instruction stringlengths 22 27.2k |
|---|
How the group of automorphisms of a compact Riemann surface acts on the first homology group of the Riemann surface?
[Here, by an automorphism of a Riemann surface, we mean conformal self homeomorphism of the Riemann surface. In the paper "Action of automorphisms of a Compact Riemann surface on the first homology group... |
I know that for a set of vectors $\{ v_{1}, v_{2}, \ldots , v_{n} \} \in \mathbb{R}^{n}$ we can show that the vectors form a basis in $\mathbb{R}^{n}$ if we show that the coefficient matrix $A$ has the property $\det(A) \neq 0$, because this shows the homogeneous system has only the trivial solution, and the non-homoge... |
I have this problem:
Let $X_n \sim \mathrm{Poisson}(1/n)$. Show that $X_n \to 0$ and $Y_n = n X_n \to 0$?
I already proved the first statement, but every time I try to show the second one I get a divergent result. Can somebody help me with this?
|
Dear all,
I'm trying to find the general second-order ODE admitting
$$x* = \alpha x$$
$$y* = \alpha^{k} y$$
and reduce it to first-order plus a quadrature.
The general solution I found by using differential invariants is
$$y'' = \frac{1}{x} + H(y, - \frac{y'}{x})$$
where H is an arbitrary function.
But I have NO clue ... |
What is the relation between the fact that $\mathbb{Q}$ is dense in $\mathbb{R}$ and the fact that the completion of $\mathbb{Q}$ is $\mathbb{R}$? Or in general, that is if $A$ is dense in a metric space $B$, what's the relation between the completion of $A$ and $B$?
|
Is the shortest formula which is provably equivalent to the Gödel sentence also the shortest undecidable formula?
I know that even if one accepts the Gödel sentence as an axiom, then yet another undecidable formula can be constructed, but it will be longer, not shorter: if we append to PA its Gödel sentence G then we c... |
I am studying a recreational probability problem (which from the comments here I discovered it has a name and long history). One way to address the paradox created by the problem is to study the median value instead of the expected value. I want to calculate the median value exactly (not only find bounds or asymptotic ... |
I need to make a constraint for the following condition:
Student 1 can only be on the team if students 2, 3, 4, and 5 are also on the team.
I'm not sure how to model this using equations. The equations must not have "and", "or", or "not equal to" signs in them. These equations are likely going to involve an M term in t... |
In Boyd's Convex Optimization text, on page 86, there is the following equation:
$$f(x) = h(g(x)) = h(g_1(x), \cdots, g_k(x))$$
Then, in order to show convexity, we take the second derivative, yielding:
$$f''(x) = g'(x)^T \nabla^2h(g(x))g'(x) + \nabla h(g(x))^T g''(x)$$, with $h:R^k \rightarrow R, g_i:R^n \rightarrow R... |
Prove: When $n > 2$,
$$n! < {\left(\frac{n+2}{\sqrt{6}}\right)}^n$$
PS: please do not use mathematical induction method.
EDIT: sorry, I forget another constraint, this problem should be solved by
algebraic mean inequality.
Thanks.
|
Suppose that $a_n > 0$ for all $n\geq 1$, and define $S_n = \sum_{i=1}^n{a_i}$. If $S_n$ is convergent, prove that
$$ \frac{n^2}{\sum_{i=1}^n{\frac{1}{a_i}}} $$
is also convergent.
Thanks.
|
A tank has $1000 m^3$ of salt solution. The salt concentration is $10\frac{kg}{m^3}$. At time zero, salt-free water starts to flow into the tank at a rate of $10\frac{m^3}{min}$. Simultaneously salt solution flows out of the tank at $10\frac{m^3}{min}$, so that the volume of the solution in the tank is always $1000 m^3... |
Long subject, so I am hoping that explains what I am asking, but still let me elaborate with an example.
