instruction stringlengths 12 30k |
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What do we count here?
$$\frac{dy}{dx} = {2\choose1}x$$
$${2\choose1} = \frac{dy/dx}{x}$$
Expanding $(x+dx)^n$ We discard the infinitesimal bits, the second term of the expansion, $\binom{n}{1}x$, seems to remain as the derivative. I wonder if there is a combinatorics connection, if it can be seen on the gra... |
Here what i did:
$$a=1 ,\quad b=1,\quad c=-x$$
$$\Delta = 1 + 4x$$
$$x < -\frac{1}{4} \rightarrow eliptic$$
$$x = -\frac{1}{4} \rightarrow parabolic$$
$$x > -\frac{1}{4} \rightarrow hyperbolic$$
Characteristic equations:
$$\frac{dy}{dx} = \frac{b \pm \sqrt{b^2 - 4ac} }{2a}= \frac{1 \pm \sqrt{1 + 4x} }{2}$$
$$... |
I have 2 vectors, a and b, from the origin to a plane.
The normal vector to the plane, is n.
(a dot n) and (b dot n) is the same, and I learned it is equal to the projection of a onto n (and b onto n).
So, this projection should be equal to the length from the origin to the point of intersection of the normal ... |
Fix $y_0$. Given $\varphi\in\mathrm{Aut}_G(Y)$, $\varphi(y_0)\in Y$. Since the action is transitive, there exists a $g\in G$ such that $gy_0=\varphi(y_0)$. Rotman wants to map it to a coset of $G_0$ in $N_G(G_0)$, namely to the coset $g^{-1}G_0$. But $g$ need not be unique. So he needs to verify two things for this f... |
Fix $y_0$. Given $\varphi\in\mathrm{Aut}_G(Y)$, $\varphi(y_0)\in Y$. Since the action is transitive, there exists a $g\in G$ such that $gy_0=\varphi(y_0)$. Rotman wants to map it to a coset of $G_0$ in $N_G(G_0)$, namely to the coset $g^{-1}G_0$. But $g$ need not be unique. So he needs to verify two things for this f... |
$$
\text{Given } \left\\{ n, \thinspace N \right\\} \in \mathbb{N}^+ \text{ and } N > n \\\\
\ \\\\
\begin{align}
p(x) &= \sum_{n}^N a_n{x^n} \\\\
\ \\\\
r(x) &= \sum_{n}^N \frac{a_n}{x^n}
\end{align}
$$
The highest power $N$ of a polynomial $p(x)$ is called its "degree". What is the highest power $n$ of... |
What is the highest power of a function called if it is negative? |
What do we count here?
$$\frac{dy}{dx} = {2\choose1}x$$
$${2\choose1} = \frac{dy/dx}{x}$$
Expanding $(x+dx)^n$ We discard the infinitesimal bits, the second term of the expansion, $\binom{n}{1}x$, seems to remain as the derivative. I wonder if there is a combinatorics connection, if it can be seen on the gra... |
$$
\text{Given } \\{n, \thinspace N\\} \in \mathbb{N}^+ \text{ and } N > n \\\\
\ \\\\
\begin{align}
p(x) &= \sum_{n}^N a_n{x^n} \\\\
\ \\\\
r(x) &= \sum_{n}^N \frac{a_n}{x^n}
\end{align}
$$
The highest power $N$ of a polynomial $p(x)$ is called its "degree". What is the highest power $n$ of $r(x)$ call... |
$$
\text{Given } \{n, \thinspace N\} \in \mathbb{N}^+ \text{ and } N > n \\\\
\ \\\\
\begin{align}
p(x) &= \sum_{n}^N a_n{x^n} \\\\
\ \\\\
r(x) &= \sum_{n}^N \frac{a_n}{x^n}
\end{align}
$$
The highest power $N$ of a polynomial $p(x)$ is called its "degree". What is the highest power $n$ of $r(x)$ called... |
$$
\text{Given } \{n, \thinspace N\} \in \mathbb{N}^+ \text{ and } N > n \\
\ \\
\begin{align}
p(x) &= \sum_{n}^N a_n{x^n} \\\\
\ \\
r(x) &= \sum_{n}^N \frac{a_n}{x^n}
\end{align}
$$
The highest power $N$ of a polynomial $p(x)$ is called its "degree". What is the highest power $n$ of $r(x)$ called?
