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According to Euclid, a point is something which has no dimensions. And we know that all curves of any type consists of points. Now this thing bothers me because if a point has no dimensions, i.e. in other words there is nothing, then how is it possible to draw any curve?
The thing I could imagine is that maybe a point... | A point in an n-dimensional Euclidean space is that point which consists of n coordinates which give a distinct location of that point.
A function in Euclidean space essentially provides a rule for defining a set coordinates in space for which that function is true.
You are correct in that the physical realization ... |
Find the range of $f(x)=(-1)^{\lfloor x \rfloor} (x-{\lfloor x \rfloor})$
I don't know how to work with this function, and how to find the range. I tried to find the domain of its inverse, but i can't do it either.
Any hints? | $$f(x)=(-1)^{\lfloor x\rfloor}\{x\},\ 0\le \{x\}\lt 1$$
Since the fractional part is being multiplied by a factor of $(-1)^{\lfloor x\rfloor}$ it would just introduce the set of numbers created by multiplying by $-1$ to the well known range of the fractional part function, thereby making the range $(-1,1)$. |
Let $G = \mathbb{R} \setminus \{-1\}$ and define the binary operation on $G$ by $a*b=a+b+ab$.
Prove $G$ is a group under this operation.
So to prove $G$ is a group, I know we have to show it is associative, has an identity element, and contains inverses for all elements.
How do I specifically show those 3 properties... | HINT
For associativity: $\forall a,b,c\in G\ \ a*(b*c)=(a*b)*c$.
For identity: show that $\exists e\in G $ such that $\forall a\in G\ e*a=a*e=a$
For inverse: show that $\forall a\in G,\ \exists a^{-1}\in G$ such that $a*a^{-1}=a^{-1}*a=e$ |
I already got tired trying to think of a function $f:\{1,2,3,4\}\rightarrow \{0,1,1,0\}$ in other words:
$$f(1)=0\\f(2)=1\\f(3)=1\\f(4)=0$$
Don't suggest division in integers; it will not pass for me. Are there ways to implement it with modulo, absolute value, and so on, without conditions? | Another one (a bit more complicated than the parabola) is:
$$f(x)=\frac{2}{\sqrt{3}}\sin\bigg(\frac{\pi}{3}(x-1)\bigg)$$
This one generates: $0,1,1,0,-1,-1,0,...$
And another simple one:
$$f(x)=1.5-\left | 2.5-x\right|$$ |
$$\lim_{n→∞} \frac{1^3 +4^3 +7^3 +... + (3n-2)^3}{[(1+4+7+...+(3n-2)]^2}$$
I'm mostly confused because I'm not sure what the raising to the second power in the denominator means. Does this mean that after I sum all of the terms of (3n-2) from n to infinity I then multiply that by itself?
Do I need to use Riemann sums... | Let $f:\mathbb R\to(\mathbb R\setminus \mathbb Q)$ be defined by
$$f(x) = \left\{
\begin{array}{lr}
\arctan(x) & \text{if }\ \arctan(x)\not\in\mathbb Q, \\
\arctan(x)+\pi & \text{if }\ \arctan(x)\in\mathbb Q.
\end{array}
\right.$$
The relevant features of $\arctan$ are that it is bounded... |
We have to find the number of options for setting pair $(a,b)$ under the terms:
$a ⊆ b ⊆\{1, 2,\ldots, n\}$
Means, they are both subsets of $\{1, 2,\ldots, n\}$ and $a⊆ b$.
I was thinking to handle the $b$ coordinate first and by that, to handle the $a$ coordinate.
But, what are the number of options for $b$?
tahnx. | Each element is either in a and b, in b but not in a, or non of both.
edit: did'nt see the first part of Frank's answer. It's perfect. |
In Secondary education in Australia, the general outline for introducing techniques to solve the quadratic equation
$$
x^2+bx+c=0
$$
is first to introduce the idea to find two numbers $p$ and $q$ such that $b=p+q$ and $c=pq$. After this, the quadratic formula is introduced, which is applicable to all quadratic equation... | To help students understand how we move from specific cases to the general case, I created a visual depiction of the process of completing the square. I used shapes analogous to Algebra Tiles, which are a good manipulative for building understanding of factoring as well. I then took the process one step further and ill... |
I have tried out to prove it by its contrapositive.
Therefore now I am proving that there is no integer whose square is congruent to $2 \pmod 3$.
Consider the case $n^2 \equiv 2 \pmod 3$
$ n^2 = 3k +2$
I am stuck to show $\sqrt{3k+2}$ is not a integer. Can anyone provide some hint to me? | Using function transformations, compress the integral by a factor of $2$ in the $x$-axis, then multiply by $2$ to get:
$$2 \int_{0}^{\pi/2} \frac{\mathrm{d}t}{3+2\cos(2t)} = 2 \int_{0}^{\pi/2} \frac{\mathrm{d}t}{4 \cos^2 t+1} = 2 \int_{0}^{\pi/2} \frac{\mathrm{\sec^2 t\ d}t}{4 + \tan^2 t+1}$$
and substituting $u = \t... |
Let $A=\{n+\frac{1}{n}:n\in\mathbb{N}\}$. Is the set $A$ closed in $\mathbb{R}$?
I think that if $n$ goes to infinity, every deleted neighborhoods of $n$ would contain the points of $A.$ So, infinity is the only limit point of $A$ and infinity is contained in the set $A.$ Hence, the set $A$ is closed? Is that right? | Use the comparison test for the tail of the sequence. By the $n$th term test, we know that $\sum_{k=1}^\infty x_n$ converges, $x_n \to 0$ as $ n\to\infty$. By the definition of convergenve, we may choose $\varepsilon =1$ and guarantee that there is some $K\in\mathbb{N}$ so that $n>K \implies |x_n|<\varepsilon=1$. Sinc... |
Consider this simple, overdetermined system of linear equations:
$$\begin{align*}
3x + y &= a\\\\ 2x + y &= b\\\\ x + y&= c
\end{align*}$$
where $a, b, c$ are arbitrary constants.
On Mathematica 13.2, if I run RowReduce[{{3, 1, a}, {2, 1, b}, {1, 1, c}}]
I get the output
{{1, 0, 0}, {0, 1, 0}, {0, 0, 1}}.
Now I... | To use "RowReduce" to solve linear equations you must append the RHS as an additional column to your matrix. The solution is then the rightmost column of the matrix in reduced row echelon form.
Here is an example:
Given the system of 3 linear equations:
a = {{2, 3, 5}, {-3, 4, -1}, {4, 1, -6}};
b = {-2, 7, 3};
Threa... |
In baby Rudin we have the definition and theorems:
Definition 1.10 An ordered set $S$ is said to have the $\it{least}$-$\it{upper}$-$\it{bound \hspace{1mm}property}$ if the following is true:
If $E\subset S$, $E$ is not empty, and $E$ is bounded above, then $sup(E)$ exists in $S$.
Theorem 1.11 Suppose $S$ is an ... | The point of the Theorem is to show that a set has the least upper bound property if and only if it has the greatest lower bound property (technically it only shows one direction; the reverse direction is almost identical).
The goal is, given an arbitrary set $B\subset S$ that has any lower bound, to show that $B$ ha... |
Integrate $$\int\dfrac{1}{\cos(x-1)\cos(x-2)\cos(x-3)}\,\textrm dx$$
My Attempt:
Using, $$\tan A-\tan B=\dfrac{\sin(A-B)}{\cos A\cdot \cos B}$$
The given integral can be transformed as
$$\int\dfrac{\tan(x-1)}{\cos(x-2)}\,\textrm dx - \int\dfrac{\tan(x-3)}{\cos(x-3)}\,\textrm dx$$
The right most integral can be calcu... | Little bit of trigonometry
$$\begin{align}
\frac{\tan(x-1)}{\cos(x-2)}
&= \frac{\tan(x-1)}{\cos((x-1) - 1)} \\
&= \frac{\tan(x-1)}{\cos(x-1)\cos 1 + \sin(x-1)\sin 1} \\
&= \frac{\tan(x-1)\sec(x-1)}{\cos 1 + \tan(x-1)\sin 1} \\
&= \frac{d\left(\sec(x-1)\right)}{\cos1 + \sin 1 \sqrt{\sec^2(x-1) - 1}} \\
&= \frac{du}{\c... |
I want to define a function for doing multiple definitions, where I apply the same function f to all values, and definitions are done not at the same time but after each other.
