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Is linear independence confined to strictly lists of vectors, or does it extend to vector spaces, subspaces, etc?
If it isn't just lists of vectors what are the most common things that can be linearly independent? |
What Things can be Linearly Independent? |
I am trying to run a montecarlo simulation based on [this][1] model in order to assess whether a loan/company will default or not. Basically the model is
$$X_i = \sqrt{1-\rho}Z_i + \sqrt{\rho} Y$$
where $$ Z_i \sim \mathcal{N}(0, 1) $$ represents the credit $i$ and $$ Y \sim \mathcal{N}(0, 1) $$
$Z_i$ are inde... |
How to estimate ECL through monte carlo simulation? |
I am trying to compute the Fourier transform
$$
\hat{g}(\xi) = \int_{\mathbb{R}} e^{-\cosh(x)}e^{-2i\pi \xi x} dx
$$
of the function $g(x)=\exp(-\cosh(x))$.
Are there any standard substitution tricks for such type of integrals? |
I'm struggling to understand a proof in Katznelson (An Introduction to Harmonic Analysis, III - Lemma 1.3)
[Screenshot of the Proof][1]
[1]: https://i.stack.imgur.com/F430J.png
Basically, given $f \in L^1(T)$, take $F(z) = e^{-f - i \tilde{f}}$. Since this is holomorphic, it's harmonic and I follow why it... |
I cannot understand how double/nested induction works. I am supposed to prove that the addition operation is both commutative (x + y = y + x) and associative (x+(y+z)) = ((x+y)+z) from the following definitions of the naturals and addition (no other axiom is given).
Naturals: Base: $0 \in \mathbb{N}$ and
Ind. $s(m)... |
How does nested induction work? |
So a buddy of mine and I are sitting in our signals class, and he asks me a question: if the coefficients of a discrete time Fourier series are period, what are their Fourier coefficients? I understand that the Fourier transform of a Fourier transform returns a translated version of the original function, is this true ... |
I'm struggling to understand a proof in Katznelson (An Introduction to Harmonic Analysis, III - Lemma 1.3)
[![enter image description here][1]][1]
Basically, given $f \in L^1(T)$, take $F(z) = e^{-f - i \tilde{f}}$. Since this is holomorphic, it's harmonic and I follow why it's bounded, $|F| \leq 1$.
My firs... |
Consider the rectangular matrices $A \in \mathbb R^{m \times n}$ and $B \in \mathbb R^{n \times m}$. Is it true that, if there exists a square and positive semi-definite matrix $X \succeq 0$ such that $AXB \succeq 0$, then there exists a square and positive definite matrix $Y \succ 0$ such that $AYB \succ 0$?
Here i... |
Is it true that $\exists X \succeq 0, AXB \succeq 0 \implies \exists Y \succ 0, AYB \succ 0$? |
What is the formula for $$\int_0^{\pi } \frac{\sin (n \sigma )}{(a-\cos \sigma )^2} \, d\sigma$$
where $ a>1 $ and $n$ is a positive integer?
To evaluate, I tried to replace $a$ with $\cosh\xi $ in the integral
$$\int_0^{\pi } \frac{\sin (n \sigma )}{(\cosh\xi-\cos \sigma )^2} \, d\sigma$$
Can someone pleas... |
I cannot understand how double/nested induction works. I am supposed to prove that the addition operation is both commutative (x + y = y + x) and associative (x+(y+z)) = ((x+y)+z) from the following definitions of the naturals and addition (no other axiom is given).
Naturals: Base: $0 \in \mathbb{N}$ and
Ind. $s(m)... |
I am new to probability and I have been trying to find a formula or an intution to solve the following problem.
For the events A, B, C ⊂ Ω the following probabilities are known:
P(A) = 0.05, P(B) = 0.1, P(A∩B) = 0.03, P(A∪C) = 0.42, P(A∩C) = 0.03, P(C\\(A∪B)) = 0.34,
P(A∩B∩C) = 0.02. Calculate the probabilities... |
It is well known that finite dimensional vector spaces (over any field $K$) canonically satisfy $V''\otimes W''\simeq (V\otimes W)''$. My question is, roughly, if this canonical map can somehow be salvaged for infinite dimensional spaces. (A more precise version is at the end).
