Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
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Proving two linear operators are equivalent iff they have the same rank From Halmos's Finite-Dimensional Vector Spaces, #7 section 51, rank and nullity. Prove that two linear operators are equivalent iff they have the same rank.
Typically when Halmos phrases exercises like that it means that they are provable, but I do... | The linear transformations $A$ and $B$ are equivalent if there exist invertible transformations $P$ and $Q$ so that (Halmos page 87, Exercise 6, (b)):
$$
A = P^{-1}BQ.
$$
If this is the case, then the rank of $A$ and $B$ are the same (Halmos, $\S50$, THM3, (10)):
$$
\rho(A) = \rho(P^{-1}BQ) = \rho(P^{-1}B)=\rho(B).
$$
... | {
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"timestamp": "2023-03-29T00:00:00",
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Probability of at least one random number out of 3 being greater than 3 other random numbers? $$\{?,?,?\mid?,?,?\}$$
There are 6 random numbers (drawn from arbitrarily large pool). What is the probability that biggest number lies in second half? Answer is $1/2$ but I tried to solve it with combination. Can you show me ... | Your problem is that the probability that the fourth number is greater than the preceding three is not $\frac 18$, it is $\frac 14$. Your calculation of $\frac 18$ assumes that the fourth number being greater than the first and the fourth number being greater than the second are independent. They are not. If the fou... | {
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"url": "https://math.stackexchange.com/questions/1170539",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Congruences- find the number of solutions
*
*Find the number of solutions of $ a_1x_1+ ... +a_nx_n =$ b (mod m).
I was thinking of denoting $ d = (a_1,a_2,...,a_n,m)$.
Then the congruence has no solutions if $d$ doesn't divide $b$. And it has solutions when $d$ divides $b$.
I was thinking of using induction, but I ... | First Question:
Theorem :
The congruence $a_1x_1+\cdots+a_nx_n\equiv b\mod m$ has exactly $gcd(a_1,\cdots,a_n,m)m^{n-1}$ when $gcd(a_1,\cdots,a_n,m)$ divides $b$, and no solution otherwise.
Proof: by induction:
*
*For $n=1$ let $d=gcd(a_1,m)$, $a=\frac{a_1}{d}$, $b'=\frac{b}{d}$ and $m'=\frac{m}{d}$ so the equat... | {
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Proof by induction that $P_n(a) \neq 0$ for $n>3$. Let $a,b,c$ be 3 non-zero coprime integers and $P_n(a)=a^n+\sum_{k=1}^{n}{{n\choose{k}}a^{n-k}(c^k-b^k)}$
Show that if $P_3(a) \neq 0$ then for all $n \geq 3, P_n(a)\neq 0$
Using mathematical induction, how can I prove it?
| Hint: $\sum_{k=1}^{n}{{n\choose{k}}a^{n-k}(c^k-b^k)} = \sum_{k=1}^{n}{{n\choose{k}}a^{n-k}c^k}-\sum_{k=1}^{n}{{n\choose{k}}a^{n-k}b^k}=(a+c)^n-(a+b)^n$
| {
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Relationship between Fibonacci's secuence and $x^2 - x - 1$. On the end of Apostol's Mathematical Analysis' first chapter, one can find the following exercise (and I paraphrase):
Prove that the $n$-th term of the Fibonacci sequence is given by $$x_n = \frac{a^n - b^n}{a-b}$$ Where $a$ and $b$ are the roots of $x^2 - x... | $$x^2-x-1=0$$
$$x^2=x+1$$
Multiply that by $x$ and you get:
$$x^3=x^2+x$$
Then replace $x^2$ by $x+1$, which yields:
$$x^3=2x+1$$
Repeat this process. Using induction, you can easily prove that:
$$x^n=x_nx+ x_{n-1} $$
Since $a$ and $b$ are the solutions of $x^2-x-1=0$ then they are also the solutions of $x^n=x_nx+1$,wh... | {
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Is $\mathbb Z_1$ a subgroup of $\mathbb Z_{10}$ $\mathbb Z_1 = \{0, 1\} \subseteq \{0, 1, 2, 3, 4, 5, 6, 7, 8, 9\} = \mathbb Z_{10}.$
Since $0 + 1 = 1$, $1 + 0 = 1$ and $1 + 1 = 0$, the identity in $\mathbb Z_1$ is $0$ and the inverse element is $1.$ Since $0 + 1 \in \mathbb Z_1$, the set $\mathbb Z_1$ is closed under ... | First of all, $\mathbb{Z}_1$ would be defined as $\{0\}$ if it were ever used. What you are referring to is $\mathbb{Z}_2$. Second, your proof is not correct. The $0$ and $1$ in the first group are completely unrelated to the $0$ and $1$ in the second group. The subset $\{0,1\}$ of $\mathbb{Z}_{10}$ is not a subgroup b... | {
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Open balls in euclidean space are homeomorphic to the whole space The following question is from Fred H. Croom's book "Principles of Topology"
Prove that each open ball $B(a,r), a\in \mathbb{R}^n, r>0$, considered as a subspace of $\mathbb{R}^n$, is homeomorphic to $\mathbb{R}^n$.
After much studying, I concluded the... | Hing: Take $B(0,1)$. The let $f(x) = \frac{x}{1-|x|}$.
| {
"language": "en",
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Is $\alpha$ algebraic or not? I am trying to find out if $\alpha = \sqrt[]{2} - i$ for $F = \mathbb{Q}$ is algebraic or not.
I found that $\alpha^2 = 1 - 2i\sqrt[]{2}$ and have been trying to construct a suitable polynomial, but I have so far been unable to. Does anyone have a nice way to do this?
| Hint: The sum of algebraic numbers is algebraic. One proof uses the resultant.
Wolfram alpha calculates the minimal polynomial of $\sqrt{2}-i$ to be $x^4-2x^2+9$. One way to come up with this on your own is to consider $1,\alpha,\alpha^2,\alpha^3,\alpha^4$ as vectors in $\mathbb{Q}[1,\sqrt{2},i]$ and find a linear comb... | {
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For what values of $k$ is $g(x)=x^3+kx^2+x$ one-to-one? I need to find for what values of $k$ $g(x)=x^3+kx^2+x$ is one-to-one. I tried finding for what values it is strictly increasing and got the derivative to be $3x^2+2kx+1>0$, but I'm not really sure where to go from there since there are two variables.
| Hints:
Your function $g(x)$, being a cubic polynomial, is one-to-one if and only if the derivative has at most one root. One root is allowed.
You can tell how many roots a quadratic equation $ax^2+bx+c=0$ has by examining its discriminant $b^2-4ac$. If the discriminant is positive, there are two roots; if zero, one roo... | {
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If Q is a matrix of orthonormal bases, is Q times its transpose anything special? I know that if Q is a matrix of orthonormal bases then Q transpose times Q is the identity matrix. Is Q times Q transpose anything special?
| It is also the identity matrix. This holds in general: if $B$ is a left inverse of a square matrix $A$, i.e. $BA=I$, then $B$ is also a right-inverse of $A$, i.e. $AB=I$.
| {
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Volume of a Solid, $x^2 - y^2 = a^2$ The question is
Find the volume of a solid rotated around the y axis, bounded by the given curves: $$x^2 - y^2 = a^2$$
$$x = a + h$$
I am lost by the number of variables in this question and the question does not tell me what kind of variables they are, only that they are both gre... | This is known as a hyperboloid of revolution of one sheet.
Forget the second equation, it only means that to get both x and y real, x should be equal or grater than a by a small amount h. $y > x$ always in the first quadrant for a quarter branch of this (rectangular) hyperbola.The limits of y are $0$ and $y_1$ in:
$$ ... | {
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In an equilateral spherical triangle, show that $\sec A=1+\sec a$ Q. In an equilateral spherical triangle, show that $\sec A=1+\sec a$
So $A$ is the vertex or the angle of the triangle and $a$ is the side of the equilateral spherical triangle.
I started off the proof by using the law of cosines:
$$\cos(a)-\cos(a)\cos(a... | Let $\,x:=\cos(a)\,$ and $\,X:=\cos(A).\,$ Then
$$ 1+\sec(a)-\sec(A) = 1+\frac1x-\frac1X = \frac{-x+X+xX}{xX}.$$
$$ \cos(a)(1-\cos(a))-(1-\cos^2(a))\cos(A)=x(1-x)-(1-x^2)X=(x-1)(-x+X+xX).$$
Thus, if $\,x\ne 1,\,$ then
$\, \cos(a)(1-\cos(a))=(1-\cos^2(a))\cos(A)\, $ implies that
$\, 1+\sec(a)=\sec(A).$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Finding Fixed Point If there is a continuous function from the closed unit disk to itself such that it is identity map on boundary, must it admit a fixed point in the interior of the disk?
| No. Consider the set of vertical line segments which are intersections of lines $x=c$ with the closed unit disc $x^2+y^2 \le 1.$ Each of these segments goes from a lower circle boundary point $P_c$ to an upper one $Q_c.$ Along each such segment we can choose a mapping which moves all the interior points but keeps the e... | {
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If $ f(x \cdot f(y) + f(x)) = y \cdot f(x) + x $, then $f(x)=x$ Let $ f : \mathbb{Q} \rightarrow \mathbb{Q} $ be a function which has the following property:
$$ f(x \cdot f(y) + f(x)) = y \cdot f(x) + x \;,\; \forall \; x, y \in \mathbb{Q} $$
Prove that $ f(x) = x, \; \forall \; x, y \in \mathbb{Q} $.
