instruction stringlengths 12 30k |
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If you define $p=X^n-1\in\mathbb C[X]$, then $p(A)=0$. This tells you that the minimal polinomial $m_A$ of $A$ divides $p$ and, in particular, that $m_A$ has all its roots simple, because the same is true of $p$.
It follows that $A$ is diagonalizable, so, up to similarity, you can suppose that it is diagonal.
C... |
**Problem:**
***From the deck of the 52 cards,cards are drawn randomly without replacement.What is the probability of drawing a king of hearts at the third attempt? If it was drawn at the 15th attempt, what is the probability ?***
----------
I have solved the first by finding P(getting the card at third at... |
If you define $p=X^n-1\in\mathbb C[X]$, then $p(A)=0$. This tells you that the minimal polinomial $m_A$ of $A$ divides $p$ and, in particular, that $m_A$ has all its roots simple, because the same is true of $p$.
It follows that $A$ is diagonalizable, so, up to similarity, you can suppose that it is diagonal.
C... |
No calculation is required and certainly no C. As you should observe when you start writing out the same product, the denominator of each next term will cancel the numerator of the previous, so the only terms left will be the denominator of the first and the numerator of the last terms. So the answer will always be $\f... |
It seems like I am facing some confusion while handling with inequalities,I was doing some tasks where it is asked to find the interval of the variable,after some steps I deduced the the necessary expression but I am facing some problem while trying to deduce the final expression.
Consider for example:
$$a^2 \g... |
Yes, $\mathbb{Z}$ is one.
Of course, as Qiaochu and Alex effectively say in the comments, this all depends on what you mean by "globally". But before I say what that means, let me point out that "locally X" is not so straightforward. In topology, we say that a space is "locally **X**" if whenever we have $p \in U$... |
To state the context, I am familiar with the Hopf-Rinow theorem.
My request is three fold,
- I would like to know of general classes of geodesically incomplete spaces. I basically want to see lots of examples for this.
- I want to know of techniques of proving and testing for geodesic incompleteness or c... |
I would like to gain intuition how "singular" a distribution can be.
Let $X \in \mathbb R^n$ be open, $\mathcal D(X)$ be the smooth test functions on $X$ with canonical LF topology, and let $\mathcal D'(X)$ be the topological dual of $D(X)$.
Let $\Psi \in D'(X)$. Can the singular support of $ \Psi$, i.e. the comp... |
If R=I+J, can we conclude that R/J is free? |
It's well known that under some "weak" hypothesis, such as finitely generation, the support of an A-module is closed in Spec(A). It is true also in the most general case? |
The support of a module is closed? |
**Problem 1:**
***A man has 5 coins,two of which are double-headed, one is double-tailed and two are normal.He shuts his eyes,picks a coin at random, and tosses it.what is the probability that the lower face as the coin is a head?***
**Problem 2:**
***The probability that a man who is 85 years. old will die be... |
Is there a procedure/algorithm for calculating sums of the form
$$ \sum_{n_1,\ldots,n_r >0} \frac{1}{L_1(n_1,\ldots,n_r)^{m_1} \ldots L_r(n_1,\ldots,n_r)^{m_r}} $$
where
$$ L_i(n_1,\ldots, n_r) := a_{1,i} n_1 + \ldots + a_{r,i} n_r $$
for some $a_{i,j} \in \mathbb{C}$. How would one go about evaluating this su... |
**Problem 1:**
***A man has 5 coins,two of which are double-headed, one is double-tailed and two are normal.He shuts his eyes,picks a coin at random, and tosses it.what is the probability that the lower face as the coin is a head?***
**Problem 2:**
***The probability that a man who is 85 years. old will die be... |
Is there a procedure/algorithm for calculating sums of the form
$$ \sum_{n_1,\ldots,n_r >0} \frac1{L_1(n_1,\ldots,n_r)^{m_1} \ldots L_r(n_1,\ldots,n_r)^{m_r}} $$
where
$$ L_i(n_1,\ldots, n_r) := a_{1,i} n_1 + \ldots + a_{r,i} n_r $$
for some $a_{i,j} \in \mathbb C$. How would one go about evaluating this sum t... |
Let $X = (X_t: t \in [0,T])$ be a stochastic process satisfying a [CIR model][1]
$$
dX_t = \beta (X_t - \gamma) dt + \sigma\sqrt{X_t} dB_t,
$$
where $B_t$ is a standard Brownian motion, $\beta$ is a negative constant, $\gamma, \sigma$ are positive constants. In order for the SDE to make sense, assume that $X_t > 0$... |
As the comments have suggested, the question is a bit under-determined, but let me supply some definitions in a plausible way and answer it based on those definitions.