As far as I see, there are (at least) ... |
What is the formula for $$\int_0^{\pi } \frac{\sin (n \sigma )}{(a-\cos \sigma )^2} \, d\sigma$$
where $ a>1 $ and $n$ is a positive integer?
To evaluate, I tried to replace $a$ with $\cosh\xi $ in the integral
$$\int_0^{\pi } \frac{\sin (n \sigma )}{(\cosh\xi-\cos \sigma )^2} \, d\sigma$$
Please note that ... |
Let's say I have `x0, x1, ..., x99` and `y0, y1, ..., y99`
X = (x0, ..., x99)
Y = (y0, ..., y99)
X_ = (x0, ..., x3)
Y_ = (y0, ..., y3)
S = CubicSpline(X, Y)
S_ = CubicSpline(X_, Y_)
I expect `S` and `S_` to be exactly match between `x1` and `x2`, but they are different in ... |
Why is the middle segment of a 4 points cubic spline not matching a 100 points cubic spline? |
I am new to probability and I have been trying to find a formula or an intution to solve the following problem.
For the events A, B, C ⊂ Ω the following probabilities are known:
$P(A) = 0.05$, $P(B) = 0.1$, $P(A∩B) = 0.03$, $P(A∪C) = 0.42$, $P(A∩C) = 0.03$, $P(C \setminus(A∪B)) = 0.34$,
$P(A∩B∩C) = 0.02$. Calcu... |
Consider two arbitrary subsets of $\mathbb R^n$, say $A$ and $B$ that aren't necessarily connected (here, I mean connected as in interligated, not in a mathematical sense) in any way possible.
Usually, when one wants to prove that $A \subset B$, what one does is to pick an arbitrary element $a \in A$ and shows that ... |
Does $\mathbf{x}^T \otimes \mathbf{y}^T$ = $\mathbf{x} \otimes \mathbf{y}$? |
Outer product of row vectors. Does $\mathbf{x}^T \otimes \mathbf{y}^T$ = $\mathbf{x} \otimes \mathbf{y}$? |
Given a real square matrix $A \in \Re^{n \times n}$, let's say its eigenvalues have a lovely property: $Re(\lambda^A_i) < 0$. We also have another diagonal real square matrix $K \in \Re^{n \times n}$ whose entries are all positive. Now, if we define $B = KA$, what is the **lower bound** of element $k_{ii}$ that guarant... |
**Important preliminary remark.**
It has to be noted that this kind of difference functional equation is linear with respect to $f$. Indeed, it can be rewritten as $Lf = g$, where $L = a_2S^2 + a_1S + a_0$, with $S(f(t)) := f(t+1)$ the shifting operator, is a linear operator and $g$ plays the role of an inhomogeneous ... |
This is more a collection of remarks than an answer, but is both long for a comment and posted in the spirit of not answering in the comments.
If (or _since_) we're splitting hairs: In my experience with introductory linear algebra books, the property of _linear independence_ generally applies to _sets_ of vectors (... |
I have the set $A =$ {$z \in \mathbb C: z = x + 0i = x, x \in \mathbb R$}. Is this set open in the complex plane?
The set $A$ contains all the points on the real axis in the complex plane. This set is open if $A$ contains all of its interior points, i.e. it contains all points safely inside the set $A$. But an open ... |
Is this set $A$ open? |
Given positive integers $n$ and $k$, set
$$ S_{n,k}=\sum_{\substack{a_1+a_2+\dots+a_k=2n\\ a_i \in 2\mathbb{N},\,i=1,\ldots,k}}\frac{(2n)!}{a_1!a_2!\dots a_k!},
$$
where $2\mathbb{N}=\{0,2,4,\ldots\}$. According to the answers of
https://math.stackexchange.com/questions/1986386/special-sum-of-multinomial-coefficie... |
Consider the case of a smooth projective variety, $D$ a divisor which is nef and big. Is it semi ample and if not is there a condition to add so this is always the case ?
