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Can we find all positive integers $a,b$ such that $a^{n}+b^{n}$ is an $(n+1)^{th}$ power? I think this question equivalent to solving the statement $$a^{n} + b^{n} = c^{n+1}$$ for $a,b,c \in \mathbb{N}$. But i don't know as to how i can solve this.
I attempted by subsituting $n$ as $n+1$ so that we get $$a^{n+1} + b^{n... |
In Sylvester's law of inertia, how does the name "law of inertia" fit with the statement of the theorem? I guess it's from physics, but I just don't see the connection.
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This is a follow-up to this question. I came across a tighter potential bound and checked it numerically for $0\le x\le 5$. I think it holds for all positive $x$, can anyone see a proof?
$$1-\exp(-4x^2/ \pi) \ge \text{erf}(x)^2$$
Note: using analysis for previous question you can show that $1-\exp(-k x^2)$ is an upper ... |
1.How many number of three digit even numbers than can be formed out of the digits 0 to 9 ?
The question seems confusing since there is no mention of whether repetition is allowed or not ?! since if repetition is allowed then the answer would be 450 (5*9*10) where as if not then the answer would be 328 (9p2+ 4*(8*8) =... |
I'm curious about the rights of ownership regarding problems presented in math textbooks. Can a math problem in general be considered intellectual property of its "inventor"? I'm assuming textual word problems can be "owned" in the same way other literature is by publishers/authors but it becomes more gray with purely ... |
I've always been taught that one way to look at complex numbers is as a Cartesian space, where the real part is the $x$ component and the imaginary part is the $y$ component.
In this sense, these complex numbers are like vectors, and they can be added geometrically like normal vectors can.
However, is there a geometric... |
I'm having difficulty justifying the change of limits in the derivation of the Riemann-Liouville derivative at xuru.org. What I don't undestand is how $\int_0^{t_2}$ becomes $\int_{t_1}^x$ in the following statement,
$\int_0^x \int_0^{t_2} f(t_1) dt_1 dt_2 = \int_0^x \int_{t_1}^x f(t_1) dt_2 dt_1$
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What is, in the language of Schemes, a $G$-galois branched cover?
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$r$ is a primitive root for the prime $p$
$x_1\equiv r^a \pmod{p}$
$x_2\equiv r^b \pmod{p}$
If $gcd(b, p-1)=1$, how can I determine $r$ if $p$, $x_2$, and $b$ are known?
|
I understand that the only free groups that are abelian are 1 and Z, hence a difference between 'free abelian groups' and 'abelian free groups'.
Can someone please tell me what are the solvable free groups?
How can I construct 'free solvable groups'?
What is known about free groups in other classes of groups like polyc... |
I'm trying to understand the approach to calculating an answer to the following question.
I need to do some processing on a list of n items.
I know that the total processing time takes $\log_2 (n)$ milliseconds.
Problem: What is the maximum number of items that can be processed in $10$ seconds (i.e., $10,000$ milliseco... |
Okay, I have this not so pretty 2nd order non-linear ODE I should be able to solve numerically.
$$f''(R) + \frac{2}{R} f'(R)=\frac{0.7}{R} \left( \frac{1}{\sqrt{f(R)}} - \frac{0.3}{\sqrt{1-f(R)}} \right),$$
$$f(1)=1.$$
The function around the origin is behaving very wildly.
I was thinking of breaking this guy up into ... |
In the course of playing around with $\sum_{n=1}^{\infty} \frac{1}{\sqrt{1+n^{n}}}$, I used w|α to obtain the power series for $f(x)=\frac{1}{\sqrt{1+x^{x}}}$
which is
\begin{align*}
\frac{1}{\sqrt{1+x^{x}}} =& \frac{1}{\sqrt{2}} - \frac{x\log(x)}{4\sqrt{2}} -\frac{x^{2}\log^{2}(x)}{32\sqrt{2}}+ \frac{5x^{3}\log^{3}(x... |
I've got a 2D cubic Bézier curve which, by construction, is a f(x) function : no loops, a single solution for each defined x.
Is there a common mean to convert this curve to a 3rd degree polynomial ? 3 should be enough since there can only be two "bumps".
Thanks!
|
The following is a geometry puzzle from a math school book. Even though it has been a long time since I finished school, I remember this puzzle quite well, and I don't have a nice solution to it.
