Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Writing $1-e^{-xy}$ as a square. Is it possible to write $1-e^{-xy} = r(x)r(y)$ for some function $r$ where $x,y$ are positive real numbers. I was just wondering to try to express that quantity like that. I tried solving the equation by Brut force but was not able to make any impact. Any suggestion will be helpful.
| Do partial differenciation on the both sides with $x$, and then assume that let $x=y$, then integrate on both sides with $x$ with limits $[o,x]$ as we know $r[o]$ so we can get the $r[x]$.
| {
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Injectivity for partially applied composition I struggle to understand the following theorem (not the proof, I can't even validate it to be true). Note: I don't have a math background.
If S is not the empty set, then (f : T → V) is injective if and only if Hom(S, f) is injective.
Hom(S, f) : Hom(S, T) → Hom(T, V)
As ... | I don’t understand how it defines the map $Hom(S,f)$. From the choice of your symbols (which refers to category theory because you are fixed the set category in which the objects are sets and the morfism are the function between these sets) I think that $Hom(S,f)$ is defined from $Hom(S,T)$ to $Hom(S,V)$ and it maps ev... | {
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Confusion on 2 factor theorem In graph theory Petersen's $2$-factor theorem states the following: Let $G$ be $2k$-regular graph, then $E(G)$ can be decomposed into the union of $k$ line-disjoint $2$-factors https://en.wikipedia.org/wiki/2-factor_theorem
Are loops allowed in this theorem ? If that is the case then in th... | If we follow the proof given on Wikipedia, then we can make it work with loops if they contribute $+2$ to the degree of their vertex. Then if we orient the graph, the oriented loop contributes $+1$ to both the in-degree and the out-degree, and the rest of the proof proceeds as usual.
We can also prove the theorem if l... | {
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Inverse trigonometric functions used for difference in angle? I was actually solving a physics question in which I got the equation 3/2 (sin i) = sin r. The graph of (r-i) against i has a constant positive slope till 45 degrees value for i. How do I get this result? The only thing I can get from the equation is that si... | We should understand refraction optics through Snell's Law. From the figure below we can see that after $i> 90^{\circ}$there is a grey area where refracted ray cannot enter.
$$ \frac{\sin i}{\sin r} = \mu = \frac{3}{2}$$
$$ \frac {\sin 90^{\circ}} {\sin{r_{critical}}} = \mu = \frac{3}{2}$$
The critical limit angle li... | {
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Solving a limit without reaching an indeterminate expression Find the limit of :
$$\lim\limits_{x \to 0^+}{(2\sqrt{x}+x)^\frac{1}{\ln x}}{}$$
I've tried to make it look like an exponent of e:
$$e^\frac{\ln (2*\sqrt{x}+x)}{\ln x}$$
but, then again I reach an indeterminate form of infinity divides by infinity.
I then tri... | Note $x^{1/\ln x} = e,$ which follows by applying $\ln$ to both sides. Thus $\sqrt x^{1/\ln x} = e^{1/2}.$
Now $x<\sqrt x$ for $0<x<1,$ and because the power $1/\ln x<0,$ we have
$$(3\sqrt x)^{1/\ln x} < (2\sqrt x+x)^{1/\ln x} < (2\sqrt x)^{1/\ln x}$$
for $x$ in this range. The left side equals $3^{1/\ln x}e^{1/2}.$ As... | {
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Find for how many values of $n$. $I_n=\int_0^1 \frac {1}{(1+x^2)^n} \, dx = \frac 14 + \frac {\pi}8$
Find for how many values of $n$. $I_n=\int_0^1 \frac {1}{(1+x^2)^n} \, dx=\frac 14 + \frac {\pi}8$
My attempt (integration by parts):
\begin{align}
I_n & = \int_0^1 \frac 1{(1+x^2)^n}\,dx = \left. \frac {x}{(1+x^2)^n}... | Note that the sequence $(I_n)$ is decreasing, hence there is at most one value of $n$ such that $I_n=\frac{1}{4}+\frac{\pi}{8}$.
| {
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Indexing of Discrete Fourier Transforms I was looking at the Discrete Fourier Tranform section in these notes and I'm very confused about how the transform is being indexed.
There, a list of the form $f(x_n)$ is given where $x_n$ takes on the values $x_n\in \{0,1,\ldots,N\}$, that is $x_n=n$. It is then claimed that ... | Adding or subtracting a multiple of $2\pi$ from $k_j$ does not affect the values of the sum $\sum_{j=0}^N f(k_j)\exp (-ix_n k_j)$ at the points $x_n$. Thus, either form could be used if all we ever going to plug for $x_n$ are integers. (Or we could randomly add $10\pi$ to all the $k_j$.)
Aliasing is the term used for t... | {
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Calculating $\int\sin(x)\ln(\sin(x))\, \mathrm{d}x$ I was solving indefinite integrals for preparing me to a math test but I found a very hard one I'm not able to solve. It is
$$\int \sin(x)\ln(\sin(x))\, \mathrm{d}x$$
I tried to solve it by substitution but I wasn't able to express the $dx$ with a smart method. The on... | In this particular case you can set $u=\cos(x)$ to get rid of $\sin(x)\mathop{dx}=-\mathop{du}$
Then use $\sin(x)=\sqrt{1-u^2}$ and the logarithm will get rid of the square root and split nicely the product $(1-u)(1+u)$.
It becomes $$\int -\frac 12\ln(1-u^2)\mathop{du}=-\frac 12\left(\int \ln(1+u)\mathop{du}+\int\ln(1-... | {
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Cyclic Cover of the Projective Line Let $Y$ be the Riemann surface defined by the equation $y^d=h(x)$ and $\pi: Y \to \mathbb{C}_\infty$ be the projection map sending $(x,y)$ to $x$. Let $\sigma: Y \to Y$ be the automorphism defined by $(x,y) \mapsto (x,\zeta y)$, where $\zeta$ is a primitive $d^{th}$ rooth of the unit... | Since $y\circ \sigma = \zeta y$, if $f\in \mathcal{M}_i$ then$f/y^i$ is $\sigma$-invariant, hence the pullback of a meromorphic function on the sphere, say $r$. Now just rewrite $f/y^i = \pi^\ast r$.
For the last part, fix a meromorphic function $f$ and define $$ f_i = \frac{1}{d} \sum\limits_{j=1}^d \zeta^{-ij}(f\circ... | {
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Expected value of the shifted inverse of a binomial random variable, and application Here is an exercise given by a colleague to a student :
Let $X\hookrightarrow B(n,p)$ and $Y=\frac{1}{X+1}$. Find ${\rm E}(Y)$.
It is not very difficult to prove that the answer is
$${\rm E}(Y) = \frac{1-q^{n+1}}{p(n+1)}$$
where $q=1... | Here is an application of your question to a setting that highly interests
me. In queueing there is the notion of utilization which is the long-run
fraction of time a server is busy serving demand. Consider a Markovian service
setting where demand arrives according to Poisson with $\lambda=1$
and there are $N+1$ server... | {
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Discriminant of Number Fields Let $K$ and $F$ be two number fields of degree $n$ and $m$ and $K\cap F = \mathbb{Q}$. Then we consider the number field $KF$ generated by $K$ and $F$. It is known that we have the relation between corresponding discriminant, i.e,
$$ d_{KF}\ \Big|\ d_K^m\cdot d_F^n. $$
If $p$ is ramified i... | Consider the fields $K=\mathbb{Q}(\sqrt[6]{2})$ and $F=\mathbb{Q}(\zeta_8\sqrt[4]{2})$, where $\zeta_8$ is the fourth root of unity. It's not hard to see that $\mathbb{Q}$ is the only real subfield of $F$ and so we have that $K \cap F = \mathbb{Q}$, as $K$ is a real subfield.
Now using PARI/GP one can calculate that $d... | {
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How to evaluate $\frac{d}{dx}\sin^2x$ I'm having trouble understanding how $\frac{d}{dx}\sin^2x =2\sin(x)\cos(x)$.
Please show as many steps of the proof as necessary so that I can apply this to other problems.
Thank you for your time~! ^_^
| $\frac{d}{dx}\sin^2x= 2\sin(x)(\sin(x))'= 2\sin(x)\cos(x)$
| {
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What is the negation of "x is odd and y is even"? I'm not sure if I find the correct answer to this question.
Using De Morgan's law;
¬(a∧b)=(¬a)∨(¬b)
so it becomes:
X is not odd OR Y is not even
or also we can say,
X is even OR Y is ODD.
Can somebody correct me if I'm wrong OR tell me if I'm right :)
| Well, not odd means even, and not even means odd. Hence, you're absolutely correct.^^
| {
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Can I prove that $f(x+1) - f(x)$ is a monotonic increasing function, given $f'(x) > x$ for every $x>0$? Given:
*
*$f$ has a derivative for every $x \in (0,\infty)$
*$f'(x) > x$ for every $x>0$
Can I prove that $f(x+1) - f(x)$ is a monotonic increasing function?
From Lagrange I know that in every $I = [x,x+1]$ wher... | No, you can't.
