Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Isoperimetric inequalities of a group How do you transform isoperimetric inequalities of a group to the of Riemann integrals of functions of the form $f\colon \mathbb{R}\rightarrow G$ where $G$ is a metric group so that being $\delta-$hyperbolic in the sense of Gromov is expressible via Riemann integration?
In other ... | You can do this using metric currents in the sense of Ambrosio-Kirchheim. This is a rather new development of geometric measure theory, triggered by Gromov and really worked out only in the last decade. I should warn you that this is rather technical stuff and nothing for the faint-hearted.
Urs Lang has a set of nice l... | {
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"url": "https://math.stackexchange.com/questions/27731",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Why do we restrict the definition of Lebesgue Integrability? The function $f(x) = \sin(x)/x$ is Riemann Integrable from $0$ to $\infty$, but it is not Lebesgue Integrable on that same interval. (Note, it is not absolutely Riemann Integrable.)
Why is it we restrict our definition of Lebesgue Integrability to absolutely... | I'll add an additional answer based on a response from my measure theory professor, as it may be of use.
Essentially, his response was that the purpose of Lebesgue Integration is to make the set of integrable functions complete. (Recall that we can form a sequence of Riemann Integrable functions that converge to a fun... | {
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Calculate the unknown I've kind of forgotten the name of the following statements:
9x = 11
10x = 9y
What are they called? And how do you solve them?
| These such statements that you have written are called algebraic equations of one and two variables (single-ly), namely $x \text{ and } y$. Both equations together are what is know as a linear system of equations with two unknown variables (those variables being $x \text{ and } y$). There are a variety ways to solve th... | {
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Death process (stochastics)
From what I understand, the question is asking me to find P(X(t) = n | X(0) = N). I know that with a linear death rate this probability is (N choose n) * [e^(-alpha*t)]^n * [1 - e^(-alpha*t)]^N-n but I don't think this is true for a constant death rate. Any help on how to approach this woul... | Some hints:
To find $P_n(t) = Pr(X(t)=n|X(0)=N)$ you need to find the probability that exactly $N-n$ deaths have happened by time $t$. This looks Poisson with a suitable parameter.
The reason $n=0$ has a different form is that there is an unstated assumption that the population cannot fall below $0$, despite the liter... | {
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the symbol for translation, transformation, or conversion What is the symbol for demonstrating syntactic conversion (transformation or translation)? For example, I want to show a calculation sequence, from $ \neg ( A \wedge B ) $ to $ \neg A \vee \neg B $. Now I just use $ \vdash $: $ \neg ( A \wedge B ) \vdash \neg A... | You could use both $\Leftrightarrow$ (\Leftrightarrow) and $\equiv$ (\equiv), meaning that the statements are logically equivalent.
| {
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expected area of a triangle determined by randomly placed points Three points are placed at independently and at random in a unit square. What is the expected value of the area of the triangle formed by the three points?
| Here's a perl script that confirms the answer Shai linked to via a Monte Carlo approach.
#!/usr/bin/perl -w
$numTrials = 1000000 ;
sub distance {
my $point1 = $_[0] ;
my $point2 = $_[1] ;
return sqrt(($x[$point1]-$x[$point2])**2 + ($y[$point1]-$y[$point2])**2) ;
}
sub heron {
my $a = $legLength[$_[0]] ;
... | {
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Finding the optimum supply quantity when there is uncertainty in forecast This is actually a quiz that will be needed in a real life food stall! I need to decide how much stock to supply for my pumpkin soup stall. I sell each soup for $5$ dollars a cup, and let's say my ingredients cost is $1$ dollar. Therefore an outc... | This appears to be an instance of the Newsvendor Model. If the probability distribution of your demand is known, a (more-or-less) closed-form solution exists in terms of your costs and profit.
| {
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exponential equation $$\sqrt{(5+2\sqrt6)^x}+\sqrt{(5-2\sqrt6)^x}=10$$
So I have squared both sides and got:
$$(5-2\sqrt6)^x+(5+2\sqrt6)^x+2\sqrt{1^x}=100$$
$$(5-2\sqrt6)^x+(5+2\sqrt6)^x+2=100$$
I don't know what to do now
| You don't have to square the equation in the first place.
Let $y = \sqrt{(5+2\sqrt{6})^x}$, then $\frac{1}{y} = \sqrt{(5-2\sqrt{6})^x}$. Hence you have $y + \frac{1}{y} = 10$ i.e. $y^2 + 1 = 10y$ i.e. $y^2-10y+1 = 0$.
Hence, $(y-5)^2 =24 \Rightarrow y = 5 \pm 2 \sqrt{6}$.
Hence, $$\sqrt{(5+2\sqrt{6})^x} = 5 \pm 2\sqrt{... | {
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A first order sentence such that the finite Spectrum of that sentence is the prime numbers The finite spectrum of a theory $T$ is the set of natural numbers such that there exists a model of that size. That is $Fs(T):= \{n \in \mathbb{N} | \exists \mathcal{M}\models T : |\mathcal{M}| =n\}$ . What I am asking for is a f... | This exists due to very general results, namely that the set of primes is rudimentary. See this excellent survey on spectra: Durand et al. Fifty Years of the Spectrum Problem: Survey and New Results. The same holds true for all known "natural" number theoretic functions. Indeed, the authors remark in section 4.2 that "... | {
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How to simplify trigonometric inequality? $| 3 ^ { \tan ( \pi x ) } - 3 ^ { 1 - \tan ( \pi x ) } | \geq 2$
| Let $y=3^{\tan(\pi x)}$ so that $3^{1-\tan(\pi x)}=\frac{3^1}{3^{\tan(\pi x)}}=\frac{3}{y}$. Now, your inequality becomes $|y-\frac{3}{y}|\ge 2$.
This can be solved with the boundary algorithm (sometimes called the test-point method)—solve the corresponding equation, $|y-\frac{3}{y}|=2$, plot those solutions on a numb... | {
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If f is surjective and g is injective, what is $f\circ g$ and $g\circ f$? Say I have $f=x^2$ (surjective) and $g=e^x$ (injective), what would $f\circ g$ and $g\circ f$ be? (injective or surjective?)
Both $f$ and $g : \mathbb{R} \to \mathbb{R}$.
I've graphed these out using Maple but I don't know how to write the proof,... | When you write $x$ in $f(x)=x^2$, it is a "dummy variable" in that you can put in anything in the proper range (here presumably the real numbers). So $f(g(x))=(g(x))^2$. Then you can expand the right side by inserting what you know about $g(x)$. Getting $g(f(x))$ is similar. Then for the injective/surjective part y... | {
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Is Lagrange's theorem the most basic result in finite group theory? Motivated by this question, can one prove that the order of an element in a finite group divides the order of the group without using Lagrange's theorem? (Or, equivalently, that the order of the group is an exponent for every element in the group?)
The... | I am late...
Here is a proposal, probably not far from Ihf's answer.
For $a \in G$ of order $p$,
define the binary relation
$x\cal Ry$ : $\exists k\in \mathbb{N} ; k<p$ such that $y=a^kx$
$\cal R$ is an equivalence relation on $G$ and sets up a partition of $G$.
A class is defined by $C_x=\left\{a^kx|k=0,1,\ldots,p-... | {
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Incremental calculation of inverse of a matrix Does there exist a fast way to calculate the inverse of an $N \times N$ matrix, if we know the inverse of the $(N-1) \times (N-1)$ sub-matrix?
For example, if $A$ is a $1000 \times 1000$ invertible matrix for which the inverse is known, and $B$ is a $1001 \times 1001$ matr... | Blockwise inverse
| {
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Simple 4-cycle permutation I call a 4-cycle permutation simple if I can write it as $(i,i+1,i+2,i+3)$ so $(2,3,4,5)$ is a simple 4-cycle but $(1,3,4,5)$ is not. I want to write $(1,2,3,5)$ as a product of simple 4-cycles. So this is what I did:
$$
(1,2,3,5)=(1,2)(1,3)(1,5)
$$
but
$$\begin{align}
(1,3)&=(2,3)(1,2)(2,... | I'm strictly shooting from the hip (i.e. this is just instinct), but it might help if you consider the following:
(1) The 4th powers of 4-cycles are unity. I.e. $(1234)^4 = e$ where $e$ is the identity.
