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If I understand correctly, most definitions of 'limits' require that the function either a) be defined in an open neighborhood around the relevant point or b) more permissively, that the relevant point is a limit point; the definition of 'continuity' is then given a special case so that functions are continuous at isol... |
I stumbled upon this number theory problem while I was solving another problem. Here is the equation: $$3^kn + 3^{k-1} + 2^m(3^{k-1} + 2h) = 2^{m+l}n$$ where $k \geq 3, h,l,m,n\in\mathbb{N}$, $n$ is odd and $n$ is not a multiple of $3$. My impression is that it does not have a solution. However, I have not progressed o... |
Background Informatiom
I would appreciate help in identifying or explaining this operation:
To calculate each of the $n$ values of $f(\Phi)$:
Sample from the distribution of each of $i$ parameters, $\phi_i$
Calculate the $i$ values of $g(\phi_{i})$
Subtract each $g_i(\phi_{i,n})$ from $g_i(\hat{\phi_i})$ (these are de... |
I've some trouble in understanding integer multiplication using FFT.
I'm using the algorithm described on wikipedia.
Here is an example of how I understand this algorithm:
$$a=173$$
$$b=95$$
Lets take $w=4$, then we have
$$a=13*2^{4\cdot 0}+10*2^{4\cdot1}$$
$$b=15*2^{4\cdot 0}+5*2^{4\cdot1}$$
In the vector notation it ... |
I have a linear operator $T_1$ which acts on the vector space of polynomials in this way:
$$T_1(p(x))=p'(x).$$
How can I find its eigenvalues and how can I know whether it is diagonalizable or not?
|
I have a waveform which I would like to take the fourier transform of. However, the simulator which generated the waveform uses an adaptive algorithm to determine the timestep for each calculation.
Now, I figure I can regularize the timestep to first order by creating a new timebase and for each point in the new base ... |
Forgive me for my notation and lack of formatting prowess, it's been a while since I've done any of this.
We have flipped a biased coin (unknown weight) $n$ times and received $k$ successes (or heads). I am trying to calculate the probability that there be a success on flip $N+1$
If we assume the bias $p$ is sampled u... |
Let $(f_n)_{n \geq 1}$ be disjointly supported sequence of functions in $L^\infty(0,1)$. Is the space $\overline{\mathrm{span}(f_n)}$ (the closure of linear span) complemented in $L^\infty(0,1)$? By complemented we mean that $L^\infty(0,1) = \overline{\mathrm{span}(f_n)} \oplus X$, where $X$ is a subspace of $L^\infty$... |
Given a pair of positive coprime integers $a_1$ and $a_2$, I'd like to determine the set of positive integers $x$ which satisfy the following system of $\text{gcd}$ equations:
\begin{eqnarray}
\gcd(x,a_1) = 1, \gcd(x - a_1,a_2) = 1, \dots, \gcd(x - a_1 \lfloor \tfrac{x}{a_1} \rfloor, a_2) = 1,
\end{eqnarray}
where $x \... |
If anyone has some insight on how to do this it would be very much appreciated.
|
Let $k$ be an algebraically closed field, and let $\mathbb{A}^n(k)$ denote the linear $n$-dimensional vector space over $k$. Suppose $X \subset \mathbb{A}^n$, $X' \subset \mathbb{A}^m$ are two varieties. We say that $X, X'$ are isomorphic if there exists a polynomial map $\phi : X \rightarrow X'$ which is bijective and... |
Some standard examples on various quals seems to be computing units/class numbers etc. of the ring $\mathbb{Q}(\alpha)$, where $\alpha$ is a root of either $X^3+aX+b$ or $X^5+aX+b$.
My questions is the following: What are some standard tricks that can be used to deduce that a particular unit is actually a fundamental u... |
I'm preparing for an exam in the signals and systems class I'm taking. One of the practice exams has a problem that requires you to take the Fourier transform of $\text{sinc}(4t)$.
From a table of Fourier transform pairs I found: $\dfrac{\omega_b}{\pi}\text{sinc}\left(\dfrac{\omega_b t}{\pi} \right)\Rightarrow \text{r... |
If $R \subseteq A \times A$ is it true that $R$ is symmetrical since $xRy$ then $yRx$ ?
