instruction stringlengths 22 27.2k |
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I was wondering if there was a "formula" or an "identity" for $\cos(x^2)$, as there is for $\cos(2x)$. My question is closely related to this one, which was only asking for $\cos(ab)$. For instance $$\cos(x^2)= \frac{\cos^3(x)+\sin^3(x)}{2\cos^2(x)+2016}$$ could be such a formula.
More precisely, I would like to know ... |
In many commutative algebra texts, Noether Normalization Lemma is proved and then Hilbert's Nullstellensatz is obtained as a corollary.
Nullstellensatz and Normalization Lemma seem to be non-trivial theorems in their own right. So, can we prove Noether Normalization Lemma after assuming Hilbert's Nullstellensatz.
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$$\lim_{n \to \infty }\int_{0}^{n}\frac{n \cdot e^{\frac{x}{n}}}{x^4+n^2}dx=?$$
I am allowed to used all the classical techniques of calculus, and this was a question from measure theory when we were studying theorem of dominated convergence, monotonic convergence. Usually I would prove that some $f_n(x)$ is rising and... |
Want to show
$$\displaystyle\bigcup^\infty_{k=1}\left(\bigcap^\infty_{n=1}A_{k,n}\right)\subset\bigcap^\infty_{n=1}\left(\bigcup^\infty_{k=1}A_{k,n}\right)$$
Note the bottoms are $k=1,n=1$ and $n=1,k=1$, rather than $k\geq n, n=1$; $n\geq k,k=1$.
I did it via the same way like $\limsup$ and $\liminf$, but failed.
$\dis... |
I have been struggling with this problem, and I think it should be possible to solve but right now I cannot find how.
Given two coordinates/points (x1,y1) and (x2,y2), the angle formed by legs d1 and d2 is 90°.
What are the lengths of d1 and d2?
Any help would be appreciated.
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The integration of Chebyshev polynomials of the first kind has the following value,
$$\int T_{n}(x) \, dx = \frac{1}{2} \, \left( \frac{T_{n+1}(x)}{n+1} - \frac{T_{n-1}(x)}{n-1} \right)$$
what happens when n=1? We have zero division.
Best, Mohd
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Consider the integral: $I$= $\int_2^{\infty} f(x)g(x) \mathrm{d}x $, where $f(x)$ and$g(x)$ are nonconstant functions. Assuming that this integral exists, does this necessarily mean that $f(x)$ and $g(x)$ are differentiable functions of $x$ ?
Personally, i think the answer is yes, since we can't evaluate $I$ by integra... |
Exercise 1.3.40.a (Fraleigh - Linear Algebra):
Prove that , if $A$ is a square matrix, then $(A^{2})^{T}=(A^{T})^{2}$ and $(A^{3})^{T}=(A^{T})^{3}$.
Solution:
In general we have:
$(A^{n})^{T}=(A^{T})^{n}$
$\Big \langle Matrix\ definition \Big \rangle$
$((A_{ij})^{n})^{T}$
$\Big \langle Matrix\ power \Big \rangle$
$((A_... |
I'm uncertain about my proof about this exercise regarding nilpotent groups. If someone could me help me out, that would be appreciated. There's a post about this problem, but it uses another definition.
A group $G$ is called nilpotent if $C^n (G)=\left\{1\right\}$ for some $n\geq 1$, where $C^1(G):=G$ and $C^{i+1}(G):... |
Suppose $R$ is a finite ring. It may be commutative or it may be not. Let $x \in R$.
Show that there exists a positive integer $n$ such that $x^n = x^{2n}$.
Is it true that $x^{k!} = x^{2k!}$, where $k = |R|$?
In the first part should we take random $x$ and consider its powers, then use pigeonhole principle, to show... |
Prove the convergence of the sequence $(z_n)$ such that :
$$ z_1=\frac{3}{2}$$ $$z_{n}=\sqrt{3z_{n-1}-2}$$ for every $ n \geq 2$.
Calculate also the limit.
