Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
absolute value random walk also a markov chain Consider the random walk $S_{n}$ for $n \geq 1$. Specifically, let $X_{1},X_{2},..$ be Independent with
$$
\mathbb{P}(X_{n}=1) = p,~~\mathbb{P}(X_{n}=-1) = 1- p =:q
$$
and $S_{n} = \sum_{k=1}^{n}X_{k}$. I have read in the book of Ross, Stochastic Processes on page 166, tha... | The probabilities in your last two displayed equations aren't quite right. It's not
$$
\mathbb P(S_n=\pm i,\ldots,|S_{j+1}|=i_{j+1},|S_j|=0)=p^{\frac{n-j}2\pm\frac i2}q^{\frac{n-j}2\mp\frac i2}
$$
but
$$
\mathbb P(S_n=\pm i,\ldots,|S_{j+1}|=i_{j+1}\;\big|\;|S_j|=0)=p^{\frac{n-j}2\pm\frac i2}q^{\frac{n-j}2\mp\frac i2}\;... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2813303",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
kernel of quotient is quotient of kernel? Let $A,B,C$ groups and $f:A\to B$ a homomorphism with $C\subset \ker f \subset A$.
Then $f$ induces a map on the quotient $A/C\to B$
Is it then true that $\ker (A/C\to B)= (\ker A\to B)/C~$?
I tried to prove it, but i am not sure if I did it right. The proof seems to tautologic... | Apart from $f(a)=1_B$ rather than $f(a)\in{\rm Ker}(A\to B)$, your proof looks correct.
When proving an equality like this, you do sometimes find that the proof looks tautological. This is because everything is just true by definition.
That is ${\rm Ker}(A/C\to B)$ and ${\rm Ker}(A\to B)/C$ both consist precisely of t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2813427",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
equation of a plane in space though 3 points We have the points $M(1,2,3),N(2,1,5), P(4,3,2)$ is it a correct way to find the equation of the plane using the determinant $\Delta$
$$\Delta=\begin{array}{|cccc|}
x & y & z & 1\\
1& 2 & 3 & 1\\
2 & 1 & 5 & 1\\
4 & 3 & 2 & 1
\end{array}=0$$
If so, what is the intuition behi... | $$\Delta=\begin{array}{|cccc|}
x & y & z & 1\\
1& 2 & 3 & 1\\
2 & 1 & 5 & 1\\
4 & 3 & 2 & 1
\end{array}=x\begin{array}{|ccc|}2&3&1\\1&5&1\\3&2&1\end{array}-y\begin{array}{|ccc|}1&3&1\\2&5&1\\4&2&1\end{array}+z\begin{array}{|ccc|}1&2&1\\2&1&1\\4&3&1\end{array}-\begin{array}{|ccc|}1&2&3\\2&1&5\\4&3&2\end{array}=0$$
Compa... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2813557",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Problems with recursive definitions I'm having a bit of trouble understanding how to go about formulating recursive definitions. This has caused trouble on the following:
$(1)$ Give a recursive definition of the set of subsentences of a sentence $\phi$ of $L_1$ — i.e. give a recursive specification of the function $\ph... | In order to define the function $\text{Sub}(ϕ) = \text { the set of subformulas of } ϕ$, we have to follow the recursive (or inductive) definition of formula :
(i) $\phi \text { is an atomic formula, i.e. }$ :
$\top, \bot, t_1=t_2 \text { and } P_n(t_1,\ldots, t_n), \text { for terms } t_i \text { and predicate sym... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2813679",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Solve $\dot X = AX$ and find the time at which the area doubles. We have a linear operator $X(t): \mathbb{E}^2 \to \mathbb{E}^2$ such that
$$\dot X = AX\quad X(0)=\mathbf{1}$$
To be clear $X$ is a $2\times 2$-matrix. An ink spot is contained in $\mathbb{E}^2$ at $t=0$. At which time $t$, will the area of the image of t... | I know the solution is $X(t) = \exp(tA)$. Now the area of a scaling $2\times 2$ matrix is the absolute value of the determinant. So we have
$$|\det(X(t))|= |\det(\exp(tA))|= |\exp(tr(tA))|= \exp(-2t)$$
For $t=0$ we have $1$. So we need to solve
$\exp(-2t) = 2$ that is $t= \frac{\ln 2}{-2}$.
Posting my working as an ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2813807",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Substitution in a double integral I have trouble in doing a substitution. That is one step in a whole demonstration. I need to compute this integral : $\epsilon << 1 $
$$ \\ \int_{ [0,1-\epsilon] } \int_{ [0,1-\epsilon] } \frac 1 {1-xy} \, dx dy$$
And for that I need to do the substitution :
$$ x = u-v \, ; \, y = ... | On the $uv$-plane, we can graph the intersection of the following inequalities
$$
\begin{align}
u-v > 0 \\
u-v < 1 - \epsilon \\
u+v > 0 \\
u+v < 1 - \epsilon
\end{align}
$$
Here is what the intersection looks like.
When $u \in \left[0, \dfrac{1-\epsilon}{2}\right]$, it follows then that $v \in [-u,u]$. Otherwise, when... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2813890",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Groups, G finite group from order 2p, p>2 prime and N is normal subgroup with index p in G , prove G cyclic i think i proved this question, but my proof isnt really elegant.
i assumed by contraditcion that there isnt exist an element from order 2p. then all the elements from lagrange are from order 2 or p.
Let N be {e,... | Since $N$ has index $p$, $G/N$ has order $p$ and is therefore cyclic. Let $gN$ generate $G/N$.
In particular $g^pN=(gN)^p=N$. Suppose $G$ is not cyclic, then the order of $g$ must be $p$ (why?).
Since $N$ is normal in $G$, $g^{-1}xg=x$ (you might be able to say this without justification, but it could do with a justifi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2813934",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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What is the meaning of "has partial derivatives everywhere"? I am reading Courant's "Differential and Integral Calculus Vol.2".
I am confused with something: Courant says that the function defined as $u(x,y)=\frac{2xy}{x^2 +y^2}$ and $u (0,0)=0$ is not continuous but has partial derivatives everywhere.
I am assuming t... | "Has partial derivatives everywhere" means quite literally that it has partial derivatives everywhere. In other words, at every point, the partial derivatives of $u$ with respect to each variable exist. In other words, for all $(x,y)\in\mathbb{R}^2$ the limits $$u_x(x,y)=\lim_{h\to 0}\frac{u(x+h,y)-u(x,y)}{h}$$ and $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2814074",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
} |
Determine the convergence being, Be,
$$f(x)=\lim_{n\rightarrow+\infty}\frac{x}{1+x^{n}},\hspace{5mm}\forall x\geq0$$
what I want is to graph that function, but as I do analytically, because for $ n = 0 $ and $ n = 1 $ are a line and a hyperbole, but then while the $ n \rightarrow \infty $ 'apparently' $ f (x) \rightarr... | Recall that for $x \ge 0$ we have
$$\lim_{n\to\infty} x^n = \begin{cases}
0, & \text{if $x < 1$} \\
1, & \text{if $x = 1$} \\
+\infty, & \text{if $x > 1$} \\
\end{cases}$$
Now clearly $$\lim_{n\to\infty} \frac{x}{1+x^n} = \begin{cases}
x, & \text{if $x < 1$} \\
\frac12, & \text{if $x = 1$} \\
0, & \text{if $x > 1$} ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2814203",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 0
} |
Suppose $f(x)$ monotonous decreases on $[0,+\infty) $ Suppose $f(x)$ monotonous decreases on $[0,+\infty) $ and $$\lim_{x\to+\infty}f(x) \text dx=0 $$
Then, proof that $\sum_{n=1}^{\infty} f(n) $ converges if and only if $\int_{0}^{\infty}f(x)\text dx $converges.
Actually I listed an inequality and almost thought it... | What you did is right. The fact that $\lim_{x\to\infty}f(x)=0$ is irrelevant for the proof that $\displaystyle\sum_{n=1}^\infty f(k)$ converges if and only if $\displaystyle\int_1^\infty f(x)\,\mathrm dx$ converges. Of course, when they converge, then $\lim_{x\to\infty}f(x)=0$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2814352",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Calculate the second derivative of this rational function I'm studying for my exam Math and I came across a problem with one exercise.
Calculate the second derivative of $$f'(x)=\frac{2x^3-3x^2}{(x^2-1)^2}$$
I just can't seem to calculate the second derivative of this rational function. If someone could help me .
I... | I split the fraction into two parts, then used the product rule on both parts.
$\dfrac{2x^3-3x^2}{(x^2-1)^2} = (2x^3*(x^2-1)^{-2}) - (3x^2*(x^2-1)^{-2})$
After product rule on both terms we have
$\dfrac{12x^3-8x^4}{(x^2-1)^3} + \dfrac{6x^2-6x}{(x^2-1)^2}$
$\dfrac{12x^3-8x^4}{(x^2-1)^3} + \dfrac{(6x^2-6x)*(x^2-1)}{(x^2-... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2814449",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Satisfying explanation of Aristotle's Wheel Paradox. The paradox:
We have a circle and there is another circle with smaller radius. They are co-centeric.
If circle make full turn without sliding, both smaller and bigger circle make full turn too. If we assume that the passed road is equal to the circumference of circl... | Its really quite simple and doesnt even need math to show..
