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Let $X$ be a connected topological space, and $Z$ be a subspace of $X$. Suppose $f:X \to Z$ is a retraction, i.e. a mapping of topological spaces such that $f \circ i = id_Z$, where $i:Z \to X$ is the inclusion.
Is it true that all fibers of $f$ are connected?
PS: By fiber, I simply mean the preimage of a point ... |
I find two contradictory definitions for "equality relation".
The first one I learnt is that [equality is what is called an equivalence relation][1], that is, it satisfies three axioms: Reflexivity, Symmetry, Transitivity.
The [second definition][2] contains an axiom of extensionality.
The third definition I ... |
I'm trying to convince myself one part of the proof for a theorem in chapter 10 in DUmmit and Foote.
>Let $R$ be a subring of $S$, let $N$ be a left $R$-module and let $\iota:N \to S\otimes_RN$ be the $R$-module homomorphism defined by $\iota(n)=1\otimes n$. Suppose that $L$ is any Left $S$-module and that $\phi:N\to ... |
I'm trying to convince myself one part of the proof for a theorem in chapter 10 in DUmmit and Foote.
>Let $R$ be a subring of $S$, let $N$ be a left $R$-module and let $\iota:N \to S\otimes_RN$ be the $R$-module homomorphism defined by $\iota(n)=1\otimes n$. Suppose that $L$ is any Left $S$-module and that $\phi:N\to ... |
I've got a problem (da Silva homework 3, question 3) that I have rewritten as the following:
Let $g:M\to M$ be a symplectomorphism such that it takes $(q,p)\to (q',p')= f(q,p), h(q,p))$.
I want to show that $\partial f/\partial p=0$ and $\partial^2h/\partial p^2=0$.
I have by the assumption that $g^{*}\alpha=... |
Let $Y_n$ be a sequence of real random variables converging in distribution to a random variable $Y$ and let $M \in \mathbb{R}$ be a fixed real number. I would like to prove that $P(Y \geq M) \geq \liminf_{n\rightarrow \infty} P(Y_n \geq M)$. This seems like it should be very easy to prove since if $x_n \rightarrow x$ ... |
I find two contradictory definitions for "equality relation".
The first one I learnt in school is that [equality is what is called an equivalence relation][1], that is, it satisfies three axioms: Reflexivity, Symmetry, Transitivity.
The [second definition][2] contains an axiom of extensionality.
The third def... |
I'm reading "Algebraic Number Theory" by Neukirch. The following paragraph is from P.70.
>To end this section, we now want to compare a Dedekind domain $\mathcal O$ to the ring
$$\mathcal O(X)=\{ \frac{f}{g}|f,g\in\mathcal O,g\not\equiv 0\mod\mathfrak p\text{ for } \mathfrak p\in X\}$$
where $X$ is a set of prime ... |
I came across the following question the other day and was wondering if my solution is the correct solution. Here is the question.
Prove that
$$
(1 + \cos \alpha + i\sin \alpha)^k + (1 + \cos \alpha - i\sin \alpha)^k = 2^{k+1} \cos \left( \frac{k\alpha}{2} \right) \cos^k \left( \frac{\alpha}{2} \right)
$$
... |
Is there a base $b\geq11$ that we obtain a perfect power after we concatenate the first $n$ positive integers of that base?
In base ten, the concatenation of first $n$ positive integers does not give any other perfect power other than $1$.
But what about on other bases?
Note that I excluding cases like $1, 12,... |
Is there a base $b\geq11$ that we obtain a perfect power after we concatenate the first $n$ positive integers of that base? |
Assume that $a = (a_0,a_1,\ldots,a_N)$ where $N \in \mathbb{N}_+$ is fixed belongs to the space $$V = \left\{a \in \{-1,1\}^{N+1} : a\neq {\bf 0},~~ \sum_{n=0}^N a_n = 0,~~a_N = 0\right\}$$ I am wondering if it is possible to bound $\sum_{n=0}^N |a_n|$ (or $\max_{0\leq n\leq N} |a_n|$ since $N$ is finite and fixed) by ... |
Assume that $a = (a_0,a_1,\ldots,a_N)$ where $N \in \mathbb{N}_+$ is fixed belongs to the space $$V = \left\{a \in (-1,1)^{N+1} : a\neq {\bf 0},~~ \sum_{n=0}^N a_n = 0,~~a_N = 0\right\}$$ I am wondering if it is possible to bound $\sum_{n=0}^N |a_n|$ (or $\max_{0\leq n\leq N} |a_n|$ since $N$ is finite and fixed) by so... |
Relate a continuous inner product to a discrete one by expanding a kernel around a Dirac delta function? |
Consider a minimax problem $$\inf_{x\in X} \sup_{y\in Y} L(x,y)$$ where $L$ is convex in $x$ and concave in $y$. We also assume that $$\inf_x \sup_y L(x,y) = \sup_y \inf_x L(x,y)$$
holds.
