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H: How do I evaluate t$\lim_{n\to\infty} \sum_{i=1}^n \left[\sqrt{1+ \frac{2i}{n}}\right]\frac{2}{n}$? (From MIT OCW 18.01 sc final Q7(a))
This was one of the questions on the final for MIT's 18.01:
$$\lim_{n\to\infty} \sum_{i=1}^n \left[\sqrt{1+ \frac{2i}{n}}\right]\frac{2}{n}$$
The answer converts it to an integral,... |
H: No. of positive integral solutions and link to coefficients in expansion
The question (from an NTA sample paper for JEE Main) -
If $p, q, r \in \Bbb N $, then the number of points having position vector $p\hat{i} + q\hat{j} +r\hat{k}$ such that $8 \leq p + q + r \leq 12$ are:
It is evident that I had to essential... |
H: localization and ideals
Statement: Every ideal J in S$^-$$^1$R is of the form S$^-$$^1$I for some ideal I in R.
Proof:
Let J = (j$_\alpha$: $\alpha$ $\in$ A) Then j$_\alpha$ = h(r$_\alpha$)h(s$_\alpha$)$^-$$^1$. Define I to be the ideal in R generated by {r$_\alpha$: $\alpha$ $\in$ A}; that is I = h$^-$$^1$(h(R)$\c... |
H: Derivation of within point scatter $W(C)$
I'm reading the book "The Elements of statistical learning". In the section about K-means clustering they derive an equation regarding the "within point scatter" which is a quantity that describes how "scattered" points are within a cluster.
\begin{aligned}
W(C) &=\frac{1}{... |
H: lagrange multiplier determinant
Can someone please explain why in my textbook they write lagrange multiplier like this :
$$\begin{vmatrix}
\frac{\partial f}{\partial x}& \frac{\partial f}{\partial y}\\
\\
\frac{\partial g}{\partial x} &\frac{\partial g}{\partial y}
\end{vmatrix}=0
$$
I don't understand where did th... |
H: Error analysis by differentiation
I've been studying physics and I found this weird differentiation.
$\ln x = \ln a + \ln b$
Now differentiating both sides,
$\dfrac{dx}x = \dfrac{da}a + \dfrac{db}b$
First of all this weird differentiation doesn't make sense to me. I understand that $(\ln x)'$ will be $\fra... |
H: Log-Likelihood ratio derivation
I am struggling to understand the derivation of the log likelihood in the proof of Lemma 1 in Kauffman14 (https://arxiv.org/pdf/1407.4443.pdf).
I will give a bit of context: the lemma is about a lower bound on the sample complexity for multi-armed bandit through a change of measure a... |
H: Proof that every codeword of binary self-orthogonal linear code has even weight
A linear code $C$ is self-orthogonal if it is contained in its dual code, that is $C\subseteq C^{\perp}$
I want to proof that every codeword of $C$ has even weight
What I got by far:
Supposing that $C$ is a binary linear code, I can co... |
H: Poincare duality for reduced homology
Reduced homology
In my understanding, the reduced homology is better-behaved than the usual singular homology because the $0$th reduced homology
counts the non-trivial "closed" $0$-chain and
reflects the idea of "orientation is the volume form".
(see below for detail)
Poincar... |
H: $\widehat{\mathbb{Z}}$-module structure
Given a torsion abelian group $A$, prove that $A$ has a unique $\widehat{\mathbf{Z}}$-module structure and that $\widehat{\mathbf{Z}}\times A\to A$ is continuous if $A$ has the discrete topology.
I proved the first part, the module structure is given by letting an element $... |
H: Finding an expression for $\dfrac{dx}{dt}$ by solving the initial value problem
The effectiveness of a police force may be measured by its clearance rate: the number of charges laid in a month divided by the total number of unsolved crimes.
In Arachnid Boy's home town, new crimes are reported roughly $20$ times ... |
H: Finding a Nash equilibrium
I'm doing the exercises at the end of the paper A Brief Introduction to the Basics of Game Theory by Matthew O. Jackson. I would be grateful if somebody could provide me with solutions to it. I'm not sure about question 2:
Two hotels are considering a location along a newly constructed h... |
H: Logic question - Statement logic
Given this statement:
"Every positive number that is smaller than $1$ is bigger than its square"
Which of these statements are true (They may be both false/right) ?
You can write the statement as:
$\forall x((x <1) \wedge (x >0) \wedge (x^2 <x))$
$\forall x((x<1) \wedge (x>0)) \r... |
H: Arrangement in a circle
Twelve politicians are seated at a round table. A committee of five is to be chosen. If each politician, for one reason or another, dislikes their immediate neighbours and refuses to serve on a committee with them, in how many ways can a complete group of five politicians be chosen?
