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H: Prove $\sum_{k=0}^n \binom{n}{k}(-1)^k \frac{x}{x+k} = \prod_{k=1}^n \frac{k}{x+k}$ and more
The current issue (vol. 120, no. 6)
of the American Mathematical Monthly
has a proof by probabilistic means
that
$$\sum_{k=0}^n \binom{n}{k}(-1)^k \frac{x}{x+k} = \prod_{k=1}^n \frac{k}{x+k}
$$
for all $x > 0$
and all
$n \... |
H: $M_1,M_2$ not homeomorphic but $G(M_1),G(M_2)$ isomorphic.
Let $M$ be a metric space, and let $G(M)$ denote the set of homeomorphisms of $M$ onto $M$. Then $G(M)$ is a group under composition. Show that if $M_1,M_2$ are homeomorphic metric spaces, then $G(M_1)$ is isomorphic to $G(M_2)$. Give an example to show th... |
H: After subdividing a painted cube, how many smaller cubes have paint on exactly 2 sides?
A solid cube of side 6 is first painted pink and then cut into smaller cubes of side 2. How many of the smaller cubes have paint on exactly 2 sides?
Answer with illustrations will be helpful for me.
Thanks in advance.
AI: Hint... |
H: Finding coordinates
The slope of the line passing through the point $(5,5)$ is $\dfrac 56$. All of the following points could be on the line except
A. $(2.5, 2) $
B. $(11, 10) $
C. $(8, 7.5) $
D. $(-1, 0) $
E. $(-7, -5)$
Will I use the slope formula: $\dfrac{y-y_2}{y_-y_1} = \dfrac{x-x_2}{x_-x_1}$? or there are ... |
H: Infinite Parity Function
I was looking at this problem, and I have a solution for a finite board with $2^n$ squares, that I want to extend to a countably infinite board.
Label the squares from $0$ to $2^n-1$. Consider the set of all squares with a $1$ at the $i$th position in their binary expansion. If the number o... |
H: Hartshorne III.7.6b) (ii) => (i) "Duality for a projective scheme)
Let X be a closed immersion of dimension n in P = *P*$^N_k$, where k is an algebraically closed field. Let $\omega_P$ denote the canonical bundle and A the local ring $\mathcal O_{P,x}$.
Then Hartshorne argues on p. 244 that the condition $$\mathcal... |
H: How to calculate $\Pr[\max(X,Y)<4]$?
Suppose the joint PDF of X,Y is $f(x,y)=1/40$ and $0 < x < 5$ and $0 < y < 8$.
How to calculate $\Pr[\max(X,Y)<4]$?
AI: Hint:
$$
P(\max(X,Y)<4)=P(X<4,Y<4)=P((X,Y)\in (-\infty,4)\times(-\infty,4)).
$$ |
H: Determining Linear Combinations
Let $\vec{u}=[2,2,3]^T$ and $\vec{v}=[3,2,1]^T$. Find a vector $\vec{w}$ that is NOT a linear combination of $\vec{u}$ and $\vec{v}$.
My work thus far/ my line of thinking:
Since the vectors are in the form $[x,y,z]^T$, that is, there are three (not sure what to call them- elements?)... |
H: $2$ out of $3$ property of the unitary group
I am trying to understand the $2$ out of $3$ property of the unitary group. I have almost got it, but I am not completely sure about the interaction between an inner product and a symplectic form to obtain an almost complex structure.
Let $V$ be a real vector space.
An ... |
H: Reference request: Tensor product of DG-modules
Let $A$ be a (skew-) commutative DG-algebra, and let $M,N$ be two DG $A$-modules.
I am looking for a reference which describes the functor $-\otimes_A -$ and its basic properties (associtivity, identity element, etc).
Assuming $A$ is a DG $K$-algebra, where $K$ is a c... |
H: How can we prove that $\displaystyle \limsup_{n \to \infty } b_n \le \limsup_{n \to \infty } a_n$
Let $(a_n)_{n\ge1}$ be a bounded sequence in $\mathbb{R}$
Set $\displaystyle B_n=\left\{\sum_{j=n}^{\infty}(\theta_j\cdot a_j)\;:\theta_j\ge 0 \;,\;\sum_{j=n}^{\infty}\theta_j=1\right\}$ for $n\ge1$
Let $(b_n)_{n\ge 1}... |
H: Trigonometry problem, using COS
Let's say two right angled triangles share a common hypotenuse which measures 10 in length and share an angle which measures $20^\circ$ in total. How do I work out the value of x (the side adjacent to the $20^\circ$ angle)? Using $\cos$ looks like the right strategy to apply but not... |
H: A question about how to take a -1 out of mutiple-valued analytic function $z^{\alpha } $ $0<\alpha<1 $
I encountered a question about multiple-valued analytic functions.
