instruction stringlengths 12 30k |
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The *vertex* of a parabola is the point on the axis of symmetry that intersects it ([Wikipedia][1]).
The point $(x,y)=\left( 0,0\right) $ is the vertex of the parabola given by $y=ax^{2}$.
By making the change of coordinates
$X=x+\dfrac{b}{2a},Y=y+\dfrac{b^{2}-4ac}{4a}$,
equation $y=ax^{2}+bx+c$ is transfor... |
On a 2D plane, line X is at x radians angle, an incoming light travels at y radians angle, how to calculate the angle of the outgoing light reflected off line X? How to do this in a way to cover all cases. |
How to calculate reflected light angle? |
An alternative approach, which isn't necessarily the *best* in and of itself, but stick with me ...
Instead of using the fact that the vertex is the point where a *vertical* line cuts the parabola exactly in half, we'll use the fact that it's also the point where a *horizontal* line *just* brushes the curve, meeting... |
On a 2D plane, line X is at x radians angle, an incoming light travels at y radians angle, how to calculate the angle of the outgoing light reflected off line X? How to do this in a way to cover all cases.
Edit: I was trying to figure out [Project Euler problem 144][1].
[1]: http://projecteuler.net/index.php?... |
I've been reading about tropical geometry and many papers reference polyhedral fans. I feel like I have a decent intuitive picture of what they are from reading articles but I still haven't been able to guess the general definition. All the ones I've encountered have been systems of linear inequalities, so that is my b... |
What is the rigorous definition of polyhedral fan? What are some good resources to learn about them? What context do they arise naturally in? |
A polyhedral cone is a subset of a real vector space which is the intersection of finitely many closed half spaces. (The defining planes of these half spaces must pass through $0$.) A fan is a finite set $F$ of polyhedral cones, all living in the same vector space, such that
(1) if $\sigma$ is a cone in $F$, and $\t... |
What is the the most _educational_ software for high school and college math?
Not the one that just gives you the answer, but has any of the following:
<ul>
</ul>
- provides step-by-step solutions;
- can verify your step-by-step solution;
- can give you hints;
- can generate problems similar to given one;
</... |
Both matrix multiplication and quaternion multiplication are non-commutative; hence the use of terms like "premultiplication" and "postmultiplication". After encountering the concept of "quaternion matrices", I am a bit puzzled as to how one may multiply two of these things, since there are at least four ways to do thi... |
If you want to avoid algebraic geometry and topology, then you will probably be forced to use algebra. This is okay, because the things you ask about have algebraic interpretations.
First of all, it probably helps to write the function field as $k(x,y),$ where $x$ and $y$ are related by some equation $f(x,y) = 0$. ... |
I need to find eigenvalues/eigenvectors of different kinds of nxn matrices. For example as listed below. What is the typical process? Should I always go by the route of finding eigenvalues by finding roots of characteristic polynomial and then getting eigenvectors by solving (A-lambda*I)*x = 0?
[,1] [,2] [,... |
I recently noticed a few things in some recent questions on MO:
1) the fundamental group of $S^2$ minus, say, 4 points, is $\langle a,b,c,d\ |\ abcd=1\rangle$.
2) The fundamental group of a torus minus a point is $\langle a,b,c\ |\ [a,b]c=1\rangle$.
I was just wondering if you have a manifold $M$, and you know... |
I am not sure if this question is appropriate here but would appreciate any help. Let $n>2$ and $x_1,\cdots,x_n$ be real numbers whose product is $1$. What is the infimum of:
$$A = \sum_{i=1}^n \frac{1}{(1+x_i)^2} $$
and
$$B = \sum_{i=1}^n \frac{1}{(1-x_i)^2} $$?
Thanks in advance! |
## Background ##
It is a standard and important fact in basic calculus/real analysis that a continuous function on a compact metric space is in fact uniformly continuous. That is, suppose (X,d) is a compact metric space and f: X → ℝ is such that for every x∈X and ε>0 there exists δ>0 suc... |
I am not sure if this question is appropriate here but would appreciate any help. Let $n>2$ and $x_1,\cdots,x_n$ be real numbers. What is the infimum of:
$$A = \sum_{i=1}^n \frac{1}{(1+x_i)^2} $$
subject to the constraint $\prod_{i=1}^nx_i=1$?
