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H: antiderivative of $\sum _{ n=0 }^{ \infty }{ (n+1){ x }^{ 2n+2 } } $
I've proven that the radius of convergence of $\sum _{ n=0 }^{ \infty }{ (n+1){ x }^{ 2n+2 } } $ is $R=1$, and that it doesn't converge at the edges.
Now, I was told that this is the derivative of a function $f(x)$, which holds $f(0)=0$.
My nex... |
H: why does the h in Torricelli's law (the form that relates height to time) go to zero rapidly?
https://class.coursera.org/calcsing-002/lecture/320
In the the above linked lecture at 5:44, we are trying to find how fast liquid leaks from a cone shaped tank. I understand the derivation but at the end it mentions that ... |
H: Number of cusps of an modular curve $X_0(N)$
Let $X_0(N) = \Gamma_0(N) / (\mathbb{H} \cup \mathbb{P}^1(\mathbb{Q}))$. A lecture note (p. 2) lists the following (very easy!) formula for the corresponding cusps:
$$\nu_\infty = \sum_{d\mid N} \varphi(\gcd(d,N/d)).$$
Unfortunately, there is no source or derivation give... |
H: Normal bundle geometry
I am wondering what a normal bundle looks like given this definition for Riemannian manifolds:
Say $M = S^2$. $T_pM$ is the tangent plane ($\mathbb R^2$) at $p \in S^2$. Lets denote the north pole by $0$ and let $S$ be a ball of some radius around $0$.
Naturally, I expect the normal vector ... |
H: Is the quotients of a group of triangular distributed numbers still following a triangular distribution?
I have a group of numbers (about 10000 numbers) between 0.8 and 1.0 which follows simple triangular distribution (for example, lower limit: 0.8, upper limit: 1.0, mode: 0.9).
If I divide 2 by each number from th... |
H: Is the number of irreducibles in any number field infinite?
Are there infinitely many irreducibles in the ring of integers of any algebraic number field ?
I tried to follow the same argument as we usually do for integers. Suppose there are finitely many irreducibles, say $p_1,\ldots ,p_n$ and let $\alpha :=1+ p_1\c... |
H: Are the number of terms in an infinite series even or odd?
This question arose after I saw a youtube-vid where Grandi's series was discussed.It seems that the sum of the series will be 0 for an even, and 1 for an odd number of terms, where a term is defined as (-1)n, n indicating the n'th term.It seems that even wh... |
H: Ambiguous notation for squared matrix
What does $A^{2}$ mean for square $A$? Is it $AA$ or $AA^{T}$? Sometimes, the result may differ.
Or there is no uniform approach?
AI: There's no ambiguity, $A^2=AA$, period. |
H: Indefinite integral of $(2x+9)e^x$
What is the indefinite integral $\displaystyle\int (2x+9)e^x\,\mathrm dx$?
Attempt:
Integration by parts seems obvious.
$u = 2x + 9, \mathrm du = 2$
$\mathrm dv = e^x, v = e^x$
$uv - \int v\,\mathrm du$
$(2x+9)e^x - \int 2e^x$
$(2x+9)e^x - 2e^x$
This is wrong but I don't see why... |
H: Convergence of a complex power series
Let $a,b,c \in \mathbb C$ with $c \in \mathbb N$. Then I have to calculate the radius of convergence of the following power series:
$$
1+ \frac{ab}{c \cdot 1!} z + \frac{a (a+1)b(b+1)}{c(c+1)2!} z^2+ \frac{a(a+1)(a+2)b(b+1)(b+2)}{c(c+1)(c+2)3!}z^3 + \cdots
$$
Using the ratio-te... |
H: Find volume on a shape with base of an ellipse
I have an ellipse with area $\pi ab$ $a = 6$, $b = 4$ these are the axis lengths. I am suppose to compute the volume of a cone of height 12.
I tried many solutions but none of them worked and I don't know why. I would type them up but I doubt much could be learned. Bas... |
H: Proving that $f(x) = \frac{x^2}{1+\sin^2(1/x)},f(0)=0$ is continuous at $0$
I'm trying to prove that
$$f\left(x\right)=\begin{cases}
\frac{x^{2}}{1+\sin^{2}\left(\frac{1}{x}\right)}, & x\neq0\\
0, & x=0
\end{cases}$$
is continuous at 0.
I know that I should show that
$$\lim_{x\rightarrow0}\frac{x^{2}}{1+\sin^{2}\... |
H: If H, N are normal subgroups of G, then do all the commutators lie in the intersection?
Okay, I know that this is elementary, but, ah, well.