Take any given point on the outside larger circle as it moves along..an do the same for the smaller one...neither one travels in a straight line from point A to point B....and they dont travel the same distance. Its seems like they do but they dont..Thats the... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2814598",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 5,
"answer_id": 4
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Is every open interval a union of half open intervals? I am reading lower limit topology on Wikipedia, which states that the lower limit topology
[...] is the topology generated by the basis of all half-open intervals $[a,b)$, where a and b are real numbers. [...] The lower limit topology is finer (has more open sets... | $$(a,b) = \bigcup_{n=1}^\infty \, \left[\, \left(1-\frac{1}{n}\right)\, a + \frac{1}{n} b ,\, b\right)
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2814703",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
"answer_count": 5,
"answer_id": 4
} |
Expressive adequacy After reading a few others posts on this site, I am still struggling in showing how sets are incomplete.
(a) Show that {$↔,\bot, ∧$} is functionally complete, but that no proper subset is.
(b) is solved! :)
(b) Assume that $c$ is a 2-place connective. Show that if either $f_c(\top, \top) = \top$ or ... | As Noah points out in the comments, you know that $c$ cannot be either the $\uparrow$ and $\downarrow$, and since those are the only two connectives that are by themselves complete, you know $c$ is not complete.
However, I doubt that this proof is 'acceptable', or at least probably not what was expected of you. Probab... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2814838",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 0
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Proving $y = x + 1$ doesn't have a quantifier-free formula in $(\mathbb{Z}, =, <)$ Assume the signature is $(\mathbb{Z}, =, <)$ with the natural interpretation, and consider whether the predicate $y = x + 1$ is representable as a quantifier-free formula in this interpretation.
First, clearly, it's representable with qu... | Quantifier-free formulas are preserved (and reflected) by embeddings. That is, if $M$ and $N$ are $L$-structures, $f\colon M\to N$ is an embedding, $\varphi(x_1,\dots,x_n)$ is a quantifier-free formula, and $a_1,\dots,a_n$ is a tuple from $M$, $$M\models \varphi(a_1,\dots,a_n) \iff N\models \varphi(f(a_1),\dots,f(a_n))... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2814942",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 1,
"answer_id": 0
} |
Understanding partial derivatives related to the Cauchy-Riemann equations So I am reading a soft introductory book on complex variables (I have complex analysis next year.) The book I am reading is called "Complex Variables Demystified". I am currently reading a chapter where the main goal is to arrive at the Cauchy-Ri... | is this correct? :)
and this? Thanks for your response by the way!
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 1
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Constrained system of linear equations I have a question that might or might not be trivial for experts from the linear programming world.
We have a system of linear equations that we want to solve:
$A\cdot x=0$, with the constraint that all variables are non-negative: $x_i \geq 0 ~\forall i$.
The system is underdeter... | If the problem is an actual numerical problem for which you have a matrix A, then you can set this problem up as a semidefinite program (see Convex Optimization by Boyd and Vandenberghe).
Such a system can be solved by good convex solvers like cvx, or sedumi and yalmip in Matlab, or Pyopt in python. In this case, si... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2815185",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 1
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Proving an analytic function $f$ is bounded on $|z|\le1/2$ independent of $f$ subject to certain conditions
Let $f:D(0,1) \to \mathbb C$ be analytic. Show that there is a constant $C$ independent of $f$ such that if $f(0)=1$ and $f(z) \notin (-\infty,0]$ for all $z \in D(0,1)$, then $|f(z)| \le C$ whenever $|z| \le 1/... | Take $f(z) = \left( \dfrac{1 + z}{1 - z} \right)^2\ (|z| < 1)$, then $f(0) = 1$ and$$
\left| f\left( \frac{1}{2} \right) \right| = \left| \frac{1 + \dfrac{1}{2}}{1 - \dfrac{1}{2}} \right|^2 = 9.
$$
Thus $9$ is indeed the tightest bound.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2815268",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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Expected value of Brownian motion when it is less that a given number:$E[W_t\mathbb{1}_{(W_t \leq a)}] $ I want to find $E[W_t\mathbb{1}_{(W_t \leq a)}] $, where $W_t$ is Brownian motion and $a \in \mathbb{R}$. I thought that since $W_t \sim N(0,t)$, that its pdf would be $\frac{1}{\sqrt{2\pi t}}e^{-x^2/2t}$, and tried... | What you have calculated is $EI_{{W_t} \leq a}$ and not $EW_tI_{{W_t} \leq a}$. Multiply the integrand by $x$ and then integrate.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2815405",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
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Shortest distance between parabola and point
Find the shortest distance between the parabola defined by $y^2 = 2x$ and a point $ E:= (1.5, 0)$.
I can't use the distance formula because I'm missing a set of points $(x, y)$ to plug into. So, instead, I have a normal that passes through the point $E$ from the parabola. ... | Given a point $P=\left(\frac{y^2}2,y\right)$ of your parabola, consider the line segment joining $P$ to $C=\left(\frac32,0\right)$. The slope of this line segment is $\frac{2y}{y^2-3}$. And the slope of the tangent to the parabola at $P$ is $\frac1y$. Since two lines are orthogonal if and only if one of them is horizon... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2815493",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 7,
"answer_id": 5
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Simplifying $e^{2x+1/2}$ So I was taking some derivatives and the question was
$e^{2x+\frac{1}{2}}$
convert this to $a\cdot b^x$ where $a$ and $b$ are constants.
This is apparently needed to take the derivative of it without using the chain rule.
Any idea how to tackle this? Tried to manipulate it but I always end ... | $$e^{2x+\frac12}=e^{2x}\cdot e^{\frac12}=\sqrt e\cdot(e^2)^x$$So $a=\sqrt e, b=e^2$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2815671",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
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Automorphism of power series Let us consider an endomorphism $\mathbb{C}[[x_1, ..., x_n]] \to \mathbb{C}[[x_1, ..., x_n]]$, $g(x_1,..,x_2) \mapsto g(f_1, ..., f_n)$ where $f_1, ..., f_n \in \mathbb{C}[[x_1, ..., x_n]]$. My homework is to show that it is an automorphism iff $ J = det(\frac{\partial f_i}{ \partial x_j})$... | I’m real handy with one-variable series, and any suggestion I give here may be off the mark for many-variable series.
But I would prepare the situation by composing with the linear inverse of the Jacobian matrix. That is, form $J(\mathbf 0)$, take the inverse of this, and then take the linear substitution whose matrix ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2815776",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Does $\sum\limits_{n=1}^\infty \ln\left(\frac{p_n}{p_n - 1}\right)$ converge? Suppose $p_n$ is the $n$-th prime number. Does $\sum\limits_{n=1}^\infty \ln\left(\frac{p_n}{p_n - 1}\right)$ converge?
Where did this question arise from:
I was trying to find $\inf_{n \in \mathbb{N}} \frac{\phi (n)}{n}$, where $\phi$ is Eul... | Note $\ln(p/(p-1)) = \ln(1/(1 - 1/p))$, so $\sum_{p\leq x} \ln(p/(p-1)) = \ln(\prod_{p\leq x} 1/(1-1/p))$. Intuitively, $\prod_p 1/(1-1/p) = \zeta(1) = \infty$, and this calculation can be justified. Thus, letting $x\rightarrow \infty$, we get $\sum_p \ln(p/(p-1)) = \ln(\prod_p 1/(1-1/p)) = \ln(\infty) = \infty$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2815851",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 2
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Prove that if ${x_1, x_2, x_3}$ are roots of ${x^3 + px + q = 0}$ then ${x_1^3+x_2^3 + x_3^3 = 3x_1x_2x_3}$ How to prove that ${x_1^3+x_2^3 + x_3^3 = 3x_1x_2x_3}$ holds in case ${x_1, x_2, x_3}$ are roots of the polynomial?
I've tried the following approach:
If $x_1$, $x_2$ and $x_3$ are roots then
$$(x-x_1)(x-x_2)(x-x... | If $ x_1,x_2,x_3 $ are roots of $ x^3+p x+q=0 $ then $ x_1+x_2+x_3 = 0 $
If $ x_1+x_2+x_3 = 0 $ then $ x_3 = -(x_1+x_2) $ and
$ x_1^3+x_2^3+x_3^3 = x_1^3+x_2^3+(-1)^3(x_1+x_2)^3 = -3(x_1^2x_2+x_1x_2^2) = -3x_1x_2(x_1+x_2) = -3x_1x_2(-x_3) = 3x_1x_2x_3 $
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2815985",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 5,
"answer_id": 3
} |
Does there exist 2 matricies, such that they can be used to transpose any n by n matrix? Ideally $\exists A, B$ to be able to transpose matrix $X \; \forall X \in M_{n\times n} $ by matrix multiplication. (Even more ideal is if there is only one matrix, $A$ that can transposes $X$ as follows: $X^T = AX$ but I'm ignorin... | If we stack row by row the $n\times n$ matrices into $n^2\times 1$ vectors, then the OP's question is:
Is the function $f:X\in M_n\rightarrow X^T\in M_n$ (we can present $f$ in the form of a $n^2\times n^2$ matrix) decomposable into a tensor product $A\otimes B^T$ ?