Let us assume we have an object as part of a container. Now the container is rotated by certain ROTATIONX, ROTATIONY, ROTATIONZ wrt the universe. Now we need to rotate the object WRT the container by rotationx, rota... |
This is actually a quiz that will be needed in a real life food stall! I need to decide how much stock to supply for my pumpkin soup stall. I sell each soup for $5$ dollars a cup, and let's say my ingredients cost is $1$ dollar. Therefore an outcome of under-forecasting is $4$ dollars per unit, while an outcome of over... |
Let $A$ be a ring, $S$ be a multiplicative subset, $M$ an $A$-module. let $\iota : A \to S^{-1}A$ be the map $a \mapsto a/1$. $\iota$ can be defined categorically as an initial object in the category of ring homomorphisms whose domain is $A$ which maps $S$ into the units (morphisms in this category are commuting triang... |
As we know,the flabby sheaf and the fine sheaf are all acyclic,but does one imply another?I need some examples to distinguish them,who can help me?
|
Probably off topic, but still.
I'm trying to solve equation of the form $P^2 = \mathrm{id}$, where $P$ is a matrix with some unknown coefficients. The problem is, Maple looses some solutions, in particular $id$ and $-id$. I checked if they are special cases of any solutions Maple found, they aren't.
The code goes like ... |
If $\displaystyle \left|\sum_{k=0}^\infty a_k\right|\leq M$ Where $M$ is a number, how can you argue that the series is convergent?
If $\displaystyle \sum_{k=0}^\infty a_k=M$ (so your series is convergent), Can you say that $\displaystyle \sum_{k=n+1}^\infty a_k$ also converges?
|
There is this programming/mathematical problem I have; it is the problem of generating a sequence of n unique random numbers, where the random number $n$ is element of of the set $S = \{0, 1, 2, \cdots N-1\}$ (Like permuting the numbers, but without using memory.)
Recently I kinda solved the problem on my own, and I go... |
Given a list of integers, how to find the sum of the differences of all possible pairs of numbers ?
For example if the number are $3,1,2$ then, answer should be $$\lvert 3-1 \rvert + \lvert 3 -2 \rvert + \lvert 1-2 \rvert
= 2 + 1 + 1
= 4 $$
Suggest a suitable algorithm for this.
|
I'm trying to find out what the graph of the function $y= \sin^2(\sqrt{x})$ looks like. Can someone please help me?
|
Possible Duplicate:
Injective and Surjective Functions
If $g(f(x))$ is one-to-one (injective) show $f(x)$ is also one-to-one given that $f$ is a function from $A$ to $B$ and $g$ a function from $B$ to $C$.
I've just started my Discrete math course and I'd like some help on this. I'm pretty sure we're supposed to use... |
I'm a techie who is looking to transition to technical writing as a career. It's been suggested that I produce documentation for an Open Source project to demonstrate my ability to do technical writing. I love Math so I thought I'd find a gap in an Open Source Math program and fill that gap.
What poorly documented Open... |
How do I do a least squares fit of the line equation $a x + b y = 1$, so that the points are as close to the line as possible? (Not just vertically close)
If I use the matrices
$$X = \left[\begin{matrix} x_{1} & y_{1} \\ x_{2} & y_{2} \\ \vdots & \vdots \\ x_{n} & y_{n} \end{matrix}\right]$$
$$Y = \left[\begin{matrix} ... |
In "Introduction to Calculus and Analysis" pages 221-223 Courant derives the following for an implicit function F(x,y)=0. Using
$dF = F_x dx + F_y dy = 0$
$dy = \frac{dy}{dx} dx = -\frac{F_x}{F_y}dx$
Also $y' = -\frac{F_x}{F_y}$
He says $f(x) = y$ therefore
$y' = \frac{F_x(x,f(x))}{F_y(x,f(x))}$
What I don't understan... |
I am really confused how to calculate the following. The answer is $40$, but what are the steps to get $40$?
$$\frac{200}{x+10} = \frac{200}{x} - 1.$$
Thank you.
|
A classical exercise in group theory is "Show that if a group has a trivial automorphism group, then it is of order $1$ or $2$." I think that the straightforward solution uses that a exponent two group is a vector space over $\operatorname{GF}(2)$, and therefore has nontrivial automorphisms as soon as its dimension is ... |
Suppose $\nu_1$ and $\nu_2$ are two complex measures, do we have $\nu_1\perp\nu_2\Leftrightarrow|\nu_1|\perp|\nu_2|$? if yes, please give a proof. If no, please give a counterexample. Thanks!