How do I show that if N and H are normal subgroups of a finite group G with coprime orders, then,
$xyx^{-1}y^{-1} \in H\cap N$ for all $x \in H, y \in N$?
I figured that, if a... |
H: If $a$ and $b$ are the roots of $0=3x^2+4x+9$, $(1+a)(1+b)$ can be expressed in the form $\large \frac uv$?
$u$ and $v$ are co prime positive integers. What is the value of $u+v$? This seems like an easy problem, but I can't figure out what I'm doing wrong.
$f(x)=3x^2+4x+9$
$f(x)=(x-a)(x-b)$
$f(-x)=(x+a)(x+b)$
$f(... |
H: Why don't the roots of this characteristic equation correspond to the given solution of this 2nd order ODE?
I am asked to solve
$$ y'' + 9y = 6\mathrm{sin}(3x) $$
using the method of undetermined coefficients.
The characteristic equation of this 2nd order ODE is
$$ \lambda^2 + 9\lambda = 0 $$
and its roots are $0$ ... |
H: Understanding summation decreasing index
I'me following some summation examples and I came to this situation
$$|4-4| + \sum_{n=1}^{\infty} |4\cdot0.1^n| = -4+4\sum_{n=0}^{\infty} 0.1^n$$
How do they get to the last result?
I thought that $|-4+4|=0$ and decreasing the index should become $\sum_{n=0}^{\infty} |4\cdot... |
H: Limit as $x\to 0$ of $x\sin(1/x)$
How to find limit as $x \to 0$ of $x\sin(1/x)$?
For $x^2\sin(1/x)$, I know it's $0$ since by the Squeeze theorem, $-x^2 \le x^2\sin(1/x) \le x^2$, but for $x\sin(1/x)$, I run into some problems when applying Squeeze theorem.
AI: To use the Squeeze Theorem, we do know that $0\leq|... |
H: What is the relationship between the lengths of the binary and decimal representations of a number?
If a is 1024 bits, then how many digits will its decimal representation have?
AI: Bits are binary numerals. If you have an 8-bit numeral, say 10110001, that is exactly an 8-bit binary numeral. In fact "bit" is an a... |
H: Winding number of image curve
How many turns does $f(z) = z^{40} + 4$ make about the origin in the complex plane after one circuit of $|z| = 2$?
AI: First, note that your curve is homotopic to the image of $|z|=2$ under $g(z)=z^{40}$ in $\mathbb C\backslash 0$. (Draw a picture if you aren't convinced.) Because the ... |
H: Does $\Gamma$ intersect $SL(2, \mathbb{R})$ transversely at $I$?
Identify the space of all $2 \times 2$ real matrices with $\mathbb{R}^4$ so that the matrix
$\left( \begin{array}{cc}
a & b\\
c & d\end{array} \right)$
corresponds to $(a, b, c, d)$.
Let $\Gamma$ denote the hyperplane in $\mathbb{R}^
4$ with equation ... |
H: Possibility of a number line that has variable density.
In my real analysis class, I have been informed that if you have a continuous line say $[0,1]$, and you do a mapping say $A\to B$ such that any element in $A$ is equal to $B^2$, where $A$ is every element in $[0,1]$ you would ultimately get a line that is den... |
H: The index of nilpotency of a nilpotent matrix
Let $A$ a matrix in $\mathcal{M}_5(\mathbb C)$ such that $A^5=0$ and $\mathrm{rank}(A^2)=2$, how prove that $A$ is nilpotent with index of nilpotency $4$? Thanks in advance.
AI: In the followings I'll use only words, but you should visualize the matrices I mention.
Jo... |
H: First derivative of holomorphic function
I want to prove that $ |f'(z)| \le \frac{1}{1-|z|}$ where $f:B(0,1) \rightarrow B(0,1)$ is a holomorphic function. My idea was to use Cauchy's integral formula. The fact that $||f||\le 1$ might be helpful too. But I don't see how to get this $1-|z|$?
AI: It is rather easy to... |
H: Taylor or Maclaurin series for the factorial function?
I am new to Taylor/Maclaurin series and want to know if there is a series representation for the factorial function?
AI: If you mean the standard factorial function $!:\mathbb{N}\rightarrow\mathbb{N}$, a Taylor series cannot exist because $\mathbb{N}$ has no ac... |
H: If F is a finite field, then $F^*$ is cyclic and $F=\Bbb{Z}_p(\alpha)$ for some $\alpha$.
From Galois Theory (Rotman):
If F is a finite field, then $F^*$ [which is the multiplicative group] is cyclic and $F=\Bbb{Z}_p(\alpha)$ for some $\alpha$.
Proof If $|F|=q$, take $\alpha$ to be a primitive (q-1)st root of uni... |
H: How to max $f(D)$ over the space where matrix $D$ is diagonal?
