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I have the following constraint functions:
$$g_{i_{min}} \leq y_{i+1}-y_{i} \leq g_{i_{max}}$$
$$y_{i_{max}}-y_{i} \geq h_{i}$$
$$v_{i_{min}} \leq \Biggl[\frac{(y_{i+1}-y_{i})^{3} (h_{i}+a)^{4}}{(h_{i}+b)^{4}}\Biggr] \leq v_{i_{max}}$$
$$Q_{i_{min}} \leq \Biggl[\frac{(y_{i+1}-y_{i})^{3} (h_{i}+a)^{10}}{(h_{i}+b)^{4}... |
I start from a simple example. I through a (possibly unfair) coin 20 times. I got "eagle" and "tail" 15 and 5 times, respectively. Now I need to estimate probabilities for eagle and tail. For eagle I get 15/20 = 0.75 and for the tail I get 0.25.
However, I know that the probability that I got are not accurate. It is st... |
I hope this question is not off-topic, but I'm currently discovering the existence of mixing processes, which attempt to express the dependence between random variables using miscellaneous coefficients ($\alpha$, $\beta$, $\phi$ and $\rho$ are the ones I've met so far).
Are there any references you would advise that su... |
Came across this math problem while programming something:
Given positive $y$, $a$, and $b$, find an integer $x$, $a\lt x\lt b$, so that $y\bmod x$ (the remainder of $y$ when divided by $x$) is positive and less than or equal to $100$,
$$0 \lt y\bmod x \leq 100.$$
I know sometimes it's not solvable depending on $y$ an... |
This is my first post here. Apologies if there are any misformats or mis-tagging.
Also if the post doesn't belong here, sorry, and I will delete it.
I derived this formula from a probability calculation. How would I go about finding the maximum value for the following?
$\frac{T}{S}\frac{S!}{(S-D)!}\frac{(T-D)!}{T!}$
w... |
Can a finite group G have a cyclic Sylow 3-subgroup of order 9, such that the intersection of the Sylow 3-subgroups of G has order exactly 3, without having non-identity normal subgroups of order coprime to 3?
In classifying the finite groups with cyclic Sylow 3-subgroups of order 9, it seems reasonable to split into... |
I am learning group theory, and while learning automorphisms, I came across conjugation as an example in many textbooks. Though the definition itself, (and when considering the case of abelian groups), it seems pretty innocent, I have to admit that I have no intuition about whats happening, or why even such a map is im... |
In other words, if X = Y in distribution, is it true that EX = EY?
I think this must be true, but I've tried to prove it a few times and I always get stuck.
Thanks in advance for any hints or reference.
|
How would one go about showing how many solutions the following congruence has?
$$x^2 + y^2 \equiv 23 \pmod{93}.$$
|
My homework question: Show that all zeros of $$p(z)=z^4 + 6z + 3$$ lie in the circle of radius $2$ centered at the origin.
I know $p(z)$ has a zero-count of $4$ by using the Fundamental Theorem of Algebra. Then using the Local Representation Theorem the $$\int \frac{n}{z+a} = 4(2 \pi i).$$ I am assuming $a=0$ since w... |
Are order and binary relation the same thing, or order is just some special binary relation? If it is the latter, what kinds of properties distinguish an order from a general binary relation?
Thanks and regards!
Sorry for the confusion already caused. What I thought was that a partial order is just some special order,... |
Equations are in $\LaTeX$ format; I'm still trying to understand how MathJax works.
Given the following integral:
$\int_0^{+\infty } \frac{1}{x \sqrt{x}} \, dx$
I'm pretty sure that does not converge.
And if the integral was a indefinite one, also I'm almost sure the result is $\ln(\sqrt x)$ using substitution.
If I a... |
This is probably a bit of a boring and ridiculously noob question, so apologies in advance, but:
Is everything in maths essentially built on adding, subtracting, multiplying and dividing? Are they the first principles on which everything else is built?
|
I remember reading in RPF's biographical book, "Surely you're joking Mr. Feynman" that he used to have timed contest about writing down the biggest number using standard symbols.
