instruction stringlengths 12 30k |
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Consider the following example,
Say you had a set of companies and for each pair of companies, you record the maximum number of emails sent by one to the other (i.e For example, when considering exchanges between companies $A$ and $B$ you pick maximum of number of emails from $A$ to $B$, and number of emails from $B... |
> **Possible Duplicate:**
> [-1 is not 1, so where is the mistake?](http://math.stackexchange.com/questions/438/1-is-not-1-so-where-is-the-mistake)
<!-- End of automatically inserted text -->
I'm trying to understand the exact point of failure in the following reasoning:
$1 = \sqrt{1} = \sqrt{(-1)(-1)} = ... |
Does 9^(2^n) + 1 always have a prime factor larger than 40? |
The $\Gamma (x)$ function has just one minimum for $x>0$ . This result uses some
properties of the gamma function:
- $\Gamma ^{\prime \prime }(x)>0$ and $\Gamma (x)>0$ for all $x>0$
- $\Gamma (1)=\Gamma (2)=1$.
Observing the following graph of $y=\Gamma (x)$ this minimum is near $x=3/2$, but likely $\min \Gam... |
The $\Gamma (x)$ function has just one minimum for $x>0$ . This result uses some
properties of the gamma function:
- $\Gamma ^{\prime \prime }(x)>0$ and $\Gamma (x)>0$ for all $x>0$
- $\Gamma (1)=\Gamma (2)=1$.
Observing the following graph (created in SWP) of $y=\Gamma (x)$ this minimum is near $x=3/2$, but ... |
In a [recent post](http://math.stackexchange.com/questions/3138/ways-for-a-continuous-function-to-not-be-smooth) I asked what are the ways a continuous function cannot be smooth. Apparently my question was not well posed, but since I am still thinking about these types of functions I will try one more time to ask my q... |
What are some examples of non smooth continuous functions without "kinks"? |
I am interested in how today's professional mathematicians view the Pythagorean theorem, in terms of how the theorem fits within the axiomatic framework of mathematics. I often come across textbooks that define length by the Pythagorean theorem, so that the theorem is in essence a definition or axiom. In more modern ... |
Where does the Pythagorean theorem "fit" within modern mathematics? |
The fact that rotation about an angle is a linear transformation is both important (for example, this is used to prove the sine/cosine angle addition formulas; see http://math.stackexchange.com/questions/1292/how-can-i-understand-and-prove-the-sum-and-difference-formulas-in-trigonometry) and somewhat intuitive geometri... |
How do you formally prove that rotation is a linear transformation? |
First how do you non-geometrically define rotation? Perhaps in the 2D case, you can define it as multiplication by a complex number.
Once you do that, use the definition of a Linear Transformation.
Show that the map T satisfies:
1) $T(a + b) = T(a) + T(b)$ for any vectors $a,b$ (or complex numbers if you chose... |
One rigorous definition of a kink that I can think of is that the left and right derivatives exist and do not agree. In that case, plenty of functions "don't have kinks" at points where they are continuous but not differentiable. $\sin \frac{1}{x}$ and the Weierstrass function "don't have kinks" in the sense that nei... |
You should be able to use a [Fully Homomorphic Encryption Scheme][1] which was recently proved to be possible by Craig Gentry from IBM (and was an important open problem in cryptography till 2009!)
Using a fully homomorphic scheme, any circuit can be evaluated homomorphically, effectively allowing computation on enc... |
First how do you non-geometrically define rotation?
Perhaps in the 2D case, you can define it as multiplication by a complex number (which is in fact true!)
Once you do that, use the definition of a Linear Transformation.
Show that the map T satisfies:
1) $T(a + b) = T(a) + T(b)$ for any vectors $a,b$ (or ... |
You should be able to use a [Fully Homomorphic Encryption Scheme][1] which was recently proved to be possible by Craig Gentry from IBM (and was an important open problem in cryptography till 2009!)
Using a fully homomorphic scheme, any circuit can be evaluated homomorphically, effectively allowing computation on enc... |
This was actually [my question][1] originally. To put the demand for intuitiveness in starker terms: **what does the arc of a circle have to do with the area of a square and the operation of reciprocation?**
If someone can show me a geometric figure with area $1 \over 1 + x^2$ that might be a good start.
