instruction stringlengths 22 15.8k |
|---|
A function $t: [a, b] \rightarrow \mathbb{R}$ is called a step function when a $k \in \mathbb{N}$ and numbers $z_0,...,z_k$ with $a = z_0 \leq z_1 \leq ... \leq z_k = b$ exist, such that for all $i \in \{1,2,...k\}$ the restriction $t |_{(z_{i-1},z_{i})}$ is constant. Let $f: [0,1] \rightarrow \mathbb{R}$ be defined by... |
If $a_{n}$ is a non-negative, decreasing sequence, we know from the integral test that if $f(n)=a_{n}$ is an integrable function, then $\sum_{n=1}^{\infty} a_{n}$ and $\int_{1}^{\infty} f(x)dx$ converge/diverge together.
When the theorem is proven, it is shown that:
$$\forall k \in \mathbb{N}, a_{k+1} \leq \int_{k}^{k+... |
Let $R$ be a PID,$M$ an $R$-module and $M$ is the direct sum of $M_1, \dots, M_k$ where $M_i \leq M$ for $1 \leq i \leq k$.
Now let $N$ be a submodule of $M$ .
Is it true that $N$ is the direct sum of $N_1,\dots,N_t$ where $N_i \lt N$ and $N_{k-t+1} \le M_i$?
|
Consider the linear space $c^{(3)}$ of all sequences $x = (x_n)_{n=1}^{\infty}$ such that $\{x_{3k+q} \}_{k=0}^{\infty}$ converges for $q = 0,1,2$. Find the dimension and a basis for $c^{(3)}/c_0$. Note that $c_0$ is the linear space of sequences that converge to $0$.
I think the dimension is $1$ using the reasoning in... |
Is there any way to tell Mathematica 7 to use "traditional" axes rather than boxing a three-dimensional graph? That is, rather than the default view produced by
Plot3D[Exp[-x^2 - y^2], {x, -2, 2}, {y, -2, 2},Boxed->False],
I would like three "axis arrows" to emanate from the origin.
|
Could someone please help?
The question reads:
For which real numbers $k$ is the zero state a stable equilibrium of the dynamic system
$x_{t+1} = Ax_t$?
$A = \begin{bmatrix} 0.1 &k \\ 0.3 & 0.3 \end{bmatrix}$
So, my thought is I need to find the eigenvalues. In order to do this I calculated the characteristic polyno... |
One of the very first number theory concepts introduced to students -- even before primeness, divisibility, etc. -- is the idea that a natural number can either be "even" (that is, evenly divisible by 2) or "odd" (all other numbers). For all intents and purposes, at that time, even and odd numbers were evenly distribu... |
Given $5x/30x^2$ I was wondering which is the correct equivalent form.
According to BEDMAS this expression is equivalent to
$5*\cfrac{x}{30}*x^2$
but, intuitively, I believe that it could also look like:
$\cfrac{5x}{30x^2}$
I asked this question on MathOverflow (which was "Off-topic" and closed) and was told it was amb... |
I calculated the cubic regression for a set of numbers following the instructions on this site:
http://epsstore.ti.com/OA_HTML/csksxvm.jsp?nSetId=71235
I am wondering how do I evaluate using the equation provided by the cubic regression without having to write down the equation on paper, and then typing in in the calcu... |
Recently I have found a need to learn more about p-adic group representations over a p-adic vector space. Generally, this motivates a study of representations $\left( V, \rho \right)$
for some group $G$ where $V$ is a vector space over $\mathbb{Q}_p$. Since I'm only familiar
with the theory for representations acting ... |
The affine line $\mathbb{A}^1$ (over $Spec R$) is, as far as I understand, $Spec R[x]$. My guess is that $0\in \mathbb{A}^1$ is given by the (opposite of the) composite homomorphism $R[x] \to R \to F$ where the first map sends a polynomial to its constant term and the second is the canonical map to the field of fractio... |
Definition:
A space $X$ is a Urysohn space iff whenever $x \neq y$ in $X$ there are nhoods of $U$ of $x$ and $V$ of $y$ such that $\overline{U} \cap \overline{V} = \emptyset$.
I want to show that every regular, T_1 space is Urysohn.