I want to maximize some function $f(D).$
Obviously if there is no constraints, I can just form matrix $G$ by $G(D)_{ij} = \frac{\partial f(D)}{\partial D_{ij}}$ and solve $G(D) = 0$ for D.
However, if D is subject to constraints, for example D has {p,... |
H: Normal Distribution and a Discrete Amount
From what I understand about normal distribution is that you make a discrete number continuous by adding .5 which every way the question asks for.
What if you were to have a discrete number with a set amount you cannot have half of.
Would you keep it discrete?
Ex) say ther... |
H: Small question regarding subspaces of order topology
When looking at $Y:=\left(0,1\right)\cup\left\{ 2\right\}
$ with the subspace topology induced by the order topology on $\mathbb{R}$ one immediately sees that $2$ is an isolated point of $Y$. Contrary to that, according to another thread I saw here when you loo... |
H: Power set and set of all mappings
I'm working with the Terence Tao's Analysis book. And I have a question in the part of set theory.
As power set axiom, Tao use the set of all function: "If X, and Y be sets. Then there exist a set which consist of all the functions from X to Y."
Using that axiom and the replaceme... |
H: Bijective map preserving inner products is linear
The question comes from Kaplansky's book Linear Algebra and Geometry on page 96 exercise 2
Let $V$ be a non-singular inner product space of characteristic $\neq2$. Let $T$ be a one-to-one map of $V$ onto itself, sending $0$ to $0$ and satisfying $(x-y, x-y) = (Tx ... |
H: Laplace-Beltrami operator on sphere.
Suppose that we have solution of
$$\delta d f = g$$
on sphere. Where $\delta d$ is Laplace-de Rham operator for functions, $f,g$ are scalar functions and $g$ has support on north hemisphere and it is non-negative there.
Than by Stokes theorem we have(I think that I have signs ... |
H: A Covering Map $\mathbb{R}P^2\longrightarrow X$ is a homeomorphism
I came across the following problem: Any covering map $\mathbb{R}P^2\longrightarrow X$ is a homeomorphism. To solve the problem you can look at the composition of covering maps
$$
S^2\longrightarrow \mathbb{R}P^2\longrightarrow X
$$
and examine t... |
H: Check that $df_x(v) = (v,v).$
Here is a proof that I am totally different from my classmates'. So I am requesting for expert reference here. Thank you. :-)
Let $f: X \rightarrow X \times X$ be the mapping $f(x) = (x,x).$ Check that $df_x(v) = (v,v).$
$\begin{eqnarray*}
df_x(v) &=& \lim_{t \rightarrow 0} \frac{f(x... |
H: When a 0-1-matrix with exactly two 1’s on each column and on each row is non-degenerated? [1]
Let $A$ be an $n\times n$ matrix with entries in the set $\{0,1\}$ which has exactly two ones in each column and two ones in each row. Give necessary and sufficient conditions for the rank of $A$ to be $n$.
AI: Let $G$ be... |
H: Primes in binary
Let $$S_n(k)=\{1\leq m\leq n: m\ \mbox{has $k$ ones in its binary representation and $m$ is prime}\}\ \\ \forall \ n\geq 2^k-1,\ k\geq1.$$ Let $\pi(x)$ be the prime number function. Then what can be said about $$f(k)=\lim_{n\rightarrow \infty} \frac{|S_n(k)|}{\pi(n)}? $$
AI: $\pi(n)$ is roughly $\f... |
H: How do you solve this word problem?
The river is flowing from point A to B at a rate of 15 miles per hour. A boat moves on still water at 45 miles per hour. If it takes David 1 hour and 15 minutes to ride the boat on the river from A to B, how long does it take him to make the return trip from B to A?
Please explai... |
H: When does $(\lim f_n)'=\lim f'_n$?
After defining
$BC^1(\mathbb R,\mathbb R):=\{f: \mathbb R \to \mathbb R \mid f \in C^1, \ \ \lVert f \rVert_\infty + \lVert f' \rVert_\infty < \infty \}$
and proving that it's complete, our lecturer made the following comment:
From the completeness of $BC^1$ immediately follow... |
H: A simple example of Lindelöf space.
Somebody can to give me a simple example of Lindelöf space?
Note. Lindelöf space is a topological space in which every open cover has a countable subcover.
AI: The natural numbers with the discrete topology.