Or, th... |
I understand the technical and logical distinction between "if" and "only if" and "if and only if". In mathematical discourse, what are some other common lexical contexts, meaningful or not, in which appears the word "only"? Personally I have always been troubled by the phrase "only if" even though I am able to parse... |
What does "only" mean? |
I understand the technical and logical distinction between "if" and "only if" and "if and only if". In mathematical discourse, what are some other common lexical contexts, meaningful or not, in which appears the word "only"? Personally I have always been troubled by the phrase "only if" even though I am able to parse... |
I understand the technical and logical distinction between "if" and "only if" and "if and only if". But I have always been troubled by the phrase "only if" even though I am able to parse and interpret it. Also in my posts on this and other sites I have frequently had to make edits to migrate the term "only" (across m... |
I understand the technical and logical distinction between "if" and "only if" and "if and only if". But I have always been troubled by the phrase "only if" even though I am able to parse and interpret it. Also in my posts on this and other sites I have frequently had to make edits to migrate the term "only" (across m... |
I understand the technical and logical distinction between "if" and "only if" and "if and only if". But I have always been troubled by the phrase "only if" even though I am able to parse and interpret it. Also in my posts on this and other sites I have frequently had to make edits to migrate the term "only" (across m... |
The Weierstrass's approximation theorem for continuous functions on a compact by using polynomials is well-known. As far as I know, there are some variants of this theorem, e.g. Stone-Weierstrass that does not refer only to polynomials as approximator functions. Where could I find these Weierstrass-like approximation t... |
The $\Gamma (x)$ function has just one minimum for $x>0$ . This result uses some
properties of the gamma function:
- $\Gamma ^{\prime \prime }(x)>0$ and $\Gamma (x)>0$ for all $x>0$
- $\Gamma (1)=\Gamma (2)=1$.
Observing the following graph (created in SWP) of $y=\Gamma (x)$ this minimum is near $x=3/2$, but ... |
Minimum of the Gamma function $\Gamma (x)$ for $x>0$. How to find $x_{\min}$? |
The $\Gamma (x)$ function has just one minimum for $x>0$ . This result uses some
properties of the gamma function:
- $\Gamma ^{\prime \prime }(x)>0$ and $\Gamma (x)>0$ for all $x>0$
- $\Gamma (1)=\Gamma (2)=1$.
Observing the following graph (created in SWP) of $y=\Gamma (x)$ this minimum is near $x=3/2$, but ... |
The Weierstrass' approximation theorem for continuous functions on a compact by using polynomials is well-known. As far as I know, there are some variants of this theorem, e.g. Stone-Weierstrass that does not refer only to polynomials as approximator functions. Where could I find these Weierstrass-like approximation th... |
The $\Gamma (x)$ function has just one minimum for $x>0$ . This result uses some
properties of the gamma function:
- $\Gamma ^{\prime \prime }(x)>0$ and $\Gamma (x)>0$ for all $x>0$
- $\Gamma (1)=\Gamma (2)=1$.
Observing the following graph (created in SWP) of $y=\Gamma (x)$ this minimum is near $x=3/2$, but ... |
I understand the technical and logical distinction between "if" and "only if" and "if and only if". But I have always been troubled by the phrase "only if" even though I am able to parse and interpret it. Also in my posts on this and other sites I have frequently had to make edits to migrate the term "only", sometime... |
If I understand your question correctly, your problem can be seen as a multi-party extension of the [Millionaires' problem][1]. In the Millionaires' problem problem, two millionaires want to know which of them is richer without revealing their wealth. There is an secure two-party protocol (due to [Andy Yao, 1982][2]), ... |
I'm reading Shelah's book "Proper and Improper Forcing" (the first two chapters were recommended for learning the basics of forcing)
Given a quasi-order $P$ we say that $\mathcal{I}$ is a *dense* subset of $P$ if
$$(\forall p \in P) (\exists q \in \mathcal{I}) (p\le q)$$
We say that it is open if for any $p,q \... |
The Weierstrass' approximation theorem for continuous functions on a compact space by using polynomials is well-known. As far as I know, there are some variants of this theorem, e.g. Stone-Weierstrass that does not refer only to polynomials as approximator functions. Where could I find these Weierstrass-like approximat... |
I'm reading Shelah's book "Proper and Improper Forcing" (the first two chapters were recommended for learning the basics of forcing)
Given a quasi-order $P$ we say that $\mathcal{I}$ is a *dense* subset of $P$ if
$$(\forall p \in P) (\exists q \in \mathcal{I}) (p\le q)$$
We say that it is open if for any $p,q \... |
In high school, we were often given questions of the form: "What is the probability of A given B?" For example:
> What is the probability that two
> people were born on the same day given
> that one was born on a Tuesday?
