Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Find G. C. F. of $(8n^3 + 8n, 2n+1)$ I'm stuck with this problem, I divided $8n^3 + 8n$ by $2n+1$ and obtained $5$, so now my G. C. F is $\gcd(2n+1, -5)$.
What's next? I can't divide $2n+1$ by $-5$.
| Well, you figured out that $\gcd(8n^3 +8n, 2n+1) = \gcd(2n+1, -5)$.
And as $\gcd(\pm a, \pm b) = \gcd(a b)$ we know $\gcd(8n^3 + 8n, 2n+1) = \gcd(2n+1,5)$.
As $5$ is prime then $\gcd(2n+1, 5)$ is either $1$ or $5$.
It is $5$ if $5|2n+1$. ANd it is $1$ if $5\not\mid 2n+1$.
And $5|2n+1 \iff$
$2n+1 \equiv 0 \pmod 5 \iff$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3433038",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Show that if $x^p-x-1$ is reducible in $F[x]$ where $F$ has characteristic $p$ then it splits in $F[x]$ into monic distinct factors. So far, I have shown that the function is separable, after all, $D_f(x)=-1$ so they are relatively prime. I am having trouble showing that it splits. Our field $F$ contains $GF(F_p)$ thus... | Hint: $x$ is a root means that $x^p = x+1$; but $(x+1)^p = x^p+1$, so $x+1$ is a root too.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Prove that $ST = TS$ if $S$ and $T$ have same eigenvectors
Suppose $V$ is finite-dimensional, $T \in \mathcal{L}(V)$ has dim $V$ distinct eigenvalues, and $S \in \mathcal{L}(V)$ has the same eigenvectors as $T$ (not necessarily with the same eigenvalues). Prove that $ST = TS$.
My general thinking is take $v \in V$ a... | Let $\lambda_1,\dots,\lambda_n$ be the distinct eigenvalues of $\textsf{T}$. Let $v_1,\dots,v_n$ be the corresponding eigenvectors. Also, let $\mu_1,\dots,\mu_n$ be the eigenvalues of $\textsf S$.
So, we have
$$\textsf{T}(v_i) = \lambda_iv_i \qquad (i=1,2,\dots,n)$$
and
$$\textsf{S}(v_i) = \mu_iv_i \qquad (i=1,2,\dot... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3433282",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Artin's theorem and Brauer's theorem on Characters
Artin: Let $\chi$ be rational valued complex character of (finite) group $G$. Then $\chi$ can be written as $\mathbb{Q}$-linear combination of characters $1_H^G$ for some cyclic subgroups $H$ of $G$.
In the theorem of Brauer, the subgroups $H$ are allowed to be elem... | The answer seems to be negative:
If we take the characters $\{ (1_H)^G \,\, | \,\, H \mbox{ is elementary subgroup of $G$} \}$, then an integer valued character of $G$ may not be integral combination of these specific induced characters.
Consider the quaternion group $G=Q_8=\langle x,y\rangle$. All subgroups of $Q_8... | {
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Is a function absolutely continuous if and only if its derivative is in $L^1$? So I was reading about the Fundamental Theorem of Calculus for Lebesgue Integrals. It said $F$ is absolutely continuous on $[a,b]$ iff $F'$ exists $a.e.$, $F'\in L^1$ and $$F(x)-F(a)=\int_a^x F'dm.$$
Well, but the Fundamental Theorem of Calc... | Correction
If a differentiable function $f$ is continuously differentiable except a measure-zero set of points on a closed interval, then Newton-Leibniz formula holds for $f$.
There is a function called Volterra's function differentiable everywhere, having a bounded derivative but its derivative is not Riemann integrab... | {
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"timestamp": "2023-03-29T00:00:00",
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Let $a_n$ and $b_n$ be sequences . if ... $\lim _{n\to \infty } |a_n - b_n| = 1$ prove that $b_n$ is Convergent Let $a_n$ and $b_n$ two sequences .
if $a_n$ is Convergent
and $\lim _{n\to \infty } |a_n - b_n| = 1$
prove that $b_n$ is also Convergent
I know that it is not true but I need to find example I think
$a... | Hint: Try $a_n=0$ and $b_n=(-1)^n$.
| {
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A subset of a set with a specified property Consider the set $A=\{1,2,3,...,2000\}$. We are asked to find a subset of maximum size so that the difference of any two members of this subset is not a prime number. I think one subset of maximum size can be $\{1,5,..., 1397\}$. i.e., numbers of the form $4k-3$ for natural $... | Here is an argument for not more than $500$ elements of $S$.
We will use your pigeonholes of $\{1,2,3,4\}, \{5,6,7,8\},...\{1997,1998,1999,2000\}$.
That is, the pigeonhole sets are of the form $\{4k+1,4k+2,4k+3,4k+4\}$ for $k=0,1,...,499$.
Let $S$ be the set with no prime differences that we are constructing.
Suppose... | {
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Solving circular arrangement problem with k identical and m distinct positions. There are 11 chairs around a circular table. In how many ways can we arrange 10 people in these seats?
If
(i) There are 11 identical chairs placed equally apart around the table
My solution is:
First fix the one person in any of the chair.... | In part I, Fix the empty chair and $10$ people can be seated in 10 identical chairs in $10P10= 10!$ ways.
In part II, by fixing one chair, you have made the chairs in a linear fashion. Then there are 11 ways a person can be seated, 10 ways the second person can be seated and so on until the last person can be seated ... | {
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Findind Hamiltonian functions for flow Consider the symplectic space $(\mathbb{R}^2, \omega_{st}=dx \wedge dy$. I want to find a Hamiltonian function, such that its time-1-flow is of the form
$\varphi(x,y)= (2x, \dfrac{1}{2} y)$
My ansatz:
Consider $\varphi_t(x,y):=(2xt, \dfrac{1}{2} y t)$. Now
$\dfrac{d}{dt} \varphi_... | Let's look for a flow of the form $$\varphi_t(x,y) = \left(2xf(t), \dfrac{y}{2f(t)}\right)$$ for some function $f(t)$. No matter what happens, this'll always be a symplectomorphism. Since $\varphi_0 = {\rm Id}_{\Bbb R^2}$, we need $f(0) = 1/2$ and $\varphi_1= \varphi$ gives $f(1) = 1$. Moreover, the group-property of t... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Counting the number of partitions of $\mathbb{R}$ into countable subsets I'm trying to determine the number of partitions of $\mathbb{R}$ into countable subsets. Obviously each partition will contain uncountably many sets and the cardinality of the set of all such partitions is bounded above by the number of relations ... | The number of such partitions is $2^{\mathfrak c}$.
First, each partition $\pi$ is a collection of disjoint subsets of $\mathbb R$, so it has size at most $\mathfrak c$, and so there are at most $|[\mathcal P(\mathbb R)]^{\le\mathfrak c}|=2^{\mathfrak c}$ partitions, where $[A]^{\le\kappa}$ denotes the collection of s... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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How can I express that 2 is the only prime number that is even using predicate logic I defined some predicates below:
$B(a)$: a is a prime number
$C(a, b)$: b is divisible by $a$
Let $a, b$ be integers greater than $1$
My attempt is below, but I am not sure whether it is correct.
$$\forall a\in \mathbb{Z}, a = 2 \Left... | This is also known as a definite descriptions in Bertrand Russell's theory of descriptions,
There is only one $x$ satisify $P$:
$$\exists x_0, P(x_0)\wedge (\forall x_1,P(x_1)\rightarrow x_0=x_1)\tag{1}$$
There is only one $x$ satisify $P$, and that $x$ satisify $Q:$
$$\exists x_0, P(x_0)\wedge (\forall x_1,P(x_1)\rig... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3434476",
"timestamp": "2023-03-29T00:00:00",
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Field extension of isomorphism of elliptic curves "If $A,B$ are elliptic curves over a field $k$ of characteristic not 2 or 3 and if $A,$ isomorphic over an extension of $k$, then they become isomorphic over an extension of $k$ of degree $\leq 6$. "
$\textbf{Q:}$ Since $A,B$ are exactly described by $g_2$ and $g_3$ or ... |
Should not $j$-invariant automatically conclude isomorphism over arbitrary extension?
No, the $j$-invariant only classifies elliptic curves over an algebraically closed field. Curves with the same $j$-invariant may not be isomorphic over $k$ as they could be twists. That is, a curve and its twist might not be isomorp... | {
"language": "en",
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Continuity of the stochastic process $X_t=\int_0^t(a+b\frac{u}{t}) \, dW_u$ I am wondering about the continuity of the stochastic process
$$X_t=\int_0^t \left(a+b\frac{u}{t}\right)\,dW_u$$
which has variance $t$ and normally distributed for
$a^2+\frac{b^2}{3}+ab=1$
The process seems to be discontinuous at $t=0$ except... | We have $$\begin{align*}X_t&=\int_0^t \left(a+b\frac{u}{t}\right)\,dW_u \\\\ &=aW_t + \frac{b}{t}\int_0^t u\, dW_u\end{align*}$$
Using Itô's formula we get:
$$\int_0^t u\, dW_u = tW_t - \int_0^t W_s\, ds$$
and so
$$X_t = (a+b)W_t - \frac{b}{t}\int_0^t W_s \, ds$$
We know: $W_s$ is continous a.s. hence from the Fundamen... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Find a triangle such that : area are natural number and sides is prime numbers Find a triangle such that :
$area=S\in\mathbb{N^{*}}$ and the sides $a,b,c$ prime numbers
I need find this triangle not by imagine I need by a prof or something
I try $2,3,5$ , $3,5,11$ , $13,11,7$ but I tried
I would like a explain to f... | One possibility is that the area is $0$, such as your $2,3,5$ attempt. In fact, $2$ together with any twin prime works.
