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I read in [On the spectral gap for finitely-generated subgroups of SU(2)](https://link.springer.com/article/10.1007/s00222-007-0072-z) that every finitely generated dense subgroup of $ SU(2) $ contains a free subgroup.
Is it true in general that every finitely generated dense subgroup of $ SU(n) $ contains a free s... |
I'm studying inclusion identities within polynomial ideals theory. More precisely, i'm interested in the correspondece of an ideal $I\subseteq \mathbb{F}[\vec{x}]$ and its associated affine variety $\mathcal{V}(I)$ for an arbitrary commutative field $\mathbb{F}$.
I've encountered some useful facts: mainly, that if ... |
Do vector minimisation problems always have an optimal solution in complete spaces? |
Helicopters Problem - What is the probability of being heard but not seen? |
Fact checking: are these inclusion relations regarding algebraic varieties of polynomial ideals correct? |
I'm studying inclusion identities within polynomial ideals theory. More precisely, i'm interested in the correspondece of an ideal $I\subseteq \mathbb{F}[\vec{x}]$ and its associated affine variety $\mathcal{V}(I)$ for an arbitrary commutative field $\mathbb{F}$.
I've encountered some useful facts: mainly, that if ... |
This is a special case of *Hewitt-Marczewski-Pondiczery theorem*, see e.g. Theorem 2.3.15 in Engelking's General Topology:
> If $d(X_s)\leq \alpha\geq\aleph_0$ for every $s\in S$ and $|S|\leq 2^\alpha$, then $d(\prod X_s)\leq\alpha$.
The $d(X)$ denotes the *[density](https://en.wikipedia.org/wiki/Cardinal_functio... |
Helicopters problem: What is the probability of being heard but not seen? |
I understand probability and statistics as a two way route. Either I begin from a random variable (r.v.) $X$ and sample from its probability distribution, or I observe a sample of values $x_1,...,x_n$ drawn from that distribution and I infer something about the distribution of the r.v. $X$. I have a conceptual problem ... |
The inclusions $\mathfrak{a} \cdot \mathfrak{b} \subseteq \mathfrak{a} \cap \mathfrak{b} \subseteq \mathfrak{a} + \mathfrak{b}$ are very well-known results; you can find them in any textbook on Commutative Algebra.
As an example where all inclusions are strict, consider, say, $\mathfrak{a} = \langle x^2y \rangle$ an... |
If $X$ is a compact, connected Hausdorff space, we know that the diagonal $\Delta_X=\{(x,x)\in X\times X\}$ is closed in $X\times X$ by Hausdorffness. But is $X\times X\setminus\Delta_X$ disconnected in general? I know that, if $X$ has a total order $<$ such that the induced order topology is contained in the original ... |
If we remove the diagonal from $X\times X$, is it necessarily disconnected? |
I'm trying to figure out how one would come up with a case for [Kronecker product](https://en.wikipedia.org/wiki/Kronecker_product) of diagonal matrices. Suppose we have a $2\times 2$ matrices
$$M_i = \begin{bmatrix}
a_i && 0\\
0 && b_i
\end{bmatrix}.$$
I want to figure out
$$M = \bigotimes_{i=1}^n M_i.$$
What w... |
Let $Y$ be the set of all non injective functions from $\mathbb{N} \to \mathbb{N}$, i.e., for each $n \in \mathbb{N}$, the cardinality $\bigl| f^{-1}({n}) \bigr|$ is different from $1$. I need to prove this cardinality equivalence.
So to do that I need to find an injective function from $Y$ to $\mathbb{N} \to \math... |
I am reading through Introduction to Algorithms by T.H.Cormen and I having a hard time understanding a point from the Djikstra algorithm correctness proof. Basically we have this image
[![enter image description here][1]][1]
With the following explanation:
> The proof of Theorem 22.6. Vertex u is selected to be ... |
I am reading through Introduction to Algorithms by T.H.Cormen and I having a hard time understanding a point from the Djikstra algorithm correctness proof. Basically we have this image
[![enter image description here][1]][1]
With the following explanation:
> The proof of Theorem 22.6. Vertex u is selected to be ... |
Let $A$ be a ring, $I \subset A$ a prime ideal and $\Phi : A \rightarrow A[x]$ the canonical inclusion homomorphism. Is $I[x]= \{ \sum_{k=0}^n a_k x^k : n\in \Bbb{N}, (a_k)_{k=0}^n \in I^n \}$ also prime in $A[x]$ ?
