Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Example of a normal extension. Can you give an example of a Normal extension which is not a splitting field of some polynomial.? I know that splitting field of a polynomial is always a normal extension but i am looking for the converse. I am sure that the normal extension will have to be infinite.
| I guess it depends on how one defines "normal" for infinite extensions. Let $$K={\bf Q}(\sqrt2,\sqrt3,\sqrt5,\sqrt7,\dots)$$ Then $K$ is normal over the rationals, in the sense that any polynomial irreducible over the rationals with a zero in $K$ splits into linear factors over $K$. But it's not a splitting field, in t... | {
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Commuting matrices and exponential function Let $A,B$ be $n\times n$ commuting matrices, that is $AB=BA$.
I also know that $\exp(Bt)=X(t)X(0)^{-1}$ where $X$ is the fundamental matrix function.
How can I show that $A\exp(Bt)=\exp(Bt)A$?
| For any $n$,
$$A\cdot(I+Bt+\frac12B^2t^2+\frac16B^3t^3+\ldots +\frac1{n!}B^nt^n)=(I+Bt+\frac12B^2t^2+\frac16B^3t^3+\ldots +\frac1{n!}B^nt^n)\cdot A $$
| {
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How to express $z^8 − 1$ as the product of two linear factors and three quadratic factors
Verify
$$(z-e^{i \theta} ) (z - e^{-i \theta} ) ≡ z^2 - 2\cos \theta + 1$$
Hence express $z^8 − 1$ as the product of two linear factors and three quadratic factors, where all
coefficients are real and expressed in a non-trigonome... | Recall the two versions of the fundamental theorem of algebra:
*
*Any complex polynomial can be factored into a product of linear terms
*Any real polynomial can be factored into a product of linear and quadratic terms (and the quadratic terms have no real roots)
In this case, we can find the factorization as a co... | {
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to find the smallest and largest number of equivalence relation in a set Let $s$ be a set of $n$ elements. The number of ordered pairs in the largest and smallest equivalence relation on set $s$ are $n^2$ and $n$.
I am able to understand the largest set of equivalence relation, but in case of smallest set of equivalenc... | $S$ is a set of $n$ elements.
$S$ is non-empty and we know, every non-empty set always contains the identity relation.
So , It can't be empty relation , I suppose since empty relation is invalidated , that explains why smallest equivalence relation on set $S$ is $n$.
i.e Empty relation is transitive and symmetric on ev... | {
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How to find the sum of the series $\sum_{k=2}^\infty \frac{1}{k^2-1}$? I have this problem :
$$S_n=\sum_{k=2}^\infty \frac{1}{k^2-1}$$
My solution
$$S_n=\sum_{k=2}^\infty \frac{1}{k^2-1} = -\frac{1}{2}\sum_{k=1}^\infty \frac{1}{k+1} -\frac{1}{k-1} = -\frac{1}{2}[(\frac{1}{3}-1)+(\frac{1}{4}-\frac{1}{2})+(\frac{1}{5}-\f... | Let $S_n=\displaystyle\sum_{k=1}^n\frac{1}{k^2-1}=\sum_{k=1}^n\frac{1}{2}\left[\frac{1}{k-1}-\frac{1}{k+1}\right]$
$\displaystyle\hspace{.2 in} =\frac{1}{2}\left[\bigg(\frac{1}{1}-\frac{1}{3}\bigg)+\bigg(\frac{1}{2}-\frac{1}{4}\bigg)+\bigg(\frac{1}{3}-\frac{1}{5}\bigg)+\cdots+\bigg(\frac{1}{n-2}-\frac{1}{n}\bigg)+\bigg... | {
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$ (1+\sin{x})^{\cos{x}} + (1+\cos{x})^{\sin{x}} > 3x $ How do I show that, for $ 0 < x < \dfrac{\pi}{4} $ (first quadrant), the inequality $ (1+\sin{x})^{\cos{x}} + (1+\cos{x})^{\sin{x}} > 3x $ is valid?
I've tried Bernoulli's, but it took me to a false inequality (though all restrictions were respected). I actually th... | Use:
$$\cos x > 1/\sqrt{2},\quad \sin x > x - x^3/6>0$$
So that we have:
$$(1+\sin{x})^{\cos{x}} + (1+\cos{x})^{\sin{x}} >
1+\left(1+\frac{1}{\sqrt{2}}\right)^{x-x^3/6}$$
Now, we can use the series expansion of $(1+a)^x$:
$$(1+a)^x = 1+ \log(1+a)x+\frac{1}{2}\log^2(1+a) x^2 + \ldots =
\sum_{n=0}^\infty \frac{1}{n!}\l... | {
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Calculation of polynomial in the finite field I'm trying to understand the McEliece cryptosystem and I'm looking to this paper http://www.mif.vu.lt/~skersys/vsd/crypto_on_codes/goppamceliece.pdf
On page 26 they are calculating syndrome and somehow they got $1 + a^{10}z$ from $
(a^8+ a^4+ a^{10}) + (a^7+ a^{11} + a)z$
C... | At the start of the section on page 25 there's a reference to Section 2.6.2. In Section 2.6.2, it's stated that $a$ (actually $\alpha$) is a primitive element satisfying $a^4+a+1 = 0$. So $a^4 = -a-1 = a+1$ (the final equality is because $-1 = 1 \mod{2}$).
Substitute $a^4 = a+1$ into $(a^8+a^4+a^{10})+(a^7+a^{11}+a)z$ ... | {
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Least possible number of squares with odd side length An $n\times(n+3)$ rectangular grid ($n>10$) is cut into some squares, with all cuts being along the grid lines. What is the least possible number of squares with odd side length?
[Source: Russian competition problem]
| OLD AND SURELY WRONG INTERPRETATION OF THE PROBLEM
If I understood correctly the problem ask for the greater common odd divisor of $n$ and $n+3$. But the parity of $n$ and $n+3$ is different, because if $n$ is even and you add 3 then $n+3$ is odd, and viceversa.
So then the problem is reduced to search the GCD of $n$ a... | {
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What's a good motivating example for the concept of a slice category? What nice example can one give a beginner to really motivate the idea of a slice category, before they've met the more general notion of a comma category?
There's the toy example of a poset category with the slices as principal ideals -- but that do... | If $X$ is a set, the slice category $\textsf{Set}/X$ can be thought of as the category of $X$-indexed collections of sets, where an object $f:Y\to X$ corresponds to the $X$-indexed collection of fibers $\{Y_x = f^{-1}\{x\}\mid x\in X\}$, and a morphism from $f:Y\to X$ to $g:Z \to X$ corresponds to a map of sets $Y_x\to... | {
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Write $\sum_{k=1}^nk\sin(kx)^2$ in closed form $\underline{Given:}$
Write in closed form $$\sum_{k=1}^nk\sin(kx)^2$$
using the fact that $$\sum_{k=1}^nku^k=\frac u{(1-u)^2}[(n)u^{n+1}(n+1)u^n+1]$$
$\underline{My\ Work:}$
I substituted $\sin(kx)^2$ for $\frac{1-\cos(2kx)}2$ which let me rewrite the summation like this: ... | We have $\dfrac{e^{2ix}}{(1-e^{2ix})^2}=\dfrac{-1}{4sin^2(x)}$ and $n(e^{2ix})^{n+1}(n+1)(e^{2ix})^n-n(e^{-2ix})^{n+1}(n+1)(e^{-2ix})^n=2in(n+1)sin(4n+2)$.
| {
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finding numbers to make both sides equal Call a triple-x number an integer $k$ such that $k=x(x+1)(x+2)$ where $x \in Z$. How many triple-x numbers are there between 0 and 100,000?
I thought by doing $8!$ and $9!$ would work to see how many combinations there would be. I am not sure how to solve this problem. Can som... | Well, there's one with $x = 0$, one with $x = 1$, and keep going until you get something bigger than $100,000$. Equivalently, you're solving the inequality $x(x+1)(x+2) < 100,000$, which can be done in various ways (though there's no really clean way of doing it).
| {
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Eulerian circuit with no isolated vertex is connected This is my first question (ever), and I am pretty new to math. So I ask for patience and understanding in advance.
So this is the proof I came up with:
Consider $G = (V,E). $
By definition of Eulerian circuit,
$$ \exists\ W: v_0, v_1,\cdots, v_k=v_o $$
An... | Your proof is generally fine; I can follow your logic. If we really want to be nitpicky:
*
*You should probably mention that we are assuming that $G = (V,E)$ has an Eulerian circuit and has no isolated vertices.
*You should say: "Since $G$ has no isolated vertices, we know that $V = \{v_0, \ldots, v_k\}$."
*When y... | {
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Tricky Substitution to get AM-GM inequality So, I'm reading the literature to find different proofs of the AM-GM inequality, the following proof quite hit me, and I don't seem to understand at all.