I don’t ... |
H: General method of evaluating $\small\sum_{n\geq 0}\left(\frac{4^n}{(2n+1)\binom{2n}{n}}\right)^2\frac{1}{n+k}$
Question: $
\mbox{How can we evaluate}\quad
\sum_{n \geq 0}\left[{4^{n} \over \left(\, 2n + 1\,\right)
\binom{2n}{n}}\right]^{2}{1 \over n + k}\quad
\mbox{for general $k$ ?.}
$
General methodology will b... |
H: How many numbers between 1 and 1,000 (both inclusive) are divisible by at least one of the prime between 1 to 50? How can I find this?
I was trying to solve a compettive programming problem in which constraints are so high so I want to deduce a formula for it so that i could do it for other ranges as well.
AI: As $... |
H: Exponential laws in modular arithmetic | disappearing mod N
Why is $(g^b \bmod N)^a \bmod N = g^{a*b} \bmod N$ ?
Specifically: Why/how does the mod N in the round brackets disappear from the first expression $(g^b \bmod N)^a \bmod N$?
I know of the exponential law that $g(^a)^b$ is equal to $g^{a*b}$ but I just do ... |
H: Evaluating $\lim_{x\to\ \infty} {x - \log(e^x + 1)}$
I stumbled upon this $(\infty-\infty)$-type limit today:
$$\lim_{x\to\ \infty} {x - \log(e^x + 1)}$$
I can't seem to be able to solve it; I tried substituting and manipulating in various ways but I still don't understand how to solve it.
Could anyone help?
AI: Re... |
H: Does the limit of a diagram with a single arrow exist?
Sorry if it’s a pointless question! I’m trying to self-learn category theory (not easy), but none of the books i’ve looked at explains this.
When they introduce limits, they give the definitions in terms of diagrams and cones and then all of them offer the same... |
H: Converse to a proposition on divisors in commutative monoids
Let $(M,*,1)$ be a commutative monoid. Define the binary relation $R$ on $M$ by $aRb$ iff there exists an $x$ in $M$ such that $a*x=b$. $R$ is the "divides" relation. Since $M$ is a commutative monoid, clearly $R$ is both reflexive and transitive. I read ... |
H: Proving consistency for an estimator. Limits and Convergence in Probability.
I need to show that $U$, as defined below, is a consistent estimator for $\mu^{2}$.
$U=\bar{Y}^{2}-\frac{1}{n}$
By the continuous mapping theorem, which states that,
$X_{n} \stackrel{\mathrm{P}}{\rightarrow} X \Rightarrow g\left(X_{n}\righ... |
H: How to derive $\frac1\pi \int_{-\pi}^{\pi}f(t)\sin nt \;\mathrm{d}t$ from $\frac{\langle\sin nx|f\rangle}{\langle \sin nx|\sin nx\rangle}$?
How to get $$\frac{\langle\sin nx|f\rangle}{\langle \sin nx|\sin nx\rangle}=\frac1\pi \int_{-\pi}^{\pi}f(t)\sin nt \;\mathrm{d}t?$$
To be specific, $\langle \sin nx|\sin nx\ran... |
H: Any multiplicative subgroup of a finite field is cyclic
I asked for minimal hints in this question. Now I've come up with a proof. Could you please verify if it is fine or contains logical mistakes?
Let $F$ be a finite field and $F^\times = F \setminus \{0\}$. Then the multiplicative group $F^\times$ is cyclic.
... |
H: A multiplicative group in which there are more than $n$ elements satisfying the equation $x^n=1$
In my proof in this question, I use the fact that a nonconstant polynomial of degree $m$ over a field has at most $m$ different roots. As such, I would like to ask for an example of a multiplicative group such that ther... |
H: When does a bounded continuous function extend continuously to its closure
Let $\Omega$ be a domain in $\mathbb{C}^n$. Let $f:\Omega\longrightarrow\mathbb{C}$ be a bounded continuous function. I wanted know if there are any necessary and sufficient conditions for $f$ to be extended continuously to $\bar{\Omega}$?
A... |
H: Finding individual PDF of two conjoined dependent variables
Given two random variables distributed using a conjoined PDF $f_{X,Y}(x,y)$, with $0 < x < y < \infty$, I should find the individual PDF for each $X$ and $Y$.
I do this by evaluating the integrals:
\begin{equation*}
f_X(x) = \int_0^x f_{X,Y}(x,y) dy
\end{e... |
H: Discontinuity of step function using open sets definition of continuity.
Definition A function $f : X \to Y$ between two topological spaces is continuous at $x$ if for any $V(f(x))$ open set containing $f(x)$ there's $U(x)$ open containing $x$ such that $f(U(x)) \subset V(f(x))$.
Consider $[0,1]$ with the subspac... |
H: A and B can do a piece of work in 9 days, B and C in 12 days, A and C in 18 days. If all of them work together, then how much time will they take?
