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H: Value of $f''(c)$
Suppose $f$ is defined in $[a,b]$ and $c\in(a,b)$ and suppose $f'$ exists in an open interval containing $c$ and that $f''(c)$ exists. Show that $$\lim_{h\rightarrow 0}\frac{f(c+2h)-2f(c+h)+f(c)}{h^2}=f''(c)$$ Give an example to show that the limit above may exist even though $f'(c)$ does not exi... |
H: Drawing points on Argand plane
The points $5 + 5i$, $1− 3i$, $− 4 + 2i$ and $−2 + 6i$ in the
Argand plane are:
(a) Collinear
(b) Concyclic
(c) The vertices of a parallelogram
(d) The vertices of a square
So when I drew the diagram, I got an rectangle in the 1st and 2nd quadrant. So, are they vertices of paralle... |
H: Find a recursive formula for the following problem
Let $a_n$ be the number of bricks in a path that is $n \geq 1$ long.
We have 3 types of bricks:
Blue: $2$ cm long
Red: $3$ cm long
Green: $1$ cm long
When a blue brick can't be placed next to a green brick.
I was trying to work this out but I'm getting into an en... |
H: How to calculate the density of $Y=X^4$
Let $X$ be a uniformly distributed variable on $[0,1]$. What is the density of $Y=X^4$? How do you calculate it? Thank you
AI: We take the slow way, by first computing the cumulative distribution function $F_Y(y)$ of $Y$. So we want $\Pr(Y\le y)$.
First do the really easy par... |
H: How to put a norm on an ultrapower of normed spaces?
I'm trying to understand how to form the ultrapower of a normed space.
I read how to construct the underlying vector space here: http://en.wikipedia.org/wiki/Ultraproduct
But I don't see an obvious norm which can be defined on the resulting equivalence classes. ... |
H: Find the standard deviation of $ \frac{\gamma}{\sqrt{2\pi\sigma}}\exp\left(-\frac{\gamma^2}{\sigma}\frac{(x-\mu)^2}{2}\right)$
Given $\frac{\gamma}{\sqrt{2\pi\sigma}}\exp\left(-\frac{\gamma^2}{\sigma}\frac{(x-\mu)^2}{2}\right)$ as a normal distribution PDF with mean $\mu$, I'd like to solve for the std deviation... |
H: Optimizing the area of a square and circle
I am suppose to optimize the sum of the area of a square and a circle with a 12cm piece of wire to have the smallest area. To me this problem seems kind of obvious. A circle is a more efficient use of space so I know that it will have more area since it doesn't waste space... |
H: Does the Convolution of Two Series Require Absolute Convergence?
Let $A=(a_n)_{n=0}^\infty$ and $B=(b_n)_{n=0}^\infty$ be sequences of real numbers. Let $C = (c_n)_{n=0}^\infty$ be the sequence such that
$$c_n = {a_0}{b_n} + {a_1}{b_{n-1}} +\cdots+ {a_n}{b_0}.$$
Let $G_A(s), G_B(s), G_C(s)$ denote the generating fu... |
H: Finding probabilities for a discrete random variable using a CDF
I have a question about notation and I want to make sure I really understand the homework.
The discrete random variable X has cdf F such that
F(x)= 0 x < 1
1/4 1 ≤ x < 3
3/4 3 ≤ x <4
1 x ≥ 4
Find P(X=2), P(X=3.5), P(... |
H: Deleted Exponential Series and Injectivity (1)?
Consider $\exp(z):=\Sigma_{n=1}^{\infty}\frac{z^n}{n!}.$ We know that the radius of convergence for this series is infinity, and hence it defines a holomorphic map which is not injective. I have following questions:
If we delete from exponential series those terms wi... |
H: Symmetry in reduced residue systems
This may be a stupid question, but it looks to me like the reduced residue systems modulo N are symmetrical about N/2; that is to say, that the there is the same number of integers not divisible by a factor of N that are smaller than N/2 as there are that are bigger. Is this a ca... |
H: Explanation of an example of linear transformation
This is an example from a text (Linear Algebra, Freidberg). I am trying to follow along, and I feel like I should know this from vector calc but I am missing something silly.
The example is:
Define the Linear transformation $T:P_2(R)\rightarrow M_{2x2}(R)$ by
$... |
H: Collection of Borel sets contains all $[a,b)$
Show that the collection of Borel sets is the smallest $\sigma$-algebra that contains intervals of the form $[a,b)$, where $a<b$.