Intuitively, most people expect that knowing someone was born on a Tuesday doesn't give yo... |
You should be able to use a [Fully Homomorphic Encryption Scheme][1] which was recently proved to be possible by Craig Gentry from IBM (and was an important open problem in cryptography till 2009!)
Using a fully homomorphic scheme, any circuit can be evaluated homomorphically, effectively allowing computation on enc... |
Minimum of the Gamma Function $\Gamma (x)$ for $x>0$. How to find $x_{\min}$? |
I'm going to use the def of "directed" that I found in Shelah's <a href = "http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.pl/1235419814">book$^1$</a> : A set $G$ is directed if it is downward closed (if $p$ in G and if $q\leq p$ then $q\in G$) **and** every two memebers of $G$ have a ... |
I'm reading Shelah's book "Proper and Improper Forcing" (the first two chapters were recommended for learning the basics of forcing)
Given a quasi-order $P$ we say that $\mathcal{I}$ is a *dense* subset of $P$ if
$$(\forall p \in P) (\exists q \in \mathcal{I}) (p\le q)$$
We say that it is open if for any $p,q \... |
I'm going to use the def of "directed" that I found in Shelah's [book](http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.pl/1235419814)<sup>1</sup>: A set $G$ is directed if it is downward closed (if $p$ in G and if $q\leq p$ then $q\in G$) **and** every two memebers of $G$ have a common... |
"Why do I always get 1 when I keep hitting the square root button on my calculator?" |
Make a picture of the usual spiral converging to the fixed point, in the style of ![alt text][1]
[1]: https://i.stack.imgur.com/WyeV5.png
Where does this little obsession aparent on this site with 'layman' explanations? :P
**NB.** The picture comes from the Wikipedia page on [fixed points](http://en.wikipe... |
Make a picture of the usual spiral converging to the fixed point, in the style of ![alt text][1]
[1]: https://i.stack.imgur.com/WyeV5.png
Where does this little obsession aparent on this site with 'layman' explanations? :P
**NB.** The picture comes from the Wikipedia page on [fixed points](http://en.wikipe... |
Make a picture of the usual spiral converging to the fixed point, in the style of ![alt text][1]
[1]: https://i.stack.imgur.com/WyeV5.png
Where does this little obsession aparent on this site with 'layman' explanations? :P
**NB.** The picture comes from the Wikipedia page on [fixed points](http://en.wikipe... |
When calculating some probabilities, I got sums of the form
$$\sum_{j=0}^c {j+a+b \choose a} p^j,$$ for integers $a, b, c > 0$.
Does someone know closed forms for these values? |
I don't have time right now to work out the exact answer, but the methods and identities in the middle third of <a href="http://www.artofproblemsolving.com/Forum/blog.php?u=14052&b=10590">this blog post</a> will give a closed form for fixed $a$. Use the geometric series formula to find $\sum_{j=0}^c p^{a+b+j}$, differ... |
I don't have time right now to work out the exact answer, but the methods and identities in the middle third of <a href="http://www.artofproblemsolving.com/Forum/blog.php?u=14052&b=10590">this blog post</a> will give a closed form for fixed $a$. Use the geometric series formula to find $\sum_{j=0}^c p^{a+b+j}$, differ... |
I subdivide a cubic bezier curve at a given *t* value using de Casteljau’s algorithm, which yields two cubic bezier curves. Afterwards I “scale” the second curve (proportionally).
