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Given the next sets: $A=\{1,2\}$ , $B=\mathbb{N}$,
a) Describe all the functions from $A$ to $B$ that are not injective.
b) Describe all the functions from $B$ to $A$ that are not onto.
c) Prove or disprove the next:
(i) The next functions exist $f\colon A\to B$ and $g\colon B\to A$ if $g\circ f$ reversal
(ii) the next... |
Let $p_n$ be the n'th non-trivial zero of the Riemann zeta function. We define the Zetor function (acronym of 'zeta' and 'zero') as follows: $$\zeta \rho (s) = \sum_{n=1}^{\infty} \frac{1}{(p_n)^s}. $$
For which values of $s$ does $\zeta \rho (s)$ converge?
Thanks,
Max
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I'm looking at Wikipedia's article on Intuitionist logic, and I can't figure out how it would prove $(p \rightarrow p)$.
Does it prove $(p \rightarrow p)$? If yes, how?
If no, is there a correct (or reasonable, if there is no "correct") axiomatization of intuitionistic logic available anywhere online?
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I was wondering today about why the word differentiable is used for describing functions that have a derivative or are differentiable.
Perhaps because originally one considered finite differences? But that seems somewhat not right, because roughly speaking a derivative measures not the difference $f(x+h)-f(x)$, but rat... |
I'm attempting to understand the proof of the statement that every finite domain $R$ is a division ring. For a fixed nonzero element $a \in R$, the left-translation map is injective since $R$ is a domain. Since $R$ is finite this map is also surjective which implies that 1 is in its image. Therefore we have the existen... |
The word problem for finite groups is decidable. Is it obvious that this is true?
In particular, I'm not entirely sure about what it means for the problem to be decidable (in this case---I think I understand what decidable means in general). I assume it means that we are given a fixed group G (do we have to assume thi... |
Let $X_1,...,X_n$ be some observations, and let $\theta$ be some parameter of the density function we want to estimate.
Then, it is well known that
$l(\theta) = n^{-1} \sum_{i=1}^n \log f(X_i ; \theta)$ is called the average log-likelihood.
What is $\mathbb{E}[\log f(X_1 ; \theta)]$ called? Meaning, "the expected value... |
Which of these two books should I read first? Is one of them more advanced than the other? I'm planning to use them to self-study geometry.
Thanks.
|
In a population that grows, would a disparity in having a boy/girl probability cause the ratio of males to females tend to the same? e.g. if the probability of having boys was $.49$ and having a girl was $.51$ would that mean a growing population would tend to be $49$% male and $51$% female?
What about a non growing po... |
For the solution of the cubic equation $x^3 + px + q = 0$ Cardano wrote it as:
$$\sqrt[3]{-\frac{q}{2} + \sqrt{\frac{q^2}{4} + \frac{p^3}{27}}}+\sqrt[3]{-\frac{q}{2} - \sqrt{\frac{q^2}{4} + \frac{p^3}{27}}}.$$
but this is ambiguous because it does not tell you which cube roots to match up. Why don't people write it thi... |
I need to compute the intersection(s) between two circular arcs. Each arc is specified by its endpoints and their height. The height is the perpendicular distance from the chord connecting the endpoints to the middle of the arc. I use this representation because it is numerically robust for very slightly bent arcs, as ... |
I am making a software that deals with employee trust. I am trying to make a math formula (no need to tell that I am bad in math :) ).
Here is the scenario:
Employee 1 trusts employee 2 = > 20% or 0.2 (average trust of employee 1 with employee 2)
Employee 2 deals with 10 customers => 13% (average - 10 customers trust ... |
[sorry for the bad English]
I am fond of astronomy and environment.
I want to try to make a "light pollution map" but I haven't my satellites... so I use as approximation of light pollution the cities' population.
Let say we have for each city $C$ citizens, each one spreads an average of $X$ Watt of electricity for lig... |
I've been reading On Numbers and Games and I noticed that Conway defines addition in his number system in terms of addition. Similarly in the analysis and logic books that I've read (I'm sure that this is not true of all such books) how addition works is assumed. From what I understand the traditional method of buildin... |
It is, of course, one of the first results in basic complex analysis that a holomorphic function satisfies the Cauchy-Riemann equations when considered as a differentiable two-variable real function. I have always seen the converse as: if $f$ is continuously differentiable as a function from $U \subset \mathbb{R}^2$ to... |
Suppose that $f$ is differentiable at
0, and that $f(0) = 0$. Prove that
$f(x) = xg(x)$ for some function $g$
which is continuous at 0.
