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How does the book *Functions of Several Variables* by Wendell Fleming compare to texts like Spivak *Calculus on Manifolds*, Munkres *Analysis on Manifolds*, C.H Edwards *Advanced Calculus of Several Variables*? I know one difference is that Fleming uses Lebesgue integration in his integration chapter. But in terms of... |
How does Fleming's Functions of Several Variables compare to other texts? |
I am trying to find the time taken by a bead to slide down the Brachistochrone curve (between 0,0 and 1,-1) under kinetic friction, but I'm getting inconsistent answers with analytical and numerical methods.
[This article on Wolfram MathWorld][1] solves the problem analytically, giving the equations:
$$
x = a[(\... |
I am trying to find the time taken by a bead to slide down the Brachistochrone curve (between 0,0 and 1,-1) under kinetic friction, but I'm getting inconsistent answers with analytical and numerical methods.
[This article on Wolfram MathWorld][1] solves the problem analytically, giving the equations:
$$
x = a[(\... |
I am trying to evaluate the trigonometric Fourier series expansions of three quasi-square waves, which are essentially the same quasi-square waveform, phase-shifted with respect to each other by multiples of $\frac{2\pi}{3}$. The three waveforms are as shown:
[Phase-shifted quasi-square waveforms][1]
$$
S_a(x) =... |
Here is something I've found on the internet
$$\begin{aligned}
f-\int f&=1\\
\left(1-\int\right)f&=1\\
f&=\left(\frac1{1-\int}\right)1\\
&=\left(1+\int+\int\int+\dots\right)1\\
&=1+\int1+\int\int1+\dots\\
&= 1+x+\frac{x^2}2+\dots\\
&= e^x
\end{aligned}$$
At first I interpreted this as a joke, but on closer in... |
Given a real number $x>1$ and a natural number $n$, what can we say about the integer part of $x^n$ in terms of $x$ and $n$?
For simplicity, let us assume $x<2$. For the first few values of $n$, $\left\lfloor x^n\right\rfloor =1$. But, once it crosses $1$, it starts growing quite rapidly. Is it possible to express $... |
I am trying to evaluate the trigonometric Fourier series expansions of three quasi-square waves, which are essentially the same quasi-square waveform, phase-shifted with respect to each other by multiples of $\frac{2\pi}{3}$. The three waveforms are as shown:
[Phase-shifted quasi-square waveforms][1]
$$
S_a(x) =... |
I've seen plenty of analysis on the derivative of a NN output with respect to a given input, but am unsure on how to compute the partial with respect to the weight of a given edge. That is, suppose our output of the network is $\hat{pred} = \sigma[v_{output}]$, i.e this is a node with no outgoing edges. What is the der... |
Consider the situation where we have two [cubic] Bézier curves with the following properties:
- They share one common point (end of curve 1 = start of curve 2)
- They have the same direction at the point where they meet (a smooth join)
- We suspect that they are likely to be the result of having split a single ... |
I am trying to find rational parametrization of the rational points on the hyperboloid $z^2=-5x^2+4y^2+9$. I found many answers about parametrization of simple hyperbolic curves, but I cannot find the case of hyperboloids. I've tried some substitutions like $\sqrt{5}x=\sqrt{4y^2+9}sin\theta,\ z=\sqrt{4y^2+9}cos\theta$ ... |
I am trying to evaluate the trigonometric Fourier series expansions of three quasi-square waves, which are essentially the same quasi-square waveform, phase-shifted with respect to each other by multiples of $\frac{2\pi}{3}$. The three waveforms are as shown:
[Phase-shifted quasi-square waveforms][1]
$$
S_a(x) =... |
I understand that the probability of absorbing eventually is one in a simple random walk with a single absorbing boundary.
However, if we depart from this simple random and consider biased random walks with unequal step sizes, the absorbing probability may not always be one. Specifically, let $S_n = X_1 + \cdots + X... |
[![enter image description here][1]][1]
[1]: https://i.stack.imgur.com/7G9Ju.jpg
I have this question but I'm stuck on the integral - usually the r gets cancelled out but this time they're both r^2? I can't tell what I'm not seeing or doing wrong. |
Suppose $A\in \mathbb{R}^{n\times n}$ is a symmetric random matrix, with iid centered Bernoulli non-diagnonal entries $A_{ij} \sim \mathrm{Bernoulli}(p) - p$, for some $p \in (0,1)$, and zeros on its diagonal. For non-diagonal entries, We have $\mathbb{E}[A_{ij}] = 0$ and $\mathrm{Var}(A_{ij}) = p(1-p)$.