PS: two complex measure $\nu_1\perp\nu_2$ if so are their real and imaginary parts, respectively.
|
Let $k,m,n\in \mathbb{N}$ where $1 < k < n-1$. Consider the equation $$\binom{n}{k} = m!$$ which can also be equivalently written as $$n!=(n-k)!k!m!$$ The only instances I found are $\binom{4}{2} = 3!$ and $\binom{10}{3} = 5!$ I do not see any pattern coming out. As I went far out, it seemed that it is hard to find oth... |
$$\sqrt{(5+2\sqrt6)^x}+\sqrt{(5-2\sqrt6)^x}=10$$
So I have squared both sides and got:
$$(5-2\sqrt6)^x+(5+2\sqrt6)^x+2\sqrt{1^x}=100$$
$$(5-2\sqrt6)^x+(5+2\sqrt6)^x+2=100$$
I don't know what to do now
|
With the initial conditions: $a>b>0$;
I need to find $$\lim_{n\to\infty}\sqrt[n]{a^n-b^n}.$$
I tried to block the equation left and right in order to use the Squeeze (sandwich, two policemen and a drunk, choose your favourite) theorem.
|
I'm trying to find some function $g(k)$ such that $$\sum_{k=0}^{\infty} g(k) \frac{(n \lambda)^k}{k!} = 0 $$ The textbook says that there is only one solution, that is $g(k)=0$ for all $k$. But I cannot see why it is so. It is also constrained that $g(k)$ depends on $k$ alone and does not depend on $\lambda$ or $n$. $n... |
I'm learning Complex Numbers by myself. I've already understood basics concepts, but now I have problems with plots/graphs.
For example, I don't know, how to plot:
(1) $|z-1|+|z+1|=4$
(2) $|z+1| - \Im z \leq 1$
(3) $\Re\left(\frac{1-z}{1+z}\right) = 1$
($z \in \mathbb{C}$)
These examples are from some problem set I'm l... |
There is a cube and an ant is performing a random walk on the edges where it can select any of the 3 adjoining vertices with equal probability. What is the expected number of steps it needs till it reaches the diagonally opposite vertex?
|
$$\int_{0}^{\frac{\pi}{4}}\ln(\cos t)dt=-\frac{{\pi}}{4}\ln2+\frac{1}{2}K$$
I ran across this integral while investigating the Catalan constant. I am wondering how it is evaluated.
I know of this famous integral when the limits of integration are $0$ and $\frac{\pi}{2}$, but when the limits are changed to $0$ and $\fr... |
I was trying to figure out, how many degrees of freedoms a $n\times n$-orthogonal matrix posses.The easiest way to determine that seems to be the fact that the matrix exponential of an antisymmetric matrix yields an orthogonal matrix:
$M^T=-M, c=\exp(M) \Rightarrow c^T=c^{-1}$
A antisymmetric matrix possesses $\frac{n(... |
Are there differences between 'counting' and 'ordering'? As such, the whole of rational number is countable, or they order-able too? In what cases counting and ordering are same or not?
|
How can I show that $ALL_{DFA}$ is in P ?
$ALL_{DFA} = \{ \langle A \rangle \mid A \text{ is a DFA and } L(A) = \Sigma^* \}$
|
Question: Find $\displaystyle \sum \limits_{n=1}^{\infty} \frac{2^{2n+1}}{5^n}$
My issue(s): How do I do this without using a calculator? I know that I have to do something with $S_n$ and $S_{n+1}$ but I'm not sure what.
|
The finite spectrum of a theory $T$ is the set of natural numbers such that there exists a model of that size. That is $Fs(T):= \{n \in \mathbb{N} | \exists \mathcal{M}\models T : |\mathcal{M}| =n\}$ . What I am asking for is a finitely axiomatized $T$ such that $Fs(T)$ is the set of prime numbers.
In other words in w... |
I have some big troubles trying to understand specific set theory stuff.
Especially when we demonstrate something about set theory we always have to keep our demonstration in set theory, typically not using second order logic.
For example to demonstrate Löwenheim-Skolem we have to quantify over formulas to explicitly b... |
Consider a game with two players A and B. Player A throws a coin with probability $p_A$ of landing heads, in which case he wins the game. If not, player B throws another coin with probability $p_B$ of landing heads, in which case B wins. If not, it is A's turn again.
