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H: Continuity - Function of 2 variables
Verify if the following function is continuous:
$$ f(x,y) = \left\{\begin{matrix}
\sqrt{1-x^2-y^2},\; if \;\;x^2 + y^2 \leq 1\\
0,if \;\; x^2 + y^2 > 1
\end{matrix}\right.$$
I think the only possible "problematic point" is where $x^2 + y^2 = 1$. So what should I do? Calculate ... |
H: $\dim U + \dim U^\perp = \dim V$. Example for being wrong when not nondegenerate
$\beta\colon V \times V \to K$ is a nondegenerate symmetric bilinear form and $U$ is a linear subspace of $V$. Then this equation is proven: $\dim U + \dim U^\perp = \dim V$.
I need to find an example of
(1) it being wrong, when $\beta... |
H: Let $\,g$ be the function defined on the set of real numbers by.....
I was thinking about the following problem which is as follows:
My Attempt: Case 1: Let $c$ be the arbitrary rational number.Let $\{x_n\}$ be the sequence of irrational numbers that converge to $c$. By density property that is $\exists$ an irra... |
H: Why do we subtract 1 when calculating permutations in a ring?
$10$ persons are to be arranged in a ring shape. Number of ways to do that is $9!.$
I wonder why we subtarct $1$ in all such cases.
I can imagine that if A,B,C,D are sitting in a row then B,C,D,A would give me a different combination but had they been ... |
H: Differentiable manifolds, uniqueness of maximal atlases and definition of smooth manifolds maps.
I have proved that given an atlas for a topological space $M$ that a maximal atlas containing $M$ is unique. But my proof would fail to generalise to the statement that a maximal atlas conatining a chart is unique. Is t... |
H: Value of $ f(2012)$
$f(x) $ is an injective function . The definition of $f(x)$ is like following:
$$
f:[0, \infty[\to \Bbb R-\{0\},
f\left(x + \frac{1}{f(y)}\right) = \frac{f(x)f(y)}{f(x) + f(y)} $$
If $f(0) = 1$ then what is the value of $ f(2012)$?
Can you help me to solve this problem ?
AI: $f\left(x+ \frac{1}{... |
H: Geometry Question - What is the length of the missing height?
Gary, $G$, can just see the top of a radio mast, $R$, over a wall $W$. Gary is $15\,\mathrm{m}$ from the wall. The wall is $45\,\mathrm{m}$ from the radio mast. The wall is $2.7\,\mathrm{m}$ high. Calculate the height of the radio mast, marked $h$ on the... |
H: Find $p$ and $q$ so that the integral converges
Find all values of $p$ and $q$ so that the below integral converges:
$$
I=\int_{0}^{1} x^p \left(\log\frac{1}{x}\right)^q\;\mathrm{d}x
$$
I tried and got the solution as:
$q\geq0$ and $p>q-1$
$-1<q<0$ and $p>-1$
Is it correct?
Solution:
After Substituting $x... |
H: How is the norm of a partition related to the norm of a vector?
Just finished a course in linear algebra, where the norm of a vector essentially was described as the length of the vector. In calculus, we just started talking about the definite integral of a function, where the norm of a partition came up, being def... |
H: Right Triangles and Altitudes
I am once again stuck on a question about geometry, this problem is about altitudes that crate right triangles:
Let there be a triangle that has side lengths of 13, 20, and 21. Given this, find the length of the altitude drawn to the side of length 21.
I have drawn the following pictur... |
H: Are these conditions enough to specify a unique number? (counting sylow p-subgroups)
Sylow's third theorem gives the two facts that the number of sylow p-subgroups $n_p$ of a group $G$, whose order we can write as $|G| = p^rm$ such that $p\not |\ m$ will satisfy both $n_p | m$ and $n_p \equiv 1\ (\mod\ p)$. I'm jus... |
H: Calculating a derivative using the chain rule vs. differentiating the composite
I have two functions $f:\mathbb{R} \rightarrow \mathbb{R}^2$ , $t \mapsto (t^3,t^2)$
and $g : \mathbb{R}^2 \rightarrow \mathbb{R}$ $(x,y) \mapsto (x^2+y^2)^{\alpha}$
Then we are asked to calculate $ (g \circ f)'(t)$ by using two differe... |
H: Intuitive explanation of sum^2
The following equation can be easily proved with induction (using $(a+b)^2=a^2+2ab+b^2$):
$$\left(\sum\limits_{i=1}^n a_i\right)^2 = \sum\limits_{i=1}^n a_i^2 + 2\sum\limits_{1\leq i<j\leq n} a_ia_j$$
Do you have some intuitive explanation why is this equation true?