I have applied induction: $$\frac{3}{2} \le z_n \le 2 \ \ \ \ \forall n \in\mathbb{N}$$
The sequence is strictly increasing and it's also bounded so it converges ... |
Let $\{X(t)\}$ be birth-death process on a finite state {0,1,2} with non negative birth rates $(\lambda_0,\lambda_1)$and death rates $(\mu_1,\mu_2)$. Suppose $\mathbb{P}(X(0)=0)=1$ and $s_0=\inf \{t\geq0:X(t)=1\}$ and $s = \inf\{t\geq s_0:X(t)=0\}$. How to calculate the $\mathbb{E}[s]$?
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Good day.
I am trying to solve the following equation:
$$\ddot{y}(x)-\frac{A}{x}\dot{y}(x)+\frac{Bx^2}{2}y(x)=0.$$
WolframAlpha says it is an Emden‐Fowler equation, but I have no idea how to solve this. Can you give me some tips?
In case of $$A=B/2=1$$ WolframALpha gives an analytical solution $$y(x)=c_1 sin\frac{x^2}{... |
I'm trying to integrate the 2-form $\omega = A(y) dx \wedge dy - dx \wedge dz + B(z)dz \wedge dy $ in the set $R_f=\{(x,y,z),\quad z=f(x,y)\quad x^2+y^2 \neq 1 \}$ with $f$ a differentiable function from $\mathbb{R}^2$ to $\mathbb{R}$ such that $f(x,y)=0$ for $(x,y)$ such that $x^2+y^2=1$.
To do this I found $\theta = ... |
We need to find the maximum point of the curve, M. I know that we must find the $dy/dx$ of the equation of the curve and set it to $0$. However, I'm having trouble differentiating the equation. I'm aware of the product/quotient rule, but didn't get the correct answer despite trying several times.
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Let $A$ be a C*-algebra. Any representation $\pi:A\to B(H)$ is uniquely extended to a $w^*$-continuous representation $\tilde{\pi}:A^{**}\to B(H)$.
Q: I am looking for an example of a faithful representation $\pi$ such that $\tilde{\pi}$ is no longer faithful.
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Let $\pi(x)$ denote the number of primes not exceeding $x$. Is $\pi(x)$ differentiable ?
My attempt: It is well known that $\log \zeta(s) = \int_{2}^{\infty} \dfrac{s\pi(x)}{x(x^s - x)} \mathrm d{x}$ where $\Re(s)\geq 2$. Since we can't integrate this by parts without differentiating $\pi(x)$, the answer seems to be y... |
I'm trying to find the $\gcd$ of
$A:= x^4 - x^3 + x^2 - x + 1$ and
$B:= x^2 + 2$.
Using the Euclidean algorithm, I've found
$$(x^4 - x^3 + x^2 - x + 1)=(x^2 - x - 1)(x^2 + 2) + (x + 3)$$
and
$$(x^2 + 2)=(x - 3)(x + 3) + 11$$
Do I then need to continue using the algorithm to get
\begin{align}
(x + 3)&=(x/11)(11) + 3\... |
The connection of the line bundle $\mathcal{O}(1)$ on $\mathbb{CP}^1$ is given by
\begin{equation}
A=\frac{i}{2}\frac{\overline{z} \, dz-z\,d\overline{z}}{1+|z|^2}
\end{equation}
This follows since the first Chern class is represented by the curvature
\begin{equation}
\frac{i}{2\pi}idA=\frac{i}{2\pi}\frac{dz\wedge d\... |
Given a decreasing sequence $a_n$ of positive real numbers, for $x>0$ define
$$
f(x)=\sum_n \min\left(\frac{x}{a_n}, \frac{a_n}{x}\right).
$$
Can $a_n$ be chosen so that $f(x)\to 0$ as $x\to 0$?
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Source: p 224, Think: A Compelling Introduction to Philosophy (1 ed, 1999) by Simon Blackburn.
I capitalised miniscules, which the author uses for variables.
I pursue only intuition; please do not answer with formal proofs. Discussing Bayes's Theorem
( $
\Pr(H|E)=\frac{\Pr(E|H)\Pr(H)}{\Pr(E)}
$ ), the author abbrevia... |
This is my definition of a smooth manifold I am supposed to work with:
Let $\mathcal{M} \subseteq \mathbb{R}^{n}$. The set $\mathcal{M}$ is a $k$-dimensional smooth submanifold of $\mathbb{R}^{n}$ if:
$\mathcal{M}$ is given locally as the zero set of a function $F\colon W \to \mathbb{R}^{n-k}$, where $W \subseteq \math... |
Derivative of $(r^2 - x^2)^{1/2}$ with respect to x? Please show the steps also.
|
I am working on trying to prove this claim:
Consider a nonzero vector v in Rn. What is the dimension of the space of all vectors in Rn that are perpendicular to v?
