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It was George W. Whitehead who first gave a correct calculation of the $2$-stem $\pi_2^s$ in *The (n+2)nd Homotopy Group of the n-Sphere*. The problem comes down to determining the suspension homomorphism $E\colon\pi_4(S^2)\rightarrow\pi_5(S^3)$, which is known to be an epimorphism with domain cyclic of order $2$ gener... |
Is there a modern explanation of how Whitehead calculated $\pi_2^s$? |
Jacobson's 1945 paper, "[The Radical and Semi-Simplicity for Arbitrary Rings][1]", has
> Theorem 10. If $\mathfrak{N}$ is a right ideal with a finite basis contained in the radical $\mathfrak{R}$ then either $\mathfrak{N}\mathfrak{R} < \mathfrak{N}$ or $\mathfrak{N} =0$
Azumaya's 1951 "[On Maximally Central Algeb... |
One way to "visualize" the 4-torus is by $$T^4=S^1\times S^1\times S^1\times S^1$$ This is an ordered set of four angles.
We have the notion of [*Betti numbers*][1] and we often say that "$b_n$ counts the number of $n$-dimensional holes." For $T^4$, it turns out that Betti numbers are $1,4,6,4,1$. Contrast this wi... |
Let $$f(x,y)=\begin{cases}\dfrac{y^4}{x^2+y^2}\text{ if } (x,y)\neq (0,0)\\
0\hspace{1.4cm}\text{ if } (x,y)=(0,0).\end{cases}$$
I can compute $\dfrac{\partial f}{\partial x}(0,0)=\lim\limits_{x\to 0}\dfrac{f(x,y)-f(0,0)}{x-0}.$
How can I calculate $\dfrac{\partial^2f}{\partial x\partial y}(0,0)$? |
how to compute multivariable derivative of order 2? |
One way to "visualize" the 4-torus is by $$T^4=S^1\times S^1\times S^1\times S^1$$ This is an ordered set of four angles.
We have the notion of [*Betti numbers*][1] and we often say that "$b_n$ counts the number of $n$-dimensional holes." For $T^4$, it turns out that Betti numbers are $b_0=1,\ b_1=4,\ b_2=6,\ b_3=... |
Let $A \subset \mathbb{R}^2$ be the set containing only the "$x$-axis" of the real 2-D plane (that is $A = \{(x, y) : x \in \mathbb{R}, y = 0\}$). Then, any open set $E$ containing $A$ must contain a set $B$ of the form $$B = \bigcup_{x \in \mathbb{R}} \{x\}\times\{(-r_x, r_x)\}.$$ It was stated in the solution to a pr... |
I am currently reading notes for a probability class I am taking and am having trouble with notation. There are two notations for a sequence of random variables: $X_n$ ~ $n \cdot Bernoulli(\frac{1}{n})$ and $X_n (\omega)$ = $n \cdot \mathbb{1}_\omega \in [0, \frac{1}{n}]$. I was told that notation one and two are diffe... |
In the Argand plane, the points Q and A represent the complex numbers 4+6i and 10+2i respectively. If A, B, C, D, E, F are the vertices, taken in clockwise order, of a regular hexagon with centre Q, find the complex number which represents B. How do I find the coordinate of B? |
I am currently reading notes for a probability class I am taking and am having trouble with notation.
Let Ω = [0, 1], and let P be the uniform probability measure on Ω so that P([a, b]) = b − a
There are two notations for a sequence of random variables:
$X_n$ ~ $n \cdot Bernoulli(\frac{1}{n})$ and $X_n (\ome... |
Consider an operator $\eta$ on the (tensor) product of Hilbert spaces $\mathcal{H}_1\otimes \mathcal{H}_2$ such that its range is of the form $V_1\otimes V_2$. Does this imply that $\eta$ must be a product operator, i.e., of the form $\eta = A\otimes B$?
**Followup Question** (in response to @David Gao's counterexam... |
Let $M$ be a compact and connected smooth manifold with a symplectic form $\omega$.
$Ham(M, \omega)$ denotes the space of hamiltonian symplectomorphisms of $(M,\omega)$.