|
$$
\text{Given } \{n, \thinspace N\} \in \mathbb{N}^+ \text{ and } N > n \\
\ \\
\begin{align}
p(x) &= \sum_{n}^N a_n{x^n} \\
\ \\
r(x) &= \sum_{n}^N \frac{a_n}{x^n}
\end{align}
$$
The highest power $N$ of a polynomial $p(x)$ is called its "degree". What is the highest power $n$ of $r(x)$ called?
|
I don't think that this is very efficient but it does determine algebraically what the algebraic form of the answer looks like.
Suppose that your matrix is $A=[a_{ij}]_{i,j=0}^n$, and that $x_i:=a_{ii}$.
Then the determinant is some polynomial
$$
\Delta=\sum_{i_1<i_2<\dots<i_k}\alpha_{i_1 i_2 \dots i_k}\ x_{i_... |
Problem:
--------
> Let $f$ be a function which has domain $D_f$ $\in$ $[-1,2]$ and range $\in$ $[0,1]$. The domain and range of a function $g$ defined by $g(x) = 1-f(x+1)$ will be?
My thinking:
------------
If the domain of $f(x)$ $\in$ $[-1,2]$, then the domain of $f(x+1)$ would $\in$ $[0,3]$ (adding 1 to bo... |
Problem:
--------
> Let $f$ be a function which has domain $D_f=[-1,2]$ and range $=[0,1]$. What are the domain and range of the function $g$ defined by $g(x) = 1-f(x+1)$?
My thinking:
------------
If the domain of $f$ is $[-1,2]$, then the domain of $x\mapsto f(x+1)$ is $[0,3]$ (adding 1 to both the extreme l... |
It is a direct consequence of the Brunn-Minkowski inequality that
\begin{equation}
|A\oplus B| - \Big(\sqrt{|A|}+\sqrt{|B|}\Big)^2 \geq |A\oplus\tilde{B}| - \Big(\sqrt{|A|}+\sqrt{|\tilde{B}|}\Big)^2,
\end{equation}
where $A,B,\tilde{B}\subset\mathbb{R}^2$ are convex and $\tilde{B}$ is homothetic to $A$, and ... |
I'm currently learning more about PDE's and have the following exercise: Consider the PDE
$$
-\Delta u+ \ln u=f\quad in\ \Omega,\
u=u_d\quad on\ \partial\Omega
$$
where $f\in L^2(\Omega)$ is nonnegative and $u_d\in W^{1,2}(\Omega)$ with $u_d\geq \varepsilon>0$ a.e. in $\Omega$. I'm now trying to prove that there... |
A coin factory produces coins with a probability of getting “tails” X that has a continuous
distribution with expectation $0.5$ and standard deviation $0.01$.
A person chooses a random coin that is produced in the factory.
Suppose that event A is dependent on the value of continuous random variable X .
In this case... |
Why $P(A|X=x) = x$ ? X that has a continuous distribution with $E(X) = 0.5$ and $σ(X) = 0.01$, A is define tails was obtained in a 1 toss of the coin |
Let $X_t$ be a Ito diffusion satisfying the SDE $dX_t=\mu(X_t)dt+\sigma(X_t)dW_t$ with $\mu$ and $\sigma$ being Lipschitz and $X_0=0$. Assume that $\sigma(x)>0$. Can we prove that $\tau=0$ almost surely where $\tau:=\inf\{t\geq0:X_t>0\}$? If not, what extra assumption needs to be made for this to be true. I know at le... |
I'm studying commutative algebra and I need to find, for the course I'm following, the primary decompositions of the following ideals, with some extra questions:
- In $\mathbb{C}[x,y]$, find, if you can, two primary decompositions of the ideal $(x^2y, xy^2)$, and find all isolated primes. Is the ideal radical? Is th... |
Finding a primary decomposition of an ideal...? |
What do we count here?
$$\frac{dy}{dx} = {2\choose1}x$$
$${2\choose1} = \frac{dy/dx}{x}$$
Expanding $(x+dx)^n$ We discard the infinitesimal bits, the second term of the expansion, $\binom{n}{1}x$, seems to remain as the derivative. I wonder if there is a combinatorics connection, if it can be seen on the gra... |
Really confused about the difference between $\forall n\in\mathbb N$ and $\bigcap_{i=1}^\infty$.