I tried to show it using that $D$ is semi ample if its stable base locus $\mathbb B(D)=\bigcap_{m\geq1}\operatorname{Bs}|mD|$ is empty, $D$ is nef... |
Nef and big implies semi ample? |
Let $f$ be a real decreasing function (resp. $g$ a real increasing function), and $(f_n)$ be a sequence of real decreasing functions (resp. $(g_n)$ be a sequence of real increasing functions) such that $$ \vert f(x)-f_n(x) \vert \leq \frac{C(x)}{\sqrt{n}}, \quad \vert g(x)-g_n(x) \vert \leq \frac{C'(x)}{\sqrt{n}}$$ wh... |
Oscar Lanzi has already given quite a nice overview over a general method to attack a certain set of polynomials trying to disprove eventual divisibility. Let's explore this idea a bit further and give a nice summary of the method:
We start with defining $q(x) := x^2+ax+b$ and examine solutions to $q(x) = x$ and $q... |
Given Hermitian Matrices $H$ and $K$ related by $AH=K$, is it necessarily true that $A$ is normal? $H$ may be assumed to be positive if that helps.
Clearly, if $A$ is a real function of $H$ defined on the spectrum of $H$, the equation $AH=K$ can hold. It also holds if $A=0$ or $A=I$ or if $A$ is any scalar matrix.
... |
Given matrices H, K that are Hermitian, with K=AH, is A necessarily normal? |
Oscar Lanzi has already given quite a nice overview over a general method to attack a certain set of polynomials trying to disprove eventual divisibility. Let's explore this idea a bit further and give a nice summary of the method:
We start with defining $q(x) := x^2+ax+b$ and examine solutions to $q(x) = x$ and $q... |
A Gaussian process is defined to be a stochastic process $X_t$ such that for every finite collection $(t_1, \ldots, t_k)$, the random variable $(X_{t_1}, \ldots, X_{t_k})$ is jointly Gaussian.
A Brownian motion is a stochastic process $B_t$ such that
1. $B_0 = 0$ a.s.
2. $B_t$ has independent increments
3. $B_t ... |
What is the difference between a Gaussian process and Brownian motion? |
I am not sure how performing a summation works with the [Levi-Civita symbol][1]. For example, if we have
$\epsilon^{ijk} x_i x_j x_k$,
how does this look once the summation is written out in components? Is it just
$\epsilon^{123} x_1 x_2 x_3 + \epsilon^{213} x_2 x_1 x_3 + \epsilon^{231} x_2 x_3 x_1 + + \eps... |
To define the property I am interested, let me introduce a notion of convolution measure on a measurable monoid $\mathcal{S}$. However, I am predominantly interested in the case where $\mathcal{S} \subseteq \mathbb{R}$ and the binary operation is just the addition ($+$).
> **Definition. (Convolution Measure)** Let $... |
Are there any interesting families of probability measures on $\mathbb{R}$ closed under convolution and "deconvolution"? |
In proposition 1.1.8 of "Model Theory: An Introduction" David Marker proves that:
>If $\cal M$ is an $\cal L$-substructure of $\cal N$, $\bar a \in M$, and $\phi(\bar v)$ is a quantifier-free formula, then $\cal M \models \phi(\bar a)\iff N\models \phi(\bar a)$
In the setting I'm working in, we have a finite langua... |
Let the function space A denote all functions f : [0, 1) → [0, 1) such that the derivative f' exists everywhere except on a set Z of Lebesgue measure zero, and such that |f'| = 1 on [0, 1) ∖ Z.
Let the function space B denote all functions f : [0, 1) → [0, 1) such that for all a, b with 0 ≤ a < b ≤ 1 we have that th... |
Are these two function spaces identical? |
Given Hermitian Matrices $H$ and $K$ related by $AH=K$, $H$ nonnegative, is it necessarily true that $A$ is normal? $H$ may be assumed to be positive if that helps.
