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H: Littlewood's 1914 proof relating to Skewes' number
From Littlewood's 1914 theorem (paraphrase):
I propose to show there are arbitrarily large values of x for which successively
$\psi(x) - x < - K\sqrt{x}\log\log\log x \tag{A}$
$ \psi(x)- x > K \sqrt{x}\log\log\log x \tag{B}$
[...2 pages later]
It suffices then, to... |
H: Prove $\sqrt{k}$ is not a rational number.
Suppose $k>1$ is an integer, and k is not a square number, then $\sqrt{k}$ is not a rational number.
Proof:
Let $\sqrt{k}=\frac{p}{q}$, and $(p,q)=1$,So $q^2|p^2$, $p\neq 1$, $k$ is not an integer.When $q=p=1$, and $k>1$.
And $q=1$, then $\sqrt{k}=p$, and $k$ is a square n... |
H: Coin Flip: "Exactly" and "At Most"
A coin is flipped $10$ times. How many outcomes have exactly three heads? How many outcomes have at most three heads?
AI: Imagine writing down a sequence of length $10$, made up of the letters H and/or T, to indicate what happened on your tosses.
Exactly $3$ heads happened precis... |
H: At least one member of a pythagorean triple is even
I am required to prove that if $a$, $b$, and $c$ are integers such that $a^2 + b^2 = c^2$, then at least one of $a$ and $b$ is even. A hint has been provided to use contradiction.
I reasoned as follows, but drew a blank in no time:
Let us instead assume that bo... |
H: some integral and series whose value is $1$.
Give me some integral and series whose value is $1$.
Where can I find a large number of these kinds of examples.
I have two examples here, but I cannot think up more...
This is geometry series, we learned in high school and before.
$$\begin{align*}\sum _{i=1}^{\infty } ... |
H: Every bounded function has an inflection point?
Hello from a first time user! I'm working through a problem set that's mostly about using the first and second derivatives to sketch curves, and a question occurred to me:
Let $f(x)$ be a function that is twice differentiable everywhere and whose domain is $ \Bbb R$.... |
H: What is the motivation for differential forms?
I am that point in my mathematical career where I am learning differential forms. I am reading from M.Spivak's Calculus on Manifolds. So far I have gone over the tensor and wedge products and their properties, defined forms, learned of their pullbacks and the propertie... |
H: Largest square written as $p^2+pq+q^2$ where $p, q$ are primes?
I got this problem from the website Brilliant, but I have doubts about the solution presented there:
$(p+q)^2-k^2=pq$
$(p+q+k)(p+q-k)=pq$
Now either $(p+q+k)=p$ and $(p+q-k)=q$ (which doesn't work), or $(p+q+k)=pq$ and $(p+q-k)=1$ . Solving the second ... |
H: Integral extensions of rings, when one of the rings is a field
The following is from page 61 of Introduction to Commutative Algebra by Atiyah & Macdonald:
Proposition 5.7. Let $A\subseteq B$ be integral domains, $B$ is integral
over $A$. Then $B$ is a field if and only if $A$ is a field.
I am curious what happe... |
H: Is there a proof that no lower bound exists for the totient function?
I read here that there is no lower bound for the totient function. Is there a proof of that?
AI: If you read carefully, the article says not that there is no lower bound, but that there is no linear lower bound — that is, there's no $c$ (and $n_0... |
H: Combination problem question.
I am working on a combination problem and I need to check if I'm doing this right.
There is a deck of cards that consist of 20 cards. There are four different colors, including 2 Green, 6 Yellow, 8 Black and 4 Red cards. Each colors are numbered, so they are distinguishable. If one dr... |
H: $p\nmid 2n-1,$ then $\sum_{k=1}^{p-1}\frac{1}{k^{2n-1}}\equiv 0 \pmod{p^3} \Leftrightarrow \sum_{k=1}^{p-1}\frac{1}{k^{2n}}\equiv 0 \pmod{p^2} $
Is it true that if $p$ is a prime and $p\nmid 2n-1,$ then
$$\sum_{k=1}^{p-1}\frac{1}{k^{2n-1}}\equiv 0 \pmod{p^3}
\hspace{12pt}\Leftrightarrow \hspace{12pt}
\sum_{k=1}... |
H: Interesting related rates question
A circle C in the xy-plane is described as follows: A point P on the circumference of C traces out the graph of $f(x) = \sqrt{x}$; the center of C is the y-intercept of the tangent line of $f(x)$ at P. If the center of C moves upwards along the y-axis at the rate of $\frac14$ cent... |
H: Factoring $a^2+b^2+c^2$?
