Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Condition on which this equation holds It is well known that for a linear transformation on an Euclidean space $V$, whose matrix under a certain basis we denote by $A$, we have
$$Ker(A)\oplus R(A)=V$$
in which $R$ denotes the row space.
Also, since the column space (denoted by $C(A)$) is exactly $Im(A)$, we have
$$di... | In general, the equality (direct sum) is not always true. However, you can do it for projection. If A satisfies $A^{2}=A$, then $V=KerA \bigoplus ImA $, a typical example of those is a diagonal $n \times n$ matrix, with 0 and 1 as eigenvalues!
| {
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Why does this work, and why is it wrong? I have devised a "proof" that $i=0$. Obviously it can't be true, but I can't see why it is wrong. I've been thinking about this for a while, and my friend and I are very confused. What is wrong with the "proof" below?
$e^{i\pi} = -1$
$e^{2i\pi} = 1$
$2i\pi = ln(1)$
$i = \frac{0}... | Now that you have proved in the second line $e^{2i\pi} = 1=e^0$, so the complex exponential function is not injective. Thus you can not define a complex log function as its inverse function.
| {
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Request for help to solve an equation with LambertW: $ (x^2-4\,x+6) e^x =y$ I want to solve the following equation:
$$ (x^2-4\,x+6) e^x =y \tag{1} $$
It looks a bit like the following equation:
$$ x e^x =y \tag{2} $$
Since the solution of equation (2) is: x=LambertW(y), I think the solution of equation (1) should als... | No, this is quite different. As far as I can tell, it can't be expressed in the form $z \exp(z) = f(y)$.
LambertW is unlikely to help here.
| {
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All real values $a$ for a $2$-dimensional vector? Find all real numbers $a$ for which there exists a $2D$, nonzero vector $v$ such that:
$\begin{pmatrix} 2 & 12 \\ 2 & -3 \end{pmatrix} {v} = a {v}$.
I substituted $v$ with $\begin{pmatrix} c \\ d \end{pmatrix}$ and multiplied to obtain the system of equations:
$2x+12y ... | you can go from $$2x+12y = kx,\, 2x-3y = ky $$ to $$\frac{y}{x} = \frac{k-2}{12} = \frac2{k+3}.$$ therefore $k$ satisfies the characteristic equation $$0=(k+3)(k-2) - 24 = k^2+k-30 = (k+6)(k-5).$$ therefore $$k = 5, -6 $$
| {
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Must PID contain 1? Must PID contain 1? My concern arises when i consider the gcd of say a and b in the PID. Since it is a PID, it is generated by one element say k. k obviously $\in (k)$. However, if PID does not contain 1, then i can't write $k = k \cdot 1$, can 1? So is it true that then k must be equal to $k = k \c... | From wikipedia:
"Integral domain" is defined almost universally as above, but there is some variation. This article follows the convention that rings have a 1, but some authors who do not follow this also do not require integral domains to have a 1
So authors who ask rings to have a $1$ ask their domains to have a $1$... | {
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Proving that the second derivative of a convex function is nonnegative My task is as follows:
Let $f:\mathbb{R}\to\mathbb{R}$ be a twice-differentiable function,
and let $f$'s second derivative be continuous. Let $f$ be convex with
the following definition of convexity: for any $a<b \in \mathbb{R}$:
$$f\left(\fr... | I would set up a proof by contradiction. Assuming a single point where $f''(x) < 0$, you can use the continuity of $f''(x)$ to find an interval $[a,b]$, where $f''(x) < 0$ throughout. The intuition is then clear, in the sense that if you draw a concave down segment, then any secant line lies below your curve. I will... | {
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Prove that $2^n\le n!$ for all $n \in \mathbb{N},n\ge4$ The problem i have is:
Prove that $2^n\le n!$ for all $n \in \mathbb{N},n\ge4$
Ive been trying to use different examples of similar problems like at:
http://web.cacs.louisiana.edu/~mgr/261/induction.html
First i show the base case $n=4$ is true.
Then assuming $2^k... | An easy and intuitive solution.
Write $k>3$ in place of $k\geq4$.
One can easily prove the base case.
Now Assume that $2^k\leq k!$
So lets prove that $2^{k+1}\leq (k+1)!$
$2^k.2\leq (k+1)k!$
Multiply both sides by $-1$ and flip the sign.
$-2^k.2> -(k+1)k!$
$k!\geq 2^k$ ->Assumption
So $-2^k.2> -(k+1)2^k$
$-2>-k... | {
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Prove that if $a,b \in \mathbb{R}$ and $|a-b|\lt 5$, then $|b|\lt|a|+5.$ I'm trying to prove that if $a,b \in \mathbb{R}$ and $|a-b|\lt 5$, then $|b|\lt|a|+5.$
I've first written down $-5\lt a-b \lt5$ and have tried to add different things from all sides of the inequality. Like adding $b+5$ to get $b\lt a+5 \lt 10+b$ b... | We have that $$|a-b| \leq 5$$
It stands that $$||a|-|b|| \leq |a-b| \\ \Rightarrow -|a-b| \leq |a|-|b| \leq |a-b|$$
From the inequalities $-|a-b| \leq |a|-|b|$ and $|a-b| \leq 5$ we get $$|b|-|a| \leq |a-b| \leq 5 \Rightarrow |b|-|a| \leq 5 \Rightarrow |b| \leq |a|+5$$
| {
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No definite integrals of trigonometry I have big problems with the following integrals:
$$\int\frac{dx}{\sin^6 x+\cos^6x}$$
$$\int\frac{\sin^2x}{\sin x+2\cos x}dx$$
It isn't nice of me but I almost have no idea, yet I tried the trigonometric substitution $\;t=\tan\frac x2\;$ , but I obtained terrible things and can't d... | HINT:
For the second one, as $\sin x+2\cos x=\sqrt5\sin\left(x+u\right)$ where $u=\arcsin\dfrac2{\sqrt5}\implies \sin u=\dfrac2{\sqrt5},\cos u=+\sqrt{1-\left(\dfrac2{\sqrt5}\right)^2}=\dfrac1{\sqrt5}$
let $x+u=y\iff x=\cdots$
$\sin^2x=\dfrac{1-\cos2x}2=\dfrac{1-\cos2\left(y-u\right)}2$
$\cos2\left(y-u\right)=\cos2y\cos... | {
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Is the set of fixed points an algebraic variety? Let $V$ be a finite dimensional $\mathbb{C}-$vector space. The linear action of its automorphism group $GL(V)$ on $V$ induces an action on the projective space $\mathbb{P}(V)$, i.e.
$$
GL(V) \times \mathbb{P}(V) \to \mathbb{P}(V), \ (A,[v] ) \mapsto [Av].
$$
I am tryin... | Let $A\in GL(V)$. Then $A$ has a finite number of eigenvalues, $\lambda_1,\ldots,\lambda_n$. For each of the $\lambda$'s, the equation you wrote is polynomial (of degree 1). Hence, the answer is yes.
As a matter of fact, we know what such a fixed locus looks like. A fixed point in $\mathbb{P}(V)$ corresponds to a $1$-d... | {
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Show that the quotient ring R/N has no non-zero nilpotent elements. An element $x$ in a ring $R$ is called nilpotent if $x^n=0$ for some $n\in \mathbb N$. Let $R$ be a commutative ring and $N=\{x\in R\mid \text{x is nilpotent}\}$.
(a) Show that $N$ is an ideal in $R$.
(b) Show that the quotient ring $R/N$ has no non-z... | You should explain what you tried for answering both these questions. Namely, the first one (a) is quite easy once you go back to the definition of an ideal.
The second one should not take much longer: assume that $x \in R/N$ is nilpotent, let $\widehat{x} \in R$ be an antecedent of $x$, and look at what the assertion ... | {
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Let G be a simple graph of order $n\geq 2$. If $|E(G)|>\binom{n-1}{2}$,then G is connected. Let G be a simple graph of order $n\geq 2$. If $|E(G)|>\binom{n-1}{2}$,then G is connected.
One of the solution I get is as shown as below:
Suppose G is not connected,
Then G is a disjoint union of two graphs $G=G_1 \cup G_2$ wh... | Note that $\binom{n-1}{2}=\binom{n}{2}-(n-1)$ and that $K_{n}$ is $n-1$-edge connected... meaning that in order to disconnect $K_{n}$ by deleting edges you must delete at least $n-1$ of them. Then since $|G|>\binom{n-1}{2}=\binom{n}{2}-(n-1)$ it follows that fewer than $n-1$ edges were deleted from $K_{n}$ to obtain $... | {
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Show that $L^{\infty}$ space does not have a countable dense set. I was able to show that when $p ≥ 1$, the $L^p$ space on the interval $[0,1]$ has a countable dense set.