A simple example: the function I want is multiDef, where
multiDef[{{a, 1}, {b, 2}, {c, a}}, f]
should be identical to the sequence of assign... | If a, b and c do not already have values, you can simply use
In[1]:= (#1 = f[#2]) & @@@ {{a, 1}, {b, 2}, {c, a}}
Out[1]= {f[1], f[2], f[f[1]]}
In[2]:= a
Out[2]= f[1]
In[3]:= b
Out[3]= f[2]
In[4]:= c
Out[4]= f[f[1]]
If they do have values then I do not recommend storing the input in a simple list such as {{a,1},...... |
Consider the finite field $\mathbb{F}_q$. Schur (1916) proved that, given $n$, when the field is sufficient large, this equation,
$$x^n+y^n= z^n$$
always has a nontrivial solution.
What conditions does the number of solutions satisfy?
I. Schur, Über die Kongruenz $x^{m} + y^{m} \equiv z^{m} \pmod{p}$, Jahresber. De... | It is more or less easy to obtain, via exponential sums, an asymptotic formula for the number $J$ of solutions of the congruence
$$x^{n}+y^{n} \equiv z^{n} \pmod{p}$$
where $$1 \leq x, y, z \leq p-1.$$
Indeed, since
\begin{eqnarray*} J &=& \sum_{x=1}^{p-1} \sum_{y=1}^{p-1} \sum_{z=1}^{p-1} \frac{1}{p}\sum_{a=0}^{p... |
Do you think the letter $\wp$ has a name? It may depend on community - the language, region, speciality, etc, so if you don't mind, please be specific about yours. (Mainly I'd like to know the English names, if any, but other information is welcome.) If yes, when and how did you come to know it? When, how, and how oft... | My own answer, about the name of the letter $\wp$. Mostly in computing, with little math. (Lengthy)
Summary
In computing, the letter $\wp$ has been plagued with a plethora of inappropriate names. I guess "Weierstrass p" is one of such names that some standard created, and has been copied without much scrutiny. I susp... |
The Question
My Understanding(final edit)
(My proof will probably suck,but it is the best I can do)
We are required to prove that $R\setminus C \in \tau$ and $R\setminus A\in\tau$ are Hausdorff but not regular.
Assumptions
1.Let X be the set,of all real numbers and
$ S=\{1/n:n \in N\}$
2.Define a set $C\subseteq... | You should perhaps start by proving that this definition of closed sets of $\mathcal{T}$ as sets of the form $A \cup S$, where $A$ is Euclidean closed and $S \subseteq T = \{\frac1n\mid n \in \Bbb N\}$ is a valid one, in that it obeys the axioms for closed sets for a topology.
Secondly note that if $O$ is Euclidean op... |
Recently I've been reading about cohomological finiteness conditions for groups, my main source being Brown's book "Cohomology of Groups".
One of the first things one learns is that a group with finite cohomological dimension necessarily is torsionfree. It is easy to come up with an example of a torsionfree group wit... | You can take a union of torsion-free word-hyperbolic groups with cohomological dimension approaching infinity to get such an example.
Word-hyperbolic groups are rank one, so one can show that any
abelian subgroup must intersect each subgroup in a cyclic group, and therefore the
subgroup cannot be an infinitely genera... |
This might be a bit of a basic question but my current understanding is that we cannot. Still it makes me wonder if we can propose a mapping between two countable sets why not?
For example why is this expression incorrect?
$$\sum_{i\in \mathbb{N}} i =\sum_{i \in \mathbb{Q}_+}i$$
Since our LHS and RHS are both positiv... | First of all, let's get the stuff about the mapping between countable sets out.
The fact that we can construct a bijective mapping between countable sets means exactly that, we can send every element of one set to one and only one in the other.
Note that this doesn't say that the mapping has anything to do with how it ... |
Find the sum $ \displaystyle\sum_{n=1}^{\infty} \frac{1}{n^{2}(2n-1)}$
$= 1 + \dfrac{1}{3\cdot 2^{2}} + \dfrac{1}{5\cdot 3^{2}} + \dfrac{1}{7\cdot 4^{2}} \ldots$
Then, I'm stuck.
I didn't find any pattern to solve.
I just need a hint. | $$\sum_{n=1}^{+\infty}\frac{1}{n^2(2n-1)}=\sum_{n=1}^{+\infty}\frac{2}{n}\left(\frac{1}{2n-1}-\frac{1}{2n}\right)=\sum_{n=1}^{+\infty}\left(\frac{4}{2n-1}-\frac{4}{2n}-\frac{1}{n^2}\right)=4\ln2-\frac{\pi^2}{6}.$$ |
I have a question about finding the sum formula of n-th terms.
Here's the series:
$5+55+555+5555$+......
What is the general formula to find the sum of n-th terms?
My attempts:
I think I need to separate 5 from this series such that:
$5(1+11+111+1111+....)$
Then, I think I need to make the statement in the paren... | $$5=5*1$$ $$5+55=5*12$$ $$5+55+555=5*123$$ $$5+55+555+5555=5*1234$$ $$5+55+555+5555+55555=5*12345$$ $$5+55+555+5555+55555+555555=5*123456$$ $$...............$$ $$...............$$ $$5+55+555+5555+55555+555555+5555555+55555555+555555555=5*123456789$$
$$5+55+555+5555+55555+555555+5555555+55555555+555555555+555555555... |
I was trying to understand how this works in order to write a proof and I am unable to. | $a\equiv b \pmod n$ means $n|a-b$; i.e., $a-b=kn$.
If $0\le a and $0\le b < n$ (i.e., $-n<-b\le0$) then $-n\lt a-b, so $k=0$, so $a-b=0$. |
$$y=\lim_{n\to \infty}\left(\sin π/2n \times \sin 2π/2n \times... \sin(n-1)π/2n\right)$$ Find y.
I took log on both the sides and I got
$$\ln y= \int_0^1 {\ln \sin πx/2\, dx}$$ but I'm stuck here. | If we wish to evaluate the limit of interest by use of Riemann sums, then we can proceed as follows. Let $y_n$ be the product defined by
$$y_n=\prod_{k=1}^{n-1}\sin\left(\frac{k\pi}{2n}\right) \tag 1$$
Taking the logarithm of both sides of $(1)$ reveals
$$\log y_n=\sum_{k=1}^{n}\log\left(\sin\left(\frac{k\pi}{2n}\... |
I'm going to ask a very silly question, so I'm begging you to be understanding if it is absolutely trivial, or if it's an exercise in some Bourbaki. I'm afraid of asking you, because the question entails this particular case: is the addition of (a finite, but variable quantity of) real numbers a continuous function?
O... | In the case where $I$ is infinite and $G$ has a non trivial topology, then the answer is no.
let $(x_\alpha) \in \prod 'G_\alpha$ and let $U$ be an open set containing $( x_\alpha)$.
U comes from an open set in the regular product topology, so we can assume that it is $\prod U_\alpha \cap \prod'G_\alpha$ where only f... |
If $\alpha, \beta, \gamma$ are supremums of $|f(x)|, |f'(x)|, |f''(x)|$ respectively over $(0,\infty)$, then what is the relationship between $\alpha, \beta, \gamma$?
It was asked by a friend like a puzzle and hence was posed as an informal question to me.
My thoughts are that there should exist a relation between th... | I suspect that it's an ordering problem with $\gamma < \beta < \alpha$.
