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H: Finding equation of position-time function
Dudley Do-Right is riding his horse at his top speed of $10m/s$ toward the bank, and is $100m$ away when the bank robber begins to accelerate away from the bank going in the same direction as Dudley Do-Right. The robber's distance, $d$, in metres away from the bank after ... |
H: Why is $V$ a neighborhood of $p$?
this is a proof for theorem that compact subsets of metric spaces are closed.
I can't understand the line: If $V$ is intersection of $V_{q1}$, $V_{q2}$, $\dots$ , $V_{qn}$,
then $V$ is a neighborhood of $p$.
I don't think being intersections of neighborhood doesn't mean it is a nei... |
H: Convergence of series, comparison test on $\frac{1}{\sqrt{n}+ \ln n}$
I am suppose to use the limit comparison test to prove divergence or convergence. There isn't really any examples in my book that show how to pick your $b_n$ so I just pick whatever works out nicely.
$$a_n = \dfrac{1}{\sqrt{n} + \ln n}$$
$$b_n = ... |
H: Compute limit with the help of MacLaurin series expansion
Compute $$\lim _{x \to 0} \dfrac{e^{2x^2}-1}{x^2}$$ with the aid of a MacLaurin series expansion.
AI: Edit: The problem has been corrected. We keep the original solution, and add a solution to the corrected problem below.
Hint: We have
$$e^{(2x)^2}=1+(2x)^2... |
H: Definition of a nilpotent group.
I saw two different definitions of a nilpotent group, but I'm not really sure how these definitions are equivalent.
The first one is from Basic Abstract Algebra (Robert Ash):
A central series for $G$ is a normal series $1 = G_0 \trianglelefteq G_1 \trianglelefteq ... \trianglelefte... |
H: Prove that $\frac{1}{\sin t} - \frac{1}{t}$ is increasing on $(0,\pi/2)$.
I would like to obtain a rigorous proof of the fact that
$$
\frac{1}{\sin t} - \frac{1}{t}
$$
is increasing on $(0,\pi/2)$. I attempted the usual by taking the derivative and seeing if it's positive, however, this led me nowhere enlightening.... |
H: for prime ideals, the intersection of the squares is the square of the intersection?
Here is something that i proved and i would appreciate feedback on my proof:
Proposition: Let $A$ be a commutative Noetherian ring and $p,q \in \operatorname{Spec}(A)$. Then $p^2 \cap q^2 = (p\cap q)^2$.
Proof: $(\Rightarrow)$: ... |
H: Differentiate a rational function
I'm having problems solving a problem. Suppose to find the slope of:
$$f(x)= \frac{x}{x-2}$$ at the given point $(3,3)$.
I got $1/(1) = 1$ (for the slope)
But my answerbook says that it's $-2$. Can someone explain what I'm doing wrong?
AI: Method one:
$$f(x) = \frac{1}{h(x)} \ \t... |
H: Math logic - determine whether an inference exist
This is the first time I see this kind of question.
Ok, I have:
$\{\neg A \vee B, B \to C, A \vee C \} \models B \vee C$
I have to determine whether an inference exist or not.
How do I do so? please help.
AI: HINTs:
Note that on the left of the double turnstile $\m... |
H: Prove that $X \nvdash \alpha\to\beta$, $X$ when maximally consistent
I'm trying to solve a sample exam and I got stuck.
Let $\alpha, \beta$ be statements and $X$ a set of statement such that $X \cup \{\beta\}$ is maximally consistent.
Given $X \cup \{\beta\} \vdash \alpha $, prove that $X \nvdash \alpha\to\neg\bet... |
H: Show that there are uncountably many intervals in R.
How can I show that there are uncountably many intervals in R?
I think contradiction would work by assuming there are countably many intervals and
using that there are countable number of rational number in R, but I'm not sure how to proceed.
AI: Recall that a se... |
H: Help with a proof in Hungerford's book
I didn't understand a detail in this proof of this theorem:
The definition of content:
I didn't understand why $C(C(f)f_1C(g)g_1)\thickapprox C(f)C(g)C(f_1g_1)$
I need help only in this part.
Thanks in advance
AI: If $p\in D[x]$ and if $a\in D$, then $C(af)=aC(f)$. (Why?) I... |
H: source to study Zariski tangent space
My algebraic geometry is more of a disconnected set of ideas; however I need to understand well the Zariski tangent space. My commutative algebra though is decent. Towards this end, I don't find Hartshorne helpful, neither Eisenbud or Shokurov-Danilov as their presentation is d... |
H: How to make this difference equation continuous?
$$ F_n=F_{n-1}-F_{n-2} $$
How can I convert this oscillating sequence into a continuous function? IE get it in terms of n.