Counterexample: $f(x)=\left\{
\begin{array}{ll}
2x,&0<x<\frac{3}{2},\\
\frac{2}{3}x^2+\frac{3}{2},&x\geq \frac{3}{2}.
\end{array}
\right.$
Note how $f'(1.1)-f'(0.1)=0$.
| {
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Solve $\cos x +\cos y - \cos(x+y)=\frac 3 2$ Solve $\cos x +\cos y - \cos(x+y)=\frac 3 2$ where $x,y\in [0,\pi]$.
I am trying to solve this but I am stuck. I know that $x=y=\pi/3$ is a solution but how do I show this is the only one? I think there are no others! Hints would be appreciated
| \begin{align}
\cos x +\cos y - \cos(x+y) &= \frac 32 \\
\cos x + \cos y - \cos x \cos y + \sin x \sin y
&= \frac 32 \\
(1 - \cos y)\cos x + \sin y \sin x &= \frac 32 - \cos y \\
\end{align}
Note that
$$\sqrt{(1-\cos y)^2 + (\sin y)^2} = \sqrt{1 - 2\cos y + \cos^2y + \sin^2 y}
= \sqrt{2(1 - \cos y... | {
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(No-)Generators of $S_4$ Let $S_4$ be the symmetric group on 4 elements and let $x=(1,2)(3,4)$ be a permutation of $S_4$. I try to proof that can be no element $y \in S_4$ such that $<x,y>$ is the whole group $S_4$.
I notice that $x \in K$ where $K$ is the Klein group. Now, I know $S_4 /K$ is isomorphic to $S_3$. How c... | You have all the ingredients for the solution. The crucial thing is that
$S_3$ is not cyclic. Let $\pi:S_4\to S_4/K$ be the projection map.
If $H=\left<x,y\right>$, then $\pi(H)=\left<\pi(y)\right>\ne S_4/K$
since $S_4/K$ is not cyclic. Therefore $H\ne S_4$.
| {
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Coprime elements generate coprime ideals Is this true in any commutative rings? I.e.
$$\gcd(a,b)=1\implies (a)+(b)=R$$
I think there must be some conditions on the ring to make this implication otherwise it does not work.
This may related to this question here.
| This is not true in arbitrary rings. For (counter)example, in a polynomial ring in two variables $R = k[x,y]$, we have $\gcd(x,y)=1$, but $(x)+(y) \neq R$.
It is true in Bézout domains. I don’t know if it’s equivalent to being a Bézout domain.
| {
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Show by mathematical induction that $(2n)! > 2^n*n!$ for all $n \geq 2$ Very stuck on this question. Here is what I have attempted.
Base case: $(2*2)! > 2^2*2!$ = $24>8$ Which is true
Hypothesis: Assume for some int k>2 that $(2k)!>2^k*k!$
Then for my inductive step, I am getting stuck. I tried,
$(2(k+1))! > 2^{k+1} *(... | $$(2(k+1))! \overset{?}{>} 2^{k+1}(k+1)!$$
$$(2k+2)(2k+1)(2k)! \overset{?}{>} \cdot2^{k+1}(k+1)!$$
$$(2k+2)(2k+1)(2k)! \overset{?}{>} 2 \cdot2^k(k+1)k!$$
Now use your IH, since $(2k)! > 2\cdot2^k k!$, you just need to show that
$$(2k+2)(2k+1) > (k+1),$$
which is obvious.
| {
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An inequality involving the logarithmic derivative of a polynomial If $P(z)=a_0+a_1z+\cdots+a_{n-1}z^{n-1}+z^n$ is a polynomial of degree $n\geq 1$ having all its zeros in $|z|\leq 1,$ then I was trying to verify the question, is it true that for all $z$ on $|z|=1$ for which $P(z)\neq 0$
$$\text{Re}\left(\frac{zP'(z... | We show that the inequality holds by induction on the degree $n$.
Base step. If $n=1$ then $P(z)=z-w$ with $|w|\leq 1$ and we have to show that for $|z|=1$ and $z\not=w$,
$$\text{Re}\left(\frac{zP'(z)}{P(z)}\right)
=\text{Re}\left(\frac{z}{z-w}\right)\stackrel{?}{\geq} \frac{1}{1+|w|}.$$
Left to the reader.
Inductive s... | {
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Is there a name for the function $1 / (1 + x)$? Does the function
$$f(x) = \frac{1}{1 + x}$$
have a recognizable name?
For example a related function with a recognizable name is the logistic function, defined by:
$$l(x) = \frac{1}{1 + e^{-x}}$$
Note: By the way I am quite happy with functions without name.... excep... | This is a homographic function.
As a curve, it is also an equilateral hyperbola.
| {
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Prove by induction that $n^4-4n^2$ is divisible by 3 for all integers $n\geq1$. For the induction case, we should show that $(n+1)^4-4(n+1)^2$ is also divisible by 3 assuming that 3 divides $n^4-4n^2$. So,
$$ \begin{align} (n+1)^4-4(n+1)&=(n^4+4n^3+6n^2+4n+1)-4(n^2+2n+1)
\\ &=n^4+4n^3+2n^2-4n-3
\\ &=n^4+2n^2+(-6n^2+6n^... | Because $$n^4-4n^2=n^4-n^2-3n^2=n(n-1)n(n+1)-3n^2$$
| {
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Prove the number of symmetric relation is $2^{\frac{n^2+n}{2}}$ If the number of set $A$ is given. let $n(A)=n$.
Prove the number of symmetric relation from $A$ to $A$ is $2^{\frac{n^2+n}{2}}$
We know that number of relations from $A$ to $A$ is $2^{n^2}$
this is obtained by, number of subsets of $A$x$A$ $=2^{n^2}$
e... | Given two distinct elements $x,y$ in $A$, you can either have both $(x,y)$ and $(y,x)$ in the relation or not. You also can have $(x,x)$ or not. The number of choices you make is the number of unordered pairs from $A$ including pairs of the same member. There are $\frac {n^2+n}2$ of those.
| {
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Notation of parametrisation of a curve
Show that $\gamma = \{r = const >0\}$ is NOT a geodesic on the plane with parametrisation $ds^2 = dr^2 + \sinh^2 d\phi^2$ of the hyperbolic plane.
I know how to work this out in principle. My confusion is with the notation
$\gamma = \{r = const >0\}$. Should I take this to mean... | You need a parametrization of the curve, which is a circle of radius $r$. In polar coordinates you can, in fact, use $t\mapsto (r(t), \phi(t)) = (r, t)$
(or $(r, ct)$ with a normalizing factor $c$ to achieve a unique speed parametrization).
Alternatively, in the standard Euclidean coordinates, you may use $(r\sin(ct), ... | {
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tan sum difference - Algebra to solve the answer
$$\tan(\alpha+\beta)=\frac{\tan\alpha+\tan\beta}{1-\tan\alpha\tan\beta}=\frac{\frac{\sqrt3}3+(-2\sqrt2)}{1-\frac{\sqrt3}3\cdot(-2\sqrt2)}$$ The exact value of $\tan(\alpha+\beta)$ is shown below: $$\tan(\alpha+\beta)=\frac{8\sqrt2-9\sqrt3}5.$$
Hello. I am having troubl... | Firstly, multiply both numerator and denominator by $3$. We have $$\frac{\frac{\sqrt3}3-2\sqrt2}{1+\frac{\sqrt3}3\cdot2\sqrt2}=\frac{\sqrt3-6\sqrt2}{3+2\sqrt6}\cdot\color{red}{\frac{3-2\sqrt6}{3-2\sqrt6}}=\frac{3\sqrt3-2\sqrt{18}-18\sqrt2+12\sqrt{12}}{9-4\cdot6}$$ so...
| {
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How many one in a million people exist? So if there are 7,632,819,325 people currently alive (According to google), then how many of those people are "One in a million"?
My math behind it was to divide the number by a million, but I just wanted to double check. I got the number 7,632; as expected- but this felt a litt... | For a single trait for a human to possess such that the odds of a single person possessing that trait are "one in a million", then the expected number of people globally to possess that trait would be around 7,632 as you state. But, that is just the expected number. Since this would be a binomial distribution, you are ... | {
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Similar matrices have same engenvalues $\implies$ we can define characteristic polynomial for any basis
This is from Linear Algebra - Hoffman and Kunze.
If $B$ is a matrix and $A$ is it's similar, what it means to say that $A$ represents $B$ in some ordered basis for $V$?
| It means that for $V$ and $W$ $n$-dimensional vector spaces over a field $F$, and $A$ and $B$ $n$ by $n$ matrices with entries in the field $F$ there are ordered bases $\alpha = (\alpha_i)_{i \in \{1,\dots,n\}}$ and $\alpha' = (\alpha'_i)_{i \in \{1,\dots,n\}}$ of $V$ and and $\beta = (\beta_i)_{i \in \{1,\dots,n\}}$ a... | {
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If the sum of the tail of a series goes to $0$, must the series converge? Suppose $\{a_n\}$ is a sequence of positive terms.