In *Understanding Analysis*, I quote from Exercise 1.2.13. that
> It is tempting to appeal to induction to conclude $$(\bigcup_{i = 1}^\infty A_i)^c = \bigcap_{i=1}^\infty A_i^c\ \ ,$$
but induction does not apply he... |
I was reading **Complex Geometry** by Daniel Huybrechts. On page 68, section 2.2 we have a proposition of holomorphic line bundle over $\mathbb P^n$,
> Proposition 2.2.6 The projection $\pi:\mathcal O(-1)\rightarrow\mathbb P^n$ is given by projecting to the first factor. Let $\{U_i\}_{i=0}^n$ be an open covering of ... |
The following identities have been suggested based on formulas in a [previous question of mine][1].
If complex $\theta_1=\cos^{-1}(p)$ and $\theta_2=\sec^{-1}(p)$, where $p\in(-1, 0) \cup (1, \infty)$, then the following relation holds:
> $$e^{i\theta_1}=\frac{1-\tan\frac{\theta_2}{2} }{1+\tan\frac{\theta_2}{2}}.... |
Let $X_n$ be the size of the $n$-th generation of a branching process, with family size probability generating function $G(s)$. Let $X_0 = 1$.
Suppose the family-mass size mass function is $P(X_1 = k) = 2^{-k-1}$ for $k=0,1,2,\dots$. What is the moment generating function of $X_n/n$? Hence, or otherwise, show that f... |
dy/dx = (2 choose 1) x=2x
So dy= 2x dx
Upon integration we get
y= x^2 +constant of integration
Is this what you are asking??? |
The two types of triangular octahedra can be characterized by the degree sequences of their vertices. The regular octahedron has six vertices with $4$ incident edges, whereas the other one has degree sequence $(3,3,4,4,5,5)$. A vertex $v$ having degree $d$ means that $v$ can “see” $d$ other vertices; that is, if you pr... |
Let $\alpha \in\Bbb R$ and let $F:\Bbb R^k \to \Bbb R$ be positive homogenous function of degree $\alpha$ i.e $ \forall x \neq 0$ in $\Bbb R^k$ and for all $\lambda >0$ we have: $F(\lambda x)=\lambda^{\alpha}F(x)$ Assume that $F$ is continuously differentiable in $\Bbb R^{k}\setminus\{0\}$ and $F(x)>0$ $\forall x \in \... |
I got this problem from my friend:
Given a complete probability space $(\Omega,\mathcal{F},\mathbb{P})$. Assume $\{X_r|r\in\mathbb{R}\}$ are i.i.d. with the uniform distribution on $(0,1)$. What is the probability of the "event" $E=\{X_s=X_t\text{ for some }s\neq t\}$?
My first thought is that this may not be an ... |
Let $\alpha \in\Bbb R$ and let $F:\Bbb R^k \to \Bbb R$ be positive homogenous function of degree $\alpha$ i.e $ \forall x \neq 0$ in $\Bbb R^k$ and for all $\lambda >0$ we have: $F(\lambda x)=\lambda^{\alpha}F(x)$ Assume that $F$ is continuously differentiable in $\Bbb R^{k}\setminus\{0\}$ and $F(x)>0$ $\forall x \in \... |
As we know,if $f(x)$ are Riemann integrable,we have
\begin{gather}
\left|\int_a^b f(x)~\mathrm{d}x\right|\leq \int_a^b |f(x)|~\mathrm{d}x.