I have the following statement in my lecture notes:
> Using Darboux’s theorem one can show that the action of $Ham(M, \omega)$ on M is trans... |
We have this question where:
"Let S be a set with |S| = 7. In how many ways can we partition the set
S into three nonempty parts?"
While our understanding of the question is not clear, we have used the "Stirling numbers of the second kind" formula to find an exact solution of 301. But we can't work around how we... |
My attempt revolves around trying to represent $(e^{z^2} - 1 - z^2)sin^3z$ as $f(z) = z^na(z)$ where $a(z)$ is analytic at $z = 0$ and $a(0)$ ≠ $0$.
- $e^{z^2} - 1 - z^2$ = $\frac{z^4}{2!} + \frac{z^6}{3!} ...$
So, we have
- ($\frac{z^4}{2!} + \frac{z^6}{3!} ...)(z - \frac{z^3}{3!} + \frac{z^5}{5!} - ...)^3$
... |
I think what is confusing you is that you are "starting out" with a planar graph of $n+1$ nodes.
But note that you are not yet saying anything about its colorability. All you're trying to show here is that $P(n) \to P(n+1)$, or:
"Any planar graph with $n$ nodes is $6$-colorable" $\to$ "Any planar graph with $n+1$... |
My attempt revolves around trying to represent $(e^{z^2} - 1 - z^2)\sin^3z$ as $f(z) = z^na(z)$ where $a(z)$ is analytic at $z = 0$ and $a(0)\neq 0$.
$$e^{z^2} - 1 - z^2 = \frac{z^4}{2!} + \frac{z^6}{3!}+ \ldots$$
So, we have
$$\left(\frac{z^4}{2!} + \frac{z^6}{3!}+ \ldots\right)\left(z - \frac{z^3}{3!} + \... |
How Do I Prove that $\mathbb{E}^d\nrightarrow (l_4)_3$? |
It is easy to decide if a given integer $n$ is the sum of two squares, and in fact there is a simple formula (based on the prime factorization) to compute the number of ways that $n$ can be written as the sum of two squares (see for example [Wikipedia][1]). But I cannot find a discussion of how two different representa... |
$\color{black}{Background:}$
$\textbf{Proposition 1:}$ LEt $L$ be a group and $G$ and $H$ be two subgroups. Then, the following statements are equivalent:
$1.$ $L,G$ and $H$ satisfy the following relations: $L=GH, G\cap H=\{1\},$ and $|G,H|=\{1\}.$
$2.$ $L$ is the direct product of $G$ and $H,$ i.e. $... |
Let $A \subset \mathbb{R}^2$ be the set containing only the "$x$-axis" of the real 2-D plane (that is $A = \{(x, y) : x \in \mathbb{R}, y = 0\}$). Then, any open set $E$ containing $A$ must contain a set $B$ of the form $$B = \bigcup_{x \in \mathbb{R}} \{x\}\times\{(-r_x, r_x)\}.$$ It was stated in the solution to a pr... |
### Problem
> Let $a>0$ and $B$ be a standard $\mathbb{R}$-valued Brownian motion. Define the stopping time $S_a:=\inf\{t\geq 0\ \vert \left\lvert B_t\right\rvert = a\}$. Compute $\mathbb{E}\left[e^{-\frac{\lambda^2}{2}S_a}\right]$ for $\lambda\in\mathbb{R}$.
### My attempt
I noticed that the expression I need... |
[Left side, white up, Red, green, white corner swapped with Red, blue, white corner][1]
[Right side, white up, Red, blue, white corner swapped with Red, green, white corner][2]
[Right, Down, Rear view of cube.][3]
[1]: https://i.stack.imgur.com/gJCZw.jpg
[2]: https://i.stack.imgur.com/mEbhA.jpg
[3]: ... |
I'm reading Spivak's Calculus on Manifolds (I'm on page 89). For a differentiable function $f : \mathbb{R}^n \to \mathbb{R}^m$, $Df(p): \mathbb{R}^n \to \mathbb{R}^m$ is a linear transformation (on the tangent space at $p \in \mathbb{R}^n$, I believe); it is just the Jacobian matrix. Spivak then defines the following l... |
I have some questions regarding two basic results (but not so trivial to prove) in measure theory. It concerns the volumes of rectangles that cover another rectangle.
