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Suppose $A$ is a non-cyclic finite abelian group and $a\in A$. Can we always find $A_1,A_2\leq A$ such that $a\in A_1$ and $A=A_1\oplus A_2$?
Equivalently, is it true that given a finite abelian $A$ and an $a\in A$, we can decompose $A$ into cyclic factors, one of which contains $a$? |
Is every element of a finite abelian group contained in a cyclic factor? |
Let $R$ be discrete valuation ring with field of fractions $K$, and $\phi:R \to A$ a *finite, local* (ie $A$ dominates $R$ in the sense that it's maximal ideal contains the image of maximal ideal of $R$) map of local rings.
**#Edit:** As Giorgio Genovesi pointed out, essentially the "interesting" case appear where $... |
Assume spaces to be *connected*. In general, the set of unbased homotopy classes $[X,Y]$ differs from the set of pointed homotopy classes $\langle X,Y\rangle$. This is captured by the $\pi_1(Y)$ action on the set $\langle X,Y\rangle$. The (computable) examples I know for which this action is non-trivial are not manifol... |
How careful do we have to be with choosing whiskers? |
Here it is:
$$\int_{K_1}^{K_2}\Big(\frac{\pi}{2}+\arcsin(\cos\varphi+0.75)+\arcsin(\sin\varphi+0.75)\Big)\,\,\mathrm{d}\varphi$$
Where $K_1=\pi+\arccos\big(\frac{3}{4}\big)$ and $K_2=\frac{3\pi}{2}-\arccos\big(\frac{3}{4}\big)$. This is coming from a circle-type question I made. The main problem is the $\arcsin(\sin\... |
How can I chain rotate and scale operations (each with different origins) while keeping each operation origin in relation to the original local component space?
Using matrix denotation (i.e. *T* for translation matrix, *R* for the rotation matrix, and *S* for the scaling matrix) I'm performing the following operatio... |
How can I chain rotate and scale operations (each with different origins) while keeping each operation origin in relation to the original local component space?
Using matrix denotation (i.e. *T* for translation matrix, *R* for the rotation matrix, and *S* for the scaling matrix) I'm performing the following operatio... |
How can I chain rotate and scale operations (each with different origins) while keeping each operation origin in relation to the original local component space?
Using matrix denotation (i.e. *T* for translation matrix, *R* for the rotation matrix, and *S* for the scaling matrix) I'm performing the following operatio... |
How can I chain rotate and scale operations (each with different origins) while keeping each operation origin in relation to the original local component space?
Using matrix denotation (i.e. *T* for translation matrix, *R* for the rotation matrix, and *S* for the scaling matrix) I'm performing the following operatio... |
How can I chain rotate and scale operations (each with different origins) while keeping each operation origin in relation to the original local component space?
Using matrix denotation (i.e. *T* for translation matrix, *R* for the rotation matrix, and *S* for the scaling matrix) I'm performing the following operatio... |
When a spherical cap is cut by a plane $y=y_1$ that is perpendicular on the plane that made the sperical cap $z=z_1$, I managed to calculate the volumes of the resulting two pieces of the spherical cap by the following integral:
$$\int\limits_{y_1}^{y_2} {\int\limits_{-\sqrt {1 - {y^2}-{z_1^2}} }^{\sqrt {1 - {y^2}-{... |
I am trying to find a causal graph from this data.
P(Z=1) = 0.25 P(Z=0) = 0.75
P(X=1 | Z =0) = 0.85 P(X=1 | Z = 1) = 0.65
P(Y=1 | Z =1, X =0) = 0.75 P(Y=1 | Z =1, X =1) = 0.25
P(Y=1 | Z =0, X =1) = 0. 5 P(Y=1 | Z =0, X =0) = 0. 5
From this data, I have calculate... |
I was wondering if it's possible to rearrange to **solve for $y$ or $x$** in the following equation containing factorials?
$x(y)! (x)! y = 1 $
-------------------
I try to solve for $y$ as
$y*y!=1/(x*x!)$
Playing around I only complicate it:
$2y(y-1)!=1/(x*x!)$
I alternatively try to guess if there'... |
How can I chain rotate and scale operations (each with different origins) while keeping each operation origin in relation to the original local component space?
