Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Given a simple graph and its complement, prove that either of them is always connected. I was tasked to prove that when given 2 graphs $G$ and $\bar{G}$ (complement), at least one of them is a always a connected graph.
Well, I always post my attempt at solution, but here I'm totally stuck. I tried to do raw algebraic m... | Let $G$ be a simple disconnected graph
$\Longrightarrow\ \exists$ atleast $2$ vertices, say $u$ and $v$ such that there does not exist a path between $u$ and $v$.
$\Longrightarrow$ All vertices of $G$ are not adjacent to both $u$ and $v$ (Why?). $u$ and $v$ are obviously not adjacent.
$\Longrightarrow$ In $\bar{G}$ all... | {
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"timestamp": "2023-03-29T00:00:00",
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Compressed sensing, approximately sparse, Power law An x in $\mathbb{R}^n$ is said to be sparse if many of it's coefficients are zeroes. x is said to be compressible(approximately sparse) if many of its coefficients are close to zero.ie Let $x=(x_1,x_2,....x_n)$. Sort the absolute values of the coefficients in decreas... | The definition looks just like the rule for determining convergent series. Using the ratio test, you can tell if a sequence of roots converges. As I recall, $\frac{1}{x}$ doesn't converge to zero. Rational values of $r$ don't converge quickly, $\frac{1}{\sqrt{x}}$, I wouldn't bet any money that's a sparse system. Are y... | {
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Getting the angles of a non-right triangle when all lengths are known I have a triangle that I know the lengths of all the sides. But I need to know the angles. Needs to work with non-right triangles as well.
I know it is possible, and I could have easily done this years ago when I was in trig, but it has completely sl... | As has been mentioned in comments, the formula that you're looking for is the Law of Cosines.
The three formulations are:
$$c^2 = a^2 + b^2 - 2ab \cos(C)$$
$$b^2 = a^2 + c^2 -2ac \cos(B)$$
$$a^2 = b^2 + c^2 - 2bc \cos(A)$$
You can use this law to find all three angles. Alternatively, once you have one of the angles, yo... | {
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Binomial coefficient $\sum_{k \leq m} \binom{m-k}{k} (-1)^k$ This is example 3 from Concrete Mathematics (Section 5.2 p.177 in 1995 edition). Although the proof is given in the book (based on a recurrent expression), I were trying to find an alternative solution, noticing that half of the terms are 0 (denote for simpl... | For a nice proof, see Arthur T. Benjamin and Jennifer J. Quinn: Proofs that Really Count, Identity 172, pp. 85-86.
Sketch of the proof:
Clearly, $\binom{m-k}k$ counts the number of ways that you can tile a $1×m$ board with $1×2$ dominoes (denoted by $d$) and $1×1$ squares (denoted by $s$) such that the number of domino... | {
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orthonormal basis in $l^{2}$ I need an orthonormal basis in $l^{2}$. One possible choice would be to take as such the sequences $\{1,0,0,0,...\}, \{0,1,0,0,...\}, \{0,0,1,0,...\}$, but I need a basis where only finitely many components of the basis vectors are zero. Does anyone know a way to construct such a basis? One... | How about this: consider $w = (1,1/2,1/3,1/4,\dots)$ (or any other vector with no no-zero entry), and complement it with all the standard basis vectors $e_1$, $e_2$,
$\dots$ to a complete set and apply Gram-Schmidt orthonormalization?
| {
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Do two injective functions prove bijection? I'm trying to prove $|A| = |B|$, and I have two injective functions $f:A \to B$ and $g:B \to A$. Is this enough proof for a bijection, which would prove $|A| = |B|$? It seems logical that it is, but I can't find a definitive answer on this.
All I found is this yahoo answer:
... | Yes this is true, it is called Cantor–Bernstein–Schroeder theorem.
| {
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Complex number: equation I would like an hint to solve this equation: $\forall n\geq 1$
$$\sum_{k=0}^{2^n-1}e^{itk}=\prod_{k=1}^{n}\{1+e^{it2^{k-1}}\} \qquad \forall t \in \mathbb{R}.$$
I went for induction but without to much success; I will keep trying, but if you have an hint...
Many thanks.
| Fix $n\in \mathbb{N}$ and $t\in \mathbb{R}$.
*
*If $t=0 \mod 2\pi$, your equality is obviously true, for it reduces to:
$$\sum_{k=0}^{2^n -1} 1 =2^n = \prod_{k=1}^n 2$$
(remember that $e^{\imath\ t}$ is $2\pi$-periodic).
*Now, assume $t\neq 0 \mod 2\pi$. Evaluate separately:
$$\begin{split}
(1-e^{\imath\ t})\ \sum... | {
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What's the difference between tuples and sequences? Both are ordered collections that can have repeated elements. Is there a difference? Are there other terms that are used for similar concepts, and how are these terms different?
| Using a basic set theoretic definition, a tuple (a, b, c, ..) represents an element of the Cartesian product of sets A x B x C ...
In a vector space the tuple represents the components of a vector in terms of basis vectors.
A sequence on the other hand represents a function (usually of the natural numbers) to some se... | {
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Double Subsequences Suppose that $\{a_{n}\}$ and $\{b_{n}\}$ are bounded. Prove that $\{a_{n}b_{n}\}$ has a convergent subsequence.
In class this is how my professor argued:
By the Bolzano-Weierstrass Theorem, there exists a subsequence $\{a_{n_k}\}$ that converges to $a$. Since $\{b_n\}$ is bounded, $\{b_{n_k}\}$ is... | The problem with your argument is that by writing $\{a_{n_k}\}$ and $\{b_{n_k}\}$ you are implicitly (and incorrectly) assuming that the convergent subsequences of $\{a_n\}$ and $\{b_n\}$ involve the same terms. In general, there is no reason why this should be the case. We need to introduce all of the subsequences tha... | {
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Partial derivative involving trace of a matrix Suppose that I have a symmetric Toeplitz $n\times n$ matrix
$$\mathbf{A}=\left[\begin{array}{cccc}a_1&a_2&\cdots& a_n\\a_2&a_1&\cdots&a_{n-1}\\\vdots&\vdots&\ddots&\vdots\\a_n&a_{n-1}&\cdots&a_1\end{array}\right]$$
where $a_i \geq 0$, and a diagonal matrix
$$\mathbf{B}=\le... | Define some variables for convenience
$$\eqalign{
P &= {\rm Diag}(\beta) \cr
B &= cP^{-1} \cr
b &= {\rm diag}(B) \cr
S &= A+B \cr
M &= AS^{-1}A \cr
}$$
all of which are symmetric matrices, except for $b$ which is a vector.
Then the function and its differential can be expressed in terms of the Frobenius (:) ... | {
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Formula for calculating residue at a simple pole. Suppose $f=P/Q$ is a rational function and suppose $f$ has a simple pole at $a$. Then a formula for calculating the residue of $f$ at $a$ is
$$
\text{Res}(f(z),a)=\lim_{z\to a}(z-a)f(z)=\lim_{z\to a}\frac{P(z)}{\frac{Q(z)-Q(a)}{z-a}}=\frac{P(a)}{Q'(a)}.
$$
In the sec... | Since the pole at $\,a\,$ is simple we have that
$$Q(z)=(z-a)H(z)\,\,,\,H(z)\,\,\text{a polynomial}\,\,,\,P(a)\cdot H(a)\neq 0\,$$
Thus, as polynomials are defined and differentiable everywhere:
$$Res_{z=a}(f)=\lim_{z\to a}\frac{P(z)}{H(z)}=\frac{P(a)}{H(a)}$$
and, of course,
$$Q'(z)=H(z)+(z-a)H'(z)\xrightarrow [z\to... | {
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Theorem formulation "Given ..., then ..." or "For all ..., ..."? When formulating a theorem, which of the following forms would be preferred, and why? Or is there another even better formulation? Are there reasons for or against mixing them in one paper?
Formulation 0: If $x\in X$, then (expression involving $x$).
Form... | I guess I'm a simpleton, I always preferred a theorem stated memorably:
Theorem 1. Even numbers are interesting.
Compared to...
Theorem 2. If $x$ is an even number, then $x$ is interesting.
...which is long-winded; or...
Theorem 3. Given an even number $x$, $x$ is interesting.
Theorem 3 also has odd consonance (writing... | {
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What is the inverse cycle of permutation? Given cyclic permutations, for example,
$σ = (123)$, $σ_{2} = (45)$,
what are the inverse cycles $σ^{-1}$, $σ_2^{-1}$?
Regards.
| Every permutation n>1 can be expressed as a product of 2-cycles.
And every 2-cycle (transposition) is inverse of itself.
Therefore the inverse of a permutations is
Just reverse products of its 2-cycles
(ab)^-1 = b^-1 a^-1
| {
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What are the probabilities of getting a "Straight flush" in a poker game? I'm not a pretty much fun of Poker, but I'd like to study that game.
What are the probabilities of getting a Straight flush in a Poker game considering this factors?