I am wondering this in order to apply it on other proof. I have search on my algebra books and notes ... |
Consider the sequence $e_n = (0,0,\ldots,0,1,0,\ldots)$ in $\ell^1$ which is weakly convergent to zero in $\ell^p$. for all $1\leq p \leq \infty.$ It is then obvious, from the theorem that sequences completely characterises weak convergence, that $\{e_n\}\cup\{ 0\}$ is weakly compact.
However, I remember somehow tha... |
Simple way to show that standard basis on $\ell^p$ is weakly pre-compact? |
Let $f$: $[a,b]\to \mathbb{R}$ bounded and Riemann-Stieltjes integrable for every monotone increasing function $\alpha$, is $f$ continous? |
I have two tensors - $A_{k}$ and $B_{lm}$. How can I find outer product of them? Outer product is defined as $$a \wedge \beta = \frac{(p + q)!}{p! q!} \text{Alt} (a \otimes \beta)$$
My result is $A_k B_{lm} + A_l B_{mk} + A_m B_{kl}$, but it's apparently wrong :( |
How can I find outer product of tensors? |
**Question**
I am **unsure of the validity or completeness of the logic** in my solution proof to Exercise 3.4.6 of Tao's Analysis I 4th Ed. I am self-teaching and the skills involved in logic, proof, quantifiers, etc are new to me.
> Prove Lemma 3.4.10. (Hint: start with the set $\{0, 1\}^X$ and apply
> the rep... |
Hello guys i am trying to solve excerise 2.7 page 14 from Gathmann notes <https://agag-gathmann.math.rptu.de/class/curves-2023/curves-2023-c2.pdf>
[![enter image description here][1]][1]
Definition
[![enter image description here][2]][2]
About (a) : Stuck here.Not sure how to use the curves not having ... |
[Wikipedia][1] defines Kummer's confluent hypergeometric function $M(a,b,z) \equiv {}_1F_1(a,b,z)$ and the first solution to the Kummer's function:
\begin{equation}
z{\frac {d^{2}w}{dz^{2}}}+(b-z){\frac {dw}{dz}}-aw=0,
\end{equation}
The second solution is given to be $z^{1-b}\;{}_1F_1(a+1-b,2-b,z)$, from where w... |
An alternative to Tricomi's confluent hypergeometric function? |
I have two tensors - $A_{k}$ and $B_{lm}$. How can I find wedge product of them? Wedge product of two tensors $\alpha$ $(p, 0)$ and $\beta$ $(q, 0)$ is defined as $$\alpha \wedge \beta = \frac{(p + q)!}{p! q!} \text{Alt} (\alpha \otimes \beta)$$, where $Alt$ is antisymmetrisation of tensor.
My result is $A_k B_{lm} ... |
[Wikipedia][1] defines Kummer's confluent hypergeometric function $M(a,b,z) \equiv {}_1F_1(a,b,z)$ and the first solution to the Kummer's function:
\begin{equation}
z{\frac {d^{2}w}{dz^{2}}}+(b-z){\frac {dw}{dz}}-aw=0,
\end{equation}
The second solution is given to be $z^{1-b}\;{}_1F_1(a+1-b,2-b,z)$, from where w... |
Is the extension by the zeros of the polynomial $f((x)=x^{2m}-p^{2n}=0$ over $\mathbb{Q}_p$ totally ramified? |
[Wikipedia][1] defines Kummer's confluent hypergeometric function $M(a,b,z) \equiv {}_1F_1(a,b,z)$ and the first solution to the Kummer's function:
\begin{equation}
z{\frac {d^{2}w}{dz^{2}}}+(b-z){\frac {dw}{dz}}-aw=0,
\end{equation}
The second solution is given to be $z^{1-b}\;{}_1F_1(a+1-b,2-b,z)$, from where w... |
I understand when $\{x_n\}$ is bounded, but not quite sure when it is not bounded. I may encounter three cases.
Case 1. Bounded below, not bounded above. In this case, $x_n$ can not converge, and we have $\lim\sup x_n=\infty$
Case 2. Not bounded below, bounded above. In this case, $x_n$ can not converge, and we h... |
I just make a quick and dirty code that uses an already found list of primes up to 1 bilion.