My attempt:
Let $x,y \in X$ be distinct points. Now consider the open set $X \setminus ... |
I have a question that is: Find a degree 3 polynomial with real coefficients whose leading coefficient is 5 that has -2, 1, and 4 as zeros?
I do not want the question answered for me what I want it explained what it is asking. I answered 5(x+2)(x-1)(x-4), but I was told I needed to finish the problem. Not exactly sure ... |
Is there an easy way to tell how many roots $f(x)$ has in $\Bbb{Q}[x]/(f)$ given the coefficients of the polynomial $f$ in $\Bbb{Q}[x]$?
Is there an easy way to find the roots as rational expressions in $x$?
The easiest example is a pure quadratic: $X^2 + 7$ for instance. If $A$ is a root, then so is $−A$. Good ole... |
Suppose we have the following lemma:
Lemma If $E_0 \hookrightarrow E$, and $E_0$ is a closed subspace then $E/E_0$ is a normed space and for $[x] \in E/E_0$ its norm is given by $||[x]|| = \text{inf}_{y \in E_0} ||x-y||$.
What is the intuition of this norm besides that it "works"? Does it just measure how close $x$ and... |
With positional notation, is there any practical usage for negative bases? I recently learned that numbers can be represented in positional notation using a negative base, but I don't see any real practical application of this. So, is there any?
|
The question is to prove Poincare Duality of the form
$(H^p_c (M))^* = H^{n-p} (M)$
using direct limits and inverse limits, where $H^p (M)$ denotes the $p$-th de Rham cohomology group of $M$. The outline of the proof has been given, which is to be followed. In particular, no homology theory of any sort can be used. (In... |
If $R$ is a ring such that $x^5=x$ for all $x\in R$, is $R$ commutative?
If the answer to the above question is yes, then what is the least positive integer $k \ge 6$, such that there exists a noncommutative ring $R$ with $x^k=x$ for all $x\in R$?
|
An equation f(x)=0 has 2 roots in the interval between 0 and 8. What is the value or a description for which the Newton Raphson method will fail to converge?
|
Oftentimes we see functions defined on open sets; I never gave this much thought, but now I'd like to get a better understanding.
Why do we define functions over open subsets of $\mathbb{R}^n$?
"Why" is not meant to be a deep philosophical question; what I mean is, for example,
When is it important to define a funct... |
As a computer science graduate who had only a basic course in abstract algebra, I want to study some abstract algebra in my free time. I've been looking through some books on the topic, and most seem to 'only' cover groups, rings and fields. Why is this the case? It seems to me you'd want to study simpler structures l... |
This is a problem (Exercise 9.2) from Apostol's Mathematical Analysis (second edition) which I couldn't solve.
$\bullet$ Define two sequences $\{f_{n}\}$ and $\{g_{n}\}$ as follows:
$f_{n}(x) = x \Bigl(1 + \frac{1}{n}\Bigr)$ for $x \in \mathbb{R}$, $n \in \mathbb{N}$
$g_{n}(x)= \begin{cases}
\frac{1}{n} & x=0 \ \te... |
I am reading a text related to SVM, and the mathematical language is giving me a little hard time.
Here training vectors xi
are mapped into a higher (maybe
infinite) dimensional space by the
function $\theta$. SVM finds a linear
separating hyperplane with the maximal
margin in this higher dimensional
space... |
In Titchmarsh's book "The theory of the Riemann Zeta function" pg. 15
where the functional equation of the zeta function is being derived,
I couldn't understand this part:
$$\frac{s}{\pi} \sum_{n=1}^{\infty} \frac{(2n\pi)^s}{n} \int_{0}^{\infty} \frac{\sin y}{y^{s+1}} dy = \frac{s}{\pi} (2\pi)^s \{-\Gamma(-s)\}\sin\fr... |
What exactly it is meant by "average increase in percentage"? I stumble upon this term in while solving this problem:
The average monthly income and expenditure of a person in the year $1995$ is $ \$ 14,000 $ and $ \$ 11,000 $ respectively and that of the year $2000$ is $ \$ 21,000 $ and $ \$ 17,600 $ respectively.Find... |
Let $(e_n)$ be the standard unit vector basis for $c_0$, let $x_1=e_1$, $x_n=e_n-e_{n-1}$ when $n\gt1$.
