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Groups can be defined abstractly as sets with a binary operation satisfying certain identities, or concretely as a collection of permutations of a set. Cayley's theorem ensures that these two definitions are equivalent: any abstract group acts as a collection of permutations of its underlying set, and this action is f... |
One can see this reference for TBL space-times.
I would like to know how the explicit expression for the function called $G$ in equations $3.108,3.108,3.110$ in the above reference is obtained.
Also it would be nice to see some further references about TBL space-times.
|
Suppose we define a topology on the real numbers as that obtained by defining open sets to be those such that the complement is finite. How does one show that this topology cannot be obtained from a distance function?
Thanks in advance.
P.S.: This is not a homework question. I am trying to self study point set topology... |
This is from Feller's Introduction to Probability Theory and Its Applications. In the context of Bernoulli trials, we define:
$$b(k;n,p) = \binom{n}{k}p^kq^{n-k},$$
$$P\{S_n \ge r\} = \sum_{v=0}^{\infty}b(r+v;n,p).$$
The latter being the probability of having at least $r$ successes. Now, supposing $r \gt np$ and knowin... |
What is the proper definition of a Countable Set?
|
I'm quite new at measure theory and all I know I read by myself, so bear with me if this question is too trivial.
Given the definition of asymptotic, can we say that for $x \rightarrow \infty$ two almost everywhere equal functions are asymptotic? I really think that this is not the case, but I haven't been able to fin... |
Let $C^0$ be the set of all of all (edit: bounded) continuous functions from $\mathbb R \to \mathbb R$ with the sup norm. Then a "path" in $C^0$ is a continuous function $f \colon [0,1] \to C^0$. The question boils down to if there is path (or continuous deformation) which transforms any one bounded continuous functi... |
It is often said that differential forms (sections of an exterior power of the cotangent bundle) are the things that you can integrate. But unless I'm being thoroughly dense differential forms are not the only things that you can integrate, c.f. the arclength form (on a 2d manifold) $ds=\sqrt{dx^2+dy^2}$, the unsigned ... |
How can we prove that it is posible to arrange numbers $1,2,3,4,\ldots, n$ in a row so that the average of any two of these numbers never appears between them?
|
I haven't studied properly the theory of infinities yet.
Let $A_0$ denote the set of natural numbers. Let $A_{i+1}$ denote the set whose elements are all the subsets of $A_i$ for $i=0,...,n,...$
I understand well that the cardinality of $A_{i+1}$ is always greater than the cardinality of $A_i$ for all $i \in \mathbb{N}... |
I would like to know whether the number of infinity sets of different cardinalities is uncountable or countable. Is the proof simple?
Thanks.
|
I'm making a game which you can see here, if you are on Windows or Linux: http://insertnamehere.org/birdsofprey/
If you click and hold your mouse on a bird, you can see I'm just swinging the bird back and forth in pendulum motion. I would like to, instead, implement a more realistic motion, where the movement of your m... |
Is there a comprehensive resource listing binomial identities? I am more interested in combinatorial proofs of such identities, but even a list without proofs will do.
|
"Life is open book."
With the advent of widely accessible, inexpensive (or even free) computational tools and Computer Algebra Systems (TI-89, Wolfram|Alpha, etc.), much of what traditionally comprises a high school math curriculum can now easily be done by almost everyone. Factoring polynomials, solving inequalities,... |
The Banach--Tarski theorem applies only in the case of three or more dimensions. In 3D, there are five regular solids, two of them being not at all obvious, and the 4D case is also interesting; but the higher-dimensional cases each yield just three "solids", only one of which isn't obvious. In dynamics, particles tend ... |
I am aware that on a finite set the number of equivalence relations is the n-th Bell's Number. On the other hand, the only reference I could find on the web for infinite sets was this: On counting equivalence relations by D.J. Baylis, in which he proves that the set of equivalence relations on $\mathbb{N}$ is uncountab... |
I'm studying for math exam and one of the questions that often appears is related to derivative of a product of two functions.
The theorem says that $(f(x)g(x))'=f'(x)g(x)+f(x)g'(x)$.
