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Is $\mathbb{R}^3\setminus\{(x,y,z)\in\mathbb{R}^3\mid x\leq 1,y\leq1,-1\leq z\leq 1\}$ a star domain? |
To answer your question: when can $(a-3)(a-7)$ be a perfect square?
Suppose
$$(a-3)(a-7)=k^2, $$
for some integer $k$. Then
$$a^2-10a+21-k^2=0.$$
This implies that
$$a = \dfrac{10\pm\sqrt{100-4(21-k^2)}}{2},$$
implying that
$$a = \dfrac{10\pm 2\sqrt{4+k^2}}{2}.$$
As $a$ has to be an integer, we look for... |
Most introductory textbooks on model theory establish the theory based on the ZF set theory (e.g. [1]). In particular, a structure is defined to be a 4-tuple of *sets*, and so on. In [2], I came to realize that this choice is unnecessary, i.e. the underlying theory ought not be ZF set theory; people suggested there are... |
To answer your question: when can $(a-3)(a-7)$ be a perfect square?
Suppose
$$(a-3)(a-7)=k^2, $$
for some integer $k$. Then
$$a^2-10a+21-k^2=0.$$
This implies that
$$a = \dfrac{10\pm\sqrt{100-4(21-k^2)}}{2},$$
implying that
$$a = \dfrac{10\pm 2\sqrt{4+k^2}}{2}.$$
As $a$ has to be an integer, we look for... |
**How to evaluate** $$\int_0^{\infty } \left(\frac{1}{(x+1)^2 \log (x+1)}-\frac{\log (x+1) \tan ^{-1}(x)}{x^3}\right) \, dx = G - \gamma + \frac{1}{4} \pi \log 2 - \frac{3}{2}.$$
I made some progress.
Integrating the second term by parts shows that negative of the integral equals
$$
-I = 1/2 + 1/2 \int_0^{\infty}... |
# Background
I am trying to understand the paper ["On $C^2$-smooth Surfaces of Constant Width" by Brendan Guilfoyle and Wilhelm Klingenberg][1]. However, despite my efforts to decipher the paper, I can't follow everything it is describing. This paper is written around complex differential geometry and topology, and ... |
How to solve this task (find modulus, real and imaginary part)?
$$z = \frac{(1+i)(1+i)^2(1+i)^3...(1+i)^{20}}{i^0+i^2+i^4+i^6+...+i^{20}}$$ |
>I hope to see how a structure can be defined without the notion of sets.
Why would you want to do so? I don't think PA is the right theory in which to do model theory. The cornerstone of model theory is Gödel's completeness theorem; its proof certainly doesn't require the full strength of ZFC. If you look at the wi... |
I have a question on the differentiable of a function and I want to be sure that my understanding of this concept is correct.
### Question:
Calculate the differentiate of $ Q: \mathbb{R}^n \rightarrow \mathbb{R}$ defined by $Q(x)=B(x;x)$ where $B: \mathbb{R}^n \times \mathbb{R}^n \rightarrow \mathbb{R} $ ... |
How do I show that $\displaystyle\sum_{x\in\text{Im}(X)}xP[X=x\mid A] = \dfrac{E[X\cdot1_A]}{P[A]}$, for $A \subseteq \Omega$ a sub-sigma algebra and $X$ a discrete random variable from $\mathcal{F} \to \mathbb{R}$?
The hint was that we should start by assuming that $X$ takes finite many values, and then extend the... |
Consider the space $H^s(\mathbb R^d)$ ($f \in L^2$ not in Schwartz class), $s \in \mathbb R$. Apply Riemann-Lebesgue Lemma to $\hat{f}$ to show that for some $s>s_0$ then there is a continuous function $g$ a.e.
