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The smallest normal set containing a subset $X$, what is a "normal set" ? |
I understand in mathematics there are many "quotienting " proceduce, is this the only reason that we consider equivariant theory for different "unequivariant" theory? Are there any more applications for equivariant theory?Thanks! |
The formula is not even close. You are right that d should have dimensions of length and the right hand side is unitless. Worse, the formula doesn't reduce properly if the errors are zero. Are you sure it isn't $d^2/\sigma_(d)^2$ on the left? That would fix the units. The cosine formula gives the distance estimate... |
Are there integer solutions to $9^x - 8^y = 1$? |
Suppose $P(x,y)$ gives transition probabilities of a random walk. I've seen $Pf=f$ being called the discrete version of <a href="http://en.wikipedia.org/wiki/Laplace's_equation">Laplace's equation</a>. In what sense are they analogous? |
Convex polyhedron $P$ is a subset of $\mathbb{R}^n$ that satisfies system of linear inequalities
\begin{align}
a_{11}x_1 + \cdots + a_{1n}x_n & \sim_1\\, c_1 \\\\
& \vdots \\\\
a_{p1}x_1 + \cdots + a_{pn}x_n & \sim_p\\, c_p,
\end{align}
where $\sim_i \in \\{\leq,\geq\\}$. It can be alternatively represented by tw... |
Convex polyhedron $P$ is a subset of $\mathbb{R}^n$ that satisfies system of linear inequalities
\begin{align}
a_{11}x_1 + \cdots + a_{1n}x_n & \sim_1\\, c_1 \\\\
& \vdots \\\\
a_{p1}x_1 + \cdots + a_{pn}x_n & \sim_p\\, c_p,
\end{align}
where $\sim_i \in \\{\leq,\geq\\}$. It can be alternatively represented by tw... |
Convex polyhedron $P$ is a subset of $\mathbb{R}^n$ that satisfies system of linear inequalities
\begin{align}
a_{11}x_1 + \cdots + a_{1n}x_n & \sim_1\\, c_1 \\\\
& \vdots \\\\
a_{p1}x_1 + \cdots + a_{pn}x_n & \sim_p\\, c_p,
\end{align}
where $\sim_i \in \\{\leq,\geq\\}$. It can be alternatively represented by tw... |
"Gromov Witten potential", when does "potential" mean here? What does the whole thing mean in physics? Thanks! |
Using generating functions one can see that the $n^{th}$ bell number, i.e., the number of all possible partitions of a set of $n$ elements, is equal to $E(X^n)$ where $X$ is a poisson random variable with mean 1. Is there a way to explain this connection intuitively? |
If not, is there only a limited rangle of polynomials for which the root can be found?
Also, if $u=x^{\frac{3}{2}}+x$, is $x$ expressible in terms of $u$? |
Are all polynomials solvable? |
I am trying to derive the following equation for an analytic function, $f$, defined on the unit disc, $D$
$f(z)=\frac{1}{\pi}\int_{D} \frac{f(w)}{(1- \bar{w}z)^{2}}dA(w)$.
{\bf Question}: Anyone have any ideas on how to derive this formula?
I've been able to verify that the above is true for $z =0$. In attem... |
Suppose that $A \in {\cal S}_+^n$ is a symmetric positive semidefinite matrix. Let $B = {\rm sign}(A)$, where the sign is taken elementwise. Is the resulting matrix $B$ always positive semidefinite?
If not, under what conditions can we say that $B \in {\cal S}_+^n$ ? |
Is a "sign matrix" obtained from a symmetric positive-semidefinite matrix itself symmetic positive-semidefinite? |
How do you show that an L^p entire (holomorphic on the complex plane) function is 0? |
I know this question is easy, but for the life of me, I cannot remember what we call this thing. Googling for this has offered no help.
