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Let $K$ be a finite extension of $\Bbb{Q}_p$.
Let $E/K$ be an elliptic curve over $K$.
Let $E(K)$ be the Mordell-Weil group of $E/K$.
Let $\widehat{E(K)}$ be the profinite completion of $E(K)$, that is, its completion with respect to finite index subgroups.
Is it true that $E(K)=\widehat{E(K)}$ ?
If we a... |
If any of the following exposition is unclear, please write a comment.
In essence, I am looking at the graph $G$ that is generated by the polynomial $q(x) = x^2+ax+b$ ($a,b \in \mathbb{Z}$) via the edge set
$$\{(n, q(n) \text{ mod } p) : n =0,\ldots,p-1\}$$
for some prime $p \in \mathbb{P}$. After some thought, it... |
Consider $\mathbb{R}^2$ with the jungle river metric $d_j$.
$$d_j(x,y) = \begin{cases} |x_2-y_2|, & \text{if $x_1 = y_1$;} \\
|x_2| + |y_2| + |x_1-y_1|, & \text{if $x_1 \neq y_1 $} \end{cases}$$
Let
$$C = \{(x, 1) : x ∈ Q\}$$
What is the closure and the interior of C?
I think the closure is C as cl(C) = C ∪ {al... |
I want to know if a given set of finite axioms *in* PA1 ( 1st order peano arithmetic ) can prove infinite sentences in PA1 such that those proofs need no other axioms except those in the given finite set ? <br/>
edit- from infinite sentences, i mean ' infinite number of sentences ' and not sentences of infinite length... |
I want to know if a given finite set of axioms *in* PA1 ( 1st order peano arithmetic ) can prove infinite sentences in PA1 such that those proofs need no other axioms except those in the given finite set ? <br/>
edit- from infinite sentences, i mean ' infinite number of sentences ' and not sentences of infinite length... |
I want to know if a given finite subset of axioms *of* PA1 ( 1st order peano arithmetic ) can prove infinite sentences in PA1 such that those proofs need no other axioms except those in the given finite set ? <br/>
edit- from infinite sentences, i mean ' infinite number of sentences ' and not sentences of infinite len... |
I want to know if a given finite subset of axioms *of* PA1 ( 1st order peano arithmetic ) can prove infinite sentences in PA1 such that those proofs need no other axioms except those in the given finite set ? <br/>
edit- from infinite sentences, i mean ' infinite number of sentences ' and not sentences of infinite len... |
First of all, let's note that L. Tu provides his introduction to the topic of cohomology in his book "Introduction to Manifolds". It might be considered a prerequisite to the sequel, i.e. the book you are using.
Secondly, I would say that your assessment of what characteristic classes are is more or less accurate. C... |
**Method 1**
Let $\epsilon>0$
Choose $\delta<\epsilon-6$. ($\delta>0$)
If $|x-4|<\delta \implies -\delta < x-4 < \delta \implies 4-\delta<x<4+\delta$,
then, $|\sqrt{x}-2| < \sqrt{x}+2 < \sqrt{4+\delta}+2 < 6+\delta < \epsilon$
Hence, by $\epsilon-\delta$ definition of limit, $\lim_{x\to 4} \sqrt{x} = 2$... |
I am seeking a small $2$-symbol Universal Turing machine $U$ such that if $T$ is a Turing machine written to the tape of $U$ somehow, and I wish for $U$ to simulate $T$ with its tape initialized to $n$, then there is some easily computable way of writing $n$ to the tape of $T$. The known smallest $2$-symbol UTMs seem t... |
Due to efforts of Abel, Ruffini, we know that there does not exist a general formula for a quintic equation. But given a specific quintic, say $x^5+5x^2-97x+1001=0$, Can there exist a root for it in radicals not necessarily involving the coefficients?
|
Show that $\lim_{x\to 4} \sqrt{x} = 2$ using $\epsilon-\delta$ definition of limit.