Is it possible to factor $a^2+b^2+c^2$ ? If we make this into only two factors, I know it has to look like this:
$(a+b+c+\cdot \cdot \cdot )(a+b+c+\cdot \cdot \cdot )$ . But I don't know how to get rid of the $2(ab+bc+ac)$ but I have no idea what else should go in the parenthesis. How can... |
H: correct understanding mathematical question
suppose that we have following question,this question is not related to itself mathematics confusion,but language problem and please help me to clarify English language terms in mathematics. question is this :
Simon arrived at work at $8:15$ A.M. and left work at $10:... |
H: proof about the sum of lim sup
I have questions about the solution below.
I couldn't understand the red lines. What is $X_nN_1$?
I'm not sure how it led to the contradiction.
Thank you!
Exercise 3: For any two real sequences $(a_n)$ and $(b_n)$, prove that
$$\limsup_{n\to\infty} (a_n + b_n) \le \limsup_{n\to\infty}... |
H: Isn't $(0)$ a prime ideal in a field?
I have read in multiple places that a field $K$ has a Krull dimension of $0$. How is this true? Isn't $(0)\subset K$ a prime ideal in $K$? Obviously $K$ is an integral domain.
Thanks in advance!
AI: The Krull dimension is defined to be the length of the longest chain of prime ... |
H: Analytic function and connected region.
We have the result.
Let $G$ be an open connected set in $\mathbb{C}$, and let $f : G \rightarrow \mathbb{C}$ be an analytic function. Then the following statements are equivalent.
*1. $f(z) = 0$ in $G$ for all $z$ *
2. There is a point $a \in G$ s.t. $f^n(a) = 0$ for each $n ... |
H: On Decompositions of Finite Group
Any finite non-cyclic abelian group $G$ can be written as product $HK$ of two proper subgroups. Here $HK=\{ hk\colon h\in H, k\in K\}$. A step further, if $G$ is a finite group such that the commutator subgroup $[G,G]$ is proper subgroup of $G$, then $G$ has a decomposition $HK$ fo... |
H: Find correlation coefficient of $f(x,y)=2$ for $0
Find the correlation coefficient for the random variables $X$ and $Y$ having joint density $f(x,y)=2$ for $0 < x \leq y<1$.
Seem like a simple problem but I'm stuck.
Since $\mbox{Corr}(X,Y) = \frac{\mbox{Cov}(X,Y)}{\sqrt{\mbox{Var}(X)}\sqrt{\mbox{Var}(Y)}}$, I fig... |
H: Continuous extension of a function
Can anybody help me with this problem?
Justify whether the following statement is true or false:
Every continuous function on $\Bbb Q\cap [0,1]$ can be extended to a continuous function on $[0,1]$.
Any help will be appreciated.
AI: See what happens if you take the function
$$f:\... |
H: How to use binomial theorem
How to use binomial theorem，From (1) to get (2)?
$$\begin{align*}\left(1+\frac{1}{n}\right)^n\tag{1}\end{align*}$$
$$\begin{align*}t_n=1+1+\frac{1}{2!}\left(1-\frac{1}{n}\right)+\frac{1}{3!}\left(1-\frac{1}{n}\right)\left(1-\frac{2}{n}\right)+\text{...}+\frac{1}{\text{n1}}\left(1-\frac{1... |
H: Derive a formula to solve a specific task
I have a specific problem.
I have 8 different variables a, b, c, d, e, f, g, h.
Each of these variables has a score out of 5, where 1 is bad and 5 is good. So a max score of 45 and a min of 0.
Of these variables I can influence the scores for 6 a, b, c, d, e, f.
I can not ... |
H: Uncountable basis and separability
We know that a Hilbert space is separable if and only if it has a countable orthonormal basis.
What I want to ask is
If a Hilbert space has an uncountable orthonormal basis, does it mean that it is not separable? Or equivalently, does it imply that the Hilbert space does not have ... |
H: Rational Solution of a System of Linear Equations
I am having a little trouble with this problem -
Let $A$ be a $m\times n$ matrix and $v$ be a $n\times 1$ matrix, both of which only has rational entries. It is known that the equation $Ax=v$ has a solution in $\mathbb R^n$. Does this imply that the given equation ... |
H: Metric of the flat torus
I am studying the flat torus $T^n=\mathbb{R}^n/\mathbb{Z}^n$. I am interested in the metric and the connection used. Unfortunately, in the books I am reading those things aren't defined. Does anyone knows this definition or a reference where I can find it? Thanks in advance for the help.