(2) This means that inverses exist. I.e. $(1234)^3 (1234)^1 = e$.
(3) And this gives a suggestion for a way of walkin... | {
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kernel maximal ideal If I have a homomorphism $\phi : R \rightarrow S$ between integral domains, how can I show that if the kernel is non-zero then it is a maximal ideal in R?
| The kernel of a ring homorphism $f\colon R\to S$ is maximal if and only if $f(R)$ is a simple ring (has no proper nontrivial two-sided ideals). In the case of a commutative ring with identity $1\neq 0$, a ring is simple if and only if it is a field.
So a homomorphism $\phi\colon R\to S$ between integral domains (which... | {
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Hints for proving some algebraic integer rings are Euclidean In my book - Algebraic Number Theory and Fermat's Last Theorem Ian Stewart, David Tall - there is the exercise:
*
*Prove that the ring of integers of $\mathbb{Q}(\zeta_5)$ (the 5'th cyclotomic ring) is Euclidean.
I can prove that the integers of $\mathbb... | HINT $\ \ $ It is norm Euclidean, i.e. the absolute value of the norm serves as a Euclidean function. For a nice survey see Lemmermeyer: The Euclidean algorithm in algebraic number fields.
| {
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Bijection between an open and a closed interval Recently, I answered to this problem:
Given $a<b\in \mathbb{R}$, find explicitly a bijection $f(x)$ from
$]a,b[$ to $[a,b]$.
using an "iterative construction" (see below the rule).
My question is: is it possible to solve the problem finding a less exotic function?
I m... | Define a bijection $f:(-1,1)\rightarrow[-1,1]$ as follows: $f(x)=2x$ if $|x|=2^{-k}$ for some $k\in\mathbb{N}$; otherwise $f(x)=x$.
| {
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Primary ideals of Noetherian rings which are not irreducible It is known that all prime ideals are irreducible (meaning that they cannot be written as an finite intersection of ideals properly containing them). While for Noetherian rings an irreducible ideal is always primary, the converse fails in general. In a recent... | The answer to the question "are primary ideals of a Noetherian domain irreducible?" is "no". For example take for domain $R=K[x,y]$ polynomials over field $K$. Ideal $I=(x^2,xy,y^2)$ is $(x,y)$-primary but reducible because $I=(x,y^2)\cap (y,x^2)$. Since $R$ is noetherian and domain we have a counterexample.
| {
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Fast growing function Aside from the power, gamma, exponential functions are there any other very fast growing functions in (semi-) regular use?
| Knuth's uparrow notation grows pretty fast. Basically, it goes as follows:
$$a\uparrow b=a^b$$
$$a\uparrow\uparrow b=a^{a^{a^{\dots}}}\bigg\}b\text{ amount of }a's$$
$$a\uparrow\uparrow\uparrow b=\underbrace{a\uparrow\uparrow(a\uparrow\uparrow (\dots(a\uparrow\uparrow a)\dots))}_{b\text{ amount of }a's}$$
etc. you get... | {
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Decomposition of $\Bbb R^n$ as union of countable disjoint closed balls and a null set This is a problem in Frank Jones's Lebesgue integration on Euclidean space (p.57),
$$\mathbb{R}^n = N \cup \bigcup_{k=1}^\infty \overline{B}_k$$
where $\lambda(N)=0$, and the closed balls are disjoint.
could any one give some hint... | This is an idea which I cannot see if it ends up working or not. Maybe someone can?
Consider the set $\mathcal S$ of all families of closed balls in $\mathbb R^n$ which are pairwise disjoint. Ordered by inclusion, this poset satisfies the hypothesis in Zorn's lemma, so there exist maximal elements $S\in\mathcal S$.
One... | {
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Limit Comparison Test Using the limit comparison test how do I find the test the convergence of the sum of $$\frac{1+2^{(n+1)}}{1+3^{(n+1)}}?$$
| The limit comparison test is a good substitute for the comparison test when the inequalities are difficult to establish; essentially, if you have a feel that the series in front of you is "essentially proportional" to another series whose convergence you know, then you can try to use the limit comparison test for it.
H... | {
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How do I equidistantly discretize a continuous function? Let's say I have a piecewise function that's continuous in an interval [1,7]:
$f(x) = \left\{
\begin{array}{lr}
-5x + 6 & : x \in [1,2)\\
-4x + 14 & : x \in [2,3)\\
-0.25x + 2.75 & : x \in [3,7]
\end{array}
\right.$
*
*How wo... | In one dimension, you could try to parametrize the curve $C(t)=\{x(t),y(t)\}$ in such a way that the "velocity" is constant, and then quantize the paramenter $t$ - that is feasible, but that would lead you to equidistant points along the curve, and that's not apparently what you are after.
Elsewhere, the problems seems... | {
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Counting number of sequences under cyclic permutation If I have say $2$ A matrices, $2$ B matrices and $1$ C matrix then I would like to know how many distinct traces of products of all of them can be formed.
Assume that $A$, $B$ and $C$ don't commute with each other.
Like $AABBC$, $CAABB$ and $BCAAB$ are distinct pr... | If you want to count the number of sequences with a fixed number of different types of letters up to cyclic permutation (words that are not guaranteed to have the same trace), then a complete answer is provided by the Polya(-Redfield) enumeration theorem, which is a corollary of Burnside's lemma.
| {
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Understanding proof of Farkas Lemma I've attached an image of my book (Theorem 4.4.1 is at the bottom of the image). I need help understanding what this book is saying.
In the first sentence on p.113:
"If (I) holds, then the primal is
feasible, and its optimal objective is
obviously zero",
They are talking about... | I'll try a very short answer to the question why a dual feasible vector $u$ must verify $b'\cdot u\geq 0$ when the primal LP has a feasible solution. Observe that any $p\geq 0$ would leave $pu$ dual feasible; as such, $b'\cdot u<0$ would make the dual objective unbounded from below, because $pb'\cdot u$ could be as lo... | {
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Why determinant of a 2 by 2 matrix is the area of a parallelogram? Let $A=\begin{bmatrix}a & b\\ c & d\end{bmatrix}$.
How could we show that $ad-bc$ is the area of a parallelogram with vertices $(0, 0),\ (a, b),\ (c, d),\ (a+b, c+d)$?
Are the areas of the following parallelograms the same?
$(1)$ parallelogram with ver... | The oriented area $A(u,v)$ of the parallelogram spanned by vectors $u,v$ is bilinear (eg. $A(u+v,w)=A(u,w)+A(v,w)$ can be seen by adding and removing a triangle) and skew-symmetric. Hence $A(ae_1+be_2,ce_1+de_2)=(ad-bc)A(e_1,e_2)=ad-bc$. (the same works for oriented volumes in any dimension)
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/29128",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Integral $\int{\sqrt{25 - x^2}dx}$ I'm trying to find $\int{\sqrt{25 - x^2} dx}$
Now I know that $\int{\frac{dx}{\sqrt{25 - x^2}}}$ would have been $\arcsin{\frac{x}{5}} + C$, but this integral I'm asking about has the rooted term in the numerator.
What are some techniques to evaluate this indefinite integral?
| Oh man, this wasn't my idea, but I found it in a hint in Schaum's Calculus 5e,
If evaluated as a definite integral, on x=0 to x=5, can consider $ \int{ \sqrt{ 25 - x^2 } dx } $ as the area under a quarter circle of radius 5.
So
$ \int_0^5{ \sqrt{ 25 - x^2 } dx } $
$ = \frac{1}{4} \pi r^2 $
$ = \frac{ 25 \pi }{4} $
This... | {
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Proving $\frac{1}{2}(n+1)<\frac{n^{n+\frac{1}{2}}}{e^{n-1}}$ without induction I want to show that $\displaystyle\frac{1}{2}(n+1)<\frac{n^{n+\frac{1}{2}}}{e^{n-1}}$. But except induction, I do not know how I could prove this?
| This is equivalent to showing $$n \leq 2 \frac{n^{n+\frac{1}{2}}}{e^{n-1}}-1 = 2\left(\frac{n^{n}n^{1/2}e}{e^n}\right)-1 $$
Let $$f(n) = 2\left(\frac{n^{n}n^{1/2}e}{e^n}\right)-n-1$$
Then $f(n) \geq 0$ (i.e. find critical points and use first derivative test).
| {
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"timestamp": "2023-03-29T00:00:00",
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Find 2d point given 2d point and transformation matrix I have a 1x3 matrix representing a point in space:
[x]
[y]
[1]
And a 3x3 matrix representing an affine 2d transformation matrix
[a][b][c]
[d][e][f]
[g][h][i]
How to I multiply the matrices so that I am given the matrix?