Let $P$ be a property of topological spaces. (Formally speaking this could be any "predicate", but it would only be asking for trouble if $P$ were n... |
The book [Irresistible Integrals by George Boros and Victor Moll](http://books.google.com/books?id=g9tH8M3NvDIC&printsec=frontcover&dq=irresistible+integrals&hl=pt-PT&ei=W8sATYbBLoSXOu2craYB&sa=X&oi=book_result&ct=result&resnum=1&ved=0CCUQ6AEwAA#v=onepage&q&f=false) on page 204 has the following identity
$\displays... |
Take the Euler scheme. Let's assume that you start with $X_0=\epsilon$. Let's take $\beta=-1$, $\gamma =1$ and $\sigma=1$ for the sake of definiteness. Then in the next step you'll have
$$X_{\Delta} = \epsilon + (1-\epsilon) \Delta + \sqrt{\epsilon} Z \Delta$$
What's the probability that this is smaller than zero... |
Is a Godel sentence logically valid? |
What you are forgetting is that you have more than *one* inequality.
The inequality $a^2\geq 8$, by *itself*, only gives you $|a|\geq \sqrt{8}=2\sqrt{2}$, or equivalently as you write:
$$a\geq 2\sqrt{2}\qquad\text{or}\qquad a\leq -2\sqrt{2}.$$
However, you have *other* inequalities that need to be satisfied as wel... |
Suppose that $R$ is a ring, $I$ and $J$ are ideals, and $R=I+J$. Can we conclude that $R/J$ is free? |
If $R=I+J$, can we conclude that $R/J$ is free? |
This might be an elementary question, but I am just beginning to learn logic theory.
From wikipedia article on Godel's incompleteness theorems
> Any effectively generated theory capable of expressing elementary arithmetic cannot be both consistent and complete. In particular, for any consistent, effectively gene... |
Is there a general way to show whether the limit
$\lim_{n \to \infty}\frac{d^{n}f(x)}{dx^n}$
converges to some expression?
What about repeatedly integrating an expression n times as $n \to \infty$? |
Is there a general way to show whether the limit
$$\lim_{n \to \infty}\frac{d^{n}f(x)}{dx^n}$$
converges to some expression?
What about repeatedly integrating an expression n times as $n \to \infty$? |
This might be an elementary question, but I am just beginning to learn logic theory.
From wikipedia article on Godel's incompleteness theorems
> Any effectively generated theory capable of expressing elementary arithmetic cannot be both consistent and complete. In particular, for any consistent, effectively gene... |
For the notion of limit, you need a notion of convergence or of "distance." Here, you are talking about *functions*, not numbers, so you need to specify how you measure the "distance" between two functions. Turns out, there are many different ways of doing it, and the answer for a particular $f(x)$ *may* depend on whic... |
First, if $Y$ takes values in $[1,5]$, then $U$ takes values in the range...? So, for $x$ in that range, you should consider the distribution function $F_U {(x)} := {\rm P}(U \le x)$. Substituting $U=2Y^2+3$ leads straightforwardly to $F_U {(x)} = F_Y {(\varphi(x))}$ for some function $\varphi(x)$, where $F_Y$ is the d... |
Is (e^z-1)/(e^z+1) analytic? |
how to linearise a square root expressions? |
How to linearise a square root expression? |
If $(A+C)^2+B$ is large compared with 1 you can divide it out to get $$\sqrt{(A+C)^2+B}\sqrt{1-\frac{2cos(\alpha-\theta)}{(A+C)^2+B}}$$ and use $\sqrt{1-x}\approx 1-\frac{x}{2}$ for $x<<1$. Does that help? |
**Problem:**
***The probability that a man who is 85 years. old will die before attaining the age of 90 is $\frac13$. A,B,C,D are four person who are 85 years old. what is the probability that A will die before attaining the age of 90 and will be the first to die ?***
|
As already mentioned in the other answers and comments, this is true if the space is assumed to be complete (see Andrey Rekalo's answer), but not necessarily otherwise (see Carl Mummert's answer). In the complete case, more can be said. If the Hilbert space dimension (cardinality of a maximal orthonormal set) is infi... |
What you are forgetting is that you have more than *one* inequality.
