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I know how to find a point on a circle given a radius and an angle, but my knowledge of trigonometry doesn't extend much further than that. My question is probably best explained diagrammatically:
Given a centre point, radius, and two points on the circle, how do I find the 'middle' point between those two points? I'm... |
Excuse my lack of knowledge and expertise in math,but to me it would came naturally that the cubic root of $-8$ would be $-2$ since $(-2)^3 = -8$.
But when I checked Wolfram Alpha for $\sqrt[3]{-8}$, real it tells me it doesn't exist.
I came to trust Wolfram Alpha so I thought I'd ask you guys, to explain the sense of ... |
Is it possible to reduce this expression further?
$$ \frac{(2x+y)(x-y) }{ (2x-y)(x+y) } .$$
|
I need help solving (b) in this math problem.
What is the relationship between the matrix f and the transformation matrix in the standard basis?
Thanks for any suggestions!
|
I want to estimate the quantile of some data. The data are so huge that they can not be accommodated in the memory. And data are not static, new data keep coming. Does anyone know any algorithm to monitor the quantile(s) of the data observed so far with very limited memory and computation? I find P2 algorithm useful. B... |
Why is my overall delta percentage of 0.9% (bottom right) less than all 5 of the values that make it up (1.7%, 3.2%, ...)? I know with the difference in magnitude of each row and column that it could skew the percentages, and I know that I'm subtracting percentages, but I'd still expect at least 1 of the values in the... |
This should be something relatively simple. I know there's a trick to this, I just can't remember it.
I have an equation
$$\frac{3x}{x-3}\geq 4.$$
I remember being shown at some point in my life that you could could multiply the entire equation by $(x-3)^2$
in order to get rid of the divisor. However,
$$3x(x-3)\geq 4... |
Let $L=\mathbb{C}(t)$ (where $t$ is an indeterminate). Let $\phi\colon L\to L$ be the $\mathbb{C}$-automorphism of $L$ given by
$$\phi(t) = \frac{3t-2}{4t-3},$$
and let $G$ be the group generated by $\phi$. Find the fixed field of $G$.
Could really use some help with this. Thanks!
|
Why is the zero set in $\mathbb{C}\times\mathbb{C}$ of a polynomial $f(x,y)$ in two complex variables always non-discrete (no zero of $f$ is isolated)?
|
I have equations for two lines, one of which is linear and the other is logarithmic, ie:
$$y = m_1 x + c_1$$
$$y = m_2 \cdot \ln(x) + c_2$$
..and I need to find out where (if at all) these lines intersect. I realise that I need to solve:
$$ m_1 \cdot x + c_1 = m_2 \cdot \ln(x) + c_2$$
..for $x$, but apart from shuffl... |
I am not that familiar with arguments using ordinals so this may be pretty simple. Let $T$ be a topology on a set $X$. For each ordinal $\alpha$ let $T_{\alpha}$ be a topology on $X$ satisfying $T\subseteq T_{\alpha+1}\subsetneq T_{\alpha}$ (notice the proper inclusion) and $T_{\alpha}=\bigcap_{\beta<\alpha}T_{\beta}$ ... |
Consider two groups $(\mathbb{Z},+)$ and $(\{1,-1,i,-i\},\cdot)$ where $i^2=-1$, show that the mapping defined by $f(n)=i^n$ for $n$ belonging to $\mathbb{Z}$ is a homomorphism from $(\mathbb{Z},+)$ onto $(\{1,-1,i,-i\},\cdot)$, and determine its kernel.
|
Let $K$ be any field and $s$ an indeterminate. Then $K(s)$ is a field extension of $K (s^n )$. Prove that $[K (s):K (s^n )]=n$. Hence show that the minimum polynomial of $s$ over $K(s^n)$ is $t^n − s^n$.
[Hint: first show that $s$ satisfies a poly of degree $n$ over $K(s^n)$; this gives $\leq$. Then show that ... |
A pair of dice is rolled repeatedly until each outcome (2 through 12) has occurred at least once. What is the expected number of rolls necessary for this to occur?
