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H: How to calculate $\sum_{i= 0}^{k-1}\left \{ \frac{ai+b}{k} \right \}$
How to calculate: $$F=\sum_{i= 0}^{k-1}\left \{ \frac{ai+b}{k} \right \}$$
where $a,b \in \mathbb Z$ and $(a,k)=1$; $\left \{ \frac{ai+b}{k} \right \} =$ fraction part of $\frac{ai+b}k$
Such problem so strange for me. Please help me solve it, giv... |
H: (simple) Expectation of random variable as a multipart function
Let the random variable $Y \in [0, \infty)$, a real number $\theta >0$, and the random variable $X$ such that $X = \theta - \min(\theta,Y)$, thus, $X \in [0, \theta]$. That is, $X = 0$ if $Y > \theta$ and $X = \theta-Y$ if $Y \leq \theta$. I would like... |
H: Symmetric Matrices Using Pythagorean Triples
Find symmetric matrices A =$\begin{pmatrix} a &b \\ c&d
\end{pmatrix}$ such that $A^{2}=I_{2}$.
Alright, so I've posed this problem earlier but my question is in regard to this problem.
I was told that $\frac{1}{t}\begin{pmatrix}\mp r & \mp s \\ \mp s & \pm r \en... |
H: Is the following set empty?
$$
sp\left \{
\begin{pmatrix}
1 \\
-1 \\
1 \\
-1
\end{pmatrix} , \begin{pmatrix}
4\\
-2 \\
4 \\
-2
\end{pmatrix} , \begin{pmatrix}
1\\
1\\
1\\
1
\end{pmatrix}
\right \} \bigcap \left \{ \begin{pmatrix}
x_1\\
x_2\\
x_3\\
x_4
\end{pmatrix} | \begin{matrix}
x_1 + x_2 = 0\\
x_... |
H: Accumulation points of accumulation points of accumulation points
Let $A'$ denote the set of accumulation points of $A$.
Find a subset $A$ of $\Bbb R^2$ such that $A, A', A'', A'''$ are all distinct.
I can find a set $A$ such that $A$ and $A'$ are distinct, but not one where $A,A',A'',A'''$ are all distinct.
AI: He... |
H: The successor of a set
Definition : The successor of a set $x$ is the set $S(x) = x \bigcup \{x\}$
Prove that $x \subseteq S(x)$ and there is no $z$ such that $ x \subset z \subset S(x)$
I really battle with proofs :(
Here is what I have:
let $ y \in x $
then $y \in S(x)$
therefore $ x \subseteq S(x) $
I think I ... |
H: $R$ is an integral domain iff $R[x_1,...,x_n]$ is a integral domain
I would like to prove that $R$ is an integral domain iff $R[x_1,...,x_n]$ is a integral domain. The converse is trivial, since R can be viewed as a subring of $R[x_1,...,x_n]$. In order to prove the first implication, I'm trying to use the same arg... |
H: A set $E$ is closed if and only if $E = E^-$
Let $E$ be a subset of a metric space $(S,d)$. I want to show that the set $E$ is closed if and only if $E = E^-$ where $E^-$ is the closure of a set $E$.
First I assumed $E = E^-$. Then since I know $E^-$ is the intersection of all closed sets containing $E$, $E^-$ mus... |
H: Does taking the direct limit of chain complexes commute with taking homology?
Suppose I have a directed system $C_i$, $i\in\mathbb{N}$ of chain complexes over free abelian groups (bounded below degree $0$)
$$C_i=0\rightarrow C^{0}_{(i)}\rightarrow C^{1}_{(i)}\rightarrow\cdots\rightarrow C^{n-1}_{(i)}\rightarrow C^n... |
H: proof of $ A\otimes M\cong M $
Let $M$ be an $A$-module. Can someone prove the $A$-module isomorphism:
$$A\otimes M \cong M?$$
(By $\otimes$ I mean tensor product.)
AI: First define a map $A\times M\rightarrow M$ such that $(a,m) \mapsto am$. This is clearly bilinear, so it induces a homomorphism $A\otimes M\righta... |
H: Cardinality of the set of surjective functions on $\mathbb{N}$?
I know that the set of all surjective mappings of ℕ onto ℕ (lets name this set as F) should have cardinality |ℝ|.
How to strictly prove that?
From the fact that cardinality of every possible function is |ℝ|, |F| <= |ℝ|.
Saw similar question on this sit... |
H: Strong mathematical induction: Prove Inequality for a provided recurrence relation $a_n$
The sequence $a_1,a_2,a_3,\dots$ is defined by: $a_1=1$, $a_2=1$, and $a_n=a_{n-1}+a_{n-2}-n+4$ for all integers $n\ge 3$. Prove using strong mathematical induction that $a_n\ge n$ for all integers $n\ge 3$.