I have written that this is also antisymmetrical if both $x\leq y$ and $y\leq x$ if $x =y$
How does I then make $R \subseteq A \times A$ transitive?
|
Let
$$
f(x, y) = \frac{x + y}{2e}
$$
and
$$
g(x, y) = \frac{(x+y)+\sqrt{x^2 + y^2 + 6xy}}{2e^{\frac{\sqrt{x^2 + y^2 + 6xy}}{2(x + y)}}}
$$
I am looking for the largest $K$ so that,
$$ \forall x, y > 0, \frac{g(x, y)}{f(x, y)} > K$$
By numerical computation I found that $K$ must be around $3.3$. But the best thing ... |
Theorem (Legendre): Let $a,b,c$ coprime positive integers, then $ax^2 + by^2 = cz^2$ has a nontrivial solution in rationals $x,y,z$ iff $\left(\frac{-bc}{a}\right)=\left(\frac{-ac}{b}\right)=\left(\frac{ab}{c}\right)=1$.
I'm trying to prove this theorem using
Lemma Modified global square theorem: The rational number $c... |
I am using matlab to solve the following ODE
$$0.02K^2 F''(K)-3.45 F'(K)-3.45=0.08 F(K) $$
subject to boundary conditions
$$F(K=0)=63.09$$ and $$ F(K=700)=0.$$
I found that I had to give bvpinit an offset such as
bvpinit(linspace(1e-3,700,10000),[1 0.1])
from the left boundary 0 because otherwise, I would get an error... |
Coonsider the polynomial $p(x) = x^2(4x^2-3)^2$.
This polynomial is special because:
All of the local maxima $M_i$ are of the form $p(M_i)=1$
All of the local minima $m_i$ are of the form $p(m_i)=0$
and $p(-1)=p(1)=1$
The polynomial has no other zeros, local maxima or local minima.
My questions are:
what conditions... |
This question is from a math-essentials booklet for physicists. The function is not analytic from the C-R conditions but that is all I know.
$ \rlap{\textbf{-------------------------------------------------------------------------------------------------}}{\mbox{Its derivative should be}}$ $2z + 2\bar{z} =4Re[z]$ So a... |
What are some of the best books on graph theory, particularly directed towards an upper division undergraduate student who has taken most the standard undergraduate courses? I'm learning graph theory as part of a combinatorics course, and would like to look deeper into it on my own. Thank you.
|
The textbook I'm using has a somewhat confusing proof of one of the directions of Heine-Borel's Theorem, namely the one that states that a closed and bounded subset of $\mathbb R^n$ is compact. It uses sequences and "Changing the centers" to 'move' the balls to $\mathbb Q^n$ and then constructs a sequence which leads ... |
I usually work in number theory so I am not familiar with Fourier transforms, I have read up on them and know the basics but it never seems to be in number theory language.
I am trying to find the transform of a primitive Dirichlet character $\chi(n) \bmod q$. I know this is a periodic function and $\chi(n)=\exp\left(\... |
How to prove the following theorem?
Let $I \in R$ be an open interval, let $c \in I$, and let $f: I-{c} \rightarrow R$ be a function. Then $\lim \limits_{x \rightarrow c} {f(x)}$ exists if for each $\epsilon > 0$, there is some $\delta > 0$ such that $x,y \in I$ and $\vert x-c \vert < \delta$ and $\vert y-c \vert < \... |
If N is characteristic in a group G and if we have that K/N is characteristic in G/N why do we have that K is characteristic in G.
|
This might be a stupid question but what is the integral of $\frac{1}{1 + 9x^2}$? I want to think it's $\tan^{-1}(3x) + c$ but the book I'm working in says $\frac{1}{3}\tan^{-1}(3x) + c$.
How did they get $1/3$?
|
Since $X$ is a compact metric space, it is sequentially compact so every sequence in $X$ has a subsequence which converges. Now, I want to show that $X^\mathbb{N}$ is sequentially compact, but I'm getting confused because we must show that the sequence of a sequence has a convergent subsequence. I know that we must use... |
You're computing an improper integral (with limits on it) to see if something converges, and you're at the part where you have something like:
$$\lim_{b\to\infty}\left(\frac{\tan^{-1}(b)^2}{2} - \frac{\tan^{-1}(1)^2}{2}\right).$$
Wouldn't it diverge because you have infinity in the equation?
|
I'd like a closed form expression $x(n)$ for the number of orbits of the symmetric group on $3$ points acting on the triples in $\{ (a,b,c) \mid a,b,c \in \Bbb{Z}, 1 \leq a,b,c \leq n, c = 2n−a−b \}$.