Notes: This is not very deep conceptually, but because of the unequal probabilities for the outcomes, it seems that the calculations involved are terribly m... |
Where could I find a formula that produces integers $n$ and $m$ such that $\binom{n}{2}/\binom{m}{2}=1/2$?
Of course, this questions can be reformulated as: How to find all the integer values of n and m such that $$2n(n-1) = m(m-1)\quad ?$$
|
How do I prove
$(2^r-1)(1-x)x^{2^{r}-2}+x^{2^{r}-1}>x^{2^{r}-r-1}$
for $\frac{1}{2}<x<1$ and $r \in \mathbb N$ $(r\neq 1)$?
|
$y = \sin(\pi x), 0 \le x \le 1$
$y' = \pi \cos(\pi x)$
$(y')^2 = \pi^2\cos^2(\pi x)$
$ds = \sqrt{1 + \pi^2 \cos^2(\pi x)}$
$r = y = \sin(\pi x)$
$S = \displaystyle\int_0^1 2 \pi\sin(\pi x)\sqrt{1 + \pi^2 \cos^2(\pi x)} dx$
I'm a little lost as to what to do next. Should I have simplified ds more or do I need to do a s... |
We know that $$\int_0^\infty \left(\frac{\sin x}{x}\right)^2 dx=\int_0^\infty \frac{\sin x}{x} \, dx=\frac{\pi}{2}.$$
How do I show that $$\int_0^\infty \left\vert\frac{\sin x}{x} \right\vert \, dx$$ converges?
|
I have read (I think) all of the previous threads on this website (and many others)
on this topic & unfortunately have not found an answer to my question. Due to the fact that I am only beginning to study the subjects of tensors, differential forms etc...
I appeal to your good nature not to respond with anything too ... |
With a perfect use of the condom you have a 2% chance to get pregnant each year. These figures seem surprisingly high to me. So I was wondering what the chance is to have a baby after a lifelong use (let's assume 25-35 years).
What if you use the pill? And with both the pill & condom? What with the "typical use" scenar... |
Consider the equation
$$\mathbf{v} = \mathbf{v}_0 + A\mathbf{x}$$
where $A$ is an $m\times n$ matrix ($m\gt n$) with entries equal to $-1$, $0$, or $1$ only. Additionally, $1$ and $-1$ only appear once in each row $A$ so the sum of a row is always zero.
$$\mathbf{x}_0 = \mathbf{0}$$
The problem is to find an $\mathbf{... |
I originally posted this over at stackoverflow and they suggested asking it over here.
link to original: https://stackoverflow.com/questions/5222781/calculating-perpendicular-and-angular-distance-between-line-segments-in-3d
Text of original:
I am working on implementing a clustering algorithm in C++. Specifically, this... |
I am having trouble proving this theorem:
For a field $k$, let $f \in k[x]$ be
irreducible, and $g \in k[x]$ be a
polynomial. Let $h\in k[x]$ be an
irreducible factor of $f\circ g$. Show
that $\deg f \mid \deg h$.
My attempt so far was to prove this:
Let $\mathrm{Sq}(n,k)$ be the set of
square $n\times n$ ... |
Lets say we are given a primal linear programming problem:
$\begin{array}{ccc}
\text{minimize } & c^{T}x & &\\
\text{subject to: } & Ax & \ge & b \\
& x & \ge & 0
\end{array}$
The dual problem is defined as:
$\begin{array}{ccc}
\text{maximize } & b^{T}y & &\\
\text{subject to: } & A^{T}y & \le & c \\
& y & \ge... |
Given $\alpha(s)$ a smooth arc-length parametrized curve, how can you write the equation for the surface $f(s,t) = \alpha(s) + t \alpha'(s)$ component-wise?
That is, I want to write $f = (f_x, f_y, f_z)$.
