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Why is the integral of $x$ from $-1$ to $1$ $x^2/2$ rather than $x^2$? |
Good question, and one that is timely for me as I'm teaching calc II next semester for the first time in several years.
An application of Taylor series that sometimes comes up in an intro to ODEs course is that of series solutions to differential equations. I usually don't spend more than a day or two on the topi... |
Same as title.
What is the difference between Godel's Completeness and Incompleteness Theorems? |
What is the difference between Godel's Completeness and Incompleteness Theorems? |
What are some good ways to motivate the material on infinite series that appears at the end of a typical American Calculus II course?
My students in this course are generally from biochemistry, computer science, economics, business, and physics (with a few humanities folks taking the course for fun) - not just math ... |
The sum of the series $\displaystyle\sum^\infty_{n=1}(\dfrac{1}{n})^{2}=\dfrac{\pi^2}{6}$ is a classical result due to Euler. Several proofs are given in the answers to [this question][1].
----------
PS. [Here][2] Robin Chapman collects 14 proofs.
----------
PPS. The improper double integral $$\int_... |
Firt,
to understand my question, first checkout this one:
http://math.stackexchange.com/questions/14573/calculating-percentages-for-taxes
Second,
consider that I'm a layman in math.
So, after trying to understand the logic used to get the final result. I was wondering:
Why multiply $20,000 by 100 and t... |
First,
to understand my question, first checkout this one:
http://math.stackexchange.com/questions/14573/calculating-percentages-for-taxes
Second,
consider that I'm a layman in math.
So, after trying to understand the logic used to get the final result. I was wondering:
Why multiply $20,000 by 100 and ... |
When you write something like $X - 17\% = 20,000$ you are mixing quantities that should not be mixed: your $X$ and your $20,000$ are in dollars, but your $17$ is in percentage points. So the expression does not really make any sense, just like $1\text{ gallon of milk} + 2\text{ plates of cookies}$ does not equal $3$. L... |
I am looking for the the method of constructing the midpoint of two points in spherical geometry. The only tools allowed for the construction are a pair of spherical compasses and a spherical ruler.
In Eclidean geometry constructing the midpoint is relatively easy. We are looking for the midpoint of points A and B. ... |
How to construct the midpoint in spherical geometry? |
First,
to understand my question, checkout this one:
http://math.stackexchange.com/questions/14573/calculating-percentages-for-taxes
Second,
consider that I'm a layman in math.
So, after trying to understand the logic used to get the final result. I was wondering:
Why multiply $20,000 by 100 and then d... |
The Rule of Three is just doing the same thing you are doing: if $\$20,000$ is $83$%, then $\frac{20000}{83}$ is $1$% of the original. Since $X$ is the original, then $\frac{X}{100}$ is **also** $1$% of the original. That means that the two quantities are equal, so we have:
$$\frac{20,000}{83} = \frac{X}{100}.$$
From... |
The "word" description is in fact the guide to a proof. The only problem is showing that the process you describe is in fact optimal. This is done by induction.
we want to prove the formulas you give are the optimal solution.
The case $n=0$ is immediate.
Now: to prove the $k+1$ case assuming the $k$th case (s... |
The "word" description is in fact the guide to a proof. The only problem is showing that the process you describe is in fact optimal. This is done by induction.
we want to prove the formulas you give are the optimal solution.
The case $n=0$ is immediate.
Now: to prove the $k+1$ case assuming the $k$th case (s... |
I haven't found any useful method to solve the following problem: Prove that if $x,y,z\in\mathbb{Z}$ and $x^3+y^3=3z^3$ then $xyz=0$.
Source: http://www.artofproblemsolving.com/Forum/viewtopic.php?f=56&t=382377 |
There's a puzzle where you have 3 houses and 3 utilities. You must draw lines so that each house is connected to all three utilities. However, I'm fairly sure that the puzzle is impossible. How is this proved? |
3 Utilities | 3 Houses puzzle? |
There's a puzzle where you have 3 houses and 3 utilities. You must draw lines so that each house is connected to all three utilities, but the lines cannot overlap. However, I'm fairly sure that the puzzle is impossible. How is this proved? |
Couldn't you view integration as a Reimann sum? That seems like a pretty direct relationship.