Similar question for:
$$B = \sum_{i=1}^n \frac{1}{(1-x_i)^2} $... |
This seems counterintuitive, but 22/7 is closer to pi than 3.14=314/100 which has a significantly greater denominator.
> Why is 22/7 a better approximation for pi than 3.14?
This has important implications: e.g. Should "pi day" by the 14th of March or 22nd of July? |
Why is 22/7 a better approximation for pi than 3.14? |
How would you find eigenvalues/eigenvectors of a nxn matrix where each diagonal entry is scalar 'd' and all other entries are '1' ? I am looking for a decomposition but cannot find anything for this. For example:
> a
[,1] [,2] [,3] [,4]
[1,] 2 1 1 1
[2,] 1 2 1 1
[3,] ... |
I need to find eigenvalues/eigenvectors of different kinds of nxn matrices. For example as listed below. What is the typical process? Should I always go by the route of finding eigenvalues by finding roots of characteristic polynomial and then getting eigenvectors by solving (A-lambda*I)*x = 0?
[,1] [,... |
Although whether \begin{aligned}P & = NP\end{aligned} is important from theoretical computer science point of view, but I fail to see any practical implication of it.
Suppose that we can prove all questions that can be verified in polynomial time have polynomial time solutions, it won't help us in finding the actual... |
Although whether $$ P = NP $$ is important from theoretical computer science point of view, but I fail to see any practical implication of it.
Suppose that we can prove all questions that can be verified in polynomial time have polynomial time solutions, it won't help us in finding the actual solutions. Conversely, ... |
Suppose we have a functional question, such as the [one here][1]. Often a key step in these problems is to try to equate or chain results together ([see my hint][2]). From looking at these questions, it seems that there is often only a finite number of ways that we can attempt to join them together. Can this fact be ex... |
I have heard that some issues in group theory prevent classifying all manifolds upto homotopy using the fundamental group invariant. Does anyone know what are those issues?
Thanks,
K. |
Why is fundamental group insufficient to classify manifolds? |
Akhil showed that the [Cardinality of set of real continuous functions][1] is the same as the continuum, using as a step the observation that functions that agree at rational points must agree everywhere, since the rationals are dense in the reals.
This isn't an obvious step, so why is it true?
[1]: http://ma... |
School starts in 1.5 weeks and I'm taking a course in Symbolic Logic. Here is a quote from the syllabus:
> You need to keep a notebook with all your homework in it. This notebook can take one of two forms. 1. A looseleaf (or some kind of removable/replaceable page notebook) or 2. A folder (aka subdirectory) on your ... |
How to do homework for a class in Symbolic Logic on my computer? |
I assume you intend for all the $x_i$ to be positive. The infimum is $\min(n/4, 1)$.
This is a good example of how to use convexity to minimize an expression.
Make the change of variables $x_i = e^{t_i}$, so $\sum t_i=0$. Define $f(t) = 1/(1+e^t)^2$. We want to minimize $\sum f(t_i)$.
Now, $f(t)$ is convex f... |
Let $X $ be a set, $|X|=n$ and $G$ be a group with a 2-transitive action on $X$.
what can be said about the size of $G$?
A simple lower bound is the number of blocks in a $\left(2,n,n^{2}\right)$ steiner system, which is $n^{2}+n$, is there a tighter lower bound? |
## Background ##
It is a standard and important fact in basic calculus/real analysis that a continuous function on a compact metric space is in fact uniformly continuous. That is, suppose (X,d) is a compact metric space and f: X → ℝ is such that for every x∈X and ε>0 there exists δ>0 suc... |
Let $X $ be a set, $|X|=n$ and $G$ be a group with a 2-transitive action on $X$.
what can be said about the size of $G$?
|
I assume you intend for all the $x_i$ to be positive. The infimum is $\min(n/4, 1)$.