AI: This sequence has exactly 6 values:
($a$,$\ $ $b$, $\ $ $b-a$, $\ $ $-a$, $\ $ $-b$, $\ $ $a-b$) in cycle.
So, there is a reason to build tr... |
H: Does $f'(x) \in o(g'(x))$ imply $f(x) \in o(g(x))$ for monotonically increasing $f$ and $g$?
The title says it all. This seems intuitively true to me, but I'm not sure how one would go about proving this.
(I'm asking because I'm trying to show that $x^n \in o(x^{n+1})$ for all natural $n$, and I was wondering if yo... |
H: Let $Q$ be a special orthogonal matrix. Show that $Q(u\times v)=Q(u)\times Q(v)$ for any vectors $u, v\in\mathbb R^3$.
Let $Q$ be a $3\times3$ special orthogonal matrix. Show that $Q(u\times v)=Q(u)\times Q(v)$ for any vectors $u, v\in\mathbb R^3$.
I have no idea how to start. I'm not sure if $Q(u)\cdot Q(V)=Q(u\... |
H: Is every point of every open set E, the subset of R^2 a limit point of E?
I think this one is really trivial.
Since E is open, for every element e in E, there is a neighborhood N of e such that N is in E.
If radius of the neighborhood of e is r, then,
for any d(n,e) such that 0
Hence, e is limit point by definitio... |
H: Question related to matrix in computer memory
I'm trying to solve the following problem from a book:
A matrix $\mathbb M$ has 3 rows and 4 columns:
$$\left[ \begin{array}{cccc}
a_{11} & a_{12} & a_{13} & a_{14}\\
a_{21} & a_{22} & a_{23} & a_{24}\\
a_{31} & a_{32} & a_{33} & a_{34}\\
\end{array} \right]$$
The 1... |
H: $W_2(f,0) = \frac{1}{2} \# f^{-1}(l)$?
There must be some rather straight forward reason for $W_2(f,0) = \frac{1}{2} \# f^{-1}(l)$ but I really get stuck with why. Could someone help me out?
AI: This follows from the proof of the Borsuk-Ulam theorem preceding this theorem. (Starting on page 91 of Guillemin and Poll... |
H: Help with the problem of conditional probability
Two friends are students of the subject of Statistics so that when one passes the notes lack other. It is known that the first will attend 80% of classes and the second at 40%, independently. What is the probability that the friends have all class notes?
AI: For any ... |
H: Is there any symmetric complex matrix not satisfying this condition
Friedberg linear algebra p.471:
"If $A$ is symmetric with entries in $\mathbb{F}$, then $||A||_{op}$ is the largest absolute value of $A$"
Here, the operator norm is defined as $||A||_{op} \triangleq \sup_{x\neq 0} \frac{||Ax||}{||x||}$.
==========... |
H: Symmetric functions possess a common zero.
So possess a common zero, means $\exists x \in S^k: f_i(x) = 0$, right?
Then I could not follow this brief proof - what is the corollary? Because the information in the proof is so little, I couldn't even guess which.
Thanks =)
Theorem. Any $k$ smooth functions $f_1,\ldot... |
H: Prove that every two sets A,B : $P(A \bigtriangleup B) \not= P(A) \bigtriangleup P(B)$
I want to show that for every 2 sets this claim are true
$$P(A \bigtriangleup B) \not= P(A) \bigtriangleup P(B)$$
Notes : P is the power set.
can I do that with truth table or it will be too primitive? if not formal way is prefer... |
H: If $\alpha +\beta = \dfrac{\pi}{4}$ prove that $(1 + \tan\alpha)(1 + \tan\beta) = 2$
If $\alpha +\beta = \dfrac{\pi}{4}$ prove that $(1 + \tan\alpha)(1 + \tan\beta) = 2$
I have had a few ideas about this:
If $\alpha +\beta = \dfrac{\pi}{4}$ then $\tan(\alpha +\beta) = \tan(\dfrac{\pi}{4}) = 1$
We also know that $... |
H: Disproving that a sequence is Cauchy's Sequence
$f_n=(-1)^nn$
I tried doing it this way :
Given $\epsilon>0$ we have for $n>m$
$|f_n-f_m|=|(-1)^nn-(-1)^mm|\le |(-1)^nn|+|(-1)^mm|=n+m>2m$
Am i good this far ? Now how do i conclude with this result that the sequence is not a cauchy sequence i am not getting the final... |
H: Does every unital ring contain all the integers?