I showed (by Holder) that for all $f \in H^s$, and $s >d/2$, then $\hat{f} \in L^1({\mathbb R}^d)$. I wan... |
I previously asked this question the other way around (given the angles of rotation, what are the resulting angles of azimuth and elevation), thinking that it would be simple to reverse it, but it's not. So now my question is what are the angles of rotation needed so that the vector normal to the surface points to a t... |
How would I explain it to someone who has not learnt anything beyond calc 1? Very vaguely:
The vector $\vec{V}$ lives inside of three-dimensional space (clear from the picture). But also clear from the picture is that $\vec{V}$ is in fact orthogonal to $\vec{U}$, meaning that $\vec{V}$ lies inside a specific 2-dimen... |
As the title suggests, I want to calculate the number of spanning trees in $K_n - e$ where $K_n$ is the complete graph on $n$ vertices and $e$ is any edge. The answer to this problem is $(n-2)*(n)^{n-3}$ and i know one method to do this. My actual question is regarding one of my approaches which is wrong. So let's pick... |
RandomGuy has given a general approach.
You can verify this particular formula by just multiplying out the the terms on the right hand side and noting that only $2^n+1$ are left.
Divisibility by 3
Option: Binomial expansion, $n$ is odd.
$(3-1)^n=\sum_{0}^{n}\binom{n}{k}3^{n-k}(-1)^k$; $\binom{n}{k}$ is a... |
***Seven points are equally distributed on a circle. How many non-congruent quadrilaterals can be drawn with vertices chosen from among the seven points?***
**My approach:**
If there are 7 points and we have to choose 4 points then there are \binom{7}{4} ways to construct a quadrilateral.
Hence, there are a to... |
Consider the sigma fields on $\mathbb{N}$: $F_n=\sigma(A_n)$, where $A_n=\{\{1\},\{2\},...\{3\}\}$ for $n\ge1$. Define $\mathcal{F}=\bigcup^\infty_{n=1}F_n$. Prove that $\mathcal{F}$ is a field but not a $\sigma$-field.
----------
**i)** For the field part, I need to show that $\mathbb{N}\in\mathcal{F}$. It is ea... |
I am looking for a reference (or a counterexample) to the following statement.
*Let $X$ be a separable metric space. Suppose that $Y$ is a completely regular topological space and $f\colon X\to Y$ is a surjective Borel function. Must $Y$ be separable too?*
This would be almost automatic if this condition forced ... |
*Seven points are equally distributed on a circle. How many non-congruent quadrilaterals can be drawn with vertices chosen from among the seven points?*
**My approach:**
If there are $7$ points and we have to choose 4 points then there are $\binom{7}{4}$ ways to construct a quadrilateral.
Hence, there are a to... |
# Background
I am trying to understand the paper ["On $C^2$-smooth Surfaces of Constant Width" by Brendan Guilfoyle and Wilhelm Klingenberg][1]. However, despite my efforts to decipher the paper, I can't follow everything it is describing. This paper is written around complex differential geometry and topology, and ... |
In one differential geometry class I'm taking, a critical point $p$ of a function $f$ on a manifold $(M,\mathcal{A})$ is defined as a point where $\forall$ coordinate chart $(U,\phi_U)\in\mathcal{A}$, $Df\circ\phi_U^{-1}(\phi_U(p))=0$. However, I don't understand why do we want to define it this way. Why can't we defin... |
A collection $\mathcal{X}$ of sets may have the property that each member of the collection has some element unique to it, more precisely, that for each element $A$ of $\mathcal{X}$, for some object $x$, for every element $X$ of $\mathcal{X}$, $x \in X$ if and only if $X = A$ (equivalently, there is no element $X$ of $... |
$
\newcommand\R{\mathbb R}
\newcommand\Ext{\mathop{\textstyle\bigwedge}}
\newcommand\doub{\mathfrak}
\newcommand\Blades[1]{\mathop{\textstyle\bigwedge^{\mkern-2mu(#1)}}}
\newcommand\MVects[1]{\mathop{\textstyle\bigwedge^{\mkern-1mu#1}}}
\newcommand\lintr{\mathbin{\lrcorner}}
$The other answers I see miss the geo... |
I found this very nice theorem in the book by Onishchik ,"Lie groups and Lie algebras I".