This is how I did it.
$A+B=9 \tag 1$
$B+C=12 \tag 2$
$A+C=18 \tag 3$
Adding (1), (2) and (3) we get:
$A+B+B+C+A+C=9+12+18$
$2(A+B+C)=39$
$A+B+C=19.5$
So, they... |
H: proving that If $F$ is countable, then $F$ may or may not be closed
In my general topology textbook there is the following exercise:
If $F$ is a non-empty countable subset of $\mathbb R$, prove that $F$ is not an open set, but that $F$ may or may not but a closed set depending on the choice of $F$.
I already prov... |
H: A question in Lesson 4 of Hoffman Kunze Linear Algebra
While self studying Linear Algebra from Hoffman Kunze I am unable to think the reasoning behind an argument in Lesson -4 of the book.
It's image :
I think there is a typo in last 3rd line: There should be $f_j$ instead of $f_{i} $ .
Now,Question : How author ch... |
H: sum and binomial coefficient induction proof
I bet the proof is simple but I have little experience with binomial coefficients and sums. I am curious about how you would solve this by induction:
$$ \sum_{i=0}^k {n\choose i} \leq n^k + 1$$
for $1 \leq k \leq n$. Where $n$ and $k$ are integers.
AI: I assume that chec... |
H: Sum of a series.
How to show that, $\sum_{n=1}^N 1/n$ $\le$ 1 + logN, for N$\ge$5
AI: hint
For any $ n\ge 2$, and any $ t\in [n-1,n] $,
$$\frac{1}{n}\le \frac 1t \;\implies$$
$$\int_{n-1}^n\frac {dt}{n}\le \int_{n-1}^n\frac{dt}{t} \;\implies$$
$$\frac{1}{n}\le \ln(n)-\ln(n-1)$$ |
H: Why can't we prove that $X= f^{-1}(f(X))$ in general?
Let $f:A \rightarrow B$ be a map. Let $X \subseteq A$, $Y \subseteq B$.
I saw this result that states $X \subseteq f^{-1}(f(X))$.
Well to prove that I said the following:
Let $x \in X$. By definition, $f(x) \in f(X)$. Again, by definition, $x \in f^{-1}(f(X))$.... |
H: Using elementary methods to prove infinitely many primes mod n
I was reading an elementary number theory text looking to enhance my knowledge and I came across the relatively simple task of proving there existed infinitely many primes of the form $4k-1$ (of course, without Dirichlet). My very elementary proof is as... |
H: Is $\frac{f'}{f}$ bounded for $f$ convex, $f>c$?
Let $c>0$, $f\colon \mathbb{R} \to [c,\infty)$ be differentiable and convex.
Do we have
$$ \left\|\frac{f'}{f}\right\|_{\infty} < \infty ?$$
This seems to be true in simple examples, but I am not sure whether this is true in general, so I would appreciate some hint o... |
H: Why divide diameter by square root of 2 to get a diameter of a circle of half the area?
I would be very grateful if you can help me with this problem.
I am trying to explain in the simplest terms possible the sequence of f-stops in photography.
The common f-stop rounded sequence is:
f/1 f/1.4 f/2 f/2.8 f/4 f/5... |
H: Looking for a function that is continuous but not sequentially weakly continuous
Let $(X, \|\cdot\|) $ be a Banach space.
A function $g:X \longrightarrow X$ is said to be sequentially weakly continuous if for every sequence $(x_n)$ in $X$ such that $x_n \rightharpoonup x$, we have $g(x_n) \rightharpoonup g(x)$.
Wha... |
H: How to investigate the surface integral $\iint_Sf(x,y,z)\,dS$?
$$\iint_Sf(x,y,z)\,dS\,,$$ where $S$ is the part of graph $z=x^2+y^2$ below the plane $z=y$.
I am wondering what is the surface mean. I can not imagine it. If I use the polar coordinates, then what is the range of each variables?
AI: Intersection of $z=... |
H: Example / Counterexample of non constant analytic function
While trying assignments of complex analysis I am unable to solve this particular question.
Does there exists a non-constant bounded analytic function on $\mathbb{C} $/{0} ?
As function is not entire so lioville theorem can't be applied . So I think the... |
H: Find the generating set of $W=\{p \in \mathbb {P}_{3}(\mathbb{R}) \mid p(2)=0\}$
so i got stuck in this question. The purpose is to find a generating set for:
$$W=\{p \in \mathbb {P}_{3}(\mathbb{R}) \mid p(2)=0\}$$
Where $ \mathbb {P}_{3}(\mathbb{R})$ is the vector space of the polynomials of degree three.
