Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Integer partition with fixed number of summands but without order For a fixed $n$ and $M$, I am interested in the number of unordered non-negative integer solutions to
$$\sum_{i = 1}^n a_i = M$$
Or, in other words, I am interested in the number of solutions with distinct numbers. For $n = 2$ and $M = 5$, I would consid... | Let the number of partitions be $P_n(M)$. By looking at the smallest number in the partition, $a_n$, we get a recurrence for $P_n(M)$:
$$
P_n(M) = P_{n-1}(M) + P_{n-1}(M-n) + P_{n-1}(M-2n) + P_{n-1}(M-3n) + ...
$$
Where $P_n(0)=1$ and $P_n(M)=0$ for $M<0$. The first term in the sum above comes from letting $a_n=0$,... | {
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"url": "https://math.stackexchange.com/questions/54635",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "14",
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"answer_id": 0
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Bipartite graph cover problem
Let $G$ be bipartite with bipartition $A$, $B$. Suppose that $C$ and $C'$ are both covers of $G$. Prove that $C^{\wedge}$ = $(A \cap C \cap C') \cup (B \cap (C \cup C'))$ is also a cover of $G$.
Does anyone know which theorem is useful for proving this statement?
| This statement is fairly easy to prove without appealing to any special theorems.
It might be useful to rewrite the statement
Every edge in $G$ has an endvertex in $C''=(A\cap C\cap C')\cup (B\cap(C\cup C')$,
which you are trying to prove, as the equivalent statement
If an edge $e\in E(G)$ has no endvertex in $B\ca... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Notation/name for the number of times a number can be exactly divided by $2$ (or a prime $p$) I am using this simple snippet of code, variants of which I have seen in many places:
for(int k = 0 ; n % 2 == 0 ; k++)
n = n / 2;
This code repeatedly divides num by 2 until it is odd and on completion k contains the num... | You might call it the "highest power of $2$ dividing $n$," but I'm not aware of any snazzy standalone term for such a thing. However, I have seen it notated succinctly as $2^k\|n$, which means $2^k$ divides into $n$ but $2^{k+1}$ does not.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/54965",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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"answer_id": 2
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non time constructible functions
A function $T: \mathbb N \rightarrow \mathbb N$ is time
constructible if $T(n) \geq n$ and there is a $TM$ $M$ that computes
the function $x \mapsto \llcorner T(\vert x\vert) \lrcorner$ in time $T(n)$. ($\llcorner T(\vert x\vert) \lrcorner$ denotes the binary representation of... | You can use the time hierarchy theorem to create a counter-example.
You know by the THT that DTIME($n$)$\subsetneq$DTIME($n^2$), so pick a language which is in DTIME($n^2$)$\backslash$DTIME($n$). For this language you have a Turing machine $A$ which decideds if a string is in the language in time $O(n^2)$ and $\omega(n... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Chord dividing circle , function Two chords $PA$ and $PB$ divide circle into three parts. The angle $PAB$ is a root of $f(x)=0$.
Find $f(x)$.
Clearly , $PA$ and $PB$ divides circle into three parts means it divides it into $3$ parts of equal areas
How can i find $f(x)$ then ?
thanks
| You may assume your circle to be the unit circle in the $(x,y)$-plane and $P=(1,0)$. If the three parts have to have equal area then $A=\bigl(\cos(2\phi),\sin(2\phi)\bigr)$ and $B=\bigl(\cos(2\phi),-\sin(2\phi)\bigr)$ for some $\phi\in\ ]0,{\pi\over2}[\ $. Calculating the area over the segment $PA$ gives the condition
... | {
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"url": "https://math.stackexchange.com/questions/55147",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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(k+1)th, (k+1)st, k-th+1, or k+1? (Inspired by a question already at english.SE)
This is more of a terminological question than a purely mathematical one, but can possibly be justified mathematically or simply by just what common practice it. The question is:
When pronouncing ordinals that involve variables, how does ... | If you want a whole lot of non-expert opinions, you can read the comments here.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
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How to solve a symbolic non-linear vector equation? I'm trying to find a solution to this symbolic non-linear vector equation:
P = a*(V0*t+P0) + b*(V1*t+P1) + (1-a-b)*(V2*t+P2) for a, b and t
where P, V0, V1, V2, P0, P1, P2 are known 3d vectors.
The interesting bit is that this equation has a simple geometrical interp... | Let's rewrite:
$$a(P_0 - P_2 + t(V_0-V_2)) + b(P_1 - P_2 + t(V_1 - V_2)) = P - P_2 - t V_2$$
which is linear in $a$ and $b$. If we let $A=P_0-P_2$ and $A'=V_0-V_2$ and $B=P_1-P_2$ and $B'=V_1-V_2$ and $C=P-P_2$ and $C'=-V_2$ then you have
$$a(A + tA') + b(B + tB') = C + tC'$$
This can be written as a matrix equation:
... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Perfect square sequence In base 10, the sequence 49,4489,444889,... consists of all perfect squares.
Is this true for any other bases (greater than 10, of course)?
| $((2*19^5+1)/3^2)=2724919437289,\ \ $ which converted to base $19$ is $88888GGGGH$. It doesn't work in base $13$ or $16$. In base $28$ it gives $CCCCCOOOOP$, where those are capital oh's (worth $24$).
This is because if we express $\frac{1}{9}$ in base $9a+1$, it is $0.aaaa\ldots$. So $\left (\frac{2(9a+1)^5+1}{3}\r... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/55394",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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"answer_id": 2
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Find the perimeter of any triangle given the three altitude lengths The altitude lengths are 12, 15 and 20. I would like a process rather than just a single solution.
| First,
$\begin{aligned}\operatorname{Area}(\triangle ABC)=\frac{ah_a}2=\frac{bh_b}2=\frac{ch_c}2\implies \frac1{h_a}&=\frac{a}{2\operatorname{Area}(\triangle ABC)}\\\frac1{h_b}&=\frac{b}{2\operatorname{Area}(\triangle ABC)}\\\frac1{h_c}&=\frac{c}{2\operatorname{Area}(\triangle ABC)} \end{aligned}$
By the already mentio... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/55440",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 5,
"answer_id": 4
} |
Uniformly continuous $f$ in $L^p([0,\infty))$ Assume that $1\leq p < \infty $, $f \in L^p([0,\infty))$, and $f$ is uniformly continuous. Prove that $\lim_{x \to \infty} f(x) = 0$ .
| Hints: Suppose for a contradiction that $f(x) \not \to 0$. Together with the definition of uniform continuity, conclude that there exist constants $\varepsilon > 0$ and $\delta = \delta(\varepsilon) > 0$ such that for any $M > 0$ there exists $x > M$ for which it holds
$$
\int_{(x,x + \delta )} {|f(y)|^p \,dy} \ge \b... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/55493",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Quasi convexity and strong duality Is there a possibility to prove the strong duality (Lagrangian duality) if the primal problem is quasiconvex?
Or is there a technique that proves that the primal problem is convex instead of quasiconvex?
| I'm not sure I understand the question clearly enough to give a precise answer, but here is a kind of meta-answer about why you should not expect such a thing to be true.
Generally speaking duality results come from taking some given objective function and adding variable multiples of other functions (somehow connected... | {
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Use of a null option in a combination with repetition problem Ok, so I am working on a combinatorics problem involving combination with repetition. The problem comes off a past test that was put up for study. Here it is:
An ice cream parlor sells six flavors of ice cream: vanilla,
chocolate, strawberry, cook... | Suppose that the problem had asked how many combinations there are with exactly $19$ scoops. That would be a bog-standard problem involving combinations with possible repetition (sometimes called a stars-and-bars or marbles-and-boxes problem). Then you’d have been trying to distribute $19$ scoops of ice cream amongst $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/55637",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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A compactness problem for model theory I'm working on the following problem:
Assume that every model of a sentence $\varphi$ satisfies a sentence from $\Sigma$. Show that there is a finite $\Delta \subseteq \Sigma$ such that every model of $\varphi$ satisfies a sentence in $\Delta$.
The quantifiers in this problem are ... | Cute, in a twisted sort of way. You are right, the quantifier structure is the main hurdle to solving the problem.
We can assume that $\varphi$ has a model, else the result is trivially true.
Suppose that there is no finite $\Delta\subseteq \Sigma$ with the desired property.
Then for every finite $\Delta \subseteq ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/55686",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
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Simple problem with pattern matching in Mathematica I'm having trouble setting a pattern for simplifying a complex expression. I've distilled the question down to the simplest case where Mathematica seems to fail.