> **Proposition 2.9.** Let $g(t,s)\in G$ be a smooth map from some domain in $\mathbb{R}^2$ into a Lie group $G$. Define the
> corresponding maps into the Lie algebra $\mathfrak{g}$ of $G$ by
$$ \xi(t,s)=\left(R_{g}^{-1}\rig... |
A collection $\mathcal{X}$ of sets may have the property that each member of the collection has some element unique to it, more precisely, that for each element $A$ of $\mathcal{X}$, for some object $x$, for every element $X$ of $\mathcal{X}$, $x \in X$ if and only if $X = A$ (equivalently, there is no element $A$ of $... |
Consider the sigma fields on $\mathbb{N}$: $F_n=\sigma(A_n)$, where $A_n=\{\{1\},\{2\},...\{3\}\}$ for $n\ge1$. Define $\mathcal{F}=\bigcup^\infty_{n=1}F_n$. Prove that $\mathcal{F}$ is a field but not a $\sigma$-field.
----------
For the $\sigma$ field part I show that $\mathcal{F}$ is a countably infinite set (... |
A $2$-cocycle of $\mathbb{R}$ (valued in the unitary group $U(1)$) is a map $c:\mathbb{R}\times \mathbb{R}\to U(1)$ such that
1. $ \ \ \ \ c(x+y,z)c(x,y)=c(x,y+z)c(y,z)$
2. $ \ \ \ \ c(x,0)=c(0,x)=1$
I'm trying to prove that every $2$-cocycle is a $2$-coboundary, i.e. there is a function $\alpha:\mathbb{R}\to... |
The functor $\Omega$ is the composite of a bunch of functors, which I will now describe. Write $\mathsf{A}$ for the diagram $2\to 3\leftarrow 1$. The inclusion $i_3\colon *\to\mathsf{A}, *\mapsto 3$ induces a left Kan extension $\mathrm{Lan}_{i_3}\colon\mathscr{C}\to\mathrm{Fun}(\mathsf{A},\mathscr{C})$ (informally sen... |
Consider the double sums :
$$A_3 = \sum_{a=1}^{\infty} \sum_{b=1}^{\infty} \frac{1}{a^3 + b^3}$$
$$A_4 = \sum_{a=1}^{\infty} \sum_{b=1}^{\infty} \frac{1}{a^4 + b^4}$$
Is there a closed form for $A_3$ or $A_4$ ?
Can we prove they are irrational ?
***
Notice that double sums over rational functions can ... |
**Hint:** you reported that you have studied geometric progressions. What is the sum of a geometric series with initial term $2^{n-1}$, ratio $\frac{-1}2$, and final term $1$? |
The following is from "Multiplicative number theory I: Classical theory" by Hugh L. Montgomery, Robert C. Vaughan:
[![enter image description here][1]][1]
I could understand the full proof but the last paragraph, i.e. to show that $|E_K(x)| \le 1/2.$
I might not receive any answer if I ask for re-write the la... |
Let $M$ be a differentiable manifold. The diffeomorphism group of $M$ is the group of all $C^{\infty}$ diffeomorphisms of $M$ to itself, denoted by $\text{Diff}(M)$. This space of diffeomorphism $\text{Diff}(M)$ is considered as a Lie Group equipped with composition of functions ($\circ$) as a group operation. Accordin... |
Why does diffeomorphism group is a manifold? |
I stumbled upon an equation that could be an identity, but I'm not sure.
$$
e^{a(\partial_x + f'(x))} e^{-a \partial_x} = e^{f(x+a) - f(x)}
$$
The operator exponential can be understood as a power series
$$
e^{a\partial_x} g(x) = (1 + a\partial_x + \frac{a^2}{2!} \partial_x^2 + \ldots) g(x) = g(x+a)
$$
... |
Do we have $\mathbb{P}(\Vert A \Vert >\lambda) \le \mathbb{P}(\langle Ax,y\rangle > \lambda)$? |
Why is the diffeomorphism group a manifold? |
A well known fact exists which is that if a multivariate normal distribution undergoes a linear transformation it's also multivariate normal.