Let us suppose there is a ring $R$ with the multiplicative identity $1$.
We know that $1+r\in R$, where $r$ is any element of the ring $R$. Does this mean $1+1$ is also part of the ring, or does $r$ have to be an element of the ring different from $1$?
Is $1+1$ cal... |
H: The value of $ A \exp\left(\frac{-1}{2\pi} \int_{-\pi}^{\pi} \ln(1+A+2BC \cos x) dx \right)$
I'm looking for the value of: $$ A \exp\left(\frac{-1}{2\pi} \int_{-\pi}^{\pi} \ln(1+A+2BC \cos x) dx \right)$$
I know we could take $y=1+A+2BC \cos x$ but changing variable in this way makes the integral from $1+ A - 2BC$... |
H: Hahn-Banach Theorem in the C*-algebra
What is the Hahn-Banach Theorem in the C*-algebra(or W*-algebra maybe)?
If B is an nondense subalgebra of C*-algebra(or W*-algebra maybe), can we get an state f of A which is always zero at the subalgebra B from the version of Hahn-Banach Theorem?
AI: Every state is a complete... |
H: Ehrenfeucht–Fraïssé game, how can I understand it?
My course of "Formal Methods" deals with Ehrenfeucht–Fraïssé games, particularly regarding the inexpressibility in FO logic.
At the moment I've fully understand what this games are and how they are supposed to work.
But I can't grasp several theorems regarding inex... |
H: Prove that : $\sqrt[4]{(a^2+1)b}+\sqrt[4]{(b^2+1)c}+ \sqrt[4]{(c^2+1)a} \le 3\sqrt[4]{2}.$
For $a,b,c\in\mathbb{R}^+$ and $a+b+c=3$ .
Prove that : $\sqrt[4]{(a^2+1)b}+\sqrt[4]{(b^2+1)c}+ \sqrt[4]{(c^2+1)a} \le 3\sqrt[4]{2}.$
AI: Oops, give a wrong answer in previous edit. Here is the corrected version.
Notice both... |
H: Strange application of the mean value theorem
I recently came across the following lemma.
Lemma. Suppose $\Omega_n^N = [-\pi n, \pi n]^N$ and $f \in L^1(\Omega_n^N)$ is $2\pi n$-periodic. Let $A$ be a subset of $[-\pi n, \pi n]$. There exists some $\alpha_N \in [-\pi n, \pi n] \setminus A$ such that \begin{equati... |
H: Electrodynamics in general spacetime
Let $M\cong\mathbb{R}^4_1$ be the usual Minkowski spacetime. Then we can formulate electrodynamics in a Lorentz invariant way by giving the EM-field $2$-form $\mathcal{F}\in\Omega^2(M)$ and reformuling the homogeneous Maxwell equations as
$$d\mathcal{F} = 0$$
Then the Poincaré l... |
H: Is there a nonnormal operator with spectrum strictly continuous?
Let $H$ be an infinite dimensional separable Hilbert space.
Definition : An operator $A \in B(H)$ is normal if $AA^{*} = A^{*}A$.
Definition : The spectrum $\sigma(A)$ of $A \in B(H)$, is the set of all $\lambda \in \mathbb{C}$ such that $A - \lambd... |
H: Name for Cartesian Product variant that does not return an empty set if one of the sets is empty
I am looking for the name of this mathematical operation that behaves very similar to Cartesian Product.
Given:
A = {1,2}
B = {3,4}
C = {5}
D = ∅
The operation (which I'll write as *), behaves like this:
A * B = {1,2... |
H: why $(r+I)(s+I) = rs + I$ in the quotient ring $R ?$
Say $R$ is a commutative ring and $I\in R$ is an ideal. Let us consider the quotient $R/I$. It is created by taking every element $a\in R$, and adding all the elements of $I$ to it. The elements of $R/I$ are of the form $a+I$ and $b+I$, $\forall a,b\in R$.
Now $... |
H: Why is $(0,1)\cup \left\{ 3 \right\}$ disconnected?
I have just learned the definition of connectedness and wikipedia gives an example of a disconnected set: $(0,1)\cup \left\{ 3 \right\}$ (https://en.wikipedia.org/wiki/Connected_space#Examples). Why is it disconnected? I need a bit clarification on this. Thanks fo... |
H: Equation of motion Pendulum using $w=e^{ix}$
I'm working with the equation of motion for a pendulum as follows:
$$x''+ \frac{g}{l} \sin (x)=0$$
Where $x$ is the angle between the pendulum and the vertical rest position.