Now let $z=(x,y)$ and $Z^*$ denote the solution set of $z$ for the minimax problem $L(z)=L(x,y)$.
Let $X^*$ denote the se... |
I am taking 4th field theory, and I am trying to solve a problem:
Find a counter example for the following false statement: If $K$ is algebraically closed and $F \subseteq K$ is a subfield of $K$, then $K$ is an algebraic closure of $F$.
I tried many examples, like $\mathbb{Q}(i)/\mathbb{Q}$, $\mathbb{C}/\mathbb{... |
I attempting to solve $x^{55} \equiv 33 \pmod{257}$. In using **Fermat's Little Theorem** and the observation that $1 = 55 \cdot (-121) + 256\cdot 26,$ I get \begin{align*}
x^{55} \equiv 33 \pmod{257} &\Rightarrow x^{55 \cdot (-121)} \equiv
33^{-121} \pmod{257} \\ & \Rightarrow x (x^{256})^{-26} \equiv 33^{-121} \p... |
Find $x$ by solving the equation
$$\frac{d (x^{2})}{dx} = 2$$
How should I write the remaining steps of the solution?
Should it be
$\implies 2x = 2$
$\implies x= 1$
**OR**
i.e. $2x=2$
i.e. $x=1$
**OR**
or $2x=2$
or $x=1$
**OR**
$\therefore 2x=2$
$\therefore x=1$
Which notatio... |
Which notation/abbreviation is appropriate to denote logical implication while writing proofs/solutions? |
I attempting to solve $x^{55} \equiv 33 \pmod{257}$. In using **Fermat's Little Theorem** and the observation that $1 = 55 \cdot (-121) + 256\cdot 26,$ I get \begin{align*}
x^{55} \equiv 33 \pmod{257} &\Rightarrow x^{55 \cdot (-121)} \equiv
33^{-121} \pmod{257} \\ & \Rightarrow x (x^{256})^{-26} \equiv 33^{-121} \p... |
I am interested about the hard problem of surface and volume integral, so can anyone suggest me a book based on the problem on surface and volume integral (containing a lot of hard problem) for practicing and different types problem solving? The book should be theoretical as well as practical, any kind of help is accep... |
So with the definition of quadratic covariation between two processes given [here][1], in the case when $Y$ is a Weiner process, is it true that
$$\lim_{||P|| \to 0} \sum_{i=1}^n (X_{(t_k + t_{k-1})/2} - X_{t_{k-1}}) (Y_{t_k} - Y_{t_{k-1}}) = \frac{1}{2} [X,Y]_t$$
when $[X,Y]_t$ exists?
I've tried breaking $[X,... |
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 \... |
For the case of an isosceles triangle, will there be a conic section that passes through the six points? Yes, this can be easily proven, more generally according to the inverse of Pascal's theorem
Note that any six points with an axis of symmetry that does not pass through any of them will lie on a conic section
... |
Suppose $1=x_1<x_2<x_3<\cdots$ is a strictly increasing sequence such that $x_{nm}=x_n x_m$ for all $n,m$. Is it true that there has to exist some $c>0$ such that $x_n=n^c$?
|
Let $f$ be a continuous weakly-increasing function that maps $\mathbb{R}_+$ to $\mathbb{R}_+$. Define a function $g$ on $\mathbb{R}_+$ as follows: $g(x)$ is the largest $y$ such that $f(y) = f(x)+1$.
