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H: A basis for a subspace of $\mathbb R^3$
I have the following question:
Find the basis of the following subspace in $\mathbb R^3$:
$$2x+4y-3z=0$$
This is what I was given. So what I have tried is to place it in to a matrix $[2,4,-3,0]$ but this was more confusing after getting the matrix $[1,2,-3/2,0]$. This was... |
H: $f(z)={2z+1\over 5z+3}$ maps
Define
$H^{+}=\{z:y>0\}$
$H^{-}=\{z:y<0\}$
$L^{+}=\{z:x>0\}$
$L^{-}=\{z:x<0\}$
$f(z)={2z+1\over 5z+3}$ maps
$1.$ $H^+\to H^+$ and $H^-\to H^-$
$2$. $H^+\to H^-$ and $H^-\to H^+$
$3.$ $H^+\to L^-$ and $H^-\to L^+$
$4.$ $H^+\to L^+$ and $H^-\to L^-$
If I take $z\in H^{+}$ then ${3\over z... |
H: eHarmony combinatoric question, probability that I should get at least 1 compatible match.
Ok.. (as I type this with a smirk on my face) - in all seriousness I am trying to figure out, given 29 degrees of compatibility and 40 million members if I should be getting at least 1 match a day. There are of course a lot ... |
H: Decompose the group $U_{60}$ as direct product of cyclic groups
Decompose the following group as a direct product of cyclic groups: $U_{60}$.
Here is what i've done so far:
$60 = 2\cdot2\cdot5\cdot3$.
Therefore the answers are
$U_{60} \cong C_2\times C_2\times C_3\times C_5$ or $U_{60}\cong C_4\times C_3\times C_5$... |
H: prove that the order of the elements of restricted direct product is finite .
if $G_i = \Bbb Z/p_i \Bbb Z$
($\Bbb Z$ means integers)
where $p_i$ is the ith integer prime , I=the positive integers
show that , every element of the restricted direct product of the $G_i$'s has finite order
my trial to solve it ,
i ... |
H: Understanding mathematics imprecisely
For a long time, it has been a complete mystery to me how any of my peers understood any math at all with anything short of filling in every detail, being careful about every set theoretic detail down to the axioms. That's a slight exaggeration, but I certainly did much worse ... |
H: Converting parametric equations to xy equation
I have these parametric equations:
$$
x=1+t \\
y=1-t
$$
How do I make this to the xy equation?
David
AI: Use the first equation to find an expression for $t$ in terms of $x$.
Substitute into the second equation to get an equation only involving $x$ and $y$.
Rearrange ... |
H: Question on the proof of a subspace of Polish space is Polish, iff it's a $G_\delta$ set.
Suppose, $X$ is a Polish space, $Y$ is a Polish subspace of $X$. $\{U_n\}_{n \in \Bbb N}$ is a basis of open sets of $X$.
Let $A = \{ x \in \overline {Y} : \forall \epsilon \exists {n}(x \in U_n \land \operatorname{diam}{(Y \c... |
H: For which $\alpha$ this sum converges? $\sum_{n=3}^{\infty} {\frac{1}{n \cdot \ln(n) \cdot \ln(\ln(n))^{\alpha}}}$
Given:
$$\sum_{n=3}^{\infty} {\frac{1}{n \cdot \ln(n) \cdot \ln(\ln(n))^{\alpha}}}$$
I am asked: For what values of $\alpha$ does this summation converge?
So I said, $f(n) = \frac{1}{n \cdot \ln(n) \cd... |
H: Criteria for a function to be plottable
Assume that $f$ is a real function. My question is how can one decide if $f$ is plottable or not? My assumption is that $f$ must be of class $C^1$, but I am not aware of such a result. My assumption is based on the fact that there are differentiable functions that can not be ... |
H: Hall $\pi$-subgroup and $HN=G$
Let $\pi$ be any set of prime numbers. A finite group $H$ is a $\pi$-group if all primes that divide $|H|$ lie in $\pi$. If $|G|<\infty$, then a Hall $\pi$-subgroup of $G$ is a $\pi$-subgroup $H$ such that $|G:H|$ is divisible by no prime in $\pi$. Let $\phi$ be a homomorphism define... |
H: Solve system of first order differential equations
I have to solve differential systems like this:
$$
\left\{
\begin{array}{c}
x' = 3x - y + z \\
y' = x + 5y - z \\
z' = x - y + 3z
\end{array}
\right.
$$
Until now I computed the eigenvalues $k = \{2,4,5\}$ by solving the equation resulted from this determinant of... |
H: Does $\int_{0}^{\infty} \frac{\sin (\tan x)}{x} \ dx $ converge?
Does $ \displaystyle \int_{0}^{\infty} \ \frac{\sin (\tan x)}{x} dx $ converge?
