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While learning topology one learns about compact set. The standard definition is:
A set $X$ is said to be compact if open cover has a finite subcover.
Since $[0,1]$ is compact, if we take a open cover for this we should be able to get a finite subcover. I know, that $(0,1)$ is not compact, so there must exists some o... |
I have a plane defined through a point P and two 3D-vectors $\overrightarrow{X}$ and $\overrightarrow {Y}$.
I wish to convert coordinates of points on this plane between local 2D-parametric and world 3D coordinate systems.
I know the conversion from 2D Parametric to 3D is
$C(u, v) = P + u\cdot \overrightarrow {X} + v\c... |
I am reading by myself this book http://tinyurl.com/37z4bbt. But to be honest, I have several problems to fully understand some part of the text. Maybe because I have not yet solid knowledge or I still need to learn more about the subject, hence asked a little help
To be exact, is this part of the construction concerni... |
Hi I have a problem with quadratic splines, I am supposed to find $
S_1 $ and $S_2$ that interpolates the following points $S(-1)=0$, $S(0)=1$, $S(1)=2$, and at the same time we want to find $S$ such that $\int_{-1}^1 \! (S(x))^2 \, \mathrm{d}x$ is minimal. The answer is on the form
$S_1(x)=a_1 x^2 +b_1x +c_1$ on $[... |
Let $S$ be a collection of disjoint sets. Let a binary operation '*' defined $\forall x,y$ each belonging to two different sets or a same set in $S$ with the property that $z=x*y$ belongs to some set in $S$ . Let $\forall A_1,A_2 \in S$ a binary operation $ \otimes$ is defined between $A_1,A_2$ as $B=A_1 \otimes A_2$ ... |
In a 2-d plane, let $A$ be the set of all points inside a circle $C$ and $P$ be the set of all points on the perimeter of the circle $C$. Is there any generalized measure in set theory which distinguishes between two such sets ?
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Suppose a parabolic mirror of a telescope is large enough (or equivalently that the camera exposure time is short enough) that you need to take into account the transit time of the light signal from hitting the mirror to reaching the focal point.
What shape should the mirror be to have the same effect as a parabolic mi... |
I know that there was a guy that could get 100 decimal digits of $\pi$ before computers were able to get thousands.
How did the guy do that?
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I am looking for a website that contains a collection of functions, more than Matlab contains, to download I use from Matlab.
The extension should be *.m or the code should appear.
Thank you!
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The original question I posted on StackOverflow. I think it's more mathematically inclined so I posted it here again. In terms of math.
There's a class of students. Each students has a score between 0 and 300 (gaussian, with some known average and standard deviation). I need to find classes of marks (like 0-100, 101-12... |
Let us assume that we are given a closed, orientable 2D manifold embedded in $R^3$, and lets call it $M$.
I think it is clear that in a coordinate neighborhood $(U, \phi)$ it is possible to at each point define a unit normal vector $N_p$ by looking at the tangent space $T_pM$ as a subspace of $T_pR^3$ and considering t... |
Let $L,K$ be number fields and $L|K$ a galois extension.
Let $(0)\neq Q$ a prime ideal in $\mathcal O_L$ (=ring of integers in $L$) and $P=Q \cap \mathcal O_K$.
$Z_Q $ denotes the decomposition field of $Q$ and $T_Q$ denotes the inertia field of $Q$.
Now put $Q' :=Q\cap Z_Q$ and $Q'' :=Q\cap T_Q$.
How does one prove, ... |
Can you partition an infinite set, into an infinite number of infinite sets?
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Update: Steve points out in comments that many direct products of residually cyclic groups will be counterexamples. Indeed, I should have spotted this: every finitely generated abelian group is residually cyclic! I should have asked about fully residually cyclic groups, ie groups $G$ such that for any finite subset $... |
How do we show the following?
Let $X$ be a topological space and let $x \in X$. Show that if $x$ has a countable neighborhood basis in $X$ then $x$ has a countable neighborhood basis in $\beta X$. Here $\beta X$ denotes the Stone–Čech compactification of $X$.
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My understanding of integration-by-parts is a little shaky. In particular, I'm not totally certain that I understand how to properly calculate the limits of integration.