If we don't want the area to be $0$, then I think the most helpful tool we have at our disposal is Heron's formula for the area:
$$
S=\sqrt{s(s-a)(s-b)(s-c)}\\
s=\frac{a+b+c}2
$$
We see that if $s$ is... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3435049",
"timestamp": "2023-03-29T00:00:00",
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How to nondimensionalize a second order differential equation
A body of mass $m$ is thrown upwards in a vertical direction from the
earth's surface with a velocity $v$. The air resistance is supposed to
be taken into account by Stoke's law $F_R= - cv$ for the flow
resistance in viscous fluids, which is reasonabl... | Calling
$$
\cases{
\tau = \frac{t}{t_o}\\
\eta = \frac{x}{x_o}
}
$$
we have
$$
\frac{d^n}{dt^n} = \frac{1}{t_o^n}\frac{d^n}{d\tau^n}
$$
and after substitution
$$
m\frac{x_o}{t_o^2}\eta''+ c\frac{x_o}{t_o}\eta' + m g = 0
$$
or
$$
\eta'' + \frac{c t_o}{m}\eta'+\frac{t_o^2 g}{x_o} = 0
$$
now determining $x_o,t_o$ such th... | {
"language": "en",
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Reference for Derivation of Higher order Runge Kutta I have a problem about determining $a_1, a_2, k_1, k_2, \ldots a_n, k_n,\ldots$
In the general form of the Higher Order Runge-Kutta below :
$$y_{r+1}=y_r+a_1k_1+a_2k_2+\cdots+a_nk_n$$
For the convenient, i'll write it down The Runge Kutta $2^{\text{nd}}$ in my book j... | A good source on Butcher tableaus and B-trees would, almost canonically, be the book by Butcher himself. For a short overview, see the three sets of slides with an introduction to B-trees, demonstration of the method for up to order 4 and outlook to implicit methods, or a historical overview
You could of course also re... | {
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How to arrive from $x^2y^2－x^2－y^2－6xy＋4$ to $(xy＋x＋y－2）(xy－x－y－2）$? How do we arrive from
$x^2y^2－x^2－y^2－6xy＋4$
to
$(xy＋x＋y－2）(xy－x－y－2）$
?
According to my book these two are equal, but I can't understand how to transform one to another.
| $$x^2y^2-x^2-y^2\color{red}{-6xy}+4=x^2y^2\color{red}{-4xy}+4-x^2\color{red}{-2xy}-y^2\\=(xy-2)^2-(x+y)^2=(xy+x+y-2)(xy-x-y-2)$$
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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A symmetric, diagonally dominant matrix A with real positive diagonal entries is positive definite Suppose $A \in \mathbb{R}^{n\times n}$ is symmetric and diagonally dominant with positive diagonal entries. I have to prove that $A$ is positive definite but without using theorems, just algebraically.
I´ve started with: ... | I guess you write $x^TAx=\sum_{i=1}^n a_{i,i}x_i^2+\sum_{i\neq j}a_{i,j}x_ix_j\ge\sum_{i=1}^n (\sum_{i\neq j}|a_{i,j}|)x_i^2-\sum_{i\neq j}|a_{i,j}||x_i||x_j| =\sum_{j>i }(|a_{i,j}|(x_i^2+x_j^2-2|x_i||x_j|))\ge 0$
| {
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How to find a substitution that transforms an equation into a particular equation. (Roots of polynomials). Find the value of $c$ so that the substitution $x=u+c$ transforms the equation $x^3-12x^2+45x-54=0$ into the equation $u^3-3u^2=0$
My first idea is to find the roots of $u^3-3u^2=0$ And getting $u=0$ and $u=3$, ho... | Here's a relatively fast approach that avoids factoring a cubic.
Notice that $u^3 - 3 u^2 = u^2 (u - 3)$ has a double root at $u = 0$, so if $x = u + c$, the given polynomial, $$p(x) := x^3 − 12 x^2 + 45 x − 54$$ has a double root at $x = (0) + c = c$.
Since $c$ is a double root, it is also a root of $$p'(x) = 3 (x^2 -... | {
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Why do we get rid of the $-$ sign in this equation I am finding interquartile ranges for the exponential function cdf as part of a project for university:
$$Q\left(\frac{1}{4}\right) = -\frac{1}{\lambda}\ln\left(1-\frac{1}{4}\right)= -\frac{1}{\lambda}\ln\left(\frac{3}{4}\right)= \frac{1}{\lambda}\left(\ln(4) - \ln(3)... | A minus sign switches the order of subtraction: ${-}(a-b) = b-a$. You can see this by writing ${-}(a-b)$ as ${-}(a + {-}{-}b)$ and distributing the negative: ${-}(a-b) = {-}(a + {-}b) = -a + {-}{-}b = b -a$.
Alternatively: Use the identity ${-}\ln(x) = \ln(1/x)$.
| {
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Riemann upper sum of Riemann integral I read this lecture note https://www.math.ucdavis.edu/~hunter/m125b/ch1.pdf. For all partitions $P$ of finite closed interval $[a,b]$, define the upper Riemann sum $f$ w.r.t. partition $P$ by $U(f; P)$. If $f$ is bounded, then $m(b-a)\leq U(f; P)$, where $m\leq f.$
why $\inf_{P} U... | The collection of partitions $\mathcal{P}$ is directed set with preorder $\supseteq$ where $P' \supseteq P$ indicates that $P'$ is a refinement of $P$, that is the set of endpoints of subintervals of $P$ is contained in the set of endpoints of subintervals of $P'$.
The net $\{U(f;P) \}$ defined on $\mathcal{P}$ is no... | {
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Solving system of Congruences with Two Variables (x and y) I know a bit about the Chinese Remainder theorem but what do I do in the case I was asked to solve a system of congruences such as this with two variables:
$3x + y = 7$ (mod 8)
$4x + 3y = 1$ (mod 8)
| The first equation is in a form suggesting substitution method as you have $y$ with coefficient $1$:
*
*$y \equiv 7-3x \pmod 8$
Plug this into the second equation and solve for $x$:
$$4x + 3(7-3x) \equiv 4x +21 -9x \stackrel{21=16+5}{\equiv} -5x +5 \equiv 1 \pmod 8$$
Now, note that the $5^2 \equiv (24+1) \equiv 1 \p... | {
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write the given third order linear equation as an equivalent system of first order equations with initial values. Write the given third order linear equation as an equivalent system of first-order equations with initial values:
$$-(‴+3\sin(t)′) = 2t$$
with $$y(3)=−2, y'(3)=3, y''(3)=0$$
Use $_1=$, $_2=′$, and $_3=″$.... | You have that :
$$\begin {cases} x_1=y \\ x_2=y'=x'_1 \\x_3=y''=x'_2 \end{cases} \implies \begin {cases} x'_1=x_2 \\ x'_2=x_3 \end{cases} $$
And as you noted you have also that
$$x'_3=-3\sin(t)x_2-2t$$
Therefore you can write the DE equation as :
$$\begin{pmatrix} x_1 \\ x_2\\x_3 \end{pmatrix}'= \begin{pmatrix}0 & 1 ... | {
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Intersection of subgroups of given orders and normality Quite confused about how to solve this question..As it would have been easier to solve if the group was given to be cyclic, but no such case here
Let $H$ and $K$ be subgroups of a group $G$ of orders $14$ and $21$ respectively. If $H \cap K \neq\{e\}$, here $e$ is... | This is not true. The issue/trick is that there is no information about the group $G$ so I can pick it to have very few normal subgroups.
For example, we can take $G=S_n$ for some $n\geq 7$ (why?). Then $G$ has very few normal subgroups (see here), and in particular none of order $7$ (why is $7$ relevant?).
For a concr... | {
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HCF $(x,y) = 16$ and LCM $(x,y) = 48000$. Then the possible number of pairs $(x,y)$ Let $S$ be the set of all ordered pairs $(x,y)$ of positive integers, with HCF $(x,y) = 16$ and LCM $(x,y) = 48000$. The
number of elements in $S$ is
My Attempt : $48000= 2^7. 3 . 5^3$ As the L.c.m contains $2^7$ as a factor and G.c.d ... | WLOG let $\dfrac xX=\dfrac yY=16$ so that $(X,Y)=1$
We have $48000\cdot16=xy=16^2XY\iff XY=3000=3\cdot2^3\cdot5^3$
So, the possible values of $X$ can be
take none of the factors $$1$$
take one of the factors $$3;2^3;5^3$$
take two of the factors $$2^3\cdot5^3;3\cdot5^3;2^3\cdot3$$
take all three of the factors $$3\cdo... | {
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Asking about $R$-submodules which are not finitely generated.