$ \displaystyle \int_{0}^{\infty} \frac{\sin (\tan x)}{x} \ dx = \int_{0}^{\frac{\pi}{2}} \frac{\sin (\tan x)}{x} \ dx + \sum_{n=1}^{\infty} \int_{\pi(n-\frac{1}{2})}^{\pi... |
H: Question about long proofs?
How do people write 50-page long proofs (and longer)? So they have a target in mind, but I can't get my head around them foreseeing that these 50 pages of work will actually lead them to exactly their target.
AI: There are two cases:
You start by noticing something, then you prove a sma... |
H: integrate function with change of variable
Find the primitive of $\;\displaystyle \int x^2 \sqrt{x+1}$ $dx$
So (...)
$u = x + 1 \quad \iff \quad u - 1 = x$
$u' = 1 \quad \iff \quad \frac{du}{dx} = 1 \rightarrow \;du = dx$
$$\int(u - 1)^2 . u^\frac12 \; du \;\;= \;\; {{(u-1)^3}\over3} \cdot {u^{3/2}\over{3/2}} + C \... |
H: The Language of the Set Theory (with ZF) and their ability to express all mathematics
Accordingly, the Language of Set Theory (in this case using $ZF$ axioms) is built up with the aim to express all mathematics. Now, I know that, for example, the construction of the numbers ($\mathbf{ \mathbb{N},\mathbb{Z}, \mathbb... |
H: homogeneous nonlinear functions
Give an example of degree one positively homogeneous function, (i.e. a function $f$, such that $\forall \alpha\ge0, f(\alpha x) =\alpha f(x)$) that is not linear, and $f: \mathbb{R} \to \mathbb{R}$.
AI: A simple example would be $f(x)=\left|x\right|$.
About the hint in my comment: F... |
H: Given a group homomorphism $f:G\to H$, if $m$ is relatively prime to $|H|$ and $x^m\in\ker f$, then $x\in \ker f$
Let $f:G\to H$ be a homomorphism, and let $m$ be an integer such that $m$ and $|H|$ are relatively prime. For any $x \in G$, if $x^m \in \ker f$, then $x \in \ker f$.
My proof step: if $x^m \in \ker... |
H: The graph of Borel measurable function whose range is a separable metrisable space
If $Y$ is separable and $f : X \to Y$ is Borel measurable, then the graph of
$f$ is Borel.
On page 14, Lemma 2.3, (iii) of this online note, given, $\{U_n\}_{n \in \Bbb N}$, a basis for the topology of a metrisable space $Y$, th... |
H: Meaning of "defined"
What are the precise meanings of terms "defined", "well defined" and "undefined", etc.? We can't define what "defined" means since then we would run into circular definitions. (If definitiveness is established in terms of something else like "existence", then again we face the same problem.)
AI... |
H: Probability Distribution of Rolling Multiple Dice
What is the function for the probability distrabution of rolling multiple (3+) dice. The function is a bell curve but I can't find the actual function for the situation. Example, what is the function for rolling 50 6 sided dice?
EDIT: A six sided die returns an inte... |
H: Morphisms between locally ringed spaces and affine schemes
I need some hints to understand the conclusion of the proof of the following lemma from the Stacks Project:
Lemma $\mathbf{6.1.}\,$ Let $X$ be a locally ringed space. Let $Y$ be an affine scheme. Let $f\in\operatorname{Mor}(X,Y)$ be a morphism of locally ... |
H: Is it possible to simplify $\frac{\Gamma\left(\frac{1}{10}\right)}{\Gamma\left(\frac{2}{15}\right)\ \Gamma\left(\frac{7}{15}\right)}$?
Is it possible to simplify this expression?
$$\frac{\displaystyle\Gamma\left(\frac{1}{10}\right)}{\displaystyle\Gamma\left(\frac{2}{15}\right)\ \Gamma\left(\frac{7}{15}\right)}$$
Is... |
H: Invariant subspaces of a linear operator that commutes with a projection
I have an assignment problem, the statement is:
Let $V$ be a vector space and $P:V \to V$ be a projection. That is, a linear operator with $P^2=P.$ We set $U:= \operatorname{im} P$ and $W:= \ker P.$ Further suppose that $T:V\to V$ is a linear... |
H: convolution of exponential distribution and uniform distribution
Given $X$ an exponentially distributed random variable with parameter $\lambda$ and $Y$ a uniformly distributed random variable between $-C$ and $C$. $X$ and $Y$ are independent. I'm supposed to calculate the distribution of $X + Y$ using convolution.... |
H: Convergence of a series of a given metric..