Clearly, if $A$ is a real function of $H$ defined on the spectrum of $H$, the equation $AH=K$ can hold. It also holds if $A=0$ or $A=I$ or if $A$ is any... |
Given matrices H, K that are Hermitian, with K=AH, H nonnegative, is A necessarily normal? |
Let the function space A denote all functions f : [0, 1) → [0, 1) such that the derivative f' exists everywhere except on a set Z of Lebesgue measure zero, and such that |f'| = 1 on [0, 1) ∖ Z.
Let the function space B denote all functions f : [0, 1) → [0, 1) such that for all a < b in [0, 1), the portion G(a,b) = {... |
Let the function space A denote all functions f : [0, 1) → [0, 1) such that the derivative f' exists everywhere except on a set Z of Lebesgue measure zero, and such that |f'| = 1 on [0, 1) ∖ Z.
Let the function space B denote all functions f : [0, 1) → [0, 1) such that for all a < b in [0, 1), the portion G<sub>f</s... |
I am currently trying to go through some literature on the classification of symmetry protected topological phases. One shortcoming in my background is that I am unable to understand the physics-motivated discussions of gauge theory in the context of condensed matter physics. What is a background gauge field, emergent ... |
Another way:
$$f(x)=\csc x+\sec x\implies f’(x)=-\csc x \cot x+\sec x\tan x=0$$
$$\implies \tan^3x=1\implies x=\frac{\pi}{4}$$
Can you finish? |
The condition (|x| < 1) for the logarithmic function log(1+x)
i understand that for logx , x needs to be positive therefore here 1+x>0 ie x>-1 but why x needs to be less than 1?
|
Let the function space $A$ denote all functions $f : [0, 1) \to [0, 1)$ such that the derivative $f'$ exists everywhere except on a set $Z$ of Lebesgue measure zero, and such that $|f'| = 1$ on $[0, 1) \setminus Z$.
Let the function space $B$ denote all functions $f : [0, 1) \to [0, 1)$ such that for all $a < b$ in ... |
I learned in school that the curve formed by the intersection of a double cone and a plane is always one of the three types of curves called "conic sections" - an ellipse, a parabola, or a hyperbola (or the degenerate forms of each - a point, a line, or a pair of lines). Apparently this has been known since at least th... |
Equation for conic section given an arbitrary cone? |
> Seeing how this can be expressed easily and in such a parsimonious manner, then is it eligible to be placed among the main basic principles of set theory, i.e. at axiomatic or near axiomatic level?
My 2c, as for one thing I am certainly not an expert of set theory:
On the "philosophical" side, I would not think... |
### Question:
Let $(E; ||.||)$ a normed vector space, given $A$ a non empty set of $\mathbb{R}^n$ we define the distance between $x \in E$ and $A$ as follow $ d_A(x) = d(x;A) = inf \{ ||x-a|| ; x \in A \}$
a)For $A \subset E$ find $ \{x \in E : d_A = 0 \}$
1- First let prove that $\bar{A} \subseteq \{... |
Let the function space $A$ denote all functions $f : [0, 1) \to [0, 1)$ such that the derivative $f'$ exists everywhere except possibly on a set $Z$ of Lebesgue measure zero such that $|f'| = 1$ on $[0, 1) \setminus Z$.
Let the function space $B$ denote all functions $f : [0, 1) \to [0, 1)$ such that for all $a < b$... |
I came across a function while trying to show that for two pairs of homotopic maps, $h,h'$ and $k,k'$, the meaningful compositions, $k\circ h, k'\circ h'$, are also homotopic. But, my question is just regarding continuity and should be independent from this topic.
Let $I=[0,1]$ and $X$, $Y$ and $Z$ be topological sp... |
Let the function space $A$ denote all functions $f : [0, 1) \to [0, 1)$ such that, for some set $Z$ of Lebesgue measure zero, the derivative $f'$ exists and $|f'| = 1$ on $[0, 1) \setminus Z$.
Let the function space $B$ denote all functions $f : [0, 1) \to [0, 1)$ such that for all $a < b$ in $[0, 1)$, the portion $... |
Let the function space $A$ denote all functions $f : [0, 1) \to [0, 1)$ such that, for some set $Z$ of Lebesgue measure zero, the derivative $f'$ exists on $[0, 1) \setminus Z$ and $|f'| = 1$ there.
