|
Question:
> As shown in the figure/picture, in isosceles $\triangle ABC,\ AB=AC.$ $H$ is a point on $AC$, take points $E$ and $F$ in turn on the extension line of $BC$, and take point $BD$ on the extension line of $CB$, so that $EF=DB$, make $EG // AC$ across point $E$ to cross the extension line of $DH$ at point ... |
Suppose we have a pairwise distinct vector of reals $[x_1,x_2,\ldots,x_n]$ and for any such vector, we create a matrix $A_n$ such that
$$a_{ij} = \frac{x_i}{x_i-x_j}$$ where $i \neq j$ and $$a_{ii} = \sum_{j\neq i} \frac{x_i}{x_i-x_j} $$
What can be said about the set of eigenvalues and eigenvectors of any $A... |
Question:
> As shown in the figure/picture, in isosceles $\triangle ABC,\ AB=AC.$ $H$ is a point on $AC$, take points $E$ and $F$ in turn on the extension line of $BC$, and take point $BD$ on the extension line of $CB$, so that $EF=DB$, make $EG // AC$ across point $E$ to cross the extension line of $DH$ at point ... |
How do I prove that a particular series converges to pi? |
Question:
> As shown in the figure/picture, in isosceles $\triangle ABC,\ AB=AC.$ $H$ is a point on $AC$, take points $E$ and $F$ in turn on the extension line of $BC$, and take point $BD$ on the extension line of $CB$, so that $EF=DB$, make $EG // AC$ across point $E$ to cross the extension line of $DH$ at point ... |
Question:
> As shown in the figure/picture, in isosceles $\triangle ABC,\ AB=AC.$ $H$ is a point on $AC$, take points $E$ and $F$ in turn on the extension line of $BC$, and take point $BD$ on the extension line of $CB$, so that $EF=DB$, make $EG // AC$ across point $E$ to cross the extension line of $DH$ at point ... |
> I think $\exp(t)$ converges for every possible input of $t$, but I haven’t found a proof yet.
For all real and complex inputs of $t$, yes. This is not something trivial to show. You need proper background in set theory, how naturals, integers, reals and complex numbers work. You need to understand not only what i... |
I have just read a [paper][1] about fractional linear functions written by Justin Lanier. I have tried to prove some observations he left in page 13. Here is the statement I want to prove:
>Given function $f(x)=\frac{ax+b}{cx+d}$ ,then
>
>$$f^{\circ n}(x)=x \iff \frac{a+d}{2\sqrt{ad-bc}}=\cos(\frac{k\pi}{n}), 1\le... |
Let $\mathcal{R}$ be an expansion of $(\mathbb{R},+,\cdot,-,<,0,1)$, and suppose that the exponential map is definable. I am asked to show that it is definable without parameters using the fact that the exponential is the unique differentiable function that verifies: $f'=f$ and $f(0)=1$.
I can express the fact that... |
How do I prove that a particular series converges to $\pi$? |
In J. E. Pin: On Reversible Automata, LATIN 92, Springer LNCS 584, 1992 the author states that "it is a well-known fact of semigroup theory that the minimal ideal of a semigroup in which the idempotents commute is actually a group whose identity is an idempotent $f$". Where can I find a proof of this fact in the litera... |
What are $0$ and $1$, the elements of $\mathbb Z/2\mathbb Z$? How do they relate to $\mathbb Z$? |
I’m in middle school (6th grade) and I have a question related to the exponential function (the teacher couldn’t help me):
We define the exponential as follows:
$$ \exp(t) = \sum_{n = 0}^{\infty} \frac{t^n}{n!}$$
For whole number inputs of $t$, this corresponds to raising $e$ to that power.
I think $\exp(t)... |
> I think $\exp(t)$ converges for every possible input of $t$, but I haven’t found a proof yet.
For all real and complex inputs of $t$, yes. This is not something trivial to show. You need proper background in set theory, how naturals, integers, reals and complex numbers work. You need to understand not only what i... |
Question:
> As shown in the figure/picture, in isosceles $\triangle ABC,\ AB=AC.$ $H$ is a point on $AC$, take points $E$ and $F$ in turn on the extension line of $BC$, and take point $BD$ on the extension line of $CB$, so that $EF=DB$, make $EG // AC$ across point $E$ to cross the extension line of $DH$ at point ... |
Under what assumptions on f can I calculate $f([a, b])$? |
suppose you have x^2 = (4x^2y^3 + 1)^2 , can you square root both sides of the equation to make it simpler? |
Can you square root both sides of an implicit equation to get the derivative? |
I've recently been studying some set theory from the book Elements of Set Theory by Herbert Enderton. This is one of the exercises.