For example, the formula I have is:
$\int_{v_1}^{v_2}{u dv} = (u_2 v_2 - u_1 v_1) - \int_{u_1}^{u_2}{v du}$
I'd like to see how to calculate $u_1$ a... |
Can you show me how to prove this::
$$66^{11 k+66} + 11^{101 k+55}\equiv \left\{\begin{array}{ll}
22 \pmod{165}& \text{if $k$ is odd,}\\
77 \pmod{165} & \text{if $k$ is even.}
\end{array}\right.$$
|
A student came to me showing a question from his exam in basic group theory, in which they are asked to prove that $\mathbb{C}^*$ modulo the subgroup of roots of unity is isomorphic to $\mathbb{R}^+$ (in both cases we mean the multiplicative groups).
Now this seems to me to be a simple error in the question. I believe ... |
I'm currently trying to do some Fourier transformations, or at least trying to understand them. The only thing I'm worried about is the complex part of the function. All I have is some basic, self thought, understanding about complex numbers.
As far as I'm concerned, a complex number exists of a real part and a complex... |
Has anyone ever come across a good cheat sheet for a list of definitions for the various algebraic structures out there, i.e. groups, fields, rings etc. Every time I come across the name of some structure, I have to look it up on wikipedia just to be sure I'm thinking of the right one, figured it would be cool to print... |
suppose, $f(x,y)$ is a bounded continuous function on $\mathbb{R}^2$. Consider
$$\lim_{ y \rightarrow y' }\> \sup_{x \in \mathbb{R}} f(x,y).$$
In how far can you switch suprema and limits, or which possible term comes closest to switching these two? Ideally, there would be something like
$$\sup_{x \in \mathbb{R}}\> \l... |
I have an idea for solving the knapsack problem, but it looks too good to be true. I would like someone to explain potential problems with this approach. I'll give an example: I want to find a subset of {2,7,11} which has elements that sum to 13. Here is the algorithm for solving this:
In binary notation 2=0010, 7=0111... |
Just a quick question about the geometry of Hilbert spaces from an intuitive standpoint. Maybe just assuming we're working with $L^2$ would simplify the situation. Basically, in something like $\mathbb{R}^2$ we have the situation that $\cos(\theta)=\frac{\langle a,b\rangle}{\vert a\vert\cdot\vert b\vert}$, and the ide... |
This is a problem that a professor proposed for the highschool mathematical olympiad in Costa Rica that we haven't been able to solve. Therefore it cannot be asked since we don't have a solution yet in general.
Let $\mathcal{F}_{k}$ for a fixed $k \in \mathbb{N}$ be the family of subsets $A_i \subset \mathbb{N}$ that... |
When ever I find definition of the quantum plane it says $A_q^2 = C\langle x,y \rangle/I$, where $I = C\langle xy-qyx \rangle$. What I want to know is whether they mean the unital free algebra or just the free algebra. In brief, is the quantum plane unital?
Moreover, when people write $A_q^N$, they mean the free (unita... |
Prove or disprove:
Let $\rho : \mathbb{N} \rightarrow \mathbb{N}$ injective. Let $(a_{n})_{n \in \mathbb{N}}$ be a sequence.
(i) If $\displaystyle\sum\limits_{n=1}^\infty a_{n}$ absolutely converges then $\displaystyle\sum\limits_{n=1}^\infty a_{\rho(n)}$ also converges absolutely.
(ii) If $\displaystyle\sum\limits_{n=... |
Let $G$ be a group and $G'$ its commutator subgroup. Let $\pi: G\to G/G'$ be the natural projection.
Statement 1: $G/G'$ is the largest Abelian quotient of $G$ in the sense that if $H\unlhd G$ and $G/H$ is Abelian, then $G'\le H$. Conversely, if $G'\le H$, then $H\unlhd G$ and $G/H$ is Abelian.
Statement 2: If $\varph... |
Show that: $\displaystyle\sum _{k=n}^{\infty } \frac{1}{k!}\leq \frac{2}{n!}$
I am clueless here, I tried to multiply both sides with $n!$, but it doesn't make things better. I know that the left one converges against $e$ for $n=0$, but I better don't want to use its numerical value.