However, when $p$ is infinite, how to prove that $L^p$ space on the interval $[0,1]$ does not have a countable dense set? I can't find some way to ap... | Consider all those elements $e_i$ whose terms are either $0$ or $1$ .They all belong to $L^\infty$ and they are uncountable having cardinality $c$
$||e_i-e_j||=1$
Now if we a countable dense set $D$ say then we should have for each $e_i$ an element $d_i$ such that $||e_i-d_i||<\epsilon $ for any $\epsilon $>0(take $\ep... | {
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behavior of function between two bounds Let $f, U, L : [0,1] \rightarrow \mathbb{R}$ be three functions with the property that
(1) U and L are continuous functions
(2) $\forall x \in [0,1]$, $L(x) \leq f(x) \leq U(x)$
(3) $f(0)=L(0)=U(0)=C$
(4) $L(x)$ and $U(x)$ are increasing.
We do not know whether $f$ is continuous.... | No Let $L(x)=x,U(x)=2x$, $f(x)=\frac{3}{2}x$ on rational number and $f(x)=\frac{4}{3}x$ on irrational number
| {
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$\sqrt{\frac{h^{2}}{4}+d^{2}-h d \cos x}-\sqrt{\frac{h^{2}}{4}+d^{2}+h d \cos x}$ equals $h\cos x$? Trying to simplify the expression, I observed: $y=\sqrt{\frac{h^{2}}{4}+d^{2}-h d \cos x}-\sqrt{\frac{h^{2}}{4}+d^{2}+h d \cos x}$ graphically equals $y=h\cos x$ when pluging in arbitrary values of $h$ and $d$. The resu... | I'm going to take $y = \sqrt{\frac{h^{2}}{4}+d^{2}+h d \cos x}-\sqrt{\frac{h^{2}}{4}+d^{2}-h d \cos x}\,\,\,\,$ and consider only positive values for $d$ and $h$ (it's identical to what you put into wolfram alpha, the $y$ in your question looks more like $- \cos$).
Now, $y$ and $\cos$ are not equal. Try adding $- \cos(... | {
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Is $f(x) = \sum^{\infty}_{n=1} \sqrt{x} e^{-n^2 x}$ continuous?. Where is bluff? I have a function defined by $f(x) = \sum^{\infty}_{n=1} \sqrt{x} e^{-n^2 x}$. The task is to check, whether $f(x)$ is continuous at $x = 0$. I have proposition of a solution and I would like someone to point out a bluff as there most like... | Note the summation when $x=0$ is infinite since it is an infinite sum of 1's, thus contradicting your argument of finite $M$.
| {
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Is $\exp(x)$ the same as $e^x$? For homework I have to find the derivative of $\text {exp}(6x^5+4x^3)$ but I am not sure if this is equivalent to $e^{6x^5+4x^3}$ If there is a difference, what do I do to calculate the derivative of it?
| While both expressions are generally the same, $\exp(x)$ is well-defined for a really large slurry of argument domains via its series: $x$ can be complex, imaginary, or even quadratic matrices.
The basic operation of exponentiation implicated by writing $e^x$ tends to have ickier definitions, like having to think about... | {
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Convergence of Cesàro means for a monotonic sequence If $(a_n)$ is a monotonic sequence and
$$
\lim_{n \to \infty} \frac{a_1 + a_2 + \cdots + a_n}{n}
$$
exists and is finite, does $a_n$ converge? If so, does it converge to the same limit?
I claimed that this was true in an old answer of mine. I think I had convinced ... | A monotonic sequence $\{a_n\}_{n=1}^\infty$ tends to some limit $A\in[-\infty,+\infty]$. If $a_n\to\text{some finite number}$ then $\dfrac{a_1+\cdots+a_n}n\to\text{that same number}$. So the only alternative (assuming, with no loss of generality, that it's nondecreasing) is
$$
a_n \to +\infty\quad \text{ and } \quad ... | {
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Find the coordinates of center for the composition of two rotations The combination of a clockwise rotation about $(0, 0)$ by $120◦$
followed by a clockwise rotation
about $(4, 0)$ by $60◦$
is a rotation. Find the coordinates of its center and its angle of rotation.
Here is my work so far:
$120◦+60◦=180◦$ which is not ... | What you wrote about the angle is a bit strange, you are saying that $a/2+b/2=(a+b)/2$... It's better to say something like , what would the tranformation look like far far away from these two points...
Of course this assumes that you know this theorem that a composition of two rotations is always a rotation. However, ... | {
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Prove: R∩R−1 is symmetric. The problem that I'm having is proving it - obviously. The only context that I am provided with is: "Prove: R∩R−1 is symmetric."
If (x,y) ∈ R then (y,x) ∈ R−1, and since it's the intersection, whatever elements are in the intersection must have both (x,y) and (y,x); making it symmetric, but h... | Choose any $(x, y) \in R \cap R^{-1}$. To show that $R \cap R^{-1}$ is symmetric, it suffices to show that $(y, x) \in R \cap R^{-1}$. Indeed, since $(x, y) \in R \cap R^{-1}$, we know that $(x, y) \in R$ and $(x, y) \in R^{-1}$. The former implies that $(y, x) \in R^{-1}$ and the latter implies that $(y, x) \in R$. He... | {
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Exponential of a matrix always converges I am trying to show that the exponential of a matrix converges for any given square matrix of size $n\times n$:
$M\mapsto e^M$ e.g. $\displaystyle e^M = \sum_{n=0}^\infty \frac{M^n}{n!}$
Can I argue that: Since $n!$ necessarily grows faster than $k^n$ will, that this converges. ... | This topic is extraordinarily well explained in the book Naive Lie Theory. Here is an extract that will answer your question.
| {
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How to compare these numbers How to compare $ 3^{\pi}$ and $\pi^{3}$
I think this can be done by taking the function $f(x)=x^{\frac{1}{x}}$ where $x>0$, on taking first derivative i got $f^1(x)=\frac{x^{\frac{1}{x}}}{x^2}({1+log(x)})$
when $f^1(x)$ equated with zero i got $x=\frac{1}{e}$.
what i do want to know is whe... | You choose the proper function for this study but I think that you made a small mistake sincd the derivative of $f(x)=x^{\frac{1}{x}}$ is $$f'(x)=x^{\frac{1}{x}} \left(\frac{1}{x^2}-\frac{\log (x)}{x^2}\right)$$ (to obtain this result, logarithmic differentiation makes life much easier).
So, the derivative cancels at ... | {
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"timestamp": "2023-03-29T00:00:00",
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How simplify and expand work in Maple When I do:
expand(sin(5*u)) (1)
The output is:
16*sin(u)*cos(u)^4 - 12*sin(u)*cos(u)^2 + sin(u) (2)
If I want it to give me an expression with merely sin(u) in it, I would do:
simplify(expand(sin(5*u)), [cos(u)^2 = 1 - sin(u)^2]) (3)
... | If you only wish to substitute tan(u)=sin(u)/cos(u) then use either the 2-argument form of the eval command (or the subs or algsubs commands).
expr := expand(tan(4*u));
3
4 tan(u) - 4 tan(u)
expr := ------------... | {
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Function on a Power Set Let $f\colon \mathcal{P}(A)\mapsto \mathcal{P}(A)$ be a function such that $U \subseteq V$ implies $f(U) \subseteq f(V)$ for every $U, V \in \mathcal{P}(A)$. Show there exists a $W \in \mathcal{P}(A)$ such that $f(W) = W$.
This is what I've been thinking:
Notice $A \subseteq A$ therefore $f(A) \... | This is the Knaster-Tarski theorem, actually. Let me give you a hint forward.
And you're essentially on the right track, but instead of constructing it transfinitely, what happens when you look at all the sets $\{B\mid f(B)\subseteq B\}$? What would their intersection be?
| {
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An example of a space $X$ which doesn't embed in $\mathbb{R}^n$ for any $n$? Apologies if this has been asked before, but couldn't find it.
The definition of embedding that I'm using is this:
Suppose $X$ and $Y$ are topological spaces. We call a function
$f:X\rightarrow Y$ an embedding if $f$ is a homeomorphism from... | consider $X = \mathbb{R}$ with discrete metric space...then $X$ cannot be embedded in $\mathbb{R^n}$ for all $n$...since $f(X)$ would be an discrete set of $\mathbb{R^n}$..but any discrete set can atmost be countable in $\mathbb{R^n}$.
| {
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"timestamp": "2023-03-29T00:00:00",
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Assuming the axiom of choice ,how to prove that every uncountable abelian group must have an uncountable proper subgroup? Assuming the axiom of choice , how to prove that every uncountable abelian group must have an uncountable proper subgroup ? Related to Does there exist any uncountable group , every proper subgroup... | Yes. Assuming the axiom of choice the answer is positive. You can find the proof in W.R. Scott's paper:
Scott, W. R. "Groups and cardinal numbers." Amer. J. Math. 74, (1952). 187-197.
The axiom of choice is used there for all manner of cardinal arithmetics.