[newX]
[newY]
[w]
| Basic matrix multiplication works here.
$$ \begin{bmatrix} c_x \\ c_y \\ c_w \end{bmatrix} = \begin{bmatrix} a & b & c \\ d & e & f \\ g & h & i \end{bmatrix} \begin{bmatrix} x \\ y \\ 1 \end{bmatrix} $$
$$ c_x = a\, x + b\, y + c $$
$$ c_y = d\, x + e\, y + f$$
$$ c_w = g\, x + h\, y + i$$
with your new point at $(x,y... | {
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two examples in analysis I want to ask for two examples in the following cases:
1) Given a bounded sequence $\{a_n\}$, $$\lim_{n\to \infty}{(a_{n+1}-a_n)}=0$$ but $\{a_n\}$ diverges.
2) A function defined on real-line $f(x)$'s Taylor series converges at a point $x_0$ but does not equal to $f(x_0)$.
Thanks for your help... | For 1):
$$
1,\frac12,0,\frac13,\frac23,1,\frac34,\frac24,\frac14,0,\frac15,\frac25,\frac35,\frac45,1,\frac56,\frac46,\frac36,\frac26,\frac16,0,\frac17,\dots
$$
I leave it to you to find an explicit formula for $a_n$.
| {
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How to prove that eigenvectors from different eigenvalues are linearly independent How can I prove that if I have $n$ eigenvectors from different eigenvalues, they are all linearly independent?
| Consider the matrix $A$ with two distinct eigen values $\lambda_1$ and $\lambda_2$. First note that the eigen vectors cannot be same , i.e., If $Ax = \lambda_1x$ and $Ax = \lambda_2x$ for some non-zero vector $x$. Well , then it means $(\lambda_1- \lambda_2)x=\bf{0}$. Since $\lambda_1$ and $\lambda_2$ are scalars , th... | {
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"answer_id": 7
} |
Recursion theory text, alternative to Soare I want/need to learn some recursion theory, roughly equivalent to parts A and B of Soare's text. This covers "basic graduate material", up to Post's problem, oracle constructions, and the finite injury priority method. Is there an alternate, more beginner friendly version of ... | A popular introductory textbook that covers both computability and complexity is Michael Sipser's "Introduction to the Theory of Computation". The computability section concludes with a nice chapter on advanced computability issues.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/29418",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 7,
"answer_id": 5
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Self-Contained Proof that $\sum\limits_{n=1}^{\infty} \frac1{n^p}$ Converges for $p > 1$ To prove the convergence of the p-series
$$\sum_{n=1}^{\infty} \frac1{n^p}$$
for $p > 1$, one typically appeals to either the Integral Test or the Cauchy Condensation Test.
I am wondering if there is a self-contained proof that th... | For every $p>1$, one can reduce the problem to the convergence of some geometric series. Then, either one takes the latter for granted or one proves it by an elementary argument. More details below.
Let $N(k)$ denote the integer part of $a^k$, where the real number $a>1$ is defined by $a^{p-1}=2$. The sum of $1/n^p$ ov... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/29450",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "184",
"answer_count": 9,
"answer_id": 7
} |
How to prove $u = \mathrm{span}\{v_{1},\dots,v_{n}\}$ The vectors $v_{1},\dots,v_{n}$ form a linearly independent set.
Addition of vector $u$ makes this set linearly dependent.
How to prove that $u \in span\{v_{1}\ldots,v_{n}\}$.
I was able to prove the reverse.
| There are $a_0, \dots, a_n$ not all zero, such that
$$a_0u + a_1v_1 + \dots a_nv_n = 0.$$
If $a_0 = 0$, the set will become independent, which is contradictory.
Clearly $a_0 \neq 0$, as $v_1, \dots, v_n$ are linearly independent, and therefore we get
$$u = -\frac{a_1}{a_0}u_1 - \dots - \frac{a_n}{a_0}u_n $$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/29500",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Is the set of all finite sequences of letters of Latin alphabet countable/uncountable? How to prove either? Today in Coding/Cryptography class, we were talking about basic definitions, and the professor mentioned that for a set $A=\left \{ \left. a, b, \dots, z \right \} \right.$ (the alphabet) we can define a set $A^{... | Suppose the 26 letters are "digits" in base-26. Any finite string of letters can be thought of as a unique positive integer in base-26. The positive integers are countable, so your set of strings must be as well.
(Actually, to be precise: letting the symbols in your set be non-zero digits establishes a bijection betw... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/29599",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
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Why is $\arctan(x)=x-x^3/3+x^5/5-x^7/7+\dots$? Why is $\arctan(x)=x-x^3/3+x^5/5-x^7/7+\dots$?
Can someone point me to a proof, or explain if it's a simple answer?
What I'm looking for is the point where it becomes understood that trigonometric functions and pi can be expressed as series. A lot of the information I fin... | Many of the functions you encounter on a regular basis are analytic functions, meaning that they can be written as a Taylor series or Taylor expansion. A Taylor series of $f(x)$ is an infinite series of the form $\sum_{i=0}^\infty a_ix^i$ which satisfies $f(x) = \sum_{i=0}^\infty a_ix^i$ wherever the series converges. ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "41",
"answer_count": 10,
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Largest eigenvalue of a real symmetric matrix If $\lambda$ is the largest eigenvalue of a real symmetric $n \times n$ matrix $H$, how can I show that: $$\forall v \in \mathbb{R^n}, ||v||=1 \implies v^tHv\leq \lambda$$
Thank you.
| Step 1: All Real Symmetric Matrices can be diagonalized in the form:
$
H = Q\Lambda Q^T
$
So, $ {\bf v}^TH{\bf v} = {\bf v}^TQ\Lambda Q^T{\bf v} $
Step 2: Define transformed vector: $ {\bf y} = Q^T{\bf v} $.
So, $ {\bf v}^TH{\bf v} = {\bf y}^T\Lambda {\bf y} $
Step 3: Expand
$ {\bf y}^T\Lambda {\bf y} = \lambda_... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/29698",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
"answer_count": 3,
"answer_id": 1
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Entire one-to-one functions are linear Can we prove that every entire one-to-one function is linear?
| I'll give the "usual" proof.
Note that by Little Picard, $f$ misses at most one point; but it is a homeomorphism onto its image, and the plane minus a point is not simply connected. Thus $f$ is onto $\mathbb{C}$, and hence bijective. Then $f$ has a holomorphic inverse, which is enough to imply $f$ is proper, that is,... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "66",
"answer_count": 6,
"answer_id": 5
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Sum of random 0 and 1 Let $x_i$ be uniformly distributed random variables in the interval $[0,a]$, with $a>0$.
Let f(x) be equal to 1 if x=0, and 0 otherwise.
Let $$S(a)=\sum_{n=1}^\infty f(x_n)$$
What is S(1)?
What is $\lim_{a->0} S(a)$ from positive side?
| First note the following equality of events:
$$
\big \{ \sum\nolimits_{n = 1}^\infty {f(X_n )} > 0 \big \} = \cup _{n = 1}^\infty \{ f(X_n ) = 1\}.
$$
Now, for each $n$,
$$
{\rm P}[f(X_n) = 1] = {\rm P}[X_n = 0] = 0.
$$
Hence,
$$
{\rm P}\big[\sum\nolimits_{n = 1}^\infty {f(X_n )} > 0\big] = {\rm P}[ \cup _... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/29859",
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"source": "stackexchange",
"question_score": "2",
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Is there a generalisation of the distribution ratio From the theory of numbers we have the
Proposition:
If $\mathfrak{a}$ and $\mathfrak{b}$ are mutually prime, then the density of primes congruent to $\mathfrak{b}$ modulo $\mathfrak{a}$ in the set of all primes is the reciprocal of $\phi (\mathfrak{a})$ where $\... | Of course, your "conjecture" is trivially true, because the set of prime divisors of $f(n)$ as $n$ ranges over the integers is determined by $f$. I guess what you meant to ask was "does this set have a density and can it be computed".