$f''(\beta) = 0 $ so $f''(\gamma) > 0$ (as $\gamma$ is a sup and $\gamma = \beta = 0$ would mean $\beta$ is a saddle-point, not, sup of $f'$). So $f' $ is increasing at $\gamma$ so $\beta > \gamma$ (as $f'(\beta) > f'(\gamma)$ )
Exact same rea... |
I'm trying to manually plot a parametric function:
$$
r=\frac{\phi}{\phi - \pi}
$$
I've translated the above to Сartesian coordinate system and found zeros for $x$ and $y$ axes.
$$
\cases{
x = r\cos\phi = \frac{\phi}{\phi - \pi}\cdot \cos\phi\\
y = r\sin\phi = \frac{\phi}{\phi - \pi}\cdot \sin\phi
}
$$
Then i want... | We can make use of the well-known fact that $\lim_{x \to 0} \frac{\sin x}{x} = 1$ and how it is proven. The main step in the prove is to note that $\cos x \le \frac{\sin x}{x} \le 1$. We will use this idea.
Now note that $\sin( \phi - \pi) = - \sin \phi $ and so we have:
$$ - \phi \le -\frac{\phi}{\phi-\pi} \sin(\phi... |
Find the sum $$\sum _{ k=1 }^{ 100 }{ \frac { k\cdot k! }{ { 100 }^{ k } } } \binom{100}{k}$$
When I asked my teacher how can I solve this question he responded it is very hard, you can't solve it. I hope you can help me in solving and understanding the question. | I am re-editing a-rodin's answer, correcting a few typos [of an earlier version, now edited].
\begin{align}
\sum\limits_{k=1}^{100} \frac {k\cdot k!}{100^k} \frac{100!}{k!(100-k)!} &= \frac{100!}{100^{100}} \sum\limits_{k=1}^{100} \frac{k\cdot100^{100-k}}{(100-k)!}\\
&= \frac{100!}{100^{100}} \sum\limits_{k=0}^{99}\fr... |
I want to make the following replacement work correctly:
Array[(2 D[xM[[#1]][t], t] * D[xM[[#2]][t], t]) &, {3,
3}] /. {xM -> {Sin[ω t], Cos[ω t], 0}}
The array is constructed with a symbolic form given in terms of time derivatives of a vector. For some reason, when I evaluate this expression, I obtain the follow... | I you only want to have a custom style for one particular hyperlink, you can just write it as a Hyperlink call.
For example,
Hyperlink["Link: Mathematica.SE",
"http://mathematica.stackexchange.com/q/76971/3066",
BaseStyle -> {"Subsubsection"},
ActiveStyle -> {Gray}]
Evaluate this and then hide or delete th... |
I am trying to plot a function which has a discontinuity at one precise point. However, when I naïvely use Plot with this function the discontinuity point does not seem to be taken into account.
For example:
Plot[If[x!= 1, 1, 0], {x, 0, 2}]
gives:
One can add a vertical line manually to highlight the discontinuity:
... | You cannot expect Plot to find isolated singular points by itself. It samples the given function at a finite number of points. It stops sampling when the new points are close to an interpolation of the previous points. If it doesn't sample your special point during this process, you won't see it in the plot. You need a... |
Well for a given kepler orbit (which is a ellipse) $0 \leq e < 1$. There are several functions to describe the motion of an object.
$$r(\nu) = \frac{a (1 - e^2)}{1 + e \cos(\nu)}$$
Where $a$ is the semi major axis, $e$ the eccentricity, $\nu$ the true anomaly. And $r$ the distance from the focus point.
One of the bas... | It seems to me that your post addresses two points
convergence of Newton method
methods converging faster than Newton method
Concerning the first point, assuming the starting point to be "reasonable", Newton method would converge without any overshoot of the solution if $$M(E_0) M''(E_0) >0$$ (if I properly remember,... |
I have constructed an exact function for counting primes in intervals and am curious to know if it 1) has any importance? 2) Has been derived already? I have no formal education in number theory, and no university affiliation, so my only option is to post on here to get any advice or feedback. $\mathbb{P}$ represents t... | Assume the following definitions where $n$ is a positive integer, $p$ is a prime, $\mu(n)$ is the Möbius function, and $M(x)$ is Mertens function:
(1) $\quad \pi(x)=\sum\limits_{n\le x}a(n)\,,\qquad a(n)=\cases{1,& $n\in\mathbb{P}$ \\ 0,& otherwise}\quad$ (see https://oeis.org/A010051 for $a(n)$)
(2) $\quad M(x)=\sum... |
I've tried to use the limit test:
$\lim_{n \rightarrow \infty} \frac{a_n}{b_n} = C$
in the series of $a_n$ and $\frac{a_n}{a_1+a_2+...+a_n}$ but it was inconclusive.
The question does not especify the sign of $a_n$ but proving it for positives would already help.
I think the sign can change as long as the partial sum... | This problem has an interesting history from the early days of real analysis attached to it. I really like this story and the proof it contains of the problem in this question is also different from the other answers posted.
In 1827 a M. Oliver published a proof in Crelle's Journal (Journal für die reine und angewandt... |
How do we evaluate this infinite product with a sum within it?
$$\large{\prod_{n=1}^{\infty}\left[1+\frac{1}{\sum_{j=1}^{n}F_j^2}\right]^{(-1)^n+1}}$$
Where $F_j$ is the Fibonacci number
If I open the product, it does not help me. I am sure there must be an equivalent form of this $1+\frac{1}{\sum_{j=1}^{n}F_j^2}$ in... | Note that by some comments, we can first rewrite $\sum_{j=1}^nF_j^2$ as $F_nF_{n+1}$, and then rewrite the product $\displaystyle\prod_{i=1}^\infty\left(1+\frac1{F_iF_{i+1}}\right)^{(-1)^i+1}$ as $\displaystyle\prod_{i=1}^\infty \left(1+\frac1{F_{2i}F_{2i+1}}\right)^2$. But now by looking at partial products we can see... |
Problem:
I'm asked to find the length of $m$, given the following diagram.
Note that $\overline{AC}$ = 1 and $\overline{CD} = 1$ and that $\overline{AB}$ is a diameter whose length is 4.
Attempt:
I know that $\angle ADB$ is a right angle and that if I can find the length $\overline{AD}$ then this will be solved. I ... | To find AG or AD draw the hight of triangle ACD to cross the side AD of triangle ABD at E.Also connect B to C. Triangle ACE is similar to triangle ABC for:
$\angle ABC=\angle CDA=arc AC/2$
So we have:
$\frac{CE}{AC}=\frac{AC}{AB}=\frac{1}{4}$ ⇒ $CE=\frac{1}{4}$
In right triangle ACE we have:
$AE^2=AC^2-CE^2=1^2-... |
I posted before about this proposition and I thought I got it right but then I was told that it is still wrong so I am really confused again..
Here is my proof
Proof: Let $A = (0,1]$
Here, since $0 ≤ a$ for all $a ∈ A$, $0$ is an lower bound
Now, suppose that there is some $b ∈ R$ such that $b$ is a lower bound fo... | Your answer is fine. Just take some care with the assumption that "since $b \in A$, b can not be a lower bound of A".
For example:
$A = [0, 1]$. The infimum of this set is also $0$ and $0 \in A$. |
Let $(I_n)_{n \geq 1}$ be a sequence such that:
$$I_n = \int_0^1 \frac{x^n}{4x + 5} dx$$
Evaluate the following limit:
$$\lim_{n \to \infty} nI_n$$
All I've been able to find is that $(I_n)$ is decreasing and converges to $0$.
Thank you! | $\newcommand{\bbx}[1]{\,\bbox[8px,border:1px groove navy]{\displaystyle{#1}}\,}
\newcommand{\braces}[1]{\left\lbrace\,{#1}\,\right\rbrace}
\newcommand{\bracks}[1]{\left\lbrack\,{#1}\,\right\rbrack}
\newcommand{\dd}{\mathrm{d}}
\newcommand{\ds}[1]{\displaystyle{#1}}
\newcommand{\expo}[1]{\,\mathrm{e}^{#1}\,}
\newc... |
Does there exist any such software/online calculator where I can enter a matrix and am able to obtain the formula for an arbitrary power of the matrix?
It would be an added benefit if the entries of the matrix are allowed to be variables.