The file can be found [here](https://sourceforge.net/projects/billion-primes/)
The code I have used is:
def sum_digit(n):
sum = 0
while (n != 0):
sum = sum + (n % 10)
... |
Let $f_- (n) = \Pi_{i=0}^n ( \sin(i) - \frac{5}{4}) $
And let
$ f_+(m) = \Pi_{i=0}^m ( \sin(i) + \frac{5}{4} ) $
It appears that
$$\sup f_- (n) \inf f_+ (m) = \frac{5}{4} $$
Why is that so ?
Notice
$$\int_0^{2 \pi} \ln(\sin(x) + \frac{5}{4}) dx = Re \int_0^{2 \pi} \ln (\sin(x) - \frac{5}{4}) dx... |
For example, I want to know the probability that a venture fund that invests in 10 startups will end up with at least 2 startups that are worth more than 5 billion.
Let's assume that the probability of investing in a startup that ends up being worth 1 billion or more is 1% and that the distribution of startup valua... |
Can I use the binomial distribution probability mass function if the data is not normal? |
I have recently come across two examples of an algebraic structure similar to a ring, but without commutative addition. It consists of a set of values and two operations $(\sigma, +, \cdot)$, with the following axioms ($a, b, c \in \sigma$):
1. $(a+b)+c=a+(b+c)$ (associativity of addition)
2. $a\cdot (b+c) = a\cdot... |
Is there a name for this algebraic structure? (ring without commutative addition) |
I have recently come across two examples of an algebraic structure similar to a ring, but without commutative addition or multiplication. It consists of a set of values and two operations $(\sigma, +, \cdot)$, with the following axioms ($a, b, c \in \sigma$):
1. $(a+b)+c=a+(b+c)$ (associativity of addition)
2. $a\c... |
Is there a name for this algebraic structure? (ring without commutative addition or multiplication) |
Let $(M, \omega)$ be a symplectic manifold. Darboux's theorem states that, for any point $p \in M$, there exists a neighborhood $U$ of $p$, a neighborhood $V \subseteq \mathbb{R}^{2n}$ of $0$, and a symplectomorphism $(U, \omega) \cong (V, \omega_\mathrm{std})$. I'm curious whether the following stronger statement woul... |
I am looking for ways to simplify the sum
$$\sum_{n=1}^\infty \frac{1}{n(n+a)^b}, \quad a\in\mathbb{R}^+, b\in\mathbb{N}.$$
The first thought I had approaching this was to use Hurwitz and/or Zeta function. How should I go about evaluating this sum? (Mathematica doesn't have any simplification, but I need at least ap... |
I have recently come across two examples of an algebraic structure similar to a ring, but without commutative addition. It consists of a set of values and two operations $(\sigma, +, \cdot)$, with the following axioms ($a, b, c \in \sigma$):
1. $(a+b)+c=a+(b+c)$ (associativity of addition)
2. $a\cdot (b+c) = a\cdot... |
Is there a name for this algebraic structure? (ring without commutative addition) |
**Fact 1**. The function $1/|\xi|^{s}$ locally integrable (in the unit ball) if and only if $s<n$.
**Fact 2**. If $f\in L^1(\mathbb{R}^n)$, then $\widehat{f}\in\mathcal{C}^{\infty}(\mathbb{R}^n)$ (bounded).
Let $H^{2}:=\left\{u\in L^2(\mathbb{R}^n): \mathcal{F}^{-1}\left((1+|\xi|^2)^2\widehat{f}(\xi)\right)\in ... |
**LATEST EDITION**
Glad to share with you that we had found, in general, that
$$\boxed{ \int_0^{\frac\pi2} {\sin^n x} \ln{(\tan x)} \,dx =\frac{\sqrt{\pi}}{4 \Gamma\left(\frac{n}{2}+1\right)} \Gamma\left(\frac{n+1}{2}\right) \left[\psi\left(\frac{n+1}{2}\right)-\psi\left(\frac{1}{2}\right)\right]}$$
where $n\in ... |
Let $A$ and $B$ be abelian varieties, and consider the natural map
$$f:(A\times B)^\vee\to A^\vee\times B^\vee$$
sending a line bundle on $A\times B$ to the restrictions to $A\times\{0\}$ and $\{0\}\times B$. I would like to check that this is an isomorphism.