I’d like to reconnect or approximate the two curves to/with a single curve in a third step. Is that possible?
[This figure][1] illustr... |
How to approximate/connect two continuous cubic bezier curves with/to a single one? |
I subdivide a cubic bezier curve at a given *t* value using de Casteljau’s algorithm, which yields two cubic bezier curves. Afterwards I “scale” the second curve (proportionally).
I’d like to reconnect or approximate the two curves to/with a single curve in a third step. Is that possible?
 is equal to (AX,A... |
I'm not sure having really understood your question, but it seems to me that mathematicians make no difference between heterogeneous and homogeneous tuples. Consider $\left(5,b,f,8,3\right)$: one can say it is an heterogeneous tuple because the set $\left\\{5,b,f,8,3\right\\}$ can be partitioned into $\left\\{b,f\right... |
Length, in Euclidean geometry, is a relation, not a number. We say that two segments are of the same length if they are congruent to each other, and congruence is one of the undefined notions of Euclidean geometry. The Pythagorean theorem comes about as soon as you decide to map lengths to positive real numbers, and it... |
A function $f$ is superadditive if $f(x) + f(y) \le f(x+y)$. The question is:
Does a real number $a$ exists such that for all
real numbers with $x, y\ \ge \ a $
$$ \Gamma(x) + \Gamma(y) \le \Gamma(x+y) \quad ?$$
|
Is the Gamma function superadditive? |
It is very true that statements and problems on conditional probability are often presented in an ambiguous way. The problem isn't specifically with the term "given", I believe, but rather with the fact that the presentation does not make it clear what the sample space is and what are the distributions (typically some ... |
This is a follow-up to the MathStackexchange question 3305 and MathOverflow question 23229.
The Bohr-Mollerup theorem states that the Gamma function is the unique function that satisfies:
$(1)\ f(x+1) = xf(x), \ (2)\ f(1) = 1, \ (3)\ \ln \circ f \ $ is convex.
Assume a function $f\colon \mathbb{R^{+}} \to \ma... |
I'm hesitant to use an answer to respond, instead of a comment, because the answer seems to merely be a matter of semantics.
Let's look at a similar question:
"What are the odds that two people are of the same gender?"
And its counterpart:
"What are the odds that two people of the same gender, if we knew th... |
This is a follow-up to the MathStackexchange question 3305 and MathOverflow question 23229.
The Bohr-Mollerup theorem states that the Gamma function is the unique function that satisfies:
$(1)\ f(x+1) = xf(x), \ (2)\ f(1) = 1, \ (3)\ \ln \circ f \ $ is convex.
Assume a function $f\colon \mathbb{R^{+}} \to \ma... |
In high school, we were often given questions of the form: "What is the probability of A given B?" For example:
> What is the probability that two
> people were born on the same day given
> that one was born on a Tuesday?
Intuitively, most people expect that knowing someone was born on a Tuesday doesn't give yo... |
Given `e1..e5` as the exponents of the base units are there any well known methods for producing simplified form?
As an example `m^2*kg*s^-1` simplifies to `W*s^2`.
I have a set of derived units as Name/exponent-vector pairs as input and it can be assumed all exponents are rational (and in fact most being integer... |
If i is the square root of -1, is the square root of i imaginary? Is it used or considered often in mathematics? How is it notated? |
what is the square root of i? |
I've heard a lot about this number e. Why is it so important? How does it fit into the 'bigger picture' of mathematics? How is it calculated and used? |
Why is the number e so important in mathematics? |
Given `e1..e5` as the exponents of the base units, are there any well known methods for producing a simplified form?
As an example `m^2*kg*s^-1` simplifies to `W*s^2`.