My idea... |
H: Variation of nested interval theorem
This is the question I'm trying to solve:
Suppose that $(u_n)^{\infty}_{n=1}$ and $(v_n)^{\infty}_{n=1}$ are two
sequences of numbers such that $u_1 < u_2 < u_3 < ...$ and $v_1 > v_2
> v_3 > ...$ Suppose also that for every $n$, $u_n < v_n$, and $\lim_{n \to \infty} (v_n - u_n)... |
H: How to differentiate the trace of a matrix times its diagonal
Let $\mathbf{\Theta}\in\mathbb{R}^{p\times p}$ be a matrix and denote $\mbox{diag}(\mathbf{\Theta})\in\mathbb{R}^{p\times p}$ the matrix that has the same diagonal as $\mathbf{\Theta}$ and every off-diagonal element zero. I am trying to calculate
$$\frac... |
H: Implicit curve/surface definition of a polynomial function that's rotated and translated
Supposing I have an $n^{th}$-order polynomial curve $$y = \sum_{i=0}^n c_ix^i$$ and an $n^{th}$-order polynomial surface $$z = \sum_{i,j\in\mathbb{Z}^+\!,\ i+j=n} c_{ij}x^iy^j.$$ Now suppose that in each case, I want to transfo... |
H: Is there a component wise characterisation of open maps just like in continuous maps?
Suppose $\mathscr{A}$ is any indexing set, and let $\prod_{\alpha\in\mathscr{A}} Y_{\alpha}$ be the product space of non-empty topological spaces $Y_{\alpha}$.
The question is this: let $X$ be any topological space. There is a ver... |
H: $\iint_{\mathbb{R}^2} \frac{1}{\sqrt{1+x^4+y^4}}$ converges or diverges?
$$\iint_{\mathbb{R}^2} \frac{1}{\sqrt{1+x^4+y^4}}$$ converges or diverges?
I've tried to change to polar coordinates but i got stuck really quick
$$\int_{0}^{2\pi}\int_{0}^{\infty}\frac{r}{\sqrt{1+r^4(1-2\sin^2(t)\cos^2(t))}}drdt$$
any hint pl... |
H: How could I solve this IVP?
$y'+y\cdot \ln^2(x)=y^2\cdot \ln^2(x)$
I tried transforming it to $y'+P(x)y=Q(x)$ but I'm not sure how
AI: Hint: $$y’=(\ln x)^2 (y^2-y)=(\ln x)^2\bigg(\left(y-\frac 12\right)^2 -\frac 14 \bigg)$$
Just separate the variables now. |
H: Solving $\int_{-\infty}^{\infty} \frac{x\sin(3x)}{x^4+1}\,dx $ without complex integration
I am looking for a way to solve :
$$\int_{-\infty}^{\infty} \frac{x\sin(3x)}{x^4+1}\,dx $$
without making use of complex integration.
What I tried was making use of integration by parts, but that didn't reach any conclusive r... |
H: Can a graph be non-planar in 3d?
I am currently reading Trudeau's introductory book on Graph Theory and have just come across the concept of planar and non-planar graphs. The definition reads: 'A graph is planar if it is isomorphic to a graph that has been drawn in a plane without edge-crossings'. My question is, i... |
H: Suppose $F(x):=\begin{cases} f(x)& x\in I\\0& x\not\in I\end{cases}$ then $F$ is piecewise constant on $J$
Suppose $f:I\to \mathbb{R}$ a piecewise constant function on the bounded interval $I$, $I\subseteq J$ a bounded interval, $F:J\to \mathbb{R}$ $F(x):=\begin{cases} f(x)& x\in I\\0& x\not\in I\end{cases}$ then ... |
H: Is $\frac{1}{n^2}\sum_{j=n}^{\infty} \frac{1}{j}$ bounded?
Repeating the title: is $\frac{1}{n^2}\sum_{j=n}^{\infty} \frac{1}{j}$ bounded?
Note that the sum starts from $n$.
(I am guessing it must not be, otherwise I will have a puzzle on a theorem I am trying to use. Nevertheless, I would like to understand this. ... |
H: Given a differentiable function $f$, prove that $\lim_{h\to 0}\frac{f\left(x + \frac{h}{2}\right)-f\left(x - \frac{h}{2}\right)}{h} = f'(x)$.
I'm trying to prove the following statement
Given a differentiable function $f:\mathbb{R}\to \mathbb{R}$, prove that $$\lim_{h \to 0} \frac{f\left(x + \frac{h}{2}\right)-f\l... |
H: Complex integral for $\frac{1}{z-z_0}$ on $γ_R=Re^{it}$, $ t∈[π,2π]$
Show that the complex integral of $\frac{1}{z-z_0}$ on $γ_R=Re^{it}$, $ t∈[π,2π]$ with $R>0$, $Im(z_0)<0$ and $z_0∈C$ is equal to $π i$ for ${R\rightarrow \infty}$
My attempt: I was able to show that $π$ is greater than the integral by using the ... |
H: Integral $\int_0^{\infty} \arctan{\left(\frac{n}{\cosh{(x)}}\right)} \mathop{dx}$
I want to evaluate the integral
$$\int_0^{\infty} \arctan{\left(\frac{n}{\cosh{(x)}}\right)} \mathop{dx}$$
I think the integral evaluates to $$\frac{\pi}{2} \ln{\left(\sqrt{n^2+1}+n\right)}$$
but I dont know how really! I think $n$ i... |
H: Upper bound of $\sum_{k=1}^n \frac{1}{\sqrt{k}}$?
