Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
How can I find two independent solution for this ODE? Please help me find two independent solutions for $$3x(x+1)y''+2(x+1)y'+4y=0$$ Thanks from a new beginner into ODE's.
| Note that both $x=0, x=1$ are regular singular points( check them!). As @Edgar assumed, let $y=\sum_1^{\infty}a_nx^n$ and by writing the equation like $$x^2y''+\frac{2}{3}xy'+\frac{4x}{3(x+1)}y=0$$ we get: $$p(x)=\frac{2}{3},\; q(x)=\frac{4x}{3(x+1)}$$ and then $p(0)=\frac{2}{3},\; q(0)=0$. Now I suggest you to set the... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/269808",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Is $X_n = 1+\frac12+\frac{1}{2^2}+\cdots+\frac{1}{2^n}; \forall n \ge 0$ bounded? Is $X_n = 1+\frac12+\frac{1}{2^2}+\cdots+\frac{1}{2^n}; \forall n \ge 0$ bounded?
I have to find an upper bound for $X_n$ and i cant figure it out, a lower bound can be 0 or 1 but does it have an upper bound?
| $$1+\frac12+\frac{1}{2^2}+\cdots+\frac{1}{2^n}=1+\frac{1}{2} \cdot \frac{\left(\frac{1}{2}\right)^{n}-1}{\frac{1}{2}-1}.$$ Now it is obviously, that it is bounded.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/269851",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Reference for an integral formula Good morning,
I'm reading a paper of W. Stoll in which the author uses some implicit facts (i.e. he states them without proofs and references) in measure theory. So I would like to ask the following question:
Let $G$ be a bounded domain in $\mathbb{R}^n$ and $S^{n-1}$ the unit sphere ... | This formula seems to be false. Consider the case of the unit disk in $\mathbb{R}^2$, $D^2$. This is obviously bounded.
$L^1(D^2 \cap L(a)) = 2$ for any $a$ in $S^1$, as the radius of $D^2$ is 1, and the intersection of the line through the origin that goes through $a$ and $D^2$ has length 2. The integral on the left ... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Abstract Algebra - Monoids I'm trying to find the identity of a monoid but all the examples I can find are not as complex as this one.
(source: gyazo.com)
| It's straightforward to show that $\otimes$ is an associative binary operation, and as others have pointed out, the identity of the monoid is $(1, 0, 1)$. However, $(\mathbb{R}^3, \otimes)$ is not a group, since for example $(0, 0, 0)$ has no inverse element.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/270019",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
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Cantor's intersection theorem and Baire Category Theorem From an old post in math stackexchange, I read a comment which goes as follows " I like to think of Baire Category Theorem as spiced up version of Cantor's Intersection Theorem". My question -----is it possible to derive the latter one using the former?
| Do you have a copy of Rudin's Principles of Mathematical Analysis? If you do, then problems 3.21 and 3.22 outline how this is done. Quoting here:
3.21: Prove: If $(E_n)$ is a sequence of closed and bounded sets in a complete metric space $X$, if $E_n \supset E_{n+1}$, and if $\lim_{n\to\infty}\text{diam}~E_n=0$, then ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/270084",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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How to find normal vector of line given point normal passes through Given a line L in three-dimensional space and a point P, how can we find the normal vector of L under the constraint that the normal passes through P?
| Let the line and point have position vectors $\vec r=\vec a+\lambda \vec b$ ($\lambda$ is real) and $\vec p$ respectively. Set $(\vec r-\vec p).\vec b=0$ and solve for $\lambda$ to obtain $\lambda_0$. The normal vector is simply $\vec a+\lambda_0 \vec b-\vec p$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/270151",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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$ \displaystyle\lim_{n\to\infty}\frac{1}{\sqrt{n^3+1}}+\frac{2}{\sqrt{n^3+2}}+\cdots+\frac{n}{\sqrt{n^3+n}}$ $$ \ X_n=\frac{1}{\sqrt{n^3+1}}+\frac{2}{\sqrt{n^3+2}}+\cdots+\frac{n}{\sqrt{n^3+n}}$$ Find $\displaystyle\lim_{n\to\infty} X_n$ using the squeeze theorem
I tried this approach:
$$
\frac{1}{\sqrt{n^3+1}}\le\frac... | Hint: use $\frac{i}{\sqrt{n^3+n}} \le \frac{i}{\sqrt{n^3+i}} \le \frac{i}{\sqrt{n^3+1}}$. I even think the Squeeze theorem can be avoided.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/270216",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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condition on $\epsilon$ to make $f$ injective
from the condition $g$ is uniformly continuous, $x$ is also U.continuous and one-one, but we dont know about $g$ is one-one or not, so $\epsilon=0$ will work?may be I am vague. Thank you.
| We have
$$
f'(x)=1+\varepsilon g'(x) \quad \forall\ x \in \mathbb{R}.
$$
For $\varepsilon \ge 0$ we have
$$
1-\varepsilon M \le f'(x)\le 1+\varepsilon M \quad \forall\ x \in \mathbb{R}.
$$
If we choose
$$
\varepsilon \in [0,1/M),
$$
then $f'>0$, i.e. $f$ is strictly increasing and therefore one-to-one.
For $\varepsilo... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/270275",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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If both roots of the Quadratic Equation are similar then prove that If both roots of the equation $(a-b)x^2+(b-c)x+(c-a)=0$ are equal, prove that $2a=b+c$.
Things should be known:
*
*Roots of a Quadratic Equations can be identified by:
The roots can be figured out by:
$$\frac{-b \pm \sqrt{d}}{2a},$$
whe... | As the two roots are equal the discriminant must be equal to $0$.
$$(b-c)^2-4(a-b)(c-a)=(a-b+c-a)^2-4(a-b)(c-a)=\{a-b-(c-a)\}^2=(2a-b-c)^2=0 \iff 2a-b-c=0$$
Alternatively, solving for $x,$ we get $$x=\frac{-(b-c)\pm\sqrt{(b-c)^2-4(a-b)(c-a)}}{2(a-b)}=\frac{c-b\pm(2a-b-c)}{2(a-b)}=\frac{c-a}{a-b}, 1$$ as $a-b\ne 0$ as ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/270344",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
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Topology - Open and closed sets in two different metric spaces I am currently working through some topology problems, and I would like to confirm my results just for some peace of mind!
Statement: Let $(X, d_x)$ be a metric space, and $Y\subset X$ a non-empty subset. Define a metric $d_y$ on $Y$ by restriction. Then, e... | This I found FALSE, example: Let $X =\mathbb{R}, Y = [0,1]$ and $A = [0, 1/2)$. $A$ is open in $Y$, howevever, $A$ is not open in $X$.
Q 3: If $A$ is closed in $X$, then $A$ is closed in $Y$. | {
"language": "en",
"url": "https://math.stackexchange.com/questions/270424",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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"answer_id": 1
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Tensor Components In Barrett Oneill's Semi-Riemann Geometry there is a definition of tensor component:
Let $\xi=(x^1,\dots ,x^n)$ be a coordinate system on $\upsilon\subset
M$. If $A \in \mathscr I^r_s (M)$ the components of $A$ relative to
$\xi$ are the real-valued functions $A _j^i, \dots j^i =A(dx^i_1,\dots
> ,d... | The is a very nice intuitive explanation of this in Penrose's Road to Reality, ch 14. As a quick summary, any vector field can be thought of as a directional derivative operator on scalar valued functions, i.e for every scalar valued smooth function $f$ and vector field $X$, define the scalar field
$$X(f) \triangleq p ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/270481",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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Integration of $x^3 \tan^{-1}(x)$ by parts I'm having problem with this question. How would one integrate $$\int x^3\tan^{-1}x\,dx\text{ ?}$$
After trying too much I got stuck at this point. How would one integrate $$\int \frac{x^4}{1+x^2}\,dx\text{ ?}$$
| You've done the hardest part. Now, the problem isn't so much about "calculus"; you simply need to recall what you've learned in algebra:
$(1)$ Divide the numerator of the integrand: $\,{x^4}\,$ by its denominator, $\,{1+x^2}\,$ using *polynomial long division *, (linked to serve as a reference).
This will give you:
$$\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/270550",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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a ideal of the ring of regular functions of an affine variety. We assume that $\Bbb k$ is an algebraically closed field.
Let $X \subset \Bbb A^n$ be an affine $\Bbb k$-variety , let's consider $ \mathfrak A_X \subset \Bbb k[t_1,...t_n]$ as the ideal of polynomials that vanish on $X$. Given a closed subset $Y\subset X$... | The Nullstellensatz is unnecessary here. If every function vanishes on a set Y (including all constant functions) then the set Y must obviously be empty; else the function f(x)=1 would be non-vanishing at some point x. Conversely if the set Y is empty then every function attains 0 at every point of Y.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Well-ordering the set of all finite sequences. Let $(A,<)$ be well-ordered set, using <, how can one define well-order on set of finite sequences?
(I thought using lexicographic order)
Thank you!
| The lexicographic order is fine, but you need to make a point where one sequence extends another -- there the definition of the lexicographic order may break down. In this case you may want to require that the shorter sequence comes before the longer sequence.