Prove that the formal series $\sum x_n$ is not weakly subseries convergent.
|
at the moment I am trying to reproduce the results of a paper.
There, it turns out that a specific physical problem is mapped onto an integral to be calculated:
$$I(\Theta; a, b) = 2\cdot\int_0^\infty \frac{a-u^2}{\left( u^2+\frac{a^2}{b-a}\right) \left(u^2+\frac{b^2}{b-a} \right) \sqrt{a - u^2 -\frac{a}{1-a/b}\sin^2(\... |
This is a question about probability theory. One often sees a probability measure characterised as being additive "for disjoint $X,Y$ we have: $P(X \cup Y) = P(X) + P(Y)$ " but sometimes as being additive like so: $P(X \cup Y) + P(X \cap Y) = P(X) +P(Y)$ . I guess what I'm asking is, how do you prove the inclusion/exc... |
I studied fourier series as an undergrad and grad. student in EE but did not fully grasp the concepts. Now that I am involved in medical imaging (MRI) understanding the basics of fourier series and transforms is very important and I am frustrated at my level of understanding. For example, a problem in a MRI physics boo... |
I am a little apprehensive to ask this question because I have a feeling it's a "duh" question but I guess that's the beauty of sites like this (anonymity):
I need to find an orthonormal eigenbasis for the $2 \times 2$ matrix $\left(\begin{array}{cc}1&1\\
1&1\end{array}\right)$. I calculated that the eigenvalues were... |
How can we prove that if $R$ is a commutative Noetherian ring, $\mathfrak{m} = (a_1,\ldots,a_n)$ is an ideal, then the completion of $R$ at $\mathfrak{m}$ is isomorphic to $R[[x_1,\ldots,x_n]]/(x_1-a_1,\ldots,x_n-a_n)$?
|
This question is about the connection between linear algebra and complex analysis. Coming from a two real dimensional domain a transformation matrix geometrically transforms a set of points (e.g. a circle) in that plane to another set of points in the two real dimensional codomain. The same holds true for a complex val... |
I read the following problem and I cannot get the right solution:
We consider a system of 12 sensors. The probability of one sensor to detect a signal with a magnitude higher than a is $p=\frac{1}{2\pi }\int_{x=a}^\infty \exp(\frac{-x^2}{2})$. What is the joint probability that 5 sensors detect this signal? It is stat... |
This is actually mentioned in my module,I am not sure how they have got this result but I think this may not be correct,I tried using wolfarm alpha which gives false or am I missing something?
I was missing approximation,it seems like data interpretation problems keeps relying on this, so,$$\frac{44}{9.3} = 4.73118 \t... |
Does the probability of winning the lottery differ between randomly generated numbers vs. selecting the same numbers every time?
Specifically. I'm interested in a breakdown of the odds per number for a given set of numbers that comprise a single US Powerball drawing (five white numbers plus the one powerball number), a... |
I know the sum of independent Gaussian distributed random variables also has a Gaussian distribution.
say we have a sequence of Gaussian random variables not necessarily independent
$ X_0, X_1, X_2, \cdots, X_{n-1}, X_n$
and say $ Z_n = X_n - X_{n-1} $ if we know for all $n \geq 1$ & the sequence
$$ Z_1,Z_2, \cdots,... |
A ring $R$ is a Boolean ring if $x^2=x$ for all $x\in R$. By Stone representation theorem
a Boolean ring is isomorphic to a subring of the ring of power set of some set.
My question is what is an example of a ring $R$ with $x^3=x$ for all $x\in R$ that is
not a Boolean ring? (Obviously every Boolean ring satisfies th... |
What is the definition of a profinite morphism in http://www.math.upenn.edu/~pop/Teaching/2010_Math624/2010_Math624PS08.pdf problem 5? This is not actually a homework of mine but I was unable to find the definition.
Snapshot:
|
Assume that the $a$'s in the following equations must be limited to a binary values of $\{0,1\}$ and the $c$'s are integer constants:
\begin{align}
2a_{34}a_{23}a_{24} &= c_1\\
a_{23} + a_{24} + a_{34} &= c_2\\
2 a_{34}a_{13}a_{14} &= c_3\\
... |
Take $X$ a set and $B_X$ the quotient of $\mathbb Z[X]$ by the ideal generated by all elements $u^3-u$ with $u\in \mathbb Z[X]$, which came up in this answer to this question.