I had some old notes from a course on abelian varieti... |
Let's break this down into an easier problem to see how to approach this. Imagine we have a d3 and roll it 2 times. What is the expected value of the number of times of each number? Well it's just simply $\frac{2}{3}$. Now, conditioned on there being exactly 1 of those 2 rolls that are 3s, we know that the roll unaccou... |
I am studying the following semi-definite problem:
$$\begin{array}{rl} \textrm{Given:} & W \in \mathbb{R}^{n \times n} \\
\textrm{Minimize:} & u_1 + u_2 + \ldots + u_n \\
\textrm{Subject to:} & \forall i \in [n]:\\
& \begin{bmatrix} X & e_i \\ e_i^\intercal & u_i \end{bmatrix} \succeq 0 \\
& (W^\intercal X W)_{i... |
Is this SDP analytically solvable? |
One of my homework problems asks us to prove that let $S$ be a subset of a ring $A$, the localization $A[S^{-1}]$ is an idempotent $A$-algebra. I don't remember my instructor defining an "idempotent $A$-algebra" in class. The assignment is due soon so I probably won't have time to ask him. Could anyone help clarify wha... |
I'm trying to find the right level of explicitness to use when writing algebra proofs. The challenge is that algebra proofs, once complete, can often devolve into simply a series of equations. But adding too much explanation can be distracting.
For example, when working on:
> If $N$ is a normal subgroup of $G$ ... |
How to write a clear proof that if $N$ is a normal subgroup whose quotient group is abelian, the commutator subgroup is a subset of $N$? |
The seesaw principle says in general the following.
> If $X,T$ are varieties with $X$ complete, and $\mathcal{L}$ a line bundle on $X\times T$, then
$$
T_{1} = \{t\in T: \mathcal{L}_{X\times\{t\}} \text{ is trivial on } X\times\{t\}\}
$$
is a closed set in $T$. Furthermore, $\mathcal{L}|_{X\times T_{1}}\simeq ... |
Ok, I do not know what your experience and knowledge are, but here are my thoughts and recommendations:
$1.$ You will never get **anywhere** without those tricks and experience. Usually, when a mathematician tries to study a conjecture or another result they always study every single thing someone else did on that t... |
I have recently come across two examples of an algebraic structure similar to a ring, but without commutative addition. It consists of a set of values and two operations $(\sigma, +, \cdot)$, with the following axioms ($a, b, c \in \sigma$):
1. $(a+b)+c=a+(b+c)$ (associativity of addition)
2. $a\cdot (b+c) = a\cdot... |
I have recently come across two examples of an algebraic structure similar to a ring, but without commutative addition. It consists of a set of values and two operations $(\sigma, +, \cdot)$, with the following axioms ($a, b, c \in \sigma$):
1. $(a+b)+c=a+(b+c)$ (associativity of addition)
2. $a\cdot (b+c) = a\cdot... |
The Gram matrix (or Gramian) of a set of vectors in an inner product space is defined as the Hermitian matrix of inner products, with entries given by $G_{ij} = \langle v_i, v_j \rangle$. When these vectors are the columns of a matrix $X$ and the vector coordinates are real numbers, the Gram matrix simplifies to $X^T X... |
Is $X^T Y X$ Appropriately Called a Gram Matrix if $Y$ is Positive Definite? |
I have recently come across two examples of an algebraic structure similar to a ring, but without commutative addition. It consists of a set of values and two operations $(\sigma, +, \cdot)$, with the following axioms ($a, b, c \in \sigma$):
1. $(a+b)+c=a+(b+c)$ (associativity of addition)
2. $a\cdot (b+c) = a\cdot... |
I'm trying to find the right level of explicitness to use when writing algebra proofs. The challenge is that algebra proofs, once complete, can often devolve into simply a series of equations. But adding too much explanation can be distracting.