A... |
H: relationship between circumference and revolution
i would like to clarify two things by this problem:first what is relationship between circumference and revolution and also revolution and distant traveled by round object.let us consider following problem:
A tire on a car rotates at $500$ RPM (revolutions per min... |
H: Theory set problem, determining the min and max number of elements from $(B \cup A) \bigtriangleup (C \cap A)$
Given that $$ |A| = 5 \\ |B| = 6 \\ |C| = 7 \\ |\Omega| = 10 \\ A \subseteq (B \cup C) $$
Determine the min and max numbers of elements from $$(B \cup A) \bigtriangleup (C \cap A)$$
I tried to solve this b... |
H: reference for operator algebra
I am taking a course on operator algebra this semester. My instructor has suggested a reference "Kadinson and Ringrose." Are there any other good/standard references for this subject that I can look up?
AI: The book by Kadison and Ringrose does not contain a number modern topics (irra... |
H: Solving logarithmic equation
I'm having trouble solving this equation. I know there is a solution as my graphics calculator can solve it, but I want to see the steps on how to get the answer.
The mathematical equation is:
$$\log_{10}n = 0.07n$$
AI: Not all equations can be solved algebraically. This equation can no... |
H: A question on divisible groups
Let $p$ be a prime and $H=\prod_{n=1}^{\infty}\mathbb Z(p^{n})$ ($\mathbb Z(p^{n})$ is the finite cyclic group of order $p^{n}$). Is $H/t(H)$ divisible ($t(H)$ denotes the maximal torsion subgroup of $H$)?
AI: Does $(1,1,1,\cdots)$ have a $p$th root modulo torsion?
If so, we have $(1,... |
H: Is there formula for this squared geometric (?) progression?
Is there non-recursive formula for the following sequence:
$$a_1=\frac12,$$
$$a_n=\frac12a_{n-1}^2+\frac12$$
If there is, how do you suggest I can determine it?
AI: Making the change of variables
$$a_n=1-2x_n$$
the recursion relation transforms into
$$x_{... |
H: $E+F=E\oplus F \Leftarrow \bigcap$ of their bases $=∅$
I'm trying to understand this theorem:
I will traduce it literally from my lecture notes:
Given n≥1 subspaces $E_1,E_2,...,E_n$ of a vector space V and considering the subspace $F=E_1+E_2+...+E_n$ if $B_1,B_2,...,B_n$ are bases of $E_1,E_2,...,E_n$ and $B_i \c... |
H: Cumulative Normal Distribution.
Let $X_1,\ldots,X_n$ be a random sample from $f(X;\theta)=\phi_{\theta,25}$, that is, $X_1,\ldots,X_n$ be normally distributed with mean $\theta$ and variance $25$.
I am not understanding how
$$sup_{\theta\leq17}[\phi(\frac{17+\frac{5}{\sqrt n}-\theta}{\frac{5}{\sqrt n}})]=\phi(1)$$
... |
H: Sequence problem, find root
the equation $x^3-5x+1=0$ has a root in $(0,1)$. Using a proper sequence for which $$|a(n+1)-a(n)|\le c|(a(n)-a(n-1)|$$ with $0<c<1$ , find the root with an approximation of $10^{-4}$.
AI: You can attempt this: define a sequence $x_n$ by
$$
x_{n+1} = \frac 15\left(1 + x_n^3\right).
$$
We... |
H: Simplify this expression that came from integration
I was doing a calculation and arrived at a term $\left[P_{l-1}(\cos(\theta)) -P_{l+1}(\cos(\theta))\right]_{0}^{\pi}$(So this is the result of an integration). Does anybody of you know how to simplify this expression? (Testing suggested to me that this one is eith... |
H: Calculate angle in triangle having 2 points and two lines
I have 2 points $B$ and $P$ and need to calculate angle $\alpha$ (maybe also I will need point $C$ and $E$)
How can I do this. I know that I can calculate point $D$ it's $(\frac{1}{2}(x_P-x_B), \frac{1}{2}(y_P-y_B))$ then calculate line that it's perpendicu... |
H: Is this integral correct?
I used substitution and got that:
$$\int_0^\pi \sin x \cdot P_n(\cos x ) \, dx=0$$
where $P_n$ is the $n$-th Legendre polynomial.
AI: Recall that Legendre polynomials are defined as orthogonal polynomials on $[-1,1]$ with weight function $w(x)=1$. In other words, we have by definition
$$(P... |
H: Pre-images of closed sets are open
Let $X$ and $Y$ be two topological spaces and let $f$ be such a map that $f^{-1}(A)$ is open in $X$ for any closed $A$. Note that if $X\stackrel{f}{\longrightarrow}Y\stackrel{g}\longrightarrow Z$ are two such maps, then $g\circ f$ is continuous. Perhaps, it is a trivial task - bu... |
H: If points $( k+3 , 2-k)$, $ (k, 1-k)$ and $(3, 4+k)$ are collinear, find the value of $k$
If the points $( k+3 , 2-k)$, $(k, 1-k)$ and $(3, 4+k)$ are collinear,
find the value of $k$.