I can write $\alpha(s) = (x(s), y(s), z(s))$, but that doesn't give much. For instance, I don't think $f_x(s,t) = ... |
Let $F$ be an arbitrary field of characteristic $0$, $K$ its algebraic closure. Define $M=\{ (x,y)\in M_n(F)×M_n(F) \mid [x,y]=0\}$ and let $N$ be the Zariski closure of $M$ in $K^{2n^2}$.
How can one show that $N$ contains the set $\{(axa^{-1},aya^{-1}) \mid (x,y)\in N, a\in \mathrm{GL}(n,K)\}$?
Thank you.
|
PROBLEM: Let $R$ be a ring with $1$ and $M$ be a unital $R$-module (i.e. $1x=x$). Let there for each submodule $M_1\neq M$ exist a submodule $M_2\neq M$, such that $M_1\cap M_2=\{0\}$. How can I prove, that $M$ is semisimple?
DEFINITIONS: A module $M$ is semisimple iff $\exists$ simple submodules $M_i\leq M$, such that... |
I know I am making a mistake somewhere but consider the following truth table:
p0 p1 p2 |
-------------------
0 0 0 | 1
0 0 1 | 0
0 1 0 | 0
0 1 1 | 1
1 0 0 | 1
1 0 1 | 1
1 1 0 | 1
1 1 1 | 1
This coresponds to the boolean formula in DNF:... |
I'm being asked to find the arc length of $y=\sin( x)$ for $[0, \frac{\pi}{2}]$ using $M_8$.
I've determined that $y\prime^2=\cos^2x$. So, using the formula for arc length, I get $\sqrt{1+\cos^2x}$ as my function.
Now, they want me to evaluate this using $M_8$, so I end up with $8$ midpoints from $\frac{\pi}{32}$ to $... |
Let $\{x_k\}$ be an infinite series. It is known that $\sum_{k=-\infty}^\infty \vert x_k\vert^2<\infty$ and $\sum_{k=-\infty}^\infty \vert x_k\vert$ does not converge. How do I prove that the Fourier transform of $\{x_k\}$, $\mathcal{F}(\omega)$ exists (or what other conditions need to be satisfied for it to exist)? I ... |
Prove $(2n+1) + (2n+3) + (2n+5) +
\cdots + (4n-1) = 3n^{2}$ for all
positive integers $n$.
So the provided solution avoids induction and makes use of the fact that $1 + 3 + 5 + \cdots + (2n-1) = n^{2}$ however I cannot understand the first step: $(2n+1) + (2n+3) + (2n+5) + \cdots + (4n-1) = (1 + 3 + 5 + \cdots + (... |
I saw some place where someone wrote
$$\frac{1}{1-f(x)} = \sum_{n=0}^{\infty} f(x)^n, \forall x \in \mathbb{C}, $$
where $f$ is some function s.t. $\vert f(x)\vert < 1, \forall x \in \mathbb{C}$.
Then he proved that the RHS $\sum_{n=0}^{\infty} f(x)^n$ absolutely converges $, \forall x \in \mathbb{C}$, in order that th... |
Let's say I have a resource, call it E.
There are four important values
Em - the maximum amount of E we can have
Ec - the current amount of E
Er - the rate at which E replenishes (this is in % of Em)
Ea - the cost amount in E of an action
Let us assume that we start at max capacity.
at t=0 : Ec = Em
at t=i : Ec = (Ec... |
Why is the area between these two functions positive?
\begin{align*}
f(x) &= (x-1)^3\\
g(x) &= (x-1).
\end{align*}
The area between $f(x)$ and $g(x)$ is in two parts.
The first part is in the 4th quadrant with lower limit $0$ and upper limit $1$, $f(x)$ is the upper function.
The second part is in the 1st quadrant wi... |
This is something that confuses me.
I have read a few mathematical texts and they often seem to use Lemma/Proposition/Theorem if they have a particular statement.
Now, which one to use? A lemma can be something you need to prove a more important theorem, but then what about Fatou's Lemma?