The proof goes like this:
$f(x+h)g(x+h)-f(x)g(x)=(f(x+h)-f(x))g(x)+(g(x+h)-g(x))f(x)$
After that we divide the equation by h and let h a... |
I know the method of solving the equation like this:
$$|2x+1|=|3x+9|$$
but the problem is if the same equation would be like this
$$| 2x+1 | = x | 3x+9 |$$
How can I solve this?
|
Can anyone help me out here? Can't seem to find the right rules of divisibility to show this:
If $a \mid m$ and $(a + 1) \mid m$, then $a(a + 1) \mid m$.
|
In a recent discussion, someone told me tuples in the modern meaning (in particular, tuples are heterogeneous: that is, different elements of a tuple can belong to different sets/have different types) first appeared in Codd's tuple calculus. I was surprised it would be so late, but searching Google Books before 1970, ... |
Given:
$A = (0,0)$
$B = (0,-10)$
$AB = AC$
Using the angle between $AB$ and $AC$, how are the coordinates at C calculated?
|
Some days ago as I had asked as to how to test the Riemann Integrability of the function. Now I was recently given this question about proving that the given function is Riemann integrable.
How can i show that this function is Riemann integrable or not in the interval $[0,1]$. I tried using partitions, but it didn't ... |
At 5am, a pedestrian starts walking from $A$ to $B$, a distance of $30$ km. At $7$ am a bicyclist whose speed is $2$ times the pedestrian's speed, started riding from $A$ to $B$, too. After some time, the bicyclist met and passed the pedestrian. Two hours after the meeting the pedestrian reached his destination ($B$).... |
I am interested in continuous functions which are not smooth, i.e. which have only finitely many derivatives (for now just taking the case $f:\mathbb{R}\rightarrow\mathbb{R}$). As far as I can tell, there are only 2 features or properties of such a function which can make it non-smooth.
The derivative of some order d... |
If I want to show a topological subspace is closed in an ambient space, does it suffice to know what happens on an open cover of the ambient space? More specifically,
Let $X$ is a topological space with a given open cover ${ U_i }$. Suppose that $Z \subset X$ is a set such that $Z \cap U_i$ is closed in $U_i$ for all ... |
An absolute value expression such as $|ax-b|$ can be rewritten in two cases as $|ax-b|=\begin{cases}
ax-b & \text{ if } x\ge \frac{b}{a} \\
b-ax & \text{ if } x< \frac{b}{a}
\end{cases}$, so an equation with $n$ separate absolute value expressions can be split up into $2^n$ cases, but is there a better way?
For examp... |
This was part of a homework problem from J.B. Conway's complex analysis text which I was assigned long ago but didn't get. A few years later I was a TA for a course where the problem was assigned. I still didn't know how to solve it, nor did any of my students. If you can give me a simple answer, my gratefulness wil... |
Let $p >5$ be a prime number. Prove that every algebraic integer of the $p$th cyclotomic field can be represented as a sum of (finitely many) distinct units of the ring of algebraic integers of the field.
Reference: http://www.artofproblemsolving.com/Forum/resources.php?c=2&cid=152&year=1977&sid=151602f87027a7ce87d3aa9... |
Serre's criterion for affineness (Hartshorne III.3.7) states that:
Let $X$ be a noetherian scheme. Suppose $H^1(X, \mathcal{F})= 0$ for every quasi-coherent sheaf on $X$. Then $X$ is affine.
There is a more general statement (EGA II.5.2.1) that reaches the same conclusion with the hypothesis that $X$ is a separated q... |
By "isomorphism" I mean any structure-preserving map with a structure-preserving inverse.
(Please accept my advance apology if this question is out of bounds. I sense that it's borderline, but I'm hoping it'll be considered in better taste than the typical "What is your favorite X?" question. I think a collection of gr... |
I was having a look at this question on SO.
From what I know, the centroid is the center of mass of an object. so, by definition its position is given by a simple mean of the positions of all the points in the object.