This is a problem from Spivak's Calculus, namely problem 27 of Chapter 10. (This is not homework, but rather self-study.) I am not sure how to go about this proof. The hint giv... |
(Motivation: I am going to be working with a high school student next week on long division, which is a subject I strongly dislike.)
Consider: $\frac{1110}{56}=19\frac{46}{56}$.
This is really a super easy problem, since once you realize $56*20=1120$ its trivial to write out $1110=56*19+46$.
You can work out the long ... |
I could swear I had a formula for this years ago in school, but I'm having trouble tracking it down. The problem:
I have $3$ red balls and $3$ black balls in a basket. I draw them out one at a time. How many different sequences of six balls can I get, i.e.
rbrbrb
rrrbbb
bbbrrr
etc...
I'm looking for a general formula ... |
Can you please let me know if my "proof" is correct: ?
Let $Y$ be a Hausdorff space and assume that each $y \in Y$ has a neighborhood $V$ such that $\overline{V}$ is regular. Prove $Y$ is regular.
Proof:
In order to show that $Y$ is regular we are going to show that for each $y \in Y$ and each open set $U$ which contai... |
What books would you recommend to learn physics, being a a Math major, from classical mechanics, electricity, etc. to modern physics?
|
I'm trying to figure out a generic way of determining a series of points on a Bézier curve where all the points are the same distance from their neighboring points. By distance I mean direct distance between the points not distance along the curve. See the image below
I've written a program that will solve this in an ... |
If we consider the unit square (i.e $[0,1] \times [0,1]$) with the lexicographic
order induced by ${\mathbb{R}}^{2}$, what are the limit points of the unit square?
My answer is: the set of limit points is equal to:
$ [0,1]^{2} \setminus ((0,1) \times \{1\}))$.
Is my answer correct? If not, how do you find them? I get c... |
I define the function $D(\mathbf M)$ for square matrices by taking a maximal set of all triples $(\mathbf v_i,k_i,n_i)_{i \in I}$ that satisfy $(\mathbf{M}-k\mathbf{I})^n\mathbf{v}_i = \mathbf{0}$ with the $\mathbf{v}_i$ linearly independent, and letting $$D(\mathbf M) = \prod_{i \in I} k_i.$$
For example the matrix
$... |
What's the general form for $x^y$?
What's the specialized form for $x^1$ and $x^0$?
|
This an interesting problem my friend has been working on for a while now (I just saw it an hour ago, but could not come up with anything substantial besides some PMI attempts).
Here's the full problem:
Let $x_{1}, x_{2}, x_{3}, \cdots x_{y}$ be all the primes that are less than a given $n>1,n \in \mathbb{N}$.
Prove t... |
From Introductory Combinatorics by Richard Brualdi
We have a chess master. He has 11 weeks to prepare for a competition so he decides that he will practice everyday by playing at least 1 game a day. To make sure that he's able to take a couple of breaks, he also decides that he wont play more than 12 games per week.
My... |
In some course notes on "evolution equations in probability spaces (etc)" I have the following definition:
A point $z \in \mathbb R^n$ is called the approximate limit of $f$ at $x$ if for every $\epsilon > 0$ the set $\{y \in \Omega : |f(y) - z| \leq \epsilon\}$ has Lebesgue density 1 at $x$. Notation: $\tilde{f}(x) :=... |
This equation:
$\hspace{1in}X=S\times(P_{FC}/P_{DC})$
Can be rewritten in terms of percentage change over a time period as follows:
$\hspace{1in}x=s+I_{FC}-I_{DC}=s-(I_{DC}-I_{FC})$
How can I actually rewrite this? I think it has something to do with logarithms because the multiplication becomes an addition and the ... |
Hi I am wondering how best to explain this, I am working with WebGL (effectively OpenGL) and I have the ray cast from clicking in 3d space. I have the far and near values of the ray cast, the camera position and the target it is looking at.
How do I work out where the ray cast starts and ends, effectively to draw a lin... |
If $\kappa$ is a measurable cardinal, and $\mathcal{U}$ is a normal ultrafilter which is $\kappa$-complete then $\mathcal{U}$ extends the club filter (i.e. every club is a member of $\mathcal{U}$).