I set up a simple rule based on a pattern:
simpRule = a b v___ - c d v___ -> e v
which works on the dire... | You need to be aware of the FullForm of the expressions you are working with to correctly use replacement rules. Consider the FullForm of the two expressions you use ReplaceAll (/.) on:
a b - c d // FullForm
-a b + c d // FullForm
(* Out *) Plus[Times[a, b], Times[-1, c, d]]
(* Out *) Plus[Times[-1, a, b], Times[c, ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/55741",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Numerical computation of the Rayleigh-Lamb curves The Rayleigh-Lamb equations:
$$\frac{\tan (pd)}{\tan (qd)}=-\left[\frac{4k^2pq}{\left(k^2-q^2\right)^2}\right]^{\pm 1}$$
(two equations, one with the +1 exponent and the other with the -1 exponent) where
$$p^2=\frac{\omega ^2}{c_L^2}-k^2$$
and
$$q^2=\frac{\omega ^2}{c_T... | The Rayleigh-Lamb equations:
$$\frac{\tan (pd)}{\tan (qd)}=-\left[\frac{4k^2pq}{\left(k^2-q^2\right)^2}\right]^{\pm 1}$$
are equivalent to the equations (as Robert Israel pointed out in a comment above)
$$\left(k^2-q^2\right)^2\sin pd \cos qd+4k^2pq \cos pd \sin qd=0$$
when the exponent is +1, and
$$\left(k^2-q^2\r... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/55872",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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Showing that $l^p(\mathbb{N})^* \cong l^q(\mathbb{N})$ I'm reading functional analysis in the summer, and have come to this exercise, asking to show that the two spaces $l^p(\mathbb{N})^*,l^q(\mathbb{N})$ are isomorphic, that is, by showing that every $l \in l^p(\mathbb{N})^*$ can be written as
$l_y(x)=\sum y_nx_n$
for... | We know that the $(e_n)$ with a $1$ at the $n$-th position and $0$s elsewhere is a Schauder basis for $\ell^p$ (which has some nice alternative equivalent definitions, I recommend Topics in Banach Space Theory by Albiac and Kalton as a nice reference about this.
So, every $x \in \ell^p$ has a unique representation by
$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/56020",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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On the growth order of an entire function $\sum \frac{z^n}{(n!)^a}$ Here $a$ is a real positive number. The result is that $f(z)=\sum_{n=1}^{+\infty} \frac{z^n}{(n!)^a}$ has a growth order $1/a$ (i.e. $\exists A,B\in \mathbb{R}$ such that $|f(z)|\leq A\exp{(B|z|^{1/a})},\forall z\in \mathbb{C}$). It is Problem 3* from ... | There is a formula expressing the growth order of entire function $f(z)=\sum_{n=0}^\infty c_nz^n\ $ in terms of its Taylor coeffitients:
$$
\rho=\limsup_{n\to\infty}\frac{\log n}{\log\frac{1}{\sqrt[n]{|c_n|}}}.
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/56068",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 3,
"answer_id": 1
} |
planar curve if and only if torsion Again I have a question, it's about a proof, if the torsion of a curve is zero, we have that
$$ B(s) = v_0,$$
a constant vector (where $B$ is the binormal), the proof ends concluding that the curve $$
\alpha \left( t \right)
$$
is such that $$
\alpha(t)\cdot v_0 = k
$$
and th... | The constant $k$ need not be $0$; that would be the case where $\alpha$ lies in a plane through the origin. You have $k=\alpha(0)\cdot v_0$, so for all $t$, $(\alpha(t)-\alpha(0))\cdot v_0=0$. This means that $\alpha(t)-\alpha(0)$ lies in the plane through the origin perpendicular to $v_0$, so $\alpha(t)$ lies in the... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/56197",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Regular curve which tangent lines pass through a fixed point How to prove that if a regular parametrized curve has the property that all its tangent lines passs through a fixed point then its trace is a segment of a straight line?
Thanks
| In his answer, user14242 used a vector multiplication for two vectors - while I believe it's only defined for 3-dim case.
If you talk not only about the 3-dim problem then just writing explicitly the equation of the tangent line you obtain
$$
r(t)+\dot{r}(t)\tau(t) = a
$$
where $a$ is a fixed point and $\tau(t)$ deno... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/56275",
"timestamp": "2023-03-29T00:00:00",
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Proving $\sum\limits_{k=1}^{n}{\frac{1}{\sqrt{k}}\ge\sqrt{n}}$ with induction I am just starting out learning mathematical induction and I got this homework question to prove with induction but I am not managing.
$$\sum\limits_{k=1}^{n}{\frac{1}{\sqrt{k}}\ge\sqrt{n}}$$
Perhaps someone can help me out I don't understand... | For those who strive for non-induction proofs...
Since $\frac 1{\sqrt k} \ge \frac 1{\sqrt n}$ for $1 \le k \le n$, we actually have
$$
\sum_{i=1}^n \frac 1{\sqrt k} \ge \sum_{i=1}^n \frac 1{\sqrt n} = \frac n{\sqrt n} = \sqrt n.
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/56335",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "16",
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Do finite algebraically closed fields exist? Let $K$ be an algebraically closed field ($\operatorname{char}K=p$). Denote
$${\mathbb F}_{p^n}=\{x\in K\mid x^{p^n}-x=0\}.$$
It's easy to prove that ${\mathbb F}_{p^n}$ consists of exactly $p^n$ elements.
But if $|K|<p^n$, we have collision with previous statement (because ... | No, there do not exist any finite algebraically closed fields. For suppose $K$ is a finite field; then the polynomial
$$f(x)=1+\prod_{\alpha\in K}(x-\alpha)\in K[x]$$
cannot have any roots in $K$ (because $f(\alpha)=1$ for any $\alpha\in K$), so $K$ cannot be algebraically closed.
Note that for $K=\mathbb{F}_{p^n}$, th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/56397",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "45",
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What is the analogue of spherical coordinates in $n$-dimensions? What's the analogue to spherical coordinates in $n$-dimensions? For example, for $n=2$ the analogue are polar coordinates $r,\theta$, which are related to the Cartesian coordinates $x_1,x_2$ by
$$x_1=r \cos \theta$$
$$x_2=r \sin \theta$$
For $n=3$, the ... | I was trying to answer exercise 9 of $ I.5. $ from Einstein gravity in a nutshell by A. Zee that I saw this question so what I am going to say is from this question.
It is said that the d-dimensional unit sphere $S^d$ is embedded into $E^{d+1}$ by usual Pythagorean relation$$(X^1)^2+(X^2)^2+.....+(X^{d+1})^2=1$$. Thus ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/56582",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "46",
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Asymptotics for prime sums of three consecutive primes We consider the sequence $R_n=p_n+p_{n+1}+p_{n+2}$, where $\{p_i\}$ is the prime number sequence, with $p_0=2$, $p_1=3$, $p_2=5$, etc..
The first few values of $R_n$ for $n=0,1,2,\dots $ are:
$10, 15, 23, 31, 41, 49, 59, 71, 83, 97, 109, 121, 131, 143, 159, 173, 18... | What this says (to me) is that there are twice as many primes in R(n) as in the natural numbers. But, since each R(n) is the sum of three primes, all of which are odd (except for R(1)), then each R(n) is odd. This eliminates half of the possible values (the evens), all of which are, of course, composite.
So, if the R(n... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/56643",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
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Solving $-u''(x) = \delta(x)$ A question asks us to solve the differential equation
$-u''(x) = \delta(x)$
with boundary conditions
$u(-2) = 0$ and $u(3) = 0$ where $\delta(x)$ is the Dirac delta function. But inside the same question, teacher gives the solution in two pieces as $u = A(x+2)$ for $x\le0$ and $u = B(x-3)... | Both describe the same type of function.
The ramp function is nothing but
\begin{align} R(x) = \begin{cases}0 & x \leq 0
\\ x & x \geq 0\end{cases} \end{align}
If you use the general solution and plug in the same boundary conditions,
\begin{align}u(-2) &= -R(-2) + C -2D = C - 2D = 0
\\
u(3) &= -R(3) + C + 3D = -3 + ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/56681",
"timestamp": "2023-03-29T00:00:00",
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Please help me to understand why $\lim\limits_{n\to\infty } {nx(1-x^2)^n}=0$ This is the exercise:
$$f_{n}(x) = nx(1-x^2)^n,\qquad n \in {N}, f_{n}:[0,1] \to {R}.$$
Find ${f(x)=\lim\limits_{n\to\infty } {nx(1-x^2)^n}}$.