Without the axiom of choice it is no longer necessary that ... | {
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A dense subspace of L^2 Let $\mathcal{H}$ be the Hilbert space of holomorphic functions defined on the unit disc $D\subset\mathbb{C}$ which is the clousure of the complex polynomial functions on the disc with respect to the inner product given by
$\langle f(x),g(x)\rangle:= \int_0^{2\pi}f(e^{i\theta})\overline{g(e^{i\t... | Polynomials are dense in $\mathcal H$. So it suffices to show that every
power $z^m$ can be approximated uniformly on $D$ by linear combinations of $1/(z-n)$. Use induction on $m$.
First of all, consider $m=0$.
$$ \eqalign{-\dfrac{n}{z-n} &= 1 + \sum_{j=1}^\infty \dfrac{z^j}{n^j} = 1 + Q_n(z) \cr
\left|Q_n(z)\r... | {
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Number of real embeddings $K\to\overline{\mathbb Q}$
How many real embeddings, $K\to\overline{\mathbb Q}$ with $K=\mathbb Q\left(\sqrt{1+\sqrt{2}}\right)$ are there ?
We set $f(x)=x^4-2x^2-1$ and if $\alpha=\sqrt{1+\sqrt{2}}$ then $f(\alpha)=0$.
Hence $\mathbb Q\left(\sqrt{1+\sqrt{2}}\right)=\mathbb Q[x]\big/(f(x))$... | We note that an embedding is a non-trivial homomorphism (therefore injective as the domain is a field)
$$\varphi:\Bbb Q[x]/((x^2-1)^2-2)\to\Bbb C.$$
Where we call it "real" if the image is contained in $\Bbb R$. However, if $\beta$ denotes a choice of root of $f(x)=(x^2-1)^2-2$, then as
$$\Bbb Q(\beta)=\{a+b\beta+c\be... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1227392",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Beta transformation is Ergodic. Let $\beta \in \mathbb{R}$ with $\beta >1$. Define $T_{\beta}:[0,1)\to [0,1)$ by:
$$T_{\beta}(x)=\beta x-[\beta x]=\{\beta x\} $$.
Consider:
$$ h(x)=\sum_{n=0}^{\infty}{\chi_{\{y:y<T_{\beta}^n(1)\}}(x)}$$
Show that the map $T_{\beta}$ preserves the measure $\mu$ defined by:
$$\mu(A)=\int... | Are you sure that $h$ looks exactly like this? I would expect $\chi$ to have coefficients like $\beta^{-n}$ which would help convergence. Also if $1$ has finite orbit (is periodic) you have to take a finite sum.
Proof of the invariance is a straightforward check using Perron-Frobenius operator. Again using PF operator... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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Expressing ${}_2F_1(a, b; c; z)^2$ as a single series Is there a way to express
$${}_2F_1\bigg(\frac{1}{12}, \frac{5}{12}; \frac{1}{2}; z\bigg)^2$$
as a single series a la Clausen? Note that Clausen's identity is not applicable here.
| Using Maple, I get
$$
\sum_{n=0}^\infty \frac{\Gamma \left( {\frac {7}{12}} \right) \Gamma \left( {\frac {11}{12}}
\right)
{\mbox{$_4$F$_3$}(\frac{1}{12},{\frac {5}{12}},-n,\frac{1}{2}-n;\,1/2,-n+{\frac {7}{12}},-n+{\frac {11}{12}};\,1)\;(4z)^n }}
{16\,\Gamma \left( -n+{\frac {11}{12}} \right) \Gamma \left( -n+{
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1227621",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 1,
"answer_id": 0
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If Riemann integrable, then it has finite number of discontinuities I know that any bounded function with a finite number of discontinuities is Riemann integrable over some interval. Is vice versa i.e.,
If a bounded function is Riemann integrable, then it has a finite number of discontinuities?
Thanks.
| No, for instance, the function
$$f(x) = \left\{ \begin{array}{ll}
e^{-x^2}, & x \notin \mathbb{Z}, \\
2, & x \in \mathbb{Z}
\end{array}\right.$$
Has countably many discontinuities, but is Riemann integrable over $\mathbb{R}$.
What is true is that a function is Riemann integrable iff the set of discontinuties is a set o... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1227713",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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Why does the residue method not work straight out of the box here? I'm trying to evaluate the integral $$I = \int_0^{\infty} \frac{\cos(x)-1}{x^2}\,\mathrm{d}x $$ The way I've done this is by rewriting $\frac{\cos(x)-1}{x^2}$ as $\Re\left[\frac{e^{iz}-1}{z^2}\right]$, and then using the residue method to get $$I = \Re\... | $\newcommand{\bbx}[1]{\,\bbox[15px,border:1px groove navy]{\displaystyle{#1}}\,}
\newcommand{\braces}[1]{\left\lbrace\,{#1}\,\right\rbrace}
\newcommand{\bracks}[1]{\left\lbrack\,{#1}\,\right\rbrack}
\newcommand{\dd}{\mathrm{d}}
\newcommand{\ds}[1]{\displaystyle{#1}}
\newcommand{\expo}[1]{\,\mathrm{e}^{#1}\,}
\new... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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"answer_id": 3
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Find eigenvalues for $T(f) = \int_{-\infty}^x tf(t)dt$ Let $V$ be the linear space of all functions continuous on $(-\infty, \infty)$ and such that that the integral $\int_{-\infty}^x tf(t)\,dt$ exists. If $f \in V$, let $g=T(f)$ be defined as $g(x) = \int_{-\infty}^x tf(t)\,dt$. Prove that every $\lambda < 0$ is an ei... | as cameron suggests, take the equation $$ \int_{-\infty}^x tf(t)\, dt = \lambda f(x) \tag 1$$
first, we will deal with the case $\lambda = 0.$
diffrentiang $(1)$ gives $xf(x) = 0$ implying $f \equiv 0$ contradicting that $f$ is an eigenfunction.
now we are assuming $\lambda \neq 0.$
differentiating $(1),$ we get $$ xf(... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How many non-negative integral solutions? How many non-negative integral solutions does this equation have?
$$17x_{17}+16x_{16}+ \ldots +2x_{2}+x_1=18^2$$
I add some conditions that bring more limitations:
$$\sum_{i=1}^{17}x_{i}=20 \quad 0 \leq x_{i} \leq 18$$
I did some calculation with them but no succeed;
do we hav... | The number of ways is the coefficient of $x^{324}$ in
$$
\begin{align}
&\left(x+x^2+x^3+\dots+x^{17}\right)^{20}\\
&=x^{20}\left(\frac{1-x^{17}}{1-x}\right)^{20}\\
&=x^{20}\sum_{k=0}^{20}\binom{20}{k}\left(-x^{17}\right)^k\sum_{j=0}^\infty\binom{-20}{j}(-x)^j\\
&=x^{20}\sum_{k=0}^{20}\binom{20}{k}\left(-x^{17}\right)^k... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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What percentage of numbers is divisible by the set of twin primes? What percentage of numbers is divisible by the set of twin primes $\{3,5,7,11,13,17,19,29,31\dots\}$ as $N\rightarrow \infty?$
Clarification
Taking the first twin prime and creating a set out of its multiples : $\{3,6,9,12,15\dots\}$ and multiplying by ... | As an addendum to mjqxxxx's excellent answer, I present a different approach which offers a minor improvement in accuracy (although a difference of $\approx 2\%$ is sufficiently large to be notable, considering how slowly the product converges at large $N$).
Let $\mathcal {P} (\mathbb{P}_2) $ represent the power set of... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
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Convergence in distributiom I need to show that for arbitrary random variables $X_n$, there exist a sequence of positive constants $a_n$ such that $a_nX_n\overset{D}\rightarrow 0$.
Thus, I need to show that $\lim_{n\rightarrow \infty} P(a_nX_n\leq x)=\begin{cases} 0\text{ if } x<0\\1 \text{ if } x>0\end{cases}$ or at l... | Since $\lim_{x\rightarrow+\infty}P\left(\left|X_{n}\right|\leq x\right)=1$ you can choose positive $a_{n}$ such that: $$P\left(\left|X_{n}\right|\leq\frac{a_{n}^{-1}}{n}\right)>1-\frac{1}{n}$$
or equivalently: $$P\left(a_{n}\left|X_{n}\right|\leq\frac{1}{n}\right)>1-\frac{1}{n}$$
Based on that for every fixed $x>0$ it ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Algorithm - Circle Overlapping Say you have a shape you want to fill up with circles, where by the circles overlap just enough to cover the whole surface area of the shape. The circles will remain as a fixed size however the shape they fill may change. Is there some sort of mathematical formula that could achieve this ... | This is a somewhat complex problem. You want to minimize the number of circles while still having the overlap. If you settle for good, but not perfect solutions, you can write algorithms to find good solutions. For instance genetic algorithms will "evolve" solutions as your shape evolves. However, if you want to do any... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1228538",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Why is the negation of $A \Rightarrow B$ not $A \Rightarrow \lnot B$? The book I am reading says that the negation of "$A$ implies $B$" is "$A$ does not necessarily imply $B$" and not "$A$ implies not $B$". I understand the distinction between the two cases but why is the first one considered true?
| Here is a more intuitive explanation. Suppose that $A$ and $B$ are unrelated. For example, $A$ could be "France is a country in Europe" and $B$ could be "I will win the lottery". It is certainly the case that we know $A$ does not imply $B$ for these sentences: knowing that France is in Europe tells me nothing about the... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Over-determined and Under-determined systems How do I show that a system is both over-determined and under-determined? I am supposed to come up with a matrix that satisfies both but am not really sure I understand what types of equations would satisfy these criteria. If anyone could give me an example and maybe a forma... | I believe that, as pointed out in Overdetermined and underdetermined systems of equation put simply, thinking of the equations in a system making up a set of requests (equations) to a certain number of people (unknowns) is helpful to understand why systems can be overdetermined or underdetermined in the first place.
In... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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integral over a subset of $\mathbb {R}^2$ is not defined while... consider the function $f(x,y)=\frac{xy}{(x^2+y^2)^2}$, we can see by some easy calculation that $\int_{-1}^1\int_{-1}^1 f(x,y)\,dx\, dy$ and $\int_{-1}^1\int_{-1}^1 f(x,y)\,dy\, dx$ exist and equals $0$.
but the function is not integrable over the square... | Note that $f(x,y)$ is homogeneous of order $-2$. That is $f(ax,ay)=a^{-2}f(x,y)$. If $f$ is not identically $0$, then the integral of $|f|$ over a unit circle is $I\ne0$. In fact,
$$
\begin{align}
I
&=\int_0^{2\pi}|\cos(\theta)\sin(\theta)|\,\mathrm{d}\theta\\
&=4\int_0^{\pi/2}\frac12\sin(2\theta)\,\mathrm{d}\theta\\
&... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1228805",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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What is the sum of this series involving factorial in denominator? $$1 + \frac{1^2 + 2^2}{2!} + \frac{{1}^2 + {2}^2 + 3^2}{3!} + \cdots$$
I can't figure out how to do summations which involve a factorial term in the denominator. Please help.
This is a past year IITJEE question by the way.
| HINT:
$$\sum_{r=1}^n\dfrac{1^2+2^2+\cdots+r^2}{r!}=\frac16\sum_{r=1}^n\dfrac{r(r+1)(2r+1)}{r!}=\frac16\sum_{r=1}^n\dfrac{r(r+1)(2r+1)}{r!}$$
Now for $r>0,$ $$\dfrac{r(r+1)(2r+1)}{r!}=\dfrac{(r+1)(2r+1)}{(r-1)!}$$
Let $(r+1)(2r+1)=2(r-1)(r-2)+a(r-1)$
Set $r=2$ to get $a$
Now, $$e^x=\sum_{r=0}^\infty\dfrac{x^u}{u!}$$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How to calculate $\lim_{n \to \infty}\sqrt[n]{\frac{2^n+3^n}{3^n+4^n}}$ I came across this strange limit whilst showing convergence of a series:
$$\lim_{n \to \infty}\sqrt[n]{\frac{2^n+3^n}{3^n+4^n}}$$
How can I calculate this limit?
| Squeeze theorem gives you the proof that the limit is $\frac{3}{4}$. Since you mentioned you were looking for another way to verify that the limit is correct, here is one way (although not rigorous like the squeeze theorem) $$\begin{align}\frac{2^n+3^n}{3^n+4^n} = \frac{2^n}{3^n+4^n}+\frac{3^n}{3^n+4^n} \\ = \left(\fra... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1229117",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 5,
"answer_id": 3
} |
What is the _simplest_ way to solve problems of this kind? Two doors with talking doorknockers - one always tells the truth and one always lies. One door leads to death other to escape. Only one question may be asked to either of the door knockers. What would that question be?
Given hint says that the question ought to... | Let $L$ be the proposition that the left door leads to escape.
You ask a question $Q$ (some proposition). Let $R$ be the truth-telling/lying status of the person you ask (so true if the person is a truth-teller, false if the person is a liar). The response from this person is the truth value of $Q \Leftrightarrow R$.
S... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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What do these notations mean, if we read those in English? If m: message, M: message space, k: key, K: keyspace, c: cipher, C: cipher space and $E_k$: encryption function, such that
$E_k(m) = c,\ m,m^* \in M,\ k\in K,\ c\in C.$
Then, what do the following notations actually mean in plain English?
\begin{eqnarray*}P[m=m... | The first reads, "given that the encryption function $E_k$ using key $k$ applied to message $m$ returns the cipher $c$, the probability that two messages, $m$ and $m^∗$, are equal is $\frac{1}{|M|}$."
The second line reads "The probability that two messages, $m$ and $m^∗$ are the same is equal to the probability that ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Proving an equivalent statement for the Stone-Weierstrass theorem In my metric space course, we were taught the Stone-Weierstrass theorem as follows
We were told however that the second condition ("A contains the constant functions") may be replaced by the condition "A contains the constant function 1". Is someone ab... | This is false. Consider the algebra generated by the functions $1$ and $x+1$ on $[0, 1]$. This algebra separates points because it contains the function $x+1$, yet every function $f$ in this algebra satisfies $f(x)\geq 1$ (because the generators $1$ and $x+1$ do, and this property is preserved under multiplication and ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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How to integrate $\int \frac{x^{\frac{k}{2}-1}}{1+x^k}dx$ How would I do the following integral?
$$\int \frac{x^{\frac{k}{2}-1}}{1+x^k}dx$$
Where $x > 0$ and $k$ is a constant greater than $0$
| Consider the integral
\begin{align}
I = \int \frac{x^{\frac{k}{2}-1}}{1+x^k}dx
\end{align}
Let $t = x^{k/2}$ for which $x = t^{2/k}$ and $dx = (2/k) t^{(2/k) - 1} \, dt$ for which the integral becomes
\begin{align}
I = \frac{2}{k} \int \frac{dt}{1 + t^{2}}.
\end{align}
This is the integral for $\tan^{-1}(t)$ leading t... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Solve $\frac{|x|}{|x-1|}+|x|=\frac{x^2}{|x-1|}$ Solve $\frac{|x|}{|x-1|}+|x|=\frac{x^2}{|x-1|}$.What will be the easiest techique to solve this sum ?
Just wanted to share a special type of equation and the fastest way to solve it.I am not asking for an answer and i have solved it in my answer given below.Thank You for ... | A shortcut formula for such sums is if $|f(x)|+|g(x)|=|f(x)+g(x)|$ then $f(x).g(x)>0$ then $[\frac{x}{x-1}][x]>=0$ which implies $x^2(x-1)>=0$.But $x^2$ is always >=0.Hence $x>1$ is the solution as well as x=0.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1229647",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 1
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Inner product on random variables
Let $(\Omega, \mathscr{F}, P)$ be a probability space and let $L^2$ denote the space of real-valued, discrete random variables with finite variance that map $\Omega$ to a set $Q$.
Define $\langle\cdot,\cdot\rangle:\Omega \to \mathbb R$ such that $\langle X,Y\rangle=E[XY]$
Is $(L^2,\la... | This is a good observation. The distinction here is that the elements of $L^2$ are not actually functions, but equivalence classes of functions. In this case, the zero element of $L^2$ is
$$\{X\in L^2 : \mathbb P(X=0)=1\}. $$
As $\langle X,X\rangle=0$ implies that $\mathbb P(X=0)=1$, positive-definiteness holds.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1229830",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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limit of sequence of quotients of sequence that converges Let
$$\lim_{n\to \infty}x_n=a$$
Prove that if
$$\lim_{n\to \infty}{x_{n+1}\over x_n}=L$$
so
$$|L|\le1$$
....
I tried for a long time but i can't prove that. please give me just a hint?
thanks
| Hint: Use argument by contradiction and the definition of the limit. Suppose $|L|>1$. For $\varepsilon_0=|L|-1>0$, $\lim_{n\to\infty}\big|\frac{x_{n+1}}{x_n}\big|=|L|>1$ implies that there is $N>0$ such that
$$ \big|\frac{x_{n+1}}{x_n}\big|>|L|-\varepsilon_0=1 \text{ whenever }n\ge N. $$
Next show $\{|x_n|\}$ is incre... | {
"language": "en",
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Is there a relationship between the existence of parallel vectors on two planes, and their line of intersection. Let me state the context first:
I have a question which goes like this:
We have two planes:
$$\pi_1 : r = (2,1,1)^{\top} + \lambda(-2,1,8)^{\top} + \mu(1, -3, -9)^{\top}$$
$$\pi_2 : r = (2,0,1)^{\top} + s(1,... | Yes, your conjecture is true. The crucial fact of note here is that a plane is an affine subspace - that is to say, it satisfies the following definition:
A subset $S$ of a vector space is an affine subspace if, for any pair of vectors $s_1,s_2\in S$ and any pair of numbers $a$ and $b$ with $a+b=1$, it holds that $as_... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Evaluate $\int \frac{dx}{1+\sin x+\cos x}$ Evaluate $$\int \frac{1}{1+\sin x+\cos x}\:dx$$
I tried several ways but all of them didn't work
I tried to use Integration-By-Parts method but it's going to give me a more complicated integral
I also tried u-substitution but all of my choices of u didn't work
Any suggestions?... | $$\int \frac{1}{1+\sin x+\cos x}\:dx\stackrel{t=\tan(x/2)}=\int\frac{dt}{1+t}=\ln |1+t|+c=\ln|1+\tan(x/2)|+c$$
As $dt=\frac12\sec^2(x/2)dx\implies 2dt=(1+\tan^2(x/2))dx\implies 2dt=(1+t^2)dx$
where:
$$\frac{1}{1+\sin x+\cos x}=\frac{1}{1+\frac{2t}{1+t^2}+\frac{1-t^2}{1+t^2}}=\frac{1+t^2}{1+t^2+2t+1-t^2}=\frac{1+t^2}{2... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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If $K:=\mathbb Q\left(\sqrt{-3}\right)$ and $R$ is the ring of integers of $K$, then $R^{\times}=\mathbb Z\big/6\mathbb Z$
How to show that if $K:=\mathbb Q\left(\sqrt{-3}\right)$ and $R$ is the ring of integers of $K$, then the group of units $R^{\times}=\mathbb Z\big/6\mathbb Z$
Now since $-3\equiv1\mod 4$ the ring... | Clearly $R^{\times}$ is an abelian group and you just found out that it has order $6$. But the only abelian group of order $6$ is the cyclic group on $6$ elements...