For a polygon, it only has mass at the vertices. So, the centroid should be given by the arithmetic me... |
Here algebraic group means affine algebraic group in both instances. Also I'm mainly interested in groups over $\mathbb{C}$. In fact I'm taking $\pi_1(G)$ to mean the fundamental group of $G_{an}$, the analytification. So I guess my question only applies to the base field being either $\mathbb{C}$ or $\mathbb{R}$. ... |
This question made me realize I had a misconception about the cyclotomic integers: I thought the units were exactly the roots of unity. There are only finitely many units but infinitely many integers so the question is impossible to solve unless there are more units. So what are the units of cyclotomic integers?
|
From Wikipedia:
First, we can use Einstein notation in
linear algebra to distinguish easily
between vectors and covectors: upper
indices are used to label components
(coordinates) of vectors, while lower
indices are used to label components
of covectors. However, vectors
themselves (not their components... |
I am aware of gauge transformations and covariant derivatives as understood in Quantum Field Theory and I am also familiar with deRham derivative for vector valued differential forms.
I thinking of the gauge field A of the gauge group G as a Lie(G) valued 1-form on the manifold.
But I can't see why under a gauge tra... |
Can anyone kindly give some reference on taking trace of vector valued differential forms?
Like if $A$ and$B$ are two vector valued forms then I want to understand how/why this equation is true?
$dTr(A\wedge B) = Tr(dA\wedge B) - Tr(A\wedge dB)$
One particular case in which I am interested in will be when $A$ is a Lie ... |
If G is a simply connected Lie Group then why is every G-bundle over an orientable 3-manifold trivial? (Why is orientability important?)
|
Is there a proof that the ratio of a circle's diameter and the circumference is the same for all circles, that doesn't involve some kind of limiting process, e.g. a direct geometrical proof?
|
Does a limit that has the value of infinite exist or not?
I've recently come across certain sources that say that if the value of a limit is infinite, then that limit does not exist. This contradicts what my calculus teachers and lecturers taught me however, that a limit doesn't exist if the right hand limit and left h... |
We have a set of items $I = \{i_1, i_2, ..., i_n\}$. Each of these items has what we call a p value, which is some real number. We want to choose a subset of $I$, call it $I'$, of size $m$ (for some $m$ with $1 \le m \le n$) such that the average of the p values of the items in $I'$ falls within some specified range,... |
Possible Duplicate:
-1 is not 1, so where is the mistake?
I'm trying to understand the exact point of failure in the following reasoning:
\begin{equation*}
1 = \sqrt{1} = \sqrt{(-1)(-1)} = \sqrt{\sqrt{-1}^2\sqrt{-1}^2} = \sqrt{(\sqrt{-1}\sqrt{-1})^2} = \sqrt{-1}\sqrt{-1} = \sqrt{-1}^2 = -1.
\end{equation*}
I've been... |
Consider the sequence defined as
$x_1 = 1$
$x_{n+1} = \sin x_n$
I think I was able to show that the sequence $\sqrt{n} x_{n}$ converges to $\sqrt{3}$ by a tedious elementary method which I wasn't too happy about.
(I think I did this by showing that $\sqrt{\frac{3}{n+1}} < x_{n} < \sqrt{\frac{3}{n}}$, don't remember exa... |
$$ \frac{4^n-1}{n-1} > 4 × 10^6 $$
What is an easy way to solve it by hand? I need to find the minimum integral value of n to satisfy this equation.
|
Suppose $f:[0,1] \to \mathbb R$ is defined as
$$
f(x) =
\begin{cases}
1, &\text{if } x \text{ is irrational; and } \\
0, &\text{if }x \text{ is rational}.
\end{cases}
$$
I know from Real Analysis that $\int_0^1 f =1$, but intuitively it seems that the "area under curve" is 0.
Is there some intuitive explan... |
How can I convert a hexadecimal number, for example 0x1A03 to its octal value?
I know that one way is to convert it to decimal and then convert it to octal
0x1A03 = 6659 = 0o15003
Is there a simple way to do it without the middle step (conversion to decimal or conversion to binary)?