The proof is as follows:
For any positive numbers: $a_1,a_2,...a_n$. We have:
$$
\dfrac{a_1+a_2+..+a_n}{n} \geq \sqrt[n]{a... | I think they mean replacing $a_k$ by $\dfrac{a_k}{\sqrt[n]{a_1a_2\cdots a_n}}$. Another way to see that is to divide both side of the original AM-GM inequality by $\sqrt[n]{a_1a_2\cdots a_n}$ to get an equivalent inequality
$$\frac1n\left(\frac{a_1}{\sqrt[n]{a_1a_2\cdots a_n}}+\frac{a_2}{\sqrt[n]{a_1a_2\cdots a_n}}+\cd... | {
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Find the minimum distance between the curves $y^2-xy-2x^2 =0$ and $y^2=x-2$ How to find the minimum distance between the curves $y^2-xy-2x^2 =0$ and $y^2=x-2$
Let $y^2-xy-2x^2 =0...(1)$ and $y^2=x-2...(2)$
In equation (1) coefficient of $x^2 =-2; y^2=1, 2xy =\frac{-1}{2}$
We know that a second degree equation where $a... | $$y^2= 2x^2+xy \iff (y-2x)(x+y)=0 \iff y = -x \text{ or } y = 2x$$
so that is a couple of lines.
Hence you want to find the shortest distance from $y=-x$ or $y=2x$ to the parabola $y^2=x-2$. Let this be denoted by the line segment with end points $(a, -a)$ or $(a, 2a)$ and $(b^2+2, b)$.
Then we need the min of $(a-b^... | {
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Different Versions of Fourier Series? What about Uniqueness? Let $f(x)$ be a function, then for its Fourier series
$$
f(x) = \frac{a_0}{2} + \sum_{n=1}^{\infty} (a_n \cos(nx) + b_n \sin(nx))
$$
I found two different definitions (both yielding different series).
i) I found
\begin{align*}
a_n & = \frac{1}{\pi} \int_0^{... | The function $f$ must be $2\pi$-periodic, otherwise the whole construction becomes meaningless (since the Fourier series is $2\pi$-periodic!)
For such a function $f$ the two expressions of $a_n$ and $b_n$ do coincide.
| {
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Is my understanding right on the divisiblity rule? For a given number and a divisor. If the prime factors of the divisor can divide a number,then can I say that the divisor will divide a number.
For example - 786
divide by 21
If I break 21 in the prime factors - 3 * 7.
So, if the number is divisible by 3 as well as 7 t... | Yes, that's right. It is a consequence of the fact that every integer can be written as a product of powers of primes in an essentially unique way (the only lack of uniqueness is in the order in which you write down the factors).
So in your example, $-786 = -2\cdot 3\cdot 131$ (and $131$ is prime), and $21 = 3\cdot 7$.... | {
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What's the dimension of $\mathbb C$ as a vector space over $\mathbb{R}$? What's the dimension of $\mathbb C$ as a vector space over $\mathbb{R}$?
I think that the answer is $2$, because $\mathbb R^2 \cong \mathbb{C}$ (since all complex numbers can be mapped to $\mathbb R^2$ and vice versa).
Is this correct, or is it a ... | Hint:
$\lbrace 1,i\rbrace$ is a basis.
As for what you have said, there is an isomorphism $$\frac{\mathbb{R}[x]}{\langle x^2+1\rangle}\simeq \mathbb{C}$$ where $\mathbb{R}[x]$ is the polynomial ring.
| {
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Entire function with vanishing derivatives? Let $f:\mathbb{C}\rightarrow\mathbb{C}$ be an entire function.
And assume that at each point, one of it's derivatives vanishes.
What can you say about $f$?
A hint suggests that $f$ must be a polynomial.
| we can also solve this without using baire category theorem..
we can use the fact that zeros of non zero analytic function is countable. Note that union of $A_{n}$ is $\mathbb{C}$ thus we must have some fix derivative identically 0. Now use power series and analytic continuation.
| {
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Prove statement about determinants. $A$ is a $3\times 3$ matrix over $\mathbb{R}$, I want to show that if $$\det(A + I_3)=\det(A+2I_3),$$ then $$2\det(A+I_3) + \det(A-I_3) + 6 = 3\det A.$$
Can you help me?
| First show that you can assume $A$ diagonal. Then the question reduces to proving a basic identity between two polynomials.
More explicitely, you can write $\det (A+ x I_3) = \sigma_3 + x \sigma_2 + x^2 \sigma_1 + x^3$, where $\sigma_3=a_1 a_2 a_3$, $\sigma_2 = a_1 a_2 + a_1 a_3 + a_2 a_3$ and $\sigma_1 = a_1 + a_2 + ... | {
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A necessary condition for a series of rational numbers to be irrational? Suppose $(q_{n})_{n\in\mathbb{Z}_{\gt 0}}$ is a decreasing sequence of positive rational numbers such that $Q:=\displaystyle{\sum_{n>0}q_{n}}$ is finite. Let's denote by $n_{i}$ and $d_{i}$ the numerator and the denominator of $q_{i}$ such that $(... | No. In fact, every positive real number $Q$ can be written this way. Here is the algorithm: pick $q_1$ to be any positive rational which lies in the interval $\left( \frac{Q}{2}, Q \right)$. Once $q_1, q_2, \dots q_n$ have been picked, pick $q_{n+1}$ to be any positive rational whose numerator and denominator are copri... | {
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Floating point arithmetic: $(x-2)^9$ This is taken from Trefethen and Bau, 13.3.
Why is there a difference in accuracy between evaluating near 2 the expression $(x-2)^9$ and this expression:
$$x^9 - 18x^8 + 144x^7 -672x^6 + 2016x^5 - 4032x^4 + 5376x^3 - 4608x^2 + 2304x - 512 $$
Where exactly is the problem?
Thanks.
| $(x-2)$ is small by definition of $x$, $(x-2)^9$ is even much smaller but can be computed with small relative error.
The single terms of the expanded polynomial are much larger and therefore you will suffer from catastrophic cancellation (see e.g. Wiki or do a web search). Example:
$$x=2 + 10^{-2} \Longrightarrow (x-2)... | {
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A unique operation on a set that makes it a group From Rotman's "Introduction to Group Theory":
Let $G$ be a group and let $X$ be a set having the same number of elements as $G$. If $f:G\rightarrow X$ is a bijection, there is a unique binary operation that can be defined on $X$ so that $X$ is a group and $f$ is an iso... | this is really just an exercise in abstraction. in essence we use the bijection to pull back the group structure on $G$ into the hitherto unstructured set $X$. we may represent the image of $x$ in $G$ as $x_G$ and if the inverse of this map is $\phi$ we may define an operation $\ast$ on $X \times X$ by:
$$
x \ast y = \... | {
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Product of inner products Is product of innerproduct again a inner product of two vectors?
For example - Is $ (< u,v >)(< x,y >) = < m,n > $? And if yes is m and n unique and how do we calculate those?
| Of course the product, which is after all a scalar (call it $S$) , can be written as an inner product of two vectors, e.g., $\vec{n} = (S, 0, 0, \ldots), \vec{m} = (1, 0, 0, \ldots)$. But the decomposition of that scalar is not unique, and in fact, you can find an infinite number of correct $\vec{m} $ cevtors for any ... | {
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Holonomy representation: is it actually a class of representations? In D. Joyce's book "Riemannian Holonomy Groups and Calibrated Geometry" (2007) the author writes that if $\nabla$ is a connection on a vector bundle $E$ (over a connected base) with the fibre $\mathbb R^k$ then the holonomy group $\mathrm{Hol}(\nabla)$... | Yes. The holonomy representation is an isomorphism class of representations.
| {
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Write an equation as a single power(Grade 11 Math, Function) $$\frac{10^{-4/5} \cdot 10^{1/15}}{10^{2/3}}$$
The answer is $10^{-7/5}$, which seems impossible to me. I get:
$10^{-4/5} \cdot 10^{-11/15}$. I see where the numerator $7$ comes from but the denominator is being a pest, and won't let me do anything because I ... | It looks like you made a sign error while combining exponents.
$$\frac1{15} - \frac23 = \frac1{15} - \frac{10}{15} = -\frac9{15}.$$
You seem to have gotten $-\frac{11}{15}$ when you should have gotten $-\frac9{15},$ perhaps by flipping the sign of $\frac1{15}.$
Of course $-\frac9{15} = -\frac35,$ and the last step is e... | {
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A compact connected solvable Lie group is a torus I am looking for the proof of the following statement.
A compact connected solvable Lie group of dimension $n\geq 1$ is a torus, i.e., it isomorphic to the product of $n$ copies of $S^1$.
A search with Google revealed that pages 51-52 of "Lie Groups and Lie Algebras I... | @Gis: I would like to add to Tim's answer the following:
*
*The lie algebra $\mathfrak{g}$ admits a bi-invariant scalar product $<\,,>$, i.e. the adjoint action $\mathrm{ad}_x$ is antisymetric $\forall x\in\mathfrak{g}$ (since $G$ is compact, it admits a bi-invariant riemannian metric).