\end{gather}
So,for Darboux integral,such as upper integral,did we also have
\begin{gather}
\left|\bar{\int_a^b} f(x)~\mathrm{d}x\right|\leq \bar{\int_a^b} |f(x)|~\mat... |
I am beginner in group theory and I have learned cyclic groups yet. When I look at the books for definition, it is said that
> Let G be a cyclic group and let a be a generator of G so that $$G= <a> = {\{ a^n | n \in Z \}}$$
I know that the definition could have been like $G={\{ an |\ n \in Z \}}$ if the operat... |
Hopefully this is not a fail. But, before I explain the math, I think it will be easier if I give a background of what I'm trying to do...
Imagine 2 line segments randomly defined in a $2D$ room. The line segments are 2D... Both are given a speed $v$ and $\omega$ (angular velocity). The point of the math, is to find... |
I am reviewing some commutative algebra notes for an upcoming course, and I found an exercise I wanted to solve, but I found out I absolutely have no idea how.
Let $K$ be a field (I think $charK \neq 2$, but the exercise doesn't outright state it), find if whether $R = k[x,y]/(2x^2 + 3y^2 - 11, x^2 - y^2 - 3)$ and $... |
Checking if a ring is Artinian...? |
I will start my question by providing some necessary context:
> Let $g(x) = c_lx^l + ... + a_1x + a$ be a polynomial of degree $l$.
>
>$g$ is said to have no missing coefficients if $c_i \neq 0$ for all $i = 0,...,l$.
>
>The coefficients of $g$ are strictly alternating if $a_0>0, a_1<0, a_2>0,...,$ or $a_0<0, a_1>... |
I am beginner in group theory and I have learned cyclic groups yet. When I look at the books for definition, it is said that
> Let G be a cyclic group and let a be a generator of G so that $$G= \langle a \rangle = {\{ a^n \mid n \in Z \}}$$
I know that the definition could have been like $G={\{ an \mid n \in Z... |
#### Background
I am following this MIT OCW course on mathematics for computer science.
In one of the recitations they come to the below result:
#### Official solution
###### Task:
A planar graph is a graph that can be drawn without any edges crossing. Also, any planar graph has a node of degree at most 5. Now... |
Can you explain to me why this proof by induction is not flawed? (Domain is graph theory, but that is secondary) |
[This][1] shows that a connected variety over a field $k$ is can be disconnected after extending the base field. I am wondering is it possible for a geometrically disconnected variety to be disconnected over the ground field. Say something disconnected over $\mathbb R$ but connected over $\mathbb C$. If not how can I p... |
So if a person has ticket $p$, the likelihood of winning the lottery is $(n-p)/n$ right? And then if $q>p$ is the chosen number, one wins $5n/(q-1)$. So you need a double sum, once over the possible ticket purchases and over the possible chosen ticket $q$. In sum you will get the following expected outcome:
$$
E[X]... |
I understand the steps to calculate the total derivative of f(x, g(x))
Related: <https://math.stackexchange.com/questions/4041538/derivative-of-fx-gx-with-respect-to-x>
I have three sub-questions related to calculating the total derivative of f(x, g(x, y)),
(1) How do I calculate its total derivative, here's ... |
I will start my question by providing some necessary context:
> Let $g(x) = c_lx^l + ... + a_1x + a$ be a polynomial of degree $l$.
>
>$g$ is said to have no missing coefficients if $c_i \neq 0$ for all $i = 0,...,l$.
>
>The coefficients of $g$ are strictly alternating if $a_0>0, a_1<0, a_2>0,...,$ or $a_0<0, a_1>... |
So if a person has ticket $p$, the likelihood of winning the lottery is $(n-p)/n$ right? And then if $q>p$ is the chosen number, one wins $5n/(q-1)$. So you need a double sum, once over the possible ticket purchases and over the possible chosen ticket $q$. In sum you will get the following expected outcome:
$$
E[X]... |
If have a point on an arc and other points related to it. I now want to move the point on the arc to a given target and rotate everything so that the arc-based-point has a vertical angle as the new position (like the if the arc-midpoint would now be just below (on y-axis) the target.