The answer to both questions is yes and the relevant result is called Chebotarev's de... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/29924",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Evaluating $\int P(\sin x, \cos x) \text{d}x$ Suppose $\displaystyle P(x,y)$ a polynomial in the variables $x,y$.
For example, $\displaystyle x^4$ or $\displaystyle x^3y^2 + 3xy + 1$.
Is there a general method which allows us to evaluate the indefinite integral
$$ \int P(\sin x, \cos x) \text{d} x$$
What about the c... | Here are some other substitutions that you can try on a rational function of trigonometric functions. We name them Bioche substitution in France.
Let $P(\sin t,\cos t)=f(t)$ where $P(x,y)$ is rational function. Let $\omega(t)=f(t)dt$.
*
*If $\omega(-t)=\omega(t)$, then $u(t)=\cos t$ might be a good substitution.
For... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/29980",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "54",
"answer_count": 3,
"answer_id": 1
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Limits: How to evaluate $\lim\limits_{x\rightarrow \infty}\sqrt[n]{x^{n}+a_{n-1}x^{n-1}+\cdots+a_{0}}-x$ This is being asked in an effort to cut down on duplicates, see here: Coping with abstract duplicate questions, and here: List of abstract duplicates.
What methods can be used to evaluate the limit $$\lim_{x\righta... | Here is one method to evaluate
$$\lim_{x\rightarrow\infty}\sqrt[n]{x^{n}+a_{n-1}x^{n-1}+\cdots+a_{0}}-x.$$
Let $Q(x)=a_{n-1}x^{n-1}+\cdots+a_{0}$ for notational convenience, and notice $\frac{Q(x)}{x^{n-1}}\rightarrow a_{n-1}$ and $\frac{Q(x)}{x^{n}}\rightarrow0$ as $x\rightarrow\infty$. The crux is the factorization ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/30040",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "65",
"answer_count": 6,
"answer_id": 3
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Are the singular values of the transpose equal to those of the original matrix? It is well known that eigenvalues for real symmetric matrices are the same for matrices $A$ and its transpose $A^\dagger$.
This made me wonder:
Can I say the same about the singular values of a rectangular matrix? So basically, are the eig... | Both eigenvalues and singular values are invariant to matrix transpose no matter a matrix is square or rectangular.
The definition of eigenvalues of $A$ (must be square) is the $\lambda$ makes
$$\det(\lambda I-A)=0$$
For $A^T$, $\det(\lambda I-A^T)=0$ is equivalent to $\det(\lambda I-A)=0$ since the determinant is inva... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/30072",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "19",
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"answer_id": 0
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when does derivative of a function coincide with derivative of Fourier series? Example:
for function $$f(x)=x^{3}(1-x)^{3}=\sum f_{s}\exp(2\pi isx)$$ Fourier series of its fourth derivative are different from derivative of its Fourier series
$$f^{(4)}(x)=-360x^{2}+360x-72=\sum g_{s}\exp(2\pi i s x)$$
with $g_{s}\neq(2... | An advertisement for the utility and aptness of Sobolev theory is the perfect connection between $L^2$ "growth conditions" on Fourier coefficients, and $L^2$ notions of differentiability, mediated by Sobolev's lemma that says ${1\over 2}+k+\epsilon$ $L^2$ differentiability of a function on the circle implies $C^k$-ness... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Show $\Sigma_{n\leq X} 1/\phi(n) \sim \log(X)*\Sigma_{k=1}^\infty \mu(k)^2/(k*\phi(k))$ I would like to show that as X approaches infinity,
$$\sum_{n\leq X} \frac{1}{\phi(n)} \sim \log(X)\cdot\sum_{k=1}^{\infty} \frac{\mu(k)^2}{k\cdot\phi(k)}.$$
I have already proven
$$\sum_{n\leq X} \frac{1}{\phi(n)} = \sum_{k\leq X} ... | Found an answer. $\sum_{t\leq X/k} 1/t = \log(X/k) + \gamma + O(1/(X/k))$ When you multiply it out, the sum can be split into four terms:
$$ \sum_{n\leq X} 1/\phi(n) = \log(X)*\sum_{k\leq X} \mu(k)^2/(k*\phi(k)) - \sum_{k\leq X} \mu(k)^2*\log(k)/(k*\phi(k)) $$
$$ + \sum_{k\leq X} \gamma*\mu(k)^2/(k*\phi(k)) + \sum_{k... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/30258",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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problem in set theory QN1 by logical argument verify that {a} is not open for any real number "a".
i guess the set is not open since there is no open interval about "a"
instead the set is said to be closed since any of its complement must be open.
am i correct?
QN2 GIVE TWO COUNTER-EXAMPLES WHICH SHOWS THAT THE IMAGE O... | Hint for Q2:
Must the image of a point in the intersection of two sets be in the intersection of the images of the two sets?
Must a point in the intersection of the images of two sets be the image of a point in the intersection of the two sets?
If the answers to these questions are different, then can you construct a c... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/30326",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Finding invertible polynomials in polynomial ring $\mathbb{Z}_{n}[x]$
Is there a method to find the units in $\mathbb{Z}_{n}[x]$?
For instance, take $\mathbb{Z}_{4}$. How do we find all invertible polynomials in $\mathbb{Z}_{4}[x]$? Clearly $2x+1$ is one. What about the others? Is there any method?
| HINT $\rm\ r_0 + r_1\ x +\:\cdots\: + r_n\ x^n\ $ is a unit in $\rm\:R[x]\:\ \iff\ r_0\:$ is a unit and $\rm\: r_i\: $ is nilpotent for $\rm\ i\ge 1\:.$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/30380",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
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An 'opposite' or 'dual' group? Let $(G, \cdot)$ be a group. Define $(G, *)$ as a group with the same underlying set and an operation $$a * b := b \cdot a.$$ What do you call such a group? What is the usual notation for it? I tried searching for 'dual group' and 'opposite group' with no results. It seems that this group... | A group is a groupoid (meaning a category all of whose arrows are invertible) with one object. If $\star$ denotes the object, group elements correspond to arrows $\star\to \star$, and multiplication $gh$ corresponds to composition $\star\xrightarrow{h} \star\xrightarrow{g} \star$. The opposite category of this category... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Continuous Collatz Conjecture Has anyone studied the real function
$$ f(x) = \frac{ 2 + 7x - ( 2 + 5x )\cos{\pi x}}{4}$$ (and $f(f(x))$ and $f(f(f(x)))$ and so
on) with respect to the Collatz conjecture?
It does what Collatz does on integers, and is defined smoothly on all
the reals.
I looked at $$\frac{ \overbrac... | Yes people have studied that, as well as extending it to the complex plane, e.g.:
Chamberland, Marc (1996). "A continuous extension of the 3x + 1 problem to the real line". Dynam. Contin. Discrete Impuls Systems. 2 (4): 495–509.
Letherman, Simon; Schleicher, Dierk; Wood, Reg (1999). "The (3n + 1)-Problem and Holomorph... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Complicated Logic Proof involving Tautology and Law of Excluded Middle I'm having great difficulty solving the following problem, and even figuring out where to start with the proof.
$$
\neg A\lor\neg(\neg B\land(\neg A\lor B))
$$
Please see the following examples of how to do proofs, I would appreciate it if you could... | The "complicated" formula :
$¬A∨¬(¬B∧(¬A∨B))$
can be re-written, due to the equivalence between $P \rightarrow Q$ and $\lnot P \lor Q$, as :
$A \rightarrow \lnot ((A \rightarrow B) \land \lnot B)$.