There are two proofs I have seen, If the transformation is invertible an application of the change of variables theorem is used.
If it's not an argument using either charac... |
I found this very nice theorem in the book by Onishchik ,"Lie groups and Lie algebras I".
> **Proposition 2.9.** Let $g(t,s)\in G$ be a smooth map from some domain in $\mathbb{R}^2$ into a Lie group $G$. Define the
> corresponding maps into the Lie algebra $\mathfrak{g}$ of $G$ by
$$ \xi(t,s)=\left(R_{g}^{-1}\rig... |
Please note I am not very confident of this solution at the moment. At best, it uses one unestablished assumption. I would greatly appreciate any feedback that can help me improve it. If it's wrong beyond repair, I'll delete it, of course.
---
For two quadrilateral to be congruent, corresponding sides and angles ... |
Hi I have this question about Lagrange multipliers and specifically when there are 2 constraints given. The standard answer to this question uses the lagrangian and 2 constraints with 2 extra variables (lambda 1 and lambda 2). I understand this method completely fine, but when I first attempted this problem, I tried co... |
> Solve
$$\label{eq_0}\tag{1}\begin{cases}
\partial_tu + (x-1) \partial_xu = 0\\
u(0,x) = x^3
\end{cases}$$
in the plane region $Q : = \{(t,x) \in \Bbb R^2 : x<1\}$
Using characteristic I found
$$\label{eq_1}\tag{2}\begin{cases}
\dot t = 1\\
\dot x = x-1
\end{cases}$$
so we have that for this ODE system... |
Consider the double sums :
$$A_3 = \sum_{a=1}^{\infty} \sum_{b=1}^{\infty} \frac{1}{a^3 + b^3}$$
$$A_4 = \sum_{a=1}^{\infty} \sum_{b=1}^{\infty} \frac{1}{a^4 + b^4}$$
Is there a closed form for $A_3$ or $A_4$ ?
Can we prove they are irrational ?
***
Notice that double sums over rational functions can ... |
I encountered a problem while working on a mathematical analysis exercise. I wish to share this problem because of its interest, while it is a simple statement I feel it is not straightforward to answer :
Given $f\in C^1([0,+\infty])$ such that $|f'(x)|\leq 1/x$ and $\lim_{R\rightarrow +\infty}\frac{1}{R}\int_0^R |... |
I was reading the Linear Algebra Done Right textbook by Axler, and an extension stemming from 5.27's proof 1 has been troubling me. This is a closer look into the book’s construction of $U$, an invariant subspace. I wanna find out if:
Given linear operator $T \in \mathcal{L}(V)$ with $\lambda$ as an eigenvalue, we d... |
There are some common meanings to 'dense' in Mathematics.
- In Topology, a subset $S\subseteq X$ of a topological space $(X, \tau)$ is dense if the intersection of every non-empty open set with $S$ is non-empty.
- In linear order theory, if $(X,<)$ is linearly ordered and $S\subseteq X$, $S$ is dense if for every... |
Where does the term 'dense' used in forcing/Martin's axiom come from? |
Let $R$ be a commutative ring and $f : R \to R^{\times}$ such that $f(x+y) = f(x)f(y)$ for all $x,y \in R.$ Define the following group $G =R \times R$ such that $$(x,y)(x', y') = (x+x', y + f(x)y').$$
This group is clearly a non abelian group in general. So my hunch is that this group can be embedded in a matrix gr... |
Is the following a matrix group? |
Please note I am not very confident of this solution at the moment. At best, it uses one unestablished assumption. I would greatly appreciate any feedback that can help me improve it. If it's wrong beyond repair, I'll delete it, of course.
As pointed out by *joriki* in the comments, a correction is needed.