So far, I've foun... | If you have $f(f(x))=x$ it means that the function is onto
Furthermore, assume that $f(y_1) = f(y_2)$ for some $y_1 \neq y_2$
$$ f(-f(y_1)-1 ) =f(-f(y_1)-1 ) $$
$$\Rightarrow -y_1 -1 = -y_2 -1 $$
$$\Rightarrow y_1 = y_2 !! $$
Thus the function is $1$ $to$ $1$, its inverse $f^{-1}$ exists,
in addition to $f(f(x))=x... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1171599",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Find generating function for sequence I am suppose to find generating function for sequence $(e_n)_0^\infty$ where $e_n$ is number of ways how to write a number $n$ as a sum of four natural odd numbers ($e_n$ is basically a number of ordered fours $(\alpha, \beta, \gamma, \delta)$ odd natural numbers that $\alpha + \be... | I’ll get you started. Consider the product
$$(x_1+x_1^3+x_1^5+\ldots)(x_2+x_2^3+x_2^5+\ldots)(x_3+x_3^3+x_3^5+\ldots)(x_4+x_4^3+x_4^5+\ldots)\;;$$
a typical term has the form $x_1^\alpha x_2^\beta x_3^\gamma x_4^\delta$, where $\alpha,\beta,\gamma$, and $\delta$ are odd positive integers. If you were to drop the subscr... | {
"language": "en",
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"question_score": "2",
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Probability of (1 + min(X,Y))/(1 + min(X,Z))? I have been trying to derive the probability $\Pr \left[ {\frac{{1 + \min \left( {X,Y} \right)}}{{1 + \min \left( {X,Z} \right)}} < c} \right]$, where X, Y, and Z are independent and follow exponential distribution with parameters $λ_x$, $λ_y$, and $λ_z$, respectively. c he... | Let $B=(\frac{1 + \min \left( {X,Y} \right)}{1 + \min \left( {X,Z} \right)} < c)$. You have define four events:
\begin{align}
A_1&=(X < Y,X < Z)\\
A_2&=(X < Y,X > Z)\\
A_3&=(X > Y,X < Z)\\
A_3&=(X > Y,X > Z)\\
\end{align}
and then wrote \begin{align}
Pr(B)&=\sum_{i=1}^4P(B|A_i)P(A_i)\\
\end{align}
For $i=1$ you then wr... | {
"language": "en",
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Help in making an animated locus is needed
I have triangular cardboard ABC, right angled at C. As shown in the attached, initially, A and B are resting on the x- and y- axes respectively. A is then allowed to slide along the x-axis with B slides accordingly along the y-axis.
The locus of C has found to be:- $9x^2 – 16... | Here's what I tried in geogebra:
*
*Create point $A$ on X-axis.
*Make circle of radius $5$ centered at $A$.
*Intersect the circle and Y-axis at point $B$.
*Make angle of $53.13^\circ$ at $A$ and $36.87^\circ$ at $B$.
*Intersect these angles at $C$.
*Join line segments $AB$,$BC$ and $CA$ and hide everything else... | {
"language": "en",
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Term for similarity transformation which is not a translation What's the best (i.e. most concise) term to refer to an orientation-preserving similarity transformation which is not a translation? Here are some descriptions I could think of, but all of them feel rather bulky. I hope they are as equivalent to one another ... | I think that what you're looking for is simply called "affinity".
From Wikipedia:
Examples of affine transformations include translation, scaling, homothety, similarity transformation, reflection, rotation, shear mapping, and compositions of them in any combination and sequence.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1172077",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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"answer_id": 2
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First order differential equation with non constant coefficients I have the following system :
$$\begin{cases}(t^2+1)x'(t)=tx+y+2t^2+1\\(t^2+1)y'(t)=-x+ty+3t\end{cases}$$
How can it be solved ?
What I have tried so far :
*
*polynomials of the first, second degree as solutions - didn't work
*One can notice that if ... | We are given:
$$\begin{cases} (t^2+1)x'(t)=tx+y+2t^2+1\\ (t^2+1)y'(t)=-x+ty+3t\end{cases}$$
From the first equation $(t^2+1)x'(t)=tx+y+2t^2+1$, we have:
$$\tag 1 y = t^2 x' + x' - t x - 2 t^2 -1$$
Taking the derivative of $(1)$, yields:
$$\tag 2 y' = t^2 x'' + x'' + t x' -x - 4 t$$
Substituting $(1)$ and $(2)$ into th... | {
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Proving a the square root of a function to be Riemann integrable How could I prove that the square root of a Riemann integrable function $f$ on a given interval, where $f(x) > 0$ is also Riemann integrable?
| The most straightforward way is Reveillark's, the validity of which follows immediately from Lebesgue's criterion for Riemann-integrability.
Here is a direct proof that doesn't appeal to this result.
Let $f \geq 0$ be Riemann-integrable on $[a,b]$. Let $\varepsilon > 0$ be given. Since $f$ is integrable, there exist s... | {
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"timestamp": "2023-03-29T00:00:00",
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Problem with a calculation using homogeneous coordinates Suppose in a complex projective plane CP2, I use homogeneous coordinates
$$(x, y, z)$$
and the following transformation:
$$A =\begin{pmatrix}
\cos{\alpha} & -\sin{\alpha} & 0 \\
\sin{\alpha} & \cos{\alpha} & 0 \\
0 & 0 & 1
\end{pmatrix}
$$
The point $I... | Projective transformations take (projective) lines to (projective) lines but they don't preserve the "midpoints of lines" or "sums of points". In fact, midpoints and "sums" are not even defined in the projective geometry.
You can just consider $\mathbb{RP}^1$, homogenous coordinates and the identity. "$(1,0)+(0,1)$" is... | {
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Global maxima of $f(x,y)=x^2y$ restricted to D Let $f(x,y) = x^2y$ and $D = \{(x,y): y\geq0 \land 2x^2+y^2 \leq a\}$ with $a>0$.
I need to find $a$ such that the global maxima of $f$ restricted to $D$ is $\frac{1}{8}$.
I found, using Lagrange multipliers, that $a$ = $\frac{3}{4}$. With that value, $f(\frac{1}{2}, \frac... | The function is monotonic in $y$, so that by definition, for each constant $x$, the maximum will be found on the upper half of the ellipse's boundary.
Thus, you can search for local maxima of:
$$f(x,y) = x^2\sqrt{a-2x^2}$$
Which will then be global maxima of $f(x,y)$ on $D$, and also happen to coincide with the local... | {
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Find two linearly independent solutions of the differential equation $(3x-1)^2 y''+(9x-3)y'-9y=0 \text{ for } x> \frac{1}{3}$ I want to find two linearly independent solutions of the differential equation
$$(3x-1)^2 y''+(9x-3)y'-9y=0 \text{ for } x> \frac{1}{3}$$
Previously I have seen that the following holds for the... | Hint
I do not know how much this could help you; so, please forgive me if this is off-topic.
If you define first $y=(3x-1)u$, $$(3x-1)^2 y''+(9x-3)y'-9y=0$$ rewrite $$(3 x-1)^2 \Big((3 x-1) u''+9 u'\Big)=0$$ for which order can be reduced and integration seems simple.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1172614",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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Are the Unit Ball and Any other Ball Topologically Equivalent How would I correctly show that the unit ball $B(0,1)\subset \mathbb{R}^n$ and the ball $B(a,r) \subset \mathbb{R}^n$ are Topologically Equivalent?
