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If we have a uniform continuous function $f: X \to Y$, then
- $f$ takes a cauchy sequence in $X$ to a cauchy sequence in $Y$.
Now is this statement true: $f$ is uniformly continuous iff, given $\epsilon > 0$, there is an $N > 0$, such that for every $x,y \in I$,where $I$ is an interval ($x \neq y$) we have $$ \B... |
I don't have an answer but only some trivial observations about your question that I post below. It got too big to be left as a comment. If I understand your question correctly, you want to compare the following two problems:
1) Let $X_i$ be i.i.d unit normals and you are interested in $X^{*} = \max(X_i, 1 \leq i \l... |
I don't have an answer but only some trivial observations about your question that I post below. It got too big to be left as a comment. If I understand your question correctly, you want to compare the following two problems:
1) Let $X_i$ be i.i.d unit normals and you are interested in $X^{*} = \max(X_i, 1 \leq i \l... |
http://en.wikipedia.org/wiki/Van_der_Waerden%27s_theorem
The usual approach is to induct on the length of the arithmetic progression, which is difficult to simplify directly to the case of two colors.
Does anyone know a different approach?
Thank you. |
Is there a simpler proof of Van Der Waerden's Theorem when there are only two colors? |
Let $f\in\mathbb{Z}[x]$ be monic and irreducible, let $K=$ splitting field of $f$ over $\mathbb{Q}$. What can we say about the relationship between $disc(f)$ and $\Delta_K$? I seem to remember that one differs from the other by a multiple of a square, but I don't know which is which. On a more philosophical note: why a... |
The two are the same if the roots of $f$ form an integral basis of the ring of integers of the splitting field (e.g. if $f$ is a cyclotomic polynomial) because then, well, they're defined by the same determinant (see <a href="http://en.wikipedia.org/wiki/Discriminant_of_an_algebraic_number_field#Examples">Wikipedia</a>... |
The two are the same if the roots of $f$ form an integral basis of the ring of integers of the splitting field (e.g. if $f$ is a cyclotomic polynomial) because then, well, they're defined by the same determinant (see <a href="http://en.wikipedia.org/wiki/Discriminant_of_an_algebraic_number_field#Examples">Wikipedia</a>... |
I know a few proofs over this theorem (where $\mathfrak{c}$ is the cardinality of $[0,1]$ and $\aleph_0$ is the cardinality of $\mathbb{N}$) where they construct two injections and then use Schröder-Bernstein (via the Cantor set or something like that).
Now I was wondering if I could do something like this:
Defin... |
Consider a function $f_1$ defined by $f_1(x)=1-x+o(x)$ and $f_1(2x)=f_1(x)^2 + 0$. It's simple to find that $f_1(x)=e^{-x}$ (for example by writing series near $x=0$).
Consider a function $f_2$ defined by $f_2(x)=2-x^2+o(x^2)$ and $f_2(2x)=f_2(x)^2-2$. It can be proven that $f_2(x)=2 \cos(x)$.
Is there any formul... |
Generalization of cos: is this function known? |
I'm reading through [wikipedia][1] about Linear Feedback Shift Registers (Specifically Fibonacci LFSRs) and the only restrictions it mentions about the taps are:
- The LFSR will only be maximum-length if the number of taps is even; just 2 or 4 taps can suffice even for extremely long sequences.
- The set of taps ... |
How are the taps of an LFSR found? |
Why $\frac{1}{\cos x}=\sum_{n=1}^\infty \frac{(-1)^n(2n-1)\pi }{x^2-\left (n-\frac{1}{2}\right )^2\pi^2}$? |
In case you don't know what I'm talking about:
http://en.wikipedia.org/wiki/Weierstrass%27s_elliptic_functions
I'm reading Farkas and Kra's "Riemann Surfaces," and they talk briefly about these guys in section 0.2. Please tell me if I am misunderstanding something or saying something wrong or funny. Let $\mathcal... |
A Torus and the Weierstrass P function?? |
In case you don't know what I'm talking about:
http://en.wikipedia.org/wiki/Weierstrass%27s_elliptic_functions
I'm reading Farkas and Kra's "Riemann Surfaces," and they talk briefly about these guys in section 0.2. Please tell me if I am misunderstanding something or saying something wrong or funny. Let $\mathcal... |
Given an infinite dimensional manifold modeled on a Banach space, what does it mean for it to have a Riemannian metric? Does it necessarily mean that it is actually a Hilbert manifold?