**Method 1**
Let $\epsilon>0$
Choose $\delta<\epsilon-6$. ($\delta>0$)
If $|x-4|<\delta \implies -\delta < x-4 < \delta \implies 4-\delta<x<4+\delta$,
then, $|\sqrt{x}-2| < \sqrt{x}+2 < \sqrt{4+\delta}+2 < 6+\delta < \ep... |
I'm studying K3 surfaces and I often encountered the fact that the tangent sheaf $\mathcal{T}$ is isomorphic to the sheaf of differentials $\Omega$. Why is this true? I guess this follows from the triviality of the canonical bundle (?). |
Show that $\lim_{x\to 4} \sqrt{x} = 2$ using $\epsilon$-$\delta$ definition of limit.
**Method 1**
Let $\epsilon>0$
Choose $\delta<\epsilon-6$. ($\delta>0$)
If $|x-4|<\delta \implies -\delta < x-4 < \delta \implies 4-\delta<x<4+\delta$,
then, $|\sqrt{x}-2| < \sqrt{x}+2 < \sqrt{4+\delta}+2 < 6+\delta < \... |
Let $g(x)$ be:
\begin{cases}
0 \space\space if\space\space x \in \mathbb Q \\
x\space\space if \space\space x \in \mathbb R - \mathbb Q \\
\end{cases}
That's what I've tried since I started:
If $g(x)$ is continuous so $\lim\limits_{x\to a}g(x) = g(a)$. By $\varepsilon - \delta$ definition:
... |
So, suppose I work in a category where fibred products exist. Let $X_1,X_2, Y$ and $Z$ objects in this category, $f_i:X_i\rightarrow Y$, and $g:Y\rightarrow Z$ be morphisms.
I am trying to show that the following square is Cartesian:
$\require{AMScd}$
\begin{CD}
X_1\times_Y X_2 @>\psi>> X_1 \times_Z X_2 \\
@Vf_... |
The least upper bound can be used to prove the intermediate value theorem (IVT) and to prove the Bolzano-Weierstrass theorem (usually through an useful lemma or the nested intervals theorem).
I've been trying to prove the IVT without the lub property, using the Bolzano-Weierstrass theorem taken as an axiom. The sket... |
Let $M$ be a complex manifold and $\Omega$ a Hermitian differential $(1,1)$-form on $M$. Is it true that for every $x \in M$ there exists an open neighborhood $x \in U \subseteq M$, a local holomorphic frame $v_1,\dots,v_n \in \Gamma(U,T^{1,0}(M))$ and some smooth functions $\Omega_1,\dots,\Omega_n \colon U \to \mathbb... |
The **least upper bound** property of the real numbers (lub) can be used to prove the intermediate value theorem (IVT) and to prove the Bolzano-Weierstrass theorem (BWT).
I've been trying to prove the IVT without the lub property, using the BWT as an axiom. The sketch of my proof goes as follows:
**Lemma**: If $u... |
Consider a graph $G=(V,E)$. An integer $k$-flow is a map $\phi: E \to \mathbb{Z}$ such that $|\phi(e)| < k$ and on each site, the sum of $\phi$ on incoming edges equals that on outgoing edges (this requires an orientation, but the orientation is actually irrelevant).
We say that a flow is *nowhere zero*, if for all ... |
How many integer $k$-flows (which are allowed to be zero)? |
Let $f = X^3 - 3X^2 + 3X - 6 \in \mathbb Q[X]$ and let $\zeta$ be a third root of unity. Let $L$ be a splitting field of $f$ over $\mathbb Q$. The goal is to find all intermediate fields $\mathbb Q \subseteq E \subseteq L$ with $[L : E] = 3$.
**What the other part already showed:** $L = \mathbb Q(\sqrt[3]{5}, \zeta)... |
I'm currently taking a course in number theory and we've just seen that any number can be written as the sum of the 4 squares, and that numbers can be written as the sum of 3 if they aren't of a certain form.