Using matrix denotation (i.e. *T* for translation matrix, *R* for the rotation matrix, and *S* for the scaling matrix) I'm performing the following operatio... |
With the aim of proving that something is uncountable, it's common to utilize the Cantor diagonal method to achieve one's goal. It's not hard to see that all of those arguments lie in proving the function given by
$$
f (\{\eta_k\}_{k\in\mathbb{N}}) = \sum^{\infty}_{k=1} \dfrac{\eta_k}{2^{k}},
$$ for all sequences $\... |
Here is an alternative way to compute the Moore-Penrose inverse $A'$ of $A$ following Penrose's argument.
---
Let's first state Penrose's generalized inverse theorem for matrices with real entries in a way that is indicative of his proof:
**Theorem:** Let $A\in\mathcal{M}(m\times n;\mathbb{R})$. Then:
1. T... |
Could you help me solve the following tricky exam problem? Let $f: [
-1, 1]\rightarrow \mathbb R$. Let $a\in [-1, 1]$. Suppose that for every non-terminating sequence {$a_n$}, $a_n\in [-1, 1]$, s.t. $a_n\rightarrow a$ the sequence {$f(a_n)$} diverges. Does such a function exist? What if the above condition is true for... |
Does a function having the property that for every $a_n\rightarrow a$ $f(a_n)$ diverges exist? |
I was wondering if it's possible to rearrange to **solve for $y$ or $x$** in the following equation containing factorials?
$$x(y)! (x)! y = 1 $$
-------------------
I try to solve for $y$ as
$$y \cdot y!= \frac{1}{x \cdot x!}$$
Playing around I only complicate it:
$$2y(y-1)!=\frac{1}{x \cdot x!}$$
... |
I am learning type theory from nlab articles.
The following is from [typed predicate logic][1].
> ### Judgments and contexts
>
> Typed predicate logic is a type theory which consists of two layers, a
> layer for types and a layer for propositions.
>
> We have the basic judgement forms of the type layer:... |
Why $\Gamma\mathrm{ ctx}$ instead of $\Gamma:\mathrm{ctx}$, etc? |
I was looking at $C^\infty$ functions that are not analytic anywhere. For that I was looking at solutions of $f'(x) = f(ax)$. There is such a function for $a = 2$. But how would you solve that differential equation in general, for any value of $a$?
Thanks! |
I was looking at a problem that deals with tetration. Here's the problem:
$$\begin{split}x^{x^{x^{\ddots}}}=3 \\ x^3=3 \Rightarrow x= \sqrt[3]{3} \end{split}$$
However, I saw that the solution is $-\frac{3}{\ln(3)}W_{-1}\left(-\frac{3}{\ln(3)}\right)\approx 2.478$ and watched some videos which explain something like ... |
The constructions of complex number that you must have seen rely on people's intuitions and experience with arithmetic, in order to avoid certain formalities that are deemed too confusing or too demotivating for most people. However, if one is very careful, one discovers that they do not hold water, just like you did: ... |
Why $\Gamma \vdash A \; \mathrm{type}$ instead of $\Gamma \vdash A :\mathrm{type}$, etc? |
> Let $p$ be an odd prime and let $\zeta$ be a $p$-th root of unity. Is it true that $$\Bigg\lvert\frac{1 - \zeta^s }{ 1 - \zeta}\Bigg\rvert \in\mathbb{Q}(\zeta)$$ for $s = 1, \ldots, p-1$?
>
> Remark: By $\lvert \cdot \rvert$ I mean the standard norm over $\mathbb{C}$.
I do not know how to prove this. I was tryi... |
Is $\bigg\lvert\frac{1 - \zeta^s }{ 1 - \zeta}\bigg\rvert \in\mathbb{Q}(\zeta)$ for $s = 1, \ldots, p-1$? |
The Cramér random model for the primes is a random subset ${{\mathcal P}}$ of the natural numbers with ${1 \not \in {\mathcal P}}, {2 \in {\mathcal P}}$, and the events ${n \in {\mathcal P}}$ for ${n=3,4,\dots}$ being jointly independent with ${{\bf P}(n \in {\mathcal P}) = \frac{1}{\log n}}$ (the restriction to ${n \g... |
I was wondering if it's possible to rearrange to **solve for $y$ or $x$** in the following equation containing factorials?
$$x(y)! (x)! y = 1 $$
-------------------
I try to solve for $y$ as
$$y \cdot y!= \frac{1}{x \cdot x!}$$
Playing around I only complicate it:
$$y^2(y-1)!=\frac{1}{x \cdot x!}$$
... |
I've seen rejections of the Axiom of Infinity. This is called finitism. Some ultrafinitists even add the negation of the Axiom of Infinity. Definitely doable.