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Say I have a numeric sequence A and a set of sequences B that vary with time.
I suspect that there is a relationship between one or more of the B sequences and sequence A, that changes in Bn are largely or wholly caused by changes in sequence A. However there is an unknown time delay between changes in A and their ef... |
The context:
I'm working on geocoded documents. This is, documents have latitude and longitude attributes, as well as some other geo attributes such as an address. Right now, I'm performing text search operations on the indexed docs and showing their geolocation on a map.
The problem:
Most of search criterias include... |
What exactly is "Approximation Theory"? If I read the wikipedia-article I doesn't get much clearer. Why are "pure" mathematicians interested in it? I see a lot of people that do harmonic analysis also do approximation theory.
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$S$ is a collection of disjoint sets. $(S,\cdot)$ is a commutative monoid and $(S,*)$ is a commutative semigroup. The identity element $e$ of $(S,\cdot)$ is the zero element of $(S,*)$.
The monoid $(S,\cdot)$ does not have a zero element. The binary operator $'*'$ is distributive over the binary operator $'\cdot'$. Wha... |
I'm currently looking at the equation for a multiclass classifier of the form
$$\sum\limits_{i=1}^{N_I} \sum\limits_{k=1,\atop k \neq y_i}^{N_K} L(1+ \mathbf{w_k}\cdot\mathbf{x_i}-\mathbf{w_{y_i}}\cdot\mathbf{x_i})$$
where
$X$ is a $N_K \times N_F$ data matrix
$y$ is a vector of class labels
$W$ is an $N_K \times N_... |
To construct a multivariate B-spline, we simply take the Kronecker tensor product between the univariate basis functions constructed for each individual dimension.
What I'd like to know is how do you construct a B-spline derivative for a multivariate function? Is it the same method of using the tensor product, but with... |
A little help here. Exercise 21, Ch. 2 from Feller's book reads
In a town a $n+1$ inhabitants, a person tells a rumor to a second person, who in turn repeats it to a third person, etc. At each step, the recipient of the rumor is chosen at random from the $n$ people available. Find the probability that the rumor will ... |
I understand Lie groups are defined by the structure constants associated with the lie brackets, which are treated as commutators in quantum mechanics, but i dont know of a math theory related to group theory to define or use an anti commutator. If Lie groups theory uses the commutator, what theory uses the anti commut... |
So i've been stuck on this problem for about an hour. I can't figure out how to do it, and help would be absolutely amazing. This is what's given:
$0 + 1 = 1$
$\forall x (x + 0 = x)$
$\forall x \forall y [x + (y+1)= (x+y)+1]$
$[0 + 1 = 1 + 0 \wedge \forall x(x + 1 = 1 + x \rightarrow (x+1)+1 = 1+(x+1)))] \rightarrow... |
I want to know why $\mathbb{Z}[x]/(1-x,p)$ is isomorphic to $\mathbb{Z}_{p}$, where $p$ is a prime integer?
Here's what I have so far, but I am unsure if I am correct. Every $f\in \mathbb{Z}[x]$ can be written as $(1-x)q+ r$ where $q\in \mathbb{Z}[x]$ and $r$ is in $\mathbb{Z}$. Does It follows that there are $p$... |
Assume I have linear equation system in $n$ variables and the solution is $x_1=x_2=\cdots = x_n$.
What properties must the matrix of the system have, or how do I prove that the system has such a solution?
|
What does $\mathbb{Z}[\omega]$ usually mean when $\omega$ is a primitive root of unity?
|
The idea of a read-only right-moving Turing Machine already exists [1].
Is there a relaxed version of this where an input-reading head only moves
right to read inputs as needed? There may be other heads doing the computation
on work tapes; the movement of these heads is unrestricted.