I think I intuitively understand it. Through orthogonal projection, one dimension collapses, resulting in a dimension of n-1. But, what would a more rigoro... |
I am out of practice with logs, but this is derived from the channel capacity theorem.
$$B\log_2\left(1 + \frac SN\right)$$
Solve for $x $
$$\log_2(2^n) = \log_2(1+x)$$
I need this equation manipulated so that $x$ is the answer.
thanks!!!
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In my real analysis text, an example of proof by induction is given by proving that for any real $x\ge 0$ and all integers $n\ge 0$ $$(1+x)^n \ge 1+nx+\frac {n(n-1)}2x^2$$
I can follow and understand the proof but then the next line after the proof is
It follows that $$(1+x)^n\ge 1+nx$$ and $$(1+x)^n \gt \frac{n(n-1)}... |
Recently I am reading Manfredo do Carmo's Differential Geometry of Curves and Surfaces. He said the $q$ is the conjugate point of $p$ with respect to a geodesic $\gamma$ joining the two points if there exists a Jacobi field which vanishes on the $p$ and $q$. (Here he assumes all the surfaces are complete).
He gives thi... |
If $$P=\begin{bmatrix}
\cos\theta & \sin\theta \\
-\sin\theta & \cos\theta
\end{bmatrix}\text{ and } A=\begin{bmatrix}1 & 1\\0 & 1\end{bmatrix}$$
And $Q=PAP'$ then $P'Q^{2009}P=?$
I noticed that $P$ is an orthogonal matrix so
$$P'=P^{-1}$$ and $P^2(\theta)=P(2\theta)$
$$Q^{2009}=P^{2009}A^{2009}(P^{-1})^{2009}$$
I... |
If we have $G = GL_2(R)$, $H = SL_2(R)$ and $A \in G$ such that $\det A = 2$.
We have to prove that $AH$ is the set of all $2 \times 2$ matrices in G that have determinant 2.
Can we simply say, oke $AH = \{AB : B \in SL_2(R)\} = \{AB : \det B=1\} = \{C=AB : \det C = \det A \cdot \det B = 2\}$ or am I missing something?... |
Source: p 224, Think: A Compelling Introduction to Philosophy (1 ed, 1999) by Simon Blackburn.
I capitalised miniscules, which the author uses for variables.
I pursue only intuition; please do not answer with formal proofs. Discussing Bayes's Theorem
( $
\Pr(H|E)=\frac{\Pr(E|H)\Pr(H)}{\Pr(E)}
$ ), the author abbrevia... |
We consider such $a\in\mathbb N$, $b\in\mathbb Z$, o numbers of the form $2^na+b$ is square to the largest possible number of values of $n=1,2,3,4,\ldots$.
It is easy to see that for $a = 60 $, $ b = -119 $ produced four squares: $2a+b=1=1^2$, $4a+b=121=11^2$, $8a+b=361=19^2$, $16a+b=841=29^2$. The next value $ 32a + ... |
I'm studying Solving Systems of Linear Equations (Fraleigh - Linear Algebra), and I'm wondering for the meaning of the grayed are which appears in the following image:
The author adds:
We know that a single linear equation in two unknowns has a line in the the plane as its solution set. Similarly, a single linear equa... |
If I have a composition of two functions:
$$y = f(g(x),h(x))$$
where both $g(x)$ and $h(x)$ are readily invertible, can I find the inverse of the composition? i.e.: Can I find $x = f^{-1}(y)$? I know this is generally possible for the the composition of one function.
Perhaps there is a special form of f that permits t... |
What is the solution to the following? It's sort of a modified/generalized geometric progression (or is there a known name for this kind?),
$$\lim_{N\to\infty}\prod_{n=0}^{N}(1+a^n), a\in(0,1)$$
|
I am learning about flow and came across the group action formal definition of flow on Wikipedia. First of all, why is it a group action of the real numbers on the set of particle positions? Is this because it corresponds to linear movement, or multiplication by a scalar? Does this definition break down if the underlyi... |
Explain how $56.42(4+3)^2=5642$ can be a true statement. I understand that we essentially need $(4+3)^2$ to act as $100$ in order for $56.42$ to become $5642$. However, if we operate in base $7$ and treat $4+3$ as $10$, then $100$ would not be the same result.
