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I have very little categorical knowledge, my question comes mostly from programming, but I'm interested in the categorical solution of this problem.
Let's assume we have a monad $(T, \eta, \mu)$
Is $T^2$ automatically a monad? Will the following (for example) $$\eta' = \eta T ∘ \eta $$ and $$\mu' = {\mu}T ∘ \mu T... |
Can a (co)monad composed with itself be a (co)monad again? |
I am going through Bartle and Sherbert's Introduction to Real Analysis and doing all the exercises. I have just tried the last of section 1.3, which requires you to prove the aforesaid statement. I think I've come up with something satisfactory, but it looks nothing like the hint the book offers at the end.
I start ... |
Is this proof that the collection F(N) of all finite subsets of N is countable correct? |
Suppose we have a countable abelian group $A$. Let $\phi$ be an automorphism of $A$.
Let $P = \{ a \in A | \exists n \in \mathbb{N}, \text{such that } \phi^{n} (a) =a \}$ be the sets of periodic points of $\phi$ in $A$.
For $a \in P$, let $m_a$ be the smallest positive integers such that $\phi^{m_a} (a) =a... |
can the period an automorphism of an abelian group be unbounded? |
Im working through the proof of the excission Axiom for singular Homology in the Bredon Homology book. I have two questions regarding this.
Bredon first defines the chain map $\gamma$ and the Chain Homotopy $T$ between $\gamma$ and the identity on the subcomplex of affine simplicies. then right after Lemma 17.1 on pa... |
Suppose we have a countable abelian group $A$. Let $\phi$ be an automorphism of $A$.
Let $P = \{ a \in A | \exists n \in \mathbb{N}, \text{such that } \phi^{n} (a) =a \}$ be the sets of periodic points of $\phi$ in $A$.
For $a \in P$, let $m_a$ be the smallest positive integers such that $\phi^{m_a} (a) =a... |
Suppose we have a countable abelian group $A$. Let $\phi$ be an automorphism of $A$.
Let $P = \{ a \in A \mid \exists n \in \mathbb{N}, \text{such that } \phi^{n} (a) =a \}$ be the sets of periodic points of $\phi$ in $A$.
For $a \in P$, let $m_a$ be the smallest positive integers such that $\phi^{m_a} (a)... |
A read a comment by a reputable user that gave me pause. The claim implied that if $f'$ is defined on a closed interval $[a,b]$, we cannot conclude that $f'$ is bounded. Taking this argument in the context of $f: \mathbb R \to \mathbb R$, I feel like this claim is incorrect, but maybe I am misunderstanding something.
... |
If $f'$ is defined on a closed interval $[a,b]$, can we conclude that $f'$ is bounded? |
If $X_t$ and $Y_t$ are both alpha mixing in the same sense for example decaying exponentially $\alpha(\ell) = O(\exp(-cn))$, $0 < c$.
Then can we say that the stacked random vector process $(X_t, Y_t)$ is also alpha mixing?
***
How about if we assume that $X_t$ and $Y_t$ are uncorrelated?
***
I am trying t... |
According to Shapiro and Wilk(1965) in lemma 3,
$W$ has lower bound: $na_1^2/(n-1).$
To find this value, they solve the problem:
$$Max\quad y'y$$
$$ s.t.\quad 1'y=0,\quad and\quad a'y = 1,\quad and \quad y_1 \leq ,\cdots,\leq y_n. $$
$$ where\qquad a'a = 1\quad and \quad -a_i = a_{n-i+1}, $$
$$ |a_1|=|a_n... |
***THIS QUESTION IS CROSS-POSTED @/FROM STACKEXCHANGE PHYSICS NETWORK**
https://physics.stackexchange.com/questions/804389/which-is-the-correct-definition-of-covectors
I apologize for cross-posting before I start. Physics forum was quite inavtive than I expected and I think this question is more about maths rather ... |
Which is the correct definition of covectors? |
I have the following function:
$$f(\omega) = arctan(\frac{-\omega*\frac{R}{L}}{-w^2 + \frac{1}{C*L}})$$
When I try to plot it (`atan(f(w)`), I get the following:
[![enter image description here][1]][1]
This is clearly not a `atan` plot. I investigated a bit and found `atan2`. When I use it instead (`atan2(-... |
> A relation is asymmetric if and only if it is both antisymmetric and irreflexive.