The answer is no and it's not a scoop ! Yet, $f$ can... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2816073",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 5,
"answer_id": 3
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What is $\lim_{n\to\infty} \left(\sum_{k=1}^n \frac1k\right) / \left(\sum_{k=0}^n \frac1{2k+1}\right)$? I have the following problem:
Evaluate
$$ \lim_{n\to\infty}{{1+\frac12+\frac13 +\frac14+\ldots+\frac1n}\over{1+\frac13 +\frac15+\frac17+\ldots+\frac1{2n+1}}} $$
I tried making it into two sums, and tried to make it... | Hint Denote the $n$th harmonic number by $$H_n := 1 + \frac{1}{2} + \frac{1}{3} + \cdots + \frac{1}{n}.$$
Then, the numerator of the given ratio is $H_n$, and the denominator can be written as
\begin{align*}
1 + \tfrac{1}{3} + \tfrac{1}{5} + \cdots + \tfrac{1}{2 n + 1}
&= \left(1 + \tfrac{1}{2} + \tfrac{1}{3} + \cdots ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2816227",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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"answer_id": 1
} |
Meaning behind Filter in Set Theory In a course in logic and set theory, we studied the concept of a Filter. We defined a filter $F \in P(S)$ on $S$ an equivalent of the following definition from Jech's Introduction to Set Theory:
(a) $S \in F$ and $\emptyset \notin F.$
(b) If $X\in F$ and $Y \in F$ then $X \cap Y \in ... | It's similar to the concept of "almost everywhere". Suppose to every subset $T\subseteq S,$ you write
$$
\mu(T) \begin{cases} =1 & \text{if } T\in F, \\ = 0 & \text{if } S\smallsetminus T\in F, \\ \text{is undefined} & \text{otherwise.} \end{cases}
$$
Then, according to the definition of "filter", you have
\begin{align... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2816362",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
} |
Proving $\sum\limits_{k=0}^{\infty}\binom {m-r+s}k\binom {n+r-s}{n-k}\binom {r+k}{m+n}=\binom rm\binom sn$
Question: How do you show the following equality holds using binomials$$\sum\limits_{k=0}^{\infty}\binom {m-r+s}k\binom {r+k}{m+n}\binom {n+r-s}{n-k}=\binom rm\binom sn$$
I would like to prove the identity using... | With OP asking for formal power series in the evaluation of
$$\sum_{k\ge 0} {m-r+s\choose k} {r+k\choose m+n}
{n+r-s\choose n-k}$$
we write
$$[z^n] (1+z)^{n+r-s} [w^{m+n}] (1+w)^r
\sum_{k\ge 0} {m-r+s\choose k}
z^k (1+w)^k
\\ = [z^n] (1+z)^{n+r-s} [w^{m+n}] (1+w)^r
(1+z+zw)^{m-r+s}
\\ = [z^n] (1+z)^{n+r-s} [w^{m+n}]... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2816488",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
"answer_count": 3,
"answer_id": 1
} |
Lipschitz constant of limit of functions part 2 This question follows from my other question Lipschitz constant of limit of functions.
Consider two metric spaces $(X,d_X)$ and $(Y,d_Y)$ and define the lipschitz constant of every continuous function $f:X\rightarrow Y$ as
$$Lip(f):=\sup\limits_{x\neq y}\frac{d_Y(f(x),f... | let $\epsilon >0$. Then $Lip(f_n) <1+\epsilon$ for $n$ sufficiently large. Hence $d_Y(f_n(x),f_n(y)) \leq (1+\epsilon ) d_X(x,y)$ for all $x,y$ for $n$ sufficiently large.. Letting $n \to \infty$ we get $d_Y(f(x),f(y)) \leq (1+\epsilon ) d_X(x,y)$ for all $x,y$. Letting $\epsilon \to 0$ we conclude that $Lip(f) \leq 1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2816629",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Explanation for Concrete Mathematics 3.38's solution I'm working on the exercises in Concrete Mathematics recently. In Exercise 3.38, one of the key points is to prove that:
For any real numbers $x,\ y \in (0,\ 1)$，$\exists n \in \mathbf{N}^+$ such that $\{nx\} + \{ny\} \geqslant 1$, where $\{x\}$ represents the fract... | Hint.
Consider the numbers represented in base $2$
$$
n = \sum_{k=0}^m a_k 2^k \\
x = \sum_{k=1}^p b_k 2^{-k}\\
y = \sum_{k=1}^q c_k 2^{-k}\\
$$
with $a_k,b_k,c_k \in\{0,1\}$ and then compare
$$
\{nx\} + \{ny\} =\frac{a_0 b_1}{2}+\frac{a_0c_1}{2}+\frac{a_1b_2}{2}+\frac{a_1 c_2}{2}+\frac{a_2 b_3}{2}+\frac{a_2c_3}{2}+\f... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2816879",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
The number of ordered pairs $(a,b)$ that are solutions for the equation $\log_{2^a}\left(\log_{2^b}\left(2^{1000}\right)\right)=1$ How many $(a,b)$ for $a,b \in \Bbb{N}$ pairs can satisfy the following equation:
$$\log_{2^a}\left(\log_{2^b}\left(2^{1000}\right)\right)=1$$
The answer is $3$, but I can't figure out how t... | I agree with your derivation
$$\log_{2^a}\left(\log_{2^b}\left(2^{1000}\right)\right)=1\iff \log_{2^b}\left(2^{1000}\right)=2^a\iff (2^b)^{2^a}=2^{1000}\iff b\cdot 2^a=1000$$
now we can have
*
*$a=1, 2^a=2, b=500$
*$a=2, 2^a=4, b=250$
*$a=3, 2^a=8, b=125$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2817135",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Simplifying nested radicals with higher-order radicals I've seen that $$\sin1^{\circ}=\frac{1}{2i}\sqrt[3]{\frac{1}{4}\sqrt{8+\sqrt{3}+\sqrt{15}+\sqrt{10-2\sqrt{5}}}+\frac{i}{4}\sqrt{8-\sqrt{3}-\sqrt{15}-\sqrt{10-2\sqrt{5}}}}-\frac{1}{2i}\sqrt[3]{\frac{1}{4}\sqrt{8+\sqrt{3}+\sqrt{15}+\sqrt{10-2\sqrt{5}}}-\frac{i}{4}\sq... | I have figured out how to find my answer using De Moivre's formula, not that the method in particular is of great importance but it is slightly alternative to InterstellarProbe's use of the definition of sine (which helped me figure this out).
De Moivre's formula is $$(\cos\theta+i\sin\theta)^n=\cos n\theta+i\sin n\the... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2817225",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
If $x$ is the only element that $x^2=e$ then $x\in Z(G)$ $G$ be a group.
$$Z(G)=\{u\in G\mid ua=au \quad \forall a\in G \}$$
If $x$ is the only element in $G$ that satisfies $x^2=e$ then $x\in Z(G)$
Attempt:
*
*$x^2=e$ then $(\forall g\in G),\; gb^2=g=b^2g$ then $gb=b^2gb^{-1}=gb=gb^{-1}\ldots$
it is not good.
*I ... | Hint: $o(gxg^{-1})=o(x)$.
$ {} {} {} $
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2817331",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Congruence system with same modulus and same variable? I have this particular problem:
$$\begin{cases}
3k \equiv 2 \pmod 8 \dots(*) \\
7k \equiv 2 \pmod 8 \dots(**)
\end{cases}
$$
I know that the solution for this is $k = 8q + 6$. I can find this easily if I solve one of the equations alone.
Now, let's assume I subtrac... | You didn't mess up anywhere, except for your interpretation of what you did. The result that $k=2q'$ (I'm using a different letter to avoid confusion with the $q$ from $k=8q+6$) is correct — the solutions $k=8q+6$ indeed satisfy this property that you found:
$$k=8q+6=2q', \quad \text{where} \quad q'=4q+3.$$
When you ha... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2817438",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 0
} |
Laplace equation in a rectangle, Dirichlet to Neumann map Consider the problem
$$\Delta u = 0,(x,y) \in \Omega=(0,1)^2 \\
u(0,y)=u(1,y)=1 \\
u(x,0)=u(x,1)=0.$$
This problem can be solved exactly. The solution is
$$u(x,y)=4\sum_{\text{odd } n} \frac{1}{n\pi} \sin(n\pi x) \left ( 1 - \frac{\sinh(n\pi y)+\sinh(n\pi(1-y))}... | The derivative series is just fine for $0 < y < 1$:
$$
u_{y}(x,y) = 4\sum_{\mbox{odd $n$}}\sin(n\pi x)\frac{\cosh(n\pi(1-y))-\cosh(n\pi y)}{\sinh(n\pi)}
$$
For $\delta < y < 1-\delta$ the fraction in the sum is bounded by $C e^{-n\pi\delta}$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2817530",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Given matrices $A$ and $B$, solve $XA = B$
Let $$A = \begin{bmatrix} 3&-7\\ 1&-2\end{bmatrix} \qquad \qquad B = \begin{bmatrix} 0&3\\ 1&-5\end{bmatrix}$$ and $X$ be an unknown $2x2$ matrix.
a. Find $A^{-1}$
b. If $XA = B$, use (a) to find $X$.