I am required to use the complex variable $w=e^{ix}$ to rewrite the equation of motion in the fo... |
H: Integrating $\int_{-1}^{1}(1+x)^m(1-x)^ndx$
Stimulated by the physics background of path integral, I want to know how to calculate this integral
$$\int_{-1}^{1}(1+x)^m(1-x)^ndx,$$
where $m$, $n$ are positive integers. I'm struggling with it.
Different kinds of approaches are welcome, thanks in advance!
AI: Let deno... |
H: Why does $L^1(\mathbb{R})$ not have the C* property?
Consider the space $L^1(\mathbb{R})$ which is a Banach-algebra when taking the convolution as the algebra product and even posses the B*-property if one takes $f^*(t)=\overline{f(-t)}$ as the involution. But $L^1(\mathbb{R})$ does not have the C* property.
What w... |
H: Calculate the limit: $\lim_{n \to \infty}\int^{1}_{0}{\frac{e^x}{1+x^n}\mbox{d}x \mbox{?}}$
How can I calculate the following limit:
$$\lim_{n \to \infty}\int^{1}_{0}{\frac{e^x}{1+x^n}\mbox{d}x \mbox{?}}$$
Is all right if I say that the initial limit is equivalent with $$\int^{1}_{0}{\lim_{n \to \infty}{\frac{e^x}{... |
H: Help for evaluating complicated integral $\int \frac 1 {x^n-x} dx$
I have this complicated integral to evaluate :
$$\int \dfrac 1 {x^n-x} dx$$
I'm struggling to evaluate this.
My attempt :
$$\int \dfrac1x \cdot \dfrac 1 {x^{n-1}-1} dx$$
Now, I try to apply integration by parts. For that, I use : $V=\large\dfrac1x$... |
H: If $(W,<)$ is a well-ordered set and $f : W \rightarrow W$ is an increasing function, then $f(x) ≥ x$ for each $x \in W$
I could use a hand understanding a proof from Jech's Set Theory.
Firstly, note that Jech defines that an increasing function $f : P \rightarrow Q$ is a function that preserves strict inequalities... |
H: Analyticity of C*-algebra valued functions
Let $\mathcal{A}$ be a unital C*-algebra and consider a function $f:\mathbb{C} \rightarrow \mathcal{A}$. What is an accessible tool to prove or disprove that $f$ is analytic, i.e. can be locally expanded in a power series of $z$?
Take as a concrete example $f_A(z) = \exp(z... |
H: Self-studying through an undergraduate math course. Need Tao-like textbooks!
I'm a physics undergraduate student who always enjoyed math, and briefly studied it at a university but for various reasons (laziness, youth) gave up and changed 'majors'. But I always wanted to go through an undergraduate math course in m... |
H: How to integrate $\int_0^x \frac{t}{(1-t)^2}dt $?
How do you integrate $\int_0^x \frac{t}{(1-t)^2}dt $?
My approach was to use partial fractions for $\frac{t}{(1-t)^2}$. I thus have to rewrite $\frac{t}{(1-t)^2}$ as $\frac{t}{(t-1)^2}$(right?) and then integrate:
$\frac{1}{(t-1)}+\frac{1}{(t-1)^2}$
But then the fir... |
H: Differential topology book
I want to self-study differential topology. I'd like to hear suggestions from you about appropriate books that I could use while studying.
Note: I have not studied differential topology before. I self-studied general topology and some algebraic topology before.
Thank you
AI: A standard in... |
H: Convergence of a series containing logarithm
I'm stuck on the following problem: for what values of $\alpha$ the following sum is convergent:$$S(\alpha)=\sum_{k=2}^{+\infty}\frac{1}{\ln(k)^\alpha}$$
If the series is convergent, there will be a constant $M$ for which
$S(\alpha\ge M)\lt\infty$, but how is it possible... |
H: If $\gcd(g(x),P_A(x)) \neq 1$ then prove that $g(A)$ is not invertible
we had this question at our exam today and my professor told me my proof is very wrong mathematically.
Given $A \in M_{n \times n } (\mathbb C)$, and a polynomial $g(x) \in \mathbb C[x]$, and that $\gcd(g(x),P_A(x)) \neq 1$. ($P_A(x)$ is the ch... |
H: Why is it considered incorrect to use the variable of integration as a boundary of integration?