It is a well known result that if the series $\displaystyle \sum_{k=1}^{\infty} a_k$ converges then $\displaystyle \lim_{m \rightarrow \infty} \displaystyle \sum_{k=m}^{\infty} a_k=0 $.
Is the c... | Yes, it is correct also when the sequence $(a_n)_n$ is not positive because $(S_n)_n$ is a Cauchy sequence
$$|S_n - S_m| = | \sum_{k=m+1}^{n} a_k|=|\sum_{k=m+1}^{\infty} a_k-\sum_{k=n+1}^{\infty} a_k|\to 0$$
as $n,m\to \infty$.
| {
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Probability to choose at least one green ball and no red balls Assume we have $n$ red balls, $n$ green balls, and unknown number of white balls.
We select each ball to a set with probability $p=\frac{1}{n}$ and not choosing it with probability $1-p=1-\frac{1}{n}$ independently with each other.
Show a constant lower bou... | Let $x=\left(1-\frac1n\right)^n$. Then you've found that the desired probability is $x(1-x)$. This has a maximum at $x=\frac12$ and monotonically increases towards that maximum. As $n\to\infty$, $x$ monotonoically increases towards $\frac1{\mathrm e}\lt\frac12$. Thus the probability monotonically increases with $n$.
| {
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Prove/disprove: $f+g$ and $f$ are differentiable at $x_0$ $\implies$ $g$ is differentiable at $x_0$
Prove/disprove: $f+g$ and $f$ are differentiable at $x_0$ $\implies$ $g$ is differentiable at $x_0$
attempt
Suppose $g$ is not differentiable at $x_0$. There are three cases:
Case I: The two following one-sided limi... | Hint: First prove that if $a(x)$ is differentiable, then $c \cdot a(x)$ is differentiable for any constant $c$. Next, prove that if $a(x)$ and $b(x)$ are differentiable, then $a(x)+b(x)$ is also differentiable.
| {
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For which values of the parameter $\beta$ is the following integral convergent For which values of the parameter $\beta$ is the following integral convergent
$$\int_0^\infty \frac{3arctan(x)}{(1+x^{\beta-1})(2+cos(x))}\,dx$$
I tried using comparison test with $g(x) = \frac{1}{x^{\beta-1}}$, however I realized that for ... | Note that the integrand has no singularities on the positive real axis, so your only concern is the behaviour at $+\infty$. As the integrand is non-negative, the integral converges if and only if
$$ f(\beta) \equiv \int \limits_1^\infty \frac{3 \arctan (x)}{(1+x^{\beta-1})(2+\cos(x))} \, \mathrm{d} x < \infty $$
holds.... | {
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Preserving addition of one in inductive step of modulo I am trying to prove the following: $$\forall m\ n : \mathbb{N}, (m + 1)\ mod\ n \neq 0 \rightarrow m\ mod\ n + 1 = (m + 1)\ mod\ n$$
So far I've done the following.
Assume arbitrary $m$ and $n$. Now assume $(m + 1)\ mod\ n \neq 0$. Proceed by cases formed by the t... | Assume $m=qn+r$ with $0\le r<n$. Then $m+1=qn+(r+1)$. Either $r+1<n$ and we are done, or $r+1=n$ and that means $m+1\equiv 0\pmod n$.
| {
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A line integral as the average of squares of distances Let $A$ be a domain in $\mathbb R^2$ whose boundary $\gamma $ is a smooth positively oriented curve and whose area is $|A|$.
Find a function $F:\mathbb R^2\to \mathbb R$ such that $$\frac{1}{|A|}\int_\gamma Fdx+Fdy$$ is the average value of the square of the dis... | For notational convenience, let me rename your domain $\Omega$, so that $\gamma = \partial \Omega$ is the positively oriented boundary of $\Omega$.
The average value of an integrable function $f : \Omega \to \mathbb{R}$ on a domain $\Omega$ is
$$
\frac{1}{\lvert \Omega \rvert} \iint_\Omega f(x,y) \, \mathrm{d}A.
$$
As... | {
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Examples of odd dimensional rationally acyclic closed manifolds. Rationally acyclic mean that the all rational homology groups are vanish except the zeroth homology group. Here closed manifold mean compact without boundary. In even dimension, we have nice examples of rationally acyclic closed connected manifolds. For e... | No! Euler characteristics is an obstruction. Any closed odd dimensional manifold has $\chi=0$ [follows from Poincaré duality]. But if it is rationally acyclic, then $\chi=1$. Contradiction.
| {
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Fulton Algebraic Curves, Exercise 2.5 Exercise 2.5 says (in part): "Let $F$ be an irreducible polynomial in $k[X,Y]$ [$k$ algebraically closed], and suppose $F$ is monic in $Y$: $F = Y^n + a_1(X)Y^{n-1} + \cdots$, with $n>0$. Let $V=V(F)\subset\mathbb{A}^2$. Show that the natural homomorphism from $k[X]$ to $\Gamma(V) ... | The condition of being one-to-one or injective is equivalent to the map being dominant, i.e., the closure of the image is the whole target variety (see Georges' answer in this thread for example). In the particular case we're dealing with you have the following composition of ring maps:
$$
k[x]\hookrightarrow k[x,y]\tw... | {
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Synthetic division: a jet flies 400 mph
A jet flies to the west for a distance of approximately 830 miles,
starting at Point A and ending at Point B. The jet is moving at 400
mph, on average. A strong wind comes from the north at 40 mph. In
minutes, how long would it take for the jet to reach its final
destina... | The plane's overall speed is $400$ mph.
Hint: Using the Pythagorean Theorem, if the plane's westward speed is $v$,
we know that $v^2+40^2=400^2$
We also know that the plane has to travel $830$ miles WEST.
| {
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What went wrong in this u-substitution in integrating functions in terms of sine? So let's say there's this thing I want to integrate in terms of sine:
$$\int \frac{1}{1+\sin{x}}\,dx$$
Since the integrand is apparently invariant when we map $x$ to $\pi-x$, we could try setting $x = \pi-u$, and consequently $u=\pi-x$, a... | You're treating $I$ as if it were a number. Are you doing a definite integral, say, from $a$ to $b$? Then you have
$$\int_a^b \frac{dx}{1+\sin x} = -\int_{\pi-a}^{\pi-b} \frac{du}{1+\sin u}.$$
This doesn't look like $I=-I$ to me.
| {
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why is it equal to $E[S^2(t_1)]E[S^2(t_2)]+E^2[S(t_1)S(t_2)]+E[N^2(t_1)]E[N^2(t_2)]+E^2[N(t_1)N(t_2)]$ $Y(t)=a[S(t)+N(t)]^2$,$S(t)$ and $N(t)$ are both Gaussian random process and WSS with zero mean,and $S(t)$ is independent of $N(t)$
\begin{align}
R_Y(t_1,t_2) & =E[Y(t_1)Y^*(t_2)] \\
& =a^2E[(S(t_1)+N(t_1))^2(S(t_2)+... | Seems to be false. Take $S(t)=X,N(t)=Y$ for all $t$ where $\{X,Y\}$ is i.i.d with standard normal distribution. Then the identity you have written does not hold.
| {
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Solution of an inequaility. I try to solve following inequailty $-3t^4-4Bt^3-2B^2t^2+(6D-2BC)t+2BD-C^2 \leq 0$ where $B$, $C$ and $D$ are real numbers.
Say $g(t)=-3t^4-4Bt^3-2B^2t^2+(6D-2BC)t+2BD-C^2$.
We observe that $g''(t)=-4(3t+B)^2$ is non-positive for all $t$. This means that $g(t)$ is concave down. Thus $g(t)$ ... | Your question is not really clear, but let's see if this helps.
Starting from your first derivative, we can find the critical points by setting:
$$-12t^3 - 12Bt^2 - 4B^2t - 6D - 2BC = 0$$
Diving by two:
$$-6t^3 - 6Bt^2 - 2B^2t - 3D - BC = 0$$
As a third degree equation, we can use Cardano's method neglecting the imagin... | {
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Intersection of all positive powers of a prime ideal in integral domain with all ideals of finite height Let $R$ be an integral domain with every prime ideal having finite height. Then is $\bigcap_{n>1} P^n$ a prime ideal of $R$ for every prime ideal $P$ of $R$ ?
The Noetherian case obvious from Krull Intersection the... | David Speyer's example of $$R = \bigcup_{n=1}^{\infty} k\left[x,\ y,\ x^{1/n!} y^{1/n!} \right]$$ for any field $k$ also works for this question. The ideal $P=(x, y, x y, x^{1/2} y^{1/2}, x^{1/3} y^{1/3}, x^{1/4} y^{1/4}, \cdots )$ is prime but $\bigcap P^n$ is not prime (see the linked answer for details).
Moreover, ... | {
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Diophantine equation $p^{2n}+2=3^m$ Let $p$ be a prime and let $m$, $n$ be positive integers.