I've seen rejections of the Axiom of Choice. That would just mean you choose to work in ZF and not ZFC, which is controversial but not unheard of.
I could... |
Finitists reject the Axiom of Infinity - are there groups who reject the others? |
Could you help me solve the following tricky exam problem? Let $f: [
-1, 1]\rightarrow \mathbb R$. Let $a\in [-1, 1]$. Suppose that for every not ultimately constant sequence {$a_n$}, $a_n\in [-1, 1]$, s.t. $a_n\rightarrow a$ the sequence {$f(a_n)$} diverges. Does such a function exist? What if the above condition is ... |
Let $d(n)$ be number of positive divisor of natural number $n$. It is known from $d(n)=O(n^{\varepsilon})$ for every $\varepsilon>0$, we can deduce that $d(n)=n^{C/(\log \log n)}$ for some constant $C$. I am curious whether there is a proof for this fact directly and deduce $d(n)=O(n^{\varepsilon})$ as a corollary. Doe... |
There is an existing theorem that states that *if polynomials have distinct degrees, they are linearly independent.*
For example, if we were to find a basis of P3 containing the *independent set* {1 + x, 1 + x^2}, we can extend the basis by adding any *distinct* vectors from the standard basis of P3 - {1, x, x^2, x^... |
Question: Is an integral domain in which every proper ideal is a product of prime ideals necessarily a Dedekind domain? Please provide a reference.
The first two sentences of the Wikipedia page for Dedekind domains are as follows: 'In abstract algebra, a Dedekind domain or Dedekind ring, named after Richard Dedekind... |
Is an integral domain with every ideal a product of prime ideals necessarily a Dedekind domain? |
I was wondering if it's possible to rearrange to **solve for $y$ or $x$** in the following equation containing factorials?
$$x(y!)(x!)y = 1 $$
-------------------
I try to solve for $y$ as
$$y \cdot y!= \frac{1}{x \cdot x!}$$
Playing around I only complicate it:
$$y^2(y-1)!=\frac{1}{x \cdot x!}$$
... |
Is it possible to solve for $x$ or $y$ in the equation $x(y!)(x!)y=1$? |
Let $X$ be a normed vector space. We define the weak-$^*$ topology on $X^*$ to be the initial topology generated by all linear functionals $ev_x: X^*\rightarrow F$ defined by
$$
f\mapsto f(x).
$$
Here is my professor's description about the subbasic open sets and basic open sets with respect to the weak-$^*$ topol... |
I was wondering if it's possible to rearrange **for $y$ or $x$** in the following equation containing factorials?
$$x(y!)(x!)y = q $$
-------------------
I try to solve for $y$ as
$$y \cdot y!= \frac{q}{x \cdot x!}$$
Playing around I only complicate it:
$$y^2(y-1)!=\frac{q}{x \cdot x!}$$
$$y^{2}... |
Is it possible to re-arrange LHS for $x$ or $y$ in the equation $x(y!)(x!)y=q$? |
Let $\Gamma$ be a consistent theory of propositional logic and $\varphi$ a propositional formula.
It is well known that $\Gamma \cup \{\neg\varphi\}$ is consistent if $\Gamma \not\vdash \varphi$.
A common procedure for proving this is to exploit deduction theorem. My question is, is there any way to show the cons... |
Solve the DE : $t\frac{\partial ^2 U}{\partial x \partial t}+2\frac{\partial U}{\partial x}=x^2$
Here's the solution:
(step 1) $\frac{\partial}{\partial x}[t\frac{\partial U}{\partial t}+2U]=x^2$ , then integrate with respect to x
(step 2) $t\frac{\partial U}{\partial t}+2U=\frac{1}{3}x^3+R(t)$ , then multipl... |
[![enter image description here][1]][1]
[1]: https://i.stack.imgur.com/xCPBQ.png
This problem is from Charles C. Pugh - Real Mathematical Analysis.
