Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Why are the Cauchy-Riemann equations in polar form 'obvious'? In my book on complex analysis I'm asked to prove the Cauchy-Riemann equations in polar form, which I did. However, at the end of the question the author asks why these relations are 'almost obvious'.
Now I get the derivation using chain rules and also the ... | A function $u+iv$ is complex differentiable if the derivative of $u$ in any direction $h$ is equal to the derivative of $v$ in direction obtained by rotating $h$ by $90$ degrees counterclockwise. This should make sense if you think of imaginary axis as real axis rotated by $90$ degrees counterclockwise.
The Cauchy-Rie... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1268368",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
If $Ha\subseteq Kb$ for some $a,b\in G$, show that $H \subseteq K$.
Let $H$ and $K$ be subgroups of a group $G$. If $Ha\subseteq Kb$ for some $a,b\in G$, show that $H \subseteq K$.
I constructed a proof by contradiction and I am wondering whether or not it is free from flaws. I thank you in advance for taking your ti... | Put $g=ba^{-1}$, then $H \subseteq Kg$. Since $1 \in H$, we have $1=kg$ for some $k \in K$. Hence $g=k^{-1}$ and $Kg=Kk^{-1}=$ (since $K$ is a subgroup )$K$, so $H \subseteq K$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1268588",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
Proving the existence of a supremum $A=\{(1+(1/n))^n \mid n\text{ is taken from positive integers}\}$
How can I prove that the set above has a supremum? I've started with an assumption that
$(1+(1/n))^n < 3$ for every positive integer n
but I could not find a way to prove it too. Could you please give me a hint?
| Outline: Using the Binomial Theorem, we find that
$$\left(1+\frac{1}{n}\right)^n \le 1+\frac{1}{1!}+\frac{1}{2!}+\cdots+\frac{1}{n!}.$$
The term $\frac{1}{3!}$ is less than $\frac{1}{2^2}$. Continue.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1268692",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Prove there exists $m$ and $k$ such that $ n = mk^2$ where $m$ is not a multiple of the square of any prime. For any positive integer $n$, prove that there exists integers $m$ and $k$ such that:
$$n = mk^2 $$
where $m$ is not a multiple of the square of any prime. (For all primes $p$, $p^2$ does not divide $m$)
I didn'... | The prime decomposition of $n = \prod p_i^{a_i}$. Set $m= \prod_{odd\: a_i} p_i$; then $\frac{n}{m}$ has only even-exponent prime factors (if any) and can be represented as $k^2$ (which may possibly be equal to $1$).
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1268774",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
Is every diagonal matrix the product of 3 matrices, $P^{-1}AP$, and why? In trying to figure out which matrices are diagonalizable, why does my textbook pursue the topic of similar matrices?
It says that "an $n \times n$ matrix A is diagonalizable when $A$ is similar to a diagonal matrix. That is, $A$ is diagonalizable... | Matrices are connected to linear maps of vector spaces, and theres a concept of a basis for vector spaces. (A basis is something so that every element is a can be written uniquely as a sum of the elements in the basis.)
Now, bases aren't unique, so if you want to see what your matrix looks like under a different basis... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1268931",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Integration in complex measure Let $v$ be a complex measure in $(X,M)$. Then $L^{1}(v)=L^{1}(|v|)$.
I have made:
$L^1(v)\subset L^1(|v|)$?.
Let $g\in L^1(v)$
As $v<<|v|$ and $|v|$ is finite measure, then for chain rule, $g.(\frac{dv}{d|v|})\in L^1(|v|)$.
As $|\frac{dv}{d|v|}|=1\;|v|-a.e$, then $|g|=|g||\frac{dv}{d|v|}|... | Let $g\in L^1(|v|).
For \;$j=r,i$,\; we \:have, v_{j} ^ {\pm} \leq v_j^{+}+v_j^{-}=|v_j| \leq |v|$, then
$g\in L^1(v_r^{+})\cap L^1(v_i^{+})\cap L^1(v_r^{-})\cap L^1(v_i^{-})$ , then $g\in L^1(v)$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1269030",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Filling in a missing portion of a truth table I have the following truth table:
$$
\boxed{
\begin{array}{c|c|c|c}
a & b & c & x \\ \hline
F & F & F & F \\
F & F & T & F \\
F & T & F & F \\
F & T & T & F \\
T & F & F & T \\
T & F & T & T \\
T & T & F & F \\
T & T &... | You need to think about what it means to find a formula $x$ that fits the truth table. The easiest way to generate such a formula is simply to write a disjunction of all the cases where you want $x$ to be true.
Essentially, $x$ will say "I am true when this combination of $A$, $B$ and $C$ occurs, or when that one occur... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1269091",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 2
} |
Question about working in modulo? This question is in essence asking for understanding of a step in Fermats theorem done Group style.
For any field the nonzero elements form a group under field multiplication. So let us take the Field $Z_p$. The group $Z_p$ - {0} form a group under field multiplication.
This is my qu... | It is a theorem that a finite subgroup of the nonzero elements of a field is cyclic, and as a corollary, the nonzero elements of a finite field form s cyclic group.
If the field is size n, the subgroup is size $n-1$, and yes, the first n powers of any element suffice to generate the cyclic subgroup generated by an ele... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1269142",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Integral over a sequence of sets whose measures $\to 0.$ If $ f \in L_p$ with $1 \leq p \leq \infty $ and ${A_n}$ is a sequence of measurable sets such that $ \mu (An) \rightarrow 0,$ then $ \int_{A_n} f \rightarrow 0$.
Can someone give me a hint?
| My proof might be a bit lengthy but I think it might be a more standard way in dealing with this kind of problem.
First by $f\in L^{p}$ and DCT ($f_n= |f|^p 1_{\{|f|>n\}}\leq |f|, f_n\rightarrow 0$ a.e. as $f \in L^p$ ), we have
$$\forall \epsilon >0, \exists N >1 , s.t. \int_{\{|f|>N\}} |f|^p <\epsilon/2.$$
Now sinc... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1269224",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
To Find Eigenvalues Find the eigenvalues of the $6\times 6$ matrix
$$\left[\begin{matrix}
0 & 0 & 0 & 1 & 0 & 0 \\
0 & 0 & 0 & 0 & 1 & 0 \\
0 & 0 & 0 & 0 & 0 & 1 \\
1 & 0 & 0 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 & 0 & 0 \\
0 & 0 & 1 & 0 & 0 & 0 \\
\end{matrix}\right]$$
The options are $1, -1, i, -i$
It is a real symmetric ma... | Here as you say it's a real symmetric matrix so all the eigen values are real. Now we know that the sum of eigen values is equal to the trace of the matrix. Here trace of the matrix is equal to $0$, so if we take only $1$ or $-1$ as the eigen value, the trace becomes non-zero, so both $1$ and $-1$ are the eigen values.... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1269342",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "21",
"answer_count": 9,
"answer_id": 2
} |
How to solve this nonstandard system of equations? How to solve this system of equations
$$\begin{cases}
2x^2+y^2=1,\\
x^2 + y \sqrt{1-x^2}=1+(1-y)\sqrt{x}.
\end{cases}$$
I see $(0,1)$ is a root.
| We solve the second system for $y$:
$$y=\frac{-x^2+\sqrt{x}+1}{\sqrt{x}+\sqrt{1-x^2}}$$
and substitute into the first equation and solve for $x$. It is then seen that $(0,1)$ is the only real-valued solution. Computer analysis finds complex solutions where $x$ is the root of a certain $16$ degree polynomial.
Edit: the ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1269444",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 3,
"answer_id": 1
} |
Convergence, finding limit $\lim_{n \to \infty} \frac{2^n}{n!}$ I just came across an exercise, however I don't know how to find the limit of $$\lim_{n \to \infty} \frac{2^n}{n!}$$ can any body help?
Of course this is not homework, I'm only trying out example myself, from https://people.math.osu.edu/fowler.291/sequenc... | Notice that
$$\frac{2^n}{n!} = {{2 *2*2*2*...*2} \over {1*2*3*4*...*n}}=2*{{2*2*...*2} \over {3*4*...*n}} \leq 2*1*1*...*{{2} \over {n}}={4 \over n}$$
So
$$0 \leq \frac{2^n}{n!} \leq {4 \over n}\xrightarrow[n\to\infty]{}0$$
And hence $\lim_{n \to \infty} \frac{2^n}{n!}=0$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1269562",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
Number of Partitions proof How do I prove that the # of partitions of n into at most k parts equals the # of partitions of n+k into exactly k parts?
I was trying to improve my ability of bijective-proofs, unfortunately I was not able to find a working bijection to transform the Ferrers Diagram.