So here is the puzzle:
The triangle $ABC$ is known to be isosceles, that is, $AC=BC$. The labelled angles are known to be $... |
A smooth surface $S$ embedded in $\mathbb{R}^3$ whose metric is inherited from $\mathbb{R}^3$
(i.e., distance measured by shortest paths on $S$) is a Riemannian $2$-manifold: differentiable
because smooth and with the metric just described. Two questions:
Are such surfaces a subset of all Riemannian $2$-manifolds? A... |
How do you prove that for all positive $a,b,c$ this formula holds true:
\begin{equation*}
\frac{a}{\sqrt{a+b}} + \frac{b}{\sqrt{b+c}} + \frac{c}{\sqrt{c+a}} > \sqrt{a+b+c}?
\end{equation*}
Any help will be invaluable.
|
Are there any odd positive numbers that satisfy the equation:
$a^2 - b^3 = 4$
?
I am certain that there are none but can't prove it. How would you prove that?
|
How does the factoring of polynomials over Galois fields work? I cannot seem to understand the basic concept.
For example: How do I factorize $x^6 - 1$ over $\operatorname{GF}(3)$? I know that the result is $(x+1)^3 (x+2)^3$, but I'm unable to compute it myself.
I've studied the articles on Wikipedia:
http://en.wikipe... |
Let $T_\varphi$ be an analytic Toeplitz operator (meaning $\varphi \in H^\infty$). Further let $K$ be a compact operator that commutes with $T_\varphi$. Now I want to show that the spectrum of $K$ only consists of $0$.
I'm trying the following things: $T_\varphi K$ is compact so we can apply the spectral theorem for co... |
$$u + v = 2$$
$$ux + vy = 1$$
$$ux^2 + vy^2 = - 1$$
$$ux^3 + vy^3 = -5$$
Can anyone give me a hint for solving this ? I'm kinda stuck.
|
Suppose you want to accumulate $12\,000$ in a $5 \%$ account by making a level deposit at the beginning of each of the next $9$ years. Find the required level payment. So this seems to be an annuity due problem. I know the following:
$ \displaystyle \sum_{k=1}^{n} \frac{A}{(1+i)^{k}} = \frac{A}{1+i} \left[\frac{1- \lef... |
I'm reading A Computational Introduction to Number Theory and Algebra, which can be found here as a free download. From the book's exercises, I'm stuck with a proof to show that $\gcd(a,b)=|a| \iff a | b$.
My current reasoning goes as follows: since $\gcd(a,b) = |a|$, we see that $a\mathbb{Z} + b\mathbb{Z}= |a|\mathbb{... |
I have not been able to solve this problem. Any insights would be appreciated!
Let $x, n > 1$ be integers. Suppose that for each $k > 1$ there exists an integer $a_{k}$
such that $x − a_k^n$ is divisible by $k$. Prove that $x = A^{n}$ for some integer $A$.
|
Suppose we rotate the graph of $y = f(x)$ about the $x$-axis from $a$ to $b$. Then (using the disk method) the volume is $$\int_a^b \pi f(x)^2 dx$$ since we approximate a little piece as a cylinder. However, if we want to find the surface area, then we approximate it as part of a cone and the formula is $$\int_a^b 2\... |
I'm having trouble wrapping my head around the following issue. My book solves a problem without using complex exponential solution like $C_1 e^{it}$ and using either $A \cos(t) + B \sin(t)$ or $A \cos(t+\phi)$ to a second-order "undamped" linear ODE. I understand that the solution can be purely real or complex. The t... |
Given two Riemannian Manifolds of dimension 2, and a point on each. If the scalar curvatures are isomorphic (as functions) in some neighbourhoods of these points, are then the manifolds necessarily locally isometric?
|
During the summer, I did an REU where we focused primarily on one-dimensional dynamics and more specifically kneading theory. One thing that I was always confused about is why the Schwarzian derivatives always seem to pop up in discussions of iterated dynamics on the real line. I understand what a Schwarzian derivative... |
Krull's Hauptidealsatz (principal ideal theorem) says that for a Noetherian ring $R$ and any $r\in R$ which is not a unit or zero-divisor, all primes minimal over $(r)$ are of height 1. How badly can this fail if $R$ is a non-Noetherian ring? For example, if $R$ is non-Noetherian, is it possible for there to be a minim... |
According to an article I'm studying ("Time series, self-similarity
and network traffic by Mark Crovella) the expectation of the square of
a time-scaled Brownian motion process $E[ B(ct)^2 ]$ where $c$ is the time
scaling is equal to $ct$.