The notes I'm reading are from [here][1] by Hunter. But I have summarized all the necessary details in this post. My question concerns the proposition... |
[Left side, white up, Red, green, white corner swapped with Red, blue, white corner][1]
[Right side, white up, Red, blue, white corner swapped with Red, green, white corner][2]
[Right, Down, Rear view of cube.][3]
[By Transposing the corners, it is possible to get all four corners in the correct orientation bu... |
I'm interested in the ring $(\mathbb Z/30\mathbb Z,+,\times)$, the elements of which I'll write down in bold. For example $$\textbf{7}=7+30\mathbb Z=\{...,-53,-23,7,37,...\}$$
in particular with $$U:=(\mathbb Z/30\mathbb Z)^\times=\{\textbf{1},\textbf{7},\textbf{11},\textbf{13},\textbf{17},\textbf{19},\textbf{23},\tex... |
I would like to know an example showing that second-order mathematical induction is stronger than first-order mathematical induction.
For a condition $P(x)$ that can be written in the language of Peano arithmetic, we can construct the set $\{x\in\mathbb{N};P(x)\}$.
Which subset of $\mathbb{N}$ cannot be construct... |
[Left side, white up, Red, green, white corner swapped with Red, blue, white corner][1]
[Right side, white up, Red, blue, white corner swapped with Red, green, white corner][2]
[Right, Down, Back view of cube.][3]
[By Transposing the corners, it is possible to get all four corners in the correct orientation bu... |
I'm interested in the ring $(\mathbb Z/30\mathbb Z,+,\times)$, the elements of which I'll write down in bold. For example $$\textbf{7}=7+30\mathbb Z=\{...,-53,-23,7,37,...\}$$
in particular with $$U:=(\mathbb Z/30\mathbb Z)^\times=\{\textbf{1},\textbf{7},\textbf{11},\textbf{13},\textbf{17},\textbf{19},\textbf{23},\tex... |
I'm interested in the ring $(\mathbb Z/30\mathbb Z,+,\times)$, the elements of which I'll write down in bold. For example $$\textbf{7}=7+30\mathbb Z=\{...,-53,-23,7,37,...\}.$$
I'm interested in particular with $$U:=(\mathbb Z/30\mathbb Z)^\times=\{\textbf{1},\textbf{7},\textbf{11},\textbf{13},\textbf{17},\textbf{19},... |
We have this question where:
"Let S be a set with |S| = 7. In how many ways can we partition the set
S into three nonempty parts?"
While our understanding of the question is not clear, we have used the "Stirling numbers of the second kind" formula to find an exact solution of 301. But we can't work around how we... |
//* Once again they want to lower a question with a piranha attack, putting negative scoring without giving reasons in the comments. Do it, it is clear that this is an illustrated Twitter, where only repetition of what is already known is allowed, with no room for innovation or creative questioning.
"Self referenc... |
I'm interested in the ring $(\mathbb Z/30\mathbb Z,+,\times)$, the elements of which I'll write down in bold. For example $$\textbf{7}=7+30\mathbb Z=\{...,-53,-23,7,37,...\}.$$
I'm interested in particular with $$U:=(\mathbb Z/30\mathbb Z)^\times=\{\textbf{1},\textbf{7},\textbf{11},\textbf{13},\textbf{17},\textbf{19},... |
*Background:*
**Proposition 1:** Let $L$ be a group and $G$ and $H$ be two subgroups. Then, the following statements are equivalent:
1. $L,G$ and $H$ satisfy the following relations: $L=GH, G\cap H=\{1\},$ and $|G,H|=\{1\}.$
2. $L$ is the direct product of $G$ and $H,$ i.e. $L\cong G\times H.$
>Proof: ... |
I'm interested in the ring $(\mathbb Z/30\mathbb Z,+,\times)$, the elements of which I'll write down in bold. For example $$\textbf{7}=7+30\mathbb Z=\{...-83,-53,-23,7,37,67...\}.$$
I'm interested in particular with $$U:=(\mathbb Z/30\mathbb Z)^\times=\{\textbf{1},\textbf{7},\textbf{11},\textbf{13},\textbf{17},\textbf... |
I have a sequence of sets $(S_k)$. What could be the condition for $S_\infty$ (the one taken at infinity) to be subset of, or be equal to, the limit $\limsup S_k$ of the sequence?