My understanding is that Hilbert Spaces have inner products, whereas Banach Spaces just have norms. If a Banach manifold has a Riema... |
A Banach Manifold with a Riemannian Metric? |
I don't have an answer but only some trivial observations about your question that I post below. It got too big to be left as a comment. If I understand your question correctly, you want to compare the following two problems:
1) Let $X_i$ be i.i.d unit normals and you are interested in $X^{*} = \max(X_i, 1 \leq i \l... |
A Conjectura de Argolo é válida? |
I don't have an answer but only some trivial observations about your question that I post below. It got too big to be left as a comment. If I understand your question correctly, you want to compare the following two problems:
1) Let $X_i$ be i.i.d unit normals and you are interested in $X^{*} = \max(X_i, 1 \leq i \l... |
Is there a good quick introduction to knot theory? I am relatively mathematically savvy so any level is appreciated. |
A good quick introduction to Knot Theory? |
Is it really true the conjecture? |
As Michael comments, Colin Adams has a well regarded text called "The Knot Book". Adams has also written a comic book about knot theory called "Why Knot?". It's very humorous but is a genuine introduction to the mathematics involved. This comic book comes with a plastic "rope" that can be knotted, unknotted, and twiste... |
I've just started learning about schemes, so maybe I'm missing something basic.
This is exercise I-24(a):
*****
Take Z = Spec$\mathbb{C}[x]$, let $X$ be the result of identifying the two closed points (x) and (x-1) of |Z|, and let $\phi: Z \to X$ be the natural projection. Let $\mathcal{O}$ be $\phi_* \mathcal{... |
I'm learning about continued fractions, and I've enjoyed them so far, but I'm unsure if I've done the following correctly. I have no real experience with analysis, so I'm not sure if my reasoning is formal enough, or correct. Any feedback would be appreciated.
Let $\xi$ be an irrational number with continued fractio... |
Can we always break an arbitrary field extension $L/K$ into an extension $F/K$ in which the only roots of unity of $F$ are those in $K$, followed by an extension $L/F$ which is of the form $L=F(\\{\omega_i\\})$ where the $\omega_i$ are roots of unity? What if we restrict to $L/K$ separable, or finite? |
I've just started learning about schemes, so maybe I'm missing something basic.
This is exercise I-24(a):
*****
Take Z = Spec$\mathbb{C}[x]$, let $X$ be the result of identifying the two closed points (x) and (x-1) of |Z|, and let $\phi: Z \to X$ be the natural projection. Let $\mathcal{O}$ be $\phi_* \mathcal{... |
I find confusing some examples I have seen. Maybe you can help me to determine what is going on with them.
A Generalized Feedback Shift Register (GFSR) sequence defines a sequence $\\{W_{i}\\}$ satisfying the equation
$$W_{k+p}=c_{0}W_{k}\bigoplus c_{1}W_{k+1}\bigoplus...\bigoplus c_{p-1}W_{k+p-1} \qquad \qquad ... |
Consider a function $f_1$ defined by $f_1(x)=1-x+o(x)$ and $f_1(2x)=f_1(x)^2 + 0$. It's simple to find that $f_1(x)=e^{-x}$ (for example by writing series near $x=0$).
Consider a function $f_2$ defined by $f_2(x)=2-x^2+o(x^2)$ and $f_2(2x)=f_2(x)^2-2$. It can be proven that $f_2(x)=2 \cos(x)$.
Is there any formul... |
Am I misunderstanding the conditions of the problem or is this a counterexample?
![Two ellipses in an X, and two circles between them][1]
[1]: https://i.stack.imgur.com/lFHHH.png
**Later:** I think the answer to your updated question is yes -- or rather, no, any subgraph induced by disjoint convex sets has n... |
As I have heard people did not trust Euler when he first discovered the formula
$$\zeta(2)=\sum_{n=1}^\infty \frac{1}{n^2}=\frac{\pi^2}{6}.$$
However, Euler was Euler and he gave other proofs.
I believe many of you know some nice proofs of this, can you please share it with us?! |
What requirements should a CRC polynomial satisfy to make the CRC catch a maximum of errors? |
What requirements should a CRC polynomial satisfy? |
I hate min and max questions.
I have a function
G(x, y) = 1/x + 4/y + 9/(4-x-y)
I need to prove that G(x,y) has a minimum value of 9
I've partialy derived them for x and y giving
(1) 1/x^2 + 9/(4-x-y)^2 = 0
(2) 4/y^2 + 9/(4-x-y)^2 = 0
This is where I get stuck I'm not entirely sure how to... |
In case you don't know what I'm talking about:
http://en.wikipedia.org/wiki/Weierstrass%27s_elliptic_functions
I'm reading Farkas and Kra's "Riemann Surfaces," and they talk briefly about these guys in section 0.2. Please tell me if I am misunderstanding something or saying something wrong or funny. Let $\wp$ be ... |
is there any way to decipher, manually of course , whether a number (may be large enough) is a prime? |
This was suggested by a problem on FreeNode's #math a little while ago...