Consider the equation
$$p^{2n}+2=3^m$$
It is easy to see that $(p,m,n)=(5,3,1)$ is a solution. Are there any other solutions?
| This isn't a full answer, but so far I've been able to show that $m$ has to be odd.
Because if $m$ is even, then $\frac{m}{2} \in \Bbb{N}$, and
$$p^{2n}+2=3^m$$
$$3^m-p^{2n} = 2$$
$$(3^\frac{m}{2}-p^n)(3^\frac{m}{2}+p^n) = 2$$
Since $\frac{m}{2}$ is a positive whole number, and $3^\frac{m}{2}+p^n > 3^\frac{m}{2}-p^n$, ... | {
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An interesting proof that $\sin^2(x) + \cos^2(x) = 1$ (using only series, no trigonometry). This question concerns an interesting proof of the fact that $\sin^2(x) + \cos^2(x) = 1$, but only using the series that defines them, not any trigonometry. So define
$$
s(x) = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \ldots
$$
and... | Your last step is unnecessarily complicated. In Step 2, you show that
$$ (s^2 + c^2)' = 0 \implies (s^2 + c^2)(x) = C, $$
where $C$ is some constant. In particular,
$$ (s^2 + c^2)(0) = C. $$
But, directly from the power series definitions of $s$ and $c$, we have
$$ s(0) = \frac{0}{1!} - \frac{0^3}{3!} + \frac{0^5}{5!... | {
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On factoring polynomials whose only coefficients are 0 and 1. I say a polynomial $P\left(z\right)=\sum_{n=0}^{d}a_{n}z^{n}$ is digital if for each $n$, $a_{n}\in\left\{ 0,1\right\}$.
Let $\alpha$ be a positive integer $\geq2$, and let $P\left(z\right)$ be a non-zero digital polynomial of degree $d$, where $d\leq\alpha... | The rational function $\, P(z)/(1-z^\alpha) \,$ is the generating function of a sequence of numbers each of which is $0$ or $1$. By construction, the sequence has a period of $\,\alpha.\,$ Let its minimal period be $\,\beta.\,$ Then $\,\beta\,$ must divide $\,\alpha\,$ because the minimal period divides all periods. Th... | {
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Prove $\{x \in \mathbb{R}:a\leq x\leq b\}=\{y\in \mathbb{R}:\exists s,t\in [0,1]\; with\; s+t=1\; and\; y=sa+tb\}$ I'm doing the $\supseteq$ direction first as I find that the easiest.
Let $A={\{x \in \mathbb{R}:a\leq x\leq b}\}$
Let $B=\{y\in \mathbb{R}:\exists s,t\in [0,1]\; with\; s+t=1\; and\; y=sa+tb\}$
Suppose y ... | The line "Suppose $\exists s,t\in[0,1]$ with $s+t=1$" appears to be your first error.
You don't need to suppose there exist such an $s$ and $t$, they clearly exist. You can let $s=\frac{1}{2}=t$, then $s,t\in[0,1]$ and $s+t=1$. Therefore there exist $s$ and $t$ that satisfy the conditions that you request.
Unfortunat... | {
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Prove $ \frac{1}{2\sqrt{1}}+\frac{1}{3\sqrt{2}}+\dots+\frac{1}{(n+1)\sqrt{n}}<2$
For any positive integer $n$ prove by induction that:
$$ \frac{1}{2\sqrt{1}}+\frac{1}{3\sqrt{2}}+\dots+\frac{1}{(n+1)\sqrt{n}}<2.$$
The author says that it is sufficient to prove that
$$ \frac{1}{2\sqrt{1}}+\frac{1}{3\sqrt{2}}+... | The idea of the author is that if you "stop" the sum at the $n$'th term you get this artificial bound of $2-$something. Then you can show by induction that this holds for every $n$. Having proven this, the assertion follows by taking the limit as $n\rightarrow \infty$ since the bound will always be less than $2$.
Long ... | {
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Probability of getting 6 k times in a row What is the probability of getting $6$ $K$ times in a row when rolling a dice N times?
I thought it's $(1/6)^k*(5/6)^{n-k}$ and that times $N-K+1$ since there are $N-K+1$ ways to place an array of consecutive elements to $N$ places.
| So, let's say that X is a random variable for tracing the number of 6s.
Let's say that C is the condition for the 6s to be consecutive.
Since the task is to find the probability of gaining at least K subsequent 6s, we can look for probability of event A-at least K subsequent 6s fell this way:
P(A)=P(X=K|C)+P(X=K+1... | {
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Find the image of $f(z) = e^{-\frac{1+z}{1-z}}$ I am trying to solve the following problem: let $f(z) = e^{-\frac{1+z}{1-z}}$, and let $\mathbb{D} = \{z: |z|<1\}$. What is the image of $\mathbb{D}$, and for each $w$ in the image, what are all of its preimages?
So far, I've noted that $-\frac{1+z}{1-z}= \frac{x^2 + y^2 ... | $f(z)=\exp(g(z))$ where $g(z)=-\frac{1+z}{1-z}$. $g$ maps the unit disk $\mathbb{D}$ onto $\mathbb{H} = \{z : \mathrm{Re}(z) < 0\}$ bijectively; (see Moebius transformation). Then $\exp$ sends $\mathbb{H}$ onto $\mathbb{D}\setminus\{0\}$, as you correctly said.
| {
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Obtain variance from density of random variable X is a random variable with the density
$$F_X(x) = \begin{cases}2 x^{-2}, & x \in (1,2)\\
0 & \operatorname{otherwise}\end{cases}$$
How can I find the
$VAR(3X^2-5)$
Normal variation is computed from equation
$E(X^2)-(EX)^2 = VAR^2(X)$
So instead of multiplication my fu... | First, using the fact that $Var(aX)=a^2Var(X)$ and $Var(X+c)=Var(X)$, note that $Var(3X^2-5)=9Var(X^2)$ and $Var(X^2)=E(X^4)-(E(X^2))^2$.
$E(X^4)=\int_1^2 x^4 .2x^{-2}=14/3$, similarly $E(X^2)=2$ and so $Var(X^2)=2/3$
and $Var(3X^2-5)=6$
Your approach is also correct but you are making a calculation error, that is $V... | {
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Stuck with integral $\int_{-\infty}^\infty \left( \frac{\sin(a t+b)}{at+b} \right)^2 \, dt$ I am stuck with the following integral:
$$\int_{-\infty}^\infty \left( \frac{\sin(a t+b)}{at+b} \right)^2 \, dt$$
I would like to show that $\varphi(t)=\frac{\sin(at+b)}{at+b}$ belongs to $L^2(\mathbb{R})$ and/or $L^1(\mathbb{R}... | Yet another strategy: once the original integral has been reduced to $\frac{2}{|a|}\int_{0}^{+\infty}\frac{\sin^2 x}{x^2}\,dx$, one may invoke $\mathcal{L}(\sin^2 x)(s)=\frac{2}{s(4+s^2)}$ and $\mathcal{L}^{-1}\left(\frac{1}{x^2}\right)(s)=s$ to further reduce it to
$$ \frac{4}{|a|}\int_{0}^{+\infty}\frac{ds}{s^2+4} = ... | {
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What is the range of convergence of $\sum_{n=0}^{\infty} {(-1)}^n\binom{1/2}{n}\frac{1}{2n+3}.$ I was fiddling with the integral
$$\int_0^1 x^2\sqrt{1-x^2} \ dx $$
and I expanded the term under square root using a binomial series. Integrating, I got the result
$$\sum_{n=0}^{\infty} {(-1)}^n\binom{1/2}{n}\frac{x^{2n+3}}... | Since the coefficient of $x^{2n+3}$ is asymptotic to a constant times $n^{-3/2}$, and $\sum_n n^{-3/2}$ converges, this does converge for $|x| \le 1$. The answer, btw, is $\pi/16$.
| {
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if $A\in M_{n×n}^{\mathbb{C}}$ and self-adjoint then $\exists t\in \mathbb{R}$ such that $A-tI$ is a negative-definite matrix I know that if $A$ is self-adjoint then all the eigenvalues of $A$ are real.And also $A$ is unitary diagonalization over the complex numbers. Therefore $A$ has a bases $B=\{v_1, v_2,..., v_n\}$ ... | Notice that $B$ is negative-definite if and only if
$$\langle Bv, v\rangle < 0$$
for all $v\neq 0$. With $B=A-tI$ we get
$$\langle Av, v\rangle -t\langle v, v\rangle < 0.$$
Now, using the operator norm (where the ambient space has the usual Euclidean norm) we have
\begin{align}
\langle Av, v\rangle -t\langle v, v\rangl... | {
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How can I solve the the following coupled linear PDEs? How can I solve the following system of linear partial differential equations or simplify them to solvable form? both Z and Y depend on x and t variables.
\begin{align}
\frac{\partial Y}{\partial t}+\frac{\partial Y}{\partial x}&=Z-Y \\
\frac{1}{c}\frac{\partial Z}... | Let
$$k=\frac{1}{c}$$
We have
\begin{cases}
\displaystyle Z(x,t)-Y(x,t)=\frac{\partial Y}{\partial x}+\frac{\partial Y}{\partial t}\\
\displaystyle Z(x,t)-Y(x,t)=-k\frac{\partial Z}{\partial t}
\end{cases}
Trivial solutions require $Z(x,t)=Y(x,t)=\operatorname{const}.$
Separating each function for nontrivial solutio... | {
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Does $(x+y)^m=x^m+y^m+z^m$ imply $(x+y+z)^m=(x+z)^m+(y+z)^m$? Let $x,y,z,m\in\mathbb{N}$, and $x,y,z,m>0$, and also $x>y$.