Please advise.
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In jug a there are $3$ black balls and $2$ red balls,In jug b there are $2$ black balls and $3$ red balls, In jug c there are $3$ black balls and $3$ red balls:
$$\text{Jug a: }\underline{\bigcirc\times3\color{red}\bigcirc\times2}\\
\text{Jug b: }\underline{\bigcirc\times2\color{red}\bigcirc\times3}\\
\text{Jug c: }\u... |
I have a question about a proof of a theorem of reduced quadratic forms. Let me explain it!
I'm going to denote the form $f=aX^2+bXY+cY^2$ as $f=(a,b,c)$. The theorem I want to prove says:
Each form $f=(a,b,c)$ is equivalent to a form $f'=(a',b',c')$ such that $|b'|\leq |a'|\leq |c'|$. The form $f'$ is said reduced for... |
I am reading a book on matrix Lie algebras (Brian Hall's). Corollary 2.30. says that if $G$ is a connected matrix Lie group, then every element $A$ of $G$ can be written in the form
$$A=e^{X_1}e^{X_2}\ldots{}e^{X_m}$$
for some $X_1$,$X_2$$\ldots$$X_m$ in the Lie algebra. Immeadiately after it is stressed that even if... |
Determine number of real roots of $f(x)=x^{5}+x^{3}-2x+1$.
I used Descartes's rule of sign to determine the number of positive and negative roots; there is only 1 negative (hence minimum 1 real root) and two positive roots. These 2 positive roots can be anything- a complex pair, or two real roots- but what about the ... |
Is there any way to find area of shaded region?
The radii of circles are $4$ and $12$ units.
|
A linear functional $T$ on $\mathcal{D}(\Omega)$ is a distribution if $\phi_n \to 0$ in $\mathcal{D}(\Omega)$ $\Rightarrow$ $T(\phi_n) \to 0$ in $\mathbb{R}$.
But I cannot find what the order of a distribution means.
|
If we have a urn with $N$ balls of two colours ($D$ red and $N-D$ black balls respectively), then probability of having $k$ red out of $n$ balls drawn at once without replacement follows the Hypergeometric distribution:
$Pr(X = k) = \dfrac{\binom{D}{k} \binom{N - D}{n-k}}{\binom{N}{n}}$
Now assume we have some a priori... |
Let $h\in L^1[0,1]$. Prove that there is a unique solution (almost everywhere) of the following integral equation:
$$f(x)=h(x)+\frac{1}{2}\int_0^x\log(1+f(y)^2)dy$$
The idea is to use the fixed-point theorem for Banach spaces.
So let's consider the map:
$$T:L^1[0,1]\to L^1[0,1],f(x)\mapsto h(x)+\frac{1}{2}\int_0^x\log(... |
Question is :
Let $n$ be a natural number not divisible by $17$. Prove that either $n^8+1$ or $n^8-1$ is divisible by $17$.
I tried to solve using Fermat theorem for a prime number $p$, and any integer $a$, $a^p-a$ is divisible by $p$.
Then $n^{17}-n$ is divisible by $17$.
Or $n(n^{16}-1)$ is divisible by $17$.
But her... |
I'm reading through Dummit and Foote Abstract Algebra, and I was looking for an explanation of the following:
Proposition 9: $F$ a field and $p(x)\in F[x]$. Then $p(x)$ has a factor of degree one if and only if $p(x)$ has a root in $F$.
Proposition 10: a polynomial of degree 2 or 3 over a field $F$ is reducible if and ... |
The Dedekind number $M(n)$ is the number of antichains in the partial order of subsets of $\{1,\dotsc,n\}$. It is only known for $0 \leq n \leq 8$.
Question. What are some known upper and lower bounds for $M(9)$?
|
Definition of $\lim_{x \to a} f(x) = L$:
$\forall \epsilon > 0, \exists \delta > 0 s.t. |f(x) - L| < \epsilon$
$ if \ 0 < |x-a| < \delta$
Question: Why can't we weaken the assumption to
$\exists N > 0$ s.t.