I am looking for an upper bound of $\sum_{k=1}^n \frac{1}{\sqrt{k}}$. Alternatively, is the sequence $\frac{1}{n\sqrt{n}}\sum_{k=1}^n \frac{1}{\sqrt{k}}$ bounded?
I am trying to use a Strong law of Large Numbers by Feller and need to show this condition.
AI: Via a c... |
H: Integrate $\int_0^1 \frac{x^2-1}{\left(x^4+x^3+x^2+x+1\right)\ln{x}} \mathop{dx}$
Insane integral $$\int_0^1 \frac{x^2-1}{\left(x^4+x^3+x^2+x+1\right)\ln x} \mathop{dx}$$
I know $x^4+x^3+x^2+x+1=\frac{x^5-1}{x-1}$ but does it help? I think $u=\ln{x}$ might be necessary some point.
AI: (edit for a little glitch in ... |
H: Integral with delta function
In an exercise, I found after some time the following integral:
$$\int_{-\infty}^\infty \mathrm{d}x\,\mathrm{d}y\,f(x,y)\int_{-1}^1 \mathrm{d}z\,\delta\left(z-\frac{2-2x-2y+xy}{xy}\right)$$
In the end I should get an integral like
$$\int_{0\leq x,y\leq 1, \, x+y\geq1}\mathrm{d}x\,\mathr... |
H: Compute the characteristic function of $Z_n=\sum\limits_{k=1}^{\xi_n}X_k$
Let $\{X_k\}_{k\ge1}$ be an i.i.d. sequence and let $\{\xi_n\}_{n\ge1}$ be a sequence of Poisson random variables with $E\xi_n=n\,\,(n=1,2,...)$. Assume independence between $\{X_k\}_{k\ge1}$ and$\{\xi_n\}_{n\ge1}$. Compute the characteristi... |
H: Enumerating open sets around elements of an uncountable set in topology - how do we justify it?
I was trying to prove for myself a basic result in point-set toplogy, namely that "Any compact subset of a Hausdorff space is also closed".
To be precise we're taking compact set here to mean - any open cover has a finit... |
H: Integral of $\sqrt{1-\|x\|^2}$
I am trying to calculate the next integral:
$$\int_{Q}\sqrt{1-\|x\|^2}dx$$
where $Q =\{x\in\mathbb{R}^n: \|x\|\leq 1\}$ and $\|x\|$ is the usual norm of $\mathbb{R}^n.$
For the cases $n = 2$ and $n = 3$ polar and spheric coordinates are useful, however, is there an easier form to comp... |
H: Sure vs almost sure convergence for a simple random variable
I thought I totally got it until I faced a simple problem and realized I'm getting a contradiction. For a sequence of independent simple rv defined on Lebesgue measure $(\Omega, \mathcal{F}, \mu)$ on $[0,1]$:
$$
X_n (\omega) = \bigg\{
\begin{array}{;r}
1 ... |
H: Prove of $\prod_{d|n} (\mu(d)(\mu(d) + 3) + 4) = 4^{d(n)}$
Found an interesting relation:
$$\prod_{d|n} (\mu(d)(\mu(d) + 3) + 4) = 4^{d(n)}$$
where $\mu(n)$ is a Möbius function and $d(n)$ is a divisors count.
I think this should be something known. The prove I know is a bit tricky and can be found in http://oeis.o... |
H: If every continuous function on a set can be extended to a continuous function on $\mathbb{R}$ then the set is closed.
Suppose $F\subseteq \mathbb{R}$ is a set such that every continuous function $f: F\rightarrow\mathbb{R}$ can be extended to a continuous function $g_f:\mathbb{R}\rightarrow\mathbb{R}$. I want to p... |
H: Evaluate $\lim_{x \to 1} \frac{\sin(3x^2-5x+2)}{x^2+x-2}$
Evaluate$$\lim_{x \to 1} \frac{\sin(3x^2-5x+2)}{x^2+x-2}$$
$(x-1)$ is a common factor for both polynomials, but I do not know how this helps because $x \to 1$ so I cannot use the fundamental trigonometric limit (after multiplying both the numerator and denom... |
H: There is more than one correct answer for function graph.
I saw question like this in Calculus book.