If $f$ is strictly-increasing, then it is possible to write $g$ as: $g(x) = f^{-1}(f(x)+1)$. If $f$ is only weakly-in... |
I come from the applied math and statistics world, but I was talking to my friend who comes from the pure math and number theory world--in particular Galois representation theory. We got together to watch the Superbowl and were talking about mathematical topics. Now I mentioned something about the confusing definition ... |
I have three questions regarding the following extract(s), I have marked red the parts for which I do not understand for later reference. *The convention followed for the Minkowski metric in these notes is $\eta_{\mu\nu}=\mathrm{diag}(1,-1,-1,-1)$*:
>The most general Lagrangian which is quadratic in vector field $A_... |
I still cannot understand why the $\exists L$ rule from sequent calculus is sound:
$$\Gamma, A[y/x] \vdash \Delta \over \Gamma, \exists xA \vdash \Delta$$
Intuitively I can explain this rule as "When $\Delta$ can be derived from assumptions $\Gamma\cup\{A[y/x]\}$, due to the existence of some object $y$ that make... |
I will denote a pair of neighbour numbers from the sequence $(a_n)$ as $x,y$ going in that order. As you already stated, once we get a pair of coprime $x,y$, the sequence is done: it strictly increases after and hence is unbounded. Indeed, $\frac{x+y}{\gcd(x,y)}= x+y > y$ and new pair numbers $y, x+y$ are also coprime.... |
[![enter image description here][1]][1]
[1]: https://i.stack.imgur.com/IGxMp.png
According to Wikipedia, for a uniform n-gonal antiprism with edge length $a$, SA = $\frac{n}{2} \left( \cot\frac{\pi}{n} + \sqrt{3} \right) a^2$ . This formula seems unnecessarily complex to me, as I deduced the surface area sho... |
Below question is from the book 'Probability course.com'. The book provides a solution using Chebyshev's inequality. Before reading that solution, I used Markov's inequality. Is my solution correct?
Question: Let $X$ be a random variable, and $X_n = X + Y_n$, where $E[Y_n] = \frac{1}{n}, V[Y_n] = \frac{\sigma^2}{n}$... |
The "intuition" is: under what condition we can derive $A(t)$ from $\exists x A$?
If we use a *term* (a "name") $t$ that is already used in the proof, we may get into trouble, because if $t$ is used already the formula that uses it "impose" a meaning on the term that **can** be not consistent with that expressed by ... |
>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'm not sure what distinction the word *locally* is trying to convey. Perhaps your friend was talking about matrix *fields*, ie continuously varying matrices? Though that doesn't seem particularly relevant to number theory.
The answer depends on how general *stretching* is. Specifically, you need to include *shearin... |
How many possible pairs in a random 5-card poker hand? |
Are symmetric matrices symmetric about a particular axis? Or is the property defined only for the matrix representation of the linear transformation? |
It appears you are not even accounting for the five cards. If I understand correctly, here's *what you are doing*.
**Step 1.** The first card can be any of the cards, so $52$ ways.
Take the $3$ other cards with the same rank / denomination out of consideration. (Why? Don't you need another card to make a pair?)
... |
In the “linear transformation” point of view, matrices represent linear transformations, by denoting where the unit vector in the $x$ direction goes in the first column and where the unit vector in the $y$ direction goes in the second. Of course for $3$ dimensions there are more basis vectors but the same applies.
T... |
As stated in the commentaries(see [here][1]), it is out of the question to prove conclusively that $$\forall x,y,z \in \mathbb R,x(y+z)=xy+xz$$On the other hand, one can [easily answer][2] by using the interpretation of xy as the area of a rectangle of sides $x$ and $y$ for $x,y>0$. This, however, seems to be OP's prob... |
In group theory, the following two classes of groups are well-known.
1. AC-groups: a group is called an *AC-group* if the centralizer of every **non-central** element is abelian.
2. CA-groups (or CT-groups): a group is called a *CA-group* if the centralizer of every **non-identity** element is abelian.
For t... |
In the “linear transformation” point of view, matrices represent linear transformations, by denoting where the unit vector in the $x$ direction goes in the first column and where the unit vector in the $y$ direction goes in the second. Of course for $3$ dimensions there are more basis vectors but the same applies.