Some CS / computer engineering things:
1. [Newton-Raphson][1] division is a clever way that you can do division with only using multiplication / addition. Telling people that that's how calculators do division might be ... |
It's not hard to see this is impossible. Connect two houses to the three utilities, and you will essentially have a square with one diagonal drawn. The two corners joined are two of the houses, the other two corners and the midpoint of the diagonal are the utilities. (The actual shape may look distorted from this, but ... |
Couldn't you view integration as a Reimann sum? That seems like a pretty direct relationship.
Some CS / computer engineering things:
1. [Newton-Raphson][1] division is a clever way that you can do division with only using multiplication / addition. Telling people that that's how calculators do division might be ... |
Here's how I **approximated** an answer:
We know player-2 usually wins. The expected value of player-2 after 9 games is 63, just over 60. So the expected number of turns the game will last is about 9.
Let's consider another version of this game which is played for 9 turns exactly and the winner is the one who got t... |
It's not hard to see this is impossible. Connect two houses to the three utilities, and you will essentially have a square with one diagonal drawn. The two corners joined are two of the houses, the other two corners and the midpoint of the diagonal are the utilities. (The actual shape may look distorted from this, but ... |
Here's how I **approximated** an answer:
We know player-2 usually wins. The expected value of player-2 after 9 games is 63, just over 60. So the expected number of turns the game will last is about 9.
Let's consider another version of this game which is played for 9 turns exactly and the winner is the one who got t... |
These are the <a href="http://en.wikipedia.org/wiki/Stirling_numbers_of_the_second_kind">Stirling numbers of the second kind</a>. I am not sure what you mean by a famous problem that this sequence solves; you seem to already have given the standard combinatorial interpretation.
The Stirling numbers of the second ki... |
Here's how I **approximated** an answer:
We know player-2 usually wins. The expected value of player-2 after 9 games is 63, just over 60. So the expected number of turns the game will last is about 9.
Let's consider another version of this game which is played for 9 turns exactly and the winner is the one who got t... |
Is a uniformly continuous function vanishing at $0$ bounded by $a|x|+c$? |
The sum of the series $\displaystyle\sum^\infty_{n=1}(\dfrac{1}{n})^{2}=\dfrac{\pi^2}{6}$ is a [classical result due to Euler][1]. Several proofs are given in the answers to [this question][2].
----------
PS. [Here][3] Robin Chapman collects 14 proofs.
----------
PPS. The improper double integral $$... |
It's not hard to see this is impossible. Connect two houses to the three utilities, and you will essentially have a square with one diagonal drawn. The two corners joined are two of the houses, the other two corners and the midpoint of the diagonal are the utilities. (The actual shape may look distorted from this, but ... |
Linear Algebra - True or False if Ay = B and Az=b then y = z? Why or why not? you may give a counter example to prove it false. |
Linear Algebra - True or False if Ay = B and Az=b then y = z? Why or why not? |
Linear Algebra - True or False if $Ay = B$ and $Az=B$ then y = z? Why or why not? you may give a counter example to prove it false. |
Linear Algebra - True or False if Ay = B and Az=B then y = z? Why or why not? |
I don't know what only one root between -1 and 0 actually means. I count these possibilities:
1. $-1<x_1=x_2<0$
2. $x_1<-1<x_2<0$
3. $-1<x_1<0<x_1$
conditions to solve 1 alone would be:
$a.f(-1)<0$,
$a.f(0)<0$ and
$\Delta >= 0$
conditions to solve 2 alone would be:
$a.f(-1)<0$,
$a.f(0)>0$,
$\Delta>0$ a... |
I am looking at the following problem:
> Evaluate $m$ in the equation $(3m-2)x^2 + 2mx + 3m = 0$ so that it has only one root between $-1$ and $0$.
I don't know what only one root between -1 and 0 actually means. I count these possibilities:
1. $-1<x_1=x_2<0$
2. $x_1<-1<x_2<0$
3. $-1<x_1<0<x_2$
co... |
I am looking at the following problem:
> Find all values of $m$ for which the equation $(3m-2)x^2 + 2mx + 3m = 0$ shas only one root between $-1$ and $0$.
