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The question is
> Examples of natural proofs by induction exercises which needs 3 or more base cases?
I think an elementary natural proof of the parity of the Somos-4
sequence is a nice exercise. Given the Somos-4 sequence of positive integers that satisfies
$$ s_n s_{n-4} = s_{n-1}s_{n-3}+s_{n_2}^2,\quad
s_... |
OP is asking for clarification about the contents of some text books , hence it is a valid query.
The hitch here is that these are not mainstream textbooks with mainstream thinking : hence there are not many comparable textbooks where we can get clarifications.
With that , these are **my equally non-mainstream ... |
I would like to numerically compute the following integral
$$
\int_a^b f(x) p(x) \mathrm{d} x
$$
for some known smooth function $f(x)$ and unknown smooth function $p(x)$.
I can estimate $p(x)$ by drawing samples from Bernoulli random variable $Y(x)$ with distribution $\operatorname{Bernoulli}(p(x))$.
As a res... |
This is likely a silly question, but I was wondering if $T$ is some nice integral transform, e.g. a Hilbert-Schmidt integral operator, with an, say, $L^2(\mathbb{R}^n)$ kernel, what then guarantees that the adjoint of $T$ is again an integral operator with a kernel? If
$$Tf(x) = \int_{\mathbb{R}^n}f(y)k(x,y)dy$$
... |
What guarantees that the adjoint of a suitable integral operator, e.g. a Hilbert-Schmidt operator, is again an integral operator with a kernel? |
Essentially, I've come across a PDE for reaction-diffusion which describes the concentration of a substance at some given point in space:
$$\frac{\partial c_a}{\partial t} = \mu_a\nabla^2 c_a + R_a(c_a,c_b,c_c)$$
Where $\mu$ describes some diffusion coefficient, $c$ refers to the concentration of one of the 3 sub... |
I am trying to do Problems Plus 4 in Chapter 7 of Stewart's calculus textbook, and I just want to make sure my work is correct:
"The centers of two disks with radius 1 are one unit apart. Find the area of the union of the two disks."
I started by trying to find the area of their intersection since the area of their... |
I'm trying to understand the following easy (not for me) exercise.
Consider $\mathbb{P}^1_k$ where $k$ is an algebraically closed field.
We have only one closed point, namely $\xi$ the one corresponding to the trivial ideal. (and this is ok)\
Then we have that $\kappa(\xi)= k(t)$
Now I need to say what are t... |
Can you solve for g(x, t) from the following equation?: $\int_{-\infty}^{\infty} g(x, t) dt $ or just simplify the integral using g(x, t). (we know f(x), but don't g(x, t), f and g are continuous)? |
A problem in _Complex Numbers_ (Andreescu, Andrica) involving a Complex Equation & a "Substitution" technique :
> Let us solve the equation $z^3 = 18 + 26i$, where $z = x + yi$ and $x$, $y$ are
integers.<br><br>
Solution given :
> We can write $(x + yi)^3 = (x + yi)^2 \times (x + yi)$ = $(x^2 − y^2 + 2xyi)... |
I have a matrix $M \in \mathbb{R}^{n \times m}$ that satisfies $M^{\top}M=D$ where $D$ is a diagonal matrix. This means that the columns of $M$ are mutually orthogonal to each other. Consider now another diagonal matrix $A \in \mathbb{R}^{n \times n}$. Is there anything I can say about the product $B=M^{\top}AM$ ? Unde... |
Of course in general the answer to my question is NO, however I am interested in two sequences of random variables that have very specific structure.
Let $a_n,b_n$ be deterministic (real-valued) sequences and let $\xi_n$ be an i.i.d sequence of random variables. Then let
$$ X_n=a_n+b_n\xi_n, \quad Y_n=a_n+b_n\xi,$$... |
If $X_n=Y_n$ in distribution and $X_n \to 0$ almost surely, then does $Y_n \to 0$ almost surely ? (For particular $X_n$ and $Y_n$) |
I'll assume that $A$, $B$ and $C$ are pairwise different and that the same holds for $D$, $E$ and $F$ (otherwise the question is meaningless).