$\forall \epsilon \in (0, N), \exists \delta > 0 s.t. |f(x) - L| < \epsilon$
$ if \ 0 < |x-a| < \delta$
?
I t... |
Why can't you use Cantor's diagonalization argument to prove that the integers are countably infinite?
i.e.
1: 12345....
2: 42345....
3: 56903...
4: 46234...
5: 23421...
etc.
Then we could create a new integer by adding 1 to each number in the diagonal, so the new integer would be 23042. Where did I go wrong in my reas... |
How many options are there to award gold, silver, and bronze medals to a group of $10$ athletes?
Is this permutation or combination, and is there repetition?
I thought this would be a combination without repetition, where I found the formula to be: $\frac{n!}{(n-r)!} = \frac{10!}{7!} = 720$ I'm doubting that this is ... |
In school i sometimes run on some excercises where you need to calculate something that has an inverse function in it but you cannot find the inverse and you need to work your way around it. I know that the inverse function is symmetrical to the original one through the $y=x$, so i figured that if you rotate the functi... |
I already know that $f_{n+m}=f_{n-1}f_m + f_nf_{m+1}$.
By letting $m=n$ it immediately follows that $f_{2n}=f_{n}(f_{n+1} + f_{n-1})$ and from that we get $f_{2n}=f^2_{n+1} - f^2_{n-1}$.
From this one should be able to see that $f_{3n}=f^3_{n+1} + f^3_n - f^3_{n-1}$, but I somehow can't figure it out.
|
Why is $\{\emptyset\}$ not a subset of $\{\{\emptyset\}\}$?
It contains this element, but why is it not a subset?
|
Give an example of a Lipschitz function $f$ on $[0,\infty)$ such that its square $f^2$ is not Lipschitz.
My example is to let $f(x) = x$ for $x \in [0, \infty$)
Then $|f(x) - f(u)|$ = $|x - u|$ $\implies$ $f$ is Lipschitz.
By definition a function $f$ is Lipschitz given a set $A \subseteq \Bbb R$ and $f: A \rightarrow ... |
Having trouble trying to differentiate this.
$y(t)=e^{it} + \alpha\int_{t}^{\infty} sin(t-s)\frac{y(s)}{s^2} ds $
$y'(t)=ie^{it}+\alpha\frac{d}{dt}(\lim_{a\to\infty}\int_{t}^{a}sin(t-s)\frac{y(s)}{s^2}ds$)
Can I bring the derivative inside the limit?
|
I'm asked to count the number of functions $f:\{1,2,3,4\} \rightarrow\{1,2,3,4,5\}$ such that $f(1)∉\{f(2),f(3),f(4)\}, f(2)\neq f(3), f(3) \neq f(4)$. How do I make the inclusion-exclusion argument needed to solve this problem?
I'd also like to know how to make a venn diagram with these sets so I can visualize them. A... |
Show that if $f$ is continuous on $[0, \infty)$ and uniformly continuous $[a, \infty)$ for some positive constant $a$, then $f$ is uniformly continuous on $[0, \infty)$.
Here is my attempt at the proof. A classmate is concerned my proof leaves a gap around $a$ that is not rigorously defined.
Proof:
Let $f: [0, \infty) ... |
Suppose you have $n$ boxes, each of which falls into one of the $k$ sizes, and you want to nest smaller ones into larger ones, such that no two boxes $A$ and $B$ are nested inside the same box, if neither $A$ contains $B$ nor $B$ contains $A$. Is there a polynomial-time algorithm to nest the boxes, such that the number... |
So I am tasked with finding whether $\frac{1}{\sin(x)}-\frac{1}{x}$ is uniformly continuous on the open interval $I=(0,1)$. To look at the "simple" ways to prove it first:
I obviously can't extend the function to $[0,1]$ since the limit at $x=0$ is not defined.
I tried to check if its derivative, $\frac{1}{x^2}-\frac{... |
Find the value of $y$ in $11y \equiv 14 \pmod{19}$.