Construct a directed graph $\Gamma$ with vertex set the set of compact convex sets in $\mathbb R^2$, and an arrow $A\to B$ if $A$ and $B$ are disjoint and there is a point $a$ in $A$ and a $t>0$ such that $a+(t,0)$ is in $B$.
I guess there ... |
What requirements should a CRC polynomial satisfy to make the CRC catch a maximum of errors?
**edit**
I'm talking about GF(2) polynomials.
|
I've just started learning about schemes, so maybe I'm missing something basic.
This is exercise I-24(a):
*****
Take Z = Spec$\mathbb{C}[x]$, let $X$ be the result of identifying the two closed points (x) and (x-1) of |Z|, and let $\phi: Z \to X$ be the natural projection. Let $\mathcal{O}$ be $\phi_* \mathcal{... |
Does Stirling's formula gives the correct number of digits of n!? |
What requirements should a CRC polynomial satisfy to make the CRC catch a maximum of errors?
**edit**
I'm talking about **GF(2) polynomials**.
As an example of the kind of requirements I'm looking for: I can imagine (but don't know for sure) that a prime polynomial catches more errors than a composite poly... |
Does Stirling's formula give the correct number of digits of n!? |
What requirements should a CRC polynomial of a given degree satisfy to make the CRC catch a maximum of errors?
**edit**
I'm talking about **GF(2) polynomials**.
As an example of the kind of requirements I'm looking for: I can imagine (but don't know for sure) that a prime polynomial catches more errors tha... |
Give ``a > b``, where ``a,b ∈ ℝ``, is it always true that ``a² > b²``? |
If a > b is a² > b² ? |
I find confusing some examples I have seen. Maybe you can help me to determine what is going on with them.
A Generalized Feedback Shift Register (GFSR) sequence defines a sequence $\\{W_{i}\\}$ satisfying the equation
$$W_{k+p}=c_{0}W_{k}\bigoplus c_{1}W_{k+1}\bigoplus...\bigoplus c_{p-1}W_{k+p-1} \qquad \qquad ... |
Lang has written several (very similar) books that discuss these topics. Basically, you're right: Riemannian metrics only make sense when the manifold is modeled on a Hilbert space. There is, however, a notion of psuedo-Riemannian manifold, which makes sense when the manifold is modeled on a self-dual Banach space. ... |
I want to count integral of f(x)=x^2 in range for example 0-10.
What should I read to do it by myself and apply it to other funcs ? |
I would like a family of functions $f_a(x)$ so that $f_a$ is $a\in\mathbb{R}$ fractionally differentiable but not $a+\epsilon$ fractionally differentiable.
Does anyone know such functions which are easily described?
Thanks in advance. |
While reading a probability paper I came across the explanation:
The set of indicator functions $I_i$ form a Boolean algebra that is in one-to-one correspondence with the quotient of the ring $\mathbb{R}[x_1, \ldots, x_n]$ with respect to the relationships $x_i(x_i - 1)=0$ where $x_i$ represents $I_i$.
I have a ... |
Greetings.
Besides Un+1 - Un ? |
Well, this is an exercise problem from *Herstein* which sounds difficult:
- How does one prove that if $|G|>2$, then $G$ has non-trivial automorphism?
The only thing, i know which connects a group with its automorphism is the theorem, $$G \cong \mathcal{I}(G)/Z(G)$$ where $\mathcal{I}(G)$ denotes the *Inner- Au... |
You can find such partial fraction expansions given as examples in many texts on complex analysis. For example, you will find a derivation of this series in Example 1 at [this link][1] to Markushevich and Silverman's book. It uses "Cauchy's theorem on partial fraction expansions", given a few pages earlier at [this li... |
Typically one considers the Hamming distance for the possible lengths of the messages, the larger the better.
See <a href="http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.5.5027&rep=rep1&type=pdf">*Koopman & Chakravarty*, Cyclic Redundancy Code (CRC) Polynomial Selection For Embedded Networks</a>. |
Well, this is an exercise problem from *Herstein* which sounds difficult:
- How does one prove that if $|G|>2$, then $G$ has non-trivial automorphism?
