instruction stringlengths 22 27.2k |
|---|
It seems like an extremely easy question, but I can't solve it. All I know is that it is a $\frac{1}{2^n}$ chance that all tosses will result in heads and $\sum_{k=0}^{n} f(k, n)$ is equal to 1 given the function.
Even though school has just started for me, this is not a homework question.
|
Can you divide a disk into $7$ pieces of equal area, with $3$ line segments? (You can surely divide it into $7$ pieces, but could those have equal areas?)
(This question was left unanswered at another forum. I can see with some visual arguments that the answer should be no, but I couldn't find a nice way to write it ... |
Suppose $a_{n}>0$ and the following series converges
$\sum_{n=1}^{\infty} a_{n}^{3}$
Does this imply that
$\sum_{n=1}^{\infty} \frac{a_{n}}{n}$
converges?
I was able to prove that the second series also converges by using the limit comparision test. Is there another way to show the second series converges (e.g. root... |
This one has been bugging me for a while - I have a number of definitions for a symmetric group, none of which I understand. Clearly, $S_n$ is a group, which I follow. Now, the wikipedia definition (most readily to hand) is:
The symmetric group on a set X is the
group whose underlying set is the
collection of all ... |
This question just reminded me of a conundrum I posed myself in my first year of university. I never did get a satisfactory answer...
Let $a_n$ be a null sequence. Does it follow that $\sum \frac{a_n}{n}$ converges?
Any ideas?
|
I don't know whether the books metioned in Best ever book on Number Theory are beyond undergraduate/high-school-olympiad level.
Please recommend your favourite.
|
Motivation: A friend asked me this question.
The Problem: Suppose you start off with a dollar. You flip a fair coin, if it lands on heads you win $50$ cents otherwise you lose $50$ cents. If after $n$ flips you have a nonzero amount of money, you win. What's the probability you win? What about the limiting case as $n$ ... |
This is the equation I'm having trouble with:
$$G \frac{M m}{r^2} = m \frac{d^2 r}{dt^2}$$
That's the non-vector form of the universal law of gravitation on the left and Newton's second law of motion on the right. I assume that upon correctly modeling and solving this, I will have a function of time that gives the dist... |
I just finished reading the Wikipedia article on the Cauchy condensation test. I understand the trapezoidal view, but apparently "the 'condensation' of terms is analogous to a substitution of an exponential function". Can anyone explain what is meant by this?
|
Can anyone explain the syntax from the following Wikipedia article on the convergence of random variables?
$$
\begin{array}{ll}
X_{n} \stackrel{d}{\rightarrow} X, & X_{n} \stackrel{D}{\rightarrow} X, \quad X_{n} \stackrel{\mathcal L}{\rightarrow} X, \quad X_{n} \stackrel{d}{\rightarrow} \mathcal{L}_{X}, \\
X_{n} \right... |
For a given $k>0$ constant, assuming that $x,y>0$. Also this equation has a particular name, or some mathematician associated with it?
|
Let $A$, $X$, and $Y$ be arbitrary topological spaces. Let $F:X\times A\rightarrow Y$ be a continuous function. Let $P$ be the map from $X\times A$ to $Y\times A$ taking $(x,a)$ to $(F(x,a),a)$. Does it follow from continuity of $F$ that $P$ is continuous?
|
I'm mainly a CS major, but I also want to learn more advanced mathematics. I see lot of cross over between CS and applied math, same can't be said about pure math.
Definition of 'advanced': Something a pure math major learns during and after late undergraduate years.
What are the fields that combines advanced mathemat... |
Is there any book or online note that covers the basics of lie groups using nonstandard analysis? Another thing I would like is to see these things in category theory (along the lines of Algebra: Chapter 0 except for differential geometry?)
|
Let $X$ be a compact metric space. Given a sequence $x_n \in X$ and an ultrafilter $\mathcal{U}$ on $\mathbb{N}$, we can define the "Banach limit" of ${x_n}$ with respect to $\mathcal{U}$. This limit is the unique element of $X$ such that every neighborhood of it contains a $\mathcal{U}$-large collection of the sequen... |
What is the difference between convergence in distribution, convergence in probability and almost sure convergence?
References
Convergence of random variables
|
I have just started auditing Linear Algebra, and the first thing we learned was how to solve a system of linear equations by reducing it to row echelon form (or reduced row echelon form). I know how to solve a system of linear equations using determinants, and so the using row echelon form seems to be very inefficient ... |
I am working a bit on a collection of Linear Algebra examples,
as well as some examples on induction. This is what is taught freshman year at our university.
I intend to release this to the public, either by selling printed copies or releasing it online.