AI: Added an illus... |
H: Integral constraints for positive function
Let $C={\cal C}([0,1],(0,\infty))$ denote the set of all continuous
maps $[0,1]\to (0,\infty)$. Let $g_1,g_2 \in C$ ; one can then define
$$
\begin{array}{rcl}
\Phi &: C& \to (0,\infty)^2 \\
f &\mapsto& \bigg(\int_{[0,1]} fg_1,\int_{[0,1]} fg_2\bigg) \\
\end{array}
$$
Obvi... |
H: Subharmonic functions in the punctured disk
I want to prove the following (exercise from Ahlfors' text):
If $\Omega$ is the punctured disk $0<|z|<1$ and if $f$ is given by $f(\zeta)=0$ for $|\zeta|=1$, $f(0)=1$, show that all functions $v \in \mathfrak B(f)$ are $\leq0$ in $\Omega$.
Here $\mathfrak B(f)$ is the c... |
H: Peculiarities about Finding Absolute Values
It is known that the formula $\sqrt{x^2}$ is equal to the value of $|x|$. In my spare time last night, I wondered about $\sqrt[3]{x^3}$. After some thought and some graphing, I came up with this:
If $x<0$, $\Im(\sqrt[3]{x^3})$
If $x=0$, $0$
If $x>0$, $\Re(\sqrt[3]{x^3... |
H: Proof of a Lemma guaranteeing the existence of the Borel-measurable functional calculus
In my lecture I had the following Lemma, which guarantees the existence of the Borel-measurable functional calculus:
Le $(H,<,>)$ be a complex Hilbert space and let $q:H\rightarrow \mathbb{C}$ be a function s.t.
1) $\vert q(x)\v... |
H: Projection matrix that sum to identity are orthogonal
Can anyone help me to show if $Z_1$, $Z_2$, and $Z_3$ are projection matrices (i.e. idempotent and symmetric) and if $Z_1 + Z_2 + Z_3 = I_n$, then we can conclude:
$$\forall i \ne j \text{ ,} Z_i Z_j = 0$$
It seems to be an easy problem but I was not able to so... |
H: Complex value of a divergent series
Given the series:
$$S=\sum_{k=1}^{\infty}\frac{2^k}{k^2}$$
the sum obviously doesn't converge.
'Maple' gives for the value of the series:
$$S(a)=\sum_{k=1}^{\infty}\frac{a^k}{k^a}$$
$S(a)=Li_a(a)$ with $Li$ polylogarithmic function.
For $a=2$, for example, $Li_2(2)\approx 2.46-2.... |
H: Proving basic lemmas about categories with finite products and terminal/initial objects.
I would expect that in any category $\mathcal{C}$ with finite products and a terminal object $1$, the isomorphism $X \times 1 \cong X$ should hold, but I have a rather hard time finding the proof of this.
In my attempt, I use t... |
H: Solution Verification - Combinatorial Card-Picking
I have a problem as such:
How many ways are there to choose nine cards out of a standard deck of 52 cards in such a way that every suit is represented in the selection at least twice?
Here's my solution:
Partition the set into the four suits - then pick two cards... |
H: Finding the image of $f(x)=\frac{1}{1+x^{2}}$
$f(x)=\frac{1}{1+x^{2}}$ and $x\geq0$
To find the image:
$y=f(x)$
$y=\frac{1}{1+x^{2}}$
$x=y^{-1}$
$x=\sqrt{y^{-1}-1}$
$y\geq1$
Then the image of $f(x)=\frac{1}{1+x^{2}}$ is $y\geq1$
Is this correct? If not, what information is missing?
AI: For $\sqrt{y^{-1}-1}$ to be ... |
H: You can write ${\left( {\frac{1}{2}} \right)^x}$ as ${2^{ - x}}$ , can the same be done with ${\left( {\frac{2}{3}} \right)^x}$?
You can write ${\left( {\frac{1}{2}} \right)^x}$ as ${2^{ - x}}$ as:
${\left( {\frac{1}{2}} \right)^x} = {({2^{ - 1}})^x} = {2^{ - x}}$
But what about ${\left( {\frac{2}{3}} \right)^x}$?
... |
H: Entire function that decays faster than exponential on reciprocals of integers is $0$
If $f(z)$ is entire and $|f(1/n)|\le e^{-n}$ for all $n\in\mathbb{N}$ then $f=0$.
My idea is to express $f$ as a power series centered at $0$ that converges on the entire complex plane, then look at
$$\left|\sum a_k/ n^k \sum n^... |
H: Intermediate Value Property
I am trying show that the function $f:[0,1]\to \mathbb{R}$ defined by $f(x)=\sin \dfrac{1}{x}$ if $x\neq 0$ and $f(0)=0$ possesses IVP. Though it looks easy, but I am not getting any clue how to start with. Any help would be appreciated.