In case this question belongs somewhere else, I would be glad to be pointed in ... | The answer is yes. We may take $b=a$ and then for the matrix
$$
A=A(a,a)=\begin{pmatrix}0 &-1&-1&-1\\
-1& 0 &-1&-1\\
1&-1&0 &-1&\\
-1&0&-1&0\end{pmatrix}
$$
we see that $A^a$ is positive (all entries positive) if and only if
$$
a=6k \quad \text{ for } \; k\in \Bbb N.
$$
So we have infinitely many such $a$, and hence ... |
$$\sum_{i=1}^n\frac{3^{i-1}}{2^{i-2}}$$
Currently, I am just iterating i from 1 to n but this seems to be unnecessary as if we could have a closed form of this sum, then we may achieve constant-time computation. | It's a sum of the geometric progression:$$2\cdot\frac{\left(\left(\frac{3}{2}\right)^n-1\right)}{\frac{3}{2}-1}$$ |
Suppose that $f=(x+1)^{100}+(x-3)^{100}$ and $g=x^2-2x-3$. What is the remainder of $\frac{f}{g}$? I know there must be something connected to the roots of $g$, but how I can use that? Also I know that the remainder's degree is less or equal to 1. I don't want the response, I want to find that by myself, hints are welc... | Complicated solution with binomial theorem:
If you write $x-1=t$ then $q(x) = t^2-4$ and $$p(x) = 2\Big[{100\choose 0}t^{100}+...+
2^{96}{100\choose 96}t^4 +2^{98}{100\choose 98}t^2 + 2^{100}\Big]$$
Now let $s=t^2$ then $q(x) = s-4= q_1(s)$ and
$$p(x) = 2\Big[{100\choose 0}s^{50}+...+
2^{96}{100\choose 96}s^2 +... |
Perhaps the "proofs" of ABC conjecture or newly released weak version of twin prime conjecture or alike readily come to your mind. These are not the proofs I am looking for. Indeed my question was inspired by some other posts seeking for a hint to understand a certain more or less well-establised proof, or some answers... | I did not initially understand the first isomorphism theorem. It was not until my first year of graduate school, when my wife wanted me to teach her some group theory, that I really hit on a natural sequence of ideas which made the first isomorphism theorem "click" for me. I talk about it here:
Does any textbook tak... |
I'm trying to solve this limit problem which asks to find what happens to the following limits as $x$ varies:
$$\lim_{n\to+\infty} \left((\sin x -1) + {1\over{n^2 +1}}\right)^{n^2}$$
My steps so far:
use exponential rule and rewrite as
$$\lim_{n\to +\infty} \exp\left(n^2 \ln \frac{n^2\sin x+\sin x-n^2}{n^2+1}\right)... | That is wrong. The $4$th roots of $\sqrt2e^{i\frac\pi4}$ are$$\sqrt[4]{\sqrt2}e^{i\frac\pi{16}},\ \sqrt[4]{\sqrt2}e^{i\left(\frac\pi{16}+\frac\pi4\right)},\ \sqrt[4]{\sqrt2}e^{i\left(\frac\pi{16}+\frac\pi2\right)}\text{ and },\ \sqrt[4]{\sqrt2}e^{i\left(\frac\pi{16}+\frac{3\pi}4\right)}.$$ |
Question: The quartic polynomial $x^4 −8x^3 + 19x^2 +kx+ 2$ has four distinct real roots denoted
$a, b, c,d$ in order from smallest to largest. If $a + d = b + c$ then
(a) Show that $a + d = b + c = 4$.
(b) Show that $abcd = 2$ and $ad + bc = 3$.
(c) Find $ad$ and $bc.$
(d) Find $a, b, c, d$ and $k$.
My attemp... | You already have lots of equations, some of them non-linear, from where hopefully the coefficients will be found:
$$\begin{align*}&a+d=4=b+c\\&ad=2\,,\,\,bc=1\end{align*}$$
so for example (as clearly all coefficients are non-zero):
$$c=\frac1b\implies 4=b+c=b+\frac1b\implies b^2-4b+1=0\implies b_{1,2}=2\pm\sqrt3\imp... |
A subset of ${\mathbb R}^2$ is open in the usual topology iff it is a (not necessarily disjoint) union of open disks. It is well-known that the plane is connected, so that there is no nontrivial partition of ${\mathbb R}^2$ into open sets.
Now say that a subset of ${\mathbb R}^2$ is weakly open iff it is a (not nec... | Your definition gives the plane the discrete topology: each singleton is the intersection of two open segments. Thus, every partition of the plane qualifies. |
1) Prove that any number that is not a power of $2$ can be expressed as a sum
of two or more consecutive positive integers, but that this is not possible for powers of $2$.
2) Which numbers can be expressed as a sum of two or more consecutive
odd positive integers?
I have found many questions the first part but am un... | Hint: For 2) you just write it as difference of two sums:
$$\sum_{i=0}^{n-1} (2i+1)-\sum_{i=0}^{m-2} (2i+1)$$,
where $m$ is the first integer and $n$ is the last integer ($n>m$).
Can you simplify it? |
Let $M\in \Bbb{R}^n$ be an open set, and let $\rho: M \to \Bbb{R}$ be continuous. Then define $v: M\to \Bbb{R}^n$ as
$$v(x) = \int_{r\in M} \frac{\rho(r)(x-r)}{\|x-r\|^n} \, dr$$
Where the above integral is taken component-wise. (if $n = 3$, we can imagine $\rho$ as charge density, and $v$ as the resulting electric fie... | The problem is that the integrand is only defined if $x\ne r$. I'm not sure how this can be treated in a mathematically clean way; in physics, the usual rule is that on the differentiation the singularity gives an additional term involving the $n$-dimensional Dirac delta $\delta^{(n)}(x-r)$.
This additional delta can ... |
Using proofs by contradiction, show that there is no smallest negative rational number and no largest positive rational number.
Assume that there is a smallest negative rational number. Therefore, the number is of the form $r = - \frac{p}{q}$, where $p$ and $q$ are positive integers. But, there is a rational number $-... | Two less "fussy" ways:
(i).$\;x\in Q\implies x\pm 1\in Q.$
And $ \;x-1
Use $x-10$.
(ii). $x\in Q\implies 2x\in Q.$
And $\;x>0\implies 2x>x, $ and $\;x<0\implies 2x |
Pretty sure this is simple but can someone please point me in the right direction.
$n=1: \frac{0.7^2}{1+0.7+0.7^2}$ =
$n=2: \frac{0.7^2+0.7^5}{1+0.7+0.7^2+0.7^3+0.7^4+0.7^5} $
This will hold for other values of n as well. i.e for each n where the numerator has the sum of all $3n-1$ powers for each n>0 and the den... | Notice
$$ \frac{ 3^n + 7^n }{3^n + 8^n} \leq\frac{3^n+7^n}{8^n} = \left( \frac{3}{8} \right)^n + \left( \frac{7}{8} \right)^n $$
Should I say more? |
A region $A$ in the first quadrant is bounded by $y=x^2$, $y=25$ and the $y$-axis. Find the value of $m$ with the property that the line $y = mx$ divides $A$ into two regions with the same area. | If you start with the geometric series: $\Sigma_{n=1}^\infty x^{n-1}=\frac{1}{1-x}$ and then integrate both sides, you should be just about there. |
Given $n = pq$, where $p$ and $q$ are prime and $\phi(n^2) = (p^2-p)(q^2-q)$ (The Euler Totient function). Is there some combination of $n$s such that $\phi(n^2)|f(n)$
$f$ can be anything (greater than$\phi(n^2)$)given it only uses $n$ and not $p$ or $q$ separately and results in an integer for the input of $n$.
I ha... | A bit more simplistically, for semiprime $n=pq$, $\phi(n^2)=p(p-1)q(q-1)=pq(pq-p-q+1)=n^2-n(p+q)+n$
In order to construct a function based solely on $n$, you are faced with explicitly knowing or discovering something about $p$ and $q$ based only on your knowledge of $n$. As poncho indicated in his answer, the problem ... |
Is it true that $\forall a,b \in \mathbb F_9$: $a \cdot b$ is a square $\iff$ $a \cdot \overline b$ is a square? | Since your group of units is cyclic, you can argue very similarly to the case of $F_p$, and the "Legendre symbol" analogue here is still a homomorphism.