Instead of the time restriction, I was wondering what if we restrict the number of characters, to say $N=10$. What is the largest number we ... |
I'm trying to reconstruct a problem I saw in the Monthly, years ago. Perhaps it'll look familiar to someone.
In the integer lattice in the plane, we grow a tree in the following natural way: Initially the tree is just the origin. At each step, we find the set of lattice points that are neighbors (distance 1) to precise... |
I'm trying to understand the proof for Lemma 4.8 in Lyndon & Schupp Combinatorial Group Theory, page 26, proposition 4.8.
In the very end of this proof, we have: $N \unlhd F$ where $F$ is a finitely generated free group. $[F:N]=p$, $p$ prime. $K \unlhd N$, $K \unlhd F $. $u,v \in N$ such that their images in $N/K$ are ... |
I have just begun delving into $p$-adic number theory. I was wondering, given a poynomial $f(x)$ with integer coefficients, what does it mean when we say, $f(x)$ has a root in $\mathbb{Z}_2$, for instance.
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The group $S_n \times S_n \times S_n$ acts on the set of Latin squares $L$ of order $n$, with $\theta:=(\alpha,\beta,\gamma)$, starting from $L$, permuting the rows by $\alpha$, permuting the columns by $\beta$ and permuting the symbols by $\gamma$. Each $\theta \in S_n \times S_n \times S_n$ is called an isotopism. ... |
The problem is "Calculating $\oint_{L} \frac{xdy - ydx}{x^2 + y^2}$, where L is a smooth, simple closed, and postively oriented curve that does not pass through the orgin".
Here is my solution:
Let $$P(x,y) = \frac{-y}{x^2 + y^2}, Q(x,y) = \frac{x}{x^2 + y^2}$$
Get $$\frac{\partial{P}}{\partial{y}} = \frac{y^2 - x^2}{... |
In this problem, I know that the hypothesis of Green's theorem must ensure that the simple closed curve is smooth, but what is smooth? Could you give a definition and an intuitive explanation?
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The orginal problem is "Calculating $\oint_{L} \frac{xdy - ydx}{x^2 + y^2}$, where L is a smooth, simple closed, and postively oriented curve that does not pass through the orgin".
But what if I modify the hypothesis and allow non-simple closed curve? I mean is there something like green formula that allows us calculua... |
Was trying to calculate $$\int_{0}^{\infty}e^{-x}\ln x dx=-\gamma$$ and I found this question:
I want to analyze $$\int\frac{e^{-x}}{x}dx$$
With $u=\displaystyle\frac{1}{x} \Rightarrow du = \displaystyle\frac{-1}{x^{2}} dx $, and $dv=e^{-x} \Rightarrow v=-e^{-x}$
Then
$$\int\frac{e^{-x}}{x}dx = \displaystyle\frac{1}... |
What are the conditions for existence of the Fourier series expansion of a function $f\colon\mathbb{R}\to\mathbb{R}$?
|
Let $L(n)=\lfloor(1+\sqrt{5})^n\rfloor$. What kind of a linear recurrence is satisfied by $L(n)$? I have no idea how to go about this, because of the presence of the greatest integer function.
Please feel free to retag it as I kept getting an error on every tag I thought was appropriate.
|
How to prove the following trigonometric identities ?
1) If $\displaystyle \tan (\alpha) \cdot \tan(\beta) = 1 \text{ then } \alpha + \beta = \frac{\pi}{2}$
I tried to prove it by using the the formula for $\tan(\alpha + \beta)$ but ain't it valid only when $\alpha + \beta \neq \frac{\pi}{2}$ ?
2) $\displaystyle\sec\t... |
Given the following series: $\sum_{n=1}^{\infty} (\sqrt[3]{n+1} - \sqrt[3]{n-1})^{\alpha}$
where $\alpha \in \mathbb{R}$. Does the series converge or diverge?