*$\mathfrak{g}=\mathfrak{Z}(... | {
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Entire functions satisfying $|f(z)|\geq |\sin z|^{10} $ for all $z\in\mathbb{C}$ How do I find all the entire functions $f(z)$ such that $|f(z)|\geq |\sin z|^{10} $ for all $z\in\mathbb{C}$
Can an entire function have essential singularity? My intuition says that "no".
But I am not sure.
Can anyone help me ?
| If $f$ has no zeros, then $g(z)=\dfrac{(\sin z)^{10}}{f(z)}$ is entire and bounded and so is a constant.
Therefore, $f$ is a constant multiple of $(\sin z)^{10}$.
If $f$ has zeros, then they must be a subset of the zeros of $\sin z$, which are $n\pi$ for $n\in\mathbb Z$.
If $f$ has finitely many zeros, you can remove t... | {
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Proof Using cartesian products Suppose that $A$, $B$, and $C$ are sets. Prove that $(A\cap B)\times C =(A\times C)\cap(B\times C)$. Prove the statement both ways or use only if and only if statements.
| The two sets $(A \cap B) \times C$ and $(A \times C) \cap (B \times C)$ are equal iff every element of one set is element of the other set and vice versa.
Consider $(A \times C) \cap (B \times C)$ first:
$$ a \in (A \times C) \cap (B \times C) \leftrightarrow a \in (A \times C) \wedge a \in (B \times C). $$
From the de... | {
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"url": "https://math.stackexchange.com/questions/996103",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
How do you solve a system of equation with a variable as an exponent? So, I've been doing some random math on my own and I ran into a problem. I was met with the system of equations of $$x=3^y$$ $$x+1=5^y$$ How can I solve this algebraically without using the graphs?
| Eliminating $x$ from the first equation and replacing in the second let you with the equation $$3^y+1=5^y$$ which is highly nonlinear (and you look for the intersection of these two functions).
You can make the problem nicer taking logarithms and get $$y\log(5)=\log(3^y+1)$$ which is now the intersection of a curve and... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/996216",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
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Periodicity over the prime indices of a multiplicative sequence implies periodicity? I have a real sequence $(a_p)$ indexed by the prime numbers which takes values -1, 0, or 1, having the property that $a_p=a_q$ whenever $p\equiv q \pmod m$, where $m$ is a fixed integer $>2$.
I'm wondering if by extending multiplicativ... | Look for sequences with values in $\{-1,0,1\}$.
Assume that $|a_p|>1$ for some integer. Then the subsequence $a_{p^n}$ tends to $\pm\infty$, which is not possible for a periodic sequence.
You may want to read about characters.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/996299",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
semantics(truth) vs formal system? my first question is can we just define semantics in logic and not define a formal system ?
why do we need a formal system to prove a proposition when for example we know the proposition is true ?
e.g. ( A ^ (A->B) ) -> B if A is true and A->B is true then B is true
and this can a... | In propositional logic we can "show" validity by truth table.
Truth table supply an algorithm to compute the truth value of every propositional formula;
in particular, truth tables can be used to tell whether a propositional expression is true for all legitimate input values, that is, logically valid [also called : a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/996389",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Indefinite Integral of a Function multiplied by Heaviside I want to do the following integral:
$$\int^\ x*H(x-a) dx $$
In mathematica I get that $$\int^\ x*H(x-a) dx = \frac{1}2*(x-a)*(x+a)*H(x-a)$$
where H(x-a) Is the heaviside function.
But by hand I can't find the same result.What I'm trying to do is that:
$$\int^... | Actually, your solution is not that far off if you keep in mind that the indefinite integral is only determined up to the addition of a constant. However, here is my way to compute the integral:
First off, we need to clarify: The Heaviside function I'll use is defined by
$$H(x) = \left\lbrace \begin{array}{cl} 1 & , ~ ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/996455",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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How to find a bound for these (simple) integrals With help of $\int_{0}^{\infty} e^{-x^2} dx = \frac{\sqrt{\pi}}{2}$ and $\int_{0}^{\infty} e^{-x} dx =1$, I would like to know how to derive the following bounds:
$$\int_{0}^m 4e^{-\frac{t^2}{8m}}dt \leq 2\sqrt{8\pi m}$$
and
$$\int_{m}^{\infty} 4 e^{-\frac{t}{8}}dt \leq ... | Hint
For the first, change variable $$\frac{t^2}{8m}=x^2$$ so $t=x \sqrt{8m}$, $dt=\sqrt{8m} dx$. So $$I=\int 4e^{-\frac{t^2}{8m}}dt=8 \sqrt{2m} \int e^{-x^2} dx$$
Do something similar for the second integral.
I am sure that you can take from here.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
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Natural Logarithm - solve the equation I am having problems understanding how to solve $e^{4x}+4e^{2x}-21 = 0$.
| Try temporarily setting $u=e^{2x}$ so it becomes $u^2+4u-21=0.$ Then solve the quadratic, discarding any negative solutions since $u=e^{2x}>0.$ After that use the $\ln$ function to get $x$ back.
For example if it came out one value of $u$ was $u=5$ then from $e^{2x}=5$ comes $2x=\ln 5$ and then $x=(1/2) \ln 5.$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/996612",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Why is $|x| \neq x$? I was just thinking about the absolute value function. Why does the following equality not hold?
$$|x| = \sqrt{x^2} = (x^{2})^{1/2} = x^1 = x$$
After all, there are clearly some values of $|x|$ that are not equal to $x$. Is this a domain/range issue?
| It should be:
$$\sqrt{x^2} = (|x|^{2})^{1/2} = |x|^1 = |x|$$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Determine whether $\phi$ is an automorphism Let $H$ be a proper subgroup of $G$ and let $\psi$ be an automorphism of $H$ other than the identity mapping. Define a mapping $\phi:G\rightarrow G$ by
$\phi(x)=$\begin{cases}\psi(x) \text{ if } x\in H\\x \text{ if } x\not\in H\end{cases}
Is $\phi(x)$ an automorphism?
| Take $g \in G\backslash H$. Every element in $H$ is a product of two elements from $G \backslash H$.
$$h = g^{-1} \cdot gh$$.
If the morphism $\phi$ conincides with $\mathbb{1}_G$ on $G \backslash H$ then it is $\mathbb{1}_G$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/996823",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
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3D rotation around arbitrary axis I have a 3D rotation matrix, R which is a combination of rotations around x-axis , y-axis and z-axis. I know how to calculate n(the arbitrary axis around which a point rotated about theta angle and this rotation is equal to rotating that point using the 3D matrix above) i.e. by finding... | I'll describe a computational method, assuming the rotation has been translated to be around an axis through the origin.
If you know the axis of rotation $A=(a,b,c)$, then you can find a vector orthogonal to this one. For example, if $a \ne 0$, $b \ne 0$, then $V=(b,-a,0)$ is such a vector. Or, if one component is $0$-... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/996913",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
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Showing a certain subspace of Hilbert space is dense Let H be the Hilbert space of square-summable sequences of reals.
A few years ago I thought I had proved that the subspace Z of real sequences with only finitely many nonzero terms, such that they sum to zero, is dense in H.
(Since then I've seen this confirmed in R... | Given $x \in H$ and $\epsilon > 0$, there is $y \in H$ with $\|x - y\| < \epsilon/2$ such that only finitely many elements of $y$ are nonzero. Suppose the sum of those nonzero elements of $y$ is $s$. Consider $z$ obtained from
$y$ by changing $n$ of its elements from $0$ to $-s/n$. Thus $z \in Z$,
and $\|z - y\|^2 ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/997021",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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Algebra (group theory) Prove, without using Cauchy’s Theorem, that any finite group $G$ of even order contains an element of order two. [Hint: Let $S = \{\,g ∈ G : g \ne g^{−1}\,\}$. Show that $S$ has even number of elements. Argue that every element not in $S$ and not equal to the identity element has order two].
My ... | In principle, this is fine - almost. However, you end with "Therefore, this is a contradiction" - but you didn't even start by making an assumption that can be contradicted! Presumably (but you should have explicitly said so!) you wanted to start from a finite group $G$ of even order that does not have an element of or... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/997099",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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An equation for the third powers of the roots of a given quadradic polynomial The roots of the equation $3x^2-4x+1=0$ are $\alpha$ and $\beta$. Find the equation with integer coefficients that has roots $\alpha^3$ and $\beta^3$.
GIVEN
SR: $\alpha + \beta = \frac43$
PR: $\alpha\beta = \frac13$
REQUIRED
SR: $\alpha^3 + \... | Using the quadratic formula, we have $$\alpha, \beta = \frac{-(-4) \pm \sqrt{(-4)^2-4(3)(1)}}{6} \\ = \frac{4 \pm \sqrt{16-12}}{6} \\ = \frac{4 \pm 2}{6} \\ = 1, \frac{1}{3} $$ Hence a polynomial you could use with roots at $\alpha^3 = 1^3 = 1$ and $\beta^3 = \left(\frac{1}{3}\right)^3 = \frac{1}{27}$ would be $$f(x) =... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/997188",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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$\frac{d^2 y}{dx^2}-2y=2\tan^3\left(x\right)$ Problem:
\begin{equation}
\frac{d^2 y}{dx^2}-2y=2\tan^3\left(x\right).