Can someone give me a hi... | I present a proof using PET (Polya Enumeration Theorem) for future reference. We will be using $Z(S_k)$, the cycle index of the symmetric group on $k$ elements. The partition into at most $k$ parts is given by $$[z^n] Z(S_k)\left(\frac{1}{1-z}\right)$$ where the fraction includes the empty term to account for slots bei... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1269644",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Elementary problems that would've been hard for past mathematicians, but are easy to solve today? I'm looking for problems that due to modern developments in mathematics would nowadays be reduced to a rote computation or at least an exercise in a textbook, but that past mathematicians (even famous and great ones such a... | I would say that computing the Fourier coefficients of a tamed function is a triviality today even at an engineering math 101 level.
Ph. Davis and R. Hersh tell the long and painful story of Fourier series.
I quote from their book:
"Fourier didn't know Euler had already done this, so he did it over. And Fourier, lik... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1269738",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "12",
"answer_count": 4,
"answer_id": 3
} |
A problem on convergent subsequences Find a sequence $\langle X_n\rangle$ such that
$$L_{X_n}=\left\{\frac{n+1}{n}:n\in\mathbb N\right\}\cup \{1\}.$$
Where $L_{X_n} = \{ p\in \mathbb{R} : \text{There exists a subsequence } \langle X_{n_k}\rangle \text{ of } \langle X_n\rangle \text{ such that} \lim X_{n_k} = p\}$.
Just... | For example $$\left( 2, 2,\frac{3}{2}, 2,\frac{3}{2} , \frac{4}{3},2,\frac{3}{2} , \frac{4}{3} ,\frac{5}{4} ,...\right)$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1269835",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
Why the $GCD$ of any two consecutive Fibonacci numbers is $1$? Note: I've noticed that this answer was given in another question, but I merely want to know if the way I'm using could also give me a proof.
I did the following:
$$F_n=F_{n-1}+F_{n-2} \\ F_n=[F_{n-2}+F_{n-3}]+[F_{n-3}+F_{n-4}]\\F_n=F_{n-2}+2F_{n-3}+F_{n-4}... | If $\,f_n = a f_{n-1}\! + f_{n-2}\,$ then induction shows $\,(f_n,f_{n-1}) = (f_1,f_0)\,$ since $\, (f_n,f_{n-1}) = (a f_{n-1}\! + f_{n-2},\,f_{n-1}) = (f_{n-2},f_{n-1}) = (f_1,f_0)\,$ by induction
Remark $\ $ Similarly one can prove much more generally that the Fibonacci numbers $\,f_n\:$ comprise a strong divisibili... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1269956",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 0
} |
What does it mean if the standard Hermitian form of complex two vectors is purely imaginary? If $v,w \in \mathbb{C}^n$, what does it mean geometrically for $\langle v , w \rangle$ to be purely imaginary?
| To understand what geometrically what it means for $\left<v,w\right>$ to be purely imaginary, it is necessary to understand geometrically how a Hermitian inner product $\left<\cdot,\cdot\right>_{\mathbb C}$ on $\mathbb C^n$ relates to the Euclidean inner product $\left<\cdot,\cdot\right>_{\mathbb R}$ on
$\mathbb R^{2n... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1270105",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 0
} |
Show that holomorphic function $f: \mathbb{C} \rightarrow \mathbb{C}$ is constant Let's $f: \mathbb{C} \rightarrow \mathbb{C}$ be a holomorphic function such that values $f$ are on line $y=ax+b$. Show that $f$ is constant.
I think I should use Cauchy-Riemann equations but I don't know what does mean that values $f$ are... | Your hypothesis reads: $$f(x+iy) = u(x,y) + i(a u(x,y) + b),$$ that is, the point $f(x+iy)$ as a point in $\Bbb R^2$ belongs to the said line. The Cauchy-Riemann equations read, abbreviating the notation: $$u_x = a u_y, \quad u_y = -au_x.$$ So $u_x = -a^2u_x \implies (1+a^2)u_x = 0$ and... oh, you can conclude now! (if... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1270305",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 5,
"answer_id": 0
} |
Algebraic Curves and Second Order Differential Equations I am curious if there are any examples of functions that are solutions to second order differential equations, that also parametrize an algebraic curve.
I am aware that the Weierstrass $\wp$ - Elliptic function satisfies a differential equation. We can then inter... | The functions $\sin(x)$ and $\cos(x)$ solve a second-order differential equation, namely
$$u'' = -u,$$
and $(\cos(x), \sin(x))$ parametrizes the algebraic curve $x^2 + y^2 = 1$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1270396",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 1,
"answer_id": 0
} |
Basic doubt about cosets Studying some basic group theory I had the following doubt: For $H$ subgroup of a finite group $G$ (doesn't matter invariance of $H$), is it true that $$|G/H|=|\{aHa^{-1}:a \in G\}| \space (\text{where G/H is the quotient of G by H})?$$
I've tried to define the most natural function between the... | No, it is not true. In fact, if $H$ is a normal subgroup of $G$, then (by the definition of normal) we have that for any $a\in G$,
$$aHa^{-1}=H$$
so that
$$\{aHa^{-1}:a\in G\}=\{H\}$$
has exactly one element, whereas there is more than one element of $G/H$ as long as $G\neq H$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1270520",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
show $\sum_{k=0}^n {k \choose i} = {n+1 \choose i+1}$ show for n $\geq i \geq 1 : \sum_{k=0}^n {k \choose i} = $ ${n+1} \choose {i+1}$
i show this with induction:
for n=i=1: ${1+1} \choose {1+1}$ = $2 \choose 2$ = 1 = $0 \choose 1$ + $1 \choose 1$ = $\sum_{k=0}^1 {k \choose 1}$
now let $\sum_{k=0}^n {k \choose i} = $ $... | We can also use the recurrence from Pascal's Triangle and telescoping series:
$$
\begin{align}
\sum_{k=0}^n\binom{k}{i}
&=\sum_{k=0}^n\left[\binom{k+1}{i+1}-\binom{k}{i+1}\right]\\
&=\sum_{k=1}^{n+1}\binom{k}{i+1}-\sum_{k=0}^n\binom{k}{i+1}\\
&=\binom{n+1}{i+1}-\binom{0}{i+1}\\
&=\binom{n+1}{i+1}
\end{align}
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1270595",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
For which values of $a,b,c$ is the matrix $A$ invertible? $A=\begin{pmatrix}1&1&1\\a&b&c\\a^2&b^2&c^2\end{pmatrix}$
$$\Rightarrow\det(A)=\begin{vmatrix}b&c\\b^2&c^2\end{vmatrix}-\begin{vmatrix}a&c\\a^2&c^2\end{vmatrix}+\begin{vmatrix}a&b\\a^2&b^2\end{vmatrix}\\=ab^2-a^2b-ac^2+a^2c+bc^2-b^2c\\=a^2(c-b)+b^2(a-c)+c^2(b-a)... | suppose the matrix is singular. then so is its transpose, which must annihilate some non-zero vector $(h,g,f)$
this gives three equations:
$$
fa^2 + ga+ h =0 \\
fb^2 + gb+ h =0 \\
fc^2 + gc+ h =0 \\
$$
we know from the algebra of fields that a quadratic equation can have at most two roots, so the three values $a,b,c$ c... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1270699",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 7,
"answer_id": 1
} |
Suppose $v_1,...,v_m$ is a linearly independent list, Will the list of vectors $Tv_1,...,Tv_m$ be a linearly independent list. Suppose $v_1,...,v_m$ is a linearly independent list of vectors in $V$ and $T∈L(V,W)$ is a linear map from V to W. Will the list of vector $Tv_1,...,Tv_m$ be a linearly independent list in W? I... | Short Answer: No, for example let $Tv=0$ for all $v\in V$.
Long Answer: If null space of $T$ is $\{0\}$ you can conclude that $Tv_1, \dots, Tv_m$ are linearly independent. Otherwise, they are linearly dependent.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1270774",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Completion with respect a collection of measures. Given a set of measures $\mathcal{M}$ on a measurable space $(\Omega,\mathscr{F})$, the $\mathcal{M}$-completion of $\mathscr{F}$ is defined as
$$ \overline{\mathscr{F}}^\mathcal{M}:=\bigcap_{\mu\in\mathcal{M}}\mathscr{F}^\mu,$$
where $\mathscr{F}^\mu$ is the completion... | The answer is no.
Example. Let $\Omega=[0,1]$.
$$\mathcal{F}=\{Y\subset\Omega : Y \subseteq [0,1]\, \& \,Y \,\text{is countable or} \, [0,1]\setminus Y\,\text{is countable}\}$$
Let $\mathcal{M}=(\delta_x: x \in [0,1]\}$ be the family of Dirac probability measures on $(\Omega,\cal{F})$. Then $\cal{N}_M=\{ \emptyset\}$.... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1270878",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Existence of a holomorphic function with the desired property Let $D=\{z\in \mathbb C:|z|<1\}$.