We have that
$$(ABC)\perp (DEF)\iff \textbf{u}\perp \textbf{v}\iff \textbf{u}\cdot\textbf{v}=0$$
Where $\textbf{u}$ and $\textbf{v}$ are non-zero vectors normal to $(ABC)... |
So I'm given the following version of the French Railway Metric:
$$
d^{'}_{f}(z_1, z_2) =
\begin{cases}
|z_1 - z_2| & \text{if } \exists \lambda \in \mathbb{R} \text{ such that } z_1 = \lambda z_2,\\
|z_1| + |z_2| & \text{otherwise. }
\end{cases}
$$
As a part of this question I have verified that this is a met... |
I made a triangular-shaped quiver q (with three vertices and three arrows) and gave representations to the quiver, that is these representations are objects of $Vect^{Free(q)}$. now I was trying to enumerate these objects(that is representation) with a functor $K:P \to Vect^{Free(q)}$ where P is a poset. Could I consid... |
What is quiver of Quivers? |
Yes, in fact $X^*\cap X=\emptyset$. Lets write down explicitly:
$$X^*=Cauchy(X)/\sim$$
where $Cauchy(X)$ denotes the set of all Cauchy sequences on $X$ and $\sim$ is some relation. In other words $X^*\subseteq \mathcal{P}(Cauchy(X))$ as sets, where $\mathcal{P}(A)$ denotes the power set of $A$. Now what is a sequ... |
So assuming $f$ is continuous on $[a,b]$ and differentiable on $(a,b)$, I'm trying to prove that if $f$ is monotone increasing (non decreasing) then is it true that $f' \ge 0$.
- $f' = \lim _{h\to 0}\frac{f(a+h)-f(a)}{h}$
To preface this, I'm quite confident in my approach to this proof but I'm not $100$% sure w... |
How to show that $\forall k>0, \exists L>0, \frac{1+kt}{1-kt} \leq e^{Lt}$ on [0,1]? |
I have written the following algorithm pseudocode to depth-first traverse an undirected connected graph $G = (V, E)$:
```
DFS(G, v):
T = (V' = {v}, E' = empty, v)
s = [] // stack with push and pop
visited = [] // array with append and size
all_visited = [] // array
for u in {u | uv in E}:
s.push(u... |
I'm working on a text-based game involving a spaceship travelling between planets. Space is represented as a 2D plane. The ship has a current position, a destination and a speed.
Every minute, the game engine updates the current position of the ship, according to its current position, current destination and speed.
... |
Let $a(x)$, $b(x)$, $c(x)$ be smooth coefficient on $\mathbb{R}$ and $u(x)$ be the solution to the linear ODE over $\mathbb{R}$:
$$ au''+bu'+cu=-1 $$
Suppose $a(x)>1$, can we have some estimate on $|u'|$ independent of the derivatives of $a(x)$, $b(x)$, $c(x)$?
I can obtain the estimate by differentiating the... |
In the book by Fulton & Harris :
> Finally, in the real case things are simpler: when we exponentiate the
adjoint representation as above, the Lie group we arrive at is already simply
connected, and so is the unique connected real Lie group with this Lie algebra.
Just to give background : It has been shown t... |
Given sequence:
$$
\begin{cases}
x_{n+1}(2\cos(\frac{\pi}{m})-x_n)=1,\forall n\geq 1\\
x_1=x\in\mathbb R,m\in\mathbb N,m\geq 2
\end{cases}
$$
Minimize $$A=\frac{1}{2}\sum_{k=1}^m (x_{k+1}-x_{k})^2$$
$A$ is actually the area bounded by the zig-zag loop between $y=\frac{1}{2\cos(\frac{\pi}{m})-x}$ and $y=x$. I tr... |
I'm trying to understand the following easy (not for me) exercise.
Consider $\mathbb{P}^1_k$ where $k$ is an algebraically closed field.
We have closed points, namely the ones corresponding to $(a_0x_1-a_1x_0)$, and $\xi$ the generic, corresponding to the trivial ideal.\
Then we have that $\kappa(\xi)= k(t)$
... |
I would like to suggest this approach.
Let's $C_C$ be the set of continuous compact supported functions and $C_C^+$ the subset of non negative functions of $C_C$.
For all $f\in C_C^+$, it's easy (integrating by parts) to find out that
$$\label{a}\tag{1}\int_0^{+\infty}F^p(x)dx=\dfrac{p}{p-1}\int_0^{+\infty}f(x)F... |
Let $f:\mathbb R\to\mathbb R$ be a continuous function such that $f>0$ for almost every $x\in\mathbb R$.