If $p$ is fix... |
Suppose $A\in \mathbb{R}^{n\times n}$ is a symmetric random matrix, with iid centered Bernoulli non-diagnonal entries $A_{ij} \sim \mathrm{Bernoulli}(p) - p$, for some $p \in (0,1)$, and zeros on its diagonal. For non-diagonal entries, We have $\mathbb{E}[A_{ij}] = 0$ and $\mathrm{Var}(A_{ij}) = p(1-p)$.
If $p$ is fix... |
We have two independent random variables:
1. r ~ U(0,1)
2. θ ~ U(0,2π)
Also we have X = r.cosθ and Y = r.sinθ.
Are X and Y independent? |
Are functions of two independent random variables also independent? |
Consider in some category
$Ker \phi \xrightarrow{\phi'}A \xrightarrow{\phi}B \xrightarrow{\phi''}Coker \phi$
How are the maps $\phi'$ and $\phi''$ defined? Does it depend on the category? I mean what are $\phi'(x)=?$ and $\phi''=?$
I am guessing the first one is the inclusion map?
I am hoping to check knowi... |
$Ker \phi \xrightarrow{\phi'}A \xrightarrow{\phi}B \xrightarrow{\phi''}Coker \phi$ How are the maps $\phi'$ and $\phi''$ defined? |
I am reading "Introduction to Topology" by Gamelin and Greene (2nd edition) and I am confused by what appears to be an inconsistency in notation for subsets and strict subsets.
The image below shows page 61 from this book. From the solid red highlighted expressions, it appears that $A \subset B$ means $A$ is a subse... |
I wanted to find the properties of a circle from 3 points that are on its circumference, I understand how the problem should be solved, but the provided code after the explanation, does not match the approach, the code is as below:
```python
def define_circle(p1, p2, p3):
"""
Returns the center and radiu... |
Say, $X_1, ..., X_n$ is Poisson($\theta$) distributed. I have to find an unbiased estimator for $\theta^2$. First I have to show that the estimator $T = \bar{(X)}$$^2$ is not unbiased:
$\mathbb{E}\bar{(X)}$$^2$ $ = \mathbb{E}(\frac{1}{n} \sum_{i=1}^{n}X_i)^2
= \frac{1}{n^2}\sum_{i=1}^{n}\mathbb{E}(\bar{X}^2) = \fra... |
You are correct. The empty set $\emptyset$ does not have any balls. Not under standard definitions anyway.
An open ball of radius $r>0$ centered around $x\in M$ is defined as
$$B(x,r)=\{y\in M\ |\ d(x,y)<r\}$$
Then an open ball (without specifying $x$ and $r$) is typically defined as $B(x,r)$ for some $x\in M$... |
Ladies/Gentlemen,
I need a formula for $$\int_0^{\pi } \frac{\sin (n \sigma )}{(a-\cos (\sigma ))^2} \, d\sigma$$ with a>1 and n positive integer. Can someone please let me know how I can get it?
Please note that I already have (if that helps) a formula for $$\int_0^{\pi } \frac{\cos (n \sigma )}{(a-\cos (\sigma )... |
I’m a little bit puzzled by geometrical proofs, like the common algebraic proof for the pythagorean theorem listed [here](https://en.wikipedia.org/wiki/Pythagorean_theorem). I understand the idea of arranging the right triangles and the area $c^2$ in a neat way to form another square and writing the area of the new squ... |
Am I missing some arguments for geometrical proofs? |
This is a question on the mechanics of backpropogation derivatives in NNs. I've seen plenty of analysis on the derivative of a NN output with respect to a given input, but am unsure on how to compute the partial with respect to the weight of a given edge. That is, suppose our output of the network is $\hat{pred} = \sig... |
I’m a little bit puzzled by geometrical proofs, like the common algebraic proof for the Pythagorean theorem listed [Wikipedia's "Pythagorean theorem" entry](https://en.wikipedia.org/wiki/Pythagorean_theorem).