This is a good example of how to use convexity to minimize an expression.
Make the change of variables $x_i = e^{t_i}$, so $\sum t_i=0$. Define $f(t) = 1/(1+e^t)^2$. We want to minimize $\sum f(t_i)$.
Now, $f(t)$ is convex f... |
What are some interesting mathematical paradoxes?
What I have in mind are things like the Banach-Tarski paradox, Paradox of Zeno of Elea, Russel's paradox, etc..
|
Mathematical paradoxes? |
The simple group of order 60 can be generated by the permutations (1,2)(3,4) and (1,3,5), but all you need to do is square the first one and it becomes the identity. Can't we find a version of the simple group where the elements of small order can be ignored?
For a group H, define Ω<sub>n</sub>(H) to be the subgrou... |
What are some interesting mathematical paradoxes?
What I have in mind are things like the Banach-Tarski paradox, Paradox of Zeno of Elea, Russel's paradox, etc..
Edit: As an additional restriction, let us focus on paradoxes that are not already in the list at:
https://secure.wikimedia.org/wikipedia/en/wiki/Cat... |
As a complement to Robin Chapman's answer:
Since G is 2-transitive, its order is divisible by n*(n-1), not just bounded below by, and so it would be interesting to bound |G|/(n*(n-1)).
If n is not a prime power, then it is quite possible for the lower bound to be huge. The smallest reasonable non-prime, n=6, has... |
## Background ##
It is a standard and important fact in basic calculus/real analysis that a continuous function on a compact metric space is in fact uniformly continuous. That is, suppose (X,d) is a compact metric space and f: X → ℝ is such that for every x∈X and ε>0 there exists δ>0 suc... |
Noncommutative algebraic geometry is a developing field. Things have not yet got the final form as in commutative geometry.
But one might wonder whether things are any better in the case of skew-fields, ie division rings, ie possibly noncommutative rings in which each nonzero element has a multiplicative inverse. ... |
One example not in the above list is [Goodstein's theorem][1], a highly nonintuitive concrete number theoretic theorem which is unprovable in Peano arithmetic (or, similarly, the Hercules vs. Hydra game). They essentially encode induction up to the ordinal $\epsilon_0 = \omega^{\omega^{\omega^{\cdot^{\cdot^{\cdot}}}}}$... |
I know a little bit of the theory of compact Riemann surfaces, wherein there is a very nice divisor -- line bundle correspondence.
But when I take up the book of Hartshorne, the notion of [Cartier divisor](http://planetmath.org/encyclopedia/CartierDivisor.html) there is very confusing. It is certainly not a direct ... |
I know a little bit of the theory of compact Riemann surfaces, wherein there is a very nice divisor -- line bundle correspondence.
But when I take up the book of Hartshorne, the notion of [Cartier divisor](http://planetmath.org/encyclopedia/CartierDivisor.html) there is very confusing. It is certainly not a direct ... |
Akhil showed that the [Cardinality of set of real continuous functions][1] is the same as the continuum, using as a step the observation that continuous functions that agree at rational points must agree everywhere, since the rationals are dense in the reals.
This isn't an obvious step, so why is it true?
[1]... |
An extreme form of constructivism is called *finitisim*. In this form, unlike the standard axiom system, infinite sets are not allowed. There are important mathematicians, such as Kronecker, who supported such a system. I can see that the natural numbers and rational numbers can easily defined in a finitist system, ... |
[GNU Emacs + AUCTeX][1] works fine for me.
[1]: http://www.youtube.com/watch?v=atIhEDQSI1U |
In [the comment thread of an answer][1], I said:
> The computable numbers are based on the intuitionistic continuum, and are not finitary.
To which T.. replied:
> Computable numbers are not based on the intuitionistic continuum.