I have a set of derived units as Name/exponent-vector pairs as input and it can be assumed all exponents are rational (and in fact most being inte... |
Pi appears a LOT in trigonometry, but only because of its 'circle-significance'. Does pi ever matter in things not concerned with circles? Is its only claim to fame the fact that its irrational and an important ratio? |
Does the number pi have any significance besides being the ratio of a circle's diameter to its circumference? |
Length, in Euclidean geometry, is a relation, not a number. We say that two segments are of the same length if they are congruent to each other, and congruence is one of the undefined notions of Euclidean geometry. The Pythagorean theorem comes about as soon as you decide to map lengths to positive real numbers, and it... |
Back in the day, I was absolutely enthralled by the study of plane curves and their properties (I have [Lockwood][1] and [Zwikker][2] to thank). I learned early on that for the purposes of generating plots on a computer (and for that matter deducing equations of "derived curves" and determining other special properties... |
Based on the previous answers, I have come to the following (possible incomplete or incorrect) conclusion: It seems that the Pythagorean theorem can be dealt with in one of two ways, either it is taken as the definition of length/distance, or it is proved by invoking the technical definitions of an inner product space.... |
Ramsey number R(3,6)=18. How to construct a graph of 17 nodes which does not contain neither a clique of order 3 or an independent set of order 6. could you show me the tactics or the adjacency matrix? |
I don't have time right now to work out the exact answer, but the methods and identities in the middle third of <a href="http://www.artofproblemsolving.com/Forum/blog.php?u=14052&b=10590">this blog post</a> will give a closed form for fixed $a$. Use the geometric series formula to find $\sum_{j=0}^c p^{a+b+j}$, differ... |
I love R.Courant and F.John's _Introduction to Calculus and Analysis_ because of its wide coverage, precise description and friendly written style.
Are there any classic textbooks like it on ODE/PDE?
thanks. |
Could you recommend some classic textbooks on ordinary/partial differential equation? |
I don't have time right now to work out the exact answer, but the methods and identities in the middle third of <a href="http://www.artofproblemsolving.com/Forum/blog.php?u=14052&b=10590">this blog post</a> will give a closed form for fixed $a$. Use the geometric series formula to find $\sum_{j=0}^c p^{a+b+j}$, differ... |
Sometimes, there are many ways to give defintion to a mathmatical principle, for example the natural base logarithm.
How about sine and cosine?
thanks. |
How many ways to define sine and cosine? |
I am a big fan of Bentley and Cooke's [Linear Algebra with Differential Equations](http://www.abebooks.com/servlet/BookDetailsPL?bi=1174162092&searchurl=an%3Dbentley%2Band%2Bcooke%26sts%3Dt%26x%3D82%26y%3D18). It was published in the early 70's, it is black and white (i.e. not flashy), and has excellent references thro... |
Good books on ODE would be:
1) Ordinary Differential Equations by Morris Tenenbaum and Harry Pollard
2) Differential Equations by George F. Simmons
3) Ordinary Differential Equations by E.Coddington
For PDE, I request you to take a look at this link: http://www.physicsforums.com/showthread.php?t=175382 |
Why we assign $1/2[f(\eplison+0) + f(\eplison-0)]$ as the value at the discontinuity point? |
Why we assign $1/2[f(\epsilon+0) + f(\epsilon-0)]$ as the value at the discontinuity point? |
Here is an answer based on Carathéodory's formulation of differentiability. Let $a\in A\subset\mathbb R$ and $f:A\to\mathbb R$. Assume there a sequence in $A\setminus\\{a\\}$ converging to $a$. Then there is at most one function $\varphi:A\to\mathbb R$ which is continuous at $a$ and satisfies $$f(x)=f(a)+(x-a)\ \varphi... |
Here is an answer based on Carathéodory's formulation of differentiability. Let $a\in A\subset\mathbb R$ and $f:A\to\mathbb R$. Assume there a sequence in $A\setminus\\{a\\}$ converging to $a$. Then there is at most one function $\varphi:A\to\mathbb R$ which is continuous at $a$ and satisfies $$f(x)=f(a)+(x-a)\ \varphi... |
Construct a string by using only n characters, the string should do not contain any repeated pattern, in other words, the string must have not any substring matching /(.+)\1/, what is the length l(n) of the longest string S(n)?
n S(n) l(n)<br/>
1 a 1<br/>
2 aba 3<br/>
3 ? ?<br/>
4 ... |
Here is an explicit example that is very concrete, but a near miss: a graph $G$ with 16 vertices containing no triangle and no independent set with six elements. I think that it is not possible to extend the graph $G$ by a single vertex preserving this property, but I would be interested if I could be proved wrong! Fo... |
First up, you should accept Ranieri's answer, which is obviously correct. The question of "slicing" seems like a temporary misunderstanding that shouldn't stand in your way. Rejection sampling works with continuous distributions.