Number of playersHow are cards dealtWho is the first player
| If you are dealt five cards, there are $4\times10 =40$ possible straight flushes ($4\times 9 =36$ if you exclude royal flushes) out of the ${52 \choose 5}= 2598960$ possible hands. So the probability is $\dfrac{40}{2598960} = \dfrac{1}{64974} \approx 0.00001539\ldots$.
The probability will increase if you can have mor... | {
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Definition of the gamma function I know that the Gamma function with argument $(-\frac{1}{ 2})$ -- in other words $\Gamma(-\frac{1}{2})$ is equal to $-2\pi^{1/2}$. However, the definition of $\Gamma(k)=\int_0^\infty t^{k-1}e^{-t}dt$ but how can $\Gamma(-\frac{1}{2})$ be obtained from the definition? WA says it does not... | The functional equation $\Gamma(z+1)=z\Gamma(z)$ allows you to define $\Gamma(z)$ for all $z$ with real part greater than $-1$, other than $z=0$: just set $\Gamma(z) = \Gamma(z+1)/z$, and the integral definition of $\Gamma(z+1)$ does converge. The value $\Gamma(\frac12)=\sqrt\pi$ is well known and can be derived from E... | {
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If $f$ continuous differentiable and $f'(r) < 1,$ then $x'=f(x/t)$ has no other solution tangent at zero to $\phi(t)=rt$ Suppose $f:\mathbb{R}\to\mathbb{R}$ is a continuous differentiable function such that $f(r)=r,$ for some $r.$ Then how to show that
If $f'(r) < 1,$ then the problem
$$x'=f(x/t)$$ has no other s... | Peter Tamaroff gave a very good hint in comments. Here is what comes out of it.
Suppose that $x$ is a solution tangent to $rt$ and not equal to it. Since solution curves do not cross, either (i) $x(t)>rt$ for all $t>0$, or (ii) $x(t)<rt$ for all $t>0$. I will consider (i), the other case being similar.
By assumption, ... | {
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Proof by exhaustion: all positive integral powers of two end in 2, 4, 6 or 8 While learning about various forms of mathematical proofs, my teacher presented an example question suitable for proof by exhaustion:
Prove that all $2^n$ end in 2, 4, 6 or 8 ($n\in\mathbb{Z},n>0$)
I have made an attempt at proving this, but... | Hint $\ $ mod $10,\:$ the powers of $\:2\:$ repeat in a cycle of length $4,\:$ starting with $2,\:$ since
$$\rm 2^{K+4} = 2^K(1+15) = 2^K + 30\cdot2^{K-1}\equiv\: 2^K\ \ (mod\ 10)\quad for\quad K\ge 1$$
Now it suffices to prove by induction that if $\rm\:f:\mathbb N\to \mathbb N = \{1,2,3\ldots\}\:$ then
$$\rm f(n+4)... | {
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Proving that a natural number is divisible by $3$ I am trying to show that $n^2 \bmod 3 = 0$ implies $n \bmod 3 = 0$.
This is a part a calculus course and I don't know anything about numbers theory. Any ideas how it can be done? Thanks!
| Hint $\rm\ (1+3k)^2 = 1 + 3\:(2k+3k^2)$
and $\rm\ \ \ (2+3k)^2 = 1 + 3\:(1+4k+3k^2)$
Said mod $3\!:\ (\pm1)^2 \equiv 1\not\equiv 0\ \ $ (note $\rm\: 2\equiv -1$)
| {
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Analysis Problem: Prove $f$ is bounded on $I$ Let $I=[a,b]$ and let $f:I\to {\mathbb R}$ be a (not necessarily continuous) function with the property that for every $x∈I$, the function $f$ is bounded on a neighborhood $V_{d_x}(x)$ of $x$. Prove that $f$ is bounded on $I$.
Thus far I have that,
For all $n∈I$ there exist... | If you cannot use the Heine-Borel theorem, argue via sup. Here's a sketch:
Let $A= \{ x \in I : f \text{ is bounded in } [a,x] \}$. Then $A$ is not empty because $a\in A$. Also, $A$ is bounded above because $A\le b$.
Prove that if $x\in A$ and $x<b$ then $x+h\in A$ for some $h>0$ using that $f$ is locally bounded at $x... | {
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The preimage of $(-\infty,a]$ under $f$ is closed for $a \in \mathbb{R}$, then $f$ is semi-continuous. So I've been thinking about this for the last two hours, but I am stuck.
Suppose $f:X \to \mathbb{R}$ where $X$ is a topological space.
$f$ is said to be semicontinuous if for any $x \in X$ and $\epsilon > 0$, there ... | I’m guessing from what you’ve written that your definition of lower semi-continuity is such that you want to start with an arbitrary $x_0\in X$ and $\epsilon>0$ and show that there is an open nbhd $U$ of $x_0$ such that $f(x)>f(x_0)-\epsilon$ for every $x\in U$. You know that for any $a\in\Bbb R$, $f^{-1}[(-\infty,a]]$... | {
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What does proving the Collatz Conjecture entail? From the get go: i'm not trying to prove the Collatz Conjecture where hundreds of smarter people have failed. I'm just curious.
I'm wondering where one would have to start in proving the Collatz Conjecture. That is, based on the nature of the problem, what's the startin... | Proving this conjecture indirectly would entail two things:
*
*Proving that there is no number n which increases indefinitely
*Proving there is no number n which loops indefinitely (besides the 4, 2, 1) loop
If one does these things then you have an answer to the collatz conjecture (and if you find a case of either... | {
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How do i scale my errorbars when i scale my data? I am plotting distributions of data with the standard deviation and median of my data. Now when i want to scale my median by a another variable, how do i need to modify the standart deviation?
| Let $X$ be some real-value random variable and $m$ be its median:
$$
\mathsf P\{X\leq m\} = \mathsf P\{X>m\}.
$$
Clearly, to scale median by the factor $\lambda> 0$ you just scale $X$ by the same factor since
$$
\mathsf P\{\lambda X\leq \lambda m\} = \mathsf P\{\lambda X>\lambda m\}.
$$
Note that although for the v... | {
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Multiplying exponents with variables inside Why is
$$(-1)^n(2^{n+2}) = (-2)^{n+2} ?$$
My thinking is that $-1^n \times 2^{n+2}$ should be $-2^{2n+2}$ but clearly this is not the case. Why is the variable essentially ignored, is there a special case of multiplication I'm unaware of?
| The exponent rules (for positive integer exponents, at any rate) are:
*
*$(a^n)^m = a^{nm}$
*$(ab)^n = a^nb^n$
*$a^na^m = a^{n+m}$.
Here, $a$ and $b$ are any real numbers, and $n$ and $m$ are positive integers. (The rules are valid in greater generality, but one has to be careful with the values of $a$ and $b$; ... | {
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Coercivity vs boundedness of operator The definition of coercivity and boundedness of a linear operator $L$ between two $B$ spaces looks similar: $\lVert Lx\lVert\geq M_1\lVert x\rVert$ and $\lVert Lx\rVert\leq M_2\lVert x\rVert$ for some constants $M_1$ and $M_2$. Thus in order to show the existence of a PDE $Lu=f$ on... | With boundedness everything is clear, because it is well known that linear operator $L$ is continuous iff $L$ is bounded. With continuity of $L$ you can solve the equation with sequential approximations. Moreover, you can apply the whole theory developed for continuous functions and, in particular, for continuous linea... | {
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Evaluate $\int\limits_0^{\frac{\pi}{2}} \frac{\sin(2nx)\sin(x)}{\cos(x)}\, dx$
How to evaluate
$$ \int\limits_0^{\frac{\pi}{2}} \frac{\sin(2nx)\sin(x)}{\cos(x)}\, dx $$
I don't know how to deal with it.
| Method 1. Let $I(n)$ denote the integral. Then by addition formula for sine and cosine,
$$\begin{align*}
I(n+1) + I(n)
&= \int_{0}^{\frac{\pi}{2}} \frac{[\sin((2n+2)x) + \sin(2nx)]\sin x}{\cos x} \; dx \\
&= \int_{0}^{\frac{\pi}{2}} 2\sin((2n+1)x) \sin x \; dx \\
&= \int_{0}^{\frac{\pi}{2}} [\cos(2nx) - \cos((2n+2)... | {
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Let $p$ be a prime. Prove that $p$ divides $ab^p−ba^p$ for all integers $a$ and $b$. Let $p$ be a prime. Prove that $p$ divides $ab^p−ba^p$ for all integers
$a$ and $b$.
| $$ab^p-ba^p = ab(b^{p-1}-a^{p-1})$$
If $p|ab$, then $p|(ab^p-ba^p)$ and also if $p \nmid ab$, then gcd$(p,a)=$gcd$(p,b)=1, \Rightarrow b^{p-1} \equiv a^{p-1} \equiv 1\pmod{p}$ (by Fermat's little theorem).
This further implies that $\displaystyle{p|(b^{p-1}-a^{p-1}) \Rightarrow p|(ab^p-ba^p)}$.