I'm trouble with a metric defined over a given set: Consider $\mathcal{P}=C_p^\infty([-\pi, \pi])$, that is, $\mathcal{P}$ is the set of all infinitely differentiable functions $f:\mathbb R\rightarrow \mathbb C$ which are $2\pi$-periodic ($f(x+2\pi)=f(x)$, for all $x\in \... |
H: If $A$ is an $n \times n$ matrix such that $A^3 = O_{3}$, show that $I - A$ is invertible with inverse $I + A + A^2$
So this question is basically a proof.
If $A$ is an $n \times n$ matrix (so square) which satisfies the condition $A^3 = O_{3}$ ($A^{3}$ gives the $3 \times 3$ zero matrix), then show that $(I - A)$ ... |
H: Prove $p^2=p$ and $qp=0$
I am not really aware what's going on in this question. I appreciate your help.
Let $U$ be a vector space over a field $F$ and $p, q: U \rightarrow U$ linear maps. Assume $p+q = \text{id}_U$ and $pq=0$. Let $K=\ker(p)$ and $L=\ker(q)$.
Prove $p^2=p$ and $qp=0$.
$$p(p+q) = p(\text{id}_U) \... |
H: Let $A_{\alpha}$ be the $\alpha$-rotation matrix. Prove $A_{\alpha}^T = (A_{\alpha})^{-1}$
Let $A_{\alpha}$ be the alpha-rotation matrix. Prove $A_{\alpha}^T = (A_{\alpha})^{-1}$
In other words, prove $A_{\alpha}$ transpose = $A_{\alpha}$ inverse.
First of all, what is a rotation-matrix? And what does that imply ... |
H: Quick Integration Clarification
$ \int_ {-\pi}^{\pi}\ f(x) \ dx $ is equivalent to integrating $f(x)$ over $-\pi < x < \pi$ (strictly less than) and also $-\pi \le x \le \pi$ (less than or equal). So I just need to reassure myself that the bounds of integration don't change when considering either less than or les... |
H: Convergence of $1+\frac{1}{2}\frac{1}{3}+\frac{1\cdot 3}{2\cdot 4}\frac{1}{5}+\frac{1\cdot 3\cdot 5}{2\cdot 4\cdot 6}\frac{1}{7}+\cdots$
Is it possible to test the convergence of $1+\dfrac{1}{2}\dfrac{1}{3}+\dfrac{1\cdot 3}{2\cdot 4}\dfrac{1}{5}+\dfrac{1\cdot 3\cdot 5}{2\cdot 4\cdot 6}\dfrac{1}{7}+\cdots$ by Gauss ... |
H: Recovering a group from its quotient group
Suppose I've a group $G$ and a normal subgroup $H$. I know the structure of $G/H$ as well as the structure of $H$. Is it possible to recover the original structure of G from this?
AI: There is one important piece of data that you have omitted: namely, if $H$ is normal in $... |
H: Convex homogeneous function
Prove (or disprove) that any CONVEX function $f$, with the property that $\forall \alpha\ge 0, f(\alpha x) \le \alpha f(x)$, is positively homogeneous; i.e. $\forall \alpha\ge 0, f(\alpha x) = \alpha f(x)$.
AI: Maybe I'm missing something, but it seems to me that you don't even need con... |
H: For the sinusoidal graph below, write the equation in form $y = a\cos\left(\frac{2\pi}{p}x\right)+b$
For the sinusoidal graph below, write the equation
$$y = a\cos \left(\frac{2\pi}{p}x\right) + b.$$
The answer I solved it to be looking at the graph is
$$y = 25\cos\left(\frac{2\pi}{12}x\right) + 30$$
Thanks any inp... |
H: Proof of Fundamental Theorem of Finite Abelian Groups?
The only proofs I've seen of this tend to involve a few intermediate results and a couple of induction proofs with some clever constructions in them. They aren't hard to follow and they're pretty short, but they do still seem to remove some of the "why it shoul... |
H: What property allows me to integrate a gaussian function?
Whenever I integrate a gaussian function, I get to a step that makes me a little uncomfortable because I don't fully understand it. The only way I know of to analytically integrate the gaussian function is to multiply two of them together, like so...
$$\int_... |
H: Let $A,B$ be elements of $M_2(\mathbb{R})$. Give an example to show that $A+B$ can be invertible if $A,B$ are both non-invertible
The goal for this problem is to show that even if two matrices $A$ and $B$ are non-invertible, $A+B$ can be invertible. I tried to show this using a proof, but I ended up actually provin... |
H: Why if the columns of $Q$ are an orthonormal basis for $F^n$ then $Q$ is unitary?
Why if the columns of $Q$ are an orthonormal basis for $F^n$ then $Q$ is unitary?
There exist any available theorems to prove it?