So I first prove that the collection $S$ of Borel sets contains those intervals. $S$ contains the closed interval $[a,b]$ and the (closed)... |
H: Three doors logic problem
Imagine three doors where behind one door $\text{A}$ there is a new car, behind door $\text{B}$ there is a goat, and behind door $\text{C}$ there is a new car and a goat.
The problem is that each door is labeled incorrectly...
If you can open only one door, is it possible to label all the... |
H: Maclaurin Series for $\ln(x+\sqrt{1+x^2})$
Is there a trick to finding the Maclaurin series for $f(x)=\ln(x+\sqrt{1+x^2})$ fast? Vaguely, I recall this being some sort of inverse hyperbolic function, but I'm not sure about which one, and what its derivatives are. This is a past exam question and I would like to kno... |
H: MLE problem - the likelihood function has no maximum.
The probability density function is:
$f(x)=e^{\theta -x}, \ 0 \le \theta \le x $
Given an n-element sample, the likelihood function is:
$$L(\theta)=\exp \left( n\theta - \sum_{i=1}^n x_i \right)$$
Since the function has no maximum, but increases with $\theta$, I... |
H: Number of distinct path in a graph with $n$ vertices
Let $T = (V , E)$ be a tree with $|V | = n\geqslant 2$. How many distinct paths are there (as sub graphs) in $T$?
I already have the answer to this question as $(n/2)$. The problem that I'm having is finding anything in the text that helps me to figure out how to... |
H: Existence of maximum area triangle among all whose vertices are in a compact subset of $\mathbb{R}^2$
Prove that among all triangles whose vertices are in a compact subset $K$ of $\mathbb{R}^2$, there exists at least one with maximum area.
I am at a loss as to how to rigorously show that you can generate a finite o... |
H: How to solve infinite square root of 1+ itself or: $\varphi=\sqrt{1+\varphi}$
How do I find $\varphi$ for $\varphi=\sqrt{1+\varphi}$ or $\varphi=\sqrt{1+\sqrt{1+\sqrt{1+\sqrt{1+\sqrt{1+...}}}}}$?
AI: The commentators have already give hint how to get the possible value of $\varphi=\varphi_\infty$, it remains to sho... |
H: Why is this composition of scheme morphisms proper?
I am learning about proper morphisms from Liu's book. I have a question about the proof of the Lemma 3.17 on page 104.
Let $A$ and $B$ be rings and suppose $\operatorname{Spec} B$ is proper over $A$. The lemma says that $B$ is finite over $A$.
The proof begins b... |
H: Proving that $f(x)=\sum^{\infty}_{n=1}\frac{1}{n^2-x^2}$ is continuous.
Prove that $f(x)=\sum^{\infty}_{n=1}\frac{1}{n^2-x^2}$ is continuous at all $x \notin \Bbb N$.
An attempt:
We should consider showing that $\sum^{\infty}_{n=1}\frac{1}{n^2-x^2}$ converges uniformly.
Also,
$$f(x)=\sum^{\infty}_{n=1}\frac{1}{n^2... |
H: Conditional probability is not a probability measure, but it does satisfy each of the requisite axioms with probability 1.
I dont quite understand what the statement in the question means, which appears in this paragraph of a textbook I am reading. How can it not be a probability measure (not even almost surely) bu... |
H: $\sum_{k=1}^{\infty}(a_k+\epsilon/2^k)$, where $a_k\ge 0$
I know that if $\sum a_k$ and $\sum b_k$ converge, then:
$$\sum_{k=1}^{\infty}(a_k+b_k)=\sum_{k=1}^{\infty}a_k+\sum_{k=1}^{\infty}b_k$$
However, as I read books on measure theory, I come across lines like this all the time:
$$\sum_{k=1}^{\infty}(m^*(E_k)+\ep... |
H: Infinite Cartesian Product minus AC example
I've been looking for an example of an empty Cartesian product whose factors are non-empty. From what I've gathered so far, this statement is equivalent to the negation of AC, ie. AC fails. So constructing an example means finding a collection of sets for which no choice ... |
H: Proof that $\mathbf{R}[\omega]_\times\mathbf{R} = [\mathbf{R}\omega]_\times$
I have to prove that
$$\mathbf{R}[\omega]_\times\mathbf{R}^\mathrm{T} = [\mathbf{R}\omega]_\times$$
Herein $\omega$ is a vector with elements. The notation $[\mathbf{a}]_\times$ is a conversion of the vector $\mathbf{a}$ to to a matrix to ... |
H: Do I need to present a formula in this proof?