For example, when working on:
> If $N$ is a normal subgroup of $G$ ... |
**Prove that in each ear decomposition of a 4-regular $n$-vertex 2-connected graph $G$, the number of ears (counting the initial cycle) is $n+1$.**
Definition of ear decomposition: $(P_0, P_1, \dots ,P_k)$ of the edge set of G such that (a) $P_0$ is a cycle of length at least 3, and (b) for $i=1,\dots,k$, $P_i$ is a... |
**LATEST EDITION**
Glad to share with you that we had found below as an answer, in general, that
$$\boxed{ \int_0^{\frac\pi2} {\sin^n x} \ln{(\tan x)} \,dx =\frac{\sqrt{\pi}}{4 \Gamma\left(\frac{n}{2}+1\right)} \Gamma\left(\frac{n+1}{2}\right) \left[\psi\left(\frac{n+1}{2}\right)-\psi\left(\frac{1}{2}\right)\righ... |
> Prove that in each ear decomposition of a $4$-regular $n$-vertex $2$-connected graph $G$, the number of ears (counting the initial cycle) is $n+1$.
Definition of ear decomposition: $(P_0, P_1, \dots ,P_k)$ of the edge set of G such that (a) $P_0$ is a cycle of length at least 3, and (b) for $i=1,\dots,k$, $P_i$ is... |
I came across the following Theorem in Folland's book (Real Analysis: Modern Techniques and their application), Theorem 8.33
1. Where do we use the fact that $f$ is continuous in this proof?
2. Why can't we state a similar theorem for $\mathbb{T}^n$ without using Sobolev space?
[![enter image description h... |
Suppose $A$ and $P$ are two nxn matrices. Is $ N(PA) \subseteq N(A)$ ? How would you prove that, or disprove that?
<br/> <br/>
I tried to prove this and realized that if I do not know if $A$ and $P$ are nonsingular, then I cannot use the simple proof utilizing $ P^{-1} $. <br/> <br/>
What I reasoned was that I have ... |
Is the nullspace of the product of two square matrices a subset of the nullspace of one of them? |
Let $f:[a,b]\rightarrow \mathbb{R}$ be a function. $f'_{+}(t)= \lim_{t'\rightarrow t^{+}}\frac {f(t')-f(t)}{t'-t}$.
Let $f:[0,1]\rightarrow \mathbb{R}$ be a concave function. Then this means $\frac{f(t)-f(0)}{t}\geq \frac{f(t')-f(0)}{t'}$ for $0<t<t'$. Then as here slope is decreasing function, $f'_{+}(t)$ is equal... |
> Consider the assymtric random walk $(S_n)_n$ on $\mathbb{Z}$. Determine $(a_n)_{n \ge 0}$ such that $\exp(\lambda S_n-a_n)_{n \ge 0}$ is a martingale. (We assume that $\lambda \in \mathbb{R}$ such that $\mathbb{E}[\exp(\lambda X_n)] < \infty$.
I tried to check the martingale property, i.e.
$$\mathbb{E}[\exp(\la... |
I started reading spivak, just today, and on page 6 of the prologue. He states that it is not possible to prove that $1\ne0$ for the real numbers, with the basic properties of the ring.
But I got the following argument:
Assuming that $1=0$, it follows that $a(1)=a(0)$ for all $a\in \mathbb{R}$, where $a(1)=a$ and... |
I have recently come across two examples of an algebraic structure similar to a ring, but without commutative addition. It consists of a set of values and two operations $(\sigma, +, \cdot)$, with the following axioms ($a, b, c \in \sigma$):
1. $(a+b)+c=a+(b+c)$ (associativity of addition)
2. $a\cdot (b+c) = a\cdot... |
I read in [On the spectral gap for finitely-generated subgroups of SU(2)](https://link.springer.com/article/10.1007/s00222-007-0072-z) that every finitely generated dense subgroup of $ SU(2) $ contains a free subgroup.
Is it true in general that every finitely generated dense subgroup of $ SU(n) $ contains a free s... |
I was looking at a problem that deals with tetration. Here's the problem:
$$\begin{split}x^{x^{x^{\ddots}}}=3 \\ x^3=3 \Rightarrow x= \sqrt[3]{3} \end{split}$$
However, I saw that the solution is $-\frac{3}{\ln(3)}W_{-1}\left(-\frac{3}{\ln(3)}\right)\approx 2.478$ and wathed some videos which explain something like t... |
I started reading spivak, just today, and on page 6 of the prologue. He states that it is not possible to prove that $1\ne0$ for the real numbers, with the basic properties of a ring.
