Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Could "$\infty$" be understood by taking the reciprocals of the Hyperreal numbers? When learning mathematics we are told that infinity is undefined. (*)
Recently I read about the infinitesimal version of Calculus and how we can in fact treat $dy/dx$ as a fraction under this approach (something we can't do with limits).... | In fact, infinite numbers are a part of the Hyperreal Numbers. Just check this out on Wikipedia.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1075326",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 0
} |
Evaluate this infinite product involving $a_k$ Let $a_0 = 5/2$ and $a_k = a_{k-1}^{2} - 2$ for $k \ge 1$ Compute:
$$\prod_{k=0}^{\infty} 1 - \frac{1}{a_k}$$
Off the bat, we can seperate $a_0$
$$= -3/2 \cdot \prod_{k=1}^{\infty} 1 - \frac{1}{a_k}$$
Lets see:
$a_0 = 5/2$
$a_1 = 25/4 - 2 = 17/4$
$a_2 = 289/16 - 2 = 257/16... | $\textbf{Hint:}$
show this by induction on $n\ge0$
$$a_n=2^{2^n}+2^{-2^n},\quad n=0,1,2\dots$$
and notice that $a_k+1=a_{k-1}^2-1=(a_{k-1}-1)(a_{k-1}+1),$
$$1-\frac{1}{a_k}=\frac{a_k-1}{a_k}=\frac{a_{k+1}+1}{a_k+1}\cdot\frac{1}{a_k}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1075415",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Checking where the complex derivative of a function exists I have the following function:
$$f(x+iy) = x^2+iy^2$$
My textbook says the function is only differentiable along the line $x = y$, can anyone please explain to me why this is so? What rules do we use to check where a function is differentiable?
I know the Cauc... | A perhaps more accessible perspective is to understand that the definition of the complex derivative, as in the real case, relies on the existence and uniqueness of a certain limit. In particular, given a point and a function, one considers evaluation of the function in a small neighbourhood of the point in question in... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1075505",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Find point on line withv given start point, distance, and line equation I have line equation $$ Ax +By + C = 0.$$
I have start point (on this line): $ P_0 = (X_0, Y_0)$.
I have distance $d$ too.
I need find point $P_2$ with distance $d$ from $P_0$ and placed on this line. I know that we have 2 points with this d... | $(A, B)$ is a normal vector for the line, therefore $v = (-B, A)$ is a direction
vector and you get all points on the line with
$$
(x, y) = P_0 + t \, v = (X_0, Y_0) + t (-B, A), \quad t \in \mathbb R.
$$
Now choose $t$ such that the length of $t(−B,A)$ is equal to the given distance $d$,
this gives the two points
$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1075564",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Calculate a limit of exponential function Calculate this limit:
$$
\lim_{x \to \infty } = \left(\frac{1}{5} + \frac{1}{5x}\right)^{\frac{x}{5}}
$$
I did this:
$$
\left(\frac{1}{5}\right)^{\frac{x}{5}}\left[\left(1+\frac{1}{x}\right)^{x}\right]^\frac{1}{5}
$$
$$
\left(\frac{1}{5}\right)^{\frac{x}{5}}\left(\frac{5}{5}\ri... | $$\lim_{x\to\infty}\left(\frac15\right)^x=0$$
$$\lim_{x\to\infty}\left(1+\frac1x\right)^x=e$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1075652",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 1
} |
What is the combination of Complex, Split-Complex and Dual Numbers If $a+bi:i^2=-1$ is a complex number, $a+cj:j^2=+1$ is a split-complex number, and $a+d\epsilon:\epsilon^2=0$ is a dual number; what is the term for the combination $a+bi+cj+d\epsilon:i^2=-1,j^2=+1,\epsilon^2=0$?
| There is no special term as far as I know. You can call them hypercomplex numbers deffiened by certain Clifford algebra $ \mathcal{Cl} (1,1,1)$ .
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1075708",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 3,
"answer_id": 0
} |
Is Dodecahedron tesselation somehow possible? In this video (at 3:25) there is an animation of planets inside a dodecahedron matrix (or any data-structure that best fit this 3d mosaic). I tried reproducing it with 12 sided dices, or in Blender, but it looked impossible, because there is a growing gap inbetween faces.
T... | As User7530 pointed out, there is - in the video - a heavy distortion of the earths near the left side of the screen and it must therefore be a hyperbolic tiling.
Also, L'Univers Chiffoné is a book that explains the possibility of living in a hyperbolic (a non-euclidian) space. its written by the french astrophysicist ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1075756",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
A question about essential ideal
Let $I$ be a nonunital C*-algebra and $I\subset B(H)$ be any nondegenerate representation and define
$$M(I)=\{T\in B(H): Tx\in I~and ~xT\in I, ~for ~all~ x\in I\}.$$
Then, how to prove $I$ sits in $M(I)$ as an essential ideal?
Definition An ideal $I\triangleleft E$ is essential ... | First of all, see this question.
If $Tx=xT=0$ for every $x\in I$ then for every $h\in H$ we have $Txh=0$.
Now the set $\{xh,\ x\in I, h\in H\}$ is dense in $H$, since the representation is non-degenerate by the question above.
Since $\ker(T)$ is closed in $H$ and contains $\{xh,\ x\in I, h\in H\}$, which is dense in ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1075843",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Find the fraction where the decimal expansion is infinite? Find the fraction with integers for the numerator and denominator, where the decimal expansion is $0.11235.....$
The numerator and denominator must be less than $100$.
Find the fraction.
I believe I can use generating functions here to get $1+x+2x^2+3x^3+5x^... | An exhaustive search (by computer) of all fractions with numerator and denominator $< 100$ shows there is only one whose decimal representation starts 0.11235:
$$
\frac{10}{89}
$$
The decimal expansion continues $\ldots 955056179775\ldots$ (I haven't bothered finding aprogram that can calculate enough that we can see t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1075901",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 4,
"answer_id": 1
} |
Notation of inf In this paper (equation 4.1) the following formula is listed:
$\inf_{u \in R} \left \{ \frac{\partial V}{\partial \boldsymbol{x}}f(\boldsymbol{x},u) \right \} < 0, \quad \forall \boldsymbol{x} \neq \boldsymbol{0} $
Now I don't understand what the term $\inf_{u \in R}$ indicates. I know that inf stands ... | The subscript gives context to the infimum. You could also write it as
$$
\inf\left\{\frac{\partial V}{\partial x} f(x,u)\ \bigg| \ u \in R\right\}
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1075979",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Prove that a matrix equals to its transpose Let $A$ be a $(n\times n)$ matrix that satisfies: $AA^t=A^tA$
Let $B$ be a matrix such that: $B=2AA^t(A^t-A)$
Prove/disprove that: $B^t=B$
I started with:
$$\begin{align}
B &=2AA^t(A^t-A) \\
B^t &=(2AA^t(A^t-A))^t
\\
B^t &=((A^t-A)^t(A^t)^tA^t2)
\\
B^t &=((A-A^t)AA^t2)
\\
B^t... | $$
\begin{eqnarray*}
B^T&{}={}&2\left(A-A^T\right)AA^T\newline
&{}={}&2\left(AAA^T-A^TAA^T\right)\newline
&{}={}&2A\left(AA^T-A^TA^T\right)\newline
&{}={}&2AA^T\left(A-A^T\right)\newline
&{}\neq{}& B\,.
\end{eqnarray*}
$$
Almost, but not quite, in general.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1076048",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 7,
"answer_id": 4
} |
Analysis: Prove divergence of sequence $(n!)^{\frac2n}$ I am trying to prove that the sequence $$a_n = (n!)^{\frac2n}$$ tends to infinity as $ n \to \infty $.
I've tried different methods but I haven't really got anywhere. Any solutions/hints?
| Let $c_n=(n!)^2$; then $\displaystyle\frac{c_{n+1}}{c_n}=\frac{((n+1)!)^2}{(n!)^2}=(n+1)^2\to\infty,$ $\;\;\;$so $(n!)^{\frac{2}{n}}=(c_n)^{\frac{1}{n}}\to\infty$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1076109",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 5,
"answer_id": 3
} |
3D coordinates of circle center given three point on the circle. Given the three coordinates $(x_1, y_1, z_1)$, $(x_2, y_2, z_2)$, $(x_3, y_3, z_3)$ defining a circle in 3D space, how to find the coordinates of the center of the circle $(x_0, y_0, z_0)$?
| In this formula
the center $O$ of
a circumscribed circle of $\triangle ABC$
is expressed as a convex combination of its vertices
in terms of
coordinates $A,B,C$ and corresponding side lengths $a,b,c$,
suitable for both 2d and 3d:
\begin{align}
O&=
A\cdot \frac{a^2\,(b^2+c^2-a^2)}{((b+c)^2-a^2)(a^2-(b-c)^2)}
\\
&+B\cdo... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1076177",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "13",
"answer_count": 11,
"answer_id": 7
} |
Counting in other bases While this could be considered opinionated to a certain degree, by setting the requirement as ease of use, is there a base that is better for performing simple math functions (+-×÷) than base ten.