AI: Show that there exists a subset $A$ of $(0,1]$... |
H: Banach Algebra: $\sigma(xy)\cup\{0\} = \sigma(yx)\cup\{0\}$
It is Rudin excercise 10.4 where we aim to prove $\sigma(xy)\cup\{0\} = \sigma(yx)\cup \{0\}$ for elements $x,y\in A$ a Banach-algebra.( $\sigma$ being the spectrum)
In (a) we prove that $e-yx$ invertible $\Leftrightarrow e-xy$ invertible.
Following the h... |
H: Is axiom of completeness an axiom?
The following statement is the axiom of completeness:
Every non empty subset of $\mathbb R$ that is bounded above has a least upper bound.
So I was wondering: is it an axiom or can it be proven? One can construct $\mathbb R$ as the set of equivalence classes of Cauchy sequence in... |
H: If $R$ is a UFD then $R[X,X^{-1}]$ is a UFD
Prove that the ring $R[X,X^{-1}]$ of Laurent polynomials over a UFD $R$ is a UFD.
I'd like someone to give a full proof.
AI: If $R$ is UFD, then $R[X]$ is UFD (see any textbook).
If $R$ is UFD and $f \in R \setminus \{0\}$, then $R[\frac{1}{f}]$ is UFD. The prime elemen... |
H: LinAlg Vector Graphing
I'm looking for a graphing calculator to graph vectors - NOT vector fields. A Google search turned up many great calculators with vector-valued function capabilities. However, I'm looking for something to help me visualize the concepts I'm learning in Linear Algebra, e.g. that combinations of... |
H: If $ a_1 a_2a_3 ...a_{20} = 2^x * y! $ Then what is the value of (x+y)?
Let us consider a series with $ a_1 = 2012 $ and $ a_n = \frac{n}{a_{n-1}} $ . If $ a_1 a_2a_3 ...a_{20} = 2^x * y! $ Then what is the value of (x+y) ?
AI: HINT:
As $a_na_{n-1}=n, a_{2r+1}a_{2r+2}=2r+2=2(r+1) $
$$\prod_{1\le r\le 20}a_r=\prod_... |
H: Expectation using a dirac measure as the probability measure
If $\varepsilon_a(dx)$ is the point mass at point $a\in\mathbb{R}$, I want to calculate $E_a[X]$ where $X$ is a real valued random variable and $E_a$ is expectation with respect to $\varepsilon_a(dx)$.
I'm not sure if my understanding is correct:
$$E_a[X]... |
H: $\sin ^6x+\cos ^6x=\frac{1}{8}\left(3\cos 4x+5\right)$, Any quick methods?
How to prove the following equation by a quick method?
\begin{eqnarray}
\\\sin ^6x+\cos ^6x=\frac{1}{8}\left(3\cos 4x+5\right)\\
\end{eqnarray}
If I use so much time to expand it and take extra care of the calculation process, I can find the... |
H: Strong explanation of Strong Form of Mathematical Induction
I don't quite understand induction well, and was wondering if you could explain to me what induction is and what the strong form of induction is.
AI: In regular induction, you assume some property holds true at some arbitrary index value, say $k$. You migh... |
H: Steady state and DE
Let $x_t = f (x_{t-1})$ a difference equation of order $1$, with $f(x) = ux(1-x)$; $u \in (0; 4)$.
Show that $x^* = 0$ is always a steady state for all $u$ .
Compute its positive steady state (meaning x? > 0) depending on .
Give two different values of $u$ such that in one case the steady s... |
H: How to simplify this equations?
A equation is given,
$$(7x-6)^3- (5x-6)^3-6x(7x-6)(5x-6)$$.
Will i use the formula $a^3 - b^3$ to simplify the above?
Any tips or solution will be appreciated.
AI: Observe that $7x-6-(5x-6)=2x$
so $6x(7x-6)(5x-6)$ can be written as $3(7x-6)(5x-6)\{7x-6-(5x-6)\}$
Use $(a-b)^3=a^3-b^3... |
H: Question on singular homology
please where i can found the prove of this:
If $X$ is a topological space and $(X_{\alpha})_{\alpha\in I}$ is the family of it's path connected components.
Prove that for each $n\in \mathbb{N}$, $$H_n(X;\mathbb{A})=\bigoplus_{\alpha\in I} H_n(X_{\alpha};\mathbb{A})$$
Please help me
th... |
H: Proving a function is onto and one to one
I'm reading up on how to prove if a function (represented by a formula) is one-to-one or onto, and I'm having some trouble understanding.