In particular, multiplying by $a$ doesn't matter. You can reduce to $$b \; \mathrm{square} \Leftrightarrow \overline{b} \; \mathrm{square}.$$ This is true, because fo... |
Recently I read some course notes and articles on Bézier curves. They all sum up the properties of Bézier curves, like the partition-of-unity property of the basis functions (Bernstein polynomials), variation diminishing property of the curve (the curve doesn't wiggle/oscillate more than the control polygon does), and ... | The convex hull property is useful for doing a quick check prior to doing some more expensive calculation. For example, suppose I need to intersect two Bezier curves. It is fairly easy to determine that their convex hulls don't overlap. So, if I find that this is the case, there is no need to try to find the intersecti... |
Lets say
$$ \text{let }A =
\begin{bmatrix}
1 & 2 & 1 \\
0 & 3 & 1 \\
4 & 1 & 2
\end{bmatrix}
$$
Find a matrix C such that:
$$ \text{let }AC =
\begin{bmatrix}
1 & 2 \\
0 & 1 \\
4 & 1
\end{bmatrix}
$$
Let X be the matrix that AC equals too
$AC = X$
Is how you would solve this.
$$... | Yes, the existence of a one-to-one and onto mapping is exactly how equality of the size of sets (the technical term is "cardinality" is defined. The (cardinal) number of negative integers is the same as the cardinal number of positive integers, and the cardinal number of negative real numbers is the same as the cardina... |
I'm having a little difficulty understanding Gram-Schmidt orthogonalization. I have a problem to apply Gram-Schmidt orthogonalization to the system of vectors $(1,1,1)^T, (1,2,1)^T$ then write the matrix of the orthogonal projection onto 2-dimensional subspace spanned by these vectors. After applying gram-schmidt I fou... | Hint; your final vectors are not correct. The point of GS it to get an orthogonal set of vectors. Are yours orthogonal? You are starting off with two non orthogonal vectors, that is
$v_1=( 1, 1, 1)$ and $v_2= ( 1, 2,1)$
The GS algorithm proceeds as follows;
let $w_1=(1,1,1)$
then we define $$w_2= v_2- \frac{\langle... |
The question asked me to prove the following equivalence.
$$（r \rightarrow p)\rightarrow (p\land q)\equiv (\lnot r \rightarrow p)\land(\lnot r \rightarrow q)\land (p \rightarrow q)$$
And I did the followings:
For L.H.S.,
$（r\rightarrow p)\rightarrow (p\land q)\\\equiv \lnot (\lnot r\lor p)\lor (p\land q)\\ \equiv (r... | After writing out the left hand side using $\;\phi \rightarrow \psi \;\equiv\; \lnot \phi \lor \psi\;$ and DeMorgan, it is an $\;\lor\;$ of $\;\land\;$'s. Similarly, the right hand side becomes a $\;\land\;$ of $\;\lor\;$'s.
How do you get from one to the other? Through distribution.
So you choose one of the two si... |
Given an $n \times n$ matrix $A$ with real entries such that $A^2=-I$, find the $\det (A)$.
My Attempt:
$|A|^2=(-1)^n\implies|A|=(-1)^{\frac n2}$
The answer given is $|A|=1$
Eigenvalues are not syllabus. So, can this question be explained without the concept of eigenvalues? Thanks.
Perhaps the condition that the e... | Let $p(t)=\det (t-A)$. Then
$$p(t)p(-t)=\det (-I-t^2)=(-1)^n(t^2+1)^n\text{.}$$
If $n=0$ then $p(t)=1$. If $n=1$ then $t^2 + 1$ splits into linear factors, a contradiction since it is irreducible.
For other $n$ note that $t^2+1$ is prime, so it divides either $p(t)$ or $p(-t)$, and thus divides both. Let $p(t)=(t^2+1)... |
The general term of a sequence is defined by $U_n=\sqrt{n+1}-\sqrt{n} $ for all positive integers of n
a) Show that $U_{n+1}
b) Show that $$\sum_{k=1}^n (\sqrt{k+1}-\sqrt{k})=\frac{n}{1+\sqrt{n+1}}$$
I don't understand any of this could someone please explain
Thanks in advance | It's a standard trick, when one has $\sqrt{\mbox{junk}}-\sqrt{\mbox{stuff}}$ to multiply by its conjugate $\sqrt{\mbox{junk}}+\sqrt{\mbox{stuff}},$ which makes it just junk $-$ stuff. To pay for this, you have to divide by the conjugate also, but that is often easier to deal with.
This trick will make part a) easy ... |
Maybe it's a known fact, but I would like to know whether the function sending a square matrix of order $n$ to its characteristic polynomial is continuous. If this is true, there are some resources where I can find this?
Thanks in advance. | The function that takes you from an $n\times n$ matrix to its characteristic polynomial is simply
$$
\begin{align}
\phi:\mathbb R^{n\times n} &\to \mathbb R[x]\\
A &\mapsto \det(A-x I_n)
\end{align}
$$
Where $I_n$ is the $n\times n$ identity matrix. As for whether this function is "continuous", well, how do you defin... |
I know that this question has been asked before but was closed due to a lack of context and I could not understand the answer provided. Currently I'm self-relearning linear algebra and this is Problem 13 of Chapter 1.2 of Gilber Strang's Introduction to Linear Algebra. I have been unable to solve this problem given the... | There is nothing wrong with your solution, but you have to take some decisions because the solution is not unique. For instance, set $w_3=0$ and $v_3=1$, and you get the result.
However, the point of linear algebra is dealing with abstract notions, such as subspaces, rather than concentrating on coordinates that depen... |
I think the question is simple, but I'm a bit confused with proving the following rule for any $u,v,w$ in a vector space $V$ (with the usual 8 axioms for vector spaces, as in it's Wikipedia's page):
$$
u = v \Rightarrow u+w = v+w
$$
I think it's related to $"="$ being a binary relation, but I don't know where to start ... | This is purely because $+$ is a function and because $u=v$ implies $(u,w)=(v,w)$. Let $f:X\to Z$ be a function and $x=y$. Then $f(x)=f(y)$.
The property is also known as $+$ being well defined. Does this help? |
There is a problem here in Q. $5$ on the last page. It states to find coordinates of point $p$.
Taking point $a=(3,2,5,1), \ b=(3,4,7,1), \ c= (5,8,9,3)$.
Also, $b$ has two coordinates in common with $a$, and $p$ lies on the same line as $a,b$.
So, those two coordinates of $p$ are same as $a,b$. Hence, $p= (3,x,y,1)$... | We must have
$$(\forall x\in \Bbb R) \;\; 2a+1\le 1-\sin(x)$$
So
$$2a+1\le \inf\{1-\sin(x),\;x\in \Bbb R\}$$
$$\implies 2a+1\le 0$$
$$\implies \max (2a+1)=0$$ |
The great mathematician Arthur Cayley seems to have said "all geometry is projective geometry" (sorry no exact source, probably it is somewhere in Felix Klein's Erlangen program).
In projective geometry, non-intersecting lines do not exist (they all meet at infinity). In hyperbolic geometry on the other hand, through ... | From the comments you included, I can see you are comparing the two geometries' synthetic axioms to see if one is a special case of the other. Of course, that is doomed to fail because the two lists contain mutually exclusive axioms about parallels (as you noticed.)
The real idea is that hyperbolic, affine and Euclide... |
Can someone explain me this:
$2^{\{1,2\}}$
I know this equals to:
$\{\varnothing, \{1\}, \{2\}, \{1, 2\}\}$
Right?
So $\{1\}$ for example is an element of $2^{\{1,2\}}$,
$\{1\} \in 2^{\{1,2\}}$
But can someone explain me to what this notation equals?
$2^{\{1,2,3\}}$
Does it equal $\{\varnothing, \{1\}, \{2\}, \{1, 2... | You're talking about powersets. A powerset of a set is the set of all possible subsets of that set. The reason $2^S$ is used is because the number of elements in the powerset is $2$ raised to the power of the number of elements in $S$.