My problem is to understand if, under these sole hypotheses, we can prove that
$(x+y)^m=x^m+y^m+z^m \Longrightarrow (x+y+z)^m=(x+z)^m+(y+z)^m.$
If yes, how can we prove it?
If not, which other ... | Expanding the given condition using binomial we get,
$$ z^m = {m\choose{1}} xy^{m-1} + {m\choose{2}} x^2y^{m-2} +...........+{m\choose{m-1}} x^{m-1}y $$
Now if we expand the claim, we get,
$$
\sum \frac{m!}{a!b!c!} x^ay^bz^c = z^m + \Bigg( {m\choose{1}} xz^{m-1} + {m\choose{2}} x^2z^{m-2} +...........+{m\choose{m-1}} ... | {
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In what the parametrisation $(x,\sqrt{1-x^2})$ of the circle is interesting? We all know the parametrisation $\gamma (t)=(\cos(t),\sin(t))$ of the circle that has the advantage to be smooth and is easy to use. Sometime, my teacher use the parametrisation $$\left\{\varphi(t)=\left(t,\sqrt{1-t^2}\right)\mid t\in [-1,1]\r... | The other parametrisation comes from pythagorean version of circle:
$ x^2+y^2=1 $
This version of circle is of form $f(x,y)=1$.
To get the parametrisation, you need to use substitution $y=f(x)$, i.e.
$f(x,f(x))=1$ and thus $x^2+f(x)^2=1$, which means $f(x)^2 = 1-x^2$ which gives $f(x)=\pm \sqrt{1 - x^2}$.
The interest... | {
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"answer_id": 2
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$a+b\sqrt{3}=\sqrt{21-12\sqrt{3}}, a,b \in \mathbb {Z}$ Find a+b So far I've reasoned that $\mathbf{a}$ and $\mathbf{b}$ can't be both negative, because $\sqrt{21-12\sqrt{3}}$ cannot be negative.
Also $\mathbf{a}$ and $\mathbf{b}$ can't be both positive, because $\sqrt{21-12\sqrt{3}}$ is from 0 to 1, thus there is no... | $$\sqrt{21-12\sqrt3}=\sqrt{12-12\sqrt3+9}=\sqrt{(2\sqrt3-3)^2}=2\sqrt3-3,$$
which gives $a=-3$,$b=2$ and $a+b=-1$.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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possible to decrypt RSA using these parameters only? If our message is 204, our public RSA-key is (e, N) = (47, 221) but the private key is unknown.
is it possible to retrieve the message without the private key and what would be the steps to do so?
| This is a special situation you can easily test. In your case a private key is just the public key:
$$204^{47} \equiv 68 \pmod {221}, \quad 68^{47} \equiv 204 \pmod {221}$$
The reason for this is, that $$47^{-1} \equiv 47 \pmod {\lambda(N)}$$
where $\lambda(N)$ is the Carmichael function. The often used private key vi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2832557",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Calculating the probability of the 'common birthday problem' differently yields a different result? Everyone is calculating the birthday problem here by multiplying the probabilities of each person's birthday. Like the following "first oe has $365/365$ the second has $364/365$..... and so on..." . And this is the com... | The birthday probability can be written as:
$$\prod_{k=1}^{n-1}(1-k/365)=\frac{364!/(365-n)!}{365^{n-1}}=\frac{\binom{365}{n}}{365^{n}/n!}.$$
To use binomial coefficients, you can reason as follows. If you have $n$ people, you need to choose $n$ distinct birthdays, which can be done in $\binom{365}{n}$ ways. The total ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2832731",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Maximum number of spanning trees of a planar graph with a fixed number of edges Let $\mathcal{G}_m$ be the set of planar graphs with exactly $m$ edges. In this question, graphs are allowed to have multiple edges and/or loops.
I want to know what the maximum number of spanning trees of any graph in $\mathcal{G}_m$ is, a... | This is a very, very tough question. It is obvious that for any graph $G$ with $n$ edges the number of spanning trees $t(G)$ does not exceed $2^n$ (each edge is either included in or excluded from a subtree; this is also the upper bound of the number of connected subgraphs of $G$). We can somewhat improve this - if $G$... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Proving the product of four consecutive integers, plus one, is a square I need some help with a Proof:
Let $m\in\mathbb{Z}$. Prove that if $m$ is the product of four consecutive integers, then $m+1$ is a perfect square.
I tried a direct proof where I said:
Assume $m$ is the product of four consecutive integers.
If $m... | Given $m$ is the product of four consecutive integers.
$$m=p(p+1)(p+2)(p+3)$$where $p$ is an integer
we need to show that $p(p+1)(p+2)(p+3)+1$ is a perfect square
Now,$$p(p+1)(p+2)(p+3)+1=p(p+3)(p+1)(p+2)+1$$
$$=(p^2+3p)(p^2+3p+2)+1$$
$$=(p^2+3p+1)(p^2+3p+2)-(p^2+3p+2)+1$$
$$=(p^2+3p+1)(p^2+3p+1+1)-(p^2+3p+2)+1$$
$$=(... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2832986",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "18",
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A measure similar to variance that's always between 0 and 1? Consider the following histogram, obtained from around 1000 measures of distance.
As you can observe, most of the data appears near the mean arond the value 5-10. I also have some isolated samples far away at values 100, 160.
1) Is there any statistical meas... | One example of such functions is the exponential family:
$$f(v) = \exp[-v^k/s^k]$$
You input variance, which is in $[0,+\infty]$ and you get out something which is $[0,1]$
*
*If variance is $0$ you get $1$ out and
*the larger variance the closer you will get to $0$.
*$s$ and $k$ are both parameters you can steer ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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On The Parametric Equation Of A Parabola Let's look at a parabola with an equation $(y-k)^2=4a(x-h)$. I'm struggling to understand why its parametric equation would be $x=h+at^2$ and $y=k+2at$. Since it is being subtracted by $h$ and $k$ respectively, why would its $x$ and $y$ values increase/have $h$ and $k$ added to ... | Let's start with the simplest case, in which $h$ and $k$ are both $0$. Then the parabola is
$$y^2 = 4ax$$
and the parametric equation is
$$y=2at, \hskip{0.5in} x=at^2$$
and you can directly verify that
$$y^2 = (2at)^2 = 4a^2t^2 = 4a(at^2) = 4ax$$
so everything works nicely.
Now let's consider the general case. We ha... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2833245",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Derivative/Gradient of log $l_1$ norm As derivative of $l_{p}$- norm is
\begin{align*}
\frac{\partial}{\partial\mathbf{x}}{||\mathbf{x}||}_{p}
&= \frac{\mathbf{x} |\mathbf{x}|^{p-2}}{{||\mathbf{x}||}_{p}^{p-1}}
\end{align*}
I want to find $\nabla log(||H||_{1})$, where $H$ is positive matrix. So, the chain rule is
\beg... | Apply the sign function element-wise to the matrix $H$
$$S = {\rm sign}(H)$$
Use this to write the Manhattan norm as
$$\eqalign{
\|H\|_1 &= S:H \cr
}$$
where the colon denotes the trace/Frobenius product, i.e.
$\,\,A:B={\rm tr}(A^TB).$
Use this to calculate the logarithmic derivative of the Manhattan norm as
$$\eqali... | {
"language": "en",
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How to determine $f^{-1}$ for $f(p) = (1 + X)p' + p$
Let $ \mathbb{R}_2[X] $ be the vector space of polynomials with real coefficient and a degree less or equal $2$.
$f$ is the map of $ \mathbb{R}_2[X] $ into $ \mathbb{R}_2[X] $ defined as:
$$ f(p) = (X + 1)p' + p $$
with $ p \in \mathbb{R}_2[X] $.
*
*Prove $f$ is... | Yor answers are correct. You should elaborate a liitle bit more why $Im(f)=\mathbb R_2[X]$.
We have $f(1)=1$, hence the first column of $A$ is $(1,0,0)^T$.
$f(X)=1+2X$, hence the second column of $A$ is $(1,2,0)^T$.