I know that $\forall x\in (0,1]$ $\Rightarrow (1-x^2) \in [0, 1) $ but I still don't know how t... | Please see $\textbf{9.10}$ at this given link: The Solution is actually worked out for a more general case and I think that should help.
*
*http://www.csie.ntu.edu.tw/~b89089/book/Apostol/ch9.pdf
| {
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Generate a random direction within a cone I have a normalized $3D$ vector giving a direction and an angle that forms a cone around it, something like this:
I'd like to generate a random, uniformly distributed normalized vector for a direction within that cone. I would also like to support angles greater than pi (but l... | So you want to uniformly sample on a spherical cap. With the notations of this link, here is some pseudo-code which performs the sampling on the cap displayed in the above link (then it remains to perform a rotation):
stopifnot(h > 0 && h < 2 * R)
xy = uniform_point_in_unit_circle()
k = h * dotproduct(xy)
s = sqrt(h * ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/56784",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "34",
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"answer_id": 5
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Is the language $L = \{0^m1^n: m \neq n - 1 \}$ context free? Consider the language: $L = \{0^m1^n : m \neq n - 1 \}$ where $m, n \geq 0$
I tried for hours and hours but couldn't find its context free grammar. I was stuck with a rule which can check on the condition $m \neq n - 1$. Would anyone can help me out? Thanks ... | The trick is that you need extra "clock" letters $a,b, \dots$ in the language with non-reversible transformations between them, to define the different phases of the algorithm that builds the string. When the first phase is done, transform $a \to b$, then $b \to c$ after phase 2, etc, then finally remove the extra let... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/56836",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Semi-partition or pre-partition For a given space $X$ the partition is usually defined as a collection of sets $E_i$ such that $E_i\cap E_j = \emptyset$ for $j\neq i$ and $X = \bigcup\limits_i E_i$.
Does anybody met the name for a collection of sets $F_i$ such that $F_i\cap F_j = \emptyset$ for $j\neq i$ but
*
*$X... | Brian M. Scott wrote:
In the topological case I'd simply call it a (pairwise) disjoint family whose union is dense in $X$; I've not seen any special term for it.
In fact, I can remember seeing it only once: such a family figures in the proof that almost countable paracompactness, a property once studied at some length... | {
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} |
$A\in M_{n}(C)$ and $A^*=-A$ and $A^4=I$ Let $A\in M_{n}(C)$ be a matrix such that $A^*=-A$ and $A^4=I$.
I need to prove that the eigenvalues of A are between $-i$ to $i$ and that $A^2+I=0$
I didn't get to any smart conclusion.
Thanks
| Do you recall that hermitian matrices ($A^*=A$) must have real eigenvalues? Similiarly, skew-hermitian matrices, i.e. $A^*=-A$, must have pure imaginary eigenvalues.
(see Why are all nonzero eigenvalues of the skew-symmetric matrices pure imaginary?)
Also, since $A$ is skew-hermitian, then $A$ is normal too, i.e. $A^*... | {
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"source": "stackexchange",
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Set Theory: Proving Statements About Sets Let $A, B,$ and $C$ be arbitrary sets taken from the positive integers.
I have to prove or disprove that: $$ \text{If }A ∩ B ∩ C = ∅, \text{then } (A ⊆ \sim B) \text{ or } (A ⊆ \sim C)$$
Here is my disproof using a counterexample:
If $A = \{ \}$ the empty set,
$B = \{2, 3\}$,
... | As another counterexample, let $A=\mathbb{N}$, and $B$ and $C$ be disjoint sets with at least one element each.
Elaborating: On the one hand, since $B$ and $C$ are disjoint,
$$
A \cap B \cap C = \mathbb{N} \cap B \cap C = B \cap C = \emptyset.
$$
On the other hand, $A = \mathbb{N}$ is not contained in the complement ... | {
"language": "en",
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Convergence of a double sum
Let $(a_i)_{i=1}^\infty$ be a sequence of positive numbers such that $\sum_1^\infty a_i < \infty$. What can we say about the double series
$$\sum_{i, j=1}^\infty a_{i+ j}^p\ ?$$
Can we find some values of $p$ for which it converges? I'm especially interested in $p=2$.
Intuitively I'm... | To sum up, the result is false in general (see first part of the post below), trivially false for nonincreasing sequences $(a_n)$ if $p<2$ (consider $a_n=n^{-2/p}$) and true for nonincreasing sequences $(a_n)$ if $p\ge2$ (see second part of the post below).
Rearranging terms, one sees that the double series converges ... | {
"language": "en",
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Determining which values to use in place of x in functions When solving partial fractions for integrations, solving x for two terms usually isn't all that difficult, but I've been running into problems with three term integration.
For example, given
$$\int\frac{x^2+3x-4}{x^3-4x^2+4x}$$
The denominator factored out to ... | To find the constants in the rational fraction $$\frac{x^2+3x-4}{x^3-4x^2+4x}=\frac{A}{x}+\frac{B}{x-2}+\frac{C}{(x-2)^2},$$
you may use any set of 3 values of $x$, provided that the denominator $x^3-4x^2+4x\ne 0$.
The "standard" method is to compare the coefficients of $$\frac{x^2+3x-4}{x^3-4x^2+4x}=\frac{A}{x}+\frac... | {
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Optimum solution to a Linear programming problem If we have a feasible space for a given LPP (linear programming problem), how is it that its optimum solution lies on one of the corner points of the graphical solution? (I am here concerned only with those LPP's which have a graphical solution with more than one corner/... | It's a special instance of the following general theorem:
The maximum of a convex function $f$ on a compact convex set $S$ is attained in an extreme point of $S$.
An extreme point of a convex set $S$ is a point in $S$ which does not lie in any open line segment joining two points of $S$. In your case, the "corner po... | {
"language": "en",
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"source": "stackexchange",
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Stalk of coherent sheaf vanishing is the following true:
if I have a coherent sheaf $F$ on a noetherian scheme $X$ with a point $x$ and the stalk $F_x$ is zero, then there is a neighborhood $U$ of $x$, such that the restriction of $F$ to $U$ is zero?
Thank you
| Yes. For the locality of the problem, you can assume that $X$ is affine: $X = \mathrm{Spec} A$ and $F = \tilde{M}$, where $A$ is a noetherian ring and $M$ is a finite $A$-module. Let $P$ a prime such that $M_P = 0$. Let $\{ x_1, \dots, x_n \}$ a set of generators of $M$ as an $A$-module. Then exist $s_i \in A \setminus... | {
"language": "en",
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Question about all the homomorphisms from $\mathbb{Z}$ to $\mathbb{Z}$ An exercise in "A first course in Abstract Algebra" asked the following:
Describe all ring homomorphisms from the ring $\mathbb{Z},+,\cdot$ to itself.
I observed that for any such ring homomorphism the following has to hold: $$\varphi(1) = \varphi(1... | I assume you are talking about Fraleigh's book. If so, he does not require that a ring homomorphism maps the multiplicative identity to itself. Follow his hint by concentrating on the possible values for $f(1)$. If $f$ is a (group) homomorphism for the group $(\mathbb{Z},+)$ and $f(1)=a$, then $f$ will reduce to multip... | {
"language": "en",
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Definition of manifold From Wikipedia:
The broadest common definition of manifold is a topological space
locally homeomorphic to a topological vector space over the reals.
A topological manifold is a topological space locally homeomorphic to
a Euclidean space.
In both concepts, a topological space is homeomorphic... | The reason to use topological vector spaces as model spaces (for differential manifolds, that is) is that you can define the differential of a curve in a topological vector space. And you can use this to define the differential of curves in your manifold, i.e. do differential geometry.
For more details see my answer he... | {
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Homology of $\mathbb{R}^3 - S^1$ I've been looking for a space on the internet for which I cannot write down the homology groups off the top of my head so I came across this:
Compute the homology of $X : = \mathbb{R}^3 - S^1$.
I thought that if I stretch $S^1$ to make it very large then it looks like a line, so $\mathb... | Consider $X = \mathbb{R}^3 \setminus (S^1 \times \{ 0 \}) \subseteq \mathbb{R}^3$, $U = X \setminus z \text{-axis}$ and $V = B(0,1/2) \times \mathbb{R}$, where $B(0, 1/2)$ is the open ball with radius $1/2$ and center in the origin of $\mathbb{R}^2$. It is clear that $\{ U, V \}$ is an open cover of $X$.