Of course the right-moving head co... |
Suppose I have a function $f \in L^2(R^n)$ with compact support, and a mollifier $\phi(x) \in C^\infty(R^n)$ with compact support s.t. $\int \phi(x) dx = \int \phi_\epsilon(x) = 1$ and $lim_{\epsilon \to 0} \phi_\epsilon(x) = \delta(x)$ where $\phi_{\epsilon}(x) = \epsilon^{-n} \phi(x/\epsilon)$ and $\delta(x)$ denotes... |
Earlier I had posted question about Borel cantelli Lemma. Thanks to Nate and Shai for your kind response. Please see my profile for the earlier questions. As you can see I do not understand the lemma well. Here are some confusions I have. I hope someone will be kind enough to help me understand the following:
Is it tru... |
I like to understand how things work. It's the only way I'll remember the rule. If anyone has a way of breaking this down into logical steps to get from one to the other... it would be greatly appreciated.
why is
$X^n\cdot X^m = X^{n+m}$ ?
|
For part b, I'm getting a little stuck. So I'm trying to show that if $(x,y,r)R(a,b,s)$ and $(a,b,s)R(x,y,r)$, then $(x,y,r)=(a,b,s)$
And so $(x,y,r)R(a,b,s)$ implies $\sqrt{(x-a)^2+(y-b)^2} \leq s-r$
and $(a,b,s)R(x,y,r)$ implies $\sqrt{(a-x)^2+(b-y)^2} \leq r-s$
After some algebra, here's what I get:
For the first o... |
Hello all this is my first math question, I hope this is the right place to post.
Anyway, this is basic stuff but I'm a bit confused here, here's the problem's legend:
"Find the equation of the plane that passes through the intersection line ($l$) of the planes $2x-y-z=2, x+y-3z+4=0$ and has a distance of 2 from the or... |
My Daughter's 4th grade math question got me thinking
Given a number of $3\!\text{ in}$ squares and $2\!\text{ in}$ squares, how many of each are needed to get a total area of $35 \! \text{ in}^2$?
Through quick trial and error (the method they wanted I believe) you find that you need $3$ $3\! \text{ in}$ squares and... |
I have read this paper and the wikipedia summary here and even this homework problem, but I do not understand how 2SAT is in P.
Here is my concern: the idea is that we start by setting a random variable. Then we set any implied variables. Repeat, and if we eventually reach a contradiction, then the problem has no solu... |
How to find the least positive root of the equation $\cos 3x + \sin 5x = 0$?
My approach so far is to represent $\sin 5x$ as $\cos \biggl(\frac{\pi}{2} - 5x\biggr)$ then the whole equation reduces to $$2\cos \biggl(\frac{\pi}{4} - x\biggr)\cdot \cos \biggl(\frac{\pi}{4} - 4x\biggr) = 0$$
From here we can write:
$$\big... |
I'm reading a paper on fluid dynamics and it references a unit vector between two particles i and j. I'm not clear what it means by a unit vector in this instance. How do I calculate the unit vector between the two particles?
|
I can't use any of the convergence tests I learned because I have no information on $f(x)$, in particular I don't know if it's continuous or positive.
The only thing I could think of was that if $\displaystyle \int_{1}^{\infty}f(x)\ \mathrm dx$ was absolutely convergent, then $|f(x)\sin x| \leq |f(x)|$ would imply by t... |
Can we alter Hilbert's axioms to have $\mathbb{Q}^3$ as a unique model?
The critical axioms seem to be the congruence axioms IV.1 and IV.4, and presumably the line completeness axiom V.2.
But how are they to be modified?
IV.1 might be replaced by requiring that there are counter-examples (irrationality of $\sqrt{2}$) ... |
It is proved in Advanced Calculus by Angus Taylor, § 20.8, that
$$\log n!=\log \left( \left( \frac{n}{e}\right) ^{n}\sqrt{2\pi n}\right)
+r_{n},$$
where
$$r_{n}=\sum_{k=1}^{\infty }S_{k}$$
with
$$S_{k}=\sum_{p=n+1}^{\infty }\frac{k}{2(k+1)(k+2)p^{k+1}}.$$
This formula for $r_{n}$ provides a method for finding the estim... |
Set $S$ is a collection of disjoint sets each having cardinality of that of $\mathbb{R}$.The cardinality of set $S$ is also that of $\mathbb{R}$. $F$ is the set which is union of all the elements of $S$. Is the cardinality of set $F$ is equal to that of $\mathbb{R}$ ?
|
Assume that Y has a beta distribution with parameters a and b. Find the density function of U = 1 - Y.
