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Once you have a formula like this, you can prove it by induction.
But that begs the question as to how you get such a formula.
In this case you might ask: (a) what's the "average" term?
and (b) how many terms are there? |
The Blue-Eyed Islander problem is one of my favorites. You can read about it [here][1] on Terry Tao's website, along with some discussion. I'll copy the problem here as well.
There is an island upon which a tribe resides. The tribe consists of 1000 people, with various eye colours. Yet, their religion forbids them... |
**HINT** Pair each summand $k$ with it's "reflection" $n+1-k$. Like the analogous proof of [Wilson's theorem][1], this method exploits the existence of nontrivial symmetry, here a reflection or involution. Such techniques often yield elegeant proofs. One should always look for innate symmetries when first pondering pro... |
**HINT** Pair each summand $k$ with it's "reflection" $n+1-k$. Like the analogous proof of [Wilson's theorem][1], this method exploits the existence of a nontrivial symmetry, here a reflection or involution. Such techniques often yield elegant proofs. One should always look for innate symmetries when first pondering pr... |
Does $S_k= \sum_{n=1}^{\infty}\sin(n^k)/n$ converge for all $k>0$?
**Motivation**: I recently learned that $S_1$ [converges](http://en.wikipedia.org/wiki/Dirichlet%27s_test). I think $S_2$ converges by the integral test. Was the question known in general?
|
Apparently 1+2+3+4+...+n = 1/2*n*(n+1).
How? What's the proof? Or maybe it is self apparent just looking at the above? Does this problem have a name and maybe a presence on the net? And what is this math called so I can add the right tags.
Thx if you can help.
Edit:
This problem is know as "The sum of the fi... |
Is are there software package that can help one find a path between two mathematical statements? |
**HINT** Pair each summand $k$ with it's "reflection" $n+1-k$. This is simply a discrete analog of the method of computing the area of the triangle under the diagonal of a square by reflecting a subtriangle through the midpoint of the diagonal to form an $n$ by $n/2$ rectangle.
Like the analogous proof of [Wilson's ... |
Let us be given a $n\times n$ matrix containing only zeros and ones.Now, the goal is to remove some 'ones' from the matrix (i.e. replace them with zeros) so that in each row and each column there is at most one 'one'. Of course, it is preferable that we leave as much 'ones' as it is possible (maximum is obviously $n$).... |
Can this algorithm fail? |
Prove that a number in the sequence $2,3,4,...,n \ (n>2$, is relatively prime to all other numbers if and only if it is a prime that exceeds $\displaystyle\frac{n}{2}$. Does such a prime always exist? |
Can this algorithm on converting a logical matrix to permutation matrix fail? |
Can this algorithm on removing 1's from a (0,1)-matrix fail? |
How is the power series expansion of an analytic function at a point constructed, without using Cauchy's theorem(or formula)?
|
What are some nice applications of algebraic topology that can be presented to beginning students? To give examples of what I have in mind: Brouwer's fixed point theorem, Borsuk-Ulam theorem, Hairy Ball Theorem, any subgroup of a free group is free.
The deeper the methods used, the better. All the above can be pro... |
What are some nice applications of algebraic topology that can be presented to beginning students? To give examples of what I have in mind: Brouwer's fixed point theorem, Borsuk-Ulam theorem, Hairy Ball Theorem, any subgroup of a free group is free.
The deeper the methods used, the better. All the above can be prove... |
It is said that there is a proof of fundamental theorem of algebra using Lie Theory. I have seen this claim at various places. But I could never find such a proof. Can anybody help me out? |
I need an algorithm to produce all strings with the following property. Here capital letter refer to strings, and small letter refer to characters. $XY$ means the concatenation of string $X$ and $Y$.
Let $\Sigma = \\{a_0, a_1,\ldots,a_n,a_0^{-1},a_1^{-1},\ldots,a_n^{-1}\\}$ be the set of usable characters. Every str... |
A fast way to generate cyclic strings with few restrictions? |
How many ways are there to choose a $2$-element subset out of an $n$-element set?