The notes gave an example using 2024
$2024=44^2+4^2+6^2+6^2$ and then leaves finding if its possible to ... |
I am dealing with an ODE of the form
$$y^{(4)}(x) = -{\sigma}^4\bigl(y(x) + f(x)\bigr)$$
over $x\in[0, L]$
where the non-homogeneous term is a centered superposition of damped waves that can be written
$$f(x) = \sum_j f_j(x) - M \quad\text{with}\quad f_j(x) = \Im(a_j\exp(iz_jx)),$$ and $M$ is the mean ... |
I’m not a mathematician and I’m working with some transforms in physical chemistry.
I use a transform to pass from the time domain of phase domain in a process that use a square wave to perturb and modulate a stimulus on a system.
In this case, the transform is
$$
A(\phi)= \int_{0}^{T} A(t)\sin(kwt + \phi)\,dt,
... |
> What is the length of time $T$ before the bowl sinks below the surface?
The **external volume of the hemisphere** is
$ V_e = {\frac 2 3} \pi (r + t)^3 $.
The **internal volume of the hemisphere** is
$ V_i = {\frac 2 3} \pi r^3 $.
The **net volume of the hemispheric bowl**
(i.e. the volume of its solid... |
You didn't specify what kind of equivalence you are talking about, but I will assume you mean topological equivalence. Regardless, if two metrics are not topologically equivalent, then they cannot be equivalent in any other reasonable sense.
Instead of defining three such metics on a set $X$ we can define three metr... |
Consider, for example, the Beta function:
$$ B(\alpha,1+\alpha) = \frac{\Gamma(\alpha)\Gamma(1+\alpha)}{\Gamma(1+2\alpha)} = \int_0^1 dt\ t^\alpha (1-t)^{\alpha-1}. $$
Expanding the integrated result in small $\alpha$, there is a leading term $1/\alpha$ that is divergent. However, if one tries to expand the integ... |
You didn't specify what kind of equivalence you are talking about, but I will assume you mean topological equivalence. Regardless, if two metrics are not topologically equivalent, then they cannot be equivalent in any other reasonable sense.
Instead of defining three such metics on a set $X$ we can define three metr... |
Let $f = X^3 - 3X^2 + 3X - 6 \in \mathbb Q[X]$ and let $\zeta$ be a third root of unity. Let $L$ be a splitting field of $f$ over $\mathbb Q$. The goal is to find all intermediate fields $\mathbb Q \subseteq E \subseteq L$ with $[L : E] = 3$.
**What the other part already showed:** $f$ is irreducible, $\alpha_i = 1 ... |
You didn't specify what kind of equivalence you are talking about, but I will assume you mean topological equivalence. Regardless, if two metrics are not topologically equivalent, then they cannot be equivalent in any other reasonable sense.
Instead of defining three such metics on a set $X$ we can define three metr... |
Let $f = X^3 - 3X^2 + 3X - 6 \in \mathbb Q[X]$ and let $\zeta$ be a third root of unity. Let $L$ be a splitting field of $f$ over $\mathbb Q$. The goal is to find all intermediate fields $\mathbb Q \subseteq E \subseteq L$ with $[L : E] = 3$.
**What the other part already showed:** $f$ is irreducible, $\alpha_i = 1 ... |
Suppose that $\varphi \colon A_1 \to A_2$ is an isogeny between two abelian varieties (of the same dimension). Is the induced map $\varphi^* \colon Pic^\circ(A_2) \to Pic^\circ(A_1)$ always injective? |
I have the following system of equations:
$$ 0 = \left[ (p_{31} - p_{11})^2 - \omega_{22} (p_{21} - p_{11})^2 \right] x^2_1 + \left[(p_{32} -p_{12} )^2 - \omega_{22} (p_{22} - p_{12})^2 \right] x_2^2+ \left[(p_{33} - p_{13})^2 - \omega_{22} (p_{23} - p_{13})^2 \right]x_3^2$$
$$0 = \left[ (p_{41} - p_{11})^2 - \om... |
Say there are $2$ players. Each is given one card, and the person with the higher number wins. The first player goes and can either bet 1 dollar or fold (giving the pot to the second player). If the first player bets 1 dollar, the second player can either call by putting 1 dollar in the pot, or fold giving the pot to t... |
What is the dual graph (in the planar sense) of a totally disconnected graph on more than one vertex? I can't figure it out, considering that the graph is planar and the dual should be still well defined... |
The **least upper bound** property of the real numbers (lub) can be used to prove the **intermediate value theorem (IVT)** and to prove the **Bolzano-Weierstrass** theorem (BWT).