Under some circumstance I have to take a -1 out of a power function $z^{\alpha } $ $0<\alpha<1$
suppose $\alpha=\frac{1}{2}$, let the bra... |
H: Usefulness of induced representations.
I am learning representation theory from Serre's book by myself. Currently I am reading about induced representations, but I don't understand the importance. The concept looks strange and the definition appears quite complicated compared to the topics discussed before it. Can ... |
H: Notation for function being differentiable at a certain point
This question describes a notation for a function $f(x)$ being (continuously) differentiable on some domain $A$.
Often, I see the requirement that some function $f(x)$ be differentiable only (or rather, at least) at a certain point $\tilde{x}$. This is u... |
H: Determine the number of iteration to find solutions accurate to within $10^{-2}$ for $f(x)=x^3-7x^2+14x-6=0$ on $[a,b]=[1,3.2]$
i got the number of iteration,$n$, to achieve the accuracy, $\epsilon=10^{-2}$ is $n=5.5\approx 6$
But in answer script, $n=8$.
My procedure is
$
\frac{(b-a)}{2^n}<\epsilon$
$\Rightarrow... |
H: Find the natural number $n$ satisfy the condition
Find the natural number $n$ satisfy the condition
$$\dfrac{1}{2}C_{2n}^1 - \dfrac{2}{3} C_{2n}^2 + \dfrac{3}{4} C_{2n}^3 - \dfrac{4}{5} C_{2n}^4 + \cdots - \dfrac{2n}{2n+1} C_{2n}^{2n} =\dfrac{1}{2013}.$$
AI: HINT:
$$\frac r{r+1}\binom {2n}r=\binom {2n}r-\frac{(2n)!... |
H: Surface integrals of second kind
In the formula for calculating surface integrals of second kind, we have:
But, this integral is denoted by $\int \int _S \vec{F}\cdot \hat{n}dS $ . So, should we always normalize the expression $ \frac{\partial \vec r }{\partial v} \times \frac{\partial \vec r}{\partial u} $ befor... |
H: Does $\lim \frac {a_n}{b_n}=1$ imply $\lim \frac {f(a_n)}{f(b_n)}=1$?
I wanted to prove the seemingly simple statement:
If $\lim \frac {a_n}{b_n}=1$ and $f$ continuous with $f(b_n)\neq0$ then $\lim \frac {f(a_n)}{f(b_n)}=1.$
I started promptly with
\begin{align}
\\ \lim \frac {f(a_n)}{f(b_n)} &= \lim \frac { f(... |
H: How could I define this $\mathrm{nw}(X)$ by using only one sentence?
A family $\mathcal N$ of subsets of a topological space $X$ is a network for $X$ if for every point $x\in X$ and any neighbourhood $U$ of $x$ there exists an $M \in \mathcal N$ such that $x\in M \subset U$.
$$\mathrm{nw}(X)=\min \{|\mathcal N|:... |
H: Anagrams with sequences inside
I need some help with this exercise:
Find the number of anagrams of the word “MONOCROMO” containing atleast one of the sequences “OMO”, “MON”, “CRO”.
Normally I know what to do, but in this one there are some cases i don't know how to handle. Please give me a hand, I need to be abl... |
H: General solution of a simple PDE
What is the general solution of this equation:
$Q \frac{\partial C}{\partial V} + r = \frac{\partial C}{\partial t}$
where Q and r are constants?
I tried to use the method of separation of variables but it doesn't work. Any help will be really appreciated.
AI: Rewriting your equatio... |
H: To what extent the statement "Data is normally distributed when mode, mean and median scores are all equal" is correct?
I read that normally distributed data have equal mode, mean and median. However in the following data set, Median and Mean are equal but there is no Mode and the data is "Normally Distributed":
$ ... |
H: Is population standard deviation useful in the computers era?
Sample standard deviation used to approximate the population standard deviation. I can imagine this was very handy when you were dealing with a large population in pre-computer era.
Now thanks to computers one can calculate the population standard deviat... |
H: Question concerning maximum increment of Brownian motion
Suppose $B$ is the standard Brownian motion, how to calculate
$$\mathbb{P}\left((\max_{0\le s\le t}B(s))-B(t)<a\right)$$
I tried $B(t)-B(s)=B(t-s)$,
$$P(B(t-s)>-a)=\int_{-a}^{\infty}\frac{1}{\sqrt{2\pi(t-s)}}e^{-\frac{y^2}{2(t-s)}}dy$$
And then I integrate i... |
H: Show that $\frac 1 2 <\frac{ab+bc+ca}{a^2+b^2+c^2} \le 1$
If $a,b,c$ are sides of a triangle, then show that $$\dfrac 1 2 <\dfrac{ab+bc+ca}{a^2+b^2+c^2} \le 1$$
Trial: $$(a-b)^2+(b-c)^2+(c-a)^2 \ge 0\\\implies a^2+b^2+c^2 \ge ab+bc+ca $$ But how I prove $\dfrac 1 2 <\dfrac{ab+bc+ca}{a^2+b^2+c^2}$ . Please help.