I recently attended a lecture about whether cover stories affect the learning of students. One of t... | Some points to consider in choosing a base:
(1) In lower bases there are very few $1$-digit facts to learn, but the numbers are longer.
(2) In higher bases, the numbers are shorter, but there are a lot more facts to learn.
(3) Certain bases give quick divisibility tests for certain numbers. For instance in base $12$, ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1076249",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Solve the equation $2x^2+5y^2+6xy-2x-4y+1=0$ in real numbers
Solve the equation $2x^2+5y^2+6xy-2x-4y+1=0$
The problem does not say it but I think solutions should be from $\mathbb{R}$. I tried to express the left sum as a sum of squares but that does not work out. Any suggestions?
| You can solve for $x$:
$(2)x^2+(6y-2)x+(5y^2-4y+1)=0\implies$
$x_{1,2}=\frac{-(6y-2)\pm\sqrt{(6y-2)^2-4\cdot2\cdot(5y^2-4y+1)}}{2\cdot2}=\frac{-6y+2\pm\sqrt{-4y^2+8y-4}}{4}=\frac{-6y+2\pm\sqrt{-4(y-1)^2}}{4}=\frac{-6y+2\pm2i(y-1)}{4}$
Then the only real solution is with $y=1$, hence $x=-1$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1076340",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 2
} |
Convergence of summable sequences If $(a_n)$ is a sequence such that
$$\lim_{n\to\infty}\frac{a_1^4+a_2^4+\dots+a_n^4}{n}=0.$$
How do I show that $\lim_{n\to\infty}\dfrac{a_1+a_2+\dots+a_n}{n}=0$?
| Since:
$$\left|\sum_{k=1}^{n} a_k\right|\leq \sqrt{n}\sqrt{\sum_{k=1}^{n} a_k^2}\leq n^{\frac{3}{4}}\left(\sum_{k=1}^{n}a_k^4\right)^{\frac{1}{4}}$$
by applying the Cauchy-Schwarz' inequality twice (or the Holder's inequality once), we have:
$$\frac{1}{n}\left|\sum_{k=1}^{n}a_k\right|\leq\left(\frac{1}{n}\sum_{k=1}^{n}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1076436",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 0
} |
No. of different real values of $x$ which satisfy $17^x+9^{x^2} = 23^x+3^{x^2}.$ Number of different real values of $x$ which satisfy $17^x+9^{x^2} = 23^x+3^{x^2}.$
$\bf{My\; Try::}$Using Hit and trial $x=0$ and $x=1$ are solution of above exponential equation.
Now we will calculate any other solution exists or not.
I... | Using derivatives, is studying functions
$ f, g: R \rightarrow R, f(x)= 9^{x^2}-3^{x^2}, g(x)=23^x-17^x$ and is found:
*
*$f$ has a minimum point in the interval $(0, 1)$ and limits to $+\infty$,$-\infty$ are equal with $+\infty$;
*$g$ has a negative minimum point and limited to $-\infty$ is $0$ and to $+\infty$ i... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1076497",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 2,
"answer_id": 1
} |
Prove that a square of a positive integer cannot end with $4$ same digits different from $0$ Prove that a square of a positive integer cannot end with $4$ same digits different from $0$.
I already proved that square of positive integer cannot end with none of digits $1,2,3,5,6,7,8,9$ using the remainder of division by ... | As $x^2=4444\equiv12 \pmod{ 16}$, so no squares can end in four $4$'s.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1076573",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
Multitangent to a polynomial function I'm trying to build some exercises on tangents of functions for beginner students in mathematical analysis. In particular I would like to suggest the study of polynomial functions $ y = p (x) $ of which is possible to determine the graph with elementary methods and also determine ... | After three solutions, I found the easy way!
Start with $g(x)=(x-p)^2(x-r)^2(x-s)^2$.
For any $m$ and $q$, consider $y=m(x-p)+q$. Add this to $g$, to get $f$, then $f$ passes through $(p,q)$ and the line $y=m(x-p)+q$ is tangent to the curve of $f$ at $p$, $r$, and $s$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1076688",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 1,
"answer_id": 0
} |
Orbits in G = $Z_6$ by listing 2 element subsets in G. 1) Let $G = \mathbb{Z}_6$. List all 2-element subsets of $G$, and show that under the regular action of G (by left addition) there are 3 orbits, 2 of length 6, one of length 3. Deduce that the stabilizers of 2-element subsets of $G$ have order 1 or 2.
2) Carry out ... | The two element subsets:
$$\{0,1\},\{0,2\},\{0,3\},\{0,4\},\{0,5\},\{1,2\},\{1,3\},\{1,4\},\{1,5\},\{2,3\},\{2,4\},\{2,5\},\{3,4\},\{3,5\},\{4,5\}.$$
Left addition: Take $\left[a\right]_6 \in \mathbb{Z}_6$ and do $a+H$ (where $H$ is a two element subset of $G$).
For example:
$$1+\{0,1\} =\{1,2\}, 1+\{1,2\} = \{2,3\},... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1076802",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Abelian group of order 99 has a subgroup of order 9
Prove that an abelian group $G$ of order 99 has a subgroup of order 9.
I have to prove this, without using Cauchy theorem. I know every basic fact about the order of a group.
I've distinguished two cases :
*
*if $G$ is cyclic, since $\mathbb Z/99\mathbb Z$ has a... | There exists an element $a$ of order $3$. If there was not every element would need to have order $11$.
But then a counting argument leads to contradiction.
We take the quotient group $ G/<a>$. Then it has order $33$. The same argument shows that there exists an element $b$ of order $3$. Take the map $p: G \rightarrow... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1076880",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 6,
"answer_id": 0
} |
Prove $4^k - 1$ is divisible by $3$ for $k = 1, 2, 3, \dots$ For example:
$$\begin{align}
4^{1} - 1 \mod 3 &=
\\
4 -1 \mod 3 &=
\\
3 \mod 3 &=
\\3*1 \mod 3 &=0
\\
\\
4^{2} - 1 \mod 3 &=
\\
16 -1 \mod 3 &=
\\
15 \mod 3 &=
\\3*5 \mod 3 &= 0
\\
\\
4^{3} - 1 \mod 3 &=
\\
64 -1 \mod 3 &=
\\
21 \mod 3 &=
\\3*7 \mod 3 &=
... | Hint:
$$4 \equiv 1 \mod 3 \Rightarrow 4^n \equiv 1^n \mod 3 \Rightarrow 4^n \equiv 1 \mod 3$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1076946",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 8,
"answer_id": 2
} |
Random variables and Linearity I have an equation $Y = 5 + 3\times X$ and I assume that $X$ is a random variable taking values from a uniform distribution. Can I consider that also $Y$ is a random variable which takes values from a uniform distribution also but in different space?
Thank you in advance
| If X is uniform on (0,1) then Y=5+3X is uniform on (5,8). More generally, for every nonzero b, Z=a+bX is uniform on (a,a+b).
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1076996",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Stopped process of Brownian motion I am baffled about the following problem:
Let $(B_t)$ be a standard Brownian motion. Let $$ \tau:= \inf\{ t \geq 0 :B_t = x \} \wedge \inf\{ t \geq 0 :B_t = -y \}$$ be a stopping time, where $x,y >0$. I am eager to know why the stopped process $(B_{t \wedge \tau})_{t \geq 0}$ is U.I..... | We know that $\mathbb{E}[\tau] < \infty$ and $B_{t \wedge \tau}$ is a martingale. It is also easy to see that $-y \leq B_{t \wedge \tau} \leq x$. Any bounded martingale is uniform integrable. Hence, $B_{t \wedge \tau}$ is UI martingale.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1077169",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
Hitting time process of Brownian motion I am stuck with this problem:
Let $(B_t)$ be a standard Brownian motion in $\mathbb{R}$. For $t \geq 0$, let
$$ H_t = \inf \{ s \geq 0 : B_s = t \}, \quad S_t = \inf \{ s \geq 0 : B_s > t \}. $$
I want to prove three things:
$1.$ Given $t \geq 0$, $H_t$ and $S_t$ are a.s. equal. ... | 1)As, Brownian paths are continuous its clear that $H_t \leq S_t$ a.s. Also, we know that $B_{\tau + s} - B_{\tau}$ is Brownian Motion for any finite stopping time $\tau$. Now, take $\tau = H_t$ which means $B_{H_t+s}-t$ is Brownian motion. Brownian motion oscillates infinitely at $0$ and hence we can find sequence of... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1077231",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Height and coheight of an ideal Given an ideal $\mathfrak{a}$, Matsumura defined the height of $\mathfrak{a}$ as:
$$\text{ht}(\mathfrak{a})=\inf_{\mathfrak{p}\in V(\mathfrak{a})}\text{ht}(\mathfrak{p})$$
He states that:
$$\text{ht}(\mathfrak{a})+\dim(A/\mathfrak{a})\leq \dim(A)$$
Any ideas on how to show this? I know I... | Hints.