Can someone please hint at how to start this question? I've so far tried finding the gradients but no luck. thanks!
AI: Hints:
T... |
H: Tackle this series
I am looking for the exact value or a smart approximation(if you have a good idea) of the following series:
$$\sum_{n=0}^\infty \frac{1}{2n+1} (P_{n+1}(0)-P_{n-1}(0))$$ where $P_n$ is the n-th Legendre polynomial and $P_{-1}(0)=-1$
AI: Let us use the relation
$$\left(\frac{P_{n+1}(x)-P_{n-1}(x)}{... |
H: Rotating a Matrix by an angle
So I have a matrix like so
\begin{pmatrix} x_0 & x_1 & x_2 & x_3 \\ y_0 & y_1 & y_2 & y_3 \end{pmatrix}
And I need rotate the matrix by an angle - for say $45$ degrees.
I read that the rotation matrix is
\begin{pmatrix} \cos(45^\circ) & \sin(45^\circ) & \\ -\sin(45^\circ) & \cos(45... |
H: Question on monotonicity and differentiability
Let $f:[0,1]\rightarrow \Re$ be continuous. Assume $f$ is differentiable almost everywhere and $f(0)>f(1)$.
Does this imply that there exists an $x\in(0,1)$ such that $f$ is differentiable at $x$ and $f'(x)<0$?
My gut feeling is yes but I do not see a way to prove it. ... |
H: Pointwise convergence domain of function series
I'm stuck at finding pointwise convergence domain of the following function series
$$\sum_{n=1}^\infty \frac{\sqrt[3]{(n+1)}-\sqrt[3]{n}}{n^x+1}$$
I tried to use d'Alembert and Weierstrass tests, but it seems to me they don't work here.
AI: We have
$$(n+1)^{1/3}-n^{1/... |
H: Proving statements like $(a\Rightarrow b) \Rightarrow (p \Rightarrow q)$.
Is there a way to simplify this sort of statement? For example, $$a \Rightarrow (b\Rightarrow c)$$ is equivalent to $$(a \wedge b) \Rightarrow c.$$ I'm looking for something similar for $$(a\Rightarrow b) \Rightarrow (p \Rightarrow q),$$ if i... |
H: Probability from chi square distribution
How do I find a probability for a chi square distribution?
I have a continous random variable from which I've got the chi square with the formula:
$$\sum \frac{(o-e)^2}{e}$$ where $o$ is the observed value and $e$ the expected value (the mean).
with that value plus the degr... |
H: Differentiate $\sin \sqrt{x^2+1} $with respect to $x$?
Differentiate $$ \sin \sqrt{x^2+1} $$ with respect to $x$?
Can someone please help me with question, im very lost.
AI: $$ \dfrac {d}{dx}\sin \sqrt{x^2+1} $$
since $\dfrac {d}{dx}\sin x=\cos x,\dfrac {d}{dx} x^n=n\cdot x^{n-1},\dfrac{d}{dx}C=0$ C is Constant
so... |
H: A combination lock that flashes red, green, or orange
Say you have a combination lock that takes in a key code in the form of 10 integers from 0 to 9 (in the right order). If you have 0-3 of the integers right, a red light will flash. If you get 4-6 of the integers right, an orange light will flash. And if you get ... |
H: Well defined, continuous and singular
Can you explain what they mean when a function is well defined, continuous and singular? I know a function is continuous when you look at the right and left hand limits and both conclude to the same number.
Am I right when I say option 5 is false? See attached picture.
AI: well... |
H: Differentiating $\tan\left(\frac{1}{ x^2 +1}\right)$
Differentiate: $\displaystyle \tan \left(\frac{1}{x^2 +1}\right)$
Do I use the quotient rule for this question? If so how do I start it of?
AI: We use the chain rule to evaluate $$ \dfrac{d}{dx}\left(\tan \frac{1}{x^2 +1}\right)$$
Since we have a function which ... |
H: Interpreting the ; in a series
This question is linked to this question.
So, suppose I set $n=5$. Given the following formula:
$$\frac{1}{n}, \dots , \frac{n-1}{n} $$
Am I suppose to get:
$$
\frac{1}{5}, \frac{2}{5}, \frac{3}{5}, \frac{4}{5} \hspace{8.2cm}(1)
$$
Or
$$
\frac{1}{5}, \frac{1}{4}, \frac{1}{3}, \fra... |
H: Loss of direction in Gauß' theorem?