Attempts to solve the problem:
1) $\lim_{n\to\infty} (\sqrt[3]{n+1} - \sqrt[3]{n-1})^{\alpha } = 0$ - not helpful.
2) Used the formula $a^{3} - b^{3} = (a-b)(a^... |
Using Lagrange's remainder, I have to prove that:
$\log(1+x) = \sum\limits_{n=1}^\infty (-1)^{n+1} \cdot \frac{x^n}{n}, \; \forall |x| < 1$
I am not quite sure how to do this. I started with the Taylor series for $x_0 = 0$:
$f(x_0) = \sum\limits_{n=0}^\infty \frac{f^{(n)}(x_0)}{n!} \cdot x^n + r_n$, where $r_n$ is the... |
Let $p,q$ belong to $\mathbb{N}$ and are relatively prime to each other. If $\alpha,\beta$ belong to $\mathbb{N}$, are also relatively prime to each other,then are $(p\beta+q\alpha)$ and $q\beta$ always relatively prime ?
|
With help from Maple, I got
$$\left(\frac{ax+by+cz}{x-y}\right)^2+\left(\frac{ay+bz+cx}{y-z}\right)^2+\left(\frac{az+bx+cy}{z-x}\right)^2-(c-a)^2-(c-b)^2$$
equal to
$$\frac{(c(x^3+y^3+z^3)+(a-c)(x^2y+y^2z+z^2x)+(b-c)(x^2z+y^2x+z^2y)-3(a+b-c)xyz)^2}{(x-y)^2(y-z)^2(x-z)^2}$$ which of course is $\ge 0$.
But with no hel... |
Recently, I consider a problem: Let $E/F$ is a Galois extension of number field, denote the idele group of $E$ (resp $F$) by $I_E$ (resp $I_F$). There is a homomorphism induced by norm map $N_{E/F}$:
$$N: I_E/E^{\ast} \to I_F/F^{\ast}$$
If there is a finite abel extension $K/E$ ,from Global Class Field Theory,there is... |
How can we prove that the product of $n$ consecutive integers is divisible by $n$ factorial?
Note: In this subsequent question and the comments here the OP has clarified that he seeks a proof that "does not use the properties of binomial coefficients". Please post answers in said newer thread so that this incorrectly-... |
How can we prove, without using the properties of binomial coefficients, the product of $n$ consecutive integers is divisible by $n$ factorial?
|
If $x_i$ is exponentially distributed $(i=1,...,n)$ with parameter $\lambda$ and $x_i$'s are mutually independent, what is the expectation of
$\left(\sum_{i=1}^n {x_i} \right)^2$
in terms of $n$ and $\lambda$ and possibly other constants?
Note: This question has gotten a statistical answer on https://stats.stackexchang... |
For example, if $f(x) = \sin x$ and $x$ is uniformly distributed on $[0, \pi]$, how is the equation found that satisfies the probability distribution function of $f(x)$? I imagine the distribution function will be greater when the derivative of $f(x)$ is closer to zero, but this is just a guess.
I apologize if this que... |
Let $(X,d)$ be a metric space and $S \subset X$. Show that $d_S(x):=\text{inf}\{d(x,s): s \in S\}=0 \Leftrightarrow x \in \overline S .$
Notes: $\overline S$ is the closure of S. Maybe you can use that a closed set is also closed for sequences in the set? I think the difficult part is when S is open, otherwise its tri... |
In my website, users can either "like" or "dislike" a posted comment. I want to put a link to sort comments by liking such that the most liked ones becomes on top.
Of course I cannot just sort by the number of likes only. I have to subtract the dislikes. But what if the difference (likes - dislikes) are the same. e.g.... |
If there is a field $F$ that is a field reduction of the real numbers, that is $F(a)=\mathbb{R}$ for some $a$, let's also denote this $F=\mathbb{R}(\setminus a)$, then given $x \in \mathbb{R}$ is there a general method to determine whether $x$ is in $F$ or $x$ is in $\mathbb{R}\setminus F$ ?