Generally speaking, if $\alpha$ and $\beta$ are two well-or... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/270640",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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What's the difference between expected values in binomial distributions and hypergeometric distributions? The formula for the expected value in a binomial distribution is:
$$E(X) = nP(s)$$
where $n$ is the number of trials and $P(s)$ is the probability of success.
The formula for the expected value in a hypergeometric ... | For either one, let $X_i=1$ if there is a success on the $i$-th trial, and $X_i=0$ otherwise. Then
$$X=X_1+X_2+\cdots+X_n,$$
and therefore by the linearity of expectation
$$E(X)=E(X_1)+E(X_2)+\cdots +E(X_n)=nE(X_1).
\tag{1}$$
Note that linearity of expectation does not require independence.
In the hypergeometric case,... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/270712",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Find intersection of two 3D lines I have two lines $(5,5,4) (10,10,6)$ and $(5,5,5) (10,10,3)$ with same $x$, $y$ and difference in $z$ values.
Please some body tell me how can I find the intersection of these lines.
EDIT: By using the answer given by coffemath I would able to find the intersection point for the above ... | The direction numbers $(a,b,c)$ for a line in space may be obtained from two points on the line by subtracting corresponding coordinates. Note that $(a,b,c)$ may be rescaled by multiplying through by any nonzero constant.
The first line has direction numbers $(5,5,2)$ while the second line has direction numbers $(5,5,-... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/270767",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "20",
"answer_count": 3,
"answer_id": 2
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Involutions of a torus $T^n$. Let $T^n$ be a complex torus of dimension $n$ and $x \in T^n$. We have a canonical involution $-id_{(T^n,x)}$ on the torus $T^n$. I want to know for which $y \in T^n$, we have $-id_{(T^n,x)}=-id_{(T^n,y)}$ as involutions of $T^n$.
My guess is, such $y$ must be a 2-torsion point of $(T^n,x... | Yes, you are right: here is a proof (I have taken the liberty of slightly modifying your notations).
Let $X=\mathbb C^n/\Lambda$ be the complex torus obtained by dividing out $\mathbb C^n$ by the lattice $\Lambda\subset \mathbb C^n$ ($\Lambda \cong \mathbb Z^{2n}$). This torus is an abelian Lie group, and this gives... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Intermediate Value Theorem and Continuity of derivative. Suppose that a function $f(x)$ is differentiable $\forall x \in [a,b]$. Prove that $f'(x)$ takes on every value between $f'(a)$ and $f'(b)$.
If the above question is a misprint and wants to say "prove that $f(x)$ takes on every value between $f(a)$ and $f(b)$", t... | This is not a misprint. You can indeed prove that $f'$ takes every value between $f'(a)$ and $f'(b)$. You cannot, however, assume that $f'$ is continuous. A standard example is $f(x) = x^2 \sin(1/x)$ when $x \ne 0$, and $0$ otherwise. This function is differentiable at $0$ but the derivative isn't continuous at it.
To ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Solve equation $\tfrac 1x (e^x-1) = \alpha$ I have the equation $\tfrac 1x (e^x-1) = \alpha$ for an positive $\alpha \in \mathbb{R}^+$ which I want to solve for $x\in \mathbb R$ (most of all I am interested in the solution $x > 0$ for $\alpha > 1$). How can I do this?
My attempt
I defined $\phi(x) = \tfrac 1x (e^x-1)$ ... | I just want to complete Hans Engler's answer. He already showed
$$x = -\frac 1\alpha -W\left(-\frac 1\alpha e^{-\tfrac 1\alpha}\right)$$
$\alpha > 0$ implies $-\tfrac 1\alpha \in \mathbb{R}^{-}$ and thus $-\tfrac 1\alpha e^{-\tfrac 1\alpha} \in \left[-\tfrac 1e,0\right)$ (The function $z\mapsto ze^z$ maps $\mathbb{R}^-... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/270961",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Finding the value of $a$ which minimizes the absolute maximum of $f(x)$ I know that this is an elementary problem in calculus and so it has a routine way of proof. I faced it and brought it up here just because it is one of R.A.Silverman's interesting problem. Let me learn your approaches like an student. Thanks.
Wha... | In my opinion, a calculus student would likely start by looking at the graph of this function for a few values of $a$. Then, they would notice that the absolute maxima occur on the endpoints or at $x=0$. So, looking at $f(-1)=f(1)=|1+a|$ and $f(0)=|a|$, I think it would be fairly easy for a calculus student to see that... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/271016",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Number of solutions for $x[1] + x[2] + \ldots + x[n] =k$ Omg this is driving me crazy seriously, it's a subproblem for a bigger problem, and i'm stuck on it.
Anyways i need the number of ways to pick $x[1]$ ammount of objects type $1$, $x[2]$ ammount of objects type $2$, $x[3]$ ammounts of objects type $3$ etc etc such... | This is a so-called stars-and-bars problem; the number that you want is
$$\binom{n+k-1}{n-1}=\binom{n+k-1}k\;.$$
The linked article has a reasonably good explanation of the reasoning behind the formula.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/271090",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Check If a point on a circle is left or right of a point What is the best way to determine if a point on a circle is to the left or to the right of another point on that same circle?
| If you mean in which direction you have to travel the shortest distance from $a$ to $b$ and assuming that the circle is centered at the origin then this is given by the sign of the determinant $\det(a\, b)$ where $a$ and $b$ are columns in a $2\times 2$ matrix. If this determinant is positive you travel in the counter... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/271149",
"timestamp": "2023-03-29T00:00:00",
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Find all the continuous functions such that $\int_{-1}^{1}f(x)x^ndx=0$. Find all the continuous functions on $[-1,1]$ such that $\int_{-1}^{1}f(x)x^ndx=0$ fof all the even integers $n$.
Clearly, if $f$ is an odd function, then it satisfies this condition. What else?
| Rewrite the integral as
$$\int_0^1 [f(x)+f(-x)]x^n dx = 0,$$
which holds for all even $n$, and do a change of variables $y=x^2$ so that for all $m$, even or odd, we have
$$ \int_0^1 \left[\frac{f(\sqrt{y})+f(-\sqrt{y})}{2 \sqrt{y}}\right] y^{m} dy = 0. $$
By the Stone Weierstrass Theorem, polynomials are uniformly dens... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/271199",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 3,
"answer_id": 2
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Scaling of random variables Note: I will use $\mathbb{P}\{X \in dx\}$ to denote $f(x)dx$ where $f(x)$ is the pdf of $X$.
While doing some homework, I came across a fault in my intuition. I was scaling a standard normally distributed random variable $Z$. Edit: I was missing the infinitesimals $dx$ and $dx/c$, so everyth... | Setting aside rigour and following your intuition about infinitesimal probabilities of finding a random variable in an infinitesimal interval, I note that the left-hand sides of your first two equations are infinitesimal whereas the right-hand sides are finite. So these are clearly wrong, even loosely interpreted. They... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/271256",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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Can this function be expressed in terms of other well-known functions? Consider the function $$f(a) = \int^1_0 \frac {t-1}{t^a-1}dt$$
Can this function be expressed in terms of 'well-known' functions for integer values of $a$? I know that it can be relatively simply evaluated for specific values of $a$ as long as they ... | Mathematica says that
$$f(a)=\frac{1}{a}\Big(\psi\left(\tfrac{2}{a}\right)-\psi\left(\tfrac{1}{a}\right)\Big)$$
where
$$\psi(z)=\frac{\Gamma'(z)}{\Gamma(z)}$$
is the digamma function.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/271323",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 1
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Mean time until adsorption for a well-mixed bounded random walk that suddenly allows for adsorption I have a random walk on some interval $[0, N]$ with probability $p$ of taking a $+1$ step, probability $(1-p)$ of taking a $-1$ step, and where we have that $p>(1-p)$. Initially the boundaries are reflecting, i.e. if th... | This is not (yet) an answer: I just try to formalise the question asked by the OP
The stationary distribution before the adsorption process sets in can be obtained from the balance equations $p \pi(k) = (1-p)\pi(k+1)$. It is given by
$$\pi(k) = \alpha^k \pi(0)= \frac{(1-\alpha) \alpha^k}{1-\alpha^{1+N}} ,\qquad \alpha ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/271378",
"timestamp": "2023-03-29T00:00:00",
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Calculate value of expression $(\sin^6 x+\cos^6 x)/(\sin^4 x+\cos^4 x)$
Calculate the value of expresion:
$$
E(x)=\frac{\sin^6 x+\cos^6 x}{\sin^4 x+\cos^4 x}
$$
for $\tan(x) = 2$.
Here is the solution but I don't know why $\sin^6 x + \cos^6 x = ( \cos^6 x(\tan^6 x + 1) )$, see this:
Can you explain to me why t... | You can factor the term $\cos^6(x)$ from $\sin^6(x)+\cos^6(x)$ in the numerator to find: $$\cos^6(x)\left(\frac{\sin^6(x)}{\cos^6(x)}+1\right)=\cos^6(x)\left(\tan^6(x)+1\right)$$ and factor $\cos^4(x)$ from the denominator to find: $$\cos^4(x)\left(\frac{\sin^4(x)}{\cos^4(x)}+1\right)=\cos^4(x)\left(\tan^4(x)+1\right)$... | {
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"timestamp": "2023-03-29T00:00:00",
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Ring of formal power series finitely generated as algebra? I'm asked if the ring of formal power series is finitely generated as a $K$-algebra. Intuition says no, but I don't know where to start. Any hint or suggestion?
| Let $A$ be a non-trivial commutative ring. Then $A[[x]]$ is not finitely generated as a $A$-algebra.