The inequality $a^2\geq 8$, by *itself*, only gives you $|a|\geq \sqrt{8}=2\sqrt{2}$, or equivalently as you write:
$$a\geq 2\sqrt{2}\qquad\text{or}\qquad a\leq -2\sqrt{2}.$$
However, you have *other* inequalities that need to be satisfied as wel... |
Is the function $\frac{e^z-1}{e^z+1}$ analytic?
I tried using $\mathop{\lim}\limits_{z\to a} \frac{f(z)-f(a)}{z-a}$, but I didnt really know what to use for $a$. |
Is $\frac{e^z-1}{e^z+1}$ analytic? |
Suppose that we have two varieties of algebras $A$ and $B$, whose operators all have arities less than some regular cardinal, and such that every $B$-algebra (please correct me if this is not the usual notation) is also an $A$-algebra. It is known that the forgetful functor from the category of $B$-algebras to that of ... |
Can nonisomorphic algebras generate isomorphic relatively free algebras? |
What is the rationale behind this? |
I'm reading the Algebra book by Knapp and he mentions in passing that an orthonormal set in an infinite dimension vector space is "never large enough" to be a vector-space basis (i.e. that every vector can be written as a **finite** sum of vectors from the basis; such bases do exist for infinite dimension vector spaces... |
Bill Dubuque has answered several questions by indicating that some form of "telescoping" is taking place. See [this](http://math.stackexchange.com/questions/13646/how-to-derive-the-following-identity) post and the links provided by Bill for more information.
I have never heard of "telescoping" until I read a few a... |
What is the rationale behind saying that $|x|=1$ implies $x=\pm 1$? |
It seems like I am facing some confusion while handling with inequalities,I was doing some tasks where it is asked to find the interval of the variable,after some steps I deduced the the necessary expression but I am facing some problem while trying to deduce the final expression.
Consider for example:
$$a^2 \g... |
What you are forgetting is that you have more than *one* inequality.
The inequality $a^2\geq 8$, by *itself*, only gives you $|a|\geq \sqrt{8}=2\sqrt{2}$, or equivalently as you write:
$$a\geq 2\sqrt{2}\qquad\text{or}\qquad a\leq -2\sqrt{2}.$$
However, you have *other* inequalities that need to be satisfied as wel... |
The category of simplicial presheaves on a small Grothendieck site $\mathcal{C}$ can be given a model structure by defining weak equivalences and cofibrations sectionwise. It's called the (global) injective model structure and has a mapping space functor $Hom$, given by $Hom(X, Y)_n = hom(X\times \Delta^n, Y)$.
Usin... |
I suppose you mean non negative $\alpha$, $\beta$? (You seem to be allowing them to be zero).
In which case, this looks like the [Frobenius Problem][1] which says that for $a,b$ such that $(a,b)=1$, every number greater than $(a-1)(b-1) - 1 $ is representable in the for $a\alpha + b \beta$ for $\alpha \ge 0, \beta \... |
I'm looking for a good book on distribution theory (in the Schwartz sense), I have the basic knowledge as given in Grafakos' Classical Fourier Analysis, but I want to know more about it. Is the reference still Laurent Schwartz' Théorie des distributions? That book is hard to obtain, it seems to be only available in Fra... |
How to decompose A if and only if B ? |
(Now that you have relaxed the last condition of paragraph 1)
I suppose you mean non negative $\alpha$, $\beta$? (You seem to be allowing them to be zero).
In which case, this looks like the [Frobenius Problem][1] which says that for $a,b$ such that $(a,b)=1$, every number greater than $(a-1)(b-1) - 1 $ is repr... |
(Now that you have relaxed the last condition of paragraph 1)
I suppose you mean non negative $\alpha$, $\beta$? (You seem to be allowing them to be zero).