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The empty set is a member of $P(\{a,b\}) \times P(\{p,q\})$. True or false?
My first instinct was false, since the empty set is a member of each power set individually, but when multiplied together, you get $(\emptyset,\emptyset)$, which I'm not sure represents the empty set. But my counter argument is that the empty... |
No subset of the real numbers is countable. True or false.
In looking at the wikipedia article for real numbers, I'm not really clear on the answer to this. They use the words computable and countable, and I'm not sure of the difference. Also, I dont think I fully understand the term countable. The definition states th... |
well i was wondering if, there s a way in which one could compare these two theories on the ground that which one will provide more accurate answers to a given problem.
Fuzzy logic always focuses on analysing between 0 and 1 and logic only dolls down to 0 and 1 , so does that imply that fuzzy logic provides us with mo... |
Imagine a two-dimensional grid consisting of 20 points along the x-axis and
10 points along the y-axis. Suppose the origin (0,0) is in the bottom-left corner and the
point (20,10) is the top-right corner. A path on the grid consists of a series of moves in
which each move is either one unit to the right or one unit up.... |
Original post on Mathoverflow here.
Given a list of identical and independently distributed Levy Stable random variables, $(X_0, X_1, \dots, X_{n-1})$, what is the is the probability that the maximum exceeds the sum of the rest? i.e.:
$$ M = \text{Max}(X_0, X_1, \dots, X_{n-1}) $$
$$ \text{Pr}( M > \sum_{j=0}^{n-1} X_... |
It's not really a typical math question. Today, while studying graphs, I suddenly got inquisitive about whether there exists a function that could possibly draw a heart-shaped graph. Out of sheer curiosity, I clicked on Google, which took me to this page.
The page seems informative, and I am glad to learn certain new t... |
Suppose a graph with 12 vertices is colored with exactly 5 colors. By the pigeonhole principle, each color appears on at least two vertices. True or false?
The correct answer is false, but I assumed it to be true. Why is this so?
|
My TI-83 calculator doesnt allow me to do this, but using Windows calculator, I can compute the factorial of say 5.8. What does this mean and how does it work?
|
This is probably simple, but I'm solving a practice problem:
$\lim_{n \to \infty}\frac{1}{n}\left( \cos{\frac{\pi}{n}} + \cos{\frac{2\pi}{n}} + \ldots +\cos{\frac{n\pi}{n}} \right)$
I recognize this as the Riemann sum from 0 to $\pi$ on $\cos{x}$, i.e. I think its the integral
$\int_0^\pi{ \cos{x}dx }$
which is 0, bu... |
I accidentally typed 4 ** 5 into Google and received 1,024 instead of 20. But 4^5 is also 1024...
Does the double multiplication sign also mean the power sign? Why two symbols?
|
The answer for this turns out to be only irreflexive. However, how is this not transitive? The definition I have for transitive states "whenever there is a
path from x to y then there must be a direct arrow from x to y". So for the above graph, if there exists a path from one point to another, then there should be a ... |
The Grassmann algebra $G$ is the algebra over a field $\mathbb{F}$ generated by the variables $e_i$ such that $e_i^2=0$ and $e_i e_j = - e_j e_i$.
I'm looking for some references on algebras $G \otimes A$ (In this case, the Grassmann algebra is over $\mathbb{C}$) where $A$ is the group algebra $\mathbb {C} \mathbb{Z}_n... |
There's one basic mathematical thing that keeps bugging me: the fact that a really simple 2D geometrical figure (like a circle) might not be a function.
I know what the definition of a function is. A circle is not a function (of one variable) because it would associate two values of the co-domain to a single value of t... |
Question: Let $V$ be a $K$-Vectorspace and $f: V \rightarrow V$ be linear. It holds that $f \circ f = f$. Show that $V = \mbox{ker}(f) \oplus \mbox{im}(f)$.