I'd appreciate help proving this; i.e.
$E[ B(ct)^2 ] = c t$
|
I'm teaching a group theory course now, and I wanted to give my students a proof that every subgroup of a cyclic group is cyclic. The easiest way I could think to do this is to say that any cyclic group corresponds to a $\mathbb{Z}/n\mathbb{Z}$ and then (implicitly) use the first isomorphism theorem (which they haven't... |
Prove the following: If $H$ is a subgroup of finite index in a group $G$, and $K$ is a subgroup of $G$ containing $H$, then $K$ is of finite index in $G$ and $[G:H] = [G:K][K:H]$.
So this is basically a bijective proof? The number of cosets of $H$ in $G$ equals the number of cosets of $K$ in $G$ times the number of cos... |
Suppose $f(x) = \begin{cases} 0 \ \ \text{if} \ x \in \mathbb{R}- \mathbb{Q} \newline \frac{1}{q} \ \ \text{if} \ x \in \mathbb{Q} \ \text{and} \ x = \frac{p}{q} \ \text{in lowest terms} \end{cases}$
(i) Is $f$ continuous on the irrationals?
(ii) Is $f$ continuous on the rationals?
For (i) you could use the sequence de... |
I'm in the process of writing my Master's Thesis on automata theory. The writing must be in English which isn't my mother tongue.
So the question is, given that this is my first time long (hundred pages) writing and since English is a second language for me, what is the book you recommend for academic writing style?... |
very sorry for the noobish post, I'm studying math on my own and lot of times im very confused about little things.
Here is my question, the original problem starts out like this: Solve the inequality $\vert x -2 \vert \geq \vert 2x -3\vert $
Here is the solved problem from my book
$\vert x -2 \vert \geq \vert 2x -... |
How does one evaluate the integral $$\int\limits_{0}^{\infty} \frac{x^{a-1} - x^{b-1}}{1-x} \ dx \quad \text{for} \ a,b \in (0,1)$$
|
Young's inequality for convolutions states that if $1 \leq p, q, r \leq \infty$ satisfy
$$\frac{1}{q} + 1 = \frac{1}{p} + \frac{1}{r}$$
for all $f \in L^p(G)$ and all $g \in L^r(G)$ where $g$ and $g'$ have the same $L^r$-norm and $g'(x) = g(x^{-1})$ ($G$ is a topological group) we have that:
$$\|f * g\|_q \leq \|g\|_r... |
Let $M$ be the mean matrix of a multitype branching process.
I am trying to figure out whether $A = (I-M)^{-1}$ has any significance.
I think $A_{ij}$ would stand for the number of times that type $j$ would be generated from type $j$, but I can't find any material on that in the web. Anyone knows anything about that?
T... |
We have two functions:
$F(x)$ and $G(x)$
Suppose the improper integral has the value 1.
$$\int_{1}^{\infty} F(x) G(x) \mathrm{d}x = 1$$
Can we find the value of the integral, where c is a constant?
$$\int_{1}^{\infty} F(x+c) G(x) \mathrm{d}x$$
I am interested in generalized answers and not specific examples.
Or, if any... |
Prove that the sequence $c_{1} = 1$, $c_{(n+1)}= 4/(1 + 5c_{n})$ , $n \geq 1$ is convergent and find its limit.
Ok so up to now I've worked out a couple of things.
$c_1 = 1$
$c_2 = 2/3$
$c_3 = 12/13$
$c_4 = 52/73$
So the odd $c_n$ are decreasing and the even $c_n$ are increasing. Intuitively, it's clear the the... |
Why does $ \text{Var}(Y) = E(\text{Var}(Y|X))+ \text{Var}(E(Y|X))$? What is the intuitive explanation for this? In laymen's terms it seems to say that the variance of $Y$ equals the expected value of the conditional variance plus the variance of the conditional expectation.
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I'm a math PhD who has been asked to interview a high school student and determine what he/she is interested in and how strong the student is.
Usually I would want them to talk as much as possible about what classes they've taken and what parts of the classes they like the best. Generally I'd like to allow them to get ... |
Need to use Fermat's Little Theorem (Let $p$ be a prime number and let $a$ be an integer. Then $a^p = a \mod p$. If $p$ does not divide $a$ then $a^p-1 \equiv 1 \mod p$.)