I'm comfortable s... |
H: Why is $\mathbb{Z}[\alpha ]$ not finitely generated as $\mathbb{Z}$-module?
Assume that $\alpha \in \mathbb{C}$ is an algebraic number which is not an algebraic integer. My question is why $\mathbb{Z}[\alpha]$ is not finitely generated as $\mathbb{Z}$-module.
Clearly there exists $ a_i \in \mathbb{Z}$, which are... |
H: Understanding Bayes' Theorem
I worked through some examples of Bayes' Theorem and now was reading the proof.
Bayes' Theorem states the following:
Suppose that the sample space S is partitioned into disjoint subsets $B_1, B_2,...,B_n$. That is, $S = B_1 \cup B_2 \cup \cdots \cup B_n$, $\Pr(B_i) > 0$ $\forall i=1,2... |
H: Factorization in Gaussian integers
Let $p$ be a natural number, suppose $p$ prime. Show that the following conditions are equivalent:
the polynomial $x^2+1\in\mathbb{Z}_p$ has roots in $\mathbb{Z}_p$
$p$ is reducible in the ring $\mathbb{Z}[i]$
there exists $a,b\in\mathbb{Z}$ such that $p=a^2+b^2$
My attempt: ... |
H: What's so well in having a least element in a set?
What's so well about the principle of well-ordering? Why is this pattern of order named like this? I am unable to trace a connection between well-ordering and a set that has a least element so I presume that there must be some mathematical becakground I'm unaware o... |
H: Correlation between multiplied numbers?
I do not have a strong math background, but I'm curious as to what this pattern is from a mathematical standpoint.
I was curious how many minutes there were in a day, so I said "24*6=144, add a 0, 1440." Then it immediately struck me that 12*12=144, 6 is half of 12, and 12 i... |
H: Can this sum ever converge?
If I have a strictly increasing sequence of positive integers, $n_1<n_2<\cdots$, can the following sum converge?
$$ \sum_{i=1}^\infty \frac{1}{n_i} (n_{i+1}-n_{i}) $$
I suspect (and would like to prove) that it always diverges. Haven't made much progress so far, though.
On a related not... |
H: Number of Sylow bases of a certain group of order 60
We have $G=\langle a,b \rangle$ where $a=(1 2 3)(4 5 6 7 8)$, $b=(2 3)(5 6 8 7)$. $G$ is soluble of order 60 so has a Hall $\pi$ subgroup for all possible $\pi\subset\{2,3,5\}$. $\langle a\rangle$ is a normal subgroup.
A Hall $\{2,3\}$ subgroup is $\langle a^5,b... |
H: Generating Laguerre polynomials using gamma functions
An exercise given by my complex analysis assistant goes as follows:
For $n \in \mathbb{N}$ and $x>0$ we define
$$P_n(x) = \frac{1}{2\pi i} \oint_\Sigma \frac{\Gamma(t-n)}{\Gamma(t+1)^2}x^tdt$$
where $\Sigma$ is a closed contour in the $t$-plane that encircl... |
H: If $17 \mid \frac{n^m - 1}{n-1}$ find the values of $n$ where $m$ is even but not divisible by $4$
Let $m, n \in \mathbb{Z}_+$ with $n > 2$, and let $\frac{n^m-1}{n-1}$ be divisible by $17$. Show that either $m$ is even:$ m \equiv 0 \mod 17$ and $n \equiv 1 \mod 17$. Find all possible values of $n$ in the cases wh... |
H: Proof that $dx/|x|$ is a Haar measure on non-zero reals?
Most importantly, what is the meaning of this notation $\lambda = dx/|x|$? How do I compute say $\lambda(0,1)$ for example?
AI: This is the Haar measure on the multiplicative group ${\bf R}^\times$ (or the group of positive reals under multiplication too, bu... |
H: Prove the equivalence of presentation of a free group with a free product
I want to prove that the following presentation of a free group (generators and relations):
$$\left(\begin{array}{c|c}
x_0,a_0&a_0a_1x_2=x_0a_1\\
x_1,a_1&a_1a_2x_0=x_1a_0\\
x_2,a_2&a_2a_0x_1=x_2a_2
\end{array}\right)$$
is equivalent to the fr... |
H: What type of input does trigonometric functions take in
I see in my Book that 45 deg is equivalent of π/4 . Ι also do the conversion if I simply convert degrees into radians like this
45* π/180 = π/4 radians
and back again
π/4 * 180/π = 45 deg
.
So I think that I grasp the Idea finally that π/4 is another way... |
H: complex analysis- Liouville's theorem
1).Let $f: \mathbb{C}\to \mathbb{C}$ be a holomorphic function. Prove that if $f(0) = f(1) =0 $ and $f:\lvert f'(z) \rvert\leq 1 $ for all $z\in \mathbb{C}$, then $f(z)=0$ for all $z\in \mathbb{C}$.