Eg: $S_k = \{x \in \mathbb{R}: x/k = 0\}$. With this, $S_k = \{0\}$ for all $k$ and the sequence converges to this set, but $S_\infty = \... |
So recently I’ve been studying integrals of products of logarithms
$$\int \prod_{n=1}^N \log(x+a_n) dx$$
Can anyone come up with general method for dealing with such integral?
Let's start simple, with N = 2.
$$ \int \frac{\log(z+m)}{z} dz = \int \frac{\log(1-u)+\log m}{u} du = -Li_2(\frac{-z}{m}) + \log m\log z + C... |
I'm interested in the ring $(\mathbb Z/30\mathbb Z,+,\times)$, the elements of which I'll write down in bold. For example $$\textbf{7}=7+30\mathbb Z=\{...-83,-53,-23,7,37,67...\}.$$
I'm interested in particular with $$U:=(\mathbb Z/30\mathbb Z)^\times=\{\textbf{1},\textbf{7},\textbf{11},\textbf{13},\textbf{17},\textbf... |
I'm trying to prove that the set of discontinuities $D$ of Thomae's Function has measure zero using the complement of $D$, which is the set of points where $f$ is continuous, namely $([0,1] \cap \mathbb{I}) \times ([0,1] \cap \mathbb{I})$. I know that $[0,1] \cap \mathbb{I}$ does not have measure zero in $\mathbb{R}$. ... |
The set of irrationals in [0,1]x[0,1] has measure zero in R^2? |
The problem statement is the same as this post:
[$L^{\infty *}$ is not isomorphic to $L^1$ ](https://math.stackexchange.com/questions/300851/l-infty-is-not-isomorphic-to-l1?noredirect=1&lq=1).
>Let $L^\infty = L^\infty(m)$, where $m$ is Lebesgue measure on $I=[0,1]$
. Show that there is a bounded linear functional $... |
What's the best way to estimate exponents by hand? Say for example (1.07)^10 ~2, or like (1.07)^15, or (1.05)^15. Is there any standard way of doing these calculations? |
I'm interested in the ring $(\mathbb Z/30\mathbb Z,+,\times)$, the elements of which I'll write down in bold. For example $$\textbf{7}=7+30\mathbb Z=\{...-83,-53,-23,7,37,67...\}.$$
I'm interested in particular with $$U:=(\mathbb Z/30\mathbb Z)^\times=\{\textbf{1},\textbf{7},\textbf{11},\textbf{13},\textbf{17},\textbf... |
I have an equation of y = floor(400(x-6)^1.1) where x is equal to or greater than 6 and increases by an increment of 1.
I want to calculate what the sum of the equations added up together might be.
For example: floor(400([x+1]-6)^1.1) + floor(400([x+2]-6)^1.1) + floor(400([x+3]-6)^1.1) ... floor(400([x+n]-6)^1.1)... |
Let $(P, \le)$ be a partially ordered set. A subset $A\subseteq P$ is a *chain* if any two of its elements are comparable. A subset $A\subseteq P$ is *cofinal* if every element of $P$ is less than or equal to some element of $A$.
> **Question:** Suppose $P$ admits some cofinal chain. Does every cofinal subset $Q$... |
The set of irrationals in $[0,1] \times [0,1]$ has measure zero in $R^2$? |
I got a little bit confused about the definition of 'integrability' of a random variable.
Recall the definition:
**Definition.** Let $(\Omega , {\mathfrak{M}}, \mathbb{P}) $ be a probability space. A random variable $X$ defined on $\Omega$ is *integrable* if $\int_{\Omega} |X| d\mathbb{P} < \infty$.
I am not s... |
I have an equation of $y = \lfloor 400(x-6)^{1.1} \rfloor$ where x is equal to or greater than 6 and increases by an increment of 1.