$\operatorname{ht}\mathfrak{p}+\dim A/\mathfrak{p}\leq \dim A$ for any prime ideal $\mathfrak p$ (why?).
$\dim A/\mathfrak a=\sup_{\mathfrak{p}\in V(\mathfrak{a})}\dim A/\mathfrak p$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1077355",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Definite integral $\int_0^{2\pi}\frac{1}{\cos^2(x)}dx$ I encountered this very simple problem recently, but I got stuck on it because I think I am missing something.
It is easy to see that indefinite integral $\int\frac{1}{\cos^2(x)}dx$ is $\tan(x)+C$. Also because $\frac{1}{\cos^2(x)}\geq0$ with the inequality being s... | This is a trigonometric analog to the classic paradox that
$$\int_{-1}^1{1\over x^2}dx={-1\over x}\Big|_{-1}^1=-1-1=-2$$
despite the fact that $1/x^2$ is strictly positive, hence its definite integral should give a positive result for the area beneath the curve. The explanation (as given by k170), lies in the fact tha... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1077481",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
Choosing a contour to integrate over. What are the guidelines for choosing a contour?
For example to integrate a real function with a singularity somewhere.
What type of contour from
Square, keyhole, circle, etc should be chosen for integration?
(1) In what cases would you choose square?
(2) In what cases would you ch... | Here are some brief general guidelines for the three contours you mentioned:
*
*The circular contour is used when we have an integral from $0$ to $2\pi$, with the integrand consisting of trigonometric functions, such as sin and cosine. Thinking about the polar coordinates, Euler's identity ($e^{ix}=\cos(x)+i\sin(x)$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1077554",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
Sum of roots: Vieta's Formula The roots of the equation $x^4-5x^2+2x-1=0$ are $\alpha, \beta, \gamma, \delta$. Let $S_n=\alpha^n +\beta^n+\gamma^n+\delta^n$
Show that $S_{n+4}-5S_{n+2}+2S_{n+1}-S_{n}=0$
I have no idea how to approach this. Could someone point me in the right direction?
| Here is a hint: multiply your equation by $x^n$. When you substitute $\alpha$ in the modified equation, what do you find? How can you relate that to the equation for $S_n$?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1077648",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 5,
"answer_id": 0
} |
If $T : F^{2 \times 2} \to F^{2\times 2}$ is $T(A) = PA$ for some fixed $2 \times 2$ matrix $P$, why is $\operatorname{tr} T = 2\operatorname{tr} P$? I am asked to prove that if $T$ is a linear operator on the space of $2 \times 2$ matrices over a field $F$ such that $T(A) = PA$ for some fixed $2 \times 2$ matrix $P$, ... | If we represent $T$ as a $4 \times 4$ matrix with respect to your basis, we have
$$
[T]_B =
\pmatrix{P_{11}&0&P_{12}&0\\0&P_{11}&0&P_{12}\\P_{21}&0&P_{22}&0\\0&P_{21}&0&P_{22}}
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1077969",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
How do I prove the circumference of the Koch snowflake is divergent? How do I prove that the circumference of the Koch snowflake is divergent?
Let's say that the line in the first picture has a lenght of $3cm$.
Since the middle part ($1cm$) gets replaced with a triangle with sidelenghts of $1cm$ each
we can assume th... | Hint:$$\frac{4^n}{3^n}=L\quad\Rightarrow\quad n=\log_{4/3}L$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1078034",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
Power series for the rational function $(1+x)^3/(1-x)^3$
Show that $$\dfrac{(1+x)^3}{(1-x)^3} =1 + \displaystyle\sum_{n=1}^{\infty} (4n^2+2)x^n$$
I tried with the partial frationaising the expression that gives me
$\dfrac{-6}{(x-1)} - \dfrac{12}{(x-1)^2} - \dfrac{8}{(x-1)^3} -1$
how to proceed further on this havi... | The most simple way to prove your identity, IMHO, is to multiply both sides by $(1-x)^3$.
This leads to:
$$ 1+3x+3x^2+x^3\stackrel{?}{=}(1-3x+3x^2+x^3)\left(1+\sum_{n\geq 1}(4n^2+2)\,x^n\right).\tag{1}$$
If we set $a_n=(4n^2+2)$, for any $n\geq 4$ the coefficient of $x^n$ in the RHS is given by $a_n-3a_{n-1}+3a_{n-2}+... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1078092",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 5,
"answer_id": 2
} |
Differentiate Archimedes's spiral I read that the only problem of differential calculus Archimedes solved was constructing the tangent to his spiral,
$$r = a + b\theta$$
I would like to differentiate it but I don't know much about differentiating polar functions and can't find this particular problem online. Without gi... | Let $r(\theta)=a+b\theta$ the equation of the Archimedean spiral.
The cartesian coordinates of a point with polar coordinates $(r,\theta)$ are
$$\left\{\begin{align}
x(r,\theta)&=r\cos\theta\\
y(r,\theta)&=r\sin\theta
\end{align}\right.
$$
so a point on the spiral has coordinates
$$\left\{\begin{align}
x(\theta)&=r(\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1078185",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 1
} |
"Almost mean" of a set of integer I have three integers $(a, b, c)$ (but this can be generalized to any size). I want to redistribute the sum $a+b+c$ as equally as possible between three variables. In our case, we have three cases:
If $a+b+c=3k$, then the solution is $(k, k, k)$.
If $a+b+c=3k+1$, then the solution is $... | This really is just the "division algorithm." Let $m$ be the sum you want, and $n$ be the number of variables.
Use the division algorithm to write $m=nq+r$ with $q$ the quotient and $0\leq r<n$ the remainder. Then you can write $m$ as the sum of $r$ instances of $q+1$ and $n-r$ instances of $q$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1078256",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
} |
Splitting up A Conditional Probability When, if ever, is it true for random variables $a,b,c,d$ that $P(a,b,c\mid d) = P(a\mid d)P(b\mid d)P(c\mid d)$? Is it related to the dependencies among the variables?
| It can be true, and yes, it is related to the dependencies among the variables. It holds when variables $a$, $b$ and $c$ are conditionally independent given $d$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1078332",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Showing $(Tp)(x) = x^2p(x)$ is a linear map (transformation) Define a linear map function $T: \mathcal{P}(\mathbb{R}) \to \mathcal{P}(\mathbb{R})$ where $\mathcal{P}(\mathbb{R})$ is the set of all polynomials with real-valued coefficients. Now let $T$ belong to the set of all possible linear maps from $\mathcal{P}(\mat... | let $p(x)=a_0+a_1x+ a_2x^2+\cdot\cdot\cdot\cdot+a_nx^n$
Now let $\lambda\in \mathbb R$ then $\lambda p(x)=\lambda a_0+(\lambda a_1)x+(\lambda a_2)x^2+\cdot\cdot\cdot\cdot+(\lambda a_n)x^n$
Then $T(\lambda p(x))=x^2\lambda a_0+(\lambda a_1)x^3+(\lambda a_2)x^4+\cdot\cdot\cdot\cdot+(\lambda a_n)x^{n+2}=\lambda x^2(a_0+a_... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1078422",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Expected value of time integral of a gaussian process While working on a problem I've stumbled upon some expected values of time integrals of Gaussian stochastic processes. Some of them were addressed this question, but I have found also this one
$$\left\langle\int _0^tdt_1\int_0^{t_1}dt_2\left(B\left(t_1\right)-B\left... | Indeed, the result is $$\int _0^t\mathrm dt_1\int_0^{t_1}\mathrm dt_2\int _0^t\mathrm dt_3\int_0^{t_3}\mathrm dt_4\left[K(t_1-t_3)-K(t_1-t_4)-K(t_2-t_3)\color{red}{+}K(t_2-t_4)\right],$$
which should be nonnegative. Note that the last sign $\color{red}{+}$ reads $-$ in your post.
Edit: One asks to compute the second mo... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1078510",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Do we really need reals? It seems to me that the set of all numbers really used by mathematics and physics is countable, because they are defined by means of a finite set of symbols and, eventually, by computable functions.
Since almost all real numbers are not computable, it seems that real numbers are a set too big a... | Suppose you have a square of area 1 cm.You want to calculate length of the diagonal.If you don't have notion of real number then you can not find length of the diagonal.Another way to say this that in a countable system you can always find some equation that has no solution in that system.For example in rational number... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1078593",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "55",
"answer_count": 13,
"answer_id": 10
} |
Easy example why complex numbers are cool I am looking for an example explainable to someone only knowing high school mathematics why complex numbers are necessary.
The best example would be possible to explain rigourously and also be clearly important in a daily day sense.
I.e. complex Laplace transform has applicatio... | Complex numbers can be used to give a very simple and short proof that the sum of the angles of a triangle is 180 degrees. Construct a triangle in the plane whose vertices are complex numbers z1.z2 and z3 in the plane.