Let $R=\mathbb{Z}[X_{1},X_{2},...,X_{n},...]$ the polynomial ring in infinitely many variables with coefficients in $\mathbb{Z}$. Find (and explain why) a $R^{R}$-submodule (being $R^R$ the regular module over $R$) which is not finitely generated.
I need h... | This is, I believe, the standard example of the fact that a submodule of a finitely generated module needs not be finitely generated.
Hint: The regular module is finitely generated (as it always is for a unital ring) by $1$. We are looking for a submodule (an ideal of $R$, basically) which is not finitely generated. $R... | {
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} |
$\int_{0}^{\infty}e^{-x}p_n(x)p_m(x)=0$ when $p_n(x)=(-1)^nn!\sum_{k=0}^{n}\binom{n}{k}\frac{(-x)^k}{k!}$ Let $n \in \mathbb N_0$. Consider the polynomials $p_n$ defined by
$$p_n(x)=(-1)^nn!\sum_{k=0}^{n}\binom{n}{k}\frac{(-x)^k}{k!}$$
I want to show that $\int_{0}^{\infty}e^{-x}p_n(x)p_m(x)=0$ for $n \neq m$.
The hi... | In order to show the orthogonality of $p_n$ and $p_m$ (with respect to the inner product $\langle f,g\rangle = \int_{0}^{+\infty}f(x)g(x)e^{-x}\,dx$) for $n\neq m$ it is enough to prove the orthogonality of $p_n(x)$ and $x^j$ for $j<\deg(p_n)=n$. Now
$$\langle p_n(x), x^j \rangle = (-1)^n n!\sum_{k=0}^{n}\binom{n}{k}\f... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3436825",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Formula for sum of combinations I am trying to find a closed-form formula or, at least, a different and more useful representation for the following sum of combinations:
\begin{equation}
\sum_{i=1}^{n}\frac{n!}{i!(n-i)!}\times\frac{(-1)^i}{i}
\end{equation}
| We have that
$$\sum_{i=1}^{n}\frac{n!}{i!(n-i)!}\times\frac{(-1)^i}{i}=\sum_{i=1}^{n}\frac{(-1)^i}{i}\binom{n}{i}$$
then refer to
*
*Sum of Pascal's triangle column
*Let $n$ be a positive integer, Prove that $\sum_{k=1}^n\frac{ (-1)^{k-1}}{k}{n \choose k} = H_n$
| {
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Area triangle (possible) The area of a right triangle is an integer greater than $ 85 $. If the hypotenuse measures $ 20 $, what is the area of this triangle?
It’s just a 3-4-5 right triangle, so the area is $\boxed{96}$.
This is true? If so, how to prove it?
But he said the area is integer, not the sides
Can you use W... | We need
$$a^2+b^2=20^2\implies a^2=400-b^2$$
which by inspection, taking $a$ and $b$ integers, leads to the unique solution $a=12$ and $b=16$ such that $\frac12 ab\ge 85$ and the area in that case is equal to $96$.
For $a$ and $b$ reals we have
$$\frac12ab=\frac12a\sqrt{400-a^2}=86 \implies a\sqrt{400-a^2}=172 $$$$\imp... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3437413",
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"source": "stackexchange",
"question_score": "5",
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Show that the smallest eigenvalue is strictly positive. I have a $2\times 2$ real, symmetric and positive definite matrix $B_{x,n}$ which depends on a point $x\in[0,1]$ and $n\in\mathbb{R}$. I want to show that for sufficiently large $n$, the smallest eigenvalue of $B_{x,n}$ is strictly positive uniformly on $x$.
To do... | Since $B$ is positive definite then $\det(B)>0$ and $\operatorname{tr}(B)>0$.
Since the determinant and the trace are continuous functions and $\displaystyle\lim_{n\rightarrow \infty}B_{x,n}=B$ then $\displaystyle\lim_{n\rightarrow \infty}\det(B_{x,n})=\det(B)$ and $\displaystyle\lim_{n\rightarrow \infty}\operatorname{... | {
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"source": "stackexchange",
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Why is $X^*$ with the weak-* topology not a locally compact vector space? I know that locally compact Hausdorff topological vector space must be isomorphic to $\mathbb{C}^d$ or $\mathbb{R}^d$ for some $d\in \mathbb{N}$.
However the Banach Alaouglu theorem says that the closed unit ball of $X^*$ is compact with respect ... | The open unit ball in the strong topology is NOT open in the weak$^*$ topology. The weak$^*$ topology is way coarser than the strong one (less open sets).
To get an idea, consider $\ell^\infty (\mathbb N)$. A local basis of open neighbourhoods in the strong topology are the sets
$$
U_n=\left\{(a_k)_{k\in\mathbb N}: |a_... | {
"language": "en",
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"source": "stackexchange",
"question_score": "5",
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Computing a sum of fractions using sigma notation Evaluate the following using sigma notation
$$\frac34 + \frac65 + \frac96 + \frac{12}{7} + \frac{15}{8}$$
For the denominator I get $\sum_{i=4}^8 i$ but how about the numerator?
| What about
$$\sum_{i=4}^8 \frac{3(i-3)}{i}=3\sum_{i=4}^81-9\sum_{i=4}^8\frac1i$$
| {
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$f : X\longrightarrow Y$ isomorphism $ \Rightarrow (?)\, \, f^*:Div(Y)\longrightarrow Div(X)$ isomorphism. Let $X$ be a complex manifold. Denote by Div$(X)$ the Weil divisors group of $X$.
We have to:
Let $f : X \longrightarrow Y$ be a holomorphic map of connected
complex manifolds and suppose that $f$ is dominant, i.e... | The inverse map gives rise to a map on Div's in the opposite direction. Since $\text{Div}(\cdot)$ is functorial, and composition of $f$ with its inverse gives the identity map, this means that the map on Div's is inverse to the original one.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Proving sequence $(\frac{e^n}{n^{100}})_{n\in\mathbb{N}}$convergence I can see that since $e^n$ grows asymptotically faster than the polynomial $n^{100}$ as n approaches to $\infty$, $e^n>0$ and $n^{100}>0$, so $\lim_{n\to\infty} \frac{e^n}{n^{100}}=\infty$.
But I'm stuck on proving the convergence of the sequence $(\f... | Let $k$ be any fixed positive integer (e.g. 100, in your example).
$$\dfrac{e^n}{n^k}=\dfrac{\sum_{i=0}^\infty n^i/i!}{n^k} > \dfrac{n^{k+1}/(k+1)!}{n^k} = \dfrac{n}{(k+1)!}\to\infty $$
A single term of the series for $e^n$ dominates the power function $n^k$
| {
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"timestamp": "2023-03-29T00:00:00",
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How to find the second derivative of y in $y^2 = x^2 + 2x$? I have a problem to solve:
use implicit differentiation to find $\frac{dy}{dx}$ and then $\frac{d^2y}{dx^2}$. Write the solutions in terms of x and y only
It means that I need to differentiate the equation one time to find $y'$ and then once more to find $y'... | From $y'y'+yy''=1$ multiply by $y^2$.
Then $(yy')^2+y^3y''=(x+1)^2+y^3y''=y^2=x^2+2x \iff y^3y''=x^2+2x-x^2-2x-1=-1$
If we continue your calculation
$y''=\dfrac 1y-\dfrac{(x+1)^2}{y^3}=\dfrac{y^2-(x+1)^2}{y^3}=\dfrac{(x^2+2x)-(x^2+2x+1)}{y^3}=\dfrac{-1}{y^3}$
Gives the same result, so I guess the textbook result is er... | {
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Prove that $4\tan^{-1}\left(\frac{1}{5}\right) - \tan^{-1}\left(\frac{1}{239}\right)= \frac{\pi}{4}$
Prove that $4\tan^{-1} \left(\dfrac{1}{5}\right) - \tan^{-1}\left(\dfrac{1}{239}\right)=\dfrac{\pi}{4}.$
I was wondering if there was a shorter solution than the method below?
Below is my attempt using what I would ca... | As advised by Maximilian Janisch, you should use the $\tan x$ formula rather $\tan^{-1}x$:
$$\tan\left[4\tan^{-1} \left(\dfrac{1}{5}\right) - \tan^{-1}\left(\dfrac{1}{239}\right)\right]=\tan\left[\dfrac{\pi}{4}\right] \iff \\
\frac{\tan\left[4\tan^{-1} \left(\dfrac{1}{5}\right)\right]-\frac1{239}}{1+\tan\left[4\tan^{-1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3438621",
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"source": "stackexchange",
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Find :$ \lim_{n\rightarrow \infty} \int_{0}^{1} f(x^n) dx $?