To prove if a function is one-to-one, it says that I have to show that for elements $a$ and $b$ in set $A$, if $f(a) = f(b)$, then $a = ... |
H: Solving for coefficients of a polynomial in terms of roots of another
This was a homework assignment a while back (already turned in) that I'm not totally sure how to approach. I think it might be simple but I can't wrap my mind around it.
If someone could give me a tip that would be great.
I'm given a polynomial $... |
H: I have ten professors, and need to pick four of them for a committee.
I have ten professors, and need to pick four of them for a committee. It is a bad idea for Hatfield and McCoy to serve together. It is a bad idea for El and Luthor to serve together. How many possible committee assignments are there?
I was thinki... |
H: How come this represents the height of a trapezoid?
Why does this formula
$$h= \frac{\sqrt{(-a+b+c+d)(a-b+c+d)(a-b+c-d)(a-b-c+d)}}{2|b-a|}$$
represent the height of a trapezoid?
Source: http://en.wikipedia.org/wiki/Trapezoid
Thanks!
AI: Draw a trapezoid with bases $a$ and $c$, so that the other two sides are $b$ an... |
H: why minimum of these functions happen at a special place?
why minimum of these functions happen at a special place? how to use derivative to find the minimum of these functions?
$$|x-1| + |x-2| + \dots + |x-9|$$ minimum is for $x = 5$
$$|x-1| + |x-2| + \dots + |x-99|$$ minimum is for $x = 50$
$$|x-1| + |x-2| + \dot... |
H: Finding a combinatorial proof of this identity: $n!=\sum_{i=0}^n \binom{n}{n-i}D_i$
Can someone prove this. Let $D_n$ be the number of derangements of $n$ objects. Find a combinatorial proof of the following identity:
$$n!=\sum_{i=0}^n \binom{n}{n-i}D_i$$
AI: You can divide all $n!$ permutations of $n$ objects ont... |
H: Compute C1, C2, C3, C4, C5
I have a question
C1 = 0, C$n$ = C$\lfloor n /2\rfloor$ + $n^2$ for all $n > 1$
Compute C1, C2, C3, C4
So what I did is:
C2 = C$\lfloor 2/2\rfloor + 2^2$ = 1 + 4 = 5 -- but that is wrong because the answer is supposed to be 4...
C4 = C$\lfloor 4/2\rfloor + 4 ^2$ = 2 + 16 = 18 -- but that... |
H: I have reached a rut in my understanding of control systems. How do I cross this?
A little background here. I'm an undergrad in the final year. I have decided academia as my career path. My grades are not high but my research caliber is good and I have ongoing projects that are promising. My field of interest is co... |
H: Notation for integer between two values
This may be a silly question, but it has been a long time since I have used set notation to any real extent. How would I write that $i$ is an integer ranging from $1$ to $N$?
My (possibly faulty) recollection is that this is expressed as $i \in \{ \mathbb{Z}: [1,N]\}$. Is t... |
H: Probability of a result from 3d6, lowest to highest
I have a fascination with tabletop sports games, and through that, I've developed an interest in probability. That said, it's not my strong suit, so I wanted to pose this question because I think involves a few different probability principles to solve it.
Here... |
H: Show that $(\phi_{n}^{(n)})^{-1}= -(\sum_{i=0}^{n-1}(\phi_{i}^{(n)})^{-1})$
So I have $n+1$ points $x_{0},x_{1},...,x_{n} \in \mathbb{R}$ and a following quasi-function:
$\phi_{j}^{(n)}=\prod_{i=0,i \neq j}^{n}(x_{j}-x_{i})$
Show that $(\phi_{n}^{(n)})^{-1}= -(\sum_{i=0}^{n-1}(\phi_{i}^{(n)})^{-1})$
AI: Hint: look... |
H: What situations/models require calculating the area under a curve?
Besides inferring distance traveled from a velocity chart,
can anyone name some graphs where you need to know the area under the curve?
For example, I know in Statistics (bell curve), the "probability density function" is used to determine what per... |
H: Inequality with the supremum
I am trying to prove the following statement for $A\subset \mathbb{R},~\epsilon>0$ with $A$ bounded above:
$\sup(A)-\epsilon<a\leq\sup(A)$, for some $a \in A$
I have tried dividing it into two cases.
Case 1: $\sup(A)\in A$. Then take $a=\sup(A)$ and we are done.
Case 2: $\sup(A)\notin ... |
H: Computing the condition to solve a set membership problem
I am stuck with the following problem for the last few weeks
Alex , Bob and Charlie stand for local election.
Given fractions
a - fraction of voters prefer Alex to Bob
b - fraction of voters prefer Bob to Charlie
c - fraction of voters prefer Charlie to Alex... |
H: Is *njwildberger* wrong about area and circumference of a circle?