Then does there exist
1.a holomorphic function $f:D\to \overline D$ with $f(0)=0$ and $|f(\frac{1}{3})|=\frac{1}{4}$ ?
2.a holomorphic function $f:D\to \overline D$ with $f(0)=0$ and $f(\frac{1}{3})=\frac{1}{2}$ ?
How to ... | For 1, use $\frac{3}{4}·\frac{1}{3} = \frac{1}{4}$ and $\lvert\frac{3}{4}\rvert < 1$.
For 2, use the open mapping theorem and the Schwarz lemma.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1271005",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Describing groups with given presentation? $\langle x,y\ |\ xy=yx,x^5=y^3\rangle$ and $\langle x,y\ |\ xy=yx,x^4=y^2\rangle$. I'm trying to describe the groups with presentations $\langle x,y\ |\ xy=yx,x^5=y^3\rangle$ and $\langle x,y\ |\ xy=yx,x^4=y^2\rangle$. I have some problems getting a good picture of what they l... | The group $\langle x,y | xy=yx, x^5=y^3 \rangle$ is the quotient of $\langle x,y | xy=yx \rangle = \mathbb{Z} \oplus \mathbb{Z}$ (with $x=(1,0)$ and $y=(0,1)$) by the (normal) subgroup generated by $5x-3y=(5,-3)$, and similarly $\langle x,y | xy=yx , x^4=y^2 \rangle$ is the quotient of $\mathbb{Z} \oplus \mathbb{Z}$ by... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1271116",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 1,
"answer_id": 0
} |
How to Separate Quasi-Linear PDE I'm attempting to solve the non-homogenous quasi-linear PDE below:
$$z\frac{\partial z}{\partial x} - \alpha y\frac{\partial z}{\partial y} = \frac{\beta}{x^3y}$$
From what I've read in texts, the general form of a quasi-linear PDE is defined as
$$a(x,y,z)\frac{\partial z}{\partial x} +... | Where does this equation come from? I'm actually leaning towards there being no solution, for two reasons - one, Maple doesn't return one, and more seriously, I've tried solving it a couple of different ways and run into seemingly insurmountable problems in all of them.
An example - If you take the second two terms $\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1271327",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
solubility of nonlinear overdetermined PDE for holonomic scaling of a frame This question is essentially a tweak of under what conditions can orthogonal vector fields make curvilinear coordinate system? .
Suppose we have a frame of vector fields $\nu_i$ on $\mathbb{R}^n$. As discussed in the linked question, these $\nu... | Without loss of of generality, we consider strictly positive functions $ f_i := \exp {g^i} $ and slove the problem for functions $g^i$.
After calculation, the equation $[\exp g^i \cdot v_i, \exp g^j \cdot v_j] = 0$
is the same as
$$ [v_i, v_j] = v_j g^i \cdot v_i - v_i g^j \cdot v_j .$$
Therefore, when the given invol... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1271393",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Right notation to recurse over a sequence or list I have a function $f(x, a)$ which is invoked over all the elements of a sequence feeding the result to the next call, with $x$ being the next element in the list and $a$ the accumulated result. What is the correct (and popular) notation to write this properly?
Lets say... | You have the concept of folding. Your definition seem to be right for me. It's the right fold. Note, that there is also left folding:
$$ F(p,a) = \begin{cases} a & ; p=[] \\ F([x_1,\ldots,x_{n-1}], f(x_n, a)) & ; p = [x_1, \ldots, x_n]; n \ge 1 \end{cases} $$
EDIT: You can also define it with the colon operator (which ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1271478",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Find integers $m$ and $n$ such that $14m+13n=7$. The Problem:
Find integers $m$ and $n$ such that $14m+13n=7$.
Where I Am:
I understand how to do this problem when the number on the RHS is $1$, and I understand how to get solutions for $m$ and $n$ in terms of some arbitrary integer through modular arithmetic, like so:
... | If $14x+13y=1$ then multiplying by $7$ gives $14(7x)+13(7y)=7.$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1271738",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 8,
"answer_id": 0
} |
using a formula to make a statement with the use of greater than, less than symbol I am trying to make statement using the greater than less, less than symbol. I am unsure if the use of $\lt\gt$ symbols have to facing the same direction in an equation.
Could you say:
$x \lt x + y \gt y$
would this mean that $x$ is les... | You are correct in that the interpretation of $x<x+y>y$ would be $x<x+y$ and $y<x+y$. This isn't the usual way such a thing is notated - sometimes people write both equations, and sometimes people write $x,y<x+y$. Still, personally I think $x<x+y>y$ has a satisfying symmetry which $x,y<x+y$ doesn't have.
And of course... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1271813",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
What can be said about the relationship between row equivalent and invertible matrices? If $rref(A)$ = $I$ and $rref(B)$ = $I$, can we say that $A$ and $B$ are row equivalent?
My intuition says Yes, but I can't really prove it. If this is true, does that mean that all invertible matrices of the same dimensions are row ... | Yes -- if you can show that the inverse of elementary row operations are also elementary row operations, the proof is essentially already done (just apply the inverse of matrix B's operations to both sides).
And yes, this fact (plus a proof that an invertible matrix's row-reduced echelon form is always the identity mat... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1271902",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Show the probability that the sum of these numbers is odd is 1/2 Setting
Let $S$ be a set of integers where at least one of the integers is odd. Suppose we pick a random subset $T$ of $S$ by including each element of $S$ independently with probability $1/2$, Show that
$$\text{Pr}\Big[\Big(\sum_{i \in T} i\Big) = \text{... | This is a proof by pairing - it establishes a bijection between sets with even sums and sets with odd sums, to show that there are the same number of subsets of each kind.
Let $U$ be the set which contains all the elements of $S$ except for the. odd integer $x$.
Every subset $T$ of $S$ either contains $x$ or it doesn't... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1271998",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 2
} |
One tailed or two tailed Ok so this the question:
An administrator at a medium-sized hospital tells the board of directors that, among patients received at the Emergency room and eventually admitted to a ward, the average length of time between arriving at Emergency and being admitted to the ward is 4 hours and 15 minu... | There is only one scenario. The administrator's statement has to be taken at face value and is the null hypothesis.
$H_0: t=4.25$ hours
The board member has an alternative hypothesis.
$H_1: t>4.25$ hours
This is one-tailed.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1272083",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Hypothesis testing rejection question If a hypothesis is rejected at the $0.025$ level of significance, then it may be rejected or not rejected at the $0.01$ level. Is this statement true?
If it is true can you explain it to me why? I cannot seem to understand why it would be true.
Also if it is any level more than $0.... | If a claim is rejected at $0.01$ significance level, it will be rejected at $0.025$, but the other way around need not be true.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1272207",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Is the "homogenous solution" to a second-order linear homogenous DE always valid? That is, given an equation
$ay''+by'+cy = 0$,
I know that solutions are of the form $e^{rt},$ where r is a constant computed from $ar^2 + br + c = 0$. For some reason, I have written down in my notes adjacently that the "homogenous solut... | An $n$th order, linear homogenous ODE has $n$ linearly independent solutions (See this for why.). For a $2$nd order linear homogenous ODE, we then expect two solutions.
In your case, the coefficients of the ODE are contants. This arises to the notable family of solutions that you've noted: $c_1e^{r_1t}+c_2e^{r_2t}$.
Yo... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1272309",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
When is shear useful? I'd never heard of the shear of a vector field until reading this article.
Shear is the symmetric, tracefree part of the gradient of a vector field.
If you were to decompose the gradient of a vector field into antisymmetric ($\propto$ curl), symmetric tracefree (shear), and tracefull(?) ($\propto$... | In the special case of two dimensions the shear operator is known as the Cauchy-Riemann operator (or $\bar \partial$ operator, or one of two Wirtinger derivatives), and is denoted
$\dfrac{\partial}{\partial \bar z}$.
It is certainly useful in complex analysis.
The $n$-dimensional case comes up in the theory of quasic... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1272399",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
Fundamental theorem of calculus related problem Let
$$F(t) = \int ^t_0 f(s,t)\, ds$$
How can I find out $F'(t)$?