I found
$$A^{-1} = \begin{bmatrix} -2&7\\ -1&3\end{bmatrix}$$
I am stuck ... | The problem with your answer lies in the fact that what we actually have is$$XA=B\iff X=BA^{-1}.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2817912",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Calculate the maximum of $f(x,y)=\left|\frac{\sin(xy)}{x\sqrt{y}}\right|$ Calculate the maximum of
$$f(x,y)=\left|\frac{\sin(xy)}{x\sqrt{y}}\right|\, , \quad \text{for} \ x\in\mathbb R\, , \, y\in\mathbb R^{+}\, .$$
I suspect that this function is unbounded. In fact:
$$f(x,y)=\left|\frac{\sin(xy)}{x\sqrt{y}}\right|= \l... | It is not true that the limi is $\infty$ for every $x$. But if you take, say, $x=1$, then what you did is correct.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2818027",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Prove that there exist only one analytic function that holds conditions
Let $f$ be a differentiable function in all $\Bbb{C}$ and numbers
$$\frac{1+i}{1},\frac{2+i}{2},\frac{3+i}{3},\frac{4+i}{4},\frac{5+i}{5},
\cdots$$ map respectively to numbers
$$\frac{1-i}{1},\frac{1-2i}{2},\frac{1-3i}{3},\frac{1-4i}{4},\fra... | Let $g$ be a function that satisfies the same conditions. Then$$(\forall n\in\mathbb{N}):f\left(1+\frac in\right)=g\left(1+\frac in\right).$$Therefore, by the identity theorem and because the limit $\lim_{n\to\infty}1+\frac in$ exists, $g=f$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2818161",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Probability that a sample is generated from a distribution
Let $f_X(x)$ and $g_{Y}(x)$ be probability mass functions of discrete random variables X and Y. Mike selects a random variable (he chooses $X$ with probability $1/2$ or $Y$ with probability $1/2$), then he generates a sample of it and gives it to us. Let $a$ b... | Consider a small interval around $a$, i.e., the observed value to be in the interval $[a -\varepsilon, a+\varepsilon]$ and then take the limit $\varepsilon \to 0$ when evaluating the ratio.
Then the ratio becomes
\begin{equation}
\mathrm{lim}_{\varepsilon \to 0}
\frac{\int^{a+\varepsilon}_{a-\varepsilon}f_X (x) dx}{\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2818318",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
$f \in \mathrm{End} (\mathbb{C^2})$ $f(e_1)=e_1+e_2$ $f(e_2)=e_2-e_1$. Eigenvalues of f and the bases of the associated eigenspaces Let $f \in \mathrm{End} (\mathbb{C^2})$ be defined by its image on the standard basis $(e_1,e_2)$:
$f(e_1)=e_1+e_2$
$f(e_2)=e_2-e_1$
I want to determine all eigenvalues of f and the bases... | If one represents the standard basis $e_1$, $e_2$ in the usual form
$e_1 = \begin{pmatrix} 1 \\ 0 \end{pmatrix}, \tag 1$
$e_2 = \begin{pmatrix} 0 \\ 1 \end{pmatrix}, \tag 2$
and writes the matrix of $f$ as
$[f] = \begin{bmatrix} \alpha & \beta \\ \gamma & \delta \end{bmatrix}, \tag 3$
then we have, since
$f(e_1) = e_1 ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2818427",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Is the series $\sum_{n=1}^{\infty} \Bigl(1-\Bigl(1-\frac{1}{n^{1+\epsilon}}\Bigr)^n\Bigr)$ convergent? While i was solving a problem in probability theory I came across the following series
$$\sum_{n=1}^{\infty} \biggl(1-\biggl(1-\frac{1}{n^{1+\epsilon}}\biggr)^n\biggr)$$
and in order to complete my solution I want to... | You can get
explicit constants
rather than big or little oh
like this:
If $0 < x < 1$,
$-\ln(1-x)
=\sum_{k=1}^{\infty} \dfrac{x^k}{k}
\gt x$
and,
$\begin{array}\\
-\ln(1-x)
&=\sum_{k=1}^{\infty} \dfrac{x^k}{k}\\
&=x+\sum_{k=2}^{\infty} \dfrac{x^k}{k}\\
&\lt x+\sum_{k=2}^{\infty} \dfrac{x^k}{2}\\
&\lt x+\dfrac{x^2}{2(1-... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2818623",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
Auto-correlation function, an inverse problem $x[n]$ is a complex function $n=0,1,2,\cdots,L-1 $
we assume $x[n]$ is periodic in its index: $x[n+L]=x[n]$
Its auto-correlation function $C[n]$ is uniquely defined as:
$$
C[n]=\sum_{i=0}^{L-1} x[i+n]x^*[i]
$$
$C[n]$ also has the periodic property: $$C[n+L]=C[n]\tag{1}$$
A... | Even up to shifts it is not unique at all. For example there is a whole collection of sequences called $m-$sequences (maximal length sequences) generated by binary linear shift registers corresponding to primitive polynomials. See the discussion on wikipedia.
There are $\phi(2^n-1)/n$ different primitive polynomials ov... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2818733",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 0
} |
Is it OK to be picky about math you find interesting? I am a layman interested in mathematics, and I would like to hear mathematicians' views on the following: Is it normal to be picky about mathematical stuff you find interesting?
I ask because 80% of math I encounter does not seem interesting to me. I want to know if... | I would say it would be unwise and rather odd for a mathematician to dismiss any part of the subject matter of mathematics as uninteresting. The subject is very large and there are deep connections between apparently very different aspects: as a simple example, consider the many different proofs of the fundamental theo... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2818839",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 6,
"answer_id": 5
} |
What is the minimum time from point A to point B? am working a bit on the theory of optimal control, and I have had a couple of doubts about how I should choose the control variable to minimize travel time.
Consider the control problem to reduce the travel time of a trolleybus, initially park at A, to a fixed pre-assig... | Instead the proposed dynamical system
$$
\begin{array}{rcl}
\dot{x}_{1} & = & x_{2}\\
\dot{x_{2}} & = & u_{1}+u_{2}
\end{array}
$$
with $0\le u_{1}\le\beta$ and $-\alpha\le u_{2}\le0$ we will consider
a simpler system with the same functionalities
$$
\begin{array}{rcl}
\dot{x}_{1} & = & x_{2}\\
\dot{x_{2}} & = & u
\end... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2818940",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Find the Pre-Image of $\ q\ $, where $\ q^{-1}(S)\ $, where $\ S=f^{-1}([0,+\infty))$
Let $\ f:\mathbb{R^2}\rightarrow\mathbb{R}\ $ be given by $\ f(x,y)=y(x-y)\ $ and let $\ q:\mathbb{R^2}\rightarrow\mathbb{R^2}\ $ be given by $\ q(r,\theta)=(r \ \text{cos}(\theta),r \ \text{sin}(\theta))$. Find the pre-image $\ q^{-... | By definition:
$$q^{-1}(S)=\{(r,\theta)\,\colon q(r,\theta)\in S\}$$
Then,
$$q(r,\theta)\in S\iff q(r,\theta)\in f^{-1}([0,+\infty))\iff f(q(r,\theta))\geq 0.$$
This yields
$$ r\sin(\theta)(r\cos(\theta)-r\sin(\theta))\geq 0$$
Calculating, I obtained that this happens if and only if
$$r^2(\sin(\theta)\cos(\theta)-\si... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2819085",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Basic Probability with random number The question I have on hand is as follows :
We draw at random a number in interval [0,1] such that each number is "equally likely". Suppose we do the experiment two times (independently), giving us two numbers in [0,1]. What is the probability that the sum of these numbers is greate... | Choosing two numbers randomly (uniformly, independently) in the unit interval $[0,1]$ is the same as choosing a single point uniformly in the unit square $[0, 1]\times [0,1]$, and looking at its first and second coordinate.
Now, take a look at that square (you can even draw it, if you want). See if you can tell which p... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2819211",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Why is the notion of analytic function so important? I think I have some understanding of what an analytic function is — it is a function that can be approximated by a Taylor power series. But why is the notion of "analytic function" so important?
I guess being analytic entails some more interesting knowledge rather t... | Being analytic, and especially being complex-analytic, is a really useful property to have, because
*
*It's very restrictive. Complex-analytic functions integrate to zero around closed contours, are constant if bounded and analytic throughout $\mathbb{C}$ (or if their absolute value has a local maximum inside a doma... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2819345",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "51",
"answer_count": 11,
"answer_id": 7
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Ring Around the Robot - Chance of ending on specific node $N$ nodes $(Node_1 .. Node_N)$ are arranged in a circle, and a robot is placed at $Node_1$. The robot moves clockwise with probability $p$ and counter-clockwise with prob. $(1-p)$. Given integers $S, B \in \mathbb{N}$, where $1<=B<=N$, what's the probability of ... | Let $X$ be the clockwise moves. The net clockwise displacement after $S$ steps is $X-(S-X)=2X-S$. Let $i$ be the position (node index) indexed $0,1\cdots, N-1$. Assume you start at $i=0$, you end in node $j$ if $2X-S = j \pmod N$
or
$$ 2X = j +S \pmod N $$
with $0\le X\le S$
Let $X_i$ be the solutions of this modular e... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2819659",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
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Is there a necessary condition for the projection of two matrices to be the same? Take $\textbf{A},\textbf{B} \in \mathbb{R}^{d \times d}$ with $\textbf{A} \neq \textbf{B}$ and $d > 1$. Let $\textbf{P}_M$ be some $d\times d$ projection matrix. Is there a necessary condition for $\textbf{P}_M \boldsymbol{A} = \textbf{P}... | Let $M_i$ denote the $i$-th column of a matrix $M$.