In this video, njwildberger says that the area and circumference of a circle are proof-less theorems. But I heard that we can derive both the area and circumference of a circle using calculus? So are the area and circumference of a circle proof-less ... |
H: Why do I get 251 square dm the correct answer is 252 square dm? (Error by a fraction)
I am sorry if I am bothering you folks, I've recently started to play with Trigonometry, it's really cool, but trying to understand what mistakes I am making, anyhow, I am guessing that I shall use the Pythagorean Theorem.
Here'... |
H: Splitting field of $x^{4 }-3$ over $\mathbb{Q}$
I'm having trouble in understanding "the form" of a splitting field.
The problem is: Construct the splitting field over $\mathbb{Q}$ of the following polynomials.
One of the polynomials is $x^{4}-3$
So, the roots of this polynomial are $\pm \sqrt[4]{3}$ and $\pm \sqrt... |
H: Prove that isomorphic rings have the same characteristic
If $\phi: A \to B$ is a ring isomorphism, I have to proof that $\operatorname{char} A= \operatorname{char} B$.
I know that an isomorphism $\phi$ is bijective and:
$$\phi(x+y)=\phi(x)+ \phi(y)$$
$$\phi(x*y)=\phi(x)*\phi(y)$$
$$\phi(1)=1$$
I have supposed that... |
H: A new restaurant has opened. They have 100 items to chose from! Of those items…
I need some help with this problem.
A new restaurant has opened. They have 100 items to chose from! Of those items…
45 are fattening
45 are gross
44 are ice-cold
10 are both fattening and gross
18 are both fattening and ice-cold
13 are ... |
H: True or False? Continuous Functions
If the function $f+g:\mathbb{R}\rightarrow \mathbb{R}$ is continuous, then the functions $f:\mathbb{R}\rightarrow \mathbb{R}$ and $g:\mathbb{R}\rightarrow \mathbb{R}$ are also continuous.
False;
Let $f(x)=\begin{cases} -1 \text{ if } x<0 \\ 1 \text{ if } x\ge 0 \end{cases}$ $\h... |
H: Is this function twice differentiable at $0$?
I have a function $f(x)$:
$$f(x)=\frac{\exp(-|x|)}{1-0.5|\tanh(2x)|}$$
If I try differentiating it in Mathematica (taking $|x|=(2\theta(x)-1)x$ where $\theta(x)$ is Heaviside step function), I get answer in terms of Heaviside functions. But looking at the derivative I c... |
H: True or False? Continuous Functions #2
If functions $f+g:\mathbb{R}$ and $g:\mathbb{R} \rightarrow \mathbb{R}$ are continuous, then so is the function $f:\mathbb{R} \rightarrow \mathbb{R}$.
I feel like this is true, but I'm not sure how to justify my answer.
AI: You have only to remember that the (algebraic) sum of... |
H: What's wrong with this computation of Laurent Series?
I was solving this problem: find the two Laurent Series representations of $f: \mathbb{C}\setminus\{0,i,-i\}\to\mathbb{C}$ given by $f(z) = 1/z(z^2+1)$ in the correct domains.
My approach was: I've used partial fractions to rewrite $f$ as
$$f(z)=\dfrac{1}{z}-\df... |
H: Can every function which can be described by words, be formulated as well?
Almost one year ago i was amused when i saw this page. It was the generation of the prime numbers using the floor function, mostly. I became more interested about the things we can do with the floor function. For instance to calculate IsPrim... |
H: Compactness of closure
Let $A$ be a subset of a metric space $X$.
Assume that each sequence in $A$ has a convergent subsequence with limit in the closure $\overline{A}$.
Does this imply that $\overline{A}$ is compact? I read it on wikipedia but I couldn't find a proof of it anywhere.
AI: Yes, that does imply that $... |
H: How can I Prove $\frac{2xy}{x+y}\leq \sqrt{xy}\leq \frac{x+y}{2}$
for $x,y>0$ prove that
$\frac{2xy}{x+y}\leq \sqrt{xy}\leq \frac{x+y}{2}$
I have tried to develop $(x+y)^2=$ and to get to an expression that must be bigger than those above
Thanks!
AI: For example
$$\frac{2xy}{x+y}\le\sqrt{xy}\iff 4x^2y^2\le xy(x^2+2... |
H: prove that $T^2=T$ diagonalizeable without using Jordan
Question:
Given that $T:V \to V$ and $T^2=T$ prove that $T$ is diagonalizable.
What I know:
$T^2-T=0=T(T-I)$.