Since I do not have experience using such material myself, ther... |
Let $m,n \geq 2$ be integers. Find all functions $f:[0,\infty) \to \mathbb{R}$ continuous at at least one point in $[0,\infty)$ and such that $$f\biggl(\frac{1}{n} \sum\limits_{i=1}^{n} x_{i}^{m} \biggr)=\frac{1}{n} \sum\limits_{i=1}^{n}(f(x_i))^{m} \quad \text{for} \ x_{i} \geq 0, i=1,2,...,n$$
I am not getting any id... |
I understand what the problem with Gimbal Lock is, such that at the North Pole, all directions are south, there's no concept of east and west. But what I don't understand is why this is such an issue for navigation systems? Surely if you find you're in Gimbal Lock, you can simply move a small amount in any direction, a... |
Let $a_n\geq 0$ be a sequence of non-negative numbers. Consider the following two statements:
$$
\text{(I)}\qquad\qquad \lim_{n\to\infty} \frac{1}{n^2}\sum_{i=1}^n a_i =0,
$$
$$
\text{(II)}\qquad\qquad\qquad \sum_{n=1}^\infty \frac{a_n}{n^2}<\infty.
$$
Questions: Does (I) imply (II)? Does (II) imply (I)? Otherwise, pl... |
Given the following:
$\sum_{i=1}^{n} a_i = 1, a_i \geq 0\;\forall i$,
$\sum_{i=1}^{n} b_i = 1, b_i \geq 0\;\forall i$,
$\sum_{i=1}^{n} |a_i - b_i| \leq e$ where $e \ll 1$,
what is the upper bound on:
\begin{equation*}
\sum_{i,j=1}^{n} |a_i\cdot a_j - b_i\cdot b_j|?
\end{equation*}
I am almost sure it is bounded b... |
On a two dimensional plane, line $X$ is at an angle of $x$ radians and an incoming light travels at an angle of $y$ radians. How can I calculate the angle of the outgoing light reflected off of the line $X$? How can I cover all possible cases?
Edit: I was trying to figure out Project Euler problem 144.
|
Imagine a process with two variables min and max, and two counters hi and lo.
We initialize min and max by selecting two random numbers (assume a uniform (0,1) distribution for convenience), and sorting them. We also set the hi and lo counters to 0.
At each step we select a new random number r from our distribution an... |
Prove (or disprove) the following statement:
For any positive integers $x,y,t$,
$$\sum_{i=1}^{t(y+1)-1} \frac{1}{t(xy+x-1)-x+i}$$
is an increasing function of $t$.
My attempts:
The statement appears to be true numerically.
Tried some obvious bounds to compare the sums for consecutive values of $t$ but didn't find one t... |
I've been reading about tropical geometry and many papers reference polyhedral fans. I feel like I have a decent intuitive picture of what they are from reading articles but I still haven't been able to guess the general definition. All the ones I've encountered have been systems of linear inequalities, so that is my b... |
What is the the most educational software for high school and college math?
Not the one that just gives you the answer, but has any of the following:
Edit: Software mentioned in answer added in brackets
provides step-by-step solutions [WolframAlpha];
can verify your step-by-step solution [empty];
can give you hints ... |
Both matrix multiplication and quaternion multiplication are non-commutative; hence the use of terms like "premultiplication" and "postmultiplication". After encountering the concept of "quaternion matrices", I am a bit puzzled as to how one may multiply two of these things, since there are at least four ways to do thi... |
I need to find eigenvalues/eigenvectors of different kinds of $n \times n$ matrices. For example, how would I determine these for the matrices listed below? What is the typical process? Should I always go by the route of finding eigenvalues by finding roots of characteristic polynomial and then getting eigenvectors by ... |
I recently noticed a few things in some recent questions on MO:
1) the fundamental group of $S^2$ minus, say, 4 points, is $\langle a,b,c,d\ |\ abcd=1\rangle$.
2) The fundamental group of a torus minus a point is $\langle a,b,c\ |\ [a,b]c=1\rangle$.
I was just wondering if you have a manifold $M$, and you know a presen... |
I am not sure if this question is appropriate here but would appreciate any help. Let $n>2$ and $x_1,\cdots,x_n$ be real numbers. What is the infimum of:
$$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} $$
Thanks in ... |
Background
It is a standard and important fact in basic calculus/real analysis that a continuous function on a compact metric space is in fact uniformly continuous. That is, suppose $(X,d)$ is a compact metric space and $f\colon X \to\mathbb R$ is such that for every $x\in X$ and $\varepsilon>0$ there exists $\delta>0... |
This seems counterintuitive, but $22/7$ is closer to $\pi$ than $3.14=314/100$ which has a significantly greater denominator.