AI: Here's one way to see it. If the columns form an orthonormal basis, you can directly check that $QQ^... |
H: Finding the rational values of constant for which these constants are roots of equation
Problem :
Determine all rational values for which $a,b,c$ are the roots of $x^3+ax^2+bx+c=0$
Solution :
Sum of the roots $a+b+c = -a$ ........(i) ( Since , as per question $a,b,c$ are roots of equation and we have to find valu... |
H: Show the points $u,v,w$ are not collinear
Consider triples of points $u,v,w \in R^2$, which we may consider as single points $(u,v,w) \in R^6$. Show that for almost every $(u,v,w) \in R^6$, the points $u,v,w$ are not collinear.
I think I should use Sard's Theorem, simply because that is the only "almost every" stat... |
H: On the Frattini Subgroup
For a prime $p$, let $H=\{x\in \mathbb{C}\colon x^{p^n}=1 \mbox{ for some } n\geq 1\}$ be the Prufer $p$-group, $C_2=\langle y\colon y^2=1\rangle$, and $G=H\oplus C_2$. Then $H$ is the unique maximal subgroup of $G$, hence it is Frattini subgroup of $G$. Let $S=\{e^{2\pi i/p^n} \colon n\geq... |
H: Why is the empty set finite?
On page 25 of Principles of Mathematical Analysis (ed. 3) by Rudin, there is the definition (excluding the irrelevant parts for this question):
Definition 2.4: For any positive integer $n$, let $J_n$ be the set whose elements are the integers $1,2,...,n$; let $J$ be the set consisting ... |
H: A cosine function has maximum value of 14 and a minmum value of 4, a period of 7, and a phase shift of 12.
A cosine function has a maximum value of 14 and a minimum value of 4, a period of 7, and a phase shift of 12. Write an equation representing this cosine function...
Could someone tell me if I'am write and if I... |
H: Solving for the volume of a tetrahedron
Can anyone explain why I was incorrect in this problem:
I used a tetrahedron solver online, http://rechneronline.de/pi/tetrahedron.php, and it says I have the right answer.
AI: Assuming that $S$ denotes the length of a side, your answer is correct. See here. |
H: Math problem using cups..
I have the following question:
Mike has 58 white cups and 198 green cups. He wants to place his cups in stacks by color so there are the same number in each stack and same color.
What is the greatest number of cups he can place in each stack?
How would I answer this guys?
AI: Edit: It se... |
H: Continuity of $f(x)=[x]+ \sqrt{x-[x]}$
Consider the function $f(x)=[x]+ \sqrt{x-[x]}, \, x\in \Bbb R$ ; where "$[ \space ]$" denotes the greatest integer function. It is obvious that if $b$ is an integer, then
$$\lim_{x\to b-} f(x)=\lim_{x\to b-}[x] + \sqrt { \lim_{x\to b-}x - \lim_{x\to b-}[x] } =b-1+\sqrt {b-(b-... |
H: Multipliciousness within an inner product space.
Question:
Let $V$ be an inner product space and $v,w\in V$. Prove that $\lvert\langle v,w\rangle\rvert=\lVert v\rVert \lVert w\rVert$ if and only if one of the vectors $v$ or $w$ is a multiple of the other.
Attempt:
Assume the identity holds and $y\neq 0$. Let
$$ a... |
H: Existence of a function satisfying given conditions
I was going through the topic of $Function$, its boundedness, continuity etc. I got a problem.
Does there exist a function defined on the closed interval $[a,b]$ which is....
1. bounded;
2. takes its maximum and minimum values;
3. takes all its values between the... |
H: (Simple) Examples on Non-Commutative Rings
Looks like it is easier to find example of commutative rings rather than
non-commutative rings. Prabably the easiest examples of the former are
$\mathbb{Z}$ and $\mathbb{Z}_n$. We can find elaborations on these two commutative rings in various
literatures including here an... |
H: Calculus the extreme value of the function $f(x,y)$
Calculus the extreme value of the $f(x,y)=x^{2}+y^{2}+xy+\dfrac{1}{x}+\dfrac{1}{y}$
pleasee help me.
AI: Use the normal derivative test. First order conditions
$$\frac{\partial f}{\partial x}=2x+y-\frac 1{x^2}=0$$
$$\frac{\partial f}{\partial y}=2y+x-\frac 1{y^2}=... |
H: How to prove $\lvert \lVert x \rVert - \lVert y \rVert \rvert \overset{\heartsuit}{\leq} \lVert x-y \rVert$?
I'm trying to show that $\lvert \lVert x \rVert - \lVert y \rVert \rvert \overset{\heartsuit}{\leq} \lVert x-y \rVert$. A hint would be nice.
AI: Observe that
$\lVert x \rVert = \lVert (x -y) +y \rVert \le... |
H: Doubt over a term in the definition
I was studying a book Handbook of Product Graphs by Richard Hammack, Wilfried Imrich, Sandi Klavžar. There is a definition of Antipodal Graph.