Indeed, the fundamental theorem of finitely generated abelian groups and $\#R^{\times} = 6$ imply that $R$ is a direct sum of primary cyclic groups. Sinc... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Scalar versus linear equation of a plane What is the difference between a scalar and a linear equation of a plane? In my textbook it says that a scalar equation is $a(x-x_1)+b(y-y_1)+c(z-z_1)=0$
and a linear equation is $ax+by+cz=d$
How do they differ in terms of what they describe?
If a line on the plane dot the norma... | The two equations describe precisely the same sets. Suppose that a non-zero vector $n = (a, b, c)$ and a point $p_{1} = (x_{1}, y_{1}, z_{1})$ are given, and let $p = (x, y, z)$ denote an arbitrary point of $\mathbf{R}^{3}$.
Expanding the "scalar" equation gives
\begin{align*}
0 &= a(x - x_{1}) + b(y - y_{1}) + c(z - z... | {
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Integration of Fundamental Solution of Laplace's equation. I am currently reading Evan's PDE and am getting hung up on many of the more "technical details". This question may be very basic (multivariable calculus). I am given that the fundamental solution of Laplace's equation is $$ \Phi(x) := \begin{cases} -\frac{1}{2... | Change to polar coordinates?
For $n \geq 3$, note
$$ \int_{B(0,\epsilon)} \frac{1}{|x|^{n-2}} \mathrm{d}x = C_n \int_0^\epsilon r^{2-n} r^{n-1} \mathrm{d}r = C_n \int_0^\epsilon r \mathrm{d}r = \frac{1}{2} C_n \epsilon^2 $$
where $C_n$ is the are of the unit sphere $\mathbb{S}^{n-1}\subset \mathbb{R}^n$.
For $n = 2$ ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1230459",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 1,
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Residue theorem Let us say we need to perform the classic integral
$$
I=\int_{-\infty}^{+\infty}dz \,\frac{e^{itz}}{z^2+1}~,
$$
where $t>0$.
What is normally done is the following. We consider the integral
$$
K=\oint dz \,\frac{e^{itz}}{z^2+1}
$$
with the contour closed on the positive imaginary part of the complex pl... | The function $\Im[z]$ is not analytic, since it does not satisfy Cauchy–Riemann equations. Therefore $e^{-\Im[z]}$ is also not analytic. The Cauchy formula cannot be applied and the second method is thus wrong. This solves the discrepancy. Finally, I thank @wisefool for his comment.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Demand $z=x+y$ and $x^2/4 + y^2/5 + z^2/25 = 1$. What is the maximum value of $f(x,y,z) = x^2+y^2+z^2$?
Demand $z=x+y$ and $x^2/4 + y^2/5 + z^2/25 = 1$. What is the maximum value of $f(x,y,z) = x^2+y^2+z^2$?
I've been attempting this with Lagrange multipliers in a few different ways. However, the resulting system of ... | You have two equations in three unknowns, so just have to choose one variable to maximize over. When you eliminate $z$ you have to do it from the second constraint as well as the objective function. Your problem becomes to maximize $2x^2+2y^2+2xy$ subject to $x^2/4+y^2/5+(x+y)^2/25=1=\frac {29}{100}x^2+\frac 6{25}y^2... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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A fair die is rolled three times, find the probability of the following events: a. All rolls show an even number of dots
b. the last two rolls show an even number of dots
c. the third roll shows an even number of dots
d. every roll shows a single dot
e. every roll shows the same number of dots
what Ive done so far:
I k... | Hint A: What's the probability of rolling an even, $P(even)$? You want this to happen all three times.
Hint B: We don't care about the first roll.
Hint C: We don't care about the first two rolls.
Hint D: What's the probability of rolling a 1, $P(1)$? You want this every time.
Hint E: The first roll doesn't matter. You ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Find the number of positive integer solutions such that $a+b+c\le 12$ and the shown square can be formed. Find the number of positive integer solutions such that $a+b+c\le 12$ and the
shown square can be formed.
$a \perp b$ and $b\perp c$.
the segments $a,b,c$ lie completely inside the square as shown.
Here is my at... | Here is a start.
Extend the line
of length $a$
by an amount $c$
and then draw a line
from the end of that
to the corner of the square
that ends the line of length $c$.
This forms a right triangle
with sides
$a+c$ and $b$
whose hypotenuse is the
diagonal of the square.
This length is
$\sqrt{(a+c)^2+b^2}$,
so the side of... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1230953",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Show that $2^{105} + 3^{105}$ is divisible by $7$ I know that $$\frac{(ak \pm 1)^n}{a}$$ gives remainder $a - 1$ is n is odd or $1$ is n is even.
So, I wrote $ 2^{105} + 3^{105}$ as $8^{35} + 27^{35}$ and then as $(7\cdot 1+1)^{35} + (7\cdot 4-1)^{35}$, which on division should give remainder of $6$ for each term and t... | Using Little Fermat, we have:
$$2^{105} +3^{105}\equiv 2^{105\bmod 6} +3^{105\bmod 6}\equiv 2^{3}+ 3^{3}\equiv 1 + 6\equiv 0 \mod 7.$$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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Solving $y'(x)-2xy(x)=2x$ by using power series I have a first order differential equation:
$y'(x)-2xy(x)=2x$
I want to construct a function that satisfies this equation by using power series.
General approach:
$y(x)=\sum_0^\infty a_nx^n$
Differentiate once:
$y'(x)=\sum_1^\infty a_nnx^{n-1}$
Now I plug in the series i... | it may be easier to see what is going on if you $\bf don't $ use the sigma notation for the sums. here is how finding solution by series works. you assume the solution is of the form $$y = a_0 + a_1x + a_2 x^2 + a_3x^3 +\cdots\\y' = a_1 + 2a_2 x + 3a_3x^2 +\cdots $$ and sub in the differential equation $y' - 2xy = 2x.$... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Solving an inequality $BIs there a way to solve $B<n!$ where $B$ is some very large real number (suppose for example $B=10^{17}$) without a calculator or gamma function?
At the very least, to find the nearest integer for $n$ to make the inequality true?
| With a log table, you could do it by hand :
Compute $\ln(B)=17 \ln(10)$
Then you just have to sum the log of the numbers :
$$\ln(n!) = \sum_{k=2}^n \ln(k)$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1231308",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Stoppage time for sequence of uniform random numbers with a recursively shrinking domain Define $x_n = U(x_{n-1})$ where $U(x)\in\lbrace 0,1,\ldots,x\rbrace$ is a uniformly distributed random integer. Given $x_0$ as some large positive integer, what is the expected value of $n$ for which $x_n=0$?
The answer I came up w... | The expected waiting time $T_k$ to get down from $k$ to $0$ is $T_0 = 0$ for the base case, and otherwise it is $T_k = 1 + H_k$, where $H_k$ is the $k$th harmonic number $1 + 1/2 + 1/3 + \cdots + 1/k$. For large $k$ this is approximately $1 + \gamma + \ln k$, where $\gamma \doteq 0.57722$ is the Euler-Mascheroni const... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Solving the indefinite integral of a trig function I'd like to ask for some feedback on my calculation. Please let me know if you spot any mistakes in my technique:
$$\int{\frac{1}{\sqrt{x}}\sin^{-1}{\sqrt{x}}}\,\,dx$$
Using substitution:
$$u = \sqrt{x},\,\,\frac{du}{dx}=\frac{1}{2\sqrt{x}},\,\,du=\frac{1}{2\sqrt{x}}d... | Instead of memorizing some formula, you can use IBP to conclude your answer. That comes more intuitively to someone in my opinion.