T... |
As stated in the commentaries(see [here][1]), it is out of the question to prove conclusively that $$\forall x,y,z \in \mathbb R,x(y+z)=xy+xz$$On the other hand, one can [easily answer][2] by using the interpretation of xy as the area of a rectangle of sides $x$ and $y$ for $x,y>0$. This, however, seems to be OP's prob... |
Are symmetric matrices symmetric about a particular axis geometrically ? Or is the property defined only for the matrix representation of the linear transformation? |
It appears you are not even accounting for the five cards. If I understand correctly, here's *what you are doing*.
**Step 1.** The first card can be any of the cards, so $52$ ways.
Take the $3$ other cards with the same rank / denomination out of consideration. (Why? Don't you need another card to make a pair?)
... |
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... |
In the “linear transformation” point of view, matrices represent linear transformations, by denoting where the unit vector in the $x$ direction goes in the first column and where the unit vector in the $y$ direction goes in the second. Of course for $3$ dimensions there are more basis vectors but the same applies.
T... |
I was given with the question below:
```
A circular cone with vertex C has a point A on the circumference of its base and
point B on the segment AC, as shown in the diagram below. The shortest
possible rope is wrapped once around the cone such that it starts at point A and
ends at point B. Assume that the base d... |
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... |
Are symmetric matrices symmetric about a particular axis geometrically ? Or is the property defined only for the matrix representation of the linear transformation?
If you consider a unit square, after transformation due to a symmetric matrix, is the resulting shape symmetric along any axis? |
In the differential form of the Gronwall's Lemma, we have the following:
$$
\frac{d}{dt} \phi(t) \leq \psi(t) \phi(t),
$$
for all $t\geq t_0$. Then, we get
$$
\phi(t) \leq \phi(t_0) exp\left(\int_{t_0}^t \psi(s)ds \right),
$$
for all $t\geq t_0$.
My remark is this inequality assumes that we can bound the f... |
Let $N_1$ and $N_2$ be two two commuting normal elements. C*-algebra generated by $N_1$ and $N_2$ is a commutative C*-algebra. So, it will be of the form $C(X)$ for $X$ some compact set in a topological space.
>Is $X$ of the form $\sigma(N_1) \times \sigma(N_2)$?
Thanks in advance for the help. |
I would like to fit a linear regression model with a lognormal distribution that has a linear expectation in the $X$, $Y$ space. So, I need a model with the following properties:
$$ Y|X \sim Lognormal $$
$$ E[Y|X] = X*β^T $$
Can you offer any suggestions? Thank you in advance. |
Let $X$ be a smooth surface and $S$ a smooth curve, over $\mathbb C$. Let $P\in X$, $s_0\in S$, and $C$ a smooth curve in $X\times S$ such that $x=(P,s_0)\in C$. Let
$$p:X\times S\longrightarrow S$$
be the projection.
Suppose that $C$ is transverse to $p^{-1}(s_0)$ at $x$.
Question: does there exist neigbourhoo... |
There's a quick way to test if there's a problem. We can just substitute some values for $k$ and $\alpha$, particularly ones where the two final answers will differ, and see which produces the correct answer. In particular, if we let $k = 1$, and for the sake of nice trig values, but $\cos(\alpha/2) < 0$, take $\alpha ... |
In the differential form of the Gronwall's Lemma, we have the following:
$$
\frac{d}{dt} \phi(t) \leq \psi(t) \phi(t),
$$
for all $t\geq t_0$. Then, we get
$$
\phi(t) \leq \phi(t_0) exp\left(\int_{t_0}^t \psi(s)ds \right),
$$
for all $t\geq t_0$.
My remark is this inequality assumes that we can bound the f... |
Let $N_1$ and $N_2$ be two distinct commuting normal elements. $C*$-algebra generated by $N_1$ and $N_2$ is a commutative $C*$-algebra. So, it will be of the form $C(X)$ for $X$ some compact set in a topological space.
>Is $X$ of the form $\sigma(N_1) \times \sigma(N_2)$?
Thanks in advance for the help.
----... |
There's a quick way to test if there's a problem. We can just substitute some values for $k$ and $\alpha$, particularly ones where the two final answers will differ, and see which produces the correct answer. In particular, if we let $k = 1$, and for the sake of nice trig values, but $\cos(\alpha/2) < 0$, take $\alpha ... |
$\sum_{n=0}^\omega 2^{\aleph_n}={\aleph_\omega}$
is this true?