Example: I have an arc-midpoint ... |
Can every (non-odd) set be partitioned into subsets of two elements? By this I mean that given a set $A$, there is some $S \subset \mathscr{P}(A)$ such that every element of $S$ has exactly two elements and every element of $A$ appears in exactly one element of $S$. For a trivial example, $\mathbb{Z}^+$ can be partit... |
Can every (non-odd) set be partitioned into subsets of two elements? |
X and Y have the joint mass function:
$$f(x,y) = \frac{C}{(x+y+1)(x+y)(x+y-1)}$$ x, y =1,2,3...
find $V = X-Y$
if $z>=0$
$$P(X-Y = z) = P(X=k+z, Y=k) (1)$$
$$=C\sum_{k=1}^{\infty}\frac{1}{(x+y+1)(x+y)(x+y-1)}(2)$$
$$=\frac{1}{2}C \{\sum_{k=1}^{\infty}\frac{1}{(2k +z-1)(2k+z)} - \frac{1}{(2k+z)(2k+z+1... |
X and Y have the joint mass function:
$$f(x,y) = \frac{C}{(x+y+1)(x+y)(x+y-1)}$$ x, y =1,2,3...
find $V = X-Y$
if $z>=0$
$$\tag 1 P(X-Y = z) = P(X=k+z, Y=k) $$
$$\tag 2=C\sum_{k=1}^{\infty}\frac{1}{(x+y+1)(x+y)(x+y-1)}$$
$$\tag 3 =\frac{1}{2}C \sum_{k=1}^{\infty}\left(\frac{1}{(2k +z-1)(2k+z)} - \fra... |
The two types of triangular octahedra can be characterized by the degree sequences of their vertices. The regular octahedron has six vertices with $4$ incident edges, whereas the other one has degree sequence $(3,3,4,4,5,5)$. A vertex $v$ having degree $d$ means that $v$ can “see” $d$ other vertices; that is, if you pr... |
I am familiar with the proof that shows the powerset of the naturals is of the same cardinality as the reals using binary representation. Here's a quick rundown of the proof:
1. Showing that $f:(-1, 1) \rightarrow \mathbb R$ where $f(x) = \frac {x}{1-x^2}$ (or any other suitable function) is bijective.
2. Showing... |
Firstly, you need to define an inner product in $T^1(V)$. As far as I am aware this is defined by summing all of the component-wise inner products for each tensor power; but, this is only guaranteed to converge in the range of the signature thanks to the factorial decay. Ilya's work that you link doesn't define the inn... |
**1.** By Girsanov's theorem, the laws of $X$ and of the solution $Y$ of $dY_t=\sigma(Y_t)dW_t$ are locally absolutely continuous, so you can assume without loss of generality that $\mu=0$.
**2.** When $\mu=0$ your process $X$ is a local martingale, hence a time change of Brownian motion—for which the conclusion is ... |
The two types of triangular octahedra can be characterized by the degree sequences of their vertices. The regular octahedron has six vertices with $4$ incident edges, whereas the other one has degree sequence $(3,3,4,4,5,5)$. A vertex $v$ having degree $d$ means that $v$ can “see” $d$ other vertices; that is, if you pr... |
I'm currently learning more about PDE's and have the following exercise: Consider the PDE
$$
-\Delta u+ \ln u=f\quad \text{in}\ \Omega,\\
u=u_d\quad \text{on}\ \partial\Omega
$$
where $f\in L^2(\Omega)$ is nonnegative and $u_d\in W^{1,2}(\Omega)$ with $u_d\geq \varepsilon>0$ a.e. in $\Omega$. I'm now trying to p... |
Show that $x^2+1$ is irreducible over the integers mod $7.$
My solution goes as follows:
>Let $R$ be a commutative ring with unit element. An element $p(x)$ in $R[x]$ is said to be an irreducible element in $R[x]$ if whenever $p(x)=a(x)b(x)$, either $a(x)$ or $b(x)$ is a unit in $R[x]$, or in other words, either... |
Let $f(x,y) = a(x^2 + y^2) +ayx + cy^2x^2 $
For a given $m$, we are told that there exists $u, v$ such that $f(u, m) = 0 $ and $f(v, m) = 0 $.
OP claims that $f(u, v) = 0 $, and that there is an easy thought unstated proof.
Presumably they are asking if we can shed light on any deeper meaning.