I read this in Wikipedia's article about binary relation. So my problem with the statement is that though I can relate asymmetry with antisymmetry but cannot see the reasoning behind a binary relation being asymmetric is also irrefle... |
According to Shapiro and Wilk(1965) in lemma 3,
$W$ has lower bound: $na_1^2/(n-1).$
To find this value, they solve the problem:
$$Max\quad y'y$$
$$ s.t.\quad 1'y=0,\quad and\quad a'y = 1,\quad and \quad y_1 \leq ,\cdots,\leq y_n. $$
$$ where\qquad a'a = 1\quad and \quad -a_i = a_{n-i+1}, $$
$$ |a_1|=|a_n... |
I have a cone, centred at the origin, aligned with the $z$ axis, and so having equation $x^2+y^2=a^2z^2$. I also have a general plane, $px+qy+rz=s$. Suppose that I know a priori that this plane and cone intersect to form a hyperbola (rather than a parabola or ellipse) in the plane.
If we then define an orthogonal co... |
Working in the first order language of set theory.
By $R$-bounded quantifiers its meant those of the form $\forall x \ R \ a \, ( \cdots) $ , or $ \exists x \ R \ a \, ( \cdots) $, and these are defined as $\forall x \, ( x \ R \ a \to \cdots)$ and $\exists x \, (x \ R \ a \land \cdots) $ respectively. Here $R$ is ... |
I was looking for a proof such as for sample variance where it's shown that expected value of sample variance with n-1 in the denominator yields the parameter. I'm not even sure what pooled sample variance / residual variance tries to estimate
$$ E[\frac{1}{n-k}\sum_{i=1}^{n}(y_i-\bar{y}_{g(i)})^2] = E[\frac{1}{n-k}\s... |
Why is the residual variance / pooled sample variance divided by n-k in ANOVA? |
$g \in L^p_{\operatorname{loc}}(\Omega) \Longleftrightarrow |g|^p \, \chi_\Omega \in L^1_{\operatorname{loc}}(\mathbb R^n) ? $ |
According to Wikipedia (https://en.wikipedia.org/wiki/Bernoulli_number) we have:
$$1^m + 2^m + ... n^m = S_m(n) = \frac{1}{m+1} \sum_{k=0}^{m} \binom{m+1}{k} B_k^{+} n^{m+1-k}$$
where $B_k^{+}$ are the (positive) Bernoulli numbers.
From plugging in a few values of m it seems that
$$\int_{-1}^{0} S_m(n) dn = -\fra... |
[enter image description here][1]
[1]: https://i.stack.imgur.com/zXj7X.png
I need to calculate this integral, but I don't think I can use Cauchy's Theorem or the Residue Theorem. And it's very complicated to calculate it directly. Is there any other method I can use to calculate this integral? |
Your argument towards anti-symmetry is not correct.
In your argument towards anti-symmetry, you state:
> $ \forall a,a\in X \quad aRa \implies \lnot (aRa) $
> But this statement seems contradictory.
No, it is not contradictory. It simply means that you cannot have $aRa$ for any $a$, for *if* you did, *then*... |
Some says covectors are linear map that maps $ V \mapsto R $ (which means it's just a row vector considering vectors are $ n $x$ 1 $ matrix and mapping is matrix multiplication), while some say it's a certain value whose components follow $ v'_i = \frac{\partial x^j}{\partial x'^i} v_j $ under coordinate transformation... |
According to Wikipedia (https://en.wikipedia.org/wiki/Bernoulli_number) we have:
$$1^m + 2^m + ... n^m = S_m(n) = \frac{1}{m+1} \sum_{k=0}^{m} \binom{m+1}{k} B_k^{+} n^{m+1-k}$$
where $B_k^{+}$ are the (positive) Bernoulli numbers.
From plugging in a few values of natural numbers m it seems that
$$\int_{-1}^{0} S... |
I have a problem with this exercise (see the text in the following link).
https://math.stackexchange.com/questions/2689105/interpolation-like-inequality-question-from-brezis-book-exercise-8-15#comment10389064_2689105
The following link practically solves it. Only one last step is missing, namely proving the ineq... |
$$\int_{\gamma_1} e^{2z} \, \mathrm dz \quad \text{for} \quad \gamma_1(t) = \cos t + it,\; t \in [0,\pi/2]$$
I need to calculate this integral, but I don't think I can use Cauchy's Theorem or the Residue Theorem. And it's very complicated to calculate it directly. Is there any other method I can use to calculate thi... |
> Assume that $a_1,a_2,\cdots,a_n$ are completely random permutations of
> numbers $1$ to n. What is the probability that none of the: $A_1 = a_1, A_2 = a_1 + a_2, \cdots, A_n = a_1 + a_2 + \cdots + a_n$ are not divisible by
> three?