I understand the idea of arranging the right triangles and the area $c^2$ in a neat way to form another squ... |
How do we know that common rearrangement proofs of the Pythagorean theorem work for any right triangle? |
This is a question on the mechanics of backpropogation derivatives in NNs. I've seen plenty of analysis on the derivative of a NN output with respect to a given input, but am unsure on how to compute the partial with respect to the weight of a given edge. That is, suppose our output of the network is $\hat{pred} = \sig... |
You are correct. The empty set $\emptyset$ does not have any balls. Not under standard definitions anyway.
An open ball of radius $r>0$ centered around $x\in M$ is defined as
$$B(x,r)=\{y\in M\ |\ d(x,y)<r\}$$
Then an open ball (without specifying $x$ and $r$) is typically defined as $B(x,r)$ for some $x\in M$... |
While studying the structure of groups of order $p^a.q^b$, I may have proved that the Fitting length of such group (which is always solvable by Burnside Theorem) is at much 2 when max$(a,b) \leq$ min$(p,q)$. Since I encounter some (hopefully minor) technical difficulties, I am not fully sure of the result.
If true, ... |
Fitting length of finite groups of order $p^a.q^b$? |
Let $n,b\in\mathbb{N}$ and $P_{n_b}$ is polynomial whose coefficients *are digits* of $n$ in base $b$.
Let $Q_{1}(x) \cdot Q_{2}(x) \cdots Q_{m}(x)$ - polynomial factorization (over integers) of $P_{n_b}$.
I'm interested in *coefficients of such factorization*, especially *union of all factor's coefficients* $\ma... |
Consider in some category
$Ker \phi \xrightarrow{\phi'}A \xrightarrow{\phi}B \xrightarrow{\phi''}Coker \phi$
How are the maps $\phi'$ and $\phi''$ defined? Does it depend on the category? I mean what are $\phi'(x)=?$ and $\phi''(x) =?$
I am guessing the first one is the inclusion map?
I am hoping to check k... |
I have the following optimization
\begin{align}
\max_{\|x\|^2\le1} \|Lx - y\|
\end{align}
where $L$ is a lower triangular and $y$ is a given vector.
Does it admit a closed-form solution?
I am interested in the general case where $L$ may be not invertible and even not square, but any progress under some assump... |
You are correct. The empty set $\emptyset$ does not have any balls. Not under standard definitions anyway.
An open ball of radius $r>0$ centered around $x\in M$ is defined as
$$B(x,r)=\{y\in M\ |\ d(x,y)<r\}$$
Then an open ball (without specifying $x$ and $r$) is typically defined as $B(x,r)$ for some $x\in M$... |
Let $M$ be an $A,B$-bimodule and $C$ a ring. Show that the functor $\text{Hom}_A(-,M)$, from the opposite category of $A,C$-bimodules to $C,B$-bimodules, is a right adjoint. [Hint: its left adjoint is essentially $\text{Hom}_B(-,M)$, with "op" in the right places.]
Following the hint, I tried to prove that $\text{H... |
Let $\Delta=\sqrt{1-k^2\sin^2x}$, $E(k)=\int_0^{\frac\pi 2}\Delta dx$ and $K(k)=\int\frac{dx}{\Delta}.$
[Start wearing purple,][1] gives a nice answer for the interesting identity $\int_0^{\frac\pi 2}\frac{dx}{\Delta^3}=\frac{E(k)}{1-k^2}.$
It is stuck in my head: Is $\int_0^{\frac\pi 2}\frac{dx}{\Delta^2}$ an el... |
Is $\int_0^{\frac\pi 2}\frac{dx}{1-k^2\sin^2x}$ an elliptic integral? |
You are correct. The empty set $\emptyset$ does not have any balls. Not under standard definitions anyway.
An open ball of radius $r>0$ centered around $x\in M$ is defined as
$$B(x,r)=\{y\in M\ |\ d(x,y)<r\}$$
Then an open ball (without specifying $x$ and $r$) is typically defined as $B(x,r)$ for some $x\in M$... |
Consider the space of symmetric matrices $symm(M)$ over reals of dimension $n \times n$. It is clear that there is a straightforward basis for this space where for any $i \ge j$ $M_{ij}(m,n) = 1$ if $m = i,j$ and $n = j,i$, otherwise 0 everywhere.
I'm trying to understand the following equation:
$$vec(M)\cdot vec... |
Prove that an (additive) functor $F$ between abelian categories (categories of modules) that admits an exact left adjoint must preserve injectives. State and prove the dual result.