I feel like this should be a really basic problem, but my standard method of attack fails: look it up in OEIS and prove... |
I am solving a problem in Mathematical Statistics by Jun Shao
Let $\{X_n \}$ be a Markov chain. Show that if $g$ is a one-to-one Borel function, then $\{g(X_n )\}$ is also a Markov chain. Give an example to show that $\{g(X_n )\}$ may not be a Markov chain in general.
I have a hard time on solving it, even though I h... |
I am currently learning about Markov chains and Markov processes, as part of my study on stochastic processes. I feel there are so many properties about Markov chain, but the book that I have makes me miss the big picture, and I might better look at some other references.
So I would like to know if there are some refe... |
I'm having trouble showing that the $L^p$ norm of the $n$-th order Dirichlet kernel is proportional to $n^{1-1/p}$.
I've tried brute force integration and it didn't work out. I would be grateful for any hints.
Thanks.
|
I'm trying to come up with a 95% confidence interval for the click-through-rate of particular advertisement. It has $x$ clicks out of $n$ impressions so far.
What's the best way to compute this, given that I expect the click-through-rate to be small? I've been told that the "usual" methods of computing a confidence in... |
I've been going through a problem solving book, and I'm a little stumped on the following question:
At each round, draw a number 1-100 out of a hat (and replace the number after you draw). You can play as many rounds as you want, and the last number you draw is the number of dollars you win, but each round costs an ext... |
its written that if you multiply the sum above by -9 and use the distributive law, all terms except "1" will cancel. I can't see that. I know that this is a divergent series. (the article I was reading was a layman's introduction to zeta regularization)
Similarly, how is $S=1+2+3+4... = -1/12$
Even if the series were... |
Apologies for a soft question. I do not have a lot of time because of my job (programmer) and my wife cannot work so I cannot quit. I also did not graduate from a very prestigious school with good grades. So grad school is out of the question for me.
I am 23 years old, I find that because I did not learn many topics w... |
Given an arbitrary arc, where you know the following values: start point $(x_0, y_0)$, end point $(x_1, y_1)$, radius ($r$) and arc direction (e.g. clockwise or counterclockwise from start to end), how can I calculate the arc's center? I know from this previous post (thanks!) that the center lies on the perpendicular ... |
I want to know if I can get some help with this proof. I tried, but I just cannot seem to get $2^{k}$. It states that,
For $k \in \mathbb{Z}_{\ge 0}$, $$\sum^{k}_{m=0}\binom{k}{m} = 2^k$$
Thank You.
|
I am an undergraduate of CS and I participate in TCS.SE. I am having trouble finding the error in a proof about the polynomial hierarchy collapsing to the first level ($NP = coNP$). I believe the reasoning to be false, because it is very simple and it would be common knowledge, however it is not.
Let $M$ be any oracle... |
I know that zeroes of holomorphic functions are isolated,and I know that if a holomorphic function has zero set whic has a limit point then it is identically zero function,i know a holomorphic function can have countable zero set, does there exixt a holomorphic function which is not identically zero, and has uncountabl... |
Let $X$ be non-empty set and $S$ be the set of all subsets of $X$, which will be a poset, under subset relation.
Let $\phi \colon S\rightarrow \mathbb{Z}$ be any function and for each $H\in S$, define $\sigma(H)=\sum_{K\leq H}\phi(K)$.
$\mu$ be the Möbius function defined by $\sum_{K\leq H} \mu(H)=\delta_{K,X}$.
How to... |
I'm having problems with exercise 1 of chapter 3 of do Carmo's "Riemannian Geometry". Here is the background:
Let $(u,v)$ be the coordinates on $\mathbb{R}^2$. Let $f,g\in C^\infty(\mathbb{R})$, and observe that $\varphi:\mathbb{R}^2\rightarrow \mathbb{R}^3$ given by $\varphi(u,v)=(f(v)\cos u,f(v)\sin u,g(v))$ is an ... |
I was given this question on a college assessment pre-test. I got the correct answer in a reasonable amount of time, but mostly because I worked backwards and double checked my answer. After I was done, I tried to find math on the net to solve it correctly, but I feel there is probably a simpler solution than this.
I... |
Find two pairs of relatively prime positive integers $(a,c)$ so that $a^2 + 5929 = c^2$.