Calculate the probability of both players winning t... |
I want to find the value of $x'$ such that all of the following constraints are satisfied:
$x' * o' < x * o$
$ o' < o $
$ x \le x'$
These are all variables but I always know the values of $x$, $o$ and $o'$. I'm sure that this is easy to solve but I have been completely unable to!
Cheers,
Pete
EDIT:
For clarification, w... |
I've been trying to prove this for a long time now, anyone willing to offer some help or get me pointed in the right direction?
$(x>0 \implies z = x) \wedge (x < 0 \implies z = -x) \implies z \ge 0$
|
The secretary problem is a well-studied optimal stopping problem with a simple solution. Suppose a set of $N$ candidates are interviewed for a secretarial position, one at a time, in random order. Each interviewee must be either accepted on the spot or rejected for good. The goal is to select the single best candida... |
I am trying to showed that if $X$ and $Y$ are path connected then $X \coprod_f Y$ is path connected. (The adjunction space). Let $A \subset X$ , and let $f:A \to Y$ be the attaching map.
(Note: there is some abuse of notation in what follows, in terms of re-use of symbols for function names for different paths)
So pat... |
I have an oracle that samples uniformly from sets of size up to around n or n2.
How do I efficiently approximate a uniform distribution on the set of non-decreasing sequences of length k whose terms are from {1,2,…,n} (using the oracle)?
This set has about (n+k−1) choose k elements, O(n^k), and is too big to directly... |
Denote by $D^iG$ the $i$th term in the lower central series of G, i.e. $D^0G = \{1\}$, $D^i=[G, D^{i-1}G]$. The claim is that if $N$ is normal in $G$, then $D^i(G/N) = (ND^i(G)/N)$.
I know we should proceed by induction, and we do thus:
For the base case, $i=0$, we have:
$$ D^0(G/N) = G/N \mbox{ and } (ND^0G)/... |
Is the area of a circle ever an integer?
I was trying to answer someone else's question on yahoo answers today and I got thumbs down from people on my answer and have come here to get a thorough proof on it because now I just must know! :)
My assertion: Suppose we have two integers a and b. We can form a rational num... |
Suppose you have a set $S\subset\mathbb{Z}[x,y].$ How can one efficiently partition the polynomials into sets such that the primes represented by the polynomials in any given set are identical? For these purposes, "efficient" can be interpreted as "subquadratic", though I'm interested in practical as well as theoretic... |
You are given positive integers N, m, and k. Is there a way to check if
$$\sum_{\stackrel{p\le N}{p\text{ prime}}}p^k\equiv0\pmod m$$
faster than computing the (modular) sum?
For concreteness, you can assume $e^k<m<N.$
I don't know of a way, but it's not obvious to me that no method exists. Fast ways to prove or refu... |
Let $L$, $M$, and $N$ are subspaces of a vector space. Prove that following is not necessarily true.
$L \cap (M + N) = (L \cap M) + (L \cap N) $
This problem is given in 'Finite dimensional vector spaces' by Halmos. I was using 'if a vector belongs to L.H.S. then it must belong to R.H.S and vice versa' argument. Neithe... |
A little reading suggests:
It is known that either $\pi + e$ or $\pi \times e$ is transcendental (or possibly both), but no proof is known that one of those two numbers in particular is transcendental.
If we just want irrationality rather than transcendence, is a proof known?
Can we prove $\pi+e$ is irrational? Can we... |
A recent answer reminded me of the gauge integral, which you can read about here.
It seems like the gauge integral is more general than the Lebesgue integral, e.g. if a function is Lebesgue integrable, it is gauge integrable. (EDIT - as Qiaochu Yuan points out, I should clarify this to mean that the set of Lebesgue int... |
Can every even integer be expressed as the difference of two primes?
If so, is there any elementary proof?
|
14% of people are left handed, if we want to find the chances of a brother and a sister both being left handed, we might be tempted to multiply .14 by .14. What is wrong with this reasoning when trying to find out the probability of both of them being left handed?
|
In Number Theory,
Given the Diffie Hellman tuple (ga,gb,gab),
Will the given info be helpful in finding the discrete log of g ,i.e a or b?