The graph of the derivative of a function is given. Sketch the graphs
of two functions that have the given derivative. (There is more than
one correct answer.)
And I think myself how can I explain exactly why ther... |
H: Is it true that $p \in \operatorname{Iso}(X)$ iff $\{p\}$ is an open set?
I have the following definition and statement in my lecture notes
Definition of isolated point:
A point $p \in E $ is called an isolated point of $E$ if there exists $U \in \mathfrak{U}_p$ ie a neighborhood of the point p, such that $U \cap E... |
H: Definition topological manifold
In the book "An introduction to manifolds" by Tu, a topological manifold is defined to be a topological space $M$ that is Hausdorff, second countable and locally Euclidean.
Does this allow things like the disjoint union of a plane and a line? Then we have a component which is locally... |
H: Evaluating $\lim_{x \to 0} \frac{\sin^3(x)\sin(\frac{1}{x})}{x^2}$
Evaluate $$\lim_{x \to 0} \frac{\sin^3(x)\sin(\frac{1}{x})}{x^2}$$
My attempt: $$\lim_{x \to 0} \frac{\sin^3(x)\sin(\frac{1}{x})}{x^2}=\lim_{x \to 0} \frac{\sin^3(x)\sin(\frac{1}{x})}{x^3}\cdot x$$
$$=\lim_{x \to 0} \sin^3(x)\cdot \frac{\sin(\frac{1... |
H: Prove that $|C|=243$ given $C=\{(A,B):A,B\subseteq S,\; A\cup B=S\}$ and $S=\{1,2,3,4,5\}$.
Problem:
Let $\displaystyle S=\{1,2,3,4,5\}$ and $\displaystyle C=\{(A,B):A,B\subseteq S,\; A\cup B=S\}$. Show that $\displaystyle |C|=243$.
I don't really know how to solve this. I know that there are $\displaystyle\sum_{i... |
H: $f,g \in k[t]$ satisfying several conditions
Let $f=f(t), g=g(t) \in k[t]$, $k$ is a field of characteristic zero.
Assume that:
(i) $k(f,g)=k(t)$.
(ii) $k(f',g')=k(t)$.
(iii) $\langle f'',g'' \rangle = k[t]$.
Is it true that (iv) $k[f',g']=k[t]$? I guess that there exists a counterexample..
For now I have checked... |
H: Common refinement with different intervals
Let $[a,b]$ be an interval. A partition of $[a,b]$ is $N+1$ points such that:
$$P:= \{x_1= a, x_2, \ldots,x_{N+1}= b \}.$$
Moreover, for an interval $[a,b]$ we can take two partitions $P^1, P^2$, and a common refinement is called when we take the union of points in $P^1, P... |
H: Second countability is invariant under orbit space of an action
I need to prove that if $X$ is a second countable space and $f:G\times X\to X$ is a left action on $X$, then its orbit space $X/G$ is also second countable. Here is my idea:
Let $\mathscr{B}$ be a countable basis for $X$, for each $B\in \mathscr{B}$ le... |
H: Why are these operations allowed when proving linear independence?
If $V, W$ are finite-dimensional vector spaces with ordered bases $\beta = \{v_1, \ldots, v_n\}$ and $\gamma = \{w_1, \ldots, w_m\}$ and our linear transformations are defined by $T_{ij}: V \to W$ where
$$T_{ij}(v_k) = \begin{cases}w_i & \text{if } ... |
H: Is $\operatorname{Iso}(E)= E \setminus\operatorname{Der}(E)$ or $\operatorname{Iso}(E)=\operatorname{Cl}(E) \setminus\operatorname{Der}(E)$?
In my lecture notes I have the following property:
$\operatorname{Iso}(E)= E \setminus\operatorname{Der}(E)$ ...(1)
but then I saw in other section, they used
$\operatorname{I... |
H: connected bipartite graph exists
Does a connected bipartite graph $G=(X \cup Y; E)$ such that $|X|=4$, $|Y|=3$, $|E|=5$ exist? Is there a way to know? Thanks!
AI: An undirected graph with $n$ nodes needs at least $n-1$ edges to be connected. Any additional structural constraints (e.g., bipartite) will not decreas... |
H: Lottery Combinations - Sum of All Numbers
In the 6/49 lottery game, there are 13,983,816 total combinations. My question is, how many combinations are there of a particular sum when adding all 6 of the 49 numbers together. For example:
6, 16, 22, 29, 36, 43 = 152 when adding all 6 numbers together.
5, 17, 22, 29, 3... |
H: Well-founded trees of any order
Suppose $T$ is a well-founded tree on $\mathbb{N}$, that is, a set of finite sequences of $\mathbb{N}$ closed under taking initial segments. Well-founded means that there is no infinite sequence $(x_n)$ such that for all $k$, $(x_1, x_2, \dots,x_k)\in T$. Put $T_0:=T$ and for any suc... |
H: If $T:\mathbb{C}\to \mathbb{C}^2$ a linear transformation?