Can you find additional pairs with $gcd(a,c) > 1$?
What I know:
$gcd(a,c) = 1$ implies that there are some $x$ and $y$ such that $ax + cy = 1$. Since $a d$oes not divide $c$, I'm guessing that $a^2$ does not divide $c^2$ as well (n... |
I have two linear, skew, 3D lines, and I was wondering how I could find a points on each of the lines whereby the distance between the two points are a particular distance apart?
I'm not after the points where the lines are the closest distance apart, nor the furthest distance apart, nor the point at which the lines cr... |
I'm a uni student and having a bit of trouble with a homework question in an engineering "Optimisation" class. I have a matrix $N$, where they ask you to find the eigenvalues and eigenvectors, which I have done.
Then they ask for you to plot the eigenvalue ($\lambda$), the eigenvector ($z$) and then $\lambda * z$ on an... |
I don't know if this is the right kind of question for here.
But, can someone help me find an english translation (a link to a pdf would be nice) of: B. Riemann, "Theorie der Abelschen Funktionen", J. Reine Angew. Math., 54 (1857)
Many Thanks.
|
Suppose that a 3-D surface has the property that $|k_1|\leq 1$ and $|k_2|\leq 1$ everywhere, where $k_1$ and $k_2$ are the principal curvatures. Prove or disprove that the curvature $k$ of a curve on that surface also satisfies $|k|\leq 1$.
|
Let $f(x)$ be a scalar function of a vector $x\in \mathbb R^n$. If we want to find $x$ such that $f(x)$ is maximized or minimized. I think we can compute the differential of $f(x)$ with respect to $x$ and then let the coefficient of $dx$ be zero. For example, I get the differential of $f(x)$ as
$$df(x)=g(x)^T dx$$
wher... |
Exercise 3.32 page 119 of Convex Optimization is concerned with the proof that if $f:\mathbb{R}\rightarrow\mathbb{R}:x\mapsto f(x)$ and $g:\mathbb{R}\rightarrow\mathbb{R}:x\mapsto g(x)$ are both convex, nondecreasing (or nonincreasing) and positive, then $h:\mathbb{R}\rightarrow\mathbb{R}:x\mapsto h(x)=f(x)g(x)$ is als... |
Possible Duplicate:
Teaching myself differential topology and differential geometry.
Dear all,
I really need some reference books about differential geometry.
Recently I am taking a course of representation theory and symmetry method for DE. I feel too difficult to read the materials without the knowledge of differe... |
I ran across this confounding limit I am wondering about. It is as follows:
$$\displaystyle \lim_{n\to \infty}\frac{1-(1-c^{n})^{2n}}{(1-c)^{2n}}, \;\ 0<c<1$$
I played around with this on Maple and found that if c is less than approximately .382 (but
greater than 0), it converges to 0. If c is greater than .382 (but ... |
If A has eigenvector $\mathbf{v}_1$ so that $A\mathbf{v}_1=\lambda_1\mathbf{v}_1$and B has eignenvector $\mathbf{v}_2$ so that $B\mathbf{v}_2=\lambda_2\mathbf{v}_2$, then what can you say about AB? can you say $AB\mathbf{v_3}=\lambda_3\mathbf{v}_3$? and what would be the relationship between $\mathbf{v}_1,\mathbf{v}_2... |
Ciao!
Let $A$ be a finite abelian group, and let
$ \psi : A \times A \to \mathbb{Q}/\mathbb{Z} $
be an alternating, non-degenerate bilinear form on $A$. Maybe I should say what I mean by these words; bilinear means it is linear in each argument separately; alternating means that $\psi(a,a) = 0$ for all $a$; non-degene... |
I met the question in my exam:
Flip one coin, the probability of head is $\frac13$, $Pr\{\text{tail}\} = \frac23$. Flip the coin twice, what are the four possible outcomes and their probabilities?
|
Possible Duplicate:
on the boundary of analytic functions
I was thinking of a problem I will feel happy if you
give hint to solve this. Suppose I have a function $f$ that is analytic
on the unit disk $D=\{z \in \mathbb{C}:|z|<1\}$ that extends continuously to $\overline{D}$. If $f$ is identically zero on some segmen... |
I have the following equation as a question for homework: $\lfloor 2x \rfloor = 2\lfloor x\rfloor$
I know what the solution is by deducting to the possibilities.