Given $f$ is continuous on $[0,1]$ , Then $$ \lim_{n\rightarrow \infty} \int_{0}^{1} f(x^n) dx $$ is
*
*$ f(1)$
*$f(0)$
*$1$
*$0$
I thinks it will be $0$ because $ \lim_{n\rightarrow \infty} \int_{0}^{1} x^n dx = \lim_{n\rightar... | The limit is $f(0)$ by DCT. A dominating integrable function is the supremum of $|f|$.
| {
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"source": "stackexchange",
"question_score": "2",
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What is the maximum possible number of edges of a graph with n vertices and k components? Following is my attempt:
For maximum possible edges $->$ all k components must be connected sub-graphs
Maximum possible edges in a graph with n vertices = ${n \choose 2}$, I thought of removing k-1 edges, but only to realise that ... | I am assuming your question is the following:
What is the maximum number of edges in a graph with $n$ vertices and $k$ connected components?
This is equivalent to maximizing the function
$$
f(x_1,...,x_k) = \sum_{i} \binom{x_i}{2} =\frac{1}{2} \sum_{i} x_i^2-x_i
$$
subject to the constraints
$$
\sum_{i}x_i = n
$$
an... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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A Theorem from Jacobson: $U^G\otimes_F V\cong (U\otimes_F V_H)^G$
Theorem: Let $G$ be a finite group, $H\le G$, $F$ a field. Let $U$ be an $F[H]$-module and $V$ an $F[G]$-module; $U^G$ is induced $F[G]$-module. Then,
$$U^G\otimes_F V\cong (U\otimes_F V_H)^G \,\,\,\, \mbox{(isomorphic as $F[G]$-modules)}$$
This the... | Yes, your proof is correct. This characterisation of induction, namely that an $F[G]$-module $W$ having an $F[H]$-submodule $U$ such that $\dim W = [G:H]\dim U$ and $U$ generates $W$ as an $F[G]$-module is very powerful and gives many other elegant proofs.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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how many $4\times 4$ matrices with specific entries Suppose we construct $4\times 4$ matrix with the following requirements:
Entry $1,1$ is $a$. We have to insert three elements $b$, one element $c$ and one element $d$. Other entries are $0$. How many such matrices are there?
Since there are three $b$ elements, out of ... | You do not need to divide by $3!$, the binomial counts unordered groups. $${15 \choose 3}\cdot 12\cdot 11$$
| {
"language": "en",
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Random walk on Non-negative integers (Invariant Dist.) I wanted to confirm my answer to the following:
Given a random walk on the non-negative integers, beginning at 0 and provided transition probabilities of $\frac{1}{n+1}$ to the right and $\frac{n}{n+1}$ to the left at some arbitrary $\ n$ we have a probability tr... | This is the right approach and the answer (up to the 0 in the last component in your last equation, which seems to be a typo) is also correct. The general form of the distribution is $\mu_n = \frac{n+1}{n!} \mu_0$, which you can prove by induction.
To make this a probability distribution, choose $\mu_0 = \frac{1}{2e}$... | {
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Show that if $n$ and $m$ are both positive, then $nm$ is positive. This problem is on Terrence Tao's Analysis I, page 36.
Let $n,m$ be natural numbers. Then $n \times m=0$ if and only if atleast one of $n,m$ is equal to zero. In particular, if $n$ and $m$ are both positive, then $nm$ is also positive.
I know that mul... | Proof(Rough sketch).
Let $n,m$ be positive natural numbers. We induct on $n$ keeping $m$ fixed.
(I) Claim. $1 \times m$ is positive.
$1 \times m := m$ by the definition of multiplication and $m$ is positive.
(II) We inductively assume that $n \times m$ is positive.
(III) We would like to show that $(n+1)\times m$ is p... | {
"language": "en",
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Is $f(x)=\begin{cases}x \ \text{if } x\in [0,1),\\3-x \ \text {if} \ x \in [1,2]\end{cases}$ continuous from $[0,2]$ to $[0,2]$? Yes/No Consider the map
$f : [0,2] \rightarrow [0,2] $ defined by $ f(x) = \begin{cases} x \ \text{if
x} \in [0,1),\\3-x \ \text {if} \ x \in [1,2]\end{cases}$.
Is $f$ is continuou... | Notice that:
$$\lim_{x \to 1^-} f(x) = \lim_{x \to 1^-} x = 1,$$
and
$$\lim_{x \to 1^+} f(x) = \lim_{x \to 1^+} 3-x = 2.$$
Since
$$\lim_{x \to 1^-} f(x) \neq \lim_{x \to 1^+} f(x),$$
we can conclude that this function is not continuous $[0, 2]$.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Contour integral of rational function to some power I am interested in the following integral $$\int_{0}^{2\pi}d\theta \left(\frac{1-|x|^2}{|1-x e^{-i\theta}|^2}\right)^{y}$$
Does it has a nice closed form expression?
I know that $$\int_{0}^{2\pi}d\theta \left(\frac{1-|x|^2}{|1-x e^{-i\theta}|^2}\right)=2\pi$$
by conto... | Assuming $|x|=r<1$, the integral is $$\int_0^{2\pi}\left(\frac{1-r^2}{1-2r\cos\theta+r^2}\right)^y d\theta.$$ It reduces to the hypergeometric function. A contour integration approach is possible here too; for $0<\Re y<1$, we integrate $\big(z(1-rz)^y(1-r/z)^y\big)^{-1}$ along the circle $|z|=1$, and squeeze the contou... | {
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Series expansion for $\arctan(1-x)$ Series expansion for $\arctan(1-x)$
I try to expand this function into its Taylor series by means of differentiating it and then integrating it terms by terms but I fail to obtain the correct result.
The derivative of $\arctan(1-x)=-\dfrac{1}{x^2-2x+2}$. By using long division, I can... | When you integrate it, you get a $+C$. To get it's value, you can substitute in $x=0$ to get that $C=\arctan(1)$.
| {
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Prove that $( R \circ S ) \cap T = \varnothing$ iff $(\mathrm{R}^{-1} \circ T) \cap S= \varnothing$. I am having a bit of a hard time proving the following statement:
Show that
$( R \circ S ) \cap T = \varnothing$ iff $(\mathrm{R}^{-1} \circ T) \cap S= \varnothing$.
I sort of understand composition of functions, in... | $( R \circ S ) \cap T = \varnothing$ iff $(\mathrm{R}^{-1} \circ T) \cap S= \varnothing \iff ( R \circ S ) \cap T \ne \varnothing$ iff $(\mathrm{R}^{-1} \circ T) \cap S \ne \varnothing$
Let R, S, T be relations on the same set A
*
*$( R \circ S ) \cap T \ne \varnothing \to (\mathrm{R}^{-1} \circ T) \cap S \ne \var... | {
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Let $\Omega$ be a half-plane and $f_n$ a sequence of holomorphic functions on $\Omega$. I want to solve the second part of this 2-part problem. The first part, which I solved, states the following:
Let $\Omega$ be a bounded region and $f_n$ a sequence of holomorphic functions on $\Omega$ and continuous on $\overline\Om... | $e^{-n(1+e^{z})} \to 0$ uniformly on the real line but it does not converge in $H(\Omega)$ where $\Omega$ is the upper half plane. [Put $z=2+i\pi$].
| {
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How do I differentiate $\sin(x)^{\cos(x^3)}$? I'm taking a calc 1 class and I have come across a function that I'm having difficulty finding answers on the web.
$y=\sin(x)^{\cos(x^3)}$
I know there's some chain rule to apply, but what do I do with the cos(x)?
I am assuming this:
$y' = \cos(x^3).(\sin(x)')^{\cos(x^3) - ... | Use that
$$y=e^{\cos (x^3)\log(\sin x)}$$
and by chain rule we have
$$y=e^{f(x)}\implies y'=f'(x)e^{f(x)}$$
| {
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"timestamp": "2023-03-29T00:00:00",
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Smallest $m$ such that, for any $m$ unit squares in an $n\times n$ grid, the centers of some four of them are vertices of a parallelogram
Consider an $n\times n$ grid formed by $n^2$ unit squares. We define the center of a unit square as the intersection of its diagonals.
Find the smallest integer $m$ such that, choos... | A partial solution
First note that if we choose the $2n-1$ squares along two adjacent edges of the grid then no parallelograms are formed and so $m$ has to be at least $2n$.
Note also that there are precisely $(2n-1)^2$ different vectors between centres of squares. One of these is the zero vector and the other vectors ... | {
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If we have an equilateral triangle with a square inscribed in it, could we prove that the triangles we get are congruent?
(I forgot to add a last point. Let $X$ be the midpoint of $\overline{GF}$.)
Could we prove that triangles $CGX$, $CFX$, $GAS$, and $FBE$ are all congruent?
| Let $M$ be the midpoint of $\overline{AB}$. As $C,G$ and $A$ are collinear, $C,X$ and $M$ are collinear and $\overline{GX}$ is parallel to $\overline{AM}$, $\triangle CXG$ must be similar to $\triangle CMA$. Hence, $\frac{GX}{XC}=\frac{AM}{MC}$. Then, observe that $XC=MC-2GX$ and, by Pythagoras', $MC=\sqrt{(2AM)^2-AM^2... | {
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Contours and Closed Path
Sorry, but I thought I am very good in complex analysis, but then I saw this question which makes me question myself "Is closed path different from contours?"
Now I know the difference between them by the logic that contours have orientation.
This problem is from Chapter 9 of Complex Analys... | It seems that a contour is defined as "made up of a finite number of smooth paths which have non-zero continuous derivatives". See p.91.