Indeed, observe that $A$ must have a maximal ideal $\mathfrak{m}$, so we have a field $k = A / \mathfrak{m}$, and if $k[[x]]$ is not finitely-generated as a $k$-algebra, then $A[[x]]$ cannot be finitely-generated as an ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/271536",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
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How to read $A=[0,1]\times[a,5]$ I have this problem: consider the two sets $A$ and $B$
$$A=[0,1]\times [a,5]$$ and $$B=\{(x,y):x^2+y^2<1\}$$
What are the values of $a$ that guarantee the existence of a hyperplane that separates $A$ from $B$.
Given a chosen value of $a$, find one of those hyperplanes.
My main problem i... | The $\times$ stands for cartesian product, i.e. $X\times Y=\{(x,y)\mid x\in X, y\in Y\}$.
Whether ordered pairs $(x,y)$ are considered a basic notion or are themselves defined (e.g. as Kurtowsky pairs) usually does not matter. See alo here.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/271670",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
} |
$10$ distinct integers with sum of any $9$ a perfect square Do there exist $10$ distinct integers such that the sum of any $9$ of them is a perfect square?
| I think the answer is yes. Here is a simple idea:
Consider the system of equations
$$S-x_i= y_i^2, 1 \leq i \leq 10\,,$$
where $S=x_1+..+x_n$.
Let $A$ be the coefficients matrix of this system. Then all the entries of $I+A$
are $1$, thus $\operatorname{rank}(I+A)=1$. This shows that $\lambda=0$ is an eigenvalue of $I... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/271727",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
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Proof of Irrationality of e using Diophantine Equations I was trying to prove that e is irrational without using the typical series expansion, so starting off $e = a/b $ Take the natural log so $1 = \ln(a/b)$ Then $1 = \ln(a)-\ln(b)$ So unless I did something horribly wrong showing the irrationality of $e$ is the sa... | One situation in which the existence of a solution
to a Diophantine equation implies an irrationality result is this:
If, for a positive integer $n$, there are
positive integers $x$ and $y$ satisfying
$x^2 - n y^2 = 1$,
then $\sqrt n$ is irrational.
I find this amusing, since this proof
is more complicated than any of... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/271792",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "14",
"answer_count": 3,
"answer_id": 1
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A problem on self adjoint matrix and its eigenvalues Let $S = \{\lambda_1, \ldots , \lambda_n\}$ be an ordered set of $n$ real numbers, not all equal, but not all necessarily distinct. Pick out the true statements:
a. There exists an $n × n$ matrix with complex entries, which is not selfadjoint, whose set of eigenv... | The general idea is to start with a diagonal matrix $[\Lambda]_{kj} = \begin{cases} 0, & j \neq k \\ \lambda_j, & j=k\end{cases}$ and then modify this to satisfy the conditions required.
1) Just set the upper triangular parts of $\Lambda$ to $i$. Choose $[A]_{kj} = \begin{cases} 0, & j>k \\ \lambda_j, & j=k \\ i, & j<k... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/271985",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
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Inequality for dense subset implies inequality for whole set? (PDE) Suppose I have an inequality that holds for all $f \in C^\infty(\Omega)$. Then since $C^\infty(\Omega)$ is dense in, say, $H^1(\Omega)$ under the latter norm, does the inequality hold for all $f \in H^1(\Omega)$ too?
(Suppose the inequality involves no... | Let $X$ be a topological space. Let $F,G$ be continuous maps from $X$ to $\mathbb{R}$. Let $Y\subset X$ be a dense subspace.
Then
$$ F|_Y \leq G|_Y \iff F \leq G $$
The key is continuity. (Actually, semi-continuity of the appropriate direction is enough.) Continuity guarantees for $x\in X\setminus Y$ and $x_\alpha \i... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/272063",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Prove that $x = 2$ is the unique solution to $3^x + 4^x = 5^x$ where $x \in \mathbb{R}$ Yesterday, my uncle asked me this question:
Prove that $x = 2$ is the unique solution to $3^x + 4^x = 5^x$ where $x \in \mathbb{R}$.
How can we do this? Note that this is not a diophantine equation since $x \in \mathbb{R}$ if you ... | $$f(x) = \left(\dfrac{3}{5}\right)^x + \left(\dfrac{4}{5}\right)^x -1$$
$$f^ \prime(x) < 0\;\forall x \in \mathbb R\tag{1}$$
$f(2) =0$. If there are two zeros of $f(x)$, then by Rolle's theorem $f^\prime(x)$ will have a zero which is a contradiction to $(1)$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/272114",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "12",
"answer_count": 4,
"answer_id": 2
} |
Distance is independent of coordinates I am asked to show $d(x,y) = ((x_2 - x_1)^2 + (y_2 -y_1)^2)^{1/2}$ does not depend on the choice of coordinates. My try is:
$V$ has basis $B = b_1 , b_2$ and $B' = b_1' , b_2'$ and $T = [[a c], [b d]]$ is the coordinate transformation matrix $Tv_{B'} = v_B$ and $x_{B'} = x_1 b'_1 ... | I would try a little bit more abstract approach. Sometimes a little bit of abstraction helps.
First, distance can be computed in terms of the dot product. So, if you have points with Cartesian coordinates $X,Y$, the distance between them is
$$
d(X,Y) = \sqrt{(X-Y)^t(X-Y)} \ .
$$
Now, if you make an orthogonal change of... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/272246",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
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A measure of non-differentiability Consider $f(x) = x^2$ and $g(x) = |x|$. Both graphs have an upward open graph, but $g(x) = |x|$ is "sharper". Is there a way to measure this sharpness?
| This may be somewhat above your pay grade, but a "measure" of a discontinuity of a function at a point may be seen in a Fourier transform of that function. For example consider the function
$$f(x) = \exp{(-|x|)} $$
which is proportional to a Lorentzian function:
$$\hat{f}(w) = \frac{1}{1+w^2} $$
(I am ignoring constan... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/272291",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
the existence of duality of closure and interior? Let the closure and interior of set $A$ be $\bar A$ and $A^o$ respectively.
In some cases, the dual is relatively easy to find, e.g. the dual of equation $\overline{A \cup B} = \bar A \cup \bar B$ is $(A \cap B)^o= A^o\cap B^o$.
However, I can't find the dual of $f(\bar... | In the duality examples that you described as "relatively easy", the key was that you get the dual of an operation by applying "complement" to the inputs and outputs. For example, writing $\sim$ for complement, we have $A^o=\sim(\overline{\sim A})$, i.e., we get the interior of $A$ by taking the complement of $A$, the... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/272359",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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How do I prove by induction that, for $n≥1, \sum_{r=1}^n \frac{1}{r(r+1)}=\frac{n}{n+1}$? Hi can you help me solve this:
I have proved that $p(1)$ is true and am now assuming that $p(k)$ is true. I just don't know how to show $p(k+1)$ for both sides?
| $$\sum_{r=1}^{n}\frac{1}{r(r+1)}=\frac{n}{n+1}$$
for $n=1$ we have $\frac{1}{1(1+1)}=\frac{1}{1+1}$ suppose that
$$\sum_{r=1}^{k}\frac{1}{r(r+1)}=\frac{k}{k+1}$$ then
$$\sum_{r=1}^{k+1}\frac{1}{r(r+1)}=\sum_{r=1}^{k}\frac{1}{r(r+1)}+\frac{1}{(k+1)(k+2)}=$$
$$=\frac{k}{k+1}+\frac{1}{(k+1)(k+2)}=\frac{k(k+2)+1}{(k+1)(k+2... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/272429",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 2
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How do I calculate the derivative of $x|x|$? I know that $$f(x)=x\cdot|x|$$ have no derivative at $$x=0$$ but how do I calculate it's derivative for the rest of the points?
When I calculate for $$x>0$$ I get that $$f'(x) = 2x $$
but for $$ x < 0 $$ I can't seem to find a way to solve the limit. As this is homework ple... | When $x<0$ replace $|x|$ by $-x$ (since that is what it is equal to) in the formula for the function and proceed.
Please note as well that the function $f(x)=x\cdot |x|$ does have a derivative at $0$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/272488",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 6,
"answer_id": 2
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Multiples of an irrational number forming a dense subset Say you picked your favorite irrational number $q$ and looking at $S = \{nq: n\in \mathbb{Z} \}$ in $\mathbb{R}$, you chopped off everything but the decimal of $nq$, leaving you with a number in $[0,1]$. Is this new set dense in $[0,1]$? If so, why? (Basically l... | A bit of a late comer to this question, but here's another proof:
Lemma: The set of points $\{x\}$ where $x\in S$, (here $\{\cdot\}$ denotes the fractional part function), has $0$ as a limit point.