Consider an object $A$ and a second object $B$(let them be groups if you so choose). We wish to consider and action of $A$ on $B$. Moreover there is a subobject $C \hookrightarrow B... |
i'm not sure what the context is for your first question, but in reply to the second one, i can mention applications in statistics that come with the word 'equivariant' attached. [they involve 'homogeneous spaces', topological spaces on which act a group of isomorphisms of interest.]
a simple example might suffice t... |
Your equation can be solved: http://www.wolframalpha.com/input/?i=solve+u%3Dx%5E%283%2F2%29%2Bx+for+x
The mathematics that decides when a polynomial is solvable by radicals is called [Galois Theory][1] and is one of the hallmarks of modern algebra.
[1]: http://en.wikipedia.org/wiki/Galois_theory |
Using generating functions one can see that the $n^{th}$ Bell number, i.e., the number of all possible partitions of a set of $n$ elements, is equal to $E(X^n)$ where $X$ is a Poisson random variable with mean 1. Is there a way to explain this connection intuitively? |
Suppose $U$ and $V$ are 2 dimensional subspaces of $\mathbb{R}^4$. How do you determine the dimension of $U \cap V$? I know that $\text{dim}(U \cup V) = \text{dim}(U)+ \text{dim}(V) - \text{dim}(U \cap V)$. So it seems that $\text{dim}(U \cap V) = 0$. |
I'm currently studying the notion of a quotient topology. The one thing I'm having trouble with understanding is what we're actually doing to the points as we're identifying them. Say we have a [0,1] x [0,1] in E^2 (euclidean 2-space) with the subspace topology and we partition X into:
1) The set {(0,0),(1,0),(0,1),(1... |
There are two things you can do with Topology: (1) try to imagine what's going on intuitively, and (2) write it down rigorously. Both are useful, necessary steps.
For instance, if I understand your first example, you have the unit square $[0,1]^2 \subset \mathbb{R}^2$ with the subspace topology. Let me write the coo... |
The formula is not even close. You are right that d should have dimensions of length and the right hand side is unitless. Worse, the formula doesn't reduce properly if the errors are zero. Are you sure it isn't $d^2/\sigma(d)^2$ on the left? That would fix the units. The cosine formula gives the distance estimate ... |
There are two things you can do with Topology: (1) try to imagine what's going on intuitively, and (2) write it down rigorously. Both are useful, necessary steps.
For instance, if I understand your first example, you have the unit square $[0,1]^2 \subset \mathbb{R}^2$ with the subspace topology. Let me write the coo... |
There are two things you can do with Topology: (1) try to imagine what's going on intuitively, and (2) write it down rigorously. Both are useful, necessary steps.
For instance, if I understand your first example, you have the unit square $[0,1]^2 \subset \mathbb{R}^2$ with the subspace topology. Let me write the coo... |
Suppose $U$ and $V$ are 2 dimensional subspaces of $\mathbb{R}^4$. How do you determine the dimension of $U \cap V$? I know that $\text{dim}(U + V) = \text{dim}(U)+ \text{dim}(V) - \text{dim}(U \cap V)$. So it seems that $\text{dim}(U \cap V) = 0$. |
There are two things you can do with Topology: (1) try to imagine what's going on intuitively, and (2) write it down rigorously. Both are useful, necessary steps.
For instance, if I understand your first example, you have the unit square $[0,1]^2 \subset \mathbb{R}^2$ with the subspace topology. Let me write the coo... |
There are two things you can do with Topology: (1) try to imagine what's going on intuitively, and (2) write it down rigorously. Both are useful, necessary steps.
For instance, if I understand your first example, you have the unit square $[0,1]^2 \subset \mathbb{R}^2$ with the subspace topology. Let me write the coo... |
Any number less than 1 can be expressed in base g as $\sum _{k=1}^\infty {\frac {D_k}{g^k}}$, where $D_k$ is the value of the $k^{th}$ digit. If we were interested in only the non-zero digits of this number, we could equivilantly express it as $\sum _{k=1}^\infty {\frac {C_k}{g^{Z(k)}}}$, where $Z(k)$ is the position o... |
As for your second example, Ross Milikan's intuition is right. Let's see how we can arrive to the conclusion that $B^n/S^{n-1} \cong S^n$ by examining what happens in low dimensions.