$\operatorname{Im}(T-I) \subseteq\operatorname{Ker}(T)$ therefore
$\dim V=\dim\operatorname{Ker}(T-I)+\dim\operatorname{Im}(T-I)\leqslant \dim\operat... |
H: Symmetric matrix with zero patern
I'm wondering if I can find some general formula for the inverse of such symmetric matrix :
$$\begin{bmatrix}1 & -k & 0 &-k\\ -k & 1 & -k & 0\\0 & -k & 1 & -k\\-k & 0 & -k & 1 \end{bmatrix}$$
which would still be valid for NxN generalization. The first line would be $$\begin{bmatri... |
H: Proof by Induction Question with regard to the Knight's Tour
I have to prove that the formula $4n^2-12n+8$ gives the number of edges on a knight graph, where n is the number of vertices horizontally and vertically and n^2 is the number of vertices.
I've proved it for $n=4$ (the smallest possible value of $n$ with w... |
H: Finding the limit of the recursive sequence $r_{n+1} = \sqrt{2 + r_n}$
In the example I am given, I am told that $r_n$ is defined as:
$$
r_n = \begin{cases}
r_0 = \sqrt{2} \\
r_{n + 1} = \sqrt{2 + r_n} \\
\end{cases}
$$
I was told to calculate $r_3$ and I found that to be:
$$
r_3 = \sqrt{2... |
H: Accumulation points of a countable set in [0,1]
Let A be a countable subset of [0,1]. Denote the set of accumulation points of A by A'. Can the set of accumulation points of A' be nonempty (i.e. can we have that A'' is nonempty)?
Thanks for your help
AI: HINT: Take $A=\Bbb Q\cap[0,1]$. |
H: Simplify $e^{\frac {(-\ln 2)}{2}}$
Simplify $\displaystyle e^{\large \frac {(-\ln2)}{2}}$
I know that $(-\ln 2)$ is $\ln\left(\frac{1}{2}\right)$ and the rule $e^{\ln x}=x$. How do I simplify with the fraction at the bottom?
AI: $$ \exp(\frac{- \ln 2}{2} ) = \exp( \ln2^{-{1/2}}) = 2^{-1/2} = \frac{1}{\sqrt{2}}$$ |
H: Derivation of Wallis's Formula
Use Euler's product formula $\Gamma (z)={1\over z}\prod_{n=1}^\infty({1+{1\over n}})^z({1+{z\over n}})^{-1}$ the fact that $\Gamma ({1\over 2})=\sqrt\pi$ to prove Wallis's Formula ${\pi\over 4}={2\over 3}\cdot {4\over 3}\cdot{4\over 5}\cdot{6\over 5}\cdots{2n\over 2n+1}\cdot{2n+2\over... |
H: How to find the probability there are 11 or more cars?
During rush hour the number of cars passing through a particular intersection has a Poisson distribution with an average of 540 per hour.
Find the probability there are 11 or more cars?
The answer is 0.006669.
I don not know how to deal with this kind of questi... |
H: Scheme-Theoretic Nakayama's Lemma
Let $X$ be a noetherian scheme and $\mathscr{F}$ a coherent $\mathscr{O}_{X}$-module. For a point $x \in X$, let $k(x)=\mathscr{O}_{X,x}/\mathfrak{m}_{x}$ be the residue field at $x$.
(a) Suppose $x \in X$ is a point such that $\mathscr{F}_{x} \otimes_{\mathscr{O}_{X,x}}k(x)=0$. S... |
H: Help on understanding Schwartz space
Can someone give an example of Schwartz space function that doesn't decay exponentially?
AI: Consider your favorite bump function construction and replace the exponential function by $$e(x)=\sum_{k=0}^\infty \frac{x^k}{\sqrt{k!}}$$ |
H: Set theory: what is other way to represent A\B?
I try to represent basic set operations using other operations with some limitations.
For example $A \setminus B$ using only $\cup$ and $\oplus$ (Symmetric difference):
$A \setminus B = A \oplus (A \oplus B) \oplus (A \cup B)$
But I stuck with the same set operation b... |
H: Prove that $b^2=a^2$
Let $G$ be a group of order $8$. Assume that there exists $a \in G$ such that $\lvert a\rvert =4$ and that no elements of $G$ has order $8$. Assume $\langle a \rangle \lhd G$, $b \notin \langle a\rangle$ and $b^2 \in \langle a\rangle$. Suppose that $\lvert b\rvert=4$, then Prove that $b^2=a^2... |
H: Maximal ideals of polynomial rings in infinitely many variables
Let $k$ be an algebraically closed field. Nullstellensatz states that the maximal ideals of the polynomial ring $R=k[X_1,\dots,X_n]$ are precisely those of the form $\langle X_1-a_1,\dots,X_n-a_n\rangle$, with $(a_1,\dots,a_n)\in k^n$.