Well, I've been studying Rudin's Principles of Mathematical Analysis and then I've thought on the following exercise: Let $A, B$ be two countable sets, then $A\times B$ is countable. My idea to prove this was to arrange the elements of $A\times B$ in a sequence. Inded, ... |
H: Uniform convergence of $\sum_{n=1}^{\infty}\ln\left(1+\frac{x^2}{n^2}\right)$
Find the convergence domain and determine if $\sum_{n=1}^{\infty}\ln(1+\frac{x^2}{n^2})$ converges uniformly on
$\mathbb{R}$
$[a,b]$ (some closed interval)
An Attempt:
Using $\ln(1+t)\leq t$, we will have:
$$\sum_{n=1}^{\infty}\ln\left(... |
H: Method for solving ODE with power series
when trying to solve second order linear homogeneous variable coefficient ODEs using a power series method, there seem to be two different general forms cropping up in my notes. The first uses an ordinary point $$x_0$$
$$y = \sum_{m = 0}^{\infty}a_m(x-x_0)^m$$
The second use... |
H: Given that $xyz=1$ , find $\frac{1}{1+x+xy}+\frac{1}{1+y+yz}+\frac{1}{1+z+xz}$?
I think I solved this problem, but I don't feel $100$ percent sure of my solution. We have:
$xy=\large {\frac 1z}$
$xz=\large \frac 1y$
$yz=\large \frac 1x$
So let's substitute these into our sum:
$\large \frac{1}{1+x+\frac 1z}+\frac{1... |
H: rigorous definition of a "logic"
It's been a couple of years since I've had a course in logic (the course was propositional and first order logic, up to Gödel's completeness theorem). I've been looking at some papers online, and they seem to talk about systems of logic like $\mathrm K$ and $\mathrm{S4}$ as "logics"... |
H: Prove this inequality by using the mean value theorem
I want to prove that $x<\frac{2x}{2-x}, \forall x \in (0,1)$, by using the mean value theorem.
So, consider $f(x)=\frac{2x}{2-x} -x$. $f(0)=0$.
$f´(x)=\frac{2x-2}{(2-x)^2} - 1$ and $f'(x)<0, \forall x \in (0,1)$. By the mean value theorem:
$$\exists c \in (0,1)~... |
H: Conditional distribution problem
Given random variables X and Y with joint density
$$f(x,y) = 2(x + y), \text{ for } 0 < y < x < 1,$$
I am trying to compute the probability that
$$P(Y < 0.5 \mid X = 0.1)$$
To that end, I computed the marginal density
$$f_X(x) = \int_0^x 2(x + y) \, \mathrm d{y} \\
= 3... |
H: Chain rule for functions of two variables
Suppose that $f(x,y)$ is a function of two variables with $f_x(0,2) = 2$ and $f_y(0,2) = -1$.
Using the chain rule compute the numerical value of $f_\theta(r\cos\theta,r\sin\theta) = 2$ at $r=2$, $\theta=\frac{\pi}{2}$.
Any hints on how to do this question would be apprecia... |
H: What is $2012^{2011}$ modulo $14$?
$$2012^{2011} \equiv x \pmod {14}$$
I need to calculate that, all the examples I've found on the net are a bit different. Thanks in advance!
AI: By the Chinese remainder theorem, knowing what something is modulo $2$ and what it is modulo $7$ is equivalent to knowing what it is mod... |
H: Prove that the additive inverse of an odd integer is an odd integer
This is a homework problem, but I don't want the answer, just a little guidance:
Prove that the additive inverse of an odd integer is an odd integer.
When approaching a problem like this, how much is it safe to assume? Is it safe to assume that ... |
H: Error in Fibonacci recurrence proof by induction?
I'm working on a problem from a number theory book (Number Theory by George E. Andrews - problem 1-1-11). The text reads:
Prove: $\displaystyle F_1F_2+F_2F_3+F_3F_4+\ldots+F_{2n-1}F_{2n}=F_{2n}^2$
I started by setting up a summation:
$$\sum_{j=1}^{n}F_{2j-1}F_{2j}... |
H: A set theory problem - proof
Prove that $X \cap (\bigcup_{i \in I} Y_i) = \bigcap_{i \in I} (Y_i \cap X) $
Is the indices finite? How do I know it isn't $1 \leq i < \infty$?
Also, isn't the RHS just $Y_1 \cap Y_2 \cap ....\cap X$? (assuming we have finite intersection).