I guess we should apply chain rule but I can't succeed. Well, perhaps, separation of variables?
| Let $$G(u,v)=\int_0^uf(s,v)\,ds$$ Note that if $u(t)=t$ and $v(t)=t$, then $F(t)=G(u,v)$. Now $$
\begin{align}
\frac{d}{dt}F(t)&=\frac{d}{dt}G(u,v)\\&=\frac{\partial G}{\partial u}\frac{du}{dt}+\frac{\partial G}{\partial v}\frac{dv}{dt}\\
&=f(u,v)\cdot1+\int_0^u\frac{\partial}{\partial v}f(s,v)\,ds\cdot1\\
&=f(t,t)+\in... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1272516",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
No. of Comparisons to find maximum in $n$ Numbers Given $n$ numbers, we want to find the maximum. In order to find the maximum in a minimal amount of comparisons, we define a binary tree s.t. we compare $n'_1=\max(n_1,n_2)$, $n'_2=\max(n_3,n_4)$; $n''_1=\max(n'_1,n'_2)$ and so on.
How many maximum comparisons are neede... | It is fairly straight-forward to prove with induction that $k-1$ comparisons are enough to find the maximum of $n_1,\dotsc,n_k$.
*
*Clearly we need no comparisons to find the maximum of $n_1$.
*Consider now the numbers $n_1,\dotsc,n_k$ and assume that we can find the maximum of $k-1$ numbers with $k-2$ comparisons.... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1272576",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Is there a concept of asymptotically independent random variables? To prove some results using a standard theorem I need my random variables to be i.i.d.
However, my random variables are discrete uniforms emerging from a rank statistics, i.e. not independent: for two $u_1$,$u_2$ knowing $u_1$ gives me $u_2$.
Yet, my i... | If $(X_n, Y_n)$ are dependent for all finite $n$, but converges in distribution (or "weakly") to some random variable $(X,Y)$ where $X$ and $Y$ is independent, then we would say that $(X_n, Y_n)$ are asymptotically independent.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1272661",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 1,
"answer_id": 0
} |
Proving direct sum when field is NOT of characteristic $2$. Let $\mathbb{F}$ be a field that is not of characteristic $2$. Define $W_1 = \{ A \in M_{n \times n} (\mathbb{F}) : A_{ij} = 0$ whenever $i \leq j\}$ and $W_2$ to be the set of all symmetric $n \times n$ matrices with entries from $\mathbb{F}$. Both $W_1$ and ... | When you take the field $F$ to be of characteristic $2$ i.e if the field is $\mathbb Z_2$ i.e the matrix is over $\mathbb Z_2$ do you see that the problem has nothing left in it?
Hint: Try to write down a matrix over $\mathbb Z_2$ and see what happens
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1272735",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 1
} |
On Learning Tensor Calculus I am highly intrigued in knowing what tensors are, but I don't really know where to start with respect to initiative and looking for an appropriate textbook.
I have taken differential equations, multivariable calculus, linear algebra, and plan to take topology next semester. Since Summer bre... | Have a look at: http://www.amazon.com/Einsteins-Theory-Introduction-Mathematically-Untrained/dp/1461407052/ref=sr_1_fkmr0_1?s=books&ie=UTF8&qid=1431089294&sr=1-1-fkmr0&keywords=relativity+theory+for+the+mathematically+unsohisticated
which is Øyvind Grøn, Arne Næss: Einstein's Theory: A Rigorous Introduction for the ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1272828",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 1
} |
Linearization of an implicitly defined function $f(x,y,z)=e^{xz}y^2+\sin(y)\cos(z)+x^2 z$
Find equation of tangent plane at $(0,\pi,0)$ and use it to approximate $f(0.1,\pi,0.1)$. Find equation of normal to tangent plane.
My attempt: I found that tangent plane is $(2\pi-1)(y-\pi)=0$ or $y=\pi$ (all partial derivatives ... | This is not an implicit function. An implicit function is like $0=e^{xz}y^2+\sin(y)\cos(z)+x^2 z$. Instead, your function is a 3D explicit function.
You are correct on your linear approximation $L(x,y,z)=f(0,\pi,0)+(2\pi-1)(y-\pi)$ and that is exactly the tangent plane $w=f(0,\pi,0)+(2\pi-1)(y-\pi)$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1272931",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Restrict $x$ in an equation, but keeping only one equation How do you put restrictions on the $x$ in an equation without writing more than one equation? This is a two part question:
*
*How to take out a section of the graph of an equation?
*How to take out everything but a section of the graph of an equation?
For... | You're on the right track. If you want to restrict the domain to $1 < x < 4$, simply multiply and divide by square root which would imply the two separate restrictions $x>1$ and $x<4$. In this case. Those would be $\sqrt{x-1}$ and $\sqrt{4-x}$. Put it all together and you have
$f(x)_{restricted} = f(x) \cdot \frac{\s... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1273015",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
} |
Logical puzzle. 3 Persons, each 2 statements, 1 lie, 1 true I got a question at university which I cannot solve. We are currently working on RSA encryption and I'm not sure what that has to do with the question.
Maybe I miss something. Anyway, here is the question:
Inspector D interviews 3 people, A, B and C. All of th... | c did it, but b doesn't know it, just a lucky guess..
Relation to RSA would be that you have one secret and one public verifiable statement. But it is somewhat abstract.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1273108",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 6,
"answer_id": 2
} |
Irreducibility and factoring in $\mathbb Z[i], \mathbb Z[\sqrt{-3}]$
In $\mathbb Z[i]$, prove that $5$ is not irreducible.
In $\mathbb Z[\sqrt{-3}]$, factor $4$ into irreducibles in two distinct ways.
I am completely stumped on how to do this. I really need all the help I can get and a possible walkthrough.
| For the first assignment you correctly found
$$5 = (1+2i)(1-2i)\ .$$
This already shows that $5$ is reducible in $\mathbb Z[i]$, as claimed.
To provide a little intuition on how to find these factors, consider the third binomial rule for the product
$$(a+bi)(a-bi) = a^2 - (bi)^2 = a^2+b^2$$
So any sum of perfect square... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1273189",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Is a topological space isomorphic to some group? I'm reading topology without tears to develop intuition before attempting Munkres. I noticed how similar a topological space was to a group in Abstract Algebra.
1.1.1 Definitions. Let $X$ be a non-empty set. A set $\tau$ of subsets of $X$ is said to be a topology on $X$... | Not in general. But you're not too far from the truth.1
Generally, if $X$ is a set, then $\mathcal P(X)$ is a group using symmetric difference as a commutative addition operator. It's even a ring when using $\cap$ for multiplication.
But in the case of $\cap$ and $\cup$ as candidates for addition we run into some probl... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1273373",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 2,
"answer_id": 1
} |
How do I show that the inversion mapping for linear transforms is continuous in the operator norm? I'm working through some analysis textbooks on my own, so I don't want the full answer. I'm only looking for a hint on this problem.
My question is related to this question, but the textbook I'm working through approaches... | Here are some algebraic manipulations that may serve as a hint for you.
$$
\|(T+S)^{-1}-S^{-1}\|=\frac{\|(T+S)^{-1}S-S^{-1}S\|}{\|S\|}=\frac{\|(T+S)^{-1}T\|}{\|S\|}=\frac{\|(T+S)^{-1}\|}{\|S\|}\|T\|
$$
From here you should be able to deduce the result using the information given to you.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1273466",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Is there a well-known type of differential equation consisting of $y$, $y'$, and $y''$ multiplied together? Is there some sort of well known type of differential equation consisting of first and second derivatives multiplied together? For example (I just made this up):
$$y''(y')^2-y-x^2=0$$
Edit: Are there any applicat... | The height profile of a viscous gravity currents on horizontal surfaces can be described using a similarity solution as
$$ (y^3y')' + 1/5(3\alpha+1)xy' - 1/5(2\alpha-1)y = 0, $$
on the domain $ 0 \leq x \leq 1$. Analytic solutions exist for $\alpha = 0$.
Is this similar enough to what you're looking for?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1273550",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Solve by using substitution method $T(n) = T(n-1) + 2T(n-2) + 3$ given $T(0)=3$ and $T(1)=5$ I'm stuck solving by substitution method:
$$T(n) = T(n-1) + 2T(n-2) + 3$$ given $T(0)=3$ and $T(1)=5$
I've tried to turn it into homogeneous by subtracting $T(n+1)$:
$$A: T(n) = T(n-1) + 2T(n-2) + 3$$
$$B: T(n+1) = T(n) + 2T(n-... | suppose we look for a constant solution $t_n = a.$ then $a$ must satisfy $a = a+2a + 3.$ we pick $a = -3/2.$ make a change a variable $$a_n = t_n + 3/2, t_n = a_n - 3/2.$$ then $a_n$ satisfies the recurrence equation $$a_n= a_{n-1}+ 2a_{n-2}, a_0 = 9/2, a_1=13/2.$$ now look for solutions $$a_n = \lambda^n \text{ wher... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1273673",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Real analysis: Show the limit $\lim nμ(E_n) = 0$. Let $(X, R, μ)$ be a measurable space. Let $f$ be a measurable and integrable function on $X$. Let $E_n = \{x ∈ X|f(x) > n\}$. Prove that $\lim nμ(E_n) = 0$.