2).True or false: If $f:\mathbb{C}\to\mathbb{C}$ is holomorphic, then f must be... |
H: Split multiplicative reduction question
Let $E/\mathbb{Q}$ be an elliptic curve and $E_{d}$ be the quadratic twist of $E$ by a squarefree integer $d$. Let $\ell$ be a prime of multiplicative reduction for $E$. If $(d, \ell) = 1$, then $\ell$ is a prime of multiplicative reduction for $E_{d}$. I read that $\ell$ is ... |
H: Quaternion group associativity
Consider the eight objects $\pm 1, \pm i, \pm j, \pm k$ with multiplication rules:
$ij=k,jk=i,ki=j,ji=-k,kj=-i,ik=-j,i^2=j^2=k^2=-1$,
where the minus signs behave as expected and $1$ and $-1$ multiply as expected. Show that these objects form a group containing exactly one involutio... |
H: Spheres in different dimension are not homotopy equivalent
Is there a way to prove that $\textbf{S}^n$ and $\textbf{S}^m$ are not homotopy equivalent if $n\neq m$ without using the machinery of homology or higher homotopy groups?
AI: Just for fun, here's an answer that doesn't use the machinery you're talking about... |
H: Definition of a dominating function and the Dominated Convergence Theorem.
I apologise if this is a rather simplistic or even silly question, but I am confused with the word "dominated" in Lebesgue's Dominated Convergence Theorem (DCT) since I can find no definition of a dominating function in the textbook I am fol... |
H: Identify all $a \mod 35$ such that $35$ is a pseudoprime to base $a$
Identify all $a \mod 35$ such that $35$ is a pseudoprime to base $a$.
By defintion, $\gcd(a,n) = 1$ for $a,n \in \mathbb{Z}$, then $a^{n-1} \equiv 1 \mod n$ means that $n$ is a pseudoprime to base $a$.
$G_{35}$ is the group of elements all copri... |
H: Finding the sum of a Taylor expansion
I want to find the following sum:
$$
\sum\limits_{k=0}^\infty (-1)^k \frac{(\ln{4})^k}{k!}
$$
I decided to substitute $x = \ln{4}$:
$$
\sum\limits_{k=0}^\infty (-1)^k \frac{x^k}{k!}
$$
The first thing I noticed is that this looks an awful lot like the series expansion of $e^x$:... |
H: Group of mappings containing an injective map is a subset of symmetric group
Let $G$ be a group of mappings on a set $X$ with respect to function composition. Show that if $G$ contains some injective function, then $G\subseteq \text{Sym}(X)$.
What I did: If $X$ is finite, then the injective mapping $g\in G$ is al... |
H: Classification of nonzero prime ideals of $\mathbb{Z}[i]$
I know the classification of Gaussian primes: let $u$ be a unit of $\mathbb{Z}[i]$. Then the following are all Gaussian primes:
1) $u(1+i)$
2) $u(a+ib)$ where $a^2+b^2=p$ for some prime number p congruent to $1$ modulo $4$
3) $uq$ where $q$ is a prime number... |
H: Algebraic expression in its most simplified form
I am trying to simplify the algebraic expression:
$$\bigg(x-\dfrac{4}{(x-3)}\bigg)\div \bigg(x+\dfrac{2+6x}{(x-3)}\bigg)$$
I am having trouble though. My current thoughts are:
$$=\bigg(\dfrac{x}{1}-\dfrac{4}{(x-3)}\bigg)\div \bigg(\dfrac{x}{1}+\dfrac{2+6x}{(x-3)}\big... |
H: Describe the graph locus represented by this equation
I want to know the shape of the region described by
$$ Im(z^2) = 4 $$
so I did the following:
$$ z=x + iy $$
$$z^2 = x^2 + 2xiy -y^2 $$then
$$Im(z^2) = 2xy
$$
then the locus is $$ 2xy = 4 $$
$$ xy = 2 $$
then it's the line $$ xy = 2 $$
what does this ... |
H: How does one show that two functors are *not* isomorphic?
Let $C$ be the category of finite-dimensional vector spaces over some field. It is easy to construct pairs of endofunctors $F, G$ of $C$, of the same variance, such that $F(V)$ and $G(V)$ have the same dimension for every $V \in C$, yet are not naturally iso... |
H: Letters for complex numbers
Suppose that I am writing a proof or some other piece of mathematical writing, and wish to introduce $n$ distinct complex numbers, for some positive integer $n$. What are the complex numbers called?