To prove $f$ is continuous, the main idea is just to prove the difference $$\begin{align*}
|f(x) - f(x_{n})|
\end{align*}$$ can be small enough. Also, we alr... |
Is there a way to continue/fix this proof of proving a function is continuous? |
Write the expansion of $\sin(A+B)$. It is given that in a triangle ABC, $\angle C=30^\text{o}$, $\sin{A} =3\sin{B}$. Show that $\cot{B}=6-\sqrt{3}$.
**My try**
The reason they have asked us to write the expansion could be just to make use of it to substitution $A+B=180^\text{o}-C$ to ,
$\sin(A+B)=\sin{A}\cos{B... |
Show that $\cot{B}=6-\sqrt{3}? |
> **Problem**: Two hallways, of widths *a* and *b* meters, meet at a right angle. What is the longest ladder (of negligible width) that can be carried around the corner horizontally?
Hi, here's my approach to this problem.
I model this as a sliding ladder against the leftmost and bottom walls (as shown). The ladd... |
> **Problem**: Two hallways, of widths *a* and *b* meters, meet at a right angle. What is the longest ladder (of negligible width) that can be carried around the corner horizontally?
Hi, here's my approach to this problem.
I model this as a sliding ladder against the leftmost and bottom walls (as shown). The ladd... |
Is $dx$ in $\int f(x)dx$ a differential of the identity function? In Zorich analysis, it is said that if $dF = f(x)dx$, then $\int dF=\int f(x)dx$. This only makes sense if $dx$ actually represents a differential, so that $f(x)dx$ is actually a function that maps each point into a linear function (i.e $\mathbb{R}\to(\m... |
Is f(x)dx in integral a differential? |
> **Problem**: Two hallways, of widths *a* and *b* meters, meet at a right angle. What is the longest ladder (of negligible width) that can be carried around the corner horizontally?
Hi, here's my approach to this problem.
I model this as a sliding ladder against the leftmost and bottom walls (as shown). The ladd... |
Show that $\cot{B}=6-\sqrt{3}$? |
$\color{blue}{\text{Required Definitions and Results:}}$
**Proposition 4.3.3 (b).** Let $x, y$ be rational. We have $|x + y| \leq |x| + |y|$.
**Proposition 4.3.3 (f).** $|x-y| = |y-x|$.
**Definition 5.1.12** (Bounded sequences). Let $M \geq 0$ be rational. A finite sequence $a_1, a_2, \ldots, a_n$ is bounded b... |
For any regular element $g$ in $\mathfrak{gl}_n(C)$, the dimension of centralizer of $g$ will be $n$. What's more, the centralizer of regular elements can be simultaneously generated by homogeneous invariant polynomial of $\mathfrak{gl}_n(C)$, hence it's isomorphic to $\exp{(\mathbb{C}^n)}$. For a reference, see lemma ... |
Consider the subset $X:=\\{(t^{2},t^{3})|t\in\mathbb{C}\\}$ in $\mathbb{C}^{2}$, and set $X_{0}:=X\backslash\\{(0,0)\\}$. Show that $$\mathcal{V}(\mathcal{J}(X_{0}))=X.$$Note: here $\mathcal{V},\mathcal{J}$ stand for the common zero locus and vanishing ideal functions, respectively.
Could you please give me a hint ... |
Consider the subset $X:=\{(t^{2},t^{3})|t\in\mathbb{C}\}$ in $\mathbb{C}^{2}$, and set $X_{0}:=X\backslash\{(0,0)\}$. Show that $$\mathcal{V}(\mathcal{J}(X_{0}))=X.$$Note: here $\mathcal{V},\mathcal{J}$ stand for the common zero locus and vanishing ideal functions, respectively.
Could you please give me a hint as t... |
Let V be the following vector space.
> $V = \{ f : \mathbb{R} \longrightarrow \mathbb{R} : \text{f = infinitely differentiable at all x, } f^{(k)}(0) = 0 \;\;, \forall \;k \in (0 \cup \mathbb{N}) \}$
How would one show that this vector space is infinite dimensional?