My attempt: So i guess that the $\oplus$ denotes a direct sum which means i have to show that (i) $V = \mbox{ker}(f) + \mbox{im}(f)$ and (ii) $\mbox{ker}(f) \cap \... |
Lennart Carleson proved Luzin's conjecture that the Fourier series of each $f\in L^2(0,2\pi)$ converges almost everywhere. Also, Richard Hunt extended the result to $L^p$ ($p>1$).
Some time ago I tried to read Carleson's paper, but I would say it is fairly hard to assimilate.
Is there an easier proof? Can someone poi... |
Given the series: $\sum_{n=0}^{\infty}(-1)^{n}(\sqrt[n]{n} - 1)^{n}$. Does the series converge?
Attempt to solution (might be incorrect):
$(\sqrt[n]{n} - 1)^{n}> (1+\frac{1}{n})^{n}$
$(1+\frac{1}{n})^{n} \to e \Rightarrow (\sqrt[n]{n} - 1)^{n}$ lower-bounded by $e$. Based on Leibniz Criterion the sequence $\{A_n\}$ (i... |
Let's consider a $3 \times 3$ matrix, $\bf A$.
Its inverse can be found by computing the adjoint of the matrix and dividing it by its determinant. Can someone please explain why this process work and how it can be visualized?
I know that the determinant can be thought of as the volume spanned by the column vectors and... |
I've recently read a article 'Theories of computational complexity'. In the article it states that with Rice's theorem one can prove that, for example, {$i∈[\mathbb N:\phi_i$ is a constant function}, is not recursive, but I can't see how. I already know the theorem, but I don't really get how I can use it. Thank you fo... |
State in words the negation of the following sentence: For every martian M, if M is green, then M is tall and ticklish.
I got the right answer to this, give or take a few words, but this is a question of form more than anything. After converting this statement to symbols and negating everything, I come up with: $\exis... |
Consider a non-empty set A containing n objects. How many relations on A are both symmetric and reflexive?
The answer to this is $2^p$ where $p=$ $n \choose 2$. However, I dont understand why this is so. Can anyone explain this?
|
Freyd-Mitchell's embedding theorem states that: if A is a small abelian category, then there exists a ring R and a full, faithful and exact functor F: A → R-Mod.
This is quite the theorem and has several useful applications (it allows one to do diagram chasing in abstract abelian categories, etc.)
I have been asked to ... |
On my mid-term exam tonight, I had a problem which went something like this:
Which algebraic structure is $(M,*)$, where $M= \left \{ \begin{bmatrix}
1 & 0\\
0 & 1
\end{bmatrix} \right., \begin{bmatrix}
0 & 1\\
-1 &0
\end{bmatrix},\begin{bmatrix}
-1 & 0\\
0 & -1
\end{bmatrix},\begin{bmatrix}
0 &-1 \\
1 ... |
If any edge from a given spanning tree T0 is contained in some minimum spanning tree T*, does this imply that T0 is also a minimum spanning tree ?
Right now, I'm trying to draw on paper some graphs to prove that it doesn't. Please correct me if it does, or help me find an example if it doesn't.
Thanks in advance.
P.S. ... |
I was wondering if the following properties of the Legendre polynomials are true in general. They hold for the first ten or fifteen polynomials.
Are the roots always simple (i.e., multiplicity $1$)?
Except for low-degree cases, the roots can't be calculated exactly, only approximated (unlike Chebyshev polynomials).
Ar... |
In an attempt to model the cassette tape spooling action, I'm hoping to solve this problem.
Required feed speed 5cm/second.
Two Spindles with starting radius of
S1: 0.8cm (empty spool)
S2: 1.8cm (full spool)
The spooled tape thickness is unknown but can be assumed to be around 15micrometer if needed.