Edit: That is can we use the solution of a Computational Diffie Hellman Problem to solve a Discrete Log. It is known that the converse is true!
|
$| 3 ^ { \tan ( \pi x ) } - 3 ^ { 1 - \tan ( \pi x ) } | \geq 2$
|
I'm having some difficulty wrapping my head around rigid motions in a plane. In particular, I'm trying to solve this following problem:
In a Euclidean plane, show that the product of two rotations around different points is equal to either a rotation around a third point or a translation. Hint: Show that it has at mos... |
Possible Duplicate:
Is there a quick proof as to why the vector space of $\mathbb{R}$ over $\mathbb{Q}$ is infinite-dimensional?
consider R as a vector space over Q then what is the dimension and how do we prove it
|
Why don't these limits exist?
$$\lim_{x\to 0}\frac1{x^2}$$
Aren't we here just approaching $x$? And, thus will not get $0$ and avoid the division on "0"?
And, what about this limit?
$$\lim_{x\to 0}\frac{|x|}{x}$$
|
From here:
Operads are a generic structure giving
a more precise definition of these
terms. An operad is an abstract set of
operations of various arities (an ugly
word to precise the number or
arguments taken by an operation: a
ternary operation is said to have
arity three), subject to relations
betwee... |
If someone could help with either of these problems that would be awesome!
$(\tan x)^2 \leq |1 - 2(\cot x)^2|$
$x^{\sin(x-a)}>1$ where $0< x < \frac{\pi}{2}$ and $a>0$
|
The set of wffs of first-order logic is recursive. Taking that checking if a wff is a sentence is straightforward I take that the set of sentences of FO is also recursive.
If we add FO the least fixed point operator, does the set of sentences stay recursive? I guess yes, but I haven't seen this proved anywhere. Where c... |
If there exists a homomorphism $f: A \rightarrow B$ between two groups $A$ and $B$, and a homomorphism $g: C \rightarrow D$ between two groups $C$ and $D$, then will there exist a homomorphism $h: A*C \rightarrow B*D$ between the free products $A*C$ and $B*D$?
|
How can one show that Dedekind's cuts are equivalent to Cantor's definition of an irrational number based on Cauchy sequences? The problem hints that it might be useful to use the lemma that, for any positive integers (p,q), there exists a positive integer M, such that Mq > p.
In the way I tried to solve it, I started... |
I have seen that $x^{2}+x+1$ and $x^{4}+x+1$ are irreducible over $\mathbb{Z}_2$ and I thought a polynomial of the form $x^{2^m}+x+1$ for $m\ge3$ would be irreducible too. However using WolframAlpha, my hunch was wrong. It could be factored over $\mathbb{Z}_2$. WolframAlpha could only generate the factors for $m=3,\ldo... |
How can the Perron-Frobenius theorem be used to show that for a connected graph, there is a simple eigenvector that is (i) real and (ii) smallest in magnitude and (iii) has an associated eigenvector that is positive?
The graph Laplacian is given as $L = D-A$, where $A$ is the non-negative adjacency matrix of the graph.... |
Self-studying some properties of the exponential-function I came to the question of ways to assign a value to the divergent sum $$s=\sum_{k=1}^{\infty}\exp(\sqrt k) $$ I have no idea how to attack this with standard methods (I do not know many).
I tried a replacement using reversing order of summation when $s$ is ... |
I've been a programmer for a good while now. Fairly experienced at a bit of math as far as coming up with algorithms and such but I am far far behind on understanding quite a deal of notation.
Here and there I run into an issue where someone will notify me that I've reinvented some piece of calculus, trig or some other... |
If I have the following two different values for x:
x = 0.022
x = 0.020
And, if I want to calculate the percentage of the difference (change) in x, will it be?
---> 0.022 - 0.020 = 0.002
---> 0.002 x 100% = 0.2%?
And, should the percentage of the difference be always positive?
Thanks.
|
At the wikipedia page for the Möbius function http://en.wikipedia.org/wiki/M%C3%B6bius_function there is an expression for the ordinary generating function for the Möbius function.