My issue is not with finding a solution. Using the Euclidean algorithm and Bezout's identity I get a final expression of: $$(11)(7)(14) - (19)(4)(14) = 14$$ which equals $$(11)(98) - (19)(56) = 14$$
which would imply $y = 98$, which checks out. But the text got a so... |
Given the figure in this image (I don't know the correct name in English. I wish to know the length of the 4th side. How to calculate it and what's the length of the 4th side?
|
Need to show that for each $n \in \mathbb{N}$, with $n \geq 2$,
$$\left(1 - \frac{1}{4}\right)\left(1 - \frac{1}{9}\right) \cdots \left(1 - \frac{1}{n^2}\right) = \frac{n + 1}{2n}$$
How to start the proof by induction? Is there any way to show this?
|
If $f_k\in L^p(\mathbb{R}^n)\cap L^r(\mathbb{R}^n) $ for some $p,r\in [1,\infty), f_k\rightarrow g$ in $L^p(\mathbb{R}^n)$ and $f_k\rightarrow h$ in $L^r(\mathbb{R}^n)$, then $g=h$ $a.e.$ in $\mathbb{R}^n$
I took $1\leq p<r$ WLOG and put $p= (1-t)1+tr$ where $t\in [0,1)$ then I applied convexity property of function $|... |
Find the polynomial $P$ of smallest degree with rational coefficients
and leading coefficient $1$ such that $$ P(49^{1/3}+7^{1/3})=4 $$
(Source:NYSML)
My attempt
Let $$ 49^{1/3}+7^{1/3}=x$$
Then,
\begin{array}
((49^{1/3})^3 &=(x-7^{1/3})^{3} \\
49 &=x^3-7-3x^2\cdot 7^{1/3}+3x \cdot 7^{2/3} \\
\end{array}
So I ha... |
An urn contains $w$ white and $b$ black balls. $n$ extractions with replacement are made. What are the probabilty of:
get $r$ white, $P(W=r)$
get $r$ white with white ball on n-th, n-th and n-th extraction (example third, sixth and ninth)
The first is easy:
$P(W=r)=\dbinom{n}{r}(\frac{w}{w+b})^r(\frac{b}{w+b})^{n-r... |
Is there anything wrong with this definition of discontinuity for a function y = f(x)?
$\forall \delta>0\, \exists \varepsilon>0$ such that $\vert x-c\vert < \delta$, but $\vert f(x) - f(c)\vert > \epsilon$.
|
let $(X, \| \|$) be the normed linear space of bounded uniformly continuous real valued functions defined on $R$. I need to prove that X is complete (under sup norm).
My attempt:
Let {$f_n$} be a Cauchy sequence in X converging to $f$ in the completion of X.
I am checking the uniform continuity of f below.
$|f(x)-f(y... |
I cannot understand or form a good intuition in my head of what a tail field is. An introduction to rigorous probability theory by Rosenthal gives the following definition:
Given a sequence of events $A_1, A_2, ...$, we define their tail field:
$$\tau = \bigcap_{n=1}^\infty\sigma(A_n, A_{n+1}, A_{n+2},\ldots) $$
So if... |
I have this matrix below and I'm trying to find it's inverse, I know I augment it with I2 but I don't know where to go from that.
\begin{bmatrix}
2&1\\
a&a
\end{bmatrix}
|
How to decide if $Q(\underline u,\underline v) $ on $\mathbb R^2$ is inner product or not if $$ (\underline u, \underline v) = \underline u^T A\underline v$$ where $$A = \begin{pmatrix}
-1 & -3 \\
5& 8 \\
\end{pmatrix}$$
Can this be opposed by considering the symmetric property only?
$$ (\underline v, \und... |
I have read the following proposition, and haven't been able to connect the convexity of $X$ to the statement's main equality. Any guidance would be much appreciated.
Define $X^\text{o}$, the polar of $X$, as follows:
$$
X^\text{o} = \{y \in \mathbb{R}^n \; | \; x \cdot y \leq 1, \; \forall x \in X \}
$$where $X$ is a ... |
A basic result in combinatorics is:
In $S_n$ there are
$(n-d)(n-2)!$ permutations $\sigma$ such that $\sigma^k(a) = b$, if $a \neq b$;
$d(n-1)!$ permutations $\sigma$ such that $\sigma^k(a) = b$, if $a = b$,
where $d$ is the number of positive divisors of $k$.