----------
To show that a vector sp... |
Experimental data show that in any triangle with sides $(a,b,c)$ and circumradius $R$, if $x > 2$ then,
$$
\frac{1}{x} - \frac{4}{x^2 - 4} < \frac{R}{a-b+Rx} + \frac{R}{b-c+Rx} - \frac{R}{c-a+Rx}
< \frac{1}{x} + \frac{4}{x^2 - 4} \tag 1
$$
Can this be proved or disproved?
**Corollary**: In the interval $(... |
Does every triangle satisfy $\frac{R}{a-b+5R} + \frac{R}{b-c+5R} > \frac{R}{c-a+5R}$? |
I've been pondering over a question sparked by two math problems: How can one assert from the outset that a problem contains hidden initial values, and what methods can be employed to discern this? My approach felt rather rudimentary; I found myself substituting various values into each new equation I derived, only to ... |
I am reading the Eisenbud, Commutative Algebra, Corollary 13.13 ( Emmy Noether ) and stuck at some statement.
First, I propose a relavant question.
**Q.** Let $R:=k[x_1 , \dots , x_d]$ with $\operatorname{char}(k)=p >0$ so that its fraction field is $K=K(R) = k(x_1, \dots ,x_d)$. Let $L'/K$ be a finite purel... |
Here is the question in my exercise.
> Let $f(x,y)$ be a function defined on a rectangle $[a,b]\times [c,d]$. Assume that for each fixed $y \in[c,d]$ the function $x\mapsto f(x,y)$ is a continuously differentiable function of $x$ on $[a,b]$; and for each fixed $x \in[a,b]$ the function $y\mapsto f(x,y)$ is a continu... |
If $x\mapsto f(x,y)$ and $y\mapsto f(x,y)$ is continuous differentiable, then whether $f(x,y)$ is continous? |
Experimental data show that in any triangle with sides $(a,b,c)$ and circumradius $R$, if $x > 2$ then,
$$
\frac{1}{x} - \frac{4}{x^2 - 4} < \frac{R}{a-b+Rx} + \frac{R}{b-c+Rx} - \frac{R}{c-a+Rx}
< \frac{1}{x} + \frac{4}{x^2 - 4} \tag 1
$$
Can this be proved or disproved?
**Corollary**: Assuming the above... |
Does every triangle satisfy $\frac{1}{a-b+\pi R} + \frac{1}{b-c+\pi R} - \frac{1}{c-a+\pi R} < \frac{1}{R}$? |
Here is the question in my exercise.
> Let $f(x,y)$ be a function defined on a rectangle $[a,b]\times [c,d]$. Assume that for each fixed $y \in[c,d]$ the function $x\mapsto f(x,y)$ is a continuously differentiable function of $x$ on $[a,b]$; and for each fixed $x \in[a,b]$ the function $y\mapsto f(x,y)$ is a continu... |
My question relates to subsection 2.2.1 of [this article][1]. This subsection recalls the work of Lindgren, Rue, and Lindström (2011) on Gaussian Markov Random Fields (GMRFs). The subsection starts with a two-dimensional regular lattice where the 4 first-order neighbours of $u_{i,j}$ are identified. The article defines... |
Notation clarification: What is the meaning of $\frac{\omega}{\mathrm{d}t}$? |
Let $(a,b,c)$ be the sides of a triangle inscribed inside a unit circle such that the vertices of the triangle are distributed uniformly on the circumference. Let $x$ be a real and let
$$
f(x) = P\left(\frac{1}{a-b+x} + \frac{1}{b-c+x} \ge \frac{1}{c-a+x}\right)
$$
The plot of $x$ vs $f(x)$ is show below along ... |
This problem is from the book "Inside Interesting Integrals" written by Prof. Paul J, Nahin, which is stated as follows,
Imagine a circle (let’s call it $C_{1}$) that has radius $a$. We then chose at random and independently, three points from the interior of that circle. These three points, if non-collinear, unique... |
This problem is from the book "Inside Interesting Integrals" written by Prof. Paul J, Nahin, which is stated as follows,
Imagine a circle (let’s call it $C_{1}$) that has radius $a$. We then chose at random (i.e., uniformly distributed) and independently, three points from the interior of that circle. These three po... |
I'm working on determining whether the following series converges or diverges:
[Image of the series][1]
I used partial fraction decomposition to turn it into this:
[Partial fraction series][2]
And I know that if I use my summation properties, I can make these two separate sums and then find whether they... |
Harmonic series in a telescoping series: diverging or converging? |
My professor gives an example to show that the completion of the codomain is necessary for the Open Mapping Theorem.