I've been trying to prove the IVT without the lub property, using the BWT as an axiom. The sketch of my proof goes as follows:
**Lemma*... |
If I have the sequence $$((k + 1) a_k)$$
Which is a strictly decreasing sequence to zero of positive real numbers. Is there any way or any criteria that the above implies that the series
$$\sum_{k=1}^{\infty}a_k$$ converges? |
Three points are uniformly thrown on a circumference of circle of radius 1, find the mathematical expectation of the area of the triangle formed by them.
I've tried to use that [formula](https://math.stackexchange.com/questions/2798370/area-of-triangle-inside-a-circle-in-terms-of-angle-and-radius): $S = \frac{1}{2}(... |
Three points are uniformly thrown on a circumference of circle of radius 1, find the mathematical expectation of the area of the triangle formed by them.
I've tried to use that [formula](https://math.stackexchange.com/questions/2798370/area-of-triangle-inside-a-circle-in-terms-of-angle-and-radius): $S = \frac{1}{2}(... |
I am attempting to solve a system of linear equations represented by a matrix, where I am given certain known values and need to solve for the unknowns. The matrix representation of my system involves resistances R1 and R2, a known voltage V1, and I need to solve for the current i1 and another voltage v2. The system is... |
Why are $|0 ⟩$ and $|1 ⟩$ perpendicular to each other? |
I am studying for my linear algebra test and came across this lemma as an exercise:
"For any non-singular matrix $X$ and positive definite matrix $A$,
$B = X^T A X$ is also positive definite."
Here I presume the dimensions of the matrix are assumed to be compatible.Since the exercise was just after the [Cholesky... |
Three points are uniformly thrown on a circumference of circle of radius 1, find the mathematical expectation of the area of the triangle formed by them.
I've tried to use that [formula](https://math.stackexchange.com/questions/2798370/area-of-triangle-inside-a-circle-in-terms-of-angle-and-radius): $S = \frac{1}{2}(... |
Why isn't adding the ways to achieve every mutually exclusive outcome giving me the denominator in the birthday problem for four people? |
Consider $\mathbb{R}^2$ with the jungle river metric $d_j$.
$$d_j(x,y) = \begin{cases} |x_2-y_2|, & \text{if $x_1 = y_1$;} \\
|x_2| + |y_2| + |x_1-y_1|, & \text{if $x_1 \neq y_1 $} \end{cases}$$
Let
$$C = \{(x, 1) : x ∈ Q\}$$
What is the closure and the interior of C?
I think the closure is C as ${\overline{C}}... |
Suppose $H$ and $K$ are subgroups of a group $G$ such that every element $g\in G$ can be uniquely written as $g = hk$ with $h\in H$ and $k\in K$. (It follows then that every element $g\in G$ can also be uniquely written as $g = kh$ with $h\in H$ and $k\in K$.)
What is the standard term for this kind of a product of... |
I can provide a partial and unfinished answer, when one of the transformations, say $T$, is injective. Then there exists a linear transformation $T^{-1}:T(X)\to X$ such that $T^{-1}Tx=x$ for each $x\in X$. Now let $V$ be any subspace of $X$. Then $T(V)=S(V)$ iff $S(V)\subset T(X)$ and $T^{-1}S(V)=V$. That is we have to... |
Three points are uniformly thrown on a circumference of circle of radius 1, find the mathematical expectation of the area of the triangle formed by them.