Does one can deduce from this that both $\inf_{x\in\mathbb R} f(x) >0$ and $\sup_{x\in\mathbb R} f(x)>0$?
If yes, how to show this? It seems reasonable to me but I do not how to approach the proof. Any hint? |
If $f:\mathbb R\to\mathbb R$ is such that $f>0$ a.e., do $\inf_{\mathbb R} f >0$ and $\sup_{\mathbb R} f>0$? |
I'm preparing a talk on Mersenne primes, Perfect numbers and Fermat primes.
In trying to provide motivation for it all I'd like to provide an application of these things.
I came up with these:
Applications of Mersenne numbers: signed/unsigned integers, towers of Hanoi
Applications of Fermat numbers: relati... |
As a student of numerical analysis, I understand that a finite difference approximation (FDM) of the derivative '$u_x$' of a desired solution '$u$' at the point $x_n$ in the domain is just a linear combination $\displaystyle \sum_{i=0}^Nc_i u_i,$ where, $u_i=u(x_i)$.
That is, $\displaystyle (u_x)_n=u_x(x_n)\approx c... |
In the book by Fulton & Harris (chapter 10, section 1) :
> Finally, in the real case things are simpler: when we exponentiate the
adjoint representation as above, the Lie group we arrive at is already simply
connected, and so is the unique connected real Lie group with this Lie algebra.
Just to give backgrou... |
As a student of numerical analysis, I understand that a finite difference approximation (FDM) of the derivative '$u_x$' of a desired solution '$u$' at the point $x_n$ in the domain is just a linear combination $\displaystyle \sum_{i=0}^Nc_i u_i,$ where, $u_i=u(x_i)$.
That is, $\displaystyle (u_x)_n=u_x(x_n)\approx c... |
**Short answer**
$x^{x^{x^{x^{x^{...}}}}}$ is not defined for $x < 0$.
**Explanation**
How is $x^{x^{x^{x^{x^{...}}}}}$ defined? It is the limit $\lim_{n\to\infty} a_n$ of the sequence
\begin{align*}a_0 & = 1 \\ a_1 & = x \\ a_2 & = x^x \\ a_3 & = x^{x^x} \\
a_4 & = x^{x^{x^x}} \\
& \ldots
\end{align*}
St... |
I just watched a [video][1] by blackpenredpen where he solved this equation. Here are the steps he took (The process transitions from one line to another. He didn't explicitly use an implies or iff sign, so I'll leave it as such):
$x^{x^{x^{\:\cdot^{\:\cdot^{\:\cdot}}}}} = 2$
$x^2 = 2$
$\left[
\begin{array}{... |
$x^{x^{x^{\:\cdot^{\:\cdot^{\:\cdot}}}}} = 2$. Why is $-\sqrt{2}$ not a solution? |
**Short answer**
$x^{x^{x^{x^{x^{...}}}}}$ is not defined for $x < 0$.
**Explanation**
How is $x^{x^{x^{x^{x^{...}}}}}$ defined? It is the limit $\lim_{n\to\infty} a_n$ of the sequence
\begin{align*}a_0 & = 1 \\ a_1 & = x \\ a_2 & = x^x \\ a_3 & = x^{x^x} \\
a_4 & = x^{x^{x^x}} \\
& \ldots
\end{align*}
St... |
I just watched a [video][1] by blackpenredpen where he solved this equation. Here are the steps he took (The process transitions from one line to another. He didn't explicitly use an implies or iff sign, so I'll leave it as such):
$x^{x^{x^{x^{x^{...}}}}} = 2$
$x^2 = 2$
$\left[
\begin{array}{l}
x = \sqrt{2}... |
$x^{x^{x^{x^{x^{...}}}}} = 2$. Why is $-\sqrt{2}$ not a solution? |
The following is from "Multiplicative number theory I: Classical theory" by Hugh L. Montgomery, Robert C. Vaughan:
[![enter image description here][1]][1]
The proof is very much ambiguous to me. For example:
1- The claim $\int_1^x u^{-\sigma} du = (x^{1- \sigma} -1)/(1- \sigma) < x^{1- \sigma}/(1- \sigma)$ is... |
In this post I will assume the NBG set theory
>For all classes $X$ let $M(X)$ mean $(\exists T)(X \in T)$ i.e. $X$ is a set and not a proper class.<br/>
Prove that $\left\{X \subseteq Y, M(Y) \right\} \vdash_{\text{NBG}} M(X)$
My ideea for the proof. We know that for all classes $X$ and $Y$ that $X \subseteq Y \im... |
[Complex Analusis]Help!How to use Laurent expansion of a function into this form? |
The following theorem appears in [The Number System, by H.A. Thurston][1]:
>*3. **Theorem:** There is just one operation $\theta$ such that, for every
$x$ and $y,$*
\begin{align*}
(i) & \theta\left(x,0\right)=x\text{ and}\\
(ii) & \theta\left(x,y^{\prime}\right)=\theta\left(x,y\right)^{\prime}.