Take $\arcsin(u)$ as the first function and $1$ as the second function. Now, using IBP,
$$I=\int\arcsin(u)\,\mathrm du=\left(\arcsin(u)\int\,\mathrm du\right)-\int\left(\frac{\mathrm d}{\ma... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1231596",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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proof related to convergence of a integral i have the following condition $$0\le f(x)\le g(x)$$ and $$\int_{a}^{b}g(x)dx$$ is convergent for any $a$ and $b$ (which means $a$ or $b$ can tend to infinity) then prove that $$\int_{a}^{b}f(x)dx$$ also is convergent. now i realize that as $\int_{a}^{b}f(x)dx$ will be bound... | Case 1
Assume $a$ and $b$ are real numbers.
Suppose that $f(x) =1$ when $x$ is irrational and $f(x) =0$ when $x$ is rational.
Take $g(x) =2$ for all $a\le x\le b$.
Then, clearly $0\le f \le g$ and $\int_a^b g(x)dx=2(b-a)$ is convergent.
But as a Riemann integral, $f$ is not integrable.
Case 2:
$b=\infty$.
Supp... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Finding the integral $\int_0^1 \frac{x^a - 1}{\log x} dx$ How to do the following integral:
$$\int_{0}^1 \dfrac{x^a-1}{\log(x)}dx$$
where $a \geq 0$?
I was asked this question by a friend, and couldn't think of any substitution that works. Plugging in a=2,3, etc in Wolfram, I get values like $\log(a+1)$, which may be t... | We can utilize
$$
\int_0^1x^t\,\mathrm{d}t=\frac{x-1}{\log(x)}
$$
combined with the substitution $x\mapsto x^{1/a}$, to get
$$
\begin{align}
\int_0^1\frac{x^a-1}{\log(x)}\,\mathrm{d}x
&=\int_0^1\frac{x-1}{\log(x)}x^{\frac1a-1}\,\mathrm{d}x\\
&=\int_0^1\int_0^1x^{\frac1a-1}x^t\,\mathrm{d}t\,\mathrm{d}x\\
&=\int_0^1\int_... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1231738",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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How to solve $\ln(x) = 2x$ I know this question might be an easy one. but it has been so long since I solved such questions and I didn't find a an explanation on the internet. I'd like if someone can remind me.
I reached that $e^{2x} = x$, but didn't know how to continue from here. I remember something that has to do ... | Draw a graph. $\log x < 2x $
A proof is by noting that $\log x < 2x$ for $x < 1$ and then differentiating both sides to see that the LHS grows slower than the RHS.
Equivalently, $e^{2x} > x$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Can a non-constant analytic function have infinitely many zeros on a closed disk? I think not, however my proof is quite sketchy so far..
My attempt: Suppose an analytic function f has infinitely many zeros on some closed disk D. Then there exists a sequence of zeros in D with a limit point in D. Thus by the identity t... | Your proof is correct, you just need to realize that when you say "has infinitely many zeros" you mean "has infinitely many points where it evaluates to $0$", so one is not talking about multiplicities here. The importance of $D$ being a closed disk is that it is then compact, and that implies the existence of a conver... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1232039",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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Prove the field of fractions of $F[[x]]$ is the ring $F((x))$ of formal Laurent series. Prove the field of fractions of $F[[x]]$ is the ring $F((x))$ of formal Laurent series.
$F[[x]]$ is contained in $F((x))$. So there's at least a ring homomorphism that is injective. Can also see it's injective because the kernel o... | The field of fractions of an integral domain is the smallest field that the domain injects into. The homomorphism that sends a power series to itself is an injective homomorphism into $F((x))$, since every power series is also a Laurent series. If $F$ is the field of fractions of $F[[x]]$, then $f$ injects into $F((x))... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Tangent line of a lemniscate at (0,0) I need to find the tangent line of the function $y=g(x)$ implicitly defined by
$(x^2+y^2)^2-2a^2(x^2-y^2)=0$
at $(0,0)$, but I don't know how.
I can't use implicit differentiation and evaluate at $(0,0)$, because when $y=0$ I can't use the Implicit Function Theorem to calculate the... | $$(x^2+y^2)^2-2a^2(x^2-y^2)=0$$
Solving for $y$ we do substitution $t=y^2$
$$x^4+x^2t+t^2-2a^2x^2+2a^2t=0$$
$$t^2+t(2x^2+2a^2)+x^4-2a^2x^2=0$$
$$t=\pm a\sqrt{4x^2+a^2}-x^2-a^2$$
As $t=-a\sqrt{4x^2+a^2}-x^2-a^2$ is not positive we get solutions
$$y=\pm \sqrt{a\sqrt{4x^2+a^2}-x^2-a^2}$$
Let $f(x)=\sqrt{a\sqrt{4x^2+a^2}-x... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Trying to show that the set of all $2$-element subsets of a denumerable set is denumerable Suppose $A$ is denumerable and put $X = \{ B : B \subset A, \; \; |B| = 2 \} $. I want to show that $X$ is denumerable as well.
My try: Let $f$ be bijection from $\mathbb{N}$ to $A$.
We know any $B \in X$ is of the form $B = \{a... | As @YuvalFilmus mentioned,$2^k 3^r=1$ if and only if $k= r=0$.
Given $A$ is denumerable, and hence so is $A\times A$, since Cartesian product of denumerable sets is denumerable.
Now, if you notice, $X=\{\{x,y\} : x,y\in A\}$ is equivalent to a subset (say Y) of $A\times A$, removing from $A\times A$
*
*$(y,x)$ if $(... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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convert riemann sum $\lim_{n\to\infty}\sum_{i=1}^{n} \frac{15 \cdot \frac{3 i}{n} - 24}{n}$ to integral notation The limit
$ \quad\quad \displaystyle \lim_{n\to\infty}\sum_{i=1}^{n} \frac{15 \cdot \frac{3 i}{n} - 24}{n} $
is the limit of a Riemann sum for a certain definite integral
$
\quad\quad \displaystyle \int_a... | Here are some alternate answers:
1) Take $a=0, b=1$; using n equal subintervals and right endpoints as sampling numbers, we get that
$\hspace{.3 in}\displaystyle \lim_{n\to\infty}\sum_{i=1}^n \left(45\cdot\frac{i}{n}-24\right)\frac{1}{n}=\int_0^1(45x-24)\;dx$
2) Take $a=0, b=3$; using n equal subintervals and right end... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Sigmoid function that approaches infinity as x approaches infinity. The function I'm looking for looks like an error function, but instead of having asymptotes $1$ and $-1$, the function I'm looking for does not have asymptote. It increases to infinity.
The derivative of this function looks like a Gaussian function; it... | I think I got the answer.
I started by assuming the function $f(x)$ derivative $f'(x)$ kinda looks like the Gaussian function, and its double derivative $f''(x)$ looks like the original function $f(x)$.
$$f''(x)=f(x)$$
So I asked Wolfram|Alpha's help (yup I cheated) and the general solution is (ignoring constants)
$$f(... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1232583",
"timestamp": "2023-03-29T00:00:00",
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Injectivity of $T:C[0,1]\rightarrow C[0,1]$ where $T(x)(t):=\int_0^t x(s)ds $ Prob. 2.7-9 in Erwin Kreyszig's "Introductory Functional Analysis with Applications": Is this map injective?
Let $C[0,1]$ denote the normed space of all (real or complex-valued) functions defined and continuous on the closed interval $[0,1]$ ... | If $\parallel x\parallel =M$, then $$ |T(x)(t)|=\bigg|\int_0^t x(s)ds \bigg|
\leq \int_0^t M \leq M $$ Hence bounded.
And $\frac{d}{dt} T(x)(t)=x(t)$ is continuous. And $T(x)(0)=0$.
If $T(x)=T(y)$ then $
\parallel T(x)- T(y)\parallel =0$ So $$ \forall
t,\ \int_0^t (x-y)(s) ds =0
$$
Assume that $t_0\in (0,1)$ with $(x-y... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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If $A $ is a square matrix of size $n$ with complex entries such that $Tr(A^k)=0 , \forall k \ge 1$ , then is it true that $A$ is nilpotent ? If $A$ is a square matrix of size $n$ with complex entries and is nilpotent , then I can show that all the eigenvalues of $A^k$ , for any $k$ , is $0$ , so $Tr(A^k)=0 , \forall... | Yes, it is true. Let $\lambda_i, i=1,\ldots, n$ denote the eigenvalues of your matrix. Then $\sum \lambda_i^k=0, k\in \mathbb{N}^*.$ This implies that $\lambda_i=0$ for all $i=1,\ldots, n$.
Just found that it is a duplicate: Traces of all positive powers of a matrix are zero implies it is nilpotent
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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I've been told that writing $x\equiv a,b,c \pmod d$ is abuse of notation, is it really? I've been told that writing $x\equiv a,b,c \pmod d$ is abuse of notation, and that I should always write:
$$
x\equiv a\pmod {d}\text{ or }x\equiv b\pmod {d}\text{ or }x\equiv c\pmod {d}
$$
How true is this?
| The acceptability of any abuse of notation depends on whether the meaning is clear. For instance if you are asked to solve the equation $x^2 - 3x + 2=0$ and write $x=1,2$ I think most everyone will know what you mean, although to be precise you should write $x = 1$ or $x=2$, or possibly even $x \in \{1,2\}$.