First, as setup, my professor provided the following equivalent formulation of an open map:
Let $T \in \mathcal{L}(X,Y)$. $T$ is an open map if there exists $A < \infty$ such that for each $y \in ... |
I'm working on determining whether the following series converges or diverges:
$$\sum_{n=1}^\infty \frac{9}{n(n+3)}$$
I used partial fraction decomposition to turn it into this:
$$\sum_{n=1}^\infty (\frac{3}{n}-\frac{3}{n+3})$$
And I know that if I use my summation properties, I can make these two separat... |
My professor gives an example to show that the completion of the codomain is necessary for the Open Mapping Theorem.
First, as setup, my professor provided the following equivalent formulation of an open map:
Let $T \in \mathcal{L}(X,Y)$. $T$ is an open map if there exists $A < \infty$ such that for each $y \in ... |
Here are some preliminary thoughts. I'm a bit tired right now, so apologies if they're half-baked.
With regards to question 1), I'm currently thinking about the following sequence of sequences:
A preliminary thought is to choose the following sequence of sequences (upper index for the `outer' sequence):
$$x^m... |
Here is the question in my exercise.
> Let $f(x,y)$ be a function defined on a rectangle $[a,b]\times [c,d]$. Assume that for each fixed $y \in[c,d]$ the function $x\mapsto f(x,y)$ is a continuously differentiable function of $x$ on $[a,b]$; and for each fixed $x \in[a,b]$ the function $y\mapsto f(x,y)$ is a continu... |
The pasting lemma states the following for a function $f\colon X\to Y$ ($X$, $Y$ topological spaces):
> If $X$ can be decomposed as a union of two closed sets $F_1$ and $F_2$ on each of which $f$'s restriction is continuous, then $f$ is continuous.
The necessity of the sets being all closed is usually shown by co... |
**DEFINITION:** The closure of $A$ in a metric space is $\overline A := \bigcap \{C : C \text{ is closed and } A \subset C \}$.
**DEFINITION:** Let $(X,d)$ be a metric space and $A \subset X$. We say $x$ is a limit point of $A$ if for all open subsets $U$ such that $x \in U$, $U \cap A \neq \emptyset$. (Sometimes, t... |
I try to compute limiting of the following integral
$$
\lim_{n\to\infty} \int_{-\infty}^\infty \frac{n^2(1-\cos(x/n))}{1+x^6}dx
$$
---
Note that for $x\ge 0$, we have $|1-\cos x|\le x^2/2$.
Then for $x\ge 0$,
$$
\frac{n^2(1-\cos(x/n))}{1+x^6}\le \frac{x^2}{2(1+x^6)}\in L^1((0,\infty))
$$
Hence,
$$
... |
How to get $\lim_{n\to\infty} \int_{-\infty}^\infty \frac{n^2(1-\cos(x/n))}{1+x^6}dx$? |
**DEFINITION:** The closure of $A$ in a metric space is $\overline A := \bigcap \{C : C \text{ is closed and } A \subset C \}$.
**DEFINITION:** Let $(X,d)$ be a metric space and $A \subset X$. We say $x$ is a limit point of $A$ if for all open subsets $U$ such that $x \in U$, $U \cap A \neq \emptyset$. (Sometimes, t... |
So I was wondering in tic tac toe in infinite dimensions where instead of a 2D grid I play on an infinite dimensional hypercube. I've come to the conclusion the words optimal strategy does not make sense (?) How? Imagine I score a win using an optimal strategy (assuming it exists). Then using the projection operator I ... |
Infinite tic tac toe complexity? |
so basically i was confronted by the following series:
$\sum_{i=0}^\infty \frac{(-1)^i}{2(2i+1)(i+1)}$
I encountered this series while working on the following integral:
$\int_0^\infty \frac{(1-cosx)}{(x^2e^x)} dx $
I first expanded the taylor series of $cos(x)$ , and then the integral just came out as a su... |
What is $\displaystyle\sum_{k=0}^\infty \frac{x^k n!}{k!(k+n)!}$?