\end{align*}
... |
**Short answer**
$x^{x^{x^{x^{x^{...}}}}}$ is not defined for $x < 0$.
**Explanation**
How is $x^{x^{x^{x^{x^{...}}}}}$ defined? It is the limit $\lim_{n\to\infty} a_n$ of the sequence
\begin{align*}a_0 & = 1 \\ a_1 & = x \\ a_2 & = x^x \\ a_3 & = x^{x^x} \\
a_4 & = x^{x^{x^x}} \\
& \ldots
\end{align*}
St... |
Consider a dynamical system defined by a 1-d unimodal map. Suppose that the topological entropy of the system is positive. Then does the time average of f converge to a constant? If so, give me any reference. |
> Let $x_1,x_2,\ldots,x_n$ be arbitrary real numbers. Prove the inequality $$\frac{x_1}{1+x_1^2} + \frac{x_2}{1+x_1^2 + x_2^2} + \cdots +
\frac{x_n}{1 + x_1^2 + \cdots + x_n^2} < \sqrt{n}$$
-----------
I found a solution here which says:
> $$\frac{x_1}{1+x_1^2} + \frac{x_2}{1+x_1^2 + x_2^2} + \cdots +\frac{x_n}{1 +... |
I'm trying to follow the proof of Vizing's Theorem given here https://math.uchicago.edu/~may/REU2015/REUPapers/Green.pdf
Unfortunately, I soon ran into trouble with this line:
> If $c_k$ is absent at $v$, then we can reassign colours $c_i$ to $\{v, w_i\}$ for $i ∈ [k]$ and we are done.
The problem is that ev... |
Can we categorify a monad $T A$ for a fixed set? |
Let $(H, \|\cdot\|)$ be a Hilbert space and let $(f_n)_n$ be a bounded sequence in $H$, i.e. there exists $C>0$ such that
$$\|f_n\|\le C \quad\forall n.$$
Let $f$ denote the weak limit of $f_n$.
Does one can conclude that $f$ is bounded in $H$? If it is true, is it bounded with the same constant $C$?
I am tryi... |
If a sequence is bounded, does its weak limit happen to be bounded? |
Improper integral with parameter. For what value of a does the integral converge? |
I encountered the ring $\mathbb{Z}/(n^p+1)\mathbb{Z}$, where $n$ is some positive integer and $p\in\{2,3,5,...\}$ a prime and I am wondering whether there is a difference in the structure when $p=2$ and $p$ an odd prime?
Independently of $n$ and $p$, if $n^p+1$ is prime, then we obtain a finite field but I am int... |
How can I compute the Laurent expansion of the function
$$f(z)=\frac{1}{(2z+1)(z-2)}$$
at $z=0$ when $\frac{1}{2}<|z|<2$
with this $\frac{1}{z^2}$ term?
[![enter image description here][1]][1]
[1]: https://i.stack.imgur.com/uQ1QQ.png |
How to use Laurent expansion of a function into this form? |
After learning about finite dimensional vector spaces, the time came for learning about infinite dimensional ones. However, these seem much less intuitive to me, and proof of that is the question I wanted to ask here.
Let's consider a infinite dimensional vector space $(V, +, \cdot)$, with no topology or any other s... |
I tried writing down a formula relating a given debt and interest to the periodic payments and number of payments.
So let's say someone starts off with a debt of $D$. The periodic interest is $r$ (for example, $0.6\% =0.006$ each month). Let's call the periodic payment $p$, and the number of payments $N$.