I know that $\mu(E_n)$ is equal to its outer measure, and it goes to $0$. But I do not know how show that and sh... | $$\lim \;n\mu(E_n) = \lim \int_{\{x:|f(x)|>n\}} n \;d\mu \leq \lim \int_{\{x:|f(x)|>n\}} |f| \;d\mu = \int_{\{x:|f(x)|=\infty\}} |f| \;d\mu=0$$ That last equality follows because $f \in L^1$ implies that $|f|<\infty$ a.e, and the second-to-last equality follows because $\{x:|f(x)|=\infty\} = \bigcap_n \{x:|f(x)|>n\}$ i... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1273754",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Math problems that are impossible to solve I recently read about the impossibility of trisecting an angle using compass and straight edge and its fascinating to see such a deceptively easy problem that is impossible to solve. I was wondering if there are more such problems like these which have been proven to be imposs... | Constructing an algorithm to solve any Diophantine equation has been proven to be equivalent to solving the halting problem, as is computing the Kolmogorov complexity (optimal compression size) of any given input, for any given universal description language.
In general I think what you're looking for is either problem... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1273867",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "17",
"answer_count": 9,
"answer_id": 0
} |
Find a sequence $a_n$ such that its sum converges, but the sum of its logarithm doesn't - Generalisation Help This question came up in a past paper that I was doing, but it seems to be a fairly common, standard question.
Give an example of a sequence $a_n$ such that $\sum(a_n)$ converges, but $\sum(\log(a_n))$ does no... | In this case, it is easy to find a simple criterion.
For $\sum \log(a_n)$ to converge, we should first have as a necessary condition $\log(a_n)$ converges to $0$ (by the limit test), and this only happens when $a_n$ converges to $1$ (and this, by turn, implies that $\sum a_n$ diverges). So, we limit our search to sequ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1273959",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Decision to play on in perfect square die game Question:You have a six-sided dice, and you will receive money that equals to sum of all the numbers you roll. After each roll, if the sum is a perfect square, the game ends and you lose all the money. If not, you can decide to keep rolling or stop the game. If your sum ... | If you count rolling a $1$ as $-35$, then your reference point is your current $35$, and you shouldn't include $35$ in the chance of further winnings.
On the other hand, you can certainly keep rolling while your score is below $43$.
So I would take the value of the next roll to be at least $(1/6)(-35)+(5/6)8$ which is ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1274046",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Contour integration of: $\int_C \frac{2}{z^2-1}\,dz$ I want to calculate this (for a homework problem, so understanding is the goal)
$$\int_C \frac{2}{z^2-1}\,dz$$ where $C$ is the circle of radius $\frac12$ centre $1$, positively oriented.
My thoughts:
$$z=-1+\frac12e^{i\theta},\,0\leq\theta\leq2\pi$$
$$z'=\frac{i}{2... | Use residue theorem to find integrals of this type. And observe that the only singular point for this analytic function in the disk above is at $z=1$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1274157",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
} |
Function whose limit does not exist at all points There are functions which are discontinuous everywhere and there are functions which are not differentiable anywhere, but are there functions with domain $\mathbb{R}$ (or "most" of it) whose limit does not exist at every point? For example, $ f:\mathbb{R}\to\mathbb{N}, ... | Things can get wild: There are functions $f:\mathbb {R}\to \mathbb {R}$ such that for every interval $I$ of positive length, $f(I) = \mathbb {R}.$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1274259",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
} |
Integration of $\int \frac{e^x}{e^{2x} + 1}dx$ I came across this question and I was unable to solve it. I know a bit about integrating linear functions, but I don't know how to integrate when two functions are divided. Please explain. I'm new to calculus.
Question: $$\int \frac{e^x}{e^{2x} + 1}dx$$
Thanks in advance.... | You may write
$$
\int \frac{e^x}{e^{2x} + 1}dx=\int \frac{d(e^x)}{(e^{x})^2 + 1}=\arctan (e^x)+C
$$ since
$$
\int \frac{1}{u^2 + 1}du=\arctan u+C.
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1274356",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Exhibit a bijective function $\Bbb Z \to \Bbb Z$ with infinitely many orbits I've the following exercise:
Give an example of a bijective function $\Bbb Z\rightarrow\Bbb Z$ with infinitely many orbits.
What would be its infinite orbits?
| HINT: Recall that there is a bijection between $\Bbb Z$ and $\Bbb{Z\times Z}$. Can you find a bijection $f\colon\Bbb{Z\times Z\to Z\times Z}$ with infinitely many infinite orbits?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1274445",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 1
} |
Prove that if $p\ge 7$ then $\exists n\in\Bbb{Z}$ such that $10^{n(p-1)}\equiv1 \mod 9p$. Prove that if $p\ge 7$ then $\exists n\in\Bbb{Z}$ such that $10^{n(p-1)}\equiv1 \mod 9p$. Edit: $p$ is prime, of course.
I tried using theorems regarding Euler, but I can't seem to arrive at something useful. I could really use yo... | By pigeon-hole, there exist $k,m$ with $10^{k(p-1)}\equiv 10^{m(p-1)}$ and wlog. $k>m$. Since $10$ is coprime to $9p$, we can cancel $10^{m(p-1)]}$ and find $10^{(k-m)(p-1)}\equiv 1\pmod{9p}$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1274520",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 3
} |
$x^9 - 2x^7 + 1 > 0$ $x^9 - 2x^7 + 1 > 0$
Solve in real numbers.
How would I do this without a graphing calculator or any graphing application? I only see a $(x-1)$ root and nothing else, can't really factor an eighth degree polynomial ...
Thanks.
| Hint:
You know that:$ x^9-2x^7+1=(x-1)(x^8+x^7-x^6-x^5-x^4-x^3-x^2+x-1)$.
Note that the second factor is positive for $x\rightarrow \pm \infty$, and is $-1$ for $x=0$ so it has at least two real roots.
You can use the Sturm theorem to find how many real roots has $x^8+x^7-x^6-x^5-x^4-x^3-x^2+x-1$, but it is very labori... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1274615",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Least Square method, find vector x that minimises $ ||Ax-b||_2^2$ Given Matrix A =
| 1 0 1 |
| 1 1 2 |
| 0 -1 -1|
and b = $[1\ \ 4\ -2]^T$
find x such that $||Ax - b||_2^2$ is minimised.
I know I have to do something along the line $A^TAx = A^Tb$
got the vector $(1/3)* [4\ 7\ 0] ^T$.
However the answer is
$x = (1/... | The vector $\begin{bmatrix} -1 \\ -1 \\1\end{bmatrix}$ is in the nullspace of $A^TA$.
So $A^TAx=A^TA\begin{bmatrix} 4/3 \\ 7/3 \\0\end{bmatrix}+\lambda A^TA
\begin{bmatrix} -1 \\ -1 \\1\end{bmatrix}=A^TA\begin{bmatrix} 4/3 \\ 7/3 \\0\end{bmatrix}+0$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1274741",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
List all the elements of ${\mathbb F_2}[X]/(X^3+X^2+1)$. I just don't get this. Looking through past papers, I came across this problem, Q. Let $F_2$ be a field with 2 elements. Let $P=x^3+x^2+1\in F_2[X]$. $I$ is the ideal of $F_2[X]$ generated by $P$. List all elements of the factor ring $F_2[X]/I$.
My answer; techni... | Use the division algorithm: For $f, g \in F[x]$, there exist $q, r \in F[x]$ such that $f(x) = q(x)g(x)+r(x)$ with $\deg r < \deg g$.