\end{equation}
using the method of undetermined coefficients or variation of parameters, with $y_p\left(x\right)=\tan\left(x\right)$.
This is what I have so far:
\begin{equation}
r^2-2=0\implies r=\pm\s... | I don't have the reputation to comment, but the general solution of the nonhomogeneous equation is just the fundamental set of solutions to the homogeneous equation added to the particular solution. So you have everything, just put it all together.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/997263",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Find the derivative of $y = x^{\ln(x)}\sec(x)^{3x}$ What is the derivative of
$$y = x^{\ln(x)}\sec(x)^{3x}$$
I tried to find the derivative of this function but somewhere along the way I seem to have gotten lost. I started off with using the product rule and then the chain rule about 4 times and things are getting m... | Try logarithmic differentiation, or somehow make use the fact that $a=e^{\ln a}$ for $a>0$. (you have to decide what $a$ should be).
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/997328",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
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Is there a formula for $k\pi ^n$, if $n$ is an odd number and $k$ is a rational number? I always find the formula for $k\pi ^n$ when $n$ is an even number and $k$ is a rational number, but I did't find for an odd number.
| The Dirichlet beta function may satisfy you :
$$\beta(x):=\sum_{n=0}^\infty \frac{(-1)^n}{(2n+1)^x}$$
with the table of values :
\begin{array} {c|c}
n&\beta(n)\\
\hline
1&\frac {\pi}4\\
2&K\\
3&\frac {\pi^3}{32}\\
4&\beta(4)\\
5&\frac {5\,\pi^5}{1536}\\
\end{array}
with $K$ the Catalan constant.
Here the $n$ even cases... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/997392",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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find common ratio of $\sum_{k=1}^\infty \frac{1}{k(k+1)}$ I have this problem, I need to find the sum.
$$\sum_{k=1}^\infty \frac{1}{k(k+1)} = \frac{1}{2}+\frac{1}{6}+\frac{1}{12}+\frac{1}{20}+\frac{1}{30}+\cdots+\frac{1}{k(k+1)}$$
The problem is that the ratio is not conclusive, Any idea how to find the ratio?
Thanks!
| $$\sum_{k=1}^\infty \frac{1}{k(k+1)} = \frac{1}{2}+(\frac{1}{2}-\frac{1}{3})+(\frac{1}{3}-\frac{1}{4})+(\frac{1}{4}-\frac{1}{5})+\cdots+(\frac{1}{k}-\frac{1}{k+1})$$ Do you see how to do it now?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/997525",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Show $1/(1+ x^2)$ is uniformly continuous on $\Bbb R$.
Prove that the function $x \mapsto \dfrac 1{1+ x^2}$ is uniformly continuous on $\mathbb{R}$.
Attempt: By definition a function $f: E →\Bbb R$ is uniformly continuous iff for every $ε > 0$, there is a $δ > 0$ such that $|x-a| < δ$ and $x,a$ are elements of $E$ im... | According to the mean value theorem,
$$
\left|\frac1{1 + x^2} - \frac1{1 + a^2}\right| = f'(c)|x-a|
$$
where $f(x)=\dfrac 1 {1+x^2}$ and $c$ is somewhere between $x$ and $a$. But $|f'(c)|\le\max |f'|$, the absolute maximum value of $|f'|$. In order for this to make sense, you need to show that $|f'|$ does have an abs... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/997602",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "16",
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"answer_id": 4
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State space and linearization I have a question about state space representation. How can I represent an equation in which I have only the second and first derivatives? For example
where $u$ is the control input.
If I put $x_1=x$ and $x_2=\dot{x}$ I will not have $x_1$ in my state space representation, and when findin... | By selecting $x_1=x$ and $x_2=\dot{x}$ you can write
$$\begin{align*}
\dot{x}_1 &= x_2 \\
\dot{x}_2 &= k_1 x_1 (1 - |x_1|) + k_2 u
\end{align*}$$
Now the fixed points are $(0, 0), (1, 0), (-1, 0)$. The Jacobian around $(1, 0)$ (as an example) is
$$ \begin{bmatrix} 0 & 1 \\ k_1(1 - 2 x_1) & 0 \end{bmatrix} |_{x_1=1} = \... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/997664",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Finding the length of a graph I am asked to find the length of the graph of the function: $\dfrac{x^2}{8} - \ln (x)$ , $x\in[1,4]$.
I am really struggling with this problem. As far as I know, we use the formula:
$L(c)= \sqrt{1 + [f'(x)]^2}$
Either I am deriving it wrong or I am missing steps because I am not getting th... | The length is
$$\int_1^4 \sqrt{1+\Bigl(\frac x4-\frac1x\Bigr)^2}\,dx
=\int_1^4\sqrt{\Bigl(\frac x4+\frac1x\Bigr)^2}\,dx\ .$$
I'm sure you can finish it from here.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Proving the existence of a homomorphism $\overline f:G/H\rightarrow G'$ such that $\overline f \pi = f.$ I'm working on a problem where I'm given that $G$ is a group, $H$ is a normal subgroup of $G,$ $f:G\rightarrow G'$ is a homomorphism, and $H\subseteq \ker(f).$ I need to show that there exists a homomorphism $\overl... | Hint: You need to check your map $\overline{f}$ is well defined. Right now it seems to depend on the choice of a representative of a coset. $G/H$ consists of equivalence classes, and equivalence classes may have different representatives. It is here that you need $H \subset ker(f)$.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Use the techniques from integration to find the volume of a cylinder of height h and radius r I was thinking of starting with a quadrilateral or rectangle.
starting with a horizontal line a $y = r$, then set the area under the curve by integrating from $0$ to $h$.
$y=r$ then $y = r^2$
$$\pi \int_{0}^{h} r^2 $$
how's t... | Actually I don't have any plotting software with me but soon I'll show you Images that'll help understanding these methods
Solution 1
Assuming that You know that Area of Circle with radius R is $\pi R^2$
Now consider a disc (differential element) of Radius $R$ and Thickness $dh$
It's volume will be $$ dv=\pi R^2 dh$$
... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Probability conundrum Good morning,
wondered if you could help me please?
I would like to work out the probability of and event happening 5 times out of 6.
all 6 events have a 1 in 60 chance of a particular outcome. I would like to know what the probability is when the event happens 6 times, and 5 of which have the sa... | Your situation correspond to binomial distribution with parameters $n=6$ and $p=1/60$ where each of 6 trials can have 60 possible outcomes.
First, let us focus on the $i$th term of the binomial expansion that corresponds that one fixed outcome (say $1$) occurs
$$
{{n}\choose{i}}p^i(1-p)^{n-i}
$$
Specifically:
$$
{{6}\c... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/998021",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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How would one solve this system $$12x^2=6z\\2y=-z\\6x-y=7$$
It's been many years since I've dealt with system equations, and now find myself in need to solve them. I am not quite sure what to do; I am interested in finding $x$ and $y$, so assuming that $z \neq 0$ (can I do this?), I isolate it and get
$$2x^2 = z = -2y ... | Given is $f(x,y)=4x^3+y^2$ with the constraint $y=6x-7$. The easiest way to solve this, is plugging in $y$ into the function, which gives
$$f(x)=4x^3+(6x-7)^2=4x^3+36x^2-84x+49$$
and we are asked to find the extrema. Differentiating:
$$f'(x)=12x^2+72x-84=0$$
Dividing by $12$ gives $f'(x)=x^2+6x-7=(x+7)(x-1)=0$. From th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/998124",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
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Does every algebra automorphism preserve augmentation ideal filtration? Let $A=\displaystyle\bigoplus_{n\geq0}A_n$ be a graded algebra and let the augmentation homomorphism $\varepsilon:A\to A_0$ be the projection. Define the augmentation ideal, denoted by $A_+$, to be the kernel of $\varepsilon$. Let $A_+^k$ to be the... | $A=k[x],\epsilon(x)=0, f(x)=x+1,f^{-1}(x)=x-1$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/998243",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
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Black Scholes PDE How to show that $V_1(S,t)=S\frac{\partial V(S,t)}{\partial S} $ satisfies Black-Scholes PDE given as $\frac{\partial V}{\partial t} + \frac{\sigma^2 S^2}{2}\frac{\partial^2V}{\partial S^2} + rS\frac{\partial V}{\partial S} -rV = 0$ ?
| Let $V(S,t)$ --- solution of the PDE.