This lemma makes me wonder:
In $S_n$ how many permuta... |
In big Rudin's book, the proof of Riesz's representation theorem quite confuses me. It states that the definition of $\mu_1(V)=\mbox{sup}\{Lf:f\prec V\}$ and $\mu_2(V)=\mbox{inf}\{\mu_1(U): V\subset U, U \ \mbox{is open}\}$ is equivalent for all $V\subset X$, where $X$ is a topological space
It is obvious that when $... |
http://www.webassign.net/zillengmath4/20.2.pdf p.2.
The conformal map $z+\frac{1}{z}$ maps circles $|z|=r$ to ellipses
and $arg(z)=\theta$ to hyperbolas.
I believe one can display both using the same equations, but I have only managed to display the ellipse and cannot understand the hyperbola.
Basically some sources (a... |
Suppose you have an array of size $2n$. There are two times $2n^2$ distinct numbers that can be put into the array without replacement, i.e. for each choice of number, there are two copies, so a maximum of two cells in the array can have the same value. What is the probability that all $2n$ numbers in the array are dis... |
How do you find the maximum of a quadratic function? Specifically,
$R(x) = -4x^2 + 4000x$
|
Three single men, two single women and two families take their places at a round table. Each of the two families consists of two parent and one child. Find the number of possible seating arrangements.
members of the same family are seated together and the two single women are separated
the seats are numbered and each... |
Let $\mathcal C\subset \mathfrak g$ be a subset in a Lie algebra $\mathfrak g$ satisfying the following two conditions:
$\mathrm{Ad}(G)\mathcal C=\mathcal C$
If $X,Y\in \mathcal C$, then $X+Y\in\mathcal C$ or $X-Y\in\mathcal C$.
(one can also assume 3. There is a proper subspace $\mathcal H\subset \mathfrak g$ such t... |
I wonder if there is a method other than solving to find out whether the solutions of a specific differential equations are bounded? I am interested in equations of the form
$$
\frac{dx_i(t)}{dt} = \sum_j A_{ij}x_j + \sum_{jk}B_{ijk}x_jx_k \\
\text{ with } B_{ijk} = 0 \text{ if any index coincides.}
$$
e.g. only equati... |
This program involves the matrix-vector computational primitive $y \leftarrow Ax$ where $x,y\in\mathbb{R}^n$ and $A\in \mathbb{R}^{n\times n}$. A is taken to be dense and banded in the two parts of the problem. Recall, there are many ways to implement the primitive. Two standard approaches are:
$$\eta_i = e_i^{T}y = e... |
Can someone help me with this? Are there direct products in the category $Ab$ (the category of the abelian groups)? If yes, please, can you give me an example? Thank you?
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Can someone explain me how we can compute RQ decomposition for a given matrix (say, $3 \times 4$). I know how to compute QR decomposition.
I know the function in MATLAB which computes this RQ decomposition. But, I want to know how we can do that on paper.
PS: The practical use of RQ decomposition is in extracting the i... |
Im a bit confused with this question.
Lets have the equation $z= x^2 + y^2$
therefore gradient f is perpendicular to surface $f=$ constant.
In my case it would be $(2x,2y,-1)$ is perpendicular to $0$?
Would really appreciate any help.
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Prove that if $a,b,c$ are non-negative real numbers, then
$(ab + bc + ca)^2 \geq 3abc(a+b+c)$.
I tried to compute from $(a-b)^2 + (b-c)^2 + (c-a)^2 \geq 0$.
|
I need to show that $$\sum\limits_{k=i}^{n}\binom{n}{k}\binom{k}{i} = \binom{n}{i}2^{n-i}$$
I know that $\displaystyle \binom{n}{k}\binom{k}{i}$ is counting the number of ways to pick $k$ elements from $n$ and then $i$ elements from those $k$, so $\displaystyle \sum\limits_{k=i}^{n}\binom{n}{k}\binom{k}{i}$ is counting... |
How would I solve the limit
$\lim_{z \rightarrow 0} \frac{ \sin ||z||_p}{||z||_p}$ ?
Note that $p \in [1,\infty]$.
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I had thought the inflection points were also critical points, though the sources I read say critical points of $f(x)$ are where: $f'(x) = 0$
http://mathworld.wolfram.com/CriticalPoi...
https://en.wikipedia.org/wiki/Critical_p...