I don't know what only one root between -1 and 0 actually means. I count these possibilities:
1. $-1<x_1=x_2<0$
2. $x_1<-1<x_2<0$
3. $-1<x_1<0<x_2... |
I am looking at the following problem:
> Find all values of $m$ for which the equation $(3m-2)x^2 + 2mx + 3m = 0$ shas only one root between $-1$ and $0$.
I don't know what only one root between -1 and 0 actually means. I count these possibilities:
1. $-1<x_1=x_2<0$
2. $x_1<-1<x_2<0$
3. $-1<x_1<0<x_2... |
Is the intersection of a decreasing sequence of countably many stationary subsets of $\omega_1$ always stationary? |
I'm preparing a talk on lattice point enumeration in polytopes (Ehrhart-Macdonald Theory), and I'd like to have an introduction with a few motivational problems/results which arise from the application of lattice point methods. (The audience consists of junior and senior faculty across the mathematical disciplines.) A ... |
Is the intersection of a decreasing sequence of countably many stationary subsets of $\omega_1$ always stationary?
It seems elementary, but I fail to find the answer in textbooks. |
You may well know this paper already but I think it is very nice:
http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.87.2267&rep=rep1&type=pdf |
Is it true or false that a group of order 12 always has a normal 2-sylow subgroup?
I have a hunch it is false.. |
The "rule of three" is an ancient ad-hoc mindless rote rule of inference that is best ignored. Instead, you should strive to learn the general principles behind it - namely, the laws of fraction arithmetic. Let's consider the example at hand. You seek the number of dollars $\rm\:X\:$ such that when decremented by $\:1... |
Consider an sequence of unique values. Each value is composed of a surname and a first name in that order of significance. Based on that, I can define a complete ordering of values in the sequence.
However, I can also usefully define less fine-grained equivalence classes of items that have a "consistent" ordering, b... |
Are there standard terms for more fine/course grained (but otherwise consistent) ways of ordering values? |
The "rule of three" is an ancient ad-hoc mindless rote rule of inference that is best ignored. Instead, you should strive to learn the general principles behind it - namely, the laws of fraction arithmetic. Let's consider the example at hand. You seek the number of dollars $\rm\:X\:$ such that when decremented by $\:1... |
This may seem like an overly trivial question, but I've just recently become confused about Langrange's 'prime' notation for derivatives (for example $f`(x)$).
I know for sure that $f`(x) = \frac{\delta f(x)}{\delta x}$.
But suppose we replace x with an expression, like 2x+1. Do we write $f`(2x+1) = \frac{\delta ... |
This may seem like an overly trivial question, but I've just recently become confused about Langrange's 'prime' notation for derivatives (for example $f`(x)$).
I know for sure that $f`(x) = \frac{\delta f(x)}{\delta x}$.
But suppose we replace x with an expression, like 2x+1. Do we write $f`(x^2+1) = \frac{\delta... |
Are there standard terms for more fine/coarse grained (but otherwise consistent) ways of ordering values? |
This may seem like an overly trivial question, but I've just recently become confused about Langrange's 'prime' notation for derivatives (for example $f'(x)$).
I know for sure that $f'(x) = \frac{\delta f(x)}{\delta x}$.
But suppose we replace x with an expression, like 2x+1. Do we write $f'(x^2+1) = \frac{\delta... |
The "rule of three" is an ancient ad-hoc mindless rote rule of inference that is best ignored. Instead, you should strive to learn the general principles behind it - namely, the laws of fraction arithmetic. Let's consider the example at hand. You seek the number of dollars $\rm\:X\:$ such that when decremented by $\:1... |
I am mostly asking about terminology here.