I know I need to find a one-to-one function $f: X\rightarrow Y$ for which $f$ and $f^{-1}$ are both continuous... | By proving $f:B(0,1)\rightarrow B(a,r)$ defined by $f:x\mapsto a+rx$ is a homeomorphism.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1172726",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Question on additivity of Riemann Integral . Given a Riemann integrable function , $f:[a,b] \rightarrow \mathbb{R}$ and a sequence ${b_k\to b}$ ( $b_k \in [a,b]$ ) is the following use of additivity of integrals correct ? $\int_a^bf(x)dx=\int_a^{b_1}f(x)dx +\int_{b_1}^{b_2}f(x)dx+ \int_{b_2}^{b_3}f(x)dx +...+\int_{b_... | The definition of the Riemann integral ensures that if $f$ is Riemann integrable on the interval $[a,b]$ then it is also Riemann integrable on any subinterval. It also ensures that $f$ is bounded, so assume $|f(x)|\leq M$ for all $x\in[a,b]$. Then, for each $k$ we have
$$\int_a^b f(x)\ dx-\int_{b_k}^bf(x)\ dx = \int_a^... | {
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How do I find the antiderivative of $sin(u)e^u$? So, I have the following indefinite integral:
$$\int sin(u)e^u \ du$$
I tried solving it using integration by parts, but it just kept repeating itself. What should I do?
| Here is a nice way to do it: by parts,
$$\int e^u\sin u\,du=e^u\sin u-\int e^u\cos u\,du\ .$$
Consider also
$$\int e^u\cos u\,du=e^u\cos u+\int e^u\sin u\,du\ .$$
Eliminating the cos integral,
$$\int e^u\sin u\,du=e^u\sin u-e^u\cos u-\int e^u\sin u\,du$$
so
$$2\int e^u\sin u\,du=e^u\sin u-e^u\cos u$$
and hence
$$\int e... | {
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Finding the largest constant $C$ such that $|\ln x−\ln y| \geq C|x−y|$ for all $x, y \in (0, 1]$ Find the greatest value of C such that
$|\ln x−\ln y|≥C|x−y|$
for any $x,y∈(0,1]$. What should my approach be? I can't think of much options.
| To find the largest $C$ so that $|\log(x)-\log(y)|\ge C|x-y|$, note that, by the Mean Value Theorem, for some $t$ strictly between $x$ and $y$
$$
\frac{\log(x)-\log(y)}{x-y}=\frac1t
$$
The minimum of $\frac1t$ on $(0,1]$ is $1$. Therefore, the largest $C$ can be is $1$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1173126",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 0
} |
What are classifying spaces actually classifying? Let $G$ be a group. When we say the classifying space of $G$ we are actually meaning the classifying space of the principal $G-$bundles because the notion of classifying spaces is about classifying the principal $G-$bundles and not about classifying groups. Is my unders... | If $G$ is a topological group, $BG$ classifies principal $G$-bundles. More precisely, for any paracompact topological space $X$, there is a natural bijection between $[X, BG]$, the set of homotopy classes of maps $X\to BG$, and $\operatorname{Prin}_G(X)$, the set of isomorphism classes of principal $G$-bundles.
There ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1173260",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
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Can a set be closed in one topology but neither open nor closed in another? Can a set be closed in one topology but neither open nor closed in another?
If we say that the complement of a open set is a closed set, i.e. if $S \subseteq X$ is open then $X \setminus S$ is closed, I argue that depending on the topology, it ... | The definition of a limit point depends on the topology too.
Recall the definition of a limit point:
A limit point of a set $S$ in a topological space $X$ is a point $x$ (which is in $X$, but not necessarily in $S$) that can be "approximated" by points of $S$ in the sense that every neighbourhood of $x$ with respect t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1173339",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 1
} |
Proving that $7^n(3n+1)-1$ is divisible by 9 I'm trying to prove the above result for all $n\geq1$ but after substituting in the inductive hypothesis, I end up with a result that is not quite obviously divisible by 9.
Usually with these divisibility induction problems, it falls apart nicely and we can easily factorise... | First, show that this is true for $n=1$:
$7^1\cdot(3\cdot1+1)-1=9\cdot3$
Second, assume that this is true for $n$:
$7^n\cdot(3n+1)-1=9k$
Third, prove that this is true for $n+1$:
$7^{n+1}\cdot(3(n+1)+1)-1=$
$7^{n+1}\cdot(3n+3+1)-1=$
$7^{n+1}\cdot(3n+1+3)-1=$
$7^{n+1}\cdot(3n+1)+7^{n+1}\cdot(3)-1=$
$7\cdot\color{red}{7^... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1173430",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 7,
"answer_id": 6
} |
Counterexamples for "every linear map on an infinite dimensional complex vector space has an eigenvalue" Every linear map on a finite dimensional complex vector space has an eigenvalue. Not so in the infinite case.
I'm interested in nice counterexamples anyone might have.
Here's one:
Consider the vector space $\mathbb... | There is an easy generalisation of the example in the question. In the polynomial ring $\Bbb C[X]$, the $\Bbb C$-linear operator $\phi: Q\mapsto PQ$ of multiplication by a fixed non-constant polynomial $P$ cannot have any eigenvalues. This is simply so since if $Q\neq0$ were an eigenvector for$~\lambda$, then one would... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1173511",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "15",
"answer_count": 5,
"answer_id": 3
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Could someone explain steps? I am learining about logarithm equations, and i can´t seem to understand how to solve such an equation, could someone help?
I must solve the equation/find $x$ for:
$$2^{2x} - 3\cdot2^x - 10=0$$
The final answer should be
$x=\dfrac{\log5}{\log2}$
| Assuming you mean $2^{2x} - 3\times2^x - 10 = 0$, then since this is a quadratic in $2^x$ we can factorise as $(2^x+2)(2^x-5)=0$. Since $2^x>0$ $\forall x$, $2^x+2\neq0$, so we must have $2^x=5$. Taking logarithms and applying the power rule, $x\log2=\log5$, whence $x=\frac{\log5}{\log2}$ as required.
If you're having... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1173604",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Limits and exponentials Asked to find $\lim_{n\to\infty}a_n$ where
$$a_n = \left(1+\dfrac1{n^2}\right)^n$$
I know that the limit = 1, and can get to this by saying $\ln a_n=n\ln\left(1+\dfrac1{n^2}\right)$ and going from there.
My question is: would it also be enough to simply direct substitute and say that…?
$$\left(1... | In short no. If you look at the limit that yields $e$ And did the same trick what would you get?
$$
\lim_{n \to \infty}\left(1+\frac{1}{n}\right)^n = e\neq \left(1+\frac{1}{\infty}\right)^\infty = 1
$$
The last equality is more to match with your statement question than being a proper definition of 1.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 6,
"answer_id": 3
} |
Convergence of $\sum^\infty_{n=1} \left(\frac 1 n-\ln(1+\frac 1 n)\right)$
Does the series: $\displaystyle\sum^\infty_{n=1} \left(\frac 1 n-\ln(1+\frac 1 n)\right)$ converge?
We know that $\ln (1+x) <x$ for $x>0$ so this series behaves like $\frac 1 n - \frac 1 n$ which is $0$, but that's just intuition.
I think the... | Notice that
$$
x - \ln(1+x) = \int_0^x \frac{t}{1+t}dt
$$
hence for all $x \geq 0$,
$$
0 \leq x - \ln(1+x) \leq \int_0^x t dt = \frac{x^2}{2}.
$$
Now, you can use the comparision test.
Edit: if you don't want integrals, then you can prove the same inequality by studying the variations of the functions
$$
f(x) = x - \l... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1173751",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Differentiable approximation $f(x) = x$ for $x>0$ and $0$ otherwise. I would like to find a twice continuously differentiable approximation of
$$f(x)= \begin{cases}
0 & x\leq 0 \\
x & x>0. \\
\end{cases}$$
Are there any approximations which are not defined piece-wise? Ideally, I'd like an approximation... | For a general method of approximation, the convolution is really usefull. Take a positive function of $C^{\infty}_c([-1,1])$, as an exemple,
$$g(x) = \exp\left({-\frac{2}{(x-1)(x+1)}}\right)$$
Let's normalise it :
$$g_{norm}(x) = \frac{g(x)}{\int_{-1}^1 g(t) dt}$$
Then we define
$$g_{\epsilon}(x) = \frac{1}{\epsilon} ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Greatest Commom Divisor In my lecture notes there is the following:
$$\newcommand{\gcd}{\operatorname{gcd}}
a, b \in \mathbb{N}, \gcd(a, b)=d \implies \exists s, t \in \mathbb{Z} : sa+tb=d \implies tb \equiv d \pmod a$$
If $\gcd(a, b)=1$ then $tb \equiv 1 \pmod a \ $ $ \ \ \ [b]_a \in \mathbb{Z}_a^{\star}$ and $[b]_a... | The point here seems to be to give a simple argument why the Euclidean algorithm terminates.
The sequences of remainders $r_i$ is strictly decreasing and $r_1 = b$. So it will stop at at most $r_b$, as the worst case would be $r_2 = b-1$, $r_3 = b-2$ and so on $r_b = b - (b-1)=1$, so then the following remainder must ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Calculate $\int_0^{\pi/2}\frac{\sin(x)\log{\sin{(x)}}}{x}\,dx$ Inspired by a question I saw these days, I try to calculate in closed form
$$\int_0^{\pi/2}\frac{\sin(x)\log{\sin{(x)}}}{x}\,dx$$
So far no fruitful idea that is worth sharing. What way would you propose? Note I prefer ways suggested, not necessarily solut... | It's not a closed form, but I hope can be useful. Using $$\log\left(\sin\left(x\right)\right)=-\log\left(2\right)-\sum_{n\geq1}\frac{\cos\left(2nx\right)}{n}$$
we have$$\int_{0}^{\pi/2}\frac{\sin\left(x\right)\log\left(\sin\left(x\right)\right)}{x}=-\log\left(2\right)\textrm{Si}\left(\frac{\pi}{2}\right)-\sum_{n\geq1}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1174054",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 1,
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How to tell where parentheses go in functional notation? The professor gave us a function $f(z) = \ln r + i \theta$ (this is for a complex analysis class). He doesn't like answering students' questions and there's no assigned textbook so I don't know where to look up such a function. How can I tell where the parenthese... | In addition to syntactic answers already discussed, you can think a little about which makes the most sense in context. For a complex analysis class, $f(z) = \ln(r) + i\theta$ is a common function to encounter. If $z$ is a complex number with modulus $r$ and principal argument $\theta$, then $f(z)$ is the principal b... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1174311",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 3,
"answer_id": 1
} |
$\mathbb Z_p^*$ is a group iff $p$ is prime I'm trying to prove $\mathbb Z_p^*$ is a group if and only if $p$ is prime. I know that if $p$ is prime $\mathbb Z_p^*$ is a group, but how can I do the converse? In another words, if the equation $ax\equiv 1 (\mod m)$ has solution for every integer $a$ which is not divisibl... | The equation in $x$ : $ax\equiv 1 \bmod n$ has a solution if and only if $(a,n)=1$. If $n$ is not prime take $a$ a proper divisor of $n$. Then $a$ is not divisible by $n$ , while $ax\equiv 1$ has no solution.