$f(X^2)=2X+3X^2$, hence the third column of $A$ is $(0,2,3)^T$.
| {
"language": "en",
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"source": "stackexchange",
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About a sum that looks like a determinant I want to prove the following equality.
$$\sum_{\sigma\in S_n}\frac{\text{sgn}(\sigma)}{|\text{Fix}(\sigma)|+1}=(-1)^{n+1}\frac{n}{n+1},$$
where $\sigma$ is a permutation on $n$ elements and $\text{sgn}, \text{Fix}$ stand for the sign of the permutation and the fixed points ... | You definitely are on the right track. The LHS can be written as
$$ \int_{0}^{1} \sum_{\sigma\in S_n}\text{sgn}(\sigma) x^{|\text{Fix}(\sigma)|}\,dx=\int_{0}^{1}\det\left(\mathbf{1}+(x-1)\mathbf{I}\right)\,dx $$
where $\mathbf{1}$ stands for the rank-1 matrix whose entries are 1s only and $\mathbf{I}$ is the identity m... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
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I got stuck at limit problem
$$\lim_{n\to\infty} n\bigg[e^{\frac x{\sqrt n}}-\frac x{\sqrt n}-1\bigg] = \frac{x^2}{2}$$
I'm not sure how to solve it.
hope somebody could help me!
_ Is there any way to see solutions for limit problem?
| By the change of variable $t:=x/\sqrt n$ you can write
$$
\lim_{n\to\infty}n\left(e^{x/\sqrt n}-\frac x{\sqrt n}-1\right)=
x^2\lim_{t\to0^+}\frac{e^t-t-1}{t^2}$$ and the dependency on $x$ is gone.
Now by L'Hospital, twice,
$$\lim_{t\to0^+}\frac{e^t-t-1}{t^2}=\lim_{t\to0^+}\frac{e^t-1}{2t}=\lim_{t\to0^+}\frac{e^t}2.$$
... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Converting between different scales I have a scale which goes from 0 to 100. Given a number on that scale (say 33) I want to find the corresponding value on another scale which goes from 25 to 100 (in this case I think the answer is 50).
Any ideas how I should go about working out the equation to calculate what the cor... | Assuming you want a linear transformation, you need to do the following:
$$y_{\text{new}}-25=\frac{100-25}{100-0}\,x.$$
Check one: $0$ should go to $25$, which it does.
Check two: $100$ should go to $100$, which it does.
Check three: the relationship is linear, which it is.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Show that the chord $DE$ bisects the chord $BC$ in a circle Suppose a circle is centered at $O$, $CD$ is a chord perpendicular to a diameter $AB$, and a chord $AE$ bisects the radius $OC$. Show that the chord $DE$ bisects the chord $BC$.
If $N$ is the middle point of $CB$, $MN$ is a middle line in $OBC$, so $MN \paral... | Let $\{F\} = \overline{AB} \cap \overline{CD}$. Note that $\angle AED = \angle ACD$ by the Inscribed Angle Theorem, and also $\triangle ACF \sim \triangle ABC$, so $\angle ACD = \angle ABC = \angle OCB$. Therefore, $\angle OCB = \angle AED$.
In other words, $\angle MEN = \angle MCN$, so $MCEN$ is cyclic. From that, $\c... | {
"language": "en",
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Suppose $b$ a random number in the interval $(-3,3)$ Which the probability of the equation $x^2+bx+1=0$ have a least a real root?
Suppose $b$ a random number in the interval $(-3,3)$ Which the probability of the equation $x^2+bx+1=0$ have a least a real root?
I don't know how to start this exercise. Can someone help ... | Hint: The polynomial $f(x) = x^2+bx+1$ has a real root if there is an $x$ s.t. $f(x) \le 0$ is satisfied [make sure you can see why].
The minimum of $f(x)$ is achieved when $df(x)/dx = 2x+b$ is 0, at $x=\frac{-b}{2}$.
The values of $b \in (-3,3)$ s.t. the inequality $f(-\frac{b}{2}) = \frac{b^2}{4} - \frac{b^2}{2}+1 =-... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Roots of a function with a logarithm / Graph of a function So I'm trying to draw the graph (manually) of a function:
$$f(x)=x-2\ln{(x^2+1)}$$
Finding the first/second derivatives is fairly easy and so is proving that there are no asymptotes, but for the life of me I can't find the roots of the function, AKA where it in... | You can't find these roots analitically because $e^x = -2(x^2+1)$ is transcedental. However, you know the only extrema of $f$ are a local maximum $x_1 >0$, (then) a local minimum $x_2<0$, and that $\lim_{x\to\pm\infty} f(x) = \pm\infty$. The sign changes, the intermediate value theorem and the fact that you found all l... | {
"language": "en",
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"source": "stackexchange",
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Application of implicit function theorem? Let $f: U \subset \mathbb{R}^2 \to \mathbb{R}$ be a continuous function in the open subset $U$ of $\mathbb{R}^2$ such that
$$(x^2+y^4)f(x,y)+f(x,y)^3 = 1, \forall (x,y) \in U$$
Show that $f$ is of class $C^1$ in $U$.
I think that is an application of implicit function theorem, ... | As usual, Ted gave a great answer (and you should accept it). I'll just nitpick a bit, which you might or might not appreciate now: being $C^1$ is a local property, so you want to check that given $(x_0,y_0)\in U$, then $f$ is of class $C^1$ in a neighbourhood of $(x_0,y_0)$.
Great, so once you apply the IFT to the fu... | {
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What is vertex degree if each vertex represents a string of $\{0,1,2\}$ and there's edge between vertices iff the strings have one digit in common?
Each vertex in graph $G$ is composed of a string of length $3$ from digits $\{0,1,2\}$. There's an edge between two vertices iff their respective strings have only one dig... | There are a total of $27$ vertices in the graph. The number of string containing only $1$ and $2$ is $8$, hence there are $19$ vertices in which $3$ appears. Similarly for $1$ and $2$.
For the vertices we have the following:
$3$ contain a single digit
$18$ contain exactly $2$ digits
$6$ contain all $3$ digits
Using the... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Separation of subsets of $\mathbb{R}^n$ with the graph of a convex function Let $A, B \subset \mathbb{R}^n$ be compact and disjoint sets. Assume that there exists a $(n-1)$-dimensional surface $S$ such that separates the $A$ and $B$, i.e. there exists $C \subset \mathbb{R}^n$ such that $\partial C = S$ and $A \subset ... | Assume that $A,\ B$ are compact and disjoint.
And assume that $S$ is convex hypersurface separating $A,\ B$ s.t.
$C$ is a convex set and $S=\partial C$. Then we can assume that
${\rm conv}\ A$ is in $C$.
Hence $({\rm conv}\ A)\bigcap B$ has measure $0$.
EXE : So there is a unit vector $v$ s.t. $$({\rm conv}\ \{ a+tv|a\... | {
"language": "en",
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Classical logic is the strongest consistent logical system I vaguely remember reading somewhere about a theorem which states that classical logic is the strongest logical system in some sense.
Unfortunately, after much search, I cannot find any reference. I’m not sure what notion of "strength" was involved here - perha... | You are probably thinking of Lindstrom's theorem which says that, among a family of abstract logics, first-order logic is the strongest that satisfies the compactness theorem and the downward Lowenheim-Skolem theorem.
| {
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Proving $\Big(a+\frac{1}{b}\Big)^2 + \Big(b+\frac{1}{c}\Big)^2 + \Big(c+\frac{1}{a}\Big)^2 \geq 3(a+b+c+1)$
Prove for every $a, b, c \in\mathbb{R^+}$, given that $abc=1$ :
$$\Big(a+\frac{1}{b}\Big)^2 + \Big(b+\frac{1}{c}\Big)^2 + \Big(c+\frac{1}{a}\Big)^2 \geq 3(a+b+c+1)$$
I tried using AM-GM or AM-HM but I can't f... | By Cauchy we have $$\Big(a+\frac{1}{b}\Big)^2 + \Big(b+\frac{1}{c}\Big)^2 + \Big(c+\frac{1}{a}\Big)^2 \geq \underbrace{{1\over 3}\bigg(a+b+c+\frac1a+\frac1b+\frac1c\bigg)^{2}}_B$$
By AM-GM and assumption $abc=1$ we have $$\frac1a+\frac1b+\frac1c \geq 3$$ so $$B \geq {1\over 3}\underbrace{\bigg(a+b+c+3\bigg)^{2}}_C$$
Le... | {
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Does the Tietze extension theorem hold for maps into R if on R we put the semi open interval topology? I guess the above is false, so I'm trying to find a counter example. The only thing I found is the whole space should not be compact. If not, the statement is true directly from the Tietze extension.
Any help will be ... | I guess you mean the topology generated by the half-open intervals $[a,b)$. It's called the lower limit topology: https://en.wikipedia.org/wiki/Lower_limit_topology
You are asking if the Tietze extension theorem still holds on a space $X$ (I assume normal) if the topology on $R$ is not the standard topology (as it is i... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2835021",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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the difference between a strip with two end glue together and a strip with two end glue together with $2\pi$ twist a strip with two end glue together is homeomorphic to a strip with two end glue together with $2\pi$ twist. I want to know why they are different in $R^{3}$ ? I think that there is a relation stronger than... | Botnh your spaces are $X= S^1\times[0,1]$, but you have two non-isotopic embeddings into $\Bbb R^3$, i.e., there is no continuos map $X\times[0,1]\to\Bbb R^3$ such that each $X\times\{t\}\to\Bbb R^3$ is an embedding$^1$ and $X\times\{0\}\to\Bbb R^3$, $X\times\{1\}\to\Bbb R^3$ correspond to the two shapes in question.