The only thing, i know which connects a group with its automorphism is the theorem, $$G/Z(G) \cong \mathcal{I}(G)$$ where $\mathcal{I}(G)$ denotes the *Inner- Au... |
Given the following game,
there is a hider and a seeker, each game consists of the hider randomly selecting one of two places to hide that is A with probability p and B with 1 - p, and the seeker randomly choses to search in A with probability q and B with 1 - q. If seeker finds hider then game ends, else both rando... |
how do i theoretically find the expected length of the following game? |
I strongly believe that even if something can be done by calculator, a high school student should understand some nuts and bolts of how it's being done.
I spent some time doing volunteer tutoring at CRLS (Cambridge Rindge and Latin School). I still remember the girl at 8th or 9th grade who was studying inequalities... |
Simple question-
Is it possible for a linear programming optimization problem possible to have a degenerate optimal solution whereas the dual has a unique optimal solution? I can't find a scenario where this is possible but intuitively it seems that it is possible. |
How to check if a sequence is upper bounded, lower bounded or bounded? |
In a 365 day year, Joe, Greg & Dean visit the park multiple times for a walk.
- Joe visits every 3rd day
- Greg visits every 5th day
- Dean visits every 7th day
- All three visit the park on the first day of the year
In the year:
- On how many days is Greg alone in the park?
- On how many days will Dean me... |
I'm self-studying field extensions. I ran over an exercise which I can't completely solve. (I haven't yet started studying Galois theory, and I think this exercise isn't meant to be solved using it, just in case):
The problem is:
a) Prove $\sqrt{2}+\sqrt[3]{5}$ is algebraic over $\mathbb{Q}$ of degree 6.
Done: I... |
x^a is fractionally differentiable at $x=0$ with $b$-fractional derivative proportional to $x^{a-b}$ for $b \leq a$. However, if $b > a$, then the fractional derivative blows up at the origin.
Does this answer your question? |
The origin of this question is the identity
$$\sum_{k=0}^n \binom{n}{k} H_k = 2^n \left(H_n - \sum_{k=1}^n \frac{1}{k 2^k}\right),$$
where $H_n$ is the $n$th harmonic number.
Dividing by $2^n$, we have
$$2^{-n} \sum_{k=0}^n \binom{n}{k} H_k = H_n - \sum_{k=1}^n \frac{1}{k 2^k}.$$
The sum on the left can now be... |
I had recently asked the question:
http://math.stackexchange.com/q/8213/2916
which received a very helpful answer. However, I'd like to ask your help again as the formulation of the problem has slightly changed and i'm lost...
**Problem:**
Given two points: $(x_l, y_l)$ and $(x_u, y_u)$
with: $x_l < x_u$ a... |
Logistic function passing through two points AND the origin? |
What is the rule that defines the relation between $a^b\text{ and }b^a$ ? |
What is the rule that defines the relation between $a^b\text{ and } b^a$ ? |
What is the remainder when $4^{44}$ is divided by 44 ? |
Solving the integral $\int_{0}^{\infty} \frac{\sin{x}}{x} \ dx = \frac{\pi}{2}$? |
I was browsing through Wikipedia for some reason or another, and it says that there is a way to do Galois theory without worrying about whether or not polynomials are irreducible. That's a bit surprising to me.
Is Wikipedia wrong (as is often the case when I read something surprising)? If not, where can I hear mor... |
I'm attempting to implement an equation (for calculating [magnetic forces between coils](http://dx.doi.org/10.1109/TMAG.2010.2049026)) that requires the use of elliptic integrals.
Unfortunately these equations require the evaluation of the elliptic integrals far outside their standard parameter range of $0\le k\le 1... |
I was always thinking about this questions when it comes to lottery. lets say you have a number range from 1 - 45. You need 5-6 numbers to win.
Let says you have 2 options
Draw 3 sets of numbers with 10 numbers on each set.
or
1 set of numbers with 19 numbers for that?
Which chance will have higher of winning... |
More choices lesser draw or more draw lesser choices? |
These probably should be comments but I don't have sufficient reputation.
Hint for problem 1: If there are an equal number of red and blue edges at each vertex, what does that tell you about the parity of the degree of each vertex?
Hint for problem 2: If you have a path from x to y passing through z, can you use ... |
You can find such partial fraction expansions given as examples in many texts on complex analysis. For example, you will find a derivation of this series in Example 1 at [this link][1] to Markushevich and Silverman's book. It uses "Cauchy's theorem on partial fraction expansions", given a few pages earlier at [this li... |
pseudocode:
void recursive('k'){ // 'k' and 'i' vertices
sumA = 0;
sumB = 0;
for each non visited 'i' neighbor do{
recursive('i');
sumA = sumA + b['i'];
sumB = sumB + max(a['i'], b['i']);
}
a['k'] = 1 + sumA;
b['k'] = sumB;
}
... |
Suppose n is an integer. What kind of bounds do we know for how close the closest prime p > n will be?