A... |
H: Sequences of i.i.d. subgaussian RVs and uniform integrability
Consider a sequence of i.i.d. subgaussian RVs $\{a_{j}\}^{n-1}_{j = 0}$; is $\{a^2_{j}\}^{n-1}_{j = 0}$ uniformly integrable (UI)?
Intuitively it appears to be so; if we take for example $a_j$ i.i.d. symmetric Bernoulli (which is subgaussian) then UI of ... |
H: Show that $2^n>1+n\sqrt{2^{n-1}}$
If $n$ be a positive integer greater than $1$, then prove that $$2^n>1+n\sqrt{2^{n-1}}$$
I found this problem under $AM \ge GM$ chapter. Help me to solve this problem using $AM \ge GM$. Thanks in advance.
AI: Hint:$1+2+2^2+\dots 2^{n-1}=2^n-1$
Edit:
Solution:
Applying A.M.-G.... |
H: What "whisker" means in box-and-whisker plot?
This is a bit off-topic but I can't help thinking about the reason behind naming box-and-whisker plot.
"Whisker" according to dictionary is "any of the long stiff hairs that grow near the mouth of a cat, mouse, etc." To me it seems very irrelevant. Why the word "whiske... |
H: In the proof for Urysohn's lemma, why isn't "$x\in {U_{r}}$, then $f(x)\leq r$, and if $x\notin {U_{r}}$, then $f(x)\geq r$" true?
This proof of Urysohn's lemma states that if $x\in \overline{U_{r}}$, then $f(x)\leq r$, and if $x\notin \overline{U_{r}}$, then $f(x)\geq r$. This portion is given on page 4.
Isn't th... |
H: How does this expression arise: $\pi(10.5) = \phi (-z_{1-\alpha} + \sqrt{n} \frac{\mu_0-\mu}{2})$?
$X_i$ is $N(\mu,\sigma^2)$ distributed and the following is given $H_0: \mu \geq 12, H_a: \mu < 12$, and $\alpha=0.01$. I'm asked to calculate $\beta=P[TII]$ if in fact $\mu=10.5$
Now this is the first step that the s... |
H: Existence of projective curve such that...
Is true that for any two different integers $d,d'>1$ there exist two projective curve, of degree $d$ and $d'$ that are not isomorphic and two projective curves that are birational ?
AI: The answer to the second question is "yes".
Consider the affine curves given by the equ... |
H: Dirac delta applied to Dirichlet function
Let's denote Dirichlet function as $d(x)$:
$$d(x)=\begin{cases}{1,x \in \mathbb{Q}\\ 0,x\not\in \mathbb{Q}}\end{cases}$$
I know that Lebesgue integral of Dirichlet function on any finite domain is zero:
$$\int_a^b d(x) dx=0$$
On the other hand, integral of Dirac function is... |
H: Trigonometry Problem Solving
How can we estimate the height (h) of a castle surrounded by a moat, using the info below?
AI: I assume that's supposed to be $50^\circ,$ not just $50$. Let $x$ be the distance (in meters) from the closer viewing point to the castle wall, so that the distance to the further viewing poin... |
H: I am not understanding what has asked to compute of the following exercise.
let $f(x)=(x+2)(x+1)x(x-1)^3(x-2)$.
To which zero of $f$ does the Bisection method converges when applied on the interval $[-3,2.5]$
Have i asked to find the root of $f(x)$ ?
AI: Do you know how to implement the bisection method? Every zero... |
H: Number of solutions for $\sum_{i=1}^{4} x_i < 22$ with condition.
I'm looking for the number of solutions to $\displaystyle\sum_{i=1}^{4} x_i < 22$ where $x_i > i$
Any help is appreciated.
I tried solving it using combinations to do $C(11,4)$ but that doesn't seem to be the right answer.
AI: Hint:
$x_1+x_2+x_3+x_4... |
H: Is there an algorithm to read this mathematical definition?
I'm reading Gemignami's Calculus and Statistics.