If $n=1$, then clearly the (unique) complex number I am interested in is called $z$. ... |
H: Proving that $\int_0^{2\pi}\frac{ 1-a\cos\theta}{1-2a\cos\theta + a²} \mathrm d\theta = 0$ for $|a|>1$
Let $|a|>1$. Show that $$\int_0^{2\pi}\frac{ 1-a\cos\theta}{1-2a\cos\theta + a²} \mathrm d\theta = 0$$
I'm also trying to solve this problem. I wasn't sure if I was supposed to start a new thread or contribute t... |
H: Is there a special term for an array consisting only of ones?
Is there a special term for an array consisting only of ones?
Sorry for the rather elementary question. I am getting into MapReduce programming and am trying to frame my code to be nice and neat.
AI: I have used, and seen used, the term "all-ones vector... |
H: easy question about conditional expectations
I got a quite easy exercise I just don't get.
Let P be a probability measure, $\frac{dQ}{dP}=Z$, $Z>0$ a.s. and $E[Z]=1$, hence Q is an equivalent proba-measure to P. Then I shall prove that for a sub-sigma field G we have:
$E_{Q}[X|G](\omega)=\frac{E_{P}[XZ|G](\omega)}{... |
H: Proof that a linear transformation is continuous
I got started recently on proofs about continuity and so on. So to start working with this on $n$-spaces I've selected to prove that every linear function $f: \mathbb{R}^n \to \mathbb{R}^m$ is continuous at every $a \in \mathbb{R}^n$. Since I'm just getting started w... |
H: How should one think to get the radius of the resulting curve?
For example, the curve C is given as the intersection between
$$
C: x²+y²+z²=1, x+y+z=0
$$
Radius: 1
Another one:
$$
C: x²+y²+z²=1, x+y=1
$$
Radius:
$$
\frac{1}{\sqrt{2}}
$$
How should I think to get these? I know they're ellipses, but substituting the... |
H: How to evaluate $\lim_{x\to 0} (1+2x)^{1/x}$
Good night guys! I'm having some trouble with this:
$$\lim_{x\to 0} (1+2x)^{1/x}$$
I know that $\lim_{x\to\infty} (1 + 1/x)^x = e$
but I don't know if i should take $h=1/(2x)$ or $h=1/x$
Can someone please help me? Thanks!
AI: We first find the limit as $x$ approaches $... |
H: Where does the function $f(x) = \frac{2x}{x - 7}$ have an increasing slope?
Where does the function $f(x) = \frac{2x}{x - 7}$ have an increasing slope?
$a. x \le 0, x > 7$
$b. x<7$
$c. x > 7$
$d. x \in \Bbb R, x \neq 7$
This question is from a test of mine in a pre-calculus course (so no calculus allowed in answe... |
H: What is the actual geometric meaning of trigonometric operations such as adding cos,sine,tan
$$\sin(\pi/4)+\cos(\pi/4)=\frac{\sqrt{2}}{2}+\frac{\sqrt{2}}{2}= \frac{2\sqrt{2}}{2}=\sqrt{2}$$
Thinking of trig components (cosine, sine) that I used to produce the result using the mechanics of algebra, makes me wonder wh... |
H: If $y$something?
For instance I know $y<x$ implies $-y > -x$ but is there a way to phrase it in terms of $y>$ something (that does not itself contain $y$)?
AI: No.
For each $x\in\Bbb R$ we have that the set $L_x:=\{y\,\mid\,y<x\}$ is downward closed meaning that $u<v\in L_x \,\implies\,u\in L_x$.
The other kind of ... |
H: Computing the expected value of a matrix?
This question is about finding a covariance matrix and I wasn't sure about the final step.
Given a standard $d$-dimensional normal RVec $X=(X_1,\ldots,X_d)$ has i.i.d components $X_j\sim N(0,1)$. Here we treat $X$ as a row vector. Now consider $Y:=\mu+XA$, where $\mu\in\mat... |
H: If $2^x=0$, find $x$.
If $2^x=0$, find $x$.
Solution: I know range of $2^x$ function is $(0,\infty)$.
So $2^x=0$ is not possible for any real value of $x$
Hence, equation is wrong. We can't find value of $x$. Am I right?
Please help me.
Can $x$ be in $[-\infty,\infty]$?
i.e is $2^x=0$ possible for $x=-\infty$?
... |
H: Boundary and closure of a set
I'm trying to show that a point is in the boundary of $E$ if and only if it belongs to the closure of both $E$ and its complement.
Let $\overline{E}$ denote the closure of $E$ and $E^\circ$ be the interior of $E$. Then the boundary is defined to be $\overline{E} \setminus E^\circ$.