For $n=1$, the (closed) unit ball is the interval $B^1 = [-1,1]$ and its boundary the "sphere" $S^0 = \left\\{ -1, 1\right}$. Now, if ... |
As for your second example, Ross Milikan's intuition is right. Let's see how we can arrive to the conclusion that $B^n/S^{n-1} \cong S^n$ by examining what happens in low dimensions.
For $n=1$, the (closed) unit ball is the interval $B^1 = [-1,1]$ and its boundary the "sphere" $S^0 = \left\\{ -1, 1\right\\}$. Now, i... |
Are there addition formulas for the Riemann Zeta function? |
As for your second example, Ross Milikan's intuition is right. Let's see how we can arrive to the conclusion that $B^n/S^{n-1} \cong S^n$ by examining what happens in low dimensions.
For $n=1$, the (closed) unit ball is the interval $B^1 = [-1,1]$ and its boundary the "sphere" $S^0 = \left\\{ -1, 1\right\\}$. Now, i... |
A map $s : \mathbb{N} \to X$ is a <em>computable sequence</em> in $(X,\nu_X)$ when there exists a computable map $f : \mathbb{N} \to \mathbb{N}$ such that $s(n) = \nu_X(f(n))$ for all $n \in \mathrm{dom}(\nu_X)$.
My best guess would be, "A map s taking N onto X is a computable sequence in the ??? (X, nu??) when ther... |
How can I read this mathematical sentence aloud in English? |
In the UK the lottery uses numbers 1 to 49 and a total of six numbers are picked. It has been said may times that there is as much chance of numbers 1 2 3 4 5 6 to be picked as any other random combination.
My question is this:
Let's say that the first 3 numbers to come out are 1,2 and 3. What are the chances of ... |
As for your second example, Ross Milikan's intuition is right. Let's see how we can arrive to the conclusion that $B^n/S^{n-1} \cong S^n$ by examining what happens in low dimensions.
For $n=1$, the (closed) unit ball is the interval $B^1 = [-1,1]$ and its boundary the "sphere" $S^0 = \left\\{ -1, 1\right\\}$. Now, i... |
As for your second example, Ross Milikan's intuition is right. Let's see how we can arrive to the conclusion that $B^n/S^{n-1} \cong S^n$ by examining what happens in low dimensions.
For $n=1$, the (closed) unit ball is the interval $B^1 = [-1,1]$ and its boundary the "sphere" $S^0 = \left\\{ -1, 1\right\\}$. Now, i... |
For instance, say $G = \langle x , y \ | \ x^{12}y=yx^{18} \rangle$. I want to know what is the [normal closure](http://groupprops.subwiki.org/wiki/Normal_subgroup_generated_by_a_subset) of $y$ in $G$.
In general, what are the standard approaches to compute the normal closure of a subset of a finitely presented gr... |
Let $X$ be a Hausdorff space and $A$ a closed subspace. Suppose the inclusion $A \hookrightarrow X$ is a cofibration. Let $f, g: X \to Y$ be maps that agree on $A$ and which are homotopic. Are they homotopic relative to $A$?
My motivation for asking this question comes from the following result:
> Let $i: A \to X... |
Are homotopic maps over a cofibration homotopic relative to the cofibration? |
As for your second example, Ross Milikan's intuition is right. Let's see how we can arrive to the conclusion that $B^n/S^{n-1} \cong S^n$ by examining what happens in low dimensions.
For $n=1$, the (closed) unit ball is the interval $B^1 = [-1,1]$ and its boundary the "sphere" $S^0 = \left\\{ -1, 1\right\\}$. Now, i... |
As for your second example, Ross Milikan's intuition is right. Let's see how we can arrive to the conclusion that $B^n/S^{n-1} \cong S^n$ examining what happens in low dimensions.