I'm a little stuck in this definition:
Definition 7: Suppose $f$ is a function from a set $T$ of real numbers into $R$. Then $f$ is said to be continuous at a point $a$ of $T$ if, given any positive number ... |
H: Limit of sequence $u_1,u_3,u_5,\dots$ with $u_{n+1}=1+\frac{1}{u_n}$
We have a sequence of numbers defined recursively by $$u_{n+1}=1+\frac{1}{u_n},$$for $n\geqslant 1$. It is also given that $u_1=1$. Find the limit $l$ of the sequence $u_1,u_3,u_5,\dots$.
So I said, $u_1,u_3,u_5,\dots$ is given by $u_{2n+1}$for... |
H: Difficulty to solve the exercise of Bisection method.
Find an approximation to $ {25}^{\frac{1}{3}}$ correct to within $10^{-4}$ using the Bisection algorithm.
How to solve it?
Where are the function and interval here?
AI: The most natural function is $f(x)=x^3-25$. You need an interval $[a,b]$ such that $f(a)$ and... |
H: Bottom to top explanation of the Mahanalobis distance?
I'm studying Pattern recognition and statistics and almost every book I open on the subject I bump into the concept of Mahanalobis distance. The books give sort of intuitive explanations, but still not good enough ones for me to actually really understand what ... |
H: $\int\sin^2(Cx)\,dx$ from a manual - need proof
In the book of quantum mechanics I came across an integral which was supposed to be from a manual ($C$ is a constant):
\begin{align}
\int\limits_{0}^d \sin^2\left( C x \right)\, d x = \left.\left(\frac{x}{2}- \frac{\sin(2Cx)}{4C}\right)\right|_0^d
\end{align}
Where ca... |
H: Compactness and Normed Linear spaces
If the set $S=\{ x \in X : ||x||=1 \}$ in the normed linear space $X$ is compact, how can it be shown that $X$ is finite dimensional?
AI: Following up on Martin's comment, if you know Riesz's lemma, you can use it, supposing that $X$ is infinite-dimensional, to inductively creat... |
H: Given that $ x + \frac 1 x = r $ what is the value of: $ x^3 + \frac 1 {x^2}$ in terms of $r$?
Given that $$ x + \cfrac 1 x = r $$
what is the value of: $$ x^3 + \cfrac 1 {x^2}$$ in terms of $r$?
NOTE: it is $\cfrac 1 {x^2}$ and not $ \cfrac 1 {x^3} $
Where I reached so far:
$$ \Big(x^3 + \cfrac 1 {x^2}\Big) + \cfr... |
H: What does ($\ln x$) or ($\log x$) mean?
How does a logarithm followed by a variable read such as ($\ln x$) or ($\log x$). Is it $\log$ times $x$ or the $\log$ of $x$? I'm a little confused by this...?
AI: "Log" is a function; hence, interpreting $\log x$ as "$\log$ times $x$" doesn't make any sense - $\log$ needs a... |
H: Vector space with multiplication
A vector space is a commutative group and I am wondering if it can be extended to be a ring by defining a multiplication. I tried $v \cdot w = (v_1 w_1, ..., v_nw_n)$ componentwise but then inverses aren't unique. Is it possible to construct a multiplication?
AI: The construction yo... |
H: Complex analysis knowledge that required to understand material in Riemann Surface
I have taken a course on complex analysis in university, at that time the instructor chose the book "Complex Analysis" by Serge Lang. Now I am participating a cemina on Riemann Surfaces which truly based on the book Lecture on Rieman... |
H: Cross product in $\mathbb R^n$
I read that the cross product can't be generalized to $\mathbb R^n$. Then I found that in $n=7$ there is a Cross product: https://en.wikipedia.org/wiki/Seven-dimensional_cross_product
Why is it not possible to define a cross product for other dimensions $ \ge 4$?
AI: The issue is the ... |
H: Hilbert polynomial of disjoint union of lines in $\Bbb{P}^3$
Let $X$ be the disjoint union of the two lines in $\Bbb{P}^3$ given by $Z(x,y)$ and $Z(z,w)$. Letting $R = k[x,y,z,w]$, I have computed the following free resolution for the homogeneous coordinate ring $S(X)$:
$$0 \to R(-4) \stackrel{\varphi_0}{\longrigh... |
H: Interchange of partial derivative and limit
Consider the following expression:
$$\frac{\partial}{\partial m} \lim_{T \rightarrow \infty} \gamma(T,m)$$
where $\gamma$ is a function of $T$ and $m$.
My question is just: can I permute the partial derivative and the limit operators? I suppose that I can, given that the ... |
H: Show that $|f(p_n)|<10^{-3}$ whenever $n>1$ but that $|p-p_n|<10^{-3}$ requires that $n>1000$.
Let $f(x)=(x-1)^{10}$.
The root of the equation , $p=1$.