Then
$$
S\frac{\partial}{\partial S}\left(\frac{\partial V}{\partial t} + \frac{\sigma^2 S^2}{2}\frac{\partial^2V}{\partial S^2} + rS\frac{\partial V}{\partial S} -rV\right) = 0
$$
$$
S\left(\frac{\partial^2 V}{\partial t \partial S} + \frac{\sigma^2 }{2}\left(2S\frac{\partial^2V}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/998341",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Norm of a functional I'm facing the problem of calculating the norm of the following functional:
$\displaystyle \phi : L_p([0,1]) \rightarrow \mathbb{R}, ~~ \phi (f) = \int\limits_{0}^{1} e^x f(x) dx $
I have no idea where to begin. I need to see the supremum of $\displaystyle \phi $ on the set $ B(0,1)= \{f \in L_p([... | for every $f\in B(0,1)$ we have
$$|\phi(f)|\leq \left(\int_0^1e^{qx}dx\right)^{1/q}||f||_p$$ where $1/p+1/q=1$, so $||\phi||\leq \frac{1}{q}(e^q-1)^{\frac1q}$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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How can I prove $(a+b+c)!>a!b!c!$ In fact, I couldn't prove the inequality because I don't know which method is used for this.
The condition for this inequality is $$(a+b+c)>1$$
| An adaptation of this answer. If you have $a$ adults, $b$ boys, and $c$ girls, then $(a+b+c)!$ counts the number of ways you can line everyone up, while $a!b!c!$ counts the number of ways you can do so with the additional requirement that all adults come before all children, and all boys before all girls. Then $(a+b+c)... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/998571",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Relationship between divergence operators defined with respect to two different volume forms. Let us assume that you have a volume form $\mu$ defined on a manifold $\mathcal{M}$. Then you can define the divergence operator with respect to this metric, such that the following relationship holds for all $\mathcal{U}\subs... | I've actually worked out the answer to the question from the following definition of the divergence : $\operatorname{div}(v)\mu = \operatorname{d}(\mathbf{i}_v \mu)$.
Short-circuiting a few computation lines, the result is : $$ \alpha \operatorname{div}_1(v) = \operatorname{div}_2(\alpha v) $$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/998655",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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The second and third Chern classes of Calabi-Yau threefolds Let $X$ be a smooth projective Calabi-Yau threefold. Then the first Chern class vanishes: $$c_1(X)=c_1(T_X)=0.$$
Is anything known about $c_2(X)$ and $c_3(X)$? What about $c_2$ of a
K$3$ surface?
(I am sorry if this is very well-known. This question is jus... | In general the top Chern class is the Euler class of the real bundle underlying the holomorphic tangent bundle, so its degree is the Euler characteristic of the manifold.
So for $X$ a $K3$ surface, we always have $c_2(X)=24$.
Unfortunately for Calabi–Yau threefolds the Euler characteristic can vary wildly, and in fact ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/998797",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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$a_n=2^n+3^n+6^n-1$. Find all positive integers that are primes to all terms of the sequence. Let the sequence $a_n=2^n+3^n+6^n-1, n\in\mathbb N_{> 0}$. Find all positive integers that are prime to all terms of this sequence. I have no idea how to approach this, but I know that I CAN'T use Euler's theorem or any genera... | Thought it could be useful to cite the solution from The IMO Compendium A Collection of Problems Suggested for The International Mathematical Olympiads: 1959-2009:
We will show that $1$ is the only such number. It is sufficient to prove that for every prime number $p$ there exists some $a_m$ such that $p \mid a_m$. Fo... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How to calculate this $\sin\frac{\pi}{9}\sin\frac{2\pi}{9}\sin\frac{4\pi}{9}$? I'm stuck with the expression
$$\sin\frac{\pi}{9}\sin\frac{2\pi}{9}\sin\frac{4\pi}{9}.$$
I have no idea how to begin, please give me a hint!
(The answer should be $\sqrt3/8$.)
| Remember that the polynomial $P(x)=3x-4x^3$ has the property $P(\sin t)=\sin 3t$. Therefore $x_1=\sin \pi/9$,
$x_2=\sin 2\pi/9$ and $x_3=-\sin4\pi/9$ are all solutions of the equation
$P(x_i)=\sin \pi/3=\sqrt3/2$. In other words they are zeros of $P(x)-\sqrt3/2$. The leading coefficient of $P$ is $-4$, so
$$
-4x^3+3x-\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/999034",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
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What to do if the critical point is not a real number? I have a function and I already differentiate it, but when I put it equals to zero I don't get a real number. What am I doing wrong?
$f(x) = x\sqrt{x^2+1}$
$f'(x) = \sqrt{x^2+1}+\frac{x^2}{\sqrt{x^2+1})}$
$f'(x) = \frac{2x^2+1}{\sqrt{x^2+1}}$
Then I equalizes it to... | This function does not have a critical point. Notice that $f'(x)=\frac{2x^{2}+1}{\sqrt{x^{2}+1}}>0$ for all $x\in\mathbb{R}$. So the function $f$ is a monotonic increasing function.
If you are doing this for finding maxima of minima, then the answer is maxima or minima do not exist on $\mathbb{R}$. However if you res... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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$X=C[0,1]$ is a Banach space, $M=\{f\in X: f(0)=0\}$, prove $M$ is closed, find explicit formula for the quotient norm, and find an isomorphism. Here is my question:
Let $X$ be a Banach space $C[0,1]$ with the supremum norm. Let $M=\{f\in X: f(0)=0\}$. Show that $M$ is closed. Find an explicit formula for the quotient ... | Continuing your solution: First I will show that $\|[f]\|=|f(0)|$. Notice that for a fixed $f\in C[0,1]$ and for any $m\in M$, we have $\|f+m\|_{\infty}=\sup_{x\in[0,1]}|f(x)+m(x)|\geq |f(0)+m(0)|=|f(0)|$. Therefore $\|[f]\|=\inf\{\|f+m\|_{\infty}:m\in M\}\geq |f(0)|$. Conversely, define the function $m(x)=f(0)-f(x)$. ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/999224",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
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$X$ is the vector space $C[0,1]$ with the norm $\|f\|_1=\int_0^1|f(t)|dt$, and $M=\{f\in X:f(0)=0\}$, show that $M$ is not closed. Here is my question:
Let $X$ be the vector space $C[0,1]$ with the norm $\|f\|_1=\int_0^1|f(t)|dt$. Let
$$M=\{f\in X:f(0)=0\}$$
Show that $M$ is not closed. Show that the “quotient norm" $i... | As for the "quotient norm", have a look at mookid's counter-example and see if you can modify it to force a violation of one of the norm axioms. I'd aim to show that $$\| [f] \|_{\hbox{am I a norm?}} = 0 \ \ \ \Longrightarrow \ \ \ f = 0$$
is violated.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/999293",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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Embedding of a ring into a ring with unity I was reading the theorem on Embedding of a ring into a ring with unity which is as follows:
Let R be ring and $R\times \mathbb Z=\{(r,n)|r\in R,n\in \mathbb Z\}$.
This is a ring with addition defined as $(r,n)+(s,m)=(r+s,n+m).$ and multiplication defined as :$(r,n).(s,m... | With your multiplication, what is your proposed unity?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/999370",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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How to show this Legendre symbol problem Let $n \in \mathbb{N}$ and $p$ an odd prime number such as $p \nmid n$.
Prove that:
$\exists x, y \in \mathbb{Z};\,\, \gcd(x, y) = 1$ such as $x^{2} + ny^{2} \equiv 0\, (\mod p) \iff \Bigg(\displaystyle \frac{-n}{p}\Bigg) = 1$, where $\Bigg(\displaystyle \frac{-n}{p}\Bigg)$ r... | Here is an outline, see if you can fill in more details.
First part: if $x^2+ny^2\equiv0\pmod p$ then $y$ is not a multiple of $p$. (If so then $x$ is a multiple of $p$, but this is impossible since $\gcd(x,y)=1$.) So $y$ has a multiplicative inverse $z$ modulo $p$, so $-n\equiv(xz)^2\pmod p$.
Converse: if $\bigl(\fr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/999472",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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If $x>y$, then $\lfloor x\rfloor\ge \lfloor y\rfloor$, formal proof For x ∈ ℝ, define by: ⌊x⌋ ∈ ℤ ∧ ⌊x⌋ ≤ x ∧ (∀z ∈ ℤ, z ≤ x ⇒ z ≤ ⌊x⌋).
Claim 1.1: ∀x ∈ ℝ, ∀y ∈ ℝ, x > y ⇒ ⌊x⌋ ≥ ⌊y⌋.
Assume, x, y ∈ ℝ # Domain assumption
Assume x > y # Antecedent assumption
Then x ≥ ⌊x⌋ # By def... | $x\geq y$ combined with $\lfloor y\rfloor\leq y$ gives $\lfloor y\rfloor\leq x$.
Here $\lfloor y\rfloor\in\mathbb Z$ so we conclude that $\lfloor y\rfloor\leq \lfloor x\rfloor$.