Recently, a question was asked about limits of integration and the OP made the following "typo":
$$I = \int_0^x f(x) \, dx$$
as opposed to
$$I = \int_0^x f(t) \, dt $$
I commented that this typically frowned upon and c... |
H: Solve equation with variables inside and outside cos
I have an issue with following problem:
A line needs to be intersected with a cosinus function.
My line would be
$\displaystyle l_1 = \binom{x_1}{y_1} + s\binom{u_1}{v_1}$
My function for a curved line in 2D is
$\displaystyle g: y_2 = d + a + a * (-cos(f * x_2))... |
H: Prove that an algorithm cannot reach a given goal
We are given an algorithm that, in each step takes a set $\left\{a, b, c\right\}$ It takes any two variables $a, b$ at random and changes them to $0.6 + 0.8b$ and $0.8a - 0.6b$. The initial value of the algorithm is $\left\{3, 4, 12\right\}$. Prove that the algorith... |
H: Problem with binomial sumation
I am trying to solve the following summation:
$\sum_{k=0}^{M-1} {M-1 \choose k} \alpha^{k} (1-\alpha)^{M-1-k} u(k)$
where:
$u(k) = 1$ , if $0 \le k < j$
$u(k) = (1-\frac{j}{k+1})$ , if $M-1 \ge k \ge j$
Is there a close expression for this summation?
AI: I don't believe there is a clo... |
H: Unambiguous terminology for domains, ranges, sources and targets.
Given a correspondence $f : X \rightarrow Y$ (which may or may not be a function) I generally use the following terminology.
$X$ is the source of $f$
$Y$ is the target
$\{x \in X \mid \exists y \in Y : xy \in f\}$ is the domain
and $\{y \in Y \mid \... |
H: Evaluating $\lim_{x\to0}(x\tan x)^x$
Any suggestions for evaluating the limit $$\lim_{x\to0}(x\tan x)^x$$
I have tried writing $\tan$ as $\dfrac{\sin}{\cos}$ and then got the Taylor series of them but it didn't lead me somewhere. Thanks a lot
AI: Assuming the limit exists and is equal to $L$, take logs:
$$\log{L} =... |
H: What would be the negation of these statements?
What would be the negation of
"No dogs have three legs".
I think "Some dogs do not have three legs"
"Some animals don't eat meat"
I think "All animals eat meat"
"I make the bread, or she does not make the bread"
I think "I do not make the bread, and she does make ... |
H: When is limit substitution valid?
This question asks to find the $\lim_{x\to0}(x\tan x)^x$ . Ron Gordon and Maisam Hedyelloo make the substitution $ x\sim \tan x$ , and it works and they get the correct answer. However, if you try to make the substitution $x \sim \arcsin x$ into $\lim_{x \to 0} \large \frac {\arcsi... |
H: working out a percentage from my email open rates
i have the following numbers:
recipients: $95$
opens: $39$
bounces: $2$
how would i get the percentage value per open?
accoridng to this post: http://www.campaignmonitor.com/blog/post/2395/all-about-email-open-rates
i would do the following:
open rate = opens / emai... |
H: Finding the limit with three variables
Here is the original problem:
$$\lim_{(x,y,z)\to (0,0,0)}{(\cos x-1)\sin(2y)(e^{3z}-1)\over x^2yz}$$
I was thinking about splitting up the limit like this:
$$\lim_{(x,y,z)\to (0,0,0)}{(\cos x-1)\over x^2}\cdot\lim_{(x,y,z)\to (0,0,0)}{\sin(2y)\over y}\cdot\lim_{(x,y,z)\to (0,... |
H: Perimeter of a rectangle with four circles in each corner
my teacher asked a question in exercise for calculate the perimeter of following:
I get result of $144\pi$ and some of my friends got $208.2$ as result (assuming $\pi\approx3.14$).
Q: what's the correct answer of this question ?
I get $144\pi$ with this:
Si... |
H: Weighted integral of random variables
Given a random zero-mean gaussian random variable $X(t)$ with parameter $t$, such that $E [X(t) X(t^\prime)] = \sigma^2 (t) \delta_{tt^\prime}$, is it possible to produce a single gaussian random variable that is the weighted sum of all $X(t)$ by a non-random function $f(t)$? I... |
H: Generating function of a counting function.