The approximates of the root, $p_n=1+\frac{1}{n}$
Show that $|f(p_n)|<10^{-3}$ whenever $n>1$ but that $|p-p_n|<10^{-3}$ requires that $n>1000$.
I started to solve ... |
H: trace of $\operatorname{End}(E)$-valued $p$-form
$\eta\,$ is $\,\operatorname{End}(E)$-valued $p$-form. $\;d_D\,$ is exterior covariant derivative. $\,E\,$ is Bundle.
How can I prove $$\,\operatorname{tr}(d_D\eta)=d(\operatorname{tr}(\eta))\quad ?$$
AI: Proofs like this entail "unpacking" the respective definitio... |
H: Infinite Sets in Compact Spaces
Given that:
$S$ is a compact set in the topological space $(X, \mathcal T)$
$T\subset S$ has no accumulation points in S
How do I show that $T$ is finite?
AI: For every $s \in S$ we find a neighbourhood $U_s$ that intersects $T$ in only at most finitely many points, so $U_s \cap T$ i... |
H: Find the least next N-digit number with the same sum of digits.
Given a number of N-digits A, I want to find the next least N-digit number B having the same sum of digits as A, if such a number exists. The original number A can start with a 0. For ex: A-> 111 then B-> 120, A->09999 B-> 18999, A->999 then B-> doesn'... |
H: Is it possible to calculate sine by hand?
Without a calculator, how can I calculate the sine of an angle, for example 32(without drawing a triangle)?
AI: You can use first order approximation $\sin(x+h)=\sin(x)+\sin'(x)h=\sin(x)+\cos(x)h$
where $x$ is the point nearest to $x+h$ at which you already know the value o... |
H: Sum of N numbers whose sum is M
In how many ways can we sum N nonnegative numbers (that is, taking values 1, 2, 3...) such that their sum is M? I found this problem doing convolution of series and combinatorics has never been my strong point.
Thank you!
Precision: The precise question is, I have a sum:
$$
\sum_{i_1... |
H: Let $p_n$ be the sequence defined by $p_n=\sum_{k=1}^n\frac{1}{k}$. Show that $p_n$ diverges even though $\lim_{n\to\infty}(p_n-p_{n-1})=0$
I have tried this as :
$$p_n=\sum_{k=1}^n\frac{1}{k}=1+\frac{1}{2}+\frac{1}{3}+\ldots+\frac{1}{n-1}+\frac{1}{n}$$
$$p_{n-1}=\sum_{k=1}^{n-1}\frac{1}{k}=1+\frac{1}{2}+\frac{1}{3... |
H: Gauß-Jordan algorithm - 'reading' the solution
Disclaimer: I'm not really sure how to do a proper coefficient-matrix in latex, if someone could edit it to look properly I'd be really thankful ;)
Given the following system of linear equations, determine the solution
set using the Gauß-Jordan-Algorithm
$$ (I):3x_1... |
H: A trouble about the topology of pointwise convergence $({\mathbb{R}}^M,\tau)$
Let $(M,d)$ be a separable metric space and $F=\{f_{\lambda}:M\longrightarrow\mathbb{R}:\lambda\in L \}\subset \mathbb{R}^M $ be an equicontinuous family of uniformly bounded functions on $M$.
How can we prove that:
$\boxed{1}\;\; \over... |
H: Index of a subgroup of $\mathbb{Z}\times\mathbb{Z}$
Let $p\in\mathbb{Z}$ be a prime and $u\in\mathbb{Z}$ be such that $u^2\equiv -1\pmod{p}$. Now define an additive subgroup $S$ of $\mathbb{Z}\times\mathbb{Z}$ by following, $$S:=\{ (a,b)\in\mathbb{Z}\times\mathbb{Z}: b\equiv ua\pmod{p}\}$$ Then what is the index of... |
H: Let $f(x)=x^3+2x^2+1,\,\,g(x)=2x^2+x+2.$ Then over $\,\left(\Bbb Z/3\Bbb Z\right)[x]\;$......
I am stuck on the following problem:
Let $f(x)=x^3+2x^2+1,\,\,g(x)=2x^2+x+2.$ Then over $\,\left(\Bbb Z/3\Bbb Z\right)[x]\;$, show that $f(x)$ is irreducible ,but $g(x)$ is not.
Can someone explain how to tackle it? Th... |
H: Random Variables from $[0,1]$ - Integration Limits
I was wondering if someone could help me understand the first steps I should take for solving the next problem:
Let $U$, $V$ be random numbers chosen independently from the interval $[0, 1]$ with uniform distribution. Find the cumulative distribution and density fo... |
H: Why are projective schemes $\mathbb P_A^n$ over a ring not affine for $n>1$?
I recently posted a very similar question, but I hid the question I really wanted answered in it. I'm posting this to make that question explicit.