Can you describe $B_X$ explicitly for small values of $|X|$?
|
What can one say about the set of all $n$-dimensional square matrices $A \in \text{GL}_n(\mathbb{C})$ that have an inverse with entries out of $\mathbb{C}$ with the properties:
unitary $:\Leftrightarrow A^*= A^{-1}$
hermitian $:\Leftrightarrow A^* = A$
where $A^*$ is the conjugate transpose of $A$.
What obviously fol... |
From Herstein's Topics in Algebra (exercise 14, section 1.2):
If $S$ is infinite and can be brought into one-to-one correspondence with the set of integers, prove that there is one-to-one correspondence between $S$ and $S \times S$.
So far I know there exists some bijection $\sigma : S \rightarrow \mathbb{Z}$. If I c... |
I'm searching for some good reading material on multifractal analysis. Preferably something accessible that doesn't put the stress too much on mathematical proofs but rather on applications. As long as it gives a good review of the status of the field, the interesting results and applications, I would be happy. Also, i... |
Let $f$ be a Dynkin diagram automorphism. Extend $f$ linearly to the root system $\Delta$.
What is a set of representatives of the orbits of $\Delta$ under $f$ ?
Thanks.
|
If I want to plot $r = 4\sin3\theta$ what is a good way to pick points? The period is $2\pi/3$ so it will repeat after that, but how do I pick points so that I get a good range of values so that I will be able to tell in which direction or how the curve is moving?
|
I'm learning for my algorithms exam and I can't derive two logarithm transformations:
$ 3^{log_{4}(n)}=n^{log_{4}(3)} $
$ log_{3}(n)=log_{3}(e)*ln(n) $
I'm not very strong in logarithms, anybody can explain how I get in both cases from the left side of the equation to the right side?
Sorry if the terminology is wrong... |
Burnside's Lemma, whose list of names is longer than the proof, says that the number of orbits of a permutation group is the average number of fixed points of its elements. It's a very elegant result, but I'm a bit disappointed by the fact that the examples given in the textbooks always amount to counting some coloring... |
Prove that three points are enough to draw/define one and only one circle, how would this be done?
|
Let $X$ be a compact topological space and $Y$ a Hausdorff space. Let $C \subseteq Y$ be closed in $Y$ and $U$ an open set in $X \times Y$ which contains $X \times C$. Prove there exists an open set $V \subseteq Y$ such that $X \times C \subseteq X \times V \subseteq U$.
Here's what I tried.
Let $b \in C$ fixed. For ea... |
I need to solve the following problem:
Suppose $f$ has the intermediate value property, i.e. if $f(a)<c<f(b)$, then there exists a value $d$ between $a$ and $b$ for which $f(d)=c$, and also has the additional property that $f^{-1}(a)$ is closed for every $a$ in a dense subset of $\mathbb{R}$, then $f$ is continuous.
... |
I'm wondering about the following:
Let $f:D \mapsto D$ be a continuous real-valued bijection from the open unit disc $\{(x,y): x^2 + y^2 <1\}$ to itself. Does f necessarily have a fixed point?
I am aware that without the bijective property, it is not necessarily true - indeed, I have constructed a counterexample withou... |
This question was posed by my high school geometry teacher, for extra credit:
Is it possible, using only a pencil, a straightedge (not a ruler) and a compass to trisect an angle of unknown value? If so, describe the technique, and mathematically prove its validity. If not, mathematically prove the impossibility.
I re... |
Let $X$ be riemann surface (not supposed compact) and $\mathcal D$ be sheaf of divisors on $X$.Remind that this means for $U\subset X$ open then $\mathcal D(U)$ is group of divisors on $U$. How to prove that $H^1(X,\mathcal D)=0$ . This is exercise in §16 of Forster book Riemann Surfaces (but is not homework for me) an... |
Possible Duplicate:
When do the multiples of two primes span all large enough natural numbers?
We have to generate a number by using only 3's and 5's.
For ex :
3 = 3
4 can't be generated
5 = 5
6 = 3 + 3
7 can't be generated
8 = 3 + 5
9 = 3 + 3 + 3
10 = 5 + 5
11 = 3 + 3 + 5
and so on...