Q.E.D.
| {
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The Fundamental Theorem of Algebra and Complex Numbers We had a quiz recently in a linear algebra course, and one of the true/false question states that
The Fundamental Theorem of Algebra asserts that addition, subtraction, multiplication and division for real numbers can be carried over to complex numbers as long as ... | The statement is false.
The Fundamental Theorem of Algebra asserts that any non-constant polynomial with complex coefficients has a root in the complex numbers. This does not state anything about the relationship between the complex numbers and the real numbers; and any proof of the FTA will certainly use the associa... | {
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Sum of two closed sets in $\mathbb R$ is closed? Is there a counterexample for the claim in the question subject, that a sum of two closed sets in $\mathbb R$ is closed? If not, how can we prove it?
(By sum of sets $X+Y$ I mean the set of all sums $x+y$ where $x$ is in $X$ and $y$ is in $Y$)
Thanks!
| It's worth mentioning that :
if one is closed + bounded, another one is closed,then the addition is closed
Since closedness can be charaterized by sequence in $\Bbb{R}^n$,if $(x_n) \in A+B$ we need to show limit of the convergence sequence still lies in it.assume $A$ is compact $B $ is closed.
Since $x_n= a_n +b_n \to... | {
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How to evaluate one improper integral Please show me the detailed solution to the question:
Compute the value of
$$\int_{0}^{\infty }\frac{\left( \ln x\right) ^{40021}}{x}dx$$
Thank you a million!
| Since this is an exercise on improper integrals, it is natural to replace the
upper and lower limits by $R$, $\frac{1}{R}$ respectively and define the integral to be the limit as $R \rightarrow \infty$ . Then write the integral as the sum of the integral from $\frac{1}{R}$ to $1$ and from $1$ to $R$. In the second inte... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/124190",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 5,
"answer_id": 4
} |
What is the difference between Green's Theorem and Stokes Theorem? I don't quite understand the difference between Green's Theorem and Stokes Theorem. I know that Green's Theorem is in $\mathbb{R}^2$ and Stokes Theorem is in $\mathbb{R}^3$ and my lecture notes give Greens Theorem and Stokes Theorem as
$$\int \!\! \int_... | Green's Theorem is a special case of Stokes's Theorem. Since your surface is in the plane and oriented counterclockwise, then your normal vector is $n = \hat{k}$, the unit vector pointing straight up.
Similarly, if you compute $\nabla \times v$, where $v\, dr = Mdx + Ndy$, you would get $\left( \frac{\partial N}{\parti... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/124262",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
"answer_count": 2,
"answer_id": 1
} |
Primes modulo which a given quadratic equation has roots Given a quadratic polynomial $ax^2 + bx + c$, with $a$, $b$ and $c$ being integers, is there a characterization of all primes $p$ for which the equation
$$ax^2 + bx + c \equiv 0 \pmod p$$
has solutions?
I have seen it mentioned that it follows from quadratic reci... | I never noticed this one before.
$$ x^3 - x - 1 \equiv 0 \pmod p $$
has one root for odd primes $p$ with $(-23|p) = -1.$
$$ x^3 - x - 1 \equiv 0 \pmod p $$
has three distinct roots for odd $p$ with $(-23|p) = 1$ and $p = u^2 + 23 v^2 $ in integers.
$$ x^3 - x - 1 \equiv 0 \pmod p $$
has no roots for odd $p$ with ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/124331",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 2,
"answer_id": 1
} |
About the exchange of $\sum$ and LM Given $f_i,g_i\in k[x_1,\cdots,x_n],1\leq i\leq s$, fix a monomial order on $k[x_1,\cdots,x_n]$, I was wondering whether there is an effective criterion to judge if this holds,$$\text{LM}(\sum_{i=1}^sf_ig_i)=\sum_{i=1}^s\text{LM}(f_ig_i),$$
where LM( ) is the leading monomial with re... | In characteristic 0, assuming all terms are non-zero so that LM is defined, this only works if $s=1$: taking the sum of coefficients on both sides of the equation you obtain the equation $1=s$. So you can never exchange a non-trivial sum and LM.
Note that only the image of LM needs to be of characteristic 0: this still... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/124401",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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There's a real between any two rationals, a rational between any two reals, but more reals than rationals? The following statements are all true:
*
*Between any two rational numbers, there is a real number (for example, their average).
*Between any two real numbers, there is a rational number (see this proof of tha... | Here's an attempt at a moral justification of this fact. One (informal) way of understanding the difference between a rational number and a real number is that a rational number somehow encodes a finite amount of information, whereas an arbitrary real number may encode a (countably) infinite amount of information. The ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/124458",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 2,
"answer_id": 1
} |
Constructive proof need to know the solutions of the equations Observe the following equations:
$2x^2 + 1 = 3^n$ has two solutions $(1, 1) ~\text{and}~ (2, 2)$
$x^2 + 1 = 2 \cdot 5^n$ has two solutions $(3, 1) ~\text{and}~ (7, 2)$
$7x^2 + 11= 2 \cdot 3^n$ has two solutions $(1, 2) ~\text{and}~ (1169, 14)$
$x^2 + 3 = 4... | All four of your equations (and many more) are mentioned in Saradha and Srinivasan, Generalized Lebesgue-Ramanujan-Nagell equations, available at http://www.math.tifr.res.in/~saradha/saradharev.pdf. The solutions are attributed to Bugeaud and Shorey, On the number of solutions of the generalized Ramanujan-Nagell equati... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/124565",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
finding a minorant to $(\sqrt{k+1} - \sqrt{k})$ Need help finding a minorant to
$(\sqrt{k+1} - \sqrt{k})$ which allows me to show that the series $\sum_{k=1}^\infty (\sqrt{k+1} - \sqrt{k})$ is divergent.
| You should observe that your series telescopes, i.e.:
$$\sum_{k=0}^n (\sqrt{k+1} - \sqrt{k}) = (\sqrt{1} -\sqrt{0}) + (\sqrt{2} -\sqrt{1}) +\cdots + (\sqrt{n}-\sqrt{n-1}) +(\sqrt{n+1}-\sqrt{n}) = \sqrt{n+1}-1\; ,$$
and therefore:
$$\sum_{k=0}^\infty (\sqrt{k+1} - \sqrt{k}) = \lim_{n\to \infty} \sum_{k=0}^n (\sqrt{k+1} ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/124708",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 5,
"answer_id": 1
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A question on pointwise convergence. The function $f_n(x):[-1,1] \to \mathbb{R}, \, \, \,f_n(x) = x^{2n-1}$ tends pointwise to the function $$f(x) = \left\{\begin{array}{l l}1&\textrm{if} \quad x=1\\0&\textrm{if} \quad -1<x<1\\-1&\textrm{if} \quad x=-1\end{array}\right.$$ but not uniformly (for obvious reasons as $f(x)... | Your argument
*
*for any $x \in [-1,1]$ and any $\epsilon > 0$ you can make $n$ large enough so that the max distance between $f_n(x)$ and $f(x)$ at that particular $x$ is less than $\epsilon$
is a great proof that for any $x$, the sequence $f_n(x)$ tends to $f(x)$. In other words, it's a proof that $f_n$ tends to... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/124754",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
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Combinatorial Probability-Rolling 12 fair dice My text says, regarding combinatorial probability, "The number of outcomes associated with any problem involving the rolling of n six-sided dice is $6^n$."
I know that in combinatorial probability $P(A)=m/n$ where $m$ is the number of ways $A$ can happen and $n$ is the num... | That is usually very good advice, at least until you simplify.
So for example, to find the probability that rolling twelve dice gives three prime numbers (not necessarily distinct) is $ \frac {3^{12}}{6^{12}}$ but that simplifies to $\frac {1}{2^{12}}$.
But it is possible to devise a problem where this advice does... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/124826",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
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Are the eigenvalues of $AB$ equal to the eigenvalues of $BA$? First of all, am I being crazy in thinking that if $\lambda$ is an eigenvalue of $AB$, where $A$ and $B$ are both $N \times N$ matrices (not necessarily invertible), then $\lambda$ is also an eigenvalue of $BA$?
If it's not true, then under what conditions i... | If $v$ is an eigenvector of $AB$ for some nonzero $\lambda$, then $Bv\ne0$
and $$\lambda Bv=B(ABv)=(BA)Bv,$$ so $Bv$ is an eigenvector for $BA$ with the same eigenvalue. If $0$ is an eigenvalue of $AB$ then $0=\det(AB)=\det(A)\det(B)=\det(BA)$ so $0$ is also an eigenvalue of $BA$.
More generally, Jacobson's lemma in op... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/124888",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "87",
"answer_count": 4,
"answer_id": 2
} |
Is this out-of-context theorem true? Can someone tell me if the following proposition is true ?