A necessary (and sufficient) condition is that for all $i$, $$A_i-B_i\in\operatorname{Ker}P_M$$
A projection is entirely determined by its image and kernel. If you know them, then this is a handy criterion.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2819747",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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Orthogonally Diagonalize a matrix with nonreal eigenvectors I am given the matrix A and asked to orthogonally diagonalize it.
$$A=\begin{pmatrix} 1 & -i \\ i & 1 \\ \end{pmatrix} $$
While doing this I got $\lambda = 0,2.$
Then I found the eigenvectors corresponding to the eigenvalues to be $$\begin{pmatrix} i \\ 1 \\ ... | It happens that $\|(a,b)\|=\sqrt{|a|^2+|b|^2}$. Therefore, the norm of both vectors that you mentioned is $\sqrt2$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2819873",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
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Lagrange multiplier when decisions variables are not in the same set
Find the maximum of $2x+y$ over the constraint set $$S = \left\{ (x,y) \in \mathbb R^2 : 2x^2 + y^2 \leq 1, \; x \leq 0 \right\}$$
I want to use Lagrange multipliers to find the optimal solution. However, Lagrange requires $\vec x \in A$. In our cas... | Formulating the problem a bit better, you have the following :
Find the maximum of $f(x,y) = 2x + y$ over $S = \{(x,y) \in \mathbb R^2 : 2x^2 + y^2 \leq 1, \; x \leq 0\}$.
Recall that one of the most important Lagrange Multiplier methods is the Kuhn-Tucker Lagrange method. The KTL method calculates the total minimum... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2819961",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
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Solve $x^2+(8y)^2=p^2(4p^2y^2+1)$ I am trying to find solutions for $x^2+(8y)^2=p^2(4p^2y^2+1)$ for integer $x,y$ where $p$ is a prime $\equiv 1 \mod 4$ that does not divide $x,y$.
I think there are no solutions but I could not prove this. Obviously $x$ is odd, and $4p^2y^2+1$ is a product of primes $\equiv 1 \mod 4$, ... | Try $p=5$, $x=691$, $y=14$. https://www.alpertron.com.ar/QUAD.HTM is a good resource.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2820047",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
PhD admission product $\lim_{n\to 0}\left(\frac21\left(\frac32\right)^2\left(\frac43\right)^3\cdots\left(\frac{n+1}{n}\right)^n\right)^{1/n}$ Hello there I saw this problem (#3) here:
http://www.sau.int/admission/2018/samplepapers/PAM.pdf
$$L=\lim_{n\to 0} \left( \frac{2}{1}\left(\frac{3}{2}\right)^2\left(\frac{4}{3}\... | $$\dfrac{n+1}{(n!)^{\frac{1}{n}}} \approx \dfrac{n+1}{\sqrt{2 \pi n}^{1/n}\dfrac n e} \to e$$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Do gradients of level curves at tangent point point at same direction? I watched the Lagrange multipliers video here and it was mentioned in minute 2:50 that the gradients of both level curves at tangent point point at the same direction
Is this guaranted that they will point always at the same direction? If it is, ca... | Good catch! The video narrator is certainly wrong when he said that "they're pointing in the same direction." Note that up to that point he was only saying that the two gradients would be proportional, which is absolutely correct. Equivalently, being proportional means that they are parallel. But they do NOT have to po... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Find the limit using delta epsilon definition. Evaluate $\lim_\limits{x \to 0}\ \dfrac{e^x-1}{e^{2x}-1}$ using $\delta - \varepsilon$ definition.
Attempt: I claim that $\lim_\limits{x \to 0}\ \dfrac{e^x-1}{e^{2x}-1} = \dfrac 12$. $\forall \varepsilon >0, \exists\delta>0$ such that
$$|\frac{e^x-1}{e^{2x}-1}-\frac12| =... | Will be easier to evaluate if you first factorise the denominator followed by the delta-epsilon evaluation
Hopefully you can proceed from there.
| {
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"url": "https://math.stackexchange.com/questions/2820415",
"timestamp": "2023-03-29T00:00:00",
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What can I say about $P\bigl[Y-X \le\frac{1}{2}\bigr]$ if $X$ and $Y$ are independent $U[0,1] $ variables? Let $X$ and $Y$ be two random independent variables with uniform distribution on $[0,1]$.
What can I say about $P\bigl[Y-X \le\frac{1}{2}\bigr]$?
I tried doing the following:
$$P\Bigl[ Y \le X + \frac{1}{2}\Bigr]... | Hint. Draw a picture of the unit square and shade the area that matters. Then you can find the answer without integrals (even without pencil and paper after you see the picture).
| {
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What's the recursive definition of the heavy binary strings? Given $Σ = \{0, 1\}$ and $w ∈ Σ^*$. The binary string $w$ is called heavy if
(the number of $1$ of $w$) - (the number of $0$ of $w$) = $1$. For example, the strings $011$, $100011110$ are heavy, while the strings $0101$, $1100$, $1100100$,
$1111111$ are not.
... | If $H$ is the set of these heavy strings, then define the basis step as $1\in H$.
Now if $x, y$ are two strings in $H$, then the concatenations $xy$ or $yx$ will have two more $1$'s than the $0$'s. So we need to add one more zero in the string $xy$ for it to be a member of $H$. So the recursive step can be given as: i... | {
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Struggling to find implicitly-defined function and its second derivative The question I am working on is as follows:
Let $y$ be implicitly defined by $$\sin(x-y) - e^{xy} + 1=0$$ and $y(0) = 0$. Find $y''(0)$.
Any help with finding the implicit function and possibly its second derivative is greatly appreciated becau... | If you derivate it once you get:
$$\cos(x-y)(1-y') - e^{xy}(xy'+y) =0$$
and for $x=0$ we get $1-y'(0) = y(0)=0$ so $y'(0)=1$ and if we derivate it second time we get:
$$-\sin(x-y)(1-y')^2 -\cos(x-y)y''-e^{xy}(xy'+y)^2 -e^{xy}(xy''+2y')=0 $$
so for $x=0$ we get:
$$ -y''(0)-y(0)^2-2y'(0)=0\implies y''(0)=-2$$
| {
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Determine the coefficient of $wx^3y^2z^2$ in $(2w -x + y -2z)^8$ They provide a similar example:
Similarly provided example
I tried to set mine up the same way, so I had
My Answer so far:
Can someone let me know if I'm even close? In the example, I have no idea no idea how the would have determined that the "6" shoul... | Hint:
We can re-write as $$(2w-x+y-2z)^8=\sum_{r=0}^8\binom8r(2w-x)^{8-r}(y-2z)^r$$
As the sum of coefficients of $w,x$ is $1+3=4,$ we need $8-r=4\iff r=?$
Now the general term of $(2w-x)^4$ is $$\binom4k(2w)^{4-k}(-x)^k$$
We need $k=3$
| {
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"timestamp": "2023-03-29T00:00:00",
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Clarifying the definition of continuity at a point in Johnsonbaugh/Pfaffenberger Foundations of Mathematical Analysis In my copy of Foundations of Mathematical Analysis, in the section on continuity, I'm not understanding definition 33.1. The definition is as follows:
Let $f$ be a function from a subset X of R into R. ... | In fact, you need $a\in X$. Otherwise, there is no meaning in the continuity of $f$ outside of $X$. Consider $X=\{0\}\cup [1,2]$. Then $0\in X$ is no accumulation point of $X$. Hence, by definition $f$ is continuous at $0$.
| {
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"timestamp": "2023-03-29T00:00:00",
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What is the relationship between singular values and eigenvalues of a matrix? Suppose I have a general $n\times n$ real matrix $A$. And suppose that $A$ has an SVD of the form $A=U^T S V$ with S of the form $I_m \oplus D$ where $I_m$ is the identity $m\times m$ matrix and $D$ is a matrix of size $n-m \times n-m$.
This... | In general the eigenvalues have no direct relation to the singular values. The only thing you can really be sure of is that the eigenvalues, in magnitude, lie in the interval $[\sigma_n,\sigma_1]$. Also each singular value of zero is in fact an eigenvalue (with the corresponding right singular vector as an eigenvector)... | {
"language": "en",
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How to prove a cone is convex and closed?
Let $A$ be a $m\times n$ matrix and consider the cones $G_0=\{d\in\mathbb R^n:Ad<0\}$ and $G'=\{d\in\mathbb R^n:Ad\le0\}$
Prove that $G'$ is a convex closed cone.