So from 8, every intege... |
I'm working on a problem that requires me to find an upper bound on the probability that the sum of independent draws from a random variable deviates from the expected value of that sum by more than a given constant. Specifically, let $X$ be a random variable and suppose that we draw $m$ values for $X$. Let $S$ be the ... |
Somebody divided $\frac{cx+d}{ax+b}$ into
$$ \frac{c}{a} + \frac{d- \frac{bc}{a}}{ax+b} .$$ For use for integrals.
Does anybody knows how was it done? Could you show me how to do something like that, please?
Thanks in advance.
|
$$\frac{{{\varphi ^n} - {{(1 - \varphi )}^n}}}{{\sqrt 5 }} = {2^m} - 1 .$$
Here $\varphi = \frac{{1 + \sqrt 5 }}{2}$ . This integer equation has no solution for $n>3$ and $m>2$. How to prove?
|
I am reading this document.
What is the meaning of $[n]$ ?
Is it power set of $\{1,2,3...n\}$?
|
Doing an exercise for exam preparation, I stumbled across the following function:
$f_n(x)= n^2x(1-nx), \quad \text{if }0 \leq x \leq \frac{1}{n} $
$f_n(x)= 0, \quad \text{if } \frac{1}{n} < x \leq 1$
The task is to find the limit of this function series and to determine whether this function converges uniformly in $... |
I read that Presburger arithmetic is decidable while Peano arithmetic is undecidable. Peano arithmetic extends Presburger arithmetic just with the addition of the multiplication operator. Can someone please give me the 'intuitive' idea behind this?
Or probably a formula in Peano arithmetic that cannot be proved. Does i... |
Do countable Hausdorff connected topological spaces exist?
|
How can I show that:
There exists a bijective $f: \mathbb{N} \to \mathbb{N}$ such that: $$\sum_{n=1}^{\infty} (-1)^{f(n)}\ln\frac{f(n)+1}{f(n)}=\ln 2010$$
I have really no idea where to begin... Taylor series doesn't seem related. I thought about the alternating series test which approximates the sum but not sure how... |
I have a home made device (think baseball) with an accelerometer and a magnetometer embedded within it and would like to capture some details of an attempted 'throw' of the device.
All I need is the speed of the device at release, and the angle from the horizontal that it was released at. To be clear I'm not talking ab... |
Suppose I have a function $F:[0,1] \times [0,\infty) \rightarrow (0,1]$ of two variables $(s,t)$ that satisfies:
(1) $F(s,0)=1$ for each $s \in [0,1]$.
(2) $\lim_{t \rightarrow \infty} F(s,t)=0$ for all $s \in [0,1]$.
(3) For each fixed $s \in [0,1]$, the function $t\mapsto F(s,t)$ is strictly decreasing.
Consider the ... |
What is the mean euclidean distane between two points on the plane which coordinates are normally distributed?
I'm assuming this would be
$\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }\int _{-\infty
}^{\infty }\int _{-\infty }^{\infty }\frac{e^{-\frac{(\text{x1}-\mu
)^2}{2 \sigma ^2}} e^{-\frac{(\text{x2... |
Does exist open domain in $\mathbb C$ who has only identity for holomorphic automorphism?
Related question: does exist open domain in $\mathbb C$ so that every holomorphic automorphism has fixed point?
These questions were inspired by (much easier!) question by mathmos6.
|
I am currently learning about polar coordinate transformation, especially for integrating over certain regions. Let's say we have to calculate
$\int_{n}{xy \; dx dy}$
Then I think the correct transformation would be (note the functional determinant $r$)
$\int_{m}{r cos \phi \cdot r sin \phi \cdot r \; d\phi dr}$
First ... |
$C_n$(the $n^{th}$ catalan number) counts the number of expressions containing $n$ pairs of parenthesis which are correctly matched
How to count the possibilities if the maximum nesting level is fixed to $d$?
|
Definition: A $n$-gon is simple if it has no self intersection and is in general position if no pair of its edges are parallel.
It is clear that by extending the edges of each simple $n$-gon in general position we would achieve a set of $n$ lines such that no pair of lines are parallel and no triple of lines intersec... |
I'm starting to learn proof and logic, all my knowledge in mathematics is: basic discrete boolean logic, introductory logic, basic Calculus, Introductory Linear Algebra.