$154$ is not prime, but $154 = 22\cdot 7$ and $23$ is prime, so $a^{22} \equiv 1 \mod 23$.
$a^{154} \equiv a^{22} \cdot a^7 \equiv 1 \cdot 6 \equiv ... |
So my brain is frazzled which is probably why this seems like a big deal to me right now, but I just can't get over this reasoning:
Suppose you have $$F = \{\text{all } 1\text{-}1, \text{increasing functions } \Bbb N \to \Bbb N\}$$
$1$-$1$: means that every value of the domain maps to some unique value of the range an... |
Prove: If $X$ is not divisible by 3 and $Y$ is not divisible by 3, then $X^2 - Y^2$ is divisible by 3.
I cannot find a good way to prove this.. Tried contrapositive and contradiction but cannot see the answer. Anyone have any ideas?
|
The Wikipedia article on long division explains the different notations. I still use the European notation I learned in elementary school in Colombia. I had difficulty adapting to the US/UK notation when I moved to the US. However, I did enjoy seeing my classmates' puzzled faces in college whenever we had a professor t... |
My question is essentially this: Why does the principle of transfinite induction not suffice to show the axiom of choice when the sets to be chosen from are indexed by a well ordered set?
I have read that one can prove the axiom of finite choice from simple induction. You induct on the size of the system of sets you ar... |
Can someone give me an explanation targeted to a high school student as to why finding thegcd of two numbers is faster using the euclidean algorithm compared to using factorization, there should be no algorithm efficiency involved, just a general explanation, something my brother in grade 9 can understand.
|
I am trying to calculate the average distance between any two nodes in an $r$-dimensional mesh.
This is a $3$-dimensional mesh with $n=3$
To choose any two points in the mesh there are $\left(n^r \left(n^r - 1\right)\right)/2$ ways of doing this.
So if we choose 2 points in the mesh $p\lbrace 1, 2, ..., r \rbrace$ and... |
Let $G$ be the group defined by generators a,b and relations $a^4=e$, $a^2b^{-2}=e$, $abab^{-1}=e$. Since the quaternion group of order $8$ is generated by elements $a,b$ satisfying the previous relations, there is an epimorphism from $G$ onto $Q_8$. Let $F$ be the free group on $\\{a,b\\}$ and $N$ the normal subgroup ... |
I was given a proof of this in class, but it does not make sense to me, can someone provide a better proof, or explain the one I have provided, thanks.
Suppose $\gcd(a,n) = 1$, where $a$ and $n$ are natural numbers. Prove that $0$, $a$, $2a$, $3a,\ldots, (n-1)a$, is complete residue system modulo $n$.
Proof given in cl... |
What is the highest power of 2 dividing 100!
This is what I have so far:
50 multiples of 2
25 multiples of 4
12 multiples of 8
6 multiples of 16
3 multiples of 32
1 multiple of 64
EDIT: giving a highest power of 2^97
am I missing anything here?
|
Kodaira defines a complex analytic family of compact complex manifolds as the data $(E,B,\pi)$, where $E$ and $B$ are complex manifolds, and $\pi$ is a surjective holomorphic submersion such that the preimage $\pi^{-1}(x)$ of any point $x \in B$ is a compact complex submanifold of $E$. (Here complex manifolds are requi... |
Calculate $1^{30} + 2^{30} + 3^{30} + \ldots + 17^{30} \mod 31$
Using Fermat's Theorem:
$$
1^{30} = 1 \mod 31,
$$
(since $31$ is prime).
This implies the above is congruent to $17 \mod 31$
This is correct, right?
|
As part of my solution to a problem, I come to a point where I need to find the solutions to
$-2\partial_{T}B\left(T\right)+\frac{3}{4}B\left(T\right)\left(A\left(T\right)^{2}+B\left(T\right)^{2}\right)=0$
$2\partial_{T}A\left(T\right)+\frac{3}{4}A\left(T\right)\left(B\left(T\right)^{2}+A\left(T\right)^{2}\right)=0$
wh... |
What are the conditions for torsion to be zero other than having a plane curve?
The only thing I can thing of is an equation that have the torsion that cancels out each other.
|
$$S = \sum_{k=1}^{\infty} \frac{\cos(\theta\log(k))}{k^a}$$
How do I go about finding the value of S, given that $\theta \to \infty$ and $0 < a < 1$.
Any special techniques that might be
helpful in calculating this sum?