For $n=1$, the (closed) unit ball is the interval $B^1 = [-1,1]$ and its boundary the "sphere" $S^0 = \left\\{ -1, 1\right\\}$. Now, if w... |
Is there any relation between homology of a space with local coefficient (in $Q$ vector space) and the homology with $Q$ coefficient?Thanks! |
As for your second example, Ross Milikan's intuition is right. Let's see how we can arrive to the conclusion that $B^n/S^{n-1} \cong S^n$ examining what happens in low dimensions.
For $n=1$, the (closed) unit ball is the interval $B^1 = [-1,1]$ and its boundary the "sphere" $S^0 = \left\\{ -1, 1\right\\}$. Now, if w... |
Let's say I am distributing a bunch of encryption keys, where if two users have the same key, they can communicate. I have 20 different keys, and I am giving each user 4. Since encryption keys don't disappear when I hand them out, it can be considered 'with replacement', but I'm not giving the same user the same key ... |
Basic probability... probability that two users share a key out of a keypool of 20? |
Let's say I am distributing a bunch of encryption keys, where if two users have the same key, they can communicate. I have 20 different keys, and I am giving each user 2. Since encryption keys don't disappear when I hand them out, it can be considered 'with replacement', but I'm not giving the same user the same key ... |
Let's say I am distributing a bunch of encryption keys, where if two users have the same key, they can communicate. I have 20 different keys, and I am giving each user 2. Since encryption keys don't disappear when I hand them out, it can be considered 'with replacement', but I'm not giving the same user the same key ... |
So I was bored and decided to figure out the indefinite integral of the absolute value function, $|x|$. Using integration by parts ($u=|x|, dv=dx$, $dx = \text{sgn}(x)=\frac{|x|}{x}$), it can be shown that $\displaystyle\int |x| dx = \frac{x |x|}{2}+C$.
Now I decided to take the integral again, finding that $\displa... |
How can I prove $\underbrace{\int \ldots \int}_{n} |x| dx = \frac{x^n |x|}{n+1}+C$? |
How can I prove $\displaystyle\underbrace{\int \ldots \int}_{n} |x| dx = \frac{x^n |x|}{n+1}+C$? |
Yes, there is a way: Use [Mathematical induction](http://en.wikipedia.org/wiki/Mathematical_induction). I don't add more details because I think it's elementary, isn't it? |
I am trying to understand left/right cosets in group theory.
Here is the example in my text:
Let $G = \lbrace 1, a, b, c, d ,e \rbrace$
Lets define the group operation $.$ by the following table, where the entry at row $x$ and column $y$ gives $x.y$
ex. $d.e = b$
<code> <b>1</b> <b>a</b> ... |
gprime, thinking about cosets in the following context may help you out. Leg $G$ be $\mathbb{R}^3$, the group operation will be vector addition. Suppose you want to know the left cosets of $H$, where $H$ is a straight line through the origin, which as you know, is a subspace. You look at the sets "$x+H$". Fix such an $... |
If your table is correct, it should be the symmetric group in three letters, with **a** and **b** the 3-cycles, and **c**, **d**, and **e** the transpositions.
Now. Your first equations is, alas, nonsense. The set that contains $1$, $a$, $b$, $c$, $d$, and $e$ is certainly not equal to the set whose elements are the... |
I am trying to understand left/right cosets in group theory.
Here is the example in my text:
Let $G = \lbrace 1, a, b, c, d ,e \rbrace$
Lets define the group operation $.$ by the following table, where the entry at row $x$ and column $y$ gives $x.y$
ex. $d.e = b$
<code> <b>1</b> <b>a</b> ... |
The fundamental idea of a coset is that of taking a subgroup and "translating" it to fill up $G$. A coset is just one such "translate." No coset but $H$ itself is a subgroup of $G$, since they don't contain $1$, but there is a lot of ground that can be covered by treating them like elements of a group or set instead.... |
If your table is correct, it should be the symmetric group in three letters, with **a** and **b** the 3-cycles, and **c**, **d**, and **e** the transpositions.