One result is that all the sets in the ultrafilter are stationary.
Now, given a stationary set $S$ such that its complemen... |
What exactly is the variation of a function ? Is it a distace or an element of some space
The total of a real valued function $f\colon [0,t] \mapsto \Re $ is as below say $\pi = \{0=t_0,t_1,\cdots , t_n=t\} $ then the $p^{th}$ variation of $f$ is
$$ \lim_{\Vert \pi\Vert \to 0}V^p(f)=\sum_{i=1}^n \vert f(t_i)-f(t_{i-1}... |
Suppose that
$$u_1 = \left(\frac{1}{\sqrt{2}},\frac{1}{\sqrt{2}},0,0,\ldots,0\right)\quad\text{and}\quad
v_1 = \left(\frac{1}{\sqrt{2}},-\frac{1}{\sqrt{2}},0,0,\ldots,0\right)$$
and
$$u_l(n) = \sum_{k=0}^{2^{l-1}-1} u_1\left(n+ \frac{kN}{2^{l-1}}\right)\quad\text{and}\quad
v_l(n) = \sum_{k=0}^{2^{l-1}-1} v_1\left(n+... |
Is there any rule for powers so that i can compare which one is greater without actually calculating? For example
54^53 and 53^54
23^26 and 26^23
3^4 and 4^3 (very simple but how without actually calculating)
|
I have an $n\times p$ matrix $Z$ with $p\gt n$.
I have a diagonal matrix $A$ with positive entries.
Is there is a known way to determine the
MP inverse of $A Z^T Z A$, if I know $A$ and the MP inverse of $Z^T Z$.
Thanks
|
In category theory, you see the idea of a class of epimorphisms being stable under pullback. For example, in a regular category, the class of regular epimorphisms is closed under pullback. Every place I've seen the notion of pullback-stability, it's always a part of a bigger definition, such as regular category, or G... |
$f_n(x) : [0,1] \rightarrow \bf R $, and $f_n$ is $\frac{1}{n}$-periodic, $\max f_n(x) = n$, $\min f_n(x) = -n$.
As $n$ and $v$ goes to infinity simultaneously, prove the convergence of
$$\lim_{n,v \rightarrow \infty} \int_0^1 f_n(x) e^{-i2\pi v x}\,dx $$
if $v,n$ are integers and $v/n$ is not an integer.
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Is this series known to converge, and if so, what does it converge to (if known)?
Where $p_n$ is prime number $n$, and $p_1 = 2$,
$$\sum\limits_{n=1}^\infty \frac{1}{p_n}$$
|
Possible Duplicate:
How can you prove that a function has no closed form integral?
When they say that, e.g., Li(x) has no closed form (for some agreed upon definition of "closed form"), do they mean that it can be proved that there isn't one, or just that none has been found yet. Most authors seem to just assert th... |
Let $f(x) = \sum_{i =0}^\infty a_i x^i$ be a power series which converges for all real $x$. Assume that $f(x)$ is not identically zero. I'm interested in the density of the zeros of $f(x)$. Let $Z$ be the set of zeros of $f(x)$. Which of the following claims about density of $Z$ are true?
Claim 1: $Z$ is nowhere dense... |
On p.33 of Sergei Matveev's "Lectures on Algebraic Topology", the relative chain group of a pair $(K,L)$, where $L \subset K$ is a subcomplex, is defined as the free abelian group on simplices with interiors in $K \setminus L$. How is this different from just the chains on $K \setminus L$? How is this the same notion a... |
Let $X$ and $Y$ be topological spaces. A function $f: X \rightarrow Y$ is defined as continuous if for each open set $U \subset Y$, $f^{-1}(U)$ is open in $X$. This definition makes sense to me when $X$ and $Y$ are metric spaces- it is equivalent to the usual $\epsilon-\delta$ definition. But why is this a good definit... |
A positive integer is $B$-smooth if and only if all of its prime divisors are less than or equal to a positive real $B$. For example, the $3$-smooth integers are of the form $2^{a} 3^{b}$ with non-negative exponents $a$ and $b$, and those integers less than or equal to $20$ are $\{1,2,3,4,6,8,9,12,16,18\}$.
In Ramanuja... |
I am reading this.