A graph $G$ is called antipodal if there exists a vertex $v$ to any vertex $u$ $\in$ $V(G)$ such that $V(G) = I(u,v)$.
What does $I(u,v... |
H: Prove this vector identity
Let $f$ and $g$: $\mathbb R^3 \rightarrow \mathbb R$ be $C^{1}$ scalar functions. Prove that
$$ \nabla \left( \frac{f}{g} \right) = \frac{1}{g^2}\left( g\nabla f - f\nabla g \right)$$ $$g \neq 0$$
AI: I will assume $f, g : \mathbb{R}^3 \to \mathbb{R}$.
We have
$$\nabla\bigg(\frac{f}{g}\b... |
H: How does the symmetric group act on tuples?
Given a set $X$, I've seen people write the action of a permutation $ \sigma \in S_{n}$ on an $n$-tuple of elements of $X$ as
$$ \;\; \sigma (x_{1},...,x_{n})=(x_{\sigma^{-1}(1)},...,x_{\sigma^{-1}(n)}). \;\;$$
But this does not seem to me to give a left action of $S_{n}... |
H: Is (im)predicativity decidable
The distinction between predicative and impredicative definitions is
important in mathematics. As first approximation, impredicativity
means circularity. Let me give you an example of an impredicative
definition.
Let $V$ be a vector-space over a field $K$, and $S \subseteq V$ a set
of... |
H: Calculate Taylor Series function and converges
Calculate the Taylor series of the function $f(x) = 3/x$ about $a = 3$.
Where (if anywhere) does the series converge to $3/x$?
AI: To calculate the Taylor series you need to find the derivatives of the function and evaluate them in the point a=3:
$$\frac{d^nf}{dx^n}=3... |
H: Counting strings of $\{ 0,1 \}$'s of length $n$ s.t. last $k$ bits are equal
I'm trying to derive a closed formula for the number of strings of $0$'s and $1$'s of length $n$ such that the last $k \leq n$ bits are all zeros or all ones, and such that there is no other place in the string with $k$ consecutive zeros o... |
H: Given $G = (V,E)$, a planar, connected graph with cycles, Prove: $|E| \leq \frac{s}{s-2}(|V|-2)$. $s$ is the length of smallest cycle
Given $G = (V,E)$, a planar, connected graph with cycles, where the smallest simple cycle is of length $s$. Prove: $|E| \leq \frac{s}{s-2}(|V|-2)$.
The first thing I thought about ... |
H: Showing $\mathrm{Var}(aX+bY) = a^2\mathrm{Var}(X)+b^2\mathrm{Var}(Y)+2ab\mathrm{Cov}(X,Y)$
I am trying to prove this equation $$\mathrm{Var}(aX+bY) = a^2\mathrm{Var}(X)+b^2\mathrm{Var}(Y)+2ab\mathrm{Cov}(X,Y),$$
where $\mathrm{Cov}(X,Y):= E(XY)-EX\cdot EX$ and $E[\cdot]$ denotes the mean of a random variable. Also... |
H: Prove that the equation $3x-\sin(2x)=0$ has only one solution
I`m trying to prove that this equation has only one solution$$3x-\sin(2x)=0$$
I need some advice how to do that, Thanks!
EDIT
AI: By intermediate value theorem it clearly has at least one solution (or just check by plugging in $x=0$). Now the derivative ... |
H: Matrix representation of the adjoint of an operator, the same as the complex conjugate of the transpose of that operator?
Since I'm not taking summer classes I decided to do some self learning on more advanced mathematics, and I've found myself stuck on this problem:
I have to show that for any operator $\hat{A}$ t... |
H: Closed formula to count number of binary numbers of length $x$ having at least $y$ $1$ bits
I'm interested in solving a sub problem of the algorithm related question from SO How many binary numbers having given constraints .... The sub problem being, having $x \geq y$
determine how many binary numbers of length $x... |
H: Evaluate $ f(a) = \int^a_{-a}|x|dx$ of $a=2$
I want to evaluate this integral $$ f(a) = \int^a_{-a}|x|dx$$between $a$ and $-a$ and I know that $\int^a_{-a}|x|dx$ is divided into two: $$\int^a_{-a}|x|dx \rightarrow a>0 = \int^a_{0}xdx $$
$$\int^a_{-a}|x|dx \rightarrow a<0 = \int^0_{-a}-xdx $$
after I evaluate it ... |
H: Approximation of an integral
)
I can't get over a step in my teacher's exercise.
$$I(x) = 2\int_{0}^{1} \frac{y^3 + 4y^2}{y^2+4y+5} dy = 2\int_{0}^{1} y dy -5\int_{0}^{1} \frac{2y}{y^2+4y+5} dy$$
Probably it's very easy..