Let the function space $B$ denote all functions $f : [0, 1) \to [0, 1)$ such that for all $a < b$ in $[0, 1)$, the por... |
I have $f(x+iy)=y^2$. This satisfies the Cauchy-Riemann equations for all $z_0=(x,0)$. Now I want to check whether $f$ is differentiable for all $(x,0)$. So I use the definition, taking the limit as $h$ tends to $0$, and letting $h=a+bi$:
$$\frac{f(z_0+h)-f(z_0)}{h}=\frac{f(x+a+bi)-f(x)}{a+ib}=\frac{b^2}{a+ib}$$
... |
> Seeing how this can be expressed easily and in such a parsimonious manner, then is it eligible to be placed among the main basic principles of set theory, i.e. at axiomatic or near axiomatic level?
My 2c, as for one thing I am certainly not an expert of set theory:
On the "philosophical" side, I would not think... |
I am trying to run a montecarlo simulation based on [this][1] model in order to assess whether a loan/company will default or not. Basically the model is
$$X_i = \sqrt{1-\rho}Z_i + \sqrt{\rho} Y$$
where $$ Z_i \sim \mathcal{N}(0, 1) $$ represents the credit $i$ and $$ Y \sim \mathcal{N}(0, 1) $$
$Z_i$ are inde... |
Let $(\Omega,\Sigma,P)$ be some probability space and $Y: \Omega \to \mathbb{R} $ be random variables. Let $x, \beta \in \mathbb{R}^n$. In a linear model we assume something like this: $E(Y|x)=\beta^Tx$. I know conditional expectation, where you condition on either an event $E \in \Sigma$ or on a sub-$\sigma$-algebra $... |
This is Exercise 3.1.13 [here][1].
Let $\gamma:[a,b] \to U \subset\mathbb{C}$ be a piecewise $C^1$, injective (or can also be closed) curve. Show that there exists a continuous function $f$ such that $\int_\gamma f(z)dz \neq 0$.
**My idea:** Use a bump function $f$ near a point $\gamma(t_0)$ to show $\int_\gamma ... |
I am asking about the properties of rank over an inverse limit.
-------------------
Suppose we have a noetherian ring $R$ with uniformizer $\pi$ and noetherian ring of formal power series $R[[X]]$. Let $\mathscr{R}$ be the $\pi$-adic completion of $R[[X]]$.
Let $S \subseteq T$ be two $R [[X]]$-modules containe... |
Would it be correct to imply from Euler's prime equality function
$x = \frac {(\log x)^{\pi(x)}}{\prod \log p}$
where $\pi(x)$ is prime counting function?
$(\sum_{i=0}^x 1/i) = (1 + \frac {1}{2} + \frac {1}{2^2} + \frac {1}{2^3} + \dots)(1 + \frac {1}{3} + \frac {1}{3^2} + \frac {1}{3^3} + \dots)(1 + \frac {1}{... |
Is the sum of a positive semidefinite matrix and positive definite matrix a positive definite matrix?
I have a positive semidefinite matrix $M\in \mathbb{R}^{n\times n}$ and the identity $I_n$, is their sum positive definite? I ask because I want to ensure their sum has an inverse. |
Sum of positive semi-definite matrix and positive definite matrix? |
This is Exercise 3.1.13 [here][1].
Let $\gamma:[a,b] \to U \subset\mathbb{C}$ be a piecewise $C^1$, injective (or can also be closed) curve. Show that there exists a continuous function $f$ such that $\int_\gamma f(z)dz \neq 0$.
**My idea:** Use a bump function $f$ near a point $\gamma(t_0)$ to show $\int_\gamma ... |
Why is the order reversed in the conjugate symmetry axiom for the inner-product? |
I have been trying to learn some Galois Theory from Weintraub's book of the same name. Weintraub uses the word “root” of a polynomial without ever defining exactly what it is.
I understand that a root of $f\in k[X]$ is some element $\alpha$ st $f(\alpha)=0$. But my problem is, to which field/ring/set is $\alpha$ th... |
Would it be correct to imply from Euler's prime equality function
$x = \frac {(\log x)^{\pi(x)}}{\prod \log p}$
where $\pi(x)$ is prime counting function?