$\bigcup \cal{A} := \{x:(\exists a \in \cal{A})x\in a\}$
Theorem:
$\bigcup (A \cup B) = \bigcup A \cup \bigcup B$
Attempt at proof:
$x \in \bigcup (A \cup B) \iff (\exists X... |
You've done IMO most part of the work, with a few technical details missing and a relatively straightforward argument missing at the end to get at the subproblem a) and then b) and c).
As has been alluded to in the comments, the intermmediate goal is to show that
$$M_n = \{a \in \mathbb N\,|\, a > 0, (a-1)^2 \le... |
Showing $E_1 \Delta N_1=E_2 \Delta N_2 \implies E_1 \Delta E_2=N_1 \Delta N_2$ in
[![Halmos Measure Theory Page 55.][1]][1]
[1]: https://i.stack.imgur.com/8cZlv.png
Not able to show this even after expanding, can someone help? |
In Baker A.'s book section 3.6 about the complexification of a real lie algebra, it briefly introduces one theorem:
> If $\mathfrak{g'}$ and $\mathfrak{g''}$ are two complexifications of $\mathfrak{g}$ then there is an isomorphism of $\mathbb{C}$-Lie algebras $\mathfrak{g'}\rightarrow\mathfrak{g''}$ which extends the ... |
What does "extend the identity function" mean? |
Consider an operator \(T_a\) acting on elements of $L^2(\mathbb{R})$, defined to symmetrically translate a function by plus and minus $a$. This operator is mathematically expressed as:
$$
(T_a f)(x) = \frac{1}{2}[f(x-a) + f(x+a)]
$$
which translates a function symmetrically by plus and minus, and its action in the ... |
For some $\epsilon > 0$, suppose $\sup_{i \in \mathbb{N}}||A_i| - |B_i|| < \epsilon$ where $\mathbb{N}$ is the set of natural numbers, do we have
$|\sup_{i \in \mathbb{N}}|A_i| - \sup_{i \in \mathbb{N}}|B_i|| < \epsilon$?
I can prove the above claim when using max not sup, but I have no idea how to prove rigorously... |
Suppose you have $x^2=(4x^2y^3 + 1)^2$, can you square root both sides of the equation to make it simpler? |
Could you help me to justify better and in detail the proof of the theorem?
This is the theorem 1.40 of the book Minimax Theorems by M. Willem
Let $N\geq 3$ y $2^{*} := 2N/(N - 2)$.
The space $\mathcal{D}^{1,2}(\mathbb{R}^N) := \left\lbrace u \in L^{2^{*}} (\mathbb{R}^N) : \nabla u\in L^2(\mathbb{R}^N)\right\rbrac... |
ould you help me to understand and complete the proof of this theorem of P.L. Lions about the existence of a minimizer? |
If $X,Y$ are sets, $Z\subset X$, and $f:Z\to Y$ is a function, $g:X\to Y$ extends $f$ (or is an extension of $f$) if $f(z)=g(z)$ for all $z\in Z$. In other words, if the restriction $g|_Z$ of $g$ to $Z$ is just $f$ again.
For a set $X$, the identity function $\text{id}_X$ on $X$ is the function $\text{id}_X:X\to X$... |
Wikipedia essentially defines an admissible trading strategy as a stochastic process $H = (H_t)_{t\geq 0}$ such that the associated value process $\int H(u) d S(u)$ is lower bounded. As I understand it, this rules out doubling strategies, since the value process is then not allowed to take arbitrary values in $(-\infty... |
Consider the configuration space $\text{Conf}_n(\mathbb R^k)$, and consider the subgroup $G=\mathbb R^k\rtimes \mathbb R^{\times}\leq \mathbb R^k\rtimes \text{GL}(k,\mathbb R)=\text{Aff}(\mathbb R^k)$ of dilations and translations in affine group, then there is a natural smooth action $G\times \text{Conf}_n(\mathbb R^k... |
### Question:
Prove that for $(v_1,v_2,...,v_{d-1}) $ with $ v_i \in \mathbb{R}^d$ and $A \in \mathbb{R}^{d \times d}$ we have that: $\wedge _{1 \leq i \leq d-1} A v_i=comat(A) \wedge _{1 \leq i \leq d-1} v_i$
*Rem:* $ \wedge $ is the cross product $ \times $ between vectors..