---
... |
I am confused about the double covers and universal cover of ${\rm SO}(p,q)$. Previously I thought that ${\rm SO}(p,q)$ had the property that it is doubly connected and that the universal cover matches the double cover. But in [comments here][1] it is said that this is not true: if $p>1,q>1$ the double cover does not m... |
Suppose we have a function $$ f(x) = \frac{(\sum_{i=1}^{n} x_i)^2}{\sum_{i=1}^{n} x_i^2} $$, where $$ x = (x_1, x_2, \ldots, x_n) $$is a vector of non-negative real numbers. We want to determine the convexity of this function. Specifically, we want to answer the question:
Is the function $$ f(x) $$ convex for $... |
Let $R$ be a commutative ring and $f : R \to R^{\times}$ such that $f(x+y) = f(x)f(y)$ for all $x,y \in R.$ Define the following group $G =R \times R$ such that $$(x,y)(x', y') = (x+x', y + f(x)y').$$
This group is clearly a non abelian group in general. So my hunch is that this group can be embedded in a matrix gr... |
If L is regular, is L' = {xz| \exists y, y \in \sum^* such that |x|=|y|=|z| and xyz \in L} regular? |
In one differential geometry class I'm taking, a critical point $p$ of a function $f$ on a manifold $(M,\mathcal{A})$ is defined as a point where $\forall$ coordinate chart $(U,\phi_U)\in\mathcal{A}$, $Df\circ\phi_U^{-1}(\phi_U(p))=0$. However, I don't understand why do we want to define it this way. Why can't we defin... |
I am confused about the double covers and universal cover of ${\rm SO}(p,q)$. Previously I thought that ${\rm SO}(p,q)$ had the property that it is doubly connected and that the universal cover matches the double cover. But in [comments here][1] it is said that this is not true: if $p>1,q>1$ the double cover does not m... |
The functor $\Omega$ is the composite of a bunch of functors, which I will now describe. Write $\mathsf{A}$ for the diagram $2\to 3\leftarrow 1$. The inclusion $i_3\colon *\to\mathsf{A}, *\mapsto 3$ induces a left Kan extension $\mathrm{Lan}_{i_3}\colon\mathscr{C}\to\mathrm{Fun}(\mathsf{A},\mathscr{C})$ (informally sen... |
I found this very nice theorem in the book by Onishchik ,"Lie groups and Lie algebras I".
> **Proposition 2.9.** Let $g(t,s)\in G$ be a smooth map from some domain in $\mathbb{R}^2$ into a Lie group $G$. Define the
> corresponding maps into the Lie algebra $\mathfrak{g}$ of $G$ by
$$ \xi(t,s)=\left(R_{g}^{-1}\rig... |
I want to design a deterministc TM for the $L$ in the title, I've seen before for $$A=\{0,1,X,Y\} \hspace{0.5cm} B=\{0,1,\$\}$$ $A$ is for the input alphabet and $B$ is for the tape alphabet. (Reference : https://www.geeksforgeeks.org/construct-turing-machine-language-l-ww-w-01/)
My question is for an alphabet $A' =... |
In the context of evaluating the line integral of a vector field $\vec{F}:\vec{x}\in\mathbb{R}^n\mapsto \vec{F}(\vec{x})\in\mathbb{R}^n$ along a regular parameterized curve $\gamma:t\in{I}\subseteq\mathbb{R}\mapsto \vec{x}(t)\in\mathbb{R}^n$, we define $\int_{\gamma}\vec{F}\cdot d\vec{x} = \int_{I}\vec{F} \left(\vec{x}... |
Must the orientation of a parameterized curve $\gamma:I\subseteq\mathbb{R}\to \mathbb{R}^n$ be in the direction of the increasing parameter? |
i was reading an Algebra textbook by Siegfried Bosch and I read that given two ideals $\mathfrak a $ and $\mathfrak b$ of a certain ring $R$;we can construct a new ideal $\mathfrak a$•$\mathfrak b $ ={$\sum_{i=1}^{<\infty}a_ib_i :a_1\in \mathfrak a,b_i\in\mathfrak b$}
.So my question is:why it was inserted $<\infty$? |