Now let's comp... | {
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prove cardinality rule $|A-B|=|B-A|\rightarrow|A|=|B|$ I need to prove this $|A-B|=|B-A|\rightarrow|A|=|B|$
I managed to come up with this:
let $f:A-B\to B-A$ while $f$ is bijective.
then define $g\colon A\to B$ as follows:
$$g(x)=\begin{cases}
f(x)& x\in (A-B) \\
x& \text{otherwise} \\
\end{cases}$$
but I'm n... | Note that
$$\begin{align}
|A| = |A \cap B| + |A \cap B^c| = |B \cap A| + |B \cap A^c| = |B|.
\end{align}$$
Here $E^c$ denotes the compliment of the event $E$ in the universal space $X$.
| {
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Incorrect manipulation of limits Here's my manipulation of a particular limit:
$\displaystyle \lim\limits_{h\rightarrow 0}\Big[\frac{f(x+h)g(x) - f(x)g(x+h)}{h}\Big]$
Using the properties of limits:
$\displaystyle \begin{align*}
&=\frac{\lim\limits_{h\rightarrow 0}\Big[f(x+h)g(x) - f(x)g(x+h)\Big]}{\lim\limits_{h\righ... | The answers already posted fully answer your question. So what follows is not an answer to your question, but it may be helpful.
Let us assume that $f$ and $g$ are differentiable at $x$. Note that
$$f(x+h)g(x) - f(x)g(x+h)= f(x+h)g(x)+(f(x)g(x)-f(x)g(x))-f(x)g(x+h).$$
We have added $0$ in the middle, which is harmles... | {
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Proof for divisibility rule for palindromic integers of even length I am studying for a test and came across this in my practice materials. I can prove it simply for some individual cases, but I don't know where to start to prove the full statement.
Prove that every palindromic integer in base $k$ with an even number ... | HINT $\rm\ \ mod\ \ x+1:\ \ f(x) + x^{n+1}\:(x^n\ f(1/x))\ \equiv\ f(-1) - f(-1)\:\equiv\: 0$
Remark $\ \ $ It is simple to verify that $\rm\ x^n\ f(1/x)\ $ is the reversal of a polynomial $\rm\:f\:$ of degree $\rm\:n\:,\:$ therefore the above is the general palindromic polynomial with even number of coefficients. ... | {
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Integrate $\int\limits_{0}^{1} \frac{\log(x^{2}-2x \cos a+1)}{x} dx$ How do I solve this: $$\int\limits_{0}^{1} \frac{\log(x^{2}-2x \cos{a}+1)}{x} \ dx$$
Integration by parts is the only thing which I could think of, clearly that seems cumbersome. Substitution also doesn't work.
| Please see $\textbf{Problem 4.30}$ in the book: ($\text{solutions are also given at the end}$)
*
*$\text{The Math Problems Notebook,}$ by Valentin Boju and Louis Funar.
| {
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Transitive graph such that the stabilizer of a point has three orbits I am looking for an example of a finite graph such that its automorphism group is transitive on the set of vertices, but the stabilizer of a point has exactly three orbits on the set of vertices. I can't find such an example. Anyone has a suggestion?... | Consider the Petersen graph. Its automorphism group is $S_5$ which acts on $2$-subsets of $\{1,\ldots,5\}$ (that can be seen as vertices of the graph). Now the result is clear.
| {
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Convergence of sequence points to the point of accumulation Wikipedia says:
Every finite interval or bounded interval that contains an infinite number of points must have at least one point of accumulation. If a bounded interval contains an infinite number of points and only one point of accumulation, then the sequenc... | What about: $$\left \{\frac{1}{n}\Bigg| n\in\mathbb Z^+ \right \}\cup\{0\}$$
This set (or sequence $a_0=0,a_n=\frac{1}{n}$) is bounded in $[0,1]$ and has only limit point, namely $0$.
A slightly more complicated example would be:
$$\left \{\frac{(-1)^n}{n}\Bigg| n\in\mathbb Z^+ \right \}\cup\{0\}$$
In the interval $[-1... | {
"language": "en",
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Feller continuity of the stochastic kernel Given a metric space $X$ with a Borel sigma-algebra, the stochastic kernel $K(x,B)$ is such that $x\mapsto K(x,B)$ is a measurable function and a $B\mapsto K(x,B)$ is a probability measure on $X$ for each $x$
Let $f:X\to \mathbb R$. We say that $f\in \mathcal C(B)$ if $f$ is c... | To make it clearer, you can re-write $ G(x) = \int\limits_X g(y)\mathbf{1}_B(y) K(x,{\rm d} y)$. Now in general $g{\mathbf 1}_B$ is not continuous anymore so as Nate pointed out you should not expect $G$ to be such.
However, if you take a continuous $g $
with $\overline{{\rm support}(g)} \subsetneq B$ then (I let you ... | {
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Improper Double Integral Counterexample Let $f: \mathbf{R}^2\to \mathbf{R}$. I want to integrate $f$ over the entire first quadrant, call $D$. Then by definition we have
$$\int \int_D f(x,y) dA =\lim_{R\to[0, \infty]\times[0, \infty]}\int \int_R f(x,y) dA$$
where $R$ is a rectangle.
I remember vaguely that the above i... | Let $f$ be 1 below the diagonal, -1 above.
| {
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Determining if a quadratic polynomial is always positive Is there a quick and systematic method to find out if a quadratic polynomial is always positive or may have positive and negative or always negative for all values of its variables?
Say, for the quadratic inequality
$$3x^{2}+8xy+5xz+2yz+7y^{2}+2z^{2}>0$$
without ... | This is what Sylvester's criterion is for. Write your quadratic as $v^T A v$ where $v$ is a vector of variables $(x_1\ x_2\ \cdots\ x_n)$ and $A$ is a matrix of constants. For example, in your case, you are interested in
$$\begin{pmatrix} x & y & z \end{pmatrix}
\begin{pmatrix} 3 & 4 & 5/2 \\ 4 & 7 & 1 \\ 5/2 & 1 & 2 ... | {
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ODE problem shooting Please help me spot my mistake:
I have an equation
$$(u(x)^{-2} + 4u'(x)^2)^{\frac{1}{2}} - u'(x)\frac{d}{du'}(u(x)^{-2} + 4u'(x)^2)^{\frac{1}{2}} = k$$ where $k$ is a constant.
I am quite sure that if I take $u(x) = \sqrt{y(x)}$ I would have the brachistochrone equation, hence I am expecting a cy... | In the line after
$$ 2k u' = \sqrt{ u^{-4} - k^2 u^{-2}} $$
(the fourth line of your computations) when you divided, you divided wrong. The integrand in the LHS should be
$$ \int \frac{u^2}{\sqrt{1-k^2 u^2}} \mathrm{d}u $$
| {
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Solve this Inequality I am not sure how to solve this equation. Any ideas
$$(1+n) + 1+(n-1) + 1+(n-2) + 1+(n-3) + 1+(n-4) + \cdots + 1+(n-n) \ge 1000$$
Assuming $1+n = a$
The equation can be made to looks like
$$a+(a-1)+(a-2)+(a-3)+(a-4)+\cdots+(a-n) \ge 1000$$
How to proceed ahead, or is there another approach to so... | Here is an example (probably not the most elegant though...) way to solve this. First either you write $(n+1)a-\frac{n(n+1)}{2}$ and replace $a$ by $n+1$ or you just realize replacing $a$ by $n+1$ in your second inequation that the left side is $\sum_{i=1}^{n+1}i=\frac{(n+1)(n+2)}{2}$.
Your inequality is then
$$(n+1)^... | {
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Normal subgroups vs characteristic subgroups It is well known that characteristic subgroups of a group $G$ are normal. Is the converse true?
| I think the simplest example is the Klein four-group, which you can think of as a direct sum of two cyclic groups of order two, say $A\oplus B$. Because it is abelian, all of its subgroups are normal. However, there is an automorphism which interchanges the two direct summands $A$ and $B$, which shows that $A$ (and $... | {
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How I can use the mean value theorem in this problem? Use the Mean Value Theorem to estimate the value of $\sqrt{80}$.
and
how should we take $f(x)$?
Thanks in advance.
Regards
| You want to estimate a value of $f(x) = \sqrt{x}$, so that's a decent place to start. The mean value theorem says that there's an $a \in (80, 81)$ such that
$$
\frac{f(81) - f(80)}{81 - 80} = f'(a).
$$
I don't know what $a$ is, but you know $f(81)$ and you hopefully know how to write down $f'$. How small can $f'(a)$ ... | {
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Some question of codimension 1 (1) "For affine variety $V$ of $\mathbb{A}^{n}$ such that its coordinate ring is UFD,
closed subvariety of $V$ which has codimension 1 is cut out by a single equation."
I looked at the proof of this statement, Where I have been using UFD in I do not know...