On the one hand, you can choose the first element of the set in $n$ ways, then the second element of the set in $n-1$ ways, then divide by $2$ because it doesn't matter which you choose first and which you choose second. This gives $\... |
Bernhard Elsner, alias MathOMan, posted this exercise in plane Geometry, [*Theorem about a circle, three chords and a midpoint*][1] on January 29th, 2010.
"Let $\mathcal{C}$ be a circle, $A,B$ two distinct points on $\mathcal{C}$ and $M$ be the midpoint of the chord $[AB]$. Take two other chords,$[PQ]$ and $[SR]$, t... |
I am auditing a Linear Algebra class, and today we were taught about the rank of a matrix. The definition was given from the row point of view" "The rank of a matrix A is the number of non-zero rows in the reduced row-echelon form of A". The lecturer then explained that if the matrix $A$ has size $m * n$, then $rank(A)... |
Is the rank of a matrix the same of its transpose? If yes, how can I prove it? |
I am auditing a Linear Algebra class, and today we were taught about the rank of a matrix. The definition was given from the row point of view:
> "The rank of a matrix A is the number
> of non-zero rows in the reduced
> row-echelon form of A".
The lecturer then explained that if the matrix $A$ has size $m * n$... |
The partial sums of the harmonic series are called "harmonic numbers." The difference between the nth harmonic number and ln(n) tends to a limit as n increases, and that limit is called [Euler's constant][1] or gamma.
There's a great book about all this called [Gamma: Exploring Euler's Constant][2].
[1]: http:... |
I am auditing a Linear Algebra class, and today we were taught about the rank of a matrix. The definition was given from the row point of view:
> "The rank of a matrix A is the number
> of non-zero rows in the reduced
> row-echelon form of A".
The lecturer then explained that if the matrix $A$ has size $m * n$... |
I need an algorithm to produce all strings with the following property. Here capital letter refer to strings, and small letter refer to characters. $XY$ means the concatenation of string $X$ and $Y$.
Let $\Sigma = \\{a_0, a_1,\ldots,a_n,a_0^{-1},a_1^{-1},\ldots,a_n^{-1}\\}$ be the set of usable characters. Every str... |
Let $p_N/q_N$ be the $N^\text{th}$ convergent of the continued fraction for some irrational number $\alpha$. It turns out that for any other approximation $p/q$ which isn't a convergent $|\alpha q - p| > |\alpha q_{N-1} - p_{N-1}|$. I'm wondering if there are any nice proofs for this result?
----------
In my bo... |
A nicer proof of Lagrange's 'best approximations' law? |
Let $p_N/q_N$ be the $N^\text{th}$ convergent of the continued fraction for some irrational number $\alpha$. It turns out that for any other approximation $p/q$ (with $q \le q_N$) which isn't a convergent $|\alpha q - p| > |\alpha q_{N-1} - p_{N-1}|$. I'm wondering if there are any nice proofs for this result?
-----... |
I like to use physical demonstrations when teaching mathematics (putting physics in the service of mathematics, for once, instead of the other way around), and it'd be great to get some more ideas to use.
I'm looking for nontrivial ideas in abstract mathematics that can be demonstrated with some contraption, constru... |
Let $p_N/q_N$ be the $N^\text{th}$ convergent of the continued fraction for some irrational number $\alpha$. It turns out that for any other approximation $p/q$ (with $q \le q_N$) which isn't a convergent $|\alpha q - p| > |\alpha q_{N-1} - p_{N-1}|$. I'm wondering if there are any nice proofs for this result?
-----... |
What might be some potential methods to encode a 100-point signal (curve) for input to a Artificial Neural Network?
Example: we have a large number of 100-pt 'curves' ranging from flat-line to approximately a half-sine wave with a wide range of max amplitude. The curves have some noise and occasional anomalous 'b... |
Encoding a Discrete Signal to an Artificial Neural Network? |
Have you ever heard of Galois theory? It is a theory that studies the solutions of equations over fields.