Although note that $\mathbb Z_p^*$ is the group in which the elements are the congruence classes relatively pri... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1174433",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Show that if $a_n> 0$ and $\lim na_n = \ell$ with $\ell \neq 0$ then the series $\sum a_n$ diverges I'm trying to understand the solution to the following problem:
Show that if $a_n> 0$ and $\lim na_n = \ell$ with $\ell \neq 0$ then the series $\sum a_n$ diverges.
I don't understand how the sign changed in the starre... | The claim $|a_n - \frac{\ell}{n}| < \frac{\epsilon}{n}$ is equivalent to
$$ -\frac{\epsilon}{n} < a_n - \frac{\ell}{n} < \frac{\epsilon}{n}. $$
In particular, the left inequality implies $(\ast)$.
Why is $|X| < Y$ equivalent to $-Y < X < Y$? By definition, $|X| = \max(X,-X)$. Now $\max(X,-X) < Y$ iff both $X < Y$ and $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1174588",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Prove that if $\lim\limits_{x\to 0}f\big(x\big(\frac{1}{x}-\big\lfloor\frac{1}{x}\big\rfloor\big)\big)$, then $\lim\limits_{x\to 0}f(x)=0$ Prove that if $\lim\limits_{x\to 0}f\bigg(x\bigg(\dfrac{1}{x}-\bigg\lfloor\dfrac{1}{x}\bigg\rfloor\bigg)\bigg)$, then $\lim\limits_{x\to 0}f(x)=0$
My attempt:
If I can show that... | Use the Squeeze Theorem. First note that
$$\frac{1}{x}-1<\left\lfloor\frac{1}{x}\right\rfloor\le\frac{1}{x}$$
from which we obtain
$$ \left|x\right|\ge \left|x\right|\left( \frac{1}{x} -\left\lfloor\frac{1}{x}\right\rfloor\right)\ge 0$$
Therefore, as $x\to 0$, the Squeeze Theorem guarantees that
$$\lim_{x\to 0} x\,\l... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1174698",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Showing that if $xf(x)=\log x$ for $x>0$, then $f^{(n)}(1)=(-1)^{n+1}n!\left(1+\frac12+\cdots+\frac{1}{n}\right)$
Let $f(x)$ be a function satisfying $$xf(x)=\log x$$ for $x>0$. Show that $$f^{(n)}(1)=(-1)^{n+1}n!\left(1+\frac12+\frac13+\cdots+\frac{1}{n}\right),$$ where $f^{(n)}(x)$ denotes the $n$th derivative of th... | Hint: First, write out the first few derivatives of each side of $$x f(x) = \log x$$ (rather than the derivatives of $f(x) = \frac{\log x}{x}$), identify the patterns of the derivatives, and prove that they are correct (using, say, induction).
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Convergence and Absolute Convergence of Arithmetic Mean of a sequence Suppose $\lim_{n\to\infty}\frac{1}{n}\sum_{i=1}^n |x_i|$ exists. Does $\lim_{n\to\infty}\frac{1}{n}\sum_{i=1}^n x_i$ exist?
How about the converse?
My thoughts:
*
*I guess for the sequence $\{x_1,-x_1,x_2,-x_2,\ldots\}$ the converse doesn't necess... | Another trivial counter example for the converse could be this:
$\lim_{n\to\infty}\frac{1}{n}\sum_{i=1}^n x_i 1\{x_i\geq 0\}$ does not exist and $\frac{1}{n}\sum_{i=1}^n |x_i| 1\{x_i\leq 0\}=\frac{1}{n}\sum_{i=1}^n x_i 1\{x_i\geq 0\}$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1175021",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
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Show that $F(x) = f(\|x\|)$ is differentiable on $\mathbb{R}^n$.
Let an even function $f:\mathbb{R}\to\mathbb{R}$ which is even and differentiable. We define $F:\mathbb{R}^n\to\mathbb{R}$ as $F(x) = f(\|x\|)$. Show that $F(x)$ is differentiable on $\mathbb{R}^n$.
My Work:
Let $x_0\in\mathbb{R}^n$.
*
*If $x_0\ne ... | While the case $x_0 \neq 0$ is correct and trivial, your proposed solution for the case $x_0=0$ is not a proof at all. A function can have a maximum or minimum at a point where it is not smooth (e.g. $x \mapsto |x|$ in $\mathbb{R}$).
Here you should use the fact that $f'(0)=0$ (because $f$ is even and differentiable at... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1175109",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Quadratic formula question: Missing multiplying factor of A? I have a very simple problem which must have a simple answer and I was wondering if anyone can point out my error.
I have the following quadratic equation to factor:
$2x^2+5x+1$
Which is of the form:
$Ax^2+Bx+C$
All I want to now do is factor this into the fo... | $Ax^2+Bx+C$ factorize as : $A(x-\alpha)(x-\beta)$ wiht $\alpha + \beta= \dfrac{-B}{A}$ and $\alpha \beta=\dfrac{C}{A}$.
obviously if $A \ne 0$, and $\alpha$ and $\beta$ are roots of the the two equations:
$$
A(x-\alpha)(x-\beta)=0 \iff Ax^2+Bx+C=0
$$
and
$$
(x-\alpha)(x-\beta)=0 \iff x^2+\dfrac{B}{A}x+\dfrac{C}{A}=0
$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1175195",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 2
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How to use Induction with Sequences? I have posted this similar question here, but with no hopes. I would just like to know:
Most of the solution I have no issue with. Look at where they say:
"Choose a representation $(n - 3^m)/2 = s_1 + ... + s_k$ in the desired form.
But to do that first, they applied induction to ... | Recall that $m=\lfloor\log_3n\rfloor$. This means that $m\le\log_3n<m+1$ and hence that $3^m\le n<3^{m+1}$. It follows that $n-3^m<3^{m+1}-3^m$ and hence that
$$\frac{n-3^m}2<\frac{3^{m+1}-3^m}2=\frac{3^m(3-1)}2=3^m\le n\;.$$
That is,
$$\frac{n-3^m}2<n\;.\tag{1}$$
Since we’re assuming in this case that $n$ is odd, we k... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1175244",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Is antisymmetric the same as reflexive? Note: The following definitions from my book, Discrete Mathematics and Its Applications [7th ed, 598].
This is my book's definition for a reflexive relation
This is my book's definition for a anti symmetric relation
Is a reflexive relation just the same as a anti symmetric rela... | Reflexivity requires all elements to be both way related with themselves. However, this does not prohibit non-equal elements from being both way related with each other.
That is: $\qquad\forall a\in A: (a,a)\in R$ from which we can prove: $$\forall a\in A\;\forall b\in A:\Big(a=b \;\to\; (a,b)\in R \wedge (b,a)\in R\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1175297",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Is the number $-1$ prime? From my understanding it's not prime because it's not greater than $0$. So my followup question is why did mathematicians exclude $-1$?
The definition of prime is having only two factors.
$-1 \cdot 1 = -1$
| In general, a definition that describes relationships between numbers without describing the domain in which such relationships are valid is incomplete.
In other words, when you say that the "definition of prime is having only two factors," you are leaving out a crucial element: for which set(s) of numbers is this rela... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1175367",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "40",
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"answer_id": 9
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If $T^2=T$ then determine whether $\ker T=\operatorname{Range}\,(T)^\perp$.
Let $T$ be linear operator on a finite dimensional inner product space $V$ such that $T^2=T$. Determine whether $\ker T=\operatorname{Range}\,(T)^\perp$.
I have proved that $\ker T=\operatorname{Range}\,(T)^\perp$ under the assumption that $T... | In fact, $\ker T = ({\cal R} T)^\bot$ iff $T$ is Hermitian.
Note that $\ker T = {\cal R} (I-T)$.
Suppose $\ker T = ({\cal R} T)^\bot$. Then
$\langle (I-T)x, Ty \rangle = 0$ for all $x,y$, hence $(I-T^*)T = 0$,
or $T= T^* T$, and so $T$ is Hermitian.
Now suppose $T$ is Hermitian.