Is it possible to calculate the radius of convergence for any of the terms in the expression?
For example what would the radius of converge... |
If I have a biconnected graph and I remove a vertex (without forgetting which vertex was removed and which vertices it was adjacent to), is there an way to check the biconnectivity of the resulting graph that is easier than checking the biconnectivity of an arbitrary graph? E.g., is there a method that in the best case... |
0
So I'm reading a paper which assumes the following statement but I would like to be able to prove it.
Let $S=Sym(\mathbb{N})$ denote the symmetric group on the set of natural numbers.
If $\emptyset\subset A \subset \mathbb{N}$ then: $$S_{A}= \{ q \in S : aq\in A,\;\forall a\in A \}$$ is a maximal subgroup of S.
Here ... |
I'm trying to integrate this:
$ U_{1}(Z_{1} + Z_{2},t) = C \int\nolimits_{0}^{\infty} (1-i\Phi) \exp[-(1+iV)g] \mathrm{d}g, $
with:
$$ \Phi = \dfrac{\theta}{t_{c}} \int_{0}^{t} \dfrac{1}{1+2t'/t_{c}} \left( 1 - \exp \left[- \dfrac{2mg}{1+2t'/t_{c}} \right] \right) \mathrm{d}t'. $$
Then, after doing $ \left|U_{1}(Z_{1}... |
Say I have $f=x^2$ (surjective) and $g=e^x$ (injective), what would $f\circ g$ and $g\circ f$ be? (injective or surjective?)
Both $f$ and $g : \mathbb{R} \to \mathbb{R}$.
I've graphed these out using Maple but I don't know how to write the proof, please help me!
|
How does one prove that:
for all $a\in G$, where $G$ is a group (not necessarily abelian) $a^{|G|} = 1_G$.
|
I've been looking at some quals problems for algebraic topology that I found online. The problem is that I don't know if I can solve them with the amount of algebraic topology that I know, but nevertheless, they seem interesting. Also I know my committee tends to ask questions about topics not on the syllabus... The pr... |
Let $K$ be a nonempty compact convex subset of $\mathbb R^n$ and let $f$ be the function that maps $x \in \mathbb R^n$ to the unique closest point $y \in K$ with respect to the $\ell_2$ norm. I want to prove that $f$ is continuous, but I can't seem to figure out how.
My thoughts: Suppose $x_n \to x$ in $\mathbb R^n$. L... |
When doing simple linear interpolation on a data set in $\mathbb{R}^2$, I just sort the points by one co-ordinate and then do piecewise interpolation between a point and its successor in the sorted data set.
How would I do this in $\mathbb{R}^3$, and generally in $\mathbb{R}^k$?
How would I know which points to "conne... |
Can anyone help me finding the $n^{th}$ derivative with respect to $x$ of the function
$$\frac{f(x)}{x-a}$$
where $f$ is infinitely differentiable, $a$ is some constant. I tried to find the first few terms but things get messy!
|
I know some nice ways to prove that $\zeta(2) = \sum_{n=1}^{\infty} \frac{1}{n^2} = \pi^2/6$. For example, see Robin Chapman's list or the answers to the question "Different methods to compute $\sum_{n=1}^{\infty} \frac{1}{n^2}$?"
Are there any nice ways to prove that $$\zeta(4) = \sum_{n=1}^{\infty} \frac{1}{n^4} = \... |
I have been struggling with this functional series. $$\sum_{n=1}^{\infty}{(-1)^{n-1}n^2x^n}$$ I need to calulate the sum.Any tips would be appreciated.
|
Motivated by this question, can one prove that the order of an element in a finite group divides the order of the group without using Lagrange's theorem? (Or, equivalently, that the order of the group is an exponent for every element in the group?)
The simplest proof I can think of uses the coset proof of Lagrange's th... |
I am interested in the following game for $n$ players:
Each of the $n$ players chooses a random integer from the interval [0,100] and the average of these numbers (call it $a$) is computed. If we denote by S the set of chosen numbers that are greater than $\frac{2a}{3}$ then we call the player that picked min(S) as the... |
Let $A = \{1,\ldots,n\}^2$, for some $n \in \mathbb{N}$. We define a partial order on $A$ by $(a,b) \leq (c,d)$ when $a \leq c$ and $b \leq d$. What is the number of ascending non-degenerate (all points are different) chains of length $k$, for relevant $k$.