The time to comp... |
How can we find $\displaystyle \lim_{y\to{b}}\frac{y-b}{\ln{y}-\ln{b}}$ without using:
(a) L'Hôpital's rule, (b) the limit $\displaystyle \lim_{h \to 0}\frac{e^h-1}{h} = 1$, and (c) the fact that $\displaystyle \frac{d}{dx}\left(e^x\right) = e^x$.
The reason for the conditions is that with this limit I'm trying to pro... |
"When you draw a circle in a plane of radius $1$ you can perfectly surround it with $6$ other circles of the same radius."
BUT when you draw a circle in a plane of radius $1$ and try to perfectly surround the central circle with $7$ circles you have to change the radius of the surround circles.
How can I find the radi... |
The set of rationals $\mathbb{Q}$ has the same cardinality as
the set of integers $\mathbb{Z}$. True or false?
This was a question on an old exam for our class. The correct answer is true. However, I did some additional reading and came across Cantor's transfinite numbers. In the book I'm reading, it says that "there a... |
Does there exist a nonzero linear transformation $T$ such that
$$\alpha^t T \alpha = 0\text{ for all $\alpha\in\mathbb{R}^n$,}$$
Where ${\alpha}^{t}$ denotes the transpose of the matrix.
Thanks! =D
|
How does one diagonalize a large sparse symmetric matrix to get the eigenvalues and the eigenvectors?
The problem is the matrix could be very large (though it is sparse), at most $2500\times 2500$. Is there a good algorithm to do that and most importantly, one that I can implement it into my own code? Thanks a lot!
|
A function $f(x)$ is defined $\forall x \in \mathbb{Q}$ and $ x \in (0,1)$ as $f(x) = q $ where $x = (p/q)$ in simplest form. Find $\sup_{\tau} \quad \min_{a} \quad f(a) + f(\tau-a)$ where $ a \in \mathbb{Q}$ and $ a \in (0,1)$ and $ \tau \in (\alpha,\beta)$ where $ 0< \alpha < \beta <2 $ and $ \alpha,\beta \in \m... |
I propose the following restriction on universal instantiation: UI may not be used to introduce new variables. The variable specified should be an "old" variable, i.e. it must already have been introduced by either an active premise or by existential instantiation (E-elimination). (An active premise is one that has not... |
See also: answers with code on GameDev.SE
How can I find out the arc length of a Bézier curve? For instance, the arc length of a linear Bézier curve is simply:
$$s = \sqrt{(x_1 - x_0)^2 + (y_1 - y_0)^2}$$
But what of quadratic, cubic, or nth-degree Bézier curves?
$$\mathbf{B}(t) = \sum_{i=0}^n {n\choose i}(1-t)^{n-i}... |
As a homework assignment, I've been given a particular subring of $M_2(\mathbb{Q})$, and asked to list all the ideals... For reference, $S$ = { set of matrices in $M_2(\mathbb{Q})$ with bottom left entry $0$ } is the subring in question.
I don't really see how to get a handle on this. Is there any sort of algorithmic w... |
I would appreciate some help with factoring a trivariate polynomial.
The polynomial in question is
$$p(x,y,z)=a_1 x^7+a_2 x^5y+a_3 x^3y^2+a_4 xy^3+a_5 x^4z+a_6 x^2yz+a_7 y^2z+a_8 xz^2,$$
where the coefficients are integers. I would like to assign values to these coefficients such that $p$ can be factored. For example, ... |
This is a follow-up to the question
on field reductions.
(EDIT: Originally this question used the notation $\{\mathbb{R}(\setminus a)\}$, but now uses $\mathbb{R}(\setminus a)$ instead.)
There isn't a unique largest subfield of $\mathbb{R}$ not containing $a$,
so let $\mathbb{R}(\setminus a)$ denote the set of maxima... |
I know that $x e^x = e$ means $x = 1$, but how do you solve for it?
|
My friend claims that one could reduce the number of states of a given turning machine by somehow blowing up the tape alphabet. He does not have any algorithm though. He only has the intuition.