... | {
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How many equivalence classes are over $4$-digit strings from $\{1,2,3,4,5,6\}$ if strings are in relation of they differ in order or are the same?
Let $A=\{1,2,3,4,5,6\}$. Let $K$ be the set of strings of length $4$
which are made up from the elements of $A$. For example, $(3,3,1,5)\in K$.
Let $E$ be a relation ove... | The equivalence class of a string $s \in K$ is determined by the number of times each digit from $\{1,2,3,4,5,6\}=:[6]$ is occurring in $s$. The set $K/_\sim$ of equivalence classes is therefore bijectively related to the multisets of cardinality $4$ on $[6]$. Counting these multisets is a stars and bars problem; the r... | {
"language": "en",
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Prove the completeness of the following metric space We know that if $(X,d)$ is a metric space then $\sigma=\frac{d}{1+d}$ is also a metric on $X$.
If $(X,d)$ is a complete metric space then how to prove that $(X,\sigma)$ is also a complete metric space ?
| To get the triangle inequality, note that the map $t \mapsto \frac{t}{t+1}$ is increasing. We have the triangle inequality for the metric $d$ already, so
\begin{equation*}
\begin{split}
&\quad \quad \quad d(x,z) \leq d(x,y) + d(y,z) \\
&\implies \frac{d(x,z)}{1+ d(x,z)} \leq \frac{d(x,y)}{1+ d(x,y)}... | {
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Maximum of expression (formula related to refraction) Let's suppose we have $n+1$ real values $d_i$ ($i=0,\ldots,n$) with $0 \le m \le d_0 \lt d1 \lt \ldots \lt d_{n-1} \lt d_n \le M$ (we don't know anymore about $d_i$), and $n$ real values $r_i \ge 1$ ($i=1,\ldots,n$) with no specified order. Then compute $c$ as follo... | Finally it turned out to be easy:
$$
c = d_0 + \sum_{i=1}^n (d_i-d_{i-1})r_i \le d_0 + \max_{i=1}^nr_i \cdot \sum_{i=1}^n (d_i-d_{i-1}) = d_0 + \max_{i=1}^nr_i \cdot (d_n-d_0) = d_0 (1 - \max_{i=1}^nr_i) + d_n \max_{i=1}^nr_i
$$
and since the first term is always $\le 0$ and the second term is always $\ge 0$ then:
$$
d... | {
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Let $A \subset X$; A retraction of $X$ onto $A$ is a continuous If $A$ is retract of $X$, then the homomorphism of fundamental groups induced by inclusion $j:A \rightarrow X$ is injective
This Lemma in Munkres has about two lines of proof as below,
If $r:A \rightarrow X$ is a retraction , then the composite map $r \c... | This is a general fact about set-maps, even: if $g \circ f$ is the identity map, then $f$ must be injective (and $g$ surjective). The proof is simple: if $f(a_1) = f(a_2)$ then hit this with $g$ on the left to get $g(f(a_1)) = g(f(a_2)$. But $g \circ f$ is the identity so $a_1=a_2$. Done.
Since you have $r \circ j ... | {
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Can someone help me do this limit? $ \lim_{n\to\infty} \frac{n!\times(2n)!}{(3n)!}$ can someone help me with this limit?
I don't know how to expand that factorial multiplication, so what I've done so far is substitute what given:
$$ \lim_{n\to\infty} \frac{n!\times(2n)!}{(3n)!}$$
$$ \lim_{n\to\infty} \frac{\infty\ti... | This is going to be a horrible overkill, but a funny one. Since
$$ \sum_{n\geq 0}\frac{n!(2n)!}{(3n)!} = \sum_{n\geq 0}(3n+1) B(n+1,2n+1)=\int_{0}^{1}\sum_{n\geq 0}(3n+1)(1-x)^{n}x^{2n}\,dx $$
equals $\phantom{}_3 F_2\left(\frac{1}{2},1,1;\frac{1}{3},\frac{2}{3};\frac{4}{27}\right)$ or $\int_{0}^{1}\frac{1+2x^2-2x^3}{(... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2836020",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 7,
"answer_id": 1
} |
How to calculate a negative feedback loop? Perhaps this could best be explained as a closed system between two people:
1) For every $1 person A receives, he will give 50% to person B and keep the rest.
2) For every 1$ person B receives, he will give 25% to person A, and keep the rest.
3) Now, person C hands person A $1... | EDIT: I misread the question, as was pointed out in the comments. However, formalizing a question like this using recurrence relations is still often a sound strategy, you just have to model it correctly. If I could recommend a general strategy to questions like this, I'd start by writing out the first few terms by han... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2836085",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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} |
why a closed set is not bounded even though it converges I looked at this question Does closed imply bounded?
according to the definition of closed set
A set $S$ in $\mathbb{R}^m$ is closed if, whenever
$\{\mathbf{x}_n\}_{n=1}^{\infty}$ is convergent sequence completely
contained in S, its limit is also contained ... | A closed set could be bounded or unbounded and a bounded set could be closed or not closed.
For example the set of integers is closed and unbounded while the interval $[0,1]$ is closed and bounded.
Convergent sequences in a set being bounded does not mean the set itself is bounded, for example in the set of natural nu... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2836205",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Can convolution be written as $(f \ast g)(t)=\int_{\tau = -\infty}^{\infty}g(t-\tau)f(\tau)\,\mathrm{d}\tau$? In every textbook that I read the definition of convolution is almost always given by
$$(f \ast g)(t)=\int_{\tau = -\infty}^{\infty} f(\tau)\,g(t-\tau)\,\mathrm{d}\tau\tag{1}$$
apart from a significant number... | Yes, this follows from change of variables. Consider the case when $f(t) = g(t) = 0$ if $t < 0$. Then, for each $t \geq 0$, using the substitution $s = t - \tau$, we obtain
\begin{equation*}
(f*g)(t) = \int_{0}^{t} f(\tau)g(t - \tau) \, d\tau = \int_{0}^{t} f(t - s) g(s) \, ds.
\end{equation*}
The same idea works in ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2836332",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Proving injectivity of a certain polynomial function So I have to find the second derivative of the inverse function of:
$$f_{(x)}=3x^4+2x^3-8x^2-20x-160 ; x\ge 2$$
which I already have done, but I still need to prove the function is injective over the given domain.
I've tried the classic $f_{(x_1)}=f_{(x_2)}~$, but I... | For $x\geq2$ we obtain:
$$(3x^4+2x^3-8x^2-20x-160)'=2(6x^3+3x^2-8x-10)=$$
$$=2(6x^3-12x^2+15x^2-30x+22x-44+34)=$$
$$=2(x-2)(6x^2+15x+22)+68>0,$$ which gives which you want.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2836419",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
} |
Finding third point of right triangle
I have the length of BC,and AC and the coordinates of points A and C. I'm trying to find the coordinates of B.
| I believe you can find the length of $AB$ (Pythagoras!). If you let the coordinates of $B$ be $(x,y),$ then since you know the other distances and coordinates, you can use the distance formula to find two equations in $x$ and $y$ (apply it to $|BA|$ and $|BC|$ in turn) that you can hopefully solve for $(x,y).$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2836592",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
} |
Algebraic manipulation in Vaughan's book on the circle method On page 125 of Vaughan's1 book on the circle method we have in Chapter 5.2 on Waring's problem for $k=1$.
I do not understand how to get from $(1)$ to $(2)$:
\begin{align}
f(z)^s &= \frac{1}{(1-z)^s}\tag{1}\\
&=\frac{1}{(s-1)!} \frac {d^{s-1}} {dz^{s-1}} \fr... | It might make more sense to look at (2) first. Here it is again with brackets to make it clear that it contains an ${(s-1)}$th-order derivative:
$$
\frac{1}{(s-1)!} \cdot \frac{d^{s-1}}{dz^{s-1}} \left[ \frac{1}{1-z} \right].
$$
So you want to differentiate the quantity in the brackets $s-1$ times. Recall that
\begin... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2837021",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Proof: Space of real matrices is the direct sum of spaces of symmetric and skew-symmetric matrices This is the problem I am trying to solve (from Artin's Algebra, Chapter 3: Vector Spaces, Section 5). I did some searching around here, and I found similar problems but no problems phrased in exactly this way:
Prove that... | You do not need to work element-wisely to get the proof done. In addition, while there may be multiple equivalent conditions to show a sum of two subspaces $V_1 + V_2$ is a direct sum, to verify that $V_1 \cap V_2$ is $\{0\}$ is the most expedient for your question. Below is a complete proof.
Let $M$ denote the space o... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2837130",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
A subset of a finite set is finite I saw this problem in a book and the solution given and thought of a simpler solution. This makes me suspicious of its validity. If someone could point out why my solution is incorrect I would appreciate it.