Proof: Given $x\in S$, Select $n$ so that $\frac{1}{n+1}\lt\{x\}\lt\frac{1}{n}$. We'll show that by selecting an appropria... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/272545",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "44",
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Modules over local ring and completion I'm stuck again at a commutative algebra question. Would love some help with this completion business...
We have a local ring $R$ and $M$ is a $R$-module with unique assassin/associated prime the maximal ideal $m$ of $R$.
i) prove that $M$ is also naturally a module over the $m$-... | Proposition. Let $R$ be a noetherian ring and $M$ an $R$-module, $M\neq 0$. Then $\operatorname{Ass}(M)=\{\mathfrak m\}$ iff for every $x\in M$ there exists a positive integer $k$ such that $\mathfrak m^kx=0$.
Proof. "$\Rightarrow$" Let $x\in M$, $x\neq 0$. Then $\operatorname{Ann}(x)$ is an ideal of $R$. Let $\mathfra... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/272621",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
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Inequality involving closure and intersection Let the closure of a set $A$ be $\bar A$. On Page 62, Introduction to Boolean Algebras,Steven Givant,Paul Halmos(2000), an exercise goes like,
Show that $P \cap \bar Q \subseteq \overline{(P \cap Q)}$, whenever $P$ is open.
I felt muddled in face of this sort of exercises... | Let $x$ in $P\cap\bar Q$. Since $x$ is in $\bar Q$, there exists a sequence $(x_n)_n$ in $Q$ such that $x_n\to x$. Since $x$ is in $P$ and $P$ is open, $x_n$ is in $P$ for every $n$ large enough, say, $n\geqslant n_0$. Hence, for every $n\geqslant n_0$, $x_n$ is in $P\cap Q$. Thus, $x$ is in $\overline{P\cap Q}$ as lim... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/272697",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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"answer_id": 1
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Minimizing a multivariable function given restraint I want to minimize the following function:
$$J(x, y, z) = x^a + y^b + z^c$$
I know I can easily determine the minimum value of $J$ using partial derivative. But I have also the following condition:
$$ x + y + z = D$$
How can I approach now?
| This is an easy example of using Lagrange multiplier.
If you reformulate your constraint as $C(x,y,z) = x+y+z-D=0$, you can define
$L(x,y,z,\lambda) := J(x,y,z)-\lambda \cdot C(x,y,z)$
If you now take the condition $\nabla L=0$ as necessary for your minimum you will fulfill
$$\frac{\partial L}{\partial x}=0 \\
\frac{\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/272769",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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Solving a system of differential equations I would like to get some help by solving the following problem:
$$
p'_1= \frac 1 x p_1 - p_2 + x$$
$$ p'_2=\frac 1{x^2}p_1+\frac 2 x p_2 - x^2 $$
with initial conditions $$p_1(1)=p_2(1)=0, x \gt0 $$
EDIT:
If I use Wolframalpha, I get
Where $u$ and $v$ are obviously $p_1$ an... | One approach is to express $p_2=-p_1'+\frac1xp_1+x$ from the first equation and substitute into the second to get:
$$-p_1''+\frac1xp_1'-\frac1{x^2}p_1+1=p_2'=\frac1{x^2}p_1+\frac2x\left(-p_1'+\frac1xp_1+x\right)-x^2$$
$$p_1''-\frac3xp_1'+\frac4{x^2}p_1=x^2-1$$
Multiplying by $x^2$ we get $x^2p_1''-3xp_1'+4p_1=x^4-x^2$,... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/272820",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
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How to do this interesting integration?
$$\lim_{\Delta x\rightarrow0}\sum_{k=1}^{n-1}\int_{k+\Delta x}^{k+1-\Delta x}x^m dx$$
How to integrate the above integral?
Edit1:
$$\lim_{\Delta x\rightarrow0}\int_{2-\Delta x}^{2+\Delta x}x^m dx$$
Does this intergral give $\space\space\space\space$ $2^m\space\space$ as the o... | For the first question:
$$
\begin{align}
\lim_{\Delta x\to0}\,\left|\,\int_k^{k+1}x^m\,\mathrm{d}x-\int_{k+\Delta x}^{k+1-\Delta x}x^m\,\mathrm{d}x\,\right|
&=\lim_{\Delta x\to0}\,\left|\,\int_k^{k+\Delta x}x^m\,\mathrm{d}x+\int_{k+1-\Delta x}^{k+1}x^m\,\mathrm{d}x\,\right|\\
&\le\lim_{\Delta x\to0}2\Delta x(k+1)^m\\
&... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/272930",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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"answer_id": 1
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Fixed: Is this set empty? $ S = \{ x \in \mathbb{Z} \mid \sqrt{x} \in \mathbb{Q}, \sqrt{x} \notin \mathbb{Z}, x \notin \mathbb{P}$ } This question has been "fixed" to reflect the question that I intended to ask
Is this set empty? $ S = \{ x \in \mathbb{Z} \mid \sqrt{x} \in \mathbb{Q}, \sqrt{x} \notin \mathbb{Z}, x \not... | To answer according to the last edit: Yes.
Let $a\in\mathbb{Z}$ and consider the polynomial $x^{2}-a$.
By the rational root theorm if there is a rational root $\frac{r}{s}$
then $s|1$ hence $s=\pm1$ and the root is an integer. So $\sqrt{a}\in\mathbb{Q}\iff\sqrt{a}\in\mathbb{Z}$ .
Since you assumed that the root is not... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 2
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How to randomly construct a square full-ranked matrix with low determinant? How to randomly construct a square (1000*1000) full-ranked matrix with low determinant?
I have tried the following method, but it failed.
In MATLAB, I just use:
n=100;
A=randi([0 1], n, n);
while rank(A)~=n
A=randi([0 1], n, n);
end
The above c... | The determinant of $e^B$ is $e^{\textrm{tr}(B)}$ (wiki) and $e^B$ is always invertible, since $e^B e^{-B}=\textrm{Id}$. So, if you have a matrix $B$ with negative trace then $\det e^B$ is positive and smaller than $1$. Using this idea I wrote the following matlab script which generates matrices with "small" determinant... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/273061",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 0
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Plot of x^(1/3) has range of 0-inf in Mathematica and R Just doing a quick plot of the cuberoot of x, but both Mathematica 9 and R 2.15.32 are not plotting it in the negative space. However they both plot x cubed just fine:
Plot[{x^(1/3), x^3},
{x, -2, 2}, PlotRange -> {-2, 2}, AspectRatio -> Automatic]
http://w... | Really funny you'd mention that... my Calc professor talked about that last semester. ;)
Many software packages plot the principal root, rather than the real root. http://mathworld.wolfram.com/PrincipalRootofUnity.html
For example, $\sqrt[3]{3}$ has three values: W|A
Mathematica uses the roots in the upper-left quadran... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/273149",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
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Generating function for the divisor function Earlier today on MathWorld (see eq. 17) I ran across the following expression, which gives a generating function for the divisor function $\sigma_k(n)$: $$\sum_{n=1}^{\infty} \sigma_k (n) x^n = \sum_{n=1}^{\infty} \frac{n^k x^n}{1-x^n}. \tag{1}$$
(The divisor function $\sig... | Switching the order of summation, we have that
$$\sum_{n=1}^{\infty}\sigma_{k}(n)x^{n}=\sum_{n=1}^{\infty}x^{n}\sum_{d|n}d^{k}=\sum_{d=1}^{\infty}d^{k}\sum_{n:\ d|n}^{\infty}x^{n}.$$ From here, applying the formula for the geometric series, we find that the above equals $$\sum_{d=1}^{\infty}d^{k}\sum_{n=1}^{\infty}x^{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/273275",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
"answer_count": 1,
"answer_id": 0
} |
Linearization of Gross-Pitaevskii-Equation Consider a PDE of the form
$\partial_t \phi = A(\partial_\xi) \phi + c\partial_\xi \phi +N(\phi)$
where $N$ is some non-linearity defined via pointwise evaluation of $\phi$.
If you want to check for stability of travelling wave solutions of PDEs you linearize the PDE at som... | Well.... first you need to specify a traveling wave solution $Q$.
Then you just take, since $N(\phi) = -i \phi(1 - |\phi|^2)$,
$$ (\partial_\phi N)(\phi) = -i(1-|\phi|^2) + 2i\phi \bar{\phi} $$
by the product rule of differential calculus. Here note $|\phi|^2 = \phi \bar\phi$.
So simplifying and evaluating it at $Q$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/273365",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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For which p the series converge? $$\sum_{n=0}^{\infty}\left(\frac{1}{n!}\right)^{p}$$
Please verify answer below
| Comparison test
$$\lim_{n\rightarrow\infty}\left(\frac{n!}{(n+1)!}\right)^{p}=\frac{1}{(n+1)^{p}}=\begin{cases}
1 & \Leftrightarrow p=0\\
0 & \Leftrightarrow p\neq0
\end{cases}$$
The series have the same convergence as $\frac{1}{n}$, so for:
*
*$p>1$ converge
*for $p<1$ don't converge
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/273497",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 2,
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How to prove that $\lVert \Delta u \rVert_{L^2} + \lVert u \rVert_{L^2}$ and $\lVert u \rVert_{H^2}$ are equivalent norms? How to prove that $\lVert \Delta u \rVert_{L^2} + \lVert u \rVert_{L^2}$ and $\lVert u \rVert_{H^2}$ are equivalent norms on a bounded domain? I hear there is a way to do it by RRT but any other wa... | As user53153 wrote this is true for bounded smooth domains and can be directly obtained by the boundary regularity theory exposed in Evans.