It says
Intuitively, an open set is a solid region minus its boundary. If we include the boundary, we get a closed set, which formally is defined as the complement of an open set.
Now, question is if a closed set includes interior points also then how can it be complement?
I know basic set theory.... |
V.I. Arnold put the following question in his Mathematical Trivium:
Can an asymptotically stable equilibrium position become unstable in the Lyapunov sense under linearization?
It puzzled me for a while, since my experience doesn't include such a pathological case - but I'm still not sure about this one.
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I am planning to write a little note detailing several proofs of Lagrange's theorem that every natural number can be written as the sum of four perfect squares. I know of three different proofs so far:
a completely elementary proof by descent.
a proof via Minkowski's theorem and lattices.
Jacobi's proof via modular ... |
All examples I have come across so far deal with norm defined on a vector space. Can norm only be defined on vector spaces?
|
I've had no luck with this one. None of the convergence tests pop into mind.
I tried looking at it in this form $\sum \sin n\frac{1+\frac{1}{2}+\cdots+\frac{1}{n}}{n}$ and apply Dirichlets test. I know that $\frac{1+\frac{1}{2}+\cdots+\frac{1}{n}}{n} \to 0$ but not sure if it's decreasing.
Regarding absolute convergenc... |
Again, I am reading this.
I am finding it a bit difficult to understand the definition of n-dimensional smooth manifold.
Now,
$\{U_a; x^1_a, x^2_a, ..., x^n_a\}$ ----(1)
Is the thing (1) a set? (I think it is not).
Is it a tuple?
Also, is $U_a$ a set or a set of sets? What is significance of the subscript $a$?
I woul... |
In the computational topology literature, the reduction algorithm for computing the Smith normal form of a boundary matrix uses the notation $b_j > 1 \: \text{ and }\: b_j\mid b_{j+1}$ in the context of the diagonal elements of the Smith matrix. Can anyone give me an idea for what it means?
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Interesting problem from a $2000$ St. Petersburg school olympiad.
There are $109$ soldiers in a camp. Every night three of them go on watch patrol. Prove that it can be arranged so that after a while, every pair of soldiers has shared watch exactly three times.
I think I'm missing the key insight here.
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given a ring $R$, a nilpotent ideal $I$ and a morphism $\phi$ of $R$-modules $M \to N$, such that $M/IM \to N/IN$ is an isomorphism.
It is easy to see that this implies $\phi$ surjective, but what about injectivity?
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Excerpt from Waterhouse, 14.4 Structure of Finite Connected groups.
Thm. Let $A$ represent a finite connected group scheme over a perfect field of characteristic $p$. Then $A$ has the form
$k[X_1, ..., X_n]/(X_1^{p^{e_1}}, \ldots, X_n^{p^{e_n}})$.
Proof. Let $I_A$ be the augmentation ideal of $A$. By connectedness, $I... |
I may be over-complicating things, but something doesn't seem right (and I swear this isn't homework, I'm friggin 30 years old :P).
I want to see what it will cost me to make a TWO liter of Coca-Cola using coke syrup and CO2 (assume the water is free). Assume CO2 costs 0.21usd per liter of water (25usd for 120 liters)... |
Let $(\Omega, \mathcal{F}, \textbf{P})$ be a probability space and $A,B, A_i$ events in $\mathcal{F}$. Prove the following properties of every probability measure:
Monotonicity: If $A \subseteq B$ then $\textbf{P}(A) \leq \textbf{P}(B)$.
Sub-additivity: If $A \subseteq \bigcup_{i} A_i$ then $\textbf{P}(A) \leq \sum_{... |
I don't know if this is a well-known fact, but I have observed that every number, no matter how large, that is equally divided by $9$, will equal $9$ if you add all the numbers it is made from until there is $1$ digit.
A quick example of what I mean:
$9*99 = 891$
$8+9+1 = 18$
$1+8 = 9$
This works even with really lon... |
This may be a simple question, but I can't make it out right now.
Let $M$ be a monoid, and $g_1, \ldots, g_n \in M$ be elements of finite order in $M$. Is $\langle g_1, \ldots, g_n \rangle$ periodic (i.e, have all its elements of finite order)?
The monoid I'm interested in is the multiplicative monoid of $n \times ... |
Dear all,
I have systems like
$(A - \lambda B) X = F$
where lambda is being updated inside a loop. I also have a limited number of eigenvectors of the matrix pair (A, B), say 40 eigenpair from a previous analysis step.