Proved is now: $$x\geq y\Rightarrow \lfloor x\rfloor\geq \lfloor y\rfloor$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/999551",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Solving for N in a binomial distribution Mid-term study...
Two dice are rolled. How many times must the dice be rolled so that the probability of getting a sum of 10 or greater on at least one roll is larger than 0.9?
So am I correct in thinking that it would be:
$$\sum_{j=1}^{n}\sum_{k=1}^{j}{n \choose k}P^k(1-P)^{j... |
So am I correct in thinking that it would be
$$\sum_{j=1}^{n}\sum_{k=1}^{j}{n \choose k}P^k(1-P)^{j-k} \geq .9$$
No, not quite.
Rolling two die such that their sum is less than 10 is a trial with probability of success $p$.
The number of successes, $N_n$, in a series of $n$ trials has a binomial probability distr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/999650",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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how to integrate $\mathrm{arcsin}\left(x^{15}\right)$? Integral by parts:
$$
I = x\sin^{-1}\left(x^{15}\right) - \int\frac{15x^{15}}{\sqrt{1-x^{30}}}dx
$$
then what?
The answer by wolfram gives an answer contains hypergeometric ${}_2F_1$ function,because it has no elementary answer. The question I want to know is, how ... | $\int\dfrac{15x^{15}}{\sqrt{1-x^{30}}}dx$
$=\int_0^x15t^{15}(1-t^{30})^{-\frac{1}{2}}~dt+C$
$=\int_0^{x^{30}}15t^\frac{1}{2}(1-t)^{-\frac{1}{2}}~d(t^\frac{1}{30})+C$
$=\dfrac{1}{2}\int_0^{x^{30}}t^{-\frac{7}{15}}(1-t)^{-\frac{1}{2}}~dt+C$
$=\dfrac{1}{2}\int_0^1(x^{30}t)^{-\frac{7}{15}}(1-x^{30}t)^{-\frac{1}{2}}~d(x^{30... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/999744",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Why is Q dense in R? Consider a topological space $S$ and an arbitrary subset $E$. Then if the closure of $E$ given by $\bar{E} = S$, we say that $E$ is dense in $S$.
How can we prove that $\mathbb{Q}^{n}$ is dense in the vector space $\mathbb{R}^{n}$ with the standard metric topology?
I intended to prove that $\mathbb... | Hint:
*
*$\mathbb{Q}$ is dense in $\mathbb{R}$.
*It suffices to show $\forall x=(x_i)\in \mathbb{R}^n$, and $x\in I=(a_1,b_1)\times (a_2,b_2)\cdots \times (a_n,b_n)\cap \mathbb{Q}^n\neq \emptyset$. That is $\exists q=(q_i)\in \mathbb{Q}^n$, s.t. $q\in I$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/999872",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 5,
"answer_id": 0
} |
Homology and Fundamental Group (Algebraic Topology - Allen Hatcher) I have some questions regarding 2 parts of the theorem of this section:
1) Having $f = \sum_{i,j}(-1)^jn_i\tau_{ij}$, it pairs $\tau_{ij}$ with opposite signs (in any way I assume), and says that the non paired $\tau_{ij}$'s are $f$.
I don't get why ne... | 1) This information is not directly for gluing, rather it is information on the orientation of the simplex. The goal is to obtain a coherently oriented two-dimensional $\Delta$ complex with boundary equal to $[f]$. Look back at the beginning of the chapter when Hatcher writes down the formula for the boundary of a 3-si... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Is $I=\langle7, 3+\sqrt{19}\rangle$ a principal ideal of $\Bbb Z[\sqrt{19}]$?
Is $I=\langle7, 3+\sqrt{19}\rangle$ a principal ideal of $\Bbb Z[\sqrt{19}]$?
I defined the norm :
$$N(a+b\sqrt{19})=(a+b\sqrt{19})(a-b\sqrt{19})=a^2-19b^2$$
Then we can see the multiplicative property :
$$N((a+b\sqrt{19})(c+d\sqrt{19}))=N(... | You know that any common divisor $z$ of $7$ and $3+\sqrt{19}$ has norm $1$, so it is a unit. Then any principal ideal $J\subseteq\mathbb{Z}[\sqrt{19}]$ containing $7$ and $3+\sqrt{19}$ also contains $1$ and is therefore equal to $\mathbb{Z}[\sqrt{19}]$. You now have to check whether $I=\mathbb{Z}[\sqrt{19}]$: If yes, t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1000046",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
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derivative of $y=\frac{x^2\sqrt{x+1}}{(x+2)(x-3)^5}$ $y=\dfrac{x^2\sqrt{x+1}}{(x+2)(x-3)^5}$
The answer is $\dfrac{x^2\sqrt{x+1}}{(x+2)(x-3)^5} \left(\dfrac{2}{x}+\dfrac{1}{2(x+1)}-\dfrac{1}{x+2}-\dfrac{5}{x-3}\right)$
I know that the quotient rule is used but I don't know how to do this problem. Would you multipl... | It's a long derivative. First use the quotient rule:
$$\frac{\frac{d}{dx}\left(x^2\sqrt{x+1}\right)(x+2)(x-3)^5- x^2\sqrt{x+1}\frac{d}{dx}\left((x+2)(x-3)^5\right)}{((x+2)(x-3)^5)^2}.$$
Then do the derivatives.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1000136",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 2
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α and ω possible limit sets of points
*
*What are all the $α$ and $ω$ possible limit sets of points for:
$$A=\begin{pmatrix}-4&-2\\3&-11 \end{pmatrix}.$$
I am not really sure what to do..
$$\dot{x}=\begin{pmatrix}-4&-2\\3&-11 \end{pmatrix}x$$
| The eigenvalues of $A$ are $-5$ and $-10$, both negative, hence:
*
*For every "initial" condition $x(0)$, $x(t)\to0$ when $t\to+\infty$
*For every "initial" condition $x(0)\ne0$, $\|x(t)\|\to+\infty$ when $t\to-\infty$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1000217",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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How to prove $\int_{0}^{-1} \frac{\operatorname{Li}_2(x)}{(1-x)^2} dx=\frac{\pi^2}{24}-\frac{\ln^2(2)}{2} $ $\def\Li{\operatorname{Li}}$
I wonder how to prove:
$$
\int_{0}^{-1} \frac{\Li_2(x)}{(1-x)^2} dx=\frac{\pi^2}{24}-\frac{\ln^2(2)}{2}
$$
I'm not used to polylogarithm, so I don't know how to tackle it. So any help... | $\def\Li{{\rm{Li}}_2}$Set $x\mapsto -x$ followed by integration by parts, we have
\begin{align}
\int_0^{-1}\frac{\Li(x)}{(1-x)^2}\,dx&=-\int_0^{1}\frac{\Li(-x)}{(1+x)^2}\,dx\qquad\Rightarrow\qquad u=\Li(-x)\,\,\mbox{and}\,\,dv=\frac{1}{(1+x)^2}\\
&=\left.\frac{\Li(-x)}{1+x}\right|_0^{1}+\int_0^{1}\frac{\ln(1+x)}{x(1+x)... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1000349",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "15",
"answer_count": 2,
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Limit of $\frac{2^n+2.71^n+\frac{4^n}{n^4}}{\frac{4^n}{n^4}+n^33^n}$ - what is wrong with my proof? Here's quite a hairy sequence, the limit of which I need to find as $n\rightarrow\infty$:
$$\frac{2^n+2.71^n+\frac{4^n}{n^4}}{\frac{4^n}{n^4}+n^33^n}$$
The Squeeze Theorem seemed like a good idea so here's what I've done... | The problem in your approach is this.
$a,b,c,d>0$, if $a>b$ and $c>d$ you cannot conclude $\frac{a}{c}>\frac{b}{d}$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1000434",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 0
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Throw a dice 4 times. What is the probability `6` be up at-least one time? First time I approach a probability question (:
Throw a dice 4 times. What is the probability 6 be up at-least one time?
Intuitively, I would say: $\frac{1}{6}\times4$.
I would explain as: If you throw one time, probability is $\frac{1}{6}$.
... | Throw a dice six times - are you certain to get at least one six?
The mean number of sixes in four throws is indeed $\frac 23$. But there are combinations with $2, 3 \text{ or } 4$ sixes, and these reduce the number with just one six (apply the same argument to six throws, where it is more intuitive).
If you get no six... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1000488",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 4,
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Set theory by Julia Robinson I used to have a set theory textbook downloaded free from the internet. I lost my laptop in the airport of a city in Eastern Europe, and it was not found (or perhaps “not found”) by the airport security. I now try to rediscover the file. I remember the author was a female mathematician from... | if you google the title of your questions, one of the links shown is to the collected works of Julia Robinson, and google books does indeed show you a discussion of Godel's incompleteness theorem on one of the pages
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1000551",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 1
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Fibers of $\operatorname{Spec}(R)\to\operatorname{Spec}(S):\mathfrak{q}\mapsto \mathfrak{q}\cap S$ are discrete? Suppose $S$ is a subring of a commutative ring $R$, such that $R$ is finitely generated as an $S$-module. I"m curious about a property of the map $\operatorname{Spec}(R)\to\operatorname{Spec}(S):\mathfrak{q}... | Since $R$ is finitely generated as an $S$-module, it has relative dimension zero. That implies if $\mathfrak{q}_1$ and $\mathfrak{q}_2$ are primes of $R$ lying over $\mathfrak{p}\in\operatorname{Spec} S$, then neither is contained in the other.