$(\sum_{i=0}^x 1/i) = (1 + \frac {1}{2} + \frac {1}{2^2} + \frac {1}{2^3} + \dots)(1 + \frac {1}{3} + \frac {1}{3^2} + \frac {1}{3^3} + \dots)(1 + \frac {1}{... |
The conjugate symmetry axiom for the inner product is as follows:
$$\overline{\langle x,y\rangle } = \langle y,x\rangle$$
It is my understanding that the conjugate is there so that the norm of a vector can be defined, but I do not understand why it is also necessary to swap the order, why not define it like thi... |
Let $(\Omega,\Sigma,P)$ be some probability space and let $Y: \Omega \to \mathbb{R} $ be a random variable. Let $x, \beta \in \mathbb{R}^n$. In a linear model we assume something like this: $E(Y|x)=\beta^Tx$. I know conditional expectation, where you condition on either an event $E \in \Sigma$ or on a sub-$\sigma$-alge... |
Given square matrices $A$, $B$ and $X$ and that
$$
AXA^T = BXB^T
$$
Does that imply $A = B$ ?
What about a specific condition, e.g. when $A, B$ are symmetric?
Sizes of $A, B$ necessarily have to be the same for the equation to be valid. |
If $AXA^T = BXB^T$, does that mean $A = B$? |
I know that the tree T has n-1 edges, one component (it is connected), and does not contain a cycle (that is, from each vertex to each there is a single path, or between every two vertices of the tree there is a single path, otherwise there would be a cycle) and we want show the equivalence between the statements: The ... |
I've been thinking about applications of [regular open sets](https://en.wikipedia.org/wiki/Regular_open_set) in topology, and I came up with the following:
If $X$ is a topological space then the set $\text{RO}(X)$ of regular open sets of $X$ forms a complete Boolean algebra. This Boolean algebra induces a [Stonean s... |
Are Stonean spaces a reflective subcategory in $\text{Top}$? |
im trying to find the number of possibilites to distrribute k balls to 2n boxes such that for every i between 1 and n the sum of the balls in the i box and the n+i box isnt equal to 6. i tried writing the equation: $$x_1+x_2+...+x_2n = k$$, but i dont know how to progress further and it seems like it should be solves w... |
Let $\, p_i\,$ be the $i$'th prime. Many proofs (e.g. below) use $\, p_{n+1}\! < 2 p_n\, $ (by [Bertrand](http://en.wikipedia.org/wiki/Bertrand%27s_postulate)) to prove that $\,p_{k+1}^2\! < p_1 p_2 \cdots p_k.\:$ It is worth remarking that this follows from a much weaker hypothesis than that of Bertrand. Rather, we ... |
I have a matrix $S$:
```
tensor([[ 0.3600, 0.2100, 0.4300, -0.0000],
[ 0.2100, 0.9300, -0.1400, 0.0000],
[ 0.4300, -0.1400, 0.7100, -0.0000],
[ 0.0000, -0.0000, -0.0000, 1.0000]])
```
and matrix $M$:
```
tensor([[0.3658, 0.3338, 0.1566, 0.1438],
[0.2850, 0.3646, 0.1580,... |
What is the property of matrix $S$ to be near-identity matrix for $M$, e.g. $SM$ is close to $M$? |
The conjugate symmetry axiom for the inner product is as follows:
$$\overline{\langle x,y\rangle } = \langle y,x\rangle$$
It is my understanding that the conjugate is there so that the norm of a vector can be defined, but I do not understand why it is also necessary to swap the order, why not define it like thi... |
Problem:
Given that $y = e^{2x}$ is a solution of
$$ (2x+1)y'' - 4(x+1)y' + 4y = 0 $$
find a linearly independent solution by reducing the order. Write the general solution.