S... |
H: Simplifying Differentiation
I'm working off my textbook and I've followed the steps easily enough until it gets to this
$ \dfrac {dy}{dx} = \dfrac{(x^2 + 1)^3}{2 \sqrt{x - 1}} + \sqrt{x-1}~(6x)~(x^2 + 1)^2$
$= \dfrac{(x^2 + 1)^2}{2\sqrt{x - 1})}[(x^2 + 1) + 12x(x - 1)]$
$= \dfrac{(x^2 + 1)^2(13x^2 - 12x + 1)}{2\sq... |
H: What is the exact definition of polynomial functions?
I'm trying to understand the difference between polynomial functions and the evaluation homomorphisms. I noticed that I don't know what's the exact definition of a polynomial function, although I've used it since I was in school.
I'm confused.
Thanks in advance
... |
H: $\liminf, \limsup$, Measure Theory, show: $\lim \int n \ln(1+(f/n)^{1/2})\mathrm{d}\mu=\infty$
Let $(X,\Omega,\mu)$ be a measure space and $M^+(X,\Omega)$ denote the set of all non-negative real valued measurable functions.
If $f \in M^+(X,\Omega)$ and $0< \int f \mathrm{d}\mu < \infty$ then $$\lim_{n\rightarrow\i... |
H: Given a triangle with points in $\mathbb{R}^3$, find the coordinates of a point perpendicular to a side
Consider the triangle ABC in $\mathbb{R}^3$ formed by the point $A(3,2,1)$, $B(4,4,2)$, $C(6,1,0)$.
Find the coordinates of the point $D$ on $BC$ such that $AD$ is perpendicular to $BC$.
I believe this uses proje... |
H: $J$ be a $3\times 3$ matrix with all entries $1$ Then $J$ is
$J$ be a $3\times 3$ matrix with all entries $1$ Then $J$ is
Diagonalizable
Positive semidefinite
$0,3$ are only eigenvalues of $J$
Is positive definite
$J$ has minimal polynomial $x(x-3)=0$ so 1, 2,3 are true , am I right?
AI: You are correct. This is ... |
H: Determining the group homomorphism in semidirect product
We know that if $N$ is a normal subgroup, $H$ is a subgroup, and $\varphi$ is the group homomorphism such that $\varphi:H\to$Aut$(N)$. And this gives a unique group, called the outer semidirect product of $N$ and $H$.
But when a group and two of its subgroups... |
H: If one of $n$ coins is fair, then find the probability that the total number of heads is even
A collection of $n$ coins is flipped. The outcomes are independent, and the $i$-th coin comes up heads with probability $\alpha_i, i=1, \dots, n$. Suppose that for some value of $j, 1 \leq j \leq n, \alpha_j=\frac{1}{2}$. ... |
H: $A$ be a $2\times 2$ real matrix with trace $2$ and determinant $-3$
$A$ be a $2\times 2$ real matrix with trace $2$ and determinant $-3$, consider the linear map $T:M_2(\mathbb{R})\to M_2(\mathbb{R}):=B\to AB$ Then which of the following are true?
$T$ is diagonalizable
$T$ is invertible
$2$ is an eigen value of ... |
H: prove $ F(x)=\int_0^\infty {\sin(tx)\over(t+1)\sqrt t} dt \in C^\infty(\mathbb R^*) $
prove that :
$$ F(x)=\int_0^\infty {\sin(tx)\over(t+1)\sqrt t} \, dt \in C^\infty(\mathbb R^*) $$
i end up proving that $F(x)\in C^ \infty(\mathbb R^{*+})$ not $\mathbb R^*$ , and i studied the case with :
$F(x)= \sqrt x G(x)$ ... |
H: Infinite series and its upper and lower limit.
I am learning analysis on my own and I am puzzled with the following question.
Consider the series $$\frac{1}{2}+\frac{1}{3}+\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{2^3}+\frac{1}{3^3}+\cdots$$ Indicate whether this series converges or diverges.
The following is what I w... |
H: Manipulating Vector Identity
I would like to expand then simplify (if possible) the following quantity.
$\nabla (a \cdot (C\, a))$
Where $a = a(x)$ is a vector valued function of $x$ and $C$ is a constant matrix.
AI: No need to use index notation here; use the chain rule.
$$\begin{align*}b \cdot \nabla [a \cdot \u... |
H: Group equals union of two subgroups
Suppose $G=H\cup K$, where $H$ and $K$ are subgroups. Show that either $H=G$ or $K=G$.
What I did: For finite $G$, if $H\neq G$ and $K\neq G$, then $|H|,|K|\le |G|/2$. But they clearly share the identity element, so $H\cup K\neq G$.
How can I extend it to $G$ infinite?
AI: We sho... |
H: Set of homomorphisms form a group?
Given vector spaces $V, W$ over field $F$, the set of all linear maps $V \to W$ forms a vector space over $F$ under pointwise addition.
Is there an analogue for groups? Can the set of all homomorphisms from groups $G \to K$ be given a group structure?