I could get the results with direct solution however due to repeated factorization of the operator m... |
As a little project for myself this winter break, I'm trying to go through as much of Enderton's Elements of Set Theory as I can. I hit a snag trying to show two forms of the Axiom of Choice are equivalent. This is exercise 31 on page 55.
The first form is:
For any relation $R$ there is a function $G\subseteq R$ with... |
It seems to my uneducated mind that if I have $\frac{n}{k}$ subproblems, each of size $\Omega (k \log k)$, that my overall problem must be at least of size $\Omega (n \log k)$, by the reasoning that if I have $\frac{n}{k}$ things, each of which is at least $k \log k$, I should be able to multiply those together to get ... |
If you don't have enough teaching experience, what kind of job you should try to find to increase the teaching experience. Now many jobs have set requirements for teaching experience. It seems contradiction, Without appropriate job we can't increase the teaching experience,but without enough teaching experience we can'... |
My question is, how can I solve the following derivative question?
$y = \ln [\ln(\ln(x^2 +1))]$
Thanks in advance
|
I have heard about a lot of fancy numerical methods for solving ODEs. I know that there are methods that (assuming sufficient smoothness) asymptotically give a low error, like the Runge-Kutta methods. These estimate the solution in a set of points $t_0$, $t_1$, etc. But what if I want to have a function that is close ... |
I have this sum:
$$\sum_{0\le y<k}\sum_{0\le x<k-y}k - \max(x,y)\ ,\qquad k\in\mathbb{N}$$
Is there a closed form for it? This is no homework, im just a highschool student whose math is too poor.
Another way to write the formula is:
$$\sum^{k-1}_{y=0}\sum^{k-y-1}_{x=0}(k-\max(x,y)),$$
If there is something unclear.
|
This question is somewhat linked to this one -- in fact, I meant to ask the following:
Let $\mathcal{M}_n$ be the multiplicative monoid of $n \times n$ matrices on $\mathbb{N}$ (including $0$). Let $M_1, \ldots, M_k \in \mathcal{M}_n$ and suppose the monoid generated by each one of them is finite. Is the monoid gene... |
$A$ an integral domain, $a,b,c\in A$. If $d$ is a greatest common divisor of $a$ and $b$, is it true that $cd$ is a greatest common divisor of $ca$ and $cb$? I know it is true if $A$ is a UFD, but can't think of a counterexample in general situation.
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I'll be finishing Calculus by Spivak somewhat soon, and want to continue into linear algebra and multivariable calculus afterwards. My current plan for learning the two subjects is just to read and work through Apostol volume II; is this a good idea, or would it be better to get a dedicated Linear Algebra book? Are the... |
Could someone suggest me how to learn some basic theory of schemes? I have two books from algebraic geometry, namely "Diophantine Geometry" from Hindry and Silverman and "Algebraic geometry and arithmetic curves" from Qing Liu. I have had difficulties to prove the equivalence of many definitions. For example Hindry and... |
Imagine I am betting on the outcome of flipping $n$ biased coins $C_i$. I know exactly what the probability of each coin landing on heads is $P(H)$ and exactly the probability of each coin landing on tails is $P(T)$.
For every flip I can place one bet on either (but never both) outcome at odds with 5% value i.e. the p... |
This is a stronger one related to the question Convergence of $\lim_{n,v \rightarrow \infty} \int_0^1 f_n (x) e^{-i2\pi v x} \mbox{d} x $.
$F_n(x) : [0,1] \rightarrow \bf R $, for $1 \leq i \leq n$, $F_n(x)= n\cdot g_{n,i}(x)$ if $x \in [\frac{i-1}{n}, \frac{i}{n})$, with $g_{n,i}$ a series of integrable functions. ... |
I have the following cost function:
$\mbox{BSP Cost}=\sum_{i=1}^{\frac{n}{G}}G^{2}\left\lceil \frac{i}{p}\right\rceil +g\left(p\right)\sum_{i=1}^{\frac{n}{G}}Gi+l\left(p\right)\frac{n}{G}$
I would like to minimize it by choosing an appropriate G (i.e., G is a function of p and n). I have simplified it to the following ... |
I am trying to verify for which $z \in \mathbb{R}$ the series $\sum _{n=1}^{\infty } \left(1-\cos \left(\frac{1}{n}\right)\right)^z$ converges. The only test that was successful for me is the Kummer Test which gave the apparently correct result that it converges if $z > \frac{1}{2}$.