This disagreement contains, I think, a good example of a philosophical question.... |
Are the computable reals finitary? |
I have seen a lot of places where the group cohomology when a group acts on a module, is extensively used. But beyond seeing the definition and some claims of partial results, I havent seen any uses of the case when we replace action on a module by action on noncommutative group. It seems hard to believe that much can ... |
What is the the most _educational_ software for high school and college math?
Not the one that just gives you the answer, but has <strike>any of</strike> the following:
_Edit: Software mentioned in answer added in brackets_
<ul>
</ul>
- provides step-by-step solutions [_WolframAlpha_];
- can verify your step-... |
Given a polynomial over a field, what are the methods to see it is irreducible? Only two comes to my mind now. First is Eisenstein criterion. Another is that if a polynomial is irreducible mod p then it is irreducible. Are there any others? |
Here are three links you may find useful:
http://wims.unice.fr/wims/wims.cgi?lang=en&+session=YV6390EF2C.1&+module=home
http://www.numbertheory.org/php/php.html
http://www.alpertron.com.ar/NUMBERT.HTM |
To better understand the Eisenstein and related irreducibility tests you should learn about Newton polygons. It's the **master theorem** behind all these related results. A good place to start is Filaseta's notes - see the links below. Note: you may need to write to Filaseta to get access to his interesting book [1] on... |
We have a pool of items. Each item has a set of characteristics. Each characteristic is either a string, integer, or floating-point number. Each item is also of a particular type. Types are hierarchical.
We also have a set of bins. Each bin has a requirement, which is a boolean expression comprised of atomic ex... |
We have a pool of items. Each item has a set of characteristics. Each characteristic is a string, integer, or floating-point number. Each item is also of a particular type. Types are hierarchical.
We also have a set of bins. Each bin has a requirement, which is a boolean expression comprised of atomic expressio... |
I would like to know how to explicitly prove that Riemann Curvature,Ricci Curvature, Sectional Curvature and Scalar Curvature are left invariant under an isometry.
I can't see this explained in most books I have looked at.
They atmost explain preservation of the connection.
I guess doing an explicit proof for... |
I am trying to figure out what the correct notation for a multiplicative algebra should be. I've seen a bunch of weird and conflicting ways for writing these things in the literature, but they are all either too specialized or bring in extra baggage.
To be more specific, given a ring [; (R, +, \times, 0, 1) ;], a... |
I am trying to figure out what the correct notation for a multiplicative algebra should be. I've seen a bunch of weird and conflicting ways for writing these things in the literature, but they are all either too specialized or bring in extra baggage.
To be more specific, given a ring $(R, +, \times, 0, 1)$, and a... |
I have seen at many places the notions that Lie Algebras are infinitesimal objects and they look really close at a point. But I never understood this. They are abstract algebraic objects different from rings in that they are equipped with a weird sort of product and a weird Jacobi identity. Any hints on how to make thi... |
The following integral,
$$ \int_0^1 \frac{x^4(1-x)^4}{x^2 + 1} \mathrm{d}x = \frac{22}{7} - \pi $$
is clearly positive, which proves that $\pi < 22/7$.
Is there a similar integral which proves $\pi > 333/106$? |
Is there an integral that proves $\pi > 333/106$? |
If your CS friends are like me, they might still find the answers above a little overwhelming. So you could start as follows:
First, show them the [Wiki's divisor](http://en.wikipedia.org/wiki/Divisor) page (it always works!). Then explain to them that by the [fundamental theorem of arithmetic](http://en.wikipedia.... |
Can every compact metric space be realized as the continuous image of a cantor set? |
The following are unrelated concepts:
1. Finitism
2. The "intuitionistic continuum" (does that mean Weyl's book? the real numbers as defined in Bishop's book on constructive analysis?)
3. Computable real numbers
For example, you can read the definition of computable real numbers under a classical or an "int... |
Below is another method for irreducibility testing - excerpted from one of my old sci.math posts.
In 1918 Stackel published the following simple observation:
**THEOREM** If $ p(x) $ is a composite integer coefficient polynomial
then $ p(n) $ is composite for all $|n| > B $, for some bound $B$,
in fact $ ... |
I am trying to figure out what the correct notation for a multiplicative algebra should be. I've seen a bunch of weird and conflicting ways for writing these things in the literature, but they are all either too specialized or bring in extra baggage.