BUT: consider the domain $\Omega=\{ (r \cos\phi,r \sin\phi); 0<r<1, 0<\phi<\omega\}$ for some $\pi<\omega<2\pi$. Then the function $u(r,\phi)=r^{\pi/\omega}\sin(\phi\pi/\omega)$ (i... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/273563",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 2,
"answer_id": 1
} |
Term for a group where every element is its own inverse? Several groups have the property that every element is its own inverse. For example, the numbers $0$ and $1$ and the XOR operator form a group of this sort, and more generally the set of all bitstrings of length $n$ and XOR form a group with this property.
These... | Another term is "group of exponent $2$".
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/274604",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 4,
"answer_id": 3
} |
Set of points reachable by the tip of a swinging sticks kinetic energy structure This is an interesting problem that I thought of myself but I'm racking my brain on it. I recently saw this kinetic energy knick knack in a scene in Iron Man 2:
http://www.youtube.com/watch?v=uBxUoxn46A0
And it got me thinking, it looks li... | One way to see the reach is to notice that if the configuration $\theta = (\theta_1,\theta_2)$ reaches some spot $x \in \mathbb{R}^2$, and $y$ is a spot obtained by rotating $x$ by $\alpha \in \mathbb{R}$, then the configuration $\theta+(\alpha, \alpha)$ will reach $y$. So, we only need to see what the minimum and maxi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/274663",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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Dual norm and distance Let $Z$ be a subspace of a normed linear space $X$ and $x\in X$ has distance $d=\inf\{||z-y||:z\in Z\}$ to $Z$.
I would like to find a function $f\in X^*$ that satifies
$||f||\le1$, $f(x)=d$ and $f(z)=0$
Is it correct that $||f||:=\sup\{|f(x)| :x\in X, ||x||\le 1\}$ because I cannot conclude from... | I'll list the ingredients and leave the cooking to you:
*
*The function $d:X\to [0,\infty)$ is sublinear in the sense used in the Hahn-Banach theorem
*There is a linear functional $\phi$ on the one-dimensional space $V=\{t x:t\in\mathbb R\}$ such that $\phi(x)=d(x)$ and $|\phi(y)|\le d(y)$ for all $y\in V$. If you... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/274792",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Ambiguous Curve: can you follow the bicycle? Let $\alpha:[0,1]\to \mathbb R^2$ be a smooth closed curve parameterized by the arc length. We will think of $\alpha$ like a back track of the wheel of a bicycle. If we suppose that the distance between the two wheels is $1$ then we can describe the front track by
$$\tau(t)... | After the which way did bicycle go book, there has been some systematic development of theory related to the bicycle problem. Much of that is either done or cited in papers by Tabachnikov and his coauthors, available online:
http://arxiv.org/find/all/1/all:+AND+bicycle+tracks/0/1/0/all/0/1
http://arxiv.org/abs/math/04... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/274849",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "32",
"answer_count": 5,
"answer_id": 2
} |
$e^{i\theta_n}\to e^{i\theta}\implies \theta_n\to\theta$ How to show $e^{i\theta_n}\to e^{i\theta}\implies \theta_n\to\theta$ for $-\pi<\theta_n,\theta<\pi.$ I'm completely stuck in it. Please help.
| Suppose $(\theta_n)$ does not converge to $\theta$, then there is an $\epsilon > 0$ and a subsequence $( \theta_{n_k} )$ such that $| \theta_{n_k} - \theta | \geq \epsilon $ for all $k$. $(\theta_{n_k})$ is bounded so it has a further subsequence $(\theta_{m_k})$ which converges to $\theta_0 \in [-\pi,\pi]$ (say) wi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/274907",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
} |
It's in my hands to have a surjective function Let $f$ be any function $A \to B$.
By definition $f$ is a surjective function if $\space \forall y \in B \space \exists \space x \in A \space( \space f(x)=y \space)$.
So, for any function I only have to ensure that there doesn't "remain" any element "alone" in the set $B$.... | There are intrinsic properties and extrinsic properties. Being surjective is an extrinsic property. If you are not given a particular codomain you cannot conclude whether or not a function is surjective.
Being injective, on the other hand, is an intrinsic property. It depends only on the function as a set of ordered p... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/274967",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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"answer_id": 1
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In every interval there is a rational and an irrational number. When the interval is between two rational numbers it is easy. But things get complicated when the interval is between two irrational numbers. I couldn't prove that.
| Supposing you mean an interval $(x,y)$ of length $y-x=l>0$ (it doesn't matter whether $l$ is rational or irrational), you can simply choose any integer $n>\frac1l$, and then the interval will contain a rational number of the form $\frac an$ with $a\in\mathbf Z$. Indeed if $a'$ is the largest integer such that $\frac{a'... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/275032",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 0
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Check convergence $\sum\limits_{n=1}^\infty\left(\sqrt{1+\frac{7}{n^{2}}}-\sqrt[3]{1-\frac{8}{n^{2}}+\frac{1}{n^{3}}}\right)$ Please help me to check convergence of $$\sum_{n=1}^{\infty}\left(\sqrt{1+\frac{7}{n^{2}}}-\sqrt[3]{1-\frac{8}{n^{2}}+\frac{1}{n^{3}}}\right)$$
| (Presumably with tools from Calc I...) Using the conjugate identities
$$
\sqrt{a}-1=\frac{a-1}{1+\sqrt{a}},\qquad1-\sqrt[3]{b}=\frac{1-b}{1+\sqrt[3]{b}+\sqrt[3]{b^2}},
$$
one gets
$$
\sqrt{1+\frac{7}{n^{2}}}-\sqrt[3]{1-\frac{8}{n^{2}}+\frac{1}{n^{3}}}=x_n+y_n,
$$
with
$$
x_n=\sqrt{1+a_n}-1=\frac{a_n}{1+\sqrt{1+a_n}},... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/275090",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
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If x,y,z are positive reals, then the minimum value of $x^2+8y^2+27z^2$ where $\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=1$ is what If $x,y, z$ are positive reals, then the minimum value of $x^2+8y^2+27z^2$ where $\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=1$ is what?
$108$ , $216$ , $405$ , $1048$
| As $x,y,z$ are +ve real, we can set $\frac 1x=\sin^2A,\frac1y+\frac1z=\cos^2A$
again, $\frac1{y\cos^2A}+\frac1{z\cos^2A}=1$
we can put $\frac1{y\cos^2A}=\cos^2B, \frac1{z\cos^2A}=\sin^2B\implies y=\cos^{-2}A\cos^{-2}B,z=\cos^{-2}A\sin^{-2}B$
So, $$x^2+8y^2+27z^2=\sin^{-4}A+\cos^{-4}A(8\cos^{-4}B+27\sin^{-4}B)$$
We nee... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/275153",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 6,
"answer_id": 1
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How can I to solve $ \cos x = 2x$? I would like to get an approx. solution to the equation: $ \cos x = 2x$, I don't need an exact solution just some approx. And I need a solution using elementary mathematics (without derivatives etc).
| Take a pocket calculator, start with $0$ and repeatedly type [cos], [$\div$], [2], [=]. This will more or less quickly converge to a value $x$ such that $\frac{\cos x}2=x$, just what you want.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/275198",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 0
} |
Spivak problem on orientations. (A comprehensive introduction to differential geometry) I have a problems doing exercise 16 of chapter 3 (p.98 in my edition) of Spivak's book.
The problem is very simple. Let $M$ be a manifold with boundary, and choose a point $p\in\delta(M)$. Now consider an element $v\in T_p M$ which ... | A change of coordinates between charts for the manifold with boundary $M$ has the form $x=(x_1, \cdots,x_n) \mapsto (\phi_1(x), \cdots,\phi_n(x))$, with $x_n, \phi_n(x)\geq0$
since $x_n,\phi_n(x)\in \mathbb H_n$.