Recall that the angle $\theta$ at a vertex like $ABC$ is the angular part $\theta$ of $(C-B)/(A-B) ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1078618",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "85",
"answer_count": 26,
"answer_id": 23
} |
Connections of theory of computability and Turing machines to other areas of mathematics The question is quite straightforward:
Could you point out some reference papers that highlight (in a way that is fairly accessible) the connections between (1) theory of computability, algorithms, and Turing machines and (2) other... | You can consider Computability and Complexity :
[for the] study and classification of which mathematical problems are computable and which are not. In addition, there is an extensive classification of computable problems into computational complexity classes according to how much computation — as a function of the siz... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1078695",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Rewriting the heat diffusion equation with temperature dependent diffusion coefficient to include joule heating. I am modelling heat flow in a solid round copper conductor with a set area. I plan to discretize and solve numerically in Python. However, I only have a curve fit for thermal conductivity and specific heat. ... | $$
\rho \cdot C(u)\frac{\partial u}{\partial t} = k(u)\cdot\frac{\partial^2u}{\partial x^2}
$$
Due to the different type of terms, I was unable to use the equation in the form from the OP. I was able to get reasonable answers after switching the terms to what is shown above.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1078748",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Basis if and only if $\varphi$ is an isomorphism Let $V$ be a finite dimensional vector space. I am aware of the theorem stating that if $\varphi \colon V \rightarrow V$ is an automorphism and $\mathcal{A}=(a_1, \ldots, a_n)$ is a basis of V then $\varphi(\mathcal{A})=(\varphi(a_1), \ldots, \varphi(a_n))$ is another ba... | Yes, you are right. If you can show that $b_1,\ldots,b_n$ is the image of a basis $a_1,\ldots,a_n$ by an isomorphism $f$, then you have shown that $\left\{b_i\right\}_{i=1,\ldots,n}$ is a basis.
This happens because since $f$ is an isomorphism, the image of $f$ has dimension $n$. But the image of $f$ is generated by $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1078816",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
How to evaluate $ \int \sec^3x \, \mathrm{d}x$? How to evaluate $ \int \sec^3x \, \mathrm{d}x$ ?
I tried integration by parts and then I got $$\int \sec^3x \, \mathrm{d}x=\sec x \tan x - \int\sec x \tan^2 x \, \mathrm{d}x.$$ Now I'm stuck solving $\int\sec x \tan^2 x \, \mathrm{d}x$? How to solve that and how to solve ... | This is a well-known problem. Using $ \left( u, v \right) = \left( \sec x, \tan x \right) $ gives $$ \begin {align*} \displaystyle\int \sec^3 \, \mathrm{d}x &= \displaystyle\int \sec x \tan x - \displaystyle\int \sec x \tan^2 x \, \mathrm{d}x \\&= \sec x \tan x - \displaystyle\int \sec^3 x \, \mathrm{d}x + \displaysty... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1078878",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 5,
"answer_id": 3
} |
Is $x\in\{\{x\}\}$ Is $x\in\{\{x\}\}$. I understand that $x\in\{x\}\in\{\{x\}\}$ does this mean $x\in\{\{x\}\}$?
Very simple just unsure about the properties of $\in$, not looking for an extravagant answer, thanks in advance for the help.
| The answer is no. $\{\{x\}\}$ has exactly one element, namely $\{x\}$.
In general, to test if $a\in b$, we do the following:
*
*Is $b$ a set?
*If so, is $a$ one of the elements in the set $b$?
$a$ can be a set itself, or not, doesn't matter.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1078981",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Vector Field in a complex projective space This question is motivated by this answer here.
Let $\mathbb{C}P^{n}$ be a complex projective space.
Let $X\in\Gamma(T\mathbb{C}P^{n})$, be a vector field. It seems, by the answer I got in the mentioned link, that the zeros of $X$ are the points $[z]\in\mathbb{C}P^{n}$ such t... | Let's try a different approach. We can think of $\Bbb CP^n$ as the usual quotient of $\Bbb C^{n+1}-\{0\}$ with the projection map $\pi(z) = [z]$. Let's think of a vector field $X$ on $U\subset\Bbb CP^n$ as being pushed down from an appropriate vector field $\tilde X$ on $\pi^{-1}(U)$ with the property that $\pi_{*z}\ti... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1079080",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Is the indicator function of the rationals Riemann integrable? $f(x) =
\begin{cases}
1 & x\in\Bbb Q \\[2ex]
0 & x\notin\Bbb Q
\end{cases}$
Is this function Riemann integrable on $[0,1]$?
Since rational and irrational numbers are dense on $[0,1]$, no matter what partition I choose, there will always be rational and irr... | Recall that a function is Riemann integrable if and only if, for any $\varepsilon > 0$, there exists a partition $P$ such that $U(f, P) - L(f, P) < \varepsilon$.
Now consider any partition $P$ of $[0,1]$. The lower sum is always zero since the infimum of the function values along any interval is zero. Further, the s... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1079172",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "15",
"answer_count": 4,
"answer_id": 2
} |
The finite field extension Let field $K$ embedded into the finite field $M$. Prove that $M = K(\theta)$ for some $\theta \in M$.
I have tried 2 ways but got stuck at both.
1) Let $|K| = p^s$ and $|M| = p^{st}$ for prime $p$ and $s, t \in \mathbb N$. And if we find the irreducible polynomial $f(x) \in K[x]$ where $deg f... | One easy approach is to appeal to the following theorem:
If $F$ is any field, and $G$ is a finite subgroup of $F^\times$, then $G$ is cyclic.
Since $M$ is finite, we can take $G=M^\times$, so there is some $\theta\in M$ such that every non-zero element of $M$ is equal to $\theta^n$ for some $n$. Clearly, this $\theta... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1079303",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
AlgebraII factoring polynomials equation: $2x^2 - 11x - 6$
Using the quadratic formula, I have found the zeros: $x_1 = 6, x_2 = -\frac{1}{2}$
Plug the zeros in: $2x^2 + \frac{1}{2}x - 6x - 6$
This is where I get lost. I factor $-6x - 6$ to: $-6(x + 1)$, but the answer says otherwise. I am also having trouble factoring ... | When you factor out the equation $2x^2-11x-6$, you get $(x-6)(2x+1)$ (David Peterson did the factoring process). This shows that the functions has two zeros in the graph. Thus, we have to sent $(x-6)(2x+1)$ equal to $0$: $$(x-6)(2x+1)=0.$$ Then we find the zeros:
$$(x-6)(2x+1)=0$$
$$x-6=0$$
$$\boxed{x=6}$$
$$2x+1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1079374",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 2
} |
To show that $f (x) = | \cos x | + |\sin x |$ is not one one and onto and not differentiable Let $f : \mathbb{R} \longrightarrow [0,2]$ be defined by $f (x) = | \cos x | + |\sin x |$. I need to show that $f$ is not one one and onto. I have only intuitive idea that $\cos x$ is even function so image of $x$ and $-x$ are ... | hints:
1) Both $\sin$ and $\cos $ share the same period. Use that to show the function is not one to one.
2) Each of $\sin $ and $\cos$ is bounded between $-1$ and $1$. Use that to show that the function is not onto by showing it is bounded between $0$ and $2$ (assuming you take as the codomain $\mathbb R$, otherwise,... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1079479",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 6,
"answer_id": 0
} |
prove that $a^2 b^2 (a^2 + b^2 - 2) \ge (a + b)(ab - 1)$ Good morning
help me to show the following inequality
for all $a$, $b$ two positive real numbers
$$a^2 b^2 (a^2 + b^2 - 2) \ge (a + b)(ab - 1)$$
thanks you
| Let $a+b=2u$ and $ab=v^2$, where $v>0$. Hence, we need to prove that $2v^4u^2-(v^2-1)u-v^6-v^4\geq0$, for which it's enough to prove that $u\geq\frac{v^2-1+\sqrt{(v^2-1)^2+8v^4(v^4+v^6)}}{4v^4}$ or $(4v^5-v^2+1)^2\geq(v^2-1)^2+8v^4(v^4+v^6)$ because $u\geq v$, or $(v-1)^2(v+1)(v^2+v+1)\geq0$. Done!
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1079550",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 0
} |
A probability function is determined on a dense set- Where is density used in the following proof? A probability function is determined on a dense set- Where is density used in the following proof?
Consider the following theorem and proof from Resnick's book A probability path. I cannot really see where the assumption ... | You’re right: as long as $D$ contains arbitrarily large reals, $F$ is right continuous. However, $F$ need not extend $F_D$ even if $D$ is dense in $\Bbb R$, so the last sentence of the statement of the lemma is a non sequitur.
To see this, let $D=\Bbb R\setminus\{1\}$, and let $F_D(x)=0$ if $x\le 0$ and $F_D(x)=1$ if $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1079661",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Finding the area under a speed time graph I recently learned about integration and I wondered how it could be applied to a speed time graph since it does not have a particular equation of a line that one can integrate. Do you split it into parts? I have heard of a type of integration which specialises in putting shapes... | This graph is a Position(time), it's antiderivative will give you the absement.