To be more specific, given a ring R and an R-module, M, with a ... |
The [Phragmen-Lindelöf theorem](http://eom.springer.de/p/p072630.htm) gives a consequence of the Riemann hypothesis, viz, the [Lindelöf hypothesis](http://en.wikipedia.org/wiki/Lindel%C3%B6f_hypothesis). As such this is weaker than Riemann hypothesis; but it is still considered that even a proof of this weaker result w... |
I am A Neophyte Amateur Mathematician. I have beed reading a lot about Journals and the topic of peer-review in Mathematics Journals. Does one have to have Professional crediendtials or have a Doctorate in order to Publish in Peer-review Mathematics Jourals or just the desire to competenyly solve Mathematical problems?... |
Does a Person Need a Mathematics Degree in order to Publish in a Mathematics jounal? |
What software do you use to **accurately draw geometry diagrams**? |
If you are a mathematician and wants to use such pictures for your notes or presentations using TeX files, use the pstricks package enhanced for use of easy by the package [LaTeXdraw](http://latexdraw.sourceforge.net/).
Then there is a very useful linux program called [xfig](http://www.google.com/url?sa=t&source=... |
What are some interesting mathematical paradoxes?
What I have in mind are things like the Banach-Tarski paradox, Paradox of Zeno of Elea, Russel's paradox, etc..
|
What are some interesting mathematical paradoxes?
What I have in mind are things like the Banach-Tarski paradox, Paradox of Zeno of Elea, Russel's paradox, etc..
Edit: As an additional restriction, let us focus on paradoxes that are not already in the list at:
https://secure.wikimedia.org/wikipedia/en/wiki/Cat... |
What is a simple example, without getting into the mess of triangulated categories, of an additive category that is not abelian? |
I'm optimizing a family of objectives over x1,...,xn with a single constraint that x's add up to 1, and when using method of Lagrange multipliers, the single multiplier always ends up 0 for members of this family...does this tell us anything interesting about these objectives? |
Meaning of Lagrange multiplier always being 0? |
I'm optimizing a family of objectives over x1,...,xn with a single constraint that x's add up to 1, and when using method of Lagrange multipliers, the single multiplier always ends up 0 for members of this family...does this tell us anything interesting about these objectives?
Edit: this is in process of showing con... |
How are operations such as the sum, the product, the quotient, exponentiation, etc. of random variables solved or approached?
This question: <a href="http://math.stackexchange.com/questions/1416/algebra-of-random-variables">Algebra of Random Variables?</a> addresses a similar problem but it starts one step further: ... |
How to work with operations/algebra of random variables? |
How are operations such as the sum, the product, the quotient, exponentiation, etc. of random variables solved or approached?
This question: <a href="http://math.stackexchange.com/questions/1416/algebra-of-random-variables">Algebra of Random Variables?</a> addresses a similar problem but starts one step further: kno... |
I'm optimizing a family of objectives over x1,...,xn with a single constraint that x's add up to 1, and when using method of Lagrange multipliers, the single multiplier always ends up 0 for members of this family...does this tell us anything interesting about these objectives?
Update, problem formulation: this is in... |
The previous version of this answer was incorrect. It has generously been accepted by Nate Eldredge on the ground that "it definitely contains the right idea". This previous version has better remain forgotten forever. The reader should consult Nate's answer, which is correct and complete. I'll just try here to spell o... |
In [the comment thread of an answer][1], I said:
> The computable numbers are based on the intuitionistic continuum, and are not finitary.
To which T.. replied:
> Computable numbers are not based on the intuitionistic continuum.