The last line of the Jacobian $Jac_a(\phi)$ at a point $a\in \partial \mathbb H_n$ has the form $(0,\cdo... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/275259",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
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} |
Calculate $\lim_{x\to 0}\frac{\ln(\cos(2x))}{x\sin x}$ Problems with calculating
$$\lim_{x\rightarrow0}\frac{\ln(\cos(2x))}{x\sin x}$$
$$\lim_{x\rightarrow0}\frac{\ln(\cos(2x))}{x\sin x}=\lim_{x\rightarrow0}\frac{\ln(2\cos^{2}(x)-1)}{(2\cos^{2}(x)-1)}\cdot \left(\frac{\sin x}{x}\right)^{-1}\cdot\frac{(2\cos^{2}(x)-1)}... | The known limits you might wanna use are, for $x\to 0$
$$\frac{\log(1+x)}x\to 1$$
$$\frac{\sin x }x\to 1$$
With them, you get
$$\begin{align}\lim\limits_{x\to 0}\frac{\log(\cos 2x)}{x\sin x}&=\lim\limits_{x\to 0}\frac{\log(1-2\sin ^2 x)}{-2\sin ^2 x}\frac{-2\sin ^2 x}{x\sin x}\\&=-2\lim\limits_{x\to 0}\frac{\log(1-2\si... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/275308",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 6,
"answer_id": 2
} |
Ultrafilters and measurability Consider a compact metric space $X$, the sigma-algebra of the boreleans of $X$, a sequence of measurable maps $f_n: X \to\Bbb R$ and an ultrafilter $U$. Take, for each $x \in X$, the $U$-limit, say $f^*(x)$, of the sequence $(f_n(x))_{n \in\Bbb N}$.
(Under what conditions on $U$) Is $f^*... | Let me first get rid of a silly case that you probably didn't intend to include. If $U$ is a principal ultrafilter, generated by $\{k\}$, then $f^*$ is just $f_k$, so it's measurable.
Now for the non-silly cases, where $U$ isn't principal. Here's an example of a sequence of measurable (in fact low-level Borel) functio... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/275365",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
$\int_{\mathbb{R}^n} dx_1 \dots dx_n \exp(−\frac{1}{2}\sum_{i,j=1}^{n}x_iA_{ij}x_j)$? Let $A$ be a symmetric positive-definite $n\times n$ matrix and $b_i$ be some real numbers How can one evaluate the following integrals?
*
*$\int_{\mathbb{R}^n} dx_1 \dots dx_n \exp(−\frac{1}{2}\sum_{i,j=1}^{n}x_iA_{ij}x_j)$
*$\in... | Let's $x=(x_1,\ldots,x_n)\in\mathbb{R}^n$. We have
\begin{align}
\int_{\mathbb{R}^n}e^{-\frac{1}{2}\langle x,x \rangle_{A}} d x =
&
\int_{\mathbb{R}^n}e^{-\frac{1}{2}\langle x,Ax \rangle} d x
\\
=
&
\int_{\mathbb{R}^n}e^{-\frac{1}{2}\langle Ux,Ux \rangle} d x
\\
=
&
\int_{\mathbb{R}^n}e^{-\frac{1}{2}\langle x,x \rangl... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/275428",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Is there a quick way to solve $3^8 \equiv x \mod 17$?
Is there a quick way to solve $3^8 \equiv x \mod 17$?
Like the above says really, is there a quick way to solve for $x$? Right now, what I started doing was $3^8 = 6561$, and then I was going to keep subtracting $17$ until I got my answer.
| When dealing with powers, squaring is a good trick to reduce computations (your computer does this too!) What this means is:
$ \begin{array}{l l l l l}
3 & &\equiv 3 &\pmod{17}\\
3^2 &\equiv 3^2 & \equiv 9 &\pmod{17}\\
3^4 & \equiv 9^2 & \equiv 81 \equiv 13 & \pmod{17}\\
3^8 & \equiv 13^2 & \equiv 169 \equi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/275501",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 5,
"answer_id": 3
} |
Order of nontrivial elements is 2 implies Abelian group If the order of all nontrivial elements in a group is 2, then the group is Abelian. I know of a proof that is just from calculations (see below). I'm wondering if there is any theory or motivation behind this fact. Perhaps to do with commutators?
Proof: $a \cdot ... | The idea of this approach is to work with a class of very small, finite, subgroups $H$ of $G$ in which we can prove commutativity. The reason for this is to be able to use the results like Cauchy's theorem and Lagrange's theorem.
Consider the subgroup $H$ generated by two distinct, nonidentity elements $a,b$ in the giv... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/275544",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "16",
"answer_count": 4,
"answer_id": 3
} |
Definition of $f+g$ and $f \cdot g$ in the context of polynomial rings? I've been asked to prove that given $f$ and $g$ are polynomials in $R[x]$, where $R$ is a commutative ring with identity,
$(f+g)(k) = f(k) + g(k)$, and $(f \cdot g)(k) = f(k) \cdot g(k)$. However, I always took these things as definition. What exa... | One construction (I do not like using constructions as definitions) of $R[x]$ is as the set of formal sums $\sum r_i x^i$ of the symbols $1, x, x^2, x^3, ...$ with coefficients in $R$. Addition here is defined pointwise and multiplication is defined using the Cauchy product rule. For every $k \in R$ there is an evaluat... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/275601",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Limit with parameter, $(e^{-1/x^2})/x^a$ How do I explore for which values of the parameter $a$
$$ \lim_{x\to 0} \frac{1}{x^a} e^{-1/x^2} = 0?
$$
For $a=0$ it is true, but I don't know what other values.
| It's true for all $a's$ (note that for $a\leqslant 0$ it's trivial) and you can prove it easily using L'Hopital's rule or Taylor's series for $\exp(-x^2)$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/275653",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 1
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Need help in describing the set of complex numbers Let set $$C=\{z\in \mathbb{C}:\sqrt{2}|z|=(i-1)z\}$$ I think C is empty , because you could put it in this way$$|z|=-\frac{(1-i)z}{\sqrt{2}}$$ but would like a second opinion.
| $$C=\{z\in \mathbb{C}:\sqrt{2}|z|=(i-1)z\}$$
let $z=a+bi,a,b\in\mathbb R$ then $$\sqrt{2}|a+bi|=(i-1)(a+bi)$$
$$\sqrt{2}|a+bi|=ai+bi^2-a-bi$$
$$\sqrt{2}|a+bi|=ai-b-a-bi$$
$$\sqrt{2}|a+bi|=-b-a+(a-b)i$$ follow that
$a-b=0$ and $\sqrt{2}|a+bi|=-b-a$ or $a=b$ and $\sqrt{2}|a+ai|=-2a\geq 0$ so $a=b\leq 0$ or $z=a+ai=a(1+... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 1
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A sequence in $C([-1,1])$ and $C^1([-1,1])$ with star-weak convergence w.r.t. to one space, but not the other The functionals
$$
\phi_n(x) = \int_{\frac{1}{n} \le |t| \le 1} \frac{x(t)}{t} \mathrm{d} t
$$
define a sequence of functionls in $C([-1,1])$ and $C^1([-1,1])$.
a) Show that $(\phi_n)$ converges *-weakly in $... | @Davide Giraduo
Regarding your derivation, i came to another result:
\begin{align*}
\int_{1/n \le |t| \le 1} \frac{x(t)}{t} \mathrm{d} t & = \int_{-1}^{-1/n} \frac{x(t)}{t} \mathrm{d} t + \int_{1/n}^1 \frac{x(t)}{t} \mathrm{d} t \\
& = ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/275789",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
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Computing $999,999\cdot 222,222 + 333,333\cdot 333,334$ by hand. I got this question from a last year's olympiad paper.
Compute $999,999\cdot 222,222 + 333,333\cdot 333,334$.
Is there an approach to this by using pen-and-paper?
EDIT Working through on paper made me figure out the answer. Posted below.
I'd now like to... | $$999,999\cdot 222,222 + 333,333\cdot 333,334=333,333\cdot 666,666 + 333,333\cdot 333,334$$
$$=333,333 \cdot 1,000,000$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/275853",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "51",
"answer_count": 6,
"answer_id": 2
} |
How to solve $|x-5|=|2x+6|-1$? $|x-5|=|2x+6|-1$.
The answer is $0$ or $-12$, but how would I solve it by algebraically solving it as opposed to sketching a graph?
$|x-5|=|2x+6|-1\\
(|x-5|)^2=(|2x+6|-1)^2\\
...\\
9x^4+204x^3+1188x^2+720x=0?$
| Consider different cases:
Case 1: $x>5$
In this case, both $x-5$ and $2x+6$ are positive, and you can resolve the absolute values positively. hence
$$
x-5=2x+6-1 \Rightarrow x = -10,
$$
which is not compatible with the assumption that $x>5$, hence no solution so far.
Case 2: $-3<x\leq5$
In this case, $x-5$ is negative,... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/275928",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 5,
"answer_id": 0
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Calculate $\underset{x\rightarrow7}{\lim}\frac{\sqrt{x+2}-\sqrt[3]{x+20}}{\sqrt[4]{x+9}-2}$ Please help me calculate this:
$$\underset{x\rightarrow7}{\lim}\frac{\sqrt{x+2}-\sqrt[3]{x+20}}{\sqrt[4]{x+9}-2}$$
Here I've tried multiplying by $\sqrt[4]{x+9}+2$ and few other method.
Thanks in advance for solution / hints us... | $\frac{112}{27}$ which is roughly 4.14815
The derivative of the top at $x=7$ is $\frac{7}{54}$
The derivative of the bottom at $x=7$ is $\frac{1}{32}$
$\frac{(\frac{7}{54})}{(\frac{1}{32})}$ is $\frac{112}{27}$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/275990",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "12",
"answer_count": 5,
"answer_id": 4
} |
Behavior of differential equation as argument goes to zero I'm trying to solve a coupled set of ODEs, but before attempting the full numerical solution, I would like to get an idea of what the solution looks like around the origin.