If $x \in \ker T$, then $\langle x, Tz ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1175555",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Integration: Area between curves Let $f(x)=x^2−c^2$ and $g(x)=c^2−x^2.$ Find $c>0$ such that the area of the region enclosed by the parabolas $f(x)$ and $g(x)$ is 9.
The question above is what I am having trouble with. In order to solve this problem I use the formula given as:
$\int_a^b f(x) - g(x) dx$
Here is what I ... | Draw a picture. I would use symmetry and say that the area is $4$ times the integral from $0$ to $c$ of $c^2-x^2$. So we want
$$\int_0^c (c^2-x^2)\,dx=\frac{9}{4}.$$
The integral is equal to $\frac{2}{3}c^3$. Now solve for $c$.
Remark: Your approach will work, except that we want $\int_{-c}^c (c^2-x^2)\,dx=\frac{9}{2}$... | {
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"timestamp": "2023-03-29T00:00:00",
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solution involving inverse of a rank-1 matrix I am looking for $\mathbf{y} \in \mathbb{R}^n$ that minimizes the following objective function that involves a real matrix $\mathbf{V} \in \mathbb{R}^{n\times n}$
\begin{equation}\tag{*}
\begin{array}{c}
\text{min} \hspace{4mm} \mathbf{y}^T \mathbf{V}\mathbf{y}
\end{array... | Of course, since $\mathbf V$ is not invertible one cannot use $\mathbf V^{-1}$. However, already the differentiation seems a problem. I've made some quick computation to get
$$
(\mathbf V^T+\mathbf V)\mathbf y=\lambda \mathbf u=[\lambda,0,\dots,0].
$$
Then $\lambda$ is the first component of the vect0r in the left:
$$
... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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A question on a problem on localization from Atiyah (3.8) I was having trouble with the following problem from Chapter $3$ of Atiyah-MacDonald
Let $S, T$ be multiplicatively closed sets in the ring $A$, such that $S\subseteq T$.
Let $\varphi : S^{−1}A \to T^{−1}A$ be the homomorphism which maps each $r/s \in S^{−1}A$ t... | For the forward direction: Let $\phi$ be a bijection and let $t\in T$. Since $t\in T$, $1/t\in T^{-1}A$. Since $\phi$ is a bijection, there is some $a/s\in S^{-1}A$ such that $\phi(a/s)=1/t$. Consider $\phi(at/s)=(a/s)t=(1/t)t=1$. Since $\phi$ is a bijection and $\phi(at/s)=1$, it follows that $at/s=1$.
On the othe... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1175979",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Martingale energy inequality I am reading a book on BMO martingales which uses a so-called energy inequality. I have not been able to find a solid reference for this. Can someone please give a reference to these inequalities. Hopefully someone has heard of them, apparently they are "well known."
| The only reference I know of is P. A. Meyer's Probability and Potentials. It is not an easy read. Chapter VII Section 6 "A few Results on Energy" can resolve your questions. Let $(X_t)$ be a right-continuous potential, i.e. there is an integrable increasing process $(A_t)$ ($A_0 = 0$, increasing paths, right continuous... | {
"language": "en",
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How do we see that the shape of an equation is closed? Actually, I would like to understand :
When we have an equation , can we decide that its shape closed curve?
For instance, for the equation:
$$((x+y)^2) + 2((x^4)/4-(x^5)/5)=C$$
For which values of C makes the shape of this equation closed?
| For any value of $C$ the graph is closed. To see this, define a function $f:\Bbb R^2\to \Bbb R$ by $f(x,y)=(x+y)^2+2\left(\dfrac{x^4}{4}-\dfrac{x^5}{5}\right)-C$. Clearly $f$ is continuous and so $f^{-1}(\{0\})=\left\{(x,y)\in\Bbb R^2:(x+y)^2+2\left(\dfrac{x^4}{4}-\dfrac{x^5}{5}\right)=C\right\}$ is a closed set in $\B... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Proof that $p$-th total variation of a brownian motion is $0$ while $p>2$ The p-th total variation is defined as $$|f|_{p,TV}=\sup_{\Pi_n}\lim_{||\Pi_n||\to n}\sum^{n-1}_{i=0}|f(x_{i+1}-f(x_{i})|^p$$
And I know how to calculate the first total variation of the standard Brownian motion. But when dealing with high order ... | Note that $2$-variation is not the same as quadratic variation. For the quadratic variation you take the limit as the partition gets finer, whereas for $p$-variation you take the supremum over all partitions.
In particular, the Brownian motion has finite quadratic variation on any finite interval but infinite $2$-varia... | {
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Clarification of Proof involving $\sum_{p \le x} \frac{1}{p}$ For fun I've been doing problems from M. Ram Murty's text "Problems in Analytic Number Theory". I recently encountered the following problem:
If
$$\lim_{x \rightarrow \infty} \frac{\pi(x)}{x/\log x } = \alpha $$
then show that
$$\sum_{p \le x} \frac{1}{p} = ... | For proving $$\sum_{p\leq x}\frac{1}{p}=\log\left(\log\left(x\right)\right)+O\left(1\right)$$
we can use the Mertens second formula$$\sum_{p\leq x}\frac{\log\left(p\right)}{p}=\log\left(x\right)+O\left(1\right).$$
So by partial summation$$\sum_{p\leq x}\frac{1}{p}=\sum_{p\leq x}\frac{1}{\log\left(p\right)}\frac{\log\... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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How to prove this simple fact without using distribution theory? Suppose function $f(x) $ is normalized to unity, i.e.,
$$ \int dx |f(x)|^2 =1 . $$
Now consider the Fourier transform of $f$, i.e.,
$$ F(k) = \int d x f(x) e^{-i k x} . $$
Here we assume that $f $ is a very well-behaved function (say, a Gaussian functi... | Assume that $f$ is real, smooth, absolutely integrable with absolutely integrable derivative. Then $f \in L^{1}\cap L^{2}$ follows. Define $g(x)=f(-x)$. Then,
$$
g^{\wedge}(s) =\overline{f^{\wedge}(s)}.
$$
The convolution $f\star g$ has Fourier transform $\sqrt{2\pi}|f^{\wedge}(s)|^{2}$. Consequently,
$$
... | {
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Number of commuting involutions in Symplectic group Definition. $x$ is called involution if it has order $2$.
Problem. I have to count maximal number of pairwise commuting involutions in $Sp(2n, F)$, $\text{char}F \neq 2$.
Attempt. Firstly, let's remember that pairwise commuting matrices are simultaneous diagonalizable... | The largest elementary abelian $2$-subgroup of ${\rm Sp}(2n,F)$ has order $2^n$, so the answer is $2^n-1$.
To see this, let $E$ be an elementary abelian $2$-subgroup. Then $E$ fixes some $1$-dimensional subspace $\langle v \rangle$ of $F^{2n}$, and hence it stabilizes the subspace $\langle v \rangle^\perp$, which has d... | {
"language": "en",
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Deduce that if $A$ is a subset of $C$, then $\sup A\leq \sup C$.
Deduce that if $A$ is a subset of $C$, then $\sup A\leq \sup C$.
How do I begin with the above proof?
I can't see why $\sup A$ must either be equal of less than $\sup C$.
Is it not possible for some $a \in A$ to be greater than some $c\in C$?
| Hint: Start with an upper bound of one of the two sets, and try to show that it is also an upper bound of the other. (You'll have to figure out which is which.)
| {
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Find the Center of Tangent Circles One of my math tests has this question.
A circle has its center at $(6,7)$ and goes through $(1,4)$. Another circle is tangent at $(1,4)$ and has the same area.
What are the possible coordinates of the second circle. Show your work or explain how you found you answer.
Could someone ... | The center $Q=(h,k)$ of the second circle lies on the line defined by $P_1=(6,7)$ and $P_2=(1,4)$. What means
$$\frac{k-4}{h-1}=\frac{4-7}{1-6}=\frac{3}{5}...(1)$$
Since these circles have the same area it follows they have equal radius. Then $$\sqrt{(h-1)^2+(k-4)^2}=\sqrt{(1-6)^2+(4-7)^2}...(2)$$
Solving the system of... | {
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Find all integral solutions to $a+b+c=abc$. Find all integral solutions of the equation $a+b+c=abc$.
Is $\{a,b,c\}=\{1,2,3\}$ the only solution? I've tried by taking $a,b,c=1,2,3$.
| a+b+c=abc then first of all (a+b+c)>=3*(abc)^1/2 so (abc)^2>=9abc so abc>=9 which shows that a+b+c>=9 which is strictly impossible to satisfy for integer.
Other way is to think that first of all 0 being obvious solution and for other solution let a+c=0 then
b=-aab
Which again gives 0 as the only solution
a+b+c=abc
(1/a... | {
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Probability of rolling all $6$ die faces I've been struggling with this for over an hour now and I still have no good results, the question is as follows:
What's the probability of getting all the numbers from $1$ to $6$ by rolling $10$ dice simultaneously?
Can you give any hints or solutions? This problem seems real... | The way I see this problem, I'd consider two finite sets, namely the set comprised by the $10$ dice (denoted by $\Theta$) and the set of all the possible outcomes (denoted by $\Omega$), in this particular situation, the six faces of the dice.