Let $m$ be odd. Let $\eta(m)$ count the number of ways we can express $m$ as a product of exactly two odd numbers, counting order. What is $$\sum_{m\text{ odd }}\eta(m)x^m\text{ ? }$$
So, as an example, $\eta(9)=3$ since $9=3\times 3=1\times 9=9\times 1$, while $\eta(p)=2... |
H: Manipulating random decimals
I've been slowly working my way into the world of AI and its representations of random. Well as you can guess this occurs from a call to random() resulting in a value between $[0, 1]$ to create a desired random effect/action.
What confuses me is that once we get a random number back, le... |
H: upper semi-continuity of a multi-valued function $T$ and lower semi-continuity of $d(x,T(x))$
Let $(X,d)$ be a complete metric space, $CB(X)$ the set of closed and bounded subsets of $X$, and $T:X\rightarrow C(X)$ be a multi-valued function.
How can you prove this:
If $T$ is upper semi-continuous then $f(x)=d(x,T(... |
H: an example of Zariski cotangent space
I am studying these very nice notes on the Zariski tangent space.
Let $f \in K[x_1,\dots,x_n]$ and let $V=Z(f)$, the zero set of $f$.
Let $P \in V$ and let $M_P$ denote the vanishing ideal of point $P$.
We define a map $d: K[V] \rightarrow K$ by sending $g \in K[V]$ to
the imag... |
H: Algorithm to find the roots of a polynomial over integer
What is the algorithm to find the roots of a polynomial over integer? I observe that one can find the roots within few seconds in Sage even when all coefficients of are very large. I have asked in https://cstheory.stackexchange.com/questions/18406/roots-... |
H: Unclear step in Eisenbud's proof of a variant of Hauptidealsatz
I need help in understanding the proof of the following
Theorem 10.1. If $\mathfrak{p}$ is a prime of a Noetherian ring $R$ minimal subject to containing $x \in R$, then its height is at most one.
In the context of $\mathfrak{q}$ being another prime,... |
H: Prove $\lim _{ n\to\infty } \sqrt [ n ]{ \sum _{ i=0 }^{ k } a_i ^n } =\max { \{{ a }_{ 1 }, \ldots ,{ a }_{ k }\} } $
We know that
$$ a_1, a_2, \ldots, a_k \ge 0 $$
Prove that:
$$\lim _{ n\to\infty } \sqrt [ n ]{ \sum _{ i=0 }^{ k } a_i ^n } =\max { \{{ a }_{ 1 }, \ldots ,{ a }_{ k }\} } $$
AI: Divide by the Maxi... |
H: Injective on a dense set
Let $f:X\to Y$ a continuous function between Hausdorff spaces such that $D\subseteq X$ is dense and $f\restriction_D$ is injective. Can we say that $f$ must be injective?
I can't find a counterexample, so I tried to prove it:
Let $x,y\in X$ such that $f(x)=f(y)$. If $x\neq y$, then there e... |
H: In the definition of Strong Feferman Set Theory, what does this phrase mean?
In this post, Mike Shulman talks about Strong Feferman Set Theory, arguing that it deals with the foundational issues raised by category theory in an especially straightforward manner.
Mike explains:
To ZFC (or your favorite set theory) w... |
H: Ratio of sum of Euler's totient to $n$: $\lim_{n \to \infty} {\log \left( \sum_{k=2}^n \varphi(k) \right) \over \log(n)}$
This is more a casual/recreational question...
It seems to me, that the limit as given in the subject line
$$\lim_{n \to \infty} {\log \left( \sum_{k=2}^n \varphi(k) \right) \over \log(n)} = \... |
H: Exponential of a matrix and related derivative
$\DeclareMathOperator{\tr}{tr}$
I have $ X \in M(n,\mathbb R) $ to be fixed. I define $ g(t) = \det(e^{tX}) $
Then the author proceeds as follows:
\begin{align}
g'(s) &= \frac {d}{dt} g(s+t)\\
&=\frac {d}{dt} \det(e^{(s+t)X}) |_{t=0}\\
&=\frac{d}{dt}(\det(e^{sX})\det(e... |
H: An Error in Landau's FOA Thm 140?
Thm 140: If $\xi > \eta$, then $\eta + \nu = \xi$ has exactly one solution $\nu$. (All lowercase Greek letters represent Dedekind cuts.)
Summary of proof and the alleged error:
Consider the set of all rational numbers of the form $X - Y$ (with $X > Y$) where $X$ is a lower number f... |
H: Is a prevariety the same as a notherian scheme?
By a pre-variety I mean a quasi-compact locally ringed space which can be covered by (a finite # of) affine varieties.