To calculate the area under this graph you can separate into these subregions:
[]
Then each area is just a simple geometric figure, for the rectangles you use the very well known formula $base\cdot height$ and for the triangles $ \dfrac {ba... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1079763",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Why the differentiation of $e^x$ is $e^x?$ $$\frac{d}{dx} e^x=e^x$$
Please explain simply as I haven't studied the first principle of differentiation yet, but I know the basics of differentiation.
| The number $e$ is defined by: $e=\sum\limits_{n=0}^\infty\dfrac{1}{n!}$ but historically I think that the definition of the $exp$ function is: $e^x=\lim\limits_{n\to+\infty}\left(1+\dfrac{x}{n}\right)^n$ and the properties of this function comes from this definition as shown in this article.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1079821",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 9,
"answer_id": 7
} |
Does equicontinuity imply uniform continuity? If $\{f_n(x)\}$ is an equicontinuous family of functions, does it follow that each function is uniformly continuous? I am a bit confused since in the Arzela-Ascoli theorem, equicontinuity is seems to mean "$\delta$ depends neither on $n$ nor $x$." For example, Wikipedia s... | Now it does not. The singleton family $F=\{f\}$, with $f(x)=x^2$ is not uniformly continuous, but it is equicontinuous.
However, if $K$ is a compact metric space and $\mathcal F\subset C(K)$ is an equicontinuous family, then $\mathcal F$ is also uniformly equicontinuous.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1079905",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 2,
"answer_id": 1
} |
How to prove that $\frac{\zeta(2) }{2}+\frac{\zeta (4)}{2^3}+\frac{\zeta (6)}{2^5}+\frac{\zeta (8)}{2^7}+\cdots=1$? How can one prove this identity?
$$\frac{\zeta(2) }{2}+\frac{\zeta (4)}{2^3}+\frac{\zeta (6)}{2^5}+\frac{\zeta (8)}{2^7}+\cdots=1$$
There is a formula for $\zeta$ values at even integers, but it invol... | $\newcommand{\angles}[1]{\left\langle{#1}\right\rangle}
\newcommand{\braces}[1]{\left\lbrace{#1}\right\rbrace}
\newcommand{\bracks}[1]{\left\lbrack{#1}\right\rbrack}
\newcommand{\dd}{\mathrm{d}}
\newcommand{\ds}[1]{\displaystyle{#1}}
\newcommand{\expo}[1]{\,\mathrm{e}^{#1}\,}
\newcommand{\half}{{1 \over 2}}
\new... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1080000",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "88",
"answer_count": 4,
"answer_id": 3
} |
Solving recursive integral equation from Markov transition probability How do I solve something like:
$$f(x) = \frac{1}{\sqrt{2 \pi}} \int_{-\infty}^\infty e^{\frac{-(y - x/2)^2}{2}}f(y)\:\mathrm{d}y$$
for $f(x)$?
Is there also a general formula that this falls under? The closest thing I found was the Fredholm integral... | We are given a discrete step continuous state transition system from state $y$ to state $x$ as
$$M[y, x] = \frac 1 {\sqrt{2\pi}} e^{-\frac 12(x - y/2)^2}$$
which has the following characteristics
*
*If you are at $x$, your destination is a normal curve from $-\infty$ to $\infty$ centered at $x/2$ with a variance of... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1080055",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
} |
Prove that a $1$-periodic function $\phi$ is constant if $\phi(\frac{x}{2}) \phi(\frac{x+1}{2})$ is a constant multiple of $\phi$ For $0 < x< \infty$, let $\phi (x)$ be positive and continuously twice differentiable satisfying:
(a) $\phi (x+1) = \phi (x)$
(b) $\phi(\frac{x}{2}) \phi(\frac{x+1}{2}) = d\phi(x),$ where ... | Taking logarithms in (b), we obtain the identity
$$\log \phi\biggl(\frac{x}{2}\biggr) + \log \phi\biggl(\frac{x+1}{2}\biggr) = \log \phi(x) + \log d.\tag{1}$$
Differentiating $(1)$ twice, that becomes
$$\frac{1}{4}\Biggl(g\biggl(\frac{x}{2}\biggr) + g\biggl(\frac{x+1}{2}\biggr)\Biggr) = g(x).\tag{2}$$
So $g$ is a conti... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1080130",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 1,
"answer_id": 0
} |
Elements with infinite roots in $p$-adic integers Let $\mathbb{Q}_p$ the $p$-adic completion of $\mathbb{Q}$ and
$$S=\{x\in\mathbb{Q}_p:1+x\mbox{ has $n^{th}$ roots in }\mathbb{Q}_p\mbox{ for infinitely many }n\in\mathbb{N}\}$$
I have to show that $p\mathbb{Z}_p\subseteq S\subseteq \mathbb{Z}_p$ where $\mathbb{Z}_p=\{x... | So let's start the easy way, if $x\not\in\Bbb Z_p$ we have $v_p(x)<0$, so that $v_p(1+x)<0$ by the strong triangle inequality. Then say $y_n^n=1+x$ for some infinite sequence, $y_n$. Then as
$$v_p(y_n)={1\over n}v_p(1+x)$$
we see that as $n\to\infty$ we have that $v_p(y_n)\to 0$. But the valuation is discrete, hence fo... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1080201",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
I can't understand a step in the proof of the associativity of matrix multiplication Matrix multiplication associativity is proven by the following reasoning:
Let there be matrices $A^{m \times n}$, $B^{n \times k}$ and $C^{k \times l}$. Then
$$
\{(AB)C\}_{ij}=\sum\limits_{p=1}^k{\{AB\}_{ip}c_{pj}
\\=\sum\limits_{p=1}^... | Here is, I think, a more intuitive way to prove it.
Letting A be a $m \times n$ matrix, it follows that B must have n rows for AB to exist. So letting B be a $n \times p$ matrix, (AB) will be an $m \times p$ matrix. For (AB)C to exist, C must have p rows, so let C be a $p\times r$ matrix.
C is a $p\times r$ matrix, B m... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1080284",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 1
} |
How close apart are two message - "Document Distance" algorithm I was looking at this algorithm that computes how close apart are two texts from one another and the formula seems a bit weird to me.
The basic steps are:
*
*For each word encountered in a text you let a vector hold its frequency. For $n$ different word... | The first formula is related to dot product $$\mathbf u\cdot \mathbf v = \|\mathbf u\|\ \|\mathbf v\|\cos \theta$$
One property (compared with Euclidean distance) is that the cosine has a upper bound. Also does not depend so much on the length of the text, because each vector is normalised to unit length in the denomin... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1080377",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
} |
Borel $\sigma$-Algebra definition.
Definition:
The Borel $\sigma$-algebra on $\mathbb R$ is the $\sigma$-algebra B($\mathbb R$) generated by the $\pi$-system $\mathcal J$ of intervals $\ (a, b]$, where $\ a<b$ in $\mathbb R$ (We also allow the possibility that $\ a=-\infty\ or \ b=\infty$) Its elements are called Bore... | Ignore the phrase "$\pi$-system" for the time being : What you are given is a collection $\mathcal{J}$ of subsets of $\mathbb{R}$ and the $\sigma$-algebra you seek is the smallest $\sigma$-algebra that contains $\mathcal{J}$. This is the definition of the Borel $\sigma$-algebra. For example $\{1\}$ is a Borel set since... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1080473",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 0
} |
Number of $ 6 $ Digit Numbers with Alphabet $ \left\{ 1, 2, 3, 4 \right\} $ with Each Digit of the Alphabet Appearing at Least Once Find the number of 6 digit numbers that can be made with the digits 1,2,3,4 if all the digits are to appear in the number at least once.
This is what I did -
I fixed four of the digits to ... | A good way (not necessarily the best way) of doing such a problem as advised by my high school teacher is to first determine the number of combinations, then follow by permuting their arrangements. For your question, there are two cases, one of the four digits repeating three times, and two of the digits repeating twic... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1080555",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 1
} |
Biholomorphic function between given set and open unit disk Let $A=\{z\in\mathbb{C}:|z|<1, |z-1|<1\}$.
From Riemann mapping theorem we know that there exist biholomorphic function between $A$ and open unit disk.
How to construct these function ?
I tried to apply Möbius transformation and classical mappings ( exponent, ... | The circles $\{|z|=1\}$ and $\{|z-1|=1\}$ intersect at two points $a,\bar a\in\mathbb{C}$, $\operatorname{Im}a>0$. The Möbius mapping
$$
\phi(z)=\frac{z-a}{z-\bar a}
$$
transforms $A$ into an angle with vertex at $0$. A power function transforms this into a half plane, and another Möbius mapping transforms it in the op... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1080617",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Integral of $\log(\sin(x))$ using contour integrals I know the integral is possible with a simple fourier series expansion of $-\log(\sin(x))$
But I am interested in complex analysis, so I want to try this.
$$I = \int_{0}^{\pi} \log(\sin(x)) dx$$
The substitution $x = \arcsin(t)$ first comes to mind.