And is there a way in ZFC to let $\infty$ range over ALL infinite ordinals (not a concrete one as in the example above) ?
$\sum_{n=0}^\infty 2^{\aleph_n}=?$
If not -why not ? And is it possible to state this in other set theoretical axiomatic ... |
Let $p$ be a prime number, $K=\mathbb{Q}(\zeta_p)$ be the cyclotomic field extension of $\mathbb{Q}$ by adding a $p$-th root of unity. Then how to compute the different ideal of the cyclotomic field extension $K/\mathbb{Q}$? |
How to compute the different ideal of the cyclotomic field extension $\mathbb{Q}(\zeta_p)/\mathbb{Q}$? |
I want to make sure I've opened up an equation correctly. The equation is as follows:
$${\mathbf{1}} + \frac{{\mathbf{1}}}{{{{\mathbf{x}}_t}}} + \cdots + \frac{{\mathbf{1}}}{{ \otimes _{k = 0}^{w - 2}{{\mathbf{x}}_{t - k}}}}$$
I opened it like the equation below:
$${\mathbf{1}} + \frac{{\mathbf{1}}}{{{{\mathbf{x}}... |
$\sum_{n=0}^\omega 2^{\aleph_n}=?$
And is there a way in ZFC to let $\infty$ range over ALL infinite ordinals (not a concrete one as in the example above) ?
$\sum_{n=0}^\infty 2^{\aleph_n}=?$
If not -why not ? And is it possible to state this in other set theoretical axiomatic systems or in the surreal nu... |
There's a quick way to test if there's a problem. We can just substitute some values for $k$ and $\alpha$, particularly ones where the two final answers will differ, and see which produces the correct answer. In particular, if we let $k = 1$, and for the sake of nice trig values, but $\cos(\alpha/2) < 0$, take $\alpha ... |
For $|x|<1$, and states $i,j\in S$ define:
$$
P_{ji} (x)=\sum_{n=0}^{\infty} p_n(j|i) x^n
$$
$$
F_{ji} (x)=\sum_{n=1}^{\infty} f_n(j|i) x^n
$$
Where $f_n (j|i)=P(\xi_n=j,\xi_k\neq j,k=1..n-1|\xi_0=i)$ (i.e. that the state j first occurs at step n) and $p_n(j|i)=P(\xi_n=j|\xi_0=i)$. Prove the following relations:... |
How can I prove this relation between the probability generating functions of a markov chain? |
$\sum_{n=0}^\omega 2^{\aleph_n}=2^{\aleph_\omega}$
Is this true?
And is there a way in ZFC to let $\infty$ range over ALL infinite ordinals (not a concrete one as in the example above) ?
$\sum_{n=0}^\infty 2^{\aleph_n}=?$
If not -why not ? And is it possible to state this in other set theoretical axiomati... |
I'm trying to find the intersection points of two spherical circles on a unit sphere.
[![two spherical circles and their intersection points][1]][1]
Each circle is defined by a normal vector *n* pointing to its center, and an angle *θ* used as radius, meaning that a point on the surface of the sphere is on the ci... |
I want to make sure I've opened up an equation correctly. The equation is as follows:
$${\mathbf{1}} + \frac{{\mathbf{1}}}{{{{\mathbf{x}}_t}}} + \cdots + \frac{{\mathbf{1}}}{{ \otimes _{k = 0}^{w - 2}{{\mathbf{x}}_{t - k}}}}$$
I opened it like the equation below:
$${\mathbf{1}} + \frac{{\mathbf{1}}}{{{{\mathbf{x}}... |
We are given that $ABC$ is equilateral so $AB=BC=CA$, and that the length of the circle is $24\pi$. What is the area of the triangle?