But $P \rightarrow Q$ is also equivalent to $\lnot (P \land \lnot Q)$; so the formula it is simply :
$A \rightarrow ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/30584",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How does a region of integration change when making a rotation change of variables? Suppose I have a p-dimensional integral:
$$\int_{0}^{\infty}\int_{0}^{\infty}\dots \int_{0}^{\infty}f(x_1,x_2,\dots,x_p)dx_1dx_2\dots dx_p$$
And then I make a rotation + translation transform:
$$W=A^{T}(X-b)$$
Question: How will the reg... | Maybe you can put
$$\int_{0}^{\infty}\int_{0}^{\infty}\dots \int_{0}^{\infty}f(x_1,x_2,\dots,x_p)dx_1dx_2\dots dx_p$$
$$ = \frac{1}{2^p} \int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\dots \int_{-\infty}^{\infty}f(|x_1|,|x_2|,\dots,|x_p|)dx_1dx_2\dots dx_p.$$
Then, if the $f$ functions are even, you can rotate away.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Find the missing term The sequence $10000, 121, 100, 31, 24, n, 20$ represents a number $x$ with respect to different bases. What is the missing number, $n$?
This is from my elementary computer aptitude paper. Is there any way to solve this quickly?
| If the base in the last term is $b$, the number is $2b$. The missing term is then in base $b-1$. Expressed in base $b-1$ the integer is $2(b-1)+2=22$ The third to last term shows that $b-2 \ge 5$ so we are safe from a carry.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How can I evaluate $\sum_{n=0}^\infty(n+1)x^n$? How can I evaluate
$$\sum_{n=1}^\infty\frac{2n}{3^{n+1}}$$?
I know the answer thanks to Wolfram Alpha, but I'm more concerned with how I can derive that answer. It cites tests to prove that it is convergent, but my class has never learned these before. So I feel that ther... | Note that $n+1$ is the number ways to choose $n$ items of $2$ types (repetitions allowed but order is ignored), so that $n+1=\left(\!\binom2n\!\right)=(-1)^n\binom{-2}n$. (This uses the notation $\left(\!\binom mn\!\right)$ for the number of ways to choose $n$ items of $m$ types with repetition, a number equal to $\bin... | {
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Prove that $a_{n}=0$ for all $n$, if $\sum a_{kn}=0$ for all $k\geq 1$ Let $\sum a_{n}$ be an absolutely convergent series such that $$\sum a_{kn}=0$$ for all $k\geq 1$. Help me prove that $a_{n}=0$ for all $n$.
Thank you!
| Let $A(p_1, \dots, p_m)$ be the set of all integers that are divisible by the first $m$ primes $p_1, \dots, p_m$.
I claim that
$$S(m) = \sum_{n \in N \setminus A(p_1, \dots, p_m)} x_n = 0.$$
If this is true, then
$$S(m) = x_1 + \sum_{n \in A} x_n = 0$$
where $A \subseteq \{p_{m+1}, p_{m+1}+1, \dots\}$.
That is,
$$|x_... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/30774",
"timestamp": "2023-03-29T00:00:00",
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Show that $\mathbb{R}^2$ is not homeomorphic to $\mathbb{R}^2 \setminus\{(0,0)\}$ Show that $\mathbb{R}^2$ is not homeomorphic to $\mathbb{R}^2 \setminus \{(0,0)\} $.
| Here's a slightly different way of looking at it that avoids fundamental groups (although has its own messy details to check). One of the spaces, upon removing a compact set, can be separated into two connected components with noncompact closure. The other can't.
| {
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"timestamp": "2023-03-29T00:00:00",
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All natural solutions of $2x^2-1=y^{15}$ How can I find all positive integers $x$ and $y$ such that $2x^2-1=y^{15}$?
PS. See here.
| I was working on Byron's suggestion before he made it but it took me a while because I'm not a real number theorist. And, I used Sage. I consider $2y^2 = x^3 + 1$ and I want to put this in a form where the coefficient of $y^2$ and $x^3$ are 1 so I multiply both sides by 1/2 first and then use the transform $(x,y) \ma... | {
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How to compare randomness of two sets of data? Given two sets of random numbers, is it possible to say that one set of random numbers has a greater degree of randomness when compared to the other?
Or one set of numbers is more random when compared to the other?
EDIT:
Consider this situation:
A hacker needs to know the ... | There are randomness tests. Some tests are powerful enough that they will distinguish a human-generated sequence of 100 heads and tails from 100 tosses of a coin with high probability. For example, the distribution of streaks tends to change radically if you reverse every other coin in the human-generated sequence, whi... | {
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"timestamp": "2023-03-29T00:00:00",
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"question_score": "2",
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Is a graph simple, given the number of vertices and the degree sequence? Does there exist a simple graph with five vertices of the following degrees?
(a) 3,3,3,3,2
I know that the answer is no, however I do not know how to explain this.
(b) 1,2,3,4,3
No, as the sum of the degrees of an undirected graph is even.
(c) 1,2... | The answer to both a, and d, is that in fact such graphs exit. It is not hard to find them.
The answer for c is that there cannot be such a graph - since there are 2 vertices with degree 4, they must be connected to all other vertices. Therefore, the vertex with degree one, is an impossibility.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/31119",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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Tips for understanding the unit circle I am having trouble grasping some of the concepts regarding the unit circle. I think I have the basics down but I do not have an intuitive sense of what is going on. Is memorizing the radian measurements and their corresponding points the only way to master this? What are some way... | It is probably useful to memorize a table like this:
\begin{align}
\theta & & \sin\theta & & \cos \theta
\\
0 & & \frac{\sqrt{0}}{2} = 0 & & \frac{\sqrt{4}}{2} = 1
\\
\frac{\pi}{6} & & \frac{\sqrt{1}}{2} = \frac{1}{2} & & \frac{\sqrt{3}}{2}
\\
\frac{\pi}{4} & & \frac{\sqrt{2}}{2} & & \frac{\sqrt{2}}{2}
\\
\frac{\pi}{3}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/31163",
"timestamp": "2023-03-29T00:00:00",
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How to calculate Jacobi Symbol $\left(\dfrac{27}{101}\right)$? How to calculate Jacobi Symbol $\left(\dfrac{27}{101}\right)$?
The book solution
$$\left(\dfrac{27}{101}\right) = \left(\dfrac{3}{101}\right)^3 = \left(\dfrac{101}{3}\right)^3 = (-1)^3 = -1$$
My solution
$$\left(\dfrac{27}{101}\right) = \left(\dfrac{101}{27... | I think it's better to make sure that the number in the lower case is a prime, since there are examples, if I remember rightly, that the Jacobi symbol is 1 but the corresponding quadratic congruence is not solvable; in addition, as already mentioned, you cannot say that $\left(\dfrac{2^b}{p}\right)\ = (-1)^{(p^2 -1)/8}... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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How to prove a number $a$ is quadratic residue modulo $n$? In general, to show that $a$ is quadratic residue modulo $n$? What do I have to show? I'm always struggling with proving a number $a$ is quadratic residue or non-quadratic residue.
For example,
If $n = 2^{\alpha}m$, where $m$ is odd, and $(a, n) = 1$.
Pro... | The correct statement is as below. Note that the special case you mention follows from the fact that $\rm\ a = b^2\ (mod\ 4\:m)\ \Rightarrow\ a = b^2\ (mod\ 4)\:,\:$ but $1$ is the only odd square $\rm\:(mod\ 4)\:,\ $ so $\rm\ a\equiv 1\ (mod\ 4)\:$
THEOREM $\ $ Let $\rm\ a,\:n\:$ be integers, with $\rm\:a\:$ coprime t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/31271",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Find equation of quadratic when given tangents? I know the equations of 4 lines which are tangents to a quadratic:
$y=2x-10$
$y=x-4$
$y=-x-4$
$y=-2x-10$
If I know that all of these equations are tangents, how do I find the equation of the quadratic?
Normally I would be told where the tangents touch the curve, but that ... | As they are symmetric around the origin, the quadratic has no linear term in $x$. So I would put $y^2=ax^2+b$ as any linear term in $y$ can be absorbed into a vertical shift or $y=ax^2+b$ to get the parabolas. Then calculate what $a$ and $b$ need to be to make them tangent. Because we incorporated the symmetry, you ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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Invariant Subspace of Two Linear Involutions I'd love some help with this practice qualifier problem:
If $A$ and $B$ are two linear operators on a finite dimensional complex vector space $V$ such that $A^2=B^2=I$ then show that $V$ has a one or two dimensional subspace invariant under $A$ and $B$.