Theorem If $u=g + i h$ is a holomorphic function in $\Omega\subseteq \mathbb{C}$ and $\Omega$ is simply connected, then $v(z)=u(w)+ \int_\gamma \,g_x(z)-ih_y (z) \,dz$ is a primitive function of $u$ (where $w\in \Omega$ is f... | I shall assume that $g$ and $h$ are realvalued. Since $u:=g+ih$ is holomorphic it follows from the the CR equations that $h_y=g_x$. Therefore for any curve $\gamma\subset\Omega$ connecting the point $z_0$ with a variable point $z$ one has
$$\int_\gamma (g_x- i h_y)\ dz=(1-i)\int_\gamma g_x\ (dx+i dy) =(1-i)\int_\gamma... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/124994",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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How to determine the limit of this sequence? I was wondering how to determine the limit of $ (n^p - (\frac{n^2}{n+1})^p)_{n\in \mathbb{N}}$ with $p>0$, as $n \to \infty$?
For example, when $p=1$, the sequence is $ (\frac{n}{n+1})_{n\in \mathbb{N}}$, so its limit is $1$.
But I am not sure how to decide it when $p \neq 1... | Given the Binomial theorem, we have (see Landau notations)
$$\begin{array}{} (n+1)^p-n^p=\Theta(n^{p-1}) & \implies 1-\left(\frac{n}{n+1}\right)^p=\Theta \left( \frac{1}{n} \right) \\ & \implies n^p - \left(\frac{n^2}{n+1}\right)^p=\Theta(n^{p-1}) \end{array}$$
Thus the limit is $0$ for $p<1$, is $1$ for $p=1$ (already... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/125043",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 5,
"answer_id": 1
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If a graph of $2n$ vertices contains a Hamiltonian cycle, then can we reach every other vertex in $n$ steps?
Problem: Given a graph $G,$ with $2n$ vertices and at least one triangle. Is it possible to show that you can reach every other vertex in $n$ steps if $G$ contains a Hamilton cycle (HC)?
EDIT: Sorry, I forgot t... | The answer is no.
Question: Let $G$ be a 3-connected, hamiltonian, planar graph with $2n$ vertices and at least one triangle. Is it true that for all vertex pairs $x,y$, that there is a walk of exactly $n$ steps from $x$ to $y$?
The following graph and vertex pair is a counter example
It is clear that the graph is pla... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/125159",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 2
} |
What are the minimal conditions on a topological space for its closure to equal the sequential closure? My question is: what are the minimal conditions on a topological space for it have the following property?
$$x\in \bar{A}\iff \exists (x_n)\subset A | x_n \to x$$
| In this paper there is the answer (section 2, on Fréchet spaces, also known as Fréchet-Urysohn spaces): Your property defines the notion of a Fréchet space and he shows that these spaces are the pseudo-open images of metric spaces. He also defines the weaker concept of a sequential space and in the follow up paper he s... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/125287",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
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How to solve $\int_0^\pi{\frac{\cos{nx}}{5 + 4\cos{x}}}dx$? How can I solve the following integral?
$$\int_0^\pi{\frac{\cos{nx}}{5 + 4\cos{x}}}dx, n \in \mathbb{N}$$
| To elaborate on Pantelis Damianou's answer
$$
\newcommand{\cis}{\operatorname{cis}}
\begin{align}
\int_0^\pi\frac{\cos(nx)}{5+4\cos(x)}\mathrm{d}x
&=\frac12\int_{-\pi}^\pi\frac{\cos(nx)}{5+4\cos(x)}\mathrm{d}x\\
&=\frac12\int_{-\pi}^\pi\frac{\cis(nx)}{5+2(\cis(x)+\cis(-x))}\mathrm{d}x\\
&=\frac12\int_{-\pi}^\pi\f... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/125399",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
"answer_count": 5,
"answer_id": 0
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annihilator is the intersection of sets If $W$ is a subspace of a finite dimensional vector space $V$ and $\{g_{1},g_{2},\cdots, g_{r}\}$ is a basis of the annihilator $W^{\circ}=\{f \in V^{\ast}| f(a)=0, \forall a \in W\}$, then $W=\cap_{i=1}^{r} N_{g_{i}}$, where for $f \in V^{\ast}$, $N_{f}=\{a \in V| f(a)=0\}$
How... | We wish to prove that
$$W = \bigcap_{i=1}^{r} N_{g_{i}}$$
Step $1$: Proving $W \subset \bigcap_{i=1}^{r} N_{g_{i}}$
Let $w \in W$. We know that the annihilator $W^{o}$ is the set of linear functionals that vanish on $W$. If $g_{i}$ is in the basis for $W^{o}$, it is certainly in $W^{o}$. Thus, the $g_{i}$ all vanish ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/125451",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
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If we define $\sin x$ as series, how can we obtain the geometric meaning of $\sin x$? In Terry Tao's textbook Analysis, he defines $\sin x$ as below:
*
*Define rational numbers
*Define Cauchy sequences of rational numbers, and equivalence of Cauchy sequences
*Define reals as the space of Cauchy sequences o... | In this hint I suggest showing from the power series that if
$$
\sin(x)=\sum_{k=0}(-1)^k\frac{x^{2k+1}}{(2k+1)!}\tag{1}
$$
and
$$
\cos(x)=\frac{\mathrm{d}}{\mathrm{d}x}\sin(x)=\sum_{k=0}(-1)^k\frac{x^{2k}}{(2k)!}\tag{2}
$$
that $\frac{\mathrm{d}}{\mathrm{d}x}\cos(x)=-\sin(x)$ and from there that
$$
\sin^2(x)+\cos^2(x)... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/125511",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "29",
"answer_count": 6,
"answer_id": 3
} |
Continuous but not Hölder continuous function on $[0,1]$ Does there exist a continuous function $F$ on $[0,1]$ which is not Hölder continuous of order $\alpha$ at any point $X_{0}$ on $[0,1]$. $0 < \alpha \le 1$.
I am trying to prove that such a function does exist. also I couldn't find a good example.
| ($1$-dimensional) Brownian motion is almost surely continuous and nowhere Hölder continuous of order $\alpha$ if $\alpha > 1/2$. IIRC one can define random Fourier series that will be almost surely continuous but nowhere Hölder continuous for any $\alpha > 0$.
EDIT: OK, here's a construction. Note that $f$ is not Hö... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/125581",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 1,
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Show $ I = \int_0^{\pi} \frac{\mathrm{d}x}{1+\cos^2 x} = \frac{\pi}{\sqrt 2}$ Show $$ I = \int_0^{\pi} \frac{\mathrm{d}x}{1+\cos^2 x} = \frac{\pi}{\sqrt 2}$$
| In case KV's solution seems a bit magical, it may be reassuring to know that there's a systematic way to integrate rational functions of trigonometric functions, the Weierstraß substitution.
With $\cos x=(1-t^2)/(1+t^2)$ and $\mathrm dx=2/(1+t^2)\mathrm dt$,
$$
\begin{eqnarray}
\int_0^\pi \frac{\mathrm dx}{1+\cos^2 x... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/125637",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 0
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Example to show the distance between two closed sets can be 0 even if the two sets are disjoint Let $A$ and $B$ be two sets of real numbers. Define the distance from $A$ to $B$ by $$\rho (A,B) = \inf \{ |a-b| : a \in A, b \in B\} \;.$$ Give an example to show that the distance between two closed sets can be $0$ even if... | Consider the sets $\mathbb N$ and $\mathbb N\pi = \{n\pi : n\in\mathbb N\}$. Then $\mathbb N\cap \mathbb N\pi=\emptyset$ as $\pi$ is irrational, but we have points in $\mathbb N\pi$ which lie arbitrarily close to the integers.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/125709",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "23",
"answer_count": 3,
"answer_id": 2
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Counting words with subset restrictions I have an alphabet of N letters {A,B,C,D...N} and would like to count how many L-length words do not contain the pattern AA.
I've been going at this all day, but continue to stumble on the same problem.
My first approach was to count all possible combinations, (N^L) and subtract ... | Call the answer $x_L$.
Then $x_L=Nx_{L-1}-y_{L-1}$, where $y_L$ is the number of allowable words of length $L$ ending in $A$.
And $y_L=x_{L-1}-y_{L-1}$.
Putting these together we get $Nx_L-x_{L+1}=x_{L-1}-(Nx_{L-1}-x_L)$, which rearranges to $x_{L+1}=(N-1)x_L+(N-1)x_{L-1}$.
Now: do you know how to solve homogeneous... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/125778",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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How to get rid of the integral in this equation $\int\limits_{x_0}^{x}{\sqrt{1+\left(\dfrac{d}{dx}f(x)\right)^2}dx}$? How to get rid of the integral $\int\limits_{x_0}^{x}{\sqrt{1+\left(\dfrac{d}{dx}f(x)\right)^2}dx}$ when $f(x)=x^2$?
| Summarising the comments, you'll get
$$
\int\limits_{x_0}^{x}{\sqrt{1+\left(\dfrac{d}{dt}f(t)\right)^2}dt}
=\int\limits_{x_0}^{x}{\sqrt{1+\left(\dfrac{d}{dt}t^2\right)^2}dt}
=\int\limits_{x_0}^{x}{\sqrt{1+4t^2}dt}
$$
To solve the last one substitute $t=\tan(u)/2$ and $dt=\sec^2(u)/2du$. Then $\sqrt{1+4t^2}= \sqrt{\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/125828",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 2
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The $n^{th}$ root of the geometric mean of binomial coefficients. $\{{C_k^n}\}_{k=0}^n$ are binomial coefficients. $G_n$ is their geometrical mean.