The equation at hand is:
$$ y''_l - (f'+g')y'_l + \biggr[ \frac{2-l^2-l}{x^2}e^{2f} -... | Your equation EQ1 forces $z_2$ to have a quadratic term. Indeed, the left-hand side of EQ1 is
$$4(1-e^2)x^3-4(f_1+2e^2f_1+g_1)x^4+O(x^5) \tag{1}$$
This is equated to $4x(f'+g')z_2$. Clearly, $z_2$ must include the term $\frac{1-e^2}{f_1+g_1}x^2$.
I used Maple to check this, including many more terms than was necessar... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
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if $f$ is entire then show that $f(z)f(1/z)$ is also entire This is again for an old exam.
Let $f$ be an entire function, show that f(z)f(1/z) is entire.
How do I go about showing the above.
Do I use the definition of analyticity?.,
Call g: f(z)f(1/z) and show that it is complex differentiable everywhere?
Edit: Well th... | As Pavel already mentioned, this is not true. In fact, the only entire functions that satisfy the stated conclusion are $f(z) = cz^n$, where $c\neq 0$.
First of all, $f$ must be a polynomial, otherwise $f(1/z)$ has an essential singularity at $z=0$. If $\deg f = n$, then $f(1/z)$ has a pole of order $n$ at the origin, ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/276120",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
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Inequality problem algebra? How would I solve the following inequality problem.
$s+1<2s+1<4$
My book answer says $s\in (0, \frac32)$ as the final answer but I cannot seem to get that answer.
| We have $$s+1<2s+1<4.$$ This means $2s+1<4$, and in particular, $2s<3$. Dividing by the $2$ gives $s<3/2$. Now, observing on the other hand that we have $s+1<2s+1$, we subtract $s+1$ from both sides and have $0<s$. This gives us a bound on both sides of $s$, i.e., $$0<s<\frac{3}{2}$$ as desired.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/276179",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
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How to solve simple systems of differential equations Say we are given a system of differential equations
$$
\left[ \begin{array}{c} x' \\ y' \end{array} \right] = A\begin{bmatrix} x \\ y \end{bmatrix}
$$
Where $A$ is a $2\times 2$ matrix.
How can I in general solve the system, and secondly sketch a solution... | If you don't want change variables then,
there is a simple way for calculate $e^A$(all cases).
Let me explain about. Let A be a matrix and $p(\lambda)=\lambda^2-traço(A)\lambda+det(A)$ the characteristic polynomial. We have 2 cases:
$1$) $p$ has two distinct roots
$2$) $p$ has one root with multiplicity 2
The case 2 is... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/276264",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
} |
Real tree and hyperbolicity I seek a proof of the following result due to Tits:
Theorem: A path-connected $0$-hyperbolic metric space is a real tree.
Do you know any proof or reference?
| I finally found the result as Théorème 4.1 in Coornaert, Delzant and Papadopoulos' book Géométrie et théorie des groupes, les groupes hyperboliques de Gromov, where path-connected is replaced with geodesic; and in a document written by Steven N. Evans: Probability and Real Trees (theorem 3.40), where path-connected is ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/276328",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
} |
Common tangent to two circles with Ruler and Compass Given two circles (centers are given) -- one is not contained within the other, the two do not intersect -- how to construct a line that is tangent to both of them? There are four such lines.
| [I will assume you know how to do basic constructions, and not explain (say) how to draw perpendicular from a point to a line.]
If you're not given the center of the circles, draw 2 chords and take their perpendicular bisector to find the centers $O_1, O_2$.
Draw the line connecting the centers. Through each center, dr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/276381",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Rayleigh-Ritz method for an extremum problem I am trying to use the Rayleigh-Ritz method to calculate an approximate solution to the extremum problem with the following functional:
$$ L[y]=\int\int_D (u_x^2+u_y^2+u^2-2xyu)\,dx\,dy, $$
$D$ is the unit square i.e. $0 \leq x \leq 1, 0 \leq y \leq 1.$
Also $u=0$ on the bou... | Your integral is in the form of
$$L(x,y,u)=\int\int_D (u_x^2+u_y^2+u^2-2xyu)\,dx\,dy$$
$$0 \leq x \leq 1, 0 \leq y \leq 1$$
Due to homogenous boundary conditions it is possible to use your approximation function
$$u(x,y)=cxy(1-x)(1-y)$$
When substituted into integral equation
$$L(x,y,u)=\int_0^1\int_0^1 (u_x^2+u_y^2+u^... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/276446",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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$x^4 + y^4 = z^2$ $x, y, z \in \mathbb{N}$,
$\gcd(x, y) = 1$
prove that $x^4 + y^4 = z^2$ has no solutions.
It is true even without $\gcd(x, y) = 1$, but it is easy to see that $\gcd(x, y)$ must be $1$
| This has been completely revised to match the intended question. The proof is by showing that there is no minimal positive solution, i.e., by infinite descent. It’s from some old notes; I’ve no idea where I cribbed it from in the first place.
Suppose that $x^4+y^4=z^2$, where $z$ is the smallest positive integer for wh... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/276515",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 1,
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Dirac Delta Function of a Function I'm trying to show that
$$\delta\big(f(x)\big) = \sum_{i}\frac{\delta(x-a_{i})}{\left|{\frac{df}{dx}(a_{i})}\right|}$$
Where $a_{i}$ are the roots of the function $f(x)$. I've tried to proceed by using a dummy function $g(x)$ and carrying out:
$$\int_{-\infty}^{\infty}dx\,\delta\big(f... | Split the integral into regions around $a_i$, the zeros of $f$ (as integration of a delta function only gives nonzero results in regions where its arg is zero)
$$
\int_{-\infty}^{\infty}\delta\big(f(x)\big)g(x)\,\mathrm{d}x = \sum_{i}\int_{a_i-\epsilon}^{a_i+\epsilon}\delta(f(x))g(x)\,\mathrm{d}x
$$
write out the Taylo... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/276583",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "49",
"answer_count": 7,
"answer_id": 4
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Is my understanding of product sigma algebra (or topology) correct? Let $(E_i, \mathcal{B}_i)$ be measurable (or topological) spaces, where $i \in I$ is an index set, possibly infinite. Their product sigma algebra (or product topology) $\mathcal{B}$ on $E= \prod_{i \in I} E_i$ is defined to be the coarsest one that can... | For the comments: I retract my error for the definition of measurability. Sorry.
For the two things generating the same sigma algebra (or topology, which is similar):
We use $\langle - \rangle$ to denote the smallest sigma algebra containing the thing in the middle. We want to show that
$$(1) \hspace{5mm}\langle \prod... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/276632",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Does a graph with $0$ vertices count as simple? Does a graph with $0$ vertices count as a simple graph?
Or does a simple graph need to have a non-empty vertex set?
Thanks!
| It is typical to refer to a graph with no vertices as the null graph. Since it has no loops and no parallel edges (indeed, it has no edges at all), it is simple.
That said, if your present work finds you writing "Such and such is true for all simple graphs except the null graph", then it could be a good idea to announc... | {
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"url": "https://math.stackexchange.com/questions/276685",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Minimal polynominal: geometric meaning I am currently studying Chapter 6 of Hoffman & Kunze's Linear Algebra which deals with characteristic values and triangulation and diagonalization theorems.
The chapter makes heavy use of the concept of the minimal polynomial which it defines as the monic polynomial of the smalle... | Consider the following matrices:
$$
A = \left(\begin{array}{cc}2&0\\0&2\end{array}\right) \ \ \text{ and } \ \
B = \left(\begin{array}{cc}2&1\\0&2\end{array}\right).
$$
The first matrix has minimal polynomial $X - 2$ and the second has minimal polynomial $(X-2)^2$. If we subtract $2I_2$ from these matrices then we ge... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
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"answer_id": 2
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Integrate $\int_{C} \frac{1}{r-\bar{z}}dz$ - conflicting answers In an homework exercise, we're asked to integrate $\int_{C} \frac{1}{k-\bar{z}}dz$ where C is some circle that doesn't pass through $k$.
I tried solving this question through two different approaches, but have arrived at different answers.
Idea: use the f... | The original function is not holomorphic since $\frac{d}{d\overline{z}}\frac{1}{k-\overline{z}}=\frac{-1}{(k-\overline{z})^{2}}$. So you cannot apply Cauchy's integral formula.
Let $C$ be centered at $c$ with radius $r$, then we have $z=re^{i\theta}+c=c+r\cos(\theta)+ri\sin(\theta)$, and its conjugate become $c+r\cos(... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/276803",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Sum of the form $r+r^2+r^4+\dots+r^{2^k} = \sum_{i=1}^k r^{2^i}$ I am wondering if there exists any formula for the following power series :
$$S = r + r^2 + r^4 + r^8 + r^{16} + r^{32} + ...... + r^{2^k}$$
Is there any way to calculate the sum of above series (if $k$ is given) ?
| I haven’t been able to obtain a closed form expression for the sum, but maybe you or someone else can do something with what follows.