Suppose $\mathfrak{q}$ is some prime over $\mathfrak{p}$ and $\mathfrak{q}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1000667",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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If $2^{\aleph_{\beta}}\geq \aleph_{\alpha}$, then $\aleph_{\alpha}^{\aleph_{\beta}}=2^{\aleph_{\beta}}$. If $2^{\aleph_{\beta}}\geq \aleph_{\alpha}$, then $\aleph_{\alpha}^{\aleph_{\beta}}=2^{\aleph_{\beta}}$.
Proof: Note that if $\beta \geq \alpha$, then we have $\aleph_{\alpha}^{\aleph_{\beta}}=2^{\aleph_{\beta}}$. ... | You don't necessarily have that $2^{\aleph_\beta}\geq\aleph_\alpha$ implies $\beta\geq\alpha$; consider that $2^{\aleph_0}\geq\aleph_1$, but $0$ certainly isn't $\geq 1$. Instead, try proving that you have both $\aleph_\alpha^{\aleph_\beta}\geq 2^{\aleph_\beta}$ and $\aleph_\alpha^{\aleph_\beta}\leq2^{\aleph_\beta}$ a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1000757",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
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How to solve $a^x=a+x$ Given the equation $a^x = a + x$, how do you express $x$ in terms of $a$?
| This equation has a solution expressed in terms of Lambert function and the solution is $$x=-a-\frac{W\left(-a^{-a} \log (a)\right)}{\log (a)}$$ In fact, any equation which can be written as $$A+Bx+C\log(D+Ex)=0$$ has explicit solution(s) in terms of Lambert function.
In the real domain, $W(t)$ only exists if $t \geq -... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1000904",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
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Is $\sqrt{m}$ irrational iff at least one prime occurs with an odd exponent in the factorisation of $m$? Thinking about it, I think I found the following criterion for irrationality of $\sqrt{m}$ if $m$ is a positive integer.
Let $p_1^{a_1}\cdots p_k^{a_k}$ be the prime factorization of $m$. Then $\sqrt{m}$ is irratio... | Yes this is true. A nice way to think about this, is that if you extend the notion of prime factorisation to allow for negative exponents, you can show that all rational numbers have a unique prime factorisation as well. Then if you square a rational number of the form $r=p_1^{a_1}\cdots p_k^{a_k}$, where $a_i\in\mathb... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1000998",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
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How many even numbers will $99$ dice show if we roll them forever under a certain condition? Consider a six-sided die with numbers from $1$ to $6$. Imagine you have a jar with $99$ of such dice. You throw all dice on the floor randomly. You look at one of the dice on the floor at a time. For each die, you do the follow... | OK, its not about Mathematica, but let's make it about using Mathematica to visualize the result, just for fun.
Analytic
The first run produces some number of nE even numbers with probability
pe0 = PDF[BinomialDistribution[99, 1/2], nE]
Any number from 0 to 99 is possible, justifying the last rule.
Irrespective of t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1001079",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Question about recurrence relation problem.
solve the following recurrence relation, subject to given initial conditions.
$a_{n+1} = 6a_n -9,$
$a_0 = 0,$
$a_1 = 3.$
Here is what I have done.
$a_{n+1} - 6a_n +9 = 0$
$a_n = r^n$
$r^{n+1} - 6r^n + 9 = 0$
$r + 6 = 0$
$r = -6$
$a_n$ = $X(-6)^n$
$a_1 = 3 = X(-6)^1$
$X = -1/... | $$\begin{align}
a_{n+1}&=6a_n-9\\
a_{n+1}-\frac95&=6a_n-\frac{54}5\\
&=6\left(a_n-\frac95\right)\\
u_{n+1}&=6u_n\\
u_n&=6u_{n-1}=6^2u_{n-2}=\cdots=6^{n-1}u_1\\
&=6^{n-1}\left(a_1-\frac95\right)\\
&=6^{n-1}\left(3-\frac95\right)\\
a_n-\frac95&=\frac65(6^{n-1})\\
&=\frac{6^n}5\\
a_n&=\frac15\left(9+6^n\right)\qquad\black... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1001193",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Prove $\int_0^1\frac{\ln2-\ln\left(1+x^2\right)}{1-x}\,dx=\frac{5\pi^2}{48}-\frac{\ln^22}{4}$ How does one prove the following integral
\begin{equation}
\int_0^1\frac{\ln2-\ln\left(1+x^2\right)}{1-x}\,dx=\frac{5\pi^2}{48}-\frac{\ln^22}{4}
\end{equation}
Wolfram Alpha and Mathematica can easily evaluate this integral... | $\newcommand{\bbx}[1]{\,\bbox[15px,border:1px groove navy]{\displaystyle{#1}}\,}
\newcommand{\braces}[1]{\left\lbrace\,{#1}\,\right\rbrace}
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"language": "en",
"url": "https://math.stackexchange.com/questions/1001269",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
"answer_count": 5,
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Algebras: Finite Additivity $\implies$ Countable Additivity (Idea?) Given an algebra $\mathcal{A}$.
Consider a set function $\mu:\mathcal{A}\to\mathbb{R}_+$:
Then countable additivity follows from finite additivity:
$$\mu(A+B)=\mu(A)+\mu(B)\implies\mu(A)=\sum_{n\in\mathbb{N}}\mu(A_n)$$
Can you explain the idea behind t... | I think there is something missing in yiur question.
To see this, consider the algebra(!) of all subsets $A \subset \Bbb{N}$ which are finite or for which $A^c$ is finite.
Define $\mu(A)=0$ if $A$ is finite and $\mu(A)=1$ otherwise. It is easy to verify that $\mu$ is finitely additive, but not countably additive.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1001457",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Proof the concave transformation of the tail distribution is always above the tail distribution I need to prove that for a given continuous non-decreasing distribution $F_X(x)$, and a concave non-decreasing distortion function $g(.)$ defined on $[0,1]$, the following holds: $$g(1-F_X(x)) \ge 1-F_X(x)$$
We know that $g... | If you're not familiar with it already, Jensen's Inequality is applicable here:
$E[g(1-\mathbf{1}_{\leq x}(X))]\leq g(1-E[\mathbf{1}_{\leq x}(X)])=g(1-F_X(x))$ [for $g$ concave].
Since $g(0)=0$ and $g(1)=1$ then
$E[g(1-\mathbf{1}_{\leq x}(X))]=E[\mathbf{1}_{\geq x}(X)]=1-F_X(x)\implies g(1-F_x)\geq 1-F_X$
$\square$..... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1001591",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Limits of Exponent Laws I have recently learned (discovered) that the exponent law $b^{mn} = {(b^m)}^n$ is not universally applicable. To demonstrate, if it were we could conclude that $(-1)^{\frac{3}{2}}$ (or by extension -1 to any power) is equal to 1.
$(-1)^{\frac{3}{2}} = (-1)^{2*\frac{3}{4}}$
= $((-1)^2)^{\frac{3... | Basically, the one thing you need to remember is : $b^m$ does not have a signification when the two following condition are simultaneously met :
-b is negative
-m is not an integer
So in your example, $(-1)^{\frac{3}{2}}$ does not have any signification. Hence obviously, you should not write it.
Just remember that and ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1001665",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Writing iterated integral of a function Write an iterated integral of a function f for the region given by a triangle with vertices at point (1,1), (1,2), (3,0).
I figured that I'm supposed to first find the equations of the three lines representing the sides of the triangle.
(1,2) and (1,1) make $x=1$
(1,1) and (3,0)... | Yep, it's as easy as that! Note that you got the slope of your second line wrong: it should be:
$$
y = \frac{-1}{2}x + \frac{3}{2}
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1001760",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Solve $\frac{1}{2}kx^{2}-cx=\frac{1}{2}ky^{2}+cy$ for $y$ I have the equation:
$\frac{1}{2}kx^{2}-cx=\frac{1}{2}ky^{2}+cy$,
where $k$ and $c$ are arbitrary constants.