**Background** A generalized linear eigenvalue problem is of the form $Ax=\lambda B x$, where $A$ and $B$ are square matrices, $\lambda$ is the eigenvalue and $x$ is the eigenvector. We can write this in the form that makes it evident that it is linear in $\lambda$: $(A-\la... |
Given a And\Or-graph induced by the transition function, each node of G corresponds to a state q belonging to set Q of the state of the Automaton, for q with $\delta(q,a)=q1*q2$, the node is a $*-node$ with two successor q1 and q2. For q={true,false}, the node q is a sink-node. Hence, if * is $\vee$ then q is $\vee-nod... |
**Let $F(n)$ be $n$-th Fibonacci number.$$F(0) = 0, F(1) = 1, F(2) = 1, F(3) = 2, F(4) = 3 \text{ and so on. }$$Given a positive integer $n \gt 2$, how to find the smallest prime number $P$ such that $P$ divides $F(n)$ but it does not divide any $F(k)$ smaller than $F(n)$.**
Now,does this problem could be solved wit... |
**Let $F(n)$ be $n$-th Fibonacci number.$$F(0) = 0, F(1) = 1, F(2) = 1, F(3) = 2, F(4) = 3 \text{ and so on. }$$Given a positive integer $n \gt 2$,Find the smallest prime number $P$ such that $P$ divides $F(n)$ but it does not divide any $F(k)$ smaller than $F(n)$ ?**
Now,my question: Is it possible to find the answ... |
**Let $F(n)$ be $n$-th Fibonacci number.$$F(0) = 0, F(1) = 1, F(2) = 1, F(3) = 2, F(4) = 3 \text{ and so on. }$$Given a positive integer $n \gt 2$,Find the smallest prime number $P$ such that $P$ divides $F(n)$ but it does not divide any $F(k)$ smaller than $F(n)$ ?**
Now,my question: Is it possible to find the answ... |
Suppose we have a Hadamard matrix $H$ with rows $H_1,H_2,\ldots$ and look at the following system
$$H_j \cdot \mathbf{x} \le 1 \ \forall j$$
$$\sum_i x_i=1$$
Is there a name for polytopes defined in this way? |
Polytopes defined by Hadamard matrices? |
Here's how I **approximated** an answer:
We know player-2 usually wins. The expected value of player-2 after 9 games is 63, just over 60. So the expected number of turns the game will last is about 9.
Let's consider another version of this game which is played for 9 turns exactly and the winner is the one who got t... |
The invariant subspace problem states something along the following lines: Let $H$ be a Banach space of dimension greater than one. Does there exist a bounded linear operator $T:H \to H$ which has a non-trivial closed $T$-invariant subspace?
Now my question is: Why is this a big problem? What does the solution give ... |
The invariant subspace problem states something along the following lines: Let $H$ be a Banach space of dimension greater than one. Does every bounded linear operator $T:H \to H$ have a non-trivial closed $T$-invariant subspace?
Now my question is: Why is this a big problem? What does the solution give us? It has be... |
Fiction "Division by Zero" By Ted Chiang
I read the fiction story <a href="http://www.fantasticmetropolis.com/i/division/full/">"Division by Zero" By Ted Chiang</a>
My interpretation is the character finds a proof that arithmetic is inconsistent.
Is there a formal proof the fiction can't come true? (I don'... |
Apparently, the solution to the [Card Doubling Paradox][1] is that a uniform probability distribution over the positive real numbers doesn't exist. Can anyone explain why this is the case and what probability distributions can exist over the positive real numbers (it seems that this would be quite limited, given that s... |
Why isn't there a uniform probability distribution over the positive real numbers? |
> Is there a finite group which is contained in infinitely many core-free finite groups as the centralizer of a central involution?
The Brauer–Fowler results show that if a finite group has no non-identity normal subgroups of odd order (so is core-free), then its order is bounded by a function of the size of a few c... |
<blockquote>Is there a formal proof the fiction can't come true?</blockquote>
No, by <a href="http://en.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_theorems#Second_incompleteness_theorem">Goedel's second incompleteness theorem</a>, formal systems can only prove their own consistency iff they are inconsistent. S... |
<blockquote>Is there a formal proof the fiction can't come true?</blockquote>
No, by <a href="http://en.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_theorems#Second_incompleteness_theorem">Gödel's second incompleteness theorem</a>, formal systems can prove their own consistency if and only if they are inconsiste... |
I was looking at movie times today and was struck by the oddly-spaced showing times. For example, at the local Loew's Theater "Tron" (127 min.) is playing on two screens at the following interlaced times: 1:00 pm, 1:45 pm, 4:00 pm, 4:45 pm, 7:00 pm, 7:45 pm, 10:00 pm and 10:45 pm. Why not space the times equally? Is th... |
<blockquote>Is there a formal proof the fiction can't come true?</blockquote>
No, by <a href="http://en.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_theorems#Second_incompleteness_theorem">Gödel's second incompleteness theorem</a>, formal systems can prove their own consistency if and only if they are inconsiste... |
I'm investigating an idea in cryptography that requires irreducible polynomials with coefficients of either 0 or 1 (e.g. over GF[2]). Essentially I am mapping bytes to polynomials. For this reason, the degree of the polynomial will always be an integer multiple of 8.