I have no idea on how to solve this question. I know that right adjoints preserve limits and left adjoints preserve colimits, so it suff... |
If I have $w = f(x,y)$ where $x = u + v$ and $y = u - v$, then why does $\frac{\partial}{\partial u}w_x = \frac{\partial w_x}{\partial x}\cdot\frac{\partial x}{\partial u} + \frac{\partial w_x}{\partial y}\cdot\frac{\partial y}{\partial u}$
Is my assumption that $\frac{\partial}{\partial u}w_x = \frac{\partial^2 w}{... |
Why is this statement true? |
Well actually I know what gradient is and what that means. My professor once said that gradient is actually a covector in passing. As I know, covector is a linear map that maps vector to a scalar but gradient maps scalar to a vector.
So I guess he meant divergence $ \nabla \cdot $, not gradient $ \nabla $.
By tha... |
Am I correct in terms of understanding gradient and covector? |
> But I’m a little confused on how you actually know that the right
> triangle you draw isn’t just the one special right triangle with which
> you can actually form such a square.
If I am not mistaken, this is the figure you are referring to.
[![enter image description here][1]][1]
Ask yourself, what is it a... |
Hungerford's book of algebra has exercise $6$ chapter $3$ section $6$ [Probably impossible with the tools at hand.]:
> Let $p \in \mathbb{Z}$ be a prime; let $F$ be a field and let $c \in
F$. Then $x^p - c$ is irreducible in $F[x]$ if and only if $x^p - c$
> has no root in $F$. [Hint: consider two cases: $\math... |
**Can I truncate the intermediate values without losing absolute accuracy?**
Short answer: **no**.
Not so short answer: Regarding add (and substraction) *absolute* errors sum up, for multiplication (also dividing) *relative* errors sum up. While only interested in a result of 20 digits (or bits in your case), you... |
Well actually I know what gradient is and what that means. My professor once said that gradient is actually a covector in passing. As I know, covector is a linear map that maps vector to a scalar but gradient maps scalar to a vector.
So I guess he meant divergence $ \nabla \cdot $, not gradient $ \nabla $.
By tha... |
> If I have $w = f(x,y)$, where $x = u + v$ and $y = u - v$, then why is the following true?
$$\frac{\partial}{\partial u}w_x = \frac{\partial w_x}{\partial x}\cdot\frac{\partial x}{\partial u} + \frac{\partial w_x}{\partial y}\cdot\frac{\partial y}{\partial u}$$
Is my assumption that $\frac{\partial}{\partial u}w... |
>Let $n\in\mathbb{N}\setminus\{1,2\}.$ Let $a_1,a_2,\ldots,a_n$ be $n$ (not necessarily distinct) real numbers each in the interval $(1,F_n),$ where $F_n$ is the $n^{\text{th}}$ Fibonacci number. Show that we can pick three numbers $a_i,a_j,a_k$ from the $n$ real numbers such that there exists a triangle with lengths $... |
>Let $n\in\mathbb{N}\setminus\{1,2\}.$ Let $a_1,a_2,\ldots,a_n$ be $n$ (not necessarily distinct) real numbers each in the interval $(1,F_n),$ where $F_n$ is the $n^{\text{th}}$ Fibonacci number. Show that we can pick three numbers $a_i,a_j,a_k$ from the $n$ real numbers such that there exists a triangle with lengths $... |
>Let $n\in\mathbb{N}\setminus\{1,2\}.$ Let $a_1,a_2,\ldots,a_n$ be $n$ (not necessarily distinct) real numbers each in the interval $(1,F_n),$ where $F_n$ is the $n^{\text{th}}$ Fibonacci number. Show that we can pick three numbers $a_i,a_j,a_k$ from the $n$ real numbers such that there exists a triangle with lengths $... |
>Let $K$ be a field of characteristic $p > 0$, and let $a ∈ K$ be an element with the following property:
$$(\forall \beta ∈ K)(\beta^p ≠ a).$$
Let $k ∈ ℕ$ be arbitrarily given. Is it then true that $X^{p^k} - a$ is irreducible in $K[X]$?
We know that in the extension $K ⊆ K[X]/(X^{p^k} - a) \cong K(\sqrt[p^k]{a})... |
An exercise from Lang.
> Let $k$ be a field. Let $n$ be a nonnegative integer.
>
> Consider the polynomial $x^{p^n}-a \in k[x]$.