In Blackburn's paper (reference below) there are some manipulations involving the geometric series
$$1 \; + \; r^{2^n} \; + \; r^{2 \cdot 2^n} \; + \; r^{3 \cdot 2^n} \; + \; \ldots \; +... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/276892",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "27",
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Matrices with columns which are eigenvectors In this question, the OP asks about finding the matrix exponential of the matrix $$M=\begin{bmatrix}
1 & 1 & 1\\
1 & 1 & 1\\
1 & 1 & 1
\end{bmatrix}.$$ It works out quite nicely because $M^2 = 3M$ so $M^n = 3^{n-1}M$. The reason this occurs is that the vector $$v = \begin{bm... | Another partial answer: One can notice that matrices of rank 1 are such examples.
Indeed, if $M$ is of rank 1, its columns are of the form $a_1C,a_2C,...,a_nC$ for a given vector $C \in \mathbb{R}^n$ ; if $L=(a_1 \ a_2 \ ... \ a_n)$, then $M=CL$.
So $MC=pC$ with the inner product $p=LC$. Also, $M^n=p^{n-1}M$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/276962",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 3,
"answer_id": 2
} |
Free group n contains subgroup of index 2 My problem is to show that any free group $F_{n}$ has a normal subgroup of index 2. I know that any subgroup of index 2 is normal. But how do I find a subgroup of index 2?
The subgroup needs to have 2 cosets. My first guess is to construct a subgroup $H<G$ as $H = <x_{1}^{2}, x... | For any subgroup $H$ of $G$ and elements $a$ and $b$ of $G$ the following statements hold.
*
*If $a \in H$ and $b \in H$, then $ab \in H$
*If $a \in H$ and $b \not\in H$, then $ab \not\in H$
*If $a \not\in H$ and $b \in H$, then $ab \not\in H$
Hence it is natural to ask when $a \not\in H$ and $b \not\in H$ impli... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/277007",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 3,
"answer_id": 2
} |
how to show $f(1/n)$ is convergent? Let $f:(0,\infty)\rightarrow \mathbb{R}$ be differentiable, $\lvert f'(x)\rvert<1 \forall x$. We need to show that
$a_n=f(1/n)$ is convergent. Well, it just converges to $f(0)$ as $\lim_{n\rightarrow \infty}f(1/n)=f(0)$ am I right? But $f$ is not defined at $0$ and I am not able to a... | The condition $|f'(x)|<1$ implies that f is lipschitz.
$$|f(x)-f(y)|\le |x-y| $$
Then $$|f(\frac{1}{n})-f(\frac{1}{m})|\le|\frac{1}{n}-\frac{1}{m}|$$
Since $x_n=\frac{1}{n}$ is Cauchy, $f(\frac{1}{n})$ also is Cauchy
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/277070",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 5,
"answer_id": 2
} |
Nilpotent Lie Group that is not simply connect nor product of Lie Groups? I have been trying to find for days a non-abelian nilpotent Lie Group that is not simply connected nor product of Lie Groups, but haven't been able to succeed.
Is there an example of this, or hints to this group, or is it fundamentally impossible... | The typical answer is a sort of Heisenberg group, presented as a quotient (by a normal subgroup)
$$
H \;=\; \{\pmatrix{1 & a & b \cr 0 & 1 & c\cr 0 & 0& 1}:a,b,c\in \mathbb R\}
\;\bigg/\;
\{\pmatrix{1 & 0 & b \cr 0 & 1 & 0\cr 0 & 0& 1}:b\in \mathbb Z\}
$$
Edit: To certify the non-simple-connectedness, note that the gr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/277118",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Arithmetic Progressions in Complex Variables From Stein and Shakarchi's Complex Analysis book, Chapter 1 Exercise 22 asks the following:
Let $\Bbb N=\{1,2,\ldots\}$ denote the set of positive integers. A subset $S\subseteq \Bbb N$ is said to be in arithmetic progression if $$S=\{a,a+d,a+2d,\ldots\}$$ where $a,d\in\Bbb... | Suppose $\mathbf{N}=S_1\cup\cdots S_k$ is a partition of $\bf N$ and $S_r=\{a_r+d_rm:m\ge0\}$. Then
$$\begin{array}{cl}\frac{z}{1-z} & =\sum_{n\in\bf N}z^n \\
& =\sum_{r=1}^k\left(\sum_{n\in S_r}z^n\right) \\
& = \sum_{r=1}^k\left(\sum_{m\ge0}z^{a_r+d_rm}\right) \\
& = \sum_{r=1}^k\frac{z^{a_r}}{1-z^{d_r}}.\end{array}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/277183",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
If $X$ is normal and $A$ is a $F_{\sigma}$-set in $X$, then $A$ is normal. How could I prove this theorem? A topological space $X$ is a normal space if, given any disjoint closed sets $E$ and $F$, there are open neighbourhoods $U$ of $E$ and $V$ of $F$ that are also disjoint. (Or more intuitively, this condition says t... | Let us begin with a lemma ( see it in the Engelking's "General Topology" book, lemma 1.5.15):
If $X$ is a $T_1$ space and for every closed $F$ and every open $W$ that contains $F$ there exists a sequence $W_1$, $W_2$, ... of open subsets of $X$ such that $F\subset \cup_{i}W_i$ and $cl(W_i)\subset W$ for $i=$ 1, 2, ..... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/277251",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 2,
"answer_id": 1
} |
Is high school contest math useful after high school? I've been prepping for a lot of high school math competitions this year. Will all the math I learn would actually mean something in college? There is a chance that all of it will be for naught, and I just wanted to know if any of you people found the math useful af... | High school math competitions require you to learn how to solve problems, especially when there is no "method" you can look up telling you how to solve these problems. Problem solving is a very desirable skill for many jobs you might someday wish to have.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/277310",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "22",
"answer_count": 5,
"answer_id": 3
} |
How to show that $\frac{x^2}{x-1}$ simplifies to $x + \frac{1}{x-1} +1$ How does $\frac{x^2}{(x-1)}$ simplify to $x + \frac{1}{x-1} +1$?
The second expression would be much easier to work with, but I cant figure out how to get there.
Thanks
| Very clever trick: If you have to show that two expressions are equivalent, you work backwards. $$\begin{align}=& x +\frac{1}{x-1} + 1 \\ \\ \\ =& \frac{x^2 - x}{x-1} +\frac{1}{x - 1} + \frac{x-1}{x-1} \\ \\ \\ =& \frac{x^2 - x + 1 + x - 1}{x - 1} \\ \\ \\ =&\frac{x^2 }{x - 1}\end{align}$$Now, write the steps backwards... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/278481",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 6,
"answer_id": 3
} |
Prove that if for every $x \in \mathbb{R}^N$ $Ax=Bx$ then $A=B$ How can I quickly prove that if for every $x \in \mathbb{R}^N$
$$Ax=Bx$$ then $A=B$ ? Where $A,B\in \mathbb{R}^{N\times N}$.
Normally I would multiply both sides by inverse of $x$, however vectors have no inverse, so I am not sure how to prove it.
| If you want to invert a matrix but all you have are vectors, put the vectors into a matrix! For example,
$$A \left( e_1 \mid e_2 \mid \cdots \mid e_n \right) = \left( A e_1 \mid A e_2 \mid \cdots \mid A e_n \right) $$
where $e_i$ is the $i$-th standard basis (column) vector. I could have chosen any vectors, but the sta... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/278555",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 5,
"answer_id": 1
} |
Trying to find $\sum\limits_{k=0}^n k \binom{n}{k}$
Possible Duplicate:
How to prove this binomial identity $\sum_{r=0}^n {r {n \choose r}} = n2^{n-1}$?
$$\begin{align}
&\sum_{k=0}^n k \binom{n}{k} =\\
&\sum_{k=0}^n k \frac{n!}{k!(n-k)!} =\\
&\sum_{k=0}^n k \frac{n(n-1)!}{(k-1)!((n-1)-(k-1))!} = \\
&n\sum_{k=0}^n \... | By convention $\binom{n}k=0$ if $k$ is a negative integer, so your last line is simply $$n\sum_{k=0}^n\binom{n-1}{k-1}=n\sum_{k=0}^{n-1}\binom{n}k=n2^{n-1}\;.$$ Everything else is fine.
By the way, there is also a combinatorial way to see that $k\binom{n}k=n\binom{n-1}{k-1}$: the lefthand side counts the ways to choose... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/278615",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 0
} |
A continuous function $f : \mathbb{R} → \mathbb{R}$ is uniformly continuous if it maps Cauchy sequences into Cauchy sequences. A continuous function $f : \mathbb{R} → \mathbb{R}$ is uniformly continuous if it maps
Cauchy sequences into Cauchy sequences.
is the above statement is true?
I guess it is not true but can't f... | The answer is no as explained by Jonas Meyer and every continuous function $f:\mathbb R\longrightarrow \mathbb R$ has this property:
If $(x_n)_{n\in\mathbb N}$ is a Cauchy sequence then $m\leq x_n \leq M, \ \ \forall \ n\in\mathbb N$ for some $m<M$. Since $f$ is uniformly continuous on $[m,M]$ the result follows.
So $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/278678",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
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