For the example: we wish to interpret $f(x) = x^4+1$ modulo $g(x) = x^3+x^2+1$. Using the division algorithm, we have $$f(x) = (x+1)g(x)+(x^2+x),$$ so $(x^4+1)+I = (x^2+x+(x+1)g(x))+I = ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1274916",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 1
} |
simplifying equations I have been trying to study analysis of algorithms with mathematical induction yet I found my algebra skills are very poor. So now I began restuddiing algebra (factoring, destributive property, simplifying, expanding) so I can handle mathematical equations better. But no matter what I do, I can't ... | Note that both expressions have the common factor $(n+1)$, which we can factor out. Hence we write $$\frac{n(n+1)}{2}+(n+1)=(n+1)\left[\frac{n}{2}+1\right]=(n+1)\left[\frac{n}{2}+\frac{2}{2}\right]=(n+1)\frac{n+2}{2}=\\=\frac{(n+1)(n+2)}{2}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1274998",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
How does $-\frac{1}{x-2} + \frac{1}{x-3}$ become $\frac{1}{2-x} - \frac{1}{3-x}$ I'm following a solution that is using a partial fraction decomposition, and I get stuck at the point where $-\frac{1}{x-2} + \frac{1}{x-3}$ becomes $\frac{1}{2-x} - \frac{1}{3-x}$
The equations are obviously equal, but some algebraic mani... | $$ -\frac{1}{x-2} = \frac{1}{-(x-2)} = \frac{1}{-x+2} = \frac{1}{2-x}$$ and
$$ \frac{1}{x - 3} = \frac{1}{-3 + x} = \frac{1}{-(3 - x)} = -\frac{1}{3-x}$$
Thus $$ -\frac{1}{x-2} + \frac{1}{x - 3} = \frac{1}{2-x} - \frac{1}{3-x} $$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1275071",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 7,
"answer_id": 6
} |
If I have the value of $\sqrt{1.3}$ could it be possible to find other square roots from that value? using the manipulation of surds? If I have the value of $\sqrt{1.3}$ could it be possible to find other square roots from that value? using the manipulation of surds?
| You can use the knowledge of $\sqrt{1.3}$ to calculate $\sqrt{x}$ only if $x = 1.3*y$, where $y$ is some other number whose square root you already know.
So $\sqrt{5.2}$ is easy, because it's just $\sqrt4*\sqrt{1.3}$, which you can calculate, assuming that you know that $\sqrt{4}=2.$
To calculate $\sqrt{2.6}$, you coul... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1275160",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
How to know if an integral is well defined regardless of path taken. I can calculate
\begin{equation*}
\int_0^i ze^{z^2} dz=\frac{1}{2e}-\frac12,
\end{equation*}
but why can I calculate this irrelevant to the path taken? Is this since it is analytic everywhere - if so, how would I go about verifying this? I can't see... | Going to the Cauchy-Riemann equations is not a good way of showing that $z\mapsto ze^{z^2}$ is differentiable everywhere, just like reasoning explicitly about $\lim_{h\to 0}\frac{f(x+h)+f(x)}{h}$ wouldn't be a good way to investigate whether the real function $x\mapsto xe^{x^2}$ is differentiable everywhere.
Instead no... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1275268",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
} |
Approximating non-rational roots by a rational roots for a quadratic equation Let $a,b,c$ be integers and suppose the equation $f(x)=ax^2+bx+c=0$ has an irrational root $r$. Let $u=\frac p q$ be any rational number such that $|u-r|<1$. Prove that $\frac 1 {q^2} \leq |f(u)| \leq K|u-r|$ for some constant $K$. Deduce tha... | A non-rational root is simply a root that is not a rational number (i.e. it is irrational).
The problem has nothing to do with the intermediate value theorem, it is an exercise in pure algebra. First, note that if $r$ is an irrational root, then the other root $r'$ must also be irrational (assume that $r'$ is rational;... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1275365",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
Elementary proof that $-1$ is a square in $\mathbb{F}_p$ for $p = 1 \mod{4}$ I am trying to proof that $-1$ is a square in $\mathbb{F}_p$ for $p = 1 \mod{4}$. Of course, this is really easy if one uses the Legendre Symbol and Euler's criterion. However, I do not want to use those. In fact, I want to prove this using a... | For $p\ge 3$, $\mathbb F_p^*$ is a cyclic group of order $p-1$. If $g$ is a generator, then $g^\frac{p-1}2= -1$. In particular $-1$ will be a square if and only if $\frac{p-1}{2}$ is even - i.e. if $p \equiv 1 \pmod 4$.
Note that this proof isn't fundamentally different from those using Fermat's little theorem. Indeed ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1275461",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 3,
"answer_id": 2
} |
Result of the limit: $\lim_{x \to \infty} \sqrt{x + \sin(x)} - \sqrt{x}$? From my calculations, the limit of
$\lim_{x \to \infty} \sqrt{x + \sin(x)} - \sqrt{x}$
Is undefined due to $sin(x)$ being a periodic function, but someone told me it should be zero.
I was just wondering if someone could please confirm what the ... | or use the binomial theorem
$$
\sqrt{x+\sin x}-\sqrt{x} = \sqrt{x} \left(\sqrt{1 + \frac{\sin x}
{x}}-1\right) \\
= \sqrt{x}\left( \frac{\sin x}x + O(\frac1{x^2}) \right) \to 0
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1275542",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 8,
"answer_id": 6
} |
Coffee Shop Meeting $A$ and $B$ decide to meet at a cafe between $5$ p.m. and $6$ p.m. They agree that the person who arrives first at the cafe would wait for exactly $15$ minutes for the other. If each of them arrives at a random time between $5$ p.m. and $6$ p.m., what is the probability that the meeting takes place?... | Let $X$ and $Y$ be the times in units of hours that $X$ and $Y$ arrive. I assume here that they are uniformly distributed on $[0,1]$ and independent. Then the meeting happens provided $|X-Y| \leq 1/4$. So the probability of the meeting is
$$\frac{\int_{|x-y| \leq 1/4,0 \leq x \leq 1,0 \leq y \leq 1} dx dy}{\int_{0 \leq... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1275612",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
} |
When should matrices have units of measurement? As a mathematician I think of matrices as $\mathbb{F}^{m\times n}$, where $\mathbb{F}$ is a field and usually $\mathbb{F} = \mathbb{R}$ or $\mathbb{F} = \mathbb{C}$. Units are not necessary.
However, some engineers have told me that they prefer matrices to have units such... | The book by George W. Hart mentioned in the comments is actually a quite good reference on the topic.
Furthermore, I also gave a talk on how a C++ library can be designed that is able to solve the topic for the general case of non-uniform physical units in vectors and matrices:
https://www.youtube.com/watch?v=J6H9Cwzyn... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1275724",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 5,
"answer_id": 2
} |
Determine values of k for a matrix to have a unique solution I have the following system and need to find for what values of $k$ does the system have
i) a unique solution
ii) no solution
iii) an infinite number of solutions
$(k^3+3k)x + (k+5)y + (k+3)z = k^5+(k+3)^2
ky + z = 3
(k^3+k^2-6k)z = k(k^2-9)$
Putting this i... |
$\textbf{HINT:}$ Check the determinant. if $\det=0$ you will have infinite solutions, also you can check the last file, if $a_{32}=0$, then check $a_{34}:$
*
*if $a_{34}\neq0$ then there are no solutions.
*if $a_{34}=0$ there are infinite solutions.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1275774",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Finding a nullspace of a matrix. I am given the following matrix $A$ and I need to find a nullspace of this matrix.
$$A =
\begin{pmatrix}
2&1&4&-1 \\
1&1&1&1 \\
1&0&3&-2 \\
-3&-2&-5&0
\end{pmatrix}$$
I have found a row reduced form of this matrix, which is:
$$A' =
\begin{pmatrix}
1&0&3&-2 \\
0&1&-2&3 ... |
Since $x_1=2x_4-3x_3$ and $x_2=2x_3-3x_4\Rightarrow$
if $(x_1,x_2,x_3,x_4)\in$ nullspace($A$): $$(x_1,x_2,x_3,x_4)=(2x_4-3x_3,2x_3-3x_4,x_3,x_4)=x_3(-3,2,1,0)+x_4(2,-3,0,1)$$ So Nullspace$(A)=\langle (-3,2,1,0),(2,-3,0,1) \rangle$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1275856",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
} |
Regularity theorem for PDE: $-\Delta u \in C(\overline{\Omega})$ implies $u\in C^1(\overline{\Omega})$? I stumbled over this question in the context of PDE theory:
Let $U$ be connected,open and bounded in $\mathbb{R}^n$ and $u \in C^0(\overline{U}) \cap C^2(U)$ and $\Delta u \in C^0(\overline{U})$ with $u|_{\partial U}... | If $U$ is a bounded domain with regular boundary then, one approach is the following: Consider the problem
$$
\left\{ \begin{array}{rl}
-\Delta u=f &\mbox{ in}\ U, \\
u=0 &\mbox{on } \partial U.
\end{array} \right.
$$
Once $f\in C(\overline{U})$, we also have that $f\in L^p(U)$ for each $p\in [1,\infty)$. T... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1276027",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
Factor Modules/Vector Spaces and its basis with canonical mappings I'm having trouble with factor modules now.
Well, specifically the following question from a past paper.
Q. T=$R^2$ i.e. the real plane, and define $f:T$->$T$ with respect to the standard basis which has the 2 by 2 matrix $$A=\left( \begin{matrix} -6 & ... | You are correct about what the kernel is.
When you say "Give the basis $C$..." I assume you mean "Give a basis $C$...". I.e. you have to find a basis.