In which case, this looks like the [Frobenius Problem][1] which says that for $a,b$ such that $(a,b)=1$, every number greater than $(a-1)(b-1) - 1 $ is repr... |
(Now that you have relaxed the last condition of paragraph 1)
I suppose you mean non negative $\alpha$, $\beta$? (You seem to be allowing them to be zero).
In which case, this looks like the [Frobenius Problem][1] which says that for $a,b$ such that $(a,b)=1$, every number greater than $(a-1)(b-1) - 1 $ is repr... |
I came across this problem. If $X1,X2..Xn$ is an iid sequence with uniform distribution on [-1,1] and we define a new random variable like this:
$Yn = \frac{\sum(Xk)}{\sum(Xk^2 + Xk^3)}$
How do I use the strong law of large numbers to show that $Yn$ is normal and find its mean and variance? The uniform distributi... |
If $(A+C)^2+B$ is large compared with 1 you can divide it out to get $$\sqrt{(A+C)^2+B}\sqrt{1-\frac{2cos(\alpha-\theta)}{(A+C)^2+B}}$$ and use $\sqrt{1-x}\approx 1-\frac{x}{2}$ for $x<<1$. If you are specifically interested in $\theta$ you can expand the cosine term and still use the approximation to the square root... |
It is a subset of the cartesian product in the category of commutative rings with unit. What is the pullback in the category of commutative $k$-algebras? Is it the same set as in rings? |
What is the pullback in the category of commutative algebras? |
Your left adjoint is not quite correct.
For example, take the case where $A$ is the variety of all groups, and $B$ is the variety of *abelian* groups. The forgetful functor from $\mathcal{A}b$ to $\mathcal{G}roup$ indeed has a left adjoint, but the left adjoint $\mathcal{F}$ takes a group $G$ to $G^{\rm ab}$, the a... |
The pullback is a subset of the cartesian product in the category of commutative rings with unit.
What is the pullback in the category of commutative $k$-algebras? Is it the same set as in rings? |
I need to find the function c(k), knowing that
$$\sum_{k=0}^{\infty} \frac{c(k)}{k!}=1$$
$$\sum_{k=0}^{\infty} \frac{c(2k)}{(2k)!}=0$$
$$\sum_{k=0}^{\infty} \frac{c(2k+1)}{(2k+1)!}=-1$$
Is it possible? |
How to solve the following system? |
I need to find the function c(k), knowing that
$$\sum_{k=0}^{\infty} \frac{c(k)}{k!}=1$$
$$\sum_{k=0}^{\infty} \frac{c(2k)}{(2k)!}=0$$
$$\sum_{k=0}^{\infty} \frac{c(2k+1)}{(2k+1)!}=1$$
Is it possible? |
I need to find the function c(k), knowing that
$$\sum_{k=0}^{\infty} \frac{c(k)}{k!}=1$$
$$\sum_{k=0}^{\infty} \frac{c(2k)}{(2k)!}=0$$
$$\sum_{k=0}^{\infty} \frac{c(2k+1)}{(2k+1)!}=1$$
$$\sum_{k=0}^{\infty} \frac{(-1)^k c(2k+1)}{(2k+1)!}=-1$$
$$\sum_{k=0}^{\infty} \frac{(-1)^k c(2k)}{(2k)!}=0$$
Is it ... |
It seems like I am facing some confusion while handling with inequalities,I was doing some tasks where it is asked to find the interval of the variable,after some steps I deduced the the necessary expression but I am facing some problem while trying to deduce the final expression.
Consider for example:
$$a^2 \g... |
What you are forgetting is that you have more than *one* inequality.
The inequality $a^2\geq 8$, by *itself*, only gives you $|a|\geq \sqrt{8}=2\sqrt{2}$, or equivalently as you write:
$$a\geq 2\sqrt{2}\qquad\text{or}\qquad a\leq -2\sqrt{2}.$$
However, you have *other* inequalities that need to be satisfied as wel... |
I came across this problem. If $X_1,X_2, \ldots, X_n$ is an iid sequence with uniform distribution on [-1,1] and we define a new random variable like this:
$Y_n = \frac{\sum X_k}{\sum(X_k^2 + X_k^3)}$
How do I use the strong law of large numbers to show that $Y_n$ is normal and find its mean and variance? The uni... |
WolframAlpha is confusing me. I'm working on two different integral problems, and with each one, for some reason, Wolfram switches the signs of the answer.