Thanks!
| Arturo's answer can be condensed to the following:
Let $U_1$, $\ldots$, $U_4$ be the eigenspaces of $A$ and $B$. Letting the simple cases aside we may assume that $U_i\oplus U_j=V$ for all $i\ne j$. We have to produce four nonzero vectors $x_i\in U_i$ that lie in a two-dimensional plane.
For $i\ne j$ denote by $P_{ij}:... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Finding best response function with probabilities (BR) given a normal-matrix representation of the game We are given players 1, 2 and their respective strategies (U, M, D for player 1, L, C, R for player 2) and the corresponding pay-offs through the following table:
$\begin{matrix}
1|2 & L & C & R\\
U & 10, 0 & 0, 1... | I think Carl already gave the right answer. Even though mixed strategies may look better than pure ones, actually they are not.
Suppose that player 1 choose a mixed strategy $\alpha_1=(a,b,c)$. Then the probability of each scenario is given by
$$\begin{matrix}
1|2 & L & C & R\\
U & a/3 & a/3 & a/3 \\
M & b/3 & b/... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Every subgroup $H$ of a free abelian group $F$ is free abelian I am working through a proof that every subgroup $H$ of a free abelian group $F$ is free abelian (for finite rank)
For the inductive step, let $\{ x_1, \ldots, x_n \}$ be a basis of $F$, let $F_n = \langle x_1,\ldots,x_{n-1} \rangle$, and let $H_n = H \cap ... | What you need is the Third Isomorphism Theorem: given a group $G,$ a normal subgroup $K$ of $G$ and a subgroup $H$ of $G$ we have that
$$
HK/K \cong H/H\cap K.
$$
You rightly guessed that the proof uses the Fundamental Isomorphism Theorem. The homomorphism $f : HK \to H/H \cap K$ defined via
$$
f(hk) = h (H\cap K)
... | {
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"timestamp": "2023-03-29T00:00:00",
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number of ordered partitions of integer How to evaluate the number of ordered partitions of the positive integer $ 5 $?
Thanks!
| Counting in binary the groups of 1s or 0s form the partitions. Half are the same so there are 2^(n-1). As to be expected this gives the same results as the gaps method, but in a different order.
Groups
0000 4
0001 3,1
0010 2,1,1
0011 2,2
0100 1,1,2
0101 1,1,1,1
0110 1,2,1
0111 1,3
Gaps
000 4
001 3,1
010 2,2
011... | {
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Looking for a function $f$ such that $f(i)=2(f(i-1)+f(\lceil i/2\rceil))$ I'm looking for a solution $f$ to the difference equation $$f(i)=2(f(i-1)+f(\lceil i/2\rceil))$$ with $f(2)=4$. Very grateful for any ideas.
PS. I've tried plotting the the initial values into "Sloan", but it doesn't seem to recognize the sequen... | Since $f$ is always positive, $f(i) \gt 2f(i-1)$ and so by induction $f(n) \gt 2^n$. $f(\lceil i/2\rceil)$ is then exponentially smaller than $f(i)$, so $2^n$ is the dominant term. Divide out $f$ by the exponential and define $g(n) = 2^{-n}f(n)$; then $g(i) = g(i-1) + 2^{1-\lfloor i/2\rfloor}g(\lceil i/2\rceil)$, wit... | {
"language": "en",
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"source": "stackexchange",
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When is the product of two quotient maps a quotient map? It is not true in general that the product of two quotient maps is a quotient maps (I don't know any examples though).
Are any weaker statements true? For example, if $X, Y, Z$ are spaces and $f : X \to Y$ is a quotient map, is it true that $ f \times {\rm id} : ... | In the category of compactly generated spaces, I think that the product of quotient maps is (always) a quotient map.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "29",
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Having trouble verifying absolute/conditional convergence for these series Greetings,
I'm having trouble applying the tests for convergence on these series; I can never seem to wrap my head around how to determine if they're absolutely convergent, conditionally convergent or divergent.
a) $\displaystyle \sum_{k=1}^{\in... | a) Since $e^x >x$ then $$\frac{\sqrt{k}}{e^{k^3}}<\frac{\sqrt{k}}{k^3}=\frac{1}{k^{5/2}}$$
But $\sum_{k=1}^\infty \frac{1}{k^{5/2}}$ is convergent ($p$-series test). Hence the original series is convergent (hence absolutely convergent since it is a positive series).
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
What is the best book to learn probability? Question is quite straight... I'm not very good in this subject but need to understand at a good level.
| I happened to take an introductory course on probability and statistics on two different universities. In one they used a horrible book, and in the other they used a truly amazing one. It's rare that a book really stands out as fantastic, but it did.
Probability and Statistics for Engineers and Scientists
by Ronald E. ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "159",
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"answer_id": 3
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How to solve implicit solution of IVP The given question is:
$$
dy/dt + 2y = 1\ ;\qquad y(0)= 5/2
$$
when i solve this i get $\ln(-4)=c$
now the problem is how to solve $\ln(-4)$?
| this is a linear differential equation and the solution of this equation is as follow:
IF= = e^∫▒〖2 dt〗
= e^2t
Now solution of equation is:
y e^2t = ∫▒〖1 .e^2t dt〗
y e^2t = e^2t/2 + c ……………………….eq(1)
Now put t=0 and y = 5/2
5/2 = 1/2+ c
C = 2
Putting this in eq(1), we get
y e^2t = e^2t/2+ 2
y=1/2+ 1/2 e^(-2t)
| {
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Are there any random variables so that E[X] and E[Y] exist but E[XY] doesn't? Are there any random variables so that E[X] and E[Y] exist but E[XY] doesn't?
| Having recently discussed here the difference between "$X$ has expectation" (in the wide sense) and "$X$ is integrable", let us give an example where $X$ and $Y$ are integrable (that is, have finite expectation) but $XY$ does not admit an expectation (that is, ${\rm E}(XY)^+ = {\rm E}(XY)^- = \infty$).
Let $Z$ be any n... | {
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Representation of integers How can I prove that every integer $n>=170$ can be written as a sum of five positive squares? (i.e. none of the squares are allowed to be zero).
I know that $169=13^2=12^2+5^2=12^2+4^2+3^2=10^2+8^2+2^2+1^2$, and $n-169=a^2+b^2+c^2+d^2$ for some integers $a$, $b$, $c$, $d$, but do I show it?
T... | Hint: let $n-169 = a^2+b^2+c^2+d^2$; if $a,b,c,d \neq 0$ then ... if $d = 0$ and $a,b,c \neq 0$ then ... if $c = d = 0$ and $a,b \neq 0$ then ... if $b = c = d = 0$ and $a \neq 0$ then ... if $a = b = c = d = 0$ then - wait, that can't happen!
| {
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What is the best way to show that no positive powers of this matrix will be the identity matrix?
Show that no positive power of the matrix $\left( \begin{array}{cc} 1 & 1 \\ 0 & 1 \end{array} \right)$ equals $I_2$.
I claim that given $A^{n}, a_{11} = 1$ and $a_{12} >0, \forall n \in \mathbb{N}$. This is the case for ... | Your solution seems OK to me. You can also find $A^n$ explicitly:
let $E=\left(\begin{array}{cc}0&1\\0&0\end{array}\right)$.