I was wondering about the following:
If I have a function $\phi:\mathbb{R}^3\rightarrow \mathbb{R}$ and I want to calculate the mean value of $E=-\nabla \phi$ over a sphere, then $E$ of course if a vector, but the mean value: $ E=-\frac{1}{V_\mathrm{sphere}}\int_V \nabla \phi=-\f... |
H: Conic section - hyperbolic path
I got that equation of path is conic section $u=\frac{1}{3c}(1+2\cos\theta)$ where $c$ is constant and one vertex of hyperbola is $(-c,0)$ and $u=r^{-1}$. So, $r=\frac{3c}{1+2\cos\theta}$. Since $e=2>1$ is eccentricity, the path is hyperbolic.
How can I find equations of asymptotes a... |
H: Tough integrals with Legendre polynomial
Does anybody here know how to integrate $\int_0^\pi P_n(\cos(x))\sin(x)\cos(x) dx$,
$\int_0^\pi P_n(\cos(x))\sin^2(x) dx$, where $P_n$ is the n-th Legendre polynomial?
They are actually extremely hard to do, as far as I see, but I pretty much need them.
AI: Rewrite the integ... |
H: Matrix determinant using Laplace method
I have the following matrix of order four for which I have calculated the determinant using Laplace's method.
$$
\begin{bmatrix}
2 & 1 & 3 & 1 \\
4 & 3 & 1 & 4 \\
-1 & 5 & -2 & 1 \\
1 & 3 & -2 & -1 \\
\end{bmatrix}
$$
Finding the determinant gives me $-726$. Now if I ... |
H: Sum $\sum_{n=0}^\infty \frac{\tan(a/2^n)}{2^n},$
$$\sum_{n=0}^\infty \frac{\tan(a/2^n)}{2^n},$$
where $a$ isn't a multiple of $\pi$. I've been going through several telescoping questions, and It seems I have hit a brick wall with this one, any help will be appreciated.
AI: Since
$$\tan\left(\frac{a}{2^n}\right)\si... |
H: Can $\{(1,2,0),(2,0,3)\}$ span $U=\{(r,s,0) \mid r,s, \in \mathbb{R}\}?$
Is it possible that $\{(1,2,0),(2,0,3)\}$ can span the subspace $U=\{(r,s,0) \mid r,s, \in \mathbb{R}\}?$
Using the definition of span, I have gotten this far:
$a(1,2,0)+b(2,0,3)=(r,s,0)$ (where $a,b,r,s \in \mathbb{R}$)
This gives the follo... |
H: Convolution doubt
Can someone explain why the general formula of the convolution is this one:
$$(f*g)(t)=\int_{-\infty}^\infty f(t-\tau)g(\tau) \, d\tau$$
But when both $f(\tau)$ and $g(\tau)$ are equal to zero for negative values of $\tau$, the convolution turns into:
$$(f*g)(t)=\int_0^t f(t-\tau)g(\tau) \, d\tau$... |
H: $\epsilon$-$\delta$ proof that $\lim_{x \to 1} \sqrt{x} = 1$
I'm trying to teach myself how to do $\epsilon$-$\delta$ proofs and would like to know if I solved this proof correctly. The answer given (Spivak, but in the solutions book) was very different.
Exercise: Prove $\lim_{x \to 1} \sqrt{x} = 1$ using $\epsilo... |
H: Prove $\exists$ neighborhood of $I \in Gl(n,\mathbb{C})$ containing no nontrivial subgroup.
Prove that there exists a neighborhood of the identity $I \in Gl(n,\mathbb{C})$ that contains no subgroup other than $\left\{ I \right\}$.
Thanks!
AI: We can prove there exists a neighborhood $V$ of $I$ in $\mathrm{GL}_{n}(\... |
H: Contrapositive: $\forall\; n > 1, n:$ composite $\implies\exists\; p$ (prime) s.t. $p \leq \sqrt n$ and $p\mid n$
Usually, I find it a cakewalk to write the contrapositive, but the following statement is quite complex for the task:
For all integers $n > 1$, if $n$ is not prime, then there exists a prime number $p$... |
H: What is the convergence speed of logistic sequence?
I am looking at the sequence $x_{n+1}=r\, x_n(1-x_n)$ where $r=1$.
Let's choose $x_1=1/2$ so as to make the sequence convergent to 0.
My question is: precisely how quickly does this sequence approach zero?