Here are my attempts:
At first, n mod $3$ should be $1$. If $n = 3k + 1$, we w... |
I have a problem with this exercise (see the text in the following link).
https://math.stackexchange.com/questions/2689105/interpolation-like-inequality-question-from-brezis-book-exercise-8-15#comment10389064_2689105
The link practically solves it. Only one last step is missing, namely proving the inequality whe... |
I am reading the Gortz's Algebraic Geometry, proof of Lemma 6.26 and trying to prove some statement in it.
First I propose a question. Let $R:=k[T_1, \dots,T_n]$ be a polynomial ring with closed point $y\in \operatorname{Spec}R$ so that $\mathfrak{p}_y$ is the maximal ideal in $R$. Let $\mathfrak{m}_y$ be the ma... |
I have the following function:
$$f(\omega) = \arctan\left(\frac{-\omega\cdot R / L}{-w^2 + 1/(C\cdot L)}\right)$$
When I try to plot it (`atan(f(w))`), I get the following:
[![enter image description here][1]][1]
This is clearly not an `atan` plot. I investigated a bit and found `atan2`. When I use it inste... |
I need to calculate the integral
$$\int_{\gamma_1} e^{2z} \, \mathrm dz \quad \text{for} \quad \gamma_1(t) = \cos t + it,\; t \in [0,\pi/2],$$
but I don't think I can use Cauchy's Theorem or the Residue Theorem. And it's very complicated to calculate it directly. Is there any other method I can use to calculate t... |
:[![enter image description here][1]][1]
[![enter image description here][2]][2]
I'm currently working through the book "Introduction to Set Theory: Third Edition" by Hrbacek and Jech and I came up with the following proof to $3.3(b)$ above (p12 in the book):
> Suppose for a contradiction that there exists a... |
Does my Proof to 3.3(b) Work? (From Introduction to Set Theory: Third Edition) |
I think this is a silly question, but suppose that $X$ is a proper geometrically integral scheme over a (not necessarily algebraically closed) field $k$, is it true that $k$ is a subfield of $\mathcal{O}_X(U)$ for all $U$?
I think the answer is yes, in particular since $X$ proper, it is of finite type, separated, a... |
Proper geometrically integral scheme $X$ over $k$ has $k$ as subfield $\mathcal{O}_X(U)$? |
I am reading the Gortz's Algebraic Geometry, proof of Lemma 6.26 and trying to prove some statement in it.
First I propose a question. Let $R:=k[T_1, \dots,T_n]$ be a polynomial ring with closed point $y\in \operatorname{Spec}R$ so that $\mathfrak{p}_y$ is the maximal ideal in $R$. Let $\mathfrak{m}_y$ be the ma... |
1.2 Numbers and the Least Upper Bound Theorem
1.2a Numbers as Infinite Decimals
There are two familiar ways of looking at numbers: as infinite decimals
and as points on a number line. The integers divide the number line into
infinitely many intervals of unit length. If we include the left
endpoint of each inte... |
I'm very confused about this in Lax's book. Why does infinite decimal representation not produce infinite 9? |
Let $k$ be a number field, $S$ a set of places of $k$, $k_S$ the maximal extension of $k$ unramified outside $S$, $\mathcal{O}_S$ the subring of $k_S$ with $\nu_{\mathfrak{p}}(\alpha)\geq 0$ for all $\mathfrak{p}\not\in S$, and $G_{S}$ the galois group $\text{Gal}(k_S/k)$. I am considering the following exercise at the... |
> Assume that $a_1,a_2,\cdots,a_n$ is a completely random permutations of numbers $1$ to $n$. What is the probability that none of the $A_1 = a_1, A_2 = a_1 + a_2, \dots, A_n = a_1 + a_2 + \cdots + a_n$ are divisible by three?