Why do we tend to convert it to Ba... |
I am having some difficulty following the concept of shrinking a set of node as given in the paper On the Bottleneck Shortest Path Problem
It says:
Shrinking a set of nodes can be done
in linear time. More precisely, given
a set S ⊆ V of nodes of an
(undirected) graph G = (V,E), one can
construct in linear tim... |
I would like help with the following problem (chapter 9, #4) from Lee's Smooth Manifolds [its not homework, I'm reading it and I got stuck on this one]
If a Lie group $G$ acts smoothly and freely on a smooth manifold $M$ and the orbit space $M/G$ has a smooth manifold structure such that the quotient map $\pi: M\to M/G... |
I'm trying to find out for which natural numbers $n$, does $9^n + 1$ have all of its prime factors less than $40$. If I can provide a positive answer to my title question, then I will have a proof that only $n = 1$ causes $9^n + 1$ to have all of its prime factors less than $40$. Thank you for any help.
|
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 likely $\min... |
In a recent post 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 question in a proper way.
I define a "kink" in the graph of a function as an abrupt, discontinuou... |
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 ... |
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 How can I understand and prove the "sum and difference formulas" in trigonometry?) and somewhat intuitive geometrically. However, even if this fact seems f... |
Somewhat prompted by the discussions of Qiaochu Yuan and Aryabhata in this question, I realized that my understanding of linear/affine transformations thus far had been built on a convoluted series of circular arguments. I will now be asking a question in order to patch the gaps in my knowledge.
Due to my innate tenden... |
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... |
How do I evaluate this function for given $s$?
$$\zeta(s) = \sum_{n=1}^\infty \frac1{n^s} = \frac{1}{1^s} + \frac{1}{2^s} + \frac{1}{3^s} + \cdots$$
|
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 refers not only to polynomials as approximator functions. Where could I find these Weierstrass-like approximation ... |
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 $p,q \in P$ we ... |
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 you any in... |
I asked myself this question when I was a young boy playing around with the calculator.
Today, I think I know the answer, but I'm not sure whether I'd be able to explain it to a child or layman playing around with a calculator.
Hence I'm interested in answers suitable for a person that, say, knows what the square root ... |
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 subdivide a cubic Bézier curve at a given t value using de Casteljau’s algorithm, which yields two cubic Bézier 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 illustrates what I’m inten... |
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 $
$$ \Gamma(x) + \Gamma(y) \le \Gamma(x+y) \quad ?$$
|
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 \mathbb{... |
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 integers and alm... |
If $i=\sqrt{-1}$, is $\large\sqrt{i}$ imaginary?
Is it used or considered often in mathematics? How is it notated?
|
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?
|
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?
|
Back in the day, I was absolutely enthralled by the study of plane curves and their properties (I have Lockwood and Zwikker 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), one sho... |
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 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.
|
Sometimes there are many ways to define a mathematical concept, for example the natural base logarithm. How about sine and cosine?
Thanks.
|
My question is why do we define $f(\xi+0)$ as $\lim\limits_{\varepsilon \rightarrow 0} f(\xi + \varepsilon^2)$ and $f(\xi-0)$ as $\lim\limits_{\varepsilon \rightarrow 0} f(\xi - \varepsilon^2)$.
Thanks.
|
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 ? ?
4 ? ?