<strike>Now. Your first equations is, alas, nonsense. The set that contains $1$, $a$, $b$, $c$, $d$, and $e$ is certainly not equal to the set whose elements... |
As for your second example, Ross Milikan's intuition is right. Let's see how we can arrive to the conclusion that $B^n/S^{n-1} \cong S^n$ examining what happens in low dimensions.
**For $n=1$**, the (closed) unit ball is the interval $B^1 = [-1,1]$ and its boundary the "sphere" $S^0 = \left\\{ -1, 1\right\\}$. Now, ... |
There are two things you can do with Topology: (1) try to imagine what's going on intuitively, and (2) write it down rigorously. Both are useful, necessary steps.
For instance, if I understand your first example, you have the unit square $[0,1]^2 \subset \mathbb{R}^2$ with the subspace topology. Let me write the coo... |
David Speyer is how I would say it in practise, in a context where I was writing it on a black/whiteboard. Here's how I would say it in a pub or walking down the street:
"Let's define a 'representation map for X' [*or your own preferred jargon*] to be just some partial function nu, from the natural numbers to X. The... |
I'm looking for a fast algorithm to perform division of large numbers (by hand). Traditional long division just isn't fast enough for my needs.
In most of these cases, I'm only looking for the modulus/remainder of the division, so if there is a fast algorithm specifically for this purpose, it would suffice. (But, if... |
Suppose we have two schemes $X, Y$ and a map $f: X\to Y$. Then we know that $Hom_X(f^*\mathcal{G}, \mathcal{F})\simeq Hom_Y(\mathcal{G}, f_*\mathcal{F})$, where $\mathcal{F}$ is an $\mathcal{O}_X$-module and $\mathcal{G}$ an $\mathcal{O}_Y$-module (and the Homs are in the category of $\mathcal{O}_X$-modules etc). This ... |
Tangent to two disks: Roots of a 4th-degree polynomial? |
As for your second example, Ross Milikan's intuition is right. Let's see how we can arrive to the conclusion that $B^n/S^{n-1} \cong S^n$ examining what happens in low dimensions.
**For $n=1$**, the (closed) unit ball is the interval $B^1 = [-1,1]$ and its boundary the "sphere" $S^0 = \left\\{ -1, 1\right\\}$. Now, ... |
I know that a universal quantifier can be distributed over conjunction and not disjunction, but I'm having a hard time wrap my head around it. Why is this the case? Is there an example of a statement that would demonstrate this principle? |
Given $(X,||.||_X), (Y,||.||_Y)$ normed linear spaces, if $(x,y)$ is an element in $X \times Y$, is it true that $||(x,y)||_{X \times Y} < \delta$ implies that $||x||_X < \delta$?
This has been true with every product norm that I have encountered, but I am not sure if it is true in general and if there is an easy wa... |
Given $(X,||.||_X), (Y,||.||_Y)$ normed linear spaces, if $(x,y)$ is an element in $X \times Y$, is it true that $||(x,y)||_{X \times Y} < \delta$ implies that $||x||_X < \delta$?
This has been true with every product norm that I have encountered, but I am not sure if it is true in general and/or if there is an easy... |
Given $(X,||.||_X)$, $(Y,||.||_Y)$ normed linear spaces, if $(x,y)$ is an element in $X \times Y$, is it true that $||(x,y)||_{X \times Y} < \delta$ implies that $||x||_X < \delta$?
This has been true with every product norm that I have encountered, but I am not sure if it is true in general and/or if there is an ea... |
I'm guessing that this is a standard enough problem, but can't seem to find an intuitive solution.
I have an undirected weighted graph $G = (V,E,f)$ with vertices $V$, edges $E$ and $f$ the weighting function for an edge.
I have a set $\mathbf{I} = \\{ I_1, ..., I_n \\}$ such that for $I \in \mathbf{I}$, $I \subs... |
I'm guessing that this is a standard enough problem, but can't seem to find an intuitive solution.
I have an undirected weighted graph $G = (V,E,f)$ with vertices $V$, edges $E$ and $f$ the weighting function for an edge.