But that substitu... | We have
\begin{align}\int_0^{\pi} \log(\sin x)\, dx &= \int_0^{\pi} \log\left|\frac{e^{ix}-e^{-ix}}{2i}\right|\, dx\\
&= \int_0^{\pi} \log\left|\frac{e^{2ix}-1}{2}\right|\, dx\\
&= \int_0^{\pi} \log|e^{2ix}-1|\, dx - \int_0^{\pi} \log 2\, dx\\
&= \frac{1}{2}\int_0^{2\pi} \log|1 - e^{ix}|\, dx - \pi\log 2\\
&= \lim_{r\t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1080710",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
} |
How can I prove $|a_{m+1}-a_{m+2}+a_{m+3}-...\pm a_{n}| \le |a_{m+1}|$? Give a sequence $(a_n)$ that satisfying:
(i) the sequence is decreasing: $$a_n \ge a_{n+1} \forall n \in \mathbb N$$
(ii) the sequence is converging to $0$: $$(a_n) \rightarrow 0$$
Problem:Prove that $|a_{m+1}-a_{m+2}+a_{m+3}-...\pm a_{n}| \le |a_{... | Let us write $n=m+p$, so that the sum has $p$ elements. We prove that
$$\left| a_{m+1}-a_{m+2} +\cdots \pm a_{m+p} \right| \leq |a_{m+1}|=a_{m+1} \tag{1} $$ for all $m,p$:
First note that the $\pm$ actually equals $(+)$ if $p$ is odd, and is $(-)$ if $p$ is even. This motivates breaking the proof into two cases:
*
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1080845",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Finding singularities in circle of convergence of $f(z)$ and showing taylor series diverges there $$f(x)=\arctan(x)$$
I know that $$\dfrac{1}{1+x^2}=1-x+x^2-x^3+\cdots=\sum_{i=0}^\infty (-x)^i$$
Also: $$\arctan(x)=\int \dfrac{1}{1+x^2}dx = \int \sum_{i=0}^\infty (-x)^i dx = x - 0.5x^2+1/3x^3+\cdots=\sum_{i=0}^\infty \... | If a power series converges in some point $x_0$, then the derivation also converges
in the point $x_0$. Here, the derivation diverges at $x_0=-1$ and $x_0=1$, so the
original power series diverges there as well.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1080921",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
Show without differentiation that $\frac {\ln{n}}{\sqrt{n+1}}$ is decreasing
Show that the function $\displaystyle \frac {\ln{n}}{\sqrt{n+1}}$ is decreasing from some $n_0$
My try: $\displaystyle a_{n+1}=\frac{\ln{(n+1)}}{\sqrt{n+2}}\le \frac{\ln{(n)+\frac{1}{n}}}{\sqrt{n+2}}$
so we want to show that $\ln{n}\cdot(\sq... | If you subtract f(n+1)-f(n), then it comes down to $\frac{(n+1)^{\sqrt{n+1}}}{(n)^{\sqrt{n+2}}}\leq 1$ , take logs and it should be easy to finish?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1081050",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
} |
Sum of the series $\sinθ\sin2θ + \sin2θ\sin3θ + \sin3θ\sin4θ + \sin4θ\sin5θ + \cdots+\sin n\theta\sin(n+1)\theta$ terms The series is given:
$$\sum_{i=1}^n \sin (i\theta) \sin ((i+1)\theta)$$
We have to find the sum to n terms of the given series.
I could took out the $2\sin^2\cos$ terms common in the series. But wha... | Hint1:
$$\sum_{i=1}^n \sin (i\theta) \sin ((i+1)\theta)
= 1/2\sum_{i=1}^n \left(\cos (\theta)-\cos ((2i+1)\theta)\right)$$
Hint2:
$$ 2\sin (\theta)\sum_{i=1}^n \cos ((2i+1)\theta) = \sum_{i=1}^n \left(\sin ((2i+2)\theta) - \sin (2i\theta)\right) $$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1081132",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 1
} |
Number of Solutions of $y^2-6y+2x^2+8x=367$? Find the number of solutions in integers to the equation
$$y^2-6y+2x^2+8x=367$$
How should I go about solving this?
Thanks!
| Note: this method requires basic knowledge of conic sections.
First of all it is equation of an ellipse,
And its vertex are $(-2,3-8\sqrt6)|(-2,3+8\sqrt6) \approx(-2,-16.6)|(-2,22.6)$
So, you can vary $y$ from $-16$ to $22$ and you will get all integral solutions.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1081212",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
} |
Simplification a trigonometric equation $$16 \cos \frac{2 \pi}{15} \cos\frac{4 \pi}{15} \cos\frac{8 \pi}{15} \cos\frac{14 \pi}{15}$$
$$=4\times 2 \cos \frac{2 \pi}{15} \cos\frac{4 \pi}{15} \times2 \cos\frac{8 \pi}{15} \cos\frac{14 \pi}{15}$$
I am intending in this way and then tried to apply the formula, $2\cos A \cos ... | Using the identity, $\cos\theta\cos2\theta\cos2^2\theta\cdots\cos2^{n-1}\theta=\dfrac{\sin2^n\theta}{2^n\sin\theta}$
By putting $n=4$ and $\theta=\dfrac{\pi}{15}$ you will get,
$- \cos\dfrac{\pi}{15}\cos \dfrac{2 \pi}{15} \cos\dfrac{4 \pi}{15} \cos\dfrac{8 \pi}{15}=-\dfrac{\sin2^4\dfrac{\pi}{15}}{2^4\sin\dfrac{\pi}{15... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1081316",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Combinatoric proof for $\sum_{k=0}^n{n\choose k}\left(-1\right)^k\left(n-k\right)^4 = 0$ ($n\geqslant5$) I'm trying to prove the following:
For every $n \ge 5$:
$$\sum_{k=0}^n{n\choose k}\left(-1\right)^k\left(n-k\right)^4 = 0$$
I've tried cancelling one $(n-k)$, and got this:
$$n\sum_{k=0}^{n-1}{n-1\choose k}\left(-1\... | It will be useful to know:
$\mathbf{ Theorem.}$ Let $p(x)= a_0+a_1x+\cdots +a_nx^n$ be $\mathit{any}$ polynomial in $\mathbb{C}[x]$ (of degree $\leq n$), then
$$ \sum_{k} {n\choose k}(-1)^k p(k)=(-1)^n n! a_n.$$
So in particular when $p$ has degree<n, then such sums are $0$.
Proof: see Graham, Knuth, Patashnik, C... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1081364",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 6,
"answer_id": 4
} |
Differential geometry problem about curves Please, help me to solve the following differential geometry problem.
Equations $F(x, y, z) = 0$ and $G(x, y, z) = 0$ denote space curve $L$.
Gradients of $F$ and $G$ are not collinear in some point $M(x_0, y_0, z_0)$ which belongs to the curve $L$.
Find:
*
*The tangent line ... | The curvature vector is defined to be the derivative of the unit tangent vector. Let us say that $\gamma(s)$ is the arc length parametrization of your curve. That is to say: $$F(\gamma(s))=0$$ $$G(\gamma(s))=0.$$ To find the tangent vector you derived and obtained: $$\nabla F(\gamma)\cdot\gamma'=0$$ $$\nabla G (\gamma)... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1081432",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Inequality in Integral Show that $\dfrac{28}{81}<\int_0^\frac{1}{3}e^{x^2}dx<\dfrac{3}{8}$.
It would be great if a solution based on the Mean Value Theorem for Integrals is posted.
| To obtain the upper bound, note that for $|x| < 1$
$$e^{x^2} = \sum_{k=0}^{\infty}\frac{x^{2k}}{k!} < \sum_{k=0}^{\infty}x^{2k} = \frac1{1-x^2}.$$
Hence,
$$\int_0^{1/3}e^{x^2}\,dx < \int_0^{1/3}\frac{dx}{1-x^2} < \frac1{3}\frac{1}{1-(1/3)^2}= \frac1{3}\frac{9}{8}= \frac{3}{8}.$$
The lower bound is easily derived by tr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1081512",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Understanding linearly independent vectors modulo $W$ We've learned in class:
Let $W \subseteq V$, a subspace. $v_1, \ldots, v_k \in V$ are said to be linearly independent modulo $W$ if for all $\alpha_1, \ldots, \alpha_k: \sum_{k=1}^n \alpha_i v_i \in W \implies \alpha_1 = \ldots = \alpha_k = 0$
Can you explain it... | Consider the quotient space $V/W.$ Then the given condition says that the image of the vectors $v_1, v_2, \cdots , v_n \in V,$ under the natural map $V \rightarrow V/W$ is also linearly ondependent.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1081593",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
Irrational number not ocurring in the period of rational numbers Write each rational number from $(0,1]$ as a fraction $a/b$ with $\gcd(a,b)=1$, and cover $a/b $ with the interval
$$
\left[\frac ab-\frac 1{4b^2}, \frac ab + \frac 1{4b^2}\right].
$$
Prove that the number $\frac 1{\sqrt{2}}$ is not covered. What I did w... | I got a good solution from one of my teachers...$$\left|\frac ab-\frac 1{\sqrt2}\right|\left(\frac ab+\frac 1{\sqrt2}\right)=\left|\frac {a^2}{b^2}-\frac 12\right|=\frac {|2a^2-b^2|}{2b^2}\gt\frac 1{2b^2}$$ Also we know that $\frac ab+\frac 1{\sqrt2}\lt2=>\left|\frac ab-\frac 1{\sqrt2}\right|\gt\frac 1{4b^2}$. Hence pr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1081688",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
showing the equation is true Am from writing my class test and I have failed to answer this question. None of my friends could help me after the test was over.the question reads:
If $P$ is the length of the perpendicular from the origin to the line which intercepts the axis at points $A$ and $B$, then show that $$\fra... | Hint:
Let $Q$ be the point of intersection of the lines $P$ and $AB$. Then, $\triangle AOQ$, $\triangle BOQ$, and $\triangle OAB$ are all similar.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1081866",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
How to compare products of prime factors efficiently? Let's say that $n$ and $m$ are two very large natural numbers, both expressed as product of prime factors, for example:
$n = 3×5×43×367×4931×629281$
$m = 8219×138107×647099$
Now I'd like to know which is smaller. Unfortunately, all I have is an old pocket calculator... | As a practical example, in a programming language like C or C++, you can easily calculate both products using floating-point arithmetic, and you can easily calculate both products modulo 2^64 using unsigned integer arithmetic.