Since $24\pi=2\pi r$ we get that $r=12$ and $OB=OC=OA=r$; $OAB$, $OAC$, $OCB$ is isosceles and because $ABC$ is equilateral the area of the triangle is $\frac{a^2\sqrt{3}}{4}$, how to... |
I was given with the question below:
```
A circular cone with vertex C has a point A on the circumference of its base and
point B on the segment AC, as shown in the diagram below. The shortest
possible rope is wrapped once around the cone such that it starts at point A and
ends at point B. Assume that the base d... |
For $|x|<1$, and states $i,j\in S$ define:
$$
P_{ji} (x)=\sum_{n=0}^{\infty} p_n(j|i) x^n
$$
$$
F_{ji} (x)=\sum_{n=1}^{\infty} f_n(j|i) x^n
$$
Where $f_n (j|i)=P(\xi_n=j,\xi_k\neq j,k=1..n-1|\xi_0=i)$ (i.e. that the state j first occurs at step n) and $p_n(j|i)=P(\xi_n=j|\xi_0=i)$. Prove the following relations:... |
> Find all strictly increasing sequences $\{a_k\}_{k=1}^\infty$ of positive integers such that $\frac{a_{nk}}{a_na_k} \in (1,2)$ for any $n,k>1$.
**Motivation**
First, I was wondering about the strictly increasing sequences of real numbers such that $\frac{a_{nk}}{a_na_k}=1$ for any integer $n,k\ge 1.$ I believe ... |
If $x+y=2$ is a tangent to the ellipse with foci $(2, 3)$ and $(3, 5)$, what is the square of the reciprocal of its eccentricity?
This could be done by the property,
The product of perpendiculars drawn from focus to the tangent is equal to $b^2$.
But I couldn't figure out where is error in my approach. |
If $x+y=2$ is a tangent to the ellipse with foci $(2, 3)$ and $(3, 5)$, what is the square of the reciprocal of its eccentricity? |
Let $p$ be a prime number, $K=\mathbb{Q}(\zeta_p)$ be the cyclotomic field extension of $\mathbb{Q}$ by adding a $p$-th root of unity.
There is a notation called [different ideal][1].
I want to know what is the different ideal of the cyclotomic field extension $K/\mathbb{Q}$?
[1]: https://en.wikipedia.org/w... |
Let $p$ be a prime number, $K=\mathbb{Q}(\zeta_p)$ be the cyclotomic field extension of $\mathbb{Q}$ by adding a $p$-th root of unity.
There is a notation called [different ideal][1], which is defined to measure the (possible) lack of duality in the ring of integers.
I want to know what is the different ideal of t... |
The following is Theorem 7.2. in Titchmarsh's book *The Theory of the Riemann Zeta-Function*:
[![enter image description here][1]][1]
I cannot figure out how the underlined equality above holds. In LHS t is greater than both n and m thus greater than max(n,m) but how it changes the sums from $\sum_{<t}$ to $\sum_... |
## Summary of wikipedia article
Let
- $f$ be "a mathematical problem",
- $\tilde f$ a numerical scheme replicating $f$,
- $x$ the "input" to the problem, and
- $\tilde x$ the "perturbed input".
Then
- $\Vert f(\tilde x)-f(x) \Vert$, is called the **condition**, describing how the problem changes subject... |
Do matrices really rotate and stretch vectors or is that definition incorrect? |
**Question:** What is a simple yet precise definition for "="?
**My try:**
I find two definitions for "equality relation" which seems to be contradictory.
The first one I learnt in school is that [equality is what is called an equivalence relation][1], that is, it satisfies three axioms: Reflexivity, Symmetry, ... |
<h1>1. Introduction</h1>
Linear regression Equations for $λ_j$ are derived in the [associated reference link here][1], as a function of table elements like the following. A table of some real sample data is found below:
$$ \text{ } $$
| $ X_{V\text{ }SD}\left(Volts\right) $ | $Y_{I\text{ }SD}(Amps) $ | $ I_{SD\te... |
Let $C([0,T];\mathbb{R}^d)$ denote the space of continuous functions with the usual supremum norm and given a path $x\in C([0,T];\mathbb{R}^d)$ let $x^s$ denote the stopped path $x(t \land s)$ with $t\in [0,T]$. Now, consider the following quotient space
$$X :=[0,T] \times C([0,T];\mathbb{R}^d) \ /\sim \ := Y/\sim$$
... |
The intuition on stretching and rotating is good. But like all intuition, one needs to question it and take it with a grain of salt. For example: does it mean that you can always write a matrix as a product of a rotation(/reflection) matrix and a stretching matrix? The answer (for square matrices) is indeed yes, and th... |
The [explanation by John Stembridge](http://www.math.lsa.umich.edu/~jrs/coxplain.html) (credit to Sabino Di Trani for linking there) is a great starting (and perhaps even end) point. I quote the beginning:
> The Lie group E8 has a "root system" associated to it that consists of 240 points in 8-dimensional space. Si... |
I'm trying to find the intersection points of two spherical circles on a unit sphere.