EDIT:
Just to give some background,
I was actually trying to figure out
$$\sum_{k=1}^{\infty} \... |
I came across this question.
Let $d(n)$ denote the number of divisors of $n$. Let $$\nu(z) = \sum\limits_{n=1}^{\infty} d(n) z^{n}$$ Whats the radius of convergence of this power series. We also have to show that $$\nu(z) = \sum\limits_{n=1}^{\infty} \frac{z^{n}}{1-z^{n}}$$
Regarding the divisor function, we have Diric... |
Am I correct to say:
A difference between an integral and a function is: that an integral can describe an area on a graph, while a function can't?
Or am I completely off course here?
|
Is there a more efficient algorithm besides Gram-Schmidt that would produce an orthonormal matrix of rank N, with first row equal to [1 1 1 1 1 ... 1] / sqrt(N)?
e.g. for N = 3, the matrix $\begin{align}
\mathsf A_3 &= \begin{bmatrix}
1/\sqrt 3 & 1/\sqrt 3 & 1/\sqrt 3 \\
2/\sqrt 6 & -1/\sqrt 6 & -1/\sqrt 6 \\
0 &... |
I don't understand what is polynomial fitting.
Can anyone explain to me how to fit a curve to given points?
|
Here's the Karnaugh map:
The answer I should be getting from the Karnaugh should be:
T = R ∙ (CGM)'
I'm really not seeing how this was arrived at through any simplification methods I've learned thus far. I can see the answers that are intended are correct (I think), though.
From what I know, the best answer I can co... |
I'm looking for an explanation on how reducing the Hamiltonian cycle problem to the Hamiltonian path's one (to proof that also the latter is NP-complete). I couldn't find any on the web, can someone help me here? (linking a source is also good).
Thank you.
|
An Exercise from Apostol's Introduction to Analytic Number Theory which I am not able to solve.
Let $\mathsf{S_{n}}$ denote the $n$-th partial sum of the series: $$\sum\limits_{r=1}^{\infty} \frac{1}{r(r+1)}$$
Prove that for every integer $k>1$, there exists integers $m$ and $n$, such that $$s_{m}-s_{n} =\frac{1}{k}$$
|
I have a problem of the following form:
minimize $\|Dx\|_2$
subject to $\|x*x\|_2 = 1$
where $x\in\mathbb R^n$, $D$ is a given diagonal matrix of positive entries, and $*$ represents convolution, i.e., $(x*x)\_n = \sum \limits_{i+j=n}x_ix_j$ and $x*x\in\mathbb R^{2n-1}$.
What approach could be used in dealing with thi... |
The scatter graph at the Wikipedia article seems to suggest so. Has anyone attacked this before? Is there a known proof?
|
Let $G=\{x,y|x^2=y^3=(xy)^3=1\}$
I would like to show that $G$ is isomorphic to $A_4.$
Let $f:\mathbf{F}_{2} \to G$ be a surjective homomorphism from the free group on two elements to $G$. Let $f$ map $x \to (12)(34)$ and $y \mapsto (123)$. I'm not sure how to show that these elements generate the kernel of $f$. If the... |
What is the least number of comparisons we need to find the median of 6 distinct numbers?
I am able to find the answer to the median of 5 distinct numbers to be 6 comparisons, and it makes sense, however in the case of 6 numbers I can't find an answer.
The best I was able to do it in by hand was 9 comparisons. Can tha... |
I'm familiar with Householder reflections; they are a simple transformation that, given a normal vector, describes reflection in the hyperplane perpendicular to that vector.
But do Householder reflections describe the complete space of matrices describing reflection about the origin, or are there other reflections that... |
I'm trying to come up with some math problems (word or otherwise) that get to the meaning of adding zero, but I'm getting stuck because it seems just too simple to me.
I have come up with questions like "John is having a birthday party and he invites five friends over but nobody shows up. How many people are at the par... |
Is it possible to have 2 different points (non-degenerate) in $n$-dimensions for any value of $n$ where $n>1$, share the same cost and both be visited by the simplex method?
|
How can I compute the pmf of a random sum ($1$ to $N$) of iid bernoulli random variables, where $N$ is distributed geometrically?
|
I was asked to find a solution to
$$\frac{\sin^2(nx)}{n^2\sin^2(x)}=2^{-1/2}$$
where $n$ is a fixed integer greater than 1.