I've been trying to use what I know to prove this statement, but I simply can't.
If $x$ is real number such that $x^4 + 2x^2 -2x \lt 0$, then $0\l... |
Suppose $A$ and $B$ are two commutative rings, and let $f:A\rightarrow B$ be a ring homomorphism. Suppose that the induced map on the spectra $f^*:Spec B\rightarrow Spec A$ is an open immersion. What does it say about $f$? does it follows that $f$ is a localization?
edit: Thanks for the answer! What about closed immers... |
I'm trying to investigate the notion of primitive spectrum and its so-called Jacobson or hull-kernel topology, but I can only find references which define it for C*-algebras: see the Wikipedia page "Spectrum of a C*-algebra" for the definition I'm talking about. It seems like this definition would make sense for any (l... |
This question was inspired by the Lagrange equation, $\frac{\partial L}{\partial q} - \frac{d}{dt}\frac{\partial L}{\partial \dot{q}} = 0$. What happens if the partial derivatives are replaced by total derivatives, leading to a situation where a function's derivative with respect to one variable is differentiated by th... |
Does someone know if the function resulting of the sum of two locally bounded functions is also locally bounded?
Thanks in advance!
|
I'm a bit confused about the following part in Sir Michael Atiyah's "K-Theory."
Let $E = X \times V$ and $F = X\times W$, and let $\phi: E\to F$ be a vector bundle homomorphism. Why is the induced map $\Phi: X \to Hom(V,W)$ continuous? $V$ and $W$ are finite dimensional vector spaces (over $\mathbb{C}$).
To be a bit ... |
Let $ A\in {\mathbb{F} }^{n\times n} $ be a fixed matrix. The set of all matrices that commute with A forms a subring of ${\mathbb{F} }^{n\times n}$.
Is any subring of ${\mathbb{F} }^{n\times n }$ (which contains the identity) of the above form?
Thanks.
|
One can't get too far in abstract algebra before encountering Zorn's Lemma. For example, it is used in the proof that every nonzero ring has a maximal ideal. However, it seems that if we restrict our focus to Noetherian rings, we can often avoid Zorn's lemma. How far could a development of the theory for just Noethe... |
Let $a, b$ and $c$ be positive integers. Recall that the greatest common divisor (gcd) function has the following representation:
\begin{eqnarray}
\textbf{gcd}(b,c) = 2 \sum_{k = 1}^{c- 1} \left \lfloor \frac{kb}{c} \right \rfloor + b + c - bc
\end{eqnarray}
as shown by Polezzi, where $\lfloor \cdot \rfloor$ denotes th... |
A few days ago I was recalling some facts about the $p$-adic numbers, for example the fact that the $p$-adic metric is an ultrametric implies very strongly that there is no order on $\mathbb{Q}_p$, as any number in the interior of an open ball is in fact its center.
I know that if you take the completion of the algebra... |
I am reading Concrete Mathematics and came across an interesting problem, number 37 of chapter 2. The answers to exercises lists no known answer to this problem:
Will all the 1/k by 1/(k+1) rectangles, for k $\ge$ 1, fit together inside a 1 by 1 square? (Recall that their areas sum to 1)
My question is: Has a soluti... |
My class is solving the Cauchy-Euler differential equation $a t^2 y'' + b t y' + c y = 0$. The solutions are powers of $t$, $y = t^r$ and then you solve for $r$ using the characteristic equation $a r^2 + (b-a) r + c = 0$. This has two roots $r = \overline{r} \pm \Delta r$ and the general solution is $y = A t^{\overli... |
I'm trying to find the general solution to
$$\frac{\text{d}y}{\text{d}x} = \frac{y-x^2}{\sin y-x}$$
Any ideas would be greatly appreciated.
Thanks!
|
In the question Math.SE #16716, Natalia asked about representing rings of matrices as centralizers of a matrix. This is an intriguing question, but had some clear problems as rings of matrices need not contain their own centralizers, which would be pretty bad.
In finite groups, subgroups that contain their own central... |
The Whitney embedding theorem states that any smooth manifold can be embedded in Euclidean space.
In the Riemannian setting this naturally leads to the question whether this can be done in such a way that the Riemannian metric is preserved, that is, whether we can find an embedding such that the inherited metric as a s... |
Looking for the proof of the lemma asserting that the conical surface (envelope) is a closed space.