I have a set $\mathbf{I} = \\{ I_1, ..., I_n \\}$ such that for $I \in \mathbf{I}$, $I \subs... |
Let (X,||.||_X), (Y,||.||_Y) be a pair of normed linear spaces, and (X x Y, ||.||_{X x Y}) the induced product space and norm.
If (x,y) is an element in X x Y, is it true that ||(x,y)||_{X x Y} < \delta implies that ||x||_{X \times Y} < \delta?
This has been true with every product norm that I have encountered, b... |
Why is the manifold structure on the tangent bundle unique? |
I'm guessing that this is a standard enough problem, but can't seem to find an intuitive solution.
I have an undirected weighted graph $G = (V,E,f)$ with vertices $V$, edges $E$ and $f$ the weighting function for an edge.
I have a set $\mathbf{I} = \\{ I_1, ..., I_n \\}$ such that for $I \in \mathbf{I}$, $I \subs... |
Matt E, in his comment, gives two statements. He shows that a universal quantifier cannot be distributed over disjunction, because if it could, a false statement would follow logically from a true one. The statement "A universal quantifier can be distributed over disjunction" is itself a universal statement, and requir... |
Let $(X,||\cdot||\_X)$, $(Y,||\cdot||\_Y)$ be a pair of normed linear spaces, and $(X \times Y, ||\cdot||\_{X \times Y})$ the induced product space and norm.
If $(x,y)$ is an element in $X \times Y$, is it true that $||(x,y)||\_{X \times Y} \lt \delta$ implies that $||x||\_{X \times Y} \lt \delta$?
This has been ... |
As I'm trying to improve my maths I've now reached the subject of differentation of more general exponential functions.
I've already studied the so called "chain rule".
Now, my book tells me:
y = a^x ==> y' = a^x * ln(a)
Then, there's the following function:
f(x) = 10^-x
Applying the above rul... |
As I'm trying to improve my maths I've now reached the subject of differentiation of more general exponential functions.
I've already studied the so called "chain rule".
Now, my book tells me:
$$y=a^x \rightarrow y^{\prime}=a^x\ln(a)$$
Then, there's the following function:
$$f(x) = 10^{-x}$$
Applying ... |
David Speyer wrote how I would say it in practise, in a context where I was writing it on a black/whiteboard. Here's how I would say it in a pub or walking down the street:
"Let's define a 'representation map for X' [*or your own preferred jargon*] to be just some partial function nu, from the natural numbers to X. ... |
So you want to find a minimum cost cut so that for each $I \in \mathbf{I}$, $I$ is 'cut' by the set of edges?
In general this is NP-Hard, even in the case when each $|I| = 2$ (i.e each set has exactly two elements). (I believe this is called as the multi-cut problem).
When $\mathbf{I} = \\{ \\{s, t\\}\\}$, this b... |
Let's say I have a bag of coins, which contains 1 quarter, 2 dimes, 3 nickles and 4 pennies. If I were to randomly pull out 3 coins, on average, how much money would I get? |
The vector space $V(n,d)$ has dimension $n$ and the vector coordinates are from set
$\Sigma$.
$d:=|\Sigma|$
How many vectors can the vector space contain? Why? |
The vector space $V(n,d)$ has dimension $n$ and the vector coordinates are from set
$\Sigma$.
$d:=|\Sigma|$
How many vectors can the vector space basis contain? Why? |
The vector space $V(n,d)$ has dimension $n$ and the vector coordinates are from set
$\Sigma$.
$d:=|\Sigma|$
How many vectors can the vector space contain? Why? |
More of a comment than an answer, but it didn't fit:
In fact, the hairy ball theorem works on any even-dimensional sphere. However, odd-dimensional spheres *do* admit nonvanishing vector fields: if you use the usual embedding of $S^{2k-1}$ in $\mathbb{R}^{2k}$, then at the point $(x_1,y_1,\ldots,x_k,y_k)$ you can p... |
**What the number of positive integers satisfying the given inequality :**
C(n + 1,n – 2) – C(n + 1,n – 1) <= 100
What I did so far :
C(n + 1,n – 2) – C(n + 1,n – 1) <= 100
=> C(n + 1,3) – C(n + 1,2) <= 100
=> n(n+1)(n-4) <= 600
Now to derive after this ?