As it turns out, there is a special type of Galois theory called Differential Galois theory, which studies fields with a differential operator on them:
http://en.wikipedia.org/wiki/Differential_galois_theory
... |
I need to find the minimum of a function f(t) = int g(t,x) dx over [0,1]. What I did in mathematica is as follows:
f[t_] = NIntegrate[g[t,x],{x,-1,1}]
FindMinimum[f[t],{t,t0}]
However mathematica halts at the first try, because NIntegrate does not work with the symbolic t. Although Plot[f[t],{t,0,1}] wor... |
The "[sum and difference][1]" formulas often come in handy, but it's not immediately obvious that they would be true.
\begin{align}
\sin(\alpha \pm \beta) &= \sin \alpha \cos \beta \pm \cos \alpha \sin \beta \\\\
\cos(\alpha \pm \beta) &= \cos \alpha \cos \beta \mp \sin \alpha \sin \beta
\end{align}
So what I ... |
How can I understand and prove the "sum and difference formulas" in trigonometry? (cos(a ± b) = ..., etc.)? |
Making explicit what is implicit in Qiaochu Yuan's comment, and demonstrating that [someone else's work][1] has failed to evade my eyes. (It is a neat article, read it.) I present this adaptation of Duval's algorithm.
Assign an order to your symbols, say $a_1, a_2, a_1^{-1}, a_2^{-1}$ let first_symbol and _last_sym... |
Is it true that sin(n<sup>k</sup>) ≠ (sin n)<sup>k</sup> for any positive integers n and integers k ≠ 1?
What if n (> 0), k (≠ 1) are rational? |
Is sin(n^k) ≠ (sin n)^k in general? |
Beginning students might appreciate applications outside mathematics. Here are a couple of current (undergraduate) projects: The topology of data (http://inperc.com/wiki/index.php?title=The_topology_of_data) and 3D image analysis (http://inperc.com/wiki/index.php?title=3D_image_analysis). The tool used is called "persi... |
I'm new at learning boolean algebra and understand the basic laws. But now I'm trying to proof:
x'z+ xyz + xy'z = z
Can someone help me with this please ?
Thanks in advance.
|
Making explicit what is implicit in Qiaochu Yuan's comment, and demonstrating that [someone else's work][1] has failed to evade my eyes. (It is a neat article, read it.) I present this adaptation of Duval's algorithm.
Assign an order to your symbols, say $a_1, a_2, a_1^{-1}, a_2^{-1}$ let first_symbol and _last_sym... |
How to proof $(f \circ\ g) ^{-1} = g^{-1} \circ\ f^{-1}$ ? |
This is straightforward. Take x in the domain of f. It goes to f(x) = y. And g takes y to z = g(y). Therefore $g^{-1}$ takes z to y and $f^{-1}$ takes y to x. Both sides of your equation takes $z$ to $x$.
Please try to think more before asking. This was not hard, was it? :) |
This is a strengthening of Chandru's question:
http://math.stackexchange.com/questions/2227/
Is there a nice and concrete example of a Fourier series that fails to converge on some "big" set of measure zero, for instance on the Cantor ternary set?
|
Making explicit what is implicit in Qiaochu Yuan's comment, and demonstrating that [someone else's work][1] has failed to evade my eyes. (It is a neat article, read it.) I present this adaptation of Duval's algorithm.
Assign an order to your symbols, say $a_1, a_2, a_1^{-1}, a_2^{-1}$ let first_symbol and _last_sym... |
The number of elements of order 2 in a group is fairly restricted: 0, odd, or infinity. All such possibilities occur already in the trivial group and in dihedral groups.
The number of elements of order 3 in a group can be shown to be similarly restricted: 0, 2 mod 6, or infinity. However, something strange happens... |
Is there a group with exactly 92 elements of order 3? |
When flipping a coin to make important decisions in life you can flip once to choose between 2 possible outcomes. (Heads I eat cake, Tails I eat chocolate!)