EDIT
A quick counterexample from myself dis... |
H: second and first derivative growth function
I have a function I would like to take the first and second derivative from
$$f(t)= a\left(1-\frac{1}{1+(b(t+i))^e+(c(t+i))^f+(d(t+i))^h)}\right)$$
I have taken the following steps
$$u(t)={\left(\mathrm{b}\, \left(\mathrm{i} + t\right)\right)}^{\mathrm{e}} + {\left(\mathr... |
H: Generalizing the Definition of Convexity
The definition of convexity can be given as:
Definition: Call a subset of $\mathbb{R} ^ k$, which will be denoted $E$, convex if given two elements of $E$, $\boldsymbol{x}$ and $\boldsymbol{y}$ and $0 < \lambda < 1$ the following holds:
$$ \lambda \boldsymbol{x} + (1 - \lamb... |
H: For a morphism of affine schemes, the inverse of an open affine subscheme is affine
This seems ridiculously simple, but it's eluding me.
Suppose $f:X\rightarrow Y$ is a morphism of affine schemes. Let $V$ be an open affine subscheme of $Y$. Why is $f^{-1}(V)$ affine?
I noted that $V$ is quasi-compact and wrote it ... |
H: Matrix of a relation on a set
If I have a Matrix $A=\begin{bmatrix} 0&0&0\\ 0 & 0 & 0 \\0&0&0\end{bmatrix}$ why is this both symmetric and anti-symmetric? If I had a Matrix $B=\begin{bmatrix} 1&1&1\\1&1&1\\1&1&1\end{bmatrix}$ would this also be symmetric and anti-symmetric?
AI: The first matrix represents the empty... |
H: You have 4 prizes, 3 tickets, n tickets- what is the probability of winning
You have bought 3 tickets in a lottery. There are n total tickets and 4 prizes. What are the odds of winning at least one prize?
I thought of it like this:
The total possible ways of extracting 4 prizes is: a= $${n\choose 4}$$
The possibili... |
H: Looking for an easy lightning introduction to Hilbert spaces and Banach spaces
I'm co-organizing a reading seminar on Higson and Roe's Analytic K-homology. Most participants are graduate students and faculty, but there are a number of undergraduates who might like to participate, and who have never taken a course i... |
H: Mean or standard deviation?
Which one of mean or standard deviation can used to solve the following problem?
A light bulb is considered defective if it lasts less than 400 hours. The following claim is made:
'Brand A light bulbs are more likely to be defective than Brand B light bulbs.'
Is the claim correct?
$$
... |
H: How to continue this problem to find distribution function of $|X-Y|$
The probability density function of
$$f(x,y) =
\begin{cases}
1/x^2, & \text{if }0 \le x \le a\text{ if }0 \le y \le a \\
0, & \text{otherwise} \\
\end{cases}
$$
How can you prove that $|X-Y|$ and $\min(X,Y)$ have the same distribution function... |
H: Transforming the Area of Integration in the Beta Function
My text derived the Beta function by change of variable and the Jacobian determinant.
$$\Gamma(x)\Gamma (y) = \int_0^{\infty} \int_0^{\infty} e^{-s-t} \space t^{x-1} \space s^{y-1} dt ds$$
Let $t = u(1-v)$ and $s = uv$. Then $ 0 \le u < {\infty}$ and $0 \le... |
H: Simplifying a Recurrence Relation
$(n_i) $ is a sequence of integers satisfying $n_{i+1}=a_{i+1}n_i+n_{i-2}$. Consider a subsequence $(n_{i_j}).$
Can $n_{j_{i+1}}$ be written in terms of $n_{j_i}$?
An attempt is to use the recurrence relation and write $$n_{j_{i+1}}=a_{j_{i+1}}n_{j_{i+1}-1}+n_{j_{i+1}-2}=a_{j_{i+1}... |
H: What is the density of $1-X^3$ if $X$ is a Cauchy random variable?
What is the density function of $Y=1-X^3$, if $X$ is a Cauchy random variable?
My approach:
$$Pr(Y<y)=Pr(1-X^{3}<y)=Pr(X<(1-y)^{-3})=\int^{y}_{-\infty}\left(1-\frac{1}{\pi(1+t^{2})}\right)^{-3}\:dt$$ - is this ok?
And then the density function would... |
H: A gambler with the devil's luck?
A gambler with $1$ dollar intends to make repeated bets of $1$ dollar until he wins $20$ dollars or is ruined. Probabilities of win/loss are $p$ and $(1-p)$, and each bet brings a gain/loss of $1$ dollar.