To get there I use the fact that th... |
A dishonest dealer marks his goods $20\%$ above the cost price. He gives a discount of $10\%$ to the customer on the marked price and makes a profit by using a false weight of $900$ gms in place of $1$ kg while buying or selling. Find the percentage profit earned by the shopkeeper.
I do tried this problem but am not qu... |
A friend sent me a link to this item today, which is billed as an "Irrational Numbers Wall Clock."
There is at least one possible mistake in it, as it is not known whether $\gamma$ is irrational.
Anyway, this started me wondering about how to improve the design of this clock from a mathematical standpoint. Here's... |
I study CS, but on the first semester I have a lot of mathematics. Of course, there is an introduction to set theory and logic. Recently, we had lectures about cardinality, different kinds of infinity, countable sets etc. I know the basics, but I don't understand many ideas as I would like to understand them.
Well, I a... |
3:30am and i was just looking at the law of large numbers and thought about the idea of a series of random variables and got myself very stuck.
when we consider the series $X'_n = X_1 + ... X_n$ and say it (divided by n) converges to $E(X_1)$, one of the definitions of a.s. convergence is based on the probability space... |
Rényi's Parking Constants comes up when one puts down unit length cars on a interval, such that the probability of covering any two interval is the same.
Are there any published results when the distribution is non-uniform?
|
I'm having trouble solving problem 12 from Section 1.2 in Hatcher's "Algebraic Topology".
Here's the relevant image for the problem:
I'm trying to find $\pi_1(R^3-Z)$, where $Z$ is the graph shown in the first figure. The answer (according to the problem statement) is supposed to be $\langle a,b,c| aba^{-1}b^{-1}cb^\e... |
I have a presentation in which I want to point out a difference between two functions.
Instead of putting the two functions on a slide and pointing to the differences, I want to do something simple.
The first function has a form of $f_1(x) = g(x)\log (h(x))$ and the second function has the form $f_2(x) = g(x)\log(r(x)h... |
Is $\ p_n^{\pi(n)} < 4^n$ where $p_n$ is the largest prime $\leq n$?
Where $\pi(n)$ is the prime counting function. Using PMT it seems asymptotically $\ p_n^{\pi(n)} \leq x^n$ where $e \leq x$
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I have no idea, how to solve this problem:
Let $\psi: (\mathbb{R} \to \mathbb{N}) \to P(P(\mathbb{R}))$ defined as:
$\psi (f) = \mathbb{R}/_{ker(f)}$
(1) Is this a surjective function?
(2)Is this an injective function?
I tried to do something, but after hours I can't see any progress... I just don't know, were/how to ... |
I do not understand the topology of a Lie group clearly. Let $G$ be a Lie group and $T_eG$ be its tangent space at the identity $e \in G$. Why $\operatorname{Aut}(T_eG)$ is an open subset of the vector space of endomorphisms of $T_eG$ (i.e., $\operatorname{End}(T_eG)$)? What does “open” mean?
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Let su(2) be the Lie algebra of SU(2), thought of as a representation of SU(2) by conjugation.
Let S be the rank-2 symmetric tensors over the complex numbers, acted on by SU(2) in the obvious way.
I have seen it asserted that the complexification of su(2) is isomorphic (as a representation) to S, and presumably there ... |
Hello
I am trying to model the relationship between two variables, say x and y. I have a number of subjects - for each subjectm I have a number of x and corresponding y, both of which are always positive. This data tends to be very sparse. There are some problem specific constraints:
1) y(0) = 0 (or very close to it... |
Consider the following function: $ f(x) = \sum_{i,j>=0} a_{ij} x^i (1-x)^j$ , where the coefficients satisfy $a_{ij} \in [0,1]$. Observe that $f(x)$ converges for $x\in [0,1]$. Is $f$ analytic over $[0,1]$? If not, how about over $[c,d]$ for $c,d \in (0,1)$?