Meaning this equation is true for any x which is between n (an integer) and $n+y$ where $0\leq y < \frac{1}{2}$
putting it simple:
\begin{align}
x \in \lbrace... |
I am wondering if the nonzero rationals with operation $x\circ y = xy/7$ a group. I think it may be and I found the identity to be 7 and found inverses for each element. Mark
|
How to solve $n$ for $n^n = 2^c$?
What's the numerical method?
I don't get this.
For $c=1000$, $n$ should be approximately $140$, right?
|
To iteratively solve large linear systems, many current state-of-the-art methods work by finding approximate solutions in successively larger (Krylov) subspaces. Are there similar iterative methods for solving systems of polynomial equations by finding approximate solutions on successively larger algebraic sets?
|
Let $M$ be an off-line Turing machine over the input alphabet $\{0,1\}^{*}$, that uses only one working tape in addition to the input tape. Construct a Turing machine $M'$, such that:
$L(M) = L(M')$
$M'$ never loops in a bounded space
(that is, $M'(w)\uparrow$ may happen
only if $M'$ visits infinitely many
cells in th... |
I'm dealing with a time-complexity problem in which I know the running time of an algorithm:
$$t = 1000 \mathrm{ms} .$$
I also know that the algorithm is upper bounded by $O(n!)$.
I want to know the approximate size of the input $n$ based on this:
$$ f(n) = n! = t $$
$$ f^{-1}(t) = n = ? $$
|
I have a linear operator $T$ which acts on the vector space of square $N\times N$ matrices in this way:
$T(A)=0.5(A-A^\mathrm{t})$
($A^\mathrm{t}$: the transpose of a matrix $A$).
I need to prove that this operator is diagonalizable.
I know that a linear transformation is diagonalizable if and only if there is a basis ... |
If $p$ is prime, determine the number of abelian groups of order $p^n$ for each $1\leq n\leq8$
(I assume that "up to isomorphism" should be included somewhere in the question for the sake of precision...)Could someone please review/confirm my work?
n = 1: $\mathbb{Z}_p $
n = 2: $\mathbb{Z}_{p^2}$ and $\mathbb{Z... |
Let $x = (a, b), y = (c, d) \in \mathbb{Z}^2$. What is the condition on $a, b, c, d$ so that ${x, y}$ is a basis?
My answer: $ad\neq bc$ and $gcd(a, c) = gcd(b, d) = 1$.
The first condition ensures that they aren't the same vector; the second ensures that we can actually "get" all of the integer values/lattice point... |
Let $G=(V,E)$ be a graph, and $A$ be its adjacency matrix. Define $n = |V|$.
Given $A$ and a natural number $m \le n$, I'm interested in the following problem:
How many simple cycles of length $m$ exist in $G$?
By simple cycle, I mean no repeated vertices along the cycle is allowed (other than the starting and ending... |
simple question.
I am looking at a presentation of the ring $k[x,yx^{-1}]$ in the following form $k[a,b]/I$ where $I$ is an ideal.
|
Can anyone suggest a collection of (solved) exercises in topology? Undergrad level, as a companion to Dugundji's Topology (although excellent it doesn't provide the solutions to the problems).
Thanks.
|
Is there an elementary(or not) proof that there are at least two consecutive primes which have difference $2n$ for every natural number $n$? i remind that Polignac's conjecture states that there should be infinite such pairs for every $n$ so it is a much more easy question to ask and of course should be valid.
In addit... |
I cannot figure out the solution to this exercise in Marker. Can someone help me?
$(Z \oplus Z, +, 0) \not\equiv (Z, +, 0)$
|
I am trying to do contour integration on $\int_c \zeta(s)\zeta(2s) \frac{x^s}{s} ds$ and
$\int_c \frac{x^s}{s^ks}ds$
where c is the line segment joining c-iT c+iT
I understand the basic theory behind it, how to find residues and Cauchy's theorem. I am just having trouble with bounding these integrals when using Cauch... |
I want to calculate how much I would pay monthly.
$X$ = Amount borrowed. $I$= Interest Rate. $Y$ = number of years
This is what I have so far, but it's wrong; I'm getting a larger value.
$$\frac{X\left(1 + \frac{I}{100}\right)^{Y}\left(\frac{I}{Y}\right)}{12\left(\left(1 + \frac{I}{100}\right)^Y - 1\right)}.$$
|
I'm trying to figure out the fee-adjusted interest rate paid when incurring a balance transfer fee on a loan, not accounting for inflation.