But I got the following argument, for set real numbers particular case:
Assuming that $1=0$, it follows that $a(1)=a(0)$ for all $... |
*I realize this matches closely with a form of the maximum modulus theorem, but my professor wants us to prove it in a non-standard way. With that said, I do wish to figure this out on my own and therefore only ask for 1 or 2 strong hints. I've been stuck on this for a long time with no clue how to start, but it's a ra... |
I have a statistical question with mean, median and percentile.
I have run my data with over thousand samples. From this run, I can see that the Average result value is higher than the median and percentile 75th value...
This seems my data is right skewed.
What else can I deduct from about my results?
Detai... |
I'm trying to determine some criteria for when the closure of a closed irreducible set $X$ of $\mathbb A^n$ is equal to the set itself. That is if $\overline{X}$ is the closure of $X$ in $\mathbb P^n$ and $X$ is a closed irreducible set of $\mathbb A^n$ then (by identifying $\mathbb A^n$ with $\mathbb P^n_{x_0,\dots,x_... |
I have recently come across two examples of an algebraic structure similar to a ring, but without commutative addition. It consists of a set of values and two operations $(\sigma, +, \cdot)$, with the following axioms ($a, b, c \in \sigma$):
1. $(a+b)+c=a+(b+c)$ (associativity of addition)
2. $a\cdot (b+c) = a\cdot... |
*I realize this matches closely with a form of the maximum modulus theorem, but my professor wants us to prove it in a non-standard way. With that said, I do wish to figure this out on my own and therefore only ask for 1 or 2 strong hints. I've been stuck on this for a long time with no clue how to start, but it's a ra... |
The *spectral radius* of a matrix is the max eigenvalue $\lambda_{max}$ of it.*Matrix norm* is a kind of norm satisfying some axioms.
**For any matrix A,is it true that $\lambda_{max}\le \Vert A\Vert$ for any matrix norm on A?**
It is easy to do with specially induced matrix norms ,which are induced from vector n... |
I'm trying to determine some criteria for when the closure of a closed irreducible set $X$ of $\mathbb A^n$ is equal to the set itself. That is if $\overline{X}$ is the closure of $X$ in $\mathbb P^n$ and $X$ is a closed irreducible set of $\mathbb A^n$ then (by identifying $\mathbb A^n$ with $\mathbb P^n_{x_0,\dots,x_... |
so recently, i have been thinking about the following question:
if there exist $\frac{{d^n}}{{dx^n}} g(x) = n \frac{{d^n}}{{dx^n}} f(x)$
can we express it as some $h(f(x))$?
so the answer is yes. We can express it as $(x-a)\frac{d}{dx}f(x)$
by transforming the function $f(x)$ into a Taylor series, and then ... |
How to extend the domain of this variable in this function? |
The *spectral radius* of a matrix is the max eigenvalue $\lambda_{max}$ of it.*Matrix norm* is a kind of norm satisfying some axioms.
**For any matrix A,is it true that $\lambda_{max}\le \Vert A\Vert$ for any matrix norm on A?**
It is easy to do with specially induced matrix norms ,which are induced from vector n... |
I wanted to complete the following sum:
$$\sum_{0\leq i <j<k} a_ib_jc_k$$
Where $a_ib_jc_k$ are all different geometric sequences with $|r|<1$. My attempt was to break up the sum into what each thing is bounded by:
$$\sum_{0\leq i <j<k} a_ib_jc_k=\sum_{i=0}^{j-1}\sum_{j=i+1}^{k-1}\sum_{k=j+1}^{\infty} a_ib_jc_k = \s... |
The Gram matrix (or Gramian) of a set of vectors in an inner product space is defined as the Hermitian matrix of inner products, with entries given by $G_{ij} = \langle \mathbf{v}_i, \mathbf{v}_j \rangle$. When these vectors are the columns of a matrix $X$ and the vector coordinates are real numbers, the Gram matrix si... |
so recently, i have been thinking about the following question:
if there exist $\frac{{d^n}}{{dx^n}} g(x) = n \frac{{d^n}}{{dx^n}} f(x)$
can we express it as some $h(f(x))$?
so the answer is yes. We can express it as $(x-a)\frac{d}{dx}f(x)$
by transforming the function $f(x)$ into a Taylor series, and then ... |
so recently, i have been thinking about the following question:
if there exist $\frac{{d^n}}{{dx^n}} g(x) = n \frac{{d^n}}{{dx^n}} f(x)$
can we express it as some $h(f(x))$?