Your path is continuous (note that $\lim_{t \to 0} \gamma(t) = \lim_{t \to 1} \gamma(t) = 0$). However it is not differentiable at $t = 0, 1$ since
$$\dfrac{t + it\sin(\pi/t)}{t-0} = ... | {
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How does Variational AutoEncoder (VAE) get mean and variance? Could somebody explain to me how VAE works in this tutorial (look only at cell which starts with class Sampling...)?
input=(batch_size=64, flatten_pixels=784) goes to Dense(64, 'relu') layer. The result goes through Dense(32) twice in parallel. One output of... | The $n=1$ case of this answer obtains the KL loss as $-\frac12(1+\ln\sigma^2-\sigma^2-\mu^2)$. This is your formula with z_mean $\mu$ and z_log_var $\ln\sigma^2$.
| {
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prove that $5^{2n+1} - 3^{2n+1} - 2^{2n+1}$ is divisible by 30 for all integers n ≥ 0. Prove that $5^{2n+1} - 3^{2n+1} - 2^{2n+1}$ is divisible by 30 for all integers n ≥ 0.
I have tried induction as follows.
Step 1:
Try n = 0, we get: $5 - 3 - 2 = 0$, which is divisible by 30.
Try n = 1, we get: $5^{3} - 3^{3} - 2^... | Let $a_n = 5^{2n+1} - 3^{2n+1} - 2^{2n+1} = 5 \cdot 25^n - 3 \cdot 9^n - 2 \cdot 4^n$. Then $a_{n+3} = 38 a_{n+2} - 361 a_{n+1} + 900 a_n$ (*). Therefore, you only need to check the claim for $n=0,1,2$, which is immediate.
(*) Because $(x-25)(x-9)(x-4) = x^3 - 38 x^2 + 361 x - 900$. The coefficients are not important h... | {
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} |
$\lambda^2$ is an eigenvalue of $A^2$. Prove that $\lambda$ or $-\lambda$ is an eigenvalue of the matrix $A$ Let $A$ be a $n\times n$ real matrix. Let $\lambda \in \mathbb{R}$ such that $\lambda^2$ is an eigenvalue of the matrix $A^2$. Prove that $\lambda$ or $-\lambda$ is an eigenvalue of the matrix $A$.
I know how t... | Let $x$ be the eigenvector associated to $\lambda^2$, consider $V=Vect(x,A,(x))$ it is stable by $A$. The matrix of $A$ in $\{x,A(x)\}$ is:
$\pmatrix{0&\lambda^2\cr 1&0}$ this implies that the characteristic polynomial of the restriction of $A$ to $V$ is $X^2-\lambda^2$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3441465",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Algebraic degree of $\cos\left(\frac{p\pi}{q}\right)$ How can we find the algebraic degree of $\cos\left(\frac{p\pi}{q}\right)$ for $p$, $q$ coprime integers?
I know that the algebraic degree of $e^{\frac{p\pi i}{q}}$ is $\phi(q)$, since cyclotomic polynomials are irreducible. I also know that $$\cos(x)=\frac{e^{ix}+e^... | Hint: Assume that $|q|\neq 1$. There is a quadratic polynomial $f(x)$ with coefficients in $\mathbb{K}:=\mathbb{Q}\Biggl(\cos\left(\frac{p\pi}{q}\right)\Biggr)$ such that $\exp\left(\frac{p\pi\text{i}}{q}\right)$ is a root of $f(x)$. Prove that $f(x)$ is irreducible over $\mathbb{K}$ by showing that $$\left[\mathbb{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3441626",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
The solution for $a+b+c+d = 4$ and $\left( \frac{1}{a^{12}} + ... + \frac{1}{d^{12}} \right) \left( 1 + 3abcd \right) = 16$ $a,b,c,d > 0$, then find the solution for
$$a+b+c+d = 4$$
and
$$ \left( \frac{1}{a^{12}} + \frac{1}{b^{12}} + \frac{1}{c^{12}} + \frac{1}{d^{12}} \right) \left( 1 + 3abcd \right) = 16$$
Attempt:... | Note that
\begin{align*}
\left(\frac{1}{a^{12}} + \frac{1}{b^{12}} + \frac{1}{c^{12}} + \frac{1}{d^{12}}\right)(1 + 3abcd)
&\geq \frac{4}{(abcd)^3}(1 + 3abcd) \\
&= \frac{4}{(abcd)^3} + \frac{12}{(abcd)^2} \\
&\geq 4 + 12 \\
&= 16
\end{align*}
where the first inequality holds by AM-GM, and the last inequality holds si... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3441805",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Boolean algebra: simplify (A+B)$'C'$(C+D) I need to solve / simplify the output of a logic circuit. The output is $(A+B)'C'(C+D)$
I wrote the truth table and it returns 0 in all outputs. I tried to solve it using laws of boolean algebra:
$(A+B)' = A'B'$
And $C′(C+D) = (C′C + C′D)$
$C′C = 0$ [Boolean law $A.A′ = 0$]
Doe... | $C'C=0$ will not imply $C'C + C'D=0$, and $C'C + C'D=1$ when $C=0,D=1.$
Here is how to simplify the expression:
\begin{align}
&(A+B)'C'(C+D)\\
&\equiv(A'B')C'(C+D)\tag*{de Morgan’s Theorem}\\
&\equiv(A'B')(C'C+C'D)\tag*{Distributive law}\\
&\equiv(A'B')(0+C'D)\tag*{Complement}\\
&\equiv(A'B')(C'D)\tag*{Identity}\\
\en... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3441994",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
Why does the boundary of mobius strip wrap twice around core circle but not any other line? A space $X$ deformation retracts onto a subspace $A$ if there exists continuous map $F:X\times [0,1]\rightarrow X$ such that $F(x,0)=x,F(x,1)\in A,F(a,t)=a$ $\forall a\in A$.
The mobius strip deformation retracts onto its core c... | The Möbius strip $M$ can be obtained by gluing two opposite edges of a rectangle by an orientation reversing homeomorpism. Explictly, we may define
$$M = [0,1 ] \times [-1,1]/\sim$$
where $(0,t) \sim (1,-t)$. Let $p : [0,1 ] \times [-1,1] \to M$ denote the quotient map.
Then you get various types of embedded circles.
T... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3442099",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Nimbers for Kayles I have a problem with calculating the nimbers for this game.
I know that for Kayles the following is true:
\begin{array}{c|c}
\hline
Hight & Nimber \\ \hline
12 & ^*4 \\ \hline
11 & ^*6 \\ \hline
10 & ^*2 \\ \hline
9 & ^*4 \\ \hline
8 & ^*1 \\ \hline
7 & ^*2 \\ \hline
6 & ^*3 \\ \hline
5 & ... | Community wiki answer so the question can be marked as answered:
A third option for the game of length $3$, in addition to leaving a game of length $1$ or a game of length $2$, is to leave two games of length $1$, with value $1\oplus1=0$, so $n(3)=\mathrm{mex}\{0,1,2\}=3$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3442249",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Verification of a basic Riemann summation problem: $f(x) = 1+x$ where $x\in [-1, 2]$ This is the very first time I solve a problem involving Riemann's summation so I would like to verify whether I get it correctly. Problem statement (hopefully, I have translated the problem statement in a comprehensible way.):
Find an... | Your work looks correct. In your particular example, $\sigma_n$ is independent of $n$, but if you had taken points on the left (or on the right) side of each partition interval, instead of in the middle, or if you had a non-linear function such as $f(x)=1+x^2$, then only in the limit as $n\to\infty$ would the Riemann ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3442422",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Derivative of a generic polynomial function Let $P:M_n(\mathbb{C})\to\mathbb{C}$ be a polynomial function, $A=(A_{ij}),E_{ji}\in M_n(\mathbb{C})$,where $E_{ji}$ is defined as matrix filled with zeros except $1$ in j-th row and i-th column and $t\in \mathbb{R}$.
I want to take derivative of $P((I+tE_{ij})A)$ with respe... | Hint
You have
$$A= \sum\limits_{k,l}A_{kl}E_{kl}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3442703",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Prove that $\mathbb{Z}$ is a UFD while $\mathbb{Z}[\sqrt{-5}]$ is not. An integral domain $R$ is called a unique factorization domain (UFD) if every nonzero, nonunit element of $R$ can be uniquely written as a product of irreducible elements, up to reordering the factorization and taking associates of the irreducible f... | For 1: the definition says "can be uniquely written", so you essentially have to prove the Fundamental Theorem of Artithmetic (not just the "uniqueness part).
For 2: are really 1,-1 and 5 irreducible?
Instead, note that $2\cdot 3=6=(1+\sqrt{-5})\cdot(1-\sqrt{-5})$
PS: Remember that irreducible elements are not units ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3442987",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
How is the subspace identified? In Linear Algebra class, I was learning about vector spaces and subspaces but can't quite grasp the concepts. We were given the following:
*
*$w = \{(a,b,c) \in R^3 : a = 1\}$
*$w = \{(a,b,c) \in R^3 : a = 0\}$
We were told that (2) is a subspace, while (1) is not because (1) is "not... | Set 1 is "not closed under addition" because vectors in this set are of the form $\left [ \begin{array}{c}
1 \\
y\\
z\\
\end{array} \right ]$. If I add any two of these together I get the result
$$
\left [ \begin{array}{c}
1 \\
a\\
b\\
\end{array} \right ] + \left [ \begin{array}{c}
1 \\
c\\
d\\
\end{array} \right ]\;... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3443123",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
How is a relation defined? I've read multiple ways of defining relations and was wondering what is generally accepted. One of these ways is as follows.