>
> Assume that $\text{char}(k) = p > 0$.
>
> Prove that either the polynomial $x^{p^n}-a$ is irreducible over the field $k$ or there exists $b\in k$ such that $a = b^p$.
How to p... |
Let $H$ be the classical Weil height for a number field (of degree $n$ say), i.e. given $\alpha$ an element of this field and $f(x) = a_0x^n + \cdots + a_n \in \mathbb{Z}[X]$ its minimal polynomial, the height is defined by
$$H(\alpha)^n = |a_0|\prod_i \max(1, |\alpha_i|)$$
where the $\alpha_i$ are the different comp... |
**Question:** I have two random variables $X\sim\text{Bernoulli}(\alpha)$ and $Y\sim\mathcal{N}(0,1)$. Is it possible to compute the mutual information $I(X;X+Y)$ analytically?
**My attempt:** By the definition of mutual information, we have
$$
I(X;X+Y)=H(X)-H(X|X+Y),
$$
where $H(X)$ can be worked out analytica... |
Given an inclusion $P\subset \Gamma$, it induces a homomorphism $\phi:\hat P\to \hat\Gamma$ on profinite completions. Then is that true that $\phi$ is injective iff given any normal subgroup of finite index $R \subset P$, there exists a subgroup of finite index $S \subset \Gamma$ such that $S\cap P \subset R$?
The ... |
I'm currently reading Riemannian Geometry by Saski, and just finished reading Proposition 4.1 regarding the connection map of a an affine connection. In particular, suppose that $M$ is a smooth manifold. Then the vertical subspace at $(p, v) \in TM$ is defined as $V_{(p, v)} = T_{(p,v)} T_p M$ (or equivalently, as the ... |
Let $E$ be a vector space.
I need to show that the union of two subspaces $U$ and $W$ is a subset of their sum, ie
$$ U \cup W \subset U+W$$
So let $$ v \in U \cup W$$
such that $ v \in U$
we have:
$$ v = v + 0$$
with $ 0 \in W$, where $ 0$ is the null vector ;
so
$ v \in U+W$;
thus the union is a subs... |
I need to show that the union of two subspaces $U$ and $W$ (of some vector space) is a subset of their sum, i.e.
$$ U \cup W \subset U+W.$$
So let $$ v \in U \cup W$$
such that $ v \in U$
we have:
$$ v = v + 0$$
with $ 0 \in W$, where $ 0$ is the null vector ;
so
$ v \in U+W$;
thus the union is a subset o... |
Setting: we are given a smooth curve $\gamma: \mathbb{R} \rightarrow \mathbb{R}^n$
Informal Question: Is it possible that $\gamma$ is a straight line on $[a,b]$, but not a straight line on $[a,b]^c$?
Formal Question: It is possible that $\gamma''(t)=0$ for all $t\in [a,b]$, while $\gamma''(t)\neq 0$ for some $t\n... |
Can a smooth curve have a segment of straight line? |
On https://openstax.org/books/calculus-volume-1/pages/2-5-the-precise-definition-of-a-limit, example 2.41, they used a Geometric Approach, which i find hard to understand.
I have 2 questions:
1.
They stated that Choose $=\min\{2−\sqrt{4−},\sqrt{4+}-2\}$, and uses $- \left(2 - \sqrt{4 - \epsilon}\right) \leq -\... |
It is known to all that for inner product space $H$, Cauchy-Schwarz inequality hold: $| \langle x,y \rangle | \leq \| x \| \cdot \| y \| \ \forall x,y\in H$, and for a normed space $X$ without inner product structure, we have two similar "Cauchy inequality":1.[real case][1](use some algebraic substitution); 2.[complex]... |
I am fairly new to proof writing so I would appreciate if you could help me out!
> Suppose that $(X^*, Y^*)$ and $(X^0, Y^0)$are saddle points of a zero-sum game with payoff matrix $A$. Show that $(X^*, Y^0) and (X^0, Y^*)$ are also saddle points.
This is what I know: If $(X^*, Y^*)$ is a saddle point for a zero... |
Why is $2sin\theta$ the upper limit for this inner integral, and why is $0$ the lower limit? |
Let $T_{n}$ be a sequence of real numbers such that $T_{n+1}\ge T_{n}^{2}+\frac{1}{5}$
Prove that $\sqrt{T_{n+5}}\ge T_{n}$.