Unfortunately, the devil is active, and ensures that every time he reaches $... |
H: To show a function is integrable
I came across this question while studying for my exam:
To show the function $f(x) = \frac{(\sin x)^2}{x^2}$ is Lebesgue integrable on [0, $\infty$). I wonder if there is any smarter way of proving it (like, using absolute continuity) without going all the way back to the definition... |
H: Jordan form of matrices
So my professor gave me this question:
$A=\begin{pmatrix}
0 & 2 & 5\\
-5 & 5 & 10\\
2 & -2 & -4 \\
\end{pmatrix}$
I had to calculate $\forall 0 < i$ $kerA^{i}$ and $ImA^{i}$
So after calculating I reached those results:
$kerA$={$\begin{pmatrix}
1 \\
5 \\
-2 \\
\end{pmatrix... |
H: Result of $ \sum_{i=1}^{\infty} \frac{1}{(2i-1)^2}$
What is the result of
$$ \sum_{i=1}^{\infty} \frac{1}{(2i-1)^2}$$
I feel that the answer should be obvious, but somehow I can't find it. The series $$ 1 + \frac{1}{9} + \frac{1}{25}\ ... $$
doesn't look familiar to any other 'known' sequence. So how would you proc... |
H: Why is $ \lim_{n \to \infty} \left(\frac{n-1}{n+1} \right)^{2n+4} = \frac{1}{e^4}$
According to WolframAlpha, the limit of
$$ \lim_{n \to \infty} \left(\frac{n-1}{n+1} \right)^{2n+4} = \frac{1}{e^4}$$
and I wonder how this result is obtained.
My approach would be to divide both nominator and denominator by $n$, yie... |
H: Do these two definitions of Disconnectedness coincide?
Here are 2 different versions of the definition of Disconnectedness that I know of, one is taught in class, and the other one is the version in Gouvêa's $p$-adic numbers - An Introduction. But I just don't seem to be able to connect the 2 definitions.
The set $... |
H: Balls arrangements in a line
We randomly arrange balls numbered 1-100 in a line. What is the probability that there is a spot which splits the balls into two groups: All the balls preceding the splitting point are placed in ascending order (regarding their number), and all the balls from that point forward are plac... |
H: $||f-f_n||_{L^1} \rightarrow 0$ but $f_n \rightarrow f$ for no $x$
Show that there are $f\in L^1(\mathbb{R}^d)$ and a sequence $\{ {f_n}\}$ with ${f_n}\in L^1(\mathbb{R}^d)$ such that $||f-f_n||_{L^1} \rightarrow 0$ but $f_n \rightarrow f$ for no $x$
Thanks.
AI: Consider the sets
\begin{array}{llllllll}
S_{1,1}:=... |
H: Can we always find an analytic function if we know countable points?
I hope to find an analytic function such that $f(n)=b_{n}$, $n\in \mathbb{N}$?
Can we always take an analytic function $f$?
AI: Yes, you can find such a function.
We can do it inductively. Start with $f_0(x) = 0$, the constant function. For conven... |
H: A Basic question on intuition of rational cut set in the construction of real numbers
The intuition for cuts presumably comes from the standard experience of approximation by terminating decimals. For example, we can approximate $\sqrt{2}$ by the sequence $1,1.4,1.41,1.414,1.4142, \dots$. Now, why a cut set for $\s... |
H: Why is $X^4+1$ reducible over $\mathbb F_p$ with $p \geq 3,$ prime
I have proven that in $\mathbb F_{p^2}^*$ exists an element $\alpha$ with $\alpha^8 = 1$.
Let $f(X) := X^4+1 \in \mathbb F_p[X]$. How can I prove that $f$ is reducible over $\mathbb F_p$?
Has $f$ a zero in $\mathbb F_p$ ?
AI: I think I know, fin... |
H: Show that $ \mathbf u^2 \mathbf v^2 = (\mathbf u \cdot \mathbf v)^2 - (\mathbf u \wedge \mathbf v)^2 $
where $ \mathbf u $ and $ \mathbf v $ are vectors. From Linear and Geometric Algebra by Alan Macdonald.
AI: You can use
$$
\begin{align}
(u\cdot v)^2 - (u\wedge v)^2
& = \frac{1}{4} \left( (uv+vu)^2 - (uv-vu)^2\ri... |
H: norm induced by inner product and triangle inequality
Let $\langle\cdot,\cdot\rangle$ be a scalar product on a space $X$, and let $\lVert \cdot\rVert$ denote the norm induced by this scalar product. I need to show that for $x,y\in X$, $\lVert x+y\rVert=\lVert x\rVert+\lVert y\rVert $ holds $\Longleftrightarrow$ $x$... |
H: How to show this identity?
Suppose $A,B,A+B$ are invertible matrices (or just elements of some arbitrary, possibly non-commutative unital ring). I know that we have:
$$
(A-A(A+B)^{-1}A)^{-1}=A^{-1}+B^{-1}
$$
This is easy to check by direct calculation.