$2^{\{1,2,3\}}$ contains all $8$ subsets of $\{1,2,3\}$:
$\{\}$
$\{1\}$
$\{2\}$
$\... |
Let $G$ and $\Gamma$ be discrete groups, and let $\phi\colon\thinspace G\to \Gamma$ be a homomorphism.
Define its cohomological dimension $\operatorname{cd}\phi$ to be the least integer $d$ such that $\phi^\ast\colon H^i(\Gamma;M)\to H^i(G;M)$ is the zero homomorphism for all $i>d$ and all $\Gamma$-modules $M$ (where ... | How about this? Let $\Gamma$ be free abelian of rank $2$. Poincare duality in the torus identifies $H^2(\Gamma;M)$ with $H_0(\Gamma;M)$, so that in particular there is a natural surjection $M\to H^2(\Gamma;M)$.
Let $F$ be the particular $\Gamma$-module $\mathbb Z\Gamma$, free module of rank one for the group ring. A ... |
I am kind of stuck with the solving process of this question:
\begin{align}
& \text{error } = 0.01 \\
\implies & \tan\theta = \text{ error }+ \sin\theta \\
\implies & \tan\theta = 0.01 + \sin\theta \\
\implies & \tan\theta - \sin\theta = 0.01 \\
\implies & \dfrac {\sin\theta}{\cos\theta} - \sin\theta = 0.01
\end{alig... | The greatest value of $x$ is $\tan^{-1} 0.275$ radians, or something infinitely smaller than that.
Plotting the graph $round(100 \tan x) - round(100 \sin x)$ on Desmos gives many solutions, but the greatest solution that satisfies $x < \pi/2$ is approximately $0.26386$.
However, $\tan 0.26386$ $\approx 0.2749933$, a... |
Background:
Let $H$ be a finitely generated commutative Hopf $k$-algebra, where $k$ is a field of non-zero characteristic. For
$$
\widehat{H}:= \text{Alg}_k\{H; k\},
$$
we recall (see Abe Chapter 4 for example) that the Hopf algebra structure of $H$ induces in a canonical way a group structure on $\widehat{H}$.
Quest... | I think that Waterhouse's "Introduction to Affine Group Schemes" (1979), section 3.2 "Comodules", might be what you're looking for.
I'm not sure why you have the constraint "non-zero characteristic" in your question; this seems irrelevant to me. |
Let $X=Spec(A)$ be a reduced normal affine scheme over an algebraically closed field $k$ of characteristic $0$, with an action of a connected reductive group $G$. Suppose
$x\in X$ is a $G$-invariant $k$-point,
$X$ contains a dense open stabilizer-free $G$-orbit (i.e. a dense open copy of $G$), and
the stabilizers of ... | Vera Serganova showed me the following (affirmative) answer. I'll use the setup from Remark 1. Note that Remark 2 shows that $X$ cannot contain a positive highest weight vector. So the following result does the job.
Proposition: Let $V$ be a representation of a reductive group $G$ and let $X$ be the closure of an orbi... |
I am trying to prove that, for linear functions on $ℝ^n$, $AB = I$, then $BA = I$.
Note: I do NOT want a proof using matrices (I already figured that out, very simple).
My attempt to prove this:
We know that $AB = I$, so I would put this in the usual function notation: $A(B(x)) = I(x) = x$. Then this would imply t... | The given proof is incorrect logically. You need a result that B is surjective if AB = I.
Since B is surjecive, for each $y \in R^n$, there exists $x \in R^n$ such that $ y = Bx$,
$$BAy = BA(Bx) = B(ABx) = BIx = Bx = y$$
so BA = I.
prove B is surjective is not hard. |
Problem: is this one-to-one, onto, or both?
$$f:\mathbb Z\to\mathbb Z; n \mapsto \left\lceil \frac n2\right\rceil$$
With help I arrived at the answer is that $f$ is onto.
However, I'm confused in the way this was shown to me. Here's what I was shown:
$f(1) = f(2)$
"Ceiling rounds to the least integer greater than o... | $f$ is onto. But $f$ is not one one as $\lceil \frac{4}{2}\rceil=\lceil \frac{3}{2}\rceil$, but $4\neq 3$. |
Parabola: the set of points in the plane that are equidistant from a line, called the directrix, and a point, called the focal point, not on the line.
Suppose we try to replicate this on a sphere: let the directrix be given as a great circle $D$. Let $C$ be a great circle perpendicular to $D$. Let $P$ be our focal poi... | We can set up the problem as an algebra problem.
Lines on the circle are great circles; we can choose our coordinates so that the line under question is the equator, and the point has coordinates $(0,a,b)$ with $a,b > 0$.
The closest point from $(x,y,z)$ to the equator lies in the same radial direction from the $z$ a... |
Suppose that $X$ is real-valued normal random variable with mean $\mu$ and variance $\sigma^2$. What is the correlation coefficient between $X$ and $X^2$? | Hint: You are trying to find:
$$\frac{E\left[\left(X^2-E\left[X^2\right]\right)\left(X-E\left[X\right]\right)\right]}{\sqrt{E\left[\left(X^2-E\left[X^2\right]\right)^2\right]E\left[\left(X-E\left[X\right]\right)^2\right]}}$$
For a normal distribution the raw moments are
$E\left[X^1\right] = \mu$
$E\left[X^2\right] =... |
In how many rotational distinct ways can we color the vertices of a cube with 2 colors and faces with 4 colors? (This can be interpreted in two ways, either you have to use exactly 4 colors or at most 4. I'm interested in solving it for both conditions).
There are questions and answers for each part separately but I c... | A mixed cycle index for faces and vertices combined can be of use in
this problem. We can then apply Burnside or Polya as desired. The
group $H$ here are the rotations permuting six faces and eight
vertices simultanteously, acting on fourteen slots for the colors. We
use $b_q$ for the cycles of the vert... |
Problem taken from walter rudin Books
Let $f$ be a continuous real function on $ℝ^1$, of which it is known
that $f'(x)$ exists for all $x ≠ 0$ and that $f'(x) → 3$ as $x → 0$.
Does it follow that $f'(0)$ exists?
My attempt: I think NO
i take $$f(x) = \left\{ \matrix{
3\sin x\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,... | A suggestion is to use the mean value theorem:
$\frac{f(h)-f(0)}{h}=f'(\xi)$
for $\xi$ between $0$ and $h$. Then pass to the limit |
Let E have finite outer measure. Show that there is an $ F_{\sigma}$ set F and $ G_{\delta }$ set G such that $ F \subseteq E \subseteq G$ and $ m^*(F) = m^*(E) = m^*(G)$.
I have already shown that $ E \subseteq G$ and $ m^*(E) = m^*(G)$. But don't know how to prove the first part. | The book "Extensions of first-order logic" by Maria Manzano http://www.amazon.com/Extensions-First-Order-Cambridge-Theoretical-Computer/dp/0521019028 might be valuable, especially if you're coming at things from a computer science perspective. |
I have the Irreducible Polynomial $g = X^4 + X + 1$ over $\mathbb{F}_2 E$, the extension of $\mathbb{F}_2 = \{0,1\}$ with root $\alpha$ of $g$.
My previous problems have asked me to find the number of elements in $E$ and see if they may all be represented in the for $a^n$ with $n \in N$ (I believe the answer to these ... | Recall that the multiplicative group of $\Bbb F_2E$ is cyclic of order 15 ( because it omits 0).
It's a convenient fact that if $r$ is a root of a polynomial over $\Bbb F_2$, then $r^2$ is also. (You prove this, or at least check that it's true for the cases you know about.)
So if you have one root of a polynomial ov... |
Below is my attempt to prove the topological version of the Bolzano-Weierstrass Theorem. Is it an effective proof? I'd appreciate any comments on it. The book gave a hint to use a nested sequence of half-intervals. The idea is pretty intuitive...I'll explain it if anyone would like me to.