Treat... |
Consider the structure $(\mathbb{N};\times,S,0,1)$, where $\times$ represents multiplication and $S$ represents successor. Is the addition function definable by a first-order formula in that structure? That is, is there a first order formula with three free variables $x,y,z$ such that the formula holds iff $x+y=z$? |
Is addition definable from successor and multiplication? |
According to Cahn's book [1] (pg. 148), given an irreducible representation (irrep) of a simple algebra of rank $r$, to construct the irreps of its regular subalgebras we can consider an extended Dynkin diagram. The weight $\Lambda$ can be written in Dynkin basis as $$\Lambda=(a_1,a_2,...,a_r)$$ before extension, where... |
When do we include singlets while branching simple algebras to regular subalgebras? |
I am learning more about how to use ultrafilters by using them to prove several of the typical results which appear as applications of propositional (or first-order) compactness.
Generally speaking, as I try to work out the proofs, I begin to see that ultrafilters produce precise descriptions/specifications of certa... |
**Question**
A homogeneous subset of ℕ is one where all the elements have the same parity. A student decided to prove that the set all homogeneous subsets of ℕ is uncountable. The student proposed that every homogeneous subset of N can be represented as an infinite binary word with all its 1s either in even position... |
Define a closed ball in a metric space $X$ with the metric $\rho$ that has centre a $x$ and radius $r$ as the set $\{y:\rho(x,y)\le r\}$ for $r\ge0$ and $x,y\in X$. Why are arbitrary unions of closed balls not a topology in $X$ but arbitrary unions of open balls are? An open ball is defined as the set $\{y:\rho(x,y)\lt... |
Why are closed balls not a topology in a metric space? |
The OP specifically asks about a 4-torus, still many of the answers are given with reference to a 2-dimensional person living in a 2-torus. It is claimed that a 2D person can not see the curved structure we associate with a torus (a donut-like shape). It seems the problem about "seeing the hole" is that it belongs to a... |
Find all triplets $(m, n, p)$, where $p$ is a prime number and $m, n ∈ \Bbb N$, such that
$p=\frac{m}{4}\sqrt{{2n-m \over 2n+m}}$
My procedure is as follows:
$p=\frac{m}{4}{\sqrt{(2n)^2-m^2}\over 2n+m}$
It can be shown that m cannot be odd, so if $m=2k$,
$p=\frac{k}{2}{\sqrt{n^2-k^2}\over n+k}$... |
Here is the definition of a continuous semigroup
Let $C$ be a subset of a Banach space $X$. A semigroup on $C$ is a group $\{S(t):t\geq 0\}$ of a self-maps defined on the subset $C$ which satisfies the following properties
* $S(0)=I,$
* $S(t+s)x=S(t)S(s)x, \forall t,s\geq 0,\; x\in C.$
A semi-group $S(t)$ is sa... |
So if a person has ticket $p$, the likelihood of winning the lottery is $(n-p)/n$ right? And then if $q>p$ is the chosen number, one wins $5n/(q-1)$. So you need a double sum, once over the possible ticket purchases and over the possible chosen ticket $q$. In sum you will get the following expected outcome:
$$
E[X]... |
I'm learning about intersection homology, and I'm trying to write a proof of the following statement:
>Take X,Y two filtered spaces with perversities $p$ and $q$ resp. , a continuous function $f:X\to Y$ such that for all stratum $S$ of Y, there is a strata $S^f$ of Y such that $f(S)\subset S^f$ and $codim_Y(S^f)\le... |
Let $f$ be integrable function. I wanted to prove that $\lim_{s \to \infty}\int_0^{2\pi}f(t)\sin(st)dt=0$.
Firstly I wanted to do it for every simple function. I started with intervals. When $f$ is a characteristic function of interval, the result follows easily.
Then my textbook says that from this result follow... |
We know that an involution is any permutation π such that π^2=id.
We also know that the number of involutions of Sn will be the number of permutations π∈Sn such that π has order 2.
But can you proof that this number will always be odd? |
I'm studying general topology and a question has come to my mind.