What if we are ... |
H: Cofactors and conjugates of $SU(3)$.
I was playing around with some equations and noticed the following: Let $A$ be an element of $SU(3)$ with components $A_{ij}$. If $C_{ij}$ is the $(i,j)$ cofactor of $A$ then $C_{ij} = \overline{A_{ij}}$.
Now this is probably not surprising. Since $A \in SU(3)$ we know that its ... |
H: A norm for Lipschitz-continuous functions
I am a first year undergrad math student, and I am struggling with a proof. Let the set $C_{\text{Lip}}:= \left \{ f:\mathbb{R}\rightarrow \mathbb{R}: f \text{ is Lipschitz continuous} \right \}$ be a Vectorspace over $\mathbb{R}$. For $f \in C_{\text{Lip}}$ we define
$$
c... |
H: proof- can NOT be a linear combination
How can I prove that $X^2-Y,X-Y^2$ CAN NOT be written as a combination of $<X^3-Y^3,X^2Y-X>$ ?
AI: Suppose $X^2-Y=p(X,Y)(X^3-Y^3)+q(X,Y)(X^2Y-X)$. Evaluating at $X=0$ gives
$Y=p(0,Y)Y^3$, which is impossible. The same technique works for $X-Y^2$. |
H: Evaluation of Standard Normal Integral
I have always wondered how we calculate the percentiles of the Standard Normal Distribution given that the CDF cannot be obtained in closed form:
$$F(x)=\int_{-\infty}^{x} \frac{1}{\sqrt{2\pi}} e^{-\frac{t^2}{2}} $$
Do we approximate the function somehow? I have seen some text... |
H: Taking the cross product of a cross product? Proving an identity that involves gradients and vectors?
Problem 20:
Solution:
I am having difficulty understanding how the boxed is not equal to 0. The derivative of 1 is equal to 0.
AI: Let's back up. What you're really having trouble with is understanding
$$\nabla ... |
H: Does the limit of this sequence exist?
While practicing for an exam, I encountered this question:
(e) For a sequence $\lbrace b_n \rbrace_{n=1}^\infty \subset \mathbb{R}$ is given: $\forall n \in \mathbb{N}$, $\: b_n < b_{n+1} < 2$.
Does $\lim_{n\to\infty} b_n$ exist? If $\lim_{n\to\infty}$ exists, give $\lim_{n\to... |
H: If a closed, smooth $m-1$ form, $\omega$ is nonzero at a point, there are local coordinates $x^i$ with $\omega = dx^2 \wedge\cdots \wedge dx^m.$
This is a problem on an old qualifying exam.
Let $\omega$ be a smooth, closed $m-1$ form on a smooth $m$-dimensional manifold $M$. If $\omega \neq 0$ at a point $p\in M$ ... |
H: Transitive closure of a union
I have a quick question regarding transitive closures. In the text I'm currently reading, (Kunen - Set Theory, 2011), the transitive closure of a set $x$ is defined as trcl$(x) = \{ a : a \in^* x \}$, where $a \in^* x$ means there is an $\in$-path from $a$ to $x$ (I think the text impl... |
H: How can I prove that 4k^2 mod 3 is always = 1
I have a statement $n \in N, \;n^2 \mod 3 = \{0, 1\}$, which basically says that any natural number $n$ when squared will have a remainder after dividing by $3$ of either $0$ or $1$.
From here I expended my proof into two cases
$n = 2k, n^2 = 4k^2$
and
$n = 2k + 1, n^2 ... |
H: Closure of the range of a compact operator
Let $X$ be an infinite-dimensional Banach space, and let $Y$ be a banach. Let $T$ be a compact operator from $X$ to $Y$, ie. if $(x_n)$ is a sequence in $X$ then there is a subsequence s.t. $T(x_{n(k)})$ converges. We wish to show that 0 is in the closure of $\{Tx,||x||=1\... |
H: What is the probability there are two or more claims?
In a group of policy holders for house insurance,the average number of claims per $100$ policies per year is $\lambda=8.0$. The number of claims for an individual policy holder is assumed to follow a Poisson distribution.
In a group of $20$ policy holders,what ... |
H: Proving a Property of a Set of Positive Integers
I have a question as such:
A set $\{a_1, \ldots , a_n \}$ of positive integers is nice iff
there are no non-trivial (i.e. those in which at least one component
is different from $0$) solutions to the equation $$a_1x_1 + \ldots +
> a_nx_n = 0$$ with $x_1 \ldots x... |
H: Converting second order equation to first order equation
How would I convert the following second order equation to a first order?
$$ u'' + 3u' - 4.5u = -2.5\sin(3t) $$
I have let $v=u'(t)$, but not sure what $v'(t)$ would look like.
$$ u' = v $$
$$ v' = ??? $$
My attempt:
$$ u'' = v' = -2.5sin3t - 3v + 4.5u$$
Sure... |
H: Is it possible for a relation to be symmetric, antisymmetric, but NOT reflexive?