I want to calculate what the sum of the equations added up together might be.
For example: $\lfloor 400([x+1]-6)^{1.1} \rfloor + \lfloor 400([x+2]-6)^{1.1} \rfloor + \lfloor 400([x+... |
Exactly $\frac{1}{5}$ people that live on earth have a condition
.There are two tests for this condition, the Z1 test and the Z2 test. When a person
goes to a doctor to test for this condition, with probability $\frac{2}{3}$ the doctor conducts Z1 on them and with probability $\frac{1}{3}$ the doctor conducts Z2 on ... |
I need to find how distant the center of an orthogonal rectangle is from the center of a circle, given a specific angle.
The dimensions of the rectangle are proportional to the circle radius, so they use a 0-1 range, where 1 is the radius of the circle. The objective is to get the position of the center of that rect... |
Let $d$ be the derivative map for de Rham cohomology. If we define $d_f=m_{-f}\circ d\circ m_f$ where $m_f(\omega)=e^f\omega$ for a smooth function $f:M\to\mathbb{R}$. Notice that $d_f^2=0$. How do we show that the cohomology defined by $d_f$ is isomorphic to the de Rham cohomology? I saw another question https://math.... |
Suppose we have two differentiable functions $f, g:\mathbb{R}\to\mathbb{R}$ such that $f(0)=0$, $g(0)=1$, $f'(x)=g(x)$ and $g'(x)=-f(x)$. Then it is possible to show not only that $f(x)=\sin(x)$ and $g(x)=\cos(x)$ satisfy these conditions, but that they are the only pair of functions that satisfy this condition. For in... |
It feels intuitive for this to be true. Let us say I have a chart from $\mathbb{R^2}$ to some section of a manifold in $\mathbb{R^3}$. If the distance in $\mathbb{R^2}$ between two points is equal to the shortest distance (locally, the geodesic) on the manifold in $\mathbb{R^3}$, does this imply the map is an isometry?... |
Does a map that preserves distance imply isometry? |
Veritasium Fourier transfer video [The Remarkable Story Behind The Most Important Algorithm Of All Time][1] explains the adding up (integral) the product of the wave in interest (blue) and a known sine (or cosine) is positive only when the known wave is part of the wave in interest.
[![enter image description here][... |
Exactly $\frac{1}{5}$ people that live on earth have a condition
.There are two tests for this condition, the Z1 test and the Z2 test. When a person
goes to a doctor to test for this condition, with probability $\frac{2}{3}$ the doctor conducts Z1 on them and with probability $\frac{1}{3}$ the doctor conducts Z2 on ... |
Given a model category $C$, I have two functors $F,G:\mathbb{N}\rightarrow C$, where see $\mathbb{N}$ as sequence category.
Question: Given a natural transformation $J:F\rightarrow G$ and suppose the $J$ induces pointwise weak equivalence then when will the Colimit be weak equivalence?
More specifically on the no... |
It is not true that rings with homeomorphic spectra are isomorphic; take any two nonisomorphic fields. When is it true that a ring homomorphism $f:A \rightarrow B$ induces a homeomorphism $f^{*}:\mathrm{Spec} B \rightarrow \mathrm{Spec} A$ given by $f^{*}(\mathfrak{p}) = f^{-1}(\mathfrak{p})$? |
All good. Perhaps, you can add more details on the "by symmetry" https://math.stackexchange.com/questions/404819/independence-of-t-and-b-t?rq=1 i.e. we have $B_t\stackrel{d}{=}-B_{t}$ and $S_{a}$ is only a function of $|B_{t}|$, which is symmetric already.
|
All good. Perhaps, you can add more details on the "by symmetry" https://math.stackexchange.com/questions/404819/independence-of-t-and-b-t?rq=1 i.e. we have $B_t\stackrel{d}{=}-B_{t}$ and $S_{a}$ is only a function of $|B_{t}|$, which is symmetric already.
One can also use pde methods to find the law for the exit t... |
The problem statement is the same as this post:
[$L^{\infty *}$ is not isomorphic to $L^1$ ](https://math.stackexchange.com/questions/300851/l-infty-is-not-isomorphic-to-l1?noredirect=1&lq=1).