Therefore, the apparent ambiguity of the problem is significantly reduced by ... | {
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A question about the additive group of the $p$-adic integers Let $J_p$ be the additive group of the $p$-adic integers. I know that it is torsion-free. I'm not pretty confortable with $p$-adic.
Is it possible to find a direct sum of infinitely many cyclic subgroup in $J_p$? (essentially I'm asking if they have finite s... | Complete rewrite:
For this, you need to look at the fraction field $\Bbb Q_p$ of the $p$-adic integer ring $\Bbb Z_p$. It’s a vector-space over $\Bbb Q$, and according to a well-known theorem depending on the Axiom of Choice, there’s a $\Bbb Q$-basis of this vector space. Since $\Bbb Q_p$ is uncountable, the basis must... | {
"language": "en",
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proving a wiki statement about uniform convergence and supremum This wiki page says that a sequence $f_n$ is uniformally convergenct to $f: S \to \mathbb{R}$ if and only if $||f_n - f||_{\sup} \to 0$. I am trying to prove this statement.
For the forward implication, we have that $f_n \to f$ uniformaly, so we have that... | It's just another way to state the definition :
This is your base definition
$$\forall \epsilon > 0, \exists N, \forall n > N, \forall x, \quad |f_n(x)-f(x)|< \epsilon$$
This is what means $\|f_n-f\|_{\sup} \to 0$ :
$$\forall \epsilon > 0, \exists N, \forall n > N, \quad \|f_n-f\|_{\sup} < \epsilon$$
And now replace $... | {
"language": "en",
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Definite integral $\int_1^2 \sqrt{1+\left(-x^{-2}+x^2/4\right)^2}\,dx$ I'm having trouble solving this integral. It relates to an arc length question. I tried Wolfram|Alpha, but when it solves it doesn't give me the option to view step-by-step.
Integral:
$$
\int_{1}^{2} \sqrt{1+\left(-x^{-2}+\frac{x^2}{4}\right)^2}\,d... | Simplifying the integrand proceed as follows.$$1+\left( -x^{-2}+\frac14x^2\right)^2=1+\left(\frac{x^4-4}{4x^2}\right)^2=\frac{(x^4+4)^2}{16x^4}$$Then after taking a square root, the integrand simplifies to $$\frac{x^4+4}{4x^2}=x^{-2}+\frac14x^2$$which is trivial to integrate.
| {
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Can the expression $D\cos(x)+Ci\sin(x)$ be rewritten in the form $R\sin(x+\theta)$? I've been trying to use the auxiliary equations method to solve the most simple SHM DE:
$$
\frac{d^2x}{dt^2}+w^2x=0
$$
The auxiliary equation has roots $\lambda=\pm iw$
Hence $x=A\exp(iwx)+B\exp(-iwx)=(A+B)\cos(wx)+i(A-B)\sin(wx)$
Letti... | You are missing that $A$ and $B$ are also complex numbers. To get a real solution, you would need $B=\bar A$ to get $\overline{x(t)}=x(t)$. Then your expression resolves to
$$
(C+iD)(\cos(wt)+i\sin(wt))+(C-iD)(\cos(wt)-i\sin(wt))=2C\cos(wt)-2D\sin(wt)
$$
and this has a classical conversion to an amplidute-phase form.
| {
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Symplectic lie algebra Can anyone explain me why, in the symplectic lie algebra, which is defined as $ sp(n)=\{X \in gl_{2n}:X^tJ+JX=0\}$ where $J=\begin{pmatrix}
0 & I \\
-I & 0 \\
\end{pmatrix} $ we can write its elements, in block form $X=\begin{pmatrix}
A & B \\
C & -A^... | Hint: The block decomposition used for $J$ (as well as the answer) suggest writing a generic matrix $X \in \mathfrak{gl}_{2n}$ as
$$X = \begin{pmatrix} A & B \\ C & D \end{pmatrix}.$$
To produce the block matrix description of $\mathfrak{sp}_{2n}$, simply substitute the block expressions for $X$ and $J$ in the definiti... | {
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How large is the set of all Turing machines? How large is the set of all Turing machines? I am confident it is infinitely large, but what kind of infinitely large is its size?
| It depends on how you define "distinct" in terms of turing machines. In general, two (one-tape) turing machines are not "distinct" if there is a component-wise bijection that goes from one's 7-tuple to the other's 7-tuple.(See http://en.wikipedia.org/wiki/Turing_machine#Formal_definition for the 7-tuple I am referring.... | {
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Defining multiplication on the tensor product of $R$-algebras. If $M$ and $N$ are $R$-algebras, then one can define a multiplication of elementary tensors as follows; $(m \otimes n) \cdot (m' \otimes n') = mm' \otimes nn'$.
My question is how can we show, using the universal property of the tensor product, that this i... | This is explained in the first paragraph of Bourbaki, A.III.4.1.
| {
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Let $N \unlhd G$ and let $N \leqslant H \leqslant G$. Show that $N \unlhd H$.
Let $N \unlhd G$ and let $N \leqslant H \leqslant G$. Show that $N \unlhd H$.
$N \unlhd G$ implies that there is some homomorphism $f$ on $G$ for which $N = ker(f)$. We want to show that there is a homormorphism $f'$ on H such that $N = ke... | Consider $H \hookrightarrow G \to G/N$
Edit: (Concerning your edit)
Your proof looks good, just a few things to mention:
*
*you don't need surjectivity of any map (it is given but not necessary). Also if a map $G \to M$ is not surjective, it will factor through a surjective map, since the image $M'$ of the map is a... | {
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Homogeneous Fredholm Integral Equation I'm having problem obtaining the solution of the homogeneous Fredholm Integral Equation of the
2nd kind, with separable kernel. I always get a zero
if I use the normal method i was taught for the nonhomogeneous type. I have an example: $$y(x) = \lambda\int_{-1}^1(x + z)y(z)dz$$
| certainly, your method of solving gives the general form $y(x)=c_1x+c_2$
Then, bringing it back into the Fredholm Integral Equation leads to the condition :
$$ y(x)=c_1x+c_2=2\lambda c_2x+\frac{2}{3}\lambda c_1$$
which implies $c_1=2\lambda c_2$ and $c_2=\frac{2}{3}\lambda c_1$
$ (\frac{4}{3}\lambda^2 -1 ) c_1=0$
So, t... | {
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Prove that if $\lim_{x\to\infty}f(x)=L$ exists and finite, and $\lim_{x\to\infty}\lfloor f(x)\rfloor$ doesn't exist then L is an integer
Let $f$ be a continuous function on $(0, \infty)$ s.t $\lim \limits_{x \to \infty}f(x) = L$ exists and finite, but $\lim \limits_{x\to \infty} \lfloor f(x) \rfloor$ doesn't exist. P... | I am going to do a proof by contradiction. Assume the opposite, that $L$ is not an integer. Note that because $L$ is not an integer, $\lfloor L\rfloor<L<\lceil L\rceil$. Because $f(x)\to L$ as $x\to\infty$, we can choose $\varepsilon=\min\{L-\lfloor L\rfloor, \lceil L\rceil-L\}$ such that there exists an $M$ such that ... | {
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Show that $\frac{3}{8}\le\int_0^{1/2}\sqrt{\frac{1-x}{1+x}}dx\le \frac{\sqrt{3}}{4}$ Show that $\displaystyle\frac{3}{8}\le\int_0^{1/2}\sqrt{\frac{1-x}{1+x}}dx\le \frac{\sqrt{3}}{4}$ without calculating the integral.
This is an exercise for training the use of mean value theorem for integral.
I tried to apply it in dif... | We have:
$$\frac{d^2}{dx^2}\sqrt{\frac{1-x}{1+x}}=\frac{(1-2x)}{(1-x)^{3/2}(1+x)^{5/2}}$$
so the integrand function is convex over $\left[0,\frac{1}{2}\right]$. That implies (see Hermite-Hadamard inequality):
$$\frac{f\left(\frac{1}{4}\right)}{2}\leq\int_{0}^{\frac{1}{2}}f(x)\,dx \leq \frac{f(0)+f\left(\frac{1}{2}\righ... | {
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Finding Projection Vector Find the projection of the vector $t = [3, 3, 3]^T$ onto the subspace spanned by the vectors $\{x, y\}$, where $x = [6; 1; -3]$, $y = [1; 0; 2]$.
I was told to look at the orthogonal basis, project the vector to each basis element, and then add them up. So do I use the projection formula for $... | The method you state is good and is basically what the other two answers recommend.
However, this involves calculating two projections. This is not onerous, but there is an easier way to project a 3-D vector onto a 2-D plane defined by two linearly independent vectors.