$xRy=C\iff C\subseteq A\times B$
and the cartesian product is defined as
$A\times B=D\iff (\forall x)(\forall y)((x,y)\in D\iff x\in A \land y\in B)$
Is this definition... | In set theory, relations are sets; thus "to be a relation" is a property of sets.
We define the pair $(x,y)$, for example with Kuratowski's definition, and then we define the cartesian product of two sets $A$ and $B$ :
$A \times B = \{ (x,y) \mid x \in A \text { and } y \in B \}$.
Finally, we define when a set is a ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3443197",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
Find the value of $x$ in this equation. $2^x -x=5$
I can't solve this equation, but I can see that $x=3$ (through trial and error) but I don't know how to attain the value $3$.
| $$2^x - x=5 \implies 2^x = 5+x$$
It is trivial to see that for $x>3$ ,
$$2^x > 5+x$$
Hence we are only left with $3$ possibilities for $x\in \mathbb {N}$ , which yields $3$ as a solution to the equation.
For $x \in \mathbb {Z}$ , we observe that for $x<-5$,
$$2^x > 5+x$$
Hence the only useful range for the solution i... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3443344",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 3
} |
Edge colouring number of a k-regular graph and relation to bridges I am researching graph theory, and am confused about this problem. I am considering a $k$-regular connected graph $G$, where $k\geq2$ and where $\chi'(G)=k$ (the edge colouring number.) I then want to be able to show that $G$ does not contain any bridge... | Note with an edge colouring $k$, you can find $k$ distinct perfect matchings, one for each colour class.
*
*Suppose a bridge $uv$ exist, then the removal of this bridge gives two distinct components, say $C_1$ and $C_2$.
*Next, we note that the union of two disjoint perfect matchings form a disjoint union of cycle ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3443474",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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$T_{a}(\text{ball}(\ell^p))$ is compact, where $T_a\colon\ell^p\to\ell^p$, $(T_a x)_{n}:=a_{n}x_{n}$ and $\mathbb{C}\ni a_n\to0$ as $n\to\infty$ Let $a\in\ell^{\infty}=\ell_{\mathbb{C}}^{\infty}$ and define $T_a\colon\ell^p\to\ell^p$, $(T_a x)_{n}:=a_{n}x_{n}$ for $1\leq p<\infty$. Suppose that $a_n\to0$. How do I prov... | We can use the following criterion of compacity in a Banach space.
A subset $A$ is precompact (namely $\bar A$ is compact) if (and only if) it is bounded, and for every $\epsilon$ there is a finite dimensional vector subspace $V$ such that for every $y\in A$ $d(y,V)\leq \epsilon$
Let $ V_n$ the subspace of sequences s... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3443642",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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spectrum of elements in $C^*$ algebra Suppose $x,y$ are two invertible positive elements in a $C^*$ algebra $A$,if $\|x\|=\|y\|$,can we compute the spectrum $\sigma(x^{-1}y)$ of $x^{-1}y$?Does there exist a relationship between the spectrum of the multiplication of two elements and the norm of elements?
| You can't expect a relation. For instance consider
$$
x=\begin{bmatrix} 1&0\\0&\tfrac1n\end{bmatrix} ,\ \ \ y=\begin{bmatrix} 1&0\\0&1\end{bmatrix} .
$$
Then $\|x\|=\|y\|=1$, and $\|x^{-1}y\|=n$. The norm only sees the maximum of the spectrum, but nothing else.
For a more dramatic example consider the block matrices ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3443865",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Show that a function is negative over its domain I would like to demonstrate that the following function is negative
\begin{equation}
f(x)=-\frac{t}{4\sqrt{x}^3}\bigg[1-\bigg(1+\sqrt{x}\bigg)^{\frac{1}{t-1}}\bigg]+\bigg(\frac{1}{1-t}\bigg)\bigg(\frac{t}{2\sqrt{x}}\bigg)^2\bigg(1+\sqrt{x}\bigg)^{\frac{2-t}{t-1}}
\end{eq... | You can show fairly simply that the first term will be negative and the second term positive with positive x and 1 > t > 0.
Because they are added, showing that the overall function is negative becomes a question of showing that the first (negative) term dominates, which you can do by setting up and solving an inequal... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3444158",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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Problematic inequality & hint I would like to ask for hint for proving following inequality:
$$x^3(1+x)+y^3(1+y)+z^3(1+z)\geq \frac{3}{4}(1+x)(1+y)(1+z)$$
for all $x>0$, $y>0$, $z>0$ such that $xyz=1$.
Generally, I tried to find some elementary solution, but even with some calculus I didn't solve it.
Edit. My attempt:
... | Since $x^3$ is convex for $x\ge 0$, we have that $x^3 \ge 1 + 3(x-1)$, using the tangent at $x=1$.(*) With this observation, it is enough to prove that
$$
\sum_{cyc} (3x-2)(1+x)\geq \frac{3}{4}(1+x)(1+y)(1+z)
$$
Expanding the terms, and using $x y z = 1$, gives the equivalent
$$
-30 + \sum_{cyc} x + 12 \sum_{cyc} x^2... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3444270",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
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Investigating the geometric patterns of $x^k+y^k=r^k$ for $k \in N$ I was investigating the graphs of equations of the form $x^k+y^k=r^k$.
I am not sure how to ask this so I will try to simplify the problem first.
For simplicity sake lets let $r=2$, now
For $k=1$, I get a line.
$x+y=2$.
For $k=2$, I get a circle wi... | For even $k$, these are Lamé Curves also known as hyperellipses.
For odd $k$, you have part of a Lamé Curve in quadrant 1 and the curve asymptotically approaches the line $y = -x$ from above in quadrants two and four. I do not know of a name for the full curves for odd $k$. (They are examples of superelliptic curves,... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3444348",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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The mean of $f$ minimizes $||f-\xi||_p$ over all $\xi\in\Bbb R$. [Reference request] I would like to know where I can find a reference for the following fact:
Let $\Omega$ be a set of finite measure, $1<p<\infty$ and $f\in L^p(\Omega,\mu)$. Denote $\bar f:=\frac 1{|\Omega|} \int_\Omega f d\mu$. Then
$$
\bar f = \arg... | This result only holds for $p = 2$. Indeed, if we denote
$$J(\xi) = \int_\Omega |f - \xi|^p \, \mathrm d\mu,$$
we get (at least formally)
$$J'(\xi) = \int_\Omega p \, |f - \xi|^{p-2} \, (f - \xi) \, \mathrm d\mu.$$
For $p = 2$, this is $0$ iff $\xi$ is the mean of $f$. But this is not true for $p \ne 2$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3444478",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
is log 0 is or is not undefined? per Figure 1.7 in pattern recognition and machine learning (free)
Plots of M = 9 polynomials fitted to the data set shown in Figure 1.2 using the regularized error
function (1.4) for two values of the regularization parameter λ corresponding to ln λ = −18 and ln λ = 0. The
case of ... |
The logarithmic function $~\log_b(x)~$ (for any positive real number $b\ne 1$) is defined only for $~x>0~$.
If possible let $~\log_b(0)=k~$, where $~k~$ is any real number. Then by the definition of logarithm, $~b^k=0~$.
Now, any ‘real’ quantity (may it be positive or negative) raised to another real quantity can n... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3444601",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
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Dividing a polygon into 6 equal regions You are given a convex polygon, ie all its internal angles are less than 180 degrees. Prove that you can always draw three straight lines through a specific point inside this polygon, such that they divide it into 6 equal (by area) regions? Bonus questions:
*
*Can you prove th... | Let's have a polygon. Let's draw a line at some angle $\theta_1$ to some fixed direction. By moving the line parallel to itself, we can make the whole polygon to lie either on one side of the line or the other. Thus, since the function “area on one side minus area on the another” is continuous, it should go through zer... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3444763",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 1,
"answer_id": 0
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Proving that an improper Riemann integral involving $f(x)$ exists given that $f(x)$ is Riemann integrable and periodic with period $1$
Given that $f(x)$ is Riemann integrable and periodic with period $1$
and $\int_{0}^{1}f\left(x\right)dx=0$, prove that
$$\int_{1}^{\infty}\frac{f\left(x\right)}{x^{s}}dx$$ exists ... | Hint:Apply Dirichlet's integral test.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3444962",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
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Dimension of affine variety mod $p$ can only increase Let $X$ be an affine variety defined by polynomials over $\mathbb{Z}$.
Reducing the polynomials modulo $p$, we obtain a variety $X_p$ defined over the finite field $\mathbb{F}_p$.
Is it true that the dimension of $X_p$ is larger or equal to the dimension of $X$?