(2) I want to see proof o... | As for (1): a closed subvariety of $V$ is defined by a prime ideal $p$ of the coordinate ring $k[V]$ of height $1$. If $k[V]$ is a UFD such a prime ideal is principal.
As for (2): the statement is false. The closed subvariety is cut out by a single equation locally but not globally as in the case of a UFD.
| {
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Example of infinite field of characteristic $p\neq 0$ Can you give me an example of infinite field of characteristic $p\neq0$?
Thanks.
| Another construction, using a tool from the formal logic: the ultraproduct.
The cartesian product of fields
$$P = {\Bbb F}_p\times{\Bbb F}_{p^2}\times{\Bbb F}_{p^3}\times\cdots$$
isn't a field ("isn't a model of..." ) because has zero divisors:
$$(0,1,0,1,\cdots)(1,0,1,0\cdots)=(0,0,0,0,\cdots).$$
The solution is takin... | {
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Prove convexity/concavity of a complicated function Can anyone help me to prove the convexity/concavity of following complicated function...? I have tried a lot of methods (definition, 1st derivative etc.), but this function is so complicated, and I finally couldn't prove... however, I plotted with many different param... | I am a newcomer so would prefer to just leave a comment, but alas I see no "comment" button, so I will leave my suggestion here in the answer box.
I have often used a Nelder-Mead "derivative free" algorithm (fminsearch in matlab) to minimize long and convoluted equations like this one. If you can substitute the constr... | {
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When can a random variable be written as a sum of two iid random variables? Suppose $X$ is a random variable; when do there exist two random variables $X',X''$, independent and identically distributed, such that $$X = X' + X''$$
My natural impulse here is to use Bochner's theorem but that did not seem to lead anywhere... | I think your characteristic function approach is the reasonable one. You take the square root of the characteristic function (the one that is $+1$ at 0), take its Fourier transform, and check that this is a positive measure. In the case where $X$ takes only finitely many values, the characteristic function is a finit... | {
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Elementary central binomial coefficient estimates
*
*How to prove that $\quad\displaystyle\frac{4^{n}}{\sqrt{4n}}<\binom{2n}{n}<\frac{4^{n}}{\sqrt{3n+1}}\quad$ for all $n$ > 1 ?
*Does anyone know any better elementary estimates?
Attempt. We have
$$\frac1{2^n}\binom{2n}{n}=\prod_{k=0}^{n-1}\frac{2n-k}{2(n-k)}... | Here is a better estimate of the quantity. I am not going to prove it. Since I know about it, hence I am sharing and will put the reference.
$$\frac{1}{2\sqrt{n}} \le {2n \choose n}2^{-2n} \le \frac{1}{\sqrt{2n}}$$
Refer to Page 590 This is the text book "Boolean functions Complexity" by Stasys Jukna.
| {
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The smallest subobject $\sum{A_i}$ containing a family of subobjects {$A_i$} In an Abelian category $\mathcal{A}$, let {$A_i$} be a family of subobjects of an object $A$. How to show that if $\mathcal{A}$ is cocomplete(i.e. the coproduct always exists in $\mathcal{A}$), then there is a smallest subobject $\sum{A_i}$ of... | You are quite right that it can't be the coproduct, since that is in general not a subobject of $A$. Here are two ways of constructing the desired subobject:
*
*As Pierre-Yves suggested in the comments, the easiest way is to take the image of the canonical map $\bigoplus_i A_i \to A$. This works in any cocomplete ca... | {
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Prove this number fact Prove that $x \neq 0,y \neq 0 \Rightarrow xy \neq 0$.
Suppose $xy = 0$. Then $\frac{xy}{xy} = 1$. Can we say that $\frac{xy}{xy} = 0$ and hence $1 = 0$ which is a contradiction? I thought $\frac{0}{0}$ was undefined.
| If xy=0, then x=0 or y=0. So, by contraposition, if is not the case that "x=0 OR y=0", then it is not the case that xy=0. By De Morgan's Laws, we have that "if it is not the case that x=0 AND it is not the case that y=0, then it is not the case that xy=0". Now we move the negations around as a "quantifier exchange" ... | {
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Is $\mathbb{Q}[2^{1/3}]$ a field? Is $\mathbb{Q}[2^{1/3}]=\{a+b2^{1/3}+c2^{2/3};a,b,c \in \mathbb{Q}\}$ a field?
I have checked that $b2^{1/3}$ and $c2^{2/3}$ both have inverses, $\frac{2^{2/3}}{2b}$ and $\frac{2^{1/3}}{2c}$, respectively.
There are some elements with $a,b,c \neq 0$ that have inverses, as $1+1*2^{1/3}... | A neat way to confirm that it is a field:
Since $\mathbb{Q}$ is a field, $\mathbb{Q}[x]$ is a PID. $\mathbb{Q}[2^{1/3}] \cong \mathbb{Q}[x] / (x^3 - 2)$. Now, $x^3 - 2$ is irreducible over $\mathbb{Q}$, since if it weren't, there would be a rational root to $x^3 - 2$. Because $\mathbb{Q}[x]$ is a PID and the polynomi... | {
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"timestamp": "2023-03-29T00:00:00",
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$f_A(x)=(x+2)^4x^4$, $m_A(x)=(x+2)^2x^2$- What can I know about $A$? $A$ is matrix under $R$ which I know the following information about it:
$f_A(x)=(x+2)^4x^4$- Characteristic polynomial
$m_A(x)=(x+2)^2x^2$- Minimal polynomial.
I'm trying to find out
(i) $A$'s rank
(ii) $\dim$ $\ker(A+2I)^2$
(iii) $\dim$ $\ker (A+2I... | If the Jordan form of A is C then let P be invertible such that $A=PCP^{-1}$ then
$$(A+2I)^2=P(C+2I)^2P^{-1}\;\rightarrow\; \dim\ker((A+2I)^2)=\dim\ker((C+2I)^2)$$
and you know exactly how $(C+2I)^2$ looks like (well, at least the part of the kernel).
The same operation should help you solve the rest of the problems
| {
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Radical ideal of $(x,y^2)$
How does one show that the radical of $(x,y^2)$ is $(x,y)$ over $\mathbb{Q}[x,y]$?
I have no idea how to do, please help me.
| Recall that the radical of an ideal $\mathfrak{a}$ is equal to the intersection of the primes containing $\mathfrak{a}$. Here, let $\mathfrak{p}$ be a prime containing $(x, y^2)$. Then $y^2 \in \mathfrak{p}$ implies $y \in \mathfrak{p}$. Can you finish it off from there?
| {
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Ratio of circumference to radius I know that the ratio of the circumference to the diameter is Pi - what about the ratio of the circumference to the radius? Does it have any practical purpose when we have Pi? Is it called something (other than 2 Pi)?
| That $\pi$ and $2 \pi$ have a very simple relationship to each other sharply limits the extent to which one can be more useful or more fundamental than the other.
However, there are probably more formulas that are simpler when expressed using $2\pi$ instead of $\pi$, than the other way around. For example, there is o... | {
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Stalks of the graph of a morphism I am interested in the graph $\Gamma_f$ of a morphism
$f: X\rightarrow Y$ between two sufficiently nice schemes $X,Y$.
One knows that it is a closed subset of $X\times Y$ (when the schemes are nice, say varieties over a field).
I would like to know the following: if you endow it with t... | Let $f:X\to Y$ be a morphism of $S$-schemes. The graph morphism $\gamma_f:X\to X\times_S Y$ is the pull-back of the diagonal morphism $\delta: Y\to Y\times_S Y$ along $f\times id_Y: X\times_S Y \to Y\times_S Y$. This implies that if $\delta$ is a closed embedding (i.e. $Y$ is separated over $S$) so is $\gamma_f$. So $\... | {
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$A\in M_n(\mathbb C)$ invertible and non-diagonalizable matrix. Prove $A^{2005}$ is not diagonalizable $A\in M_n(\mathbb C)$ invertible and non-diagonalizable matrix. I need to prove that $A^{2005}$ is not diagonalizable as well. I am asked as well if Is it true also for $A\in M_n(\mathbb R)$. (clearly a question from... | As luck would have it, the implication: $A^n$ diagonalizable and invertible $\Rightarrow A$ diagonalizable, was discussed in XKCD forum recently. See my answer there as well as further dicussion in another thread.
| {
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A sufficient condition for linearity? If $f$ is a linear function (defined on $\mathbb{R}$), then for each $x$, $f(x) – xf’(x) = f(0)$. Is the converse true? That is, is it true that if $f$ is a differentiable function defined on $\mathbb{R}$ such that for each $x$, $f(x) – xf’(x) = f(0)$, then $f$ is linear?
| If $f\in C^2$ (it is twice differentiable, and $f''$ is continuous), then the answer is yes; I don't know if it's necessarily true without this hypothesis.