### Answer:
1- First reme... |
### The Given Approach
The solution given in the question makes a mistake in the first step. It appears that the solution implies that
$$ 2^{\sin(x)} = \mathrm{e}^{-\sin(x)}, $$
which is not correct. Everything after this is based on the faulty assumption, which implies that the given approach is not going to w... |
When is the quotient of $\operatorname{GL}_n(\mathbb{R})$ by a discrete subgroup compact? |
Question:
> As shown in the diagram, in isosceles $\triangle ABC,\ AB=AC.$ $H$ is a point on $AC$, take points $E$ and $F$ in turn on the extension line of $BC$, and take point $BD$ on the extension line of $CB$, so that $EF=DB$, make $EG // AC$ across point $E$ to cross the extension line of $DH$ at point $G$, co... |
**Why do we need admissibility?**
The problem is that you need extra conditions for the first fundamental theorem of asset pricing to hold; the existence of an equivalent martingale measure is not enough if you don't restrict the set of trading strategies you can use. Here, I give an example of an arbitrage strategy... |
$$ I = \int_{0}^{\infty} \frac{\sqrt{2} \left( e^{-2vz - 2b\sinh(z)}\cos\left(\pi\left(\frac{1}{4} + v\right)\right) + e^{2vz - 2b\sinh(z)}\sin\left(\pi\left(\frac{1}{4} + v\right)\right) \right)}{\sqrt{e^z - e^{-z}}} \, \mathrm{d}z $$
Going through Desmos, the function does converge.$f(z)$, the integrand, is holomorp... |
To quote different Mathematics StackExchange users:
> One suggestion for finding a list of some (not all) problems in a given book one should attempt: search professors' websites for course information on courses they've taught previously. Ideally, you'll find a course that (a) uses the book you're interested in or ... |
I am thinking about the computation of the class group and the Picard group for the case of
* Number fields over $\mathbb{Q}$ and $\mathbb{F}_p(t)$
* Complex varieties
I would like to know what kinds of algorithms exist for the computation of these two groups. Evidently Mathematica has some tools in this regard... |
### The Given Approach
The solution given in the question makes a mistake in the first step. It appears that the solution implies that
$$ 2^{\sin(x)} = \mathrm{e}^{-\sin(x)}, $$
which is not correct. Everything after this is based on the faulty assumption, which implies that the given approach is not going to w... |
I am unable to prove the following lemma of Cauchy's integral theorem for simple closed curve.
>**Lemma**: Let $R$ be a simply connected region. If $f(z)$ be analytic in $R-\{a\}$ and is continuous on $R$ then for all simple closed curves $\gamma$ in $R$ $$\int_\gamma f(z)dz = 0$$
This result really surprised me... |
I am a little bit confused about the difference between the H-Adaptive Integration and adaptive quadrature algorithms, AFAIK:
- Both break down the entire region of integration into smaller sub-regions.
- Both apply a quadrature rule: Each sub-region is tackled using a pre-defined numerical integration rule (like M... |
How to calculate $\int_{C_a} \frac{e^z}{(z-a)(z-b)}dz$? |
I am trying to show the following:
$$\mathbb{P}[a<X<b]\leq\mathbb{P}[a<X]\mathbb{P}[X<b]$$
where $X$ is any random variable and $a,b$ are constants. I feel I am missing the obvious here, but does the below proof hold?
$$\mathbb{P}[a<X<b]=\mathbb{P}[a<X|X < b]\mathbb{P}[X<b]\leq \mathbb{P}[a<X]\mathbb{P}[X<b]... |
I am calculating a closed form expectation and I encountered the [Tricomi's confluent hypergeometric function](https://en.wikipedia.org/wiki/Confluent_hypergeometric_function#:~:text=Tricomi's%20(confluent%20hypergeometric)%20function%20U,function%20of%20the%20second%20kind.)
(aka confluent hypergeometric function of... |
I'm currently looking at an old exam and I have encountered the following task:
$ A=\left( \begin{array}{rrr}
2 & -1 \\
-1 & 2 \\
\end{array}\right)$ $b=\left( \begin{array}{rrr}
1 \\
1 \\
\end{array}\right) $
Alongside this there's an image with a coordinate system and contour lines. The exact solution... |
Consider two bounded sequences $\{|A_i|\}_{i = 1}^{\infty}$ and $\{|B_i|\}_{i = 1}^{\infty}$, for some $\epsilon > 0$, suppose $\sup_{i \in \mathbb{N}}||A_i| - |B_i|| < \epsilon$ where $\mathbb{N}$ is the set of natural numbers, do we have
$|\sup_{i \in \mathbb{N}}|A_i| - \sup_{i \in \mathbb{N}}|B_i|| < \epsilon$?