I know how to do then when they give the density function of Y, but i'm confused here.
Thanks!
|
In the following picture, the author of the Field and Wave Electromagnetics shows the geometrical meaning of the direction of the gradient. That is, only by following the direction of the normal vector to the curve at that pointer could the rate of change be the maximum.
But what about the geometrical interpretation ... |
I have another question about correctness of my proof. We know, that $N$ and $G/N$ are p-groups, than $G$ is p-group.
Proof: $|N|=p^m$ and thus $|G|=p^m q$. Suppose that $(p^m,q)=1$, than in $G$ exists element of order $k$, such that $k | q$ and $(k,p^m)=1$ - suppose it's $b \in G$. Element $bN \in G/N$ is also of orde... |
My question is similar to this, but not identical. I believe the following to be true, but I'd like a reference.
Given (quasicoherent?) sheaves of $\mathcal O_X$ modules $E$ and $F$ on a projective variety $X$,
$$
H^n(X, E\otimes F) \cong \bigoplus_{p+q=n} H^p(X, E) \otimes H^q(X, F).
$$
Is this true, and what is a ... |
I'm reading High Performance JavaScript, and I think the graphs in one chapter are just plain wrong. Here is one on Google Books.
The y axis is "Times faster", and it runs from -1.5 to +4.0. Now, I would have thought that "1 times faster" means "no faster", "2 times faster" means "twice as fast", and "0.5 times faster"... |
Let $a_n$ be a sequence in $\mathbb{R}$, $\phi: \mathbb{N} \rightarrow \mathbb{N}$ a monotonically increasing function with $\phi (0)=0$. Show that
$\sum _{n=0}^{\infty } a_n$ converges $\Rightarrow \sum _{i=0}^{\infty } {(\sum _{j=\phi (i)}^{\phi (i+1)-1 }a_{j})}$ converges.
My notes: Can you use that ${(\sum _{j=\ph... |
I'm given the following exercise:
$\iint\limits_D \exp(x^{2}+y^{2})dA$
And I dont even know where to start, any chance someone could give me a hint?
D is a half circle, given by:
$9\le x^{2}+y^{2}\le 16, y\ge 0$
Wheres A, I don't know what it is, its not given in the exercise. - Thats partly whats bugging me.
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I'm looking for an example of a non-commutative ring, $R$, with identity s.t $R$ does not contain a non-trival 2 sided ideal and $R$ is not a division ring
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$-5 = 2x + 5$
I want to "re-start" my mathematical education. I am needing it more and more at work. A co-worker asked me to break that down (he couldn't remember either).
Where should I start? What books?
Thanks!
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My son is taking algebra and I'm a little rusty. Not using a calculator or the internet, how would you find the roots of $2x^4 + 3x^3 - 11x^2 - 9x + 15 = 0$. Please list step by step. Thanks, Brian
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What is the probability of getting $3$ heads in a row? Would it be $\frac 18$?
assuming the coin is a fair one.
|
What are some motivating consequences of the axiom of choice (or its omission)? I know that weak forms of choice are sometimes required for interesting results like Banach-Tarski; what are some important consequences of a strong formulation of the axiom of choice?
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According to Wikipedia, a (non-commutative) ring $R$ is local if and only if there do not exist two proper (principal) (left) ideals $I_1, I_2$ such that $R = I_1 + I_2$. It is easy to see that local rings have this property, but I'm having trouble showing the other way: if there do not exist two proper co-maximal (pri... |
This argument appeared in a proof in a paper (I am paraphrasing), and I am not sure why it is true. It should be rather simple.
Let $(\Omega,P)$ be some sample space. Let $X$ be a random variable between $0$ and $1$. Let $U \subseteq \Omega$ and let
$U^\prime = ${$ \omega \colon X(\omega) \ge E[X \mid U] - 1/m$}.