Then $A=I+E$ and $E^2=0$. So $(I+nE)(I+E)=I+(n+1)E$ and so, by induction, $A^n=I+nE\ne I$ for $n\ge1$.
| {
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"timestamp": "2023-03-29T00:00:00",
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Probability distribution for the remainder of a fixed integer In the "Notes" section of Modern Computer Algebra by Joachim Von Zur Gathen, there is a quick throwaway remark that says:
Dirichlet also proves the fact, surprising at first sight, that for fixed $a$ in a division the remainder $r = a \operatorname{rem} b$,... | sos440's answer is correct, but I think it makes the calculation look unnecessarily complicated. The boundaries where the remainder switches between being greater or less than $b/2$ are $a/b=n/2$, that is $b=2a/n$, for $n>2$. If we choose $b$ as a real number uniformly distributed over $[0,a]$, we can calculate the pro... | {
"language": "en",
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"source": "stackexchange",
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Alternative definition for topological spaces? I have just started reading topology so I am a total beginner but why are topological spaces defined in terms of open sets? I find it hard and unnatural to think about them intuitively. Perhaps the reason is that I can't see them visually. Take groups, for example, are rel... | From Wikipedia:
In topology and related branches of mathematics, the Kuratowski closure axioms are a set of axioms which can be used to define a topological structure on a set. They are equivalent to the more commonly used open set definition. They were first introduced by Kazimierz Kuratowski, in a slightly different ... | {
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Basic question about tensor products I almost feel embarrassed to ask this, but I am trying to learn about tensor products (for now over Abelian groups). Here is the definition given:
Let $A$ and $B$ be abelian groups. Their tensor product, denoted by $A \otimes B$, is the abelian group having the following presentatio... | You are missing the relations $(na,b)=(a,nb)=n(a,b)$ for all $n\in\mathbb{Z}$ (recall that abelian groups are $\mathbb{Z}$-modules). Thus
$$a\otimes 0_B=a\otimes (0_{\mathbb{Z}}\cdot 0_B)=(0_{\mathbb{Z}}\cdot a)\otimes 0_B=0_A\otimes 0_B=0.$$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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a question on notation for function spaces
If $X$ is some topological space, such
as the unit interval $[0,1]$, we can
consider the space of all continuous
functions from $X$ to $R$. This is a
vector subspace of $R^X$ since the sum
of any two continuous functions is
continuous and scalar multiplication
i... | This means the space of all functions from $X$ to $R$. Without regard for any structure. Set-theoretic ones.
| {
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Finding x in $a^{x} \bmod b = c$ when values a,b, and c are known? If values $a$, $b$, and $c$ are known, is there an efficient way to find $x$ in the equation: $a^{x} \bmod b = c$?
E.g. finding $x$ in $128^{x}\bmod 209 = 39$.
| A better reference than Wikipedia for the discrete logarithm problem is Andrew Sutherland's 2007 MIT Thesis Order Computations in Generic Groups. Here's an excerpt from p. 14 that provides a concise summary of the current state of knowledge.
| {
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Localization at a prime ideal in $\mathbb{Z}/6\mathbb{Z}$ How can we compute the localization of the ring $\mathbb{Z}/6\mathbb{Z}$ at the prime ideal $2\mathbb{Z}/\mathbb{6Z}$? (or how do we see that this localization is an integral domain)?
| One simple way to compute this is to exploit the universal property of localization. By definition $\rm\ L\ =\ \mathbb Z/6_{\:(2)}\ =\ S^{-1}\ \mathbb Z/6\ $ where $\rm\ S\ =\ \mathbb Z/6 \ \backslash\ 2\ \mathbb Z/6\ =\ \{\bar 1, \bar 3, \bar 5\}\:.\: $ Hence, since the natural map $\rm\ \mathbb Z/6\ \to\ \mathbb Z/2\... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Are calculus and real analysis the same thing?
*
*I guess this may seem stupid, but
how calculus and real analysis are
different from and related to each
other?
I tend to think they are the same
because all I know is that the
objects of both are real-valued
functions defined on $\mathbb{R}^n$,
and their topics are co... | *
*A first approximation is that real analysis is the rigorous version of calculus. You might think about the distinction as follows: engineers use calculus, but pure mathematicians use real analysis. The term "real analysis" also includes topics not of interest to engineers but of interest to pure mathematicians.
*... | {
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What are good resources to self-teach mathematics? I am teaching myself mathematics using textbooks and I'm currently studying the UK a-level syllabus (I think in the USA this is equivalent to pre-college algebra & calculus). Two resources I have found invaluable for this are this website (http://math.stackexchange.com... | Yes, this site as well as wolfram|alpha are both excellent resources for teaching yourself math!
In addition, I would suggest looking at this site. It provides tons of great math videos, if you are like me and too lazy to read your book sometimes. :) KhanAcademy is also good, but I do prefer the latter. If you can affo... | {
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Using the Partial Summation Formula Partial Summation formula:
Consider $\sum a_n$ and $\sum b_n$. If $A_n= \sum _{k=1}^{n} a_k$, then
$\sum _{k=1}^{n} a_kb_k = A_nb_{n+1}- \sum _{k=1}^{n} A_k(b_{k+1}-b_k)$
So $\sum _{k=1}^{\infty} a_kb_k$ converges if both
$\sum _{k=1}^{\infty} A_k(b_{k+1}-b_k)$
and $\{A_nb_{n+1... | Off the top of my head, partial summation is not what I would use to solve your first problem. Instead note that for all $n$
$\sqrt{c_n c_{n+1}} \leq \sqrt{ (\max c_n, c_{n+1})^2} = \max c_n, c_{n+1} \leq c_n + c_{n+1}$.
By a direct comparison, this implies $\sum_n \sqrt{ c_n c_{n+1}}$ converges.
Is there some reaso... | {
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What's the opposite of a cross product? For example, $a \times b = c$
If you only know $a$ and $c$, what method can you use to find $b$?
| The name "product" for the cross product is unfortunate. It really should not be thought of as a product in the ordinary sense; for example, it is not even associative. Thus one should not expect it to have properties analogous to the properties of ordinary multiplication.
What the cross product really is is a Lie brac... | {
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"timestamp": "2023-03-29T00:00:00",
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Faithful representations and character tables Suppose an n-dimensional irreducible complex representation is not faithful. Then a non-identity element gets mapped to the identity matrix in $GL_n(\mathbb{C})$ so that the value of its associated character on the conjugacy class of this element is $n$. Thus, $n$ appears... | If $\chi$ is the character, and $\chi(g)=\chi(1)=n$ for some group element $g$, then $\rho(g)$ is an $n\times n$ matrix $A$ whose eigenvalues are all complex numbers of modulus 1 and whose trace is $n$ (here $\rho$ is the representation whose character is $\chi$). Also, some power of $A$ is the identity. Can you see ho... | {
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Write $\sum_{1}^{n} F_{2n-1} \cdot F_{2n}$ in a simpler form, where $F_n$ is the n-th element of the Fibonacci sequence? The exercise asks to express the following:
$\sum_{1}^{n} F_{2n-1} \cdot F_{2n}$
in a simpler form, not necessarily a closed one. The previous problem in the set was the same, with a different expres... | You want to watch those indices. I think you mean $\sum_{k=0}^n F_k^2 = F_n F_{n+1}$ and $\sum_{k=1}^n F_{2k-1} F_{2k}$. Hmm, it looks to me like this one can be expressed as a linear combination of $n$, $1$ and a certain Fibonacci number...
| {
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Finding probability of an unfair coin An unfair coin is tossed giving heads with probability $p$ and tails with probability $1-p$. How many tosses do we have to perform if we want to find $p$ with a desired accuracy?
There is an obvious bound of $N$ tosses for $\lfloor \log_{10}{N} \rfloor$ digits of $p$; is there a be... | This is a binomial distribution. The standard deviation on the number of heads is $\sqrt{Np(1-p)}$, so leaving aside the difference between your measured $p$ and the real $p$ you need $N \gt \frac{p(1-p)}{accuracy^2}$
| {
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"timestamp": "2023-03-29T00:00:00",
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Inverse Image as the left adjoint to pushforward Assume $X$ and $Y$ are topological spaces, $f : X \to Y$ is a continuous map. Let ${\bf Sh}(X)$, ${\bf Sh}(Y)$ be the category of sheaves on $X$ and $Y$ respectively. Modulo existence issues we can define the inverse image functor $f^{-1} : {\bf Sh}(Y) \to {\bf Sh}(X)$ t... | A functor which is a left adjoint preserves colimits (see for instance Mac Lane, "Categories for the working mathematician", chapter V, section 5 "Adjoints on Limits"); particularly, stalks.
| {
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"timestamp": "2023-03-29T00:00:00",
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"question_score": "10",
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Why is Harish-Chandra's last name never used? This is only barely a math question but I don't know where else to ask. I've always wondered about Harish-Chandra's name. The Wikipedia article seems to mention "Mehrotra" as a last name but only in passing, and it's not even used in the page's title. Did he simply not use ... | A link to a biography by Roger Howe now shows up on Wikipedia, and it has this to say:
about the name harish-chandra: indian names do not necessarily follow
the western two-part pattern of given name, family name. a person may
often have only one name, and this was the case with harish-chandra,
who in his you... | {
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"answer_id": 0
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how to prove this inequality? Given $x>0$, $y>0$ and $x + y =1$, how to prove that $\frac{1}{x}\cdot\log_2\left(\frac{1}{y}\right)+\frac{1}{y}\cdot\log_2\left(\frac{1}{x}\right)\ge 4$ ?
| Hint 1: Rewrite this inequality as:
$$-x\log_2 x - (1-x)\log_2 (1-x) \geq 4 x (1-x)$$
Both sides of the inequality define concave functions on the interval $[0,1]$. Plot them. Can you show that the graph of the second is always lying below the graph of the other?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/32971",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 1
} |
Solving differential equation Below is my work for a particular problem that is mixing me up, since no matter how many times, I can't get my answer to match the book solution.