Prove
$$\lim\limits_{n\to\infty}{G_n}^{1/n}=\sqrt{e}$$
| In fact, we have
$$ \lim_{n\to\infty}\left[\prod_{k=0}^{n}\binom{n}{k}\right]^{1/n^2} = \exp\left(1+2\int_{0}^{1}x\log x\; dx\right) = \sqrt{e}.$$
This follows from the identity
$$\frac{1}{n^2}\log \left[\prod_{k=0}^{n}\binom{n}{k}\right] = 2\sum_{j=1}^{n}\frac{j}{n}\log\left(\frac{j}{n}\right)\frac{1}{n} + \left(1+\fr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/125890",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
"answer_count": 5,
"answer_id": 3
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Homotopy inverses need not induce inverse homomorphisms Let $f:X \rightarrow Y$ and $g : Y \rightarrow X$ be homotopy inverses, ie. $f \circ g$ and $g\circ f$ are homotopic to the identities on $X$ and $Y$. We know that $f_*$ and $g_*$ are isomorphisms on the fundamental groups of $X$ and $Y$. However, it is my underst... | If $f,g$ are pointed maps (which is necessary so that $f_*,g_*$ make sense): No, they induce inverse homomorphsism.
Homotopic maps induce the same maps on homotopy groups, in particular fundamental groups. This means that we have a functor $\pi_1 : \mathrm{hTop}_* \to \mathrm{Grp}$. Every functor maps two inverse isomo... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/125959",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Distance between bounded and compact sets Let $(X,d)$ be a metric space and define for $B\subset X$ bounded, i.e.
$$\operatorname{diam}(B)= \sup \{ d(x,y) \colon x,y\in B \} < \infty,$$
the measure
$$\beta(B) = \inf\{r > 0\colon\text{there exist finitely many balls of radius r which cover } B\},$$
or equivalently,
$$... | Let $d_0=d_H(B,K(X))$. So for $d>d_0$ we can find compact $K$ with $d_H(B,K)<d$. In particular, $B \subseteq \cup_{k\in K} B(k,d)$.
As $K$ is compact, for any $\epsilon>0$ we can find $k_1,\cdots,k_n\in K$ with $K\subseteq \cup_i B(k_i,\epsilon)$. For $b\in B$, we can find $k\in K$ with $d(b,k)<d$. Then we can find... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/126031",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
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Divisor/multiple game Two players $A$ and $B$ play the following game:
Start with the set $S$ of the first 25 natural numbers: $S=\{1,2,\ldots,25\}$.
Player $A$ first picks an even number $x_0$ and removes it from $S$: We have $S:=S-\{x_0\}$.
Then they take turns (starting with $B$) picking a number $x_n\in S$ which i... | Second player (B) wins.
Consider the following pairing:
$2,14$
$3,15$
$4,16$
$5,25$
$6,12$
$7,21$
$8,24$
$9,18$
$10,20$
$11,22$
The left out numbers are $1,13,17,19,23$.
Now whatever number player one (A) picks, the second player (B) picks the paired number from the above pairings.
Ultimately, player one (A) will be o... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/126098",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
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Inequality involving the regularized gamma function Prove that $$Q(x,\ln 2) := \frac{\int_{\ln 2}^{\infty} t^{x-1} e^{-t} dt}{\int_{0}^{\infty} t^{x-1} e^{-t} dt} \geqslant 1 - 2^{-x}$$ for all $x\geqslant 1$.
($Q$ is the regularized gamma function.)
| We have
$$
\frac{\int_{\ln 2}^{\infty} t^{x-1} e^{-t} \,dt}{\int_{0}^{\infty} t^{x-1} e^{-t} \,dt} = \frac{\int_{0}^{\infty} t^{x-1} e^{-t} \,dt - \int_{0}^{\log 2} t^{x-1} e^{-t} \,dt}{\int_{0}^{\infty} t^{x-1} e^{-t} \,dt} = 1 - \frac{\int_{0}^{\log 2} t^{x-1} e^{-t} dt}{\int_{0}^{\infty} t^{x-1} e^{-t} \,dt},
$$
s... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/126156",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 2,
"answer_id": 0
} |
Calculate the slope of a line passing through the intersection of two lines Let say I have this figure,
I know slope $m_1$, slope $m_1$, $(x_1, y_1)$, $(x_2, y_2)$ and $(x_3, y_3)$. I need to calculate slope $m_3$. Note the line with $m_3$ slope will always equally bisect line with $m_1$ slope and line with $m_2$.
|
We understand that:
$$m_1=\tan(\alpha)$$
$$m_2=\tan(\beta),$$
Then:
$$
m_3=\tan\left(\frac{\alpha+\beta}2\right).
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/126237",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Solving $217 x \equiv 1 \quad \text{(mod 221)}$ I am given the problem:
Find an integer $x$ between $0$ and $221$ such that
$$217 x \equiv 1 \quad \text{(mod 221)}$$
How do I solve this? Unfortunately I am lost.
| In this special case, you can multiply the congruence by $-1$ and you'll get
$$4x\equiv 220 \pmod{221}.$$
(Just notice that $-217 \equiv 4 \pmod{221}$ and $-1\equiv220\pmod{221}$.)
This implies that $x\equiv 55 \pmod{221}$ is a solution. (And since $\gcd(4,221)=1$, there is only one solution modulo $221$.)
In general,... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/126286",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Help me understand a 3d graph I've just seen this graph and while it's isn't the first 3d graph I've seen, as
a math "noob" I never thought how these graphs are plotted. I can draw 2d graphs on paper by marking the input and output values of a function. It's also easy for me to visualize what the graph I'm seeing says ... | Set your function equal to a given constant, this give you a function you are used to, and varying the height (ie what you set your function equal to) gives you the graph (2d) of the surface intersected with planes parallel to the xy-plane. Its essentially the same as a contour map of a mountain.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/126401",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Is there an abelian category of topological groups? There are lots of reasons why the category of topological abelian groups (i.e. internal abelian groups in $\bf Top$) is not an abelian category. So I'm wondering:
Is there a "suitably well behaved" subcategory of $\bf Top$, say $\bf T$, such that $\bf Ab(T)$ is an ab... | This was alluded to in the comments and may not be what you're looking for, but it surely deserves mention that you can take $\mathbf{T}$ to be the category of compact Hausdorff spaces. The category $\mathbf{Ab}(\mathbf{T})$ is the the category of compact abelian groups, which is equivalent to $\mathbf{Ab}^{op}$ and h... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/126537",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "15",
"answer_count": 2,
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re-writing a $\min(X,Y)$ function linearly for LP problem I am trying to formulate an LP problem. In the problem I have a $\min(X,Y)$ that I would like to formulate linearly as a set of constraints. For example, replacing $\min(X,Y)$ with some variable $Z$, and having a set of constraints on Z.
I believe that there a... | You could use $\min(x,y) = \frac{1}{2}(x + y - |x - y|)$ where $|x - y|$ can be replaced by the variables $z_1 + z_2$ with constraints $z_i \ge 0$ for $i=1,2$ and $z_1 - z_2 = x - y$. $z_1$ and $z_2$ are, therefore, the positive or the negative part of $|x-y|$.
Edit:
For the reformulation to work, you must ensure that... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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finding final state of numbers after certain operations There are $N$ children sitting along a circle, numbered $1,2,\dots,n$ clockwise. The $i$-th child has a piece of paper with number $a_i$ written on it. They play the following game:
In the first round, the child numbered $x$ adds to his number the sum of the numbe... | In principle there is, but in practice I doubt that there’s anything very useful.
It really suffices to solve the problem when $i=1$, since for any other value of $i$ we can simply relabel the children. If we start at position $1$, we can define $a_{kn+i}$ to child $i$’s number after $k$ rounds have been played. Then t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/126677",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Field extension, primitive element theorem I would like to know if it is true that
$\mathbb{Q}(\sqrt{2}-i, \sqrt{3}+i) = \mathbb{Q}(\sqrt{2}-i+2(\sqrt{3}+i))$.
I can prove, that $\mathbb{Q}(\sqrt{2}-i, \sqrt{3}+i) = \mathbb{Q}(\sqrt{2},\sqrt{3},i)$, so the degree of this extension is 8. Would it be enough to show that ... | Let $\alpha=\sqrt{2}-i+2(\sqrt{3}+i)$.