This disagreement contains, I think, a good example of a philosophical question:... |
The previous version of this answer was incorrect. It has generously been accepted by Nate Eldredge on the ground that "it definitely contains the right idea". This previous version has better remain forgotten forever. The reader should consult Nate's answer, which is correct and complete. I'll just try here to spell o... |
The previous version of this answer was incorrect. It has generously been accepted by Nate Eldredge on the ground that "it definitely contains the right idea". This previous version has better remain forgotten forever. The reader should consult [Nate's answer](http://math.stackexchange.com/questions/1618/is-the-natural... |
A prime number not equal to 2 and 5 can't have last digit equal to 2,4,5,6 and 8.
Is it true that this is the only restriction on last digits of prime numbers?
I mean if its true that for any sequence of digits with last digit not equal to 2,4,5,6 and 8
there exists a prime number with given sequence of digits?
|
There have already been a lot of good advices. Let me stress the importance of chosing the journal right. There are too difficulties here:
1. The journal is interested in your subject.
2. Your paper fits the quality standards of the journal.
As for (1) there is an easy algorithm: you sure cite other works in y... |
What is the probability of getting yahtzee using N dices with X sides in Y throws?
Which side of the dice one gets yahtzee with doesn't matter (e.g. it doesn't matter if it is ones or twos etc.). I also assume perfect strategy is used, that is, after each throw one saves the number it is the most of (and if equal, j... |
What is the probability of getting yahtzee? |
What is your question? This is well-known stuff. It appears for instance in section 4.5 of Apostol's *Introduction to Analytic Number Theory*, which also contains an upper bound. You could at least say where you got yours from. (This should have been a comment, not an answer, but I don't have enough reputation for addi... |
We should have $\mathbb{Q} = \lim_\rightarrow \frac 1n \mathbb{Z}$ but a few things are confusing me. Since the index category is a set, we should get the coproduct: $\bigsqcup \frac 1n \mathbb{Z}$. But in this coproduct we have $1/2 \neq 2/4$ which clearly is wrong. This make me suspect that the index category is not ... |
It's easy for scalars, $(\exp(a(x)))' = a' e^a$. But can anything be said about matrices? Do $A(x)$ and $A'(x)$ commute such that $(\exp(A(x)))' = A' e^A = e^A A'$ or is this only a special case? |
Is there a general formula for the derivative of $\exp(A(x))$ when $A(x)$ is a matrix? |
I'm working on a problem from a past exam and I'm stuck, so I'm asking for help.
Here it is: A=(1/2)*[1,1,1,1;1,1,-1,-1;1,-1,1,-1;1,-1,-1,1], find $\mathbf A^{-1}$.
My problem isn't the inverse matrix itself. We just get the determinant, see if it's zero or not, get the adjoint matrix and divide it by determinan... |
How to get inverse of this matrix using least amount of space? |
I'm not sure a 'power equation' is the right name for the equation I'd like to know more about (and, specifically, about its solutions), but I don't know the 'proper' way to name it.
The simplest 'power polynomial' is : $P(x) = x^x $. The simplest 'power equation is: (1) $x^x = c$ for some $c \in \mathbb{R} $. Wha... |
I'm working on a problem from a past exam and I'm stuck, so I'm asking for help.
Here it is:
[1, 1, 1,1]
[1,1,-1,-1]
A=(1/2)*[1,1,-1,-1]
[1,1,-1,-1]
find $\mathbf A^{-1}$.
My problem isn't the inverse matrix itself. We just... |
I'm working on a problem from a past exam and I'm stuck, so I'm asking for help.
Here it is: $A = \frac12 \left[\begin{array}{rrrr}
1 & 1 & 1 & 1 \\\\
1 & 1 & -1 & -1 \\\\
1 & -1 & 1 & -1 \\\\
1 & -1 & -1 & 1
\end{array}\right]$ find $\mathbf A^{-1}$.
My problem isn't the inverse matrix itself. We just... |
I'd like to write down proofs. I use first-order logic and natural deduction. Formulas tend to be long, too long for LaTeX. Writing formulas with the computer is also a slow process. Handwriting is a lot faster and more natural, but I can't trust paper. Time, water, dirt, fire are its enemies. I could write down someth... |
How to write down proofs? |
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