Given an open cover, $U_i$ let $U_n$ be some open set such that $n\in U_... |
H: Cardinality of tautologies for propositional logic
I'm wondering how many tautologies there are in propositional logic. I'm thinking that it must be at least countable, since ($P_{1} \wedge P_{2} \wedge \cdots P_{n}) \models P_{i}$ should be a tautology for any natural number $n$ and any $i \in \{1,2,\cdots,n\}$, w... |
H: Check the convergence of the series of matrices
$$ \sum_{k=1}^{\infty} ( 1/ k^2 ) A ^k $$ where
A =\begin{bmatrix}-1 & 1 \\ 0 & -1 \end{bmatrix}
AI: Hints.
Let $J=\pmatrix{0&1\\ 0&0}$. Then $A=-I+J$. Since $J^2=0$, in the binomial expansion of $A^k = (-I+J)^k$, only two terms remain.
The value of $\sum_{k=1}^... |
H: Reference for the Law of the Unconscious Statistician?
Does anyone know of a reference (a book or journal article) for the Law of the Unconscious Statistician?
AI: Robert Israel gives a reference here. It is Ross' book "Introduction to Probability Models", but it is only in Ed. 1-3. |
H: Is it possible to have some number sequences that have no formula to solve them?
I'm by no means advanced at mathematics, but I'm trying to figure out a formula to get the nth value of the following sequence: $1,4,10,20,35,56,84$.
I'm using 'difference' tables to try and come up with a formula and I'm currently at ... |
H: Show the given space is uncountable.
Let $X$ be a compact Hausdorff space without any isolated point. Show that $X$ is uncountable.
As $X$ is compact Hausdorff, it is normal. then for any two distinct points $x$ and $y$, we have a continuous map $f$ from $X$ to $[0,1]$ such that $f(x)=0$ and $f(y)=1.$
As $X$ is co... |
H: What is the inverse of the function $x \mapsto \frac{ax}{\sqrt{a^2 - |x|^2}}$?
Let $B_a$ be the open ball{$x: |x|^2<a$} in $\mathbb{R}^k, |x|^2 = \sum x_i^2$.
What is the inverse of the function $x \mapsto \frac{ax}{\sqrt{a^2 - |x|^2}}$?
Here I want some justification to equate $|x|^2$ with $x^2$, I don't know ... |
H: Induction and convergence of an inequality: $\frac{1\cdot3\cdot5\cdots(2n-1)}{2\cdot4\cdot6\cdots(2n)}\leq \frac{1}{\sqrt{2n+1}}$
Problem statement:
Prove that $\frac{1*3*5*...*(2n-1)}{2*4*6*...(2n)}\leq \frac{1}{\sqrt{2n+1}}$ and that there exists a limit when $n \to \infty $.
, $n\in \mathbb{N}$
My progress
LHS ... |
H: Number of connected sets on $\mathbb{R}^n$
Let $\mathcal{A}_n$ denote the family of all connected sets over $\mathbb{R}^n$. If $n=1$ we have that the cardinality of $\mathcal{A}_1$ is the cardinality of the continuum. But if $n>1$ I've only shown that $\mathcal{A}_2$ is at least the cardinality of the continuum. I... |
H: Proof Help: In a group $G$, there exists a $g$ such that $g^2 = e$
I'm working through an Abstract Algebra book to teach myself, and came across the problem:
Prove: If $G$ is a finite group of even order, then there exists a $g\in G$ such that $g^2 = e$ and $g \ne e$.
(In this book, $e$ is used as the identity el... |
H: Proving that a graph is NOT bipartite
For any $n,k\in \mathbb{N}$ where $n\geq k$ let $G_{n,k}$ be the graph where $V(G_{n,k})$ is the set of all $k$-subsets of $[n]$, and two subsets $S,T$ are adjacent iff $|S\cap T|=1$.
Prove that when $n\geq 3k-3 > 0$, $G_{n,k}$ is not bipartite.
This is homework, so I don't wan... |
H: Why is normal distribution more accurate than binomial distribution?
I'm having a tough time understanding this.
This is what I am told about comparing the two:
The probability that Saredo is late for school is 0.6.
What is the probability that in one month she is late 9 times?
Remember that one month would includ... |
H: Is it solvable? Six venn diagram problem. Very Complicated
It classifies 10000 people as
young or old
male or female
married or single
Of these 10000 people, 3000 are young, 4600 are male, 7000 are married
1320 = young and male, 3010 = married and male, 1400 = young and married
600 = young and married and male.
... |
H: Interval iff image is interval
Show that a nonempty set $E$ of real numbers is an interval if and only if every continuous real-valued function on $E$ has an interval as its image.
Suppose $E$ is an interval. For any $a,b\in E$ with $a<b$, we have that any $c\in(a,b)$ belongs to $E$. Let $f$ be a continuous real-... |
H: Does $S(x) = x+1$ always hold in Peano arithmetic?