1) Find the cross-product of the two vectors that ... | {
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Proving that the products GCDs of the coefficients of two polynomials is equal to the GCD of their product's coefficients? Assume that $p(x)=a_nx^n+\dots+a_0$ and $q(x)=b_nx^n+\dots+b_0$ where the coefficients are integers. Let $y$ be the gcd of $a_n,\dots,a_0$ and let $z$ be the gcd of $b_n,\dots,b_0$. How does one pr... | Write $p(x)=y P(x)$ and $q(x)=zQ(x)$, where $P(x)$ and $Q(x)$ are primitive polynomials (i.e. the g.c.d. of their coefficients is $1$). Thus
$$p(x)q(x)=yz P(x)Q(x)$$
and it is enough to show $P(x)Q(x)$ is primitive. . It not, there exists a prime number $a$ that divides all its coefficients. Reduce the coefficients ... | {
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Euclid's proof of infinitude of primes. http://en.wikipedia.org/wiki/Euclid's_theorem
I just read Euclid's proof for the existence of infinitely many primes (I have never used his proof earlier to prove this). It seems to me that he assumes it exist a finite number and he constructs the number $x=p_1 \cdot... \cdot p_n... | Theorem: Any $n\in N$ with $n>1$ is divisible by a prime. Proof: Among the divisors of $n$ that are greater than $1$ (noting that $n$ itself is a divisor of $n$), there is a LEAST one, $M.$ Now $M$ is prime because $1<M, $ and if $M=A B$ with $A>1<B$ then $A$ would be a divisor of $n$ with $1<A<M$, which is absurd by... | {
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Is the pushforward measure a categorical-theoretic pushout? Given two measurable spaces $(X,\mathscr{F}),(Y,\mathscr{G}),$ $f:X \to Y$ measurable and $\mu:\mathscr{F} \to [0,\infty)$ a measure, the pushforward of $f_*(\mu):\mathscr{G} \to [0, \infty)$ is defined as $f_*(\mu)(G)=\mu f^{-1}(G).$
Is this terminology coin... | Let me elaborate on the comment by @tomasz. Consider a category $\mathtt{Mes}$ where objects are measurable spaces (sets endowed with $\sigma$-algebras) and morhpisms are measurable maps. For each object $M$ you can construct another object $\mathcal M(M)$ whose elements are measures on $M$, and the $\sigma$-algebra is... | {
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Subgroups of a permutation group The permutation group $S_{4}$ is defined as the group of all possible permutations of [1234].
i) Find the number of subgroups of $S_{4}$ that have order 2.
ii) A: { [1234], [2143], [3412], [4321] } and B: { [1234], [1243], [2134], [2143] }. Which of A and B are subgroups of $S_{4}$?
Try... | Hints:
i) If a subgroup has order two then there are exactly one trivial and one non-trivial element. Furthermore the nontrivial element $a\in G$ must fulfill $a^2=e$. So just check with of the elements of $S_4$ fulfill this requirement.
ii) Just check the subgroup axioms for $A$ and $B$.
| {
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How long would it take to guess an arbitrary integer or real number? Let's say two mathematicians play a game. One of them picks an arbitrary element from a countably infinite set (perhaps the integers, as per the title), and the other one guesses what it is. The second player has as many guesses as they need, and afte... | In the first case, one constructs a bijection $f$ from the natural numbers into said countable set. Then you guess $f(1),f(2), \dots$ and eventually (after finitely many steps) you guess correctly.
In the second case, with probability 100% you will always guess incorrectly, so usually the game will go on forever. (You ... | {
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Property on homomorphism between finite groups I was reading some lecture notes on homomorphism between finite groups and intuitively it appeared to me that for a $\phi$ an homomorphism $G \rightarrow H$, we should have:
$$|\,\textrm{Im}(\phi)\,| = \frac{|\,G\,|}{|\,\textrm{ker}(\phi)\,|}$$
where $| . |$ stands for or... | ker($\phi$) is a normal subgroup of G, so G/ker($\phi$) is well defined and |G/ker($\phi$)|=|G|/|ker($\phi$)|.
Thanks to "the first isomorphism theorem", there exists an isomorphism between Im($\phi$) and G/Ker($\phi$) moreover |Im($\phi$)|=|G|/|Ker($\phi$)|
| {
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Relative primes What is the number of integers between 1 and 60 that are relatively prime to 60? I know that the answer is 16, but how do I go about finding the relative primes using a quick process?
| You can just use Euler's totient function. Given a prime $p$, $\phi(p^\alpha) = (p - 1)p^{\alpha - 1}$. The function is multiplicative, so $\phi(p^\alpha q^\beta) = \phi(p^\alpha) \phi(q^\beta)$.
Since $60 = 2^2 \times 3 \times 5$, you calculate $$\phi(60) = \phi(4) \phi(3) \phi(5) = 2 \times 2 \times 4 = 16.$$
To veri... | {
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"answer_count": 4,
"answer_id": 3
} |
Solving a non-linear, multivariable system of equations I'm researching the mathematics behind GPS, and at the moment I'm trying to get my head around how to solve the following system of equations:
$\sqrt{(x-x_1)^2+(y-y_1)^2+(z-z_1)^2}=r_1$
$\sqrt{(x-x_2)^2+(y-y_2)^2+(z-z_2)^2}=r_2$
$\sqrt{(x-x_3)^2+(y-y_3)^2+(z-z_3)^... | I think that you can make the problem simpler the solution of which not requiring Newton-Raphson or any root finder method; the solution is explicit and direct (it does not require any iteration).
Start squaring the expressions to define $$f_i={(x-x_i)^2+(y-y_i)^2+(z-z_i)^2}-r_i^2=0$$ Now, develop $(f_2-f_1)$ and $(f_... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1179036",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Solving 2nd order ODE How would you this ODE:
$$x''+x+\cos^3t=0$$
The homogenous solution is $x=C\cos t$ so $C\cos t$ is term is part of the solution. But how would you go from there?
| The solution of the ode
$$ x''+x=0 $$
should be
$$ x(t) = c_1\cos(t)+c_2\sin(t). $$
Added: For the particular solution you can use variation of parameters method.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1179159",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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Definition of previsible processes? Definition from my textbook:
A stochastic process $X = (X_n, n \in \mathbb{N}_0)$ is called predictable (or previsible) with respect to the filtration $\mathbb{F} = (\mathcal{F}_n, n \in \mathbb{N}_0 )$ if $X_0$ is constant and if, for every $n \in \mathbb{N}$
$$X_n \text{ is } \... | Technically, you can always predict in terms of the best available information but you will NOT necessarily know the value of $X_{n+1}$ if you know the values of $X_0, \ldots, X_n$. This is easy to show since based on the definition of a previsible process, if you are at $(n-1)^{th}$ index of the filtration (i.e. $\mat... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1179291",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
To show that $7 \Bbb Z$ and $16 \Bbb Z$ are isomorphic as groups but not isomorphic as rings. To show that $7 \Bbb Z$ and $16 \Bbb Z$ are isomorphic as groups but not isomorphic as rings.
I have done the first part but finding difficult to show that they are not isomorphic as rings??
| $n \mathbf{Z}$ is generated by $n$, so that a group morphism $n \mathbf{Z} \to G$ is characterized by the value it takes on $n$. Therefore, the group morphism $7 \mathbf{Z} \to 16 \mathbf{Z}$ sending $6$ to $17$ is an isomorphism, and its inverse is defined by $16\mapsto 7$. Now take $f : 7 \mathbf{Z} \to 16 \mathbf{Z}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1179423",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
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Prove that odd polynomials $f(x)$ of degree $\leq 10$ with $f(-1) = 0$ form a vector space.
Let $P(X)$ be the usual vector space of polynomials in $x$ with real coefficients.
Let $U$ denote the subset of $P(X)$ consisting of those elements $f(x)$ which have degree less than or equal to $10$, satisfy $f(-x)=-f(x)$ for ... | $f(-x)=-f(x)$ means that the subspace $U$ consists solely of odd functions. The subspace $V$ of odd functions in $P(X)$ with degree less than or equal to 10 has basis $x,x^{3},x^{5},x^{7},x^{9}$. Indeed, all these functions are linearly independent, odd and have degree less than 10. Moreover, they span $V$: If $g \in P... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1179580",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 1
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Evaluate $\lim_{t \to \infty} \frac{(\sqrt t+ t^2)}{(4t - t^2)}$. Why is the limit as $t \to \infty = 0$? Would like some feedback on my process. Also, would like to gain a Calculus I understanding of limits.
A question many are having a problem explaining to me is why is the limit as x approaches infinity designated ... | $$\lim_{t \to \infty} \dfrac{(\sqrt t+ t^2)}{(4t - t^2)}$$
$$=\lim_{t \to \infty} \dfrac{(1+t^{3/2})}{(4\sqrt t - t^{3/2})}$$
$$=\lim_{t \to \infty} \dfrac{(1+t^{3/2})}{\sqrt t(4 - t)}$$
$$=\lim_{t \to \infty} \dfrac{1}{\sqrt t(4 - t)} + \lim_{t \to \infty} \dfrac {t^{3/2}}{\sqrt t(4 - t)}$$
$$=0 + \lim_{t \to \infty} ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1179682",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 0
} |
If $p$ is prime and congruent to $1$, then show $((\frac{p-1}{2})!)^2 \equiv -1 \pmod p$ I got another one.