Why is $22/7$ a better approximation for $\pi$ than $3.14$?
This has important implications: e.g. Should "$\pi$-day" be the $14^{th}$ of March or $22^{nd}$ of July?
|
How would you find eigenvalues/eigenvectors of a $n\times n$ matrix where each diagonal entry is scalar $d$ and all other entries are $1$ ? I am looking for a decomposition but cannot find anything for this. For example:
$\begin{pmatrix}2&1&1&1\\1&2&1&1\\1&1&2&1\\1&1&1&2\end{pmatrix}$
|
Although whether $$ 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, if ... |
If $k_1,k_2,...,k_n$ are principal curvatures of a hypersurface in $\mathbb{R}^{n+1}$ then one can apparently locally parametrize the manifold as $(x_1,x_2,...,x_n,y)$ such that,
$y = \frac{1}{2}(k_1 x_1^2 +...+k_n x_n^2) + O(\vert x \vert ^3) $
I would like to know how this is proven.
I guess the coordinates are obta... |
Suppose we have a functional question, such as the one here. Often a key step in these problems is to try to equate or chain results together (see my hint). 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 examined mor... |
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.
|
Possible Duplicate:
Subgroup of $\mathbb{R}$ either dense or has a least positive element?
Let $(\mathbb{R},+)$ be the group of Real Numbers under addition. Let $H$ be a proper subgroup of $\mathbb{R}$. Prove that either $H$ is dense in $\mathbb{R}$ or there is an $a \in \mathbb{R}$ such that $H=\{ na : n=0, \pm{1},... |
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 comp... |
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 $G$?
|
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/Category:Mat... |
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 $Ω_n(H)$ to be the subgroup g... |
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. Algeb... |
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 there is very confusing. It is certainly not a direct sum of points; perhaps it is better to understand it in terms ... |
Let's take field $(K, +,\times,0,1)$. Let's take set $S(K) \subset K$ such that:
\[ \forall~e \in K,\exists~a \in S(K),\exists~p \in K: e = a \times p \times p \]
\[ \lnot \exists~a, b \in S(K), \exists~p \in K, (a \times p \times p = b \land a \neq b) \]
For example $S(\mathbb{R}) = \\{-1, 1\\}$, $S(\mathbb{C}) = \\{1... |
In the comment thread of an answer, 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 comput... |
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 ... |
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?
|
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 expressions ... |
Was reading "Taylor's Expansion" when i came across this interesting problem, interesting for me.
Suppose that $f^{n+2}$ is continuous on $[0,x]$. Prove that there is a $\theta \in (0,1)$ such that
\begin{align}
f(x) &= f(0) + \frac{f'(0)}{1!}x + \cdots + \frac{f^{n-1}(0)}{(n-1)!}x^{n-1} \\
& \quad + \frac{f^{n}\left... |
Before starting algebraic geometry, I got some understanding of Compact Riemann Surfaces which is more or less rigorous; and then I attended a couple of lectures on analytic geometry. I did not quite fully grasp the analytic geometry rigorously; but it stuck in my mind and this previous experiences helped me greatly wh... |
These are two famous identities of Ramanujan. Where can I find the proofs of them:
$ \displaystyle \frac{1}{\pi} = \frac{2\sqrt{2}}{9801} \sum\limits_{k=0}^{\infty} \frac{(4k!)(1103 + 26390k)}{(k!)^{2} (396)^{4k}}$
$\displaystyle \int\limits_{0}^{\infty} \frac{1 + x^{2}/(b+1)^{2}}{1+x^{2}/a^{2}} \times \frac{1+ x^{2}... |
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 the se... |
The correspondence between algebraic and analytic geometry is thought-provoking. The GAGA makes this precise to some extent. But there is more of this analogy. As I checked an article on analytic space, most of the notions there seems to have an algebraic analogue. I had been wondering why people studied analytic geome... |
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 fin... |
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 $\pi > 333/106$?
|
Can every compact metric space be realized as the continuous image of a cantor set?
|
The Phragmén–Lindelöf theorem gives a consequence of the Riemann hypothesis, viz., the Lindelöf hypothesis. As such this is weaker than Riemann hypothesis; but it is still considered that even a proof of this weaker result will be a breakthrough.