If $f(x)-xf'(x)=f(0)$ for all $x$, then
$$f(x)=xf'(x)+f(0),$$
so that
$$f'(x)=f'(x)+xf''(x)$$
$$0=xf''(x)$$
This shows that $f''(x)=0$ for all $x\neq0$, but becaus... | {
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Prove that all even integers $n \neq 2^k$ are expressible as a sum of consecutive positive integers How do I prove that any even integer $n \neq 2^k$ is expressible as a sum of positive consecutive integers (more than 2 positive consecutive integer)?
For example:
14 = 2 + 3 + 4 + 5
84 = 9 + 10 + ... + 15
n = sum (k +... | (The following is merely a simplification of @Arturo Magidin's proof. So, please do not upvote my post.) Suppose $S=k+(k+1)+\ldots+\left(k+(n-1)\right)$ for some $k\ge1$ and $n\ge2$. Then
$$
S = nk + \sum_{j=0}^{n-1} j = nk+\frac{n(n-1)}{2} = \frac{n(n+2k-1)}{2}.
$$
Hence $S\in\mathbb{N}$ can be written as a consecut... | {
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Formula for $1^2+2^2+3^2+...+n^2$ In example to get formula for $1^2+2^2+3^2+...+n^2$ they express $f(n)$ as:
$$f(n)=an^3+bn^2+cn+d$$ also known that $f(0)=0$, $f(1)=1$, $f(2)=5$ and $f(3)=14$
Then this values are inserted into function, we get system of equations solve them and get a,b,c,d coefficients and we get that... | There are several ways to see this:
*
*As Rasmus pointed one out in a comment, you can estimate the sum by an integral.
*Imagine the numbers being added as cross sections of a quadratic pyramid. Its volume is cubic in its linear dimensions.
*Apply the difference operator $\Delta g(n)=g(n+1)-g(n)$ to $f$ repeatedly... | {
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Unique ways to assign P boxes to C bags? How many ways can I arrange $C$ unlabeled bags into $P$ labeled boxes such that each box receives at least $S$ bags (where C > S and C > P)? Assume that I can combine bags to fit a box. I have just learnt that there are $\binom{C-1}{P-1}$ unique ways to keep P balls into C bags.... | First reduce the number of bags by subtracting the required minimum number in each bag. Using your notation: $C' = C-SP$. Now, you're freely place the remaining $C'$ items into $P$ boxes. Which is
$(C'+P-1)!/C'!(P-1)!$
Take your example: 4 Boxes, 24 Bags and each Box should get 6 Bags.
$C'= 24-6*24 = 0$,
$(0+4-1)!... | {
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Maximizing the sum of two numbers, the sum of whose squares is constant
How could we prove that if the sum of the squares of two numbers is a constant, then the sum of the numbers would have its maximum value when the numbers are equal?
This result is also true for more than two numbers. I tested the results by taki... | Here's the pedestrian, Calculus I method:
Let $x$ and $y$ be the two numbers. The condition "the sum of the squares is a constant" means that there is a fixed number $c$ such that $x^2+y^2=c$. That means that if you know one of the two numbers, say, $x$, then you can figure out the absolute value of the other one: $y^2... | {
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How to calculate number of lumps of a 1D discrete point distribution? I would like to calculate the number of lumps of a given set of points.
Defining "number of lumps" as "the number of groups with points at distance 1"
Supose we have a discrete 1D space in this segment
For example N=15
. . . . . . . . . ... | If you are looking for a general formula that, given the points, returns the number of clusters (I think that "cluster" is a more common name that "lump", in this context), I'm afraid you won't find it. The problem is quite complicated, and there are many algorithms (google for hierachical clustering, percolation). For... | {
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How do Equally Sized Spheres Fit into Space? How much volume do spheres take up when filling a rectangular prism of a shape?
I assume it's somewhere in between $\frac{3}{4}\pi r^3$ and $r^3$, but I don't know where.
This might be better if I broke it into two questions:
First: How many spheres can fit into a given spac... | You can read a lot about these questions on Wikipedia.
Concerning the random version, there are some links at Density of randomly packing a box. The accepted answer links to a paper that "focuses on spheres".
| {
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Throw a die $N$ times, observe results are a monotonic sequence. What is probability that all 6 numbers occur in the sequence? I throw a die $N$ times and the results are observed to be a monotonic sequence. What is probability that all 6 numbers occur in the sequence?
I'm having trouble with this. There are two cases:... | I have a slightly different answer to the above, comments are very welcome :)
The number of monotonic sequences we can observe when we throw a dice $N$ times is 2$ N+5\choose5$-$6\choose1$ since the six sequences which consist of the same number repeatedly are counted as both increasing and decreasing (i.e. we have cou... | {
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Inserting numbers to create a geometric progression Place three numbers in between 15, 31, 104 in such way, that they would be successive members of geometric progression.
PS! I am studying for a test and need help. I would never ask anyone at math.stackexchange.com to do my homework for me. Any help at all would be s... | There is no geometric progression that contains all of $15$, $31$, and $104$, let alone also the hypothetical "extra" numbers.
For suppose that $15=kr^a$, $31=kr^b$, and $104=kr^c$ where $a$, $b$, $c$ are integers ($r$ need not be an integer, and $a$, $b$, $c$ need not be consecutive). Then $31=kr^ar^{b-a}=15r^{b-a}$.... | {
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Algebraic proof that collection of all subsets of a set (power set) of $N$ elements has $2^N$ elements In other words, is there an algebraic proof showing that $\sum_{k=0}^{N} {N\choose k} = 2^N$? I've been trying to do it some some time now, but I can't seem to figure it out.
| I don't know what you mean by "algebraic". Notice that if $N$ is $0$, we have the empty set, which has exactly one subset, namely itself. That's a basis for a proof by mathematical induction.
For the inductive step, suppose a set with $N$ elements has $2^N$ subsets, and consider a set of $N+1$ elements that results f... | {
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Minimal area of a surface that splits a cube into two equal-size pieces I had read the following problem and its solution from one source problem which was the following:
You want to cut a unit cube into two pieces each with volume 1/2. What dividing surface, which might be curved, has the smallest surface area?
The au... | We know from the isoperimetric inequality that locally the surface must be a sphere (where we can include the plane as the limiting case of a sphere with infinite radius). Also, the surface must be orthogonal to the cube where they meet; if they're not, you can deform the surface locally to reduce its area. A sphere or... | {
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Equation of the tangents
What is the equation of the tangent to $y=x^3-6x^2+12x+2$ that is parallel to the line $y=3x$ ?
I have no idea, how to solve, no example is given in the book!
Appreciate your help!
| Look at a taylor series (ref 1, 2) of your function about a point $x_c$ of order 1 (linear).
Then find which $x_c$ produces a line parallel to $3x$, or has a slope of $3$.
Hint! There are two solutions actually.
| {
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Why is the set of commutators not a subgroup? I was surprised to see that one talks about the subgroup generated by the commutators, because I thought the commutators would form a subgroup. Some research told me that it's because commutators are not necessarily closed under product (books by Rotman and Mac Lane popped ... | See the exercise 2.43 in the book "An introduction to the theory of group"- Joseph Rotman (4th ed.).
He also had made a nice remark:
The first finite group in which product of two commutators is not a commutator has order 96.
| {
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"timestamp": "2023-03-29T00:00:00",
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QQ plot explanation The figure shows the Q-Q plot of a theoretical and empirical standardized Normal distribution generated through the $qqnorm()$ function of R statistical tool.
How can I describe the right tail (top right) that does not follow the red reference line? What does it mean when there is a "trend" that ru... | It means that in the right tail, your data do not fit normal well, specifically, there are far less numbers there would be in a normal sample. If the black curved up, there would be more than in a typical normal sample.
You can think of the black curve as a graph of a function that , if applied to your data, would mak... | {
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Finite differences of function composition I'm trying to express the following in finite differences:
$$\frac{d}{dx}\left[ A(x)\frac{d\, u(x)}{dx} \right].$$
Let $h$ be the step size and $x_{i-1} = x_i - h$ and $x_{i+ 1} = x_i + h$
If I take centered differences evaluated in $x_i$, I get:
$\begin{align*}\left\{\frac{... | While the approach of robjohn is certainly possible, it is often better to take the approach suggested by the original poster:
$$
\left\{\frac{d}{dx}\left[ A(x)\frac{d\, u(x)}{dx}\right]\right\}_i
= \frac{A_{i+1/2}\left[\frac{u_{i+1}-u_{i}}{h}\right] - A_{i-1/2}\left[\frac{u_{i}-u_{i-1}}{h}\right]}{h}
$$
As he note... | {
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How to obtain the Standard Deviation of a ratio of independent binomial random variables? X and Y are 2 independent binomial random variables with parameters (n,p) and (m,q) respectively.