Bolzano-Weierstrass Theor... | Choose so that $A\cap[a_n,b_n]$ is infinite. It is not hard to argue this is possible at each step. |
I was surprised to learn from John Stillwell's comment in answer to the
question,
"Can the unsolvability of quintics be seen in the geometry of the icosahedron?",
that
There is not a single picture in the whole...
of Felix Klein's 1884 book,
Lectures on the Icosahedron and the Solution of Equations of the Fifth Degr... | Some other books of Klein (Fricke-Klein on authomorphic functions, his Lectures on hypergeometric functions) have a lot of excellent
pictures. From reading many old mathematical books I cannot conclude that there was any trend
or fashion that changed with time or from country to country.
The explanation why some books... |
Solve:
$$ \sin x + \sin 3x + \sin 5x = 0. $$
Attempt at a solution: applying formulas for summation of sine we get after a series of operations: $ \sin x(8 \cos x \cos 2x \cos x + 1) = 0$ equaling sine to $0$ we get one solution $180k$. comparing the other factor we eventually get $\cos^2 x = 0.25 $ this in turn p... | Yes, expressed in degrees, you've covered all the bases.
(Nit pick: why not specify that $k \in \mathbb Z$?)
Nice work on solving the equation. |
I have a matrix $M$, whose dimension I am unsure of, which has only $\lbrace0,1\rbrace$ entries. I would like to generate all the possible matrices that result from changing (some subset) of the $1$'s to $-1$'s. For example, if $M=\lbrace\lbrace0,1\rbrace,\lbrace 1,0\rbrace\rbrace$, then I would like to generate the l... | For example
m = {{0, 1}, {1, 0}};
ReplacePart[m, #] & /@ Thread[Rest[Subsets@Position[m, 1]] -> -1]
(* {{{0, -1}, {1, 0}}, {{0, 1}, {-1, 0}}, {{0, -1}, {-1, 0}}} *) |
With base $16$, the digits are denoted as $0, 1,\ldots, 9, A,\ldots, F$. Let $n = AB3$. Rewrite $n$ with bases $10$, $2$.
I have no clue what this question means and how I should attempt to do this. | Hint: a number $n$ in base 10 can be written as follow: $$n=a_i \cdot 10^{i} + a_{i-1} \cdot 10^{i-1} \cdot \dots \cdot a_1 \cdot 10 + a_0 \cdot 10^0 $$ where $a_i \in \{0, 1, \dots, 9\}$. Similar to this case... |
I am doing this Probability MOOC from edX. I can't understand concept of number of events.
How come D,E and F are events. They don't make any sense. I mean, practically, if I have to choose one item from a set of 3 items. I will just choose one. So I get 3 choices. Why remove 1 item and then group others into 2 and... | I think you are conflating the definition of an event with the definition of an outcome. You are correct that there are only three outcomes in this sample space.
However, an event is a set of [possibly multiple] outcomes. The solution enumerates all possible sets of the three outcomes. |
Here, $\mathbb{N}$ is the set of natural numbers. I was reading why there is no bijection from $\mathbb{N}^\mathbb{N}$ to $\mathbb{N}$, and I understood we took any subset from $\mathbb{N}^\mathbb{N}$ and proved it not bijective to $\mathbb{N}$ by Cantor's Diagonalization argument.
What am I unable to understand is, ... | It might be best to approach this by contradiction. In particular, if we assume $\lim_{x\to\infty}f(x)\neq 0$, then either the limit is nonzero or $f$ decreases without bound. It should be possible to show in each case that the integral does not converge. |
Let $n \geq 1$ and $N \geq 1$ be integers. I am interested in the sum $$\sum_{k=0}^{N} \binom{k + n-1}{n - 1}$$ I don't know how to tackle this. I've tried using the definition of $\binom{n}{k}$ but did not get anywhere.
Could anyone suggest a method of attack for evaluating this sum? | We use the coefficient of operator $[z^k]$ to denote the coefficient of $z^k$ in a series. This way we can write for instance
\begin{align*}
\binom{n}{k}=[z^k](1+z)^n\tag{1}
\end{align*}
We obtain
\begin{align*}
\color{blue}{\sum_{k=0}^N\binom{k+n-1}{n-1}}&=\sum_{k=0}^N[z^{n-1}](1+z)^{k+n-1}\tag{2}\\
&=[z^{n-1}](1+z... |
Assume that $M$ is a submanifold of $\mathbb{R}^n$ with the following property:
For every $x\in M$ and $y \in \mathbb{R}^n \setminus M$, the open straight line $\{tx+(1-t)y\mid t \in (0,1)\}$ does not intersect $M$.
Does this imply that $M$ is a whole affine sub space? | Yes. First, it is not hard to show that $M$ contains lines passing through each pair of its points. Now pick $ a \in M $. We will show that $L = M - a $ (sum of two sets in Minkowski's sense)is a vector subspace of $R^n.$
Proof: Note that $L$ contains $0$ and it contains all line passing through its points. Now take... |
I am reading Applied linear algebra: the decoupling principle by Lorenzo Adlai Sadun (btw very recommendable!)
On page 69 it gives an example where a real, square matrix $A=[(a,-b),(b,a)]$ is raised to the k'th power: $$A^k.(1,0)^T$$ The result must be a real vector. Nevertheless it seems easier to do the calculation... | We have that
\begin{align}
A=\begin{pmatrix}a & -b\\
b &a
\end{pmatrix}
\end{align}
with $a$ and $b$ real numbers. In space $L(\mathbb{C}^2)$ this matrix has 2 eigen-values $a+bi$ and $a-bi$ with eigen vectors $\mathbf{b}_1=(1,-i)$ and $\mathbf{b}_2=(1,i)$.
We have $(1,0)=\mathbf{e}_1=(\mathbf{b}_1+\mathbf{b}_2)... |
If the linear system $Ax=0$ ($A_{3\times 3}(\mathbb{R})$) has infinitely many solutions, does it mean that the system $Ax=b$ is consistent $\forall b\in\mathbb{R^3}$. If not, find the set of $b\in\mathbb{R^3}$ for which the system $Ax=b$ is consistent.
What characterization of Kronecker Capelli's theorem can prove or ... | Start by expanding the integrand.
$$\begin{align}
I&=\int \left(2x^2e^{x^2}+e^{x^2}\right)\,dx\\
&=\int 2x^2e^{x^2}\,dx+\int {e^{x^2}}\,dx\tag{a}\\
&=x e^{x^2}-\int 2x^2 e^{x^2}\,dx+\int 2x^2 e^{x^2}\,dx\\
&=x e^{x^2}+C
\end{align}$$
$(\text{a})$: Use integration by parts on the second integral with $u=e^{x^2}$ and $... |
I'm trying to close some outstanding gaps in my understanding of a technique for proving an angle is irrational. While the rest of the details of the proof aren't important for my question, the short version of the proof is that if $\theta$ is a rational angle then $2 \cos(\theta)$ is an algebraic integer.
My gap in ... | " the short version of the proof is that if θ is a rational angle then 2cos(θ) is an algebraic integer."
My first answer was completely wrong. Sorry.
But appearently this is not an if and only if statement.
If $2\cos(\pi\theta)$ is an algebraic integer then $ \theta$ is rational.
But the converse is not true. $\... |
Suppose I say "In triangle ABC, point P lies in the interior of line segment AB."
Does that mean that P lies on AB or that it lies inside the triangle?
Thank you.
Edit: I saw this in 2019 AMC 10B Question 16.
2019 AMC 10B Questions
Sorry I cannot upload the screenshot of the question, it says something went wrong. | The only reasonable meaning is that $P$ lies on $AB$, but it's not one of the endpoints. |
A die is thrown until 5 followed by 3 appears. Like 156234753
What is the expected number of throws?
I've been trying to solve this problem but can't find anything similar.
Any help?
My solution:
Let $\displaystyle X$ be the random variable of the qustion (number of throws until 53).
Let the event $\displaystyle ... | As already suggested in a comment the system has two states: either we just have thrown a $5$ (and waiting for a 3) or any other number ($X$). The expected number of throws till the consecutive '53' is therefore:
$$\begin{align}
t_x&=1+\frac56t_x+\frac16t_5\\
t_5&=1+\frac46t_x+\frac16t_5
\end{align}$$
Solving the syste... |
How do I go about finding for which values of $a$ the $\lim_{x\to a}f(x) = \lfloor x \rfloor$ doesn't exist?