In some books, they seem to implicitly say that $S(x) = x+1$ holds always in Peano arithmetic. But does it really hold in all cases, even in non-standard ones? The standard model of course satisfies this, but non-standard ones don't seem obvious.
Also, in Peano ari... |
H: Solve for the domain of x
Solve for the domain of x
$$y=\sqrt{\text{Cos}\left[x^2\right]}\tag1$$
my answer:where is wrong?
$\text{Cos}\left[x^2\right]\geq 0$, so $x^2\in [2k \pi -\pi /2,2k \pi +\pi /2]\Rightarrow x\in \left[\sqrt{2k \pi -\pi /2},\sqrt{2k \pi +\pi /2}\right]$
The answer in my book
a little strange... |
H: Is there a way to standardize the Poisson distribution?
For example, a variable of Normal distribution, $T$, with mean $\mu$ and variance $\sigma^2$ can be standardized into $S$ like this:
$$
S=\frac{T-\mu}{\sigma}\;\Longrightarrow\;F(x)=\Phi\left(\frac{x-\mu}{\sigma}\right)
$$
My question is, for the Poisson distr... |
H: Word problem (food for thought)
I thought of this question today as I was coming home from work in my car (probably because of my parents' anniversary).
This problem assumes the parents of everyone in the world got married and everyone in the future gets married at some point in time (a.k.a. it's hypothetical). If ... |
H: What is an effective means to get senior high school students to write their complete working out as part of their answer.
In Australia and in the International Baccalaureate (2 systems I have worked in), for better or worse, mathematics is assessed by criteria. This increases the importance of students to express... |
H: Given a matrix C, calculate $e^C e^{C^T}$ and $e^{-C}$.
Given:
$$\mathcal C=\pmatrix{-5&3&-3\\2&-4&1\\2&-2&-1}$$
Calculate: $e^C e^{C^T}$
Calculate: $e^{-C}$
I am using the formula $\mathcal f(C)=\mathcal S f(\lambda)\mathcal S^{-1}$, therefore I used Matlab to calculate $det(\lambda I-\mathcal C)=0$ to get my eige... |
H: Vector spaces and direct sums.
What is the relationship between $S \oplus T$ and $T \oplus S$? Is the direct sum operation commutative? Formulate and prove a similar statement concerning associativity. Is there an "identitiy" for direct sum? What about "negatives"?
$S \oplus T = T \oplus S$ because elements of $S... |
H: find the power representation of $x^2 \arctan(x^3)$
Wondering what im doing wrong in this problem im ask to find the power series representation of
$x^2 \arctan(x^3)$
now i know that arctan's power series representation is this
$$\sum_{n=0}^\infty (-1)^n \frac{x^{2n+1}}{2n+1} $$
i could have sworn that for solvin... |
H: Filling in 'x' in a log function
if $3^5=x$ (exponential equation) converts to log form gives $log_3x=5$
that makes sense.
$$
3^5 = 243 \Rightarrow x=243
$$
So if I take the log form again: $log_3x=5$ and replace $x$ with $243$. I then take the log of $243$, expecting to get $5$?? But instead, I get $2.3856$??
Can ... |
H: Is it true that $cl_{\beta X} Z(f) = Z(\hat f)$?
Suppose $f\in C^* (X)$ and let $\hat f$ be its continuous extension to $\beta X$. It is clear that
$cl_{\beta X} Z(f)=cl_{\beta X} f^{-1}(${$0$}$)\subseteq \hat f ^{-1}(${$0$}$)= Z(\hat f)$, but I am having some difficulty showing the reverse inclusion.
AI: It is no... |
H: Proof of a Property of Vertical Asymptotes
I'm trying to understand a proof in my Calculus textbook of the following theorem:
Let the functions $f$ and $g$ be continuous on an interval containing
$c$. If $f(c) \neq 0$, $g(c) = 0$, and there is an open interval
containing $c$ such that $g(x) \neq 0$ for all $x ... |
H: Slightly confused about the definition of upper limits and lower limits.
I'm reading "The way of Analysis" by Strichartz, and the following is the definition of an upperlimit.
The upper limit (limsup) of a sequence $\{x_j\}$ is the extended real number
$$\limsup _{k \to \infty} x_k = \lim_{k \to \infty }\sup_{j... |
H: recurrence relation question
How can I build a recurrence equation if there isn't an $n$-variable? Example: $a_n = 3$. Also, how would I start making a recurrence equation for $a_n = 2n + 3$?
AI: For the first one, you could use $a_0=3$ and $a_n=a_{n-1}$. For the second, put $a_0=3$ and $a_n=2+a_{n-1}$.