... |
I have been studying the polylogarithm function and came across its relation with Bernoulli polynomials, as Wikipedia site asserts:
> For positive integer polylogarithm orders $s$, the Hurwitz zeta function $\zeta(1−s, x)$ reduces to Bernoulli polynomials, $\zeta(1−n, x) = −B_n(x) / n$, and Jonquière's inversion for... |
I read in one russian book about complex vectors. There was formula for product of two complex vectors. I was confused because I didn't find what this formula is.
[![enter image description here][1]][1]
$(a_1,a_2)$ and so on is scalar product of two vectors.
What this formula for product of two complex vectors... |
What this formula for product of two complex vectors is? |
Let $G$ be a group and $H$ be a normal subgroup of $G$ such that $G/H$ is abelian. Prove trat $G' \subseteq H$.
I start by taking an arbitrary element $g' \in G'$. Then I can write $g'$ as an arbitrary product of elements of $G'$, that is: $g' = g_1^{-1}h_1^{-1}g_1h_1 \ldots g_n^{-1}h_n^{-1}g_nh_n$. Now, how do I s... |
A bijective linear (antilinear) operator $A$ on a Hilbert space $\mathcal{H}$ is called unitary (untiunitrary) if $\langle A\psi |A\phi \rangle =\langle \psi |\phi \rangle$ (resp. $\langle A\psi |A\phi \rangle =\langle \phi |\psi \rangle$) for all $\phi ,\psi \in \mathcal{H}$.
By the relation $\langle \psi |A^{\dagg... |
Claim: If for any $\epsilon >0$, there exists a $\delta >0$, such that for any |x-x_0| < $\delta$, it holds
$$
(L-\epsilon)^k \leq f(x) \leq (L+\epsilon)^k,
$$
where $k$ is a positive number. Then we have
$$
\lim_{x \rightarrow x_0} f(x) =L.
$$
-------------------------------------
The following is my... |
In a calculus of variations problem, I am given $V = \mathcal C([0,1], \mathbb R)$ to be a normed function space. I want to show that the set $$A = \{x\in V \quad \vert \quad x(0) = x(1) = 0, \quad \max_{s\in [0,1]} ||x(s)|| \leq 1\}$$ is closed (and also be able to show it for different sets).
Unfortunately my ba... |
I'm currently looking at an old exam and I have encountered the following task:
$ A=\left( \begin{array}{rrr}
2 & -1 \\
-1 & 2 \\
\end{array}\right)$ $b=\left( \begin{array}{rrr}
1 \\
1 \\
\end{array}\right) $
Alongside this there's an image with a coordinate system and contour lines. The exact solution... |
I'm currently looking at an old exam and I have encountered the following task:
$ A=\left( \begin{array}{rrr}
2 & -1 \\
-1 & 2 \\
\end{array}\right)$ $b=\left( \begin{array}{rrr}
1 \\
1 \\
\end{array}\right) $
Alongside this there's an image with a coordinate system and contour lines. The exact solution... |
I'm currently looking at an old exam and I have encountered the following task:
$ A=\left( \begin{array}{rrr}
2 & -1 \\
-1 & 2 \\
\end{array}\right)$ $b=\left( \begin{array}{rrr}
1 \\
1 \\
\end{array}\right) $
Alongside this there's an [image][1] with a coordinate system and contour lines. The exact sol... |
I am unable to prove the following lemma of Cauchy's integral theorem for simple closed curve.
>**Lemma**: Let $R$ be a simply connected region. If $f(z)$ be analytic in $R-\{a\}$ and is continuous on $R$ then for all simple closed curves $\gamma$ in $R$ $$\int_\gamma f(z)dz = 0$$
This result really surprised me... |
I have been practicing functional equations recently and decided to do a functional equation that involves derivatives:$$\text{Find all functions }\mathbb R\to\mathbb R\text{ such that }\dfrac d{dx}f^2(x)=f(2x)$$Now a solution that can be found right away is the identity function $I(x)=x$, and the other solution I was ... |
Are there any other functions which satisfy $\frac d{dx}f^2(x)=f(2x)$ other than $f(x)=\sin(x)$ and $f(x)=x$? |
Question 1: Consider here the two functions $f: \mathbb R\rightarrow \mathbb R$
and $g: \mathbb R \rightarrow \mathbb R$ defined by $f(t)=t^2$ and $g(t)=t^2 -t +1$. Show that $f, g$ are not solutions of a same ODE on $\mathbb R$ of the form $y'=F(t,y)$, with $F: \mathbb R^2 \rightarrow \mathbb R$ a function.
I k... |
I'm studying a group in GAP, and I have a list of relators of this finited presented group. My trouble is, this group has too many relations (actually more than 500), and GAP gives me these relations (maybe not) randomly. I want to write the presentation of this group more elegantly, is there a way GAP give me these re... |
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