How do I go about simplifying this and solving for $y$ in terms of $x$, excluding the obvious solution $y=-x$
| Treat $x$ as a constant as well. We could use the standard quadratic formula, but since you noticed that $y = -x$ is a solution, let's try factoring instead. We obtain:
\begin{align*}
0
&= \tfrac{1}{2}ky^{2} - \tfrac{1}{2}kx^{2} + cy + cx \\
&= \tfrac{1}{2}k(y^{2} - x^{2}) + c(y + x) \\
&= \tfrac{1}{2}k(y - x)(y + x) +... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1001876",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 0
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How to determine if set of vectors is a basis for W Consider the subspace $$
W =\left\{ \begin{bmatrix} x_{1} \\ x_{2} \\ x_{3}\end{bmatrix} \in \mathbb{R}^3 \,| x_{1}+x_{2}+x_{3} = 0 \right\}
$$
Is the set S a basis for W? $$
S= \left\{ \begin{bmatrix} -1 \\ -1 \\ 2 \end{bmatrix} , \begin{bmatrix} -3 \\ 2 \\ 1\end{bm... |
$S$ is a basis for $W$ since: (2) span $W$; (1) is a linearly independent set.
To prove (1) you just have to solve: $\alpha (-1,-1,2)+\beta(-3,2,1)=0$ for $\alpha$ and $\beta$ to get $\alpha=0=\beta$.
To prove (2): Let $(x_1,x_2,x_3)\in W$ (i.e. $x_1+x_2+x_3=0$) you have to find $\alpha$ and $\beta$ such that $\alp... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1001960",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
taking the inverse of power series I am working with solution to near regular singular points.
I started with:
$$y_1(x)=x^\frac{1}{2}\left[1-\frac{3}{4}x+\frac{9}{64}x^2-\frac{3}{256}x^3+\cdots\right] $$
Then I squared it:
$$y_1^2(x) = x\left(1-\frac{3}{2}x+\frac{27}{32}x^2-\frac{15}{64}x^3+\cdots\right)$$
Why is the i... | If $$y=\sqrt{x} \left(1-\frac{3 x}{4}+\frac{9 x^2}{64}-\frac{3 x^3}{256}+\cdots\right)$$ then effectively $$y^2=x \left(1-\frac{3 x}{2}+\frac{27 x^2}{32}-\frac{15 x^3}{64} +\cdots\right)$$ Now you want to compute $\frac{1}{y^2 }$. You can write $$\frac{x}{y^2 }=\frac{1}{1-\frac{3 x}{2}+\frac{27 x^2}{32}-\frac{15 x^3}{6... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1002059",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Free Group Norms
Hello everyone, I'm trying to solve this problem, but I'm stuck... i don't quite understand the definition of the norm, If you guys can give me a better explanation, I would appreciate it, Thanks
| You have to calculate how many groups elements are represented by words at length at most $N$ in the group generators.
Let $G$ be free abelian with generators $a,b$. How many elements have shortest representative of length $n$? For $n=0$ there is just $1$. For $n>0$, we have $a^n,b^n,a^{-n},b^{-n}$, which is $4$. We al... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1002165",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
After removing any part the rest can be split evenly. Consequences?
Let $S$ be a finite collection of real numbers (not necessarily distinct). If any element of $S$ is removed then the remaining real numbers can be divided into two collections with same size and same sum ; then is it true that all elements of $S$ are ... | Yes, all elements of $S$ must be equal.
Clearly, $S$ has an odd number of elements; say $S=\{x_1,\dots,x_{2n+1}\}$. The condition "if any element of $S$ is removed then the remaining real numbers can be divided into two collections with same size and same sum" means that the column vector $\mathbf X=[x_1,\dots,x_{2n+1}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1002267",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 1
} |
Number of positive integral solutions to $x+y+z+w=20$ with $xWhat is the number of positive unequal integral solution of the equation $x+y+z+w=20$, if $\,x<y<z<w\,$ and $\,x,y,z,w\ge1\;?$
How to solve this question?
| For a count of the solutions without restriction on the order by inclusion-exclusion, see this answer. The result is $552$; then the number of ordered solutions is obtained as $\frac{552}{4!}=23$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1002344",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 1
} |
Integral equation and metric spaces Let $C([0,\frac{\pi }{2}])$ be the set off all continuous functions defined on $[0,\frac{\pi }{2}]$ . Prove that this integral equation $$
f(t) = \int\limits_0^{\frac{\pi }{2}} {\arctan } (\frac{{f(s)}}{2} + t)\,ds
$$ has an unique solution on $C([0,\frac{\pi }{2}])$.
Any ideas ? I j... | Daniel is right, Banach's fixed point theorem is the way to go. Defining $A$ as he did, we see that $A: C([0,\pi/2]) \rightarrow C([0,\pi/2])$. In addition,
$||Af-Ag||_{C([0,\pi/2])} = \sup_{t \in [0,\pi/2]} \left|\int_0^{\pi/2} \arctan\left(\frac{f(s)}{2} +t\right) - \arctan\left(\frac{g(s)}{2} + t\right) ds\right|... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1002447",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
How prove this $x^3+y^3+z^3+3\ge 2(x^2+y^2+z^2)$ Question:
let $x,y,z>0$ and such $xyz=1$, show that
$$x^3+y^3+z^3+3\ge 2(x^2+y^2+z^2)$$
My idea: use AM-GM inequality
$$x^3+x^3+1\ge 3x^2$$
$$y^3+y^3+1\ge 3y^2$$
$$z^3+z^3+1\ge 3z^2$$
so
$$2(x^3+y^3+z^3)+3\ge 3(x^2+y^2+z^2)$$
But this is not my inequality,so How pro... | Here is a possible solution: (although it is not the most elegant one)
I will employ Mixing Variables technique here. Since the inequality is symmetric, WLOG let $x=\min(x,y,z)$. Therefore $t^2:=yz \ge 1$.
Let
$$f(x,y,z)=x^3+y^3+z^3-2(x^2+y^2+z^2)$$
I wish to show
$$f(x,y,z)\ge f(x,\sqrt{yz},\sqrt{yz}) = f(\frac1{t^2... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1002529",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "14",
"answer_count": 4,
"answer_id": 3
} |
2(n-1) induction There are $n$ cities and every pair of cities is connected by exactly one direct one-way road. Now more one-way roads have been added between some cities so that between some pairs of cities there may be two direct roads between them, for example, there may now be a road going directly from some city A... | Yes, you're on the right track.
Hint: Your first question will rule out one city as a potential dead end. What would happen if you never asked about that city again, and simply pretended it didn't exist?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1002645",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Mathematical trivia (i.e. collections of anecdotes and miscellaneous (recreational) mathematics) Can you suggest some books on mathematical trivia?
I use the word "trivia" with a double meaning in this case:
*
*curious anecdotes that enlighten what the real life of mathematicians is like (like the ones in Mathemati... | My suggestions:
1."The Penguin book of curious and interesting mathematics" by David G. Wells;
2."The Colossal Book of Mathematics" by Martin Gardner;
3."Maths Facts, Fun, Tricks and Trivia" by Paul Swan;
4."Math hysteria" by Ian Stewart;
5."In Mathematical Circles: A Selection of Mathematical Stories and Anecdotes: Qu... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1002768",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "17",
"answer_count": 6,
"answer_id": 0
} |
$\lim_{x\rightarrow\infty}(\frac{x+1}{x-1})^{\sqrt{x^2-1}}$ I'm trying to determine $\lim_{x\rightarrow\infty}(\frac{x+1}{x-1})^{\sqrt{x^2-1}}$ using L'Hopital's Rule.
I can clearly see that $\lim_{x\rightarrow\infty}(\frac{x+1}{x-1})^{\sqrt{x^2-1}} = \frac{\infty}{\infty},$ so we can use L'Hoptial's Rule. I'm having ... | Use that if $\displaystyle\lim_{x\to+\infty}f(x)=L$ then $\displaystyle\lim_{x\to+\infty}\ln(f(x))=\ln(L)$, for $f(x),L>0$.
And that $\ln(a^b)=b\ln(a)$ for $a,b>0$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1002826",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 5,
"answer_id": 1
} |
Show that if $f$ is complex differentiable and in some region $U$ it is $f'=0$, then $f$ is constant I have to show show that if $f$ is complex differentiable and in some region $U$ it is $f'=0$, then $f$ is constant. How can one prove it?
| Let $a\in U$ and $V=\{z\in U / f(z) =f(a)\}$. Show that $V$ is open and closed in $U$.
Let $b $ in $ V$ hence $ b$ in $ U$. There is $ r>0$ st the ball $B(b,r)$ is in $U$.Fix $z$ in this ball. The map $g(t)=f(tb+(1-t)z)$ is dif and $g'=0$ then $g(0)=g(1)$. This show the opnes.
$V$ is closed in $U$ because $f$ is contin... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1002946",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
quick way to prove $\mathbb{Z}_2 \times \mathbb{Z}_3$ is isomorphic to $\mathbb{Z}_6$ I want to show that $\mathbb{Z}_2 \times \mathbb{Z}_3$ is isomorphic to $\mathbb{Z}_6$. The number of elements and their orders are equal but I don't see a way to prove that the groups are isomorphic except for writing down the Cayley... | Hint:
What is the subgroup generated by $ (1_{\Bbb Z_2},1_{\Bbb Z_3}){}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1003041",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
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