Question:
What is the strongest known result yet on the Lindelöf hypothe... |
I am a neophyte amateur mathematician. I have been reading a lot about journals and the topic of peer-review in mathematics journals. Does one have to have professional credentials or have a Doctorate in order to publish in peer-reviewed mathematics journals or just the desire to compently solve mathematical problems? ... |
What software do you use to accurately draw geometry diagrams?
|
Morera's theorem is an underappreciated theorem in complex analysis. I have been struck by the simplicity of its proof and some clever applications of it and I had been interested in finding out more of such. Please contribute examples. One example is the Weierstrass theorem that if a sequence of holomorphic functions ... |
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 $x_1,\ldots,x_n$ 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: the answer was -- it m... |
How are operations such as the sum, the product, the quotient, exponentiation, etc. of random variables solved or approached?
This question addresses a similar problem but starts one step further: knowing how to find the distribution of a function of random variables. I'm asking for the previous step.
Then, my question... |
I recently read a very good inequality concerning the no of primes $\pi(x)$:
$$\pi(n)>\frac{1}{6}\frac{n}{\log n}\mathrm{\ for\ }n\ge 2$$
Are any other such elementary inequalities concerning the primes?
|
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 dig... |
What is the probability of getting a yahtzee using $N$ dice with $X$ sides in $Y$ throws in a single round?
Which side of the dice appears on the yahtzee with doesn't matter (i.e. it doesn't matter if I throw ones, or twos, etc.). I also assume perfect strategy is used; that is, after each throw, one saves the number a... |
We should have $\displaystyle\mathbb{Q} = \lim_{\rightarrow} \frac 1n \mathbb{Z}$ 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 c... |
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?
|
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 get the determ... |
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} $. What are... |
Give an example of sequence of Measurable functions defined on some measurable subset $E$ of $\mathbb{R}$ such that $f_{n} \to f$ pointwise almost everywhere on $E$ but $$\int\limits_{E} f \ dm \not\leq \lim_{n} \inf \int\limits_{E} f_{n} \ dm$$
|
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... |
We know that if $ \displaystyle d(n)= \sum\limits_{d \mid n} 1$, then we have
$$ \sum\limits_{n \leq x} d(n)= x\log{x} + (2C-1)x + \mathcal{O}(\sqrt{x})$$
I have referred Apostol's "Analytic Number theory" and i understood the first half of the proof where the error term is $\mathcal{O}(x)$, but please tell me as to h... |
Recently I found this post on Reddit. It describes the following algorithm to find e:
Here is an example of e turning up
unexpectedly. Select a random number
between 0 and 1. Now select another
and add it to the first. Keep doing
this, piling on random numbers. How
many random numbers, on average, do
you ... |
Is there a isomorphism between the additive group of real algebraic numbers and the multiplicative group of positive real algebraic numbers, which is order preserving.
|
How can I determine the size of the largest collection of $k$-element subsets of an $n$-element set such that each pair of subsets has at most $m$ elements in common?
|
From: Moduli space
we see that moduli of curves is a very algebro-geometric topic.
It is easy to understand its relevance and importance in algebraic geometry. But the mind boggles when we try to imagine how on earth such a topic from pure and abstruse mathematics is relevant in physics.
I will be thankful if somebod... |
I am trying to brush up my graph theory skills. I have not done any in over 4 years and i am rusty...If someone could help me out with this simple proof i would appreciate it.
Prove that for any graph $G$ of order at least 2, the degree sequence has at least one pair of repeated entries.
So the degree sequence if a lis... |
Here is a time sample: $Q = \{(t_i, x_i) | 0 \leq x_i \leq x_{i+1}, 1 \leq i \leq n\}$
and rules:
(1) $T_1 \leq t_{i+1} - t_i < T_2$ where $T_1, T_2 > 0$
(2) $x_i$ comes with error:
$x_i = \lfloor x(t_i) \rfloor$ where $\lfloor x \rfloor$ - whole part of $x$, $x(t_i)$ - voluntary unknown law of object motion.
How we c... |
The reason I ask this is because difficult mathematical proofs are just not plain self-evident. You would need a few years of intensive study before you can get to the point of understanding the topics and the proofs. The problem is that for a difficult math theorem, maybe only a handful of mathematicians have studied ... |
We know that there exist real functions which are continuous at each irrational and dis-
continuous at each rational number.
But does there exist a function $f: \mathbb{R} \to \mathbb{R}$ that is differentiable at every irrational and discontinuous at every rational?
|
For any natural number $n > 1$, define $E(n)$,to be the highest exponent to
which a prime divides it. For instance, $E(12)=E(36)=2$. Show that $$\lim_{N \to \infty} \frac{1}{N} \sum\limits_{n=2}^{N} E(n)$$ exists and find its value
|
We know that $\displaystyle\zeta(2)=\sum\limits_{n=1}^{\infty} \frac{1}{n^2} = \frac{\pi^2}{6}$ and it converges.