>Let $L^\infty = L^\infty(m)$, where $m$ is Lebesgue measure on $I=[0,1]$
. Show that there is a bounded linear functional $... |
Veritasium Fourier transfer video [The Remarkable Story Behind The Most Important Algorithm Of All Time][1] explains the adding up (integral) the product of the wave in interest (blue) and a known sine (or cosine) is positive only when the known wave is part of the wave in interest.
[![enter image description here][... |
In this [other question](https://math.stackexchange.com/q/4838311/909869) I found an approximation as follows: $$\ln(x+1) \approx \dfrac{\ln(x)}{1-x^{-\frac{1}{\ln(2)}}}$$
By differentiating both sides, since just one logarithmic terms remains you could find the following approximation:
$$\ln(x)\approx \dfrac{\ln(2... |
> Consider an information source that produces numbers k in the set SX =
> {1, 2, 3, 4}. Find and plot the PMF in the following cases:
>
> (a) p<sub>k</sub> = p<sub>1</sub>/k, for k = 1, 2, 3, 4. Hint: Find
> p<sub>1</sub>.
I attempted to solve this problem by doing the following.
p<sub>1</sub>/1+ p<sub>1</s... |
What's the best way to estimate exponents by hand? Say for example $(1.07)^{10}$ $\sim2$, or like $(1.07)^{15}$, or $(1.05)^{15}$. Is there any standard way of doing these calculations? |
This question is in the spirit of [this question][1] I asked earlier.
[1]: https://math.stackexchange.com/q/4886446/1067109
Let $f, g:\mathbb{R}\to\mathbb{R}$ be differentiable functions such that $f(0)=0$, $g(0)=1$, $f'(x)=g(x)$, and $g'(x)=-f(x)$. Is it possible to show that both $f$ and $g$ are periodic ju... |
Is every element of $SL_3(\mathbb{F}_p[x])$ a torsion element?
I'm thinking of examples of torsion elements:
Any generator (since a generator p times is the identity.)
Any element that has a rotation matrix as one of its factors.
And there are a bunch more examples.
Then I can't think of any element that does ... |
Torsion elements of $SL_3(\mathbb{F}_p[x])$? (Quick question) |
I'm trying to understand exercise 6.7 in this article:
https://www.gameludere.it/2019/05/19/funzioni-generatrici-esponenziali-e-applicazioni/#1_Le_funzioni_generatrici_esponenziali
But there is a step I don't understand.
That is, he has this generating function:
$$G_{r}(x)=\sum_{n=0}^{\infty}{n \brace r} \frac{... |
I'm studying generating functions.
I'm trying to understand exercise 6.7 in this article:
https://www.gameludere.it/2019/05/19/funzioni-generatrici-esponenziali-e-applicazioni/#1_Le_funzioni_generatrici_esponenziali
But there is a step I don't understand.
That is, he has this generating function:
$$G_{r}(x)=\s... |
We are trying to solve the following Optimization Problem
$$
\begin{aligned}
& \min _{\left\{y_{i j}^{m}\right\}} \sum_{i \in I_{m}} f_{i j}^{m}\left(y_{i j}^{m}\right)+\sum_{j \in J} \Phi^{m}\left(z_{i j}^{m}\right) \\
& \text { subject to } y_{i j}^{m}-z_{i j}^{m}=0 \quad \forall i, j, m
\end{aligned}
$$
... |
Is every element of $SL_3(\mathbb{F}_p[x])$ a torsion element?
Here are my thoughts:
First of all, the group is noncommutative, so a torsion element is an element of finite order.
I'm thinking of examples of torsion elements:
Any generator (since a generator p times is the identity.)
Any element that has a r... |
In the book on Infinitesimal Differential Geometry (http://www.iam.fmph.uniba.sk/amuc/_vol-73/_no_2/_giordano/giordano.pdf, p.4)
there is such term as "standard infinitesimals". This infinitesimals can be represented as Cauchy sequence following way:
$x(t)=y(t)+o(t)$ for $t$->$0$
Let $x(t)=t$ and $y(t)=sin(t)$.