I was wondering, this seems to be the same in scheme language as a Noetherian scheme, or am I overseeing something?
AI: The typical example of a pre-... |
H: Find the value of $\large i^{i^{.^{.^.}}}$
Find the value of $\large i^{i^{.^{.^.}}}$ ?
How should we start to solve it ?
Also you can see this one if it helps.
Thanks
AI: Let $z=i^{i^{.^{.^{.}}}}$. Then, as Hagen von Eitzen pointed out, $i^{z}=z$. Then $1=z\,i^{-z}=z \, e^{-i\pi z/2} $. It follows that
$$-\frac{i... |
H: Induction Using Multi-Index
Does anyone know how to use induction in the context of multi-indices?
I know the induction is done on the multi-index length, the main problem is how to split a multi-index of length $n+1$ into one of length $n$ and another of length $1$.
For example, I have the following problem: Prov... |
H: Payment problem
I'm looking for an algorithm, which can solve the following problem:
There is a basket, containing (n) products, paired with a value, which shows how much money is required to cover them.
E.g basket(n=3):
{apple,500€},{banana,400€},{cherry,300€}
In our purse we have different type of vouchers(m pcs)... |
H: What is the proper terminology for these two types of multiplication?
((QUESTION REWORKED))
First question on this site, and I apologize if this question has been answered. I searched and searched and the fact I don't know the basic terminology is hindering me from finding the answer.
This question stems from a cur... |
H: Change of variables with a square
Can someone help me understand this a bit better:
$\int (x-y)^2 dx = \int(y-x)^2dx$
as $(y-x)^2 = (x-y)^2$. Now, if I make the change $z = x-y$ in the one on the LHS I get:
$\int z^2 dz$
as $dz = dx$. Now, if I make the change $z=y-x$ in the one on the RHS I get:
$\int z^2 (-dz)$... |
H: A name for set of disjoint intervals
What's in a name? That which we call a rose by any other name would smell as sweet.
William Shakespeare
I'm looking for a short name for the phenomenon collection of disjoint intervals.
I currently use selection, which I'm not very satisfied with, and I wonder if there is a be... |
H: Diffeomorphisms preserving harmonic functions
I'm looking for smooth maps $ \Bbb R^n \to \Bbb
R^n $ with the property that, whenever $ h $ is a harmonic function ($\Delta h=0$), $ h\circ f $ is also harmonic. Is there a nice characterization of these functions? I'm especially interested in those function which ar... |
H: Why does every undirected graph have at least one cut of size $|E| / 2$?
In an undirected graph $G = (V, E)$, a global maximum cut in $G$ is a pair $(S, V - S)$ with the largest possible number of edges with one endpoint in $S$ and another endpoint in $V - S$ (this quantity is called the size of the cut).
There is ... |
H: Baby Rudin vs. Abbott
I am considering Stephen Abbott's Understanding Analysis and Walter Rudin's Principles of Mathematical Analysis. I am looking for a comparison between the two that addresses both of the following matters,
Is one of the two substantially more mathematically rigorous than the other?
Does one i... |
H: Let $R$ be a ring with every element but $1$ having a left quasi-inverse. Then $R-\{1\}$ is a group under $a*b=a+b-ba$.
This question is related to exercise 1.51 from Rotman's "Introduction to the Theory of Groups".
An element $a$ in a ring $R$ (with unit element $1$) has a left quasi-inverse if there exists an el... |
H: Regarding 3-fold connected coverings of the $S^1 \vee \mathbb{R} P^2$
As in the question, I need to determine all of the 3-fold connected coverings of the wedge of the unit circle and the real projective plane.
Here's what I think: I know that the fundamental group of $S^1$ is isomorphic to the set of integers $\... |
H: Continuing direct product on a subcategory
Let $F$ is a full subcategory of a category $G$, both categories having binary direct product.
Is it always true that there is such a binary direct product in $G$ that it is a continuation of a binary direct product in $F$?
Hm, can it be generalized for infinitary direct p... |
H: Practice Problem Books
The Analysis I/II/III (Differentiation and Continuity/Sequence and Series/Integration) published by AMS. The first one is this. It's a problem-solution book. I found it excellent because of the quality of questions there.
I am looking for similar book (complete problem (only) book with comple... |
H: the representation on the regular representation is faithful
I am reading the proof of the following proposition.
Proposition. As algebras, $\mathbb{C} G \cong \bigoplus_i \mathrm{End}(W_i),$ where $G$ is a finite group and $W_i$ are irreducible representation of $G$.