Here are my attempts: Since $1+\dots+n = n(n+1)/2$, we have that $n \mod 3 = 1$. Let $n = ... |
I am struggling to find a quick an intuitive solution to this problem:
You parked your car on Monday and didn't pay for parking. You leave your car there until the end of Tuesday. You know that you have the same probability of getting a ticket on Monday and Tuesday. You also know that if you get a ticket on Monday, th... |
The sum of the following Bosonic operators: $$ \sum_{<ij>}a_{i}^{+}a_{j}+a_{j}^{+ }a_{i} $$ where i is the lattice site index(i=1 to N, it has periodic boundary condition, $$ r_{i+N}=r_{i}$$, j: nearest neighbour, and it can be rewritten as: $$ (1/2) \times \sum_{i,i+\delta}a_{i}^{+}a_{i+\delta}+a_{i+\delta}^{+}a_{i} ... |
Some says covectors are linear map that maps $ V \mapsto R $ (which means it's just a row vector considering vectors are $ n $x$ 1 $ matrix and mapping is matrix multiplication), while some say it's a certain value whose components follow $ v'_i = \frac{\partial x^j}{\partial x'^i} v_j $ under coordinate transformation... |
> Assume that $a_1,a_2,\cdots,a_n$ is a completely random permutations of numbers $1$ to $n$. What is the probability that none of the $n$ partial sums $A_1 = a_1$, $A_2 = a_1 + a_2, \dots$, and $A_n = a_1 + a_2 + \cdots + a_n$ are divisible by three?
Here are my attempts: Since $1+\dots+n = n(n+1)/2$, we have that ... |
>Find the best possible upper bound on $P(X \geq 100)$ if $g_X(10)=10$ ($g_X$ is a generating function), where $X$ is a discrete random variable.
I started by writing the equation to find possible generating function $g_X(10)=(1-p)\cdot10^0+p\cdot10^{100} = 10$.
Its solution is $p = \frac{9}{10^{100}-1}$, hence ... |
Let $G$ be a locally profinite group and $H \leq G$ an subgroup. If $(\sigma, W)$ is a smooth representation of $G$ then its *classical* induction is the pair $(\Sigma, \text{IND}_H^G(W))$ where
$$\text{IND}_H^G(W)= \{ f: G \rightarrow W: f(hg) = \sigma(h)f(g) \text{ for } g \in G, h \in H \}, $$
and $\Sigma: G \... |
My question is on Exercise 7.3.8 of *Chaos and Nonlinear Dynamics* (2nd ed) by Strogatz:
> **7.3.8.** Recall the system $\dot{r} = r(1-r^2) + \mu r \cos \theta, \; \dot{\theta} = 1$ of Example 7.3.1. Using the computer, plot the phase portrait for various values of $\mu > 0$. Is there a critical value $\mu_c$ at whi... |
Does $\sum^\infty_{n=1} \frac{\textrm{sin}^2(n) + n^n}{1 + n^2e^nn!}$ converge? |
Let $G$ be a locally profinite group and $H \leq G$ an subgroup. If $(\sigma, W)$ is a smooth representation of $G$ then its *classical induction* is the pair $(\Sigma, \text{IND}_H^G(W))$ where
$$\text{IND}_H^G(W)= \{ f: G \rightarrow W: f(hg) = \sigma(h)f(g) \text{ for } g \in G, h \in H \}, $$
and $\Sigma: G \... |
> Prove that a set of lines, each of which intersect the others, are either coplanar or share a single common point.
>
> Source: Hadamard
This is a surprisingly difficult problem, because it requires ping-ponging back and forth between two dissimilar concepts: _lines being coplanar_ and _lines sharing a common poi... |
The following identities have been suggested based on formulas in a [previous question of mine][1].
If complex $\theta_1=\cos^{-1}(p)$ and $\theta_2=\sec^{-1}(p)$, where $p\in(-1, 0) \cup (1, \infty)$, then the following relation holds:
> $$e^{i\theta_1}=\frac{1-\tan\frac{\theta_2}{2} }{1+\tan\frac{\theta_2}{2}}.... |
I am studying the following proposition:
>Let $P_0,P_1$ be probability measures on $(X,\Sigma)$ having real-valued densities $f_0,f_1$ with respect to measure $\mu$. Suppose there exist a measurable function $T:X\to \mathbb R$ and a function $g:\mathbb R \to [0,\infty]$ such that
>
>1. $\frac{f_1(x)}{f_0(x)}=g(T(x... |
First, this not a question from any academic resource, I thought about it and I'm just wondering if it can be solved.
The question :
In how many ways we can arrange $n$ men and $m$ women in a circle (when $n ≥ 3$ , $m ≥ 2$ and $ n > m$ ) such that between every $2$ men we can have at most $m -1$ women?