Ultimatel... |
H: Proving $(xyz)' = x'+y'+z'$
I'm trying to prove that $(xyz)' = x'+y'+z'$ using theorems/axioms.
I tried this but I just want to make sure if its the correct route or if I've done something "illegal"/wrong.
(xyz)' = [(xy)z]' by associativity
= [(x*y)'+z'] by DeMorgan's Law
= [(x'+y') + z'] by DeMorgan... |
H: question about Graph Theory notation
I'm just starting to learn graph theory. I have two questions about notation:
1). For a graph $G$ we denote the vertex set $V$ and the edge set $E$ by $G=(V,E)$.
So we have a graph $G=$ ({$v_{1},v_{2},...,v_n$}, {$e_{1}e_{2},...e_{n}$}).
My textbook presents the edge set as $E... |
H: Conversion between numeral systems.
I would like to know if there is a "standard" for converting a number of base N to a number of base N. For example, 117 decimal to 728 of "213 base". Any random base to any random base, not only decimal to binary, etc. I would like to know if there is some obscure law/rule/trick ... |
H: how come this summation after produkt?
I am really stuck in this step. I hope, the context does not matter here, so i didnot provide what this is about. I am trying to get ML-Estimator. but the problem is, as i see in my textbook, how they changed the produkt to summation, why it became $\sum$
$$L(a;X)=\prod_{i=1}^... |
H: Is every Tichonov space necessarily homeomorphic to a subset of a compact Hausdorff space?
My textbook says "A Tichonov space is homeomorphic to a subset of a compact Hausdorff space."
Doesn't the subset also have to be compact Hausdorff?
Motivation:- For the subset of the compact Hausdorff space to be homeomorphi... |
H: Confusing about sequence
when define a function, for example
$f(n)=1^2+2^2+\text{...}+(n-1)^2$
then what is $f(2n)$ ?
simply substitute $2n-1$ for $n-1$?
or $f(2n)=2^2+4^2+\text{...}(2n-2)^2$
or other?
What's the relation between
$1^2+2^2+\text{...}(n-1)^2$,
$1^2+2^2+\text{...}(2n)^2$,
$1^2+3^2+\text{...}(2n-1)^2$... |
H: How can I show that the conditional expectation $E(X\mid X)=X$?
I tried to show that $E(X\mid X=x)=x$, which would lead me to get $E(X\mid X)=X$ but I am having trouble doing so. I know that the definition of conditional expectation (continuous case) is: $$E(X\mid Y=y)=\int_{-\infty}^{\infty}x f_{X\mid Y}(x\mid y)\... |
H: Is $R \setminus P$ a multiplicative subset?
Let $S$ be a subset of the ring $R$; we say that $S$ is multiplicative if
(a) $0 \notin S$,
(b) $1 \in S$, and
(c) whenever $a,b\in S$, we have $ab \in S$.
We can merge (b) and (c) by stating that $S$ is closed under multiplication, if we regard $1$ as the empty pr... |
H: Prove that $(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{d})^2\ge \frac{1}{a^2}+\frac{4}{a^2+b^2}+\frac{12}{a^2+b^2+c^2}+\frac{18}{a^2+b^2+c^2+d^2}$
Let $a,b,c,d$ be positive numbers. Show that
$$\left(\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}+\dfrac{1}{d}\right)^2\ge \dfrac{1}{a^2}+\dfrac{4}{a^2+b^2}+\dfrac{12}{a... |
H: Can the ceiling function be used to prove the Archimedean property?
Recall the following definition of the Archimedean property:
For each $x \in \Bbb{R}$, there exists some $n \in \Bbb{N}$ such that $n>x$.
My textbook proves this by invoking the completeness axiom. My first instinct however was to use the ceiling... |
H: The final number after $999$ operations.
I wanted to know, let the numbers $1,\frac12,\frac13,\dots,\frac1{1000}$ be written on a blackboard. One may delete two arbitrary numbers $a$ and $b$ and write $a+b+ab$ instead. After $999$ such operations only one number is left. What is this final number.
I tried, let $*$ ... |
H: What's definition $(\ker \varphi)_P$ the stalk of kernel presheaf
If $\varphi : \mathcal{F} \to \mathcal{G}$ a morphism of sheaves then what definition of $(\ker \varphi)_P$? I digested that an element is $\langle U,r \rangle $ where $r \in \mathcal{F}(U)$ and $\varphi(U)(r) = 0$ and pairs are identified as in usua... |
H: Is $L^2(0,T;H_n)$ compactly embedded in $L^2(0,T;H)$?
Let $H$ be a separable Hilbert space with basis $h_i.$ Let
$$H_n := \text{span}\{h_1,...,h_n\}.$$
Questions:
1) Is $L^2(0,T;H_n)$ compactly embedded in $L^2(0,T;H)$?