Since $\alpha\in\mathbb{Q}(\sqrt{2}-i,\sqrt{3}+i)$, it follows that $\mathbb{Q}(\alpha)=\mathbb{Q}(\sqrt{2}-i,\sqrt{3}+i)$ if and only if their degrees over $\mathbb{Q}$ are equal. The degree $[\mathbb{Q}(\alpha):\mathbb{Q}]$ is equal to the degree of the monic irreducible of $\al... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Cancelling summands in a direct sum decomposition Let $M$ be a Noetherian and Artinian module. Suppose that:
$$\bigoplus_{i=1}^{q} A_{i} \oplus \bigoplus_{i=1}^{t} B_{i} \cong \bigoplus_{i=1}^{q} A_{i} \oplus \bigoplus_{i=1}^{r} C_{i}$$
where all $A_{i},B_{i},C_{i}$ are indecomposable submodules of $M$.
Can we always ... | Cancellation means that for modules $M,N,P$ over a ring $R$ (not assumed commutative) we have the implication
$$M\oplus N\cong M\oplus P \implies N\cong P$$
Cancellation holds for modules that are only assumed artinian (which of course answers your question in the affirmative) thanks to a theorem by Camps and Dick... | {
"language": "en",
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Computing the best constant in classical Hardy's inequality Classical Hardy's inequality (cfr. Hardy-Littlewood-Polya Inequalities, Theorem 327)
If $p>1$, $f(x) \ge 0$ and $F(x)=\int_0^xf(y)\, dy$ then
$$\tag{H} \int_0^\infty \left(\frac{F(x)}{x}\right)^p\, dx < C\int_0^\infty (f(x))^p\, dx $$
unless $f \equiv 0$. The... |
We have the operator $T: L^p(\mathbb{R}^+) \to L^p(\mathbb{R}^+)$ with $p \in (1, \infty)$, defined by$$(Tf)(x) := {1\over x} \int_0^x f(t)\,dt.$$Calculate $\|T\|$.
For the operator $T$ defined above, the operator norm is $p/(p - 1)$. We will also note that this is also a bounded operator for $p = \infty$, but not fo... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/126889",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "33",
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Simple Logic Question I've very little understanding in logic, how can I simply show that this is true:
$$((X \wedge \neg Y)\Rightarrow \neg Z) \Leftrightarrow ((X\wedge Z)\Rightarrow Y)$$
Thanks a lot.
| You want to show that
$$((X \wedge \neg Y)\Rightarrow \neg Z) \Leftrightarrow ((X\wedge Z)\Rightarrow Y).$$
It is hard to know without context what "show" might mean. For example, we could be working with a specific set of axioms. Since an axiom system was not specified, I will assume we are looking for a precise but ... | {
"language": "en",
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"source": "stackexchange",
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The sum of an Irreducible Representation I was hoping someone could help me with the following question.
Let $\rho$ be an irreducible presentation of a finite group $G.$ Prove
\begin{equation*}
\sum_{g \in G} \rho(g) = 0
\end{equation*}
unless $\rho$ is the trivial representation of degree $1$.
I think I have to use ... | Let $t=\sum_{g\in G}\rho(g)$, which is an linear endomorphism of $V$. The subset $t(V)$ of $V$ is a $G$-submodule of $V$, as you can easily check. Moreover, $G$ acts trivially on all elements of $t(V)$.
If $V$ is irreducible, then either $t(V)=0$ or $t(V)=V$. In the first case, we have that in fact $t=0$. In the second... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/127002",
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"source": "stackexchange",
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Complex Analysis: Liouville's theorem Proof I'm being asked to find an alternate proof for the one commonly given for Liouville's Theorem in complex analysis by evaluating the following given an entire function $f$, and two distinct, arbitrary complex numbers $a$ and $b$:
$$\lim_{R\to\infty}\oint_{|z|=R} {f(z)\over(z-a... | You can use the $ML$ inequality (with boundedness of $f$) to show $\displaystyle \lim_{R\rightarrow \infty} \oint_{|z|=R} \frac{f(z)}{(z-a)(z-b)}dz = 0$.
Combining this with your formula using the Cauchy integral formula, you get $$ 0 = 2\pi i\bigg(\frac{f(b)-f(a)}{b-a}\bigg)$$ from which you immediately conclude $f(b)... | {
"language": "en",
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"source": "stackexchange",
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Integration Problem Proof ($\sin x$) Problem: Integration of $\displaystyle\int_{-1}^1 {\sin x\over 1+x^2} \; dx = 0 $
(according to WolframAlpha Definite Integral Calculator)
But I don't understand how. I tried to prove using integration by parts.
Here's the work:
$$
\int_{-1}^1 {\sin x\over 1+x^2} \; dx = \int_{-1... | You don’t have to do any actual integration. Let $$f(x)=\frac{\sin x}{1+x^2}\;$$ then $$f(-x)=\frac{\sin(-x)}{1+(-x)^2}=\frac{-\sin x}{1+x^2}=-f(x)\;,$$ so $f(x)$ is an odd function. The signed area between $x=-1$ and $x=0$ is therefore just the negative of the signed area from $x=0$ to $x=1$, and the whole thing cance... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/127122",
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Proving an asymptotic lower bound for the integral $\int_{0}^{\infty} \exp\left( - \frac{x^2}{2y^{2r}} - \frac{y^2}{2}\right) \frac{dy}{y^s}$ This is a follow up to the great answer posted to https://math.stackexchange.com/a/125991/7980
Let $ 0 < r < \infty, 0 < s < \infty$ , fix $x > 1$ and consider the integral
$$ I... | I think that if you make the change of variables $y = \lambda z$ with $\lambda = x^{\frac 2 {r+1} }\;i.e. \frac {x^2} {\lambda^{2r}} = \lambda^2$ you convert it into $\lambda ^{s-1} \int e^{-\lambda^2 \frac 12(z^{-2r} + z^2)} \frac {dz}{z^s}$ which looks like a fairly normal laplace type expansion.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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two subgroups of $S_{n}$ and $S_{m}$ If $H\subseteq S_{n}$ and $K\subseteq S_{m}$ how can I then show that I can think of $H\times K$ as it was a subgroup of $S_{m+n}$?
| In hopes of getting this off the Unanswered list, here’s a hint expanding on Jyrki’s first comment.
$K$ is a group of permutations of the set $\{1,\dots,m\}$, so each $k\in K$ is a bijection $$k:\{1,\dots,m\}\to\{1,\dots,m\}\;.$$ For each $k\in K$ let $$\hat k:\{n+1,\dots,n+m\}\to\{n+1,\dots,n+m\}:n+i\mapsto n+k(i)\;,$... | {
"language": "en",
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Root Calculation by Hand Is it possible to calculate and find the solution of $ \; \large{105^{1/5}} \; $ without using a calculator? Could someone show me how to do that, please?
Well, when I use a Casio scientific calculator, I get this answer: $105^{1/5}\approx " 2.536517482 "$. With WolframAlpha, I can an even more... | Another way of doing this would be to use logarithm, just like Euler did:
$$
105^{1/5} = \mathrm{e}^{\tfrac{1}{5} \log (105)} = \mathrm{e}^{\tfrac{1}{5} \log (3)}
\cdot \mathrm{e}^{\tfrac{1}{5} \log (5)} \cdot \mathrm{e}^{\tfrac{1}{5} \log (7)}
$$
Use $$\log(3) = \log\left(\frac{2+1}{2-1}\right) = \log\left(1+\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/127310",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "92",
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Prove that $||x|-|y||\le |x-y|$ I've seen the full proof of the Triangle Inequality
\begin{equation*}
|x+y|\le|x|+|y|.
\end{equation*}
However, I haven't seen the proof of the reverse triangle inequality:
\begin{equation*}
||x|-|y||\le|x-y|.
\end{equation*}
Would you please prove this using only the Triangle Inequal... | For all $x,y\in \mathbb{R}$, the triangle inequality gives
\begin{equation}
|x|=|x-y+y| \leq |x-y|+|y|,
\end{equation}
\begin{equation}
|x|-|y|\leq |x-y| \tag{1}.
\end{equation}
Interchaning $x\leftrightarrow y$ gives
\begin{equation}
|y|-|x| \leq |y-x|
\end{equation}
which when rearranged gives
\begin{equation}
-\left... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/127372",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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If a holomorphic function $f$ has modulus $1$ on the unit circle, why does $f(z_0)=0$ for some $z_0$ in the disk? I don't understand the final step of an argument I read.
Suppose $f$ is holomorphic in a neighborhood containing the closed unit disk, nonconstant, and $|f(z)|=1$ when $|z|=1$. There is some point $z_0$ in ... | If not, consider $g(z)=\frac 1{f(z)}$ on the closure of the unit disc. We have $|g(z)|=1$ if $|z|=1$ and $|g(z)|>1$ if $|z|<1$. Since $g$ is holomorphic on the unit disk, the maximum modulus principle yields a contradiction.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Is the solution to a driftless SDE with Lipschitz variation a martingale? If $\sigma$ is Lipschitz, with Lipschitz constant $K$, and $(X_t)_{t\geq 0}$ solves
$$dX_t=\sigma(X_t)dB_t,$$ where $B$ is a Brownian motion, then is $X$ a martingale? I'm having difficulty getting past the self-reference here. I tried showing th... | Yes.
$$[X]_t = \int_0^t\sigma(X_u)^2du,$$
so
$$\begin{align} \mathbb{E}([X]_t) \le \int_0^t \mathbb{E}\left[(x_0 + K|X_u-x_0|)^2\right]du.\\
\end{align}
$$
$X$ is locally bounded in $L^2$. See, for example, Karatzas and Shreve equation 5.2.15 (p. 289). So it follows easily that $\mathbb{E}([X]_t)<\infty$, for each $t... | {
"language": "en",
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Find the radius of the circle? Two Circle of an equal of an radii are drawn , without any overlap , in a semicircle of radius 2 cm.