When to pick Proposition or Th... |
Is there a general result about the existence of $($non-trivial$)$ solutions of the diophantine equation:
$$Ax^2 + By^2 = Cz^2$$
for $A,B,C$ known positive integers, pair-wise relatively prime?
What if we know $C$ is $1$ or $3?$
|
Question: if f is analytical and $ |f(z)| < M $ for $ |z| =< R $ find an upper bound for $ |f('n)(z)|$ in $ |z| =< \rho < R $ (where $f('n)$ means the nth derivative of $f$).
So I cited Cauchy's Inequality, to find the bound $\frac{n!M_R} {R^n}$ , but since we have the additional limit of $\rho$ I think we should be a... |
To motivate this question, let me begin with a picture:
Each letter labels a "blob" of this quartic Julia set. (is there a technical term for these parts?). Because of resolution limitations I haven't been able to color and label every blob.
The only answer to the the MO question Symmetries of Julia sets for $z^2 - c... |
My problem is how to somehow relate the the gcd and congruence. I know that $(a,b) = ax + by$. I also know that $a \equiv b \pmod n$ means $n\mid a-b$. Any hints?
Thanks!
|
$T(1) = 1 , T(n) = 2T(n/2) + n^3$? Divide and conquer, need help, I dont know how to solve it?
|
I recently saw a conjecture on a blog ( http://blog.tanyakhovanova.com/?p=311 ) which the author refers to as the 86 conjecture. The conjecture claims that all powers of 2 greater than $2^{86}$ have a zero in their base 10 representation.
At first glance this seems like a numerical curiosity, and I wasn't sure that th... |
Can anyone point me to any reference on the following inequality:
Let $c>0, a,b \in \Bbb{R}$
$$
\|a+b\|^2 \leq (1+c)\|a\|^2+(1+{\textstyle\frac{1}{c}})\|b\|^2
$$
I'm not even sure if "Bohr's Inequality" is this inequality's name, since I can't find it anywhere in the internet.
I'm having a bad day and I got stuck in ... |
I've seen one proof of $\mathrm{psl}_2(\mathbf Z) \cong C_2 * C_3$ based on the Ping-pong lemma
But what I'm looking for is how this result was historically obtained first.
I've checked Coxeter & Moser's "Generators and relations for discrete groups" and they (§7.2) do things that I find weird, they let the matrices ac... |
I have seen a picture for $dV$ so that $dV = r^{2} \sin(\theta)\,dr\,d\theta\,d\phi$. But how can I deduce things like $dA$ and $dV$? In a simpler coordinate (not sure about the name), $dA = r \,dr\,d\phi$, again no idea how to derive it (just given to me to memorize it). So explain what is $dA$ in polar coordinates?
|
I came across the following sentence in an article:
$f$ is injective on $L^1(\mathbb{R})$
where $f$ is some function of $g$. Now I have a vague idea that injective= one-to-one, and can visualize it in $L^1$. Does it mean anything to say injective on $L^n$? Can someone illustrate it with an example in $L^2$?
|
Suppose you have an exponential generating function
$$
F(x)=F_0+F_1x+F_2\frac{x^2}{2!}+\cdots+F_n\frac{x^n}{n!}+\cdots
$$
and you want to get only the even terms
$$
F_e(x)=F_0+F_2\frac{x^2}{2!}+F_4\frac{x^4}{4!}+\cdots
$$
Is it possible to write $F_e(x)$ in terms of $F(x)$? I noticed that taking the derivative of $... |
This is a homework exercise, so please don't post full solutions to the question below.
Grothendieck (I believe) introduced several axioms an abelian category A voluntarily could satisfy. In particular, we have the two following:
AB5 A is cocomplete and filtered colimits of exact sequences are exact.
2 AB5* A is compl... |
I've been trying to understand de la Vallee Poussin's "Demonstration Simplifiee du Theorem de Dirichlet sur la Progression Arithmetique" and I've got stuck at the following step on pg 18 where Poussin takes the logarithmic derivative of:
$$\Sigma_{n}\frac{\chi(n)}{n^{s}} = \prod_{q}(1-\frac{\chi(q)}{q^{s}})^{-1}$$
in ... |
What property of a cryptographic function is described below?