At first g... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Isosceles trapezoid with inscribed circle The area an isosceles trapezoid is equal to $S$, and the height is equal to the half of one of the non-parallel sides. If a circle can be inscribed in the trapezoid, find, with the proof, the radius of the inscribed circle. Express your answer in terms of $S$ only.
I labeled th... |
Let the radius of the circle be $r$; then the height of the isosceles trapezoid is $2r$, and the length of a lateral side would be $4r$.
The four right triangles with $OB$ and $OC$ as hypotenuses are congruent. The four right triangles with $OA$ and $OD$ as hypotenuses are also congruent. Therefore the lengths marke... | {
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An example of a group with $1+np$ Sylow $p$-subgroups I want to find an example of a group $G$ with $1+p$ Sylow $p$-subgroups, where $p$ is a fixed prime. My problem is I don't know a lot of known Sylow subgroups, and the answer of this depends on $p$.
Then I'd like to know if this can be done easily for $1+np$.
Any su... | For $p=2$, the Klein's group $\{1,3,5,7\}$ (mod 8 multiplication) has
3 subgroups (they are the only proper subgroups).
For $p=3$, and $p=5$, the examples are $S_4$ and $S_5$ respectively.
| {
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What's the relationship between continuity property of Lebesgue measure and continuity on a metric space? This is a topic from Lebesgue measure in $\textit {Carothers' Real Analysis}$:
I know how to prove Theorem 16.23. However, I can not figure out why he names this property as continuity? Besides what's the relatio... | Not clearly. Continuity property of Lebesgue measure is just a nature term defined that measure of a limit is equal to limit of the measure. And with this, we can take limit in parentheses out (like continuity in a metric space that $f$ is continuous <=> $limit_{n->+∞} f(x_n) = f(limit_{n->+∞} x_n$) whenever $limit_{n-... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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How can you derive $\sin(x) = \sin(x+2\pi)$ from the Taylor series for $\sin(x)$? \begin{eqnarray*}
\sin(x) & = & x - \frac{x^3}{3!} + \frac{x^5}{5!} - \ldots\\
\sin(x+2π) & = & x + 2\pi - \frac{(x+2π)^3}{3!} + \frac{(x+2π)^5}{5!} - \ldots \\
\end{eqnarray*}
Those two series must be equal, but how can you show that by ... | A fairly easy way to introduce $\pi$ in trigonometric functions defined by series is:
*
*Define
$$e^z=\sum_{n=0}^{\infty} \frac{z^n}{n!}$$
Then use Euler's formula to define $\sin$ and $\cos$:
$$\sin z=\sum_{n=0}^\infty(-1)^n\frac{z^{2n+1}}{(2n+1)!}$$
$$\cos z=\sum_{n=0}^\infty(-1)^n\frac{z^{2n}}{(2n)!}$$
*Then ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Are all continuous one one functions differentiable? I was reading about one one functions and found out that they cannot have maxima or minima except at endpoints of domain. So their derivative , if it exists, must not change it sign , i.e. , the function should be either strictly increasing or strictly decreasing. F... |
$x^{1/3}$ is not differentiable at $0$. See its graph above. It's qualitatively different from the example given by 5xum.
The Cantor function $ +\, x$ is an example of a function that's continuous and one-to-one, but non-differentiable at uncountably many points.
There's a limit to how bad an example can get. The set ... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Proving algebraic equations with circle theorems
I got as far as stating that OBP=90˚ (as angle between tangent and radius is always 90˚), and thus CBO=90˚- 2x. CBO=OCB as they are bases in a isosceles. COB=180-90-2x-90-2x. But after this, i am clueless.
I am stuck with this Question. It is from a GCSE Further Maths p... | Angle BOD = 180 − y
Angle OCD = x
Angle OBC = 90 − 2x
Angle BCO= 90 − 2x
Angle BOD reflex = 360 - (90 − 2x) − (90 − 2x) − x − x = 180 + 2x.
180 − y + 180 + 2x = 360,
thus y = 2x
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1234168",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
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Expression for $dW$ for a 3D position dependent force $\vec{F}(\vec{r})$. I was looking at the derivation of the infinitesimal element of work done for a 3d position dependent force and I couldn't get over the switching of $\text{d}\vec{v}$ and $\text{d}\vec{r}$ in the third line and how the author went from the penult... | This is the well-known derivation of the kinetic energy formula. You'll find it easier to work in scalars initially to see what's happening - so let's make the assumption that the force is always in the direction of motion (thereby obviating the need for the dot products).
The derivation is a "shortcut" application of ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1234249",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Why $K = (X_1, X_2, ...)$, the ideal generated by $X_1, X_2, ...$ not finitely generated as a R-module?
Let $R = \mathbb{Z}[X_1, X_2, \dots]$ be the ring of polynomials in countably many variables over $\mathbb Z$. Why $K = (X_1, X_2, ...)$, the ideal generated by $X_1, X_2, ...$ is not finitely generated as an $R$-mo... | $\renewcommand{\phi}[0]{\varphi}$First note that it follows from the universal property of polynomial rings that for each $t$, there is a (unique) homomorphism of rings
$$
\phi_{t} : \mathbb{Z}[X_1, X_2, \dots] \to \mathbb{Z}[X_t, X_{t+1}, \dots]
$$
which maps an integer to itself, $X_{i}$ to zero, for $i < t$, and $X_... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1234351",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
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If the equation $|x^2+4x+3|-mx+2m=0$ has exactly three solutions then find value of m. Problem :
If the equation $|x^2+4x+3|-mx+2m=0$ has exactly three solutions then find value of $m$.
My Approach:
$|x^2+4x+3|-mx+2m=0$
Case I : $x^2+4x+3-mx+2m=0$
$\Rightarrow x^2+ x (4-m) + 3+2m=0 $
Discriminant of above qudratic ... | $$m(x-2)=|(x+3)(x+1)|\ge0$$
If $m=0,$ there are two real solutions
Else $m(x-2)=|(x+3)(x+1)|=0$ has no solution
So, $$m(x-2)=|(x+3)(x+1)|>0$$
Now $|(x+3)(x+1)|=-(x+3)(x+1)$ if $-3\le x\le-1$
$=+(x+3)(x+1)$ otherwise
If $m>0,x-2>0\iff x>2\implies m(x-2)=x^2+4x+3$ which has exactly two solutions
If $m<0,x-2<0\iff x<2$
If... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1234482",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 6,
"answer_id": 4
} |
What is $\left[\frac{1}{2}(p-1)\right]! \;(\text{mod } p)$ for $p = 4k+1$? Theorem #114 in Hardy and Wright says if $p = 4k+3$ then
$$ \left[\frac{1}{2}(p-1)\right]! \equiv (-1)^\nu \mod p$$
where $\nu = \# \{ \text{non residues mod } p\text{ less than }p/2\}$.
*
*Is there corresponding result for $p = 4k+1$?
... | In $\mathbb{F}_p^*$ there are exactly $\frac{p-1}{2}$ quadratic residues, and if $p\equiv 1\pmod{4}$, $-1$ is a quadratic residue, hence the quadratic residues are symmetrically distributed around $\frac{p}{2}$, so the number of quadratic residues less than $\frac{p}{2}$ is just $\frac{p-1}{4}$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1234680",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Ring homomorphism from $M_3(\mathbb{R})$ into $\mathbb{R}$ I was working on this problem
Find all ring homomorphisms from $M_3(\mathbb{R})$ into $\mathbb{R}$.
My attempt:-
I found that if we have any ring homomorphism $\phi$, then $\ker(\phi)$ should be either zero or the entire ring (since $M_3(\mathbb{R})$ is simp... | There is no injective ring homomorphism since every matrix of the form $AB-BA$ must be mapped into $0$. To conclude, it is well known that there exist some $AB-BA \neq 0$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1234820",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 2
} |
Prove that the greatest lower bound of $F$ (in the subset partial order) is $\cap F$. This is one of the question I'm working on:
Suppose $A$ is a set, $F \subseteq \mathbb{P(A)}$, and $F \neq
\emptyset$. Then prove that the greatest lower bound of $F$ (in the
subset partial order) is $\cap F$.
Now this is my att... | HINT: Show it directly from the definition of greatest lower bound. Suppose that $L$ is a lower bound for $F$. Then $L\subseteq X$ for each $X\in F$. What can you say about the relationship between $L$ and $\bigcap F$?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1234921",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
What are the three subgroups of $\mathbb{Z}_4\times\mathbb{Z}_6$ of 12 elements? My formatting didn't work in the title, here is the question again:
What are the three subgroups of $\mathbb{Z}_4\times\mathbb{Z}_6$ of 12 elements?