Lets see that $G'=\overline{ G'}.$ Note that this contention $G'\subset\overline{ G'}$ is always true. Let's see ... | And to show that $G'$ is convex just take two points $x,y \in G'$, the segment between these points must lie in $G'$. Let $0\leq \alpha\leq 1$, study the point $p = \alpha x + (1-\alpha)y$:
$$ Ap = A(\alpha x + (1-\alpha)y) = \alpha Ax + (1-\alpha)Ay \leq 0 $$
the last inequality is true because $\alpha, 1-\alpha \geq ... | {
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How can $\int_0^x\lfloor t \rfloor^2dt$ be written as $\sum_{j=1}^{\lfloor x - 1 \rfloor} j^2 + q^2r$ Question 6(c) from Section 1.15 Exercises of Apostol's Calculus is the following:
Find all $x > 0$ for which $\int_0^x\lfloor t \rfloor^2dt = 2(x-1).$
A particular piece of reference material solves the problem in th... | Note that $$\int_{0}^{x}\lfloor t \rfloor^2 dt = \int_{0}^{\lfloor x \rfloor}\lfloor t \rfloor^2 dt + \int_{\lfloor x \rfloor}^{x}\lfloor t \rfloor^2 dt.$$
The first sum is given by the summation, and the second term is
$$\int_{\lfloor x \rfloor}^{x}\lfloor t \rfloor^2 dt = \int_{\lfloor x \rfloor}^{x}\lfloor x \rfloo... | {
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BlackJack Card Probability when Counting Cards In a single deck blackjack game - if you're not counting cards - the probability that the next card will be a 10/J/Q/K is 16/52.
I'm trying to figure out how to adjust the probabilities when you are counting cards. For those that might not be familiar, a common card count... | As in my answer to your later question
$\qquad$BlackJack Card Counting Probabilities
defining $f(a,b,c)$ as the number of $52$-card subsets of the $104$-card deck consisting of
*
*$a$ low cards$\;(2,3,4,5,6)$.$\\[4pt]$
*$b$ neutral cards$\;(7,8,8)$.$\\[4pt]$
*$c$ high cards$(10,\text{J},\text{Q},\text{K},\text{... | {
"language": "en",
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Is $\int_{-2}^3\frac{1}{x^3}dx=\frac5{72}$ or not defined? If we divide it into two parts such that $$I=\int_{-2}^0\frac{1}{x^3}dx+\int_0^3\frac{1}{x^3}dx$$
And then use substitution $x=-t$ we get $$I=\int_2^3\frac1{x^3}dx=\frac5{72}$$
However, If we use limits on both part separately, they both diverge, so the integra... | You are right that the integral is not defined (because of the singularity), so you can't trust what the anti-derivative tells you. It's even worse with $$\int_{-1}^{1}\frac{1}{x^2}dx=-2\tag{1}$$
The area is clearly positive ($+\infty$). If you were to break up $(1)$ you would get $$\int_{-1}^{1}\frac{1}{x^2}dx=2\int_{... | {
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Let $M$ be a orientable 2-closed surface,prove $H^1(M)$ is direct sum of an even number of $\Bbb Z$ Let $M$ be a orientable 2-closed surface,prove $H^1(M)$ is direct sum of an even number of $\Bbb Z$
Could anyone give some hints?
| Hint: Think about how you would triangulate such a surface. Think about how you can triangulate a $1$-holed torus, $2$-holed torus, $\dots$ , $n$-holed torus. This of course relies on what Georges mentioned in the comments; the fact that $M$ is just a sphere with $n$ handles.
Then think about what this triangulation i... | {
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How does squaring both sides of an equation lead to extraneous solutions? Let’s say I have $x = x + 1$, which is a false statment for real $x$; why can I solve for real $x$ when I square both sides of the equation, giving $x^2=(x+1)^2$?
| An equality $e_1=e_2$, where $e_1$ and $e_2$ are expressions whose value is a number, means that the two expressions denote the same number.
You would of course expect that you if you apply the same function to two different representations of the same underlying number that you would therefore get the same result in b... | {
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Find logarithm fit to two points Say I have an equation $f(x) = \log_b(ax+1)$, where $a$ and $b$ are constants. If I have two distinct points $(x_1,y_1)$ and $(x_2, y_2)$, where $x_2 > x_1$ and $y_2 > y_1$, how can I find values for $a$ and $b$ such that $f(x_1) = y_1$, and $f(x_2) = y_2$?
| Let $y_1=\log_{b}ax_1+1$ and $y_2=\log_{b}ax_2+1$ therefore$$b^{y_1}=ax_1+1\\b^{y_2}=ax_2+1$$then we have $$x_2b^{y_1}-x_1b^{y_2}=x_2-x_1$$ which doesn't have any analytic answer so doesn't for $a$
| {
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Why does the identity $\mathbb{E}(X) = \mathbb{E}\left(\int \mathbb{1}_{u \leq X}du\right)$ hold? I'm reading on Hoeffding's covariance identity, the proof of which is neatly covered here, or, in a similar manner, in this MSE post, but I can't seem to fully understand the trick/property used there.
I.e., assume $(X_1,... | What underlies the equality $\mathbb E(X) = \mathbb E(\int \mathbb 1_{u\le X}\,du)$ is, intuitively, the way one thinks of the Lebesgue integral as coming from partitioning the $y$-axis, whereas the Riemann integral comes from partitioning the $x$-axis.
Think of a reasonable function $f(x)$ (say continuous, but that's ... | {
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Convergence of $a_n=(1-\frac12)^{(\frac12-\frac13)^{...^{(\frac{1}{n}-\frac{1}{n+1})}}}$ let us moving to telescopic sum using exponent ,Assume we have this sequence: $a_n=(1-\frac12)^{(\frac12-\frac13)^{...^{(\frac{1}{n}-\frac{1}{n+1})}}}$ with $n\geq1$ , this sequence can be written as power of sequences : ${x_n} ^ ... | $\forall n\in N^{*}:u_{n}=(1-\frac{1}{2})^{(\frac{1}{2}-\frac{1}{3})^{(\frac{1}{3}-\frac{1}{4})^{...(\frac{1}{n}-\frac{1}{n+1})}}}\gt 0$
$u_{1}=v_{1}=1-\frac{1}{2}=\frac{1}{2}$
$u_{2}=v_{1}^{v_{2}},u_{3}=v_{1}^{v_{2}^{v_{3}}},...,u_{n}=v_{1}^{v_{2}^{v_{3}^{....v_{n}}}},v_{n}=\frac{1}{n}-\frac{1}{n+1}=\frac{1}{n(n+1)}$
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2822112",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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A urn contains blue balls and red balls. I need to find probabiltiy of drawing more blue balls than red balls
A urn contains 5 identical blue balls and 4 identical red balls. Taking 5 balls at random from the urn what is the probability that the number of blue balls be greater than the number of red balls?
My first ... | Three approaches:
(1) This can be viewed as a hypergeometric distribution. The urn contiains
four red balls and five blue balls. Let $X$ be the number of red balls
among five balls drawn at random without replacement. To draw more blue balls
then red you need to evaluate $P(X \le 2).$ In R statistical software
this ca... | {
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"timestamp": "2023-03-29T00:00:00",
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If $F_0\cap G_0=\emptyset$ then $x$ is a local minimum of function Consider the theorem:
Consider the following linear optimization problem $$\max 2x_1+3x_2$$ $$\text{s.t.} x_1+x_2\le8\\ -x_1+2x_2\le4\\ x_1,x_2\ge0$$
a) For each extreme point verify if necessary condition of theorem is satisfied.
b)Find the optimal s... | Notice that $d_1+d_2<0$ and $-d_1+d_2<0$ implies that $d_2<-|d_1|$, or $-d_2>|d_1|$. Hence $-2d_1-3d_2>-2d_1+3|d_1|\geqslant0$ for all $d_1\in\mathbb{R}$.
You can also draw a picture (good for these 2d geometric problems):
| {
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How to find the minimum of $f(x)=\frac{4x^2}{\sqrt{x^2-16}}$ without using the derivative?
Find the minimum of function $$f(x)=\frac{4x^2}{\sqrt{x^2-16}}$$ without using the derivative.
In math class we haven't learnt how to solve this kind of problems (optimization) yet. I already know that is solvable using derivat... | It is the same as finding the minimum of $\frac{4z}{\sqrt{z-16}}$ for $z>16$, or the minimum of $\frac{16 t}{\sqrt{t-1}}$ for $t>1$, or the minimum of $\frac{16(u+1)}{\sqrt{u}}$ for $u>0$, or the minimum of $16\left(v+\frac{1}{v}\right)$ for $v>0$. It is clearly $\color{red}{32}$ by the AM-GM inequality.
| {
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"source": "stackexchange",
"question_score": "2",
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Finding a negative power of $i$ How to find the value of $i$ when it has negative power? When solving for $i$ with positive power, I use something like
$$i^{101} = (i^{2})^{50}\times i = (-1)^{50} \times i = 1 \times i = i.$$
But how to solve for negative power of $i$ such as $i^{-10}$?
Can anyone explain what to do in... | By your analogy, $$i^{-10}=\frac{1}{i^{10}}=\frac{1}{i^8*i^2}=\frac{1}{(i^{2})^4* i^2}.$$ Since $i^2=-1$, it follows that $$i^{-10}=\frac{1}{(-1)^4*-1}=\frac{1}{-1}=-1.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2822529",
"timestamp": "2023-03-29T00:00:00",
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Is a Contraction also a Contraction under equivalent metrics?
Definition of a Contraction. Let $(X, d)$ be a metric space. Then a map $T : X → X$ is called a contraction on $X$ if there exists $q ∈ [0, 1)$ such that $d(T(x),T(y)) \le q d(x,y)$ for all $x, y$ in $X$.