I based... |
How can I compute the Laurent expansion of the function
$$f(z)=\frac{1}{(2z+1)(z-2)}$$
at $0$ when $\frac{1}{2}<|z|<2$
with this $\frac{1}{z^2}$ term?
[![enter image description here][1]][1]
[1]: https://i.stack.imgur.com/uQ1QQ.png |
I've seen in some books of Analysis the notation $\mathcal{L}(\mathcal{L}(V), V) \cong \mathcal{L}_2(V)$ and I don't find anything about this vector space. What is it? Is there a name for it? Also, if I wrote this wrong, I aks for someone to correct me. |
Is there a name for this vector space? |
**Hint** $ $ Verify both sides are solutions of the recurrence $\rm\: f(n) - f(n\!-\!1) = (-1)^n 4n,\,\ f(1) =4,$ therefore they are equal for all $\rm\:n\:$ by a simple induction (which amounts to the *uniqueness* theorem for such recurrences -- see my [many posts on *telescopy*](https://math.stackexchange.com/search?... |
After learning about finite dimensional vector spaces, the time came for learning about infinite dimensional ones. However, these seem much less intuitive to me, and proof of that is the question I wanted to ask here.
Let's consider a infinite dimensional vector space $(V, +, \cdot)$, with no topology or any other s... |
Given that X and Y have joint probability generating function(pgf)
$$G_{X,Y}(s,t)=e^{\alpha(s-1)+\beta(t-1)+\gamma(st-1)}$$
the question ask me to find the marginal distribution of $X$,$Y$ and the distribution of $X+Y$
For marginal distribution of $X$, when $t=1$ $$G_{X,Y}(s,1)=G_X(s)=e^{(\alpha+\gamma)(s-1)}$$ wh... |
I noticed some books define the Riemman curvature tensor with opossite signs on all 3 terms. I wonder what happens to the Ricci curvature, the scalar curvature, and the Einstein tensor when using one or other definition? Would they also change sign? And if I change the signs of the metric from mostly plus to mostly mi... |
Whathappens to the Ricci curvature, the scalar curvature, and Einstein tensor if I change the signs in the def of the Riemann curvature tensor? |
I don't think I could ever say I understand ZFC if I don't get to the bottom of this question.
It all started when we had extensionality and comprehension. Then Russell finds a paradox and here comes ZFC to the rescue. We keep extensionality and we introduce a series of axioms that build the sets we need and whatever ... |
Do we really need the axiom of regularity? |
How to solve the following mixed exponential inequality: $3^x + x < 4$? (from Spivak's Calculus) |
I begin by two examples showing that the solutions of
$$AL+L^T=0\tag{1}$$
can depend on parameters.
a) If we take
$$A=\pmatrix{1&0&0\\0&1&1\\0&0&-1},$$
then the general solution of (1) is :
$$L=\pmatrix{0&a&0\\-a&b&-2b\\0&-2b&4b}.$$
b) If $$A=\pmatrix{1&0&0&0\\0&0&1&0\\0&1&0&0\\0&0&0&1}$$ (in fac... |
I wonder, if given a graph $A$ represented by an adjacency matrix $C$ and one would invert $C$, is there a proof that shows what the structure of $A^{-1}$ would look like? For example looking at the structure of Molecules using the Hückel matrix. |
Let $F$ be a multiplicative generalized cohomology theory.
Let $\sigma^0 \in F^0(pt) \simeq \tilde{F}^0(S^0)$, and let for any $n\in \mathbb{Z}$, $\sigma^n \in \tilde{F}^n(S^n)$ be the element associated to $\sigma^0$ under the suspension isomorphism.
In the famous paper of Atiyah-Bott-Shapiro (Clifford Module... |
After learning about finite dimensional vector spaces, the time came for learning about infinite dimensional ones. However, these seem much less intuitive to me, and proof of that is the question I wanted to ask here.
Let's consider a infinite dimensional vector space $(V, +, \cdot)$, with no topology or any other s... |
I am trying to find a compact expression for $\sum_{k=0}^{n}\binom{n}{k} \frac{(-1)^{k}}{(N-k)^2}$, where $N > n \geq 0$. Maple simplifies it using hypergeometric functions, which is not very useful. Any idea whether this can be simplified ? and if yes, how ? Thanks a lot! |
(I do not understand well where the [Stone-Von Neumann Theorem][1] enters the issue, maybe the right theorem is the [Stone Theorem][2], not the SvN one?)