(trials, probability parameter)
| There probably isn't a closed-form formula for this.
But $X$ has mean $np$ and standard deviation $\sqrt{np(1-p)}$, and $Y$ has mean $mq$ and standard deviation $mq(1-q)$.
Now you need a simple fact: if $X$ has mean $\mu$ and standard deviation $\sigma$, then $\log X$ has mean approximately $\log \mu$ and standard devi... | {
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Checking whether a point lies on a wide line segment I know how to define a point on the segment, but I have this piece is a wide interval. The line has a width.
I have $x_1$ $y_1$, $x_2$ $y_2$, width and $x_3$ $y_3$
$x_3$ and $x_4$ what you need to check.
perhaps someone can help, and function in $\Bbb{C}$ #
| Trying to understand your question, perhaps this picture might help.
You seem to be asking how to find out whether the point $C$ is inside the thick line $AB$.
You should drop a perpendicular from $C$ to $AB$, meeting at the point $D$. If the (absolute) length of $CD$ is more than half the width of the thick line the... | {
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How to compute that the unit digit of $\frac{(7^{6002} − 1)}{2}$? The mother problem is:
Find the unit digit in LCM of $7^{3001} − 1$ and $7^{3001} + 1$
This problem comes with four options to choose the correct answer from,my approach,as the two number are two consecutive even numbers hence the required LCM is $$\f... | We look directly at the mother problem. Exactly as in your approach, we observe that we need to evaluate
$$\frac{(7^{3001}-1)(7^{3001}+1)}{2}$$
modulo $10$.
Let our expression above be $x$. Then $2x= (7^{3001}-1)(7^{3001}+1)$. We will evaluate $2x$ modulo $20$.
Note that $7^{3000}$ is congruent to $1$ modulo $4$ and ... | {
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Determinantal formula for reproducing integral kernel How do I prove the following?
$$\int\det\left(K(x_{i},x_{j})\right)_{1\leq i,j\leq n}dx_{1} \cdots dx_{N}=\underset{i=1}{\overset{n}{\prod}}\left(\int K(x_{i},x_{i})\;dx_{i}-(i-1)\right)$$
where
$$K(x,y)=\sum_{l=1}^n \psi_l(x)\overline{\psi_l}(y)$$
and
$$\{\psi_l(x)... | (This is too long to fit into a comment.)
Note that the product in the integrand of the right side foils out as
$$\sum_{A\subseteq [n]} (-1)^{n-|A|}\left(\prod_{i\not\in A} (i-1)\right)\prod_{j\in A}K(x_j,x_j).$$
(Here $[n]=\{1,2,\dots,n\}$.) On the other hand, we can use Leibniz formula to expand the determinant in t... | {
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Finding limit of a quotient with two square roots: $\lim_{t\to 0}\frac{\sqrt{1+t}-\sqrt{1-t}}t$ Find
$$
\lim_{t\to 0}\frac{\sqrt{1+t}-\sqrt{1-t}}{t}.
$$
I can't think of how to start this or what to do at all. Anything I try just doesn't change the function.
| HINT $\ $ Use the same method in your prior question, i.e. rationalize the numerator by multiplying both the numerator and denominator by the numerator's conjugate $\rm\:\sqrt{1+t}+\sqrt{1-t}\:.$ Then the numerator becomes $\rm\:(1+t)-(1-t) = 2\:t,\:$ which cancels with the denominator $\rm\:t\:,\:$ so $\rm\:\ldots$
... | {
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Calculating points on a plane In the example picture below, I know the points $A$, $B$, $C$ & $D$. How would I go about calculating $x$, $y$, $z$ & $w$ and $O$, but as points on the actual plane itself (e.g. treating $D$ as $(0, 0)$, $A$ as $(0, 1)$, $C$ as $(1, 0)$ and $B$ as $(1, 1)$.
Ultimately I need to be able to... | This should be done in terms of plane projective geometry. This means you have to introduce homogeneous coordinates. The given points $A=(a_1,a_2)$, $\ldots$, and $D=(d_1,d_2)$ have "old" homogeneous coordinates $(a_1,a_2, 1)$, $\ldots$, and $(d_1,d_2,1)$ and should get "new" homogeneous coordinates $\alpha(0,1,1)$, $... | {
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Fibonacci divisibilty properties $ F_n\mid F_{kn},\,$ $\, \gcd(F_n,F_m) = F_{\gcd(n,m)}$ Can any one give a generalization of the following properties in a single proof? I have checked the results, which I have given below by trial and error method. I am looking for a general proof, which will cover the all my results ... | The general proof of this is that the fibonacci numbers arise from the expression
$$F_n \sqrt{-5} = (\frac 12\sqrt{-5}+\frac 12\sqrt{-1})^n - (\frac 12\sqrt{-5}-\frac 12\sqrt{-1})^n$$
Since this is an example of the general $a^n-b^n$, which $a^m-b^m$ divides, if $m \mid n$, it follows that there is a unique number, ge... | {
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Method to reverse a Kronecker product Let's say I have two simple vectors: $[0, 1]$ and $[1, 0]$.
Their Kronecker product would be $[0, 0, 1, 0]$.
Let's say I have only the Kronecker product. How can I find the two initial vectors back?
If my two vectors are written as : $[a, b]$ and $[c, d]$, the (given) Kronecker pro... | I prefer to say the Kronecker of two vectors is reversible, but up to a scale factor. In fact, Niel de Beaudrap has given the right answer. Here I attempt to present it in a concise way.
Let $a\in\mathbb{R}^N$ and $b\in\mathbb{R}^M$ be two column vectors. The OP is: given $a\otimes b$, how to determine $a$ and $b$?
Not... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/60399",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "22",
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Fourier transform solution of three-dimensional wave equation One of the PDE books I'm studying says that the 3D wave equation can be solved via the Fourier transform, but doesn't give any details. I'd like to try to work the details out for myself, but I'm having trouble getting started - in particular, what variable... | You use the Fourier transform in all three space variables. The wave equation $\frac{\partial^2 u}{\partial t^2} = c^2 \left( \frac{\partial^2 u}{\partial x_1^2} + \frac{\partial^2 u}{\partial x_2^2} + \frac{\partial^2 u}{\partial x_3^2}\right)$ becomes
$ \frac{\partial^2 U}{\partial t^2} = - c^2 (p_1^2 + p_2^2 + p_3^... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/60461",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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What is the term for a factorial type operation, but with summation instead of products? (Pardon if this seems a bit beginner, this is my first post in math - trying to improve my knowledge while tackling Project Euler problems)
I'm aware of Sigma notation, but is there a function/name for e.g.
$$ 4 + 3 + 2 + 1 \longri... | The name for
$$
T_n= \sum_{k=1}^n k = 1+2+3+ \dotsb +(n-1)+n = \frac{n(n+1)}{2} = \frac{n^2+n}{2} = {n+1 \choose 2}
$$
is the $n$th triangular number. This picture demonstrates the reasoning for the name:
$$T_1=1\qquad T_2=3\qquad T_3=6\qquad T_4=10\qquad T_5=15\qquad T_6=21$$
$\hskip1.7in$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/60578",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "99",
"answer_count": 4,
"answer_id": 3
} |
There exists $C\neq0$ with $CA=BC$ iff $A$ and $B$ have a common eigenvalue Question:
Suppose $V$ and $W$ are finite dimensional vector spaces over $\mathbb{C}$. $A$ is a linear transformation on $V$, $B$ is a linear transformation on $W$. Then there exists a non-zero linear map $C:V\to W$ s.t. $CA=BC$ iff $A$ and $B$... | Here is a simple solution for the if condition.