My question: Does a contraction remain a contractio... | user357151 showed this isn't true for equivalent metrics in general.
However, if we restrict ourselves to metrics induced by equivalent norms, we get an interesting relation.
Consider $T:V \to V$, where $V$ is a normed vector space with two equivalent norms, say $||\cdot||_1$ and $||\cdot||_2$. Then, there exists some... | {
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Induced action is proper discontinuous Let $f:X\to Y$ be a surjective map, and let $G$ act on $X$ such that for each $g\in G$ and $x,x'\in X$ $f(x')=f(x)$ implies $f(g\cdot x)=f(g\cdot x')$. Further assume that the group action on $X$ is proper discontinuous.
In the above situation, we get an induced action on $Y$. Bu... | Consider $\mathbb{R}\times S^1$ where $S^1$ is the quotient of $\mathbb{R}$ by the action of $\mathbb{Z}$ defined by $n.x=x+n$ We denote by $p:\mathbb{R}\rightarrow S^1$ the quotient map. Consider $\mathbb{R}\times S^1$ endowed with the action of $\mathbb{Z}$ defined by $n.(x,y)=(x+n,p(y+nc))$ where $c$ is irrational n... | {
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Loewner order in terms of eigenvalues Suppose that $A \succeq B$, where $A$ and $B$ are real symmetric matrices, so that $A - B$ is positive semidefinite, equivalently, $A - B$ has nonnegative eigenvalues.
Is it always true that $\lambda_i(A) \geq \lambda_i(B)$ (assuming that eigenvalues are ordered)?
| Just so that this question not remains formally unanswered: as @julian pointed out in the comments by the min-max-theorem we have that for all $k \in \{ 1, \ldots, d \}$
$$
\lambda_k(A)
= \min_{\substack{U \subset \mathbb C^d, \\ \dim(U) = k}} \max_{x \in U \setminus \{ 0 \}} \frac{x^{\mathsf{T}} A x}{x^{\mathsf{T}} x}... | {
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$\frac{a^2} {1+a^2} + \frac{b^2} {1+b^2} + \frac{c^2} {1+c^2} = 2.$ Prove $\frac{a} {1+a^2} + \frac{b} {1+b^2} + \frac{c} {1+c^2} \leq \sqrt{2}.$ $a, b, c ∈ \mathbb{R}+.$
WLOG assume $a \leq b \leq c.$ I tried substitution: $x=\frac{1} {1+a^2}, y=\frac{1} {1+b^2}, z=\frac{1} {1+c^2},$ so $x \geq y \geq z$ and $(1-x)+(1... | Let $$B:=\frac{1} {1+a^2} + \frac{1} {1+b^2} + \frac{1} {1+c^2}$$
From: $$A:=\frac{a^2} {1+a^2} + \frac{b^2} {1+b^2} + \frac{c^2} {1+c^2} = 2$$
we get $A+B =3$ so $B =1$.
Now by Cauchy inequality we have $$A\cdot B \geq \big(\underbrace{\frac{a} {1+a^2} + \frac{b} {1+b^2} + \frac{c} {1+c^2}}_{C}\big)^2$$
So we have $C^... | {
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"url": "https://math.stackexchange.com/questions/2822937",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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Directional derivative, gradient and metric Considering the general expression of gradient with the directional derivative operator on $f$ function along $\vec{v}$ vector :
$$df(v)=\langle\text{grad}(f),v\rangle = g^{ij}\partial_{i} f v_{j}=\partial_{i} f v^{i}$$
taking $\partial_{i} = \dfrac{\partial}{\partial x^{i}}$... | If you are in a flat Euclidean space $\Bbb R^n$ and you are using the euclidean metric then the metric tensor is $g_{ij}=\delta_{ij}$ as well
$g^{ij}=\delta^{ij}$. So the law of raising indexes gives that $\partial^i=\partial_i$.
| {
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"answer_id": 2
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probability game , What is the probability of winning? In a certain game, you perform three tasks sequentially. First, you flip quarter, and if you get heads you win the game. If you get tails, then you move to the second task. The second task is rolling a single die. If you roll a six, you win the game. If you roll... | You win if and only if you satisfy one of the three mutually exclusive situations:
*
*(1) You win on the first task by flipping a head
*(2) You lose on the first task by flipping a tail followed by winning on the second task by rolling a six
*(3) You lose on the first task by flipping a tail followed by losing on ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2823200",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Probability path - Exercise 6.14 : on almost sure divergence
Let $\{X_n\}$ be independent with $P(X_n = n^2) = \frac{1}{n}$ and $P(X_n = -1) = 1 - \frac{1}{n}$.
Show that $\sum_{n=1}^{\infty}X_n = -\infty$ almost surely.
I found that $E[X_n] = n + \frac{1}{n} - 1 \to \infty$. So intuitively it has to tend to $\infty... | Fix $n$. Let $S_n=X_1+\ldots+X_n$ and let $E_i$ be the event that $X_{i+1},\ldots,X_n$ are all equal to $-1$. Then $Pr[E_i]=\prod_{j=i+1}^{n}\left(1-\dfrac{1}{j} \right)=\dfrac{i}{n}$. Now let $i =\lfloor\sqrt{2n} \rfloor$, and note that if $E_i$ is false, then $S_n > 2n-n=n$. The probability that $S_n>n$ is thus at le... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2823311",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
Invalid syllogism passes Gensler's star test. Why? According to Gensler (2017):
An instance of a letter is distributed in a wff if it occurs just after “all” or anywhere after “no” or “not.” (p. 0008)
He then defines the star test as follows:
Star premise letters that are distributed and conclusion letters that aren... | See H.Gensler, Introduction to Logic, 2nd ed.,2017, page 9 :
More precisely, a syllogism is a vertical sequence of one or more wffs in which
each letter occurs twice and the letters “form a chain” (each wff has at least one letter in common with the wff just below it, if there is one, and the first wff has at least ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2823424",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Solving a 2nd order ODE: $\frac{d^2}{dx^2}y(x)=\left(C+(1+x^2)^{-1}\right)y(x)$. I would like to solve the following ode:
$$\frac{d^2}{dx^2}y(x)=\left(C+(1+x^2)^{-1}\right)y(x),\quad x\in\mathbb{R};$$
with boundary condition $y(0)=1$. $C$ is just some constant.
I am very stuck with this. Does anyone have any suggestion... | I am afraid that a closed form solution could not exist and that, provided a second boundary condition, numerical method would be required.
Even if $C=0$ the solution is far away to be simple since given by
$$y=\, _2F_1\left(-\frac{\sqrt{5}+1}{4} ,\frac{\sqrt{5}-1}{4}
;\frac{1}{2};-x^2\right)+c_1\, x \,\, _2F_1\lef... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2823522",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
A representation of $S^3$ I am intending to learn low dimensional topology from Saveliev's Book "Lectures on the Topology of 3-manifolds" by myself.
At the very beginning, he gives a Heegaard splitting of $S^3$ stating that
"the sphere $S^3$ is represented as the result of revolving the 2-sphere $S^2=R^2\cup\{\infty\}$... | Here the technical mechanism of ‘’revolving’’ means that for each element of the 2-sphere $S^2$ (or any other surface $F$), there is a circle $S^1$ attached. This kind of space is called ‘’circle bundle’’ over the 2-sphere (or over $F$, respectively).
For the hypersphere $S^3$ it happens that can be fibered as
$S^1\h... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2823637",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 1,
"answer_id": 0
} |
How do I apply integrating factor to solve this differential equation? $$x \frac{dy}{dx} + y = -2x^6y^4$$
I tried to find the general solution by dividing both sides by $x$ or $x^6$ but no solution I could get.. Do I even solve it with integrating factor?
| Contrary to other answers, you CAN find an integrating factor and manipulate your ODE via substitutions.
We have the ODE :
$$xy' + y = -2x^6y^4$$
Dividing both sides by $-\frac{1}{3}xy^4$, we yield :
$$-\frac{3y'}{y^4} - \frac{3}{xy^3} = 6x^5$$
Let $v(x) = \frac{1}{y^3(x)}$ and then this gives $v'(x) = -\frac{3y'(x)}{y... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2823820",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 6,
"answer_id": 2
} |
$\mathbb{Z} \times \mathbb{Z}$ and application of isomorphism theorems Let $G = \mathbb{Z} \times \mathbb{Z}$ with group law given by addition. Let $H$ be the subgroup generated by the element $(2,3)$. Then $G/H$ is isomorphic to $\mathbb{Z}$.
Is $G/H$ also isomorphic to $\mathbb{Z}_2\times \mathbb{Z}_3$
using the hom... | You have that $(2,3)H$ and $(4,3)H$ are both send to the same element using the homomorphism. However it's not true that $(2,3)H = (4,3)H$, as this would mean that $(2,0) \in H$, which isn't the case.