I've tried to use that [formula](https://math.stackexchange.com/questions/2798370/area-of-triangle-inside-a-circle-in-terms-of-angle-and-radius): $S = \frac{1}{2}(... |
Suppose $F_X(x)$ is CDF of random variable $X$. I am going to find the value of $x$ such that the following equation is satisfied:
$$
ax+bxF_X(x)+ c \int_{x}^{\infty} \left[ 1-F_X(y) \right] \text{d}y = 0
$$
where $a,b,c$ are known parameters.
Is there any way to solve this equation? |
How to solve this integral equation involving CDF of a random variable? |
I am attempting to solve a system of linear equations represented by a matrix, where I am given certain known values and need to solve for the unknowns. The matrix representation of my system involves resistances $R_1$ and $R_2$, a known voltage $V_1$, and I need to solve for the current $i_1$ and another voltage $v_2$... |
To be able to write inequalities with $\nabla f$ and for it to coexist and be summed with $\rho \|x - y\|$, it would need to have real values to begin with, in which case the left-side's norm is just the absolute value and it's fairly straightforward to go from the first inequality to the second.
The problem with th... |
> Let $n ≥ 5$. Show that, among the ${s_n}$ space parts, there
are at least $ {2n-3\over 4} $ tetrahedra
; where ${s_n}$ denotes the number of parts the space is divided by $n$ planes in general position
what I didn't understand is the following
>Interpretation of the numerator of the fraction $ {2n-3\ove... |
Given an **inner product** on a vector space I can always define a norm and a **metric** (and a topology using that metric).
Is the converse true?
Given a distance on a vector space, can I define a norm and an inner product?
And what about a metric space that is not a vector space?
Thanks |
Can we derive a norm and an inner product from a metric? |
If any of the following exposition is unclear, please write a comment.
In essence, I am looking at the graph $G$ that is generated by the polynomial $q(x) = x^2+ax+b$ ($a,b \in \mathbb{Z}$) via the edge set
$$\{(n, q(n) \text{ mod } p) : n =0,\ldots,p-1\}$$
for some prime $p \in \mathbb{P}$. After some thought, it... |
Why isn't adding the ways to achieve every mutually exclusive outcome giving me the denominator in the birthday problem for four people?
$$\binom{4}{2} \cdot 365 \cdot 364 +\binom{4}{3} \cdot 365 \cdot 364+365 \cdot 364 \cdot 363 \cdot 362+365-365^4 \neq 0$$
:
Exactly Two share a Birthday + Exactly Three share a... |
Edit: I spent some time trying to generalise this formula for any build plate $l$ and $w$ and have realised a very small error in one of the quadratic equations below which I will correct tomorrow because I've spent a whole 10h on this today...
I have come across an interesting problem whilst looking to estimate how... |
Suppose $H$ and $K$ are subgroups of a group $G$ such that every element $g\in G$ can be uniquely written as $g = hk$ with $h\in H$ and $k\in K$. (It follows then that every element $g\in G$ can also be uniquely written as $g = kh$ with $h\in H$ and $k\in K$.)
What is the standard term for this kind of a product of... |
[I am refering to this document.][1]
**The relevant part is:**
[![enter image description here][2]][2]
**My question:**
I understand that `(a or b) and (not(a) or c)` implies `b or c` but what really confuses me is the statement
"The produced resolvents $S{'} = S_x \circ S_{\overline{x}}$ ***replace*** the... |
I'll expand on my comment above, but the comment gives the (local) intuition. Consider the map $\pi\colon V-\{0\}\to\Bbb P(V)$. Then for any $\xi\in W$, $\pi_{*\xi}\colon T_\xi V\to T_{\pi(\xi)}\Bbb P(V) = T_W\Bbb P(V)$. Note that $\pi_{*\xi}(W)=0$, so $\pi_*$ induces a surjective map (hence isomorphism) $T_\xi V/W = V... |
Let $(M,g)$ be a Riemannian manifold of dimension $n$ and let $g_{ij}$ be the matrix of the components of $g$ in coordinates. Since $g_{ij}$ is symmetric and positive-definite, it can diagonalized by $g_{ij} = PDP^{T}$, with $D = \text{diag}(\lambda_1, \ldots, \lambda_n)$ diagonal and $P$ orthogonal. For $s \in \mathbb... |
The **least upper bound** property of the real numbers (lub) can be used to prove the **intermediate value theorem (IVT)** and to prove the **Bolzano-Weierstrass** theorem (BWT).