From my numerical experiments $\lim_{n\to\infty}n\,x_n=1$ ... |
H: Binomial Sum: An In-Depth Analysis into the Relatedness of Two Equivalences
How is it that
$$n(1+x)^{n-1}=\sum_{k=1}^n C(n,k)kx^{k-1}?$$
How can this be used to show that
$$n2^{n-1}=\sum_{k=1}^nkC(n,k)?$$
AI: Hint: We have by the Binomial Theorem:
$$(1+x)^n=\sum_{k=0}^n \binom{n}{k}x^k.$$
Differentiate both sides w... |
H: Weak maximum principle for the p-Laplacian
For the equation $\Delta_p u = 0 $ in $U$ ($U$ open and bounded), does a weak maximum principle hold? (The maximum and minimum occur on $\partial U$)? If yes, someone can indicate a book with the theorem?
Thanks in advance ( my english is horrible, sorry ... )
AI: Yes. S... |
H: Question about coordinate change
Say $f$ is a function $f: \mathbb R^2 \to \mathbb R$. Can someone show me an example of such an $f$ with the property that $(\partial / \partial x)^2 f(x,y) = 0$ and $(\partial / \partial y)^2 f(x,y) = 0$ in one example of coordinates and $(\partial / \partial x)^2 f(x,y) \neq 0$ an... |
H: Why does DP solve a problem in polynomial time whereas brute force is exponential?
I am just learning DP, so maybe this is a noob doubt. I've read (while trying to understand the difference between DP and greedy approach - and I am still not fully clear) that DP goes through all possible solutions to a problem and ... |
H: Regarding Limit/continuity/convergence
let $$f_n(x)=\begin{cases} 1-nx&\text{when }x\in[0,1/n]\\0&\text{when }x\in [1/n,1]\end{cases}$$
Which of the following is correct?
$\lim_{ n\to\infty} f_n(x)$ defines a continuous function on $[0,1]$
$\{f_n\}$ converges uniformly on $[0,1]$
$\lim_{n\to\infty} f_n(x)=0$ for a... |
H: Fitting a sinusoidal function to three known points separated by $30$ degrees
I have three data points measured at $-30$, $0$, and $30$ degrees, respectively. I would like to fit these points to a sinusoidal function of the form:
$$f(\theta)≈A\sin(\theta + B) + C$$
Is this possible? If so, what would be the best ap... |
H: Chance that first 6 characters of a SHA-1 hash matching another SHA-1 hash?
Just what the question says -- what is the chance that the first six characters of a SHA-1 hash will match the first six characters of any given SHA-1 hash?
AI: SHA-1 produces a 120 bit value, no characters.
However, such hashes are often c... |
H: You roll a die until the sum of all your rolls is greater than 13. What number are you most likely to land on, on the last roll?
So I was thinking of doing this recursively: $f(x,i)$ is equal to the probability of rolling greater than $x$ and landing on $i$ on the last roll. $f(0,i) = 1/6$ for $i = \{1,2,..,6\}$. $... |
H: Is there an accepted symbol for irrational numbers?
$\mathbb Q$ is used to represent rational numbers. $\mathbb R$ is used to represent reals.
Is there a symbol or convention that represents irrationals.
Possibly $\mathbb R - \mathbb Q$?
AI: Customarily, the set of irrational numbers is expressed as the set of all ... |
H: Positive integer multiples of an irrational mod 1 are dense
I'm not sure how to solve this one.
Thank you!
$2.$ For any $\alpha\in \mathbb R$ we define $$\lfloor \alpha \rfloor = \max_{n\in\mathbb Z}\{\,n\mid n\leq \alpha\,\}$$ and $$\alpha\bmod 1 = \alpha - \lfloor \alpha \rfloor$$ Let $\alpha$ be irrational.
(a)... |
H: How to calculate the height of a circular segment based on the area.
Given an area of a circular segment, how can one find the height of the circular segment?
In the image below, assume the area of the green segment is known. How can one find the value of h?
I have also seen this problem described as the Quarter T... |
H: Indefinite integral of $\log(\sin(x))$
I'm computing the indefinite integral of $\log(\sin(x))$; this is the my solution with integration by substitution:
$$
\begin{align}
&\int\log(\sin(x))dx\\
= &\int\log(y)\frac{1}{\cos(x)}dy \\
= &\frac{1}{\cos(x)}\int\log(y)dy \\
= &\frac{1}{\cos(x)}(y\log(y)-y) \\
= &\tan(x)\... |
H: Ideals in ring extensions
Let $R$ be a ring, commutative with $1$, subring of a ring $R'$. Let $\mathfrak{p}$ be an ideal of $R$. Let's denote by $\mathfrak{p}R'$ the extended ideal, i.e. the ideal generated by $\mathfrak{p}$ in $R'$.