But I say it's not possible. Else one could arbitrarily keep decreasing the states via the same algorithm and arrive at some ... |
Define $A$ as a nonempty set, $\mathcal{B}:=\{f: A \rightarrow \mathbb{R}: f(A) \text{is bounded} \} ,d_\infty:=\text{sup}\{|f(x)-g(x)|:x \in A\}$.
For which $A$ is $\overline {B_1(0)} \subset \mathcal{B}$ compact?
Notes: ${B_1(0)}$ is the unit sphere. I already proved that $(\mathcal{B},d_\infty)$ is a metric space.
... |
I want to ask, how can I simplify this formula ?
$ e^{-i0.5t}+e^{i0.5t} $
I know that it can be simplify to $\cos(0.5t)$, but I don't know how :/
|
I wonder if someone could shed some light in the following question
Let $(x,y)$ denote the greatest common divisor of $x$ and $y$, and let $x_1,y_1,x_2,y_2$ be integers.
Is the following statement true?
If $(x_1,y_1)=(x_2,y_2)=(x_1^2+y_1^2,x_2^2+y_2^2)=1$, then $$(x_1x_2\pm y_1y_2, x_1y_2\mp x_2y_1)=1$$
If not, what ... |
Let C be a collection of subsets of $\mathbb{N}$ such that $A,B\in C \Rightarrow A \not\subseteq B$. Can C be uncountable?
|
Let $T$ be a bounded linear operator on $L^2(\mathbb R)$. So, let us now assume that $T$ commutes with the translations $\tau_x$. How do I now show that $T$ is given by a convolution with respect to a distribution?
By the way, I know I can probably find the proof somewhere in one of Stein's books, but I would like to p... |
I have a Hermitian matrix $A$ that satisfies some symmetries which I can express via $AS = SA$ for a unitary matrix $S$. Now I am interested in the eigenvectors of $A$, but I want that these eigenvectors also respect my symmetries ( compare to Bloch waves in physics where the eigenvectors are chosen to reflect the tran... |
According to the precedence of logical connectives, operator $\rightarrow$ gets higher precedence than $\leftrightarrow$ operator. But what about associativity of $\rightarrow$ operator?
The implies operator ($\rightarrow$) does not have the associative property. That means that $(p \rightarrow q) \rightarrow r$ is not... |
For a given matrix $A$ the question "is there a vector (or are there vectors)
whose direction remain the same when it is multiplied with the matrix"
leads to the well-known eigenvalue problem $A\mathbf{v}=\lambda\mathbf{v}$,
where $\mathbf{v}$ is an eigenvector of the matrix. Since a vector is fully
described by its di... |
I'm trying to wrap my head around this:
$a \equiv b^{-1}\pmod{y}$
$x \equiv c^{-1} \pmod{y}$
I know all values but $x$ and $y$. Is it possible to solve this problem for $x$?
Can anyone point out a good resource on this kind of problems?
|
For a given group of data, I have both a mean and variance (already calculated). Looking over the raw spread of data, and its numerous outliers, I expect that variance will be high.
My question is -- now that I have my variance calculated, what do I need to compare it to to determine if the variance is significant (i.e... |
I have skimmed this video that I found on mathoverflow:
http://tube.sfu-kras.ru/video/407?playlist=397
At about 15:05 the lecturer wrote down an equality
$\sum F(m_1, \ldots, m_m)z^{m_1}\ldots z^{m_m} = \frac{1}{det(zI - A)}$, where F is given in terms of coefficients of the $m$-by-$m$ matrix $A$. The definition of $F$... |
I know this is a trivial problem but am stuck with this thing.
$A \in \mathbb{R}^{m \times m}$ is a symmetric positive definite matrix and $C \in \mathbb{R}^{m \times p}$
How do I show that the matrix $$B = \begin{bmatrix} A & C \\ C^T & 0 \end{bmatrix}$$ is indefinite?