I am confused about the double covers and universal cover of ${\rm SO}^+(p,q)$, where by this notation I mean the connected component with the identity. Previously I thought that ${\rm SO}^+(p,q)$ had the property that it is doubly connected and that the universal cover matches the double cover. But in [comments here][... |
Given the family has **two kids**, and **hiddenly** adopting binary biological sex (each kid is either a boy (B) or a girl (G)), the **sample space** is
$$\Omega=\{(G,G), (G,B), (B,G), (B,B) \}$$
with the following probability measure:
$$\mathbb P ((G,G))=p^2, \mathbb P ((G,B))=\mathbb P ((G,B))=p(1-p), \mathb... |
I want to design a deterministc TM for the $L$ in the title, I've seen before for $$A=\{0,1,X,Y\} \hspace{0.5cm} B=\{0,1,\$\}$$ $A$ is for the input alphabet and $B$ is for the tape alphabet. (Reference : https://www.geeksforgeeks.org/construct-turing-machine-language-l-ww-w-01/)
My question is for an alphabet $A' =... |
> Solve
$$\label{eq_0}\tag{1}\begin{cases}
\partial_tu + (x-1) \partial_xu = 0\\
u(0,x) = x^3
\end{cases}$$
in the plane region $Q : = \{(t,x) \in \Bbb R^2 : x<1\}$
Using characteristic I found
$$\label{eq_1}\tag{2}\begin{cases}
\dot t = 1\\
\dot x = x-1
\end{cases}$$
so we have that for this ODE system... |
This is from High dimensional probability by Vershynin.
In the proof of Grothendieck's inequality, for $u,v \in \mathbb{R}^n$, we find functions $\phi, \varphi \colon \mathbb{R} \to l^2$ such that $\frac{2}{\pi}\arcsin<\phi(u), \varphi(v)> = \beta<u,v>$(where $l^2$ is the Hilbert space of square-summable sequences)... |
Consider the following Lyapunov candidate function:
$$ V(x,y,q_i) = \frac{1}{2}x^2 + \frac{1}{2}y^2 + \sum_{i=1}^{n}\ln\big(\cosh(q_{i})\big) \tag1$$
where $q_{i} = \sqrt{(x-x_{i})^2 + (y-y_{i})^2} - L, \ L>0 \ (\text{constant})$, $n\geq1$ is an arbitrary positive integer number. The symbols $x,y\in \mathbb{R} $ ... |
Question from an old test:
>Find an example of a periodic Markov chain with $3$ states, with a stationary distribution $( \frac{1}{6}, \frac{1}{3}, \frac{1}{2})$.
I can see only two ways to connect the states.
[![Graphs][1]][1]
A triangle won't work because if we connect it in the way shown in (A), the st... |
A very nice and helpful result is that for convex functions, the integral of a subgradient is always a subgradient of the integral, without any domination assumption needed. More precisely, let $T$ be a measurable space, $\mu$ be a (positive) measure on $T$ and $f:T\times\mathbb R^d\to\mathbb R^d$ be a mapping such tha... |
Consider $X = \text{number of defective items in bought items}$.
Is the probability that $X \geq a$ given $X = a$, always 1: $P(X \geq a| X = a)=1$.
I was wondering if the above holds because $A$ (event that $X \geq a$) will always occur if $B$ (event that $X = a$) occurs, therefore, making the probability 1.
... |
Is this conditional probability always equal to 1? |
Consider the following Lyapunov candidate function:
$$ V(x,y,q_i) = \frac{1}{2}x^2 + \frac{1}{2}y^2 + \sum_{i=1}^{n}\ln\big(\cosh(q_{i})\big) \tag1$$
where $q_{i} = \sqrt{(x-x_{i})^2 + (y-y_{i})^2} - L, \ L>0 \ (\text{constant})$, $n\geq1$ is an arbitrary positive integer number. The symbols $x,y\in \mathbb{R} $ ... |
Let $p$ be an element of a commutative ring with unity. The following definition is natural:
> $p$ is *maximal under division* if its only divisors (up to equivalence) are $1$ and itself. That is, for all $a$, if $(p) \subseteq (a)$ then $(a) = (1)$ or $(a) = (p)$.