Help please...
This is not a homework assignment, but just a ... |
Is there any relationship between disorder of a permutation and order of a permutation?
|
For a set $S$ with $n$ elements, the notation for a combination $\binom{n}{k}$, or $C(n, k)$, indicates the number of combinations of $k$ elements from $S$, but how does one indicate the actual set created from combinations of $k$ elements from $S$? That is, $\binom{n}{k}$ is the size of the set I'd like to represent.
... |
Thomson et al. provide a proof that $\lim_{n\rightarrow \infty} \sqrt[n]{n}=1$ in this book (page 73). It has to do with using an inequality that relies on the binomial theorem:
I have an alternative proof that I know (from elsewhere) as follows.
Proof.
\begin{align}
\lim_{n\rightarrow \infty} \frac{ \log n}{n} = 0
\... |
$G$ is a bipartite graph with $2m$ nodes on the left $(u_0..u_{2m-1})$, and $2^{m}$ nodes on the right $(v_0..v_{2^{m}-1})$.
There is an edge (connection) between $u_i$ and $v_j$ iff $(i+1)$'th digit of the binary representation of $j$ is $0$ or $(i-m+1)$'th bit is $1$. In other words, each of the $2m$ nodes on the l... |
How can one simply see that Ky Fan $k$-norm satisfies the triangle inequality? (The Ky Fan $k$-norm of a matrix is the sum of the $k$ largest singular values of the matrix)
|
I'm trying to better understand the concept of differentiable submanifold. However, it looks like many different definitions are adopted by various authors and so I'm trying to keep myself in sync by proving them equivalent. Now I'm stuck with the following two.
Let $M$ denote a $n$-differentiable manifold and let $M' ... |
I was playing around with the functions $2^x$ and $\log_2(x)$. As they are the inversions of each other, I thought there was a simple number $a$ for which $2^x - a$ touches $\log_2(x)$.
Using trial-and-error, for $a = 2$ looked like the functions just touched each other, but on closer inspection they did cross each oth... |
Will you provide a simple application of Sard's theorem using some specific functions?
|
Let $\Omega$ be a finite probability space and define the Total Variation as $TV(\mu,\nu):=\sup\{|\mu(A)-\nu(A)|: A\subseteq\Omega\}$. One can also derive that
$TV(\mu,\nu)=\frac{1}{2}\sum_{x\in\Omega}|\mu(x)-\nu(x)|$
I know how to derive this fact by showing $\leq$ and $\geq$, however, I am interested if there is a di... |
I have an exam on $p$-adic integers (and a bit on Galois Theory) that my professor said would be very computational, but he never does any examples of the theorems he proves in class. He said the exam would be mostly, if not entirely computational, and I am trying to hypothesize what kinds of questions the professor c... |
A question about a proof of the Cayley-Hamilton theorem using Zariski topology.
"The set $C$ of all matrices of size $n \times n$ (over an algebraically closed field $k$) with distinct eigenvalues is dense in the Zariski topology".
Can we argue as follows?
Since non-empty open sets in the Zariski topology are dense in ... |
Suppose we take the circle $S_1$ and three points on this circle, which defines a triangle. By moving the points continuously on $S_1$, we obtain a continuous transformation of the triangle.
I was wondering what is the structure of this group of transformations ?
|
So the problem is to find all points $(x,y)$ on the real plane such that $f(x,y) = \cos^2(x+t) + 2\sin(x+t)\cos(y) - \frac{(\cos y - 1)^2}{2} - \sin(x) \lt .5$ for all real $t$. I'm not sure where to start with this, I don't think I fully understand the equation... how can the result be in terms of $f(x,y)$ when $t$ is... |
Does there exist a fast way to calculate the inverse of an $N \times N$ matrix, if we know the inverse of the $(N-1) \times (N-1)$ sub-matrix?
For example, if $A$ is a $1000 \times 1000$ invertible matrix for which the inverse is known, and $B$ is a $1001 \times 1001$ matrix obtained by adding a new row and column to $... |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.