B = F(A)
A = F(B)
|
Can someone help me understand the idea behind constructing a formula for the following:
For $n\in\mathbb{N}$, $n\geq 2$, find and prove a formula for:
$$\prod_{i=2}^n \left(1 - \frac{1}{i^2}\right).$$
Please, Please, I need to know HOW to do this not just how to solve this question.
Confused about:
- Why does I s... |
Is this a valid proof for $(2n+1,3n+1)=1$?
$\exists \ d \in \mathbb{Z}$
$d | 2n+1$ means $2n+1 \equiv 0$ (mod $d$)
$d | 3n+1$ means $3n+1 \equiv 0$ (mod $d$)
Multiply 1. by $3$ and 2. by $2$:
$6n+3 \equiv 0$ (mod $d$)
$6n+2 \equiv 0$ (mod $d$)
Subtract the second from the first to get:
$ 1 \equiv 0$ (mod $d$), which ... |
I'd like to solve this one similarly to my previous question: Is this a Valid proof for $(2n+1,3n+1)=1$?
I did find a somewhat related post that uses a different method: How to show that $\gcd(n! + 1, (n + 1)! + 1) \mid n$?
So how would I go about this? I can write the above like:
$\exists \ d \ \in \mathbb{Z}$
$n!+1 ... |
I have the following parametric equations:
$$x = \sin\theta,\qquad y = \cos\theta,\qquad 0 \leq \theta \leq \pi.$$
The corresponding Cartesian equation is $x^2 + y^2 = \sin^2\theta + \cos^2\theta = 1$.
So it would be $x^2 + y^2 = 1$.
How would I develop the curve for the equation?
Should I plot points? I know that ... |
find the value of R so that the line through (3, 13) and (R, -5) has a slope of 8
|
Show that $13$ divides $2^{70} + 3^{70}$.
My main problem here is trying to figure out how to separate the two elements in the sum, and then use Fermat's little theorem. So how can I separate the two?
Thanks!
|
In S. Wagon's "The Banach-Tarski Paradox," amenable groups are defined on p. 12 as follows:
[amenable] groups bear a left-invariant, finitely additive measure of total measure one that is defined on all subsets.
He defines $SO_2$ to be the group of rotations of the unit circle, which he has used to show that $S_1$, t... |
Working my way through a combinatorics text and I'm hung up on a couple of questions:
1.) Let $p=p_1 p_2\cdots p_n$ be a permutation. An inversion of $p$ is a pair of entries $(p_i,p_j)$ so that $i<j$ but $p_i>p_j$. Let us a call a permutation even (resp. odd) if it has an even (resp. odd) number of inversion. Prove... |
I would like to show that $a^n \mid b^n$ implies $a \mid b$
I thought I could convert it to congruences and work backwards, but as far as I remember, $a \equiv b \pmod{m}$ implies $a^n \equiv b^n \pmod{m}$, not the opposite, unless $m$ is prime. Is that right?
So I am not sure how to approach this one. Any ideas?
|
Let $X,Y$ denote (real) vector spaces. The vector space of $n$-linear maps $X^n \to Y$ will be denoted by $L^n(X,Y)$. Unless I'm much mistaken
$$L(X,L(X,Y)) \ \ \ L^2(X,Y) \ \ \ L(X \otimes X,Y)$$
are pairwise isomorphic. One could even view the three as pairwise naturally isomorphic functors from vector spaces to vec... |
How to add two integers in their Cantor expansion?
|
Given the following problem:
integrate $\cos^3(2x)$
I was given the solution
\begin{align*}
\int\cos^3(2x)\, \mathrm dx &= \int\cos^2(2x) \cos 2x\, \mathrm dx = \int(1-\sin^2 2x)\cos 2x\, \mathrm dx\\
&= \frac12\int (1-u^2)\, \mathrm du = \cdots
\end{align*}
but the problem is I am stuck on the 1/2. Where di... |
I need to simplify $\sin^3(x)+\cos^2(x)\sin(x)$:
First thing I noticed was the pythagorean identity.
$$\sin(x)\sin^2(x)+\cos^2(x)\sin(x) \rightarrow \sin(x)(1)\sin(x)\rightarrow\sin^2(x),$$
but this doesn't work obviously.