Quadratic residues are completely new to me...
Thanks!
| Take the congruences, $p-1 \equiv -1 \pmod p$, $\text{ }\text{ } p-2 \equiv -2 \pmod p$
and so on upto, $\frac{p+1}2 \equiv -\frac{p-1}2 \pmod p$.
Multiplying and rearranging, $$(p-1)!\equiv 1\cdot (-1)\cdot 2 \cdot (-2) ...\frac{p-1}2 \cdot (-\frac{p-1}2) \equiv (-1)^{\frac{p-1}2}[(\frac{p-1}{2})!]^2 \pmod p$$
Thus, b... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1179835",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 0
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Product of functions in $H^1(B)$ where $B \subset \mathbb{R}^2$ I'm rather new to Sobolev spaces and finding myself rather deficient of intuition. So when given a problem like the below where I need to "prove or disprove", I'm finding myself stuck.
Suppose $B$ is a ball in $\mathbb{R}^2$ and $u,v$ are in the Sobolev sp... | Let me add the counterexample I gave in the comments for completeness and with further explanations:
Following Alt (Funktionalanaysis, section 8.7), we have the charaterization:
$$|x|^\rho \in H^{m,p}(B_1) \iff \rho > m -\frac{n}{p} \tag{1}$$
From this characterization it is easy to obtain counterexamples. We are inter... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1179914",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
How to show $n^5 + 29 n$ is divisible by $30$ Show that $n^5 + 29 n$ is divisible by $30$.
Attempt:
$n^4 ≡ 1 \pmod 5$ By Fermat Little Theorem
| You can prove this by induction. Case $n = 1$ is obviously true.
Hint: Use the binomial theorem in the inductive step. Also, a number is divisible by $30$ if and only if it is divisible by its prime factors $2$, $3$, and $5$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1180122",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 8,
"answer_id": 5
} |
Why is $\deg(f)$ well-defined? If $M$ and $N$ are boundaryless, compact, connected, oriented $n$-manifolds, and $f:M\to N$ is smooth, then if $\omega_0$ is a $n$-form on $N$ and $\int_N\omega_0\neq 0$, there is a number $a$ such that $\int_M f^*\omega_0=a\int_N\omega_0$.
In fact, if $\omega$ is any $n$-form on $N$, the... | Suppose $M$ is oriented, compact and $\partial N=\emptyset$. All $n$-forms on $N$ are closed, and by Stokes theorem, $\int_N\partial \eta=0$ for any exact form, so that the map
$$\int_N:\Omega^n(N)\to\Bbb R$$ automatically factors through the isomorphism
$$\int_N:H^n_{dR}(N)\xrightarrow{\sim\,}\mathbb R\,.$$
The conclu... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1180207",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 0
} |
A functional problem on page 268 in the book GTM 95 probability
I have noticed that if
$$
\lim_{c\downarrow 0}\int_{-\infty}^{\infty} e^{c|x|}P(dx) < \infty
$$
the system $\{1, x, x^2, \cdots \}$ is complete in $L^2$.
So I want to know which theorem or result was used here?
| This seems to be more of a problem in complex analysis (+ a bit of
Fourier analysis) than in functional analysis.
Your assumption on $\mathbb{P}$ implies that $\int e^{c\left|x\right|}d\mathbb{P}\left(x\right)<\infty$
for some $c>0$. This easily implies $e^{\frac{c}{2}\left|x\right|}\in L^{2}\left(\mathbb{P}\right)$.
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1180290",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Are there some techniques for checking whether a statement implies another without truth tables? Are there some techniques for checking whether a statement implies another without truth tables?
For example, I was asked whether $P\Longrightarrow P_{1}$ given the following statements:
$$P: [p \land (q \land r)]\lor \neg[... | You could write a proof instead. Suppose $P$.
If $p\land q\land r$, then $p\land(q\lor r)$. Therefore $P_1$ tautologically.
Otherwise $\neg[p \lor (q \land r)]\Leftrightarrow \neg p \land \neg(q\land r)$. Then $p$ being true contradicts $P$, so $P\Rightarrow P_1$ in that case. In the case that $p$ is false, $P$ reduce... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1180379",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 4,
"answer_id": 3
} |
Finding a Matrix from Determinants I've stumbled upon this problem on my homework, and I have no clue how to do it, and haven't found any help online:
If I'm understanding this correctly, then $det(M) = ad - cb + eh - gf$ ?
What I don't get is how to find M from knowing this information.
Are there any suggestions on h... | It's unclear what the dimension of $M$ should be, but here is a start to a solution where $M$ is a $2\times 2$ matrix:
$$\begin{pmatrix}a&??\\1&d\end{pmatrix}$$
Try to finish it from here (i.e., find what goes in the entry labeled $??$).
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1180460",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
what is this sort of function called? I am doing an assignment but I do not know how to do this problem. I have the following:
$$
f(x)=
\begin{cases}
0 & \text{for $x<0$},\\
x & \text{for $x\geq 0$}.
\end{cases}
$$
We are meant to determine whether it is continuous or not. I do not know why there is like a bracket with... | This is an example of a piecewise function.
The big bracket means that the function is evaluated differently depending on what domain $x$ is in.
When $x < 0$ then $f(x) = 0$.
When $x \geq 0$ then $f(x) = x$.
Continuous functions are functions that do not have any gaps.
This function is continuous.
An example of a fu... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1180571",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 6,
"answer_id": 3
} |
Examples of Separable Spaces that are not Second-Countable In this post I give an example of a separable space that is not second-countable. What are other good examples?
| $\pi$-Base is an online encyclopedia of topological spaces inspired by Steen and Seebach's Counterexamples in Topology. It currently lists twenty-one separable spaces that are not second countable. You can learn more about any of these spaces by viewing the search result.
Appert Space
Arens-Fort Space
Deleted Diameter ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1180664",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 1
} |
Finding the basis of $\mathbb R^5/W.$ I need help finding out the basis in the following question :
Let $~~W=\big<[1~~2~~1~~0~~1]^t~,[1~~0~~1~~1~~1]^t~,[1~~2~~1~~3~~1]^t\big >~$ be a subspace of $\mathbb R^5$ . Find a basis of $\mathbb R^5/W.$
I can't figure out the basis , kindly help with some hint on how to pro... | Row reduction of the matrix:
$$\begin{bmatrix}
1&1&1&x\\
2&0&2&y\\
1&1&1&z\\
0&1&3&t\\
1&1&1&u\end{bmatrix}
\rightsquigarrow
\begin{bmatrix}
1&1&1&x\\
0&1&3&t\\
0&0&6&y-2x+2t\\
0&0&0&z-x\\
0&0&0&u-x
\end{bmatrix}$$
shows $W$ has dimension $3$ and is defined by the equations $\,x=z=u,\,$ hence $\,\mathbf R^5/W\,$ has di... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1180743",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Congruence of square-free numbers Let $m$ be a square-free number. I want to prove that, given some $b \in \mathbb{Z}^+$, where $\gcd(b,m)>1$, that
$\qquad b^{\,c\,\phi(m) + 1} \equiv b \pmod m,\ $ for some $c \in \mathbb{Z}^+$.
I've verified this with a bunch of different values of $m$ and $b$, but I'm really stuck ... | Let $b=p_1^{a_1}p_2^{a_2}\ldots p_k^{a_k}$, where $p_1, p_2, \ldots p_k$ are prime divisors of b. We'll prove there exist $c$ (actually every natural number works), such that:
$$p_1^{a_1(c\phi(m)+1)} \equiv p_1^{a_1} \pmod {m}$$
$$p_2^{a_2(c\phi(m)+1)} \equiv p_2^{a_2} \pmod {m}$$
$$\cdots$$
$$p_k^{a_k(c\phi(m)+1)} \eq... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1180947",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Write down each of the terms in the expansion of $\sin(x^2)-(\sin(x))^2$. Write down each of the terms in the expansion of $\sin(x^2)-(\sin(x))^2$. Taylor's Theorem applies at the point $a=0$ and with $n=4$.
Got no idea how to proceed. My lecture notes have one example that I barely understand. I'd really appreciate a... | we will keep track of upto powers of $x^4.$ $$\sin(x) = x - \frac 16 x^3 + \cdots, (\sin x)^2 = x^2 - \frac 13 x^4 + \cdots, \sin(x^2) = x^2+ 0x^4 + \cdots $$ therefore $$\sin (x^2) - (\sin x)^2 = \frac 13 x^4 + \cdots $$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1181055",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 3
} |
Is there a relationship between Turing's Halting theorem and Gödel Incompleteness Turing's proof that a Halting oracle is impossible and Gödel's proof that and omega-consistent first order theory of arithmetic must be incomplete are similar in that they use self-referential arguments. Is there an interesting relationsh... |
Turing's proof that a Halting oracle is impossible and Gödel's proof that and omega-consistent first order theory of arithmetic must be incomplete are similar in that they use self-referential arguments. Is there an interesting relationship between them.
Well, Gödel's theorem is a simple consequence of Turing's proof... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1181151",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 1,
"answer_id": 0
} |
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