AI: The set of all homomorph... |
H: For any arrangment of numbers 1 to 10 in a circle, there will always exist a pair of 3 adjacent numbers in the circle that sum up to 17 or more
I set out to solve the following question using the pigeonhole principle
Regardless of how one arranges numbers $1$ to $10$ in a circle, there will always exist a pair of ... |
H: Help Understanding Complex Roots
I was reading a graphical explanation of complex roots, and between Figures 7 and 8 I became confused.
The roots appear in the imaginary plane, but I don't understand why the original function must be inverted before the graphical representation fits. I realize that inversion is... |
H: Non-self-mapping automorphism implies abelian
Suppose $\sigma\in\text{Aut}(G)$. If $\sigma^2=1$ and $x^{\sigma}\neq x$ for $1\neq x\in G$, show that if $G$ is finite, it must be abelian.
There's a hint to show that the set $\{x^{-1}x^{\sigma}\mid x\in G\}$ is the whole group $G$. I have already proved this hint ($x... |
H: If $x$ and $y$ are in this given sequence, can $2^x+2^y+1$ be prime?
The sequence:
$3, 11, 13, 17, 19, 29, 37, 41, 53, 59, 61, 67, 83, 97, 101, 107, 113, 131, 137, 139, 149, 163, 173, 179, 181, 193, 197, 211, 227, 257, 269, 281, 293, 313, 317, 347, 349, 353, 373, 379, 389, 401, 409, 419, 421, 443, 449, 461, 467, 49... |
H: How to find remainder?
$$a=r\mod (r+1) \ \ \forall r\in\{2,3,4,\dots,9\}$$
Then how do we find $'x'$ if $$a=x\mod 11$$
I get $$2a=9\mod11$$ but that does not help.
Please keep solution simple , i don't now number theory.
The above is the crux what I got from the question:
Let $n_1,n_2,... $ be an increasing sequ... |
H: $A\ne 0:V\to V$ be linear,real vec space $V$
$A\ne 0:V\to V$ be linear,real vec space $V$, $\dim V=n$,$V_0=A(V),\dim V_0=k<n$ and for some $\lambda\in\mathbb{R}, A^2=\lambda A$
Then
$\lambda=1$
$|\lambda|^n=1$
$\lambda$ is the only eigen value of $A$
There is a subspace $V_1\subseteq V$ such that $Ax=0\forall x\i... |
H: this is regarding exponentials distribution
In an office building, the lift breaks down randomly at a mean rate of 3 times per week. The random variable X represents the time in days between successive lift breakdowns.
(i) Calculate the probability that the time interval between successive lift breakdowns is
betwee... |
H: The value of a limit of a power series: $\lim\limits_{x\rightarrow +\infty} \sum_{k=1}^\infty (-1)^k \left(\frac{x}{k} \right)^k$
What is the answer to the following limit of a power series?
$$\lim_{x\rightarrow +\infty} \sum_{k=1}^\infty (-1)^k \left(\frac{x}{k} \right)^k$$
AI: A simple calculation shows that
\beg... |
H: 2-colorable belongs to $\mathsf P$
To show that 2-colorable belongs to $\mathsf P$, I have a straightforward mental description in mind that I don't think will be considered as a formal proof. Hence I am interested to know how this must be said as an answer to this question. Here's what I think: Say we have 2 color... |
H: Ask book to deeply understand partially ordered sets
I learn little about ordering and poset before, but I think it's not enough and want to learn more about ordering and Poset. Can anyone please recommend some best books to learn about this topic. I really appreciate it. Thanks so much
AI: E.Harzheim, Ordered Sets... |
H: I am confused with the def of derivative.
The problem given :
Let $f:\Bbb R\rightarrow \Bbb R$ and $F:\Bbb R^2\rightarrow \Bbb R$ be differentiable and satisfy $F(x,f(x))=0$ and $\displaystyle \frac {dF}{dy}$ is not zero.
Prove that $f'(x)= -\dfrac {dF}{dx}/\frac {dF}{dy}$ where $y=f(x)$
I am so confused here.
Sinc... |
H: A simple riddle related to addition of odd numbers
I'm not sure if this type of question can be asked here, but if it can then here goes:
Is it possible to get to 50 by adding 9 positive odd numbers? The odd numbers can be repeated, but they should all be positive numbers and all 9 numbers should be used.
PS : The ... |
H: Quotient of abelian groups of rank $2$
Let $A, B$ be abelian groups, $B$ is contained in $A$, both $A$ and $B$ are assumed to have rank $2$. Is there a standard way to show that the quotient group $A/B$ is finite? I think there exists some general theorem about modules finitely generated over a PID, but i can't fin... |
H: trigonometric identity related to $ \sum_{n=1}^{\infty}\frac{\sin(n)\sin(n^{2})}{\sqrt{n}} $
As homework I was given the following series to check for convergence:
$ \displaystyle \sum_{n=1}^{\infty}\dfrac{\sin(n)\sin(n^{2})}{\sqrt{n}} $
and the tip was "use the appropriate identity".