Given ${f}''(x)= x^{-\frac{3}{2}}$ where $f'(4)= 2$ and $f(0)= 0$, solve the differential equation.
$$f'(x)= \int x^{-\frac{3}{2}} \Rightarrow \... | Nothing to worry about! There is a minor slip, $-2(4)^{-1/2}=-2/2=-1$. You got $-4$ instead.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/33033",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
I haven't studied math in 12 years and need help wrapping my mind back around it I was never fabulous at Algebra and have always studied the arts. However, now I have to take Math 30 pure 12 years after I finished my last required high school math class.
If anyone has thoughts on how to help me re-learn some of what I ... | Practise, Practise, Practise.
Maths is not a spectator sport and you only get the hang of it by experimenting with it yourself. Even if after reading a question you think "I can do that" don't skip it - you might find it was more complicated than you first thought, and if not, you will gain confidence by doing it.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/33167",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 1
} |
An inequality on a convex function An exercise in my textbook asked to prove the following inequality, valid for all $a,b,c,d \in R $
$$\left(\frac{a}{2} + \frac{b}{3} + \frac{c}{12} + \frac{d}{12}\right)^4 \leq \frac{a^4}{2} + \frac{b^4}{3} + \frac{c^4}{12} + \frac{d^4}{12}$$
There is a straightforward proof using Con... | By Holder
$$\frac{a^4}{2} + \frac{b^4}{3} + \frac{c^4}{12} + \frac{d^4}{12}=\left(\frac{a^4}{2} + \frac{b^4}{3} + \frac{c^4}{12} + \frac{d^4}{12}\right)\left(\frac{1}{2} + \frac{1}{3} + \frac{1}{12} + \frac{1}{12}\right)^3\geq$$
$$\geq\left(\frac{|a|}{2} + \frac{|b|}{3} + \frac{|c|}{12} + \frac{|d|}{12}\right)^4\geq\l... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/33225",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 3,
"answer_id": 2
} |
For a covariance matrix, what would be the properties associated with the eigenvectors space of this matrix? I want to know, since the covariance matrix is symmetric, positive, and semi-definite, then if I calculate its eigenvectors what would be the properties of the space constructed by those eigenvectors (correspond... | The eigenvectors correspond to the principal components and the eigenvalues correspond to the variance explained by the principal components.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/33344",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Prove that if $p$ is an odd prime that divides a number of the form $n^4 + 1$ then $p \equiv 1 \pmod{8}$ Problem
Prove that if $p$ is an odd prime that divides a number of the form $n^4 + 1$ then $p \equiv 1 \pmod{8}$
My attempt was,
Since $p$ divides $n^4 + 1 \implies n^4 + 1 \equiv 0 \pmod{p} \Leftrightarrow n^4 \... | If $$p \mid (n^k+1), $$
$$n^k \equiv -1 \pmod{p}$$
$$n^{2k} \equiv 1 \pmod{p}$$
If$$ \operatorname{ord}_pn=d,$$ then $d \mid 2k$.
If $d \mid k$, then $$n^k\equiv 1 \pmod{p}$$ $\Rightarrow$ $$-1\equiv 1 \pmod{p}$$
$\Rightarrow p\mid 2$ which is impossible as $p$ is odd prime $\Rightarrow d\nmid k$.
If $(k,2)=1$ i.e.,... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/33392",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "15",
"answer_count": 3,
"answer_id": 2
} |
Confusing question: try and prove that $x -\tan(x) = (2k+1)\frac{\pi}{2}$ has no solution in $[\frac{3\pi}{4},\frac{5\pi}{4}]$ I am trying to show that $x - \tan(x) = (2k+1)\frac{\pi}{2}$ has no solution in $[\frac{3\pi}{4},\frac{5\pi}{4}]$. However, I seem to be stuck as I don't know where to begin.
The only sort of ... | Let $f(x) = x -tan(x)$. Taking the first derivative we see that this function is constantly decreasing. The derivative will be negative everywhere besides $x = \pi$. Hence the values live in the interval $[f(\frac{5\pi}{4}),f(\frac{3\pi}{4})]$. Computing those values you see that for every choice of $k$, the number $(2... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/33457",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
The locus of the intersection point of two perpendicular tangents to a given ellipse
For a given ellipse, find the locus of all points P for which the two tangents are perpendicular.
I have a trigonometric proof that the locus is a circle, but I'd like a pure (synthetic) geometry proof.
| If all you want is a proof that the locus is a circle, we may assume that the ellipse is given by
$$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1.$$
Ignoring vertical tangents for now,
if a line $y=mx+k$ is tangent to the ellipse, then plugging in this value of $y$ into the equation for the ellipse gives
$$\frac{x^2}{a^2} + \... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/33520",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "30",
"answer_count": 4,
"answer_id": 1
} |
Finding distribution functions of exponential random variables Find the distribution functions of X+Y/X and X+Y/Z, given that X, Y, and Z have a common exponential distribution.
I think the main thing is that I wanted to confirm the distribution I got for X+Y. I'm doing the integral, and my calculus is a little rusty. ... | Thanks so much for your help. I'm still having some trouble, however. Currently for (X+Y)/X I have denoted X+Y = t, adn the distribution of T is \int \boldsymbol{\alpha e^{-\alpha x}(1+\alpha x + \frac{(\alpha x)^2}{2})}. Then doing the integral caculations for the distribution, I get \int \boldsymbol{\alpha e^{-\alpha... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/33683",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 3
} |
CDF of a ratio of exponential variables Let $X$ and $Y$ be independent exponential variables with rates $\alpha$ and $\beta$, respectively. Find the CDF of $X/Y$.
I tried out the problem, and wanted to check to see if my answer of: $\frac{\alpha}{ \beta/t + \alpha}$ is correct, where $t$ is the time, which we need in o... | Here is a one-line proof.
$$
\mathbb P(X/Y \le t) = \mathbb P(Y \ge X/t) = \mathbb E[\exp(-\beta X/t)] = \text{MGF}_X(-\beta/t) = (1 - (-\beta/t)1/\alpha)^{-1} = \frac{\alpha}{\alpha + \beta/t} = \frac{\alpha t}{\alpha t + \beta}.
$$
N.B.: For the MGF of an exponential variable, see this table.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/33778",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
"answer_count": 4,
"answer_id": 3
} |
Calculate relative contribution to percent change Let me use a simple example to illustrate my problem. First, assume we are calculating rate $r$ at time $t$ such that $r_t=\frac{x_t}{y_t}$. Furthermore, each measure has two component parts: $x = x_A +x_B$ and $y = y_A + y_B$. We can thus calculate percent change $c$ f... | this might be a bit old but I was looking for something similar and I found these two articles which may help someone with a similar question (or so I hope):
Contribution to Growth: growth here could be taken as a change rate
calculating contribution percent change: author proposes a couple of ways to determine how a v... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/33889",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
meaning of $GF(2)[x]/(x^3-1)$ What does $GF(2)[x]/(x^3-1)$ mean? I know $GF(2)$ is the Galois field with 2 elements, but what does the forward slash mean? Also, what's the meaning of the entire expression?
Thanks!
| $\mbox{GF}(2)$ is the finite field with 2 elements (one of the rare instances in mathematics where the common name for an object is kind of larger than the object itself).
$\mbox{GF}(2)[x]$ is the ring of polynomials in the variable $x$ with coefficients in that field. If you're not sure what this means, you should pr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/33945",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
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