Does there exists a bijective map $f:\mathbb{N} \to \mathbb{N}$ such that the sum $$\sum\limits_{n=1}^{\infty} \frac{f(n)}{n^2}$$ converges.
If our $s=2$ was not fixed, then can we have a function such ... |
Does there exist a finite group $G$ and a normal subgroup $H$ of $G$ such that $\lvert\operatorname{Aut}{(H)}\rvert>\lvert\operatorname{Aut}{(G)}\rvert$?
|
Problem: Let $n$ be an integer and $p$ a prime dividing $5(n^2-n+\frac{3}{2})^2-\frac{1}{4}$. Prove that $p \equiv 1 \pmod{10}$.
The polynomial can be re-written as $(\sqrt{5}(n^2-n+\frac{3}{2})-\frac{1}{2})(\sqrt{5}(n^2-n+\frac{3}{2})+\frac{1}{2})$. If this vanishes mod $p$ then $5$ is a quadratic residue mod $p$, whi... |
Sometimes one encounter the requirement that the objects of a category needs to be a set. What if it was not, could you provide examples of what could go wrong? (One example per answer).
|
I have a data set $2\; 3\; 3\; 4\; 4\; 4\; 4$
I want to find the number of unique numbers of $3$ digit numbers that can be formed using this.
I was thinking of doing $\large{\frac{^7P_3}{4!\times 2!}}$, but this doesn't seem right.
|
Is possible to construct infinitely differentiable functions that interpolate through arbitrary points, the way polynomial splines do? If so, do they have a name and is there an algorithm for constructing the curves.
I suppose a infinite degree polynomial spline might qualify but I was hoping there might be something m... |
If $a < b$, which of the following are always correct?
1) $a^3< a^2b$
The answer to this question would be true because there is theorem that "If $a < b$ and we add the same positive number to both sides of the inequalitie, this would not effect the inequality."
Now the second problem is interesting:
2) $a^2 < ab$
Now ... |
I have a partition of a positive integer $(p)$. How can I prove that the factorial of $p$ can always be divided by the product of the factorials of the parts?
As a quick example $\frac{9!}{(2!3!4!)} = 1260$ (no remainder), where $9=2+3+4$.
I can nearly see it by looking at factors, but I can't see a way to guarantee it... |
Now, I may have only slept two hours last night and would currently struggle to discern a 'proof' by induction of FLT from a piece of genuine mathematics, but that doesn't stop mathematics from bugging me. At present I am puzzled by something I saw on MO this morning...
The linked question concerns the Hilbert cube $[0... |
Statement For every $n > 1$ there is always at least one prime $p$ such that $n < p < 2n$.
I am curious to know that if I replace that $2n$ by $2n-\epsilon$, ($\epsilon>0$) then what is the $\inf (\epsilon)$ so that the inequality still holds, meaning there is always a prime between $n$ and $2n-\epsilon$
|
As a teenager I was given this problem which took me a few years to solve. I'd like to know if this hae ever been published. When I presented my solution I was told that it was similar to one of several he had seen.
The problem:
For an $n$ dimensional space, develop a formula that evaluates the maximum number of $n$ ... |
Let $G$ be a graph such that $\delta(G) \geq k$.
Prove that $G$ has a path of length at least $k$.
Solution: We know that
$\delta(G) = \min\lbrace \deg(v) \mid v \in V(G) \rbrace$
If $\delta(G) = k$ then there exists some $v \in V(G)$ such that $\deg(v) = k$. This means all other vertices $u \in V(G)$ have $\deg(u)... |
Ax = b. I need a way to analyze a square matrix A to see if its solution vector x will always be positive when b is positive.
This question arises from solving the radiosity equation:
(source: siggraph.org)
I'm interested to know when A is incorrect, which would be when x has negative values even though b is positiv... |
Suppose $G$ is a locally compact abelian group, with Haar Measure $\mu$, then is $\mu(E)=\mu(E^{-1})$ for all subsets $E$ of $G$?
I have seen that this is true for all Borel subsets of $G$, but I am in need of a proof and also an example showing why it doesn't hold for other subsets.
|
Supose we have a set $S$ of natural numbers. We define:
$$ f(x) = \begin{cases} {x \over 2} & x \text{ is even } \\ {x - 1 \over 2} & x \text{ is odd }\end{cases} $$
Now let $a_0 = a_1 = 1$ and $a_n = a_{n-1} + a_{n-2}$ for $n \notin S$ and $a_n = f(a_{n-1} + a_{n-2})$ otherwise for $ n > 1$.
I want to calculate the r... |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.