The question is, how can I reasonably arrive at the simplified... |
H: Finding the complementary language of a given language
I'm trying to figure out what's the complementary language of:
L = {w#w : w∈{a,b}*, |w| = k}
I think it's the language of all the words w#w where |w|!=k.
I think my answer is not correct. How should I think about this? And what is the correct answer?
AI: Hint: ... |
H: Prove that the polynomial divided by a fraction of the power of n is equal to the sum of fractions of any constans and successive powers of
Let n≥1 and n is integer. P(x) - polynomial and $deg P(x)<n$. Prove if $ a \in \Bbb R/{0} $ then:
$ \frac{P(x)}{(ax+b)^n} = \frac{c_1}{ax+b} + \frac{c_2}{(ax+b)^2}+...+\frac{c... |
H: Establishing equality between two functions $\mathbb{Z}_5[x]$; $a(x) = x^5 + 1$ and $b(x) = x-4$
Consider the following two polynomials in $\mathbb{Z}_5[x]$:*
$$
a(x) = x^5 + 1
$$
$$
b(x) = x - 4
$$
You may check that $a(0) = b(0), a(1) = b(1), \ldots, a(4) = b(4)$, hence $a(x)$ and $b(x)$ are equal functions
from... |
H: I'm looking for a function $f(x,y,z)$, which has partial derivatives only in single point
Function must be defined in $\mathbb R$. I know that Dirichlet function is involved somehow, but i still can't find out an example.
AI: As usually, let $D(x)$ be the characteristic function of the set of all rational numbers $... |
H: looking for a combinatorial interpretation
given positive integers $n,m$ does the fraction
$$
\frac{(nm)!}{n!^mm!}
$$
count something? Namely does it correspond to the number of possibilities to do something?
AI: This counts the number of partions of $nm$ items into $m$ groups of $n$ items.
$$\frac{(nm)!}{(n!)^m}=\... |
H: How to check if function has global extremes?
I need to determine if function:
$f(x,y)=x+2y-2\log(xy) $
has global minimum/maximum.
I've found local minimum at $(2,1)$, but that's not any proof of global minimum.
AI: Put $x=1/y$. You get $f(x,1/x)=x+2/x$. Approach $x$ to zero from below and from above. What do you ... |
H: Show that $e^x \geq (3/2) x^2$ for all non-negative $x$
I am attempting to solve a two-part problem, posed in Buck's Advanced Calculus on page 153. It asks "Show that $e^x \geq \frac{3}{2}x^2$ $\forall x\geq 0$. Can $3/2$ be replaced by a larger constant?"
This is after the section regarding Taylor polynomials, so... |
H: Why is the determinant the volume of a parallelepiped in any dimensions?
For $n = 2$, I can visualize that the determinant $n \times n$ matrix is the area of the parallelograms by actually calculating the area by coordinates. But how can one easily realize that it is true for any dimensions?
AI: If the column vect... |
H: Showing that if $\lim_{x\to\infty}xf(x)=l,$ then $\lim_{x\to\infty}f(x)=0.$
Let $l \in\mathbb{R} $ and $ f:(0,\infty) \to \Bbb R $ be a function such that $ \lim_{x\to \infty} xf(x)=l $ .
Prove that $ \lim_{x\to \infty} f(x)=0 $.
Any help would be appreciated - I found this difficult to prove
AI: $\lim\limits_{x\t... |
H: How to examine if multivariable functions are differentiable?
How to examine if functions:
$f(x,y)=|x+y|$
and
$g(x,y)=\sqrt{|xy|}$
are diffirentiable in points: $(0,0)$ for $f(x,y)$ and $(0,1)$ for $g(x,y)$
AI: Normally, if you suspect a function to be differentiable, the easiest is to show that its partial derivat... |
H: Simple operator proof
Let $A$ and $B$ be linear operators on vector space $\nu$. There exist an operator $X$ on $\nu$ for which $AX = B$ holds, only if $\text{img}(B) \subseteq \text{img}(A)$.
I would like to see the proof of this statement.
AI: Assume that such an $X$ exists, and let $y \in \mathrm{img}(B)$ so tha... |
H: Does every function have a value of 0 for the nth derivative?
Just out of curiosity, and for no other reason then I've been staring at the ceiling wondering, does every function have a value of zero at the nth derivative? And if it does, is there a function that can be used to tell you at which derivative the value... |
H: Prime ideal in the ring of polynomials
I'm trying to do the following:
Let $R = K[X,Y,Z]$ and $\mathfrak{p}$ = $(X+Y,Z^{2}-X)$. Show that $\mathfrak{p}$
is prime and find the transcendence degree of $R/\mathfrak{p}$.