I think I can show that the answer is yes for the $[c,d]$ cas... |
Firstly, let's establish what exactly I mean by these symbols. Let $\omega_0 = \{ 0, 1, 2, \ldots \}$, where $0, 1, 2, \ldots$ are the usual von Neumann representations of the natural numbers. Let $n$ be a finite natural number. For each $n$, define $\omega_{n+1} = \sup S$, where $S$ the image of the set $R$ under the ... |
How to embed a matrix, for example, a $3x3$ singular matrix in to $R^9$?
How to compute the induced metric? Is it just the Frobenius norm of the matrix?
Many Thanks.
sam
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I am referring to the this wiki page, which states that the way to find the volume of a Polyhedron can be done via Divergence Theorem:
$$\int \nabla \cdot \vec{F} dV=\oint \vec{F} \cdot \hat{n} dS$$
By choosing
$$ \vec{F}=\frac{\vec{x}}{3}$$
One can then obtain the volume formula for a polyhedron with $k$ faces:
$$V = ... |
If I have $22x = 23y = 24z$ and $x+y+z = 3865$ how to obtain the values of $x$,$y$ and $z$ ?
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It seems like I am having troubles while dealing with dishonest merchants! Here is another problem:
A cloth merchant says that due to slump in the market, he sells the cloth at $10\%$ loss but he uses a false meter scale and actually gains $15\%$.Find the actual length of the scale.
Any hints, solution, explanation are... |
$1$.The compound interest on certain sum for $2$ years is $\$410$ and simple interest is $\$400$. Find the rate of interest per-year.
$2$.A sum of money invested at compound interest amounts in 3 years to $\$ 2400$ an in $4$ years to $ \$ 2520$. Find the rate of interest per-year.
Solving them using the traditional com... |
This may be a simple question:
Find all subrings of $\mathbb{Z}^2$.
|
Let $G$ and $H$ be two Lie groups and $\rho: G \to H$ be a homomorphism. How to differentiate $\rho$ to obtain a Lie algebra homomorphism $d\rho_e: T_eG \to T_eH$?
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Definition of lattice require that any two elements of lattice should have LUB and GLB, while complete lattice extends it to, every subset should have LUB and GLB. But by induction , it is possible to show that if any two elements have LUB and GLB then every subset should have LUB and GLB. I read somewhere that the dif... |
What is cardinality of $P(\mathbb{R})$? And $P(P(\mathbb{R}))$?
P is a Power Set, $\mathbb{R}$ is set of real numbers.
In general - how can find cardinality of different sets? Is/are there a good method(s)? And is there some kind of handout, that provides list of cardinality of popular sets ($\mathbb{R}$, $\mathbb{Q}$,... |
The standard proof of the Prime Number Theorem requires taking into consideration that there are no zeroes of the Riemann Zeta function in which the real part equals one. But consider the following argument:
The probability that a number less than X is prime $\pi(X)/X$ is approximately $\Pi_{p<X} (1-1/p)$ (this is for ... |
It's been a really long time since I've done calculus or any other kind of math beyond tip calculation. I was given a spreadsheet that calculates and plots a growth curve over time based on a handful of inputs. It goes out for one year. I'd like to be able to make educated (though not necessarily precise) prediction... |
According to Tarski's Fixed Point Theorem, for a complete lattice $L$, and monotone function $f:L \rightarrow L$, the set of fixed points of $f$ forms complete lattice.
Definition of $lfp(f)$ and $gfp(f)$ is given in terms of set $\text{Red}(f)$ and $\text{Ext}(f)$. Where $\text{Red}(f)$ is set of all those points $l'... |
Wikipedia : In mathematics, an algebraic number is a number that is a root of a non-zero polynomial in one variable with rational (or equivalently, integer) coefficients.
Question : Show that the Set of algebraic number in $\mathbb{C}$ is countable ?
Hint : if $\mathbb{F(z) = a_ 0 + a_1z + .... + a_nz^n}$ we denote th... |
It seems to me that a = x (mod m) can mean either that a is the remainder of x divided by m or that the remainders of a and x are the same when divided by m (eg. with respect to mod m). For example: 2 = 6 (mod 4) would be the first meaning I discussed, but 6 = 2 (mod 4) would be the second. It seems that I must be co... |
I have been given the following pdf :
fT (t; B, C) = ( exp(-t/C) - exp(-t/B) ) / ( C - B ) , (t>0)
where the overall mean is B+C.
I am unsure as to how to write the log likelihood function of B and C.
The next part of the Q asks me to derive the equations that would have to be solved in order to find the max likeliho... |
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