The proof goes as follows.
A representation $G ... |
H: Differences between $C_c^\infty[0,T]$ and $C_c^\infty(0,T)$
I believe it is true that:
If $f \in C_c^\infty(0,T)$, then $f(T)=f(0)=0$.
$C_c^\infty(0,T) \subset C_c^\infty[0,T]$
$C^\infty(0,T) \subset C_c^\infty[0,T]$
If $f \in C_c^\infty[0,T]$, it doesn't necessarily mean that $f(T)=f(0)=0$.
Finally, what happen... |
H: One Point Derivations on locally Lipschitz functions
Let $A$ be the algebra of $\mathbb{R}\to\mathbb{R}$ locally Lipschitz functions. What is the vector space of derivations at $0$? The proof that for continuous functions there aren't really any doesn't seem to work in this case.
I was thinking about trying to defi... |
H: How to prove this property of floor function?
$
\left\lfloor { - x} \right\rfloor = - \left\lfloor x \right\rfloor
$
if $x \in \mathbb{Z}$ and
$ \left\lfloor { - x}\right\rfloor = - \left\lfloor x \right\rfloor -1 $ otherwise
This is an exercise from Tom Apostol's book "Calculus Volume I" section 1.11 number 4. H... |
H: Usually, main results are called theorems, while smaller results are called propositions. Is there a name for super-immediate results?
In mathematical papers, main results are called theorems, while less central results are called propositions. But sometimes, there is a result that is so immediate, it doesn't even ... |
H: Different direct product in a category and its full subcategory
A question related to Continuing direct product on a subcategory.
Let $F$ is a full subcategory of a category $G$.
I denote $\operatorname{Ob}X$ the set of objects of a category $X$.
Is it possible the following?
There are binary direct products $\tim... |
H: Finding a planar graph satisfying these properties
I need to construct a 3-regular connected planar graph with a planar embedding where each face has degree 4 and 6. In addition, each vertex must be incident with exactly one face of degree 4.
seems to fit the description well if extended infinitely but unfortunate... |
H: When can the inverse Fourier tranform be represented in terms of integral?
From Wikipedia
Use the convention for the Fourier transform that
$$
(\mathcal{F}f)(\xi):=\int_{\mathbb{R}^n} e^{-2\pi iy\cdot\xi} \, f(y)\,dy.
$$
Furthermore, we assume that the Fourier transform is also integrable.
The most common ... |
H: Are my answers right here about true and false statements?
Every integer is a rational number -> false -- correct?
Let r = true; s = true. Is this statement true or false? $$\lnot [r \lor (\lnot s \lor r)];\quad$$ -> true -- correct?
Let p = true; r = false; q = false. Is the following statement true or false?
$$... |
H: Riccati differential equation $y'=x^2+y^2$
$$y'=x^2+y^2$$ I know, that this is a kind of Riccati equation, but is it possible to solve it with only simple methods? Thank you
AI: It can be solved (or, rather, transformed into a recognizable form) using simple methods, but the result can only be expressed in terms of... |
H: Geometric Distribution Probability Problem
Suppose that $X$ is a geometric random variable with parameter (probability of success) $p$.
Show that $\Pr(X > a+b \mid X>a) = \Pr(X>b)$
First I thought I'd start by calculating $\Pr(X>n)$ where $n=a+b$:
$$\Pr(X > n) = p_{n+1} + p_{n+2} + \cdots = ?\tag{1}$$
But I don't... |
H: is this the right truth table?
When I filled out the table I tried my best to figure it out. But If I made any mistakes please help me correct them. Thanks!
sorry 5th one should be false
AI: EDIT: UPDATE
Now your table is mostly correct.
Check your truth value assignment columns; we need to cover all possible $2... |
H: Prove that if A is symmetric and invertible, then$ (A^{-1})^t = (A^t)^{-1}$.
I have a problem. Prove that if A is symmetric and invertible, then $(A^{-1})^t = (A^t)^{-1}$.
This is what I have done, please tell me if it is correct:
Say $C=A^{-1}$. Then, $C^t = ((C^{-1})^t)^{-1} = ((C^{-1})^{-1})^t = C^t$.
AI: If $A$... |
H: Prove that if a product$ AB$ of $n\times n$ matrices is invertible, so are the factors $A$ and $B$.
I'm really struggling with "simple" proofs, and I could only come up with this. I googled for the solution and know how to do it now, but I first want to know if there's anything wrong with my solution?
$$
(AB)^{-1}=... |
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