Let $A$ be a nonzero commutative ring with unit. Define $\mathbb P_A^n$ to be the scheme $\... |
H: Confusing Trigonometry Problem
Lets say at an intersection the words "STOP HERE" are painted on the road in red letters 2.5m high. It is important that drivers using this lane can read the letters. How can I find the angle subtended by the letters to the eyes of a driver 20m from the base of the letters and 1.25m a... |
H: for $ F(x,y) = 10$, what is $ y'$?
For an input $x$ and output $y$ of a system it is know that $x,y$ always satisfy
$$ F(x,y) = 10 $$
At a certain point, $x=1$ and $y=1$. The question is how $y$ responds to a small decrease in $x$, e.g. to $ 0.999$ (that is, what is $y'(x)$ of the function $y(x)$ near $x=1$.
Now th... |
H: Prove: If in all subgraphs of $G$ there is a vertex of degree $<2$ then $G$ is a forest
I need help proving this:
Given a graph $G$, prove that if in all subgraphs of $G$ there is a vertex of degree less than $2$ ($1$ or $0$) then $G$ is a forest.
AI: Have you tried to prove the contrapositive? Like Damian said, a... |
H: Easy way to compute the area between $f(x)=x$ and $g(x)=x^2\ln(x)$
Is there an easy to compute the area between $f(x)=x$ and $g(x)=x^2\ln(x)$ without refering to the Lambert W-function?
AI: By letting $x_0$ be the positive point that satisfies $x_0 \log x_0 = 1$, we get that the relevant integral equals a polynomia... |
H: $E$ is closed $\iff\partial E$ (boundary of set $E$) $\subseteq E$
I am studying topology of euclidean space from William Wade's text book.
I saw this question. But I cannot come up with any ideas.
Please show me the solution in an instructive an clear way.
Thank you for yourhelp.
$E$ is closed $\iff\partial E$ ... |
H: What's the meaning of computing an integral at a given point?
Let $f$ be a function. If one finds $\displaystyle \frac{\mathrm d}{\mathrm dx}f$ and computes it at $x=a$, then one gets the rate of change of $f$ at $a$. That can be useful in some situations. But if one finds $\int f \space \mathrm dx$ and computes it... |
H: Alternative unconditional form of $\sqrt{n -\sqrt{n -\sqrt{n -\cdots}}}$?
Consider $a_n$, where
$$\begin{align} a_n &=\small{\sqrt{n -\!\!\!\sqrt{n -\!\!\!\sqrt{n -\!\!\sqrt{n -\!\!\sqrt{n -\!\!\sqrt{n -\!\sqrt{n - \cdots}}}}}}}}\end{align}$$
Using a recursive solution, such that:
$$a_n = f(n) = \sqrt{n - f(n)}$$
i... |
H: Learning to read complex math formulas
could anybody point me to a book or article where I could learn how to read formulas like this one:
I have no idea what that means.
AI: The following points may be helpful:
$i$ is used to index the various numerical values $x_i$ you have. Usually, unless specified otherwise... |
H: Compute this limit $\lim_{x\to0}\frac{\sin(x^2+\frac{1}{x})-\sin\frac{1}{x}}{x}$ using L'Hôpital's rule
I have asked this problem before, but I can't understand the explanation, I couldn't understand how the sin multiply for cos, and too multiply for A + and - B: $$\sin(A)-\sin(B)=2\sin\left(\frac{A-B}{2}\right)\co... |
H: Is $\ln(x)$ uniformly continuous?
Let $x\in[1,\infty)$. Is $\ln x$ uniformly continuous? I took this function to be continuous and wrote the following proof which I'm not entirely sure of.
Let $\varepsilon>0 $, $x,y\in[1, ∞)$ and $x>y$.
Then, $\ln x< x$ and $\ln y< y$ and this follows that $0<|\ln x-\ln y|<|x-y|$ ... |
H: Can I always extend a selfadjoint Operator in $L^2$?
Assume that we have a self-adjoint operator $T\colon D \to D$ where $D \subset L^2$ is some finite dimensional subspace. Can I conclude that than a self-adjoint operator $S \colon L^2 \to L^2$ exists with $S=T$ on $D$ ?
Hope someone can help me.
Best regards,
Ada... |
H: hard time with series convergence or divergence
I'm having real hard time with this series
I can't prove that the series converges and also I can't prove that the series diverges:
$$\sum_{k=1}^\infty\frac{\sin^2(n)}{n}.$$
any help would be appreciated.
AI: An idea:
$$\sin^2n=\frac12(1-\cos 2n)$$
Now, using Dirichle... |
H: If $\gamma$ is an uncountable ordinal, then $\gamma$ contains uncountably many successor ordinals.