Suppose that a strongly continuous one-parameter group of unitaries $\mathbb{C} \ni z \mapsto U_z$ exists in the complex Hilbert space $\cal H$ with the properti... |
Let $(H, \|\cdot\|)$ be a Hilbert space and let $(f_n)_n$ be a bounded sequence in $H$, i.e. there exists $C>0$ such that
$$\|f_n\|\le C \quad\forall n.$$
Let $f$ denote the weak limit of $f_n$.
Does one can conclude that $f$ is bounded in $H$? If it is true, is it bounded with the same constant $C$?
I am tryi... |
Poisson distribution expresses the probability that a specific number of discrete independent events happen over a fixed time interval, as long as the events are sufficiently rare.
When searching for derivation of Poisson distribution from the first principles online, I saw all of them make the following extra assum... |
Let $a(x)$, $b(x)$, $c(x)$ be smooth coefficient on $\mathbb{R}$ and $u(x)$ be the solution to the linear ODE over $\mathbb{R}$:
$$ au''+bu'+cu=-1 $$
Suppose $a(x)>1$, can we have some estimate on $|u'|$ independent of the derivatives of $a(x)$, $b(x)$, $c(x)$? Or in simplest setting, how can we estimate the deri... |
I have an expression of the form
$$
\tag{1}
A(\chi) = \int_{0}^\infty\sum_{i=0}^\infty (-1)^{i+1}\frac{(2i+1)\chi}{\sqrt{t}}\exp\left(\frac{-(2i+1)^2\chi^2}{t}\right)\mathrm{d}t.
$$
If I am not mistaken, this expression is equal to
$$
\tag{2}
A(\chi) = \frac{1}{4}\int_{0}^\infty \frac{\chi}{\sqrt{t}}\frac... |
After learning about finite dimensional vector spaces, the time came for learning about infinite dimensional ones. However, these seem much less intuitive to me, and proof of that is the question I wanted to ask here.
Let's consider a infinite dimensional vector space $(V, +, \cdot)$, with no topology or any other s... |
After learning about finite dimensional vector spaces, the time came for learning about infinite dimensional ones. However, these seem much less intuitive to me, and proof of that is the question I wanted to ask here.
Let's consider a infinite dimensional vector space $(V, +, \cdot)$, with no topology or any other s... |
(I do not understand well where the [Stone-Von Neumann Theorem][1] enters the issue, maybe the right theorem is the [Stone Theorem][2], not the SvN one?)
Suppose that a strongly continuous one-parameter group of unitaries $\mathbb{C} \ni z \mapsto U_z$ exists in the complex Hilbert space $\cal H$ with the proper... |
How can I compute the Laurent expansion of the function
$$f(z)=\frac{1}{(2z+1)(z-2)}$$
around $0$ when $\frac{1}{2}<|z|<2$
with this $\frac{1}{z^2}$ term?
[![enter image description here][1]][1]
[1]: https://i.stack.imgur.com/uQ1QQ.png |
In addition to Isaacs' Theorem that Arturo mentioned, below is a concrete example of the application of this theorem. It appeared in the paper *On Commutators in Groups*, by L-C. Kappe, R. F. Morse, Groups St Andrews $2005$, Vol. 2, p.541. However, their calculation in Example 4.5 is wrong and we present the improved o... |
Are there some good overviews of basic facts about stochastic integrals and stochastic calculus?
These can be in the form of resources (preferably accessible online) as well as directly writing out these results.
If possible, it would be helpful to link to proofs of the results. These can be proofs externally or ... |
After learning about finite dimensional vector spaces, the time came for learning about infinite dimensional ones. However, these seem much less intuitive to me, and proof of that is the question I wanted to ask here.
Let's consider a infinite dimensional vector space $(V, +, \cdot)$, with no topology or any other s... |
Both sides are solutions of $\,f(n\!+\!1)-f(n) = (n\!+\!1)(n\!+\!1)!,\ f(0) = 0,\:$ so they are equal by the *uniqueness theorem* for first-order recurrences (fundamental theorem of difference calculus), which has a trivial [one-line inductive proof](https://math.stackexchange.com/a/239339/242) (expressed in the langua... |
**Abel's Test:** Let $f_n(x)$ be a non-increasing sequence of functions such that $0 \le f_n(x) \le M$ for all $x \in [a,b]$. If $\sum a_n$ converges then $\sum a_nf_n(x)$ converges uniformly in $[a,b]$.