Although closely related, in a ring with zero di... |
While playing with Pascal's triangle, I observed that $\binom{4}{k } \mod 2 =0$ for $k=1,2,3$,and $\binom{8}{k } \mod 2 =0$ for $k=1,2,3,4,5,6,7$ This made me curious about the values of $n>1$ and $m>1$ that satisfy $\binom{n}{k } \mod m =0$ for all $k=1,2,3,\dots,n-1$ I asked my friend to write a $C++$ code that coul... |
There must be an error in your implementation of the 2nd method (since the ellipse you get by combining the two constraints does not have any point at distance 1 from the origin, whereas you claim to have found one!)
It is not completely clear from your question what you did.
Going to only one constraint would logi... |
In groups, we would define product of two normal subgroups something like $$K\cdot H =\{h\cdot k : h\in H, k\in K\}$$ However, there's a constraint in rings, we need ideals to be additive subgroups and in particular, closed under addition.
What more elements should we include in the set $P=\{ab:a\in \mathfrak a, b\i... |
I am looking at Jech's proof of properties of Namba forcing. The conditions consist of perfect subtrees of $\omega_2^{<\omega}$, where perfect means every node has $\omega_2$ many (not necessarily immediate) successors. The following is a key step in showing the forcing doesn't add reals (so $\omega_1$ is preserved): s... |
I'm trying to read through HoTT, but I'm confused by the path induction principle, it seems too strong at the first glance. I tried "proving" that all suitable paths commute, and it looks like I kind of managed to do it. Where did I make a mistake?
Given a type $A$ with a judgmentally unique term $a: A$ (for any $x:... |
Why aren't $\infty$-groupoids commutative in HoTT? |
In groups, we would define product of two normal subgroups something like $$K\cdot H =\{h\cdot k : h\in H, k\in K\}$$ However, there's a constraint in rings, we need ideals to be additive subgroups and in particular, closed under addition.
What more elements should we include in the set $P=\{ab:a\in \mathfrak a, b\i... |
I'm confused by the reasoning for the answer to subproblem **(c)** from Blitzstein and Hwang,chapter 3, problem 27:
> A company with n women and m men as employees is deciding which employees to promote.
> (a) Suppose for this part that the company decides to promote t employees, where 1 <= t <= n + m, by choos... |
I'm writing solutions for students who are taking a competition exam and I took problems from old [purple comet][1] competition problems. This problem is the last one from the 2004 middle school contest and says:
In the addition problem WHITE+WATER=PICNIC each distinct letter represents a different digit. Find the n... |
I found this very nice theorem in the book by Onishchik ,"Lie groups and Lie algebras I".
> **Proposition 2.9.** Let $g(t,s)\in G$ be a smooth map from some domain in $\mathbb{R}^2$ into a Lie group $G$. Define the
> corresponding maps into the Lie algebra $\mathfrak{g}$ of $G$ by
$$ \xi(t,s)=\left(R_{g}^{-1}\rig... |
Let $R$ be a commutative ring (not necessarily Noetherian) of finite Krull dimension. Suppose that for any prime ideal $p$, $\dim(R) = \dim(R/p) + \mathrm{height}(p)$ (call this condition $DIM$) .