Thank you.
|
does there exist the notion of a non-integer power of a matrix? This seems to be accessible via semigroup-theory, yet I have not seen an actual definition so far.
I am not too firm at this right now, but I am curious. Can you give me a sketch of the definition and provide with some introductory information?
|
I have been stuck on this one for hours.
Let $x$, $y$, $z$ be non-negative real numbers.
Also we know $x + z \leq 2$.
Prove the following:
$(x - 2y + z)^2 \geq 4xz - 8y$.
Apparently this can be proven with or without AGM, which is $xy \leq \left(\frac{x + y}{2}\right)^2$.
This is what I have done so far:
\begin{alig... |
Suppose I am given the following inductive definition$^1$: Let $T$ be the smallest family of sets $\{T_1,T_2,T_3,\ldots\}$ such that
$0 \leq k < n \Rightarrow k \in T_n$
$t_1 \in T_n \wedge n > 0 \Rightarrow \lambda.t_1 \in T_{n - 1}$
$t_1 \in T_n \wedge t_2 \in T_n \Rightarrow (t_1 ~ t_2) \in T_n$
The elements of $T... |
As above, I'm trying to find all linear operators $L: C([0,1]) \to C([0,1])$ which satisfy the following 2 conditions:
I) $Lf \, \geq \, 0$ for all non-negative $f\in C([0,1])$.
II) $Lf = f$ for $f(x)= 1$, $f(x)=x$, and $f(x)=x^2$.
I'm honestly not sure where to start here - I'm struggling to use these conditions to... |
Let $c$ be an integer, not necessarily positive and not a square. Let $R=\mathbb{Z}[\sqrt{c}]$
denote the set of numbers of the form $$a+b\sqrt{c}, a,b \in \mathbb{Z}.$$
Then $R$ is a subring of $\mathbb{C}$ under the usual addition and multiplication.
My question is: if $R$ is a UFD (unique factorization domain), do... |
Let $X$ be a smooth projective curve of genus $g$. Let $\Omega$ be the sheaf of differentials. Mumford (in Abelian Varieties, sec. 2.6, in proving the theorem of the cube) asserts that there is an effective divisor of degree $g$ on $X$ such that $H^0(X, \Omega \otimes L(-D))=0$.
This should be easy, but I'm missing th... |
Recently I have started reading about some of the history of mathematics in order to better understand things.
A lot of ideas in algebra come from trying to understand the problem of finding solutions to polynomials in terms of radicals, which is solved by the Abel-Ruffini theorem and Galois theory.
I was wondering if ... |
I'm looking for ideas for a 15-hour mathematical enrichment course in a Chinese high school. What (fairly) elementary subject would you suggest as a topic for such a course?
Background/considerations:
My students are generally quite good at math, but many of them have not been exposed to rigorous or abstract mathemati... |
Does anyone know a non trivial(that we cannot define on every set) topology defined on the power set of an uncountable set?
|
I have following problem:
We throw a fair dice time after time.What is time expectancy to get first time 66 i.e two six in the row for the first time. I probably should use this http://en.wikipedia.org/wiki/Expected_value#Discrete_distribution_taking_only_non-negative_integer_values though I don't understand how. ... |
As we know, Lie group is a special smooth manifold. I want to find some geometric property, which is only satisfied by the Lie group. I only found one property: parallelizability. Can you show me something more?
|
Is there a classification of all finite indecomposable p-groups?
(something like: those are exactly the quaternions, primary cylic groups, some dihedral groups, etc.).
|
Problem: Given that $f$ is differentiable at $[0,1]$ and $f(0)=f(1)=0$. If $ \forall x\in (0,1)$ $|f''(x)|\leq A$ show that $\forall x \in [0,1]$ $|f'(x)| \leq (A/2) $.
My attempt was to to develop a Taylor series for $f(x)$ and $f'(x)$ around point $c \in (0,1)$ where $f'(c)=0$.
|
Two tangent vectors at a point on a surface are
$T_1 = 4i + 2j + 3k$
and $T_2 = -2i - 3j + 1k$
Using the property of the dot product of two normal vectors, determine the unit vector normal to the surface at the point
|
Intuitively what is meant by a left invariant vector field on a manifold?
|
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.