2) Is $L^2(0,T;H_n^*)$ compactly embedded in $L^2(0,T;H^*)$?
I have no idea how to even begin. ... |
H: Image of a normal Hall Subgroup under an automorphism
Let $G$ be a group such that $|G|= n = md$, where $\gcd(m,d)=1$.
Let $N$ be a normal subgroup of $G$ with order $m$.
Further, let us define a subgroup $H$ of order $d$. I managed to prove that, $H \cap N= [e] \implies HN = G$, and $HN \cong H \times N$.
Now, l... |
H: Convergence of $\sum_{n=0}^\infty z^{2^n}$
Let formally $f(z) := \sum_{n=0}^\infty z^{2^n}$. What is the raduis of convergence of this series ?
AI: There is no need to think of how to apply any special test or theorem to this. Just think about how the terms grow, and how it compares to the most basic series you've ... |
H: Evaluating $\int{{x^2 -1}\over{x^3 \sqrt{2x^4-2x^2+1}}} \mathrm dx$
How to evaluate:
$$\int{{x^2 -1}\over{x^3 \sqrt{2x^4-2x^2+1}}} \mathrm dx$$
AI: HINT:
First put $x^2=y$ in
$$\int{{x^2 -1}\over{x^3 \sqrt{2x^4-2x^2+1}}} \mathrm dx=\int{{x^2 -1}\over{2x^4 \sqrt{2x^4-2x^2+1}}} \mathrm 2xdx$$
to get $$\int \frac{y-1}... |
H: The General Validity of $ab=cd$ Implies $ba=dc$.
I am doing some research of my own and I have a brief question and I don't recall studying this particular thing. Is it generally true that if a set coupled with a binary operation is closed under inverses and has an identity then $ab=cd$ implies $ba=dc$ for all $a,... |
H: Is there a way to prove a boolean operator isn't universal?
In boolean algebra, I could prove an operator is universal by implementing a NAND or NOR gate with it. But is there a way to prove a boolean operator isn't universal?
I would like to know a general method that should work for every (or almost every) incomp... |
H: How to define a incomplete metric on $\mathbb{S}^1$?
Let $\mathbb{S}^1=\{x\in\mathbb{R}^2:\ \|x\|_2=1\}$.
My question is: Is it possible to define a incomplete metric on $\mathbb{S}^1$, i.e. a metric such that $\mathbb{S}^1$ is not complete.
Thank you.
AI: Let $f \colon (0,1) \to \mathbb S^1$ be a bijection. Define... |
H: Confusion about Spec of quotient ring
Consider the ring $A := \dfrac{\mathbb C[x]}{(x(x-1)(x-2))}$. According to some sources (cf. Vakil) $sp(Spec (A))$ should be just the three points $\{0,1,2\}.$ It seems right, because $A$ is the ring of regular functions on these three points. However, I have the impression tha... |
H: Product of two compact spaces is compact
I read the proof that uses tube lemma and I do not have any problem with it but I cannot see what is wrong with the proof that first came to my mind:
Let $X$ and $Y$ be compact spaces. Let $\mathcal{A}$ be an open covering of $X\times Y$. Then, $\bigcup_{U\in \mathcal{A}}{U... |
H: Questions about the composition of two dominant rational maps.
I am reading the lecture notes. On page 4, I have some difficulty in understanding the proof of the fact that $g \circ f$ exists for two dominant rational maps $f, g$.
Let $f: V \to W$ and $g: W \to U$ be dominant rational maps. It is said that "if we... |
H: How to teach the division algorithm?
What is the best way to introduce the division algorithm? Are there real life examples of an application of this algorithm. At present I state and prove the division algorithm and then do some numerical examples but most of the students find this approach pretty dry and boring. ... |
H: Qustions about the maps $\mathcal{O}(Y) \to \mathcal{O}_P \to K(Y)$.
I am reading the book Algebraic Geometry by Robin Hartshorne. I am trying to understand the maps $\mathcal{O}(Y) \to \mathcal{O}_P \to K(Y)$ on Page 16.
Are there some functions in $K(Y)$ but not in $\mathcal{O}_P$? Are there some functions in $... |
H: Meromorphic function on Riemann surface
I've got this exercise I can't solve. May someone help me? Thank you.
Let $X, Y, Z$ be homogeneous coordinates on complex projective plan and let $C=\{[X:Y:Z] |X^{4}+XY^{3}+Z^{4}=0\}$. Consider the meromorphic function $f=\frac{X}{Y}$ defined on $C$.