If you can estimate the rounding error in the floating-point arithmetic products, and prove ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1081963",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "13",
"answer_count": 3,
"answer_id": 2
} |
What parts of a pure mathematics undergraduate curriculum have been discovered since $1964?$ What parts of an undergraduate curriculum in pure mathematics have been discovered since, say, $1964?$ (I'm choosing this because it's $50$ years ago). Pure mathematics textbooks from before $1964$ seem to contain everything in... | As elementary number theory is still considered to be a pure mathematics course, much has entered this field which is currently being applied. In 1964, there was no Diffie-Hellman nor RSA public key cryptography; nor were elliptic curves being used for digital signatures, key agreements, or to generate ``random'' numbe... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1082010",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "123",
"answer_count": 19,
"answer_id": 18
} |
Equation with three variables I am confused as how to solve an equation with three squared variables to get its integer solutions? As:
$$x^2+y^2+z^2=200$$
Thanks!
| If you are doing it to find solution to your previous question about the area of that triangle then you should use heron's formula and simplify it in terms of $x, y, z$ like,
$\sqrt{s(s-a)(s-b)(s-c)}=\dfrac{1}{4}\sqrt{(x+y+z)(x+y-z)(x-y+z)(-x+y+z)}$, which will simplify into,
$\dfrac{1}{4}\sqrt{-(x^4+y^4+z^4)+2(x^2y^2+... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1082092",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
How are Zeta function values calculated from within the Critical Strip? We note that for $Re(s) > 1$
$$ \zeta(s) = \sum_{i=1}^{\infty}\frac{1}{i^s} $$
Furthermore
$$\zeta(s) = 2^s \pi^{s-1} \sin \left(\frac{\pi s}{2} \right) \Gamma(1-s) \zeta(1-s)$$
Allows us to define the zeta function for all values where
$$Re(s) < ... | An extension of the area of convergence can be obtained by rearranging the original series. The series
$$\zeta(s)=\frac{1}{s-1}\sum_{n=1}^\infty \left(\frac{n}{(n+1)^s}-\frac{n-s}{n^s}\right)$$
converges for $\Re s > 0$. See here.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1082139",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 7,
"answer_id": 6
} |
How to prove that $\sin(\theta)= 2\sin(\theta/2)\cos(\theta/2)$? $$\sin (\theta) = 2 \sin \left(\frac{\theta}{2}\right) \cos \left(\frac{\theta}{2}\right)$$
How? Please help. Thanks in advance.
| If you know complex numbers, you can also use:
$e^{i\theta} = \cos(\theta)+i\sin(\theta)$ with $e^{i\theta} = (e^{i\frac{\theta}{2}})^2$ (this "trick" yields De Moivre's formula).
Look at the imaginary part of that expression.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1082266",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
} |
Number of valid NxN Takuzu Boards a.k.a 0h h1 (details inside)? Takuzu a logic puzzle which has a $N \times N$ grid filled with $0$'s and $1$'s following these rules:
*
*Every row/column has equal number of $0$'s and $1$'s
*No two rows/columns are same
*No three adjacent (all three horizontal or all three vertical... | This is a hard problem and I doubt it's been studied seriously. However, my computer tells me that for the first few boards, you can get:
*
*2 x 2 ... 2
*4 x 4 ... 72
*6 x 6 ... 4,140
*8 x 8 ... 4,111,116
After that my algorithm is too slow to keep going.
I searched on OEIS for any variation of this sequence bu... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1082341",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 1,
"answer_id": 0
} |
The locus of points $z$ which satisfy $|z - k^2c| = k|z - c|$, for $k \neq 1$, is a circle Use algebra to prove that the locus of points z which satisfy $|z - k^2c| = k|z - c|$, for $k \neq 1$ and $c = a + bi$ any fixed complex number, is a circle centre $O$.
Give the radius of the circle in terms of $k$ and $|c|$.
I s... | Hint: If you square both sides, and expand out (initially writing things in terms of your variables and their complex conjugates), you will get a lot of useful cancellation.
Here is the beginning of such a calculation, to help you out.
Squaring the left hand side yields:
$$(z-k^2c)(\overline{z}-k^2\overline{c})=|z|^2+k... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1082442",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 1
} |
Integration by differentiating under the integral sign $I = \int_0^1 \frac{\arctan x}{x+1} dx$ $$I = \int_0^1 \frac{\arctan x}{x+1} dx$$
I spend a lot of my time trying to solve this integral by differentiating under the integral sign, but I couldn't get something useful. I already tried:
$$I(t) = \int_0^1 e^{-tx}\frac... | I have an answer, though it is without using differentiation under integration
$$I=\int_{0}^1 \frac{\tan^{-1}x}{1+x}dx=\int_{0}^{\pi/4}\frac{\theta \sec^2\theta}{1+\tan \theta}d\theta\\ =\int_{0}^{\pi/4}\frac{(\pi/4-\theta)\sec^2(\pi/4-\theta)}{1+\frac{1-\tan\theta}{1+\tan\theta}}d\theta\\=\int_{0}^{\pi/4}\frac{(\pi/4-... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1082491",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 7,
"answer_id": 0
} |
If $(a,4)=2=(b,4)$, prove $(a+b,4)=4$. I'm almost embarrassed to be asking about a problem such as this one (exercise 12 in Niven 1.2), but here goes:
Given $(a,4)=(b,4)=2$, show that $(a+b,4)=4$.
I have plenty of tricks for working with gcds multiplicatively, however I have honestly no idea how to attack this proble... | $2$ divides both of $a$ and $b$, but $4$ doesn't, then $a,b$ are congruent with $2$ (mod $4$), then $a+b$ is a multiple of $4$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1082565",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 5,
"answer_id": 2
} |
Find the value of : $\lim_{ x \to \infty} \left(\sqrt{x+\sqrt{x+\sqrt{x}}}-\sqrt{x}\right)$ I need to calculate the limit of the function below:
$$\lim_{ x \to \infty} \left(\sqrt{x+\sqrt{x+\sqrt{x}}}-\sqrt{x}\right)$$
I tried multiplying by the conjugate, substituting $x=\frac{1}{t^4}$, and both led to nothing.
| Multiply both numerator and denominator by $\sqrt{x+\sqrt{x+\sqrt{x}}}+\sqrt{x}$
You will get $$\dfrac{\sqrt{x+\sqrt{x}}}{\sqrt{x+\sqrt{x+\sqrt{x}}}+\sqrt{x}}$$
Divide both numerator and denominator by $\sqrt{x}$
$$\dfrac{\sqrt{1+\dfrac{1}{\sqrt{x}}}}{\sqrt{1+\sqrt{\dfrac{1}{x}+\dfrac{1}{x\sqrt{x}}}}+1}$$
On finding th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1082649",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "12",
"answer_count": 6,
"answer_id": 0
} |
What is the difference between a function and a formula? I think that the difference is that the domain and codomain are part of a function definition, whereas a formula is just a relationship between variables, with no particular input set specified.
Hence, for two functions $f$ and $g$, $f(x)$ can be equal to $g(x)$ ... | A function is a map from one set to another. They are the same if both the domain and the 'formula' are the same. A formula on the other hand is a word physicists and chemists like to use for a function that expresses a relation between variables that arise in nature.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1082718",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 4,
"answer_id": 2
} |
How can I use these two bijections to form a bijection $\mathbb{R}^{\mathbb{N}} \to \mathbb{R}$? Build a bijection $\mathbb{R}^{\mathbb{N}} \to \mathbb{R}$ by using the two following known biections $\varphi:{\mathbb{N}} \to {\mathbb{N}} \times {\mathbb{N}}$ and $\psi:{\mathbb{R}} \to \{0,1\}^{{\mathbb{N}}}$.
Edit.