If I prove that $\mathfrak{p}$ is prime the question is over just using the fact that the quoti... |
H: Figure out rate of change of speed for fixed distance
Say I have two points. I know the distance between those points. And let's say I have a bobbin that's starting at point A at a fixed speed of 100 jiggers/second
By point B, I want my bobbin to only be traveling 20 jiggers/second.
What equation can I use to figur... |
H: The wedge product
I have seen the wedge-product as being defined in differential geometry in the definition of a differential form or p-form. Now in the course we have proven the basic properties of this product and how to take the differential.
Now when we apply this to the differentiation of a function in on a cu... |
H: Calculate the highest possible chunk size
I need to find the higest possible value to multiply with in the following senario.
I have a collection of items, all items are of the same type and thay have a fixed
number of columns.
The problem is that i need to find out what the higest possible value that i can
multipl... |
H: How to evaluate a $A_{x\times y}B_{y\times z}$ where $ A$ and $B$ are matrices, $x\neq z$
I know how to evaluate a Cx,y * Dy,x (C rows are equals to D columns), but how do I evaluate a matrix multiplication in which the involved matrices (A and B) have respectively different number of rows and columns?
Here is the ... |
H: Prove that $q(a_i)\in \{a_1,..., a_n\}$
Let $p(x)$ and $q(x)$ be polynomials with rational coefficients such that $p(x)$ is irreducible over $\mathbb{Q}$. Let $a_1,..., a_n\in \mathbb{C}$ be the complex roots of $p$, and suppose that $q(a_1)=a_2$. Prove that $q(a_i)\in \{a_1,..., a_n\}$ for all $i\in {2, 3,..., n}$... |
H: Faithful group actions and dimensions
Just a quick question. I'm trying to understand the answer to one of my previous questions. The precise problem I want to show is as follows.
Let $G$ be a group acting faithfully on a manifold $X$. If $G$ is such that $\dim G$ makes sense (for example, $G$ is also a manifo... |
H: Prove if $|z|=|w|=1$, and $1+zw \neq 0$, then $ {{z+w} \over {1+zw}} $ is a real number
If $|z|=|w|=1$, and $1+zw \neq 0$, then $ {{z+w} \over {1+zw}} \in \Bbb R $
i found one link that had a similar problem.
Prove if $|z| < 1$ and $ |w| < 1$, then $|1-zw^*| \neq 0$ and $| {{z-w} \over {1-zw^*}}| < 1$
AI: HINT:
Le... |
H: How to build a $K(G,1)$ space for every group $G$?
I am reading the book Algebraic Topology by Allen Hatcher.
And in page 89 when he explains how to build a $K(G,1)$ space for every group $G$:
He builds a $\Delta$-complex with the $n$-simplices $\left[g_{0},\dots,g_{n}\right]
$ of elements of $G$, and he attaches ... |
H: Prove that in a tree with maximum degree $k$, there are at least $k$ leaves
Prove that in a tree with a maximum degree for each vertex is $k$, there are at least $k$ leaves.
So I said:
$2|E| = \sum_{v \in V} {\deg(v)} \leq k $ which is, if we say that we have AT MOST $k-1$ leaves (I used the contradiction method... |
H: Doubt on Binomial Series
Sorry to ask this simple question, but I am finding it hard to get a convincing answer
As we all know, binomial series is defined as
$$(1 + x)^k = 1 + \frac{k}{1!}x + \frac{k(k-1)}{2!} x^2 + \dots,$$
where $k$ is a real number and $-1 < x < 1$.
Why the restriction $-1 < x < 1$? Even withou... |
H: Numerical Methods Texts, mid-level.
what are the 3-4 numerical analysis/methods texts, that you value as best?
I would prefer a mid-level treatment, balanced between theory and applications.
Right now, I'm using the book by Sauer. It is pretty good, but I wish to couple it with a deeper treatment.
Many Thanks.
AI:... |
H: In $S_9$: for given $\sigma$ is there $\tau$ with $\tau^2=\sigma, \; \tau^3 = \sigma$?
Struggling with these:
Let $\sigma$ in $S_9$ be given by $\sigma=(8\,9)(5\,6\,7\,1\,2\,3\,4)$.
1: Is there a $\tau\in S_9$ with $\tau^2=\sigma\,?\;$ Tip: think of $ \epsilon ( \sigma)$.
2: Is there a $\tau\in \langle\sigma \rangl... |
H: Prove that the matrix is totally unimodular
Is there any (theoretic) way I can prove the matrix is totally unimodular? I have tested it by Matlab and know it is TU, however I cannot prove it.