... |
Let $f: D \to \mathbb{R}\ $ be a function on non-singular, convex domain $D \subseteq \mathbb{R}^d$ 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 ... |
In the Portuguese book Análise Matemática (Mathematical Analysis) by C. Sarrico, it is proved there exist
$$\displaystyle\lim_{n\rightarrow +\infty}f_{n}(x)=\lim_{n\rightarrow +\infty }e^{\log f_{n}(x)},\qquad(1)$$
where
$$f_{n}(x)=\dfrac{n!n^{x}}{x(x+1)(x+2)\cdots (x+n)}.\qquad(2)$$
And it is stated there that
$$\disp... |
Determine the Euler characteristic of the surface
$$ M=\left\{(x,y,z); \sqrt{x^2+y^2}=1+z^{2n}, 0< z< 1\right\} $$
|
I'm struggling with this former Putnam Exam problem:
Suppose $f$ and $g$ are nonconstant, differentiable, real-valued functions on $R$. Furthermore, suppose that for each pair of real numbers $x$ and $y$, $f(x + y) = f(x)f(y) - g(x)g(y)$ and $g(x + y) = f(x)g(y) + g(x)f(y)$. If $f'(0) = 0$, prove that $(f(x))^2 + (g(... |
Let $f : [0, 1] \to \mathbb{R}$ be a 'well behaved' function and define $c_n$ by the Riemann sums
$$
c_n := \frac{1}{n} \sum_{k = 1}^n f\left(\frac{k - \theta}{n}\right)
$$
for some given $\theta \in [0, 1]$. Then obviously $c_n$ will converge to the integral of $f$. Under which conditions on $f$ will this convergence ... |
Walking on a lattice. The number of various paths from $(0,0)$ to $(m,n)$ using north and east steps is binomial coefficient
$C(m+n,m)$
if he needs to go back $(0,0)$ using south and west steps, and doesn't pass by the passed points. Then what is the number of various loops walking from $(0,0)$ to $(m,n)$ then return... |
A continuous function does not always map open sets to open sets, but a continuous function will map compact sets to compact sets. One could make list of such preservations of topological
properties by a continuous function $f$:
$$ f( \mathrm{open} ) \neq \mathrm{open} \;,$$
$$ f( \mathrm{closed} ) \neq \mathrm{close... |
I was given this question (there were prior questions defining the equivelence relation ~)
Let $E$ be the set of equivelence classes of $\sim$. Define $f$ from $E$ to $\mathbb R$,
$$f:=\{([(x,y)],r)|x=yr,\: x,y,r\text{ in }\mathbb R\}.$$
I'm trying to understand what it means; is the following correct?
$f:[(x,y)] \map... |
I have a programming problem I'm trying to solve where, in order to be efficient, I have to find the probability of some events occuring.
Lets say I have 3 transactions. At each transaction there is a 5% chance that the transaction will be declined. I want to know what are the chances of 0, 1, 2 and 3 of the transacti... |
Suppose $f$ and $g$ are are two polynomials with complex coefficents (i.e $f,g \in \mathbb{C}[x]$).
Let $m$ be the order of $f$ and let $n$ be the order of $g$.
Are there some general conditions where
$fg= \alpha x^{n+m}$
for some non-zero $\alpha \in \mathbb{C}$
|
A couple of years ago I found the following continued fraction for $\frac1{e-2}$:
$$\frac{1}{e-2} = 1+\cfrac1{2 + \cfrac2{3 + \cfrac3{4 + \cfrac4{5 + \cfrac5{6 + \cfrac6{7 + \cfrac7{\cdots}}}}}}}$$
from fooling around with the well-known continued fraction for $\phi$. Can anyone here help me figure out why this equalit... |
I've got a question in my task sheet. The question is as follows.
$$
\frac{43\cdot93\cdot47\cdot97}{3007}=X
$$
Find the exact value of $X$. I've tried a lot, but couldn't find easier way to do it without calculator, which of course, is not allowed in exam. There are no options, they're just asking the value of $X$.
Wou... |
I have a problem I tried to solve, but couldn't because I don't know the method to solve it and I've never come across such problem.
Here's the problem.
$(PQ)²=RSP$
Where $P, Q, R, S$ are distinct single digit natural numbers, then $R=$?
We need to find out the value of $R$, and given option are
a) $1$
b) $2$
c) $3$
d)... |
Let $X = \mathbb{R}^n$ and let $Z$ be a closed subspace of $X$. The inclusion map $i: Z \hookrightarrow X$ induces a map $C_0(X) \to C_0(Z)$ (via pullback). My question is simple: what is the kernel? The answer should either be $C_0(X - Z)$ or $C(X - Z)$ and I think I know which one is correct. But I need a sanity ... |
Here's one more question I need help with.