Numerically, there's a solution just above 1/n so I decided to find this one (rather than one of the other roots). I hoped to find an obvious pattern that I could use to find some sort of closed-... |
A function $f: \mathbb{R} \to \mathbb{R}$ is said to have the intermediate-value property if for any $a$, $b$ and $\lambda \in [f(a),f(b)]$ there is $x \in [a,b]$ such that $f(x)=\lambda$.
A function $f$ is injective if $f(x)=f(y) \Rightarrow x=y$.
Now it is the case that every injective function with the intermediate-... |
This is a follow-up to a question posted recently.
Let
$$s_n = \sum_{r=1}^{n} \frac{1}{r(r+1)},$$
where we take $s_0 = 0$.
The problem I am interested in is this: For fixed $k \geq 2$, find all solutions $(m,n)$ in nonnegative integers to the Diophantine equation $$s_m - s_n = \frac{1}{k}.$$
Current state of knowl... |
I had already posted this on mathoverflow and was advised to post the same here. So here it goes:
$X=\{(x,y,z)|x^2+y^2+z^2\le 1$ and $z≥0\}$ i.e. $X$ is the top half of a $3$-Disk.
$Z=X/E$, where $E$ is the equivalence relation on the the plane $z = 0$ which is as follows:
$(x,y,0)∼(−x,−y,0)$.
I was told that this spac... |
Although this is not actually homework, it's an exercise from the book;
I want to rewrite the variance,
\begin{equation*}
var[f]=E[(f(x)-E[f(x)])^2]
\end{equation*}
as
\begin{equation*}
E[f(x)^2]-E[f(x)]^2.
\end{equation*}
I'm not sure how it reduces, and it has had me stumped for a while. The answer would be nice... |
There is an intricate connection between Hurwitz Zeta and the (traditional) polygamma function:
$$\psi_n(z)=(-1)^{n+1}n!\zeta(n+1,z)$$
If to use a generalization for Bernoulli numbers, this can be considered a formula, connecting polygamma and Bernoulli polynomials of negative order:
$$\psi_n(z)=(-1)^{n+1}n!\frac{B_{-... |
If s and t are two nodes in a graph.
How to prove that the maximum number of edge-disjoint cut sets which divide s and t equals the length of the shortest path from s to t?
|
A very simple solution to the diffusion equation is $u(x,t)=x^2+2 t$
My question:
How can this be a solution to the diffusion equation when nothing really diffuses, but just stays the same - see plot: Here
I can see that it is a solution but can't believe my eyes... Is there some kind of intuition other than: "it satis... |
I'm currently reading Hilton & Stammbach's A First Course in Homological Algebra, and the following point has stumped me:
In section 1.8, they construct co-free modules ("left moodule" over some ring) as essentially coming from the right adjoint to the forgetful functor from $\Lambda$-Modules to Abelian Groups. On the ... |
Quaternions came up while I was interning not too long ago and it seemed like no one really know how they worked. While eventually certain people were tracked down and were able to help with the issue, it piqued my interest in quaternions.
After reading many articles and a couple books on them, I began to know the form... |
Can any of you guys think of a topological space $(X,\tau)$ and a family of subsets {As}${s \in S}$ of $X$ such that for a certain $x \in X$ you can find a subset $V$ such that $x \in V$ and {$s \in S: V \cap A_{s} \neq \emptyset$} is finite, whereas for every $W \subseteq X$, with $x \in W$, we have that {$s \in S: W ... |
In my attempts to derive the closed form for the arclength of the hyperbola, I wound up with the following integral:
$$\int\frac{\sqrt{1-m\;\sin^2 u}}{\sin^2 u}\mathrm{d}u$$
I am aware that such integrals are expressible as combinations of elliptic integrals and trigonometric functions, and Mathematica does return a re... |
Possible Duplicate:
Boolean Algebra, Simplification: Don't know the method used
Based on a previous question, the answer I got for the "sum of products" method was:
RC' + RCG' + RCGM'
Just for completeness, can someone help walk me through to how this was arrived at? Thanks again in advance!
|
Can someone help direct me towards information on calculating the polynomial remainder?
I know how to do the basics, but what if I don't have access to the polynomial in expanded form?