Thank you :)
AI: Observe that:
$$\begin{align*}
2\int_{0}^{1} \frac{y^3 + 4y^2}{y^2+4y+5} dy
... |
H: Questions about the concept of Structure, Model and Formal Language
When we start to define mathematical logic (specifically, propositional, first order, and second order logic) we start defining the concept of a language. At the begining this is done in a purely syntactical way. So far, so good.
Now, in this form... |
H: Galois group $\operatorname{Gal}(f/\mathbb{Q})$ of the polynomial $f(x)=(x^2+3)(x^2-2)$
Find the Galois group $\operatorname{Gal}(f/\mathbb{Q})$ of the polynomial $f(x)=(x^2+3)(x^2-2)$.
Any explanations during the demonstration, will be appreciated. Thanks!
AI: The splitting field of $f$ is $\mathbb{Q}(\sqrt{2}, ... |
H: A property of completely separable mad families
A family of sets $\mathcal{A}\subset[\omega]^\omega$ is called almost disjoint (a.d.) iff $\forall a,b\in\mathcal{A}(a\neq b\rightarrow |a\cap b|<\omega)$ and $\mathcal{A}$ is infinite (as such families turn out to be not that interesting if they are finite).
For an a... |
H: Is $\frac{\sin x^2}{\sin^2 x}$ uniformly continuous on $(0,1)$? Is my proof correct?
I'm said to check if $\dfrac{\sin x^2}{\sin^2 x}$ is uniformly continuous on $(0,1)$
I can see that $\dfrac{\sin x^2}{\sin^2 x}$ is continuous on $(0,1].$ So $\displaystyle\lim_{x\to1-}\dfrac{\sin x^2}{\sin^2 x}$ is finite.
I need ... |
H: What does square brackets around a polynomial mean?
EDIT: The first paragraph has been indicated as inaccurate, please see this answer.
When we have something like $\mathbb{Z}_2[x]/(x^2 + x + 1)$, we
understand that this means the set of polynomials over indeterminate
$x$ where the coefficients are drawn from $\mat... |
H: What is wrong with this fake proof that $\lim\limits_{n\rightarrow \infty}\sqrt[n]{n!} = 1$?
$$\lim_{n\rightarrow \infty}\sqrt[n]{n!}=\lim_{n\rightarrow \infty}\sqrt[n]{1}*\sqrt[n]{2}\cdots\cdot\sqrt[n]{n}=1\cdot1\cdot\ldots\cdot1=1$$
I already know that this is incorrect but I am wondering why. It probably has som... |
H: What's the difference between $(\mathbb Z_n,+)$ and $(\mathbb Z_n,*)$?
I noticed that $(\mathbb Z_n,+)$ and $(\mathbb Z_n,*)$ are not the same thing. For example 2 is not invertible in $(\mathbb Z_6,*)$. However, since $\bar2+\bar4=\bar0$, thus it is invertible in $(\mathbb Z_6,+)$.
I found the following problems:... |
H: What is the result of (λx.λy.x + ( λx.x+1) (x+y)) ( λz.z-4 5) 10?
Could you explain what should I do about
λx.λy.x
part?
Thanks.
AI: The parentheses are pretty bad, here is my guess on what the expression should be in order to evaluate to a number in the end:
$(\lambda x.\lambda y.(x + ( \lambda x.x+1) (x+y)))~(... |
H: Mean value theorem for essentially bounded functions
I have the problem with the following:
Let $f \in L^{\infty}(\mathbb{R}_+)$ is the mean value theorem in the following form:
Let $0 \leq a < b < \infty$, then
$\int_{a}^{b} |f| \ d\mu = m(b-a)$, for some $m \leq \mathrm{ess} \sup |f|$.
valid for $f$?
We can see... |
H: Calculate the matrices of $R$ and $R\circ R$ with respect to the basis $(e_1,e_2,e_3,e_4)=(1,x,\frac{1}{2}x^2,\frac{1}{6}x^3)$
I am unsure how to calculate the basis matrices of the linear map defined below. I appreciate your help.
Let $V=\mathbb{Q}[x]_{\le3}$ be the set of polynomials over $\mathbb{Q}$ of degree a... |
H: using gamma function to simplify integration
I have to evaluate $\int_0^1 x^2 \ln(\frac1x)^3 $.I used gamma function and used substitution $t=\ln (\frac {1}{x})^3$.
In this I get to integrate from $1$ to $-\infty$ with a minus sign outside.Because of this minus sign by interchanging upper and lower limit
I get to... |
H: How to test the convergence of the series $\sum_{n=1}^\infty n^{-1-1/n}$?