$T/ \text{ker} f$ is one-dimensional since $\dim(T/S) = \dim(T) - \dim(S)$. Any nonzero vector of a one-dimensional space constitutes a basis. (I.e. pick any vector $v... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1276096",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Conditional Probability Four fair 6-sided dice are rolled. The sum of the numbers shown on the dice is 8. What is the probability that 2's were rolled on all four dice?
The answer should be 1/(# of ways 4 numbers sum to 8)
However, I can't find a way, other than listing all possibilities, to find the denominator.
| I proved a relevant theorem here ( for some reason this was closed )
Theorem 1: The number of distinct $n$-tuples of whole numbers whose components sum to a whole number $m$ is given by
$$ N^{(m)}_n \equiv \binom{m+n-1}{n-1}.$$
to apply this to your problem , you need to take into account that the theorem allows a min... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1276177",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
Perfect matching problem We have a graph $G = (V,E)$. Two players are playing a game in which they are alternately selecting edges of $G$ such that in every moment all the selected edges are forming a simple path (path without cycles).
Prove that if $G$ contains a perfect matching, then the first player can win. A play... | Bipartition the vertices according to the perfect matching into sets $A$ and $B$. The first player plays an edge of the matching; the second player cannot. However, since this is not a cycle, we are at a new vertex, so the first player can again play an edge from the matching. He can continue this way until all of the ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1276237",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
What is this problem asking? (limits of functions) I was asked to prove or find a counterexample of the statement "Suppose that $f: \mathbb{R}\to \mathbb{R}$, and let $x_k=f(\dfrac{1}{k})$. Then $\lim_{k\to\infty} f(x_k)=\lim_{t\to0} f(t)$." I'm not sure what the statement is saying. If anyone could explain it, it woul... | There are two different types of limits being used in the statement
$$\lim_{k\to\infty}f(x_k)=\lim_{t\to 0} f(t).$$
On the left we have a sequence:
$$f(x_1), f(x_2), f(x_3), \ldots$$
to which we might apply the definition of $x_k$ to rewrite as
$$f(f(1)), f(f(1/2), f(f(1/3)), \ldots$$
The limit of this sequence is giv... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1276330",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Consider $n, k ∈ \mathbb{N}$. Prove that $\gcd(n, n + k)\mid k$
Consider $n, k\in\mathbb N$. Prove that $\gcd(n, n + k)\mid k$
Here is my proof. Is it correct?
A proposition in number theory states the following:
$$\forall a, b \in \mathbb{Z}, \ \gcd(n,m)\mid an+bm$$
A corollary (consequence) to this is that if $... | Even easier $d=\gcd(n,n+k)$
*
*if $d=1$ then $d \mid k$
*if $d>1$ then $d \mid n$ and $d \mid n+k$ or $d\cdot q_1=n$ and $d\cdot q_2=n+k$. Altogether $d\cdot q_2 = d\cdot q_1 +k$ or $d(q_1-q_2)=k \Rightarrow d \mid k$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1276513",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 1
} |
For what values of k is this singular matrix diagonalizable? So the matrix is the following:
\begin{bmatrix}
1 &1 &k \\
1&1 &k \\
1&1 &k
\end{bmatrix}
I've found the eigan values which are $0$ with an algebraic multiplicity of $2$ and $k+2$ with an algebraic multiplicity of $1$.
However I'm having trouble with t... | Hint: Substitute $r = -k$ and $t = 0$ to get the first eigenvector Wolfram alpha is giving you, and $r = -1$ and $t = 1$ to get the second one. Wolfram alpha isn't giving you the entire two-dimensional eigenspace, but only an eigenbasis.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1276602",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
$y^5 =(x+2)^4+(e^x)(ln y)−15$ finding $\frac{dy}{dx}$ at $(0,1)$
Unsure what to do regarding the $y^5$.
Should I convert it to a $y$= function and take the $5$ root of the other side. Then differentiate? Any help would be great thanks.
| Unfortunately we cannot rearrange this to find $y$. Fortunately we can still use implicit differentiation to find a derivative. Take the derivative of all terms w.r.t. $x$, using the chain rule and treating $y$ as a function of $x$ - we obtain
$$
5y^4\frac{\mathrm{d}y}{\mathrm{d}x} = 4(x+2)^3 + \mathrm{e}^x\ln y +\frac... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1276690",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
How many 1's can a regular 0,1-matrix contain? A matrix of order $n$ has all of its entries in $\{0,1\}$. What is the maximum number of $1$ in the matrix for which the matrix is non singular.
| It should be $n^2-n+1$. Let $v=e_1+\cdots+e_n$, where $\{e_1, \cdots, e_n\}$ is the standard orthonormal basis. Then the matrix with row vectors $\{v, v-e_1, \cdots, v-e_{n-1}\}$ is a matrix containing the maximum number of 1 for which it is nonsingular.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1276789",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 2,
"answer_id": 1
} |
Proving that there exists a horizontal chord with length $1/n$ for a continuous function $f: [0,1] \to \mathbb R$ Given a continuous function $f: [0,1] \to \mathbb R$ and that the chord which connects $A(0, f(0)), B(1, f(1))$ is horizontal then prove that there exists a horizontal chord $CD$ to the graph $C_f$ with le... | Intervals with length $\frac{1}{k}$ and $\frac{1}{k+1}$ overlap pretty bad, so I do not think it is a good idea to use induction on $k$. Instead, just take $h_n:[0,1-1/n]\to\mathbb{R}$:
$$ h_n(x) = f\left(x+\frac{1}{n}\right)-f(x) $$
that is obviously continuous. If $h_n$ is zero somewhere in $I=[0,1-1/n]$ we have noth... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1276889",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Identity of tensor products over an algebra I hope this question hasn't been asked before, but I wasn't sure what to search so as to check. Suppose $R$ is a commutative ring with 1 and that $A$, $B$ are unital $R$-algebras. Suppose further that $M$, $M'$ are $A$-modules and $N$, $N'$ are $B$-modules. Is the following i... | For $M,M',N,N'$ free there is an obvious, well defined isomorphism. In fact, if $M=A^{(I)}$, $M'=A^{(I')}$, $N=B^{(J)}$, $N'=B^{(J')}$, they are both naturally isomorphic to $A\otimes_R B^{(I\times I'\times J\times J')}$.
In the general case, take free resolutions of $M,M',N,N'$, they induce free resolutions of both $(... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1277000",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
} |
Factorising polynomials over $\mathbb{Z}_2$ Is there some fast way to determine whether a polynomial divides another in $\mathbb{Z}_2$? Is there some fast way to factor polynomials in $\mathbb{Z}_2$ into irreducible polynomials?
Is there a fast way to find $n$ such that a given polynomial in $\mathbb{Z}_2$ divides $x^n... | To check for divisibility is easy: since $\mathbb Z_2$ is a discrete valuation ring, for any two polynomials $f, g$, $f$ divides $g$ iff it divides it over the fraction field $\mathbb Q_2$ and the content of $f$ divides the content of $g$. Testing this is just Euclidean division, which is “fast” enough by any definitio... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1277055",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
$N^2 = T^*T$, $N$ is non-negative, and $T$ is invertible, how to prove $N$ is also invertible? I'm reading Hoffman's "Linear Algebra", and met this line that I couldn't figure it out.
In $\S9.5$ Spectral Theory, pp 342, he mentioned:
Let $V$ be a finite-dimensional inner product space and let $T$ be any
linear opera... | If $T$ is invertible then $\ker T=\{0\}$. The equation
$$
\|N\alpha\|^2 = \langle N\alpha,N\alpha\rangle = \langle T\alpha,T\alpha\rangle=
\|T\alpha\|^2
$$
implies $\ker N = \ker T$. Hence $\ker N=\{0\}$, and since it is a linear mapping from the finite-dimensional $V$ to $V$, it is invertible.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1277153",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
How to do this integral. I need to do this:
$$\int_0^\infty e^{ikt}\sin(pt)dt$$
I already have the solution, I'm just clueless on how to actually calculate it. I've tried several changes,integration by parts and extending it to a $(-\infty,\infty)$ integral so it'd be a simple Fourier transform, but I can't solve it.
I... | Under the condition that all integrals exist ($|\operatorname{Im}(p)|<\operatorname{Im}(k)$)
$$\begin{align}
\int_0^\infty e^{ikt}\sin(pt)\,dt&=\frac{1}{2\,i}\int_0^\infty e^{ikt}(e^{ipt}-e^{-ipt})\,dt\\
&=\frac{1}{2\,i}\Bigl(\frac{e^{i(k+p)t}}{i(k+p)}-\frac{e^{i(k-p)t}}{i(k-p)}\Bigr|_0^\infty\Bigl)\\
&=\frac12\Bigl(\f... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1277240",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
What is the practical meaning of derivatives? I mean practically integration means sum of all components, and the integral can be visualized as the area below a curve.