EDIT: Assume that $R'$ is a free $R$-module of finite rank $n$ and let $x_1,\ldo... |
H: A simple circle problem
There is a big circle of radius 20cm and a smaller circle 100 cm away from it of radius 5cm now imagine these two to be 2 tires connected by a chain , where the bigger one completes one rotation how many rotation will small complete??
Any idea how to Solve this??
AI: When the chain moves b... |
H: How to write formula for bracketed function
I am a Java programmer with little theoretical math experience. I've written a plugin for a program, part of which runs a "bracketed" formula (I don't know what else to call it). I have been asked to release the formula to an audience that may understand it better in ma... |
H: How to prove $f'(a)=0$?
Let $f:I\to\mathbb{R}^n$ be a differentiable function, where $I\subset\mathbb{R}$ is a interval.
For each $c\in \mathbb{R}^n$, define $X_c=\{x\in I;\;\;f(x)=c\}$.
The problem asks to show that if there exists $c\in \mathbb{R}^n$ such that $a\in I\cap \left (X_c\right)'$, then $f'(a)=0$.
I do... |
H: Solutions to $Ax=x$, where $x=(1,1,1,......1)$, for $A\in GL(n,\mathbb{C})$
I know that the set of solutions $A\in GL(n,\mathbb{C})$ satisfies $\det(A-I)=0 $.
I was wondering if there was an easy to way determine what subgroup, call it $H$, this is. How many conjugacy classes would this group (up conjugation by el... |
H: Sum of a geometric series $\sum_0^\infty \frac{1}{2^{1+2n}}$
$$\sum_0^\infty \frac{1}{2^{1+3n}}$$
So maybe I have written the sequence incorrectly, but how do I apply the $\frac{1}{1 - r}$ formula for summing a geometric sequence to this? When I do it I get something over one which is wrong because this is suppose ... |
H: Centre of mass moves with constant velocity
The centre of mass of the Newton $n$-body problem is given by $$S=\frac{1}{M} \sum m_ix_i$$ with $M=\sum m_i$.
Show that it moves with contant speed and hence has no acceleration.
I don't understand as if I differentiate, I'll surely just get $$S'=\frac{1}{M} \sum m_ix'_i... |
H: For what variables these equalities are satisfied?
Assume:
$$
P \subseteq \{1,2,\dots,N\},\quad |P| = K, \qquad x \in \mathbb{R}_+^K , \qquad w = e^{-j\frac{2\pi}N}
$$
Consider:
$$
h_{P,X}(l) = \sum_{i=1}^K \sum_{j=1}^K x_ix_jw^{(p_i-p_j)l}
$$
Now suppose we want to find $(P,X)$'s that satisfy:
$$
h_{P,X}(1) = h_{... |
H: Show that $f(x) = x^p -x -1 \in \Bbb{F}_p[x]$ is irreducible over $\Bbb{F}_p$ for every $p$.
Let $p$ be a prime.
a) Show that $f$ has no roots in $\Bbb{F}_p$.
Let $F^*$ be the multiplicative group of $\Bbb{F}_p$. Then, by lagrange's thoerem for all nonzero $\alpha \in \Bbb{F}_p$, $\alpha^{p-1} = 1 \implies \alpha... |
H: Clarification regarding a group theory proof
In a group we have $abc = cba$. If $c \neq 1$, is the group abelian? See the following link.
(I am new to this site but it is my understanding that you cannot PM authors, correct? Which is too bad because it means I have to open this thread)
In regards to Math Gem's ans... |
H: An ideal in the ring of infinitely differentiable functions
Well, I was just doing an elementary exercise, but am a tad bit skeptical about how I've gone about it. It goes as follows:
Let $R$ be the ring of infinitely differentiable functions defined on, say, the open interval $-1 < t <1$.
Let $J_n$ be the set of ... |
H: How to split the rent if two roommates live there from the beginning and a third one joins in the middle of the month?
I've been thinking over it and I can't figure it out.
Consider the rent of the house is $X$. Now there are two roommates from the beginning and a third one joins in the middle of the month. Now ho... |
H: Calculating the area of a triangle
Consider the circle of radius $1$ and center in $x=1$, $y=1$. Let $p$ be the point in the circle more close to the origin. Suppose that $p$ is the centroid of a triangle with vertex in $(0,1)$, $(2,0)$ and some point $(x,y)$.