Then ... |
Starting with the famous Basel problem, Euler evaluated the Riemann zeta function for all even positive integers and the result is a compact expression involving Bernoulli numbers. However, the evaluation of the zeta function at odd positive integers (in terms of getting a closed form sum) is still open. There has been... |
Given the following formula:
$$\bigg[\forall x P(x,x) \wedge \forall x \forall y \forall z\bigg( P(x,y) \wedge P(y,z) \Rightarrow P(x,z)\bigg) \wedge \forall x \forall y (P(x,y) \vee P(y,x)) \bigg] \Longrightarrow \exists y \forall x P(y,x)
$$
It is known that this formula is always true if x,y,z are in any domain w... |
Motivated by my ignorance here, if $X$ is a projective toric variety, is
$$H^m(X, \mathcal O_X) \cong
\begin{cases}
0 & m > 0 \\
\mathbb C & m = 1
\end{cases}
$$
as for $\mathbb P^n$?
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If $F_{1}$ is a subfield of $F_{2}$, is $GL(N,F_{1})$ contained in $GL(N,F_{2})$? I am particular interested in the case of when $F_{1} =\mathbb{Q}(\omega)$ and $F_{2} = \mathbb{C}$. Here $\omega$ is a complex root of unity.
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I am trying to better understand how unique factorization of algebraic integers in an algebraic number ring implies that the class number of that number ring is 1.
I am asking for some examples of this to get me started.
|
I need to re-arrange the following equation to solve for $d$:
$$L = 4\pi C_x d^2$$
Thank you.
|
So one way to define a group presentation is to say, well, let's generate the free group with some number of generators, and then quotient by saying certain elements (relators) cancel just as $aa^{-1}$ does (and other things, you have to take the normal subgroup generated by the elements, but the basic idea works, I th... |
My research has brought me to the following, very general problem.
Given a fixed, but arbitrary, natural number, $\displaystyle v$, consider the following family of polynomials: The $\displaystyle (n-1)^{th}$ derivative of
$$\displaystyle (1-x^2)^{v+n} \ \ \forall n \in \mathbb{N} $$
I would like to prove (or dispr... |
If $X$ is a locally compact and metrizable space such that its Alexandroff compactification is not first countable. Does this imply that no other compactification of $X$ can be first countable? Why?
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There are ten people in a room:
Person $1$ knows $9$ people. Person $2$
knows $8$ people. etc. Person $9$ knows $1$
person.
How many people does person $10$ know?
What is the answer and what is the principle behind this question?
|
(Assume a unit speed parametrization)
Prove that an admissible curve $c:(a,b) \to \mathbb R^3$ (hence $c^{\prime} \times c^{\prime\prime}$ is never zero) with a zero torsion is contained in a plane.
I understand why this should be true, if there is no torsion element to the curve then $c$ can't twist into the third d... |
The following is a multiplication problem in traditional base ten notation. Each letter represents a different digit. What digit does each letter represent? How did you get your foot in the door?
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A newspaper launches a competition. It said that readers should submit one number between $1$ and $1000$. A £$2000$ prize would be awarded to the person that got the closest to $2/3$ of the mean of all the entries.
What would a smart entry be? What is the underlying principle behind this type of question. I know it ... |
I have this question:
$a \equiv 12^{772} \pmod{71}$, when $0 \leq a < 71$
and I am having troubles getting it started.
$\frac{772}{71}$ is $10$ with a remainder of $62$;
How do I do this question?
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Is there an infinite set of strings whose Kolmogorov complexities are computable?
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Assume $f:(a,b) \to \mathbb R^3$ is an admissible unit speed curve (hence $f^{\prime} \times f^{\prime\prime}$ is never zero)
If $f$ lies on the sphere with center $a$ and radius $r$ prove that
$f = a - (1/\kappa) \mathbf N - (1/\kappa)^{\prime} (1/\tau) \mathbf B$
With $|f^{\prime}| = 1$,
$\mathbf T = f^{\prime}$
... |
Is there any way to find a function, even just similar, from a set of values?
I get these pairs of values from two sensors and would like to find a simple function that describes the relationship between these pairs of numbers and then to estimate the values without having to take each time.