I rearranged the first expression in terms of $T_{n}$, but seemingly to no avail. Are there any other pointers for solving this problem? |
Why is $2\sin\theta$ the upper limit for this inner integral, and why is $0$ the lower limit? |
I am in a pickle, and I would genuinely appreciate it if you could guide me.
I spent 2 years to find a way to develop a ratio scale, and I did it; however, it cannot be used the way it is.
The scale consists of 17 values 1/9, 1/8 , 1/7, 1/6, 1/5 , 1/4, 1/3, 1/2, 0 , 2, 3, 4, 5, 6,7, 8, 9, and I need to transform ... |
Suppose $X$ is a finite set and $f:X\rightarrow \mathbb{R}$ satisfies $\sum_{x\in X}f(x)=0$. Let $p\in\Delta(X)$ be a probability measure on $X$. Does the following statement hold?
$$
\sum_{x\in X} f(x)(p(x))^2 \ge 0 \Rightarrow \sum_{x\in X} f(x)p(x) \ge 0.
$$
I find it hard to find any counterexamples. I know t... |
You are correct. The empty set $\emptyset$ does not have any balls. Not under standard definitions anyway.
An open ball of radius $r>0$ centered around $x\in M$ is defined as
$$B(x,r)=\{y\in M\ |\ d(x,y)<r\}$$
Then an open ball (without specifying $x$ and $r$) is typically defined as $B(x,r)$ for some $x\in M$... |
Hint:
$u:=\sqrt{x}, v:=\sqrt{y}$ where $u,v \ge 0;$
Consider
$0 \le u,v <1;$
Numerator : $u^2v^2 \le uv \le (u+v)^2.$
Can you finish? |
Below is a more complete and rigorous proof than the other existing answers.
## Abstract
We will prove the formula for the number of $r$-combinations $_nC_r$ by showing that $_nP_r = r! \cdot {_nC_r}$. We show the latter by first proving $_nP_r \ge r! \cdot {_nC_r}$, and then showing that $_nP_r \not> r! \cdot {_... |
Two variable change every year, e.g. $a_1=500$, $a_2=600$ and $b_1=1000$ and $b_2=800$. Their ratio change is $$YoY=a_2/b_2-a_1/b_1=600/800-500/1000=0.25$$
The negative change in the denominator is a negative event and I want to reflect this, so e.g. in the example above $YoY$ is smaller due to the change in the denom... |
I want to show that tanh(x) is strictly increasing using tanh'(x). I know that
$$ tan(x) = \frac{sinh(x)}{cosh(x)}$$
$$ tan'(x) = \frac{sinh'(x)}{cosh'(x)}$$
$$=\frac{cosh^2(x)-sinh^2(x)}{cosh^2(x)}$$
$$= \frac{1}{cosh^2(x)}$$
And I know that $\frac{1}{cosh^2(x)}$ = $1-tanh^2(x)$. I also am aware that if a functio... |
Below is a more complete and rigorous proof than the other existing answers.
## Abstract
We will prove the formula for the number of $k$-combinations $_nC_k$ by showing that $_nP_k = k! \cdot {_nC_k}$. We show the latter by first proving $_nP_k \ge k! \cdot {_nC_k}$, and then showing that $_nP_k \not> k! \cdot {_... |
Two variables change every year, e.g. $a_1=500$, $a_2=600$ and $b_1=1000$ and $b_2=800$. Their ratio change is $$YoY=a_2/b_2-a_1/b_1=600/800-500/1000=0.25$$
The negative change in the denominator is a negative event and I want to reflect this, so e.g. in the example above $YoY$ is smaller if the change in the denomina... |
Out of the different [divisibility tricks][1] there's a really simple rule that works for more than one divisor: The digit sum. Specifically, if your digit sum is a multiple of $1,3$ or $9$ then so was the original number (although the $1$ case is trivial it'll make sense to include it for later).
One can then ask: ... |
I have the following question:
Let $a\in \mathbf{R}$ sucht that $X = L[a] \cap 2^{\omega}$ is uncountable. Why is $X$ is a $\Sigma_2^1$ set?
Thank you very much for any help in advance. |
Out of the different [divisibility tricks][1] there's a really simple rule that works for more than one divisor: The digit sum. Specifically, if your digit sum is a multiple of $1,3$ or $9$ then so was the original number (although the $1$ case is trivial it'll make sense to include it for later).