If $A$ is a set $\{2,4,6,8\}$, and we are asked to give a relation on $A$ that is:
symmetric, antisymmetric, but not reflexive, is this possible?
If we were to say $\{(2,2),(4,4)\}$, it would indeed be symmetric and antisymmetric, but ... |
H: Why is the empty family linearly independent?
Why is the empty family linearly independent?
How can you prove this? Is it right to say that it can not be written as a linear combination of the others vectors?
AI: Any subset of a linearly independent set is linearly independent. This is almost obvious for non empty... |
H: Decreasing sequence of measurable functions
Suppose $f_1(x),f_2(x),\ldots:[0,1]\rightarrow\mathbb{R}$ are measurable functions such that $f_1(x)\geq f_2(x)\geq\ldots$. (infinite sequence) and $\lim_{n\rightarrow\infty}f_n(x)=0$. Is it true that $\lim_{n\rightarrow \infty}\int_0^1 f_n(x)dx=0$?
I wanted to apply the ... |
H: Greatest lower bound and meet-semilattice of set
I am wondering if I am understanding this concept correctly.
A partially ordered set $S$ is called a meet-semilattice if, for all $x,y \in S$, a unique greatest lower bound of $x$ and $y$ exists.
So let's say we have the set $S = \{1,2,3,12,36\}$, and we want to ask ... |
H: Integrals of probability density functions and their inter-relationships
I am a little bit confused about the relationship between marginal probability density functions (pdfs), joint pdfs (jpdfs), and conditional pdfs (cpdfs), and their integrals. Let me define the following pdfs: $f_X\left(x\right), f_Y\left(y\ri... |
H: The closure of the complement of $A \subseteq \mathbb{R}^d$ with Lebesgue measure zero is $\mathbb{R}^d$?
I have been working on an excercise in measure theory for a few hours now, and although I have learned a lot, the answer to this problem avoids me. It concerns proving the following assertion:
Let $A \subsete... |
H: Integral Of $\int \frac{2\cdot \cos^2(x)}{x^2}dx$
I`m trying to integrate the following:
$$\int\frac{2\cdot \cos^2(x)}{x^2}dx$$
what I did first is:
$$\int \frac{2\cdot (\frac{1}{2}+\frac{cos2x}{2})}{x^2}dx=\int \frac{1+cos2x}{x^2}dx$$
now what? any suggestions? thanks!
AI: Using a simple integration by parts, we a... |
H: Question about basic exponential/logarithm properties
Solve for $k$: $$e^{k/2}=a$$
Solution: $$e^{2k}=a$$
$$ k/2 = \mathbf{ln}a$$
$$ k=2\mathbf{ln}a$$
$$= \mathbf{ln}a^2$$
My question is: why does $2\mathbf{ln}a = \mathbf{ln}a^2$? Why can you transfer the $2$ to be an exponent of $a$?
AI: Because logarithms map... |
H: Intersection of a plane with an infinite right circular cylinder by means of coordinates
So, I started studying analytic geometry and I must say I'm finding it much harder than "classic" geometry, because of the equations without help from diagrams... Still, I wanted to see how to use it to get alternative proofs o... |
H: Functions are big only over exponentially small sets
Given $f:[0,1]\rightarrow\mathbb{R}$. Suppose $\mu(\{x\in[0,1]\mid |f_n(x)|>1/n\})\leq 1/2^n$ for all integers $n\geq 1$. Is it true that $\lim_{n\rightarrow\infty}f_n(x)=0$ for almost every $x$?
I tried to apply the Chebyshev's inequality, but it doesn't work ou... |
H: Why is $2^x>2x$ when $x>2$?
I came across this inequality while trying to prove that a function was increasing for $x>2$. I checked it graphically and it's true but I don't have a particularly good proof of it.
So, can you tell me why $2^x>2x$ when $x>2$?
AI: You can just take the derivative: $\frac d{dx}(2^x-2x)... |
H: How to factor these monomials?
This is the original problem:
$x^3+x^2y+xy^2+y^3$
Answer:
$(x+y)(x^2+y^2)$
I understand that the answer is correct, but I can't figure out how to get to it.
AI: $$\underbrace{x^3+x^2y}_{\text{Group these two}}+\overbrace{xy^2+y^3}^{\text{Group these two}} = x^2(x+y) + (x+y)y^2 = (x+y)... |
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