If $T:\mathbb{C}\to \mathbb{C}^2$ is the function given as $T\begin{pmatrix}x+\imath y\end{pmatrix} = \begin{pmatrix}x-\imath y\\x+\imath y\end{pmatrix}$ where $\vec{v}_1,\vec{v}_2\in \mathbb{C}$ and $\lambda \in \mathbb{C}$
is a linear transfomation?
I th... |
H: Basis for Product Topology ..
Problem- Prove that ${P}$ is a basis for $\prod X_\alpha$, where $P$ is the product basis:
$$P=\Big\{\prod\limits_{\alpha \in I} U_\alpha\Big\}$$
Where each $U_\alpha$ is open set in the space $(X_\alpha)_{\tau_\alpha}$ and $U_\alpha=X_\alpha$ for all but finitely many $\alpha \in I$.
... |
H: Show that a set is closed, bounded and not compact in $\mathbb{R}^\infty$.
Let $e_i=(0,\dots,0,1,0,\dots,0,\dots)$, where 1 appears in the $i$th place. Let $X$ be the set of all the points $e_i$. Show that $X$ is closed, bounded and non-compact.
It is bounded because for any $x\in X$, $X\subseteq B(x,1)$ and it is ... |
H: Word Problem: Roger bought some pencils and erasers at the stationery store
Roger bought some pencils and erasers at the stationery store. If he
bought more pencils than erasers, and the total number of the pencils
and erasers he bought is between 12 and 20 (inclusive), which of the
following statements must be tr... |
H: Instability in Calculating Mahalanobis Distance
I am trying to calculate Mahalanobis distance from a point to a cluster of points. The code below does that.
import tensorflow.keras.backend as K
import pandas as pd
import scipy as sp
import numpy as np
def mahalanobis(x=None, data=None, cov=None):
x_minus_mu = ... |
H: Inverse of "diagonal block" matrix
Let
$$A = \begin{bmatrix} A_{11} & \cdots & A_{1m} \\ \vdots & \ddots & \vdots \\ A_{m1} & \cdots & A_{mm} \end{bmatrix}$$ be a block matrix where each matrix $A_{ij} \in \mathbb{R}^{n\times n}$ is diagonal. What is $A^{-1}$?
It seems that it's possible to iteratively apply the u... |
H: In what sense is $\Pi x: A.B$ the same as $B[x := a_1] \times B[x := a_2]$ when A is a finite type with two elements $a_1$ and $a_2$
This is in the context of the Type Theory system $\lambda P$ as presented in Chapter 5 of "Type Theory and Formal Proof: An Introduction" by Rob Nederpelt and Herman Guevers.
Since I ... |
H: Fundamental set in the space of bounded sequences
Definition: Set $S$ fundamental set in Banach space $X$ if $\overline{Lin(S)}=X$.
If $e_n=(0,\ldots ,0,1,0,\ldots)$ is a sequence that has $0$ everywhere, except on the $n$-th place and $e=(1,1,1,\ldots)$ is a constant sequence, then the set $S=\{e_n|n\in \mathbb{N}... |
H: Ramification in a splitting field
This is part of an exercise I'm doing for self study. Here, $K = \mathbb Q(\alpha)=\mathbb Q[X]/(X^5-X+1)$, and $L$ is the splitting field.
"Using the fact that any extension of local fields has a unique maximal unramified subextension, prove that for any monic irreducible polynomi... |
H: Problem with General Contour integral
I am trying to calculate a contour integral $$\oint_{\Gamma}\frac{z^{\alpha}e^z}{z-b}\,dz$$ where $\Gamma$ is the counterclockwise path from $\sigma-i\infty$ to $\sigma+i\infty$ that loops back around and closes on a semicircle toward the left. In this case, $\sigma$ is an arbi... |
H: Group / field extension solvability in the case of $x^3 - 2$
Whenever a polynomial is solvable by radicals, the Galois group of its splitting field must be a solvable group.
A group $G$ is solvable if there are subgroups $H_0, H_1, \dots , H_n$ such
that
$$\ 1 = H_0 ≤ H_1 ≤ H_2 ≤ \cdots ≤ H_n = G$$
with the propert... |
H: Equivalence of optimization problems involving trace and Frobenius norm of PSD matrices
An optimization problem involving symmetric PSD matrices $A,B,C \in \Re^{n \times n}$ is
$\min\limits_{A,B,C}\ Tr(AB) + ||A-C||^2_{F}$ , s.t. $A \succeq 0$.
An equivalent optimization problem holding matrices $B$ and $C$ constan... |
H: $x^{5x}=y^y$, $x, y \in \mathbb{Z}^+$, find largest value of $x$.