How to test the convergence of the series $\displaystyle\sum_{n=1}^\infty\frac{1}{n^{1+1/n}}?$
Help me. I'm clueless.
AI: You for $a_n = \frac1n$ and $b_n = \frac{1}{n^{1+1/n}}$ that
$$
\lim_{n\to\infty}\frac{a_n}{b_n}=\lim_{n\to\infty}\f... |
H: Can you tell which are correct terms of the sum of solution of the integral $\int (x^2+1)^n dx $?
According to WolframAlpha,
$$\int (x^2+1)^n dx = x \cdot _2F_1(\frac{1}{2},-n;\frac{1}{3};-x^2)$$
and
$$_2F_1(\frac{1}{2},-n;\frac{1}{3};-x^2)= \sum_{n=0}^{\infty} \frac{1}{3}(-n)\frac{(-x^2)^n}{n!}.$$
Can you tell w... |
H: Properties of an alternating bilinear form its coordinate matrix
I found that I lack many basic knowledge about linear algebra, so read the wiki article about Bilinear Forms. Especially this Paragraph. I tried to proof of "Every alternating form is skew-symmetric." was quite easy. And I found a counter-example for ... |
H: Showing that the matrix transformation $T(f) = x*f'(x)+f''(x)$ is linear
I want to show that the following matrix transformation is linear.
$T(f) = x*f'(x)+f''(x)$
I know I have to show that $T(f+g) = T(f) + T(g)$ but I don't understand what $T(f+g)$ will look like.
Is $T(f+g) = x*(f'(x)+g'(x))+(f''(x)+(g''(x))$ th... |
H: A question on inverse functions
Let $f:\mathbb R \to \mathbb R$ is a strictly increasing function and $f^{-1}$ is its inverse function. It satisfies:
$f(x_1)+x_1=a$; $f^{-1}(x_2)+x_2=a$.
What is the value of $x_1+x_2$? Thanks for your help.
AI: It seems the following. Since the function $f$ is strictly increasing... |
H: If every irreducible element in $D$ is prime, then $D$ has the unique factorization property.
Suppose every irreducible element in a domain $D$ is prime.
I'm trying to prove this implication:
In a integral domain $D$, if $a=c_1c_2...c_n$ and $a=d_1d_2...d_m$
($c_i,d_i$ irreducible), then $n=m$ and up to order $c... |
H: Every function is the sum of an even function and an odd function in a unique way
It is known that every function $f(x)$ defined on the interval $(-a,a)$ can be represented as the sum of an even function and an odd function. However
How do you prove that this representation is unique?
Thanks for your help.
AI: If... |
H: $f'(x)=f(x)$ and $f(0)=0$ implies that $f(x)=0$ formal proof
How can I prove that if a function is such that $f'(x)=f(x)$ and also $f(0)=0$ then $f(x)=0$ for every $x$. I have an idea but it's too long, I want to know if there is a simple way to do it. Thanks!
Obviously in a formal way.
AI: An implicit assumption i... |
H: List the set of points of discontinuity of piecewise function
List the set of points of discontinuity of $f:(0,\infty)\to\mathbb R$, defined by
$$f(x)=\begin{cases}x-[x]\text{ if [x] is even}\\1-x+[x]\text{ if [x] is odd}\end{cases}$$
AI: We see $$x\in(0,1), \text{floor}(x)=0 ~~~~~\text{so}~~~f(x)=x$$ $$x\in[1,2), ... |
H: Why $L$ is the eigenspace of $L_A$?
$A=\frac{1}{\sqrt{5}}\begin{pmatrix} 1&2\\2&-1 \end{pmatrix}$
Let $L_A$ be a reflection of $R^2$ about a line $L$ through the origin.
Then $L$ is the one dimensional eigenspace of $L_A$ corresponding to the eigenvalue 1.
I know the eigenvalues are 1,-1. But why it should be 1?
A... |
H: Numbers of students registered for various courses
I have a probability problem that I’m struggling with. Any help would be appreciated.
Here is the problem:
A class has 28 students. In this class, each student is registered in 2 languages courses (at most) chosen among English, German or Spanish. The number of stu... |
H: How to simplify $a^n - b^n$?
How to simplify $a^n - b^n$?
If it would be $(a+b)^n$, then I could use the binomial theorem, but it's a bit different, and I have no idea how to solve it.
Thanks in advance.
AI: If you are looking for this???
$$
a^n-b^n=(a-b)\Big(\sum_{i=0}^{n-1}a^{n-1-i}b^i\Big)
$$ |
H: Is this series convergent? $\sum_{i=1}^{\infty} \frac{(\log n)^2}{n^2}$
$\sum_{i=1}^{\infty} \frac{(\log n)^2}{n^2}$
I guess it is convergent, so I apply comparsion test for this.