You can also flip twice to choose between 4 outcomes. (Heads-Heads, Tails-Tails, Heads-Tails, Tails-Heads)
Can you use a coin to choose evenly between three ... |
Is it possible to 'split' coin flipping 3 ways? |
I'm looking for the equation of a family of functions that roughly resembles the sketch below (with apologies for the crudeness of said sketch):
<img src="http://img202.imageshack.us/img202/6907/functionsketch.png" />
Properties I'm looking for:
* f(x) approaches asymptote y=y1 as x approaches -inf
* f(x) app... |
I'm looking for the equation of a family of functions that roughly resembles the sketch below (with apologies for the crudeness of said sketch):
<img src="http://img202.imageshack.us/img202/6907/functionsketch.png" />
Properties I'm looking for:
* $\lim_{x\to-\infty}f(x)=y_1$ (i.e. approaches asymptote y=y<sub... |
Beginning students might appreciate applications outside mathematics. Here are a couple of current (undergraduate) projects: [The topology of data][1] and 3D image analysis. The tool used is called "persistent homology".
[1]: http://inperc.com/wiki/index.php?title=The_topology_of_data "The topology of data"
... |
I heard that one can construct field based on $p^n$ elements where $p$ is prime. I tried with $p = n = 2$. It seemed easy as there are 2 groups which $\\{0,1,2,3\\}$ can form and 1 group formed by $\\{1,2,3\\}$. However each time I get into contradiction. Is there anything I missed or it is not possible to form $p^n$ e... |
I've come across statements in the past along the lines of "function f has no closed form integral", which I assume means that there is no combination of the operations:
* addition/subtraction
* multiplication/division
* raising to powers and roots
* trigonometric functions
* exponential functions
* logarithmic... |
How can I find the number of *k*-permutations of *n* objects, where there are *x* types of objects, and r<sub>1</sub>, r<sub>2</sub>, r<sub>3</sub> ... r<sub>x</sub> give the number of each type of object?
Example:
> I have 20 letters from the alphabet. There are some duplicates - 4 of them are *a*, 5 of them are... |
I am learning the axioms of Zermelo-Fraenkel (ZF) set theory.
One axiom schema basically says that given any set S and any formula phi(x), there is a set T consisting of all those elements x of S such that phi(x).
I find this axiom schema unsatisfying because it only guarantees that subsets of S _definable by a ... |
Is the Subset Axiom Schema in ZF necessary? |
Let U be a unitary operator from Hilbert space H to Hilbert space K.
How do you call a *-homomorphism f from B(H) to B(K), defined by $f(a) = U a U^{-1}$?
I'm interested both in a symbol, which can be used in formulas, and a name for it, which can be used in texts or in speech. |
How did the definition of Algebraic objects like group, ring and field come up? When groups were first introduced, were they given the 4 axioms as we give now. And what made Mathematicians to think of something like this.
|
I'm certainly no historian, so let me give my impression of the history in a CW answer and others can improve on this answer.
Groups were understood as permutations of objects long before they were understood as abstract groups, just as manifolds were understood as nice subsets of $\mathbb{R}^n$ long before they wer... |
Let U be a unitary transformation from Hilbert space H to Hilbert space K.
How do you call a *-homomorphism f from B(H) to B(K), defined by $f(a) = U a U^{-1}$?
I'm interested both in a symbol, which can be used in formulas, and a name for it, which can be used in texts or in speech. |
*OP's [equivalent question on Stack Overflow](http://stackoverflow.com/questions/3474961/find-minimum-of-a-function-defined-by-integration-in-mathematica) was already answered. This is [the accepted answer by Andrew Moylan](http://stackoverflow.com/questions/3474961/find-minimum-of-a-function-defined-by-integration-in-... |
So, maybe I should write this as a response. So, let's take the notions in order. First, I will reserve the word "set" for objects in the theory, and use "collection" for the more intuitive notion.