1. [http://www.wolframalpha.com/input/?i=integral+dx/(x^.5(x-1))][1]
2. [http://www.wolframalpha.com/input/?i=integral+(x-1)/(x^2-4)][2]
For the first one, t... |
Is it correct to say that the number of circles in an Apollonian gasket is countable becuase we can form a correspondence with a Cantor set, as their methods of construction are similar?
What about if we apply the Apollonian gasket construction inside of a fractal like the Koch snowflake? (I think that will still be... |
Is the number of circles in the Apollonian gasket countable? |
Suppose I have acquired 100,000 customers for a transaction-based business at 12 months into the business.
Assuming there is one fixed-price product/service worth $10 that each customer makes 4 purchases of across a 6 month period and then ceases to be a customer, how can I calculate my total revenue by month 12?
... |
Is it correct to say that the number of circles in an Apollonian gasket is countable becuase we can form a correspondence with a Cantor set, as their methods of construction are similar?
What about if we apply the Apollonian gasket construction inside of a fractal like the Koch snowflake? (I think that will still be... |
Suppose I have acquired 100,000 customers for a transaction-based business at 12 months into the business and their number has increased at a steady rate of growth month by month (open to suggestions regarding this rate.)
Assuming there is one fixed-price product/service worth $10 that each customer makes 4 purchas... |
How to find shortest distance between two skew lines in 3D? |
So, I am a graduate student who is certain that he does not want to do analysis (I think...). What are the most exciting fields in mathematics right now? It seems to me that very generally, they are algebraic geometry and algebraic topology, with the latter being specifically concerned with homotopy theory (something... |
Well I only took one course on elementary geometry so I don't know much about this, but we used Edwin Moise's book [Elementary Geometry From an Advanced Standpoint](http://www.amazon.com/Elementary-Geometry-Advanced-Standpoint-3rd/dp/0201508672/ref=sr_1_1?s=books&ie=UTF8&qid=1291941012&sr=1-1). It is very well written ... |
WolframAlpha is confusing me. I'm working on two different integral problems, and with each one, for some reason, Wolfram switches the signs of the answer.
1. [http://www.wolframalpha.com/input/?i=integral+dx/(x^.5(x-1))][1]
2. [http://www.wolframalpha.com/input/?i=integral+(x-1)/(x^2-4)][2]
For the first one, t... |
Is there a general way to show whether the limit $$\lim_{n \to \infty}\frac{\mathrm d^n}{\mathrm dx^n} f(x)$$ converges to some expression?
What about repeatedly integrating an expression $n$ times as $n \to \infty$? |
Does $\lim_{n \to \infty}\frac{\mathrm d^n}{\mathrm dx^n} f(x)$ converge? |
It seems like I am facing some confusion while handling with inequalities,I was doing some tasks where it is asked to find the interval of the variable,after some steps I deduced the the necessary expression but I am facing some problem while trying to deduce the final expression.
Consider for example:
$$a^2 \g... |
I've been spinning my wheels on this for an embarrassing amount of time. This is not for an assignment, I'm just studying.
> Three balls are placed at random in
> three boxes. There is no restriction
> on the number of balls per box; list
> the 27 possible outcomes of this
> experiment.
I could only come up w... |
Wow, this is great! I would love to expound a bit on some of the things you have mentioned. While I agree with Ryan that what is exciting and hot is entirely subjective, I too find homotopy theory and algebraic geometry enthralling. I really only know things about homotopy theory though, so I can only talk on that.
... |
Wow, this is great! I would love to expound a bit on some of the things you have mentioned. While I agree with Ryan that what is exciting and hot is entirely subjective, I too find homotopy theory and algebraic geometry enthralling. I really only know things about homotopy theory though, so I can only talk on that.