This is where I am stuck, I just can't see any way to simplify.
|
I've been working my way through Enderton's Elements of Set Theory for a while, and I feel I have a decent grasp on some of the basics of elementary set theory. My question is, where should I look to next in set theory? What is a good book for set theory that may be considered 'the next step up'?
If it helps any, my ba... |
Question:
Consider a square matrix of order $m$.
At each step you can move one step to
the right or one step to the top. How
many possibilities are to reach $(m,m)$
from $(0,0)$?
I think it is just counting the Central binomial coefficients.
Am I right? If not what is be the correct answer and why?
|
Let $R$ be a commutative ring with identity. I've been trying to prove the following:
If $S \subset R$ is a maximal multiplicative set, then $R \setminus S$ is a prime ideal of $R$.
I have spent some time in it, but nothing is coming to my mind. Any idea/solution will be appreciated.
|
I read that if $F$ is the field of algebraic numbers over $\mathbb{Q}$, then every polynomial in $F[x]$ splits over $F$. That's awesome! Nevertheless, I don't fully understand why it is true. Can you throw some ideas about why this is true?
|
I have differential equation $x\cdot\sqrt{4+y^2}\,dx + y\cdot\sqrt{1+x^2}\,dy = 0$, which is simple to solve on paper, but I have problem solving it with Maple (I use 10th, tried on 14th also). When applying dsolve it complains that it is not ODE.
Any suggestions?
Thank you in advance
|
Follow up from: https://stackoverflow.com/questions/5225061/exponential-decay-surrounding-bounding-box
How can I manipulate a simple logistic function like this one:
So that the lower bound is at 0, and the (-3, 3) range is roughly doubled and depends on some value (the lower edge of the bounding box as described in t... |
In a standard deck, there are 13 cards in each suit,such as hearts. For simplicity we'll number them from 1 to 13. This question is about these 13 cards.
Start with the cards in the usual order. This time you are allowed to divide the deck into three piles, consisting of the top n cards, the next m cards, and the rema... |
Very simple question but I keep getting this wrong!
If you have two points e.g. $A(13, 6$) & $B(11, 12)$,
Using the gradient formula $m = \frac{(y_2 - y_1)}{(x_2 - x_1)}$ how do you know which of $A$ or $B$ corresponds to $(x_1, y_1)$?
I can work this out by drawing a diagram of the line. If the slope is negative, the ... |
I just opened vol.1 of the Bourbaki treatise to take a look at how they define mathematical structure. I was amazed by its sheer complexity. Can you recommend an introductory text that wouldn't require as much effort to understand?
Also, a few related soft questions:
1) Is category theory more general than this theory?... |
I have recently come across this problem trying to visualise a certain economic model and I'm finding the solution is just beyond my reach. As far as I can tell there is a simplification of the problem which is easier and would still be good to have answered.
There are two moving point particles A and B on a Cartesian ... |
How many circles (radius $r$) are needed to cover circle whose radius is $2$ times bigger (radius $2r$).
I think we need to use area which is $S=\pi R^2$ but I don't really know what to do.
|
I've got a question regarding the constructible universe and I'm a bit confused about the Condensation Lemma for the universe constructible from some set $A$. Help will be greatly appreciated:
Let's assume that we have $(M,E)$ a model of ZF (or maybe of ZFC) but not necessarily standard or transitive. Since it's a mode... |
So, I had a quiz last night in my discrete structures class and this question came up:
"There are 3 processors in a computer. The instructions to a computer are labeled 1,2,3,4,5,6,7,8,9,10 each taking 1 second to execute. The relation between a pair of instructions (x,y) is given by R:'x' is executed at the same time ... |
Let $x_i (i=1,...,n, n>d)$ be a unit vector in $R^d$. $c_i>0$ is a positive real scalar. How to prove the following fact?