I know that, given a surjective map (between sets) and a topology on its domain, among the topologies on the codomain that make the map continuous, there exists a finest one (in the course, we call it the quotient topology induced by the map and the d... |
Find all triplets $(m, n, p)$, where $p$ is a prime number and $m, n ∈ \Bbb N$, such that
$p=\frac{m}{4}\sqrt{{2n-m \over 2n+m}}$
My procedure is as follows:
$p=\frac{m}{4}{\sqrt{(2n)^2-m^2}\over 2n+m}$
It can be shown that m cannot be odd, so if $m=2k$,
$p=\frac{k}{2}{\sqrt{n^2-k^2}\over n+k}$... |
This is a follow-up to my previous model theory question, here: https://math.stackexchange.com/questions/4886153/is-addition-definable-from-successor-and-multiplication. I asked whether addition is definable by a first-order formula in the structure $(\mathbb{N};\times,S,0,1)$ in that question, and found that that ques... |
Is addition a term-function in this structure? |
I need to determine if a series $$\sum\limits_{n=1}^{\infty} \frac{(z-1+i)^{2n-1}}{5^n(n+1)ln^3(n+1)}$$ converges at a point $z_1 = -1$
After substituting the point, I got: $$ \sum\limits_{n=1}^{\infty} \frac{(i - 2)^{2n-1}}{5^n(n+1)ln^3(n+1)} $$
And I do not know what to do next. I have heard that it is necessary ... |
Is addition by a specific nonzero natural number a term function in this structure? |
Why is $[X,Y]_{\mathfrak g} = -[X,Y]$? |
Yes, this is true:
Let $C$ be a cofinal chain in $P$ and $Q$ a cofinal subset of $P$.
Obviously, $P$ and $Q$ are upward directed.
W.l.o.g, we may assume that $\emptyset \neq C$ has no maximum, hence $C$ contains a cofinal subset order-isomorphic to a regular cardinal.
Hence, we may assume that $C = \{c_\alpha: ... |
Suppose we have $n\times n$ positive definite matrix $S$ and $n\times n$ positive semi-definite matrix $Y$. Let $R$ be a diagonal matrix of indicators, such that WLOG $RSR$ is a principle submatrix of $S$ padded by zeros. Does there exist a matrix $\Psi$ such that
$$
tr(Y[RSR]^+)=tr(\Psi S^{-1})
$$
where $\psi$... |
Does this matrix exist? A problem related to the trace of a pseudoinverse |
Let $\mathcal{C}=\{(x,y,z)\in \mathbb{R}^3 | x^2+y^2=z^2\}$. Prove that $\mathcal{C}$ is not locallly euclidean.
I consider $\mathbf{x}=(0,0,0)$ and $U$ an open neighbourood of $\mathbf{x}$ in $\mathcal{C}$. I suppose that exists $\varphi:U\to D^n$ homeomorfism. So, it's true that $U\setminus \{\mathbf{x}\}\cong_{... |
Let $\pmb M$ be a set with $n$ elements and $\pmb {P(M)}$ the set which contains all the subsets of $\pmb M$. Find the functions $f : \pmb {P(M)} \rightarrow \{0,1,2,...,n\} $ which check the following properties :
a)$f(\pmb A) \ne 0$, for $\pmb A \ne \emptyset$
b)$f(\pmb A \cup \pmb B) = f(\pmb A \cap \pmb B) +... |
The question is as it is from the title; let $A(t) = \begin{pmatrix} 1 & t \\ 0 & 1 \end{pmatrix}$, and let $\pi: \mathbb{R} \rightarrow GL(\mathbb{C}^{2})$ be a representation of the additive group $\mathbb{R}$ such that $\pi: t \rightarrow A(t)$, and I want to show that such representation does not have an invariant ... |
Suppose we have $n\times n$ positive definite matrix $S$ and $n\times n$ positive semi-definite matrix $Y$. Let $R$ be a diagonal matrix of indicators, such that WLOG $RSR$ is a principle submatrix of $S$ padded by zeros. Does there exist a matrix $\Psi$ such that
$$
tr(Y[RSR]^+)=tr(\Psi S^{-1})
$$
where $\psi$... |
Does this matrix exist? A fun puzzle related to the trace of a pseudoinverse |
This is a second part of a two-part question.