I would like to build a table of such irreducible... |
I was looking at movie times today and was struck by the oddly-spaced showing times. For example, at the local Loew's Theater "Tron" (127 min.) is playing on two screens at the following interlaced times: 1:00 pm, 1:45 pm, 4:00 pm, 4:45 pm, 7:00 pm, 7:45 pm, 10:00 pm and 10:45 pm. Why not space the times equally? Is th... |
I was looking at movie times today and was struck by the oddly-spaced showing times. For example, at the local Loew's Theater "Tron" (127 min.) is playing on two screens at the following interlaced times: 1:00 pm, 1:45 pm, 4:00 pm, 4:45 pm, 7:00 pm, 7:45 pm, 10:00 pm and 10:45 pm. Why not space the times equally? Is th... |
In numerical analysis, I was asked whether Bairstow's algorithm convergence rate is quadratic.
My initial feeling was that it does, since it is essentially Newton's method for a system of non-linear equations, and Newton's method converges quadratically in one dimension (When f is from R to R).
A quick search on ... |
According to a paper "Optimal Irreducible Polynomials for GF(2m) Arithmetic" by M. Scott,
"it is in practise always possible to chooose as an irreducible polynomial either a trinomial... or a pentanomial." <a href="http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.75.455"> [PDF link]</a>
In random number gen... |
I was looking at movie times today and was struck by the oddly-spaced showing times. For example, at the local Loew's Theater "Tron" (127 min.) is playing on two screens at the following interlaced times: 1:00 pm, 1:45 pm, 4:00 pm, 4:45 pm, 7:00 pm, 7:45 pm, 10:00 pm and 10:45 pm. Why not space the times equally? Is th... |
Let me try to split the question in a few parts,
- In what sense and why is it impossible to have a bilinear form of a non-trivial signature on a complex vector space?
(Is this related to anotherfact that I want to understand that any complex square matrix can be similarity transformed to a symmetric compl... |
This question came up when trying to trying to find a 3x3 real matrix A such that (1) Ax is nonzero for nonzero x and (2) Ax is orthogonal to x for any x in R^3. We know such a matrix cannot exist because A must have an eigenvalue (thus there is some x such that either Ax = 0 or Ax is parallel to x). However,
> Is... |
Geometric argument that operators on R^3 have an eigenvalue? |
You can find a nice concise exposition of Newton's method for algebraic systems of equations in the paper below. There it is applied to the algebraic equations that arise from (multivariate) polynomial factorization - but the ideas are quite general (Newton-Hensel methods are fundamental for effective algebraic computa... |
According to a paper "Optimal Irreducible Polynomials for GF(2m) Arithmetic" by M. Scott,
"it is in practise always possible to chooose as an irreducible polynomial either a trinomial... or a pentanomial." <a href="http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.75.455"> [PDF link]</a>
In random number gen... |
This question came up when trying to trying to find a 3x3 real matrix A such that (1) Ax is nonzero for nonzero x and (2) Ax is orthogonal to x for any x in R^3. We know such a matrix cannot exist because A must have an eigenvalue (thus there is some x such that either Ax = 0 or Ax is parallel to x). However,
> Is... |
If $A\colon\mathbb{R}^3\to\mathbb{R}^3$ was a linear map with no eigenvectors, then $x\mapsto Ax/\Vert Ax\Vert$ would give a map on the unit sphere with no fixed points and without taking any point to its antipode, contradicting the [hairy ball theorem][1].