I'd especially appreciate an answer that pushes me in the right direction of proving a good bound for this without showing the proof fully.
Context: I ran into this while coming up to a solution for http://stack... |
**Quick formulation of the problem:**
Given two points: $(x_l, y_l)$ and $(x_u, y_u)$
with: $x_l < x_u$ and $y_l < y_u$,
and given lower asymptote=0 and higher asymptote=1,
what's the logistic function that passes through the two points?
__Explanatory image:__
![logistic][1]
__Other details:__
I... |
Hopefully I understand your question properly. To make sure, here's how I understand it:
There's a lottery, where 5 distinct random numbers will be chosen from 1-45 inclusive. If on your lottery card, you have all 5, you win.
**Options:**
- a) draw 3 cards with 10 random numbers per card
- b) draw 1 card wi... |
Is it really correct my conjecture? |
Does Stirling's formula give the correct number of digits of n!? |
I had recently asked the question:
http://math.stackexchange.com/q/8213/2916
which received a very helpful answer. However, I'd like to ask your help again as the formulation of the problem has slightly changed and i'm lost...
**Problem:**
Given two points: $(x_l, y_l)$ and $(x_u, y_u)$
with: $x_l < x_u$ a... |
While reading a probability paper titled [Gröbner bases and factorisation in discrete probability and Bayes](http://www.springerlink.com/content/j83542h24q561674/) (can't find free version, sorry), I came across the explanation:
The set of indicator functions $I_i$ form a Boolean algebra that is in one-to-one corre... |
I just read this document - [Jacobi's Four Square Theorem](http://www.math.ohio-state.edu/~econrad/Jacobi/sumofsq/sumofsq.html). It shows how to count the number of representations of a number as the sum of four squares.
I can follow the proof but currently it just appears like a brilliant coincidence, so I am wonde... |
What else can the elliptic integral count? |
What could be the possible way to find the `Nth term` of following series where the sign toggles after each triangular number?
`1 -2 -3 4 5 6 -7 -8 -9 -10 11 12 13 14 15 -16 -17 ....`
The series cannot be in a Geometric Progression because there are 4 distinct triangular numbers in the above series. |
I'm attempting to implement an equation (for calculating [magnetic forces between coils](http://dx.doi.org/10.1109/TMAG.2010.2049026)) that requires the use of elliptic integrals.
Unfortunately these equations require the evaluation of the elliptic integrals far outside their standard parameter range of $0\le m\le 1... |
How to check if a sequence is bounded from above, bounded from below or bounded? |
I'm attempting to implement an equation (for calculating [magnetic forces between coils](http://dx.doi.org/10.1109/TMAG.2010.2049026), eqs (22–24) in the linked paper) that requires the use of elliptic integrals.
Unfortunately these equations require the evaluation of the elliptic integrals far outside their standar... |
Consider the set $F_n: = [n]^{[n]}$ of all functions $f:[n] \rightarrow [n]$, $[n] = \lbrace 1,2,...,n\rbrace$. It is well known that $|F_n| = n^n$.
> But what is the number of isomorphism types?
Can someone give the generating function? |
How do I make x = out of this equation?
1/x^2 + 9/(4-x-y)^2 = 0
I need to get this into x="blah"?
Also. How do you guys get that funky formatting? |
You ask for a function. A function is a mapping from one set to
another set but you neglect to specify which sets. Your list of
$x$-values consists of small nonnegative integers together with
an ellipsis. From this, one might surmise that your intended
domain is $\mathbb{N}_0$, the set of nonnegative integers. Like... |
How do I make $x =$ out of this equation?
$$1/x^2 + 9/(4-x-y)^2 = 0$$
I need to get this into $x=$"blah"?
Also. How do you guys get that funky formatting? |
How do I make $x =$ out of this equation?
$$-1/x^2 + 9/(4-x-y)^2 = 0$$
I need to get this into $x=$"blah"?
Also. How do you guys get that funky formatting? |
Students should learn *math*. You know, the stuff that most of us *do*. When we use the word "math" we certainly don't have in our minds adding, multiplying, factoring polynomials, or even calculus. So why do we teach children that this is what math is?
This view has been in several published places like Devlin's *T... |
claims arrive in an insurance portfolio, according to a homogenous Poission process X(t), t>=0. Assume that each of the 12 months in a year has exactly 30 days. Calculate the value of the following conditional probability:
P{X(2) = 3, X(3) = 5 | X(4) = 5}
where X(k) is the number of claims during the first k months o... |
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