One can then ask: ... |
This problem is in my book:
>Show that as $z$ traverses a small circle in the complex plane in the
positive (counterclockwise) direction, the corresponding point P on
the sphere traverses a small circle in the negative (clockwise) direction with respect to someone standing at the center of the circle and with b... |
How to prove that:rotation in complex plane corresponding to opposite rotation in Riemann's sphere (Complex Analysis by Theodore W. Gamelin 1.3.3)? |
This problem is in my book:
>Show that as $z$ traverses a small circle in the complex plane in the
positive (counterclockwise) direction, the corresponding point P on
the sphere traverses a small circle in the negative (clockwise) direction with respect to someone standing at the center of the circle and with b... |
As I understand, Cantor’s diagonal set can be empty (diagonal equal to zeros), that is, there could be a mapping from the the Natural Numbers to the Power Set of the Natural Numbers in which the empty set is not mapped. The function is still not a bijection but there is only one element not mapped (one set that is ther... |
Do Cantor’s theorem rely on the Empty Set being in the power set of a set? |
I have a number that is perfect square, for example 36, and I have all prime divisors of this number 2,2,3,3. How many ways do we have to get 36 from divisors if we need to multiply k(from 2 to 1000) elements? (repetition is allowed, order matters) For example, if k = 3: 1,6,6 - one way to get 36. In addition, we have ... |
$Z$ definition? (Hussemoller, fibre bundles) |
Problem. *Let $C$ be a smooth Jordan curve, and $ \gamma: [-1,1]\to\mathbb R^2,$ a segment crossing $C$ perpendicularly at $x_0\in C$, with $\gamma(0)=x_0$. Show that there exists an $\varepsilon>0,\,$ such that the segments
$$
\ell_-=\big\{\gamma(t): t\in (-\varepsilon,0)\big\}\quad\text{and}\quad
\ell_+=\big\{\gam... |
This problem is in my book:
>Show that as $z$ traverses a small circle in the complex plane in the
positive (counterclockwise) direction, the corresponding point P on
the sphere traverses a small circle in the negative (clockwise) direction with respect to someone standing at the center of the circle and with b... |
While studying the structure of groups of order $p^a.q^b$, I may have proved that the Fitting length of such group (which is always solvable by Burnside Theorem) is at much 2 when max$(a,b) \leq$ min$(p,q)$. Since I encounter some (hopefully minor) technical difficulties, I am not fully sure of the result.
If true, ... |
I was going through my lecture notes and was wondering about the connection between the diagonalized form of a matrix and it's spectral projectors:
Let $ A \in M_{n\times n}(F) $ with spectrum $\sigma(A) = \{\lambda _1,...,\lambda_k\}. A$ is diagonalizable if and only if there exist matrices $\{G_1,...,G_k\}$ such t... |
I have the following question:
Let $a\in \mathbf{R}$ sucht that $X = L[a] \cap 2^{\omega}$ is uncountable. Why is $X$ is a $\Sigma_2^1$ set?
$L[a]$ is the inner model that can be built by constructibility relative to $a$.
Thank you very much for any help in advance. |
$x = 2\lambda x^3$
Either $x = 0$ or $\lambda =\frac {1}{2x}$.
You have investigated what happens if $\lambda = \frac 1{2x}$ but what about $x = 0?$
|
So I am studying [this proof of Quillen's theorem](https://mandal.ku.edu/talks/ProjCI.pdf). (page no. 27)
Here they say for a commutative ring $A$ we are given $As+At=1$.
So we get $as+bt=1$ for some $a,b\in A$.
But, in the proof they used for $k\in \mathbb{N}$ and $as^k+bt^k=1$ for some $a,b\in A$
Can any... |
I would like to calculate the following expectations of the output of a Neural network. I make the hypothesis that the $E[\beta_i] = 0$ and also I am assuming that the activation $h_j$ is independent of the weights $W_{ij}$. I could write the expectation of the output as follows:
$$E\left[ \left( \beta_i + \sum_{j=1... |
Let $G$ be a **non-discrete** Lie Group acting on a Riemannian manifold $M$ by its isometries. When $G$ is discrete, there are plenty of literature on Fundamendal Domains (FD), but **a quick literature search shows none on FD when $G$ is not discrete, and is say a Lie Group of manifold dimension at least one**.
**Wh... |
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