In the first part I was asked to prove that any group of order $5784 = 2^3 \cdot 3 \cdot 241$ has subgroups of the following *indexes*: 3, 6, 8, 12, and 24.
I proved that part by showing:
- G has subgroups of orders 2, 2^2, 2^3, 3, 241 (by Sylow theorem).
- More... |
I am asked to construct two 3-manifolds M1,M2 both covered by two open sets U,V(different for each manifold) s.t the intersection is diffeomorphic to S^1 x R^2 but the second de Rham group of M1 and M2 is trivial and not trivial respectively.
My thought so far is to take M1=S^2 x R and M2=S^1 x R^2 but i cant work out... |
Let $\pmb M$ be a set with $n$ elements and $\pmb {P(M)}$ the set which contains all the subsets of $\pmb M$. Find the functions $f : \pmb {P(M)} \rightarrow \{0,1,2,...,n\} $ which check the following properties :
a)$f(\pmb A) \ne 0$, for $\pmb A \ne \emptyset$
b)$f(\pmb A \cup \pmb B) = f(\pmb A \cap \pmb B) +... |
Suppose we have $n\times n$ positive definite matrix $S$ and $n\times n$ positive semi-definite matrix $Y$. Let $R$ be a diagonal matrix of indicators, such that WLOG $RSR$ is a principle submatrix of $S$ padded by zeros. Does there exist a matrix $\Psi$ such that
$$
tr(Y[RSR]^+)=tr(\Psi S^{-1})
$$
where $\psi$... |
Given a small number $\varepsilon >0$ and a constant $1/3\le \alpha < 1 $, I am looking for the smallest possible number $x^*$ such that for all real $x\ge \max\{x^*,3\}$, we have
$$\frac{x}{(\log x)^{\alpha}} \ge \varepsilon^{-2\alpha}. \tag1$$
Equivalently, the quantity I am looking for is (an upper bound of) ... |
We know that an involution is any permutation $\pi$ such that $\pi^2=id$.
We also know that the number of involutions of $S_n$ will be the number of permutations $\pi\in S_n$ such that $\pi$ has order $2.$
But can you prove that this number will always be odd? |
Consider a sufficiently regular and decaying function $f\in \mathcal{S}(\mathbb{R})$ and define the Fourier transform as $$
\mathcal{F}(f)(\xi)=\int e^{-ix\xi}f(x)dx \quad \hbox{ and } \quad \mathcal{F}^{-1}(f)(x)=\int e^{ix\xi}f(\xi)d\xi.
$$
With these definitions I would like to calculate $\mathcal{F}_{\mu\mapsto\... |
Suppose we have $n\times n$ positive definite matrix $S$ and $n\times n$ positive semi-definite matrix $Y$. Let $R$ be a diagonal matrix of indicators, such that WLOG $RSR$ is a principle submatrix of $S$ padded by zeros. Does there exist a matrix $\Psi$ such that
$$
tr(Y[RSR]^+)=tr(\Psi S^{-1})
$$
where $\psi$... |
Suppose we are given several vectors $\alpha_1,\dots,\alpha_m$ in an $n$-dimensional space. The convex cone spanned by them is defined to be the set
\begin{equation}
C=\{\sum_{i=1}^mt_i\alpha_i|\ t_i\ge0,\forall i\}.
\end{equation}
However, computing the condition of this cone by hand is usually hard. Besides, I ... |
Let $\pmb M$ be a set with $n$ elements and $\pmb {P(M)}$ the set which contains all the subsets of $\pmb M$. Find the functions $f : \pmb {P(M)} \rightarrow \{0,1,2,...,n\} $ which check the following properties :
a)$f(\pmb A) \ne 0$, for $\pmb A \ne \emptyset$
b)$f(\pmb A \cup \pmb B) = f(\pmb A \cap \pmb B) +... |
I stumbled upon this problem while running Mariokart tournaments during my undergrad. In a normal competition where there are 1v1 matchups, a round robin tournament is always done in $m(n) = \frac{1}{2}n(n-1)$ matches, as each participant must play every other participant once, and a matchup of $A \rightarrow B$ also s... |
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