First, suppose that $|S|=1$ and $T\subset S$. Then $T=\emptyset$ or $S$. Hen... | I would guess that your book's proof is more complicated for one or more of the following reasons:
*
*Cardinality has not been defined yet
*Your book is using a definition of finiteness that requires more work than this
*The authors intend to later address constructive set theory in which the result can fail for... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2837203",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Generating Functions written as product of geometric series I am reading Richard Stanley's "Topics in Algebraic Combinatorics" and just before the notes for Chapter 8, he was discussing generating functions for plane partitions and solid partitions. It is claimed there that:
It is easy to see that for any integer sequ... | Expaning the factors on the right-hand side as power series shows that they affect the coefficients $a_n$ only for $n\ge i$. Thus the $b_i$ can be iteratively determined for $i=1,2,3,\ldots$ , with each $b_i$ chosen such $a_i$ comes out right, without messing up $a_n$ for $n\lt i$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2837285",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Topology: Homeomorphism between finite complement topology in $\mathbb{R}$ and one of its subspaces My class notes say that because $U=\mathbb{R}\backslash\{x_1,x_2,..,x_n\}$ has the same cardinality than $\mathbb{R}$, there exists a homeomorphism between:
$(U,T_{cof})$ and $(\mathbb{R},T_{cof})$, where $T_{cof}$ is th... | The following is easy to prove:
Proposition 1: Let $X$ be a set endowed with the cofinite topology. The subspace topology on any $Y \subset X$ is identical to the cofinite topology on $Y$.
Proof: Exercise.
Proposition 2: Let $X$ and $Y$ be two cofinite topological spaces. Then any injective mapping $f: X \to Y$ is cont... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2837404",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 2
} |
Calculate $\sum_{n=1}^\infty\frac{1}{n^2}$ and $\sum_{n=1}^\infty\frac{1}{n^4}$ I need to do that using
$$\sum_{n \in \mathbb{Z}}\frac{1}{(z-n)^2}=\left(\frac{\pi}{\sin \pi z}\right)^2$$
I've already prove that this is true. The thing is that this function in meromorphic and it's poles are $\mathbb{Z}$. So the natural... | For the first sum:
For $z = \frac12$ we have
$$\pi^2 = \frac{\pi^2}{\sin^2\frac{\pi}2} = \sum_{n\in\mathbb{Z}}\frac1{\left(n-\frac12\right)^2} = 4\sum_{n\in\mathbb{Z}}\frac1{(2n-1)^2} = 8\sum_{\substack{k\in\mathbb{N} \\ k \text{ odd}}} \frac1{k^2}$$
$$\sum_{\substack{k\in\mathbb{N} \\ k \text{ even}}} \frac1{k^2} = \s... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2837543",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Number of different sums with k numbers from {1, 5, 10, 50}
Say we have $k$ numbers, each of which belongs to the set $S = \{1, 5, 10, 50\}$
How many different sums can be created by adding these numbers?
If $k = 1$, the are four different sums.
Also, if $k = 2$, there are ten:
$$\begin{align} 1 + 1 = 2 \quad 1 +... | Imagine that we have $k$ baskets, labeled $50,\ 10,\ 5,\ 1.$ If we distribute $k$ balls into the baskets, the number of balls in each basket indicates how many summands of each value to take. The number of ways of distributing the balls can be computed with stars and bars. It is the binomial coefficient $$\binom{k+3... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2837653",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
Neural Networks and the Chain Rule With neural networks, back-propagation is an implementation of the chain rule. However, the chain rule is only applicable for differentiable functions. With non-differentiable functions, there is no chain rule that works in general. And so, it seems that back-propagation is invalid... | The answer to this question might be more clear now with the following two papers:
*
*Kakade and Lee (2018)
https://papers.nips.cc/paper/7943-provably-correct-automatic-sub-differentiation-for-qualified-programs.pdf
*Bolte and Pauwels (2019) https://arxiv.org/pdf/1909.10300.pdf
As you say, it is wrong to use the ch... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2837737",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 1,
"answer_id": 0
} |
Equation involving floor function and fractional part function How to solve $\frac{1}{\lfloor x \rfloor} + \frac{1}{\lfloor 2x \rfloor} = \{x\} + \frac{1}{3}$ , where $\lfloor \rfloor$ denotes floor function and {} denotes fractional part. I did couple of questions like this by solving for {x} and bounding it from 0 ... | Notice that (1) the values $\lfloor x \rfloor, \lfloor 2 x \rfloor$ depend only on the value of the integer $\lfloor 2 x \rfloor$ and (2) $x \geq 1$. On the other hand, for, e.g., $x \geq 5$, we have $\frac{1}{\lfloor x \rfloor} + \frac{1}{\lfloor 2 x \rfloor} \leq \frac{1}{5} + \frac{1}{10} = \frac{3}{10} < \frac{1}{3... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2837861",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 1
} |
What's an $\mathcal O_X$-algebra when $X= \operatorname{Spec} R$? Take $X= \operatorname{Spec}R$, $R$ a commutative ring with unit. What is an $\mathcal O_X$-algebra in that case? Is there more than just ordinary $R$-algebras?
Thank you in advance.
| In the following thread
Noetherian $R$-algebra corresponds to a coherent sheaf of rings on $\operatorname{Spec}(R)$
you find the following: Let $X:=Spec(A)$ with $A$ a commutative unital ring. I believe the following "Theorem" holds:
"Theorem". There is an equivalence of categories between the category of sheaves of qu... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2837923",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
The set $A=\{0\} \cup \{\frac 1n \mid n \in \mathbb N\}$ For the set $A=\{0\} \cup \{\frac 1n \mid n \in \mathbb N\}$, I understand that $\{\frac 1n \mid n \in \mathbb N\}$ is open and closed in $A$ because it is a union of all the connected components $\{\frac 1n\}$ in $A$ for all $n \in \mathbb N$. Even though $\{0\}... | Viewing A as a subspace of R, since {0} is closed,
within A, B = A - {0} is open. B is not closed within A
because 0 is an adherance point of B that is not in B.
Using the clumbsy definition of closed, B is not closed
within A because 0 is a limit point of B that is not in B.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2838037",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
How to minimize $x+4+\frac{1}{2x-1}$? How can I minimize $x+4+\frac{1}{2x-1}$ for $x > \frac{1}{2}$? I have tried using AM-GM to get the inequality
$2\sqrt{\frac{x+4}{2x-1}} \leq x+4+\frac{1}{2x-1}$.
Then, using that AM = GM when the terms are equal, I got that $x = \frac{-7+\sqrt{89}}{4}$. However, when I used calculu... | This is not an answer but an illustration.
I agree with dxiv that OP found a lower bound instead of greatest lower bound. Actually, when we split an expression into 2 and then use A.M.$\geq $G.M., we can create many lower bound, which may be dependent on a variable. In order to get the greatest lower bound a constant,... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2838097",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Singularity of the product of two rectangular matrices? Let $A$ be an $m \times n$ matrix and $B$ be an $n \times m$ matrix where $m<n$.
Then can we say that the product $AB_{m \times m}$ is always singular or always non-singular?
Also, can we say that $BA_{n \times n}$ is always singular or non-singular?. Does this c... | $\textbf{BA is always Singular}$:
Result: $A$ is $m \times n$ and $m<n$ implies $Ax=0$ has non zero solution $x_0$.
Proof:
View the matrix $A$ as a linear map $A:\Bbb{F}^n \rightarrow \Bbb{F}^m, x \mapsto Ax.$
By dimension theorem, $$dim\;\Bbb{F}^n=rank\;A+null\;A $$
$$n\leq m+null\;A$$
So, $null\;A \geq n-m >0$,.... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2838189",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
} |
What does "copies" of a Ring/Module etc. mean Could someone explain me what the word "copies" in Terms of Rings/Vectorspaces/Modules means?
E.g in the context "For a ring R, the smallest subring containing 1 is called the characteristic subring of R. It can be obtained by adding copies of 1 and −1 together many times i... | "Copies" does not really stand for a mathematical thing. "Direct sum of $n$ copies of $M$" is just the best way English and several indo-european languages have at their disposal to state concisely that a construction such as $X_1\oplus X_2\oplus\cdots \oplus X_n$ (i.e. "direct sum of $n$ objects") is applied to the ca... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2838285",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Understanding Proof of Cauchy-Schwarz Inequality The following is part of the proof for the Cauchy-Schwarz inequality from Appendix C (Linear Spaces Review) of Introduction to Laplace Transforms and Fourier Series, Second Edition, by Phil Dyke:
I'm struggling to understand the following:
*
*$- \lambda \langle \alph... | The term-by-term steps going to the second line of the expansion are
$$
\lambda \langle\alpha b,a \rangle \to \lambda \alpha \langle b,a\rangle\\
\lambda \langle a,\alpha b \rangle \to \lambda \bar \alpha \langle a,b \rangle
$$
From there, we note that $\langle a,b \rangle = \alpha$, and $\langle b,a \rangle = \bar \al... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2838392",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
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