I'm trying to use Dirichlet... |
H: What is the difference between a surface and the graph of a function?
When I was studying a book, Elementary classical analysis (Jerrold.E.Marsden), there was a confusing sentence.
"The unit sphere $x^2+y^2+z^2=1$ in $\mathbb R^3$ is a surface of the form $F(x,y,z)=c$ which is not the graph of a function."
What do... |
H: Given a spanning set, what is the span of the 'transpose' of the set?
Given $$sp\left \{
\begin{pmatrix}
a_1\\
a_2\\
a_3
\end{pmatrix}
,\begin{pmatrix}
b_1\\
b_2\\
b_3
\end{pmatrix}
,\begin{pmatrix}
c_1\\
c_2\\
c_3
\end{pmatrix}
\right \} = \mathbb{R}^3$$
What is
$$sp\left \{
\begin{pmatrix}
a_1\\
b_1\\... |
H: Prime divisibility in a prime square bandtwidth
I am seeking your support for proving (or fail) formally the following homework:
Let $p_j\in\Bbb P$ a prime, then any $q\in\Bbb N$ within the interval $p_j<q<p_j^2$ is prime, if and only if all:
$$p_{i}\nmid q$$ with $1\le i<j$
There should be a simple sieving argumen... |
H: Simple Expect Value Exercise
Question: We have $9$ coins, $1$ of them is false (lighter). We divide them up in pairs (with one left) and weigh them (that is taking two in a balance and seeing if one of them is lighter). What is the expected value of number of weighings to find the false one?
I suspect this is a rat... |
H: Quasicompact over affine scheme
Let $X$ be a scheme and $f : X \rightarrow \mathrm{Spec}\, A$ a quasicompact morphism. Are there any easy conditions on $A$ under which we can say that $X$ is quasicompact?
Quasicompact morphism means only that there is an affine cover $\cup_{i \in I} \mathrm{Spec}\, A_i$ where $f^{-... |
H: Counterexample to inverse Leibniz alternating series test
The alternating series test is a sufficient condition for the convergence of a numerical series. I am searching for a counterexample for its inverse: i.e. a series (alternating, of course) which converges, but for which the hypothesis of the theorem are fals... |
H: Uniform convergance for $f_n(x)=x^n-x^{2n}$
the function $f_n(x)=x^n-x^{2n}$ converge to $f(x)=0$ in $(-1,1]$. Intuativly the function does not converge uniformally in (-1,1]. How can I prove it?
I tried using the definition $\lim \limits_{n\to\infty}\sup \limits_{ x\in (-1,1]}|f_n(x)-f(x)|$ function is continial f... |
H: Laurent Series Expansion Problems
Expand in a laurent Series :
1- $f_{1} (z) = \frac{z^{2} - 2z +5 }{(z^{2}+1) (z-2)}$
in the ring : $1 < |z| < 2 $
2- $ f_{2} (z) = \frac{1 }{(z-3) (z+2)}$
In :
$i. 2 < |z| < 3
\\ ii. 0 < |z+2| < 5$
I managed to solve the second one but not sure if it is correct
For i. $2 < |z| ... |
H: Show that $\left( \frac{q}{p} \right) \equiv q^{(p-1) / 2} \mod p$, where $\left( \frac{q}{p} \right)$ is the Legendre Symbol
Show that if $p$ is any odd prime then
$$\left( \frac{q}{p} \right) \equiv q^{\frac{p-1}{2}} \mod p.$$
stating any theory that you use. In particular, you may assume the existence of a prim... |
H: Absolute convergance of function series
The question is for which values of $x\in \mathbb R$, the following series absolute/conditionally converge: $$\sum_{n=1}^{\infty}\frac{x^n}{(1+x)(1+x^2)...(1+x^n)}$$ I have no idea how to solev it except M-test of wirestrass but I don't know how to bound it . Forgive me if ta... |
H: Topology on Integers such that set of all Primes is open
In my topology homework we are asked to describe a topology on the Integers such that:
set of all Primes is open.
for each $x\in\mathbb Z$, the set $\{x\}$ is not open.
$\forall x,y \in\mathbb Z$ distinct, there is an open $U\ni x$ and an open $V\ni y$ such ... |
H: Subrings and homomorphisms of unitary rings
Let $(R,+,\cdot)$ be a ring (in the definition i use multiplication is associative operation and it's not assumed that there is unity in the ring).
I've seen two definitons of subring.