Note that "$A$ is a subset of $B$" is an abbreviation for two statements: (i) $A$ is a set; and (ii) for every $x$, if ... |
I am tutoring several talented students, middle school level and early high school level, in mathematics. I am always looking for new sources from which to draw questions. Can anyone recommend books, web-sites, etc. with a interesting questions?
I know of the ArtofProblemSolving.com. I am currently using the books "... |
How do you calculate the following summation? $\sum_{i=k}^n i$ |
I've already taken my calculus sequence and I'm interested in brushing up and staying sharp on the basics. What books are available to help me? |
What are the recommended textbooks for introductory calculus? |
Familiar Exercise: Suppose $G$ is a finite group and $T$ is an automorphism of $G$ which sends more than three quarters of elements of $G$ onto their inverses, then prove that $G$ is abelian.
The group of Quarternions is an example of a group which sends exactly $\displaystyle \frac{3}{4}$ elements onto their invers... |
Having picked up a rudimentary understanding of tensors from reading mechanics papers and Wikipedia, I tend to think of rank-2 tensors simply as square matrices (along with appropriate transformation rules). Certainly, if the distinction between vectors and dual vectors is ignored, a rank 2 tensor $T$ seems to be simpl... |
Is it misleading to think of rank-2 tensors as matrices? |
Elementary Analysis : The theory of calculus by Kenneth Ross is a beautiful written book. Link is here : http://www.springer.com/mathematics/analysis/book/978-0-387-90459-7
Diferential and Integral Calculus by R.Courant is again a very good book which one can learn.
http://books.google.co.in/books?id=eyC1nk9-YjkC... |
Familiar Exercise: Suppose $G$ is a finite group and $T$ is an automorphism of $G$ which sends more than three quarters of elements of $G$ onto their inverses, then prove that $G$ is abelian.
The group of Quarternions is an example of a group which sends exactly $\displaystyle \frac{3}{4}$ elements onto their invers... |
The statements of Poincare duality for manifolds and Serre Duality for coherent sheaves on algebraic varieties or analytic spaces look tantalizingly similar. I have heard tangential statements from some people that there is indeed some connection between the two. But I was never able to figure it out on myself. For ins... |
This answer builds on damiano's comment.
Because of Wedderburn's theorem, we know that there are no finite-dimensional non-commutative division algebras over $\mathbb{F}_p$. In particular, every quaternion algebra over $\mathbb{F}_p$ is split, i.e., isomorphic to the algebra $M_2(\mathbb{F}_p)$ of $2 \times 2$ matr... |
Let me address the question: "what made Mathematicians ... think of something like this"?
The answer is: Galois, in studying the problem of factorization of polynomials, realized that
reasoning about the symmetries of the roots could be more powerful a tool than studying explicit formulas (which, loosely speaking, ... |
I've already taken my calculus sequence and I'm interested in brushing up and staying sharp on the basics.
I'm particularly interested in studying for my IEEE Certified Software Development Associate exam - 10% of the exam is mathematics (calculus, differential equations, and statistics). Calculus is one of the math... |
I've already taken my calculus sequence and I'm interested in brushing up and staying sharp on the basics. So far, my calculus background is limited to single-variable calculus, which I applied in my physics sequence and engineering statistics. I learned a little bit of vector math in physics as well, but I'm not stron... |
A friend of mine is subediting a book. The author of the book died in 1999. The author, at some point, uses the word "trillion" which is, unfortunately, an ambiguous word in the UK: when I was at school it used to mean $10^{18}$ but nowadays it means $10^{12}$. My friend is faced with the following paragraph:
'Accor... |
What I think the author intends in the first part is that some piece of genetic code gives you $8$ distinct amino acids and you want to count the number of ways to rearrange them to get a protein, which is where the $8! \approx 40000$ number comes from. Since there are $20$ amino acids, the best you can do for $30$ am... |
What I think the author intends in the first part is that some piece of genetic code gives you $8$ distinct amino acids and you want to count the number of ways to rearrange them to get a protein, which is where the $8! \approx 40000$ number comes from. Since there are $20$ amino acids, the best you can do for $30$ am... |
I need an algorithm to produce all strings with the following property. Here capital letter refer to strings, and small letter refer to characters. $XY$ means the concatenation of string $X$ and $Y$.