Let $x$ and $y$ be positive integers satisfying $x^{5x} = y^y$. What is the largest possible value for $x$?
I'm stuck on this question in an Olympiad past paper. Anyone have any ideas about this one?
AI: First, we have:
$$x^{5x}=y^y<y^{5y} \implies x... |
H: Find value of $\dfrac{(1+\tan^2\frac{5\pi}{12})({1-\tan^2\frac{11\pi}{12}})}{\tan\frac{\pi}{12}\tan\frac{17\pi}{12}}$
My attempt :
$$\dfrac{\left(1+\tan^2\dfrac{5\pi}{12}\right)\left(1-\tan^2\dfrac{\pi}{12}\right)}{\tan\dfrac{\pi}{12}\tan\dfrac{5\pi}{12}}$$
Change into variable form
$$\dfrac{(1+a^2)(1-b^2)}{ab}$$
$... |
H: Euler function theorem
I cant figure out the way to solve this question,
Let n be a positive integer and {d1,d2,...,dr} be the whole positive divisor of n. Show that enter image description here then holds.
For example, when n = 12,
φ(1) + φ(2) + φ(3) + φ(4) + φ(6) + φ(12) = 1 + 1 + 2 + 2 + 2 + 4 = 12
It certainl... |
H: Continuous injective map from real rational numbers to real irrational numbers
Does there exist any continuous one-to-one map from $\mathbb{Q}$ to $\mathbb{R}-\mathbb{Q}$?
If there does exist one injective map with the above condition, then $|\mathbb{Q}|\leq |\mathbb{R}-\mathbb{Q}|$. But, then we can't go on with t... |
H: PIDs are not Artinian?
In my notes there is the following statement:
Let $A$ be a PID, then $A$ as an $A$-module is trivially Noetherian but not Artinian. In fact, take a prime element $p$ in $A$, then we have the chain $$(p)\supset (p^2) \supset (p^3) \supset \dots$$
There are a few things that I don't understan... |
H: Calculating $ \lim_{x\rightarrow\infty}\frac{\delta(0)}{x} $
Does this limit go to $1$? I am not sure how to calculate it, because it contains the Dirac-delta generalized function on $ 0 $.
$$ \lim_{x\rightarrow\infty}\frac{\delta(0)}{x} $$
I came across this limit by trying to "evaluate" the following expression:
... |
H: Expected number of cards in original position in a shuffled deck of $52$ cards?
Assume is shuffle is quite good that it randomizes the card order.
We know that E = $ \sum_{X=1}^n X*P(X) $
We are already know that n=52 and that there are 52! ways to arrange the cards.
So probability that exactly 1 card is in correc... |
H: Integral of Error Function
Is there a way to approximate this integral with a constant expressed in terms of $\delta$
$$\int_{0}^{1} e^{-\left(\frac{x^2}{2\delta^2 }\right)} dx$$
Thanks
AI: Let $\frac{x}{\delta}=y$
Now your integral becomes
$$\delta \sqrt{2\pi}\int_{0}^{\frac{1}{\delta}}\frac{1}{\sqrt{2\pi}}e^{-\fr... |
H: How to solve integral with variable limits?
In one of the questions that I'm solving, I have got an integral like this
$$k(w) = \int_{w-1}^{w}f(x)dx$$
where the function $f(x)$ is defined in this way
$$f(x)=
\begin{cases}
x,& \text{if } 0\leq x \leq 1\\
2-x,& \text{if } 1\leq x \leq 2\\
0, ... |
H: What is the cardinality of $\{f:\mathbb{N}\to\mathbb{N}\ |\ \forall n f(n)\not = n\}$
I think the set $A= \{f:\mathbb{N}\to\mathbb{N}\ |\ \forall n f(n)\not = n\}$ has the size of the continuum. Well $A\subseteq \omega^{\omega}$ so $|A|\leq|\omega^{\omega}|=2^{\omega}$. But I couldn't prove the reverse inequality ... |
H: Inequality involving an improper integral
To prove: $\displaystyle \int_x^\infty \exp \left(-\frac{t^2}{2}\right) \, dt < \frac{1}{x}\exp\left(-\frac{x^2}{2}\right)$, $\quad x>0$
We know
$$\int_x^\infty t^{-2} \exp\left(-\frac{t^2}{2}\right) \, dt \leq \exp\left(-\frac{x^2}{2}\right)\int_x ^\infty t^{-2} \, dt = ... |
H: Dimension of the annihilator (Linear algebra done right, 3.106)
$V$ is finite-dim vector space, and $\text{U}$ its subspace, and $\text{V}'$ and $\text{U}'$ are their dual counterparts. $\text{U}^0 = \{ \phi \in \text{V}': \phi(u) = 0 $ for all u $ \in \text{U} \}$, then it's proved in Axler's 3.106 that:
$\text{di... |
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