Let $\lambda$ be the common eigenvalue. Let $u$ be a right eigenvector for $B$, that is
$$Bu= \lambda u$$
and $v$ be a left eigenvector for $A$, that is
$$v^TA= \lambda v^T$$
Then $C =uv^T$ is a non-zero $n \times n$ matrix which works:
$$CA = u v^T A = \lambda u v^T =... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/60641",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
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} |
Can someone explain consensus theorem for boolean algebra In boolean algebra, below is the consensus theorem
$$X⋅Y + X'⋅Z + Y⋅Z = X⋅Y + X'⋅Z$$
$$(X+Y)⋅(X'+Z)⋅(Y+Z) = (X+Y)⋅(X'+Z)$$
I don't really understand it? Can I simplify it to
$$X'⋅Z + Y⋅Z = X' \cdot Z$$
I don't suppose so. Anyways, why can $Y \cdot Z$ be removed... | Something like the following:
$X \cdot Y + X' \cdot Z + Y \cdot Z $ =
$X \cdot Y + X' \cdot Z + (X + X') \cdot Y \cdot Z $ =
$X \cdot Y + X \cdot Y \cdot Z + X' \cdot Z + X' \cdot Y \cdot Z$ =
$X \cdot (Y + Y \cdot Z) + X' \cdot (Z + Y \cdot Z)$ =
$X \cdot Y + X' \cdot Z$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/60713",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 4,
"answer_id": 0
} |
Orthogonal mapping $f$ which preserves angle between $x$ and $f(x)$ Let $f: \mathbf{R}^n \rightarrow \mathbf{R}^n$ be a linear orthogonal mapping such that
$\displaystyle\frac{\langle f(x), x\rangle}{\|fx\| \|x\|}=\cos \phi$, where $\phi \in [0, 2 \pi)$.
Are there such mapping besides $id$, $-id$ in case whe $n$ is od... | Your orthogonal transformation $f$ can be "complexified" to a unitary transformation $U$ on ${\mathbb C}^n$ such that $U(x + i y) = f(x) + i f(y)$ for $x, y \in {\mathbb R}^n$. Being normal, $U$ can be diagonalized, and its eigenvalues have absolute value 1.
The only possible real eigenvalues are $\pm 1$, in which ca... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/60756",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
$x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$ show that $x=-c/b$ when $a=0$ OK, this one has me stumped. Given that the solution for $ax^2+bx+c =0$ $$x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}\qquad(*)$$
How would you show using $(*)$ that $x=-c/b$ when $a=0$
(Please dont use $a=0$ hence, $bx+c=0$.)
| One of the two solutions approaches $-\frac{c}{b}$ in the limit as $a\rightarrow 0$. Assuming $b$ is positive, apply L'Hospital's rule to $$x=\frac{-b+\sqrt{b^2-4ac}}{2a}$$ If $b$ is negative, work with the other solution.
(And as $a\rightarrow 0$, the second solution approaches $\pm\infty$.)
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/60792",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 3
} |
One last question on the concept of limits I read through this post on the notion of limits Approaching to zero, but not equal to zero, then why do the points get overlapped?
But there's one last question I have. [Trust me, it'll be my last ever question on limits!]
It clearly says as you get closer and closer to a ce... | An important point to consider is that you cannot actually "plug in" $2$ and get $4$.
When $x = 2$, you will have a division by zero, since $x - 2 = 0$.
When calculating $\frac{x^2 - 4}{x - 2}$ for $x = 2$ using a computer, you may get $4$ due to rounding errors when you have set $x$ to a value very close to 2. This is... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/60966",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 5,
"answer_id": 3
} |
Nested Square Roots $5^0+\sqrt{5^1+\sqrt{5^2+\sqrt{5^4+\sqrt\dots}}}$ How would one go about computing the value of $X$, where
$X=5^0+ \sqrt{5^1+\sqrt{5^2+\sqrt{5^4+\sqrt{5^8+\sqrt{5^{16}+\sqrt{5^{32}+\dots}}}}}}$
I have tried the standard way of squaring then trying some trick, but nothing is working. I have also look... | The trick is to pull out a $\sqrt{5}$ factor from the second term:
$$
\frac{\sqrt{5^1+ \sqrt{5^2 + \sqrt{5^4 + \sqrt{5^8 + \cdots}}}}}{\sqrt{5}}
= \sqrt{1 + \sqrt{1 + \sqrt{1 + \sqrt{1 + \cdots}}}},
$$
which I call $Y$ for convenience. To see why this is true, observe that
$$ \begin{align*} \frac{\sqrt{5^1+x}}{\sqrt... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/61012",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "23",
"answer_count": 1,
"answer_id": 0
} |
Canonical to Parametric, Ellipse Equation I've done some algebra tricks in this derivation and I'm not sure if it's okay to do those things.
$$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$$
$$\frac{x^2}{a^2} + \frac{y^2}{b^2} = \cos^2\theta + \sin^2\theta$$
Can I really do this next step?
$$\frac{x^2}{a^2} = \cos^2\theta\qua... | The idea behind your argument is absolutely fine. Any two non-negative numbers $u$ and $v$ such that $u+v=1$ can be expressed as $u=\cos^2\theta$, $v=\sin^2\theta$ for some $\theta$. This is so obvious that it probably does not require proof. Set $u=\cos^2\theta$. Then $v=1-\cos^2\theta=\sin^2\theta$.
The second di... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/61071",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 0
} |
Series used in proof of irrationality of $ e^y $ for every rational $ y $ Following from the book "An Introduction to the Theory of Numbers" - Hardy & Wright I am having trouble with this proof. The book uses a familiar proof for the irrationality of e and continues into some generalizations that lose me.
In the follo... | For your first question, this follows from the binomial theorem
$$x^n(1-x)^n=x^n\sum_{m=0}^{n}{n\choose m}(-1)^mx^m=\sum_{m=0}^{n}{n\choose m}(-1)^{m}x^{m+n}=\sum_{m=n}^{2n}{n\choose m-n}(-1)^{m-n}x^m$$
where the last equality is from reindexing the sum. Then let $c_m={n\choose m-n}(-1)^m$, which is notably an integer.... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/61116",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
Proof for $\max (a_i + b_i) \leq \max a_i + \max b_i$ for $i=1..n$ I know this question is almost trivial because the truth of this statement is completely intuitive, but I'm looking for a nice and as formal as possible proof for $$\max (a_i + b_i) \leq \max a_i + \max b_i$$ with $i=1,\ldots,n$
Thanks in advance,
Fede... | For any choice of $j$ in $1,2,\ldots,n$, we have that
$$a_j\leq\max\{a_i\}\quad\text{and}\quad b_j\leq\max\{b_i\}$$
by the very definition of "$\max$". Then, by additivity of inequalities, we have that for each $j$,
$$a_j+b_j\leq\max\{a_i\}+\max\{b_i\}.$$
But because this is true for all $j$, it is true in particular ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/61241",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
standard symbol for "to prove" or "need to show" in proofs? Is there a standard symbol used as shorthand for "to prove" or "need to show" in a proof? I've seen "N.T.S." but was wondering if there is anything more abstract — not bound to English.
| I routinely use WTS for "Want to Show" - and most teachers and professors that I have come across immediately understood what it meant. I do not know if this is because they were already familiar with it, or if it was obvious to them. But I still use it all the time.
I got this from a few grad students at my undergrad,... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/61287",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 3
} |
Does the Schur complement preserve the partial order? Let
$$\begin{bmatrix}
A_{1} &B_1 \\ B_1' &C_1
\end{bmatrix} \quad \text{and} \quad \begin{bmatrix}
A_2 &B_2 \\ B_2' &C_2
\end{bmatrix}$$
be symmetric positive definite and conformably partitioned matrices. If
$$\begin{bmatrix}
A_{1} &B_1 \\ B_1' &C_1
\end{bma... | For a general block matrix $X=\begin{pmatrix}A&B\\C&D\end{pmatrix}$, the Schur complement $S$ to the block $D$ satisfies
$$
\begin{pmatrix}A&B\\C&D\end{pmatrix}
=\begin{pmatrix}I&BD^{-1}\\&I\end{pmatrix}
\begin{pmatrix}S\\&D\end{pmatrix}
\begin{pmatrix}I\\D^{-1}C&I\end{pmatrix}.
$$
So, when $X$ is Hermitian,
$$
\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/61417",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 2
} |
Motivation of the Gaussian Integral I read on Wikipedia that Laplace was the first to evaluate
$$\int\nolimits_{-\infty}^\infty e^{-x^2} \, \mathrm dx$$
Does anybody know what he was doing that lead him to that integral? Even better, can someone pose a natural problem that would lead to this integral?
Edit: Many of the... | The integral you gave, when taken as a definite integral :
$\int^{x_2}_{x_1} e^{-x^2} dx$
When scaled by $\frac {1}{\pi^{0.5}}$
is/describes the univariate probability density of a normally-distributed trandom variable $X$ with mean=0 and standard deviation ${\frac {1}{2^{0.5}}}$, i.e. This means that the numerical va... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/61480",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "16",
"answer_count": 6,
"answer_id": 4
} |
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