First, ... |
In how many ways we can arrange $n$ men and $m$ women in a circle such that between every $2$ men we can have at most $m -1$ women? |
Got asked this question on an interview, and somehow I just wasn't satisfied with the proof given to me. Would love some help, although with my intuition I cannot guarantee if there's an analytic way to do this!
Consider the following step function:
$f: \mathbb{N} \rightarrow \mathbb{N}, f(n) = \lfloor \frac{n^{2}}... |
I am interested in the asymptotics for large order and large argument for the modified Bessel function of first kind
$$I_{\frac{N-1}{2}} (Nz) $$
where z is a fixed positive real numer z>0 and N is natural. I want to pass N to $+\infty$.
I checked Nist and Abramowitz and Stegun, but I did not find anything useful. D... |
I have been reading a bit about formal languages to understand exactly what we mean by "first-order" when talking about things such as the transfer principle. If we use the language of set theory $\displaystyle S=\left\langle \in \right\rangle$, then all first-order statements we can create right now consist of logical... |
Which statements count as first-order? |
> Prove that a set of lines, each of which intersect the others, are either coplanar or share a single common point.
>
> Source: Hadamard
This is a surprisingly difficult problem, because it requires ping-ponging back and forth between two dissimilar concepts: _lines being coplanar_ and _lines sharing a common poi... |
Lemma 1 :$$\displaystyle{L1:\frac{{L{i_n}\left( {a \cdot z} \right)}}{z} = \frac{d}{{dz}}\left( {L{i_{n + 1}}\left( {a \cdot z} \right)} \right)}$$ and $$\displaystyle{\frac{{Log\left( {1 - a \cdot z} \right)}}{z} = - \frac{d}{{dz}}\left( {L{i_2}\left( {a \cdot z} \right)} \right)} $$http://mathworld.wolfram.com/Polylo... |
I am a bit new to dynamical systems and don't know my way around terminology, so have had a hard time answering this for myself.
I know the basics of theory for 2D linear, time-invariant systems, i.e.,
$$
\dot{x}=a_1x+a_2y \\
\dot{y}=b_1x+b_2y
$$
I know that there are explicit exponential solutions, and a ton o... |
Are there general solutions to quadratic, 2D, continuous, time-invariant dynamical systems? |
Oscar Lanzi has already given quite a nice overview over a general method to attack a certain set of polynomials trying to disprove eventual divisibility. Let's explore this idea a bit further and give a nice summary of the method:
We start with defining $q(x) := x^2+ax+b$ and examine solutions to $q(x) = x$ and $q... |
I am not sure how performing a summation works with the [Levi-Civita symbol][1]. For example, if we have
$\epsilon^{ijk} x_i x_j x_k$,
how does this look once the summation is written out in components? Is it just
$\epsilon^{123} x_1 x_2 x_3 + \epsilon^{213} x_2 x_1 x_3 + \epsilon^{231} x_2 x_3 x_1 + + \eps... |
Let $f$ be a function from $A$ to $B$.
Let $f$ be a surjection onto $B$.
I strongly think that for any $Y \subset B$, then $f(f^-1(Y))=Y$.
I also think the surjection condition is necessary and no other condition is necessary (e.g. $f$ might not be an injection).
I have both the proof (hopefully correct) and ... |
The point of distinguishing a covector from a vector is to avoid any dependence on coordinates. Why do this? Because the geometric and physical properties of space should not depend on coordinates (e.g., whether you use inches or meters to determine distance or position).
One way to do this is to write everything in... |
Reading Burton's Elementary Number Theory, I try to make my ideas clear by attempting at drawing a Venn diagram involving the sets " divisors of number $a$" , " divisors of number $b$", " set of all numbers of the form $ ax+by$ ( with $x,y \in \mathbb Z$)".
_________________________________
Let $a, b$ be intege... |
For any given $j,k\in\mathbb{N}$, with $j\leq k$, how is it possible to prove the following equivalence
$$
\frac{2^j(2k-j)!}{(k-j)!}=\sum_{l=j}^k\frac{2^ll(2k-l-1)!}{(k-l)!}?
$$
My attempt is:
$$
\frac{2^j(2k-j)!}{(k-j)!}=\sum_{l=j}^k\frac{2^ll(2k-l-1)!}{(k-l)!}\\
2^j\binom{2k-j}{k-j}k!=\sum_{l=j}^k\binom{2k-l-1... |
Show that $p(x)=x^5+x^3+x+1$ is irreducible.