Suppose $\gamma$ is an uncountable ordinal, i.e. $\gamma$ has uncountably many elements.
We write
$$
\gamma = \{0\} \cup\{\alpha \in \gamma: \alpha \text{ is a successor ordinal}\} \cup\{\alpha \in \gamma: \alpha \tex... |
H: For a group $G$: show that $p$ is a divisor of $\# \mathcal{Z}$
Given a prime number $p$, an integer $n>0$, and a group $G$, where $\#G=p^n$.
Let $\mathcal{Z}(G)$ the center of the group: $\mathcal{Z}(G)G= \{a\in G; xa=ax, \text{ }\forall x\in G \}$
Now I have to show that $p$ is a divisor of $\# \mathcal{Z}(G)$.... |
H: Plase explain me the statment related to interior point in $\Bbb R^n$
I wrote such definition at the class. I understand the definition. But also my teacher said " needs provoing!" I underlined it with blue pencil. I could not understand what my teacher wants us and why he wrote such proof requirment. Please expl... |
H: Why does $(-2^2)^3$ equal $-64$ and not $64$?
The title says it all. Why does $(-2^2)^3$ equal $-64$ and not $64$? This was on my algebra final, and I am completely stuck on how it works.
AI: The negative sign ($-$) applies to the quantity $2^2$, so that $-2^2$ means $-(2^2)=-4$, not $(-2)^2=4$. $$(-2^2)^3=(-4)^3=-... |
H: Number of indecomosbale $\mathbb{Z}_p[G]$ modules finite
Is there a theorem like those of Jones, which tells if the number of different $\mathbb{Z}_p[G]$ modules is finite, where $G$ is a finite group and $\mathbb{Z}_l$ the $p$-adic ring?
AI: Yes, Heller and Reiner proved this in the early 60s.
Proposition: $\hat{\... |
H: Logistic functions - how to find the growth rate
We have the formule for a model with logistic growth:
$$ N_t = N_{t-1} + g\, N_{t-1}\left( 1 -\dfrac{N_{t-1}}{K}\right)$$
where $g$ defines the growth rate and $K$ is the carrying capacity.
Let's say we have the following data:
$N_0 = 10$,
$N_1 = 18$,
$N_2 = 29$, ... |
H: A question on faithfully flat extension
This question arose while reading page 116 of Red Book by Mumford.
Let $B$ be a faithfully flat extension of $A$. Can I claim that $b \otimes 1 = 1 \otimes b$ in $B\otimes_A B$ if and only if $b\in A$?
AI: You are asking if the sequence
$0 \to A \to B \to B\otimes_A B$, whe... |
H: Why do we define a linear transformation to have the property that $f(cW)=c f(W)$?
Why we define a lin tranfs to have the property that $f(cW)=c f(W)$ ?
let $V,T$ be any two vector spaces and
let $f:V\rightarrow T$ be a linear transformation between $V $and $T $
why do we assume this condition ?
i think you woul... |
H: Question about Switching Between Random Variables
Find the density function of $Y = aX$, where $a > 0$, in terms of the density function of $X$.
Show that the continuous random variables $X$ and $-X$
have the same distribution function if and only if $f_X(x) = f_X(-x)$ for all $x \in \mathbb R$.
I got the first p... |
H: Is there a tree $T$ such that $\text{diam}(T) \geq k$, where $k$ is the number of vertices with degree less than 3?
Let $T$ be an undirected tree, let $d$ be the diameter of $T$, and let $s$ be the number of vertices in $T$ with degree less than 3. Recall the diameter of a graph is the length of the longest shortes... |
H: $\sum_{n=1}^{\infty }\left(\frac{2n+5}{7n+6}\right)^{n\log(n+1)} $ converges or diverges?
I am trying to determine whether this series converges or diverges: $\sum_{n=1}^{\infty }\left(\frac{2n+5}{7n+6}\right)^{n\log(n+1)}$.
Here is my solution: I called: $a_{n}=\left(\frac{2n+5}{7n+6}\right)^{n\log(n+1)}$. Then, I... |
H: Does $\alpha + \beta = \alpha$ imply $\beta \le \aleph_0$
Just like in title, my question is : Does $\alpha + \beta = \alpha$ imply $\beta \le \aleph_0$ where, $\alpha$ and $\beta$ are cardinals?
P.S. I actualy have to prove $\alpha + \beta = \alpha$ $\iff$ $\alpha + \aleph_0 \cdot \beta = \alpha$. Any ideas?