**Abel's Test corollary:** Let $f_n(x)$ be a monotonic functions such that $0 \le f_n(x) \le M$ for all $x \in... |
conditions of dirichlet and dini's thereom be simply monotonic? |
I was talking to a professor, and we both were stumbled on the problem of: /sqrt{/sqrt{x^{2}}-1}=2
My professor's thought process was that you would immediately square both sides to get /sqrt{x^{2}-1}=4 then solve to get x= +/- 5.
My thought process was that you could simplify the /sqrt{x^{2}} first which would ... |
Why are there two different ways to solve the same problem but two different answers? |
[OEIS A169945](https://oeis.org/A169945) introduces the concept of _thickness_ of a polynomial as the magnitude of the largest coefficient in the expansion of the square of the polynomial. Considering the $2^{n+1}$ polynomials $p(x)$ with coefficients $0$ or $1$ and degree at most $n$, it denotes by $T(n,k)$, the numbe... |
In Munkres' Analysis on Manifolds, page 57 Theorem 7.1(the Chain Rule) it states: ...For this purpose, let us introduce the function $F(\mathbf h)$ defined by setting $F(\mathbf 0)=\mathbf 0$ and
\begin{equation*}
F(\mathbf h)=\frac{[\Delta(\mathbf h)-Df(\mathbf a)\cdot \mathbf h]}{|\mathbf h|} for \ 0<|\mathbf h|<\d... |
**Abel's Test:** Let $f_n(x)$ be a non-increasing sequence of functions such that $0 \le f_n(x) \le M$ for all $x \in [a,b]$. If $\sum a_n$ converges then $\sum a_nf_n(x)$ converges uniformly in $[a,b]$.
**Abel's Test corollary:** Let $f_n(x)$ be a monotonic functions such that $0 \le f_n(x) \le M$ for all $x \in... |
Are there some good overviews of basic facts about Stochastic Integrals and Stochastic Calculus?
These can be in the form of resources (preferably accessible online) as well as directly writing out these results.
If possible, it would be helpful to link to proofs of the results. These can be external proofs or pr... |
I was talking to a professor, and we both were stumbled on the problem of: $\sqrt{\sqrt{x^{2}}-1}=2$
My professor's thought process was that you would immediately square both sides to get $\sqrt{x^{2}-1}=4$ then solve to get $x= \pm 5$
My thought process was that you could simplify the $\sqrt{x^{2}}$ first which... |
Are your $v_i$ vectors in your vector space? It's up to you: you can impose the rule that only finitely many coordinates are non-zero or allow any number of non-zero coordinates.
**Finitely many non-zero coordinates**
Your $e_i$ vectors form a basis but your $v_i$ vectors are not in the space.
**Any numbe... |
When studying *Elementary Differential Equations* by William, I found trouble understanding
[Theorem 5.1.5][1]
It says the two solutions are linearly independent iff their Wronskian is never zero, but I think they can still be linearly independent even if the Wronskian is zero for some $x$.
In the proof, when ... |
When studying *Elementary Differential Equations* by William, I found trouble understanding
[Theorem 5.1.5][1]
It says the two solutions are linearly independent iff their Wronskian is never zero, but I think they can still be linearly independent even if the Wronskian is zero for some $x$.
In the proof, when ... |
I found how to demonstrate the Taylor Series with integral remainder.
$$ f(x) = \sum_{k=0}^n \Biggl[\frac{f^{(k)}(a)}{k!}(x-a)^n \Biggr] + \int_a^x \frac{f^{(n +1)}(t)}{n!}(x-t)^n dt $$
But I do you pass to this formula to the formula with the $o(x^n)$ term ?
$$ f(x) = \sum_{k=0}^n \Biggl[\frac{f^{(k)}(a)}{k!}... |
How do you demonstrate each $o(x^n)$ term on Taylor Series? |
In addition to Isaacs' Theorem that Arturo mentioned, below is a concrete example of the application of this theorem. It appeared in the paper *On Commutators in Groups*, by L-C. Kappe, R. F. Morse, Groups St Andrews $2005$, Vol. 2, p.541. However, their calculation in Example 4.5 is wrong and we present the improved o... |
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