N... | Let me state the following result (Lemma 05F7 in the Stacks Project):
Let $f:X\rightarrow S$ be a morphism of finite type where $S$ is an
irreducible scheme (with generic point $\eta$). If $n=\mathrm{dim}X_\eta$
then there exists a nonempty open $U\subseteq S$ such that for
all $s\in U$ one has $\mathrm{dim} X_s=n$.
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3445066",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
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Integrating function $h:=\lambda(A \cap (B-x))$ for $\lambda(A),\lambda(B)<\infty$ Given two sets $A,B$ with finite measure let $h:=\lambda(A \cap (B-x))$. Show this function is integrable and calculate its integral.
I thought about using the following identity, $\lambda(A)+\lambda(B)=\lambda(A \cup B)+\lambda(A \cap B... | $\lambda (A\cap (B-x)=\int 1_{A}(y)1_B(x+y)dy$
Use Tonelli's theorem to show that $\int_{\Bbb{R}} \int_{\Bbb{R}}F(x,y)dydx<\infty$ where $F(x,y)=1_{A}(y)1_B(x+y)$ is non-negative.
The integral is equal to $m(A)m(B)$ by translation invarianve of the Lebesgue measure.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3445256",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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Riemann sum of $\int_1^2 {1\over x^2} dx$. I've spent quite a time solving the following problem:
Evaluate using Riemann's sum:
$$
I = \int_1^2{1\over x^2} dx
$$
I was first trying the following approach, which didn't work since the summation seems undoable to me:
$$
\Delta x = {1\over n}\\
I = \lim_{n\to\infty}\su... | I've just tried one more technique while solving a similar problem and it seems to work fine. Let's split the interval $[1, 2]$ with points $x_0, x_1, \dots, x_n$ so that they form a geometric progression. Let $q$ denote the denominator of geometric progression. So the interval is split by the points: $q, q^2, \dots, q... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3445371",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
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Proof of closed-form solution of the difference of two factorial series Context
I'm working on a problem tangentially related to the Kepler Equation 1. The details are very much in the weeds, and I'm not in a position to explain at this time exactly how I have arrived at Equation 1. Yet, I believe that the following ho... | Starting from @marty cohen's answer and fimplifying, we have
$$f_1(k)=\sqrt{\pi }\,\frac{ \Gamma (k+1)}{\Gamma \left(k+\frac{1}{2}\right)}-1$$
$$f_2(k)=\frac{\Gamma \left(k+\frac{3}{2}\right)}{\sqrt{\pi }\, \Gamma (k+1)}-\frac{1}{2}$$
$$f_1(k)-\pi f_2(k)=\frac \pi 2-1+\sqrt \pi\left(\frac{\Gamma (k+1)}{\Gamma \left(k+\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3445441",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
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Prove that $\sum_{i=1}^{n}\frac{1}{\left(n+i\right)^{2}}\sim\frac1{2n}$ I would like a proof of the asymptotic relationship $$\sum_{i=1}^{n}\frac{1}{\left(n+i\right)^{2}}\sim\frac1{2n}$$
without assuming that the sum is a Riemann sum.
This problem arose from Question 1909556, which asks about the Riemann sum of $\int_1... | A more elementary approach.
$$\frac{1}{2n}=\sum_{i=1}^{n}\frac{1}{(n+i)(n+i-1)}$$ because the sum telescopes to $\frac{1}{n}-\frac{1}{2n}.$
So:
$$\begin{align}\frac{1}{2n}-\sum_{i=1}^{n}\frac{1}{\left(n+i\right)^{2}}&=\sum_{i=1}^{n}\left(\frac{1}{(n+i)(n+i-1)}-\frac{1}{(n+i)^2}\right)\\
&=\sum_{i=1}^{n}\frac{1}{(n+i)^2... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3445583",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 4,
"answer_id": 0
} |
how to calculate number of states for this logistics problem? Consider a logistics problem with 3 cities, 5 trucks and 3 packages. Each truck can be at any of the locations. A package can either be at one of the locations or in one of the trucks. What would the number of states be if an atomic representation is used?
w... | For Atomic Presentation, the total number of states would at least be $n^{p+t}$, where $n$ is the number of cities, $p$ is the number of packages, and $t$ is the number of trucks. In this case, the least states are: $3^{5+3} = 6561$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3445650",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
What is the significance of Topological Entropy? I'm taking a course on Dynamical systems and we learnt about topological entropy. I get that topological entropy is a quantity which measures how quickly distinct points separate asymptotically, and that it is a topological invariant (which makes it good). But how does k... | Perhaps the primary value is that it is related to other invariants outside of just topological dynamics, as expressed in the so-called Variational Principle, which says:
If $T : X \to X$ is a continuous self-map of a compact Hausdorff space $X$ then its topological entropy $h(T)$ is the equal to the supremum of the m... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3445780",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 0
} |
Finding the angles and area of unusual shapes My friend sent me the following geometry problems. I think I have the first one, but I think the 2nd and 3rd are unsolvable, although I could be missing something.
My attempt:
*
*I'm pretty sure this one is 1080. I'm having a hard time writing out my explanation here, b... | 2) Note $ \angle ABD = 90 -25 =65$, which yields $x=65 -25 =40$ due to isosceles triangle.
3) The shaded area is equal to the area of the quadrilateral minus the areas of the two small right triangles. Thus,
$$A= \frac 12\cdot 4(x+4) + \frac12 \cdot 6(2+y) -\frac12 \cdot 4x -\frac12\cdot 6y=14$$
where $x$ and $y$ are t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3445927",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
} |
Difference between domain and co-domain in sets? Let's say I have a subset of the Cartesian plane, for example:
$\{(x, y) \in R \times R: 2x+3 > 5\}$.
If I am asked to find the co-domain of the following set, how would I do so? I know how to find the domain, which is done by finding all possible $(x,y)$ ordered pairs a... | Normally, domain and co-domain are defined on functions, but we'll use your textbook's definition for domain and co-domain on relations.
In your relation, note that your set describes only the $x$-values, $2x+3>5$, i.e. $x>1$. So the $x$-values that satisfy this condition is your domain, as defined in your textbook. Ho... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3446005",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Find the largest divisor of 6006006006 that does not exceed 60,000 factoring it, we have
$6 * 1001*1000001 = 2*3*7*11*13* (100^3+1) = 2*3*7*11*13*101*9901$
with this prime factorization, how do you check the largest divisor without too much guessing and checking?
| One way is to look for targets to get close to and just use your own numerical nous. Here, the obvious targets are the given $60000$, and $6006006006/60000 \approx 100100$.
$2\cdot3\cdot9901=59406$.
For this instance, start with $60000=10000\cdot6$ and notice that $9901$ is a little less than $10000$, then look for $6$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3446116",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
$x^4 + 1020x^3 - 4x^2 + 2039x+1$ is divisible by 1019. Give all integers x in the range [1,2018] such that $x^4 + 1020x^3 - 4x^2 + 2039x+1$
is divisible by 1019.
Applying mod 1019 to the coefficients, we have
$x^4 + x^3 - 4x^2 + x + 1$ which is equal to $(x-1)^2(x^2+3x+1)$
I tried making 1 of the factors equal to a mul... | As $1019$ is prime, one of the factors must be $\equiv 0\pmod{1019}$. For $(x-1)$ this obviously means that $x=1$ or $x=1020$.
For the factor $x^2+3x+1$, we'd expect two solutions $x=\frac{-3\pm\sqrt{5}}2$ - but what is $\sqrt 5$ in modular arithmetic? Any $y$ with $y^2\equiv 5\pmod{1019}$. Fortunately, just playing wi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3446386",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Can anyone explain to me why $(34)(123) = (124)$? $$\begin{align}
(34)H&= \{(34)(1),(34)(123),(34)(132)\}\\
& = \{(34),(124),(1432)\}.
\end{align} $$
Can anyone explain to me why $(34)(123) = (124)$?
I don't understand coset multiplying can you help me with this
| If $\sigma=(123)$, $\tau=(34)$, and we compose right-to-left (as is consistent with your other compositions), then
$$\begin{align}
1 &\stackrel{\sigma}{\mapsto} 2 \stackrel{\tau}{\mapsto}2, \\
2&\stackrel{\sigma}{\mapsto} 3 \stackrel{\tau}{\mapsto} 4, \\
4 &\stackrel{\sigma}{\mapsto} 4 \stackrel{\tau}{\mapsto} 3, \\
3 ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3446529",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Prove that exist $x\in \left\{ 1,...,14 \right\}$ such that $\sigma(x)=x$, where $\sigma\in S_{14}$ and $|\sigma|=28$?
Let $\sigma\in S_{14}$ which is an even permutation of the order of $28$. Prove that exist $x\in \left\{ 1,...,14 \right\}$ such that $\sigma(x)=x$.
My try:
We know that the permutation order is equa... | You already have the prime factorisation of $28$. To get an element of order $28$, you need to partition $14$ into divisors of $28$ (namely, $1$, $2$, $4$, $7$, and $14$) so that their LCM is $28$.${}^\dagger$ So, what are the partitions of $14$ into those divisors, potentially including $1$, $4$, and $14$, such that t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3446663",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
} |
Prove that $\sqrt{2a^2+b+1}+\sqrt{2 b^2+a+1}\geq 4$ when $a+b=2$. Suppose that $a,b$ are non-negative numbers such that $a+b=2$. I want to prove $$\sqrt{2 a^2+b+1}+\sqrt{2 b^2+a+1}\geq 4.$$
My attempt: I tried proving $$2a^2+b+1\geq 4$$ but this seems wrong (try $a=0,b=1$).