I know that this group does not have order 24 since $\gcd(4, 6) \ne 1$, but I am at a loss... | Think of homomorphisms $\mathbb Z_4 \times \mathbb Z_6 \to \mathbb Z_2$. If you choose them surjective, the kernel will have order $12$. Conversely, a subgroup of index $2$ is always normal, so corresponds to such an homomorphism.
A morphism $G \times H \to K$ corresponds to a pair of homomorphisms $G \to K$ and $H \t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1235004",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Question about $M/GI/ \infty $ queue Consider an $M/GI/ \infty $ queue with the following service time distribution: the service time is $1/\mu_i$ with probabbility $p_i$, and $\sum_{i=1}^kp_i=1$ and $\sum_{i=1}^kp_i/\mu_i=1/\mu$. In other words the service time consists of a mixture of $K$ deterministic service times.... | It seems that the departure process is indeed a Poisson process. See for example the first line of the paper: Newell, G. F. "The $M/G/\infty$ Queue." SIAM Journal on Applied Mathematics 14.1 (1966): 86-88.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1235089",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
How to integrate $\frac{1}{(1+a\cos x)}$ from $-\pi$ to $\pi$ How to solve the following integration?$$\int_{-\pi}^\pi\frac{1}{1+a \cos x}$$
| You can look at my earlier answer here (on my previous avatar).
Below is another method. We have
$$I = \int_{-\pi}^{\pi} \dfrac{du}{1+a\cos(u)} = 2 \int_0^{\pi} \dfrac{du}{1+a \cos(u)} = 2\int_0^{\pi/2} \dfrac{du}{1+a\cos(u)} + 2\int_{\pi/2}^{\pi} \dfrac{du}{1+a\cos(u)}$$
Hence,
$$\dfrac{I}2 = \int_0^{\pi/2} \dfrac{du}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1235200",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Matrix representation for linear transformation on $\mathbb{R}^{3}$ I am trying to figure out how to solve this problem:
Find a matrix representation for the following linear transformation on $\mathbb{R}^{3}$: A clockwise rotation by $60^{\circ}$ around the $x$-axis.
The answer is:
$$
\begin{bmatrix} 1 & 0 & 0 \\ 0 & ... | Thinking about the mechanical process of rotating a physical object in three dimensions, it is clear that this involves going around an axis. Then focussing on the plane perpendicular to the axis it is the usual 2d-rotation.
So a 3d-rotation means a choice of axis (line) and then quantum of rotation (when measured in r... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1235297",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
locally path connectedness While studying covering spaces , hatcher mentioned the "shrinking wedge of circles" this space is locally path connected as I was told , but I wasn't able to prove it nor to see it, it looks like comb space to me which is not locally path connected , can anyone help me prove it locally path c... | Let $X$ be the shrinking wedge of circles and let $P\in X$ be the wedge point. For any $Q\in X$ with $Q\ne P$ there is an open neighborhood $U_Q$ containing $Q$ such that $U_Q\cap X$ is homeomorphic to the open interval $(0,1)$ which is path connected.
Now, try to show that if $B$ is any open ball centered at $P$, and... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1235506",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 1
} |
Why is ${\bf N}\otimes\bar{\bf N} \cong{\bf 1}\oplus\text{(the adjoint representation)}$? I just watched this lecture and there Susskind says that
$${\bf N}\otimes\bar{\bf N} ~\cong~{\bf 1}\oplus\text{(the adjoint representation)}$$
for the Lie group $G= SU(N)$. Unfortunately, he does not offer any explanation for thi... | We have $\mathfrak{su}(n)\otimes\mathbb{C}\cong\mathfrak{sl}(n,\mathbb{C})$ acting on $N=\mathbb{C}^n$. Since $N\otimes N^\ast\cong\mathrm{End}(N)$ as representations, this is the representation of $\mathfrak{sl}(n,\mathbb{C})$ on $\mathfrak{gl}(n,\mathbb{C})$, which internally decomposes as
$$ \mathfrak{gl}(n,\mathbb{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1235583",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
How to estimate the axis of symmetry for an even function with error? I have a situation here, where, for an unknown $t$, and an unknown but nice* real function $f$, for which $x\rightarrow f(x-t)$ is even, I measure $f(x) + \epsilon_x$, where $\epsilon_x$ is some kind of more or less random error, hopefully small.
Now... | Since it looks like you are going to measure some physical quantity, it seems fair to assume that $x=t$ is a local maximum or minimum for $f$. This would not be the case with a function like $$f(x)=x^3{\sin{\frac{1}{x}}}$$ that is even, differentiable but doesn't exhibit a maximum or a minimum at $x=0$.
If you have a r... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1235710",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
How many solutions of equation How many solutions of equation
$x_1+x_2+x_3+x_4=n$ in $N_0$ such that $x_1\leq x_2\leq x_3 \leq x_4$?
I found solutions of $x_1+x_2+x_3=n$ in $N_0$ , $x_1\leq x_2\leq x_3 $ in the following way :
Let $S$ set of all solutions and $A_k$ set of all solutions of equation $x_1+x_2+x_3+x_4=... | Sometimes a little research can help. The formula you posted was discovered by Jon Perry in 2003.
The generating function for this problem is:
$$g(x) = \frac{1}{(1-x) \left(1-x^2\right) \left(1-x^3\right) \left(1-x^4\right)} $$
There does not seem to be something simple for your question but Michael Somos comes up wit... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1235833",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Convergence of spectrum with multiplicity under norm convergence This article by Joachim Weidmann claims that, if a sequence $A_n$ of bounded operators in a Hilbert space converges in norm topology, i.e., $\|A_n - A\| \rightarrow 0$, then "isolated eigenvalues $\lambda$ of $A$ of finite multiplicity are exactly the lim... | I'll give here another proof which makes the (admittedly, strong) additional assumption that the operators are compact and self-adjoint. For such operators, we have a very useful tool in the Courant-Fischer min-max principle, which we will use here in the form
$$
\lambda_k(A) = \min_{\dim V=k-1} \max_{x \in V^\bot, \... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1235948",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Show that $a(-1) = (-1)a = -a $. In a ring $R$ with identity 1, show that $$a(-1) = (-1)a = -a \qquad\forall\, a \in R$$ I have started with $a + (-a) = 0$ but cant proceed from here.
| We have $\space[1+(-1)]=0$.
Therefore $a\cdot [1+(-1)]=a\cdot 0=0$.
By left distributive law,
$$a\cdot 1+a\cdot (-1)=0$$
$$a+a\cdot (-1)=0.$$
Now $-a \in R$. Adding $-a$ to both sides, we get
$$(-a)+[a+a\cdot (-1)]=(-a)+0,$$
or
$$[(-a)+a]+a\cdot (-1)=-a\quad [\text{associative property}]$$
or
$$0+a\cdot (-1)=-a,$$
or
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1236015",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
Eigen values of a transpose operator Let $T$ be linear operator on $M_{nxn}(R)$ defined by $T(A)=A^t.$
Then $\pm 1$ is the only eigen value.
My try :
Let $n=2,$ then $[T]_{\beta}$ = $
\begin{pmatrix}
a & 0 & 0 & 0 \\
0 & c & 0 & 0 \\
0 & 0 & b & 0 \\
0 & 0 & 0 & d
\end{pmatrix}$
Then the corresponding eige... | If $T(A)=\lambda A$, then $T^2(A)=\lambda^2 A$. But $T^2=I$ which only has an eigenvalue of 1. So $\lambda^2=1$ or $\lambda=-1,1$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1236152",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
$n \times m$ matrix conversion? Is it possible to convert an $n\times m$ matrix $A$ such that
$$ A=CB $$
where $B$ is a $1\times m$ matrix which contains all elements of $A$, and $C$ is a $n\times 1$ matrix. I'm assuming no since this might give a special case of matrices.. but i am not so sure. If this is not possible... | Regarding your intial question. No, it isn't be possible:
Consider an nxm matrix of random values - it has n.m independent points of data. Where as a 1xm and a nx1 matrix together have only n+m independent points of data.
There's less data being stored so the two cannot be equivalent.
I don't think I can answer the res... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1236240",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
} |
$f$ convex and concave, then $f=ax+b$ Let $f$ be a real function defined on some interval $I$.
Assuming that $f$ both convex and concave on $I$, i.e, for any $x,y\in I$ one has
$$f(\lambda x+(1-\lambda)y)=\lambda f(x)+(1-\lambda)f(y),\, \, \lambda\in (0,1) .$$
I would like to show that $f$ is of the form
$f=ax+b$ for... | Let $x<y$ be in $I$. Then for $z=\lambda x + (1-\lambda)y\in [x,y]$,
$$f(z)=f(\lambda x + (1-\lambda)y)=\lambda f(x)+(1-\lambda)f(y)=\frac{f(y)-f(x)}{y-x}(z-x)+f(x)$$
Hence, $f(z)=az+b$ for some $a,b$ on every closed interval in $I$. Every interval can be expressed as the limit of a non-decreasing sequence of closed in... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1236356",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 0
} |
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