Is there a similar intuition or geometric meaning of the derivative?
| Interestingly the history of Calculus is a bit backwards from the way it is taught in schools. It began with the investigation into finding areas of certain geometric shapes, a field of Mathematics called Quadrature. This led the way to integration theory. One landmark result which is by Fermat, when he demonstrated: $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1277323",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
problem of numeric sequence Give an example of a sequence $(A_n)$ which is not convergent but the sequence $(B_n)$ defined by $\displaystyle B_n = \frac{A_1+A_2+\cdots+A_n}{n}$ is convergent.
| You can also consider the alternating $0,1$ sequence i.e.
$$A_n = 0 \mbox{ if $n$ is odd} $$
$$A_n =1 \mbox{ if $n$ is even }$$
Then $B_n \to 1/2$ but as $A_n$ is oscillatory, it does not converge. For more details, refer Cesaro sums and Cesaro means.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1277415",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Extending an automorphism to the integral closure I need some help to solve the second part of this problem. Also I will appreciate corrections about my solution to the first part. The problem is the following.
Let $\sigma$ be an automorphism of the integral domain $R$. Show that $\sigma$ extends in an unique way to t... | The uniqueness is fairly trivial. Since suppose $\bar{\sigma}$ is an extension then it must be defined as you said. Indeed $\bar{\sigma}(r/s)=\bar{\sigma}(r\cdot s^{-1})=\bar{\sigma}(r)\cdot\bar{\sigma}(s)^{-1}=\sigma(r)\cdot\sigma(s)^{-1}=\sigma(r)/\sigma(s)$. And indeed $\bar{\sigma}$ is an automorphism if $\sigma$ i... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1277499",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
proof that $\frac{x^p - 1}{x-1} = 1 + x + \dots + x^{p-1}$ is irreducible I am reading the group theory text of Eugene Dickson. Theorem 33 shows this polynomial is irreducible
$$ \frac{x^p - 1}{x-1} = 1 + x + \dots + x^{p-1} \in \mathbb{Z}[x]$$
He shows this polynomial is irreducible in $\mathbb{F}_q[x]$ whenever $p$ ... | To clarify an issue that came up in the comments, by Gauss's lemma a monic polynomial is irreducible over $\mathbb{Q}[x]$ iff it's irreducible over $\mathbb{Z}[x]$, and by reduction $\bmod p$, if a polynomial is irreducible $\bmod p$ for any particular prime $p$, then it's irreducible over $\mathbb{Z}[x]$. So once you ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1277577",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 1,
"answer_id": 0
} |
Calculate fractional part of square root without taking square root Let's say I have a number $x>0$ and I need to calculate the fractional part of its square root:
$$f(x) = \sqrt x-\lfloor\sqrt x\rfloor$$
If I have $\lfloor\sqrt x\rfloor$ available, is there a way I can achieve (or approximate) this without having to c... | After a bit more reading on Ted's Math World (first pointed out by @hatch22 in his answer), I came across a quite beautiful way to perform this calculation using the Pythagorean theorem:
Given some estimate $y$ of the square root (in the case here we have$\lfloor\sqrt{x}\rfloor$), if we let the hypotenuse of a right tr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1277652",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 2
} |
Finding the integral I've come across the following integration problem online:
$$\int{\frac{x^3}{x^4-2x+1}}\,\,dx$$
This is beyond my current knowledge of integrals, but it seems to be a very interesting integration problem. Would anyone be able to give a tip on how to being solving the expression?
If there are any t... | HINT:You can solve the intgral in this way:
$$\int{\frac{x^3}{x^4-2x+1}}\,\,dx=$$
$$\frac{1}{4}\int{\frac{4x^3+2-2}{x^4-2x+1}}\,\,dx=$$
$$\frac{1}{4}\int{\frac{4x^3-2}{x^4-2x+1}}\,\,dx+\frac{1}{4}\int{\frac{2}{x^4-2x+1}}\,\,dx=$$
$$\frac{1}{4}ln|x^4-2x+1|+\frac{1}{4}\int{\frac{2}{x^4-2x+1}}\,\,dx=$$
and then you have t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1277740",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Notation for a vector with constant equal components of arbitrary dimension Is there a standard notation to specify vectors of the form $(c,c,\ldots,c)$, where $c$ is some constant?, e.g. suppose I have the vector $(c,c,c,c,c)$ which has 5 components. Is there a standard (short hand) way to write such a vector? I am de... | You can use $\overrightarrow{(c)}_n$ or $\overrightarrow{c}$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1277793",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Fermats Little Theorem: How $(3^7)^{17}$ will leave the same remainder when divided by $17$ as $3^7$? As it is used in this explanation, how and why $(3^7)^{17}$ will leave the same remainder when divided by $17$ as $3^7$?
Thanks!
| Because $a^p \equiv a \pmod p$ (where $p$ is a prime) by Fermat's little theorem. (Think $3^7$ as $a$.)
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1277858",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 2
} |
Find area of solid using cylindrical shell method Here's my problem, I have to use the shell method to find the area bounded between f(x)=4x-x^2 and y=3 by rotating about y=2. I understand I have to first solve for x. Which is weird because I have two x variables with different powers. So does this mean I am going to h... | You only need one of those equations since both regions we want to rotate are equal regions if that makes sense. So you should solve that equation above and get $x =\pm \sqrt{4-y}+2$. Well use the right hand side of the parabola in rotate it. We will also use the axis of symmetry of our parabola to help guide us to wha... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1277957",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Inequality used in the proof of Kolmogorov Strong Law of Large Numbers I'm trying show that convergence follows.
$$\sum_{k \geq 1} \frac{\sigma^2_k}{k^2} < \infty \Rightarrow \lim_{M \rightarrow \infty}\frac{1}{M^2}\sum_{k \leq M} \sigma^2_k=0.$$
Let's consider $D_k = \sum_{n \geq k} \displaystyle\frac{\sigma_n^2}{n^2... | \begin{align*}
\sum_{k=1}^M k^2(D_k-D_{k+1})
&= \sum_{k=1}^M k^2 D_k - \sum_{k=1}^M k^2 D_{k+1} \\
&= \sum_{k=1}^M k^2 D_k - \sum_{k=2}^{M+1} (k-1)^2 D_k \\
&= D_1 + \sum_{k=2}^M (k^2-(k-1)^2) D_k - M^2 D_{M+1} \\
&= D_1 + \sum_{k=2}^M (2k-1) D_k - M^2 D_{M+1} \\
&= \sum_{k=1}^M (2k-1) D_k - M^2 D_{M+1} \\
&\le \sum_{k... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1278042",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
} |
What is the probability your friend wins on his n'th toss? You and your friend roll a sixth-sided die. If he lands even number then he wins, and if you land odds then you win. He goes first, if he wins on his first roll then the game is over. If he doesn't win on his first roll, then he pass the die to you. repeat unti... | Both of you have the same probability of winning and losing on a single roll, $\frac{1}{2}$
The probability that he wins in his first toss is $\frac{1}{2}$.
He wins in his second toss when he and you both don't win your first tosses and he wins in his second toss. The probability that this happens is $\frac{1}{2} \cdot... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1278158",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Is Summation a linear operator Say I have the following recursive function, where superscript does not denote a power, but instead denotes a point in time. And I wonder if it's linear for the points in space (denoted by subscript i). $T$ means the tempurature, where $t$ means the actual time. (So timesteps in the recur... | Write $T_i=(T_i^0,\dots,T_i^m)\in\mathbb{R}^{m+1}$ (I'm assuming "temperature" is a real variable...). Your map, for fixed $t$, is equivalent to $$f(T_i,t)=[\begin{array}{ccc}r_1(t)&\cdots&r_m(t)\end{array}]MT_i,$$ where $M$ is the $m\times (m+1)$ matrix defined by the equation $$M_{ij} =
\cases{-1 & i=j\\
1 & j=i+1\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1278274",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Let A be a $n\times n$ non-singular symmetric matrix with entries in $(0,\infty)$. Let $A$ be a $n\times n$ non-singular symmetric matrix with entries in $(0,\infty)$. Then we can conclude that
(a) $|A| > 0 (|A|$ denotes the determinant of A).
(b) $A$ is a positive definite matrix.
(c) $B = A^2$ is a positive defini... | For a, b and d your approach is right you have a counterexample. Let's take a look at c. $A$ symmetric non singular and therefore is diagonalisable with real nonzero eigenvalues. $A^2$ will then have positive eigenvalues (the squares of those of $A$) on top of being symmetric so it is positive definite. Non singular is... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1278375",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
} |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.