Answer:
Let $y = e^{2x}v$. We have:
\begin{align*}
\dfrac{dy}{dx} &= e^{2x} \dfrac{dv}{dx} + 2 e^{2x} v \\
\dfrac{d^2y}{dx^2} &=
... |
In Introduction to Set Theory by Hrbacek and Jech, the Recursion Theorem is stated as:
For any set $A$, any $a\in A$, and any function $g: A \times \mathbb{N} \rightarrow A$, there exists a unique infinite sequence $f: \mathbb{N} \rightarrow A$ such that
(a)$f(0)=a$;
(b)$f(n+1) = g(f(n) , n)$ for all $n\in... |
Could someone please give me a hint for this problem?
The following sequence is exact:
$0 \rightarrow A \xrightarrow{f} B \xrightarrow{g} C \xrightarrow{h} D \xrightarrow{i} E \rightarrow \cdots$.
Prove: If $A, B, D$, and $E$ are free, then so is $C$.
This is my attempt:
$0 \rightarrow A \xrightarrow... |
The condition (|x| < 1) for the logarithmic function log(1+x)
i understand that for logx , x needs to be positive therefore here 1+x>0 ie x>-1 but why x needs to be less than 1?
|
So I made some **grave mistake** here: Still I want to show the post and add the real solution at the end:
With Gauss Law: The integral over the divergence on a volume is equal to the integral of that field on the boundary:
(Also see: [Divergence Theorem - Wikipedia](https://en.wikipedia.org/wiki/Divergence_the... |
I am currently working on the following problem :
> Let $X$ be a topological space such that its suspension $\Sigma X$ is a n-manifold. Show that $H_k(X) \cong H_k(\mathcal{S}^{n-1})$ for all $k$.
I already proved that $H_k(X) \cong H_{k+1}(\Sigma X)$ for all $k$, using Mayer-Vietoris. I thought it could help me... |
If the suspension of a topological space X is a n-manifold, why does X have to be a homology sphere? |
Could someone please give me a hint for this problem?
The following sequence is exact:
$0 \rightarrow A \xrightarrow{f} B \xrightarrow{g} C \xrightarrow{h} D \xrightarrow{i} E \xrightarrow{j} F \to 0$.
Prove: If $A, B, D, E$, and $F$ are free, then $C$ is torsion-free.
This is my attempt:
$0 \rightar... |
Could someone please give me a hint for this problem?
The following sequence is exact:
$0 \rightarrow A \xrightarrow{f} B \xrightarrow{g} C \xrightarrow{h} D \xrightarrow{i} E \xrightarrow{j} F \to 0$.
Prove: If $A, B, D, E$, and $F$ are free modules, then $C$ is torsion-free.
This is my attempt:
$0 ... |
The following is from Vakil's notes.
> Given a locally free sheaf $\mathcal{F}$ with rank $n$, and a trivializing open neighborhood of $\mathcal{F}$ (an open cover $\{U_i\}$ such that over each $U_i$, $\mathcal{F}|_{U_i} \cong \mathcal{O}^{\oplus n}_{U_i}$ as $\mathcal{O}$-modules) we have transition functions $g_{i... |
Given a twice differentiable function $f : [0,\infty) \to \mathbb{R}^n$, if $\int_{0}^{\infty} |f(s)|^2 ds < \infty$, does that imply $\lim_{s \to \infty} f'(s) = 0$ where $f'$ is $\frac{df}{ds}$ and $|\cdot|$ represents the Euclidean norm?
|
If a function is in L2, does its derivative have limit at infinity? |
Refer to [This book][1] for this discussion.
\
**NOTE**: By Fourier Sine Transform, I am **NOT** referring to complex part of complex Fourier Transform (which is $0$ for even functions), but the single sided sine transform as defined below, which can be non-zero for even functions too.
Definitions from the book... |
So I made some **grave mistake** here: Still I want to show the post and add the real solution at the end:
With Gauss Law: The integral over the divergence on a volume is equal to the integral of that field on the boundary:
(Also see: [Divergence Theorem - Wikipedia](https://en.wikipedia.org/wiki/Divergence_the... |
Is it possible to go over all lines of an infinite grid with a pencil without lifting it or going over a drawn line ?
The pencil can cross through a segment already drawn but cannot go over an already drawn line.
After doodling around, I have the feeling it is not possible.
If indeed it is not possible what demo... |
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