If the APR is 4%, the loan 5000, the transfer 2% and you pay it off in 200 days (54.8% of the year when rounded to 2 decimals), I see it this way:
(APR: 4% Loan: 5,000 Transfer f... |
If I have a field $K$ and an extension $L$ of $K$ such that all (non-constant)
polynomials in $K[X]$ have a root in $L$, is the set of algebraic elements
of $L$ over $K$ (the sub-field of all the elements of $L$ which are roots of a polynomial in $K[X]$) algebraically closed ?
Do you have a counterexample ?
|
I've been wrestling with this question for quite some time now, and the result was like 20 leaves of paper packed with scribbling...anyway, here's the question:
I need to find the solution to the following source excited Helmholtz equation:
$$(\nabla^2 + k^2)G_1(r,a) = -\delta(r-a)$$
With the boundary condition:
$$\lef... |
$$f(x)=\frac{x^2+1}{x^3-x}$$
$$f^{(100)}(x)=?$$
I tried differnetiating once and twice, but did not see any pattern emerging and can't guess what the 100th derivative should be.
EDIT
so decomposing this as $$f(x)=-\frac{1}{x}+\frac{1}{x+1}+\frac{1}{x-1}
$$ does the job. Thanks for the hints! (edit: Sivaram has a com... |
If $\mathbf{A}_{n\times n}$ is a positive semi-definite matrix with eigenvalues $\{\alpha_k\},\ k\in\{1,...,n\}$, and $\mathbf{B}_{m\times n}$ is an arbitrary matrix with singular values $\{\beta_k\},\ k\in\{1,...,\min(m,n)\}$, can anything be said about the singular values $\{\gamma_k\},\ k\in\{1,...,\min(m,n)\}$ of t... |
This question may be a bit vague, but neverthless, i would like to see an answer. Wikipedia tells me that:
In abstract algebra, localization is a systematic method of adding multiplicative inverses to a ring.
It is clear that for integral domains, we have the Field of Fractions, and we work on it. It's obvious that a... |
The following theorem is an alternative definition of limits. In this theorem, you don't need to know the value of $\lim \limits_{x \rightarrow c} {f(x)}$ in order to prove the limit exists.
Let $I \in R$ be an open interval, let $c \in I$, and let $f: I-{c} \rightarrow R$ be a function. Then $\lim \limits_{x \rightar... |
Let $k$ be a field, not necessarily algebraically closed, not necessarily of characteristic 0 (actually, the example I have in mind is $k=F_2$). Let $V,W$ be varieties over $k$, and $W\to V$ a morphism defined over $k$, such that every fiber is isomorphic to an affine space of the same dimension.
What conditions imply... |
Let $G$ and $H$ be finite abelian groups such that $G \times G \cong H \times H$.
Then $G \cong H$.
I was going to just write the hypothesis as $G^2 \cong H^2$ and take square roots on both sides, but I don't think that would suffice (and neither would saying "true" a la Myself!)...
The hypothesis tells us that th... |
Consider the category of all open sets of a given topological space where the morphism are inclusions,why one can see a Presheaf as a contravariant functor?
|
This is not really significant to what I am trying to do right now but I'm just wondering...
Why is it that the discriminant for the cyclotomic polynomials $\Phi_{1}(x)$ and $\Phi_{2}(x)$ is 1?
Using WolframAlpha, the discriminant for other linear polynomials is also 1. Why is that?
Thank you.
|
if $f$ is integrable, i wish to show
$\frac{n}{2} \int_{-1/n}^{1/n} (f(x+y) - f(x))dy \to 0 $ as $ n \to \infty $
this looks to be very intuitive, but im having trouble proving it formally. any tips?