[1]: http://en.wikipedia.org/w/index.php?title=Hairy_ba... |
Let me try to split the question in a few parts,
- I would like to understand the claim that all non-degenerate bilinear symmetric forms are equivalent over the complex while for the reals they can be distinguished by the signature. Hence there is just one Clifford algebra over $\mathbb{C}^n$ but over $\mathbb{R}^n... |
I'm not so well up on my math, but I was wondering if the set of "decimal expressible" real numbers $R$ between $[0,1]$ is countable?
Wikipedia gives the definition:
> A set $S$ is called countable if there exists an injective function $f$ from $S$ to the natural numbers $\mathbb{N}$
It would seem to my mathem... |
Decimal expressible real numbers between [0,1] countable? |
I'm not so well up on my math, but I was wondering if the set of "decimal expressible" real numbers $R$ between $[0,1]$ is countable?
Wikipedia gives the definition:
> A set $S$ is called countable if there exists an injective function $f$ from $S$ to the natural numbers $\mathbb{N}$
It would seem to my mathem... |
No they are not countable. Any real number can be expressed in decimal notation, and this representation may be infinite (non-terminating). And that's where your problem lies: how can you reverse an infinite decimal? |
> Is there a commutative integral domain R in which:
>
> * every nonzero prime ideal Q is maximal, and
> * there are maximal ideals Q with R/Q of sizes 3, 11, and 27?
This doesn't happen with number rings of Galois extensions of <b>Q</b>, as far as I know, since what happens to 3 seriously happens to 3. You cann... |
No they are not countable. Any real number can be expressed in decimal notation, and this representation may be infinite (non-terminating). And that's where your problem lies: how can you reverse an infinite decimal?
In my interpretation of your question there is no difference between what you are calling "decimal e... |
I'm not so well up on my math, but I was wondering if the set of "[EDIT: *finitely*] decimal expressible" real numbers $R$ between $[0,1]$ is countable?
Wikipedia gives the definition:
> A set $S$ is called countable if there exists an injective function $f$ from $S$ to the natural numbers $\mathbb{N}$
It woul... |
Your intuition is correct in the following sense: the set of "decimal expressible" real numbers in [0, 1] _such that the decimal expression has only finitely many nonzero digits_ is **countable**, and the "injective function" you gave works. Don't worry about "mathematical terminology for reversing a number"; to show ... |
Your intuition is correct: the set of "decimal expressible" real numbers in [0, 1] such that the decimal expression has only finitely many nonzero digits is **countable**, and the "injective function" you gave works.
However, if you allow decimal expressions with an infinite number of decimals (e.g. 0.333...), then ... |
This question came up when trying to trying to find a 3x3 real matrix A such that (1) Ax is nonzero for nonzero x and (2) Ax is orthogonal to x for any x in R^3. We know such a matrix cannot exist because A must have an eigenvalue (thus there is some x such that either Ax = 0 or Ax is parallel to x). However,
> Is... |
(Finitely) decimal expressible real numbers between [0,1] countable? |
Suppose that $g$ is a bijection on the real line, and $g^{-1} \circ f \circ g$ is a $C^\infty$ function whenever $f$ is $C^\infty$. It seems howlingly obvious that this can only happen if $g$ is itself $C^\infty$. But I can't figure out how to prove it.
Can you help me, Internets?
(Context: I want to show that... |
If $g^{-1} \circ f \circ g$ is $C^\infty$ whenever $f$ is $C^\infty$, must $g$ be $C^\infty$? |
Suppose that $g$ is a bijection on the real line, and $g^{-1} \circ f \circ g$ is a $C^\infty$ function whenever $f$ is $C^\infty$. It seems howlingly obvious that this can only happen if $g$ is itself $C^\infty$. But I can't figure out how to prove it.
Can you help me, Internets?
(Context: I want to show that... |
The infinite series...
$\pi/4 = 1 - 1/3 + 1/5 - 1/7 ...$
...is very intriguing to me and seems like a crazy coincidence (its relationship to $\pi$). Is it actually crazy or does it have an easy to explain, logical reasoning behind it that would make it seem not so magical? |
Why does this converge to $\pi/4$? |
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