**My attempt**: First, I try to shifting Eisenstein's up to $10$ i.e $p(x+a)$, where $a=1,\dots, 10$ using calculator. I did not continue because if I am being honest, you will get bored after a several failures. Also, please, I ask you to do not recommend rational root t... |
**Question**: Prove that $f = X^{3} + 2$ is irreducible in $\mathbb{F}_{49}[X]$. Is $f$ irreducible in $\mathbb{F}_{7^{n}}$ for all even $n$?
**My attempt**: Notice that $\deg(f) = 3$, therefore $f$ is irreducible in $\mathbb{F}_{49}[X]$ if and only if $f$ has no roots. We will first look at $f$ in $\mathbb{F}_{7}[X... |
I read a book about orthogonal polynomials written by Gabor Szego. There is a conclusion:
Let the real-valued functions $f_0(x),f_1(x),\cdots,f_l(x), l $ finite or infinite, be of the class $L_\alpha^2[a,b]$ and linearly independent. By orthogonalization, there exists one orthogonal set $\phi_0(x),\phi_1(x),\cdots,... |
Your proof cannot be right. You claim that if $x$ is *any* upper bound of $A$, then $x=\sup A$. So if, say, $A=\{-2,3,15\}$ and if you take $x=10^{100}$ then since $x$ is an upper bound of $A$, $\sup A=10^{100}$. You don't believe that, do you? One of the problems with your proof takes place when you write “taking $y=\... |
>Consider a function $f:[0,1]\to\mathbb{R}$ satisfying
$$\int_{\ 0}^1\left(f\left(x\right)\left(x-f\left(x\right)\right)\right)dx=\frac{1}{12}$$
Then find the value of $f'(1)$
After banging my head on the table for half an hour, I assumed the function to be of the form $\frac{x}n$ where $n\in\mathbb{R}$. Solving... |
For an oral exam I want to learn the proof of the Galois solvability criterion (Galois 1831).
I have already seen the proof online, I tried to understand it but it is incredibly long.
Could someone explain the proof idea to me by dividing the proof in some steps?
Since it is an oral exam I don't have to rewri... |
I'm currently engaged in a problem that involves covering a convex polygon with circles, with constant radius of 'r'. The challenge lies in finding an optimal solution for this task. I know this problem is NP-Hard and I got to search for approximation algorithms but I cant find many papers revolving around this problem... |
Fractional linear transformation is a map from extended complex plane to itself, defined by:
\begin{equation}
z\to \frac{az+b}{cz+d}
\end{equation}
with $ad-bc\ne0$.
Wikipedia says that *"a linear fractional transformation is a transformation that is represented by a fraction whose numerator and denominator ar... |
Why the name linear fractional map? |
I'm reading the proof of Proposition $2$ of the Appendix [here][1].
The proposition 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 Pòlya urn random process having $S\cdot Id$ as replacement matrix and $(\alpha_... |
I'm reading the proof of Proposition $2$ of the Appendix [here][1].
The proposition 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 Pòlya urn random process having $S\cdot Id$ as replacement matrix and $(\alpha_... |
Let $\xi^*,\varsigma,\eta:\mathbb{R}\rightarrow (0,\infty)$. You may assume any degree of smoothness required of these functions.
Assume that $\xi^{*}(t)$ and $\varsigma(t)$ have finite positive limits as $t\rightarrow\infty$ and as $t\rightarrow -\infty$.
Define $\delta:\mathbb{R}\rightarrow\mathbb{R}$ by:
$$\d... |
Your argument towards anti-symmetry is not correct.
In your argument towards anti-symmetry, you state:
> $ \forall a,a\in X \quad aRa \implies \lnot (aRa) $
> But this statement seems contradictory.
No, it is not contradictory. It simply means that you cannot have $aRa$ for any $a$, for *if* you did, *then*... |
Terry Tao [RMT book](https://terrytao.files.wordpress.com/2011/02/matrix-book.pdf
) has the following formula for joint moment of freely independent random variables $X,Y$ in Section 2.5
$$\tau(XYXY)=\tau(X)^2\tau(Y^2)+\tau(X^2)\tau(Y)^2-\tau(X)^2\tau(Y)^2$$
Exercise 2.5.17 then asks to prove that this is possib... |
Given $\mathcal{G}$ a sigma-field and $X$ a $\mathcal{G}$-measurable random variable, on an intuitive level, why do we need $Y$ to be independent of $\mathcal{G}$ to pull out $X$ from the conditional expectation? $$E[f(X,Y)|\mathcal{G}]=E[f(x,Y)]|_{x=X}$$ I understand that if $X$ is $\mathcal{G}$-measurable, it means i... |
Let $M$ be a finitely generated noetherian module over noetherian ring $R$.