EDIT: added the n infront of the integral, thanks Jonas
|
I have begun working with Mathematica for some Graph Theory, and I want to compute the number of spanning trees of all cubic graphs up to 12 vertices. I have found that
GraphData["Cubic", 12]
will give me the cubic graphs with 12 vertices. But how do I compute the number of spanning trees in each of these graphs. A goo... |
I would really appreciate if you could help out with the following question
Let $W_1,W_2,\ldots$ be the event times in a Poisson process $\{X(t);t\ge0\}$ of rate $\lambda$, and let $f(w)$ be an arbitrary function. Verify that
$$
\mathbb{E}\left[\sum_{i=1}^{X(t)}f(W_i)\right]=\lambda\int_0^tf(w)\,dw.
$$
(originally ... |
From what I understand, Black-Scholes equation in finance is used to price options which are a contract between a potential buyer and a seller. Can I use this mathematical framework to "buy" a stock? I do not have the choice using options in the market I am dealing with -- I either buy something or I don't. So I was wo... |
Suppose that V and W are vector spaces (say over the reals) and that T is a linear surjection from V onto W with kernel K. By picking a basis for W (using choice) and pulling it back through T to a linearly independent set in V (again using choice) it is not difficult to get a linear embedding i of W into V satisfying ... |
My friend and I have been stumped on this problem for a little while and I thought asking for tips couldn't hurt (we did ask the teacher, but we got other problems after)
Here is the question :
Let $\{X_n\}_{n \geq 1}$ be a sequence of random variables defined on the same probability space $(\Omega, F, \mathbb{P})$ wit... |
Hey guys, simple question in linear algebra.
I want to find the nth exponent of this matrix:
$$
\left[
\begin{array}{cc}
1 & 1 \\
0 & 2
\end{array}
\right]
$$
I'm trying to use the diagonal algorithm, first by finding the eigenvalues, so I get $t_1=1$, and $t_2=2$, and than I realize that the matrix is not diag... |
I was skimming through the topology book recommended through this question and I came across a question that I apparently solved incorrectly. It's question 2.5 on page 12.
Repeated here:
Let $X$ be $\mathbb{R}$, and let $\Omega$ consists [sic] of the empty set and complements of all finite subsets of $\mathbb{R}$. Is ... |
I would appreciate any possible help for this question because I have no clue what to do! Thanks so much!
Consider a population made of a fixed number (N) of people. At time t=0 there is only one infected individual and N-1 susceptible people in the population. When you get infected, you remain in the infected state fo... |
If I put 4 green rabbits and 1 red rabbit into a hat, that's 5 rabbits!
I then make 5 attempts at pulling the red rabbit out of the hat.
If I get a green rabbit I put it back in the hat.
On the first attempt I have a 1 in 5 chance of red rabbit success.
If the first attempt fails, on the second attempt I still have a 1... |
Can anyone please tell me if there is any other proof for the cardinality of all mappings, that is not by induction, i.e., not this one (http://www.proofwiki.org/wiki/Cardinality_of_Set_of_All_Mappings) ?
Thanks
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How to prove that any vector bundle with a fiber metric g admits a metric connection? It seems I should use partition of unity, but I have no idea how to proceed.
Also it seems there are two definitions of connection on a vector bundle E, one is that
$\nabla: \Gamma(TM) \times \Gamma(E) \to \Gamma$(E),
and the other o... |
I have no clue how to proceed with this question. Please help me in deriving the differential equations! Thanks very much... I really appreciate it :)
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define $$h(x)=\int_0^{2\pi}f(x-y)g(y)dy=f*g(x)$$
if $f,g \in L^2$ are $2\pi$ periodic, show that h is continuous on $[0,2\pi)$
so let $x_n \to x$, then
$$|h(x)-h(x_n)|=|\int f(x-y)g(y)-\int f(x_n-y)g(y)|$$
i tried to manipulate this but couldnt get anywhere. I was also thinking of using Holder's inequality somehow but ... |
How does one find the center of the group G/Z(G)?
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I have tried $\gcd(0,8)$ in a lot of online gcd (or hcf) calculators, but some say $\gcd(0,8)=0$, some other gives $\gcd(0,8)=8$ and some others give $\gcd(0,8)=1$. So really which one of these is correct and why there are different conventions?
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I have known for a while that $\mathbb{R}P^1 \simeq S^1$. Indeed, visually it is easy to see why this is the case (any two antipodal points define a line through the origin)
Say we now forget everything we know, and let $F$ be a division ring and let $n \ge 0$. We definie an equivalence relation of $F - \{ 0 \}$ by $x ... |
I posted this on cs theory yesterday but did not get an answer and hence I am posting here.
I have a low rank matrix given as $AB^T$ where $A,B \in \mathbb{R}^{n \times p}$ and $p \ll n$. (I know $A$ and $B$ separately)
One efficient way to get the Singular Value Decomposition is to do a reduced QR on $A$ and $B$ i.e. ... |
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