M... | Hint: Note that $\mathbb R^\omega=(2^\omega)^\omega=2^{\omega^2}=2^\omega=\mathbb R$. If you can find a bijection for each equality, composing them should give you your desired bijection.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1082818",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 2,
"answer_id": 1
} |
A continuous function on $[0,1]$ not of bounded variation I'm looking for a continuous function $f$ defined on the compact interval $[0,1]$ which
is not of bounded variation.
I think such function might exist. Any idea?
Of course the function $f$ such that
$$
f(x) =
\begin{cases}
1 & \text{if $x \in [0,1] \cap \math... | Consider any continuous function passing through the points $(\frac1{2n},\frac1n)$ and $(\frac1{2n+1},0)$, e.g. composed of linear segments. It must have infinite variation because $\sum\frac1n=\infty$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1082897",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "27",
"answer_count": 4,
"answer_id": 2
} |
Sum of a series of a number raised to incrementing powers How would I estimate the sum of a series of numbers like this: $2^0+2^1+2^2+2^3+\cdots+2^n$. What math course deals with this sort of calculation? Thanks much!
| late to the party but i think it's useful to have a way of getting to the general formula.
this is a geometric serie which means it's the sum of a geometric sequence (a fancy word for a sequence where each successive term is the previous term times a fixed number). we can find a general formula for geometric series fol... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1082963",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 6,
"answer_id": 5
} |
Recommended Books for AIME/USAMO Preparation? What books would you guys recommend to learn number theory, geometry, combinatorics, and algebra at a level appropriate for the AIME and/or USAMO? With the month of February approaching, the month the AMC (American Mathematics Competition) series takes place in, I'm realizi... | I find Larson "Problem Solving Through Problems" a great book. It's problem after problem organized by approaches and topics. Many of the solutions are particularly clever. Yet he breaks the thought process into steps so you can really gain an insight as to how to solve problems that look pretty challenging at first gl... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1083036",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 1
} |
Do there exist nontrivial global solutions of the PDE $ u_x - 2xy^2 u_y = 0 $? Consider the following PDE,
$$ u_x - 2xy^2 u_y = 0 $$
Does there exist a non-trivial solution $u\in \mathcal{C}^1(\mathbb{R}^2,\mathbb{R})$?
It is clear that all solutions for $u\in \mathcal{C}^1( \mathbb{R}^2_+,\mathbb{R})$ are given by $... | The only potential problem is at $y = 0$. For continuity we need $f(t)$ to go to a limit as $t \to \pm \infty$, and then we can define $u(x,0)$ as that limit (let's call it $c$). For partial differentiability wrt $y$ we need existence of
$$ \lim_{y \to 0} \dfrac{u(x,y) - u(x,0)}{y} = \lim_{y \to 0} \dfrac{f(x^2 - 1/y... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1083113",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Simulating elastic collision I wrote a simple program where i can move around some objects. Every object has a bounding box and I use hooke's law to apply forces to the colliding objects. On every tick, I calculate the forces, divide them by the masses, multiply by elapsed time to get the velocities and then move the o... | You want to implement a coefficient of restitution. The common way of doing this with a penalty (spring-based) method is to modify your force depending on the relative velocity. You don't list how you estimate the closest distance between the two objects, but the main idea is to multiply the force (exerted on both obje... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1083195",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Functional inequalities Let x,y,z be the lengths of the sides of a triangle, and let$$f(x,y,z)=\left|\frac {x-y}{x+y}+\frac {y-z}{y+z}+\frac {z-x}{z+x}\right|.$$ Find the upper limit of $f(x,y,z)$. I simply used the fact that $|x-y|\le z$ and the other 3 to prove that $f(x,y,z)\le \frac 18=0.125$. But the answer given ... | Solution:let $x\ge y\ge z$, First Note
$$I=\dfrac{x-y}{x+y}+\dfrac{y-z}{y+z}+\dfrac{z-x}{z+x}=\dfrac{(x-y)(x-z)(y-z)}{(x+y)(y+z)(x+z)}$$
then let $$x=c+b,y=c+a,z=a+b,c\ge b\ge a>0$$
so
$$I=\dfrac{(c-b)(c-a)(b-a)}{(2a+b+c)(2b+a+c)(2c+a+b)}<\dfrac{(c-b)cb}{(b+c)(2b+c)(2c+b)}=\dfrac{1}{F}$$
then we only find $F$ minimum
l... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1083250",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Finding the inverse of a non linear function Let $F:\mathbb{R}^2\to \mathbb{R}^2$ be the diffeomorphism given by $$F(x,y)=(y+\sin x, x) $$ Find $F^{-1}$.
I know that the answer is $F^{-1}(x,y)=(y,x-\sin y$), this can be shown to be true by taking $F\circ F^{-1}(x,y)=F(F^{-1}(x,y))=F(y,x-\sin y)=(x-\sin y+\sin y,y)=(x,y... | Just found the answer:
*
*Replace $f\left(x,y\right)$ by $\left(u,v\right)$ resulting in:
$x+y+1=u$ and $x-y-1=v$
*Switch $x$ and $u$ and switch $y$ and $v$ resulting in: $u+v+1=x$
and $u-v-1=y$
*Solve for $u$ and $v$ resulting in: $u=\frac{1}{2}x+\frac{1}{2}y$
and $v=\frac{1}{2}x-\frac{1}{2}y-1$
*Replace $\left(... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1083342",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Recurrence Relation Involving the gamma Function I'm having some doubts about my approach to the following problem. I am given that the function $k(z)$ is defined such that,
$$k(z)=\Gamma\left(\frac{1}{2}+z\right)\Gamma\left(\frac{1}{2}-z\right)\cos{\pi z}$$
I'm required to find the recurrence relation linking $k(z+1)$... | You made a sign error.
\begin{align}
\frac{k(n+1)}{k(n)}
&=\frac{\Gamma\left(\frac{1}{2}+n+1\right)\Gamma\left(\frac{1}{2}-n\color{red}{-1}\right)(-\cos(\pi n))}{\Gamma\left(\frac{1}{2}+n\right)\Gamma\left(\frac{1}{2}-n\right)\cos(\pi n)}\\
&=\frac{\left(\frac{1}{2}+n\right)\Gamma\left(\frac{1}{2}+n\right)\frac{\Gamma\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1083502",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
} |
Find the number of polynomial zeros of $z^4-7z^3-2z^2+z-3=0$.
Find the number of solutions of $$z^4-7z^3-2z^2+z-3=0$$ inside the unit disc.
The Rouche theorem fails obviously. Is there any other method that can help?
I have known the answer by Matlab, but I have to prove it by complex analysis.
Thanks!
| Note that $|z^4-2z^2+z-3|<7$ if $|z|=1$. The triangle inequality only says "$\leq$" but equality can only occur if $z^4$, $-2z^2$, $z$ and $-3$ all have the same argument. So it suffices to check $z=-1$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1083540",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 1
} |
proof: primitive pythagorean triple, a or b has to be divisible by 3 I'm reading "A friendly introduction to number theory" and I'm stuck in this exercise, I'm mentioning this because what I need is a basic answer, all I know about primitive pythagorean triplets is they satisfy $a^2 + b^2 = c^2$ and a, b and c has no c... | \begin{array}{|c|c|}
\hline
n \pmod 3 & n^2 \pmod 3 \\
\hline
0 & 0 \\
1 & 1 \\
2 & 1 \\
\hline
\end{array}
If neither $a$ nor $b$ is a multiple of $3$, then $a^2 + b^2 \equiv c^2 \pmod 3$ becomes $1 + 1 \equiv c^2 \pmod 3$, which simplifies to $c^2 \equiv 2 \pmod 3$; which has no solution.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1083592",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 1
} |
Is $\ln(1+\frac{1}{x-1}) \ge \frac{1}{x}$ for all $x \ge 2$? Plotting both functions $\ln(1+\frac{1}{x-1})$ and $\frac{1}{x}$ in $[2,\infty)$ gives the impression that $\ln(1+\frac{1}{x-1}) \ge \frac{1}{x}$ for all $x \ge 2$.
Is it possible to prove it?
| hint :
$$f(x)=ln(1+\frac{1}{x-1})-\frac{1}{x}\\x≥2\\ f'<0$$f(x) is decreasing function ,but f(x) is above the x axis $$\\f(2)>0,f(\infty)>0\\f(x)>0\\so\\ln(1+\frac{1}{x-1})-\frac{1}{x}≥0\\ln(1+\frac{1}{x-1})≥\frac{1}{x}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1083668",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 3
} |
A union in the proof of Egorov's theorem Egorov's Theorem: Let $(X,M,\mu)$ be a finite measure space and $f_n$ a sequence of measurable functions on $X$ that converges pointwise a.e. on $X$ to a function $f$ that is finite a.e. on $X$. Then for each $\epsilon>0$, there is a measurable subset $X_{\epsilon}$ of $X$ for w... | An arguably more efficient way to see this is to keep in mind the close relation between the logical quantifiers $\forall,\exists$ and the set operations $\cap,\cup$, respectively. In this regard, an argument showing $\bigcup_n X_n^m=X_0$ might go like this:
Let $x\in X_0$. Since $f_n(x)\to f(x)$, $\forall\epsilon>0,\e... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1083733",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.