My question is: is there any way to calculate the area... |
H: Finding an $n$ so the sequence $\left\{\frac{1}{n}\right\}_{n = 1}^\infty$ satisfies $|a_n| < 10^{-4}$
How to find $n$ so that $\left\{\frac{1}{n}\right\}_{n = 1}^\infty$ satisfies
$$|a_n| < 10^{-4}$$
I can't find this formula in my book anywhere. It seems like it would be very time consuming to just plug in number... |
H: Reference Request for The Study of Abelian Groups
So I finished Lang's Algebra and after reading this partial Structure Theorem for abelian torsion groups that are not finitely generated , I've gotten interested in abelian groups, in particular infinite abelian groups and structure theorems. Can anyone recommend a ... |
H: Factor groups of matrices
Let $G=GL(2,\mathbb R)\oplus GL(2,\mathbb R)$ and let $H=\{(A,B)\in G\mid \det(A)=\det(B)\}$. Prove that $G/H \simeq (\mathbb R^*,\times)$.
I'm guessing I should use: "Let $G$ be a group and let $H$ be a normal subgroup of $G$. The set $G/H = \{aH\mid a \in G\}$ is a group under the opera... |
H: SPECGRAM return value
I was studying this code:
fm = 8000;
dt = 1/fm; % dt=0.000125
t = [1:dt:5];
y = sin(2*pi*200*t);
tw = 0.05;
ws = 2 .^ round( log2( tw*fm ) ); % ws=512
o = ws/2; % o=128
w = hanning(ws);
[ X, f, tj ] = specgram( y, ws, fm, w, o );
What X represents is an array of Spectrums, one per "... |
H: $\lim_{(x,y)\to (0,0)} \frac{x^m y^n}{x^2 + y^2}$ exists iff $m+ n > 2$
I would like to prove, given $m,n \in \mathbb{Z}^+$, $$\lim_{(x,y)\to (0,0)}\frac{x^ny^m}{x^2 + y^2} \iff m+n>2.$$
(My gut tells me this should hold for $m,n \in \mathbb{R}^{>0}$ as well.) The ($\Rightarrow$) direction is pretty easy to show by... |
H: Infinite series convergence value problem
Is it possible to find the convergent value of the series: $\sum\limits_{x=0}^{\infty}(x+3) \cdot a^x $ where $a$ is a constant less than 1?
I thought about expanding:
$\sum\limits_{x=0}^{\infty}(x+3) \cdot a^x =\sum\limits_{x=0}^{\infty}x \cdot a^x + 3 \sum\limits_{x=0}... |
H: Determining if a Linear Transformation is Surjective
I am aware that to check if a linear transformation is injective, then we must simply check if the kernel of that linear transformation is the zero subspace or not. If the kernel is the zero subspace, then the linear transformation is indeed injective.
Is there a... |
H: injection $\mathbb{N}\times\mathbb{N}\to\mathbb{N}$
Today a friend of mine told me a nice fact, but we couldn't prove it. The fact is that there is an injection $\mathbb{N}\times\mathbb{N}\to\mathbb{N}$ defined by the fomula $(m,n)\mapsto (m+n)^{\max\{m,n\}}$, where $\mathbb{N}$ denotes the natural numbers.
How to ... |
H: What is this Weierstrass' proof of uniqueness of $\mathbb{R}$ and $\mathbb{C}$ algebras?
I'm reading Derbyshire's Unknown Quantity.
It's an interesting exercise to enumerate and classify all possible algebras. Your results will depend on what you are willing to allow. The narrowest case is that of commutative, ass... |
H: Clarifying a proof of $\limsup (a_n+b_n) \le \limsup a_n + \limsup b_n$
Could you help me understand the solution below?
"otherwise we clearly have the equality" -> why? It's not clear to me. :(
"The inequality is trivially satisfied" -> why? even if the right side is +infinity, what if the left side is also +infi... |
H: Find force required for a launch between two points
Let me start by saying this is within a game environment, so gravity isn't 9.81m/s^2, and the unit of measure for distance will be "blocks".
I'm attempting to find the amount of force needed in order to launch a player into the air, from point A to point B. For my... |
H: When does an analytic function grow faster than a polynomial?
Suppose $f$ is an analytic function with power series expansion $f(z)=\sum_{n=0}^{\infty} a_nz^n$, and $p = \sum_{n=0}^{d}b_nz^n$ is a polynomial. If $f$ is a polynomial of degree larger than $d$, then $|f|$ grows faster than $|p|$, but the situation is ... |
H: an injection into $\mathbb{N}$
Is that true that the map $f\colon \{(m,n)\in\mathbb N^2:m\le n\}\to\mathbb N$ defined by $(m,n)\mapsto (m+n)^{\max\{m,n\}}$ is an injection? If it is, how to prove that? I have asked a similar question but it appeared to be very easy. My original struggling in the previous question w... |
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