so the answer is yes. We can express it as $(x-a)\frac{d}{dx}f(x)$
by transforming the function $f(x)$ into a Taylor series, and then ... |
so recently, i have been thinking about the following question:
if there exist $\frac{{d^n}}{{dx^n}} g(x) = n \frac{{d^n}}{{dx^n}} f(x)$
can we express it as some $h(f(x))$?
so the answer is yes. We can express it as $(x-a)\frac{d}{dx}f(x)$
by transforming the function $f(x)$ into a Taylor series, and then ... |
a) Show that the equation and boundary conditions $\frac{d}{dx}(x^2\frac{dy}{dx})+\lambda xy=0, y(1)=0, y'(2)=0$ form a regular Sturm-Liouville system. Show further that the system can be written as a constrained variational problem with functional $S[y]=\int_{1}^{2}x^2y'^2dx, y(1)=0$, and constraint $C[y]=\int_{1}^{2}... |
How should I derive the relationship $\lambda_{k}=\int_{1}^{2}x^2y_{k}'^2dx, k=1, 2, ...$? |
How to construct a complex function series that converges uniformly on a closed region but its derivative does not converge uniformly on the boundary? |
Why is the shell method not
$$\lim_{n \rightarrow \infty} \sum_{k=1}^n 2\pi\left(\frac{(b-a)}{n}\right)\cdot f\left((k-1)\cdot\frac{(b-a)}{n}\right)\cdot \frac{1}{n} + \pi f\left((k-1)\cdot\frac{(b-a)}{n}\right) \cdot \left(\frac{1}{n}\right)^2.$$?
This means the shells are concentric circles going outward, the ... |
I'm trying to classify the various topological concepts about connectedness. According to 3 assertions ((Locally) path-connectedness implies (locally) connectedness. Connectedness together with locally path-connectedness implies path-connectedness.), we can draw this diagram:
+--------------------------+
|C... |
Its an if and only if but I have one direction down. So suppose $(X,\mathbf{A},\mu)$ is a measure space and $X$ contains sets of arbitrarily small positive measure. Choose $A_1 \in \mathbf{A}$ such that $\mu(A_1) \in (0,1]$ And inductively for $n$ choose $A_n \in \mathbf{A}$ such that
$$\mu(A_n) \in (0,\frac{1}{3}\... |
Suppose there is a 6 sided die where each side is equally likely.
I want to to answer the following question: On average, how many times do I have to roll the dice before observing the following sequence (in this order): 1,2,3,4,5,6,5,4,3,2,1
I thought I could model this problem as a Discrete Markov Chain using ... |
I’m trying to prove that n to the second power < 2 to the power of n. I see this proof question from the example of 2.5.3 in the link[enter link description here][1]
I understand that when you prove an inductive case where P(k+1) is true, you have first to assume the base case where k to the power of n < 2 to the kt... |
How can you prove an inductive case of “n to the second power < 2 to the nth power”? |
I wanted to complete the following sum:
$$\sum_{0\leq i <j<k} a_ib_jc_k$$
Where $a_ib_jc_k$ are all different geometric sequences with $|r|<1$. My attempt was to break up the sum into what each thing is bounded by:
$$\sum_{0\leq i <j<k} a_ib_jc_k=\sum_{i=0}^{j-1}\sum_{j=i+1}^{k-1}\sum_{k=j+1}^{\infty} a_ib_jc_k = \s... |
Let $f:[a,b]\rightarrow \mathbb{R}$ be a function. $f'_{+}(t)= \lim_{t'\rightarrow t^{+}}\frac {f(t')-f(t)}{t'-t}$.
Let $f:[0,1]\rightarrow \mathbb{R}$ be a concave function. Then this means $\frac{f(t)-f(0)}{t}\geq \frac{f(t')-f(0)}{t'}$ for $0<t<t'$. Then as here slope is decreasing function, $f'_{+}(t)$ is equal... |
Let $R \subset S$ a flat, injective extension of commutative rings, $M \subset N $ an inclusion of $R$-modules. We identify $M \otimes_R 1_R \subset N \otimes_R 1_R \subset N \otimes S$ as $R$-submodules.
**Question:** Is there an immediate argument to show that $M \otimes_R 1_R = (N \otimes_R 1_R) \cap M \otimes_R... |
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