Samir has some amount of rupees with him, the difference between $45.45$% and 36.36% of which is Rs. $789435689$. Find the amount.
a) $8683792579$
b) $8673792579$
c) $7573781479$
d) $88667292579$
e) None of these
Now, Samir is the name of Indian male, rupees is Indian currenc... |
A function $f : \mathbb{R} \to \mathbb{R}$ is convex (or "concave up") provided that for all $x,y \in \mathbb{R}$ and $t \in [0,1]$,
$$f(tx + (1-t)y) \le tf(x) + (1-t)f(y).$$
Equivalently, a line segment between two points on the graph lies above the graph, the region above the graph is convex, etc. I want to know why... |
Some years ago I was interested in the following Markov chain
whose state space is the positive integers. The chain begins at state "1",
and from state "n" the chain next jumps to a state uniformly
selected from {n+1,n+2,...,2n}.
As time goes on, this chain goes to infinity, with occasional
large jumps. In any case, t... |
Is there a characterization of the nonconstant entire functions $f$ that satisfy $|f(z)|=1$ for all $|z|=1$?
Clearly, $f(z)=z^n$ works for all $n$. Also, it's not difficult to show that if $f$ is such an entire function, then $f$ must vanish somewhere inside the unit disk. What else can be said about those functions?... |
I read this fascinating article:
http://www.newscientist.com/article/mg20627550.200-enter-the-matrix-the-deep-law-that-shapes-our-reality.html
Unfortunately all the other papers I find googling are just not tangible to me :-(
Could anyone please point me to some material that bridges the gap from this popular science e... |
Consider the following expression.
$1631310734315390891207403279946696528907777175176794464896666909137684785971138$
$2649033004075188224$
This is a $98$ decimal digit number.
This can be represented as $424^{37}$ which has just 5 digits.
or consider this number:
$16907351492333570491078177094338635851326626269508214... |
From a St. Petersburg school olympiad, 11th grade.
Prove or disprove: a non constant polynomial $P$ with non-negative integer coefficients is uniquely determined by its values $P(2)$ and $P(P(2))$.
|
I had asked this question sometime ago here. Now I have a question which I think is related to it.
Let $f$ be an increasing function (continuous, of course!) with $f(1)=0$.
Consider the sequence $s_{n}= ( \sum\limits_{k=1}^{n} f(k) - \int\limits_{1}^{n} f(x) dx )$.
When does $s_{n}$ converge?
|
Suppose we have a tetrahedron defined by points $(0,0,0),(1,1,0),(0,1,1),(1,0,1)$. Now define surface by $(a,b,a + b - 2ab)$ for $a,b$ between $0$ and $1$. Let $E_1$ be the set of points inside the tetrahedron and $E_2$ be the set of points on that surface, $A$ the area of that surface and $d$ be Euclidian distance
Is ... |
I’m interested in some sequences which have no square factor.
$$
s_i =\begin{cases}
4 & i=0; \\\\
s_{i-1}^2 - 2 & \text{otherwise}
\end{cases}$$
This is Lucas–Lehmer primality test sequence. A003010 in OEIS.
when ${s_i}$ has a square factor? I had checked that the sequence has no square factor for i=1,2,...,7
|
Prove that if $\{n_k\}$ is a strictly increasing sequence of positive integers, then the sum of the series $$\sum_{k=1}^{\infty} \frac{2^{n_k}}{(n_{k})!}$$ is an irrational number.
This is just a problem, given by a friend of mine, when we were doing some problems in Analysis. And if I remember correctly, he mentioned... |
Definition: A subgroup of a group is termed strongly potentially characteristic if there is an embedding of the bigger group in some group such that, in that embedding both the group and the subgroup become characteristic.
One can refer to this link.
Although, I have somewhat understood the definition, I would like t... |
If I create a $10$ digit password with the following requirements:
At least one uppercase letter A-Z : $26$
At least one lowercase letter a-z : $26$
At least one digit from $0-9$ : $10$
At least one common symbol $(\#,\$,\%,$ etc) : $32$
By inclusion-exclusion, I can calculate I have ~ $3.2333\mathtt E+19$ possible c... |
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