I've been trying to prove the IVT without the lub property, using the BWT as an axiom. The sketch of my proof goes as follows:
**Lemma*... |
I'll expand on my comment above, but the comment gives the (local) intuition. Consider the map $\pi\colon V-\{0\}\to\Bbb P(V)$. Then for any $\xi\in W$, $\pi_{*\xi}\colon T_\xi V\to T_{\pi(\xi)}\Bbb P(V) = T_W\Bbb P(V)$. Note that $\pi_{*\xi}(W)=0$, so $\pi_*$ induces a surjective map (hence isomorphism) $T_\xi V/W = V... |
I'll expand on my comment above, but the comment gives the (local) intuition. Consider the map $\pi\colon V-\{0\}\to\Bbb P(V)$. Then for any $\xi\in W$, $\pi_{*\xi}\colon T_\xi V\to T_{\pi(\xi)}\Bbb P(V) = T_W\Bbb P(V)$. Note that $\pi_{*\xi}(W)=0$, so $\pi_*$ induces a surjective map (hence isomorphism) $T_\xi V/W = V... |
The **least upper bound** property of the real numbers (lub) can be used to prove the **intermediate value theorem (IVT)** and to prove the **Bolzano-Weierstrass** theorem (BWT).
I've been trying to prove the IVT without the lub property, using the BWT as an axiom. The sketch of my proof goes as follows:
**Lemma*... |
$\DeclareMathOperator \erf{erf}$
Wolfram Alpha gives the following [$\erf^{-1}(z)$ series](https://www.wolframalpha.com/input?i=series+expansion+inverf%28x%29):
>$$\sum_{n=1}^\infty\frac{z^n}{2\pi n}\int_0^{2\pi}e^{it}\erf(e^{it})^{-n}dt$$
which can be derived via [Lagrange inversion](https://en.m.wikipedia.org/... |
How helpful is $f^{-1}(z)=\frac1{2\pi i}\oint\ln(1-\frac z{f(w)})dw$, or the method to find it, in deriving integral representations of $f^{-1}(z)$? |
I'll expand on my comment above, but the comment gives the (local) intuition. Consider the map $\pi\colon V-\{0\}\to\Bbb P(V)$. Then for any $\xi\in W$, $\pi_{*\xi}\colon T_\xi V\to T_{\pi(\xi)}\Bbb P(V) = T_W\Bbb P(V)$. Note that $\pi_{*\xi}(W)=0$, so $\pi_*$ induces a surjective map (hence isomorphism) $T_\xi V/W = V... |
In the theory of CW complexes, the *incidence coefficient* between a cell $\sigma$ of dimension $k$ and a cell $\tau$ of dimension $k - 1$ of a complex is defined as the topological degree of the map $\phi$ that attaches $\tau$ into $\sigma$, or 0 if $\tau$ is not contained in $\sigma$.
If we write the incidence numbe... |
This will only be a partial answer aiming to give a strong necessary condition in the case where $f$ is assumed to be uniformly continuous over $\mathbb{R}$, making use of the theory of tempered distributions and the fact that the Fourier transform acts "nicely" on delay equations. We will show that uniformly continuou... |
Given a (say real) random matrix $M=(M_{i,j})_{1\leq i, j \leq N}$, the moments method consists in computing (the limits in $N$ of) the quantities $$ \mathbb{E} \left(\mathrm{tr} M^k\right)^{1/k}, $$
where $\mathrm{tr}$ is the normalised trace and $k$ a natural number. Computing such limits in the case of Hermitian ma... |
My question is to do with parametrisation of arclengths.