Let $\Sigma = \\{a_0, a_1,\ldots,a_n,a_0^{-1},a_1^{-1},\ldots,a_n^{-1}\\}$ be the set of usable characters. Every str... |
Is there any overarching reason why, after excluding the infinite classes of finite simple groups
(cyclic, alternating, Lie-type),
what remains---the *sporadic*, exceptional finite simple groups, is in fact a finite list (just 26)? In some sense, the prime numbers can be viewed as "sporadic," but there is an infinit... |
Why are there only a finite number of sporadic simple groups? |
Are you still stuck? I'll use your terminology, which seems to be used in electrical engineering.
1) x'z + xyz + xy'z
2) x'z + xz(y+y')
Since y+y' = 1, and xy1 = xy then
3) x'z + xz
4) (x'+x)z
5) z
Hope that helps. |
How should I rotate a point (x,y) to location (a,b) on the co-ordinate by any angle?
![alt text][1]
[1]: http://i.imgur.com/LQiiA.png |
Are you still stuck? I'll use your terminology, which seems to be used in electrical engineering.
1) x'z + xyz + xy'z
2) x'z + xz(y+y')
Since y+y' = 1, and xz1 = xz then
3) x'z + xz
4) (x'+x)z
5) z
Hope that helps. |
I have a function that (more or less) is supposed to select a small number m of random number from the range [1,n] (for some n >> m) and I need to test that it work. Is there an easy to implement test that will give good confidence that it is working correctly? (For reference the chi squared test is not simple enough.) |
Is there a simple test for uniform distributions? |
How to prove $(f \circ\ g) ^{-1} = g^{-1} \circ\ f^{-1}$ ? |
Here's a question. Have there been any theoretical results showing that if P != NP, there must exist some problems in NP which are not NP-complete and which are not in P either? Just curious because I've never seen this question addressed. |
Assuming P != NP, do we know whether there are problems which are in NP, not in P and are not NP complete ? |
Yes, this is known as Ladner's theorem, proved by Richard Ladner in 1975. The class of such problems are called [NP-intermediate](http://en.wikipedia.org/wiki/NP-Intermediate). [Here's one proof](http://books.google.com/books?id=47GwSJfb1mQC&pg=PA38) that does "cut and paste" between some problem in P and some NP-compl... |
A friend of mine is subediting a book. The author of the book died in 1999. The author, at some point, uses the word "trillion" which is, unfortunately, an ambiguous word in the UK: when I was at school it used to mean $10^{18}$ but nowadays it means $10^{12}$. My friend is faced with the following paragraph:
'Accor... |
I'd like to prove that $\forall x\left(x\not\in x\right)$ in the context of Morse-Kelley set theory.
Let's call $A=\left\\{y:y\not\in y\right\\}$. I can easily prove that $A\not\in A$. In fact, if you suppose that $A\in A$, it follows that $A\not\in A$ by definition of class comprehension. Since this is a contradict... |
How to prove that nothing is a member of itself? |
[*Calculus: Single Variable, Course Advantage Edition* by Hughes-Hallett, Gleason, et al](http://bcs.wiley.com/he-bcs/Books?action=index&itemId=0471448761&itemTypeId=BKS&bcsId=1404), aka the "Harvard Consortium" book, is perhaps the standard "reform calculus" book. |
The Sum of 2 sets of Measure zero might well be very large for example, the sum of $x$-axis and $y$-axis, is nothing but the whole plane. Similarly one can ask this question about Cantor sets:
If $C$ is the cantor set, then what is the measure of $C+C$? |
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