-1 -1 -1 -1 0 0 0 0 0 0 0 0
0 0 0 0 -1 -1 -1 -1 0 0 0 0
0 0 0 0 0 0 0 0 -1 -1 -1 -1
1 1 1 1 0 ... |
H: generalisations of Lagrange's four-square theorem
For which positive integers $a, b, c, d$, any natural number $n$ can be represented as
$$n=ax^2+by^2+cz^2+dw^2$$
where $ x, y,z,w$ are integers?
Lagrange's four-square theorem states that $(a,b,c,d)=(1,1,1,1)$ works. Ramanujan proved that there are exactly $54$ ... |
H: The minimal polynomial of a primitive $p^{m}$-th root of unity over $\mathbb{Q}_p$
Proposition 7.13 of Neukirch's ANT states that for a primitive $p^{m}$-th root of unity $\zeta$ ($p$ prime) the extension $\mathbb{Q}_{p}(\zeta)/\mathbb{Q}_{p}$ is totally ramified of degree $(p-1)p^{m-1}$. In the proof he shows that... |
H: Chaotic iterative example needed
I'm using a very simple numerical method to find solutions to an equation. Start with an equation $\operatorname{f}(x)=0$ that you need to solve. Rearrange to give $x=\operatorname{g}(x)$ and then use the recurrence relation $x_{n+1} = \operatorname{g}(x_n)$ to hopefully tend toward... |
H: Test to see if a degree $\leq4$ polynomial is factorable
I'm in the middle of a programming project and we'd like to have tests to determine if polynomials in $\mathbb{Z}[x]$ of degrees up to 4 are factorable over $\mathbb{Q}$. A test that computes the discriminant (or something similar) and then asks if that is a ... |
H: Is the notation $[x,\to[$ common?
I recently started reading Topology and Groupoids by Ronald Brown and this notation came up. The notations is $$[x,\to[ \; =\{z \mid x \leq z\}$$ and a similar notation for other type of intervals. I have never seen this before, and I was baffled wondering if this was a funny $\LaT... |
H: Prove this block matrices are similar
Prove that the block matrices
$
\left(
\begin{array}{cc}
AB & 0\\
B & 0\\
\end{array}
\right)
$
and
$
\left(
\begin{array}{cc}
0 & 0\\
B & BA\\
\end{array}
\right)
$
are similar.
Where $\mathbf{K}$ is any field, $A\in \mathbf{K}^{m\times n}$, $B\in \mathbf{K}^{n\times m}$
and b... |
H: Notation of "defined for all complex numbers except the negative integers and zero"
The Euler and Weierstrass forms of the gamma function are :
$$\mathop{\mathrm{\Gamma}}\left(z\right)=\frac{1}{z}\prod^{\infty}_{n=1}\frac{\left(1+\frac{1}{n}\right)^{z}}{\left(1+\frac{z}{n}\right)}=\frac{e^{-\gamma z}}{z}\prod^{\inf... |
H: distinguishable and indistinguishable people and ticket offices
In how many ways we can arrange p people in the queue to the 5 ticet offices
a) people are distinguishable ticket offices are distinguishable
b) people are distinguishable ticket offices are indistinguishable
c) people are indistinguishable ticet offic... |
H: Characterization of the Subsets of Euclidean Space which are Homeomorphic to the Space Itself
I have no real experience in topology (although I have done a course in metric spaces) but in the course of a project I am doing it has become useful to produce (if possible) a characterization of the subsets of arbitrary ... |
H: Resolvent lemma
I would like to proof a lemma that I am quite sure should be correct as I found it somewhere, I am writing a thesis about quantum walks and need this to get through an article.
Let $X$ be Banach space, $A \in B(X)$, $\lambda \in \rho(A)$ (the resolvent set of the operator) I need to proof that:
$B... |
H: What is the maximum of a set of random variables?
I'm a bit confused by a problem I'm supposed to solve:
Given a sample of random variables $(X_1,...,X_n)$, uniformly distributed in the interval $[0,2\vartheta]$, $\vartheta>0$ and unknown. What value does the parameter $\gamma$ have to take for the estimator $$\the... |
H: Stochastic Automaton accepting every word with same probability
I am looking for a stochastic automaton, which induces the same probability $c \in [0,1]$ for all words in $\Sigma^*$, where $\Sigma$ is some finite alphabet.
A stochastic automaton over an alphabet $\Sigma$ is a tuple $\mathcal A$= { $\mathcal Q$, $\S... |
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