As part of a course on mechanics (with mathematical focus), I have covered intrinsic coordinates in a plane curve. In class and homeworks etc, we covered ideas such as determining the shape of a chain hanging under its own weight.
We also learned arclength... |
I am reading a book called ‘A Book of Set Theory’, by Charles C. Pinter. Quite early on he mentions the class axiom, which goes like ‘If S(x) is any statement about an object x, there exists a class which consists of all those elements x which satisfy S(x)’. The author says that this axiom helps us avoid semantic parad... |
Why does the class axiom suffice? |
If I have the following strictly decreasing sequence of positive real numbers $$((k + 1) a_k) \in c_0$$
where
$$c_0= \bigg\{ \left \{ x_n \right \}_{n\in \mathbb{N}} \subseteq\mathbb{R} \colon \text{lim}_{n\to \infty}x_n=0 \bigg\}$$
Is there any way or any criteria that the above implies that the series
$$\sum... |
Please tell me how to calculate this limit:
$$
\lim_{x \to 0}\frac{2x^7-\displaystyle{\int_{x^2}^{x^3} \text{sin}(t^2)dt}}{\text{tg}(x^6)}
$$
I want to apply L'Hopital's rule, but the problem is that the numerator contains an integer with variable limits...
*Try expanding the integral in a series? Or apply the... |
Consider some human action, C. It could be to commit a crime like launching a cyber-attack or embezzling funds.
How likely is it that there is an actor that intends to attempt C in the next year? Call it P(A)
If an attempt is made how likely is it that the attempt will succeed? Call it P(S).
Assume that a cre... |
If one event influences the other, are they independant? |
If one event influences the other, are they independent? |
**HINT**
Another possible approach. Let $|x - 4| < \delta_{\varepsilon}$. Then one concludes that:
\begin{align*}
|\sqrt{x} - 2| = \frac{|x - 4|}{\sqrt{x} + 2} < \frac{|x - 4|}{2} < \frac{\delta_{\varepsilon}}{2} < \varepsilon
\end{align*}
Can you proceed from here? |
Let $R$ be a ring with $1$. Let $M$ be a left $R$-module. For $r \in Z(R)$ the map $m^M_r: M \to M, x \mapsto rx$ is $R$-linear. A functor $F: \text{R-Mod} \to \text{R-Mod}$ preserves multiplication, if $F(m^M_r) = m^{F(M)}_r$ for all left $R$-modules $M$ and all $r \in Z(R)$.
Why does for an $(R,R)$-bimodule $M$ t... |
Tensoring and Hom functors preserve multiplication? |
Please tell me how to calculate this limit:
$$
\lim_{x \to 0}\frac{2x^7-\displaystyle{\int_{x^2}^{x^3} \text{sin}(t^2)dt}}{\text{tg}(x^6)}
$$
I want to apply L'Hopital's rule, but the problem is that the numerator contains an integral with variable limits...
*Try expanding the integral in a series. Or apply th... |
Let $R$ be a ring with $1$. Let $M$ be a left $R$-module. For $r \in Z(R)$ the map $m^M_r: M \to M, x \mapsto rx$ is $R$-linear. A functor $F: \text{R-Mod} \to \text{R-Mod}$ preserves multiplication, if $F(m^M_r) = m^{F(M)}_r$ for all left $R$-modules $M$ and all $r \in Z(R)$.
Why does for an $(R,R)$-bimodule $M$ t... |
The question I am trying to solve is below. I have proven it is an order but am unsure how to draw the diagram for it. Can someone point me in the right direction?
Let $A = \{1, 2, 3, 4\}$, and let $R$ be a binary relation on $A \times A$ given by: $((a, b),(c, d)) \in R$ if and only if $a$ divides $c$ and $b$ divid... |
I found an equivalent formulation of the ABC Conjecture in *[The ABC's of Number Theory][1]* by Prof. Noam D. Elkies of Harvard University which states that:
(Masser-Oesterl´e [Oe]): For every real $\epsilon > 0$ there exists $\kappa(\epsilon) > 0$ such that
$$N(ABC) > \kappa(\epsilon) \cdot C^{1−\epsilon}$$
h... |
I'm having trouble understanding what does the covariant derivative do in a coordinate system where we have changing basis vectors. I always thought it was giving us the coordinates while taking into account the change in basis vectors $\vec{a}_i(\xi^1,\xi^2,...,\xi^n)$. I tried to visualize it with the following scena... |
I have searched for numerical methods to solve IVP with irregular singular points but didn't find any.