1) non-empty subset $S \subset R$ is called subring of ring $(R,+,\cdot)$ iff $(S,+,\c... |
H: Listing subgroups of a group
I made a program to list all the subgroups of any group and I came up with satisfactory result for $\operatorname{Symmetric Group}[3]$ as
$\left\{\{\text{Cycles}[\{\}]\},\left\{\text{Cycles}[\{\}],\text{Cycles}\left[\left(
\begin{array}{cc}
1 & 2 \\
\end{array}
\right)\right]\right\},\... |
H: Shortcut in calculating examples of elements of a given order?
My question is:
Find all possible orders of elements of the group of units $G_{31}$. Give an example of an elememt of each possible order.
I did the question, but I felt I did it a long way. As $31$ is prime, elements of $G_{31} = \{1, 2, \cdots 30\}$... |
H: Equation $f(x,y) f(y,z) = f(x,z)$
How to solve the functional equation $f(x,y) f(y,z) = f(x,z)$?
AI: Set $g(x)=f(x,0)$ and $h(z)=f(0,z)$. Then, we have $f(x,z)=f(x,0)f(0,z)=g(x)h(z)$ for all $x$ and $z$. Apply this to the original equation to obtain $g(x)h(y)g(y)h(z)=g(x)h(z)$.
There are three possibilities now:
... |
H: why is an annulus close to it's boundary when it's boundary curves are close?
This is the motivating question for the rather vague question here: when is the region bounded by a Jordan curve "skinny"?
Suppose we are given two Jordan curves in the plane, one inside the other and each contained in an epsilon neighbor... |
H: Inequality for embedding in Sobolev space
For $\Omega=(0,1). $Prove that there exists $M>0$ such that
$$||u||_{C^0(\overline{\Omega})}\le M||u||_{H^1(\Omega)}$$
for all $u\in H^1(\Omega).$
AI: Assume that $u(0)=0$. Then
$$
u(x)=\int_0^x u'(s)ds=\int_0^1 \textbf{1}_{(0,x)}u'(s)ds.
$$
Now you use Cauchy-Schwarz ineq... |
H: Axiom of Regularity - Transitive set
I just managed to confuse myself completely while studying for Set Theory.
We have the Axiom of regularity:
$$\forall S (S\not= \emptyset \rightarrow (\exists x\in S)(S\cap x=\emptyset))$$
Now a set is transitive, if $x\in T$ implies $x\subset T$.
I don't understand anymore how ... |
H: Comparison between the limits of two real functions
I know
If $f:D(\subset\mathbb R)\to\mathbb R,c$ is a limit point of $D,$ and $f(x)\ge(\text{resp.}\le)~a~\forall~x\in D-\{c\},$ then $\displaystyle\lim_{x\to c}f(x)\ge(\text{resp.}\le)~a.$ (Provided the limit exists)
If $f,g,h:D(\subset\mathbb R)\to \mathbb R,c$ ... |
H: Deducing a result about entire functions
We need to show that for an entire function $f$ on $\mathbb{C}$, there are constants $a_1,...,a_n$ such that $\int^{2 \pi}_{0}|f(re^{i \theta})|^2 d\theta=2\pi\sum_{n=0}^{\infty}|a_n|^2r^{2n}$.
Thoughts so far are that we can find a power series expansion about $0$ by Taylor... |
H: Books on computational complexity
Can anyone recommend a good book on the subjects of computability and computational complexity? What are the de facto standard texts (say, for graduate students) in this area?
I've heard a thing or two on these subjects from "CS people" back at the university. With lots of hand-wav... |
H: What is rigorous notation for functions?
I have seen many ways to denote a function: $f(x)=x^2, y=x^2, f: x\mapsto x^2$ and so on. What is exact notation for functions? Please include lethal doses of rigor, set theory, and of course notational exactness.
Note: I am very familiar with functions in general. I just kn... |
H: $\frac{a}{1-a}+\frac{b}{1-b}+\frac{c}{1-c}+\frac{d}{1-d}+\frac{e}{1-e}\ge\frac{5}{4}$
I tried to solve this inequality:
$$\frac{a}{1-a}+\frac{b}{1-b}+\frac{c}{1-c}+\frac{d}{1-d}+\frac{e}{1-e}\ge\frac{5}{4}$$
with
$$a+b+c+d+e=1$$
I am stuck at this. I don't want the full solution, a hint would be enough.
AI: Hint:
... |
H: Approximation of beam
Assume that there is a simply supported beam subjected to concentrated moments $M_0$ at each end. The governing equation is
$$EI\frac{d^2y}{dx^2}-M(x)=0$$
with the boundary conditions $y(0)=0$ and $y(H)=0$. I know that there is an exact solution in the form of $y(x)=\frac{M_0\,x}{2EI}(x-H)$ bu... |
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