... |
How is it possible that Cauchy sequence represents two different numbers? |
In the book on Infinitesimal Differential Geometry (http://www.iam.fmph.uniba.sk/amuc/_vol-73/_no_2/_giordano/giordano.pdf, p.4)
there is such term as "standard infinitesimals". This infinitesimals can be represented as Cauchy sequence following way:
$x(t)=y(t)+o(t)$ for $t\to0$
Let $x(t)=t$ and $y(t)=\sin(t)$.
... |
Let $X$ be the collections of all non-empty compact subsets of $\mathbb{R}^2$, which has the Euclidean metric. Let $(X,d)$ be a metric space, where $d$ is the Hasudorff metric.
Is $X$ separable?
Furthermore, let $Y$ be the collections of all finite subsets of $X$. Let $(Y,d)$ be a metric space, where $d$ is the ... |
It is not true that rings with homeomorphic spectra are isomorphic; take any two nonisomorphic fields. It is also an exercise in Atiyah-Macdonald that the induced map on spectra is dense if and only if the kernel is contained in the nilradical of the source.
When is it true that a ring homomorphism $f:A \rightarrow... |
> Let $$f(x,y)=\begin{cases} y^2\ln\left(1+\dfrac{x^2}{y^2}\right)\text{
if }y\neq 0\\ 0\hspace{2.7cm}\text{ if }y=0.\end{cases}$$ Prove that
> $\dfrac{\partial^2f}{\partial x\partial y}$ and
> $\dfrac{\partial^2f}{\partial y\partial x}$ are discontinuous at
> $(0,0)$.
My attempt is to show $\lim\limits_{(a,b)... |
> Let $$f(x,y)=\begin{cases} y^2\ln\left(1+\dfrac{x^2}{y^2}\right)\text{
if }y\neq 0\\ 0\hspace{2.7cm}\text{ if }y=0.\end{cases}$$ Prove that
> $\dfrac{\partial^2f}{\partial x\partial y}$ and
> $\dfrac{\partial^2f}{\partial y\partial x}$ are discontinuous at
> $(0,0)$.
My attempt is to show $\lim\limits_{(a,b)... |
So I was goofing around in desmos when I plugged the function \
$f(x)=x^x$\
And noticed that desmos shows only the positive part of the graph. Why does this happen. The function has only real values. Can you explain why it doesn't work? |
> Let $$f(x,y)=\begin{cases} y^2\ln\left(1+\dfrac{x^2}{y^2}\right)\text{
if }y\neq 0\\ 0\hspace{2.7cm}\text{ if }y=0.\end{cases}$$ Prove that
> $\dfrac{\partial^2f}{\partial x\partial y}$ and
> $\dfrac{\partial^2f}{\partial y\partial x}$ are discontinuous at
> $(0,0)$.
My attempt is to show $\lim\limits_{(a,b)... |
I am reading an interesting paper [One of the numbers ζ(5), ζ(7), ζ(9), ζ(11) is irrational](https://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=rm&paperid=427&option_lang=eng) by V.V. Zudilin. We fix odd numbers $q$ and $r$, $q\geq r+4$ and a tuple $\eta_0,\eta_1,...,\eta_q$ of positive integer parameters satis... |
Consider the following questions, the whole exercise is dedicated to determing the square root of negative numbers, after introducing the complex numbers. Eg.
$$\sqrt{-144}$$
Solutions for the whole exercise only give one solution for each problem, the above being $12i$.
However, this is providing to be rathe... |
So I was goofing around in desmos when I plugged the function \
$f(x)=x^x$\
And noticed that desmos shows only the positive part of the graph. Why does this happen. The function has real values. Can you explain why it doesn't work? |
*Too long for comments.*
Using q-Pochhammer symbols amke some inexpected factors to appear in the problem.