[![two spherical circles and their intersection points][1]][1]
Each circle is defined by a normal vector *n* pointing to its center, and an angle *θ* used as radius, meaning that a point on the surface of the sphere is on the ci... |
The trick is to unwrap the side of the cone. Since the side is $AC=12$, you get a circular sector with the same radius. To find the angle, the length is $2\pi \frac 62=6\pi$. Then the angle of the circular sector is then $\frac{6\pi}{12}=\frac{\pi}2$. So point $A$ is at a distance $12$ from $C$, point $B$ is at a dista... |
The "intuition" is: under what condition we can derive $A(t)$ from $\exists x A$?
If we use a *term* (a "name") $t$ that is already used in the proof, we may get into trouble, because if $t$ is used already the formula that uses it "impose" a meaning on the term that **can** be not consistent with that expressed by ... |
**Question:** What is a simple yet precise definition for "=" as a binary relation?
**My try:**
I find two definitions for "equality relation" which seems to be contradictory.
The first one I learnt in school is that [equality is what is called an equivalence relation][1], that is, it satisfies three axioms: Re... |
The intuition on stretching and rotating is good. But like all intuition, one needs to question it and take it with a grain of salt. For example: does it mean that you can always write a matrix as a product of a rotation(/reflection) matrix and a stretching matrix? The answer (for square matrices) is indeed yes, and th... |
<h1>1. Introduction</h1>
Linear regression Equations for $λ_j$ are derived in [the associated reference link here][1], as a function of table elements like the following. A table of some real sample data is found below:
$$ \text{ } $$
| $ X_{V\text{ }SD}\left(Volts\right) $ | $Y_{I\text{ }SD}(Amps) $ | $ I_{SD\te... |
**Question:** What is a simple yet precise definition for "=" as a binary relation?
**My try:**
I find two definitions for "equality relation" which seems to be contradictory.
The first one I learnt in school is that [equality is what is called an equivalence relation][1], that is, it satisfies three axioms: Re... |
<h1>1. Introduction</h1>
Linear regression Equations for $λ_j$ are derived here:
> https://math.stackexchange.com/questions/4836964/how-is-it-possible-to-optimize-the-shichman-hodges-slope-parameters-from-the-lef
as a function of table elements like the following. A table of some real sample data is found belo... |
In the “linear transformation” point of view, matrices represent linear transformations, by denoting where the unit vector in the $x$ direction goes in the first column and where the unit vector in the $y$ direction goes in the second. Of course for $3$ dimensions there are more basis vectors but the same applies.
T... |
Let $F_1 = (2,3), F_2 = (3,5)$. At a point $P = (x,y)$ on the line $x+y=2$, the normal to this line bisects the angle between $PF_1$ and $PF_2$, that is, the line acts as a mirror for the ray coming out of $F_1$ towards the line, and reflects it at $P$ back into the ellipse to pass through $F_2$. The point $P$ can be... |
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... |
I am interested in the following question:
> A real plane algebraic curve $f(x,y)=0$ of odd degree $d>1$ has 3-fold rotational symmetry, then $3\mid d$.
I think when converting to polar coordinates it should only contain $r$ and $\cos(3\theta),\sin(3\theta)$, so $f$ is a polynomial in $\frac 1 4 (x^3-3xy^2), \frac ... |
Is it true that if the probability density function of a continuous random variable is an even function, then the continuous random variable is symmetric? |
I have been trying to understand some elementary set theory recently, and am trying to understand how the real number line can be defined using the set of rational numbers. In particular, I am trying to understand Dedekind cuts. I understand the definition of a Dedekind cut, and I have no trouble identifying if a cut i... |
<p>I am interested in the following question:</p>
<blockquote>
<p>A bounded real plane algebraic curve <span class="math-container">$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... |
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