$\frac{log^2}{n^2} < \frac{n^2}{n^2} = 1$
So it is bounded by 1 and hence it is convergent.
Can I do in this way? It kind of make sens... |
H: A question on a periodic function
Let $f(x)$ be a bounded real function on $\mathbb R$ and for any $x \in \mathbb R$
$$
f(x+\frac{13} {42})+f(x)=f(x+\frac16)+f(x+\frac17) \tag1.
$$
What is the fastest way to compute the period of the function?
Thanks for your help.
AI: Without any extra assumptions on $f$, you can ... |
H: How to prove two subspaces are complementary
To give some context, I'm continuing my question here.
Let $U$ be a vector space over a field $F$ and $p, q: U \rightarrow U$ linear maps. Assume $p+q = \text{id}_U$ and $pq=0$. Let $K=\ker(p)$ and $L=\ker(q)$. From the previous question, it is proven that $p^2=p$, $qp=0... |
H: Representing a real valued function as a sum of odd and even functions
With $f(x)$ being a real valued function we can write it as a sum of an odd function $m(x)$ and an even function $n(x)$:
$f(x)=m(x)+n(x)$
Write an equation for $f(-x)$ in terms of $m(x)$ and $n(x)$:
My attempt using the properties - even ... |
H: How to integrate a function of $y$ over a polygon
I am given the coordinates of the vertices of a polygon and I need to integrate a function of $f$ (shown in the picture) to solve my problem. How can I do it? Btw. I will use it in a software which I am developing as a term project. So, I am looking for a programmab... |
H: Integral inequality, $f$ continuous, increasing function
Let $f$ be a continuous, increasing function on $[a,b]$, $c$ is the middle of $[a,b]$.
Prove that $\frac{f(a)+f(c)}{2} \le \frac{1}{b-a} \cdot \int _a ^b f(x)dx \le \frac{f(b)+f(c)}{2} $ .
Could you help me with that?
I thought I may use intermediate value t... |
H: Example of a $\kappa$-long sequence of disjoint club subsets of regular cardinal $\kappa$
I'm self-studying set theory and got stuck on this exercise:
Let $\kappa$ be a regular cardinal. Give an example of a sequence $\langle C_\alpha\mid\alpha<\kappa\rangle$ such that $C_\alpha$ is club in $\kappa$ for every $\al... |
H: How to check if $x^TCx\geq0$?
I have the next 3x3 block matrix C, where each block is a square matrix.
$$
C = \begin{bmatrix}
0 & A & B \\
0 & A+K_1 & B \\
0 & A & B+K_2
\end{bmatrix},
$$
where $K_i$ is also a squared diagonal matrix (I have freedom choosing their elements). I want to be sure that $x^TCx \geq 0$ ch... |
H: Finding a basis with Change of Basis
Find the coordinate vector for v relative to the basis S = {v1, v2, v3} for $R^3$.
$$v = (2,-1,3);$$
$$v1 = (1,0,0); v2 = (2,2,0); v3 = (3,3,3);$$
So I did and I got the coordinate vector space as (v)s = (3, -2, 1). I do not believe that this answer is wrong. So now I am a... |
H: What happens outside radius of convergence
A real power series $\sum_{n=0}^\infty a_n z^n$ has radius of convergence $R$. I am able to prove that for any real number $r>R$, the sequence $|a_n|r^n$ must be unbounded. Must it also tend to $\infty$? Please give me some hints, thanks.
AI: Let $a_n=\frac{1}{n!}$ for $n$... |
H: Give an equation of the plane parallel to $3x-12y+4z=0$ and tangent to the surface $x^2+y^2+z^2=676$
Give an equation of the plane parallel to $3x-12y+4z=0$ and tangent to the surface $x^2+y^2+z^2=676$
What I tried:
the normal vector is:
$$
[3,-12, 4]
$$
From the equation:
$$
[2x_0, 2y_0, 2z_0]
$$
It seems, that th... |
H: Given $\int _0 ^{+ \infty} \frac{1}{(1+x)^a} =1, \ \ a =?$
Given $\int_0^{+ \infty} \frac{1}{(1+x)^a} =1$ what is the value of $a$?.
I know that $\int_0^{+ \infty} \frac{1}{(1+x)^2} =1$.
Are there any other solutions?
Could you help me?
AI: Possible solution: $$\int_0^{+ \infty} \frac{1}{(1+x)^a} =1$$
$$\therefore... |
H: Proof that stochastic process on infinite graph ends in finite step.
Infinite Graph
Let $G$ be an infinite graph that is constructed this way: start with two unconnected nodes $v_1$ and $u_1$. We call this "level 1".
Create two more unconnected nodes $v_2$ and $u_2$. Connect $v_1$ to both of them with directed edge... |
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