Hence $G/H$ isn't isomorphic to $\mathbb{Z}_2 \times \mathbb{Z}_3$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2823928",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 5,
"answer_id": 2
} |
Distance from a convex set to a point Let $Y \in \mathbb{R}^n$ be a nonempty convex set such that $0 \notin Y$ and fix $y_1,\dots,y_n$ in $Y$, where $n \ge 2$. I know that there exist $i,j$ such that $\Vert y_i \Vert > \Vert y_j\Vert$. Define $ C = C(y_1,\dots,y_n)$, i.e. the set of convex combinations. Moreover, I kn... | If I am understanding correctly, this seems quite trivial if the aformentioned distance function, $d$, computes the distance between the origin and the cet $C$ by finding a point $x$ in $C$ with minimal $L_2$-norm, i.e., $$ d(\overset{\to}{0}, C) = \min_{x \in C} \left| \left| x\right| \right|_2.$$
If that's the case, ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2824104",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
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Let $X_1$ and $X_2$ be two random independent variables with Poisson distribution $\lambda = 1$ Let $X_1$ and $X_2$ be two random independent variables with Poisson distribution $\lambda = 1$. Denoting by $Y = min\{X_1,X_2\}$, I want to calculate $P[Y \geq 1]$.
This is what I did:
$$ P[Y \geq 1] = 1 - P[Y \leq 1] $$
I ... | The quickest solution is $P(Y\ge 1)=P(X_1,\,X_2\ge 1)=(1-e^{-1})^2$. Note that $P(Y\ge 1)=1-P(Y=0)$, because the distribution of $Y$ is discrete.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2824222",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Symmetric Bilinear Form: index of bilinear = the number of positive eigenvalues Problem
Let $b:\mathbb{R}^n \times \mathbb{R}^n \rightarrow \mathbb{R}$ a symmetric bilinear form and $A$ the (transformation) matrix of $b$. Further, let $\mu$ be the number of positive roots (counted by their multiplicity). Proof that $\t... | You should stress (and justify) that the eigen-values of a symmetrical matrix are always real scalars.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2824308",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Are Kan extensions extensions in the traditional sense? Suppose $A$ is a full subcategory of $B$. Given $F:A\to Z$, is it true that the left Kan extension agrees with $F$ on $A$, ie. for all $a\in A$, $\mathrm{Lan}_i F(a)\simeq F(a)$, where $i:A\to B$ is the inclusion?
| You might be interested in these notes I wrote a few months ago. When I say "It turns out that this definition, albeit correct, it too general" I mean precisely what is contained in Kevin's answer!
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2824432",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Prove $\operatorname{Cov}(\overline{X_n}, X_j - \overline{X_n}) = 0$ for independent normally distributed random variables My homework states the following problem:
Let $X_1, \dots, X_n$ be independent $N(\mu, \sigma^2)$ distributed random variables, $\overline{X_n}$ be the sample mean and $S_n^2$ the empirical varian... | You have \begin{align}\text{Cov}(\bar X_n,X_j-\bar X_n)&=\text{Cov}(X_j,\bar X_n)-\text{Var}(\bar X_n)\\&=\text{Cov}\left(X_j,\frac{1}{n}\sum_{i=1}^nX_i\right)-\text{Var}(\bar X_n)\\&=\frac{1}{n}\sum_{i=1}^n\text{Cov}(X_i,X_j)-\text{Var}(\bar X_n)\\&=\frac{1}{n}\left(\text{Var}(X_i)+\sum_{i\ne j}\text{Cov}(X_i,X_j)\rig... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2824547",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Prove that $\sqrt n$ is irrational unless $n = m^2$ for some natural number $m$ (from Spivak Calculus 3.ed., §2, Ex 17b). I've looked up the solution to this problem in the Spivak Caluclus Answers Book and found the following proof:
If $\sqrt n = a/b$, then $nb^2 = a^2$, so the factorization into primes of $nb^2$ and o... | When it is said that every prime appears an even number of times in the factorization of $n$, this is counting each prime with multiplicity.
In your example $18 = 3 \cdot 3 \cdot 2$, $3$ is listed twice. So at the end, the number of primes counted with multiplicity in the factorization of $18$ is odd.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2824640",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Spectrum of $C(K) \oplus C(K')$ Let $K$, $K'$ be compact Hausdorff spaces. How does the spectrum of the $C^*$-algebra given by the direct sum $C(K) \oplus C(K')$ look like?
| Note that the spectrum of a C$^*$-algebra $A$ is the topological space of all nonzero $*$-homomorphisms $A\to\mathbb C$.
For the direct sum ,the map $\alpha:C(K)\oplus C(K')\to C(K\sqcup K')$ given by $$\alpha(f,g)(k)=\begin{cases} f(k),&\ k\in K\\ \ \\ g(k),&\ k\in K'\end{cases}$$ is a $*$-isomorphism.
For the tensor... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2824801",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Roots of a complex equation are outside the unit disc using the triangle inequality Problem: Use the triangle inequality to show that the roots of the complex equation $$z^4+z+4=0$$ has roots all outside the unit disc $|z|\le1$
My Thought Process: Clearly I need to use the triangle inequality and this would be a proof ... | If $z^4+z+4=0$, then $z^4+z=-4$, so you'd need $\lvert z^4+z \rvert = 4 $. But you've shown that $\lvert z^4+z \rvert \leq 2$ for $\lvert z \rvert \leq 1$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2824963",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
3rd Factorial Moment of X ~ geo(p) I'm working through Pitman's Probability (1993) problem 23b, page 221 (for knawledge not school).
Specifically, I am having trouble calculating
$E[(X)_3] = G^{(3)}(1)$, with $G^{(k)}(z) = \sum_{i=k}^{\infty}(P(X=i)\cdot(i)_k\cdot z^{i-k}$), where $G(z)$ is the probability generating... | In general, if $P(z)$ is the generating function of a random variable $X$ with probability mass function $\{p_k\}$, then since $0\leqslant p_k\leqslant 1$ and $\sum_{k=1}^\infty p_k=1$, it follows from dominated convergence that
$$
\frac{\mathsf d}{\mathsf dz} P(z) = \frac{\mathsf d}{\mathsf dz} \sum_{k=1}^\infty p_kz^... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2825061",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Help with finding Limit What is the limit of $$\lim_{n \to \infty} \left(\frac{n!}{n^n}\right)^{\frac{3n^3+4}{4n^4-1}}$$
Does any one can help, I am not sure how to solve this.
| hint: Use the famous Sterling inequality:
$$ \sqrt{2\pi n}\left(\dfrac{n}{e}\right)^n \le n! \le \sqrt{2\pi n}\left(\dfrac{n}{e}\right)^n\cdot e^{\frac{1}{12n}}$$
and use the Squeeze lemma to find the limit. Can you manage to take it from here?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2825213",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 0
} |
Fundamental Integral theorem for functionals Given a functional $L: X\to \mathbb{R}$. It is possible for two $x,x_h \in X$, to write
$$L(x)-L(x_h)=\int_0^1 L'(x_h+s(x-x_h))(x-x_h) ds$$
where $L'(\cdot)(v)$ is the directional in $v$. Why does this have to be tested with $x-x_h$ ?
Greetings.
| Let
$\gamma(s) = x_h + (x - x_h)s, \; s \in [0, 1]; \tag 1$
we note that
$\gamma(0) = x_h, \; \gamma(1) = x_h + (x - x_h) = x; \tag 2$
then
$L(x) - L(x_h) = L(\gamma(1)) - L(\gamma(0)) = \displaystyle \int_0^1 \dfrac{dL(\gamma(s))}{ds} \; ds; \tag 3$
now by the chain rule,
$\dfrac{dL(\gamma(s))}{ds} = L'(\gamma(s)) \df... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2825329",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Chain rule for Hessian. How to compute $D^2 f^\alpha$ How does the chain rule generalize to the Hessian matrix. In particular, how can we compute $$D^2 f^\alpha,$$
where $f:\mathbb{R}^N \to \mathbb{R}$, $N>1$, and $\alpha >0$?
| There is nothing tricky about this--you can just use the ordinary single-variable chain rule to compute each partial derivative. If $\partial_i$ denotes the derivative with respect to the $i$th variable, then $$\partial_i(f^\alpha)=\alpha f^{\alpha-1}\partial_i(f)$$ (this is literally nothing but the fact that for a f... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2825422",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
How do you change the order of integration without sketching? Specifically, for a double integral $$\int_a^b \int_{g_1(x)}^{g_2(x)} f(x,y) \, dy \, dx$$ how would you change the order of integration without having to sketch it out? I came across this while researching which talks about the use of the Heaviside function... | I consider it similar
to reversing the order of summation
in a double sum.
I'm going to
try to think this through
logically.
In this case,
$\int_a^b \int_{g_1(x)}^{g_2(x)} f(x,y) \, dy \, dx$,
$g_1(x) \le y \le g_2(x)$.
Therefore,
assuming that
$g_1$ and $g_2$
are strictly monotonic increasing
and therefore have an inv... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2825515",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
How to explain irrational numbers to laymen? I am trying to describe how irrational numbers, which are all modeled as a series of fractions, can themselves not be fractions, and are instead part of a unique group of "decimal numbers" outside of fractions, called the irrational numbers. I am confused atm.
From Wikipedi... | Representations that use ... to denote an infinite sequence often trick the mind into thinking the infinite sequence will behave like a finite one. They don't.
The fraction series are a neat way to give you an idea of the value of an irrational. If cut the sequence somewhere and compute the result of that finite sum, y... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2825625",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
"answer_count": 8,
"answer_id": 5
} |
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