... |
Suppose I have acquired 103,304 customers for a transaction-based business at 12 months into the business and their number has increased at a monthly rate of 40% (from month 1, in which 2,551 customers were acquired). Assume 103,304 represents the total number of new customers (active or dead) acquired over the 12 mont... |
Expanding on Jack's answer, there is a whole exciting field known as "chromatic homotopy theory". I don't know much about it, I'll say a few slogans since I noticed you also asked the question "geometry or topology". Hopefully none of them are wrong. (If so, somebody please correct me!)
Chromatic homotopy theory ... |
I'm reading the Algebra book by Knapp and he mentions in passing that an orthonormal set in an infinite dimension vector space is "never large enough" to be a vector-space basis (i.e. that every vector can be written as a **finite** sum of vectors from the basis; such bases do exist for infinite dimension vector spaces... |
What you are forgetting is that you have more than *one* inequality.
The inequality $a^2\geq 8$, by *itself*, only gives you $|a|\geq \sqrt{8}=2\sqrt{2}$, or equivalently as you (almost) write:
$$a\geq 2\sqrt{2}\qquad\text{or}\qquad a\leq -2\sqrt{2}.$$
**Added:** Note that you took $a^2\geq 8$, a *weak* inequality... |
The permenant can be interpreted as the number of perfect matchings in bipartite graphs.
> Is there a similar graphical interpretation of determinant? |
Graphical interpretation of determinant? |
Graph-theoretic interpretation of determinant? |
The permanent can be interpreted as the number of perfect matchings in bipartite graphs.
> Is there a similar graph-theoretic interpretation of the determinant? |
A question about the previous answer (it seems that I can't comment). In the estimate, $|f^{(k+1)}(x)| \leq (\frac{1}{1-r})^{k+1}$, is there a missing $k!$ term as in the original poster's computation of the derivative? If so, how does this affect the answer? |
I'm reading the Algebra book by Knapp and he mentions in passing that an orthonormal set in an infinite dimension vector space is "never large enough" to be a vector-space basis (i.e. that every vector can be written as a **finite** sum of vectors from the basis; such bases do exist for infinite dimension vector spaces... |
Well I only took one course on elementary geometry so I don't know much about this, but we used Edwin Moise's book [Elementary Geometry From an Advanced Standpoint](http://www.amazon.com/Elementary-Geometry-Advanced-Standpoint-3rd/dp/0201508672/ref=sr_1_1?s=books&ie=UTF8&qid=1291941012&sr=1-1). It is very well written ... |
My undergraduate number theory class decided to dip into a bit of algebraic geometry to finish up the semester. I'm having trouble understanding this bit of information that the instructor presented in his notes.
Here it is in paraphrase (assume we are over an abstract field k)
We take a polynomial in k, $f =Y^... |
The term "point at infinity" is not actually a well-defined term from the point of view of your projective model. You should think of it this way:
Suppose you are given a homogeneous polynomial and you are interested in its zero set as a subset of projective space.
The projective space is made up of several so-cal... |
The term "point at infinity" is not actually a well-defined term from the point of view of your projective model. You should think of it this way:
Suppose you are given a homogeneous polynomial and you are interested in its zero set as a subset of projective space.
The projective space is made up of several so-cal... |
Here's another way to think about the "line at infinity" and the "points at infinity"...
Think of the usual $XY$-plane as sitting inside of $3$-space, but instead of it sitting in its usual place, $\{(x,y,0) : x,y\in\mathbb{R}\}$, shift it up by $1$ so that it sits as the $z=1$ plane.
Now, you are sitting at the ... |
According to wikipedia the [right quotient of a regular language][1] with ANY other language is regular. I have not been able to find a proof of this fact. All the sources talk about quotient with another regular language. Can somebody point me to a proof?
P.S. Definition from Wikipedia: The right quotient (or simpl... |
I have two faces ($S_1$ and $S_2$), each bounded by a series of Vertex $V$. These two faces may or may not intersect. The question is how to find the polyhedrons that are formed by the intersection of the two?
An example of this is shown below:
![alt text][1]
In the above example, there should be two polyhedron... |
I have two faces ($S_1$ and $S_2$), each bounded by a series of Vertex $V$. These two faces may or may not intersect. The question is how to find the polyhedrons that are formed by the intersection of the two?
An example of this is shown below:
![alt text][1]
In the above example, there should be two polyhedron... |
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