$$f_n(a) = \prod_{i=0}^n ( \sin(i) +a)$$
$$f_n(a) = i\,\frac {(-1)^n}{2^{n+1} }\,e^{-\frac{i}{2} n (n+\pi +1)}\left(i \left(a-\sqrt{a^2-1}\right);e^i\right){}_{n+1} \left(i
\left(a+\sqrt{a^2-1}\right... |
We have just started covering the limit of sequences and I've stumbled upon this limit in our uni's excercises:
$$\lim_{n\rightarrow \infty} {\sqrt[n]{ln(n)}}$$
I've considered solving it using the fact that $\lim_{n\rightarrow \infty} {\sqrt[n]{a}}=1$ for $a>0$. And since we're dealing with natural numbers, with... |
Is this a correct approach to calculating $\lim_{n\rightarrow \infty} {\sqrt[n]{ln(n)}}$? |
There is always one principal root. The radical "$\sqrt{}$" always indicates the positive root if the roots are real. e.g. $\sqrt {144} = 12.$ It does not equal $-12.$ Yet, $(-12)^2 = 144$
Or, we could say $x^2 = 144$ has two solutions $x = \pm 12.$
How does this work with principal roots? If we take the ... |
I saw this proof in Harvey Friedman [notes][1] titled THE AXIOMATIZATION OF SET THEORY BY EXTENSIONALITY, SEPARATION, AND REDUCIBILITY.
This a quote from page 5.
> LEMMA 1.2. Let $\varphi$ be a formula in $L(\in)$ whose free variables
> are among $x_1,…,x_n , n \geq 0$. let $\varphi^W$ be the result of
> rela... |
I’m hoping at some point to begin working through Cocchiarella and Freund’s book *Modal Logic: Its Syntax and Semantics*. Have any of you used this book before? What would you say are its prerequisites, if any? I’ve heard that it’s [overly technical][1] and tends not to sufficiently explain or justify its idiosyncrasie... |
Prerequisites for Cocchiarella modal logic book? |
While I find information on Modified Newton's method, [I have read this post from mathstackexchange.](https://math.stackexchange.com/questions/312819/rate-of-convergence-of-modified-newtons-method-for-multiple-roots)
In short, a modified Newton's method, $x_{n+1} = x_n - k\frac{f(x_n)}{f'(x_n)} ,$ such that ,in the ... |
Given polynomials of the form $$(1+x+x^2+x^3+\cdots+x^k)^n $$ We can calcualte the coefficients by writing it in the form $$(1-x^{k+1})^n \over (1-x)^n$$ and using the power series $(1-x)^{-n}$, as has been explained nicely on SE [here](https://math.stackexchange.com/questions/3656079/coefficient-of-xi-in-xx2-xkn?) and... |
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Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of... |
While I find information on Modified Newton's method, [I have read this post from mathstackexchange.](https://math.stackexchange.com/questions/312819/rate-of-convergence-of-modified-newtons-method-for-multiple-roots)
In short, a modified Newton's method, $x_{n+1} = x_n - k\frac{f(x_n)}{f'(x_n)} ,$ such that ,in the ... |
this is question from my country university exam entrance in Indonesia called utbk I'm very confused about the sufficiency data
Is the set {x| x<−1orx>5
, x is a real number} the solution set for |m−3x|>n
? Decide whether statements (1) and (2) below are sufficient to answer the question.
(1) m=6 and n=-9
(2... |
after many researches on the subject, I can't find any convincing argument anywhere, so I come to you about this problem which has been brought by some of my high school students.
Let $ABC$ be a random triangle and $I$ his incenter. It is known that $AB=c$, $AC=b$ and $BC=a$. I'm looking for a clean way to express ... |
Here are the first $40$ integers for which $\dfrac{3^n+1}{n+1}$ is an integer. I've denoted their residue mod $72$ by color and a symbol.
$$\begin{array}{r}\dagger\,\color{violet}0,&\bullet\,\color{brown}1,&*\,\color{red}3,&27,&531,\\
1\,035,&4\,635,&6\,363,&11\,475,&19\,683,\\
4\,131,&80\,955,&*\,\color{red}{266\,4... |
Lets say we have a series $a_n(-1)^n$
- lets assume we are finding the remainder of the series
Looking at the series $\sum_1^{\infty}a_n(-1)^n$ I know that if the series converges to S that the nth partial sum is $S_n$
- Cool, but the nth partial sum does not give us the sum of the whole series
I underst... |
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