How can I prove the above result?
| By Minkowski (triangle inequality) we obtain:
$$\sum_{cyc}\sqrt{2a^2+b+1}=\sum_{cyc}\sqrt{2a^2+\frac{b(a+b)}{2}+\frac{(a+b)^2}{4}}=$$
$$=\frac{1}{2}\sum_{cyc}\sqrt{9a^2+4ab+3b^2}=\frac{1}{2}\sum_{cyc}\sqrt{7a^2+b^2+8}\geq$$
$$\geq\frac{1}{2}\sqrt{7(a+b)^2+(b+a)^2+8(1+1)^2}=4.$$
We see that our inequality is true for al... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3446873",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 6,
"answer_id": 1
} |
Finding the amount of possibilities to split two parties in pairs I have two parties. Each party contains $3$ people. I'm trying to figure out how many possibilities there are to divide those two parties into pairs so each pair contains a person from party A and a person from party B.
I think the answer is $6$. But I'... | You are correct that there are six ways to match people from party $A$ with people from party $B$ in pairs.
Line up the people in party $A$ in some order, say alphabetically. There are three ways to match a person from party $B$ with the first person in line, two ways to match one of the remaining people from party $B... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3447118",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
If $D\in [AB$ s.t. $AD=BC$ and $\angle{ADC}=\frac{3}{4}\cdot \angle{ABC}$ find $\angle {A}$. Let $\triangle {ABC}$ s.t. $AB=AC$ and $\angle{A}>90$. If $D\in [AB$ s.t. $AD=BC$ and $\angle{ADC}=\frac{3}{4}\cdot \angle{ABC}$ find $\angle {A}$.
My idea: I denote $\angle B=4x$, then I apply "Sine theorem" in $\triangle {AB... |
Let $E$ on $BC$ such that $CE\cong AC\cong AB$, and denote $\angle ABC = \alpha$.
*
*$\triangle BDE$ is isosceles, therefore $\angle BDE =\angle BED= \frac{\alpha}2$.
*Since $\angle ADC = \frac34\alpha$, we have $\angle EDC = \frac14 \alpha$.
*Note that $\angle DEC = 180^\circ -\frac12\alpha$, so that $\triangle ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3447224",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Determine if the fuction satisfy a local or a uniform Lipschitz condition The question: Determine if $f(t,y) = \frac{t^2y}{1+y^2}$ satisfies a local or a uniform Lipschitz condition.
My thoughts: Well from my work, I have
$$ |f(t,x) - f(t,y)| \leq K |x-y|. $$
I can determine what $K$ is by the mean value theorem. For... | $Df(t,y)=\begin{pmatrix}
\frac{2ty}{1+y^{2}} &\frac{1-y^2}{(y^2+1)^2}
\end{pmatrix}$ so $\|Df(x,y)\|\le \sqrt{4t^2+1}.$
Let $(t,y),(t_1,y_1),(t_2,y_2)\in \overline J\times \overline U$, where $J,U$ is some bounded open sets in $\mathbb R$
Then, the MVT (in two variables) and compactness show that
$|f(t_1,y_1)-f(t_2,... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3447354",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Show that function $E$ is non-multiplicative. Definition 1:
A function $f$ is said to be non-multiplicative if
$$f(ab)\ne f(a)f(b)$$
for all coprime integers $a,b>1$.
Definition 2:
We define the function $E$ as
$$E(n)= n+1-\tau (n)- \phi(n)$$
$$=\sum_{(n,d)\notin\{1,d\} \\ \ \ \ \ 1<d<n}1.$$
Here, $(n,d)$ denotes gcd$(... | First recall that $n = \sum_{d | n} \phi(d)$, so using this we have
$$E(n) = \sum_{d | n} \phi(d) - \sum_{d | n} 1 - (\phi(n) - 1) = \sum_{d | n,\, d < n} (\phi(d) - 1)$$
Now note that (i) for coprime $a, b$, and $d | ab$, we have a unique decomposition $d = d_1 d_2$ for $d_1 | a$ and $d_2 | b$, and that (ii) for copr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3447525",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 1,
"answer_id": 0
} |
Are trigonometry functions Ratios or Distance? Consider a point, say $S(2,3)$. Now here $3$ indicate that $S$ is $3$ units away from x axis. Right?
Now consider what Wikipedia says:
The trigonometric functions cos and sin are defined, respectively, as
the x- and y-coordinate values of point A.
This definition of ... | This is a good question. I think it best always to regard the values of $\sin$, $\cos$, and so on as ratios. What allows these values seemingly to be defined as distances in your quotation from Wikipedia is that that definition refers to the unit circle—a circle whose radius is $1$. A similar definition that works f... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3447626",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 7,
"answer_id": 2
} |
Is my understanding of cardinality of sets of strings correct? Is my understanding of cardinality of sets of strings correct?
1) A finite set of symbols (containing $x$ symbols) and strings of finite length $n$ gives us $x^n$ finitely many strings.
2) A finite set of symbols (containing $x$ symbols) and strings of coun... | Apart from the issue about $x$ pointed out in the comments, your assertions are correct.
For the last one, if $X$ is a finite set of cardinality $x$, let $X^n$ denote the set of words of length $n$ on the alphabet $X$ (a finite set of cardinality $x^n$, as you observed). Note that, for $n = 0$, $X^0$ is the singleton ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3447817",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Line $11x+3y-48 = 0$ tangent a graph $f(x) = \frac{4x + 3}{3x - 6}$ at $(a,b)$ Line $11x+3y-48 = 0$ tangent a graph $f(x) = \frac{4x + 3}{3x - 6}$ at (a,b) when $a<b$
$a-b = ...$
Find gradient of the line, df(x) / dx
$\frac{(4x+3)3 - (3x-6)(4)} {(3x-6)^2}$
Which the same as $ -11/3$
$(12x+9 - 12x + 24 ) 3= -11(3x-6)^2$... | Rewrite
$$f(x)=\frac43+\frac{11}{3x-6}.$$
Now
$$f'(x)=-\frac{11}{(3x-6)^2}\cdot3$$
and $f'(x)=-11/3$ gives $(3x-6)=\pm1$, that is $x=7/3$ and $x=5/3$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3447965",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
In how many ways can four men and four women be seated at a round table if no two men are to be in adjacent seats? In how many ways can four men and four women be seated at a round table if no two men are to be in adjacent seats?
Please use principle of inclusion exclusion to solve.
My approach:
Let $S_i$ represent the... | The error you made was counting the number of consecutive men rather than the number of pairs of adjacent men. Let $|A_i|$ be the number of pairs of adjacent men.
Let Angela be one of the women. Seat Angela. We will use her as our reference point. Relative to her, we can arrange the other seven people in $7!$ ways ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3448150",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Characterizing convex hull by set of affinely independent vectors While the statement seems intuitively plausible, I am currently struggling to see a proof that any convex hull "spanned" by a set of $n+1$ affinely independent $n$-vectors, $\{x_1,x_2,...,x_{n+1}\}$, is uniquely characterized by this set of vectors.
That... | Let use now use the barycentric coordinates associated with the simplex $\Delta$. For simplicity, let us write $s_j,t_j$ instead of $x_j-O,z_j-O$ for the vectors. Note that, thanks to the affine linear independence of the two sets, these new sets of vectors are linearly independent.
Since the two set have the same conv... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3448432",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Prove that $x^2+px+p^2$ is a factor $(x+p)^n-x^n-p^n$, if $n$ be odd and not divisible by $3$. Question:
Prove that $x^2+px+p^2$ is a factor of $(x+p)^n-x^n-p^n$, if $n$ is odd and is not divisible by $3$.
My approach:
$$(x+p)^n-x^n-p^n=\sum_{r=0}^n\limits {n\choose r} x^{n-r}p^r-x^n-p^n$$ What can I do after that?
| You need to show every root of $x^2 + px + p^2$ is a root of $(x + p)^n - x^n - p^n$. The quadratic formula gives that $x = p\omega$ or $p\bar{\omega}$
where $w = -{1 \over 2} + {\sqrt{3} \over 2}i$ is a complex third root of unity. So what you need to show is that for the values of $n$ in question
that
$$(p\omega + p... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3448544",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 0
} |
Combinatorics and Expected Value There are 15 candidates running for a given Senate seat, comprised of 10 men and 5 women. There are 32 polls, in which any of the candidates are equally likely to be ranked first, independently of the other polls (meaning the position of each candidate on any of the polls is purely rand... | Looking at this problem, we can identify a discrete random variable.
Let $X$ be the number of polls where a woman ranks first. $X$ is a binomial variable with the distribution $X$~$Bin(32,\frac{1}{3})$, since the chance of a woman being ranked first is $\frac{5}{15}=\frac{1}{3}$.
Then the expected value of a binomial ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3448628",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
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