PS: This is... |
What the number of positive integers satisfying the given inequality ? |
**What number of positive integers satisfying the given inequality :**
C(n + 1,n – 2) – C(n + 1,n – 1) <= 100
What I did so far :
C(n + 1,n – 2) – C(n + 1,n – 1) <= 100
=> C(n + 1,3) – C(n + 1,2) <= 100
=> n(n+1)(n-4) <= 600
Now to derive after this ?
PS: This is not... |
What number of positive integers satisfying the given inequality ? |
Given finite-dimensional vector spaces $V,W$, there is an isomorphism $\text{Hom}(V,W) \rightarrow V^* \otimes W$. In particular, any linear map $\phi : V \rightarrow W$ has a tensor expansion $\sum v^*_i \otimes w_i$ where $v^*_i \in V^*, w_i \in W$.
For example, if one chooses dual bases $\{x_i\}, \{e_i\}$ of $V$,... |
Given finite-dimensional vector spaces $V,W$, there is an isomorphism $\text{Hom}(V,W) \rightarrow V^* \otimes W$. In particular, any linear map $\phi : V \rightarrow W$ has a tensor expansion $\sum v^*_i \otimes w_i$ where $v^*_i \in V^*, w_i \in W$.
For example, if one chooses dual bases $\\{x_i\\}, \\{e_i\\}$ of ... |
Let $T_f g = f \cdot g$ where $f, g$ are in $H^2(\mathbb{D})$ and so is their product (where $H^2$ is the Hardy space on the open unit disk). Now $T_f$ is a bounded operator.
I want to show this by showing that $f \in H^\infty$. So I try to write $f = G_1 h_1$ and $g = G_2 h_2$ where $G_i$ are outer functions and $h... |
Given finite-dimensional vector spaces $V,W$, there is an isomorphism $\text{Hom}(V,W) \rightarrow V^* \otimes W$. In particular, any linear map $\phi : V \rightarrow W$ has a tensor expansion $\sum v^*_i \otimes w_i$ where $v^*_i \in V^*, w_i \in W$.
For example, if one chooses dual bases $\\{x_i\\}, \\{e_i\\}$ of ... |
**What number of positive integers satisfying the given inequality :**
C(n + 1,n – 2) – C(n + 1,n – 1) <= 100
What I did so far :
C(n + 1,n – 2) – C(n + 1,n – 1) <= 100
=> C(n + 1,3) – C(n + 1,2) <= 100
=> n(n+1)(n-4) <= 600
Now to derive after this ?
PS: This is not... |
Let $(X,||\cdot||\_X)$, $(Y,||\cdot||\_Y)$ be a pair of normed linear spaces, and $(X \times Y, ||\cdot||\_{X \times Y})$ the induced product space and norm.
If $(x,y)$ is an element in $X \times Y$, is it true that $||(x,y)||\_{X \times Y} \lt \delta$ implies that $||x||\_{X \times Y} \lt \delta$, where $(||x||\_{X... |
I'm trying to show that the Möbius strip with boundary circle identified to a point is homeomorphic to P^2 (real projective space). I get geometrically why this is so, but, how does one generally construct maps between these spaces to show that they're homeomorphic? |
Möbius strip with edge identified - constructing map? |
Let $(X,||\cdot||\_X)$, $(Y,||\cdot||\_Y)$ be a pair of normed linear spaces, and $(X \times Y, ||\cdot||\_{X \times Y})$ the induced product space and norm.
If $(x,y)$ is an element in $X \times Y$, is it true that $||(x,y)||\_{X \times Y} \lt \delta$ implies that $||x||\_{X \times Y} \lt \delta$, where $(||x||\_{X... |
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