AI: I... |
H: Calculating $\lim_{x \to a} \frac{x^2 - (a+1)x + a}{x^3-a^3}$ using L'Hospital
I tried to calculate this limit:
$$\lim_{x \to a} \frac{x^2 - (a+1)x + a}{x^3-a^3}$$
Using L'Hospital's rule I get:
$$\lim_{x \to a} \frac{2x - (a+1)}{3x^2} = \frac{2a - (a+1)}{3a^2} = 0$$
But actually the limit is
$$\lim_{x \to a} \frac... |
H: Reference request for complex variables
My curriculum for math has the first chapter on complex variables. It is as stated below:
Functions of complex variables:
Continuity and derivability of a function
Analytic functions
Necessary condition for $f(z)$ to be analytic, sufficient conditions (without
proof)
Cauchy... |
H: Do endomorphisms of quotients always lift?
Let $H \to G$ be an injective homomorphism of Abelian groups and let $\varphi$ be an endomorphism of $H$. Must $\varphi$ extend to an endomorphism of $G$? The answer is no; a counterexample is the endomorphism of the subgroup $2\mathbb{Z} \times \mathbb{Z}_2$ of $\mathbb{Z... |
H: $-1 = 0$ by integration by parts of $\tan(x)$
I had a calculus final yesterday, and in a question we had to find a primitive of $\tan(x)$ in order to solve a differential equation.
A friend of mine forgot that such a primitive could easily be found, tried to integrate $\tan(x)$ by parts... and then arrived to the r... |
H: Choosing random number $[1,n]$. What is the expected value of $f(x) = x^2$?
We have just started learning discrete probability and this question came up:
We choose a random number from $[1,n]$, and we let be $f(x) = x^2$ and $g(x) = 2^{-x}$.
I) What is the expected value of $f$?
II) What is the expected value of $... |
H: Regular Pentagon is the Unique Largest Two-Distance Set in the Plane
A two-distance set is a collection of points for which only two distinct distances appear among pairs of points. (That is, the distance between any pair of points is either $x$ or $y$, and these values may be whatever you want.)
The unique (up to ... |
H: Two tangent closed discs connected
Let $X$ be a connected subset of a metric space $M$. Show that $X^0$ (the interior of $X$) is not necessarily connected.
So the example I'm thinking of is $X$ being two closed discs, tangent at a point. The interior is clearly not connected, since there exist two disjoint, non-e... |
H: Calculus Complicated Substitution Derivative
When,
$$y=6u^3+2u^2+5u-2 \ , \ u= \frac{1}{w^3+2} \ , \ w=\sin x -1 $$find what the derivative of $ \ y \ $equals when $ \ x = \pi \ . $
Tried it many times, still can't seem to get the right answer (81)
AI: $$\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dw}\cdot\frac{dw}{... |
H: Weak topology on an infinite-dimensional normed vector space is not metrizable
I've been pondering over this problem for a while now, but I can't come up with a proof or even a useful approach...
Let $X$ be am infinite-dimensional normed vector space over $\mathbb{K}$ (that is either $\mathbb{R}$ or $\mathbb{C}$).
... |
H: Characteristic Primes of repunits
First off, we're working in base 10. A repunit is a number of form $111111...1$. ( n ones)
For some integer sequence $(a_n)$, a charateristic prime $p$ of $a_n$ is a prime which divides $a_n$, but none of $a_1,...,a_n$. Does every repunit have a charteristic prime?
AI: The answe... |
H: In a random graph of $n$ vertices, what is the expected value of the number of simple paths?
I am very new to discrete probabilty and was asked this question:
In a random graph $G$ on $n$ vertices (any edge can be in the graph with probabilty of $\frac{1}{2}$,) what is the expected value of the number of paths bet... |
H: Possible mistake in exercise in Hartshorne exercise II.2.18b
I'm trying to solve Exercise II.2.18b in Hartshorne, and I've constructed what appears to be a counterexample to its statement. Can someone tell me where I've gone wrong?
The statement is as follows. Let $\phi : A \rightarrow B$ be a ring homomorphism, ... |
H: Find all different integer exponents
Find all different integers that satisfy the following equality:
$m(\sin^{n}x + \cos^{n} x- 1) = n(\sin^{m}x + \cos^{m}x - 1), (\forall) x\in\mathbb{R}.$
Case1: $m$ is odd, $n$ is even, then put $x=180^0 => m=n=0 =>$ contradiction.
Case2: $m$ and $n$ is odd, then put $x=180^0 =... |
H: Simple partial differentiation
I have a simple partial differentiation question here, given:
$u = x^2 - y^2$ and $v = x^2 -y$, find $x_u$ and $x_v$ in terms of $x$ and $y$.
What is the easiest way to go about this?
Thanks
AI: HINT: Think $x = x(u,v)$ and $y= y(u,v)$ and use implicit differentiation.
First we consi... |
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