Does that mean that for any inclusion of prime ideals $p \subsetneq q$, there is a longest chain (a chain of length $\di... |
The following is from "Multiplicative number theory I: Classical theory" by Hugh L. Montgomery, Robert C. Vaughan:
[![enter image description here][1]][1]
My question is that how Eq. D.10 is derived from the previous one? (The authors assume probably that $\text{var}$ is equal to or greater than the last two inte... |
My question is, I hope, simple. I have a smooth, properly embedded one dimensional submanifold $M \subset R^n, \; n > 2$ (i.e. a smooth curve, which is actually the level set of a smooth function $\phi : R^n \rightarrow R^{n-1}$). I know from separate considerations that the projection $\pi$ of $M$ onto its first two c... |
In the book *One Parameter Semigroups for Linear Evolution Equations*, the authors provide some definitions as follows:
> (Page 124). [For $A \in \mathcal{L}(X)$, define] For each $n \in \mathbb{N}, x \in D(A^n)$ we define the *n-norm* $$\lVert x\rVert _n: = \lVert A^nx \rVert $$ and call $$X_n : = (D(A^n), \lVe... |
$x^n/1+x^n$ is monotonic , $x \in R $ , $n \in N$
tried solving it by differentiating but not getting feasible result,
also is there any other way of computation through which this can be solved easily?
any hint would be appreciated |
Given a function $f: \mathbb{R}^3 \to \mathbb{R}^3 $ that maps a vector field $A$ into a vector field $B$ and that is **non-injective**, meaning that elements in the codomain can potentially be hit twice.
What would be an analytical solution of calculating the average vector in the codomain?
Pardon my inaccurate... |
In groups, we would define product of two normal subgroups $H$ and $K$ something like $$H\cdot K =\{h\cdot k : h\in H, k\in K\}$$ However, there's a constraint in rings, we need ideals to be additive subgroups and in particular, closed under addition.
What more elements should we include in the set $P=\{ab:a\in \mat... |
$x^n/1+x^n$ is monotonic , $x \in R^+ $ , $n \in N$
tried solving it by differentiating but not getting feasible result,
also is there any other way of computation through which this can be solved easily?
any hint would be appreciated |
I found very understandable proofs why $\nabla \cdot (\nabla \times H) = 0$ might be zero. But I do not derive any understanding out of it.
Imagine I have any source field. Why cannot I use Stokes Theorem on it and construct a field which would have the source field as curl? |
$x^n/(1+x^n)$ is monotonic , $x \in R^+ $ , $n \in N$
tried solving it by differentiating but not getting feasible result,
also is there any other way of computation through which this can be solved easily?
any hint would be appreciated |
I'm studying for my complex analysis qual, and I've found a problem that I haven't seen solved here yet:
Take $\lambda$ to be purely imaginary. Prove that $z-\lambda=\frac{1}{3}e^{z^2}$ has exactly one solution in the strip
$S:=\{x+iy:x\in [-1,1]\}$ .
I've tried showing that the roots are all in the disk $|\lamb... |
Rouché's Theorem: How many roots does lambda-z=1/3(e^z^2) have in the strip? |
Let $p$ be an element of a commutative ring with unity. The following definition is natural:
> $p$ is *minimal under division* if its only divisors (up to equivalence) are $1$ and itself. That is, for all $a$, if $(p) \subseteq (a)$ then $(a) = (1)$ or $(a) = (p)$.
Although closely related, in a ring with zero di... |
Suppose we have a function $$ f(x) = \frac{(\sum_{i=1}^{n} x_i)^2}{\sum_{i=1}^{n} x_i^2} ,$$ where $x = (x_1, x_2, \ldots, x_n) $ is a vector of non-negative real numbers. We want to determine the convexity of this function. Specifically, we want to answer the question:
Is the function $ f(x) $ convex for $ x \geq ... |
A [Surface of Constant Width][1] is a 3D surface with the special property that any two parallel planes which are tangent to it are always a constant distance apart, no matter the relative rotations of the surface and paired-planes. I want to construct a trio of parametric equations $\bigl(X(u, v), Y(u, v), Z(u, v)\big... |
Can the construction of this 2D Curve of Constant Width be adapted to a 3D Surface of Constant Width? |
My question is based on exercise II.6.9 in Hartshorne's *Algebraic Geometry*.
Suppose that $X$ is a projective curve over $k$ (an algebraically closed field). Let $\tilde{X}$ be its normalization and $\pi: \tilde{X}\rightarrow X$ be the projection map. For each point $P\in X$, let $\mathscr{O}_P$ be its local ring ... |
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