1) Calculate zeroes and p... |
H: If $a,b$ are roots for $x^2+3x+1=0$.Calculating $(\frac{a}{b+1})^2 +(\frac{b}{a+1})^2$
If $a,b$ are roots for the equation $x^2+3x+1=0$.How to calculate $$\left(\frac{a}{b+1}\right)^2 +\left(\frac{b}{a+1}\right)^2$$
AI: Because $x^2+3x+1=0$, we have $x^2=-3x-1$ and also $x^2+2x+1=-x$, for $x=a,b$. Hence $$\left(\fr... |
H: convergence of a series $a_1 + a_1 a_2 + a_1 a_2 a_3 +\cdots$
Suppose all $a_n$ are real numbers and $\lim_{n\to\infty} a_n$ exists.
What is the condition for the convergence( or divergence ) of the series
$$ a_1 + a_1 a_2 + a_1 a_2 a_3 +\cdots $$
I can prove that $ \lim_{n\to\infty} |a_n| < 1 $ ( or > 1 ) guarante... |
H: Continuity proof via one-sided derivatives
$f\left(x\right)$ is defined on $[a,b]$, differentiable on $\left(a,b\right)$ and has one-sided derivatives at points a and b. How to prove that $f\left(x\right)$ is continuous on $[a,b]$?
The function is continuous on $\left(a,b\right)$ due to necessary condition of di... |
H: finding value of formula.
I am little bit confusing how to calculate $δβ/δρ$ value if I have set of values like this.
I have the values of $β$ and $ρ$ like this.
$$ \begin{array}{l|l}
β & ρ\\ \hline
0,324 & 0,687\\
0,322 & 0,695\\
0,319 & 0,721\\
0,317 & 0,759\\
0,316 & ... |
H: Linear transformation $f$
I am tring to solve the following task:
Linear transformation $f: \mathbb R^2 \rightarrow \mathbb R^2$ is given by $f(\begin{bmatrix} x_1\\ x_2 \end{bmatrix}) =
\begin{bmatrix} 2x_1-x_2\\ x_1+x_2 \end{bmatrix}$.
Answer true or false to the following questions:
a) in some basis of trans... |
H: Archimedes' derivation of the spherical cap area formula
Archimedes derived a formula for the area of a spherical cap.
so Archimedes says that the curved surface area of a spherical cap is equal to the area of a circle with radius equal to the distance between the vertex at the curved surface and the base of the ... |
H: Are $\mathbb{C} \otimes _\mathbb{R} \mathbb{C}$ and $\mathbb{C} \otimes _\mathbb{C} \mathbb{C}$ isomorphic as $\mathbb{R}$-vector spaces?
Are $\mathbb{C} \otimes _\mathbb{R} \mathbb{C}$ and $\mathbb{C} \otimes _\mathbb{C} \mathbb{C}$ isomorphic as $\mathbb{R}$-vector spaces?
I am having a very hard time at digest... |
H: $f_k$ differentiable$ \implies f(x)=\sum_{k=1}^{n}f_k(x_k)$ is differentiable
Let $f_k:(a,b)\rightarrow\mathbb{R}$ differentiable functions by $1\leq k\leq n$. Let $f:(a,b)^n\rightarrow\mathbb{R}$ defined by $$f(x)=\sum_{k=1}^{n}f_k(x_k)$$ Prove that $f$ is differentiable and calculate its derivative at any point ... |
H: How to calculate the inverse of a point with respect to a circle?
The theory said:
The inverse of a point $P$, with respect to a circle centered at $O$ and has a radius $r$, is the point $P'$ such that
The three points $O$, $P$ and $P'$ are colinear.
$OP \times OP'=r^2$
But I don't figure out how to calculate ... |
H: Find $S$ where $S=\sqrt[3] {5+2 \sqrt {13}}+\sqrt[3]{5-2 \sqrt {13}}$, why am I getting an imaginary number?
$\large S=\sqrt[3] {5+2 \sqrt {13}}+\sqrt[3]{5-2 \sqrt {13}}$
Multiplying by conjugate:
$\large S=\dfrac {-3}{\sqrt[3] {5+2 \sqrt {13}}-\sqrt[3]{5-2 \sqrt {13}}}$
From the original:
$\large S-2\sqrt[3]{5-2 ... |
H: Show that $(x_n)$ converge to $l$.
Let $(x_n)$ be a sequence of reals. Show that if every subsequence $(x_{n_k})$ of $(x_n)$ has a further subsequence $(x_{n_{k_r}})$ that converge to $l$, then $(x_n)$ converge to $l$.
I know the fact that subsequence of $(x_n)$ converge to the limit same as $(x_n)$ does, but I'm... |
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