I have a question asking about the critical points of a cubic and the minimum maximum v... |
Suppose $P(|X| < 1) = 1$ and $P(|Y| = 2) = 1$.
Then which of the following is true?
The standard deviation of $X$ is smaller than that of $Y$.
The mean of $X$ is smaller than that of $Y$.
The variance of $X$ is larger than that of $Y$.
The median of $X$ is equal to the median of $Y$.
None of the above is correct.
Sin... |
I think something like the following statement is true, but I don't recall a reference.
Suppose $f(x,y)\in k[[x,y]]$ is power series with no constant or linear terms. Then, if the quadratic terms factor into a product of linear terms, the entire power series factors.
I know I'm missing a condition, because $y^2-x^3$ i... |
Solve the system of equations $$\begin{cases}163-400z\sin{x}&=0\\-135z+85\cos{x}+61&=0\end{cases}$$
What is the best way of going about this?
I rearranged the second equation for $z$ and then substituted it into the first one to find $$163-400\left(\frac{1}{135}(85\cos{x}+61)\right)\sin{x}=0$$ $$\implies 22005 - 34000\... |
In another thread it was proved that transposition as a linear map is diagonalizable. This, however, does not hold when we are working over a field of characteristic 2. I suppose the proof of this can be a bit tricky. Any hints?
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I know this may sound too simple or maybe too absurd to discuss, but I am having a hard time visualizing a point in space!
In Euclid's Elements a 'Point' is defined as Something which has no part. Now, any geometrical figure viz. Line Segment,Triangle,Square etc. can be said to be composed of points. No matter how smal... |
Can someone tell me how to calculate the order of a) $LU$ decomposition as well as b) Gaussian elimination of a square matrix $A$? I am at a loss ...
Given:: $A$ is a $n\times n$ matrix and ultimate aim is to find a solution set of $Ax=b$. I couldn't solve the first part but for the 2nd part, I think the number of ba... |
I wanted to find the approximate average of all real points in $(x)^{x}$ from $[-4,-2]$. This means I am ignoring all real inputs that give a complex output and need average to be a real number.
To first solve this I found the following defined sets of $x^x$ when $x<0$.
$$x=-\frac{2m}{2k+1}|m,k\in\mathbb{Z}$$
$$x=-\f... |
I think it is true that any power of a logarithm, no matter how big, will eventually grow slower than a linear function with positive slope.
Is it true that for any exponent $m>0$ (no matter how big we make $m$), the function $f(x)$ $$f(x)=(\ln x)^m$$
will eventually always be less than $g(x) = x$ ?
I am pretty sure th... |
$C$ is a convex object in the plane. $D_1,\dots,D_n$ are pairwise-disjoint convex subsets of $C$ such that $D_1\cup\dots\cup D_n \subsetneq C$, like this:
Is it always possible to partition $C$ to $n$ convex subsets, $E_1,\dots,E_n$, such that $E_1\cup\dots\cup E_n = C$, and for every $i=1,\dots,n$: $D_i \subseteq E_i... |
I have the following topology over $\mathbb R$
$$ T = \{\emptyset\} \cup \{G\subseteq \mathbb R: \mathbb Q \setminus G \text{ is finite}\} $$
How could I study the closure of $\mathbb Q$ and $\mathbb R\setminus \mathbb Q$? Thanks in advance
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I am currently reading a proof in which a fact is used without proof:
For a Banach space $X$ and a bounded linear operator $T: X \to X$,
$$ \lim_{n \to \infty} \| T^n \|^{\frac{1}{n}} = \inf_{n \in \mathbb{N}} \| T^n \|^{\frac{1}{n}}. $$
Do we have to show that the sequence $(\| T^n \|^{\frac{1}{n}} )_{n
\in
\ma... |
Why is this limit non-existant?
$\lim_{(x,y) \rightarrow (0,0)} \frac{ e^{x+y} - x - y}{\sqrt{x^2 + y^2}}$
I can't seem to find $2$ different paths that would show it is non-existant.
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Is there any requirement that the two operations of a ring have to be related to each other, excluding the requirement of distributivity? We all know from grade school that multiplication of integers is repeated addition, and we also know that the integers form a ring under addition and multiplication. Are there any ... |
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