For example, $\frac{1}{1-x}$ represents the polynomial $\sum_{i=0}^\infty{x^i}$. I want to be able to find the remainder using an expr... |
How does one prove the following limit?
$$
\lim_{n \to \infty}
\sqrt{1 + 2 \sqrt{1 + 3 \sqrt{1 + \cdots \sqrt{1 + (n - 1) \sqrt{1 + n}}}}}
= 3.
$$
|
Given two values $a$ and $b$, how should one go about solving the following inequality for $r$:
$$a \le b^r -r .$$
Applying $\log_b$ on both sides of the inequality doesn't help me much since that yields the following:
$$\log_b a \le \log_b (b^r-r) .$$
I know that
$$\log_z x - \log_z y = \frac{\log_z x}{\log_z y} .$$... |
A subset $S$ of a metric space $X$ is totally bounded if for any $r>0$, $S$ can be covered by a finite number of $X$-balls of radius $r$.
A metric space $X$ is totally bounded if it is a totally bounded subset of itself.
For example, bounded subsets of $\mathbb{R}^n$ are totally bounded.
Are there any interesting neces... |
Use the Lagrange remainder to determine how many terms of the Taylor series of cos expanded
at 0 are needed to estimate $cos(\frac{21\pi}{5})$ . Accuracy to 3 decimal places
So I have a lot of rough work on paper. I know math.stackexchange dosn't really answer assignment questions but I am looking for a head start on e... |
Everything that follows takes place in the Borel $\sigma$-algebra with Lebesgue measure.
The Lebesgue mean of $f$ at $x$ is defined as $\displaystyle \lim_{\epsilon\to 0} \int_{x-\epsilon}^{x+\epsilon} \frac{f}{2\epsilon}$. The function defined by $[f](x) =$ "Lebesgue mean of $f$ at $x$" is equal to $f$ almost everywhe... |
What is the name of the following matrix?
$$\begin{pmatrix} a & b & 0 \\
c & d & 0 \\
0 & a & b\\
0& c& d& \\
b & 0 & a \\
d & 0 & c\end{pmatrix}$$
It looks like a Block Toeplitz matrix, but usually one defines those by full shifts by (in this case) $2 \times 2$ matrices. In particular, I'm interested in solving li... |
Let $\Sigma$ be a $n-1$ dimensional space-like submanifold of a $n+1$ dimensional space-time $(V,g)$ and let $x \in \Sigma$. Then $(T_x \Sigma)^\perp$ is of dimension $2$ and is time-like. Such a $2$-plane as $(T_x \Sigma)^\perp$ cuts the future lightcone at $x$ along two linearly independent future null directions. Le... |
If a space has curvature, then the curvature can be seen intrinsically by finding sums of angles in triangles made of geodesics. Under general relativity, space-time is curved on local scales. On global scales, experiments have determined that space-time is almost totally flat.
If one were to, for each of three points ... |
More specifically, assume that $d$ is taken from $[1, 2^n]$ and $m$ is taken from $[1, n]$. What is an upper bound on the probability that $d$ is a multiple of $m$?
|
The Complex plane, set of all $z=x+iy$ where $x$ and $y$ are real, surface area equals cross product of $x$ and $y$ equals aleph-something (that's not the question). Projecting the plane onto a sphere, and adding the complex infinity to the set (without a value for $\arg(z)$), gives the Riemann sphere, which, being a s... |
True or False?
If $\displaystyle \sum_{n=1}^{\infty}a_{n}^2$ is convergent, and$\displaystyle \sum_{n=1}^{\infty} (\frac{\pi}{2} - b_{n})$ is divergent, then $\displaystyle {a_{n}(\sin(b_{n})-1)}$ is a convergent sequence.
So we know that $(a_{n})^2$ is convergent.
Therefore, $\lim_{n \to \infty} a_{n}^2 = 0$
I'm not ... |
Based on a previous question (and this ought to be the last of this sequence I promise), I just need to know one more thing. According to this truth table:
I'm told the answer I should be getting by using the sum of products method on the truth table should be:
T = R ∙ (RCGM)'
Unlike before with the Karnaugh map, I ... |
I have a text book which gives two diagrams of beads suspended from strings with different angles, and worked examples to calculate the tension in the strings.
The first bead is suspended by $2$ strings attached to a rod, both strings at angles to the vertical (the bead is at the bottom, the rod is across the top). The... |
Suppose we look at $k$-digit strings with digits between $1$ and $n$. How many distinct classes of strings are there when digit identity and order doesn't matter?
More formally, what is the number of equivalence classes over strings $d_1,\ldots,d_k$ with $d_i\in \{1,\ldots, n\}$ where strings $d,\hat{d}$ are considered... |
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