What is a proper method I can use to solve such equations?
For example, we could consider the following:
$$
\begin{cases}
y''+\dfrac{y'}{x^2}+y=x^3+6x+3\\
y(0)=0, \\y'(0)=0.
\end{cases}
$$
I'm not interes... |
What are proper numerical methods to solve IVP at irregular singular point? |
Suppose you have the coordinates of three points $A,B,C$. You want to construct a plane $\pi$ such that
$ d(\pi, A) = r_1 $
$ d(\pi, B) = r_2 $
$ d(\pi , C ) = r_3 $
where $d(\pi,v) $ is the perpendicular distance function, which returns the distance between point $v$ and the plane $\pi$.
What are the... |
Let $g(x)$ be:
\begin{cases}
0 \space\space \text{if}\space\space x \in \mathbb Q \\
x\space\space \text{if} \space\space x \in \mathbb R - \mathbb Q \\
\end{cases}
That's what I've tried since I started:
If $g(x)$ is continuous so $\lim\limits_{x\to a}g(x) = g(a)$. By $\varepsilon - \delta$ ... |
I'm looking for some function that describes the variance of model weights when trained with Stochastic Gradient Descent for $m$ independent minibatches.
I can apply central limit theorem to a single minibatch of SGD training to get the result that the gradient estimate (and therefore the updated model) has variance... |
The sequence given by:
$$x_n=(1^{-1/q},2^{-1/q}-1^{-1/q}, 3^{-1/q}-2^{-1/q}, \dots)$$
That is,
$$\sum_{n=1}^{\infty}n^{-1/q}-(n-1)^{-1/q}$$
Is this sequence in the sequence space $\ell_p$ ? where for $1<p< \infty$ and $q>1$:
$$\ell_p= \left \{ \left \{ x_n \right \}_{n\in \mathbb{N}} \subseteq\mathbb{R} \colo... |
Let the *multiplicative closure* operator $h$ be defined on $\mathcal{P}(\mathbb{N})$ by setting $hA$ equal to the smallest set containing $A$ that is closed under multiplication.
In L. F. Meyers' solution to Problem E $1941$ on pp. $408$-$9$ of the April $1968$ *American Mathematical Monthly* (https://www.jstor.org/s... |
For non-square $n \times m$ real matrices, when will $\| A \hat x\|$ be constant for all unit vectors $\hat x$?
If $A$ is square, the answer is: If and only if $$A = \lambda I Q$$ where $\lambda$ is a scalar and $Q$ an orthogonal matrix. What about non-square matrices?
**Background & context:** This question cam... |
For non-square $n \times m$ matrices, when will $\| A \hat x\|$ be constant for all unit vectors $\hat x$? |
> let $ f(x)=ax^2+bx+c\in \Bbb R[x] $ be an irreducible polynomial
.Prove that the field $ \Bbb {R[x]\over <f(x)>} $ is isomorphic to the field of the complex number
Knowing that $ \Bbb {R[x]\over <f(x)>}$ =\{$h+gv|h,g\in \Bbb R;av^2+bv+c$}
.How can I detect an element $\alpha$ $\in\Bbb {R[x]\over <f(x)>}$ such t... |
The number field $E=\mathbb Q(\sqrt{\sqrt{2}-3})$ is a CM-field since it is a totally imaginary extension of the totally real field $\mathbb Q(\sqrt 2)$.
> Is there a way to construct an abelian surface defined over $E$ with CM by $E$?
I suppose the problem is intractable in general if $E$ is an arbitrary CM-fie... |
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