I have a "black box" sensor... |
I was just reading a proof of the dimension theorem in Steven Roman's Advanced Linear Algebra. In addressing the cases of infinite bases, Roman proceeds to show that if $\mathcal{B}$ and $\mathcal{C}$ are bases of a space $V$, then $|\mathcal{B}|\leq |\mathcal{C}|$, working up to an application of the BSC-Theorem. Anyw... |
Suppose I have a function $f$ that is infinitely differentiable on some interval $I$.
When I construct a Taylor series $P$ for it, using some point $a$ in $I$, does $f(x) = P(x)$ for all $x$ in $I$?
I'm confused as to whether Taylor series approximate (or equal - I'm not about that either) functions on one point, or on... |
On an infinite grid of ideal one-ohm resistors, what's the equivalant resistance between two nodes a knights move away?
(please fix the tags, I didn't really know where to put it)
|
I have been interested in learning formal math formally enough so that I could write a proof assistant using some simple parsing tools, and explain to someone else with little math knowledge how to write one. Naturally, it would be a very clunky and tedious assistant, but this is the standard I have set for myself for ... |
The usual statement of the Lebesgue Decomposition Theorem says that given two $\sigma$-finite measures $\mu$ and $\nu$ on a measure space, we can decompose $\nu = \nu_1 + \nu_2$, where $\nu_1$ is absolutely continuous with respect to $\mu$ and $\nu_2$ and $\mu$ are mutually singular.
Wikipedia (link text) says that th... |
I understand that $\pi$ and $e$ are transcendental and that these are not simple facts. I mean, I have been told that these results are deep and difficult, and I am happy to believe them. I am curious what types of techniques are used and just how difficult of a problem it is. Would this result be a reasonable capstone... |
Does anybody know where I can find a general form, in terms of n, of the sum $\left(\sum\limits_{i=1}^n a_i\right)^n$. What I mean is, there appears to be some sort of pattern, if you look at $\left(\sum\limits_{i=1}^2 a_i\right)^2$ you get $a_1^2+2a_1a_2+a_2^2$ and if you let $n=3$ then you get $a_1^3+a_2^3+a_3^3+3a_... |
this step in the proof is confusing me:
$$\sum_1^\infty {\frac{4^{n}}{3^{n-1}}}\qquad \longrightarrow \qquad\sum_1^\infty 4\left(\frac43\right)^{n-1}$$
please explain how/why this happened?
cheers,
gregg
|
I'm trying to figure out whether my hardware function for mapping operating system pages to DDR memory controllers is injective:
$$F(x) = a x \pmod{b},\text{ where $x$ in $[0,b)$, $a$ and $b$ are co-prime}$$
My hunch is that it is injective and I think it has to do with the Chinese Remainder Theorem, though it may be s... |
I am not getting this theorem:
Angle between chord AB and tangent at A is the same as subtended by segment AB at any point on the circumference.
How to prove this theorem?
|
in functional analysis, you encounter the terms 'adjoint' and 'formal adjoint'.
What does 'formal' in that case mean? It Sounds like a hint that 'formal adjoints' lack a certain property to make them a 'true' adjoint.
I have nowhere found a definition, and would be eager to know.
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And what about the other direction? If $(u_{n})$ is increasing, what about $(a_{n})$?
I'm guessing the former is true since we know that if $(a_{n})$ converges, $(u_{n})$ converges to the same limit. That tells me that around infinity the sequences behave roughly the same. I tried proof by induction but got stuck.
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Guys is my logic correct? If Earth travels around the Sun in $29.78$ km/s and I have to calculate how long will it take to travel $1$km I did following:
$1s / 29.78 = 0.03358$ (to $5$ d.p)
I think it's ok but I would like to get confirmation on this.
Thank you.
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How do i show that $f_{1}(x)=1$, $f_{2}(x)=e^{x}$ and $f_{3}(x)=\sin{x}$ are linearly independent, as elements of the vector space, of continuous functions $\mathcal{C}[0,1]$.
So for showing these elements are linearly independent, one needs to show that if $$ a_{1} \cdot 1 + a_{2} \cdot e^{x} + a_{3} \cdot \sin{x}=0$... |
What is wrong with this proof?
Is $\pi=4?$
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