I've tried using $x-1 \lt \lfloor x \rfloor \le x$, but it seems I'm lacking some fundamental way of thinking about the general picture...
I'm a newbie in the subject, so a meticulous answer would be much apprec... | I was looking for an answer involving the epsilon-delta notation, and I should have explicitly said so. I think I've found it, so here it is:
Starting off, we have $ \lfloor a \rfloor - \varepsilon < \lfloor x \rfloor < \lfloor a\rfloor+\varepsilon $. It follows that $\lfloor x\rfloor$ is
at least: the closest intege... |
what are the number of tangents that can be drawn from the point $(\frac{-1}{2},0)$ to the curve $y=e^{\{x\}}$.Here { } denotes the fractional part function
what I have done:Since we cannot differentiate the fractional part function I removed the fractional part function as follows
y=$e^x$, $ x\in [0,1)$
y=$e^{x-1... | The function is continuous and differentiable, except at integer values, where it is continuous and differentiable on the right only. At any point $x=x_0$, the (right-hand) derivative is $e^{\{x_0\}}$, and this can be taken for the slope of the tangent.
We have the tangent equation
$$y=e^{\{x_0\}}(x-x_0)+e^{\{x_0\}}=... |
I have a question about the process to find the mass of a sphere with a varying radial density in respect to the radius.
It's something really simple, but I would like someone to explain it me.
Say that the density varies with: $\rho(r)=a-br$
So when you go from $m=\rho V$
Why is it that it changes to $dm=\rho (r) ... | There actually is a sense in which you can say $dm=Vd\rho$ instead of $dm=\rho dV$.
In the usual interpretation, $dm=\rho dV$ means the infinitesimal bit of mass is equal to the density of an infinitesimal volume, times that extra little bit of volume. The infinitesimal volumes are added up to reach the total volume, ... |
What is the size of the smallest set of vertices in planar graph when removed the remaining graph has only one face (outer face)?
Is this parameter well studied? Does it have a name? A case of special interest is the class of bipartite 2-connected graphs. | The minimum number of vertices one can remove in order to make the graph acyclic (which is the only meaning I can give to "one outer face") is called the decycling number of a graph. There is a famous conjecture here by Albertson and Berman, and independently Akiyama and Watanabe which says that the decycling number of... |
The generalize rule for a transitive relation is
a -> b
b -> c
therefor
a -> c
If an element has less than 3 elements, can it still be transitive? If so, does that provide any useful information? | It may or may not be transitive. It depends on the set and the relation.
Let S={} then R={} and it is trivially transitive.
Let S={a} then there are two relations R1={}, R2={(a,a)} and both are transitive trivially.
Let S={a,b} then there are 16 relations
R1={} --> Transitive
R2={(a,a)} ... |
By work of Deligne and others (I am following Deligne-Milne's notes which I just began to read: http://www.jmilne.org/math/xnotes/tc.pdf) we know that a given affine group scheme G can be recovered from their category of representations Rep(G) (thought of as a neutral Tannakian category). If G-->G' is a morphism Prop. ... | Given two elements $V,W$ of $Rep_G$, we can compute $Hom_{K} (V, W)$. It is just the maximal subobject of $V^{\vee} \otimes W$ that comes by pullback from a representation of $G'$. (Because it is the maximal $K$-invariant subrepresentation of that)
I think $Rep_{K}$ is essentially the universal abelian category contai... |
There are multiple answers to this question on this website, but they do not address the question that I have.
To be proved: $A = \{4n + 3 \mid n \in \mathbb N\}$ contains infinitely many prime numbers. Hint:... Consider the number $N = 4p_1...p_m-1 = 4(p_1...p_m-1)+3$. Argue that $N$ must contain a factor $4q+3$, usi... | We have that
$$(4a+1)(4b+1)=16ab+4a+4b+1=4(4ab+a+b)+1$$
and so, taking $c=4ab+a+b,$ we see that $(4a+1)(4b+1)$ is of the form $4c+1.$
Suppose $N>1$ is an integer such that every prime factor of $N$ is of the form $4a+1$ $(a\in\mathbb{N}).$
Can you see how the above hint shows that $N$ must be a number of the form ... |
$g(x)=\frac{x^{1/4}}{x^3+1}$, find $g'(x)$. Use the quotient rule.
My attempt was:
\begin{align}g'(x)&=\frac{\frac{1}{4}(x^{-3/4})(x^3+1)-(3x^2)(x^{1/4})}{(x^3+1)^2}\\
g'(x)&=\frac{(x^3+1)-(3x^2)(x^{1/4})}{4(x^{3/4})(x^3+1)^2}\end{align}
I am $100\ \%$ sure of the denominator which is the easy part, but or the numer... | The claim that the statement "$X\neq \bigcup_{x\in X}{x}$ is false" is not true. What I believe you mean is $X\neq \bigcup_{x\in X}{\{x\}}$.
For example, take $X=\{\{0\},\{1\}\}$. Then $\bigcup{X}=\{0,1\}\neq X$.
However, the claim that "$X\neq \bigcup_{x\in X}{x}$ is false" is true. To see this, notice that $\{y\... |
I want to solve $\int_0^{2\pi}x^4\cos(x) dx$ without having to apply integration by parts 4 times. My attempt: $$ \int_0^{2\pi} x^4 \cos(x) dx= \int_0^{2\pi} \frac{\partial^4}{\partial a^4} \cos(ax) dx = \frac{\partial^4}{\partial a^4} \int_0^{2\pi} \cos(ax) dx = 0 $$
1) Why is this wrong? (I guess it's about switchi... | Following my comment above: the function $f(x,a) = \cos(a x)$ is regular enough (continuously differentiable, at least) for your approach of switching integral and differentiation to be justified. The error is in the last step: $g\colon a\mapsto\int_0^{2\pi} f(x,a)dx = \int_0^{2\pi} \cos ax dx$ is a function of $a$, wh... |
What books/notes should one read to learn model theory? As I do not have much background in logic it would be ideal if such a reference does not assume much background in logic. Also, as I am interested in arithmetic geometry, is there a reference with a view towards such a topic? | There are a bunch of suggestions on Peter Smith's blog. |
Is there any conditions for an ideal $I$ that assures the canonical map $A\to A/I$ is flat?
Here's my try:
(1)It's obvious when $I=(0)$ or $I=A$.
(2)Since $I\otimes_A A/I=0$, it can't be faithfully flat unless $I=0$.
(3)If $I$ contains non zero divisor $a\in I$, then multiplying $a$ is injective as a map $\lambda_... | Suppose that $R/I$ is a flat $R$-module. Then, I claim that $R \to R/I$ must be a localization.
Indeed, define $S$ to be $\{s \in R\mid \exists t \in R \, (st-1) \in I\}$. Then, clearly, there is an induced surjective ring homomorphism $\varphi:S^{-1}R \to R/I$.
To show that $\varphi:S^{-1}R \to R/I$ is also injectiv... |
If either point is above or on the y=1/4 line, or below or on the y=-1/4 line, then the two points are definitely on the same side of the x-axis. For the other possible points I know I need to calculate portions of the areas of circles with radius=1/4; I'm just not sure how to go about that. | Let $(X_1, Y_1)$ and $(X_2, Y_2)$ be the Cartesian coordinates of the two random points with $D$. Then $X_1$, $X_2$, $Y_1$ and $Y_2$ and independent and uniformly distributed over $(-1,1)$ interval.
The event of interest is $A = \{Y_1 Y_2 > 0\}$, conditioned on another event $$B = \{(X_1-X_2)^2 + (Y_1-Y_2)^2 < \tfrac{... |
What is the precise difference or relation between these terms in logic:
Inference, Reasoning, Deduction, and Induction? | The difference between deductive argument and inductive argument is as follow,
Deductive: the conclusion of a deductive argument follows necessarily from the premisses( it follow with certainty. It is this case that we say a deductive argument is EXPLICATION(i.e it does not need any other premiss, "more" or "less" to... |
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