Fact: There exist some vectors $x_i$ such that $\sum_{i=1}^n c_i x_i=0$ if and only if $c_i\le\sum_{j\neq i}c_j, \forall i$.
The necessity is easy to prove. (My proof of the necessity: If $\sum_{i... |
Measure theory was established on naive set theory(Not totally sure). But after Russell discovered the paradox named by him, set theory was reconstructed in the sense of axiomatization.
My question is in the first chapter of many measure theory textbooks, there is a set theory introduction, most likely describing the n... |
I have this question:
Let $f$ be a Polynomial so that $\mathbb{Q}[x] \ni f=ax+b$.
With the initial information: $f(\sqrt{2})=0$,
How can someone prove that $f=0$?
Thanks.
|
How can we prove that all change-of-basis matrices are invertible? The trivial case when it's a change of basis for $\mathbb{R^{n}}$ is easily demonstratable using, for example, determinants. But I am struggling to rigorously show this for all bases, for example for a two-dimensional subspace of $\mathbb{R^{4}}$. I am ... |
I need to find integer x with which function's y gets lowest integer values
$$f(x)=\frac{x^2-x-17}{x-2}$$
I tried to find derivative, but it never equals 0.
Other steps was to change expression
$$f(x)=\frac{x^2-x-17}{x-2}=1+\frac{(x-5)(x+3)}{x-2}=x-2+\frac{3(x-7)}{x-2}$$
But didn't notice any solutions
|
Hello I'm having trouble showing the following:
Let $u$ be a positive measure. If $\int_E f\, du= \int_E g\, du$ for all measurable $E$ then $f=g$ a.e.
I was trying to argue by contradiction: if $f\neq g$ a.e. then there must exist some set $E=\{x: f(x)\neq g(x)\}$ such that $u(E) \gt 0$. Then let $E^+=\{x: f(x)\gt g(x... |
"The ordered pair $(a,b)$ is defined to be the set $\{\{a\},\{a,b\}\}$." ~ Hungerford's Algebra (p.6)
I think this is the first time that i've seen this definition. I've read the wiki page. Is it defined this way, as opposed to a definition relating to functions as in a Cartesian product, because this definition is c... |
I am looking for some good text/reference on complex Fourier series resp. Fourier analysis for complex (in particular holomoprhic) functions (of one variable). The more it contains on this particular subject, the better.
Background: For my diploma thesis, I need in particular to understand asymptotics of the Fourier co... |
If I am correct in stating that a factorial of a number ( of entities ) is the number of ways in which those entities can be arranged, then my question is as simple as asking - how do you conceive the idea of arranging nothing ?
Its easy to conceive of a null element in the context of arrays, for example - so you say ... |
Can someone give an insight on the following problem? I'm not sure how to start the problem. It's a practice problem for "mean value theorem" and "Taylor's Theorem" so I'm assuming they might be necessary for the proof. Thanks!
Let $f: \mathbb{R} \to \mathbb{R}$ be a function. Suppose that $f$ is differentiable, that $... |
What is difference between "recognizable" and "decidable" in context of Turing machines?
|
Say $z \in \mathbb{C}$ and $\bar{z}$ the complex conjugate (i.e. with $\bar{z} z = \left|z \right|^2$).
Can a function of $z$ and $\bar{z}$ be analytical?
Example: $f(z,\bar{z}) = Az^3 + B \bar{z} z$
I thought no, because the partial derivatives will depend on the direction in the complex plane (i.e. the phase of the l... |
Is there any theorem that apllies to non simple cycles and chromatic number?
For example we know that x(G)=2 if G does not contains odd cycles.
What about a non-simple cycle that contains odd cycles.