If these are the largest possible circles that the semicircle can accomodate , then
what is the radius of each of the circles?
Thanks in advance.
| Due to symmetry two circles in a semicircle is the same problem as one in a quatercircle or four in a full circle. If we look at a quatercircle originating at the origin, with radius $r$ and completely contained in the first quadrant, then the circle has to be centered at a point $(c,c)$, touching the x axis, the y axi... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Proving that a set is countable by finding a bijection
$Z$ is the set of non-negative integers including $0$. Show that $Z \times Z \times Z$ is countable by constructing the actual bijection $f: Z\times Z\times Z \to \mathbb{N}$ ($\mathbb{N}$ is the set of all natural numbers). There is no need to prove that it is a ... | If you don't need an actual "formula", then you can write
$$
\mathbb{Z}\times\mathbb{Z}\times\mathbb{Z} = \bigcup_{n=0}^\infty \{ (x,y,z)\in \mathbb{Z}^3 : |x|+|y| +|z| = n \}
$$
and then rely on the fact that each term in this union is a finite set.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/127695",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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"answer_id": 2
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Do countable unital rings with uncountably many distinct right ideals have uncountably many maximal right ideals? Suppose we are given a countable unital ring $R$ with uncountably many distinct right ideals. Does it follow from this that $R$ has uncountably many maximal right ideals?
| No.
Take $R= \mathbb Q[X_0,X_1,...,X_n,...]/\langle X_iX_j\mid i,j\in \mathbb N\rangle=\mathbb Q[x_0,x_1,...,x_n,...]$
The only maximal ideal (actually only prime ideal !) is
$\langle x_0,x_1,...,x_n,...\rangle$ but $R$ has a family of distinct ideals indexed by the uncountably many subsets $P\subset \mathbb N$, ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
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Use of determinants I have been teaching myself maths (primarily calculus) throughout this and last year, and was stumped with the use of determinants. In the math textbooks I have, they simply show how to compute a determinant and some properties about them (i.e. Cramer's Rule), but not why they are used and how they... | Here's another example of use of determinants:
Let $F$ be a field, let $K$ be a field containing $F$, and finite-dimensional as a vector space over $F$. Let $\alpha$ be an element of $K$. The map $T:K\to K$ given by $T(x)=\alpha x$ is a linear transformation. Given a basis for $K$ as a vector space over $F$, one can fi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/127834",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "38",
"answer_count": 3,
"answer_id": 2
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What is standard coordinates? What is meant by the word standard in "Euclidean space is special in having a standard set of global coordinates."?
Then "A manifold in general does not have standard coordinates"
This makes me think standard means something else then 'most common used'.
Is R^n special in any sense, as a m... | Usually when we write "$\mathbb{R}^n$" we are thinking of an explicit description of it as $n$-tuples of real numbers. This description "is" the standard set of global coordinates, namely the coordinate functions $x_i$. But this description isn't part of $\mathbb{R}^n$ "as a manifold", in that it contains more informat... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/127981",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
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What is the correct terminology for Permutation & Combination formulae that allow repeating elements. Let me explain by example.
Q: Given four possible values, {1,2,3,4}, how many 2 value permutations are there ? | In "permutations", the order matters. In "combinations", the order does not matter.
The basic rules of counting are the Product Rule and the Sum Rule. See here, for example.
*
*Permutations with repetitions allowed:
If you have $n$ objects, and you want to count how many permutations of length $m$ there are: there a... | {
"language": "en",
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Where is the highest point of $f(x)=\sqrt[x]{x}$ in the $x$-axis? I mean, the highest point of the $f(x)=\sqrt[x]{x}$ is when $x=e$.
I'm trying to calculate how can I prove that or how can it be calculated.
| Well, write $$f(x) = e^{\frac{1}{x}\ln(x)}$$ and differentiate and set equal to 0 to get:
$$\dfrac{d}{dx}(e^{\frac{1}{x}\ln(x)})=\bigg(-\frac{1}{x^2}\ln(x)+\frac{1}{x^2}\bigg)e^{\frac{1}{x}\ln(x)}=0$$
Which implies (after dividing by the exponential term) that
$$\frac{1}{x^2}(1-\ln(x))=0$$
Whence $1=\ln(x)$ or $x=e$.
N... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/128114",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
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Prove that the Lie derivative of a vector field equals the Lie bracket: $\frac{d}{dt} ((\phi_{-t})_* Y)|_{t=0} = [X,Y]$ Let $X$ and $Y$ be vector fields on a smooth manifold $M$, and let $\phi_t$ be the flow of $X$, i.e. $\frac{d}{dt} \phi_t(p) = X_p$. I am trying to prove the following formula:
$\frac{d}{dt} ((\phi_{-... | Here is a simple proof which I found in the book "Differentiable Manifolds: A Theoretical Phisics Approach" of G. F. T. del Castillo. Precisely it is proposition 2.20.
We denote $(\mathcal{L}_XY)_x=\frac{d}{dt}(\phi_t^*Y)_x|_{t=0},$ where $(\phi^*_tY)_x=(\phi_{t}^{-1})_{*\phi_t(x)}Y_{\phi_t(x)}.$
Recall also that $(Xf)... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/128195",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "20",
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Non-principal Ideals in a Complete Lattice Given a complete lattice is it possible to have orderideals which are not principal? Can one not always just join together every element of the ideal to get its maximal, generating element? What about for frames?
Thanks!
| Short answer is no. In order to get a counterexample, consider the Boolean algebra of subsets of the natural numbers $\mathcal P(\mathbb N)$ and let $FIN$ denote the ideal of finite subsets of $\mathbb N$. Observe that $\mathcal P(\mathbb N)$ is a complete lattice and $FIN$ is a not principal ideal.
| {
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"answer_id": 0
} |
Ideal not finitely generated Let $R=\{a_0+a_1 X+a_2 X^2 +\cdots + a_n X^n\}$, where $a_0$ is an integer and the rest of the coefficients are rational numbers.
Let $I=\{a_1 X+a_2 X^2+\cdots +a_n X^n\}$ where all of the coefficients are rational numbers.
Prove that I is an ideal of R.
Show further that I is not finitely ... | This ring is an example of a Bézout domain that is not a unique factorization domain (since not all nonzero noninvertible elements decompose into irreducibles in the first place; for instance $X$ does not). The wikipedia page gives a proof of the Bézout property, namely that any finitely generated ideal is in fact a pr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/128300",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 1
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Help Calculating a certain integral I study an article, and I got stuck on a problem of calculating an integral. Whatever I do, I do not get the result mentioned there. The notations are $u,\tilde u$ are functions defined on $\Omega \subset \Bbb{R}^N$ with values in $\Bbb{R}^n$:
$$ \eta_\varepsilon = \int_\Omega \tilde... | Independent of the details of your calculation, the book's answer can't be right since $|B_\epsilon|$ clearly goes as $\epsilon\omega_N$ and not as $\epsilon^{(N-1)/N}\omega_{N-1}$. It looks as if they were calculating the integral over the sphere rather than the ball, but I don't see why they would do that. Are you su... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/128363",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
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How to determine the number of directed/undirected graphs? I'm kind of stuck on this homework problem, could anyone give me a springboard for it? If we have $n\in\mathbb{Z}^+$, and we let the set of vertices $V$ be a set of size $n$, how can we determine the number of directed graphs/undirected graphs/graphs with loops... | A start: We will show how to count labelled, loopless, undirected graphs. There are $\binom{n}{2}$ ways to choose a set $\{u,v\}$ of two vertices. For every such set, we say yes or no depending on whether we have decided to join $u$ and $v$ by an edge. Alternately, but somewhat less concretely, let $P$ be the set of a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/128439",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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} |
Prove Algebric Identity
Possible Duplicate:
Value of $\sum\limits_n x^n$
Given $a\in\mathbb{R}$ and $0<a<1$
let $(X_n)$ be a sequence defined: $X_n=1+a+a^2+...+a^n$, $\forall n\in\mathbb{N}$.
How do I show that $X_n=\frac{1-a^{n+1}}{1-a}$
Thanks.
| $$\begin{align*}
(1-a)(1+a+a^2+\dots+a^n)&=(1+a+\dots+a^n)-a(1+a++\dots+a^n)\\
&=(1+\color{red}{a+\dots+a^n})-(\color{red}{a+a^2++\dots+a^n}+a^{n+1})\\
&=1-a^{n+1}
\end{align*}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/128517",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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Limit and continuity For this question, should I use differentiation method or the integration method ?