If we abuse the notation a little bit, it is $\displaystyle\sum_{k=0}^\infty \binom n {-k}(-x)^k$. What is its value and how to tackle it algebraically? |
so basically i was confronted by the following series:
$\sum_{i=0}^\infty \frac{(-1)^i}{2(2i+1)(i+1)}$
I encountered this series while working on the following integral:
$\int_0^\infty \frac{(1-cosx)}{(x^2e^x)} dx $
I first expanded the taylor series of $cos(x)$ , and then the integral just came out as a su... |
What is $\displaystyle\sum_{k=0}^\infty \frac{x^k n!}{k!(k+n)!}$?
If we abuse the notation a little bit, it is $\displaystyle\sum_{k=0}^\infty \binom n {-k}(-x)^k$. What is its value and how to tackle it algebraically?
It is also of probabilistic meanings, in fact the question itself comes from computing $\mathb... |
The vertices of a triangle are three unifomly random points on a unit circle. The side lengths are, in random order, $a,b,c$.
>Show that $P(ab>c)=\frac12$.
The result is strongly suggested by simulations, and by my attempt shown below.
The simplicity of the result suggests that there may be an intuitive explan... |
Is it possible to evaluate $\int_{0}^{\frac{\pi}{2}} e^{-(\pi \tan(x) - 1)^2} \, dx$? |
Finding a line in $R^{3}$ through A and B is simple enough. The problem that then arises is that since point C is not on the plane, it is not simply a case of finding two vectors in the plane etc.
I would suggest elimination of wrong answers is the approach intended by the way the question is written.
Have you tr... |
It turns out that the familiar result that surjective monotones $\mathbb R\to\mathbb R$ are continuous extends to general LOTS (linearly ordered topological spaces):
> **Theorem**. If $X$, $Y$ are LOTS with $Y$ being dense and complete, then any surjective monotone $X\to Y$ is continuous.
(The same proof from $\m... |
Let V = $[ f : \mathbb{R} \longrightarrow \mathbb{R} ]$ be an infinite vector space over field $\mathbb{R}$.
Find a subset $W$ of $V$, such that:
> 1. For arbitrary distinct $f_{1}, f_{2}, ... , f_{n} \in W$, set [$\{f_{1}, f_{2}, ... , f_{n} \}$] is linearly independent.
> 2. For any $x \in \mathbb{R}$, and a... |
I have a doubt concerning Lemma 3.7.4 from Topics in Algebra by I. N. Herstein.
The statement of the Lemma is:
> Let $R$ be a Euclidean ring. Then every element in $R$ is either a unit in $R$ or can be written as the product of a finite number of prime elements of $R$.
My question is that $0$ is an element of ... |
I've been trying to solve the following differential equation:
$$y + \sqrt{(y')^2 + 1} = k$$
where $ k $ is a constant. Despite my efforts, I haven't been able to find a closed-form solution for it. It seems to be a rather tricky equation due to its nonlinearity and the presence of the square root term. I've attempte... |
Does this work? $$\left(a^{\frac{2}{3}} + b^{\frac{2}{3}}\right)^{\frac{3}{2}} > \left(a^2+b^2\right)^{\frac{1}{2}} \Longleftrightarrow \left(a^{\frac{2}{3}} + b^{\frac{2}{3}}\right)^{3} > \left(a^2+b^2\right) \Longleftrightarrow$$$$ a^2 + b^2 + 3(ab)^{\frac{2}{3}}\left(a^{\frac{2}{3}} + b^{\frac{2}{3}}\right) > a^2+b^... |
Does this work? $$\left(a^{\frac{2}{3}} + b^{\frac{2}{3}}\right)^{\frac{3}{2}} > \left(a^2+b^2\right)^{\frac{1}{2}} \Longleftrightarrow \left(a^{\frac{2}{3}} + b^{\frac{2}{3}}\right)^{3} > \left(a^2+b^2\right) \Longleftrightarrow$$$$ a^2 + b^2 + 3(ab)^{\frac{2}{3}}\left(a^{\frac{2}{3}} + b^{\frac{2}{3}}\right) > a^2+b^... |
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