I can show that there exists a vector $v$ such t... |
Let $P$ and $Q$ be complex polynomials with $deg(P) < deg(Q)$.
Let $Q(z) = (z-z_1)^{k_1} (z-z_2)^{k_2}...(z-z_m)^{k_m}, z_i \in \mathbb{C} \text{ and } k_i \in \mathbb{N}$ be a complete decomposition. Then, I have to prove that there exists a unique exposition of the following form:
$\frac{P(z)}{Q(z)} = \sum\limits_{i... |
If a cyclic subgroup has a proper subgroup of infinite order, how many finite subgroups does it have? Explain. Any help appreciated! Thank you.
|
Euler's formula for connected planar graphs (i.e. a single connected component) states that $v-e+f=2$. State the generalization of Euler's formula for planar graphs with $k$ connected components (where $k\geq1$).
The correct answer is $v-e+f=1+k$, but I'm not understanding the reasoning behind it. Anyone care to share... |
I'm trying to help my son study 8th grade math (in Texas). He's having trouble with the "Nth rule". Is anyone familiar with this terminology? I think that it may be related to the Nth term in a series, but I also think that it has a more specific meaning.
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What's the largest regular tetrahedron (having side length $x$) you can fit inside a sphere with a unit radius?
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At Wikipedia's Erlangen program I read that "quite often, it appears there are two or more distinct geometries with isomorphic automorphism groups". Some examples are given.
But what are examples of geometries nevertheless being
uniquely defined (up to isomorphism)
by the isomorphism class of their
automorphism... |
I'm attempting to understand how B-Spline derivatives work.
To do this, i've generated some data in MATLAB to plot sin(x) over 0 - 2*pi. I then use the splinetool to fit an order 2 spline to the data which it fits perfectly. However, when I look at the 1st derivative of this sine wave data it's max is 0.017 and it's m... |
Hecke operators are things that act on modular forms and give rise to a lot of interesting arithmetical results:
http://en.wikipedia.org/wiki/Hecke_operator
On the other hand on the wikpedia page for Hecke algebras, which should naively be an algebra of Hecke operators, the term seems to acquire very different meanings... |
Problem:
Let $\omega\in\Omega^r(M^n)$ suppose that $\int_\sum \omega = 0$ for every oriented smooth manifold $\sum \subseteq M^n$ that is diffeomorphic to $S^r$. Show that $d\omega = 0$.
Proof:
Assume $d\omega \neq 0$. Then there exists $v_1, \ldots, v_{r+1}\in T_pM$ such that $d\omega_p(v_1, \ldots, v_{r+1}) \neq 0$.... |
Recall that a model category is a complete and cocomplete category with classes of morphisms called cofibrations, fibrations, and weak equivalences. These are closed under composition and satisfy certain axioms, such as lifting properties. Furthermore, according to Mark Hovey (but this apparently varies in the literatu... |
This is some sort of standard puzzle, which many do it using trial and error or brute-force method. The question goes like this, given the numbers, 11,13,31,33,42,44,46 what is choices of numbers (a number can be chosen more than once) so that sum adds up to 100.
Is there a formula like way or approach to this other th... |
Are there any interesting examples of second order stationary processes on ${\mathcal R}^2$ or ${\mathcal R}^3$ that are not isotropic? The book I am looking at has no examples.
Update : I asked this question in mathoverflow, apparently such examples are not easy to come by.
Update: Processes with anisotropic variogram... |
Could you prove or disprove the following statement?
Let
$f\colon[0,1]^2\rightarrow \mathbb R$
be a continuous function. Then
there are continuous functions
$g,\ h\colon [0,1]\rightarrow \mathbb R$ and
$\Phi\colon \mathbb R \to \mathbb R$
such that $$ f(x,y) = \Phi(g(x) + h(y)).$$
(This problem popp... |
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