$\lim_{x\to \infty} (\frac{x}{x+2})^{x/8}$
this is what i got so far:
Note: $\lim \limits_{n\to\infty} [1 + (a/n)]^n = e^{\underline{a}}\ldots\ldots (1)$
$$
L = \lim \left[\frac{x}{x+2}\right]^{x/8} = \lim\left[\fr... | You are probably intended to use the fact that
$$\lim_{t\to\infty}\left(1+\frac{1}{t}\right)^t=e.$$
A manipulation close to what you were doing gets us there.
We have
$$\left(1+\frac{2}{x}\right)^{x/8}=\left(\left(1+\frac{2}{x}\right)^{x/2}\right)^{1/4}.$$
Let $t=\frac{x}{2}$. Then $\frac{2}{x}=\frac{1}{t}$. Note tha... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/128568",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
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Does the group of Diffeomorphisms act transitively on the space of Riemannian metrics? Let $M$ be a smooth manifold (maybe compact, if that helps). Denote by $\operatorname{Diff}(M)$ the group of diffeomorphisms $M\to M$ and by $R(M)$ the space of Riemannian metrics on $M$. We obtain a canonical group action
$$ R(M) \t... | This map will not be transitive in general. For example, if $g$ is a metric and $\phi \in Diff(M)$ then the curvature of $\phi^* g$ is going to be the pullback of the curvature of $g$. So there's no way for a metric with zero curvature to be diffeomorphic to a manifold with non-zero curvature. Or for example, if $g$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/128651",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
"answer_count": 3,
"answer_id": 0
} |
What matrices preserve the $L_1$ norm for positive, unit norm vectors? It's easy to show that orthogonal/unitary matrices preserve the $L_2$ norm of a vector, but if I want a transformation that preserves the $L_1$ norm, what can I deduce about the matrices that do this? I feel like it should be something like the col... | The matrices that preserve the set $P$ of probability vectors are those whose columns are members of $P$. This is obvious since if $x \in P$, $M x$ is a convex combination of the columns of $M$ with coefficients given by the entries of $x$. Each column of $M$ must be in $P$ (take $x$ to be a vector with a single $1$ ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/128702",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
"answer_count": 2,
"answer_id": 0
} |
Summing numbers which increase by a fixed amount (arithmetic progression) An auditorium has 21 rows of seats. The first row has 18 seats, and each succeeding row has two more seats than the previous row. How many seats are there in the auditorium?
Now I supposed you could use sigma notation since this kind of problem r... | You could write the total using sigma notation as $$\sum_{k=0}^{20}(18+2k)\,$$ among many other ways, but I’m pretty sure that what’s wanted here is the actual total. You can add everything up by hand, which is a bit tedious, or you can use the standard formula for the sum of an arithmetic progression, if you know it, ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/128786",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 6,
"answer_id": 1
} |
Maximum area of rectangle with fixed perimeter. How can you, with polynomial functions, determine the maximum area of a rectangle with a fixed perimeter.
Here's the exact problem—
You have 28 feet of rabbit-proof fencing to install around your vegetable garden. What are the dimensions of the garden with the largest ar... | The result you need is that for a rectangle with a given perimeter the square has the largest area. So with a perimeter of 28 feet, you can form a square with sides of 7 feet and area of 49 square feet.
This follows since given a positive number $A$ with $xy = A$ the sum $x + y$ is smallest when $x = y = \sqrt{A}$.
Yo... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/128825",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
"answer_count": 4,
"answer_id": 2
} |
inequality $(a+c)(a+b+c)<0$, prove $(b-c)^2>4a(a+b+c)$
If $(a+c)(a+b+c)<0,$
prove $$(b-c)^2>4a(a+b+c)$$
I will use the constructor method that want to know can not directly prove it?
| Consider the quadratic
$$ f(x) = ax^2 - (b-c)x + (a+b+c) $$
$$f(1)f(0) = 2(a+c)(a+b+c) \lt 0$$
Thus if $a \neq 0$, then this has a real root in $(0,1)$ and so
$$(b-c)^2 \ge 4a(a+b+c)$$
If $(b-c)^2 = 4a(a+b+c)$, then we have a double root in $(0,1)$ in which case, $f(0)$ and $f(1)$ will have the same sign.
Thus $$(b-c)^... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/128898",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Disprove Homeomorphism I have a problem that puzzles me. I need to show that the two sets
$A = \{(x,y) \in \mathbb{R}^2 \, \, \vert \, \, |x| \leq 1 \}$
and
$B = \{(x,y) \in \mathbb{R}^2 \, \, \vert \, \, x \geq 0 \}$
are not homeomorphic; but I'm not able to figure out how start or what I need to arrive at.
| What about this: if A and B are not homeomorphic, then there exists a continuous bijection $ f: \, A \rightarrow B $ such that $ f^{-1} $ is not continuous. Then look at the function $f(x,y) = \left( \tan\left( \frac{(1+x) \pi}{4} \right), y \right)$. Problem at $ x = 1 $ ?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/128952",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 3
} |
Evaluating $\int \dfrac {2x} {x^{2} + 6x + 13}dx$ I am having trouble understanding the first step of evaluating
$$\int \dfrac {2x} {x^{2} + 6x + 13}dx$$
When faced with integrals such as the one above, how do you know to manipulate the integral into:
$$\int \dfrac {2x+6} {x^{2} + 6x + 13}dx - 6 \int \dfrac {1} {x^{2}... | Just keep in mind which "templates" can be applied. The LHS in your second line is "prepped" for the $\int\frac{du}{u}$ template. Your choices for a rational function with a quadratic denominator are limited to polynomial division and then partial fractions for the remainder, if the denominator factors (which it always... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/129093",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 0
} |
Question about primes in square-free numbers For any prime, what percentage of the square-free numbers has that prime as a prime factor?
| Let $A(n)=\{\mathrm{squarefree~numbers~\le n}\}$ and $B_p(n)=\{x\in A(n); p\mid x\}$.
Then the asymptotic density of $B_p$ in $A$ is $b_p = \lim_{n\rightarrow \infty} |B_p(n)|/|A(n)|$. (It seems from the comments that this is not what @RudyToody is looking for, but I thought it's worth writing up anyway.) Let the densi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/129148",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 1,
"answer_id": 0
} |
Cross product in complex vector spaces When inner product is defined in complex vector space, conjugation is performed on one of the vectors. What about is the cross product of two complex 3D vectors?
I suppose that one possible generalization is
$A\otimes B \rightarrow \left ( A\times B \right )^*$ where $\times$ ... | Yes, this is correct definition. If $v$, $w$ are perpendicular vectors in $\Bbb C^3$ (according to hermitian product) then $v,w,v\times w$ form matrix in $SU_3$.
We can define complex cross product using octonion multiplication (and vice versa). Let's use Cayley-Dickson formula twice: $$(a+b^\iota)(c+d^\iota)=ac-\bar ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/129227",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
"answer_count": 3,
"answer_id": 1
} |
What's the meaning of a set to the power of another set? ${ \mathbb{N} }^{ \left\{ 0,1 \right\} }$ and ${ \left\{ 0,1 \right\} }^{ \mathbb{N} }$ to be more specific, and is there a countable subset in each one of them? How do I find them?
| The syntax $X^Y$ where $X$ and $Y$ are sets means the set of functions from $Y$ to $X$.
Recall for the following that the von Neumann ordinal $2 = \left\{0,1\right\}$.
We often identify the powerset $\mathcal{P}(X)$ with the set of functions $2^X$, since we can think of the latter set as the set of characteristic func... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/129301",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "24",
"answer_count": 2,
"answer_id": 1
} |
Dual of a finite dimensional algebra is a coalgebra (ex. from Sweedler) Let $(A, M, u)$ be a finite dimensional algebra where $M: A\otimes A \rightarrow A$ denotes multiplication and $u: k \rightarrow A$ denotes unit.
I want to prove that $(A^*, \Delta, \varepsilon) $ is a colagebra where
$\Delta: A^*\rightarrow A^* \... | Given $M^*c^*=:d^* \in (A \otimes A)^*$, $\rho^{-1}(d^*)=d_1^* \otimes d_2^* \in A^* \otimes A^*$, where $d_1^*(a)=d^*(a \otimes 1)$ and $d_2^*(a)=d^*(1 \otimes a)$. Notice that $\rho (d_1^* \otimes d_2^*)=d^*$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/129361",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
"answer_count": 1,
"answer_id": 0
} |
Square Root Of A Square Root Of A Square Root Is there some way to determine how many times one must root a number and its subsequent roots until it is equal to the square root of two or of the root of a number less than two?
sqrt(16)=4
sqrt(4)=2
sqrt(2) ... 3
--
sqrt(27)=5.19615...
sqrt(5.19615...)=2.27950...
sqrt(2.2... | Let's do this by example, and I'll let you generalize.
Say we want to know about $91$, one of my favorite numbers because it is the lowest number that I think most people might, at first thought, say is prime even though it isn't (another way of saying it isn't divisible by the 'easy-to-see' primes).
Well, I note that ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/129446",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 2
} |
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