E.g A square and a triangle having a common vertex is (n=6,3-chromatic)
A petagon and a triangle having a common v... |
Let $\{a_k\}$ be an infinite sequence, with $\sum_{k=\infty}^\infty \vert a_k\vert^2<\infty$. Let $f(\omega)=\sum_{k=-\infty}^\infty a_k e^{ik\omega}$ be its Fourier series. By Plancheral's theorem, $f$ is bounded. How do I get a reasonable bound for $|f|$, i.e., if $|f|\leq g$, how do I find $g$?
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(Someone may please change the title if they can think of a better one)
We have a Hilbert Space $\mathcal{H}$ that consists of all functions $\psi(x)$ such that
$\int_{-\infty}^{\infty} |\psi(x)|^2 dx \lt \infty $
Now I have to show that there are functions in $\mathcal{H}$ such that $Q\phi(x)=x\psi(x)$ is not in $\m... |
For an assignment, I've been asked to find the $\frac{dy}{dx}$ for the formula $$2 \cos(4x) \sin(9y)=7$$
The major problem I'm having with understanding implicit differentiation is understanding what distinguishes a typical $\frac{d}{dx}$ from $\frac{dy}{dx}$ An example I came across gave the process of $$\frac{d}{dx}... |
How can I prove, that a polynomial function $$f(x) = \sum_{0\le k \le n}a_k x^k\qquad n\in\mathbb N,\ a_k\in\mathbb C$$ is zero for at most $n$ different values of $x$, unless all $a_0,a_1,\ldots,a_n$ are zero?
|
If $w(X)\leq n$ ($n$ is finite), and if $B_1$ is a base of $X$ such that $|B_1|\leq n$, then for any base $B$ of $X$ we have $B_1$ is contained in $B$.
Can you help me in proving this fact?
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I am wondering if there are some studies about the number of reversals of direction to return to the starting point in random walk (either symmetric or non-symmetric), for example, its distribution and expectation etc.
|
Given the formula $x^{2}y^{2}-8x=3$, find the second derivative.
I calculated the first derivative as $$-\frac{xy^{2}+4}{x^{2}y}$$
Working from that, I calculated the second derivative starting with
$$\frac{([x^{2}y)\frac{d}{dx}(-xy^{2}+4)]-[(-xy^{2}+4)\frac{d}{dx}(x^{2}y)]}{(x^{2}y)^{2}}$$
The left $\frac{d}{dx}$ was ... |
I have two boolean matrices:
A = |1 1 0|
|0 1 0|
|0 0 1|
and
B = |1 0 0|
|1 1 1|
|0 0 1|
What is the result of A x B and what are the steps needed to attain the result?
Note: My textbook says that the answer to the above is:
A x B = |1 1 1|
|1 1 1|
|0 0 1|
and that A * B is not equal... |
First off, is this question appropriate for this Stack Exchange? If not, I'll remove it.
Now to my question:
Is there an easy way to apply the trend line formula from a chart to any given X value in Excel?
For example, I want to get the Y value for a given X = $2,006.00. I've already taken the formula and retyped it ou... |
A problem in Lee's Introduction to Topological Manifolds got me thinking about this question. I can easily construct a locally euclidean space that is not second countable, by taking a disjoint union of $\mathbb R^n$ over an uncountable index set, but this is not connected of course. Similarly one can easily find conne... |
The group of units of the rings $\text{GF}(p^r)$ and $\mathbb{Z}/p^r\mathbb{Z}$ are both cyclic (except for the exception of prime powers are not cyclic when $p=2$ and $r\ge 3$). This is a strong result which I have been using a lot but I don't understand it properly.
I would like to know more alternative proofs of the... |
I would like to know the value of the following improper integral:
$$\int \limits_{-\infty}^{\infty}\frac{2x}{1+x^2}dx$$
as the function $f(x)=\displaystyle\frac{2x}{1+x^2}$ satisfies $f(-x)=-f(x)$ can I immediately conclude that the integral is equal to zero?
Thank you in advance for any suggestion.
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