Second, it seems you are attempting to randomly generate individuals from a population ... |
Let $f: D \to \mathbb{R}\ $ be a function on convex domain $D \subseteq \mathbb{R}^2$ and let us assume the second-order derivatives of $f$ exist. It is well known that $f$ is convex if and only if its Hessian $\nabla^2 f(x)$ is positive semi-definite for all $x \in D$. It is also known that if $\nabla^2 f(x)$ is posit... |
In the Portuguese book *Análise Matemática* (Mathematical Analysis) by C. Sarrico, it is proved there exist
$\displaystyle\lim_{n\rightarrow +\infty}f_{n}(x)=\lim_{n\rightarrow +\infty }e^{\log f_{n}(x)}$ (1),
where
$f_{n}(x)=\dfrac{n!n^{x}}{x(x+1)(x+2)\cdots (x+n)}$. (2)
And it is stated there that
$\d... |
Two ways of defining the gamma function $\Gamma (x)$. How does one show they are equivalent? |
Construct a string by using only n characters, the string should do not contain any repeated pattern, in other words, the string must have not any substring matching /(.+)\1/, what is the length l(n) of the longest string S(n)?
n S(n) l(n)<br/>
1 a 1<br/>
2 aba 3<br/>
3 ? ?<br/>
4 ... |
Construct a string by using only n characters, the string should do not contain any consecutive repeated pattern, in other words, the string must have not any substring matching /(.+)\1/, what is the length l(n) of the longest string S(n)?
n S(n) l(n)<br/>
1 a 1<br/>
2 aba 3<br/>
3 ? ... |
I'm struggling with this former Putnam Exam problem:
Suppose $f$ and $g$ are nonconstant, differentiable, real-valued functions on $R$. Furthermore, suppose that for each pair of real numbers $x$ and $y$, $f(x + y) = f(x)f(y) - g(x)g(y)$ and $g(x + y) = f(x)g(y) + g(x)f(y)$. If $f'(0) = 0$, prove that $(f(x))^2 + ... |
Uniqueness of $\sin x$ and $\cos x$? (Putnam Exam problem) |
A theorem of Thue's (not Thue's theorem!) states that for $n = 3$ (and therefore, also for larger $n$) an infinite sequence of characters (chosen out of $n$) without repeated sub-sequences (a.k.a square-free) exists.
Edit: here's a slightly related reference: J.D. Currie's paper ['There are ternary circular square-... |
Construct a string by using only n characters, the string should do not contain any consecutive repeated pattern, in other words, the string must have not any substring matching /(.+)\1/, what is the length l(n) of the longest string S(n)?
n S(n) l(n)
1 a 1
2 aba 3
3 ? ... |
Let $f: D \to \mathbb{R}\ $ be a function on convex domain $D \subseteq \mathbb{R}^2$ and let us assume the second-order derivatives of $f$ exist. It is well known that $f$ is convex if and only if its Hessian $\nabla^2 f(x)$ is positive semi-definite for all $x \in D$. It is also known that if $\nabla^2 f(x)$ is posit... |
A theorem of Thue's (not Thue's theorem!) states that for $n = 3$ (and therefore, also for larger $n$) an infinite sequence of characters (chosen out of $n$) without repeated sub-sequences (a.k.a square-free) exists.
Edit: here's a slightly related reference: J.D. Currie's paper ['There are ternary circular square-... |
Let $f : [0, 1] \to \mathbb{R}$ be a 'well behaved' function and define $c_n$ by the Riemann sums
$$
c_n := \frac{1}{n} \sum_{k = 1}^n f(\frac{k - \theta}{n})
$$
for some given $\theta \in [0, 1]$. Then obviously $c_n$ will converge to the integral of $f$. Under which conditions on $f$ will this convergence be mono... |
When do Riemann sums converge monotonously? |
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