Let $\phi$ be an automorphism of $M$.
Let $P = \{ a \in M \mid \exists n \in \mathbb{N}, \text{such that } \phi^{n} (a) =a \}$ be the sets of periodic points of $\phi$ in $M$.
For $a \in P$, let $m_a$ be the smallest positive in... |
can the period of an automorphism of a finitely generated module be unbounded? |
Legendre equation is
$$ (1-x^2) y'' - 2xy' + \lambda y=0$$
We are interested in finding solutions in the range $[-1,1]$. We seek solutions around the ordinary point $x=0$
$$ \sum_{n=0}^\infty c_n x^n $$
Since legendre equation has regular singular points at $-1,1$ , this series is guaranteed to converge in $|x|... |
>I know that the infinite union of compact sets is not necessarily compact, so I would like to know where that fails and why the fact that the sequence of the centres of the sets that bound each $K_n$ fixes that.
Let's see. So why would an infinite sequence of compact sets not be compact? By the Heine Borel theorem,... |
I have read about the forgetful functor $U$ from
$\textsf{Mon}(\mathcal{C})$ to $\mathcal{C}$ (see e.g. [this nLab page][1]). I think I have read somewhere that this functor is _monoidal_, but I cannot find the source for it anymore and now I doubt that I remember this correctly.
I can see that the forgetful functo... |
Is the category of monoids $\textsf{Mon}(\mathcal{C})$ in a monoidal category $\mathcal{C}$ itself monoidal? |
I'm working through exercises 50 and 51 in Baez, Muniain, _Gauge Fields, Knots and Gravity_. Let $ M $ be a manifold and suppose we can write $ M $ as a product $ M = \mathbb R\times S $ of a "time" $ 1 $-dimensional manifold $ \mathbb R $ with some "space" manifold $ S $.
Take a $ 2 $-form $ F $ on $ M $. Exercise ... |
First I know that this question has all ready be asked for exemple [here][1] but for linear operator with only one variable here I want to show it for linear operator **with two variables**.
### Question:
Prove that a linear operator $\phi$ is continuous iff verifying $|| \phi (v;w)|| \leq M ||v|| ||w|| $
... |
This is part of a proof that I saw in lectures but the lecturer quickly skipped any calculation and just stated this result. It's not obvious to me that this must be true from the formula, although plotting in desmos does convince me. I believe this is true for all $n$.
I think to show this, we need to use $1-p \leq... |
First I know that this question has all ready be asked for exemple [here][1] but for bilinear operator with only one variable here I want to show it for linear operator **with two variables**.
### Question:
Prove that a bilinear operator $\phi$ is continuous iff verifying $|| \phi (v;w)|| \leq M ||v|| ||w|| $ ... |
**Question**: Prove that $f = X^{3} + 2$ is irreducible in $\mathbb{F}_{49}[X]$. Is $f$ irreducible in $\mathbb{F}_{7^{n}}$ for all even $n$?
**My attempt**: Notice that $\deg(f) = 3$, therefore $f$ is irreducible in $\mathbb{F}_{49}[X]$ if and only if $f$ has no roots. We will first look at $f$ in $\mathbb{F}_{7}[X... |
I am not sure how performing a summation works with the [Levi-Civita symbol][1]. For example, if we have
$\epsilon^{ijk} x_i x_j x_k$,
how does this look once the summation is written out in components? Is it just
$\epsilon^{123} x_1 x_2 x_3 + \epsilon^{213} x_2 x_1 x_3 + \epsilon^{231} x_2 x_3 x_1 + + \eps... |
Given positive integers $n$ and $k$, set
$$ S_{n,k}=\sum_{\substack{a_1+a_2+\dots+a_k=2n\\ a_i \in 2\mathbb{N},\,i=1,\ldots,k}}\frac{(2n)!}{a_1!a_2!\dots a_k!},
$$
where $2\mathbb{N}=\{0,2,4,\ldots\}$. According to the answers of
https://math.stackexchange.com/questions/1986386/special-sum-of-multinomial-coefficie... |
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